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\begin{document}
\begin{abstract} Various equivalent conditions for a semigroup or a resolvent generated by a Markov process to be of Feller type are given. \end{abstract}
\maketitle \thispagestyle{empty}
The Feller property of the semigroup generated by a Markov process plays a prominent role in the theory of stochastic processes. This is mainly due to the fact that if the Feller property holds true, then --- under the additional assumption of right continuity of the paths --- the simple Markov property implies the strong Markov property (e.g., \cite[Theorem~III.3.1]{ReYo91} or \cite[Theorem~III.15.3]{Wi79}).
However, in many instances it is of advantage to consider the associated resolvent instead of the semigroup. Therefore we present in this note a result which states various forms of the equivalence of the Feller property as expressed in terms of the semigroup or of the resolvent. The material here seems to be quite well-known, and our presentation of it owes very much to~\cite{Ra56} --- most notably the inversion formula for the Laplace transform, equation~\eqref{inv_L} in connection with lemma~\ref{lem_5}. On the other hand, we were not able to locate a reference where the results are collected and stated in the form of the theorem given below.
Assume that $(E,d)$ is a locally compact separable metric space with Borel
$\sigma$--algebra denoted by $\mathcal{B}(E)$. $B(E)$ denotes the space of bounded measurable real valued functions on $E$, $\ensuremath C_0(E)$ the subspace of continuous functions vanishing at infinity. $B(E)$ and $\ensuremath C_0(E)$ are equipped with the sup-norm $\|\,\cdot\,\|$.
The following definition is as in~\cite{ReYo91}: \begin{definition} \label{def_1}
A \emph{Feller semigroup} is a family $U=(U_t,\,t\ge 0)$ of positive
linear operators on $\ensuremath C_0(E)$ such that
\begin{enum_a}
\item $U_0=\text{id}$ and $\|U_t\|\le 1$ for every $t\ge 0$;
\item $U_{t+s} = U_t\comp U_s$ for every pair $s$, $t\ge 0$;
\item $\lim_{t\downarrow 0} \|U_t f - f\|=0$ for every $f\in \ensuremath C_0(E)$.
\end{enum_a} \end{definition}
Analogously we define \begin{definition} \label{def_2}
A \emph{Feller resolvent} is a family $R=(R_\lambda,\,\lambda>0)$ of positive
linear operators on $\ensuremath C_0(E)$ such that
\begin{enum_a}
\item $\|R_\lambda\|\le \lambda^{-1}$ for every $\lambda>0$;
\item $R_\lambda - R_\mu = (\mu-\lambda) R_\lambda\comp R_\mu$ for every
pair $\lambda$, $\mu>0$;
\item $\lim_{\lambda\to\infty}\|\lambda R_\lambda f - f\|=0$ for every $f\in\ensuremath C_0(E)$.
\end{enum_a} \end{definition}
In the sequel we shall focus our attention on semigroups $U$ and resolvents $R$ associated with an $E$--valued Markov process, and which are \emph{a priori} defined on $B(E)$. (In our notation, we shall not distinguish between $U$ and $R$ as defined on $B(E)$ and their restrictions to $\ensuremath C_0(E)$.)
Let $X = (X_t,\,t\ge 0)$ be a Markov process with state space $E$, and let $(P_x,\,x\in E)$ denote the associated family of probability measures on some measurable space $(\Omega,\mathcal{A})$, so that in particular $P_x(X_0=x) = 1$. $E_x(\,\cdot\,)$ denotes the expectation with respect to $P_x$. We assume throughout that for every $f\in B(E)$ the mapping \begin{equation*}
(t,x) \mapsto E_x\bigl(f(X_t)\bigr) \end{equation*} is measurable from $\mathbb{R}_+\times E$ into $\mathbb{R}$. The semigroup $U$ and resolvent $R$ associated with $X$ act on $B(E)$ as follows. For $f\in B(E)$, $x\in E$, $t\ge 0$, and $\lambda>0$ set \begin{align}
U_t f(x) &= E_x\bigl(f(X_t)\bigr), \label{eq_1}\\
R_\lambda f(x) &= \int_0^\infty e^{-\lambda t} U_t f (x)\,dt. \label{eq_2} \end{align} Property~(a) of Definitions~\ref{def_1} and \ref{def_2} is obviously satisfied. The semigroup property, (b) in definition~\ref{def_1}, follows from the Markov property of $X$, and this in turn implies the resolvent equation, (b) of definition~\ref{def_2}. Moreover, it follows also from the Markov property of $X$ that the semigroup and the resolvent commute. On the other hand, in general neither the property that $U$ or $R$ map $\ensuremath C_0(E)$ into itself, nor the strong continuity property (c) in Definitions~\ref{def_1}, \ref{def_2} hold true on $B(E)$ or on $\ensuremath C_0(E)$.
If $W$ is a subspace of $B(E)$ the resolvent equation shows that the image of $W$ under $R_\lambda$ is independent of the choice of $\lambda>0$, and in the sequel we shall denote the image by $RW$. Furthermore, for simplicity we shall write $U\ensuremath C_0(E)\subset \ensuremath C_0(E)$, if $U_t f\in\ensuremath C_0(E)$ for all $t\ge 0$, $f\in\ensuremath C_0(E)$.
\begin{theorem*} \label{thm} The following statements are equivalent: \begin{enum_a}
\item $U$ is Feller.
\item $R$ is Feller.
\item $U\ensuremath C_0(E)\subset\ensuremath C_0(E)$, and for all $f\in\ensuremath C_0(E)$, $x\in E$,
$\lim_{t\downarrow 0} U_t f(x) = f(x)$.
\item $U\ensuremath C_0(E)\subset\ensuremath C_0(E)$, and for all $f\in\ensuremath C_0(E)$, $x\in E$,
$\lim_{\lambda\rightarrow \infty} \lambda R_\lambda f(x) = f(x)$.
\item $R\ensuremath C_0(E)\subset\ensuremath C_0(E)$, and for all $f\in\ensuremath C_0(E)$, $x\in E$,
$\lim_{t\downarrow 0} U_t f(x) = f(x)$.
\item $R\ensuremath C_0(E)\subset\ensuremath C_0(E)$, and for all $f\in\ensuremath C_0(E)$, $x\in E$,
$\lim_{\lambda\rightarrow \infty} \lambda R_\lambda f(x) = f(x)$. \end{enum_a} \end{theorem*}
We prepare a sequence of lemmas. The first one follows directly from the dominated convergence theorem:
\begin{lemma} \label{lem_3} Assume that for $f\in B(E)$, $U_t f \rightarrow f$ as $t\downarrow 0$. Then $\lambda R_\lambda f \rightarrow f$ as $\lambda\to+\infty$. \end{lemma}
\begin{lemma} \label{lem_4} The semigroup $U$ is strongly continuous on $RB(E)$. \end{lemma}
\begin{proof} If strong continuity at $t=0$ has been shown, strong continuity at $t>0$ follows from the semigroup property of $U$, and the fact that $U$ and $R$ commute. Therefore it is enough to show strong continuity at $t=0$.
Let $f\in B(E)$, $\lambda>0$, $t>0$, and consider for $x\in E$ the following computation \begin{align*}
U_t R_\lambda f(x) &- R_\lambda f(x)\\
&= \int_0^\infty e^{-\lambda s} E_x\bigl(f(X_{t+s})\bigr)\,ds
- \int_0^\infty e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds\\
&= e^{\lambda t} \int_t^\infty e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds
- \int_0^\infty e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds\\
&= \bigl(e^{\lambda t}-1\bigr) \int_t^\infty e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds
- \int_0^t e^{-\lambda s} E_x\bigl(f(X_s)\bigr)\,ds\\ \end{align*} where we used Fubini's theorem and the Markov property of $X$. Thus we get the following estimation \begin{align*}
\bigl\|U_t R_\lambda f - R_\lambda f\bigr\|
&\le \biggl(\bigl(e^{\lambda t} - 1\bigr)\int_t^\infty e^{-\lambda s}\,ds
+\int_0^t e^{-\lambda s}\,ds\biggl)\, \|f\|\\
&= \frac{2}{\lambda}\, \bigl(1-e^{-\lambda t}\bigr)\,\|f\|, \end{align*} which converges to zero as $t$ decreases to zero. \end{proof}
For $\lambda>0$, $t\ge 0$, $f\in B(E)$, $x\in E$ set \begin{equation} \label{inv_L}
U^\lambda_t f(x) = \sum_{n=1}^\infty \frac{(-1)^{n+1}}{n!}
\,n\lambda\, e^{n\lambda t}\, R_{n\lambda} f(x). \end{equation}
Observe that, because of $n\lambda\|R_{n\lambda} f\| \le \|f\|$, the last sum converges in $B(E)$.
For the proof of the next lemma we refer the reader to~\cite[p.~477~f]{Ra56}:
\begin{lemma} \label{lem_5} For all $t\ge 0$, $f\in RB(E)$, $U^\lambda_t f$ converges in $B(E)$ to $U_t f$ as $\lambda$ tends to infinity. \end{lemma}
\begin{lemma} \label{lem_6} If $U_t\ensuremath C_0(E) \subset \ensuremath C_0(E)$ for all $t\ge 0$, then $R_\lambda\ensuremath C_0(E)\subset \ensuremath C_0(E)$, for all $\lambda>0$. If $R_\lambda\ensuremath C_0(E)\subset \ensuremath C_0(E)$, for some $\lambda>0$, and $R_\lambda\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, then $U_t\ensuremath C_0(E) \subset \ensuremath C_0(E)$ for all $t\ge 0$. \end{lemma}
\begin{proof} Assume that $U_t\ensuremath C_0(E) \subset \ensuremath C_0(E)$ for all $t\ge 0$, let $f\in\ensuremath C_0(E)$, $x\in E$, and suppose that $(x_n,\,n\in\mathbb{N})$ is a sequence converging in $(E,d)$ to $x$. Then a straightforward application of the dominated convergence theorem shows that for every $\lambda>0$, $R_\lambda f(x_n)$ converges to $R_\lambda f(x)$. Hence $R_\lambda f\in \ensuremath C_0(E)$.
Now assume that that $R_\lambda\ensuremath C_0(E)\subset \ensuremath C_0(E)$, for some and therefore for all $\lambda>0$, and that $R_\lambda\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$. Consider $f\in R\ensuremath C_0(E)$, $t>0$, and for $\lambda>0$ define $U^\lambda_t f$ as in equation~\eqref{inv_L}. Because $R_{n\lambda}f\in\ensuremath C_0(E)$ and the series in formula~\eqref{inv_L} converges uniformly in $x\in E$, we get $U^\lambda_t f\in\ensuremath C_0(E)$. By lemma~\ref{lem_5}, we find that $U^\lambda_t f$ converges uniformly to $U_t f$ as $\lambda\to+\infty$. Hence $U_t f\in\ensuremath C_0(E)$. Since $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, $U_t$ is a contraction and $\ensuremath C_0(E)$ is closed, we get that $U_t\ensuremath C_0(E)\subset\ensuremath C_0(E)$ for every $t\ge 0$. \end{proof}
The following lemma is proved as a part of Theorem~17.4 in~\cite{Ka97} (cf.\ also the proof of Proposition~2.4 in~\cite{ReYo91}).
\begin{lemma} \label{lem_7} Assume that $R\ensuremath C_0(E)\subset \ensuremath C_0(E)$, and that for all $x\in E$, $f\in\ensuremath C_0(E)$, $\lim_{\lambda\to\infty} \lambda R_\lambda f(x) = f(x)$. Then $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$. \end{lemma}
If for all $f\in\ensuremath C_0(E)$, $x\in E$, $U_t f(x)$ converges to $f(x)$ as $t$ decreases to zero, then similarly as in the proof of lemma~\ref{lem_3} we get that $\lambda R_\lambda f(x)$ converges to $f(x)$ as $\lambda\to+\infty$. Thus we obtain the following
\begin{corollary} \label{cor_8} Assume that $R\ensuremath C_0(E)\subset \ensuremath C_0(E)$, and that for all $x\in E$, $f\in\ensuremath C_0(E)$, $\lim_{t\downarrow 0} U_t f(x) = f(x)$. Then $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$. \end{corollary}
Now we can come to the
\begin{proof}[Proof of the theorem] We begin by proving the equivalence of statements~(a), (b), (d), and~(f):
\noindent ``(a)\space$\Rightarrow$\space(b)'' Assume that $U$ is Feller. From lemma~\ref{lem_6} it follows that $R_\lambda\ensuremath C_0(E)\subset\ensuremath C_0(E)$, $\lambda>0$. Let $f\in\ensuremath C_0(E)$. Since $U$ is strongly continuous on $\ensuremath C_0(E)$, lemma~\ref{lem_3} implies that $\lambda R_\lambda f$ converges to $f$ as $\lambda$ tends to $+\infty$. Hence $R$ is Feller.
\noindent ``(b)\space$\Rightarrow$\space(f)'' This is trivial.
\noindent ``(f)\space$\Rightarrow$\space(d)'' By lemma~\ref{lem_7}, $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, and therefore lemma~\ref{lem_6} entails that $U\ensuremath C_0(E)\subset\ensuremath C_0(E)$.
\noindent ``(d)\space$\Rightarrow$\space(a)'' By lemmas~\ref{lem_6} and \ref{lem_7}, $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, and therefore by lemma~\ref{lem_4}, $U$ is strongly continuous on $\ensuremath C_0(E)$. Thus $U$ is Feller.
Now we prove the equivalence of~(a), (c), and (e):
\noindent ``(a)\space$\Rightarrow$\space(c)'' This is trivial.
\noindent ``(c)\space$\Rightarrow$\space(e)'' This follows directly from Lemma~\ref{lem_6}.
\noindent ``(e)\space$\Rightarrow$\space(a)'' By corollary~\ref{cor_8}, $R\ensuremath C_0(E)$ is dense in $\ensuremath C_0(E)$, hence it follows from lem\-ma~\ref{lem_6} that $U\ensuremath C_0(E)\subset\ensuremath C_0(E)$. Furthermore, lemma~\ref{lem_4} implies the strong continuity of $U$ on $R\ensuremath C_0(E)$, and by density therefore on $\ensuremath C_0(E)$. (a) follows. \end{proof}
\noindent \textbf{Acknowledgement.} J.~P. thanks Florian Werner for a useful discussion.
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arXiv
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Evaluation of a laying-hen tracking algorithm based on a hybrid support vector machine
Cheng Wang1,
Hongqian Chen2, 3,
Xuebin Zhang1 and
Chaoying Meng1Email author
Received: 8 April 2016
Behavior is an important indicator reflecting the welfare of animals. Manual analysis of video is the most commonly used method to study animal behavior. However, this approach is tedious and depends on a subjective judgment of the analysts. There is an urgent need for automatic identification of individual animals and automatic tracking is a fundamental part of the solution to this problem.
In this study, an algorithm based on a Hybrid Support Vector Machine (HSVM) was developed for the automated tracking of individual laying hens in a layer group. More than 500 h of video was conducted with laying hens raised under a floor system by using an experimental platform. The experimental results demonstrated that the HSVM tracker outperformed the Frag (fragment-based tracking method), the TLD (Tracking-Learning-Detection), the PLS (object tracking via partial least squares analysis), the MeanShift Algorithm, and the Particle Filter Algorithm based on their overlap rate and the average overlap rate.
The experimental results indicate that the HSVM tracker achieved better robustness and state-of-the-art performance in its ability to track individual laying hens than the other algorithms tested. It has potential for use in monitoring animal behavior under practical rearing conditions.
Locomotion tracking
Support vector machine
The behavior of animals is an important indicator of their welfare [1, 2]. Animal behavior is typically monitored through manual observation which requires substantial manpower and cannot always guarantee accuracy [3]. The demand for methods to automatically monitor animal behavior and track their movement has recently been increasing thereby promoting the initiation of related research [4].
Previous studies of animal behavior have focused on two main objectives, namely the identification of specific behavior and the tracking of animal movement. With respect to behavioral identification, the appearance of animals varies widely depending on their location which renders image processing and interpretation very difficult [5]. Some researchers have identified the behavior of animal groups through visual techniques such as monitoring the weight distribution in poultry flocks [6, 7], the spatial distribution of pigs [8, 9], the distribution of broilers [10], and the trajectory of a flock of poultry [11].
Monitoring the behavior of a particular animal in a group requires information obtained from tracking the specific animal and this can be achieved by limiting the animal's activity to ensure that it remains in an appropriate location without other animals in its vicinity. This idea has been applied to monitor a pig's weight [12] and back fat levels [13] and to monitor a laying hen's activities [14].
With respect to motion tracking, Computer Vision Technology was first used in 1997 to track animal behavior [15]. In 1998, Sergean et al. [16] developed a tracking system using color information and segmented individual birds using contour information. Currently, Ellipse Fitting is the most common approach used to track laying hens. Fujii et al. [17] used a method based on particle filters for tracking multiple hens. However, the particle filters lost track of the hens when sudden quick movements were made. The method which was proposed by Kashiha [18] had a superior performance for tracking individual laying hens in an image area but was unable to identify and track an individual laying hen in a flock. To solve this problems, Nakarmi et al. [19] installed a RFID (Radio Frequency Identification) antenna array at the bottom of a cage and attached RFIDs to the feet of hens' to determine their location for further tracking in the distance image. Although this method can achieve suitable tracking results, it is very limited in its application. It is not conducive to practical application and wearing the RFID can lead to discomfort for the hens which in turn may alter their behavior.
To address the challenges discussed above, a new laying-hen tracking algorithm, based on the Hybrid Support Vector Machine (HSVM) model has been proposed as a method to track a single hen within a flock raised under a floor system in real time with high robustness. The objective of this experiment was to compare the ability of this method to track individual laying hens in a flock with 5 other commonly used algorithms.
Experimental pen design and setup
This study was approved by the Animal Care and Use Committee of China Agricultural University (Beijing, China). As tracking targets, six 20-week-old Hyline Brown laying hens weighing an average of 1.4 kg were selected for study. The hens were allowed a 2 wk acclimation period before commencing data collection.
A 1.2 m × 1.5 m pen (Fig. 1a, b) was constructed to house the birds (Fig. 1c). On two sides of the pen, LED lighting was used to illuminate the test area from 0500 h to 2100 h every day to ensure that the intensity of illumination in the pen region was approximately 15–20 lux. The hens were fed twice a day at 0900 h and 1700 h and their eggs were collected at 1700 h every day. Manure was removed daily and the barn temperature was maintained about 20 °C.
A schematic drawing and photograph of the experimental pen and observation objects. (a) Photograph (b) Schematic (c) Observation objects
The height of the cameras used to collect video (Launch, LC5505E7-C83R) was set at 2.2 m. Videos were operated from 0500 h to 2100 h. Over 500 h of video were obtained during the subsequent 30 d. Ten 3-min fragments out of the 500 h of video were randomly chosen to validate the tracking algorithm and 778 images in the video fragments were randomly chosen and manually labeled.
The tracking algorithm consisted of three steps including initialization, tracking and updating. For initialization, the contour area of the target was manually marked and the rotation method was used to obtain the size of the minimum outer rectangle of the contour area. This minimum outer rectangle was represented as T0{w0,h0,a0,c0}, where w0 corresponded to the width of T0, h0 represented the height of T0, a0 was the angle between T0 and the x-axis, and c0 was the center of T0. This rectangle was the initial tracking rectangle and the width and height of each sample was consistent with it.
Binary HSVM model (HSVMb)
The HSVM model consisted of a one-class model, a binary classification model and a regression model. Around the initial tracking rectangle, the three types of HSVM were sampled as follows. Firstly, the Binary Classification Support Vector Machine (HSVMb) model was established [20]. The binary model is often used for the tracking-by-detection strategy [21, 22] used in object tracking. However, this method results in a fuzzy boundary between positive and negative samples. To handle this problem, the regression model aids in locating the target more accurately to avoid drift.
For the HSVMb, the positive and negative samples were expressed as {xi, yi}, where yi ∈{+1,0} was the label of sample xi. If yi = 1, xi was a positive sample, and x0 denoted the sample in the initial tracking rectangle. l(xi) denoted the location of sample xi, and l(x0) denoted the location of T0. The distance-based rule was used to select training samples [21, 23]. If ||l(xi)-l(x0)|| < d1, yi = 1, and if d2 < ||l(xi)-l(x0)|| < d3, yi = 0, (Fig. 2a) where
$$ {d}_2=\raisebox{1ex}{$\sqrt{W^2+{H}^2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.,{d}_1=\sqrt{20},\kern0.5em {d}_3=2\sqrt{W^2+{H}^2}\left(W={w}_0,\ H={h}_0\right) $$
Example of sampling by the three types of support vector machines. (a) SVMb (b) SVMr (c) SVMo
To extract the histogram of orientation gradients, 50 positive and 50 negative samples were randomly selected according to the above rules. In the HSVM, the window size for the histogram of orientation gradient was 16 × 16 pixels and the cell size was 4 × 4 pixels. One block consisted of 4 cells and strided each cell once with 9 orientations. All of the samples selected for feature extraction were normalized to the size of the window. With the features and training pairs {xi, yi}, the binary HSVM model was obtained. The confidence score of a new candidate sample xi was calculated by:
$$ con{f}_b(x)={\displaystyle \sum_i{a}_i{y}_i{k}_b\left({x}_i,x\right)} $$
where ai was the Lagrange Multiplier and kb(xi, x) was the Kernel Trick [24].
Regression SVM model (HSVMr)
For HSVMr, all of the samples satisfying d1 < ||l(xi)-l(x0)|| < d2 were selected as training samples (Fig. 2b). The bounding box overlap area ratio was chosen to generate the regression function value yi of sample xi, which has been widely used to evaluate the accuracy of object detection [23]:
$$ {y}_i=\frac{area\left({x}_o\cap {x}_i\right)}{area\left({x}_o\cup {x}_i\right)} $$
where x0 denoted the initial tracking rectangle. Following this principle, 50 training samples were randomly selected to obtain the regression HSVM model. For any candidate in region x, its confidence score confr(x) was calculated as follows:
$$ con{f}_r(x)={\displaystyle \sum_i\left({a}_i-{a}_i^{*}\right){k}_r}\left({x}_i^T,x\right) $$
where ai and ai * were the Lagrange Multipliers and kr(xi T, x) was the Kernel Trick [24].
One-class support vector machine (HSVMo)
The one-class HSVM was the third model. The one-class model can be considered as an appearance model and can distinguish between individual layers [24]. Consequently, during the tracking stage, the confidence score of the candidate samples, chosen according to the tracking strategy used, was calculated using the HSVM model after feature extraction. The candidate region corresponding to the highest score was the tracking result of the current frame (Fig. 2c). After obtaining the tracking result for the current frame, we decided whether or not it was necessary to re-sample for model re-training in order to adapt to changes in target appearance.
One difference between the HSVMo and the first two models was that it used the entire tracking result region of each previous frame as the training sample. The confidence score of a candidate sample xi was calculated as follows:
$$ con{f}_o(x)={\displaystyle \sum_i{a}_i}{k}_o\left({x}_i,x\right) $$
where ai was the Lagrange Multiplier and ko(xi, x) was the Kernel Trick [24].
After obtaining these three sub-models, the confidence score of a candidate sample xi was calculated by
$$ conf(x)=\raisebox{1ex}{$\left({w}_o\ast conf{n}_o(x)+{w}_r\ast conf{n}_r(x)+{w}_b\ast conf{n}_b(x)\right)$}\!\left/ \!\raisebox{-1ex}{${w}_o+{w}_r+{w}_b$}\right. $$
where confno(x), confnr(x), and confnb(x) were the results after normalizing confo(x), confr(x), and confb(x) into the range [0,1]. wo, wr, and wb, corresponded to the weights of each sub-model, respectively. The weights of each sub-model determined the relative contribution of each HSVM. HSVMb, adopted the binary classification, and was robust to changes in bird pose and therefore it worked the best for monitoring preening and flapping of wings for example. HSVMr effectively solved the drift problem. It had the best results for when the test hens were close to each other. HSVMo was not sensitive to a fast-changing background and therefore had good performance to monitor sudden movements from the hens [24]. Considering the adaptation of the different support vector machines to different scenarios and the results of repeated attempts, wo, wr, and wb were set to 0.3, 0.6, and 0.1.
In the tracking phase, the candidate samples were obtained around the tracking object. The model scoring was applied to select the best tracking results. The specific process was as follows:
The tracking result of the previous frame was set as the initial target area To{wo, ho, ao, and co} of the current frame.
co was set as the center of rotation. The target area was rotated h times in clockwise and counterclockwise directions, respectively. Each rotation was deflected by k degrees. If the coordinate of point X was (x,y) before the rotation, it became (x',y') after the rotation and the mapping formula was
$$ y\hbox{'}=\left(x-{x}_0\right)\times sin\left({a}_0+{\left(-1\right)}^ih\times k\right)+\left(y-{y}_0\right)\times cos\left({a}_0+{\left(-1\right)}^ih\times k\right)+{y}_0 $$
$$ x\hbox{'}=\left(x-{x}_0\right)\times cos\left({a}_0+{\left(-1\right)}^ih\times k\right)-\left(y-{y}_0\right)\times sin\left({a}_0+{\left(-1\right)}^ih\times k\right)+{x}_0 $$
where the coordinate of c0 was (x0,y0). If the rotation direction was clockwise, i = 1; otherwise i =2.
There were a total of 2×h + 1 candidate regions. After the features were extracted from these regions, the HSVM model was used to calculate their confidence score. The candidate region with the highest score was chosen as the best tracking region Ta{wa,ha,aa,ca}, with respect to the angle. In the current experiment, h was set to 5 and k was set to 3;
Ta was expanded m times to obtain the shift search area Tm{wm,hm,am,cm}, where wm = m×wa,hm = m×ha,am = aa,and cm = ca. The search box Ts{ws, hs, as, cs} was used to search the entire shift search area, where the initial value of the search box was ws = wa, hs = ha, and as = aa. If the coordinate of ca was (xa,ya) and the coordinate of cs was (xs,ys), then
$$ {x}_s=-0.1\times W\times cos{\alpha}_a+0.1\times H\times sin{\alpha}_a+{x}_a $$
$$ {y}_s=-0.1\times W\times sin{\alpha}_a-0.1\times H\times cos{\alpha}_a+{y}_a $$
The search box maintained the same size and angle during the search process, while displacing it by M and N steps in the indicated direction along the width and height of the search area, respectively. When the search box was moved i times along the width and j times along the height, ws, hs, and as remained unchanged, and the coordinates of cs were calculated as follows:
$$ x\hbox{'}=\left(-\left(m-1\right)\times \frac{1}{2}\times {W}_a+\raisebox{1ex}{$\left(m-1\right)$}\!\left/ \!\raisebox{-1ex}{$M\times {W}_a\times i$}\right.\right)\times \cos {a}_a-\left(-\left(m-1\right)\times \frac{1}{2}\times {H}_a+\raisebox{1ex}{$\left(m-1\right)$}\!\left/ \!\raisebox{-1ex}{$N\times {H}_a\times j$}\right.\right)\times \sin {a}_a+{x}_a $$
$$ y\hbox{'}=\left(-\left(m-1\right)\times \frac{1}{2}\times {W}_a+\raisebox{1ex}{$\left(m-1\right)$}\!\left/ \!\raisebox{-1ex}{$M\times {W}_a\times i$}\right.\right)\times \sin {a}_a+\left(-\left(m-1\right)\times \frac{1}{2}\times {H}_a+\raisebox{1ex}{$\left(m-1\right)$}\!\left/ \!\raisebox{-1ex}{$N\times {H}_a\times j$}\right.\right)\times \cos {a}_a+{y}_a $$
Thus, there were a total of M×N regions. After extracting the features of these regions and scoring them using the HSVM model, the candidate region with the highest score was selected as the best region, with respect to displacement (which was an initial target region of tracking). In this study, m = 1.2, M = 5, and N = 5.
The steps (b) and (c) were alternated until the two adjacent quasi-tracking areas coincided. At this time, the corresponding tracking box became the tracking area of this frame image (Fig. 3).
Schematic diagram of the tracking process. The tracking object is indicated by an ellipse; the blue box represents the best tracking area of the current step; the orange box represents the location of the tracking box in previous steps; the red dashed boxes represent the candidate regions. The best region is selected from the candidate regions
Because histogram of orientation gradient feature extraction is relatively time-consuming, the displacement and angle of laying hens were tracked separately. Firstly, the algorithm tracked the change in the angle and subsequently the change in the displacement, and was iterated until there was no more movement. In this way, the number of sampling iterations was effectively reduced. This method had no significant impact on the final results and effectively improved the real-time performance of the algorithm. For instance, in an iterative process, the number of sampling iterations of the tracking strategy was M×N + 2H + 1, while this number increased to (M×N)×(2H + 1) if the displacement and angle were tracked simultaneously.
Because a hen uses a non-rigid body motion, its appearance may change significantly during movement, especially if it turns, or if some of its body is partially obscured. To accommodate the hens' changing appearance during movement, the model must be updated.
The degree of change in appearance had to be calculated after the end of each frame of video tracking to determine if it required updating [24]:
$$ d\left({x}_{cur},{x}_j\right)=1- \max \frac{\left\langle {x}_{cur},{x}_j\right\rangle }{\left\Vert {x}_{cur}\right\Vert \bullet \left\Vert {x}_j\right\Vert } $$
In the above formula, xcur was the characteristic value of the tracking result of the current frame and xj was the characteristic value of the previous tracking results of each frame.
If d(xcur, xj) was less than a pre-set value (0.05 in our experiment), the data was re-sampled and then retrained for the model. The re-sampling rules were as follows:
For the binary HSVM, the image area corresponding to the image tracking box was taken as a positive sample to be inserted into the queue of 40 positive samples using the first-in-first-out strategy. Sometimes the target was blocked for a considerable duration of time and all of the positive samples corresponded to the blocked target. This could have resulted in drift problems. This problem was solved by reserving the 10 initial positive samples. In addition, 50 negative samples were randomly selected to replace all of the former negative samples.
For the regression HSVM, the sampling method for positive samples was the same as the method for the binary HSVM. For negative samples, 20 negative samples were randomly selected to replace the original negative samples.
For the one-class HSVM, the sampling method for positive samples was the same as the method for the binary HSVM. The whole algorithm process is shown in (Fig. 4).
The flow chart of the HSVM tracker
The two most important criteria for the evaluation of algorithm tracking methods are real-time operation and robustness. The HSVM was implemented in OpenCV on a personal computer with a 3.50GHz Intel® Core™ i2-4150. It achieved an average speed of about 9.1 frames per second.
One Hyline Brown hen was chosen from the 6 observation objects as the tracking target. HSVM was compared with 5 other algorithms including Frag [25], TLD [26], PLS [27] (these three algorithms can all be downloaded from the homepage of the original author), MeanShift, and the Particle Filter Algorithm (these two are widely used classical algorithms). Each of these algorithms were used to track the target hen in the experimental video. Three experiments with 3 different randomly-selected tracking targets were conducted and the 6 algorithms were compared in these 3 experiments. The results are shown in (Fig. 5).
Experimental results of the six algorithms: (a) HSVM; (b) TLD; (c) Frag; (d) Particle filter; (e) MeanShift; (f) PLS
To assess the robustness of the algorithm, the overlap rate (OR) was used to quantify the tracking accuracy. The overlap rate was calculated as:
$$ OR=\frac{area\left({R}_t{\displaystyle \cap {R}_l}\right)}{area\left({R}_t{\displaystyle \cup {R}_l}\right)} $$
where Rt represented the results of the tracking and Rl represented the ground truth.
The overlap rates were calculated for the 6 aforementioned algorithms (Fig. 6). The vertical axis of the statistical graph represented the overlap rate. Higher overlap scores indicated more accurate tracking while an overlap rate of 0 indicated that the algorithm completely lost the tracking targets. Figure 6a shows that for most frames, HSVM maintained an overlap rate of approximately 0.8.
Overlap rate of the six algorithms: (a) HSVM; (b) TLD; (c) Frag; (d) Particle filter; (e) MeanShift; (f) PLS. The vertical axis of the statistical graph represents the overlap rate. Higher overlap rate scores indicate more accurate tracking, while an overlap rate of 0 indicates that the algorithm completely lost the tracking targets. The horizontal axis of the statistical graph represents the frame number of the images which have been labeled
An aggregation of the laying hens occurred during the 430th–600th frames. The hens' mutual occlusion sent the overlap rate on a downward trend but the algorithm self-adjusted to recover an overlap rate of approximately 0.8. Figure 6b shows the statistical graph for the TLD algorithm.
The overlap rate curve dropped significantly at the beginning, indicating that the drift of the tracking box increased until the tracking box missed the target. The tracking box only rebounded to the target for a short period of time in the middle part of the frames. Figure 6c shows the graph for the Frag algorithm. The overlap rate curve decreased until the overlap rate was approximately 0.4 because the target hen kept changing its direction of movement. The curve then maintained this value for some time. After the 430th frame, the overlap rate curve declined again until the tracking box missed the target because of the aggregation of hens.
The Particle Filter Algorithm lost and retrieved the target frequently during the tracking process. As a result, the value of its overlap rate varied between 0 and 0.5, as shown in Fig. 6d, but it quickly recovered the target hen each time it lost it. Figure 6e shows that the MeanShift Algorithm tracking boxes expanded easily when the target hen got close to other laying hens resulting in the decline of the overlap rate curve. When the hens aggregated around the 430th frame, the tracking box simply expanded instead of losing the target. Therefore, after the 430th frame, the overlap rate curve did not suffer an obvious drop. The tracking box lost the target and stayed on the flock of hens when the target hen left the flock. Subsequently, the tracking box was transferred to other laying hens until the target hen and tracking box coincided again. The overlap rate curve of the PLS algorithm showed relatively stable performance, overall, and the value of overlap rate was approximately 0.6. Even so, the curve began to decline around the 430th frame until the tracking box lost the target.
From the figures described above, each algorithm adapted to different situations in the movement of laying hens. The average overlap rate is shown in Table 1 according to the different scenarios in the 778 images.
Average overlap rate for the six algorithms conducted for different scenarios
Average overlap rate
HSVM
MeanShift
Change of direction
Two hens mutual occlusion
Preening
Multi hens mutual occlusion
Table 1 shows that HSVM obtained a higher average overlap rate than the other algorithms both with respect to the total average overlap rate and for the different particular scenarios. The value of the overall average overlap rate was 35 % higher than the highest value among the other algorithms. When tracking a single target in a multi-hen mutual occlusion situation (the most challenging scenario), HSVM's average overlap rate was 68 %, which was 41 % higher than the highest value attained for the other algorithms. HSVM was relatively stable with the average overlap rate maintained between 68 and 79 % across the specific cases and the overall average. The PLS algorithm attained the best performance among the contrast algorithms because the PLS was able to model the correlation of target appearance and class labels due to its capacity for both dimensionality reduction and classification [27]. The value of the average overlap rate for the changing of direction, two hens' mutual occlusion, and preening scenarios was 55, 61 and 62 % respectively. However, PLS performed poorly in handling the heavy occlusion, which can easily and quickly change the appearance of targets [28]. In the situation of multiple hens' mutual occlusion, PLS lost the target hen for some frames resulting in a drop in the average overlap rate to 23 %. For the situation of multiple hens' mutual occlusion, the best performance (excluding that of HSVM) was achieved by the Particle Filter Algorithm, whereby the average overlap rate only reached 27 %.
The TLD algorithm used the optical flow method to track the object, which meant the following three conditions had to be satisfied. First, the change of luminance in the different frames should be very small. Secondly, the content of two adjacent frames should change very slowly. Finally, the projections of nearby image points were nearby points and shared similar speed [29]. The lighting in our hen house was not uniform and could not be kept stable. Moreover, hens often made sudden and quick movements such that the average overlap rate of TLD was only 11 %. The reason is that the true target was blurred, and it was difficult for the TLD to distinguish it from the background [30].
The MeanShift tracker had the advantage of low complexity, but it also failed with fast motion, illumination changes, cluttered background and occlusion [31, 32]. The average overlap rate of the MeanShift tracker was only 17 % higher than that of TLD. The Particle Filter Algorithm tracked the object by predicting its location in the next frame. It worked well when the object was briefly blocked. However, if the occlusions lasted for a longer duration, the tracking was more likely to fail [33]. Furthermore, the Particle Filter Algorithm lost the target during quick or sudden movements [17]. Thus, the average overlap rate of the Particle Filter Algorithm was similar to that of the MeanShift tracker. The Frag can cope with many different situations due to the use of local appearance models [34]. But Frag performed poorly in this experiment because it could not handle drastic appearance changes [35–37], so the average overlap rate of Frag was only 28 %.
Figure 6 and Table 1 demonstrate that our HSVM tracker was superior to the classical methods and existing state-of-the-art methods, with respect to better coverage and robustness on the testing sequences.
HSVM owes its success to the following aspects. First, the algorithm used histogram of orientation gradient features to detect laying hens and this effectively described the contour of the laying hens. Secondly, a new type of tracking strategy that accounted for the laying hens' displacement and their body angle improved the tracking accuracy. Third, although the histogram of orientation gradient feature extraction was time-consuming, the algorithm still had a good real-time performance by optimizing the tracking process and reducing the number of sampling iterations.
Although the HSVM algorithm showed impressive potential, there are still areas that need improvement. The histogram of orientation gradient feature was based on the object edge gradient (Fig. 7). Thus, if the tracking object is significantly occluded for a long time, the HSVM algorithm may also lose track of the object. In this experiment, the stocking density was not too high, and this situation happened only a few times in the videos. In further research, the stocking density will be increased to explore approaches to improve the robustness of the algorithm.
Visualization results of the histogram of orientation gradient feature used in the HSVM track
In this paper, a laying hen tracking algorithm based on the HSVM was developed to track a single hen within a flock of hens under a floor system. The experimental results showed that the algorithm achieved better robustness and real-time performance than other comparable algorithms, indicating that HSVM has a substantial practical value in the field. Because it does not require the support of a sensor, the HSVM had better application prospects. With the tracking approach, we can classify the laying hens' behavior to achieve automatic recognition. To improve the average overlap rate in future work, we will investigate a method to adjust the size of the tracking box based on the size change of the moving tracking targets.
The authors sincerely thank Dr. Phil Thacker (University of Saskatchewan) for his help in polishing this manuscript.
We greatly appreciate the financial support provided by the Key Projects in the National Science & Technology Pillar Program during the Twelfth Five-year Plan Period (No. 2014BAD08B05). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
The datasets during and/or analyzed during the current study available from the corresponding authors on reasonable request.
MC conceived and designed the experimental plan. CW and XZ performed the experiments, and HC analyzed the data. CW wrote the paper. All authors read and approved the final manuscript.
College of Information and Electrical Engineering, China Agricultural University, Beijing, 100083, China
Network Center, China Agricultural University, Beijing, 100083, China
College of Water Resources and Civil Engineering, China Agricultural University, Beijing, 100083, China
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CommonCrawl
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What would the equilibrium temperature be at the poles in a world without seasonality?
Inspired by: How does Antarctica stay frozen?
If the Earth was in a fixed solstice state - northern winter and southern summer (e.g. the axis obliquity rotated with the Earth's orbit), what would the equilibrium temperature be at each pole? Assume daily cycle and convection and so on operate as usual, just that it's permanent midnight in the Arctic, and permanent midday in the Antarctic.
Bonus questions: Would the equilibrium temperatures be significantly different if it was northern summer and southern winter (e.g. would land masses and orography have much effect, relative to the solar imbalance)?
hypothetical poles axial-obliquity seasons
naught101naught101
$\begingroup$ You would have to run a climate model while forcing the sunlight from the solstice state. $\endgroup$ – Communisty Sep 19 '18 at 11:22
If we consider that by "Assume daily cycle and convection and so on operate as usual" you meant that all heat transport from/to the pole remain as it is today. Then, we can do a back of the envelope calculation. This calculation will at least give you an order-of-magnitude answer, and we can then consider everything that would affect the result.
Currently both poles radiate more energy than they receive trough direct solar irradiation. This is because they are colder than the rest of the planet. Therefore, there is a neat heat transport from the equatorial latitudes to the poles. That heat travels on atmospheric and oceanic currents. This figure nicely summarize this radiation balance:
Figure from here ©The COMET Program
If we consider that the heat transport depicted at the bottom of the figure remains the same. The only change in the energy budget would be from solar irradiation, and once they start warming the only way to release the additional heat and reach equilibrium would be trough infrared radiation (black body radiation).
The poles receive zero energy on the equinox and the winter solstice and 12.64 kWh/m$^2$ per day on the summer solstice. Therefore a rough yearly average would be 3.16 kWh/m$^2$ per day. So, if we were locked on a summer solstice state you would receive four time more energy, and none if locked in a winter solstice state.
Summer solstice locked pole
Let's consider the summer solstice case first. If we call $E_1$ the current amount of outgoing radiation from the pole, and $T_1$ the current temperature. The Stefan–Boltzmann law states that
$E_1 = \sigma T_1^4$
Analogously, if $E_2$ and $T_2$ represent the summer solstice locked state, you have
Considering that $E_2 = 4 E_1$
You can write that
$4 = \frac{T_2^4}{T_1^4}$
$T_2 = \left(\frac{1}{4}\right)^4 T_1 = 1.41\, T_1$
Therefore the temperature would increase 41%. It doesn't sounds much, but considering that the temperatures above are in Kelvin, it es a lot! In fact, with the current mean temperature in the north pole of -16 °C (257K), it would go up to +89°C!! And the South pole would go from the current -49°C (224K) to +42°C.
This diverges from any posible reality because with 89°C at the north pole there would not be a net influx of energy from the equatorial latitudes as we are assuming. Instead, atmospheric and oceanic currents would take a lot of that heat down south and the temperature would equilibrarte at a much lower value.
Note that in the above calculation I've ignored the energy flux due to advection in atmospheric/oceanic currents. I did so because that flux is about five time smaller than the flux from solar radiation, so it won't change much the results and ignoring it simplifies the math. You can do the calculation considering those flows if you want.
Winter solstice locked pole
In the case of a pole locked to a winter solstice. There would be no incoming solar radiation at all. Therefore, all incoming energy would be transported by currents from equatorial latitudes. In that way, we can see in the above figure that the north pole receives about 150 W/m$^2$, and the south pole about 100 W/m$^2$. The equilibrium temperature would be reached when they radiate the same amount. So, using Stefan–Boltzmann law again we can do some rough calculations. Let's call the south pole temperature $T_{SP}$ and $T_{NP}$ for the north pole. Then:
$T_{SP} = \left(\frac{E_{SP}}{\sigma}\right)^{1/4} = \left(\frac{100 \,W/m^2}{5.67\times 10^{-8}}\right)^{1/4} = 205 K= -68°C$
And for the north pole
$T_{NP} = \left(\frac{E_{NP}}{\sigma}\right)^{1/4} = \left(\frac{150 \,W/m^2}{5.67\times 10^{-8}}\right)^{1/4} = 227 K= -46°C$
But again, with such cold temperatures the actual inflow of energy would be grater and the equilibrium temperature would not be that extreme. In any case, that condition would be favourable to the growth of a massive ice sheet in the corresponding polar area, in the same way I describe in this question and answer.
As a final thought. For this condition to happen in real life you would need something analogous to Tidal locking to lock the period of Earth's precession (currently about 26,000 years) to exactly one year. And don't know of any mechanism that could achieve that.
Camilo RadaCamilo Rada
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CommonCrawl
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Consider the following LP. Apply surplus variables & initial tableau. Then use revised simplex method to obtain the tableau for basic variables
Consider this LP problem.
\begin{array}{cccll} \min & Z= & 8x & +10y+25z & \\ \text{s.t.} & & 2x & \phantom{+10y}+ 2z & \ge 60 \\ & & 2x & +4y+5z & \ge 70 \\ & & & \phantom{+}3y+z & \ge 27 \\ & & & x,y,z & \ge 0 \end{array}
I am confused as to why I have to do a revised simplex method here since the basic matrix has the identity form already.
But if I still apply the "rule" we learned in class to obtain the tableau, the tableau looks the same as our original tableau. Where am I doing wrong?
linear-programming
GNUSupporter 8964民主女神 地下教會
Jordy NelsonJordy Nelson
Add surplus variables $s_1,s_2,s_3$ into the LP.
\begin{array}{rrrrrll} \min & Z= & 8x & +10y & + 25z & & \\ \text{s.t.} & & 2x & & + 2z & - s_1 & = 60 \\ & & 2x & +4y & + 5z & - s_2 & = 70 \\ & & & 3y & + z & - s_3 & = 27 \end{array}
$$x,y,z,s_1,s_2,s_3 \ge 0$$
Initially, we have $s_1,s_2,s_3$ as basic variables. Multiplying the above three equations by $-1$ gives basic matrix $I_3$, but $s_1,s_2,s_3$ are all negative, so it's a basic infeasible solution. To find a BFS (basic feasible solution), you need the revised simplex method. In the intial simplex tableau below, we transform $\min Z$ to $\max Z = -8x - 10y - 25z$ without changing any constraints. (I note $x,y,z$ as $x_1,x_2,x_3$ respectively.)
\begin{array}{rrrrrrr|r} & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & \\ \hline s_1 & -2 & 0 & -2 & 1 & 0 & 0 & -60 \\ s_2 & -2 & -4 & -5 & 0 & 1 & 0 & -70 \\ s_3 & 0 & -3 & -1 & 0 & 0 & 1 & -27 \\ \hline & 8 & 10 & 25 & 0 & 0 & 0 & 0 \\ \end{array}
Using the two-phase method or the big-M method to find a basic feasible solution is much slower than the dual simplex method.
\begin{array}{rrrrrrr|r} & x_1 & x_2 & x_3 & s_1 & s_2 & s_3 & \\ \hline s_1 & -2 & 0 & -2 & 1 & 0 & 0 & -60 \\ s_2 & -2 & -4^* & -5 & 0 & 1 & 0 & -70 \\ s_3 & 0 & -3 & -1 & 0 & 0 & 1 & -27 \\ \hline & 8 & 10 & 25 & 0 & 0 & 0 & 0 \\ \hline\hline s_1 & -2^* & 0 & -2 & 1 & 0 & 0 & -60 \\ x_2 & 1/2 & 1 & 5/4 & 0 & -1/4 & 0 & 35/2 \\ s_3 & 3/2 & 0 & 11/4 & 0 & -3/4 & 1 & 51/2 \\ \hline & 3 & 0 & 25/2 & 0 & 5/2 & 0 & -175 \\ \hline\hline x_1 & 1 & 0 & 1 & -1/2 & 0 & 0 & 30 \\ x_2 & 0 & 1 & 3/4 & 1/4 & -1/4 & 0 & 5/2 \\ s_3 & 0 & 0 & 5/4 & 3/4 & -3/4^* & 1 & -39/2 \\ \hline & 0 & 0 & 19/2 & 3/2 & 5/2 & 0 & -265 \\ \hline\hline x_1 & 1 & 0 & 1 & -1/2 & 0 & 0 & 30 \\ x_2 & 0 & 1 & 1/3 & 0 & 0 & -1/3 & 9 \\ s_2 & 0 & 0 & -5/3 & -1 & 1 & -4/3 & 26 \\ \hline & 0 & 0 & 41/3 & 4 & 0 & 10/3 & -330 \end{array}
Therefore, the optimal solution is $(x^*,y^*,z^*)=(30,9,0)$.
GNUSupporter 8964民主女神 地下教會GNUSupporter 8964民主女神 地下教會
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Area of a surface by rotating the curve about the x-axis.
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CommonCrawl
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\begin{document}
\title{Implementing quantum algorithms on temporal photonic cluster states}
\author{Daiqin Su} \email{[email protected]} \author{Krishna Kumar Sabapathy}
\author{Casey R. Myers}
\author{Haoyu Qi}
\author{Christian Weedbrook}
\author{Kamil Br\'adler}
\affiliation{Xanadu, 372 Richmond Street West, Toronto, Ontario M5V 1X6, Canada }
\begin{abstract} { Implementing quantum algorithms is essential for quantum computation. We study the implementation of three quantum algorithms by performing homodyne measurements on a two-dimensional temporal continuous-variable cluster state. We first review the generation of temporal cluster states and the implementation of gates using the measurement-based model. Alongside this we discuss methods to introduce non-Gaussianity into the cluster states. The first algorithm we consider is Gaussian Boson Sampling in which only Gaussian unitaries need to be implemented. Taking into account the fact that input states are also Gaussian, the errors due to the effect of finite squeezing can be corrected, provided a moderate amount of online squeezing is available. This helps to construct a large Gaussian Boson Sampling machine. The second algorithm is the continuous-variable Instantaneous Quantum Polynomial circuit in which one needs to implement non-Gaussian gates, such as the cubic phase gate. We discuss several methods of implementing the cubic phase gate and fit them into the temporal cluster state architecture. The third algorithm is the continuous-variable version of Grover's search algorithm, the main challenge of which is the implementation of the inversion operator. We propose a method to implement the inversion operator by injecting a resource state into a teleportation circuit. The resource state is simulated using the Strawberry Fields quantum software package.} \end{abstract}
\maketitle
\tableofcontents
\section{Introduction}
Measurement-based (or one way) quantum computation is a particular model of quantum computation \cite{Raussendorf2001}. It is based on a multipartite entangled resource state called a cluster state \cite{Briegel2001}, and local measurements. For continuous-variable (CV) measurement-based quantum computation \cite{Menicucci2006, Loock2007example, Gu2009}, the cluster state is a highly entangled multimode Gaussian state and the required local measurements are homodyne and non-Gaussian measurements. One of the main challenges of measurement-based quantum computation is to generate a scalable and universal cluster state. Several ways of generating CV cluster states have been proposed \cite{Zhang2006, Loock2007, Menicucci2008, Menicucci2010, Menicucci2011} and some of them have been experimentally realised \cite{Yokoyama2013, Yoshikawa2016, Chen2014}. In particular, the temporal CV cluster state architecture is advantageous in terms of the scalability \cite{Alexander2018Universal} because it requires only a small number of optical elements. A one-dimensional temporal CV cluster state with 10,000 entangled modes \cite{Yokoyama2013}, as well as a one-million-mode version \cite{Yoshikawa2016}, have been experimentally generated. However, measuring the one-dimensional cluster states can only implement single-mode unitaries. To implement arbitrary unitaries, two-dimensional temporal cluster states are required \cite{Menicucci2011}. A method to generate two-dimensional temporal cluster states has been proposed by Menicucci \cite{Menicucci2011}. Given the successful generation of the large one-dimensional temporal cluster states, the experimental realisation of two-dimensional temporal cluster states can be expected in the near future.
Our work in this paper is based on two-dimensional temporal cluster states. The implementation of a set of universal gates (phase shift, squeezing gate, cubic phase gate, beam splitter, {\rm etc.}) via homodyne measurements and non-Gaussian resource states on a two-dimensional temporal cluster state were discussed in Refs. \cite{Alexander2017MBLO, Alexander2018Universal}. This constitutes the first step towards a universal measurement-based quantum computation. The natural next step is to implement some particular algorithms based on this set of universal gates. In this work we focus on implementing three important quantum algorithms: Gaussian Boson Sampling \cite{Hamilton2017}, continuous-variable Instantaneous Quantum Polynomial (CV-IQP) circuit \cite{Douce2017CVIQP} and the CV Grover's search algorithm \cite{Pati2000}.
In Gaussian Boson Sampling a set of squeezed states are injected into a linear multimode interferometer and the output state is measured by photon number resolved detectors (PNR) to obtain the photon statistics. It is evident that only Gaussian states and Gaussian unitaries are involved, and the only non-Gaussian element is the photon number detection. This makes the implementation relatively easy: only Gaussian gates are required. In addition, for the Gaussian unitaries and Gaussian states the errors due to the effect of finite squeezing can be corrected \cite{Su2018EC}, provided a moderate amount of online squeezing is available. It is therefore possible, in principle, to conceive a Gaussian Boson Sampling machine with a large number of modes.
For the CV-IQP circuit \cite{Douce2017CVIQP}, non-Gaussian gates are required. In particular, we consider commuting unitaries that are functions of the position quadrature. The lowest order non-Gaussian gate is the cubic phase gate, which has been studied extensively \cite{Krishna2018ON}. We summarise various implementations of the cubic phase gate and explore which are better suited for measurement-based quantum computation. The direct implementation of higher-order non-Gaussian gates is more challenging. However, they can be decomposed into cubic phase gates and Gaussian gates \cite{BraunsteinLloyd1999}.
The continuous-variable Grover's search algorithm \cite{Grover1997, Pati2000} is another algorithm that requires non-Gaussian gates. In this case the so-called ``Grover diffusion operator'' is challenging and a direct discrete-variable analog cannot be used. Instead this operator must be implemented via non-Gaussian gate teleportation. We consider two methods to implement the algorithm logic, using one and two continuous variable qumodes. We show in both cases that the Grover diffusion operator reduces to a sequence of higher-order quadrature phase gates.
The paper is organised as follows: in Sec \ref{sec:1Dcluster}, we review the generation of one-dimensional temporal cluster states and the implementation of single-mode unitaries. In particular, we focus on the implementation of the cubic phase gate. We also discuss several methods to introduce non-Gaussianity into the temporal cluster state. In Sec. \ref{sec:2Dcluster}, we summarise the generation of the universal two-dimensional cluster states and the implementation of two-mode unitaries, such as the beam splitter. Sec. \ref{sec:GBS} discusses the implementation of the Gaussian Boson Sampling, Sec. \ref{sec:IQP} discusses the implementation of CV-IQP and Sec. \ref{sec:CVGrover} discusses the implementation of CV Grover's search algorithm. We conclude in Sec. \ref{sec:conclusion}.
\section{One-dimensional temporal cluster state}\label{sec:1Dcluster}
\subsection{Gate teleportation and basic elements of graphical representation}\label{sec:gate-teleportation}
CV quantum teleportation \cite{Pirandola2015}, or CV gate teleportation \cite{Weedbrook2012RMP}, is the fundamental building block of CV measurement-based quantum computation. To implement a measurement-based Gaussian unitary, the input mode is coupled with one of the two modes of a two-mode squeezed state, the outputs of which are detected with two homodyne detectors, as shown in Fig. \ref{fig:gate-teleportation}. The input state is then teleported to another mode of the two-mode squeezed state, with an additional Gaussian unitary acting on it. The implemented Gaussian unitary depends on the measurement quadratures of the homodyne detection. By changing the measurement quadratures, an arbitrary single-mode Gaussian unitary can be implemented. Here the two-mode squeezed state plays the role of a resource state for gate teleportation. The implementation of non-Gaussian gates will be discussed in Sec. \ref{sec:cubic}.
\begin{figure}
\caption{Gate teleportation circuit. Two single-mode squeezed vacuum states are generated by squeezing the vacuum using two single-mode squeezers, the action of which is represented by the single-mode squeezing operator $S (r, \theta)$ with $\theta= 0$ and $\pi$, respectively. A two-mode squeezed vacuum
state is produced after the two single-mode squeezed states pass through a beam splitter $B(\frac{\pi}{4})$ (a $50:50$ beam splitter). The input mode (with input state $| \psi_{\rm in} \rangle$) couples with one of the two modes of the two-mode squeezed state via a beam splitter $B(\frac{\pi}{4})$, the outputs of which are detected by two homodyne detectors $D_1$ and $D_3$. The homodyne measurement outcomes ${m}_1$ and ${m}_3$ are used to displace the output state in the other mode of the two-mode squeezed state. The displacement operator is denoted as $D (\boldsymbol m)$ where $\boldsymbol m= (m_1, m_3)$. A unitary is implemented on the input state that depends on the measurement quadratures of the homodyne detectors. }
\label{fig:gate-teleportation}
\end{figure}
\begin{figure*}
\caption{ Elements of the simplified graphical representation of the CV cluster state and measurement-based quantum computation \cite{Alexander2016}. The shaded green dot represents an input mode, the solid black dot represents a general optical mode and the grey square represents an output mode. A blue link between two optical modes represents two-mode entanglement. Two-mode entanglement is generated by applying a $50:50$ beam splitter (BS) to two squeezed pulses with orthogonal squeezing directions. An ellipse encircling two optical modes represents applying a $50:50$ beam splitter to them. An ellipse filled with light blue represents applying a $50:50$ beam splitter and then performing homodyne measurements. }
\label{fig:graph-elements}
\end{figure*}
Our main interest in this paper is the CV cluster state, which is the resource state for universal measurement-based quantum computation. By mentioning a CV cluster state we mean a pure entangled multimode Gaussian state, although non-Gaussian CV cluster states are also available \cite{Walschaers2018nonGaussianCluster}, but are not conventional. It is convenient to represent CV cluster states graphically \cite{Menicucci2011graph}. It is also possible to represent the measurements, input and output modes graphically. We follow Ref. \cite{Alexander2016} and introduce the graphical representation of the CV cluster state and measurement-based quantum computation. Fig. \ref{fig:graph-elements} shows some of the basic graph elements that we are going to use (more elements will be introduced in the following sections). As an example, Fig. \ref{fig:gate-teleportation-graph} (a) shows a graphical representation of the gate teleportation in Fig. \ref{fig:gate-teleportation}, and Fig. \ref{fig:gate-teleportation-graph} (b) shows the corresponding gate model circuit.
\begin{figure}
\caption{ (a) Graphical representation of the gate teleportation shown in Fig. \ref{fig:gate-teleportation}. All relevant elements are introduced in Fig. \ref{fig:graph-elements}.
(b) An equivalent gate model circuit. $| \psi_{\rm in} \rangle$, $| \psi_{\rm out} \rangle$ and $\hat U$ are the input state, output state and implemented unitary respectively. }
\label{fig:gate-teleportation-graph}
\end{figure}
\subsection{Generation of one-dimensional temporal cluster state}
Fig. \ref{fig:standard-cluster-state} shows the graphical representation of a standard one-dimensional and a two-dimensional CV cluster states: each node corresponds to a single optical mode and the links represent correlations between different optical modes. We use different notations in Fig. \ref{fig:standard-cluster-state}, as compared to Fig. \ref{fig:graph-elements}, to show the differences between the standard cluster states and the temporal cluster states discussed in this paper.
\begin{figure}
\caption{ Standard CV cluster states: each circle (node) represents a single optical mode and the link between the modes represents the correlation (entanglement). We introduce different notations here to distinguish them from the temporal CV cluster states discussed in this paper. (a) One-dimensional cluster state. (b) Two-dimensional cluster state. }
\label{fig:standard-cluster-state}
\end{figure}
In this section, we focus on the one-dimensional temporal CV cluster state \cite{Yokoyama2013}, which is generated by the optical setup shown in Fig. \ref{fig:1D-cluster-setup}. Optical parametric oscillators continuously generate pairs of single-mode squeezed vacuum states that are squeezed in orthogonal directions, e.g., one in position quadrature and the other in momentum quadrature. These squeezed pulses are then injected into the optical setup in Fig. \ref{fig:1D-cluster-setup}, which consists of two $50:50$ beam splitters and a delay loop \cite{Yokoyama2013}. The first beam splitter $B_1$ is used to generate a two-mode squeezed vacuum state. The delay loop delays the bottom mode by $\Delta t$, exactly the same as the time interval between two adjacent pairs of single-mode squeezed states. This is to make sure the top mode (non-delay mode) interferes with the delayed mode of an earlier two-mode squeezed state. If the output modes are not detected by the homodyne detectors, a temporal CV cluster state is generated, the graphical representation of which is shown in Fig. \ref{fig:1D-cluster-state}. In contrast to the spatial cluster states, the entanglement of the temporal cluster states is present between optical modes appearing at different times.
From Fig. \ref{fig:1D-cluster-state} (b) we see that the produced temporal cluster state has a width of two nodes. However, it still has dimension of one because it can only be used to implement single-mode unitaries \cite{Menicucci2011}, as will be clear in the next section. To compare with the standard one-dimensional cluster states as in Fig. \ref{fig:standard-cluster-state} (a), and for ease of representation as the dimension increases, we will use a simplified graphical representation instead of ones as shown in Fig. \ref{fig:1D-cluster-state} (b). The one-dimensional temporal cluster state is represented by Fig. \ref{fig:1D-cluster-graph}, where an ellipse and the two optical modes that it encircles are together defined as a macronode \cite{Alexander2016}. The macronode can be considered as an analogue to the node in the standard cluster states shown in Fig. \ref{fig:standard-cluster-state}. However, one has to keep in mind that the ellipse represents a beam splitter transformation on the two modes, as defined in Fig. \ref{fig:graph-elements}.
\begin{figure*}
\caption{ Optical setup that generates one-dimensional temporal cluster states \cite{Menicucci2011}. A series of pairs of single-mode squeezed pulses, with orthogonal squeezing directions, are produced and injected into the optical setup with a repetition time interval $\Delta t$. The evolution of a single pair of single-mode squeezed pulses is illustrated. At time $t_0$, a pair of squeezed pulses are injected into the setup. After passing through the first $50:50$ beam splitter $B_1$, a two-mode squeezed state is produced at $t_1$. The top mode keeps moving while the bottom mode is delayed by the delay loop. When the top mode arrives at the second $50:50$ beam splitter $B_2$ at $t_2$, it couples with the delayed mode of an earlier pair of entangled modes. They hit the homodyne detectors at $t_3$. This process continues until we stop injecting squeezing pulses. In producing the temporal cluster states, we can choose not to detect the optical modes. }
\label{fig:1D-cluster-setup}
\end{figure*}
\begin{figure}
\caption{ One-dimensional temporal cluster state that is produced by the optical setup in Fig. \ref{fig:1D-cluster-setup} \cite{Menicucci2011, Menicucci2011graph}. (a) A series of two-mode squeezed vacuum states are generated with time interval $\Delta t$. One of the two modes of each two-mode squeezed state is delayed by $\Delta t$, and couples with the non-delayed mode of the latter two-mode squeezed state via a $50:50$ beam splitter (represented by an ellipse). (b) Graphical representation of the one-dimensional temporal cluster state. Note that the one-dimensional cluster state has a width of two nodes. We basically follow \cite{Menicucci2011} except colouring the nodes. The colour (blue and orange) of the links indicate the signs of the weights. }
\label{fig:1D-cluster-state}
\end{figure}
\begin{figure*}
\caption{ Simplified graphical representation of the one-dimensional temporal cluster state \cite{Alexander2016}.
}
\label{fig:1D-cluster-graph}
\end{figure*}
\subsection{Implementation of single-mode Gaussian gates} In this subsection we recall the measurement-based implementation of Gaussian unitaries in a one-dimensional temporal cluster architecture \cite{Alexander2017MBLO, Alexander2016}. One can implement arbitrary single-mode Gaussian gates by directly choosing the measurement quadratures of the homodyne detection. Homodyne measurements $\hat{p}(\theta_1) = m_1$ and $\hat{p}(\theta_3)=m_3$, as shown in Fig. \ref{fig:gate-teleportation}, result in the implementation of the Gaussian unitary operator $\hat{A}(\theta_1,\theta_3,m_1,m_3)$ given by \cite{Alexander2016} \begin{align} &\hat{A}(\theta_1,\theta_3,m_1,m_3) \nonumber\\ &= D\left[-i\frac{e^{i\theta_1}m_3+e^{i\theta_3}m_1}{\sin(\theta_1-\theta_3)} \right] R(\theta_+) S({\rm ln} \tan{\theta_-}) R(\theta_+), \end{align} where $S(r) = \exp[-r(\hat{a}^2-\hat{a}^{\dagger 2})/2]$, $R(\theta) = \exp(i\theta \hat{a}^{\dag}\hat{a})$, $D(\alpha) = \exp(\alpha \hat{a}^{\dag} - \alpha^* \hat{a})$, $\theta_{\pm} = (\theta_3\pm \theta_1)/2$. The operator is implemented on the immediate next macronode of the one-dimensional cluster that follows the macronode on which the measurement is performed. Note that while $\theta_1,\theta_3$ are the parameters under our control that we can choose depending on the kind of unitary we want to implement, there is an additional displacement factor that is unavoidable and has to be accounted for in the feedforward. This covers all the Gaussian elements (up to displacement correction operators).
\subsection{Implementing single-mode non-Gaussian operations}\label{sec:cubic} With regard to non-Gaussian elements, such as the cubic phase gate, one needs to use an additional optical setup where the non-Gaussianity is injected into the cluster through a resource state. This is achieved through a gate-teleportation circuit where the second homodyne detector in Fig. \ref{fig:gate-teleportation} (corresponding to outcome ${m}_1$) is replaced by an optical setup given by \begin{align} \label{psub0}
\mbox{ \Qcircuit @C=0.7em @R=2.5em { \lstick{{\rm to~ cluster}}& \qw&\multigate{1}{B\left(\frac{\pi}{4}\right)} & \qw &\measureD{\mbox{$\Pi_{p_{\theta}}$}} & \cw ~~~~~~m_1\\
\lstick{\ket{\phi_r}}& \qw&\ghost{B\left(\frac{\pi}{4}\right)} & \qw &\measureD{\mbox{$\Pi_x$}} & \cw ~~~~~~~m_e\,. }} \end{align} In Eq. \eqref{psub0}, $\ket{\phi_r}$ is any suitable resource state, $\Pi_x$ and $\Pi_{p_{\theta}}$ are homodyne measurements of the quadratures $\hat{x}$
and $\hat{p}_{\theta} = \hat{p} \cos{\theta} + \hat{q} \sin{\theta}$. For a resource state \begin{align} \ket{\phi_r} = \int \mathrm d x \phi_r(x) \ket{x} = \phi(\hat{x}) \ket{0}_p, \end{align} with $\ket{x}$ the position eigenstate and $\ket{0}_p = \int \mathrm d x \ket{x}$ is the zero-momentum eigenstate. The outcome operator that applies to the next node of the cluster is given by \cite{Alexander2018Universal} \begin{align} \hat{L}(\phi_r,\theta,\bs{m}) & = Z([\sqrt{2} m_3 - m_e]/2) \hat{M}(\theta,m_1) \hat{\mathcal{E}}(\phi_r,m_e)\nonumber\\ &\quad\times X(-m_3) S(\ln{2}),\nonumber\\ \hat{M}(\theta,m_1) & = X(-2m_1 \,{\rm sec}\theta) R(-\pi/2) S(-\ln{2}) P({\rm tan}\theta),\nonumber\\ \hat{\mathcal{E}}(\phi_r,m_e) & = \sqrt{2} X(-m_e) S(\ln{\sqrt{2}}) \phi_r(\sqrt{2}m_e-\hat{x}), \label{e4} \end{align} where $\boldsymbol m = (m_3, m_1, m_e)$, and $X(s) = e^{-i s \hat p}$ and $Z(s) = e^{i s \hat x}$ are the displacement operators in $\hat x$ and $\hat p$, respectively. We now consider the simplest case where the resource state is an ideal cubic phase state and the implemented gate is the ideal cubic phase gate. \\
\noindent \textbf{Ideal cubic phase gate.} To achieve universal quantum computation using CV quantum systems, non-Gaussian gates are required \cite{BraunsteinLloyd1999}. A non-Gaussian gate corresponds to a Hamiltonian consisting of degree greater than quadratic in position and momentum quadrature operators (or annihilation and creation operators). The simplest and most widely used non-Gaussian gate is the cubic phase gate \cite{gkp}, which is defined as \begin{eqnarray} V (\gamma) =\exp \big(i \gamma \hat x^3/3 \big), \end{eqnarray} where $\gamma$ is the gate strength. Correspondingly, we can define the idealised (and unnormalisable) cubic phase states by applying $\hat V (\gamma)$ to the zero-momentum eigenstate, namely, \begin{eqnarray}
| \phi_{\gamma} \rangle = V (\gamma) \ket{0_p} = \int {\rm d} x ~ e^{i \gamma x^3/3} | x \rangle. \end{eqnarray}
A direct implementation of the cubic phase gate requires strong nonlinearity, such as in a nonlinear crystal, which is very challenging for current experimental techniques. Another way to implement the cubic phase gate is based on the adaptive non-Gaussian measurement (AnGM) \cite{angm} method with a resource state, e.g., the cubic phase state. This method either requires Gaussian feedforward or post selection, the latter implies that it is nondeterministic. However, this method can be well fitted into the measurement-based quantum computation, which itself is based on teleportation.
\begin{figure}
\caption{ Teleportation circuit that implements a cubic phase gate \cite{Alexander2018Universal}. A cubic phase circuit (the part in the orange shaded box) consists of a beam splitter
$B \big({\rm -}\frac{\pi}{4} \big)$ (a 50:50 beam splitter with an additional $\pi$ phase shift), two homodyne detectors that measure quadratures $\hat x_e$ and $\hat p_{\theta_1}$, and an input cubic phase state $| \phi_{\gamma} \rangle$. The whole teleportation circuit is a generalisation of the gate teleportation circuit Fig. \ref{fig:gate-teleportation}, which implements a single-mode Gaussian unitary, by replacing one of the homodyne detectors by the cubic phase circuit. Note that another homodyne detector measures the position quadrature $\hat x_3$. }
\label{fig:cubic-phase-single}
\end{figure}
While there are many types of teleportation circuits, we focus on a particular one which is shown in Fig. \ref{fig:cubic-phase-single} \cite{angm,Alexander2018Universal}. This is basically a generalisation of the circuit in Fig. \ref{fig:gate-teleportation} that implements measurement-based Gaussian unitaries. The ideal cubic phase state
$| \phi_{\gamma} \rangle$ is used as a resource state to implement the cubic phase gate. In the infinite squeezing limit, the circuit in Fig. \ref{fig:cubic-phase-single} implements a unitary \cite{Alexander2018Universal} \begin{eqnarray}\label{eq:cubic-phase-gate} \hat L (\gamma, \sigma, \boldsymbol m) = Z \big(\sqrt{2} m_3 \big) X (\kappa) R \bigg( -\frac{\pi}{2}\bigg) P (\tau) V \big( -2 \sqrt{2} \gamma \big), \nonumber\\ \end{eqnarray} where \begin{eqnarray} \tau &=& 4 \sigma + 4\gamma \big(m_3 + \sqrt{2} m_e \big), \nonumber\\ \kappa &=& -2m_1 \sqrt{1+\sigma^2} - 2\sigma \big(\sqrt{2}m_3 + m_e \big) \nonumber\\ && - \sqrt{2} \gamma \big( m_3 + \sqrt{2} m_e \big)^2, \nonumber\\ \sigma &=& \tan \theta_1, \end{eqnarray} and the overall phase has been neglected. Here $ P(\tau) = \exp \big(i \tau \hat x^2/2 \big)$ is known as the shear operator. From Eq. \eqref{eq:cubic-phase-gate} it is evident that a cubic phase gate is implemented, as well as a series of Gaussian unitaries: displacements, phase shift and local shear. To implement only a cubic phase gate, Gaussian feedforward corrections need to be applied after the homodyne detection. \\
\noindent \textit{Simplification using adaptivity.} At every step of the computation and hence the measurement outcome, there are measurement dependent displacements that are produced. These can be accumulated and corrected at the end of the computation. With the cubic phase gate, one can adjust the angle of the homodyne measurement to account for any quadratic phase gates that results from commuting the displacements across the cubic gate implementation operator as in Eq. \eqref{eq:cubic-phase-gate}, which was already shown in Ref. \cite{Alexander2018Universal}.
\subsubsection{Resource states for approximate/weak cubic phase gate}\label{Sec:WeakCubicPhase} There are various resource states that have been proposed in the literature (for a review see Ref. \cite{Krishna2018ON}). We briefly go through a few of the proposals that are suited for implementation in the cluster architecture. Note from Eq. \eqref{e4} \cite{Alexander2018Universal} that the effective non-Gaussian operator $ \phi_r(\sqrt{2}m_e-\hat{x})$ is in principle similar to the implementation of a GKP circuit \cite{Krishna2018ON,gkp}, which implements the transformation $\phi_r(\hat{x}+q)$, where $q$ is the output of a homodyne measurement in the GKP circuit. The advantage of the AnMG \cite{angm}, when compared to the GKP method, is that the quadratic feedforward corrections that are required can be incorporated into the measurement, whereas for the GKP method. It would require a dynamic quadratic phase gate as a feedforward correction. Both methods require a measurement dependent displacement operation. \\
\noindent \textbf{ON states.} We briefly introduce the use of ON states \cite{Krishna2018ON} as a resource for implementing weak quadrature phase gates given by $\exp\big(i \gamma \hat x^N\big)$, where $N$ is the order of the gate. A general ON state is defined as \begin{align}
\ket{ON} = (1+|a|^2)^{-1/2} \left(\ket{0} + a \ket{N} \right), ~a \in \mathbb{C}\label{eqn:ON}, \end{align} where the Fock excitation $N$ is also the order of the gate we want to implement.
With respect to the implementation of a cubic phase gate we consider as resource an ON state with $N=3$. The $\ket{03}$ state is defined as $\ket{\phi_r} = c_a (\ket{0} + a \ket{3}), \, c_a = (1 + |a|^2)^{-1/2}$. The ON states also serve as a transparent example of how the non-Gaussianity of the resource states in its wave-function gets transferred to the corresponding non-Gaussian operator that is applied to the respective node of the cluster. Choosing $a = -i\gamma \sqrt{3}/2$, $|\gamma| \ll 1$, and following the steps mentioned in Ref. \cite{Krishna2018ON} we have that \begin{align} \phi_r(\kappa-\hat{x}) = \hat{A}_{\kappa} Z(3\gamma (\kappa^2-1/2) ) P(-3\gamma \kappa) V(\gamma), \label{e9} \end{align} where $\hat{A}_{\kappa} = \exp [-(\kappa -\hat{x})^2/2] $ is an unavoidable measurement dependent Gaussian noise factor that results from the choice of resource state (and can be interpreted as a type of finite squeezing effect). Note that we have also neglected the phase factors and overall scaling constants in Eq. \eqref{e9}. We can then substitute the above expression into Eq. \eqref{e4} to get the final effective operation of using the $\ket{ON} =\ket{03}$ state as the ancilla non-Gaussian state in the measurement. The Gaussian elements can all be commuted through the non-Gaussian operation to the left to be corrected at the end of the commutation. We can implement stronger cubic gates by repeating this procedure. This method in principle can be used for higher-order gates using suitable ON states with higher values of $N$. Also a generalised version to implement higher-order non-linear gates using the measurement based computation has been recently proposed in Ref. \cite{h-order}. \\
\noindent \textbf{GKP state.} The GKP state was the first proposal for the approximate preparation of the cubic phase state \cite{gkp}. Here, one arm of a two mode squeezed state is displaced along the momentum quadrature and then measured using a photon number resolving detector. Then depending on the measurement outcome, the other arm is squeezed to generate the cubic phase state of interest, as shown in Eq. \eqref{eq:GKPstate} below: \begin{align}\label{eq:GKPstate}
\mbox{ \Qcircuit @C=0.5em @R=2em { \lstick{\ket{0}}& \qw&\gate{S}& \multigate{1}{B(\pi/4)} & \qw & \gate{Z(w)}&\measureD{\mbox{$\Pi_{n}$}} & \ustick{m} \controlo \cw \cwx[1] \\
\lstick{\ket{0}}& \qw&\gate{S^{-1}}&\ghost{B(\pi/4)} & \qw & \qw&\qw &\gate{S(m)} & \rstick{\ket{\phi_r} ~.} \qw }} \end{align} This is a possible candidate for use as an ancilla resource, as shown in Fig.~\ref{fig:cubic-phase-single}. It turns out that the approximation works well only in the limit of large initial squeezing, displacement and photon-measurement outcome. A later analysis of this procedure put forth some of the experimental challenges of this method \cite{ghose-sanders}.
Since the resource state directly approximates the cubic phase state, its use in the generalised measurement in the cluster will implement an operation that approximates Eq. \eqref{eq:cubic-phase-gate}. \\
\noindent \textbf{MFF state.} The resource state proposed by Marek-Filip-Furasawa (MFF) in Refs. \cite{angm, mff1} is of the form $\ket{\phi_r} = {S}\big(1+i\gamma \hat{x}^3 \big)\ket{0}$, where $\big(1+i\gamma \hat{x}^3 \big)$ is the first-order Taylor expansion of a cubic phase gate $\exp \big(i \gamma \hat{x}^3 \big)$ and $S$ is the squeezing operator. To implement higher-order expansions one has to then generate the appropriate resource state by applying the suitable Taylor expansion operator on vacuum. \\
\noindent \textbf{Gaussian optimised state} \cite{angm}. Here, the authors proposed a state which is a superposition of the first four Fock basis states after performing a Gaussian operation on their test states, i.e., a state of the form \begin{align} \ket{\phi_r} = U_G \left[ \sum_{i=0}^3 c_i \ket{i}\right]. \end{align} This use of Gaussian optimised non-Gaussian state preparation was earlier proposed in Ref. \cite{gaussopt}. The reason that the superposition was truncated at $n=3$ is because it is possible to experimentally prepare these states in the lab \cite{akira-upto3}. The optimisation was performed by minimising the cost function which was the variance of the non-linear quadrature (NLQ) operator \cite{angm} \begin{align} \hat{p}_{\rm NLQ} = \hat{p}^2 - 3 \gamma \hat{x}^2, \end{align} and the mean $\langle\hat{p}_{\rm NLQ}\rangle$.
\subsubsection{Additional methods for non-Gaussian operations in the cluster} Non-Gaussianity is an important resource to have in the cluster state with regard to universal quantum computation \cite{Walschaers2018nonGaussianCluster, Ohliger2010limitation}. We now present two ways to inject non-Gaussianity into the cluster. The first method we explore is what we deem the cluster-gate model, where an off-cluster gate model is used to implement non-Gaussian operations on particular cluster modes in case it is advantageous over the measurement-based method discussed earlier. The second method is to use the already existing optical elements in the cluster architecture but switching the homodyne detectors for photon number-resolving detectors. The final possible location non-Gaussian resources could be incorporated to generate a non-Gaussian cluster state is at the level of the source. We however explore this avenue elsewhere. This subsection entirely focuses on the one-dimensional cluster but the methods can be suitably generalised to the two-dimensional cluster where one can `double' the effects by repeating these methods on the remaining modes in the cluster generation circuit. \\
\begin{figure}
\caption{Wigner function for an input displaced squeezed state with $(\alpha,\xi) = (0.02+i\,0.84,0.5)$ and the two-mode squeezed state with squeeze parameter $r=0.68$, and the PNR measurements $(\kappa_1,\kappa_2)$ reading $(3,2)$, respectively. The Wigner function was computed using the Strawberry Fields software \cite{SFpaper}.
}
\label{fig:pnrsq}
\end{figure}
\begin{figure}
\caption{Wigner function for a coherent state with $\alpha = 0.02+i\,0.84$ and the squeezing parameter of the two-mode squeezed state given by $r=0.68$, with PNR measurements $(\kappa_1,\kappa_2)$ reading $(4,1)$, respectively. The Wigner function was computed using the Strawberry Fields software \cite{SFpaper}. }
\label{fig:pnrcoh}
\end{figure}
\noindent \textbf{Hybrid cluster-gate model.} It is possible that certain realistic implementations of the cubic phase gate may be noisy or challenging to implement using the ancilla-assisted measurement-based model. One could in principle use a switch to take out a particular mode and perform a gate directly and feed it back into the cluster as shown in the circuit below: \begin{align}
\mbox{ \Qcircuit @C=0.7em @R=2.0em { \lstick{{\rm switch~out} ~~\ket{\psi}} & \qw&\multigate{1}{~U~} & \qw && ~~~~~~ T\ket{\psi}~{\rm to~switch~in}\\
\lstick{\ket{\phi_r}}& \qw&\ghost{~U~} & \qw &\measureD{\mbox{$\Pi$}} & \cw ~~~~~~~~~m ~~~~. }} \end{align}
With this scheme, one could in principle apply several variations of the implementation of the cubic phase gate mentioned in Table II of Ref. \cite{Krishna2018ON}. In the event the gate model produces better quality/fidelity of the cubic phase gate, this hybrid approach could prove advantageous.
\begin{figure*}
\caption{ Optical setup that implements gate teleportation, or a series of single-mode unitaries \cite{Alexander2017MBLO}. A switch has been added as compared to Fig. \ref{fig:1D-cluster-setup}, which allows for injecting input states and reading out output states. The states of the switch ({\bf s$_1$}, {\bf s$_2$} and {\bf s$_3$}) have to be set accordingly during the computational process. The homodyne measurement outcomes ${m}_1$ and ${m}_2$ are used to do feedforward. There are two ways of doing feedforward: {\bf f$_1$} and {\bf f$_2$}. {\bf f$_1$} means doing feedforward after every measurement step, while {\bf f$_2$} means doing feedforward at the end of computation. If only Gaussian unitaries are implemented, both ways are allowable. However, if non-Gaussian unitaries are implemented one has to do {\bf f$_1$}. }
\label{fig:1D-cluster-computation-setup}
\end{figure*}
In the ancilla-based computation, the gate is applied by using a switch at one of the homodyne detectors and applying a general adaptive non-Gaussian measurement that implements a non-Gaussian operation on a cluster node. In the proposed hybrid model, we use the switch of Fig. \ref{fig:1D-cluster-computation-setup} first in mode {\bf s$_3$}, then we apply the gate and feed it into the cluster using mode {\bf s$_2$} of the switch. Then one can use the cluster and perform the measurement-based computation on the modes. \\
The current state-of-the-art generation of squeezed pulses is at a repetition rate in the GHz range \cite{Mangold2014}. Other experimental elements that would have an effect on the rate at which computation can be performed include optical switches, adaptive homodyne measurements, and non-Gaussian resource generation elements such as photon number resolving detectors. \\
\noindent \textbf{PNR induced non-Gaussian operations.} While it is necessary to have precise control over the gates for specific algorithms where the gate order is crucial, it turns out that having any non-linearity in the cluster would still be advantageous for other applications such as quantum machine learning \cite{Biamonte2017}. To this end, we obtain the operator that is implemented when the homodyne measurements are replaced by PNR detectors using a switch between the two types of measurements.
\begin{figure}
\caption{Measurement probabilities $P(n_1,n_2)$ for input displaced squeezed state with $(\alpha,\xi) = (0.02+i\,0.84,0.5)$ and the two-mode squeezed state with squeeze parameter $r=0.68$. Measurement outcomes with $\kappa_1=\kappa_2$ are more favorable. The probability was computed using the Strawberry Fields software \cite{SFpaper}. }
\label{fig:pnrsqprob}
\end{figure}
\begin{figure}
\caption{Measurement probabilities $P(n_1,n_2)$ for an input coherent state with $\alpha = 0.02+i\,0.84$ and the squeezing parameter of the two-mode squeezed state given by $r=0.68$. Measurement outcomes are observed to be skewed away from $\kappa_1 = \kappa_2$. The probability was computed using the Strawberry Fields software \cite{SFpaper}. }
\label{fig:pnrcohprob}
\end{figure}
The basic one-dimensional cluster circuit with the homodyne measurements replaced by the photon-number resolving (PNR) detectors is given by \begin{align} \label{psub}
\mbox{ \Qcircuit @C=0.45em @R=1.5em { \lstick{\ket{\psi_{\rm in}}} & \qw &\qw&\qw&\multigate{1}{B(\pi/4)}& \measureD{\Pi_n} & \cw ~~~~ \kappa_1\\ \lstick{\ket{0}} &\gate{S(r)}&\multigate{1}{B(\pi/4)} &\qw& \ghost{B(\pi/4)}&\measureD{\Pi_n}& \cw ~~~~ \kappa_2\\ \lstick{\ket{0}}&\gate{S(r)^{-1}}& \ghost{B(\pi/4)}& \qw&\qw&\qw & ~~~~\widehat{T}_{\kappa_1,\kappa_2} \ket{\psi_{\rm in}}.\\ }} \end{align} The single-mode squeezers $S(r)$ and the $50:50$ beam splitter $B(\pi/4)$ action on the two vacuum states produces the standard two-mode squeezed state $\ket{\psi_{\rm sq}(r)}$. So we need to evaluate \begin{align} &\widehat{T}_{\kappa_1,\kappa_2} \ket{\psi_{\rm in}} = [\bra{\kappa_1} \bra{\kappa_2} ][ B(\pi/4) \otimes \mathbb{I}][\ket{\psi_{\rm in}} \ket{\psi_{\rm sq}(r)}], \nonumber \\ & \ket{\psi_{\rm sq}(r)} = {\rm sech\,r} \sum_{j=0}^{\infty} ({\rm tanh\,r})^n \ket{jj}. \label{pnrops} \end{align}
The matrix elements of the beam splitter were previously derived in \cite{kraus10} and employed in \cite{nong} to study Gaussian and non-Gaussian channels. We have that \begin{align} B(\pi/4) = \sum_{m_1=0}^{\infty} \sum_{m_2 =0}^{\infty} \sum_{n_1 =0}^{\infty} \sum_{n_2 =0}^{\infty} C^{m_1m_2}_{n_1n_2} \ket{m_1 m_2} \bra{n_1n_2}, \end{align} where \begin{align} C^{m_1m_2}_{n_1n_2} &= \sqrt{\frac{m_1!m_2!}{n_1!n_2!}} \sum_{r=0}^{n_1} \sum_{j=0}^{n_2} \binom{n_1}{r} \binom{n_2}{j} (-1)^{n_2-j} \nonumber\\ & \times 2^{-(n_1+n_2)} \delta_{m_2,r+j} \,\,\delta_{m_1,n_1+n_2-r-j}. \label{coeff} \end{align} Substituting Eq. \eqref{coeff} into Eq. \eqref{pnrops} and after some simplification we get that \begin{align} \ket{\psi_{\rm out}} &= \widehat{T}_{\kappa_1,\kappa_2} \ket{\psi_{\rm in}} \propto \sum_{n=0}^{\kappa_1+\kappa_2} \sum_{r=r_{\rm min}}^{r_{\rm max}} \sum_{j=0}^n ({\rm tanh\,r})^n (-1)^{n-j} \nonumber\\ & \times \binom{\kappa_1+\kappa_2 -n}{r} \binom{n}{\kappa_2-r} \braket{\kappa_1+\kappa_2 -n}{\psi_{\rm in}} \ket{n}, \end{align} where \begin{align} r_{\rm min} &= \max\{0,\kappa_2-n\}, \, \nonumber \\ r_{\rm max} &= \max\{\kappa_2,\kappa_1+\kappa_2 -n\}. \end{align} The final physical state is then given by normalising $\ket{\psi_{\rm out}}$.
We now provide some examples of the resulting action of the PNR measurements on different input modes. The Wigner function of the output for measurement outcomes $(\kappa_1,\kappa_2) = (3,2)$ for an input coherent state is given in Fig. \ref{fig:pnrsq} and outcomes $(4,1)$ for a squeezed state in Fig. \ref{fig:pnrcoh}.
In Fig. \ref{fig:pnrsqprob} we also plot the probability of obtaining the various Fock outcomes for the input squeezed state. We observe that for a fixed total number of photon detections, the measurement outcomes $\kappa_1 = \kappa_2$ are more favorable. Similarly, we plot the probability distribution for various PNR measurement outcome pairs for the input coherent state in Fig.\,\ref{fig:pnrcohprob}. We find that the measurements tend to be skewed away from $\kappa_1= \kappa_2$.
\subsection{Teleportation along a one-dimensional temporal CV cluster state} The main purpose of generating CV temporal cluster states is to achieve quantum computation, e.g., to implement unitary transformations. To do that we have to inject input states into the cluster and readout the output states after performing some unitary transformations. The one-dimensional CV temporal cluster states are not universal because they can only be used to implement single-mode unitaries. However, it is instructive to introduce how to inject input states, implement single-mode unitaries via homodyne measurements and readout output states based on one-dimensional cluster states. The generalisation to universal temporal CV cluster states (at least two-dimensional cluster) is then straightforward.
The strategy is to add a switch after the delay loop, as shown in Fig. \ref{fig:1D-cluster-computation-setup}. The switch has three states, denoted as {\bf s$_1$}, {\bf s$_2$} and {\bf s$_3$}. At state {\bf s$_1$}, the delay loop is connected to the beam splitter $B_2$, and the input and output wires are disconnected to the optical setup. At state {\bf s$_2$}, the input wire is connected to the beam splitter $B_2$ and input states can be injected into the optical setup. At state {\bf s$_3$}, the output wire is connected to the delay loop and optical fields from the delay loop can be readout or detected. The states of the switch are set accordingly during the process of computation.
Measurement-based quantum computation is based on the gate teleportation, which has been discussed in Sec. \ref{sec:gate-teleportation}. The gate teleportation protocol described in Fig. \ref{fig:gate-teleportation} can be implemented in the optical setup in Fig. \ref{fig:1D-cluster-computation-setup}. This is achieved by injecting only one pair of single-mode squeezed pulses and setting the switch in the state {\bf s$_2$} to the let the input mode couple with the top mode. After performing a feedforward to the delayed mode using the homodyne measurement outcomes, the state of the delayed mode is transformed into the input state applied by a particular unitary.
A sequence of gate teleportations, thus a sequence of single-mode unitaries, can be implemented by injecting a series of pairs of single-mode squeezed states and appropriately setting the states of the switch. Fig. \ref{fig:four-unitary-teleportation} shows an example of implementing four single-mode uintaries, and Table \ref{tab:teleportation-four-unitary} lists the states of the switch and relevant actions during the computation process. If only Gaussian unitaries are implemented, the feedforward can be done at the end. We next move on to the generation of two-dimensional cluster states and performing computation using them.
\begin{figure}
\caption{Implementation of four single-mode unitaries using one-dimensional temporal cluster states \cite{Alexander2017MBLO}. (a) Four single-mode unitaries $\hat U_0$, $\hat U_1$, $\hat U_2$ and $\hat U_3$ are implemented sequentially via four steps of homodyne measurements. $T_i$ ($i = 0, 1, 2, 3$) is the time of performing the beam splitter transformation and homodyne measurements, and is also used as the notation of the macronode. We assume that $T_i < T_{i+1}$, which means the time direction is from left to right and is different from that shown in Fig. \ref{fig:1D-cluster-state}. (b) The corresponding gate model circuit. }
\label{fig:four-unitary-teleportation}
\end{figure}
\begin{table*}[tp] \caption{Process of implementing four single-mode uintaries.} \label{tab:teleportation-four-unitary}\centering \begin{center}
\begin{tabular}{| c | c | l |}
\hline
Time (macronode) & ~State of switch~ & Description \\ \hline
$T_0$ & {\bf s$_2$} & Inject input state/Homodyne measurements/Feedforward to delayed mode \\ \hline
$T_1$ & {\bf s$_1$} & Homodyne measurements/Feedforward to delayed mode \\ \hline
$T_2$ & {\bf s$_1$} & Homodyne measurements/Feedforward to delayed mode \\ \hline
$T_3$ & {\bf s$_3$} & Homodyne measurements/Feedforward to output mode/Readout output state \\
\hline
\end{tabular} \end{center} \end{table*}
\section{Two-dimensional temporal cluster states}\label{sec:2Dcluster}
\subsection{Generation of two-dimensional temporal cluster state}
Two-dimensional temporal cluster states can be generated by injecting four squeezed pulses into the optical setup shown in Fig. \ref{fig:2D-cluster-setup}. The first (top) pair of single-mode squeezed pulses are injected into an optical circuit (leading to the beam splitter $B_3$) that is the same as Fig. \ref{fig:1D-cluster-setup} and a one-dimensional temporal cluster state is generated. The second (bottom) pair of single-mode squeezed pulses are injected into an optical part (leading to the beam splitter $B_4$) that is also the same as Fig. \ref{fig:1D-cluster-setup} but with a delay $M\Delta t$, with $M$ a positive integer. This gives rise to another one-dimensional temporal cluster state. Finally, the beam splitters $B_5$ and $B_6$ lead to the second dimension of a two-dimensional temporal cluster state. The depth of the two-dimensional cluster state depends on $M$, namely, the length of the second delay loop.
\begin{figure*}\label{fig:2D-cluster-setup}
\end{figure*}
A complete graphical representation of the two-dimensional temporal cluster state was developed in Ref. \cite{Menicucci2011}. A simplified version was discussed in Ref. \cite{Alexander2016} and we will use the simplified version in this paper. To discuss the graphical representation, we first introduce additional graph elements, as shown in Fig. \ref{fig:4mode-macronode}. As before, the black dots represents the optical modes, in particular, the optical modes before entering the beam splitters $B_3$ and $B_4$, as shown in Fig. \ref{fig:2D-cluster-setup}. The black circle represents the action of the four beam splitters $B_3$, $B_4$, $B_5$ and $B_6$. If the black circle is filled with colours (light blue in Fig. \ref{fig:4mode-macronode}), it means homodyne measurements are performed to the four modes. The black circle and the four modes it encircles are defined as a macronode. With these graph elements in hand, together with those introduced in Fig. \ref{fig:graph-elements}, we can draw a graph for any two-dimensional temporal cluster state. Fig. \ref{fig:2D-cluster-state-graph} shows an example with $M=5$.
\begin{figure}
\caption{ Additional graph elements. (a) The black dots represent optical modes as before and the labels show their correspondence in Fig. \ref{fig:2D-cluster-setup}. The black circle represents the action of four beam splitters $B_3$, $B_4$, $B_5$ and $B_6$. (b) The black circle filled with light blue represents the action of the four beam splitters followed by homodyne measurements. }
\label{fig:4mode-macronode}
\end{figure}
\begin{figure*}
\caption{ Graphical representation of a two-dimensional temporal cluster state \cite{Alexander2017MBLO} with depth five ($M=5$). $T_i$ denotes the time to perform the four-beam-splitter transformation and it also acts as the label of the corresponding macronode. It is assumed that $T_i < T_{i+1}$ and $T_{i+1} - T_i = \Delta t$. Note that the last mode of each column should be connected to the first mode of the next column, which is not plotted in the figure for convenience. }
\label{fig:2D-cluster-state-graph}
\end{figure*}
\subsection{Measurement-based two-mode Gaussian unitary}\label{sec:MBGU}
Four-mode homodyne detection, as shown in Fig. \ref{fig:2D-cluster-setup}, can implement an arbitrary two-mode Gaussian unitary by appropriately choosing the measurement quadratures (angles). A detailed discussion was given in Ref. \cite{Alexander2016}. Fig. \ref{fig:teleportation-graph-2mode} shows the graphical representation of the four-mode homodyne measurement. In the limit of infinite squeezing in the source squeezed pulses, the two-mode unitaries can be implemented perfectly. However, for physical squeezed pulses, the amount of squeezing is finite. This results in errors when implementing unitaries via homodyne measurements. If the input states are known, the error due to the effect of finite squeezing can be corrected \cite{Su2018EC}. For simplicity, we first ignore the finite-squeezing effect and work in the infinite squeezing limit.
\begin{figure}
\caption{ Two-mode Gaussian unitaries via four-mode homodyne measurements \cite{Alexander2017MBLO} . (a) Two input modes (modes $a$ and $b$) and two optical modes $c$ and $d$ couple via four beam splitters, and are detected by homodyne detectors. Depending on the measurement angles of the homodyne detectors, a two-mode Gaussian unitary is implemented. The output state comes out via the output modes $c'$ and $d'$. (b) Gate model representation of the corresponding two-mode Gaussian unitary. }
\label{fig:teleportation-graph-2mode}
\end{figure}
The four homodyne measurements implement a Gaussian unitary $\hat G_{jk}$ \cite{Alexander2016}, \begin{eqnarray} \hat G_{jk} (\boldsymbol m, \boldsymbol \theta) &=& \hat B_{jk}^{\dag}(\pi/4) \hat A_j (m_1, m_3, \theta_1, \theta_3) \nonumber\\ &&\times \hat A_k (m_2, m_4, \theta_2, \theta_4) \hat B_{jk}(\pi/4), \end{eqnarray} where $\boldsymbol m = (m_1, m_2, m_3, m_4)$, $\boldsymbol \theta = (\theta_1, \theta_2, \theta_3, \theta_4)$ and \begin{eqnarray} \hat A_j (m_h, m_l, \theta_h, \theta_l) = D_j (m_h, m_l, \theta_h, \theta_l) \hat U_j (\theta_h, \theta_l). \end{eqnarray} $D_j (m_h, m_l, \theta_h, \theta_l)$ is the phase-space displacement operator \begin{eqnarray}
D_j (m_h, m_l, \theta_h, \theta_l) = \hat D \bigg[ \frac{-i e^{i\theta_l} m_h - i e^{i \theta_h} m_l}{\sin \big( \theta_h - \theta_l \big)} \bigg] \end{eqnarray} and $\hat U_j (\theta_h, \theta_l)$ is a single-mode unitary \begin{align} \hat U_j (\theta_h, \theta_l) &= R_j(\theta_{h,l}^+) S_j \big[ {\rm ln} \big(\tan \theta_{h,l}^- \big) \big] R_j(\theta_{h,l}^+), \end{align} where $\theta_{h,l}^{\pm} = \big(\theta_h \pm \theta_l\big)/2$. The displacements can be corrected by applying feedforward corrections conditioned on the homodyne measurement outcomes, so the implemented two-mode Gaussian unitary is \begin{eqnarray}\label{eq:MB-2mode-unitary} \hat U_{jk} (\boldsymbol \theta) = \hat B_{jk}^{\dag}(\pi/4) \hat U_j (\theta_1, \theta_3) \hat U_k (\theta_2, \theta_4) \hat B_{jk}(\pi/4), \end{eqnarray} which is completely determined by the homodyne measurement angles.
In the case that $\theta_1 = \theta_2$ and $\theta_3 = \theta_4$, the two same single-mode unitaries commute with the beam splitter operator such that a pair of same single-mode unitaries is implemented, namely, \begin{eqnarray} \hat U_{jk} (\theta_1, \theta_3) = \hat U_j (\theta_1, \theta_3) \hat U_k (\theta_1, \theta_3). \end{eqnarray}
\noindent {\bf Phase shifts}: If further $\theta_1 = \theta_3 = \theta$, then a pair of same phase shifts is implemented, $\hat U_{jk} (\theta) = R_j(2\theta) R_k(2\theta)$. \\
\noindent {\bf Squeezers}: If further $\theta_1 = -\theta_3 = \theta$, then a pair of same single-mode squeezers is implemented, $\hat U_{jk} (\theta) = S_j \big[ {\rm ln} (\tan \theta) \big] S_k \big[ {\rm ln} (\tan \theta) \big]$.\\
To get rid of the beam splitters in Eq. \eqref{eq:MB-2mode-unitary}, the implemented single-mode unitaries in two modes have to be the same. This constraint is less desirable because generally we want to implement different single-mode unitaries in different modes. This difficulty can be overcome by sequentially implementing the two-mode unitary Eq. \eqref{eq:MB-2mode-unitary} such that the beam splitters between two neighbouring unitaries cancel. The resulting two-mode unitary is a sequence of different single-mode unitaries in each mode sandwiched by two beam splitters. The two beam splitters can be further cancelled by another two steps of homodyne measurement (one before and the other after the sequence of two-mode unitaries) because the homodyne measurements can also induce a beam splitter transformation.
In the case of $\theta_3 = \theta_1 - \pi/2$ and $\theta_4 = \theta_2 - \pi/2$, the two-mode unitary Eq. \eqref{eq:MB-2mode-unitary} is \begin{eqnarray} &&\hat U_{jk} (\theta_1, \theta_2) \nonumber\\ &=& \hat B_{jk}^{\dag}(\pi/4) R_j \bigg(2\theta_1 + \frac{\pi}{2} \bigg) R_k \bigg(2\theta_2 + \frac{\pi}{2} \bigg) \hat B_{jk}(\pi/4) \nonumber\\ &=& R_j (\theta^+_{1,2}) R_k(\theta^+_{1,2}) \bigg[ R_j \bigg(\frac{\pi}{2} \bigg) \hat B_{jk}(\theta^-_{1,2}) R_k \bigg(\frac{\pi}{2} \bigg) \bigg]. \end{eqnarray}
Therefore a variable beam splitter is implemented with some additional phase shifts.
\subsection{Measurement-based non-Gaussian unitary}\label{sec:CubicPhase}
\begin{figure*}
\caption{ Implementing cubic phase gates in two-dimensional temporal cluster states. The cubic phase circuit (the orange shaded box) is fitted into a two-dimensional temporal cluster state in one measurement step. The whole circuit implements a cubic phase gate, a Gaussian unitary and two $50:50$ beam splitters. }
\label{fig:cubic-phase-two}
\end{figure*}
For the two-dimensional temporal cluster states that we are interested in, four homodyne measurements in each step implement a two-mode Gaussian unitary (or two single-mode Gaussian unitaries). To include the cubic phase gate such that universal quantum computation is possible, a switch and a cubic phase circuit (orange shaded part in Fig. \ref{fig:cubic-phase-single}) can be introduced to the optical setup in Fig. \ref{fig:2D-cluster-setup}. When we implement Gaussian unitaries, the switch is set to connect the homodyne detector $D_1$. When we want to implement a cubic phase gate, the switch is set to connect the cubic phase circuit. Fig. \ref{fig:cubic-phase-two} shows an equivalent circuit in one measurement step when implementing a cubic phase gate (and another single-mode Gaussian unitary).
In realistic cases, the amount of squeezing in the pulses that are used to produce the temporal cluster states is finite. This finite squeezing gives rise to errors in the implemented unitaries. For Gaussian unitaries (and Gaussian input states), the errors due to the effect of finite squeezing can be corrected by using the information of the input states \cite{Su2018EC}. An error correction scheme for non-Gaussian gates (and states) has not been developed. Furthermore, the ideal cubic phase state is unphysical since it requires infinite energy. The ideal cubic phase state can be approximated by some non-Gaussian states, e.g., ON states from Sec. \ref{Sec:WeakCubicPhase} (a superposition of the vacuum and Fock states) \cite{Krishna2018ON}. In this case additional errors are introduced.
In next three sections we consider the implementation of three important quantum algorithms using temporal cluster states.
\section{Gaussian Boson Sampling}\label{sec:GBS}
We are now ready to discuss the implementations of quantum algorithms using two-dimensional temporal cluster states. We will focus on three important quantum algorithms: Gaussian Boson Sampling \cite{Hamilton2017}, CV-IQP \cite{Douce2017CVIQP} and CV Grover's search algorithm \cite{Pati2000}. To clearly illustrate the main steps of these quantum algorithms, we consider algorithms with only a few quantum modes. The generalisation to algorithms with a large number of modes is straightforward.
In the original implementation of the Boson Sampling algorithm \cite{Aaronson2011BS}, single photons are injected into a linear optical network, which consists of beam splitters and phase shifters, and the output state is detected by single photon detectors to obtain the photon number distribution. The key observation is that the sampling probability of a certain output photon pattern is related to the permanent of the submatrices of a unitary matrix determined by the linear optical network \cite{Aaronson2011BS}. Given that estimating the permanent of a matrix is hard for a classical computer, while sampling the photon number distribution can be done efficiently, it is believed that Boson Sampling may achieve quantum supremacy before a universal quantum computer is built \cite{Boixo2018}. The original Boson Sampling scheme has also been experimentally implemented \cite{Tillmann2013, Spring2013, Crespi2013, Broome2013}.
Several variants of the original Boson Sampling have been proposed. The single photon inputs can be generated by heralding the same number of two-mode squeezed vacuum states, which is known as Scattershot Boson Sampling \cite{Lund2014SBS}. Another variant is to replace the single photon input stats by Gaussian input states \cite{Sahleh2015}. One important example is to inject squeezed coherent states into a linear network and the output photon number distribution is related to the vibronic spectra of molecules \cite{Huh2015, {huh2017vibronic}}. Another example is to directly inject squeezed vacuum states into the linear network \cite{Hamilton2017}. The probability of the output photon number distribution is shown to be related to the hafnian of the submatrices of a matrix determined by the linear optical network \cite{Hamilton2017}. In particular, the number of perfect matchings of an undirected graph can be estimated using the Gaussian Boson Sampling \cite{Bradler2017}.
\begin{figure}
\caption{ Gaussian Boson Sampling with four modes. We assume that all phases are included in the squeezers and beam splitters. }
\label{fig:4mode-GBS-circuit}
\end{figure}
In contrast to the original Boson Sampling, the only non-Gaussian element in Gaussian Boson Sampling is the photon number detection. This is due to the fact that both the input states and unitaries are Gaussian. In this paper, we consider the implementation of Gaussian Boson Sampling using the two-dimensional temporal cluster states. The implementation with linear optics using two-dimensional temporal cluster states has been discussed in Ref. \cite{Alexander2017MBLO}, as well as the original Boson Sampling. The main challenge there is the noise due to the effect of finite squeezing in the cluster states. We find here that the Gaussianity of the states and unitaries in the Gaussian Boson Sampling can lead to experimental advantages because one can correct the error due to the effect of finite squeezing \cite{Su2018EC}.
\begin{figure*}
\caption{ Measurement-based Gaussian Boson Sampling. The blue shaded macronode implements two identical single-mode squeezers, the green shaded macronode implements a beam splitter and the yellow shaded macronode implements two phase shifts. The macronode that is not shaded implements a single-mode phase shift. }
\label{fig:4mode-GBS-graph}
\end{figure*}
To clearly illustrate how to implement Gaussian Boson Sampling in the two-dimensional temporal cluster states, we first ignore the noise due to the effect of finite squeezing, namely, we work in the infinite squeezing limit. Any linear optical network can be decomposed into a series of beam splitters and phase shifters \cite{clements2016optimal}, the implementations of which via measuring the cluster states have been discussed in Sec. \ref{sec:MBGU}. Given a linear optical network in the gate model, it is straightforward to map it to the measurement-based model. Without loss of generality, we consider a Gaussian Boson Sampling with four modes. The gate model circuit is given by Fig. \ref{fig:4mode-GBS-circuit} and the implementation in the temporal cluster states is shown in Fig. \ref{fig:4mode-GBS-graph}. In Fig. \ref{fig:4mode-GBS-graph} we assume that the input states are vacuum states and the squeezed states (with the same amount of squeezing) are generated via homodyne measurements. One can also prepare the input states beforehand and tune the amount of squeezing according to specific algorithms.
After the linear optical network the output modes are detected by photon number resolution (PNR) detectors, giving the photon number distribution. Table \ref{tab:GBS} provides a detailed description of steps for the implementation of Gaussian Boson Sampling via homodyne measurements.
Generalisation of the above implementation to a linear optical network with a large number of modes is straightforward if the squeezing is infinite. However, the amount of squeezing of a physical squeezed state is always finite and this results in errors when implementing a unitary via homodyne measurements \cite{Alexander2014}. The overall noise increases as the number of measurement steps grows. Ref. \cite{Alexander2017MBLO} showed that to implement a Boson Sampling with 6 photons requires about 20 dB of squeezing in the cluster states, which is very challenging for the state-of-the-art technologies. For measurement-based Gaussian Boson Sampling, both the states and unitaries are Gaussian and the errors due to the effect of finite squeezing can be corrected if a moderate amount of online squeezing can be achieved \cite{Su2018EC}. Online squeezing has been demonstrated in several experiments \cite{Miwa2014, Marshall2016} and so it is promising to include the finite-squeezing error correction on a linear network with a higher number of modes.
\begin{table*}[tp] \caption{Process of implementing four-mode Gaussian Boson Sampling.} \label{tab:GBS}\centering \begin{center}
\begin{tabular}{| c | c | l |}
\hline
Time (macronode) & ~States of switches~ & Operations \\ \hline
$T_1$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_2$ & (${\rm \bf s}_2$, $\bar {\rm \bf s}_2$) & Inject input states/Homodyne measurements/Squeezing unitary \\ \hline
$T_3$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_4$ & (${\rm \bf s}_2$, $\bar {\rm \bf s}_2$) & Inject input states/Homodyne measurements/Squeezing unitary \\ \hline
$T_5$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & None \\ \hline
$T_6$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_7$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & Homodyne measurements/Beam splitter unitary \\ \hline
$T_8$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & Homodyne measurements/Phase shifts \\ \hline
$T_9$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & Homodyne measurements/Beam splitter unitary \\ \hline
$T_{10}$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_{11} \rightarrow T_{25}$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_1$) & Homodyne measurements/Two-mode or single-mode unitaries \\ \hline
$T_{26}$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_3$) & Readout output state/Photon number detection \\ \hline
$T_{27}$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_3$) & Readout output state/Photon number detection \\ \hline
$T_{28}$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_3$) & Readout output state/Photon number detection \\ \hline
$T_{29}$ & (${\rm \bf s}_1$, $\bar {\rm \bf s}_3$) & Readout output state/Photon number detection \\
\hline
\end{tabular} \end{center} \end{table*}
\section{Continuous-variable Instantaneous Quantum Polynomial circuits}\label{sec:IQP}
The instantaneous quantum polynomial (IQP) computation is a particular kind of quantum computation that consists of only commuting gates. It was shown that sampling the output probability distributions of the IQP computation cannot be efficiently achieved by classical computers \cite{Bremner2011}. IQP circuits have been extended to the continuous-variable domain, denoted as CV-IQP, by using squeezed states and homodyne detection \cite{Douce2017CVIQP}. A particular implementation of a CV-IQP circuit is to include unitaries that are only functions of position quadratures, e.g., $e^{i f(\hat x)}$ where the function $f(\hat x)$ is an arbitrary polynomial of $\hat x$. If $f(\hat x)$ is a polynomial up to a quadratic function of $\hat x$, then the unitary is Gaussian and can be implemented efficiently. If $f(\hat x)$ is a cubic function of $\hat x$, then the unitary represents a cubic phase gate, the implementation of which has been discussed in detail in Sec. \ref{sec:CubicPhase}. If $f(\hat x)$ is a higher-order polynomial of $\hat x$, the direct implementation of the unitary is in principle possible but is challenging \cite{Krishna2018ON}. The strategy is to decompose the unitary into Gaussian gates and cubic phase gates \cite{BraunsteinLloyd1999}. The decomposition requires unitaries that are also functions of $\hat p$, e.g., the Fourier transform ${F}$. Therefore, after the decomposition the equivalent circuit does not look like a CV-IQP circuit due to the presence of non-commuting unitaries. However, if each higher-order unitary is considered as a whole as if it wasn't decomposed, then the circuit is still a CV-IQP circuit.
The CV-IQP circuit can be decomposed into three parts \cite{Arrazola2017}: momentum-squeezed vacuum states as inputs, a sequence of commuting unitaries and homodyne detectors. Fig. \ref{fig:CV-IQP} (a) shows an example of a typical four-mode CV-IQP circuit up to third order unitaries. In this paper, we are interested in implementing the CV-IQP circuits using the two-dimensional temporal cluster states and homodyne measurements. It is straightforward to conceive a measurement-based implementation given a CV-IQP circuit like Fig. \ref{fig:CV-IQP} (a). However, we find it very helpful to rearrange those commuting unitaries. The strategy is to move all controlled-Z gates $C_Z$ to be directly after the squeezed vacuum states and move all momentum displacement operators $Z$ to be directly before the homodyne detectors, as shown in Fig. \ref{fig:CV-IQP} (b). The momentum displacements can be absorbed into the feedforward which we have to do before the homodyne measurements, therefore we only need to consider a series of controlled-Z gates, which are Gaussian unitaries, and a series of cubic-phase gates. The main advantage of this rearrangement is that before the action of the cubic phase gates, both the state and unitaries are Gaussian. When implementing the controlled-Z gates using cluster states, we can correct the errors due to the effect of finite squeezing \cite{Su2018EC}. Since the finite squeezing noise increases as the number of implemented gates increases, this rearrangement therefore can significantly reduce the noise.
A direct implementation of the controlled-Z gate is challenging, however it can be decomposed into two single-mode squeezers with equal squeezing parameters and squeezing angles, and a beam splitter \cite{Loock2007}. Two identical single-mode squeezers and a beam splitter can be implemented directly via homodyne measurements, as discussed in Sec. \ref{sec:MBGU}. Fig. \ref{fig:4mode-IQP-graph} shows an implementation of the four-mode CV-IQP circuit shown in Fig. \ref{fig:CV-IQP} via measuring the two-dimensional temporal cluster states. Table \ref{tab:CV-IQP} describes the detailed process of implementing the CV-IQP circuit of Fig. \ref{fig:CV-IQP}. If higher-order unitaries are required, e.g., $e^{i \lambda \hat x^4}$, we have to first decompose it into series of Gaussian gates and cubic phase gates and then implement it just as in Fig. \ref{fig:4mode-IQP-graph}. Generalisation to a CV-IQP circuit with a larger number of modes is straightforward.
\begin{figure*}
\caption{ CV-IQP circuit. (a) An example of a four-mode CV-IQP circuit. The four squeezers generate four single-mode momentum-squeezed states. The commuting gates include displacement $Z$, cubic phase gate $V$ and controlled-Z gate $C_Z$. The output modes are detected by homodyne detectors $D_i$, $i=1,..,4$. (b) The commuting gates are rearranged such that all controlled-Z gates are right after the single-mode squeezers, all displacements are right before the homodyne detectors and the cubic phase gates are in between. }
\label{fig:CV-IQP}
\end{figure*}
\begin{figure*}
\caption{ Measurement-based four-mode CV-IQP circuit. The blue shaded macronode implements two identical single-mode squeezers, the green shaded macronode implements a beam splitter (basically a $50:50$ beam splitter), the pink shaded macronode implements a cubic phase gate and an identity gate, the yellow shaded macronode implements two phase shifts which is not explicitly labelled. The macronode that implements a single mode gate is not filled with color, which we assume implements a phase shift. In the first column of macronodes, the four input modes (green circles) are prepared in the momentum squeezed states. In the first four columns, the two macronodes in a black dashed box together implements a controlled-Z gate. From the fifth to the seventh column, the three macronodes in a black dotted box together implements a cubic phase gate and an identity gate. In the last column, the four output modes (grey squares) are detected by homodyne detectors. }
\label{fig:4mode-IQP-graph}
\end{figure*}
\begin{table*}[tp] \caption{Process of implementing four-mode CV-IQP circuit.} \label{tab:CV-IQP}\centering \begin{center}
\begin{tabular}{| c | c | l |}
\hline
Time (macronode) & ~States of switches~ & Operations \\ \hline
$T_1$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_2$ & ($\boldsymbol{\rm s}_2$, $\bar {\boldsymbol{\rm s}}_2$) & Inject input states/Homodyne measurements/Squeezing unitary \\ \hline
$T_3$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol {\rm s}}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_4$ & ($\boldsymbol{\rm s}_2$, $\bar {\boldsymbol{\rm s}}_2$) & Inject input states/Homodyne measurements/Squeezing unitary \\ \hline
$T_5$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & None \\ \hline
$T_6 \rightarrow T_{10}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Two beam splitters and phase shifts \\ \hline
$T_{11} \rightarrow T_{15}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Squeezers and phase shifts \\ \hline
$T_{15} \rightarrow T_{20}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & One beam splitter and phase shifts \\ \hline
$T_{21} \rightarrow T_{25}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Two beam splitters and phase shifts \\ \hline
$T_{26}$ & ($\boldsymbol{\rm s}_1$, $\bar{\boldsymbol {\rm s}}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_{27}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Inject cubic phase state/Homodyne measurements/Cubic phase gate \\ \hline
$T_{28}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Homodyne measurements/Phase shifts \\ \hline
$T_{29}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Inject cubic phase state/Homodyne measurements/Cubic phase gate \\ \hline
$T_{30}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_{31} \rightarrow T_{35}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Two beam splitters and phase shifts \\ \hline
$T_{36}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_3$) & Readout output state/Homodyne detection \\ \hline
$T_{37}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_3$) & Readout output state/Homodyne detection \\ \hline
$T_{38}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_3$) & Readout output state/Homodyne detection \\ \hline
$T_{39}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_3$) & Readout output state/Homodyne detection \\
\hline
\end{tabular} \end{center} \end{table*}
\section{Continuous-variable Grover search algorithm}\label{sec:CVGrover}
One of the first quantum algorithms to show a speed up compared to any classical counterpart was Grover's quantum search algorithm \cite{Grover1997}. In this algorithm an unsorted list of $N$ entries could be searched to find an unmarked item with $O(\sqrt{N})$ steps, instead of the best classical case which requires $O(N)$ steps. This quantum search algorithm, originally proposed for discrete-variable quantum systems~\cite{Grover1997}, was generalised to continuous variables by Pati \textit{et al.} \cite{Pati2000}. It was argued that the CV proposal may be superior to any discrete-variable implementation for large database searches, due to the considerable amount of information that could in principle be encoded with small CV systems.
In the original CV quantum search algorithm proposal \cite{Pati2000}, the information of a database with $N$ entries $\{1, 2, \ldots, N\}$ is encoded into $n$ continuous variables or modes state $\ket{x}=\ket{x_{1}, x_{2}, \ldots, x_{n}}$, by dividing a compact subspace of the $n$-dimensional state space into $N$ equal subvolumes $\Delta x$. In the case when $N=4$ and $n=1$, the one-dimensional state space will be divided into four equal regions, as shown in Fig. \ref{fig:Pati1D2D}~(a); if $N=4$ and $n=2$, the two-dimensional state space will be divided into four equal regions, as shown in Fig. \ref{fig:Pati1D2D}~(b). An alternative method of encoding information into a single continuous variable was presented by Adcock~\textit{et al.} \cite{Adcock2009}, in the context of the Deutsch-Jozsa algorithm. In this case each subregion in the one-dimensional state space corresponds to a single bit of information. In this paper we use the encoding of Pati \textit{et al.} \cite{Pati2000}.
\begin{figure}
\caption{Geometry for encoding $N$ qubits into $n$ continuous variables: (a) State space division for $N=4$ and $n=1$; (b) State space division for $N=4$ and $n=2$. }
\label{fig:Pati1D2D}
\end{figure}
The basic elements required for Grover's search algorithm with discrete-variables~\cite{NielsenBook}, shown in Fig. \ref{fig:GroverCircuit}, are an oracle, which recognises solutions to the searching problem, Hadamard gates and the Grover diffusion operator. We assume that the oracle is a given black box operation and concentrate on the Hadamard gate and Grover diffusion operator. In continuous variables the analog to the Hadamard gate is the Fourier gate \cite{Pati2000}, which can easily be implemented optically with a $\pi/2$ phase shift, where ${F}\ket{x}$ is an eigenstate of the conjugate quadrature. The action of the Fourier gate is \begin{equation} {F}\ket{x}=\frac{1}{\sqrt{\pi^{n}}}\int \mathrm d y ~e^{2ixy} \ket{y}, \end{equation} where $xy=x_{1}y_{1}+\cdots +x_{n}y_{n}$, $\ket{y}=\ket{y_{1},y_{2},\cdots , y_{n}}$ and both $x$ and $y$ are in the position basis.
\begin{figure}
\caption{Circuit diagram for Grover's search algorithm~\cite{Grover1997, NielsenBook}. }
\label{fig:GroverCircuit}
\end{figure}
In continuous variables the Grover diffusion operator is a selective inversion operator \cite{Pati2000} defined as \begin{equation} \hat{I}_{x}=\mathbb{I}-2P_{\Delta x}, \end{equation} where $\mathbb{I}$ is the identity operator and $P_{\Delta x}$ is the projection operator defined as \begin{equation} P_{\Delta x}=\int_{x_{0}-\Delta x/2}^{x_{0}+\Delta x/2} \mathrm d x' \ket{x'}\bra{x'}\label{Eqn:PDeltax}, \end{equation} defining a projection operator for a subvolume $\Delta x$ centered at $x_0$. The main challenge with implementing a CV version of Grover's search algorithm is the implementation of this selective inversion operator $\hat{I}_{x}$.
The obvious first attempt to implement this selective inversion operator $\hat{I}_{x}$ would be to use the results of Lloyd and Braunstein \cite{BraunsteinLloyd1999}: any Hamiltonian consisting of an arbitrary polynomial of the conjugate operators $\hat{x}$ and $\hat{p}$ can be constructed with only Gaussian operations and a single operator with order greater than 2 in $\hat{x}$ or $\hat{p}$, such as the cubic phase gate. The selective inversion operator can be rewritten as $\hat{I}_{x}=e^{\ln(\mathbb{I}-2P_{\Delta x})}=e^{P_{\Delta x}\ln(-1)}=e^{i\pi P_{\Delta x}}$. However it is not immediately obvious that the projector $P_{\Delta x}$ can be written as a polynomial of $\hat{x}$ and $\hat{p}$ operators.
If we consider a particular geometrical representation for the selective inversion operator $\hat{I}_{x}$ by choosing the number of encoding continuous variables $n$, we can write $\hat{I}_{x}$ as a function in state space: $\hat{I}_{x} = f(\hat{x}_{1}, \ldots, \hat{x}_{n})$. The form of this function is a direct consequence of the projector $P_{\Delta x}$: it is a step function with positive constant values for $N-1$ database entries/state space regions and a negative constant value for the flagged region. The operator $P_{\Delta x}$ can be expressed as a function of $f(\hat{x}_{1}, \ldots, \hat{x}_{n})$: $P_{\Delta x}=\frac{1}{2}(\mathbb{I}-f(\hat{x}_{1}, \ldots, \hat{x}_{n}))$. Since $\hat{I}_{x}=e^{i\pi P_{\Delta x}}$, we can re-write the relation between $\hat{I}_{x}$ and $f(\hat{x}_{1}, \ldots, \hat{x}_{n})$ as: \begin{equation} \hat{I}_{x}=i e^{-i\frac{\pi}{2}f(\hat{x}_{1}, \ldots, \hat{x}_{n})}.\label{eqn:Ifun} \end{equation}
The function $f(\hat{x}_{1}, \ldots, \hat{x}_{n})$ is not necessarily a polynomial function in state space variables $\hat{x}_{1}, \ldots, \hat{x}_{n}$. In the case that $n=1$, $\hat{I}_{x}$ will be a top hat function, shown in Fig. \ref{fig:Pati1DFun}~(a). In the case that $n=2$, $\hat{I}_{x}$ will be a two-dimensional step function, shown in Fig. \ref{fig:Pati2DFun}~(a). If the function $f(\hat{x}_{1}, \ldots, \hat{x}_{n})$ could be decomposed into a polynomial function of position operators $\hat{x}_{i}$, then we could implement the operator $\hat{I}_{x}$ via the gate teleportation circuits, similar to that in Fig. \ref{fig:cubic-phase-single}. The gate teleportation would then reduce to a state preparation problem.
\begin{figure}
\caption{(a) Selective inversion operator $\hat{I}_{x}$ for $N=4$ and $n=1$. $\hat{I}_{x}$ function $f(x_{1})$ expanded in the Fock basis states up to (b) 5 photons; (c) 10 photons; (d) 20 photons.}
\label{fig:Pati1DFun}
\end{figure}
\begin{figure}
\caption{(a) Selective inversion operator $\hat{I}_{x}$ for $N=4$ and $n=2$. $\hat{I}_{x}$ function $f(x_{1}, x_{2})$ expanded in the Fock basis states up to (b) 3 photons ; (c) 5 photons; (d) 10 photons in each mode.}
\label{fig:Pati2DFun}
\end{figure}
We assume the function $f(\hat{x}_{1}, \ldots, \hat{x}_{n})$ is a well behaved function such that $f(\hat{x}_{1}, \ldots, \hat{x}_{n})\ket{x_{1},\ldots, x_{n}}=f(x_{1}, \ldots, x_{n})\ket{x_{1},\ldots, x_{n}}$ and expand it as a sum over the orthonormal Fock state wavefuctions $\psi_{n}(x_{i})=\langle x_{i}\ket{n}$~\cite{ArfkenBook}: \begin{equation} f(x_{1}, \ldots, x_{n})=\sum_{k_{1}, \ldots, k_{n}=0}^{\infty}c_{k_{1}, \ldots, k_{n}}\psi_{k_{1}}(x_{1})\cdots \psi_{k_{n}}(x_{n})\label{Eqn:function}. \end{equation}
Given the chosen geometry resulting from the number of encoding continuous variables $n$, we can truncate the sum in Eq. \eqref{Eqn:function} to a ensure convergence for the given number of entires to be searched over $N$ . If we consider the $n=1$ case, the function becomes \begin{align} f(x) &=\sum_{k=0}^{\infty}c_{k}\psi_{n}(x) \nonumber\\ &\approx \sum_{k=0}^{m}\left(\sqrt{\pi}2^{k}k!\right)^{-\frac{1}{2}}e^{-\frac{x^{2}}{2}}\mathcal{H}_{k}(x)\nonumber\\ &\approx \sum_{j=0}^{p}\sum_{k=0}^{m}\frac{1}{\sqrt{\sqrt{\pi}2^{k}k!}j!}\biggl(\frac{-x^{2}}{2}\biggr)^{j}\mathcal{H}_{k}(x)\label{eqn:functionpoly}, \end{align} where $m$ is the order of the highest Fock state to retain and $\mathcal{H}_{k}(x)$ is the Hermite polynomial, a known polynomial function in $x$ for a given $k$. In the last equality, we have expanded the Gaussian function $e^{-x^2/2}$ and truncated at the order $x^{2p}$, such that a significantly good approximation is obtained. Given the truncation of the sum in Eq. \eqref{eqn:functionpoly} to $m p$ terms, there will only be terms of order $x^{m+2p}$ and lower. When we combine this with Eq. \eqref{eqn:Ifun} this implies that the selective inversion operator $\hat{I}_{x}$ will take the form $\hat{I}_{x}\approx i e^{-i\frac{\pi}{2}\sum_{q=0}^{m+2p}c_{q}\hat{x}^{q}}$. From this point it is clear that we can decompose $\hat{I}_{x}$ into a sequence of higher-order quadrature phase gates \cite{h-order}.
\begin{figure}\label{fig:1D2DSF}
\end{figure}
\begin{figure*}
\caption{Measurement-based two mode CV-Grover search algorithm. The yellow shaded macronodes implement Fourier or inverse Fourier transforms and the green shaded macronodes implement the selective inversion operator $\hat{I}_{x}$. The input states (two green circles) are two independent single-mode squeezed states (squeezed in $x$ quadrature), which are the approximations of the position eigenstates. After two applications of $\mathcal{C}$ (separated by two red dashed lines), the outputs are detected by homodyne detectors. In this case $N=4$, so four applications of the selective inversion operator are required~\cite{Pati2000}. }
\label{fig:GroverTemporal}
\end{figure*}
\begin{table*}[tp] \caption{Process of implementing two-mode Grover's search algorithm.} \label{tab:CV-Grover}\centering \begin{center}
\begin{tabular}{| c | c | l |}
\hline
Time (macronode) & ~States of switches~ & Operations \\ \hline
$T_1$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Homodyne measurements/Phase shift \\ \hline
$T_2$ & ($\boldsymbol{\rm s}_2$, $\bar {\boldsymbol{\rm s}}_2$) & Inject input states/Homodyne measurements/Fourier transform \\ \hline
$T_3$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol {\rm s}}_1$) & None \\ \hline
$T_4 \rightarrow T_{6}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Projection and phase shifts \\ \hline
$T_{7} \rightarrow T_{9}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Inverse Fourier transfom and phase shifts \\ \hline
$T_{10} \rightarrow T_{12}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Projection and phase shifts \\ \hline
$T_{13} \rightarrow T_{15}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Fourier transform and phase shifts \\ \hline
$T_{16} \rightarrow T_{18}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Projection and phase shifts \\ \hline
$T_{19} \rightarrow T_{21}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Inverse Fourier transform and phase shifts \\ \hline
$T_{22} \rightarrow T_{24}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & Projection and phase shifts \\ \hline
$T_{25}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_1$) & None \\ \hline
$T_{26}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_3$) & Readout output state/Homodyne detection \\ \hline
$T_{27}$ & ($\boldsymbol{\rm s}_1$, $\bar {\boldsymbol{\rm s}}_3$) & Readout output state/Homodyne detection \\
\hline
\end{tabular} \end{center} \end{table*}
We show two explicit examples for approximating the selective inversion operator $\hat{I}_{x}$: the one dimensional case $f(x_{1})$ in Fig. \ref{fig:Pati1DFun} and the two dimensional case $f(x_{1}, x_{2})$ in Fig. \ref{fig:Pati2DFun}. The state required to be produced for the gate teleportation can be written as a sum of Fock states $\ket{\psi_{\text{prep}}}_{1}=\sum_{k=0}^{m}c_{k}\ket{k}$ for the one-dimensional case. In Fig. \ref{fig:Pati1DFun} we can see how well the approximation in Eq.~(\ref{eqn:functionpoly}) holds for the $m=5$, 10 and 20 photon cases. For the two-dimensional state space case the state required to be produced for the gate teleportation can be written as $\ket{\psi_{\text{prep}}}_{2}=\sum_{k, l=0}^{p, q}c_{k, l}\ket{k}\ket{l}$. In Fig. \ref{fig:Pati2DFun} we can see how well the approximation in Eq.~(\ref{eqn:functionpoly}) holds for the $(p,q)=(3,3)$, $(5,5)$ and $(10,10)$ photon cases.
The preparation states $\ket{\psi_{\text{prep}}}_{1}$ and $\ket{\psi_{\text{prep}}}_{2}$ can be constructed with an optimisation and machine learning algorithm specifically designed for photonic quantum information tasks, i.e. the Strawberry Fields quantum software package \cite{SFpaper}. In Fig. \ref{fig:1D2DSF} we show how well the optimisation routine can approximate the desired preparation states. The gates used to construct these states are Gaussian gates and the Kerr gate. Note that the gate sequences for the construction of both of these states contain operators in both $\hat{x}_{i}$ and the conjugate variable $\hat{p}_{i}$. This means the gate sequence generated with the optimisation routine \cite{SFpaper} cannot be directly used for gate teleportation.
Provided that the state $\ket{\psi_{\text{prep}}}$ has been appropriately prepared, the selective inversion operator can be implemented via teleportation in a similar way to the method described for the cubic phase gate in Sec. \ref{sec:cubic}. To complete the Grover search algorithm, one needs to construct a compound search operator $\mathcal{C}$~\cite{Pati2000}, defined as \begin{eqnarray} \mathcal{C} = - \hat{I}_{x_i} {F}^{\dag} \hat{I}_{x_f} {F}, \end{eqnarray} where $\hat{I}_{x_i}$ is the projection operator to the initial state and $\hat{I}_{x_f}$ is the projection operator to the final (target) state. The target state can be selected with high probability with approximately $\sqrt{N}$ applications of $\mathcal{C}$. As an example, we consider the implementation of the Grover search algorithm with two-dimensional temporal cluster states for the case when $N=4$, as shown in Fig. \ref{fig:GroverTemporal}. The details of the implementation process is given in Table \ref{tab:CV-Grover}. In this particular case we would require two applications of $\mathcal{C}$, and four applications of the selective inversion operator $\hat{I}_{x}$~\cite{Pati2000}.
\section{Conclusion}\label{sec:conclusion}
We discussed in detail the implementations of three important quantum algorithms using the two-dimensional temporal cluster states: Gaussian Boson Sampling, CV-IQP and the CV Grover's search algorithm. We reviewed and summarised the simplified graphical representation and generation of one-dimensional and two-dimensional CV cluster states, and the implementation of basic Gaussian and non-Gaussian gates (phase shifter, squeezer, beam splitter, cubic phase gate, {\rm etc.}) by homodyne measurements on the temporal cluster.
For Gaussian Boson Sampling, only Gaussian unitaries are required. Although the implementation of the original Boson Sampling using temporal cluster states has been discussed in Ref. \cite{Alexander2017MBLO}, we emphasise that implementing Gaussian Boson Sampling using temporal cluster states shows advantages in an experimental realisation. The is because the states and unitaries for Gaussian Boson Sampling are Gaussian, meaning the errors due to finite squeezing in the cluster states can be corrected \cite{Su2018EC}. This could lead to the implementation of Gaussian Boson Sampling with a large number of modes, and thus may potentially help to achieve quantum supremacy \cite{Boixo2018, Harrow2017, Preskill2018, Vankov2018, Chen2018}.
For the CV-IQP circuit, we had to introduce non-Gaussian gates. The fundamental non-Gaussian gate we discussed was the cubic phase gate. In the CV-IQP circuit, all unitaries commute. We thus rearranged the gates such that all controlled-Z gates were to the right of the squeezers, all displacements were directly before the homodyne detectors and the non-Gaussian gates in between. The advantage of this rearrangement is similar to that we obtained for the Gaussian Boson Sampling: before the non-Gaussian gates the states and gates are Gaussian and thus the errors due to finite squeezing can be corrected. This would increase the number of modes we can prepare given the overall noise tolerance.
For the CV Grover's search algorithm we also require non-Gaussian gates. In this case we need to implement a selective inversion operator that can be reduced to a state teleportation problem, which we show is ultimately equivalent to a sequence of higher-order quadrature phase gates. We consider implementing the inversion operator with both a single and two continuous variable qumodes. In both cases we explicitly consider the state that would be required for gate teleportation. This state is simulated with the Strawberry Fields quantum software package \cite{SFpaper}. \\
\noindent {\bf Acknowledgements:} We would like to thank Joshua Izaac for help with data visualisation and Juan Miguel Arrazola $\&$ Thomas Bromley for valuable, patient explanations regarding the Strawberry Fields software package.
\end{document}
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arXiv
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Cross hole sonic test results for analysis of pile load test
Swapan Kunar Bagui1,
S. K. Puri1 &
K. Subbiah1
Quality of concrete for pile can be checked using Cross-hole Sonic Logging (CSL) Test. A processing method wide-band CSL data is presented herein. First Time Arrival (FTA) is an important consideration. In pile capacity analysis or CSL analysis, it is assumed that pile cross section is uniform with uniform value of elastic modulus of concrete but in real practice both are non-uniform. The procedure identifies the location accuracy and further characterizes the features of the defect. FTA is used to find out the location of the distress in the pile. This method identifies the exact location of any void or defect inside the rebar cage of a drilled shaft. This method provides a significant improvement to current techniques used in quality control during construction of bridges. In this present paper, the analysis has been carried out based on uniform and non-uniform values of pile cross section and E value of concrete. Cross hole sonic and pile load test using O-Cell were carried out on same pile at 7 and 28 days of concreting. Same pipes were used for base grout after cross hole sonic test. These results were used to analyze O-cell test results based on a case study and presented in this paper. The distribution of skin frication and skin friction force has also been presented herein with both uniform and non-uniform cross section and E values of concrete. Based on the field test results and analysis a simplified methodology, has been proposed in this paper, for development of Equivalent Top Down Loading with consideration of elastic shortening of pile and surrounding soil for both cases i.e., uniform and non-uniform E values and pile cross sections.
Generally the rivers widen near sea area than that at the origin of the river at hill. Navigation vehicles use the river and therefore wide spans of bridges are required at these locations. Soils near sea area generally have poor bearing capacity. Due to heavy load transfer, open foundation is not a feasible solution and hence deep pile foundations are required. Several methods of construction exist today for the construction of drilled shafts (Reese and O'Neill 1999). The most widely used method today is the wet method, i.e., shafts are cast under wet conditions using bentonite slurry in order to keep the borehole open vertical during drilling to avoid collapsing of soil and preventing proper casting of the concrete. Several defects in form of necking /bulging may be occurred. These defects occur due to failure of soft vertical strata after excavation of soil /during lowering of cage (Sarhan et al. 2003) and these defects occupying up to 15% of the cast in situ pile section could remain undetected. Camp et al. (2007) recorded that, out of 441 piles tested on different projects in South Carolina, around more than 70% of the projects had at least one pile containing a defect. Such defects in the integrity of the cast in situ pile can affect their pile capacity and ability to transmit the design loads and hence quality control of pile construction is a major important critical factor. Several methods are currently used to perform Non Destructive Testing (NDT) of deep foundations. CSL is a common and most reliable technique among the most common methods of NDT testing. Complete guideline is available in ASTM D 6760 (2008) It is observed that defects near the top of a drilled shaft will significantly affect its structural capacity due to unsound concrete. CSL is one of the most reliable techniques for determining the integrity of cast in-situ pile.
It is a NDT method which consists of ultrasonic signal transmission through the pile between two similar water filled tubes. The tubes are kept to the cage of reinforcement and cast forever into the pile. The total number of tube varies from two to four or more.
A transmitter and a receiver are dropped to the lowest point of the pile in separate tubes. Quantities of the signal transmission are captured @ 5 cm interval during rising of transmitter and receiver. Shapes are collected from all permutations of pipes. Flaws should be addressed if these are indicated in more than 50% of the profiles (Likins et al. 2007). Defects must be addressed if these are indicated in more than one profile.
Li et al. 2005 stated that defects present in a pile may not be encountered during a Cross-Hole Sonic Logging (CSL) Test. Even when defects are indeed encountered, they may not always be detected. This paper proposes a probabilistic analysis procedure for evaluating the reliability of CSL, thereby providing a theoretical supplement to existing experimental evaluations of CSL. The reliability of this integrity testing method is represented by the inspection probability, which is expressed as the product of the encountered probability and the detection probability. Mathematical models to calculate the encountered probability are formulated and a detection probability model is suggested based on existing CSL test data. Several examples are presented to illustrate the proposed procedure. The results indicate that there exists a minimum detectable defect size below which the defect cannot be inspected. For a given number of access tubes, the minimum detectable defect size, as a percentage of the pile cross-sectional area, decreases with the pile diameter. The encountered probability can be taken as an index to determine the required number of access tubes. When the target encountered probability is specified as 0.95, three and four access tubes will be sufficient to encounter defects larger than 15 and 5% of the pile cross-sectional area, respectively.
Iskander et al. 2003 presented the results of nondestructive integrity tests (NDTs) and axial static load tests on drilled shafts constructed in varved clay at the National Geotechnical Experimentation Site in Amherst, Mass. The shafts were constructed with built-in defects to study: (1) the effectiveness of conventional NDT methods in detecting construction defects and (2) the effect of defects on the capacity of drilled shafts. Defects included voids and soil inclusions occupying 5–45% of the cross section as well as a soft bottom. Nine organizations participated in a blind defect prediction symposium, using a variety of NDT techniques. Most participants located defects that were larger than 10% of the cross sectional area. However, false positives and inability to locate smaller defects and multiple defects in the same shaft were encountered. Static load tests indicated that (1) minor defects had little or no effect on skin friction; (2) a soft bottom resulted in a 33% reduction in end bearing relative to a sound bottom; and (3) reloading resulted in a 20–30% reduction in the geotechnical capacity.
Summary of literature review
From past study, it is observed that very limited studies were carried out on cross hole sonic test and its further uses to determine pile capacity. Therefore, there is a need of this study and it is also necessary to compare the test results obtained from conventional pile load test.
Basic principles of techniques
Cross-hole Sonic Logging (CSL) is a derivative of the Ultra-sonic Pulse Velocity (UPV) test. The sound wave velocity in concrete (V) is a function of the density and Young Modulus of concrete (Finno and Osborn 1997) as presented in the following Equation.
$$ V=\sqrt{\frac{E\left(1-\mu \right)}{\rho \left(1+\mu \right)\left(1-2\mu \right)}} $$
E = Young's modulus (MPa);
p = Density (kilogram per cubic meter); and.
μ = Poisson's ratio.
The CSL method is an ultrasonic test that involves measuring the propagation time of ultrasonic signals between two probes in vertical tube /ducts in a shaft. These tubes were casted into a shaft during construction of cast in-situ pile.
The difference in signal coming time permits one to calculate and identify areas of low compactness of concrete. For the case of good concrete, pulse velocity is on the order of 4000 m/sec, depending on its ingredients. Concrete consists of soil, gravel, betonies or honeycombing which causes lower propagation velocity so that the presence of these irregularities is immediately observable. Signal reduction is sign of unsound quality zones because more energy is transmitted through sound concrete than through poor concrete.
The dynamic Young's modulus of elasticity (E) of the concrete may be determined from the pulse velocity and the dynamic Poisson's ratio using the following relationship (I S: 13311 1992):
$$ E=\frac{\rho \left(1+\mu \right)\left(1-2\mu \right){V}^2}{\left(1-\mu \right)} $$
E = Young's Modulus of concrete in MPa;
ƿ = Density in kg/m3; and.
V = Pulse velocity in km/second.
The value of the dynamic Poisson's ratio varies from 0.20 to 0.35, with recommended value of 0.25 is adopted for calculation in this present study.
A typical defect identification and first time arrival is shown in Fig. 1.
Defect and FTA Graph
Objectives and approach
The purpose of the present study is to present a method developed to detect the exact location of the defects after performing the CSL test at 7 days of concreting to determine E value of concrete and non-uniform cross sections at different depths of drilled shaft. To accomplish these objectives, a drilled shaft sample of around 100 m long pile with 2.5 m diameter was constructed with arrangement of cross hole sonic pipe arrangement (Four steel Pipes of 40 mm diameter). and O-Cell arrangement for CSL test and O-Cell Test. O-Cell test was conducted at 28 days of concreting.
The objectives of this present paper is also to determine unit skin friction for both uniform and non-uniform values of E and pile cross-section and preparation of equivalent top down loading graph and recommendation of pile capacity for both cases.
Quantity of concrete used
A pile of 100 m long and 2.5 m diameter was constructed. Excavation of soil was carried out using rotary Equipment with reverse air circulation method. Length of excavation was recorded using boring pipe of length segment of 2.5–3.0 m as shown Fig. 2 with sequence of numbering as shown in Fig. 2 so that exact depth of pile can be constructed. Length of excavation, after excavation of soil, before concrete works and after pouring each 49.1 m3 with wire rope arrange for cross checking the depth of concreting. Volume of concreting recorded using wire rope method as shown in Fig. 2.
Checking Excavation and Concrete Pouring Depth
Concrete volume is noted after 10 m depth of pouring and presented in Table 1 and comparison of concrete volume with theoretical volume is presented in Fig. 3.
Table 1 Volume and area of concrete executed
Theoretical and Actual Volume of Concrete
Cross hole test method
The principle of cross hole sonic testing has been applied for quality control of deep excavation. Although the sensitivity of the sensor is well known and often causes some difficulties in achieving reasonable results, the lack of a better solution has led to the method becoming well accepted and used worldwide where excavations must be supported by a fluid (Clean Water or like as water). In order to improve the actual state-of-the-art, Bauer has developed the Sonic Meter "RSM-SY8" which, amongst many other features self-calibrates over depth and thus gives more reliable results this Sonic meter follow the code ASTM- D 6760–02.
In order to test the ultrasonic integrity test of a completed bored pile, the ultrasonic signal of compression waves is analyzed between four parallel pipes, as shown in Fig. 4. Four Pipes are used to conduct this test.
Setting for Ultrasonic Integrity Testing of a Bored Pile
Pre - preparation and execution of ultrasonic testing of bored piles
Special steel pipes (inner Ø 40 mm, outer Ø 44 mm) were installed on the reinforcement cage 2.5 m diameter of pile. The number and the position of the pipes recorded on the drawings. In no instance should the pipes be allowed to reduce the clear distances required for concrete flow. Since the initiation of the concrete pour is often critical to the quality of the whole placement it is recommended to curtail the pipes 50 cm above the cut of level of pile.
Pipes must be watertight at their base and joints. A reduction in the inner diameter is avoided, an additional sealing over the joints may affect the ultrasonic signal. Therefore, the position of the joints should also be recorded. Steel pipes with welded caps and crimped connections have proved effective.
Before concreting the pipes were be closed at their top.
After concreting, and before testing, the following steps are adopted:
Cutting the testing pipes 5 cm–10 cm above the cut of level of pile.
Measuring and recording of the actual distances between pipes at the accessible top of the pile.
Checking the continuity of the pipes by using a steel bolt.
Measuring and recording the maximum depth of each pipe.
Filling of the pipes with clean water.
Closing the pipes until testing is commenced.
Typical photograph showing the progress of the test is shown in Fig. 5.
Progress of Cross Hole Sonic Test
The testing commences after lowering both sensor to the designated lower reference elevation. The sensor are slowly (0.5 m/s) pulled upwards with a rope system. The same elevation of the sensor was verified frequently and any deviations recorded. The technical measurement regulations should follow the relevant standards (eg ASTM D6760–02).
In suspicious areas repeated or additional measurements were carried out.
Apparatus for test
Apparatus for Allowing Internal Inspection (Steel Pipe): Before concreting, access pipe typically have an internal diameter from 38 to 50 mm.
Apparatus for Determining Physical Test Parameters
Weighted Measuring Tape
Apparatus for Obtaining Measurements:
Sensor (Probes): Probes were allowed a generated or detected pulse within 100 mm of the bottom of the Steel pipe. The weight of each probe was in all cases to be sufficient to allow it to sink under its own weight in the Steel pipe. The probe housing was waterproof to at least 1.5 times the maximum depth of testing.
Transmitter Probe—the transmitter probe was generated an ultrasonic pulse with a minimum frequency of 30,000 Hz.
Receiver Probe—the receiver probe was a similar size and compatible design to the transmitter probe and used to detect the arrival of the ultrasonic pulse generated by the transmitter probe.
Signal Transmission Cables—the signal cables used to deploy the probes and transmit data from the probes were sufficiently robust to support the probes' weight. The cable was to be abrasion resistant to allow repeated field use and maintain flexibility in the range of anticipated temperatures. All cable connectors or splices were watertight
Apparatus of Wave Machine (RSM-SY8) for Processing Data—The apparatus for processing the data was a digital computer or microprocessor capable of analyzing all data to identify at least the first arrival and energy of the transmitted ultrasonic pulse at the receiver probe for each depth interval. The data were then be compiled into a single ultrasonic profile for each duct pair.
Apparatus Wave Machine (RSM-SY8) for Display of Measured Data—the apparatus was capable of displaying the raw receiver ultrasonic pulses to confirm data quality during acquisition. After data acquisition, the apparatus was capable of displaying the raw data of each ultrasonic pulse along the entire pile length. The apparatus displayed the processed ultrasonic profile.
Start test on selected pile
Check that the apparatus is functioning correctly prior to mobilizing to site.
Date of Testing—The tests were performed no sooner than 7 days after casting of concrete.
Preparing Steel pipe for Testing—the steel pipe was exposed and the protective top caps removed. Use a weighted measuring tape to measure and record the length of each pipe to the nearest 10 mm. The pipes were filled to the top with clean water.
Steel Pipes Documentation—Assign a systematic reference label to each pipe and prepare a reference sketch of the pipe layout using the magnetic compass or a site plan diagram. The as-built details of the pipe layout were recorded including measuring the center-to-center separations of the exposed pipes, make sequence number as clock wise like as 1–2, 2–3. 3–4, 4–1 or 1–2, 2–3, 3–4, 4–1, 1–3, 2–4, and measuring the pipe length exposed above the concrete. Typical diagram is shown in Fig. 6.
Positions of Pipes
Sensor (Probe) Preparation—to obtain a good acoustic coupling between the sensors (probes) and the water in the pipes, the sensors (probes) were clean and free from all contaminants.
Check that test equipment and sensors (probes) are functioning correctly prior to actual testing by placing the sensors (probes) in two adjacent water filled access ducts of one pile just below the level of the shaft concrete and verifying that ultrasonic pulses are received in the recording apparatus.
Obtaining Measurements with the Apparatus:
Pay due regard to safety and any special instructions or manufacturer's procedures pertaining to the particular apparatus employed.
Document the pair of access ducts being tested. Place the sensor (probe) cable pulley guides into the pipe. Insert the transmitter and receiver sensor into these pipes ensuring that the cables are engaged over the respective cable pulley guides fixed at the pipes tops.
Zero the depth-measuring device if required by the recording apparatus.
Carefully lowered the sensor (probes) down the pipes at a steady rate not exceeding 0.5 m/s, always keeping them at the same level, until one sensor (probe) reached the bottom of the pipe. Temporarily secure the cables at that level with the cables remaining in equal tension.
Sensor cable connect the Wave Machine, then power on & select program. Input the data such as Pile information (diameter, length of pipe, concrete grade etc) and steel pipe (access duct) information (inner diameter, outer diameter, length of pipe, bottom level, through speed, lift up speed etc).
Adjust the test apparatus, if necessary, selecting the power settings required for the pipe separation distance and concrete characteristics encountered such that an ultrasonic pulse with good amplitude can be consistently obtained in a portion of pile shaft of good quality.
Begin recording the ultrasonic pulses as the sensor (probes) are raised. Lift all probes by steadily pulling the sensor (probe) cables simultaneously at a speed of ascent slow enough to capture one ultrasonic pulse for each depth interval specified. If an ultrasonic pulse is not obtained for any depth interval, then the sensor (probes) was lowered past that depth and the test repeated until all depth intervals have an associated ultrasonic pulse.
Data quality checks
After completing data acquisition, view the ultrasonic profile obtained. Check the ultrasonic profile quality.
Compare the length of the measured ultrasonic profile with the measured access duct length. In comparing these measurements a correction should be made to account for the length between the bottom of the probe assembly to the exact point of the transmitter and receiver on the probe. The difference between the corrected measurements shall not exceed 1% of the measured length or 0.25 m, whichever is larger.
Ensure that the captured data is labeled with the pile identification, identification of the all access ducts for the data set, date of test, identification of the test operator, and any further necessary project information such as site and location details as requested by the specified. Store the data and information safely.
Completing the test
If the ultrasonic profile indicates an exception, then the suspect exception zone may be further investigated by special test procedures such as fan shaped tests, tests with the sensors (probes) raised at a fixed offset distance, or other tomographical techniques. The sensors (probes) shall be lowered to a depth of at least 1 m below the anomaly and raised to a depth of at least 1 m above the anomaly.
Cross-Hole Sonic Logging test was conducted just 1 day before conducting O-Cell load test. Based on ultrasonic pulse velocity, density of concrete and Poisson ratio of concrete, E Values at different depths were calculated using Eq. 2 and presented in Table 2 and Fig. 7.
Table 2 E Value at different depth
Wave Speed VS Sound Speed
Strains were measured during O-Cell load test which was conducted at 28 days of concreting. Strains were recorded for each load increment and plotted and shown in Fig. 8.
Strain Recorded during O-Cell Load Test
Unconfined compressive strength at 28 days, Fc was tested in the project site laboratory and test results was found to be 52.1 MPa and this strength was converted to cube strength by multiplying a factor, 0.8. Unit weight of concrete was found to be 2405 kg/m3. E value is calculated using following ACI formula as mentioned below:
E = 0.043 ɣ 1.5 fc 0.5 = 0.043 × 24051.5 × 42.31.5 = 32,992 MPa.40 mm diameter, 32 bar used and equivalent cross section is found = (3.1.14/4) × 2500 × 2500 + [280/(3 × 13–1)(3.1.14/4) × 40 × 40 × 32 = 5,102,341 mm2.
Forty millimeters diameter steel with 32 numbers were used in pile caging and pile stiffness, AE was calculated and found to be 168,340 MN. Unit skin friction has been determined from O-Cell test results and cross hole sonic logging test and presented in Table 3. Distribution of skin friction for the case of CSL is presented in Fig. 9. Similar graph can be plotted based on O-Cell Test Results. Strains of two adjacent strain gauges are considered and converted to compressive load using following formula, load = εAE(ε = Strain, AE = stiffness' of pile) as obtained from Eq. 3. The differences in compressive load and self-weight are used to determine skin friction from the theory of force equilibrium.
Table 3 Comparison of unit skin friction
Unit Skin Shear from Cross Hole Sonic Test Results
Result interpretation
The explanation of CSL results requires understanding of the proficiencies and restrictions of the method. Pile integrity calculation from CSL is based on the FAT of the signal. The pulse velocity in concrete can be determined by dividing the distance between the pipes by the FAT.
Quality of concrete is determined based on rating mentioned in Table 4 (I S:1311–1992). Considering Tables 2 and 4, it is observed that good quality of concrete was used in the construction of pile. Hence pile integrity is found to be satisfactory.
Table 4 Velocity criterion for quality of concrete
The cross-sectional area of the concrete directly influences the stiffness of the test pile. To estimate the cross-sectional area of the concrete, the volume of pouring concrete was recorded along the depth of the shaft, as shown in Fig. 3. Based on volume of concrete pouring, cross section has been estimated 10 m interval and presented in Table 1.
Ultrasonic Pulse velocity has been determined @ 10 m interval taking average velocity from CSL Test and E value concrete is also determined @ 10 m interval using Eq. 2 and presented in Table 2. E value is comparing with ACI Method and it is observed that E value obtained from CSL is slightly higher than ACI Method. Uniform stiffness used for O-Cell calculation and non-uniform stiffness obtained from CSL is used to determine unit shear skin friction. The friction values for both cases are presented in Table 3 and presented in Fig. 9 for the case of CSL Method.
Friction force has been determined both cases and it was found to be 32.5 MN from O-Cell load case and 34.8 MN for the case of CSL method. Design capacity of pile is found to be 30.5 MN for the case of O-Cell results and 31.1 MN for the case of CSL. Both capacities are comparable and only 2% variations are noticed. Therefore, it may be concluded that proposed method may be considered for O-Cell test result analysis.
Construction of equivalent top down load
The bidirectional test results are considered as upward and downward load-settlement curves with respect to Osterberg-cell. In the case of upward deflection is less, extrapolation is required using hyperbolic curve. It is expected that the upward load-displacement behavior is governed by the skin resistance of the test pile above the O-cell and that the downward behavior is governed by the skin friction and toe resistances below the O-cell. The original method for constructing Equivalent Top Load (ETL) curves suggested by Osterberg (1995). In the initial development, the uphill and downhill settlement curves are adding combing the upward and down- ward loads for same settlement. The upward and downward force represents the skin friction force (Qs) and toe bearing capacity (Qb) above and below the Osterberg-cell respectively, noting that Qb also includes the skin friction i.e., by the adjacent soil below the O-cell. Elastic shortening (δc) of the foundation material is considered in estimating pile capacity.
The actual field O-Cell tested curve with the smaller displacement should be extrapolated to generate a resulting curve up to the applied load. Hyperbolic curve fitting method can be used.
The revised simplified method is based on settlement at the head consists of both the toe settlement and the elastic shortening of the foundation material. In Osterberg-cell tests, the settlement of the lowest plate considers the base settlement and the flexible shortening of the foundation below the O-cell. Hence, there is no need of elastic shortening consideration below the Osterberg-cell but foundation section above the O-cell, the elastic shortening is required for determination of equivalent top-down load test mostly exceeds that in O-cell tests (England 2009). Elastic shortening has been determined using following Equation:
$$ \delta =\frac{C{Q}_s}{AE}\times L $$
δ=Elastic Shortening;
L = Length of pile above O-Cell;
AE = Stiffness of Pile;
Qs = Upward force; and.
C = Factor, 0.5 for rectangular distribution of skin shear and 0.33 for triangular distribution.
Load deflection curve was prepared and calculated based on following steps:
Choose a known deflection and find out downward force and upward force and add both to get top down load for summing up both deflection .
Similarly choose a second deflection and calculate load deflection as mentioned in Step 1. Generally upward movement is less and this maximum upward deflection and load will be considered until downward deflection will more than upward maximum deflection.
Beyond this deflection as mentioned in Step 2, it will calculates as per method as mentioned Step1.
Load deflection curves are prepared and presented in Fig. 10.
Actual Load Settlement Curve from O-Cell Test Results
Length of pile above top of the O-Cell is 80 m. Elastic shortening above O-Cell is considered. It is found rectangular skin shear developed as shown Fig. 9 and C value is taken 0.5 and finally Equivalent Top Load presented in Fig. 11.
Equivalent Top Load
Movement after elastic shortening for the case of non-uniform values of E and cross section are also calculated and elastic shortening is more than 0.1 mm i.e., close to value of uniform case i.e., results obtained from O-Cell results. Therefore, proposed method is comparable with conventional method.
Cross-hole sonic logging test is conducted to determine quality of concrete for deep pile and identifies the defect location based on ultrasonic pulse velocity. Generally unit skin friction can be determined CSL and strain captured during O-Cell pile load test. This paper presents this method and compares skin friction force /ultimate pile capacity and compares O-Cell test result. It is found that both methods are comparable and analysis may be carried out using proposed method. Based on the results and analysis presented herein, following conclusions may be drawn:
Actual cross section can be determined based on concrete consumed during concreting. This is carried out using wire rope method.
Cross-hole sonic logging test results can be used to determine pile capacity when same pile is tested using O-Cell.
E value of concrete can be determined using formula as mentioned in Eq. 2 and it may be used for O-Cell test results analysis.
Quality of concrete in pile will be determined based on ultrasonic pulse velocity and quality of concrete grade as mentioned in Table 4.
Strains at different depths of pile can be found from strain gauge reading. Uniform pile cross section - A, E value and non-uniform cross sections and non-uniform E values can be used to determine unit shear skin friction as proposed method mentioned in this paper.
Skin friction forces obtained by both methods may be used for ultimate capacity of pile.
Proposed methodology is found to be suitable and may be used for O-Cell test results analysis and comparison.
Determination of Equivalent top down load has been simplified and same may be used for analysis.
CSL:
Cross-hole Sonic Logging
FTA:
First Time Arrival
NDT:
ASTM:
American Society for Testing Material
C T:
Cross-hole Tomography
MN:
Mega Newton
ETL:
ASTM D6760 (2008) Standard test method for integrity testing of concrete deep foundations by ultrasonic crosshole testing
Camp WM III, Holley DW, Canivan GJ (2007) Cross-hole Sonic Logging (CSL) of South Carolina drilled shafts: a five year summary. GeoDenver 2007 New Peaks in Geotechnica, Denver (CD Rom)
England M (2009) Review of methods of analysis o test results from Bi-directional static load tests. In: Proceedings of deep foundationson bored and auger piles. Taylor & Francis, Milton Park, pp 235–239
Richard J. Finno, Peter W. Osborn: Final reports of project: non-destructive evaluation of a deep foundation test. Infrastructure Technology Institute (ITI) at the Northwestern University National Geotechnical Experimentation Site, 1997
I S: 13311 (Part 1) (1992) Non-destructive testing of concrete -methods of test, part 1 ultrasonic pulse velocity
Iskander M, Roy D, Kelley S, Ealy C (2003) Drilled shaft defects: detection, and effects on capacity in Varved clay. J Geotech Geoenviron 129(12):1128–1137
Li DQ, Zhang LM, Tang WH (2005) Reliability evaluation of cross-hole sonic logging for bored pile integrity. J Geotech Geoenviron 131(9):1130–1138 Online publication date: September 01, 2005
Likins GE, Rausche F, Webster K, Klesney A (2007) Defect analysis for CSL testing. Geo-Denver 2007 New Peaks in Geotechnica, Denver (CD Rom)
Osterberg JO (1995) FHWA-SA-94-035. The Osterberg cell for load testing drilled shaft and driven piles. Federal Highway Administration, Washington D. C.
Reese, L. C., O'Neill, M. W., (1999). "Drilled shafts: construction procedures and design methods". Publication no. FHWA-IF-99-025, Federal Highway
Sarhan HA, O'Neill MW, Tabsh S (2003) Response of cast in situ pile with minor flaws to axial and lateral loads. Eng Struct 25(1):47–56
Data share
No objection for data sharing.
No research funding is available.
I CT (I) Pvt. Limited, A 11 Green Park, New Delhi, India
Swapan Kunar Bagui, S. K. Puri & K. Subbiah
Swapan Kunar Bagui
S. K. Puri
K. Subbiah
S K BAGUI: Main Author-Collection of field data, literature review, preparation of document. S K PURI: Guide team member for document preparation, cost variation of project, estimation of base grout quantity. K Subbiah: Document collection from Institute, Methodology for field testing. The author(s) read and approved the final manuscript.
Correspondence to Swapan Kunar Bagui.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Bagui, S.K., Puri, S.K. & Subbiah, K. Cross hole sonic test results for analysis of pile load test. ABEN 1, 15 (2020). https://doi.org/10.1186/s43251-020-00017-4
Osterberg cell
Bidirectional load test
Pile integrity test
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CommonCrawl
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What does it mean to do multi-dimensional processing with tensors in tensor cores?
In some tweets about NeurIPS 2018, this video from NVIDIA appeared. At around 0.37, she says:
If you think about the current computations in our deep learning systems, they are all based on Linear Algebra. Can we come up with better paradigms to do multi-dimensional processing? can we do truly tensor-algebraic techniques in our tensor cores?
I was wondering what she is talking about. I'm not an expert so I'd like to understand better this specific point.
machine-learning deep-learning ai-design linear-algebra
wrong_pathwrong_path
Both PR and Theoretically Valid
Although Anima Anandkumar's presentation is a puff piece for NVidia, her representation is not contrary to theory.
... next level [above NVidia's GPU success] ... means new algorithmic research. So, if you think about the current computations in our deep learning systems, they are all based on linear algebra. Can we come up with better paradigms to do multi-dimensional processing. Can we do truly tensor algebraic techniques in our tensor cores, and what kind of new architectures will this realize?
A tensor is the extension of the idea of functions and vectors. Vectors emerged allow the encapsulation of multiple variables of the same type into a single unit to express multidimensional properties such as position or force. Variables that are not encapsulations of multiple variables are then referred to as scalars. When such multidimensional properties are compositions of functions rather than variables because their values are dependent upon other variables, they are vector fields.
To more cleanly model electromagnetism and gravity in mathematics, higher levels of variable and functional encapsulation were required. Scalar variables and functions that return scalar variables are thus tensors of rank one (1). Vectors and vector fields are tensors of rank two (2), and higher ranks are extensions of the pattern created by scalars and vectors. A matrix containing functions can be represented as a tensor of rank three (3), encapsulating its elements into a single unit.
Digital Processing and Integration Trend
CUDA cores, which are the signal processing units used in current generation NVidia GPUs (graphical processing units) can be used for production of two or three dimensional rendering or leveraged to produce parallel processing of signals through an artificial network. This is following the trend to delegate to digital circuitry the most significant bottlenecks in algorithm execution. VLSI (very large scale integration) technology is the logical result of this trend.
Industrial and military calculators with tests and branching running on rows of racks of relays
Add speed with tubes and load programs with punch cards
Reduce power and improve reliability and speed with transistors, magnetic core memory, and paper tape
Integrated circuits on motherboards with vinyl tape
Handheld programmable calculators for students
Microprocessors with magnetic disks
Floating point, digital signal, and graphics processors to extend microprocessors
Multiple cores (repeating large scale circuit patterns on a single substrate)
Re-purposing and extending graphics rendering circuits to offload artificial network computations to GPUs
With all these, speed, size, energy thrift, and convenience are not really a shift in the original paradigm that started with Norbert Wiener, Claude Shannon, Alan Turing, John von Neumann, and others. In fact, computing is still catching up in several ways with the perspectives of these pioneers and a long way off from producing in VLSI common concepts in science. Software is the solution, which is why it is called soft, meaning flexible, not necessarily weak. However, flexibility sacrifices speed and capability, thus the trend above.
The Parallel Computing Challenge
Part of what Anima Anandkumar is stating that the signal paths in current VLSI processors are still a much lower level of abstraction than the ideas of mathematicians, physicists, and AI engineers. Scientific theory describes probability, statistical distribution, expectation, force, loss, gain, pain, reward, memory, momentum, semantics, combination, and correlation at a much higher level than current digital circuitry.
The use of sequential algorithms is the splaying out in time what could be a massively parallel operation. The serial algorithm limits the rate of processing. The finding of ways of dealing with things in parallel in mathematics can be done with a pencil. In computers, paralleling algorithms and finding parallel processing structures in VLSI form that are as flexible as software in some respects is much more challenging and thus far behind.
This is one problem that has been the focus of research in at least twenty large corporate and government labs since for half a century and has been the intention of VLSI from the onset. The work is not specific to NVidia. It is not a new problem and the approach to solving it has been along this paradigm.
Add abstraction and encapsulation in the mind.
Express it in mathematics.
Write it as a serial algorithm, leveraging whatever parallel constructs are supported by programming languages and libraries that can leverage VLSI parallelism or computing clusters.
The last seventy years of computing machinery development has been bringing hardware, operating systems, and software closer to the level of mathematical expressions that were twenty to two hundred years old. That may change, and everyone wants to ride the new wave.
Anima Anandkumar and her counterparts in IBM, Intel, Google, Microsoft, the U.S. Navy, Amazon, Alibaba, and the other corporate and government laboratories do not state (because it is either classified or company confidential) what they intend to do to further the parallelization of computing. Whatever they do along those particular lines would not be a paradigm shift but rather a next step along the current paradigm.
Enters Corporate Strategy
They would also not state what they may be doing that is not in that paradigm. They are constrained to only give hints without theoretical substance. If they were working on a chip that exhibited what the human brain exhibits when neurons grow and connect according to DNA based propensity, they wouldn't say that in a technically precise way. Corporate secrecy is part of the global economic game play, not tipping the hand. They have their game face on.
The idea of moving from linear to non-linear is a good public relations theme and when used for puff pieces is not conclusively technical. The pitch usually goes along this line.
What they did was very linear. We are moving into a non-linear space.
It is an attempt to claim that what was done was primitive and the impending game changing advancement is coming from the speaker and their people. Sometimes it works to create a temporary hike in the value of tradable securities, which that company needs as of this writing. If there is a true change in the game, it will be known when it is released. Those who have worked in laboratories for years know to wait until something is released that demonstrates, when the example code is configured and ran, what game change actually occurred, if any. Or they develop the game changer themselves, which is why it is a lab.
Ambiguity of 'Linear' Even in Mathematics
Also note that curved lines are still lines and the term linear can mean two things depending on the context.
Conformant to the linear equation $\vec{Y} = V \vec{X}$, such that it graphs as a line, plane, or higher dimensional flat surface, with constant gradient and without curvature.
Conformant to the principles of linear algebra, which would include spaces, eigenvalues, regression with high-order polynomials, and a number of constructs that involve varying gradient and curvature.
Predicting Paradigm Shifts
The skepticism developed after watching tech company PR for some time does not necessarily dismiss game changing technology advancements of the past and potentials of the future. Any open source project team, individual, corporation, or government lab might do something that shifts the paradigm, usually over a period of years. Classic examples:
$F = m a$
Oxygen (that air is not an element)
Alternating current energy transmission
Internal combustion
Computing examples:
First transistor
LISP and FORTRAN
C and UNIX
Shifts do not need to be as far reaching and game changing as these to have an impact. Who will achieve the next with regard to AI concepts, information structure, algorithm, execution environment is not something very easy to predict. Consider those that lived before any of the above and try to imagine them trying to predict that the projects of Isaac Newton, Antoine Lavoisier, Michael Faraday, Nicola Tesla, Norbert Wiener, Claude Shannon, Ken Thompson, Dennis Ritchie, or any of the others were the seeds of the next paradigm shift.
There is research into analog artificial networks, neuromorphic hardware, semantic modelling, graph algorithms, and other potential game changers, each of which has impressive conceptual foundations and is discussed in some of the Q&A here. These are a few.
What topologies are largely unexplored in machine learning?
Novelty Search Mutation Algorithm
Can deep networks be trained to prove theorems?
If digital values are mere estimates, why not return to analog for AI?
Are commercially available neural ICs digital?
What of these may point to the beginnings of paradigm shift cannot be known, and even if an idea first posted here or referenced from here is the seed, it may not be known later. The multilayer perceptron may be the seed of future streets and highways being dominated by automated vehicles in 2090, but no one in a million people will realize in seventy years that the trend toward AV research was seeded by MLP enthusiasm from the first decade of this century.
Gaps to be Filled and NVidia is a Contendere
All this aside, we use NVidia hardware every day for robotics and analysis, so they have credibility resulting from past success. If they produced a chip that does something remarkably clever before Intel or the IBM-MIT collaboration, it would be a small surprise, but not it is not a completely unbelievable possibility. Certainly the comprehension of Hilbert spaces, semantics, and topology are limited in the computer science field, and a paradigm shift toward greater comprehension of them or some new thing that is not even part of mathematical thought today would add some needed diversity to the computing industry.
Douglas DaseecoDouglas Daseeco
Not the answer you're looking for? Browse other questions tagged machine-learning deep-learning ai-design linear-algebra or ask your own question.
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BMC Bioinformatics
Machine learning predicts nucleosome binding modes of transcription factors
K. C. Kishan2 na1,
Sridevi K. Subramanya1 na1,
Rui Li2 &
Feng Cui1
BMC Bioinformatics volume 22, Article number: 166 (2021) Cite this article
Most transcription factors (TFs) compete with nucleosomes to gain access to their cognate binding sites. Recent studies have identified several TF-nucleosome interaction modes including end binding (EB), oriented binding, periodic binding, dyad binding, groove binding, and gyre spanning. However, there are substantial experimental challenges in measuring nucleosome binding modes for thousands of TFs in different species.
We present a computational prediction of the binding modes based on TF protein sequences. With a nested cross-validation procedure, our model outperforms several fine-tuned off-the-shelf machine learning (ML) methods in the multi-label classification task. Our binary classifier for the EB mode performs better than these ML methods with the area under precision-recall curve achieving 75%. The end preference of most TFs is consistent with low nucleosome occupancy around their binding site in GM12878 cells. The nucleosome occupancy data is used as an alternative dataset to confirm the superiority of our EB classifier.
We develop the first ML-based approach for efficient and comprehensive analysis of nucleosome binding modes of TFs.
The nucleosome is the basic repeating structural and functional unit of chromatin, which consists of 147 base pairs of DNA wrapped around eight histones. Although nucleosomes cover most of the genome, their locations on DNA are not random. A long-standing goal in chromosome biology is to understand which factors control nucleosome positioning and how nucleosomes interact with those factors to regulate gene expression. One of the determinants of nucleosome positioning is transcription factors (TFs) including activators and components of the preinitiation complex [1].
The traditional view on nucleosome-TF interactions is that TFs displace nucleosomes to gain access to their cognate binding sites. The binding of TFs to a nucleosome results in a ternary structure that is relatively unstable [2] because the TFs have higher binding affinities to free DNA than nucleosomal DNA. This difference in binding affinities leads to the destabilization of the ternary structure. On the other hand, a subgroup of TFs, known as pioneer TFs [3], can interact with nucleosomal DNA, open chromatin, and establish developmental competence. We and other groups have shown that the rotational setting of binding sites in a nucleosome is a critical determinant for their accessibility to these pioneer TFs [4,5,6]. These studies show that pioneer TFs are able to bind nucleosomal DNA while most TFs appear not to have this ability.
A recent study systematically explored the modes of interactions between nucleosomes and 220 TFs that represent diverse structural families [7]. This study has identified several binding modes for the TFs including gyre spanning (GS), oriented binding (OB), end binding (EB), periodic binding (PB) and dyad binding (DB). Importantly, these modes are not mutually exclusive, meaning that a TF may have multiple nucleosome binding modes. These data clearly show that the binary classification of TFs based on their ability to bind nucleosomal DNA is not enough to capture the diversity of the interaction landscape between TFs and the nucleosome.
However, there are substantial experimental challenges to determine nucleosome binding preferences for thousands of TFs in different species. Thus, efficient computational methods capable of determining the preferences are needed. Here, we present ProtGauss, a machine learning (ML) model to predict nucleosome binding modes of TFs based on Gaussian representation for protein sequences. Our model differs from other ML methods by (1) using the Gaussian representation of sequences to capture the diversity of the subsequence representations via a covariance matrix and (2) designing a kernel function to capture the similarity between sequence features represented by Gaussian distributions. These differences are important because general ML methods often take vector representations of the sequences as input, which is obtained by computing the average of subsequence features from ProtVec [8]. The average representation however fails to capture the variation of subsequence features, and therefore is not expressive enough to represent an arbitrarily long protein sequence.
In this study, we focused on 167 TFs that have at least one nucleosome binding mode (or label) measured experimentally (see "Methods"). We used nested cross-validation to train and evaluate the model. The inner cross-validation is used to optimize the hyperparameters of the models and the outer cross-validation estimates the performance of the model with optimal hyperparameters. The nested cross-validation eliminates the bias introduced by simple cross-validation and can thus alleviate the overfitting problem. In this multi-label classification problem, the ProtGauss model outperformed several fine-tuned off-the-shelf ML methods including logistic regression, support vector machine, k-nearest neighbours, and random forest.
Then we built binary classifiers for individual binding modes and found that the classifier for the EB mode is the best. Our EB classifier was superior to the ML methods with the area under precision-recall (minor-AUPR) curve achieving 75%. We further showed that the EB mode of TFs is related to decreased nucleosome occupancy around their bind sites in GM12878 cells mapped by ChIP-seq (ChIP-sequencing) and MNase-seq (micrococcal nuclease digestion with deep sequencing). The nucleosome occupancy profiles around TF binding sites were used as an alternative dataset to confirm the superiority of the EB classifier compared to other ML methods.
Using the EB classifier, we predicted that a vast majority (88–99%) of TFs in five model organisms (yeast, nematode, fruit fly, mouse and human) have this binding mode with mammalian TFs being the lowest and the yeast TFs being the highest. Our prediction showed that several TFs in the SOX family including SOX2 and SOX11 do not have the EB mode, consistent with experimental studies [9]. Overall, this work represents the first systematic analysis of nucleosome binding mode of TFs using a computational method.
Different components of the proposed machine learning method for predicting binding modes of transcriptions factors to nucleosomes are described below. Figure 1 shows the overall block diagram of ProtGauss. To learn feature matrix \(X\), ProtVec [8] method is trained on the subsequences of fixed length \(l_{s}\). Feature matrix \(X\) is projected to mean vector \(\mu\) and covariance matrix \(\Sigma\) of Gaussian distribution. Then, the kernel matrix is defined using the similarity between Gaussian distributions and a multi-label classifier is trained to model nucleosome-binding preferences of TFs.
A model to predict nucleosome-TF binding patterns. The sequence \(s_{i}\) is projected to a multivariate Gaussian distribution with mean \(\tilde{\mu }_{i}\) and covariance matrix \(\tilde{\Sigma }_{i}\). Similarity between these multivariate Gaussian distributions is computed to form kernel matrix and a multi-label classifier is trained to model binding preferences using the kernel matrix
Learning features from protein sequences
Amino acid sequences of TFs are the input to the proposed model. Breaking the sequences into fixed length overlapping subsequences (i.e. biological words) is the simplest and most common technique in bioinformatics to learn features [10, 11]. If the length of the sequence is \(l\) and the length of the subsequence is \(l_{s}\), the sequences can be broken into \(l - \left( {l_{s} - 1} \right)\) subsequences. For simplicity, let the number of subsequences be represented by \(L\).
Given that a primary protein sequence can be split into overlapping subsequences of length \(l_{s}\), we used ProtVec to extract features from the protein sequence. ProtVec embeds each subsequence to d-dimensional vector that characterizes the biophysical and biochemical properties of sequences. In particular, each subsequence of length \(l_{s}\) is represented by a continuous vector of dimension \(d = 100\). The parameter \(d\) is a hyperparameter for ProtVec method. For the sequence that is split into \(L\) subsequences, ProtVec model learns a matrix \(X\) of shape \(L \times d\) where each row represents the feature learnt for a subsequence. Figure 2 illustrates the details on how the sequences are projected to embedding matrix \(X\) using ProtVec.
A workflow to represent an amino acid sequence as a matrix of ProtVec embeddings. A sequence \(s_{i}\) is split into the subsequences with length \(l_{s} = 3\) and the embeddings of these subsequences from the ProtVec model are used to obtain a feature matrix \(X\)
Gaussian representation of sequences
A sequence of length \(l\) is represented as the matrix \(X\) of shape \(L \times d.\) The feature matrix \(X\) contains \(d\) dimensional feature for \(L\) subsequences. A simple approach to learn representation for sequences from the subsequence feature matrix \(X\) is to compute the mean of the representation of subsequences [8]. However, the mean of the subsequence matrix blurs the representation of the subsequences and may not be good enough to represent the sequence. To address this challenge, we proposed a novel approach to represent the sequence \(s\) as a multivariate Gaussian distribution and the biological words (subsequences) are assumed to be generated from that distribution (that is, the subsequence representations are the samples from this distribution) [12]. To the best of our knowledge, our proposed model is the first work to represent protein sequences as Gaussian distribution. Specifically, we consider the embeddings of all subsequences present in the sequence as the independent and identically distributed (i.i.d) samples drawn from the distribution:
$$x \sim N\left( {\mu , \Sigma } \right)$$
where \(x\) represents the ProtVec representation for subsequence sampled from multivariate Gaussian distribution with mean \(\mu\) and covariance matrix \(\Sigma\).
Given the feature matrix \(X\), the mean vector and the covariance matrix are set to their Maximum Likelihood estimates, given by the empirical mean \(\tilde{\mu }\) and the empirical covariance matrix \(\tilde{\Sigma }\) respectively. Specifically, the sample mean of the sequence corresponds to the mean of the subsequence representations, i.e. the vectors of the subsequences in the sequence are added and normalized by the number of subsequences. For a sequence \(s\) with \(L\) subsequences, the mean of the distribution is given by:
$$\tilde{\mu } = \frac{1}{L}\mathop \sum \limits_{i = 1}^{L} x_{i}$$
The empirical covariance matrix is then defined as:
$$\tilde{\Sigma } = \frac{1}{L}\mathop \sum \limits_{i = 1}^{L} (x_{i} - \tilde{\mu })(x_{i} - \tilde{\mu })^{T}$$
Since the sequence is represented as a multivariate Gaussian distribution with sample mean \(\tilde{\mu }\) and the empirical covariance matrix \(\tilde{\Sigma }\), the problem of classifying binding preferences based on the sequence transforms to classifying based on the distribution [12].
Measuring similarity between sequences
To classify the multivariate Gaussian representation of protein sequences, we proceed by defining the similarity between the multivariate Gaussian distributions of two sequences for prediction of nucleosome binding modes. Here, the similarity measure between two distributions is proposed. The similarity between mean vectors \(\tilde{\mu }_{i}\) and \(\tilde{\mu }_{j}\) can be computed using cosine similarity as:
$$sim\left( {\tilde{\mu }_{i} , \tilde{\mu }_{j} } \right) = \frac{{\tilde{\mu }_{i} \cdot \tilde{\mu }_{j} }}{{||\tilde{\mu }_{i} ||_{2} ||\tilde{\mu }_{j} ||_{2} }}$$
where \(\parallel \cdot \parallel_{2}\) denotes the Euclidean norm of the vectors. Similarly, the similarity between the covariance matrices \(\widetilde{{\tilde{\Sigma }}}_{i}\) and \(\tilde{\Sigma }_{j}\) can be computed as:
$$sim\left( {\tilde{\Sigma }_{i} , \tilde{\Sigma }_{j} } \right) = \frac{{\mathop \sum \nolimits_{{}}^{{}} \tilde{\Sigma }_{i} \odot \tilde{\Sigma }_{j} }}{{||\tilde{\Sigma }_{i} ||_{F} ||\tilde{\Sigma }_{j} ||_{F} }}$$
where \(\odot\) is the element-wise multiplication between the matrices and \(\parallel \cdot \parallel_{F}\) represents the Frobenius norm of the matrix. Then, the similarity between two sequences \(s_{i}\) and \(s_{j}\) is measured as the convex combination of the similarity between their mean vectors \(\tilde{\mu }_{i}\) and \(\tilde{\mu }_{j}\) and their covariance matrices \(\tilde{\Sigma }_{i}\) and \(\tilde{\Sigma }_{j}\). Therefore, the similarity between two sequences \(s_{i}\) and \(s_{j}\) is given by:
$$sim\left( {s_{i} , s_{j} } \right) = \alpha \cdot sim\left( {\tilde{\mu }_{i} , \tilde{\mu }_{j} } \right) + \left( {1 - \alpha } \right) \cdot sim\left( {\tilde{\Sigma }_{i} , \tilde{\Sigma }_{j} } \right)$$
where \(\alpha \in \left( {0, 1} \right)\) is the hyperparameter that controls the relative importance of similarity between mean vectors and the similarity between the covariance matrices. A kernel matrix \(K\) is defined where \(K_{ij} = sim\left( {s_{i} , s_{j} } \right)\).
Training our proposed model involves two steps: (a) training ProtVec to learn representation of protein sequences and (b) converting these representations to Gaussian distributions and training Support Vector Machine (SVM) with similarity kernel between Gaussian distributions. First, we collected 561,568 sequences from Swiss-Prot database (UniProt release 2019_11) and trained the ProtVec model on these sequences. The trained ProtVec model was used to obtain the feature matrix \(X_{i}\) for each sequence \(s_{i}\). Second, the Gaussian representation for each sequence is obtained from their respective feature matrix and the similarities between sequences is computed to define the kernel for Support Vector Machine (SVM) [13]. Since the TFs can have multiple binding preferences, it is a multi-label classification task. One-vs-Rest classifier is used to train on multi-label classification problems. We used the scikit-learn library in Python to implement the SVM classifier and nested cross-validation with 10 outer and 10 inner cross-validation to report the results.
The two metrics were considered for performance evaluation of our proposed method as well as other baselines in this task. (1) Accuracy is also known as subset accuracy, that measures the percentages of test TFs that were correctly predicted (i.e. a TF is correctly predicted if the set of predicted binding preferences of this TF exactly matches with the set of its ground-truth preferences; in this case, the ground-truths are nucleosome binding preferences of TFs that are measured experimentally [7]). (2) Micro-averaged area under the precision-recall curve (micro-AUPR) [14] combines the predictions across all binding preferences into a vector, and then the area under the precision-recall curve is computed based on that vector. We further considered Matthews correlation coefficient (MCC) as a measure of the quality of binary classifications to compare several methods with nucleosome occupancy data. This metric addresses the concern of an imbalanced testing set and obtains more reliable performance. MCC can be computed as
$$MCC = \frac{TP \times TN - FP \times FN}{{\surd \left( {TP + FP} \right)\left( {TP + FN} \right) \left( {TN + FP} \right)\left( {TN + FN} \right)}}$$
For multi-label classification, ranking-based metrics such as micro-AUPR is an appropriate metric for class imbalance scenarios [15]. In this work, the class imbalance is extreme i.e. the EB mode has 121 positives, the gyre spanning mode has only 3 positives, the dyad binding mode has 10 positives, and the orientational binding mode has 12 positives out of 167 TFs (Table 1). Since micro-AUPR gives equal importance for all samples across classes, classes with relatively few positive samples should not influence the overall AUPR score of the model if the model is performing well on other common binding preferences.
Table 1 Datasets used in the study
When we formulate multi-label classification as multiple binary classification, accuracy is not an appropriate metric to compare different models. For example, there are 10 positive and 157 negative cases for the dyad binding mode. A simple majority classifier predicts all the TFs to have no dyad preference, the accuracy of the classifier will be 94.01% (= 157/167). In contrast, PR curves are appropriate to classify such binding preferences and therefore we chose micro-AUPR for comparison.
For the baseline methods, the mean of the feature matrix \(X \in R^{L \times d}\) from ProtVec was taken across subsequences as: \(x = \frac{1}{L}\sum\nolimits_{i}^{L} {X_{i} \in R^{d} }\) to obtain the feature vector. The feature vector \(x \in R^{d}\) is used to train other baselines for performance comparison. Our proposed model was compared with Logistic Regression (LR), Support Vector Machine (SVM) [13], K-nearest neighbours (kNN) [16], and Random Forest (RF) [17, 18]. All of these methods were implemented using the scikit-learn library in Python. Table 2 shows the list of settings for tuning parameters of these baselines. We adopted grid search with nested cross-validation to find the optimal parameters and evaluate the performance of the baselines.
Table 2 The parameters and set of values for various off-the-shelf baselines
TF datasets
Seven nucleosome-TF interaction patterns for 195 TFs from diverse structural families were determined experimentally in a prior study (Table S5 in [7]). The interaction modes are not mutually exclusive, and a given TF can have more than one binding modes. These experimentally determined binding modes are used as ground truths of the present study. For multi-label classification, it is important for TFs to belong to at least one class. However, there are 28 TFs that have none of these patterns. Thus, 167 (= 195 − 28) TFs have at least one binding mode. Moreover, 24 TFs have ChIP-seq data from GM12878 cells in ENCODE that were used for testing the model (see below) and 21 of them belong to the 167 TFs. As a result, the remaining 146 (= 167 − 21) TFs were used to train the model (Additional file 1: Supplementary Table S1). The full-length sequences of TFs were taken from Animal TFDB 3.0 [19].
The tested ProtGauss model was applied to all TFs from five model species, including 1,664 TFs from human (Additional file 1: Supplementary Table S2), 1,636 TFs from mouse (Additional file 1: Supplementary Table S3), 651 TFs from fruit fly (Additional file 1: Supplementary Table S4), 748 TFs from nematode (Additional file 1: Supplementary Table S5), and 296 TFs from yeast (Additional file 1: Supplementary Table S6). The full-length sequences of the TFs, except those from yeast, were downloaded from Animal TFDB 3.0 [19] in the FASTA format. The human TF protein sequences file contained a total of 1,675 sequences which included repeated TF entries for "HOPX" (10 times), "ZNF177" (2 times), and "LCOR" (4 times). Note that HOPX and LCOR each have two unique sequences. Thus, these three TFs have five unique sequences that were retained. The remaining entries (11 sequences) were omitted, which results in the human dataset containing 1664 (= 1675 − 11) TF sequences. Among these 1,664 TFs, two sequences have no TF name. Their names (H3BRB8/H3BSE6 and PAWR) were extracted using the Ensembl ID provided in the fasta file. The DNA-binding domains of the TFs were also retrieved from the same file. However, some of the entries had "Miscellaneous" as a domain. Thus, Pfam batch sequence search [20] utility from EMBL-EBI was used to determine the unknown TF domains in batch. The same process was incorporated to extract the unknown TF domains in the other 4 species. Finally, yeast DNA-binding TFs were taken from literature [21] and their full-length sequences were retrieved from UniProt.
Nucleosome (MNase-seq) datasets
The MNase-seq short reads for in vivo nucleosomes in the GM12878 cell line (hg19) were downloaded from the University of California Santa Cruz (UCSC) Genome Browser HTTP server [22]. A total of nine BAM files that are in Gene Expression Omnibus (GEO) ID GSM920558 with names wgEncodeSydhNsomeGm12878AlnRepX.bam, where X is between 1 and 9 were downloaded, merged, and sorted based on chromosomes using SAMtools [23]. The reads in each chromosome file were extended to 147 bp in the 5′ to 3′ direction.
The normalized nucleosome occupancy of a TF was calculated at each nucleotide position in the genome by dividing the total number of nucleosomal DNA sequences surrounding the ChIP peak position by the average number of nucleosomal sequences across the genome as described in the paper [24]. The normalized values were smoothed with a 61-bp window.
ChIP-seq datasets
Out of the 1,664 human TFs, 106 TFs have ChIP-seq data available in the GM12878 cell line, which were deposited in ENCODE database [25]. The GM12878 cell line was chosen because (1) it is a Tier 1 cell line in the ENCODE project in which a number of ChIP-seq datasets are available and (2) it is derived from normal cells that have no global aberrant epigenetic modifications and chromatin reorganization, which are often observed in cancer chromatin [26].
The ChIP-seq peaks (i.e., the 'optimal' set) of the 106 TFs in the human genome hg19 were downloaded from ENCODE. Among 106 TFs, 24 TFs have known E-MI penetration (lig147) values [7] (Additional file 1: Supplementary Table S7) and 82 TFs do not have E-MI values (Additional file 1: Supplementary Table S8). E-MI stands for enriched-sequence-based mutual information, which provides information about the relative location of TF binding in nucleosomal DNA [7]. TFs with E-MI < 20 indicate that the TFs prefer to bind nucleosome ends, whereas TFs with E-MI > 20 indicate that the TFs do not have this preference. Note that there is no overlap between the 106 TFs with the 146 TFs used to train the ProtGauss model. The overlap between the 106 TFs and the 195 TFs with E-MI values [7] is the set of 24 TFs that was used as the test set.
Calculation of nucleosome occupancy around TF binding sites
For a given TF with ChIP-seq data available in the GM12878 cell line, the nucleosome occupancy profile was calculated around the ChIP peak centre (± 1000 bp). Two groups of human TFs in GM12878 cells were used to calculate the nucleosome occupancy profiles: one contains 24 TFs with both ChIP-seq data and known EMI penetration (lig147) values (Additional file 1: Supplementary Table S7), and the other includes 82 TFs with ChIP-seq data but not EMI penetration values (Additional file 1: Supplementary Table S8). For TFs in both groups, the nucleosome occupancy patterns around ChIP peak center were divided into three types: 'peak-at-centre', 'dip-at-centre' and 'questionable' (see detailed below). Note that 5 TFs have the 'questionable' pattern (Additional file 1: Supplementary Table S8), resulting in 101 (= 24 + 82 − 5) TFs that have either 'peak-at-centre' or 'dip-at-centre' nucleosome occupancy patterns (Additional file 1: Supplementary Table S7 and S8).
Extensive exploration of subsequence length \(l_{s}\)
ProtVec method learns the representation of sequence by breaking the sequence into subsequences (i.e. biological works) of length \(l_{s}\). The length \(l_{s}\) is a hyperparameter that plays a key role in learning representation of the protein sequences and thus significantly impacts the performance of proposed model that uses features learned by ProtVec as input.
Thus, to determine the optimal subsequence length \(l_{s}\), the impact of the subsequence length on the performance of the proposed model was systematically evaluated. This is because the length of subsequence plays an important role to learn the representation using ProtVec (see "Methods"). For this experiment, two different settings were considered for subsequence lengths of 3, 4, 5, or 6 amino acids. First, multi-label classification was performed to compare various subsequence lengths (Fig. 3a). Second, binary classification was performed for individual binding modes (i.e., DB, EB, GB, OB and PB, as well as nucleosome stability) across subsequence lengths (Fig. 3b, Additional file 2: Supplementary Figure S1-S3). For both experiments, micro-AUPR was used as a comparison metric since it indicates how the model performs overall for all binding preferences and is not sensitive to the predictive performance for individual binding preferences.
Performance comparison of the model trained on full-length sequences of TFs. a Micro-AUPR and accuracy comparison between models trained on all binding patterns with different subsequence lengths. b Micro-AUPR and accuracy comparison between models trained with subsequences with the length of 4 residues for individual binding patterns
For the multi-label classification task, the model trained with the subsequence length \(l_{s} = 4\) achieved superior performance compared to other subsequence lengths. In particular, the model trained using \(l_{s} = 4\) achieved 0.61 on micro-AUPR and 0.224 on accuracy. The model trained with \(l_{s} = 3\) achieved similar performance i.e. 0.613 on micro-AUPR and 0.176 on accuracy. Furthermore, we observed that using longer subsequence length (5 or 6) substantially decreases the performance of the model (Fig. 3a).
For the binary classification task, among the five binding modes and nucleosome stability, we observed that the classifier performed best for the EB mode compared to other nucleosomes-binding modes (Fig. 3b, Additional file 2: Supplementary Figure S1–S3). The performance of the binary classifiers generally follows a decreasing trend with increasing subsequence lengths from 3 to 6, in which the classifier with subsequence length 4 outperforms other subsequence lengths (Fig. 3b, Additional file 2: Supplementary Figure S1–S3). Thus, we selected the subsequence length \(l_{s} = 4\) for the following experiments.
Optimization of the hyper-parameter \(\alpha\)
To optimize the hyper-parameter \(\alpha\) in Eq. 6, the impact of different \(\alpha\) values on the performance of our model was measured by two metrics, micro-AUPR (Fig. 4a) and accuracy (Fig. 4b). An appropriate value of \(\alpha\) that controls the relative importance of the mean and covariance matrix is crucial for our model. For this experiment, the model was trained with subsequence length \(l_{s} = 4\). The model achieved the best micro-AUPR when the value of \(\alpha\) is 0.3. Furthermore, when the similarity between covariance matrices was removed (with \(\alpha\) = 1), our method considers only the similarity between the mean vectors of sequences and the performance dropped significantly (Fig. 4a, b). These results support the idea of representing the protein sequences as Gaussian distributions instead of representing them as mean vectors of subsequence representation.
Impact of \(\alpha\) on the performance of proposed methods measured by various performance metrics such as a micro-AUPR, b accuracy. \(\alpha \in \left[ {0, 1} \right]\) controls the impact of similarity between mean vectors and covariance matrices. If \(\alpha = 0\), only the similarity between covariance matrices is considered. In contrast, the only similarity between mean vectors is considered when \(\alpha = 1\)
Comparison between ProtGauss and off-the-shelf methods on binding preferences of TFs
To evaluate the performance of our model on all nucleosome-binding preferences of a given TF, we compared it with four off-the-shelf ML methods that are used as baselines (see "Methods"). Because the off-the-shelf ML methods are often not tuned for our prediction tasks, to get a fair comparison with our model, we fine-tuned the baselines using a set of parameter values (Table 2). For this experiment, we used subsequence length \(l_{s} = 4\) together with \(\alpha\) = 0.3 for our method, and compared with other methods trained with optimized parameters. We found that our ProtGauss model outperformed the baselines in two metrics, micro-AUPR, and accuracy (Table 3). Specifically, our model achieved 0.61 on micro-AUPR and 0.224 on accuracy. By contrast, an optimized SVM model rendered 0.544 on micro-AUPR and 0.155 on accuracy. Compared with the SVM model, the ProtGauss model achieved improvements of 12.1% on micro-AUPR, and 44.5% on accuracy. The results showed that the proposed similarity-based kernel achieves significant improvement over the SVM classifier that does not consider the variance of the features. Overall, we demonstrated the effectiveness of our proposed method and the benefit of using multivariate Gaussian distributions to represent sequences for measuring similarity between a pair of sequences.
Table 3 Performance comparison of our proposed method (in bold) with other baselines for all binding mode data
Binary classifier for the EB mode outperforms those for other binding modes
To identify which individual binding mode the ProtGauss model performs the best, we built a binary classifier for each binding mode and nucleosome stability. The performance of the classifiers trained on the full-length sequences of TFs was evaluated with the subsequence length \(l_{s} = 4\) together with \(\alpha\) = 0.3. It has been shown that the highest micro-AUPR was achieved for the EB mode (Fig. 3b), indicating that the ProtGauss classifier for the EB mode prediction outperforms the classifiers for other binding modes.
To check if the ProtGauss classifier for the EB mode is better than other fine-tuned ML models, we applied these models to the 24 TFs with E-MI data (Additional file 1: Supplementary Table S7). We found that our model achieved the highest micro-AUPR (0.75), accuracy (0.776) and MCC (0.352), compared to other models (Table 4). This result indicates that, in addition to the multi-label classification problem (Table 3), our model outperforms other models in the binary classification problem (Table 4).
Table 4 Performance comparison of the proposed method (in bold) and various baselines for end binding mode data
End preference of TFs is related to low nucleosome occupancy around TF binding sites
For a given TF, its genome-wide binding sites are measured by ChIP-seq assays, whereas nucleosome locations across the genome can be determined by MNase-seq assays. Nucleosome occupancy reflects the fraction of cells from a population in which a given region of DNA is occupied by a histone octamer [1]. TF binding sites at nucleosomal DNA ends become more accessible due to a process known as "breathing" [27,28,29], in which DNA is detached from histones. Thus, if a TF preferentially binds to the ends of a nucleosome, its binding sites are likely to have a relatively lower nucleosome occupancy, compared to a TF that binds to the central region of a nucleosome.
To link the nucleosome EB mode of TFs measured in vitro with nucleosome occupancy around TF binding sites measured in vivo, we assessed 106 TFs in GM12878 cells, which have both ChIP-seq and MNase-seq data (Additional file 1: Supplementary Table S7 and S8). For each TF, the average nucleosome occupancy was calculated for the genomic regions around the centre of ChIP fragments. Three types of nucleosome occupancy profiles were identified: (1) dip at centre (Fig. 5a); (2) peak at centre (Fig. 5b); and (3) questionable, in which no clear dip or peak is shown (Fig. 5c). Out of the 106 TFs, 86, 15 and 5 TFs has the 'dip-at-centre', 'peak-at-centre' and 'questionable' profiles, respectively (Additional file 1: Supplementary Table S7 and S8).
Profiles of nucleosome occupancy around a RFX5 ChIP clusters showing dip at centre (dip), b NFATC1 ChIP clusters showing peak at centre (peak), and c FOXK2 ChIP clusters showing no clear peak or dip at centre (questionable)
Detailed analysis of the 24 (out of 106) TFs with both ChIP-seq data and E-MI penetration (lig147) values (Additional file 1: Supplementary Table S7) revealed a clear trend (Fig. 6). That is, TFs with the EB mode, which have EMI penetration values < 20 as defined in [7], tend to have the 'dip-at-centre' nucleosome occupancy profile. By contrast, TFs without the end preference (with E-MI penetration values > 20) tend to have the 'peak-at-centre' nucleosome occupancy profile. The higher E-MI penetration value a TF has, the more likely it has the 'peak-at-centre' profile. Our data establish a clear correlation relationship between nucleosome occupancy profiles around TF binding sites and the EB mode of TFs. The nucleosome occupancy of TFs can be used as an alternative dataset to test our ProtGauss model as well as other baselines.
Categorization of nucleosome occupancy profiles for 24 TFs with known E-MI penetration (lig147), in terms of peak (red), dip (green), and questionable (orange) nucleosome occupancy profiles. TFs with an E-MI penetration (lig147) less than 20 are defined as having end preference (7)
Use of nucleosome occupancy data as an alternative test dataset
To further illustrate our ProtGauss model outperforms fine-tuned baselines, we used the 86 TFs with the 'dip-at-centre' profile and 15 TFs with the 'peak-at-centre' profiles as an alternative test set. The 5 TFs with the 'questionable' profile are not included. Note that the 101 (= 86 + 15) TFs are not in the training set. The performance on this test set indicates the ability of various models to be generalized to new sequences that were not the part of the training procedure. We found that our model is superior to other models in the alternative test set (Table 5). We also provided a summary of prediction results on the nucleosome occupancy data as a confusion matrix (Table 6). Specifically, our method identified more true positives and true negatives combined (i.e., 85 + 3 = 88) and less false positives and false negatives combined (i.e., 1 + 12 = 13), compared to other methods. The second best model is KNN (Table 5). It identifies all TFs to have end preference; it has 86 true positives and true negatives combined (i.e., 86 + 0 = 86) and 15 false positives and false negatives combined (i.e., 15 + 0 = 15). Furthermore, random forest achieves similar performance to kNN but identifies 81 true positives and true negatives combined (i.e., 78 + 3 = 81) and 20 false positives and false negatives combined (i.e. 8 + 12 = 20). These results indicate that our ProtGauss model not only correctly predicts end binding modes of TFs but also reduces the number of incorrect predictions.
Table 5 Performance comparison of proposed method (in bold) and other baselines for nucleosome occupancy data
Table 6 Comparison of confusion matrix for proposed method (in bold) and other baselines on nucleosome occupancy data
Predicted nucleosome-binding modes in TFs of eukaryotes
To gain insights into the EB modes of eukaryotic TFs, we applied the ProtGauss model to thousands of TFs in five model species including yeast, nematode, fruit fly, mouse and human (Table 7). Examination of the fraction of TFs with predicted end preference reveals an interesting trend in eukaryotes. That is, the fraction of TFs with predicted EB mode achieves the highest in yeast (98.98%) and becomes lower in higher eukaryotes such as nematode (95.59%), fruit fly (96.01%), mouse (86.43%) and human (88.34%). This result indicates that compared to yeast, higher eukaryotes like mammals contain more TFs that do not have end binding preference and potentially target the central region of nucleosomes.
Table 7 Summary of EB mode prediction for transcription factors (TF) in different species
Human TFs not having end preference are enriched in the SOX and HOX families
To characterize the human TFs with no EB preferences, we first focused on the 13 pioneer TFs [3] that are capable of binding nucleosomal DNA (Additional file 1: Supplementary Table S9). It was found that 9 out of the 13 TFs are predicted to have end preference including OCT4/POU5F1 and p53. This result is consistent with previous studies that these two proteins interact with the end of nucleosomal DNA [30, 31].
A detailed analysis of the 1,664 human TFs showed that the TFs from a protein family are likely to share the same DNA-binding domain (Additional file 1: Supplementary Table S10). Grouping the TFs based on their DNA-binding domains rendered 74 unique domain families, in which 10 domain families have at least 20 TFs (Additional file 1: Supplementary Table S11). Analysis of these 10 domain families revealed that the HMG domain (52 TFs) and the homeobox domain (198 TFs) stand out, with 11 and 67 TFs not having the EB mode, respectively (Additional file 1: Supplementary Table S12).
Further analysis showed that the SOX family, one of HMG-containing TF families, contains a large fraction of TFs (10 out of 19, 53%) not having the end preference, including SOX2 and SOX11 (Additional file 1: Supplementary Table S12). This result is consistent with the cryo-electron microscopy (cryo-EM) structure of SOX2 or SOX11 in complex with a nucleosome, in which SOX2 and SOX11 interact with nucleosomal DNA at the superhelical location 2, which is close to the center (dyad) of the nucleosome [9]. On the other hand, TFs without the EB mode are also enriched in the HOX family, one of the homeobox-containing TF families, with 23 out of 39 (59%) TFs having no end preference (Additional file 1: Supplementary Table S13). These data are in accordance with recent work showing that the HOX family proteins have strong binding selectivity to less accessible chromatin regions [32]. Note that most members of two well-established pioneer TF families, FOX and GATA, are predicted to have the EB mode, with only 17 (out of 49) and 0 (out of 10) TFs not having the EB mode, respectively (Additional file 1: Supplementary Table S13), indicating that TFs from these two families tend to interact with the end of nucleosomal DNA.
In this paper, we develop a novel sequence-based machine learning model, ProtGauss, to predict nucleosome binding modes of TFs identified in previous studies [7]. Our model splits a protein sequence into overlapping \(l_{s}\)-length subsequences, and the embeddings of these subsequences are learned with the ProtVec model [8] to obtain a feature matrix. With extensive exploration of subsequence lengths, we found that the length of 4 amino acids is optimal for predicting nucleosome binding modes of TFs. We also tuned the hyper-parameter α to achieve a high performance. With the optimal subsequence length and α, our model outperformed four off-the-shelf machine learning methods.
For comparison with the four machine learning algorithms, we used two different metrics that capture the performance of the models from different perspectives. Accuracy is not appropriate for this task because of the imbalanced experimental data (Table 1). Also, accuracy is a strict metric that requires all the binding preferences to be predicted correctly to classify TFs as correctly classified. Thus, other metrics such as micro-averaged AUPR score are better alternatives for imbalanced data and multi-label classification. Moreover, micro-AUPR is used to compare the performance of different configurations of our models because micro-AUPR is not sensitive to the performance of the model on individual classes. Furthermore, in case of binary classification, we used MCC to compare models for predicting end binding preference. Since the number of positive samples are relatively larger than negative samples, micro-AUPR may be biased and may lead to overestimated performance for binary classification.
With the Gaussian representation of the sequences and the multi-label binding preferences, we adopted nested cross-validation to train and evaluate our proposed approach. We note that our model is robust with the random splitting of the dataset into folds over multiple runs. Furthermore, nested cross-validation allows us to tune the hyper-parameters using inner cross-validation and evaluate the performance of optimized models using outer cross-validation to alleviate the problem of overfitting.
We built binary classifiers based on full length sequences of TFs to identify which binding mode our method works best, and found that the classifier for the EB mode is superior to other classifiers. Testing this classifier with 24 TFs in the test set and 101 TFs in the alternative test set based on nucleosome occupancy data showed that the ProtGauss model outperformed 4 baselines (Tables 4, 5). We further applied the classifier for end preference to thousands of TFs in five model organisms. Based on the results, we proposed a model for the EB mode of TFs (Fig. 7). That is, more than 88% of all TFs are likely to bind to the ends of a nucleosome or free DNA. This model is consistent with a well-known observation that most TFs are located in genomic regions with low nucleosome occupancies [33]. Interestingly, the fraction of TFs with end preference is decreased in higher eukaryotes, compared to yeast. Furthermore, we found that most known pioneer factors have the EB mode, and the TFs having no end preference are enriched in the SOX and HOX protein families.
A model for the end binding mode of TFs to a nucleosome. Over 88% of TFs in five model species are predicted to bind nucleosomal DNA ends or free DNA, whereas less than 12% of the TFs are predicted not to have the end binding preference. The fraction of the TFs binding to nucleosomal DNA ends or free DNA is the highest in yeast (~ 99%) and the lowest in mammals (~ 88%)
These observations have important implications. First, the number of TFs predicted not having the EB mode is increased in higher eukaryotes suggesting that these TFs may play an important role in differentiation and development. Second, the vast majority of known pioneer TFs (9 out of 13) have the EB mode, suggesting that the end preference is one of the main features of pioneer TFs. Third, many SOX and HOX family proteins have no end preference and can potentially target the central region of a nucleosome [7], suggesting that these TFs possess specific structural motifs that allow them to recognize cognate binding sites located inside of a nucleosome. It remains to be determined if they represent a new class of TFs that function differently from known pioneer TFs.
It is intriguing that some TFs (e.g., SOX14) are classified as having end-binding preferences and other TFs in the same protein family (e.g., SOX2 and SOX12) are predicted to bind internal sequence of a nucleosome. In our view, TFs in the same protein family may work in a coordinated manner. The TFs that are able to bind the central region of a nucleosome may act as a pioneer TFs to open chromatin. After that, this TF may be replaced by other family members that can only bind nucleosome ends or free DNA to initiate a developmental process. One example supporting this interpretation comes from the GATA family. GATA1/2/3 factors are required for the differentiation of mesoderm-derived tissues, including the hematopoietic system [34]. In particular, GATA2 is uniquely induced by BMP4 signaling during the establishment of hematopoietic stem/progenitor cells (HSPC) [35,36,37] and is suppressed during the differentiation of HSPC to proerythroblasts (ProE). This suppression is mediated by the displacement of GATA2 from its upstream enhancer by GATA1, a process referred to as the 'GATA switch' [38].
ProGauss is a powerful machine learning method that learns features from protein sequences by mapping short subsequence representations as Gaussian distributions. The similarities between these distributions are used to define the kernel for training a SVM classifier. This method has been successfully applied to predict nucleosome binding modes of TFs, outperforming four other machine learning approaches. A binary classifier for the EB mode of TFs was applied to TFs in five model species, and it was found that about 88% of human TFs have this mode. Human TFs not having this mode are enriched in SOX and HOX TF families. Understanding whether these TFs have pioneering activities will shed new light on mechanisms underlying chromatin opening and developmental competence.
Availability and requirements
Project name: ProtGauss
Project home page: https://github.com/kckishan/ProtGauss
Operating system: Ubuntu 18.04.5 LTS (Bionic Beaver)
Programming languages: Python3.6.9
Other requirements: argparse 1.1, gensim 3.8.1, numpy 1.16.1, pandas 1.0.0, pickle 4.0, prettytable 0.7.2, pyfasta 0.5.2, scikit-learn 0.23.1, texttable 1.6.2, tqdm 4.40.2
License: GNU General Public License
ProtGauss is available in the GitHub repository (https://github.com/kckishan/ProtGauss).
Minor-AUPR:
Area under precision-recall
TF:
Transcription factor
EB:
End binding
MCC:
Matthews correlation coefficient
SVM:
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The authors are grateful for the support from Research Computing at the Rochester Institute of Technology for providing computational resources and support that have contributed to the research results reported in this publication.
National Institutes of Health grant (grant number GM116102) to FC and National Science Foundation grant (grant number NSF 1850492) to RL. Funding for open access charge: NIH/GM116102.
K.C. Kishan and SrideviK. Subramanya contribute equally to the paper
Thomas H. Gosnell School of Life Sciences, Rochester Institute of Technology, 1 Lomb Memorial Drive, Rochester, NY, 14623, USA
Sridevi K. Subramanya & Feng Cui
Golisano College of Computing and Information Sciences, Rochester Institute of Technology, 20 Lomb Memorial Drive, Rochester, NY, 14623, USA
K. C. Kishan & Rui Li
K. C. Kishan
Sridevi K. Subramanya
Feng Cui
KK and SKS developed software. KK, SKS, RL and FC analysed the data. KK, SKS, RL and FC drafted and edited the manuscript. All authors read and approved the final manuscript.
Correspondence to Feng Cui.
The authors declare that they have no conflicts of interest.
Additional file 1.
Supplementary Tables S1 to S13.
Supplementary Figures S1 to S3.
Kishan, K.C., Subramanya, S.K., Li, R. et al. Machine learning predicts nucleosome binding modes of transcription factors. BMC Bioinformatics 22, 166 (2021). https://doi.org/10.1186/s12859-021-04093-9
Nucleosome binding modes
Submission enquiries: [email protected]
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CommonCrawl
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Let $a,$ $b,$ $c$ be complex numbers such that
\begin{align*}
ab + 4b &= -16, \\
bc + 4c &= -16, \\
ca + 4a &= -16.
\end{align*}Enter all possible values of $abc,$ separated by commas.
Adding the equations, we get
\[ab + ac + bc + 4(a + b + c) = -48.\]Multiplying the equations by $c,$ $a,$ $b,$ respectively, we get
\begin{align*}
abc + 4bc &= -16c, \\
abc + 4ac &= -16a, \\
abc + 4ab &= -16b.
\end{align*}Adding all these equations, we get
\[3abc + 4(ab + ac + bc) = -16(a + b + c).\]Then
\begin{align*}
3abc &= -4(ab + ac + bc) - 16(a + b +c) \\
&= -4(ab + ac + bc + 4(a + b + c)) \\
&= (-4)(-48) = 192,
\end{align*}so $abc = \boxed{64}.$
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Math Dataset
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\begin{document}
\title[Proof of a conjecture of Dahmen and Beukers]{Proof of a conjecture of Dahmen and Beukers\\ on counting integral Lam\'{e} equations\\ with finite monodromy} \author{Zhijie Chen} \address{Department of Mathematical Sciences, Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China} \email{[email protected]} \author{Ting-Jung Kuo} \address{Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan } \email{[email protected]} \author{Chang-Shou Lin} \address{Center for Advanced Study in Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan } \email{[email protected]}
\begin{abstract} In this paper, we prove Dahmen and Beukers' conjecture that the number of integral Lam\'{e} equations with index $n$ modulo scalar equivalence with the monodromy group dihedral $D_{N}$ of order $2N$ is given by \[L_{n}(N)=\frac{1}{2}\left( \frac{n(n+1)\Psi(N)}{24}-\left( a_{n} \phi(N)+b_{n}\phi(\tfrac{N}{2}) \right) \right) +\frac{2} {3}\varepsilon_{n}(N).\] Our main tool is the new pre-modular form $Z_{r,s}^{(n)}(\tau)$ of weight $n(n+1)/2$ introduced by Lin and Wang \cite{LW2} and the associated modular form $M_{n,N}(\tau):=\prod_{(r,s)}Z_{r,s}^{(n)}(\tau)$ of weight $\Psi(N)n(n+1)/{2}$, where the product runs over all $N$-torsion points $(r,s)$ of exact order $N$. We show that this conjecture is equivalent to the precise formula of the vanishing order of $M_{n,N}(\tau)$ at infinity: \[v_{\infty}(M_{n,N}(\tau))=a_{n}\phi(N)+b_{n}\phi( N/2).\] This formula is extremely hard to prove because the explicit expression of $Z_{r,s}^{(n)}(\tau)$ is not known for general $n$. Here we succeed to prove it by using certain Painlev\'{e} VI equations. Our result also indicates that this conjecture is intimately connected with the problem of counting pole numbers of algebraic solutions of certain Painlev\'{e} VI equations. The main results of this paper has been announced in \cite{Lin-CDM}.
\end{abstract} \maketitle \tableofcontents
\section{Introduction}
The main purpose of this paper is to prove a conjecture about counting the number of the equivalent class of integral Lam\'{e} equations with $D_{N}$ (i.e. dihedral of order $2N$) as its monodromy group, which was proposed by S. Dahmen and F. Beukers \cite{Dahmen0} and partially confirmed in \cite{Dahmen0,Dahmen,LW2}. The problem of counting integral Lam\'{e} equations with finite monodromy is a long-standing open problem that was already studied in
\cite{BD,Chi,Lit} before Dahmen-Beukers conjecture \cite{Dahmen0}. We will see that this problem is closely related to finding a precise formula of counting pole numbers of algebraic solutions of certain Painlev\'{e} VI equations (see Remark \ref{remark1-8}), which will be studied in another work. The connection between the Lam\'{e} equation and Painlev\'{e} VI equation was already studied in \cite{LPU} from the view point of moduli spaces, which can not be applied to prove Dahmen-Beukers conjecture. Our idea is different from \cite{LPU}. Our proof of this conjecture is to apply the new pre-modular form $Z_{r,s}^{(n)}(\tau)$ (as a function of $\tau\in\mathbb{H}=\{\tau | \operatorname{Im}\tau>0\}$) introduced in \cite{LW2} to connect the Lam\'{e} equation with Painlev\'{e} VI equation. This $Z_{r,s}^{(n)}(\tau)$ is called \emph{pre-modular} in \cite[Definition 0.1]{LW2} because
$Z_{r,s}^{(n)}(\tau)$ is a \emph{modular form of weight $n(n+1)/2$} with respect to the principal congruence subgroup $\Gamma(N)=\{A\in SL(2,\mathbb{Z})|A\equiv I_{2}\operatorname{mod}N\}$ for any given $N$-torsion point $(r,s)\in \mathcal{Q} (N)$, where $\mathcal{Q}(N)$ is the set of $N$-torsion points of exact order $N$ defined by \begin{equation} \mathcal{Q}(N):=\left \{ \left. \left( \tfrac{k_{1}}{N},\tfrac{k_{2}} {N}\right) \right \vert \gcd(k_{1},k_{2},N)=1,0\leq k_{1},k_{2}<N\right \} .\label{N-torsion-1} \end{equation} Consequently, \begin{equation} M_{n,N}(\tau):=\prod_{(r,s)\in \mathcal{Q}(N)}Z_{r,s}^{(n)}(\tau) \label{modular-S1L} \end{equation} is a \emph{modular form of weight} $\frac{n(n+1)}{2}\Psi(N)$ with respect to $SL(2,\mathbb{Z})$, where \begin{equation}
\Psi(N):=\# \mathcal{Q}(N)=N^{2}\prod_{p|N,\text{ }p\text{ prime}}\left( 1-\tfrac{1}{p^{2}}\right) .\label{Euler-2} \end{equation} This $M_{n,N}(\tau)$ is difficult to study due to the absence of the explicit expressions for general $n\geq 5$. Our main result of this paper is to calculate the vanishing order $v_{\infty}(M_{n,N}(\tau))$ of this modular form $M_{n,N}(\tau)$ at infinity.
\begin{theorem}\label{thm-vanish-order} For any $n\in \mathbb{N}$ \textit{and} $N\in \mathbb{N}_{\geq3}$, there holds \[v_{\infty}(M_{n,N}(\tau))=a_{n}\phi(N)+b_{n}\phi \left( N/2\right),\] where \begin{align}
\phi(N) & :=\# \{k\in \mathbb{Z}|\gcd(k,N)=1,0\leq k<N\} \label{Euler}\\
& =N\prod_{p|N,\text{ }p\text{ prime}}\left( 1-\tfrac{1}{p}\right) \nonumber \end{align} is the Euler function (set $\phi(N)=0$ if $N\not \in \mathbb{N}$), and \begin{equation} a_{2n}=a_{2n+1}=n(n+1)/2,\text{ \ }b_{2n}=b_{2n-1}=n^{2}. \label{iv-23-5} \end{equation} \end{theorem} We will see in \S 1.2 that Theorem \ref{thm-vanish-order} is actually equivalent to the validity of Dahmen-Beukers conjecture. Our proof of Theorem \ref{thm-vanish-order} relies on two key steps: Step 1 is the formula of $PL_{n}(N)$ obtained by Dahmen \cite{Dahmen} (See (\ref{iv-24-1}) below for the definition of $PL_{n}(N)$); Step 2 is the asymptotics of $Z_{r,s}^{(n)}(\tau)$ as $\tau\to \infty$. We note that Step 1 is algebraic and Step 2 is analytic. Therefore, the complete proof of Theorem \ref{thm-vanish-order} has to apply both the algebraic theory and the analytic theory of the integral Lam\'{e} equation.
Throughout the paper, we use the notations $\omega_{0}=0$, $\omega_{1}=1$,
$\omega_{2}=\tau$, $\omega_{3}=1+\tau$ and $\Lambda_{\tau}=\mathbb{Z+Z}\tau$, where $\tau \in \mathbb{H}=\{ \tau|\operatorname{Im}\tau>0\}$. Define $E_{\tau }:=\mathbb{C}/\Lambda_{\tau}$ to be a flat torus and $E_{\tau
}[2]:=\{ \frac{\omega_{k}}{2}|0\leq k\leq3\}+\Lambda_{\tau}$ to be the set consisting of the lattice points and 2-torsion points in $E_{\tau}$. For $z\in \mathbb{C}$ we denote $[z]:=z \ (\operatorname{mod} \Lambda_{\tau}) \in E_{\tau}$. For a point $[z]$ in $E_{\tau}$ we often write $z$ instead of $[z]$ to simplify notations when no confusion arises.
Let $\wp(z)=\wp(z|\tau)$ be the Weierstrass $\wp$-function with periods $\Lambda_{\tau}$, defined by \[
\wp(z|\tau):=\frac{1}{z^{2}}+\sum_{\omega \in \Lambda_{\tau}\backslash \{0\}}\left( \frac{1}{(z-\omega)^{2}}-\frac{1}{\omega^{2}}\right) , \]
and $e_{k}(\tau):=\wp(\frac{\omega_{k}}{2}|\tau)$ for $k\in \{1,2,3\}$. Let
$\zeta(z)=\zeta(z|\tau):=-\int^{z}\wp(\xi|\tau)d\xi$ be the Weierstrass zeta function with two quasi-periods: \begin{equation}
\eta_{1}(\tau)=\zeta(z+1|\tau)-\zeta(z|\tau),\text{ \ }\eta_{2}(\tau
)=\zeta(z+\tau|\tau)-\zeta(z|\tau), \label{quasi} \end{equation}
and $\sigma(z)=\sigma(z|\tau)$ be the Weierstrass sigma function defined by $\frac{\sigma^{\prime}(z)}{\sigma(z)}:=\zeta(z)$. Remark that $\zeta(z)$ is an odd meromorphic function with simple poles at the lattice points $\Lambda_{\tau}$, while $\sigma(z)$ is an odd holomorphic function with simple zeros at the lattice points $\Lambda_{\tau}$. We take \cite{Lang} as our general reference on elliptic functions.
\subsection{Integral Lam\'{e} equation and Dahmen-Beukers conjecture}
Consider the classical Lam\'{e} equation (\cite{Halphen,Ince,Poole,Whittaker-Watson}): \begin{equation}
y^{\prime \prime}(z)=\left[ n(n+1)\wp(z|\tau)+B\right] y(z), \label{iv-21-1} \end{equation} where $n>0$ and $B\in \mathbb{C}$ are called \emph{index} and \emph{accessory parameter} respectively. Equation (\ref{iv-21-1}) was introduced by the French mathematician Gabriel Lam\'{e} in 1837. Since then, the Lam\'{e} equation has led to an abundance of applications in physics, especially in the context of completely integrable models such as generalized Calogero-Moser type systems; see e.g. \cite{AMM,Cal,Its,Krichever,OP,Perelomov} and references therein. In this paper, we only consider the \textit{integral} Lam\'{e} equation, i.e.
$n\in \mathbb{N}$ is a \textit{positive integer}. Then the corresponding Lam\'{e} potential $n(n+1)\wp(z|\tau)$ is also known as the simplest elliptic solutions of stationary KdV hierarchy (see e.g. \cite{GH-Book,Gesztesy-Weikard}).
The Lam\'{e} equation possesses symmetry from $SL(2,\mathbb{Z})$ action. Given any $\gamma=\bigl(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\bigr) \in SL(2,\mathbb{Z})$, we let $\tau^{\prime}=\gamma \cdot \tau:=\frac{a\tau+b}{c\tau +d}$ and make a scalar change of the independent variable $z\mapsto(c\tau+d)z$, then (\ref{iv-21-1}) is transformed to \begin{equation}
y^{\prime \prime}(z)=\left[ n(n+1)\wp(z|\tau^{\prime})+(c\tau+d)^{2}B\right] y(z). \label{iv-22-1} \end{equation} These scalar changes of variable induce a natural \emph{equivalence relation} on the space of all Lam\'{e} equations; we call such two Lam\'{e} equations (\ref{iv-21-1}) and (\ref{iv-22-1}) \emph{scalar equivalent}.
By letting $x=\wp(z|\tau)$, we obtain the \emph{algebraic form} of Lam\'{e} equations \begin{equation} p(x)\frac{d^{2}y}{dx^{2}}+\frac{1}{2}p^{\prime}(x)\frac{dy}{dx}-\left( n(n+1)x+B\right) y=0\;\text{ on }\;\mathbb{CP}^{1}, \label{iv-21-2} \end{equation} where $p(x):=4x^{3}-g_{2}(\tau)x-g_{3}(\tau)$ and $g_{2}(\tau), g_{3}(\tau)$ are the well-known invariants of the elliptic curve $E_{\tau}$, given by \[
\wp'(z|\tau)^{2}=4\prod_{k=1}^{3}(\wp(z|\tau)-e_{k}(\tau))=4\wp(z|\tau)^{3}
-g_{2}(\tau)\wp(z|\tau)-g_{3}(\tau). \] Equivalently, two Lam\'{e} equations of the algebraic form (\ref{iv-21-2}) are \emph{scalar equivalent }if one can be transformed into the other by changing
variable $x\mapsto ax$ for some $a\in \mathbb{C}\backslash \{0\}$.
Classically, people are interested in (\ref{iv-21-2}) with \emph{algebraic solutions} only, or equivalently with a \emph{finite monodromy group}; see \cite{Bal,BD,Beukers-Waall,Chi,Lit,Lit1,Maier,vdW} and references therein. It is known that for an integral Lam\'{e} equation, if the monodromy group is finite, then the monodromy group of (\ref{iv-21-2}) is a dihedral group $D_{N}$ for some $N\in \mathbb{N}_{\geq3}$; see e. g. \cite[Theorem 1.5]{Bal} or \cite[Corollary 3.4]{Beukers-Waall}. \emph{Let $L_{n}(N)$ denote the number of Lam\'{e} equations (\ref{iv-21-2}) modulo scalar equivalence with the monodromy group dihedral $D_{N}$}. It is well known (cf. \cite{Chi,Beukers-Waall,Lit,vdW} ) that $L_{n}(N)$ is \emph{finite} for given $n\in \mathbb{N}$ and $N\in \mathbb{N}_{\geq3}$. The issue is how to compute it (cf. \cite{BD,Beukers-Waall,Chi,Dahmen0,Lit,vdW}). In 2003 Dahmen and Beukers proposed the following conjecture (see \cite[Conjecture 73]{Dahmen0}):
\noindent \textbf{Dahmen-Beukers conjecture.} \textit{For every} $n\in \mathbb{N}$ \textit{and} $N\in \mathbb{N}_{\geq3}$, \begin{equation} L_{n}(N)=\frac{1}{2}\left( \frac{n(n+1)\Psi(N)}{24}-\left( a_{n} \phi(N)+b_{n}\phi(\tfrac{N}{2}) \right) \right) +\frac{2} {3}\varepsilon_{n}(N), \label{Dahmen-conjecture-1} \end{equation} \textit{where $(a_n, b_n)$ is given in (\ref{iv-23-5}), $\Psi(N)$ is in (\ref{Euler-2}), $\phi(N)$ is in (\ref{Euler}) and} \begin{equation} \varepsilon_{n}(N)=\left \{ \begin{array} [c]{l} 1\text{ \ \textit{if} }N=3\text{ \textit{and} }n\equiv1\operatorname{mod}3,\\ 0\text{ \ \textit{otherwise}.} \end{array} \right. \label{iv-23-1} \end{equation}
Denote $PL_{n}(N)$ to be the number of Lam\'{e} equations (\ref{iv-21-2}) modulo scalar equivalence with the \emph{projective} monodromy group dihedral $D_{N}$. It is not difficult to see that when the monodromy group $M\simeq D_{N}$, then the projective monodromy group $PM\simeq D_{N/2}$ if $N$ is even and $PM\simeq D_{N}$ if $N$ is odd. Thus (cf. \cite[(4.1)]{Dahmen}) \begin{equation} PL_{n}(N)=\left \{ \begin{array} [c]{l} L_{n}(N)+L_{n}(2N)\text{ \ if }N\text{ is odd,}\\ L_{n}(2N)\text{ \ if }N\text{ is even.} \end{array} \right. \label{iv-24-1} \end{equation} On the other hand, the well-known Klein theorem asserts that any second order Fuchsian differential equation on $\mathbb{CP}^{1}$ has finite projective monodromy group if and only if it is a pull-back of a hypergeometric equation belonging to the basic Schwarz list. Thus every Lam\'{e} equation (\ref{iv-21-2}) with \emph{projective} monodromy group dihedral $D_{N}$ is such a pull-back by \emph{Belyi functions }(see \cite{Chi,Dahmen,Lit,Lit1}). By means of \emph{Grothendieck correpondence} between \emph{Belyi pairs} and \emph{dessin d'enfants}, Litcanu \cite{Lit} first showed how $PL_{n}(N)$ can be counted by using the combinatorics of dessins. Later by applying this algebraic approach, Dahmen \cite{Dahmen} proved
\noindent{\bf Theorem A.} \cite{Dahmen} \begin{equation} PL_{n}(N)=\left \{ \begin{array} [c]{l} 0\text{ \ \ \ \ \ \textit{if} }N\in \{1,2\} \text{,}\\ \frac{n(n+1)}{12}\left( \Psi(N)-3\phi(N)\right) +\frac{2}{3}\varepsilon _{n}(N)\text{ \textit{otherwise},} \end{array} \right. \label{iv-25-1} \end{equation} \emph{where $\varepsilon_{n}(N)$ is defined in (\ref{iv-23-1}), $\Psi(N)$ is in (\ref{Euler-2}) and $\phi(N)$ is in (\ref{Euler}).}
\emph{Consequently, the formula (\ref{Dahmen-conjecture-1}) holds for all $n\in \mathbb{N}$ and $4|N$.}
Theorem A, which confirms the conjecture for the case $n\in \mathbb{N}$ and $4|N$, will play an important role in our proof of Theorem \ref{thm-vanish-order}. Besides, the conjecture for the case $n\in \{1,2,3,4\}$ and $N\in \mathbb{N}_{\geq3}$ was proved in \cite{Dahmen, LW2}. See \S \ref{Dahmen's} for more details on these results. As far as we know, the general case $n\geq5$ and $4\nmid N$ still remains open.
\subsection{Dahmen-Beukers conjecture and Pre-modular forms}
The aforementioned pre-modular form $Z_{r,s}^{(n)}(\tau)$ is holomorphic in $\tau$ for any real pair $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$.
For $n\in \{1,2,3\}$, $Z_{r,s}^{(n)}(\tau)$ was constructed in \cite{Dahmen0} by using the classical Frobenius-Stickelberger formula (e.g. \cite[p.458]{Whittaker-Watson}). However, this idea can not work for $n\geq4$. Based on the earlier works \cite{CLW,LW}, Wang and the third author \cite{LW2} succeeded to prove the existence of such pre-modular forms. We will briefly review this theory in \S \ref{Dahmen's}.1. Furthermore, we will see in Theorem \ref{simple-zero con} that $Z_{r,s}^{(n)}(\tau)$ has at most \emph{simple} zeros. This simple zero property plays a fundamental role in the connection of Theorem \ref{thm-vanish-order} and Dahmen-Buekers conjecture. Indeed, this connection has already been pointed out by Dahmen \cite[Lemma 65]{Dahmen0}.
As mentioned before, $Z_{r,s}^{(n)}(\tau)$ is a modular form of weight $n(n+1)/2$ with respect to $\Gamma(N)$ for $(r,s)\in \mathcal{Q}(N)$, and $M_{n,N}(\tau)$ defined in (\ref{modular-S1L}) is a modular form of weight $\frac{n(n+1)}{2}\Psi(N)$ with respect to $SL(2,\mathbb{Z})$. Since $Z_{r,s}^{(n)}(\tau)$ satisfies the hypothesis of \cite[Lemma 65]{Dahmen0} which relates Theorem \ref{thm-vanish-order} with Dahmen-Beukers conjecture, then \cite[Lemma 65]{Dahmen0} says that the simple zero property of $Z_{r,s}^{(n)}(\tau)$ implies \[L_n(N)=\frac12\left( \frac{n(n+1)\Psi(N)}{24}-v_{\infty}(M_{n,N}(\tau)) \right) +\frac{2} {3}\varepsilon_{n}(N). \] We will recall the precise statement of \cite[Lemma 65]{Dahmen0} in Lemma \ref{lem-C} below. This identity clearly shows that Theorem \ref{thm-vanish-order} is equivalent to the validity of Dahmen-Beukers conjecture, i.e.
\begin{theorem}[=Theorem \ref{thm-vanish-order}]\label{thm-Dahmen-Conjecture} The formula (\ref{Dahmen-conjecture-1}) holds for all $n\in \mathbb{N}$ and $N\in \mathbb{N}_{\geq3}$. \end{theorem}
In order to prove Theorem \ref{thm-vanish-order}, we need to study the asymptotics of $Z_{r,s} ^{(n)}(\tau)$ as $\tau \rightarrow \infty$. For our purpose, it suffices to consider $\tau \in F_{2}$, where $F_{2}$ is a fundamental domain of $\Gamma(2)$ defined by \[
F_{2}:=\{ \tau \in \mathbb{H}\text{ }|\text{ }0\leq \operatorname{Re}
\tau<2,\text{ }|\tau-1/2|\geq1/2,\text{ }|\tau-3/2|>1/2\}. \] Denote $q=e^{2\pi i\tau}$. Clearly $q\rightarrow0$ as $F_{2}\ni \tau \rightarrow \infty$. Then we have the following result for $Z_{r,s}^{(n)}(\tau)$.
\begin{theorem} \label{weak} Given any $n\geq1$ and $(r,s)\in \mathbb{C}^{2}\backslash \frac {1}{2}\mathbb{Z}^{2}$. Then as $F_{2}\ni \tau \rightarrow \infty$ the following hold. \begin{itemize} \item[(1)] $\lim_{F_{2}\ni \tau \rightarrow \infty}Z_{r,s}^{(n)}(\tau)$ exists and is not zero as long as $\operatorname{Re}s\in (0,1/2) \cup(1/2,1)$. \item[(2)] There exist $\tilde{a}_n, \tilde{b}_n\in \mathbb{Z}_{\geq 0}$ satisfying $2\tilde{a}_n+\tilde{b}_n=2a_n+b_n$ such that \[
Z_{r,0}^{(n)}(\tau)=\alpha_{0}^{(n)}(r)q^{\tilde{a}_{n}}+O(|q|^{\tilde{a}_{n}+1}), \] \[ Z_{r,\frac{1}{2}}^{(n)}(\tau)=\beta_{0}^{(n)}(r)q^{\frac{\tilde{b}_{n}}{2}
}+O(|q|^{\frac{\tilde{b}_{n}+1}{2}}), \] where both $\alpha_{0}^{(n)}(r)$ and $\beta_{0}^{(n)}(r)$ are holomorphic in $\mathbb{C}\backslash \mathbb{Z}$ and have no rational zeros in $(0,1/2)\cup (1/2,1)$. Here $(a_{n},b_{n})$ is given by (\ref{iv-23-5}). \item[(3)] For $(r,s)=(\frac14,0)$,
\[Z_{\frac14,0}^{(n)}(\tau)=c_n q^{a_n}+O(|q|^{a_n+1}),\quad c_n\neq 0.\] In particular, $\tilde{a}_n=a_n$ and so $\tilde{b}_n=b_n$. \end{itemize}
\end{theorem}
It is easy to see that Theorem \ref{thm-vanish-order} is a consequence of Theorem \ref{weak}. The proof of Theorem \ref{weak} is not trivial at all due to the absence of explicit expressions of $Z_{r,s}^{(n)}(\tau)$ for general $n\geq 5$. Here we state Theorem \ref{weak} (2) and (3) separately to emphasize that the proof of Theorem \ref{weak} (2) can not yield $(\tilde{a}_n,\tilde{b}_n)=(a_n, b_n)$, and we have to compute $Z_{\frac14,0}^{(n)}(\tau)$ as stated in (3) to prove $(\tilde{a}_n,\tilde{b}_n)=(a_n, b_n)$. We will apply Painlev\'{e} VI equation to prove Theorem \ref{weak} (1) \& (3), while Theorem A plays a key role in the proof of Theorem \ref{weak} (2). As an application, we recall the classical modular function $j(\tau)$ of $SL(2,\mathbb{Z} )$: \[ j(\tau):=1728\frac{g_{2}(\tau)^{3}}{g_{2}(\tau)^{3}-27g_{3}(\tau)^{2} }=1728\frac{g_{2}(\tau)^{3}}{\Delta(\tau)}. \]
\begin{corollary}\label{degree} Assume either $N>3$ or $n\not \equiv 1\operatorname{mod}3$. Then \[ M_{n,N}(\tau)=C_{n,N}\Delta(\tau)^{k(n,N)}\ell_{n,N}(j(\tau))^{2}, \] where $k(n,N)=\frac{n(n+1)}{24}\Psi(N)$, $\ell_{n,N}$ is a monic polynomial and $C_{n,N}$ is a non-zero constant. Furthermore, $\deg \ell_{n,N}=L_{n}(N)$, where $L_{n}(N)$ is precisely the number in Dahmen-Beukers conjecture. \end{corollary}
In general, the polynomial $\ell_{n,N}$ is very difficult to compute even for small $n$ and $N$. In our joint work with Wang \cite{CKLW}, we studied the case $n=1$ and computed $\ell_{1,N}$ for $N\leq9$ by applying Painlev\'{e} VI equation. In general, we conjectured that $\ell_{1,N}(j)$ \emph{is irreducible in} $\mathbb{Q}[j]$. Once this conjecture can be proved, all the zeros of $\ell_{1,N}(j)$ should not be algebraic integers provided $N\geq5$, which implies that all the corresponding $\tau$'s are transcendental.
\subsection{Painlev\'{e} VI equation} Since it is impossible to write down the explicit expression of $Z_{r,s}^{(n)}(\tau)$ for general $n$, Theorem \ref{weak} can not be proved by direct computations. In this paper, we overcome this difficulty by establishing a precise connection between $Z_{r,s}^{(n)}(\tau)$ and Painlev\'{e} VI equation.
The well-known Painlev\'{e} VI equation with free parameters $(\alpha,\beta,\gamma,\delta)$ (denoted it by PVI$(\alpha,\beta,\gamma,\delta)$) reads as \begin{align} \frac{d^{2}\lambda}{dt^{2}}= & \frac{1}{2}\left( \frac{1}{\lambda}+\frac {1}{\lambda-1}+\frac{1}{\lambda-t}\right) \left( \frac{d\lambda}{dt}\right) ^{2}-\left( \frac{1}{t}+\frac{1}{t-1}+\frac{1}{\lambda-t}\right) \frac{d\lambda}{dt}\nonumber \\ & +\frac{\lambda(\lambda-1)(\lambda-t)}{t^{2}(t-1)^{2}}\left[ \alpha +\beta \frac{t}{\lambda^{2}}+\gamma \frac{t-1}{(\lambda-1)^{2}}+\delta \frac{t(t-1)}{(\lambda-t)^{2}}\right] . \label{46-0} \end{align} Due to its connection with many different disciplines in mathematics and physics, PVI (\ref{46-0}) has been extensively studied in the past several decades. See \cite{Boalch,Dubrovin-Mazzocco,Guzzetti,Guzzetti1,Hit1,Hit2,GP,Lisovyy-Tykhyy,Y.Manin,Okamoto2,Okamoto1} and references therein.
One of the fundamental properties for PVI (\ref{46-0}) is the so-called \emph{Painlev\'{e} property} which says that any solution $\lambda(t)$ of (\ref{46-0}) has neither movable branch points nor movable essential singularities; in other words, for any $t_{0}\in \mathbb{C}\backslash \{0,1\}$, either $\lambda(t)$ is holomorphic at $t_{0}$ or $\lambda(t)$ has a pole at $t_{0} $. Moreover, $\lambda(t)$ has at most simple poles if $\alpha \not =0$ (see Theorem \ref{thm-2A} in \S \ref{expression-completely}).
By the Painlev\'{e} property, it is reasonable to lift PVI (\ref{46-0}) to the covering space $\mathbb{H=}\{ \tau|\operatorname{Im}\tau>0\}$ of $\mathbb{C}\backslash \{0,1\}$ by the following transformation: \begin{equation} t=\frac{e_{3}(\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau)},\text{ \ }
\lambda(t)=\frac{\wp(p(\tau)|\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau)}. \label{II-130} \end{equation} Then it is well known (cf. \cite{Babich-Bordag,Y.Manin,Painleve}) that $p(\tau)$ satisfies the elliptic form of Painlev\'{e} VI equation (denoted it by EPVI$(\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})$): \begin{equation} \frac{d^{2}p(\tau)}{d\tau^{2}}=\frac{-1}{4\pi^{2}}\sum_{k=0}^{3}\alpha_{k} \wp^{\prime}\left( \left. p(\tau)+\tfrac{\omega_{k}}{2}\right \vert \tau \right) , \label{124-0} \end{equation} with parameters given by \[ \left( \alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3}\right) =\left( \alpha,-\beta,\gamma,\tfrac{1}{2}-\delta \right) . \]
The Painlev\'{e} property implies that $\wp(p(\tau)|\tau)$ is
single-valued meromorphic in $\mathbb{H}$. This is an advantage of making the transformation (\ref{II-130}).
In this paper, we only consider the elliptic form (\ref{124-0}) with parameters \begin{equation} (\alpha_{0},\alpha_{1},\alpha_{2},\alpha_{3})=(\tfrac{1}{2}(n+\tfrac{1} {2})^{2},\tfrac{1}{8},\tfrac{1}{8},\tfrac{1}{8}),\text{ }n\in \mathbb{N}\cup \{0\}, \label{parameter-n-1} \end{equation} or equivalently PVI (\ref{46-0}) with parameter \begin{align}\label{parameter-n} (\alpha,\beta,\gamma,\delta)= ( \tfrac{1}{2}(n+\tfrac{1}{2} )^{2},\tfrac{-1}{8},\tfrac{1}{8},\tfrac{3}{8}), \end{align} which is related to the integral Lam\'{e} equation. For any monodromy data $(r,s)\in\mathbb{C}^2\setminus\frac{1}{2}\mathbb{Z}^2$, we proved in \cite{Chen-Kuo-Lin} that there is \emph{a unique solution} $p_{r,s}^{(n)}(\tau)$ of EPVI$(\tfrac{1}{2}(n+\tfrac{1} {2})^{2},\tfrac{1}{8},\tfrac{1}{8},\tfrac{1}{8})$ associated with this $(r,s)$; we will recall this statement in Theorem \ref{thm-II-8}. To study the asymptotics of $Z_{r,s}^{(n)}(\tau)$, we need the following "explicit" expression of
$\wp(p_{r,s}^{(n)}(\tau)|\tau)$.
\begin{theorem} \label{thm-II-18 copy(1)}Fix any $n\geq0$ and $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$. Then $p_{r,s}^{(n)}(\tau)$ is expressed by \[
\wp(p_{r,s}^{(n)}(\tau)|\tau)=\frac{P_{n}(Z_{r,s}(\tau);r+s\tau,\tau)} {Z_{r,s}^{(n-1)}(\tau)Z_{r,s}^{(n+1)}(\tau)}, \] where $Z_{r,s}^{(-1)} (\tau)=Z_{r,s}^{(0)}(\tau):=1$,
\[Z_{r,s}(\tau):=\zeta(r+s\tau|\tau)-r\eta_{1}(\tau)-s\eta_{2}(\tau),\]
and $P_{n}(\cdot;r+s\tau,\tau)$ are polynomials with coefficients being rational functions of $\wp(r+s\tau|\tau)$, $\wp^{\prime
}(r+s\tau|\tau)$ and $e_{k}(\tau)$, $k\in \{1,2,3\}$.
Furthermore, any two of
$P_{n}(Z_{r,s}(\tau);r+s\tau,\tau)$, $Z_{r,s}^{(n-1)}(\tau)$,
$Z_{r,s}^{(n+1)}(\tau)$ have no common zeros in $\tau$ provided $r+s\tau \not \in E_{\tau}[2]$. \end{theorem}
\begin{remark} In the seminal work \cite{Hit1}, Hitchin proved his famous formula for EPVI$(\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8})$: \begin{equation}
\wp(p^{(0)}_{r,s}(\tau)|\tau)=\wp(r+s\tau|\tau)+\frac{\wp^{\prime}(r+s\tau|\tau)} {2Z_{r,s}(\tau)}. \label{0II-1} \end{equation} Therefore, Theorem \ref{thm-II-18 copy(1)} is a generalization of Hitchin's formula to the general case $n\geq 1$. Theorem \ref{thm-II-18 copy(1)} plays a crucial role in our proof of Theorem \ref{weak} (1) \& (3). We believe that Theorem \ref{thm-II-18 copy(1)} has potential applications to some other problems of Painlev\'{e} VI equations. \end{remark}
\begin{remark} \label{remark1-8} We will see in \S \ref{expression-completely} that $p_{r,s}^{(n)}(\tau)$ can be obtained from $p_{r,s}^{(0)}(\tau)$ via the famous Okamoto transformation \cite{Okamoto1}. Since the Okamoto transformation preserves algebraic solutions, it is known (cf. \cite{Mazzocco,CKLW}) that $\wp(p_{r,s}^{(n)}
(\tau)|\tau)$ gives an algebraic solution of PVI$(\tfrac{1}{2}(n+\tfrac{1} {2})^{2},\tfrac{-1}{8},\tfrac{1}{8},\tfrac{3}{8})$ if and only if
$(r,s)\in \mathcal{Q}(N)$ for some $N\ge3$. Moreover, after analytic continuation, $\wp(p_{r,s}^{(n)}(\tau)|\tau)=\wp(p_{1-r,1-s}^{(n)}
(\tau)|\tau)$ gives the same algebraic solution for all $(r,s)\in \mathcal{Q}(N)$ (i.e. this algebraic solution has $\Psi(N)/2$ branches) if $N$
is odd; $\wp(p_{r,s}^{(n)}(\tau)|\tau)$ gives three different algebraic solutions for all $(r,s)\in \mathcal{Q}(N)$ (i.e. each algebraic solution has $\Psi(N)/6$ branches) if $N$ is even. See also \cite[Lemma 3.3] {Chen-Kuo-Lin2016}. Since Theorem \ref{thm-II-18 copy(1)} shows that zeros of $Z_{r,s}^{(n-1)}(\tau)$, $Z_{r,s}^{(n+1)}(\tau)$ give poles of $\wp
(p_{r,s}^{(n)}(\tau)|\tau)$, the zero numbers of $M_{n\pm1,N}(\tau)$ and hence Dahmen-Beukers conjecture, are closely related to counting pole numbers of algebraic solutions of PVI$(\tfrac{1}{2}(n+\tfrac{1}{2})^{2},\tfrac{-1}{8},\tfrac{1} {8},\tfrac{3}{8})$. This precise relation for the case $n=1$ has been discussed in \cite{Chen-Kuo-Lin2016}, where an explicit formula of the pole number of algebraic solutions for PVI$(\tfrac{9}{8},\tfrac{-1}{8},\tfrac{1} {8},\tfrac{3}{8})$ was obtained via $M_{2,N}(\tau)$. We will study the general case $n\ge2$ elsewhere. \end{remark}
\begin{remark} \label{remarkk} The proof of Theorem \ref{thm-II-18 copy(1)} will be given in \S \ref{expression-completely}. Since we have no explicit expressions of $Z_{r,s}^{(n\pm1)}(\tau)$, the key point is how to prove that $Z_{r,s}^{(n\pm1)}(\tau)$ appears in the denominator of
$\wp(p_{r,s}^{(n)}(\tau)|\tau)$. Our basic idea is following. It is well-known that solutions of Painlev\'{e} VI equation govern the isomonodromic deformation of some linear ODEs. We already proved in \cite{Chen-Kuo-Lin} that the monodromy of the associated linear ODE (see GLE (\ref{89-1}) in \S \ref{expression-completely})
corresponding to $p_{r,s}^{(n)}(\tau)$ is generated by (\ref{iiii}) via this $(r,s)$; see Theorem \ref{thm-II-8} in \S \ref{expression-completely}. If $\tau \to \tau_{0}$ with $\tau_{0}$ being any pole of $\wp(p_{r,s}^{(n)}(\tau)|\tau)$, we proved in \cite{Chen-Kuo-Lin0} that GLE (\ref{89-1}) converges to the Lam\'{e} equation $y^{\prime \prime
}=[m(m+1)\wp(z|\tau_{0})+B]y$ with $m\in \{n\pm1\}$ and its monodromy also generated by (\ref{iiii}) via this $(r,s)$, which finally indicates that either $Z_{r,s}^{(n-1)} (\tau_{0})=0$ or $Z_{r,s}^{(n+1)}(\tau_{0})=0$. Hence, the zeros of $Z_{r,s}^{(n-1)}(\tau)Z_{r,s}^{(n+1)}(\tau)$ coincide with the poles of
$\wp(p_{r,s}^{(n)}(\tau)|\tau)$; see Theorem \ref{q-n=z-n} in \S \ref{expression-completely}. \end{remark}
The paper is organized as follows. In \S \ref{Dahmen's}, We give a detailed explanation of the deep connection between $Z_{r,s}^{(n)}(\tau)$ and Dahmen-Beukers conjecture. We also review Damhen's and Chou's proofs of this conjecture for $n\leq 4$. In \S 3 we give a complete proof of Dahmen-Beukers conjecture by applying Theorem \ref{weak}. In \S \ref{expression-completely}, first we apply the Okamoto transformation to generalize Hitchin's formula via an induction approach. The induction process is technical and will be given in Appendix A. Then we establish the precise connection between $Z_{r,s}^{(n)}(\tau)$ and Painlev\'{e} VI equation, and apply it to prove Theorem \ref{thm-II-18 copy(1)} and Theorem \ref{simple-zero con}. Finally in \S \ref{sec-asymptotics}, we apply the theory of Painlev\'{e} VI equation to study the asymptotics of $Z_{r,s}^{(n)}(\tau)$ and prove Theorem \ref{weak}.
\section{Pre-modular form and Dahmen-Beukers conjecture}
\label{Dahmen's}
\subsection{Connection between $Z_{r,s}^{(n)}(\tau)$ and the conjecture} In this section, we give a detailed explanation that \emph{counting integral Lam\'{e} equations with monodromy group dihedral $D_{N}$ is equivalent to counting zeros of $Z_{r,s}^{(n)}(\cdot)$ for $(r,s)\in \mathcal{Q}(N)$}. This is why $Z_{r,s}^{(n)}(\tau)$ can be applied to study Dahmen-Beukers conjecture.
First we briefly review the theory of $Z_{r,s}^{(n)}(\tau)$ from \cite{CLW,LW2}. The Lam\'{e} equation (\ref{iv-21-1}) is associated with a hyperelliptic curve $\bar{Y}_{n}\subset$ Sym$^{n}E_{\tau}:=E_{\tau}^{n}/S_{n}$ (i.e. the $n$-th symmetric product of $E_{\tau}$) defined by \[ Y_{n}=Y_{n}(\tau):=\left \{ \{[a_{i}]\}_{i=1}^{n}\left \vert \begin{array} [c]{c} \sum_{j\not =i}\left( \zeta(a_{i}-a_{j})+\zeta(a_{j})-\zeta(a_{i})\right) =0\\ \lbrack a_{i}]\not =0,[a_{i}]\not =[a_{j}],\text{ }i=1,\cdots,n \end{array} \right. \right \} , \] and $\bar{Y}_{n}$ is the closure of $Y_{n}$ in Sym$^{n}E_{\tau}$. Historically $\bar{Y}_{n}$ is also known as the \emph{Lam\'{e} curve}. It was proved in \cite{CLW,LW2} that the following hold:
(a). $\bar{Y}_{n}(\tau)$ is a hyperelliptic curve with arithmetic genus $n$
and $\bar{Y}_{n}(\tau)=Y_{n}(\tau)\cup \{ \{0,\cdots,0\} \}$. The affine part $Y_{n}(\tau)\simeq \{(B,C)|C^{2} =\ell_{n}(B)\}$, where $\ell_{n}(B)$ is the so-called Lam\'{e} polynomial of degree $2n+1$, and the branch points of $Y_{n}(\tau)$ are precisely those $\{[a_{i}]\}_{i=1}^{n}\in Y_{n}(\tau)$ such that $\{[a_{1}],\cdot \cdot \cdot,[a_{n}]\}=\{-[a_{1} ],\cdot \cdot \cdot,-[a_{n}]\}$.
(b). Let $K(\bar{Y}_{n}(\tau))$ be the field of rational functions on $\bar {Y}_{n}(\tau)$. Set $\sigma_{n}$ to be the addition map from $\bar{Y}_{n} (\tau)$ onto $E_{\tau}$: \[ \sigma_{n}(\boldsymbol{a}):=\sum_{i=1}^{n}[a_{i}],\text{ \ }\forall \boldsymbol{a}=\{[a_{1}],\cdot \cdot \cdot \lbrack a_{n}]\} \in \bar{Y}_{n} (\tau). \] Then the degree of $\sigma_{n}$ is $\frac{n(n+1)}{2}$. Consequently, $K(\bar{Y}_{n}(\tau))$ is a finite extension of $K(E_{\tau})$ and \[ \left[ K(\bar{Y}_{n}(\tau)):K(E_{\tau})\right] =\tfrac{n(n+1)}{2}. \]
(c). Define $\boldsymbol{z}_{n}: \bar{Y}_{n}(\tau)\to\mathbb{C}\cup \{ \infty \}$ by: \begin{equation}\label{z--n} \boldsymbol{z}_{n}(\boldsymbol{a}):=\zeta \left( \sum \nolimits_{i=1}^{n} a_{i}\right) -\sum \nolimits_{i=1}^{n}\zeta(a_{i}). \end{equation} Then $\boldsymbol{z}_{n}\in K(\bar{Y}_{n}(\tau))$ is a primitive generator of the finite field extension of $K(\bar{Y}_{n}(\tau))$ over $K(E_{\tau})$, and there is a minimal polynomial \begin{equation} W_{n}(X)=W_{n}(X;\sigma_{n},\tau)\in \mathbb{Q}[g_{2}(\tau),g_{3}(\tau
),\wp(\sigma_{n}|\tau),\wp^{\prime}(\sigma_{n}|\tau)][X] \label{k-ll} \end{equation} of the field extension $K(\bar{Y}_{n})$ over $K(E_{\tau})$ which defines the covering map $\sigma_{n}$ between algebraic curves. Define \begin{equation}
Z_{r,s}(\tau):=\zeta(r+s\tau|\tau)-r\eta_{1}(\tau)-s\eta_{2}(\tau). \label{Hecke} \end{equation} Then $W_{n}(X)$ is characterized by the following properties:
\begin{theorem} \cite[Theorem 3.2]{LW2} \label{thm-5A}
\begin{itemize} \item[(1)] $W_{n}(X;\sigma_{n},\tau)$ is a monic polynomial of $X$-degree $\frac{1}{2}n(n+1)$ such that \[ W_{n}(\boldsymbol{z}_{n}(\boldsymbol{a});\sigma_{n}(\boldsymbol{a}),\tau)=0. \] Moreover, $W_{n}(X;\sigma_{n},\tau)$ is weighted homogeneous with $X$, $\wp(\sigma_{n})$, $\wp^{\prime}(\sigma_{n})$, $g_{2}$, $g_{3}$ being of weight $1$, $2$, $3$, $4$, $6$ respectively.
\item[(2)] Fix any $\tau$. For each $a\in E_{\tau }\backslash E_{\tau}[2]$ being outside the branch loci of $\sigma _{n}:\bar{Y}_{n}\rightarrow E_{\tau}$, $W_{n}(X;a,\tau)$ has $\frac{1}{2}n(n+1)$ distinct zeros. Furthermore, for any zero $X_{0}$ of $W_{n}(X;a,\tau)$, there exists $\boldsymbol{a}\in \bar{Y}_{n}$ such that $a=\sigma _{n}(\boldsymbol{a})$ and $X_{0}=\boldsymbol{z}_{n}(\boldsymbol{a})$.
\item[(3)] For any given $(r,s)$, the function $Z_{r,s} ^{(n)}(\tau):=W_{n}(Z_{r,s}(\tau);r+s\tau,\tau)$ is a pre-modular form of weight $\frac{1}{2}n(n+1)$ with $Z_{r,s}(\tau)$, $\wp(r+s\tau)$, $\wp^{\prime}(r+s\tau)$, $g_{2}$, $g_{3}$ being of weight $1$, $2$, $3$, $4$, $6$ respectively. \end{itemize} \end{theorem}
Since for any $(m_{1},m_{2})\in \mathbb{Z}^{2}$, $\wp(r+s\tau|\tau)=\wp
(m_{1}-r+(m_{2}-s)\tau|\tau)$, $\wp^{\prime}(r+s\tau|\tau)=-\wp^{\prime}
(m_{1}-r+(m_{2}-s)\tau|\tau)$ and $Z_{r,s}(\tau)=-Z_{m_{1}-r,m_{2}-s}(\tau)$, we see from Theorem \ref{thm-5A} that \begin{equation} Z_{r,s}^{(n)}(\tau)=(-1)^{\frac{n(n+1)}{2}}Z_{m_{1}-r,m_{2}-s}^{(n)}(\tau). \label{iv-20} \end{equation}
Let $\boldsymbol{a}\in Y_{n}(\tau)$ and suppose that $\boldsymbol{a}$ satisfies \begin{equation} \sigma_{n}(\boldsymbol{a})=r+s\tau \text{ \ and }\sum_{i=1}^{n}\zeta (a_{i})=r\eta_{1}(\tau)+s\eta_{2}(\tau) \label{v-10} \end{equation} for some $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$. Then \[ Z_{r,s}(\tau)=\zeta \left( \sum \nolimits_{i=1}^{n}a_{i}\right) -\sum \nolimits_{i=1}^{n}\zeta(a_{i})=\boldsymbol{z}_{n}(\boldsymbol{a}). \] Together with Theorem \ref{thm-5A} we obtain $Z_{r,s}^{(n)}(\tau)=0$. Thus, the zero of $Z_{r,s}^{(n)}(\tau)$ encodes the identity (\ref{v-10}). We summarize the properties of $Z_{r,s}^{(n)}(\tau)$ in the following result.
\begin{theorem} \cite[Theorem 4.3]{LW2} \label{thm-A} \begin{itemize} \item[(1)] For $\bigl(\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\bigr) \in SL(2,\mathbb{Z})$, there holds \[Z_{ar-bs,ds-cr}^{(n)}(\tfrac{a\tau+b}{c\tau+d})=(c\tau+d)^{n(n+1)/2}Z_{r,s}^{(n)}(\tau).\] In particular, for $(r,s)\in\mathcal{Q}(N)$, $Z_{r,s}^{(n)}(\tau)$ is a modular form of weight $n(n+1)/2$ with respect to $\Gamma(N)$. \item[(2)] For any pair $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, $Z_{r,s} ^{(n)}(\tau)=0$ for some $\tau \in \mathbb{H}$ if and only if there is a unique $\boldsymbol{a}=\{[a_{1}],\cdot \cdot \cdot,[a_{n}]\} \in Y_{n} (\tau)$ such that (\ref{v-10}) holds. \end{itemize} \end{theorem}
For any $\{a_{1},\cdots,a_{n}\} \subset \mathbb{C}\setminus\{0\}$, we consider the classical \emph{Hermite-Halphen} ansatz \[ y_{\{a_{i}\}}(z):=e^{z\sum_{i=1}^{n}\zeta(a_{i})}\prod_{i=1}^{n}\frac {\sigma(z-a_{i})}{\sigma(z)}. \] By (\ref{quasi}) and the transformation law of $\sigma(z)$: \begin{equation} \sigma(z+\omega_{k})=-e^{\eta_{k}(z+\frac{\omega_{k}}{2})}\sigma(z),\text{ \ }k=1,2, \label{82-2} \end{equation} we see that $y_{\{a_{i}\}}(z)$ and $y_{\{b_{i}\}}(z)$ are \emph{linearly dependent} if $\{[a_{1}]$, $\cdots$, $[a_{n}]\}=\{[b_{1}]$, $\cdots$, $[b_{n}]\}$, so up to a non-zero constant $y_{\{a_{i}\}}(z)$ is uniquely determined by $\{[a_{1}]$, $\cdots$, $[a_{n}]\}$. We recall the following classical result.
\begin{proposition} [cf. \cite{CLW,Halphen}] \label{prop-B} $y_{\{a_{i}\}}(z)$ is a solution to the integral Lam\'{e} equation (\ref{iv-21-1}) for some $B\in \mathbb{C}$ if and only if $\{[a_{1}]$, $\cdots$, $[a_{n}]\} \in Y_{n}$ and \begin{equation} B=B_{\boldsymbol{a}}:=(2n-1)\sum_{i=1}^{n}\wp(a_{i}). \label{B-expression} \end{equation} Moreover, $\{[a_{1}]$, $\cdots$, $[a_{n}]\} \in Y_{n}$ is not a branch point (equivalently, $\ell_{n}(B)\not =0$) if and only if $y_{\{a_{i}\}}(z)$ and $y_{\{-a_{i}\}}(z)$ are linearly independent.\end{proposition}
By (\ref{82-2}) we have \[ y_{\{a_{i}\}}(z+1)=e^{-2\pi is}y_{\{a_{i}\}}(z)\;\text{ and }\;y_{\{a_{i} \}}(z+\tau)=e^{2\pi ir}y_{\{a_{i}\}}(z), \] where $(r,s)\in \mathbb{C}^{2}$ is uniquely determined by (note $\tau \eta_{1}-\eta_{2}=2\pi i$) \[ r+s\tau=\sum_{i=1}^{n}a_{i}\text{ \ and }r\eta_{1}+s\eta_{2}=\sum_{i=1} ^{n}\zeta(a_{i}). \] Clearly $y_{\{-a_{i}\}}(z)$ is also a solution of (\ref{iv-21-1}) if $y_{\{a_{i}\}}(z)$ is. Since the monodromy representation of (\ref{iv-21-1}) is a group homomorphism $\rho:\pi _{1}(E_{\tau})\rightarrow SL(2,\mathbb{C})$, the monodromy group of (\ref{iv-21-1}) with respect to $(y_{\{a_{i}\}}(z),y_{\{-a_{i} \}}(z))$ is generated by \begin{equation} \rho(\ell_{1})= \begin{pmatrix} e^{-2\pi is} & 0\\ 0 & e^{2\pi is} \end{pmatrix}\; \text{ and }\;\rho(\ell_{2})= \begin{pmatrix} e^{2\pi ir} & 0\\ 0 & e^{-2\pi ir} \end{pmatrix} \label{iiii} \end{equation} provided $\ell_{n}(B)\not =0$, where $\ell_{n}(B)$ is the Lam\'{e} polynomial recalled in (a), and $\ell_{i}$ denote the two fundamental cycles $z\to z+\omega_i$ of $E_{\tau}$.
Therefore, Theorem \ref{thm-A} says that $Z_{r,s}^{(n)}(\tau)=0$ for some $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$ if and only if there is a non-branch point $\boldsymbol{a}\in Y_{n}(\tau)$ such that the monodromy of the Lam\'{e} equation (\ref{iv-21-1}) with $B=B_{\boldsymbol{a}}$
is generated by (\ref{iiii}) via this $(r,s)$. In particular, if $(r,s)=(\frac{k_{1}}{N},\frac{k_{2}}{N} )\in \mathcal{Q}(N)$ is a $N$-torsion point of exact order $N$ and $\tau$ is any zero of $Z_{r,s}^{(n)}(\cdot)$, then the monodromy group of the Lam\'{e} equation (\ref{iv-21-1}) with some $B$ is \emph{finite} with order $N$, or equivalently the corresponding algebraic form (\ref{iv-21-2}) has \emph{dihedral $D_{N}$} as its monodromy group; see \cite{Dahmen0} for a proof. In conclusion, {\it counting integral Lam\'{e} equations (modulo scalar equivalence) with monodromy group dihedral $D_{N}$ is equivalent to counting zeros of the modular form $M_{n,N}(\cdot)=\prod_{(r,s)\in \mathcal{Q}(N)}Z_{r,s}^{(n)}(\cdot)$ (modulo $SL(2,\mathbb{Z})$ action)}.
\subsection{The previous partial answer \cite{Dahmen0,Dahmen,LW2}}
In 2007 Dahmen \cite{Dahmen} studied this conjecture by considering the \emph{projective} monodromy group of integral Lam\'{e} equations. Recall \S 1.1 that the number of integral Lam\'{e} equations modulo scalar equivalence with (resp. projective) monodromy group dihedral $D_{N}$ is denoted by $L_{n}(N)$ (resp. $PL_{n}(N)$). Then (cf. \cite[(4.1)]{Dahmen}) \begin{equation} PL_{n}(N)=\left \{ \begin{array} [c]{l} L_{n}(N)+L_{n}(2N)\text{ \ if }N\text{ is odd,}\\ L_{n}(2N)\text{ \ if }N\text{ is even.} \end{array} \right. \label{iv-24} \end{equation} As recalled in Theorem A, Dahmen obtained the explicit formula \begin{equation} PL_{n}(N)=\left \{ \begin{array} [c]{l} 0\text{ \ \ if }N\in \{1,2\} \text{,}\\ \frac{n(n+1)}{12}\left( \Psi(N)-3\phi(N)\right) +\frac{2}{3}\varepsilon _{n}(N)\text{ \ otherwise,} \end{array} \right. \label{iv-25} \end{equation} where $\phi(N)$, $\Psi(N)$ and $\varepsilon_{n}(N)$ are defined in (\ref{Euler}), (\ref{Euler-2}) and (\ref{iv-23-1}). A immediate consequence of (\ref{iv-24})-(\ref{iv-25}) is that formula (\ref{Dahmen-conjecture-1}), i.e. Dahmen-Beukers conjecture, holds for any $n\in \mathbb{N}$ and
$N\in \mathbb{N}_{\geq3}$ satisfying $4|N$.
On the other hand, the following result is the key lemma in \cite{Dahmen0} to prove the conjecture for $n\in \{1,2,3\}$ and $N\in\mathbb{N}_{\geq 3}$.
\begin{lemma} (\cite[Lemma 65]{Dahmen0}) \label{lem-C} Let $n\geq 1$ and $N\geq 3$. Suppose for any $N$-torsion point $(r,s)\in\mathcal{Q}(N)$, there exists a modular form $f_{r,s}^{(n)}(\tau)$ of weight $n(n+1)/2$ with respect to $\Gamma(N)$ such that Theorem \ref{thm-A} (1)-(2) holds for $f_{r,s}^{(n)}(\tau)$ and $(r,s)\in\mathcal{Q}(N)$. Consider the modular form \begin{equation} M_{n,N}(\tau):=\prod_{(r,s)\in \mathcal{Q}(N)}f_{r,s}^{(n)}(\tau) \label{modular-SL} \end{equation} and define \begin{equation} U_{n}(N):=\frac{1}{2}\left( \frac{n(n+1)\Psi(N)}{24}-v_{\infty}\left( M_{n,N}(\tau)\right) \right) +\frac{2}{3}\varepsilon_{n}(N). \label{U-n-value} \end{equation} Then $L_{n}(N)\leq U_{n}(N)$. Moreover, $L_{n}(N)=U_{n}(N)$ if and only if $f_{r,s}^{(n)}(\tau)$ has only simple zeros in $\mathbb{H}$ for all $(r,s)\in \mathcal{Q}({N})$.\end{lemma}
Theorem \ref{thm-A} shows that the new pre-modular form $Z_{r,s}^{(n)}(\tau)$ introduced by Wang and the third author \cite{LW2} satisfies all the assumptions of Lemma \ref{lem-C}. Therefore, the conclusion of Lemma \ref{lem-C} holds for all $n\geq1$ by letting $f_{r,s}^{(n)}(\tau)=Z_{r,s}^{(n)}(\tau)$. Indeed, by applying Lemma \ref{lem-C} to $Z_{r,s}^{(n)}(\tau)$, the conjecture was proved in \cite{Dahmen0,Dahmen,LW2} for $n\in\{1,2,3,4\}$. To explain how to apply Lemma \ref{lem-C}, we reproduce their proofs below.
For $n\in \{1,2,3,4\}$, the explicit expressions of $Z_{r,s}^{(n)}(\tau)$ are known as shown in \cite{Dahmen0,LW2}. Denote
$Z=Z_{r,s}(\tau)$, $\wp=\wp(r+s\tau|\tau)$ and $\wp^{\prime
}=\wp^{\prime}(r+s\tau|\tau)$ for convenience. Then \begin{equation} Z_{r,s}^{(1)}(\tau)=Z_{r,s}(\tau), \quad Z_{r,s}^{(2)}(\tau)=Z^{3}-3\wp Z-\wp^{\prime}, \label{z-n-1} \end{equation} \begin{align}\label{z-n-4} Z_{r,s}^{(3)}(\tau)= & Z^{6}-15\wp Z^{4}-20\wp^{\prime}Z^{3}+\left( \tfrac{27}{4}g_{2}-45\wp^{2}\right) Z^{2}\\ & -12\wp \wp^{\prime}Z-\tfrac{5}{4}(\wp^{\prime})^{2},\nonumber \end{align} {\allowdisplaybreaks \begin{align*} Z_{r,s}^{(4)}(\tau)= & Z^{10}-45\wp Z^{8}-120\wp^{\prime}Z^{7}+(\tfrac {399}{4}g_{2}-630\wp^{2})Z^{6}-504\wp \wp^{\prime}Z^{5}\\ & -\tfrac{15}{4}(280\wp^{3}-49g_{2}\wp-115g_{3})Z^{4}+15(11g_{2}-24\wp ^{2})\wp^{\prime}Z^{3}\\ & -\tfrac{9}{4}(140\wp^{4}-245g_{2}\wp^{2}+190g_{3}\wp+21g_{2}^{2} )Z^{2}\\ & -(40\wp^{3}-163g_{2}\wp+125g_{3})\wp^{\prime}Z+\tfrac{3}{4}(25g_{2} -3\wp^{2})(\wp^{\prime})^{2}. \end{align*} } Hence the vanishing order $v_{\infty}(M_{n,N}(\tau))$ at infinity for $n\in\{1,2,3,4\}$ can be calculated explicitly: \begin{equation} v_{\infty}(M_{n,N}(\tau))=a_{n}\phi(N)+b_{n}\phi \left( {N}/{2}\right), \label{1-18} \end{equation} where $a_{n}$ and $b_{n}$ are given by (\ref{iv-23-5}). Remark that the computation of (\ref{1-18}) is not simple because the expression of $Z_{r,s}^{(n)}(\tau)$ for $n=3,4$ is already very complicated. We refer \cite{Dahmen0,LW2} for the actual computations of (\ref{1-18}).
Now by applying Lemma \ref{lem-C} and (\ref{1-18}), the following result was proved in \cite{Dahmen0,Dahmen,LW2}.
\begin{theorem}\cite{Dahmen0,Dahmen,LW2} Dahmen-Beukers conjecture holds for $n\in \{1,2,3,4\}$. \end{theorem}
\begin{proof} For $n\leq 4$, Lemma \ref{lem-C} and (\ref{1-18}) yield \begin{align*} L_{n}(N) & \leq U_{n}(N)\\ & =\frac{1}{2}\left( \frac{n(n+1)\Psi(N)}{24}-\left( a_{n}\phi(N)+b_{n} \phi (\tfrac{N}{2}) \right) \right) +\frac{2}{3}\varepsilon _{n}(N). \end{align*} This, together with (\ref{iv-24})-(\ref{iv-25}), easily implies that for odd $N$, \[ PL_{n}(N)=L_{n}(N)+L_{n}(2N)\leq U_{n}(N)+U_{n}(2N)=PL_{n}(N), \]
so $L_{n}(N)=U_{n}(N)$ and $L_{n}(2N)=U_{n}(2N)$ if $N$ is odd. If $4|N$ then $L_{n}(N)=U_{n}(N)$ follows directly from (\ref{iv-24})-(\ref{iv-25}). This proves Dahmen-Beukers conjecture for $n\in \{1,2,3,4\}$.\end{proof}
A consequence of this proof and Lemma \ref{lem-C} is that for $n\in \{1,2,3,4\}$, $Z_{r,s}^{(n)}(\tau)$ has at most simple zeros in $\mathbb{H}$ if $(r,s)\in \mathcal{Q}(N)$.
\section{Proof of Dahmen-Beukers conjecture}
In this section, we acknowledge the validity of Theorem \ref{weak} (1) \& (3) and apply it to give proofs of Theorem \ref{weak} (2), Theorem \ref{thm-vanish-order} and Dahmen-Beukers conjecture.
The first step of our proof is \begin{theorem} [=Theorem \ref{simple-zn}]\label{simple-zero con}For any $n\geq1$ and $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, $Z_{r,s} ^{(n)}(\tau)$ has at most simple zeros in $\mathbb{H}$. \end{theorem}
Theorem \ref{simple-zero con} for $n=1$ was proved in our previous joint work with Wang \cite{CKLW}. For any $n\geq 1$, Theorem \ref{simple-zero con} is actually a consequence of Theorem \ref{thm-II-18 copy(1)}, which will be proved in \S \ref{expression-completely}. Thanks to Theorem \ref{simple-zero con}, we can apply Lemma \ref{lem-C} to obtain
\begin{corollary} \label{coro-con}Let $n\in \mathbb{N}$ \textit{and} $N\in \mathbb{N}_{\geq3}$. Then $L_{n}(N)=U_{n}(N)$, where $U_n(N)$ is defined in (\ref{U-n-value}). \end{corollary}
Corollary \ref{coro-con} shows that Dahmen-Beukers conjecture (i.e. Theorem \ref{thm-Dahmen-Conjecture}) is equivalent to Theorem \ref{thm-vanish-order} as we discussed in \S 1. For the rest of this section, we will give a complete proof of Theorem \ref{thm-vanish-order} by assuming Theorem \ref{weak} (1) \& (3).
\begin{remark} \label{remark-Dahmen}As mentioned before, in both Dahmen's and Chou's proof
for $n\in \{1,2,3,4\}$, since the explicit expression of $Z_{r,s}^{(n)}(\tau)$ is known, they could prove the identity (\ref{1-18}) first and then obtained the simple zero property of $Z_{r,s}^{(n)}(\tau)$ for a \textbf{rational pair} $(r,s)$ as a byproduct. However, this approach can not work for general $n\geq5$ because it is impossible to write down the explicit expression of $Z_{r,s}^{(n)}(\tau)$ for general $n$. In this paper, we exploit an \textbf{opposite idea}, namely we prove the simple zero property of $Z_{r,s}^{(n)}(\tau)$ (i.e. Theorem \ref{simple-zero con}) first, and then apply it to prove the identity (\ref{1-18})! \end{remark}
First we have the following result, which proves Theorem \ref{weak}-(2).
\begin{theorem} \label{modular-zero}Fix any $n\in \mathbb{N}$ and recall $a_n, b_n$ in (\ref{iv-23-5}). Then there exist $\tilde{a} _{n},\tilde{b}_{n}\in \mathbb{N\cup \{}0\mathbb{\}}$ independent of $N$ and satisfying \begin{equation} 2\tilde{a}_{n}+\tilde{b}_{n}=2a_{n}+b_{n}, \label{iv-39-1} \end{equation} such that the following hold. \begin{itemize} \item[(1)] \begin{equation} Z_{r,0}^{(n)}(\tau)=\alpha_{0}^{(n)}(r)q^{\tilde{a}_{n}}+\sum_{k=1}^{\infty }\alpha_{k}^{(n)}(r)q^{\tilde{a}_{n}+k},\quad \alpha_{0}^{(n)}(r)\not\equiv 0, \label{z-nn} \end{equation} \begin{equation}\label{z-0nn} Z_{r,\frac{1}{2}}^{(n)}(\tau)=\beta_{0}^{(n)}(r)q^{\frac{\tilde{b}_{n}}{2} }+\sum_{k=1}^{\infty}\beta_{k}^{(n)}(r)q^{\frac{\tilde{b}_{n}+k}{2}},\quad \beta_{0}^{(n)}(r)\not\equiv 0, \end{equation} where both $\alpha_{0}^{(n)}(r)$ and $\beta_{0}^{(n)}(r)$ are holomorphic in $r\in\mathbb{C}\setminus\mathbb{Z}$ and have \emph{no} rational zeros in $(0,1/2)\cup(1/2,1)$. \item[(2)] For any $N\in \mathbb{N}_{\geq3}$, \begin{equation} v_{\infty}( M_{n,N}(\tau))=\tilde{a}_{n}\phi(N)+\tilde{b}_{n} \phi (N/2). \label{iv-39} \end{equation} \end{itemize} \end{theorem}
\begin{proof} {\bf Step 1.} We prove (\ref{z-nn})-(\ref{z-0nn}).
Recall the following $q=e^{2\pi i\tau}$ expansions (see e.g. \cite[Appendix A]{LW2}): \begin{equation} g_{2}(\tau)=\frac{4}{3}\pi^{4}+320\pi^{4}\sum_{n=1}^{\infty}\sigma_{3} (n)q^{n}, \label{iv-34} \end{equation} \[ g_{3}(\tau)=\frac{8}{27}\pi^{6}-\frac{448}{3}\pi^{6}\sum_{n=1}^{\infty} \sigma_{5}(n)q^{n}, \]
where $\sigma_{k}(n):=\sum_{d|n}d^{k}$, and \begin{equation}
\wp^{\prime}(r|\tau)=-2\pi^{3}\cot(\pi r)-2\pi^{3}\cot^{3}(\pi r)+16\pi ^{3}\sum_{n,m=1}^{\infty}n^{2}\sin(2\pi nr)q^{nm}, \label{iv-35-2} \end{equation} \[
\wp(r|\tau)=\pi^{2}\cot^{2}(\pi r)+\frac{2}{3}\pi^{2}+8\pi^{2}\sum _{n,m=1}^{\infty}n(1-\cos(2\pi nr))q^{nm}, \] \begin{equation} Z_{r,0}(\tau)=\pi \cot(\pi r)+4\pi \sum_{n,m=1}^{\infty}\sin(2n\pi r)q^{nm}. \label{iv-35} \end{equation} Note from Theorem \ref{thm-5A} that \[ Z_{r,s}^{(n)}(\tau)\in \mathbb{Q}\left[ g_{2}(\tau),g_{3}(\tau),\wp
(r+s\tau|\tau),\wp^{\prime}(r+s\tau|\tau)\right] [Z_{r,s}(\tau)]\text{.} \] Applying (\ref{iv-34})-(\ref{iv-35}), there exist $\tilde{a}_{n}\in \mathbb{N}\cup \{0\}$ and holomorphic functions $\alpha_{k}^{(n)}(r)$ in $r\in \mathbb{C}\setminus\mathbb{Z}$ which have poles at most at $r\in \mathbb{Z}$ and $\alpha_{0}^{(n)}(r)\not \equiv 0$, such that \begin{equation} Z_{r,0}^{(n)}(\tau)=\alpha_{0}^{(n)}(r)q^{\tilde{a}_{n}}+\sum_{k=1}^{\infty }\alpha_{k}^{(n)}(r)q^{\tilde{a}_{n}+k}. \label{iv-32} \end{equation} Similarly, by applying (see \cite[Appendix A]{LW2}) \[ \wp^{\prime}\left( \left. r+\tfrac{\tau}{2}\right \vert \tau \right) =16\pi^{3}\sum_{n,m=1}^{\infty}n^{2}\sin(2\pi nr)q^{n(m-\frac{1}{2})}, \] \[ \wp \left( \left. r+\tfrac{\tau}{2}\right \vert \tau \right) =-\frac{\pi^{2} }{3}+8\pi^{2}\sum_{n,m=1}^{\infty}q^{nm}-8\pi^{2}\sum_{n,m=1}^{\infty} n\cos(2\pi nr)q^{n(m-\frac{1}{2})}, \] \begin{equation} Z_{r,\frac{1}{2}}(\tau)=4\pi \sum_{n,m=1}^{\infty}\sin(2\pi nr)q^{n(m-\frac {1}{2})}, \label{iv-35-1} \end{equation} there exist $\tilde{b}_{n}\in \mathbb{N}\cup \{0\}$ and holomorphic functions $\beta_{k}^{(n)}(r)$ in $r\in \mathbb{C}$ satisfying $\beta_{0}^{(n)}(r)\not \equiv 0$, such that \begin{equation} Z_{r,\frac{1}{2}}^{(n)}(\tau)=\beta_{0}^{(n)}(r)q^{\frac{\tilde{b}_{n}}{2} }+\sum_{k=1}^{\infty}\beta_{k}^{(n)}(r)q^{\frac{\tilde{b}_{n}+k}{2}}. \label{iv-33} \end{equation}
{\bf Step 2.} We prove $2\tilde{a}_n+\tilde{b}_n=2a_n+b_n$.
By Theorem \ref{weak}-(1) which will be proved by applying Painlev\'{e} VI equation in \S \ref{sec-asymptotics}, we have $v_{\infty}(Z_{r,s}^{(n)}(\tau))=0$ for any $s\in (0,1/2)\cup(1/2,1)$, so we see from (\ref{modular-SL}) that \begin{equation} v_{\infty}\left( M_{n,N}(\tau)\right) =\sum_{(r,0)\in \mathcal{Q} (N)}v_{\infty}(Z_{r,0}^{(n)}(\tau))+\sum_{(r,\frac{1}{2})\in \mathcal{Q} (N)}v_{\infty}(Z_{r,\frac12}^{(n)}(\tau)). \label{iv-30} \end{equation} Furthermore, Theorem A says that (\ref{Dahmen-conjecture-1}) holds for any
$N$ satisfying $4|N$. This together with (\ref{U-n-value}) and Corollary \ref{coro-con} implies \begin{align} v_{\infty}\left( M_{n,N}(\tau)\right) &=a_{n}\phi(N)+b_{n}\phi \left( \tfrac{N}{2}\right)\nonumber\\
&=(2a_n+b_n)\phi(\tfrac{N}{2}), \text{ for any }N\text{ satisfying }4|N, \label{iv-39-3} \end{align}
where we used $\phi(N)=2\phi(N/2)$ for $4|N$ to obtain the second equality.
On the other hand, since zeros of a meromorphic function are isolated, it follows that $\alpha_{0}^{(n)}(r)$ has only finitely many zeros in $[0,1]$. Denote all its rational zeros in $(0,1)$ by \[ O_{1}=\left \{ \left. \frac{l_{1,k}}{m_{1,k}}\in(0,1)\right \vert \gcd (l_{1,k},m_{1,k})=1,k\leq j_{1}\right \} . \] Again denote all rational zeros in $(0,1)$ of $\beta_{0}^{(n)}(r)$ by \[ O_{2}=\left \{ \left. \frac{l_{2,k}}{m_{2,k}}\in(0,1)\right \vert \gcd (l_{2,k},m_{2,k})=1,k\leq j_{2}\right \} . \]
Now for any $N$ such that $4|N$ and $N>\max \{m_{i,k}|i=1,2,k\leq j_{i}\}$, we have $r\not \in O_{1}$, i.e. $\alpha_{0}^{(n)}(r)\not =0$ and so $v_{\infty }(Z_{r,0}^{(n)}(\tau))=\tilde{a}_{n}$ for any $(r,0)=(\frac{k_{1}}{N} ,0)\in \mathcal{Q}(N)$; also $r\not \in O_{2}$, i.e. $\beta_{0}^{(n)} (r)\not =0$ and so $v_{\infty}(Z_{r,\frac{1}{2}}^{(n)}(\tau))=\tilde{b}_{n}/2$ for any $(r,\frac{1}{2})=(\frac{k_{1}}{N},\frac{1}{2})\in \mathcal{Q}(N)$. Therefore, \begin{align} v_{\infty}(M_{n,N}(\tau)) &=\sum_{(r,0)\in \mathcal{Q}(N)} \tilde{a}_{n}+\sum_{(r,\frac{1}{2})\in \mathcal{Q}(N)}\tfrac{\tilde{b}_{n}} {2}=\tilde{a}_{n}\phi(N)+\tilde{b}_{n}\phi \left( \tfrac{N}{2}\right)\nonumber\\ &=(2\tilde{a}_n+\tilde{b}_n)\phi(\tfrac{N}{2}), \label{iv-31} \end{align}
where we used $\phi(N)=2\phi(N/2)$ for $4|N$ again. Clearly (\ref{iv-39-3})-(\ref{iv-31}) imply \begin{equation} 2\tilde{a}_{n}+\tilde{b}_{n}=2a_{n}+b_{n}. \label{iv-36} \end{equation}
{\bf Step 3}. We prove (\ref{iv-39}), and both $\alpha_{0}^{(n)}(r)$ and $\beta_{0}^{(n)}(r)$ have no rational zeros in $(0,1/2)\cup(1/2,1)$.
Note (\ref{iv-39-3}) and (\ref{iv-36}) already prove (\ref{iv-39}) for $4|N$. So it suffices to consider $4\nmid N$. Fix any odd $N\geq3$. Then $\phi(2N)=\phi(N)$ and $\phi(N/2)=0$. It follows from (\ref{iv-32}) and (\ref{iv-33}) that $v_{\infty}(Z_{r,0} ^{(n)}(\tau))\geq \tilde{a}_{n}$ for any $(r,0)\in \mathcal{Q}(N)$ and $v_{\infty}(Z_{r,\frac{1}{2}}^{(n)}(\tau))\geq \tilde{b}_{n}/2$ for any $(r,\frac{1}{2})\in \mathcal{Q}(N)$, i.e. \begin{equation} v_{\infty}\left( M_{n,N}(\tau)\right) \geq \tilde{a}_{n}\phi(N)+\tilde{b} _{n}\phi \left( \tfrac{N}{2}\right) . \label{iv-37} \end{equation} Similarly, \begin{equation} v_{\infty}\left( M_{n,2N}(\tau)\right) \geq \tilde{a}_{n}\phi(2N)+\tilde {b}_{n}\phi(N). \label{iv-38} \end{equation} On the other hand, by denoting the RHS of (\ref{Dahmen-conjecture-1}) by $\overline{L}_{n}(N)$, i.e. \begin{equation} \overline{L}_{n}(N):=\frac{1}{2}\left( \frac{n(n+1)\Psi(N)}{24}-\left( a_{n}\phi(N)+b_{n}\phi(\tfrac{N}{2}) \right) \right) +\frac {2}{3}\varepsilon_{n}(N), \label{iv-26} \end{equation} it is easy to derive from (\ref{iv-24})-(\ref{iv-25}) and (\ref{iv-26}) that \begin{equation*} L_{n}(N)+L_{n}(2N)=\overline{L}_{n}(N)+\overline{L}_{n}(2N)\;\text{ if }\;N\text{ is odd.} \end{equation*} This, together with Corollary \ref{coro-con} which says $L_n(N)=U_{n}(N)$, implies \begin{equation*} U_{n}(N)+U_{n}(2N)=\overline{L}_{n}(N)+\overline{L}_{n}(2N)\;\text{ if }\;N\text{ is odd,} \end{equation*} and so \begin{align*} & v_{\infty}\left( M_{n,N}(\tau)\right) +v_{\infty}\left( M_{n,2N} (\tau)\right) \\ = & a_{n}\phi(N)+b_{n}\phi \left( \tfrac{N}{2}\right) +a_{n}\phi (2N)+b_{n}\phi(N)\\ = & (2a_{n}+b_{n})\phi(N), \;\text{ if }\;N\text{ is odd.} \end{align*} Therefore, we see from (\ref{iv-36}) that both (\ref{iv-37}) and (\ref{iv-38}) must be identities, namely (\ref{iv-39}) holds for all $N\geq3$. Furthermore, \[v_{\infty}(Z_{r,0}^{(n)}(\tau))=\tilde{a}_n,\quad v_{\infty}(Z_{r,\frac12}^{(n)}(\tau))=\tilde{b}_n/2\] for any $r\in (0,1/2)\cup(1/2,1)\cap \mathbb{Q}$, namely both $\alpha_{0}^{(n)}(r)$ and $\beta_{0}^{(n)}(r)$ have no rational zeros in $(0,1/2)\cup(1/2,1)$. This completes the proof. \end{proof}
\begin{corollary} \label{a-n=b-n}Assume Theorem \ref{weak}-(3), then (\ref{z-nn}) implies $\tilde{a}_n=a_n$ and so $\tilde{b}_n=b_n$ by Theorem \ref{modular-zero}. \end{corollary}
Now we are in a position to prove Dahmen-Beukers conjecture.
\begin{proof} [Proof of Theorems \ref{thm-vanish-order} and \ref{thm-Dahmen-Conjecture}] Obviously Theorem \ref{thm-vanish-order} follows directly from Theorem \ref{modular-zero} and Corollary \ref{a-n=b-n}, and Theorem \ref{thm-Dahmen-Conjecture}, i.e. Dahmen-Beukers conjecture, follows from Theorem \ref{thm-vanish-order} and Corollary \ref{coro-con}. \end{proof}
We conclude this section by proving Corollary \ref{degree}.
\begin{proof}[Proof of Corollary \ref{degree}]
Consider the case either $N>3$ or $n\not \equiv 1\operatorname{mod}3$, i.e. $\varepsilon_{n}(N)=0$. Recall that $M_{n,N}(\tau)$ is a modular form of weight $\frac{n(n+1)}{2} \Psi(N)$. Then $k(n,N):=\frac{n(n+1)}{24}\Psi(N)\in \mathbb{N}$ and $\frac{M_{n,N}(\tau )}{\Delta(\tau)^{k(n,N)}}$ is invariant under $SL(2,\mathbb{Z})$. Observe from (\ref{iv-20}) that any zero of $\frac{M_{n,N}(\tau)}{\Delta(\tau)^{k(n,N)}}$ must be doubled. Since $\frac{M_{n,N}(\tau)}{\Delta(\tau)^{k(n,N)}}$ has no poles in $\mathbb{H}$, we conclude that \begin{equation} \frac{M_{n,N}(\tau)}{\Delta(\tau)^{k(n,N)}}=C_{n,N}\ell_{n,N}(j(\tau ))^{2}\label{j-polynomial} \end{equation} for some monic polynomial $\ell_{n,N}$ of $j$ and non-zero constant $C_{n,N}$. Recall the $q$-expansions (cf. \cite[p.193]{Husemoller}): \[ \Delta(\tau)=(2\pi)^{12}q\prod_{n=1}^{+\infty}(1-q^{n})^{24}, \] \[ j(\tau)=\tfrac{1}{q}+744+196884q+21493760q^{2}+\cdots. \] By comparing the leading term of $q$-expansions in (\ref{j-polynomial}), we easily obtain \[ \deg \ell_{n,N}=\tfrac{1}{2}\left( k(n,N)-v_{\infty}\left( M_{n,N} (\tau)\right) \right) =L_{n}(N). \] The proof is complete.\end{proof}
\section{Generalization of Hitchin's formula}
\label{expression-completely}
From now on, we apply Painlev\'{e} VI equation to prove Theorem \ref{weak} (1) \& (3) and Theorem \ref{simple-zero con}. In this section, we generalize Hitchin's formula to the general case $n\in\mathbb{N}$ and prove Theorem \ref{thm-II-18 copy(1)} and Theorem \ref{simple-zero con}.
\subsection{The Hamiltonian system and asymptotics at poles} It is well known (cf. \cite{GP}) that PVI (\ref{46-0}) is equivalent to the following Hamiltonian system \begin{equation} \frac{d\lambda(t)}{dt}=\frac{\partial K}{\partial \mu},\text{ \ }\frac{d\mu (t)}{dt}=-\frac{\partial K}{\partial \lambda},\label{aa} \end{equation} where $K=K(\lambda,\mu,t)$ is given by \begin{equation} K=\frac{1}{t(t-1)}\left \{ \begin{array} [c]{l} \lambda(\lambda-1)(\lambda-t)\mu^{2}+\theta_{0}(\theta_{0}+\theta_{4} )(\lambda-t)\\ -\left[ \begin{array} [c]{l} \theta_{1}(\lambda-1)(\lambda-t)+\theta_{2}\lambda(\lambda-t)\\ +(\theta_{3}-1)\lambda(\lambda-1) \end{array} \right] \mu \end{array} \right \} ,\label{98} \end{equation} and the relation of parameters is given by \begin{equation} \left( \alpha,\beta,\gamma,\delta \right) =\left( \tfrac{1}{2}\theta_{4} ^{2},\,-\tfrac{1}{2}\theta_{1}^{2},\, \tfrac{1}{2}\theta_{2}^{2},\, \tfrac{1} {2}\left( 1-\theta_{3}^{2}\right) \right) ,\label{46-2} \end{equation} \begin{equation} 2\theta_{0}+\theta_{1}+\theta_{2}+\theta_{3}+\theta_{4}=1.\label{46-3} \end{equation} Here we recall the following classical result concerning the asymptotics of $\lambda(t)$ at poles.
\begin{proposition} \cite[Proposition 1.4.1]{GP} \label{thm-2A} Assume $\theta_{4}\not =0$. Then for any $t_{0} \in \mathbb{C}\backslash \{0,1\}$, there exist two $1$ -parameter families of solutions $\lambda(t)$ of PVI (\ref{46-0}) such that \begin{equation} \lambda(t)=\frac{\xi}{t-t_{0}}+h+O(t-t_{0})\text{ as }t\rightarrow t_{0}, \label{II-132} \end{equation} where $h\in \mathbb{C}$ can be taken arbitrary and \begin{equation} \xi=\xi(\theta_{4},t_{0})\in \left \{ \pm \tfrac{t_{0}(t_{0}-1)}{\theta_{4} }\right \} . \label{II-133} \end{equation} Furthermore, these two $1$-parameter families of solutions give all solutions of PVI (\ref{46-0}) which has a pole at $t_{0} $.\end{proposition}
\begin{definition} When $\theta_{4}>0$, we call that a pole $t_{0}\in \mathbb{C}\backslash \{0,1\}$ of $\lambda(t)$ is a positive pole (resp. a negative pole) if the residue is $\frac{t_{0}(t_{0} -1)}{\theta_{4}}$ (resp. $-\frac{t_{0}(t_{0}-1)}{\theta_{4}}$). \end{definition}
Here we have the following simple but interesting observation, which gives a criterion to determine the type of poles. This result is important when we want to connect $Z_{r,s}^{(n)}(\tau)$ with PVI.
\begin{proposition} \label{Prop-II-1}Assume $\theta_{4}>0$. Let $t_{0}\in \mathbb{C}\backslash \{0,1\}$ and $\lambda(t)$ be a solution of PVI (\ref{46-0}) which has a pole at $t_{0}$. Then $\lambda \mu$ is holomorphic at $t_{0}$ and more precisely, \begin{equation} \lim \limits_{t\rightarrow t_{0}}\lambda \mu=\left \{ \begin{array} [c]{l} -\theta_{0}\text{\ \ \ if }t_{0}\text{ is a negative pole,}\\ -(\theta_{0}+\theta_{4})\text{\ \ \ if }t_{0}\text{ is a positive pole,} \end{array} \right. \label{II-136} \end{equation} where $\mu(t)$ is defined by the first equation of the Hamiltonian system (\ref{aa}). \end{proposition}
\begin{proof} By Proposition \ref{thm-2A} we have \begin{equation} \lambda(t)=\frac{\xi}{t-t_{0}}\left( 1+O(t-t_{0})\right) \text{ as }t\rightarrow t_{0}, \label{II-137} \end{equation} \begin{equation} \lambda^{\prime}(t)=-\frac{\xi}{(t-t_{0})^{2}}\left( 1+O(t-t_{0})\right) \text{ as }t\rightarrow t_{0}. \label{II-138} \end{equation} Since the first equation of the Hamiltonian system (\ref{aa}) reads{\allowdisplaybreaks \begin{align} \lambda^{\prime}(t)=\frac{\partial K}{\partial \mu}= & \frac{1} {t(t-1)}\big[2\lambda(\lambda-1)(\lambda-t)\mu-\theta_{1}(\lambda -1)(\lambda-t)\label{II-139}\\ & -\theta_{2}\lambda(\lambda-t)-(\theta_{3}-1)\lambda(\lambda -1)\big],\nonumber \end{align} }we see that $\mu(t)$ is meromorphic in a neighborhood of $t_{0}$ and so does $\lambda \mu$. Denote \begin{equation} \lim \limits_{t\rightarrow t_{0}}\lambda(t)\mu(t)=L\in \mathbb{C}\cup \{ \infty \}. \label{II-140} \end{equation} If $L=\infty$, then by substituting (\ref{II-137})-(\ref{II-138}) into (\ref{II-139}), we easily obtain a contradiction. Thus $L$ is finite, namely $\lambda \mu$ is holomorphic at $t_{0}$. Again by substituting (\ref{II-137} )-(\ref{II-138}) and (\ref{II-140}) into (\ref{II-139}) and comparing the coefficients of the term $(t-t_{0})^{-2}$, we easily obtain \[ 2L=\theta_{1}+\theta_{2}+\theta_{3}-1-\frac{t_{0}(t_{0}-1)}{\xi}. \] This, together with (\ref{II-133}) and (\ref{46-3}), proves (\ref{II-136}). \end{proof}
\subsection{The Okamoto transformations} From now on we consider the elliptic form (\ref{124-0}) with parameters \[ (\alpha_{0},\alpha_1, \alpha_2, \alpha_3) =(\tfrac{1}{2}(n+\tfrac{1}{2})^{2},\tfrac{1}{8},\tfrac{1}{8},\tfrac{1}{8}), \quad n\in\mathbb{Z}_{\geq 0}, \] or equivalently PVI (\ref{46-0}) with parameters \[ (\alpha,\beta,\gamma,\delta)= ( \tfrac{1}{2}(n+\tfrac{1}{2} )^{2},\tfrac{-1}{8},\tfrac{1}{8},\tfrac{3}{8}). \] It is known that solutions of PVI$(\frac{1}{2}(n+\frac12)^2,\frac{-1}{8},\frac{1}{8},\frac {3}{8})$ could be obtained from solutions of PVI$(\frac{1}{8},\frac{-1}{8},\frac{1}{8},\frac {3}{8})$ (i.e. $n=0$) via the Okamoto transformation \cite{Okamoto1}. First we recall the explicit form of the Okamoto transformations. By (\ref{46-2})-(\ref{46-3}), it is convenient to think of the parameter space of PVI (\ref{46-0}) (equivalently the Hamiltonian system (\ref{aa})-(\ref{98})) as an affine space \[ \mathcal{K}=\left \{ \theta=(\theta_{0},\theta_{1},\theta_{2},\theta _{3},\theta_{4})\in \mathbb{C}^{5}\text{ }:\text{ }2\theta_{0}+\theta _{1}+\theta_{2}+\theta_{3}+\theta_{4}=1\right \} . \] An Okamoto transformation $\kappa$ maps solutions $\lambda(t)$ of PVI (\ref{46-0}) (resp. solutions $(\lambda(t),\mu(t))$ of the Hamiltonian system (\ref{aa})) with parameter $\theta$ to solutions $\kappa(\lambda)(t)$ of PVI (\ref{46-0}) (resp. solutions $(\kappa(\lambda)(t),\kappa(\mu)(t))$ of (\ref{aa})) with new parameter $\kappa(\theta)\in \mathcal{K}$. The list of the Okamoto transformations $\kappa_{j}(0\leq j\leq4)$ is given in the Table 1 (cf. \cite{Tsuda-Okamoto-Sakai}).\begin{table}[tbh] \caption{Okamoto transformations} \centering \par \begin{tabular}
[c]{|l|c|c|c|c|c|l|c|c|}\hline & $\theta_{0}$ & $\theta_{1}$ & $\theta_{2}$ & $\theta_{3}$ & $\theta_{4}$ & $t$ & $\lambda$ & $\mu$\\ \hline $\kappa_{0}$ & $-\theta_{0}$ & $\theta_{1}+\theta_{0}$ & $\theta_{2} +\theta_{0}$ & $\theta_{3}+\theta_{0}$ & $\theta_{4}+\theta_{0}$ & $t$ & $\lambda+\frac{\theta_{0}}{\mu}$ & $\mu$\\ \hline $\kappa_{1}$ & $\theta_{0}+\theta_{1}$ & $-\theta_{1}$ & $\theta_{2}$ & $\theta_{3}$ & $\theta_{4}$ & $t$ & $\lambda$ & $\mu-\frac{\theta_{1}} {\lambda}$\\ \hline $\kappa_{2}$ & $\theta_{0}+\theta_{2}$ & $\theta_{1}$ & $-\theta_{2}$ & $\theta_{3}$ & $\theta_{4}$ & $t$ & $\lambda$ & $\mu-\frac{\theta_{2}} {\lambda-1}$\\ \hline $\kappa_{3}$ & $\theta_{0}+\theta_{3}$ & $\theta_{1}$ & $\theta_{2}$ & $-\theta_{3}$ & $\theta_{4}$ & $t$ & $\lambda$ & $\mu-\frac{\theta_{3} }{\lambda-t}$\\ \hline $\kappa_{4}$ & $\theta_{0}+\theta_{4}$ & $\theta_{1}$ & $\theta_{2}$ & $\theta_{3}$ & $-\theta_{4}$ & $t$ & $\lambda$ & $\mu$\\ \hline \end{tabular} \end{table}
These five transformations $\kappa_{j}$ $(0\leq j\leq4)$, which satisfy $\kappa_{j}\circ \kappa_{j}=Id$, generate the affine Weyl group of type $D_{4}^{(1)}$: \[ W(D_{4}^{(1)})=\left \langle \kappa_{0},\kappa_{1},\kappa_{2},\kappa_{3} ,\kappa_{4}\right \rangle . \] Define \begin{equation} \kappa_{5}:=\kappa_{0}(\kappa_{3}\kappa_{2}\kappa_{1}\kappa_{0})^{2}\kappa _{4}, \label{II-120} \end{equation} A straightforward computation shows \begin{equation} \kappa_{5}(\theta)=(\theta_{0}-1,\theta_{1},\theta_{2},\theta_{3},\theta _{4}+2). \label{II-121} \end{equation} Note that for PVI$(\frac{1}{2}(n+\frac12)^2,\frac{-1}{8},\frac{1}{8},\frac {3}{8})$, the corresponding $\theta^n:=\theta\in \mathcal{K}$ is given by \begin{equation} \theta^{n}:=\left( -\tfrac{n+1}{2},\text{ }\tfrac{1}{2},\text{ }\tfrac{1} {2},\text{ }\tfrac{1}{2},\text{ }n+\tfrac{1}{2}\right),\quad\text{i.e. }\; \theta_4=n+\tfrac12>0. \label{I1I-175} \end{equation} Letting $\kappa^{0,1}:=\kappa_{0}\circ \kappa_{3} \circ \kappa_{2}\circ \kappa_{1}$, we have $\kappa^{0,1}(\theta^{0})=\theta^{1}$. Define \[\kappa^{0,n}:=\begin{cases} (\kappa_{5})^m \circ \kappa^{0,1},\quad \text{if $n=2m+1$ odd},\\ (\kappa_{5})^m \quad \text{if $n=2m$ even}, \end{cases}\] then it follows from $\kappa_5(\theta^n)=\theta^{n+2}$ that $\kappa^{0,n}(\theta^0)=\theta^n$ for any $n$. Thus there exist two \emph{rational functions} $R_n (\cdot,\cdot,\cdot)$ and $\tilde{R}_n(\cdot,\cdot,\cdot)$ of three independent variables with coefficients in $\mathbb{Q}$ such that for any solution $(\lambda^{(0)}(t),\mu^{(0)}(t))$ of the Hamiltonian system (\ref{aa}) with parameter $\theta^{0}$, $(\lambda^{(n) }(t),\mu^{(n)}(t))$ given by \begin{equation} \lambda^{(n)}(t):=\kappa(\lambda^{(0)})(t)=R_{n }(\lambda^{(0)}(t),\mu^{(0)}(t),t), \label{II-128} \end{equation} \begin{equation} \mu^{(n)}(t):=\kappa(\mu^{(0)})(t)=\tilde{R}_{n }(\lambda^{(n)}(t),\mu^{(0)}(t),t), \label{II-128-0} \end{equation} is a solution of the Hamiltonian system (\ref{aa}) with parameter $\theta^{n}$, or equivalently, $\lambda^{(n)}(t)$ is a solution of PVI$(\frac{1}{2}(n+\frac12)^2,\frac{-1}{8},\frac{1}{8},\frac {3}{8})$.
Now we recall Hitchin's formula.
\noindent \textbf{Theorem B. (Hitchin \cite{Hit1})} \emph{For any pair $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, define $p^{(0)}_{r,s}(\tau)$ by \begin{equation}
\wp(p^{(0)}_{r,s}(\tau)|\tau):=\wp(r+s\tau|\tau)+\frac{\wp^{\prime}(r+s\tau|\tau)}
{2(\zeta(r+s\tau|\tau)-r\eta_{1}(\tau)-s\eta_{2}(\tau))}. \label{II-1} \end{equation}
Then $p^{(0)}_{r,s}(\tau)$ is a solution of EPVI$(\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8})$; or equivalently, \[\lambda^{(0)}_{r,s}(t):=\frac{\wp(p^{(0)}_{r,s}(\tau)|\tau)-e_1(\tau)} {e_2(\tau)-e_1(\tau)}\] is a solution of PVI$(\frac{1}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$.}
\noindent \textbf{Notation}: \emph{Let $\lambda^{(n)}(t)$ be a solution of PVI$(\frac{1}{2}(n+\frac12)^2,\frac{-1}{8},\frac{1}{8},\frac {3}{8})$ and $p^{(n)}(\tau)$ be the corresponding solution of EPVI$(\frac{1}{2}(n+\frac12)^2,\frac{1}{8},\frac{1}{8},\frac {1}{8})$. We denote them by $\lambda^{(n)}_{r,s}(t)$ and $p^{(n)}_{r,s}(\tau)$ respectively if them come from the solution $\lambda_{r,s}^{(0)}(t)$ of PVI$(\frac{1}{8},$ $\frac{-1}{8},\frac{1}{8},\frac{3}{8})$ and the corresponding $p_{r,s}^{(0)}(\tau)$ of EPVI$(\frac{1}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8})$ stated in Theorem B via (\ref{II-128}), i.e. \begin{equation}
\frac{\wp(p_{r,s}^{(n)}(\tau)|\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1} (\tau)}=\lambda^{(n)}_{r,s}(t)=R_n\left(\lambda_{r,s}^{(0)}(t),\mu_{r,s}^{(0)}(t),t\right), \label{comred} \end{equation} where $\mu_{r,s}^{(0)}(t)$ is the corresponding $\mu^{(0)}(t)$ of $\lambda_{r,s}^{(0)}(t)$. We also use notation $\mu_{r,s}^{(n)}(t)$ via (\ref{II-128-0}).}
\begin{remark} \label{identify} Clearly for any $m_{1},m_{2}\in \mathbb{Z}$, $\pm p_{r,s}^{(n)}(\tau)+m_{1}+m_{2}\tau$ is also a solution of EPVI$(\frac{1}{2}(n+\frac12)^2,\frac{1}{8},\frac{1}{8},\frac {1}{8})$. Since they all give the same $\lambda_{r,s}^{(n)}(t)$ via (\ref{II-130}), in this paper we always identify all these solutions $\pm p_{r,s}^{(n)}(\tau)+m_{1}+m_{2}\tau$ with the same one $p_{r,s}^{(n)}(\tau)$. \end{remark}
\subsection{Generalization of Hitchin's formula}
In this section, we exploit the Okamoto transformation to study the explicit expression of $\wp(p_{r,s}^{(n)}(\tau)|\tau)$, which can be seen as a generalization of Hitchin's formula (\ref{II-1}). First we recall the Okamoto transformation $\kappa_{5}$ defined in (\ref{II-120}).
\begin{lemma} \label{lemII-2-5}Let $(\lambda(t),\mu(t))$ be a solution of the Hamiltonian system (\ref{aa}) with parameter $\theta \in \mathcal{K}$. Then $(\tilde {\lambda}(t),\tilde{\mu}(t)):=(\kappa_{5}(\lambda)(t),\kappa_{5}(\mu)(t))$, which is a solution of the Hamiltonian system (\ref{aa}) with new parameter \[ \tilde{\theta}=(\tilde{\theta}_{0},\tilde{\theta}_{1},\tilde{\theta} _{2},\tilde{\theta}_{3},\tilde{\theta}_{4}):=\kappa_{5}(\theta)=(\theta _{0}-1,\theta_{1},\theta_{2},\theta_{3},\theta_{4}+2), \] are expressed as{\allowdisplaybreaks \begin{align*} \tilde{\lambda}(t) & =\hat{\lambda}(t)+\frac{1-\theta_{0}}{\tilde{\mu} (t)},\\ \tilde{\mu}(t) & =\hat{\mu}(t)+\frac{\theta_{0}+\theta_{1}-1}{\hat{\lambda }(t)}+\frac{\theta_{0}+\theta_{2}-1}{\hat{\lambda}(t)-1}+\frac{\theta _{0}+\theta_{3}-1}{\hat{\lambda}(t)-t}, \end{align*} }where{\allowdisplaybreaks \begin{align*} \hat{\lambda}(t) & =\bar{\lambda}(t)+\frac{1+\theta_{4}}{\hat{\mu} (t)},\text{ \ \ \ }\bar{\lambda}(t)=\lambda(t)+\frac{\theta_{0}+\theta_{4} }{\mu(t)},\\ \hat{\mu}(t) & =\mu(t)-\frac{\theta_{0}+\theta_{1}+\theta_{4}}{\text{\ } \bar{\lambda}(t)}-\frac{\theta_{0}+\theta_{2}+\theta_{4}}{\text{\ } \bar{\lambda}(t)-1}-\frac{\theta_{0}+\theta_{3}+\theta_{4}}{\text{\ } \bar{\lambda}(t)-t}. \end{align*} } \end{lemma}
\begin{proof} The proof is just a straightforward computation via Table 1. \end{proof}
Since the Okamoto transformation is invertible, it is known from $\kappa^{0,n}(\theta^0)=\theta^n$ that $(\lambda _{r,s}^{(n)}(t),\mu_{r,s}^{(n)}(t))$ can be transformed into $(\lambda_{r,s}^{(m)}(t),\mu_{r,s}^{(m)}(t))$ via Okamoto transformations for any $n\not =m$. In the following, we often omit the subscripts $r,s$ for convenience. The following result gives the explicit expression of $(\lambda^{(n)}(t),\mu ^{(n)}(t))$ in terms of $(\lambda^{(n-1)}(t),\mu^{(n-1)}(t))$.
\begin{lemma} \label{thm-II-16}Under the above notations, for $n\geq1$ there holds: \begin{align} \mu^{(n)}= & \mu^{(n-1)}-\frac{n}{2}\bigg(\frac{1}{\lambda^{(n-1)} +\frac{n-1}{2\mu^{(n-1)}}}\label{II-500}\\ & +\frac{1}{\lambda^{(n-1)}+\frac{n-1}{2\mu^{(n-1)}}-1}+\frac{1} {\lambda^{(n-1)}+\frac{n-1}{2\mu^{(n-1)}}-t}\bigg),\nonumber \end{align} \begin{equation} \lambda^{(n)}=\lambda^{(n-1)}+\frac{n-1}{2\mu^{(n-1)}}+\frac{n+1}{2\mu^{(n)}}. \label{II-501} \end{equation}
\end{lemma}
\begin{proof} We prove these two formulae via induction.
\textbf{Step 1.} We consider $n=1$.
Recalling Table 1 and $\kappa^{0,1}:=\kappa_{0}\circ \kappa_{3} \circ \kappa_{2}\circ \kappa_{1}$, we have $\kappa^{0,1}(\theta^{0})=\theta^{1}$. Using the expressions of $\kappa_{j}$ given in Table 1, a straightforward calculation gives \begin{equation} \mu^{(1)}=\mu^{(0)}-\frac{1}{2}\left( \frac{1}{\lambda^{(0)}}+\frac {1}{\lambda^{(0)}-1}+\frac{1}{\lambda^{(0)}-t}\right) , \label{II-502} \end{equation} \begin{equation} \lambda^{(1)}=\lambda^{(0)}+\frac{1}{\mu^{(1)}}. \label{503} \end{equation} This proves (\ref{II-500})-(\ref{II-501}) for $n=1$.
\textbf{Step 2.} Assume that (\ref{II-500})-(\ref{II-501}) hold for $n=m-1$, where $m\geq2$, we claim that (\ref{II-500})-(\ref{II-501}) hold for $n=m$.
Since $\kappa_{5}(\theta^{m-2})=\theta^{m}$, we apply Lemma \ref{lemII-2-5} to obtain \begin{equation} \lambda^{(m)}=\hat{\lambda}+\frac{m+1}{2\mu^{(m)}},\text{ }\mu^{(m)}=\hat{\mu }-\frac{m}{2}\Big(\frac{1}{\hat{\lambda}}+\frac{1}{\hat{\lambda}-1}+\frac {1}{\hat{\lambda}-t}\Big), \label{II-504} \end{equation} where {\allowdisplaybreaks \begin{align} \bar{\lambda} & =\lambda^{(m-2)}+\frac{m-2}{2\mu^{(m-2)}},\label{II-506}\\ \hat{\lambda} & =\bar{\lambda}(t)+\frac{2m-1}{2\hat{\mu}(t)}, \label{II-505} \\ \hat{\mu} & =\mu^{(m-2)}-\frac{m-1}{2}\left( \frac{1}{\text{\ }\bar {\lambda}(t)}-\frac{1}{\text{\ }\bar{\lambda}(t)-1}-\frac{1}{\text{\ } \bar{\lambda}(t)-t}\right) . \label{II-508} \end{align} }Since (\ref{II-500})-(\ref{II-501}) hold for $n=m-1$, it is easy to see from (\ref{II-506}) and (\ref{II-508}) that $\hat{\mu}=\mu^{(m-1)}$. Substituting $\hat{\mu}=\mu^{(m-1)}$ and (\ref{II-506}) into (\ref{II-505}), we have \[ \hat{\lambda}=\lambda^{(m-2)}+\frac{m-2}{2\mu^{(m-2)}}+\frac{2m-1} {2\mu^{(m-1)}}=\lambda^{(m-1)}+\frac{m-1}{2\mu^{(m-1)}}. \] Consequently, (\ref{II-504}) implies that (\ref{II-500})-(\ref{II-501}) hold for $n=m$. This completes the proof. \end{proof}
To state our main results of this section, we give some general settings. Fix any $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$ and let \begin{equation}
a(\tau)=r+s\tau,\text{ \ }Z_{r,s}(\tau)=\zeta(a(\tau)|\tau)-r\eta_{1} (\tau)-s\eta_{2}(\tau). \label{II-46-7} \end{equation} Note from Theorem B that \begin{equation}
\lambda_{r,s}^{(0)}(t)=\frac{\wp(p_{r,s}^{(0)}(\tau)|\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau)}
=\frac{\wp(a(\tau)|\tau)+\frac{\wp^{\prime}\left( a(\tau)|\tau \right) }{2Z_{r,s}(\tau)}-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau)}. \label{II-126} \end{equation} Then \cite[(4.21)]{Chen-Kuo-Lin} shows that the corresponding $\mu_{r,s}^{(0)}(t)$ is given by \begin{equation}
\mu_{r,s}^{(0)}(t)=\frac{e_{2}(\tau)-e_{1}(\tau)}{2(\wp(p_{r,s}^{(0)}(\tau)|\tau)-\wp(a(\tau)|\tau
))}=\frac{(e_{2}(\tau)-e_{1}(\tau))Z_{r,s}(\tau)}{\wp^{\prime}(a(\tau)|\tau)}. \label{II-127} \end{equation} We rewrite them as follows: \begin{equation} \lambda^{(0)}(t)=\lambda_{r,s}^{(0)}(t)=q_{0}(Z_{r,s}(\tau)),\text{ \ } \mu^{(0)}(t)=\mu_{r,s}^{(0)}(t)=p_{0}(Z_{r,s}(\tau)), \label{II-46} \end{equation} where \begin{equation} q_{0}(X):=\frac{R_{0}(X)}{Q_{0}(X)},\text{ \ \ }p_{0}(X):=\frac{Q_{0} (X)}{G_{0}(X)}, \label{II-513} \end{equation} \begin{equation}
Q_{0}(X)=X,\text{ \ }G_{0}(X)=\frac{\wp^{\prime}(a(\tau)|\tau)}{e_{2} (\tau)-e_{1}(\tau)}, \label{II-511} \end{equation} \begin{equation}
R_{0}(X)=\frac{\wp(a(\tau)|\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau)}
X+\frac{1}{2}\frac{\wp^{\prime}(a(\tau)|\tau)}{e_{2}(\tau)-e_{1}(\tau)}. \label{II-512} \end{equation} In the following, we fix any $\tau \in \mathbb{H}$ such that $a(\tau )\not \in E_{\tau}[2]$ and let $t=t(\tau)$. Clearly $R_{0}(X)$, $Q_{0}(X)$ and $G_{0}(X)$ satisfy
\begin{itemize} \item[(1-$0$)] $Q_{0}(X)$ is a polynomial of degree $1$; $R_{0}(X)$ is a polynomial of degree $1$; $G_{0}(X)$ is a non-zero constant.
\item[(2-$0$)] any two of $\{Q_{0}(X),G_{0}(X),R_{0}(X)\}$ has no common zeros.
\item[(3-$0$)] $Q_{0}(X)|(R_{0}(X)-\frac{1}{2}G_{0}(X))$.
\item[(4-$0$)] $\deg(R_{0}(X)-Q_{0}(X))=\deg(R_{0}(X)-tQ_{0}(X))=\deg R_{0}(X)=1$. \end{itemize}
\noindent Remark that $a(\tau)\not \in E_{\tau}[2]$ is the \emph{key} assumption, because these properties can \emph{not} hold if $a(\tau)\in E_{\tau}[2]$.
For convenience, we define \begin{align} H(x;y) & :=x(x-y)(x-ty),\label{II-510}\\ H^{\prime}(x;y) & :=\frac{\partial H}{\partial x} =x(x-y)+x(x-ty)+(x-y)(x-ty).\nonumber \end{align} Clearly we have \begin{equation} H(bx;by)=b^{3}H(x;y),\text{ \ \ }H^{\prime}(bx;by)=b^{2}H^{\prime}(x;y). \label{II-510-1} \end{equation} Now starting from $Q_{0}(X)$, $G_{0}(X)$ and $R_{0}(X)$ defined in (\ref{II-511})-(\ref{II-512}) and setting $Q_{-2}(X)=Q_{-1}(X)=1$, we can define $Q_{n}(X)$, $G_{n}(X)$, $R_{n}(X)$ for $n\geq1$ by induction as follows: \begin{equation} \varphi_{n-1}(X):=R_{n-1}(X)Q_{n-2}(X)+\tfrac{n-1}{2}G_{n-1}(X), \label{II-515} \end{equation} \begin{equation} G_{n}(X):=\frac{H(\varphi_{n-1}(X);Q_{n-3}(X)Q_{n-2}(X)Q_{n-1}(X))} {G_{n-1}(X)Q_{n-3}(X)^{3}}, \label{II-516} \end{equation} {\allowdisplaybreaks \begin{align} Q_{n}(X):= & \frac{H(\varphi_{n-1}(X);Q_{n-3}(X)Q_{n-2}(X)Q_{n-1} (X))}{G_{n-1}(X)^{2}Q_{n-3}(X)^{2}}\nonumber \\ & -\frac{n}{2}\frac{H^{\prime}(\varphi_{n-1}(X);Q_{n-3}(X)Q_{n-2} (X)Q_{n-1}(X))}{G_{n-1}(X)Q_{n-3}(X)^{2}}\nonumber \\ = & \frac{Q_{n-3}(X)G_{n}(X)}{G_{n-1}(X)}-\frac{n}{2}\frac{H^{\prime }(\varphi_{n-1}(X);Q_{n-3}(X)Q_{n-2}(X)Q_{n-1}(X))}{G_{n-1}(X)Q_{n-3}(X)^{2}}, \label{II-517} \end{align} }{\allowdisplaybreaks \begin{equation} R_{n}(X):=\frac{\varphi_{n-1}(X)Q_{n}(X)+\frac{n+1}{2}Q_{n-3}(X)G_{n} (X)}{Q_{n-3}(X)Q_{n-1}(X)}. \label{II-518} \end{equation} We will prove that }$Q_{n}(X)$, $G_{n}(X)$, $R_{n}(X)$ are all \emph{polynomials}; see Theorem \ref{thm-II-17} below.
Recalling $(p_{0}(X),q_{0}(X))$ defined in (\ref{II-46})-(\ref{II-513}), we exploit (\ref{II-500})-(\ref{II-501}) to define rational function pairs $(p_{n}(X),q_{n}(X))$ for all $n\geq1$ by induction: \begin{align} p_{n}(X):= & p_{n-1}(X)-\frac{n}{2}\bigg(\frac{1}{q_{n-1}(X)+\frac {n-1}{2p_{n-1}(X)}}\label{II-500-1}\\ & +\frac{1}{q_{n-1}(X)+\frac{n-1}{2p_{n-1}(X)}-1}+\frac{1}{q_{n-1} (X)+\frac{n-1}{2p_{n-1}(X)}-t}\bigg),\nonumber \end{align} \begin{equation} q_{n}(X):=q_{n-1}(X)+\frac{n-1}{2p_{n-1}(X)}+\frac{n+1}{2p_{n}(X)}. \label{II-501-1} \end{equation} Here is the first main result of this section.
\begin{theorem} \label{thm-II-17}Under the assumption $a(\tau)\not \in E_{\tau}[2]$ and the above notations, for any $n\geq1$ we have{\allowdisplaybreaks \begin{align} p_{n}(X) & =\frac{Q_{n-2}(X)Q_{n-1}(X)Q_{n}(X)}{G_{n}(X)},\label{II-514}\\ q_{n}(X) & =\frac{R_{n}(X)}{Q_{n-2}(X)Q_{n}(X)}, \label{II-514-1} \end{align} }where $Q_{-2}(X)=Q_{-1}(X)=1$, $Q_{0}(X)$, $R_{0}(X)$, $G_{0}(X)$ are given by (\ref{II-511})-(\ref{II-512}), and $Q_{n}(X)$, $G_{n}(X)$, $R_{n}(X)$ are given by induction in (\ref{II-515})-(\ref{II-518}). Furthermore, $Q_{n}(X)$, $G_{n}(X)$, $R_{n}(X)$ satisfy
\begin{itemize} \item[(1-$n$)] $Q_{n}(X)$ is a polynomial of degree $\frac{(n+1)(n+2)}{2}$;
$G_{n}(X)$ is a polynomial of degree $\frac{3n(n+1)}{2}$; $R_{n}(X)$ is a polynomial of degree $n(n+1)+1$. Their coefficients are all rational functions of $e_{1}(\tau)$, $e_{2}(\tau)$, $e_{3}(\tau)$, $\wp(a(\tau)|\tau)$ and
$\wp^{\prime}(a(\tau)|\tau)$ with coefficients in $\mathbb{Q}$.
\item[(2-$n$)] any two of $\{Q_{n-2}(X),Q_{n-1}(X),Q_{n}(X),G_{n}(X)\}$ have no common zeros; any two of $\{Q_{n-2}(X),Q_{n}(X),R_{n}(X)\}$ have no common zeros; $G_{n-1}(X)$ and $G_{n}(X)$ have no common zeros.
\item[(3-$n$)] $Q_{n-2}(X)|\varphi_{n}(X)$ and $Q_{n}(X)|(R_{n}(X)Q_{n-1} (X)-\frac{n+1}{2}G_{n}(X))$.
\item[(4-$n$)] $\deg(R_{n}-Q_{n-2}Q_{n})=\deg(R_{n}-tQ_{n-2}Q_{n})=\deg R_{n}=n(n+1)+1.$ \end{itemize} \end{theorem}
\begin{remark} The proof of Theorem \ref{thm-II-17} is technical and will be given in Appendix \ref{appendix-A}. Although the expressions (\ref{II-515})-(\ref{II-518}) of $Q_{n}(X)$, $G_{n}(X)$, $R_{n}(X)$ defined by induction look complicated, they turn out to be very useful. For example, they will be applied to calculate the asymptotics of $Z_{r,s}^{(n)}(\tau)$ in \S \ref{sec-asymptotics}. \end{remark}
As an application of Theorem \ref{thm-II-17}, we have the following important result, which is a generalization of Hitchin's formula (\ref{II-1}) to solutions of PVI$(\tfrac{1}{2}(n+\tfrac{1}{2})^{2},\tfrac{-1}{8},\tfrac{1} {8},\tfrac{3}{8})$. Since for any fixed $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, Property (1-$n$) shows that the coefficients depend on $\tau$ as $\tau \in \mathbb{H}$ deforms, we also denote $Q_{n}(X)$,
$G_{n}(X)$, $R_{n}(X)$ by $Q_{n}(X|\tau)$, $G_{n}(X|\tau)$ and $R_{n}(X|\tau)$ respectively.
\begin{theorem} \label{thm-II-18}Fix any $(r,s)\in \mathbb{C}^{2}\backslash \frac{1} {2}\mathbb{Z}^{2}$ and let $a(\tau)=r+s\tau$, $Z_{r,s}(\tau)=\zeta
(a(\tau)|\tau)-r\eta_{1}(\tau)-s\eta_{2}(\tau)$ as before. Then for any $n\geq0$ the following hold:
\begin{itemize} \item[(i)] $\lambda_{r,s}^{(n)}(t)$, $\mu_{r,s}^{(n)}(t)$ and $p_{r,s} ^{(n)}(\tau)$ are expressed by \begin{equation}
\lambda_{r,s}^{(n)}(t)=q_{n}(Z_{r,s}(\tau))=\frac{R_{n}(Z_{r,s}(\tau)|\tau
)}{Q_{n-2}(Z_{r,s}(\tau)|\tau)Q_{n}(Z_{r,s}(\tau)|\tau)}, \label{II-535} \end{equation} \begin{align} \mu_{r,s}^{(n)}(t) & =p_{n}(Z_{r,s}(\tau))\label{II-536}\\
& =\frac{Q_{n-2}(Z_{r,s}(\tau)|\tau)Q_{n-1}(Z_{r,s}(\tau)|\tau)Q_{n}
(Z_{r,s}(\tau)|\tau)}{G_{n}(Z_{r,s}(\tau)|\tau)},\nonumber \end{align} \begin{equation}
\wp(p_{r,s}^{(n)}(\tau)|\tau)=\frac{(e_{2}(\tau)-e_{1}(\tau))R_{n}
(Z_{r,s}(\tau)|\tau)}{Q_{n-2}(Z_{r,s}(\tau)|\tau)Q_{n}(Z_{r,s}(\tau)|\tau )}+e_{1}(\tau). \label{II-537} \end{equation}
\item[(ii)] Let $t_{0}=t(\tau_{0})$ such that $a(\tau_{0})\not \in E_{\tau _{0}}[2]$, then $t_{0}$ is a positive pole of $\lambda_{r,s}^{(n)}(t)$ if and only if $Q_{n-2}(Z_{r,s}(\tau_{0})|\tau_{0})=0$; $t_{0}$ is a negative pole of
$\lambda_{r,s}^{(n)}(t)$ if and only if $Q_{n}(Z_{r,s}(\tau_{0})|\tau_{0})=0$. In particular, both $\lambda_{r,s}^{(0)}(t)$ and $\lambda_{r,s}^{(1)}(t)$ have no positive poles in $\mathbb{C}\backslash \{0,1\}$ provided $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$. \end{itemize} \end{theorem}
\begin{remark} For Hitchin's formula (\ref{II-1}), the result that any zero $\tau_{0}$ of $Z_{r,s}(\tau)$ gives a negative pole $t(\tau_{0})$ of $\lambda_{r,s} ^{(0)}(t)$ was proved by Hitchin in \cite[Proposition 9]{Hit1}. Theorem \ref{thm-II-18}-(ii), which extends this result to the general case $n\geq1$, will be applied to establish the relation $Q_{n}(Z_{r,s}(\tau
)|\tau)=\mathfrak{q}_{n}(\tau)Z_{r,s}^{(n+1)}(\tau)$ for some non-zero constant $\mathfrak{q}_{n}(\tau)$ in \S \ref{expression-completely}.4. \end{remark}
\begin{proof} [Proof of Theorem \ref{thm-II-18}](i) For the formulae (\ref{II-535} )-(\ref{II-536}), the case $n=0$ follows from (\ref{II-46}), and the general case $n\geq1$ follows directly from Lemma \ref{thm-II-16} and (\ref{II-500-1} )-(\ref{II-501-1}). Clearly (\ref{II-535}) implies (\ref{II-537}). This proves (i).
(ii) Let $t_{0}=t(\tau_{0})$ such that $a(\tau_{0})\not \in E_{\tau_{0}}[2]$. Then $a(\tau)\not \in E_{\tau}[2]$ for $\tau$ in a small neighborhood $U$ of
$\tau_{0}$. Consequently, $Q_{n-2}(X|\tau)$, $Q_{n}(X|\tau)$ and $R_{n}
(X|\tau)$ are all well-defined polynomials for each fixed $\tau \in U$ and holomorphically depend on $\tau \in U$ (because the coefficients of these polynomials are rational functions of $e_{1}(\tau)$, $e_{2}(\tau)$,
$e_{3}(\tau)$, $\wp(a(\tau)|\tau)$ and $\wp^{\prime}(a(\tau)|\tau)$ and take finite values provided $a(\tau)\not \in E_{\tau}[2]$). Therefore, by property (2-$n$) in Theorem \ref{thm-II-17}, we conclude that $t_{0}$ is a pole of $\lambda_{r,s}^{(n)}(t)$ if and only if \[
\text{either }Q_{n-2}(Z_{r,s}(\tau_{0})|\tau_{0})=0\text{ \ \ or \ \ }
Q_{n}(Z_{r,s}(\tau_{0})|\tau_{0})=0. \]
If $Q_{n-2}(Z_{r,s}(\tau_{0})|\tau_{0})=0$, then properties (2-$n$)-(3-$n$) in Theorem \ref{thm-II-17} imply \[ \lim_{t\rightarrow t_{0}}\lambda_{r,s}^{(n)}\mu_{r,s}^{(n)}=\frac
{R_{n}(Z_{r,s}(\tau_{0})|\tau_{0})Q_{n-1}(Z_{r,s}(\tau_{0})|\tau_{0})}
{G_{n}(Z_{r,s}(\tau_{0})|\tau_{0})}=\frac{-n}{2}=-(\theta_0+\theta_4), \] where we use (\ref{I1I-175}) to obtain the last equality. This, together with Proposition \ref{Prop-II-1}, shows that
$t_{0}$ is a positive pole. If $Q_{n}(Z_{r,s}(\tau_{0})|\tau_{0})=0$, then properties (2-$n$)-(3-$n$) in Theorem \ref{thm-II-17} imply \[ \lim_{t\rightarrow t_{0}}\lambda_{r,s}^{(n)}\mu_{r,s}^{(n)}=\frac
{R_{n}(Z_{r,s}(\tau_{0})|\tau_{0})Q_{n-1}(Z_{r,s}(\tau_{0})|\tau_{0})}
{G_{n}(Z_{r,s}(\tau_{0})|\tau_{0})}=\frac{n+1}{2}=-\theta_0. \] By Proposition \ref{Prop-II-1} again, we see that $t_{0}$ is a negative pole. Since $Q_{-2}=Q_{-1}\equiv1$, we see that the assertion (ii) holds. This completes the proof. \end{proof}
\begin{corollary} \label{thm-II-18-1}For any $n\geq0$ and fixed real pair $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, $Q_{n}(Z_{r,s}(\tau
)|\tau)$ has only simple zeros as a holomorphic function of $\tau \in \mathbb{H}$. \end{corollary}
\begin{proof} For any $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, $a(\tau)\not \in E_{\tau}[2]$ for all $\tau \in \mathbb{H}$, which implies that
$Q_{n}(X|\tau)$ is a well-defined polynomial for each fixed $\tau$ and holomorphically depend on $\tau \in \mathbb{H}$. Thus $Q_{n}(Z_{r,s}(\tau
)|\tau)$ is holomorphic as a function of $\tau \in \mathbb{H}$. Let $\tau_{0}$
be any a zero of $Q_{n}(Z_{r,s}(\tau)|\tau)$, then $t(\tau_{0})$ is a pole of $\lambda_{r,s}^{(n)}(t)$. By Theorem \ref{thm-2A} we know that $t(\tau_{0})$ is a simple pole of $\lambda_{r,s}^{(n)}(t)$. Since $t^{\prime}(\tau_{0})\not =0$
and Property (2-$n$) in Theorem \ref{thm-II-17} shows $R_{n}(Z_{r,s}(\tau _{0})|\tau_{0})\not =0$, we conclude that $\tau_{0}$ is a simple zero of
$Q_{n}(Z_{r,s}(\tau)|\tau)$. \end{proof}
Now we give two examples. For convenience we write $\wp=\wp(a(\tau)|\tau)$, $Z=Z_{r,s}(\tau)$ and $e_{k}=e_{k}(\tau)$.
\begin{theorem} \label{thm-expression-1}Let $n=1$ in Theorem \ref{thm-II-18}. Then \begin{align}\label{expression-1-0} \lambda_{r,s}^{(1)}(t) =\frac{(\wp-e_{1})Z^{3}+\frac{3\wp^{\prime}}{2}Z^{2}+\frac {6\wp^{2}+6e_{1}\wp-g_{2}}{2}Z+\frac{\wp+2e_{1}}{2}\wp^{\prime}}{(e_{2}-e_{1})(Z^{3}-3\wp Z-\wp^{\prime})}, \end{align} \[ \mu_{r,s}^{(1)}(t)=\frac{2(e_{2}-e_{1})Z(Z^{3}-3\wp Z-\wp^{\prime})} {2\wp^{\prime}Z^{3}+(12\wp^{2}-g_{2})Z^{2}+6\wp\wp^{\prime}Z+(\wp^{\prime})^{2}}, \] \begin{equation}
\wp(p_{r,s}^{(1)}(\tau)|\tau)=\wp+\frac{3\wp^{\prime}Z^{2}+\left( 12\wp^{2}-g_{2}\right) Z+3\wp\wp^{\prime}}{2(Z^{3}-3\wp Z-\wp^{\prime})}. \label{expression-1} \end{equation} That is, when $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, (\ref{expression-1}) gives solutions of EPVI$(\frac{9}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8})$; or equivalently, (\ref{expression-1-0}) gives solutions of PVI$(\frac{9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. \end{theorem}
\begin{proof} Recalling (\ref{II-511}), (\ref{II-512}) and (\ref{II-515})-(\ref{II-518}), a direct computation gives \[ G_{1}(X)=\frac{\wp^{\prime}X^{3}+(6\wp^{2}-\frac{g_{2}}{2})X^{2} +3\wp\wp^{\prime}X+\frac{(\wp^{\prime})^{2}}{2}}{4(e_{2}-e_{1})^{2}}, \] \[ Q_{1}(X)=\frac{X^{3}-3\wp X-\wp^{\prime}}{4(e_{2}-e_{1})}, \] \[ R_{1}(X)=\frac{(\wp-e_{1})X^{3}+\frac{3\wp^{\prime}}{2}X^{2}+\frac {6\wp^{2}+6e_{1}\wp-g_{2}}{2}X+\frac{\wp+2e_{1}}{2}\wp^{\prime}}{4(e_{2}-e_{1})^{2}}. \] Then this theorem follows readily from (\ref{II-535})-(\ref{II-537}). \end{proof}
Formula (\ref{expression-1}), which is a generalization of Hitchin's formula (\ref{II-1}) to PVI$(\frac{9}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$, was first obtained by Takemura \cite{Takemura} via a different method. The following formula (\ref{expression-2}) for PVI$(\frac{25}{8},\frac{-1}{8},\frac{1}{8},\frac {3}{8})$ can not be found in the literature and is new.
\begin{theorem} \label{thm-expression-2}Let $n=2$ in Theorem \ref{thm-II-18}. Then \begin{equation}
\wp(p_{r,s}^{(2)}(\tau)|\tau)=\wp+\frac{\Xi^{(2)}(Z)}{8Z Z_{r,s}^{(3)}(\tau)}, \label{expression-2} \end{equation} where $Z_{r,s}^{(3)}(\tau)$ is given in (\ref{z-n-4}) and {\allowdisplaybreaks \begin{align*} \Xi^{(2)}(Z)= & 28\wp^{\prime}Z^{6}+\left( 288\wp^{2}-24g_{2}\right) Z^{5}+300\wp\wp^{\prime}Z ^{4}\\ & +\left( 640\wp^{3}-88g_{2}\wp-52g_{3}\right) Z^{3}\\ & +(180\wp^{2}-3g_{2})\wp^{\prime}Z^{2}+24\wp(\wp^{\prime})^{2}Z+(\wp^{\prime})^{3}. \end{align*} }That is, when $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, (\ref{expression-2}) gives solutions of EPVI$(\frac{25}{8},\frac{1}{8},\frac{1}{8},\frac{1}{8})$. \end{theorem} Theorem \ref{thm-expression-2} can be proved similarly as Theorem \ref{thm-expression-1}; we just need to calculate $G_{2}$, $Q_{2}$, $R_{2}$ by using Theorem \ref{thm-II-17}. We omit the details here.
Now we turn to the special case $(r,s)=(\frac{1}{4},0)$ and have the following simple observation.
\begin{theorem} \label{r=1/4}For any $n\geq0$, there holds \begin{equation} \lambda_{\frac{1}{4},0}^{(n)}(t)=\frac{(-1)^{n}}{2n+1}t^{\frac{1}{2}},\text{ \ }\mu_{\frac{1}{4},0}^{(n)}(t)=\frac{1}{4\lambda_{\frac{1}{4},0}^{(n)} (t)}=(-1)^{n}\frac{2n+1}{4}t^{-\frac{1}{2}}. \label{vi-0} \end{equation} In particular, $t=1$ is not a branch point of $\lambda_{\frac{1}{4},0} ^{(n)}(t)$. \end{theorem}
\begin{remark} A simple computation shows that $\frac{(-1)^{n}}{2n+1}t^{\frac{1}{2}}$ is really
a solution of
PVI$(\frac12(n+\frac12)^2,\frac{-1}{8},\frac{1}{8},\frac{3}{8})$ for \emph{any} $n\geq0$. Hitchin \cite{Hit2} already knew that $t^{\frac{1}{2}}$ is a solution of PVI$(\frac{1}{8},\frac{-1}{8},\frac{1}{2k^{2}},\frac{1} {2}-\frac{1}{2k^{2}})$ for any $k\in \mathbb{N}$ and hence a solution of PVI$(\frac{1}{8},\frac{-1}{8},\frac{1}{8},\frac{3}{8})$. Here we need to prove $t^{\frac{1}{2}}=\lambda_{\frac{1}{4},0}^{(0)}(t)$. Formula (\ref{vi-0}) will play a key role in our proof of Theorem \ref{weak}-(3) in \S \ref{sec-asymptotics}. \end{remark}
\begin{proof} [Proof of Theorem \ref{r=1/4}]In this proof we always consider $(r,s)=(\frac {1}{4},0)$ and write $\lambda^{(n)}(t)=\lambda_{\frac{1}{4},0}^{(n)}(t)$, $\mu^{(n)}(t)=\mu_{\frac{1}{4},0}^{(n)}(t)$ for convenience. First we prove (\ref{vi-0}) for $n=0$. Recalling the Hamiltonian system (\ref{aa})-(\ref{98}) and (\ref{I1I-175}), we know that $(\lambda^{(0)},\mu^{(0)})$ satisfies \begin{equation} \frac{d\lambda^{(0)}}{dt}=\frac{(2\lambda^{(0)}\mu^{(0)}-\frac{1}{2} )(\lambda^{(0)}-1)(\lambda^{(0)}-t)+\frac{1}{2}(t-1)\lambda^{(0)}}{t(t-1)}, \label{vi-2} \end{equation} \begin{equation} \frac{d\mu^{(0)}}{dt}=-\frac{[3(\lambda^{(0)})^{2}-2(t+1)\lambda^{(0)} +t](\mu^{(0)})^{2}-(\lambda^{(0)}-t)\mu^{(0)}}{t(t-1)}. \label{vi-3} \end{equation}
Recall the addition formula \[ \zeta(u+v)+\zeta(u-v)-2\zeta(u)=\frac{\wp^{\prime}(u)}{\wp(u)-\wp(v)}. \] Letting $u=\frac{1}{4}$ and $v=\frac{\omega_{1}}{2}=\frac{1}{2}$ leads to \begin{align*} \frac{\wp^{\prime}(\frac{1}{4})}{\wp(\frac{1}{4})-e_{1}} & =\zeta \left( 3/4\right) +\zeta \left( -1/4\right) -2\zeta \left( 1/4\right) \\ & =\eta_{1}-4\zeta \left( 1/4\right) =-4Z_{\frac{1}{4},0}(\tau). \end{align*} This identity, together with $a(\tau)=r+s\tau=\frac{1}{4}$ and (\ref{II-46}
)-(\ref{II-512}), easily implies $4R_{0}(Z_{\frac{1}{4},0}(\tau)|\tau
)=G_{0}(Z_{\frac{1}{4},0}(\tau)|\tau)$ and so \begin{equation} \lambda^{(0)}(t)\mu^{(0)}(t)\equiv \tfrac{1}{4}. \label{vi-1} \end{equation} Inserting (\ref{vi-1}) into (\ref{vi-2}) gives $\frac{d\lambda^{(0)}} {dt}=\frac{\lambda^{(0)}}{2t}$ and then $\frac{d\mu^{(0)}}{dt}=\frac {-1}{8t\lambda^{(0)}}$. Substituting this and (\ref{vi-1}) into (\ref{vi-3}) we easily obtain \[ (\lambda^{(0)}(t))^{2}\equiv t. \] This proves (\ref{vi-0}) for $n=0$ (In \S \ref{sec-asymptotics} we will prove $\lambda^{(0)}(t)\rightarrow1$ as $t=t(\tau)\rightarrow1$ by letting $F_{2} \ni \tau \rightarrow \infty$, where $F_{2}$ is a fundamental domain of $\Gamma(2)$ defined in (\ref{funde}); see (\ref{iv-53})). Finally, formula (\ref{vi-0}) for all $n\geq1$ can be proved via Lemma \ref{thm-II-16} by induction. Since the proof is trivial, we omit the details here. This completes the proof. \end{proof}
\subsection{Connection between PVI and $Z_{r,s}^{(n)}(\tau)$}
In this section, we establish the precise connection of PVI (\ref{46-0}) and the pre-modular form $Z_{r,s}^{(n)}(\tau)$. Consequently, we give the proof of Theorem \ref{thm-II-18 copy(1)} and Theorem \ref{simple-zero con}.
Let $n\geq1$. Recall Theorem \ref{thm-II-17} that $Q_{n}(X)$ is a polynomial of degree $\frac{(n+1)(n+2)}{2}$ with coefficients being rational functions of
$e_{k}(\tau)$'s, $\wp(a(\tau)|\tau)$ and $\wp^{\prime}(a(\tau)|\tau)$, provided that $a(\tau)\not \in E_{\tau}[2]$. In this section we denote it by $Q_{n}(X)=Q_{n}(X;a(\tau),\tau)$. Then we have
\begin{theorem} \label{q-n=z-n}Let $\sigma_{n+1}=a(\tau)\not \in \Lambda_{\tau}$ in (\ref{k-ll}). Then for any $n\geq0$ there holds \begin{equation} Q_{n}(X;a(\tau),\tau)=\mathfrak{q}_{n}(\tau)W_{n+1}(X;a(\tau),\tau), \label{caxi} \end{equation} where $\mathfrak{q} _{n}(\tau)\neq 0$ denotes the coefficient of the leading term $X^{\frac{(n+1)(n+2)}{2}}$ of the polynomial $Q_{n} (X;a(\tau),\tau)$, and its expression will be given in Lemma \ref{coefficient-Qn}.
\end{theorem}
To prove Theorem \ref{q-n=z-n},
we need to apply the monodromy theory of the associated linear ODE for $p_{r,s}^{(n)}(\tau)$. For EPVI$(\frac{1}{2}(n+\frac12)^2,\frac{1}{8},\frac{1}{8},\frac {1}{8})$, one choice of its associated linear ODE is the generalized Lam\'{e} equation (denoted it by GLE$(n,p,A,\tau)$) \begin{equation} y^{\prime \prime}=\left[ \begin{array} [c]{l} n(n+1)\wp(z)+\frac{3}{4}(\wp(z+p)+\wp(z-p))\\ +A(\zeta(z+p)-\zeta(z-p))+B \end{array} \right] y=:I(z)y, \label{89-1} \end{equation} where $\pm p\not \in E_{\tau}[2]$ are always assumed to be \emph{apparent singularities} (i.e. non-logarithmic), which is equivalent to (see \cite[Lemma 2.1]{Chen-Kuo-Lin0}) \begin{equation} B=A^{2}-\zeta(2p)A-\tfrac{3}{4}\wp(2p)-n(n+1)\wp ( p) . \label{101} \end{equation}
Fix any base point $q_{0}\in E_{\tau}\backslash(\{ \pm \lbrack p]\} \cup E_{\tau}[2])$. The monodromy representation of GLE (\ref{89-1}) is a homomorphism $\rho:\pi_{1}( E_{\tau}\backslash ( \{ \pm \lbrack p] \} \cup E_{\tau}[2]),q_{0}) \rightarrow SL(2,\mathbb{C})$. Since $n\in \mathbb{Z}_{\geq 0}$ and the local exponents of (\ref{89-1}) at $0$ are $-n$ and $n+1$, the local monodromy matrix at $0$ is $I_{2}$. Thus the monodromy representation is reduced to $\rho:\pi _{1}( E_{\tau}\backslash \{ \pm \lbrack p]\} ,q_{0}) \rightarrow SL(2,\mathbb{C})$. Let $\gamma_{\pm}\in \pi_{1}( E_{\tau }\backslash(\{ \pm \lbrack p]\} \cup E_{\tau}[2]),q_{0}) $ be a simple loop encircling $\pm p$ counterclockwise respectively, and $\ell_{j}\in \pi _{1}( E_{\tau}\backslash(\{ \pm \lbrack p]\} \cup E_{\tau}[2]),q_{0} ) $, $j=1,2$, be two fundamental cycles of $E_{\tau}$ connecting $q_{0}$ with $q_{0}+\omega_{j}$ such that $\ell_{j}$ does not intersect with $L+\Lambda_{\tau}$ (here $L$ is the straight segment connecting $\pm p$) and satisfies \[ \gamma_{+}\gamma_{-}=\ell_{1}\ell_{2}\ell_{1}^{-1}\ell_{2}^{-1}\text{ in } \pi_{1}( E_{\tau}\backslash\{ \pm \lbrack p] \} ,q_{0}) . \] Since the local exponents of (\ref{89-1}) at $\pm p$ are $ \{-\frac{1}{2}, \frac{3}{2}\}$ and $\pm p\not \in E_{\tau}[2]$ are apparent singularities, we always have $\rho(\gamma_{\pm})=-I_{2}$. Denote by $N_{j}=\rho(\ell_j)$ the monodromy matrix along the loop $\ell_{j}$ of GLE (\ref{89-1}) with respect to any linearly independent solutions. Then the monodromy group of GLE (\ref{89-1}) is generated by $\{-I_{2},N_{1},N_{2}\}$, i.e. is always \emph{abelian and so reducible}. It is known (cf. \cite{CKL1}) that expect finitely many $A$'s for given $(\tau, p)$, $N_1$ and $N_2$ can be diagonalized simultaneously, and more precisely, there exists $(r,s)\in\mathbb{C}^2\setminus\frac12\mathbb{Z}^2$ such that \[N_{1}= \begin{pmatrix} e^{-2\pi is} & 0\\ 0 & e^{2\pi is} \end{pmatrix},\quad N_{2}= \begin{pmatrix} e^{2\pi ir} & 0\\ 0 & e^{-2\pi ir} \end{pmatrix}.\]
Let $U$ be an open subset of $\mathbb{H}$ such that $p(\tau)\not \in E_{\tau}[2]$ for any $\tau \in U$. Then we proved in \cite{Chen-Kuo-Lin0} that $p(\tau)$ \emph{is a solution of EPVI$(\frac{1}{2}(n+\frac12)^2,\frac{1}{8},\frac{1}{8},\frac {1}{8})$ if and only if there exist }$A(\tau)$\emph{ (and the corresponding $B(\tau)$ via (\ref{101}))
such that the associated GLE$(n,p(\tau),A(\tau),\tau)$ is monodromy preserving as $\tau \in U$ deforms}. We refer the reader to \cite{Chen-Kuo-Lin0} for the more general statement, where we proved that $(p(\tau),A(\tau))$ sloves a new Hamiltonian system which is equivalent to EPVI. Here we need the following result \cite{Chen-Kuo-Lin}.
\begin{theorem}(\cite[Theorem 5.3]{Chen-Kuo-Lin}) \label{thm-II-8} For $n\in\mathbb{Z}_{\geq 0}$, let $p^{(n)}(\tau)$ be a solution to EPVI$(\frac{1}{2}(n+\frac12)^2,\frac{1}{8},\frac{1}{8},\frac {1}{8})$. Then the following hold:
\begin{itemize} \item[(1)] For any $\tau$ satisfying $p^{(n)} (\tau)\not \in E_{\tau}[2]$, the monodromy group of the associated GLE$(n$, $ p^{(n)}(\tau), A^{(n)}(\tau), \tau)$ is generated by \begin{equation} \rho(\gamma_{\pm})=-I_{2},\text{ }N_{1}= \begin{pmatrix} e^{-2\pi is} & 0\\ 0 & e^{2\pi is} \end{pmatrix} \text{, }N_{2}= \begin{pmatrix} e^{2\pi ir} & 0\\ 0 & e^{-2\pi ir} \end{pmatrix}
\label{II-101} \end{equation} if and only if $(r,s) \in \mathbb{C}^{2}\backslash \frac {1}{2}\mathbb{Z}^{2}$ and $p^{(n)}(\tau)=p_{r,s}^{(n) }(\tau)$ in the sense of Remark \ref{identify}.
\item[(2)] $\wp(p^{(n)}_{r_{1},s_{1}}(\tau)|\tau)\equiv \wp
(p_{r_{2},s_{2}}^{(n)}(\tau)|\tau)\Longleftrightarrow(r_{1},s_{1}) \equiv \pm(r_{2},s_{2}) \operatorname{mod}$ $\mathbb{Z}^{2}$. \end{itemize} \end{theorem}
Now we are in a position to prove Theorem \ref{q-n=z-n}.
\begin{proof}[Proof of Theorem \ref{q-n=z-n}] It is known in \cite{LW2} that $W_{1}(X)=X$, so (\ref{II-511}) gives $Q_{0}(X)=W_{1}(X)$. Therefore, we only need to prove this theorem for any fixed $n\geq1$. Fix any $\tau_{0}\in \mathbb{H}$ and any $a_{0}\not \in E_{\tau_{0}}[2]$ being outside the branch loci of $\sigma_{n+1}:\bar{Y} _{n+1}(\tau_{0})\rightarrow E_{\tau_{0}}$. Let $X_{0}$ be any zero of $W_{n+1}(X;a_{0},\tau_{0})$. Our goal is to prove $Q_{n}(X_{0};a_{0},\tau _{0})=0$.
By Theorem \ref{thm-5A}-(2), there is $\boldsymbol{a}=\{[a_{1}],\cdots,[a_{n+1}]\} \in \bar{Y}_{n+1}(\tau_{0})$ such that $a_{0}=\sigma_{n+1}(\boldsymbol{a})$ and $X_{0}=\boldsymbol{z}_{n+1}(\boldsymbol{a})$. Clearly $\boldsymbol{a}$ is not a branch point of $\bar{Y}_{n+1}(\tau_{0})$ because $a_{0}\notin E_{\tau_0}[2]$. Recalling (\ref{z--n}), we define $(r,s)$ by \begin{align} r+s\tau_{0} & =\sum_{i=1}^{n+1}a_{i}=a_{0}\notin E_{\tau_0}[2],\label{kkll-1}\\
r\eta_{1}(\tau_{0})+s\eta_{2}(\tau_{0}) & =\zeta(a_{0}|\tau_{0})-X_{0}
=\sum_{i=1}^{n+1}\zeta(a_{i}|\tau_{0}).\nonumber \end{align} Since $\tau_{0}\eta_{1}(\tau_{0})-\eta_{2}(\tau_{0})=2\pi i$, $(r,s)$ is uniquely determined and $(r,s)\not \in \frac{1}{2}\mathbb{Z}^{2}$. Clearly (\ref{kkll-1}) implies \begin{equation} a_{0}=r+s\tau_{0}\text{ \ \ and \ \ }X_{0}=Z_{r,s}(\tau_{0}). \label{k-il} \end{equation} By Proposition \ref{prop-B} and (\ref{iiii}), $y_{\{a_{i}\}}(z)$ and $y_{\{-a_{i}\}}(z)$ are linearly independent solutions to the integral Lam\'{e} equation \begin{equation}
y^{\prime \prime}=[ (n+1)(n+2)\wp(z|\tau_{0})+B_{0}] y, \label{III-67} \end{equation}
where $B_{0}=(2n+1)\sum_{i=1}^{n+1}\wp(a_{i}|\tau_{0})$, and the monodromy group of (\ref{III-67}) with respect to $(y_{\{a_{i}\}}(z),y_{\{-a_{i} \}}(z))$ is generated by \begin{equation} \rho(\ell_{1})= \begin{pmatrix} e^{-2\pi is} & 0\\ 0 & e^{2\pi is} \end{pmatrix}\; \text{ and }\;\rho(\ell_{2})= \begin{pmatrix} e^{2\pi ir} & 0\\ 0 & e^{-2\pi ir} \end{pmatrix}. \label{III-66-3} \end{equation}
On the other hand, by \cite[Lemma 3.1]{Chen-Kuo-Lin0} we see that given any $\tilde{h}\in\mathbb{C}$,
EPVI$(\tfrac{1}{2}(n+\tfrac{1}{2})^{2},\frac{1}{8},\frac{1}{8},\frac{1}{8})$ has a solution $p(\tau)$ such that $p(\tau_{0})=0$ and \begin{equation} p(\tau)=c_{0}(\tau-\tau_{0})^{\frac{1}{2}}(1+\tilde{h}(\tau-\tau_{0} )+O(\tau-\tau_{0})^{2})\text{ as }\tau \rightarrow \tau_{0}, \label{515-5} \end{equation} where $c_{0}^{2}=i\frac{2n+1}{2\pi}$ (note $\pm p(\tau)$ corresponds to the same $\lambda(t)$, so $c_{0}$ is determined up to a sign $\pm$). Consequently, the corresponding \[
\lambda(t)=\frac{\wp(p(\tau)|\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau)} \] is a solution of PVI$(\tfrac{1}{2}(n+\tfrac{1}{2})^{2},\frac{-1}{8},\frac {1}{8},\frac{3}{8})$ and has a \emph{negative pole} at $t(\tau_{0})$; see \cite[Lemma 3.1]{Chen-Kuo-Lin0}. Furthermore, it follows from \cite[Theorem 1.5]{Chen-Kuo-Lin0} that the potential $I(z;\tau)$ of the associated GLE$(n,p(\tau),A(\tau),\tau)$ \begin{align} y^{\prime \prime} & =\left[ \begin{array} [c]{l}
n(n+1) \wp(z|\tau) +\frac{3}{4}( \wp( z+p(\tau)|\tau) +\wp( z-p(\tau)|\tau) )\\
+A(\tau)\left( \zeta( z+p(\tau)|\tau) -\zeta( z-p(\tau)|\tau) \right) +B(\tau) \end{array} \right] y\nonumber \\ & =:I(z;\tau)y \label{60-4} \end{align}
which is monodromy preserving as $\tau$ deforms, converges to $(n+1)(n+2)\wp(z|\tau_{0} )+\tilde{B}$ uniformly for $z$ bounded away from lattice points $\Lambda_{\tau_{0}}$ as $\tau \rightarrow \tau_{0}$, where \[ \tilde{B}:=\lim_{\tau \rightarrow \tau_{0}}B(\tau)=2\pi ic_{0}^{2}\left( 4\pi i\tilde{h}-\eta_{1}(\tau_{0})\right). \] Now letting \[ \tilde{h}=\frac{(2n+1)\eta_{1}(\tau_{0})-B_{0}}{4\pi i(2n+1)} \]
in (\ref{515-5}), we obtain $\tilde{B}=B_0$, namely the potential $I(z;\tau)$ of GLE (\ref{60-4}) converges to exactly the potential $(n+1)(n+2)\wp(z|\tau_{0})+B_{0}$ of the Lam\'{e} equation (\ref{III-67}) as $\tau \rightarrow \tau_{0}$. Then we proved in \cite[Theorem 6.2]{Chen-Kuo-Lin} that the monodromy of GLE (\ref{60-4}) is also given by \[N_1=\begin{pmatrix} e^{-2\pi is} & 0\\ 0 & e^{2\pi is} \end{pmatrix},\quad N_2=\begin{pmatrix} e^{2\pi ir} & 0\\ 0 & e^{-2\pi ir} \end{pmatrix},\]
the same as the Lam\'{e} equation (\ref{III-67}). From here and Theorem \ref{thm-II-8}, we obtain $\wp(p(\tau)|\tau)=\wp
(p_{r,s}^{(n)}(\tau)|\tau)$ and so $\lambda(t)=\lambda_{r,s}^{(n)}(t)$. Recalling that $t(\tau_{0})$ is a negative pole of $\lambda(t)$, we conclude from Theorem \ref{thm-II-18} that $Q_{n}(Z_{r,s}(\tau_{0});r+s\tau_{0},\tau_{0})=0$. Then by (\ref{k-il}) we finally obtain $Q_{n}(X_{0};a_{0},\tau_{0})=0$.
Therefore, we have proved that \[ W_{n+1}(X;a_{0},\tau_{0})=0\text{ \ }\Rightarrow \text{ \ }Q_{n}(X;a_{0} ,\tau_{0})=0. \] Since $\deg_{X}Q_{n}=\deg_{X}W_{n+1}={(n+1)(n+2)}/{2}$ and Theorem \ref{thm-5A}-(2) says that $W_{n+1}(X;a_{0},\tau_{0})$ has ${(n+1)(n+2)}/{2}$ distinct zeros, we conclude that \begin{equation} Q_{n}(X;a_{0},\tau_{0})=\mathfrak{q}_{n}(\tau_{0})W_{n+1}(X;a_{0},\tau_{0}) \label{ivi} \end{equation}
for any $\tau_{0}$ and $a_{0}\not \in E_{\tau_{0}}[2]$ being outside the branch loci of $\sigma_{n+1}:\bar{Y}_{n+1}(\tau_{0})\rightarrow E_{\tau_{0}}$. By $W_{n+1}(X;\sigma_{n+1},\tau)\in \mathbb{Q}[\wp(\sigma_{n+1}|\tau)$,
$\wp^{\prime}(\sigma_{n+1}|\tau)$, $g_{2}(\tau)$, $g_{3}(\tau)][X]$ and the fact that coefficients of $Q_{n}(X;a(\tau),\tau)$ are rational functions of
$e_{1}(\tau)$, $e_{2}(\tau)$, $e_{3}(\tau)$, $\wp(a(\tau)|\tau)$ and
$\wp^{\prime}(a(\tau)|\tau)$ provided $a(\tau)\not \in E_{\tau}[2]$, it follows from (\ref{ivi}) and analytic continuation that (\ref{caxi}) holds for any $\tau$ and $\sigma_{n+1}=a(\tau)\not \in \Lambda_{\tau}$. This completes the proof. \end{proof}
A direct consequence of Theorem \ref{thm-5A}-(3), Theorem \ref{q-n=z-n} and Corollary \ref{thm-II-18-1} is (note from Lemma \ref{coefficient-Qn} that $\mathfrak{q}_{n}(\tau)\not =0$ for any $\tau \in \mathbb{H}$)
\begin{theorem} \label{simple-zn}For any $n\geq1$ and $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$, there holds \[ Z_{r,s}^{(n)}(\tau)=\frac{Q_{n-1}(Z_{r,s}(\tau);r+s\tau,\tau)}{\mathfrak{q} _{n-1}(\tau)}\text{ \ for any }\tau \in \mathbb{H}. \] In particular, for any $(r,s)\in \mathbb{R}^{2}\backslash \frac{1}{2} \mathbb{Z}^{2}$, $Z_{r,s}^{(n)}(\tau)$ has only simple zeros as a holomorphic function of $\tau \in \mathbb{H}$. \end{theorem}
In general, it is too difficult to write down the explicit formula of
$Z_{r,s}^{(n)}(\tau)$ and so it seems impossible to prove the simple zero property directly. Theorem \ref{simple-zn}, which answers an open question raised in \cite{Dahmen0,LW2}, is a beautiful application of Painlev\'{e} VI equation to pre-modular forms.
Now we are in the position to prove Theorem \ref{thm-II-18 copy(1)}.
\begin{proof} [Proof of Theorem \ref{thm-II-18 copy(1)}]Clearly Theorem \ref{thm-II-18 copy(1)} follows readily from Theorems \ref{thm-II-17}, \ref{thm-II-18} and \ref{simple-zn}. \end{proof}
\section{Asymptotics of pre-modular forms}
\label{sec-asymptotics}
In this section, we study the asymptotics of $Z_{r,s}^{(n)}(\tau)$ and give the proof of Theorem \ref{weak} (1) \& (3).
\subsection{The case $\operatorname{Re}s\in(0,\frac{1}{2})$}
First we study the asymptotic behaviors of solution $\lambda _{r,s}^{(n)}(t)$ in terms of free parameters $(r,s)$ at the branch point $t=1$. By Theorem \ref{thm-II-8}-(2), we can always assume $\operatorname{Re} s\in \lbrack0,\frac{1}{2}]$ for any solution $\lambda_{r,s}^{(n)}(t)$. In this section we only consider the case $\operatorname{Re}s\in(0,\frac{1}{2})$. Since $t=t(\tau)=\frac{e_{3} (\tau)-e_{1}(\tau)}{e_{2}(\tau)-e_{1}(\tau)}$ maps the fundamental domain \begin{equation}
F_{2}:=\{ \tau \in \mathbb{H}\text{ }|\text{ }0\leq \operatorname{Re}
\tau<2,\text{ }|\tau-1/2|\geq1/2,\text{ }|\tau-3/2|>1/2\} \label{funde} \end{equation} of $\Gamma(2)$ onto $\mathbb{C}\backslash \{0,1\}$, without loss of generality we may only consider $\tau \in F_{2}$. Then $t\rightarrow1$ if $\tau \rightarrow \infty$. The first result of this section is
\begin{theorem} \label{solution-asymptotic}Fix $n\in \mathbb{N\cup \{}0\mathbb{\}}$ and $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$ such that $\operatorname{Re}s\in(0,\frac{1}{2})$. Then as $F_{2}\ni \tau \rightarrow \infty$ there holds \begin{equation} \lambda_{r,s}^{(n)}(t)=1+C^{(n)}(s)e^{2\pi ir}\left( \frac{1-t}{16}\right)
^{2s}+o\left( |1-t|^{2s}\right) , \label{iv6} \end{equation} where $t=t(\tau)\rightarrow1$ and \begin{equation} C^{(n)}(s)=\left \{ \begin{array} [c]{l} \frac{8s}{2s-1}\prod_{k=0}^{m-1}\frac{(s+k)(s+k+\frac{1}{2})} {(s-k-1)(s-k-\frac{3}{2})}\text{ if }n=2m,\\ \frac{8s^{2}}{(2s-1)(s-1)}\prod_{k=0}^{m-1}\frac{(s+k+\frac{1}{2} )(s+k+1)}{(s-k-\frac{3}{2})(s-k-2)}\text{ if }n=2m+1. \end{array} \right. \label{iv7} \end{equation} Here we set $\prod_{k=0}^{-1}\ast=1$. \end{theorem}
\begin{remark} If we use the notation $(\alpha)_{m}:=\alpha(\alpha+1)\cdots(\alpha+m-1)$, then (\ref{iv7}) can be written as \[ C^{(n)}(s)=\left \{ \begin{array} [c]{l} \frac{8s}{2s-1}\cdot \frac{(s)_{m}(s+\frac{1}{2})_{m}}{(1-s)_{m}(\frac{3} {2}-s)_{m}}\text{ if }n=2m,\\ \frac{8s^{2}}{(2s-1)(s-1)}\cdot \frac{(s+\frac{1}{2})_{m}(s+1)_{m}}{(\frac {3}{2}-s)_{m}(2-s)_{m}}\text{ if }n=2m+1. \end{array} \right. \] Theorem \ref{solution-asymptotic} for the simplest case $n=0$ was known, see e. g. \cite[Appendix A]{CKLW}, where the remaining case $s\in \{0,\frac{1} {2}\}$ was also discussed. We can compare Theorem \ref{solution-asymptotic} with Guzzetti's works \cite{Guzzetti,Guzzetti1} for asymptotics of generic solutions $\lambda(t)$ to Painlev\'{e} VI equation with generic parameters: \[
\lambda(t)=1+C(1-t)^{2s}+o\left( |1-t|^{2s}\right) \] as $t\rightarrow1$ along some special paths. See \cite{Guzzetti,Guzzetti1} for his precise statements. It does not seem that he wrote down the explicit formula for the coefficient $C$ in terms of the monodromy data $(r,s)$ for our concerning case PVI$(\frac12(n+\frac12)^2,\frac{-1}{8},\frac18,\frac38)$. \end{remark}
We prove Theorem \ref{solution-asymptotic} by applying Theorems \ref{thm-II-17} and \ref{thm-II-18}. To this goal, we have to study the asymptotic behaviors of $R_{n}(X)-Q_{n-2}(X)Q_{n}(X)$ and $Q_{n}(X)$ with $X=Z_{r,s}(\tau)$. To do this, we recall the following $q=e^{2\pi i\tau}$ expansion for
$\wp(z|\tau)$ (see e. g. \cite[p.46]{Lang}): For $|q|<|e^{2\pi iz}|<|q|^{-1}$, \begin{align}
& \wp(z|\tau)\label{iv-8}\\ = & -\frac{\pi^{2}}{3}-4\pi^{2}\left[ \frac{e^{2\pi iz}}{(1-e^{2\pi iz})^{2}}+\sum_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}\left( e^{2\pi inz}+e^{-2\pi inz}-2\right) \right] .\nonumber \end{align} Now we put \begin{equation} z=a(\tau)=r+s\tau \text{ \ and denote }x=e^{2\pi i(r+s\tau)}. \label{iv-9} \end{equation} Since $\operatorname{Re}s\in(0,\frac{1}{2})$ and $\tau \in F_{2}$, we have $q\rightarrow0$ as $\tau \rightarrow \infty$ and \[
|q|^{-1}>|x|=e^{-2\pi(\operatorname{Im}r+\operatorname{Im}s\operatorname{Re}
\tau)}|q|^{\operatorname{Re}s}>|q|^{\frac{1}{2}}\text{\ \ and }|q|^{\frac
{1}{2}}=o(|x|) \]
for $|\tau|$ large. Hence (\ref{iv-8})-(\ref{iv-9}) give \[
\wp(a(\tau)|\tau)=-\tfrac{\pi^{2}}{3}-4\pi^{2}x+o(|x|), \] \[
\wp^{\prime}(a(\tau)|\tau)=-8\pi^{3}ix+o(|x|). \] Letting $z=\frac{1}{2},\frac{\tau}{2},\frac{1+\tau}{2}$ in (\ref{iv-8}) respectively, we easily obtain \begin{equation}
e_{1}(\tau)=\tfrac{2\pi^{2}}{3}+16\pi^{2}q+O(|q|^{2}), \label{iv-50} \end{equation} \[
e_{2}(\tau)=-\tfrac{\pi^{2}}{3}-8\pi^{2}q^{\frac{1}{2}}-8\pi^{2}q+O(|q|^{\frac {3}{2}}), \] \begin{equation}
e_{3}(\tau)=-\tfrac{\pi^{2}}{3}+8\pi^{2}q^{\frac{1}{2}}-8\pi^{2}q+O(|q|^{\frac {3}{2}}), \label{iv-51} \end{equation} and so \begin{equation} t-1=\frac{e_{3}(\tau)-e_{2}(\tau)}{e_{2}(\tau)-e_{1}(\tau)}=-16q^{\frac{1}{2}
}+O(|q|)=o(|x|), \label{iv-10} \end{equation} \begin{equation}
e_{2}(\tau)-e_{1}(\tau)=-\pi^{2}+o(|x|). \label{iv-10-1} \end{equation} In fact, it is easy to obtain the well-known $q$-expansion of $t$: \begin{align} t= & 1-16q^{\frac{1}{2}}+128q-704q^{\frac{3}{2}}+3072q^{2}-11488q^{\frac {5}{2}}\label{t-expansion}\\ & +38400q^{3}-117632q^{\frac{7}{2}}+335872q^{4}+\cdots.\nonumber \end{align} On the other hand, it was proved in \cite[(5.3)]{CKLW} that \[
Z_{r,s}(\tau)=\pi i(2s-1)-2\pi ix+o(|x|). \] These, together with (\ref{II-511})-(\ref{II-512}), easily imply \begin{equation}
G_{0}(Z_{r,s}(\tau)|\tau)=8\pi ix+o(|x|), \label{iv-17} \end{equation} \begin{equation}
Q_{0}(Z_{r,s}(\tau)|\tau)=\pi i(2s-1)-2\pi ix+o(|x|), \label{iv-18} \end{equation} \begin{align}
& R_{0}(Z_{r,s}(\tau)|\tau)-tQ_{0}(Z_{r,s}(\tau)|\tau) \nonumber\\ =&R_{0}(Z_{r,s}
(\tau)|\tau)-Q_{0}(Z_{r,s}(\tau)|\tau)+o(|x|)
=8\pi isx+o(|x|). \label{iv-19} \end{align} Starting from these formulae, we want to prove
\begin{lemma} \label{Q-asymp}Given any $n\geq0$, there holds \[
Q_{n}(Z_{r,s}(\tau)|\tau)=\check{Q}_{n}(s)+o(1)\text{ \ as }F_{2}\ni \tau \rightarrow \infty, \] where $\check{Q}_{n}(s)$ is a polynomial of $s$ defined as follows:
\begin{itemize} \item[(1)] if $n=2m$ with $m\in \mathbb{N}\cup \{0\}$, \begin{align} \check{Q}_{n}(s):= & \left[ \pi i(2s-1)\right] ^{m+1}\prod_{k=0} ^{m-1}\left[ (s+k)\left( s+k+\tfrac{1}{2}\right) \right. \nonumber \\ & \left. (s-k-1)\left( s-k-\tfrac{3}{2}\right) \right] ^{m-k}; \label{Q-expansion} \end{align}
\item[(2)] if $n=2m+1$ with $m\in \mathbb{N}\cup \{0\}$, \begin{align} \check{Q}_{n}(s):= & \left[ \pi i(2s-1)\right] ^{m+1}s^{m+1} (s-1)^{m+1}\prod_{k=0}^{m-1}\left[ \left( s+k+\tfrac{1}{2}\right) \right. \nonumber \\ & \left. (s+k+1)\left( s-k-\tfrac{3}{2}\right) (s-k-2)\right] ^{m-k}. \label{Q-expansion2} \end{align}
\end{itemize} \end{lemma}
\begin{proof}
In this proof we write $Q_{n}=Q_{n}(Z_{r,s}(\tau)|\tau)$, $G_{n}=G_{n}
(Z_{r,s}(\tau)|\tau)$ and $R_{n}=R_{n}(Z_{r,s}(\tau)|\tau)$ for convenience. We want to prove \begin{equation}
Q_{n}=\check{Q}_{n}+o(1),\text{ \ \ }G_{n}=\check{G}_{n}x+o(|x|), \label{iv-11} \end{equation} \begin{equation}
R_{n}-tQ_{n-2}Q_{n}=R_{n}-Q_{n-2}Q_{n}+o(|x|)=\check{R}_{n}x+o(|x|) \label{iv-12} \end{equation} by induction, where $\check{G}_{n}$ and $\check{R}_{n}$ are polynomials of $s$ defined as follows: if $n=2m$ with $m\in \mathbb{N\cup \{}0\mathbb{\}}$, \begin{align} \check{G}_{n}:= & 8\pi i\left[ \pi i(2s-1)\right] ^{3m}\prod_{k=0} ^{m-1}\left[ (s+k)^{3(m-k)}\left( s+k+\tfrac{1}{2}\right) ^{3(m-k)-1}\right. \nonumber \\ & \left. (s-k-1)^{3(m-k)-2}\left( s-k-\tfrac{3}{2}\right) ^{3(m-k-1)} \right] , \label{G-expansion} \end{align} \begin{align} \check{R}_{n}:= & 8\pi i\left[ \pi i(2s-1)\right] ^{2m}s^{2m+1}\left( s+\tfrac{1}{2}\right) ^{2m}\nonumber \\ & \prod_{k=1}^{m-1}\left[ (s+k)\left( s+k+\tfrac{1}{2}\right) (s-k)\left( s-k-\tfrac{1}{2}\right) \right] ^{2(m-k)}; \label{R-expansion} \end{align} if $n=2m+1$ with $m\in \mathbb{N}\cup \{0\}$, \begin{align} \check{G}_{n}:= & 8\pi i\left[ \pi i(2s-1)\right] ^{3m+1}s^{3m+2} (s-1)^{3m}\prod_{k=0}^{m-1}\left[ \left( s+k+\tfrac{1}{2}\right) ^{3(m-k)}\right. \nonumber \\ & \left. (s+k+1)^{3(m-k)-1}\left( s-k-\tfrac{3}{2}\right) ^{3(m-k)-2} (s-k-2)^{3(m-k-1)}\right] , \label{G-expansion2} \end{align} \begin{align} \check{R}_{n}:= & 8\pi i\left[ \pi i(2s-1)\right] ^{2m}s^{2m+3}\left( s+\tfrac{1}{2}\right) ^{2m}(s+1)^{2m}(s-1)^{2m}\nonumber \\ \prod_{k=1}^{m-1} & \left[ \left( s+k+\tfrac{1}{2}\right) (s+k+1)\left( s-k-\tfrac{1}{2}\right) (s-k-1)\right] ^{2(m-k)}. \label{R-expan2} \end{align} Here we set $\prod_{k=i}^{j}\ast=1$ for $j<i$ as before. Remark by the definitions of $\check{Q}_{n}$, $\check{G}_{n}$ and $\check{R}_{n}$, we have $\check{Q}_{n}\check{G}_{n}\check{R}_{n}\not =0$ because of $\operatorname{Re}s\in(0,\frac{1}{2})$. Furthermore, a direct computation shows that (set $\check{Q}_{-2}=\check{Q}_{-1}=1$) \begin{equation} \check{R}_{n}\check{Q}_{n-1}=s\check{G}_{n}, \label{iv-13} \end{equation} \begin{equation} \check{G}_{n}\check{Q}_{n-3}^{2}=\left( s+\tfrac{n-1}{2}\right) ^{2}\check {G}_{n-1}\check{Q}_{n-2}\check{Q}_{n-1}, \label{iv-14} \end{equation} \begin{equation} \check{Q}_{n}\check{Q}_{n-3}=\left( s+\tfrac{n-1}{2}\right) \left( s-\tfrac{n+1}{2}\right) \check{Q}_{n-2}\check{Q}_{n-1}, \label{iv-15} \end{equation} \begin{equation} \check{R}_{n}\check{Q}_{n-3}^{2}=s\left( s+\tfrac{n-1}{2}\right) ^{2} \check{G}_{n-1}\check{Q}_{n-2}, \label{iv-15-1} \end{equation} hold for any $n\geq1$. We remark that these four identities are the key points of this proof. By (\ref{iv-17})-(\ref{iv-19}) we see that (\ref{iv-11} )-(\ref{iv-12}) hold for $n=0$.
\textbf{Step 1}. We prove (\ref{iv-11})-(\ref{iv-12}) for $n=1$.
Recalling (\ref{II-515}) that $\varphi_{0}=R_{0}$, we deduce from (\ref{iv-18})-(\ref{iv-19}) that \[
H(\varphi_{0};Q_{0})=-64\pi^{3}i(2s-1)s^{2}x^{2}+o(|x|^{2}), \] \[
H^{\prime}(\varphi_{0};Q_{0})=-16\pi^{2}(2s-1)sx+o(|x|). \] Together with (\ref{iv-17}) and (\ref{II-516})-(\ref{II-518}), we easily obtain \[
G_{1}=\frac{H(\varphi_{0};Q_{0})}{G_{0}}=-8\pi^{2}(2s-1)s^{2}x+o(|x|)=\check
{G}_{1}x+o(|x|), \] \[ Q_{1}=\frac{G_{1}-\frac{1}{2}H^{\prime}(\varphi_{0};Q_{0})}{G_{0}}=\pi i(2s-1)s(s-1)+o(1)=\check{Q}_{1}+o(1), \] \[
R_{1}-Q_{1}=\frac{(R_{0}-Q_{0})Q_{1}+G_{1}}{Q_{0}}=8\pi is^{3}x+o(|x|)=\check
{R}_{1}x+o(|x|), \] \[
R_{1}-tQ_{1}=R_{1}-Q_{1}+o(|x|)=\check{R}_{1}x+o(|x|). \] Therefore, (\ref{iv-11})-(\ref{iv-12}) hold for $n=1$.
\textbf{Step 2}. Assume that (\ref{iv-11})-(\ref{iv-12}) hold for any $\ell \leq n-1$ where $n\geq2$, we prove (\ref{iv-11})-(\ref{iv-12}) for $n$.
By our assumption that (\ref{iv-11})-(\ref{iv-12}) hold for $n-1$ and (\ref{iv-13}), we have \begin{align*}
(R_{n-1}-Q_{n-3}Q_{n-1})Q_{n-2} & =\left( \check{R}_{n-1}x+o(|x|)\right) \left( \check{Q}_{n-2}+o(1)\right) \\
& =s\check{G}_{n-1}x+o(|x|). \end{align*} Recalling (\ref{II-515}) that $\varphi_{n-1}=R_{n-1}Q_{n-2}+\frac{n-1} {2}G_{n-1}$, we obtain{\allowdisplaybreaks \begin{align*} & \varphi_{n-1}-tQ_{n-3}Q_{n-2}Q_{n-1}\\
= & \varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1}+o(|x|)\\
= & (R_{n-1}-Q_{n-3}Q_{n-1})Q_{n-2}+\tfrac{n-1}{2}G_{n-1}+o(|x|)\\
= & \left( s+\tfrac{n-1}{2}\right) \check{G}_{n-1}x+o(|x|), \end{align*} }and so \[ \varphi_{n-1}=Q_{n-3}Q_{n-2}Q_{n-1}+o(1)=\check{Q}_{n-3}\check{Q}_{n-2} \check{Q}_{n-1}+o(1). \] Applying (\ref{II-516}) and (\ref{iv-14}) leads to{\allowdisplaybreaks \begin{align*} G_{n} & =\frac{H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})}{G_{n-1}Q_{n-3}^{3}}\\ & =\frac{[\check{Q}_{n-3}\check{Q}_{n-2}\check{Q}_{n-1}+o(1)][(s+\frac
{n-1}{2})\check{G}_{n-1}x+o(|x|)]^{2}}{[\check{G}_{n-1}x+o(|x|)][\check {Q}_{n-3}+o(1)]^{3}}\\ & =\frac{(s+\frac{n-1}{2})^{2}\check{G}_{n-1}\check{Q}_{n-2}\check{Q}_{n-1}
}{\check{Q}_{n-3}^{2}}x+o(|x|)=\check{G}_{n}x+o(|x|). \end{align*} }Furthermore, \begin{align*} H^{\prime} & (\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})=2\varphi_{n-1}
(\varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1})+o(|x|)\\ & =2\left( s+\tfrac{n-1}{2}\right) \check{G}_{n-1}\check{Q}_{n-3}\check
{Q}_{n-2}\check{Q}_{n-1}x+o(|x|). \end{align*} Then we derive from (\ref{II-517}) and (\ref{iv-15}) that{\allowdisplaybreaks \begin{align*} Q_{n}= & \frac{Q_{n-3}G_{n}}{G_{n-1}}-\frac{n}{2}\frac{H^{\prime} (\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})}{G_{n-1}Q_{n-3}^{2}}\\ = & \frac{(\check{Q}_{n-3}+o(1))\left( \frac{(s+\frac{n-1}{2})^{2}\check
{G}_{n-1}\check{Q}_{n-2}\check{Q}_{n-1}}{\check{Q}_{n-3}^{2}}x+o(|x|)\right)
}{\check{G}_{n-1}x+o(|x|)}\\ & -n\frac{\left( s+\frac{n-1}{2}\right) \check{G}_{n-1}\check{Q}
_{n-3}\check{Q}_{n-2}\check{Q}_{n-1}x+o(|x|)}{(\check{G}_{n-1}x+o(|x|))(\check {Q}_{n-3}+o(1))^{2}}\\ = & \left( s+\tfrac{n-1}{2}\right) \left( s-\tfrac{n+1}{2}\right) \frac{\check{Q}_{n-2}\check{Q}_{n-1}}{\check{Q}_{n-3}}+o(1)=\check{Q} _{n}+o(1). \end{align*} }So (\ref{iv-11}) holds for $n$. Finally,{\allowdisplaybreaks \begin{align*} & \left( \varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1}\right) Q_{n}+\tfrac{n+1} {2}Q_{n-3}G_{n}\\
= & \left[ \left( s+\tfrac{n-1}{2}\right) \check{G}_{n-1}x+o(|x|)\right] \\ & \times \left[ \left( s+\tfrac{n-1}{2}\right) \left( s-\tfrac{n+1} {2}\right) \frac{\check{Q}_{n-2}\check{Q}_{n-1}}{\check{Q}_{n-3}}+o(1)\right] \\ & +\tfrac{n+1}{2}\left[ \check{Q}_{n-3}+o(1)\right] \left[ \frac {(s+\frac{n-1}{2})^{2}\check{G}_{n-1}\check{Q}_{n-2}\check{Q}_{n-1}}{\check
{Q}_{n-3}^{2}}x+o(|x|)\right] \\ = & s\left( s+\tfrac{n-1}{2}\right) ^{2}\frac{\check{G}_{n-1}\check
{Q}_{n-2}\check{Q}_{n-1}}{\check{Q}_{n-3}}x+o(|x|). \end{align*} }This, together with (\ref{II-518}) and (\ref{iv-15-1}), gives{\allowdisplaybreaks \begin{align*}
R_{n}-tQ_{n-2}Q_{n} & =R_{n}-Q_{n-2}Q_{n}+o(|x|)\\ & =\frac{\left( \varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1}\right) Q_{n}+\frac
{n+1}{2}Q_{n-3}G_{n}}{Q_{n-3}Q_{n-1}}+o(|x|)\\ & =s\left( s+\tfrac{n-1}{2}\right) ^{2}\frac{\check{G}_{n-1}\check{Q}_{n-2}
}{\check{Q}_{n-3}^{2}}x+o(|x|)\\
& =\check{R}_{n}x+o(|x|), \end{align*} }namely (\ref{iv-12}) holds for $n$. The proof is complete. \end{proof}
As an application of Lemma \ref{Q-asymp}, we prove Theorem \ref{solution-asymptotic}.
\begin{proof} [Proof of Theorem \ref{solution-asymptotic}]Recall from (\ref{II-535}) in Theorem \ref{thm-II-18} that \[
\lambda_{r,s}^{(n)}(t)-1=\frac{R_{n}(Z_{r,s}(\tau)|\tau)-Q_{n-2}(Z_{r,s}
(\tau)|\tau)Q_{n}(Z_{r,s}(\tau)|\tau)}{Q_{n-2}(Z_{r,s}(\tau)|\tau
)Q_{n}(Z_{r,s}(\tau)|\tau)}. \] Furthermore, (\ref{iv-9}) and (\ref{iv-10}) imply \[ x=e^{2\pi ir}q^{s}=e^{2\pi ir}\left( \frac{1-t}{16}\right) ^{2s}
+o(|1-t|^{2s})\text{ as }\tau \rightarrow \infty. \] Then Theorem \ref{solution-asymptotic} follows readily from (\ref{iv-11} )-(\ref{iv-12}) and the definitions (\ref{Q-expansion}), (\ref{Q-expansion2}), (\ref{R-expansion}), (\ref{R-expan2}) of $\check{Q}_{n}$, $\check{R}_{n}$. \end{proof}
To obtain the asymptotics of $Z_{r,s}^{(n)}(\tau)$, we need to calculate the coefficient $\mathfrak{q}_n(\tau)$ of the leading term $X^{\frac{(n+1)(n+2)}{2}}$ of $Q_{n}(X)$.
\begin{lemma} \label{coefficient-Qn}Denote the coefficient of the leading term $X^{\frac{(n+1)(n+2)}{2}}$ of $Q_{n}(X)$ by $\mathfrak{q}_{n}=\mathfrak{q} _{n}(\tau)$. Then for all $n\geq0$, there holds \begin{equation} \mathfrak{q}_{n}(\tau)=\left \{ \begin{array} [c]{l} 2^{-\frac{n(n+2)}{2}}(e_{2}(\tau)-e_{1}(\tau))^{-\frac{n(n+2)}{4}}\text{ \ if }n\text{ even,}\\ 2^{-\frac{(n+1)^{2}}{2}}(e_{2}(\tau)-e_{1}(\tau))^{-\frac{(n+1)^{2}}{4}}\text{ \ if }n\text{ odd.} \end{array} \right. \label{iv-1} \end{equation}
\end{lemma}
\begin{proof} Denote the coefficient of the leading term $X^{\frac{3n(n+1)}{2}}$ of $G_{n}(X)$ by $\mathfrak{g}_{n}$, and the coefficient of the leading term $X^{n(n+1)+1}$ of $R_{n}(X)$ by $\mathfrak{r}_{n}$. We prove (\ref{iv-1}) and the following two formulae \begin{equation} \mathfrak{g}_{n}=\left \{ \begin{array} [c]{l} 2^{-\frac{3n^{2}}{2}}(e_{2}(\tau)-e_{1}(\tau))^{-\frac{3n^{2}}{4}-1}
\wp^{\prime}(a(\tau)|\tau)\text{ \ if }n\text{ even,}\\ 2^{-\frac{3n^{2}+1}{2}}(e_{2}(\tau)-e_{1}(\tau))^{-\frac{3n^{2}+1}{4}-1}
\wp^{\prime}(a(\tau)|\tau)\text{ if }n\text{ odd,} \end{array} \right. \label{iv-2} \end{equation} \begin{equation} \mathfrak{r}_{n}=\left \{ \begin{array} [c]{l} 2^{-n^{2}}(e_{2}(\tau)-e_{1}(\tau))^{-\frac{n^{2}}{2}-1}\left( \wp
(a(\tau)|\tau)-e_{1}(\tau)\right) \text{ \ if }n\text{ even,}\\ 2^{-n^{2}-1}(e_{2}(\tau)-e_{1}(\tau))^{-\frac{n^{2}+1}{2}-1}\left( \wp
(a(\tau)|\tau)-e_{1}(\tau)\right) \text{ if }n\text{ odd} \end{array} \right. \label{iv-3} \end{equation} by induction. Recall $Q_{-1}(X)\equiv1$. Then (\ref{iv-1}) holds for $Q_{-1} $.
\textbf{Step 1}. We consider $n=0,1$. This follows directly from (\ref{II-511})-(\ref{II-512}) and Theorem \ref{thm-expression-1}.
\textbf{Step 2.} Assume that (\ref{iv-1})-(\ref{iv-3}) hold for all $0\leq m\leq n-1$ where $n\geq2$. We prove that (\ref{iv-1})-(\ref{iv-3}) also hold for $n$. Without loss of generality, we may assume that $n$ is even. The case that $n$ is odd can be proved similarly and we omit the details here.
Recalling property (4-$(n-1)$), we have \begin{align*} \deg R_{n-1}Q_{n-2} & =\deg(R_{n-1}-tQ_{n-3}Q_{n-1})Q_{n-2}\\ & =\deg(R_{n-1}-Q_{n-3}Q_{n-1})Q_{n-2}\\ & =\tfrac{3n(n-1)}{2}+1>\deg G_{n-1}. \end{align*} Then the leading term of $\varphi_{n-1}=R_{n-1}Q_{n-2}+\frac{n-1}{2}G_{n-1}$ comes from that of $R_{n-1}Q_{n-2}$, so the coefficient of the leading term of $\varphi_{n-1}$ is (using (\ref{iv-1})-(\ref{iv-3}) and recalling $n$ even) \begin{align} \mathfrak{r}_{n-1}\mathfrak{q}_{n-2} & =\frac{\wp(a)-e_{1}}{2^{(n-1)^{2} +1}(e_{2}-e_{1})^{\frac{(n-1)^{2}+1}{2}+1}}\cdot \frac{1}{2^{\frac{n(n-2)}{2} }(e_{2}-e_{1})^{\frac{n(n-2)}{4}}}\nonumber \\ & =\frac{\wp(a)-e_{1}}{2^{\frac{3n^{2}-6n}{2}+2}(e_{2}-e_{1})^{\frac {3n^{2}-6n}{4}+2}}. \label{iv-5} \end{align} Similarly, the coefficient of the leading term of $\varphi_{n-1} -Q_{n-3}Q_{n-2}Q_{n-1}=(R_{n-1}-Q_{n-3}Q_{n-1})Q_{n-2}+\frac{n-1}{2}G_{n-1}$ is \[ (\mathfrak{r}_{n-1}-\mathfrak{q}_{n-3}\mathfrak{q}_{n-1})\mathfrak{q} _{n-2}=\frac{\wp(a)-e_{2}}{2^{\frac{3n^{2}-6n}{2}+2}(e_{2}-e_{1} )^{\frac{3n^{2}-6n}{4}+2}}, \] and the coefficient of the leading term of $\varphi_{n-1}-tQ_{n-3} Q_{n-2}Q_{n-1}$ is (using $t=\frac{e_{3}-e_{1}}{e_{2}-e_{1}}$) \[ (\mathfrak{r}_{n-1}-t\mathfrak{q}_{n-3}\mathfrak{q}_{n-1})\mathfrak{q} _{n-2}=\frac{\wp(a)-e_{3}}{2^{\frac{3n^{2}-6n}{2}+2}(e_{2}-e_{1} )^{\frac{3n^{2}-6n}{4}+2}}. \] Consequently, we see from (\ref{II-510}) and $(\wp^{\prime})^{2}=4(\wp -e_{1})(\wp-e_{2})(\wp-e_{3})$ that the coefficient $\mathfrak{h}_{n-1}$ of the leading term of $H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})$ is \[ \mathfrak{h}_{n-1}=\frac{\wp^{\prime}(a)^{2}}{2^{\frac{9(n^{2}-2n)}{2} +8}(e_{2}-e_{1})^{\frac{9(n^{2}-2n)}{4}+6}}. \] Therefore, (\ref{II-516}) leads to \[ \mathfrak{g}_{n}=\frac{\mathfrak{h}_{n-1}}{\mathfrak{g}_{n-1}\mathfrak{q} _{n-3}^{3}}=\frac{\wp^{\prime}(a)}{2^{\frac{3n^{2}}{2}}(e_{2}-e_{1} )^{\frac{3n^{2}}{4}+1}}, \] which proves (\ref{iv-2}) for $n$. Recalling from (\ref{II-530-2} )-(\ref{II-530-3}) that \[ \deg H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})>\deg G_{n-1}H^{\prime} (\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1}), \] we see from (\ref{II-517}) that{ \[ \mathfrak{q}_{n}=\frac{\mathfrak{h}_{n-1}}{\mathfrak{g}_{n-1}^{2} \mathfrak{q}_{n-3}^{2}}=\frac{1}{2^{\frac{n(n+2)}{2}}(e_{2}-e_{1} )^{\frac{n(n+2)}{4}}}, \] which proves (\ref{iv-1}) for }$n$. Finally, we see from \[ \deg \varphi_{n-1}Q_{n}=2n^{2}+2>\deg Q_{n-3}G_{n} \] and (\ref{II-518}), (\ref{iv-5}) that \[ \mathfrak{r}_{n}=\frac{\mathfrak{r}_{n-1}\mathfrak{q}_{n-2}\cdot \mathfrak{q}_{n}}{\mathfrak{q}_{n-3}\mathfrak{q}_{n-1}}=\frac{\wp(a)-e_{1} }{2^{n^{2}}(e_{2}-e_{1})^{\frac{n^{2}}{2}+1}}, \] which proves (\ref{iv-3}) for $n$. The proof is complete. \end{proof}
Now we can prove Theorem \ref{weak}-(1) as stated in the following theorem. We set $\prod_{k=0}^{-1}\ast=1$ for convenience.
\begin{theorem}[=Theorem \ref{weak}-(1)] \label{asymp-Zn copy(1)}Given any $n\geq1$ and $(r,s)\in \mathbb{C}^{2}\backslash \frac{1}{2}\mathbb{Z}^{2}$ such that $\operatorname{Re}s\in(0,\frac{1}{2})$, there holds \[ Z_{r,s}^{(n)}(\tau)=\check{Z}^{(n)}(s)+o(1)\text{ as }F_{2}\ni \tau \rightarrow \infty, \] where $\check{Z}^{(n)}(s)$ is a polynomial of $s$ given as follows:
\begin{itemize} \item[(1)] if $n=2m+1$ with $m\in \mathbb{N}\cup \{0\}$, \begin{align*} \check{Z}^{(n)}(s) = & (2\pi)^{2m(m+1)}\left[ \pi i(2s-1)\right] ^{m+1}\prod_{k=0}^{m-1}\left[ (s+k)\left( s+k+\tfrac{1}{2}\right) \right. \\ & \left. (s-k-1)\left( s-k-\tfrac{3}{2}\right) \right] ^{m-k}; \end{align*}
\item[(2)] if $n=2m$ with $m\in \mathbb{N}$, \begin{align*} \check{Z}^{(n)}(s) = & (-1)^{m^{2}}(2\pi)^{2m^{2}}\left[ \pi i(2s-1)\right] ^{m}s^{m}(s-1)^{m}\prod_{k=0}^{m-2}\left[ \left( s+k+\tfrac{1}{2}\right) \right. \\ & \left. (s+k+1)\left( s-k-\tfrac{3}{2}\right) (s-k-2)\right] ^{m-1-k}. \end{align*}
\end{itemize}
\noindent In particular, \begin{equation} \lim_{F_{2}\ni \tau \rightarrow \infty}Z_{r,s}^{(n)}(\tau)\not =0\text{ as long as }\operatorname{Re}s\in (0,1/2) \cup(1/2,1). \label{n1on-zero} \end{equation}
\end{theorem}
\begin{proof} Since $e_{2}(\tau)-e_{1}(\tau)=-\pi^{2}+o(1)$ as $F_{2}\ni \tau \rightarrow \infty$, Theorem \ref{asymp-Zn copy(1)} is a direct consequence of Lemmas \ref{Q-asymp}-\ref{coefficient-Qn}, Theorem \ref{simple-zn} and (\ref{iv-20}). \end{proof}
\subsection{The case $s=0$}
In this section, we give the proof of Theorem \ref{weak}-(3). When $F_{2}\ni \tau \rightarrow \infty$, we use notation \[
f(\tau)\sim q^{a}\text{ to denote }f(\tau)=Cq^{a}+o(|q|^{a}) \] with some constant $C\not =0$.
\begin{proof}[Proof of Theorem \ref{weak}-(3)] By (\ref{iv-39-1}) it suffices for us to prove $\tilde{a}_{n}=a_{n}$ for all $n$, which holds true already for $n\leq4$. For general $n$, Theorem \ref{modular-zero} shows that we can determine $\tilde{a}_{n}$ by computing the asymptotics of $Z_{\frac{1}{4},0}^{(n)}(\tau)$ as $F_{2}\ni \tau \rightarrow \infty$. Therefore in this proof, we always consider $(r,s)=(\frac{1}{4},0)$. We also write $Q_{n}=Q_{n}(Z_{\frac{1}{4},0}
(\tau)|\tau)$, $G_{n}=G_{n}(Z_{\frac{1}{4},0}(\tau)|\tau)$, $R_{n}
=R_{n}(Z_{\frac{1}{4},0}(\tau)|\tau)$, $\varphi_{n}=\varphi_{n}(Z_{\frac{1}
{4},0}(\tau)|\tau)$, $\lambda^{(n)}=\lambda_{\frac{1}{4},0}^{(n)}(t(\tau))$ and $\mu^{(n)}=\mu_{\frac{1}{4},0}^{(n)}(t(\tau))$ for convenience. We want to prove for any $n\geq0$ that \begin{equation} G_{n}\sim q^{\frac{n(n+1)}{2}-a_{n}},\text{ }\varphi_{n}\sim q^{\frac {n(n+1)}{2}-a_{n}}\text{ and }Q_{n}\sim q^{a_{n+1}} \label{iv-45} \end{equation} as $F_{2}\ni \tau \rightarrow \infty$, where \begin{equation} a_{2k}=a_{2k+1}=k(k+1)/2\text{ \ for all }k\in \mathbb{N\cup \{}0\mathbb{\}}. \label{iv-45-1} \end{equation} Once (\ref{iv-45}) is proved, we immediately obtain from Theorem \ref{simple-zn} that \[ Z_{\frac{1}{4},0}^{(n)}(\tau)=\frac{Q_{n-1}}{\mathfrak{q}_{n-1}}\sim q^{a_{n} }\text{ \ for all }n\geq1 \] and so $\tilde{a}_{n}=a_{n}$ for all $n$.
Now we turn to prove (\ref{iv-45}) by induction.
\textbf{Step 1.} We prove (\ref{iv-45}) for $n=0$.
Recall from (\ref{iv-35-2})-(\ref{iv-35}) that \[
\wp^{\prime}\left( \tfrac{1}{4}|\tau \right) =-4\pi^{3}+16\pi^{3}q+16\pi
^{3}q^{2}+O(|q|^{3}), \] \[
\wp \left( \tfrac{1}{4}|\tau \right) =\frac{5}{3}\pi^{2}+8\pi^{2}q+40\pi
^{2}q^{2}+O(|q|^{3}), \] \[
Q_{0}=Z_{\frac{1}{4},0}(\tau)=\pi+4\pi q+4\pi q^{2}+O(|q|^{3}). \] Together with (\ref{iv-50})-(\ref{iv-51}), (\ref{II-46})-(\ref{II-512}) and (\ref{II-515}), a straightforward computation leads to \[
G_{0}=4\pi-32\pi q^{\frac{1}{2}}+O(|q|), \] \[
\varphi_{0}=R_{0}=\pi-8\pi q^{\frac{1}{2}}+O(|q|), \] \begin{equation}
\lambda_{0}=1-8q^{\frac{1}{2}}+O(|q|). \label{iv-53} \end{equation} In particular, (\ref{iv-45}) holds for $n=0$.
\textbf{Step 2.} We claim that for any $n\geq0$, there hold \begin{equation} R_{n}Q_{n-1}=\frac{1}{4}G_{n}, \label{iv-56} \end{equation} \begin{align} & Q_{n-2}Q_{n-1}Q_{n}\label{iv-65-1}\\ = & (-1)^{n}\frac{2n+1}{4}G_{n}\left( 1+8q^{\frac{1}{2}}+32q+96q^{\frac
{3}{2}}+O(|q|^{2})\right) ,\nonumber \end{align} \begin{equation} \varphi_{n}=R_{n}Q_{n-1}+\frac{n}{2}G_{n}=\frac{2n+1}{4}G_{n}. \label{iv-57} \end{equation} Remark that (\ref{iv-56})-(\ref{iv-57}) are the key points of this proof.
Indeed, recall Theorem \ref{r=1/4} that \begin{equation} \lambda^{(n)}=\frac{(-1)^{n}}{2n+1}t^{\frac{1}{2}},\ \mu^{(n)}=\frac {1}{4\lambda^{(n)}}=(-1)^{n}\frac{2n+1}{4}t^{-\frac{1}{2}}. \label{iv-54} \end{equation} It follows from (\ref{t-expansion}) that \begin{equation} \mu^{(n)}=(-1)^{n}\frac{2n+1}{4}\left( 1+8q^{\frac{1}{2}}+32q+96q^{\frac
{3}{2}}+O(|q|^{2})\right) . \label{iv-55} \end{equation} Since Theorem \ref{thm-II-18} gives \[ \lambda^{(n)}=\frac{R_{n}}{Q_{n-2}Q_{n}},\text{ }\mu^{(n)}=\frac {Q_{n-2}Q_{n-1}Q_{n}}{G_{n}}, \] it follows from (\ref{iv-54}) and (\ref{iv-55}) that (\ref{iv-56}) and (\ref{iv-65-1}) hold. Finally, (\ref{II-515}) and (\ref{iv-56}) yield (\ref{iv-57}).
\textbf{Step 3.} Assume that (\ref{iv-45}) holds for any $k\leq n-1$ for some $n\geq1$, we prove (\ref{iv-45}) for $n$. We consider two cases separately.
\textbf{Case 1}. $n=2m$ for some $m\in \mathbb{N}$.
Then (\ref{t-expansion}) and (\ref{iv-65-1})-(\ref{iv-57}) with $n-1$ imply \[ \varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1}=\frac{2n-1}{2}G_{n-1}\left(
1+O(|q|^{\frac{1}{2}})\right) , \] \[ \varphi_{n-1}-tQ_{n-3}Q_{n-2}Q_{n-1}=\frac{2n-1}{2}G_{n-1}\left(
1+O(|q|^{\frac{1}{2}})\right) . \] It follows from (\ref{II-510}) that \[ H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})=\frac{(2n-1)^{3}}{16}G_{n-1}^{3}\left(
1+O(|q|^{\frac{1}{2}})\right) , \] \[ H^{\prime}(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})=\frac{(2n-1)^{2}}{2}
G_{n-1}^{2}\left( 1+O(|q|^{\frac{1}{2}})\right) . \] Therefore, we derive from (\ref{II-516})-(\ref{II-517}) and (\ref{iv-45} )-(\ref{iv-45-1}) that{\allowdisplaybreaks \begin{align*} G_{n} & =\frac{H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})}{G_{n-1}Q_{n-3}^{3}
}=\frac{\frac{(2n-1)^{3}}{16}G_{n-1}^{2}( 1+O(|q|^{\frac{1}{2}})) }{Q_{n-3}^{3}}\\ & \sim \frac{q^{n(n-1)-2a_{n-1}}}{q^{3a_{n-2}}}=q^{\frac{3m^{2}+m}{2} }=q^{\frac{n(n+1)}{2}-a_{n}}, \end{align*} }{\allowdisplaybreaks \begin{align*} Q_{n} & =\frac{Q_{n-3}^{3}G_{n}-\frac{n}{2}H^{\prime}(\varphi_{n-1} ;Q_{n-3}Q_{n-2}Q_{n-1})}{G_{n-1}Q_{n-3}^{2}}\\
& =\frac{\frac{(2n-1)^{3}}{16}G_{n-1}^{2}(1+O(|q|^{\frac{1}{2}
})) -\frac{n(2n-1)^{2}}{4}G_{n-1}^{2}( 1+O(|q|^{\frac{1}{2} })) }{G_{n-1}Q_{n-3}^{2}}\\
& =-\frac{(2n+1)(2n-1)^{2}G_{n-1}( 1+O(|q|^{\frac{1}{2}})) }{16Q_{n-3}^{2}}\\ & \sim \frac{q^{\frac{n(n-1)}{2}-a_{n-1}}}{q^{2a_{n-2}}}=q^{\frac{m(m+1)}{2} }=q^{a_{n+1}}, \end{align*} and so }$\varphi_{n}\sim q^{\frac{n(n+1)}{2}-a_{n}}${ by (\ref{iv-57}). }This proves (\ref{iv-45}) for even $n$.
\textbf{Case 2}. $n=2m+1$ for some $m\in \mathbb{N\cup \{}0\mathbb{\}}$. Differently from Case 1, there are \emph{cancellations} in the following computations.
Again by (\ref{t-expansion}) and (\ref{iv-65-1})-(\ref{iv-57}) with $n-1$, we have{\allowdisplaybreaks \begin{align*} & \varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1}\\ = & \frac{2n-1}{4}G_{n-1}-\frac{2n-1}{4}G_{n-1}\left( 1+8q^{\frac{1}{2}
}+32q+96q^{\frac{3}{2}}+O(|q|^{2})\right) \\ = & -2(2n-1)q^{\frac{1}{2}}G_{n-1}\left( 1+4q^{\frac{1}{2}}
+12q+O(|q|^{\frac{3}{2}})\right) , \end{align*} }{\allowdisplaybreaks \begin{align*} & \varphi_{n-1}-tQ_{n-3}Q_{n-2}Q_{n-1}\\ = & \frac{2n-1}{4}G_{n-1}-\left( 1-16q^{\frac{1}{2}}+128q-704q^{\frac{3}
{2}}+O(|q|^{2})\right) \\ & \times \frac{2n-1}{4}G_{n-1}\left( 1+8q^{\frac{1}{2}}+32q+96q^{\frac{3}{2}
}+O(|q|^{2})\right) \\
= & 2(2n-1)q^{\frac{1}{2}}G_{n-1}\left( 1-4q^{\frac{1}{2}}+12q+O(|q|^{\frac {3}{2}})\right) . \end{align*} }Consequently, \[ H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})=-(2n-1)^{3}qG_{n-1}^{3}\left(
1+O(|q|)\right) , \] {\allowdisplaybreaks \begin{align*} & H^{\prime}(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})\\ = & \varphi_{n-1}(\varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1}+\varphi_{n-1} -tQ_{n-3}Q_{n-2}Q_{n-1})\\ & +(\varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1})(\varphi_{n-1}-tQ_{n-3}Q_{n-2} Q_{n-1})\\
= & -8(2n-1)^{2}qG_{n-1}^{2}(1+O(|q|)). \end{align*} }Therefore,{\allowdisplaybreaks \begin{align*} G_{n} & =\frac{H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})}{G_{n-1}Q_{n-3}^{3}
}=\frac{-(2n-1)^{3}qG_{n-1}^{2}\left( 1+O(|q|)\right) }{Q_{n-3}^{3}}\\ & \sim \frac{q^{1+n(n-1)-2a_{n-1}}}{q^{3a_{n-2}}}=q^{\frac{3m^{2}+5m+2}{2} }=q^{\frac{n(n+1)}{2}-a_{n}}, \end{align*} }{\allowdisplaybreaks \begin{align*} Q_{n} & =\frac{Q_{n-3}^{3}G_{n}-\frac{n}{2}H^{\prime}(\varphi_{n-1} ;Q_{n-3}Q_{n-2}Q_{n-1})}{G_{n-1}Q_{n-3}^{2}}\\
& =\frac{-(2n-1)^{3}qG_{n-1}^{2}\left( 1+O(|q|)\right) +4n(2n-1)^{2}
qG_{n-1}^{2}\left( 1+O(|q|)\right) }{G_{n-1}Q_{n-3}^{2}}\\
& =\frac{(2n+1)(2n-1)^{2}qG_{n-1}\left( 1+O(|q|)\right) }{Q_{n-3}^{2}}\\ & \sim \frac{q^{1+\frac{n(n-1)}{2}-a_{n-1}}}{q^{2a_{n-2}}}=q^{\frac {(m+1)(m+2)}{2}}=q^{a_{n+1}}, \end{align*} and so }$\varphi_{n}\sim q^{\frac{n(n+1)}{2}-a_{n}}${ by (\ref{iv-57}). }This proves (\ref{iv-45}) for odd $n$.
The proof is complete. \end{proof}
\begin{remark} During the proof of Theorem \ref{weak}-(3), we can actually compute explicitly the coefficient of the leading term $q^{a_{n+1}}$ of $Q_{n}(Z_{\frac{1}{4}
,0}(\tau)|\tau)$ by induction. We leave the details to the interested reader. In particular, applying Lemma \ref{coefficient-Qn} and Theorem \ref{simple-zn} one can obtain the explicit asymptotics \[ Z_{\frac{1}{4},0}^{(n)}(\tau)=(-1)^{C_{n}}16^{a_{n}}\pi^{\frac{n(n+1)}{2}
}\prod_{k=1}^{n-1}(2k+1)^{n-k}q^{a_{n}}+O(|q|^{a_{n}+1}), \] where \[ C_{n}=\left \{ \begin{array} [c]{l} \frac{n^{2}-1}{4}\text{ \ if \ }n\text{ is odd,}\\ \frac{n^{2}}{4}\text{ \ if \ }n\text{ is even.} \end{array} \right. \] The interesting thing is that \emph{every odd} positive integer will appear in the coefficient of the leading term $q^{a_{n}}$ as $n$ increases. This phenomenon is mysterious to us; it indicates that the pre-modular form $Z_{r,s}^{(n)}(\tau)$ should possess many interesting unknown properties that are worthy to study in a future work. \end{remark}
\appendix
\section{Proof of Theorem \ref{thm-II-17}}
\label{appendix-A}
\begin{proof} [Proof of Theorem \ref{thm-II-17}]We prove this theorem by induction. By using (\ref{II-500-1}) and (\ref{II-501-1}), the proof of (\ref{II-514} )-(\ref{II-514-1}) and (\ref{II-515})-(\ref{II-518}) is simple. The difficult part is to prove the validity of properties (1-$n$)-(4-$n$).
\textbf{Step 1.} we consider $n=1$.
By (\ref{II-513}), (\ref{II-510}) and (\ref{II-500-1}), we have \begin{align} p_{1} & =p_{0}-\frac{1}{2}\left( \frac{1}{q_{0}}+\frac{1}{q_{0}-1}+\frac {1}{q_{0}-t}\right) \nonumber \\ & =Q_{0}\frac{H(R_{0};Q_{0})-\frac{1}{2}G_{0}H^{\prime}(R_{0};Q_{0})} {G_{0}H(R_{0};Q_{0})}=\frac{Q_{0}Q_{1}}{G_{1}}, \label{II-521} \end{align} where \begin{align} G_{1} & =\frac{H(R_{0};Q_{0})}{G_{0}},\label{II-519}\\ Q_{1} & =\frac{H(R_{0};Q_{0})-\frac{1}{2}G_{0}H^{\prime}(R_{0};Q_{0})} {G_{0}^{2}}=\frac{G_{1}-\frac{1}{2}H^{\prime}(R_{0};Q_{0})}{G_{0}}. \label{II-520} \end{align} Since $G_{0}(X)$ is a non-zero constant, by properties (1-$0$), (4-$0$) and (\ref{II-510}), it is easy to see that $G_{1}$ and $Q_{1}$ are both polynomials of degree $3$.
If $Q_{0}(X)=0$, then property (3-$0$) gives $R_{0}(X)=\frac{1}{2} G_{0}(X)\not =0$, which implies \[ H(R_{0}(X);Q_{0}(X))=\frac{1}{8}G_{0}(X)^{3},\text{ \ }H^{\prime} (R_{0}(X);Q_{0}(X))=\frac{3}{4}G_{0}(X)^{2}. \] From here, it is easy to see that \begin{equation} G_{1}(X)=\frac{1}{8}G_{0}(X)^{2}\not =0,\text{ \ }Q_{1}(X)=-\frac{1}{4} G_{0}(X)\not =0. \label{II-522} \end{equation}
If both $G_{1}(X)=0$ and $Q_{1}(X)=0$, then (\ref{II-519})-(\ref{II-520}) give \[ H(R_{0}(X);Q_{0}(X))=H^{\prime}(R_{0}(X);Q_{0}(X))=0, \] which implies $R_{0}(X)=Q_{0}(X)=0$, a contradiction with property (2-$0$). Thus we have proved that any two of $\{Q_{-1},Q_{0},Q_{1},G_{1}\}$ have no common zeros. Clearly, $G_{1}$ and $G_{0}$ have no common zeros.
By (\ref{II-513}), (\ref{II-501-1}) and (\ref{II-521}) we have \[ q_{1}=q_{0}+\frac{1}{p_{1}}=\frac{R_{0}Q_{1}+G_{1}}{Q_{0}Q_{1}}=\frac{R_{1} }{Q_{1}}, \] where \begin{equation} R_{1}=\frac{1}{Q_{0}}(R_{0}Q_{1}+G_{1}). \label{II-523} \end{equation} Recall $Q_{0}(X)=X$. If $Q_{0}(X)=0$, then $R_{0}(X)=\frac{1}{2}G_{0}(X)$ and (\ref{II-522}) gives $R_{0}(X)Q_{1}(X)+G_{1}(X)=0$. Thus, $R_{1}$ is a polynomial of degree $3$, namely property (1-$1$) holds.
If $Q_{1}(X)=0$, then $Q_{0}(X)\not =0$ and $G_{1}(X)\not =0$, which implies $R_{1}(X)=\frac{G_{1}(X)}{Q_{0}(X)}\not =0$. Thus, any two of $\{Q_{-1} ,Q_{1},R_{1}\}$ have no common zeros, namely property (2-$1$) holds.
Define $\varphi_{1}=R_{1}Q_{0}+\frac{1}{2}G_{1}$ by (\ref{II-515}). Then property (3-$1$) follows directly from $Q_{-1}=1$ and (\ref{II-523}).
Finally, by property (4-$0$) and \[ R_{1}-tQ_{1}=\frac{(R_{0}-tQ_{0})Q_{1}+G_{1}}{Q_{0}}, \] it is easy to see that property (4-$1$) holds. This completes the proof for $n=1$.
\textbf{Step 2.} Assume that this theorem holds for $(p_{m},q_{m})$, where $1\leq m$ $\leq n-1$ and $n\geq2$, we prove that it also holds for $(p_{n},q_{n})$.
By (\ref{II-500-1}) and (\ref{II-515}) we have{\allowdisplaybreaks \begin{align*} p_{n}= & \frac{Q_{n-3}Q_{n-2}Q_{n-1}}{G_{n-1}}-\frac{n}{2}\Bigg(\frac {1}{\frac{R_{n-1}}{Q_{n-3}Q_{n-1}}+\frac{(n-1)G_{n-1}}{2Q_{n-3}Q_{n-2}Q_{n-1} }-1}\\ & +\frac{1}{\frac{R_{n-1}}{Q_{n-3}Q_{n-1}}+\frac{(n-1)G_{n-1}}{2Q_{n-3} Q_{n-2}Q_{n-1}}}+\frac{1}{\frac{R_{n-1}}{Q_{n-3}Q_{n-1}}+\frac{(n-1)G_{n-1} }{2Q_{n-3}Q_{n-2}Q_{n-1}}-t}\Bigg)\\ = & \frac{Q_{n-3}Q_{n-2}Q_{n-1}}{G_{n-1}H(\varphi_{n-1};Q_{n-3} Q_{n-2}Q_{n-1})}\times \\ & \Big(H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})-\frac{n}{2}G_{n-1}H^{\prime }(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})\Big)\\ = & \frac{Q_{n-2}Q_{n-1}Q_{n}}{G_{n}}, \end{align*} }where $G_{n}$ and $Q_{n}$ are given by (\ref{II-516}) and (\ref{II-517}) respectively. This proves (\ref{II-514}). By (\ref{II-501-1}) we have{\allowdisplaybreaks \[ q_{n}=\frac{R_{n-1}}{Q_{n-3}Q_{n-1}}+\frac{(n-1)G_{n-1}}{2Q_{n-3} Q_{n-2}Q_{n-1}}+\frac{(n+1)G_{n}}{2Q_{n-2}Q_{n-1}Q_{n}}=\frac{R_{n}} {Q_{n-2}Q_{n}}, \] }where $R_{n}$ is given by (\ref{II-518}). This proves (\ref{II-514-1}).
Now we need to prove properties (1-$n$)-(4-$n$). Since the proof is long, we divide it into several steps.
\textbf{Step 2-1.} We prove that $G_{n}$ and $Q_{n}$ are polynomials.
Since properties (2-$(n-1)$)-(3-$(n-1)$) show that $G_{n-1}$ and $Q_{n-3}$
have no common zeros and $Q_{n-3}|\varphi_{n-1}$, it suffices to prove \begin{equation}
G_{n-1}|H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1}),\text{ \ \ }G_{n-1}|H_{n}, \label{II-526} \end{equation} where \begin{equation} H_{n}:=\frac{H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})}{G_{n-1}}-\frac{n} {2}H^{\prime}(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1}). \label{II-526-1} \end{equation}
By the formulae (\ref{II-516})-(\ref{II-518}) for $n-1$, we have \begin{equation} G_{n-2}G_{n-1}=\frac{H(\varphi_{n-2};Q_{n-4}Q_{n-3}Q_{n-2})}{Q_{n-4}^{3} }=H\Big(\frac{\varphi_{n-2}}{Q_{n-4}};Q_{n-3}Q_{n-2}\Big), \label{II-524-0} \end{equation} \begin{equation} G_{n-2}Q_{n-1}=Q_{n-4}G_{n-1}-\frac{n-1}{2}H^{\prime}\Big(\frac{\varphi_{n-2} }{Q_{n-4}};Q_{n-3}Q_{n-2}\Big), \label{II-524-1} \end{equation} \begin{equation} R_{n-1}Q_{n-2}=\frac{\varphi_{n-2}}{Q_{n-4}}Q_{n-1}+\frac{n}{2}G_{n-1}. \label{II-524} \end{equation} Notice that $\frac{\varphi_{n-2}}{Q_{n-4}}$ is a polynomial by property (3-$(n-2)$). Substituting (\ref{II-524}) into (\ref{II-515}) leads to \begin{align} \varphi_{n-1} & =\frac{\varphi_{n-2}}{Q_{n-4}}Q_{n-1}+\frac{2n-1}{2} G_{n-1}\label{II-525}\\ & \equiv \frac{\varphi_{n-2}}{Q_{n-4}}Q_{n-1}\text{ \ \ }\operatorname{mod} G_{n-1}.\nonumber \end{align} Here for polynomials $f_{j}(X)$, the notation $f_{1}\equiv f_{2}
\operatorname{mod}f_{3}$ means $f_{3}|(f_{1}-f_{2})$. Therefore, we derive from (\ref{II-510}), (\ref{II-524-0}) and (\ref{II-525}) that{\allowdisplaybreaks \begin{align} & H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})\label{II-527}\\ \equiv & \frac{2n-1}{2}G_{n-1}H^{\prime}\Big(\frac{\varphi_{n-2}}{Q_{n-4} }Q_{n-1};Q_{n-3}Q_{n-2}Q_{n-1}\Big)\nonumber \\ & +H\Big(\frac{\varphi_{n-2}}{Q_{n-4}}Q_{n-1};Q_{n-3}Q_{n-2}Q_{n-1} \Big)\text{ \ \ \ }\operatorname{mod}G_{n-1}^{2}\nonumber \\ = & \left( \frac{2n-1}{2}H^{\prime}\Big(\frac{\varphi_{n-2}}{Q_{n-4} };Q_{n-3}Q_{n-2}\Big)+G_{n-2}Q_{n-1}\right) Q_{n-1}^{2}G_{n-1},\nonumber \end{align} }which proves the first assertion in (\ref{II-526}). Furthermore, by (\ref{II-526-1}), (\ref{II-527}), (\ref{II-525}) and (\ref{II-524-1}), we have{\allowdisplaybreaks \begin{align*} H_{n}\equiv & \left( \frac{2n-1}{2}H^{\prime}\Big(\frac{\varphi_{n-2} }{Q_{n-4}};Q_{n-3}Q_{n-2}\Big)+G_{n-2}Q_{n-1}\right) Q_{n-1}^{2}\\ & -\frac{n}{2}H^{\prime}\Big(\frac{\varphi_{n-2}}{Q_{n-4}}Q_{n-1} ;Q_{n-3}Q_{n-2}Q_{n-1}\Big)\text{ \ \ }\operatorname{mod}G_{n-1}\\ = & \left( \frac{n-1}{2}H^{\prime}\Big(\frac{\varphi_{n-2}}{Q_{n-4} };Q_{n-3}Q_{n-2}\Big)+G_{n-2}Q_{n-1}\right) Q_{n-1}^{2}\\ = & Q_{n-4}Q_{n-1}^{2}G_{n-1}, \end{align*}
}namely $G_{n-1}|H_{n}$. This proves (\ref{II-526}).
\textbf{Step 2-2.} We prove that $R_{n}$ is a polynomial.
Since properties (2-$(n-1)$)-(3-$(n-1)$) show that $Q_{n-3}|\varphi_{n-1}$ and $Q_{n-3}$ has no common zeros with $Q_{n-1}$, it suffices to prove \begin{equation} Q_{n-1}\left \vert \varphi_{n-1}Q_{n}+\frac{n+1}{2}Q_{n-3}G_{n}.\right. \label{II-528} \end{equation}
Recall from (\ref{II-525}) that \[ \varphi_{n-1}\equiv \frac{2n-1}{2}G_{n-1}\text{ }\operatorname{mod}Q_{n-1}. \] This, together with the formulae (\ref{II-516})-(\ref{II-517}) and (\ref{II-510}), implies{\allowdisplaybreaks \begin{align*} & \left( \varphi_{n-1}Q_{n}+\frac{n+1}{2}Q_{n-3}G_{n}\right) Q_{n-3} ^{2}G_{n-1}\\ \equiv & \left( \frac{2n-1}{2}G_{n-1}Q_{n}+\frac{n+1}{2}Q_{n-3}G_{n}\right) Q_{n-3}^{2}G_{n-1}\text{ }\operatorname{mod}Q_{n-1}\\ = & \frac{3n}{2}H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})\\ & -\frac{n(2n-1)}{4}G_{n-1}H^{\prime}(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})\\ \equiv & \frac{3n}{2}\left( \frac{2n-1}{2}G_{n-1}\right) ^{3} -\frac{3n(2n-1)}{4}G_{n-1}\left( \frac{2n-1}{2}G_{n-1}\right) ^{2}\text{ }\operatorname{mod}Q_{n-1}\\ = & 0. \end{align*}
}This proves $Q_{n-1}|(\varphi_{n-1}Q_{n}+\frac{n+1}{2}Q_{n-3}G_{n} )Q_{n-3}^{2}G_{n-1}$. Since property (2-$(n-1)$) shows that any two of $\{Q_{n-1},G_{n-1},Q_{n-3}\}$ have no common zeros, we obtain (\ref{II-528} ).
\textbf{Step 2-3.} We prove that $\deg Q_{n}=\frac{(n+1)(n+2)}{2}$, $\deg G_{n}=\frac{3n(n+1)}{2}$ and $\deg R_{n}=n(n+1)+1$. Therefore, property (1-$n$) holds.
Recall from properties (1-$(n-1)$) and (4-$(n-1)$) that $\deg G_{n-1} =\frac{3n(n-1)}{2}$ and{\allowdisplaybreaks \begin{align} \deg(R_{n-1}-Q_{n-3}Q_{n-1}) & =\deg(R_{n-1}-tQ_{n-3}Q_{n-1}) \label{II-530} \\ & =\deg R_{n-1}=n(n-1)+1.\nonumber \end{align} }Moreover, $\deg Q_{m}=\frac{(m+1)(m+2)}{2}$ for any $-2\leq m\leq n-1$ by our induction. Hence, \[ \deg Q_{n-3}Q_{n-2}Q_{n-1}=\deg R_{n-1}Q_{n-2}=\frac{3n(n-1)}{2}+1>\deg G_{n-1}, \] by which and (\ref{II-515}), we have {\allowdisplaybreaks \begin{align*} & \deg(\varphi_{n-1}-tQ_{n-3}Q_{n-2}Q_{n-1})\\ = & \deg \left( R_{n-1}-tQ_{n-3}Q_{n-1}\right) Q_{n-2}=\frac{3n(n-1)}{2}+1, \end{align*} }and similarly,{\allowdisplaybreaks \begin{align} & \deg(\varphi_{n-1}-Q_{n-3}Q_{n-2}Q_{n-1})\label{II-530-2}\\ = & \deg \varphi_{n-1}=\deg Q_{n-3}Q_{n-2}Q_{n-1}=\frac{3n(n-1)} {2}+1,\nonumber \end{align} }namely{\allowdisplaybreaks \begin{align} \deg H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1}) & =3\deg \varphi_{n-1} ,\nonumber \\ \deg H^{\prime}(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1}) & \leq2\deg \varphi_{n-1}. \label{II-530-3} \end{align} }From here and (\ref{II-516})-(\ref{II-518}), it is easy to see that $\deg G_{n}=\frac{3n(n+1)}{2}$, $\deg Q_{n}=\frac{(n+1)(n+2)}{2}$ and $\deg R_{n}=n(n+1)+1$.
\textbf{Step 2-4.} We prove that property (4-$n$) holds.
It is easy to see that{\allowdisplaybreaks \begin{align*} \deg R_{n-1}Q_{n-2}Q_{n} & =\deg Q_{n-3}Q_{n-2}Q_{n-1}Q_{n}\\ & >\deg G_{n-1}Q_{n}=\deg G_{n}Q_{n-3}\text{,} \end{align*} }so we derive from (\ref{II-518}) and (\ref{II-530}) that{\allowdisplaybreaks \begin{align*} & \deg(R_{n}-tQ_{n-2}Q_{n})\\ = & \deg(R_{n-1}Q_{n-2}Q_{n}-tQ_{n-3}Q_{n-2}Q_{n-1}Q_{n})-\deg Q_{n-3}Q_{n-1}\\ = & \deg(R_{n-1}-tQ_{n-3}Q_{n-1})\deg Q_{n-2}Q_{n}-\deg Q_{n-3}Q_{n-1}\\ = & \deg R_{n-1}\deg Q_{n-2}Q_{n}-\deg Q_{n-3}Q_{n-1}=\deg R_{n}. \end{align*} }Similarly, $\deg(R_{n}-Q_{n-2}Q_{n})=\deg R_{n}$.
\textbf{Step 2-5.} We prove that $G_{n-1}$ and $G_{n}$ have no common zeros.
Recall from (\ref{II-516}) and (\ref{II-527}) that \[ Q_{n-3}^{3}G_{n}\equiv \left( \tfrac{2n-1}{2}H^{\prime}\Big(\tfrac {\varphi_{n-2}}{Q_{n-4}};Q_{n-3}Q_{n-2}\Big)+G_{n-2}Q_{n-1}\right) Q_{n-1}^{2}\operatorname{mod}G_{n-1}. \] If both $G_{n}(X)=0$ and $G_{n-1}(X)=0$, then property (2-$(n-1)$) shows that $G_{n-2}(X)\not =0$, $Q_{n-1}(X)\not =0$ and $Q_{n-3}(X)\not =0$. Thus, \begin{equation} \frac{2n-1}{2}H^{\prime}\Big(\frac{\varphi_{n-2}(X)}{Q_{n-4}(X)} ;Q_{n-3}(X)Q_{n-2}(X)\Big)+G_{n-2}(X)Q_{n-1}(X)=0. \label{II-531} \end{equation} On the other hand, (\ref{II-524-1}) and $n\geq2$ give \[ H^{\prime}\Big(\frac{\varphi_{n-2}(X)}{Q_{n-4}(X)};Q_{n-3}(X)Q_{n-2} (X)\Big)=\frac{-2}{n-1}G_{n-2}(X)Q_{n-1}(X)\not =0, \] which yields a contradiction with (\ref{II-531}).
\textbf{Step 2-6.} We prove that any two of $\{Q_{n-2},Q_{n-1},Q_{n},G_{n}\}$ have no common zeros.
If $Q_{n-2}(X)=0$, then property (2-$(n-1)$) gives $Q_{n-1}(X)\not =0$, $Q_{n-3}(X)\not =0$ and $G_{n-1}(X)\not =0$. By (\ref{II-515}) we have $\varphi_{n-1}(X)=\frac{n-1}{2}G_{n-1}(X)$, which implies{\allowdisplaybreaks \begin{align} H(\varphi_{n-1}(X);Q_{n-3}(X)Q_{n-2}(X)Q_{n-1}(X)) & =\frac{(n-1)^{3}} {8}G_{n-1}(X)^{3},\label{II-532}\\ H^{\prime}(\varphi_{n-1}(X);Q_{n-3}(X)Q_{n-2}(X)Q_{n-1}(X)) & =\frac {3(n-1)^{2}}{4}G_{n-1}(X)^{2}.\nonumber \end{align} }From here and (\ref{II-516})-(\ref{II-517}), it is easy to see that{\allowdisplaybreaks \begin{align} G_{n}(X) & =\frac{(n-1)^{3}}{8Q_{n-3}(X)^{3}}G_{n-1}(X)^{2}\not = 0,\label{II-533}\\ Q_{n}(X) & =-\frac{(2n+1)(n-1)^{2}}{8Q_{n-3}(X)^{2}}G_{n-1}(X)\not = 0.\nonumber \end{align} }This proves that $Q_{n-2}$ has no common zeros with any of $\{G_{n} ,Q_{n},Q_{n-1}\}$.
If $Q_{n-1}(X)=0$, then property (2-$(n-1)$) implies both $Q_{n-3}(X)\not =0$ and $G_{n-1}(X)\not =0$. By (\ref{II-525}) we have $\varphi_{n-1} (X)=\frac{2n-1}{2}G_{n-1}(X)$. Similarly as (\ref{II-532})-(\ref{II-533}), we obtain{\allowdisplaybreaks \begin{align*} G_{n}(X) & =\frac{(2n-1)^{3}}{8Q_{n-3}(X)^{3}}G_{n-1}(X)^{2}\not =0,\\ Q_{n}(X) & =-\frac{(n+1)(2n-1)^{2}}{8Q_{n-3}(X)^{2}}G_{n-1}(X)\not =0. \end{align*} }This proves that $Q_{n-1}$ has no common zeros with any of $\{G_{n},Q_{n}\}$.
If both $Q_{n}(X)=0$ and $G_{n}(X)=0$, then $Q_{n-1}(X)Q_{n-2}(X)\not =0$ and $G_{n-1}(X)\not =0$. Recalling from property (3-$(n-1)$) that $\frac {\varphi_{n-1}}{Q_{n-3}}$ is a polynomial, it is easy to see from (\ref{II-516})-(\ref{II-517}) that{\allowdisplaybreaks \begin{align*} H\Big(\frac{\varphi_{n-1}(X)}{Q_{n-3}(X)};Q_{n-1}(X)Q_{n-2}(X)\Big) & =0,\\ H^{\prime}\Big(\frac{\varphi_{n-1}(X)}{Q_{n-3}(X)};Q_{n-1}(X)Q_{n-2}(X)\Big) & =0, \end{align*} }which implies $Q_{n-1}(X)Q_{n-2}(X)=0$, a contradiction. This proves that $Q_{n}$ has no common zeros with $G_{n}$.
\textbf{Step 2-7.} We prove that any two of $\{Q_{n-2},Q_{n},R_{n}\}$ have no common zeros. Then property (2-$n$) holds.
If $Q_{n}(X)=0$, then $Q_{n-1}(X)\not =0$ and $G_{n}(X)\not =0$. Since $\frac{\varphi_{n-1}}{Q_{n-3}}$ is a polynomial by property (3-$(n-1)$), we have \[ R_{n}(X)=\frac{n+1}{2}\frac{G_{n}(X)}{Q_{n-1}(X)}\not =0. \] This proves that $Q_{n}$ has no common zeros with $R_{n}$.
If $Q_{n-2}(X)=0$, as in Step 2-6 we have $Q_{n}(X)\not =0$, $Q_{n-1} (X)\not =0$, $Q_{n-3}(X)\not =0$ and $\varphi_{n-1}(X)=\frac{n-1}{2} G_{n-1}(X)\not =0$. Then (\ref{II-516})-(\ref{II-518}) imply{\allowdisplaybreaks \begin{align} & Q_{n-3}(X)Q_{n-1}(X)R_{n}(X)\label{II-534}\\ = & \frac{n-1}{2}G_{n-1}(X)Q_{n}(X)+\frac{n+1}{2}Q_{n-3}(X)G_{n} (X)\nonumber \\ = & n\frac{H(\frac{n-1}{2}G_{n-1}(X);0)}{G_{n-1}(X)Q_{n-3}(X)^{2}} -\frac{n(n-1)}{4Q_{n-3}(X)^{2}}H^{\prime}\Big(\frac{n-1}{2}G_{n-1} (X);0\Big)\nonumber \\ = & -\frac{n(n-1)^{3}}{16Q_{n-3}(X)^{2}}G_{n-1}(X)^{2}\not =0,\nonumber \end{align} }namely $R_{n}(X)\not =0$. This proves that $Q_{n-2}$ has no common zeros with $R_{n}$.
\textbf{Step 2-8.} We prove $Q_{n-2}|\varphi_{n}$ and $Q_{n}|(R_{n} Q_{n-1}-\frac{n+1}{2}G_{n})$. Then property (3-$n$) holds.
Since $\frac{\varphi_{n-1}}{Q_{n-3}}$ is a polynomial by property (3-$(n-1)$), by (\ref{II-518}) we have \[ R_{n}Q_{n-1}-\frac{n+1}{2}G_{n}=\frac{\varphi_{n-1}}{Q_{n-3}}Q_{n}. \]
This proves $Q_{n}|(R_{n}Q_{n-1}-\frac{n+1}{2}G_{n})$.
Recall from (\ref{II-515}) that $\varphi_{n-1}\equiv \frac{n-1}{2}G_{n-1}$ $\operatorname{mod}Q_{n-2}$. This, together with $\varphi_{n}=R_{n} Q_{n-1}+\frac{n}{2}G_{n}$ and (\ref{II-516})-(\ref{II-518}), implies{\allowdisplaybreaks \begin{align*} & \varphi_{n}G_{n-1}Q_{n-3}^{3}\\ = & \left( R_{n}Q_{n-1}+\frac{n}{2}G_{n}\right) G_{n-1}Q_{n-3}^{3}\\ = & \left( \varphi_{n-1}Q_{n}+\frac{2n+1}{2}Q_{n-3}G_{n}\right) G_{n-1}Q_{n-3}^{2}\\ \equiv & \left( \frac{n-1}{2}G_{n-1}Q_{n}+\frac{2n+1}{2}Q_{n-3}G_{n}\right) G_{n-1}Q_{n-3}^{2}\operatorname{mod}Q_{n-2}\\ = & \frac{3n}{2}H(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})\\ & -\frac{n(n-1)}{4}G_{n-1}H^{\prime}(\varphi_{n-1};Q_{n-3}Q_{n-2}Q_{n-1})\\ \equiv & \frac{3n}{2}\left( \frac{n-1}{2}G_{n-1}\right) ^{3}-\frac {3n(n-1)}{4}G_{n-1}\left( \frac{n-1}{2}G_{n-1}\right) ^{2}\operatorname{mod} Q_{n-2}\\ = & 0. \end{align*} }Together with property (2-$(n-1)$) that any two of $\{Q_{n-3},G_{n-1}
,Q_{n-2}\}$ have no common zeros, we conclude that $Q_{n-2}|\varphi_{n}$.
Therefore, the proof is complete. \end{proof}
{\bf Acknowledgements} The research of Z. Chen was supported NSFC (Grant No. 12071240).
\end{document}
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Results for 'Intuitionism'
Bibliography: Moral Intuitionism in Meta-Ethics
Bibliography: Intuitionism and Constructivism in Philosophy of Mathematics
1. Intuitionistic sentential calculus with iden-tity.Intuitionistic Sentential Calculus - 1990 - Bulletin of the Section of Logic 19 (3):92-99.details
Intuitionistic Logic in Logic and Philosophy of Logic
Ethical Intuitionism.Michael Huemer - 2005 - Palgrave Macmillan.details
This book defends a form of ethical intuitionism, according to which (i) there are objective moral truths; (ii) we know some of these truths through a kind of immediate, intellectual awareness, or "intuition"; and (iii) our knowledge of moral truths gives us reasons for action independent of our desires. The author rebuts all the major objections to this theory and shows that the alternative theories about the nature of ethics all face grave difficulties.
Moral Intuitionism in Meta-Ethics
Moral intuitionism and disagreement.Brian Besong - 2014 - Synthese 191 (12):2767-2789.details
According to moral intuitionism, at least some moral seeming states are justification-conferring. The primary defense of this view currently comes from advocates of the standard account, who take the justification-conferring power of a moral seeming to be determined by its phenomenological credentials alone. However, the standard account is vulnerable to a problem. In brief, the standard account implies that moral knowledge is seriously undermined by those commonplace moral disagreements in which both agents have equally good phenomenological credentials supporting their (...) disputed moral beliefs. However, it is implausible to think that commonplace disagreement seriously undermines moral knowledge, and thus it is implausible to think that the standard account of moral intuitionism is true. (shrink)
Moral Disagreement in Meta-Ethics
Moral Skepticism in Meta-Ethics
Revisionary intuitionism.Michael Huemer - 2008 - Social Philosophy and Policy 25 (1):368-392.details
I argue that, given evidence of the factors that tend to distort our intuitions, ethical intuitionists should disown a wide range of common moral intuitions, and that they should typically give preference to abstract, formal intuitions over more substantive ethical intuitions. In place of the common sense morality with which intuitionism has traditionally allied, the suggested approach may lead to a highly revisionary normative ethics.
Moral Intuition in Normative Ethics
Ethical Intuitionism: Re-Evaluations.Philip Stratton-Lake (ed.) - 2002 - Oxford University Press UK.details
Ethical Intuitionism was the dominant moral theory in Britain for much of the 18th, 19th and the first third of the twentieth century. However, during the middle decades of the twentieth century ethical intuitionism came to be regarded as utterly untenable. It was thought to be either empty, or metaphysically and epistemologically extravagant, or both. This hostility led to a neglect of the central intuitionist texts, and encouraged the growth of a caricature of intuitionism that could easily (...) be rejected before moving on to 'more serious' philosophical theories. More recently, however, this hostility towards ethical intuitionism has subsided. A wide range of moral philosophers, from Aristotelians, to rule-consequentialists, to expressivists, Kantians and deontologists, are beginning to look to the ethical intuitionists's work as a positive resource. It is, therefore, a good time to get clear on what it was that intuitionists said, and re-evaluate their contribution to our understanding of morality. This volume is the first serious engagement with ethical intuitionism in the light of contemporary developments in ethical theory. It contains essays by eminent moral philosophers working in very different traditions whose aim is to clarify and assess ethical intuitionism. Issues addressed include whether the plurality of basic principles intuitionists adhere to can be grounded in some more fundamental principle; the autonomy of ethics and self-evidence; moral realism and internalism; and the open question argument and naturalism. (shrink)
Autonomy, Misc in Social and Political Philosophy
Henry Sidgwick in 19th Century Philosophy
Intuitionism an Introduction.Arend Heyting - 1956 - Amsterdam, Netherlands: North-Holland.details
Areas of Mathematics in Philosophy of Mathematics
Intuitionism.A. Heyting - 1966 - Amsterdam: North-Holland Pub. Co..details
Intuitionism and Constructivism in Philosophy of Mathematics
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Social intuitionists answer six questions about morality.Jonathan Haidt & Fredrik Bjorklund - 2008 - In W. Sinnott-Armstrong (ed.), Moral Psychology Vol. 2. MIT Press.details
We review the state of the art in moral psychology to answer 6 questions: 1) Where do moral beliefs and motivations come from? 2) How does moral judgment work? 3) What is the evidence for the social intuitionist model? 4) What exactly are the moral intuitions? 5) How does morality develop? And 6) Why do people vary in their morality? We describe the intuitionist approach to moral psychology. The mind makes rapid affective evaluations of everything it encounters, and these evaluations (...) (intuitions) shape and push subsequent moral reasoning. This approach to moral judgment has a variety of implications for moral philosophy and for the law in that it questions common assumptions about the reliability and causal efficacy of private, conscious reasoning. (shrink)
Moral Emotivism and Sentimentalism in Meta-Ethics
Psychology of Ethics in Normative Ethics
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Brouwerian intuitionism.Michael Detlefsen - 1990 - Mind 99 (396):501-534.details
The aims of this paper are twofold: firstly, to say something about that philosophy of mathematics known as 'intuitionism' and, secondly, to fit these remarks into a more general message for the philosophy of mathematics as a whole. What I have to say on the first score can, without too much inaccuracy, be compressed into two theses. The first is that the intuitionistic critique of classical mathematics can be seen as based primarily on epistemological rather than on meaning-theoretic considerations. (...) The second is that the intuitionist's chief objection to the classical mathematician's use of logic does not center on the use of particular logical principles (in particular, the law of excluded middle and its ilk). Rather on the role the classical mathematician assigns (or at least extends) generally (i.e. regardless of the particular principles used) to the use of logic in the production mathematical proofs. Thus, the intuitionist critique of logic that we shall be presenting is far more radical than that which has commonly been presented. -/- Concerning the second, more general, theme, my claim is this: some restriction of the role of logical inference in mathematical proof such as that mentioned above is necessary if one is to account for the seeming difference in the epistemic conditions of provers whose reasoning is based on genuine insight into the subject-matter being investigated, and would-be provers whose reasoning is based not on such insight, but rather on principles of inference which hold of every subject-matter indifferently. (shrink)
Epistemology of Mathematics, Misc in Philosophy of Mathematics
Mathematical Intuition in Philosophy of Mathematics
Mathematical Proof in Philosophy of Mathematics
Philosophy of Mathematics, Misc in Philosophy of Mathematics
Intuitionism and the Modal Logic of Vagueness.Susanne Bobzien & Ian Rumfitt - 2020 - Journal of Philosophical Logic 49 (2):221-248.details
Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with (...) the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist's characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator 'it is clearly the case that'. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson's objections to intuitionistic treatments of vagueness. (shrink)
Higher-Order Vagueness in Philosophy of Language
Intuitionistic Theories of Vagueness in Philosophy of Language
Modal Logic in Logic and Philosophy of Logic
Sorites Paradox in Philosophy of Language
Intuitionism.David Kaspar - 2012 - Continuum.details
Thinking about morality -- Story of contemporary intuitionism -- Moral knowledge -- New challenges to intuitionism -- Grounds of morality -- Right and the good reconsidered -- Intuitionism's rivals -- Being moral: how and why.
Basic Intuitionistic Conditional Logic.Yale Weiss - 2019 - Journal of Philosophical Logic 48 (3):447-469.details
Conditional logics have traditionally been intended to formalize various intuitively correct modes of reasoning involving conditional expressions in natural language. Although conditional logics have by now been thoroughly studied in a classical context, they have yet to be systematically examined in an intuitionistic context, despite compelling philosophical and technical reasons to do so. This paper addresses this gap by thoroughly examining the basic intuitionistic conditional logic ICK, the intuitionistic counterpart of Chellas' important classical system CK. I give ICK both worlds (...) semantics and algebraic semantics, and prove that these are equivalent. I give a Gödel-type embedding of ICK into CK and a Glivenko-type embedding of CK into ICK. I axiomatize ICK and prove soundness, completeness, and decidability results. Finally, I discuss extending ICK. (shrink)
Logic of Conditionals in Philosophy of Language
Ethical intuitionism and the linguistic analogy.Philipp Schwind - 2018 - Canadian Journal of Philosophy 48 (2):292-311.details
It is a central tenet of ethical intuitionism as defended by W. D. Ross and others that moral theory should reflect the convictions of mature moral agents. Hence, intuitionism is plausible to the extent that it corresponds to our well-considered moral judgments. After arguing for this claim, I discuss whether intuitionists offer an empirically adequate account of our moral obligations. I do this by applying recent empirical research by John Mikhail that is based on the idea of a (...) universal moral grammar to a number of claims implicit in W. D. Ross's normative theory. I argue that the results at least partly vindicate intuitionism. (shrink)
John Rawls in 20th Century Philosophy
Moral Pluralism in Normative Ethics
Moral Psychology, Misc in Normative Ethics
Pluralistic Deontological Theories in Normative Ethics
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Intuitionistic logic with strong negation.Yuri Gurevich - 1977 - Studia Logica 36 (1-2):49 - 59.details
This paper is a reaction to the following remark by grzegorczyk: "the compound sentences are not a product of experiment. they arise from reasoning. this concerns also negations; we see that the lemon is yellow, we do not see that it is not blue." generally, in science the truth is ascertained as indirectly as falsehood. an example: a litmus-paper is used to verify the sentence "the solution is acid." this approach gives rise to a (very intuitionistic indeed) conservative extension of (...) the heyting logic satisfying natural duality laws. (shrink)
Intuitionistic Epistemic Logic, Kripke Models and Fitch's Paradox.Carlo Proietti - 2012 - Journal of Philosophical Logic 41 (5):877-900.details
The present work is motivated by two questions. (1) What should an intuitionistic epistemic logic look like? (2) How should one interpret the knowledge operator in a Kripke-model for it? In what follows we outline an answer to (2) and give a model-theoretic definition of the operator K. This will shed some light also on (1), since it turns out that K, defined as we do, fulfills the properties of a necessity operator for a normal modal logic. The interest of (...) our construction also lies in a better insight into the intuitionistic solution to Fitch's paradox, which is discussed in the third section. In particular we examine, in the light of our definition, De Vidi and Solomon's proposal of formulating the verification thesis as Φ → ¬¬KΦ. We show, as our main result, that this definition excapes the paradox, though it is validated only under restrictive conditions on the models. (shrink)
Formal Epistemology in Epistemology
Intuitionism in the Philosophy of Mathematics: Introducing a Phenomenological Account.Philipp Berghofer - 2020 - Philosophia Mathematica 28 (2):204-235.details
The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians (...) such as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)
Husserl: Philosophy of Mathematics in Continental Philosophy
Full intuitionistic linear logic.Martin Hyland & Valeria de Paiva - 1993 - Annals of Pure and Applied Logic 64 (3):273-291.details
In this paper we give a brief treatment of a theory of proofs for a system of Full Intuitionistic Linear Logic. This system is distinct from Classical Linear Logic, but unlike the standard Intuitionistic Linear Logic of Girard and Lafont includes the multiplicative disjunction par. This connective does have an entirely natural interpretation in a variety of categorical models of Intuitionistic Linear Logic. The main proof-theoretic problem arises from the observation of Schellinx that cut elimination fails outright for an intuitive (...) formulation of Full Intuitionistic Linear Logic; the nub of the problem is the interaction between par and linear implication. We present here a term assignment system which gives an interpretation of proofs as some kind of non-deterministic function. In this way we find a system which does enjoy cut elimination. The system is a direct result of an analysis of the categorical semantics, though we make an effort to present the system as if it were purely a proof-theoretic construction. (shrink)
Proof Theory in Logic and Philosophy of Logic
Ethical Intuitionism and Moral Skepticism.Clayton Littlejohn - 2011 - In Jill Graper Hernandez (ed.), The New Intuitionism.details
In this paper, I defend a non-skeptical intuitionist approach to moral epistemology from recent criticisms. Starting with Sinnott-Armstrong's skeptical attacks, I argue that a familiar sort of skeptical argument rests on a problematic conception of the evidential grounds of our moral judgments. The success of his argument turns on whether we conceive of the evidential grounds of our moral judgments as consisting entirely of non-normative considerations. While we cannot avoid skepticism if we accept this conception of our evidential grounds, that's (...) because accepting this conception of our evidential grounds is tantamount to accepting the skeptic's conclusion. We have nothing to fear from arguments for skepticism from skepticism. (shrink)
Evidence and Knowledge in Epistemology
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Elements of Intuitionism.Michael Dummett - 1977 - Oxford University Press.details
This is a long-awaited new edition of one of the best known Oxford Logic Guides. The book gives an introduction to intuitionistic mathematics, leading the reader gently through the fundamental mathematical and philosophical concepts. The treatment of various topics, for example Brouwer's proof of the Bar Theorem, valuation systems, and the completeness of intuitionistic first-order logic, have been completely revised.
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Intuitionistic Propositional Logic with Galois Negations.Minghui Ma & Guiying Li - 2023 - Studia Logica 111 (1):21-56.details
Intuitionistic propositional logic with Galois negations ( $$\mathsf {IGN}$$ ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair $$(\lnot,{\sim })$$ and dual Galois pair $$(\dot{\lnot },\dot{\sim })$$ of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for $$\mathsf {IGN}$$ are developed. A Hilbert-style axiomatic system $$\mathsf {HN}$$ is given for $$\mathsf {IGN}$$, and Galois negation logics are defined as extensions of $$\mathsf {IGN}$$. We give the bi-tense (...) logic $$\mathsf {S4N}_t$$ which is obtained from the minimal tense extension of the modal logic $$\mathsf {S4}$$ by adding tense operators. We give a new extended Gödel translation $$\tau $$ and prove that $$\mathsf {IGN}$$ is embedded into $$\mathsf {S4N}_t$$ by $$\tau $$. Moreover, every Kripke-complete Galois negation logic L is embedded into its tense companion $$\tau (L)$$. (shrink)
Mathematical Logic in Philosophy of Mathematics
Intuitionistic Completeness and Classical Logic.D. C. McCarty - 2002 - Notre Dame Journal of Formal Logic 43 (4):243-248.details
We show that, if a suitable intuitionistic metatheory proves that consistency implies satisfiability for subfinite sets of propositional formulas relative either to standard structures or to Kripke models, then that metatheory also proves every negative instance of every classical propositional tautology. Since reasonable intuitionistic set theories such as HAS or IZF do not demonstrate all such negative instances, these theories cannot prove completeness for intuitionistic propositional logic in the present sense.
Intuitionistic Type Theory.Per Martin-Löf - 1984 - Bibliopolis.details
Intuitionistic hybrid logic.Torben Braüner & Valeria de Paiva - 2006 - Journal of Applied Logic 4 (3):231-255.details
Hybrid logics are a principled generalization of both modal logics and description logics, a standard formalism for knowledge representation. In this paper we give the first constructive version of hybrid logic, thereby showing that it is possible to hybridize constructive modal logics. Alternative systems are discussed, but we fix on a reasonable and well-motivated version of intuitionistic hybrid logic and prove essential proof-theoretical results for a natural deduction formulation of it. Our natural deduction system is also extended with additional inference (...) rules corresponding to conditions on the accessibility relations expressed by so-called geometric theories. Thus, we give natural deduction systems in a uniform way for a wide class of constructive hybrid logics. This shows that constructive hybrid logics are a viable enterprise and opens up the way for future applications. (shrink)
Moral Intuitionism Defeated?Nathan Ballantyne & Joshua C. Thurow - 2013 - American Philosophical Quarterly 50 (4):411-422.details
Walter Sinnott-Armstrong has developed and progressively refined an argument against moral intuitionism—the view on which some moral beliefs enjoy non-inferential justification. He has stated his argument in a few different forms, but the basic idea is straightforward. To start with, Sinnott-Armstrong highlights facts relevant to the truth of moral beliefs: such beliefs are sometimes biased, influenced by various irrelevant factors, and often subject to disagreement. Given these facts, Sinnott-Armstrong infers that many moral beliefs are false. What then shall we (...) think of our own moral beliefs? Either we have reason to think some of our moral beliefs are reliably formed or we have no such reason. If the latter, our moral beliefs are unjustified. If we have reason to think some moral beliefs are reliably formed, then those beliefs are not non-inferentially justified, because then we'll have reason to accept something—namely, that they are reliably formed—that entails or supports those beliefs. But then, either way, our moral beliefs are not non-inferentially justified, and so moral intuitionism is false. This paper takes issue with Sinnott-Armstrong's interesting and widely discussed argument, which we here call the Empirical Defeat Argument (EDA). According to us, the EDA does not defeat moral intuitionism. In section 1, we will set out the argument, briefly reviewing the rationale Sinnott-Armstrong offers for the premises. Then, in section 2, we identify a critical but dubious epistemological assumption concerning the nature of defeat that undergirds the argument. Finally, in section 3, we will defend our challenge to the EDA by answering two objections. (shrink)
Debunking Arguments about Morality in Meta-Ethics
Inquisitive Intuitionistic Logic.Wesley H. Holliday - forthcoming - In Nicola Olivetti & Rineke Verbrugge (eds.), Advances in Modal Logic, Vol. 11. London: College Publications.details
Inquisitive logic is a research program seeking to expand the purview of logic beyond declarative sentences to include the logic of questions. To this end, inquisitive propositional logic extends classical propositional logic for declarative sentences with principles governing a new binary connective of inquisitive disjunction, which allows the formation of questions. Recently inquisitive logicians have considered what happens if the logic of declarative sentences is assumed to be intuitionistic rather than classical. In short, what should inquisitive logic be on an (...) intuitionistic base? In this paper, we provide an answer to this question from the perspective of nuclear semantics, an approach to classical and intuitionistic semantics pursued in our previous work. In particular, we show how Beth semantics for intuitionistic logic naturally extends to a semantics for inquisitive intuitionistic logic. In addition, we show how an explicit view of inquisitive intuitionistic logic comes via a translation into propositional lax logic, whose completeness we prove with respect to Beth semantics. (shrink)
Logical Consequence and Entailment in Logic and Philosophy of Logic
Nonclassical Logic, Misc in Logic and Philosophy of Logic
Semantics for Modal Logic in Logic and Philosophy of Logic
Semi-intuitionistic Logic.Juan Manuel Cornejo - 2011 - Studia Logica 98 (1-2):9-25.details
The purpose of this paper is to define a new logic $${\mathcal {SI}}$$ called semi-intuitionistic logic such that the semi-Heyting algebras introduced in [ 4 ] by Sankappanavar are the semantics for $${\mathcal {SI}}$$ . Besides, the intuitionistic logic will be an axiomatic extension of $${\mathcal {SI}}$$.
Intuitionism and conservatism.Mark T. Nelson - 1990 - Metaphilosophy 21 (3):282-293.details
I define ethical intuitionism as the view that it is appropriate to appeal to inferentially unsupported moral beliefs in the course of moral reasoning. I mention four common objections to this view, including the view that all such appeals to intuition make ethical theory politically and noetically conservative. I defend intuitionism from versions of this criticism expressed by R.B. Brandt, R.M. Hare and Richard Miller.
Reflective Equilibrium in Meta-Ethics
Perceptual Intuitionism.Robert Cowan - 2015 - Philosophy and Phenomenological Research 90 (1):164-193.details
In the recent metaethical literature there has been significant interest in the prospects for what I am denoting 'Perceptual Intuitionism': the view that normal ethical agents can and do have non-inferential justification for first-order ethical beliefs by having ethical perceptual experiences, e.g., Cullison 2010, McBrayer 2010, Vayrynen 2008. If true, it promises to constitute an independent a posteriori intuitionist epistemology, providing an alternative to intuitionist accounts which posit a priori intuition and/or emotion as sources of non-inferentially justified ethical beliefs. (...) As it is formulated, it is plausible that a necessary con- dition for the view is the truth of Ethical Perception: normal ethical agents can and do have perceptual experiences as of the instantiation of ethical properties. In this paper a sophisticated and promising account of Ethical Perception is offered. Extant objections are shown to fail. However, it will be argued that it is far from obvious that the account of Perceptual Intuitionism which emerges constitutes an independent alternative to other intuitionist accounts. This is because we have reason to think that ethical perceptual experience may be epistemically dependent on other epistemic sources, e.g. a priori intuition or emotion. (shrink)
Moral Epistemology, Misc in Meta-Ethics
Intuitionistic ε- and τ-calculi.David Devidi - 1995 - Mathematical Logic Quarterly 41 (4):523-546.details
There are several open problems in the study of the calculi which result from adding either of Hilbert's ϵ- or τ-operators to the first order intuitionistic predicate calculus. This paper provides answers to several of them. In particular, the first complete and sound semantics for these calculi are presented, in both a "quasi-extensional" version which uses choice functions in a straightforward way to interpret the ϵ- or τ-terms, and in a form which does not require extensionality assumptions. Unlike the classical (...) case, the addition of either operator to intuitionistic logic is non-conservative. Several interesting consequences of the addition of each operator are proved. Finally, the independence of several other schemes in either calculus are also proved, making use of the semantics supplied earlier in the paper. (shrink)
Intuitionistic validity in T-normal Kripke structures.Samuel R. Buss - 1993 - Annals of Pure and Applied Logic 59 (3):159-173.details
Let T be a first-order theory. A T-normal Kripke structure is one in which every world is a classical model of T. This paper gives a characterization of the intuitionistic theory T of sentences intuitionistically valid in all T-normal Kripke structures and proves the corresponding soundness and completeness theorems. For Peano arithmetic , the theory PA is a proper subtheory of Heyting arithmetic , so HA is complete but not sound for PA-normal Kripke structures.
An intuitionistic defence of Berkeley's master argument.Conor McGlynn - 2019 - Analysis 79 (2):236-242.details
Berkeley's 'master argument' for idealism has been the subject of extensive criticism. Two of his strongest critics, A.N. Prior and J.L. Mackie, argue that due to various logical confusions on the part of Berkeley, the master argument fails to establish his idealist conclusion. Prior argues that Berkeley's argument 'proves too little' in its conclusion, while Mackie contends that Berkeley confuses two different kinds of self-refutation in his argument. This paper proposes a defence of the master argument based on intuitionistic argument. (...) It begins by giving a brief exposition of the master argument and Prior's and Mackie's criticism. The following section explains why we might read the master argument along intuitionistic lines. The final section demonstrates that, according to intuitionistic logic, Berkeley's argument withstands the criticisms of Prior and Mackie. (shrink)
Berkeley: Immaterialism in 17th/18th Century Philosophy
Social Intuitionism and the Psychology of Moral Reasoning.Hanno Sauer - 2011 - Philosophy Compass 6 (10):708-721.details
Rationalism about the psychology of moral judgment holds, among other things, that the justifying moral reasons we have for our judgments are also the causally effective reasons for why we make those judgments. This can be called the 'effectiveness'-thesis regarding moral reasoning. The theory that best exemplifies the thesis is the traditional conscious reasoning-paradigm. Current empirical moral psychology, however, poses a serious challenge to this thesis: it argues that in fact, emotional reactions are necessary and sufficient to account for moral (...) judgment, and that typically, moral reasoning is a matter of mere confabulation. In this survey, the empirical challenge to this thesis made by the 'social intuitionist' model of moral judgment and reasoning is discussed. The model claims that moral reasoning is essentially ineffective and, psychologically speaking, a matter of mere post hoc-rationalizations of cognitively impenetratable gut reactions. Several interpretations of this evidence are discussed and it is shown that there is room for a psychology of moral reasoning that can account for the available empirical evidence and yet does not have to give up the most central elements of a normative picture of moral reasoning. (shrink)
Moral Cognitivism in Meta-Ethics
Moral Judgment, Misc in Meta-Ethics
Moral Noncognitivism in Meta-Ethics
Moral Reasoning and Motivation, Misc in Meta-Ethics
Intuitionistic Fixed Point Theories for Strictly Positive Operators.Christian Rüede & Thomas Strahm - 2002 - Mathematical Logic Quarterly 48 (2):195-202.details
In this paper it is shown that the intuitionistic .xed point theory equation image for α times iterated fixed points of strictly positive operator forms is conservative for negative arithmetic and equation image sentences over the theory equation image for α times iterated arithmetic comprehension without set parameters.This generalizes results previously due to Buchholz [5] and Arai [2].
Ethical Intuitionism: A Structural Critique.Danny Frederick - 2016 - Journal of Value Inquiry 50 (3):631-47.details
Ethical intuitionists regard moral knowledge as deriving from moral intuition, moral observation, moral emotion and inference. However, moral intuitions, observations and emotions are cultural artefacts which often differ starkly between cultures. Intuitionists attribute uncongenial moral intuitions, observations or emotions to bias or to intellectual or moral failings; but that leads to sectarian ad hominen attacks. Intuitionists try to avoid that by restricting epistemically genuine intuitions, observations or emotions to those which are widely agreed. That does not avoid the problem. It (...) also limits epistemically genuine intuitions, observations or emotions to those with meagre content, and the intuitionists offer no plausible explanation for how inference from such insubstantial propositions can engender substantial moral knowledge. Instead of moral knowledge, intuitionism offers the prospect of mutual name-calling between intellectually stagnant groups. I criticise and reject the principle of phenomenal conservatism, to which intuitionists sometimes appeal. (shrink)
Dual Intuitionistic Logic and a Variety of Negations: The Logic of Scientific Research.Yaroslav Shramko - 2005 - Studia Logica 80 (2-3):347-367.details
We consider a logic which is semantically dual (in some precise sense of the term) to intuitionistic. This logic can be labeled as "falsification logic": it embodies the Popperian methodology of scientific discovery. Whereas intuitionistic logic deals with constructive truth and non-constructive falsity, and Nelson's logic takes both truth and falsity as constructive notions, in the falsification logic truth is essentially non-constructive as opposed to falsity that is conceived constructively. We also briefly clarify the relationships of our falsification logic to (...) some other logical systems. (shrink)
Intuitionistic Quantum Logic of an n-level System.Martijn Caspers, Chris Heunen, Nicolaas P. Landsman & Bas Spitters - 2009 - Foundations of Physics 39 (7):731-759.details
A decade ago, Isham and Butterfield proposed a topos-theoretic approach to quantum mechanics, which meanwhile has been extended by Döring and Isham so as to provide a new mathematical foundation for all of physics. Last year, three of the present authors redeveloped and refined these ideas by combining the C*-algebraic approach to quantum theory with the so-called internal language of topos theory (Heunen et al. in arXiv:0709.4364). The goal of the present paper is to illustrate our abstract setup through the (...) concrete example of the C*-algebra M n (ℂ) of complex n×n matrices. This leads to an explicit expression for the pointfree quantum phase space Σ n and the associated logical structure and Gelfand transform of an n-level system. We also determine the pertinent non-probabilisitic state-proposition pairing (or valuation) and give a very natural topos-theoretic reformulation of the Kochen–Specker Theorem.In our approach, the nondistributive lattice ℘(M n (ℂ)) of projections in M n (ℂ) (which forms the basis of the traditional quantum logic of Birkhoff and von Neumann) is replaced by a specific distributive lattice $\mathcal{O}(\Sigma_{n})$ of functions from the poset $\mathcal{C}(M_{n}(\mathbb{C}))$ of all unital commutative C*-subalgebras C of M n (ℂ) to ℘(M n (ℂ)). The lattice $\mathcal{O}(\Sigma_{n})$ is essentially the (pointfree) topology of the quantum phase space Σ n , and as such defines a Heyting algebra. Each element of $\mathcal{O}(\Sigma_{n})$ corresponds to a "Bohrified" proposition, in the sense that to each classical context $C\in\mathcal{C}(M_{n}(\mathbb{C}))$ it associates a yes-no question (i.e. an element of the Boolean lattice ℘(C) of projections in C), rather than being a single projection as in standard quantum logic. Distributivity is recovered at the expense of the law of the excluded middle (Tertium Non Datur), whose demise is in our opinion to be welcomed, not just in intuitionistic logic in the spirit of Brouwer, but also in quantum logic in the spirit of von Neumann. (shrink)
Category Theory in Philosophy of Mathematics
Quantum Logic in Logic and Philosophy of Logic
An Intuitionistic Reformulation of Mally's Deontic Logic.Gert-Jan C. Lokhorst - 2013 - Journal of Philosophical Logic 42 (4):635-641.details
In 1926, Ernst Mally proposed a number of deontic postulates. He added them as axioms to classical propositional logic. The resulting system was unsatisfactory because it had the consequence that A is the case if and only if it is obligatory that A. We present an intuitionistic reformulation of Mally's deontic logic. We show that this system does not provide the just-mentioned objectionable theorem while most of the theorems that Mally considered acceptable are still derivable. The resulting system is unacceptable (...) as a deontic logic, but it does make sense as a lax logic in the modern sense of the word. (shrink)
Deontic Logic in Logic and Philosophy of Logic
Intuitionism and Formalism.L. E. J. Brouwer - 1913 - Bulletin of the American Mathematical Society 20:81-96.details
Intuitionistic logic and modality via topology.Leo Esakia - 2004 - Annals of Pure and Applied Logic 127 (1-3):155-170.details
In the pioneering article and two papers, written jointly with McKinsey, Tarski developed the so-called algebraic and topological frameworks for the Intuitionistic Logic and the Lewis modal system. In this paper, we present an outline of modern systems with a topological tinge. We consider topological interpretation of basic systems GL and G of the provability logic in terms of the Cantor derivative and the Hausdorff residue.
Intuitionism Disproved?Timothy Williamson - 1982 - Analysis 42 (4):203--7.details
Perennial philosophers' hopes are unlikely victims of swift, natural deduction. Yet anti-realism has been thought one. Not hoping for anti-realism myself I here show it, lest it be underestimated, to survive the following argument, adapted from W. D.Hart pp. 156, 164-5; he credits first publication to Fitch).
Dual-Intuitionistic Logic.Igor Urbas - 1996 - Notre Dame Journal of Formal Logic 37 (3):440-451.details
The sequent system LDJ is formulated using the same connectives as Gentzen's intuitionistic sequent system LJ, but is dual in the following sense: (i) whereas LJ is singular in the consequent, LDJ is singular in the antecedent; (ii) whereas LJ has the same sentential counter-theorems as classical LK but not the same theorems, LDJ has the same sentential theorems as LK but not the same counter-theorems. In particular, LDJ does not reject all contradictions and is accordingly paraconsistent. To obtain a (...) more precise mapping, both LJ and LDJ are extended by adding a "pseudo-difference" operator which is the dual of intuitionistic implication. Cut-elimination and decidability are proved for the extended systems and , and a simply consistent but -inconsistent Set Theory with Unrestricted Comprehension Schema based on LDJ is sketched. (shrink)
Cooperative Intuitionism.Stephen Ingram - 2020 - The Philosophical Quarterly 70 (281):780-799.details
According to pluralistic intuitionist theories, some of our moral beliefs are non-inferentially justified, and these beliefs come in both an a priori and an a posteriori variety. In this paper I present new support for this pluralistic form of intuitionism by examining the deeply social nature of moral inquiry. This is something that intuitionists have tended to neglect. It does play an important role in an intuitionist theory offered by Bengson, Cuneo, and Shafer-Landau (forth), but whilst they invoke the (...) social nature of moral inquiry in order to argue that ordinary moral intuitions are trustworthy, my argument focuses on what I will call the 'frontiers' of moral inquiry. I will show that inclusive and cooperative dialogue is necessary at moral inquiry's frontiers, and that intuitionists can expect such dialogue to result in both a priori and a posteriori moral beliefs. (shrink)
Empathy and Sympathy in Normative Ethics
Feminist Epistemology in Epistemology
Moral Nonnaturalism in Meta-Ethics
Moral Rationalism in Meta-Ethics
Intuitionistic N-Graphs.M. Quispe-Cruz, A. G. de Oliveira, R. J. G. B. de Queiroz & V. de Paiva - 2014 - Logic Journal of the IGPL 22 (2):274-285.details
The geometric system of deduction called N-Graphs was introduced by de Oliveira in 2001. The proofs in this system are represented by means of digraphs and, while its derivations are mostly based on Gentzen's sequent calculus, the system gets its inspiration from geometrically based systems, such as the Kneales' tables of development, Statman's proofs-as-graphs, Buss' logical flow graphs, and Girard's proof-nets. Given that all these geometric systems appeal to the classical symmetry between premises and conclusions, providing an intuitionistic version of (...) any of these is an interesting exercise in extending the range of applicability of the geometric system in question. In this article we produce an intuitionistic version of N-Graphs, based on Maehara's LJ' system, as described by Takeuti. Recall that LJ' has multiple conclusions in all but the essential intuitionistic rules, e.g., implication right and negation right. We show soundness and completeness of our intuitionistic N-Graphs with respect to LJ'. We also discuss how we expect to extend this work to a version of N-Graphs corresponding to the intuitionistic logic system FIL (Full Intuitionistic Logic) of de Paiva and Pereira and sketch future developments. (shrink)
Intuitionistic Logic according to Dijkstra's Calculus of Equational Deduction.Jaime Bohórquez V. - 2008 - Notre Dame Journal of Formal Logic 49 (4):361-384.details
Dijkstra and Scholten have proposed a formalization of classical predicate logic on a novel deductive system as an alternative to Hilbert's style of proof and Gentzen's deductive systems. In this context we call it CED (Calculus of Equational Deduction). This deductive method promotes logical equivalence over implication and shows that there are easy ways to prove predicate formulas without the introduction of hypotheses or metamathematical tools such as the deduction theorem. Moreover, syntactic considerations (in Dijkstra's words, "letting the symbols do (...) the work") have led to the "calculational style," an impressive array of techniques for elegant proof constructions. In this paper, we formalize intuitionistic predicate logic according to CED with similar success. In this system (I-CED), we prove Leibniz's principle for intuitionistic logic and also prove that any (intuitionistic) valid formula of predicate logic can be proved in I-CED. (shrink)
Intuitionistic notions of boundedness in ℕ.Fred Richman - 2009 - Mathematical Logic Quarterly 55 (1):31-36.details
We consider notions of boundedness of subsets of the natural numbers ℕ that occur when doing mathematics in the context of intuitionistic logic. We obtain a new characterization of the notion of a pseudobounded subset and we formulate the closely related notion of a detachably finite subset. We establish metric equivalents for a subset of ℕ to be detachably finite and to satisfy the ascending chain condition. Following Ishihara, we spell out the relationship between detachable finiteness and sequential continuity. Most (...) of the results do not require countable choice. (shrink)
Intuitionism and subjectivism.Mark T. Nelson - 1991 - Metaphilosophy 22 (1-2):115-121.details
I define ethical intuitionism as the view that it is appropriate to appeal to inferentially unsupported moral beliefs in the course of moral reasoning. I mention four common objections to this view, including the view that all such appeals to intuitionism collapse into "subjectivism", i.e., that they make truth in ethical theory depend on what people believe. I defend intuitionism from versions of this criticism expressed by R.M. Hare and Peter Singer.
Ethical Intuitionism and Ethical Naturalism.Nicholas Sturgeon - 2002 - In Philip Stratton-Lake (ed.), Ethical Intuitionism: Re-evaluations. Oxford University Press.details
Moral Naturalism in Meta-Ethics
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Rational intuitionism.Philip Stratton-Lake - 2012 - In Roger Crisp (ed.), The Oxford Handbook of the History of Ethics. Oxford University Press. pp. 337-357.details
In this paper I give a critical overview of the views of the main Rational Intuitionists from 18th to 20th century.
Mathematical Intuitionism and Intersubjectivity. A Critical Exposition of Arguments for Intuitionism.Tomasz Placek - 1999 - Bulletin of Symbolic Logic 8 (4):518-520.details
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CommonCrawl
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-functions
Calculation of the conditional distribution
MCMC algorithm
Conditional mean
Samples from the conditional distribution
Stopping criteria
Running the conditional distribution estimation task
In the graphical user interface
In the output folder
The conditional distribution represents the uncertainty of the individual parameter values. The conditional distribution estimation task permits to sample from this distribution. The samples are used to calculate the condition mean, or directly as estimators of the individual parameters in the plots to improve their informativeness [1]. They are also used to compute the statistical tests.
The conditional distribution is \(p(\psi_i|y_i;\hat{\theta})\) with \(\psi_i\) the individual parameters for individual i, \(\hat{\theta}\) the estimated population parameters, and \(y_i\) the data (observations) for individual i. The conditional distribution represents the uncertainty of the individual's parameter value, taking into account the information at hand for this individual:
the observed data for that individual,
the covariate values for that individual,
and the fact that the individual belongs to the population for which we have already estimated the typical parameter value (fixed effects) and the variability (standard deviation of the random effects).
It is not possible to directly calculate the probability for a given \(\psi_i\) (no closed form), but is possible to obtain samples from the distribution using a Markov-Chain Monte-Carlo procedure (MCMC).
MCMC methods are a class of algorithms for sampling from probability distributions for which direct sampling is difficult. They consist of constructing a stochastic procedure which, in its stationary state, yields draws from the probability distribution of interest. Among the MCMC class, we use the Metropolis-Hastings (MH) algorithm, which has the property of being able to sample probability distributions which can be computed up to a constant. This is the case for our conditional distribution, which can be rewritten as:
$$p(\psi_i|y_i)=\frac{p(y_i|\psi_i)p(\psi_i)}{p(y_i)}$$
\(p(y_i|\psi_i)\) is the conditional density function of the data when knowing the individual parameter values and can be computed (closed form solution). \(p(\psi_i)\) is the density function for the individual parameters and can also be computed. The likelihood \(p(y_i)\) has no closed form solution but it is constant.
In brief, the MH algorithm works in the following way: at each iteration k, a new individual parameter value is drawn from a proposal distribution for each individual. The new value is accepted with a probability that depends on \(p(\psi_i)\) and \(p(y_i|\psi_i)\). After a transition period, the algorithm reaches a stationary state where the accepted values follow the conditional distribution probability \(p(\psi_i|y_i)\). For the proposal distribution, three different distributions are used in turn with a (2,2,2) pattern (setting "Number of iterations of kernel 1/2/3" in Settings > Project Settings): the population distribution, a unidimensional Gaussian random walk, or a multidimensional Gaussian random walk. For the random walks, the variance of the Gaussian is automatically adapted to reach an optimal acceptance ratio ("target acceptance ratio" setting in Settings > Project Settings).
The draws from the conditional distribution generated by the MCMC algorithm can be used to estimate any summary statistics of the distribution (mean, standard deviation, quantiles, etc). In particular we calculate the conditional mean by averaging over all draws:
$$ \hat{\psi}_i^{mean} = \frac{1}{K}\sum_{k=1}^{K}\psi_i^{k}$$
The standard deviation of the conditional distribution is also calculated.
Among all samples from the conditional distribution, a small number (between 1 and 10, see "Simulated parameters per individual" setting) is kept to be used in the plots. These samples are unbiased estimators and they present the advantage of not being affected by shrinkage, as shown for example on the documentation of the plot "distribution of the individual parameters".
Shrinkage and the use of random samples from the conditional distribution are explained in more details here.
At iteration k, the conditional mean is calculated for each individual by averaging over all k previous iterations. The average conditional means over all individuals (noted E(X|y)), and the standard deviation of the conditional means over all individuals (noted sd(X|y)) are calculated and displayed in the pop-up window. The algorithm stops when, for all parameters, the average conditional means and standard deviations of the last 50 iterations ("Interval length" setting) do not deviate by more than 5% (2.5% in each direction, "relative interval" setting) from the average and standard deviation values at iteration k.
During the evaluation of the conditional distribution, the following plot pop-ups, displaying the average conditional means over all individuals (noted E(X|y)), and the standard deviation of the conditional means over all individuals (noted sd(X|y)) for each iteration of the MCMC algorithm.
The convergence criteria described above means that the blue line, which represents the average over all individuals of the conditional mean, must be within the tube. The tube is centered around the last value of the blue line and spans over 5% of that last value. The algorithm stops when all blue lines are in their tube.
Dependencies between tasks:
The "Population parameters" task must be run before launching the conditional distribution task.
The conditional distribution task is recommended before calculating the log-likelihood task without the linearization method (i.e log-likelihood via importance sampling).
The conditional distribution task is necessary for the statistical tests.
The samples generated during the conditional distribution task will be reused for the Standard errors task (without linearization).
In the Indiv.Param section of the Results tab, a summary of the estimated conditional mean is given (min, max and quartiles) as shown in the figure below.
To see the estimated parameter value for each individual, the user can click on the [INDIV. ESTIM.] section. Notice that the user can also see them in the output files, which can be accessed via the folder icon at the bottom left. Notice that there is a "Copy table" icon on the top of each table to copy them in Excel, Word, … The table format and display will be kept.
After having run the conditional distribution task, the following files are available:
summary.txt: contains the summary statistics (as displayed in the GUI)
IndividualParameters/estimatedIndividualParameters.txt: the individual parameters for each subject-occasion are displayed. The conditional mean (*_mean) and the standard deviation (*_sd) of the conditional distribution are added to the file.
IndividualParameters/estimatedRandomEffects.txt: the individual random effects for each subject-occasion are displayed. Those corresponding to the conditional mean (*_mean) are added to the file, together with the standard deviation (*_sd).
IndividualParameters/simulatedIndividualParameters.txt: several simulated individual parameters (draws from the conditional distribution) are recorded for each individual. The rep column permits to distinguish the several simulated parameters for each individual.
IndividualParameters/simulatedRandomEffects.txt: the random effects corresponding to the simulated individual parameters are recorded.
More details about the content of the output files can be found here.
To change the settings, you can click on the settings button next the conditional distribution task.
Interval length (default: 50): number of iterations over which the convergence criteria is checked.
Relative interval (default: 0.05): size of the interval (relative to the current average or standard deviation) in which the last "interval length" iterations must be for the stopping criteria to be met. A value at 0.05 means that over the last "interval length" iterations, the value should not vary by more than 5% (2.5% in each direction).
Simulated parameters per individual (default: via calculation): number of draws from the conditional distribution that will be used in the plots. The number is calculated as min(10, idealNb) with idealNb = max(500 / number of subject , 5000 / number of observations). This means that the maximum number is 10 (which is usually the case for small data sets). For large data sets, the number may be reduced, but the number of individual times the number of simulated parameters should be at least 500, and the number of observations times the number of simulated parameters should be at least 5000. This ensures to have a sufficiently large but not unnecessarily large number of dots in the plots such as Observations versus predictions or Correlation between random effects.
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CommonCrawl
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\begin{definition}[Definition:Limit Superior/Definition 1]
Let $\sequence {x_n}$ be a bounded sequence in $\R$.
Let $L$ be the set of all real numbers which are the limit of some subsequence of $\sequence {x_n}$.
From Existence of Maximum and Minimum of Bounded Sequence, $L$ has a maximum.
This maximum is called the '''limit superior'''.
It can be denoted:
:$\ds \map {\limsup_{n \mathop \to \infty} } {x_n} = \overline l$
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\begin{definition}[Definition:Unsatisfiable]
Let $\mathcal L$ be a logical language.
Let $\mathscr M$ be a formal semantics for $\mathcal L$.
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\begin{definition}[Definition:Hausdorff Space/Definition 2]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a '''Hausdorff space''' or '''$T_2$ space''' {{iff}} each point is the intersection of all its closed neighborhoods.
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BMC Ecology and Evolution
The composition of braconid wasp communities in three forest fragments in a tropical lowland forest of Panama
Louise A. Rodríguez ORCID: orcid.org/0000-0002-0343-46431,3 &
Enrique Medianero ORCID: orcid.org/0000-0002-8430-90342,3
BMC Ecology and Evolution volume 22, Article number: 98 (2022) Cite this article
In the last 171 years, the forests along the eastern bank of the Panama Canal have been pressured by anthropic activities. Studies of the influence of habitat fragmentation on braconid wasp communities in Central America is scarce, showing the existing information gap on these communities required to implement strategic plans for ecosystem sustainability and conservation. This study investigated how fragmentation affects braconid wasp communities in three areas in Panama City: Metropolitan Natural Park, Albrook and Corozal. Two permanent Malaise Traps were installed in the center of each fragment and were reviewed weekly from May 2019 to March 2020. Alpha and beta diversity indices and the similarity index were used to demonstrate the composition of braconid wasp communities in three forest fragments.
A similarity of 94% was estimated for the subfamily composition and 74% was estimated for the morphospecies composition of wasp community in the fragments studied. Wasp subfamily and morphospecies assemblages were more similar between fragments of Albrook and Metropolitan Natural Park. Richness and abundance of braconid wasps observed were statistically different between the fragments studied.
Richness, abundance, and composition of braconid wasps differ among habitat fragments with high similarity between subfamilies and morphospecies. Therefore, the fragments studied can be used as stepping stones to maintain remaining populations of braconid wasp communities. Monitoring is recommended to assess the effect of fragmentation on the remaining forests.
Habitat fragmentation represents one of the most serious threats to global biodiversity [1,2,3,4,5,6]. Global assessments have shown that habitat fragmentation is provoking a decrease in population size and an increased risk of disappearance of many flora and fauna [7, 8]. More than 50% of tropical forests have been degraded globally and with the use of satellite images, it is revealed that almost 43% of the terrestrial surface has been converted from its natural state for anthropogenic purposes [9, 10]. From 2000 to 2010, there was a net loss of forest cover of 7 million hectares (ha) per year in the tropical countries of the world [11]. In Panama, the forest cover has decreased from 5,245,000 ha in 1947 to 2,481,658 ha in 2019, which represents 47% forest cover loss [12]. If the fragmentation process continues at an exponential rate, the world's tropical forests could disappear completely [13].
Habitat fragmentation often occurs due to some disturbance mechanism (agriculture, deforestation, urbanization, fires, etc.) as a result of topographic differences [14]. As the world's human population continually increases, urban areas are growing rapidly and threaten the habitat of native species of flora and fauna. Studies investigating the influence of habitat fragmentation on biodiversity and the risk of species extinction are of the utmost importance and have been a main focus of biodiversity conservation research [15,16,17,18,19,20]. For instance, the abundance of dung beetle decreased with increased urban land cover [21]. Both common and rare species of social wasps are threatened by forest fragmentation in Central Amazon [22].
In tropical forests, evidence indicates that one of the taxa that respond faster to environmental changes are insects [23, 24]. Insects play important roles in almost all trophic levels; therefore, it is important to understand the response of these organisms to fragmentation. Within this taxon, parasitoids as a group may be used to assess the effects of habitat fragmentation because they play an ecological role in regulating populations of other insects due to prey denso-dependence [13]. González and Ruíz [25] proposed the use of braconid wasps (parasitoids) as indicators of biological diversity in deciduous forests and in evaluating and monitoring the effects of anthropogenic activities on ecosystems. Braconid wasps are regulatory agents of various groups of herbivorous insects that indicate the presence or absence of other species through the food chain [26]. Most braconid wasps are endo and exo parasitoids which feed on the larval stages of Coleoptera, Diptera and Lepidoptera [27]. This makes braconid wasps good biological indicators of habitat disturbances [25].
Smith and Mayfield [28] studied the taxonomic and functional diversity of bees visiting flowers of three tree species in small and large tropical forest fragments in tropical Australian landscapes. Species and functional diversity were found to differ significantly between small and large fragments. There was less taxonomic diversity of bees visiting flowers in small fragments. Additionally, native eusocial stingless bees were not common on small remains despite the presence of floral resources similar to those sampled on large remains [28].
In a similar study, Ruiz-Guerra et al. [29] studied the abundance, species richness, similarity, and prevalence of braconid parasitoid wasps for four types of land use (secondary forest, plantations, live fences, and pastures) and preserved tropical humid forest remnants in southern Mexico. Species richness and abundance were found to be higher in preserved and secondary forests than in other land use types.
To establish more direct links with ecosystem processes, there is a need to investigate patterns of functional trait and taxonomic diversity with biological indicators. Fragmentation experiments are useful tools used to provide clear evidence of the strong and typically degrading impacts of biodiversity loss [30]. Studies on the influence of habitat fragmentation on braconid wasp communities in lowland forests of Central America are scarce, portraying the existing information gap on these communities required to implement strategic plans for ecosystem sustainability and conservation. It is expected that landscapes with an intermediate degree of fragmentation will cause separation of braconid wasp communities which is reflected by a low similarity between communities. This research study contributes to the construction of baseline data by evaluating the vulnerability of braconid wasp communities leading to strategic plans for the sustainability and conservation of ecosystems. We sought to determine how habitat fragmentation may affect braconid wasp communities in fragmented lowland forest locations in Panama using similarity and fragmentation indices.
A total of 1697 individual wasps belonging to 77 morphospecies and 16 subfamilies were recorded. Of the 1697 individuals, approximately 39% were collected in the PNM fragment, 36% in the fragment of COR and 25% in the fragment of ALB (Table 1). Among the three fragments, Rogadinae was found to be the most abundant subfamily with 456 individual wasps, followed by Alysiinae with 391 individual wasps, Adeliinae with 254 individual wasps, Doryctinae with 168 individual wasps, Aphidiinae with 154 individual wasps and Microgastrinae with 108 individual wasps. In the PNM fragment, 664 individual wasps and 14 subfamilies were observed (Table 1); with the most abundant subfamily being Adeliinae with 119 individual wasps, followed by Doryctinae with 114 individual wasps, Rogadinae with 106 individual wasps, Alysiinae with 89 individual wasps and Microgastrinae with 85 individual wasps (Fig. 1). In the COR fragment, 603 individual wasps and 13 subfamilies were observed (Table 1); with the most abundant subfamily being Alysiinae with 268 individual wasps, followed by Rogadinae with 160 individual wasps, Aphidiinae with 53 individual wasps and Adeliinae with 48 individual wasps (Fig. 1). In the ALB fragment, 430 individual wasps and 16 subfamilies were observed (Table 1); with the most abundant subfamily being Rogadinae with 190 individual wasps, followed by Adeliinae with 87 individual wasps, Alysiinae with 34 individual wasps and Aphidiinae with 27 individual wasps (Fig. 1).
Table 1 Numbers of subfamilies, morphospecies and individuals of braconid wasps in the three fragments studied during the years 2019–2020
Variation in individual abundance of braconid wasp in the three habitat fragments studied in lowland tropical forests of Panama. Each panel shows data for a habitat fragment and is grouped by subfamily. Stacked bar plots show the dominant subfamilies within each habitat fragment
Among the three fragments, the most abundant morphospecies belonged to the subfamilies Rogadinae (M 112 y M1), Adeliinae (M 117), Microgastrinae (M 118), Aphidiinae (M 130) and Alysiinae (M 152), respectively (Table 2). The aforementioned morphospecies account for approximately 65% of the total individual wasp separated as morphospecies. In the PNM fragment, the most abundant morphospecies belonged to the subfamilies Rogadinae (M 117) and Microgastrinae (M 118) with 80 individual wasps for each, followed by Alysiinae (M 152) with 75 individual wasps, Doryctinae (M 128) with 73 individual wasps and Aphidiinae (M 130) with 73 individual wasps (Table 2). In the COR fragment, the most abundant morphospecies belonged to the subfamilies Rogadinae (M 112, M 1 and M 29) with 387 individual wasps and Aphidiinae (M 113) with 32 individual wasps (Table 2). In the ALB fragment, the most abundant morphospecies belonged to the subfamilies Rogadinae (M 1, M117, M5 and M2) with 215 individual wasps and Adelinae (M 116) with 21 individual wasps (Table 2).
Table 2 Abundance and relative abundance of morphospecies in the three fragments studied during the years 2019–2020
The Margalef index value indicated a higher species richness in the PNM, followed by ALB and COR, respectively (Table 3). The range of D´ values were close to 1 in the three fragments studied, which represents a lower degree of dominance/predominance of part of one or two species, which is also interpreted as a high diversity (Table 3). The values found with this index showed that the fragments with the highest diversity are ALB and PNM and that of COR showed a lower diversity. The H´ index indicated that the fragments with the greatest diversity were ALB and PNM and the lowest value was found in the COR fragment. According to the range of J´values all the fragments were proportional to the diversity and those that presented a more equitable distribution/pairing were ALB and PNM (Table 3). The fragment that presented less equality was COR.
Table 3 Indices of α diversity of braconid wasps in three habitat fragments in Panama
According to the Whittaker index, the highest turnover value in the composition of morphospecies recorded was between the COR and PNM fragments, followed by COR and ALB, and finally the ALB and PNM fragments (Table 4). The highest turnover value in the composition of subfamilies recorded was between the COR and PNM fragments, followed by COR and ALB, and finally the ALB and PNM fragments (Table 5).
Table 4 Whittaker's β diversity index of morphospecies of braconid wasps in three habitat fragments in Panama
Table 5 Whittaker's β diversity index of subfamilies of braconid wasps in three habitat fragments in Panama
The Bray–Curtis similarity analysis showed a morphospecies similarity and variation of approximately 0.5706 among the three fragments studied (Fig. 2). In the dendrogram it can be seen that there are two groupings where the morphospecies composition is more similar between the COR and ALB fragments with 0.51 (Fig. 2). The Bray–Curtis similarity analysis showed a similarity and variation of subfamilies of approximately 0.6496 among the three fragments studied (Fig. 3). In the dendrogram it can be seen that there are two groupings where the subfamily composition is more similar between the PNM and ALB fragments with 0.63 (Fig. 3).
Similarity dendrogram (Bray–Curtis) of the Braconidae morphospecies found in the three fragments studied
Similarity dendrogram (Bray–Curtis) of the Braconidae subfamilies found in the three fragments studied
The results obtained by means of the Kruskal–Wallis test determined that richness (H = 8.22, gl = 2, p = 0.015) and abundance (H = 12.95, gl = 2, p < 0.05) of braconid subfamilies observed were statistically different between the fragments. The results obtained by means of the Kruskal–Wallis test determined that richness (H = 6.053, gl = 2, p < 0.05) and abundance (H = 12.95, gl = 2, p < 0.05) of braconid morphospecies observed were statistically different between the fragments.
In the correspondence analysis (CA), 100% cumulative percentage was explained for both morphospecies and subfamilies in the first two axes. A total inertia of 0.286 was calculated for braconid subfamilies and a total inertia of 0.469 for braconid morphospecies. Wasp assemblages (both morphospecies and subfamilies) were more similar between the ALB and PNM fragment. The data obtained demonstrated distinct clustering in the three fragments, as depicted in Fig. 4 for subfamilies. The data also demonstrated that the subfamily Euphorinae was only present in the fragment of ALB and Meteorinae was present in the remnants of ALB and COR.
Correspondence analysis (CA) to compare the similarity of braconid wasp communities between habitat remnants of Corozal (COR), Albrook (ALB) and Metropolitan Natural Park (PNM). The two replicates of each habitat remnant are enveloped to make similarities among habitat remnants more apparent
Of the 16 subfamilies, 12 were shared between all fragments studied, two between the ALB and PNM fragments, and one between the ALB and COR fragments (Fig. 5). One subfamily was registered in the fragment of ALB which was not shared between the fragments studied. Of the 77 morphospecies, 29 were shared between all fragments, 11 between the ALB and PNM fragments, three between the COR and PNM fragments and three between the ALB and COR fragments (Fig. 6). Nine registered morphospecies were only found in the ALB fragment, nine different morphospecies were only found in the COR fragment and 13 different morphospecies were found in the PNM fragment, all of which were not shared between the fragments.
Venn diagram of the number of shared subfamilies of braconid wasps in the three fragments studied in lowland tropical forests of Panama: Corozal (COR), Albrook (ALB) and Metropolitan Natural Park (PNM). Of the 16 subfamilies, 12 were shared between the three habitat fragments studied
Venn diagram of the number of shared morphospecies of braconid wasps in the three fragments studied in lowland tropical forests of Panama: Corozal (COR), Albrook (ALB) and Metropolitan Natural Park (PNM). Of the 77 morphospecies, 29 were shared between the three habitat fragments studied
According to the Diserud-Odegaard Index, a similarity of 0.9418 (94%) was estimated for subfamily composition in the fragments studied. Likewise, a similarity of 0.7401 (74%) was estimated for morphospecies composition in the fragments studied. These results depicted similar subfamily and morphospecies composition within and between the fragments studied.
Braconid wasp communities in fragmented lowland tropical forests of Panama are still very similar, both at the subfamily and morphospecies level. The fragments studied are considered to be areas large enough to maintain the biodiversity of wasp communities. The results of this study are consistent with those obtained by Valdés [31] and Ruiz-Guerra et al. [29]. Ruiz-Guerra et al. [29] demonstrated that Braconidae communities are very similar in the remnants of conserved tropical forests and secondary forests of Mexico. Valdés [31] depicted that butterfly communities are very similar in lowland tropical forests of Panama. In the study by Valdés [31], three of the four habitat fragments compared correspond to those used in the present study. The results of Valdés [31], indicated that when these three sites were compared, a similarity of 97% was obtained for the butterfly community. The proposed explanation stems from the species' dispersal ability; since these taxa have the ability to fly, these organisms are able to move freely between fragments and as a result maintain high similarities between communities.
Richness and abundance of braconid subfamilies were statistically different between the fragments studied. The higher richness of braconid wasps found in the fragment of ALB can be explained by the heterogeneous vegetation, composed of open grassland, stubble, border vegetation and secondary forests in this fragment. The higher abundance and heterogeneity in the fragment of PNM can be explained by its proximity to a source; such as the remainder of the PNM. This is seen with the island-continent model, where local populations may be unequal in size and longevity, with one large fragment from which dispersers migrate to other fragments [32, 33]. The PNM habitat fragment is a part of the PNM; however, it has been isolated due to the development of highways. The habitat fragment however, is still considered a part of the PNM; where the remainder of the park is considered as the continent/source and the habitat fragment as the island/sink. Both fragments offer a wide range of microhabitats for organisms, allowing the survival of more individuals and increasing the availability and diversity of hosts [22]..The low richness in the fragment of COR demonstrates the introduction of biodiversity loss due to the process of fragmentation. This highlights the importance of maintaining continuous forests close to other remnants and the need to conserve fragments which provide various habitats for maintenance of species diversity.
The theory provided by the finding of braconid wasp communities is that of metapopulations [34], which assumes that species are distributed over a heterogenous space and not all territories are habitable for each species; as seen with the subfamilies of Euphorinae, Braconinae, Helconinae and Meterorinae. The fragments studied were divided into remnants/patches at a given moment primarily due to the effects of urbanization and industrialization, which separated the populations of braconid wasps forming their metapopulations. The model of the theory that supports this finding is that of patched populations [32, 34], where there are similar patches without clear distinction between sources and sinks. Two suggestions for which there is no distinction may be that the fragments have the same probability of being colonized by braconid wasps due to their dispersal ability and all fragments studied were of similar size. The fragments studied are considered to be areas large enough to keep the metapopulations of wasps in equilibrium. However, it would be necessary to prove how the dispersal dynamics of braconid wasps is carried out among fragments to prove that the fragments are indeed patched populations and determine the minimum critical size of the ecosystem to preserve the diversity and composition of species.
The results of this study also indicate that the communities of wasps (Hymenoptera: Braconidae) in the lowland forest fragments of Panama are mainly constituted by the subfamilies of Rogadinae, Alysiinae, Adeliinae, Doryctinae, Aphidiinae and Microgastrinae. The presence of these subfamilies is noteworthy since it implies interspecific relationships by parasitism between the species and other arthropods and implies that the fragments are diverse. The Microgastrinae, Adeliinae and Rogadinae subfamilies indicate interspecific relationships with Lepidoptera as they are endo parasitoids of larvae of the order Lepidoptera [35]. The Alysiinae subfamily indicates interspecific relationships with Diptera as they are endo parasitoids of the larvae or eggs of the order Diptera [35]. The Aphidiinae subfamily indicates interspecific relationships with the Stenorrhyncha order, particularly the Aphididae family [35]. The Doryctinae subfamily indicates both interspecific relationships and diversification in the fragments since they are ecto parasitoids of the larvae of the orders Coleoptera, Lepidoptera, Hymenoptera, Embioptera and Phytophagia [35]. The Doryctinae subfamily has been reported to dominate in a dry tropical forest in Mexico [36] and in a semi-deciduous forest in Venezuela [37].
There are two major limitations in this study that could be addressed in future research. First, although the appropriate trap was used, it is recommended to complement the research with another technique. An interesting technique for this community may be light traps [38, 39]. Species that are nocturnal are always attracted to light. Secondly, although the research demonstrated the composition of braconid wasp communities in three forest fragments, the research did not demonstrate the effect of habitat fragmentation on species communities. This can be demonstrated by comparing fragmented landscapes to a continuous forest or a gradient of fragment size.
Taking this into account, effective planning for the conservation and preservation of the fragments studied require learning and adaptation. A substantial set of theoretical and practical guides have been developed to maintain biodiversity and ecosystem function [40], as well as operational models that use these guides for systematic conservation ([33]; Fig. 7). With the aid of the operational model for conservation planning by Knight et al. [41], it is suggested that the fragments be considered as stepping stones. Stepping stones can improve the persistence of metapopulations allowing the flow of individuals between fragments, ensuring the exchange of stochastic local extinction and recolonization [42]. The preservation of these fairly conserved fragments, with heterogenous vegetation, can favor the presence of organisms by offering various microhabitats to ensure the viability of metapopulations. Baum et al. [43] showed that a matrix can determine if, and to what extent, corridors and stepping stones, increase the connectivity of a landscape for the survival of species using Prokelisia crocea and Spartina pectinata as indicators. Spatial configuration is particularly important for regional dynamics and must be taken into account in management plans. Small and medium fragments play a fundamental role; in such a way that it is necessary to identify and preserve these fragments.
Operational model for conservation planning, incorporating assessment, and management phases (Grantham et al. 2010; Knight et al. 2006)
The results of this study indicate that the richness, abundance, and composition of braconid wasps differ among habitat fragments in lowland tropical forests of Panama. Forest fragments have relatively high similarity of braconid wasp subfamilies and morphospecies. This finding was interpreted as an indication that a species' dispersal ability plays a major role in its survival.
Annual inventories to assess the real change in braconid wasp communities or other organisms could provide critical monitoring of the effects of habitat fragmentation. These results can help to increase the understanding of the influence of habitat fragmentation on braconid wasp communities in Panama to develop successful conservation strategies. This would make it possible to address the question of the effect of fragmentation on the braconid wasp species.
The three selected study sites in Panama City were Metropolitan Natural Park (PNM), public land in the town of Albrook (ALB) and public land in the town of Corozal (COR) (Fig. 8). Each fragment was bordered and labeled; as depicted in the map constructed with Google Earth. The PNM fragment is situated 8°59′41.55′′N and 79°32′35.22′′ W with an area of approximately 18.12 ha and a perimeter of 1.756 km. Vegetation is characterized as a mixture of tropical humid forest and lowland tropical dry forest, with few areas of stubble and grasslands, and a well-defined stratum. The ALB fragment is situated 8°58′37.49′′N and 79°33′43.82′′W with an area of approximately 34.79 ha and a perimeter of 5.003 km. Vegetation is characterized as heterogenous, composed of open grasslands, stubble and secondary forests. The COR fragment is situated 8°59′19.34′′N and 79°34′11.83′′W with an area of approximately 56.31 ha and a perimeter of 3.028 km. Vegetation is characterized as herbaceous with late secondary forest and some open areas. In this study, the habitat fragmentation definition defined by Saunders et al. [44] was used, where they described a fragment as any patch of native vegetation around which most of the original vegetation has been removed. For this reason, well-preserved fragments that are separated by a matrix of urban areas were considered as fragments in Panama City. The selected fragments are close in proximity to each other, which guarantees that the results obtained are effects of the fragmentation process of an original matrix and not the natural result caused by the distances between them. The distance between the fragments of ALB and COR is 1.129 km. The distance between the fragments of COR and PNM is 2.565 km. The distance between the fragments of ALB and PNM is 2.329 km.
Map of the Pacific Basin of the Panama Canal where the habitat fragments studied in lowland tropical forests of Panama are located: Corozal (COR), Albrook (ALB) and Metropolitan Natural Park (PNM). Google Earth was used in order to construct the map
The study sites form part of a biological corridor that runs along the eastern bank of the Panama Canal [45]. In accordance with the UNESCO classification, the three study sites are characterized as lowland tropical semi-deciduous forests. The three sites present annual average temperatures of 26.4 °C, with an annual average precipitation between 1501 to 1800 mm and altitudes between 20 to 150 masl. In the last 171 years, the forests along the eastern bank of the Panama Canal have been pressured by anthropic activities [46]. One of the first anthropic impact along the banks of the Panama Canal was the construction of Panama's railway in 1850. The second anthropic impact along the banks of the Panama Canal was the construction of the French Canal in 1881. Nevertheless, the greatest anthropic impact started in the 1900's when the forests along the eastern bank of the Panama Canal were intervened by military bases [46]. During this time, the remaining fragments maintained advanced secondary forest vegetation. In the last 50 years, these sites have been anthropically pressured due to the construction of neighborhoods along the banks of the Panama Canal [46]. The COR fragment has been used for Panama government security training activities. The ALB fragment has minimal human traffic; however, it contains a water reservoir at the peak of the fragment. The PNM fragment is along a hiking trail in a protected area but with visitor traffic [47].
Sampling protocol
Two permanent lightweight Malaise Trap, Townes Style separated by approximately 0.2 km were installed in the center of each fragment and wasp samples were collected weekly from May 2019 to March 2020. The traps were made with organza fabric, with dimensions of 5.8 ft tall by 5.4 ft long, and contained a polyethylene collector bottle with 95% alcohol at its highest end [48]. This is a trap designed to collect fast-flying insects whose behavior is to fly upwards when it touches a surface. The Malaise Trap, Townes Style has been widely used in braconid wasps diversity studies in Central America and the world [49,50,51,52,53]. A reasonable flat, log-less area of approximately 2 m2 was chosen for the placement location of each trap in the forest. The collection bottle was placed facing the magnetic north and half-filled with 95% ethanol. The traps were placed in open areas which served as insect corridors. To exclude the edge effect, a safety distance of approximately 0.2 km was defined from the border of each fragment towards the center. Samples were collected every 7 days; resulting in a sampling effort of 44 weeks per trap for the period between 2019 and 2020.
Samples were taken to the Master's degree laboratory at the University of Panama, where braconid specimens were separated from the rest of the sample and placed in vials of one dran with 95% ethanol. All braconid individuals were mounted on entomological pins, number 2 or 3. Braconid individuals were first sorted as morphospecies (M) using the method of Oliver and Beattie [54]. Oliver and Beattie [54] demonstrated that comparisons can be made using morphospecies assemblages as long as each contain a unique identification. Samples were then identified to subfamily level using the Sharkey and Campos taxonomic key from the book Aguilar et al. [35] and the manual by Sharkey et al. [26].
In order to measure the diversity of subfamilies and morphospecies over a special scale, alpha (α) and beta (β) diversity were calculated. α diversity is the species richness of a particular community that is consider to be homogeneous. Currently there are many indices to measure α diversity. In this study, to determine the α of each fragment, the Simpson dominance index (D′), Shannon-Weiner Index (H′), Equity index or Pielou equity (J′) and Margalef diversity index were used. D′ takes into account the most important species without considering the rest of the species.
$${D}^{^{\prime}}=1-\sum ({{p}_{i})}^{2},$$
where pi is the proportional abundance of species i (the number of individuals of species i divided by the total number of individuals in the sample).
H' combines information on species richness and equity in what is called diversity or heterogeneity [55, 56]. The average degree of uncertainty is measured in predicting the species to which a given randomly chosen individual within a biotic community belongs.
$$H^{\prime}= -\sum_{i=1}{p}_{i} ln {p}_{i},$$
where ln is the natural logarithm and pi is the proportional abundance of species i (the number of individuals of species i divided by the total number of individuals in the sample).
J′ is the maximum possible diversity for a given number of species that occurs if all species are present in equal numbers. Its value ranges from 0 to 1, so that 1 corresponds to situations where all species are equally abundant.
$${J}^{^{\prime}}=\frac{H{^{\prime}}}{{H}^{^{\prime}}max},$$
where H' max is the lnS, H' is the Shannon-Weiner index.
Margalef diversity index estimates the biodiversity of a community based on the numerical distribution of the individuals of the different species based on the number of individuals in the analyzed sample.
$${D}_{Mg}=\frac{S-1}{\mathrm{ln}N},$$
where S is the number of species and N is the total number of individuals.
The β diversity is the diversity of species between communities. It is the degree of species replacement or biotic change through environmental gradients [57, 58]. The indices used to determine β diversity are approaches based on pairwise similarities. To determine the β of each fragment, the Whittaker and Bray–Curtis Indices were used. The Bray Curtis Index is a modified version of the Sorensen Index [59]
$${S}_{S}=\frac{2a}{2a+b+c},$$
where a is the number of species present in both samples, b is the number of species found in community A and c is the number of species found in community B. Whittaker index estimates the degree of species replacement or biotic change through environmental gradients [60]. The α and β diversity indices were estimated using the EstimateS and the "vegan" package from RStudio.
For analysis of the data, all samples were pooled monthly, resulting in one sample per site. A Kruskal–Wallis test was performed using the program STATISTICA [61], to determine differences between the number of individuals registered in each study site. Correspondence analysis (CA) calculated by the program XL-Stat [62] was used to characterize the braconid community based on the number of individuals and site. A comparison of the composition of the braconid community was calculated using formula for the Multiple Similarity Index by Diserud and Odegaard [63], as given by the Eq. (2):
$${\text{C}}_{\text{S}}^{\text{T}}\text{=}\frac{\text{T}}{\text{T-1}}\left(\text{1} - \frac{{\text{S}}_{\text{T}}}{\sum_{\text{i}}{{\text{a}}}_{\text{i}}}\right),$$
where ai is the number of species in site Ai, T is the number of sites and ST = \(\sum_{i}{a}_{i}-\sum_{i<j}{a}_{ij}+ \sum_{i<j<k}{a}_{ijk}\dots ,\) aij is the number of species shared by sites Ai and Aj; and aijk is the number of species shared by sites Ai, Aj and Ak, etc. The multiple similarity index takes into consideration the information of species shared by two or more sites and avoids the problem of covariance between pairwise similarities [63], reducing the probability of Type II errors. The multiple similarity index avoids the loss of information concerning the number of species shared among three or more sites and the lack of independence between pairwise similarities due to the repetition of each site in several pairs as seen with β diversity indices [63]. The calculation of this index was done manually.
ALB:
CA:
Corresponding analysis
FRAG:
Fragmentation Index
Morphospecies
M_#:
Morphospecies number
PNM:
Metropolitan Natural Park
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To the University of Panama, Secretaría Nacional de Ciencia Tecnología e Innovación (SENACYT) and Sistema Nacional de Investigación (SNI) for providing funds and resources to carry out research activities. To MIAmbiente (Ministry of Environment) and Metropolitan Natural Park for issuing collection permits. To Marcelo Mack and JeanCarlos Abrego for their help with sample collection.
This research was funded by Secretaría Nacional de Ciencia, Tecnología e Innovación (SENACYT), Sistema Nacional de Investigación (SNI) and the University of Panama (UP). The funding bodies played no role in the design of the study collection, analysis and interpretation of data and in writing the manuscript.
Programa de Maestría en Ciencias Biológicas, Universidad de Panamá, Campus Octavio Méndez Pereira, Avenida Transístmica, Panama City, Panama
Louise A. Rodríguez
Departamento de Ciencias Ambientales and Programa de Maestría en Entomología, Universidad de Panamá, Campus Octavio Méndez Pereira, Avenida Transístmica , Panama City, Panama
Enrique Medianero
Miembro del Sistema Nacional de Investigación (SNI-SENACYT), Panama City, Panama
Louise A. Rodríguez & Enrique Medianero
LAR and EM contributed to the study conception and design. Material preparation, data collection and analysis were performed by LAR. The first draft of the manuscript was written by LAR. EM and LAR commented on previous versions of the manuscript and have substantively revised it. EM and LAR read and approved the final manuscript.
Correspondence to Enrique Medianero.
A scientific collecting permit was obtained from the Ministry of Environment to collect specimens. A letter of authorization was also obtained from the manager of Metropolitan Natural Park to collect specimens inside the park.
Consent to publication
Rodríguez, L.A., Medianero, E. The composition of braconid wasp communities in three forest fragments in a tropical lowland forest of Panama. BMC Ecol Evo 22, 98 (2022). https://doi.org/10.1186/s12862-022-02051-4
Braconidae
Forest fragments
Parasitoid
Similarity Index
Biodiversity, Ecosystem Services and Sustainable Development
Submission enquiries: [email protected]
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\begin{document}
\title{When geometric phases turn topological}
\author{P.{} Aguilar} \email{[email protected]} \affiliation{Instituto de Ciencias Nucleares \\
Universidad Nacional Aut\'onoma de M\'exico\\
PO Box 70-543, 04510, CDMX, M\'exico.}
\author{C.{} Chryssomalakos} \email{[email protected]} \affiliation{Instituto de Ciencias Nucleares \\
Universidad Nacional Aut\'onoma de M\'exico\\
PO Box 70-543, 04510, CDMX, M\'exico.}
\author{E.{} Guzm\'an-Gonz\'alez} \email{[email protected]} \affiliation{Instituto de Ciencias Nucleares \\
Universidad Nacional Aut\'onoma de M\'exico\\
PO Box 70-543, 04510, CDMX, M\'exico.}
\author{L.{} Hanotel} \email{[email protected]} \affiliation{Instituto de Ciencias Nucleares \\
Universidad Nacional Aut\'onoma de M\'exico\\
PO Box 70-543, 04510, CDMX, M\'exico.}
\author{E.{} Serrano-Ens\'astiga} \email{[email protected]} \affiliation{Institut f\"ur Theoretische Physik \\
Universit\"at T\"ubingen\\
72076, T\"ubingen, Germany}
\begin{abstract}
\noindent Geometric phases, accumulated when a quantum system traces a cycle in quantum state space, do not depend on the parametrization of the cyclic path, but do depend on the path itself. In the presence of noise that deforms the path, the phase gets affected, compromising the robustness of possible applications, \emph{e.g.}, in quantum computing. We show that for a special class of spin states, called anticoherent, and for paths that correspond to a sequence of rotations in physical space, the phase only depends on topological characteristics of the path, in particular, its homotopy class, and is therefore immune to noise. \end{abstract}
\maketitle
\section{Introduction} \label{Intro}
Cyclic evolution of a quantum system gives rise to a geometric phase $\varphi_\text{geo}$, invariant under time-reparametrizations of the path $C(t)$ taken by the state of the system in the corresponding projective Hilbert space $\mathbb{P}(\mathcal{H}) \equiv \mathbb{P}$~\cite{Ber:84,Sim:83,Aha.Ana:87,Ber:89,Sam.Bha:88,Muk.Sim:93}. Motivated by this, proposals have been made to use such phases in quantum computation, the invariance mentioned translating, in this case, in noise resilience (see, \hbox{\em e.g.{}},~\cite{Zan.Ras:99,Jon.Ved.Eke.Cas:00,Pac.Zan:01,Ota.Kon:09}). However, whatever noise exists, affects not only the time-parametrization of $C(t)$, but also its form, and this residual effect will, in general, leave its imprint on $\varphi_\text{geo}$, to the detriment of the robustness of the computation. As a concrete mental picture, consider a spin-1/2 particle in the presence of a time-varying magnetic field $\mathbf{B}(t)=\mathbf{B}_0(t)=B \hat{\mathbf{n}}_0(t)$, starting out, at $t=0$, in the eigenstate $\ket{\hat{\mathbf{n}}_0(0) +}$ and following adiabatically $\mathbf{B}_0(t)$ for $t>0$. For a cyclic evolution, $\hat{\mathbf{n}}_0(T)=\hat{\mathbf{n}}_0(0)$, the phase $\varphi_\text{geo}$ is proportional to the area enclosed by the curve $C_0(t)=\hat{\mathbf{n}}_0(t)$ on the unit sphere. In the presence of noise in the field, $\mathbf{B}(t)=\mathbf{B}_0(t)+\epsilon \mathbf{B}_1(t)$, the component of $\mathbf{B}_1$ along the tangent to $C_0(t)$ induces reparametrization of $C_0(t)$ and, hence, leaves $\varphi_\text{geo}$ invariant, but the normal component changes the shape of $C_0(t)$ and modifies, in general, $\varphi_\text{geo}$. The statistics of $\varphi_\text{geo}$ were first computed analytically in~\cite{DeC.Pal:03}, considering $\mathbf{B}_1$ as a classical stochastic field, with subsequent numerical~\cite{Fil:08} and experimental~\cite{Fil.etal:09} confirmations of the results, while the subtler case of $\mathbf{B}_1$ being a quantum operator was treated in~\cite{Agu.Chr.Guz:16}, with some qualitative differences showing up (see also~\cite{Fue.Car.Bos.Ved:02} for earlier work treating $\mathbf{B}$ as a quantum field, \cite{Liu.Fen.Wan:11,Lar:12,Wan.Wei.Lia:15} for subsequent theoretical debate, and the recent experimental observation reported in~\cite{Gas.Ber.Abd.Pec.Fil.Wal:16}).
Our focal point in this letter is to outline a scenario in which the presence of parametric noise has no effect whatsoever on the geometric phase. The mechanism presented, and possible generalizations to the non-abelian case~\cite{Wil.Zee:84,Ana:88}, could be of interest in holonomic quantum computing~\cite{Zan.Ras:99}. The setup is best presented via Majorana's stellar representation and involves anticoherent spin states --- both concepts are now briefly explained.
In a little known 1932 paper~\cite{Maj:32}, Majorana pointed out that spin-$s$ quantum states can be labeled by a constellation of $2s$ points on the unit sphere. The recipe given is simple as it is cryptic: expand the state in question $\ket{\psi}$ in the $S_z$ eigenbasis, $\ket{\psi}=\sum_{k=-s}^s c_k \ket{s,k}$, and use the expansion coefficients to write down a polynomial in an auxiliary variable $\zeta$, \begin{equation} \label{Ppsidef} P_{\psi}(\zeta) = \sum_{k=0}^{2s} (-1)^{2s-k} c_k \sqrt{\binom{2s}{k}} \zeta^k \, , \end{equation} the roots of which, stereographically projected from the south pole onto the unit sphere $S^2$, supply the \emph{Majorana constellation} of $\ket{\psi}$. When $\ket{\psi}$ is transformed in Hilbert space by the unitary representation $D(R)$ of a rotation $R \in SO(3)$, the constellation rotates by $R$ on $S^2$. Additional information about the Majorana constellation may be found in~\cite{Ben.Zyc:17,Chr.Guz.Ser:18} while interesting applications appear in, \hbox{\em e.g.{}},~\cite{Bar.Tur.Dem:06,Bar.Tur.Dem:07,Mak.Suo:07}.
A \emph{spin coherent} state $\ket{\mathbf{n}}$, where $\mathbf{n}$ lives on the 2-sphere $S^2$, is an eigenstate of $\mathbf{n} \cdot \mathbf{S}$ with eigenvalue $s$ ($\mathbf{S}$ denotes the spin-$s$ representation of the generators of $\mathfrak{su}(2)$)~\cite{Per:86,Rad:71,Chr.Guz.Ser:18}. Spin coherent states maximize the modulus of the spin expectation value and are, in many respects, the ``most classical'' spin states available. Their Majorana constellation consists in $2s$ coincident stars in the direction $\mathbf{n}$ --- intuitively, they are the most directional states possible. In~\cite{Zim:06}, Zimba considered the natural question of which spin states should be declared the ``most quantum'', or, as he aptly named them, ``anticoherent'' --- one would expect these to correspond to constellations spread out ``as uniformly as possible'' over $S^2$. Analytically, the natural requirement, adopted by Zimba, for an anticoherent state $\ket{\psi}$ is that its spin expectation value vanish, $\bra{\psi} \mathbf{n} \cdot \mathbf{S} \ket{\psi}=0$, for all $\mathbf{n}$ in $S^2$. At around the same time, those same states showed up in the classification of the different phases of bosonic condensates~\cite{Bar.Tur.Dem:06,Bar.Tur.Dem:07}, as well as the study of the so called inert states~\cite{Mak.Suo:07}.
\section{When geometric phases turn topological} \label{Wgptt}
The formulation of geometric phases we adopt here is the one put forth in~\cite{Muk.Sim:93}, which generalizes the original setup of~\cite{Ber:84} to the case of non-cyclic, non-adiabatic evolutions. Given a curve $C(t)=[\psi(t)]$ in $\mathbb{P}$, with $0 \leq t \leq 1$, a geometric phase $\varphi_\text{geo}$ may be associated to it via \begin{align} \label{gpMS} \varphi_\text{geo} &= \varphi_\text{tot}+\varphi_{\text{dyn}} \nonumber \\ &= \arg \braket{\psi(0)}{\psi(1)} + i\int_\mathcal{C} dt \, \bra{\psi(t)} \partial_t \ket{\psi(t)} \, , \end{align} where $\mathcal{C}(t) \equiv \ket{\psi(t)}$ is an arbitrary lift of $C(t)$ in the Hilbert space $\mathcal{H}$, and the two terms on the right hand side are known as the \emph{total} and \emph{dynamical} phase, respectively. It can be shown easily that $\varphi_\text{geo}$ does not depend on the lift, and is, therefore, a property of $C(t)$. It is also easy to show that $\varphi_\text{geo}$ does not change under a reparametrization of $C(t)$, $C(t) \rightarrow C'(t)=C(s(t))$, with $s(t)$ a monotonically increasing function of $t$. Under more general perturbations of $C(t)$ though, that affect the locus of points the curve passes through, $\varphi_\text{geo}$ does change --- when such perturbations are infinitesimal, and leave the endpoints fixed, only the second term of~(\ref{gpMS}), \hbox{\em i.e.{}}, the dynamical phase, contributes to the change of $\varphi_\text{geo}$.
We specify now the above to the case in which $C(t)$ is the path taken by a spin-$s$ state under a sequence of rotations $R(t) \in SO(3)$, $C(t)=[\psi(t)]$, with \begin{equation} \label{psitRpsi0} \ket{\psi(t)}=D(R(t)) \ket{\psi(0)} \, , \end{equation} where $D(R(t))$ is the spin-$s$ irreducible representation of $R(t)$. Taking into account that for $D(R(t))=e^{-i \mathbf{m}(t) \cdot \mathbf{S}}$, \begin{equation} \label{derDRt} \partial_t D(R(t)) = -i (\tilde{\mathbf{m}}(t)\cdot \mathbf{S}) \, D(R(t)) \end{equation} holds, with $\tilde{\mathbf{m}}(t)$ a complicated function of $\mathbf{m}(t)$, the details of which are not essential to our argument, $\varphi_\text{dyn}$ in the \hbox{r.h.s.{}}{} of~(\ref{gpMS}) becomes \begin{equation} \varphi_\text{dyn} = \int_{t_\text{i}}^{t_\text{f}} dt \, \bra{\psi(t)} \tilde{\mathbf{m}}(t) \cdot \mathbf{S} \ket{\psi(t)} = 0 \label{phidynzero} \, , \end{equation} the second equality only being valid when $\ket{\psi(0)}$, and, hence, $\ket{\psi(t)}$, are anticoherent. Then $\varphi_\text{geo}$ in~(\ref{gpMS}) reduces to $\varphi_\text{tot}$, depending only on the endpoints $\ket{\psi(0)}$, $\ket{\psi(1)}$ of the lift $\mathcal{C}(t)$. It is important to keep in mind that $\mathcal{C}(t)$ is not an arbitrary lift of $C(t)$, rather, it is uniquely determined by~(\ref{psitRpsi0}) once the phase of, say, $\ket{\psi(0)}$ has been chosen. It is also clear that $\varphi_\text{geo}$ is independent of this latter choice, as the phase change $\ket{\psi(0)}\rightarrow e^{i \alpha} \ket{\psi(0)}$ implies $\ket{\psi(1)}\rightarrow e^{i \alpha} \ket{\psi(1)}$, and $\arg \braket{\psi(0)}{\psi(1)}$ remains invariant.
We consider now anticoherent states with Majorana representations that have a non-trivial rotation symmetry group $\Gamma$, which we assume to be a discrete subgroup of $SO(3)$. This latter assumption only simplifies the presentation --- our results below easily extend to the special case in which all stars lie on a diameter of $S^2$, so that the symmetry group has a continuous $U(1)$ component. In the presence of such discrete symmetries, there are open curves $R(t)$ in $SO(3)$ that give rise, via~(\ref{psitRpsi0}), to closed curves $[\psi(t)]$ in $\mathbb{P}$. Take, for example, a curve $R(t)$ that starts, at $t=0$, at the identity $e$ of $SO(3)$, and ends, at $t=1$, at the rotation $R_m \in \Gamma$. Since the Majorana constellation determines the state up to phase, and $R_m$ is a symmetry of the constellation, we have (putting $\ket{\psi(0)} \equiv \ket{\Psi}$), \begin{equation} \label{Rspsi} D(R_m) \ket{\Psi}=e^{i \alpha_m} \ket{\Psi} \, , \end{equation} so that $\pket{\psi(1)}=\pket{\psi(0)} =\pket{\Psi}$, \hbox{\em i.e.{}}, $\pket{\psi(t)}$ is a closed curve in $\mathbb{P}$. For such a curve, (\ref{gpMS}), (\ref{phidynzero}) imply that $\varphi_\text{geo}=\alpha_m$, \emph{i.e.}, \emph{regardless of the details of the curve $R(t)$, the geometrical phase only depends on its endpoints.}
In fact, $\pket{\psi(t)}$ lies in the subset of $\mathbb{P}$ given by the orbit $\mathcal{O}_{\pket{\Psi}}$ of $\pket{\Psi}$ under the action of $SO(3)$, so that $C(t)$ is a closed curve in $\mathcal{O}_{\pket{\Psi}} \subset \mathbb{P}$. The $SO(3)$ orbit of a state consists of all states that share the same shape of their Majorana constellations, but differ in its orientation in space. For a general state $\pket{\Psi}$, this orbit is a copy of $SO(3)$, but in the presence of symmetries, codified by $\Gamma$, it reduces to the quotient space $SO(3)/\Gamma$, in which two rotations $R$, $R'$ are identified if there exists a symmetry rotation $R_m \in \Gamma$ such that $R'=R R_m$. One may visualize $\mathcal{O}_{\pket{\psi}}$ as a certain subset of $SO(3)$ in an infinite number of ways --- a canonical choice is to define a biinvariant distance function $D(g_1,g_2)$ in $SO(3)$, given by $D(g_1,g_2)=\text{Tr} (g_1 g_2^{-1})$ and then assign to each symmetry rotation $R_m \in \Gamma$ a ``cell'' $C_m$ consisting of all group elements in $SO(3)$ for which the closest symmetry rotation is $R_m$, \hbox{\em i.e.{}}, \begin{equation} \label{celldef} \min_i(D(g,R_i))=D(g,R_m) \Rightarrow g \in C_m \, . \end{equation} This assignment divides the whole $SO(3)$ in cells, excluding those group elements that lie on the interface between two (or more) cells, \hbox{\em i.e.{}}, are equidistant from two (or more) symmetry rotations. These latter group elements can also be ``distributed'' among a cell and its neighbors in some canonical way --- the orbit $\mathcal{O}_{\pket{\psi}}$ may then be identified with any of the cells $C_m$.
The identifications among points of $SO(3)$ mentioned above, that give rise to $\mathcal{O}_{\pket{\Psi}}$, endow the latter with a complicated topology, a signature feature of which is that for two given closed curves $C_1(t)$, $C_2(t)$ in $\mathcal{O}_{\pket{\Psi}}$, both of which start, at $t=0$, and end, at $t=1$, at the same point, there may not exist a continuous map (\emph{homotopy}) that brings one into the other, fixing all along the endpoints. One then says that the two curves belong to different \emph{homotopy classes}, denoted by $[C_1]$, $[C_2]$, respectively, and the set of all such classes forms the \emph{fundamental group} $\pi_1(\mathcal{O}_{\pket{\Psi}})$ of $\mathcal{O}_{\pket{\Psi}}$, in which the group multiplication is given by concatenation of representative curves, \hbox{\em i.e.{}}, $[C_1] \cdot [C_2]=[C_1 \cdot C_2]$, where $(C_1 \cdot C_2)(t)$ is the curve that first goes through $C_1$ (for $0 \leq t \leq 1/2$), and then through $C_2$ (for $1/2 < t \leq 1$). It follows from our discussion above that \emph{the geometric phase acquired by an anticoherent spin state $\pket{\psi(t)}$, going around a curve $C$ in $SO(3)$ that projects to a loop $\tilde{C}$ in its orbit space $\mathcal{O}_{\pket{\psi}}$, is constant on the homotopy class of $\tilde{C}$}.
The question that naturally arises now is how many homotopy classes are there, for a given discrete symmetry group $\Gamma$, and how they combine among themselves, in other words, what is the structure of the fundamental group $\pi_1(SO(3)/\Gamma)$? The following theorem, the proof of which may be found in~\cite{Mer:79}, addresses just that: \begin{theorem} \label{merminthm} Let $G$ be a connected, simply connected continuous group. Let $H$ be any subgroup of $G$. Let $H_0$ be the set of points in $H$ that are connected to the identity by continuous paths lying entirely in $H$. Then $H_0$ is a normal subgroup of $H$, and the quotient group $H/H_0$ is isomorphic to the fundamental group $\pi_1(G/H)$ of the coset space $G/H$. \end{theorem} Since $SO(3)$ is not simply connected, as assumed of the group $G$ in the theorem, we have to pass to its universal cover $SU(2)$. To each $SO(3)$ rotation matrix there correspond two $SU(2)$ matrices, that only differ in an overall sign. To each continuous path $R(t)$ in $SO(3)$, that starts at the identity, there corresponds a unique path $\mathcal{R}(t)$ in $SU(2)$, that also starts at the identity (there is of course a second path, that starts at minus the identity). Finally, the symmetry group $\Gamma$ gets lifted to $\Gamma^C$ in $SU(2)$, which has twice as many elements. Applying now theorem~\ref{merminthm} we conclude that $\pi_1(SU(2)/\Gamma^C) \sim \Gamma^C$, since $H_0$ in the theorem contains just the identity in our case. The result makes perfect sense intuitively: one expects that homotopy classes correspond somehow to curves that start at the identity in $SO(3)$ and get to any of the symmetry rotations $R_m$ in $\Gamma$. However, because $SO(3)$ is not simply connected, there are two homotopically inequivalent such curves, for each $R_m$ --- the corresponding doubling-up of the homotopy classes is exactly captured by the doubled-up $\Gamma^C$. Taking into account that the geometric phase corresponding to the product of two homotopy classes is the sum of the geometric phases corresponding to the factors, so that the corresponding phase factors simply multiply, we may summarize our findings in the following \begin{theorem} \label{thmsumm} Consider an anticoherent state $\ket{\psi}$, the Majorana constellation of which has rotational symmetry group $\Gamma \subset SO(3)$. Then the geometric phase factors $e^{i\alpha_m}$ acquired by the rotated state $\ket{\psi(t)}=D(R(t))\ket{\psi}$, when $R(t)$ traces a path in $SO(3)$, starting at the identity $R_0$ and ending at $R_m =R_{\mathbf{n},\phi} \in \Gamma$, provide a 1-dimensional representation of the fundamental group $\pi_1(SO(3)/\Gamma)$ of the orbit space of $\ket{\psi}$, the latter group being isomorphic to the lift $\Gamma^C \!$ of $\Gamma$ in $SU(2)$. \end{theorem}
As a concrete example of the above general setup, consider the spin-2 anticoherent state $\ket{\phi_\text{tetra}}=(1,0,0,\sqrt{2},0)/\sqrt{3}$, expressed in the $S_z$ eigenbasis $(2,1,0,-1,-2)$, with Majorana constellation given by a regular tetrahedron --- the corresponding rotation symmetry group $\Gamma_{\text{tetra}}$ contains 12 elements, shown in Fig.~\ref{fig:GammaPlot}, while the cell $C_0$, surrounding the identity $R_0$ in $SO(3)$, is shown in Fig.~\ref{fig:curves1}.
\begin{figure}
\caption{ Plot of the twelve elements of $\Gamma_\text{tetra}$ in the axis-angle parametrization of $SO(3)$. To help visualize their position, the twelve points have been organized as follows: the vertices of the two tetrahedra give the four symmetry rotations by $2\pi/3$ and their inverses, while the vertices of the two triangles give the three rotations by $\pi$ --- these latter appear twice, as pairs of antipodal points, which are identified. The identity $R_0$ is at the origin and cannot be seen in the figure.
}
\label{fig:GammaPlot}
\end{figure}
\begin{figure}\label{fig:curves1}
\end{figure}
The geometric phases accumulated as a result of symmetry rotations of the tetrahedral state, as well as some other interesting cases, are summarized in table~\ref{geophaseplatonic}.
\begin{table}
\begin{tabular}{| l | c | c | c | c | c |} \hline & $\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$2$\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$ & $\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$3$\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$ & $\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$4$\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$ & $\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$5$\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$ & $\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$$2s$$\phantom{\rule[-.2ex]{.5ex}{2.5ex}}$ \\ \hline spin $s$, m=0 $\rule[-.2ex]{0ex}{2.5ex}$& $s \pi$ & - & - & - & - \\ \hline spin $s$, GHZ $\rule[-1.1ex]{0ex}{3.4ex}$& - & - & - & - & $\pi$ \\ \hline Tetrahedron & $0$ & $\phantom{\rule[-1.1ex]{0ex}{3.4ex}} \frac{2\pi}{3}$ & - & - & - \\ \hline $\rule[-.2ex]{0ex}{2.5ex}$Cube & $0$ & $0$ & $0$ & - & - \\ \hline Octahedron $\rule[-.2ex]{0ex}{2.5ex}$& $\pi$ & $0$ & $\pi$ & - & - \\ \hline Dodecahedron $\rule[-.2ex]{0ex}{2.5ex}$& 0 & 0 & - & 0 & - \\ \hline Icosahedron $\rule[-.2ex]{0ex}{2.5ex}$& 0 & 0 & - & 0 & - \\ \hline \end{tabular} \caption{Absolute values of geometric phases corresponding to symmetry rotations of the spin $s$, $m=0$ and GHZ states, and those corresponding to the platonic solids (inverse rotations give opposite phases). The rotations are specified by their order (top row), which, in this case, uniquely defines them.} \label{geophaseplatonic} \end{table}
These phases can, in principle, be computed by applying the appropriate rotation matrix to the corresponding state, but, in fact, can be shown to only depend on the number of stars, in the constellation of $\ket{\psi}$, pointing in the direction of the rotation axis, and the corresponding rotation angle~\cite{Bar.Tur.Dem:07}, thus reducing their computation to simple star gazing.
Some remarks are due at this point: \begin{enumerate} \item The geometric phase of $m=0$ spin-$s$ states reported in~\cite{Rob.Ber:94} is a special case of our general setting, in which the Majorana constellation consists of $s$ points in a direction $\hat{\mathbf{n}}$, and another $s$ points in the antipodal direction --- the corresponding symmetry rotation exchanges the two groups of points. Note that the symmetry group in this case has a continuous component (rotations around $\hat{\mathbf{n}}$). \item The Majorana constellation of the spin-$s$ GHZ state $\ket{\psi_{\text{GHZ}}} \sim \ket{s,s}+\ket{s,-s}$ consists in $2s$ equidistant points along the equator --- the symmetry rotation is around the $z$-axis by an angle of $\pi/s$. \item The above phases are insensitive to perturbations of the path taken in $SO(3)$, but are still affected by end-point imprecision. However, the effect is weak, being at most quadratic in the rotation error, \hbox{\em i.e.{}}, assuming a rotation \begin{equation} \label{errorrot} R=e^{-i\epsilon \hat{\mathbf{n}} \cdot \mathbf{S}} R_m \end{equation} is applied to $\ket{\psi}$, where $R_m$ is a symmetry rotation and the prefactor is due to noise, it is easily seen that \begin{equation} \label{errorphase} \bra{\psi} R \ket{\psi} = e^{i \alpha_m}(1+\mathcal{O}(\epsilon^2)) \, , \end{equation} with $\alpha_m$ as in~(\ref{Rspsi}). \item A further virtue of the use of anticoherent states in the setup we consider, is that the vanishing of the spin expectation value implies the absence of dynamical phase, when the states are rotated with magnetic fields --- generic states, on the contrary, do accumulate dynamical phase, which requires further processing for its elimination (see, \hbox{\em e.g.{}},~\cite{Fal.Faz.Pal.Sie.Ved:00,Jon.Ved.Eke.Cas:00}). Thus, if a beam of spins in an anticoherent state is split into two, and each of these secondary beams is subjected to a different symmetry rotation, say, $R_1$, $R_2$ respectively, when the two beams are brought back together to interfere, the pattern observed will only depend (apart from noise effects) on the difference $\alpha_1-\alpha_2$. \item Zimba gives the following generalization of the concept of anticoherence~\cite{Zim:06}: a state $\ket{\psi}$ is $k$-anticoherent if $k$ is the largest integer such that $\bra{\psi} (\hat{\mathbf{n}} \cdot \mathbf{S})^r \ket{\psi}$ is independent of $\hat{\mathbf{n}}$ for $r=1, \ldots, k$. For a $k$-anticoherent state, the error in the phase in~(\ref{errorphase}) is independent of $\hat{\mathbf{n}}$ up to $\mathcal{O}(\epsilon^k)$. For example, the tetrahedral state $\ket{\phi_{\text{tetra}}}$ turns out to be 2-anticoherent. Accordingly, if a rotation $R$ as in~(\ref{errorrot}) is applied, with $\epsilon=.1$ (about 6 degrees), the error in the phase obtained will be of the order of $10^{-2}$, with $\hat{\mathbf{n}}$-dependent part of the order of $10^{-3}$. \end{enumerate} We are currently working on a generalization of the above to the non-abelian case and plan to report our findings in this direction in a forthcoming publication.
\end{document}
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Laurie Brown (physicist)
Laurie Mark Brown (born 1923, New York City) is an American theoretical physicist and historian of quantum field theory and elementary particle physics.[1]
Biography
Brown studied at Cornell University, where in 1951 he received his Ph.D. under Richard Feynman.[2] Since 1950 he has been on the faculty of the physics department of Northwestern University, where he became a tenured professor and eventually retired as professor emeritus. For the academic year 1952–1953 he was at the Institute for Advanced Study. For the academic years 1958–1959 and 1959–1960 he was a Fulbright Scholar in Italy. In 1966 he was an IEA professor at the University of Vienna. From 1960 to 1970 he served as a consultant for Argonne National Laboratory and the Laboratory's Accelerator Committee.
Brown is one of the leading science historians for the development of quantum field theory and elementary particle physics, especially in the era after 1945. During the 1990s one focus of his work was the history of modern physics in Japan.
He was the editor for Feynman's Thesis: A New Approach to Quantum Physics (2005),[3] Selected Papers of Richard Feynman, with Commentary (2000),[4] and (with John Rigden as co-editor) Most of the Good Stuff: Memories of Richard Feynman (1993).
Brown was one of the founders of the Forum on History of Physics of the American Physical Society and was the chair of the Forum in 1984 and again in 1989. He is a member of the American Association for the Advancement of Science and a member of the History of Science Society. In 1961 he was elected a Fellow of the American Physical Society.
Selected publications
• Brown, Laurie M. (1958). "Two-component fermion theory". Phys. Rev. 111 (3): 957–965. Bibcode:1958PhRv..111..957B. doi:10.1103/PhysRev.111.957.
• as editor with Lillian Hoddeson: The birth of particle physics (International Symposium on the history of particle physics, Fermilab 1980). Cambridge University Press. 1983.; Brown, Laurie M.; Hoddeson, Lillian (1986). pbk. edition. ISBN 9780521338370.
• with Donald F. Moyer: Brown, Laurie M.; Moyer, Donald F. (1984). "Lady or tiger?—The Meitner–Hupfeld effect and Heisenberg's neutron theory". American Journal of Physics. 52 (2): 130–136. Bibcode:1984AmJPh..52..130B. doi:10.1119/1.13920.
• with Max Dresden and Lillian Hoddeson: Pions to Quarks: Particle physics in the 1950s (based on a Fermilab symposium). Cambridge University Press. 1989.[5]
• with Tian Yu Cao: Brown, Laurie M.; Cao, Tian Yu (1991). "Spontaneous breakdown of symmetry: Its rediscovery and integration into quantum field theory". Historical Studies in the Physical and Biological Sciences. 21 (2): 211–235. doi:10.2307/27757663. JSTOR 27757663.
• as editor: Renormalization: from Lorentz to Landau and beyond. Springer. 1993.
• as editor with Abraham Pais and Brian Pippard: Twentieth Century Physics. Vol. 3 vols. (2nd ed.). IOP. 1995.[6]
• with Helmut Rechendberg: Origin of the concept of nuclear forces. Taylor and Francis. 1996. ISBN 9780750303736.
• as editor with Lillian Hoddeson, Michael Riordan, and Max Dresden: The rise of the standard model. A history of particle physics from 1964 to 1979. Cambridge University Press. 1997. ISBN 9780521578165.
• Brown, Laurie M. (2002). "The Compton Effect as One Path to QED". Studies in History and Philosophy of Modern Physics. 3 (2): 211–249. Bibcode:2002SHPMP..33..211B. doi:10.1016/S1355-2198(02)00005-9.
• Brown, Laurie M. (2006). "Paul A. M. Dirac's Principles of Quantum Mechanics". Physics in Perspective. 8 (4): 381–407. Bibcode:2006PhP.....8..381B. doi:10.1007/s00016-006-0276-4. S2CID 120303937.
References
1. Laurie Brown, Professor Emeritus, Department of Physics and Astronomy, Northwestern University
2. Feynman, R. P.; Brown, L. M. (15 January 1952). "Radiative corrections to Compton Scattering" (PDF). Phys. Rev. 85 (2): 231–244. Bibcode:1952PhRv...85..231B. doi:10.1103/PhysRev.85.231.
3. Feynman, R. P. (2005). Brown, Laurie M. (ed.). Feynman's Thesis: A New Approach to Quantum Theory. World Scientific. Bibcode:2005ftna.book.....B. ISBN 978-981-256-366-8.; pbk. World Scientific. 2005. ISBN 978-981-256-380-4.
4. Feynman, R. P. (2000). Brown, Laurie M. (ed.). Selected Papers of Richard Feynman: With Commentary. World Scientific. ISBN 978-981-02-4130-8.; Feynman, Richard Phillips (2000). pbk. ISBN 978-981-02-4131-5.
5. Dilworth, C. (27 July 1990). "Review of Pions and Quarks by L. M. Brown, M. Dresden, & L. Hoddeson". Science. 249 (4967): 426–427. Bibcode:1990Sci...249..426B. doi:10.1126/science.249.4967.426. PMID 17755946.
6. Mackay, A. (1996). "Review of Twentieth Century Physics ed. by L. M. Brown, A. Pais, & B. Pippard" (PDF). Science Progress. 79 (2): 159–162.
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Lie algebra
In mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space ${\mathfrak {g}}$ together with an operation called the Lie bracket, an alternating bilinear map ${\mathfrak {g}}\times {\mathfrak {g}}\rightarrow {\mathfrak {g}}$, that satisfies the Jacobi identity. Otherwise said, a Lie algebra is an algebra over a field where the multiplication operation is now called Lie bracket and has two additional properties: it is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors $x$ and $y$ is denoted $[x,y]$. The Lie bracket does not need to be associative, meaning that the Lie algebra can be non associative. Given an associative algebra (like for example the space of square matrices), a Lie bracket can be and is often defined through the commutator, namely defining $[x,y]=xy-yx$ correctly defines a Lie bracket in addition to the already existing multiplication operation.
"Lie bracket" redirects here. For the operation on vector fields, see Lie bracket of vector fields.
Lie groups and Lie algebras
Classical groups
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
Simple Lie groups
Classical
• An
• Bn
• Cn
• Dn
Exceptional
• G2
• F4
• E6
• E7
• E8
Other Lie groups
• Circle
• Lorentz
• Poincaré
• Conformal group
• Diffeomorphism
• Loop
• Euclidean
Lie algebras
• Lie group–Lie algebra correspondence
• Exponential map
• Adjoint representation
• Killing form
• Index
• Simple Lie algebra
• Loop algebra
• Affine Lie algebra
Semisimple Lie algebra
• Dynkin diagrams
• Cartan subalgebra
• Root system
• Weyl group
• Real form
• Complexification
• Split Lie algebra
• Compact Lie algebra
Representation theory
• Lie group representation
• Lie algebra representation
• Representation theory of semisimple Lie algebras
• Representations of classical Lie groups
• Theorem of the highest weight
• Borel–Weil–Bott theorem
Lie groups in physics
• Particle physics and representation theory
• Lorentz group representations
• Poincaré group representations
• Galilean group representations
Scientists
• Sophus Lie
• Henri Poincaré
• Wilhelm Killing
• Élie Cartan
• Hermann Weyl
• Claude Chevalley
• Harish-Chandra
• Armand Borel
• Glossary
• Table of Lie groups
Algebraic structure → Ring theory
Ring theory
Basic concepts
Rings
• Subrings
• Ideal
• Quotient ring
• Fractional ideal
• Total ring of fractions
• Product of rings
• Free product of associative algebras
• Tensor product of algebras
Ring homomorphisms
• Kernel
• Inner automorphism
• Frobenius endomorphism
Algebraic structures
• Module
• Associative algebra
• Graded ring
• Involutive ring
• Category of rings
• Initial ring $\mathbb {Z} $
• Terminal ring $0=\mathbb {Z} _{1}$
Related structures
• Field
• Finite field
• Non-associative ring
• Lie ring
• Jordan ring
• Semiring
• Semifield
Commutative algebra
Commutative rings
• Integral domain
• Integrally closed domain
• GCD domain
• Unique factorization domain
• Principal ideal domain
• Euclidean domain
• Field
• Finite field
• Composition ring
• Polynomial ring
• Formal power series ring
Algebraic number theory
• Algebraic number field
• Ring of integers
• Algebraic independence
• Transcendental number theory
• Transcendence degree
p-adic number theory and decimals
• Direct limit/Inverse limit
• Zero ring $\mathbb {Z} _{1}$
• Integers modulo pn $\mathbb {Z} /p^{n}\mathbb {Z} $
• Prüfer p-ring $\mathbb {Z} (p^{\infty })$
• Base-p circle ring $\mathbb {T} $
• Base-p integers $\mathbb {Z} $
• p-adic rationals $\mathbb {Z} [1/p]$
• Base-p real numbers $\mathbb {R} $
• p-adic integers $\mathbb {Z} _{p}$
• p-adic numbers $\mathbb {Q} _{p}$
• p-adic solenoid $\mathbb {T} _{p}$
Algebraic geometry
• Affine variety
Noncommutative algebra
Noncommutative rings
• Division ring
• Semiprimitive ring
• Simple ring
• Commutator
Noncommutative algebraic geometry
Free algebra
Clifford algebra
• Geometric algebra
Operator algebra
Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings (Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras.
In physics, Lie groups appear as symmetry groups of physical systems, and their Lie algebras (tangent vectors near the identity) may be thought of as infinitesimal symmetry motions. Thus Lie algebras and their representations are used extensively in physics, notably in quantum mechanics and particle physics.
An elementary example (that is not derived from an associative algebra) is the space of three dimensional vectors ${\mathfrak {g}}=\mathbb {R} ^{3}$ with the Lie bracket operation defined by the cross product $[x,y]=x\times y.$ This is skew-symmetric since $x\times y=-y\times x$, and instead of associativity it satisfies the Jacobi identity:
$x\times (y\times z)\ =\ (x\times y)\times z\ +\ y\times (x\times z).$
This is the Lie algebra of the Lie group of rotations of space, and each vector $v\in \mathbb {R} ^{3}$ may be pictured as an infinitesimal rotation around the axis $v$, with velocity equal to the magnitude of $v$. The Lie bracket is a measure of the non-commutativity between two rotations: since a rotation commutes with itself, we have the alternating property $[x,x]=x\times x=0$.
History
Lie algebras were introduced to study the concept of infinitesimal transformations by Marius Sophus Lie in the 1870s,[1] and independently discovered by Wilhelm Killing[2] in the 1880s. The name Lie algebra was given by Hermann Weyl in the 1930s; in older texts, the term infinitesimal group is used.
Definitions
Definition of a Lie algebra
A Lie algebra is a vector space $\,{\mathfrak {g}}$ over some field $F$ together with a binary operation $[\,\cdot \,,\cdot \,]:{\mathfrak {g}}\times {\mathfrak {g}}\to {\mathfrak {g}}$ called the Lie bracket satisfying the following axioms:[lower-alpha 1]
• Bilinearity,
$[ax+by,z]=a[x,z]+b[y,z],$
$[z,ax+by]=a[z,x]+b[z,y]$
for all scalars $a$, $b$ in $F$ and all elements $x$, $y$, $z$ in ${\mathfrak {g}}$.
• Alternativity,
$[x,x]=0\ $
for all $x$ in ${\mathfrak {g}}$.
• The Jacobi identity,
$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\ $
for all $x$, $y$, $z$ in ${\mathfrak {g}}$.
Using bilinearity to expand the Lie bracket $[x+y,x+y]$ and using alternativity shows that $[x,y]+[y,x]=0\ $ for all elements $x$, $y$ in ${\mathfrak {g}}$, showing that bilinearity and alternativity together imply
• Anticommutativity,
$[x,y]=-[y,x],\ $
for all elements $x$, $y$ in ${\mathfrak {g}}$. If the field's characteristic is not 2 then anticommutativity implies alternativity, since it implies $[x,x]=-[x,x].$[3]
It is customary to denote a Lie algebra by a lower-case fraktur letter such as ${\mathfrak {g,h,b,n}}$. If a Lie algebra is associated with a Lie group, then the algebra is denoted by the fraktur version of the group: for example the Lie algebra of SU(n) is ${\mathfrak {su}}(n)$.
Generators and dimension
Elements of a Lie algebra ${\mathfrak {g}}$ are said to generate it if the smallest subalgebra containing these elements is ${\mathfrak {g}}$ itself. The dimension of a Lie algebra is its dimension as a vector space over $F$. The cardinality of a minimal generating set of a Lie algebra is always less than or equal to its dimension.
See the classification of low-dimensional real Lie algebras for other small examples.
Subalgebras, ideals and homomorphisms
The Lie bracket is not required to be associative, meaning that $[[x,y],z]$ need not equal $[x,[y,z]]$. However, it is flexible. Nonetheless, much of the terminology of associative rings and algebras is commonly applied to Lie algebras. A Lie subalgebra is a subspace ${\mathfrak {h}}\subseteq {\mathfrak {g}}$ which is closed under the Lie bracket. An ideal ${\mathfrak {i}}\subseteq {\mathfrak {g}}$ is a subalgebra satisfying the stronger condition:[4]
$[{\mathfrak {g}},{\mathfrak {i}}]\subseteq {\mathfrak {i}}.$
A Lie algebra homomorphism is a linear map compatible with the respective Lie brackets:
$\phi :{\mathfrak {g}}\to {\mathfrak {g'}},\quad \phi ([x,y])=[\phi (x),\phi (y)]\ {\text{for all}}\ x,y\in {\mathfrak {g}}.$ :{\mathfrak {g}}\to {\mathfrak {g'}},\quad \phi ([x,y])=[\phi (x),\phi (y)]\ {\text{for all}}\ x,y\in {\mathfrak {g}}.}
As for associative rings, ideals are precisely the kernels of homomorphisms; given a Lie algebra ${\mathfrak {g}}$ and an ideal ${\mathfrak {i}}$ in it, one constructs the factor algebra or quotient algebra ${\mathfrak {g}}/{\mathfrak {i}}$, and the first isomorphism theorem holds for Lie algebras.
Since the Lie bracket is a kind of infinitesimal commutator of the corresponding Lie group, we say that two elements $x,y\in {\mathfrak {g}}$ commute if their bracket vanishes: $[x,y]=0$.
The centralizer subalgebra of a subset $S\subset {\mathfrak {g}}$ is the set of elements commuting with $S$: that is, ${\mathfrak {z}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}\ \mid \ [x,s]=0\ {\text{ for all }}s\in S\}$. The centralizer of ${\mathfrak {g}}$ itself is the center ${\mathfrak {z}}({\mathfrak {g}})$. Similarly, for a subspace S, the normalizer subalgebra of $S$ is ${\mathfrak {n}}_{\mathfrak {g}}(S)=\{x\in {\mathfrak {g}}\ \mid \ [x,s]\in S\ {\text{ for all}}\ s\in S\}$.[5] Equivalently, if $S$ is a Lie subalgebra, ${\mathfrak {n}}_{\mathfrak {g}}(S)$ is the largest subalgebra such that $S$ is an ideal of ${\mathfrak {n}}_{\mathfrak {g}}(S)$.
Examples
For ${\mathfrak {d}}(2)\subset {\mathfrak {gl}}(2)$, the commutator of two elements $g\in {\mathfrak {gl}}(2)$ and $d\in {\mathfrak {d}}(2)$:
${\begin{aligned}\left[{\begin{bmatrix}a&b\\c&d\end{bmatrix}},{\begin{bmatrix}x&0\\0&y\end{bmatrix}}\right]&={\begin{bmatrix}ax&by\\cx&dy\\\end{bmatrix}}-{\begin{bmatrix}ax&bx\\cy&dy\\\end{bmatrix}}\\&={\begin{bmatrix}0&b(y-x)\\c(x-y)&0\end{bmatrix}}\end{aligned}}$
shows ${\mathfrak {d}}(2)$ is a subalgebra, but not an ideal. In fact, every one-dimensional linear subspace of a Lie algebra has an induced abelian Lie algebra structure, which is generally not an ideal. For any simple Lie algebra, all abelian Lie algebras can never be ideals.
Direct sum and semidirect product
For two Lie algebras ${\mathfrak {g^{}}}$ and ${\mathfrak {g'}}$, their direct sum Lie algebra is the vector space ${\mathfrak {g}}\oplus {\mathfrak {g'}}$consisting of all pairs ${\mathfrak {}}(x,x'),\,x\in {\mathfrak {g}},\ x'\in {\mathfrak {g'}}$, with the operation
$[(x,x'),(y,y')]=([x,y],[x',y']),$
so that the copies of ${\mathfrak {g}},{\mathfrak {g}}'$ commute with each other: $[(x,0),(0,x')]=0.$
Let ${\mathfrak {g}}$ be a Lie algebra and ${\mathfrak {i}}$ an ideal of ${\mathfrak {g}}$. If the canonical map ${\mathfrak {g}}\to {\mathfrak {g}}/{\mathfrak {i}}$ splits (i.e., admits a section), then ${\mathfrak {g}}$ is said to be a semidirect product of ${\mathfrak {i}}$ and ${\mathfrak {g}}/{\mathfrak {i}}$, ${\mathfrak {g}}={\mathfrak {g}}/{\mathfrak {i}}\ltimes {\mathfrak {i}}$. See also semidirect sum of Lie algebras.
Levi's theorem says that a finite-dimensional Lie algebra is a semidirect product of its radical and the complementary subalgebra (Levi subalgebra).
Derivations
A derivation on the Lie algebra ${\mathfrak {g}}$ (or on any non-associative algebra) is a linear map $\delta \colon {\mathfrak {g}}\rightarrow {\mathfrak {g}}$ that obeys the Leibniz law, that is,
$\delta ([x,y])=[\delta (x),y]+[x,\delta (y)]$
for all $x,y\in {\mathfrak {g}}$. The inner derivation associated to any $x\in {\mathfrak {g}}$ is the adjoint mapping $\mathrm {ad} _{x}$ defined by $\mathrm {ad} _{x}(y):=[x,y]$. (This is a derivation as a consequence of the Jacobi identity.) The outer derivations are derivations which do not come from the adjoint representation of the Lie algebra. If ${\mathfrak {g}}$ is semisimple, every derivation is inner.
The derivations form a vector space $\mathrm {Der} ({\mathfrak {g}})$, which is a Lie subalgebra of ${\mathfrak {gl}}({\mathfrak {g}})$; the bracket is commutator. The inner derivations form a Lie subalgebra of $\mathrm {Der} ({\mathfrak {g}})$.
Examples
For example, given a Lie algebra ideal ${\mathfrak {i}}\subset {\mathfrak {g}}$ the adjoint representation ${\mathfrak {ad}}_{\mathfrak {g}}$ of ${\mathfrak {g}}$ acts as outer derivations on ${\mathfrak {i}}$ since $[x,i]\subset {\mathfrak {i}}$ for any $x\in {\mathfrak {g}}$ and $i\in {\mathfrak {i}}$. For the Lie algebra ${\mathfrak {b}}_{n}$ of upper triangular matrices in ${\mathfrak {gl}}(n)$, it has an ideal ${\mathfrak {n}}_{n}$ of strictly upper triangular matrices (where the only non-zero elements are above the diagonal of the matrix). For instance, the commutator of elements in ${\mathfrak {b}}_{3}$ and ${\mathfrak {n}}_{3}$ gives
${\begin{aligned}\left[{\begin{bmatrix}a&b&c\\0&d&e\\0&0&f\end{bmatrix}},{\begin{bmatrix}0&x&y\\0&0&z\\0&0&0\end{bmatrix}}\right]&={\begin{bmatrix}0&ax&ay+bz\\0&0&dz\\0&0&0\end{bmatrix}}-{\begin{bmatrix}0&dx&ex+yf\\0&0&fz\\0&0&0\end{bmatrix}}\\&={\begin{bmatrix}0&(a-d)x&(a-f)y-ex+bz\\0&0&(d-f)z\\0&0&0\end{bmatrix}}\end{aligned}}$
shows there exist outer derivations from ${\mathfrak {b}}_{3}$ in ${\text{Der}}({\mathfrak {n}}_{3})$.
Split Lie algebra
Let V be a finite-dimensional vector space over a field F, ${\mathfrak {gl}}(V)$ the Lie algebra of linear transformations and ${\mathfrak {g}}\subseteq {\mathfrak {gl}}(V)$ a Lie subalgebra. Then ${\mathfrak {g}}$ is said to be split if the roots of the characteristic polynomials of all linear transformations in ${\mathfrak {g}}$ are in the base field F.[6] More generally, a finite-dimensional Lie algebra ${\mathfrak {g}}$ is said to be split if it has a Cartan subalgebra whose image under the adjoint representation $\operatorname {ad} :{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})$ :{\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})} is a split Lie algebra. A split real form of a complex semisimple Lie algebra (cf. #Real form and complexification) is an example of a split real Lie algebra. See also split Lie algebra for further information.
Vector space basis
For practical calculations, it is often convenient to choose an explicit vector space basis for the algebra. A common construction for this basis is sketched in the article structure constants.
Definition using category-theoretic notation
Although the definitions above are sufficient for a conventional understanding of Lie algebras, once this is understood, additional insight can be gained by using notation common to category theory, that is, by defining a Lie algebra in terms of linear maps—that is, morphisms of the category of vector spaces—without considering individual elements. (In this section, the field over which the algebra is defined is supposed to be of characteristic different from two.)
For the category-theoretic definition of Lie algebras, two braiding isomorphisms are needed. If A is a vector space, the interchange isomorphism $\tau :A\otimes A\to A\otimes A$ is defined by
$\tau (x\otimes y)=y\otimes x.$
The cyclic-permutation braiding $\sigma :A\otimes A\otimes A\to A\otimes A\otimes A$ is defined as
$\sigma =(\mathrm {id} \otimes \tau )\circ (\tau \otimes \mathrm {id} ),$
where $\mathrm {id} $ is the identity morphism. Equivalently, $\sigma $ is defined by
$\sigma (x\otimes y\otimes z)=y\otimes z\otimes x.$
With this notation, a Lie algebra can be defined as an object $A$ in the category of vector spaces together with a morphism
$[\cdot ,\cdot ]:A\otimes A\rightarrow A$
that satisfies the two morphism equalities
$[\cdot ,\cdot ]\circ (\mathrm {id} +\tau )=0,$
and
$[\cdot ,\cdot ]\circ ([\cdot ,\cdot ]\otimes \mathrm {id} )\circ (\mathrm {id} +\sigma +\sigma ^{2})=0.$
Examples
Vector spaces
Any vector space $V$ endowed with the identically zero Lie bracket becomes a Lie algebra. Such Lie algebras are called abelian, cf. below. Any one-dimensional Lie algebra over a field of characteristic different from 2 is abelian, by the alternating property of the Lie bracket.
Associative algebra with commutator bracket
• On an associative algebra $A$ over a field $F$ with multiplication $(x,y)\mapsto xy$, a Lie bracket may be defined by the commutator $[x,y]=xy-yx$. With this bracket, $A$ is a Lie algebra.[7] The associative algebra A is called an enveloping algebra of the Lie algebra $(A,[\,\cdot \,,\cdot \,])$. Every Lie algebra can be embedded into one that arises from an associative algebra in this fashion; see universal enveloping algebra.
• The associative algebra of the endomorphisms of an F-vector space $V$ with the above Lie bracket is denoted ${\mathfrak {gl}}(V)$.
• For a finite dimensional vector space $V=F^{n}$, the previous example is exactly the Lie algebra of n × n matrices, denoted ${\mathfrak {gl}}(n,F)$ or ${\mathfrak {gl}}_{n}(F)$,[8] and with bracket $[X,Y]=XY-YX$ where adjacency indicates matrix multiplication. This is the Lie algebra of the general linear group, consisting of invertible matrices.
Special matrices
Two important subalgebras of ${\mathfrak {gl}}_{n}(F)$ are:
• The matrices of trace zero form the special linear Lie algebra ${\mathfrak {sl}}_{n}(F)$, the Lie algebra of the special linear group $\mathrm {SL} _{n}(F)$.[9]
• The skew-hermitian matrices form the unitary Lie algebra ${\mathfrak {u}}(n)$, the Lie algebra of the unitary group U(n).
Matrix Lie algebras
A complex matrix group is a Lie group consisting of matrices, $G\subset M_{n}(\mathbb {C} )$, where the multiplication of G is matrix multiplication. The corresponding Lie algebra ${\mathfrak {g}}$ is the space of matrices which are tangent vectors to G inside the linear space $M_{n}(\mathbb {C} )$: this consists of derivatives of smooth curves in G at the identity:
${\mathfrak {g}}=\{X=c'(0)\in M_{n}(\mathbb {C} )\ \mid \ {\text{ smooth }}c:\mathbb {R} \to G,\ c(0)=I\}.$
The Lie bracket of ${\mathfrak {g}}$ is given by the commutator of matrices, $[X,Y]=XY-YX$. Given the Lie algebra, one can recover the Lie group as the image of the matrix exponential mapping $\exp :M_{n}(\mathbb {C} )\to M_{n}(\mathbb {C} )$ defined by $\exp(X)=I+X+{\tfrac {1}{2!}}X^{2}+\cdots $, which converges for every matrix $X$: that is, $G=\exp({\mathfrak {g}})$.
The following are examples of Lie algebras of matrix Lie groups:[10]
• The special linear group ${\rm {SL}}_{n}(\mathbb {C} )$, consisting of all n × n matrices with determinant 1. Its Lie algebra ${\mathfrak {sl}}_{n}(\mathbb {C} )$consists of all n × n matrices with complex entries and trace 0. Similarly, one can define the corresponding real Lie group ${\rm {SL}}_{n}(\mathbb {R} )$ and its Lie algebra ${\mathfrak {sl}}_{n}(\mathbb {R} )$.
• The unitary group $U(n)$ consists of n × n unitary matrices (satisfying $U^{*}=U^{-1}$). Its Lie algebra ${\mathfrak {u}}(n)$ consists of skew-self-adjoint matrices ($X^{*}=-X$).
• The special orthogonal group $\mathrm {SO} (n)$, consisting of real determinant-one orthogonal matrices ($A^{\mathrm {T} }=A^{-1}$). Its Lie algebra ${\mathfrak {so}}(n)$ consists of real skew-symmetric matrices ($X^{\rm {T}}=-X$). The full orthogonal group $\mathrm {O} (n)$, without the determinant-one condition, consists of $\mathrm {SO} (n)$ and a separate connected component, so it has the same Lie algebra as $\mathrm {SO} (n)$. See also infinitesimal rotations with skew-symmetric matrices. Similarly, one can define a complex version of this group and algebra, simply by allowing complex matrix entries.
Two dimensions
• On any field $F$ there is, up to isomorphism, a single two-dimensional nonabelian Lie algebra. With generators x, y, its bracket is defined as $\left[x,y\right]=y$. It generates the affine group in one dimension.
This can be realized by the matrices:
$x=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad y=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right).$
Since
$\left({\begin{array}{cc}1&c\\0&0\end{array}}\right)^{n+1}=\left({\begin{array}{cc}1&c\\0&0\end{array}}\right)$
for any natural number $n$ and any $c$, one sees that the resulting Lie group elements are upper triangular 2×2 matrices with unit lower diagonal:
$\exp(a\cdot {}x+b\cdot {}y)=\left({\begin{array}{cc}e^{a}&{\tfrac {b}{a}}(e^{a}-1)\\0&1\end{array}}\right)=1+{\tfrac {e^{a}-1}{a}}\left(a\cdot {}x+b\cdot {}y\right).$
Three dimensions
• The Heisenberg algebra ${\rm {H}}_{3}(\mathbb {R} )$ is a three-dimensional Lie algebra generated by elements x, y, and z with Lie brackets
$[x,y]=z,\quad [x,z]=0,\quad [y,z]=0$.
It is usually realized as the space of 3×3 strictly upper-triangular matrices, with the commutator Lie bracket and the basis
$x=\left({\begin{array}{ccc}0&1&0\\0&0&0\\0&0&0\end{array}}\right),\quad y=\left({\begin{array}{ccc}0&0&0\\0&0&1\\0&0&0\end{array}}\right),\quad z=\left({\begin{array}{ccc}0&0&1\\0&0&0\\0&0&0\end{array}}\right)~.\quad $
Any element of the Heisenberg group has a representation as a product of group generators, i.e., matrix exponentials of these Lie algebra generators,
$\left({\begin{array}{ccc}1&a&c\\0&1&b\\0&0&1\end{array}}\right)=e^{by}e^{cz}e^{ax}~.$
• The Lie algebra ${\mathfrak {so}}(3)$ of the group SO(3) is spanned by the three matrices[11]
$F_{1}=\left({\begin{array}{ccc}0&0&0\\0&0&-1\\0&1&0\end{array}}\right),\quad F_{2}=\left({\begin{array}{ccc}0&0&1\\0&0&0\\-1&0&0\end{array}}\right),\quad F_{3}=\left({\begin{array}{ccc}0&-1&0\\1&0&0\\0&0&0\end{array}}\right)~.\quad $
The commutation relations among these generators are
$[F_{1},F_{2}]=F_{3},$
$[F_{2},F_{3}]=F_{1},$
$[F_{3},F_{1}]=F_{2}.$
The three-dimensional Euclidean space $\mathbb {R} ^{3}$ with the Lie bracket given by the cross product of vectors has the same commutation relations as above: thus, it is isomorphic to ${\mathfrak {so}}(3)$. This Lie algebra is unitarily equivalent to the usual Spin (physics) angular-momentum component operators for spin-1 particles in quantum mechanics.
Infinite dimensions
• An important class of infinite-dimensional real Lie algebras arises in differential topology. The space of smooth vector fields on a differentiable manifold M forms a Lie algebra, where the Lie bracket is defined to be the commutator of vector fields. One way of expressing the Lie bracket is through the formalism of Lie derivatives, which identifies a vector field X with a first order partial differential operator LX acting on smooth functions by letting LX(f) be the directional derivative of the function f in the direction of X. The Lie bracket [X,Y] of two vector fields is the vector field defined through its action on functions by the formula:
$L_{[X,Y]}f=L_{X}(L_{Y}f)-L_{Y}(L_{X}f).\,$
• Kac–Moody algebras are a large class of infinite-dimensional Lie algebras whose structure is very similar to the finite-dimensional cases above.
• The Moyal algebra is an infinite-dimensional Lie algebra that contains all classical Lie algebras as subalgebras.
• The Virasoro algebra is of paramount importance in string theory.
Representations
Main article: Lie algebra representation
Definitions
Given a vector space V, let ${\mathfrak {gl}}(V)$ denote the Lie algebra consisting of all linear endomorphisms of V, with bracket given by $[X,Y]=XY-YX$. A representation of a Lie algebra ${\mathfrak {g}}$ on V is a Lie algebra homomorphism
$\pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V).$ :{\mathfrak {g}}\to {\mathfrak {gl}}(V).}
A representation is said to be faithful if its kernel is zero. Ado's theorem[12] states that every finite-dimensional Lie algebra has a faithful representation on a finite-dimensional vector space.
Adjoint representation
For any Lie algebra ${\mathfrak {g}}$, we can define a representation
$\operatorname {ad} \colon {\mathfrak {g}}\to {\mathfrak {gl}}({\mathfrak {g}})$
given by $\operatorname {ad} (x)(y)=[x,y]$; it is a representation on the vector space ${\mathfrak {g}}$ called the adjoint representation.
Goals of representation theory
One important aspect of the study of Lie algebras (especially semisimple Lie algebras) is the study of their representations. (Indeed, most of the books listed in the references section devote a substantial fraction of their pages to representation theory.) Although Ado's theorem is an important result, the primary goal of representation theory is not to find a faithful representation of a given Lie algebra ${\mathfrak {g}}$. Indeed, in the semisimple case, the adjoint representation is already faithful. Rather the goal is to understand all possible representation of ${\mathfrak {g}}$, up to the natural notion of equivalence. In the semisimple case over a field of characteristic zero, Weyl's theorem[13] says that every finite-dimensional representation is a direct sum of irreducible representations (those with no nontrivial invariant subspaces). The irreducible representations, in turn, are classified by a theorem of the highest weight.
Representation theory in physics
The representation theory of Lie algebras plays an important role in various parts of theoretical physics. There, one considers operators on the space of states that satisfy certain natural commutation relations. These commutation relations typically come from a symmetry of the problem—specifically, they are the relations of the Lie algebra of the relevant symmetry group. An example would be the angular momentum operators, whose commutation relations are those of the Lie algebra ${\mathfrak {so}}(3)$ of the rotation group SO(3). Typically, the space of states is very far from being irreducible under the pertinent operators, but one can attempt to decompose it into irreducible pieces. In doing so, one needs to know the irreducible representations of the given Lie algebra. In the study of the quantum hydrogen atom, for example, quantum mechanics textbooks give (without calling it that) a classification of the irreducible representations of the Lie algebra ${\mathfrak {so}}(3)$.
Structure theory and classification
Lie algebras can be classified to some extent. In particular, this has an application to the classification of Lie groups.
Abelian, nilpotent, and solvable
Analogously to abelian, nilpotent, and solvable groups, defined in terms of the derived subgroups, one can define abelian, nilpotent, and solvable Lie algebras.
A Lie algebra ${\mathfrak {g}}$ is abelian if the Lie bracket vanishes, i.e. [x,y] = 0, for all x and y in ${\mathfrak {g}}$. Abelian Lie algebras correspond to commutative (or abelian) connected Lie groups such as vector spaces $\mathbb {K} ^{n}$ or tori $\mathbb {T} ^{n}$, and are all of the form ${\mathfrak {k}}^{n},$ meaning an n-dimensional vector space with the trivial Lie bracket.
A more general class of Lie algebras is defined by the vanishing of all commutators of given length. A Lie algebra ${\mathfrak {g}}$ is nilpotent if the lower central series
${\mathfrak {g}}>[{\mathfrak {g}},{\mathfrak {g}}]>[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}]>[[[{\mathfrak {g}},{\mathfrak {g}}],{\mathfrak {g}}],{\mathfrak {g}}]>\cdots $
becomes zero eventually. By Engel's theorem, a Lie algebra is nilpotent if and only if for every u in ${\mathfrak {g}}$ the adjoint endomorphism
$\operatorname {ad} (u):{\mathfrak {g}}\to {\mathfrak {g}},\quad \operatorname {ad} (u)v=[u,v]$
is nilpotent.
More generally still, a Lie algebra ${\mathfrak {g}}$ is said to be solvable if the derived series:
${\mathfrak {g}}>[{\mathfrak {g}},{\mathfrak {g}}]>[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]>[[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]],[[{\mathfrak {g}},{\mathfrak {g}}],[{\mathfrak {g}},{\mathfrak {g}}]]]>\cdots $
becomes zero eventually.
Every finite-dimensional Lie algebra has a unique maximal solvable ideal, called its radical. Under the Lie correspondence, nilpotent (respectively, solvable) connected Lie groups correspond to nilpotent (respectively, solvable) Lie algebras.
Simple and semisimple
Main article: Semisimple Lie algebra
A Lie algebra is "simple" if it has no non-trivial ideals and is not abelian. (This implies that a one-dimensional—necessarily abelian—Lie algebra is by definition not simple, even though it has no nontrivial ideals.) A Lie algebra ${\mathfrak {g}}$ is called semisimple if it is isomorphic to a direct sum of simple algebras. There are several equivalent characterizations of semisimple algebras, such as having no nonzero solvable ideals.
The concept of semisimplicity for Lie algebras is closely related with the complete reducibility (semisimplicity) of their representations. When the ground field F has characteristic zero, any finite-dimensional representation of a semisimple Lie algebra is semisimple (i.e., direct sum of irreducible representations). In general, a Lie algebra is called reductive if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive.
Cartan's criterion
Cartan's criterion gives conditions for a Lie algebra to be nilpotent, solvable, or semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on ${\mathfrak {g}}$ defined by the formula
$K(u,v)=\operatorname {tr} (\operatorname {ad} (u)\operatorname {ad} (v)),$
where tr denotes the trace of a linear operator. A Lie algebra ${\mathfrak {g}}$ is semisimple if and only if the Killing form is nondegenerate. A Lie algebra ${\mathfrak {g}}$ is solvable if and only if $K({\mathfrak {g}},[{\mathfrak {g}},{\mathfrak {g}}])=0.$
Classification
The Levi decomposition expresses an arbitrary Lie algebra as a semidirect sum of its solvable radical and a semisimple Lie algebra, almost in a canonical way. (Such a decomposition exists for a finite-dimensional Lie algebra over a field of characteristic zero.[14]) Furthermore, semisimple Lie algebras over an algebraically closed field have been completely classified through their root systems.
Relation to Lie groups
Main article: Lie group–Lie algebra correspondence
Although Lie algebras are often studied in their own right, historically they arose as a means to study Lie groups.
We now briefly outline the relationship between Lie groups and Lie algebras. Any Lie group gives rise to a canonically determined Lie algebra (concretely, the tangent space at the identity). Conversely, for any finite-dimensional Lie algebra ${\mathfrak {g}}$, there exists a corresponding connected Lie group $G$ with Lie algebra ${\mathfrak {g}}$. This is Lie's third theorem; see the Baker–Campbell–Hausdorff formula. This Lie group is not determined uniquely; however, any two Lie groups with the same Lie algebra are locally isomorphic, and in particular, have the same universal cover. For instance, the special orthogonal group SO(3) and the special unitary group SU(2) give rise to the same Lie algebra, which is isomorphic to $\mathbb {R} ^{3}$ with the cross-product, but SU(2) is a simply-connected twofold cover of SO(3).
If we consider simply connected Lie groups, however, we have a one-to-one correspondence: For each (finite-dimensional real) Lie algebra ${\mathfrak {g}}$, there is a unique simply connected Lie group $G$ with Lie algebra ${\mathfrak {g}}$.
The correspondence between Lie algebras and Lie groups is used in several ways, including in the classification of Lie groups and the related matter of the representation theory of Lie groups. Every representation of a Lie algebra lifts uniquely to a representation of the corresponding connected, simply connected Lie group, and conversely every representation of any Lie group induces a representation of the group's Lie algebra; the representations are in one-to-one correspondence. Therefore, knowing the representations of a Lie algebra settles the question of representations of the group.
As for classification, it can be shown that any connected Lie group with a given Lie algebra is isomorphic to the universal cover mod a discrete central subgroup. So classifying Lie groups becomes simply a matter of counting the discrete subgroups of the center, once the classification of Lie algebras is known (solved by Cartan et al. in the semisimple case).
If the Lie algebra is infinite-dimensional, the issue is more subtle. In many instances, the exponential map is not even locally a homeomorphism (for example, in Diff(S1), one may find diffeomorphisms arbitrarily close to the identity that are not in the image of exp). Furthermore, some infinite-dimensional Lie algebras are not the Lie algebra of any group.
Real form and complexification
Given a complex Lie algebra ${\mathfrak {g}}$, a real Lie algebra ${\mathfrak {g}}_{0}$ is said to be a real form of ${\mathfrak {g}}$ if the complexification ${\mathfrak {g}}_{0}\otimes _{\mathbb {R} }\mathbb {C} \simeq {\mathfrak {g}}$ is isomorphic to ${\mathfrak {g}}$.[15] A real form need not be unique; for example, ${\mathfrak {sl}}_{2}\mathbb {C} $ has two real forms ${\mathfrak {sl}}_{2}\mathbb {R} $ and ${\mathfrak {su}}_{2}$.[15]
Given a semisimple finite-dimensional complex Lie algebra ${\mathfrak {g}}$, a split form of it is a real form that splits; i.e., it has a Cartan subalgebra which acts via an adjoint representation with real eigenvalues. A split form exists and is unique (up to isomorphisms).[15] A compact form is a real form that is the Lie algebra of a compact Lie group. A compact form exists and is also unique.[15]
Lie algebra with additional structures
A Lie algebra can be equipped with some additional structures that are assumed to be compatible with the bracket. For example, a graded Lie algebra is a Lie algebra with a graded vector space structure. If it also comes with differential (so that the underlying graded vector space is a chain complex), then it is called a differential graded Lie algebra.
A simplicial Lie algebra is a simplicial object in the category of Lie algebras; in other words, it is obtained by replacing the underlying set with a simplicial set (so it might be better thought of as a family of Lie algebras).
Lie ring
A Lie ring arises as a generalisation of Lie algebras, or through the study of the lower central series of groups. A Lie ring is defined as a nonassociative ring with multiplication that is anticommutative and satisfies the Jacobi identity. More specifically we can define a Lie ring $L$ to be an abelian group with an operation $[\cdot ,\cdot ]$ that has the following properties:
• Bilinearity:
$[x+y,z]=[x,z]+[y,z],\quad [z,x+y]=[z,x]+[z,y]$
for all x, y, z ∈ L.
• The Jacobi identity:
$[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0\quad $
for all x, y, z in L.
• For all x in L:
$[x,x]=0\quad $
Lie rings need not be Lie groups under addition. Any Lie algebra is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator $[x,y]=xy-yx$. Conversely to any Lie algebra there is a corresponding ring, called the universal enveloping algebra.
Lie rings are used in the study of finite p-groups through the Lazard correspondence. The lower central factors of a p-group are finite abelian p-groups, so modules over Z/pZ. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives. The Lie ring structure is enriched with another module homomorphism, the pth power map, making the associated Lie ring a so-called restricted Lie ring.
Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and then reducing modulo p to get a Lie algebra over a finite field.
Examples
• Any Lie algebra over a general ring instead of a field is an example of a Lie ring. Lie rings are not Lie groups under addition, despite the name.
• Any associative ring can be made into a Lie ring by defining a bracket operator
$[x,y]=xy-yx.$
• For an example of a Lie ring arising from the study of groups, let $G$ be a group with $[x,y]=x^{-1}y^{-1}xy$ the commutator operation, and let $G=G_{0}\supseteq G_{1}\supseteq G_{2}\supseteq \cdots \supseteq G_{n}\supseteq \cdots $ be a central series in $G$ — that is the commutator subgroup $[G_{i},G_{j}]$ is contained in $G_{i+j}$ for any $i,j$. Then
$L=\bigoplus G_{i}/G_{i+1}$
is a Lie ring with addition supplied by the group operation (which is abelian in each homogeneous part), and the bracket operation given by
$[xG_{i},yG_{j}]=[x,y]G_{i+j}\ $
extended linearly. The centrality of the series ensures that the commutator $[x,y]$ gives the bracket operation the appropriate Lie theoretic properties.
See also
• Adjoint representation of a Lie algebra
• Affine Lie algebra
• Anyonic Lie algebra
• Automorphism of a Lie algebra
• Chiral Lie algebra
• Free Lie algebra
• Index of a Lie algebra
• Lie algebra cohomology
• Lie algebra extension
• Lie algebra representation
• Lie bialgebra
• Lie coalgebra
• Lie operad
• Particle physics and representation theory
• Lie superalgebra
• Poisson algebra
• Pre-Lie algebra
• Quantum groups
• Moyal algebra
• Quasi-Frobenius Lie algebra
• Quasi-Lie algebra
• Restricted Lie algebra
• Serre relations
• Symmetric Lie algebra
• Gelfand–Fuks cohomology
Remarks
1. Bourbaki (1989, Section 2.) allows more generally for a module over a commutative ring; in this article, this is called a Lie ring.
References
1. O'Connor & Robertson 2000
2. O'Connor & Robertson 2005
3. Humphreys 1978, p. 1
4. Due to the anticommutativity of the commutator, the notions of a left and right ideal in a Lie algebra coincide.
5. Jacobson 1962, p. 28
6. Jacobson 1962, p. 42
7. Bourbaki 1989, §1.2. Example 1.
8. Bourbaki 1989, §1.2. Example 2.
9. Humphreys 1978, p. 2
10. Hall 2015, §3.4
11. Hall 2015, Example 3.27
12. Jacobson 1962, Ch. VI
13. Hall 2015, Theorem 10.9
14. Jacobson 1962, Ch. III, § 9.
15. Fulton & Harris 1991, §26.1.
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• Boza, Luis; Fedriani, Eugenio M.; Núñez, Juan (2001-06-01). "A new method for classifying complex filiform Lie algebras". Applied Mathematics and Computation. 121 (2–3): 169–175. doi:10.1016/s0096-3003(99)00270-2. ISSN 0096-3003.
• Bourbaki, Nicolas (1989). Lie Groups and Lie Algebras: Chapters 1-3. Springer. ISBN 978-3-540-64242-8.
• Erdmann, Karin; Wildon, Mark (2006). Introduction to Lie Algebras. Springer. ISBN 1-84628-040-0.
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• Hall, Brian C. (2015). Lie groups, Lie algebras, and Representations: An Elementary Introduction. Graduate Texts in Mathematics. Vol. 222 (2nd ed.). Springer. doi:10.1007/978-3-319-13467-3. ISBN 978-3319134666. ISSN 0072-5285.
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• Kac, Victor G.; et al. Course notes for MIT 18.745: Introduction to Lie Algebras. Archived from the original on 2010-04-20.{{cite book}}: CS1 maint: bot: original URL status unknown (link)
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• O'Connor, J.J; Robertson, E.F. (2000). "Biography of Sophus Lie". MacTutor History of Mathematics Archive.
• O'Connor, J.J; Robertson, E.F. (2005). "Biography of Wilhelm Killing". MacTutor History of Mathematics Archive.
• Popovych, R.O.; Boyko, V.M.; Nesterenko, M.O.; Lutfullin, M.W.; et al. (2003). "Realizations of real low-dimensional Lie algebras". J. Phys. A: Math. Gen. 36 (26): 7337–60. arXiv:math-ph/0301029. Bibcode:2003JPhA...36.7337P. doi:10.1088/0305-4470/36/26/309. S2CID 9800361.
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• Varadarajan, Veeravalli S. (2004). Lie Groups, Lie Algebras, and Their Representations (1st ed.). Springer. ISBN 978-0-387-90969-1.
External links
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Bayesian Gaussian regression analysis of malnutrition for children under five years of age in Ethiopia, EMDHS 2014
Seid Mohammed1 &
Zeytu G. Asfaw2
Archives of Public Health volume 76, Article number: 21 (2018) Cite this article
The term malnutrition generally refers to both under-nutrition and over-nutrition, but this study uses the term to refer solely to a deficiency of nutrition. In Ethiopia, child malnutrition is one of the most serious public health problem and the highest in the world. The purpose of the present study was to identify the high risk factors of malnutrition and test different statistical models for childhood malnutrition and, thereafter weighing the preferable model through model comparison criteria.
Bayesian Gaussian regression model was used to analyze the effect of selected socioeconomic, demographic, health and environmental covariates on malnutrition under five years old child's. Inference was made using Bayesian approach based on Markov Chain Monte Carlo (MCMC) simulation techniques in BayesX.
The study found that the variables such as sex of a child, preceding birth interval, age of the child, father's education level, source of water, mother's body mass index, head of household sex, mother's age at birth, wealth index, birth order, diarrhea, child's size at birth and duration of breast feeding showed significant effects on children's malnutrition in Ethiopia. The age of child, mother's age at birth and mother's body mass index could also be important factors with a non linear effect for the child's malnutrition in Ethiopia.
Thus, the present study emphasizes a special care on variables such as sex of child, preceding birth interval, father's education level, source of water, sex of head of household, wealth index, birth order, diarrhea, child's size at birth, duration of breast feeding, age of child, mother's age at birth and mother's body mass index to combat childhood malnutrition in developing countries.
Malnutrition remains one of the most common causes of morbidity and mortality among under five years old children throughout the World [1]. Worldwide, over 10 million children under the age of 5 years die every year from preventable and treatable illnesses despite effective health interventions. At least half of these deaths are caused by malnutrition. The 2011 Ethiopian DHS report shows that 29% of children under age five are underweight (have low weight-for-age), and 9% are severely underweight.
The term "malnutrition" is sometimes also used synonymously for undernutrition. However, strictly speaking, malnutrition includes both undernutrition as well as over nutrition, Fig. 1. Undernutrition may be defined as insufficient intake of energy and nutrients to meet an individual's needs to maintain good health [2, 3]. Undernutrition is classified into type I and type II nutrient deficiencies [4]. In this paper, we have concerned on the type II nutrient deficiencies. Type II nutrients include protein, energy, zinc, magnesium, potassium and sodium. When there is a deficiency in one of the type II or growth nutrients, the person stops growing [5].
General framework for the study on under five years old children malnutrition, EMDHS 2014
There are three kinds of type II undernutrition in children: stunting, underweight and wasting [6]. In nutrition, anthropometric data collected in the Ethiopian mini demographic and health survey (EMDHS) are used to calculate three indices of nutritional status such as height-for-age, weight-for-age and weight-for-height. These three indices are measured through Z-scores. Z-scores represents the number of standard deviations by which an individual child's anthropometric index differs from the median of the World Health Organization international growth reference population [7].
Weight-for-age (Underweight) is a composite index of height-for-age (Stunted) and weight-for-height (Wasted). A child can be underweight for his/her age because he or she is stunted, wasted, or both. Weight-for-age is an overall indicator of a population's nutritional health. Children with weight-for-age Z-scores below minus two standard deviations from the median of the reference population are considered as underweight. Furthermore, children with Z-scores below minus three standard deviations from the median of the reference population are considered to be severely underweight, while children with Z-scores between minus three and minus two standard deviations are known to be moderately underweight [8].
Weight-for-age value for a child i is determined using a Z-score (Z i ) which is defined as:
$$\begin{array}{*{20}l} Z_{i}= \frac{{AI}_{i} - MAI}{\sigma} \end{array} $$
where AI i represents child's anthropometric indicator (weight at a certain age) for the ith child, i=1,2,....n, MAI is median of the reference population and σ is standard deviation (SD) of the reference population.
Authors are interested in modeling the various possible factors and their contribution for the high prevalence of malnutrition problems. To expand authors understanding of the most common and consistent factors on the risk of childhood malnutrition, it is necessary to consider expected determinants for malnutrition using Bayesian approach. Thus, the present study focuses on the identification of the high risk factors of malnutrition and test different statistical models for childhood malnutrition and, thereafter weighing the preferable model through model comparison criteria.
Study sample and setting
The data sets used in the present study were obtained from the Ethiopian Mini Demography Health Survey, EMDHS (2014). The survey drew a representative sample of women of reproductive age (15-49), by administering a questionnaire and making an anthropometric assessment of women and their children that were born within the previous five years [9].
For the 2014 EMDHS, a representative sample is approximately 4893 children aged less than 59 months with complete anthropometric measurements of underweight [8]. In the present study, data are presented for 3115 of these children considering that values had missed for malnutrition (underweight) as well as it's determinants.
The causes of children malnutrition are multiple. Our analysis started with a large number of covariates including a set of socio-economic, demographic, health and environmental characteristics that are considered as the most important determinants of children's malnutrition as suggested by previous studies ([10–12]).
Response variable
In our application, malnutrition (underweight) was considered as the response variable. Z-score (in a standardized form) was used as a continuous variable to maximize the amount of information available in the data set.
We have considered both continuous and categorical variables as expected determinants of children malnutrition.
Continuous covariates
Child's age in months (Chag)
Mother's age at birth (MAB)
Mother's body mass index (BMI)
Categorical Covariates (as factor coding)
Sex of child (Chsex: female or male)
Mother's current work status (MWsts: no or yes)
Mother's education level (MED: no formal education, primary or secondary and above)
Father's education level (FED: no formal education, primary or secondary and above)
Locality where child lives (Residence: rural or urban)
Wealth index (Welnx: poor, medium or rich)
Duration of breast feeding (Brstfdg: never breast fed, fed but no currently breast feeding or still breast feeding)
Sex of household head (HHsex: female or male)
Age of household head in years (HHage: 15-38, 39-63 or above 63)
Birth order (Border: 1-4, 5-9 or 10 and above)
Preceding birth interval in months (PresBint: less than 24, 24-47 or 48 and above)
Child's size at birth (Chsize: small, average or large)
Sources of drinking water (Water: not improved or improved)
Toilet facility (Toilet: no facility or have facility)
Had diarrhea recently (Diarhea: no or yes)
Ever had vaccination (Vacination: no or yes)
Whether mother take drug for intestinal parasites during pregnancy (Drug: no or yes)
The statistical analysis employed in the present study is based on Bayesian approaches which allow a flexible framework for realistically complex models. These approaches allow us to analyze usual linear effects of categorical covariates and non linear effects of continuous covariates within a unified semi-parametric Bayesian framework for modeling and inference. Basically, we are interested in model fitting of Gaussian linear regression model to identify those variables which have linear effects on the children's malnutrition. Extending to additive Gaussian regression model to find out those variables which have non linear effect on children malnutrition. Moreover, we have considered the semi-parametric regression model to look at both effects. Finally, we employed the model comparison Criterion to choose the preferable model for the data analysis.
Gaussian linear regression model
Consider the normal linear regression model in which a response variable y is related to one or more explanatory variables. For a random sample of n individuals, the model becomes:
$$ \eta_{i} =W_{i}^{'}\nu + V_{i}^{'}\gamma $$
Here, W i =(wi1,....,w ip ) is a vector of continuous covariates. ν=ν1,.....,ν p is a vector of regression coefficients for the continuous covariates. V i =(vi1,....,v ik ) is a vector of categorical covariates. γ=γ0,γ1,.....,γ k is a vector of regression coefficients for the categorical covariates. p=1,2,3;k=1,2,....,17 and i=1,2,...,3115.
And also, this model can be written as:
$$\eta_{i}=X_{i}\beta $$
where: X i =(W i ,V i ) and β=(ν,γ).
Gaussian semi-parametric regression model
The assumption of a parametric linear predictor for assessing the influence of covariate effects on responses seems to be rigid and restrictive in practical application situation and also in many real statistically complex situation since their forms can not be predetermined a priori. Besides, practical experience has shown that continuous covariates often have nonlinear effects. In our study, for the continuous covariates in the data set, the assumption of a strictly linear effect on the predictor may not be appropriate, i.e. some effects may be of unknown nonlinear form (such as, mother's age and mother's BMI) as suggested by Khaled [12] and Mohammed [13].
Hence, it is necessary to seek for a more flexible approach for estimating the continuous covariates by relaxing the parametric linear assumptions. This in turn allows continuous covariates to follow their true functional form. This can be done using an approach referred to as nonparametric regression model. To specify a non parametric regression model, an appropriate smooth function that contains the unknown regression function needs to be chosen.
The semi-parametric regression model is obtained by extending model (1) as follows:
$$ \eta_{i} = f_{1}(w_{i1})+....+f_{p}(w_{ip}) + V_{i}^{'}\gamma $$
Here, i=1,2,...,n and p=3f i (w i ) are smooth functions of the continuous covariates and \(V_{i}^{'}\gamma \) represents the strictly linear part of the predictor.
It is based on the posterior distribution. Basic statistics like mean, mode, median, variance and quartiles are used to characterize the posterior distribution. The joint conjugate prior for (β,σ2) has the structure [14]:
$$p\left(\beta,\sigma^{2}\right)= p\left(\beta|\sigma^{2}\right) p\left(\sigma^{2}\right) $$
Then, the posterior distribution is given by:
$$ p\left(\beta,\sigma^{2}|y\right) \propto p\left(y|\beta,\sigma^{2}\right) p\left(\beta|\sigma^{2}\right)p\left(\sigma^{2}\right) $$
where the conditional prior for the parameter vector β is the multivariate Gaussian distribution with mean \(\hat {\beta }\) and covariance matrix σ2V β [14]:
$$\beta|\sigma^{2} \sim N_{p}\left(\hat{\beta}, \sigma^{2} V_{\beta}\right) $$
and to obtain the prior for σ2, now we integrate β out of the joint posterior to get the marginal posterior for σ2 [14]:
$$\pi\left(\sigma^{2}\right)= \int \pi\left(\beta,\sigma^{2}\right) d\beta $$
Then, the marginal posterior distribution of σ2 becomes inverted gamma, which is clearly
$$IG(a, b) $$
In Bayesian approach, the vector of unknown parameters to be estimated is θ=(β,σ2). Therefore, we need to choose prior distributions for these parameters. If prior information is scarce, a large value for the variance parameter should be chosen, so that the prior distribution is flat. This type of prior is called non informative prior. On the other hand, if the analyst has considerable information about the coefficient β, he/she should choose a small value for the variance parameter.
For our specific application in model (1), due to the absence of any prior knowledge we use a noncommittal or vague priors π(ν)∝constant and π(γ)∝constant for the parameters of fixed (linear) effects. For each regression coefficient, the prior distribution is a very broad normal distribution, with a mean of zero and a standard deviation that is extremely large relative to the scale of the data. The same assumption is made for the prior on the intercept. Finally, the prior on the standard deviation of the predicted value is merely a uniform distribution extending from zero to an extremely large value far beyond any realistic value for the scale of the data. In the specific analysis demonstrated in this section of our article, the data were standardized so that the prior would be broad regardless of the original scale of the data. The results were then simply algebraically transformed back to the original scale. For the standardized data, the prior on the intercept and regression coefficients was a normal distribution with mean at zero and large standard deviation (example; 1000). This normal distribution is virtually flat over the range of possible intercepts and regression coefficients for standardized data.
To begin, we will choose a non-informative (vague) prior [14]. But in model (2), the parameters of interest f j is considered as random variables and have to be supplemented with appropriate prior assumptions. Several alternatives are available as smoothness priors for the unknown functions f j (w j ). Among the others, random walk priors [14], Bayesian Penalized-Splines [15], Bayesian smoothing splines [16] are the most commonly used. In the present study, the Bayesian smoothing spline was used by taking cubic P-spline with second order random walk priors [17, 18].
By defining an additional hyperprior for the variance parameters the amount of smoothness can be estimated simultaneously with the regression coefficients. We assign the conjugate prior for \(\tau ^{2}_{j}\) which is an inverse gamma prior with hyper parameters a j and b j , i.e \(\tau ^{2}_{j} \sim IG(a_{j}, b_{j})\). Common choices for a j and b j are a j = 1 and b j small, e.g. b=0.005orb j =0.0005. Alternatively we may set a j =b j , e.g. a j =b j =0.001. Based on experience from extensive simulation studies the researcher use a j =b j =0.001 as the standard choice. Since the results may considerably depend on the choice of a j and b j some sort of sensitivity analysis is strongly recommended. For instance, the models under consideration could be re-estimated with (a small) number of different choices for a j and b j .
Model comparison and selection
Model selection is the task of selecting the best model from a set of candidate models based on the performance of each model.
The next question is why should we consider model selection? There are several reasons. First, people tend to believe or can understand simpler models with fewer predictors and less complicated structure. Second, one can certainly add more and more features into the model without screening and get better and better fit, till perfect fit, but the problem is over fitting. Note that the authors want to find the best-predicting model not the best fitting model.
Model comparison is required for a diversity of activities, including variable selection in regression, determination of the number of components or the choice of parametric family. In frequentest approach, we can also perform the familiar statistical test via the anova function. As with frequentest analogues, Bayesian model comparison will not inform about which model is true, but rather about the preference for the model given the data and other information [14].
The models proposed in the present study are quite general and the model building process can be quite challenging. Currently, an automated procedure for Bayesian model selection is not available. However, a few recommendations are possible:
Users should try to incorporate everything that is theoretically possible.
Different Bayesian models could be compared via the Deviance Information Criterion (DIC) [19].
In the present study, AIC (Akaike Information Criterion) is used to compare the linear frequent and the linear Bayesian approach. Then, we compared the additive frequent and the Bayesian approach by using the GCV (Generalized Cross-Validation) score.
The classical approach to model comparison involves a trade-off between how well the model fits the data and the level of complexity. Spiegelhalter et al. [19] devised a selection criterion which was based on Bayesian measures of model complexity and how good a fit the model is for the data. The measure of complexity which we adopted in this work is suggested by [19].
A widely used statistic for comparing models in a Bayesian framework is the DIC. DIC is a hierarchical modeling generalization of the AIC (Akaike information criterion) and BIC (Bayesian information criterion). It is particularly useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by Markov chain Monte Carlo (MCMC) simulation. The idea is that models with smaller DIC should be preferred to models with large DIC, Fig. 2.
Chart that approximating the Posterior marginal distribution through BayesX
Descriptive analysis
In the present study, the response variable malnutrition seems reasonable to assume at least approximately Gaussian (normal) distributed since it has a continuous Z-score value. Then it can be reasonably approximated by a Gaussian distribution that can be observed from the histogram plot in Fig. 3 and Additional file 1. In Fig. 4, the scatter plot of malnutrition vs each continuous covariates such as child age in months, mother's age at birth and mother's body mass index showed that there is no definite pattern of relationships respectively. To overcome this problem, we deployed a non parametric method to explore relationships among covariates (see Fig. 5).
Histogram for underweight showing a normal distribution in under five years old children malnutrition, EMDHS 2014
Scatter plots that represent the relationship between each continuous covariates with under five years old children malnutrition, EMDHS 2014
The Non Linear Effects of Continuous Variables on under five years old children malnutrition, EMDHS 2014
The main purpose of the present descriptive analysis was to describe the variation among the categorical explanatory variables with regard to children malnutrition in Ethiopia through percentage value.
Table 1 showed that the proportion of children's malnutrition decreases as the age of head of household, child's birth order and father's as well as mother's education level increases. The proportion of underweight children is approximately nine times higher for those born to uneducated father's than for those whose father's have more than secondary education (59.3% versus 6.7%). Children born from mothers in the poorest wealth quintile are more than twice as likely to be malnourished as children born from mothers in the richest wealth quintile (57.3% compared with 26.1%). The proportion of children malnutrition, as can be seen in Table 1, differs by type of place of residence: urban and rural. From Table 1 we observed that children reside in rural areas were more likely to be malnourished. On the other hand, children ever had vaccination were apparently more often affected by malnutrition than those never got vaccination but there was no consistent trend in the pattern of malnutrition with respect to children got vaccination. With regards to underweight children, female children are slightly more likely to be malnourished than male children (52% versus 48%).
Table 1 Distribution of categorical variables vs under five years old children malnutrition, EMDHS 2014
Regarding Child's birth interval in month, the lowest prevalence of all child's underweight status was observed among children whose birth interval is less than 24 months (21%), Table 1. As opposed to the highest prevalence of all child's underweight status was recorded from children whose birth interval is between 24 and 47 (54.9%). Also, children reported as small or average at birth are much more likely to be malnourished (34.8% and 38.5%, respectively) than those reported as large at birth (26.7%).
Inferential analysis
In this section, the statistical procedure was used in combination with the BayesX stepwise selection method. This enabled us to select different covariates which contribute to malnutrition. Table 2 gives results for the fixed effects on the malnutrition of children under five years old in Ethiopia. The output gives posterior means, posterior median along with their standard deviations and 95% credible intervals.
Table 2 Results of fixed effects estimation results of parametric coefficients
Since the 95% credible interval do not include zero, father's education level, place of residence (rural), sex of the head of household (male), child's sex (female), source of water (not improved), diarrhea (had diarrhea), drug (never took drug for intestinal parasites during pregnancy), children wealth index, birth order, preceding birth interval, duration of breast feeding and size of child at birth were found statistically significant at 5% significance level. But, age of household was found statistically insignificant.
Figure 5 displays nonlinear effects and estimated functions of mother's age at birth in year, child's age in month and mother's body mass index for under five years old child data. The shaded region represents twice the point wise asymptotic standard errors of the estimated curve.
The panels in Fig. 5 show an interval marked as HDI, which stands for highest density interval. Points inside an HDI have higher probability density (credibility) than points outside the HDI, and the points inside the 95% HDI include 95% of the distribution. Thus, the 95% HDI includes the most credible values of the parameter. The 95% HDI is useful both as a summary of the distribution and as a decision tool. Specifically, the 95% HDI can be used to help decide which parameter values should be deemed not credible, that is, rejected. This decision process goes beyond probabilistic Bayesian inference, which generates the complete posterior distribution, not a discrete decision regarding which values can be accepted or rejected. One simple decision rule is that any value outside the 95% HDI is rejected. In particular, if we want to decide whether the regression coefficients are nonzero, we consider whether zero is included in the 95% HDI.
In the present study, all continuous variables shows significant effect on underweight status of children under age of five years old. Here we can see in Fig. 5, the positive and negative linear effects on malnutrition at lower level of mother's body mass index and age of child respectively. And in addition, mother's age at birth seems have a slight positive linear effect on the malnutrition of children.
Figure 5 showed the nonlinear effects of child's age in month shows that the children face a risk of suffering from malnutrition during the first 30 months of their life, and then it is slight thereafter.
We can use Akaike Information Criterion (AIC), Generalized Cross-Validation (GCV) and Deviance Information Criterion (DIC) as a comparative measure to choose among different models, with lower being better [14].
The core point here is to select the better model with respect to their AIC value. Based on Table 3, it is evident that the Bayesian linear regression model has smaller AIC value than the frequent linear model.
Table 3 Cumulative information for all models
As illustrated the GCV value of semi parametric regression model in Table 3, the Bayesian approach with small value than that of the frequent approach which still is the one that can be selected.
Next, we focused on the comparison of model 1 with model 2 as well as model 2 with model 4 based on the detected results in relative to the frequent and Bayesian approach, respectively. Since model 1 and model 3 are included under the frequent approach. Since ANOVA function is an automatic functioning machine, we used ANOVA function as a comparing system of model 1 and model 3 and thus, model 3 was found to have a better fit.
As DIC is a criteria used as a comparing tool for Bayesian approach, model 2 and model 4 can be compared using the descripted DIC value in Table 3. Consequently, the models with a better fit of less DIC value are preferable models. Based on its performance, model 2 was chosen as a suitable model to identify the most determinants of childhood malnutrition.
The study aimed at examining the major influential factors behind children's (under five year of age) malnutrition. The status of child malnutrition in the country was measured as underweight. The study showed, all Children 3115 (31.7%) were affected by malnutrition. For our study, suitably fitting (Bayesian Gaussian linear regression) model was chosen as a suitable model to identify determinants to childhood malnutrition in the Ethiopian context. The finding revealed that the covariates such as sex of child, preceding birth interval, age of child, father's education level, source of water (the condition of an availability of water), mother's body mass index, household head's sex, mother's age at birth, wealth index, birth order, diarrhea, child's size at birth and duration of breast feeding were identified as statistically significant factors; whereas age of head of household was found to be statistically insignificant.
The results indicated that the variables such as access to health care, for children's mothers who have not taken drug during pregnancy, had significant effects on malnutrition status of children. It was therefore implied that taking drug during pregnancy (by mothers) was more effective against underweight of children.
It is a well known fact that breast feeding had a greater influence over the growth of a child which is also confirmed by our study. Furthermore, our study revealed that diarrhea practice and duration of breast feeding also contributed significantly for children's malnutrition which fell in line with the results recorded by Bete - Israel [20].
The living conditions along with the area of living (being in and out of an urban area) could determine the child's malnutrition status. Problems such as poor health care access, lack of sufficiently (accessible) toilet supply, lack of modern source power like stove, cylinder and lack of awareness on the how of curing the available source of water for using it to their personal hygiene was assumed to be the risk factors of malnutrition status [21]. Our study indicated that the place of residence (rural) was associated with significant effects of malnutrition (underweight). This finding evens the finding(s) in earlier (previous) studies [22, 23]. The education attainment of fathers was also associated with significant effects to malnutrition, as of our finding. Similarly, a study [24] concluded that it (the factor in point) had an association with childhood malnutrition.
A household's source of drinking water has been shown to be associated with malnutrition of a child in Nigeria (weight-for-age) in separate analysis [12], and that this study has also emphasized the significant of this factor of risk of malnutrition. More, it is associated with malnutrition of a child in that it impacted a risk of childhood diseases such as diarrhea, and is affective indirectly as a 'measure of wealth' and availability of water. This result quite consistent with some studies [11, 23, 25] but not persistent with other finding [26, 27].
Malnutrition in women is assessed using BMI. Parents with low BMI values are malnourished and are therefore likely to have undernourished and weak children. At the same time, very high BMI values indicate poor quality of the food and hence, may also imply weakness of the children [12]. The patterns of mother's body mass index (top of Fig. 5) showed that the higher impact of BMI through the interval between 15-25, indicates that there was poor quality of food for mothers. When the BMI of non pregnant women falls below the suggested cut-off point, which is less than 18.5\(\frac {kg}{m^{2}}\), malnutrition is indicated. Women who are underweight may have complications during childbirth and may deliver a child who can be underweight [6]. Our study finding indicated that there exist an association between the BMI of the mother and child's acquiring of malnutrition. This finding is of not surprise and it correspondence with the results found by others on studies analyzing the childhood malnutrition like [23–25].
Determinants that explain the cause of malnutrition in Ethiopian children community have been explored using different General additive models and Bayesian approaches. By using model comparison criteria, Gaussian linear model in Bayesian approaches was the suitable best fitted model. The findings of the present analysis indicated that sex of child, preceding birth interval, father's education level, source of water, head of household's sex, wealth index, birth order, diarrhea, child's size at birth and duration of breast feeding are important determinants of childhood malnutrition. The age of child, mother's age at birth and mother's body mass index could also be important factors with a non linear effect for the child's malnutrition in Ethiopia. Thus, a special emphasis need to be given on these factors to combat childhood malnutrition in developing countries.
AIC:
Akaike information criterion
DIC:
Deviance information criterion
EMDHS:
Ethiopian mini demographic health survey
GCV:
Generalized cross-validation
MCMC:
Markov chain Monte Carlo
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Authors acknowledge Ethiopian Central Statistical Agency (Addis Ababa) and School of Mathematical and Statistical Modeling, Hawassa University.
This work was financially supported by the School of Mathematical and Statistical Modeling, Hawassa University.
Availability of data and material
The analysis in this study is based on data available from the Ethiopian Demographic and Health Survey.
Ethics approval and consent to participant
Department of Statistics, Aksum University, Aksum, Ethiopia
Seid Mohammed
School of Mathematical and Statistical Sciences, College of Natural and Computational Sciences, Hawassa University, Hawassa, Ethiopia
Zeytu G. Asfaw
Both authors SM and ZGA generated the idea, the corresponding author SM contributed in the data analysis and interpretation, ZGA contributed as an advisory. Both authors read and approved the final manuscript.
Correspondence to Seid Mohammed.
Additional file 1
Histogram from Z-score value for underweight showing a normal distribution in under five years old children malnutrition, EMDHS 2014. (DOCX 35.9 kb)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Mohammed, S., Asfaw, Z. Bayesian Gaussian regression analysis of malnutrition for children under five years of age in Ethiopia, EMDHS 2014. Arch Public Health 76, 21 (2018). https://doi.org/10.1186/s13690-018-0264-6
Accepted: 14 February 2018
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Akbar Rahimi1Email author
https://doi.org/10.1186/s13717-016-0069-x
Due to the health effects caused by airborne pollutants in urban areas, forecasting of air quality parameters is one of the most important topics of air quality research. During recent years, statistical models based on artificial neural networks (ANNs) have been increasingly applied and evaluated for forecasting of air quality.
The development of ANN and multiple linear regressions (MLRs) has been applied to short-term prediction of the NO2 and NO x concentrations as a function of meteorological conditions. The optimum structure of ANN was determined by a trial and error method. We used hourly NO x and NO2 concentrations and metrological parameters, automatic monitoring network during October and November 2012 for two monitoring sites (Abrasan and Farmandari sites) in Tabriz, Iran.
Designing of the network architecture is based on the approximation theory of Kolmogorov, and the structure of ANN with 30 neurons had the best performance. ANN trained by scaled-conjugate-gradient (trainscg) training algorithm has implemented to model. It also demonstrates that MLP neural networks offer several advantages over linear MLR models. The results show that the correlation coefficient (R 2) values are 0.92 and 0/94 for NO2 and NO x concentrations, respectively. But in MLR model, R 2 values were 0.41 and 0.44 for NO2 and NO x concentrations, respectively.
This work shows that MLP neural networks can accurately model the relationship between local meteorological data and NO2 and NO x concentrations in an urban environment compared to linear models.
Air pollution prediction
NO x
By inducing oxidative stress, air pollutants may lead to allergic inflammation and induce acute asthma exacerbations (Sfetsos and Vlachogiannis 2010). Because air pollutants can harm human health (Hong et al. 2011), the forecasting of air pollutant concentrations has received much attention. The air pollution forecasting is a complex issue but is closely related to human health and the environment (Wang et al. 2015). Nitrogen oxides (NO x = NO + NO2) are emitted into the urban atmosphere primarily from vehicle exhausts. Primary NO x emissions are mostly in the form of nitric oxide (NO) which then reacts with ozone (O3) to form nitrogen dioxide (NO2) (Gardner and Dorling 1999). Many works have been carried out to determine the factors which control NO x and NO2 concentrations in order to enable the development of tools to aid in the forecasting of pollutant concentrations. One approach to predict future concentrations is to use a detailed atmospheric diffusion model. Such models aim to resolve the underlying physical and chemical equations controlling pollutant concentrations and therefore require detailed emissions data and meteorological fields. Collet and Oduyemi (1997) provide a detailed review of these particular types of models. The second approach is to devise statistical models which attempt to determine the underlying relationship between a set of input data (predictors) and targets (predicted). Regression modeling is an example of such a statistical approach and has been applied to air quality modeling and prediction in a number of studies (Shi and Harrison 1997; Ziomass et al. 1995). One of the limitations imposed by linear regression models is that they will underperform when used to model nonlinear systems (Gardner and Dorling 1998). The highly nonlinear processes of pollutant concentration genesis and its dynamics are only partially known, and they need complex computer modeling and simulation to obtain a reliable prediction (Brunelli et al. 2008).
Artificial neural network (ANN) modeling has proven to be a reliable air pollution time series modeling tool (Gardner and Dorling 1998; Niska et al. 2004). It can be used to derive nonlinear functions relating the concentrations of pollutants to meteorological and source characteristics. The accuracy of the trained neural network model in predicting the concentrations of an unseen data set reflect the extent to which the input parameters have captured the emission and dispersion pattern that have resulted from the observed concentrations (Elangasinghe et al. 2014). ANNs can model nonlinear systems and have been employed for modeling of systems (Chelani et al. 2002). Gardner and Dorling (1999) used a multilayer perceptron (MLP) approach, for modeling of NO and NO2 concentrations in London. They found out that the temporal variation of emissions could be represented by using the input variables of time of day and day of week. In addition, simple meteorological input variables are used, providing some indication of atmospheric stability, without the need for processing of the measured meteorological data. In the air quality forecasting, the selection of optimal input subset (Jiang et al. 2004) becomes especially a tedious task due to large number of measurements from heterogeneous sources and their nonlinear interactions. Applications of ANNs in the atmospheric sciences generally give better results than linear methods (Gardner and Dorling 1998).
The findings of numerous research studies also exhibit that the performance of ANNs is generally superior in comparison to traditional statistical methods, such as multiple regression, classification and regression trees, and autoregressive models (Gardner and Dorling 2000; Chaloulakou et al. 2003a; Grivas and Chaloulakou 2006; Palani et al. 2008; Elangasinghe et al. 2014). In this paper, we used ANN for forecasting air pollution in new geographic location (Tabriz) with deferent climate condition to confirm previous studies. So, in this work we developed a model that could make accurate short-term (hourly) predictions and since the relationship between NOx and NO2 and meteorology in Tabriz using ANN.
The study was carried out in the city of Tabriz in the northwest area of Iran. Tabriz is the center of east Azerbaijan province and is located in within the 46.17° east longitudinal and 35.05° north latitudinal position (Mojtabazadeh 2005). In the mid-twentieth century, Tabriz is selected as one of the industrial poles in Iran. Establishment of heavy industrial centers in west and southwest is the main factor in Tabriz air pollution (Sadr Mousavi and Rahimi 2008). The city is increasingly faced with development and population growth (Breuste and Rahimi 2015) and the most densely populated in northwest of Iran (Fig. 1).
The study area map (author's illustration)
Artificial neural networks (ANNs) are able to approximate accurately complicated nonlinear input–output relationships. Like their physics-based numerical model counterparts, ANNs require training or calibration. After training, each application of the trained ANN is an estimation of a simple algebraic expression with known coefficients and is executed practically instantaneously. The ANN technique is flexible enough to accommodate additional constraints that may arise in the application (Palani et al. 2008). ANNs need a considerable amount of historical data to be trained; upon satisfactory training, an ANN should be able to provide output for previously "unseen" inputs (Palani et al. 2008, Antanasijević et al. 2013). The selection of input variables for an ANN forecasting model is a key issue, since irrelevant or noisy variables may have negative effects on the training process, resulting to an unnecessarily complex model structure and poor generalization power (Voukantsis et al. 2011).
ANNs represent complex, nonlinear functions with many parameters that are adjusted (calibrated or trained) in such a way that the ANN's output becomes similar to measured output on a known data set. ANNs need a considerable amount of historical data to be trained; upon satisfactory training, an ANN should be able to provide output for previously "unseen" inputs. The main differences between the various types of ANNs involve network architecture and the method for determining the weights and functions for inputs and neurodes (training). The multilayer perceptron (MLP) neural network has been designed to function well in modeling nonlinear phenomena. A feed-forward MLP network consists of an input layer and output layer with one or more hidden layers in between. Each layer contains a certain number of artificial neurons (Palani et al. 2008).
The general procedure for the ANN simulation includes the following steps:
Representation of input and output vectors.
Representation of the transfer function.
Selection of the network structure.
Selection of the random weights.
Selection of the learning procedure.
Presentation of the test pattern and prediction or validation set of data for generalization.
Multilayer perceptron (MLP) is a feed-forward layered network with one input layer, one output layer, and some hidden layers. Figure 2 shows a MLP with one hidden layer. The task of every node is computing a weighted sum of its inputs and passing the sum through a soft nonlinearity. This soft nonlinearity or activity function of neurons should be no decreasing and differentiable. The most popular function is unipolar sigmoid Eq. (1):
The structure of a three-layer MLP (author's illustration)
$$ f\left(\theta \right)=\frac{1}{1+{e}^{-\kern0.5em \theta }} $$
The task of the network is vector mapping, i.e., by inserting the input vector, X q , the network will answer with the vector Z q in its output (for q = 1,…,Q). The goal is to adapt the parameters of the network in order to bring the actual output Z q close to corresponding desired output d q , (for q = 1,…,Q). The most popular method for training MLP is back propagation algorithm. Back propagation is based on minimization of a suitable error or cost function. Total sum squared error (TSSE) is considered as the cost function Eq. (2).
$$ \mathrm{TSSE}={\displaystyle \sum_q{\displaystyle \sum_k{\left({d}_k^q-{z}_k^q\right)}^2}}\kern0.5em \mathrm{f}\mathrm{o}\mathrm{r}\kern0.5em \left(q=1,\dots Q\right) $$
where \( {d}_k^q \) and \( {z}_k^q \) are the components of desired and actual output vectors, respectively. Training can be carried out in two modes: pattern mode and batch mode. Pattern mode is preferred because of easier implementation and less demand on memory. In the pattern mode, the correction of weights is made immediately after the error is detected; but in the batch mode, the individual error for all patterns are accumulated and then the accumulated error for entire training set` used for the correction of weights. In forward pass, the network outputs are computed by proceeding implementation forward through the network, layer by layer form Eqs. (3) and (4):
$$ \left\{\begin{array}{l}{\mathrm{net}}_j={\displaystyle \sum_j{x}_i{w}_{ij}}\\ {}{y}_i=\frac{1}{1+{e}^{-{\mathrm{net}}_j}}\end{array}\right.,\begin{array}{cc}\hfill j=1,\dots, {l}_2\hfill & \hfill \hfill \end{array} $$
$$ \left\{\begin{array}{l}{\mathrm{net}}_k={\displaystyle \sum_j{y}_j{u}_{jk}}\\ {}{z}_k=\frac{1}{1+{e}^{-{\mathrm{net}}_k}}\end{array}\right.,\begin{array}{cc}\hfill k=1,\dots {l}_3\hfill & \hfill \hfill \end{array} $$
where w ij is the connection weight between node i and j and u jk is the connection weight between node j and k, respectively. l 2 and l 3 are the number of neurons in hidden and output layers. In backward pass, the error gradients versus weights values, i.e., \( \frac{\partial E}{\partial {w}_{ij}} \) (for i = 1,…l 1, j = 1,…l 2) and \( \frac{\partial E}{\partial {u}_{jk}} \) (for j = 1,..l 2, k = 1,…l 3), are computed layer by layer starting from the output layer and proceeding backwards. Then the connection weights between nodes of different layers are updated by Eqs. (5) and (6):
$$ {u}_{jk}\left(n+1\right)={u}_{jk}(n)-\eta \times \frac{\partial E}{\partial {u}_{jk}}+\alpha \kern0.5em \left({u}_{jk}(n)-{u}_{jk}\left(n-1\right)\right) $$
$$ {w}_{ij}\left(n+1\right)={w}_{ij}(n)-\eta \times \frac{\partial E}{\partial {w}_{ij}}+\alpha \kern0.5em \left({w}_{ij}(n)-{w}_{ij}\left(n-1\right)\right) $$
where η is the learning rate adjusted between 0 and 1, α is the momentum factor in the interval [0,1] and is used to speed up the convergence as well as alleviating the local minima problem. The decision to stop training is based on some test result on the network, which is carried out every N epoch after TSSE becomes smaller than a threshold value (Vakil-Baghmisheh and Pavešic 2003; Rahimi 2016).
Hourly air pollution concentration data were collected from the Department of the Environment, automatic monitoring network during October and November 2012 for two monitoring sites (Abrasan and Farmandari sites) in Tabriz. Both sites represent the most polluted parts of the city which are located in the busiest part in the Tabriz. The data from both sites were combined to produce one series of data.
Hourly meteorological data were obtained for the same period from the department of Tabriz Met. Office. Meteorological data were selected to be used in this study, since these are the best representative for the whole of the urban area and also contain relevant derived atmospheric turbulence parameters. The meteorological variables in this work were as follows:
Wind direction (degree)
Precipitation (mm.)
Vapor pressure (mbar)
Relative humidity (percent)
Total radiation (J)
Barometric pressure (mbar)
In order to be used with the MLP, meteorological data and concentration data were normalized, respectively, into the range 0–1 and 0.2–0.8. This was carried out by determining the maximum and minimum values of each variable over the whole data period and calculating normalized variables (Gardner and Dorling 1999). The available data set was separated into 745 training sets, 405 validation data set, and 232 testing data set.
Choice of network structure
Feed-forward neural networks have been used in this study. The architecture of a net is established base on the numbers of neurons in the input and the output layers and the number of the hidden layers and/or the number of neurons in each hidden layer depends on the kind of the modeled system. Designing of the network architecture is based on the theory of Kolmogorov (Kolmogorov 1957). According to this theory, a feed-forward neural network, containing at least one hidden layer with (2N + 1) neurons, is able to approximate any continuous function converting the N-dimensional input vector into the M-dimensional output vector. This theory does not describe precisely a net architecture, but it is rather a starting point to optimization procedure.
In order to determine the optimum number of hidden nodes, a series of topology was used. the number of nodes was varied from 21 to 35. The starting point for nodes in this paper was based on Kolmogorov theory.
Each topology was repeated three times to avoid random correlation due to the random initialization of the weight. Figure 3 illustrates the relation between the network error and number of neuron in hidden layer. The root mean square error (RMSE) was used as the error function. The R 2 of each output was calculated by Eq. (7):
Effect of the number of neurons in hidden layer on the performance of the neural network in prediction of NO2 and NO x concentrations at test set (author's illustration)
$$ {R}^2=\left(\frac{{\displaystyle {\sum}_i\left[\left({x}_i-\overline{x}\right)\left({y}_i-\overline{y}\right)\right]}}{{\left[{\left({\displaystyle {\sum}_i{x}_i-\overline{x}}\right)}^2{\left({\displaystyle {\sum}_i{y}_i-\overline{y}}\right)}^2\right]}^{1/2}}\right) $$
where x i is original target vector, \( \overline{x} \) is the mean of target vector, y i is the predicted vector, \( \overline{y} \) is the mean of predicted vector, and j is an index of data (Zupan and Gasteiger 1999).
Neural model development
Particularly, this step is crucial for a robustness and accuracy of the developed neural model. The following procedure was carried out for selection of input variables:
In the first model configuration, meteorological variables are tested as input. So, wind speed, wind direction, precipitation, vapor pressure, air temperature, relative humidity, barometric pressure, and total radiation were used one by one as input of network, and the NO2 and NO x concentrations were used for output. Then, a progressive increase of the number of the input variables was carried out, in order to increase the number of model parameters. The criterion for increasing the number of variables was the value of the correlation coefficient R 2 and RMSE. Thus, if the increase of a given input variable resulted in a decrease in the value of RMSE and increase in the value of (R 2), the variable was added in the model. If not, it was increased (RMSE) and decreased (R 2), the procedure was repeated with another variables, because the selection of input variables has significant effect on performance of networks. The network structure was selected 8-30-1. It was found that there is a good agreement between prediction and real data.
In the case of prediction of NO2 and NO x concentrations, we added NO (ANN predicted) and O3 concentrations to input variable set. With this modification in input variables set, significance increasing in regression coefficient (R 2) was observed. It can be due to interaction between this species. This interaction can be described by following chemical reaction set Eqs. (8) to (10), (Gürmen and Fogler 2006):
$$ {\mathrm{NO}}_2+hv\to \mathrm{NO}+\mathrm{O} $$
$$ \mathrm{O}+{\mathrm{O}}_2\to {\mathrm{O}}_3 $$
$$ {\mathrm{O}}_3+\mathrm{NO}\to {\mathrm{NO}}_2+{\mathrm{O}}_2 $$
RMSE in the selected network is 0.0046 and 0.0038; R 2 is 0.92 and 0.94 for NO2 and NO x , respectively. Table 1 shows the effect of input variable selection on network performance.
The effect of different inputs on optimized network performance
Meteorological parameter
NO x and NO2
Meteorological parameter + O3+ NO (predicted)
Meteorological parameter + O3
Meteorological parameter + O3 + SO2
Figures 4 and 5 present the comparisons of prediction results on the testing data during October and November 2012, for NO2 and NO x concentrations, respectively. It is shown that the prediction results generated by the MLP model are getting closer to the actual data. Multiple linear regression models were developed in this work for result comparison. The best model with the lowest RMSE 3.6 and 2.94 and the highest R 2 is 0.41 and 0.44 for NO2 and NO x concentrations, respectively. The results in this work exhibit that the performance of ANNs is generally superior for air pollution modeling in comparison to multiple linear regression (MLR) as a traditional statistical method.
Comparison of observed and calculated values for NO2 (author's illustration)
Comparison between observed and calculated values of NO x (author's illustration)
Importance analyses
The Garson method (Garson 1991) is shown by Olden and Jackson (Olden and Jackson 2002) and is based in the partition of the neural weights of the hidden and output layers. This method determines the relative importance (I) of jth input neuron in the output neuron. This relative importance is defined as:
$$ {I}_j=\frac{{\displaystyle {\sum}_{m=1}^{N^h}\left(\frac{\left|{W}_{jm}^{ih}\right|}{{\displaystyle {\sum}_{k=1}^{m={N}^h}\left|{W}_{km}^{jh}\right|}}*\left|{W}_{mn}^{ho}\right|\right)}}{{\displaystyle {\sum}_{k=1}^{k={N}^i}\left\{{\displaystyle {\sum}_{m=1}^{m={n}^h}\left(\frac{\left|{W}_{jm}^{ih}\right|}{{\displaystyle {\sum}_{k=1}^{m={N}^h}\left|{W}_{km}^{jh}\right|}}\right)*\left|{W}_{mn}^{ho}\right|}\right\}}} $$
where N h is the number of neuron in hidden layer, N i is the number of weight for each neuron in hidden layer and W mn is the weight of nth neuron in output layer.
The use of Garson method in this work reveals that the NO (ANN predicted) concentration, relative humidity, and air temperature are the best important variables in NO2 and NO x concentration prediction (Fig. 6).
Calculated importance (%) for each input variables in the prediction NO2 and NO x concentration (author's illustration)
Why the air pollution forecasting is important?
Air pollution is rapidly increasing due to various human activities, and it is the introduction into the atmosphere of chemicals, particulates, or biological materials that cause discomfort, disease, or death to humans, damage other living organisms such as food crops, or damage the natural environment or built environment. Indeed, air pollution is one of the important environmental problems in metropolitan and industrial cities (Garcia Nieto and Alvarez Antَn 2014). Stoves in homes, vehicles, factories, and fires are different sources of air pollution. Both ambient (outdoor) and household (indoor) pollution exert many harmful effects on either human health or the environment (Bedoui et al. 2016). Increasing air pollution has become a global problem that is triggering both official anxiety and public concern. As reported in an assessment by the World Health Organization (WHO 2014), air pollution has become the largest single environmental health risk in many parts of the world, and around seven million people died from air pollution exposure in 2012, equivalent to one in eight of the total global deaths (Xie et al. 2016).
Air pollution in all major cities of Iran has reached a dangerous and alarming level. Air pollution poses a dire risk to Iranians today. The consequences can be measured in the numbers of pollution-related deaths, the number of school and work days lost to pollution, and additional health challenges experienced by children, the elderly, and people with heart or lung conditions (Khani 2016). These are drastic times for Iran's big cities such as Tabriz.
The public is informed of air quality index (AQI) calculated from air pollutants concentrations forecasted and associated health risks through government announcements (Zhang et al. 2012). Therefore, an accurate and reliable model for forecasting air pollutant concentrations is important since it can provide advanced air pollution information at an early stage such that guiding the works of air pollution control and public health protection (Bai et al. 2016).
In recent years, many research efforts have been made to develop the air quality prediction models. Atmospheric dispersion models used to predict the ground level concentration of the air pollutants around the sources (Kesarkar et al. 2007; Bhaskar et al. 2008; Singh et al. 2012) require precised knowledge of several source parameters and the meteorological conditions (Collett and Oduyemi 1997; Gardner and Dorling 1998).
Linear and nonlinear methods for air pollution forecasting
In recent decades, air pollution has been considered a serious threat to the environment, the quality of life, and the health of people around the world and forecasting of air quality parameters is the common goal for a great number of researches due to the diseases caused by the different gas pollutants. In recent time, there have been many attempts to analyze the concentration of air pollutants and explore them to build short-term forecast of concentrations. Linear and nonlinear models were developed, however, there was no significance difference noted between nonlinear and linear models (Pires et al. 2008a; Pires et al. 2008b).
The statistical models attempt to determine the underlying relationship between a set of input data and targets. Several linear (multiple linear regression, principal component regression, partial least squares regression) and nonlinear (multivariate polynomial regression, artificial neural networks, support vector machines) regression models are now available, which have the ability to relate the input and output variables (Singh et al. 2012). Although linear regression modeling finds some applications in the air quality prediction (Shi and Harrison 1997), it generally does not permit for consideration of complex and nonlinearity in data (Gardner and Dorling 1998). Partial least squares (PLS) is a multivariate regression method that projects the input–output data down into a latent space, extracting a number of principal factors with an orthogonal structure, while capturing most of the variance in the original data. Multivariate polynomial regression (MPR) captures nonlinearities in data to some extent and is considered a low-order nonlinear method (Singh et al. 2010). ANN, which has the capabilities of nonlinear mapping, self-adaption, and robustness, has proved its superiority and is widely used in forecasting fields. Recently, various structures of the ANN have been developed for improving the forecasting performances of air pollutant concentrations (Bai et al. 2016). Results in this work confirm that the ANN in air pollution forcasting generally gives better results than linear methods.
Tabriz air pollution resource and the role of metrological parameter in pollution exacerbating
Environment pollution is a challenge to the modern society, especially in developing countries for example Iran. In the beginning of the century, industrialization and expansion of the factories became the main concern of Iranian big cities. The city of Tabriz in Iran that could hold a large population in it turned to become as one of the industrial poles in the country. Within the years 1967–1975, Tabriz city was the subject of new changes and developments. But in the process of industrialization and installation of the manufacturing sites and factories, some decision makers and executive managers did not try to take the geographical and topographical conditions of the city into their considerations; therefore, the city of Tabriz became more and more polluted and the people's social health and hygiene were endangered (Mojtabazadeh 2005). Establishment of high industrial factories in the west and southwest of Tabriz, such as chemical and petrochemical industries, thermal power plant, and oil refinery, and blowing of wind from west and southwest transferred their pollution to the inner city (Sadr Mousavi and Rahimi 2008). In recent decade, changing the patterns of vehicle use, particularly in urban areas, and increasingly use of private cars instead of urban public transport cause that the vehicles are a significant source of emissions into the atmosphere and Tabriz air pollution. So, industrial factories and vehicles are the main air pollution factors in Tabriz now.
The result of the past study in Tabriz air pollution indicates that the metrological parameters and, especially, wind blowing are the main variables in intensification and alteration of Tabriz air pollution (Sadr Mousavi and Rahimi 2008, 2010, Mojtabazadeh 2005). But, the results in this paper show that the relative humidity and temperature are the main metrological variables in the prediction of NO2 and NO x concentrations and the wind importance for NO2 and NO x modeling approximately is 7%. While in Sadr Mousavi and Rahimi (2008) studies, wind speed and wind direction important in CO concentration modeling are 19.17 and 14.12%, respectively. Also, based on this work results, we cannot conclude like Mojtabazadeh (2005) results that the main metrological variable in Tabriz air pollution is wind blowing.
Fluctuations of hourly NO2 and NO x concentrations in Tabriz atmosphere for the period of October and November 2012 were studied. It is found that ANN is a useful tool for the short-term prediction of NO2 and NO x concentrations. An ANN trained by scaled-conjugate-gradient (trainscg) training algorithm has been implemented to model NO2 and NO x concentrations. The optimum structure of ANN was determined by obtaining a minimum RMSE for the test set. It was found that the structure of ANN with 30 neurons in the hidden layer has the best performance. It is also demonstrated that MLP neural networks had advantages over traditional MLR models.
This work shows that MLP neural networks can accurately model the relationship between local meteorological data and NO2 and NO x concentrations in an urban environment and the performance of ANNs is generally superior in comparison to traditional statistical methods, such as multiple regressions. So, this paper confirms the Gardner and Dorling 2000; Chaloulakou et al. 2003b; Grivas and Chaloulakou 2006; Palani et al. 2008; Elangasinghe et al. 2014 studies in new geographic location. The fluctuation of NO2 and NO x concentrations in this work could be influenced by local meteorological factors such as relative humidity and temperature.
The author declares that he has no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Department of Landscape Engineering, Agricultural Faculty, University of Tabriz, Bolvare 29 Bahman, Tabriz, Iran
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\begin{document}
\title{Distinguishing numbers of finite $4$-valent vertex-transitive graphs}
\begin{abstract} The distinguishing number of a graph $G$ is the smallest $k$ such that $G$ admits a $k$-colouring for which the only colour-preserving automorphism of $G$ is the identity. We determine the distinguishing number of finite $4$-valent vertex-transitive graphs. We show that, apart from one infinite family and finitely many examples, they all have distinguishing number $2$. \end{abstract}
\section{Introduction} All graphs in this paper will be finite. A \emph{distinguishing colouring} of a graph is a colouring which is not preserved by any non-identity automorphism. The \emph{distinguishing number} $D(G)$ of a graph $G$ is the least number of colours needed for a distinguishing colouring of the vertices of $G$. These concepts were first introduced by Albertson and Collins \cite{albertsoncollins} and have since received considerable attention.
It is an easy observation that a graph has distinguishing number $1$ if and only if its automorphism group is trivial. Hence, by \cite{erdosrenyi} almost all graphs have distinguishing number $1$. This obviously is not true for vertex-transitive graphs which always have non-trivial automorphisms. However, it seems that the vast majority of vertex-transitive graphs still have the lowest possible distinguishing number $2$. Hence let us call a vertex-transitive graph \emph{exceptional} if its distinguishing number is not equal to $2$.
One of the most interesting results concerning distinguishing numbers of vertex-transitive graphs is that, apart from the complete and edgeless graphs, there are only finitely many exceptional vertex-primitive graphs~\cite{Cameron,seress}. It is only natural to ask whether something similar holds for vertex-transitive graphs as well. As a first step, H\"uning et al.\ recently determined the exceptional $3$-valent vertex-transitive graphs and their distinguishing numbers.
\begin{theorem}\cite[Corollary~2.2]{HuningEtAl}\label{theo:cubic} The exceptional connected $3$-valent vertex-transitive graphs are \begin{enumerate} \item $K_4$ and $K_{3,3}$, with distinguishing number $4$, and \item $Q_3\cong K_4\times K_2$ and the Petersen graph, with distinguishing number $3$. \end{enumerate} \end{theorem}
This result shows that there are only finitely many connected $3$-valent vertex-transitive exceptional graphs. This is not true for $4$-valent graphs, as shown by the following family of graphs. For $n\geq 3$, the \emph{wreath graph $W_n$} is the lexicographic product $C_n[2K_1]$ of a cycle of length $n$ with an edgeless graph of order $2$, see Figure \ref{fig:wreath}.
\begin{figure}
\caption{The wreath graph $W_{10}$}
\label{fig:wreath}
\end{figure}
It is easy to see that wreath graphs form an infinite family of connected exceptional $4$-valent vertex-transitive graphs, thus providing a negative answer to~\cite[Question~2]{HuningEtAl}. Our main result shows that this is the only such family, that is, apart from the wreath graphs, there are only finitely many connected exceptional $4$-valent vertex-transitive graphs.
\begin{theorem}\label{thm:main} The exceptional connected $4$-valent vertex-transitive graphs are \begin{enumerate} \item $K_5$ and $K_{4,4} \cong W_4$, with distinguishing number $5$, and \item $K_3 \square K_3$, $K_4 \square K_2$, $K_5\times K_2$ and $W_n$ for some $n\geq 3$, $n\neq 4$, with distinguishing number $3$. \end{enumerate} \end{theorem}
In particular, there is no example with distinguishing number $4$. This leads us to the following question.
\begin{question} For $\Delta\geq 5$, is there a connected $\Delta$-valent vertex-transitive graph $G$ with $D(G)=\Delta$? \end{question}
More generally, one could ask about ``gaps'' in the set of distinguishing numbers of connected $\Delta$-valent vertex-transitive graphs, as a subset of $\{2,\ldots,\Delta+1\}$.
Using lexicographic products, it is not hard to construct infinite families of connected exceptional vertex-transitive graphs with fixed valency. \begin{example}
Let $H_1$ be a connected vertex-transitive graph of valency $\Delta_1$ and let $H_2$ be a vertex-transitive graph of valency $\Delta_2$ on $n_2$ vertices. Then the lexicographic product $H_1[H_2]$ is connected, has valency $\Delta_1 n_2 + \Delta_2$ and its distinguishing number is at least $D(H_2)+1$. For an infinite family of examples that are not lexicographic products, note that, for every $n\geq 3$ and every $d\geq 2$, the graph $(C_n[d^2K_1])\square K_2$ has valency $2d^2+1$ and distinguishing number strictly greater than $d$. \end{example}
We hence pose the following (informal) problem. \begin{problem} Is there a ``natural small family'' $\mathcal{F}$ of exceptional graphs such that, for every positive integer $k$, all but finitely many $k$-valent connected exceptional vertex-transitive graphs are contained in $\mathcal{F}$? \end{problem}
\section{Definitions and auxiliary results}
Throughout this paper, all graphs are assumed to be finite and simple. Graph theoretic notions that are not explicitly defined will be taken from \cite{diestelbook}.
An \emph{automorphism} of a graph is an adjacency preserving permutation of its vertices. The group of all automorphisms of a graph $G$ is denoted by $\operatorname{Aut} G$. We say that a graph is \emph{vertex-transitive} if its automorphism group is transitive (that is, for every pair of vertices, there exists an automorphism mapping the first to the second).
An \emph{arc} in a graph $G$ is an ordered pair of adjacent vertices, or equivalently, a walk of length $2$ in $G$. An \emph{$s$-arc} is a non-backtracking walk of length $s$ in $G$, i.e.\ a sequence of vertices $v_0, \dots, v_s$ where $v_i$ is adjacent to $v_{i+1}$ for $0 \leq i \leq s-1$, and $v_{i-1} \neq v_{i+1}$ for $1 \leq i \leq s-1$. The automorphism group $\operatorname{Aut} G$ acts on the set of edges, arcs, and $s$-arcs of $G$ in an obvious way. Call a graph \emph{edge-transitive}, \emph{arc-transitive}, or \emph{$s$-arc-transitive}, if the action of $\operatorname{Aut} G$ on edges, arcs, or $s$-arcs is transitive, respectively. Analogously define \emph{arc-regular} and \emph{$s$-arc-regular}.
The \emph{local group} at a vertex $v$ is the permutation group induced by the stabiliser of $v$ acting on its neighbourhood $N(v)$. Note that, for vertex-transitive graphs, this does not depend on the choice of $v$ (up to permutation equivalence). We say that a graph is locally $\Gamma$, if the local group is isomorphic to $\Gamma$.
A graph $G$ is called \emph{$k$-connected} if it remains connected after removing any set of at most $k-1$ vertices and all incident edges, and \emph{$k$-edge connected} if it remains connected after removing any set of $k$ edges. The following result about the connectivity of vertex-transitive graphs is due to Watkins~\cite{connectivity}.
\begin{lemma} \label{lem:connectivity} A vertex-transitive graph with valency $r$ is at least $\frac{2r}3$-connected. \end{lemma}
If we impose additional properties on the set of vertices to be removed, then we can remove much larger sets without disconnecting the graph. The following lemma follows easily from results in \cite{cyclicconnectivity}.
\begin{lemma} \label{lem:cyclicconnectivity} If $G$ is a $k$-valent vertex-transitive graph with $k \geq 4$ and girth $g \geq 5$, then there is a $g$-cycle $C$ in $G$ such that $G-C$ is $2$-edge connected. \end{lemma}
\begin{proof} By \cite[Theorem 4.5]{cyclicconnectivity}, there is a $g$-cycle $C$ such that the edges with one endpoint in $C$ and the other endpoint in $H:= G - C$ form a minimum (w.r.t.\ cardinality) cut separating two cycles in $G$. Assume that $H$ was not $2$-edge connected and let $e$ be a cut-edge of $H$. Let $A$ and $B$ be the two components of $H - e$. By \cite[Lemma 3.3]{cyclicconnectivity}, the minimum degree of $H$ is $2$, so $A$ and $B$ each contain at most one vertex of degree $1$, and thus there are cycles in both components. Now either the cut separating $A \cup C$ from $B$, or the cut separating $B \cup C$ from $A$ contains strictly fewer edges than the cut separating $C$ from $H$, contradicting the minimality. \end{proof}
We will also need the notion of \emph{distinguishing index} $D'(G)$ of a graph $G$, which is the least number of colours needed for a distinguishing colouring of the edges of $G$. Here are a few results giving upper bounds on $D'(G)$. The first two are Theorems~2.8 and 3.2 in~\cite{Monika}.
\begin{theorem} \label{monikabound} Let $G$ be a connected graph that is neither a symmetric nor a bisymmetric tree. If the maximum degree $\Delta(G)$ of $G$ is at least 3, then $D'(G)\leq \Delta(G) - 1$ unless G is $K_4$ or $K_{3,3}$. \end{theorem}
\begin{theorem} \label{traceable} If $G$ is a graph of order at least $7$ with a Hamiltonian path, then $D'(G)\leq 2$. \end{theorem}
\begin{lemma}\label{lem:linegraph} If $G$ is a connected graph on $5$ or more vertices, then $\operatorname{Aut} L(G)$ is permutationally equivalent to $\operatorname{Aut} G$ with its natural action on $E(G)$. Furthermore, in this case $D'(G) \leq D(G)$, unless $G$ is a tree. \end{lemma} \begin{proof}
The first part is a variant of Whitney's theorem due to Jung \cite{junglinegraph}, the second part follows by applying \cite[Theorem 1.3]{lehnersmith} to a distinguishing colouring with $D(G)$ colours. \end{proof}
In the remainder of this section, we discuss some known results on distinguishing numbers and determine the distinguishing numbers of several graphs that will occur in the proof of Theorem~\ref{thm:main}. The following lemma gives a general bound on distinguishing numbers and was independently proved in \cite{deltabound1} and \cite{deltabound2}.
\begin{lemma} \label{degreebound}
If $G$ is a connected graph with maximum degree $\Delta$, then $D(G) \leq \Delta + 1$, with equality if and only if $G$ is either $C_5$, or $K_n$ or $K_{n,n}$ for some $n \geq 1$. \end{lemma}
For $n\geq 2$, we define a family of graphs $C_{n,K_{3,3}}$ as follows. For $1 \leq i \leq n$, let $H_i$ be disjoint copies of $K_{3,3}$ with bipartition $V(H_i) = X_i \cup Y_i$. Let $C_{n,K_{3,3}}$ be the graph obtained from this collection by adding a matching between $X_i$ and $Y_{i+1}$ for $1 \leq i \leq n-1$, and between $X_n$ and $Y_1$, see Figure \ref{fig:k33cycle}.
\begin{figure}\label{fig:k33cycle}
\end{figure}
\begin{lemma} \label{lem:specialgraphs} The following graphs have distinguishing number at most $2$: \begin{enumerate}[label=(\arabic*),leftmargin=*] \item \label{itm:linegraph-not-exceptional} The line graph of every non-exceptional $3$-valent graph; \item \label{itm:linegraph-pet-q3-wn} The line graphs of the following graphs: the Petersen graph, $Q_3$, $K_3 \square K_3$, $K_5 \times K_2$, and $W_n$ for every $n \geq 3$; \item \label{itm:bipcomphae} The bipartite complement of the Heawood graph; \item \label{itm:4hypercube} The $4$-dimensional hypercube $Q_4$; \item \label{itm:46cage} The $(4,6)$-cage, and \item \label{itm:cycleofk33} The graph $C_{n,K_{3,3}}$ for $n \geq 2$. \end{enumerate} \end{lemma}
\begin{proof} Lemma~\ref{lem:linegraph} immediately implies \ref{itm:linegraph-not-exceptional}.
For \ref{itm:linegraph-pet-q3-wn}, it suffices to observe that all the base graphs have at least $7$ vertices and a Hamiltonian path, and then apply Theorem~\ref{traceable} and Lemma~\ref{lem:linegraph} .
For \ref{itm:bipcomphae} note that the bipartite complement of the Heawood graph has the same automorphism group as the Heawood graph and thus also the same distinguishing number. By Theorem~\ref{theo:cubic}, this distinguishing number is $2$.
\ref{itm:4hypercube} follows from \cite{hypercubedistinguishing}, where distinguishing numbers of all hypercubes were determined.
For the proof of \ref{itm:46cage} first note that the $(4,6)$-cage is bipartite and any two vertices in each of its parts have exactly one neighbour in common. Let $v$ be any vertex, let $v_i$ for $1 \leq i \leq 4$ be the neighbours of $v$, and let $v_{ij}$ $1 \leq i \leq 3$ be the neighbours of $v_i$.
Colour $v$ white, for $1 \leq i \leq 4$ colour $v_i$ black, and colour $v_{ij}$ black if $i < j$ and white otherwise. Finally colour the common neighbours of $v_{22}$ and $v_{32}$, and $v_{22}$ and $v_{33}$ black and all other vertices at distance $3$ from $v$ white, see Figure \ref{fig:46cage}.
\begin{figure}
\caption{Colouring of the $(4,6)$-cage, all vertices at distance $3$ from $v$ not shown in the picture are coloured white.}
\label{fig:46cage}
\end{figure}
Let $\gamma$ be a colour preserving automorphism. Then $\gamma$ must fix $v$, since it is the only white vertex with $4$ black neighbours. Furthermore $\gamma$ must fix all neighbours of $v$ since they have a different number of black neighbours. It must also fix the two black vertices at distance $3$ from $v$ for the same reason. Now it is easy to see that $\gamma$ has to fix all vertices at distance $2$ from $v$ and hence it is the identity.
For \ref{itm:cycleofk33}, consider the colouring shown in Figure~\ref{fig:k33cycle}. Note that the automorphism group has two orbits on edges: those that belong to a copy of $K_{3,3}$, and those that don't, which we call matching edges. There is a unique matching edge both of whose endpoints are coloured white. Every colour preserving automorphism must fix this edge and the matching it is contained in. The colours on the remaining edges in this matching make sure that every colour preserving automorphism must fix this matching pointwise, and thus must fix every matching between two copies of $K_{3,3}$ setwise. It is now easy to see that a colour preserving automorphism fixes all vertices of $C_{6,K_{3,3}}$. Finally note that this colouring can be generalised to a colouring of $C_{n,K_{3,3}}$ for any number $n \geq 2$. \end{proof}
\section{The proof of Theorem~\ref{thm:main}}
In this section, we prove our main result. Determining the distinguishing numbers of the exceptional graphs is straightforward and will be left to the reader.
To show that the remaining graphs have distinguishing number $2$, we distinguish cases according to the local group of $A:=\operatorname{Aut} G$. Define the \emph{type} of an edge $uv$ as the size of the orbit of $u$ under the action of the local group at $v$. By the orbit-stabiliser lemma, this is the index of $A_{uv}$ in $A_v$. Since by vertex transitivity $|A_v|=|A_u|$, this also shows that the type is well-defined, i.e.\ it does not depend on the endpoint of the edge.
Note that since the orbits of the local group at $v$ partition the neighbourhood of $v$ the types of edges incident to $v$ correspond to a partition of $4$. Since $G$ is vertex-transitive, this partition is the same for every vertex. Since the only partitions of $4$ that do not contain a part of size $1$ are $(2,2)$ and $(4)$, we split up the proof of Theorem \ref{thm:main} into the following three cases: \begin{enumerate} \item There are edges of type $1$. This case is treated in Section \ref{sec:type1}. \item All edges have type $2$. This is treated in Section~\ref{sec:type2}. \item[4.] All edges have type $4$, and hence $G$ is arc-transitive. For this case, see Section~\ref{sec:arctransitive}. \end{enumerate}
\subsection{Graphs with edges of type 1} \label{sec:type1}
Let $G_{t \geq 2}$ be the graph obtained from $G$ by removing all edges of type $1$. Note that the components of $G_{t \geq 2}$ form a system of imprimitivity for $A$. We will need the following results.
\begin{lemma} \label{lem:uniquetype1} Assume that every vertex of $G$ is incident to a unique type $1$-edge, $G_{t \geq 2}$ is not connected, and any two components of $G_{t \geq 2}$ are connected by at most one type $1$-edge. Then $G$ has a distinguishing $2$-colouring. \end{lemma} \begin{proof} Let $k$ be the number of vertices in a component of $G_{t \geq 2}$. Consider the graph $H$ obtained from $G$ by contracting every component of $G_{t \geq 2}$ to a single vertex. By our assumptions, $H$ is a $k$-regular graph and it follows from Lemma~\ref{degreebound} that its distinguishing number is at most $k+1$. Let $c'$ be a distinguishing colouring of $H$ with colours $\{0,1,\dots,k\}$. We now colour $G$ in the following way: in every component of $G_{t \geq 2}$, we colour as many vertices black as the colour of the corresponding vertex of $H$ suggests.
Since $c'$ is distinguishing, any automorphism which preserves the resulting colouring has to fix all components of $G_{t \geq 2}$ setwise. As every type 1 edge is uniquely identified by the components it connects, each type 1 edge and hence also every vertex must be fixed by every colour-preserving automorphism. \end{proof}
\begin{lemma} \label{lem:deletetype1} Let $G$ be a connected vertex-transitive graph. Assume that $G_{t \geq 2}$ is not connected, let $H$ be a component of $G_{t \geq 2}$ and let $v \in H$. If $H$ admits a $2$-colouring $c'$ such that the only automorphism of $H$ fixing $v$ and preserving $c'$ is the identity, then $G$ has a distinguishing $2$-colouring. \end{lemma}
\begin{proof} Denote the components of $G_{t \geq 2}$ by $H_1, \dots, H_R$. Note that each $H_i$ is isomorphic to $H$. Let $v_1 \in H_1$. Note that the graph obtained from $G$ by contracting the components $H_1, \dots, H_R$ is connected and vertex-transitive and thus at least $2$-connected. Hence $G - H_1$ is connected, and thus $(G - H_1) + v_1$ is connected as well.
For $i\in\{2,\ldots,R\}$, pick some shortest path from $H_i$ to $v_1$ in $(G - H_1) + v_1$ and let $v_i$ and $e_i$ be the first vertex and edge of this path, respectively. Without loss of generality we may assume that the number of black vertices in $c'$ is not exactly one---otherwise change the colour of $v$ to obtain a colouring with this property. Let $\pi_i \colon H \to H_i$ be an isomorphism which maps $v$ to $v_i$. Such an isomorphism exists because $G$ (and thus also $H$) is vertex-transitive. Now define a colouring $c$ of $G$ by \[ c(x) = \begin{cases} \mathrm{black} & \mathrm{if}\; x = v_1,\\ \mathrm{white} & \mathrm{if}\; x \in H_1 - v_1,\\ c'(\pi_i^{-1}(x)) & \mathrm{if}\; x \in H_i \;\mathrm{for}\; i \neq 1. \end{cases} \] Let $\gamma$ be an automorphism of $G$ preserving $c$. We show that $\gamma$ fixes every vertex and thus $c$ is distinguishing.
First, note that $\gamma$ must fix $v_1$, since $v_1$ is the only black vertex in $H_1$ which in turn is the only component with a unique black vertex.
Next we show that, for $i \neq 1$, every $H_i$ must be fixed pointwise by $\gamma$. Assume not. Let $H_i$ be a component such that the distance from $H_i$ to $v_1$ is minimal, among the components that are not fixed pointwise. The endpoint $u_i$ of $e_i$ which does not lie in $H_i$ is either $v_1$, or it lies in some component $H_j$ which is closer to $v_1$. Hence $u_i$ is fixed by $\gamma$. Since $e_1$ has type $1$, $\gamma$ must also fix $v_i$ and thus induce an automorphism of $H_i$. By hypothesis, this induced automorphism is trivial and thus $\gamma$ fixes $H_i$ pointwise.
Finally, let $x \in H_1 - v_1$. Then $x$ is incident to an edge of type 1 which connects $H_1$ to a different component $H_i$. Since the other endpoint of this edge is fixed by $\gamma$, the same must be true for $x$. \end{proof}
\begin{corollary} \label{cor:deletetype1} Let $G$ be a connected, vertex-transitive graph and let $H$ be a component of $G_{t \geq 2}$. If $H$ has a distinguishing $2$-colouring, then so does $G$. \end{corollary}
\begin{proof} If $H$ is the only component of $G_{t \geq 2}$, then a distinguishing colouring of $H$ is also distinguishing for $G$, otherwise apply Lemma \ref{lem:deletetype1}. \end{proof}
\begin{theorem} \label{thm:type1edges} Let $G$ be a connected $4$-valent vertex-transitive graph containing edges of type $1$. Then $D(G) = 2$, unless $G$ is $K_4 \square K_2$. \end{theorem}
\begin{proof} If all edges are of type $1$, then $A_v=1$ and thus colouring one vertex black and all other vertices white yields a distinguishing colouring.
Next assume that the local group has two orbits of size $1$ and one orbit of size $2$. In this case $G_{t \geq 2}$ is a union of cycles. If there is only one such cycle, then it must have length $6$ or more, and hence $G$ is $2$-distinguishable by Corollary \ref{cor:deletetype1}. If there is more than one, then the conditions of Lemma \ref{lem:deletetype1} are satisfied.
Finally consider the case where the local group has one orbit of size $1$ and one orbit of size $3$. All components of $G_{t \geq 2}$ are isomorphic to some $3$-regular vertex-transitive graph $G'$. Also note that the induced action of $A$ on $G'$ is arc-transitive.
If $G'$ has distinguishing number $2$, then we can apply Corollary \ref{cor:deletetype1} to obtain a distinguishing $2$-colouring of $G$. By Theorem~\ref{theo:cubic}, the only other possibility is that $G'$ is isomorphic to one of $K_4, K_{3,3}, Q_3$ or the Petersen graph.
If $G_{t \geq 2}$ is connected, then $G$ is obtained from $G'$ by adding edges of type $1$. Since $A$ is arc-transitive on $G'$, no edge of type $1$ can connect two neighbours (in $G'$) of the same vertex. Otherwise any two neighbours of this vertex would have to be connected by an edge, contradicting the fact that each vertex of $G$ is adjacent to only one edge of type $1$. Hence an edge of type $1$ can't connect vertices at distance at most $2$ in $G'$. This rules out $K_4, K_{3,3}$ and the Petersen graph as possibilities for $G'$, since they have diameter at most $2$. The only way to add edges with respect to this constraint in the cube $Q_3$ yields $G=K_{4,4}$ which does not contain edges of type $1$.
Thus we can assume that $G_{t \geq 2}$ is not connected. Both the Petersen graph and $Q_3$ have colourings satisfying the condition of Lemma \ref{lem:deletetype1}, see Figure~\ref{fig:cubepetersenvertexstabiliser}. Hence if $G'$ is one of them, then $G$ has a distinguishing $2$-colouring.
\begin{figure}
\caption{Colourings satisfying the condition of Lemma \ref{lem:deletetype1}, $v$ is the square vertex.}
\label{fig:cubepetersenvertexstabiliser}
\end{figure}
We may thus assume that  $G'$ is either $K_4$ or $K_{3,3}$. By Lemma \ref{lem:uniquetype1} we may assume that there is a pair of components of $G_{t \geq 2}$ connected by multiple type $1$ edges. Since $G$ is vertex-transitive and each vertex is incident to a unique edge of type 1, the number of type 1 edges between any pair of adjacent components of $G_{t \geq 2}$ is independent of the choice of the pair. Furthermore, recall that $A$ acts arc-transitively on $G'$. Hence if two adjacent vertices in a component $H$ are both adjacent to the same component $H'$ (via type 1 edges), then all vertices of $H$ are adjacent to $H'$. For $G' = K_4$, this is the only possibility, and the resulting graph is $G=K_4 \square K_2$. For $G' = K_{3,3}$, the above observation tells us that all vertices in the same bipartite class of a component send their type 1 edges to the same component, and hence $G = C_{n,K_{3,3}}$ (see Figure~\ref{fig:k33cycle}) for some $n\geq 2$, which has distinguishing number $2$. \end{proof}
\subsection{Graphs with only edges of type 2} \label{sec:type2} In this section, we assume that all edges of $G$ are of type $2$. This implies that $A$ has two orbits on arcs and therefore at most two orbits on edges. We distinguish two subcases according to whether $G$ is edge-transitive or not.
\subsubsection{Edge-transitive case} \begin{theorem} Let $G$ be a connected $4$-valent graph that is vertex- and edge-transitive but not arc-transitive. Then $D(G) = 2$. \end{theorem} \begin{proof} In this case, $A$ has two orbits on arcs and each arc is in a different orbit than its inverse arc. By removing one of the two orbits, $G$ becomes an arc-transitive directed graph in which every vertex has in- and out-degree $2$. There is some $s \geq 1$ such that $A$ acts regularly on directed $s$-arcs (see for example \cite[Lemma 5.4(v)]{PotVer}).
Let $P= (v_0, \dots, v_s)$ be a directed $s$-arc in $G$. Suppose for a contradiction that there is an arc from $v_s$ to $v_0$. Clearly, in this case $s \geq 2$, as $G$ does not contain any $2$-cycles. There is an automorphism fixing $(v_0,\ldots,v_{s-1})$ pointwise, but not fixing $v_s$. Therefore, the second out-neighbour $v'_s \neq v_s$ of $v_{s-1}$ must also have $v_0$ as an out-neighbour. By directed $2$-arc-transitivity we conclude that for any vertex $v_i$ on $P$, the out-neighbours of $v_i$ are exactly the in-neighbours of $v_{i+2}$, so the digraph is a directed wreath graph and $G$ is arc-transitive, which gives the desired contradiction.
We may thus assume that there is no arc from $v_s$ to $v_0$. Colour the vertices of $P$ black and the remaining vertices white. Note that $v_0$ is the unique black vertex with no black in-neighbour. Hence $v_0$ and thus all of $P$ must be fixed by any colour-preserving automorphism. By $s$-arc-regularity, this implies that the colouring is distinguishing and $G$ has distinguishing number $2$. \end{proof}
\subsubsection{Non-edge-transitive case} If $G$ is not edge-transitive, then there must be $2$ orbits on edges each of which forms a disjoint union of cycles. Denote the two subgraphs induced by the edge orbits by $G_1$ and $G_2$. By transitivity, all cycles in $G_1$ have the same length, the same is true for $G_2$.
We will inductively construct a distinguishing colouring from partial colourings of $G$. Let $\tilde c$ be a partial colouring of $G$ with domain $\tilde V \subseteq V$, that is, $\tilde c$ is a function from $\tilde V$ to some set $C$ of colours. An \emph{extension} of $\tilde c$ is a colouring $c$ of $G$ such that $c$ and $\tilde c$ coincide on $\tilde V$.
\begin{lemma} \label{lem:cyclecolourinduction} Let $G$ be a connected $4$-valent vertex-transitive but not edge-transitive graph and assume that all edges have type $2$. Let $G_1$ and $G_2$ be the subgraphs induced by the two edge orbits. Let $V'$ be a set of vertices of $G$ and let $C$ be a cycle in $G_1$ which is disjoint from $V'$ and contains a neighbour $v$ of some vertex in $V'$. Then there is a cycle $D$ in $G_1$ which is disjoint from $V'$ (possibly $D = C$) and a partial $2$-colouring $\tilde c$ of $G$ with domain $C \cup D$ such that \begin{itemize} \item $C$ and $D$ both contain either $1$ or $2$ black vertices, and \item if $\gamma \in \operatorname{Aut} G$ fixes $V'$ pointwise and fixes any extension of $\tilde c$, then $\gamma$ fixes $V' \cup C \cup D$ pointwise. \end{itemize} \end{lemma}
\begin{proof} Call a vertex $u$ a twin of $v$ if there is an automorphism in the stabiliser of $V'$ that moves $u$ to $v$. Note that $v$ has at most one twin, since there is an edge in $G_2$ connecting $v$ to some $w$ in $V'$, and $w$ has only one other neighbour in $G_2$.
If $v$ has no twin then every automorphism that fixes $V'$ pointwise must fix $v$. Set $D=C$, colour $v$ and one of its neighbours on $C$ black and colour the remaining vertices of $C$ white. Then every automorphism which fixes $V'$ as well as an extension of this colouring must fix $v$ and its black neighbour and thus also fixes $C$.
Next assume that $v$ has a twin that lies on $C$. Again let $D = C$ and colour $v$ and one of its neighbours in $C$ black, but make sure that the black neighbour of $v$ is not a twin of $v$. The same argument as above tells us that this colouring has the desired properties.
Finally assume that $v$ has a twin $u$ that lies outside of $C$. Let $D$ be the cycle in $G_1$ containing $u$ and observe that $D$ is also disjoint from $V'$. Colour $v$ and one of its neighbours in $C$ black, colour one of the neighbours of $u$ in $D$ black, and colour the remaining vertices of $C \cup D$ white. Any automorphism that fixes $V'$ as well as an extension of this colouring must fix $u$ and $v$ and their respective black neighbours, whence we have found the desired colouring. \end{proof}
\begin{theorem} \label{thm:type2edges} Let $G$ be a connected $4$-valent vertex-transitive but not edge-transitive graph and assume that all edges have type $2$. Then $D(G) = 2$. \end{theorem}
\begin{proof}
Let $G_1$ and $G_2$ be the subgraphs induced by the two edge orbits respectively and without loss of generality assume that cycles in $G_1$ are at least as long as cycles in $G_2$.
If $G_1$ consists of a single cycle then this cycle must have length at least $6$. Hence there is a distinguishing $2$-colouring of $G_1$ which must also be distinguishing $2$-colouring of $G$. Hence we may assume that $G_1$ consists of more than one cycle.
If cycles in $G_1$ have length at least $4$, then let $C_1$ be a cycle in $G_1$ and let $v_1$ be a vertex on this cycle. Now inductively apply Lemma~\ref{lem:cyclecolourinduction}. For the first step, let $V' = \{v_1\}$. In each step, pick a cycle $C \neq C_1$ which contains a $G_2$-neighbour of $V'$, colour it according to the lemma and add the vertices of $C \cup D$ to $V'$. The graph obtained from $G$ by contracting every cycle in $G_1$ is connected and vertex-transitive. Hence, by Lemma \ref{lem:connectivity} it is $2$-connected and remains connected after removing $C_1$. In particular, the above colouring procedure assigns colours to all vertices except those in $C_1$. Finally colour $v_1$ and its neighbours on $C_1$ black, and colour the rest of $C_1$ white.
We claim that the resulting colouring is distinguishing. Clearly, every colour-preserving automorphism must fix $v_1$ since it is the only black vertex both of whose neighbours in $G_1$ are black (recall that $C_1$ is the only cycle in $G_1$ containing $3$ black vertices). Using Lemma~\ref{lem:cyclecolourinduction} inductively, we see that every colour-preserving automorphism must fix every cycle pointwise, except possibly $C_1$. Hence the colouring is distinguishing unless the two neighbours of $v_1$ in $G_1$ have the same $G_2$-neighbourhood. In this case, by vertex-transitivity any two vertices at distance $2$ in $G_1$ have the same $G_2$-neighbourhood. If cycles in $G_1$ have length $5$ or more, this implies that vertices have degree at least $3$ in $G_2$ which is a contradiction. If cycles in $G_1$ have length $4$, then so do cycles in $G_2$ and $G$ is a graph obtained by identifying antipodal points of $4$-cycles, i.e., a wreath graph, which contradicts the assumption that $G$ is not edge transitive.
It remains to deal with the case when both $G_1$ and $G_2$ are disjoint unions of $3$-cycles. Let $H$ be the graph with vertices these $3$-cycles, with two such $3$-cycles being adjacent in $H$ if they share a vertex in $G$. It is easy to see that $H$ is regular of valency 3 and $G=L(H)$. By Theorem~\ref{monikabound}, we have $D(G)=D'(H)\leq 2$, unless $H$ is $K_4$ or $K_{3,3}$. Finally, note that $L(K_4)\cong W_3$ while $L(K_{3,3})\cong K_3 \square K_3$.
\end{proof}
\subsection{Arc-transitive graphs} \label{sec:arctransitive}
We first prove a few lemmas to show that we can restrict ourselves to graphs with girth~$4$.
\begin{lemma}\label{lem:girth3} Let $G$ be a connected $4$-valent arc-transitive graph. If $G$ has girth $3$, then $G$ is either $K_5$ or $W_3$, or the line graph of a $3$-valent arc-transitive graph. \end{lemma}
\begin{proof} Follows from ~\cite[Theorem~5.1(1)]{Girth4}). \end{proof}
\begin{lemma} \label{lem:girth-46cage} Let $G$ be a connected graph of minimal valency at least $3$ and girth $g \geq 5$. If $G$ is $s$-arc-transitive, then $s \leq g-3$, unless $G$ is a Moore graph of girth $5$, or the incidence graph of a projective plane. \end{lemma}
\begin{proof} Assume for a contradiction that $G$ is $(g-2)$-arc-transitive. Let $C = (v_0, \dots, v_{g-1})$ be a cycle of length $g$. Note that $(v_0,\dots,v_{g-2})$ is a $(g-2)$-arc and that its endpoints have a common neighbour. By $(g-2)$-arc-transitivity, every $(g-2)$-arc has this property.
Let $v_{g-2}'$ be a neighbour of $v_{g-3}$ outside of $C$. Then $(v_0, \dots , v_{g-3}, v_{g-2}')$ is a $(g-2)$-arc, whence $v_{g-2}'$ and $v_0$ have a common neighbour $v_{g-1}'$. Now the closed walk $(v_0,v_{g-1},v_{g-2},v_{g-3},v_{g-2}',v_{g-1}',v_0)$ shows that $g \leq 6$.
If $g=5$, then the fact that the endpoints of every $3$-arc have a common neighbour implies that $G$ has diameter $2$ and is thus a Moore graph.
If $g=6$, then an analogous argument as above yields that $G$ has diameter $3$. If $G$ was not bipartite, then for $v\in V$ there would be an edge connecting two vertices $x$ and $y$ at the same distance from $v$, and since $g=6$ we have $d(x,v) = d(y,v) = 3$. But then there is a $4$-arc from $v$ to $x$ whence by the above argument $v$ and $x$ have a common neighbour, contradicting $d(x,v)=3$.
Hence $G$ is bipartite and every vertex at distance $2$ from a given vertex $v$ has a unique common neighbour with $v$. It follows that $G$ is the incidence graph of a projective plane. \end{proof}
\begin{lemma} \label{lem:girth5d2} Let $G$ be a connected $4$-valent arc-transitive graph of girth at least $5$, then $D(G) = 2$. \end{lemma}
\begin{proof} Let $g$ be the girth of $G$ and let $s$ be such that $G$ is $s$-arc-transitive but not $(s+1)$-arc-transitive. Note that there is no $4$-valent Moore graph, and that there is a unique $4$-valent graph that is the incidence graph of a projective plane, namely the $(4,6)$-cage. By Lemmas~\ref{lem:specialgraphs} and \ref{lem:girth-46cage} we may thus assume that $s \leq g - 3$.
By Lemma \ref{lem:cyclicconnectivity}, there is a cycle $C = (v_0, \dots, v_{g-1})$ such that $G-C$ is $2$-edge connected. Let $P=(v_{s+1},v_{s},\ldots,v_1)$ and let $X$ be its pointwise stabiliser. Note that $P$ is an $s$-arc and thus $X$ is not transitive on $N(v_1)\setminus \{v_2\}$ (otherwise $G$ would be $(s+1)$-arc-transitive). Let $v_0'$ be a neighbour of $v_1$ that is in a different orbit than $v_0$ under $X$.
Note that the subgraph induced by the vertices $\{v_0',v_0,v_1,\dots, v_{g-2}\}$ is a tree since any additional edge between these vertices would give a cycle of length less than $g$. Denote this tree by $T$ and let $H$ be the subgraph obtained from $G$ by removing all vertices of $T$. Observe that $v_0'$ has degree at most $3$ in $G-C$. If $H$ is not connected, then there is one component of $H$ that is connected to $v_0'$ by a unique edge. Removing that edge from $G-C$ would disconnect it, contradicting the fact that $G-C$ is $2$-edge connected. It follows that $H$ is connected.
Colour all vertices of $T$ black and colour $v_{g-1}$ white. Inductively colour the vertices of $G$ as follows: Let $x$ be a vertex at minimal distance to $v_{g-1}$ in $H$ that has not been coloured yet. If $x$ is fixed by the pointwise stabiliser in $A$ of all previously coloured points, then colour it white. Otherwise colour it black.
We claim that this colouring is distinguishing. First note that if an automorphism fixes two neighbours $u$ and $w$ of a vertex $v$, then it must also fix $v$, since otherwise the image of $v$ would also be a common neighbour of $u$ and $w$ contradicting $g \geq 5$. Note that this implies that all vertices in $H$ with a neighbour outside of $H$ are coloured white. Indeed, at the time such a vertex $x$ is considered for colouring, two of its neighbours are already coloured: its predecessor on a shortest $v_{g-1}$-$x$-path in $H$ and its neighbour outside of $H$. Hence by the previous observation, $x$ is coloured white.
Next we show that $v_1$ is the only black vertex with three black neighbours. By the above observations it is the only such vertex in $T$. Now let $x$ be a black vertex in $H$. Then at most one neighbour of $x$ was coloured before $x$ (otherwise we would have coloured $x$ white). Furthermore, if $P$ is a shortest $v_{g-1}$-$x$-path in $H$, then $P \cup C$ contains an $s$-arc ending in $x$. Hence the pointwise stabiliser of $x$ and all vertices coloured before $x$ does not act transitively on the remaining neighbours of $x$, whence at most one of them will be coloured black.
Let $\gamma$ be a colour preserving automorphism. The above discussion shows that $\gamma$ must fix $v_1$. Furthermore all neighbours of $T$ are white, so $\gamma$ must preserve $T$ setwise. Since there is no automorphism of $G$ that fixes $(v_1,\dots,v_{g-2})$ and moves $v_0$ to $v_0'$, $\gamma$ must fix $T$ pointwise. Finally assume that there is a vertex in $H$ that is not fixed by $\gamma$ and let $x$ be the first such vertex that was coloured in the inductive procedure. Clearly, $x$ is coloured black. Let $y$ be the neighbour of $x$ on a shortest $v_{g-1}$-$x$-path $P$, and let $S$ be an $s$-arc contained in $C \cup P$. Then $S$ is pointwise stabilised by $\gamma$, and since the orbit of $x$ under the pointwise stabiliser of $S$ is not a singleton, it contains exactly one other element $x'$. Every automorphism that fixes $x$ and $S$ also fixes $x'$ and vice versa. Hence at most one of $x$ and $x'$ can be coloured black and thus neither of them can be moved by $\gamma$. \end{proof} \begin{figure}
\caption{The tree $T$ in the proof of Lemma \ref{lem:girth5d2}.}
\label{fig:girth5d2}
\end{figure}
Next we give some results for the case when $G$ has girth exactly $4$. Note that in this case, there must be vertices at distance $2$ from each other with $2$ or more common neighbours. The following two lemmas follow from results in~\cite{Girth4}.
\begin{lemma} \label{lem:girth4manyneighbours} Let $G$ be a connected $4$-valent arc-transitive graph. If there are two vertices at distance $2$ with $3$ or more common neighbours, then $G$ is isomorphic to either $K_5\times K_2$ or $W_n$ for some $n \geq 4$. \end{lemma}
\begin{proof} If there are vertices with $4$ common neighbours, then by \cite[Lemma 4.3]{Girth4}, $G$ is a wreath graph. Otherwise, Subcase II.A of the proof \cite[Theorem 3.3]{Girth4} implies that $G\cong K_5\times K_2$. \end{proof}
\begin{lemma}\label{2ATGirth4} Let $G$ be a connected $4$-valent $2$-arc-transitive graph. If $G$ has girth $4$ but no two vertices at distance $2$ have more than $2$ common neighbours, then $G$ is isomorphic to either $Q_4$, or the bipartite complement of the Heawood graph. \end{lemma}
\begin{proof} By $2$-arc-transitivity, every edge is contained in at least three $4$-cycles. Subcase II.B of the proof of \cite[Theorem 3.3]{Girth4} then implies that $G$ is isomorphic to one of the two graphs as claimed. \end{proof}
The hardest case to deal with is when the graph is locally $D_4$. In this case, we take advantage of the following structural property. Note that $D_4$ in its natural action on $4$ points admits a unique system of imprimitivity with $2$ blocks of size $2$. We say that a $2$-arc $(v_0,v_1,v_2)$ is \emph{straight}, if $\{v_0,v_2\}$ is a block with respect to the local group at $v_1$, and \emph{crooked} otherwise. Note that, of the three $2$-arcs starting with a given arc, one is straight and two are crooked. Further note that fixing a crooked $2$-arc fixes all neighbours of its midpoint. Finally, note that $A$ acts transitively on crooked $2$-arcs of $G$. Call a cycle in $G$ \emph{straight}, if all sub-arcs of length $2$ are straight.
\begin{theorem} \label{thm:arctransitive} Let $G$ be a connected $4$-valent arc-transitive graph, then $D(G) = 2$ unless $G$ is $K_5$, $K_3\square K_3$, $K_5 \times K_2$, or $W_n$ for some $n \geq 3$. \end{theorem}
\begin{proof} By Lemmas \ref{lem:girth3}, \ref{lem:girth5d2}, as well as Lemma \ref{lem:specialgraphs}, we can assume that $G$ has girth $4$. By Lemma \ref{lem:girth4manyneighbours}, we can assume that no two vertices have more than two common neighbours.
Since $G$ is arc-transitive, the local group must be a transitive subgroup of $S_4$. If the local group is $2$-transitive, then $G$ is $2$-arc-transitive and this case is handled with Lemmas \ref{2ATGirth4} and \ref{lem:specialgraphs}.
If the local group is $C_4$ or $V_4$, then $G$ is arc-regular. One can then colour one vertex $v$ and three of its neighbours black, and colour the remaining vertices white. Any colour preserving automorphism must fix the arc from $v$ to its unique white neighbour, thus the colouring is distinguishing.
The last remaining case is that $G$ is locally $D_4$. Suppose first that $G$ contains a $4$-cycle that is not straight. Let $(u,v,w,x)$ be a $4$-cycle of $G$ such that $(u,v,w)$ is a crooked $2$-arc.
We claim that any automorphism fixing $u$ and all of its neighbours must be the identity. By arc-transitivity and connectedness it is enough to show that such an automorphism must fix all neighbours of $v$. Since no pair of vertices has more than two common neighbours, $u$ and $w$ are the only two common neighbours of $v$ and $x$. In particular, if an automorphism fixes $w$ and all its neighbours, then it must also fix $u$. Hence it fixes a crooked $2$-arc with midpoint $v$, and thus it fixes $v$ and all of its neighbours, thus proving our claim.
Let $y$ be the unique vertex such that $(v,w,y)$ is a straight $2$-arc, and let $P=(u,v,w,y)$. Suppose that $y$ is adjacent to $u$. Let $u'$ be the unique vertex other than $u$ such that $(u',v,w)$ is crooked. Note that there is an automorphism fixing $v$ and $w$ (and thus $y$) and mapping $u$ to $u'$, and thus $y$ is adjacent to $u'$, and $v$ and $y$ have at least $3$ common neighbours ($u$, $u'$, and $w$), contradicting an earlier hypothesis. We conclude that $y$ is not adjacent to $u$ and thus the induced subgraph on $P$ is a path of length $3$. Colour $P$ black and colour the remaining vertices white. Since $(u,v,w)$ is crooked, but $(v,w,y)$ is straight, every colour preserving automorphism fixes $P$ pointwise, and thus it fixes $v$ and all its neighbours. Hence, by the above claim, this colouring is distinguishing.
From now on, we can assume that all $4$-cycles of $G$ are straight. Let $\mathcal C$ be the set of all $4$-cycles. Note that every edge is contained in a unique straight $4$-cycle, whence $\mathcal C$ forms a partition of $E(G)$. Furthermore, any two elements of $\mathcal C$ intersect in at most one vertex, since otherwise there would be vertices with $3$ or more common neighbours.
Now consider the auxiliary graph $G'$ with vertex set $\mathcal C$ and an edge between two vertices if the $4$-cycles have a vertex in common. Note that $G'$ is a $4$-valent graph on $|\mathcal C| = \frac{|E(G)|}4 = \frac{|V(G)|}2$ vertices.
Note that $A$ has a natural induced action on $G'$, and this is easily seen to be locally $D_4$. Furthermore any distinguishing colouring of $L(G')$ corresponds to a distinguishing colouring of $G$. By Lemma \ref{lem:linegraph} and the above observations $D(G') \geq D(L(G')) \geq D(G)$. Hence if $D(G') = 2$, then $D(G) = 2$ and we are done. By induction, we may thus assume that $G'$ is one of $K_5$, $K_3\square K_3$, $K_5 \times K_2$, or $W_n$ for some $n \geq 3$. If $G'\neq K_5$, then by Lemma~\ref{lem:specialgraphs} \ref{itm:linegraph-pet-q3-wn}, we have $D(L(G'))=2$ and we are done. Finally note that $G' = K_5$ is not possible, since $A$ induces a transitive, locally $D_4$ action on $G'$, but $K_5$ admits no such action. \end{proof}
\noindent\textsc{Acknowledgements.} We would like to thank the anonymous referees for a number of helpful suggestions.
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arXiv
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Matching-range-constrained real-time loop closure detection with CNNs features
Dongdong Bai1,2,
Chaoqun Wang1,2,
Bo Zhang1,2,
Xiaodong Yi1,2 &
Yuhua Tang1,2
Robotics and Biomimetics volume 3, Article number: 15 (2016) Cite this article
The loop closure detection (LCD) is an essential part of visual simultaneous localization and mapping systems (SLAM). LCD is capable of identifying and compensating the accumulation drift of localization algorithms to produce an consistent map if the loops are checked correctly. Deep convolutional neural networks (CNNs) have outperformed state-of-the-art solutions that use traditional hand-crafted features in many computer vision and pattern recognition applications. After the great success of CNNs, there has been much interest in applying CNNs features to robotic fields such as visual LCD. Some researchers focus on using a pre-trained CNNs model as a method of generating an image representation appropriate for visual loop closure detection in SLAM. However, there are many fundamental differences and challenges involved in character between simple computer vision applications and robotic applications. Firstly, the adjacent images in the dataset of loop closure detection might have more resemblance than the images that form the loop closure. Secondly, real-time performance is one of the most critical demands for robots. In this paper, we focus on making use of the feature generated by CNNs layers to implement LCD in real environment. In order to address the above challenges, we explicitly provide a value to limit the matching range of images to solve the first problem; meanwhile we get better results than state-of-the-art methods and improve the real-time performance using an efficient feature compression method.
A simultaneous localization and mapping systems (SLAM) algorithm aims to map an unknown environment while simultaneously localizing the robot. Loop closure detection (LCD) is the technique of determining whether a mobile robot is back to a previously visited location, and it is critical for building a consistent map of the environment by correcting the localization errors that accumulate over time. Therefore, LCD is considered one of the most essential techniques in SLAM. To develop a LCD algorithm, one class of popular and successful techniques is based on matching the current view of the robot with those corresponding to previously visited locations in the robot map. In this case, LCD becomes an image matching problem, which typically includes two steps, image description and similarity measurement.
The state-of-the-art algorithms take advantage of the bag-of-words (BoW) model [1–3] to describe images. The BoW model clusters the visual feature descriptors in images, builds the dictionary, and then finds the corresponding words of each image. The BoW model is commonly used in visual features like SIFT [4], Surf [5] which have achieved great success in the past years. Despite significant progress in visual LCD, challenges still remain especially in dynamical and large-scale environment. As robots aim at long-term autonomous operations in the long period of time, such as days, weeks, or months, they are faced with environment that can undergo dramatic condition change and viewpoint change over time. Unfortunately, the hand-craft methods can not deal with these situations very well. Recent progress in the computer vision and machine learning community has shown that the features generated by convolutional neural networks (CNNs) outperform other methods in a variety of visual recognition, classification, and detection applications [6]. CNNs have been demonstrated to be versatile and transferable, that is to say, even though they were trained on a very specific target task, they can be successfully used for solving different problems and may outperform traditional hand-craft features [7, 8].
However, when we actually use these features generated by CNNs layers in a practical environment, two challenges appear. Firstly, the adjacent images in the dataset of LCD might have more resemblance than the images that really form the loop closure, so the algorithm tends to identify the adjacent images as loop closure, which is certainly not preferred. Secondly, the feature matching is computationally intensive because the dimension of features generated by CNNs may be very large, and LCD may have to compare the current image to a large amount of pre-captured images in order to decide whether the robot returns to previously visited positions. This can not satisfy strong request for real-time performance in robotic applications.
Against the background, in this paper we provide two solutions to address the above two challenges. Firstly, we explicitly provide matching range of candidate images to prevent images matching with their adjacent images. Meanwhile, we get better performance than state-of-the-art algorithms by adapting the matching range. Secondly, we provide a efficient feature compression method to reduce the dimension of feature generated by CNNs layers, which boosts real-time performance with marginal performance loss.
The rest of this paper is organized as follows. "Related work" section gives a brief introduction to the related work on LCD, CNNs, and datasets used in our subsequent experiments. In "Matching-range-constrained visual loop closure detection" section we present the details of Places CNNs model and how it is used to generate image descriptors. "Real-time large-scale visual loop closure detection" section shows algorithm and experiment results on the compression algorithm to realize real-time LCD. Finally, we conclude the paper in "Conclusions" section with a short discussion and future work.
Loop closure detection
The focus of research in LCD has recently moved from recognizing previously visited place without significant appearance changes [9, 10] to more realistic dynamical environment. Methods that address the LCD problem span from matching sequences of images [11, 12], transforming images to becoming invariant against common scene changes such as shadows [13, 14], learning how environments change over time and predicting these changes in image space [15−17], building up LCD hypotheses over time [18, 19], and building a map of experiences that cover the different appearances of a place over time [20].
Deep convolutional neural network based feature
Previous works mostly relied on the hand-crafted traditional features or operated on the raw pixel levels [21]. These hand-crafted features are designed by experts having a lot of domain-specific knowledge. However, robot may be faced with a variety of complex and changeable environments during the process of localization. So it is very challenging for any people to take all factors affecting the performance of visual LCD into consideration.
Recently, there has been a trend in exploiting features generated by CNNs in computer vision, especially in the field of object recognition and detection [6]. A comprehensive evaluation further demonstrates the advantages of deep CNNs features with respect to shallow hand-crafted feature for image classification [22]. The advantage is that researchers will be free from mastering the knowledge of specific domains and the CNNs architecture can be used for many different domains, especially in visual systems with minor changes. CNN is a well-known architecture proposed by LeCun et al. [23] to recognize hand-written digits. Several research groups have recently shown that CNNs outperform classical approaches for object classification or detection that are based on hand-crafted features [8, 24]. The open-source software Caffe [25] provides pre-trained CNNs architectures for a variety of recognition tasks, which greatly reduces the difficulty in deploying and training CNNs for different tasks.
Hou et al. [7] were the pioneers to consider using features generated by CNNs layers for visual LCD. They use a public pre-trained CNNs model, Places CNNs, trained on the scene-centric dataset Places [26] with over 2.5 million images of 205 scene categories, as an efficient whole-image descriptor generator for LCD. They comprehensively compared the performance of Places CNNs model's all layers by using the euclidean distance as the similarity measurement. Their work demonstrated that the pool5 layer provides the best image descriptors in terms of both detection accuracy and dimension of feature among all Places CNNs descriptors.
Experiments are conducted on two publicly available datasets with known frame correspondences, City Centre and New College built by Cummins et al. [9], firstly used by their loop closure detection algorithm called FAB-MAP. These two datasets are viewpoint change datasets widely used in visual SLAM research and in LCD in particular. The two datasets contain 2474 and 2146 images, respectively. Images are numbered sequentially in the order of collection. The camera was mounted on a pan-tilt and collects images from the left and right of the robot. Image collection was triggered every 1.5 m (on the basis of odometry) by the robot when it is driven through an outdoor urban environment with stable lighting conditions. The vehicle is in motion while the images are collected, so the robot travels same distance between the collection of the right and left images. Obviously, these two datasets exhibit strong viewpoint change. Ground truths in terms of true loop closures are also available. Details of these two datasets are available online.Footnote 1
Matching-range-constrained visual loop closure detection
In this section, we first explain the reason for setting the matching range and choose the matching range by evaluating the precision–recall performance.
Image descriptor
In our experiment, we use the Places CNNs model trained on a scene-centric database [26] and constructed by Caffe [25]. The architecture of this pre-trained CNNs model is briefly summarized in Table 1. This CNNs model is a multi-layer neural network that mainly consists of three types of layers: five convolutional layers, three max-pooling layers and three fully connected layers. Note that a max-pooling layer only follows the first, second, and fifth convolutional layer but not the third and fourth convolutional layers.
Table 1 Architecture of the Places CNNs model and the dimension of the feature of each layer
Image matching
By using pre-trained CNNs model, we can create an image descriptor from each layer in the CNNs. That is, when we provide an image to CNNs, the output of each layer of CNNs is considered as a feature vector u of the image. In addition, we normalize these feature vectors to be unit vector U. We adapt the precision−recall curve as the performance evaluation criteria and euclidean distance for similarity measurement. The precision−recall curve is a standard evaluation method widely used in pattern recognition and in LCD particularly. To produce the precision−recall curve of a given image descriptor from CNNs layers, we compute feature vector U of the current view of robot and then find its nearest neighbor in the robot map that corresponds to previously visited locations according to the euclidean distance. Then we set a threshold on the euclidean distance to determine whether loop closure can be accepted, and we get precision and recall pairs by comparing our results with the ground truth of the dataset after all images in the dataset are considered. Finally, we can produce the precision−recall curve by varying the value of the threshold.
Matching-range setting
The feature provided by pool5 is an efficient whole-image descriptor in the application of visual LCD, the robot collects one image every 1.5 m, so the pool5 may generate more resemblance descriptors for adjacent images than the images forming actual loop closure by the euclidean distance measurement. For example, the distance between the 1399th and 1401th image is 0.2342 and the distance between 349th and 1401th is 0.3302; obviously the 1399th image is more similar to 1401th image than the 349th image. The algorithm without setting the matching range would believe the 1399th and 1401th image form the loop closure (these images are all chosen from City Centre and are shown in Fig. 1). But the distance between the collection of the 1399th image and the 1401th image is only 3 m, due to the robot collected images every 1.5 m, while the ground truths reveal the image really matching the 1401th image is 349th image. In this situation, the algorithm will get a number of errors and greatly degrade its performance.
Three example images of City Centre dataset. a NO.0349, b NO.1399, c NO.1401
In order to deal with this problem, we provide a constraint to limit the matching range of images for the current position to determine the loop closure image. Concretely, if the current image number is N and the number of excluded images is L, the algorithm will only determine whether loop closure appears only from the image number 1 to image number N − L. If the range is set properly, the distance (on the basis of odometry) between the image N − L and image N is long enough so that the difference of descriptors generated by pool5 between the image N and the image N − L is distinguishable for the LCD algorithm. In this case, the above problems can be properly addressed. The precision−recall curve for different matching ranges on the City Centre and New College dataset is shown in Fig. 2 given different L values. Specifically, the L = 0 curve represents the case where the current image is compared with all previous images, and the resulted precision−recall performance is very poor due to high resemblance with the adjacent images, which do not form loop closures.
Visual loop closure detection precision−recalls on City Centre dataset and New College dataset use feature generated by pool5 with different value L. a Precision−recalls on City Center dataset, b precision−recalls on New College dataset
For the sake of evaluating the effects of matching ranges on LCD performance, we implement the algorithm with different L values on the City Centre and New College dataset. Figure 2 shows the resulting precision−recall curves for various experimental settings. The experiments show that performance improves as the L value increases at the beginning and then degrades drastically as shown in L = 1200 in Fig. 2a. The reason of this phenomenon is that, at first, the increase in the number of excluded images can reduce likelihood of occurrence of above problem; however, if the algorithm excludes too many images will result in a small matching range for an image, which may lead to the case that no images in the matching range can really form loop closure with current image revealed by the ground truth, but the LCD algorithm will still identify a "wrong" image for the current image to form loop closure based on euclidean distance.
As shown in Fig. 2a, in the City Center dataset, L = 800 achieves the best precision−recall performance. If the precision is guaranteed to be 100 %, the achievable recall is approximate 10 % better than the benchmark proposed in [7] and 35 % better than the benchmark FAB-MAP proposed in [9]. Figure 2b illustrates the effects of selected L values on the performance in the New College dataset, where the L = 100 is optimal. If the precision is guaranteed to be 100 %, the algorithm achieves comparable recall performance with FAB-MAP [9]. Therefore, the optimal searching range settings represented by L depends on the dataset, which reflects the explored environments and the chosen routing of the robots. Hence, we are investigating the methodology of training L via on-line observations, which is left for our future works.
Real-time large-scale visual loop closure detection
The results have demonstrated that by setting proper matching ranges, the features from layer pool5 is robust against viewpoint change on City Centre and New College dataset. However, computing the euclidean distance between many 9216 dimension pool5 feature vectors may become a computationally intensive operation and a bottleneck of the real-time performance. Therefore, directly using the features extracted by pool5 may not satisfy real-time demand of robotic application, especially in large-scale scenes. Hence, the high-dimensional CNNs features may be compressed as in [27]. In this section, we explore the power of locality-sensitive hash (LSH) functions to address the challenge. The results demonstrate that speed-ups of four to seven times can be achieved with negligible performance degradation in terms of the precision−recall metric.
Feature compression
As the dimension of feature extracted by pool5 is very large, we naturally think of compressing the feature to low-dimensional vectors to accelerate LCD algorithm. We adopt the LSH method proposed by Charikar [28], which uses random hyperplanes to generate an LSH function. This algorithm can preserve the cosine similarity between vectors and shapely reduces the dimension of vectors, which in turns greatly reduces the time of calculating our feature distance similarity matrix.
However, the compression may be achieved at cost of performance degradation on detection accuracy. Therefore, we generate compression features for all images in City Centre and New College dataset with various compression ratios to find the trade-off between real-time and precision−recall performance.
The benchmark we adopted is non-compressed cosine similarity, which calculates the normalized inner product of two original features expressed as
$$\cos (\theta (u,v))=\frac{|u\cdot\,v|}{\sqrt{|u||v|}}, $$
where \(\theta (u,v)\) is the angle between the vectors u and v. \(|u\cdot\,v|\) is the inner product of u and v, and |u| and |v| represent the length of vectors u and v, respectively. Following the random hyperplanes strategy in [29], the LSH proposes to use a collection of random vectors in a k-dimensional vector space. We first generate a spherically symmetric random vector r of unit length from this k-dimensional space. We then define a hash function, \(h_{r}\), as:
$$h_{r}(u)=\left\{ \begin{array}{ll} 1 &\quad r\cdot\,u\ge 0\\ 0 &\quad r\cdot\,u<0\\ \end{array}\right.,$$
For vectors u and v, we have
$$ Pr[h_{r}(u)=h_{r}(v)]=1-\frac{\theta (u,v)}{\pi }, $$
which is proved by Goemans and Williamson [30] and reveals that the probability of a random hyperplane separating two vectors is proportional to the angle between the two vectors (i.e., \(\theta (u,v)\)). From Eq. (3) we may infer that
$$ \cos (\theta (u,v))=\cos ((1-Pr[h_{r}(u)=h_{r}(v)])\pi ), $$
According to Eq. (4), we can get \(Pr[h_{r}(u)=h_{r}(v)]\) by the calculation of hamming distanceFootnote 2 between the vectors u and v. It is noted that the cosine similarity measurement is also equivalent to the euclidean distance measurement for normalized vectors. Hence, the hamming distance measurement is equivalent to euclidean distance measurement used in our previous experiments. In addition, the computation of hamming distance between two bit vectors has many advantages in comparison with euclidean distance, such as time and memory saving. Intuitively, Eq. (4) is stochastic, and based on numerical evaluation, we should generate sufficient random hyperplane to achieve more satisfactory approximation. As we generate a larger number d of random vectors, the hamming distance may estimate the euclidean distance between two vectors more accurately, however, at the cost of increasing the amount of computation. Hence, we should choose a proper d to strike a beneficial trade-off between the approximation accuracy and the computation complexity.
Algorithm implementation
In the previous subsection, we introduced the algorithm for feature compression. The complete algorithm for the LCD is as follows:
Firstly, we produce n feature vectors generated by CNNs layer of pool5 (for City Centre and New College dataset, n is equal to 2474 and 2146, respectively) using Caffe [25].
Secondly, we generate d \((d<k)\) unit random vectors \(\{r_{0},r_{1},\ldots ,r_{d}\}\). Each \(r_{i}\) has k elements (for feature extracted by pool5, k = 9216), and each elements is sampled from a Gaussian function with mean 0 and variance 1. We then put the d vector \(\{r_{0},r_{1},\ldots ,r_{d}\}\) together into a matrix D of dimension \(k\times d\).
Thirdly, we produce the inner product between D and every feature vector v with dimension of k to get vector \(u=D^{T}v\). Then, for every vector u, we use the function \(h_{r}(u)\) [as Eq. (2)] to produce compressed feature \(\bar{u}\) as: \(\bar{u}=\{h_{r1}(u),h_{r2}(u),\ldots ,h_{rd}(u)\}\). Hence, each compressed feature is represented by a bit stream of length d. The time complexity of steps 2 and 3 is O(nkd) time.
For every image in the City Centre and New College dataset, we produce its compression feature vector \(\bar{u}\) and then find its nearest neighbor from the previous images in the dataset by calculating the hamming distance of their compressed features. The time complexity of this step is \(O(n^{2}d)\).
Overall, the time complexity of the total algorithm is \(O(nkd+n^{2}d)\) to calculate the full similarity matrix. However, the time complexity of using the cosine similarity to produce the full similarity matrix would be \(O(n^{2}k)\). For large-scale LCD, n must be very large, so we can get great speed-up factor from the algorithm. Further more, with the increase in n, the algorithm will achieve greater speed-up factor.
In this paper, we implement the algorithm and compare the visual LCD performance achieved with the compressed feature vectors of pool5 of different lengths d = (128, 256, 512, 1024, 2048) bits on the City Centre and New College datasets in Fig. 3. Since the hamming distance over bit vectors is more computationally efficient, the best-matching image among 2474 candidates can be found within 5.67 ms on a standard desktop machine with a 3.60-GHz CPU with four cores and 8GB memory. This corresponds to a speed-up factor of ~4 compared to the algorithm that uses the euclidean distance over the original pool5 features, which required 21.3476 ms for each candidate. Calculating the hashes requires 1.0913ms using a non-optimized Python implementation. Table 2 summarizes the required time for the main algorithmic steps. We can see that the LSH enables real-time visual LCD using CNNs-based features on large-scale places.
Hamming distance over the original feature vectors of 9126 generated by pool5 can be closely approximated by the hamming distance over bit vectors of length 1024 with marginal precision loss. a City Centre dataset, b New College dataset
Table 2 Runtime comparison between original features and compressed features
Our paper presented a thorough investigation on the effects of matching range for CNNs-based LCD. By proper selecting the matching range, we get better performance than the state-of-the-art algorithms on the City Centre and New College dataset. In addition, we provide acceleration solutions for large-scale real-time LCD by using a specialized LSH method to compress high-dimensional CNNs features.
Note that our study is still preliminary at this point since the matching range is dataset-sensitive and chosen manually in this paper. In our future works, we will employ machine learning algorithm to enable the LCD algorithm autonomously finding the optimal matching ranges for various datasets. Also, we may use the point location in equal balls algorithm [31] to accelerate the process of finding most similar images for current viewpoint. Also, we plan to train CNNs specifically for the task of LCD under dynamic environment to get a better descriptor.
http://www.robots.ox.ac.uk/~mobile/IJRR_2008_Dataset/.
Hamming distance is the number of bits which differ between two binary strings. Thus, the hamming distance between two strings A and B is \(\sum _{i}|A_{i}-B_{i}|\).
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DB and CW have jointly carried out matching-range-constrained visual LCD system. DB designed the real-time large-scale visual LCD and drafted the manuscript. BZ developed the methodology and revised the manuscript. XY and YT supervised the study. All authors read and approved the final manuscript.
This work is supported by HPCL Project Fund No. 201502-02, NUDT Key Project Fund No. 434513322412, the NSFC Fund No. 61303068, and the Project Fund No. ZDYYJCYJ20140601
College of Computer, National University of Defense Technology, Changsha, China
Dongdong Bai, Chaoqun Wang, Bo Zhang, Xiaodong Yi & Yuhua Tang
State Key Laboratory of High Performance Computing, National University of Defense Technology, Changsha, China
Dongdong Bai
Chaoqun Wang
Xiaodong Yi
Yuhua Tang
Correspondence to Bo Zhang.
Bai, D., Wang, C., Zhang, B. et al. Matching-range-constrained real-time loop closure detection with CNNs features. Robot. Biomim. 3, 15 (2016). https://doi.org/10.1186/s40638-016-0047-x
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Real-time Computing and Robotics
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Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne).[1] Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem.
Formally, a scheme is a topological space together with commutative rings for all of its open sets, which arises from gluing together spectra (spaces of prime ideals) of commutative rings along their open subsets. In other words, it is a ringed space which is locally a spectrum of a commutative ring.
The relative point of view is that much of algebraic geometry should be developed for a morphism X → Y of schemes (called a scheme X over Y), rather than for an individual scheme. For example, in studying algebraic surfaces, it can be useful to consider families of algebraic surfaces over any scheme Y. In many cases, the family of all varieties of a given type can itself be viewed as a variety or scheme, known as a moduli space.
For some of the detailed definitions in the theory of schemes, see the glossary of scheme theory.
Development
The origins of algebraic geometry mostly lie in the study of polynomial equations over the real numbers. By the 19th century, it became clear (notably in the work of Jean-Victor Poncelet and Bernhard Riemann) that algebraic geometry was simplified by working over the field of complex numbers, which has the advantage of being algebraically closed.[2] Two issues gradually drew attention in the early 20th century, motivated by problems in number theory: how can algebraic geometry be developed over any algebraically closed field, especially in positive characteristic? (The tools of topology and complex analysis used to study complex varieties do not seem to apply here.) And what about algebraic geometry over an arbitrary field?
Hilbert's Nullstellensatz suggests an approach to algebraic geometry over any algebraically closed field k: the maximal ideals in the polynomial ring k[x1,...,xn] are in one-to-one correspondence with the set kn of n-tuples of elements of k, and the prime ideals correspond to the irreducible algebraic sets in kn, known as affine varieties. Motivated by these ideas, Emmy Noether and Wolfgang Krull developed the subject of commutative algebra in the 1920s and 1930s.[3] Their work generalizes algebraic geometry in a purely algebraic direction: instead of studying the prime ideals in a polynomial ring, one can study the prime ideals in any commutative ring. For example, Krull defined the dimension of any commutative ring in terms of prime ideals. At least when the ring is Noetherian, he proved many of the properties one would want from the geometric notion of dimension.
Noether and Krull's commutative algebra can be viewed as an algebraic approach to affine algebraic varieties. However, many arguments in algebraic geometry work better for projective varieties, essentially because projective varieties are compact. From the 1920s to the 1940s, B. L. van der Waerden, André Weil and Oscar Zariski applied commutative algebra as a new foundation for algebraic geometry in the richer setting of projective (or quasi-projective) varieties.[4] In particular, the Zariski topology is a useful topology on a variety over any algebraically closed field, replacing to some extent the classical topology on a complex variety (based on the topology of the complex numbers).
For applications to number theory, van der Waerden and Weil formulated algebraic geometry over any field, not necessarily algebraically closed. Weil was the first to define an abstract variety (not embedded in projective space), by gluing affine varieties along open subsets, on the model of manifolds in topology. He needed this generality for his construction of the Jacobian variety of a curve over any field. (Later, Jacobians were shown to be projective varieties by Weil, Chow and Matsusaka.)
The algebraic geometers of the Italian school had often used the somewhat foggy concept of the generic point of an algebraic variety. What is true for the generic point is true for "most" points of the variety. In Weil's Foundations of Algebraic Geometry (1946), generic points are constructed by taking points in a very large algebraically closed field, called a universal domain.[4] Although this worked as a foundation, it was awkward: there were many different generic points for the same variety. (In the later theory of schemes, each algebraic variety has a single generic point.)
In the 1950s, Claude Chevalley, Masayoshi Nagata and Jean-Pierre Serre, motivated in part by the Weil conjectures relating number theory and algebraic geometry, further extended the objects of algebraic geometry, for example by generalizing the base rings allowed. The word scheme was first used in the 1956 Chevalley Seminar, in which Chevalley was pursuing Zariski's ideas.[5] According to Pierre Cartier, it was André Martineau who suggested to Serre the possibility of using the spectrum of an arbitrary commutative ring as a foundation for algebraic geometry.[6]
Origin of schemes
Grothendieck then gave the decisive definition of a scheme, bringing to a conclusion a generation of experimental suggestions and partial developments.[7] He defined the spectrum X of a commutative ring R as the space of prime ideals of R with a natural topology (known as the Zariski topology), but augmented it with a sheaf of rings: to every open subset U he assigned a commutative ring OX(U). These objects Spec(R) are the affine schemes; a general scheme is then obtained by "gluing together" affine schemes.
Much of algebraic geometry focuses on projective or quasi-projective varieties over a field k; in fact, k is often taken to be the complex numbers. Schemes of that sort are very special compared to arbitrary schemes; compare the examples below. Nonetheless, it is convenient that Grothendieck developed a large body of theory for arbitrary schemes. For example, it is common to construct a moduli space first as a scheme, and only later study whether it is a more concrete object such as a projective variety. Also, applications to number theory rapidly lead to schemes over the integers which are not defined over any field.
Definition
An affine scheme is a locally ringed space isomorphic to the spectrum Spec(R) of a commutative ring R. A scheme is a locally ringed space X admitting a covering by open sets Ui, such that each Ui (as a locally ringed space) is an affine scheme.[8] In particular, X comes with a sheaf OX, which assigns to every open subset U a commutative ring OX(U) called the ring of regular functions on U. One can think of a scheme as being covered by "coordinate charts" which are affine schemes. The definition means exactly that schemes are obtained by gluing together affine schemes using the Zariski topology.
In the early days, this was called a prescheme, and a scheme was defined to be a separated prescheme. The term prescheme has fallen out of use, but can still be found in older books, such as Grothendieck's "Éléments de géométrie algébrique" and Mumford's "Red Book".[9]
A basic example of an affine scheme is affine n-space over a field k, for a natural number n. By definition, An
k
is the spectrum of the polynomial ring k[x1,...,xn]. In the spirit of scheme theory, affine n-space can in fact be defined over any commutative ring R, meaning Spec(R[x1,...,xn]).
The category of schemes
Schemes form a category, with morphisms defined as morphisms of locally ringed spaces. (See also: morphism of schemes.) For a scheme Y, a scheme X over Y (or a Y-scheme) means a morphism X → Y of schemes. A scheme X over a commutative ring R means a morphism X → Spec(R).
An algebraic variety over a field k can be defined as a scheme over k with certain properties. There are different conventions about exactly which schemes should be called varieties. One standard choice is that a variety over k means an integral separated scheme of finite type over k.[10]
A morphism f: X → Y of schemes determines a pullback homomorphism on the rings of regular functions, f*: O(Y) → O(X). In the case of affine schemes, this construction gives a one-to-one correspondence between morphisms Spec(A) → Spec(B) of schemes and ring homomorphisms B → A.[11] In this sense, scheme theory completely subsumes the theory of commutative rings.
Since Z is an initial object in the category of commutative rings, the category of schemes has Spec(Z) as a terminal object.
For a scheme X over a commutative ring R, an R-point of X means a section of the morphism X → Spec(R). One writes X(R) for the set of R-points of X. In examples, this definition reconstructs the old notion of the set of solutions of the defining equations of X with values in R. When R is a field k, X(k) is also called the set of k-rational points of X.
More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, an S-point of X means a morphism Spec(S) → X over R. One writes X(S) for the set of S-points of X. (This generalizes the old observation that given some equations over a field k, one can consider the set of solutions of the equations in any field extension E of k.) For a scheme X over R, the assignment S ↦ X(S) is a functor from commutative R-algebras to sets. It is an important observation that a scheme X over R is determined by this functor of points.[12]
The fiber product of schemes always exists. That is, for any schemes X and Z with morphisms to a scheme Y, the fiber product X×YZ (in the sense of category theory) exists in the category of schemes. If X and Z are schemes over a field k, their fiber product over Spec(k) may be called the product X × Z in the category of k-schemes. For example, the product of affine spaces Am and An over k is affine space Am+n over k.
Since the category of schemes has fiber products and also a terminal object Spec(Z), it has all finite limits.
Examples
Here and below, all the rings considered are commutative:
• Every affine scheme Spec(R) is a scheme.
• A polynomial f over a field k, f ∈ k[x1, ..., xn], determines a closed subscheme f = 0 in affine space An over k, called an affine hypersurface. Formally, it can be defined as
$\operatorname {Spec} k[x_{1},\ldots ,x_{n}]/(f).$
For example, taking k to be the complex numbers, the equation x2 = y2(y+1) defines a singular curve in the affine plane A2
C
, called a nodal cubic curve.
• For any commutative ring R and natural number n, projective space Pn
R
can be constructed as a scheme by gluing n + 1 copies of affine n-space over R along open subsets. This is the fundamental example that motivates going beyond affine schemes. The key advantage of projective space over affine space is that Pn
R
is proper over R; this is an algebro-geometric version of compactness. A related observation is that complex projective space CPn is a compact space in the classical topology (based on the topology of C), whereas Cn is not (for n > 0).
• A homogeneous polynomial f of positive degree in the polynomial ring R[x0, ..., xn] determines a closed subscheme f = 0 in projective space Pn over R, called a projective hypersurface. In terms of the Proj construction, this subscheme can be written as
$\operatorname {Proj} R[x_{0},\ldots ,x_{n}]/(f).$
For example, the closed subscheme x3 + y3 = z3 of P2
Q
is an elliptic curve over the rational numbers.
• The line with two origins (over a field k) is the scheme defined by starting with two copies of the affine line over k, and gluing together the two open subsets A1 − 0 by the identity map. This is a simple example of a non-separated scheme. In particular, it is not affine.[13]
• A simple reason to go beyond affine schemes is that an open subset of an affine scheme need not be affine. For example, let X = An − 0, say over the complex numbers C; then X is not affine for n ≥ 2. (The restriction on n is necessary: the affine line minus the origin is isomorphic to the affine scheme Spec(C[x, x−1]). To show that X is not affine, one computes that every regular function on X extends to a regular function on An, when n ≥ 2. (This is analogous to Hartogs's lemma in complex analysis, though easier to prove.) That is, the inclusion f: X → An induces an isomorphism from O(An) = C[x1, ...., xn] to O(X). If X were affine, it would follow that f was an isomorphism. But f is not surjective and hence not an isomorphism. Therefore, the scheme X is not affine.[14]
• Let k be a field. Then the scheme $ \operatorname {Spec} \left(\prod _{n=1}^{\infty }k\right)$ is an affine scheme whose underlying topological space is the Stone–Čech compactification of the positive integers (with the discrete topology). In fact, the prime ideals of this ring are in one-to-one correspondence with the ultrafilters on the positive integers, with the ideal $ \prod _{m\neq n}k$ corresponding to the principal ultrafilter associated to the positive integer n.[15] This topological space is zero-dimensional, and in particular, each point is an irreducible component. Since affine schemes are quasi-compact, this is an example of a quasi-compact scheme with infinitely many irreducible components. (By contrast, a Noetherian scheme has only finitely many irreducible components.)
Examples of morphisms
It is also fruitful to consider examples of morphisms as examples of schemes since they demonstrate their technical effectiveness for encapsulating many objects of study in algebraic and arithmetic geometry.
Arithmetic surfaces
If we consider a polynomial $f\in \mathbb {Z} [x,y]$ then the affine scheme $X=\operatorname {Spec} (\mathbb {Z} [x,y]/(f))$ has a canonical morphism to $\operatorname {Spec} \mathbb {Z} $ and is called an Arithmetic surface. The fibers $X_{p}=X\times _{\operatorname {Spec} (\mathbb {Z} )}\operatorname {Spec} (\mathbb {F} _{p})$ are then algebraic curves over the finite fields $\mathbb {F} _{p}$. If $f(x,y)=y^{2}-x^{3}+ax^{2}+bx+c$ is an Elliptic curve then the fibers over its discriminant locus generated by $\Delta _{f}$ where
$\Delta _{f}=-4a^{3}c+a^{2}b^{2}+18abc-4b^{3}-27c^{2}$
[16] are all singular schemes. For example, if $p$ is a prime number and
$X=\operatorname {Spec} \left({\frac {\mathbb {Z} [x,y]}{(y^{2}-x^{3}-p)}}\right)$
then its discriminant is $-27p^{2}$. In particular, this curve is singular over the prime numbers $3,p$.
Motivation for schemes
Here are some of the ways in which schemes go beyond older notions of algebraic varieties, and their significance.
• Field extensions. Given some polynomial equations in n variables over a field k, one can study the set X(k) of solutions of the equations in the product set kn. If the field k is algebraically closed (for example the complex numbers), then one can base algebraic geometry on sets such as X(k): define the Zariski topology on X(k), consider polynomial mappings between different sets of this type, and so on. But if k is not algebraically closed, then the set X(k) is not rich enough. Indeed, one can study the solutions X(E) of the given equations in any field extension E of k, but these sets are not determined by X(k) in any reasonable sense. For example, the plane curve X over the real numbers defined by x2 + y2 = −1 has X(R) empty, but X(C) not empty. (In fact, X(C) can be identified with C − 0.) By contrast, a scheme X over a field k has enough information to determine the set X(E) of E-rational points for every extension field E of k. (In particular, the closed subscheme of A2
R
defined by x2 + y2 = −1 is a nonempty topological space.)
• Generic point. The points of the affine line A1
C
, as a scheme, are its complex points (one for each complex number) together with one generic point (whose closure is the whole scheme). The generic point is the image of a natural morphism Spec(C(x)) → A1
C
, where C(x) is the field of rational functions in one variable. To see why it is useful to have an actual "generic point" in the scheme, consider the following example.
• Let X be the plane curve y2 = x(x−1)(x−5) over the complex numbers. This is a closed subscheme of A2
C
. It can be viewed as a ramified double cover of the affine line A1
C
by projecting to the x-coordinate. The fiber of the morphism X → A1 over the generic point of A1 is exactly the generic point of X, yielding the morphism
$\operatorname {Spec} \mathbf {C} (x)\left({\sqrt {x(x-1)(x-5)}}\right)\to \operatorname {Spec} \mathbf {C} (x).$
This in turn is equivalent to the degree-2 extension of fields
$\mathbf {C} (x)\subset \mathbf {C} (x)\left({\sqrt {x(x-1)(x-5)}}\right).$
Thus, having an actual generic point of a variety yields a geometric relation between a degree-2 morphism of algebraic varieties and the corresponding degree-2 extension of function fields. This generalizes to a relation between the fundamental group (which classifies covering spaces in topology) and the Galois group (which classifies certain field extensions). Indeed, Grothendieck's theory of the étale fundamental group treats the fundamental group and the Galois group on the same footing.
• Nilpotent elements. Let X be the closed subscheme of the affine line A1
C
defined by x2 = 0, sometimes called a fat point. The ring of regular functions on X is C[x]/(x2); in particular, the regular function x on X is nilpotent but not zero. To indicate the meaning of this scheme: two regular functions on the affine line have the same restriction to X if and only if they have the same value and first derivative at the origin. Allowing such non-reduced schemes brings the ideas of calculus and infinitesimals into algebraic geometry.
• For a more elaborate example, one can describe all the zero-dimensional closed subschemes of degree 2 in a smooth complex variety Y. Such a subscheme consists of either two distinct complex points of Y, or else a subscheme isomorphic to X = Spec C[x]/(x2) as in the previous paragraph. Subschemes of the latter type are determined by a complex point y of Y together with a line in the tangent space TyY.[17] This again indicates that non-reduced subschemes have geometric meaning, related to derivatives and tangent vectors.
Coherent sheaves
Main article: Coherent sheaf
A central part of scheme theory is the notion of coherent sheaves, generalizing the notion of (algebraic) vector bundles. For a scheme X, one starts by considering the abelian category of OX-modules, which are sheaves of abelian groups on X that form a module over the sheaf of regular functions OX. In particular, a module M over a commutative ring R determines an associated OX-module ~M on X = Spec(R). A quasi-coherent sheaf on a scheme X means an OX-module that is the sheaf associated to a module on each affine open subset of X. Finally, a coherent sheaf (on a Noetherian scheme X, say) is an OX-module that is the sheaf associated to a finitely generated module on each affine open subset of X.
Coherent sheaves include the important class of vector bundles, which are the sheaves that locally come from finitely generated free modules. An example is the tangent bundle of a smooth variety over a field. However, coherent sheaves are richer; for example, a vector bundle on a closed subscheme Y of X can be viewed as a coherent sheaf on X which is zero outside Y (by the direct image construction). In this way, coherent sheaves on a scheme X include information about all closed subschemes of X. Moreover, sheaf cohomology has good properties for coherent (and quasi-coherent) sheaves. The resulting theory of coherent sheaf cohomology is perhaps the main technical tool in algebraic geometry.[18][19]
Generalizations
Considered as its functor of points, a scheme is a functor which is a sheaf of sets for the Zariski topology on the category of commutative rings, and which, locally in the Zariski topology, is an affine scheme. This can be generalized in several ways. One is to use the étale topology. Michael Artin defined an algebraic space as a functor which is a sheaf in the étale topology and which, locally in the étale topology, is an affine scheme. Equivalently, an algebraic space is the quotient of a scheme by an étale equivalence relation. A powerful result, the Artin representability theorem, gives simple conditions for a functor to be represented by an algebraic space.[20]
A further generalization is the idea of a stack. Crudely speaking, algebraic stacks generalize algebraic spaces by having an algebraic group attached to each point, which is viewed as the automorphism group of that point. For example, any action of an algebraic group G on an algebraic variety X determines a quotient stack [X/G], which remembers the stabilizer subgroups for the action of G. More generally, moduli spaces in algebraic geometry are often best viewed as stacks, thereby keeping track of the automorphism groups of the objects being classified.
Grothendieck originally introduced stacks as a tool for the theory of descent. In that formulation, stacks are (informally speaking) sheaves of categories.[21] From this general notion, Artin defined the narrower class of algebraic stacks (or "Artin stacks"), which can be considered geometric objects. These include Deligne–Mumford stacks (similar to orbifolds in topology), for which the stabilizer groups are finite, and algebraic spaces, for which the stabilizer groups are trivial. The Keel–Mori theorem says that an algebraic stack with finite stabilizer groups has a coarse moduli space which is an algebraic space.
Another type of generalization is to enrich the structure sheaf, bringing algebraic geometry closer to homotopy theory. In this setting, known as derived algebraic geometry or "spectral algebraic geometry", the structure sheaf is replaced by a homotopical analog of a sheaf of commutative rings (for example, a sheaf of E-infinity ring spectra). These sheaves admit algebraic operations which are associative and commutative only up to an equivalence relation. Taking the quotient by this equivalence relation yields the structure sheaf of an ordinary scheme. Not taking the quotient, however, leads to a theory which can remember higher information, in the same way that derived functors in homological algebra yield higher information about operations such as tensor product and the Hom functor on modules.
See also
• Flat morphism, Smooth morphism, Proper morphism, Finite morphism, Étale morphism
• Stable curve
• Birational geometry
• Étale cohomology, Chow group, Hodge theory
• Group scheme, Abelian variety, Linear algebraic group, Reductive group
• Moduli of algebraic curves
• Gluing schemes
Citations
1. Introduction of the first edition of "Éléments de géométrie algébrique".
2. Dieudonné 1985, Chapters IV and V.
3. Dieudonné 1985, sections VII.2 and VII.5.
4. Dieudonné 1985, section VII.4.
5. Chevalley, C. (1955–1956), Les schémas, Séminaire Henri Cartan, vol. 8
6. Cartier 2001, note 29.
7. Dieudonné 1985, sections VII.4, VIII.2, VIII.3.
8. Hartshorne 1997, section II.2.
9. Mumford 1999, Chapter II.
10. Stacks Project, Tag 020D.
11. Hartshorne 1997, Proposition II.2.3.
12. Eisenbud & Harris 1998, Proposition VI-2.
13. Hartshorne 1997, Example II.4.0.1.
14. Hartshorne 1997, Exercises I.3.6 and III.4.3.
15. Arapura 2011, section 1.
16. "Elliptic curves" (PDF). p. 20.
17. Eisenbud & Harris 1998, Example II-10.
18. Dieudonné 1985, sections VIII.2 and VIII.3.
19. Hartshorne 1997, Chapter III.
20. Stacks Project, Tag 07Y1.
21. Vistoli 2005, Definition 4.6.
References
• Arapura, Donu (2011), "Frobenius amplitude, ultraproducts, and vanishing on singular spaces", Illinois Journal of Mathematics, 55 (4): 1367–1384, doi:10.1215/ijm/1373636688, MR 3082873
• Cartier, Pierre (2001), "A mad day's work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry", Bulletin of the American Mathematical Society, 38 (4): 389–408, doi:10.1090/S0273-0979-01-00913-2, MR 1848254
• Dieudonné, Jean (1985), History of Algebraic Geometry, Wadsworth, ISBN 978-0-534-03723-9, MR 0780183
• Eisenbud, David; Harris, Joe (1998). The Geometry of Schemes. Springer-Verlag. ISBN 978-0-387-98637-1. MR 1730819.
• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• Hartshorne, Robin (1997) [1977]. Algebraic Geometry. Springer-Verlag. ISBN 978-0-387-90244-9. MR 0463157.
• Igor R. Shafarevich (2013). Basic Algebraic Geometry 2: Schemes and Complex Manifolds. Springer-Verlag. ISBN 978-3642380099. MR 0456457.
• Qing Liu (2002). Algebraic Geometry and Arithmetic Curves. Oxford University Press. ISBN 978-0-19-850284-5. MR 1917232.
• Mumford, David (1999). The Red Book of Varieties and Schemes: Includes the Michigan Lectures (1974) on Curves and Their Jacobians. Lecture Notes in Mathematics. Vol. 1358 (2nd ed.). Springer-Verlag. doi:10.1007/b62130. ISBN 978-3-540-63293-1. MR 1748380.
• Vistoli, Angelo (2005), "Grothendieck topologies, fibered categories and descent theory", Fundamental Algebraic Geometry, Providence, RI: American Mathematical Society, pp. 1–104, arXiv:math/0412512, Bibcode:2004math.....12512V, MR 2223406
External links
• David Mumford, Can one explain schemes to biologists?
• The Stacks Project Authors, The Stacks Project
• https://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ - the comment section contains some interesting discussion on scheme theory (including the posts from Terence Tao).
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• France
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\begin{document}
\title{On Lie Groups and The Theory of Complex Variables} \date{\today}
\begin{thanks} {I am much indebted to Godofredo Iommi EcheverrÃa for a careful reading of a first version of this note.} \end{thanks}
\author[G.~Iommi Amun\'ategui]{Godofredo Iommi Amun\'ategui} \address{Instituto de F\'isica, Pontificia Universidad Cat\'olica de Valpara\'iso (P.UCV)} \email{\href{[email protected]}{[email protected]}}
\begin{abstract} In this note we envisage the relation existing between the Lie Groups and the Theory of Complex Variables. In particular, it is shown that the dimensions of the irreducibles representations of $SU(N)$ may be written in terms of the Eisenstein integers and an identity is built up between the imaginary parts of the dimensions of the irreducible representations of the Lie Groups $SU(3)$ and $Sp(4)$. \end{abstract}
\maketitle
\section{Introduction} In a work dealing with the classes of binary quadratic forms with complex integral coefficients, G. Eisenstein \cite{d} introduced the numbers $a+b \omega$, where $a$ and $b$ are real integers and $\omega$ is an imaginary cube roof of unity.
D. Speiser \cite[Fig.7 and Fig. 22]{s} noticed a curious connection between Lie Groups of rank $2$ and the theory of complex variables. In particular, he pointed out that in the lattice formed by the dimension function of $SU(3)$, the values are arithmetical means of their closed neighbors, fact which is reminiscent of harmonic functions. Consequently, he proposed the following identity for the dimension $\text{Dim}(p_1, p_2)$ of an irreducible representation $(p_1, p_2)$ of $SU(3)$\footnote{To obtain equation \eqref{spe} use must be made of $z= \left(p_1 + \frac{p_2}{2} \right) +i p_2 \frac{\sqrt{3}}{2}.$}: \begin{equation} \label{spe} \text{Dim}(p_1, p_2)= \frac{1}{2} p_1 p_2 (p_1 + p_2) = \frac{1}{3 \sqrt{3}} \text{Im}z^3. \end{equation}
In Section \ref{sec:2} we shall clarify the meaning of this equation and subsequently we shall write the irreducible representation of $SU(N)$ in terms of the Eisenstein numbers. In Section \ref{sec:3} an identity is built up between the imaginary parts of the dimensions of the irreducible representations of the Lie Groups $SU(3)$ and $Sp(4)$.
\section{Eisenstein integers and the dimensions of the irreducible representations of the unitary groups} \label{sec:2}
The Eisenstein integers numbers are $a+b \omega$, where $\omega= \frac{-1 + i \sqrt{3}}{2}$ is one of the cubic roots of unity, and the others are $1$ and $ \omega^2=\frac{-1 - i \sqrt{3}}{2}$. These numbers form a triangular lattice \cite[Fig 8.10 (a), p. 221)]{cg} and \cite[pp. 2--5]{c}.
\begin{lemma} The dimensions of the irreducible representations $(p_1, p_2)$ of $SU(3)$ may be written through Eisenstein numbers as: \begin{equation} \label{2.1} \text{Dim}(p_1,p_2)= \frac{\text{Im}(a+ b \omega )^3}{3 \sqrt{3}}. \end{equation} \end{lemma}
The imaginary part of $(a+b\omega)^3$ is: \begin{equation} \label{2.3} \text{Im}(a+ b \omega )^3=3ab(a\omega + b \omega^2) = i 3 \sqrt{3} \frac{(a-b)ab}{2}. \end{equation}
The number $N(a,b)= \frac{1}{3 \sqrt{3}} \text{Im}(a+b \omega)^3 =\frac{(a-b)ab}{2}$ may be assigned to each Eisenstein lattice point. If we set $a=p_1 +p_2$ and $b=p_2$, we recover $D(p_1, p_2)$.
Long ago H. Weyl \cite{w} obtained the branching law for the groups of linear transformations. Hereafter we shall restrict his result to the case of unitary groups. For briefness we omit the details and we state the reduction $SU(N)\rightarrow SU(N-1)$: the $SU(N)$ irreducible representation $(p_1, p_2, \dots, p_{N-1})$ reduces into $SU(N-1)$ irreducible representations according to the formula
\begin{align} \left[ \left( p_1, p_2, \dots, p_{N-1}\right) \right] = \nonumber \sum_{k_{1,1}=1}^{p_1} \sum_{k_{1,2}=1}^{p_2} \cdots &\\ \sum_{k_{1,N-1}=1}^{p_{N-1}} \left(p_{1}-k_{1,1}+ k_{1,2}, p_2 -k_{1,2}+ k_{1,3}, \dots, p_{N-2} - k_{1, N-2} + k_{1, N-1} \right).\label{2.4} \end{align}
In order to write the dimension of $(p_1, p_2, \dots, p_{N-1})$ in terms of Eisenstein numbers the reduction chain $SU(N) \rightarrow SU(N-1) \rightarrow \dots \rightarrow SU(3)$ must be considered, i.e. the $SU(3)$ content of $(p_1, p_2, \dots, p_{N-1})$ must be displayed. As an example, let us examine the case of an irreducible representation $SU(5)$. Taking into account the chain $SU(5) \rightarrow SU(4) \rightarrow SU(3)$, the dimension of $(p_1, p_2, \dots, p_{N-1})$ may be expressed as follows:
\begin{eqnarray*} \label{2.5} \text{Dim}(p_1, p_2, p_3, p_4)= \sum_{k_{1,1}=1}^{p_1} \sum_{k_{1,2}=1}^{p_2} \sum_{k_{1,3}=1}^{p_3} \sum_{k_{1,4}=1}^{p_4} \sum_{k_{2,1}=1}^{p_{1}-k_{1,1}+k_{1,2}} \sum_{k_{2,2}=1}^{p_{2} -k_{1,2}+ k_{1,3}} \sum_{k_{2,3}=1}^{p_{3}-k_{1,3}+k_{1,4}} &\\ \frac{1}{3 \sqrt{3}} I_m\Big((a-b-k_{1,1}+ k_{1,2}-k_{2,1}+ k_{2,2}) + &\\ \frac{1}{2} (b-k_{1,2}+k_{1,3}-k_{2,2}+k_{2,3}) +i \frac{\sqrt{3}}{2} (b-k_{1,2}+k_{1,3}-k_{2,2}+ k_{2,3}) \Big)^3 \end{eqnarray*}
Although such identities acquire an involved aspect their underlying structure is transparent. The general result is:
\begin{lemma} \label{lem:2} The dimension of the irreducible representation $(p_1, p_2, \dots, p_{N-1})$ of $SU(N)$ may be written in terms if the Eisenstein numbers as: \begin{eqnarray*} \label{2.6} \text{Dim}(p_1, p_2, p_3, p_4)= &\\ \sum_{k_{1,1}=1}^{p_1} \cdots \sum_{k_{1, N-1}=1}^{p_{N-1}} \sum_{k_{2,1}=1}^{p_1-k_{1,1}+k_{1,2}} \cdots &\\ \sum_{k_{2,N-2}=1}^{p_{N-2}-k_{1, N-2}+k_{1,N-1}} \sum_{k_{3,1}=1}^{p_{1} -k_{1,1}+ k_{1,2}-k_{2,1}+k_{2,2}} \cdots \sum_{k_{3, N-3}=1}^{p_{1}-k_{1,N-3}+k_{1,N-2}-k_{2, N-3}+ k_{2, N-1}} \cdots &\\ \frac{1}{3 \sqrt{3}} I_m\Big((a-b-k_{1,1}+ k_{1,2}-k_{2,1}+ k_{2,2} -k_{3,1}+ k_{3,2}- \dots) + &\\ \frac{1}{2} (b-k_{1,2}+k_{1,3}-k_{2,2}+k_{2,3}-k_{3,2}+k_{3,3}-\dots ) &\\ +i \frac{\sqrt{3}}{2} (b-k_{1,2}+k_{1,3}-k_{2,2}+ k_{2,3}- k_{3,2}+k_{3,3}- \dots) \Big)^3. \end{eqnarray*} \end{lemma}
Remark that, for $N>3$, this formula consists of $\frac{1}{2} (N+2)(N-3)$ summations.
\section{Concerning an identity on the complex plane}\label{sec:3}
In this section our purpose is to build up an identity between the imaginary parts of the dimensions of the irreducible representations of the Unitary Group $SU(3)$ and the dimensions of the irreducible representations of the Symplectic Group $Sp(4)$. These Groups are subgroups of the Unitary Group $SU(4)$. The Group $SU(4)$ has rank $3$, hence its lattice is $3$-dimensional and to each lattice point $(p_1, p_2 , p_3 )$ corresponds an irreducible representation whose dimension is given by : \begin{equation} \label{3} \text{Dim} ( p_1,p_2 ,p_3 )= \frac{1}{2! 3!} p_1 p_2 p_3 (p_1 + p_2)(p_2 + p_3)(p_1 +p_2 + p_3 ) \end{equation} where $p_1, p_2$ and $p_3$ are positive integers.
In order to construct such an identity ,we shall follow a procedure whose main steps are:
\begin{enumerate} \item[(a)] \label{a}The branching rule for the reduction $SU(4) \rightarrow SU(3)$.
\item[(b)] The branching rule for the reduction $SU(4) \rightarrow Sp(4) $ and the geometrical transformation which allows us to express the $Sp (4)$ lattice point $( q_1, q_2)$ by means of the complex variable $z'$.\label{b} \end{enumerate} The final move is nothing but an adequate combination of $(a)$ and $(b)$.
For $N=4$, Weyl's branching rule \ref{2.4} may be written as: \begin{equation} \label{3.2} [(p_1, p_2, p_3)] = \sum_{a_1=1}^{p_1}\sum_{a_2=1}^{p_2}\sum_{a_3=1}^{p_3} (p_1 -a_1 +a_2, p_2-a_2+a_3), \end{equation} where $( p_1 , p_2 , p_3 )$ corresponds to an irreducible representation of $SU(4)$ in the Group lattice. The square brackets indicate the $SU(3)$ content of $(p_1 ,p_2 ,p_3 )$.
At this point it seems in order to recall that the Symplectic Group $Sp (4)$ consists of the subset of unitary matrices in $4$ dimensions $(4=2l)$ which leaves invariant a skew symmetric bilinear form: \begin{equation*} \sum_{i=1}^2 (x_i y_{i+2}-x_{i+2}y_i). \end{equation*}
The existence of a non-degenerate skew symmetric form requires an even number of dimensions. Besides, let us remark that the reduction $SU(4) \rightarrow Sp(4)$ has been solved using a geometrical method \cite{i}. This reduction admits three cases which take into account the $Sp(4)$ lattice diagram symmetry:
\begin{equation} \label{3.3} p_1 < p_3, \quad [(p_1, p_2, p_3)] = \sum_{\rho=0}^{p_1-1}\sum_{\lambda=0}^{p_2-1} (p_1 + p_3-1-2p, 1+\rho+ \lambda) \end{equation}
\begin{equation} \label{3.4} p_1 = p_3, \quad [(p_1, p_2, p_3)] = \sum_{\rho=0}^{p_1-1}\sum_{\lambda=0}^{p_2-1} (2 p_1 -1-2 \rho, 1 + \rho + \lambda) \end{equation} (or the same expression with $p_3$ instead of $p_1$). \begin{equation} \label{3.5} p_1 > p_3, \quad [(p_1, p_2, p_3)] = \sum_{\rho=0}^{p_3-1}\sum_{\lambda=0}^{p_2-1} (p_1 + p_3-1-\rho, 1 + \lambda + \rho). \end{equation}
A geometric consideration of the $2-$dimensional lattice of $Sp(4)$ is crucial to find a transformation which gives room to an identity between the irreducible representations of $Sp(4)$ and the imaginary part of a complex expression. To achieve our goal let us envisage the $Sp (4)$ lattice point $(q_1 ,q_2 )$ in such a manner that the $q_1-$axis coincides with the $x-$axis of the complex plane. We get:
\begin{equation} \label{3.6}
z'= -\frac{q_1}{2} +i \left(\frac{q_1}{2} + q_2 \right). \end{equation} From this transformation and the dimension formula for the irreducible representations of $Sp (4)$ \begin{equation} \label{3.7} \text{Dim} (q_1 ,q_2 ) = \frac{1}{6} q_1 q_2 (q_1 + q_2)(q_1 + 2 q_2). \end{equation} we deduce that \begin{equation} \label{3.8} \text{Dim} ( q_1 , q_2) = \text{Im} \frac{z'^4}{6}. \end{equation}
Let $ [(p_1, p_2, p_3)]_{SU(3)}$ and $ [(p_1, p_2, p_3)]_{Sp(4)}$ denote respectively \eqref{3.2}, \eqref{3.3},\eqref{3.4} and \eqref{3.5}. We may state the identity on the complex plane in the symbolic form:
\begin{lemma}\label{I} \begin{equation*} \text{Im} \frac{z^3}{3 \sqrt{3}} [(p_1, p_2, p_3)]_{SU(3)} = \text{Im} \frac{z'^4}{6} [(p_1, p_2, p_3)]_{Sp(4)}. \end{equation*} where to each resulting term of the decompositions must be applied the corresponding coordinate transformation. \end{lemma}
To illustrate Lemma \ref{I}, let us work out the reduction corresponding to the irreducible representation $(5,3,2)$ of dimension $1000$ of $SU(4)$ : \begin{align*} \text{Im} \frac{z^3}{3 \sqrt{3}} \left( \sum_{a_1=1}^{5}\sum_{a_2=1}^{3}\sum_{a_3=1}^{2} (5 -a_1 +a_2, 3-a_2+a_3) \right)^3 = &\\ \text{Im} \frac{z'^4}{6} \left( \sum_{\rho=0}^{1}\sum_{\lambda=0}^{2} (6-2\rho, 1 + \lambda + \rho) \right)^4. \end{align*}
Through algebraic manipulations, finally, we obtain:
\begin{align*} \label{9}
\frac{1}{3 \sqrt{3}} \text{Im} \Big( \Big[\left(\frac{5}{2} + i3 \frac{\sqrt{3}}{2} \right)^3
+ 2 \left(\frac{7}{2} + i3 \frac{\sqrt{3}}{2} \right)^3 + 2\left(\frac{9}{2} + i3 \frac{\sqrt{3}}{2} \right)^3 &\\ +2 \left(\frac{11}{2} + i3 \frac{\sqrt{3}}{2} \right)^3 + 2\left(\frac{13}{2} + i3 \frac{\sqrt{3}}{2} \right)^3 + \left(\frac{15}{2} + i3 \frac{\sqrt{3}}{2} \right)^3 \Big] &\\ +\Big[\left(3 + i 2 \sqrt{3} \right)^3 + \left(4 + i 2 \sqrt{3} \right)^3 + \left(5 + i 2 \sqrt{3} \right)^3 + \left(6 + i 2 \sqrt{3} \right)^3 + \left(7 + i 2 \sqrt{3} \right)^3 \Big]+ &\\ \Big[ \left(3 + i \sqrt{3} \right)^3 + 2\left(4 + i \sqrt{3} \right)^3 + 2\left(5 + i \sqrt{3} \right)^3 + 2\left(6 + i\sqrt{3} \right)^3+ 2\left(7 + i \sqrt{3} \right)^3 + &\\ \left(8 + i \sqrt{3} \right)^3\Big]+ \Big[\left(\frac{7}{2} + i \frac{\sqrt{3}}{2} \right)^3 +\left(\frac{9}{2} + i \frac{\sqrt{3}}{2} \right)^3 +\left(\frac{11}{2} + i \frac{\sqrt{3}}{2} \right)^3+\left(\frac{13}{2} + i \frac{\sqrt{3}}{2} \right)^3 &\\ +\left(\frac{15}{2} + i \frac{\sqrt{3}}{2} \right)^3\Big] \Big) =&\\ \frac{\text{Im}}{6} \Big(\Big[(4i-3)^4 + (5i-3)^4 +(6i-3)^4 +(4i-2)^4 +(5i-2)^4+(6i-2)^4 \Big] \Big). \end{align*}
Dimensional verification: \begin{align*} \text{Dim} [(5,3,2)]_{SU(3)}=[6+30+54+84+120+81] + [10+24+42+64+90] +&\\ [8+30+48+70+96+63]+ [6+10+15+21+28]. \end{align*}
\begin{align*} \text{Dim} [(5,3,2)]_{Sp(4)}=[56+160+324+64+140+256]. \end{align*}
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arXiv
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\begin{definition}[Definition:Irreducible Representation]
Let $G$ be a group.
Let $V$ be a $G$-representation.
Then $V$ is '''irreducible''' {{iff}} the only subrepresentations are $V$ and $0$.
Category:Definitions/Representation Theory
\end{definition}
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ProofWiki
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\begin{document}
\title{Supplementary materials:\\ Programmable interference between two microwave quantum memories } \author{Yvonne Y. Gao} \email[corresponding author:]{[email protected]} \affiliation{Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA} \affiliation{Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA} \author{B.J. Lester} \affiliation{Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA} \affiliation{Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA} \author{Yaxing Zhang} \affiliation{Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA} \affiliation{Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA} \author{C. Wang} \affiliation{University of Massachusetts, Amherst, MA 01003-9337 USA} \author{S. Rosenblum} \affiliation{Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA} \affiliation{Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA} \author{\\L. Frunzio} \affiliation{Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA} \affiliation{Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA}
\author{Liang Jiang} \affiliation{Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA} \affiliation{Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA} \author{S.M. Girvin} \affiliation{Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA} \affiliation{Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA} \author{R.J. Schoelkopf}
\affiliation{Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA} \affiliation{Yale Quantum Institute, Yale University, New Haven, Connecticut 06511, USA} \date{\today}
\pacs{} \maketitle
\section{Device architecture and system parameters} \begin{figure}
\caption{\textbf{A cartoon showing the top view of the 3D double-cavity cQED system}. The center transmon ancilla provides nonlinear coupling between Alice and Bob. The package can accommodate two additional transmon ancillae. In this experiment, only one (qA) is included in the device. The RF drives are coupled to the system through the drive port of qC.}
\label{sfig:cartoon}
\end{figure}
Our cQED system includes two three dimensional (3D) superconducting microwave cavities, Alice and Bob, two quasi-planar readout resonators, RA and RC, and two transmon-type superconducting qubits, qA and qC. All components are housed in a single block of high-purity (4N) aluminum in the structure shown in Fig.~\ref{sfig:cartoon}. Alice and Bob act as quantum memories that are capable of coherently storing quantum information in bosonic states. They are formed by 3D coaxial transmission lines that are short-circuited on one end and open-circuited on the other by virtue of a narrow circular waveguide \cite{reagor2016}. The resonance frequency of the cavities' fundamental modes are determined by the lengths of the transmission lines (4.8 and 5.6 mm respectively for Alice and Bob). An elliptical tunnel is machined between Alice and Bob, allowing the insertion of a chip containing the transmon ancilla and readout resonator into the cavities. Two additional tunnels are machined on either side of Alice and Bob to allow the incorporation of additional transmon and readout channels. Each mode is coupled to the fridge input/output lines via standard SMA couplers.
The whole package is chemically etched after machining to improve the surface quality of the cavities. The superconducting transmons are fabricated on sapphire substrates using electron-beam lithography and the standard shadow-mask evaporation of Al/AlOx/Al Josephson junction. During the same fabrication process, a separate strip of the tri-layer film is also deposited. Together with the wall of the tunnel, it forms a hybrid planar-3D $\lambda$/2 stripline resonator that is capacitively coupled to the transmon. This design combines the advantages of both precise, lithographic control of critical dimensions and the low surface/radiation loss of 3D structures \cite{axline2016}. The chip containing these structures is inserted into the tunnel such that the transmon antenna(a) protrudes into the cavities to the desired capacitive coupling to Alice and Bob.
This system is an extension of the device used in Ref.~\cite{wang2016}. The additional tunnels allow individual ancillae, accompanied by their respective readout resonator, to couple to each cavity in order to provide fast, independent cavity manipulations and tomography. Only one additional ancilla (qA) is required for this particular experiment. It couples to only Alice and is used perform differential phase shifts (DPS) in the Mach-Zehnder interometer described in the main text. The parameters of all relevant components are summarized in Table~\ref{table:params_multi_ancillae}.
\begin{table*}[!htb] \centering
\begin{tabular}{c c c c c c } \hline\hline\\[-2ex]
&\,\,\, Frequency\,\,\,\, & Nonlinear &interactions: \\
& \,\,\, $\omega/2\pi$\,\,\,\, & Alice & Bob & qA & qC \\ \hline\\[-2ex] Alice\, &5.554\,GHz\, & 4\,kHz\, & $\lesssim$1\,kHz\, & 1.01\,MHz\,\,\, &0.62\,MHz\, \\ Bob\, & 6.543\,GHz\, & $\lesssim$1\,kHz\, & 2\,kHz\, & ($\sim$ 0)\,\,\, &0.26\,MHz\, \\ qA\, & 4.989\,GHz\, & 1.01\,MHz\, & ($\sim$ 0)\, & 185\,MHz\,\,\, & N.A\, \\ qC\, & 5.901\,GHz\, & 0.62\,MHz\, &0.26\,MHz\, & N.A\,\, & 74\,MHz\, \\ RA\, & 7.724\,GHz\, & (4\,kHz)\, & ($\sim$ 0)\, & 1.2\,MHz\,\, \, &($\sim$ 0)\, \\ RC\, & 8.062\,GHz\, & ($\sim$ 2\,kHz)\, & ($\sim$ 1\,kHz)\, & ($\sim$ 0)\,\,\, &1.1\,MHz\, \\[0.5ex]
\hline \end{tabular} \caption{\textbf{Hamiltonian parameters of all cQED components.} Values that are within a parenthesis are estimated/simulated parameters. Some non-linear couplings, such as $\chi$ between qA and Bob, are omitted because they are too small to be simulated and measured.}\label{table:params_multi_ancillae} \end{table*}
We highlight the large detuning between Alice and Bob, which results in a small cross-Kerr ($\chi_{ab} \lesssim$1\,kHz) between them. This ensures that there is minimal unwanted interactions between the two modes in the absence of the RF drives, therefore allowing the operation to have a large on-off ratio. Further, it is also important that both Alice and Bob have relatively weak self-Kerr, which is the non-linearity inherited from their couplings to the ancillae. Large self-Kerr would cause imperfections in the interference between Alice and Bob, especially at larger photon numbers. Finally, we comment on the choice of the resonance frequency of qC. It is placed in between the frequencies of Alice and Bob so that it has comparable, but different, dispersive coupling strength to each cavity. This allows us to use qC as a meter to efficiently probe the joint photon number state, $|n, m\rangle_{AB}$, in Alice and Bob by measuring its transition frequency $\omega_{ge}$, which is given by $\omega_{\mathrm{ge}} = \omega^{00}_{\mathrm{ge}} - n_{A}\chi_{\mathrm{ac}} - m_{B}\chi_{\mathrm{bc}}$.
We characterize the coherence of each component in the system using standard cQED measurements. The results are summarized in Table~\ref{table:t1t2s}. \begin{table}[!htb] \centering
\begin{tabular}{c c c c c} \hline\hline\\[-2ex]
& T$_{1}$ ($\mu$s)\,\, & T$_{2}$ ($\mu$s)\,\, & T$_{2E}$ ($\mu$s)\,\, & Population\\ \hline\\[-2ex] Alice\, & 400-500\,\,\, & 400-600\,\,\, & N.A\,\, & $<$1\% \\ Bob\, & 400-500\,\,\, &400-600\,\,\, & N.A\,\, & $<$1\%\\ qA\, & 70\,\,\, & 15-20\,\, \, & 30-40\,\, & 2-3\%\\ qC\, & 50\,\,\, & 10-15\,\, \, & 25-40\,\, & 2-3\% \\
\hline \end{tabular} \caption{\textbf{Coherence properties of the the system.} The device exhibits some fluctuations in its coherence times. In particular, the relatively low $T_2$ of qA and qC are likely a result of low-frequency mechanical vibrations.}\label{table:t1t2s} \end{table}
\section{The driven Josephson circuit Hamiltonian} Similar to the treatment in~\cite{leghtas2015}, the full Hamiltonian describing the system consisting of Alice, Bob, qC, and two RF drive tones applied to qC can be written as \begin{align}\label{eq:hamiltonian} \hat{H}/\hbar = &\omega_{a}\hat{a}^{\dagger}\hat{a} + \omega_{b}\hat{b}^{\dagger}\hat{b} + \omega_{c}\hat{c}^{\dagger}\hat{c} - \frac{E_{\mathrm{J}}}{\hbar}(\cos{\hat{\varphi}} + \frac{\hat{\varphi}^2}{2}) \nonumber\\ &+ 2\mathrm{Re}[\epsilon_{1}e^{-i\omega_{\mathrm{1}}t} + \epsilon_{2}e^{-i\omega_{\mathrm{2}}t}](\hat{c}^{\dagger} + \hat{c}) \end{align} where $\omega_{k}$ is the frequency of each mode, $k$, and $\hat{\varphi}$ is the flux across the junction, which can be decomposed into a linear combination of the phase across each mode: \begin{equation} \hat{\varphi} = \sum_{k=a,b,c} {\phi}_{k}(\hat{k}^{\dagger}+\hat{k}) \end{equation} This Hamiltonian captures the behavior of the system when irradiated by two drives with complex amplitudes, $\epsilon_{1}$ and $\epsilon_{2}$, and frequencies $\omega_{1}$ and $\omega_{2}$, respectively. In our particular configuration, the drive tones are predominantly coupled to qC, as assumed in Eq.~\ref{eq:hamiltonian}. Further, we assume that the drive tones are stiff, i.e the annihilation and creation of a photon from the drive does not cause any change to the mode. Therefore, they can simply be treated as classical drives.
We eliminate the fastest time scales corresponding to the resonance frequencies of each mode using the following unitary transformation \begin{equation} \hat{U} = e^{-i\omega_{\mathrm{ge}}t\hat{c}^{\dagger}\hat{c}}e^{-i\omega_{a}t\hat{a}^{\dagger}\hat{a}}e^{-i\omega_{b}t\hat{b}^{\dagger}\hat{b}} \end{equation} Then we make a displacement transformation for qC such that $\hat{c}\rightarrow \hat{c} + \xi_1 e^{-i\omega_1 t}+\xi_2 e^{-i\omega_2 t}$.
In this new frame, we express the amplitudes of the drives, $\xi_{1(2)}$, as a function of the amplitudes of the drive tones and their respective detunings from the $|g\rangle-|e\rangle$ transition frequency of qC, $\omega_{\mathrm{ge}}$: \begin{equation} \xi_{1(2)} = -\frac{i \epsilon_{1(2)}}{(\tilde{\kappa}/2 + i (\omega_{\mathrm{ge}} - \omega_{1(2)})} \end{equation} where $\tilde{\kappa}$ is the effective decay rate of the mode to which the drives couple to primarily. In this case, it is the decay associated with qC, which is at least an order of magnitude smaller than the coupling rates associated with the driven interaction.
Now, we derive the effective Hamiltonian by expanding the cosine potential in Eq. (1) to the 4th order and perform the standard rotating wave approximations (RWA). As the frequency matching condition is satisfied when $\omega_2 - \omega_1 = \omega_b - \omega_a$, the only 4th order, non-rotating terms are: \begin{align} \hat{H} &= \hat{H}^{1(2)}_{ss} + \hat{H}_{\mathrm{Kerr}} + \hat{H}_{\mathrm{int}}\\ \hat{H}_{\mathrm{int}} &= -E_{\mathrm{J}}\phi^2_{c}\phi_{a} \phi_{b}(\xi_{1}\xi_2^{*}\hat{a}^{\dagger}\hat{b} + \xi_{1}^{*}\xi_2\hat{a}\hat{b}^{\dagger}) \\
\hat{H}^{1(2)}_{\mathrm{ss}}&\approx -E_{\mathrm{J}}\phi^4_{c} |\xi_{1(2)}|^{2}\hat{c}^{\dagger}\hat{c} = -2\alpha|\xi_{1(2)}|^{2}\hat{c}^{\dagger}\hat{c}\\ \hat{H}_{\mathrm{Kerr}} & = -\sum_{k=a,b,c} \frac{E_{\mathrm{J}}{\phi}_{k}^4}{4}\hat{k}^{\dagger}\hat{k}^{\dagger}\hat{k}\hat{k} - E_{\mathrm{J}}{\phi}_{a}^2\phi_{b}^2\hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b} \nonumber\\ &\quad - E_{\mathrm{J}}{\phi}_{a}^2\phi_{c}^2\hat{a}^{\dagger}\hat{a}\hat{c}^{\dagger}\hat{c}- E_{\mathrm{J}}{\phi}_{b}^2\phi_{c}^2\hat{b}^{\dagger}\hat{b}\hat{c}^{\dagger}\hat{c} \end{align}
where $H_{\mathrm{int}}$ is the desired interaction term. $\hat{H}_{\mathrm{Kerr}}$ describes the self-Kerr and cross-Kerr coupling terms~\cite{nigg2012}, which do not depend on the RF drives. They are calibrated independently using methods developed in Ref.~\cite{kirchmair2013}. Finally, $H^{1(2)}_{\mathrm{ss}}$ captures the Stark shift of the resonance frequency of qC in the presence of each RF drive. In the limit of $|\omega_{1,2} - \omega_{\mathrm{ge}}|\gg\alpha$, the Stark shift only depends on the normalized drive strengths, $\xi_1$ and $\xi_2$, as well as $\alpha = \frac{1}{2}E_{\mathrm{J}}\phi^4_{c}$ (Eq.~S6). For the drive configuration we have chosen in this experiment, the detuning between $\omega_1$ and $\omega_{\mathrm{ge}}$ is only a factor of $\sim$2 larger than $\alpha$. Thus, we must take into account the other slowly rotating terms in the cosine expansion which have the form $\alpha\xi_{1}e^{-i\delta t}(\hat{c}^{\dagger})^2\hat{c}$ + h.c, where $\delta = \omega_1 - \omega_{\mathrm{ge}}\approx157\,$MHz. We treat this perturbatively and to the leading order in the drive power, this term results in a modification factor $\delta/(\delta + \alpha)$ to the Stark shift which must be take into account in our calibration process. Further, such non-resonant terms also introduce a correction factor to the coupling strength $\tilde{g} =g(1+ 2\alpha/(\delta + (\omega_2 - \omega_\mathrm{ge}) + (\omega_a-\omega_{\mathrm{ge}}) + (\omega_b - \omega_{\mathrm{ge}})))^{-1}$~\cite{zhang2018}. We measure $\alpha$ independently using simple spectroscopic methods. It is also worth noting that the drives are applied adiabatically with a smooth ring-up and ring down time of $\sim 100$\,ns to ensure that no additional spectral components are present in the drive tones.
\begin{figure}\label{sfig:pump_cal}
\end{figure}
Due to the difference in the detuning to qC from each RF drive, the resulting Stark shifts differ quite significantly in this configuration (Fig.~\ref{sfig:pump_cal}). For the further detuned drive (magenta), the Stark shift exhibits linear dependence on the drive power, in good agreement with the model using 4th order expansion. However, the drive tone that is spectrally much closer to qC, the resulting Stark shift deviates from the predicted linear trend at higher powers. Such deviation becomes strong when the Stark shift becomes comparable to the detuning of the drive from $\omega_{\mathrm{ge}}$. We verified that the observed AC Stark shift is well captured by our model that keeps up to quartic terms in the expansion of cosine potential in Eq. (1) and treats the drives non-perturbatively; see Ref.[7]. Here, by using the full DAC range of our RF input signal, we can tune the effective drive ampliudes up to $\xi_1 \approx 0.75$ and $\xi_2 \approx 0.25$. In practice, we operate at where the dependence on the drive power is roughly linear (black crosses) and the effects of six order terms are negligible. From this calibration, the effective coupling strengths, $g$, are computed for a chosen set of $\xi_{1(2)}$ from the single photon dynamics under $U_{\mathrm{BS}}$ and compared against the theory prediction as shown in Fig.\,2(c) of the main text.
The particular choice of drive frequencies is used in order to avoid exciting any higher order transitions of qC. In principle, we could move the frequencies of both drives together such that the two drives are roughly equally detuned from qC. However, this requires one tone to be placed below the $\omega_{\mathrm{ge}}$, which makes the spectral overlap with the $|e\rangle-|f\rangle$ or $|f\rangle-|h\rangle$ transitions more likely.
\section{drive-assisted absorption of cavity photons}\label{loss} The transmon, qC, that supplies the non-linearity needed for the controlled beamsplitter operation between cavities typically has much shorter lifetime than the cavities. As a result, hybridization of the cavities with the qC leads to a shorter cavity lifetime. To leading order in the transmon-cavity coupling, the rate of cavity decay inherited from the qC via the ``inverse Purcell effect" \cite{reagor2016} is given by Fermi's golden rule: \begin{equation}
\kappa_\gamma = -2 |\lambda|^2 {\rm Im} \frac{1}{(\omega_\mathrm{cav}-\omega_\mathrm{ge}) + i\gamma/2}, \end{equation}
where $\lambda$ is the coupling strength between the cavity with frequency $\omega_\mathrm{cav}$ and the transmon qubit. For large detuning between transmon and the cavity $|\omega_\mathrm{cav}-\omega_\mathrm{ge}|\gg \gamma$, we have $\kappa_\gamma \approx |\lambda/(\omega_\mathrm{cav}-\omega_\mathrm{ge})|^2 \gamma$. This decay rate can be thought of as the result of hybridization between a cavity excitation (photon) and an excitation of the transmon. Such inherited decay can be largely suppressed by using a large $|\omega_\mathrm{cav}-\omega_\mathrm{ge}|$, as is the case in the present experiment.
\begin{figure}\label{sfig:bs_decays}
\end{figure}
In the presence of the RF drives, a cavity photon combined with drive photons can also hybridize the excitations of qC from the ground state to higher excited states due to multi-photon resonances. Taking into account these processes yields a generalized formula for $\kappa_\gamma$ \begin{align}\label{eq:kappa_gamma}
\kappa_\gamma = -2 \sum_{K K'}|M_{K K'}|^2 {\rm Im} \frac{1}{(\omega_\mathrm{cav}-\nu_{KK'})+i\gamma_{KK'}/2}. \end{align}
Each term in the summation refers to a process in which one cavity photon, $K$ photons from the low frequency drive and $K'$ photons from the high frequency one, excite qC from the ground state to the $(K+K'+1)$-th excited state. Hence, the resonance frequency $\nu_{KK'}$ is given by the relation \[ \nu_{K K'} + K\omega_\mathrm1 + K'\omega_\mathrm2 = \mathcal E_{K+K'+1}-\mathcal E_0, \] where $\mathcal E_K$ is the AC-Stark-shifted energy of the $K$-th excited state of the transmon. $M_{KK'}$ and $\gamma_{KK'}$ are the matrix element and width of the same resonance process, respectively. In the weak drive regime, we have from perturbation theory that \[ |M_{KK'}|\propto|\xi_1|^{|K|}|\xi_2|^{|K'|},\]\[ \gamma_{KK'}\approx (K+K'+1)\gamma.\] To go beyond the weak drives, we have developed a Floquet theory to study the dynamics of the driven non-linear transmon which will be described in depth in a separate theory analysis~\cite{zhang2018}.
Experimentally, we observe a consistent degradation of the BS quality as the drive powers become higher. To investigate qualitatively, we extract the coherence time scales of a single excitation evolving under the engineered $\hat{U}_{\mathrm{BS}}$ at each $|\xi_1||\xi_2|$ as described in the main text. As we sweep the drive powers in each of these experiments, we adjust the drive frequencies accordingly to ensure that the resonance condition is still satisfied. As show in Fig.~\ref{sfig:bs_decays}, both $\tau_{1}$ and $\tau_{\phi}$ remains basically independent of $|\xi_1||\xi_2|$ at low drive powers but a significant reduction is observed when $|\xi_1||\xi_2|\gtrsim 0.15$. This is not surprising since when the coupling is weak, the systems is essentially limited by its intrinsic decoherence time scales, i.e. cavity photon loss and dephasing. In this regime, the transmon excited levels are only virtually involved in the process and hence, its decoherence properties do not influence the quality of the operation. However, as we increase the drive powers the transmon's participation increases. More specifically, for the frequency configuration used in the experiment,participation of transmon's $|f\rangle$ state increases most strongly as drive strengths increase. This is because $\nu_{10}$ becomes increasingly close to $\omega_a$ due to AC Stark shift. In other words, the process in which one cavity photon at $\omega_a$ together with one drive photon at $\omega_1$ excites the transmon from $|g\rangle$ to $|f\rangle$ becomes less virtual. This subjects the system to the much less favorable decoherence time scales of the ancilla, which is more than an order of magnitude faster than that of cavities. It also results in a increase in the probability of finding the transmon in the excited states as shown in Fig.~\ref{sfig:bs_decays}(c).
\begin{figure}\label{sfig:p1001}
\end{figure}
\begin{figure*}\label{sfig:a2b1}
\end{figure*}
The interference experiments described in the main text are performed at $|\xi_1||\xi_2|\approx0.1$. This particular power is chosen as a compromise between minimizing qC participation and maximizing the effective coupling strength. At this drive power, we are able to perform a relatively fast BS operation ($T_{\mathrm{BS}}\lesssim$ 5\,$\mu$s $\ll \kappa_{a (b)}$) while still ensuring that qC remains largely in its ground state during the process. We show in Fig.~\ref{sfig:p1001} the oscillation of both $P_{10}$ and $P_{01}$, which have equal contrast and are exactly out of phase with each other. This implies that the excitation is strictly confined to the Hilbert space of Alice and Bob, with minimal participation of the excited levels of qC. The maximum contrast of the $P_{10}$ oscillation is $\approx$ 0.81. This is a result of the imperfections in both the state preparation and the joint photon number measurement which requires a long (4.8\,$\mu$s), spectrally-selective $\pi$ pulse on qC. These two effects can be calibrated using a Rabi experiment after preparing the initial state $|1, 0\rangle_{\mathrm{AB}}$. The measured contrast is $82\pm2\%$, consistent with that of the $P_{10}$ oscillation. In the main text, we have normalized the results of joint photon number measurements by this independently extracted scaling factor. The sum of $P_{10}$ and $P_{01}$ gives the overall decay envelope, which is consistent with the average intrinsic photon loss rate of Alice and Bob, as shown in Fig.~2(b) of the main text.
\section{Multiphoton interference}
One advantage of using cavity states as memories is the availability of large Hilbert space. This allows the efficient encoding of quantum information using a variety of different schemes such as the Binomial code and the cat code. To demonstrate that our engineered beamsplitter operation is compatible multiphoton states, we now perform the same type of interference studies with more excitations.
In the first example, we initialize the system in $|2, 1\rangle_\mathrm{AB}$. Due to the absence of an independent ancilla that couples to Bob, it is rather cumbersome to prepare a Fock state in Bob directly using OCT pulses. To overcome this, we instead first initialize the system in $|1, 0\rangle_{AB}$. Subsequently, we use the engineered bilinear coupling to perform a SWAP operation and transfer this excitation from Alice to Bob. Finally, we complete the preparation by putting two photons in Alice using a numerically optimized OCT pulse. The entire process takes $\sim$ 5 $\mu$s and produces the desired state with $\sim$ 85\% fidelity with some spurious populations in $|0, 0\rangle_\mathrm{AB}$, $|1, 0\rangle_\mathrm{AB}$, and $|2, 0\rangle_\mathrm{AB}$. However, since the interference conserves total photon number and joint parity, the spurious populations do not change the statistics of the outcomes of $|2, 1\rangle_\mathrm{AB}$ undergoing $\hat{U}(\theta)$. Instead, they result in a deterministic reduction of the measurement contrast.
The probability of each possible outcome is measured with a spectrally-selective pulse on qC as shown in Fig.~\ref{sfig:a2b1}(a). Even with only three excitations, the interference dynamics is already rather complex. As we extend our study to include more excitations, it can quickly become a non-trivial problem to predict the dynamics. This type of multi-excitation interference is often referred to as the generalized HOM interference, which has been studied extensively theoretically~\cite{lim2005, tillmann2013, khalid2017}. This simple illustration indicates that our implementation has the potential to realize such complex multipartite interference.
An example of a multiphoton state that can be prepared trivially in the cavities is the coherent state. Here we prepare the system in $|\alpha, 0\rangle_\mathrm{AB}$, with $\alpha = \sqrt{2}$ and measure the population after the operation. We shown in Fig.~\ref{sfig:coherent_state}, the measurement of the initial state, the outcome after a single BS and that after two consecutive BS, which is equivalent to a SWAP operation. Since we only have independent tomography capability on Alice, we chose to measure its Wigner function after each operation using qA and using qC to probe the population in Bob. We can fit both measurements to extract the photon numbers in each cavity. After a single BS, we observe that the coherent state has been reduced to half its original size in Alice. The corresponding measurement of Bob shows that it now contains the same coherent state with $\bar{n}\approx 1$. When we implement the BS operation twice, Alice is fully evacuated to the vacuum state and Bob contains $\bar{n} \approx 2$. This demonstrates that the operation has indeed fully transferred the excitations from Alice to Bob.
\begin{figure}
\caption{\textbf{Behaviour of a coherent state after the BS operation}. The system is prepared in $|\alpha, 0\rangle_\mathrm{AB}$ with $\alpha=\sqrt{2}$. Top row: measured Wigner function for A under different conditions (a) initial state; (b) 50:50 BS, (c) two 50:50 BS. Bottom row: corresponding photon population measured in B for each conditions respectively}
\label{sfig:coherent_state}
\end{figure}
This is a interesting illustration because it highlights the behavior of a semi-classical state under a 50:50 BS. It simply becomes two separable coherent states each of half the size of the initial state. No entanglement is created between Alice and Bob because the BS operation is linear. Further, the presence of larger photon number states also highlights the effects of non-linearities in our system during the operation. We can see this effect from Fig.~\ref{sfig:coherent_state}(b) where the coherent state in Alice after a single BS is distorted due to self-Kerr~\cite{kirchmair2013}. Such non-linear effects can cause complications as we move towards interference between more excitations. Therefore, it is desirable to realize an effective coupling rate much faster than the self-Kerr. More over, the coherent state can simply be considered as a weighted superposition of Fock states. Thus, another implication of this experiment is that our engineered operation is fully capable of handling not only multiple excitations but more importantly, the superpositions of different photon number states. This property is highly desirable since continuous-variable based quantum error correction schemes, such as the cat code~\cite{mirrahimi2014}, are a promising route towards realizing error-protected logical qubits~\cite{ofek2016}. In fact, the engineered bilinear coupling between high-Q modes is an important ingredient in constructing logical gates between multiple bosonic logical qubits encoded superconducting cavities~\cite{mirrahimi2014}.
\section{Non-idealities at large photon numbers } So far, we have shown that techniques described in this work are compatible with multi-photon states. However, there are additional sources of imperfections when higher photon number states are present. Additionally, we have utilized the engineered BS and the parity mapping protocol described in Ref.~\cite{sun2014} to probe the quantum state overlap between Alice and Bob (Fig.~4(b)). Experimentally, we observe a reduced contrast of the measured $\langle \hat{P_A}\rangle$ as the photon number population increases. This arises from both the errors in the parity measurement as well as imperfections in the BS operation at higher photon numbers.
\begin{figure}\label{sfig:overlap_contrast}
\end{figure}
A major source of error in the parity measurement at higher photon numbers is the bandwidth of the transmon pulses used in the parity mapping sequence. Ideally, the transmon rotations should have infinite bandwidth such that they are completely independent of cavity photon number. However, this would result in spectral overlaps with other transmon transitions which would cause leakage errors. In light of these two conflicting requirements, we chose to use a Gaussian pulse with $\sigma = 2\pi\cdot 20$\,MHz for the transmon rotations with standard first order DRAG corrections~\cite{motzoi2009} as a compromise. The measured overlap between $|\psi\rangle_{A} = e^{i\phi_A}|\alpha\rangle$ and $|\psi\rangle_{B} = e^{i\phi_{B}}|\alpha\rangle$ at $\phi_{A} = \phi_{B}$ as a function of the $|\alpha|^2$ is shown in Fig.~\ref{sfig:overlap_contrast}. The maximum contrast at $\alpha = 0$ is $\sim$\,94\%, limited by $\sim$\,2\% readout errors and $\sim$\,4\% parity mapping errors due to transmon decoherence.
The effects of higher photon number states on the BS operation are three-fold. Firstly, because of the non-linearities of cavities, the frequencies of transitions between excited states of the cavities are detuned from the frequency matching condition for the BS operation, therefore causing reduction in the BS fidelity. Such detunings are proportional to $N\chi_{aa}$, $N\chi_{bb}$, and $N\chi_{ab}$, with $\chi_{aa}$, $\chi_{bb}$ being the self-Kerr of Alice and Bob, and $\chi_{ab}$ the cross-Kerr. We estimate an upper bound on the resulting infidelity to SWAP a Fock state $|N\rangle$ to be proportional to $N[N(\chi_{aa} + \chi_{ab} + \chi_{ab})/g]^2$. Although the intrinsic self-Kerr is relatively small for both Alice and Bob, it is enhanced when the drives are present. This is consistent with distortion observed in Fig.~\ref{sfig:coherent_state}(b). We extract the self-Kerr based on the Wigner tomography of a coherent state after a single BS to be $\chi_{aa}/2\pi \approx 8$\,kHz and $\chi_{bb}/2\pi \approx 5$\,kHz. This drive-induced non-linearity is very sensitive to the exact power and frequency configurations of the drives. A more thorough theoretical and experimental analysis of the change of self-Kerr as a function of drive powers and frequencies is currently underway. With these values of self-Kerr and the independently measured cavity decoherence times, we simulate the maximum overlap measured at $\phi_A = \phi_B$ and find that it is consistent with the measured reduction up to $|\alpha|^2\sim 2$. The additional reduction could be due to the limited spectral bandwidth of the parity measured described in the previous paragraph and other decay mechanisms discussed below.
Secondly, the decay rates for transitions between the $N$-th and $(N-1)$-th level of the cavities grow as $N$ increases. If the cavity only suffers from linear decay (one photon loss) which is the dominant decay mechanism, the rate of transitions between higher levels increases linearly in $N$. The resulting BS infidelity will scale as $N\kappa/g$ where $\kappa$ is the linear decay rate of the cavities. However, there can be additional loss mechanisms that do not scale as $N$ due to the coupling to a non-linear transmon mode. This leads to the nonlinear decay of the memory modes, i.e., the total rate of decay depends on the instantaneous energy of the cavities: $\kappa_\mathrm{total} = \kappa + N\kappa_{\mathrm{NL}}$. One example of nonlinear decay is two-photon loss where cavities decay by emitting two photons at a time. In general, as the transmon-cavity detuning is large compared to their coupling strength, such non-linear decay is typically weaker than the transmon-induced linear decay and the self-Kerr of the cavities. However, in the presence of drives, the cavity frequencies can be close to certain two-photon transitions where the non-linear decay rates will be enhanced. In this case, the decay rate of transitions between the $N$-th and $(N-1)$-th level of cavities grows as $N^2$ rather than $N$, resulting in an increase in the BS infidelity proportional to $N^2\kappa_{\mathrm{NL}}/g$.
Thirdly, the transmon is more likely to be excited at large cavity photon number. When the frequency of one of the cavities is close to the resonance frequencies $\nu_{KK'}$ in Eq.~\ref{eq:kappa_gamma}, the probability of exciting the transmon by absorbing both cavity and pump photons increases. The excitation rate is proportional to the cavity photon number $N$. This results in additional non-idealities in the operation when large photon number states are present. In practice, all three of the above-mentioned effects, and potentially other imperfections not considered here, contribute to the degradation in the BS quality. Additional experimental investigations are currently underway in order to provide a more thorough and quantitative analysis.
\section{Experiment wiring} The details of the room temperature control configuration and the fridge wirings are shown in Fig~\ref{sfig:wiring}. The device is housed inside a Cryoperm magnetic shield and thermalized to the base plate of a dilution refrigerator with a base temperature of $\sim$15\,mK. We use commercial low-pass (LPF) and custom infrared (Ecco) filters along the microwave lines to reduce stray radiation and photon shot noise. Two Josephson parametric converters (JPCs) are also installed on the base plate. They are connected to output ports of the two readout resonators via circulators and provide near-quantum-limited amplification of the output signal, which is further amplified by commercial HEMT amplifiers at the 4K stage. This gives us the capability to perform efficient single-shot readout of both transmon ancillae independently.
RF signals used to control each mode are IQ-modulated by DAC outputs from an integrated FPGA system at room temperature. The signal is then mixed with their respective local oscillators (LOs) and then amplified by standard room temperature amplifiers (ZVE and MTQ). Fast switches are placed on the input lines before the signal is transmitted to the fridge in order suppress the unwanted RF power during idle times. The same LO generator is used for the displacement and drive tones for each cavity mode to eliminate relative drifts of their phases. The two drives used to enable the bilinear coupling are combined after their respective amplification and filtered carefully at room temperature. Bandpass filters (BPF) are employed on this line to reduce the spectral width of the noise sent into the fridge. In particular, we suppress the noise near qC resonance frequency by $\gtrsim$ 50 dB to prevent spurious population of the mode. Fast switches are use on all room temperature input lines before they go in to the fridge to prevent excess power input during idle times.
A dedicated input line is used to bring the drives to the sample at base so that a special attenuation configuration can be used. In this case, the drive line (purple) contains a custom 10\,dB reflective attenuator~\cite{andrew2018} at base. It reflects a large fraction of the input power, which gets dissipated at the upper stages of the fridge where more cooling power is available. This allows us to send large amount RF power without heating up the base plate significantly. The drive tones are then combined with the qC input line via a directional coupler, which has a 6 dB insertion loss. With this configuration, we are able to introduce a relative strong coupling and run the experiments described in the main text at a reasonable duty cycle without causing any measurable heating to the system. \\
\begin{figure*}
\caption{\textbf{Room temperature RF controls and fridge line configurations}}
\label{sfig:wiring}
\end{figure*}
\end{document}
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Philosophy of Action > Decision Theory > Game Theory > Convention and Coordination
Convention and Coordination
Evolutionary Game Theory (171)
Game-Theoretic Principles (26)
Game Theory and Ethics (54)
Game Theory and Political Philosophy (67)
Prisoner's Dilemma (277)
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Game Theory, Misc (104)
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The College of Idaho
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Game theory and scalar implicatures.Daniel Rothschild - 2013 - Philosophical Perspectives 27 (1):438-478.details
Convention and Coordination in Philosophy of Action
Conversational Implicature in Philosophy of Language
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Conditionals all the way down.Matheus Silva - manuscriptdetails
It is usually accepted that unconditional statements are clearer and less problematic than conditional ones. This article goes against this popular belief by advancing the contrarian hypothesis that all unconditional statements can be reduced to conditional ones due to the way our assumptions support our assertions. In fact, considering the coherentist process by which most of our different beliefs mutually support themselves, the only genuine example of unconditional statements are cases of self-justified beliefs, but these examples are controversial and few (...) and far between. The distinction between unconditional and conditional statements is similar to the distinction between assumptions and premises in that is a largely conventional idealisation that results from our attempts to limit epistemic complexity. (shrink)
Assertion in Philosophy of Language
Epistemic Regress in Epistemology
Evidence in Epistemology
Foundationalism and Coherentism in Epistemology
Logical Consequence and Entailment in Logic and Philosophy of Logic
Semantics-Pragmatics Distinction in Philosophy of Language
A tragic coalition of the rational and irrational: a threat to collective responses to COVID-19.Marinus Ferreira, Marc Cheong, Colin Klein & Mark Alfano - 2022 - Philosophical Psychology.details
There is not as much resistance to COVID-19 mitigation as there seems, but there are structural features that make resistance seem worse than it is. Here we describe two ways that the problem seeming to be worse than it is can make it worse. First, visible hesitation to implement COVID-19 responses signals to the wider society that mitigation measures may not succeed, which undermines people's conditional willingness to join in on those efforts. Second, our evaluations of others' willingness to implement (...) these measures are informed by our attempts to mind-read them. Yet attempts to mind-read groups often mislead us, because groups invariably act from diverse motives whereas mind-reading works best when identifying relatively stable and consistent motivations. This means that a small minority of people refusing to implement measures can have an outsized prominence that prompts mind-reading to diagnose widespread hesitation. These two factors form a feedback loop with each other: we see some people's hesitation, which prompts us to mind-read other people as being more uncertain about the responses than they actually are, which undermines our confidence in the responses, which in turn encourages others to mind-read this hesitation, which further undermines confidence. (shrink)
Mindreading in Philosophy of Cognitive Science
Trust in Normative Ethics
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Epistemic Foundations of Salience-Based Coordination.Vojtěch Zachník - 2021 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 4 (28):819-844.details
This paper aims to assess current theoretical findings on the origin of coordination by salience and suggests a way to clarify the existing framework. The main concern is to reveal how different coordination mechanisms rely on specific epistemic aspects of reasoning. The paper highlights the fact that basic epistemic assumptions of theories diverge in a way that makes them essentially distinctive. Consequently, recommendations and predictions of the traditional views of coordination by salience are, in principle, based on the processes related (...) to the agent's presumptions regarding the cognitive abilities of a co-player. This finding implies that we should consider these theories as complementary, and not competitive, explanations of the same phenomenon. -/- . (shrink)
Salience in Philosophy of Mind
What Is Minimally Cooperative Behavior?Kirk Ludwig - 2020 - In Anika Fiebich (ed.), Minimal Cooperation and Shared Agency. Cham, Switzerland: Springer. pp. 9-40.details
Cooperation admits of degrees. When factory workers stage a slowdown, they do not cease to cooperate with management in the production of goods altogether, but they are not fully cooperative either. Full cooperation implies that participants in a joint action are committed to rendering appropriate contributions as needed toward their joint end so as to bring it about, consistently with the type of action and the generally agreed upon constraints within which they work, as efficiently as they can, where their (...) contributions are sensitive to information (where available) about how others are contributing in the sense that they adjust as needed their contributions in light of information about how others are contributing to ensure effective pursuit of their joint end, where this includes rendering aid to other participants if needed, insofar as they are able. Full cooperation entails those cooperating are engaged in a joint intentional action. Some prominent studies of joint intentional action focus exclusively on cases of full cooperation (notably that of Michael Bratman (2014)). But not all joint intentional action is fully cooperative. One example is the work slowdown. Another example is provided by competitive games like chess and football, or sports like boxing and wrestling, where participants are clearly not intending to contribute to the pursuit of all of the goals of the others engaged in the activity, even when those goals are internal to the type of activity in question, but instead intend actively to frustrate some of them. In this paper, I provide a taxonomy of forms of non-cooperative behavior within the context of behavior that is still to some degree cooperative, and I argue that the minimal conditions of joint intentional action define minimal cooperative behavior, that is, that minimally joint intentional action is per se minimally cooperative behavior. I define in precise terms what that comes to, and how it is possible in cases in which it seems that one or more participants are in one or more ways acting so as to frustrate the contributions of other participants to their joint action. (shrink)
Action Theory, Misc in Philosophy of Action
Collective Action in Philosophy of Action
Collective Intentions in Philosophy of Action
The limits of conventional justification: inductive risk and industry bias beyond conventionalism.Miguel Ohnesorge - 2020 - Frontiers in Research Metric and Analytics 14.details
This article develops a constructive criticism of methodological conventionalism. Methodological conventionalism asserts that standards of inductive risk ought to be justified in virtue of their ability to facilitate coordination in a research community. On that view, industry bias occurs when conventional methodological standards are violated to foster industry preferences. The underlying account of scientific conventionality, however, is problematically incomplete. Conventions may be justified in virtue of their coordinative functions, but often qualify for posterior empirical criticism as research advances. Accordingly, industry (...) bias does not only threaten existing conventions but may impede their empirically warranted improvement if they align with industry preferences. My empiricist account of standards of inductive risk avoids such a problem by asserting that conventional justification can be pragmatically warranted but has, in principle, only a provisional status. Methodological conventions, therefore, should not only be defended from preference-based infringements on their coordinative function but ought to be subjected to empirical criticism. (shrink)
Applications of Science in General Philosophy of Science
Medical Methodology in Philosophy of Science, Misc
Science and Values in General Philosophy of Science
What Makes a Kind an Art-kind?Michel-Antoine Xhignesse - 2020 - British Journal of Aesthetics 60 (4):471-88.details
The premise that every work belongs to an art-kind has recently inspired a kind-centred approach to theories of art. Kind-centred analyses posit that we should abandon the project of giving a general theory of art and focus instead on giving theories of the arts. The main difficulty, however, is to explain what makes a given kind an art-kind in the first place. Kind-centred theorists have passed this buck on to appreciative practices, but this move proves unsatisfactory. I argue that the (...) root of this dissatisfaction stems not from the act of kicking the can down the road, but from not kicking it far enough. The missing ingredient, I argue, is a notion of convention which does the work of marking the difference between art and non-art for a given physical medium. (shrink)
Aesthetics, General Works in Aesthetics
Approaches to Social Ontology, Misc in Social and Political Philosophy
Ontology of Social Domains, Misc in Social and Political Philosophy
The Artworld in Aesthetics
The Definition of Art in Aesthetics
Conspiring with the Enemy: The Ethic of Cooperation in Warfare.Yvonne Chiu - 2019 - New York, NY, USA: Columbia University Press.details
*North American Society for Social Philosophy (NASSP) Book Award 2019.* -/- *International Studies Association (ISA) - International Ethics Section Book Award 2021.* -/- Although military mores have relied primarily on just war theory, the ethic of cooperation in warfare (ECW)—between enemies even as they are trying to kill each other—is as central to the practice of warfare and to conceptualization of its morality. Neither game theory nor unilateral moral duties (God-given or otherwise) can explain the explicit language of cooperation in (...) developing and enforcing principles of military ethics and the law of armed conflict. -/- The ethic of cooperation is borne of various motivations: reciprocity, self-preservation, and efficiency, to be sure, but also a sense of warrior honor and concern with human rights. This shared morality can persist despite making it more difficult for one side or the other to win and, unfortunately, its well-meaning motivations often lead to unintended tragic consequences. -/- This book explores three manifestations of this significant yet overlooked ethic of cooperation in warfare: (1) for a "fair fight," (2) to protect classes of individuals (e.g., non-combatants or prisoners of war), and (3) to end the war quickly. Such cooperation can take unexpected forms, from ad hoc decisions on the battlefield to institutionalization in international law, and is the source of some critical tensions in one of the most significant developments in warfare in recent years: namely, how to handle terrorism or other forms of warfare that lie outside the purview of international law. -/- Each type of ECW raises questions internal to that ethic, such as inconsistencies in the concept of "parity" across different weapons bans, contradictions within the warrior ethic that heavily influence—and therefore confuse—notions of the "fair fight," the disconnect between what protections a person receives and his responsibility for the war (e.g., political leaders), or the limited decisiveness of outcomes generated by very short wars. -/- Their simultaneous application also generates significant tensions and raises questions about the proper relationship of ECW to the immediate goal of war itself, which is to win, and thus yield either a political settlement or a justicial decision. For example, the ECWs for a "fair fight" and to protect classes of individuals can make it harder to win the war, but even more concerning is that they can also kill more people, which in the latter case contravenes its very purpose. -/- Human history is in some ways the story of trying to concurrently wage and tame war, and the architecture of warfare itself is informed by the ECW, in particular: (a) the political nature of war, (b) the abdication from jus ad bellum judgments in order to concentrate on justice within war (jus in bello), and (c) the ways in which modern nation-states collude to define "legitimacy" in war. -/- The combination of these three features leave questions of justicial right and responsibility for war disturbingly unresolved, it also generates new challenges in a geopolitical context in which cooperative and non-cooperative (e.g. contemporary terrorism) forms of warfare clash. (shrink)
Deontological Moral Theories in Normative Ethics
Game Theory and Ethics in Philosophy of Action
Game Theory and Political Philosophy in Philosophy of Action
Global Justice in Social and Political Philosophy
Human Rights in Social and Political Philosophy
International Justice in Social and Political Philosophy
International Order in Social and Political Philosophy
Military Ethics in Applied Ethics
Moral Norms in Meta-Ethics
States and Nations in Social and Political Philosophy
Virtue Ethics in Normative Ethics
War and Violence in Social and Political Philosophy
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Explaining Prosocial Behavior: Team Reasoning or Social Influence?Cedric Paternotte - 2019 - In Michiru Nagatsu & Attila Ruzzene (eds.), Contemporary Philosophy and Social Science: An Interdisciplinary Dialogue. pp. 93 - 102.details
Prisoner's Dilemma in Philosophy of Action
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Permissive Metaepistemology.David Thorstad - 2019 - Mind 128 (511):907-926.details
Recent objections to epistemic permissivism have a metaepistemic flavor. Impermissivists argue that their view best accounts for connections between rationality, planning and deference. Impermissivism is also taken to best explain the value of rational belief and normative assessment. These objections pose a series of metaepistemic explanatory challenges for permissivism. In this paper, I illustrate how permissivists might meet their explanatory burdens by developing two permissivist metaepistemic views which fare well against the explanatory challenges.
Bayesian Reasoning in Philosophy of Probability
Epistemic Norms in Epistemology
Epistemic Permissivism in Epistemology
Evidentialism in Epistemology
Formal Epistemology, Misc in Epistemology
Moral Expressivism in Meta-Ethics
Are There Distinctively Moral Reasons?Andrew T. Forcehimes & Luke Semrau - 2018 - Ethical Theory and Moral Practice 21 (3):699-717.details
A dogma of contemporary normative theorizing holds that some reasons are distinctively moral while others are not. Call this view Reasons Pluralism. This essay looks at four approaches to vindicating the apparent distinction between moral and non-moral reasons. In the end, however, all are found wanting. Though not dispositive, the failure of these approaches supplies strong evidence that the dogma of Reasons Pluralism is ill-founded.
Internalism and Externalism about Reasons in Philosophy of Action
Moral Responsibility in Meta-Ethics
Reasons and Oughts in Philosophy of Action
Review of Herbert Gintis's Individuality and Entanglement: The Moral and Material Bases of Social Life. Princeton: Princeton University Press, 2017, 357 pp. [REVIEW]Michiru Nagatsu - 2018 - Erasmus Journal for Philosophy and Economics 11 (1):117-124.details
In his own words, Herbert Gintis's latest book is "an analysis of human nature and a tribute to its wonders" (3).1More prosaically, it is a collection of essays, some of which are original and others published elsewhere. Instead of being structured around topics in decision and game theory,like his previous book (2009), this book develops interrelated themes, such as the evolutionary origins of moral sense, its central role in political games, and the socially entangled nature of human rationality and individuality. (...) Some chapters develop Gintis'svision of the unified behavioral sciences by model-building demonstrations; others do so by reflecting on history and methodology. (shrink)
Evolutionary Game Theory in Philosophy of Action
Ontology of Social Domains in Social and Political Philosophy
Philosophy of Sociology, Misc in Philosophy of Social Science
Rational Choice Theory in Philosophy of Social Science
Social Ontology, Misc in Social and Political Philosophy
Cristina Bicchieri, Norms in the Wild. How to Diagnose, Measure, and Change Social Norms. [REVIEW]Cedric Paternotte - 2018 - Oeconomia 8:267 - 272.details
Values and Norms in Normative Ethics
Coordination technology for active support networks: context, needfinding, and design.Stanley J. Rosenschein & Todd Davies - 2018 - AI and Society 33 (1):113-123.details
Coordination is a key problem for addressing goal–action gaps in many human endeavors. We define interpersonal coordination as a type of communicative action characterized by low interpersonal belief and goal conflict. Such situations are particularly well described as having collectively "intelligent", "common good" solutions, viz., ones that almost everyone would agree constitute social improvements. Coordination is useful across the spectrum of interpersonal communication—from isolated individuals to organizational teams. Much attention has been paid to coordination in teams and organizations. In this (...) paper we focus on the looser interpersonal structures we call active support networks, and on technology that meets their needs. We describe two needfinding investigations focused on social support, which examined four application areas for improving coordination in ASNs: academic coaching, vocational training, early learning intervention, and volunteer coordination; and existing technology relevant to ASNs. We find a thus-far unmet need for personal task management software that allows smooth integration with an individual's active support network. Based on identified needs, we then describe an open architecture for coordination that has been developed into working software. The design includes a set of capabilities we call "social prompting", as well as templates for accomplishing multi-task goals, and an engine that controls coordination in the network. The resulting tool is currently available and in continuing development. We explain its use in ASNs with an example. Follow-up studies are underway in which the technology is being applied in existing support networks. (shrink)
Communication in Social Sciences
Deliberation in Philosophy of Action
Software in Philosophy of Computing and Information
Review of Francesco Guala "Understanding Institutions". [REVIEW]Christopher Clarke - 2017 - British Journal for the Philosophy of Science.details
Institutions in Social and Political Philosophy
Normativity and Naturalism in Value Theory, Miscellaneous
Philosophy of Social Science, General Works in Philosophy of Social Science
Limited Conventions about Morals.Marinus Ferreira - 2017 - Dissertation, University of Aucklanddetails
n this thesis I describe how conventions specify how to put normative principles into practice. I identify a class of recurring situations where there are some given normative principles in effect, but they underdetermine what each individual should do, and what is best for an individual depends on what others do. I demonstrate that in such cases, whenever the community develops a response that repeatedly brings them to as good an outcome as is available according to their principles, that response (...) is a Lewisian convention where the benefit of an outcome to each individual is measured by the extent to which it conforms to the principles they subscribe to. Since these conventions are constrained by the normative principles, I call them limited conventions. They are supplements to the principles, and are ineradicably involved in moral action insofar as the abovementioned cases of moral underdetermination are in play. That has the consequence that in these cases the only reliable way to follow your principles is to follow the relevant conventions. As examples of this mechanism I offer a conventionalist analysis of authority, such that the commands of an authority is normative when they instantiate a limited convention, and of the variation in understandings of virtue and vice across societies, such that the evaluative vocabulary of each society is a set of different limited conventions about how to express in word and deed the evaluative points of the virtues and vices in question. Finally, I discuss how conventions and similar forms of guidance provide a way for individuals to participate in their community's moral life without having a full understanding of the principles that underlie it, or even if they are profoundly ignorant or outright mistaken about the demands of morality. (shrink)
Explanation of Action in Philosophy of Action
Internalism and Externalism about Moral Judgment in Meta-Ethics
Moral Reasoning and Motivation in Meta-Ethics
Moral Uncertainty in Meta-Ethics
The Logic of Joint Ability in Two-Player Tacit Games.Peter Hawke - 2017 - Review of Symbolic Logic 10 (3):481-508.details
Logics of joint strategic ability have recently received attention, with arguably the most influential being those in a family that includes Coalition Logic (CL) and Alternating-time Temporal Logic (ATL). Notably, both CL and ATL bypass the epistemic issues that underpin Schelling-type coordination problems, by apparently relying on the meta-level assumption of (perfectly reliable) communication between cooperating rational agents. Yet such epistemic issues arise naturally in settings relevant to ATL and CL: these logics are standardly interpreted on structures where agents move (...) simultaneously, opening the possibility that an agent cannot foresee the concurrent choices of other agents. In this paper we introduce a variant of CL we call Two-Player Strategic Coordination Logic (SCL2). The key novelty of this framework is an operator for capturing coalitional ability when the cooperating agents cannot share strategic information. We identify significant differences in the expressive power and validities of SCL2 and CL2, and present a sound and complete axiomatization for SCL2. We briefly address conceptual challenges when shifting attention to games with more than two players and stronger notions of rationality. (shrink)
Doxastic and Epistemic Logic in Logic and Philosophy of Logic
Team Reasoning: Theory and Evidence.Jurgis Karpus & Natalie Gold - 2017 - In Julian Kiverstein (ed.), The Routledge Handbook of Philosophy of the Social Mind. New York, USA: Routledge. pp. 400-417.details
The chapter reviews recent theoretical and empirical developments concerning the theory of team reasoning in game theoretic interactions.
Game-Theoretic Principles in Philosophy of Action
Normative and Descriptive Game Theory in Philosophy of Action
The Topology of Communities of Trust.Mark Alfano - 2016 - Russian Sociological Review 15 (4):30-56.details
Hobbes emphasized that the state of nature is a state of war because it is characterized by fundamental and generalized distrust. Exiting the state of nature and the conflicts it inevitably fosters is therefore a matter of establishing trust. Extant discussions of trust in the philosophical literature, however, focus either on isolated dyads of trusting individuals or trust in large, faceless institutions. In this paper, I begin to fill the gap between these extremes by analyzing what I call the topology (...) of communities of trust. Such communities are best understood in terms of interlocking dyadic relationships that approximate the ideal of being symmetric, Euclidean, reflexive, and transitive. Few communities of trust live up to this demanding ideal, and those that do tend to be small (between three and fifteen individuals). Nevertheless, such communities of trust serve as the conditions for the possibility of various important prudential epistemic, cultural, and mental health goods. However, communities of trust also make possible various problematic phenomena. They can become insular and walled-off from the surrounding community, leading to distrust of out-groups. And they can lead their members to abandon public goods for tribal or parochial goods. These drawbacks of communities of trust arise from some of the same mecha-nisms that give them positive prudential, epistemic, cultural, and mental health value – and so can at most be mitigated, not eliminated. (shrink)
The Unilateralist's Curse and the Case for a Principle of Conformity.Nick Bostrom, Thomas Douglas & Anders Sandberg - 2016 - Social Epistemology 30 (4):350-371.details
In some situations a number of agents each have the ability to undertake an initiative that would have significant effects on the others. Suppose that each of these agents is purely motivated by an altruistic concern for the common good. We show that if each agent acts on her own personal judgment as to whether the initiative should be undertaken, then the initiative will be undertaken more often than is optimal. We suggest that this phenomenon, which we call the unilateralist's (...) curse, arises in many contexts, including some that are important for public policy. To lift the curse, we propose a principle of conformity, which would discourage unilateralist action. We consider three different models for how this principle could be implemented, and respond to an objection that could be raised against it. (shrink)
Agent Causation in Philosophy of Action
Social norms and unthinkable options.Ulf Hlobil - 2016 - Synthese 193 (8):2519–2537.details
We sometimes violate social norms in order to express our views and to trigger public debates. Many extant accounts of social norms don't give us any insight into this phenomenon. Drawing on Cristina Bicchieri's work, I am putting forward an empirical hypothesis that helps us to understand such norm violations. The hypothesis says, roughly, that we often adhere to norms because we are systematically blind to norm-violating options. I argue that this hypothesis is independently plausible and has interesting consequences. It (...) implies, e.g., that some experimental paradigms for investigating social norms have hitherto unnoticed shortcomings. (shrink)
Ethics and Cognitive Science, Misc in Normative Ethics
Explaining Universal Social Institutions: A Game-Theoretic Approach.Michael Vlerick - 2016 - Topoi 35 (1):291-300.details
Universal social institutions, such as marriage, commons management and property, have emerged independently in radically different cultures. This requires explanation. As Boyer and Petersen point out 'in a purely localist framework would have to constitute massively improbable coincidences' . According to Boyer and Petersen, those institutions emerged naturally out of genetically wired behavioural dispositions, such as marriage out of mating strategies and borders out of territorial behaviour. While I agree with Boyer and Petersen that 'unnatural' institutions cannot thrive, this one-sided (...) explanation of universal social institutions in terms of genetic human nature is unsatisfactory. Drawing on the literature on multi-level selection and gene-culture coevolution, I argue that universal social institutions are first and foremost the products of cultural selection. They occupy fitness peaks in the landscape of cultural possibilities, much in the same way that biological adaptations occupy fitness peaks in the landscape of biological possibilities. To show this, I use game-theory. By modelling the domains of social interaction in which marriage, commons management, and property emerged as Prisoner's dilemma situations, it becomes clear how an institutional framework allows the group to move to an interactive equilibrium with a larger payoff. Institutions do so by incentivising all parties to adopt a cooperative strategy. They are culturally selected ways of optimising genetically constrained domains of human social interaction. (shrink)
Philosophy of Social Science, Miscellaneous in Philosophy of Social Science
The Tragedy of the Commons as a Voting Game.Luc Bovens - 2015 - In Martin Peterson (ed.), The Prisoner's Dilemma. Classic philosophical arguments. Cambridge University Press. pp. 156-176.details
The Tragedy of the Commons is often associated with an n-person Prisoner's Dilemma. But it can also have the structure of an n-person Game of Chicken, an Assurance Game, or of a Voting Games (or a Three-in-a-Boat Game). I present three historical stories that document tragedies of the commons, as presented in Aristotle, Mahanarayan and Hume and argue that the descriptions of these historical cases align better with Voting Games than with any other games.
Philosophy of Economics, Misc in Philosophy of Social Science
Theory in Economics in Philosophy of Social Science
Punishing Atypical Dirty Hands.Fausto Corvino - 2015 - International Journal of Applied Philosophy 29 (2):281-297.details
Should those who get dirty hands always be punished in the same way? Must their punishment be regardless of the background elements that determined the DH dilemma, which has polluted their morality? This paper holds that common arguments in favour of punishing DH overlook the important difference between classic DH dilemmas that are structurally inescapable and those that are caused by a collective action problem. My thesis emphasizes that in talking about DH, our analysis should go beyond the structure of (...) the dilemma. We should also take into serious consideration the background dynamics that made the choice between two evils inevitable. (shrink)
Political Theory in Social and Political Philosophy
Punishment in Applied Ethics
The Minimalistic Definition of Conventions: One Step beyond Millikan's Approach.Vojtech Zachnik - 2015 - Organon F: Medzinárodný Časopis Pre Analytickú Filozofiu 22 (3):378-394.details
The study proposes a new approach towards a social phenomenon called convention and submits a minimalistic definition of convention, which provides a promising basis for future analysis unburdened by contra-Lewisian objections. The definition itself, based on the insights of Ruth Millikan in the study Language Conventions Made Simple, represents a simple and efficient means of delimiting essential components of conventional behaviour (stripped of most of the controversial issues from previous debates on Lewis's notion) solely by means of the role of (...) precedent and its ability to reproduce. Yet, it is argued that a few additional conditions are required for a valid and distinct notion of conventionality: namely, the inclusion of a coordination aspect and an extension of the concept of precedent. The final version of the definition, thereafter, meets intuitive requirements of conventionality (e.g., arbitrariness) and has the generality to embrace different types of conventions. (shrink)
Promises in Normative Ethics
Social Norms, The Invisible Hand, and the Law.Jonny Anomaly & Geoffrey Brennan - 2014 - University of Queensland Law Journal 33 (2).details
Economics in Social Sciences
Invisible Hand Explanations in Philosophy of Social Science
Legal Authority in Philosophy of Law
Nature of Law, Misc in Philosophy of Law
Normativity of Law in Philosophy of Law
Propositional Content in Signalling Systems.Jonathan Birch - 2014 - Philosophical Studies 171 (3):493-512.details
Skyrms, building on the work of Dretske, has recently developed a novel information-theoretic account of propositional content in simple signalling systems. Information-theoretic accounts of content traditionally struggle to accommodate the possibility of misrepresentation, and I show that Skyrms's account is no exception. I proceed to argue, however, that a modified version of Skyrms's account can overcome this problem. On my proposed account, the propositional content of a signal is determined not by the information that it actually carries, but by the (...) information that it would carry at the nearest separating equilibrium of the underlying evolutionary dynamics. I show that this amended account yields reasonable ascriptions of false propositional content in a well-known formal model of the evolution of communication , and close with a discussion of the serious but perhaps not insuperable difficulties we face in applying the account to examples of signalling in the real world. (shrink)
Evolution of Phenomena, Misc in Philosophy of Biology
Information-Based Accounts of Mental Content in Philosophy of Mind
Misinformation in Philosophy of Computing and Information
Naturalism and Intentionality in Philosophy of Mind
Theories of Representation in Philosophy of Mind
Disagreement about Taste: Commonality Presuppositions and Coordination.Teresa Marques & Manuel García-Carpintero - 2014 - Australasian Journal of Philosophy 92 (4):701-723.details
The paper confronts the disagreement argument for relativism about matters of taste, defending a specific form of contextualism. It is first considered whether the disagreement data might manifest an inviariantist attitude speakers pre-reflectively have. Semantic and ontological enlightenment should then make the impressions of disagreement vanish, or at least leave them as lingering ineffectual Müller-Lyer-like illusions; but it is granted to relativists that this does not fully happen. López de Sa's appeal to presuppositions of commonality and Sundell's appeal to metalinguistic (...) disagreement are discussed, and it is argued that, although they help to clarify the issues, they do not fully explain why such impressions remain under enlightenment. To do it, the paper develops a suggestion that other writers have made, that the lingering impression of disagreement is a consequence of a practical conflict, appealing to dispositions to practical coordination that come together with presuppositions of commonality in axiological matters. (shrink)
Contextualism about Truth in Philosophy of Language
Philosophy of Food and Drink, Miscellaneous in Philosophy of Biology
Relativism about Truth in Philosophy of Language
Value Relativism in Value Theory, Miscellaneous
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On the Emergence of Descriptive Norms.Ryan Muldoon, Chiara Lisciandra, Cristina Bicchieri, Stephan Hartmann & Jan Sprenger - 2014 - Politics, Philosophy and Economics 13 (1):3-22.details
A descriptive norm is a behavioral rule that individuals follow when their empirical expectations of others following the same rule are met. We aim to provide an account of the emergence of descriptive norms by first looking at a simple case, that of the standing ovation. We examine the structure of a standing ovation, and show it can be generalized to describe the emergence of a wide range of descriptive norms.
Team Reasoning and Shared Intention.Abraham Sesshu Roth - 2014 - In Anita Konzelmann Ziv & Hans Bernhard Schmid (eds.), Institutions, Emotions, and Group Agents. Springer. pp. 279-295.details
Collective Intentionality in Philosophy of Mind
The self organization of human interaction.Rick Dale, Riccardo Fusaroli, Nicholas Duran & Daniel Richardson - 2013 - Psychology of Learning and Motivation 59.details
We describe a "centipede's dilemma" that faces the sciences of human interaction. Research on human interaction has been involved in extensive theoretical debate, although the vast majority of research tends to focus on a small set of human behaviors, cognitive processes, and interactive contexts. The problem is that naturalistic human interaction must integrate all of these factors simultaneously, and grander theoretical mitigation cannot come only from focused experimental or computational agendas. We look to dynamical systems theory as a framework for (...) thinking about how these multiple behaviors, processes, and contexts can be integrated into a broader account of human interaction. By introducing and utilizing basic concepts of self-organization and synergy, we review empirical work that shows how human interaction is flexible and adaptive and structures itself incrementally during unfolding interactive tasks, such as conversation, or more focused goal-based contexts. We end on acknowledging that dynamical systems accounts are very short on concrete models, and we briefly describe ways that theoretical frameworks could be integrated, rather than endlessly disputed, to achieve some success on the centipede's dilemma of human interaction. (shrink)
You better play 7: mutual versus common knowledge of advice in a weak-link experiment.Giovanna Devetag, Hykel Hosni & Giacomo Sillari - 2013 - Synthese 190 (8):1351-1381.details
This paper presents the results of an experiment on mutual versus common knowledge of advice in a two-player weak-link game with random matching. Our experimental subjects play in pairs for thirteen rounds. After a brief learning phase common to all treatments, we vary the knowledge levels associated with external advice given in the form of a suggestion to pick the strategy supporting the payoff-dominant equilibrium. Our results are somewhat surprising and can be summarized as follows: in all our treatments both (...) the choice of the efficiency-inducing action and the percentage of efficient equilibrium play are higher with respect to the control treatment, revealing that even a condition as weak as mutual knowledge of level 1 is sufficient to significantly increase the salience of the efficient equilibrium with respect to the absence of advice. Furthermore, and contrary to our hypothesis, mutual knowledge of level 2 induces, under suitable conditions, successful coordination more frequently than common knowledge. (shrink)
Game Theory in Philosophy of Action
Social Norms: Repeated Interactions, Punishment, and Context Dependence.Jonathan Grose & Cedric Paternotte - 2013 - Public Reason 5 (1):3-13.details
Philosophy of Social Science, Misc in Philosophy of Social Science
A new Debate on an Old Question. Introductory note to 'Can the Social Contract be Signed by an Invisible Hand'.Bernd Lahno - 2013 - RMM 4:39-43.details
Moral Contractarianism in Normative Ethics
Imitation and conventional communication.Richard Moore - 2013 - Biology and Philosophy 28 (3):481-500.details
To the extent that language is conventional, non-verbal individuals, including human infants, must participate in conventions in order to learn to use even simple utterances of words. This raises the question of which varieties of learning could make this possible. In this paper I defend Tomasello's (The cultural origins of human cognition. Harvard UP, Cambridge, 1999, Origins of human communication. MIT, Cambridge, 2008) claim that knowledge of linguistic conventions could be learned through imitation. This is possible because Lewisian accounts of (...) convention have overstated what one must know to participate in conventions; and because the required knowledge could be learned imitatively. The imitation claim that I defend is consistent with what we know about both the proliferation of conventional behaviours in human children, who are skilful imitators, and the comparative absence of such behaviours in non-human great apes, who are poor at imitative learning. (shrink)
Evolution of Language in Philosophy of Language
Linguistic Convention in Philosophy of Language
Social norms and game theory: harmony or discord?Cédric Paternotte & Jonathan Grose - 2013 - British Journal for the Philosophy of Science 64 (3):551-587.details
Recent years have witnessed an increased number of game-theoretic approaches to social norms, which apparently share some common vocabulary and methods. We describe three major approaches of this kind (due to Binmore, Bicchieri and Gintis), before comparing them systematically on five crucial themes: generality of the solution, preference transformation, punishment, epistemic conditions and type of explanation. This allows us to show that these theories are, by and large, less compatible than they seem. We then argue that those three theories struggle (...) to account for three phenomena pertaining to social norms (namely context-dependence, conflicting norms and self-evidence), with which any complete game-theoretic account should in principle be able to deal. (shrink)
Game Theory, Misc in Philosophy of Action
Coming to terms: Quantifying the benefits of linguistic coordination.Riccardo Fusaroli, Bahador Bahrami, Karsten Olsen, Andreas Roepstorff, Geraint Rees, Chris Frith & Kristian Tylén - 2012 - Psychological Science 23 (8):931-939.details
Sharing a public language facilitates particularly efficient forms of joint perception and action by giving interlocutors refined tools for directing attention and aligning conceptual models and action. We hypothesized that interlocutors who flexibly align their linguistic practices and converge on a shared language will improve their cooperative performance on joint tasks. To test this prediction, we employed a novel experimental design, in which pairs of participants cooperated linguistically to solve a perceptual task. We found that dyad members generally showed a (...) high propensity to adapt to each other's linguistic practices. However, although general linguistic alignment did not have a positive effect on performance, the alignment of particular task-relevant vocabularies strongly correlated with collective performance. In other words, the more dyad members selectively aligned linguistic tools fit for the task, the better they performed. Our work thus uncovers the interplay between social dynamics and sensitivity to task affordances in successful cooperation. (shrink)
Joint Attention in Philosophy of Mind
Identification in Games: Changing Places.Darrell Patrick Rowbottom - 2012 - Erkenntnis 77 (2):197-206.details
This paper offers a novel 'changing places' account of identification in games, where the consequences of role swapping are crucial. First, it illustrates how such an account is consistent with the view, in classical game theory, that only outcomes (and not pathways) are significant. Second, it argues that this account is superior to the 'pooled resources' alternative when it comes to dealing with some situations in which many players identify. Third, it shows how such a 'changing places' account can be (...) used in games where some of the players identify with one another, but others do not. Finally, it illustrates how the model can handle the notion that identification comes in degrees. (shrink)
Hume's Natural History of Justice.Mark Collier - 2011 - In C. Taylor & S. Buckle (eds.), Hume and the Enlightenment. Pickering & Chatto. pp. 131-142.details
In Book III, Part 2 of the Treatise, Hume presents a natural history of justice. Self-interest clearly plays a central role in his account; our ancestors invented justice conventions, he maintains, for the sake of reciprocal advantage. But this is not what makes his approach so novel and attractive. Hume recognizes that prudential considerations are not sufficient to explain how human beings – with our propensities towards temporal discounting and free-riding – could have established conventions for social exchange and collective (...) action in commercial societies. This leads him to develop an innovative account of the role that emotional aversions play in establishing trust between strategically rational agents. (shrink)
Evolution of Morality in Normative Ethics
Hume: Justice in 17th/18th Century Philosophy
Hume: Philosophy of Mind in 17th/18th Century Philosophy
Justice in Social and Political Philosophy
Neuroethics in Applied Ethics
Property Rights in Social and Political Philosophy
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A dynamic logic of agency II: Deterministic dla {\mathcal{dla}} , coalition logic, and game theory.Emiliano Lorini - 2010 - Journal of Logic, Language and Information 19 (3):327-351.details
We continue the work initiated in Herzig and Lorini (J Logic Lang Inform, in press) whose aim is to provide a minimalistic logical framework combining the expressiveness of dynamic logic in which actions are first-class citizens in the object language, with the expressiveness of logics of agency such as STIT and logics of group capabilities such as CL and ATL. We present a logic called ( Deterministic Dynamic logic of Agency ) which supports reasoning about actions and joint actions of (...) agents and coalitions, and agentive and coalitional capabilities. In it is supposed that, once all agents have selected a joint action, the effect of this joint action is deterministic. In order to assess we prove that it embeds Coalition Logic. We then extend with modal operators for agents' preferences, and show that the resulting logic is sufficiently expressive to capture the game-theoretic concepts of best response and Nash equilibrium. (shrink)
Review of Brian Skyrms, Signals: Evolution, Learning, and Information. [REVIEW]Cedric Paternotte - 2010 - Notre Dame Philosophical Reviews 2010 (11).details
Rule-following and Coordination: A Game-theoretic Perspective.Giacomo Sillari - 2010 - Rivista di Filosofia 101 (3):355-386.details
Norms as reasons for Action.Bernd Lahno - 2009 - Archiv für Rechts- und Sozialphilosophie 95 (4):563-578.details
Social norms are based on social standards. The relevant standards come in two forms. Compliance with social standards of evaluation may be understood as goal-oriented behavior under the constraints of external and internal sanctions. Compliance with norms, which directly refer to specific ways of conduct, may not. Therefore, although norm-guided behavior may be consistent with utility maximizing, no satisfying account of norm compliance can be given within a Rational Choice framework or any other framework solely based on instrumental rationality.
Collective Intentions and Game Theory.Raimo Tuomela - 2009 - Journal of Philosophy 106 (5):292-300.details
Activity and Convention.Richard Alterman - 2008 - Topoi 27 (1-2):127-138.details
This paper develops Lewis' notion of convention within a framework that mixes cognitive science with some more social theories of activity like distributed cognition and activity theory. The close examination of everyday situations of convention-based activity will produce some interesting issues for a cognitive theory of behavior. Uncertainty, dynamics, and the complexities of the performance of convention-based activities that are distributed over time and/or place, are driving factors in the analysis that is presented. How the actors reason and manage their (...) collaboration is characterized as pragmatic action. During the course of recurrent activities, the participants adapt previously learned convention-based activities to new circumstances. The coordinating representations that are a part of the design of the context mediate parts of the activity. As they act, the participants learn. (shrink)
Social Ontology as Convention.Mark H. Bickhard - 2008 - Topoi 27 (1-2):139-149.details
I will argue that social ontology is constituted as hierarchical and interlocking conventions of multifarious kinds. Convention, in turn, is modeled in a manner derived from that of David K. Lewis. Convention is usually held to be inadequate for models of social ontologies, with one primary reason being that there seems to be no place for normativity. I argue that two related changes are required in the basic modeling framework in order to address this (and other) issue(s): (1) a shift (...) to an intentional model—among other reasons, in order to account for normativity—and (2) moving away from the belief-desire, propositional attitude, framework for understanding the intentional realm toward an interactive, pragmatic model of intentionality. These shifts provide natural approaches to: (1) understanding the normativities of social realities; (2) the sense in which social ontology is often constituted in implicit relations among the participants rather than elaborated and iterated explicit beliefs and desires; (3) and language. (shrink)
Do Conventions Need to Be Common Knowledge?Ken Binmore - 2008 - Topoi 27 (1-2):17-27.details
Do conventions need to be common knowledge in order to work? David Lewis builds this requirement into his definition of a convention. This paper explores the extent to which his approach finds support in the game theory literature. The knowledge formalism developed by Robert Aumann and others militates against Lewis's approach, because it shows that it is almost impossible for something to become common knowledge in a large society. On the other hand, Ariel Rubinstein's Email Game suggests that coordinated action (...) is no less hard for rational players without a common knowledge requirement. But an unnecessary simplifying assumption in the Email Game turns out to be doing all the work, and the current paper concludes that common knowledge is better excluded from a definition of the conventions that we use to regulate our daily lives. (shrink)
Common Knowledge of Rationality in Extensive Games.Boudewijn de Bruin - 2008 - Notre Dame Journal of Formal Logic 49 (3):261-280.details
We develop a logical system that captures two different interpretations of what extensive games model, and we apply this to a long-standing debate in game theory between those who defend the claim that common knowledge of rationality leads to backward induction or subgame perfect (Nash) equilibria and those who reject this claim. We show that a defense of the claim à la Aumann (1995) rests on a conception of extensive game playing as a one-shot event in combination with a principle (...) of rationality that is incompatible with it, while a rejection of the claim à la Reny (1988) assumes a temporally extended, many-moment interpretation of extensive games in combination with implausible belief revision policies. In addition, the logical system provides an original inductive and implicit axiomatization of rationality in extensive games based on relations of dominance rather than the usual direct axiomatization of rationality as maximization of expected utility. (shrink)
The Unconventional, but Conventionalist, Legacy of Lewis's "Convention".Olivier Favereau - 2008 - Topoi 27 (1-2):115-126.details
The philosopher David Lewis is credited by many social scientists, including mainstream economists, with having founded the modern (game-theoretical) approach to conventions, viewed as solutions to recurrent coordination problems. Yet it is generally ignored that he revised his approach, soon after the publication of his well-known book. I suggest that this revision has deep implications (probably not perceived by Lewis himself) on the analytical links between coordination, uncertainty and rationality. Thinking anew about these issues leads me to map out an (...) alternative social scientific research programme. The traditional ontological equipment of methodological individualism should be reinforced in order to admit the existence of an "intersubjective" world beside the two familiar worlds: the "objective" world of observable things, and the "subjective" world of expectations and individual beliefs. In particular, language becomes necessary to understand coordination via conventions, rather than the other way round. That has led a group of institutionalist economists and pragmatist sociologists to develop an enlarged model of rationality, no longer isolated from questions of coordination and values. This model is the basis for the "Economics of Conventions". (shrink)
Collective Mentality in Philosophy of Mind
Social convention revisited.Margaret Gilbert - 2008 - Topoi (1-2):5-16.details
This article will compare and contrast two very different accounts of convention: the game-theoretical account of Lewis in Convention, and the account initially proposed by Margaret Gilbert (the present author) in chapter six of On Social Facts, and further elaborated here. Gilbert's account is not a variant of Lewis's. It was arrived at in part as the result of a detailed critique of Lewis's account in relation to a central everyday concept of a social convention. An account of convention need (...) not be judged by that standard. Perhaps it reveals the nature of an important phenomenon. Looked at in that light, these very different accounts are not incompatible. Indeed, neither should be ignored if one is seeking to understand the way in which human beings arrive at some degree of social order. (shrink)
Philosophy of Language, Misc in Philosophy of Language
Social Practices in Social and Political Philosophy
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CommonCrawl
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CLRS Solutions 24.1 The Bellman-Ford algorithm
24.1 The Bellman-Ford algorithm 24.1 The Bellman-Ford algorithm Table of contents
Run the Bellman-Ford algorithm on the directed graph of Figure 24.4, using vertex $z$ as the source. In each pass, relax edges in the same order as in the figure, and show the $d$ and $\pi$ values after each pass. Now, change the weight of edge $(z, x)$ to $4$ and run the algorithm again, using $s$ as the source.
Using vertex $z$ as the source:
$d$ values:
$$ \begin{array}{cccccc} s & t & x & y & z \\ \hline \infty & \infty & \infty & \infty & 0 \\ 2 & \infty & 7 & \infty & 0 \\ 2 & 5 & 7 & 9 & 0 \\ 2 & 5 & 6 & 9 & 0 \\ 2 & 4 & 6 & 9 & 0 \end{array} $$
$\pi$ values:
$$ \begin{array}{cccccc} s & t & x & y & z \\ \hline \text{NIL} & \text{NIL} & \text{NIL} & \text{NIL} & \text{NIL} \\ z & \text{NIL} & z & \text{NIL} & \text{NIL} \\ z & x & z & s & \text{NIL} \\ z & x & y & s & \text{NIL} \\ z & x & y & s & \text{NIL} \end{array} $$
Changing the weight of edge $(z, x)$ to $4$:
$$ \begin{array}{cccccc} s & t & x & y & z \\ \hline 0 & \infty & \infty & \infty & \infty \\ 0 & 6 & \infty & 7 & \infty \\ 0 & 6 & 4 & 7 & 2 \\ 0 & 2 & 4 & 7 & 2 \\ 0 & 2 & 4 & 7 & -2 \end{array} $$
$$ \begin{array}{cccccc} s & t & x & y & z \\ \hline \text{NIL} & \text{NIL} & \text{NIL} & \text{NIL} & \text{NIL} \\ \text{NIL} & s & \text{NIL} & s & \text{NIL} \\ \text{NIL} & s & y & s & t \\ \text{NIL} & x & y & s & t \\ \text{NIL} & x & y & s & t \end{array} $$
Consider edge $(z, x)$, it'll return $\text{FALSE}$ since $x.d = 4 > z.d + w(z, x) = -2 + 4$.
Prove Corollary 24.3.
Suppose there is a path from $s$ to $v$. Then there must be a shortest such path of length $\delta(s, v)$. It must have finite length since it contains at most $|V| - 1$ edges and each edge has finite length. By Lemma 24.2, $v.d = \delta(s, v) < \infty$ upon termination.
On the other hand, suppose $v.d < \infty$ when $\text{BELLMAN-FORD}$ terminates. Recall that $v.d$ is monotonically decreasing throughout the algorithm, and $\text{RELAX}$ will update $v.d$ only if $u.d + w(u, v) < v.d$ for some $u$ adjacent to $v$. Moreover, we update $v.\pi = u$ at this point, so $v$ has an ancestor in the predecessor subgraph. Since this is a tree rooted at $s$, there must be a path from $s$ to $v$ in this tree. Every edge in the tree is also an edge in $G$, so there is also a path in $G$ from $s$ to $v$.
Given a weighted, directed graph $G = (V, E)$ with no negative-weight cycles, let $m$ be the maximum over all vertices $v \in V$ of the minimum number of edges in a shortest path from the source $s$ to $v$. (Here, the shortest path is by weight, not the number of edges.) Suggest a simple change to the Bellman-Ford algorithm that allows it to terminate in $m + 1$ passes, even if $m$ is not known in advance.
Modify the Bellman-Ford algorithm so that it sets $v.d$ to $-\infty$ for all vertices $v$ for which there is a negative-weight cycle on some path from the source to $v$.
BELLMAN-FORD'(G, w, s)
INITIALIZE-SINGLE-SOURCE(G, s)
for i = 1 to |G.V| - 1
for each edge (u, v) ∈ G.E
RELAX(u, v, w)
for each edge(u, v) ∈ G.E
if v.d > u.d + w(u, v)
for each vertex v ∈ marked vertices
FOLLOW-AND-MARK-PRED(v)
if v != NIL and v.d != -∞
v.d = -∞
FOLLOW-AND-MARK-PRED(v.π)
After running $\text{BELLMAN-FORD}'$, run $\text{DFS}$ with all vertices on negative-weight cycles as source vertices. All the vertices that can be reached from these vertices should have their $d$ attributes set to $-\infty$.
Let $G = (V, E)$ be a weighted, directed graph with weight function $w : E \rightarrow \mathbb R$. Give an $O(VE)$-time algorithm to find, for each vertex $v \in V$, the value $\delta^*(v) = \min_{u \in V} \{\delta(u, v)\}$.
if v.d > min(w(u, v), w(u, v) + u.d)
v.d = min(w(u, v), w(u, v) + u.d)
v.π = u.π
Suppose that a weighted, directed graph $G = (V, E)$ has a negative-weight cycle. Give an efficient algorithm to list the vertices of one such cycle. Prove that your algorithm is correct.
Based on exercise 24.1-4, $\text{DFS}$ from a vertex $u$ that $u.d = -\infty$, if the weight sum on the search path is negative and the next vertex is $\text{BLACK}$, then the search path forms a negative-weight cycle.
Previous 23-4 Alternative minimum-spanning-tree algorithms
Next 24.2 Single-source shortest paths in directed acyclic graphs
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CommonCrawl
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Abstract: The method of Galerkin approximations is employed to prove the existence of a strong global (in time) solution of a doubly nonlinear parabolic equation in an unbounded domain. The second integral identity is established for Galerkin approximations, and passing to the limit in it an estimate for the decay rate of the norm of the solution from below is obtained. The estimates characterizing the decay rate of the solution as $x\to\infty$ obtained here are used to derive an upper bound for the decay rate of the solution with respect to time; the resulting estimate is pretty close to the lower one.
Keywords: doubly nonlinear parabolic equation, rate of decay of the solution, lower estimate, existence of a strong global (in time) solution.
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CommonCrawl
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Category Archives: Diamond
Highly specialized; only a subset of mathematicians who specialize in the area will have the background to follow these posts.
The Springer Correspondence, Part I: The Flag Variety
Posted on January 11, 2015 by Maria Gillespie
In prior posts, we've seen that the irreducible representations of the symmetric group $S_n$ are in one-to-one correspondence with the partitions of $n$, and the Schur functions give an elegant encoding of their characters as symmetric polynomials. Now we can dive a bit deeper: a geometric construction known as the Springer resolution allows us to obtain all the irreducible representations of $S_n$ geometrically, and as a side bonus give natural graded representations that will allow us to define a $q$-analog of the Schur functions known as the Hall-Littlewood polynomials.
Quite a mouthful of terminology. Let's start at the beginning.
The Classical Flag Variety
When you think of a flag, you might imagine something like this:
Roughly speaking, a flag on a flagpole consists of:
a point (the bulbous part at the top of the pole),
a line passing through that point (the pole),
a plane passing through that line (the plane containing the flag), and
space to put it in.
Mathematically, this is the data of a complete flag in three dimensions. However, higher-dimensional beings would require more complicated flags. So in general, a complete flag in $n$-dimensional space $\mathbb{C}^n$ is a chain of vector spaces of each dimension from $0$ to $n$, each containing the previous:
$$0=V_0\subset V_1 \subset V_2 \subset \cdots \subset V_n=\mathbb{C}^n$$
with $\dim V_i=i$ for all $i$.
(Our higher-dimensional flag-waving imaginary friends are living in a world of complex numbers because $\mathbb{C}$ is algebraically closed and therefore easier to work with. However, one could define the flag variety similarly over any field $k$.)
Variety Structure
Now that we've defined our flags, let's see what happens when we wave them around continuously in space. It turns out we get a smooth algebraic variety!
Indeed, the set of all possible flags in $\mathbb{C}^n$ forms an algebraic variety of dimension $n(n-1)$ (over $\mathbb{R}$), covered by open sets similar to the Schubert cells of the Grassmannian. In particular, given a flag $\{V_i\}_{i=1}^n$, we can choose $n$ vectors $v_1,\ldots,v_n$ such that the span of $v_1,\ldots,v_i$ is $V_i$ for each $i$, and list the vectors $v_i$ as row vectors of an $n\times n$ matrix. We can then perform certain row reduction operations to form a different basis $v_1^\prime,\ldots,v_n^\prime$ that also span the subspaces of the flag, but whose matrix is in the following canonical form: it has $1$'s in a permutation matrix shape, $0$'s to the left and below each $1$, and arbitrary complex numbers in all other entries.
For instance, say we start with the flag in three dimensions generated by the vectors $\langle 0,2,3\rangle$, $\langle 1, 1, 4\rangle$, and $\langle 1, 2, -3\rangle$. The corresponding matrix is $$\left(\begin{array}{ccc} 0 & 2 & 3 \\ 1 & 1 & 4 \\ 1 & 2 & -3\end{array}\right).$$ We start by finding the leftmost nonzero element in the first row and scale that row so that this element is $\newcommand{\PP}{\mathbb{P}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\ZZ}{\mathbb{Z}}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\Fl}{F\ell}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\inv}{inv}1$. Then subtract multiples of this row from the rows below it so that all the entries below that $1$ are $0$. Continue the process on all further rows:
$$\left(\begin{array}{ccc} 0 & 2 & 3 \\ 1 & 1 & 4 \\ 1 & 2 & -3\end{array}\right) \to \left(\begin{array}{ccc} 0 & 1 & 1.5 \\ 1 & 0 & 2.5 \\ 1 & 0 & -6\end{array}\right)\to \left(\begin{array}{ccc} 0 & 1 & 1.5 \\ 1 & 0 & 2.5 \\ 0 & 0 & 1\end{array}\right)$$
It is easy to see that this process does not change the flag formed by the partial row spans, and that any two matrices in canonical form define different flags. So, the flag variety is covered by $n!$ open sets given by choosing a permutation and forming the corresponding canonical form. For instance, one such open set in the $5$-dimensional flag variety is the open set given by all matrices of the form
$$\left(\begin{array}{ccccc}
0 & 1 & \ast & \ast & \ast \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & \ast & 0 \\
0 & 0 & 0 & 1 & 0 \end{array}\right)$$
We call this open set $X_{45132}$ because it corresponds to the permutation matrix formed by placing a $1$ in the $4$th column from the right in the first row, in the $5$th from the right in the second row, and so on. The maximum number of $\ast$'s we can have in such a matrix is when the permutation is $w_0=n(n-1)\cdots 3 2 1$, in which case the dimension of the open set $X_{12\cdots n}$ is $n(n-1)/2$ over $\CC$ — or $n(n-1)$ over $\RR$, since $\CC$ is two-dimensional over $\RR$. In general, the number of $\ast$'s in the set $X_w$ is the inversion number $\inv(w)$, the number of pairs of entries of $w$ which are out of order.
Finally, in order to paste these disjoint open sets together to form a smooth manifold, we consider the closures of the sets $X_w$ as a disjoint union of other $X_w$'s. The partial ordering in which $\overline{X_w}=\sqcup_{v\le w} X_v$ is called the Bruhat order, a famous partial ordering on permutations. (For a nice introduction to Bruhat order, one place to start is Yufei Zhao's expository paper on the subject.)
Intersection Cohomology
Now suppose we wish to answer incidence questions about our flags: which flags satisfy certain given constraints? As in the case of the Grassmannians, this boils down to understanding how the Schubert cells $X_w$ intersect. This question is equaivalent to studying the cohomology ring of the flag variety $\Fl_n$ over $\mathbb{Z}$, where we consider the Schubert cells as forming a cell complex structure on the flag variety.
The cohomology ring $H^\ast(\Fl_n)$, as it turns out, is the coinvariant ring that we discussed in the last post! For full details I will refer the interested reader to Fulton's book on Young tableaux. To give the reader the general idea here, the Schubert cell $X_w$ can be thought of as a cohomology class in $H^{2i}(\Fl_n)$ where $i=\inv(w)$. We call this cohomology class $\sigma_w$, and note that for the transpositions $s_i$ formed by swapping $i$ and $i+1$, we have $\sigma_{s_i}\in H^2(\Fl_n)$. It turns out that setting $x_i=\sigma_i-\sigma_{i+1}$ for $i\le n-1$ and $x_n=-\sigma_{s_{n-1}}$ gives a set of generators for the cohomology ring, and in fact $$H^\ast(\Fl_n)=\mathbb{Z}[x_1,\ldots,x_n]/(e_1,\ldots,e_n)$$ where $e_1,\ldots,e_n$ are the elementary symmetric polynomials in $x_1,\ldots,x_n$.
Posted in Diamond, Gemstones | 1 Reply
Digging deeper: The isotypic components
Posted on July 27, 2014 by Maria Gillespie
In last week's post, we made use of the coinvariant ring $$\mathbb{C}[x_1,\ldots,x_n]/I$$ where $I=(p_1,\ldots,p_n)$ is the ideal generated by the positive-degree homogeneous $S_n$-invariants (symmetric polynomials). We saw that this was an $S_n$-module with Hilbert series $(n)_q!$, and claimed that it was the regular representation.
Let's see why that is, and see if we can understand where the irreducible components occur.
More precisely, our goal is to understand the series $$\sum_{d} H_{\chi^\mu}(d)q^d$$ where $H_{\chi^\mu}(d)$ is the number of copies of the $\mu$th irreducible representation of $S_n$ occurring in the $d$th degree component of $\mathbb{C}[x_1,\ldots,x_n]/I$. In Stanley's paper on invariants of finite groups, he states without proof the answer as the following "unpublished result of Lusztig":
Let $G$ be the group of all $m \times m$ permutation matrices, and let $\chi$ be the irreducible character of $G$ corresponding to the partition $\mu$ of $m$. Then $H_{\chi}(n)$ is equal to the number of standard Young tableaux $T$ of shape $\mu$ such that $n$ is equal to the sum of those entries $i$ of $T$ for which $i$ appears in a column to the left of $i+1$.
To prove this, let's start with the identity we showed last time, using boldface $\mathbf{x}$ to denote $x_1,\ldots,x_n$ as a shorthand:
$$\mathbb{C}[\mathbf{x}]=\Lambda_{\mathbb{C}}(\mathbf{x})\otimes_{\mathbb{C}[S_n]}\mathbb{C}[\mathbf{x}]/I$$
Since $\Lambda_{\mathbb{C}}(\mathbf{x})$, the ring of symmetric functions, consists entirely of copies of the trivial representation of $S_n$, we see that the irreducible components of type $\chi^\mu$ in degree $d$ on the right hand side come from those of that type in $\mathbb{C}[\mathbf{x}]/I$ of degree $d-k$, tensored with the trivial representations in degree $k$ in $\Lambda$, for some $k$. Moreover, there are $p_n(d)$ copies of the trivial representation in the $d$th degree in $\Lambda$ for all $d$, where $p_n(d)$ is the number of partitions of $d$ into parts of size at most $n$. (One can use the elementary or power sum symmetric function bases to see this.) From this we obtain the following series identity:
$$\left(\sum \left\langle (\mathbb{C}[\mathbf{x}])_d,\chi^\mu \right\rangle q^d\right)= \left(\sum p_n(d)q^d\right)\cdot \left(\sum H_{\chi^\mu}(d) q^d\right)$$
To simplify the left hand side, we can use the generalized version of Molien's theorem for isotypic components (see here.) This gives us
$$\sum \left\langle (\mathbb{C}[\mathbf{x}])_d,\chi^\mu \right\rangle q^d=\frac{1}{n!}\sum_{\pi\in S_n}\frac{\overline{\chi^\mu}(\pi)}{\prod (1-q^{c_i(\pi)})}$$ where the $c_i(\pi)$'s are the cycle lengths of $\pi$.
(See this post for details on the above simplification in the case of the trivial character. The case of a general $\chi^\mu$ is analogous.)
If we group the permutations $\pi$ in the sum above according to cycle type (i.e. by conjugacy class), and use the fact that characters of $S_n$ are integers and hence $\overline{\chi^\mu}=\chi^\mu$, we have $$\sum \left\langle (\mathbb{C}[\mathbf{x}])_d,\chi^\mu \right\rangle q^d=\frac{1}{n!}\sum_{\lambda\vdash n} \frac{n!}{z_\lambda}\frac{\chi^\mu(\lambda)}{\prod_i (1-q^{\lambda_i})}.$$ Here $z_\lambda$ are the numbers such that $n!/z_\lambda$ is the size of the conjugacy class corresponding to the partition $\lambda$. It is not hard to see that this simplifies to a specialization of the power sum symmetric functions: $$\sum \frac{\chi^\mu(\lambda)}{z_\lambda} p_\lambda(1,q,q^2,\ldots)$$
Finally, by the representation-theoretic definition of the Schur functions, we see that this is simply $$s_\mu(1,q,q^2,\ldots).$$
Substituting for the left hand side of our original equation, we now have $$s_\lambda(1,q,q^2,\ldots)=\left(\sum p_n(d) q^d\right)\cdot \left(\sum H_{\chi^\mu}(d) q^d\right).$$ We can simplify this further by using the series identity $$\sum p_n(d) q^d=\frac{1}{(1-q)(1-q^2)\cdots(1-q^n)}.$$ In addition, there is a well-known identity (see also Enumerative Combinatorics vol. 2, Proposition 7.19.11) $$s_\mu(1,q,q^2,\ldots)=\frac{\sum_T q^{\operatorname{maj} T}}{(1-q)(1-q^2)\cdots (1-q^n)}$$ where the sum ranges over all standard Young tableaux $T$ of shape $\mu$, and where $\operatorname{maj} T$ denotes the sum of all entries $i$ of $T$ that occur in a higher row than $i+1$ (written in English notation).
This does it: putting everything together and solving for $\sum H_{\chi^\mu}(d) q^d$, we obtain $$\sum H_{\chi^\mu}(d) q^d=\sum_{T}q^{\operatorname{maj} T},$$ which is just about equivalent to Lusztig's claim. (The only difference is whether we are looking at the rows or the columns that $i$ and $i+1$ are in. There must have been a typo, because the two are not the same $q$-series for the shape $(3,1)$. Replacing "column to the left of" by "row above" or replacing $\mu$ by its conjugate would fix the theorem statement above.)
One final consequence of the formulas above is that it is easy to deduce that the ring of coinvariants, $\mathbb{C}[\mathbf{x}]/I$, is isomorphic to the regular representation of $S_n$. Indeed, setting $q=1$ we see that the total number of copies of the irreducible representation corresponding to $\mu$ is equal to the number of standard Young tableaux of shape $\mu$, giving us the regular representation.
Acknowledgments: The above techniques were shown to me by Vic Reiner at a recent conference. Thanks also to Federico Castillo for many discussions about the ring of coinvariants.
Posted in Diamond | 2 Replies
Molien's Theorem and symmetric functions
Posted on August 21, 2013 by Maria Gillespie
My colleague David Harden recently pointed me to Molien's theorem, a neat little fact about the invariant polynomials under the action by a finite group. It turns out that this has a nice interpretation in the case of the symmetric group $S_n$ that brings in some nice combinatorial and group-theoretic arguments.
The general version of Molien's theorem can be stated thus: Suppose we have a finite subgroup $G$ of the general linear group $GL_n(\mathbb{C})$. Then $G$ acts on the polynomial ring $R=\mathbb{C}[x_1,\ldots,x_n]$ in the natural way, that is, by replacing the column vector of variables $(x_1,\ldots,x_n)$ with their image under left matrix multiplication by $G$.
Let $R^G$ be the invariant space under this action. Then $R$ is graded by degree; that is, for each nonnegative integer $k$, the space $R_k^G$ of $G$-invariant homogeneous polynomials of degree $k$ are a finite dimensional subspace of $R^G$, and $R^G$ is the direct sum of these homogeneous components. What is the size (dimension) of these homogeneous components?
If $d_k$ denotes the dimension of the $k$th piece, then Molien's theorem states that the generating function for the $d_k$'s is given by
$$\sum_{k\ge 0} d_k t^k=\frac{1}{|G|}\sum_{M\in G} \frac{1}{\det(I-tM)}$$ where $I$ is the $n\times n$ identity matrix.
There is a nice exposition of the proof (in slightly more generality) in this paper of Richard Stanley, which makes use of some basic facts from representation theory. Rather than go into the proof, let's look at the special case $G=S_n$, namely the set of all permutation matrices in $GL_n$.
Specialization at $G=S_n$
In this case, the invariant space $R^{S_n}$ is simply the space of symmetric polynomials in $x_1,\ldots,x_n$, and the $k$th graded piece consists of the degree-$k$ homogeneous symmetric polynomials. But we know exactly how many linearly independent homogeneous symmetric polynomials of degree $k$ there can be – as shown in my previous post, the monomial symmetric polynomials $m_\lambda$, where $\lambda$ is any partition of $k$, form a basis of this space in the case that we have infinitely many variables. Since we only have $n$ variables, however, some of these are now zero, namely those for which $\lambda$ has more than $n$ parts. The nonzero $m_\lambda$'s are still linearly independent, so the dimension of the $k$th graded piece in this case is $p(k,n)$, the number of partitions of $k$ into at most $n$ parts.
Notice that by considering the conjugate of each partition, we see that the number of partitions of $k$ into at most $n$ parts is equal to the number of partitions of $k$ that use parts of size at most $n$. It is not hard to see that the generating function for $p(k,n)$ is therefore
$$\sum_{k\ge 0}p(k,n)t^k=\frac{1}{(1-t)(1-t^2)\cdots (1-t^n)}.$$
Molien's theorem says that this generating function should also be equal to $$\frac{1}{n!}\sum_{M\in S_n}\frac{1}{\det(I-tM)}$$ where we use the somewhat sloppy notation $M\in S_n$ to indicate that $M$ is an $n\times n$ permutation matrix. What are these determinants? Well, suppose $M$ corresponds to a permutation with cycle type $\lambda$, that is, when we decompose the permutation into cycles the lengths of the cycles are $\lambda_1,\ldots,\lambda_r$ in nonincreasing order. Then notice that, up to simultaneous reordering of the rows and columns, $I-tM$ is a block matrix with blocks of sizes $\lambda_1,\ldots,\lambda_r$. The determinant of a block of size $\lambda_i$ is easily seen to be $1-t^{\lambda_i}$. For instance
$$\det \left(\begin{array}{ccc} 1 & -t & 0 \\ 0& 1 & -t \\ -t & 0 & 1\end{array}\right)=1-t^3,$$ and in general, the determinant of such a block will have contributions only from the product of the 1's down the diagonal and from the product of the off-diagonal $-t$'s; all other permutations have a $0$ among the corresponding matrix entries. The sign on the product of $t$'s is always negative since either $\lambda_i$ is odd, in which case the cyclic permutation of length $\lambda_i$ is even, or $\lambda_i$ is even, in which case the permutation is odd. Hence, the determinant of each block is $1-t^{\lambda_i}$, and the entire determinant is
$$\det (I-tM)=\prod_i (1-t^{\lambda_i}).$$
So, our summation becomes
$$\frac{1}{n!}\sum_{\pi\in S_n} \frac{1}{\prod_{\lambda_i\in c(\pi)} (1-t^{\lambda_i})}$$ where $c(\pi)$ denotes the cycle type of a permutation $\pi$. Already we have an interesting identity; we now know this series is equal to
$$\frac{1}{(1-t)(1-t^2)\cdots (1-t^n)}.$$ But can we prove it directly?
It turns out that the equality of these two series can be viewed as a consequence of Burnside's Lemma. In particular, consider the action of the symmetric group on the set $X$ of weak compositions of $k$ having $n$ parts, that is, an ordered $n$-tuple of nonnegative integers (possibly $0$) that add up to $k$. Then Burnside's lemma states that the number of orbits under this action, which correspond to the partitions of $k$ having at most $n$ parts, is equal to
$$\frac{1}{n!}\sum_{\pi \in S_n} |X^\pi|$$ where $X^\pi$ is the collection of weak compositions which are fixed under permuting the entries by $\pi$. We claim that this is the coefficient of $t^k$ in
$$\frac{1}{n!}\sum_{\pi\in S_n} \frac{1}{\prod_{\lambda_i\in c(\pi)} (1-t^{\lambda_i})}$$ hence showing that the two generating functions are equal.
To see this, note that if $\pi\in S_n$ has cycle type $\lambda$, then $X^\pi$ consists of the weak compositions which have $\lambda_1$ of their parts equal to each other, $\lambda_2$ other parts equal to each other, and so on. Say WLOG that the first $\lambda_1$ parts are all equal, and the second $\lambda_2$ are equal, and so on. Then the first $\lambda_1$ parts total to some multiple of $\lambda_1$, and the next $\lambda_2$ total to some multiple of $\lambda_2$, and so on, and so the total number of such compositions of $k$ is the coefficient of $t^k$ in the product $$\frac{1}{\prod_{\lambda_i\in c(\pi)} (1-t^{\lambda_i})}.$$ Averaging over all $\pi\in S_n$ yields the result.
Posted in Diamond | Leave a reply
A q-analog of the decomposition of the regular representation of the symmetric group
Posted on March 11, 2013 by Maria Gillespie
It is a well-known fact of representation theory that, if the irreducible representations of a finite group $\DeclareMathOperator{\maj}{maj} \DeclareMathOperator{\sh}{sh} G$ are $V_1,\ldots,V_m$, and $R$ is the regular representation formed by $G$ acting on itself by left multiplication, then
$$R=\bigoplus_{i=1}^{m} (\dim V_i) \cdot V_i$$ is its decomposition into irreducibles.
I've recently discovered a $q$-analog of this fact for $G=S_n$ that is a simple consequence of some known results in symmetric function theory.
In Enumerative Combinatorics, Stanley defines a generalization of the major index on permutations to standard tableaux. For a permutation $$w=w_1,\ldots,w_n$$ of $1,\ldots,n$, a descent is a position $i$ such that $w_i>w_{i+1}$. For instance, $52413$ has two descents, in positions $1$ and $3$. The major index of $w$, denoted $\maj(w)$, is the sum of the positions of the descents, in this case $$\maj(52413)=1+3=4.$$
To generalize this to standard Young tableaux, notice that $i$ is a descent of $w$ if and only if the location of $i$ occurs after $i+1$ in the inverse permutation $w^{-1}$. With this as an alternative notion of descent, we define a descent of a tableau $T$ to be a number $i$ for which $i+1$ occurs in a lower row than $i$. In fact, this is precisely a descent of the inverse of the reading word of $T$, the word formed by reading the rows of $T$ from left to right, starting from the bottom row.
As an example, the tableau $T$ below has two descents, $2$ and $4$, since $3$ and $5$ occur in lower rows than $2$ and $4$ respectively:
So $\maj(T)=2+4=6$. Note that its reading word $5367124$, and the inverse permutation is $5627134$, which correspondingly has descents in positions $2$ and $4$.
(This is a slightly different approach to the major index than taken by Stanley, who used a reading word that read the columns from bottom to top, starting at the leftmost column. The descents remain the same in either case, since both reading words Schensted insert to give the same standard Young tableau.)
Now, the major index for tableaux gives a remarkable specialization of the Schur functions $s_\lambda$. As shown in Stanley's book, we have $$s_\lambda(1,q,q^2,q^3,\ldots)=\frac{\sum_{T} q^{\maj(T)}}{(1-q)(1-q^2)\cdots(1-q^n)}$$ where the sum is over all standard Young tableaux $T$ of shape $\lambda$. When I came across this fact, I was reminded of a similar specialization of the power sum symmetric functions. It is easy to see that
$$p_\lambda(1,q,q^2,q^3,\ldots)=\prod_{i}\frac{1}{1-q^{\lambda_i}},$$ an identity that comes up in defining a $q$-analog of the Hall inner product in the theory of Hall-Littlewood symmetric functions. In any case, the power sum symmetric functions are related to the Schur functions via the irreducible characters $\chi_\mu$ of the symmetric group $S_n$, and so we get
\begin{eqnarray*}
p_\lambda(1,q,q^2,\ldots) &=& \sum_{|\mu|=n} \chi_{\mu}(\lambda) s_{\mu}(1,q,q^2,\ldots) \\
\prod_{i} \frac{1}{1-q^{\lambda_i}} &=& \sum_{\mu} \chi_{\mu}(\lambda) \frac{\sum_{T\text{ shape }\mu} q^{\maj(T)}}{(1-q)(1-q^2)\cdots(1-q^n)} \\
\end{eqnarray*}
This can be simplified to the equation: \begin{equation}
\sum_{|T|=n} \chi_{\sh(T)}(\lambda)q^{\maj(T)} = \frac{(1-q)(1-q^2)\cdots (1-q^n)}{(1-q^{\lambda_1})(1-q^{\lambda_2})\cdots(1-q^{\lambda_k})}
\end{equation} where $\sh(T)$ denotes the shape of the tableau $T$.
Notice that when we take $q\to 1$ above, the right hand side is $0$ unless $\lambda=(1^n)$ is the partition of $n$ into all $1$'s. If $\lambda$ is not this partition, setting $q=1$ yields $$\sum \chi_\mu(\lambda)\cdot f^{\mu}=0$$ where $f^\mu$ is the number of standard Young tableaux of shape $\mu$. Otherwise if $\lambda=(1^n)$, we obtain $$\sum \chi_\mu(\lambda)\cdot f^{\mu}=n!.$$ Recall also that $f^\mu$ (see e.g. Stanley or Sagan) is equal to the dimension of the irreducible representation $V_\mu$ of $S_n$. Thus, these two equations together are equivalent to the fact that, if $R$ is the regular representation,
$$\chi_R=\sum_\mu (\dim V_\mu) \cdot \chi_{\mu}$$ which is in turn equivalent to the decomposition of $R$ into irreducibles.
Therefore, Equation (1) is a $q$-analog of the decomposition of the regular representation. I'm not sure this is known, and I find it's a rather pretty consequence of the Schur function specialization at powers of $q$.
EDIT: It is known, as Steven Sam pointed out in the comments below, and it gives a formula for a graded character of a graded version of the regular representation.
Summary: Symmetric functions transition table
Posted on August 4, 2012 by Maria Gillespie
Over the last few weeks I've been writing about several little gemstones that I have seen in symmetric function theory. But one of the main overarching beauties of the entire area is that there are at least five natural bases with which to express any symmetric functions: the monomial ($m_\lambda$), elementary ($e_\lambda$), power sum ($p_\lambda$), complete homogeneous ($h_\lambda$), and Schur ($s_\lambda$) bases. As a quick reminder, here is an example of each, in three variables $x,y,z$:
$m_{(3,2,2)}=x^3y^2z^2+y^3x^2z^2+z^3y^2x^2$
$e_{(3,2,2)}=e_3e_2e_2=xyz(xy+yz+zx)^2$
$p_{(3,2,2)}=p_3p_2p_2=(x^3+y^3+z^3)(x^2+y^2+z^2)^2$
$h_{(2,1)}=h_2h_1=(x^2+y^2+z^2+xy+yz+zx)(x+y+z)$
$s_{(3,1)}=m_{(3,1)}+m_{(2,2)}+2m_{(2,1,1)}$
Since we can usually transition between the bases fairly easily, this gives us lots of flexibility in attacking problems involving symmetric functions; it's sometimes just a matter of picking the right basis.
So, to wrap up my recent streak on symmetric function theory, I've posted below a list of rules for transitioning between the bases. (The only ones I have not mentioned is how to take a polynomial expressed in the monomial symmetric functions $m_\lambda$ in terms of the others; this is rarely needed and also rather difficult.)
Elementary to monomial:
$$e_\lambda=\sum M_{\lambda\mu} m_\mu$$
where $M_{\lambda\mu}$ is the number of $0,1$-matrices with row sums $\lambda_i$ and column sums $\mu_j$.
Elementary to homogeneous:
$$e_n=\det \left(\begin{array}{cccccc}
h_1 & 1 & 0 & 0 &\cdots & 0 \\
h_2 & h_1 & 1 & 0 & \cdots & 0 \\
h_3 & h_2 & h_1 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
h_{n-1} & h_{n-2} & h_{n-3} & h_{n-4} & \ddots & 1 \\
h_n & h_{n-1} & h_{n-2} & h_{n-3} & \cdots & h_1
\end{array}\right)$$
Elementary to power sum:
$$e_n=\frac{1}{n!}\det\left(\begin{array}{cccccc}
p_1 & 1 & 0 & 0 &\cdots & 0 \\
p_2 & p_1 & 2 & 0 & \cdots & 0 \\
p_3 & p_2 & p_1 & 3 & \cdots & 0 \\
p_{n-1} & p_{n-2} & p_{n-3} & p_{n-4} & \ddots & n-1 \\
p_n & p_{n-1} & p_{n-2} & p_{n-3} & \cdots & p_1
Elementary to Schur:
$$e_\lambda=\sum_{\mu} K_{\mu'\lambda}s_\mu$$
where $K_{\lambda\mu}$ is the number of semistandard Young tableau of shape $\lambda$ and content $\mu$.
Homogeneous to monomial:
$$h_\lambda=\sum N_{\lambda\mu} m_\mu$$
where $N_{\lambda\mu}$ is the number of matrices with nonnegative integer entries with row sums $\lambda_i$ and column sums $\mu_j$.
Homogeneous to elementary:
$$h_n=\det\left(\begin{array}{cccccc}
e_1 & 1 & 0 & 0 &\cdots & 0 \\
e_2 & e_1 & 1 & 0 & \cdots & 0 \\
e_3 & e_2 & e_1 & 1 & \cdots & 0 \\
e_{n-1} & e_{n-2} & e_{n-3} & e_{n-4} & \ddots & 1 \\
e_n & e_{n-1} & e_{n-2} & e_{n-3} & \cdots & e_1
Homogeneous to power sum:
$$h_n=\frac{1}{n!}\det\left(\begin{array}{cccccc}
p_1 & -1 & 0 & 0 &\cdots & 0 \\
p_2 & p_1 & -2 & 0 & \cdots & 0 \\
p_3 & p_2 & p_1 & -3 & \cdots & 0 \\
p_{n-1} & p_{n-2} & p_{n-3} & p_{n-4} & \ddots & -(n-1) \\
Homogeneous to Schur:
$$h_{\lambda}=\sum_\mu K_{\mu\lambda}s_\mu$$
Power sum to monomial:
$$p_\lambda=\sum_{\mu} R_{\lambda\mu}m_\mu$$
where $R_{\lambda\mu}$ is the number of ways of sorting the parts of $\lambda$ into a number of ordered blocks in such a way that the sum of the parts in the $j$th block is $\mu_j$.
Power sum to elementary:
Newton-Gerard identities:
$$p_n=\det\left(\begin{array}{cccccc}
2e_2 & e_1 & 1 & 0 & \cdots & 0 \\
3e_3 & e_2 & e_1 & 1 & \cdots & 0 \\
(n-1)e_{n-1} & e_{n-2} & e_{n-3} & e_{n-4} & \ddots & 1 \\
ne_n & e_{n-1} & e_{n-2} & e_{n-3} & \cdots & e_1
Power sum to homogeneous:
$$p_n=(-1)^{n-1}\det\left(\begin{array}{cccccc}
2h_2 & h_1 & 1 & 0 & \cdots & 0 \\
3h_3 & h_2 & h_1 & 1 & \cdots & 0 \\
(n-1)h_{n-1} & h_{n-2} & h_{n-3} & h_{n-4} & \ddots & 1 \\
nh_n & h_{n-1} & h_{n-2} & h_{n-3} & \cdots & h_1
Power sum to Schur:
Let $\chi^\lambda$ be the $\lambda$th character of the symmetric group $S_n$ where $n=|\lambda|$, and write $\chi^\lambda(\mu)$ to denote the value of $\chi^{\lambda}$ at any permutation with cycle type $\mu$. Then for any partition $\mu\vdash n$, we have:
$$p_\mu=\sum_{\lambda\vdash n} \chi^\lambda(\mu) s_\lambda$$ Alternatively: $$p_n=s_{(n)}-s_{(n-1,1)}+s_{(n-2,1,1)}-s_{(n-3,1,1,1)}+\cdots+(-1)^{n}s_{(1,1,\ldots,1)}$$
Schur to monomial:
$$s_{\lambda}=\sum_{\mu\vdash |\lambda|} K_{\lambda \mu}m_\mu$$
Schur to elementary:
(Dual Jacobi-Trudi Identity.)
$$s_{\lambda/\mu} = \det \left(e_{\lambda'_i-\mu'_j-i+j}\right)_{i,j=1}^n$$
Schur to homogeneous:
(Jacobi-Trudi Identity.)
$$s_{\lambda/\mu} = \det \left(h_{\lambda_i-\mu_j-i+j}\right)_{i,j=1}^n$$
Schur to power sum:
$$s_\lambda=\sum_{\nu\vdash |\lambda|} z_\nu^{-1} \chi^{\lambda}(\nu) p_\nu$$
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18.12.2018 - Henrik Gustafsson (Stanford): Vertex Operators, Solvable Lattice Models and Metaplectic Whittaker functions
Abstract: In this talk, based on joint work with Ben Brubaker, Valentin Buciumas and Daniel Bump, I will explain new connections relating metaplectic Whittaker functions and certain solvable lattice models with operators on a q-deformed fermionic Fock space. We will discuss the locality properties of these operators which agree with those of a q-deformed vertex operator algebra, and review the underlying quantum groups that are part of the above connections. In the process we also obtain a Fock space operator description of ribbon symmetric functions, or LLT polynomials, introduced by Lascoux, Leclerc and Thibon.
11.12.2018 - Arash Ardehali (Uppsala): Microscopic structure of black holes from asymptotic analysis of special functions.
Abstract: After a concise introduction to quantum field theory from a combinatorial perspective, it will be explained how ideas from analytic combinatorics are helping to extract information about the microscopic structure of black holes via asymptotic analysis of various quantum field theory generating functions.
28.11.2018 - Federico Zerbini (IPhT): A class of non-holomorphic modular forms from string theory.
Abstract: Modular graph functions were discovered by physicists studying genus-one string-theory amplitudes. They generalize special values of non-holomorphic Eisenstein series, and they display striking properties, which led to conjecture that they should be related to iterated Eichler integrals of Eisenstein series. In this talk we will introduce modular graph functions and iterated Eichler integrals, and describe the state of the art.
21.11.2018 - Brandon Williams (TU Darmstadt): Hilbert modular forms and Borcherds products.
Abstract: I will talk about how Borcherds products can be used to compute graded rings of Hilbert modular forms.
14.11.2018 - Pär Kurlberg (KTH): Class numbers and class groups for definite binary quadratic forms.
Abstract: Gauss made the remarkable discovery that the set of integral binary quadratic forms of fixed discriminant carries a composition law, i.e., two forms can be "glued together" into a third form. Moreover, as two quadratic forms related to each other via an integral linear change of variables can be viewed as equivalent, it is natural to consider equivalence classes of quadratic forms. Amazingly, Gauss' composition law makes these equivalence classes into a finite abelian group - in a sense it is the first abstract group "found in nature". Extensive calculations led Gauss and others to conjecture that the number h(d) of equivalence classes of such forms of negative discriminant d tends to infinity with |d|, and that the class number is h(d)=1 in exactly 13 cases: d is in { -3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163 }. While this was known assuming the Generalized Riemann Hypothesis, it was only in the 1960's that the problem was solved by Alan Baker and by Harold Stark. We will outline the resolution of Gauss' class number one problem and survey some known results regarding the growth of h(d). We will also consider some recent conjectures regarding how often a fixed abelian group occur as a class group, and how often an integer occurs as a class number. In particular: do all abelian groups occur, or are there "missing" class groups?
31.10.2018 - Martín Sombra (ICREA & Universitat de Barcelona): The zero set of the independence polynomial of a graph.
Abstract: In statistical mechanics, the independence polynomial of a graph $G$ arises as the partition function of the hard-core lattice gas model on $G$. The distribution of the zeros of these polynomials when $G \to \infty$ is relevant for the study of this model and, in particular, for the determination of its phase transitions. In this talk, I will review the known results on the location of these zeros, with emphasis on the case of rooted regular trees of fixed degree and varying depth $k \ge 0$. Our main result states that for these graphs, the zero sets of their independence polynomials converge as $k \to \infty$ to the bifurcation measure of a certain family of dynamical systems on the Riemann sphere. This is ongoing work with Juan Rivera-Letelier (Rochester).
10.10.2018 - Jonathan Nilsson (Chalmers/GU): Representation theory for Lie algebras of vector fields.
Abstract: Let A be the algebra of polynomial functions on a fixed affine algebraic variety. The Lie algebra V of polynomial vector fields on the variety is isomorphic to the Lie algebra of derivations of A. This Lie algebra does generically not admit a Cartan subalgebra, so the classical root-weight approach to representation theory fails here. Instead I will discuss some new classes of modules that have the property that they admit compatible A- and V-module structures. To concretize the general theory I will also show explicitly what happens in a simple nontrivial case, namely when the variety is the 2-sphere. The talk is based on joint work with Yuly Billig and Vyacheslav Futorny.
26.09.2018 - Christian Johansson (Chalmers/GU): Mod p completed cohomology of locally symmetric varieties.
Abstract: This talk is about singular cohomology of (certain) manifolds, with coefficients in F_p (p some fixed prime). More precisely, the manifolds in question are certain locally symmetric varieties. I will discuss how one can prove that the direct limit of such cohomology groups vanish above the middle degree as one goes up a certain tower of locally symmetric varieties. This confirms many cases of a conjecture of Calegari and Emerton, which is of central importance in the p-adic part of Langlands program. This is joint work in progress with David Hansen.
14.09.2018, Rainer Dietmann (Royal Holloway, University of London): Lines on cubic hypersurfaces.
Abstract: Wooley has shown that every rational cubic hypersurface defined by a cubic form in at least 37 variables contains a rational line. In joint work with Julia Brandes we were able to improve this in the generic case of non-singular cubic forms, reducing the required number of variables to 31, applying recent work of Browning, Dietmann and Heath-Brown on intersections of cubic and quadric hypersurfaces. Time permitting, we also briefly want to discuss the related problem of finding lines on cubic hypersurfaces defined over p-adic fields.
12.09.2018 - Daniele Casazza (ICMAT, Spain): On the factorization of p-adic L-series.
Abstract: In this talk we present the basic ideas concerning p-adic L-functions and how they can be used to deduce interesting arithmetic information. After a general overview and the presentation of some notable example, we will focus more in detail on the case of interest for us. We will explain how one can construct a new p-adic L-function using Saito-Kurokawa lifts and use it to deduce some interesting arithmetic information.
16.08.2018 - Yoonbok Lee (Incheon National University): The a-values of the Riemann zeta function near the critical line.
Abstract: We study the value distribution of the Riemann zeta function near the line $\Re s = 1/2$. We find an asymptotic formula for the number of $a$-values in the rectangle $ 1/2 + h_1 / (\log T)^\theta \leq \Re s \leq 1/2+ h_2 /(\log T)^\theta $, $T \leq \Im s \leq 2T$ for fixed $h_1, h_2>0$ and $ 0 < \theta <1/13$. To prove it, we need an extension of the valid range of Lamzouri, Lester and Radziwill's recent results on the discrepancy between the distribution of $\zeta(s)$ and its random model. We also propose the secondary main term for the Selberg's central limit theorem by providing sharper estimates on the line $\Re s = 1/2 + 1/(\log T)^\theta $. This is a joint work with Junsoo Ha in KIAS.
31.05.2018 - Chunhui Liu (Kyoto University): Counting rational points in arithmetic varieties by the determinant method.
Abstract: By the slope method in Arakelov geometry, we can construct a family of hypersurfaces which cover the rational points of bounded height on an arithmetic variety but don't contain the generic point of this variety. By estimating some invariants of Arakelov geometry, we can control the number and the maximal degree of this family of auxiliary hypersurfaces explicitly. In this talk, I will explain the method of studying the problem of counting rational points by the approach of Arakelov geometry.
29.05.2018 - Manish M. Patnaik (University of Alberta): Metaplectic covers of Kac-Moody groups and Whittaker functions.
Abstract: We will describe how to construct metaplectic covers of Kac-Moody groups, generalizing a classical construction of Matsumoto. In the case of non-archimedean local fields, we will then explain how to formulate Whittaker functions on such groups, and compute them via a Casselman-Shalika type formula. Joint work with A. Puskas.
18.05.2018 - Will Sawin (ETH-ITS): The circle method and free rational curves on hypersurfaces.
Abstract: In joint work in progress with Tim Browning, we study a system of two of Diophantine equations over F_q[t], and use the circle method to show a relationship between their numbers of solutions. As a consequence, we bound the dimension of the singular locus of the moduli space of rational curves on a smooth projective hypersurface. I will explain how these problems are related and what techniques we use to get the best bound.
26.04.2018 - Sofia Tirabassi (University of Bergen): Fourier--Mukai partners of Enriques and bielliptic surfaces in positive characteristic.
Abstract: We show that Enriques and bielliptic surfaces in positive characteristic do not have any non-trivial Fourier--Mukai partners. This is joint work with K. Honigs and M. Lieblich.
24.04.2018 - Wanmin Liu (IBS Center for Geometry and Physics): Classification of full exceptional collections of line bundles on three blow-ups of P^3.
Abstract: A fullness conjecture of Kuznetsov says that if a smooth projective variety X admits a full exceptional collection of line bundles of length l, then any exceptional collection of line bundles of length l is full. In this talk, we show that this conjecture holds for X as the blow-up of P^3 at a point, a line, or a twisted cubic curve, i.e. any exceptional collection of line bundles of length 6 on X is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such X. This is a joint work with Song Yang and Xun Yu at Tianjin University.
12.04.2018 - Dennis Eriksson (Chalmers/GU): Transfinite diameters in the Berkovich setting.
Abstract: The classical transfinite diameter is a notion of size of a compact set in the complex plane, given by a well-defined limiting process. It generalizes the notion of diameter of a disc. The definition and existence of the limit were further generalized and studied on more general complex manifolds by Berman-Boucksom. In this talk, I will explain how these results can be extended to the non-archimedean/Berkovich setting. A key ingredient is given by Knudsen-Mumfords version of the Grothendieck-Riemann-Roch theorem. This is joint work with Sébastien Boucksom.
23.03.2018 - Sebastián Herrero (Chalmers/GU): A Jensen–Rohrlich type formula for the hyperbolic 3-space.
Abstract: The classical Jensen's formula is a well-known theorem of complex analysis which characterizes, for a meromorphic function f on the unit disc, the value of the integral of log|f(z)| on the unit circle in terms of the zeros and poles of f inside the unit disc. An important theorem of Rohrlich establishes a version of Jensen's formula for modular functions f with respect to the full modular group PSL_2(Z) and expresses the integral of log|f(z)| over the corresponding modular curve in terms of special values of Dedekind's eta function.
In this talk I will present a Jensen–Rohrlich type formula for certain family of functions defined in the hyperbolic 3-space which are automorphic for the group PSL_2(O_K) where O_K denotes the ring of integers of an imaginary quadratic field. This is joint work with Ö. Imamoglu (ETH Zurich), A.-M. von Pippich (TU Darmstadt) and Á. Tóth (Eotvos Lorand Univ.).
09.03.2018 - Julia Brandes (Chalmers/GU): Optimal mean value estimates beyond Vinogradov's mean value theorem.
Abstract: Mean values for exponential sums play a central role in the study of diophantine equations. In particular, strong upper bounds for such mean values control the number of integer solutions of the corresponding systems of diagonal equations. Since the groundbreaking resolution of Vinogradov's mean value theorem by Wooley and Bourgain, Demeter and Guth, we can now prove optimal upper bounds for mean values connected to translation-dilation-invariant systems. This has inspired Wooley's call for a "Big Theory of Everything", a challenge to establish optimal mean value estimates for any mean values associated with systems of diagonal equations.
We establish optimal bounds for a family of mean values that are not of Vinogradov type. This is the first time bounds of this quality have been obtained for non-translation-dilation-invariant systems. As a consequence, we establish the analytic Hasse principle for the number of solutions of certain systems of quadratic and cubic equations in fewer variables than hitherto thought necessary. This is joint work with Trevor Wooley.
23.02.2018 - Jakob Palmkvist (Chalmers/GU): Generators and relations for (generalized) Cartan superalgebras.
Abstract: In Kac's classification of finite-dimensional Lie superalgebras, the contragredient ones can be constructed from Dynkin diagrams similar to those of the simple finite-dimensional Lie algebras, but with additional types of nodes. For example, A(0,n-1)=sl(1|n) can be constructed by adding a "gray'' node to the Dynkin diagram of A_{n-1}=sl(n), corresponding to an odd null root. The Cartan superalgebras constitute a different class, where the simplest example is W(n), the derivation algebra of the Grassmann algebra on n generators. I will in my talk present a novel construction of W(n), from the same Dynkin diagram as A(0,n-1), but with additional generators and relations. I will then generalize this result to the exceptional Lie algebras E_n, which can be extended to infinite-dimensional Borcherds superalgebras, in the same way as A_{n-1} can be extended to A(0,n-1). In this case, the construction leads to so called tensor hierarchy algebras, which seem to provide an underlying algebraic structure of certain supergravity models.
08.02.2018 - Håkan Granath (Stockholms universitet): Heun functions and quaternionic modular forms.
Abstract: The Shimura curve of discriminant 10 is uniformized by a subgroup of an arithmetic (2,2,2,3) quadrilateral group. Hence its uniformization is related to a class of special functions called Heun functions. In the talk I will give an introduction to these concepts, present the differential structure of the ring of modular forms for the Shimura curve, and show how one can relate the ring generators to explicit Heun functions for the quadrilateral group. Furthermore I will describe how the Picard-Fuchs equation of the associated family of abelian surfaces has solutions that are modular forms. As an application of these explicit identifications, I will describe how the exceptional sets of the associated Heun functions can be determined, and, time permitting, say a few words on how exceptional values can be computed. The talk is based on joint work with Srinath Baba.
02.02.2018 - Martin Raum (Chalmers/GU): Unifying relaxed notions of modular forms.
Abstract: Elliptic modular forms are functions on the complex upper half plane that are invariant under a certain action of the special linear group with integer entries. Their history comprises close to two centuries of amazing discoveries and application: The proof of Fermat's Last Theorem is probably the most famous; The theory of theta functions is among its most frequently employed parts. During the past decade it has been à la mode to study relaxed notions of modularity. Relevant keywords that we will discuss are mock modular forms and higher order modular forms. We have witnessed their application, equally stunning as surprising, to conformal field theory, string theory, combinatorics, and many more areas. In this talk, we suggest a change of perspective on such generalizations. Most of the novel variants of modular forms (with one prominent exception) can be viewed as components of vector-valued modular forms. This unification draws its charm from the past and the future. On the one hand, we integrate results by Kuga and Shimura that hitherto seemed almost forgotten. On the other hand, we can point out connections, for example, between mock modular forms and so-called iterated integrals that have not yet been noticed. This is joint work with Michael Mertens.
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Big Ideas Learning, LLC | Seventh Grade
Home Reports Center Math Seventh Grade Big Ideas Math: Modeling Real Life
Big Ideas Math: Modeling Real Life - Seventh Grade
There are no above grade level assessment items for Grade 7. Examples of assessment items which assess grade-level standards include:
Chapter 1, Quiz 2, Item 9, students use a vertical number line that shows elevations of a submarine after certain events to determine the distance the submarine rises after diving and the original elevation of the submarine. (7.NS.1.c)
Chapter 3, Test A, Item 13, students factor a linear expression in order to determine the length of a square patio that has a perimeter of 16x + 12 feet. (7.EE.1)
Chapter 3, Performance Task, Item 1, students write and simplify expressions from information provided in a diagram and a table. They describe and explain what they notice about the two expressions. (7.EE.1-2)
Chapter 5, Test A, Item 6, students find the density of a substance in grams per millimeter by examining a graph. (7.RP.2.d)
Course Benchmark 2, Item 30, students find the actual perimeter and area of a square using information about the scale drawing of a square. (7.G.1)
Chapter 8, Alternative Assessment, Item 1, students are given the scenario about finding out how the residents in their town feel about opening a new gas station. Students describe how to conduct a survey so that the sample is biased, and unbiased survey of 200 people. They project how many residents out of 6200 will support the gas station if 80 out of 200 supported it. (7.SP.1-2)
The instructional materials reviewed for Big Ideas Math: Modeling Real Life Grade 7 meet expectations for spending a majority of instructional time on major work of the grade. This includes all the clusters in 7.RP.A, 7.NS.A, and 7.EE.A, B.
The supporting domain Statistics and Probability enhances focus and coherence to major standard/clusters of the grade, especially domains 7.NS and 7.RP. For example:
In Chapter 5, Section 5.2, Solve Problems Involving Scale Drawings of Geometric figures (7.G.1) is connected to the major work of analyzing proportional relationships (7.RP.A). Students write and solve a proportion using the scale and ratios of the lengths of a drawing.
In Chapter 7, Section 7.1, 7.SP.5 is connected to 7.RP.A as students work with probability as the ratio of desired outcomes to possible outcomes, and examine the probability between 0 and 1 including 0 and 1 of an event. Relative frequency is also defined as a ratio. For example, in Problem 4, students describe the likelihood of each event when making three-point shots or missing the shots.
In Chapter 7, Section 7.3, Compound Events, connects 7.G.8a with 7.RP.3 when students determine probability by computation of rational numbers, and representing answers as fractions and percents. For example, Problem 4 expresses the probability as 1/6 or 16 2/3%.
In Chapter 7, Section 7.3, Probability of Compound Events, 7.SP.8 is connected to the major work of solving real-world problems with rational numbers involving the four operations, 7.NS.3. Students solve simple and compound probabilities using rational numbers in various forms.
Chapter 8, Section 8.1, Example 3 utilizes proportions to solve a problem to make projections for modeling real world problems. After randomly surveying 75 students, students use the results to estimate the number of students from the total population of 1200. Cluster 7.SP.A supports 7.RP.3.
In Chapter 8, Section 8.2, Self-Assessment, Problem 4, students apply and extend previous understandings of operations with fractions (7.NS.A) to draw inferences about a population (7.SP.A). Students find the means of three samples of the number of hours music students practice each week, and use the means to make one estimate for the mean number of practice hours. The calculations result in a rational number that, when converted to a decimal, results in a repeating decimal, which they make sense of in order to answer the question about the number of hours music students practice each week (7.NS.2).
Chapter 9, Section 9.5, Problem Solving with Angles, 7.G.5 is connected to the major work of solving word problems leading to equations, 7.EE.4.a as students write and solve equations to find the missing angle using properties of supplementary, complementary, adjacent, and vertical angles.
Laurie's Notes, "Preparing to Teach" describe connections between content from prior grades and lessons to the current learning. For example, in Chapter 4, Section 4, "Students should know how to graph numbers on a number line and how to solve one-variable inequalities using whole numbers. In the exploration, students will be translating inequalities from verbal statements to graphical representations and symbolic sentences."
Chapter Overviews describe connections between content from prior and future grades to the current learning, and the progression of learning that will occur. For example, Chapter 5, "Laurie's Notes: Chapter Overview", "The study of ratios and proportions in this chapter builds upon and connects to prior work with rates and ratios in the previous course." This supports Standard 6.RP. In Sections 5.1 and 5.2, students decide whether two quantities are in a proportional relationship using ratio tables. This supports Standard 7.RP.2.a and uses unit rates involving rational numbers. During Sections 5.3, 5.4, and 5.5, students write, solve, and graph proportions. This supports Standard 7.RP.2.a-7.RP.3, "Graphing proportional relationships enables students to see the connection between the constant of proportionality and equivalent ratios", but the term "Slope", Standards 8.EE.5-6, is not included. In Section 5.6, students work with scale drawings, which supports Standard 7.G.1.
Each chapter's Progressions page contains two charts. "Through the Grades", lists the relevant portions of standards from prior and future grades (grades 6 and 8) that connect to the grade 7 standards addressed in that chapter. For example, Chapter 4, Sections 4.1-4.2, students use algebra tiles to review the process of solving one-step equations. This is identified as revisiting work from a prior grade-level in the "Chapter Exploration and supports grade-level work in section 4.3 of solving equations of the form px + q =r and p(x + q) =r. This supports Standard 7.EE.4a.
Each lesson presents opportunities for students to work with grade-level problems. However, "Scaffolding Instruction" notes suggest assignments for students at different levels of proficiency (emergent, proficient, advanced). These levels are not defined, nor is there any tool used to determine which students fall into which level. In the Concepts, Skills and Problem Solving section at the end of each lesson problems are assigned based on these proficiencies, therefore, not all students have opportunities to engage with the full intent of grade-level standards. For example:
In the Teacher Edition, Chapter 6, Section 6.5, the assignments for proficient and advanced students includes a reasoning task in which students determine the price of a drone that is discounted 40%, and then discounted an additional 60% a month later. This reasoning task is omitted from the assignments for emerging students.
In the Teacher Edition, Chapter 9, Section 9.2, the assignments for advanced students include a critical thinking task in which students determine how increasing the radius of a circle impacts the area of the circle. This critical thinking task is omitted from the assignments for emerging and proficient students.
Each section within a chapter includes problems where the publisher states, "students encounter varying "Depth of Knowledge" levels, reaching higher cognitive demand and promoting student discourse". In Chapter 8, Section 8.1, students examine a sample of a population for validity. This supports Standard 7.SP.1 and use a random sample to draw inferences about a population which supports Standard 7.SP.2.
In "Exploration 1" students "make conclusions about the favorite extracurricular activities of students at their school" by first identifying the population and samples of the population, (DOK Level 1) and then by evaluating the differences between two samples and evaluating their conclusions for validity and explain their thinking, (DOK Level 3).
Problem 2 students compare two samples to determine which sample is unbiased, (DOK Level 2).
In Chapter 4, Section 4.6, students roll two different colored dice with negative and positive numbers on each cube. When the students roll a pair of dice, they write an inequality to represent them. Then they roll one die and multiply each side of the inequality to represent them. They are then asked if the original inequality is still true. Finally, they are asked to make conjectures about how to solve an inequality of the form ax <b for x when a>0 and when a<0. These conjectures will help to develop the key idea(s) for the section which is to write and solve inequalities using multiplication and division. This supports standard 7.EE.4.b.
In Chapter 6, students use a percent model to justify their answers, instead of assessing the reasonableness of answers using mental computation and estimation strategies. Mental computation and estimation are strategies specifically called for in standard 7.EE.3.
Materials explicitly relate grade-level concepts to prior knowledge from earlier grades. At the beginning of each section in Laurie's Notes, there is a heading marked "Preparing to Teach", which includes a brief explanation of how work in prior courses relates to the work involved in that lesson. In some cases it outlines what happened in prior courses, but is not specific to which grade or course this happens. For example:
In Chapter 1, Section 1.1, it states that in prior courses students were introduced to integers, absolute value, and number lines. For example, "It is important that students review these foundational skills because they are necessary for adding and subtracting rational numbers." In Chapter 1, Section 1.1, students review the concept of absolute value (6.NS.7). This leads into Section 1.2 where students begin adding integers (7.NS.1.b).
In Chapter 3, Section 3.3 states that students have used the distributive property in previous courses. It adds, "They will extend their understanding to include algebraic expressions involving rational numbers. This property is very important to algebraic work in future courses". In Chapter 3, Section 3.3, Exploration 1, students build upon their experience with the distributive property to include rational numbers. In Example 1, students apply the distributive property to simplify expressions.
In Chapter 5, Section 5.2, the Preparing to Teach notes, explain the connection between students' prior work with ratios (describing ratio relationships, completing tables), (6.RP.A), and the content in Section 5.2, stating, "In this lesson, they will extend their work with ratios to include fractions, making connections to their recent work with fractions." In Section 5.1, students complete ratio tables, and write and interpret ratios, but now with fractions, forming a bridge to upcoming work of finding and using unit rates involving rational numbers (7.RP.1).
In Chapter 6, Section 6.1, Preparing to Teach, notes state students "should know how to solve simple percent problems, and how to use ratio tables, Standard 6.RP.3." The remainder of Chapter 6, "will build upon this understanding to write and solve percent proportions." (7.RP.3)
In the Resources by Chapter book, each chapter has a few questions that are named as "Prerequisite Skills Practice". The intent is for practice from prior knowledge. There is no mention of previous grade knowledge or previous lesson knowledge.
In Chapter 5, Algebraic Expressions and Properties, 6.EE, Apply and extend previous understandings of arithmetic to algebraic expressions is directly related to the Chapter 5 learning goals of, "Evaluate algebraic expressions given values of their variables (Section 5.1), Write algebraic expressions and solve problems involving algebraic expressions (Section 5.2), Identify equivalent expressions and apply properties to generate equivalent expressions (Section 5.3), Identify equivalent expressions and apply properties to generate equivalent expressions (Section 5.4), and Factor numerical and algebraic expressions (Section 5.5).
In Chapter 3, students engage simultaneously in Standards 7.NS.A and 7.EE.A, as they simplify, add, subtract, factor and expand linear expressions involving positive and negative number coefficients. For example, in Section 3.1, Try It, Problem 9, students simplify 2s - 9s + 8t - t. In Section 3.3, Try It, Problem 5, students use the distributive property to simplify the expression -3/2 (a - 4 - 2a).
In Chapter 4, students use operations with integers, Cluster 7.NS.A to solve problems using numerical and algebraic expressions and equations, Cluster 7.EE.B.
In Chapter 5, Domain 7.RP connects ratio with computations with rational numbers 7.NS, as students explore rates and unit rates. For example, in Section 5.6, students analyze proportional relationships and use them to solve real-world problems.
Chapter 6, the problems and activities provide connections between the skills and understandings of Cluster 7.EE.B to those of Cluster 7.RP.A as students write proportions and equations to represent and solve percent problems, and to write equations to solve problems involving discounts and markups. In Section 6.3, Practice, Problem 23, students write and solve an equation to determine the percent of sales tax on a model rocket costing $24 with a sales tax of $1.92.
Chapter 8, Section 8.4, students use random sampling to draw inferences about a population, connecting 7.SP.A with drawing informal comparative inferences about two populations, 7.SP.B.
In Chapter 1, Section 2, Exploration 1 (7.NS.1.d), students are taught to add integers with chips and using number lines. "Write an addition expression represented by the number line. Then find the sum." After these examples, students are asked to use conceptual strategies (number line or chips).
In Chapter 3, Lesson 2, Exploration 1, students use algebra tiles to model a sum of terms equal to zero and simplify expressions. In the Concepts, Skills and Problem Solving section, students have two additional problems where they use algebra tiles to simplify expressions. (7.EE.1)
Chapter 1, Section 4, "Subtracting Integers," Exploration 1 asks students to work with partners and use integer counters to find the differences and sums of several problems with two different representations. For example, "4 - 2" and "4 + (-2)"; "-3 - 1" and "-3 + (-1)" and "13 - 1". Student pairs are asked to generate a rule for subtracting integers. Students who can't generate a rule are prompted to use a number line. After working independently students share their rule with a partner and discuss any discrepancies. (7.NS.1)
Chapter 4, Section 1 "Solving Equations Using Addition or Subtraction" Exploration 1, students are asked, "Write the four equations modeled by the algebra tiles. Explain how you can use algebra tiles to solve each equation." (7.EE.3)
The instructional materials do not always provide students opportunities to independently demonstrate conceptual understanding throughout the grade-level. The shift from conceptual understanding, most prevalent in the Exploration Section, to procedural understanding occurs within the lesson. The Examples and "Concepts, Skills, and Problem Solving" sections have a focus that is primarily procedural with limited opportunities to demonstrate conceptual understanding. For example:
In Chapter 3, Section 2, only Problems 8 and 9 ask students to demonstrate conceptual understanding. For example, Problems 10-17 ask students to "Find the Sum." Problem 10: "(n+8) + (n-12)"; Problem 16: "(6-2.7h) + (-1.3j-4)." Problems 19-26 ask students to "Find the difference." Problem 19: "(-2g+7) - (g+11)"; Problem 26: "(1-5q) - (2.5s+8) - (O.5q+6)". (7.EE.1)
In Chapter 2, Section 2, Concepts, Skills & Problem Solving, the majority of the questions require procedural knowledge and do not ask students to demonstrate conceptual understanding. For example, Problems 13-28 ask students to "Find the quotient, if possible", such as Problem 16: "-18 ÷ (-3)"; and Problem 22: "-49 ÷ (-7)". (7.NS.1)
The instructional materials for Big Ideas Math: Modeling Real Life Grade 7 meet expectations that they attend to those standards that set an expectation of procedural skill. The instructional materials attend to operations with rational numbers (7.NS.A), using the properties of operations to generate equivalent expressions (7.EE.1), and solving real-life and mathematical problems using numerical and algebraic expressions (7.EE.B). For example:
In Chapter 1, Lesson 5, students subtract rational numbers. Examples 1-3 provide step-by-step explanations of the procedural skill of rational numbers. In the Concept, Skills, and Problem Solving section, students have many opportunities to demonstrate their skill of subtracting rational numbers. (7.NS.1)
In Chapter 2, Lesson 1, students multiply rational numbers. Examples 1-3 provide step-by-step explanations of the procedural skill of multiplying rational numbers. In the Concept, Skills, and Problem Solving section, students have many opportunities to demonstrate their skill of multiplying rational numbers. (7.NS.2)
In Chapter 3, Lesson 4, students factor expressions. Examples 1-3 provide step-by-step explanations of the procedural skill of factoring an expression. In the Concept, Skills, and Problem Solving section, students have many opportunities to demonstrate their skill of factoring an expression. (7.EE.1)
In Chapter 4, Lesson 1, students solve equations using addition and subtraction. Examples 1-3 provide step-by-step explanations of the procedural skill of solving an equation using addition and subtraction. In the Concept, Skills, and Problem Solving section, students have many opportunities to demonstrate their skill of solving an equation. (7.EE.4.a)
In each lesson there is a "Review & Refresh" section, which provides additional practice for skills previously taught. Within these sections are further opportunities to practice the procedural skills. For example:
In Chapter 2, Lesson 2, there are four problems requiring multiplication of rational numbers. For example: "Problem 1: 8 x 10; Problem 2: -6(9); Problem 3: 4(7); Problem 4: -9(-8)". (7.NS.2)
In Chapter 3, Lesson 4, there are three problems requiring simplifying expressions. For example: "Problem 1: 8(k-5); Problem 2: -4.5(-6+2d); Problem 3: -1/4(3g-6-5g)". (7.EE.1)
In Chapter 4, Lesson 1, there are four problems asking students to factor out the coefficient of the variable term. For example: "Problem 1: 4x-20; Problem 2: -6y-18; Problem 3: -2/5w + 4/5; Problem 4: 0.75z - 6.75". (7.EE.4.a)
Chapter 5, Lesson 1, Example 3, Modeling Real Life, "You mix 1/2 cup of yellow paint for every 3/4 cup of blue paint to make 15 cups of green paint. How much yellow paint do you use?" Students are given two methods to solve the questions with both methods being explained and answered. For example, "Method 1: The ratio of yellow paint to blue paint is 1/2 to 3/4. Use a ratio table to find an equivalent ratio in which the total amount of yellow paint and blue paint is 15 cups." [A completed ratio table with annotated description as to how it was filled out is included.] "Method 2: You can use the ratio of yellow paint to blue paint to find the fraction of the green paint that is made from yellow paint. You use 1/2 cup of yellow paint for every ¾ cup of blue paint, so the fraction of the green paint that is made from yellow paint is 2/5 [included equation and solution]. So, you use 2/5 ⋅ 15 = 6 cups of yellow paint." (7.RP.1)
Chapter 1, Lesson 1, Example 3, Modeling Real Life, "A moon has an ocean underneath its icy surface. Scientists run tests above and below the surface. [Table Provided] The table shows the elevations of each test. Which test is deepest? Which test is closest to the surface?" The explanation from this point provides students with step-by-step directions on how to solve the problem. "To determine which test is deepest, find the least elevation. Graph the elevations on a vertical number line. [Vertical line provided.] The number line shows that the salinity test is deepest. The number line also shows that the atmosphere test and the ice test are closest to the surface. To determine which is closer to the surface, identify which elevation has a lesser absolute value. Atmosphere: ∣0.3∣ = 0.3 Ice: ∣−0.25∣ = 0.25 So, the salinity test is deepest and the ice test is closest to the surface." (7.NS.1)
Chapter 2, Lesson 1, Problem 17, "On a mountain, the temperature decreases by 18°F for each 5000-foot increase in elevation. At 7000 feet, the temperature is 41°F. What is the temperature at 22,000 feet? Justify your answer." (7.NS.3, multi-step, routine)
Chapter 3, Lesson 4, Problem 41, Dig Deeper, "A square fire pit with a side length of s feet is bordered by 1-foot square stones as shown. [Diagram provided] a. How many stones does it take to border the fire pit with two rows of stones? Use a diagram to justify your answer." (routine) "b. You border the fire pit with n rows of stones. How many stones are in the nth row? Explain your reasoning." (non-routine) (7.EE.3)
Chapter 6, Lesson 3, Problem 32, Dig Deeper, "At a restaurant, the amount of your bill before taxes and tip is $19.83. A 6% sales tax is applied to your bill, and you leave a tip equal to 19% of the original amount. Use mental math to estimate the total amount of money you pay. Explain your reasoning. (Hint: Use 10% of the original amount.)" (7.RP.3, routine)
In Chapter 4, Lesson 3, Solving Two-Step Equations, students begin with an Exploration example that uses algebra tiles to show the steps for solving an equation and the relationship to the properties of equality. These examples show the conceptual solving of an equation through models. The lesson shifts to a procedural steps of solving two step equations with Examples 1: "-3x + 5 = 2" and Example 2: "x/8- 1/2 = -7/2". Example 3 is a procedural example of solving two step equations by combining like terms "3y - 8y = 25". The lesson progresses to independent application of the skill in Concepts, Skills, and Problem Solving. Students solve equations procedurally.
Chapter 6, Lesson 1, Fractions, Decimals and Percents, students begin the lesson with an Exploration activity where they compare numbers in different forms based on a variety of strategies. Example 1, presents a conceptual model of a decimal using a hundredth grid, and how to convert a decimal to a percent. Example 2, shows the students how to procedurally build on what they have learned to convert a fraction to a decimal to a percent using division. The lesson then moves to independent practice in Concepts, Skills, and Problem Solving where students procedurally convert between decimals, percents, and fractions.
Chapter 7, Lesson 2, Experimental and Theoretical Probability, students' learning begins with an Exploration activity in which students conduct two experiments to find relative frequencies (Flip a Quarter and Toss and Thumbtack) to understand the concept behind probability. The lesson moves on to Example 1, Finding an Experimental Probability by utilizing a formula. "$$P(event) =\frac {number of times the event occurs}{total number of trials}$$", and Example 2, Finding a Theoretical Probability, by utilizing the formula "$$P (event)= \frac{number of favorable outcomes}{number of possible outcomes}$$". Example 3, shows the steps for applying each formula to compare probabilities. The bar growth shows the results of rolling a number cube 300 times. How does the experimental probability of rolling an odd number compare with the theoretical probability?" The independent practice in Concepts, Skills, and Problem Solving has the students finding an experimental probability and theoretical probability based on an event.
Chapter 9, Lesson 1, Circles and Circumference, begins with Exploration 1, where students use a compass to draw circles and conceptually see the length of the diameter and circumference. Exploration 2, continues to explore diameter and circumference through hands on modeling. The lesson continues with three examples showing the steps of applying the formula for finding radius, circumference, and perimeter of a circle. The independent work of the students is within the Concepts, Skills, and Problem Solving in which students are asked to procedurally solve for the radius, diameter, circumference and perimeter.
Chapter 1, Lesson 4, Subtracting Rational Numbers, Exploration 1 (MP2), students work with a partner in answering the following questions: a. Choose a unit fraction to represent the space between the tick marks on each number line. "What expressions involving subtraction are being modeled? What are the differences? b. Do the rules for subtracting integers apply to all rational numbers? Explain your reasoning. You have used the commutative and associative properties to add integers. Do these properties apply in expressions involving subtraction? Explain your reasoning." MP2 is identified in the teaching notes, "The number line helps students see that the rules for subtracting rational numbers shouldn't be different from the rules for subtracting integers."
Chapter 8, Lesson 1, Samples and Populations, Example 2 (MP3), students are given the scenario, "You want to know how the residents of your town feel about adding a new landfill. Determine whether each conclusion is valid." Students are provided with information about the survey. MP3 is identified in the teaching notes, "Ask a volunteer to read part (a). Then ask whether the conclusion is valid. Students should recognize that the sample is biased because the survey was not random—you only surveyed nearby residents. Ask a volunteer to read part (b). Then ask whether the conclusion is valid. Students should recognize that the sample is random and large enough to provide accurate data, so it is an unbiased sample."
Chapter 5, Lesson 4, Writing and Solving Proportions, Example 3 (MP1), students are provided with two examples of solving proportions using cross products. MP1 is identified in the teaching notes, "As you work through the problems with students, share with them the wisdom of analyzing the problem first to decide what method makes the most sense."
The MPs are identified in the digital Student Dashboard under Student Resources, Standards for Mathematical Practice. This link takes you to the same information found in the Teacher Edition. For example:
Chapter 9, Lesson 1, Circles and Circumference, Exploration 2 - Exploring Diameter and Circumference, students work with a partner and find the circumference and diameter of a circular base. They determine whether the circumference or diameter is greater and by how much. "Math Practice - Calculate Accurately," students are asked, "What other methods can you use to calculate the circumference of a circle? Which methods are more accurate?"
Chapter 6, Lesson 1, Fractions, Decimals, and Percents, Concepts, Skills & Problem Solving, Problem 39, "MP Problem Solving", "The table shows the portion of students in each grade that participate in School Spirit Week. Order the grades by portion of participation from least to greatest."
Chapter 2, Lesson 4, Multiplying Rational Numbers, Concept Skills, & Problem Solving, Problems 10-12. "MP Reasoning", "Without multiplying, tell whether the value of the expression is positive or negative. Explain your reasoning."
MP7 and MP8 are under-identified in the series, both are identified in four of the ten chapters.
The instructional materials do not present opportunities for students to engage in MP1: Make Sense of Problems and Persevere in Solving Them, MP4: Model with mathematics, and MP5: Use appropriate tools strategically.
Chapter 2, Lesson 3, Laurie's Notes, Example 1, "Mathematically proficient students are able to plan a solution. Choosing between methods may help students be more efficient and accurate when writing fractions as decimals. Complete part (a) as a class. The first step is to write the mixed number as an equivalent improper fraction. Then divide the numerator by the denominator. Point out that the negative sign is simply placed in the answer after the calculations are complete. Discuss the Another Method note with students. Point out that to find an equivalent fraction with a denominator that is a power of 10, you multiply the numerator and denominator by powers of 2 or 5. This is not possible for repeating decimals. Complete part (b) as a class. Remind students to always divide the numerator by the denominator, regardless of the size of the numbers!" In Example 1, the solution is provided for students and therefore they do not have to persevere in solving the problem.
Chapter 5, Lesson 5, Laurie's Notes, Example 3, "Ask students to explain why the graph represents a ratio relationship and to identify the unit rate. Plotting the ordered pairs confirms that x and y are proportional. 'What is the constant of proportionality?' 16. 'What is the equation of the line?' y = 16x. Students can use the equation to find the area cleaned for any amount of time." Students are analyzing a given model, not using a model to solve a problem.
Chapter 7, Lesson 3, Laurie's Notes, Example 1, "The tree diagram helps students visualize the 8 outcomes in the sample space." Students are provided with a worked out example, and do not create a tree diagram as a way to model a problem independently.
MP5: While the Dynamic Student Edition includes tools for students, the instructional materials present few opportunities for students to choose their own tool, therefore, the full meaning of MP5 is not being attended to. For example:
Chapter 8, Lesson 2, Laurie's Notes, Example 2, "Students can use calculators to quickly find the mean of each sample." Teachers direct students to use calculators.
Chapter 7, Lesson 2, Laurie's Notes, Exploration 1, "Combine the results for each experiment. As the data are gathered and recorded, several students with calculators can summarize the results." Students are not selecting their own tool in this example.
"You be the Teacher", found in many lessons, presents opportunities for students to critique the reasoning of others, and construct arguments. Examples of where students engage in the full intent of MP3 include the following:
Chapter 4, Lesson 2, Problem 28, You Be the Teacher, "Your friend solves the equation -4.2x=21. Is your friend correct? Explain your reasoning." The student work is provided to examine.
Chapter 6, Lesson 1, Problem 20, You Be the Teacher, "Your friend uses the percent proportion to answer the question below. Is your friend correct? Explain your reasoning. '40% of what number is 34?'" The student work is provided to examine.
The Student Edition labels MP3 as "MP Construct Arguments," however, these activities do not always require students to construct arguments. In the Student Edition, "Construct Arguments" was labeled only once for students and "Build Arguments" was labeled once for students. For example:
Chapter 2, Lesson 1, Construct Arguments, students construct viable arguments by writing general rules for multiplying (i) two integers with the same sign and (ii) two integers with different signs. Students are prompted to "Construct an argument that you can use to convince a friend of the rules you wrote in Exploration 1(c)."
Chapter 8, Lesson 4, Exploration 1, Build Arguments is identified in the Math Practice blue box with the following question, "How does taking multiple random samples allow you to make conclusions about two populations?"
In Chapter 1, Lesson 4, Subtracting Integers, students are shown an example of subtracting integers. In Laurie's notes, teachers are prompted, "Ask students if it is possible to determine when the difference of two negative numbers will be positive and when the difference of two negative numbers will be negative."
In Chapter 5, Lesson 2, Example 1, students find a unit rate based on given information. In Laurie's notes, teachers are prompted, "There are several ways in which students may explain their reasoning. Take time to hear a variety of approaches." This is labeled as MP3, but there is no support for teachers to assist students in constructing a viable argument or critiquing the thoughts of others.
Chapter 1, Lesson 2, Example 2, The Teacher's Guide is noted with MP3 with the following directions, "'When you add two integers with different signs, how do you know if the sum is positive or negative?' Students answered a similar question in Example 1, but now they should be using the concept of absolute value, even if they don't use the precise language. You want to hear something about the size of the number, meaning its absolute value." There is no reference to MP3 in the Student Edition in this Lesson.
The materials attend to the vocabulary at the beginning of each chapter in the Getting Ready section. For example, in the Getting Ready section for Chapter 3, students read, "The following vocabulary terms (like terms, linear expression, factoring an expression) are defined in this chapter. Think about what each term might mean and record your thoughts." In Laurie's Notes for the chapter, teachers are provided with the following notes regarding the vocabulary: "A. These terms represent some of the vocabulary that students will encounter in Chapter 3. Discuss the terms as a class. B. Where have students heard the word like terms outside of a math classroom? In what contexts? Students may not be able to write the actual definition, but they may write phrases associated with like terms. C. Allowing students to discuss these terms now will prepare them for understanding the terms as they are presented in the chapter. D. When students encounter a new definition, encourage them to write in their Student Journals. They will revisit these definitions during the Chapter Review."
Key vocabulary for a section is noted in a box in the margins of the student textbook, along with a list of pages where the students will encounter the vocabulary. Vocabulary also appears in some of the Key Ideas boxes. For example, in Chapter 6, Lesson 4, the Key Idea box contains the definition for percent of change, percent of increase, and percent of decrease with an equation of how to find each.
Each chapter has a review section that includes a list of vocabulary important to the unit and the page number the students will find the terms. For example, in Chapter 4, Review, teachers are given the prompt: "As a review of the chapter vocabulary, have students revisit the vocabulary section in their Student Journals to fill in any missing definitions and record examples of each term." In the Student Edition, the terms and page number are provided and students are asked to "Write the definition and give an example of each vocabulary term." Additionally, there is a Graphic Organizer Section where students need to create a "Summary Triangle" for each concept.
The Chapter 4, Laurie's Notes, Chapter 4 Overview states, "Be sure to use precise language when discussing multiplying or dividing an inequality by a negative quantity. Use language such as, "The direction of the inequality symbol must be reversed." Simply saying, "switch the sign" is not precise."
In Chapter 7, Chapter Exploration includes a list of vocabulary words related to probability. Laurie's Notes (page T-282) guides teachers to have students use contextual clues and record notes and definitions related to the mathematical terms throughout the chapter.
In Chapter 9, Section 9.4, Laurie's Notes, "Motivate, guides teachers to play a game that will help students remember vocabulary and their meanings relating to triangles."
In Chapter 2, Lesson 1, Laurie's Notes remind teachers that "students should say, "Negative 5 times negative 6 equals 30". Teachers are advised to respond to students saying, "minus 5", by reminding them that minus represents an operation.
In Chapter 8, Lesson 1, Laurie's Notes, teachers are asked to discuss the following, "Define unbiased sample and biased sample. Give a few examples of each. Then ask students to write the definitions in their own words and share an example of each type of sample. The size of a sample can have a great influence on the results. A sample that is not large enough may not be unbiased and a sample that is too large may be too cumbersome to use. As a rule of thumb, a sample of 30 is usually large enough to provide accurate data for modest population sizes."
In Chapter 7, Lesson 1, Laurie's Notes, teachers are asked to "Discuss the vocabulary words: experiment, outcomes, event, and favorable outcomes. You can relate the vocabulary to the exploration and to rolling two number cubes. 'What does it mean to perform an experiment at random?' All of the possible outcomes are equally likely. Ask students to identify the favorable outcomes for the events of choosing each color of marble. green (2), blue (1), red (1), yellow (1), purple (1) Be sure students understand that there can be more than one favorable outcome. 'What are some other examples of experiments and events? What are the favorable outcomes for these events?' Sample answer: An experiment is rolling a number cube with the numbers 1–6. An event is rolling a number greater than 4, with favorable outcomes of 5 and 6."
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what is transistor biasing
One of the few examples is "TR One, one transistor radio" TR One, Ch 9 with an amplified AM (amplitude modulation) detector. As the value of β and the value of VBE are not same for every transistor, whenever a transistor is replaced, the operating point tends to change. Meaning is current conduction takes place only due to one type of carrier electron or holes. produce the desired amplification or switching effect. An example of an audio amplifier stage using base-biasing is "Crystal radio with one transistor . Therefore, the operating point needs to be stabilized i.e. Transistor Biasing Transistor Biasing is the process of setting a transistors DC operating voltage or current conditions to the correct level so that any AC input signal can … As temperature increases, the values of ICE, β, VBE gets affected. 8 (ii). transistor) has the characteristics shown in Fig. The below figure shows a transistor amplifier that is provided with DC biasing on both input and output circuits. The stability factor should be as low as possible so that the collector current doesn't get affected. Biasing is process of applying potential (DC) across any electronic equipment in order to make it operate as we require(i.e in our region of interest). It is convenient to use the existing VCC supply instead of a new bias supply. You may not understand this not, but we're going The given DC voltage and currents are so chosen that the transistor remains in active region for entire input AC cycle. Fixed bias with emitter resistor It is easy to understand that depending on how they are put together, two basic types of transistors can result, as shown in Figure 1. Below is a typical BJT receiving base bias: VBB is the base supply voltage, which is used to give the transistor sufficient current to turn the transistor on. A similar circuit is shown in the figure below. Hence operating point should be made independent of the temperature so as to achieve stability. If a signal of very small voltage is given to the input of BJT, it cannot be amplified. stability against variations in β that may exist from one transistor to the next. Let us have a look at the factors that affect the stabilization of operating point. The circuit which provides transistor biasing is called as Biasing Circuit. Fig. For bipolar junction transistors and FET transistors, there are a plethora of different methods available to ensure Here the base-emitter junction of the transistor is forward biased by the voltage drop across RB which is the result of IB flowing through it. What is Transistor Biasing: Transistor Biasing is the process of setting a transistors DC operating voltage or current conditions to the correct level so that any AC input signal can be amplified correctly by the transistor.. Collector Feedback Biasing a Transistor: Collector Feedback Biasing circuit. In electronics, biasing is the setting of initial operating conditions (current and voltage) of an active device in an amplifier. Biasing a transistor is done when you are using it to amplify a signal and is a method of keeping it conducting when the signal is not present. One way to bias a BJT transistor is a method called voltage divider bias. various regions in order This leads to a constant value for IB resulting in a fixed operating point due to which the circuit is named as f… Transistor Working Principle: Transistor is nothing but a transferred resistor (Transistor = transferred + resistor). Stabilization of the operating point has to be achieved due to the following reasons. In this video, the basic of the transistor biasing like what is load line, what is Q-point, What is biasing, why BJT requires biasing is explained. An anti-parallel diode is called transistor, which means two diodes are connected reversely to form a new electronics component, such components are called Transistor. FIGURE 1 – Schematic symbol, physical representation and diode model of NPN transistor. It is economical to minimize the DC source to one supply instead of two which also makes the circuit simple. It is understood that IC should be kept constant in spite of variations of ICBO or ICO. The self-destruction of such an unstabilized transistor is known as Thermal run away. Fig. To achieve this, biasing circuits are introduced. Whatever be the application, a stabilized DC bias is a must for proper transistor functioning. The transistor acts exactly like a dead phone when there is no DC bias. Your email address will not be published. The BJT should be in the active region, to be operated as an amplifier. Then only the AC input signal can be amplified by the transistor correctly. The collector leakage current ICBO is greatly influenced by temperature variations. Hey, thanks for the A2A! The operating point shifts due to change in temperature. It denoted by S. By definition, the rate of change of collector current IC with respect to the collector leakage current ICO at constant β and IB is called Stability factor. Bipolar Junction Transistor Biasing Transistors are the most important semiconductor active devices essential for almost all circuits. The general expression of stability factor for a CE configuration can be obtained as under. . The basic purpose of transistor biasing is to keep the base-emitter junction forward biased and collector- base junction reverse biased at any instant of the applied signal. BJT takes the conduction in electron and holes. The process of making the operating point independent of temperature changes or variations in transistor parameters is known as Stabilization. Many electronic devices, such as diodes, transistors and vacuum tubes, whose function is processing time-varying signals, also require a steady (DC) current or voltage at their terminals to operate correctly.This current or voltage is a bias. The input voltage should exceed cut-in voltage for the transistor to be ON. In the transistor amplifier circuits drawn so far biasing was done with the aid of a battery VBB which was separate from the battery VCC used in the output circuit. The simplest biasing applies a base-bias resistor between the base and a base battery VBB. A transistor is based in order to make the emitter base junction forward biased and collector base junction reverse biased, so that it maintains in active region, to work as an amplifier. Accordingly, the two types of a junction transistor are PNP and NPN. Transistor biasing can be defined as the proper flow of zero signal collector current and the maintenance of proper collector-emitter voltage during the passage of signal. Once the stabilization is achieved, the values of IC and VCE become independent of temperature variations or replacement of transistor. In other words, midpoint biasing provides the largest possible output. 8 (i) shows the base resistor transistor circuit. PNP transistor works when the emitter-base junction is forward biased while collector-base junction is reverse biased. Biasing of the bipolar junction transistor (BJT) is the process of applying external voltages to it. the intended output effect. Base bias the simplest way to bias a BJT transistor. Although transistor switching circuits operate without bias, it is unusual for analog circuits to operate without bias. The device (i.e. Base bias ensures that the voltage fed to the base, VBB, is the correct voltage, which then supplies the correct current so that the BJT has enough base current to switch the transistor on. They are used as electronic switches, amplifiers, etc in circuits. TRANSISTOR BIASING Questions and Answers pdf free Download :: Post Views: 184. Transistor biasing is the controlled amount of voltage and current that must go to a transistor for it to However, in modern transistors, I CBO is usually less than 100 nA and its effect on the bias is negligible if V BB >> I CBO R B. Q10. The transistor biasing the process of setting the DC voltage and the current in the transistor to the correct level so that the AC input signal can be properly amplified. Transistor biasing is the controlled amount of voltage and current that must go to a transistor for it to produce the desired amplification or switching effect. What is Transistor Biasing: Transistor Biasing is the process of setting a transistors DC operating voltage or current conditions to the correct level so that any AC input signal can be amplified correctly by the transistor.. Voltage Divider Bias of a BJT Transistor: Voltage Divider Transistor Biasing circuit. Name * Email * Website. The biasing circuit shown by Figure 1 has a base resistor RB connected between the base and the VCC. The process of making the operating point independent of temperature changes or variations in transistor parameters is known as Stabilization. This classification is based on the polarity in the structure. Hope it helps! The reason is that midpoint biasing allows optimum operation of the amplifier. There are oth… Graphical Analysis of Self-Biased JFET. Let us understand these concepts in detail. In order to avoid thermal runaway and the destruction of transistor, it is necessary to stabilize the operating point, i.e., to keep IC constant. In the previous chapter, we explained how a transistor acts as a good amplifier, if both the input and output sections are biased. However, one application of fixed bias is to achieve crude automatic gain control in the transistor by feeding the base resistor from a DC signal derived from the AC output of a later stage. Biasing is the process of providing DC voltage which helps in the functioning of the circuit. Hence DC biasing is needed. Determine V CC, R C and R B . Posted on by Leave a comment. The gain of a transistor can vary significantly between different batches, which results in widely different operating points for sequential units in serial production or after replacement of a transistor. A good biasing circuit helps in the stabilization of operating point. Without appropriate transistor biasing, the transistor may not function at all or amplify very poorly, such as In this video, the basics of the transistor biasing are explained with the help of load line and Q-point. Leave a Reply Cancel reply. It means that, if the Base voltage is zero or less than 0.7 V, the current cannot flow and it acts as an open circuit. So, Base voltage is a minimum of 0.7 V in reverse bias to conduct the transistor. Transistors may be NPN, PNP, FET, JFET, etc which have different functions in electronic circuits. Hence it is necessary to stabilize the operating point. The flow of collector current and also the collector leakage current causes heat dissipation. What Are Methods Of Transistor Biasing? to function properly and amplify signals to the correct level. If appropriate DC voltages and currents are given through BJT by external sources, so that BJT operates in active region and superimpose the AC signals to be amplified, then this problem can be avoided. Note the resistor from the base to the battery terminal. VCC). that transistors are biased correctly to produce proper amplification and/or switching. The main factor that affect the operating point is the temperature. " crystal radio, Ch 9. To come out of this, the biasing conditions are set so that zero signal collector current IC = 1 mA. Differentiating above expression with respect to IC, we get, $$1 = \beta \frac{d I_B}{d I_C} + (\beta + 1)\frac{d I_{CO}}{dI_C}$$, $$1 = \beta \frac{d I_B}{d I_C} + \frac{(\beta + 1)}{S}$$, Since $\frac{d I_{CO}}{d I_C} = \frac{1}{S}$, $$S = \frac{\beta + 1}{1 - \beta \left (\frac{d I_B}{d I_C} \right )}$$. You can't build a transistor out of two diodes, but using two diodes helps to explain how the transistor biasing works. When a transistor is used as an amplifier, it is always designed for midpoint bias. Above is an example of BJT biasing. In order to use the BJT for any application like amplification, the two junctions of the transistor CB and BE should be properly biased according to the required application. Figure 1Two basic types of transistors: (a) PNP and (b) NPN. The extent to which a biasing circuit is successful in maintaining this is measured by Stability factor. A good biasing circuit helps in the stabilization of operating point. However, in the interest of simplicity and economy, it is desirable that transistor circuit should have a single source of supply—the one in the output circuit (i.e. Therefore, it's very important that a transistor is biased correctly for it to produce If the operating point is not stabilized, there occurs a cumulative effect which increases this heat dissipation. For a transistor to be operated as a faithful amplifier, the operating point should be stabilized. For a transistor to … The biasing in transistor circuits is done by using two DC sources V BB and V CC. This is the major difference between the bipolar junction transistor and field effect transistor. to go over each of the methods, so you'll have a clearer idea. The proper flow of zero signal collector current and the maintenance of proper collectoremitter voltage during the passage of signal is known as Transistor Biasing. In other words, transistors must be fed the correct or appropriate levels of voltages and/or currents to their various regions in order to function properly and amplify signals to the correct level. Base Resistor method; Collector to Base bias; Biasing with Collector feedback resistor; Voltage-divider bias RB is a resistance value that is used to provid… This controlled amount of voltage and/or currents fed to the Advertisement. From the figure, the mathematical expression for IBis obtained as Here the values of VCC and VBE are fixed while the value for RB is constant once the circuit is designed. This point is … The BJT transistor must receive the appropriate level of base current IB, collector Once the stabilization is achieved, the values of IC and VCEbecome independent of temperature variations or replacement of transistor. PNP Transistor as Open Switch. The emitter terminal is formed by P-type semiconductor thus, for forward biasing the P-type terminal should be connected with positive terminal and N-type with negative terminal. What is Field Effect Transistor: Field effect transistor is a unipolar device. So the main problem which affects the operating point is temperature. Hence we can understand that any change in collector leakage current changes the collector current to a great extent. Transformer Coupled Class A Power Amplifier. In other words, transistors must be fed the correct or appropriate levels of voltages and/or currents to their S=1 is the ideal value. 8. Lecture Series on Basic Electronics by Prof. T.S.Natarajan, Department of physics, IIT Madras For more Courses visit http://nptel.iitm.ac.in The bias circuit stabilizes the operating point of the transistor for variations in transistor characteristics and operating temperature. it is necessary to keep IC constant. Required fields are marked * Comment. Fixed Bias • The fixed-bias configuration is the simplest of transistor biasing arrangements, but it is also quite unstable •For most configurations the dc analysis begins with a determination of the base current •For the dc analysis of a transistor network, all capacitors are replaced by an open-circuit equivalent Fixed Base Bias Method: Fixed Base bias or Base resistor method is the basic type of transistor bias … . Need for biasing a transistor are: To make operating point to be at the centre of load line for faithful amplification. 6..Explain Emitter feedback bias method or Fixed bias with emitter resistor. voltage VCC, collector current IC, and the resistance values of the resistors help to provide correct bias levels and provide The commonly used methods of transistor biasing are. Biasing a transistor is applying a suitable DC voltage across the transistor terminals to operate the transistor in the desired region. Hence the stability factor S depends on β, IB and IC. We use a transfer characteristic curve of junction field … Bipolar transistors are biased to operate correctly. transistor is used as a switch. The DC operating voltage or current conditions of a transistor is set to get the correct level. $S = \frac{d I_C}{d I_{CO}}$ at constant IB and β. Because, for a BJT, to amplify a signal, two conditions have to be met. 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\begin{document}
\begin{titlepage}
\title{Implicit Manifold Reconstruction\thanks{A preliminary version appears in Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms, 2014, 161--173.}} \author{Siu-Wing Cheng\footnote{Supported by Research Grants Council, Hong Kong, China (project~no.~612109). Department of Computer Science and Engineering, HKUST, Hong Kong.} \and Man-Kwun Chiu\footnote{Supported by JST ERATO Grant Number JPMJER1201, Japan and ERC StG 757609. Institut f\"ur Informatik, Freie Universit\"at Berlin, Berlin, Germany.}}
\date{}
\maketitle
\begin{abstract}
Let ${\cal M} \subset \mathbb{R}^d$ be a compact, smooth and boundaryless manifold with dimension $m$ and unit reach. We show how to construct a function $\varphi: \real^d \rightarrow \real^{d-m}$ from a uniform $(\eps,\kappa)$-sample $P$ of $\cal M$ that offers several guarantees. Let $Z_\varphi$ denote the zero set of $\varphi$. Let $\widehat{{\cal M}}$ denote the set of points at distance $\eps$ or less from $\cal M$. There exists $\eps_0 \in (0,1)$ that decreases as $d$ increases such that if $\eps \leq \eps_0$, the following guarantees hold. First, $Z_\varphi \cap \widehat{\cal M}$ is a faithful approximation of $\cal M$ in the sense that $Z_\varphi \cap \widehat{\cal M}$ is homeomorphic to $\cal M$, the Hausdorff distance between $Z_\varphi \cap \widehat{\cal M}$ and $\cal M$ is $O(m^{5/2}\eps^{2})$, and the normal spaces at nearby points in $Z_\varphi \cap \widehat{\cal M}$ and $\cal M$ make an angle $O(m^2\sqrt{\kappa\eps})$. Second, $\varphi$ has local support; in particular, the value of $\varphi$ at a point is affected only by sample points in $P$ that lie within a distance of $O(m\eps)$. Third, we give a projection operator that only uses sample points in $P$ at distance $O(m\eps)$ from the initial point. The projection operator maps any initial point near $P$ onto $Z_\varphi \cap \widehat{\cal M}$ in the limit by repeated applications.
\end{abstract}
\end{titlepage}
\section{Introduction}
Sensory devices and numerical experiments may generate numerous data points in $\real^d$ for some large $d$ due to the large number of attributes of the data that are being monitored. It is often believed that the data points are governed by some hidden processes with fewer controlling parameters, and therefore, the data points may lie in some $m$-dimensional manifold $\cal M$ for some $m \ll d$. This motivates the study of manifold reconstruction.
In computational geometry, there are several known results that offer provably faithful reconstructions in the sense that the reconstruction is topologically equivalent to $\cal M$, the Hausdorff distance between the reconstruction and $\cal M$ decreases as the sampling density increases, and the angular error between the tangent spaces at nearby points in the reconstruction and $\cal M$ decreases as the sampling density increases. These include the weighted cocone complex by Cheng, Dey and Ramos~\cite{CDR05}, the weighted witness complex by Boissonnat, Guibas and Oudot~\cite{bgo}, and the tangential Delaunay complex by Boissonnat and Ghosh~\cite{bg}. These reconstructions are $m$-dimensional simplicial complexes with the given sample points as vertices.
The corresponding reconstruction algorithms have to deal with the challenging issue of ``sliver removal'' in high dimensions.
Solutions of partial differential equations on manifolds are required in quite a few areas such as biology~\cite{memoli}, image processing~\cite{turk,witkin}, weathering~\cite{dorsey}, and fluid dynamics~\cite{myers,myersb}. The underlying manifold is often specified by a point cloud. It has been reported~\cite{liang} that local reconstructions of a manifold in the form of zero level sets of local functions are preferred for solving partial differential equations on the manifold.
Several numerical methods for solving partial differential equations on level sets have been developed~\cite{bertalmio,greer,liang,ruuth}.
In this paper, we propose an implicit reconstruction for manifolds with arbitrary codimension in $\real^d$. Let $\cal M$ be a compact, smooth, and boundaryless manifold with unit reach. Let $P$ be a \emph{uniform $(\eps,\kappa)$-sample} of $\cal M$, that is, every point in ${\cal M}$ is at distance $\eps$ or less from some point in $P$ and the number of sample points inside any $d$-ball of radius $\eps$ is at most some constant $\kappa$. We assume that the following information is specified in the input: (i) the manifold dimension $m$, (ii) a \emph{neighborhood radius} $\gamma = 4\eps$, and (iii) approximate tangent spaces at points in $P$ such that the true tangent space at each point in $P$ makes an angle at most $m\gamma$ with the given approximate tangent space at that point. There are many algorithms for estimating the manifold dimension (e.g.~\cite{CC09,CWW,HA,LB,TSL00}). When the sample points satisfy some local uniformity condition (e.g., a constant upper bound on the number of sample points inside any ball of radius $\eps$ centered in $\cal M$), the neighborhood radius $\gamma$ can be set by measuring the maximum distance from a sample point to its $k$th nearest neighbor for some appropriate $k$. If the sample points are drawn from an independent and identical distribution on $\cal M$, a recently proposed reach estimator can be used to set $\gamma$~\cite{aamari}. There are many algorithms for estimating tangent spaces (e.g.~\cite{BSW,CC16,GW,LMR12,SSM98}), which give an $O(\eps)$ angular error.
We use the conditions of $\gamma = 4\eps$ and angular error at most $m\gamma$ in order to keep the number of unknown parameters small. One may worry about satisfying these two conditions simultaneously, but it is not a concern as we explain below. Suppose that the estimation algorithms return an angular error bound of $c\eps$ for some known constant $c \geq 1$ and a value $\ell$ such that $\eps \leq \ell = O(\eps)$. We can set $\gamma = \max\{4\ell,c\ell\}$. Then, the angular error is at most $c\eps \leq c\ell \leq m\gamma$. Moreover, letting $c' = \max\{\frac{\ell}{\eps},\frac{c\ell}{4\eps}\}$, the input sample can be viewed as a uniform $(\eps',\kappa')$-sample, where $\eps' = c'\eps = \gamma/4$ and $\kappa' = (2c'+1)^d \kappa$, because a packing argument shows that if any $d$-ball of radius $\eps$ contains at most $\kappa$ sample points, then any $d$-ball of radius $c'\eps$ contains at most $(2c'+1)^d \kappa$ sample points.
Our main result is a formula for a function $\varphi: \real^d \rightarrow \real^{d-m}$ using the $(\eps,\kappa)$-sample $P$ and the neighborhood radius $\gamma$ such that
the zero set of $\varphi$ near $\cal M$ forms a reconstruction of $\cal M$. Let $Z_\varphi$ denote the zero set of $\varphi$. Let $\widehat{\cal M}$ denote the set of points at distance $\eps$ or less from $\cal M$. We prove that there exists $\eps_0 \in (0,1)$ that decreases as $d$ increases such that if $\eps \leq \eps_0$, the following guarantees hold. First, $Z_\varphi \cap \widehat{\cal M}$ is a faithful approximation of $\cal M$ in the sense that $Z_\varphi \cap \widehat{\cal M}$ is homeomorphic to $\cal M$, the Hausdorff distance between $Z_\varphi \cap \widehat{\cal M}$ and $\cal M$ is $O(m^{5/2}\gamma^{2}) = O(m^{5/2}\eps^2)$, and the normal spaces at nearby points in $Z_\varphi \cap \widehat{\cal M}$ and $\cal M$ make an angle $O(m^2\sqrt{\kappa\gamma}) = O(m^2\sqrt{\kappa\eps})$. Second, $\varphi$ has local support; in particular, the value of $\varphi$ at a point is affected only by sample points in $P$ that lie within a distance of $m\gamma$. Third, we give a projection operator that only uses sample points in $P$ at distance $m\gamma$ from the initial point. The projection operator maps any initial point near $P$ onto $Z_\varphi \cap \widehat{\cal M}$ in the limit by repeated applications.
Implicit surfaces in three dimensions have been extensively studied, particularly in computer graphics and solid modeling (e.g.~\cite{alexa,carr,hoppe,levin}). Two functions have been defined in~\cite{DS,Kol} and shown to give faithful reconstruction of the underlying surface in three dimensions. In $\real^d$, a function is defined in~\cite{bf} and shown to give faithful reconstruction of $(d-1)$-dimensional manifold. There seems to be no prior work with provable guarantees on implicit reconstructions of manifolds in $\real^d$ with codimension less than $d-1$. In the computer graphics community, similar functions have been proposed as projection operators by Adamson and Alexa~\cite{AA} for designing a complex of surface patches connected via vertices and curves in three dimensions. Each surface patch is the set of stationary points under a projection operator. For each surface patch, some input points with prescribed tangent spaces are given for defining the corresponding projection operator, but these input points need not form an $\eps$-sample of the resulting surface patch. It is discussed how to generalize the framework to $\real^d$ for a complex of submanifolds. However, no mathematical guarantee was provided in~\cite{AA} for $\real^3$ or $\real^d$.
Although the zero set of our function $\varphi$ has a subset near $\cal M$ that is a faithful reconstruction, $\varphi$ should not be confused to be an smooth implicit function as in the Implicit Function Theorem. If the normal bundle of $\cal M$ is topologically non-trivial, one cannot define a smooth implicit function whose zero set is a faithful reconstruction of $\cal M$.
We provide the definition of our function $\varphi$ in the next section. Afterwards, we give the proofs of the theoretical guarantees.
\section{Function formulation} \label{sec:def}
We use lowercase and uppercase letters in {\sf mathsf} font to denote column vectors and matrices, respectively. A point is always specified as a column vector. Given a matrix $\mat{K}$, we use $\col{\mat{K}}$ to denote the column space of $\mat{K}$. We call the unit eigenvectors of a square matrix corresponding to the $k$ largest (resp. smallest) eigenvalues the $k$ \emph{most dominant} (resp. \emph{least dominant}) unit eigenvectors.
Recall that $\gamma = 4\eps$ is the input neighborhood radius. We will make use of a weight function $\omega : \real^d \rightarrow \real$ defined as \begin{eqnarray*} \omega(\mat{x},\mat{p}) = \frac{h(\norm{\mat{x}-\mat{p}})} {\sum_{\mat{q} \in P} h(\norm{\mat{x}-\mat{q}})}, \end{eqnarray*} where \[ h(s) = \left\{\begin{array}{ll} \displaystyle \left(1-\frac{s}{m\gamma}\right)^{2m} \left(\frac{2s}{\gamma}+ 1\right), & \mbox{if $s \in [0, m\gamma]$}, \\ [1.5em] 0, & \mbox{if $s > m\gamma$}. \end{array}\right. \] Note that $h$ is differentiable in $(0,\infty)$ and $h'(s) = 0$ for $s \geq m\gamma$. This weight function is inspired by the Wendland functions~\cite{wendland}.
Since approximate tangent spaces at the sample points are specified in the input, we can assume that a $d \times m$ matrix $\mat{T}_\mat{p}$ is given for each $\mat{p} \in P$ such that $\mat{T}_\mat{p}$ has orthogonal unit columns and $\col{\mat{T}_\mat{p}}$ is the approximate tangent space at $\mat{p}$. Define the following matrix and vector space for each point $\mat{x} \in \real^d$: \begin{center} \begin{tabular}{lcp{4.5in}} $\mat{C}_{\mat{x}}$ & = & $\sum_{\mat{p} \, \in P} \omega(\mat{x},\mat{p}) \cdot \mat{T}_{\mat{p}}^{} \cdot \mat{T}_{\mat{p}}^t$, \\ $L_\mat{x}$ & = & space spanned by the $(d-m)$ least dominant unit eigenvectors of $\mat{C}_{\mat{x}}$. \\ \end{tabular} \end{center} The $(d-m)$ least dominant unit eigenvectors of $\mat{T}_\mat{p}^{} \cdot \mat{T}_\mat{p}^t$ span an approximate normal space of $\cal M$ at $\mat{p}$. So $L_\mat{x}$ is the ``weighted average'' of the approximate normal spaces at the sample points near $\mat{x}$.
Define a class $\Phi$ of functions $\varrho: \real^d \rightarrow \real^{d-m}$ as follows: \begin{quote} $\displaystyle \Phi = \left\{\varrho \, :\, \varrho(\mat{x}) = \sum_{\mat{p}\, \in P} \omega(\mat{x},\mat{p}) \cdot \mat{B}_{\varrho,\mat{x}}^t \cdot (\mat{x}-\mat{p})\right\}$, where $\mat{B}_{\varrho,\mat{x}}$ is any $d \times (d-m)$ matrix with linearly independent columns such that $\col{\mat{B}_{\varrho,\mat{x}}} = L_\mat{x}$. \end{quote} Evaluating $\varrho(\mat{x})$ requires only the sample points at distance $m\gamma$ or less from $\mat{x}$, and $\omega$ gives more weight to sample points nearer $\mat{x}$. Different choices of $\mat{B}_{\varrho,\mat{x}}$ at each $\mat{x} \in \real^d$ give rise to different functions in $\Phi$. A natural choice is a $d \times (d-m)$ matrix consisting of $d-m$ orthogonal unit vectors that span $L_\mat{x}$.
We denote the corresponding function in $\Phi$ by $\varphi$ and so \[ \varphi(\mat{x}) = \sum_{\mat{p}\, \in P} \omega(\mat{x},\mat{p}) \cdot \mat{B}_{\varphi,\mat{x}}^t \cdot (\mat{x}-\mat{p}). \]
We will show that every function in $\Phi$ has the same zero set. $Z_\varphi$ as a whole is not a good reconstruction of $\cal M$. Indeed, by definition, $\varphi(\mat{x}) = 0$ for any $\mat{x} \in \real^d$ at distance $m\gamma$ or more from $\cal M$. We focus on the subset $\widehat{\cal M}$ of $\real^d$ (i.e., the set of points at distance $\eps$ or less from $\mani$). We show that $Z_\varphi \cap \widehat{\cal M}$ is a faithful reconstruction of $\cal M$.
\section{Preliminaries} \label{sec:prelim}
\subsection{Definitions}
Given a matrix or vector, the corresponding italic lowercase letter with subscripts denotes an element. For example, $k_{ij}$ denotes the $(i,j)$ entry of a matrix $\mat{K}$ and $v_i$ denotes the $i$-th coordinate of a vector $\mat{v}$. We use $\mat{I}_j$ to denote a $j \times j$ identity matrix and $\mat{0}_{i,j}$ an $i \times j$ zero matrix. The 2-norms of $\mat{v}$ and $\mat{K}$ are $\norm{\mat{v}} = \left(\sum_i v_i^2 \right)^{1/2}$ and $\norm{\mat{K}} = \max\left\{\, \norm{\mat{K} \mat{v}} : \norm{\mat{v}} = 1 \,\right\}$.
We use $B(\mat{x},r)$ to denote the geometric $d$-ball centered at $\mat{x}$ with radius $r$. We use $\angle (\mat{v},E)$ to denote the angle between a vector $\mat{v}$ and its projection in an affine subspace $E$. The angle $\angle (E,F)$ between two affine subspaces $E$ and $F$, where $\mathrm{dim}(E) \leq \mathrm{dim}(F)$, is $\max\{ \angle (\mat{v},F) : \mbox{vector $v$ in $E$}\}$.
The normal space of $\mani$ at a point $\mat{z}$, denoted $N_\mat{z}$, is the linear subspace of $\real^d$ that comprises of all vectors normal to $\mani$ at $\mat{z}$. Each vector in $N_\mat{z}$ has $d$ coordinates although $N_\mat{z}$ has dimension $d-m$. The tangent space of $\mani$ at $\mat{z}$, denoted $T_\mat{z}$, is the orthogonal complement of $N_\mat{z}$.
The medial axis of $\mani$ is the closure of the set of points in $\real^d$ that have two or more closest points in $\mani$. The \emph{local feature size} at a point $\mat{z} \in \mani$ is the distance from $\mat{z}$ to the medial axis. We assume that the reach or minimum local feature size of $\cal M$ is 1.
Let $\nu$ denote the nearest point map. That is, for every point $\mat{x}$ that does not belong to the medial axis of $\mani$, $\nu(\mat{x})$ is the point in $\mani$ nearest to $\mat{x}$.
\subsection{Basic results} \label{sec:basics}
We need the following basic results on $\eps$-sampling theory, matrices, and linear subspaces.
\begin{lemma}{\em (\cite{CDR05,GW})} \label{lem:basic} \hspace{.2in} \begin{emromani} \item For all $\mat{y},\mat{z} \in \mani$, if $\norm{\mat{y}-\mat{z}} \leq \xi$ for some $\xi < 1$, $\mat{y}$ is at distance $\xi^2/2$ or less from $\mat{z} + T_\mat{z}$.
\item For all $\mat{y},\mat{z} \in \mani$, if $\norm{\mat{y}-\mat{z}} \leq \xi$ for a small enough $\xi$, then $\angle (N_\mat{y}, N_\mat{z}) \leq 4\xi$.
\end{emromani} \end{lemma}
\begin{lemma}
\label{lem:ball}
Let $P$ be a uniform $(\eps,\kappa)$-sample of $\mani$. For any $\mat{x} \in \mathbb{R}^d$ and any $t \in \bigl[1,\frac{1}{\sqrt{2\eps}}\bigr]$, $| P \cap B(\mat{x}, t \eps) | \leq (4t+1)^m\kappa$. \end{lemma}
\begin{proof} We first show an upper bound on the minimum number of balls with radii $\eps$ such that their union contains $\mani \cap B(\mat{x}, t \eps)$, which will imply the desired result. We pick a maximal set $S$ of points in $\mani \cap B(\mat{x}, t \eps)$ such that any two of them are at distance $\eps$ or more apart.
It implies that $\mani \cap B(\mat{x}, t \eps) \subseteq \cup_{\mat{z} \in S} B(\mat{z}, \eps)$. Otherwise there exists a point $\mat{z} \in \mani \cap B(\mat{x}, t \eps)$ such that the distance between $\mat{z}$ and $S$ is larger than $\eps$, then we can get a larger set by adding $\mat{z}$ to $S$, a contradiction to the definition of $S$. Let $S'$ denote the projection of $S$ onto $\mat{x} + T_{\nu(\mat{x})}$. By Lemma~\ref{lem:basic}(i), the distance between any two points in $S'$ is at least $\eps - (t\eps)^2 \geq \eps/2$ when $t \leq \frac{1}{\sqrt{2\eps}}$. Thus, any two balls centered at points in $S'$ with radius $\eps/4$ are interior-disjoint.
Since the projection of $\mani \cap B(\mat{x}, t \eps)$ into $\mat{x} + T_{\nu(\mat{x})}$ is contained in $(\mat{x} + T_{\nu(\mat{x})}) \cap B(\mat{x}, t \eps)$, $|S'|$ is no more than the size of a maximal packing of interior-disjoint $m$-dimensional balls with radius $\eps/4$ in $(\mat{x} + T_{\nu(\mat{x})}) \cap B(\mat{x}, t \eps + \eps/4)$, which
is at most the volume of $(\mat{x} + T_{\nu(\mat{x})}) \cap B(\mat{x},t\eps + \eps/4)$ divided by $(\eps/4)^m V_m$, where $V_m$ is the volume of a unit $m$-ball. Thus, $|S| = |S'| \leq \frac{(t\eps + \eps/4)^m}{(\eps/4)^m} = (4t+1)^m$.
Then, $| P \cap B(\mat{x}, t \eps) | \leq (4t+1)^m\kappa$ by the definition of uniform $(\eps,\kappa)$-sampling. \end{proof}
Partition a square matrix $\mat{K}$ into blocks: \[ \begin{pmatrix} \mat{K}_{11} & \cdots & \mat{K}_{1r} \\ \vdots & \ddots & \vdots \\ \mat{K}_{r1} & \cdots & \mat{K}_{rr} \end{pmatrix} \] The matrices $\mat{K}_{ii}$ are square, but they may have different dimensions. For $j \not= i$, $\mat{K}_{ij}$ may be square or rectangular. For any $i,j,k \in [1,r]$, $\mat{K}_{ik}$ and $\mat{K}_{jk}$ have the same number of columns and $\mat{K}_{ij}$ and $\mat{K}_{ik}$ have the same number of rows. Each row of blocks $\begin{pmatrix}\mat{K}_{i1} & \cdots & \mat{K}_{ir}\end{pmatrix}$ defines a \emph{generalized gershgorin set} $G_i$ as follows. Let $n_i$ be the dimension of $\mat{K}_{ii}$. \[ G_i = \left\{\mu \in \real : \frac{1}{\norm{(\mat{K}_{ii} - \mu\mat{I}_{n_i})^{-1}}} \leq \sum_{j \not= i} \norm{\mat{K}_{ij}}\right\} \] It follows that the numbers in $G_i$ are at least the smallest eigenvalue of $\mat{K}_{ii}$ minus $\sum_{i \not= j} \norm{\mat{K}_{ij}}$ and at most the maximum eigenvalue of $\mat{K}_{ii}$ plus $\sum_{i \not= j} \norm{\mat{K}_{ij}}$. The eigenvalues of $\mat{K}_{ii}$ are defined to be in $G_i$ using a continuity argument~\cite{gc}.
\begin{lemma}{\em (\cite{gc})} \label{lem:gc} Consider any partition of a square matrix $\mat{K}$ into blocks. Every eigenvalue of $\mat{K}$ lies in some generalized gershgorin set $G_i$ with respect to this partition. Moreover, if a generalized gershgorin set $G_i$ is disjoint from the union of the other generalized gershgorin sets, then $G_i$ contains exactly $n_i$ eigenvalues of $\mat{K}$, where $n_i$ is the dimension of $\mat{K}_{ii}$. \end{lemma}
\begin{lemma}{\em (\cite{golub})} \label{lem:matrix} Let $(\mat{U} \,\,\, \mat{V})$ be a $d \times d$ orthogonal matrix, where $\mat{U}$ is $d \times r$ and $\mat{V}$ is $d \times (d-r)$. Let $\mat{K}$ be a $d \times r$ matrix with orthogonal unit columns. Then, $\angle (\col{\mat{U}},\col{\mat{K}}) = \arcsin(\norm{\mat{V}^t \cdot \mat{K}})$. \end{lemma}
\begin{lemma}{\em (\cite[Lemma~1.1]{EI94})} \label{lem:slant} Let $\mat{M}_1$ be an $s \times s$ real symmetric matrix with eigenvalues $\lambda_1,\ldots,\lambda_s$ in an arbitrary order. Let $\mat{v}_i$ denote a unit eigenvector of $\mat{M}_1$ corresponding to $\lambda_i$. If $\mat{M}_1 + \mat{M}_2$ is a real symmetric matrix, $\sigma$ is an eigenvalue of $\mat{M}_1 + \mat{M}_2$, and $\mat{e}$ is a unit eigenvector of $\mat{M}_1 + \mat{M}_2$ corresponding to $\sigma$, then for every $r \in [1,s-1]$, the angle between $\mat{e}$ and the space spanned by $\{\mat{v}_1,\ldots,\mat{v}_r\}$ is at most $\arcsin\left(\norm{\mat{M}_2}/\min_{i \in [r+1,s]}
|\lambda_i - \sigma|\right)$. \end{lemma}
\begin{lemma} \label{lem:choice} Let $V$ and $W$ be two linear subspaces of the same dimension $k$ in $\real^d$ such that $\theta = \angle (V,W) < \pi/2$. \begin{emromani}
\item For each orthonormal basis $\{\mat{v}_1,\ldots,\mat{v}_k\}$ of $V$, there exists an orthonormal basis $\{\mat{w}_1,\ldots,\mat{w}_k\}$ of $W$ such that $\angle (\mat{v}_i,\mat{w}_i) \leq \theta$ for $i \in [1,k]$ and $\angle (\mat{v}_i, \mat{w}_j-\mat{v}_j) \in \left[\frac{\pi-\theta}{2},\frac{\pi+\theta}{2}\right]$ for $i,j \in [1,k]$.
\item If $k > d/2$, then there exist orthonormal bases $\{\mat{v}_1,\ldots,\mat{v}_k\}$ and $\{\mat{w}_1,\ldots,\mat{w}_k\}$ of $V$ and $W$, respectively, such that $\mat{v}_i = \mat{w}_i$ for $i \in [1,2k-d]$, $\angle (\mat{v}_i,\mat{w}_i) \leq \theta$ for $i \in [1,k]$, and $\angle (\mat{v}_i, \mat{w}_j-\mat{v}_j) \in \left[\frac{\pi-\theta}{2},\frac{\pi+\theta}{2}\right]$ for $i,j \in [1,k]$. Hence, for any distinct $i$ and $j$, if $i \in [1,2k-d]$ or $j \in [1,2k-d]$, then $\mat{v}_i \perp \mat{w}_j$.
\end{emromani} \end{lemma} \begin{proof} We make use of principal angles and principal vectors~\cite{bjorck,galantai,miao}. Pick unit vectors $\mat{a}_1 \in V$ and $\mat{b}_1 \in W$ that minimizes $\angle(\mat{a}_1,\mat{b}_1)$. For $i \in [2,k]$, pick unit vectors $\mat{a}_i \in V$ and $\mat{b}_i \in W$ that minimizes $\angle(\mat{a}_i,\mat{b}_i)$ subject to $\mat{a}_i \perp \mat{a}_j$ and $\mat{b}_i \perp \mat{b}_j$ for all $j \in [1,i-1]$. The angles $\angle(\mat{a}_1,\mat{b}_1),\ldots,\angle(\mat{a}_k,\mat{b}_k)$ are called the principal angles. The vectors $\{\mat{a}_1,\ldots,\mat{a}_k\}$ and $\{\mat{b}_1,\ldots,\mat{b}_k\}$ are called principal vectors. Note that $\{\mat{a}_1,\ldots,\mat{a}_k\}$ and $\{\mat{b}_1,\ldots,\mat{b}_k\}$ are orthonormal bases of $V$ and $W$, respectively. The alternative definition of principal angles in~\cite{galantai} implies that for $i \in [1,k]$, $\theta_i \leq \theta = \angle (V,W)$. It is also known that $\mat{a}_i \perp \mat{b}_j$ for $i \not= j$~\cite{bjorck,galantai}.
Consider (i). Given an orthonormal basis $\{\mat{v}_1,\ldots,\mat{v}_k\}$ of $V$, for each $i \in [1,k]$, $\mat{v}_i = \sum_{r=1}^k c_{ir}\mat{a}_r$ for some real coefficients $c_{ir}$'s. Correspondingly, define $\mat{w}_i = \sum_{r=1}^k c_{ir}\mat{b}_r$. Note that $\norm{\mat{w}_i} = (\sum_{r=1}^k c_{ir}^2)^{1/2} = \norm{\mat{v}_i} = 1$. Also, for $i \not= j$, $\mat{w}_i^t\mat{w}_j^{} = \sum_{r=1}^k c_{ir}c_{jr} = \mat{v}_i^t\mat{v}_j^{} = 0$. So $\{\mat{w}_1,\ldots,\mat{w}_k\}$ is an orthonormal basis of $W$.
For $i \in [1,k]$, $\mat{v}_i^t\mat{w}_i^{} = \sum_{r=1}^k c_{ir}^2 \mat{a}_r^t\mat{b}_r^{} \geq \cos\theta$ because $\angle(\mat{a}_r,\mat{b}_r) \leq \theta$ and $\sum_{r=1}^k c_{ir}^2 = \norm{\mat{v}_i} = 1$. It follows that $\angle(\mat{v}_i,\mat{w}_i) \leq \theta$. Since $\mat{v}_i$ and $\mat{w}_i$ are unit vectors and $\angle (\mat{v}_i,\mat{w}_i) \leq \theta$, $\mat{v}_i + \mat{w}_i$ is an angle bisector between $\mat{v}_i$ and $\mat{w}_i$. Hence, $\angle (\mat{v}_i,\mat{v}_i + \mat{w}_i) \leq \theta/2$. It suffices to show that for any $i,j \in [1,k]$, $\mat{v}_i + \mat{w}_i \perp \mat{w}_j - \mat{v}_j$, which then implies that $\left|\frac{\pi}{2} - \angle (\mat{v}_i,\mat{w}_j-\mat{v}_j) \right| \leq \angle (\mat{v}_i,\mat{v}_i + \mat{w}_i) \leq \theta/2$, completing the proof of (i). To see that $\mat{v}_i + \mat{w}_i \perp \mat{w}_j - \mat{v}_j$, we check $(\mat{v}_i + \mat{w}_i)^t \cdot (\mat{w}_j - \mat{v}_j) = \sum_{r=1}^k (c_{ir}\mat{a}_r + c_{ir}\mat{b}_r)^t \cdot \sum_{r=1}^k (c_{jr}\mat{b}_r - c_{jr}\mat{a}_r)$. Recall that $\mat{a}_r$ and $\mat{b}_r$ are unit vectors and for $r \not=s$, $\mat{a}_r \perp \mat{a}_s$, $\mat{b}_r \perp \mat{b}_s$, and $\mat{a}_r \perp \mat{b}_s$. Therefore, $\sum_{r=1}^k (c_{ir}\mat{a}_r + c_{ir}\mat{b}_r)^t \cdot \sum_{r=1}^k (c_{jr}\mat{b}_r - c_{jr}\mat{a}_r) = 0$.
Consider (ii). Since $k > d/2$, the dimension of $V \cap W$ is at least $2k-d$. Pick an arbitrary subset $\{\mat{u}_1,\ldots,\mat{u}_{2k-d}\}$ of the orthonormal basis of $V \cap W$. Set $\mat{v}_i = \mat{w}_i = \mat{u}_i$ for $i \in [1,2k-d]$. Complete $\{\mat{v}_1,\ldots,\mat{v}_{2k-d}\}$ arbitrarily to an orthonormal basis $\{\mat{v}_1,\ldots,\mat{v}_k\}$ of $V$. Then, we construct $\mat{w}_j$ as the same way as in (i) for $j \in [2k-d+1,k]$. \end{proof}
\begin{lemma} \label{lem:angle} Let $E_1$ and $E_2$ be two $k$-dimensional linear subspaces. Let $\{\mat{u}_1,\ldots,\mat{u}_k\}$ be a basis of $E_1$ consisting of unit vectors such that for any distinct $i,j \in [1,k]$, $\angle (\mat{u}_i,\mat{u}_j) \in [\pi/2-\phi,\pi/2+\phi]$ for some $\phi \in \left[0,\arcsin\left(\frac{1}{k}\right)\right)$. For any $\theta \in \left[0,\arcsin\left(\sqrt{\frac{1}{k}-\sin\phi}\right)\right)$, if $\angle (\mat{u}_i,E_2) \leq \theta$ for all $i \in [1,k]$, then $\angle (E_1,E_2) \leq \arctan\Bigl(\frac{\sqrt{k}\sin\theta}{\sqrt{1-k\sin^2\theta-k\sin\phi}}\Bigr)$. \end{lemma} \begin{proof} Orient space such that $E_2$ is spanned by the first $k$ coordinate axes of $\real^d$. Then, for all $i \in [1,k]$, we can write \[ \mat{u}_i = \begin{pmatrix} \mat{v}_i \\ \mat{w}_i \end{pmatrix}, \] where $\mat{v}_i$ consists of the first $k$ coordinates and $\mat{w}_i$ consists the remaining $d-k$ coordinates. Note that \[ \begin{pmatrix} \mat{0}_{k,1} \\ \mat{w}_i \end{pmatrix} \perp E_2 \quad \mbox{and} \quad \begin{pmatrix} \mat{v}_i \\ \mat{0}_{d-k,1} \end{pmatrix} \in E_2. \]
Since $\angle (\mat{u}_i,E_2) \leq \theta$ by assumption, we have $\norm{\mat{w}_i} \leq \sin\theta$. As a result, $\norm{\mat{v}_i} \in [\cos\theta, 1]$. For any $i \not= j$, we have \begin{eqnarray*} & & \begin{pmatrix} \mat{v}_i^t \,\,\,\, \mat{w}_i^t \end{pmatrix} \cdot \begin{pmatrix} \mat{v}_j \\ \mat{w}_j \end{pmatrix} \in \left[\cos\left(\frac{\pi}{2}+\phi\right),\cos\left(\frac{\pi}{2}-\phi\right)\right] \\ & \Rightarrow & \mat{v}_i^t\cdot \mat{v}_j + \mat{w}_i^t\cdot\mat{w}_j \in [-\sin\phi,\sin\phi] \\
& \Rightarrow & |\mat{v}_i^t \cdot \mat{v}_j| \leq \norm{\mat{w}_i} \cdot \norm{\mat{w}_j} + \sin\phi \leq \sin^2\theta + \sin\phi. \end{eqnarray*} Let $\mat{n}$ be a vector in $E_1$ that makes the angle $\angle(E_1,E_2)$ with $E_2$. By flipping the orientation of any $\mat{u}_i$'s if necessary, we can ensure that $\mat{n}$ is a convex combination of $\{\mat{u}_1,\ldots,\mat{u}_k\}$, i.e., $\mat{n} = \displaystyle \sum_{i=1}^k \lambda_i \begin{pmatrix} \mat{v}_i \\ \mat{w}_i \end{pmatrix}$ for some $\lambda_i$'s in $[0,1]$ such that $\sum_{i=1}^k \lambda_i = 1$. Note that flipping the orientation of any $\mat{u}_i$ preserves the angle $\angle (\mat{u}_i,E_2)$ and the fact that for any distinct $i,j \in [1,k]$, $\angle (\mat{u}_i,\mat{u}_j) \in [\pi/2-\phi,\pi/2+\phi]$. Hence, \begin{eqnarray*} \angle (E_1,E_2) & = &
\arctan\left(\frac{\|\sum_{i=1}^k \lambda_i \mat{w}_i\|}
{\|\sum_{i=1}^k \lambda_i \mat{v}_i\|}\right) \;\; \leq \;\; \arctan\left(\frac{\sum_{i=1}^k \lambda_i\norm{\mat{w}_i}} {\sqrt{\sum_{i=1}^k\sum_{j=1}^k \lambda_i\lambda_j \cdot \mat{v}_i^t \cdot \mat{v}_j}} \right) \\ & \leq & \arctan\left(\frac{\sin\theta} {\sqrt{\cos^2\theta \sum_{i=1}^k \lambda_i^2 - (\sin^2\theta+\sin\phi) \sum_{i\not=j}\lambda_i\lambda_j}} \right) \\ & = & \arctan\left(\frac{\sin\theta} {\sqrt{\sum_{i=1}^k \lambda_i^2 - (\sin^2\theta+\sin\phi) \left(\sum_{i=1}^k\lambda_i\right)^2 }} \right) \\ & \leq & \arctan\left(\frac{\sqrt{k}\sin\theta} {\sqrt{1 - k\sin^2\theta-k\sin\phi}} \right). \end{eqnarray*} The last step uses the fact that $\sum_{i=1}^k\lambda_i^2$ is minimized when $\lambda_i = 1/k$ for all $i$. \end{proof}
\section{Accuracy of $L_\mat{x}$}
The main result of this section is Lemma~\ref{lemma::normal_angle} below: for every point $\mat{z} \in \mani$ and every point $\mat{x}$ near $\mat{z}$, $N_\mat{z}$ is approximated by $L_\mat{x}$. We need the following technical result. Recall that $\nu$ is the nearest point map.
\begin{lemma} \label{lem:C} Let $\mat{x}$ be a point at distance $2\eps$ or less from $\mani$. Assume a coordinate frame such that the columns of $\begin{pmatrix} \mat{I}_m \\ \mat{0}_{d-m,m} \end{pmatrix}$ form an orthonormal basis of $T_{\nu(\mat{x})}$. Partition $\mat{C}_\mat{x}$ into $\begin{pmatrix} \mat{C}_{11} & \mat{C}_{12} \\ \mat{C}_{21} & \mat{C}_{22} \\ \end{pmatrix}$, where $\mat{C}_{11}$ is $m \times m$, $\mat{C}_{12}$ is $m \times (d-m)$, $\mat{C}_{21}$ is $(d-m) \times m$, and $\mat{C}_{22}$ is $(d-m) \times (d-m)$. Then, $\norm{\mat{C}_{12}}$ and $\norm{\mat{C}_{21}}$ are $O(m\gamma)$, $\norm{\mat{C}_{22}}$ is $O(m^2\gamma^2)$, and the smallest eigenvalue of $\mat{C}_{11}$ is at least $1 - O(m^2\gamma^2)$. \end{lemma} \begin{proof} Consider any sample point $\mat{p} \in P$. Partition $\mat{T}_\mat{p}$ into $\begin{pmatrix} \mat{Y}_\mat{p} \\ \mat{Z}_\mat{p} \end{pmatrix}$, where $\mat{Y}_\mat{p}$ is $m \times m$ and $\mat{Z}_\mat{p}$ is $(d-m) \times m$. For all $\mat{p} \in P \cap B(\mat{x},m\gamma)$, \[ \norm{\mat{p}-\nu(\mat{x})} \leq \norm{\mat{p}-\mat{x}} + \norm{\mat{x}-\nu(\mat{x})} \leq m\gamma + 2\eps < (m+1)\gamma. \] Then, $\angle (T_\mat{p}, T_{\nu(\mat{x})}) \leq 4(m+1)\gamma$ by Lemma~\ref{lem:basic}(ii).
Since $\begin{pmatrix} \mat{I}_m \\ \mat{0}_{d-m,m} \end{pmatrix}$ and $\begin{pmatrix} \mat{0}_{m,d-m} \\ \mat{I}_{d-m} \end{pmatrix}$ form a $d \times d$ orthogonal matrix, we obtain \begin{eqnarray*} \arcsin(\norm{\mat{Z}_\mat{p}}) & = & \arcsin(\norm{(\mat{0}_{d-m,m} \,\, \mat{I}_{d-m}) \cdot \mat{T}_\mat{p}}) \\ & = & \angle (T_{\nu(\mat{x})}, \col{\mat{T}_\mat{p}}) \quad\quad\quad\quad\quad\quad\quad (\because \text{Lemma~\ref{lem:matrix}}) \\ & \leq & \angle (T_\mat{p}, T_{\nu(\mat{x})}) + \angle (T_\mat{p},\col{\mat{T}_\mat{p}}) \\ & \leq & 4(m+1)\gamma + m \gamma. \end{eqnarray*}
(We use the assumption that the input approximate tangent spaces have angular errors at most $m\gamma$. Although an angular error of $O(m\gamma)$ also works, an exact bound of $m\gamma$ makes explicit the input requirement for constructing the formula of $\varphi$.) Hence, we have \begin{equation} \forall \, \mat{p} \in P \cap B(\mat{x},m\gamma), \quad \norm{\mat{Z}_\mat{p}} = O(m\gamma).
\label{eq:Z} \end{equation}
Because $\omega(\mat{x},\mat{p})$ vanishes for all $\mat{p} \not\in B(\mat{x},m\gamma)$, $\mat{C}_{12} = \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \omega(\mat{x},\mat{p}) \cdot \mat{Y}_\mat{p}^{} \cdot \mat{Z}_\mat{p}^t$. Since the columns in $\mat{T}_\mat{p}$ have unit 2-norm, we get $\norm{\mat{Y}_\mat{p}} \leq 1$. Thus, \[ \norm{\mat{C}_{12}} =
\left\| \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \omega(\mat{x},\mat{p}) \cdot
\mat{Y}_{\mat{p}}^{} \cdot \mat{Z}_{\mat{p}}^t \right\| \leq \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \omega(\mat{x},\mat{p}) \cdot \norm{\mat{Z}_\mat{p}} = O(m\gamma).
\] Similarly, \[ \norm{\mat{C}_{21}} =
\left\| \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \omega(\mat{x},\mat{p}) \cdot
\mat{Z}_{\mat{p}}^{} \cdot \mat{Y}_{\mat{p}}^t \right\| \leq \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \omega(\mat{x},\mat{p}) \cdot \norm{\mat{Z}_\mat{p}} = O(m\gamma),
\] \[ \norm{\mat{C}_{22}} =
\left\|\sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \omega(\mat{x},\mat{p}) \cdot
\mat{Z}_\mat{p}^{} \cdot \mat{Z}_\mat{p}^t\right\| \leq \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \omega(\mat{x},\mat{p}) \cdot \norm{\mat{Z}_\mat{p}^{}}^2 = O(m^2\gamma^2).
\]
Since $\mat{T}_\mat{p}^t \cdot \mat{T}_\mat{p}^{} = \mat{Y}_\mat{p}^t \cdot \mat{Y}_\mat{p}^{} + \mat{Z}_\mat{p}^t \cdot \mat{Z}_\mat{p}^{}$, the minimum eigenvalue of $\mat{Y}_\mat{p}^t \cdot \mat{Y}_\mat{p}^{}$ is at least the minimum eigenvalue of $\mat{T}_\mat{p}^t \cdot \mat{T}_\mat{p}^{}$ minus $\norm{\mat{Z}_\mat{p}^t \cdot \mat{Z}_\mat{p}^{}}$. Therefore, \begin{equation} \text{minimum eigenvalue of $\mat{Y}_\mat{p}^t \cdot \mat{Y}_\mat{p}^{} \geq 1 - O(m^2\gamma^2)$}.
\label{eq:Y} \end{equation} $\mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t$ has the same eigenvalues as $\mat{Y}_\mat{p}^t \cdot \mat{Y}_\mat{p}^{}$. The smallest eigenvalue of a real symmetric matrix $\mat{M}$ is $\min_{\mat{v} \not= \mat{0}} (\mat{v}^t \cdot \mat{M} \cdot \mat{v})/\norm{\mat{v}}^2$. Then, using the relation $\mat{C}_{11} = \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \omega(\mat{x},\mat{p}) \cdot \mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t$, we conclude that the smallest eigenvalue of $\mat{C}_{11}$ is at least the sum of the smallest eigenvalues of $\omega(\mat{x},\mat{p}) \cdot \mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t$. This sum is at least $1 - O(m^2\gamma^2)$ by \eqref{eq:Y}.
\end{proof}
We are ready to show that the angle between $L_\mat{x}$ and any nearby normal space of $\mani$ is $O(m\sqrt{m}\gamma)$.
\begin{lemma} \label{lemma::normal_angle} For every point $\mat{z} \in \mani$ and every point $\mat{x} \in B(\mat{z},2\eps)$, $\angle(L_\mat{x},N_\mat{z}) = O(m\sqrt{m}\gamma)$. \end{lemma} \begin{proof} Adopt a coordinate frame such that the columns of $\begin{pmatrix} \mat{I}_m \\ \mat{0}_{d-m,m} \end{pmatrix}$ form an orthonormal basis of $T_{\nu(\mat{x})}$. Let $\mat{A}_\mat{x}$ be the $d \times m$ matrix whose columns are the $m$ most dominant unit eigenvectors of $\mat{C}_\mat{x}$. Thus, $\col{\mat{A}_\mat{x}}$ is the orthogonal complement of $L_\mat{x}$. Let $\mat{e} = \begin{pmatrix} \mat{v} \\ \mat{w} \end{pmatrix}$ be any column vector of $\mat{A}_\mat{x}$, where $\mat{v}$ consists of the first $m$ coordinates and $\mat{w}$ consists of the last $d-m$ coordinates. Then, $\angle (\mat{e},T_{\nu(\mat{x})}) = \arctan(\norm{\mat{w}}/\norm{\mat{v}})$.
We show that $\angle (\mat{e},T_{\nu(\mat{x})}) = O(m\gamma)$. Partition $\mat{C}_\mat{x}$ into $\begin{pmatrix} \mat{C}_{11} & \mat{C}_{12} \\ \mat{C}_{21} & \mat{C}_{22} \\ \end{pmatrix}$, where $\mat{C}_{11}$ is $m \times m$, $\mat{C}_{12}$ is $m \times (d-m)$, $\mat{C}_{21}$ is $(d-m) \times m$, and $\mat{C}_{22}$ is $(d-m) \times (d-m)$. Let $\sigma$ be the eigenvalue of $\mat{C}_\mat{x}$ corresponding to $\mat{e}$. Then, \[ \mat{C}_\mat{x} \, \mat{e} = \begin{pmatrix} \mat{C}_{11} & \mat{C}_{12} \\ \mat{C}_{21} & \mat{C}_{22} \\ \end{pmatrix} \begin{pmatrix} \mat{v} \\ \mat{w} \end{pmatrix} \,\, = \,\, \sigma \begin{pmatrix} \mat{v} \\ \mat{w} \end{pmatrix}, \] which implies that \[ \norm{\mat{w}} = \norm{(\sigma\mat{I}_{d-m} - \mat{C}_{22})^{-1}\mat{C}_{21}\mat{v}} \leq \norm{(\sigma\mat{I}_{d-m} - \mat{C}_{22})^{-1}} \cdot \norm{\mat{C}_{21}}. \] Following the definition of generalized gershgorin sets (Section~\ref{sec:prelim}), define \begin{eqnarray*} G_1 & = & \left\{\mu \in \real : \frac{1}{\norm{(\mat{C}_{11} - \mu \mat{I}_m)^{-1}}} \leq \norm{\mat{C}_{12}}\right\}, \\ G_2 & = & \left\{\mu \in \real: \frac{1}{\norm{(\mat{C}_{22} - \mu \mat{I}_{d-m})^{-1}}} \leq \norm{\mat{C}_{21}}\right\}. \end{eqnarray*} The numbers in $G_1$ are at least the minimum eigenvalue value of $\mat{C}_{11}$ minus $\norm{\mat{C}_{12}}$, which is at least $1 - O(m\gamma + m^2\gamma^2)$ by Lemma~\ref{lem:C}.
The numbers in $G_2$ are at most $\norm{\mat{C}_{22}} + \norm{\mat{C}_{21}} = O(m\gamma + m^2\gamma^2)$ by Lemma~\ref{lem:C}. Since every number in $G_1$ is greater than any number in $G_2$, by Lemma~\ref{lem:gc}, $G_1$ contains the $m$ largest eigenvalues of $\mat{C}_\mat{x}$. Thus, $\sigma$ belongs to $G_1$ and $\sigma \geq 1 - O(m\gamma + m^2\gamma^2)$ which is asymptotically greater than $\norm{\mat{C}_{22}} = O(m^2\gamma^2)$ (Lemma~\ref{lem:C}). Therefore, \[ \norm{(\sigma\mat{I}_{d-m} - \mat{C}_{22})^{-1}} \leq \frac{1}{1- O(m\gamma + m^2\gamma^2)}. \] By Lemma~\ref{lem:C}, $\norm{\mat{C}_{21}} = O(m\gamma)$, and therefore, \[ \norm{\mat{w}} \leq \norm{(\sigma\mat{I}_{d-m}-\mat{C}_{22})^{-1}} \cdot \norm{\mat{C}_{21}} \leq \frac{ O(m\gamma)}{1 - O(m\gamma + m^2\gamma^2)} = O(m\gamma). \] As a result, $1 \geq \norm{\mat{v}} \geq 1 - \norm{\mat{w}} \geq 1-O(m\gamma)$. Thus, $\angle(\mat{e},T_{\nu(\mat{x})}) = \arctan(\norm{\mat{w}}/\norm{\mat{v}}) = O(m\gamma)$.
Since $\mat{e}$ is any column vector of $\mat{A}_\mat{x}$, the angle bound in the previous paragraph applies to all column vectors of $\mat{A}_\mat{x}$. We can apply Lemma~\ref{lem:angle} with $E_1 = \col{\mat{A}_\mat{x}}$, $E_2 = T_{\nu(\mat{x})}$, $\{\mat{u}_1,\ldots,\mat{u}_m\}$ equal to the columns of $\mat{A}_\mat{x}$, $\phi = 0$, $k = m$, and $\theta$ equal to the $O(m\gamma)$ bound on $\angle (\mat{e},T_{\nu(\mat{x})})$. Then, \[ \angle (\col{\mat{A}_\mat{x}},T_{\nu(\mat{x})}) \leq \arctan\left(\frac{O(m\sqrt{m}\gamma)}{\sqrt{1-O(m^3\gamma^2)}}\right) = O(m\sqrt{m}\gamma). \] Since $\norm{\nu(\mat{x}) - \mat{z}} \leq \norm{\mat{x} - \nu(\mat{x})} + \norm{\mat{x} - \mat{z}} \leq 4\eps$, Lemma~\ref{lem:basic}(ii) implies that $\angle (T_{\nu(\mat{x})},T_\mat{z}) \leq 16\eps$. Hence, \begin{eqnarray*} \angle (L_\mat{x},N_\mat{z}) & = & \angle (\col{\mat{A}_\mat{x}},T_\mat{z}) \\ & \leq & \angle (\col{\mat{A}_\mat{x}},T_{\nu(\mat{x})}) + \angle (T_{\nu(\mat{x})},T_\mat{z}) \\ & = & O(m\sqrt{m}\gamma).
\end{eqnarray*} \end{proof}
\section{Projection into $L_\mat{x}$}
For every point $\mat{z} \in \mani$ and every unit vector $\mat{n} \in N_\mat{z}$, we want to bound the instantaneous change in the normalized projection of $\mat{n}$ in $L_\mat{x}$ as $\mat{x}$ moves. If we view the projection as a map $f$, this is equivalent to analyzing the Jacobian of $f$ which is given in Lemmas~\ref{lemma::gradient} and~\ref{cor:gradient} below. To this end, some technical results are needed. First, we need to study the variation of $\mat{C}_\mat{x}$ as $\mat{x}$ moves (Lemma~\ref{lem:delta_C}). Second, we need to bound the turn of $L_\mat{x}$ if $\mat{x}$ moves slightly (Lemma~\ref{lemma::normal_change}).
Let $\delta_k > 0$ denote an arbitrarily small change in the coordinate $x_k$ of $\mat{x}$. Define \[ \Delta h(\norm{\mat{x}-\mat{p}}) = \frac{\partial h(\norm{\mat{x}-\mat{p}})}{\partial x_k} \cdot \delta_k. \] For simplicity, we omit the dependence of $\Delta h(\norm{\mat{x}-\mat{p}})$ on $k$ in the notation.
\begin{lemma} \label{lem:delta_C} Let $\mat{x}$ be a point at distance $2\eps$ or less from $\mani$. Assume a coordinate frame such that the columns of $\begin{pmatrix} \mat{I}_m \\ \mat{0}_{d-m,m} \end{pmatrix}$ form an orthonormal basis of $T_{\nu(\mat{x})}$. Define the $d \times d$ matrix $\Delta\mat{C}_\mat{x} = \begin{pmatrix} \dfrac{\partial c_{ij}}{\partial x_k} \cdot \delta_k \end{pmatrix}$, where $c_{ij}$ is the $(i,j)$ entry of $\mat{C}_\mat{x}$. The following properties hold when $\delta_k$ is small enough. \begin{emromani}
\item $\displaystyle \norm{\Delta\mat{C}_\mat{x}} \leq \frac{O(m\gamma) \cdot
\sum_{\mat{p} \in P} |\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}$.
\item The $m$ largest eigenvalues of $\mat{C}_\mat{x} + \Delta\mat{C}_\mat{x}$ are at least $\displaystyle 1 - O(m\gamma) - \frac{O(m\gamma) \cdot \sum_{\mat{p}
\in P} |\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}$.
\end{emromani} \end{lemma}
\begin{proof} Using standard calculus, we obtain \[ \Delta \mat{C}_\mat{x} = \frac{1}{(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}))^2} \left(\sum_{\mat{p},\mat{q} \in P} h(\norm{\mat{x}-\mat{p}}) \cdot \Delta h(\norm{\mat{x}-\mat{q}}) \cdot \left( \mat{T}_\mat{q}^{} \cdot \mat{T}_\mat{q}^t - \mat{T}_\mat{p}^{} \cdot \mat{T}_\mat{p}^t \right) \right). \] Partition $\mat{C}_\mat{x}$ and $\Delta\mat{C}_\mat{x}$ as follows: \[ \mat{C}_\mat{x} = \begin{pmatrix} \mat{C}_{11} & \mat{C}_{12} \\ \mat{C}_{21} & \mat{C}_{22} \\ \end{pmatrix}, \quad\quad\quad \Delta\mat{C}_\mat{x} = \begin{pmatrix} \Delta\mat{C}_{11} & \Delta\mat{C}_{12} \\ \Delta\mat{C}_{21} & \Delta\mat{C}_{22} \\ \end{pmatrix} \] where $\mat{C}_{11}$ and $\Delta\mat{C}_{11}$ are $m \times m$, $\mat{C}_{12}$ and $\Delta\mat{C}_{12}$ are $m \times (d-m)$, $\mat{C}_{21}$ and $\Delta\mat{C}_{21}$ are $(d-m) \times m$, and $\mat{C}_{22}$ and $\Delta\mat{C}_{22}$ are $(d-m) \times (d-m)$.
For every sample point $\mat{p} \in P$, partition $\mat{T}_\mat{p}$ into $\mat{T}_\mat{p} = \begin{pmatrix} \mat{Y}_\mat{p} \\ \mat{Z}_\mat{p} \end{pmatrix}$, where $\mat{Y}_\mat{p}$ is an $m \times m$ matrix and $\mat{Z}_\mat{p}$ is a $(d-m) \times m$ matrix.
By \eqref{eq:Z} and \eqref{eq:Y}, for every sample point $\mat{p} \in P \cap B(\mat{x},m\gamma)$, $\norm{\mat{Z}_\mat{p}} = O(m\gamma)$ and the eigenvalues of $\mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t$ are at least $1 - O(m^2\gamma^2)$. Moreover, \[ \norm{\mat{Y}_\mat{p}^{}\cdot\mat{Y}_\mat{p}^t} = \norm{\mat{Y}_\mat{p}^t\cdot\mat{Y}_\mat{p}^{}} = \norm{\mat{T}_\mat{p}^t\cdot\mat{T}_\mat{p}^{} - \mat{Z}_\mat{p}^t\cdot\mat{Z}_\mat{p}^{}} \leq \norm{\mat{T}_\mat{p}}^2 + \norm{\mat{Z}_\mat{p}}^2 = 1 + O(m^2\gamma^2), \] which also implies that \[ \norm{\mat{Y}_\mat{p}} = 1 + O(m^2\gamma^2). \] Because for any real symmetric matrix $\mat{M}$, $\norm{\mat{M}} = \max_{\mat{v} \not= \mat{0}} \left(\mat{v}^t \cdot \mat{M} \cdot \mat{v}\right)/\norm{\mat{v}}^2$, we conclude that $\norm{\mat{Y}_\mat{q}^{} \cdot \mat{Y}_\mat{q}^t - \mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t}$ is at most the maximum eigenvalue of $\mat{Y}_\mat{q}^{} \cdot \mat{Y}_\mat{q}^t$ minus the minimum eigenvalue of $\mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t$. Therefore, \[ \begin{array}{lll} \norm{\mat{Y}_\mat{q}^{} \cdot \mat{Y}_\mat{q}^t - \mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t} & \leq & 1 + O(m^2\gamma^2) - (1 - O(m^2\gamma^2)) \,\, = \,\, O(m^2\gamma^2). \end{array} \] Moreover, \[ \begin{array}{lll} \norm{\mat{Y}_\mat{q}^{} \cdot \mat{Z}_\mat{q}^t - \mat{Y}_\mat{p}^{} \cdot \mat{Z}_\mat{p}^t} & \leq & \norm{\mat{Y}_\mat{q}} \cdot \norm{\mat{Z}_\mat{q}} + \norm{\mat{Y}_\mat{p}} \cdot \norm{\mat{Z}_\mat{p}} \,\, = \,\, O(m\gamma), \\ \norm{\mat{Z}_\mat{q}^{} \cdot \mat{Z}_\mat{q}^t - \mat{Z}_\mat{p}^{} \cdot \mat{Z}_\mat{p}^t} & \leq & \norm{\mat{Z}_\mat{q}}^2 + \norm{\mat{Z}_\mat{p}}^2 \,\, = \,\, O(m^2\gamma^2). \end{array} \] On the other hand, for every sample point $\mat{p} \in P \setminus B(\mat{x},m\gamma)$, \[ h(\norm{\mat{x}-\mat{p}}) = 0, \quad\quad\quad \Delta h(\norm{\mat{x}-\mat{p}}) = 0. \] Consequently, \begin{eqnarray} \norm{\Delta\mat{C}_{11}} & = &
\frac{\left\|\sum_{\mat{p},\mat{q} \in P} h(\norm{\mat{x}-\mat{p}}) \cdot \Delta h(\norm{\mat{x}-\mat{q}}) \cdot (\mat{Y}_\mat{q}^{} \cdot \mat{Y}_\mat{q}^t -
\mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t)\right\|} {(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}))^2} \nonumber \\ & \leq & \frac{\sum_{\mat{p},\mat{q} \in P} h(\norm{\mat{x}-\mat{p}}) \cdot
|\Delta h(\norm{\mat{x}-\mat{q}})| \cdot \norm{\mat{Y}_\mat{q}^{} \cdot \mat{Y}_\mat{q}^t - \mat{Y}_\mat{p}^{} \cdot \mat{Y}_\mat{p}^t}} {(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}))^2} \nonumber \\ & = & \frac{ O(m^2\gamma^2) \cdot \sum_{\mat{p}
\in P} |\Delta h(\norm{\mat{x}-\mat{p}})|} {\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}. \label{eq:C11} \end{eqnarray} By symmetry, \begin{eqnarray} \norm{\Delta\mat{C}_{12}} = \norm{\Delta\mat{C}_{21}}
& = & \frac{\left\|\sum_{\mat{p},\mat{q} \in P} h(\norm{\mat{x}-\mat{p}}) \cdot \Delta h(\norm{\mat{x}-\mat{q}}) \cdot (\mat{Y}_\mat{q}^{} \cdot \mat{Z}_\mat{q}^t -
\mat{Y}_\mat{p}^{} \cdot \mat{Z}_\mat{p}^t)\right\|} {(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}))^2} \nonumber \\ & \leq & \frac{ \sum_{\mat{p},\mat{q} \in P}
h(\norm{\mat{x}-\mat{p}}) \cdot |\Delta h(\norm{\mat{x}-\mat{q}})| \cdot \norm{\mat{Y}_\mat{q}^{} \cdot \mat{Z}_\mat{q}^t - \mat{Y}_\mat{p}^{} \cdot \mat{Z}_\mat{p}^t}} {(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}))^2} \nonumber \\ & = & \frac{ O(m\gamma) \cdot \sum_{\mat{p} \in P}
|\Delta h(\norm{\mat{x}-\mat{p}})|} {\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}, \label{eq:C12} \end{eqnarray} Similarly, \begin{eqnarray} \norm{\Delta\mat{C}_{22}} & = &
\frac{\left\| \sum_{\mat{p},\mat{q} \in P} h(\norm{\mat{x}-\mat{p}}) \cdot \Delta h(\norm{\mat{x}-\mat{q}}) \cdot (\mat{Z}_\mat{q}^{} \cdot \mat{Z}_\mat{q}^t -
\mat{Z}_\mat{p}^{} \cdot \mat{Z}_\mat{p}^t)\right\|} {(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}))^2} \nonumber \\ & \leq & \frac{ \sum_{\mat{p},\mat{q} \in P} h(\norm{\mat{x}-\mat{p}}) \cdot
|\Delta h(\norm{\mat{x}-\mat{q}})| \cdot \norm{\mat{Z}_\mat{q}^{} \cdot \mat{Z}_\mat{q}^t - \mat{Z}_\mat{p}^{} \cdot \mat{Z}_\mat{p}^t}} {(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}))^2} \nonumber \\
& = & \frac{O(m^2\gamma^2) \cdot \sum_{\mat{p} \in P} |\Delta h(\norm{\mat{x}-\mat{p}})|} {\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}. \label{eq:C22} \end{eqnarray}
From the discussion of generalized gershgorin sets (Section~\ref{sec:basics}), we have \begin{eqnarray} \norm{\Delta\mat{C}_\mat{x}} & \leq & \max\{\norm{\Delta\mat{C}_{11}} + \norm{\Delta\mat{C}_{12}}\,,\, \norm{\Delta\mat{C}_{21}} + \norm{\Delta\mat{C}_{22}}\}. \label{eq:Cx} \end{eqnarray} The correctness of (i) is then proved by plugging into \eqref{eq:Cx} the inequalities \eqref{eq:C11}, \eqref{eq:C12}, and \eqref{eq:C22}.
Define the following generalized gershgorin sets: \begin{eqnarray*} G_1 & = & \left\{\mu : \frac{1}{\norm{(\mat{C}_{11} + \Delta\mat{C}_{11} - \mu \mat{I}_m)^{-1}}} \leq \norm{\mat{C}_{12} + \Delta\mat{C}_{12}}\right\}, \\ G_2 & = & \left\{\mu : \frac{1}{\norm{(\mat{C}_{22} + \Delta\mat{C}_{22}- \mu \mat{I}_{d-m})^{-1}}} \leq \norm{\mat{C}_{21}+\Delta\mat{C}_{21}}\right\}. \end{eqnarray*} We give a lower bound for the values in $G_1$ and an upper bound for the values in $G_2$.
Consider $G_1$. The minimum eigenvalue of $\mat{C}_{11}+\Delta\mat{C}_{11}$ is at least the minimum eigenvalue of $\mat{C}_{11}$ minus $\norm{\Delta\mat{C}_{11}}$. Therefore, by Lemma~\ref{lem:C} and \eqref{eq:C11}, \[ \text{minimum eigenvalue of $\mat{C}_{11}+\Delta\mat{C}_{11} \geq 1 - O(m^2\gamma^2) - \frac{O(m^2\gamma^2) \cdot \sum_{\mat{p} \in P}
|\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}$}. \] On the other hand, by Lemma~\ref{lem:C} and \eqref{eq:C12}, \begin{eqnarray} \norm{\mat{C}_{12} + \Delta\mat{C}_{12}} & \leq & \norm{\mat{C}_{12}} + \norm{\Delta\mat{C}_{12}} \nonumber \\ & \leq & O(m\gamma) + \frac{O(m\gamma) \cdot \sum_{\mat{p} \in P}
|\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}. \label{eq:C1212} \end{eqnarray} The values in $G_1$ are at least the minimum eigenvalue value of $\mat{C}_{11} + \Delta\mat{C}_{11}$ minus $\norm{\mat{C}_{12} + \Delta\mat{C}_{12}}$. Therefore, \begin{equation} \min\{\mu : \mu \in G_1\} \geq 1 - O(m\gamma) - \frac{O(m\gamma) \cdot \sum_{\mat{p} \in P}
|\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}. \label{eq:G1} \end{equation}
Consider $G_2$. By Lemma~\ref{lem:C} and \eqref{eq:C22}, \begin{eqnarray*} \norm{\mat{C}_{22} + \Delta\mat{C}_{22}} & \leq & \norm{\mat{C}_{22}} + \norm{\Delta\mat{C}_{22}} \\ & \leq & O(m^2\gamma^2) + \frac{O(m^2\gamma^2) \cdot \sum_{\mat{p} \in P}
|\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}. \end{eqnarray*} By symmetry and \eqref{eq:C1212}, \begin{eqnarray*} \norm{\mat{C}_{21} + \Delta\mat{C}_{21}} = \norm{\mat{C}_{12} + \Delta\mat{C}_{12}} & \leq & O(m\gamma) + \frac{O(m\gamma) \cdot \sum_{\mat{p} \in P}
|\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}. \end{eqnarray*} The values in $G_2$ are at most $\norm{\mat{C}_{22} + \Delta\mat{C}_{22}} + \norm{\mat{C}_{21} + \Delta\mat{C}_{21}}$. Therefore, \begin{equation} \max\{\mu : \mu \in G_2\} = O(m\gamma) + \frac{O(m\gamma) \cdot \sum_{\mat{p}
\in P} |\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}. \label{eq:G2} \end{equation}
It follows from \eqref{eq:G1} and \eqref{eq:G2} that $G_1$ and $G_2$ are disjoint because every number in $G_2$ is much smaller than those in $G_1$. Lemma~\ref{lem:gc} implies that $G_1$ contains the $m$ largest eigenvalues of $\mat{C}_\mat{x} + \Delta\mat{C}_\mat{x}$. The correctness of (ii) then follows from \eqref{eq:G1}. \end{proof}
We need another technical result on bounding $|\Delta h(\norm{\mat{x}-\mat{p}})|$ from above and $h(\norm{\mat{x}-\mat{q}})$ from below, where $\mat{q}$ is the nearest sample point to $\nu(\mat{x})$.
\begin{lemma}\label{lemma::in_out_ratio} Let $\mat{x}$ be any point at distance $2\eps$ or less from $\mani$. \begin{emromani}
\item For all $\mat{p} \in P$, $\displaystyle |\Delta h(\norm{\mat{x}-\mat{p}})| \leq \left(1 - \frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m-1} \cdot O\left(\frac{m\delta_k}{\gamma}\right)$.
\item $h(\norm{\mat{x}-\mat{q}}) > 0.06$, where $\mat{q}$ is the nearest sample point to $\nu(\mat{x})$.
\end{emromani} \end{lemma} \begin{proof} Consider (i). Since $\Delta h(\norm{\mat{x}-\mat{p}}) = 0$ for any $\mat{p} \in P \setminus B(\mat{x},m\gamma)$, we only need to consider the case of $\norm{\mat{x}-\mat{p}} \leq m\gamma$. Taking derivative gives \begin{eqnarray*}
|\Delta h(\norm{\mat{x}-\mat{p}})| & \leq & 2m\left(1 - \frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m-1} \left(\frac{2\norm{\mat{x}-\mat{p}}}{\gamma} + 1\right) \cdot
\frac{|x_k-p_k|}{m\gamma\norm{\mat{x}-\mat{p}}} \cdot \delta_k + \\ & & \left(1 - \frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m} \cdot
\frac{2|x_k-p_k|}{\gamma\norm{\mat{x}-\mat{p}}} \cdot \delta_k \\ & \leq & \left(1 - \frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m-1} \cdot O\left(\frac{m\delta_k}{\gamma}\right), \end{eqnarray*} establishing the correctness of (i).
Consider (ii). As $P$ is a uniform $(\eps,\kappa)$-sample, $\norm{\mat{q}-\nu(\mat{x})} \leq \eps$. Therefore, $\norm{\mat{q}-\mat{x}} \leq \norm{\mat{x}-\nu(\mat{x})} + \norm{\nu(\mat{x})-\mat{q}} \leq 3\eps$. Then, \begin{eqnarray*} h(\norm{\mat{x}-\mat{q}}) & = & \left(1-\frac{\norm{\mat{x}-\mat{q}}}{m\gamma}\right)^{2m} \left(\frac{2\norm{\mat{x}-\mat{q}}}{\gamma}+1\right) \\ & \geq & \left(1-\frac{3\eps}{m\gamma}\right)^{2m} \\ & = & \left(1 - \frac{3}{4m}\right)^{2m}. \end{eqnarray*} The minimum of $\left(1- \frac{3}{4m}\right)^{2m}$ is achieved at $m=1$, and it is equal to 0.0625.
\end{proof}
The following lemma allows us to ignore the contribution of the points near the boundary of $B(\mat{x}, m\gamma)$ in $\frac{\sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma)} |\Delta h(\norm{\mat{x}-\mat{p}})|}{\sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma)} h(\norm{\mat{x}-\mat{p}})}$.
\begin{lemma}
\label{lem:center}
Let $\mat{x}$ be any point at distance $2\eps$ or less from $\mani$. Let $P$ be a uniform $(\eps,\kappa)$-sample of $\mani$. Let $r = \sqrt{m} \eps/3$. Then,
\[
\sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma)} \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1} \leq (23\kappa+1) \cdot \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma - r)} \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1}.
\] \end{lemma} \begin{proof} Observe that \begin{eqnarray*} \sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma) } \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1} & = & \sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma-r) } \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1} + \\ & & \sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma) \setminus B(\mat{x}, m\gamma-r) } \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1}. \end{eqnarray*} We prove the lemma by bounding the two terms on the right hand side above.
We show a lower bound for the first term. As $P$ is a uniform $(\eps,\kappa)$-sample, there exists some point $\mat{q} \in P$ such that $\norm{\mat{q}-\nu(\mat{x})} \leq \eps$. Therefore, $\norm{\mat{q}-\mat{x}} \leq \norm{\mat{x}-\nu(\mat{x})} + \norm{\nu(\mat{x})-\mat{q}} \leq 3\eps \leq m\gamma - r$. Then, \begin{eqnarray*}
\sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma-r) } \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1}
& \geq & \left(1-\frac{\norm{\mat{x}-\mat{q}}}{m \gamma}\right)^{2m-1} \\
& \geq & \left(1 - \frac{3\eps}{m\gamma}\right)^{2m-1} \\
& \geq & \left(1 - \frac{3}{4m}\right)^{2m}. \end{eqnarray*} The quantity $\left(1 - \frac{3}{4m}\right)^{2m}$ achieves its minimum of 1/16 when $m=1$. Hence, \[ \sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma-r) } \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1}\geq \frac{1}{16}. \]
We show an upper bound for the second term. For any point $\mat{p} \in B(\mat{x}, m\gamma) \setminus B(\mat{x}, m\gamma-r)$, $\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1}$ achieves its maximum of $\bigl(\frac{r}{m \gamma}\bigr)^{2m-1} = \bigl(\frac{1}{12\sqrt{m}}\bigr)^{2m-1}$ when $\norm{\mat{x}-\mat{p}} = m\gamma-r$. By Lemma~\ref{lem:ball}, $|P \cap B(\mat{x}, m\gamma) \setminus B(\mat{x}, m\gamma-r)| \leq |P \cap B(\mat{x},m\gamma)| \leq (4m\gamma/\eps + 1)^m \kappa$. Therefore, \begin{eqnarray*} \sum_{\mat{p} \in P \cap B(\mat{x}, m\gamma) \setminus B(\mat{x}, m\gamma-r) } \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m \gamma}\right)^{2m-1} & \leq & (16m+1)^m\kappa \left(\frac{1}{12\sqrt{m}}\right)^{2m-1} \\ & \leq & (17)^m \kappa \sqrt{m}/ 12^{2m-1} \\
& \leq & 17 \kappa/12. \end{eqnarray*}
Therefore, the second term is at most the first term multiplied by $23\kappa$.
\end{proof}
We bound the turn of $L_\mat{x}$ when $\mat{x}$ moves slightly in the next result.
\begin{lemma} \label{lemma::normal_change} For every point $\mat{x}$ at distance at most $2\eps$ from $\mani$ and for every vector $\Delta \mat{x} \in N_{\nu(\mat{x})} \cup T_{\nu(\mat{x})}$, if $\norm{\Delta\mat{x}}$ is small enough and $\mat{x} + \Delta\mat{x}$ is at distance $2\eps$ or less from $\mani$, then $\angle (L_\mat{x}, L_{\mat{x} + \Delta\mat{x}}) = O(\kappa m^2 \,\norm{\Delta\mat{x}})$. \end{lemma} \begin{proof} Adopt a coordinate frame such that the columns of $\begin{pmatrix} \mat{I}_m \\ \mat{0}_{d-m,m} \end{pmatrix}$ form an orthonormal basis of $T_{\nu(\mat{x})}$, and $\Delta\mat{x}$ points in the direction of the $x_k$-axis for some $k \in [1,d]$. Let $\delta_k = \norm{\Delta\mat{x}}$.
Every entry of $\mat{C}_{\mat{x}+\Delta\mat{x}}$ is some algebraic function in $\delta_k$. By Taylor's Theorem, the $(i,j)$ entry of $\mat{C}_{\mat{x}+\Delta\mat{x}}$ is equal to the $(i,j)$ entry of $\mat{C}_\mat{x} + \Delta\mat{C}_\mat{x}$ plus or minus an $O(\delta_k^2)$ term. Therefore, \[ \mat{C}_{\mat{x} + \Delta\mat{x}} = \mat{C}_\mat{x} + \Delta\mat{C}_\mat{x} + \mat{Z}, \] where $\mat{Z}$ is a $d \times d$ matrix in which every entry is $\pm O(\delta_k^2)$. It follows that \begin{equation} \norm{Z} = O(d\delta_k^2). \label{eq:Z2} \end{equation} Since $\mat{Z} = \mat{C}_{\mat{x}+\Delta\mat{x}} - (\mat{C}_\mat{x} + \Delta\mat{C}_\mat{x})$, $\mat{Z}$ is real symmetric.
Let $\mat{e}$ be one of the $m$ most dominant unit eigenvectors of $\mat{C}_{\mat{x}+\Delta\mat{x}}$. Let $\sigma$ be the eigenvalue of $\mat{C}_{\mat{x}+\Delta\mat{x}}$ corresponding to $\mat{e}$. Therefore, \[ \mat{C}_{\mat{x} + \Delta\mat{x}} \cdot \mat{e} = (\mat{C}_\mat{x} + \Delta\mat{C}_\mat{x} + \mat{Z}) \cdot \mat{e} = \sigma\mat{e}. \] Let $\mat{A}_\mat{x}$ be the $d \times m$ matrix consisting of the $m$ most dominant unit eigenvectors of $\mat{C}_\mat{x}$. So $\col{\mat{A}_\mat{x}}$ is the linear subspace spanned by these eigenvectors. Let $\Lambda$ be the set of the $d-m$ smallest eigenvalues of $\mat{C}_\mat{x}$. We apply Lemma~\ref{lem:slant} with $\mat{M}_1 = \mat{C}_\mat{x}$, $\mat{M}_2 = \Delta\mat{C}_\mat{x} + \mat{Z}$, and $r = m$: \begin{eqnarray} \angle (\col{\mat{A}_\mat{x}},\mat{e}) & \leq & \arcsin\left(\frac{\norm{\Delta\mat{C}_\mat{x} + \mat{Z}}}
{\min_{\lambda \in \Lambda} |\lambda - \sigma|}\right) \nonumber \\ & \leq & \arcsin\left(\frac{\norm{\Delta\mat{C}_\mat{x}} + \norm{\mat{Z}}}
{\min_{\lambda \in \Lambda} |\lambda - \sigma|}\right). \label{eq:angle} \end{eqnarray}
We bound $\angle (\col{\mat{A}_\mat{x}},\mat{e})$ by showing an upper bound for $\norm{\Delta\mat{C}_\mat{x}}$ and a lower bound for $|\lambda - \sigma|$.
For all $\mat{p} \in P \setminus B(\mat{x},m\gamma)$, $h(\norm{\mat{x}-\mat{p}}) = \Delta h(\mat{x}-\mat{p}) = 0$. Then, Lemmas~\ref{lem:delta_C}(i), \ref{lemma::in_out_ratio}(i) and~\ref{lem:center} imply that \begin{eqnarray*} \norm{\Delta\mat{C}_\mat{x}} & \leq & \frac{O(m^2\delta_k) \cdot \sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m-1}} {\sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)} \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m} \left(\frac{2\norm{\mat{x}-\mat{p}}}{\gamma} + 1\right)} \\ & \leq & \frac{O(\kappa m^2\delta_k) \cdot \sum_{\mat{p} \in
P \cap B(\mat{x},m\gamma - r)}
\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m-1}} {\sum_{\mat{p} \in P \cap B(\mat{x},m\gamma -r)}
\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m}
\left(\frac{2\norm{\mat{x}-\mat{p}}}{\gamma} + 1\right)}, \end{eqnarray*} where $r = \sqrt{m} \eps/3$. In the denominator, $\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right) \left(\frac{2\norm{\mat{x}-\mat{p}}}{\gamma} + 1\right)$ is at its minimum of $\frac{2\sqrt{m}\eps}{3\gamma} - \frac{2\eps^2}{9\gamma^2} + \frac{\eps}{3\sqrt{m}\gamma} = \Omega(\sqrt{m})$ when $\norm{\mat{x}-\mat{p}} = m\gamma - r$. It follows that \begin{equation} \norm{\Delta\mat{C}_\mat{x}} = O(\kappa m^{3/2}\delta_k). \label{eq:Cx2} \end{equation}
Lemmas~\ref{lem:gc} and~\ref{lem:C} imply that \begin{equation} \max\{\lambda : \lambda \in \Lambda\} = O(m\gamma). \label{eq:lambda} \end{equation} We write $\mat{C}_\mat{x} + \Delta\mat{C}_\mat{x}$ as the sum $\mat{C}_{\mat{x}+\Delta\mat{x}} + (-\mat{Z})$ and apply Weyl's inequality~\cite[Theorem~3.3.16]{horn} to conclude that the eigenvalue $\sigma$ is at least the $m$-th largest eigenvalue of $\mat{C}_\mat{x} + \Delta\mat{C}_\mat{x}$ minus the largest eigenvalue of $-\mat{Z}$. Then, by Lemma~\ref{lem:delta_C}(ii) and \eqref{eq:Z2}, \[ \sigma \geq 1 - O(m\gamma) - \frac{O(m\gamma) \cdot
\sum_{\mat{p} \in P} |\Delta h(\norm{\mat{x}-\mat{p}})|} {\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})} - O(d\delta_k^2). \] Together with \eqref{eq:lambda}, we obtain \[
\min_{\lambda \in \Lambda} |\lambda - \sigma| \geq 1 - O(m\gamma) - \frac{O(m\gamma)
\cdot \sum_{\mat{p} \in P} |\Delta h(\norm{\mat{x}-\mat{p}})|} {\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})} - O(d\delta_k^2). \] As $\delta_k$ approaches zero, both $\Delta h(\norm{\mat{x}-\mat{p}})$ and $O(d\delta_k^2)$ approach zero. But $\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}) > 0.06$ by Lemma~\ref{lemma::in_out_ratio}(ii). Therefore, for a sufficiently small $\delta_k$, \begin{equation}
\exists\, \text{a constant $\eta > 0$ such that $\min\{\lambda \in \Lambda : |\lambda - \sigma|\} \geq \eta$}. \label{eq:lambda-sigma} \end{equation}
Plugging \eqref{eq:Z2}, \eqref{eq:Cx2} and \eqref{eq:lambda-sigma} into \eqref{eq:angle} gives \[ \angle (\col{\mat{A}_\mat{x}},\mat{e}) \leq \arcsin\left(\frac{O(\kappa m^{3/2}\delta_k) + O(d\delta_k^2)} {\eta}\right) = O(\kappa m^{3/2}\delta_k). \] Since $\mat{e}$ is any one of the $m$ most dominant unit eigenvectors of $\mat{C}_{\mat{x} + \Delta\mat{x}}$, the angle bound $O(\kappa m^{3/2}\delta_k)$ holds for all the $m$ most dominant unit eigenvectors of $\mat{C}_{\mat{x} + \Delta\mat{x}}$. Then, by Lemma~\ref{lem:angle}, $\col{\mat{A}_\mat{x}}$ makes an $O(\kappa m^2\delta_k)$ angle with the space spanned by the $m$ most dominant unit eigenvectors of $\mat{C}_{\mat{x} + \Delta\mat{x}}$. It follows that $\angle (L_\mat{x},L_{\mat{x} + \Delta\mat{x}}) = O(\kappa m^2\delta_k)$. \end{proof}
Next, we need a technical result on the angle between a vector in some linear subspace to its projection in another linear subspace.
\begin{lemma} \label{lem:proj-angle} Let $E_1$ and $E_2$ be two $(d-m)$-dimensional linear subspaces that make an angle $\phi < \pi/2$. Let $\mat{n}$ be a unit vector in $\real^d$. Let $\mat{u}_i$ be the projection of $\mat{n}$ in $E_i$ for $i \in [1,2]$. Let $\{\mat{v}_1,\ldots,\mat{v}_{d-m}\}$ and $\{\mat{w}_1,\ldots,\mat{w}_{d-m}\}$ be bases of $E_1$ and $E_2$, respectively, that satisfy either Lemma~\ref{lem:choice}(i) or Lemma~\ref{lem:choice}(ii). Let $\alpha_1 = \sum_{i=1}^{d-m} (\mat{n}^t\mat{v}_i^{})^2$ and let $\alpha_2 = \sum_{i=d-2m+1}^{d-m} ((\mat{w}_i-\mat{v}_i)^t \mat{n})^2$. If $\alpha_1 > \alpha_2 + (2m^2\phi^2)/\cos\phi$, then \[ \frac{\mat{u}_1^t\mat{u}_2^{}}{\norm{\mat{u}_1}\norm{\mat{u}_2}} \geq \sqrt{1-\frac{\alpha_2}{\alpha_1}}\cos\phi - \frac{2m^2\phi^2}{\sqrt{\alpha_1^2-\alpha_1\alpha_2}}. \] \end{lemma} \begin{proof} By Lemma~\ref{lem:choice}, \begin{eqnarray} \forall\, i \in [1,d-2m], & &\mat{v}_i = \mat{w}_i, \label{eq:choice-1} \\ \forall\, i \in [1,d-m], & &\angle (\mat{v}_i,\mat{w}_i) \leq \phi, \label{eq:choice-2} \\ \forall\, i,j, \in [1,d-m], & &\angle (\mat{v}_i,\mat{w}_j-\mat{v}_j) \in \left[(\pi-\phi)/2,(\pi+\phi)/2\right]. \label{eq:choice-3} \end{eqnarray} If $m \geq d/2$, then \eqref{eq:choice-1} is vacuous because $[1,d-2m]$ is an empty range. There is no harm done as $d-m \leq d/2$ in this case and Lemma~\ref{lem:choice}(i) is applicable, leading to \eqref{eq:choice-2} and \eqref{eq:choice-3} only. If $m < d/2$, then Lemma~\ref{lem:choice}(ii) is applicable, leading to \eqref{eq:choice-1}, \eqref{eq:choice-2} and \eqref{eq:choice-3}.
Since $\mat{u}_i$ is the projection of $\mat{n}$ into $E_i$, we have \begin{eqnarray} \mat{u}_1 & = & (\mat{v}_1 \,\, \cdots \,\, \mat{v}_{d-m}) (\mat{v}_1 \,\, \cdots \,\, \mat{v}_{d-m})^t \mat{n}, \label{eq:u_1} \\ \mat{u}_2 & = & (\mat{w}_1 \,\, \cdots \,\, \mat{w}_{d-m}) (\mat{w}_1 \,\, \cdots \,\, \mat{w}_{d-m})^t \mat{n}. \label{eq:u_2} \end{eqnarray}
We first bound $\mat{u}_1^t \mat{u}_2^{}$ from below. Standard algebra gives \begin{eqnarray} \mat{u}_1^t \mat{u}_2^{} & = & \sum_{i \in [1,d-m]} \mat{n}^t\mat{v}_i^{}\mat{v}_i^t\mat{w}_i^{}\mat{w}_i^t\mat{n} + \sum_{\substack{i\not=j, \\ i,j \in [1,d-m]}} \mat{n}^t\mat{v}_i^{}\mat{v}_i^t\mat{w}_j^{}\mat{w}_j^t\mat{n}. \label{eq:u_1u_2} \end{eqnarray} We analyze the second term in \eqref{eq:u_1u_2}. By \eqref{eq:choice-1}, if $i \not= j$ and $i$ or $j$ belongs to $[1,d-2m]$, then $\mat{v}_i \perp \mat{w}_j$. It implies that $\mat{v}_i^t\mat{w}_j^{} = 0$ in the second term in \eqref{eq:u_1u_2} whenever $i$ or $j$ belongs to $[1,d-2m]$. The remaining case is that both $i$ and $j$ belong to $[d-2m+1,d-m]$.
Define a vector $\mat{h}_i$ for $i \in [1,d-m]$ as follows: \[ \forall\,i \in [1,d-m],\quad \mat{h}_i = \mat{w}_i - \mat{v}_i. \] It follows from \eqref{eq:choice-2} that \begin{equation} \norm{\mat{h}_i} = 2\sin\frac{\angle(\mat{v}_i,\mat{w}_i)}{2} \leq \phi. \label{eq:h_i} \end{equation} We rewrite \eqref{eq:u_1u_2} using $\mat{w}_i = \mat{v}_i + \mat{h}_i$ for $i \in [d-2m+1,d-m]$: \begin{eqnarray} \mat{u}_1^t\mat{u}_2^{} & = & \sum_{i \in [1,d-m]} \mat{n}^t\mat{v}_i^{}\mat{v}_i^t\mat{w}_i^{}\mat{w}_i^t\mat{n} \quad + \sum_{\substack{i \not= j, \\ i,j \in [d-2m+1,d-m]}} \mat{n}^t\mat{v}_i^{}\mat{v}_i^t(\mat{v}_j^{}+\mat{h}_j^{})(\mat{v}_j^{} +\mat{h}_j^{})^t\mat{n} \nonumber \\ & = & \sum_{i \in [1,d-m]} \mat{n}^t\mat{v}_i^{}\mat{v}_i^t\mat{w}_i^{}\mat{w}_i^t\mat{n} \quad + \sum_{\substack{i \not= j, \\ i,j \in [d-2m+1,d-m]}} (\mat{n}^t\mat{v}_i^{}\mat{v}_i^t\mat{h}_j^{}\mat{v}_j^t\mat{n} + \mat{n}^t\mat{v}_i^{}\mat{v}_i^t\mat{h}_j^{}\mat{h}_j^t\mat{n}). \label{eq:u_1u_2-1} \end{eqnarray} Notice that if $m \geq d/2$, then $d-2m+1 \leq 1$, which implies that $[d-2m+1,d-m]$ acts as the range $[1,d-m]$. In this case, Lemma~\ref{lem:choice}(i) is applicable and so \eqref{eq:choice-1} is vacuous, meaning that there is no simplification from \eqref{eq:u_1u_2} to \eqref{eq:u_1u_2-1}.
By \eqref{eq:choice-2}, we get \begin{equation} \forall\, i \in [1,d-m], \quad \mat{v}_i^t\mat{w}_i^{} \geq \cos\phi. \label{eq:v_iw_i} \end{equation} Moreover, \begin{eqnarray} \forall\, i,j \in [1,d-m], \quad \mat{v}_i^t\mat{h}_j^{} & = & \norm{\mat{v}_i} \norm{\mat{h}_j} \cos(\angle(\mat{v}_i,\mat{h}_j)) \nonumber \\ & = & \norm{\mat{h}_j} \cos(\angle(\mat{v}_i,\mat{w}_j-\mat{v}_j)) \nonumber \\ & \stackrel{\eqref{eq:choice-3}}{\geq} & -\norm{\mat{h}_j}\sin(\phi/2) \nonumber \\ & \stackrel{\eqref{eq:h_i}}{\geq} & -\phi\sin(\phi/2). \label{eq:v_iw_j} \end{eqnarray} By substituting \eqref{eq:v_iw_i} and \eqref{eq:v_iw_j} into the first and second terms in \eqref{eq:u_1u_2-1}, respectively, we obtain \begin{eqnarray*} \mat{u}_1^t\mat{u}_2^{} & \geq & \cos\phi \left(\sum_{i \in [1,d-m]} \mat{n}^t\mat{v}_i^{}\mat{w}_i^t\mat{n}\right) \quad - \quad \phi\sin\frac{\phi}{2} \left( \sum_{\substack{i \not= j,\\ i,j \in [d-2m+1,d-m]}} (\mat{n}^t\mat{v}_i^{}\mat{v}_j^t\mat{n} + \mat{n}^t\mat{v}_i^{}\mat{h}_j^t\mat{n})\right) \\ & = & \cos\phi \left(\sum_{i \in [1,d-m]} \mat{n}^t\mat{v}_i^{}\mat{w}_i^t\mat{n}\right) \quad - \quad \phi\sin\frac{\phi}{2} \left( \sum_{\substack{i \not= j,\\ i,j \in [d-2m+1,d-m]}} \mat{n}^t\mat{v}_i^{}\mat{w}_j^t\mat{n} \right). \end{eqnarray*} Both $\mat{n}^t\mat{v}_i$ and $\mat{w}_j^t \mat{n}$ are at most 1, which implies that $\mat{n}^t\mat{v}_i^{}\mat{w}_j^t\mat{n} \leq 1$. Therefore, \[ \mat{u}_1^t\mat{u}_2^{} \, \geq \, \cos\phi \left(\sum_{i \in [1,d-m]} \mat{n}^t\mat{v}_i^{}\mat{w}_i^t\mat{n}\right) - m^2\phi^2. \]
Recall from the lemma statement that $\alpha_1 = \sum_{i=1}^{d-m} (\mat{n}^t\mat{v}_i^{})^2$ and $\alpha_2 = \sum_{i=d-2m+1}^{d-m} (\mat{h}_i^t \mat{n})^2$. We define one more quantity: \[ \alpha_3 = \sum_{i=d-2m+1}^{d-m} \mat{n}^t\mat{v}_i^{}\mat{h}_i^t\mat{n}. \] Standard algebraic manipulation shows that $\alpha_1 + \alpha_3 = \sum_{i\in [1,d-m]} \mat{n}^t\mat{v}_i^{}\mat{w}_i^t\mat{n}$, and therefore, \[ \mat{u}_1^t\mat{u}_2^{} \, \geq \, (\alpha_1+\alpha_3)\cos\phi - m^2\phi^2. \] By definition, \begin{eqnarray*} \norm{\mat{u}_1} & = & \sqrt{\sum_{i \in [1,d-m]} (\mat{n}^t\mat{v}_i)^2} \,\,\, = \,\,\, \sqrt{\alpha_1}, \\ \norm{\mat{u}_2} & = & \sqrt{\sum_{i \in [1,d-m]} (\mat{n}^t\mat{w}_i)^2} \\ & \stackrel{\eqref{eq:choice-1}}{=} & \sqrt{\sum_{i \in [1,d-2m]} (\mat{n}^t\mat{v}_i)^2 \quad + \sum_{i \in [d-2m+1,d-m]} (\mat{n}^t\mat{w}_i)^2 } \\ & = & \sqrt{\sum_{i \in [1,d-2m]} (\mat{n}^t\mat{v}_i)^2 \quad + \sum_{i \in [d-2m+1,d-m]} (\mat{n}^t(\mat{v}_i + \mat{h}_i))^2 } \\ & = & \sqrt{\sum_{i \in [1,d-m]} (\mat{n}^t\mat{v}_i^{})^2 + \sum_{i \in [d-2m+1,d-m]} (2\mat{n}^t\mat{v}_i^{}\mat{h}_i^t\mat{n} + (\mat{h}^t_i\mat{n})^2)} \\ & = & \sqrt{\alpha_1 + 2\alpha_3 + \alpha_2}. \end{eqnarray*} Consequently, \begin{eqnarray} \frac{\mat{u}_1^t\mat{u}_2^{}}{\norm{\mat{u}_1}\norm{\mat{u}_2}} & \geq & \frac{(\alpha_1+\alpha_3)\cos\phi - m^2\phi^2}{\norm{u_1}\norm{u_2}} \nonumber \\ & = & \frac{(\alpha_1+\alpha_3)\cos\phi - m^2\phi^2} {\sqrt{\alpha_1} \cdot \sqrt{\alpha_1+2\alpha_3+\alpha_2}}. \label{eq:u_1u_2-2} \end{eqnarray}
Treating $\alpha_3$ as a free variable while fixing the other values, we can apply standard calculus to show that the right hand side of \eqref{eq:u_1u_2-2} is minimized when $\alpha_3 = -\alpha_2-\frac{m^2\phi^2} {\cos\phi}$ under the condition that $\alpha_1 > \alpha_2 + \frac{2m^2\phi^2}{\cos\phi}$. (This condition ensures that the denominator $\sqrt{\alpha_1^2 + 2\alpha_1\alpha_3 + \alpha_1\alpha_2}$ is real and positive.) This condition is assumed to be satisfied in the lemma statement. Substituting $\alpha_3 = -\alpha_2-\frac{m^2\phi^2}{\cos\phi}$ into \eqref{eq:u_1u_2-2} gives \begin{eqnarray*} \frac{\mat{u}_1^t\mat{u}_2^{}}{\norm{\mat{u}_1}\norm{\mat{u}_2}} & \geq & \frac{(\alpha_1-\alpha_2)\cos\phi - 2m^2\phi^2}{\sqrt{\alpha_1\left(\alpha_1 - \alpha_2 - 2m^2\phi^2/\cos\phi\right)}} \\ & \geq & \frac{(\alpha_1-\alpha_2)\cos\phi - 2m^2\phi^2}{\sqrt{\alpha_1^2 - \alpha_1\alpha_2}} \\ & = & \sqrt{1-\frac{\alpha_2}{\alpha_1}}\cos\phi - \frac{2m^2\phi^2}{\sqrt{\alpha_1^2-\alpha_1\alpha_2}}. \end{eqnarray*} \end{proof}
We are ready to bound the instantaneous change in the normalized projection of a normal vector of $\mani$ into $L_\mat{x}$ as $\mat{x}$ moves, which is the main result of this section.
\begin{lemma}\label{lemma::gradient} Let $\mat{z}$ be any point in $\mani$. Let $\mat{n}$ be any unit vector in $N_\mat{z}$. Define the function $f: B(\mat{z},2\eps) \rightarrow L_\mat{x}$ such that $f(\mat{x})$ is the normalized projection of $\mat{n}$ into $L_\mat{x}$, i.e., $f(\mat{x})$ is the unit vector in $L_\mat{x}$ parallel to the projection of $\mat{n}$ in $L_\mat{x}$. For every point $\mat{x}$ in the interior of $B(\mat{z},2\eps)$ and every $k \in [1,d]$, $\norm{\partial f(\mat{x})/\partial x_k} = O(\kappa m^{3})$. \end{lemma} \begin{proof} Let $\mat{x}$ be a point in the interior of $B(\mat{z},2\eps)$. Consider any index $k \in [1,d]$. Let $\Delta\mat{x}$ be a vector parallel to the $x_k$-axis such that $\mat{x}+\Delta\mat{x} \in B(\mat{z},2\eps)$ and $\delta_k = \norm{\Delta\mat{x}}$ is arbitrarily small. Let $\phi$ denote the angle $\angle (L_\mat{x},L_{\mat{x}+\Delta\mat{x}})$. By Lemma~\ref{lemma::normal_change}, $\phi = O(\kappa m^2\,\delta_k)$. Since $\phi < \pi/2$, there are orthonormal bases of $L_\mat{x}$ and $L_{\mat{x}+\Delta\mat{x}}$ that satisfy either Lemma~\ref{lem:choice}(i) or Lemma~\ref{lem:choice}(ii). Let $\{\mat{v}_1,\ldots,\mat{v}_{d-m}\}$ and $\{\mat{w}_1,\ldots,\mat{w}_{d-m}\}$ be such orthonormal bases of $L_\mat{x}$ and $L_{\mat{x}+\Delta\mat{x}}$, respectively. We want to apply Lemma~\ref{lem:proj-angle}, so we need to verify that $\alpha_1 > \alpha_2 + (2m^2\phi^2)/\cos\phi$, where $\alpha_1 = \sum_{i=1}^{d-m} (\mat{n}^t\mat{v}_i)^2$ and $\alpha_2 = \sum_{i=d-2m+1}^{d-m} ((\mat{w}_i-\mat{v}_i)^t\mat{n})^2$.
First, $\alpha_2 \leq \sum_{i=d-2m+1}^{d-m} \norm{\mat{w}_i-\mat{v}_i}^2$. Since $\angle (\mat{v}_i,\mat{w}_i) \leq \phi$ for $i \in [d-2m+1,d-m]$ by Lemma~\ref{lem:choice}, we obtain $\norm{\mat{w}_i-\mat{v}_i} = 2\sin\frac{\angle(\mat{v}_i,\mat{w}_i)}{2} \leq \phi$. It follows that \[ \alpha_2 \leq m\phi^2 = O(\kappa^2m^5\delta_k^2). \]
Second, observe that $\alpha_1 = \left\|(\mat{v}_1 \, \cdots \, \mat{v}_{d-m})
(\mat{v}_1 \, \cdots \, \mat{v}_{d-m})^t \mat{n}\right\|^2$, where $(\mat{v}_1 \, \cdots \, \mat{v}_{d-m}) (\mat{v}_1 \, \cdots \, \mat{v}_{d-m})^t \mat{n}$ is the projection of $\mat{n}$ into $L_\mat{x}$. Therefore, $\alpha_1 \geq \cos^2 (\angle(L_\mat{x},N_\mat{z}))$. Then, Lemma~\ref{lemma::normal_angle} implies that \[ \alpha_1 \geq \cos^2 (O(m\sqrt{m}\,\gamma)) \geq 1 - O(m^3\gamma^2). \] As $\alpha_2 + \frac{2m^2\phi^2}{\cos\phi}$ approaches zero as $\delta_k \rightarrow 0$, we get $\alpha_1 > \alpha_2 + \frac{2m^2\phi^2}{\cos\phi}$. Then, by Lemma~\ref{lem:proj-angle}, \[ \frac{\mat{u}_1^t\mat{u}_2^{}}{\norm{\mat{u}_1}\norm{\mat{u}_2}} \geq \sqrt{1-\frac{\alpha_2}{\alpha_1}}\cos\phi - \frac{2m^2\phi^2}{\sqrt{\alpha_1^2-\alpha_1\alpha_2}}, \] where $\mat{u}_1$ and $\mat{u}_2$ are the projections of $\mat{n}$ into $L_\mat{x}$ and $L_{\mat{x}+\Delta\mat{x}}$, respectively.
Finally, \begin{eqnarray*}
\left\| \frac{\partial f(\mat{x})}{\partial x_k} \right\|^2 & = & \lim_{\delta_k \rightarrow 0} \, \frac{1}{\delta_k^2} \left(\frac{\mat{u}_2}{\norm{\mat{u}_2}} - \frac{\mat{u}_1}{\norm{\mat{u}_1}}\right)^t \left(\frac{\mat{u}_2}{\norm{\mat{u}_2}} - \frac{\mat{u}_1}{\norm{\mat{u}_1}}\right) \\ & = & \lim_{\delta_k \rightarrow 0} \, \frac{1}{\delta_k^2} \left(2 - \frac{2\mat{u}_1^t\mat{u}_2^{}} {\norm{\mat{u}_1}\norm{\mat{u}_2}}\right) \\ & \leq & \lim_{\delta_k \rightarrow 0} \, \frac{1}{\delta_k^2} \left(2 - 2\sqrt{1-\frac{\alpha_2}{\alpha_1}}\cos\phi + \frac{4m^2\phi^2}{\sqrt{\alpha_1^2-\alpha_1\alpha_2}}\right) \\ & \leq & \lim_{\delta_k \rightarrow 0} \, \frac{1}{\delta_k^2} \left(2 - 2\left(1-\frac{\alpha_2}{\alpha_1}\right)\cos\phi + \frac{4m^2\phi^2}{\sqrt{\alpha_1^2 - \alpha_1\alpha_2}}\right). \end{eqnarray*} We have shown earlier that $\alpha_2 \leq m\phi^2$ and $\alpha_1 \geq 1-O(m^3\gamma^2)$. Using these relations and the facts that $\cos\phi \geq 1-\phi^2/2$ and $\phi = O(\kappa m^2\,\delta_k)$, we obtain \begin{eqnarray*}
\left\| \frac{\partial f(\mat{x})}{\partial x_k} \right\|^2 & \leq & \lim_{\delta_k \rightarrow 0} \, \frac{1}{\delta_k^2} \left(2 - 2\left(1 - \frac{m\phi^2}{\alpha_1} \right)\left(1 - \frac{\phi^2}{2}\right) + \frac{4m^2\phi^2}{\sqrt{\alpha_1^2 - \alpha_1m\phi^2}} \right) \\ & = & \lim_{\delta_k \rightarrow 0} \, \frac{1}{\delta_k^2} \left(2 - 2\left(1 - \frac{m\phi^2}{\alpha_1} - \frac{\phi^2}{2} + \frac{m\phi^4}{2\alpha_1}\right) + \frac{4m^2\phi^2}{\sqrt{\alpha_1^2 - \alpha_1 m\phi^2}} \right) \\ & \leq & \lim_{\delta_k \rightarrow 0} \, \frac{1}{\delta_k^2} \left( O\left(\frac{\kappa^2 m^5 \delta_k^2}{\alpha_1}\right) + \frac{O(\kappa^2 m^6 \delta_k^2)}{\sqrt{\alpha_1^2 - O(\alpha_1 \kappa^2 m^5 \delta_k^2)}} \right) \\ & = & O(\kappa^2m^6). \end{eqnarray*} \end{proof}
We use Lemma~\ref{lemma::gradient} to bound $\norm{\mat{J}_f(\mat{x})}$. Multiplying the bound in Lemma~\ref{lemma::gradient} by $\sqrt{d}$ already gives a bound. We give a tighter analysis that yields a bound independent of $d$.
\begin{lemma} \label{cor:gradient} Let $\mat{z}$ be any point in $\mani$. Let $\mat{J}_f$ be the Jacobian of the function $f : B(\mat{z},2\eps)\rightarrow L_\mat{x}$ defined in Lemma~\ref{lemma::gradient}. For any point $\mat{x}$ in the interior of $B(\mat{z},2\eps)$, $\norm{\mat{J}_f(\mat{x})} = O(\kappa m^{3})$. \end{lemma} \begin{proof} Fix a unit vector $\mat{n} \in N_\mat{z}$ as required in the definition of $f$ in Lemma~\ref{lemma::gradient}. Let $\mat{x}$ be a point in the interior of $B(\mat{z},2\eps)$. Let $\mat{R}$ be any $d \times d$ orthogonal matrix. Apply the orthogonal transformation induced by $\mat{R}$ to $\real^d$. Then define the function $g : B(\mat{z}',2\eps) \rightarrow L_{\mat{x}'}$, where $\mat{z}' = \mat{R} \cdot \mat{z}$ and $\mat{x}' = \mat{R} \cdot \mat{x}$, such that $g(\mat{x}')$ is the normalized projection of $\mat{R} \cdot \mat{n}$ into $L_{\mat{x}'}$.
First, we show that $f(\mat{x}) = \mat{R}^t \cdot g(\mat{x}')$. Let $\ell$ be the length of the projection of $\mat{n}$ into $L_\mat{x}$. Let $\mat{Q}$ be any $d \times (d-m)$ matrix whose columns form an orthonormal basis of $L_\mat{x}$. It follows from the definition of $f$ that $f(\mat{x}) = \frac{1}{\ell} \cdot \mat{Q}\cdot \mat{Q}^t\cdot \mat{n}$. Since an orthogonal transformation preserves lengths, $\ell$ is also the length of the projection of $\mat{R}\cdot\mat{n}$ into $L_{\mat{x}'}$. Then, $g(\mat{x}') = \frac{1}{\ell} \cdot \mat{R}\cdot\mat{Q}\cdot\mat{Q}^t\cdot\mat{R}^t \cdot \mat{R}\cdot\mat{n} = \frac{1}{\ell}\cdot\mat{R}\cdot\mat{Q}\cdot\mat{Q}^t\cdot\mat{n}$, which implies that $f(\mat{x}) = \mat{R}^t \cdot g(\mat{x}')$.
We show that $\mat{J}_f(\mat{x}) = \mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R}$. Let $\Delta\mat{x}$ be an arbitrarily short vector. By Taylor's Theorem, \begin{equation} f(\mat{x}+\Delta\mat{x}) = f(\mat{x}) + \mat{J}_f(\mat{x}) \cdot \Delta\mat{x} + \mat{e}_f, \label{eq:cor:gradient-1} \end{equation} where $\mat{e}_f/\norm{\Delta\mat{x}}$ converges to the zero vector as $\norm{\Delta\mat{x}} \rightarrow 0$. Similarly, \begin{equation} g(\mat{R}\cdot\mat{x}+\mat{R}\cdot\Delta\mat{x}) = g(\mat{x}') + \mat{J}_g(\mat{x}') \cdot \mat{R}\cdot\Delta\mat{x} + \mat{e}_g, \label{eq:cor:gradient-2} \end{equation} where $\mat{e}_g/\norm{\mat{R}\cdot\Delta\mat{x}}$ converges to the zero vector as $\norm{\mat{R}\cdot\Delta\mat{x}} \rightarrow 0$. Since $\mat{R}$ is fixed, it means that $\mat{e}_g/\norm{\Delta\mat{x}}$ tends to the zero vector as $\norm{\Delta\mat{x}} \rightarrow 0$. We multiply both sides of \eqref{eq:cor:gradient-2} by $\mat{R}^t$ and then subtract the resulting equation from \eqref{eq:cor:gradient-1}. Some terms cancel each other because $f(\mat{x} + \Delta\mat{x}) = \mat{R}^t \cdot g\left(\mat{R}\cdot(\mat{x} + \Delta\mat{x})\right)$ and $f(\mat{x}) = \mat{R}^t \cdot g(\mat{x}') = \mat{R}^t \cdot g\left(\mat{R} \cdot \mat{x}\right)$. We obtain \[ \left(\mat{J}_f(\mat{x}) - \mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R} \right) \cdot \Delta\mat{x} = \mat{R}^t\cdot\mat{e}_g - \mat{e}_f. \] Therefore, \[
\left\|\left(\mat{J}_f(\mat{x}) -
\mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R}\right) \cdot \Delta\mat{x}\right\| \leq
\left\|\mat{R}^t\cdot\mat{e}_g\right\|
+ \left\|\mat{e}_f\right\|. \] We are free to choose the direction of $\Delta\mat{x}$. We choose it
such that $\left\|(\mat{J}_f -
\mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R}) \cdot \Delta\mat{x}\right\| = \left\|\mat{J}_f(\mat{x}) -
\mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R}\right\| \cdot \left\|\Delta\mat{x}\right\|$, i.e., $\Delta\mat{x}$ is an eigenvector corresponding to the largest eigenvalue of $\mat{J}_f(\mat{x}) - \mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R}$. Then, \[
\left\|\mat{J}_f(\mat{x}) - \mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R}\right\| \leq
\frac{\left\|\mat{R}^t\mat{e}_g\right\|}{\left\|\Delta\mat{x}\right\|} +
\frac{\left\|\mat{e}_f\right\|}{\left\|\Delta\mat{x}\right\|}. \] Since the right hand side tends to zero as $\norm{\Delta\mat{x}} \rightarrow 0$, we conclude that \[ \lim_{\norm{\Delta\mat{x}} \rightarrow 0} \norm{\mat{J}_f(\mat{x}) - \mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R}} = 0, \] which implies that $\mat{J}_f(\mat{x}) = \mat{R}^t \cdot \mat{J}_g(\mat{x}') \cdot \mat{R}$.
By definition, $\norm{\mat{J}_f(\mat{x})} = \norm{\mat{J}_f(\mat{x}) \cdot \mat{v}}$ for some unit vector $\mat{v}$. We choose $\mat{R}$ to be the $d \times d$ orthogonal matrix such that $\mat{R} \cdot\mat{v} = (1,0,\ldots,0)^t$. Then, $\norm{\mat{R} \cdot \mat{J}_f(\mat{x}) \cdot \mat{v}} = \norm{\mat{R} \cdot \mat{J}_f(\mat{x}) \cdot \mat{R}^t \cdot \mat{R} \cdot \mat{v}} = \norm{\mat{J}_g(\mat{x}') \cdot (1,0,\ldots,0)^t}$, which is the 2-norm of the first column of $\mat{J}_g(\mat{x}')$. Lemma~\ref{lemma::gradient} is independent of the coordinate frame. So we can apply Lemma~\ref{lemma::gradient} to $g$ and conclude that the 2-norm of the first column of $\mat{J}_g(\mat{x}')$ is $O(\kappa m^{3})$. As a result, $\norm{\mat{R} \cdot \mat{J}_f(\mat{x}) \cdot \mat{v}} = O(\kappa m^{3})$. Since multiplying any vector with an orthogonal matrix preserves the 2-norm of the vector, we conclude that $\norm{\mat{J}_f(\mat{x})} = \norm{\mat{J}_f(\mat{x}) \cdot \mat{v}} = \norm{\mat{R} \cdot \mat{J}_f(\mat{x}) \cdot \mat{v}} = O(\kappa m^{3})$. \end{proof}
\section{Faithful reconstruction}
In this section, we prove our main result that $Z_\varphi \cap \widehat{\mani}$ is a faithful reconstruction of $\mani$. Recall the class $\Phi$ of functions $\varrho: \real^d \rightarrow \real^{d-m}$: \begin{quote} $\displaystyle \Phi = \left\{ \varrho : \varrho(\mat{x}) = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{B}^t_{\varrho,\mat{x}} \cdot (\mat{x}-\mat{p}) \right\}$, where $\mat{B}_{\varrho,\mat{x}}$ is any $d \times (d-m)$ matrix with linearly independent columns such that $\col{\mat{B}_{\varrho,\mat{x}}} = L_{\mat{x}}$. \end{quote}
We claim that the choice of $\mat{B}_{\varrho,\mat{x}}$ has no impact on the zero-set $Z_\varrho$ as long as the columns of $\mat{B}_{\varrho,\mat{x}}$ are linearly independent. In this section, we will prove some useful properties of functions in $\Phi$. These properties will allow us to show that $Z_\varphi \cap \widehat{\mani}$ is a faithful approximation of $\mani$.
We will study properties of $Z_\varphi \cap \widehat{\mani}$ by analyzing $Z_\varrho \cap \widehat{\mani}$ for another function $\varrho \in \Phi$ conveniently chosen for the analysis. Since we will conduct some local analysis, we are only concerned with functions that are defined near some chosen points in $\mani$. This motivates us to define for every point $\mat{z} \in \mani$ the following class $\Phi_{\mat{z}}$ of functions: \begin{quote} $\displaystyle \Phi_{\mat{z}} = \left\{ \varrho : \varrho: B(\mat{z},2\eps) \rightarrow \real^{d-m},\,\, \varrho(\mat{x}) = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{B}^t_{\varrho,\mat{x}} \cdot (\mat{x}-\mat{p}) \right\}$, where $\mat{B}_{\varrho,\mat{x}}$ is any $d \times (d-m)$ matrix with linearly independent columns such that $\col{\mat{B}_{\varrho,\mat{x}}} = L_{\mat{x}}$. \end{quote} $\Phi_{\mat{z}}$ is a local version of $\Phi$. The next result shows that functions in $\Phi_\mat{z}$ with overlapping domains have consistent zero sets.
\begin{lemma} \label{lem:agree}
Let $\mat{y}$ and $\mat{z}$ be two arbitrary points in $\mani$ that are not necessarily distinct. For every point $\mat{x} \in B(\mat{y},2\eps) \cap B(\mat{z},2\eps)$, if there exists $\varrho \in \Phi_{\mat{y}}$ such that $\varrho(\mat{x}) = \mat{0}_{d-m,1}$, then for every $\varrho \in \Phi_{\mat{y}} \cup \Phi_{\mat{z}}$, $\varrho(\mat{x}) = \mat{0}_{d-m,1}$. \end{lemma} \begin{proof} Take two functions $\varrho, \bar{\varrho} \in \Phi_{\mat{y}} \cup \Phi_{\mat{z}}$. Fix a point $\mat{x} \in B(\mat{y},2\eps) \cap B(\mat{z},2\eps)$. By definition, $\varrho(\mat{x}) = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{B}_{\varrho,\mat{x}}^t \cdot (\mat{x}-\mat{p})$ and $\bar{\varrho}(\mat{x}) = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{B}_{\bar{\varrho},\mat{x}}^t \cdot (\mat{x}-\mat{p})$. The columns of $\mat{B}_{\varrho,\mat{x}}$ and $\mat{B}_{\bar{\varrho},\mat{x}}$ form two bases of $L_\mat{x}$, which means that there is a $(d-m) \times (d-m)$ invertible matrix $\mat{R}$ such that $\mat{R} \cdot \mat{B}_{\varrho,\mat{x}}^t = \mat{B}_{\bar{\varrho},\mat{x}}^t$. If $\varrho(\mat{x}) = \mat{0}_{d-m,1}$, then $\bar{\varrho}(\mat{x}) = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{R} \cdot \mat{B}_{\varrho,\mat{x}}^t \cdot (\mat{x}-\mat{p}) = \mat{R} \cdot \varrho(\mat{x}) = \mat{0}_{d-m,1}$. \end{proof}
We define a particular function $\varrho_{\mat{z}} \in \Phi_\mat{z}$ to analyze the properties of $Z_\varphi \cap \widehat{\mani}$ in a small neighborhood of $\mat{z}$.
\begin{definition} \label{df:local} Let $\mat{z}$ be any point in $\mani$. Let $\{\mat{v}_1,\ldots, \mat{v}_{d-m}\}$ be any set of unit vectors forming a basis of $N_\mat{z}$. For $i \in [1,d-m]$, let $f_{\mat{v}_i}$ be the function that maps every $\mat{x}$ in $B(\mat{z},2\eps)$ to the normalized projection of $\mat{v}_i$ in $L_\mat{x}$. Define a {\bf canonical function} $\varrho_\mat{z} : B(\mat{z},2\eps) \rightarrow \real^{d-m}$ with respect to $\mat{z}$ and $\{\mat{v}_1,\ldots,\mat{v}_{d-m}\}$ such that for all $\mat{x} \in B(\mat{z}, 2\eps)$, $\varrho_\mat{z}(\mat{x}) = \sum_{\mat{p}\, \in P} \omega(\mat{x},\mat{p}) \cdot [f_{\mat{v}_1}(\mat{x}),\ldots,f_{\mat{v}_{d-m}}(\mat{x})]^t \cdot (\mat{x}-\mat{p})$. \end{definition}
We show that whenever $\eps$ is sufficiently small, $\varrho_\mat{z}$ belongs to $\Phi_{\mat{z}}$ and $\varrho_{\mat{z}}$ is continuous in the interior of $B(\mat{z}, 2\eps)$.
\begin{lemma}\label{lemma:local} Let $\varrho_\mat{z}$ be the canonical function with respect to a point $\mat{z} \in \mani$ and some set of unit vectors $\{\mat{v}_1,\ldots,\mat{v}_{d-m}\}$ forming a basis of $N_\mat{z}$ for which there exists some $\phi \in \left[0,\arcsin\left(\frac{1}{3d-3m}\right)\right)$ such that for any distinct $i,j \in [1,d-m]$, $\angle (\mat{v}_i,\mat{v}_j) \in [\pi/2-\phi,\pi/2+\phi]$. There exists $\eps_0 \in (0,1)$ that decreases as $d$ increases such that for every point $\mat{z} \in \mani$, if $\eps \leq \eps_0$, then $\varrho_\mat{z} \in \Phi_{\mat{z}}$ and $\varrho_\mat{z}$ is continuous in the interior of $B(\mat{z}, 2\eps)$. \end{lemma} \begin{proof} To show that $\varrho_\mat{z} \in \Phi_{\mat{z}}$, it suffices to prove that $\{f_{\mat{v}_1}(\mat{x}),\ldots,f_{\mat{v}_{d-m}}(\mat{x})\}$ form a basis of $L_\mat{x}$, which boils down to showing that $\{f_{\mat{v}_1}(\mat{x}),\ldots,f_{\mat{v}_{d-m}}(\mat{x})\}$ are linearly independent.
Since $\angle (L_\mat{x},N_\mat{z}) = O(m\sqrt{m}\,\gamma)$ by Lemma~\ref{lemma::normal_angle}, we get $\angle (f_{\mat{v}_i}(\mat{x}),\mat{v}_i) = O(m\sqrt{m}\,\gamma)$. Assume to the contrary that $f_{\mat{v}_1}(\mat{x}), \ldots, f_{\mat{v}_{d-m}}(\mat{x})$ are linearly dependent. Then, \[ \angle \left(f_{\mat{v}_1}(\mat{x}), \col{(f_{\mat{v}_2}(\mat{x}) \, \cdots \, f_{\mat{v}_{d-m}}(\mat{x}))} \right) = 0. \] Since $\angle \left(\mat{v}_i, \col{(f_{\mat{v}_2}(\mat{x}) \, \cdots \, f_{\mat{v}_{d-m}}(\mat{x}))}\right) = O(m\sqrt{m}\,\gamma)$ for all $i \in [2,d-m]$, Lemma~\ref{lem:angle} implies that \[ \angle \left(\col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}, \col{(f_{\mat{v}_2}(\mat{x}) \, \cdots \, f_{\mat{v}_{d-m}}(\mat{x}))} \right) = O\left(m\sqrt{dm-m^2}\,\gamma\right). \] By triangle inequality, $\angle \left(\mat{v}_1, \col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}\right) \leq \angle \left(\mat{v}_1, f_{\mat{v}_1}(\mat{x})\right) + \angle \left(f_{\mat{v}_1}(\mat{x}), \col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}\right)$. The dimension of $\col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}$ is at least the dimension of $\col{(f_{\mat{v}_2}(\mat{x}) \, \cdots \, f_{\mat{v}_{d-m}}(\mat{x}))}$. Thus, \begin{eqnarray*} \angle \left(f_{\mat{v}_1}(\mat{x}), \col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}\right) & \leq & \angle \left(f_{\mat{v}_1}(\mat{x}), \col{(f_{\mat{v}_2}(\mat{x}) \, \cdots \, f_{\mat{v}_{d-m}}(\mat{x}))}\right) + \\ & & \angle \left(\col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}, \col{(f_{\mat{v}_2}(\mat{x}) \, \cdots \, f_{\mat{v}_{d-m}}(\mat{x}))}\right). \end{eqnarray*} Combining the above observations, we obtain \begin{eqnarray*} \angle (\mat{v}_1, \col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}) & \leq & \angle \left(\mat{v}_1, f_{\mat{v}_1}(\mat{x})\right) + \angle \left(f_{\mat{v}_1}(\mat{x}), \col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}\right) \\ & \leq & \angle \left(\mat{v}_1, f_{\mat{v}_1}(\mat{x})\right) + \angle \left(f_{\mat{v}_1}(\mat{x}), \col{(f_{\mat{v}_2}(\mat{x}) \, \cdots \, f_{\mat{v}_{d-m}}(\mat{x}))}\right) + \\ & & \angle \left(\col{(\mat{v}_2 \, \cdots \, \mat{v}_{d-m})}, \col{(f_{\mat{v}_2}(\mat{x}) \, \cdots \, f_{\mat{v}_{d-m}}(\mat{x}))}\right) \\ & = & O(m\sqrt{dm-m^2}\,\gamma). \end{eqnarray*}
Recall that $\gamma = 4\eps \leq 4\eps_0$. Assume that $\eps_0 < \frac{1}{Cm\sqrt{dm-m^2}}$ for some appropriate constant $C \geq 1$. Then $\angle (\mat{v}_1,\col{\mat{v}_2 \cdots \mat{v}_{d-m}}) < \pi/6$. Note that $\eps_0$ decreases as $d$ increases. Let $\mat{u}$ be the normalized projection of $\mat{v}_1$ in $\col{\mat{v}_2 \cdots \mat{v}_{d-m}}$. It means that \[ \mat{v}_1^t \cdot \mat{u} > \cos(\pi/6) = \sqrt{3}/2. \]
We can write $\mat{u} = \sum_{i=2}^{d-m} \lambda_i \mat{v}_i$ for some $\lambda_i$. Let $k = \argmax_{i=[2,d-m]} |\lambda_i|$. We take the dot product of $\mat{u}$ and $\sign(\lambda_k)\mat{v}_k$. This dot product is equal to $|\lambda_k|\norm{\mat{v}_k}^2 + \sign(\lambda_k)\sum_{i\neq k} \lambda_i \mat{v}_i^t \cdot \mat{v}_k$ and it is at most 1 as $\mat{u}$ and $\mat{v}_k$ are unit vectors. Since $\angle (\mat{v}_i,\mat{v}_j) \in \bigl[\frac{\pi}{2}-\phi,\frac{\pi}{2}+\phi\bigr]$, the projection of $\mat{v}_j$ in the direction of $\mat{v}_i$ has magnitude at most $\sin\phi$. It follows that \begin{eqnarray*}
1 & \geq & |\lambda_k| - \sum_{i\neq k} |\lambda_i| \mat{v}_i^t \cdot \mat{v}_k\\ & \geq &
|\lambda_k| - (d-m-2) |\lambda_k| \sin \phi. \end{eqnarray*}
We get $|\lambda_k| \leq 1/ (1 - (d-m-2)\sin\phi) < 1.5$ because $\sin\phi < \frac{1}{3d-3m}$ by assumption of the lemma. Thus,
\[
\mat{v}_1^t \cdot \mat{u} = \sum_{i=2}^{d-m} \lambda_i \mat{v}_1^t \cdot \mat{v}_i \leq \sin\phi \cdot \sum_{i=2}^{d-m} |\lambda_i| < 1.5(d-m)\sin\phi < 0.5. \] This is a contradiction because we have derived earlier that $\mat{v}_1^t \cdot \mat{u} > \sqrt{3}/2$. We conclude that $\{f_{\mat{v}_1}(\mat{x}),\ldots,f_{\mat{v}_{d-m}}(\mat{x})\}$ are linearly independent, and therefore, $\varrho_\mat{z} \in \Phi_{\mat{z}}$.
By Lemma~\ref{lemma::gradient}, for $i \in [1,d-m]$, $f_{\mat{v}_i}$ is differentiable and hence continuous in the interior of $B(\mat{z},2\eps)$. Because $\varrho_\mat{z}$ is a sum of products of continuous functions, $\varrho_\mat{z}$ is also continuous in the interior of $B(\mat{z},2\eps)$ \cite[Ch~2:~Corollary~3.7]{mendelson}. \end{proof}
Next, we show that the gradient of $\varrho_\mat{z}$ varies monotonically.
\begin{lemma}\label{lemma::unique} Let $\mat{z}$ be any point in $\mani$. Let $\mat{v}_i$ be any unit vector in $N_\mat{z}$. For any $\mat{x} \in B(\mat{z},2\eps)$, let $\varrho_{\mat{z},i}(\mat{x}) = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})$. Let $\tau$ be any value greater than $1$. For every $t \geq 1$ and every point $\mat{x} \in B(\mat{z}, t\eps^{\tau})$, \begin{itemize}
\item $\left\|\nabla \varrho_{\mat{z},i}(\mat{x})\right\| \in \left[1 - O(t\kappa\sqrt{m}\eps^{\tau-1}+ \kappa m^{4}\gamma), 1 + O(t\kappa\sqrt{m}\eps^{\tau-1} + \kappa m^{4}\gamma)\right]$ and
\item $\mat{v}_i^t \cdot \nabla\varrho_{\mat{z},i}(\mat{x}) \geq 1 - O(t\kappa\sqrt{m}\eps^{\tau-1} + \kappa m^{4}\gamma)$.
\end{itemize} \end{lemma} \begin{proof} From the definition of $\varrho_{\mat{z},i}(\mat{x}) = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})$, we obtain \begin{eqnarray}
\left\|\nabla \varrho_{\mat{z},i}(\mat{x})\right\| & \leq & \sum_{\mat{p} \in P} \left(\omega(\mat{x},\mat{p}) + \omega(\mat{x},\mat{p}) \cdot \norm{\mat{J}_{f_{\mat{v}_i}}(\mat{x})} \cdot \norm{\mat{x}-\mat{p}} \right) + \nonumber \\
& & \left\|\sum_{\mat{p} \in P}
\nabla \omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})\right\|. \label{eq:unique-1} \end{eqnarray} Consider the first term in \eqref{eq:unique-1}. By Lemma~\ref{cor:gradient}, $\norm{\mat{J}_{f_{\mat{v}_i}}(\mat{x})} = O(\kappa m^{3})$. For any $\mat{p} \not\in B(\mat{x},m\gamma)$, $\omega(\mat{x},\mat{p})$ vanishes. If $\mat{p} \in B(\mat{x},m\gamma)$, then \begin{equation} \norm{\mat{J}_{f_{\mat{v}_i}}(\mat{x})} \cdot \norm{\mat{x}-\mat{p}} = O(\kappa m^{4}\gamma). \label{eq:unique} \end{equation} Therefore, \begin{equation} \sum_{\mat{p} \in P} \left(\omega(\mat{x},\mat{p}) + \omega(\mat{x},\mat{p}) \cdot \norm{\mat{J}_{f_{\mat{v}_i}}(\mat{x})} \cdot \norm{\mat{x}-\mat{p}} \right) \leq 1 + O(\kappa m^{4}\gamma). \label{eq:unique-2} \end{equation} Consider the second term in \eqref{eq:unique-1}. For any point $\mat{p} \not\in B(\mat{x},m\gamma)$, $\nabla \omega(\mat{x},\mat{p})$ is a zero vector. If $\mat{p} \in B(\mat{x},m\gamma)$, then $\norm{\mat{p}-\nu(\mat{x})} \leq \norm{\mat{p}-\mat{x}} + \norm{\mat{x}-\nu(\mat{x})} \leq m\gamma + t \eps^{\tau} = O(m\gamma)$. By Lemma~\ref{lem:basic}(i), $\mat{p} - \nu(\mat{x})$ makes an angle $\pi/2 - O(m\gamma)$ with $N_{\nu(\mat{x})}$. It follows from Lemma~\ref{lemma::normal_angle} that $\mat{p} - \nu(\mat{x})$ makes an angle $\pi/2 - O(m\sqrt{m}\,\gamma)$ with $L_\mat{x}$. Therefore, the projection of $\mat{p} - \nu(\mat{x})$ onto $L_\mat{x}$ has length less than $O(m\sqrt{m}\,\gamma) \cdot O(m\gamma) = O(m^{5/2}\gamma^2)$. Since
$f_{\mat{v}_i}(\mat{x})$ is a unit vector in $L_\mat{x}$, the projection $\mat{p} -\nu(\mat{x})$ in $L_\mat{x}$ has length at least $\left|f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{p}-\nu(\mat{x}))\right|
\geq \left|f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{p}-\mat{x})\right| - \norm{\mat{x}-\nu(\mat{x})}$. Therefore, \begin{equation}
\left| f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})\right| \leq \norm{\mat{x}-\nu(\mat{x})} + O(m^{5/2}\gamma^2) \leq t\eps^{\tau} + O(m^{5/2}\gamma^2). \label{eq:unique-0} \end{equation} We conclude that \begin{equation}
\left\|\sum_{\mat{p} \in P}
\nabla \omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})\right\|
\leq O(t\eps^{\tau} + m^{5/2}\gamma^2) \cdot \sum_{\mat{p} \in P} \left\|\nabla\omega(\mat{x},\mat{p})\right\|. \label{eq:unique-3} \end{equation} Since \[ \nabla\omega(\mat{x},\mat{p}) = \frac{\left(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})\right) \frac{\mathrm{d}h(\norm{\mat{x}-\mat{p}})}{\mathrm{d} \norm{\mat{x}-\mat{p}}} \cdot \frac{\mat{x}-\mat{p}}{\norm{\mat{x}-\mat{p}}} - h(\norm{\mat{x}-\mat{p}}) \sum_{\mat{p} \in P} \frac{\mathrm{d}h(\norm{\mat{x}-\mat{p}})}{\mathrm{d} \norm{\mat{x}-\mat{p}}} \cdot \frac{\mat{x}-\mat{p}}{\norm{\mat{x}-\mat{p}}}} {\left(\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})\right)^2}, \] we obtain \[ \sum_{\mat{p} \in P} \norm{\nabla\omega(\mat{x},\mat{p})} \leq \frac{2\,
\sum_{\mat{p} \in P} |\mathrm{d}h(\norm{\mat{x}-\mat{p}})/\mathrm{d}
\norm{\mat{x}-\mat{p}}|} {\sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}})}. \]
By Lemma~\ref{lemma::in_out_ratio}(i), differentiating $h(\norm{\mat{x}-\mat{p}})$ with respect to $\norm{\mat{x}-\mat{p}}$ gives \[
\left|\frac{\mathrm{d}h(\norm{\mat{x}-\mat{p}})}{\mathrm{d}
\norm{\mat{x}-\mat{p}}}\right| \leq O\left(\frac{m}{\gamma}\right) \cdot \left(1- \frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m-1}. \] On the other hand, \[ \sum_{\mat{p} \in P} h(\norm{\mat{x}-\mat{p}}) = \sum_{\mat{p} \in P} \left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma} \right)^{2m}\left(\frac{2\norm{\mat{x}-\mat{p}}}{\gamma}+1\right). \] For all $\mat{p} \in P \setminus B(\mat{x},m\gamma)$,
$h(\norm{\mat{x}-\mat{p}}) = 0$ and $\left|\frac{\mathrm{d}h(\norm{\mat{x}-\mat{p}})}{\mathrm{d}
\norm{\mat{x}-\mat{p}}}\right| = 0$. Then, \begin{eqnarray*}
\sum_{\mat{p} \in P} \norm{\nabla\omega(\mat{x},\mat{p})}
& \leq & \frac{O\left(\frac{m}{\gamma}\right) \sum_{\mat{p} \in
P \cap B(\mat{x},m\gamma)}
\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m-1}} {\sum_{\mat{p} \in P \cap B(\mat{x},m\gamma)}
\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m}
\left(\frac{2\norm{\mat{x}-\mat{p}}}{\gamma} + 1\right)}. \end{eqnarray*} Let $r = \sqrt{m}\eps/3$. By Lemma~\ref{lem:center}, \begin{eqnarray*}
\sum_{\mat{p} \in P} \norm{\nabla\omega(\mat{x},\mat{p})}
& \leq & \frac{O\left(\frac{\kappa m}{\gamma}\right) \cdot \sum_{\mat{p} \in
P \cap B(\mat{x},m\gamma - r)}
\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m-1}} {\sum_{\mat{p} \in P \cap B(\mat{x},m\gamma -r)}
\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right)^{2m}
\left(\frac{2\norm{\mat{x}-\mat{p}}}{\gamma} + 1\right)}. \end{eqnarray*} In the denominator, the term $\left(1-\frac{\norm{\mat{x}-\mat{p}}}{m\gamma}\right) \left(\frac{2\norm{\mat{x}-\mat{p}}}{\gamma} + 1\right)$ achieves its minimum
$\frac{2\sqrt{m}\eps}{3\gamma} - \frac{2\eps^2}{9\gamma^2} + \frac{\eps}{3\sqrt{m}\gamma} = \Omega(\sqrt{m})$ when $\norm{\mat{x}-\mat{p}} = m\gamma - r$.
It follows that \begin{equation}
\sum_{\mat{p} \in P} \left\|\nabla\omega(\mat{x},\mat{p})\right\| = O(\kappa\sqrt{m}/\gamma). \label{eq:unique-4} \end{equation}
Substituting \eqref{eq:unique-4} into \eqref{eq:unique-3} gives \begin{equation}
\left\|\sum_{\mat{p} \in P}
\nabla \omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})\right\| = O(t\kappa \sqrt{m}\eps^{\tau-1} + \kappa m^{3}\gamma). \label{eq:unique-5} \end{equation} By substituting \eqref{eq:unique-2} and \eqref{eq:unique-5} into \eqref{eq:unique-1}, we have \[
\left\|\nabla \varrho_{\mat{z},i}(\mat{x})\right\| \leq 1 + O(t \kappa \sqrt{m}\eps^{\tau-1} + \kappa m^{4}\gamma), \] establishing the upper range limit for
$\left\|\nabla \varrho_{\mat{z},i}(\mat{x})\right\|$. Symmetrically, \begin{eqnarray*} \norm{\nabla \varrho_{\mat{z},i}(\mat{x})} & \geq & \sum_{\mat{p} \in P} \left(\omega(\mat{x},\mat{p}) - \omega(\mat{x},\mat{p}) \cdot \norm{\mat{J}_{f_{\mat{v}_i}}(\mat{x})} \cdot \norm{\mat{x}-\mat{p}} \right) - \\
& & \left\|\sum_{\mat{p} \in P}
\nabla \omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})\right\|. \end{eqnarray*} By \eqref{eq:unique} and \eqref{eq:unique-5}, we have \[ \norm{\nabla \varrho_{\mat{z},i}(\mat{x})} \geq 1 - O(\kappa m^{4}\gamma) - O(t \kappa \sqrt{m}\eps^{\tau-1} + \kappa m^{3}\gamma) = 1 - O(t \kappa \sqrt{m}\eps^{\tau-1} + \kappa m^{4}\gamma), \] establishing the lower range limit for
$\left\|\nabla \varrho_{\mat{z},i}(\mat{x})\right\|$.
Observe that \begin{eqnarray*} \mat{v}_i^t \cdot \nabla\varrho_{\mat{z},i}(\mat{x}) & = & \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{v}_i^t \cdot f_{\mat{v}_i}(\mat{x}) + \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{v}_i^t \cdot \mat{J}_{f_{\mat{v}_i}}(\mat{x})^t \cdot (\mat{x}-\mat{p}) + \\ & & \sum_{\mat{p} \in P} \mat{v}_i^t \cdot \nabla\omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p}). \end{eqnarray*} Therefore, \begin{eqnarray*} \mat{v}_i^t \cdot \nabla\varrho_{\mat{z},i}(\mat{x}) & \geq & \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{v}_i^t \cdot f_{\mat{v}_i}(\mat{x}) -
\left|\sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{v}_i^t \cdot J_{f_{\mat{v}_i}}(\mat{x})^t \cdot (\mat{x}-\mat{p})\right| - \\
& & \left|\sum_{\mat{p} \in P}
\mat{v}_i^t \cdot \nabla\omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p}) \right|. \end{eqnarray*} Since $\angle (f_{\mat{v}_i}(\mat{x}),\mat{v}_i)$ is $O(m\sqrt{m}\,\gamma)$ by Lemma~\ref{lemma::normal_angle}, we get $\mat{v}_i^t \cdot f_{\mat{v}_i}(\mat{x}) \geq 1 - O(m^3\gamma^2)$, which implies that $\sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{v}_i^t \cdot f_{\mat{v}_i}(\mat{x}) \geq 1-O(m^3\gamma^2)$. The second term is at most $\sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \norm{\mat{J}_{f_{\mat{v}_i}}(\mat{x})} \cdot \norm{\mat{x}-\mat{p}} \leq O(\kappa m^{4}\gamma)$ by \eqref{eq:unique}. The third term is at most $\sum_{\mat{p} \in P} \norm{\nabla\omega(\mat{x},\mat{p})} \cdot
|f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})|$, which is $O(t \kappa \sqrt{m}\eps^{\tau-1} + \kappa m^{3}\gamma)$ by \eqref{eq:unique-0} and \eqref{eq:unique-4}. As a result, $\mat{v}_i^t \cdot \nabla \varrho_{\mat{z},i}(\mat{x}) \geq 1 - O(t \kappa \sqrt{m}\eps^{\tau-1} + \kappa m^{4}\gamma)$. \end{proof}
The next result shows that every point $z$ in $\mani$ is near $Z_{\varrho_\mat{z}}$.
\begin{lemma}\label{lemma::exist} Let $\varrho_\mat{z}$ be the canonical function with respect to a point $\mat{z} \in \mani$ and an orthonormal basis $\{\mat{v}_1,\ldots, \mat{v}_{d-m}\}$ of $N_\mat{z}$.
There exists $\eps_0 \in (0,1)$ and $c_m \geq 1$ such that if $\eps \leq \eps_0$, then $Z_{\varrho_\mat{z}} \cap B(\mat{z}, c_m\gamma^{2}) \cap (\mat{z} + N_\mat{z}) \not= \emptyset$ and $Z_{\varrho_\mat{z}} \cap ( B(\mat{z}, 2\eps) \setminus B(\mat{z}, c_m\gamma^{2})) \cap (\mat{z} + N_\mat{z}) = \emptyset$. The value $\eps_0$ decreases as $d$ increases, and $c_m$ is linear in $m^{5/2}$. \end{lemma} \begin{proof} We first show that $Z_{\varrho_\mat{z}} \cap ( B(\mat{z}, 2\eps) \setminus B(\mat{z}, c_m\gamma^{2})) \cap (\mat{z} + N_\mat{z})$ is empty. For all $i \in [1,d-m]$ and all point $\mat{x} \in B(\mat{z},2\eps)$, let $\varrho_{\mat{z},i} = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p})$.
We claim that there exists a value $c_m \geq 1$ that is linear in $m^{5/2}$ such that for every $\mat{x} \in B(\mat{z}, 2\eps) \cap (\mat{z} + N_\mat{z})$ and every $i \in [1,d-m]$, if $\mat{v}_i^t \cdot (\mat{x}-\mat{z}) \geq c_m\gamma^{2}$, then $\varrho_{\mat{z},i}(\mat{x}) > 0$. We ignore all $\mat{p} \in P \setminus B(\mat{x},m\gamma)$ because $\omega(\mat{x},\mat{p}) = 0$ in this case, so such points have no influence over $\varrho_\mat{z}(\mat{x})$. $P \cap B(\mat{x},m\gamma)$ is non-empty because, by uniform $(\eps,\kappa)$-sampling, there is a point $\mat{q} \in P$ such that $\norm{\mat{q}-\mat{z}} \leq \eps$ which implies that $\norm{\mat{q}-\mat{x}} \leq \norm{\mat{x}-\mat{z}} + \norm{\mat{q}-\mat{z}} \leq 3\eps \leq m\gamma$. For every $\mat{p} \in P \cap B(\mat{x},m\gamma)$, \[ \mat{v}_i^t \cdot (\mat{x}-\mat{p}) \geq \mat{v}_i^t
\cdot (\mat{x}-\mat{z}) - |\mat{v}_i^t \cdot (\mat{z}-\mat{p})|. \] The first term is bounded from below as $\mat{v}_i^t \cdot (\mat{x}-\mat{z}) \geq c_m\gamma^2$ by assumption. Consider the second term. Since $\norm{\mat{p}-\mat{z}} \leq \norm{\mat{p}-\mat{x}} + \norm{\mat{x}-\mat{z}} \leq m\gamma+ 2\eps < (m+1)\gamma$, Lemma~\ref{lem:basic}(i) implies that
the second term $|\mat{v}_i^t \cdot (\mat{z}-\mat{p})|$ is at most $(m+1)^2\gamma^2/2$.
It follows that \[ \mat{v}_i^t \cdot (\mat{x}-\mat{p}) \geq c_m \gamma^{2} - (m+1)^2\gamma^2/2. \] For $i \in [1,d-m]$, define $\mat{h}_i(\mat{x}) = f_{\mat{v}_i}(\mat{x}) - \mat{v}_i$. Lemma~\ref{lemma::normal_angle} implies that \[ \norm{\mat{h}_i(\mat{x})} \leq 2\sin\frac{\angle (L_\mat{x},N_\mat{z})}{2} = O(m\sqrt{m}\,\gamma). \] Observe that \begin{eqnarray*} f_{\mat{v}_i}(\mat{x})^t \cdot (\mat{x}-\mat{p}) & = & \mat{v}_i^t \cdot (\mat{x}-\mat{p}) + \mat{h}_i(\mat{x})^t \cdot (\mat{x}-\mat{p}) \\ & \geq & c_m \gamma^{2} - (m+1)^2\gamma^2/2 - \norm{\mat{h}_i(\mat{x})} \cdot \norm{\mat{x}-\mat{p}} \\ & \geq & c_m \gamma^{2} - (m+1)^2\gamma^2/2 - O(m^{5/2}\gamma^2) \\ & > & 0, \end{eqnarray*} whenever $c_m$ is a large enough value that is linear in $m^{5/2}$. As a result, $\varrho_{\mat{z},i}(\mat{x}) > 0$. This proves our claim.
We can symmetrically show that if $\mat{v}_i^t \cdot (\mat{x}-\mat{z}) \leq -c_m\gamma^2$, then $\varrho_{\mat{z},i}(\mat{x}) < 0$. Thus, $\varrho_{\mat{z},i}^{-1}(0) \cap B(\mat{z},2\eps) \cap (\mat{z} + N_\mat{z})$ lies in a $(d-m)$-dimensional slab $S_{\mat{v}_i} \subset \mat{z} + N_{\mat{z}}$ that is bounded by two $(d-m-1)$-dimensional flats orthogonal to $\mat{v}_i$ and at distance $c_m\gamma^2$ from $\mat{z}$. It follows that $(Z_{\varrho_\mat{z}} \cap ( B(\mat{z}, 2\eps) \cap (\mat{z} + N_\mat{z})) \setminus S_{\mat{v}_i} = \emptyset$. By Lemma~\ref{lem:agree}, $Z_{\varrho_\mat{z}}$ is identical for any choice of the orthonormal basis $\{\mat{v}_1,\ldots, \mat{v}_{d-m}\}$ of $N_\mat{z}$. It means that we can set $\mat{v}_i$ to be any unit vector $\mat{v} \in N_\mat{z}$ and the proof above still works. Observe that $\bigcap_{\mat{v} \in N_\mat{z}} S_\mat{v} = B(\mat{z},c_m\gamma^2) \cap (\mat{z} + N_\mat{z})$. Hence, $Z_{\varrho_\mat{z}} \cap ( B(\mat{z}, 2\eps) \setminus B(\mat{z}, c_m\gamma^2)) \cap (\mat{z} + N_\mat{z}) = \emptyset$.
To establish that $Z_{\varrho_\mat{z}} \cap B(\mat{z}, c_m\gamma^{2}) \cap (\mat{z} + N_\mat{z}) \not= \emptyset$, it suffices to show that $\bigcap_{i=1}^{d-m} \varrho_{\mat{z},i}^{-1}(0)$ contains a point in $\bigcap_{i=1}^{d-m} S_{\mat{v}_i}$. This is because $\bigcap_{i=1}^{d-m} S_{\mat{v}_i}$ is contained in $B(\mat{z},c_m\sqrt{d-m}\gamma^2)$, and for $\eps_0 \leq 1/(16c_m\sqrt{d-m})$, we have $B(\mat{z},c_m\sqrt{d-m}\gamma^2) \subseteq B(\mat{z},\eps)$ as $c_m\sqrt{d-m}\gamma^2 \leq 16c_m\sqrt{d-m}\,\eps^2 \leq 16c_m\sqrt{d-m}\, \eps_0\eps$. Then, the fact that $Z_{\varrho_{\mat{z}}} \cap (B(\mat{z},2\eps) \setminus B(\mat{z},c_m\gamma^2)) \cap (\mat{z} + N_\mat{z}) = \emptyset$ implies that $\bigcap_{i=1}^{d-m} \varrho^{-1}_{\mat{z},i}(0)$ contains a point in $B(\mat{z},c_m\gamma^2) \cap (\mat{z} + N_\mat{z})$.
In fact, we choose an even smaller $\eps_0$ such that $\sqrt{\eps_0} \leq 1/(16c_m\sqrt{d-m})$, which gives $c_m\sqrt{d-m} \gamma^2 \leq \eps^{3/2}$. This will allow us apply Lemma~\ref{lemma::unique} later. The exponent 3/2 is an arbitrary choice. Any number greater than 1 will do.
Let $C = \bigcap_{i=1}^{d-m} S_{\mat{v}_i}$. It is a $(d-m)$-dimensional cube that lies in $\mat{z} + N_{\mat{z}}$, has $\mat{z}$ as its center, and has side length $2c_m\gamma^{2} $. The facets of $C$ are orthogonal to the directions $\mat{v}_1,\ldots,\mat{v}_{d-m}$.
Adopt a coordinate frame such that $\mat{v}_1,\ldots,\mat{v}_{d-m}$ are the first $d-m$ coordinate axes of $\real^d$. For $i \in [1,d-m]$, define $H_i$ to be the set of maximal line segments that lie inside $C$ and are parallel to the direction $\mat{v}_i$.
First, we claim that every line segment $l \in H_i$ intersects $\varrho_{\mat{z},i}^{-1}(0)$ at exactly one point. We have shown earlier that $\varrho_{\mat{z},i}$ has opposite signs at the endpoints of $l$. So $l \cap \varrho_{\mat{z},i}^{-1}(\mat{0}) \not= \emptyset$. Suppose to the contrary that $l \cap \varrho_{\mat{z},i}^{-1}(0)$ contains two distinct points $\mat{y}_1$ and $\mat{y}_2$. So $\mat{y}_1 - \mat{y}_2$ is parallel to $\mat{v}_i$. Assume without loss of generality that $\mat{y}_1 - \mat{y}_2$ has the same orientation as $\mat{v}_i$. By Lemma~\ref{lemma::unique}, $(\mat{y}_1-\mat{y}_2)^t \cdot \nabla\varrho_{\mat{z},i}(\mat{x}) > 0$ for every $\mat{x} \in B(\mat{z},c_m\sqrt{d-m}\gamma^2) \subseteq B(\mat{z},\eps^{3/2})$. But then $\varrho_{\mat{z},i}(\mat{x})$ increases strictly monotonically from $\mat{y}_2$ to $\mat{y}_1$, which implies that $\varrho_{\mat{z},i}(\mat{y}_1) > 0$. This is a contradiction because $\mat{y}_1 \in \varrho_{\mat{z},i}^{-1}(0)$, thereby establishing our claim.
Define a function $g_i : C \rightarrow [-c_m\gamma^{2}, c_m\gamma^{2}]$ such that $g_i(\mat{x}) = b_{i,\mat{x}}$, where \begin{itemize}
\item $(x_1,\ldots,x_{i-1},b_{i,\mat{x}},x_{i+1},\ldots,x_d) \in C$ and
\item $\varrho_{\mat{z},i}(x_1,\ldots,x_{i-1},b_{i,\mat{x}},x_{i+1},\ldots,x_d) = 0$.
\end{itemize} Our claim in the previous paragraph ensures the existence and uniqueness of $b_{i,\mat{x}}$. We show that $g_i$ is continuous. Since $\varrho_{\mat{z},i}$ is continuous, $\varrho_{\mat{z},i}^{-1}(0)$ is compact~\cite[Ch~3:~Theorem 5.4,~Ch~5:~Theorem 2.11]{mendelson}, which implies that for any interval $[a,b] \subset \real$, $\varrho_{\mat{z},i}^{-1}(0) \cap \{\mat{x} \in C : x_i \in [a,b]\}$ is compact. Let $\pi_i$ be the function that projects points in $C$ onto the linear subspace spanned by $\{\mat{v}_1,\ldots,\mat{v}_{i-1},\mat{v}_{i+1},\ldots,\mat{v}_{d-m}\}$. Since $\pi_i$ is continuous, its image is compact and so is the following product~\cite[Ch~5:~Theorem 2.9 \& Theorem 4.2]{mendelson}: \[ \pi_i\left(\varrho_{\mat{z},i}^{-1}(0) \cap \{\mat{x} \in C : x_i \in [a,b]\}\right) \times [-c_m\gamma^{2}, c_m\gamma^{2}]. \] Observe that this product is homeomorphic to $g_i^{-1}([a,b])$. Therefore, $g_i^{-1}([a,b])$ is compact for any interval $[a,b] \subset \real$, which implies that $g_i$ is continuous~\cite[Ch~2:~Theorem 6.10]{mendelson}.
Define a function $g : C \rightarrow C$ such that \[ g(\mat{x}) = \left(g_1(\mat{x}), \ldots, g_{d-m}(\mat{x})\right)^t. \] The function $g$ is continuous as each $g_i$ is continuous. Notice that $\varrho_{\mat{z},i}^{-1}(0) \cap C$ is the subset of $C$ that satisfy the equation $g_i(x_1,\ldots,x_i,\ldots,x_d) = x_i$. Since $\varrho_\mat{z}(\mat{x}) = (\varrho_{\mat{z},1}(\mat{x}), \ldots, \varrho_{\mat{z},d-m}(\mat{x}))^t$, we conclude that $Z_{\varrho_\mat{z}} \cap C$ is the subset of $C$ that satisfy the equation $g(\mat{x}) = \mat{x}$. By the Brouwer fixed-point theorem~\cite[Ch~4:~Theorem 4.6]{mendelson}, there is indeed such a point in $C$.
\end{proof}
Recall that $\nu$ is the map that sends every point in $\real^d$ to its nearest point in $\mani$. We need to show that $Z_\varphi \cap \widehat{\mani}$ is compact in order to prove that $Z_\varphi \cap \widehat{\mani}$ and $\mani$ are homeomorphic.
\begin{lemma}\label{lemma::compact} $Z_{\varphi} \cap \widehat{\mani}$ is compact. \end{lemma} \begin{proof} By Lemmas~\ref{lem:agree} and~\ref{lemma:local}, for any point $\mat{z} \in \mani$, $Z_{\varphi}$ agrees locally with $Z_{\varrho_\mat{z}}$ where $\varrho_{\mat{z}}$ is the canonical function with respect to $\mat{z}$ and any orthonormal basis of $N_\mat{z}$. Our strategy is to construct a finite number of such $Z_{\varrho_{\mat{z}}}$'s and prove that each is compact. The lemma then follows as a finite union of compact sets is compact.
Take a maximal set $Y$ of points in $\widehat{\mani}$ such that any two of them are at distance $\eps^{\tau}$ or more apart.
It implies that any two balls centered at points in $Y$ with radius $\eps^{\tau}/2$ are interior-disjoint. Since $\widehat{\mani}$ is the product of $\mani$ and a ball of radius $\eps$, $\widehat{\mani}$ is compact~\cite[Ch~5:~Theorem~4.2]{mendelson}.
It follows that $|Y|$ is finite. The maximality
also implies that $\widehat{\mani} \subseteq \bigcup_{\mat{y} \in Y} B(\mat{y},\eps^{\tau})$. The intersection $Z_\varphi \cap \bigcup_{\mat{y} \in Y} B(\mat{y},\eps^{\tau})$ is equal to $\bigcup_{\mat{y} \in Y} Z_\varphi \cap B(\mat{y},\eps^{\tau})$ which is a subset of $\bigcup_{\mat{y} \in Y} Z_\varphi \cap B(\nu(\mat{y}),\eps^{\tau} + \eps)$ because $\norm{\mat{y} - \nu(\mat{y})} \leq \eps$. By Lemmas~\ref{lem:agree} and~\ref{lemma:local}, $Z_\varphi \cap B(\nu(y), \eps^\tau + \eps) = Z_{\varrho_{\nu(y)}} \cap B(\nu(y), \eps^\tau + \eps)$. Therefore, \[ Z_\varphi \cap \widehat{\mani} \subseteq Z_\varphi \cap \bigcup_{\mat{y} \in Y} B(\mat{y},\eps^\tau) \subseteq \bigcup_{\mat{y} \in Y} Z_{\varrho_{\nu(y)}} \cap B(\nu(y),\eps^\tau + \eps). \] As $\varrho_{\nu(\mat{y})}$ is continuous in the interior of $B(\nu(y),2\eps)$ by Lemma~\ref{lemma:local}, $Z_{\varrho_{\nu(y)}} \cap B(\nu(y),\eps^{\tau}+\eps)$ is compact~\cite[Ch~3:~Theorem 5.4,~Ch~5:~Theorem 2.11]{mendelson}. It implies that the finite union $\bigcup_{\mat{y} \in Y} Z_{\varrho_{\nu(y)}} \cap B(\nu(y),\eps^{\tau} + \eps)$ is also compact. Finally, observe that \[ Z_\varphi \cap \widehat{\mani} = \left(\bigcup_{\mat{y} \in Y} Z_{\varrho_{\nu(y)}} \cap B(\nu(y),\eps^{\tau}+\eps)\right) \cap \widehat{\mani}, \] which is compact because it is the intersection of two compact subsets in $\real^d$. \end{proof}
We are ready to prove the faithful approximation of $\mani$ by $Z_\varphi \cap \widehat{\mani}$.
\begin{theorem}
\label{thm:main}
Let $\mani$ be an $m$-dimensional compact smooth manifold in $\real^d$.
Let $P$ be a uniform $(\eps,\kappa)$-sample of $\mani$ for some constant $\kappa \geq 1$.
We assume that $\mani$ has unit reach, $m$ is known,
a neighborhood radius $\gamma = 4\eps$, and approximate tangent spaces
with angular errors
at most $m\gamma$ are specified at the points in $P$.
Let $\widehat{\mani}$ be the set of
points within a distance $\eps$ from $\mani$. We can construct a function $\varphi: \real^d \rightarrow \real^{d-m}$
for which there exists $\eps_0 \in (0,1)$ that decreases as $d$ increases such that
the following properties hold whenever $\eps \leq \eps_0$.
\begin{emromani}
\item The restriction of the nearest point map to $Z_\varphi \cap
\widehat{\mani}$ is a homeomorphism between $Z_\varphi \cap \widehat{\mani}$ and $\mani$.
\item The Hausdorff distance between $Z_\varphi \cap \widehat{\mani}$ and $\mani$
is $O(m^{5/2}\gamma^{2}) = O(m^{5/2}\eps^2)$.
\item For all $\mat{x} \in Z_\varphi \cap \widehat{\mani}$, $N_{\nu(\mat{x})}$
makes an $O(m^2\sqrt{\kappa\gamma}) = O(m^2\sqrt{\kappa\eps})$ angle with the normal space of $Z_\varphi$ at
$\mat{x}$.
\end{emromani} \end{theorem}
\begin{proof} Consider (i). Let $\mu$ denote the restriction of $\nu$ to $Z_\varphi \cap \widehat{\mani}$. First, we show that $\mu$ is injective. Suppose to the contrary that there are two points $\mat{y}_1, \mat{y}_2 \in Z_\varphi \cap \widehat{\mani}$ such that $\mu(\mat{y}_1)$ and $\mu(\mat{y}_2)$ are the same point $\mat{z} \in \mani$. Then, $\mat{y}_1$ and $\mat{y}_2$ belong to $\mat{z} + N_\mat{z}$, which implies that $\mat{y}_1-\mat{y}_2 \in N_\mat{z}$. Note that $\mat{y}_1$ and $\mat{y}_2$ lie in $B(\mat{z},\eps)$. By Lemmas~\ref{lem:agree} and~\ref{lemma:local}, $Z_\varphi \cap B(\mat{z},\eps) = Z_{\varrho_{\mat{z}}} \cap B(\mat{z},\eps)$. Then, Lemma~\ref{lemma::exist} implies that $\mat{y}_1$ and $\mat{y}_2$ belong to $B(\mat{z}, t\gamma^2)$ for some large enough $t$ that is linear in $m^{5/2}$. By Lemma~\ref{lemma::unique}, we can define $\mat{v}_1 = \mat{y}_1-\mat{y}_2$ and get $(\mat{y}_1-\mat{y}_2)^t \cdot \nabla \varrho_{\mat{z},1}(\mat{x}) >0$ for all $\mat{x} \in B(\mat{z}, t\gamma^2)$ when $\eps_0$ is sufficiently small. But then $\varrho_{\mat{z},1}(\mat{x})$ increases strictly monotonically from $\mat{y}_2$ to $\mat{y}_1$, which implies that $\varrho_{\mat{z},1}(\mat{y}_1) > 0$. This is a contradiction because $\mat{y}_1$ belongs to $Z_\varphi$ and hence $Z_{\varrho_\mat{z}}$ by Lemmas~\ref{lem:agree} and~\ref{lemma:local}. This proves that $\mu$ is injective.
Next, we show that $\mu$ is surjective. Let $\mat{z}$ be any point in $\mani$. It follows from Lemmas~\ref{lem:agree},~\ref{lemma:local}, and~\ref{lemma::exist} that there exists a point $\mat{y} \in Z_\varphi \cap \widehat{\mani} \cap (\mat{z} + N_\mat{z})$. We show that $\mu$ must map $\mat{y}$ to $\mat{z}$. Suppose that $\mu$ maps $\mat{y}$ to another point $\mat{z}_2 \in \mani$, i.e. $\norm{\mat{y}-\mat{z}_2} < \norm{\mat{y} - \mat{z}}$. We grow a ball $B$ tangent to $\mani$ at $\mat{z}$ by moving its center linearly from $\mat{z}$ towards $\mat{y}$. When $B$ is tiny, it touches $\mani$ only at $\mat{z}$. When the center of $B$ reaches $\mat{y}$, $B$ contains both $\mat{z}$ and $\mat{z}_2$. Thus, the radius of the growing $B$ must become the local feature size of $\mani$ at $\mat{z}$ before or when its center reaches $\mat{y}$. Recall that the reach of $\mani$ is assumed to be 1. Thus, $\norm{\mat{y}-\mat{z}} \geq 1 > \eps$. This contradicts the fact that $\mat{y} \in \widehat{\mani} \cap (\mat{z} + N_\mat{z})$, thereby proving that $\mu$ is surjective.
Since $Z_\varphi \cap \widehat{\mani}$ avoids the medial axis, the restriction $\mu$ is continuous. Therefore, $\mu$ is a continuous bijection from $Z_\varphi \cap \widehat{\mani}$ to $\mani$. The spaces $\mani$ and $Z_\varphi \cap \widehat{\mani}$ are compact by assumption and Lemma~\ref{lemma::compact}, respectively, so we conclude from the existence of $\mu$ that $\mani$ and $Z_\varphi \cap \widehat{\mani}$ are homeomorphic~\cite[Ch~5:~Theorem 2.14]{mendelson}. This proves the correctness of (i).
Consider (ii). By Lemmas~\ref{lem:agree},~\ref{lemma:local}, and~\ref{lemma::exist}, for any point $\mat{z} \in \mani$, there exists a point $\mat{x} \in Z_\varphi$ within a distance of $c_m\gamma^2$, where $c_m \geq 1$ is some value linear in $m^{5/2}$. Therefore, $c_m\gamma^2 = O(m^{5/2}\eps_0\eps) < \eps$ for a small enough $\eps_0$. So $\mat{x} \in Z_\varphi \cap \widehat{\mani}$. It follows that the directed Hausdorff distance from $\mani$ to $Z_\varphi \cap \widehat{\mani}$ is $O(m^{5/2}\gamma^2)$. Conversely, for any point $\mat{x} \in Z_\varphi \cap \widehat{\mani}$, $\norm{\nu(\mat{x}) - \mat{x}} \leq \eps$ and $x \in \nu(x) + N_{\nu(\mat{x})}$. By Lemmas~\ref{lem:agree},~\ref{lemma:local}, and~\ref{lemma::exist}, $Z_\varphi \cap (B(\nu(\mat{x}), 2\eps) \setminus B(\nu(\mat{x}),c_m\gamma^2)) \cap (\nu(x) + N_{\nu(\mat{x})})$ is empty. So $\norm{\nu(\mat{x}) - \mat{x}} \leq c_m\gamma^2 = O(m^{5/2}\gamma^2)$. It follows that the directed Hausdorff distance from $Z_\varphi \cap \widehat{\mani}$ to $\mani$ is $O(m^{5/2}\gamma^2)$.
Consider (iii). By Lemma~\ref{lemma::unique}, for every point $\mat{x} \in Z_\varphi \cap \widehat{\mani}$ and every unit vector $\mat{v}_1 \in N_{\nu(\mat{x})}$, $\norm{\nabla\varrho_{\nu(\mat{x}),1}(\mat{x})} \leq 1 + O(\kappa m^4\gamma)$ and $\mat{v}_1^t \cdot \nabla\varrho_{\nu(\mat{x}),1}(\mat{x}) \geq 1 - O(\kappa m^4 \gamma)$. Thus, \[ \angle (\mat{v}_1, \nabla\varrho_{\nu(\mat{x}),1}(\mat{x})) \leq \arccos\left(\frac{\mat{v}_1^t \cdot \nabla\varrho_{\nu(\mat{x}),1}(\mat{x})} {\norm{\nabla\varrho_{\nu(\mat{x}),1}(\mat{x})}}\right) \leq \arccos\left(\frac{1-O(\kappa m^4\gamma)} {1 + O(\kappa m^4\gamma)}\right) = O(m^2\sqrt{\kappa\gamma}). \] The vector $\nabla\varrho_{\nu(\mat{x}),1}(\mat{x})$ belongs to the normal space of $Z_\varphi$ at $\mat{x}$. (Recall that $Z_\varphi$ agrees with $Z_{\varrho_{\nu(\mat{x})}}$ locally.) Thus, the angle between $N_{\nu(\mat{x})}$ and the normal space of $Z_\varphi$ at $\mat{x}$ is $O(m^2\sqrt{\kappa\gamma})$. \end{proof}
\cancel{ On the surface, the statement of Lemma~\ref{lem:main} suggests that we should make $\delta$ close to 1 in order to achieve a Hausdorff distance close to $\eps^2$ between $\mani$ and $Z_\varphi \cap \mani_\delta$ and an angular error close to $O(\eps^{1/2})$ between the normal spaces. The disadvantage is that we focus on a much smaller neighborhood $\mani_\delta$ around $\mani$ when $\delta$ is close to 1. Conversely, when $\delta$ approaches zero, a much larger neighborhood around $\mani$ is considered, but the Hausdorff distance approaches $\eps$ and the angular error approaches $O(1)$. In order to get a stronger result, we apply Lemma~\ref{lem:main} for $\delta$ as well as $\delta' = 1-\delta$.
Without loss of generality, we assume that $\delta \leq \delta' = 1-\delta$. It means that we pick $\delta$ from the interval $(0, 1/2]$ instead of $(0, 1)$. As stated in Lemmas~\ref{lemma::exist} and~\ref{lem:main}, the threshold $\eps_0$ required for $\delta' = 1-\delta$ being greater than 1/2 is smaller (i.e., higher sampling density) than for $\delta$ being less than 1/2. So we assume that $\eps_0$ is small enough for $\delta'$, which is also small enough for $\delta$. Since $\delta$ is at most 1/2, $\mani_{1-\delta} \subseteq \mani_{\delta}$ and hence $Z_\varphi \cap \mani_{1-\delta} \subseteq Z_\varphi \cap \mani_{\delta}$. By Lemma~\ref{lem:main}(i), both $Z_\varphi \cap \mani_{1-\delta}$ and $Z_\varphi \cap \mani_{\delta}$ are homeomorphic to $\mani$. This implies that $Z_\varphi \cap \mani_{1-\delta} = Z_\varphi \cap \mani_{\delta}$. Therefore, $Z_\varphi \cap \mani_{\delta}$ enjoys the properties in Lemma~\ref{lem:main} for $Z_\varphi \cap \mani_{1-\delta}$. This allows us to obtain the main theorem of this paper stated below. }
\cancel{ On the surface, the statement of Lemma~\ref{lem:main} suggests that we should make $\delta$ close to 1 in order to achieve a Hausdorff distance close to $\eps^2$ between $\mani$ and $Z_\varphi \cap \mani_\delta$ and an angular error close to $O(\eps^{1/2})$ between the normal spaces. Conversely, if $\delta$ approaches zero, the Hausdorff distance approaches $\eps$ and the angular error approaches $O(1)$. As stated in Lemmas~\ref{lemma::exist} and~\ref{lem:main}, the value $\eps_0$ required for $\delta$ being greater than 1/2 is smaller (i.e., higher sampling density) than for $\delta$ being less than 1/2. If $\delta$ is at most 1/2, then $\mani_{1-\delta} \subseteq \mani_{\delta}$ and hence $Z_\varphi \cap \mani_{1-\delta} \subseteq Z_\varphi \cap \mani_{\delta}$. (Assume that the threshold $\eps_0$ is small enough for $1-\delta$, which is also small enough for $\delta$.) Nonetheless, the restriction of the nearest point map is a continuous bijection from both $Z_\varphi \cap \mani_{1-\delta}$ and $Z_\varphi \cap \mani_{\delta}$ to $\mani$. This implies that $Z_\varphi \cap \mani_{1-\delta} = Z_\varphi \cap \mani_{\delta}$. Therefore, $Z_\varphi \cap \mani_{\delta}$ enjoys the properties in Lemma~\ref{lem:main} for $Z_\varphi \cap \mani_{1-\delta}$. This allows us to obtain the main theorem of this paper stated below. }
\section{Projection operator}
Our proof of convergence will make use of the property that $\mat{B}_{\varphi,\mat{x}}$ is a $d \times (d-m)$ matrix with orthogonal unit columns such that $\col{\mat{B}_{\varphi,\mat{x}}} = L_\mat{x}$. Such a matrix can be obtained by an eigen-decomposition of $\mat{C}_\mat{x}$.
We rewrite $\varphi(\mat{x}) = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{B}_{\varphi,\mat{x}}^t \cdot (\mat{x}-\mat{p}) = \mat{B}_{\varphi,\mat{x}}^t \cdot (\mat{x} - \mat{a}_\mat{x})$, where $\mat{a}_\mat{x} = \sum_{\mat{p} \in P} \omega(\mat{x},\mat{p}) \cdot \mat{p}$. Intuitively, as $\varphi(\mat{a}_\mat{x}) = \mat{0}$, we want to move the current point $\mat{x}_i$ closer to $\mat{a}_\mat{x}$. We also want to move directly onto $Z_{\varphi}$ without much drifting. Therefore, it is desirable to move $\mat{x}_i$ within the affine subspace $\mat{x}_i + L_{\mat{x}_i}$ which is roughly normal to $Z_\varphi$. The projection follows an iterative scheme: \[ \mat{x}_{i+1} = \mat{x}_i + \mat{B}_{\varphi,\mat{x}_i}^{} \cdot \mat{B}_{\varphi,\mat{x}_i}^t \cdot ( \mat{a}_{\mat{x}_i} - \mat{x}_i). \] Note that $\mat{B}_{\varphi,\mat{x}_i}^{} \cdot \mat{B}_{\varphi,\mat{x}_i}^t \cdot ( \mat{a}_{\mat{x}_i} - \mat{x}_i)$ is the projection of the vector $\mat{a}_{\mat{x}_i} - \mat{x}_i$ into $L_{\mat{x}_i}$. The iterative scheme moves the current point $\mat{x}_i$ by this projected vector to the new point $\mat{x}_{i+1}$. In other words, $\mat{x}_{i+1}$ is the projection of $\mat{a}_{\mat{x}_i}$ onto the affine subspace $\mat{x}_i + L_{\mat{x}_i}$.
We prove two technical results in order to establish the proof of convergence. The first one shows that any initial point near $\mani$ is moved to within an $O(m^{7/2}\gamma^2)$ distance from $\mani$ after a single iteration. Let $\tilde{\mat{x}}_i$ denote the nearest point in $Z_\varphi$ to $\mat{x}_i$. The second result shows that $\norm{\mat{x}_{i+1} - \tilde{\mat{x}}_i} \ll \norm{\mat{x}_i - \tilde{\mat{x}}_i}$, which implies that $\norm{\mat{x}_{i+1} - \tilde{\mat{x}}_{i+1}} \ll \norm{\mat{x}_i - \tilde{\mat{x}}_i}$.
\begin{lemma} \label{lem:first_itr} Let $P$ be a uniform $(\eps,\kappa)$-sample of $\mani$.
For every point $\mat{x}$ within a distance $m\gamma$ from $P$ and every $d \times (d-m)$ matrix $\mat{B}_{\varphi,\mat{x}}$ that satisfies $\col{\mat{B}_{\varphi,\mat{x}}} = L_\mat{x}$, we have $\norm{\mat{y} - \nu(\mat{x})} = O(m^{7/2}\gamma^2)$, where $\mat{y} = \mat{x} + \mat{B}_{\varphi,\mat{x}}^{} \cdot \mat{B}_{\varphi,\mat{x}}^t \cdot (\mat{a}_\mat{x} - \mat{x})$. \end{lemma} \begin{proof} For every sample point $\mat{p} \in B(\mat{x},m\gamma)$, $\norm{\mat{p} - \nu(\mat{x})} \leq \norm{\mat{p} - \mat{x}} + \norm{\mat{x} - \nu(\mat{x})} = O(m\gamma)$. By Lemma~\ref{lem:basic}(i), the distance between $\mat{p}$ and $\nu(\mat{x}) + T_{\nu(\mat{x})}$ is $O(m^2\gamma^2)$. As $\mat{a}_\mat{x}$ is convex combination of all $\mat{p} \in B(\mat{x},m\gamma)$, the distance between $\mat{a}_\mat{x}$ and $\nu(\mat{x}) + T_{\nu(\mat{x})}$ is also $O(m^2\gamma^2)$.
Let $\hat{\mat{a}}_\mat{x}$ be the projection of $\mat{a}_\mat{x}$ into $\nu(\mat{x}) + N_{\nu(\mat{x})}$. The vector $\hat{\mat{a}}_\mat{x} - \mat{a}_\mat{x}$ is parallel to $T_{\nu(\mat{x})}$, so $\hat{\mat{a}}_\mat{x}$ is also at distance $O(m^2\gamma^2)$ from $\nu(\mat{x}) + T_{\nu(\mat{x})}$. As $\hat{\mat{a}}_\mat{x} \in \nu(\mat{x}) + N_{\nu(\mat{x})}$, the vector $\hat{\mat{a}}_\mat{x} - \nu(\mat{x})$ is orthogonal to $T_{\nu(\mat{x})}$, which implies that $\norm{\hat{\mat{a}}_\mat{x} - \nu(\mat{x})} = O(m^2\gamma^2)$. Therefore, it suffices to prove that $\norm{\hat{\mat{a}}_\mat{x} - \mat{y}} = O(m^{7/2}\gamma^2)$ as $\norm{\mat{y} - \nu(\mat{x})} \leq \norm{\hat{\mat{a}}_\mat{x} - \mat{y}} + \norm{\hat{\mat{a}}_\mat{x} - \nu(\mat{x})} = \norm{\hat{\mat{a}}_\mat{x} - \mat{y}} + O(m^2\gamma^2)$.
\begin{figure}\label{fg:proj-1}
\end{figure}
Refer to Figure~\ref{fg:proj-1}(a). By construction, $\hat{\mat{a}}_\mat{x} \in \nu(\mat{x}) + N_{\nu(\mat{x})}$. Also, $\mat{x} - \nu(\mat{x}) \in N_{\nu(\mat{x})}$, implying that $\mat{x} \in \nu(\mat{x}) + N_{\nu(\mat{x})}$. Therefore, $\angle \mat{x}\,\hat{\mat{a}}_\mat{x}\,\mat{a}_\mat{x} = \pi/2$. From the previous discussion, $\mat{y}$ is the projection of $\mat{a}_\mat{x}$ onto $\mat{x} + L_\mat{x}$. So $\angle \mat{x}\,\mat{y}\,\mat{a}_\mat{x} = \pi/2$. As a result, $\mat{x}$, $\mat{y}$, $\hat{\mat{a}}_\mat{x}$, and $\mat{a}_\mat{x}$ lie on a $(d-1)$-dimensional sphere $S$ that has $\mat{x}\,\mat{a}_\mat{x}$ as a diameter. Since $\mat{a}_\mat{x}$ is a convex combination of all $\mat{p} \in P \cap B(\mat{x},m\gamma)$, we have $\norm{\mat{a}_\mat{x} - \mat{x}} \leq m\gamma$. Thus, $\mathrm{radius}(S) = O(m\gamma)$.
Since $\angle \mat{x}\,\hat{\mat{a}}_\mat{x}\,\mat{a}_\mat{x} = \pi/2$, we have $\norm{\hat{\mat{a}}_\mat{x} - \mat{x}}^2 + \norm{\hat{\mat{a}}_\mat{x} - \mat{a}_\mat{x}}^2 = \norm{\mat{a}_\mat{x} - \mat{x}}^2$. It follows that $\norm{\hat{\mat{a}}_\mat{x} - \mat{x}} \geq \norm{\mat{a}_\mat{x} - \mat{x}}/2$ or $\norm{\hat{\mat{a}}_\mat{x} - \mat{a}_\mat{x}} \geq \norm{\mat{a}_\mat{x} - \mat{x}}/2$. We prove that $\angle \hat{\mat{a}}_\mat{x}\,\mat{x}\,\mat{y} = O(m^{5/2}\gamma)$ if $\norm{\hat{\mat{a}}_\mat{x} - \mat{x}} \geq \norm{\mat{a}_\mat{x} - \mat{x}}/2$. Let
$\{\mat{v}_1,\ldots,\mat{v}_{d-m}\}$ and $\{\mat{w}_1,\ldots,\mat{w}_{d-m}\}$ be orthonormal bases of $N_{\nu(\mat{x})}$ and $L_{\mat{x}}$, respectively, that satisfy Lemma~\ref{lem:choice}. Note that $\hat{\mat{a}}_\mat{x} - \mat{x} \in N_{\nu(\mat{x})}$ and $\mat{y}-\mat{x} \in L_\mat{x}$. Refer to Lemma~\ref{lem:proj-angle}. Let $(\mat{a}_\mat{x}-\mat{x})/\norm{\mat{a}_\mat{x}-\mat{x}}$ be the unit vector $\mat{n}$, let $\hat{\mat{a}}_\mat{x} - \mat{x}$ be the vector $\mat{u}_1$, let $\mat{y} - \mat{x}$ be the vector $\mat{u}_2$ as specified in Lemma~\ref{lem:proj-angle}, and let $\phi = \angle (L_\mat{x},N_{\nu(\mat{x})}) = O(m\sqrt{m}\,\gamma)$ by Lemma~\ref{lemma::normal_angle}. We need to show that the values $\alpha_1$ and $\alpha_2$ defined in Lemma~\ref{lem:proj-angle} satisfy the assumption that $\alpha_1 > \alpha_2 + (2m^2\phi^2)/\cos\phi$.
By Lemma~\ref{lem:choice}, $\angle (\mat{v}_i,\mat{w}_i) \leq \phi$ for $i \in [1,d-m]$, which implies that $\norm{\mat{v}_i-\mat{w}_i} \leq 2\sin(\phi/2) \leq \phi$. By definition, $\alpha_2 = \sum_{i=d-2m+1}^{d-m} ((\mat{w}_i-\mat{v}_i)^t \mat{n})^2$, and therefore, $\alpha_2 \leq \sum_{i=d-2m+1}^{d-m} \norm{\mat{w}_i-\mat{v}_i}^2 \leq m\phi^2 = O(m^4\gamma^2)$. By definition, $\alpha_1$ is the squared norm of the projection of $\mat{n} = (\mat{a}_\mat{x}-\mat{x})/\norm{\mat{a}_\mat{x}-\mat{x}}$ onto $N_{\nu(\mat{x})}$. Since $\hat{\mat{a}}_\mat{x} - \mat{x}$ is the projection of $\mat{a}_\mat{x}-\mat{x}$ onto $N_{\nu(\mat{x})}$, we get $\alpha_1 = \norm{\hat{\mat{a}}_\mat{x}-\mat{x}}^2/\norm{\mat{a}_\mat{x}-\mat{x}}^2 \geq 1/4$ because $\norm{\hat{\mat{a}}_\mat{x} - \mat{x}} \geq \norm{\mat{a}_\mat{x}-\mat{x}}/2$ by assumption. This shows that $\alpha_1 > \alpha_2 + (2m^2\phi^2)/\cos\phi$. Then, Lemma~\ref{lem:proj-angle} implies that $\angle \hat{\mat{a}}_\mat{x}\,\mat{x}\,\mat{y} = \angle (\mat{u}_1,\mat{u}_2) \leq \arccos\left(\sqrt{1-\frac{\alpha_2}{\alpha_1}}\cos\phi - \frac{2m^2\phi^2}{\sqrt{\alpha_1^2-\alpha_1\alpha_2}}\right)$. One can verify that the right hand side is $\arccos(1-O(m^5\gamma^2))$ and so $\angle \hat{\mat{a}}_\mat{x}\,\mat{x}\,\mat{y} = O(m^{5/2}\gamma)$.
Similarly, we can prove that $\angle \hat{\mat{a}}_\mat{x}\,\mat{a}_\mat{x}\,\mat{y} =O(m^{5/2}\gamma)$ if $\norm{\hat{\mat{a}}_\mat{x} - \mat{a}_\mat{x}} \geq \norm{\mat{a}_\mat{x} - \mat{x}}/2$. We conclude that $\angle \hat{\mat{a}}_\mat{x}\,\mat{x}\,\mat{y} = O(m^{5/2}\gamma)$ or $\angle \hat{\mat{a}}_\mat{x}\,\mat{a}_\mat{x}\,\mat{y} =O(m^{5/2}\gamma)$.
Without loss of generality, assume that $\angle \hat{\mat{a}}_\mat{x}\,\mat{a}_\mat{x}\,\mat{y} = O(m^{5/2}\gamma)$. Consider the circumcircle of $\hat{\mat{a}}_\mat{x}\,\mat{a}_\mat{x}\,\mat{y}$. Let $\mat{o}$ be its center. Refer to Figure~\ref{fg:proj-1}(b). The angle $\angle \hat{\mat{a}}_\mat{x}\,\mat{o}\,\mat{y} = 2\angle \hat{\mat{a}}_\mat{x}\,\mat{a}_\mat{x}\,\mat{y}$. Then, $\norm{\hat{\mat{a}}_\mat{x} - \mat{y}} = 2\norm{\mat{o} - \mat{y}}\,\sin(\angle \hat{\mat{a}}_\mat{x}\,\mat{o}\,\mat{y}/2) \leq \mbox{radius}(S) \cdot O(m^{5/2}\gamma) = O(m^{7/2}\gamma^2)$. \end{proof}
Next, we prove that $\mat{x}_{i+1}$ is much closer to $Z_\varphi$ than $\mat{x}_i$.
\begin{lemma} \label{lem:iterate} Let $P$ be a uniform $(\eps,\kappa)$-sample of $\mani$. There exists $\eps_0 \in (0,1)$ that decreases as $d$ and $\kappa$ increase such that if $\eps \leq \eps_0$, then for any point $\mat{y}$
at distance $O(m^{7/2}\gamma^2)$ or less from $\mani$, we have $\norm{\mat{y}' - \tilde{\mat{y}}} \leq \gamma^{1/4} \cdot \norm{\mat{y} - \tilde{\mat{y}}}$, where $\tilde{\mat{y}}$ is the nearest point in $Z_\varphi \cap \widehat{\mani}$ to $\mat{y}$ and $\mat{y}' = \mat{y} + \mat{B}_{\varphi,\mat{y}}^{} \cdot \mat{B}_{\varphi,\mat{y}}^t \cdot ( \mat{a}_{\mat{y}} - \mat{y})$. \end{lemma} \begin{proof} Let $\mat{z} = \nu(\mat{y})$. For $i \in [1,d-m]$, let $\mat{v}_i$ be the unit vector in $N_{\mat{z}}$ such that $\mat{B}_{\varphi,\mat{y}} = (f_{\mat{v}_1}(\mat{y}), \ldots, f_{\mat{v}_{d-m}}(\mat{y}))$ consists of orthogonal unit column vectors. By Lemma~\ref{lemma::normal_angle}, $\angle (L_{\mat{y}},N_{\mat{z}}) = O(m\sqrt{m}\,\gamma)$, so for any distinct $i,j \in [1,d-m]$, $\angle (\mat{v}_i, \mat{v}_j) = \pi/2 \pm O(m\sqrt{m}\,\gamma)$. This allows us to prove as in the proof of Lemma~\ref{lemma:local} that $\{\mat{v}_1,\ldots,\mat{v}_{d-m}\}$ are linearly independent and hence they form a basis of $N_{\mat{z}}$.
Let $\varrho_{\mat{z}}$ be the canonical function with respect to $\mat{z}$ and the basis $\{\mat{v}_1,\ldots,\mat{v}_{d-m}\}$ of $N_\mat{z}$.
Since $\norm{\mat{y}-\tilde{\mat{y}}}$ is at most $\norm{\mat{y}-\mat{z}}$ plus the distance from $\mat{z}$ to $Z_\varphi \cap \widehat{\mani}$, by Theorem~\ref{thm:main}, we have $\norm{\mat{y}-\tilde{\mat{y}}} \leq O(m^{7/2}\gamma^2) + O(m^{5/2}\gamma^2) = O(m^{7/2}\gamma^2)$. So $\norm{\tilde{\mat{y}}-\mat{z}} \leq \norm{\mat{y} - \tilde{\mat{y}}} + \norm{\mat{y}-\mat{z}} = O(m^{7/2}\gamma^2)$. Therefore, \[ \mbox{segment $\mat{y}\,\tilde{\mat{y}}$ is contained in $B(\mat{z}, tm^{7/2}\gamma^2)$ for some constant $t$}, \] implying that $\varrho_\mat{z}(\mat{x})$ is defined for any point $\mat{x}$ in the segment $\mat{y}\,\tilde{\mat{y}}$ as long as $\eps_0 < 1/(8tm^{7/2})$ so that $tm^{7/2}\gamma^2 \leq 16tm^{7/2}\eps_0\eps < 2\eps$. By Lemmas~\ref{lem:agree} and~\ref{lemma:local}, $\varrho_\mat{z}^{-1}(0)$ agrees with $Z_\varphi$ within $B(\mat{z},tm^{7/2}\gamma^2)$. Then, the following relations follow from Lemma~\ref{lemma::normal_angle}, Lemma~\ref{lemma::unique}, Theorem~\ref{thm:main}, and the facts that $\angle (\mat{v}_i,f_{\mat{v}_i}(\mat{y})) = O(m\sqrt{m}\,\gamma)$ for any $i \in [1,d-m]$, and $\angle (\mat{v}_i,f_{\mat{v}_j}(\mat{y})) = \pi/2 \pm O(m\sqrt{m}\,\gamma)$ for any distinct $i,j \in [1,d-m]$. \begin{itemize}
\item For all $i \in [1,d-m]$ and all $\mat{x} \in B(\mat{z},tm^{7/2}\gamma^2)$, $\norm{\nabla\varrho_{\mat{z},i}(\mat{x})} \in \left[1-O(\kappa m^4\gamma),1+O(\kappa m^4\gamma)\right]$.
\item For all distinct indices $i,j \in [d-m]$ and for all pair of points $\mat{x},\mat{x}' \in B(\mat{z}, tm^{7/2}\gamma^2)$, $\nabla\varrho_{\mat{z},i}(\mat{x})^t \cdot \nabla\varrho_{\mat{z},j}(\mat{x}') = \pm O(\kappa m^4\gamma)$.
\item For all $i \in [d-m]$, $f_{\mat{v}_i}(\mat{y})^t \cdot \nabla\varrho_{\mat{z},i}(\mat{y}) \in \left[1 - O(\kappa m^4\gamma),1 + O(\kappa m^4\gamma)\right]$.
\item For all distinct $i,j \in [d-m]$, $f_{\mat{v}_i}(\mat{y})^t \cdot \nabla\varrho_{\mat{z},j}(\mat{y}) = \pm O(\kappa m^4\gamma)$.
\end{itemize}
We first prove lower and upper bounds on $\norm{\varrho_\mat{z}(\mat{y})}$. Since $\tilde{\mat{y}}$ is the nearest point in $Z_\varphi \cap \widehat{\mani}$ to $\mat{y}$, the vector $\mat{y}-\tilde{\mat{y}}$ belongs to the normal space of $Z_\varphi$ at $\tilde{\mat{y}}$. Recall that $Z_{\varrho_\mat{z}}$ agrees with $Z_\varphi$ locally, so the normal space of $Z_\varphi$ at $\tilde{\mat{y}}$ is spanned by $\{\nabla\varrho_{\mat{z},1}(\tilde{\mat{y}}), \ldots, \nabla\varrho_{\mat{z},d-m}(\tilde{\mat{y}})\}$. Let $\mat{u} = \sum_{i=1}^{d-m} \lambda_i\cdot\nabla\varrho_{\mat{z},i}(\tilde{\mat{y}})$ denote the unit vector $(\mat{y}-\tilde{\mat{y}})/\norm{\mat{y} - \tilde{\mat{y}}}$. Standard vector calculus gives \begin{eqnarray} \varrho_{\mat{z}}(\mat{y}) & = & \left(\bigintsss_{0}^1 \left(\nabla \varrho_{\mat{z},1}(\tilde{\mat{y}}+r\mat{u}), \ldots, \nabla \varrho_{\mat{z},d-m}(\tilde{\mat{y}}+r\mat{u})\right)^t \cdot (\mat{y}-\tilde{\mat{y}}) \;\; \mbox{d}r \right) \nonumber \\ & = & \norm{\mat{y}-\tilde{\mat{y}}} \cdot \bigintsss_{0}^1 \left(\nabla \varrho_{\mat{z},1}(\tilde{\mat{y}}+r\mat{u}), \ldots, \nabla \varrho_{\mat{z},d-m}(\tilde{\mat{y}}+r\mat{u})\right)^t \cdot \left(\sum_{i=1}^{d-m} \lambda_i\cdot\nabla\varrho_{\mat{z},i}(\tilde{\mat{y}})\right) \; \mbox{d}r \nonumber \\ & = & \norm{\mat{y}-\tilde{\mat{y}}} \cdot \begin{pmatrix} \lambda_1 + \sum_{i=1}^{d-m} (\pm\lambda_i) \cdot O(\kappa m^4\gamma) \\ \vdots \\ \lambda_{d-m} + \sum_{i=1}^{d-m} (\pm\lambda_i) \cdot O(\kappa m^4\gamma) \end{pmatrix}. \label{eq:iterate} \end{eqnarray} Hence, \begin{equation}
\sum_{i=1}^{d-m}\lambda_i^2 - O(\kappa m^4\gamma) \left(\sum_{i=1}^{d-m}|\lambda_i|\right)^2 \leq \frac{\norm{\varrho_{\mat{z}}(\mat{y})}^2}{\norm{\mat{y}-\tilde{\mat{y}}}^2} \leq
\sum_{i=1}^{d-m}\lambda_i^2 + O(\kappa m^4\gamma) \left(\sum_{i=1}^{d-m}|\lambda_i|\right)^2. \label{eq:iterate-2} \end{equation}
We claim that if $\eps_0$ is small enough, then \begin{equation}
\forall\, i \in [1,d-m], \quad |\lambda_i| \leq 1 + O((d-m)\kappa m^4\gamma). \label{eq:iterate-3} \end{equation}
Let $k = \argmax_{i=[1,d-m]} |\lambda_i|$. We take the dot product of $\sum_{i=1}^{d-m} \lambda_{i} \cdot \nabla\varrho_{\mat{z},i}(\tilde{\mat{y}})$ and $\nabla\varrho_{\mat{z},k}(\tilde{\mat{y}})$ or $-\nabla\varrho_{\mat{z},k}(\tilde{\mat{y}})$ depending on whether $\lambda_{k}$ is non-negative or negative, respectively. This dot product is at most $1 + O(\kappa m^4\gamma)$ as $\norm{\nabla\varrho_{\mat{z},k}(\tilde{\mat{y}})} = 1 + O(\kappa m^4\gamma)$. On the other hand, for each $i \not= k$, $\lambda_{i} \cdot \nabla\varrho_{\mat{z},i}(\tilde{\mat{y}})^t \cdot \nabla\varrho_{\mat{z},k}(\tilde{\mat{y}})$ contributes
$\pm|\lambda_{i}| \cdot O(\kappa m^4\gamma)$. It follows that \begin{eqnarray*}
& & |\lambda_{k}|\left(1 - O(\kappa m^4\gamma)\right) - O(\kappa m^4\gamma) \sum_{i\not=k} |\lambda_{i}| \leq 1
+ O(\kappa m^4\gamma) \\
& \Rightarrow & \left(1 - O((d-m)\kappa m^4\gamma))\right)|\lambda_{k}| \leq 1 + O(\kappa m^4\gamma) \\
& \Rightarrow & |\lambda_{k}| \leq 1 + O((d-m)\kappa m^4\gamma)). \end{eqnarray*}
Since $|\lambda_k| = \max_i |\lambda_i|$, it establishes our claim. \cancel{ Let
$\pi(1),\ldots,\pi(d-m)$ be the permutation such that $|\lambda_{\pi(1)}| \geq
|\lambda_{\pi(2)}| \geq \ldots \geq |\lambda_{\pi(d-m)}|$. We take the dot product of $\sum_{i=1}^{d-m} \lambda_{\pi(i)} \cdot \nabla\varrho_{\mat{z},\pi(i)}(\tilde{\mat{y}})$ and $\nabla\varrho_{\mat{z},\pi(1)}(\tilde{\mat{y}})$ or $-\nabla\varrho_{\mat{z},\pi(1)}(\tilde{\mat{y}})$ depending on whether $\lambda_{\pi(1)}$ is non-negative or negative, respectively. This dot product is at most $1 + O(\kappa m^4\gamma)$ as $\norm{\nabla\varrho_{\mat{z},\pi(1)}(\tilde{\mat{y}})} = 1 + O(\kappa m^4\gamma)$. On the other hand, for each $i \not= 1$, $\lambda_{\pi(i)} \cdot \nabla\varrho_{\mat{z},\pi(i)}(\tilde{\mat{y}})^t \cdot \nabla\varrho_{\mat{z},\pi(1)}(\tilde{\mat{y}})$ contributes
$\pm|\lambda_{\pi(i)}| \cdot O(\kappa m^4\gamma)$. It follows that \begin{eqnarray*}
& & |\lambda_{\pi(1)}|\left(1 - O(\kappa m^4\gamma)\right) - O(\kappa m^4\gamma) \sum_{i\not=1} |\lambda_{\pi(i)}| \leq 1 + O(\kappa m^4\gamma) \\
& \Rightarrow & \left(1 - O((d-m)\kappa m^4\gamma))\right)|\lambda_{\pi(1)}| \leq 1 + O(\kappa m^4\gamma) \\
& \Rightarrow & |\lambda_{\pi(1)}| \leq 1 + O((d-m)\kappa m^4\gamma)). \end{eqnarray*} In general, suppose that we consider $\lambda_{\pi(k)}$. As before, we get \[
|\lambda_{\pi(k)}|\left(1-O(\kappa m^4\gamma)\right) - O(\kappa m^4\gamma)
\sum_{i\not=k} |\lambda_{\pi(i)}| \leq 1 + O(\kappa m^4\gamma). \]
Inductively, we have shown that $|\lambda_{\pi(j)}| \leq 1 + O((d-m)\kappa m^4\gamma)$ for $j \in [1,k-1]$. Therefore, \begin{eqnarray*}
& & |\lambda_{\pi(k)}|\left(1-O(\kappa m^4\gamma)\right) - O((k-1)\kappa m^4\gamma)) -
O((d-m-k)\kappa m^4\gamma)|\lambda_{\pi(k)}| \\ & \leq & 1 + O(\kappa m^4\gamma). \end{eqnarray*}
Hence, $|\lambda_{\pi(k)}| \leq 1 + O((d-m)\kappa m^4\gamma))$, establishing our claim. }
Since $\sum_{i=1}^{d-m} \lambda_i\cdot\nabla\varrho_{\mat{z},i}(\tilde{\mat{y}})$ is a unit vector, we get \[
\left\|\sum_{i=1}^{d-m} \lambda_i \cdot \nabla\varrho_{\mat{z},i}(\tilde{\mat{y}}) \right\|^2 = \sum_{i=1}^{d-m} \lambda_i^2\cdot\norm{\nabla\varrho_{\mat{z},i}(\tilde{\mat{y}})}^2 + \sum_{i\not=j}\lambda_i\lambda_j\cdot\nabla\varrho_{\mat{z},i}(\mat{y})^t \cdot \nabla\varrho_{\mat{z},j}(\mat{y}) = 1, \] which implies that \[ 1 - O(\kappa m^4\gamma) -
O(\kappa m^4\gamma)\sum_{i\not=j}|\lambda_i\lambda_j| \leq \;\; \sum_{i=1}^{d-m} \lambda_i^2 \;\; \leq 1 + O(\kappa m^4\gamma) +
O(\kappa m^4\gamma)\sum_{i\not=j}|\lambda_i\lambda_j|. \]
Using the above relations concerning $\lambda_i$'s, we get an upper bound of the right hand side of \eqref{eq:iterate-2} as follows. \begin{eqnarray*} \sum_{i=1}^{d-m} \lambda_i^2 + O(\kappa m^4\gamma) \left(\sum_{i=1}^{d-m}
|\lambda_i|\right)^2 & \leq & 1 + O(\kappa m^4\gamma) + O(\kappa m^4\gamma)\sum_{i\not=j}|\lambda_i\lambda_j| \\ & \leq & 1 + O(\kappa m^4\gamma) + O(\kappa m^4\gamma) \cdot (d^2 + O(d^2(d-m)\kappa m^4\gamma)) \\ & \leq & 1 + O(d^2\kappa m^4\gamma). \end{eqnarray*}
Symmetrically, we get a lower bound of the left hand side of \eqref{eq:iterate-2}: \[
\sum_{i=1}^{d-m} \lambda_i^2 - O(\kappa m^4\gamma) \left(\sum_{i=1}^{d-m} |\lambda_i|\right)^2 \geq 1 - O(d^2\kappa m^4\gamma). \] Thus, we simplify \eqref{eq:iterate-2} to \begin{equation} (1 - O(d^2\kappa m^4\gamma)) \cdot \norm{\mat{y}-\tilde{\mat{y}}}^2 \leq \norm{\varrho_{\mat{z}}(\mat{y})}^2 \leq (1 + O(d^2\kappa m^4\gamma)) \cdot \norm{\mat{y}-\tilde{\mat{y}}}^2. \label{eq:iterate-4} \end{equation} In other words, $\norm{\varrho_{\mat{z}}(\mat{y})}$ is a good approximation of the distance from $\mat{y}$ to the zero-set of $\varrho_{\mat{z}}$.
Next, we give a lower bound on $\cos\angle \mat{y}'\,\mat{y}\,\tilde{\mat{y}}$. Consider the dot product $(\mat{y}'-\mat{y})^t \cdot (\tilde{\mat{y}}-\mat{y})$. By expanding $\mat{B}_{\varphi,\mat{y}}^t \cdot (\mat{a}_\mat{y}-\mat{y})$, we get \[ \mat{y}'-\mat{y} = \mat{B}_{\varphi,\mat{y}} \cdot \mat{B}_{\varphi,\mat{y}}^t \cdot (\mat{a}_\mat{y} - \mat{y}) = \mat{B}_{\varphi,\mat{y}} \cdot (-\varrho_{\mat{z}}(\mat{y})). \] Since $\mat{B}_{\varphi,\mat{y}}$ consists of orthogonal unit column vectors, we get \begin{equation} \norm{\mat{y}'-\mat{y}} = \norm{\mat{B}_{\varphi,\mat{y}} \cdot (-\varrho_{\mat{z}}(\mat{y}))} = \norm{\varrho_{\mat{z}}(\mat{y})}. \label{eq:iterate-6} \end{equation} Therefore, \begin{eqnarray} (\mat{y}'-\mat{y})^t \cdot (\tilde{\mat{y}}-\mat{y}) & = & \norm{\varrho_{\mat{z}}(\mat{y})} \cdot \norm{\mat{y}-\tilde{\mat{y}}} \cdot \cos \angle \mat{y}'\,\mat{y}\,\tilde{\mat{y}} \nonumber \\ & \leq & \sqrt{1 + O(d^2\kappa m^4\gamma)} \cdot \norm{\mat{y}-\tilde{\mat{y}}}^2 \cdot \cos \angle \mat{y}'\,\mat{y}\,\tilde{\mat{y}}. \label{eq:iterate-5} \end{eqnarray} Recall that $\sum_{i=1}^{d-m}\lambda_i \cdot \nabla\varrho_{\mat{z},i}(\tilde{\mat{y}})$ is the unit vector $(\mat{y}-\tilde{\mat{y}})/\norm{\mat{y}-\tilde{\mat{y}}}$. By expanding $(\mat{y}'-\mat{y})^t \cdot (\tilde{\mat{y}}-\mat{y})$, we get \begin{eqnarray*} (\mat{y}'-\mat{y})^t \cdot (\tilde{\mat{y}}-\mat{y}) & = & \left(\mat{B}_{\varphi,\mat{y}} \cdot \varrho_{\mat{z}}(\mat{y})\right)^t \cdot \norm{\mat{y}-\tilde{\mat{y}}} \cdot \sum_{i=1}^{d-m} \lambda_i \cdot \nabla\varrho_{\mat{z},i}(\tilde{\mat{y}}) \\ & = & \left(\sum_{i=1}^{d-m} \varrho_{\mat{z},i}(\mat{y}) \cdot f_{\mat{v}_i}(\mat{y})\right)^t \cdot \norm{\mat{y}-\tilde{\mat{y}}} \cdot \sum_{i=1}^{d-m} \lambda_i \cdot \nabla\varrho_{\mat{z},i}(\tilde{\mat{y}}) \\ & = & \sum_{i=1}^{d-m} \sum_{j=1}^{d-m} \varrho_{\mat{z},i}(\mat{y}) \cdot \lambda_j \cdot f_{\mat{v}_i}(\mat{y})^t \cdot \nabla\varrho_{\mat{z},j}(\tilde{\mat{y}}) \cdot \norm{\mat{y}-\tilde{\mat{y}}} \\ & = & \sum_{i=1}^{d-m} \varrho_{\mat{z},i}(\mat{y}) \cdot \norm{\mat{y}-\tilde{\mat{y}}} \cdot \beta_i, \end{eqnarray*} where $\beta_i = \lambda_i + \sum_{i=1}^{d-m} (\pm\lambda_i) \cdot O(\kappa m^4\gamma)$ for $i \in [1,d-m]$. Note the similarity between the $\beta_i$'s and the vector in \eqref{eq:iterate}. Therefore, $\norm{\mat{y}-\tilde{\mat{y}}} \cdot \beta_i = \varrho_{\mat{z},i}(\mat{y}) + \norm{\mat{y}-\tilde{\mat{y}}} \cdot \sum_{i=1}^{d-m} (\pm\lambda_i) \cdot O(\kappa m^4\gamma) \geq
\varrho_{\mat{z},i}(\mat{y}) - O((d-m)\kappa m^4\gamma))\cdot\norm{\mat{y}-\tilde{\mat{y}}}$ as $|\lambda_i| \leq 1 + O((d-m)\kappa m^4\gamma)$. Hence, \begin{eqnarray*} (\mat{y}'-\mat{y})^t \cdot (\tilde{\mat{y}}-\mat{y}) & \geq & \sum_{i=1}^{d-m} \varrho_{\mat{z},i}(\mat{y})^2 - O((d-m)\kappa m^4\gamma)) \cdot \norm{\mat{y}-\tilde{\mat{y}}} \cdot
\sum_{i=1}^{d-m} |\varrho_{\mat{z},i}(\mat{y})| \\ & \geq & \norm{\varrho_{\mat{z}}(\mat{y})}^2 - O((d-m)\kappa m^4\gamma) \cdot \norm{\mat{y}-\tilde{\mat{y}}} \cdot \sqrt{d-m} \cdot \norm{\varrho_{\mat{z}}(\mat{y})} \\ & \geq & \norm{\varrho_{\mat{z}}(\mat{y})}^2 - O((d-m)^{3/2}\kappa m^4\gamma) \cdot \norm{\mat{y}-\tilde{\mat{y}}} \cdot \norm{\varrho_\mat{z}(\mat{y})}. \end{eqnarray*}
Substituting \eqref{eq:iterate-4} into the above, we get \[ (\mat{y}'-\mat{y})^t \cdot (\tilde{\mat{y}}-\mat{y}) \geq \bigl(1 - O(d^2\kappa m^4\gamma)\bigr) \cdot \norm{\mat{y}-\tilde{\mat{y}}}^2. \] Combining \eqref{eq:iterate-5} with the above inequality gives \[ \cos\angle\mat{y}'\,\mat{y}\,\tilde{\mat{y}} \geq 1 - O(d^2\kappa m^4\gamma). \]
Finally, consider triangle $\mat{y}'\mat{y}\,\tilde{\mat{y}}$. By the cosine law, we have \[ \norm{\mat{y}'-\tilde{\mat{y}}} = \bigl(\norm{\mat{y}'-\mat{y}}^2 + \norm{\mat{y}-\tilde{\mat{y}}}^2 - 2 \norm{\mat{y}'-\mat{y}}\, \norm{\mat{y}-\tilde{\mat{y}}}\, \cos\angle\mat{y}'\mat{y}\,\tilde{\mat{y}}\bigr)^{1/2}. \] By \eqref{eq:iterate-4} and \eqref{eq:iterate-6}, $\norm{\mat{y}'-\mat{y}}^2 \leq (1+O(d^2\kappa m^4\gamma)) \cdot \norm{\mat{y} - \tilde{\mat{y}}}^2$. Therefore, \begin{eqnarray*} \norm{\mat{y}' - \tilde{\mat{y}}} & \leq & \norm{\mat{y}-\tilde{\mat{y}}} \cdot \left(2 + O(d^2\kappa m^4\gamma) - 2\left(1 - O(d^2\kappa m^4\gamma)\right)\left(1 - O(d^2\kappa m^4\gamma)\right)\right)^{1/2} \\ & \leq & O(dm^2\sqrt{\kappa\gamma}) \cdot \norm{\mat{y}-\tilde{\mat{y}}} \\
& \leq & \gamma^{1/4} \cdot \norm{\mat{y}-\tilde{\mat{y}}} \end{eqnarray*} whenever $\eps_0$ is small enough so that $\gamma^{1/4} = O(\eps^{1/4}) = O(\eps_0^{1/4})$ cancels the $O(dm^2\sqrt{\kappa})$ factor. This requires $\eps_0$ to decrease as $d$ and $\kappa$ increase. \end{proof}
By combining Lemmas~\ref{lem:first_itr} and~\ref{lem:iterate}, we prove that the projection operator will bring an initial point to a point in $Z_\varphi \cap \widehat{\mani}$ in the limit.
\begin{theorem} Let $\varphi$ be the function for a uniform $(\eps,\kappa)$-sample of an $m$-dimensional compact smooth manifold $\mani$ in $\real^d$ as specified in Theorem~\ref{thm:main}.
Define the projection operator $\mat{x}_{i+1} = \mat{x}_i + \mat{B}_{\varphi,\mat{x}_i} \cdot \mat{B}_{\varphi,\mat{x}_i}^t \cdot (\mat{a}_{\mat{x}_i} - \mat{x}_i)$, where $\mat{a}_{\mat{x}_i} = \sum_{\mat{p} \in P} \omega(\mat{x}_i,\mat{p}) \cdot \mat{p}$.
There exists $\eps_0 \in (0,1)$ that decreases as $d$ and $\kappa$ increase such that if $\eps \leq \eps_0$, then for any initial point $\mat{x}_0$ at distance $m\gamma$ or less from some sample point, where $\gamma$ is the input neighborhood radius, the following properties hold. \begin{itemize}
\item $\lim_{i \rightarrow \infty} x_i \in Z_\varphi \cap \widehat{\mani}$, where $\widehat{\mani}$ is the set of points within a distance of $\eps$ from $\mani$.
\item For all $i > 0$, $\norm{\mat{x}_i - \nu(\mat{x}_0)} = O(m^{7/2}\gamma^2) = O(m^{7/2}\eps^2)$.
\end{itemize} \end{theorem} \begin{proof} For any point $\mat{x}$, let $\tilde{\mat{x}}$ denote the nearest point in $Z_\varphi \cap \widehat{\mani}$ to $\mat{x}$. By Lemma~\ref{lem:first_itr}, $\norm{\mat{x}_1 - \nu(\mat{x}_0)} = O(m^{7/2}\gamma^2)$. Let $\mat{b}$ be the nearest point in $Z_\varphi \cap \widehat{\mani}$ to $\nu(\mat{x}_0)$. Since $\norm{\mat{b} - \nu(\mat{x}_0)} = O(m^{5/2}\gamma^2)$ by Theorem~\ref{thm:main}, triangle inequality implies that for a small enough $\eps_0$, \begin{align*} \norm{\mat{x}_1-\tilde{\mat{x}}_1} \leq \norm{\mat{x}_1 - \mat{b}} & \leq \norm{\mat{b} - \nu(\mat{x}_0)} + \norm{\mat{x}_1 - \nu(\mat{x}_0)} \\ & \leq O(m^{5/2}\gamma^2) + O(m^{7/2}\gamma^2) \\ & = O(m^{7/2}\gamma^2). \end{align*} Since $\norm{\mat{x}_1 - \nu(\mat{x}_0)} = O(m^{7/2}\gamma^2)$, Lemma~\ref{lem:iterate} is applicable to $\mat{x}_1$. It ensures that $\norm{\mat{x}_2 - \tilde{\mat{x}}_2} \leq \norm{\mat{x}_2-\tilde{\mat{x}}_1} \leq \gamma^{1/4} \cdot \norm{\mat{x}_1 - \tilde{\mat{x}}_1} = O(m^{7/2}\gamma^{9/4})$, which is smaller than $O(m^{7/2}\gamma^2)$ and so Lemma~\ref{lem:iterate} is applicable to $\mat{x}_2$. Repeating this argument gives \[ \norm{\mat{x}_i - \tilde{\mat{x}}_i} \leq \norm{\mat{x}_i - \tilde{\mat{x}}_{i-1}} = O(m^{7/2}\gamma^{(7+i)/4}). \] This proves that $\lim_{i \rightarrow \infty} x_i \in Z_\varphi \cap \widehat{\mani}$. By triangle inequality, \begin{align*} \norm{\mat{x}_i - \mat{x}_{i-1}} & \leq \norm{\mat{x}_i - \tilde{\mat{x}}_{i-1}} + \norm{\mat{x}_{i-1} - \tilde{\mat{x}}_{i-1}} \\ & = O(m^{7/2}\gamma^{(7+i)/4}) + O(m^{7/2}\gamma^{(6+i)/4}) \\ & = O(m^{7/2}\gamma^{(7+i)/4}). \end{align*} Therefore, for a small enough $\eps_0$, \begin{align*} \norm{\mat{x}_i - \nu(\mat{x}_0)} & \leq \sum_{j=2}^i \norm{\mat{x}_j-\mat{x}_{j-1}} + \norm{\mat{x}_1-\nu(\mat{x}_0)} \\ & < \sum_{j=2}^i O(m^{7/2}\gamma^{(7+j)/4}) + O(m^{7/2}\gamma^2) \\ & = O(m^{7/2}\gamma^2). \end{align*} \end{proof}
\section{Conclusion}
We define a function $\varphi$ from a uniform $(\eps,\kappa)$-sample of a compact smooth manifold $\mani$ in $\real^d$ such that the zero-set of $\varphi$ near $\mani$ is a faithful reconstruction of $\mani$. Moreover, we give a projection operator that will yield a point on the zero-set near $\mani$ in the limit by iterative applications. More work is needed to improve the angular error of $O(m^2\sqrt{\kappa\eps})$, which is weaker than the $O(\eps)$ angular error offered by provably good simplicial reconstructions. It would also be desirable for $\eps$ to depend on $m$ only instead of $d$. Another natural question is how to deal with non-smooth manifolds and non-manifolds.
\paragraph{Acknowledgment}
The authors would like to thank the anonymous reviewers for helpful comments, pointing out mistakes in an earlier version that we subsequently corrected, and suggesting the removal of some slack in the bounds on Hausdorff distance and angular error.
\end{document}
|
arXiv
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\begin{document}
\title{ extbf{The hamiltonicity\of essentially 9-connected line
graphs} \footnotetext[1]{Department of Mathematics and European Centre of
Excellence NTIS (New Technologies for the Information Society),
University of West Bohemia, Univerzitn\'{\i}~8, 306~14~Plze\v{n},
Czech Republic. Supported by project 17-04611S of the Czech Science
Foundation. E-mail:
\{\texttt{kaisert,vranap}\}\texttt{@kma.zcu.cz}.}
\begin{abstract}
Yang et al. proved that every 3-connected, essentially 11-connected
line graph is Hamilton-connected. This was extended by Li and Yang
to 3-connected, essentially 10-connected graphs. Strengthening their
result further, we prove that 3-connected, essentially 9-connected
line graphs are Hamilton-connected. We use a method based on
quasigraphs in combination with the discharging technique. The
result extends to claw-free graphs. \end{abstract}
\section{Introduction} \label{sec:introduction}
A conjecture of Thomassen~\cite{Tho-reflections} states that every 4-connected line graph is hamiltonian. Motivated by this conjecture, Lai et al.~\cite{LSWZ-every} studied the hamiltonicity of line graphs in relation to their essential connectivity (the definition is recalled later in this section). They proved the following result:
\begin{theorem}\label{t:lai}
Every 3-connected, essentially 11-connected line graph is
hamiltonian. \end{theorem}
In fact, Yang et al.~\cite{YLLG-collapsible} prove that under the assumption of Theorem~\ref{t:lai}, the graph is \emph{Hamilton-connected}, i.e., any pair of its vertices is joined by a Hamilton path. The result was recently improved by Li and Yang~\cite{LY-every} who showed that 3-connected, essentially 10-connected graphs are Hamilton-connected.
This partially answered a question of Lai et al.~\cite{LSWZ-every} whether the constant in Theorem~\ref{t:lai} can be replaced by a smaller one. They note that the least possible value is 4, which is improved to 5 in \cite{YXLG-hamiltonicity}. We add that there are 3-connected, essentially 4-connected line graphs that do not even have a Hamilton path, such as the line graph of a graph obtained by adding one pendant edge to each vertex of the graph in Figure~\ref{fig:example}a.
Regarding the assumption of 3-connectedness in Theorem~\ref{t:lai}, we remark that there are 2-connected line graphs of arbitrary essential connectivity that do not have any Hamilton path. A class of examples can be constructed by replacing each edge of the complete graph on 4 vertices by an odd number of internally disjoint paths of length 3, as shown in Figure~\ref{fig:example}b.
\begin{figure}
\caption{(a) A graph used to construct a 3-connected, essentially
4-edge-connected line graph without a Hamilton path. (b) A graph
$G$ such that $L(G)$ is 2-connected and essentially $k$-connected
(for $k=9$) and has no Hamilton path.}
\label{fig:example}
\end{figure}
The main result of the present paper is a further strengthening of Theorem~\ref{t:lai} as follows:
\begin{theorem}\label{t:main}
Every 3-connected, essentially 9-connected line graph is
Hamilton-connected. \end{theorem}
Our approach uses a strengthening of the main result of~\cite{KV-hamilton}, proved in a companion paper~\cite{KV-quasi}. In particular, we use a reduction to hypergraphs, which is described in Section~\ref{sec:hyper}. Preliminaries on the Hamilton connectivity of line graphs (and the implications on the hypergraph side of the problem) are given in Section~\ref{sec:hamilt-conn}.
Section~\ref{sec:quasi} describes quasigraphs, another crucial component of the approach of~\cite{KV-hamilton,KV-quasi}, and states the main technical tool of this paper, Theorem~\ref{t:skeletal} proved in~\cite{KV-quasi}. A counting argument based on the outcome of Theorem~\ref{t:skeletal} is given in Section~\ref{sec:counting}.
Section~\ref{sec:structural} is essentially a study of small configurations in the hypergraph corresponding to a graph satisfying the hypothesis of Theorem~\ref{t:main}. It prepares the way for the application of a discharging argument in Section~\ref{sec:discharging}. The final section is devoted to an extension of the result to claw-free graphs and concluding remarks.
In the remainder of this section, we recall the necessary terminology and notation. For graph-theoretical concepts not explained here, see, for example, Diestel~\cite{Die-graph}. Unless otherwise noted, all graphs in this paper are loopless and may contain parallel edges. When speaking of a line graph of a graph $G$, it is understood that $G$ may contain parallel edges (i.e., $G$ may be a multigraph), but $L(G)$ is by definition a simple graph.
Let $G$ be a graph. We write $V(G)$ and $E(G)$ for the set of vertices and the set of edges of $G$, respectively. For $i\geq 0$, $V_i(G)$ is the set of vertices of degree $i$ in $G$.
A vertex-cut or edge-cut $X$ is called \emph{essential} if $G-X$ has at least two components which are \emph{nontrivial} (i.e., contain more than one vertex). The graph $G$ is \emph{essentially
$k$-connected} if it has more than $k$ vertices and contains no essential vertex-cut of size less than $k$. Similarly, $G$ is \emph{essentially $k$-edge-connected} if it contains no essential edge-cut of size less than $k$. It is not hard to see that $L(G)$ is $k$-connected if and only if $G$ is essentially $k$-edge-connected and $\size{E(G)} > k$.
We extend the above definitions as follows. An edge-cut $X$ in a graph $G$ is \emph{$r$-essential} ($r\geq 0$) if there at least two components of $G-C$, each of which contains at least $r$ edges. The graph $G$ is \emph{$r$-essentially $k$-edge-connected} ($k\geq 1$) if it has no $r$-essential edge-cuts of size less than $k$. Thus, `$0$-essentially $k$-edge-connected' is the same as `$k$-edge-connected', while `$1$-essentially $k$-edge-connected' is the same as `essentially $k$-edge-connected'.
We note the following easy observation: \begin{observation}\label{obs:2-ess}
The line graph $L(G)$ of a graph $G$ is essentially $k$-connected if
and only if $G$ is $2$-essentially $k$-edge-connected and
$\size{E(G)} > k$. \end{observation}
The \emph{length} of a path is the number of its edges. The degree of a vertex $v$ in a graph $G$ is denoted by $d_G(v)$. Given a set of vertices $X\subseteq V(G)$, we let $\bd G X$ denote the set of edges of $G$ with exactly one endvertex in $X$. Furthermore, we extend the notation for the degree of a vertex and set $d_G(X) = \size{\bd G X}$.
Besides graphs, we will also consider \emph{3-hypergraphs}, that is, hypergraphs with all hyperedges of size 2 or 3 (called \emph{2-hyperedges} or \emph{3-hyperedges} accordingly). For a hypergraph $H$, $V(H)$ and $E(H)$ denote its vertex set and its hyperedge set, respectively. The symbol $V_i(H)$ denotes the set of vertices of degree $i$ ($i\geq 0$). In addition, for $i\in\Setx{2,3}$, we let $E_i(H)$ denote the set of $i$-hyperedges of $H$. We define a graph $G(H)$ whose vertex set is $V(H) \cup E_3(H)$, with vertices $u,v$ joined by an edge if either $u,v$ are neighbours in $H$, or $v\in E_3(H)$ and $u$ is a vertex contained in $v$.
For $X\subseteq V(H)$, we let $\bd H X$ be the set of hyperedges of $H$ intersecting $X$ but not contained in it, and define $d_H(X) = \size{\bd H X}$ as in the graph case.
If $e$ is a hyperedge of $H$ and $v$ is a vertex contained in $e$, then the \emph{detachment} of $e$ from $v$ is the operation which removes $e$ from $H$ and, in case $\size e=3$, replaces it with $e-\Setx v$.
\section{Reduction to hypergraphs} \label{sec:hyper}
From this point on, let $G$ be a graph whose line graph satisfies the hypothesis of Theorem~\ref{t:main}. By Observation~\ref{obs:2-ess} and the preceding discussion, $G$ is essentially 3-edge-connected and 2-essentially 9-edge-connected.
We begin by transforming $G$ to a graph $G^0$ called the \emph{core} of $G$. Recall that the \emph{suppression} of a vertex $v$ of degree 2 is the contraction of one of its incident edges (discarding any loop that may result). The graph $G^0$ is obtained from $G-V_1(G)$ by suppressing all the vertices of degree 2. By our assumption that $G$ is essentially 3-edge-connected, $G-V_1(G)$ has minimum degree at least 2 unless $G$ is a star. (Note also that that $V_2(G-V_1(G)) = V_2(G)$.) Clearly, either $G^0$ is \emph{trivial} (i.e., it has only one vertex), or $G^0$ has minimum degree at least 3. This observation is strengthened in Lemma~\ref{l:g0} below.
Given a set $X\subset V(G^0)$, we define $\xcore X$ as the union of $X$ with the set of vertices $y\in V(G)-V(G^0)$ such that $N_G(y) \subseteq X$.
\begin{observation}
\label{obs:cuts}
For any $X\subseteq V(G^0)$, it holds that $d_{G^0}(X) = d_G(\xcore
X)$. Furthermore, if $\bd{G^0} X$ is an $r$-essential edge-cut in
$G^0$ ($r \geq 0$), then $\bd G {\xcore X}$ is an $r$-essential
edge-cut in $G$. \end{observation}
\begin{lemma}
\label{l:g0}
If $L(G)$ is 3-connected and essentially 9-connected, then $G^0$ has
the following properties:
\begin{enumerate}[\quad(i)]
\item $G^0$ is 3-edge-connected,
\item $G^0$ is essentially 4-edge-connected,
\item $G^0$ is 2-essentially 9-edge-connected.
\end{enumerate} \end{lemma} \begin{proof}
Part (i) was proved by Shao~\cite{Sha-claw}
(see~\cite[Lemma~2.2(i)]{LSWZ-every}). Part (iii) of the lemma
follows from Observation~\ref{obs:cuts}. We prove part (ii). For
contradiction, let $F$ be an essential edge-cut in $G^0$ of size at
most 3. Let $K$ and $L$ be two components of $G^0-F$ containing at
least one edge each. Since $F$ is not 2-essential, one of them (say,
$K$) contains exactly one edge. The assumption that $\size F \leq 3$
implies that one of the two vertices of $K$ has degree at most 2, a
contradiction with part (i). \end{proof}
The vertices of $G$ which are not vertices of $G^0$ are called \emph{transient}. A vertex $v$ of $G^0$ with $d_{G^0}(v) = 3$ is said to be \emph{protected} if it is adjacent in $G$ to a transient vertex.
We now turn $G^0$ into a 3-hypergraph $H^0$. Let $W_3$ be the set of vertices of degree 3 in $G^0$, and let $W_3^\times$ be the subset of $W_3$ consisting of the protected vertices. We choose a maximal independent subset $W$ of $W_3-W_3^\times$. The 3-hypergraph $H^0$ is constructed from $G^0-W$ by adding, for each $w\in W$, a hyperedge $h(w)$ consisting of the neighbours of $w$ in $G^0$ provided that there are at least two such neighbours; if there are exactly two, then $\size{h(w)} = 2$. The procedure is illustrated in Figure~\ref{fig:hyper}. In all the figures in this paper, a 3-hyperedge is represented by three lines meeting at a point which is not marked as a vertex.
\begin{figure}
\caption{The transformation of $G^0$ to $H^0$. (a) Part of the graph
$G^0$ with vertices in $W$ filled white. (b) The corresponding
part of $H^0$, where the three incident lines without a common
vertex mark represent a 3-hyperedge. (c) The set $\Setx{x,y}^+$
includes all of the vertices of $W$ shown in the picture.}
\label{fig:hyper}
\end{figure}
The vertices in $W$ are called \emph{temporary}, the other vertices of $G^0$ \emph{permanent}. Thus, the set of permanent vertices is the vertex set of $H^0$. Note that all protected vertices are permanent.
\begin{lemma}\label{l:permanent}
Every edge of $G$ has a permanent endvertex. \end{lemma} \begin{proof}
Assume that an edge $e$ of $G$ has endvertices $u$ and $v$, and that
$u$ is not permanent. If $u$ is transient, then $v$ is protected and
hence permanent. Otherwise, $u$ is temporary, in which case $v$ is
neither transient (as this would make $u$ permanent) nor temporary
(since temporary vertices form an independent set). Hence, $v$ is
permanent. \end{proof}
Similarly to the set $\xcore X$ above, we introduce a set $\xhyper Y \subseteq V(G^0)$, where $Y\subseteq V(H^0)$. The definition is illustrated in Figure~\ref{fig:hyper}c. The set $\xhyper Y$ is defined as the union of $Y$ with the set of all the vertices $w\in W$ such that in $G^0$, $w$ is incident with at least two edges to $Y$ (possibly parallel). As in Observation~\ref{obs:cuts}, we have $d_{H^0}(Y) = d_{G^0}(\xhyper Y)$. Since $G^0$ is 3-edge-connected, we obtain the following observation:
\begin{observation}\label{obs:3ec}
The hypergraph $H^0$ is 3-edge-connected. \end{observation}
A more detailed study of the properties of $H^0$ is undertaken in Section~\ref{sec:structural}. The results will be used in the design of a discharging procedure in Section~\ref{sec:discharging}.
We add one more definition. To each edge $e$ of $G$, we want to assign a hyperedge $k(e)$ of $H^0$ which `corresponds' to $e$ in $H^0$. First, let us define a value $k_1(e)$ which is either an edge of $G^0$, or the empty set: \begin{equation*}
k_1(e) =
\begin{cases}
\emptyset & \text{if $e$ is incident with $V_1(G)$,}\\
e' & \text{if $e$ is incident with a transient vertex $z\notin V_1(G)$}\\
& \text{\quad and $e'$ is obtained by suppressing $z$,}\\
e & \text{otherwise.}
\end{cases} \end{equation*} Observe that $k_1(e)$ is well-defined, because the assumption that $G$ is essentially 3-edge-connected implies that $V_1(G) \cup V_2(G-V_1(G))$ is an independent set.
Next, we associate a hyperedge $k_2(f)$ of $H^0$ with each edge $f$ of $G^0$. In the definition, we allow $f$ or $k_2(f)$ to be the empty set. \begin{equation*}
k_2(f) =
\begin{cases}
\emptyset & \text{if $f = \emptyset$}, \\
h(w) & \text{if $f$ is incident with a temporary vertex $w$,} \\
f & \text{otherwise.}
\end{cases} \end{equation*} Finally, for an edge $e$ of $G$, we define \begin{equation*}
k(e) = k_2(k_1(e)). \end{equation*}
\section{Hamilton connectivity of line graphs} \label{sec:hamilt-conn}
For the cases where the core $G^0$ is small, we will be able to verify Theorem~\ref{t:main} directly using the following result \cite[Lemma~3.3]{LLS-s-hamiltonian}: \begin{lemma}\label{l:trees}
If $L(G)$ is 3-connected and $G^0$ contains two edge-disjoint
spanning trees, then $L(G)$ is Hamilton-connected. \end{lemma} Lemma~\ref{l:trees} will be used in conjunction with the characterization of graphs with two disjoint spanning trees, which follows from a more general result of Tutte~\cite{Tut-problem} and Nash-Williams~\cite{NW-edge}: \begin{theorem}[Tutte and Nash-Williams]\label{t:tnw}
The graph $G^0$ has two edge-disjoint spanning trees if and only if
for every partition $\mathcal P$ of $V(G^0)$, the number of edges of
$V(G^0)$ with endvertices in different classes of $\mathcal P$ is at least
$2(\size\mathcal P - 1)$. \end{theorem}
Using Lemma~\ref{l:trees} and Theorem~\ref{t:tnw}, we obtain the following: \begin{lemma}\label{l:small}
If $G^0$ has at most 5 vertices, then $L(G)$ is Hamilton-connected. \end{lemma} \begin{proof}
We use Theorem~\ref{t:tnw} to show that $G^0$ admits two
edge-disjoint spanning trees. Let $\mathcal P$ be a partition of
$V(G^0)$ and let $F$ be the set of edges with endvertices in
different classes of $\mathcal P$. By Lemma~\ref{l:g0}(i), $G^0$ is
3-edge-connected and thus $\size F \geq 3\size\mathcal P/2$. Since
$\size\mathcal P\leq 5$, we have $\lceil 3\size\mathcal P/2\rceil \geq
2(\size\mathcal P-1)$. The lemma follows from Theorem~\ref{t:tnw} and
Lemma~\ref{l:trees}. \end{proof} For the purposes of this paper, Lemma~\ref{l:small} enables us to restrict ourselves to the case that $G^0$ has at least 6 vertices.
We recall a well-known necessary and sufficient condition for the line graph $L(G)$ to be Hamilton-connected. Let $e_1,e_2$ be edges of $G$. A trail $T$ in $G$ is an \emph{$(e_1,e_2)$-trail} if its first edge is $e_1$ and its last edge is $e_2$. The trail $T$ is \emph{internally dominating} if every edge of $G$ is incident with an internal vertex of $T$. Similarly, $T$ is \emph{internally spanning} if every vertex of $G$ appears as an internal vertex of $T$. The following is a folklore analogue of Harary and Nash-Williams' characterization of hamiltonian line graphs (cf.~\cite[Theorem~1.5]{LLZ-eulerian}).
\begin{theorem}\label{t:ham-conn-preimage}
Let $G$ be a graph with at least 3 edges. Then $L(G)$ is
Hamilton-connected if and only if for every pair of edges $e_1,e_2\in
E(G)$, $G$ has an internally dominating $(e_1,e_2)$-trail. \end{theorem}
We will infer the existence of internally dominating trails in $G$ using hypergraphs obtained by a small modification of $H^0$ (the 3-hypergraph associated with $G$ as in Section~\ref{sec:hyper}). First, we define a 3-hypergraph $H^e$ and a pair of vertices $a_1,a_2$ of $H^e$. The edges $e_i$ and the vertices $a_i$ ($i=1,2$) will be considered fixed throughout the paper.
For $i=1,2$, let $a_i$ be a permanent vertex of $e_i$ (which exists by Lemma~\ref{l:permanent}). If possible, we choose $a_1$ and $a_2$ so as to be distinct. The hypergraph $H^e$ is obtained from $H^0$ by detaching $k(e_1)$ from $a_1$ and, subsequently, detaching $k(e_2)$ from $a_2$. We also define $G^e$ as the graph $G(H^e)$ corresponding to $H^e$. Note that every permanent vertex of $G$ is a vertex of $G^e$.
We say that a trail $T$ in a graph $G'$ \emph{spans} a set $X\subseteq V(G')$ if $X\subseteq V(T)$. If the first and last vertices of $T$ are $a$ and $b$, respectively, we say that $T$ is an \emph{$ab$-trail}. The case $a=b$ is allowed in this definition.
The following lemma provides a bridge between spanning $a_1a_2$-trails in $G(H^e)$ and internally dominating $(e_1,e_2)$-trails in $G$:
\begin{lemma}\label{l:join}
If $G^e$ admits an $a_1a_2$-trail spanning $V(H^e)$, then $G$
contains an internally dominating $(e_1,e_2)$-trail. \end{lemma} \begin{proof}
Let $T^e$ be an $a_1a_2$-trail in $G^e$ spanning $V(H^e)$. Let $T$
be the corresponding $a_1a_2$-trail in $G$ (that is, whenever $T^e$
uses an edge $f$ obtained by suppressing a vertex $w$, $T$ uses the
two edges incident with $w$). Since $a_i$ is an endvertex of $e_i$
($i=1,2$), we can construct an $(e_1,e_2)$-trail $T'$ in $G$ by
prepending $e_1$ and appending $e_2$ to $T$. Since $T$ spans all
permanent vertices and every edge of $G$ has a permanent endvertex,
$T'$ is internally dominating. \end{proof}
In view of Lemma~\ref{l:join}, proving Theorem~\ref{t:main} reduces to finding an $a_1a_2$-trail spanning $V(H^e)$ in $G^e$ for each choice of $e_1,e_2$. A basic tool for this is Proposition~\ref{p:qt-join} in Section~\ref{sec:quasi}.
\section{Quasigraphs} \label{sec:quasi}
Our proof relies on a strengthening of the so-called Skeletal Lemma~\cite[Lemma 17]{KV-hamilton}. The required stronger version is proved in~\cite{KV-quasi}. The formulation and proof of this result use the language and some theory of quasigraphs; in this section, we recall just the bare minimum allowing us to state Theorem~\ref{t:skeletal}.
Recall that a \emph{3-hypergraph} is a hypergraph whose edges have size 2 or 3. Let $H$ be a 3-hypergraph.
A \emph{quasigraph} in $H$ is a mapping $\pi$ that assigns to each hyperedge $e$ of $H$ either a subset of $e$ of size 2, or the empty set. The hyperedges $e$ with $\pi(e)\neq \emptyset$ are said to be \emph{used} by $\pi$.
Let $\pi$ be a quasigraph in $H$. We let $\pi^*$ denote the graph on $V(H)$, obtained by considering the pairs $\pi(e)$ ($e\in E(H)$) as edges whenever $\pi(e)\neq\emptyset$. If $\pi^*$ is a forest, then $\pi$ is said to be \emph{acyclic}. If $\pi^*$ is the union of a circuit and a set of isolated vertices, then $\pi$ is a \emph{quasicycle}. The hypergraph $H$ is \emph{acyclic} if there exists no quasicycle in $H$.
Let $X\subseteq V(H)$. We say that the quasigraph $\pi$ is \emph{connected} on $X$ if the induced subgraph of $\pi^*$ on $X$ is connected. A somewhat more involved notion is anticonnectedness: we say that $\pi$ is \emph{anticonnected} on $X$ if for each nontrivial partition $\mathcal R$ of $X$, there is a hyperedge $f$ such that $f$ intersects at least two classes of $\mathcal R$ and $\pi(f)$ is contained in one of them.
Let $\mathcal P$ be a partition of $V(H)$. If $e\in E(H)$, then $e/\mathcal P$ is defined as the set of all classes of $\mathcal P$ intersected by $e$. If there is more than one such class, then $e$ is said to be \emph{$\mathcal P$-crossing}. The hypergraph $H/\mathcal P$ has vertex set $\mathcal P$ and its hyperedges are all the sets of the form $e/\mathcal P$, where $e$ is a $\mathcal P$-crossing hyperedge of $H$. A quasigraph $\pi/\mathcal P$ in this hypergraph is defined by setting, for every $\mathcal P$-crossing hyperedge $e$ of $H$, \begin{equation*}
(\pi/\mathcal P)(e/\mathcal P) =
\begin{cases}
\pi(e)/\mathcal P & \text{if $\pi(e)$ is $\mathcal P$-crossing,}\\
\emptyset & \text{otherwise.}
\end{cases} \end{equation*}
The \emph{complement} $\overline\pi$ of $\pi$ is the subhypergraph of $H$ (on the same vertex set) consisting of the hyperedges not used by $\pi$.
A partition $\pi$ of $V(H)$ is \emph{$\pi$-skeletal} if both of the following conditions hold: \begin{enumerate}[(1)]
\item for each $X\in\mathcal P$, $\pi$ is both connected on $X$ and
anticonnected on $X$,
\item the complement of $\pi/\mathcal P$ in $H/\mathcal P$ is acyclic. \end{enumerate}
For our purposes, the version of the Skeletal Lemma stated below in Theorem~\ref{t:skeletal} needs to take care of a particular configuration we call `bad leaf' since it presents a problem in our computations. Let us describe this configuration.
Recall that $\pi$ is a quasigraph in a 3-hypergraph $H$. Assume now that $\pi$ is acyclic. In each component of the graph $\pi^*$, we choose an arbitrary root and orient all the edges of $\pi^*$ toward the root. A hyperedge $e$ of $H$ is \emph{associated with} a vertex $u$ if it is used by $\pi$ and $u$ is the tail of $\pi(e)$ in the resulting oriented graph. Thus, every vertex has at most one associated hyperedge, and conversely, each hyperedge is associated with at most one vertex.
A vertex $u$ of $H$ is a \emph{bad leaf} for $\pi$ (and the given choice of the roots of the components of $\pi^*$) if all of the following hold: \begin{enumerate}[\quad(i)] \item $u$ is a leaf of $\pi^*$, \item $u$ is incident with exactly three hyperedges, exactly one of
which has size 3 (say, $e$), and \item $e$ is associated with $u$. \end{enumerate}
\begin{figure}
\caption{A bad leaf $u$.}
\label{fig:bad}
\end{figure}
Bad leaves can be eliminated at the cost of performing certain local modifications in the hypergraph. More precisely, if $u$ is a vertex of the $3$-hypergraph $H$ incident with exactly two hyperedges of size 2 and exactly one hyperedge of size 3, then a \emph{switch at $u$} is the operation depicted in Figure~\ref{fig:switch}. (We remark that in~\cite{KV-quasi}, the switch operation acts on quasigraphs in $H$ as well, but this is not necessary for our purposes.)
We say that a $3$-hypergraph $\tilde H$ is \emph{related} to $H$ if it can be obtained from $H$ by a finite sequence of switches at suitable vertices. Note that, in this case, $G(\tilde H)$ is isomorphic to $G(H)$.
\begin{figure}
\caption{(a) A 3-hypergraph $H$ with a vertex $u$ suitable for a
switch. (b) The hyperedges incident with $u$ in the 3-hypergraph
resulting from the switch. The other hyperedges are not modified.}
\label{fig:switch}
\end{figure}
We can now finally state the main technical result mentioned above, a strengthening of the Skeletal Lemma proved in~\cite{KV-quasi}:
\begin{theorem}
\label{t:skeletal}
Let $H$ be a 3-hypergraph. There exists a hypergraph $\tilde H$
related to $H$ and an acyclic quasigraph $\sigma$ in $\tilde H$ such
that $\sigma$ has no bad leaves (for any choice of the roots of the
components of $\sigma^*$) and $V(\tilde H)$ admits a
$\sigma$-skeletal partition $\mathcal S$. \end{theorem}
Theorem~\ref{t:skeletal} will be used in conjunction with the following result, implied by a special case of Lemma 28 in~\cite{KV-hamilton}: \begin{proposition}\label{p:qt-join}
Let $H$ be a 3-hypergraph and $b_1,b_2$ vertices of $H$. If $H$
admits an acyclic quasigraph that is both connected and
anticonnected on $V(H)$, then the graph $G(H)$ contains a
$b_1b_2$-trail spanning $V(H)$. \end{proposition} Proposition~\ref{p:qt-join} will be useful in Section~\ref{sec:final}, where we infer that the assumption of the proposition is satisfied, which will enable us to apply Lemma~\ref{l:join}.
\section{Counting the hyperedges} \label{sec:counting}
Recall that $G$ is a graph satisfying the assumptions of Theorem~\ref{t:main}, and that $H^0$ is a 3-hypergraph associated with the core $G^0$ of $G$. Additionally, $e_1,e_2$ are fixed edges of $G$, $a_1,a_2$ are their permanent endvertices, and $H^e$ is a modification of $H^0$ defined in Section~\ref{sec:hamilt-conn}.
By Theorem~\ref{t:skeletal}, there is a 3-hypergraph $\tilde H$ related to $H^e$ and an acyclic quasigraph $\sigma$ in $\tilde H$ with no bad leaves such that $V(\tilde H)$ admits a $\sigma$-skeletal partition $\mathcal S$. Our ultimate use of $\sigma$ is to find a connected $X(e_1,e_2)$-join in the graph $G(H^e)$ to be used in Lemma~\ref{l:join}. Since $G(\tilde H)$ is isomorphic to $G(H^e)$ if $\tilde H$ and $H^e$ are related, we may assume without loss of generality that $\tilde H = H^e$.
As we will see in Section~\ref{sec:final}, the proof of Theorem~\ref{t:main} is simple in the case that $\size\mathcal S = 1$. In the following calculations, we therefore assume $\mathcal S \geq 2$.
Recall that for a $3$-hypergraph $H$, an edge $e$ of $H$ and a partition $\mathcal P$ of $V(H)$, the notation $e/\mathcal P$ and $H/\mathcal P$ has been defined in Section~\ref{sec:quasi}.
Let us write $d^0(P)$ for the degree of a vertex $P\in\mathcal S$ in the hypergraph $H^0/\mathcal S$.
We proceed as in Section 8 of~\cite{KV-hamilton}. We set $\tau = \sigma/\mathcal S$. Let $n = \size\mathcal S$ and let $m$ denote the number of hyperedges of $H^e/\mathcal S$. For $k\in\Setx{2,3}$, let $m_k$ be the number of $k$-hyperedges of $H^e/\mathcal S$ used by $\sigma$, and let $\overline m_k$ denote the number of $k$-hyperedges of $\overline\tau$. Since $\mathcal S$ is $\sigma$-solid, the graph $\tau^*$ is acyclic. It has $n$ vertices and $m_2 + m_3$ edges, and hence \begin{equation}
\label{eq:1}
m_2 + m_3 \leq n - 1. \end{equation} Similarly, the complement $\overline\tau$ is an acyclic hypergraph. Consider the graph $G(\overline\tau)$, defined in Section~\ref{sec:introduction}. Since $\overline\tau$ is acyclic, so is $G(\overline\tau)$. As $G(\overline\tau)$ has $n+\overline m_3$ vertices and $\overline m_2 + 3\overline m_3$ edges, we get \begin{equation}
\label{eq:2}
\overline m_2 + 2\overline m_3 \leq n - 1. \end{equation} Moreover, by the assumption, either $\tau^*$ or $G(\overline\tau)$ is disconnected and therefore it has at most $n-2$ edges. Adding \eqref{eq:1} to \eqref{eq:2} and using this fact, we find \begin{equation}
\label{eq:3}
m \leq 2n-3-\overline m_3. \end{equation}
For any hypergraph $H$, let $s(H)$ be the sum of vertex degrees in $H$. Let us set \begin{equation*}
\varepsilon = s(H^0/\mathcal S) - s(H^e/\mathcal S) \end{equation*} and observe that by the definition of the hypergraph $H^e$, $\varepsilon \leq 4$. Furthermore, we have \begin{align*}
s(H^e/\mathcal S) &= 2(m_2 + \overline m_2) + 3(m_3 + \overline
m_3) = 2m + (m_3 + \overline m_3) \\
&\leq 4n - 6 + m_3 - \overline m_3, \end{align*} where the last inequality follows by using \eqref{eq:3} to substitute for $m$. Substituting $s(H^0/\mathcal S) - \varepsilon$ for $s(H^e/\mathcal S)$, we obtain \begin{equation}
\label{eq:s-h0}
s(H^0/\mathcal S) \leq 4n-6 + (m_3 - \overline m_3) + \varepsilon. \end{equation}
The quasigraph $\tau$ in $H^e/\mathcal S$ determines an (acyclic) quasigraph $\tau^0$ in $H^0/\mathcal S$ in a natural way. Let $m^0_3$ be the number of 3-hyperedges of $H^0/\mathcal S$ used by $\tau^0$ and observe that $m^0_3 \geq m_3$. Furthermore, let $\widetilde m^0_i$ ($i\in\Setx{2,3}$) be the number of $i$-hyperedges of $H^0/\mathcal S$, whether used by $\tau^0$ or not.
Inequality~\eqref{eq:s-h0} has two corollaries. Firstly, using the fact that $\varepsilon \leq 4$ and $m^0_3 \geq m_3$, and ignoring the $\overline m_3$ term, we find \begin{equation}
\label{eq:main}
\sum_{P\in V(H^0/\mathcal S)}(d^0(P) - 4) - m^0_3 \leq -2. \end{equation}
For the second corollary, note that since $s(H^0/\mathcal S) = 2\widetilde m^0_2 + 3\widetilde m^0_3$ and $m_3 - \overline m_3 \leq \widetilde m^0_3$, \eqref{eq:s-h0} implies \begin{equation}
\label{eq:small}
\widetilde m^0_2 + \widetilde m^0_3 \leq 2n-3 + \frac\varepsilon 2. \end{equation}
For later use, we record an observation concerning classes $X$ of $\mathcal S$ such that $\size{X^+} \geq 1$; let us call such classes \emph{nontrivial}. \begin{observation}
\label{obs:nontriv}
If $X$ is a nontrivial class of $\mathcal S$, then $G^0[X^+]$ is not a
matching. \end{observation} \begin{proof}
Suppose that $X\in\mathcal S$ is nontrivial. We prove that $G^0[X^+]$ has
at least two incident edges. If $\size X > 1$, then this follows
from the fact that $\sigma$ is both connected and anticonnected on
$X$ by the choice of $\mathcal S$. On the other hand, if $\size X = 1$,
then some vertex of $X^+-X$ is incident with at least two edges to
$X$ and the assertion also holds. \end{proof}
\section{Structural observations} \label{sec:structural}
We continue to use the notation and assumptions of Section~\ref{sec:counting}. The objective of this and the following section is to rule out most cases in the proof of Theorem~\ref{t:main} by establishing the following: \begin{proposition}\label{p:S4}
If $G^0$ has at least 6 vertices, then the partition $\mathcal S$ has at
most 4 classes. \end{proposition}
For the sake of a contradiction, let us assume that $\size\mathcal S \geq 5$. We now prove several claims concerning the structure of the hypergraph $H^0/\mathcal S$ in this case.
\begin{lemma}\label{l:path}
Suppose that $G^0$ has at least 6 vertices. Then the following hold:
\begin{enumerate}[\quad(i)]
\item $G^0$ contains no path of length 2 with two vertices of degree
3 and one vertex of degree at most 4,
\item no permanent vertex of degree 3 in $G^0$ is adjacent to a
permanent vertex of degree at most 4.
\end{enumerate} \end{lemma} \begin{proof}
(i) We prove that if $x_1x_2x_3$ is a path in $G^0$, then
$\sum_{i=1}^3 d_{G^0}(x_i) \geq 11$. Suppose the contrary. Define
$X=\Setx{x_1,x_2,x_3}$. Then $d_{G^0}(X) \leq 6$. Since $G^0-X$ must
be a matching on at least 3 vertices, and $G^0$ has minimum degree
at least 3, we have $d_{G^0}(X) \geq 7$. This is a contradiction.
(ii) Let $x,y$ be permanent vertices of $G^0$ such that $d_{G^0}(x)
= 3$ and $d_{G^0}(y) \leq 4$. By part (i), the vertex $x$ has no
temporary neighbour. Therefore, $x$ must have a transient neighbour
in $G$. Since $G$ is 2-essentially 9-edge-connected,
$G^0-\Setx{x,y}$ must be a matching and we obtain a similar
contradiction as in the proof of (i). \end{proof}
\begin{figure}
\caption{Forbidden configurations in $H^0/\mathcal S$. Figure (a)
illustrates part (i) of Lemma~\ref{l:forb}, (b) and (c) relate to
part (ii). The gray regions represent the classes $P$ and $Q$ of
$\mathcal S$, dashed lines represent optional hyperedges. The dotted
line in figure (a) means that the hyperedge can be of size 2 or
3.}
\label{fig:forb}
\end{figure}
\begin{lemma}\label{l:forb}
Let $P,Q$ be neighbouring vertices of the hypergraph $H^0/\mathcal S$. If
$\size{\mathcal S} \geq 5$ and $G^0$ has at least 6 vertices, then the
following holds:
\begin{enumerate}[\quad(i)]
\item if $d_{H^0/\mathcal S}(P) = 3$, then $d_{H^0/\mathcal S}(Q) \geq 7$,
\item if $d_{H^0/\mathcal S}(P) = 4$ and $P$ is incident with a 3-hyperedge of
$H^0/\mathcal S$, then $d_{H^0/\mathcal S}(Q) \geq 6$.
\end{enumerate} \end{lemma} \begin{proof}
We prove (i). Since $G^0$ is essentially 4-edge-connected, the class
$P$ is trivial by Observation~\ref{obs:nontriv}; say, $P =
\Setx{u}$. Being a permanent vertex of degree 3 in $G^0$, the vertex
$u$ is either protected, or adjacent to a temporary vertex.
We set
\begin{equation*}
X =
\begin{cases}
(P\cup Q)^+ \cup \Setx{z} & \text{if $u$ is adjacent to a
temporary vertex $z$,} \\
(P\cup Q)^+ & \text{otherwise}.
\end{cases}
\end{equation*}
Suppose that $u$ is adjacent to a temporary vertex $z$. By
Lemma~\ref{l:path}(i), $u$ is not adjacent to any other temporary
vertex, which implies
\begin{equation}
\label{eq:8}
d_{G^0}(X) \leq 8.
\end{equation}
Since $G^0[X]$ is not a matching, $G^0-X$ must be a matching as
$G^0$ is 2-essentially 9-edge-connected.
A similar argument shows that $G^0-X$ is a matching just as well if
$u$ is protected. In particular, in either case, no temporary vertex
of $G^0$ has two neighbours outside $X$.
Enumerate the classes of $\mathcal S$ other than $P$ and $Q$ as
$Y_1,\dots,Y_k$. By Observation~\ref{obs:nontriv}, each $Y_i$ is a
trivial class, say $Y_i = \Setx{y_i}$.
Since $k\geq 3$ and $d_{G^0}(y_i) \geq 3$ for $1\leq i \leq k$,
inequality~\eqref{eq:8} implies that $G^0$ contains an edge joining
two of the vertices $y_i$ --- say, $y_1$ and $y_2$. Since
$d_{G^0}(\Setx{y_2,\dots,y_k}) \geq 3$, we have
$d_{G^0}(\Setx{y_1,y_2})\leq 5$, so without loss of generality,
$d_{G^0}(y_1) = 3$ and $d_{G^0}(y_2) \leq 4$. This is a
contradiction with Lemma~\ref{l:path}(ii).
Part (ii) can be proved using a minor modification of the above
argument, which we leave to the reader. \end{proof}
\section{Discharging} \label{sec:discharging}
We continue the discussion of Section~\ref{sec:structural} by using a discharging-type argument to prove Proposition~\ref{p:S4}. The discharging process takes place in the hypergraph $H^0/\mathcal S$. Recall our hypothesis that the partition $\mathcal S$ has at least 5 classes.
In Section~\ref{sec:counting}, we defined $\tau^0$ as the quasigraph in $H^0/\mathcal S$ corresponding to the quasigraph $\tau = \sigma/\mathcal S$ in $H^e/\mathcal S$. The notion of a hyperedge associated with a vertex will be carried over from $H^e$ to $H^0/\mathcal S$: by definition, a hyperedge $e/\mathcal S$ of $H^0/\mathcal S$ is associated with $P\in\mathcal S$ if the corresponding hyperedge of $H^e$ is associated with a vertex of $H^e$ contained in $P$. Note that the definition makes sense thanks to the fact that $\sigma[P]$ is a quasitree (and hence $\sigma[P]^*$ is connected).
We write $V_i = V_i(H^0/\mathcal S)$. Furthermore, $V_i^\triangle$ denotes the subset of $V_i$ consisting of vertices which have an associated 3-hyperedge and $V^\triangle$ is the union of all $V_i^\triangle$. Given a vertex $P$ of $H^0/\mathcal S$, the symbol $\mnbr P$ denotes the multiset consisting of vertices $Q$ such that $P$ and $Q$ are contained in a hyperedge of $H^0/\mathcal S$, with one occurrence of $Q$ for each such hyperedge.
We begin by assigning charges to the vertices and hyperedges of $H^0/\mathcal S$, guided by inequality~\eqref{eq:main}: \begin{itemize} \item each vertex $P$ will get a charge of $d^0(P) - 4$ units, \item a 3-hyperedge of $H^0/\mathcal S$ will get a charge of $-1$ if it is
used by $\tau^0$ and a zero charge otherwise, \item 2-hyperedges of $H^0/\mathcal S$ get zero charge. \end{itemize} By~\eqref{eq:main}, the total charge is negative. At the same time, the only elements of $H^0/\mathcal S$ with negative charge are 3-vertices and 3-hyperedges used by $\tau^0$.
As usual in discharging arguments, we will describe rules for the redistribution of charge which keep the total charge unchanged and (given the assumptions about the graph $G$) make all the individual charges non-negative. This contradiction will show that $\sigma$ is actually a quasitree with connected complement.
\begin{figure}
\caption{The rules for the redistribution of charge. Thick lines
represent the quasigraph $\tau^0$, with the association of a
hyperedge to a vertex shown by thick arrows. Gray arrows indicate
the flow of the stated amount of charge. The degree of $P$ is not
represented.}
\label{fig:rules}
\end{figure}
Charge will only be sent by vertices, the recipient may be either a vertex or a hyperedge. Let $P$ be a vertex of $H^0/\mathcal S$ (that is, $P\in\mathcal S$). The rules (more than one of which may apply) are in the following list. See Figure~\ref{fig:rules} for a schematic representation. \begin{enumerate}[\quad(D1)] \item if $P\in V^\triangle$, then $P$ sends its associated hyperedge 1
unit of charge, \item if $P$ has an associated 2-hyperedge whose head is a degree 3
vertex $Q$, then $P$ sends 1 unit of charge to $Q$, \item if $P$ has a neighbour $Q$ in $V_4^\triangle$, then $P$ sends
$Q$ a charge of $1/5$ for each common hyperedge, \item if $P$ has a degree 3 neighbour $Q$, then $P$ sends $Q$ a charge
of $1/3$ for each common hyperedge which is not associated with $P$. \end{enumerate}
We claim that after the redistribution of charge, all vertices and hyperedges of $H^0/\mathcal S$ will have nonnegative charge. This is clear for 2-hyperedges and for 3-hyperedges not used by $\tau^0$. Furthermore, each 3-hyperedge used by $\tau^0$ will obtain 1 unit of charge by rule (D1), making the resulting charge zero.
Let us therefore investigate the ways a vertex $P$ may be discharged. Suppose first that the degree $d^0(P)$ of $P$ is at least 7. Since at most one unit of charge is transferred from $P$ based on rules (D1) and (D2), the total transfer from $P$ is at most \begin{equation*}
1 + (d^0(P) - 1)\cdot\frac13 \leq d^0(P)-4, \end{equation*} where the inequality follows from the assumption that $d^0(P) \geq 7$. Since the right hand side is the original charge of the vertex, the resulting charge is nonnegative.
We may thus assume that $d^0(P) \leq 6$. By Lemma~\ref{l:forb}(i), no neighbour $P'$ of $P$ in $H^0/\mathcal S$ has $d^0(P') = 3$. Suppose that $d^0(P) = 6$. If $P$ is discharged according to rule (D1) or (D2), then rule (D4) does not apply to $P$, and the transfer from $P$ is at most $1 + 5 \cdot 1/5 = 2$, the initial charge of $P$. On the other hand, if none of (D1) and (D2) apply, then $P$ sends at most $6 \cdot 1/3 = 2$ units of charge as well.
If $d^0(v) = 5$, then $P$ has no neighbour in $V_4^\triangle$ (Lemma~\ref{l:forb}(ii)), which rules out the use of (D3) for the discharging of $P$. Furthermore, the applicability of (D1), (D2) and (D4) is mutually exclusive. This means that $P$ only sends at most a charge of 1 unit, which equals its initial charge.
We are left with the case that $d^0(P) \leq 4$. Suppose that $d^0(P) = 4$ (so its initial charge is zero). Lemma~\ref{l:forb} implies that no neighbour of $P$ in $H^0/\mathcal S$ is contained in $V_3\cup V_4^\triangle$. Thus, if $P$ sends any charge at all, it must be according to rule (D1). In this case, $P\in V_4^\triangle$ and according to rule (D3), $P$ receives a charge of $1/5$ from each of the five vertices in $\mnbr P$, so its resulting charge is $-1 + 5 \cdot 1/5 = 0$.
It remains to consider the case that $d^0(P) = 3$. By Lemma~\ref{l:g0}(ii), $P$ contains a single vertex $v$ of $H^0$. By the property (Q2) of $\sigma$, $v$ is not a leaf of $\sigma^*$ in $H^e$, and therefore $P$ is not a leaf of $(\tau^0)^*$ in $H^0/\mathcal S$. Let $e$ be an edge of $(\tau^0)^*$ with $P$ as its head, and let the tail of $e$ be denoted by $t$.
Similarly to the discussion in the preceding cases, $P$ is not discharged according to any of the rules (D2), (D3) or (D4). We distinguish two cases.
If $P\notin V_3^\triangle$, then it does not send any charge to its neighbours or incident hyperedges. By Lemma~\ref{l:forb}, at most one of the four vertices in $\mnbr P$ has an associated 3-hyperedge $f$ such that the head of $\tau^0(f)$ is $P$. Each of the remaining vertices sends either $1/3$ or $1$ unit of charge to $P$ (according to rule (D4) or (D2), respectively), which accounts for a resulting charge of at least $-1 + 3 \cdot 1/3 = 0$.
If $P\in V_3^\triangle$, then $P$ is discharged according to (D1), which decreases its charge from $-1$ to $-2$. On the other hand, Lemma~\ref{l:forb} implies that $e$ is a 2-hyperedge, so $t$ sends one unit of charge to $P$ according to rule (D2). Furthermore, as above, $P$ gets at least a charge of $1/3$ from each of the three remaining vertices in $\mnbr P$. Hence, the new charge is nonnegative again. This concludes the analysis. The contradiction establishes Proposition~\ref{p:S4}.
\section{Completing the proof of Theorem~\ref{t:main}} \label{sec:final}
In this section, we prove Theorem~\ref{t:main}. Before doing so, we narrow down the set of possible cases by proving that the size of the partition $\mathcal S$ is not greater than 2.
Recall from Section~\ref{sec:counting} the notation $\widetilde m^0_i$ for $\size{E_i(H^0/\mathcal S)}$ (where $i\in\Setx{2,3}$) and inequality~\eqref{eq:small}: \begin{equation*}
\widetilde m^0_2 + \widetilde m^0_3 \leq 2n-3 + \varepsilon/2, \end{equation*} where $n=\size{\mathcal S}$ and $\varepsilon$ is the difference of the sum of vertex degrees in $H^0/\mathcal S$ and in $H^e/\mathcal S$. Since $\varepsilon\leq 4$ and, by Proposition~\ref{p:S4}, $n \leq 4$, we have \begin{equation}
\label{eq:few}
H^0/\mathcal S \text{ has at most 7 hyperedges}. \end{equation}
Before stating the next proposition, we recall that nontrivial class of $\mathcal S$ is a class $X$ such that $\size{X^+} \geq 1$. \begin{proposition}\label{p:S2}
If $G^0$ has at least 6 vertices, then $n\leq 2$. Moreover, if $n =
2$, then the following hold:
\begin{enumerate}[\quad(i)]
\item one of the classes of $\mathcal S$ is a trivial class $\Setx x$,
\item the degree of $x$ in $H^0$ is 3,
\item $x$ is incident with the hyperedges $k(e_1)$ and $k(e_2)$ of
$H^0$,
\item the size of $k(e_1)$ and $k(e_2)$ is 2.
\end{enumerate} \end{proposition} \begin{proof}
The proof consists of a series of claims.
\setcounter{claim}{0}
\begin{claim}
The partition $\mathcal S$ has at most one nontrivial class.
\end{claim}
\begin{claimproof}
By Observation~\ref{obs:nontriv}, if $\mathcal S$ has two nontrivial
classes $X,Y$, \eqref{eq:few} implies that $\bd{G^0}{X^+}$ is a
2-essential edge-cut in $G^0$ of size at most 7.
\end{claimproof}
\begin{claim}
If $\mathcal S$ has a nontrivial class, then $H^0/\mathcal S$ contains no
3-hyperedge.
\end{claim}
\begin{claimproof}
Let $X\in\mathcal S$ be nontrivial and let $e/\mathcal S$ be a 3-hyperedge of
$H^0/\mathcal S$ (where $e$ is a hyperedge of $H^0$). We can choose two
vertices of $e/\mathcal S$ (say $Y_1,Y_2$) distinct from $X$. For
$i=1,2$, let $y_i$ be the vertex of $e$ in $Y_i$, and let $w$ be
the vertex of $G^0$ such that $e = h(w)$. Then the edge-cut
$\bd{G_0}{X^+}$ of size at most 7 separates $X^+$ from the path
$y_1wy_2$ in $G^0$ and is therefore 2-essential, a contradiction.
\end{claimproof}
\begin{claim}
The partition $\mathcal S$ has exactly one nontrivial class.
\end{claim}
\begin{claimproof}
For contradiction, suppose that all the classes of $\mathcal S$ are
trivial. Recall the parameter $\widetilde m^0_3$, introduced above
equation~\eqref{eq:main}. Since we assume that $G^0$ has at least
6 vertices, we must have $\widetilde m^0_3 \geq 2$. We let the
vertices of $H^0$ be denoted by $x,y,z$ or $x,y,z,u$ depending on
whether $n$ equals 3 or 4. Furthermore, the 3-hyperedges of $H^0$
are denoted by $h(w_i)$, where $1\leq i\leq \widetilde m^0_3$ and
all the $w_i$'s are temporary vertices of $G^0$.
If $H^0$ contains two 3-hyperedges intersecting in exactly two
vertices (say, $e=xyz$ and $f=xyu$), then the edge-cut
$\bd{G^0}{\Setx{x,z}^+}$ is a 2-essential edge-cut of size at most
7, a contradiction. Thus, all the 3-hyperedges of $H^0$ contain the
same triple of vertices, say $\Setx{x,y,z}$.
Suppose now that $\widetilde m^0_3 \geq 3$. By~\eqref{eq:few}, we may assume
that $x$ is incident with only at most three 2-hyperedges of
$H^0$. But then the size of the 2-essential edge-cut
$\bd{G^0}{\Setx{x,w_1,w_2}^+}$ in $G^0$ is at most $8$, a
contradiction.
We conclude that $\widetilde m^0_3 = 2$, which implies $n=4$. If $uz$ is a
2-hyperedge in $H^0$, then the edge-cut $\bd{G^0}{\Setx{x,w_1,y}^+}$
in $G^0$ separates the paths $xw_1y$ and $uzw_2$ and is therefore
2-essential. In addition, its size is at most 8. This contradiction
concludes the proof of the claim.
\end{claimproof}
\begin{claim}\label{cl:two-classes}
The partition $\mathcal S$ has at most two classes.
\end{claim}
\begin{claimproof}
Suppose that $n > 2$. By the above claims, $\mathcal S$ has a single
nontrivial class $P$ and all the hyperedges of $H^0$ are of size
2. Let the vertices of $H^0$ comprising the nontrivial parts of
$\mathcal S$ be denoted by $x,y$ or $x,y,z$ depending on $n$.
Suppose first that $n=4$. Since the degree of each of the vertices
$x,y,z$ in $H^0$ is at least 3, \eqref{eq:few} implies that at least
two 2-hyperedges have both endvertices in the set
$\Setx{x,y,z}$. Consequently, the edge-cut $\bd{G^0}{P^+}$ of size
at most 7 is 2-essential.
We infer that $n=3$. By a similar argument, the vertices $x$ and $y$
must be adjacent vertices of degree 3 in $G^0$. Since they are
permanent and there is no temporary vertex, they must be
protected. However, if $x$ is adjacent to a transient vertex, then
it is easy to show that $\bd{G^0}{P^+}$ is a 2-essential 4-edge-cut
in $G^0$.
\end{claimproof}
To finish the proof of the proposition, it remains to establish
properties (ii)--(iv). Let us write $\mathcal S = \Setx{P,\Setx x}$, where
$P$ is the nontrivial class. By inequality~\eqref{eq:small} and the
fact that $G^0$ is 3-edge-connected, $x$ has degree 3 in $H^0$ and
$\varepsilon=4$. The latter fact means that $k(e_1)$ and $k(e_2)$ are
2-hyperedges of $H^0$ incident with $x$. The proof is complete. \end{proof}
Having established Proposition~\ref{p:S2}, we can now prove the main result of this paper. \begin{proof}[Proof of Theorem~\ref{t:main}]
If the graph $G^0$ has at most 5 vertices, then $L(G)$ is
Hamilton-connected by Lemma~\ref{l:small}. Assume thus that
$\size{V(G^0)} \geq 6$. By Proposition~\ref{p:S2}, we have $n \leq
2$.
If $n=1$, then by the choice of $\mathcal S$, $\sigma$ is an acyclic
quasigraph in $H^e$ that is both connected and anticonnected on
$V(H^e)$. Proposition~\ref{p:qt-join} implies that $G(H^e)$ admits
an $a_1a_2$-trail spanning $V(H^e)$. By Lemma~\ref{l:join}, it
follows that $G$ admits an internally dominating
$(e_1,e_2)$-trail. Since the choice of $e_1$ and $e_2$ is arbitrary,
$L(G)$ is Hamilton-connected by Theorem~\ref{t:ham-conn-preimage}.
The discussion in the case that $n=2$ is only slightly more
complicated. By Proposition~\ref{p:S2}, $\mathcal S$ contains a trivial
class $\Setx{x}$, $x$ has degree 3 in $H^0$ and is incident in $H^0$
with the 2-hyperedges $k(e_1)$ and $k(e_2)$. Let $P$ denote the
other class of $\mathcal S$ and let $y_1$ and $y_2$ be the endvertex of
$k(e_1)$ and $k(e_2)$, respectively, in $P$. Furthermore, let $f$ be
the third 2-hyperedge of $H^0$ incident with $x$, and let $y_3$ be
its endvertex in $P$.
By the definition of the temporary vertices $a_1$ and $a_2$, we may
assume without loss of generality that $a_1 = x$ and $a_2 = y_2$. To
find an $xy_2$-trail in $G(H^e)$ spanning $V(H^e)$,
we proceed as follows.
Denoting the hypergraph obtained from $H^e$ by removing $x$ by
$H_1$, we observe that $\sigma$ determines an acyclic quasigraph in
$H_1$ that is both connected and anticonnected on $V(H_1)=P$. By
Proposition~\ref{p:qt-join}, $G(H_1)$ admits an $y_3y_2$-trail $T_1$
spanning $P$. Adding the edge $f$ to the beginning of $T_1$, we
obtain an $xy_2$-trail in $G(H^e)$ spanning $V(H^e)$ as desired. \end{proof}
\section{Claw-free graphs} \label{sec:claw}
As with Theorem~\ref{t:lai}, Theorem~\ref{t:main} can be extended to claw-free graphs. The procedure is the same as that used in \cite[Section 11]{KV-hamilton}. We use the \emph{$M$-closure} introduced in~\cite{RV-line}, namely the following result~\cite[Theorem 9]{RV-line}:
\begin{theorem}\label{t:closure}
If $G$ is a connected claw-free graph, then there is a well-defined
graph $cl^M(G)$ with the following properties:
\begin{enumerate}[\quad(i)]
\item $G$ is a spanning subgraph of $cl^M(G)$,
\item $cl^M(G)$ is the line graph of a multigraph,
\item $cl^M(G)$ is Hamilton-connected if and only if $G$ is
Hamilton-connected.
\end{enumerate} \end{theorem}
In the context of the present paper, the reference to multigraphs in Theorem~\ref{t:closure} is not necessary, since parallel edges in graphs are allowed by default.
By condition (i) in Theorem~\ref{t:closure}, the connectivity of $cl^M(G)$ is greater than or equal to that of $G$. As shown by the following observation, the closure operation does not decrease the essential connectivity either.
\begin{lemma}\label{l:ess}
If $G$ is essentially $k$-connected, then $cl^M(G)$ is also
essentially $k$-connected. \end{lemma} \begin{proof}
Clearly, $\size{V(cl^M(G))} > k$. For contradiction, let $X$ be a
minimal essential vertex-cut in $cl^M(G)$ of size less than
$k$. Since $X$ is not essential in $G$, there is a component $K$ of
$cl^M(G)-X$ such that $K$ contains some edges, but $G[V(K)]$ is
edgeless.
The operation $cl^M$, as defined in~\cite[Section~4]{RV-line},
consists of a sequence of local completions at suitable vertices
$x_1,\dots,x_\ell$. Here, the \emph{local completion} at $x_i$ is
the addition of all possible edges joining the neighbours of
$x_i$. Let $e$ be an edge of $K$ and let $Y = X \cap
\Setx{x_1,\dots,x_\ell}$. In $G$, all the vertices with a neighbour
in $V(K)$ are contained in $X$. The fact that $K$ contains at least
one edge implies that $Y\neq\emptyset$. Let us say that $x_i\in Y$,
where $1\leq i\leq\ell$. By the minimality of $X$, $x_i$ has a
neighbour (in $cl^M(G)$) in some component $L$ of $cl^M(G)-X$ other
than $K$. Although the edge between them could be added by a local
completion at some vertex $x_j$ ($1\leq j\leq\ell$), this can only
happen if some vertex of $Y$ has a neighbour in $V(L)$ prior to the
local completion. We conclude that some vertex of $Y$, say $x_k$,
has a neighbour in $V(L)$ in $G$. But then the local completion at
$x_k$ adds an edge between a neighbour of $x_k$ in $V(K)$ and a
neighbour of $x_k$ in $V(L)$, contradicting the assumption that $K$
and $L$ are different components of $cl^M(G)-X$. \end{proof}
Using Lemma~\ref{l:ess}, we find that $cl^M(G)$ is a 3-connected, essentially 9-connected line graph. Thus, $cl^M(G)$ is Hamilton-connected by Theorem~\ref{t:main}. Condition (iii) of Theorem~\ref{t:closure} implies that $G$ is Hamilton-connected.
\section{Conclusion} \label{sec:conclusion}
We have shown that every 3-connected, essentially 9-connected claw-free graph is Hamilton-connected, and that this assertion is false with 9 replaced by 4. The obvious question is left unresolved: what is the least value of $k$ such that 3-connected, essentially $k$-connected claw-free graphs are Hamilton-connected (or hamiltonian)? This remains an interesting problem for further investigation.
\end{document}
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\begin{document}
\begin{abstract} In \cite{baker-ozel}, by using Fredholm index we developed a version of Quillen's geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In this work, by using Quinn's Transversality Theorem \cite{Quinn}, it will be shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It will be shown that the Thom isomorphism in this theory will be satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps will be proved. After we discuss the relation between this theory and classical cobordism, we describe some applications to the complex cobordism of flag varieties of loop groups and we do some calculations. \end{abstract} \maketitle \section{Preliminaries.}
\subsection{Complex Cobordism.} Complex bordism theory $MU$ was originally defined by geometric means as bordism classes of maps of stably complex manifolds. For a space $X$, $MU_q(X)$ is the set of equivalence classes of maps $M\stackrel{f}\rightarrow X$ where $M$ is a closed, stably almost complex manifold. It means that $M$ is a compact smooth manifold without boundary of dimension $q$ with $TM$ stably complex.
Two such maps $(M,f)$ and $(N,g)$ are bordant iff their topological union extends to a map $W\rightarrow X$ of a compact stably almost complex manifold $W$ of dimension $q + 1$, whose boundary is the union of $M$ and $N$. Here, the stably almost structures on $M$ and $N$ induced by the embedding into $W$ and the original ones are required to be equivalent. The Abelian group structure on $MU_* (X)$ is given by the operation of disjoint union. In \cite{thom}, Rene Thom gave the homotopy theoretic construction of this group. The details can be found in \cite{stong}.
The dual cohomology theory $MU$ called complex cobordism was a given a geometric description by D. Quillen. His detailed construction can be found in \cite{Quillen}. For a manifold $X$ of dimension $n$ an element in $MU^{n-q}(X)$ is represented by a smooth proper map $M\rightarrow X$ of a (not necessiraly compact) manifold $M$ of dimension $q$ together with an equivalence class of complex orientations. Two such maps are cobordant iff they are bordant as maps with complex orientations.
Now we recall the basic properties of multiplicative generalized cohomology theories with complex orientation.
A multiplicative cohomology theory $h$ is a functor from topological spaces to graded rings satisfying the Eilenberg-Steenrod axioms. Details can be found in \cite{Dyer} and \cite{adams}. A multiplicative cohomology theory $h$ is complex oriented if the complex vector bundles are oriented for $h$. In a complex oriented cohomology theory the Euler class and the Chern classes of complex vector bundles are defined and satisfy the usual properties. Examples of complex oriented cohomolgy theories are ordinary cohomology, complex $K$-theory, elliptic cohomology and complex cobordism.
Complex cobordism is the universal complex oriented cohomology theory and so for any such theory $h$ we have canonical map $MU\rightarrow h$.
A complex orientation of a proper map of smooth manifolds $f: M\rightarrow N$ is a factorization $$ M \stackrel{i}\rightarrow \xi \stackrel{\pi}\rightarrow N $$ where $\xi$ is a complex vector bundle and $i$ is an embedding with a stably complex normal bundle.
A compact manifold is said to be complex oriented iff the tangent bundle is stably complex.
All details about complex cobordism and multiplicative complex oriented generalized cohomology theories can be found in \cite{adams},\cite{Dold},\cite{Dyer} and \cite{Quillen}.
\subsection{Infinite Dimensional Manifolds and Pressley-Segal Stratifications} The general reference for the section is \cite{segal-pressley}.
$LG$ is the group of all smooth maps from $\cdot^1$ to compact semi-simple Lie group $G$. Its essential homogeneous spaces are infinite flag manifold $LG/T$ and based loop space $\Omega G$ where $T$ is a maximal torus of $G$. We are interested to infinite flag manifold $LG/T$. The cell complexes and the dual stratifications of $LG/T$ are analogically like as $G/T$. Also we are interested to Grassmannians $\operatorname{Gr} (H)$ of infinite dimensional complex separable Hilbert space $H$. Its stratifications can be found in \cite{segal-pressley}. \section{The Fredholm Index and Complex Cobordism of Hilbert Manifolds.} In \cite{Quillen}, Quillen gave a geometric interpretation of cobordism groups which suggests a way of defining the cobordism of separable Hilbert manifolds equipped with suitable structure. In order that such a definition be sensible, it ought to reduce to his for finite dimensional manifolds and smooth maps of manifolds and be capable of supporting reasonable calculations for important types of infinite dimensional manifolds such as homogeneous spaces of free loop groups of finite dimensional Lie groups.
\subsection{Cobordism of separable Hilbert manifolds.} By a manifold, we mean a smooth manifold modelled on a separable Hilbert space; see Lang \cite{lang} for details on infinite dimensional manifolds. The facts about Fredholm map can be found in \cite{conway}. \begin{defn}\label{zirto} Suppose that $f: X\rightarrow Y$ is a proper Fredholm map with even index at each point. Then $f$ is an \emph{admissible complex orientable map} if there is a smooth factorization $$f: X\stackrel{\tilde{f}}\rightarrow \xi \stackrel{q}\rightarrow Y,$$ where $q:\xi \rightarrow Y$ is a finite dimensional smooth complex vector bundle and $\tilde{f}$ is a smooth embedding endowed with a complex structure on its normal bundle $\nu (\tilde{f})$. \par A complex orientation for a Fredholm map $f$ of odd index is defined to be one for the map $ (f,\varepsilon): X\rightarrow Y\times \mathbb R $ given by $(f,\varepsilon) (x) = (f(x),0)$ for every $x \in X$. At $x\in X$, $\operatorname{index} (f, \varepsilon)_x = (\operatorname{index} f_x) - 1$. Also the finite dimensional complex vector bundle $\xi$ in the smooth factorization will be replaced by $\xi \times \mathbb R$. \end{defn} Suppose that $f$ is an admissible complex orientable map. Then since the map $f$ is the Fredholm and $\xi$ is a finite dimensional vector bundle, we see $\tilde{f}$ is also a Fredholm map. By the surjectivity of $q$, $$\operatorname{index} \tilde{f} = \operatorname{index} f - \operatorname{dim} \xi.$$ Before we give a notion of equivalence of such factorizations $\tilde{f}$ of $f$, we want to give some definitions. \begin{defn} Let $X$, $Y$ be the smooth separable Hilbert manifolds and $F: X \times \mathbb R \rightarrow Y$ a smooth map. Then we will say that $F$ is an \emph{isotopy} if it satisfies the following conditions. \begin{enumerate} \item For every $t \in \mathbb R$, the map $F_t$ given by $F_t (x) = F(x, t)$ is an embedding. \item There exist numbers $t_0 < t_1$ such that $F_t = F_{t_0}$ for all $t \leqslant t_0$ and $F_t = F_{t_1}$ for all $t \geqslant t_1$. \end{enumerate} The closed interval $[t_0, t_1]$ is called a \emph{proper domain} for the isotopy. We say that two embeddings $f: X \rightarrow Y$ and $g: X \rightarrow Y$ are \emph{isotopic} if there exists an isotopy $F_t: X\times \mathbb R \rightarrow Y$ with proper domain $[t_0, t_1]$ such that $f = F_{t_0}$ and $g = F_{t_1}$. \end{defn} \begin{prop}(see \cite{lang})\label{lang} The relation of isotopy between smooth embeddings is an equivalence relation. \end{prop} \begin{defn} Two factorizations $f: X\stackrel{\tilde{f}}\rightarrow \xi \stackrel{q}\rightarrow Y$ and $f: X\stackrel{\tilde{f'}}\rightarrow \xi' \stackrel{q'}\rightarrow Y$ are \emph{equivalent} if $\xi$ and $\xi'$ can be embedded as subvector bundles of a vector bundle $\xi''\rightarrow Y$ such that $\tilde{f}$ and $\tilde{f'}$ are isotopic in $\xi''$ and this isotopy is compatible with the complex structure on the normal bundle. That is, there is an isotopy $F$ such that for all $t \in [t_0, t_1]$, $F_t: X \rightarrow \xi''$ is endowed with a complex structure on its normal bundle which matches that of $\tilde{f}$ and $\tilde{f'}$ in $\xi''$ at $t_0$ and $t_1$ respectively. \end{defn} By Proposition \ref{lang}, we have \begin{prop} The relation of equivalence of admissible complex orientability of proper Fredholm maps between separable Hilbert manifolds is an equivalence relation. \end{prop} This generalizes Quillen's notion of complex orientability for maps of finite dimensional manifolds. We can also define a notion of cobordism of admissible complex orientable maps between separable Hilbert manifolds. First we recall some ideas on the transversality.
\begin{defn}Let $f_1: M_1 \rightarrow N, f_2: M_2 \rightarrow N$ be smooth maps between Hilbert manifolds. Then $f_1$ and $f_2$ are \emph{transverse} at $y \in N$ if \cdot df_1 (T_{x_1} M_1) + df_2 (T_{x_2}M_2) = T_y N \cdot whenever $f_1 (x_1) = f_2(x_2) = y$. The maps $f_1$ and $f_2$ are said to be \emph{transverse} if they are transverse at every point of $N$. \end{defn} \begin{lem} Smooth maps $f_i: M_i \rightarrow N (i = 1,2)$ are transverse if and only if $f_1\times f_2 : M_1 \times M_2 \rightarrow N\times N$ is transverse to the diagonal map $\Delta : N\rightarrow N\times N$. \end{lem} \begin{defn} Let $f_1: M_1 \rightarrow N, f_2: M_2 \rightarrow N$ be transverse smooth maps between smooth Hilbert manifolds. The \emph{topological pullback} \cdot M_1 \prod_N M_2 = \cdot (x_1 , x_2) \in M_1 \times M_2 : f_1(x_1) = f_2(x_2)\cdot \cdot is a submanifold of $M_1 \times M_2$ and the diagram \cdot \begin{CD} M_1 \prod_N M_2 @>{f_2}^*(f_1)>> M_2 \cdot @VV{f_1}^*(f_2)V @VVf_2V \cdot M_1 @>f_1>> N \end{CD} \cdot is commutative, where the map ${f_i}^* (f_j)$ is pull-back of $f_j$ by $f_i$. \end{defn}
\begin{defn} Let $f_i: X_i \rightarrow Y (i= 0,1)$ be admissible complex oriented maps. Then $f_0$ is \emph{cobordant} to $f_1$ if there is an admissible complex orientable map $h: W \rightarrow Y\times \mathbb R$ such that the maps $\varepsilon_i: Y\rightarrow Y \times \mathbb R \cdot\text{given by} \cdot\varepsilon_i (y) = (y,i)$ for $i= 0, 1 $, are transverse to $h$ and the pull-back map ${\varepsilon_i}^* h$ is equivalent to $f_i$. The cobordism class of $f: X\rightarrow Y$ will be denoted by $[X, f]$. \end{defn} \begin{prop}\label{grem} If $f: X\rightarrow Y$ is an admissible complex orientable map and $g: Z\rightarrow Y$ a smooth map transverse to $f$, then the pull-back map \cdot {g^*}(f): Z\prod_Y X\rightarrow Z \cdot is an admissible complex orientable map with finite dimensional pull-back vector bundle \cdot g^*(\xi) = Z\prod_Y \xi = \cdot(z,v)\in Z\times \xi : g(z) = q(v)\cdot \cdot in the factorization of $g^* (f)$, where $q: \xi\rightarrow Y$ is the finite-dimensional complex vector bundle in the factorization of $f$ as in Definition \ref{zirto}. \end{prop} The next result was proved in \cite{ozel} by essentially the same argument as in the finite dimensional situation using the Implicit Function Theorem \cite{lang}. \begin{thm} Cobordism is an equivalence relation. \end{thm}
\begin{defn} For a separable Hilbert manifold $Y$, $\mathcal U^d (Y)$ is the set of cobordism classes of the admissible complex orientable proper Fredholm maps of index $-d$. \end{defn}
In the above definition, instead of proper maps, closed maps could be used for infinite dimensional Hilbert manifolds, because of the following result of Smale \cite{smale}. \begin{thm}\label{martin} When $X$ and $Y$ are infinite dimensional, every closed Fredholm map $X\rightarrow Y$ is proper. \end{thm} My next result is the following.
\begin{thm}\label{iain} If $f: X\rightarrow Y$ is an admissible complex orientable Fredholm map of index $d_1$ and $g: Y\rightarrow Z$ is an admissible complex orientable Fredholm map of index $d_2$, then $g\circ f: X\rightarrow Z$ is an admissible complex orientable Fredholm map with index $d_1 + d_2$. \end{thm}
Let $g:Y \rightarrow Z$ be an admissible complex orientable Fredholm map of index $r$. By Theorem \ref{iain}, we have \emph{push-forward, or Gysin map} \cdot g_{*}: \mathcal U^d (Y) \rightarrow \mathcal U^{d + r} (Z) \cdot given by $g_{*} ([X, f]) = ([X, g \circ f])$.
We show in \cite{ozel} that it is well-defined. If $g': Y\rightarrow Z$ is a second map cobordant to $g$ then ${g'}_* = g_*$; in particular, if $g$ and $g'$ are homotopic through proper Fredholm maps they induce the same Gysin maps. Clearly, we have $(h\circ g)_{*} = h_{*}g_{*}$ for admissible complex orientable Fredholm maps $h,g$ and $\operatorname{Id}_{*} = \operatorname{Id}$.
The graded cobordism set $\mathcal U^* (Y)$ of the separable Hilbert manifold $Y$ has a group structure given as follows. Let $[X_1 , f_1]$ and $[X_2 , f_2]$ be cobordism classes. Then $[X_1 , f_1] + [X_2 , f_2]$ is the class of the map $f_1 \sqcup f_2: X_1 \sqcup X_2 \rightarrow Y$, where $X_1 \sqcup X_2$ is the topological sum (disjoint union) of $X_1$ and $X_2$. We show in \cite{ozel} that this sum is well-defined. As usual, the class of the empty set $\emptyset$ is the zero element of the cobordism set and the negative of $[X,f]$ is itself with the opposite orientation on the normal bundle of the embedding $\tilde{f}$. Then we have \begin{thm} The graded cobordism set $\mathcal U^* (Y)$ of the admissible complex orientable maps of $Y$ is a graded abelian group. \end{thm} Now we define relative cobordism . \begin{defn} If $A$ is a finite dimensional submanifold of $Y$, the relative cobordism set $\mathcal U^* (Y,A)$ is the set of the admissible complex orientable maps of $Y$ whose images lie in $Y-A$. \end{defn} More generally, \begin{thm}\label{relative} Let $A$ be a finite dimensional submanifold of $Y$. Then the relative cobordism set $\mathcal U^* (Y,A)$ is a graded abelian group and there is a homomorphism $\kappa^* :\mathcal U^* (Y,A)\rightarrow \mathcal U^* (Y)$ by $\kappa^*[M\stackrel{h}\rightarrow Y] =[M\stackrel{h}\rightarrow Y]$ with $h(M)\subseteq Y-A$. \end{thm}
If our cobordism functor $\mathcal U^*( \cdot)$ of admissible complex orientable Fredholm maps is restricted to finite dimensional Hilbert manifolds, it agrees Quillen's complex cobordism functor $MU^* (\cdot)$. \begin{thm} For finite dimensional separable Hilbert manifolds $A\subseteq Y$, there is a natural isomorphism $$ \mathcal U^*(Y,A)\cong MU^*(Y,A). $$ \end{thm} \section{Transversal approximations, contravariance and cup products.} We would like to define a product structure on the graded cobordism group $\mathcal U^* (Y)$. Given cobordism classes $[X_1 , f_1] \in \mathcal U^{d_1}(Y_1)$ and $[X_2 , f_2]\in \mathcal U^{d_2}(Y_2)$, their external product is \cdot [X_1 , f_1]\times [X_2 , f_2 ] = [X_1 \times X_2 ,f_1 \times f_2] \in \mathcal U^{d_1 + d_2} (Y_1\times Y_2). \cdot Although there is the external product in the category of cobordism of separable Hilbert manifolds, we can not necessarily define an internal product on $U^{*}(Y)$ unless $Y$ is a finite dimensional manifold. However,if admissible complex orientable Fredholm map $f_1 \times f_2$ is transverse to the diagonal imbedding $\Delta:Y\rightarrow Y\times Y$, then we do have an internal (cup) product \cdot [X_1 , f_1] \cup [X_2 , f_2] = \Delta^* [X_1\times X_2, f_1\times f_2]. \cdot If $Y$ is finite dimensional, then by Thom's Transversality Theorem in \cite{thom}, every complex orientable map to $Y$ has a transverse approximation, hence the cup product $\cup$ induces a graded ring structure on $\mathcal U^* (Y)$. The unit element $1$ is represented by the identity map $Y\rightarrow Y$ with index $0$. However F. Quinn \cite{Quinn} proved the generalization of Thom's Transversality Theorem for separable Hilbert manifolds using smooth transversal approximations of Sard functions in fine topology.
Details about the fine topology, jets and smooth maps space $C^m(W, N)$,$C^{\infty}(W, N)$ can be found in \cite{michor}. In this topology, the derivatives of the difference function between the function $g$ and its approximation $g'$ are bounded. We would like to interpret this approximation in the fine topology. We need some notation to describe this situation.
\begin{defn} Let $X$ and $Y$ be smooth manifolds. A \emph{k-jet from} $X$ \emph{to} $Y$ is an equivalence class $[f, x]_k$ of pairs $(f, x)$ where $f: X\rightarrow Y$ is a smooth mapping, $x \in X$. The pairs $(f, x)$ and $(f', x')$ are \emph{equivalent} if $x = x'$, $f$ and $f'$ have same Taylor expansion of order $k$ at $x$ in some pair of coordinate charts centered at $x$ and $f (x)$ respectively. We will write $J^k f (x)=[f, x]_k $ and call this the \emph{k-jet of} $f$ at $x$. \end{defn} There is an equivalent definition of this equivalence relation: $[f, x]_k = [f', x']_k$ if $x=x'$ and $T^k_x f= T^k_x f'$ where $T^k$ is the $k$th tangent mapping. \begin{defn} For a topological space $X$,a covering of $X$ is \emph{locally finite} if every point has a neighborhood which intersects only finitely many elements of the covering. \end{defn} Approximation $g'$ of $g$ in the smooth fine topology means the following. Let $\{L_i\cdot_{i \in I}$ be a locally finite cover of $W$. For every open set $L_i$, there is a bounded continuous map $\varepsilon_i: L_i \rightarrow [0, \infty)$ such that for every $x \in L_i$ and $k > 0$, \cdot
||J^k g (x) - J^k g' (x) || < \varepsilon_i (x). \cdot
\begin{defn} Let $E$ be a Banach space. We say that a collection $\mathcal S$ of smooth functions $\alpha: E \rightarrow \mathbb R$ is a \emph{Sard class} if it satisfies the following conditions: \begin{enumerate} \item for $r \in \mathbb R$, $y \in E$ and $\alpha \in \mathcal S$, then the function $x \rightarrow \alpha(rx + y)$ is also in the class $\mathcal S$, \item if $\alpha_n \in \mathcal S$, then the rank of differential $D_x (\alpha_1, \ldots, \alpha_n)$ is constant for all $x$ not in some closed finite dimensional submanifold of $E$. \end{enumerate} \end{defn} \begin{defn}\label{Sard} Let $\mathcal S$ be a Sard class on $E$, $U$open in $E$, and $M$ a smooth Banach manifold. We define $\mathcal S(U, M)$ to be the collection of \emph{Sard functions} $f: U\rightarrow M$ such that for each $x \in U$ there is a neighbourhood $V$ of $x$, functions $\alpha_1, \ldots, \alpha_n \in \mathcal S$, and a smooth map $g:W\rightarrow M$, where $W$ open in $\mathbb R^n$ contains
$(\alpha_1, \ldots, \alpha_n)(V)$, all such that $f|V= g \circ
(\alpha_1, \ldots, \alpha_n)|V$. \end{defn} \begin{defn} The \emph{support} of a function $f: X\rightarrow \mathbb R$ is the closure of the set of points $x$ such that $f(x)\neq 0$. \end{defn} From \cite{Quinn}, we have \begin{thm} $E$ admits a Sard class $\mathcal S$ if $\mathcal S(E, \mathbb R)$ contains a function with bounded nonempty support. In particular, the separable Hilbert space admits Sard classes. \end{thm} \begin{defn} A \emph{refinement} of a covering of $X$ is a second covering, each element of which is contained in an element of the first covering. \end{defn} \begin{defn} A topological space is \emph{paracompact} if it is Hausdorff, and every open covering has a locally finite open refirement. \end{defn} \begin{defn} A smooth \emph{partition of unity} on a manifold $X$ consists of an covering $\{U_i\cdot$ of $X$ and a system of smooth functions $\psi_i: X\rightarrow \mathbb R$ satisfying the following conditions. \begin{enumerate} \item $\forall x \in X$, we have $\psi_i (x) \geqslant 0$;\cdot \item the support of $\psi_i$ is contained in $U_i$;\cdot \item the covering is locally finite;\cdot \item for each point $x \in X$, we have \cdot \displaystyle\sum_{i} \psi_i (x) = 1. \cdot \end{enumerate} \end{defn} \begin{defn} A manifold $X$ will be said to \emph{admit partitions of unity} if it is paracompact, and if, given a locally finite open covering $\{U_i\cdot$, there exists a partition of unity $\cdot\psi_i\cdot$ such that the support of $\psi_i$ is contained in some $U_i$. \end{defn} From \cite{lang}, we have \begin{thm} Let $X$ be a paracompact smooth manifold modelled on a separable Hilbert space $H$. Then $X$ admits smooth partitions of unity. \end{thm} From \cite{Eells-McAlpin}, \begin{thm} On a separable Hilbert manifold the functions constructed using the partitions of unity form a Sard class. \end{thm}
The following result was proved by F. Quinn . \begin{thm}\label{kirmizi} Let $H$ be the smooth separable Hilbert space and let $U$ be an open set in $H$. If $f: W\rightarrow N$ is a smooth proper Fredholm map, then smooth maps transversal to $f$ are dense in $\mathcal S(U, N)$ with the $C^0$ fine topology. \end{thm} We will require the Open Embedding Theorem of Eells $\cdot$ Elworthy \cite{Eells-Elw}. \begin{thm}\label{open embedding} Let $M$ be a smooth manifold modelled on the separable infinite dimensional Hilbert space $H$. Then $M$ is diffeomorphic to an open subset of $H$. \end{thm} \begin{thm} The space $\mathcal S(U, N)$ is dense in $C^0 (U, N)$ in the $C^0$ fine topology. \end{thm} Using these techniques, Quinn proves the following result. \begin{cor}\label{canalici} Let $M$ be a smooth separable Hilbert manifold. If $f: W\rightarrow N$ is a smooth proper Fredholm map, then smooth maps transversal to $f$ are dense in $C^0(M,N)$ in the $C^0$ fine topology. \end{cor}
From \cite{Eells-Elw}, we have \begin{thm}\label{homotopy} Let $X$ and $Y$ be two smooth manifold modelled on the separable infinite dimensional Hilbert space $H$. If there is a homotopy equivalence $\varphi: X\rightarrow Y$, then $\varphi$ is homotopic to a diffeomorphism. \end{thm} Now I will try to tell a detailed explanation of Quinn's very technical work in Theorem \ref{kirmizi}.
Let $f: W\rightarrow N$ be a smooth proper Fredholm map and let $g \in \mathcal S(U, N)$ specified in Theorem \ref{kirmizi}. Now we are given an arbitrary $C^0$ fine neighborhood of $g$ in which we want to find a map transversal to $f$. Since some neighborhood of the image of $g$ is metrizable, we can impose a metric on it and then we can choose a function $\varepsilon: U \rightarrow (0,1)$ so that the $\varepsilon$-neighborhood of $g$ in $\mathcal S(U, N)$ lies in the given $C^0$ fine neighborhood. Now we explain the Smale decomposition of the smooth proper Fredholm map $f: W\rightarrow N$. If $x \in N$, then there is a coordinate neighborhood $\Theta : U\thickapprox H \times \cdot^k$ about $x$ such that $$ f^{-1}(u)\supseteqq \displaystyle\bigcup_{i=1}^n V_i \supseteqq f^{-1}({\Theta}^{-1}(H \times \cdot^k)) $$ for some sets $V_i$ such that $\Psi_i : V_i \thickapprox H\times W_i, W_i$ open in $\cdot^m$, and $\Theta \circ f \circ \Psi_{i}^{-1} = (\pi,f_i): H\times W_i \rightarrow H\times \cdot^k$, where $\cdot^k$ is the open unit ball in $\cdot^k$.
Using separability of $U$, we cover $g(u)$ by $N-f(W)$ and a countable number of sets of the form $\Theta_{i}^{-1}(H_i \times \cdot^{k_i})$, where $\Theta_i : u_i \thickapprox H_i \times \cdot^{k_i}$ are coordinate neighborhoods as given in the Smale decomposition of $f$. We denote the corresponding coordinate neighborhoods in the domain by $(V_{i_j}, \Psi_{i_j})$,where $\Psi_{i_j}:V_{i_j}\thickapprox H_i \times W_i$, $W_i$ open in $\cdot^{m_i}$. Let $\{Y_i\cdot$ be a locally finite refinement of $\{g^{-1}(\Theta_{i}^{-1}(H_i \times \cdot^{k_i})), g^{-1}(N-f(W))\cdot$, and let $\{Z_i\cdot$ be a subcover such that $Y_i \supseteqq \bar{Z_i}$.
Inductively we will get an approximation of $g$. Let $g_0 = g$. Given $g_i$, let $g_{i+1}$ be the approximation defined by Quinn applied the situation $U_1 =U_2 = Y_i, U_3= Z_i$, $W$ the disjoint union of of the $W_{i_j}$ over $j$, and the $C^0$ fine topology neighborhood so small that $g_{i+1} = g_i + \frac{\varepsilon}{2^{i+1}}y_i$, and $g_{i+1}(Y_i) \subseteq \Theta_{i}^{-1}(H_i \times \cdot^{k_i})$, where $y_i \in \cdot^{k_i}$. Since $\{Y_i\cdot$ is locally finite, $g' = \displaystyle\lim_{i\rightarrow\infty}g_i$ is well-defined. $g'$ is an $\varepsilon$-approximation of $g$ and $g'\in\mathcal S (U, N)$ and it is transverse to $f$ everywhere. It is interesting that this approximation can be done in the $C^r$ fine topology. For separable Hilbert manifolds, it can be even done in the smooth $C^{\infty}$ fine topology. In this case, $g' \in \overline{\mathcal S(U, N)}$. \begin{thm}\label{hist} Let $U$ be an open set in separable infinite dimensional Hilbert space $H$ and let $f: M\rightarrow N$ be a proper Fredholm map between separable infinite dimensional Hilbert manifolds $M$ and $N$. Then the set of maps transverse to $f$ is dense in the closure of Sard function space $\overline{\mathcal S(U, N)}$ in the $C^{\infty}$ fine topology. \end{thm}
By Corollary \ref{canalici}, a smooth map (even continuous map) $g: Z\rightarrow Y$ can be deformed to a smooth map $g': Z\rightarrow Y$ by a small correction until it is transverse to an admissible complex orientable map $f: X \rightarrow Y$. It is obvious that they are homotopic each other. By definition of Cobordism and Proposition \ref{grem}, the cobordism functor is contravariant for any smooth map between separable Hilbert manifolds. \begin{thm}\label{son} Let $f: X\rightarrow Y$ be an admissible complex oriented map and let $g: Z\rightarrow Y$ be a smooth (may be continuous) map. Then the cobordism class of the pull-back $Z \prod_Y X \rightarrow Z$ depends only on the cobordism class of $f$, hence there is a map $g^*: \mathcal U^d (Y)\rightarrow \mathcal U^d (Z)$ given by $$g^* [X, f] =g'^* [X, f] = [Z\prod_Y X, {g'}^*(f)],$$ where $g'$ is a smooth $\varepsilon$-approximation of g which is transverse to $f$. Moreover, $g^*$ depends only on the homotopy class of $g$. \end{thm} Now we give the functorial property of $\mathcal U^*$ theory. \begin{thm} Let $X,Y$ and $Z$ be separable Hilbert manifolds. If $Z\stackrel{\alpha} \rightarrow Y \stackrel{\beta}\rightarrow X$ are smooth functions, then $$ (\beta \circ \alpha)^* = {\alpha}^* {\beta}^*: \mathcal U^d(X)\rightarrow \mathcal U^d(Z). $$ The identity map $\operatorname{Id} : X\rightarrow X$ induces the identity endomorphism $\operatorname{Id}^* :\mathcal U^d(X)\rightarrow \mathcal U^d(X)$ for every $d$. \end{thm} \begin{proof} After making them transverse where appropriate we consider the following commutative diagram \cdot \begin{CD} M'' @>{\alpha}'>> M' @>{\beta}'>> M \cdot @VVf''V @VVf'V @VVfV \cdot Z @>\alpha >> Y @>\beta >> X, \end{CD} \cdot where both squares are pullback diagrams with transverse $(\beta, f)$ and $(\alpha, f')$; the outer square is then also pullback with transverse $(\beta \circ\alpha, f)$. Given an admissible complex oriented map $f: M\rightarrow X$, we show that the maps $M'\stackrel{f'}\rightarrow Y$ and $M''\stackrel{f''}\rightarrow Z$ are admissible complex orientable and $\alpha^* \beta^* [M\stackrel{f}\rightarrow X] =(\beta \alpha )^*[M\stackrel{f}\rightarrow X]$. It is clear that $M''= Z\displaystyle\prod_Y M' = Z\displaystyle\prod_Y (Y \displaystyle\prod_X M)$ as well as $Z\displaystyle\prod_X M$ so that $\alpha^* \beta^* [M\stackrel{f}\rightarrow X] =(\beta \alpha )^*[M\stackrel{f}\rightarrow X]$ by the following commutative diagram \cdot \begin{CD} M'' @>{\alpha}'>> M' @>{\beta}'>> M \cdot @VV\tilde{f''}V @VV\tilde{f'}V @VV\tilde{f}V \cdot \xi'' @>\alpha >> \xi' @>\beta >> \xi\cdot @VVq''V @VVq'V @VVqV \cdot Z @>\alpha >> Y @>\beta >> X, \end{CD} \cdot where the finite dimensional vector bundle $\xi''= Z\displaystyle\prod_Y \xi' = Z\displaystyle\prod_Y (Y \displaystyle\prod_X \xi)$ as well as $Z\displaystyle\prod_X \xi$. \end{proof}
Let turn back the interior(cup) products in $\mathcal U^*$. Given cobordism classes $[X_1,f_1] \in \mathcal U^{d_1}(Y_1)$ and $[X_2 , f_2]\in \mathcal U^{d_2}(Y_2)$, their external product is \cdot [X_1 , f_1]\times [X_2 , f_2 ] = [X_1 \times X_2 ,f_1 \times f_2] \in \mathcal U^{d_1 + d_2} (Y_1\times Y_2). \cdot If admissible complex orientable Fredholm map $f_1 \times f_2$ is transverse to the diagonal imbedding $\Delta:Y\rightarrow Y\times Y$, then we do have an internal (cup) product \cdot [X_1 , f_1] \cup [X_2 , f_2] = \Delta^* [X_1\times X_2, f_1\times f_2]. \cdot If the diagonal imbedding $\Delta:Y\rightarrow Y\times Y$ is not transverse to smooth proper Fredholm map $f_1 \times f_2:X_1 \times X_2 \rightarrow $, by Quinn's transversality Theorem, we can find a smooth $\varepsilon$-approximation $\Delta'$ of $\Delta$ which is transverse to $f_1\times f_2$. Then \begin{thm} If $[X_1,f_1] \in \mathcal U^{d_1}(Y_1)$ and $[X_2 , f_2]\in \mathcal U^{d_2}(Y_2)$,internal(cup) product \cdot [X_1 , f_1] \cup [X_2 , f_2] = \Delta^* [X_1\times X_2, f_1\times f_2] = \Delta'^* [X_1\times X_2, f_1\times f_2]\in \mathcal U^{d_1 + d_2} (Y) \cdot where $\Delta'$ is a smooth $\varepsilon$-approximation of $\Delta$ which is transverse to $f_1\times f_2$. \end{thm} The cup product is well-defined and associative.
Then, $\mathcal U^* (\cdot)$ is a multiplicative contravariant functor for smooth functions on the separable Hilbert manifolds. The question of whether it agrees with other cobordism functors such as representable cobordism seems not so easily answered and there is also no obvious dual bordism functor.
In this section, we show how to define Euler classes in complex cobordism for finite dimensional complex vector bundles over separable Hilbert manifolds. In order to do this, we use Sard classes. We know from \cite{janich} that global sections of a vector bundle on a smooth separable Hilbert manifold can be constructed using partitions of unity, then all sections are Sard. Given a smooth vector bundle $\pi: E\rightarrow B$ over a separable Hilbert manifold $B$, we know from Theorem \ref{open embedding}, that $B$ can be embedded as a open subset of a separable Hilbert space $H$. By Theorem \ref{hist}, we have \begin{cor} Let $\pi: E\rightarrow B$ be a finite dimensional complex vector bundle over a separable Hilbert manifold $B$ and let $i: B\rightarrow E$ be the zero-section. Then there is an approximation $\tilde{i}$ of $i$ with $\tilde{i}$ transverse to $i$. \end{cor} Then, we define the Euler class of a finite dimensional complex vector bundle on a separable Hilbert manifold. Note that Theorem \ref{son} implies that this Euler class is a well-defined invariant of the bundle $\pi$. \begin{defn} Let $\pi: \xi \rightarrow B$ be a finite dimensional complex vector bundle of dimension $d$ on a separable Hilbert manifold $B$ with zero-section $i: B\rightarrow \xi$. The \emph{$\mathcal U$-theory Euler class} of $\xi$ is the element $$ \chi(\pi) =i^* i_* (1)\in \mathcal U^{2d}(B). $$ \end{defn} We have the following projection formula for the Gysin map. \begin{thm}\label{projection formula} Let $f: X\rightarrow Y$ be an admissible complex orientable map and let $\pi: \xi\rightarrow Y$ be a finite dimensional smooth complex vector bundle of dimension d. Then \cdot \chi (\xi) \cup [X, f] = f_* \chi (f^* \xi). \cdot \end{thm} \begin{proof} Let $s$ be a smooth section of $\pi$ transverse to the zero section $i: Y\rightarrow \xi$. Then $Y'= \{y \in Y: s(y) = i(y)\cdot$ is a submanifold of complex codimension $d$ and $\chi(\xi) = [Y',j]$, where $j: Y'\rightarrow Y$ is the inclusion. Setting $$ X' = f^{-1} Y' = \{x\in X : s(f(x))= i(f(x))\cdot, $$ which is also a submanifold of $X$ of complex codimension $d$, we have \begin{align} \chi (\xi) \cup [X,f] &= [Y',j] \cup [X,f]\notag\cdot
& = [X',f_{|X'}].\notag \end{align} Now we determine $f_* \chi (f^* E)$. By transversality theorem, $f$ can be deformed to a smooth map $f'$ such that the composite section $s\circ f':X\rightarrow f'^*$ is transverse to the zero section and they agree on $X'$, hence by definition we have $\chi
(f^* \xi) = [X',j]$ where $j:X'\rightarrow X$ is the inclusion. Hence, $f_*\chi (f^* \xi) = [X',f_{|X'}]$ by definition of the Gysin map $f_*$. \end{proof} Now we need a useful lemma from \cite{Quinn}. \begin{lem}\label{tub} A smooth split submanifold of a smooth separable Hilbert manifold has a smooth tubular neighborhood. \end{lem}
Let $\pi: \xi \rightarrow X$ be a finite dimensional complex vector bundle of dimension $d$ on a separable Hilbert manifold $X$ with zero-section $i: X\rightarrow \xi$. The map $i$ is proper so that we have the Gysin map $$ i_*: \mathcal U^{j}(X)\rightarrow \mathcal U^{j+2d}(\xi,\xi-U) $$ where $U$ is a smooth neighborhood of the zero section.
The map $\pi$ is not proper. However if $U$ is contained in a tube
$U^r$ of finite radius $r$, then $\pi_{|\bar{U}}$ is proper and we can define $$ \pi_*:\mathcal U^{j+2d}(\xi,\xi-U)\rightarrow \mathcal U^{j}(X). $$ Since $\pi i= \operatorname{Id}$ we have $\pi_* i_* = \operatorname{Id}$. The composite map $i\pi $ is homotopic to $\operatorname{Id}_{\xi}$. If $U = U^{\circ}$ is itself a tube, the homotopy moves on $U$ and we have \emph{Thom isomorphism} $$ \mathcal U^{j+2d}(\xi,\xi-U)\cong \mathcal U^{j}(X). $$
\section{The relationship between $\mathcal U$-theory and $MU$-theory.} In this section we consider the relationship between $\mathcal U$-theory and $MU$-theory. Later we discuss the particular cases of Grassmannians and $LG/T$. \par First we discuss the general relationship between $\cdot^*(\cdot)$ and $MU^*(\cdot)$. Let $X$ be a separable Hilbert manifold. For each proper smooth map $f: M\longrightarrow X$ where $M$ is a finite dimensional manifold, there is a pullback homomorphism $$f^*: \cdot^*(X)\longrightarrow\cdot^*(M)=MU^*(M).$$ If we consider all such maps into $X$, then there is a unique homomorphism $$ \rho :\cdot^*(X)\longrightarrow\displaystyle\lim_{\overleftarrow{M\downarrow X}}MU^*(M), $$ where the limit is taken over all proper smooth maps $M\rightarrow X$ from finite dimensional manifolds, which form a directed system along commuting diagrams of the form \cdot \begin{CD} M_1 @>f>> M_2 \cdot @VVV @VVV \cdot X @>=>> X \end{CD} \cdot and hence give rise to an inverse system along induced maps $f^* : MU^* (M_2)\rightarrow MU^* (M_1)$ in cobordism.
Let $X$ be a separable Hilbert manifold. Each of the following conjectures appears reasonable and is consistent with examples we will discuss later. We might also hope that surjectivity could be replaced by isomorphism, but we do not have any examples supporting this. \begin{conj} \label{conj:InvLimSurj} $\rho$ is always a surjection. \end{conj} \begin{conj} If $\cdot^{\operatorname{ev}}(X)=0$ or $\cdot^{\mathrm{odd}}(X)=0$, $\rho$ is a surjection. \end{conj} \begin{conj} If $MU^{\operatorname{ev}}(X)=0$ or $MU^{\mathrm{odd}}(X)=0$, $\rho$ is a surjection. \end{conj}
Now we discuss some important special cases. Let $H$ be a separable complex Hilbert space, with $H^n$ ($n\geq1$) an increasing sequence of finite dimensional subspaces with $\operatorname{dim} H^n = n$ with $H^\infty=\displaystyle\bigcup_{n\geqslant 1} H^n$ dense in $H$. We use a theorem of Kuiper \cite{Kuiper}. \begin{thm}\label{Kuiper} The unitary group $U(H)$ of a separable Hilbert space $H$ is contractible. \end{thm}
Let $\operatorname{Gr}_n(H)$ be the space of all $n$-dimensional subspaces of $H$, which is a separable Hilbert manifold. Then $$ \operatorname{Gr}_n(H^\infty) = \displaystyle\bigcup_{k\geqslant n} \operatorname{Gr}_n (H^k) $$ is a dense subspace of $\operatorname{Gr}_n (H)$ which we will take it to be a model for the classifying space $BU(n)$. \begin{thm} \label{ProjHilbSpace} The natural embedding $\operatorname{Gr}_n (H^\infty)\rightarrow \operatorname{Gr}_{n}(H)$ is a homotopy equivalence, and the natural $n$-plane bundle $\xi_n\rightarrow \operatorname{Gr}_n(H)$ is universal. \end{thm} \begin{proof} By a theorem of Pressley and Segal \cite{segal-pressley}, the unitary group $U(H)$ acts on $\operatorname{Gr}(H)$ transitively and hence $U(H)$ acts on $\operatorname{Gr}_n (H)$ transitively. Let $H^n$ be an $n$-dimensional subspace of infinite dimensional separable Hilbert space $H$ and let $H'$ be the orthogonal complement of $H^n$ in $H$. The stabilizer group of $H^n$ is $U(H^n) \times U(H')$ which acts freely on the contractible space $U(H)$. Hence \begin{align} \operatorname{Gr}_{n}(H)& = U(H)/(U(H^n) \times U(H'))\notag\cdot &= B(U(H^n) \times U(H'))\notag\cdot &= BU(H^n) \times BU(H').\notag \end{align} By Kuiper's Theorem \ref{Kuiper}, $U(H')$ is contractible, hence so is $BU(H')$. Hence $$ \operatorname{Gr}_{n} (H) \simeq BU(H^n) = BU(n). $$ On the other hand, \cdot \operatorname{Gr}_n (H^\infty) = \displaystyle\bigcup_{k\geqslant n} U(H^k)/(U(H^n) \times U(H''))\subseteq \operatorname{Gr}_n(H), \cdot where $H''$ is the orthogonal complement of $H^n$ in $H^k$.
By the construction, the natural $n$-plane bundle $\xi_n\rightarrow \operatorname{Gr}_n(H)$ is universal. Also, the natural bundle $\xi_n^\infty\rightarrow \operatorname{Gr}_n(H^\infty)$ is classified by the inclusion $\operatorname{Gr}_n(H^\infty)\rightarrow \operatorname{Gr}_n(H)$ and since the latter is universal, this inclusion is a homotopy equivalence. \end{proof}
In particular, the inclusion of the projective space $$ P(H^\infty)=\displaystyle\bigcup_{n\geqslant 1} P(H^n) \subseteq P(H) $$ is a homotopy equivalence. \begin{thm} The natural homomorphism $$ \rho: \cdot^*(P(H))\longrightarrow\displaystyle\lim_{\overleftarrow{n}} MU^*(P(H^n))=MU^*(P (H^\infty)) $$ is surjective. \end{thm} \begin{proof} We will show by induction that $$ \cdot^*(P(H))\xrightarrow{i_n^*}MU^*(P(H^{n+1})) $$ is surjective for each $n$. It will suffice to show that $x^i\in \operatorname{im} i_n^*$ for $i=0,\ldots,n$. For $n=0$, this is trivial.
Now we verify it for $n=1$. By Theorem \ref{ProjHilbSpace}, since the natural line bundle $\lambda\rightarrow P(H)\simeq P(H^\infty)$ is universal, the following diagram commutes for each $n\geqslant 1$ \cdot \begin{CD} \eta_n =i_n^*(\lambda) @>i_n^*>> \lambda\cdot @VVi_n^*(\lambda)V @VVV\cdot \mathbb C P^n = P(H^{n+1}) @>i_n>> P(H), \end{CD} \cdot where $i_n: \mathbb C P^{n} =P(H^{n+1}) \rightarrow P(H)$ is the inclusion map. By the compatibility of induced bundles, for $n\geqslant 1$ and the generator $x = \chi(\eta_n)\in MU^*(P(H^{n+1}))$, there exists an Euler class $\tilde{x} = \chi(\lambda) \in \mathcal U^2(P(H))$ satisfying $i_n^*(\tilde{x}) = x$, where $i_n: P(H^{n+1}) \rightarrow P(H)$ is the inclusion map.
Assume that $i_n^*$ is surjective. Then there are elements $$ y_i\in \cdot^{2i}(P(H)),\cdot i=0,\ldots,n, $$ such that $$ i_n^*y_i = x^i \in MU^{2i} (P(H^{n+1})). $$ Also, $$ i_{n+1}^* y_i = x^i + z_i x^{n+1} \in MU^{2i}(P(H^{n+2})) $$ where $z_i \in MU_{2(n+1-i)}$.
In particular, let $y_{n} = [W, f] \in \cdot^{2n}(P(H))$. Then the following diagram commutes \cdot \begin{CD} f^*(\lambda) @>f^*>> \lambda\cdot @VVV @VVV\cdot W @>f>> P(H) \end{CD} \cdot and there is an Euler class $\chi(f^*(\lambda)) = [W', g] \in \cdot^2(W)$. Now by Theorem \ref{projection formula}, $$ y_{n+1} = f_* \chi(f^*(\lambda)) \in \cdot^{2n+2}(P(H)) $$ satisfies $$ i_{n+1}^* y_{n+1} = x^{n} \chi (\eta_n) =x^{n+1}. $$
Hence, $\operatorname{im} i_{n+1}^*$ contains the $MU^*$-submodule generated by $x^i$ $(i=0,\ldots,n)$ and so $i_{n+1}^*$ is surjective. This completes the induction.
This shows that the induced homomorphism $$ \rho:\cdot^*(P(H))\longrightarrow\displaystyle\lim_{\overleftarrow{n}} MU^*(P(H^n))=MU^*(P (H^\infty)) $$ is surjective. \end{proof}
Note that it is also possible to prove this result by using the projective spaces $P({H^n}^{\bot}) \subseteq P(H)$ to realize cobordism classes restricting to the classes $x^n$ in $MU^*(P (H^\infty))$.
Next we discuss some geometry of Grassmannians from Pressley $\cdot$ Segal \cite{segal-pressley}, whose ideas and notation we assume. We take for our separable Hilbert space $H = L^2 (\cdot^1 ; \cdot)$ and let $H_+$ to be the closure of the subspace of $H$ containing the functions $z^n : z\rightarrow z^n (n\geqslant 0)$. Then $$ \operatorname{Gr}_0 (H) = \displaystyle\lim_{\overrightarrow{k\geqslant 1}} \operatorname{Gr} (H_{-k,k}), $$ where $\operatorname{Gr} (H_{-k,k})$ is the Grassmannian of the finite dimensional vector space $$ H_{-k,k} = z^{-k}H_+ / z^k H_+. $$ $\operatorname{Gr}_0 (H)$ is dense in $\operatorname{Gr} (H)$ and is also known to be homotopic to the classifying space of $K$-theory, $BU\times \cdot$. \begin{thm} For $n\geqslant 1$, the natural homomorphism $$ \rho: \mathcal U^*(\operatorname{Gr}_{n}(H))\rightarrow MU^*(Gr_n(H)) $$ is surjective. \end{thm} \begin{proof} For $k\geqslant n$, the inclusion $i: \operatorname{Gr}_n(H_{-k,k})\rightarrow \operatorname{Gr}_n(H)$ induces a contravariant map $$\cdot^*(Gr_n(H)) \rightarrow \cdot^*(\operatorname{Gr}_n(H_{-k,k}) =MU^*(\operatorname{Gr}_n(H_{-k,k})).$$ For $k\geqslant n$, since $C_S \subseteq \operatorname{Gr}_n(H_{-k,k})$ is transverse to $\Sigma_S$, there exists a stratum $\Sigma_{S'}$ such that \cdot \sigma_{S',k}= [\operatorname{Gr}_n (H_{-k,k}) \cap \Sigma_{S'} \rightarrow \operatorname{Gr}_n(H_{-k,k})] \in MU^*(\operatorname{Gr}_n(H_{-k,k})) \cdot are the classical Schubert cells. By an argument using the Atiyah-Hirzebruch spectral sequence and results on Schubert cells in cohomology \cite{milnor-stasheff}, the cobordism classes $\sigma_{S', k}$ provide generators for the $MU^*$-module $MU^*(\operatorname{Gr}_n(H_{-k,k}))$. Then $i^*$ is surjective. For each $k$, $$MU^{\mathrm{odd}}(\operatorname{Gr}_n(H_{-k,k}))= 0,$$ hence \begin{align} \cdot^*(\operatorname{Gr}_n(H))\rightarrow& \displaystyle\lim_{\overleftarrow{k}}MU^*(\operatorname{Gr}_n(H_{-k,k}))\notag\cdot &= MU^*(\operatorname{Gr}_n(H^\infty))\notag\cdot &\cong MU^*(\operatorname{Gr}_n(H))\notag \end{align} is surjective. \end{proof}
\begin{thm} For a compact connected semi-simple Lie group $G$, $$ \rho: \cdot^*(LG/T)\rightarrow MU^*(LG / T) $$ is surjective. \end{thm} \begin{proof} As $LG/T$ has no odd dimensional cells,the Atiyah- Hirzebruch spectral sequence for $MU^*(LG/T)$ collapses. Hence it suffices to show that the composition \cdot \cdot^*(LG/T) \rightarrow MU^*(LG/T) \rightarrow H^*(LG/T, \mathbb Z) \cdot is surjective. Since $H^*(LG/T,\mathbb Z)$ is generated by the Schubert classes $\varepsilon^w (w \in W)$ dual to the Schubert cells $C_w$, and $\Sigma_w$ is dual to $C_w$, the image of the stratum $\Sigma_w$ under the composition map gives $\varepsilon^w$, establishing the desired surjectivity. \end{proof}
Similarly, we have \begin{thm} For a compact connected semi-simple Lie group $G$, $$ \rho : \cdot^*(\Omega G) \rightarrow MU^*(\Omega G) $$ is surjective. \end{thm}
\section{Cobordism classes related to Pressley-Segal stratifications and some calculations.} In this section, we show that the stratifications introduced by Pressley $\cdot$ Segal \cite{segal-pressley} give rise some further interesting cobordism classes in $\mathcal U^*(LG /T)$. \par The Grassmannian $\operatorname{Gr} H$ of \cite{segal-pressley} is a separable Hilbert manifold, and the stratum $\Sigma_S \subseteq \operatorname{Gr} H$ is a locally closed contractible complex submanifold of codimension $\ell (S)$ and the inclusion map $\Sigma_S \rightarrow \operatorname{Gr}(H)$ is a proper Fredholm map with index $- \ell (S)$. \par Therefore, we have \begin{thm} The stratum $\Sigma_S\rightarrow \operatorname{Gr} (H)$ represents a class in $\mathcal U^{2 \ell (S)} (\operatorname{Gr} (H))$. \end{thm} These strata $\Sigma_S$ are dual to the Schubert cells $C_S$ in the following sense; \begin{enumerate} \item the dimension of $C_S$ is the codimension of $\Sigma_S$ and \item $C_S$ meets $\Sigma_S$ transversely in a single point, and meets no other stratum the same codimension. \end{enumerate} \par The loop group $LG$ acts via the adjoint representation on the Hilbert space $$ H_{\mathfrak g} = L^2 (\cdot^1 ; {\mathfrak g}_{\cdot}), $$ where ${\mathfrak g}_{\cdot})$ is the complexified Lie algebra of $G$. If $\operatorname{dim} G = n$, we can identify $H_{\mathfrak g}$ with $H^n$ and since the adjoint representation is unitary for a suitable Hermitian inner product, this identifies $LG$ with a subgroup of $LU(n)$. Then \cite{segal-pressley} shows how to identify the based loop group $\Omega G$ with a submanifold of $\Omega U(n)$, which can be itself identified with a submanifold of $\operatorname{Gr} (H_{\mathfrak g})$.
Then $\Omega G$ inherits a stratification with strata $\Sigma_{\lambda}$ indexed by homomorphisms $\lambda : \cdot\rightarrow T$. Each stratum $\Sigma_{\lambda} \subseteq \Omega G$ is a locally closed contractible complex submanifold of codimension $d_{\lambda}$, and the inclusion map $\Sigma_{\lambda}\rightarrow \Omega G$ is an admissible Fredholm map. Then \begin{thm} For each $\lambda$, the inclusion $\Sigma_{\lambda}\rightarrow \Omega G$ represents a class in $\mathcal U^{2 d_{\lambda}} (\Omega G)$. \end{thm}
If we restrict to the inverse limit $MU^*(\Omega G)$, these stratum $\Sigma_{\lambda}$ provide the basis elements of $MU^{2d_{\lambda}}(\Omega G)$. Since $H^*(\Omega G)$ is even graded and $MU^*$ is also even graded, the Atiyah-Hirzebruch spectral sequence $$ H^*(\Omega G; MU^*)\Rightarrow MU^*(\Omega G) $$ collapses, hence we have an isomorphism $$ \mathcal U^* (\Omega G) /{\operatorname{ker} \rho} \cong MU^*(\Omega G) \cong H^*(\Omega G)\otimes MU^*. $$ For $G= SU(2)$, we have $$ \mathcal U^* (\Omega SU(2)) /{\operatorname{ker} \rho} \cong MU^*(\Omega SU(2)) \cong \Gamma_{\cdot}(\gamma)\otimes MU^*, $$ where $\Gamma_{\cdot}(\gamma)$ is a divided power algebra with the $\cdot$-module basis $\gamma^{[n]}$ in each degree $2n$ for $n\geqslant 1$.
\par Such stratifications also exist for the homogeneous space $LG /T$. \begin{thm}For $w \in \widetilde{W}$, the inclusion $\Sigma_{w}\rightarrow LG/T$ represents a class in $\mathcal U^{2 \ell (w)} (L G / T)$. \end{thm}
Similarly, if we restrict to the inverse limit $MU^*(LG/T)$, we have an isomorphism $$ \mathcal U^* (LG/T) /{\operatorname{ker} \rho} \cong MU^*(LG/T) \cong H^*( LG/T)\otimes MU^*. $$ For $G= SU(2)$, we have $$ \mathcal U^* (LSU(2)/T) /{\operatorname{ker} \rho} \cong MU^*(LSU(2)/T) \cong \Gamma_{\cdot}(x_0, x_1)/I_{\cdot}\otimes MU^*, $$ where the ideal $I_{\cdot}$ is given by $$ I_{\cdot} = \left(x_0^{[n]} x_1^{[m]} - \binom{n + m - 1}{m} x_0^{[n + m]} - \binom{n + m - 1}{n} x_1^{[n + m]}: \cdot m,n \geqslant 1\right) $$ and which has the $\cdot$-module basis $\{x_0^{[n]}, x_1^{[n]}\cdot$ in each degree $2 n$ for $n \geqslant 1$.
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BMC Genomics
Algorithms for differential splicing detection using exon arrays: a comparative assessment
Karin Zimmermann1,
Marcel Jentsch2,
Axel Rasche3,
Michael Hummel4 &
Ulf Leser1
BMC Genomics volume 16, Article number: 136 (2015) Cite this article
The analysis of differential splicing (DS) is crucial for understanding physiological processes in cells and organs. In particular, aberrant transcripts are known to be involved in various diseases including cancer. A widely used technique for studying DS are exon arrays. Over the last decade a variety of algorithms for the detection of DS events from exon arrays has been developed. However, no comprehensive, comparative evaluation including sensitivity to the most important data features has been conducted so far. To this end, we created multiple data sets based on simulated data to assess strengths and weaknesses of seven published methods as well as a newly developed method, KLAS. Additionally, we evaluated all methods on two cancer data sets that comprised RT-PCR validated results.
Our studies indicated ARH as the most robust methods when integrating the results over all scenarios and data sets. Nevertheless, special cases or requirements favor other methods. While FIRMA was highly sensitive according to experimental data, SplicingCompass, MIDAS and ANOSVA showed high specificity throughout the scenarios. On experimental data ARH, FIRMA, MIDAS, and KLAS performed best.
Each method shows different characteristics regarding sensitivity, specificity, interference to certain data settings and robustness over multiple data sets. While some methods can be considered as generally good choices over all data sets and scenarios, other methods show heterogeneous prediction quality on the different data sets. The adequate method has to be chosen carefully and with a defined study aim in mind.
Alternative Splicing is an important mechanism for providing the protein diversity essential for eukaryotes [1]. One of the central roles of different isoforms is the development of tissue specific properties [2]. Due to its high complexity, the alternative splicing machinery is strongly susceptible to errors leading to aberrant isoforms with a lack of, or sometimes even opposing, function to the protein intended [3]. One possibility to capture such alterations is provided by exon arrays. In comparison to their more coarse-grained predecessors, the gene arrays, they offer an exon-based resolution [4]. This possibility led to wide-spread usage, reflected by over 15,000 samples across many different tissues deposited into GEO [5].
The detection of altered expression on the exon level is more challenging than gene based analyses. On one hand, changes in expression levels might be more subtle, which makes it harder to distinguish signal from noise. On the other hand, changes in the expression of the gene has to be taken into account to avoid false positives as well as false negatives. To accomplish this task, exon expression is usually normalized to the corresponding gene expression. Figure 1 visualizes a situation where a comparison only on exon level would lead to the opposite of the desired result, as the only exon differentially spliced would gain the lowest evidence for DS.
Differential exon expression. The second left exon in tissue A is differentially spliced. A comparison on exon level only would lead to the opposite of the desired result, as the only exon differentially spliced would gain the lowest evidence for DS.
Besides using exon arrays, the challenge of DS detection also can be addressed with next generation sequencing (NGS), i.e. RNA-seq [6]. However, the NGS approach suffers from two main disadvantages. First, sequencing is still pricey; sequencing a sample is about four times more expensive than an exon array ($500 (Affymetrix) vs. $2000 (paired end, several providers)). Second, there are many more institutions having the know-how and the equipment, and, more importantly, the downstream analysis experience, to conduct exon array experiments than ones who have the facilities for NGS.
Besides, the wealth of existing expression data from exon arrays constitutes an important basis for many scientific questions. This led to a variety of algorithms for differential splicing detection developed over time. Different approaches were taken to solve the task. Most of the methods, such as MIDAS [7], use a statistical approach. Other methods combine statistics with the exploitation of the preprocessing results (e.g. FIRMA [8]) or with a refined probe selection procedure (e.g. MADS [9]). ANOSVA's [10] strong point is its independence from transcript annotation which makes it applicable to poorly annotated data. Moreover, it is designed to be very specific, which is confirmed by our evaluations. SplicingCompass, a graphical approach based on angles between exons, inherently distinguishes between differences in gene and exon expression. ARH [11] is specifically designed to be robust with respect to the number of exons per gene. These differences make it impossible to compare methods analytically, which calls for careful empirical studies to identify the best tool for a given scenario.
Here, we report on, to our knowledge, the most comprehensive comparative assessment of algorithms for DS detection on exon arrays. We compared and evaluated nine different methods for the detection of differential splicing from exon arrays. We discerned the performance and challenges for each method over a range of different parameters. Using a comprehensive artificial dataset we compared the impact of different expression levels, numbers of exons per gene, different amounts of differentially spliced samples per condition as well as the influence of different group sizes. Additionally, we applied all methods to two well studied and partly RT-PCR validated cancer data sets [12,13].
We included, to our knowledge, all published methods where an implementation was available: MADS, MIDAS, SI [7], PAC [7,14], ANOSVA, ARH, SplicingCompass [15] and FIRMA. We furthermore incorporated KLAS [16], a novel method introduced in this work. Note that we did not use FIRMA for evaluation on artificial data as we used the model proposed by the authors of [8] (on the basis of which FIRMA was developed) for the generation of our data. However, we applied FIRMA to the two experimental data sets. We had to skip methods with no implementation, like Remas [17].
In the following we give a brief description of each method; for details we refer the reader to the original publications.
The Splicing Index (SI) is similar to the fold change (FC) often used on the gene level. As opposed to the FC, exon expression is first normalized to the corresponding gene expression before calculating the ratio between two conditions. ARH, an information theoretical approach based on Shannon's entropy, computes the splicing deviation between conditions for every exon and transforms it into a probability for differential splicing. A gene-wise entropy computed from the probabilities is used as final quantification of DS. As with SI, MIDAS uses gene level normalized exon values. Unlike SI, a statistical test determines whether a significant difference, i.e. DS, is observed. MADS takes advantage of an elaborate gene signal estimator for probe-wise SI computation and assesses its significance with a t-test. The final p-value for an exon aggregates the singular probe-level p-values. As the input to all other methods compared is based on exon level, we adopted MADS to work on this level as well for comparability reasons. We will therefore refer to this modified method as MADS'. The underlying assumption in PAC is the proportionality of exon expression to its corresponding gene expression. Deviation from exon to gene expression results in low correlation and therefore indicates DS. ANOSVA detects DS by applying statistical tests to the parameters of a fitted exon expression model. SplicingCompass, originally developed for NGS data, can easily be adapted to exon array data. The idea is to access the significance of difference between angles spanned by exon tuples in one condition compared to the ones in the other condition [15]. FIRMA deduces scores for DS by searching for a high difference between estimated and observed expression.
KLAS is a novel method and is therefore described in more detail. It uses a similar approach as ARH, but relies on the Kullback-Leibler divergence in the last step. The Kullback-Leibler divergence is an indicator for the variety of two probability distributions. For each condition c i ∈{c 1…c n } the deviation d of the expression of every exon e from its gene g as in Equations 1 and 2 is computed.
$$\begin{array}{@{}rcl@{}} &d_{e,c_{1}}=x_{e,c_{1}}-\textbf{x}_{c_{1}} \end{array} $$
((1))
$$\begin{array}{@{}rcl@{}} &p_{e,g}=\frac{2^{d_{e,c_{i}}}}{\sum\limits_{e}2^{d_{e,c_{i}}}} \end{array} $$
$$\begin{array}{@{}rcl@{}} &Q_{c_{1}}=\frac{{quant}_{0.75}\left(d_{e,c_{1}}\right)}{{quant}_{0.25}\left(d_{e,c_{1}}\right)} \end{array} $$
$$\begin{array}{@{}rcl@{}} &kl(c_{1},c_{2})= \end{array} $$
$$\begin{array}{@{}rcl@{}} &Q_{c_{1}} \sum \limits_{e} p_{e,c_{1}}log \frac{p_{e,c_{1}}}{p_{e,c_{2}}} + Q_{c_{2}} \sum \limits_{e} p_{e,c_{2}}log \frac{p_{e,c_{2}}}{p_{e,c_{1}}} \end{array} $$
These deviations are turned into a probability distribution per gene and condition, such that the contribution of every exon to the expression of the gene can be denoted by Equation 3. This is a major difference to ARH, which assesses one probability distribution for both conditions based on the deviation from the median exon ratio between conditions. To account for the deviation within a gene, the interquartile range (see Equations 4 and 5) is computed, equivalently to ARH, yet here is used to compare two conditions based on a modified Kullback-Leibler divergence as formulated in Equation 7 instead of the Entropy corrected by its theoretical maximum as for ARH. The main difference between KLAS and ARH thus is the level at which the entropy, respectively the Kullback-Leibler divergence, (i.e. relative Entropy), is computed. While entropy is a feature of one probability distribution, the Kullback-Leibler divergence is an indicator for the variety of two probability distributions. The comparable performance to ARH ascertains the information theory as adequate tool for robust predictions. Where ARH is constrained to case control studies the approach to establish the probability distribution within the samples allows extension of the analyses to more than two conditions.
The performance of each method for differential splicing detection is influenced by many factors. A detailed analysis of the properties inherent to the different methods can only be achieved by using specifically designed artificial test data. To this end, we generated a range of synthetic data sets using the model from [8] applying multiple parameter allocations in many combinations (Table 1) using default settings, i.e., cmean=7/10 is chosen for low/high expression. We chose the model of [8] because it is the most fine-grained model we are aware of.
Table 1 Parameters: Values used for the different parameters tested
Specifically, we studied the influence of the number of exons per gene (enum∈{10,30}), the expression intensity (expr∈{h i g h,l o w}), the number of samples (snum∈{15:15,15:5}) per group as well as the percentage of differentially spliced samples (pcnt∈{60,100}). The combination of these four parameters with two allocations each led to a total of 16 scenarios yielding a detailed insight that is important when choosing the adequate method for a given dataset or for a certain purpose.
In each scenario we generated 200 simulated genes. While 100 genes were specific to the parameter criteria in addition to displaying differential splicing events (true positives (TP)) the remaining 100 genes, designed as true negatives (TN), show no altered exon expression. Thus, probably the most challenging of the 16 data sets for a DS detection method (see also Table 1) consisted of (1) one condition containing 15 samples and a second condition containing only 5, (2) low expression intensity, (3) only 60% of the samples in a group exhibiting differential splicing and (4) a high number of exons per gene.
It is undoubtedly more demanding to detect DS in a small group where not all samples display the event than in a large group under the same condition. Concerning the scenarios with an imbalance in group size, we therefore switched the DS event containing group for half of the TP genes. Thus, in settings with one condition containing 15 samples, the other one 5 samples and DS was only simulated in 60% of the samples, half of the TP genes show the DS event in the small group and half of them in the large group.
In addition to the synthetic data sets, we evaluated all methods including FIRMA on two well studied cancer data sets. We declare that we used no primary material from human or animals. All exon array data used were already published and are publicly available as stated in the corresponding articles. The first is provided by Affymetrix [12] and consists of 20 arrays, 10 colon cancer samples as well as their paired control. DS results were partly validated by RT-PCR. As a positive control (TP) we used all 18 probe set IDs indicated in the section 'differentially spliced between tissue types' and one additional probe set from the section 'previously reported splicing events in colon cancer' (see supplementary material [12]) that was positively validated. The negative control (TN) was formed by the 10 probe set IDs in the section 'alternatively spliced but not differential between tissue types' (see supplementary material [12]). Mapping to our data (we used only core exons and the human genome version 19) led to 12 TP and 8 TN probe sets corresponding to 10 (TP) and 8 (TN) genes respectively. We also applied all methods on a lung cancer data set [13] consisting of 36 paired samples, 18 normal and 18 NSCLC. The study provides validation data for 3 TN and 19 TP examples of DS.
Preprocessing and normalization of the cancer data sets was performed as proposed in [21].
For the evaluation we determined, for each scenario, accuracy (ACC) or AUC in the cases where no binary classification was applicable. Furthermore, we quantified sensitivity and specificity for more fine-grained insights. Note, that in the case of binary classification (DS event / no DS event) accuracy corresponds to the area under the curve (AUC).
Some of the methods produce p-values indicating the certainty of a DS event taking place, while PAC, KLAS, ARH and SI output a heuristic score. To achieve comparability and avoid cutoff problems, we also derived a p-value for all score-based methods using an exact Monte Carlo permutation test [18]. Applied to the scores, a gene wise p-value is computed with a significance level of α=0.05. Nevertheless, we quantify performance on the basis of scores as well.
As stated, score based methods exhibit the difficulty of choosing a cutoff at which a result is believed to be relevant. There are best practices for some methods (SI is mostly used with a cutoff of 1.5 [19] or 2 [20]) or recommendations for others (ARH = 0.03 [11]) yet no appropriate value is known for PAC and KLAS. We therefore add a second evaluation for the score-based methods only based on AUC. No binary classification, as in the p-value-based evaluation, is applied in this case.
Firstly, we report on the results for simulated data. The examined parameters (section "Synthetic data") were evaluated by p-value for all methods as well as by score for the score based methods only (for results see Additional file 1). Analysis of variance was applied to determine the significance of parameter influence (see also Additional file 1: Section "Significance of parameter influence" and Table 1).
Subsequently, the results on the colon and lung cancer data sets were reported with a focus on the RT-PCR validated results. As in the case of simulated data, accuracy, sensitivity and specificity was used to evaluate performance.
P-value based evaluation
An overview on the accuracy over all scenarios was visualized in Figure 2 using hierarchical clustering (euclidean distance, complete linkage) of methods as well as scenarios. The method performing best for one scenario was indicated by an asterisk (multiple maxima per column are possible). The most striking observation was the clear superiority of MADS', which performed equally well independent of data-imposed challenges.
P-value based accuracy, i.e. binary AUC for all scenarios. Asterisks indicate highest values per scenario, multiple maxima are possible. Column names encode scenarios in the order expression.exons.percent.samples, thus H.10.100.5 describes the scenario with high expression, 10 exons per gene, 100 percent spliced samples in the respective group and 5 versus 15 samples per group.
While most of the methods achieved good results in the 'easy' cases of equal group size and consistent splicing events, accuracy dropped quickly when sample sizes in groups diverged, less samples per condition were spliced, or expression intensity decreased. MADS' is closely followed by ARH, SI, SplicingCompass and KLAS, which showed similar behaviour (Figure 2).
The third-best method cluster consists of ANOSVA and MIDAS. The two performed well in the easy scenarios of sufficient sample numbers and 100% AS events in one group. As circumstances got more challenging, a rapid decay in accuracy could be observed.
MADS'. This algorithm showed a unique performance not only concerning efficiency but also in the sensitivity to parameter influences (see Additional file 2: Figure S2). The most obvious interference was incurred by the expression level. While in the high expression range almost no FP were observed, FP rate increased significantly in the scenarios with low expression. A second observation correlating with the expression level was the dependence on the number of exons contained in a gene. In low expression ranges MADS' performed consistently better in scenarios with a high number of exons per gene, while in high-expression scenarios it performed better with a low number of exons per gene.
ARH, SI, SplicingCompass and KLAS. The four methods behaved similarly in terms of classifying the genes actually spliced differentially (Additional file 2: Figure S2). All showed a clear performance advantage in the case of high expression also sharing the outliers: in the scenarios with 60% DS events and low sample size, genes containing the DS event in the small sample group were not classified correctly (see red squares in upper left area, Additional file 2: Figure S2). All other methods performed homogeneously bad or well irrespective of the fact that the DS event was not contained in the majority class. While ARH displayed a rather homogeneous response for the control genes, SI was strongly impacted by the number of samples per group. SplicingCompass displayed the lowest number of FPs in this group, as the consideration of all pairwise angles requires relatively high effect sizes. Systematic influences observable by Additional file 2: Figure S2 were exon number and percentage of samples displaying differential splicing.
ANOSVA, MIDAS and PAC. These methods formed the third method-cluster showing results very similar to each other throughout all scenarios. While ANOSVA and MIDAS were highly specific, ANOSVA educed not a single FP at the cost of a slightly lower sensitivity compared to MIDAS (Figure 2, Additional file 2: Figure S4). The most obvious difference between the two was the difficulty of ANOSVA to deal with a high number of exons. MIDAS, on the other hand, performed independently of this parameter. As expected from a statistical method, the parameter impacting the performance most was the percentage of samples displaying the DS event in one group. Both methods failed to detect the DS event in most of the TP cases.
Thus, if avoiding false positives is of high importance, MIDAS and even more, ANOSVA, are a suitable choice. PAC failed to detect most of the positive events and also led to some FPs independently of the underlying scenarios.
For more details on the significance of parameter influence see Additional file 1: Section "Significance of parameter influence".
Sensitivity and specificity
Depending on the aim of a potential study it can be important to choose a method explicitly focusing on high sensitivity or high specificity. While the earlier assures the correct detection of a sample having a certain property, the later describes the ability to not detect samples not having this property, i.e. a certain disease. High sensitivity is required in all areas of diagnostics; when it comes to biomarker detection, a high specificity might be of higher interest. As biomarkers are usually used for screening of large populations for preventive reasons, a high number of false positives could lead to an increased workload of testing or unnecessary treatment [22].
Average sensitivity and specificity over all scenarios is displayed in Figure 3 and Additional file 2: Figure S4. High specificity values for ANOSVA, PAC and MIDAS came at the cost of sensitivity. While SI and KLAS presented very similar values - with KLAS showing a slightly better result - ARH was more focused on specificity. In between performed SplicingCompass with very high specificity yet lower sensitivity. Additional file 2: Figure S4 gives a scenario-wide overview on specificity and sensitivity. Sensitivity was clearly dominated by MADS', followed by KLAS, exposing its strength in this category in comparison to its cluster mates.
Sensitivity and specificity averaged over all scenarios.
We applied all 9 methods - including FIRMA - to two partly RT-PCR validated data sets, one from colon cancer and one from lung cancer. First we investigated the overall predictions of every method to assess the number of prognosticated differential DS events. Second, we compared the predictions based on TPs and TNs confirmed by RT-PCR. The p-value cutoff is set to 0.05.
Colon cancer data MADS' predicted the highest number of DS events (>13000) (Additional file 2: Figure S5). ARH, KLAS and ANOSVA produced approximately the same gene number (about 2000) while slightly differing in the gene set. SI and FIRMA proposed about 1000 DS genes while PAC, MIDAS and Splicing Compass showed the most conservative result (less than 500 genes). Thus, MADS' was an outlier in the number of predicted DS events, claiming the sought event in over 70% of the genes.
When considering only the validated results ARH and FIRMA appeared as the most accurate methods (see Figure 4) closely followed by MIDAS. KLAS and ANOSVA displayed relatively good results whereas the remaining three methods showed either a high specificity at the cost of sensitivity (MADS') or a high sensitivity with a sacrifice of specificity (SplicingCompass, PAC), see Additional file 2: Figure S6.
Accuracy computed on the RT-PCR validated results for the colon cancer data set (left) and the lung cancer data set (right).
Lung cancer data. Again, MADS' predicted the highest number of DS events (>10000) (Additional file 1: Figure S5). ARH, KLAS, FIRMA, and ANOSVA predicted about 3000 DS events with considerable overlap in the gene set as shown in Additional file 2: Figure S8. SI nominated about 2000, MIDAS and Splicing Compass 1000 and PAC showed the most conservative result with less than 200 genes.
As the data set provided such a high verification rate, number of TN examples was very low (we used non-verified events as TN). Under such circumstances accuracy is not a good measure for performance, and we thus focused on sensitivity and specificity instead (Additional file 2: Figure S7). SplicingCompass, ANOSVA, KLAS, FIRMA, and ARH were the methods performing best. According to accuracy, FIRMA, KLAS and ARH achieved the highest values when ignoring MADS' due to its high prediction rate. Similarly, sensitivity was also dominated by FIRMA, KLAS and ARH while considering only methods with non-zero specificity values. When focusing on specificity, SplicingCompass was the clear winner followed by ANOSVA, KLAS, ARH and SI, all ranging on the second place.
Though a variety of methods for the detection of DS based on exon array data has been developed over time, no broad evaluation concerning their advantages and drawbacks in regard to (combined) influences of properties such as the number of samples, expression intensity or exon number has been performed yet. In this work we evaluated the impact of an extensive set of parameter combinations on the performance of eight methods. Additionally, we assessed all methods and a ninth one with respect to validated experimental data. In contrast to related work which focused on the comparison based on experimental data [11] and thus on fixed scenarios, we also exploited simulated data sets to study the (combined) influence of various properties of differentially spliced genes and their measurements in exon arrays.
A rank comparison of accuracy-based results is shown in Table 2, putting results on synthetic and on real data sets side-by-side together with the ranking reported in [11].
Table 2 Result summary and comparison
Here, we present the outcome of synthetic, colon cancer and lung data. Concerning the accuracy based results, some methods ranked consistently low (ANOSVA, SI, SplicingCompass and PAC), others consistently high (ARH, FIRMA, KLAS and MADS') while MIDAS included a positive outlier. Nevertheless, results on experimental data should be handled with care due to the unbalanced nature and small size of the evaluation data in these data sets. Recall that accuracy is highly susceptible to a diverging number of positive and negative examples. Especially in the case of MADS', which predicted a high number of DS genes, combined with an disproportionate high number of positive examples in the lung cancer data set this is an issue.
Algorithmic performance explained
Clearly, the performance of different algorithms was influenced differently by the various parameters and the different data properties, such as effect size and variance. To shed more light into the cause of these differences, we here sought to explain differences in the method's performance in terms of their underlying mathematical formulation of the problem.
Exon number and DS exons per gene
Two methods - ANOSVA and SplicingCompass - were significantly affected by the number of exons per gene, i.e. they display a better performance in the low exon number scenario. This is remarkable, as a major concern of most other algorithms is a rising number of FPs with increasing exon number due to parallel tests. In the case of ANOSVA, the reason is that, the higher the number of exons, the more improbable it becomes to obtain significant predictions for TPs as the number of DS exons remains constant. This is underpinned by the observation, that predictions were better in the second half (genes 50 to 100) of TPs, where two instead of one exon is modeled as DS. The same reason applies to SplicingCompass, a statistics-based method, which accesses the difference between exon angles within and between groups. The higher the number of exons - while the number of DS events is constant - the lower is the ratio of angles representing a DS event. This impedes the detection of differences between groups.
Interestingly, MIDAS, also a statistical method, was not affected by these parameters. Unlike SplicingCompass and ANOSVA, MIDAS directly takes into account the gene expression normalized exon expression, i.e. effect size, and applies a separate test for every exon. The number of exons per gene is thus not as important. In contrast, SplicingCompass and ANOSVA operate on a gene based level.
Sample number and variance
For any method based on statistical tests, one expects that a higher number of samples improves performance as it increases test power. As expected, this behaviour was observed for ANOSVA and MIDAS, both inherently statistical methods. However, the same (positive) effect also could be observed for SI, ARH and KLAS, which do not perform tests. The explanation is that all these three methods use permutation tests, which become more stable with increasing numbers of samples. The effect was the strongest for SI with those genes which were not differentially spliced (see Additional file 2: Figure S2).
Expression level and effect size
All methods were significantly affected by the expression level: The lower, the worse were the results. This is to be expected, as low expressions means a less clear separation between signal and noise. As expression decreases, also the variance decreases, which in turn makes it more probable to confuse spurious 'effects' as splicing events.
MADS' for example showed this behaviour for the non DS genes, by producing a high number of FPs in the low expression scenarios which is not visible for similar methods, like for instance MIDAS. While MIDAS computes an exon level SI and subsequently applies statistical testing, MADS' produces a gene wise aggregate as final p-value. The approach of MADS' is thus more sensitive and yields performance improvements but can, on the other hand, also be too sensitive for other scenarios (e.g., see Additional file 2: Figure S2).
The rather simple splicing index performed well in most of the scenarios, although this method does not consider variance and does not perform any kind of deviation correction. However, this is due to the structure of the generated data, while various influences alter the challenges imposed by the data, the one affecting SI most - a small number of rather drastic outliers - was not contained in the scenarios. Thus the focus on effect size led to remarkable results.
Percent of spliced samples
The greatest impact due to this parameter is observed for statistical methods, i.e. ANOSVA, MIDAS and SplicingCompass. As they are by design susceptible to variance, fluctuations like in the case of decreased sample ratio with DS events per group (i.e. a lower percentage of DS samples) lowers performance as increased variance prevents effects from being significant.
Effect size, variance and gene level correction
As already mentioned in the previous paragraph, statistical methods in general are rather conservative in predicting DS events. One root of this behaviour is their test-basis, but other effects come on top. MIDAS uses gene-normalized expression values instead of exon expression values and thus requires a fairly great effect as the normalization is rather drastic. ANOSVA applies an ANOVA on a so-called interaction term derived from a fitted linear model which further smoothes away differences. Other methods are less strict in these regards. For instance, ARH uses the median exon ratio between groups for correcting for the underlying gene expression. Compared to MIDAS, which directly uses exon to gene ratio, the approach of ARH often results in a less pronounced correction which better preserves effect strength. Splicing Compass accesses the difference between exon angles within and between groups. It does not perform any explicit gene level correction, but implicitly all pairwise angels are considered, resulting in an indirect and rather weak form of normalization. Again, this helps this method to increase its sensitivity.
The ambivalence of MADS'
Combining the results of simulated and experimental data completes the picture of MADS'. While leading performance for simulated data, MADS' seemed to overrate DS events in the experimental settings. The excellent performance in the artificial scenario reflects the strong sensitivity of the method: relatively 'hard' scenarios are still positively identified, settings in which other methods clearly voted against an DS event. According to our experiments MADS' can not be recommended for the pure prediction of DS events, but we consider it highly suitable for ranking DS candidates because genes with a very low MADS' p value very likely show differential splicing.
Comparison to related work
A comparison of MIDAS, FIRMA, SPLICE [23], ARH, PAC, SI, ANOSVA, MADS', and correlation [24] has been performed previously [11]. However, the evaluation of Rasche et al. used only a single scenario by benchmarking on different tissue data, while our main interest lies in the susceptibility of the methods to different data properties. Furthermore, [11] focused on ranking performance and evaluated based on AUC instead of accuracy, sensitivity and specificity. Using AUC avoids the problem of choosing a cutoff, but precisely the proper selection of a cutoff decides on the usefulness of a method in reality. Due to such differences, a comparison of our results with those from [11] should be interpreted carefully as the two measures quantify a different matter. Rank product of the methods led to the order as indicated in the last column of Table 2. The most striking difference is the good performance of PAC. PAC strongly depends on the gene estimate and the exon estimate used. Furthermore, we compute p-values from PAC scores, which were much more susceptible to noise than for example the SI and therefore had difficulties leading to significant results.
Further comparative work was done by Laajala et al. [25]. Though focusing on preprocessing, they implicitly compared FIRMA, SI and MIDAS, indicating that MIDAS develops its strength with growing number of DS exons.
Which method for which data?
Depending on the research question and the experimental data, different methods pose an appropriate choice. As practically all methods showed a significant dependency on the expression level and the amount of DS samples per class the two parameters are of no help for method selection. If sample number is low and / or imbalanced, SplicingCompass is the most reasonable choice according to our evaluation. Independence on the number of exons is best achieved by ARH, while KLAS, SI and MIDAS pose similarly good choices. High specificity throughout the data sets was provided by ARH, SplicingCompass and MIDAS. When it comes to the most sensitive methods FIRMA, ARH and KLAS fulfill the task best. As validation of results is expensive and time-intensive most studies are interested in high sensitivity and specificity as well as in robustness of the method. According to our evaluation, ARH meets these requirements best.
Over time a variety of methods for the detection of DS has been published, each of them with different characteristics regarding sensitivity, specificity, interference to certain data settings and robustness over multiple data sets. While some methods, such as ARH, can be considered as generally good choices over all data sets and scenarios, other methods show heterogeneous prediction quality on the different data sets. The adequate method has to be chosen carefully and with a defined study aim in mind.
To avoid an unmanageable flood of data scenarios we restricted our simulations to cases, where one and two exons are differentially spliced per gene. Naturally, this does not represent the spectrum of actually occurring DS events. Thus, based on our study, an important question to address in future work would be the susceptibility of the methods to the number of DS exons per gene. Further improvement could be provided by varying the noise level in data generation to assess method robustness.
An important topic when discussing exon arrays is its replaceability with RNA-seq. Next generation sequencing is the younger technology and therefore under constant development. While claimed to be the more accurate technology, it still displays difficulties in certain areas such as high FP and FN values in low expression ranges [26]. Therefore, we should probably see this technologies as complementary rather than preferable.
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This work was funded by BioBankCloud, TRR54, and the Max-Planck-Society.
Department of Computer Science, Knowledge Management in Bioinformatics, Humboldt Universitaet zu Berlin, Rudower Chaussee 25, Berlin, 12489, Germany
Karin Zimmermann & Ulf Leser
Department of Mathematics and Computer Science, Freie Universitaet Berlin, Berlin, Germany
Marcel Jentsch
Department of Vertebrate Genomics, Max Planck Institute for Molecular Genetics, Ihnestr. 63-73, Berlin, 14195, Germany
Axel Rasche
Institut fuer Pathologie CBF, Charite - Universitaetsmedizin Berlin, Hindenburgdamm 30, Berlin, 12200, Germany
Michael Hummel
Karin Zimmermann
Ulf Leser
Correspondence to Karin Zimmermann.
KZ, MJ and UL conceived the research. KZ carried out the experiments, analysed the results and drafted the manuskript. MJ developed KLAS, MJ and AR implemented the DS methods. AR, MH, and UL helped to revise the manuscript and to interpret the data. All the authors read and approved the final manuscript for publication.
Additional file 1
Supplementary materials.
Supplementary figures.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Zimmermann, K., Jentsch, M., Rasche, A. et al. Algorithms for differential splicing detection using exon arrays: a comparative assessment. BMC Genomics 16, 136 (2015). https://doi.org/10.1186/s12864-015-1322-x
Alternative splicing
Differential splicing
Exon arrays
Method comparison
Parameter influence
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\begin{document}
\title{Lipschitz continuous solutions of the Vlasov-Maxwell systems with a conductor boundary condition}
\author{Yunbai Cao} \address{Department of Mathematics, Rutgers University, Piscataway, NJ 08854; email: [email protected]} \author{Chanwoo Kim} \address{Department of Mathematics, University of Wisconsin-Madison, Madison, WI 53706; email: [email protected]}
\begin{abstract} We consider relativistic plasma particles subjected to an external gravitation force in a $3$D half space whose boundary is a perfect conductor. When the mean free path is much bigger than the variation of electromagnetic fields, the collision effect is negligible. As an effective PDE, we study the relativistic Vlasov-Maxwell system and its local-in-time unique solvability in the space-time locally Lipschitz space, for several basic mesoscopic (kinetic) boundary conditions: the inflow, diffuse, and specular reflection boundary conditions. We construct weak solutions to these initial-boundary value problems and study their locally Lipschitz continuity with the aid of a weight function depending on the solutions themselves. Finally, we prove the uniqueness of a solution, by using regularity estimate and realizing the Gauss's law at the boundary within Lipschitz continuous space. \end{abstract}
\maketitle
\subsubsection*{\textbf{\large{Introduction}}}
Plasma is the most abundant form of ordinary matter in universe, being mostly associated with stars. The Sun, our nearest star, is composed of 92.1$\%$ hydrogen and 7.8$\%$ helium by number, and 0.1$\%$ of heavier elements. At the central core, hydrogen burns into helium (so-called the p-p chain of reactions starting from the fusion of two protons into a nucleus of deuterium), which is the major reaction that drives the sun’s radiance (see the famous B$^2$FH paper \cite{B2FH} for details).
\begin{wrapfigure}{r}{0.25\textwidth}
\centering
\includegraphics[width=0.90\linewidth]{exobase}
\tiny{Figure 1. The different transition regions in a stellar (\cite{PP})}
\label{fig:wrapfig}$^1$ \end{wrapfigure}
At the upper atmosphere of the Sun (solar corona), electrons and protons escape from the solar corona (upper atmosphere), while traces of heavier elements have been identified (\cite{PLL}). This emission of plasma particles is called solar wind. The solar corona can be decomposed according to the Knudsen number of plasma. At low altitude the collision is dominant (Knudsen number $\ll $ density scale height), and hence the particles are assumed to be in hydrostatic/hydrodynamic equilibrium of MHD-type systems. Above this regime (exosphere), the collision rate between particles is assumed to be negligibly small: Knudsen number is about the density scale height of the Sun, which is an order of 100km. These two extreme Knudsen number regimes are separated by a narrow transition regime which is called the exobase (See Figure 1 adopted from \cite{PP}\footnote{permission to use the figure granted by \url{https://www.agu.org/Publish-with-AGU/Publish/Author-Resources/Policies/Permission-policy\#repository}} ). Above the exobase, there have been extensive research activities on the solar wind using collisionless Boltzmann equation (e.g. linear steady Vlasov model), which has been called the exospheric solar wind models. In the early 60s, Chamberlain suggested the ``solar breeze model" that the radial expansion of the solar corona results from the thermal evaporation of the hot coronal protons out of the gravitational field of the Sun \cite{Chamberlain}. In this model the ambient polarization electric field is implemented as a well-known Pannekoek-Rosseland (PR) electric field \cite{Pannekoek, Rosseland}, which will be discussed at \eqref{PR}. \textit{In this paper, we are interested in a kinetic description of the exospheric solar wind using the initial-boundary value problem of the full relativistic Vlasov-Maxwell system subjected to ambient polarization electric field, geomagnetic field, and gravitation.}
When the collision effect is negligible, the master equation describing dynamics of two species plasma (an average of 95\% of the solar wind ions are protons \cite{PLL}) is the relativistic Vlasov-Maxwell system (RVM) \begin{equation} \label{VMfrakF_I} \begin{split}
\partial_t f_\pm + \hat v_\pm \cdot \nabla_x f_\pm + \mathfrak F_\pm \cdot \nabla_v f_\pm = 0, & \ \ \text{ in } \mathbb{R}_+ \times \Omega \times \mathbb R^3 ,
\\ f_\pm(0,x,v) = f_{0,\pm}(x,v) , & \ \ \text{ in } \Omega \times \mathbb R^3.
\end{split} \end{equation} Here $f_\pm = f_\pm(t,x,v) \ge 0 $ represents the density distribution functions for the proton $(+)$ and electron $(-)$ respectively. The relativistic velocity is \begin{equation} \label{rel_v_I}
\hat v_\pm = \frac{v}{\sqrt{m_\pm^2 + |v|^2 / c^2}}, \end{equation} where $m_\pm$ is the magnitude of the masses of protons and electrons, and $c$ is the speed of light.
The Lorentz force $\mathfrak F_\pm$ consists of self-consistent field electromagnetic fields plus given polarization electric field $E_{\text{ext}}$, geomagnetic field $B_{\text{ext}}$, and gravitation:
\begin{equation} \label{frakF_I} \mathfrak F_\pm = e_\pm \left(E+ E_{\text{ext}} + \frac{\hat v_\pm}{c} \times (B + B_{\text{ext}}) \right) - m_\pm g \mathbf e_3, \end{equation} The self-consistent fields $E(t,x)$, $B(t,x)$ are coupled with $f_\pm$ through the inhomogeneous Maxwell equations \begin{equation} \label{Maxwell_I} \begin{split}
\partial_t E & = c \nabla_x \times B - 4 \pi J, \, \ \ \ \nabla_x \cdot E = 4\pi \rho, \ \ \text{ in } \mathbb{R}_+ \times \Omega,
\\ \partial_t B & = - c \nabla_x \times E, \, \ \ \ \nabla_x \cdot B = 0, \ \ \text{ in } \mathbb{R}_+ \times \Omega, \end{split} \end{equation} with initial conditions \begin{equation} \label{BEinitdata_I} E(0,x) = E_0(x), \ \ B(0,x) = B_0(x), \ \ \text{ in } \Omega. \end{equation} Here, the electric density and current are defined as \begin{equation} \label{rhoJ_I} \rho = \int_{\mathbb R^3} ( e_+f_+ + e_- f_-) \mathrm{d} v, \ \ \ J = \int_{\mathbb R^3} (\hat v_+ e_+ f_+ + \hat v_- e_- f_-) \mathrm{d} v. \end{equation}
Due to its importance, there have been extensive studies on the global regularity of the Cauchy problem of RVM. Here, we only overview papers relevant to our approach, and we refer to \cite{LS, LSt2, DL} for a more complete list of references. In a classical solution context, Glassey and Strauss first studied a continuation criterion of the relativistic Vlasov-Maxwell system in the whole space $\mathbb R^3$ in \cite{GS}, using so-called the Glassey-Strauss representation, which is a crucial tool in our analysis of this paper. It was shown that classical solution exists for all time as long as the velocity support of the particle density function $f$ is compact. Later Klainerman and Staffilani prove the result using a different method in \cite{KS}. The work of \cite{GS} leads to substantial developments in \cite{GSc1, GSc2, GSc3, GS4, GS2, GS3}. Notably in \cite{GSc1, GSc2, GSc3}, Glassey and Schaeffer proved that in the two-dimensional and two-and-a-half dimensional case, for regular initial data with compact velocity support, the system has unique global in time solution. More recently, in \cite{LS} Luk and Strain proved a new continuation criterion for the system by showing that the classical solution exists for all time if the velocity support of $f$ is bounded after projecting to any two-dimensional plane. Then in \cite{LSt2}, they improve the result of \cite{GSc1, GSc2, GSc3} the two-dimensional and two-and-a-half dimensional case by only requiring the initial data to have polynomial decay in velocity space. In addition, they showed that in the three-dimensional case, a regular solution can be extended by assuming a bound on a certain moment of $f$. In a weak solution context, the global weak solutions of the RVM system were obtained in \cite{DL} using a velocity averaging lemma, and the questions of its uniqueness and global regularity are still open.
In many applications of plasma models, the particles are in contact with a different phase through a sharp interface, which can be considered as a (either solid or moving) boundary. In the solar wind model, under the top of exobase, the space is filled with fully ionized plasma particles with a very short mean free path, which can be considered as a perfect conductor. We set the altitude of the exobase $x_3=0$ and consider the upper half space \begin{equation} \label{domain} \Omega = \mathbb R^3_+ := \{ (x_1, x_2, x_3) \in \mathbb R^3: x_3 > 0 \} . \end{equation} At the top of the exobase, we assume a perfect conductor boundary condition for the self-consistent electromagnetic fields. Denote by $n$ the outward unit normal of $\Omega$ (which is $n=-\mathbf e_3$ for our case); $[V]$ the jump of $V$ across $\partial\Omega$: $[V](x_1,x_2)= \lim_{x_3 \downarrow 0}V(x_1,x_2,x_3)- \lim_{x_3 \uparrow 0}V(x_1,x_2,x_3)$. Then from $\partial_t B = - \nabla_x \times E$ and $\nabla_x \cdot B = 0$, we derive the jump conditions (see \cite{CKKRM} for the details) \[ \label{jump_EB} n \times [E] = 0 , \ \ \ n \cdot [B] =0. \] In other words, the tangential electric fields $E_1$, $E_2$, and the normal magnetic field $B_3$ are continuous across the interface $\partial\Omega$. Therefore, we obtain boundary conditions for a perfect conductor of the solutions $(E,B)$ to \eqref{VMfrakF}-\eqref{rhoJ}. \begin{equation} \label{E12B3bc_I} E_1 = E_2 = 0, \, B_3 = 0 , \text{ on } \ \mathbb{R}_+ \times \partial \Omega. \end{equation} The initial-boundary value problem of the Vlasov-Maxwell system with the perfect conductor boundary condition has been studied by Guo in \cite{Guo93} for general domains with boundary. By approximating the phase space via a sequence of domains and linear systems and using the compactness result of \cite{DL}, he establishes a global existence of weak solutions for RVM with the perfect conductor boundary condition for various boundary conditions of $f$. The regularity question is highly nontrivial since the stability of the ballistic trajectory depends on the sign of the normal component of the field at the boundary. As a matter of fact, in \cite{Guo96}, he constructs an example of the RVM system such that the solution immediately does not belong to $C^1$. Under the favorite sign condition of the field at the boundary, in \cite{Guo95}, he constructs regular solutions for a 1D model of the Vlasov-Maxwell system on a half line. In the proof, he introduces an important weight function $\alpha$ and establishes a crucial velocity lemma. This technique motivates us to define the kinetic weight function in \eqref{alphadef} and build a weighted regularity estimate for $f$ along with it for the RVM system with boundary. We will discuss the role of kinetic weight in Definition \ref{def:alpha} and its remarks. There are several interesting related research lines. Here we only list some of them for readers' convenience: stationary solutions of the RVM (\cite{R}), initial-boundary value problem of Maxwell system in time-dependent domains (\cite{CS}), an inverse boundary value problem of Maxwell's equations (\cite{OPS}), a dielectric boundary problem \cite{BronoldFehske}, and a non-perfect conductor boundary problem (\cite{Dantas,Matus}).
Now we consider the gravitation (and the gravitation constant $g>0$), an ambient polarization electric field $E_{\text{ext}}$, and geomagnetic field $B_{\text{ext}}$ near the exobase. As we are only interested in the dynamics near the exobase, we can assume that $E_{\text{ext}}$ and $B_{\text{ext}}$ take forms of \begin{equation} \label{EBext_I} E_{\text{ext}} = E_e \mathbf e_3 \text{ and } B_{\text{ext}} = B_e \mathbf e_3, \end{equation} where $E_e, B_e$ are the magnitude of the fields, and $\mathbf e_3$ is a unit vector $(0 \ 0 \ 1)^T$. In the early 1920's Pannekoek and Rosseland independently calculated an electric potential of a Sun-like gaseous star, which consists of fully ionized matter in isothermal equilibrium (temperature$=T$). Recall that, for the two-species model, we have $e_{\pm}$ and $m_{\pm}$ be the charge and mass of negative/positive ions, respectively. Pannekoek and Rosseland conclude that the gravitational constant $g>0$ and the polarization electrical field $E_{\text{ext}}= E_e (0 \ 0 \ 1)^T$ satisfy the following condition at the exobase: \begin{equation} \label{PR_field}
\frac{E_{e}}{-g} = - \frac{m_+- m_-}{ e_+ + |e_-|}. \end{equation} For electron/proton gaseous star ($m_+>1800 m_-$ and $e_+=1=-e_-$), this identity implies that the polarization electrical field is \textit{upward}. Moreover, from $e_+ E_e= \frac{1}{2}(m_+ - m_-) g $, we derive the \textit{Pannekoek-Rosseland condition}: \begin{equation} \label{PR}
m_+ g >2e_+ E_{e}. \end{equation} This condition crucially implies that the gravitation effect dominates ambient electromagnetic one so that the acceleration of particles would be \textit{attractive to the boundary}. We will explain the importance of the Pannekoek-Rosseland condition qualitatively when defining the kinetic weight in Definition \ref{def:alpha} and its remarks. It might be worth mentioning another important physical domain with boundary in plasma physics which is a fusion reactor such as tokamak. In a lab on the earth fusion can happen above 100 million Celsius (much higher than Sun's temperature) and no boundary materials can effectively withstand direct contact with such heat. To solve this problem, scientists have devised plasma held inside a doughnut-shaped magnetic field: if a confining external magnetic field is large enough, the plasma is localized away from the boundary. In other words, the acceleration of particles due to this external electromagnetic field is \textit{repellent to the boundary}, which is the exact opposite effect of gravitation/polarization electric field satisfying the Pannekoek-Rosseland condition. In some sense, one can reduce the initial-boundary value problem to the Cauchy problem when the confining magnetic field is dominant (\cite{Zhang2,JSW}).
Finally we consider a boundary condition of density distribution of plasma particles on the incoming phase boundary $\gamma_- := \{(x,v) \in \partial \Omega \times \mathbb R^3: v_3>0 \}.$ In addition let $\gamma_+ := \{(x,v) \in \partial \Omega \times \mathbb R^3: v_3<0 \}$ and $\gamma_0 := \{(x,v) \in \partial \Omega \times \mathbb R^3: v_3=0 \}$ denote the outgoing phase boundary and grazing phase boundary, respectively. In this paper we consider the following three simple physical boundary conditions, which were originally proposed by James Clerk Maxwell \cite{Maxwell}. An inflow boundary condition (inflow BC) is given by a prescribed date $g_\pm : \mathbb{R}_+ \times \gamma_- \to \mathbb R$:
\begin{equation} \label{inflow_I}
f_\pm(t,x,v) = g_\pm(t,x,v), \ \ \text{on} \ \mathbb{R}_+ \times \gamma_-.
\end{equation} A diffuse boundary condition (diffuse BC) takes the form of
\begin{equation} \label{diffuseBC_I}
f_\pm(t,x,v ) =
\frac{1}{2\pi T_w^2}e^{- \frac{|v|^2}{2 T_w}}
\int_{u_3 < 0 } - f_\pm(t,x,u) \hat u_{\pm,3} du, \ \ \text{on} \ \mathbb{R}_+ \times \gamma_-,
\end{equation}
where $T_w(x)$ is a positive smooth prescribed boundary temperature. As we are interested in a short-time dynamics from now on we assume the isothermal case $T_w(x)= 1$ for the sake of simplicity. We also have a generalized diffuse boundary condition \cite{CKLi}. Finally, a specular reflection boundary condition (specular BC) is given by
\begin{equation} \label{spec_I}
f_\pm(t,x,v_\parallel, v_3 ) = f_\pm(t,x, v_\parallel, -v_3 ) \ \ \text{on} \ \mathbb{R}_+ \times \gamma_-.
\end{equation} For the diffuse BC and specular BC, the boundary conditions enjoy a null flux condition:
$\int_{\mathbb R^3} f_\pm(t,x,v) \hat v_{\pm,3} dv = 0 \text{ for } x \in \partial \Omega$ , which implies a conservation of mass for a strong solution of RVM
$\int_{\Omega \times \mathbb R^3} f_\pm(t,x,v) dv dx = \int_{\Omega \times \mathbb R^3} f_\pm(0,x,v) dv dx \text{ for all } t \ge 0.$
One of the advantages of kinetic theory is that we can devise different boundary conditions from the microscopic interaction law of particles and boundaries. For example in a recent solar wind model, a non-Maxwellian inflow boundary condition is used to explain coronal heating phenomena (\cite{PP}).
Stability of the RVM system has also been studied extensively. Notably, in \cite{LS1, LS2, LS3}, the authors study the spatially inhomogeneous equilibrium in domains without boundary. A sharp criterion for spectral stability was given in \cite{LS3} and the nonlinear stability is studied in \cite{LS2}. In the case of bounded domains, stability analysis of the system was carried out in \cite{NS1} when the domain is a $2$D disk with perfect conducting boundary which reflects particles specularly. And then later the authors consider the case when the domain is a $3$D solid torus. More recently, the stability analysis was generalized to any axisymmetric domains in \cite{Zhang1}.
\hide
In this paper, we consider a plasma particle dynamics near above the exobase using the two species relativistic Vlasov-Maxwell system subjected to the solar gravitation and given geoelectricmagentic fields. Our geoelectric fields only agree with the Pannekoek-Rosseland condition at the top of the exobase (not for all altitude)
$(E_{\text{ext}}, B_{\text{ext}})$, which only satisfy so-called Pannekoek-Rosseland condition at the top of the exobase (\ref{PR}).
the Pannekoek-Rosseland electric field
Pannekoek-Rosseland
The first one is the solar breeze model. It was pro- posed by Chamberlain (1960) who suggested that the protons with a velocity exceeding the critical escape velocity evapo- rate like neutral particles escape out of a planetary atmosphere (Jeans 1923, Brandt Chamberlain 1960). Thus Chamberlain suggested that the radial expansion of the solar corona results from the thermal evaporation of the hot coronal protons out of the gravitational field of the Sun.
As indicated above, the present work is based on the ki- netic/exospheric model of the ion-exosphere originally devel- oped by Lemaire-Scherer (1970; 1971a) for geomagnetic field lines open to the magnetospheric tail. This model, initially ded- icated to the study of the polar wind, had subsequently been applied to model the solar wind (Lemaire-Scherer 1971b), with the assumption of Maxwellian VDF’s for the protons and electrons at the top of the collision-dominated part of the corona.
In zero order kinetic approximations, or exospheric models, two separate regions are considered: first, the collision-dominated barosphere at low altitude, in which the particles are assumed to be in hydrostatic/hydrodynamic equilibrium, and secondly, an exosphere in which the collision rate between particles is assumed to be negligibly small. These two extreme Knudsen number regimes are separated by a surface which is called the exobase.
To obtaina kineticdescriptionof the solar windphenomenas,everaaluthorshaveapplie exosperic theoreis o the collisionless region of the solar corona
Among the kinetic approaches, the purely collisionless one is generally called the exospheric approach. Two classes of exo- spheric solar wind models have been developed during the last fourty years.
An average of 95 of the solar wind ions are protons. Helium is the most abundant heavy ion with 3.2 in average in the slow solar wind and 4.2 in the high speed solar wind [Schwenn, 1990]. However, this fraction is variable, espe- cially in the slow speed solar wind, and helium concentra- tion can sometimes be as high as 10 of the total ions concentration. Oxygen, carbon, neon, nitrogen, silicon, magnesium, iron, sulfur, and other heavy minor ions are also detected in much smaller amounts (around 1 all together).
three internal (thermonuclear, radiative (energy is transported mainly by radiative diffusion), and convective) zones, the solar surface (photosphere), the lower (chromosphere) and upper atmosphere (corona),
The solar interior further consists of a radiative zone, where energy is transported mainly by radiative diffusion
Markus J. Aschwanden, Chapter 11 - The Sun, Editor(s): Tilman Spohn, Doris Breuer, Torrence V. Johnson, Encyclopedia of the Solar System (Third Edition), Elsevier, 2014, Pages 235-259,
= A kinetic model of the solar wind with Kappa distribution functions in the corona M. Maksimovic1, V. Pierrard2, and J.F. Lemaire2 Astron. Astrophys. 324, 725–734 (1997) ==
\unhide
\subsubsection*{\textbf{\large{Main Theorems}}}
As a major goal of this paper, we construct a weak solution of RVM in a locally Lipschitz space, in which we can guarantee a \textit{uniqueness!} The major difficulty is that the density distribution $f_{\pm}$ is singular at the grazing set $\gamma_0$ in general. Notably, a solution is discontinuous at the grazing set $\gamma_0$ (\cite{Kim}), and a derivative $\nabla_{x,v} f_{\pm}$ blows up at $\gamma_0$ (\cite{Guo95, GKTT1}). If a trajectory emanating from the grazing set can propagate inside the domain (either if the domain is not convex or the field is repellent to the boundary) then such singularities propagate inside the domain and the regularity of solutions become restrictive (\cite{Kim, Guo96, GKTT2, KimLee}). In particular, following the proof of Guo-Kim-Tonon-Trescases \cite{GKTT1}, we can deduce that a global $H^1(\Omega)$ bound is not possible for solutions $f_\pm$ of \eqref{VMfrakF}-\eqref{rhoJ} \& \eqref{E12B3bc}, in general. Unfortunately, such low regularity hardly guarantees uniqueness due to the nonlinear term $\mathfrak{F}_\pm \cdot \nabla_v f_{\pm}$. To overcome such obstacle, we adopt a kinetic weight function $\alpha_\pm(t,x,v)$ in the regularity estimate of a locally Lipschitz space, inspired by \cite{Guo96, GKTT1}.
\begin{definition}[Kinetic weight]\label{def:alpha}Recall the Lorentz force $\mathfrak{F}_{\pm}$ in \eqref{frakF} with $(E_{\text{ext}}, B_{\text{ext}})$ in \eqref{EBext}. We define
\begin{equation} \label{alphadef}
\begin{split}
\alpha_\pm (t, x_\parallel, x_3, v) := & \sqrt{(x_3)^2+(\hat{v}_{\pm,3})^2 -2 \mathfrak F_{\pm,3}(t,x_\parallel, 0 ,v) \frac{x_3}{\langle v_\pm \rangle}}
\\ = & \sqrt{(x_3)^2+(\hat{v}_{\pm,3})^2 +2\bigg( m_\pm g - e_\pm \Big(E_3 +E_e + \frac{1}{c} (\hat v_\pm \times B)_3 \Big)_{x_3=0} \bigg) \frac{x_3}{\langle v _\pm\rangle}},
\end{split}
\end{equation} where we have used that $( \hat v_{\pm } \times B_{\text{ext}} )_3 = 0$ for \eqref{EBext}.
\end{definition} \begin{remark}
Clearly, $\alpha_\pm$ is well-defined when $-\mathfrak{F}_{\pm,3} (t, x_\parallel, 0, v)$ is positive. In this paper, we assume this condition on the initial data at the boundary:
\begin{equation} \label{E0B0g}
m_\pm g -e_\pm \Big(E_{0,3}(x) +E_e + \frac{1}{c} (\hat v_\pm \times B_0(x))_3 \Big)_{x_3=0} > c_1, \ \ \text{for some } \ c_1>0.
\end{equation}
\end{remark}
\begin{remark}
The condition \eqref{E0B0g} is not very restrictive under the Pannekoek-Rosseland condition \eqref{PR}. Note that $-\mathfrak{F}_{\pm,3} (t, x_\parallel, 0, v)$ equals
\begin{equation} \notag \underbrace{\big( m_\pm g - e_\pm E_e \big)} \frac{x_3}{\langle v _\pm\rangle}
- e_\pm \Big(E_{0,3} + ( \frac{\hat v_\pm}{c} \times B_0)_3 \Big)_{x_3=0} \frac{x_3}{\langle v _\pm\rangle}.
\end{equation}
If the Pannekoek-Rosseland condition \eqref{PR} holds then the underlined coefficients of the first term, which corresponds to the net force at the equilibrium, has lower bounds:
\begin{equation} \notag
\big( m_+ g - e_+ E_e \big) > \frac{m_+g }{2}, \ \ \ \big( m_- g - e_- E_e \big)>|e_-| E_e.
\end{equation}
From \eqref{PR_field}, we know that both lower bounds are of the same size. If $E_{0,3}|_{x_3=0}$ and $B_{0,1}|_{x_3=0}$, $B_{0,2}|_{x_3=0}$ are smaller than such lower bounds then the condition \eqref{E0B0g} holds. It is the case when the initial state of plasma is either close to the neutral state or vacuum at the boundary. \end{remark}
\begin{remark}Since being introduced in \cite{Guo95}, such weight function $\alpha$ and its variants have served important roles in the regularity analysis for various kinetic equations with boundary such as \cite{CAO2, CAO3, CAO1, CK, CKL, CKLi, GKTT1, HV}. Notably in \cite{GKTT1}, an $\alpha$-weighted $C^1$ solution for the Boltzmann equation was constructed in convex domains. In \cite{CKL}, the authors used a different version of kinetic weight to construct the global strong solution to the Vlasov-Poisson-Boltzmann (VPB) system in convex domains with diffuse BC. The result was generalized to the two-species case in \cite{CAO3}, and to the case of the presence of the external field in \cite{CAO2}. A generalized diffuse boundary condition (namely the Cercignani-Lampis boundary condition) for the VPB system is studied in \cite{CKLi}. A survey on the recent development in this direction can be found in \cite{CK}.
\end{remark}
Although the problem has been set already (RVM system \eqref{VMfrakF_I}-\eqref{rhoJ_I} under the perfect conductor boundary condition of electromagnetic field \eqref{E12B3bc_I} ), we list them here redundantly for the sake of the reader's convenience: Let $\Omega$ the half space \eqref{domain}. We read the RVM system \begin{equation} \label{VMfrakF} \begin{split}
\partial_t f_\pm + \hat v_\pm \cdot \nabla_x f_\pm + \mathfrak F_\pm \cdot \nabla_v f_\pm = 0, & \ \ \text{ in } \mathbb{R}_+ \times \Omega \times \mathbb R^3 ,
\\ f_\pm(0,x,v) = f_{0,\pm}(x,v) , & \ \ \text{ in } \Omega \times \mathbb R^3.
\end{split} \end{equation} with the relativistic velocity, Lorentz force, and the external fields \begin{align}
\hat v_\pm &= {v}\Big/ {\sqrt{m_\pm^2 + |v|^2 / c^2}},\label{rel_v}\\ \mathfrak F_\pm &= e_\pm \left(E+ E_{\text{ext}} + \frac{\hat v_\pm}{c} \times (B + B_{\text{ext}}) \right) - m_\pm g \mathbf e_3,\label{frakF}\\ E_{\text{ext}} &= E_e \mathbf e_3 \text{ and } B_{\text{ext}} = B_e \mathbf e_3. \label{EBext} \end{align} The Maxwell's equations solve \begin{equation} \label{Maxwell} \begin{split}
\partial_t E & = c \nabla_x \times B - 4 \pi J, \, \ \ \ \nabla_x \cdot E = 4\pi \rho, \ \ \text{ in } \mathbb{R}_+ \times \Omega,
\\ \partial_t B & = - c \nabla_x \times E, \, \ \ \ \nabla_x \cdot B = 0, \ \ \text{ in } \mathbb{R}_+ \times \Omega, \end{split} \end{equation} \begin{equation} \label{BEinitdata} E(0,x) = E_0(x), \ \ B(0,x) = B_0(x), \ \ \text{ in } \Omega. \end{equation} where the electric density and current are defined as \begin{equation} \label{rhoJ} \rho = \int_{\mathbb R^3} ( e_+f_+ + e_- f_-) \mathrm{d} v, \ \ \ J = \int_{\mathbb R^3} (\hat v_+ e_+ f_+ + \hat v_- e_- f_-) \mathrm{d} v. \end{equation} Finally we impose the perfect conductor boundary condition \begin{equation} \label{E12B3bc} E_1 = E_2 = 0, \, B_3 = 0 , \text{ on } \ \mathbb{R}_+ \times \partial \Omega, \end{equation} and consider the inflow BC, diffuse BC, and specular BC on the incoming boundary $\gamma_-$: \begin{align}
f_\pm(t,x,v) = g_\pm(t,x,v) \ \ &\text{on} \ \mathbb{R}_+ \times \gamma_-,
\label{inflow}\\
f_\pm(t,x,v ) =
\frac{1}{2\pi T_w^2}e^{- \frac{|v|^2}{2 T_w}}
\int_{u_3 < 0 } - f_\pm(t,x,u) \hat u_{\pm,3} du \ \ &\text{on} \ \mathbb{R}_+ \times \gamma_-,
\label{diffuseBC}\\
f_\pm(t,x,v_\parallel, v_3 ) = f_\pm(t,x, v_\parallel, -v_3 ) \ \ &\text{on} \ \mathbb{R}_+ \times \gamma_-
.\label{spec}
\end{align}
We define a notation of weak solutions to this initial-boundary value problem. \begin{definition}[Definition 1.5 of \cite{Guo93}] \label{weaksoldef}
Let $f_\pm \in L^1_{\text{loc} } ((0,T) \times \Omega \times \mathbb R^3 ) \cap L^1_{\text{loc} } ((0,T) \times \gamma_+ ) $, $f_{0, \pm } \in L^1_{\text{loc}}( \Omega \times \mathbb R^3 ) $, $g \in L^1_{\text{loc} } ((0,T) \times \gamma_- )$. Let $E, B \in L^1_{\text{loc}}( (0,T) \times \Omega ) $, $E_0, B_0 \in L^1_{\text{loc} }(\Omega ) $. Then $(f_\pm, E, B)$ is a weak solution of \eqref{VMfrakF}-\eqref{rhoJ} under the perfect conductor boundary condition of electromagnetic field \eqref{E12B3bc} and different boundary conditions for $f_{\pm}$ \eqref{inflow}, \eqref{diffuseBC}, or \eqref{spec}, if for any test functions
\[
\begin{split}
& \phi(t,x,v) \in C_c^\infty([0,T) \times \Omega \times \mathbb R^3 ), \text{ with } \text{ supp } \phi \subset \{ [0, T) \times \bar \Omega \times \mathbb R^3 \} \setminus \{ (\{0\} \times \gamma ) \cup (0,T) \times \gamma_0 \}, \text{ and }
\\ & \Psi(t,x) \in C_c^\infty([0,T) \times \bar \Omega ; \mathbb R^3 ), \ \Phi(t,x) \in C_c^\infty([0,T) \times \Omega ; \mathbb R^3 ),
\end{split}
\]
we have
\begin{equation} \label{weakf}
\begin{split}
& \iint_{\Omega \times \mathbb R^3 } f_{0, \pm} \phi(0) dv dx + \int_0^T \iint_{\Omega \times \mathbb R^3 } (\partial_t \phi + \nabla \phi \cdot \hat v + \mathfrak F_{\pm} \cdot \nabla_v \phi ) f_\pm dv dx dt
\\ = & \int_0^T \int_{\gamma_+ } \phi f_\pm \hat v_3 d v dS_x + \underbrace{ \int_0^T \int_{\gamma_- } \phi f_\pm \hat v_3 d v dS_x }_{\eqref{weakf}_{\text{BC} } },
\end{split}
\end{equation}
and
\begin{equation} \label{Maxweak1}
\int_0^T \int_\Omega E \cdot \partial_t \Psi dx dt - \int_\Omega \Psi(0,x) \cdot E_0 dx = - \int_0^T \int_\Omega (\nabla_x \times \Psi) \cdot B dx dt + 4\pi \int_0^T \int_\Omega \Psi \cdot J dx dt,
\end{equation}
\begin{equation} \label{Maxweak2}
\int_0^T \int_\Omega B \cdot \partial_t \Phi dx dt + \int_\Omega \Phi(0,x) \cdot B_0 dx = \int_0^T \int_\Omega (\nabla_x \times \Phi) \cdot E dx dt,
\end{equation}
and
\begin{equation} \label{nablaEBweak}
\nabla \cdot E = 4 \pi \rho, \ \nabla \cdot B = 0 \text{ in the sense of distributions in } (0,T) \times \Omega \times \mathbb R^3.
\end{equation}
Here, the boundary term of \eqref{weakf} is determined by different boundary conditions:
\[
\begin{split}
\eqref{weakf}_{\text{BC} } =
\begin{cases}
\int_0^T \int_{\gamma_- } \phi g_\pm \hat v_3 \, d v dS_x, \text{ for the inflow BC } \eqref{inflow},
\\ \int_0^T \int_{\gamma_+ } \left( - \frac{1}{2\pi T_w^2} \int_{u_3 > 0 } e^{- \frac{|u|^2}{2 T_w}} \phi (t,x,u )\hat u_3 du \right) \hat v_3 f_\pm \, dv dS_x, \text{ for the diffuse BC } \eqref{diffuseBC},
\\ \int_0^T \int_{\gamma_+} \phi(t,x,v_\parallel, - v_3 ) f_\pm \hat v_3 \ dv dS_x, \text{ for the specular BC } \eqref{spec}.
\end{cases}
\end{split}
\]
\end{definition}
Now we state the main theorems.
\begin{theorem} [inflow BC] \label{main1} Suppose the initial datum $f_{0,\pm}$ satisfies, for some $\delta>0$, \begin{equation} \label{f0bdd} \begin{split}
& \| \langle v \rangle^{4 + \delta } f_{0,\pm} \|_{L^\infty(\Omega \times \mathbb R^3) } + \| \langle v \rangle^{5 + \delta } \nabla_{x_\parallel} f_{0,\pm} \|_{L^\infty(\Omega \times \mathbb R^3) }
\\ & + \| \langle v \rangle^{5 + \delta } \alpha_\pm \partial_{x_3} f_{0,\pm} \|_{L^\infty(\Omega \times \mathbb R^3) } + \| \langle v \rangle^{5 + \delta } \nabla_v f_{0,\pm} \|_{L^\infty(\Omega \times \mathbb R^3) } < \infty,
\end{split}
\end{equation}
and the inflow boundary datum $g_\pm$ satisfies
\begin{equation} \label{inflowdata}
\| \langle v \rangle^{5 + \delta } \partial_t g_\pm \|_{L^\infty( (0 , \infty) \times \gamma_-) } + \| \langle v \rangle^{5 + \delta } \nabla_{x_\parallel} g_\pm \|_{L^\infty (0 , \infty) \times \gamma_-) } + \| \langle v \rangle^{5 + \delta } \nabla_{v} g_\pm \|_{L^\infty( (0 , \infty) \times \gamma_-) } < \infty. \end{equation} Moreover, $E_0,B_0, g$ satisfies \eqref{E0B0g}, and the compatibility conditions \begin{equation} \label{EBintialC} \begin{split} \nabla \cdot E_0 = 4 \pi \rho_0 , \, \nabla \cdot B_0 = 0, & \text{ in } \Omega, \\ E_{0, \parallel } = 0, \, B_{0,3} = 0 , & \text{ on } \partial \Omega, \end{split} \end{equation}
and \begin{equation} \label{E0B0bdd}
\| E_0 \|_{C^2(\Omega ) } + \| B_0 \|_{C^2(\Omega) } < \infty. \end{equation} Then there exists a unique solution $f_\pm(t,x,v), E(t,x,v), B(t,x,v) $ for $0 \le t \le T$ with $T \ll 1 $ to RVM for the inflow BC \eqref{inflow} in the sense of Definition \ref{weaksoldef}, such that, \begin{equation} \label{inflowfreg} \begin{split}
\sup_{0 \le t \le T} \Big( & \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_\pm(t) \|_{L^\infty(\Omega \times \mathbb R^3 )} + \| \langle v \rangle^{5+\delta} \alpha_\pm \partial_{x_3} f_\pm(t) \|_{L^\infty(\Omega \times \mathbb R^3 )} \\
& + \| \langle v \rangle^{5+\delta} \nabla_{v} f_\pm(t) \|_{L^\infty(\Omega \times \mathbb R^3 )} \Big) < \infty,
\end{split} \end{equation} and \begin{equation} \label{inflowEBreg}
\sup_{0 \le t \le T} \left( \| \nabla_x E(t) \|_{L^\infty( \Omega ) } + \| \nabla_x B(t) \|_{L^\infty( \Omega ) } \right) < \infty. \end{equation} \end{theorem}
\begin{theorem} [diffuse BC] \label{main2} Suppose $f_{0,\pm}$ satisfies \eqref{f0bdd}, and $E_0, B_0, g$ satisfy \eqref{E0B0g}, \eqref{EBintialC}, and \eqref{E0B0bdd}. Then there exists a unique solution $f_\pm(t,x,v), E(t,x,v), B(t,x,v) $ for $0 \le t \le T$ with $T \ll 1 $ to RVM for the diffuse BC \eqref{diffuseBC} in the sense of Definition \ref{weaksoldef}, such that both \eqref{inflowfreg} and \eqref{inflowEBreg} hold. \hide\begin{equation} \label{inflowfreg}
\sup_{0 \le t \le T} \left( \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_\pm(t) \|_{L^\infty(\Omega \times \mathbb R^3 )} + \| \langle v \rangle^{5+\delta} \alpha_\pm \partial_{x_3} f_\pm(t) \|_{L^\infty(\Omega \times \mathbb R^3 )} + \| \langle v \rangle^{5+\delta} \nabla_{v} f_\pm(t) \|_{L^\infty(\Omega \times \mathbb R^3 )} \right) < \infty, \end{equation} and \begin{equation} \label{inflowEBreg}
\sup_{0 \le t \le T} \left( \| \nabla_x E(t) \|_{L^\infty( \Omega ) } + \| \nabla_x B(t) \|_{L^\infty( \Omega ) } \right) < \infty. \end{equation}\unhide \end{theorem}
\begin{theorem} [specular BC] \label{main3} Suppose $f_{0,\pm}$ satisfies \begin{equation} \label{f0spec} \begin{split}
\| \langle v \rangle ^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha_\pm \langle v \rangle } } } \nabla_{x} f_{0,\pm} \|_\infty + \| \langle v \rangle^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha_\pm \langle v \rangle } } } \nabla_v f_{0,\pm} \|_\infty < \infty, \end{split} \end{equation} for some $C> 0$. $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{EBintialC}, and \eqref{E0B0bdd}. Then there exists a unique solution $f_\pm(t,x,v)$, $E(t,x,v)$, $B(t,x,v) $ for $0 \le t \le T$ with $T \ll 1 $ to the RVM with system \eqref{spec} in the sense of Definition \ref{weaksoldef}, such that \begin{equation} \label{specfbd}
\sup_{0 \le t \le T} \left( \| \langle v \rangle^{4+\delta} \nabla_{x} f_\pm (t) \|_{L^\infty(\Omega \times \mathbb R^3 )} + \| \langle v \rangle^{4+\delta} \nabla_{v} f_\pm (t) \|_{L^\infty(\Omega \times \mathbb R^3 )} \right) < \infty, \end{equation} and \eqref{inflowEBreg} holds.
\end{theorem}
\begin{remark}
A large class of functional spaces satisfy the condition \eqref{f0bdd}. Indeed any function, whose weak derivatives $\nabla_{x,v} f$ are bounded in $L^\infty(\Omega \times \mathbb R^3)$, and decays fast enough as $|v| \to \infty$, belongs to the space of \eqref{f0bdd}. Actually $\partial_{x_3} f_{0, \pm}$ is allowed to be singular at the grazing set $\gamma_0$. \end{remark}
\begin{remark}
As far as the authors know, Theorem \eqref{main1}- \eqref{main3} provide the first unique solvability of the RVM system when the physical boundary has a global effect (cf. \cite{Zhang2,JSW}). The time span $T$ of existence depends on the size of the initial data $f_0$, $E_0$, $B_0$ and their derivatives, and $c_1$ in \eqref{E0B0g}. \end{remark}
\begin{remark}We prove the weighted regularity estimate using a Lagrangian approach of \cite{GKTT1}. We take a direct differentiation to the Lagrangian solution along the generalized characteristics. The generalized characteristics depend on the boundary condition.
\end{remark}
\begin{remark} Here we require that the initial data $f_0$ to vanish exponentially fast towards the grazing set \eqref{specfbd}. This allows us to establish the regularity estimate for specular BC \eqref{specfbd} with the $W^{1,\infty}$ field. We prove this theorem in section \ref{chapspec}. \end{remark}
\subsubsection*{\textbf{\large{Difficulties and Key Ingredients}}} \label{diffidea} The problem in this paper is a coupled system of hyperbolic equations and kinetic Vlasov equation with characteristic boundary condition: the problem suffers \textit{a loss of derivative of wave equation} (cf. \cite{CS}) and \textit{the boundary singularity of Vlasov equation} (cf. \cite{GKTT1, Guo95}) at the same time. The key difficulty in the construction of a \textit{unique} solutions of the RVM system with physical boundary conditions is a control of the nonlinear term $\mathfrak{F}_\pm \cdot \nabla_v f_\pm$. We overcome this difficulty by establishing a regularity estimate for both the electromagnetic field $E$ and $B$, and the density distribution $f_\pm$ using the Glassey-Strauss representation. We detail several key difficulties along the road. In this section and the rest of the paper, for sake of simplicity, we will consider the one-species relativistic Vlasov-Maxwell-system since the analysis of the two-species case does not process essential difference from that of the one species case: \begin{equation} \label{VMfrakF1} \begin{split}
\partial_t f + \hat v \cdot \nabla_x f + \mathfrak F \cdot \nabla_v f = 0, & \text{ in } \mathbb{R}_+ \times \Omega \times \mathbb R^3 ,
\\ f(0,x,v) = f_{0}(x,v) , & \text{ in } \Omega \times \mathbb R^3,
\end{split}
\end{equation} We also set all the charge and mass of the plasma $f$ equal to one, so here $\hat v = \frac{1}{\sqrt{1 +|v|^2 } } $, and \begin{equation} \label{frakF1} \mathfrak F = \left(E + E_{\text{ext}}+ \frac{\hat v}{c} \times (B + B_{\text{ext}}) \right) - g \mathbf e_3, \end{equation} and $E,B$ satisfies the Maxwell equations \eqref{Maxwell}, with \begin{equation} \label{rhoJ1} \rho = \int_{\mathbb R^3} f dv, \, J = \int_{\mathbb R^3} \hat v f dv. \end{equation}
\subsubsection*{$\bullet$\textbf{Wave equation and the Neumann BC}} From the Maxwell's equations \eqref{Maxwell}, we have the inhomogenous wave equations for $E$ and $B$: \begin{align} \partial_t^2 E - \Delta_x E = -4\pi \nabla_x \rho - 4\pi \partial_t J, \ &\text{ in } \ \mathbb{R}_+ \times \Omega,\label{wave_eq_E}\\ \partial_t^2 B - \Delta_x B = 4 \pi \nabla_x \times J, \ &\text{ in } \ \mathbb{R}_+ \times \Omega,\label{wave_eq_B} \end{align} with the boundary condition $E_1 = E_2 = 0, B_3 = 0$ on $\partial\Omega$ in \eqref{E12B3bc} and the initial condition \begin{equation} \label{initialC} \begin{split}
E|_{t=0} = E_0,\ \ \partial_t E|_{t=0} = \partial_t E_0:= \nabla_x \times B_0 - 4\pi J|_{t=0}, \text{ in } \Omega
,
\\
B|_{t=0} = B_0,\ \ \partial_t B|_{t=0} = \partial_t B_0:= - \nabla_x \times E_0, \text{ in } \Omega
. \end{split} \end{equation} The boundary conditions of $E_3, B_\parallel$ components are not a priori given, which causes some trouble handling weak solutions based on the Glassey-Strauss representation. Of course, if the fields $E,B \in C^2 (\Omega) \cap C^1( \bar \Omega)$, and $\rho \in C^1(\Omega ) \cap C(\bar \Omega)$, then from the Maxwell's equations \eqref{Maxwell} and the perfect conductor boundary condition \eqref{E12B3bc}, we deduce the Neumann boundary condition \begin{equation} \label{E3B1B2bc} \partial_3 E_3 = 4\pi \rho, \partial_3 B_1 = 4\pi J_2, \ \partial_3 B_2 = - 4\pi J_1 \ \text{ on } \mathbb{R}_+ \times \partial \Omega. \end{equation} \hide\begin{equation}
\partial_{3} E_3 |_{\partial \Omega} = 4 \pi \rho |_{\partial \Omega } + \partial_1 E_1 |_{\partial \Omega } + \partial_2 E_2 |_{\partial \Omega } = 4 \pi \rho|_{\partial \Omega}, \end{equation} and \begin{equation} \begin{split}
\partial_3 B_2 |_{\partial \Omega} = & \partial_2 B_3 |_{\partial \Omega} - \partial_t E_1 |_{\partial \Omega} - 4\pi J_1 |_{\partial \Omega} = - 4 \pi J_1 |_{\partial \Omega},
\\ \partial_3 B_1 |_{\partial \Omega} = & \partial_1 B_3 |_{\partial \Omega} + \partial_t E_2 |_{\partial \Omega} + 4\pi J_2 |_{\partial \Omega} = - 4 \pi J_2 |_{\partial \Omega}. \end{split} \end{equation} Therefore $E_3$ and $B_\parallel$ satisfy the following Neumann boundary condition \begin{equation} \label{E3B1B2bc} \partial_3 E_3 = 4\pi \rho, \partial_3 B_1 = 4\pi J_2, \ \partial_3 B_2 = - 4\pi J_1 \ \text{ on } \partial \Omega. \end{equation}\unhide
One of the main goals in this paper is to equip a solution space in which we can realize the Neumann BC \eqref{E3B1B2bc} in a suitable sense and hence guarantee a unique solvability. Indeed we can justify the Neumann boundary condition \eqref{E3B1B2bc} in a weak solution formulation testing against smooth test functions that do not vanish at the boundary $\partial \Omega$ in Lemma \ref{Maxtowave}, and prove the uniqueness of weak solution.
We then carefully show in Lemma \ref{wavetoMax} that, assuming the continuity equation \[ \nabla \cdot J + \partial_t \rho = 0, \] and some compatibility conditions of the initial datum \eqref{EBintialC}, the weak solution of wave equations with boundary conditions \eqref{wave_eq_E}-\eqref{E3B1B2bc} is indeed a solution to the Maxwell equations. This equivalence allows us to solve the RVM system by looking for solutions to the wave equations with boundary conditions \eqref{wave_eq_E}, \eqref{wave_eq_B}, which is the first step of our analysis.
\subsubsection*{$\bullet$\textbf{Glassey-Strauss Representation in the half space}} The wave equations \eqref{wave_eq_E}, \eqref{wave_eq_B} suffer from the ``loss of derivatives'' of $(E,B)$ with respect to the regularity of the source terms $\rho$ and $J$. As Glassey mentions in his book \cite{Glassey}, the key idea of the Glassey-Strauss representation is replacing the derivatives $\partial_t, \nabla_x$ by a geometric operator $T$ in \eqref{def:T1} and a kinetic transport operator $S$ in \eqref{def:S1}: \begin{equation} \label{pxptST} \begin{split}
\partial_t = \frac{S- \hat{v} \cdot T}{1+ \hat v \cdot \o}, \ \
\partial_i = \frac{\o_i S}{ 1+ \hat v \cdot \o} + \left( \delta_{ij} - \frac{\o_i \hat{v}_j}{1+ \hat v \cdot \o}\right) T_j,
\end{split}\end{equation} while, for $\o= \o(x,y) = \frac{y-x}{|y-x|}$, \begin{align}
T_i &:= \partial_i - \o_i \partial t, \label{def:T1}\\
S &:= \partial_t + \hat v \cdot \nabla_x. \label{def:S1} \end{align} Note that \begin{equation} \label{T=y}
T_j f (t- |y-x| , y, v ) = \partial_{y_j } [ f(t- |y-x|, y, v ) ], \end{equation} which is a tangential derivative along the surface of a backward light cone \cite{Glassey}. On the other hand, the Vlasov equation \eqref{VMfrakF1} implies that \begin{equation} \label{S=Lf_v} Sf= - \nabla_v \cdot [ (E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}})- g \mathbf e_3)f]. \end{equation} Therefore, in \cite{GS,Glassey}, they can take off the derivatives $T_j$, $S$ from $f$ using the integration by parts within the Green's formula of \eqref{wave_eq_E}--\eqref{wave_eq_B} by connecting the source terms to $f$ via \eqref{rhoJ1}.
For our problem, we derive the Glassey-Strauss representation in the presence of a boundary. For $E_\parallel$ and $B_3$, to solve the wave equation with the Dirichlet boundary condition \eqref{E12B3bc}, we employ the odd extension of the initial data and the forcing term into the lower half space $\mathbb R^3_- : = \{ (x_1, x_2, x_3 ) \in \mathbb R^3: x_3 < 0 \} $. Solving the whole space wave equation with the oddly extended data gives us the solution that satisfies \eqref{E12B3bc}. On the other hand, for $E_3$ and $B_\parallel$, we decompose the solution into two parts: one with the Neumann boundary condition of \eqref{E3B1B2bc} and the zero forcing term and initial data, and the other part satisfying \eqref{wave_eq_E}-\eqref{initialC} with the zero Neumann boundary condition. For the first part, we find out the expression using the fundamental solution of the Helmholtz equation. And for the second part, we use the even extension to get the solution with the zero Neumann boundary condition.
For the expression in the lower half space, we introduce
\begin{equation} \label{def:o-} \bar \o = \begin{bmatrix} \o_1 & \o_2 & - \o_3 \end{bmatrix} ^T, \end{equation} and \begin{equation} \label{def:T-} \begin{split}
\bar{T}_3 f &= -\partial_{y_3} [f(t-|y-x|, y_\parallel, - y_3, v)] = \partial_{y_3} f - \bar{\omega}_3 \partial_t f ,
\\ \bar T _i f &= \partial_{y_i} [f(t-|y-x|, y_\parallel, - y_3, v)] = \partial_{y_i} f - \bar{\omega}_i \partial_t f \, \text{ for } \ i=1,2. \end{split} \end{equation} Then through direct calculation we obtain an explicit expressions of $E$ and $B$ by solving the wave equations \eqref{wave_eq_E}--\eqref{initialC} under the boundary condition \eqref{E12B3bc} and \eqref{E3B1B2bc} in Proposition \ref{Eiform} and Proposition \ref{Biform} respectively.
\subsubsection*{$\bullet$\textbf{Weighted $W^{1,\infty}$ estimate of $f$ and the regularity of the fields}} An intrinsic feature of the transport equation in domains with boundary is the singular behavior of its derivatives: the solution of a linear transport equation with physical boundaries is known to not have high regularity \cite{GKTT1}. However, to get the unique solvability, one must control $\nabla_v f $ effectively. This in turn requires the control of spatial derivatives of the distribution function and the spatial derivatives of electromagnetic fields. But due to the characteristic boundary, $f$ does not have high enough regularity to achieve the required regularity for $E$ and $B$ directly from the hyperbolic equations. We explain the ideas of the paper and the methods we use to overcome the difficulties in the rest of this section and the next.
Let's consider a solution of the RVM system with inflow boundary data \eqref{VMfrakF1}-\eqref{rhoJ1}, \eqref{inflow}. The \textit{characteristics (trajectory)} is determined by the Hamilton ODEs, \begin{equation} \label{HamiltonODE} \begin{split}
\frac{d}{ds} X(s;t,x,v)&= \hat{V}(s;t,x,v),\\
\frac{d}{ds} V(s;t,x,v)&= \mathfrak F (s,X(s;t,x,v),V(s;t,x,v)). \end{split} \end{equation} We define \textit{the backward exit time} $t_{\mathbf{b}}(t,x,v)$ as \begin{equation} \label{tb} t_{\mathbf{b}} (t,x,v) := \sup \{s \geq 0 : X(\tau;t,x,v) \in \Omega \ \ \text{for all } \tau \in (t-s,t) \}. \end{equation} Furthermore, we define $x_{\mathbf{b}} (t,x,v) := X(t-t_{\mathbf{b}}(t,x,v);t,x,v)$, and $v_{\mathbf{b}}(t,x,v) := V(t-t_{\mathbf{b}}(t,x,v);t,x,v)$. We can solve the Vlasov equation \eqref{VMfrakF1} with the inflow boundary condition \eqref{inflow} as \[ f(t,x,v) = g( t -t_{\mathbf{b}}(t,x,v) , X(t-t_{\mathbf{b}}(t,x,v);t,x,v), V(t-t_{\mathbf{b}}(t,x,v);t,x,v) ) \text{ for } t \ge t_{\mathbf{b}}(t,x,v). \] From some direct computations (see \eqref{pxbvb} and \eqref{pxiF}), the derivatives of $f$ have a bound in general as \[ \nabla_x f(t,x,v) \sim \nabla_x t_{\mathbf{b}}(t,x,v), \] which can be further bounded from the direct computation of the characteristics (see \eqref{pxitb}) as \begin{equation} \label{dtbbd}
\nabla_x t_{\mathbf{b}}(t,x,v) \lesssim \frac{1 + \sup_{0 \le s \le t} \| \nabla_x \mathfrak F(s) \|_\infty }{\hat{v}_{\mathbf{b},3}}. \end{equation} The formation of such singularity motivates us to introduce the following notion. As a first order approximation of $\hat{v}_{\mathbf{b},3}(t,x,v)$, we define the kinetic weight \begin{equation} \label{alphadef} \begin{split}
\alpha(t, x_\parallel, x_3, v) := & \sqrt{(x_3)^2+(\hat{v}_{3})^2 -2 \mathfrak F_3(t,x_\parallel, 0 ,v) \frac{x_3}{\langle v \rangle}}
\\ = & \sqrt{(x_3)^2+(\hat{v}_{3})^2 -2\left( E_3(t,x_\parallel, 0 ) + E_e + (\hat v \times B)_3(t,x_\parallel, 0 ) - g \right) \frac{x_3}{\langle v \rangle}}. \end{split} \end{equation} Note that $\alpha = \hat v_3$ on $\partial \Omega$. Crucially $\alpha$ is effectively invariant along the characteristics, thanks to the velocity lemma (Lemma \ref{vlemma}). This allows us to prove an $\alpha$-weighted bound on the derivatives of $f$, more specifically, we prove that for any $0 < \delta < 1$, \begin{equation} \langle v \rangle^{5 + \delta} \alpha \nabla_x f (t) \in L^\infty( \Omega \times \mathbb R^3 ), \text{ for } 0< t < T. \end{equation}
On the other hand, due to the generic singularity \eqref{dtbbd}, to close the estimate we need to bound $\nabla_x E$, $\nabla_x B$ by $ \langle v \rangle^{5 + \delta} \alpha \nabla_x f $ in the generalized Glassey-Strauss representation. By taking the derivatives directly to the formulas of $E$ and $B$, in Lemma \ref{EBW1inftylemma} we achieve the bound \[
| \nabla_x E | + | \nabla_x B | \lesssim ``\text{ initial data }" + \sup_{0 \le t \le T} \| \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f (t) \|_\infty \int_{ \Omega \cap \{ |x| < T \}} \int_{ \mathbb R^3} \frac{1}{ \langle v \rangle^{4 + \delta} \alpha (t,x,v) } dv dx. \] Then from the local-to-nonlocal estimate (Lemma \ref{1alphaintv}), we derive \[ \int_{ \mathbb R^3} \frac{1}{ \langle v \rangle^{4 + \delta} \alpha (t,x,v) } dv \lesssim \ln(1+ \frac{1}{x_3} ) \in L^1_{\text{loc} } (\Omega), \] and we are able to close the estimate and conclude $E, B \in W^{1,\infty}((0,T) \times \Omega ).$
In the construction of solution, we study a sequence of solutions $(f^\ell, E^\ell, B^\ell )$ and pass the limit. To achieve a uniform estimate, we use a weight $\alpha^\ell(t,x,v)$, which is the same form of \eqref{alphadef} with exchanging $E,B$ to $E^\ell, B^\ell$. Since $\alpha^\ell$ depends on $E^\ell, B^\ell$ and hence $f^\ell$, when passing the limit of the sequence $\{ \alpha^{\ell-1} \partial_{x_3} f^\ell \}_{\ell = 1}^\infty$, we need to verify that \begin{equation} \label{alphp3flim} \alpha^{\ell-1} \partial_{x_3} f^\ell \overset{\ast}{\rightharpoonup} \alpha \partial_{x_3} f \text{ in } L^\infty((0,T) \times \Omega \times \mathbb R^3 ). \end{equation}
Obviously this convergence is nontrivial since the norm itself is nonlinear. To obtain this, we observe that since we can bound $\nabla E^\ell , \nabla B^\ell$ pointwisely, they have traces and a strong convergence \[
E^\ell |_{\partial \Omega} \to E |_{\partial \Omega} , \ B^\ell|_{\partial \Omega} \to B |_{\partial \Omega}. \] Thus we can prove that a strong convergence $\alpha^{\ell-1} \to \alpha $ in $L^\infty$. On the other hand, using a positive lower bound of $\alpha^\ell$ away from the grazing set, we obtain a upper uniform bound of $\partial_{x_3} \alpha^{\ell-1}- \partial_{x_3} \alpha$ locally. We then achieve the desired convergence \eqref{alphp3flim} using uniform bound of $f^\ell$. We refer to Lemma \ref{fEBregin} for more details.
Among other boundary conditions, we find that the specular boundary condition suffers most. Due to the lack of higher regularity of the fields (e.g. compare to \cite{CAO1} where the field is $C^2$), we can only derive an exponential-in-$\alpha$ singularity of the derivative of trajectory \begin{equation}\label{lemma_Dxv1}
\begin{split}
| \partial_{\mathbf e } X_{\mathbf{cl}}(s;t,x,v)| & \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } } ,
\\ | \partial_{\mathbf e } V_{\mathbf{cl}}(s;t,x,v)| & \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } }.
\end{split} \end{equation} Clearly, such a strong singularity can be harmful in our analysis based on the Glassey-Strauss representation. We study the specular BC problem with great care. Details are presented in section \ref{chapspec}.
\subsubsection*{$\bullet$\textbf{A Priori $L^\infty$ estimate of $\nabla_v f$ and uniqueness}} A simple Gronwall's inequality implies \begin{equation} \label{fgdiff}
\| \langle v \rangle^{4+ \delta } (f- g ) (t) \|_\infty \lesssim \| \langle v \rangle^{4+ \delta } (f- g )(0) \|_\infty + \sup_{0 \le t \le T } \| \langle v \rangle^{4+ \delta } \nabla_v f(t) \|_\infty \int_0^t \|( E_f - E_g + B_f - B_g)(s) \|_\infty. \end{equation} For constructing a solution and proving its uniqueness, we establish an effective stability estimate of the difference of solutions $f-g$, and $E_f - E_g$, $B_f - B_g$. To control the nonlinear term of the equation of $f-g$, $(E_f - E_g + B_f - B_g) \cdot \nabla_v f $, we establish an estimate of $\nabla_v f $. From the Lagrangian view point along the characteristics \eqref{HamiltonODE}, we have \[ \begin{split} & \nabla_v f(t,x,v) \\ & \sim \nabla_{x} f_0(X(0;t,x,v) , V(0;t,x,v) ) \cdot \nabla_v X(0;t,x,v) + \nabla_{v} f_0(X(0;t,x,v) , V(0;t,x,v) ) \cdot \nabla_v V(0;t,x,v). \end{split} \] Clearly effective control of $\nabla_x E$, $\nabla_x B$ is necessary. Now we crucially use our estimate of $\alpha \partial_{x_3} f $ to obtain bounds for $\nabla_x E$, $\nabla_x B$ in the Glassey-Strauss representation, which in turn gives the a priori $L^\infty$ estimate of $\nabla_v f$. Using this $\nabla_v f$-bound and a pointwise bound from the Glassey-Strauss representation (Lemma \ref{EBlinflemma})
\[
\| E_{f-g}(t) \|_\infty + \| E_{f-g}(t) \|_\infty \lesssim \sup_{0 \le s \le t } \| \langle v \rangle^{4+ \delta } ( f - g)(s) \|_\infty, \] we achieve
an $L^\infty$ stability as $
\sup_{0 \le s \le t } \| \langle v \rangle^{4+ \delta } (f- g ) (s) \|_\infty \lesssim e^{Ct } \| \langle v \rangle^{4+ \delta } (f- g )(0) \|_\infty.$
\hide
On the other hand, notice that $E_f - E_g = E_{f-g}$, $B_f - B_g = B_{f-g}$ solves the Maxwell equations \begin{equation} \label{Maxwellfg} \begin{split} \partial_t E_{f-g} & = \nabla_x \times B_{f-g} - 4 \pi J_{f-g}, \, \nabla_x \cdot E_{f-g} = 4\pi \rho_{f-g}, \\ \partial_t B_{f-g} & = - \nabla_x \times E_{f-g}, \, \nabla_x \cdot B_{f-g} = 0. \end{split} \end{equation} Then Lemma \ref{Maxtowave} implies that $E_{f-g}$ and $B_{f-g}$ solves the wave equation \begin{equation} \begin{split} \partial_t^2 E_{f-g} - \Delta_x E_{f-g} = & -4\pi \nabla_x \rho_{f-g} - 4\pi \partial_t J_{f-g}, \\ \partial_t^2 B_{f-g} - \Delta_x B_{f-g} = &4 \pi \nabla_x \times J_{f-g}, \end{split} \end{equation} with \begin{equation} \begin{split} & E_{f-g, \parallel } =0, B_{f-g,3} = 0, \text{ on } \partial \Omega \\ & \partial_3 E_{f-g, 3} = 4 \pi \rho_{f-g}, \partial_3 B_{f-g, 1 } = 4 \pi J_{f-g,2} , \partial_3 B_2 = -4 \pi J_{f-g,1}, \text{ on } \partial \Omega, \end{split} \end{equation} in the sense of \eqref{waveinner}, \eqref{waveD_weak}. Therefore, from Glassey-Strauss representation taking into account of the boundary, $E_{f-g}$ and $B_{f-g}$ has the form in Proposition \ref{Eiform} and Proposition \ref{Biform} with $f$ changes to $f-g$ everywhere. This allows us to bound (same as in Lemma \ref{EBlinflemma}) \[
\| E_{f-g}(t) \|_\infty + \| E_{f-g}(t) \|_\infty \lesssim \sup_{0 \le s \le t } \| \langle v \rangle^{4+ \delta } ( f - g)(s) \|_\infty. \] Together with \eqref{fgdiff}, we use Gronwall's inequality to get the $L^\infty$ stability \[
\sup_{0 \le s \le t } \| \langle v \rangle^{4+ \delta } (f- g ) (s) \|_\infty \lesssim e^{Ct } \| \langle v \rangle^{4+ \delta } (f- g )(0) \|_\infty. \] \unhide
\hide
\subsubsection*{$\bullet$\textbf{Specular BC}} Among other boundary conditions, we find that the specular boundary condition suffers most. Due to the lack of higher regularity of the fields (e.g. compare to \cite{CAO1} where the field is $C^2$), we can only derive an exponential-in-$\alpha$ singularity of the derivative of trajectory \begin{equation}\label{lemma_Dxv1}
\begin{split}
| \partial_{\mathbf e } X_{\mathbf{cl}}(s;t,x,v)| & \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } } ,
\\ | \partial_{\mathbf e } V_{\mathbf{cl}}(s;t,x,v)| & \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } }.
\end{split} \end{equation} We achieve
The proof of the theorem relies on the estimate of taking the derivatives to $f$ under the generalized characteristics for the specular reflection, i.e. the ``specular cycles", $X_{\mathbf{cl}}(s;t,x,v) , V_{\mathbf{cl}}(s;t,x,v) $ as in \eqref{cycles}. One has \begin{equation} \label{pef1} \begin{split}
\partial_{\mathbf e } f(t,x,v ) & = \partial_{\mathbf e} (f(0, X_{\mathbf{cl}}(0;t,x,v) , V_{\mathbf{cl}}(0;t,x,v) )
\\ & = \nabla_x f_0 \cdot \partial_{\mathbf e } X_{\mathbf{cl}}(0;t,x,v) + \nabla_v f_0 \cdot \partial_{\mathbf e } V_{\mathbf{cl}}(0;t,x,v). \end{split} \end{equation}
Note that here we have a high singularity in terms of $\alpha$ due to the fact of lacking higher regularity of the fields (e.g. compare to \cite{CAO1} where the field is $C^2$).
The proof of the theorem relies on the estimate of taking the derivatives to $f$ under the generalized characteristics for the specular reflection, i.e. the ``specular cycles", $X_{\mathbf{cl}}(s;t,x,v) , V_{\mathbf{cl}}(s;t,x,v) $ as in \eqref{cycles}. One has \begin{equation} \label{pef1} \begin{split} \partial_{\mathbf e } f(t,x,v ) & = \partial_{\mathbf e} (f(0, X_{\mathbf{cl}}(0;t,x,v) , V_{\mathbf{cl}}(0;t,x,v) ) \\ & = \nabla_x f_0 \cdot \partial_{\mathbf e } X_{\mathbf{cl}}(0;t,x,v) + \nabla_v f_0 \cdot \partial_{\mathbf e } V_{\mathbf{cl}}(0;t,x,v). \end{split} \end{equation} Then from the crucial estimate of the derivatives of the generalized characteristics in Lemma \ref{dXVcl}, for $\partial_{\mathbf e } \in \{ \nabla_x , \nabla_v \}$, we have \begin{equation}\label{lemma_Dxv1} \begin{split}
| \partial_{\mathbf e } X_{\mathbf{cl}}(s;t,x,v)| & \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } } ,
\\ | \partial_{\mathbf e } V_{\mathbf{cl}}(s;t,x,v)| & \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } }. \end{split} \end{equation} Note that here we have a high singularity in terms of $\alpha$ due to the fact of lacking higher regularity of the fields (e.g. compare to \cite{CAO1} where the field is $C^2$). From \eqref{lemma_Dxv1} and the strong decay of the initial data $f_0$ towards the grazing set in \eqref{f0spec}, we can control $\nabla_x f, \nabla_v f $ in an $L^\infty$ space. Details are presented in section \ref{chapspec}.
\unhide
\subsubsection*{\textbf{{Acknowledgements.}}} This project is supported in part by National Science Foundation under Grant No. 1900923 and 2047681 (NSF-CAREER). CK was supported by Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea (2021H1D3A2A01039047), and he thanks Professor Seung Yeal Ha for the kind hospitality of the BP program.
\tableofcontents
\section{Uniqueness of the Maxwell equations} In this section, we consider the uniqueness of solution to the Maxwell equations in $(0,T) \times \Omega$ in a presence of free charge: \begin{align}
\label{Maxwell1} \partial_t E & = \nabla_x \times B - 4 \pi J, \\ \label{Maxwell2} \partial_t B & = - \nabla_x \times E, \\ \label{Maxwell3} \nabla_x \cdot E& = 4 \pi \rho, \\ \label{Maxwell4} \nabla_x \cdot B &= 0, \end{align} with initial condition: \begin{equation} \label{EBat0} E(0,x) = E_0(x), \ B(0,x) = B_0(x) \text{ in } \Omega, \end{equation} and the perfect conductor boundary condition: \begin{equation} \label{percond} E_1 = E_2 = 0 \text{ on } \partial \Omega, \ B_3 = 0 \text{ on } \partial \Omega. \end{equation} It is worth to recall that the boundary conditions for $E_3$ and $B_1, B_2$ are not given originally.
\begin{definition}\label{sol_M} For given $E_0, B_0, \rho , j$, we say functions \begin{equation} \label{EBspace} E(t,x), B(t,x) \in W^{1,\infty}((0,T) \times \Omega), \end{equation} is a solution to the equations \eqref{Maxwell1}-\eqref{percond} if \eqref{Maxwell1}-\eqref{Maxwell4} holds almost everywhere in $(0,T) \times \Omega$, and \eqref{EBat0}, \eqref{percond} holds in the sense of trace. \end{definition} \begin{remark} Traces of $W^{1,\infty}((0,T) \times \Omega)$ are well-defined in a classical sense since any uniformly continuous function in space and time can be extended up to $[0,T] \times \bar\Omega$.
\end{remark}
The goal of this section is to prove the following uniqueness result: \begin{theorem}\label{EBunique} Suppose $E_0(x), B_0(x) \in W^{1,p}(\Omega) $, and $\nabla_x \rho, \nabla_x J, \partial_t J \in L^\infty((0,T); L_{\text{loc}}^p(\Omega))$ for some $p>1$, and \begin{equation} \label{conteq} \nabla \cdot J = - \partial_t \rho. \end{equation} Then a solution $E(t,x), B(t,x) \in W^{1,\infty}((0,T) \times \Omega)$ to the equations \eqref{Maxwell1}-\eqref{percond} in the sense of Definition \ref{sol_M} is unique.
\end{theorem}
The key of proof is to realize $E_3, B_1, B_2$ as weak solutions of inhomengenous wave equations with the Neumann boundary condition from a weak solution of Maxwell equations in the sense of Definition \ref{sol_M}.
\begin{definition}\label{def:weak_wave} Given any $u_0, u_1 : \Omega \to \mathbb R $, $G: (0,T) \times \Omega \to \mathbb R$, and $g : (0,T) \times \partial \Omega \to \mathbb R$, we define a function $u(t) \in W^{1,p}(\Omega)$ for $t \in (0,T)$, $p > 1$ to be a weak solution of the inhomengenous wave equation with Neumann boundary condition: \begin{equation} \label{waveNeu} \begin{split} \partial_t^2 u - \Delta_x u &= G,
\\ -\partial_{x_3} u(t,x) |_{\partial \Omega } &= g, \\ u(0,x) = u_0, \, \partial_t u(0,x ) & = u_1, \end{split} \end{equation} if for any $\phi \in C_c^\infty ([0,T) \times \bar \Omega )$, we have \begin{equation} \label{waveinner} \begin{split} &\langle u, \phi \rangle_N \\ : = & \int_{\Omega} ( u_1 (x) \phi(0,x) - u_0 (x) \partial_t \phi(0,x) ) dx + \int_0^T \int_\Omega u(t,x) \left( \partial_t^2 \phi(t,x) - \Delta_x \phi(t,x) \right) dx dt \\ & - \int_0^T \int_{\partial \Omega} u(t,x_\parallel, 0 ) \partial_{x_3} \phi(t,x_\parallel, 0 ) dx_\parallel dt + \int_0^T \int_{\partial \Omega } g(t,x_\parallel ) \phi(t, x_\parallel, 0 ) dx_\parallel dt - \int_0^T \int_\Omega G \phi \, dx dt \\ = & 0 , \end{split}
\end{equation} and each terms in \eqref{waveinner} are all bounded. Note that since $u(t) \in W^{1,p}(\Omega)$, it has a trace $u(t) |_{\partial \Omega} \in L^p (\partial \Omega)$. Also, note that $\text{supp} ( \phi ) \subset [0,T) \times \bar \Omega $ is compact, but $ \phi |_{t = 0 } \neq 0$, and $\phi |_{\partial \Omega } \neq 0 $ in general.
We also define a weak solution of the Dirichlet boundary problem: \begin{equation} \label{waveD} \begin{split} \partial_t^2 u - \Delta_x u &= G,
\\ u(t,x) |_{\partial \Omega } &= g, \\ u(0,x) = u_0, \, \partial_t u(0,x ) & = u_1,
\end{split}
\end{equation} if for any $\phi \in C_c^\infty((0,T) \times \bar \Omega) $, with $\phi |_{\partial \Omega} = 0$, we have \begin{equation} \label{waveD_weak} \begin{split} \langle u, \phi \rangle _{D} := & \int_{\Omega} ( u_1 (x) \phi(0,x) - u_0 (x) \partial_t \phi(0,x) ) dx + \int_0^T \int_\Omega u(t,x) \left( \partial_t^2 \phi(t,x) - \Delta_x \phi(t,x) \right) dx dt \\ & - \int_0^T \int_{\partial \Omega } g(t,x_\parallel ) \partial_{x_3} \phi(t, x_\parallel, 0 ) dx_\parallel dt - \int_0^T \int_\Omega G \phi \, dx dt \\ = & 0, \end{split} \end{equation} and each terms in \eqref{waveD_weak} are all bounded. \end{definition}
\begin{lemma} \label{Maxtowave} Suppose $E(t,x), B(t,x) \in W^{1,\infty}((0,T) \times \Omega)$ is a solution to the equations \eqref{Maxwell1}-\eqref{percond} in the sense of Definition \ref{sol_M}, and $E_0(x), B_0(x) \in W^{1,p}(\Omega) $, $\nabla_x \rho, \nabla_x J, \partial_t J \in L^\infty((0,T); L_{\text{loc}}^p(\Omega))$ for some $p>1$. Then $E_1, E_2, B_3$ solve the wave equation with the Dirichlet boundary condition \eqref{waveD} in the sense of \eqref{waveD_weak} with \begin{align}
u_0 = E_{0,i}, \ u_1 = \partial_t E_{0,i} : = (\nabla_x \times B)_i - 4 \pi J_{0,i} , \ G = -4\pi \partial_{x_i} \rho - 4 \pi \partial_t J_i, \ g = 0 , \ \ \text{for} \ E_i, i =1,2, \label{E12sol} \\
u_0 = B_{0,3}, \ u_1 = \partial_t B_{0,3} := - (\nabla_x \times E_0)_3, \ G = 4 \pi (\nabla_x \times J )_3, \ g = 0, \ \ \text{for} \ B_3, \label{B3sol}
\end{align} respectively.
Moreover, $E_3, B_1, B_2$ solve the wave equation with the Neumann boundary condition \eqref{waveNeu} in the sense of \eqref{waveinner} \text{ with } \begin{align}
u_0 = E_{0,3}, \ u_1 = \partial_t E_{0,3} : = - \partial_2 B_{0,1} + \partial_1 B_{0,2} - 4 \pi J_{0,3} , \ G = -4\pi \partial_{x_3} \rho - 4 \pi \partial_t J_3, \ g = - 4\pi \rho, \ \ \text{for} \ E_3, \label{E3sol} \\
u_0 = B_{0,i}, \ u_1 = \partial_t B_{0,i} := - (\nabla_x \times E_0)_i, \ G = 4 \pi (\nabla_x \times J )_i, \ g = (-1)^{i+1} 4 \pi J_{\underline i}, \ \ \text{for} \ B_i, \ i=1,2, \label{B12sol}
\end{align}
respectively.
\end{lemma} \begin{proof}\textit{Step 1. }We first consider $E_3, B_1, B_2$. Let $\phi(t,x) \in C^\infty_c( [0,T) \times \bar \Omega ) $. Since $E(t,x), B(t,x) \in W^{1,\infty}((0,T) \times \Omega)$, we can multiply $\partial_{x_3} \phi $ to equation \eqref{Maxwell3} and integrate over $(0,T) \times \Omega$ to get \begin{equation} \int_0^T \int_\Omega ( \nabla_x \cdot E ) \partial_3 \phi dx dt = \int_0^T \int_\Omega 4 \pi \rho \partial_3 \phi dx dt, \end{equation} then from integration by parts, we get \begin{equation} \label{E3phi1} \begin{split} &\int_0^T \int_\Omega - ( E_1 \partial_1 \partial_3 \phi + E_2 \partial_2 \partial_3 \phi + E_3 \partial_3^2 \phi ) dx dt - \int_0^T \int_{\partial \Omega} E_3 \partial_3 \phi dx_\parallel dt \\ = & \int_0^T \int_\Omega - 4 \pi \partial_3 \rho \phi dx dt - \int_0^T \int_{\partial \Omega } 4 \pi \rho \phi dx_\parallel dt. \end{split} \end{equation}
Next, multiplying $\partial_1 \phi $ to $\partial_t B_2 = - (\partial_3 E_1 - \partial_1 E_3) $ and integrate over $(0,T) \times \Omega$ we get \begin{equation} \label{ptB2p1phi} \int_0^T \int_\Omega \partial_t B_2 \partial_1 \phi dx dt = \int_0^T \int_\Omega - (\partial_3 E_1 - \partial_1 E_3) \partial_1 \phi dx dt. \end{equation} From integration by parts, the LHS of \eqref{ptB2p1phi} equals to \[ \begin{split} & \int_0^T \int_\Omega - B_2 \partial_1 \partial_t \phi dx - \int_\Omega B_2(0,x) \partial_1 \phi(0,x) dx \\ = & \int_0^T \int_\Omega \partial_1 B_2 \partial_t \phi dt dx + \int_\Omega \partial_1 B_2(0,x) \phi(0,x) dx. \end{split} \] And the RHS of \eqref{ptB2p1phi} equals to \[ \int_0^T \int_\Omega (E_1 \partial_3 \partial_1 \phi - E_3 \partial_1^2 \phi )dx dt, \]
where we've used that $E_1 |_{\partial \Omega} = 0 $. Thus we get \begin{equation} \label{E3phi2} \begin{split}
\int_0^T \int_\Omega \partial_1 B_2 \partial_t \phi dt dx + \int_\Omega \partial_1 B_2(0,x) \phi(0,x) dx = \int_0^T \int_\Omega (E_1 \partial_3 \partial_1 \phi - E_3 \partial_1^2 \phi )dx dt.
\end{split}
\end{equation} Similarly, multiplying $- \partial_2 \phi $ to $\partial_t B_1 = - (\partial_2 E_3 - \partial_3 E_2) $ and integrate over $(0,T) \times \Omega$, using integration by parts and that $E_2 |_{\partial \Omega} = 0 $, we get \begin{equation} \label{E3phi3} \begin{split}
- \int_0^T \int_\Omega \partial_2 B_1 \partial_t \phi dt dx - \int_\Omega \partial_2 B_1(0,x) \phi(0) dx = \int_0^T \int_\Omega (E_2 \partial_3 \partial_2 \phi - E_3 \partial_2^2 \phi )dx dt.
\end{split} \end{equation}
Finally, multiplying $ - \partial_t \phi$ to $\partial_t E_3 = \partial_1 B_2 - \partial_2 B_1 - 4\pi J_3 $ and integrate over $(0,T) \times \Omega$ we get \begin{equation}
- \int_0^T \int_\Omega \partial_t E_3 \partial_t \phi dx dt = - \int_0^T \int_\Omega ( \partial_1 B_2 - \partial_2 B_1 - 4\pi J_3 ) \partial_t \phi dx dt. \end{equation} From integration by parts, this gives \begin{equation} \label{E3phi4} \begin{split} & \int_0^T \int_\Omega E_3 \partial_t^2 \phi dx dt + \int_\Omega E_3(0) \partial_t \phi(0) dx \\= & - \int_0^T \int_\Omega ( \partial_1 B_2 - \partial_2 B_1 ) \partial_t \phi dx dt - \int_0^T \int_\Omega 4 \pi ( \partial_t J_3 \phi ) dx dt - \int_\Omega 4 \pi J_3(0,x) \phi(0,x) dx. \end{split} \end{equation}
Adding up \eqref{E3phi1}, \eqref{E3phi2}, \eqref{E3phi3}, and \eqref{E3phi4}, we get \begin{equation} \begin{split} & \int_0^T \int_\Omega ( \partial_t^2 \phi - \Delta_x \phi )E_3 dx dt + \int_\Omega \left( E_3(0,x) \partial_t \phi(0) + (\partial_2 B_1(0,x) - \partial_1 B_2(0,x) + 4\pi J_3(0,x) ) \phi(0,x) \right) dx \\ = & \int_0^T \int_\Omega ( - 4 \pi \partial_3 \rho - 4\pi \partial_t J_3 ) \phi dx dt + \int_0^T \int_{\partial \Omega} E_3 \partial_3 \phi dx_\parallel dt - \int_0^T \int_{\partial \Omega } 4 \pi \rho \phi dx_\parallel dt. \end{split} \end{equation} This proves \eqref{E3sol}.
\textit{Step 2. }Next, we do the same procedure for $B_1, B_2$. Multiply $\partial_t B_1 = - \partial_2 E_3 + \partial_3 E_2 $ by $ - \partial_t \phi $ and integrate over $(0,T) \times \Omega$, we get \[ - \int_0^T \int_\Omega \partial_t B_1 \partial_t \phi dx dt = - \int_0^T \int_\Omega ( - \partial_2 E_3 + \partial_3 E_2 ) \partial_t \phi dx dt. \]
From integration by parts and that $E_2 |_{\partial \Omega } = 0$, this gives \begin{equation} \label{B1phi1} \int_0^T \int_\Omega B_1 \partial_t^2 \phi dx dt + \int_\Omega B_1(0,x) \partial_t \phi(0,x) dx = - \int_0^T \int_\Omega ( E_3 \partial_2 \partial_t \phi - E_2 \partial_3 \partial_t \phi ) dx dt. \end{equation} Multiply $\partial_1 \phi $ to \eqref{Maxwell4} and integrate over $(0,T) \times \mathbb R^3$, we get \[ \int_0^T \int_\Omega (\nabla_x \cdot B) \partial_1 \phi dx dt = 0. \]
From integration by parts and that $B_3 |_{\partial \Omega } = 0$, this gives \begin{equation} \int_0^T \int_\Omega ( - B_1 \partial_1^2 \phi - B_2 \partial_2\partial_1 \phi - B_3 \partial_3 \partial_1 \phi ) dx dt = 0. \end{equation} Multiply $- \partial_2 \phi $ to $ \partial_t E_3 = \partial_1 B_2 - \partial_2 B_1 - 4 \pi J_3$ and integrate over $(0,T) \times \Omega$, we get \[ - \int_0^T \int_\Omega \partial_t E_3 \partial_2 \phi dx dt = - \int_0^T \int_\Omega ( \partial_1 B_2 - \partial_2 B_1 - 4 \pi J_3 ) \partial_2 \phi dx dt. \] From integration by parts, this gives \begin{equation}
\int_0^T \int_\Omega E_3 \partial_t \partial_2 \phi dx dt - \int_\Omega \partial_2 E_3(0,x) \phi(0,x) dx = \int_0^T \int_\Omega ( B_2 \partial_1 \partial_2 \phi - B_1 \partial_2^2 \phi - 4\pi \partial_2 J_3 \phi ) dx dt. \end{equation} Multiply $ \partial_3 \phi $ to $\partial_t E_2 = - (\partial_1 B_3 - \partial_3 B_1 ) - 4 \pi J_2$ and integrate over $(0,T) \times \Omega$, we get \[
\int_0^T \int_\Omega \partial_t E_2 \partial_3 \phi dx dt = \int_0^T \int_\Omega ( - \partial_1 B_3 + \partial_3 B_1 - 4 \pi J_2 ) \partial_3 \phi dx dt. \]
From integration by parts and that $ E_2 |_{\partial \Omega } = 0$, we have \begin{equation} \label{B1phi4} \begin{split}
&- \int_0^T \int_\Omega E_2 \partial_t \partial_3 \phi dx dt + \int_\Omega \partial_3 E_2(0,x) \phi (0,x) dx
\\ & = \int_0^T \int_\Omega ( B_3 \partial_1 \partial_3 \phi - B_1 \partial_3^2 \phi + 4 \pi \partial_3 J_2 \phi ) dx dt - \int_0^T \int_{\partial \Omega} B_1 \partial_3 \phi dx_\parallel dt + \int_0^T \int_{\partial \Omega } 4 \pi J_2 \phi dx_\parallel dt.
\end{split} \end{equation}
Adding up \eqref{B1phi1}--\eqref{B1phi4}, we get \begin{equation} \begin{split} & \int_0^T \int_\Omega ( \partial_t^2 \phi - \Delta_x \phi ) B_1 + \int_\Omega \left(B_1(0,x) \partial_t \phi(0,x) + (\partial_2 E_3(0,x) - \partial_3 E_2(0,x) )\phi(0,x) \right) dx \\ = & \int_0^T \int_\Omega ( 4 \pi \partial_2 J_3 - 4 \pi \partial_3 J_2 ) dx dt + \int_0^T \int_{\partial \Omega} B_1 \partial_3 \phi dx_\parallel dt- \int_0^T \int_{\partial \Omega } 4 \pi J_2 \phi dx_\parallel dt. \end{split} \end{equation} Using the same argument we also derive \[ \begin{split} & \int_0^T \int_\Omega ( \partial_t^2 \phi - \Delta_x \phi ) B_2 + \int_\Omega \left(B_2(0,x) \partial_t \phi(0,x) + (- \partial_1 E_3(0,x) - \partial_3 E_1(0,x) )\phi(0,x) \right) dx \\ = & \int_0^T \int_\Omega ( - 4 \pi \partial_1 J_3 + 4 \pi \partial_3 J_1 ) dx dt + \int_0^T \int_{\partial \Omega} B_2 \partial_3 \phi dx_\parallel dt + \int_0^T \int_{\partial \Omega } 4 \pi J_1 \phi dx_\parallel dt. \end{split} \] This proves \eqref{B12sol}.
\textit{Step 3. }Next, we consider $E_1, E_2, B_3$. Let $\phi(t,x) \in C_c^\infty([0,T) \times \bar \Omega ) $ such that $\phi |_{\partial \Omega } = 0$. Note that this implies $\partial_1 \phi|_{\partial \Omega} = \partial_2 \phi|_{\partial \Omega} = 0 $. Multiply $\partial_{x_1} \phi $ to equation \eqref{Maxwell3} and integrate over $(0,T) \times \Omega$ to get \[ \int_0^T \int_\Omega ( \nabla_x \cdot E ) \partial_1 \phi dx dt = \int_0^T \int_\Omega 4 \pi \rho \partial_1 \phi dx dt, \]
then from integration by parts and that $\partial_2 \phi |_{\partial \Omega} = 0$, we get \begin{equation} \label{E1phi1} \int_0^T \int_\Omega (-E_1 \partial_1^2 \phi - E_2 \partial_2 \partial_1 \phi - E_3 \partial_3 \partial_1 \phi ) dx dt = - \int_0^T \int_\Omega 4 \pi \partial_1 \rho \phi dx dt. \end{equation} Next, multiply $\partial_2 \phi $ to $\partial_t B_3 = - (\partial_1 E_2 - \partial_2 E_1 ) $ and integrate over $(0,T) \times \Omega$ we get \[ \int_0^T \int_\Omega \partial_t B_3 \partial_2 \phi dx dt = \int_0^T \int_\Omega - (\partial_1 E_2 - \partial_2 E_1) (\partial_2 \phi ) dx dt . \] From integration by parts, \begin{equation} \label{E1phi2} \begin{split} \int_0^T \int_\Omega \partial_2 B_3 \partial_t \phi dx dt + \int_\Omega \partial_2 B_3(0,x) \phi(0,x) dx = \int_0^T \int_\Omega ( E_2 \partial_1 \partial_2 \phi - E_1 \partial_2^2 \phi ) dx dt. \end{split} \end{equation} Multiply $- \partial_3 \phi $ to $\partial_t B_2 = \partial_1 E_3 - \partial_3 E_1 $ and integrate over $(0,T) \times \Omega$ we get \[ - \int_0^T \int_\Omega \partial_t B_2 \partial_3 \phi dx dt = - \int_0^T \int_\Omega (\partial_1 E_3 - \partial_3 E_1) (\partial_3 \phi ) dx dt . \]
From integration by parts and that $E_1 |_{\partial \Omega} = \phi |_{\partial \Omega} = 0$, we have \begin{equation} \label{E1phi3} - \int_0^T \int_\Omega \partial_3 B_2 \partial_t \phi dx dt - \int_\Omega \partial_3 B_2(0,x) \phi(0,x) dx dt = \int_0^T \int_\Omega ( E_3 \partial_1 \partial_3 \phi - E_1 \partial_3^2 \phi ) dx dt. \end{equation} Then multiplying $ - \partial_t \phi$ to $\partial_t E_1 = \partial_2 B_3 - \partial_3 B_2 - 4\pi J_1 $ and integrate over $(0,T) \times \Omega$, we get \[
- \int_0^T \int_\Omega \partial_t E_1 \partial_t \phi dx dt = - \int_0^T \int_\Omega ( \partial_2 B_3 - \partial_3 B_2 - 4\pi J_1 ) \partial_t \phi dx dt. \] From integration by parts, this gives \begin{equation} \label{E1phi4} \begin{split} & \int_0^T \int_\Omega E_1 \partial_t^2 \phi dx dt + \int_\Omega E_1(0) \partial_t \phi(0) dx \\= & - \int_0^T \int_\Omega ( \partial_2 B_3 - \partial_3 B_2 ) \partial_t \phi dx dt - \int_0^T \int_\Omega 4 \pi ( \partial_t J_1 \phi ) dx dt - \int_\Omega 4 \pi J_1 (0,x) \phi(0,x) dx. \end{split} \end{equation}
Adding up \eqref{E1phi1}-\eqref{E1phi4}, we get \[ \begin{split} & \int_0^T \int_\Omega ( \partial_t^2 \phi - \Delta_x \phi )E_1 dx dt + \int_\Omega \left( E_1(0,x) \partial_t \phi(0) + (\partial_2 B_3(0,x) - \partial_3 B_2(0,x) + 4\pi J_1(0,x) ) \phi(0,x) \right) dx \\ = & \int_0^T \int_\Omega ( - 4 \pi \partial_1 \rho - 4\pi \partial_t J_1 ) \phi dx dt. \end{split} \] Using the same argument, we derive \[ \begin{split} & \int_0^T \int_\Omega ( \partial_t^2 \phi - \Delta_x \phi )E_2 dx dt + \int_\Omega \left( E_2(0,x) \partial_t \phi(0) + (- \partial_1 B_3(0,x) + \partial_3 B_1(0,x) + 4\pi J_2(0,x) ) \phi(0,x) \right) dx \\ = & \int_0^T \int_\Omega ( - 4 \pi \partial_2 \rho - 4\pi \partial_t J_2 ) \phi dx dt. \end{split} \] This proves \eqref{E12sol}.
\textit{Step 4. }Next, multiply $\partial_t B_3 = - \partial_1 E_2 + \partial_2 E_1 $ by $ - \partial_t \phi $ and integrate over $(0,T) \times \Omega$, we get \[ - \int_0^T \int_\Omega \partial_t B_3 \partial_t \phi dx dt = - \int_0^T \int_\Omega ( - \partial_1 E_2 + \partial_2 E_1 ) \partial_t \phi dx dt. \] From integration by parts, \begin{equation} \label{B3phi1}
\int_0^T \int_\Omega B_3 \partial_t^2 \phi dx dt + \int_\Omega B_3(0,x) \partial_t \phi(0,x) dx = \int_0^T \int_\Omega ( - E_2 \partial_1 \partial_t \phi + E_1 \partial_2 \partial_t \phi ) dx dt . \end{equation} Multiply $\partial_3 \phi $ to \eqref{Maxwell4} and integrate over $(0,T) \times \mathbb R^3$, we get \[ \int_0^T \int_\Omega (\nabla_x \cdot B) \partial_3 \phi dx dt = 0. \]
From integration by parts and that $B_3 |_{\partial \Omega } = 0$, this gives \begin{equation} \label{B3phi2} \int_0^T \int_\Omega ( - B_1 \partial_1\partial_3 \phi - B_2 \partial_2\partial_3 \phi - B_3 \partial_3^2 \phi ) dx dt = 0. \end{equation} Multiply $- \partial_1 \phi $ to $ \partial_t E_2 = - \partial_1 B_3 + \partial_3 B_1 - 4 \pi J_2$ and integrate over $(0,T) \times \Omega$, we get \[ - \int_0^T \int_\Omega \partial_t E_2 \partial_1 \phi dx dt = - \int_0^T \int_\Omega ( -\partial_1 B_3 + \partial_3 B_1 - 4 \pi J_2 ) \partial_1 \phi dx dt. \]
From integration by parts and that $\partial_1 \phi |_{\partial \Omega } = 0$, this gives \begin{equation} \label{B3phi3}
\int_0^T \int_\Omega E_2 \partial_t \partial_1 \phi dx dt - \int_\Omega \partial_1 E_2(0,x) \phi(0,x) dx = \int_0^T \int_\Omega ( - B_3 \partial_1^2 \phi + B_1 \partial_3 \partial_1 \phi - 4\pi \partial_1 J_2 \phi ) dx dt. \end{equation} Multiply $ \partial_2 \phi $ to $\partial_t E_1 = \partial_2 B_3 - \partial_3 B_2 - 4 \pi J_1$ and integrate over $(0,T) \times \Omega$, we get \[
\int_0^T \int_\Omega \partial_t E_1 \partial_2 \phi dx dt = \int_0^T \int_\Omega ( \partial_2 B_3 - \partial_3 B_2 - 4 \pi J_1 ) \partial_2 \phi dx dt. \]
From integration by parts and that $\partial_2 \phi |_{\partial \Omega } = 0$, we have \begin{equation} \label{B3phi4} \begin{split} - \int_0^T \int_\Omega E_1 \partial_t \partial_2 \phi dx dt + \int_\Omega \partial_2 E_1(0,x) \phi (0,x) dx
= \int_0^T \int_\Omega ( - B_3 \partial_2^2 \phi + B_2 \partial_3 \partial_2 \phi + 4 \pi \partial_2 J_1 \phi ) dx dt.
\end{split} \end{equation}
Adding up \eqref{B3phi1}-\eqref{B3phi4}, we get \[ \begin{split} & \int_0^T \int_\Omega ( \partial_t^2 \phi - \Delta_x \phi ) B_3 + \int_\Omega \left(B_3(0,x) \partial_t \phi(0,x) + (\partial_1 E_2(0,x) - \partial_2 E_1(0,x) )\phi(0,x) \right) dx \\ = & \int_0^T \int_\Omega ( 4 \pi \partial_1 J_2 - 4 \pi \partial_2 J_1 ) dx dt . \end{split} \] This proves \eqref{B3sol}.\end{proof}
Next, we prove the uniqueness of wave equation with Neumann BC \eqref{waveNeu} and Dirichlet BC \eqref{waveD}. \begin{lemma} \label{wavesol}
Suppose $u(t,x), \tilde u(t,x)$ are weak solutions with the Neumann BC \eqref{waveNeu} with the same $u_0$, $u_1$, $G$, and $g$ in the sense of weak formulation \eqref{waveinner}. Then $u(t,x) = \tilde u(t,x)$. \end{lemma} \begin{proof} It suffices to show that if $u$ is the solution of \eqref{waveNeu} with \begin{equation} \label{waveNeu0} \begin{split} u_0 = u_1 = G = g = 0, \end{split} \end{equation} in the sense of \eqref{waveinner}, then for any $\psi \in C_c^\infty((0,T) \times \Omega )$, \begin{equation} \label{intupsi} \int_0^T \int_\Omega u(t,x) \psi(t,x) dx dt = 0. \end{equation}
Let $\tilde \psi (t,x) = \psi(T-t, x )$, then $\tilde \psi \in C_c^\infty((0,T) \times \Omega )$. We consider the function $\tilde v(t,x) : (0,T) \times \mathbb R^3 \to \mathbb R$ given by \begin{equation} \label{vtest} \begin{split}
\tilde v(t,x) := & \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 > 0 \} }\frac{ \tilde \psi (t - |y-x|, y_\parallel, y_3 ) }{|y-x| } dy
\\ & + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 < 0 \} }\frac{ \tilde \psi ( t - |y-x|, y_\parallel, - y_3 ) }{|y-x| } dy. \end{split} \end{equation} Then the function $\tilde v(t,x)$ is a weak solution of the wave equation in $(0,T) \times \mathbb R^3$ \begin{equation} \label{tildevwave} \begin{split} (\partial_t^2 - \Delta_x) \tilde v(t,x) = & \mathbf 1_{x_3 > 0 } \tilde \psi(t,x) + \mathbf 1_{x_3 < 0 } \tilde \psi(t, \bar x) \\ \tilde v(0,x) = & 0, \ \partial_t \tilde v(0,x) = 0, \end{split} \end{equation} where $x= (x_1, x_2, -x_3)$. And since $\psi \in C_c^\infty((0,T) \times \Omega )$, the function $ \mathbf 1_{x_3 > 0 } \tilde \psi(t,x) + \mathbf 1_{x_3 < 0 } \tilde \psi(t, \bar x) $ is smooth in $(0,T) \times \mathbb R^3$. Thus $\tilde v $ is smooth. Moreover, for some small $\delta > 0$, \begin{equation} \label{vtsupp} \tilde v(s,x) = 0 \text{ for } s \in [ 0,\delta ), \end{equation} and a direct computation yields \begin{equation} \label{p3tildev} \partial_{x_3} \tilde v = 0 \text{ on } (0,T) \times \partial \Omega. \end{equation}
Now, let $v(t,x) : [0,T) \times \bar \Omega \to \mathbb R^3$ be given by \begin{equation} \label{defv} v(t,x) = \tilde v(T-t,x ). \end{equation} Then $v(t,x)$ is smooth, and by \eqref{vtsupp}, $v(t,x) \in C_c^\infty([0,T) \times \bar \Omega ) $. Moreover, from \eqref{tildevwave}, \eqref{p3tildev}, \begin{equation} \label{wavevneu} \begin{split} (\partial_t^2 - \Delta_x )v(t,x) = & \tilde \psi(T-t, x ) \text{ in } (0,T) \times \Omega, \\ \partial_{x_3} v(t,x) = & 0 \text{ on } (0,T) \times \partial \Omega. \end{split}
\end{equation} Now, since $u$ is the solution of \eqref{waveNeu} with $u_0 = u_1 = G = g = 0$, and $v(t,x) \in C_c^\infty([0,T) \times \bar \Omega ) $, $\partial_{x_3} v|_{\partial \Omega } = 0$, from \eqref{waveinner} we have \begin{equation} \label{inner0} \begin{split} 0 = & \int_{\Omega} ( u_1 (x) v(0,x) - u_0 (x) \partial_t v(0,x) ) dx + \int_0^T \int_\Omega u(t,x) \left( \partial_t^2 v(t,x) - \Delta_x v(t,x) \right) dx dt \\ & - \int_0^T \int_{\partial \Omega } u(t,x_\parallel ) \partial_{x_3} v(t, x_\parallel, 0 ) dx_\parallel dt + \int_0^T \int_{\partial \Omega } g(t,x_\parallel ) v(t, x_\parallel, 0 ) dx_\parallel dt - \int_0^T \int_\Omega G v \, dx dt
\\ = & \int_0^T \int_\Omega u (t,x) \tilde \psi (T-t, x ) dx dt \\ = & \int_0^T \int_\Omega u(t,x) \psi (t,x) dx dt. \end{split} \end{equation} Thus, we proved \eqref{intupsi} and this conclude the lemma. \end{proof}
We also prove a similar version of the lemma that will be used later. \begin{lemma} \label{wavesol2}
Let $u :(0,T) \times \Omega \to \mathbb R$ be a function such that for any $\phi \in C_c^\infty([0,T) \times \bar \Omega )$, with $\partial_{x_3} \phi |_{\partial \Omega} = 0$, \begin{equation} \label{waveuN0} \int_0^T \int_\Omega u ( \partial_t^2- \Delta_x ) \phi dx dt = 0, \end{equation} then $u =0$. \end{lemma} \begin{proof} Take any $\psi \in C_c^\infty ((0,T) \times \Omega ) $. Let $\tilde \psi (t,x) = \psi(T-t, x )$, and define $\tilde v (t,x) $ in the same way as \eqref{vtest}. Then define $v(t,x)$ as in \eqref{defv}. Then, as showed in \eqref{vtest}-\eqref{wavevneu}, $v(t,x) \in C_c^\infty([0,T) \times \bar \Omega ) $, \[ \begin{split}
(\partial_t^2 - \Delta_x )v(t,x) = \psi(t, x ) \text{ in } (0,T) \times \Omega, \text{ and } \partial_{x_3} v |_{\partial \Omega } = 0 . \end{split} \] Therefore, from \eqref{waveuN0} \[
\int_0^T \int_\Omega u \psi dx dt = \int_0^T \int_\Omega u ( \partial_t^2- \Delta_x ) v dx dt = 0. \] Thus $u = 0$. \end{proof}
\begin{lemma} \label{wavesolD} Suppose $u(t,x), \tilde u(t,x)$ are weak solutions with the Dirichlet BC \eqref{waveD} with the same $u_0$, $u_1$, $G$, and $g$ in the sense of weak formulation \eqref{waveD_weak}. Then $u(t,x) = \tilde u(t,x)$. \end{lemma} \begin{proof} It suffices to show that if $u$ is the solution of \eqref{waveD} with \begin{equation} \label{waveD0} \begin{split} u_0 = u_1 = G = g = 0, \end{split} \end{equation} in the sense of \eqref{waveD_weak}, then for any $\psi \in C_c^\infty((0,T) \times \Omega )$, \begin{equation} \label{intupsiD} \int_0^T \int_\Omega u(t,x) \psi(t,x) dx dt = 0. \end{equation} Now, let $\tilde \psi(t,x) = \psi (T-t,x)$, and define the function $\tilde w(t,x) : (0,T) \times \mathbb R^3 \to \mathbb R^3$ as \begin{equation}
\tilde w(t,x) : = \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 > 0 \} }\frac{ \tilde \psi (t - |y-x|, y_\parallel, y_3 ) }{|y-x| } dy - \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 < 0 \} }\frac{ \tilde \psi ( t - |y-x|, y_\parallel, - y_3 ) }{|y-x| } dy, \end{equation} and let $w(t,x) : [0,T) \times \bar \Omega \to \mathbb R^3$ be given by \begin{equation} w(t,x) = \tilde w(T-t,x ). \end{equation} Then from direct computation and using the same argument as \eqref{vtest}-\eqref{wavevneu}, we get $w(t,x) \in C_c^\infty([0,T) \times \bar \Omega ) $, and \begin{equation} \label{wavewneu} \begin{split} (\partial_t^2 - \Delta_x )w(t,x) = & \tilde \psi(T-t, x ) = \psi(t,x) \text{ in } (0,T) \times \Omega, \\ w(t,x) = & 0 \text{ on } (0,T) \times \partial \Omega. \end{split} \end{equation} Therefore from \eqref{waveD_weak} and \eqref{waveD0}, we have \[
0 = \int_0^T \int_\Omega u (\partial_t^2 - \Delta_x )w dx dt = \int_0^T \int_\Omega u \psi dx dt. \] This proves \eqref{intupsiD}.\end{proof}
Next, we show that the Lipschitz solutions of the wave equations solves the Maxwell equations if the continuity equation {conteq} and some initial compatibility condition are satisfied. \begin{lemma} \label{wavetoMax}
Suppose $E(t,x), B(t,x) \in W^{1, \infty}((0,T) \times \Omega ) $, $\nabla_x \rho, \partial_t J, \nabla_x J \in L^\infty((0,T); L_{\text{loc}}^{p}( \Omega )) $, with
\begin{equation}
\nabla \cdot J = - \partial_t \rho.
\end{equation}
Assume
\begin{equation} \label{weakEass}
\begin{split}
& E_1, E_2 \text{ solves } \eqref{waveD} \text{ with } \eqref{E12sol}, \text{ and } E_3 \text{ solves } \eqref{waveNeu}
\text{ with } \eqref{E3sol},
\\ & B_3 \text{ solves } \eqref{waveD} \text{ with } \eqref{B3sol}, \text{ and } B_1, B_2 \text{ solves } \eqref{waveNeu}
\text{ with } \eqref{B12sol}.
\end{split}
\end{equation}
Further we assume compatibility conditions
\begin{align}
\nabla \cdot E_0 = 4 \pi \rho_0, \ \nabla_x \cdot B_0 =0, \text{ in } \Omega, \label{GaussE0} \\
E_{0,1} = E_{0,2} = B_{0,3} = 0 \text{ on } \partial \Omega. \label{Dirt0}
\end{align}
Then we have
\begin{equation}
\begin{split}
\partial_t E & = \nabla_x \times B - 4 \pi J, \, \nabla_x \cdot E = 4\pi \rho,
\\ \partial_t B & = - \nabla_x \times E, \, \nabla_x \cdot B = 0.
\end{split}
\end{equation} \end{lemma} \begin{proof}
Let's first prove $ \nabla \cdot E =4 \pi \rho$.
In the view of Lemma \ref{wavesolD}, it suffices to show that for any $\phi(t,x) \in C_c^\infty( [0,T) \times \bar \Omega ) $ with $\phi |_{\partial \Omega } = 0 $, we have
\begin{equation} \label{GaussEweak}
\int_0^T \int_\Omega ( \nabla \cdot E - 4 \pi \rho ) (\partial_t^2 - \Delta_x) \phi dx dt = 0.
\end{equation}
Now by direct computation with integration by parts, we have
\[
\begin{split}
& \int_0^T \int_\Omega ( \nabla \cdot E - 4 \pi \rho ) (\partial_t^2 - \Delta_x) \phi dx dt
\\ = & \int_0^T \int_\Omega - \sum_{i=1}^3 E_i (\partial_t^2 - \Delta_x) (\partial_{x_i} \phi ) dx dt - \int_0^T \int_{\partial \Omega } E_3 (\partial_t^2 - \Delta_x) \phi \ dx_\parallel dt - \int_0^T \int_{\Omega } 4 \pi \rho (\partial_t^2 - \Delta_x) \phi dx dt
\\ = & \int_\Omega ( - \partial_t E_{0} \cdot \nabla \phi(0,x) + E_{0} \cdot \nabla \partial_t \phi(0,x) ) dx + 4 \pi \int_0^T \int_\Omega \sum_{i=1}^3 ( \partial_{x_i} \rho + \partial_t J_i ) (\partial_{x_i} \phi ) dx dt
\\ & - \int_0^T \int_{\partial \Omega } (E_3 \partial_{x_3}^2 \phi - 4 \pi \rho \partial_{x_3} \phi ) \ d x_\parallel dt - \int_0^T \int_{\partial \Omega } E_3 (- \partial_{x_3}^2 ) \phi \ dx_\parallel dt - \int_0^T \int_{\Omega } 4 \pi \rho (\partial_t^2 - \Delta_x) \phi dx dt
\\ = & \int_\Omega ( - \partial_t E_{0} \cdot \nabla \phi(0,x) + E_{0} \cdot \nabla \partial_t \phi(0,x) ) dx - 4 \pi \int_0^T \int_\Omega \rho \Delta_x \phi dx dt - 4\pi \int_0^T \int_\Omega \sum_{i=1}^3 J_i \partial_t \partial_{x_i } \phi dx dt
\\ & - 4\pi \int_\Omega \sum_{i=1}^3 J_{0,i} \partial_{x_i} \phi(0,x) dx - \int_0^T \int_{\Omega } 4 \pi \rho (\partial_t^2 - \Delta_x) \phi dx dt
\\ = & \int_\Omega ( - \partial_t E_{0} \cdot \nabla \phi(0,x) + E_{0} \cdot \nabla \partial_t \phi(0,x) ) dx + 4\pi \int_0^T \int_\Omega \nabla \cdot J \partial_t \phi dx dt
\\ & + 4\pi \int_\Omega J_0 \cdot \nabla \phi(0,x) dx - 4 \pi \int_0^T \int_\Omega \rho \partial_t^2 \phi dx dt
\end{split}
\]
where in the second equality we've used \eqref{weakEass}.
Now, using $\nabla \cdot J = - \partial_t \rho$, \eqref{GaussE0}, and integration by parts we obtain
\[
\begin{split}
& \int_0^T \int_\Omega ( \nabla \cdot E - 4 \pi \rho ) (\partial_t^2 - \Delta_x) \phi dx dt
\\ = & \int_\Omega ( - \partial_t E_{0} \cdot \nabla \phi(0,x) + E_{0} \cdot \nabla \partial_t \phi(0,x) ) dx
\\ & - 4\pi \int_0^T \int_\Omega \partial_t \rho \partial_t \phi dx dt + 4\pi \int_\Omega J_0 \cdot \nabla \phi(0,x) dx - 4 \pi \int_0^T \int_\Omega \rho \partial_t^2 \phi dx dt
\\ = & \int_\Omega ( - \partial_t E_{0} + 4 \pi J_0 ) \cdot \nabla \phi(0,x) dx - \int_\Omega ( \nabla \cdot E_{0} - 4 \pi \rho_0 ) \partial_t \phi(0,x) ) dx
\\ = & \int_\Omega - (\nabla_x \times B_0) \cdot \nabla \phi(0,x) dx = 0.
\end{split}
\]
This proves \eqref{GaussEweak}.
Next, let's show that $ \partial_t E_1 = (\nabla_x \times B)_1 - 4 \pi J_1 $. It suffices to prove for any $\phi(t,x) \in C_c^\infty( [0,T) \times \bar \Omega ) $ with $ \phi |_{\partial \Omega } = 0 $, we have
\begin{equation} \label{Ampere1}
\int_0^T \int_\Omega (\partial_t E_1 - ( \nabla_x \times B)_1 + 4 \pi J_1 )(\partial_t^2 - \Delta_x) \phi dx dt = 0.
\end{equation}
Using \eqref{weakEass} and integration by parts, we compute
\[
\begin{split}
& \int_0^T \int_\Omega (\partial_t E_1 - ( \nabla_x \times B)_1 + 4 \pi J_1 )(\partial_t^2 - \Delta_x) \phi dx dt
\\ = & - \int_0^T \int_\Omega E_1 ( \partial_t^2 - \Delta_x ) (\partial_t \phi )dx dt - \int_\Omega E_{0,1} (\partial_t^2 - \Delta_x) \phi(0,x) dx + \int_0^T \int_\Omega 4\pi J_1 (\partial_t^2 - \Delta_x) \phi dx dt
\\ & + \int_0^T \int_\Omega B_3 ( \partial_t^2 - \Delta_x ) (\partial_2 \phi ) dx dt - \int_0^T \int_\Omega B_2 ( \partial_t^2 - \Delta_x ) (\partial_3 \phi ) dx dt - \int_0^T \int_{\partial \Omega } B_2(\partial_t^2 - \Delta_x ) \phi d x_\parallel dt
\\ = & \int_\Omega \left( E_{0,1} \partial_t^2 \phi(0,x) - \partial_t E_{0,1} \partial_t \phi(0,x) \right) dx + \int_0^T \int_\Omega 4 \pi ( \partial_1 \rho + \partial_t J_1 ) (\partial_t \phi ) dx dt - \int_\Omega E_{0,1} \partial_t^2 \phi (0,x) dx + \int_\Omega \Delta_x E_{0,1} \phi (0,x) dx
\\ & + \int_0^T \int_\Omega 4\pi J_1 (\partial_t^2 - \Delta_x) \phi dx dt - \int_\Omega B_{0,3} (\partial_t \partial_2 \phi(0,x) )dx + \int_\Omega \partial_t B_{0,3 } \partial_2 \phi(0,x) dx + \int_0^T \int_\Omega 4 \pi (\nabla_x \times J )_3 (\partial_2 \phi ) dx dt
\\ & + \int_\Omega B_{0,2} ( \partial_t \partial_3 \phi(0,x) ) dx - \int_\Omega \partial_t B_{0,2} \partial_3 \phi(0,x) dx - \int_0^T \int_\Omega 4 \pi (\nabla_x \times J)_2 (\partial_3 \phi ) dx dt - \int_0^T \int_{\partial \Omega } B_2 \partial_3^2 \phi dx_\parallel dt
\\ & - \int_0^T \int_{\partial \Omega} 4 \pi J_1 \partial_3 \phi d x_\parallel dt + \int_0^T \int_{\partial \Omega } B_2 \partial_3^2 \phi dx_\parallel dt
\\ = & - \int_\Omega \partial_t E_{0,1} \partial_t \phi(0,x) dx + \int_\Omega 4 \pi \rho_0 \partial_1 \phi(0,x) dx + \int_0^T \int_\Omega 4 \pi \partial_t \rho \partial_1 \phi dx dt - \int_\Omega 4 \pi J_{0,1} \partial_t \phi(0,x) dx + \int_\Omega E_{0,1} \Delta_x \phi(0,x) dx
\\ & - \int_0^T \int_\Omega 4 \pi J_1 \Delta_x \phi dx dt + \int_\Omega ( B_{0,2} \partial_t \partial_3 \phi - B_{0,3} \partial_t \partial_2 \phi (0,x) ) dx - \int_\Omega ( - \partial_t B_{0,3} \partial_2 \phi(0,x) + \partial_t B_{0,2} \partial_3 \phi(0,x) )dx
\\ & + \int_0^T \int_\Omega 4 \pi (- J_2 \partial_1 \partial_2 \phi + J_1 \partial_2^2 \phi - J_3 \partial_1 \partial_3 \phi + J_1 \partial_3^2 \phi ) dx dt + \int_0^T \int_{\partial \Omega } 4 \pi J_1 \partial_3 \phi dx_\parallel dt- \int_0^T \int_{\partial \Omega} 4 \pi J_1 \partial_3 \phi d x_\parallel dt.
\end{split}
\]
Then from \eqref{conteq} and integration by parts,
\[
\begin{split}
& \int_0^T \int_\Omega 4 \pi \partial_t \rho \partial_1 \phi dx dt - \int_0^T \int_\Omega 4 \pi J_1 \Delta_x \phi dx dt+ \int_0^T \int_\Omega 4 \pi (- J_2 \partial_1 \partial_2 \phi + J_1 \partial_2^2 \phi - J_3 \partial_1 \partial_3 \phi + J_1 \partial_3^2 \phi ) dx dt
\\ = & - \int_0^T \int_\Omega 4 \pi \nabla \cdot J \partial_1 \phi dx dt - \int_0^T \int_\Omega 4 \pi J_1 \Delta_x \phi dx dt+ \int_0^T \int_\Omega 4 \pi (- J_2 \partial_1 \partial_2 \phi + J_1 \partial_2^2 \phi - J_3 \partial_1 \partial_3 \phi + J_1 \partial_3^2 \phi ) dx dt
\\ = & \int_0^T \int_\Omega 4 \pi ( J_1 \partial_1^2 \phi + J_2 \partial_2 \partial_1 \phi + J_3 \partial_3 \partial_1 \phi ) dx dt - \int_0^T \int_\Omega 4 \pi J_1 \Delta_x \phi dx dt
\\ & + \int_0^T \int_\Omega 4 \pi (- J_2 \partial_1 \partial_2 \phi + J_1 \partial_2^2 \phi - J_3 \partial_1 \partial_3 \phi + J_1 \partial_3^2 \phi ) dx dt
\\ = & 0.
\end{split}
\]
Thus,
\begin{equation}
\begin{split}
& \int_0^T \int_\Omega (\partial_t E_1 - ( \nabla_x \times B)_1 + 4 \pi J_1 )(\partial_t^2 - \Delta_x) \phi dx dt
\\ = & \int_\Omega ((-\partial_t E_{0,1} - 4\pi J_{0,1} ) \partial_t \phi(0,x) + B_{0,2} \partial_3 \partial_t \phi(0,x) - B_{0,3} \partial_2 \partial_t \phi(0,x) ) dx
\\ & + \int_\Omega ( -4\pi \partial_1 \rho_0 \phi(0,x) + E_{0,1} \Delta_x \phi(0,x) + \partial_t B_{0,3} \partial_2 \phi(0,x) - \partial_t B_{0,2} \partial_3 \phi(0,x) ) dx.
\end{split}
\end{equation}
From \eqref{weakEass}, we have $ -\partial_t E_{0,1} - 4\pi J_{0,1} - \partial_3 B_{0,2} + \partial_2 B_{0,3} = 0 $, and $\partial_t B_0 = - \nabla_x \times E_0$. Together with \eqref{GaussE0}, \eqref{Dirt0}, we use integration by parts to get
\[
\begin{split}
& \int_\Omega ((-\partial_t E_{0,1} - 4\pi J_{0,1} ) \partial_t \phi(0,x) + B_{0,2} \partial_3 \partial_t \phi(0,x) - B_{0,3} \partial_2 \partial_t \phi(0,x) ) dx
\\ & + \int_\Omega ( -4\pi \partial_1 \rho_0 \phi(0,x) + E_{0,1} \Delta_x \phi(0,x) + \partial_t B_{0,3} \partial_2 \phi(0,x) - \partial_t B_{0,2} \partial_3 \phi(0,x) ) dx
\\ = & 0
\end{split}
\]
Thus, we conclude \eqref{Ampere1}. And from the same argument we can show that $ \partial_t E_2 = (\nabla_x \times B)_2 + 4 \pi J_2 $.
Next, let's prove $\partial_t E_3 = ( \nabla_x \times B)_3 - 4 \pi J_3$. In the view of Lemma \ref{wavesol2}, it suffices to show that for any $\psi \in C_c^\infty( [0,T) \times \bar \Omega ) $ with $\partial_{x_3} \psi |_{\partial \Omega } = 0 $, we have
\begin{equation} \label{Ampere3}
\int_0^T \int_\Omega (\partial_t E_3 - ( \nabla_x \times B)_3 + 4 \pi J_3 )(\partial_t^2 - \Delta_x) \psi dx dt = 0.
\end{equation}
Using \eqref{weakEass} and integration by parts, we compute
\[
\begin{split}
& \int_0^T \int_\Omega (\partial_t E_3 - ( \nabla_x \times B)_3 + 4 \pi J_3 )(\partial_t^2 - \Delta_x) \psi dx dt
\\ = & - \int_0^T \int_\Omega E_3 ( \partial_t^2 - \Delta_x ) (\partial_t \psi )dx dt - \int_\Omega E_{0,3} (\partial_t^2 - \Delta_x) \psi(0,x) dx + \int_0^T \int_\Omega 4\pi J_3 (\partial_t^2 - \Delta_x) \psi dx dt
\\ & + \int_0^T \int_\Omega B_2 ( \partial_t^2 - \Delta_x ) (\partial_1 \psi ) dx dt - \int_0^T \int_\Omega B_1 ( \partial_t^2 - \Delta_x ) (\partial_2 \psi ) dx dt
\\ = & \int_\Omega \left( E_{0,3} \partial_t^2 \psi(0,x) - \partial_t E_{0,3} \partial_t \psi(0,x) \right) dx + \int_0^T \int_\Omega 4 \pi ( \partial_3 \rho + \partial_t J_3 ) (\partial_t \psi ) dx dt + \int_0^T \int_{\partial \Omega } 4 \pi \rho \partial_t \psi dx_\parallel dt
\\ & - \int_\Omega E_{0,3} \partial_t^2 \psi (0,x) dx + \int_\Omega \Delta_x E_{0,3} \psi (0,x) dx + \int_{\partial \Omega } 4 \pi \rho_0 \psi(0,x_\parallel ) dx_\parallel + \int_0^T \int_\Omega 4\pi J_3 (\partial_t^2 - \Delta_x) \psi dx dt
\\ & - \int_\Omega B_{0,2} (\partial_t \partial_1 \psi(0,x) )dx + \int_\Omega \partial_t B_{0,2 } \partial_1 \psi(0,x) dx + \int_0^T \int_\Omega 4 \pi (\nabla_x \times J )_2 (\partial_1 \psi ) dx dt - \int_0^T \int_{\partial \Omega } (-4 \pi J_1 \partial_1 \psi ) dx_\parallel dt
\\ & + \int_\Omega B_{0,1} ( \partial_t \partial_2 \psi(0,x) ) dx - \int_\Omega \partial_t B_{0,1} \partial_2 \psi(0,x) dx - \int_0^T \int_\Omega 4 \pi (\nabla_x \times J)_1 (\partial_2 \psi ) dx dt + \int_0^T \int_{\partial \Omega } ( 4 \pi J_2 \partial_2 \psi ) dx_\parallel dt
\\ = & - \int_\Omega \partial_t E_{0,3} \partial_t \psi(0,x) dx + \int_0^T \int_\Omega 4 \pi \partial_t \rho \partial_3 \psi dx dt + \int_\Omega 4 \pi \rho_0 \partial_3 \psi(0,x) dx - \int_\Omega J_{0,3} \partial_t \psi (0,x) dx
\\ & + \int_\Omega E_{0,3} \Delta_x \psi (0,x) dx + \int_{\partial \Omega } 4 \pi \rho_0 \psi(0,x_\parallel ) dx_\parallel - \int_0^T \int_\Omega 4 \pi J_3 \Delta_x \psi dx dt + \int_\Omega ( \partial_1 B_{0,2} - \partial_2 B_{0,1} ) \partial_t \psi(0,x) dx
\\ & + \int_\Omega (- \partial_t B_{0,1} \partial_2 \psi(0,x) + \partial_t B_{0,2} \partial_1 \psi(0,x) ) dx + \int_0^T \int_\Omega 4\pi (J_3 \partial_2^2 \psi - J_2 \partial_3 \partial_2 \psi - J_1 \partial_3 \partial_1 \psi + J_3 \partial_1^2 \psi) dx dt.
\end{split}
\]
From \eqref{conteq} and integration by parts,
\[
\begin{split}
& \int_0^T \int_\Omega 4 \pi \partial_t \rho \partial_3 \psi dx dt - \int_0^T \int_\Omega 4 \pi J_3 \Delta_x \psi dx dt+ \int_0^T \int_\Omega 4 \pi (J_3 \partial_2^2 \psi - J_2 \partial_3 \partial_2 \psi - J_1 \partial_3 \partial_1 \psi + J_3 \partial_1^2 \psi) dx dt
\\ = & - \int_0^T \int_\Omega 4 \pi \nabla \cdot J \partial_3 \psi dx dt - \int_0^T \int_\Omega 4 \pi J_3 \Delta_x \psi dx dt+ \int_0^T \int_\Omega 4 \pi (J_3 \partial_2^2 \psi - J_2 \partial_3 \partial_2 \psi - J_1 \partial_3 \partial_1 \psi + J_3 \partial_1^2 \psi) dx dt
\\ = & \int_0^T \int_\Omega 4 \pi ( J_1 \partial_1 \partial_3 \psi + J_2 \partial_2 \partial_3 \psi + J_3 \partial_3^2 \psi ) dx dt - \int_0^T \int_\Omega 4 \pi J_3 \Delta_x \psi dx dt
\\ & + \int_0^T \int_\Omega 4 \pi (J_3 \partial_2^2 \psi - J_2 \partial_3 \partial_2 \psi - J_1 \partial_3 \partial_1 \psi + J_3 \partial_1^2 \psi) dx dt
\\ = & 0.
\end{split}
\]
Thus,
\begin{equation}
\begin{split}
& \int_0^T \int_\Omega (\partial_t E_3 - ( \nabla_x \times B)_3 + 4 \pi J_3 )(\partial_t^2 - \Delta_x) \psi dx dt
\\ = & \int_\Omega (-\partial_t E_{0,3} - 4\pi J_{0,3} + \partial_1 B_{0,2} - \partial_2 B_{0,1} ) \partial_t \psi(0,x) dx
\\ & + \int_\Omega ( -4\pi \partial_3 \rho_0 \psi(0,x) + E_{0,3} \Delta_x \psi(0,x) - \partial_t B_{0,1} \partial_2 \psi(0,x) + \partial_t B_{0,2} \partial_1 \psi(0,x) ) dx.
\end{split}
\end{equation}
From \eqref{weakEass}, we have $ -\partial_t E_{0,3} - 4\pi J_{0,3} + \partial_1 B_{0,2} - \partial_2 B_{0,1} = 0 $, and $\partial_t B_0 = - \nabla_x \times E_0$. Together with \eqref{GaussE0}, \eqref{Dirt0}, we use integration by parts to get
\[
\begin{split}
& \int_\Omega (-\partial_t E_{0,3} - 4\pi J_{0,3} + \partial_1 B_{0,2} - \partial_2 B_{0,1} ) \partial_t \psi(0,x) dx
\\ & + \int_\Omega ( -4\pi \partial_3 \rho_0 \psi(0,x) + E_{0,3} \Delta_x \psi(0,x) - \partial_t B_{0,1} \partial_2 \psi(0,x) + \partial_t B_{0,2} \partial_1 \psi(0,x) ) dx
\\ & = 0.
\end{split}
\]
This concludes \eqref{Ampere3}.
Next, we prove $ \partial_t B_1 = - (\nabla_x \times E)_1$. From Lemma \ref{wavesol2}, it suffices to prove that for any $\psi(t,x) \in C_c^\infty( [0,T) \times \bar \Omega ) $ with $ \partial_{x_3} \psi |_{\partial \Omega } = 0 $, we have
\begin{equation} \label{Faraday1}
\int_0^T \int_\Omega (\partial_t B_1 + (\nabla_x \times E)_1 ) (\partial_t^2 - \Delta_x ) \psi dx dt = 0.
\end{equation}
Using \eqref{weakEass} and integration by parts, we compute
\[
\begin{split}
& \int_0^T \int_\Omega (\partial_t B_1 + (\nabla_x \times E)_1 ) (\partial_t^2 - \Delta_x ) \psi dx dt
\\ = & - \int_0^T \int_\Omega B_1 ( \partial_t^2 - \Delta_x ) (\partial_t \psi )dx dt - \int_\Omega B_{0,1} (\partial_t^2 - \Delta_x) \psi(0,x) dx
\\ & - \int_0^T \int_\Omega E_3 ( \partial_t^2 - \Delta_x ) (\partial_2 \psi ) dx dt + \int_0^T \int_\Omega E_2 ( \partial_t^2 - \Delta_x ) (\partial_3 \psi ) dx dt
\\ = & \int_{\Omega} ( B_{0,1} \partial_t^2 \psi(0,x) - \partial_t B_{0,1} \partial_t \psi(0,x) ) dx + \int_0^T \int_{\partial \Omega } 4 \pi J_2 \partial_t \psi dx_\parallel dt - \int_0^T \int_\Omega 4 \pi (\nabla_x \times J )_1 \partial_t \psi dx dt
\\ & - \int_\Omega B_{0,1} (\partial_t^2 - \Delta_x) \psi(0,x) dx
\\ & + \int_\Omega (E_{0,3} \partial_t \partial_2 \psi(0,x) - \partial_t E_{0,3} \partial_2 \psi(0,x) ) dx + \int_0^T \int_{\partial \Omega } 4 \pi \rho \partial_2 \psi dx_\parallel dt + \int_0^T \int_\Omega 4 \pi ( \partial_3 \rho + \partial_t J_3 ) \partial_2 \psi dx dt
\\ & + \int_\Omega (-E_{0,2} \partial_t \partial_3 \psi(0,x) + \partial_t E_{0,2} \partial_3 \psi (0,x) ) dx - \int_0^T \int_\Omega 4 \pi ( \partial_2 \rho + \partial_t J_2 ) \partial_3 \psi dx dt
\\ = & - \int_\Omega \partial_t B_{0,1} \partial_t \psi (0,x) dx + \int_0^T \int_\Omega 4\pi (J_3 \partial_2 \partial_t \psi - J_2 \partial_3 \partial_t \psi ) dx dt + \int_\Omega B_{0,1} \Delta_x \psi(0,x) dx + \int_{ \partial \Omega } 4\pi J_{0,2} \psi (0,x_\parallel ) dx_\parallel
\\ & + \int_\Omega (- \partial_2 E_{0,3} + \partial_3 E_{0,2} ) \partial_t \psi(0,x) dx + \int_\Omega (- \partial_t E_{0,3 } \partial_2 \psi(0,x) + \partial_t E_{0,2 } \partial_3 \psi (0,x) ) dx
\\ & + \int_{\Omega} 4\pi ( - J_{0,3} \partial_2 \psi(0,x) + J_{0,2} \partial_3 \psi(0,x) ) dx + \int_0^T \int_\Omega 4\pi (- J_3 \partial_t \partial_2 \psi + J_2 \partial_t \partial_3 \psi ) dx dt
\\ = & \int_\Omega (-\partial_t B_{0,1} - \partial_2 E_{0,3} + \partial_3 E_{0,2} ) \partial_t \psi(0,x) dx
\\ & + \int_\Omega (- \partial_t E_{0,3} \partial_2 \psi(0,x) + \partial_t E_{0,2} \partial_3 \psi(0,x) + B_{0,1} \Delta_x \psi(0,x) - 4\pi J_{0,3} \partial_2 \psi(0,x) + 4\pi J_{0,2} \partial_3 \psi(0,x) ) dx.
\end{split}
\]
From \eqref{weakEass}, we have $ -\partial_t B_{0,1} - \partial_2 E_{0,3} + \partial_3 E_{0,2} = 0$, and $\partial_t E_0= \nabla_x \times B_0 - 4\pi J_0$. Together with \eqref{GaussE0}, we use integration by parts to get
\[
\begin{split}
& \int_\Omega (-\partial_t B_{0,1} - \partial_2 E_{0,3} + \partial_3 E_{0,2} ) \partial_t \psi(0,x) dx
\\ & + \int_\Omega (- \partial_t E_{0,3} \partial_2 \psi(0,x) + \partial_t E_{0,2} \partial_3 \psi(0,x) + B_{0,1} \Delta_x \psi(0,x) - 4\pi J_{0,3} \partial_2 \psi(0,x) + 4\pi J_{0,2} \partial_3 \psi(0,x) ) dx
\\ &= 0.
\end{split}
\]
This proves \eqref{Faraday1}. Using the same argument we also prove $ \partial_t B_2 = - (\nabla_x \times E)_2$.
Next, we prove $ \partial_t B_3 = - (\nabla_x \times E)_3$. From Lemma \ref{wavesolD}, it suffices to prove that for any $\phi \in C_c^\infty( [0,T) \times \bar \Omega ) $ with $ \phi |_{\partial \Omega } = 0 $,
\begin{equation} \label{Faraday3}
\int_0^T \int_\Omega (\partial_t B_3 +(\nabla_x \times E)_3 ) (\partial_t^2 - \Delta_x) \phi dx dt = 0.
\end{equation}
Using \eqref{weakEass} and integration by parts, we compute
\[
\begin{split}
& \int_0^T \int_\Omega (\partial_t B_3 +(\nabla_x \times E)_3 ) (\partial_t^2 - \Delta_x) \phi dx dt
\\ = & - \int_0^T \int_\Omega B_3 ( \partial_t^2 - \Delta_x ) (\partial_t \phi )dx dt - \int_\Omega B_{0,3} (\partial_t^2 - \Delta_x) \phi(0,x) dx
\\ & - \int_0^T \int_\Omega E_2 ( \partial_t^2 - \Delta_x ) (\partial_1 \phi ) dx dt + \int_0^T \int_\Omega E_1 ( \partial_t^2 - \Delta_x ) (\partial_2 \phi ) dx dt
\\ = & \int_{\Omega} ( B_{0,3} \partial_t^2 \phi(0,x) - \partial_t B_{0,3} \partial_t \phi(0,x) ) dx - \int_0^T \int_\Omega 4 \pi (\nabla_x \times J )_3 \partial_t \phi dx dt - \int_\Omega B_{0,3} (\partial_t^2 - \Delta_x) \phi(0,x) dx
\\ & + \int_\Omega (E_{0,2} \partial_t \partial_1 \phi(0,x) - \partial_t E_{0,2} \partial_1 \phi(0,x) ) dx + \int_0^T \int_\Omega 4 \pi ( \partial_2 \rho + \partial_t J_2 ) \partial_1 \phi dx dt
\\ & + \int_\Omega (-E_{0,1} \partial_t \partial_2 \phi(0,x) + \partial_t E_{0,1} \partial_2 \phi (0,x) ) dx - \int_0^T \int_\Omega 4 \pi ( \partial_1 \rho + \partial_t J_1 ) \partial_2 \phi dx dt
\\ = & - \int_\Omega \partial_t B_{0,3} \partial_t \phi (0,x) dx + \int_0^T \int_\Omega 4\pi (J_2 \partial_1 \partial_t \phi - J_1 \partial_2 \partial_t \phi ) dx dt + \int_\Omega B_{0,3} \Delta_x \phi(0,x) dx
\\ & + \int_\Omega (- \partial_1 E_{0,2} + \partial_2 E_{0,1} ) \partial_t \phi(0,x) dx + \int_\Omega (- \partial_t E_{0,2 } \partial_1 \phi(0,x) + \partial_t E_{0,1} \partial_2 \phi(0,x) ) dx
\\ & + \int_0^T \int_\Omega 4 \pi (- J_2 \partial_t \partial_1 \phi + J_1 \partial_t \partial_2 \phi ) dx dt + \int_\Omega 4\pi (- J_{0,2} \partial_1 \phi(0,x) + J_{0,1} \partial_2 \phi(0,x) ) dx
\\ = & \int_\Omega ( - \partial_t B_{0,3} - \partial_1 E_{0,2} + \partial_2 E_{0,1} ) \partial_t \phi(0,x) dx
\\ & + \int_\Omega ( B_{0,3} \Delta_x \phi(0,x) - \partial_t E_{0,2} \partial_1 \phi(0,x) + \partial_t E_{0,1} \partial_2 \phi(0,x) - 4\pi J_{0,2} \partial_1\phi(0,x) + 4\pi J_{0,1} \partial_2 \phi(0,x) )dx
\end{split}
\]
From \eqref{weakEass}, we have $ -\partial_t B_{0,3} - \partial_1 E_{0,2} + \partial_2 E_{0,1} = 0$, and $\partial_t E_0= \nabla_x \times B_0 - 4\pi J_0$. Together with \eqref{GaussE0}, we use integration by parts to get
\[
\begin{split}
& \int_\Omega ( - \partial_t B_{0,3} - \partial_1 E_{0,2} + \partial_2 E_{0,1} ) \partial_t \phi(0,x) dx
\\ & + \int_\Omega ( B_{0,3} \Delta_x \phi(0,x) - \partial_t E_{0,2} \partial_1 \phi(0,x) + \partial_t E_{0,1} \partial_2 \phi(0,x) - 4\pi J_{0,2} \partial_1\phi(0,x) + 4\pi J_{0,1} \partial_2 \phi(0,x) )dx
\\ & = 0.
\end{split}
\]
This proves \eqref{Faraday3}.
Lastly, we prove $\nabla_x \cdot B = 0$. From Lemma \ref{wavesol2}, it suffices to prove that for any $\psi(t,x) \in C_c^\infty( [0,T) \times \bar \Omega ) $ with $ \partial_{x_3} \psi |_{\partial \Omega } = 0 $, we have
\begin{equation} \label{GaussM1}
\int_0^T \int_\Omega (\nabla_x \cdot B ) (\partial_t^2 - \Delta_x ) \psi dx dt = 0.
\end{equation}
Using \eqref{weakEass}, integration by parts, and \eqref{Dirt0}, we compute
\[
\begin{split}
& \int_0^T \int_\Omega (\nabla_x \cdot B ) (\partial_t^2 - \Delta_x ) \psi dx dt
\\ = & - \int_0^T \int_\Omega \sum_{i=1}^3 B_i (\partial_t^2 - \Delta_x ) \partial_{x_i} \psi dx dt
\\ = & \int_\Omega \sum_{i=1}^3 ( B_{0,i} \partial_{x_i} \partial_t \psi(0,x) -\partial_t B_{0,i} \partial_{x_i} \psi(0,x) ) dx - \int_0^T \int_\Omega \sum_{i=1}^3 4 \pi (\nabla_x \times J)_i \partial_{x_i} \psi dx dt
\\ & + \int_0^T \int_{\partial \Omega } 4\pi J_2 \partial_{x_1} \psi dx_\parallel dt + \int_0^T \int_{\partial \Omega } - 4\pi J_1 \partial_{x_2} \psi dx_\parallel dt
\\ = & \int_\Omega (- \nabla_x \cdot B_0 ) \partial_t \psi(0,x) dx - \int_\Omega \left( \sum_{i=1}^3 \partial_t B_{0,i} \partial_{x_i} \psi(0,x) \right) dx.
\end{split}
\]
From \eqref{GaussE0} we have $\nabla_x \cdot B_0 = 0$. And from \eqref{weakEass}, $\partial_t B_0 = - \nabla_x \times E_0$, so from integration by parts and \eqref{Dirt0}, we have
\[
\int_\Omega \left( \sum_{i=1}^3 \partial_t B_{0,i} \partial_{x_i} \psi(0,x) \right) dx = 0.
\]
This proves \eqref{GaussM1}.\end{proof}
Now we are ready to prove Theorem \ref{EBunique}.
\begin{proof}[\textbf{Proof of Theorem \ref{EBunique}}]
The proof is a direct consequence of previous lemmas. Let $(E,B) \in W^{1,\infty}((0,T) \times \Omega )$, and $(\tilde E, \tilde B) \in W^{1,\infty}((0,T) \times \Omega )$ be two solutions of \eqref{Maxwell1}-\eqref{percond}.
We consider $E_1- \tilde E_1$, $E_2 - \tilde E_2$, and $B_3 - \tilde B_3$. From Lemma \ref{Maxtowave}, both $E_i$ and $\tilde E_i$ satisfy \eqref{E12sol} for $i=1,2$, and both $B_3$ and $\tilde B_3$ satisfy \eqref{B3sol}. Therefore, from Lemma \ref{wavesolD}, we have
\begin{equation}
E_1 = \tilde E_1, \ E_2 = \tilde E_2, \ B_3 = \tilde B_3.
\end{equation}
And for $E_3 - \tilde E_3$, $B_1 - \tilde B_1$, and $B_2 - \tilde B_2$. From Lemma \ref{Maxtowave}, we have both $E_3$ and $\tilde E_3$ satisfy \eqref{E3sol}, and both $B_i$ and $\tilde B_i$ satisfy \eqref{B12sol} for $i = 1,2$. Therefore, from Lemma \ref{wavesol}, we deduce that
\begin{equation}
E_3 = \tilde E_3, \ B_1 = \tilde B_1, \ B_2 = \tilde B_2.
\end{equation}
Thus, we get $E = \tilde E$, $B = \tilde B$, and this concludes the uniqueness of the solution. Now this solution should solve the Maxwell equation by Lemma \ref{wavetoMax}. \end{proof}
\section{Glassey-Strauss Representation of $E$ and $B$}
In this section, we give a representation of the field $E$ and $B$ by solving the wave equations \eqref{wave_eq_E}, \eqref{wave_eq_B}, under the boundary condition \eqref{E12B3bc} and \eqref{E3B1B2bc}.
We first consider the electronic field $E$. The tangential component $E_\parallel = (E_1, E_2)$ satisfies \begin{equation} \label{waveEparallel} \begin{split} \partial_t^2 E_\parallel - \Delta_x E_\parallel = G_\parallel:= -4 \pi \nabla_\parallel \rho - 4\pi \partial_t J_\parallel , \\
E_\parallel |_{t=0}= E_{0\parallel}, \ \partial_t E_\parallel |_{t=0} = \partial_t E_{0\parallel} , \end{split} \end{equation} and \begin{equation} \label{Dirichlet} E_\parallel = 0 \ \ \text{ on } \ \partial \Omega. \end{equation}
Define \begin{equation} \label{def:bar} \begin{split} \bar{x}= (x_\parallel, -x_3) \ \ \ \text{for} \ \ x= (x_\parallel, x_3)= (x_1, x_2, x_3). \end{split} \end{equation} To solve the Dirichlet boundary condition, we employ the odd extension of the data: for $i=1,2$, and $x \in \mathbb{R}^3$, \begin{equation} \begin{split}\label{odd_ext} G_i(t,x_\parallel, x_3) = & \mathbf 1_{x_3 > 0 } G_i(t,x ) - \mathbf 1_{x_3 < 0 } G_i(t,\bar x), \\ E_{0i} ( x_\parallel, x_3) = & \mathbf 1_{x_3 > 0 } E_{0i} ( x) - \mathbf 1_{x_3 < 0 } E_{0i}( \bar x), \\ \partial_t E_{0i} ( x_\parallel, x_3) = & \mathbf 1_{x_3 > 0 } \partial_t E_{0i} ( x) - \mathbf 1_{x_3 < 0 } \partial_t E_{0i}( \bar x). \end{split} \end{equation} Then the weak solution of $E_\parallel(t,x)$ to \eqref{waveEparallel} with data \eqref{odd_ext} in the whole space $\mathbb{R}^3$ takes a form of, for $i=1,2,$ \begin{align}
&E_i(t,x ) = \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 > 0 \} } \left( t \partial_t E_{0i}( y ) + E_{0i}(y ) + \nabla E_{0i} (y ) \cdot (y-x) \right) dS_y \notag \\ & \ \ \ \ \ + \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 < 0 \} } \big( - t \partial_t E_{0i}( \bar y ) - E_{0i} ( \bar y)
- \nabla E_{0i} (\bar y) \cdot (\bar y - \bar x) \big)
dS_y \notag
\\ & \ \ \ \ \ + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 > 0 \} }\frac{ G_i( t - |y-x|, y ) }{|y-x| } dy \label{Eexpanmajor1}\\
&\ \ \ \ \ + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 < 0 \} }\frac{ - G_i( t - |y-x|, \bar y ) }{|y-x| } dy, \label{Eexpanmajor2} \end{align}
where $B(x,t)= \{ y \in \mathbb{R}^3: |x-y| <t\}$ and $\partial B(x,t)= \{ y \in \mathbb{R}^3: |x-y| =t\}$. The above form is then a (weak) solution of \eqref{waveEparallel} and \eqref{Dirichlet}. Next, we express \eqref{Eexpanmajor1} and \eqref{Eexpanmajor2} using the Glassey-Strauss representation \cite{GS} in $\Omega$. Define
\begin{equation} \label{pxptST} \begin{split}
\partial_t = \frac{S- \hat{v} \cdot T}{1+ \hat v \cdot \o}, \ \ \partial_i = \frac{\o_i S}{ 1+ \hat v \cdot \o} + \left( \delta_{ij} - \frac{\o_i \hat{v}_j}{1+ \hat v \cdot \o}\right) T_j, \end{split}\end{equation}
while, for $\o= \o(x,y) = \frac{y-x}{|y-x|}$,
\begin{align} T_i &:= \partial_i - \o_i \partial t, \label{def:T}\\ S &:= \partial_t + \hat v \cdot \nabla_x. \label{def:S} \end{align} Note that \begin{equation} \label{T=y}
T_j f (t- |y-x| , y, v ) = \partial_{y_j } [ f(t- |y-x|, y, v ) ], \end{equation} and the Vlasov equation \eqref{VMfrakF1} implies that
\begin{equation} \label{S=Lf_v}
Sf= - \nabla_v \cdot [ (E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}})- g \mathbf e_3)f].
\end{equation}
From \eqref{rhoJ1} and \eqref{pxptST}, \begin{align}
&\eqref{Eexpanmajor1}=- \int_{B(x;t) \cap \{y_3 > 0\}} \frac{( \partial_i \mathbf{\rho} +\partial_t {J}_i )(t-|y-x|,y )}{|y-x|} \mathrm{d} y
\notag\\ &\ \ =- \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} (\partial_i f + \hat{v}_i \partial_t f) (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}\notag
\\ &\ \ =- \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \frac{\o_i + \hat{v}_i}{1+ \hat{v} \cdot \o}(Sf) (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} \notag
\\ & \ \ \ \ \ - \int_{ \substack{ B(x;t) \\ \cap \{y_3 > 0\} }
} \int_{\mathbb{R}^3} \left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right) T_j f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}. \notag
\end{align} Here, we followed the Einstein convention (when an index variable appears twice, it implies summation of that term over all the values of the index) and will do throughout this section.
Then replace $T_j f$ with \eqref{T=y} and apply the integration by parts to get the last term equals \begin{equation} \label{upperT1} \begin{split}
&- \int_{ \partial B(x;t) \cap \{y_3>0 \}} \o_j \left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right) f(0, y,v) \mathrm{d} v \frac{\mathrm{d} S_y}{|y-x|}
\\ &+ \int_{B(x;t) \cap \{y_3= 0\}} \int_{\mathbb{R}^3} \left( \delta_{i3} - \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+\hat{v} \cdot \o} \right) f (t-|y-x|,y_\parallel, 0 ,v) \mathrm{d} v \frac{ \mathrm{d} y_\parallel}{|y-x|} \\
&+ \int_{\substack{ B(x;t) \\ \cap \{y_3 > 0\} }} \int_{\mathbb{R}^3}
\frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{ (1+ \hat v \cdot \o )^2}
f (t-|y-x|,y ,v) \mathrm{d} v \frac{\mathrm{d} y}{|y-x|^2}. \end{split} \end{equation} where we have used that, from \cite{GS,Glassey}, \begin{equation} \notag
\frac{\partial}{\partial y_j}\left[\frac{1}{|y-x|}\left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right]= \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{|y-x|^2 (1+ \hat v \cdot \o )^2}. \end{equation}
In order to express \eqref{Eexpanmajor2} in the lower half space we modify the idea of Glassey-Strauss slightly. Define
\begin{equation} \label{def:o-}
\bar \o = \begin{bmatrix} \o_1 & \o_2 & - \o_3 \end{bmatrix} ^T.
\end{equation} We use the same $S$ of \eqref{def:S} but \begin{equation} \label{def:T-} \begin{split}
\bar{T}_3 f &= -\partial_{y_3} [f(t-|y-x|, y_\parallel, - y_3, v)] = \partial_{y_3} f - \bar{\omega}_3 \partial_t f ,
\\ \bar T _i f &= \partial_{y_i} [f(t-|y-x|, y_\parallel, - y_3, v)] = \partial_{y_i} f - \bar{\omega}_i \partial_t f \, \text{ for } \ i=1,2. \end{split} \end{equation} Then we get \begin{align} \partial_t &= \frac{S- \hat{v} \cdot \bar{T} }{1+ \hat{v} \cdot \bar{\omega} }, \label{ptST-}\\
\partial_{y_i} &= \bar{T}_i + \bar{\o}_i \frac{S- \hat{v} \cdot \bar{T} }{1+ \hat{v} \cdot \bar{\omega} } = \frac{ \bar{\o}_i S }{1+ \hat{v} \cdot \bar{\omega} } + \bar{T}_i - \bar{\o}_i \frac{ \hat{v} \cdot \bar{T} }{1+ \hat{v} \cdot \bar\omega }.
\label{pxST-} \end{align}
Therefore, we derive \hide\begin{equation} \begin{split}
\partial_i + \hat{v}_i \partial_t =& \frac{\o_i + \hat{v}_i }{1+ \hat{v}_\parallel \cdot \omega_\parallel - \hat{v}_3 \o_3}S
+ T^-_i - (\o_i + \hat{v}_i )\frac{ \hat{v}_\parallel \cdot T^-_\parallel - \hat{v}_3 T_3^-}{1+ \hat{v}_\parallel \cdot \omega_\parallel - \hat{v}_3 \o_3},
\\
\partial_3 + \hat{v}_3 \partial_t = & \frac{-\o_3 + \hat{v}_3 }{1+ \hat{v}_\parallel \cdot \omega_\parallel - \hat{v}_3 \o_3}S - T^-_3 - (-\o_3 + \hat{v}_3 )\frac{ \hat{v}_\parallel \cdot T^-_\parallel - \hat{v}_3 T_3^-}{1+ \hat{v}_\parallel \cdot \omega_\parallel - \hat{v}_3 \o_3}. \end{split} \end{equation} Or \unhide \begin{equation} \label{ST_lower} \begin{split}
\partial_i + \hat{v}_i \partial_t
= \frac{ \bar{\o}_i+ \hat{v}_i }{1+
\hat v \cdot \bar\o
}S
+ \left( \delta_{ij} - \frac{ \bar{\o}_i \hat{v}_j + \hat{v}_i \hat{v}_j }{1+ \hat v \cdot \bar\o }\right) \bar{T}_j . \end{split}\end{equation}
Now we consider \eqref{Eexpanmajor2}. From \eqref{ST_lower}, \begin{align}
&\eqref{Eexpanmajor2}= \int_{B(x;t) \cap \{y_3 <0\}} \int_{\mathbb{R}^3} (\partial_i f + \hat{v}_i \partial_t f) (t-|y-x|,\bar y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}\notag
\\& \ \ = \int_{B(x;t) \cap \{y_3 <0\}} \int_{\mathbb{R}^3} \frac{ \bar{\o}_i + \hat{v}_i }{1+ \hat v \cdot \bar{\o} }(Sf) (t-|y-x|, \bar y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} \notag
\\& \ \ + \int_{\substack{ B(x;t) \\ \cap \{y_3 < 0\}
} } \int_{\mathbb{R}^3} \Big( \delta_{ij} - \frac{ \bar{\o}_i \hat{v}_j+ \hat{v}_i \hat{v}_j}{1+ \hat v \cdot \bar{\o}} \Big) \bar{T}_j f (t-|y-x|, \bar y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}. \notag
\end{align}
Applying \eqref{def:T-} and the integration by parts, we derive that the last term equals \begin{equation} \label{GSlowerhalf2} \begin{split}
& \int_{ \partial B(x;t) \cap \{ y_3 < 0 \}} \int_{\mathbb{R}^3} \bar{\o}_j \Big( \delta_{ij} - \frac{ \bar{\o} _i \hat{v}_j + \hat{v}_i \hat{v}_j}{1+ \hat v \cdot \bar{\o} } \Big) f(0, \bar y ,v) \mathrm{d} v \frac{\mathrm{d} S_y}{|y-x|}
\\ &+ \int_{B(x;t) \cap \{y_3 =0\}} \int_{\mathbb{R}^3} \iota_3 \Big( \delta_{i3} - \frac{\bar{\o}_i \hat{v}_3 + \hat{v}_i \hat{v}_3}{1+ \hat v \cdot \bar{\o} } \Big) f(t-|y-x|,y_\parallel, 0 ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}\\
&- \int_{B(x;t) \cap \{y_3 <0\}} \int_{\mathbb{R}^3}
\frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \bar{\o}_i ) }{ (1+ \hat v \cdot \bar{\o} )^2}
f(t-|y-x|, \bar y ,v) \mathrm{d} v\frac{ \mathrm{d} y}{|y-x|^2}, \end{split} \end{equation} where we have utilized the notation \begin{equation} \label{iota} \iota_i = +1 \ \ \text{ for } \ i=1,2, \ \ \iota_3=-1, \end{equation} and the direct computation \begin{equation} \label{DyT} \iota_j \frac{\partial}{\partial y_j}\left[
\frac{1}{|y-x|}
\Big(
\delta_{ij} - \frac{ \iota_i \o_i \hat{v}_j + \hat{v}_i \hat{v}_j}{1+ \hat v \cdot \bar \o}
\Big)\right]
= \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \bar{\o}_i ) }{|y-x|^2 (1+ \hat v \cdot \bar{\o} )^2}.
\end{equation} \hide
Now we compute $\sum_j \iota_j \frac{\partial}{\partial y_j}\left[ \frac{1}{|y-x|} \Big( \delta_{ij} - \frac{ \iota_i \o_i \hat{v}_j + \hat{v}_i \hat{v}_j}{1+ \hat v \cdot \o^-} \Big)\right] $. Note that \[
\partial_{y_j} |y-x|= \frac{(y-x)_j}{|y-x|}, \ \ \
\partial_{y_j} \o_i = \frac{1}{|y-x|} \left(
\delta_{ij} - \frac{(y-x)_i (y-x)_j}{|y-x|^2} \right)
\] We have
where we have computed \begin{equation} \begin{split}
\sum_{j} \bigg\{& - \iota_j |y-x|^{-2} \o_j \Big(
\delta_{ij} - \frac{ \iota_i \o_i \hat{v}_j + \hat{v}_i \hat{v}_j}{1+ \hat v \cdot \o^-}
\Big)\\
&+ \iota_j |y-x|^{-1} (-1)^2 [1+ \hat v \cdot \o^- ]^{-2}
\left(\sum_k
\hat{v}_k \iota_k \partial_{y_j} \o_k
\right)
\left( \iota_i \o_i \hat{v}_j + \hat{v}_i \hat{v}_j \right)\\
&+ \iota_j |y-x|^{-1} (-1) [1+ \hat v \cdot \o^-]^{-1}
\iota_i \partial_{y_j} \o_i \hat{v}_j \bigg\}\\
\end{split} \end{equation} \begin{equation} \begin{split}
= \frac{1}{|y-x|^2 [1+ \hat v \cdot \o^-]^2}\sum_{j} \bigg\{&
-[1+ \hat v \cdot \o^-]^2 \delta_{ij} \iota_j \o_j
+ [1+ \hat v \cdot \o^-] \{ \iota_i \o_i \iota_j\hat{v}_j + \hat{v}_i \iota_j\hat{v}_j\} \o_j \\
&+ \{ \iota_i \o_i \iota_j\hat{v}_j + \hat{v}_i \iota_j \hat{v}_j\}
\sum_k \iota_k \hat{v}_k (\delta_{kj} - \o_k \o_j) \\
&- [1+ \hat v \cdot \o^-] \iota_j \hat{v}_j ( \iota_i\delta_{ij} - \iota_i \o_i \o_j )
\bigg\}\\ \end{split} \end{equation} Now the summation equals \begin{equation} \begin{split}
& (\hat v \cdot \o^- ) ^2\hat{v}_i - (\hat v \cdot \o^-) ^2\hat{v}_i + |\hat{v}|^2 \hat{v}_i\\ &- (\hat v \cdot \o^-)^2 \iota_i \o_i + (\hat v \cdot \o^-)^2 \iota_i \o_i + (\hat v \cdot \o^-) \hat{v}_i
+|\hat{v}|^2 \iota_i \o_i -(\hat v \cdot \o^-)^2 \iota_i \o_i - (\hat v \cdot \o^-) \hat{v}_i + (\hat v \cdot \o^-)^2 \iota_i \o_i \\ &- 2(\hat v \cdot \o^-) \iota_i \o_i +(\hat v \cdot \o^-) \iota_i \o_i - (\iota_i)^2 \hat{v}_i + (\hat v \cdot \o^-) \iota_i \o_i \\ &- \iota_i \o_i \\
=& |\hat{v}|^2 \hat{v}_i + |\hat{v}|^2 \iota_i \o_i- (\iota_i)^2 \hat{v}_i - \iota_i \o_i . \end{split} \end{equation}
\unhide
Next, we consider the normal components of the Electronic field $E_3$. From \eqref{wave_eq_E}, \eqref{initialC}, and \eqref{E3B1B2bc}, we have \begin{equation} \label{waveE3} \begin{split} \partial_t^2 E_3 - \Delta_x E_3 = G_3:= -4 \pi \partial_3 \rho - 4\pi \partial_t J_3 ,
\\ E_3 |_{t=0}= E_{03}, \ \partial_t E_3 |_{t=0} = \partial_t E_{03} , \end{split} \end{equation} and \begin{equation}
\partial_3 E_3 = 4 \pi \rho \ \ \text{ on } \ \ \partial \Omega. \label{Neumann} \end{equation} It is convenient to decompose the solution into two parts: one with the Neumann boundary condition of \eqref{waveE3} and the zero forcing term and initial data \begin{equation} \label{waveE3b} \begin{split} \partial_t^2 w - \Delta_x w = 0 \ \ &\text{ in } \ \ \Omega,
\\ w |_{t=0} = 0, \ \partial_t w |_{t=0} = 0 \ \ &\text{ in } \ \ \Omega, \\ \partial_3 w= 4 \pi \rho \ \ &\text{ on } \ \ \partial \Omega, \end{split}\end{equation} and the other part $\tilde E_3$ with the initial data of \eqref{waveE3} and the zero Neumann boundary condition. We achieve it by the even extension trick. Recall $\bar x$ in \eqref{def:bar}. For $x \in \mathbb{R}^3$, define \begin{equation} \label{wavetildeE32} \begin{split} G_3(t,x) = & \mathbf{1}_{x_3 > 0 } G_3(t,x ) + \mathbf{1}_{x_3 < 0 } G_3(t,\bar x ) , \\ E_{03} (x) = & \mathbf 1_{x_3 > 0 } E_{03 } (x ) + \mathbf 1_{x_3 < 0 } E_{03 } ( \bar x ), \\ \partial_t E_{03} (x) = & \mathbf 1_{x_3 > 0 } \partial_t E_{03} ( x ) + \mathbf 1_{x_3 < 0 } \partial_t E_{03}( \bar x ). \end{split} \end{equation} The weak solution $\tilde E_3$ to \eqref{waveE3} with the data \eqref{wavetildeE32} in the whole space $\mathbb{R}^3$ take a form of \begin{align} \tilde E_3(t,x ) &= \frac{1}{4 \pi t^2} \int_{ \substack{\partial B(x;t)\\ \cap \{ y_3 > 0 \}} } \left( t \partial_t E_{03}( y ) + E_{03}(y ) + \nabla E_{03} (y ) \cdot (y-x) \right) dS_y\notag \\ & + \frac{1}{4 \pi t^2} \int_{ \substack{\partial B(x;t) \\ \cap \{ y_3 < 0 \} }} \big( t \partial_t E_{03}( \bar y ) + E_{03} (\bar y)
+ \nabla E_{03} ( \bar y ) \cdot (\bar y - \bar x ) \big) dS_y\notag
\\ & + \frac{1}{4 \pi } \int_{ \substack{ B(x;t) \\ \cap \{ y_3 > 0 \}} }\frac{ G_3( t - |y-x|, y ) }{|y-x| } dy \label{tildeE3formula1}\\
& + \frac{1}{4 \pi } \int_{\substack{ B(x;t) \\ \cap \{ y_3 < 0 \} }}\frac{ G_3( t - |y-x|, \bar y ) }{|y-x| } dy. \label{tildeE3formula2} \end{align} Following the same argument to expand \eqref{Eexpanmajor1} and \eqref{Eexpanmajor2}, we derive that \begin{equation} \label{E3expression1} \begin{split} &\eqref{tildeE3formula1}\\
& = - \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \frac{\o_3 + \hat{v}_3}{1+ \hat{v} \cdot \o}(Sf) (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}
\\ & + \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3}\frac{ (|\hat{v}|^2-1 )(\hat{v}_3 + \o_3 ) }{ (1+ \hat v \cdot \o )^2} f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|^2}
\\ & - \int_{ \partial B(x;t) \cap \{ y_3>0 \}} \o_j \left( \delta_{3j} - \frac{(\o_3 + \hat{v}_3)\hat{v}_j}{1+\hat{v} \cdot \o} \right) f(0, y,v) \mathrm{d} v \frac{\mathrm{d} S_y}{|y-x|}
\\ & + \int_{B(x;t) \cap \{y_3= 0\}} \int_{\mathbb{R}^3} \left( 1 - \frac{(\o_3 + \hat{v}_3)\hat{v}_3}{1+\hat{v} \cdot \o} \right) f (t-|y-x|,y_\parallel, 0 ,v) \mathrm{d} v \frac{ \mathrm{d} y_\parallel}{|y-x|}, \end{split} \end{equation} \begin{equation} \begin{split}\label{E3expression2} & \eqref{tildeE3formula2}\\
&= - \int_{B(x;t) \cap \{y_3 <0\}} \int_{\mathbb{R}^3} \frac{ \bar{\o}_3 + \hat{v}_3 }{1+ \hat v \cdot \bar{\o} }(Sf) (t-|y-x|, \bar y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} \\ &+ \int_{B(x;t) \cap \{y_3 <0\}} \int_{\mathbb{R}^3}
\frac{ (|\hat{v}|^2-1 )(\hat{v}_3 + \bar{\o}_3 ) }{ (1+ \hat v \cdot \bar{\o} )^2}
f(t-|y-x|,\bar y ,v) \mathrm{d} v\frac{ \mathrm{d} y}{|y-x|^2} \\
&- \int_{\partial B(x;t) \cap \{ y_3 < 0 \}} \int_{\mathbb{R}^3} \bar{\o}_j \Big( \delta_{3j} - \frac{ \bar{\o} _3 \hat{v}_j + \hat{v}_3 \hat{v}_j}{1+ \hat v \cdot \bar{\o} } \Big) f(0,\bar y ,v) \mathrm{d} v \frac{\mathrm{d} S_y}{|y-x|}
\\ &+ \int_{B(x;t) \cap \{y_3 =0\}} \int_{\mathbb{R}^3} \Big( 1- \frac{ (\bar{\o}_3 + \hat{v}_3 ) \hat{v}_3}{1+ \hat v \cdot \bar{\o} } \Big) f(t-|y-x|,y_\parallel, 0 ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}. \end{split} \end{equation} Note that the weak derivative $\partial_3$ to the form of $\tilde{E}_3$ solves the linear wave equation \eqref{waveE3} with oddly extended forcing term and the initial data in the sense of distributions. Thus it satisfies \begin{equation} \label{p3tildeE3=0} \partial_3 \tilde E_3=0 \ \ \text{ on } \ \ \partial \Omega. \end{equation}
Now we consider \eqref{waveE3b}. We assume $\rho(t,x)$ for all $t\leq 0$, which implies $w(t,x)=0$ for all $t\leq 0$. Define the Laplace transformation: \begin{equation} W(p, x) = \int^\infty_{- \infty} e^{-p t} w (t,x ) dt, \ \ \ \ R(p, x) = \int^\infty_{- \infty} e^{-p t} \rho (t,x ) dt .\label{LaplaceT} \end{equation} Then $W$ solves the Helmholtz equation with the same Neumann boundary condition: \begin{equation} \label{Helmholtz} \begin{split} p^2 W - \Delta_x W = 0 & \ \ \text{ in } \ \Omega, \\ \partial_n W = 4 \pi R & \ \ \text{ on } \ \partial \Omega. \end{split}
\end{equation} The solution for $( p^2 - \Delta _x ) \Phi(x) = \delta(x) $ in $\mathbb R^3$ is known as $ \frac{1}{4\pi } \frac{e^{\pm p |x| }}{ |x| }$. We choose \begin{equation} \label{fundaHelmPhi}
\Phi(x) = \frac{1}{4\pi } \frac{e^{- p |x| }}{ |x| }. \end{equation} We have the following identities: \begin{lemma} \label{Helmu}{\it Suppose $u \in C^2(\bar \Omega )$ is an arbitrary function. For a fixed $x \in \Omega$ and $\Phi$ in \eqref{fundaHelmPhi}, we have \begin{equation} \label{uPhiintrep} \begin{split} u(x) &= \int_\Omega \Phi(y-x) ( p^2- \Delta_x ) u (y) dy \\ &+ \int_{\partial \Omega } \left[ \Phi(y-x) \partial_n u(y) - u(y) \partial_n \Phi(y-x) \right] dS_y.\end{split} \end{equation}} \end{lemma} \begin{proof}The proof is rather standard. Fix $x \in \Omega$. Let $0 < \e \ll 1 $, and $B(x,\e)$ be a ball centered at $x$ with radius $\e$ such that $B(x, \e ) \subset \Omega$. Let $V_\e = \Omega - B(x,\e)$. Then, by the integration by parts, \begin{equation} \notag \begin{split} & - \int_{V_\e} \Phi(y-x ) ( \Delta_y - p^2 ) u(y) dy \\
&= \int_{ \partial \Omega } u(y) \partial_n \Phi(y-x ) dS_y + \int_{\partial B(x,\e) } u(y) \partial_n \Phi(y-x ) dS_y \\ & \ \ - \int_{\partial \Omega} \Phi(y-x ) \partial_n u(y)dS_y - \int_{\partial B(x, \e) } \Phi(y-x ) \partial_n u(y)dS_y. \end{split} \end{equation} From \eqref{fundaHelmPhi}, $
\int_{\partial B(x, \e) } \Phi(y-x ) \partial_n u(y)dS_y \lesssim 4\pi \e^2 \frac{e^{|p| \e }}{4 \pi \e } \to 0,$ as $\e \to 0.$ And by direct computation, \begin{equation} \notag \begin{split}
&\int_{\partial B(x,\e) } u(y) \partial_n \Phi(y-x ) dS_y = \int_{\partial B(x,\e) } u(y) \frac{-(y-x)}{|y-x| } \cdot \nabla \Phi(y-x ) dS_y
\\ = & \frac{1}{4 \pi } \int_{\partial B(x,\e) } u(y) \frac{-(y-x)}{|y-x| } \cdot ( - i p |y-x | - 1 ) \frac{ e^{- i p |y-x | } (y-x) }{|y-x |^3 } dS_y
\\ = & \frac{1}{4 \pi } \int_{\partial B(x,\e) } \left( - (- p |y-x| - 1 ) \frac{ e^{- p|y-x | }}{|y-x |^2 } \right) u(y) dS_y
\\ = & \left( ( 1 - (- p) \e ) e^{- p \e} \right) \left( \frac{1}{4 \pi \e^2 } \int_{\partial B(x,\e) } u(y) dS_y \right)
\to u(x), \text{ as } \e \to 0. \end{split} \end{equation} Combining all together, and letting $\e \to 0$ we get \eqref{uPhiintrep}.\end{proof}
Next for $x \in \Omega$, let $\phi^x(y)$ be the function such that \begin{equation} \label{phixyforO} \begin{split} (\Delta_y - p^2 ) \phi_N^x(y) = 0 \ \ &\text{ in } \ \ \Omega, \\ \partial_n \phi_N^x(y) = \partial_n \Phi(y-x ) \ \ &\text{ on } \ \ \partial \Omega. \end{split} \end{equation} The integration by parts implies \begin{equation} \label{phixyintbyparts} \begin{split} 0 = & \int_{\Omega} ( \Delta_y - p^2 ) \phi_N^x(y) u(y) dy \\ = & \int_{\Omega } ( \Delta_y - p^2 )u(y) \phi_N^x(y) dy + \int_{\partial \Omega } [ \partial_n \phi_N^x(y) u(y) - \phi_N^x(y) \partial_n u(y) ] dS_y \\ = & \int_{\Omega } ( \Delta_y - p^2 )u(y) \phi_N^x(y) dy + \int_{\partial \Omega } [ \partial_n \Phi(y-x ) u(y) - \phi_N^x(y) \partial_n u(y) ] dS_y. \end{split} \end{equation} By adding \eqref{phixyintbyparts} to \eqref{uPhiintrep}, we derive that \begin{equation} \label{urepPhi2} \begin{split} u(x) = & - \int_{\Omega} \left( \Phi(y-x) -\phi_N^x(y) \right) ( \Delta_y - p^2 )u(y) dy\\ & + \int_{\partial \Omega } ( \Phi(y-x) -\phi_N^x(y) ) \partial_n u(y) dS_y. \end{split} \end{equation} For the half space $\Omega = \mathbb R_+^3$, we have, with $\bar x$ in \eqref{def:bar}, \begin{equation} \label{phiNxyformhalf} \phi_N^x(y) = - \Phi(y- \bar x ) . \end{equation} \hide Since $x,y \in \mathbb R_+^3$, so $y - \tilde x \neq 0$, and $(\Delta_y - p^2 ) ( - \Phi(y- \tilde x )) = 0$ for all $y \in \mathbb R_+^3$. Also, by direct computation \[
\partial_{y_3} \Phi(y- x ) = \frac{1}{4 \pi } \left( \frac{(\pm p |y-x| -1 ) e^{\pm p |y-x | } (y_3 - x_3 ) }{ |y-x |^3 } \right). \]
For $y \in \partial \Omega$, $y_3 = 0$ and $|y - x | = |y - \tilde x | $, thus \begin{equation} \begin{split}
\partial_{y_3} \phi_N^x(y) |_{y_3 = 0 } = - \partial_{y_3} \Phi(y- \tilde x ) |_{y_3 = 0 } = & - \frac{1}{4 \pi } \left( \frac{(\pm p |y- \tilde x| -1 ) e^{\pm p |y-\tilde x | } (y_3 + x_3 ) }{ |y-\tilde x |^3 } \right) |_{y_3 = 0 }
\\ = & - \frac{1}{4 \pi } \left( \frac{(\pm p |y- x| -1 ) e^{\pm p |y- x | } x_3 }{ |y- x |^3 } \right)
\\ = & \partial_{y_3} \Phi(y- x ) |_{y_3 = 0 }, \end{split} \end{equation} and we proved \eqref{phiNxyformhalf}. \unhide
Finally, we derive that, from \eqref{urepPhi2} and \eqref{phiNxyformhalf}: \begin{lemma} \label{Helmu2} {\it For $\Omega = \mathbb{R}^3_+$, and $\Phi$ in \eqref{fundaHelmPhi}, \begin{equation} \begin{split}\label{formula:u} u(x) =& - \int_{\Omega} \left( \Phi(y-x) + \Phi(y-\bar x) \right) ( \Delta_y - p^2 )u(y) dy\\ & + \int_{\partial \Omega } ( \Phi(y-x) + \Phi(y-\bar x) ) \partial_n u(y) dS_y. \end{split} \end{equation}} \end{lemma}
By applying \eqref{phiNxyformhalf} to \eqref{Helmholtz}, we derive that \begin{equation} \begin{split}\label{form_W} W(p,x) &= \int_{\partial \Omega } ( \Phi(y-x) + \Phi(y-\bar x) ) 4 \pi R (y) d S_y \\ &= 2 \int_{\mathbb R^2 }\frac{e^{-p ( (y_1 - x_1)^2 +(y_2 - x_2 )^2 + x_3^2 )^{1/2}}}{ (y_1 - x_1)^2 +((y_2 - x_2 )^2 + x_3^2 )^{1/2}} R(y_1, y_2 ) d y_1 dy_2 . \end{split} \end{equation} Using the inverse Laplace transform, we derive that
\begin{equation} \label{wexpression1} \begin{split} &w(t,x) = \frac{1}{2\pi } \int_{-\infty}^\infty e^{(p_1 + i p_2 )t} W( p_1 + i p_2 ,x ) dp_2\\
& \ \ = \frac{1}{ \pi } \int_{-\infty}^\infty d p_2 \ e^{(p_1 + i p_2 )t} \int_{\mathbb R^2 }d y_1 dy_2\frac{e^{ -( p_1 + i p_2) ( (y_1 - x_1)^2 +(y_2 - x_2 )^2 + x_3^2 )^{1/2}}}{ (y_1 - x_1)^2 +((y_2 - x_2 )^2 + x_3^2 )^{1/2}} \\ & \ \ \ \ \ \ \times \int_{-\infty}^{\infty} ds \ e^{-( p_1 + i p_2) s } (- \rho(s,y_1,y_2 )) . \end{split} \end{equation} Finally, we derive that, using the identity $\int_{-\infty}^\infty e^{i p_2 t } d p_2 = 2 \pi \delta(t) $, \begin{equation} \label{wexpression2} \begin{split}
&w(t,x) = \frac{- 1}{ \pi } \int_{\mathbb R^2 } \int_\mathbb{R} \int_\mathbb{R} \frac{e^{( p_1 + i p_2 )(t -s - \sqrt{| y_\parallel - x_\parallel |^2 + x_3^2 } ) }}{ \sqrt{| y_\parallel - x_\parallel |^2 + x_3^2 } } \rho( s, y_\parallel ) dp_2 ds dy_\parallel
\\ &= - 2 \int_{\mathbb R^2 } \int_\mathbb{R} \frac{e^{p_1 ( t-s - \sqrt{| y_\parallel - x_\parallel |^2 + x_3^2 } ) } \delta(t-s - \sqrt{| y_\parallel - x_\parallel |^2 + x_3^2 } ) }{ \sqrt{| y_\parallel - x_\parallel |^2 + x_3^2 } } \rho ( s, y_\parallel ) ds dy_\parallel \\ &= - 2 \int_{
\sqrt{| y_\parallel - x_\parallel |^2 + x_3^2 }<t
} \frac{\rho (t - \sqrt{| y_\parallel - x_\parallel |^2 + x_3^2 },y_\parallel ) }{ \sqrt{| y_\parallel - x_\parallel |^2 + x_3^2 }} dy_\parallel.
\end{split} \end{equation}
Collecting the terms, we conclude the following formula: \begin{proposition} \label{Eiform}
\begin{align}
\label{Eesttat0pos} &E_i(t,x) \\ & = \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 > 0 \} } \left( t \partial_t E_{0,i}( y_\parallel, y_3 ) + E_{0,i} (y_\parallel, y_3) + \nabla E_{0,i} (y_\parallel, y_3) \cdot (y-x) \right) dS_y \\ \label{Eesttat0neg} & + \frac{1}{ 4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 < 0 \} } \iota_i \big( - t \partial_t E_i(0, y_\parallel, - y_3 ) - E_i (0,y_\parallel, -y_3) \\ \notag & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad - \nabla_\parallel E_i (0,y_\parallel, - y_3) \cdot (y_\parallel-x_\parallel) + \partial_3 E_i (0,y_\parallel, - y_3) \cdot (y_3-x_3) \big) dS_y
\\ \label{Eestbulkpos} & + \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3}\frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{|y-x|^2 (1+ \hat v \cdot \o )^2} f(t-|y-x|,y ,v)\mathrm{d} v \mathrm{d} y
\\ \label{Eestbulkneg} &- \int_{B(x;t) \cap \{y_3 <0\}} \int_{\mathbb{R}^3} \iota_i \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \iota_i \o_i ) }{|y-x|^2 (1+ \hat v \cdot \o^-)^2}f(t-|y-x|,y_\parallel, -y_3 ,v) \mathrm{d} v \mathrm{d} y
\\ \label{EestSpos} &- \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \frac{\o_i + \hat{v}_i}{1+ \hat v \cdot \o }(Sf) (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}
\\ \label{EestSneg} &+\int_{B(x;t) \cap \{y_3 <0\}} \int_{\mathbb{R}^3} \iota_i \frac{ \iota_i \o_i + \hat{v}_i }{1+ \hat v \cdot \o^- }(Sf)(t-|y-x|,y_\parallel, -y_3 ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}
\\ \label{Eestbdrypos} & + \int_{B(x;t) \cap \{y_3= 0\}} \int_{\mathbb{R}^3} \left( \delta_{i 3 } - \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) f (t-|y-x|,y_\parallel, 0 ,v) \mathrm{d} v\frac{ \mathrm{d} y_\parallel}{|y-x|}
\\ \label{Eestbdryneg} & - \int_{B(x;t) \cap \{y_3 =0\}} \int_{\mathbb{R}^3} \iota_i \left( \delta_{i 3 } - \frac{ \iota_i \o_i \hat{v}_3 + \hat{v}_i \hat{v}_3}{1+ \hat v \cdot \o^-} \right) f(t-|y-x|,y_\parallel, 0 ,v) \mathrm{d} v \frac{ \mathrm{d} y_\parallel }{|y-x|}
\\ \label{Eestinitialpos} &- \int_{\partial B(x;t) \cap \{ y_3>0 \} } \int_{\mathbb R^3} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) f(0,y,v) \mathrm{d} v \frac{\mathrm{d} S_y}{|y-x|}
\\ \label{Eestinitialneg} & + \int_{\partial B(x;t) \cap \{ y_3<0 \}} \iota_i \int_{\mathbb{R}^3} \sum_j \iota_j \omega_j \Big( \delta_{ij} - \frac{ \iota_i \o_i \hat{v}_j + \hat{v}_i \hat{v}_j}{1+ \hat v \cdot \o^-} \Big) f(0,y_\parallel, -y_3 ,v) \mathrm{d} v \frac{\mathrm{d} S_y}{|y-x|}
\\ \label{Eest3bdrycontri} & - \delta_{i3} \int_{ B(x;t) \cap \{y_3 =0\}} \int_{\mathbb R^3 } \frac{ 2 f (t - |y-x |, y_\parallel, 0, v )}{|y-x | } dv d S_y . \end{align} \end{proposition}
Next, we solve for $B$. For $B_1, B_2$ we have, for $i =1,2,$ \begin{equation} \label{wavetildeB}
\begin{split} \partial_t^2 B_i - \Delta_x B_i = & 4 \pi ( \nabla_x \times J)_i:= H_i \ \ \text{ in } \ \ \Omega, \\ \partial_{x_3} B_1 = & 4 \pi J_2 , \ \partial_{x_3} B_2 = 4 \pi J_1 \ \ \text{ on } \ \ \partial \Omega, \\ B_i(0,x) = & B_{0 i}, \partial_t B_i(0,x) = \partial_t B_{0 i} \ \ \text{ in } \ \ \Omega. \end{split} \end{equation}
To solve \eqref{wavetildeB} we write $B_i = \tilde B_i + B_{bi} $ with $\tilde B_i$ satisfies the wave equation in $(0, \infty) \times \mathbb R^3 $ with even extension in $x_3$: \begin{equation} \label{wavetildeBi} \begin{split} \partial_t^2 \tilde B_i - \Delta_x \tilde B_i = & \mathbf 1_{x_3 > 0 } H_i(t,x) + \mathbf 1_{x_3 < 0 } H_i(t,\bar x), \\ \tilde B_i(0,x) = & \mathbf 1_{x_3 > 0 } B_{0i} (x) + \mathbf 1_{x_3 < 0 } B_{0i} (\bar x), \\ \partial_t \tilde B_i (0 , x) = & \mathbf 1_{x_3 > 0 } \partial_t B_{0i} (x) + \mathbf 1_{x_3 < 0 } \partial_t B_{0i}(\bar x). \end{split} \end{equation} And $B_{bi}$ satisfies \begin{equation} \label{Bibeq} \begin{split} \partial_t^2 B_{bi} - \Delta_x B_{bi} = 0 & \text{ in } \Omega, \\ B_{bi} (0,x) = 0, \partial_t B_{bi} = 0 & \text{ in } \Omega, \\ \partial_{x_3} B_{b1} = 4 \pi J_2, \ \partial_{x_3} B_{b2} = - 4 \pi J_1 & \text{ on } \Omega. \end{split} \end{equation} Then from \eqref{wavetildeBi}, \[ \label{tildeE3formula} \begin{split} & \tilde B_i (t,x ) \\ = & \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 > 0 \} } \left( t \partial_t B_{0i} (y ) + B_{0i}(y) + \nabla B_{0i} (y) \cdot (y-x) \right) dS_y \\ \end{split} \] \[ \begin{split}
& + \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 < 0 \} } \big( t \partial_t B_{0i}(\bar y ) + B_{0i}(\bar y) + \nabla B_{0i} (\bar y) \cdot (\bar y - \bar x) \big) dS_y
\\ & + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 > 0 \} }\frac{ H_i ( t - |y-x|, y ) }{|y-x| } dy + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 < 0 \} }\frac{ H_i( t - |y-x|, \bar y ) }{|y-x| } dy. \end{split} \] Applying \eqref{wexpression2} to \eqref{Bibeq}, \begin{equation} \begin{split}
B_{bi}(t,x) = (-1)^i 2 \int_{ B(x;t) \cap \{y_3 =0\}} \int_{\mathbb R^3 } \frac{ \hat v_{\underline{i} } f (t - |y-x |, y_\parallel, 0, v )}{|y-x | } dv d S_y, \end{split} \end{equation} where we have used the notation \begin{equation} \label{def_under_i}
\underline i = \begin{cases} 2, \text{ if } i=1, \\ 1, \text{ if } i=2. \end{cases}
\end{equation} Thus, \begin{equation} \label{Bi12rep} \begin{split} B_i(t,x) = & \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 > 0 \} } \left( t \partial_t B_{0i} ( y ) + B_{0i}(y) + \nabla B_{0i} (y) \cdot (y-x) \right) dS_y \\ & + \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 < 0 \} } \big( t \partial_t B_{0i}(\bar y ) + B_{0i}(\bar y) + \nabla B_{0i} (\bar y) \cdot (\bar y - \bar x) \big) dS_y
\\ & + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 > 0 \} }\frac{ H_i ( t - |y-x|, y ) }{|y-x| } dy + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 < 0 \} }\frac{ H_i( t - |y-x|, \bar y ) }{|y-x| } dy
\\ & + (-1)^i 2 \int_{ B(x;t) \cap \{y_3 =0\}} \int_{\mathbb R^3 } \frac{ \hat v_{\underline i } f (t - |y-x |, y_\parallel, 0, v )}{|y-x | } dv d S_y. \end{split} \end{equation}
On the other hand, $B_3(t,x)$ satisfies \[ \begin{split} \partial_t^2 B_3 - \Delta_x B_3 = & 4 \pi (\nabla_x \times J )_3 := H_3 \text{ in } \Omega, \\ B_3(0,x) = & B_{03}, \partial_t B_3(0,x) = \partial_t B_{03} \text{ in } \Omega, \\ B_3 = & 0 \text{ on } \partial \Omega. \end{split} \] Using the odd extension in $x_3$: \[ \begin{split} H_3(t,x) = & \mathbf 1_{x_3 > 0 } H_3(t,x) - \mathbf 1_{x_3 < 0 } H_3(t,\bar x), \\ B_{03} (x) = & \mathbf 1_{x_3 > 0 } B_{03} (x) - \mathbf 1_{x_3 < 0 } B_{03} (\bar x), \\ \partial_t B_{03} (0 , x) = & \mathbf 1_{x_3 > 0 } \partial_t B_{03} (x) - \mathbf 1_{x_3 < 0 } \partial_t B_{03}(\bar x), \end{split} \] we get the expression for $B_3$: \begin{equation} \label{B3rep} \begin{split} B_3(t,x) = & \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 > 0 \} } \left( t \partial_t B_{03} (y ) + B_{30}(y) + \nabla B_{03}(y) \cdot (y-x) \right) dS_y \\ & - \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 < 0 \} } \big( t \partial_t B_{03}( \bar y ) + B_{03} (\bar y) + \nabla B_{03} (\bar y) \cdot (\bar y - \bar x ) \big) dS_y
\\ & + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 > 0 \} }\frac{ H_3 ( t - |y-x|, y ) }{|y-x| } dy \\&- \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 < 0 \} }\frac{ H_3( t - |y-x|, \bar y ) }{|y-x| } dy. \end{split} \end{equation} Combining \eqref{Bi12rep} and \eqref{B3rep}, we get for $i=1,2,3$, \begin{align}
B_i(t,x) =
\notag & \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 > 0 \} } \left( t \partial_t B_{0i} ( y ) + B_{0i}(y) + \nabla B_{0i} (y) \cdot (y-x) \right) dS_y \\ \notag & + \frac{\iota_i}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 < 0 \} } \big( t \partial_t B_{0i}( \bar y ) + B_{0i}(\bar y) + \nabla B_{0i} (\bar y) \cdot (\bar y - \bar x ) \big) dS_y
\\ \label{Bexpanmajor1} & + \frac{1}{4 \pi } \int_{B(x;t) \cap \{ y_3 > 0 \} }\frac{ H_i ( t - |y-x|, y ) }{|y-x| } dy
\\ \label{Bexpanmajor2} & + \frac{\iota_i}{4 \pi } \int_{B(x;t) \cap \{ y_3 < 0 \} }\frac{ H_i( t - |y-x|, \bar y ) }{|y-x| } dy
\\ \notag & + (-1)^i 2( 1- \delta_{i3} ) \int_{ B(x;t) \cap \{y_3 =0\}} \int_{\mathbb R^3 } \frac{ \hat v_{\underline i } f (t - |y-x |, y_\parallel, 0, v )}{|y-x | } dv d S_y.
\end{align}
Using \eqref{pxptST}, we have \begin{align}
\notag \eqref{Bexpanmajor1} = & \int_{B(x;t) \cap \{ y_3 > 0 \} } \int_{\mathbb R^3} \frac{( \nabla_x f \times \hat v )_i ( t - |y-x|, y, v ) }{|y-x| } dy \\ \notag
= & \int_{B(x;t) \cap \{ y_3 > 0 \} } \int_{\mathbb R^3} \frac{ (\o \times \hat v)_i }{ 1+ \hat{v} \cdot \o} S f ( t - |y-x|, y ,v ) dv \frac{ dy} {|y-x|}
\\ \label{BupperT} & + \int_{B(x;t) \cap \{ y_3 > 0 \} } \int_{\mathbb R^3} \left( (T \times \hat v)_i - \frac{ (\o \times \hat v )_i \hat v \cdot T }{1+ \hat{v} \cdot \o}\right) f ( t - |y-x|, y, v ) dv \frac{ dy} {|y-x|}.
\end{align}
And for \eqref{BupperT}, we replace $T_j f$ with \eqref{T=y} and apply the integration by parts to get \eqref{BupperT} equals \begin{equation} \label{BupperT1} \begin{split}
& \int_{ \partial B(x; t) \cap \{ y_3 > 0 \} } \int_{\mathbb R^3} ( \o \times \hat v)_i \left( 1 - \frac{ \hat v \cdot \o }{1+ \hat{v} \cdot \o}\right) f ( 0, y, v ) dv \frac{ dS_y} {t}
\\ & + \int_{ B(x;t) \cap \{ y_3 = 0 \} } \int_{\mathbb R^3} \left( - (e_3 \times \hat v)_i + \frac{ ( \o \times \hat v )_i }{1+ \hat{v} \cdot \o} ( \hat v \cdot e_3 ) \right) f ( t- |y-x|, y_\parallel, 0, v ) dv \frac{ dy_\parallel} {|y-x|}
\\ & + \int_{B(x;t) \cap \{ y_3 > 0 \} } \int_{\mathbb R^3} \frac{ ( \o \times \hat v )_i \left( 1 - |\hat v |^2 \right) }{( 1 + \hat v \cdot \o )^2|y-x |^2} f ( t - |y-x|, y, v ) dv dy . \end{split} \end{equation} where we have used that, from \cite{GS,Glassey}, \begin{equation} \notag \begin{split}
& \partial_{y_j } \left( \frac{ ( \o \times \hat v ) \hat v_j }{ ( 1 + \hat v \cdot \o )|y-x | } \right)
\\ = & \frac{ ( \o \times \hat v ) \left( - ( \o \cdot \hat v )( 1 + \o \cdot \hat v ) - ( \o \cdot \hat v ) ( 1 + \o \cdot \hat v ) - |\hat v |^2 + ( \o \cdot \hat v )^2 \right) }{( 1 + \hat v \cdot \o )^2|y-x |^2}
\\ = & \frac{ ( \o \times \hat v ) \left( - 2 ( \o \cdot \hat v ) - |\hat v |^2 - ( \o \cdot \hat v )^2 \right) }{( 1 + \hat v \cdot \o )^2|y-x |^2},
\end{split} \end{equation} and \begin{equation} \notag \begin{split}
& - \nabla_{y } ( \frac{1}{ |y-x| } ) \times \hat v + \partial_{y_j } \left( \frac{ ( \o \times \hat v ) \hat v_j }{ ( 1 + \hat v \cdot \o )|y-x | } \right)
\\ & = \frac{ ( \o \times \hat v ) \left( ( 1 + \hat v \cdot \o )^2 - 2 ( \o \cdot \hat v ) - |\hat v |^2 - ( \o \cdot \hat v )^2 \right) }{( 1 + \hat v \cdot \o )^2|y-x |^2}
= \frac{ ( \o \times \hat v ) \left( 1 - |\hat v |^2 \right) }{( 1 + \hat v \cdot \o )^2|y-x |^2}. \end{split} \end{equation} Now we consider \eqref{Bexpanmajor2}. From \eqref{ST_lower}, \begin{align} \notag \eqref{Bexpanmajor2}
& = \iota_i \int_{B(x;t) \cap \{ y_3 < 0 \} } \int_{\mathbb R^3} \frac{( \nabla_x f \times \hat v )_i ( t - |y-x|, \bar y, v ) }{|y-x| } dy \\ \notag
= & \iota_i \int_{B(x;t) \cap \{ y_3 < 0 \} } \int_{\mathbb R^3} \frac{ (\bar \o \times \hat v )_i }{ 1+ \hat{v} \cdot \bar \o } S f ( t - |y-x|, \bar y ,v ) dv \frac{ dy} {|y-x|}
\\ \label{BlowerT} & + \iota_i \int_{B(x;t) \cap \{ y_3 < 0 \} } \int_{\mathbb R^3} \left( ( \bar T \times \hat v )_i - \frac{ ( \hat v \cdot \bar T ) ( \bar \o \times \hat v )_i }{1+ \hat{v} \cdot \bar \o}\right) f ( t - |y-x|, \bar y, v ) dv \frac{ dy} {|y-x|}. \end{align}
And for \eqref{BlowerT}, applying \eqref{def:T-} and the integration by parts, we derive that \eqref{BlowerT} equals \begin{equation} \label{HSlowerhalf2} \begin{split}
& \iota_i \int_{ \partial B(x; t) \cap \{ y_3 < 0 \} } \int_{\mathbb R^3} (\bar \o \times \hat v)_i \left( 1 - \frac{ \hat v \cdot \bar \o }{1+ \hat{v} \cdot \bar \o }\right) f ( 0, \bar y, v ) dv \frac{ dS_y} {t}
\\ + & \iota_i \int_{ B(x;t) \cap \{ y_3 = 0 \} } \int_{\mathbb R^3} \left( -(e_3 \times \hat v )_i + \frac{ ( \bar \o \times \hat v )_i }{1+ \hat{v} \cdot \bar \o} ( \hat v \cdot e_3 ) \right) \frac{ f ( t- |y-x|, y_\parallel, 0, v ) }{|y-x|} dv d y_\parallel
\\ + & \iota_i \int_{B(x;t) \cap \{ y_3 < 0 \} } \int_{\mathbb R^3} \left( \frac{ ( \bar \o \times \hat v )_i \left( 1 - |\hat v |^2 \right) }{( 1 + \hat v \cdot \bar \o )^2|y-x |^2} \right) f ( t - |y-x|, \bar y, v ) dv dy, \end{split} \end{equation} where we have used the direct computation \begin{equation} \notag \begin{split}
\iota_j \partial_{y_j } \left( \frac{ ( \bar \o \times \hat v ) \hat v_j }{ ( 1 + \hat v \cdot \bar \o )|y-x | } \right) = & \frac{ ( \bar \o \times \hat v ) \left( - ( \hat v \cdot \bar \o )( 1 + \hat v \cdot \bar \o ) - \hat v \cdot \bar \o - | \hat v |^2 \right) }{( 1 + \hat v \cdot \bar \o )^2|y-x |^2}
\\ = & \frac{ ( \bar \o \times \hat v ) \left( - 2 ( \hat v \cdot \bar \o ) - |\hat v |^2 - ( \hat v \cdot \bar \o )^2 \right) }{( 1 +\hat v \cdot \bar \o )^2|y-x |^2},
\end{split} \end{equation} and \begin{equation} \notag \begin{split}
&- \overline{ \nabla_y (|y-x|^{-1} ) } \times \hat v + \iota_j \partial_{y_j } \left( \frac{ ( \bar \o \times \hat v ) \hat v_j }{ ( 1 + \hat v \cdot \bar \o )|y-x | } \right)
\\ & = \frac{ ( \bar \o \times \hat v ) \left( ( 1 + \hat v \cdot \bar \o )^2 - 2 ( \hat v \cdot \bar \o ) - |\hat v |^2 - ( \hat v \cdot \bar \o )^2 \right) }{( 1 + \hat v \cdot \bar \o )^2|y-x |^2}
= \frac{ ( \bar \o \times \hat v ) \left( 1 - |\hat v |^2 \right) }{( 1 + \hat v \cdot \bar \o )^2|y-x |^2}. \end{split} \end{equation}
Collecting the terms, we conclude the following formula: \begin{proposition} \label{Biform}
\begin{align} \label{Besttat0pos}& B_i(t,x ) \\ = & \frac{1}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 > 0 \} } \left( t \partial_t B_{0,i}( y_\parallel, y_3 ) + B_{0,i} (y_\parallel, y_3) + \nabla B_{0,i} (y_\parallel, y_3) \cdot (y-x) \right) dS_y \\ \notag & + \frac{\iota_i}{4 \pi t^2} \int_{\partial B(x;t) \cap \{ y_3 < 0 \} } \big( t \partial_t B_{0,i}( y_\parallel, - y_3 ) + B_{0,i} (y_\parallel, -y_3) \\ \label{Besttat0neg} & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \nabla_\parallel B_{0,i} (0,y_\parallel, - y_3) \cdot (y_\parallel-x_\parallel) - \partial_3 B_{0,i} (0,y_\parallel, - y_3) \cdot (y_3-x_3) \big) dS_y
\\ \label{Bestbulkpos} & + \int_{B(x;t) \cap \{ y_3 > 0 \} } \int_{\mathbb R^3} \frac{ ( \o \times \hat v )_i \left( 1 - |\hat v |^2 \right) }{( 1 + \hat v \cdot \o )^2|y-x |^2} f ( t - |y-x|, y_\parallel, y_3, v ) dv dy
\\ \label{Bestbulkneg} & + \int_{B(x;t) \cap \{ y_3 < 0 \} } \int_{\mathbb R^3} \iota_i \frac{ ( \o^- \times \hat v )_i \left( 1 - |\hat v |^2 \right) }{( 1 + \hat v \cdot \o^- )^2|y-x |^2} f ( t - |y-x|, y_\parallel, - y_3, v ) dv dy
\\ \label{BestSpos} & + \int_{B(x;t) \cap \{ y_3 > 0 \} } \int_{\mathbb R^3} \frac{(\o \times \hat v)_i }{ 1+ \hat{v} \cdot \o} S f ( t - |y-x|, y_\parallel, y_3 ,v ) dv \frac{ dy} {|y-x|}
\\ \label{BestSneg} & + \int_{B(x;t) \cap \{ y_3 < 0 \} } \int_{\mathbb R^3} \iota_i \frac{ (\o^- \times \hat v)_i }{ 1+ \hat{v} \cdot \o^-} S f ( t - |y-x|, y_\parallel, - y_3 ,v ) dv \frac{ dy} {|y-x|}
\\ \label{Bestbdrypos} & + \int_{ B(x;t) \cap \{ y_3 = 0 \} } \int_{\mathbb R^3} \left( -(e_3 \times \hat v)_i + \frac{ ( \o \times \hat v)_i \hat v_3 }{1+ \hat{v} \cdot \o} \right) f ( t- |y-x|, y_\parallel, 0, v ) dv \frac{ dy_\parallel} {|y-x|}
\\ \label{Bestbdryneg} & + \int_{ B(x;t) \cap \{ y_3 = 0 \} } \int_{\mathbb R^3} \iota_i \left( - (e_3 \times \hat v )_i + \frac{ ( \o^- \times \hat v)_i \hat v_3 }{1+ \hat{v} \cdot \o^-} \right) f ( t- |y-x|, y_\parallel, 0, v ) dv \frac{ dy_\parallel} {|y-x|} \\ \label{Bestinitialpos} & + \int_{\partial B(x;t) \cap \{ y_3>0 \} } \int_{\mathbb R^3} \left( \frac{ ( \o \times \hat v)_i }{1+ \hat{v} \cdot \o}\right) f ( 0, y_\parallel, y_3, v ) dv \frac{ dS_y} {t} \\ \label{Bestinitialneg} & + \int_{\partial B(x;t) \cap \{ y_3 < 0 \} } \int_{\mathbb R^3} \iota_i \left( \frac{ ( \o^- \times \hat v )_i }{1+ \hat{v} \cdot \o^- }\right) f ( 0, y_\parallel, - y_3, v ) dv \frac{ dS_y} {t}
\\ \label{Bestbdrycontri} & +(-1)^i 2 ( 1 - \delta_{i3} ) \int_{ B(x;t) \cap \{y_3 =0\}} \int_{\mathbb R^3 } \frac{ \hat v_{\underline{i} } f (t - |y-x |, y_\parallel, 0, v )}{|y-x | } dv d S_y.
\end{align} \end{proposition}
\section{Regularity estimate of the field} With the formula for $E$ as in \eqref{Eesttat0pos}--\eqref{Eest3bdrycontri}, and $B$ as in \eqref{Besttat0pos}--\eqref{Bestbdrycontri}, we have the estimate of the fields. \begin{lemma} \label{EBlinflemma} There exists a $0 < T \ll 1 $ such that for any $t \in [0, T]$, we have \begin{equation} \label{EBlinftybdd}
\| E(t) \|_\infty + \| B(t) \|_\infty \lesssim \| E_0 \|_\infty + \| B_0 \|_\infty + t \left( \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty + \| E_0 \|_{C^1} + \|B_0 \|_{C^1 } \right). \end{equation} \end{lemma} \begin{proof} Using the expression for $E_i(t,x)$ in \eqref{Eesttat0pos}-\eqref{Eest3bdrycontri}, we have \[ \begin{split}
| \eqref{Eesttat0pos} | \le & \frac{1}{ t^2} \int_{\partial B(x;t) \cap \{ y_3 > 0 \} } t | \partial_t E_{0,i}( y ) + | E_{0,i} (y) | + | \nabla E_{0,i} (y ) | | (y-x) | dS_y
\\ \lesssim &\| E_0 \|_\infty + t \left( \| \partial_t E_0 \|_\infty + + \| \nabla_x E_0 \|_\infty \right) \lesssim \| E_0 \|_\infty + t \| E_0 \|_{C^1}. \end{split} \] And the same estimate can be made for \eqref{Eesttat0neg}. Thus \begin{equation} \label{Elinftyest1}
| \eqref{Eesttat0pos} | + | \eqref{Eesttat0neg} | \lesssim \| E_0 \|_{C^1}. \end{equation} Next, from \cite{GS2} we have \begin{equation} \label{vdecaybasic}
\frac{1}{1 + \hat v \cdot \o } \le \frac{ 2 (1+|v|^2 )}{ 1 + | v \times \o |^2 } \le 2( 1 + |v|^2 ), \text{ and } | \o + \hat v |^2 \le 2 ( 1 + \hat v \cdot \o ),
\end{equation} and since $ 1 - |\hat{v}|^2 = \frac{1}{ 1 + |v|^2 }$, \begin{equation} \label{vkerneldecay}
| \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{ (1+ \hat v \cdot \o )^2} | \le \frac{ \sqrt 2 }{ 1+ |v|^2 } \frac{1}{ (1 + \hat v \cdot \o)^{3/2} } \le 4 \sqrt{ 1 + |v|^2 }. \end{equation} Thus, \[ \begin{split}
| \eqref{Eestbulkpos} | \le & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \left| \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{|y-x|^2 (1+ \hat v \cdot \o )^2} \right| | f(t-|y-x|,y ,v)| \mathrm{d} v \mathrm{d} y
\\ \le & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \frac{4 \sqrt{ 1 + |v|^2 } }{|y-x|^2} | f(t-|y-x|,y ,v)| \mathrm{d} v \mathrm{d} y
\\ \lesssim & \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \int_{B(x;t) \cap \{y_3 > 0\}} \frac{1}{|y-x |^2 } \int_{\mathbb{R}^3} \frac{1}{ 1 + |v|^{3 + \delta } } dv dy
\\ \lesssim & t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{split} \] We have the same estimate for \eqref{Eestbulkneg}, and thus \begin{equation} \label{Elinftyest2}
| \eqref{Eestbulkpos} | + | \eqref{Eestbulkneg} | \lesssim t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{equation}
Next, from the equation \eqref{VMfrakF1} and the definition of $Sf$, we have \[ Sf = - (E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 ) \cdot \nabla_v f. \] From integration by parts in $v$ and the fact that $ \nabla_v \cdot ( \hat v \times ( B + B_{\text{ext}})) =0$, \begin{equation} \label{EestSposibp} \begin{split} & \eqref{EestSpos}
\\ = & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \frac{\o_i + \hat{v}_i}{1+ \hat v \cdot \o }((E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 ) \cdot \nabla_v f) (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}
\\ = & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \nabla_v \left( \frac{\o_i + \hat{v}_i}{1+ \hat v \cdot \o } \right) \cdot (E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 ) f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} \\ = & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i ( v ,\o ) \cdot (E + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 )
f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}, \end{split} \end{equation} where \begin{equation} \label{calSi} \mathcal S^E_i(\o, v ) = \nabla_v \left( \frac{\o_i + \hat{v}_i}{1+ \hat v \cdot \o } \right) = \frac{ (e_i - \hat v_i \hat v ) ( 1 + \hat v \cdot \o ) - ( \o_i + \hat v_i ) ( \o - (\o \cdot \hat v ) \hat v ) }{\langle v \rangle ( 1 + \hat v \cdot \o )^2 }. \end{equation} By writing \begin{equation} \label{ominusdotexp} \o - (\o \cdot \hat v ) \hat v = \o(1 + \hat v \cdot \o ) - ( \hat v \cdot \o )( \o + \hat v ), \end{equation} we have from \eqref{vdecaybasic}, \begin{equation} \label{calSiest} \begin{split}
| \mathcal S^E_i(\o, \hat v ) | \le & | \frac{ (e_i - \hat v_i \hat v ) }{ \langle v \rangle ( 1 + \hat v \cdot \o )} | + | \frac{ \o ( \o_i + \hat v_i ) }{ \langle v \rangle ( 1 + \hat v \cdot \o )} | + | \frac{ ( \o_i + \hat v_i ) ( \hat v \cdot \o )( \o + \hat v ) }{ \langle v \rangle ( 1 + \hat v \cdot \o )^2} |
\\ \le & 2 \sqrt{ 1 + |v|^2 } + 2 \sqrt{ 1 + |v|^2 } + + 8 \sqrt{ 1 + |v|^2 }
\\ = & 12 \sqrt{ 1 + |v|^2 }. \end{split} \end{equation} Thus, \[ \begin{split}
& |\eqref{EestSpos}|
\\ \le & \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty
\\ & \times \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} | E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g \mathbf e_3 |( t-|y-x | , y ) \frac{1}{ 1 + |v|^{3 + \delta }} dv \frac{dy}{|y-x | }
\\ \lesssim & \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \sup_{ 0 \le t \le T} \left( \| E(t) \|_\infty + \| B(t) \|_\infty + g + E_e + |B_e| \right) \int_{B(x;t) \cap \{y_3 > 0\}} \frac{1}{|y-x | } dy
\\ \lesssim & t^2 \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \sup_{ 0 \le t \le T} \left( \| E(t) \|_\infty + \| B(t) \|_\infty + g + E_e + |B_e| \right). \end{split} \] Applying the same estimate to \eqref{EestSneg} we get \begin{equation} \label{Elinftyest3}
|\eqref{EestSpos}| + |\eqref{EestSneg}| \lesssim t^2 \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \sup_{ 0 \le t \le T} \left( \| E(t) \|_\infty + \| B(t) \|_\infty + g + E_e+ |B_e| \right). \end{equation} Next, we have from \eqref{vdecaybasic}, \begin{equation} \label{vdecaybasic2}
| \frac{(\o_i + \hat{v}_i)}{1+ \hat v \cdot \o } | \le 2 \sqrt{ 1 + |v|^2}. \end{equation} So \[ \begin{split}
| \eqref{Eestbdrypos} | \le & \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3}
\int_{\mathbb{R}^3} | \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } | | f (t- ( |z_\parallel | + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) | \mathrm{d} v\frac{ \mathrm{d} z_\parallel}{(|z_\parallel |^2 + x_3^2)^{1/2}}
\\ \lesssim & \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \int_{ \sqrt{ |z_\parallel |^2 + x_3^2 } < t } \frac{1}{(|z_\parallel |^2 + x_3^2)^{1/2}} \int_{\mathbb{R}^3} \frac{1}{1 + |v|^{3 + \delta } } dv \mathrm{d} z_\parallel
\\ \lesssim & t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{split} \] And we have the same estimate for \eqref{Eestbdryneg}, thus \begin{equation} \label{Elinftyest4}
| \eqref{Eestbdrypos} | + | \eqref{Eestbdryneg} | \lesssim t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{equation} Again from \eqref{vdecaybasic2}, \[ \begin{split}
| \eqref{Eestinitialpos} | \le & \int_{|x-y| = t , \ y_3>0} | \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) f(0, y,v) | \mathrm{d} v \frac{\mathrm{d} S_y}{|y-x|}
\\ \lesssim & \frac{1}{t} \| \langle v \rangle^{4+\delta} f(0) \|_\infty \int_{|x-y| = t , \ y_3>0} \int_{\mathbb R^3 } \frac{1}{1 + |v|^{3 + \delta } } dv dS_y
\\ \lesssim & t \| \langle v \rangle^{4+\delta} f(0) \|_\infty. \end{split} \] Using the same estimate for \eqref{Eestinitialneg}, we get \begin{equation} \label{Elinftyest5}
| \eqref{Eestinitialpos} | + | \eqref{Eestinitialneg} | \lesssim t \| \langle v \rangle^{4+\delta} f(0) \|_\infty. \end{equation} Next, we have \begin{equation} \label{Elinftyest6} \begin{split}
| \eqref{Eest3bdrycontri} | \le & \int_{ B(x;t) \cap \{y_3 =0\}} \int_{\mathbb R^3 } \frac{ 8 \pi | f (t - |y-x |, y_\parallel, 0, v ) | }{|y-x | } dv d S_y
\\ \lesssim & \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \int_{ \sqrt{ |z_\parallel |^2 + x_3^2 } < t } \frac{1}{(|z_\parallel |^2 + x_3^2)^{1/2}} \int_{\mathbb R^3 } \frac{1}{1 + |v|^{4 + \delta } } dv \mathrm{d} z_\parallel
\\ \lesssim &t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{split} \end{equation}
Next, we go to estimate for $B(t,x)$ in \eqref{Besttat0pos}--\eqref{Bestbdrycontri}. Similar to \eqref{Elinftyest1}, we have \begin{equation} \label{Blinftyest1}
| \eqref{Besttat0pos} | + | \eqref{Besttat0neg} | \lesssim \| B_0 \|_{C^1 }. \end{equation} From \eqref{vdecaybasic} we have \begin{equation} \label{vdecayBbulk}
| \frac{ ( \o \times \hat v ) \left( 1 - |\hat v |^2 \right) }{( 1 + \hat v \cdot \o )^2} | \le 2 | \frac{ \o \times v }{ (1 + |v|^2 )^{3/2} ( 1 + \hat v \cdot \o )^2 } | \le 8 \frac{ | \o \times v | \sqrt{ 1 + |v|^2 } }{ (1 + | \o \times v |^2 )^2 } \le 8 \sqrt{ 1 + |v|^2 }. \end{equation} Thus \[ \begin{split}
| \eqref{Bestbulkpos} | + | \eqref{Bestbulkneg} | \lesssim & \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \int_{B(x;t)} \frac{1}{|y-x |^2 } \int_{\mathbb{R}^3} \frac{1}{ 1 + |v|^{3 + \delta } } dv dy
\\ \lesssim & t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{split} \] Next, using the equation \eqref{VMfrakF1} and the definition of $Sf$, from integration by parts in $v$, \begin{equation} \label{BestSposibp} \begin{split} & \eqref{BestSpos}
\\= & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \frac{(\o \times \hat v )_i}{1+ \hat v \cdot \o }((E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 ) \cdot \nabla_v f) (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}
\\ = & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \nabla_v \left( \frac{(\o \times \hat v )_i}{1+ \hat v \cdot \o } \right) \cdot (E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 ) f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} \\ = & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^B_i ( v ,\o ) \cdot (E + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 )
f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}, \end{split} \end{equation} where by direct calculation we have \begin{equation} \label{calBSirep} \begin{split}
\mathcal S^B_i(v,\o ) = \nabla_v \left( \frac{( \o \times \hat v )_i }{ 1 + \hat v \cdot \o } \right) = & \nabla_v \left( \frac{( \o \times v )_i }{ \sqrt{ 1+ |v|^2 } + v \cdot \o } \right)
\\ = & \frac{ \nabla_v [ (\o \times v)_i ] }{\sqrt{1+|v|^2 }+ v \cdot \o } + \frac{ (\o \times v)_i ( \hat v + \o ) }{ ( \sqrt{1+|v|^2 } + v \cdot \o )^2 }
\\ = & \frac{ \nabla_v [ (\o \times v)_i ] }{\sqrt{1+|v|^2 } (1 + \hat v \cdot \o ) } + \frac{ (\o \times v)_i ( \hat v + \o ) }{ ( \sqrt{1+|v|^2 } (1 + \hat v \cdot \o ) )^2 }. \end{split} \end{equation} Therefore from \eqref{vdecaybasic}, we have \begin{equation} \label{calBSiest} \begin{split}
| \mathcal S^B_i(v,\o ) | \le \frac{ 2}{\sqrt{1+|v|^2 } (1 + \hat v \cdot \o ) } + 8 \frac{ | \o \times v| \sqrt{1+|v|^2 } }{ (1 + | \o \times v |^2 ) ^2 } \le 12 \sqrt{1 + |v|^2 }. \end{split} \end{equation} So we can use the same argument as in \eqref{EestSposibp}--\eqref{Elinftyest3} with \eqref{BestSposibp} and \eqref{calBSiest} to get \[ \begin{split}
& |\eqref{BestSpos}| + |\eqref{BestSneg}|
\\ \lesssim & \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \sup_{ 0 \le t \le T} \left( \| E(t) \|_\infty + \| B(t) \|_\infty + g + E_e + |B_e| \right) \int_{B(x;t)} \frac{1}{|y-x | } dy
\\ \lesssim & t^2 \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \sup_{ 0 \le t \le T} \left( \| E(t) \|_\infty + \| B(t) \|_\infty + g + E_e + |B_e| \right). \end{split} \] Next, again from \eqref{vdecaybasic}, \begin{equation} \label{vdecaybasic3}
| \frac{ ( \o \times \hat v)_i }{1+ \hat{v} \cdot \o} | \le 2 \sqrt{1+ |v|^2 }. \end{equation} So similar to \eqref{Elinftyest4}, \eqref{Elinftyest5} we get \begin{equation}
| \eqref{Bestbdrypos} | + | \eqref{Bestbdryneg} | \lesssim t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty, \end{equation} and \begin{equation}
| \eqref{Bestinitialpos} | + | \eqref{Bestinitialneg} | \lesssim t \| \langle v \rangle^{4+\delta} f(0) \|_\infty. \end{equation} And finally same as \eqref{Elinftyest6}, we have \begin{equation} \label{Blinftyest6}
| \eqref{Bestbdrycontri} | \lesssim t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{equation}
Combining \eqref{Elinftyest1}, \eqref{Elinftyest2}, \eqref{Elinftyest3}, \eqref{Elinftyest4}, \eqref{Elinftyest5}, \eqref{Elinftyest6}, and \eqref{Blinftyest1}--\eqref{Blinftyest6}, we get \begin{equation} \label{EBinftybd}
\begin{split}
& \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty
\\ \lesssim & \| E_0 \|_\infty + \|B_0 \|_\infty + t( \| E_0 \|_{C^1 } + \| B_0 \|_{C^1 } )
\\ & + t \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty \left( 1 + t ( \sup_{ 0 \le t \le T} \left( \| E(t) \|_\infty + \| B(t) \|_\infty \right) + g + E_e + |B_e| )\right). \end{split} \end{equation} Taking $0 < T \ll1 $, we conclude \eqref{EBlinftybdd}.
\end{proof}
Next we estimate the derivatives of the fields.
\begin{lemma} \label{EBW1inftylemma} With the formula $E(t,x)$ as in \eqref{Eesttat0pos}--\eqref{Eest3bdrycontri}, and $B(t,x)$ as in \eqref{Besttat0pos}--\eqref{Bestbdrycontri}, there exists a $T \ll 1 $ such that for any $t \in [0, T]$, \begin{equation} \label{nablaxparaE} \begin{split}
\| \nabla_{x_\parallel} E(t) \|_\infty + \| \nabla_{x_\parallel} B(t) \|_\infty \lesssim & \| E_0 \|_{C^2} + \| B_0 \|_{C^2} + \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f(t) \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty , \end{split} \end{equation} \begin{equation} \label{lambdanablaxE} \begin{split}
\| \partial_{x_3} E(t) \|_\infty + \| \partial_{x_3} B(t) \|_\infty \lesssim & \| E_0 \|_{C^2} + \| B_0 \|_{C^2} + \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5 + \delta } \alpha \partial_{x_3} f(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f(t) \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty, \end{split} \end{equation} and \begin{equation} \label{ptEBest} \begin{split}
\| \partial_t E (t) \|_\infty + \| \partial_t B (t) \|_\infty \lesssim & \| E_0 \|_{C^2}+ \| B_0 \|_{C^2} + \sup_{ 0 \le t \le T} \left( \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty + \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f(t) \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{split} \end{equation}
\end{lemma}
\begin{proof}
We take derivative to $\frac{ \partial}{\partial_{x_k } } E_i(t,x)$ in \eqref{Eesttat0pos}-\eqref{Eest3bdrycontri} and estimate each term.
First, by using the change of variables $z = y - x$ and spherical coordinate for $z$, we have \begin{equation} \label{Eestat00} \begin{split}
\eqref{Eesttat0pos} = & \frac{1}{4\pi t^2 } \int_{ \{ |z| = t , x_3 + z_3 > 0 \} } \left( t \partial_t E_{0,i}( x + z ) + E_{0,i} (x+z) + \nabla E_{0,i} ( x+ z) \cdot z \right) dS_z
\\ = & \frac{1}{4\pi t^2 } \int_{ \{ t \cos \phi > - x_3 \} } \int_0^{2 \pi } \left( t \partial_t E_{0,i}( x + z ) + E_{0,i} (x+z) + \nabla E_{0,i} ( x+ z) \cdot z \right) t^2 \sin \phi d\theta d\phi.
\end{split} \end{equation} Thus \begin{equation} \begin{split}
\frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0pos} = & \frac{1}{4\pi t^2 } \int_{ \{ |z| = t , x_3 + z_3 > 0 \} } \left( t \partial_{x_k } \partial_t E_{0,i}( x + z ) + \partial_{x_k } E_{0,i} (x+z) + \nabla \partial_{x_k } E_{0,i} ( x+ z) \cdot z \right) dS_z \\ & + \frac{1}{4 \pi t^2 } \delta_{k3 } \int_0^{2\pi } ( t \partial_t E_{0,i}( x_\parallel + z_\parallel, 0 ) + E_{0,i} (x_\parallel + z_\parallel, 0 ) + \nabla E_{0,i} ( x_\parallel + z_\parallel, 0 ) \cdot z ) \\ & \quad \quad \quad \quad \quad \quad \times \left( \frac{d}{d x_3} \cos^{-1} \left( \frac{-x_3}{t} \right) \right) \left( \sin \left( \cos^{-1} \left( \frac{-x_3}{t} \right) \right) \right) t^2 d \theta
\\ = & \frac{1}{4\pi t^2 } \int_{ \{ |z| = t , x_3 + z_3 > 0 \} } \left( t \partial_{x_k } \partial_t E_{0,i}( x + z ) + \partial_{x_k } E_{0,i} (x+z) + \nabla \partial_{x_k } E_{0,i} ( x+ z) \cdot z \right) dS_z \\ & + \frac{1}{4 \pi t} \delta_{k3 } \int_0^{2\pi } ( t \partial_t E_{0,i}( x_\parallel + z_\parallel , 0 ) + E_{0,i} (x_\parallel + z_\parallel , 0) + \nabla E_{0,i} ( x_\parallel + z_\parallel , 0) \cdot z ) d\theta. \end{split} \end{equation} For $i = 1,2$, $E_{0,i} ( x_\parallel + z_\parallel , 0 ) = 0$, thus we have \[
| \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0pos}_{i=1,2} | \lesssim t \| \nabla_x \partial_t E_0 \|_\infty + \| \nabla_x E_0 \|_\infty + t \| \nabla_x ^2 E_0 \|_\infty + \| \partial_t E_0 \|_\infty + \| \nabla_x E_0 \|_\infty \lesssim \| E_0 \|_{C^2 }. \] And we apply the same estimate for $ \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0neg}_{i=1,2} $ to obtain \[
| \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0pos}_{i=1,2} | + | \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0neg}_{i=1,2} | \lesssim \| E_0 \|_{C^2 }. \] For $i=3$, we use the cancellation for $ \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0pos}_{i=3} + \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0neg}_{i=3} $ at $y_3 = 0$ to get \begin{equation} \label{pxE03posneg} \begin{split} & \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0pos}_{i=3} + \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0neg}_{i=3}
\\ = & \frac{1}{4\pi t^2 } \int_{ \{ |z| = t , x_3 + z_3 > 0 \} } \left( t \partial_{x_k } \partial_t E_{0,3}( x + z ) + \partial_{x_k } E_{0,3} (x+z) + \nabla \partial_{x_k } E_{0,3} ( x+ z) \cdot z \right) dS_z
\\ & + \frac{1}{4\pi t^2 } \int_{ \{ |z| = t , x_3 + z_3 < 0 \} } \iota_k \left( t \partial_{x_k } \partial_t E_{0,3}( x + z ) + \partial_{x_k } E_{0,3} (x+z) + \nabla_\parallel \partial_{x_k } E_{0,3} ( x+ z) \cdot z_\parallel - \partial_{x_3} \partial_{x_k } E_{0,3} ( x+ z) \cdot z_3 \right) dS_z \\ & + \frac{2}{4\pi t^2 } \delta_{k 3} \int_{0}^{2 \pi } \left( \partial_{x_3} E_{0, 3 } ( x_\parallel + z_\parallel, 0 ) \right) \frac{-x_3 }{t} t^2 d\theta. \end{split}
\end{equation} Thus $| \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0pos}_{i=3} | + | \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0neg}_{i=3} | \lesssim \| E_0 \|_{C^2 }$, and therefore \begin{equation} \label{Eestat01}
| \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0pos} | + | \frac{ \partial}{\partial_{x_k } } \eqref{Eesttat0neg} | \lesssim \| E_0 \|_{C^2 }. \end{equation}
Next, using the change of variables $ z = y-x$ we have \begin{equation} \label{Eestbulkpos1} \begin{split}
\frac{ \partial}{\partial_{x_k } } \eqref{Eestbulkpos} =& \int_{ \{ |z| < t \} \cap \{z_3 > -x_3 \} } \int_{\mathbb{R}^3} \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{|z|^2 (1+ \hat v \cdot \o )^2} \partial_{x_k}f (t-|z| ,z+x ,v) \mathrm{d} v \mathrm{d} z
\\& + \delta_{k3} \int_{ \{ |z| < t \} \cap \{z_3 = -x_3 \} } \int_{\mathbb{R}^3}\frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{|z|^2 (1+ \hat v \cdot \o )^2} f(t-|z|,z+x ,v) \mathrm{d} v \mathrm{d} z_\parallel
\\ = & \underbrace{ \int_{ \{ |y-x| < t \} \cap \{y_3 > 0 \} } \int_{\mathbb{R}^3} \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{|y-x|^2 (1+ \hat v \cdot \o )^2} \partial_{x_k}f (t-|y-x| ,y ,v) \mathrm{d} v \mathrm{d} y }_{\eqref{Eestbulkpos1}_1}
\\& + \underbrace{ \delta_{k3} \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \int_{\mathbb{R}^3}\frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{ ( |y_\parallel - x_\parallel |^2 + x_3^2 ) (1+ \hat v \cdot \o )^2} f(t-|y-x|, y_\parallel, 0 ,v) \mathrm{d} v \mathrm{d} y_\parallel}_{\eqref{Eestbulkpos1}_2}. \end{split} \end{equation} Using \eqref{vkerneldecay}, we have for $k =1,2$, \begin{equation} \label{Eestbulkpospara2} \begin{split}
& \int_{ \{ |y-x| < t \} \cap \{y_3 > 0 \} } \int_{\mathbb{R}^3} \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{|y-x|^2 (1+ \hat v \cdot \o )^2} \partial_{x_k}f (t-|y-x| ,y ,v) \mathrm{d} v \mathrm{d} y
\\ \lesssim & \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty \int_{ \{ |y-x| < t \} } \int_{\mathbb{R}^3} \frac{ 1 }{|y-x|^2} ( 1 + |v| )^{-3-\delta} \mathrm{d} v \mathrm{d} y
\\ \lesssim & \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty. \end{split} \end{equation} For $k= 3$, we have for any $ 1 < p < \frac{3}{2}$, from \eqref{vkerneldecay}, and Lemma \ref{1alphaintv} which will be proved in the next section, \begin{equation} \label{Eestbulkpos2} \begin{split}
& \int_{ \{ |y-x| < t \} \cap \{y_3 > 0 \} } \int_{\mathbb{R}^3} \frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{|y-x|^2 (1+ \hat v \cdot \o )^2} \partial_{x_3}f (t-|y-x| ,y ,v) \mathrm{d} v \mathrm{d} y
\\ \lesssim & \sup_{0 \le t \le T} \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \int_{ \{ |y-x| < t \} \cap \{y_3 > 0 \} } \int_{\mathbb{R}^3} \frac{1}{|y-x |^2 } \frac{ ( 1 + |v| )^{-4-\delta} }{ \alpha(t- |y-x |, y, v ) } dv dy
\\ \lesssim & \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right) \int_{ \{ |y-x| < t \} \cap \{y_3 > 0 \} } \frac{1}{|y-x|^2}\left(1 + \ln (1 + \frac{1}{y_3}) \right) dy
\\ \lesssim & \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right) \int_{ \{ |y-x| < t \} \cap \{y_3 > 0 \} } \left( \frac{1}{|y-x|^{2p } } + | \ln ( 1 + \frac{1}{y_3} ) |^{\frac{p}{p-1} } \right) dy
\\ \lesssim & \sup_{0 \le t \le T} \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty .
\end{split} \end{equation}
We leave the estimate of $\eqref{Eestbulkpos1}_2$ together with the estimate of $\partial_{x_k} \eqref{Eestbdrypos}$ later.
Next, from \eqref{EestSposibp}, and using the change of variables $z = y -x $ and taking $\frac{\partial}{\partial_{x_k } } $ derivative to \eqref{EestSpos} we have \begin{equation} \label{EestSpos1} \begin{split}
\frac{ \partial }{\partial_{x_k } } \eqref{EestSpos} = & \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i (v ,\o ) \cdot ( \partial_{x_k } E + \hat v \times \partial_{x_k } B) f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}
\\ & + \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i (v ,\o ) \cdot ( E + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 ) \partial_{x_k} f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|}
\\ & + \delta_{k3} \int_{B(x;t) \cap \{y_3 = 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i (v ,\o ) \cdot ( E + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3) f (t-|y-x|,y_\parallel, 0 ,v) \mathrm{d} v \frac{ \mathrm{d} y_\parallel }{|y-x|}. \end{split} \end{equation} Thus for $k=1,2$, from \eqref{calSiest} we have, \begin{equation} \label{EestSpospara2} \begin{split}
& | \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i (v ,\o ) \cdot ( \partial_{x_k } E + \hat v \times \partial_{x_k } B) f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} |
\\ \lesssim & \sup_{0 \le t \le T} \| ( 1 + |v|^{4 +\delta } ) f(t) \|_\infty \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb R^3 } | ( \partial_{x_k } E + \hat v \times \partial_{x_k } B) | \left( \frac{1}{ 1 + |v|^{3 + \delta } } \right) dv \frac{ \mathrm{d} y}{|y-x|}
\\ \lesssim & \sup_{0 \le t \le T} \| ( 1 + |v|^{4 +\delta } ) f(t) \|_\infty \left( \sup_{0 \le t \le T} \| \nabla_{x_\parallel} E(t) \|_\infty + \sup_{0 \le t \le T} \| \nabla_{x_\parallel} B(t) \|_\infty \right) \times \int_{B(x;t) } \frac{dy}{|y-x| }
\\ \lesssim & t^2 \sup_{0 \le t \le T} \| ( 1 + |v|^{4 +\delta } ) f(t) \|_\infty \left( \sup_{0 \le t \le T} \| \nabla_{x_\parallel} E(t) \|_\infty + \sup_{0 \le t \le T} \| \nabla_{x_\parallel} B(t) \|_\infty \right). \end{split} \end{equation} Similarly, for $k=3$, \begin{equation} \label{EestSpos2} \begin{split}
& | \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i (v ,\o ) \cdot ( \partial_{x_k } E + \hat v \times \partial_{x_3 } B) f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} |
\\ \lesssim & t^{2 } \sup_{0 \le t \le T} \| ( 1 + |v|^{4 +\delta } ) f(t) \|_\infty \left( \sup_{0 \le t \le T} \| \partial_{x_3} E(t) \|_\infty + \sup_{0 \le t \le T} \| \partial_{x_3} B(t) \|_\infty \right), \end{split} \end{equation} for any $0 < \delta \ll 1 $. Thus for $ t \ll 1$, the terms from \eqref{EestSpospara2}, \eqref{EestSpos2} will be absorbed into the LHS. From \eqref{calSiest}, we have for $k=1,2$, \begin{equation} \label{EestSpospara3} \begin{split}
& | \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i (v ,\o ) \cdot ( E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3) \partial_{x_k} f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} |
\\ \lesssim & \left( \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty + |B_e| + g \right) \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty \right)
\\ & \quad \quad \times \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \frac{(1+ |v|^{-3 - \delta } )}{|y-x|} dv dy
\\ \lesssim & \left( \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty + |B_e| +g \right) \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty \right). \end{split} \end{equation} And, for $k=3$, \begin{equation} \label{EestSpos3} \begin{split}
& | \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i (v ,\o ) \cdot ( E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g \mathbf e_3) \partial_{x_3} f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} |
\\ \lesssim & \left( \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty +|B_e| + E_e +g \right) \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right)
\\ & \quad \quad \quad \times \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \frac{1}{ |y-x| } \left( \frac{ \langle v \rangle^{ -4- \delta} }{ \alpha(t- |y-x |, y, v ) } \right) dv dy
\\ \lesssim & \left( \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty + |B_e| +E_e+g \right)
\\ & \times \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right) \int_{B(x;t) \cap \{y_3 > 0\}} \frac{1}{ |y-x| } | \ln (y_3 ) | dy
\\ \lesssim & \left( \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty +|B_e| + E_e +g \right) \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right)
\\ &\quad \quad \times \int_{B(x;t) \cap \{y_3 > 0\}} \left( \frac{1}{ |y-x|^2 } + \left( \ln (y_3 ) \right)^2 \right) dy
\\ \lesssim & \left( \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty \right) \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right). \end{split} \end{equation} And \begin{equation} \label{EestSpos4} \begin{split}
& | \int_{B(x;t) \cap \{y_3 = 0\}} \int_{\mathbb{R}^3} \mathcal S^E_i (v ,\o ) \cdot ( E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g \mathbf e_3) f (t-|y-x|,y_\parallel, 0 ,v) \mathrm{d} v \frac{ \mathrm{d} y_\parallel }{|y-x|} |
\\ \lesssim & \left( \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty + |B_e| + E_e +g \right) \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty
\\ & \quad \quad \times \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 ) < t^2 \} } \frac{1}{ ( |y_\parallel - x_\parallel |^2 + x_3^2 )^{1/2} } dy_\parallel
\\ \lesssim & \left( \sup_{0 \le t \le T} \| E(t) \|_\infty + \sup_{0 \le t \le T} \| B(t) \|_\infty + |B_e| +E_e + g \right) \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta } ) f(t) \|_\infty. \end{split} \end{equation} Thus combining \eqref{EestSpos1}, \eqref{EestSpospara2}, \eqref{EestSpos2}, \eqref{EestSpospara3}, \eqref{EestSpos3}, \eqref{EestSpos4}, we get \[ \begin{split}
| \nabla_{x_\parallel} \eqref{EestSpos} | \lesssim & t^{2} \left( \sup_{0 \le t \le T} \| \nabla_{x_\parallel} E(t) \|_\infty + \sup_{0 \le t \le T} \| \nabla_{x_\parallel} B(t) \|_\infty \right) + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty
\\ | \frac{ \partial }{\partial_{x_3 } } \eqref{EestSpos} | \lesssim & t^{2} \left( \sup_{0 \le t \le T} \| \partial_{x_3} E(t) \|_\infty + \sup_{0 \le t \le T} \| \partial_{x_3} B(t) \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right) + \sup_{0 \le t \le T} \| (1 + |v|^{4 + \delta }) f(t) \|_\infty. \end{split} \]
By the same argument we get the same estimate for $ | \frac{ \partial}{\partial_{x_k } } \eqref{EestSneg} | $. Therefore \begin{equation} \label{EestSfinal} \begin{split}
| \nabla_{x_\parallel} \eqref{EestSpos} | + | \nabla_{x_\parallel} \eqref{EestSneg} | \lesssim & t^{2} \left( \sup_{0 \le t \le T} \| \nabla_{x_\parallel} E(t) \|_\infty + \sup_{0 \le t \le T} \| \nabla_{x_\parallel} B(t) \|_\infty \right) + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty
\\ | \frac{ \partial }{\partial_{x_3 } } \eqref{EestSpos} | + | \frac{ \partial }{\partial_{x_3 } } \eqref{EestSneg} | \lesssim & t^{2} \left( \sup_{0 \le t \le T} \| \partial_{x_3} E(t) \|_\infty + \sup_{0 \le t \le T} \| \partial_{x_3} B(t) \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right) + \sup_{0 \le t \le T} \| (1 + |v|^{4 + \delta }) f(t) \|_\infty. \end{split} \end{equation}
Next, using the change of variables $z_\parallel = y_\parallel - x_\parallel $ we have \begin{equation} \label{Eestbdrypos1} \begin{split} & \frac{ \partial }{\partial_{x_k } } \eqref{Eestbdrypos}
\\ = & \frac{ \partial }{\partial_{x_k } } \left( \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3}
\int_{\mathbb{R}^3} \left( \delta_{i3 } - \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v\frac{ \mathrm{d} z_\parallel}{(|z_\parallel |^2 + x_3^2)^{1/2}} \right)
\\ = & ( 1 - \delta_{k3} ) \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \left( \delta_{i3 } -\frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) \partial_{x_k } f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v\frac{ \mathrm{d} z_\parallel}{(|z_\parallel |^2 + x_3^2)^{1/2}}
\\ & + \delta_{k3 } \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \partial_{x_3} \left( \frac{ 1}{(|z_\parallel |^2 + x_3^2)^{1/2}} \left( \delta_{i3 } -\frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) \right) f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v \ \mathrm{d} z_\parallel
\\ & + \delta_{k3} \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \left( \delta_{i3} - \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) \partial_{t } f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v\frac{ - x_3 }{(|z_\parallel |^2 + x_3^2)} \mathrm{d} z_\parallel
\\ & - \delta_{k3} \int_{ \sqrt{t^2 - |z_\parallel |^2 } = x_3} \int_{\mathbb{R}^3} \left( \delta_{i3} \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) f (0 ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v \frac{x_3 }{\sqrt{t^2 - x_3^2} } \left( \frac{ z_\parallel }{\sqrt{t^2-x^2}} \cdot \frac{z_\parallel}{ | z_\parallel | } \right) \frac{ \mathrm{d} S_{ z_\parallel } }{t}. \end{split} \end{equation}
The first term is only contribute as the tangential derivative, from \eqref{vdecaybasic2}, \begin{equation} \label{Eestbdrypos2} \begin{split}
& | ( 1 - \delta_{k3} ) \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \left( \delta_{i3 } -\frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) \partial_{x_k } f (t- ( |z_\parallel | + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v\frac{ \mathrm{d} z_\parallel}{(|z_\parallel |^2 + x_3^2)^{1/2}} |
\\ \lesssim & \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel } f(t) \|_\infty \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb R^3} \frac{ ( 1 + |v|^{4 + \delta} )^{-1}}{(|z_\parallel |^2 + x_3^2)^{1/2} } dv d z_\parallel
\\ \lesssim & \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel } f(t) \|_\infty \int_{r^2 + x_3^2 < t^2 } \frac{ r }{ (r^2 + x_3^2 )^{1/2} } dr \lesssim \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel } f(t) \|_\infty.
\end{split} \end{equation} For the second term, recall $\eqref{Eestbulkpos1}_2$ using the identity (\cite{CKKRM}) \[
\frac{ (|\hat{v}|^2-1 )(\hat{v}_i + \o_i ) }{ ( |y_\parallel - x_\parallel |^2 + x_3^2 ) (1+ \hat v \cdot \o )^2} = \left. \sum_{j=1}^3 \frac{\partial}{\partial y_j}\left[\frac{1}{|y-x|}\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] \right|_{y_3 = 0 }, \] we have \begin{equation} \label{Eestbdrypos2.5} \begin{split}
& | \eqref{Eestbulkpos1}_2 + \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \partial_{x_3} \left( \frac{ 1}{(|z_\parallel |^2 + x_3^2)^{1/2}} \left( \delta_{i3 } -\frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) \right) f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v \ \mathrm{d} z_\parallel |
\\ = & | \delta_{k3} \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \int_{\mathbb{R}^3} \left. \sum_{j=1}^3 \frac{\partial}{\partial y_j}\left[\frac{1}{|y-x|}\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] f (t-|y-x|,y ,v) \right|_{y_3 = 0 } \mathrm{d} v \mathrm{d} y_\parallel
\\ & + \delta_{k3} \int_{ \{ | y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \int_{\mathbb{R}^3} \partial_{x_3} \left( \frac{ \left( \delta_{i3 } -\frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right)}{(|y_\parallel - x_\parallel |^2 + x_3^2)^{1/2}} \right) f (t- ( |y_\parallel - x_\parallel |^2 + x_3^2 )^{1/2} ,y_\parallel, 0 ,v) \mathrm{d} v \ \mathrm{d} y_\parallel |
\\ = & | \delta_{k3} \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \int_{\mathbb{R}^3} \left. \sum_{j=1}^2 \frac{\partial}{\partial y_j}\left[\frac{1}{|y-x|}\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] f (t-|y-x|,y ,v) \right|_{y_3 = 0 } \mathrm{d} v \mathrm{d} y_\parallel
\\ & + \delta_{k3} \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \int_{\mathbb{R}^3} \left. \left( \frac{\partial}{\partial y_3}\left[\frac{1}{|y-x|}\left( \delta_{i3} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] f (t-|y-x|,y ,v) \right) \right|_{y_3 = 0 } \mathrm{d} v \mathrm{d} y_\parallel
\\ & + \delta_{k3} \int_{ \{ | y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \int_{\mathbb{R}^3} \partial_{x_3} \left( \frac{ 1}{(|y_\parallel - x_\parallel |^2 + x_3^2)^{1/2}} \left( \delta_{i3 } -\frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) \right)
\\ & \quad \quad \times f (t- ( |y_\parallel - x_\parallel |^2 + x_3^2 )^{1/2} ,y_\parallel, 0 ,v) \mathrm{d} v \ \mathrm{d} y_\parallel |
\\ = & | \delta_{k3} \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \int_{\mathbb{R}^3} \sum_{j=1}^2 \frac{\partial}{\partial y_j}\left[\frac{1}{(|y_\parallel-x_\parallel |^2 + x_3^2 )^{1/2}}\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right]
\\ & \quad \quad \times f (t-(|y_\parallel-x_\parallel |^2 + x_3^2 )^{1/2},y_\parallel, 0 ,v) \mathrm{d} v \mathrm{d} y_\parallel |, \end{split}
\end{equation} where we've used the cancellation $ \left. \frac{\partial}{\partial y_3}\left[\frac{1}{|y-x|}\left( \delta_{i3} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right] \right) \right|_{y_3 = 0 } = - \partial_{x_3} \left( \frac{ 1}{(|y_\parallel - x_\parallel |^2 + x_3^2)^{1/2}} \left( \delta_{i3 } -\frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) \right) $. Thus from integration by parts and \eqref{vdecaybasic}, \begin{equation} \label{Eestbdrypos2.6} \begin{split}
& | \delta_{k3} \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \int_{\mathbb{R}^3} \sum_{j=1}^2 \frac{\partial}{\partial y_j}\left[\frac{1}{|y-x|}\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] f (t-|y-x|,y ,v) \mathrm{d} v \mathrm{d} y_\parallel |
\\ \lesssim & | - \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \left[\frac{| |y_\parallel -x_\parallel | }{(|y_\parallel -x_\parallel |^2 + x_3^2 ) }\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] \partial_t f (t-|y-x|,y ,v) \mathrm{d} v \mathrm{d} y_\parallel
\\ & - \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \left[\frac{1 }{(|y_\parallel -x_\parallel |^2 + x_3^2 )^{1/2} }\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] \nabla_{x_\parallel} f (t-|y-x|,y ,v) \mathrm{d} v \mathrm{d} y_\parallel
\\ & + \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} = t \} } \sum_{j=1}^2 \frac{\o_j }{t } \left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right) f (0,y ,v) \mathrm{d} v \mathrm{d} S_{y_\parallel } |
\\ \lesssim & | \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \left[\frac{| |y_\parallel -x_\parallel | }{(|y_\parallel -x_\parallel |^2 + x_3^2 ) }\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] \partial_t f (t-|y-x|,y ,v) \mathrm{d} v \mathrm{d} y_\parallel |
\\ + & \sup_{ 0 \le t \le T } \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel} f(t) \|_\infty \int_{ |z_\parallel |^2 + x_3^2 < t^2 } \frac{1}{ ( |z_\parallel |^2 + x_3^2 )^{1/2} } \int_{\mathbb R^3 } \frac{1}{ 1 + |v|^{3 + \delta } } dv d z_\parallel
\\ & + \| ( 1 +|v|^{4 + \delta } ) f_0 \|_\infty \int_{\mathbb R^3 } \frac{1}{ 1 + |v|^{3 + \delta } } dv
\\ \lesssim & | \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \left[\frac{| |y_\parallel -x_\parallel | }{(|y_\parallel -x_\parallel |^2 + x_3^2 ) }\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] \partial_t f (t-|y-x|,y ,v) \mathrm{d} v \mathrm{d} y_\parallel |
\\ & + \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel} f(t) \|_\infty + \| ( 1 +|v|^{4 + \delta } ) f_0 \|_\infty. \end{split} \end{equation}
Now from \eqref{VMfrakF1}, we write $\partial_t f = - \hat v \cdot \nabla_x f -(E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g \mathbf{e}_3 ) \cdot \nabla_v f $. Then using $\alpha = \hat v_3$ on $\partial \Omega$, integration by parts in $v$, and that $ \nabla_v \cdot ( \hat v \times B) =0$, we get \begin{equation} \label{Eestbdrypos3} \begin{split}
& | \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \partial_{t } f (t- ( |z_\parallel | + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v \frac{ x_3 }{(|z_\parallel |^2 + x_3^2)} \mathrm{d} z_\parallel |
\\ \le & | \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \left( \hat v \cdot \nabla_x f (t- ( |z_\parallel | + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \right) \mathrm{d} v \frac{ x_3 }{(|z_\parallel |^2 + x_3^2)} \mathrm{d} z_\parallel |
\\ & + | \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o }
\\ & \quad \quad \times \left( (E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g \mathbf{e}_3 ) \cdot \nabla_v f (t- ( |z_\parallel | + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \right) \mathrm{d} v \frac{ x_3 }{(|z_\parallel |^2 + x_3^2)} \mathrm{d} z_\parallel |
\\ \lesssim & \sup_{0 \le t \le T} \left( \| ( 1 + |v|^{5 + \delta} ) \alpha \partial_{x_3 } f(t) \|_\infty + \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel} f(t) \|_\infty \right) \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb R^3} \langle v \rangle^{-4-\delta} \frac{ x_3 }{(|z_\parallel |^2 + x_3^2)} dv \mathrm{d} z_\parallel
\\ & + | \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} \nabla_v \left( \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) (E + E_{\text{ext}} + \hat v \times B - g \mathbf{e}_3 ) f (t- ( |z_\parallel | + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v
\\ & \quad \times \frac{ x_3 }{(|z_\parallel |^2 + x_3^2)} \mathrm{d} z_\parallel |
\\ \lesssim & \sup_{0 \le t \le T} \| ( 1 + |v|^{5 + \delta} ) \left( \| ( 1 + |v|^{5 + \delta} ) \alpha \partial_{x_3 } f(t) \|_\infty + \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel} f(t) \|_\infty \right)
\\ & + \left( \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} )f(t) \|_\infty \right) \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb R^3} (1 + |v|^{3 + \delta})^{-1} \frac{ x_3 }{(|z_\parallel |^2 + x_3^2)} dv \mathrm{d} z_\parallel
\\ \lesssim & \sup_{0 \le t \le T} \left( \| ( 1 + |v|^{5 + \delta} ) \alpha \partial_{x_3 } f(t) \|_\infty + \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel} f(t) \|_\infty \right) + \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} )f(t) \|_\infty , \end{split} \end{equation} where we've used from \eqref{calSi}, \eqref{calSiest}, and \eqref{vdecaybasic2} that \[
\left| \nabla_v \left( \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } \right) \right| = \left| \mathcal S^E_i( \o, \hat v ) \hat v_3 + \frac{(\o_i + \hat{v}_i)}{1+ \hat v \cdot \o } \nabla_v \hat v_3 \right| \le 14 \sqrt{1 + |v|^2}. \] By the same argument we have \begin{equation} \label{Eestbdrypos3.5} \begin{split}
& | \int_{ \{ (| y_\parallel - x_\parallel |^2 + |x_3|^2 )^{1/2} < t \} } \left[\frac{| |y_\parallel -x_\parallel | }{(|y_\parallel -x_\parallel |^2 + x_3^2 ) }\left( \delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+\hat{v} \cdot \o} \right)\right] \partial_t f (t-|y-x|,y ,v) \mathrm{d} v \mathrm{d} y_\parallel |
\\ \lesssim & \sup_{0 \le t \le T} \left( \| ( 1 + |v|^{5 + \delta} ) \alpha \partial_{x_3 } f(t) \|_\infty + \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel} f(t) \|_\infty \right) + \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} )f(t) \|_\infty. \end{split} \end{equation}
We also have \begin{equation} \label{Eestbdrypos4} \begin{split}
& | \int_{ \sqrt{t^2 - |z_\parallel |^2 } = x_3} \int_{\mathbb{R}^3} \frac{(\o_i + \hat{v}_i)\hat{v}_3}{1+ \hat v \cdot \o } f (0 ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v \frac{x_3 }{\sqrt{t^2 - x_3^2} } \left( \frac{ z_\parallel }{\sqrt{t^2 - x_3^2}} \cdot \frac{z_\parallel}{ | z_\parallel | } \right) \frac{ \mathrm{d} S_{ z_\parallel } }{t} |
\\ \lesssim & \| \langle v \rangle^{4+\delta} f_0 \|_\infty \int_{ |z_\parallel | = \sqrt{t^2 -x_3^2 } } \int_{\mathbb R^3} (1 + |v|^{3 + \delta})^{-1} \frac{x_3}{ \sqrt{ t^2 - x_3^2 } } dv \frac{ \mathrm{d} S_{ z_\parallel } }{t}
\\ \lesssim & \| \langle v \rangle^{4+\delta} f_0 \|_\infty \left( \frac{x_3}{t } \right) \lesssim \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{split} \end{equation} Thus from \eqref{Eestbdrypos1}, \eqref{Eestbdrypos2}, \eqref{Eestbdrypos2.5}, \eqref{Eestbdrypos2.6}, \eqref{Eestbdrypos3}, \eqref{Eestbdrypos3.5}, \eqref{Eestbdrypos4}, and together with \eqref{Eestbulkpos1}--\eqref{Eestbulkpos2}, we have \[ \begin{split}
| \nabla_{x_\parallel} \eqref{Eestbulkpos} + \nabla_{x_\parallel} \eqref{Eestbdrypos} | \lesssim & \sup_{0 \le t \le T} \left\{ \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty \right\},
\\ | \frac{ \partial}{\partial_{x_3 } } \eqref{Eestbulkpos} + \frac{ \partial}{\partial_{x_3 } } \eqref{Eestbdrypos} | \lesssim & \sup_{0 \le t \le T} \left\{ \|\langle v \rangle^{4+\delta} f(t) \|_\infty + \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3 } f(t) \|_\infty + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty \right\}. \end{split} \] By the same argument we get the same estimate for $ \frac{ \partial}{\partial_{x_k} } \eqref{Eestbulkneg} + \frac{\partial}{\partial_{x_k } } \eqref{Eestbdryneg} $. Thus \begin{equation} \label{Eestbdryfinal} \begin{split}
& | \nabla_{x_\parallel} \eqref{Eestbulkpos} + \nabla_{x_\parallel} \eqref{Eestbdrypos} |+ | \nabla_{x_\parallel} \eqref{Eestbulkneg} + \nabla_{x_\parallel} \eqref{Eestbdryneg} | \lesssim \sup_{0 \le t \le T} \left\{ \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty \right\},
\\ & | \frac{ \partial}{\partial_{x_3 } } \eqref{Eestbulkpos} + \frac{ \partial}{\partial_{x_3 } } \eqref{Eestbdrypos} | + | \frac{ \partial}{\partial_{x_3 } } \eqref{Eestbulkneg} + \frac{ \partial}{\partial_{x_3 } } \eqref{Eestbdryneg} |
\\ \lesssim & \sup_{0 \le t \le T} \left\{ \|\langle v \rangle^{4+\delta} f(t) \|_\infty + \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3 } f(t) \|_\infty + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty \right\}. \end{split} \end{equation} Next, by using the change of variables $z = y - x$ and spherical coordinate for $z$, we have \begin{equation} \label{Eestinitalpos0} \begin{split}
\eqref{Eestinitialpos} = & \int_{|z| = t , \ z_3>-x_3} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) f(0, z+x ,v) \mathrm{d} v \frac{\mathrm{d} S_z}{t}
\\ = & \int_{ \ t \cos \phi >-x_3} \int_0^{2\pi} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) f(0, z+x ,v) \mathrm{d} v \frac{ t^2 \sin \phi \, \mathrm{d} \theta \mathrm{d} \phi }{t}. \end{split} \end{equation} Thus \begin{equation} \label{pEestinitialpos} \begin{split} & \frac{ \partial}{\partial_{x_k } } \eqref{Eestinitialpos} \\ = &- \int_{ \ t \cos \phi >-x_3} \int_0^{2\pi} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) \partial_{x_k} f(0, z+x ,v) ( t \sin \phi ) \, \mathrm{d} v \mathrm{d} \theta \mathrm{d} \phi \\ & - \delta_{k3} \int_0^{2\pi} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) f(0, z_\parallel +x_\parallel, 0 ,v) \left( \frac{d}{d x_3} \cos^{-1} \left( \frac{-x_3}{t} \right) \right) \left( t \sin \left( \cos^{-1} \left( \frac{-x_3}{t} \right) \right) \right) \, \mathrm{d} v \mathrm{d} \theta \\ = & - \int_{ \ t \cos \phi >-x_3} \int_0^{2\pi} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) \partial_{x_k} f(0, z+x ,v) ( t \sin \phi ) \, \mathrm{d} v \mathrm{d} \theta \mathrm{d} \phi \\ & - \delta_{k3} \int_0^{2\pi} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) f(0, z_\parallel +x_\parallel, 0 ,v) \frac{-1}{\sqrt{ 1- \left( \frac{x_3}{t} \right)^2 } } \frac{-1}{t} \left( t \sqrt{ 1- \left( \frac{x_3}{t} \right)^2 } \right) \, \mathrm{d} v \mathrm{d} \theta.
\end{split} \end{equation} So from \eqref{vdecaybasic2}, for $k=1,2$, \begin{equation} \begin{split}
& | \int_{ \ t \cos \phi >-x_3} \int_0^{2\pi} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) \partial_{x_k} f(0, z+x ,v) ( t \sin \phi ) \, \mathrm{d} v \mathrm{d} \theta \mathrm{d} \phi |
\\ \lesssim & \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_0 \|_\infty \int_{ \ t \cos \phi >-x_3} \int_0^{2\pi} \int_{\mathbb R^3} \langle v \rangle^{-4-\delta} ( t \sin \phi ) \, \mathrm{d} v \mathrm{d} \theta \mathrm{d} \phi
\\ \lesssim & \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_0 \|_\infty. \end{split} \end{equation} And for $k=3$, \begin{equation} \begin{split}
& | \int_{ \ t \cos \phi >-x_3} \int_0^{2\pi} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) \partial_{x_3} f(0, z+x ,v) ( t \sin \phi ) \, \mathrm{d} v \mathrm{d} \theta \mathrm{d} \phi |
\\ \lesssim & \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_\infty \int_{ \ t \cos \phi >-x_3} \int_0^{2\pi} \int_{\mathbb R^3} \frac{\langle v \rangle^{-4-\delta} }{\alpha(0, z +x ,v )} ( t \sin \phi ) \, \mathrm{d} v \mathrm{d} \theta \mathrm{d} \phi
\\ \lesssim & \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_\infty \int_{ \ t \cos \phi >-x_3} t \sin \phi | \ln ( t \cos \phi + x_3 ) | d\phi
\\ \lesssim & \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_\infty \int_0^{t + x_3} | \ln (s) | ds \lesssim \| \langle v \rangle^{5+\delta} \alpha f_0 \|_\infty, \end{split} \end{equation} and \begin{equation} \begin{split}
| \int_0^{2\pi} \sum_j \o_j & \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) f(0, z_\parallel +x_\parallel, 0 ,v) \frac{-1}{\sqrt{ 1- \left( \frac{x_3}{t} \right)^2 } } \frac{-1}{t} \left( t \sqrt{ 1- \left( \frac{x_3}{t} \right)^2 } \right) \, \mathrm{d} v \mathrm{d} \theta |
\\ = & | \int_0^{2\pi} \sum_j \o_j \left(\delta_{ij} - \frac{(\o_i + \hat{v}_i)\hat{v}_j}{1+ \hat v \cdot \o }\right) f(0, z_\parallel +x_\parallel, 0 ,v) \, \mathrm{d} v \mathrm{d} \theta | \lesssim \| \langle v \rangle^{4+\delta} f(0) \|_\infty. \end{split} \end{equation} Therefore, we have \[ \begin{split}
| \nabla_{x_\parallel} \eqref{Eestinitialpos} | \lesssim & \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_0 \|_\infty,
\\ | \frac{ \partial}{\partial_{x_3 } } \eqref{Eestinitialpos} | \lesssim & \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_\infty + \| \langle v \rangle^{4+\delta} f(0) \|_\infty. \end{split} \] And by the same argument we have the same estimate for $\frac{ \partial}{\partial_{x_k } } \eqref{Eestinitialneg}$. Thus \begin{equation} \label{Eestinitialfinal} \begin{split}
| \nabla_{x_\parallel} \eqref{Eestinitialpos} | + | \nabla_{x_\parallel} \eqref{Eestinitialneg} | \lesssim & \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_0 \|_\infty,
\\ | \frac{ \partial}{\partial_{x_3 } } \eqref{Eestinitialpos} | + | \frac{ \partial}{\partial_{x_3 } } \eqref{Eestinitialneg} | \lesssim & \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_\infty + \| \langle v \rangle^{4+\delta} f(0) \|_\infty. \end{split} \end{equation} Finally, we estimate $\frac{\partial}{\partial x_k } \eqref{Eest3bdrycontri} $. We have \[ \begin{split}
& | \frac{\partial}{\partial x_k } \eqref{Eest3bdrycontri} |
\\ & \le 2 | \frac{\partial}{\partial x_k } \int_{ \sqrt{ |z_\parallel |^2 + x_3^2 } < t } \int_{\mathbb R^3 } \frac{ f (t - \sqrt{ |z_\parallel |^2 + x_3^2 }, x_\parallel + z_\parallel, 0, v )}{ \sqrt{ |z_\parallel |^2 + x_3^2 } } dv d z_\parallel |
\\ & \lesssim ( 1 - \delta_{k3} ) \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} | \partial_{x_k } f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) | \mathrm{d} v\frac{ \mathrm{d} z_\parallel}{(|z_\parallel |^2 + x_3^2)^{1/2}}
\\ & + \delta_{k3 } \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} | f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v \ \partial_{x_3} \left( \frac{ 1}{(|z_\parallel |^2 + x_3^2)^{1/2}} \right) | \mathrm{d} z_\parallel
\\ & + \delta_{k3} \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} | \partial_{t } f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v\frac{ - x_3 }{(|z_\parallel |^2 + x_3^2)} | \mathrm{d} z_\parallel
\\ & - \delta_{k3} \int_{ 0}^{2 \pi } \int_{\mathbb{R}^3} | \frac{ | f (0 ,z_\parallel + x_\parallel, 0 ,v) | }{t} \mathrm{d} v \frac{x_3 }{\sqrt{t^2 - x_3^2} } \sqrt{t^2 - x_3^2} d\theta. \end{split} \] Similar to the estimate in \eqref{Eestbdrypos1}-\eqref{Eestbdrypos4}, we get \[
\int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} | \partial_{x_k } f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) | \mathrm{d} v\frac{ \mathrm{d} z_\parallel}{(|z_\parallel |^2 + x_3^2)^{1/2}} \lesssim \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel } f(t) \|_\infty, \] \[ \begin{split}
& \int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} \int_{\mathbb{R}^3} | f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v \ \partial_{x_3} \left( \frac{ 1}{(|z_\parallel |^2 + x_3^2)^{1/2}} \right) | \mathrm{d} z_\parallel
\\ & + \int_{ 0}^{2 \pi } \int_{\mathbb{R}^3} | \frac{ | f (0 ,z_\parallel + x_\parallel, 0 ,v) | }{t} \mathrm{d} v \frac{x_3 }{\sqrt{t^2 - x_3^2} } \sqrt{t^2 - x_3^2} d\theta \lesssim \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty. \end{split} \] And \[ \begin{split}
\int_{ \sqrt{t^2 - |z_\parallel |^2 } > x_3} & \int_{\mathbb{R}^3} | \partial_{t } f (t- ( |z_\parallel |^2 + x_3^2 )^{1/2} ,z_\parallel + x_\parallel, 0 ,v) \mathrm{d} v\frac{ - x_3 }{(|z_\parallel |^2 + x_3^2)} | \mathrm{d} z_\parallel
\\ \lesssim & \left( \| ( 1 + |v|^{5 + \delta} ) \alpha \partial_{x_3 } f(t) \|_\infty + \| ( 1 + |v|^{4 + \delta} ) \nabla_{x_\parallel} f(t) \|_\infty \right) + \sup_{0 \le t \le T} \| ( 1 + |v|^{4 + \delta} )f(t) \|_\infty.
\end{split} \] Therefore \begin{equation} \label{E3estbdryfinal} \begin{split}
| \nabla_{x_\parallel} \eqref{Eest3bdrycontri} | \lesssim & \sup_{0 \le t \le T} \left\{ \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty \right\},
\\ | \frac{\partial}{\partial x_3} \eqref{Eest3bdrycontri} | \lesssim & \sup_{0 \le t \le T} \left\{ \|\langle v \rangle^{4+\delta} f(t) \|_\infty + \| \langle v \rangle^{5+\delta} \alpha \nabla_{x } f(t) \|_\infty + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty \right\}.
\end{split} \end{equation}
Next, we estimate the $\partial_{x_k}$ derivatives to $B$. Using the same argument as in \eqref{Eestat00}--\eqref{Eestat01}, we get \begin{equation} \label{Bestat01}
| \frac{ \partial}{\partial_{x_k } } \eqref{Besttat0pos} | + | \frac{ \partial}{\partial_{x_k } } \eqref{Besttat0neg} | \lesssim \| E_0 \|_{C^2 }. \end{equation} Next, using the decay of the kernel in \eqref{vdecayBbulk}, and following the same argument as in \eqref{Eestbulkpos1}--\eqref{Eestbulkpos2}, and \eqref{Eestbdrypos1}--\eqref{Eestbdryfinal},
we obtain \begin{equation} \label{Bestbulkfinal} \begin{split}
& | \nabla_{x_\parallel} \eqref{Bestbulkpos} + \nabla_{x_\parallel} \eqref{Bestbdrypos} | + | \nabla_{x_\parallel} \eqref{Bestbulkneg} + \nabla_{x_\parallel} \eqref{Bestbdryneg} | \lesssim \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty
\\ & | \frac{ \partial}{\partial_{x_3 } } \eqref{Bestbulkpos} + \frac{ \partial}{\partial_{x_3 } } \eqref{Bestbdrypos} | + | \frac{ \partial}{\partial_{x_3 } } \eqref{Bestbulkneg} + \frac{ \partial}{\partial_{x_3 } } \eqref{Bestbdryneg} |
\\ \lesssim & \sup_{0 \le t \le T} \| ( 1 + |v|^{5 + \delta } ) \alpha \partial_{x_3} f (t) \|_\infty + \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty | . \end{split} \end{equation} Next, from \eqref{BestSposibp} we have \begin{equation} \label{BestSposibp} \begin{split}
\frac{ \partial}{\partial x_k } \eqref{BestSpos} = \frac{ \partial}{\partial x_k } \left( \int_{B(x;t) \cap \{y_3 > 0\}} \int_{\mathbb{R}^3} \mathcal S^B_i ( v ,\o ) \cdot (E + E_{\text{ext}} + \hat v \times ( B + B_{\text{ext}}) - g\mathbf e_3 )
f (t-|y-x|,y ,v) \mathrm{d} v \frac{ \mathrm{d} y}{|y-x|} \right), \end{split} \end{equation} where $\mathcal S^B_i (v, \o )$ as in \eqref{calBSirep}. Then using the bound for $\mathcal S^B_i (v, \o )$ in \eqref{calBSiest} and applying the same argument as in \eqref{EestSpos1}--\eqref{EestSfinal}, we obtain \begin{equation} \label{BestSfinal} \begin{split}
| \nabla_{x_\parallel} \eqref{BestSpos} | + | \nabla_{x_\parallel} \eqref{BestSneg} | \lesssim & t^{2} \left( \sup_{0 \le t \le T} \| \nabla_{x_\parallel} E(t) \|_\infty + \sup_{0 \le t \le T} \| \nabla_{x_\parallel} B(t) \|_\infty \right) + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty
\\ | \frac{ \partial }{\partial_{x_3 } } \eqref{BestSpos} | + | \frac{ \partial }{\partial_{x_3 } } \eqref{BestSneg} | \lesssim & t^{2} \left( \sup_{0 \le t \le T} \| \partial_{x_3} E(t) \|_\infty + \sup_{0 \le t \le T} \| \partial_{x_3} B(t) \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty \right) + \sup_{0 \le t \le T} \| (1 + |v|^{4 + \delta }) f(t) \|_\infty. \end{split} \end{equation} Next, using the same argument as in \eqref{Eestinitalpos0}--\eqref{Eestinitialfinal} together with \eqref{vdecaybasic3}, we have \begin{equation} \label{Bestinitialfinal} \begin{split}
| \nabla_{x_\parallel} \eqref{Bestinitialpos} | + | \nabla_{x_\parallel} \eqref{Bestinitialneg} | \lesssim & \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_0 \|_\infty,
\\ | \frac{ \partial}{\partial_{x_3 } } \eqref{Bestinitialpos} | + | \frac{ \partial}{\partial_{x_3 } } \eqref{Bestinitialneg} | \lesssim & \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_\infty + \| \langle v \rangle^{4+\delta} f(0) \|_\infty. \end{split} \end{equation} Finally, similar \eqref{E3estbdryfinal}, we have \begin{equation} \label{Bestbdryfinal} \begin{split}
| \nabla_{x_\parallel} \eqref{Bestbdrycontri} | \lesssim & \sup_{0 \le t \le T} \left\{ \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty \right\},
\\ | \frac{\partial}{\partial x_3} \eqref{Bestbdrycontri} | \lesssim & \sup_{0 \le t \le T} \big\{ \|\langle v \rangle^{4+\delta} f(t) \|_\infty ( 1+ \sup_{0 \le t \le T} \left( \| E(t) \|_\infty + \| B(t) \|_\infty \right) )
\\ & \quad \quad \quad \quad + \| \langle v \rangle^{5+\delta} \alpha \nabla_{x } f(t) \|_\infty + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty \big\}.
\end{split} \end{equation}
Collecting \eqref{Eestbulkpos1}, \eqref{EestSfinal}, \eqref{Eestbdryfinal}, \eqref{Eestinitialfinal}, and \eqref{E3estbdryfinal}, and \eqref{Eestat01}--\eqref{Bestbdryfinal}, and letting $T \ll 1$, we get \begin{equation} \label{dxEBfinal} \begin{split}
\| \nabla_{x_\parallel} E \|_\infty + \| \nabla_{x_\parallel} B \|_\infty \lesssim & \|E_0\|_{C^2} + \|B_0\|_{C^2} + \sup_{0 \le t \le T} \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty .
\\ \| \partial_{x_3} E \|_\infty + \| \partial_{x_3} B \|_\infty \lesssim & \|E_0\|_{C^2} + \|B_0\|_{C^2} + \sup_{0 \le t \le T} \| \left( \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t) \|_\infty + \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f (t) \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \| \langle v \rangle^{4+\delta} f(t) \|_\infty . \end{split} \end{equation} This concludes \eqref{nablaxparaE} and \eqref{lambdanablaxE}.
For $\partial_t E$, and $\partial_t B$, from the Maxwell equations \eqref{Maxwell}, we have \[
\| \partial_t E \|_\infty \le \| \nabla_x B \|_\infty + \| \langle v \rangle^{4+\delta} f \|_\infty, \ \| \partial_t B \|_\infty \le \| \nabla_x E \|_\infty. \] So from \eqref{nablaxparaE}, \eqref{lambdanablaxE}, we get \eqref{ptEBest}.
\end{proof}
\section{Estimates on trajectories}
We have the following crucial lemma:
\begin{lemma}[Velocity lemma] \label{vlemma} Let $\alpha$ be defined as in \eqref{alphadef}. Suppose \begin{equation} \label{vlfieldass} \begin{split}
\sup_{ 0 \le t \le T} & \left( \| E(t) \|_\infty + \| B(t) \|_\infty \right.
\\ & \left. + \| \partial_t E_3(t) \|_\infty +\| \partial_t (\hat v \times B)_3(t) \|_\infty + \| \nabla_x E_3 (t) \|_\infty + \| \nabla_x (\hat v \times B)_3 (t) \|_\infty \right) + g + B_e < C. \end{split}
\end{equation} And for all $t, x_\parallel$, \begin{equation} \label{signcondition} g - E_e - E_3(t,x_\parallel, 0 ) - (\hat v \times B)_3(t,x_\parallel, 0 ) > c_0 \text{ for some } c_0 > 0.
\end{equation} Then for any $(t,x,v) \in (0, T) \times \Omega \times \mathbb R^3$, with the trajectory $X(s;t,x,v)$ and $V(s;t,x,v)$ satisfies \eqref{HamiltonODE}, \begin{equation} \label{alphaest}
e^{-10\frac{C}{c_0}|t-s| } \alpha(t,x,v) \le \alpha(s, X(s;t,x,v) , V(s;t,x,v) ) \le e^{10\frac{C}{c_0}|t-s| } \alpha(t,x,v). \end{equation}
\end{lemma}
\begin{proof} Note that \begin{eqnarray*} \frac{\partial \hat{v}_3}{\partial v_3 } = \frac{1}{\langle v \rangle} - \frac{ ( \hat{v}_3)^2 }{\langle v \rangle}, \ \nabla_v \left( \frac{1}{\langle v \rangle } \right) = - \frac{\hat v }{ \langle v \rangle^2 }. \end{eqnarray*} By direct computation, \begin{align}
[ \partial_t & + \hat v \cdot \nabla_x + \mathfrak F \cdot \nabla_v ] (\alpha^2 ) \notag
\\ \label{diffalpha2main} = & - 2 \frac{ \hat v_3}{\langle v \rangle} \mathfrak F_3(t, x_\parallel, 0,v ) + 2 \frac{\hat v_3}{\langle v \rangle} \mathfrak F_3(t,x, v ) \\ \notag & + 2 \left( \partial_t \mathfrak F_3(t,x_\parallel, 0 ,v) \right) \frac{x_3} {\langle v \rangle } + 2 \hat v_3 x_3
- ( 2 \hat v_\parallel \cdot \nabla_{x_\parallel} \mathfrak F_3(t, x_\parallel, 0,v ) ) \frac{x_3}{\langle v \rangle} \\ \notag & - 2 \frac{ (\hat v_3)^3}{\langle v \rangle } \mathfrak F_3(t,x, v ) -2 (\hat v_3)^2 \frac{ \hat v_\parallel}{\langle v \rangle} \cdot \mathfrak F_\parallel (t,x,v) - 2 \frac{x_3}{\langle v \rangle } \mathfrak F(t,x,v) \cdot \nabla_v \mathfrak F_3(t,x_\parallel, 0,v) \\ \label{diffalph2arest} & + 2 x_3 \frac{ \hat v }{\langle v \rangle^2} \cdot \mathfrak F(t,x,v) \mathfrak F_3(t,x_\parallel, 0, v ).
\end{align} Using the fundamental theorem of calculus \begin{equation} \notag \eqref{diffalpha2main} = 2 \frac{ \hat v_3}{\langle v \rangle} \left( \int_0^{x_3} \partial_{x_3} \mathfrak F_3 (t,x_\parallel, s, v ) ds \right). \end{equation} Since $\mathfrak F_3 = E_3 + E_e + \hat v _1 B_2 - \hat v_2 B_1 - g $, and since $\hat v \cdot \mathfrak F = \hat v \cdot ( E + E_{\text{ext}}+ \hat v \times (B_3 + B_{\text{ext}} ) - g \mathbf e_3 ) = \hat v \cdot E - ( g - E_e) \hat v_3 $, we have \begin{equation} \label{diffalpha3} \begin{split} [ \partial_t & + \hat v \cdot \nabla_x + \mathfrak F \cdot \nabla_v ] (\alpha^2 ) \\ = & 2 \frac{ \hat v_3}{\langle v \rangle} \left( \int_0^{x_3} \partial_{x_3} E_3 (t,x_\parallel, s ) + \hat v_1 \partial_{x_3} B_2( t, x_\parallel, s) - \hat v_2 \partial_{x_3} B_1 (t,x_\parallel, s ) ds \right) \\ & + 2 \frac{x_3}{\langle v \rangle} \left( ( \partial_{t} - \hat v_\parallel \cdot \nabla_{x_\parallel } ) \left( E_3 (t,x_\parallel, 0 ) + \hat v_1 B_2( t, x_\parallel, 0) - \hat v_2 B_1 (t,x_\parallel, 0) \right) \right) + 2 \hat v_3 x_3 \\ & - 2 \frac{ (\hat v_3 )^2}{\langle v \rangle} \left( \hat v \cdot E(t,x,v) - (g - E_e) \hat v_3 \right) \\ & - 2\frac{x_3}{\langle v \rangle} \left( E + E_{\text{ext}}+ (\hat v \times (B + B_{\text{ext}}) - g \mathbf e_3 ) \right) \cdot \nabla_v ( \hat v _1 B_2(t,x_\parallel,0) - \hat v_2 B_1(t,x_\parallel,0) ) \\ & + 2 x_3 \frac{ \hat v \cdot E - (g - E_e) \hat v_3 }{\langle v \rangle^2} \left( E_3(t,x_\parallel,0 ) + \hat v _1 B_2 (t,x_\parallel,0 ) - \hat v_2 B_1 (t,x_\parallel,0 ) - g \right) \\ & \underbrace{ - 2 \frac{x_3}{\langle v \rangle} ( \hat v \times B_{\text{ext}} ) \cdot \nabla_v ( \hat v _1 B_2(t,x_\parallel,0) - \hat v_2 B_1(t,x_\parallel,0) ) }_{\eqref{diffalpha3}_1}. \end{split} \end{equation} Now from the assumptions \eqref{vlfieldass} and \eqref{signcondition}, all the terms on the RHS of \eqref{diffalpha3} except $\eqref{diffalpha3}_1$ can be bounded by \begin{equation} \label{diffalpha31} \frac{C_1}{c_0} \left( c_0 \frac{ x_3 }{ \langle v \rangle} + (x_3)^2 + (\hat v_3)^2 \right),
\end{equation} where $C_1 = \sup_{ 0 \le t \le T} \left( \| E(t) \|_\infty + \| B(t) \|_\infty + \| E_3 (t) \|_{W^{1,\infty} } + \| B_1 (t) \|_{W^{1,\infty} } + \| B_2(t) \|_{W^{1,\infty} } \right) + E_e + g $. And from direct computation, \[ \begin{split} \eqref{diffalpha3}_1 = & - 2 B_e \frac{x_3}{ \langle v \rangle^2 } \begin{bmatrix} \hat v_2 \\ - \hat v_1 \\ 0 \end{bmatrix} \cdot \left( \begin{bmatrix} 1 - \hat v_1^2 \\ - \hat v_1 \hat v_2 \\ - \hat v_1 \hat v_3 \end{bmatrix} B_2 (t,x_\parallel, 0 ) - \begin{bmatrix} - \hat v_2 \hat v_1 \\ 1 - \hat v_2 ^2 \\ - \hat v_2 \hat v_3 \end{bmatrix} B_1(t,x_\parallel, 0 ) \right) \\ = & - 2 B_e \frac{x_3}{ \langle v \rangle^2 } \left( \hat v_2 B_2 (t,x_\parallel, 0 ) + \hat v_1 B_1(t,x_\parallel, 0 ) ,\right) \end{split} \] thus from \eqref{signcondition} \begin{equation} \label{diffalpha32}
| \eqref{diffalpha3}_1 | \le \frac{B_e}{c_0} c_0 \frac{x_3}{\langle v \rangle }. \end{equation} Combining \eqref{diffalpha31} and \eqref{diffalpha32}, we get
\begin{equation} \label{diffalpha21} \begin{split}
| [ \partial_t & + \hat v \cdot \nabla_x + \mathfrak F \cdot \nabla_v ] (\alpha^2 ) |
\le 10 (\frac{C_1 + B_e}{c_0} ) c_0 \frac{ x_3 }{ \langle v \rangle} + 8 C_1 \left( (x_3)^2 + (\hat v_3)^2 \right). \end{split} \end{equation}
From the expression of $\alpha$ in \eqref{alphadef} and the assumption \eqref{signcondition}, this yields \begin{equation} \label{tranalpha}
| [ \partial_t + \hat v \cdot \nabla_x + \mathfrak F \cdot \nabla_v ] (\alpha^2 ) | \le 20 (\frac{C_1 + B_e}{c_0} ) \alpha^2, \end{equation}
Thus along the characteristics, by the Gr\"owall's inequality we get
\begin{equation} e^{ - 20 (\frac{C_1 + B_e}{c_0} )|t-s| } \alpha^2(t,x,v) \le \alpha^2(s, X(s;t,x,v) , V(s;t,x,v) ) \le e^{20 (\frac{C_1 + B_e}{c_0} )|t-s| } \alpha^2(t,x,v) . \end{equation} Taking square root we get \eqref{alphaest}.
\end{proof}
\begin{lemma} \label{1alphaintv} Let $\alpha$ be defined as in \eqref{alphadef}. Then for any $(t,x) \in [0, T) \times \Omega$, we have \begin{equation} \label{alphavint}
\int_{\mathbb R^3} \frac{\mathbf 1_{|v| \le M } }{ \alpha(t, x, v ) } dv \le 4 M^3 \ln \left( 1 + \frac{1}{x_3} \right), \end{equation} and \begin{equation} \label{alphavdecayint}
\int_{\mathbb R^3 } \frac{1}{ 1 + |v|^{4 + \delta} } \frac{ 1 }{ \alpha(t, x, v )} dv \le C_\delta \ln \left( 1+ \frac{1}{x_3} \right). \end{equation} \end{lemma}
\begin{proof} From \eqref{alphadef} we have \[ \begin{split}
\int_{\mathbb R^3} \frac{\mathbf 1_{|v| \le M } }{ \alpha(t, x, v ) } dv = & \int_{ |v| \le M } \left( (x_3)^2+(\hat{v}_{3})^2 -2\left( E_3(t,x_\parallel, 0 ) + E_e+ (\hat v \times B)_3(t,x_\parallel, 0 ) - g \right) \frac{x_3}{\langle v \rangle} \right)^{-1/2} dv
\\ \le & \int_{|v| \le M } \frac{2}{ x_3 + \frac{| v_3 | }{\langle v \rangle } } dv
\\ \le & \int_{ |v_3| \le M } \frac{2 M^2 }{ x_3 + \frac{| v_3 | }{ M } } dv_3 \\ = & 4 M^3 \ln \left( x_3 + \frac{v_3}{M} \right) \big \rvert_0^M = 4M^3 \ln \left( 1 + \frac{1}{x_3} \right). \end{split} \] Now, for \eqref{alphavdecayint}, we have \begin{equation} \label{alphavdecayintest1} \begin{split}
\int_{\mathbb R^3 } \frac{1}{ 1 + |v|^{4 + \delta} } \frac{ 1 }{ \alpha(t, x, v )} dv \le & \int_{\mathbb R^3 } \frac{1}{ 1 + |v|^{4 + \delta} } \frac{2}{x_3 + \frac{| v_3 | }{\langle v \rangle } } dv
\\ \lesssim & \int_{\mathbb R^3 } \frac{1}{ \left( 1 + |v|^{3 + \delta} \right) \left(x_3 + |v_3| \right) } dv. \end{split} \end{equation} Using the spherical coordinate $v = (r, \theta, \phi )$ we have $dv = r^2 \sin \phi \, dr d\theta d\phi$, $ v_3 = r \cos \phi$, and \begin{equation} \label{alphavdecayintest2} \begin{split}
& \int_{\mathbb R^3 } \frac{1}{ \left( 1 + |v|^{3 + \delta} \right) \left(x_3 + |v_3| \right) } dv
\\ = & 4 \pi \int_0^\infty \int_0^{\pi /2 } \frac{r^2 \sin \phi }{ ( 1 + r^{3 + \delta } )( x_3 + r \cos \phi )} d \phi dr
\\ = & 4 \pi \int_0^1 \int_0^{\pi /2 } \frac{r^2 \sin \phi }{ ( 1 + r^{3 + \delta } )( x_3 + r \cos \phi )} d \phi dr + 4 \pi \int_1^\infty \int_0^{\pi /2 } \frac{r^2 \sin \phi }{ ( 1 + r^{3 + \delta } )( x_3 + r \cos \phi )} d \phi dr.
\end{split} \end{equation} Using change of variables $ r \cos \phi = u $, $ - r \sin \phi \, d \phi = du$, we have \begin{equation} \label{alphavdecayintest3} \begin{split} & \int_0^1 \int_0^{\pi /2 } \frac{r^2 \sin \phi }{ ( 1 + r^{3 + \delta } )( x_3 + r \cos \phi )} d \phi dr \\ \le & \int_0^1 \int_0^{\pi /2 } \frac{r^2 \sin \phi }{ x_3 + r \cos \phi } d \phi dr \\ = & \int_0^1\int_0^{r } \frac{r}{ x_3 + u} d u dr = \int_0^1 r \ln \left(1 + \frac{r}{x_3} \right) dr < \ln \left(1 + \frac{1}{x_3} \right). \end{split} \end{equation} And using change of variables $ \cos \phi = u$, $- \sin \phi \, d\phi = du$, we have \begin{equation} \label{alphavdecayintest4} \begin{split} & \int_1^\infty \int_0^{\pi /2 } \frac{r^2 \sin \phi }{ ( 1 + r^{3 + \delta } )( x_3 + r \cos \phi )} d \phi dr
\\ \le & C_\delta \int_0^{\pi /2 } \frac{\sin \phi }{x_3 + \cos \phi } d\phi
\\ = & C_\delta \int_0^1 \frac{1}{x_3 + u } du = C_\delta \ln \left( 1+ \frac{1}{x_3} \right). \end{split} \end{equation} Combining \eqref{alphavdecayintest1}, \eqref{alphavdecayintest2}, \eqref{alphavdecayintest3}, and \eqref{alphavdecayintest4} we conclude \eqref{alphavdecayint}.
\end{proof}
We have the following estimate on the backward exit time $t_{\mathbf{b}}$ for the trajectory.
\begin{lemma} Let $(t,x,v) \in (0, T) \times \Omega \times \mathbb R^3$, and the trajectory $X(s;t,x,v)$ and $V(s;t,x,v)$ satisfies \eqref{HamiltonODE}. Extending $E(t ) = E_0$, $B(t) = B_0$ for $t < 0 $. Suppose for all $t,x,v$, \begin{equation} \label{gbig10}
g - E_e - E_3(t,x_\parallel, 0 ) - (\hat v \times B)_3(t,x_\parallel, 0 ) > c_0,
\end{equation} then there exists a $C$ depending on $T$, $g$, $B_e$, $\| E \|_{W^{1,\infty}((0,T) \times \Omega )} $, $\| B \|_{W^{1,\infty}((0,T) \times \Omega )} $ such that \begin{equation} \label{tbbdvb}
\frac{ t_{\mathbf{b}}(t,x,v) }{ \sup_{t-t_{\mathbf{b}} < s < t } \sqrt{ 1 + |V(s)|^2 }} \le \frac{C}{c_0} {\hat v_{\mathbf{b}}}_{,3}. \end{equation} If $t -t_{\mathbf{b}}(t,x,v) \le 0$, then \begin{equation} \label{tbdV0}
\frac{ t }{ \sup_{0 \le s \le t } \sqrt{ 1 + |V(s)|^2 }} \le \frac{C}{c_0} \alpha(0,X(0),V(0)). \end{equation}
\end{lemma} \begin{proof} We first prove \eqref{tbbdvb}. For any $(t,x,v) \in (0, T) \times \Omega \times \mathbb R^3$ with $t -t_{\mathbf{b}}(t,x,v) > 0$, we have \[
{v_{\mathbf{b}}}_{,3} - v_3 = \int_{t-t_{\mathbf{b}}}^t - \mathfrak F_3(s, X(s;t,x,v), V(s;t,x,v) ) ds. \] From \eqref{gbig10} this implies \begin{equation} \label{tbvbbd1}
\int_{t-t_{\mathbf{b}}}^t c_0 ds < \int_{t-t_{\mathbf{b}}}^t - \mathfrak F_3(s, X(s;t,x,v), V(s;t,x,v) ) ds \le | {v_{\mathbf{b}}}_{,3}| + | v_3 | \end{equation} On the other hand, from \eqref{alphaest}, \begin{equation} \label{tbvbbd2}
| {v_{\mathbf{b}}}_{,3}| + | v_3 | = \langle v_{\mathbf{b}} \rangle | \hat{{v_{\mathbf{b}}}}_{,3}| + \langle v \rangle | \hat v_3 | \le \sup_{t-t_{\mathbf{b}} < s < t } \langle V(s) \rangle \left( \hat{{v_{\mathbf{b}}}}_{,3} + \alpha (t,x,v) \right) < C \sup_{t-t_{\mathbf{b}} < s < t } \langle V(s) \rangle \hat{{v_{\mathbf{b}}}}_{,3}. \end{equation} Combining \eqref{tbvbbd1} and \eqref{tbvbbd2} we get \[ t_{\mathbf{b}} c_0 < C \sup_{t-t_{\mathbf{b}} < s < t } \langle V(s) \rangle \hat{{v_{\mathbf{b}}}}_{,3} . \] This implies \eqref{tbbdvb}.
For $t-t_{\mathbf{b}}(t,x,v) \le 0$, using the same argument we have, \[ \begin{split}
c_0 t < \int_0^t - \mathfrak F_3(s,X(s) ,V(s) ) ds \le & |V_3(0) | + |v_3|
\end{split} \] and from \eqref{alphaest}, \[
|V_3(0) | + |v_3| \le \sup_{t-t_{\mathbf{b}} < s < t } \langle V(s) \rangle \left( \alpha(0,X(0), V(0) ) + \alpha (t,x,v) \right) < C \sup_{t-t_{\mathbf{b}} < s < t } \langle V(s) \rangle \alpha(0,X(0), V(0) ), \] thus we get \[ c_0 t < C \sup_{t-t_{\mathbf{b}} < s < t } \langle V(s) \rangle \alpha(0,X(0), V(0) ), \] and this yields \eqref{tbdV0}.
\end{proof}
\begin{lemma} Suppose \[
\sup_{0 \le t \le T} \| \nabla_x E(t) \|_{\infty } + \sup_{0 \le t \le T} \| \nabla_x B(t) \|_{\infty } < \infty. \] Then for any $s, t \in (0,T)$, we have \begin{equation} \label{pxviXVest} \begin{split}
& | \partial_{x_i} X(s;t,x,v) | \lesssim e^{C_1 |t-s| } ,
\\ & | \partial_{x_i} V (s;t,x,v)| \lesssim (t-s) e^{C_1 |t-s| } ,
\\ & | \partial_{v_i} X(s;t,x,v) | \lesssim \frac{|t-s|}{\langle V(s) \rangle } e^{C_1 |t-s| } ,
\\ & | \partial_{v_i} V (s;t,x,v)| \lesssim e^{C_1 |t-s| }, \end{split} \end{equation} and for $i \neq j $, \begin{equation} \label{pxiXj} \begin{split}
| \partial_{x_i} X_j(s;t,x,v) | \lesssim e^{C_1 |t-s| } \frac{ |t-s |^2}{\langle V(s) \rangle } . \end{split}
\end{equation} where $C_1 = \left( \sup_{0 \le t \le T} \left( \| \nabla_x E(t) \|_\infty + \| \nabla_x B(t) \|_\infty + \| E(t) \|_\infty + \| B(t) \|_\infty \right) + g +|B_e|\right)^2$. \end{lemma} \begin{proof} The expressions of $X(s;t,x,v)$ and $V(s;t,x,v) $ are \begin{equation} \begin{split} X(s;t,x,v) = & x-(t-s) \hat{v} + \int^t_s\int^t_\tau \hat{\mathfrak F } _{}(\tau^\prime, X (\tau^\prime), V(\tau^\prime) ) \mathrm{d} \tau^\prime \mathrm{d} \tau, \\ V(s;t,x,v) = & v - \int_s^t \mathfrak F (\tau, X(\tau) , V(\tau) ) \mathrm{d} \tau. \end{split} \end{equation}
We denote \begin{equation} \label{hatF} \begin{split}
\frac{d}{ds} \hat{V}(s)&= \frac{1}{\sqrt{1+ |V(s)|^2}} \frac{d}{ds} V(s)
- \frac{1}{\sqrt{1+ |V(s)|^2}}\hat{V}(s) \cdot \frac{d}{ds}V(s) \hat{V}(s) \\
&=\frac{1}{\sqrt{1+ |V(s)|^2}} \mathfrak F (s,X(s),V(s))
- \frac{1}{\sqrt{1+ |V(s)|^2}}\hat{V}(s) \cdot \mathfrak F (s,X(s),V(s)) \hat{V}(s)
\\ &:= \hat{\mathfrak F }_{} (s,X(s),V(s)). \end{split}\end{equation}
By direct computation we get \begin{equation} \label{pxipviXV} \begin{split} \partial_{x_i} X(s;t,x,v) = & e_i + \int^t_s\int^t_\tau [ \nabla_x \hat{\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{x_i} X(\tau' ) + \nabla_v \hat{\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{x_i} V(\tau' ) ] \mathrm{d} \tau^\prime \mathrm{d} \tau, \\ \partial_{v_i} X(s;t,x,v) = & -(t-s) \partial_{v_i} \hat v + \int^t_s\int^t_\tau [ \nabla_x \hat{\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{v_i} X(\tau' ) + \nabla_v \hat{\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{v_i} V(\tau' ) ] \mathrm{d} \tau^\prime \mathrm{d} \tau, \\ \partial_{x_i} V(s;t,x,v) = & - \int_s^t [ \nabla_x {\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{x_i} X(\tau ) + \nabla_v {\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{x_i} V(\tau ) ] \mathrm{d} \tau \\ \partial_{v_i} V(s;t,x,v) = & e_i - \int_s^t [ \nabla_x {\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{v_i} X(\tau ) + \nabla_v {\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{v_i} V(\tau ) ] \mathrm{d} \tau. \end{split} \end{equation} Since $\mathfrak F = E + E_{\text{ext}} + \hat v \times ( B +B_{\text{ext}}) - g \mathbf e_3$, we have \begin{equation} \label{nablaxfrakF}
| \nabla_x \mathfrak F | \le | \nabla_{x} E | + | \nabla_{x} B | . \end{equation} And since $\partial_{v_i} \hat v = \frac{ e_i}{\langle v \rangle } - \frac{\hat v_i \hat v }{\langle v \rangle} $, \begin{equation} \label{nablavfrakF}
| \nabla_v \mathfrak F | \lesssim \frac{1}{ \sqrt{ 1 + |v|^2 } } (| B | + |B_e|) . \end{equation} From the expression of $\hat{\mathfrak F}$ in \eqref{hatF}, we have from \eqref{nablaxfrakF}, \begin{equation} \label{nablaxhatF}
| \nabla_x \hat{\mathfrak F} | \le \frac{2}{ \sqrt{ 1 + |v|^2 }} | \nabla_x \mathfrak F | \lesssim \frac{1}{ \sqrt{ 1 + |v|^2 }} \left( | \nabla_{x} E | + | \nabla_{x} B | \right), \end{equation} and from \eqref{nablavfrakF}, \begin{equation} \label{nablavhatF} \begin{split}
| \nabla_v \hat{\mathfrak F} | \le & \left| \nabla_v( \frac{1}{ \sqrt{1 +|v|^2}} )(\mathfrak F - \hat v \cdot \mathfrak F \cdot \hat v ) \right| + \left| \frac{1}{ \sqrt{1 +|v|^2}} \left( \nabla_v \mathfrak F - \nabla_v ( \hat v \cdot \mathfrak F \cdot \hat v ) \right) \right|
\\ \lesssim & \frac{1}{ 1 +|v|^2} | \mathfrak F | + \frac{1}{ \sqrt{ 1 +|v|^2}}| \nabla_v \mathfrak F |
\\ \lesssim & \frac{1}{ 1 +|v|^2} \left( | E | + |B| + g + |B_e| \right) \end{split} \end{equation} From \eqref{pxipviXV} and Fubini's theorem, \[ \begin{split} \partial_{x_i} X(s) = & e_i + \int_s^t \int_s^{\tau'} [ \nabla_x \hat{\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{x_i} X(\tau' ) + \nabla_v \hat{\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{x_i} V(\tau' ) ] \mathrm{d} \tau \mathrm{d} \tau' \\ = & e_i + \int_s^t (\tau' -s ) [ \nabla_x \hat{\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{x_i} X(\tau' ) + \nabla_v \hat{\mathfrak F } _{}(\tau^\prime ) \cdot \partial_{x_i} V(\tau' ) ] \mathrm{d} \tau'. \end{split} \] Therefore, from \eqref{nablaxhatF} and \eqref{nablavhatF} we have \begin{equation} \label{pxiXsest1} \begin{split}
| \partial_{x_i} X(s) | \le & 1 + (t-s) \int_s^t \left( | \nabla_x \hat{\mathfrak F } (\tau ) | | \partial_{x_i} X(\tau ) | + | \nabla_v \hat{\mathfrak F } (\tau ) | | \partial_{x_i} V(\tau )| \right) \mathrm{d} \tau
\\ \lesssim & 1 + (t-s) \left( \| \nabla_x E \|_\infty + \| \nabla_x B \|_\infty \right) \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } | \partial_{x_i} X(\tau ) | d\tau
\\ & \quad + (t-s) \left( \| E \|_\infty + \| B \|_\infty + g +|B_e| \right) \int_s^t \frac{1}{ 1 +|V(\tau) |^2 } | \partial_{x_i} V(\tau ) | d\tau. \end{split} \end{equation} Thus \begin{equation} \label{pxiXsest2} \begin{split}
& \langle V(s) \rangle | \partial_{x_i} X(s) |
\\ \lesssim & \langle V(s) \rangle + (t-s) \left( \| \nabla_x E \|_\infty + \| \nabla_x B \|_\infty \right) \frac{ \sup_{0 \le s \le t } \langle V(s) \rangle}{ \inf_{0 \le s \le t } \langle V(s) \rangle } \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } \langle V(\tau) \rangle | \partial_{x_i} X(\tau ) | d\tau
\\ & \quad + (t-s) \left( \| E \|_\infty + \| B \|_\infty + g +|B_e| \right) \frac{ \sup_{0 \le s \le t } \langle V(s) \rangle}{ \inf_{0 \le s \le t } \langle V(s) \rangle } \int_s^t \frac{1}{ \sqrt{ 1 +|V(\tau) |^2 }} | \partial_{x_i} V(\tau ) | d\tau
\\ \lesssim & \langle V(s) \rangle + C_1 \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } \left( \langle V(\tau) \rangle | \partial_{x_i} X(\tau ) | + | \partial_{x_i} V(\tau ) | \right) d\tau, \end{split}
\end{equation} where $C_1 = \left( \sup_{0 \le t \le T} \left( \| \nabla_x E(t) \|_\infty + \| \nabla_x B(t) \|_\infty + \| E(t) \|_\infty + \| B(t) \|_\infty \right) + g + | B_e| \right)^2 $. From \eqref{pxipviXV}, \eqref{nablaxfrakF}, and \eqref{nablavfrakF}, \begin{equation} \label{pxiVsest2} \begin{split}
| \partial_{x_i} V(s) | \lesssim & \left( \| \nabla_x E \|_\infty + \| \nabla_x B \|_\infty \right) \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } \langle V(\tau) \rangle | \partial_{x_i} X(\tau ) | d \tau
\\ & + ( \| B \|_\infty + |B_e| ) \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } | \partial_{x_i} V(\tau ) | \mathrm{d} \tau
\\ \lesssim & C_1 \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } \left( \langle V(\tau) \rangle | \partial_{x_i} X(\tau ) | + | \partial_{x_i} V(\tau ) | \right) d\tau. \end{split} \end{equation} Combine \eqref{pxiXsest2} and \eqref{pxiVsest2} we have \begin{equation} \label{pxiXspxiVsest}
\langle V(s) \rangle | \partial_{x_i} X(s) | + | \partial_{x_i} V(s) | \lesssim \langle V(s) \rangle + C_1 \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } \left( \langle V(\tau) \rangle | \partial_{x_i} X(\tau ) | + | \partial_{x_i} V(\tau ) | \right) d\tau. \end{equation} So from Gronwall's inequality, \begin{equation} \label{pxiXspxiVsest2}
\langle V(s) \rangle | \partial_{x_i} X(s) | + | \partial_{x_i} V(s) | \lesssim \langle V(s) \rangle e^{C_1 \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } d\tau } \lesssim e^{C_1|t-s| } \langle V(s) \rangle, \end{equation} Next, using the same argument as \eqref{nablaxfrakF}--\eqref{pxiXspxiVsest}, from \eqref{pxipviXV} we get \[ \begin{split}
\langle V(s) \rangle | \partial_{v_i} X(s) | + | \partial_{v_i} V(s) | \lesssim 1 + C_1 \int_s^t \frac{1}{\sqrt{ 1 +|V(\tau) |^2 } } \left( \langle V(\tau) \rangle | \partial_{x_i} X(\tau ) | + | \partial_{x_i} V(\tau ) | \right) d\tau. \end{split}
\] Again by Gronwall's inequality, \begin{equation}
\langle V(s) \rangle | \partial_{v_i} X(s) | + | \partial_{v_i} V(s) | \lesssim e^{C_1 |t-s| } . \end{equation} Thus, \begin{equation} \label{pviXspviVsest2}
| \partial_{v_i} X(s;t,x,v) | \lesssim \frac{e^{C_1 |t-s|} }{ \langle V(s) \rangle } , \, | \partial_{v_i} V (s;t,x,v)| \lesssim e^{C_1 |t-s|} . \end{equation} Now plug \eqref{pxiXspxiVsest2}, \eqref{pviXspviVsest2} back to \eqref{pxipviXV} and using \eqref{nablaxhatF}, \eqref{nablavhatF}, and \eqref{pviXspviVsest2}, we have \begin{equation} \label{pviXspviVsest3} \begin{split}
| \partial_{v_i} X (s;t,x,v)| & \lesssim \frac{ |t-s|}{\langle v \rangle } + e^{C_1 |t-s| } \int_s^t \int_\tau^t \frac{1}{ \langle V(\tau' ) \rangle^2 } d\tau' d\tau
\\ & \lesssim \frac{|t-s|}{\langle V(s) \rangle } e^{C_1 |t-s| }, \end{split} \end{equation} and \begin{equation} \begin{split}
| \partial_{x_i} V(s;t,x,v) | & \lesssim e^{C_1 |t-s| } \int_s^t \frac{1}{ \langle V(\tau) \rangle } \langle V(\tau) \rangle \mathrm{d} \tau \lesssim |t-s| e^{C_1 |t-s| } \end{split} \end{equation} From \eqref{pxiXspxiVsest2}, \eqref{pviXspviVsest2}, and \eqref{pviXspviVsest3} we conclude \eqref{pxviXVest}. Finally, for $i \neq j$, from \eqref{pxipviXV}, \eqref{nablaxhatF}, and \eqref{nablavhatF}, \[
| \partial_{x_i } X_j (s;t,x,v) | \lesssim e^{C_1 |t-s| } \int_s^t \int_\tau^t \frac{1}{ \langle V(\tau' ) \rangle } d\tau' d\tau \lesssim e^{C_1 |t-s| } \frac{ |t-s |^2}{\langle V(s) \rangle }. \]
\end{proof}
\section{$W^{1,\infty}$ estimate of inflow problem }
In this section, we prove an a priori estimate for the inflow problem \eqref{VMfrakF1}, \eqref{inflow}.
From \eqref{HamiltonODE}, we have \begin{equation} \label{Frep} f(t,x,v) = \mathbf 1_{t_{\mathbf{b}} \ge t } f( 0, X(0), V(0)) + \mathbf 1_{t_{\mathbf{b}} < t } g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) . \end{equation}
From \eqref{HamiltonODE}, we have \begin{equation} \label{Xiformula} \begin{split} X_{i}(s;t,x,v) &= x_i - \int^t_s \hat{V}_i(\tau;t,x,v) \mathrm{d} \tau \\ &=x_i - \int^t_s \left\{ \hat{v}_i - \int^t_\tau \hat{\mathfrak F }_i (\tau^\prime) \mathrm{d} \tau^\prime\right\} \mathrm{d} \tau\\ &= x_i-(t-s) \hat{v}_i + \int^t_s\int^t_\tau \hat{\mathfrak F } _{i}(\tau^\prime, X (\tau^\prime), V(\tau^\prime) ) \mathrm{d} \tau^\prime \mathrm{d} \tau. \end{split} \end{equation} Set $s=t-t_{\mathbf{b}}$ so that $X_3(t-t_{\mathbf{b}};t,x,v)=0$. Then
\begin{equation} \begin{split} t_{\mathbf{b}} \hat{v}_3&=x_3+ \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau \hat{\mathfrak F } _{3}(\tau^\prime) \mathrm{d} \tau^\prime \mathrm{d} \tau\\
\end{split} \end{equation} By taking derivatives we obtain \begin{equation} \begin{split} \partial_x x_3+ \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau\partial_x [\hat{\mathfrak F } _{3}(\tau^\prime, X(\tau^\prime), V(\tau^\prime))] \mathrm{d} \tau^\prime \mathrm{d} \tau =& \Big( \hat{v}_3 - \int^{t}_{t-t_{\mathbf{b}}} \hat{\mathfrak F }_{3} (\tau^\prime) \mathrm{d} \tau^\prime \Big) \partial_x t_{\mathbf{b}} \\ =& \ { \hat v_{\mathbf{b}}}_{,3} \partial_x t_{\mathbf{b}}, \end{split} \end{equation} and hence \begin{equation} \label{pxitb} \begin{split} \partial_{x_i} t_{\mathbf{b}} = & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ \partial_{x_i} x_3+ \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau \partial_{x_i} [\hat{\mathfrak F } _{3}(\tau^\prime, X(\tau^\prime), V(\tau^\prime))] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} \\ = & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ \partial_{x_i} x_3+ \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} . \end{split} \end{equation}
Similarly, \begin{equation} \begin{split} \partial_{v_i} x_3 - t_{\mathbf{b}} \partial_{v_i} \hat v_3 + \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau\partial_{v_i} [\hat{\mathfrak F } _{3}(\tau^\prime, X(\tau^\prime), V(\tau^\prime))] \mathrm{d} \tau^\prime \mathrm{d} \tau =& \Big( \hat{v}_3 - \int^{t}_{t-t_{\mathbf{b}}} \hat{\mathfrak F }_{3} (\tau^\prime) \mathrm{d} \tau^\prime \Big) \partial_{v_i} t_{\mathbf{b}} \\ =& \ { \hat v_{\mathbf{b}}}_{,3} \partial_{v_i} t_{\mathbf{b}}, \end{split} \end{equation} Thus \begin{equation} \label{pvitb}
\begin{split} \partial_{v_i} t_{\mathbf{b}} = & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ - t_{\mathbf{b}} \partial_{v_i} \hat v_3 + \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau \partial_{v_i} [\hat{\mathfrak F } _{3}(\tau^\prime, X(\tau^\prime), V(\tau^\prime))] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} \\ = & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ - t_{\mathbf{b}} \partial_{v_i} \hat v_3 +\int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} . \end{split} \end{equation}
And we have \begin{equation} \label{pxbvb} \begin{split} \partial_{x_i} x_{\mathbf{b}} = & e_i - (\partial_{x_i} t_{\mathbf{b}} ) \hat v_{\mathbf{b}} +\int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \\ \partial_{x_i} v_{\mathbf{b}} = & - (\partial_{x_i } t_{\mathbf{b}} ) \mathfrak F(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) - \int_{t-t_{\mathbf{b}}}^t [ \nabla_x {\mathfrak F } _{3}(\tau) \cdot \partial_{x_i} X(\tau) + \nabla_v {\mathfrak F } _{3}(\tau ) \cdot \partial_{x_i} V(\tau) ] \mathrm{d} \tau \\ \partial_{v_i} x_{\mathbf{b}} = & - (\partial_{v_i} \hat v ) t_{\mathbf{b}} - ( \partial_{v_i} t_{\mathbf{b}} ) \hat v_{\mathbf{b}} +\int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \\ \partial_{v_i} v_{\mathbf{b}} = & e_i - (\partial_{v_i } t_{\mathbf{b}} ) \mathfrak F(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) - \int_{t-t_{\mathbf{b}}}^t [ \nabla_x {\mathfrak F } _{3}(\tau) \cdot \partial_{v_i} X(\tau) + \nabla_v {\mathfrak F } _{3}(\tau ) \cdot \partial_{v_i} V(\tau) ] \mathrm{d} \tau. \end{split} \end{equation}
We have the following calculations for the derivatives of $f$ in \eqref{Frep}. \begin{equation} \label{pxiF} \begin{split}
& \partial_{x_i} f(t,x,v) \\ = & \mathbf 1_{t_{\mathbf{b}} > t } \{ \nabla_x f_0( X(0), V(0)) \cdot \partial_{x_i} X(0) + \nabla_v f_0(X(0), V(0)) \cdot \partial_{x_i} V(0) \} \\ & + \mathbf 1_{t_{\mathbf{b}} < t } \{- \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \partial_{x_i} t_{\mathbf{b}} +\nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \partial_{x_i} x_{\mathbf{b}} +\nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \partial_{x_i} v_{\mathbf{b}} \} \\ = & \mathbf 1_{t_{\mathbf{b}} > t } \{ \nabla_x f_0( X(0), V(0)) \cdot \partial_{x_i} X(0) + \nabla_v f_0(X(0), V(0)) \cdot \partial_{x_i} V(0) \} \\ & + \mathbf 1_{t_{\mathbf{b}} < t } \bigg( - \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ \partial_{x_i} x_3+ \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} \\ & \quad + \nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \cdot \left\{ e_i - (\partial_{x_i} t_{\mathbf{b}} ) \hat v_{\mathbf{b}} +\int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} \\ & \quad + \nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \cdot \left\{ - (\partial_{x_i } t_{\mathbf{b}} ) \mathfrak F(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) - \int_{t-t_{\mathbf{b}}}^t [ \nabla_x {\mathfrak F } _{3}(\tau) \cdot \partial_{v_i} X(\tau) + \nabla_v {\mathfrak F } _{3}(\tau ) \cdot \partial_{v_i} V(\tau) ] \mathrm{d} \tau \right\} \bigg) , \end{split} \end{equation} and \begin{equation} \label{pviF} \begin{split}
& \partial_{v_i} f(t,x,v) \\ = & \mathbf 1_{t_{\mathbf{b}} > t } \{ \nabla_x f_0( X(0), V(0)) \cdot \partial_{v_i} X(0) + \nabla_v f_0(X(0), V(0)) \cdot \partial_{v_i} V(0) \} \\ & + \mathbf 1_{t_{\mathbf{b}} < t } \{- \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \partial_{v_i} t_{\mathbf{b}} +\nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \partial_{v_i} x_{\mathbf{b}} +\nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \partial_{v_i} v_{\mathbf{b}} \} \\ = & \mathbf 1_{t_{\mathbf{b}} > t } \{ \nabla_x f_0( X(0), V(0)) \cdot \partial_{v_i} X(0) + \nabla_v f_0(X(0), V(0)) \cdot \partial_{v_i} V(0) \} \\ & + \mathbf 1_{t_{\mathbf{b}} < t } \bigg( - \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ - t_{\mathbf{b}} \partial_{v_i} \hat v_3 +\int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} \\ & \quad + \nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \cdot \left\{ - (\partial_{v_i} \hat v ) t_{\mathbf{b}} - ( \partial_{v_i} t_{\mathbf{b}} ) \hat v_{\mathbf{b}} + +\int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} \\ & \quad + \nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) \cdot \left\{ e_i - (\partial_{v_i } t_{\mathbf{b}} ) \mathfrak f(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) - \int_{t-t_{\mathbf{b}}}^t [ \nabla_x {\mathfrak F } _{3}(\tau) \cdot \partial_{v_i} X(\tau) + \nabla_v {\mathfrak F } _{3}(\tau ) \cdot \partial_{v_i} V(\tau) ] \mathrm{d} \tau \right\} \bigg) \end{split} \end{equation}
\begin{proposition} \label{inflowprop} Let $(f,E,B)$ be a solution of \eqref{VMfrakF1}, \eqref{inflow}, \eqref{Maxwell}. Suppose the fields satisfies \eqref{gbig10}, and \[
\sup_{0 \le t \le T} \left(\| \nabla_{x} E(t) \|_\infty + \| \nabla_{x} B(t) \|_\infty \right) < \infty. \] And assume that for $\delta > 0$, \[ \begin{split}
\| \langle v \rangle^{5 + \delta } \nabla_{x_\parallel} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha \partial_{x_3} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta } \nabla_v f_0 \|_\infty & < \infty,
\\ \| \langle v \rangle^{5 + \delta } \partial_t g \|_\infty + \| \langle v \rangle^{5 + \delta } \nabla_{x_\parallel} g \|_\infty + \| \langle v \rangle^{5 + \delta } \nabla_{v} g \|_\infty & < \infty. \end{split} \] then for $0 < T \ll 1$ small enough, we have \begin{equation} \label{alphavnablafbd} \begin{split}
& \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha \partial_{x_3} f(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \nabla_{v} f(t) \|_\infty \right)
< \infty. \end{split} \end{equation}
\end{proposition} \begin{proof} For notational simplicity, we assume that the lower order terms of $E$ and $B$ are smaller than the higher order terms: \begin{equation} \begin{split}
& \sup_{0 \le t \le T} \| E(t) \|_\infty + g + |B_e| \lesssim \sup_{0 \le t \le T } \| \nabla_{x} E(t) \|_\infty ,
\\ & \sup_{0 \le t \le T} \| B(t) \|_\infty + g + |B_e| \lesssim \sup_{0 \le t \le T } \| \nabla_{x} B(t) \|_\infty . \end{split} \end{equation}
From \eqref{pxiF}, \eqref{pviF}, we have \begin{equation} \label{nablaxfest} \begin{split}
& | \partial_{x_i} f(t,x,v) |
\\ \le & \mathbf 1_{t_{\mathbf{b}} > t } \{ | \nabla_x f_0( X(0), V(0)) | | \partial_{x_i} X(0) | + | \nabla_v f_0(X(0), V(0))| | \partial_{x_i} V(0) | \}
\\ & + \mathbf 1_{t_{\mathbf{b}} < t } \bigg( | \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \left| \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ \partial_{x_i} x_3+ \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} \right|
\\ & \quad + | \nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \left| e_i - (\partial_{x_i} t_{\mathbf{b}} ) \hat v_{\mathbf{b}} +\int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right|
\\ & \quad + | \nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} )| \left| - (\partial_{x_i } t_{\mathbf{b}} ) \mathfrak F(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) - \int_{t-t_{\mathbf{b}}}^t [ \nabla_x {\mathfrak F } _{3}(\tau) \cdot \partial_{x_i} X(\tau) + \nabla_v {\mathfrak F } _{3}(\tau ) \cdot \partial_{x_i} V(\tau) ] \mathrm{d} \tau \right| \bigg). \end{split} \end{equation} \begin{equation} \label{nablavfest} \begin{split}
& | \partial_{v_i} f(t,x,v) |
\\ \le & \mathbf 1_{t_{\mathbf{b}} > t } \{ | \nabla_x f_0( X(0), V(0)) | | \partial_{v_i} X(0) | + | \nabla_v f_0(X(0), V(0))| | \partial_{v_i} V(0) | \}
\\ & + \mathbf 1_{t_{\mathbf{b}} < t } \bigg( | \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \left| \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ - t_{\mathbf{b}} \partial_{v_i} \hat v_3 + \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right\} \right|
\\ & \quad + | \nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \left| -(\partial_{v_i} \hat v ) t_{\mathbf{b}} - (\partial_{v_i} t_{\mathbf{b}} ) \hat v_{\mathbf{b}} +\int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau [ \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} V(\tau') ] \mathrm{d} \tau^\prime \mathrm{d} \tau \right|
\\ & \quad + | \nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} )| \left| e_i - (\partial_{v_i } t_{\mathbf{b}} ) \mathfrak F(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) - \int_{t-t_{\mathbf{b}}}^t [ \nabla_x {\mathfrak F } _{3}(\tau) \cdot \partial_{v_i} X(\tau) + \nabla_v {\mathfrak F } _{3}(\tau ) \cdot \partial_{v_i} V(\tau) ] \mathrm{d} \tau \right| \bigg). \end{split} \end{equation}
From \eqref{nablaxhatF}, \eqref{nablavhatF}, \eqref{pxviXVest}, \eqref{pxitb}, and \eqref{pvitb}, we have \begin{equation} \label{pxitbest} \begin{split}
| \partial_{x_i} t_{\mathbf{b}} | \le & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ | \partial_{x_i} x_3 | + \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau | \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{x_i} V(\tau') | \mathrm{d} \tau^\prime \mathrm{d} \tau \right\}
\\ \le & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ | \partial_{x_i} x_3 | + \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau | \nabla_x \hat{\mathfrak F } _{3}(\tau' )| | \partial_{x_i} X(\tau') | + | \nabla_v \hat{\mathfrak F } _{3}(\tau' ) | | \partial_{x_i} V(\tau') | \mathrm{d} \tau^\prime \mathrm{d} \tau \right\}
\\ \lesssim & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left( | \partial_{x_i} x_3 | + \frac{ C t_{\mathbf{b}}^2}{\langle v \rangle} ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) \right), \end{split} \end{equation} and \begin{equation} \label{pvitbest} \begin{split}
| \partial_{v_i} t_{\mathbf{b}} | \le & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ | t_{\mathbf{b}} \partial_{v_i} \hat v_3 | + \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau | \nabla_x \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} X(\tau') + \nabla_v \hat{\mathfrak F } _{3}(\tau' ) \cdot \partial_{v_i} V(\tau') | \mathrm{d} \tau^\prime \mathrm{d} \tau \right\}
\\ \le & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left\{ | t_{\mathbf{b}} \partial_{v_i} \hat v_3 | + \int^{t}_{t-t_{\mathbf{b}}} \int^t_\tau | \nabla_x \hat{\mathfrak F } _{3}(\tau' )| | \partial_{v_i} X(\tau') | + | \nabla_v \hat{\mathfrak F } _{3}(\tau' ) | | \partial_{v_i} V(\tau') | \mathrm{d} \tau^\prime \mathrm{d} \tau \right\}
\\ \lesssim & \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } \left( \frac{t_{\mathbf{b}}}{\langle v \rangle } + \frac{ C t_{\mathbf{b}}^2}{\langle v \rangle^2} ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) \right). \end{split} \end{equation}
Thus from \eqref{nablaxfest} and \eqref{pxitbest}, for $i = 1,2$, \begin{align}
& \notag | \langle v \rangle^{4+\delta} \partial_{x_i} f(t,x,v) |
\\ \label{vnablaparaf0} & \le \langle v \rangle^{4+\delta} \big( | \nabla_{x_\parallel} f_0( X(0), V(0)) | | \partial_{x_i} X_\parallel (0) |+ | \partial_{x_3} f_0( X(0), V(0)) | | \partial_{x_i} X_3 (0) |
\\ \notag &\quad \quad \quad \quad \quad \quad \quad + | \nabla_v f_0(X(0), V(0))| | \nabla_x V(0) | \big)
\\ \label{vnablaparaptg} & + \langle v \rangle^{4+\delta} | \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \frac{C t_{\mathbf{b}}^2}{{ \hat v_{\mathbf{b}}}_{,3} \langle v \rangle } ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } )
\\ \label{vnablaparaxg} & + \langle v \rangle^{4+\delta} | \nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \left( 1 + \left( \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } + 1 \right) \frac{ C t_{\mathbf{b}}^2}{\langle v \rangle} ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) \right)
\\ \notag & + \langle v \rangle^{4+\delta} | \nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} )| C \left( \frac{t_{\mathbf{b}}^2}{{ \hat v_{\mathbf{b}}}_{,3} \langle v \rangle } | \mathfrak F(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) | + t_{\mathbf{b}} \right) ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) . \end{align} From \eqref{pxviXVest}, \eqref{pxipviXV}, and \eqref{tbdV0}, we have \begin{equation} \label{nablaparaf0est} \begin{split}
| \eqref{vnablaparaf0} | \le & C \langle v \rangle^{4+\delta} \left( | \nabla_{x_\parallel} f_0( X(0), V(0)) | + t | \partial_{x_3} f_0( X(0), V(0)) | + (1+|v|) | \nabla_v f_0(X(0), V(0))| \right)
\\ \le & ( C^2 +1 ) \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{5+\delta} \nabla_v f_0 \|_{L^\infty_{x} } \right). \end{split} \end{equation} And \begin{equation} \label{nablaparagest} \begin{split}
& |\eqref{vnablaparaptg}| +| \eqref{vnablaparaxg} | + | \eqref{vnablaparaxg} |
\\ \le & ( C + t_{\mathbf{b}} C ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) )
\\ & \times \left( \| \langle v \rangle^{4+\delta} \partial_t g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{4+\delta} \nabla_x g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{4+\delta} \nabla_v g \|_{L^\infty_{t,x} } \right) . \end{split} \end{equation}
And from \eqref{nablaxfest} and \eqref{pxitbest}, for $i=3$, \begin{align}
& \notag | \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f(t,x,v) |
\\ \label{alphavnablaf0} & \le \langle v \rangle^{5+\delta} \alpha(t,x,v) \big( | \nabla_{x_\parallel} f_0( X(0), V(0)) | | \partial_{x_3} X_\parallel (0) |+ | \partial_{x_3} f_0( X(0), V(0)) | | \partial_{x_3} X_3 (0) |
\\ \notag &\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + | \nabla_v f_0(X(0), V(0))| | \nabla_x V(0) | \big)
\\ \label{alphavptg} & + \langle v \rangle^{5+\delta} | \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \frac{\alpha(t,x,v)}{{ \hat v_{\mathbf{b}}}_{,3} } ( 1 + \frac{ C t_{\mathbf{b}}^2}{\langle v \rangle } ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} }) )
\\ \label{alphavnablaxg} & + \langle v \rangle^{5+\delta} | \nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \frac{\alpha(t,x,v)}{{ \hat v_{\mathbf{b}}}_{,3} } ( 1 + \frac{ C t_{\mathbf{b}}^2}{\langle v \rangle } ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) )
\\ \label{alphavnablavg} & + \langle v \rangle^{5+\delta} | \nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} )| \frac{\alpha(t,x,v)}{{ \hat v_{\mathbf{b}}}_{,3} } | \mathfrak F(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) | ( 1 + t_{\mathbf{b}} C ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) ) . \end{align} Now from \eqref{pxviXVest} and the velocity lemma \eqref{alphaest}, \begin{equation} \label{alphavnablaf0est} \begin{split}
|\eqref{alphavnablaf0} | \le & C \frac{ \langle v \rangle^{5+\delta} \alpha(t,x,v) }{ ( 1 +|V(0)| )^{5 + \delta } \alpha(0,X(0),V(0))} \big( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{5+\delta} \nabla_{x_\parallel} f_0 \|_{L^\infty_{x} }
\\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + \| \langle v \rangle^{5+\delta} \alpha \nabla_v f_0 \|_{L^\infty_{x} } \big)
\\ \le & C \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{5+\delta} \nabla_{x_\parallel} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{5+\delta} \alpha \nabla_v f(0) \|_{L^\infty_{x} } \right), \end{split} \end{equation} and \begin{equation} \label{alphavptgxgvgest} \begin{split}
& |\eqref{alphavptg} | + |\eqref{alphavnablaxg} | + |\eqref{alphavnablavg} |
\\ \le & \frac{ \langle v \rangle^{5+\delta} \alpha(t,x,v) }{ ( 1 + |v_{\mathbf{b}}| )^{5 + \delta } \hat{ {v_{\mathbf{b}}}_{,3} } } ( 1 + t_{\mathbf{b}} C ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) )
\\ & \times \left( \| \langle v \rangle^{5+\delta} \partial_t g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_x g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_v g \|_{L^\infty_{t,x} } \right)
\\ \le & ( C + t_{\mathbf{b}} C ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) )
\\ & \times \left( \| \langle v \rangle^{5+\delta} \partial_t g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_x g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_v g \|_{L^\infty_{t,x} } \right). \end{split} \end{equation}
Also, similarly from \eqref{nablavfest} and \eqref{pvitbest}, \begin{align}
& \notag | \langle v \rangle^{5+\delta} \partial_{v_i} f(t,x,v) |
\\ \notag & \le \langle v \rangle^{5+\delta} \big( | \nabla_{x_\parallel} f_0( X(0), V(0)) | | \partial_{v_i} X_\parallel (0) |+ | \partial_{x_3} f_0( X(0), V(0)) | | \partial_{v_i} X_3 (0) |
\\ \notag &\quad \quad \quad \quad \quad \quad \quad + | \nabla_v f_0(X(0), V(0))| | \nabla_{v_i} V(0) | \big)
\\ \notag & + \langle v \rangle^{5+\delta} | \partial_t g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \frac{t_{\mathbf{b}}}{{ \hat v_{\mathbf{b}}}_{,3} \langle v \rangle } \times ( 1 + \frac{t_{\mathbf{b}}}{\langle v \rangle} C ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) )
\\ \notag & + \langle v \rangle^{5+\delta} | \nabla_x g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \left( \frac{t_{\mathbf{b}}}{\langle v \rangle} + \left( \frac{1}{{ \hat v_{\mathbf{b}}}_{,3} } + 1 \right) \frac{t_{\mathbf{b}}^2}{\langle v \rangle} C ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) \right)
\\ \notag & + \langle v \rangle^{5+\delta} | \nabla_v g(t - t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} )| \left(( 1 + \frac{t_{\mathbf{b}}}{{ \hat v_{\mathbf{b}}}_{,3} \langle v \rangle } | \mathfrak F(t-t_{\mathbf{b}},x_{\mathbf{b}},t_{\mathbf{b}}) | + t_{\mathbf{b}} C ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) \right)
\\ \notag & \le ( C_2^2 +1 ) \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{5+\delta} \nabla_v f_0 \|_{L^\infty_{x} } \right)
\\ \notag & \quad + ( C + t_{\mathbf{b}} C ( \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} } ) )
\\ \label{dvfestfinal} & \quad \quad \times \left( \| \langle v \rangle^{5+\delta} \partial_t g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_x g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_v g \|_{L^\infty_{t,x} } \right) . \end{align}
Now from \eqref{lambdanablaxE} we have \[ \begin{split}
& \| \nabla_{x} E \|_{L^\infty_{t,x} } + \| \nabla_{x} B \|_{L^\infty_{t,x} }
\\ & \lesssim \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t,x,v) \|_{L^{\infty}_{t,x} } + \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f(t,x,v) \|_{L^{\infty}_{t,x} } +C, \end{split} \]
thus combining \eqref{nablaparaf0est}, \eqref{nablaparagest}, \eqref{alphavnablaf0est}, \eqref{alphavptgxgvgest}, and \eqref{dvfestfinal}, and by choosing $0 < T \ll 1 $ small enough
we have \begin{equation} \label{dxfestfinalin} \begin{split}
& \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t,x,v) \|_{L^{\infty}_{t,x} } + \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f(t,x,v) \|_{L^{\infty}_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_{v} f(t,x,v) \|_{L^{\infty}_{t,x} }
\\ \le & 2 \left( \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{5+\delta} \nabla_{x_\parallel} f_0 \|_{L^\infty_{x} } + \| \langle v \rangle^{5+\delta} \nabla_v f_0 \|_{L^\infty_{x} } \right)
\\ & + C_1 \left( \| \langle v \rangle^{5+\delta} \partial_t g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_x g \|_{L^\infty_{t,x} } + \| \langle v \rangle^{5+\delta} \nabla_v g \|_{L^\infty_{t,x} } \right)
\\ & + \frac{1}{2} \left( \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t,x,v) \|_{L^{\infty}_{t,x} } + \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f(t,x,v) \|_{L^{\infty}_{t,x} } \right) < \infty.
\end{split} \end{equation} This conclude \eqref{alphavnablafbd}.
\end{proof}
We state and prove a variation of Ukai's trace theorem in \cite{Ukai}. \begin{lemma} \label{traceUkai} Suppose $f \in L^\infty( (0,T) \times \Omega \times \mathbb R^3 )$, and $\mathfrak F \in W^{1,\infty}((0,T) \times \mathbb R^3 ) $ satisfy \begin{equation} \label{transopass} \partial_t f + \hat v \cdot \nabla_x f + \mathfrak F \cdot \nabla_v f = h \in L^\infty((0,T) \times \Omega \times \mathbb R^3 ). \end{equation} Then $f \in L^\infty((0,T) \times (\gamma \setminus \gamma_0 ) ) $, and \begin{equation} \label{tracelinftyUkai}
\sup_{0 \le t \le T } \| f(t) \|_{L^\infty( \gamma \setminus \gamma_0 ) } \le \sup_{0 \le t \le T } \| f(t) \|_{L^\infty( \Omega \times \mathbb R^3 ) } .
\end{equation} \end{lemma} \begin{proof} Denote the characteristics $X(s;t,x,v), V(s;t,x,v)$ which solves \begin{equation} \label{charaholder} \begin{split} \frac{d}{ds} X(s;t,x,v) = & \hat V(s;t,x,v), \\ \frac{d}{ds} V(s;t,x,v) = & \mathfrak F(s, X(s;t,x,v) ,V(s;t,x,v) ), \end{split} \end{equation} and $X(t;t,x,v) = x $, $V(t;t,x,v) = v$. Then since $\mathfrak F \in W^{1,\infty}((0,T) \times \mathbb R^3 ) $, the characteristics \eqref{charaholder} is H\"older continuous. From \eqref{transopass}, for almost every $(t,x,v) \in (0,T) \times \gamma_+$, and $ \max\{ 0, t-t_{\mathbf{b}}(t,x,v) \} $, \[ f(t,x,v) = f(s, X(s;t,x,v) , V(s;t,x,v) ) + \int_s^t h(\tau; X(\tau;t,x,v) , V(\tau;t,x,v) ) d\tau. \] Thus \[
\sup_{0 < t < T} \| f(t) \|_{L^\infty(\gamma_+ ) } \le \sup_{0 < t < T} \| f(t) \|_{L^\infty(\Omega \times \mathbb R^3 ) } + (t-s) \| h \|_{L^\infty((0,T) \times \Omega \times \mathbb R^3 ) }. \] Since $ t -s > 0$ can be arbitrarily small, we have \[
\sup_{0 < t < T} \| f(t) \|_{L^\infty(\gamma_+ ) } \le \sup_{0 < t < T} \| f(t) \|_{L^\infty(\Omega \times \mathbb R^3 ) }. \] Now, for $(x,v) \in \gamma_-$ and $s \in (t, \max \{ T, t_{\mathbf{f}}(t,x,v) \} ) $, we have \[ f(t,x,v) = f(s, X(s;t,x,v) , V(s;t,x,v) ) - \int_t^s h(\tau; X(\tau;t,x,v) , V(\tau;t,x,v) ) d\tau. \] Using the same argument we get \[
\sup_{0 < t < T} \| f(t) \|_{L^\infty(\gamma_- ) } \le \sup_{0 < t < T} \| f(t) \|_{L^\infty(\Omega \times \mathbb R^3 ) }. \] This proves \eqref{tracelinftyUkai}.
\end{proof}
Next, we prove a trace theorem for the derivatives of $f$.
\begin{lemma} \label{tracepf} Let $(f,E,B)$ be a solution of \eqref{VMfrakF1}, \eqref{inflow}, \eqref{Maxwell}. Suppose \begin{equation} \label{nablaEBassin}
\sup_{0 \le t \le T} \left(\| \nabla_{x} E(t) \|_\infty + \| \nabla_{x} B(t) \|_\infty \right) < \infty, \end{equation} \begin{equation}
\| \langle v \rangle^{5 + \delta } \partial_t g \|_\infty + \| \langle v \rangle^{5 + \delta } \nabla_{x_\parallel} g \|_\infty + \| \langle v \rangle^{5 + \delta } \nabla_{v} g \|_\infty < \infty, \end{equation} and \begin{equation} \label{diffusefinalin}
\sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha \partial_{x_3} f(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \nabla_{v} f(t) \|_\infty \right) < \infty.
\end{equation} Then \begin{equation} \label{pfgammapbd}
\sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f (t) \|_{L^\infty(\gamma \setminus \gamma_0) } + \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t,x,v) \|_{L^\infty(\gamma \setminus \gamma_0 ) } + \| \langle v \rangle^{5 + \delta } \nabla_{v} f (t) \|_{L^\infty(\gamma \setminus \gamma_0) } \right) < \infty
\end{equation} \end{lemma}
\begin{proof} The proof uses similar argument as Ukai's proof of a trace theorem in \cite{Ukai}. Next, notice that for $ \partial_{\mathbf e} \in \{ \nabla_{x_\parallel} ,\nabla_v \} $, and $p = 4 + \delta, 5+ \delta$, we have \begin{equation} \partial_t ( \langle v \rangle^{p} \partial_{\mathbf e} f ) + \hat v \cdot \nabla_x ( \langle v \rangle^{p}\partial_{\mathbf e} f )+ \mathfrak F \cdot \nabla_v ( \langle v \rangle^{p}\partial_{\mathbf e} f ) = - \langle v \rangle^{p}\partial_{\mathbf e} \hat v \cdot \nabla_x f - \langle v \rangle^{p} \partial_{\mathbf e } \mathfrak F \cdot \nabla_v f - \mathfrak F \cdot \nabla_v ( \langle v \rangle^{p} ) \partial_{\mathbf e} f. \end{equation} Then for almost every $(x,v) \in \gamma_+$, and $ s \in ( \max\{ 0 , t- t_{\mathbf{b}}(t,x,v) \} , t ) $, we have \begin{equation} \label{pefexp1} \begin{split} \langle v \rangle^{p} \partial_{\mathbf e} f (t,x,v) = & \langle V(s;t,x,v) \rangle^{p}\partial_{\mathbf e} f (s,X(s;t,x,v) ,V(s;t,x,v)) \\ & + \int_s^t \left( - \langle v \rangle^{p} \partial_{\mathbf e} \hat v \cdot \nabla_x f - \langle v \rangle^{p} \partial_{\mathbf e} \mathfrak F \cdot \nabla_v f - \mathfrak F \cdot \nabla_v ( \langle v \rangle^{p} ) \partial_{\mathbf e} f \right) (\tau, X(\tau;t,x,v), V(\tau;t,x,v) ) d \tau. \end{split} \end{equation} Thus, from \eqref{nablaEBassin} and \eqref{diffusefinalin}, \begin{equation} \label{pefbd1} \begin{split}
& | \langle v \rangle^{p} \partial_{\mathbf e} f (t,x,v) |
\\ \le & \sup_{0 \le s \le t} \| \langle v\rangle^p \partial_{\mathbf e} f(s) \|_{\infty} + \frac{ C (t-s) }{\alpha(t,x,v) } \bigg( ( \sup_{0 \le t \le T} (\| \nabla_{x} E(t) \|_\infty + \| \nabla_{x} B(t) \|_\infty ) + g + |B_e| )
\\ & \quad \quad \quad \quad \times \sup_{ 0 \le s \le t } \left( \| \langle v \rangle^{4 + \delta}\nabla_{x_\parallel } f(t) \|_\infty + \| \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f(t) \|_\infty + \| \langle v \rangle^{5 + \delta} \nabla_v f(t) \|_\infty \right) \bigg), \end{split} \end{equation} since we can choose $s $ close enough to $t$ such that \[ \begin{split}
& \frac{ C (t-s) }{\alpha(t,x,v) } \bigg( ( \sup_{0 \le t \le T} (\| \nabla_{x} E(t) \|_\infty + \| \nabla_{x} B(t) \|_\infty ) + g + |B_e|)
\\ & \quad \quad \quad \quad \times \sup_{ 0 \le s \le t } \left( \| \langle v \rangle^{4 + \delta}\nabla_{x_\parallel } f(t) \|_\infty + \| \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f(t) \|_\infty + \| \langle v \rangle^{5 + \delta} \nabla_v f(t) \|_\infty \right) \bigg)< \e \ll 1, \end{split} \] we have \[
| \langle v \rangle^{p} \partial_{\mathbf e} f (t,x,v) | \le \sup_{0 \le s \le t} \| \langle v\rangle^p \partial_{\mathbf e} f(s) \|_{\infty} + \e, \] for any $\e > 0$. Therefore, we get \begin{equation} \label{xparavfbdpin} \begin{split}
& \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f (t) \|_{L^\infty(\gamma_+) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f (t) \|_\infty,
\\ & \sup_{0 \le t \le T} \| \langle v \rangle^{5 + \delta } \nabla_{v} f (t) \|_{L^\infty(\gamma_+) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{5 + \delta } \nabla_{v} f (t) \|_\infty. \end{split} \end{equation} Similarly, since \[ \begin{split} & (\partial_t + \hat v \cdot \nabla_x +\mathfrak F \cdot \nabla_v ) ( \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f ) \\ & = - \langle v \rangle^{5+\delta} \alpha \partial_{x_3 } \mathfrak F \cdot \nabla_v f - \mathfrak F \cdot \nabla_v ( \langle v \rangle^{5+\delta} ) \alpha \partial_{x_3} f - [(\partial_t + \hat v \cdot \nabla_x + \mathfrak F \cdot \nabla_v ) \alpha ] \langle v \rangle^{5+\delta} \partial_{x_3} f \\ & = : G_{\alpha} (t,x,v). \end{split} \] Then for almost every $(x,v) \in \gamma_+$, and $ s \in ( \max\{ 0 , t- t_{\mathbf{b}}(t,x,v) \} , t ) $, we have \begin{equation} \label{ap3fexp} \begin{split} & \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t,x,v) \\ = & \langle V(s;t,x,v) \rangle^{5 +\delta} \alpha \partial_{x_3} f (s, X(s;t,x,v) , V(s;t,x,v) ) + \int_s^t G_\alpha(\tau , X(\tau; t,x,v) , V(\tau; t,x,v) ) d\tau. \end{split} \end{equation}
Since \[ \begin{split}
| G_\alpha (t,x,v) | \le & C \big( (( \sup_{0 \le t \le T} \| \nabla_{x} E(t) \|_\infty + \| \nabla_{x} B(t) \|_\infty ) + g + |B_e|)
\\ & \quad \quad \quad \quad \times \sup_{ 0 \le s \le t } \left( \| \langle v \rangle^{4 + \delta}\nabla_{x_\parallel } f(t) \|_\infty + \| \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f(t) \|_\infty + \| \langle v \rangle^{5 + \delta} \nabla_v f(t) \|_\infty \right) \big), \end{split} \] using the same argument as \eqref{pefbd1}--\eqref{xparavfbdpin} , we obtain \begin{equation} \label{x3fbdpin}
\sup_{0 \le t \le T} \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t,x,v) \|_{L^\infty(\gamma_+ ) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t,x,v) \|_\infty. \end{equation}
Now, for $(x,v) \in \gamma_-$, and any $s \in (t, \max \{ T, t_{\mathbf{f}}(t,x,v) \} ) $ we have the same formula \eqref{pefexp1} and \eqref{ap3fexp} for $ \langle v \rangle^p \partial \mathbf e f $ and $\langle v \rangle ^{5 +\delta}\alpha \partial_{x_3} f $ respectively. Therefore by the same argument, we get \begin{equation} \label{xparavfbdmin} \begin{split}
& \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f (t) \|_{L^\infty(\gamma_-) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f (t) \|_\infty,
\\ & \sup_{0 \le t \le T} \| \langle v \rangle^{5 + \delta } \nabla_{v} f (t) \|_{L^\infty(\gamma_-) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{5 + \delta } \nabla_{v} f (t) \|_\infty,
\\ & \sup_{0 \le t \le T} \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t,x,v) \|_{L^\infty(\gamma_-) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t,x,v) \|_\infty. \end{split} \end{equation}
Combining \eqref{xparavfbdpin}, \eqref{x3fbdpin}, and \eqref{xparavfbdmin}, we conclude \eqref{pfgammapbd}.
\end{proof}
\section{Local existence} We prove the local existence for RVM system with the inflow boundary condition in this section. We recursively define a sequence of functions: \[ f^0(t,x,v) = f_0(x,v), \ E^0(t,x) = E_0(t,x), \ B^0(t,x) = B_0(x). \] For $\ell \ge 1$, let $f^\ell$ be the solution of \begin{equation} \label{fellseqin} \begin{split} \partial_t f^\ell + \hat v \cdot \nabla_x f^\ell + \mathfrak F^{\ell-1} \cdot \nabla_v f^\ell = & 0, \text{ where } \mathfrak F^{\ell-1} = E^{\ell-1} + E_{\text{ext}} + \hat v \times ( B^{\ell-1} + B_{\text{ext}} ) - g \mathbf e_3, \\f^\ell(0,x,v) = & f_0(x,v) ,
\\ f^\ell(t,x,v) |_{\gamma_-} = & g(t,x,v). \end{split} \end{equation} Let $\rho^\ell = \int_{\mathbb R^3 } f^\ell dv, j^\ell = \int_{\mathbb R^3 } \hat v f^\ell dv$. Let \begin{equation} \label{ElBlin} E^\ell = \eqref{Eesttat0pos} + \dots + \eqref{Eest3bdrycontri}, \ B^\ell = \eqref{Besttat0pos} + \dots + \eqref{Bestbdrycontri}, \text{ with } f \text{ changes to } f^\ell. \end{equation}
We prove several uniform-in-$\ell$ bounds for the sequence before passing the limit.
\begin{lemma} Suppose $f_0$ satisfies \eqref{f0bdd}, $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{E0B0bdd}, then there exits $M_1, M_2$, and $c_0$, such that for $0 < T \ll 1 $, \begin{equation} \label{fellboundin} \begin{split}
\sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } f^\ell(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} \right) < & M_1, \
\\ \sup_\ell \sup_{0 \le t \le T} \left( \| E^\ell (t) \|_\infty + \| B^\ell (t) \|_\infty \right) + |B_e| + E_e + g < & M_2, \\ \inf_{\ell} \inf_{ t ,x_\parallel} \left( g - E_e - E^\ell_3(t,x_\parallel, 0 ) - (\hat v \times B^\ell)_3(t,x_\parallel, 0 ) \right) > & c_0. \end{split} \end{equation} \end{lemma}
\begin{proof} Let $\ell \ge 1$. By induction hypothesis we assume that \begin{equation} \label{inductfEBin} \begin{split}
\sup_{ 0 \le i \le \ell } \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta} f^{\ell-i}(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} \right) < & M_1,
\\ \sup_{ 0 \le i \le \ell } \sup_{0 \le t \le T} \left( \| E^{\ell-i} (t) \|_\infty + \| B^{\ell-i} (t) \|_\infty \right) + |B_e| + E_e + g < & M_2.
\end{split} \end{equation}
Denote the characteristics $(X^\ell, V^\ell)$ which solves \begin{equation} \label{XV_ell} \begin{split} \frac{d}{ds}X^\ell(s;t,x,v) &= \hat V^\ell(s;t,x,v),\\ \frac{d}{ds} V^\ell(s;t,x,v) &= \mathfrak F^\ell (s, X^\ell(s;t,x,v),V^\ell(s;t,x,v) ).\end{split} \end{equation}
First, we note that for any $ 0 s < t$, since \[ V^{\ell } (s; t,x,v) = v - \int_{s}^{ t} \mathfrak F^{\ell } (\tau, X^{\ell}(\tau), V^{\ell}(\tau) ) d\tau, \] from \eqref{inductfEBin}, we have \[
|v| - (t-s)M_2 \le | V^{\ell } (s; t, x, v) | \le |v | + (t- s)M_2. \] Thus \begin{equation} \label{Vsvin} \left( 1 + (t-s)(M_2+g) \right)^{-1} \langle v \rangle \le \langle V^{\ell } (s; t, x, v ) \rangle \le \left( 1 + (t-s)(M_2+g) \right) \langle v \rangle. \end{equation}
From \eqref{fellseqin} we have for any $(t,x,v) \in (0,T) \times \bar \Omega \times \mathbb R^3$, \begin{equation} \begin{split} f^{\ell+1}(t,x,v) = & \mathbf 1_{ {t_{\mathbf{b}}}^\ell \le 0} f^{\ell+1}( 0, X^\ell(0), V^\ell(0)) + \mathbf 1_{{t_{\mathbf{b}}}^\ell \ge 0 } g ({t_{\mathbf{b}}}^\ell , X^\ell({t_{\mathbf{b}}}^\ell; t,x,v), V^\ell({t_{\mathbf{b}}}^\ell;t,x,v) ).
\end{split} \end{equation} And \eqref{Vsvin} gives \begin{equation} \label{fnmiterate2in} \begin{split}
& \langle v \rangle^{4 + \delta } |f^{\ell+1}(t,x,v) |
\\ \le & \mathbf 1_{ {t_{\mathbf{b}}}^\ell \le 0} (1+ T (M_2+g))^{4 + \delta} | \langle V^\ell(0) \rangle^{4 + \delta } f^{\ell+1}( 0, X^\ell(0), V^\ell(0)) | \\ & + \mathbf 1_{ {t_{\mathbf{b}}}^\ell \ge 0 } (1+ T (M_2+g))^{4 + \delta} \langle V^\ell( {t_{\mathbf{b}}}^\ell) \rangle^{4 + \delta} g ({t_{\mathbf{b}}}^\ell , X^\ell({t_{\mathbf{b}}}^\ell; t,x,v), V^\ell({t_{\mathbf{b}}}^\ell;t,x,v) ). \end{split} \end{equation} Thus, by choosing $T \ll 1 $ we have \begin{equation} \label{fell1finalin}
\sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } f^{\ell+1}(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3 ) } < M_1. \end{equation} Now from \eqref{ElBlin}, using the same argument in the proof of Lemma \ref{EBlinflemma}, we have \begin{equation} \label{BEellinftyestinflow1}
\begin{split}
& \sup_{0 \le t \le T} \| E^{\ell+1} (t) \|_\infty + \sup_{0 \le t \le T} \| B^{\ell+1} (t) \|_\infty
\\ \le & C( \| E_0 \|_{\infty } + \| B_0 \|_{\infty } ) + C T ( \| E_0 \|_{C^1 } + \| B_0 \|_{C^1 } )
\\ &+ C T \sup_{ 0 \le t \le T} \| \langle v \rangle^{4+\delta} f^\ell (t) \|_\infty \left( 1 + T ( \sup_{ 0 \le t \le T} \left( \| E^\ell(t) \|_\infty + \| B^\ell(t) \|_\infty \right) + g + |B_e| ) \right)
\\ \le & C( \| E_0 \|_{C^1 } + \| B_0 \|_{C^1 } ) + C T M_1 \left( 1 + T (M_2+g + |B_e|) \right). \end{split}
\end{equation} Letting $M_2 = (C+1) ( \| E_0 \|_{C^1 } + \| B_0 \|_{C^1 } ) + |B_e| + E_e + g $ and $T \ll 1 $, we get \begin{equation} \label{EBlifinalin}
\sup_{0 \le t \le T} \| E^{\ell+1} (t) \|_\infty + \sup_{0 \le t \le T} \| B^{\ell+1} (t) \|_\infty + |B_e| + E_e + g< M_2. \end{equation} Next, from \eqref{E0B0g} and the proof of Lemma \ref{EBlinflemma}, by letting $c_0 = \frac{c_1}{2}$ in \eqref{E0B0g}, we get \[ \begin{split}
\inf_{ t ,x_\parallel} \left( g - E_e - E^{\ell+1}_3(t,x_\parallel, 0 ) - (\hat v \times B^{\ell+1})_3(t,x_\parallel, 0 ) \right) > & 2 c_0 - C T M_1 \left( 1 + T (M_2+g + |B_e|) \right). \end{split} \] By choosing $T \ll 1$ small enough, we have \begin{equation} \label{BEellinftyestinflow4}
\inf_{ t ,x_\parallel} \left( g - E_e - E^{\ell+1}_3(t,x_\parallel, 0 ) - (\hat v \times B^{\ell+1})_3(t,x_\parallel, 0 ) \right) > c_0. \end{equation} Combining with \eqref{fell1finalin} and \eqref{EBlifinalin}, we conclude \eqref{fellboundin} by induction.
\end{proof}
Next, we consider the derivative of the sequences. Define $\alpha^{\ell} $ as \begin{equation} \label{alphan} \begin{split} \alpha^\ell(t, x, v ) = \sqrt{(x_3)^2+(\hat{v}_{3})^2 -2\left( E^\ell_3(t,x_\parallel, 0 ) + E_e + (\hat v \times B^\ell)_3(t,x_\parallel, 0 ) - g \right) \frac{x_3}{\langle v \rangle}}. \end{split} \end{equation} We have the following estimate. \begin{lemma} Suppose $f_0$ satisfies \eqref{f0bdd}, \eqref{inflowdata}, $E_0$, $B_0$ satisfy \eqref{E0B0bdd}, then there exits $M_3, M_4$ such that for $0 < T \ll 1 $, \begin{equation} \label{flElBldsqbdin} \begin{split}
&\sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_\infty \right)
\\& + \sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right) < M_3 ,
\\ & \sup_\ell \sup_{0 \le t \le T} \left( \| \partial_t E^\ell(t) \|_\infty + \| \partial_t B^\ell(t) \|_\infty + \| \nabla_{x } E^\ell(t) \|_\infty +\| \nabla_{x } B^\ell(t) \|_\infty \right) < M_4. \end{split} \end{equation} \end{lemma} \begin{proof} The proof is essentially the same as the proof of Proposition \ref{inflowprop}.
From the uniform estimate \eqref{fellboundin}, and from the velocity lemma (Lemma \ref{vlemma}), we have for some $C>0$,
\begin{equation} e^{-C|t-s| } \alpha^\ell(t,x,v) \le \alpha^\ell(s, X^\ell(s;t,x,v) , V^\ell(s;t,x,v) ) \le e^{C|t-s| } \alpha^\ell(t,x,v), \ \text{ for all } \ell. \end{equation} Therefore, following the same proof of Proposition \ref{inflowprop} and Lemma \ref{tracepf}, we get \begin{equation} \label{} \begin{split}
& \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 +\delta } \nabla_{x_\parallel} f^{\ell+1} (t) \|_\infty + \| \langle v \rangle^{5 +\delta } \alpha^\ell \partial_{x_3} f ^{\ell+1} (t) \|_\infty + \| \langle v \rangle^{5 +\delta } \nabla_{v} f^{\ell+1} (t) \|_\infty \right)
\\ + & \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right)
\\ \le & C_k \bigg( \left( \| \langle v \rangle^{4 +\delta } \nabla_{x_\parallel} f_0 \|_\infty + \| \| \langle v \rangle^{5 +\delta } \alpha \partial_{x_3} f_0 \|_\infty + \| \| \langle v \rangle^{5 +\delta } \nabla_{v} f_0 \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } g (t) \|_{L^\infty(\gamma_-)} + \| \langle v \rangle^{5 + \delta } \nabla_v g(t) \|_{L^\infty(\gamma_-} + \| \langle v \rangle^{5 + \delta } \partial_t g(t) \|_{L^\infty(\gamma_-)} \right) \bigg). \end{split} \end{equation} Thus, by choosing $M_3 \gg 1$, we conclude \begin{equation} \label{dfelluniin} \begin{split}
& \sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_\infty \right) \\& + \sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right)< M_3. \end{split} \end{equation} From this, we use the same argument to get \eqref{dxEBfinal} in the proof of Lemma \ref{EBW1inftylemma} and obtain \[ \begin{split}
\sup_{0 \le t \le T} & \left( \| \partial_t E^{\ell+1}(t) \|_\infty + \| \partial_t B^{\ell+1}(t) \|_\infty + \| \nabla_{x } E^{\ell+1}(t) \|_\infty +\| \nabla_{x } B^{\ell+1}(t) \|_\infty \right)
\\ \le & TC \sup_{1 \le i \le \ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{i}(t) \|_\infty + \| \partial_t B^{i}(t) \|_\infty + \| \nabla_{x } E^{i}(t) \|_\infty +\| \nabla_{x } B^{i}(t) \|_\infty \right)
\\ & C \left( \| E_0 \|_{C^2 } + \| B_0 \|_{C_2} \right) + C \sup_{0 \le t \le T} \left( \| \langle v \rangle ^{4 + \delta } \nabla_{x_\parallel } f^{\ell +1}(t) \|_\infty + \| \langle v \rangle^{\ell + \delta } \alpha^{n} \partial_{x_3 } f^{\ell +1}(t) \|_\infty \right)
\\ & + C \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right)
\\ & + C \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } f^{\ell+1}(t) \|_\infty + \| E^{\ell +1} (t) \|_\infty + \| B^{\ell +1} (t) \|_\infty \right). \end{split} \] From \eqref{fellboundin} and \eqref{dfelluniin}, this gives \[ \begin{split}
\sup_{\ell} \sup_{0 \le t \le T} & \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right)
\\ \le & TC \sup_{ \ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right)
\\ & + C \left( \| E_0 \|_{C^2 } + \| B_0 \|_{C_2} \right)+ C(M_1 +M_2 +M_3). \end{split} \] Therefore, by choosing $M_4 \gg 1$ and $T \ll 1$, we get \begin{equation}
\sup_{\ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right) < M_4. \end{equation} Together with \eqref{dfelluniin}, we conclude \eqref{flElBldsqbdin}. \end{proof}
We have the following trace properties for $E^\ell$ and $B^\ell$:
\begin{lemma} \label{EBelltrace} Suppose $E^\ell, B^\ell$ satisfies \eqref{flElBldsqbdin}. Then for any $\ell \ge 1$, $0 < t < T$, \begin{equation} \label{EBpO} E^\ell(t,\cdot ,0) \in L^\infty(\partial \Omega ), \ B^\ell(t,\cdot,0) \in L^\infty(\partial \Omega ). \end{equation} \end{lemma} \begin{proof} From \eqref{flElBldsqbdin} we have \[ E^\ell(t,x) \in W^{1,\infty}((0,T) \times \Omega), \ B^\ell (t,x) \in W^{1,\infty}((0,T) \times \Omega), \] in particular, from the Morrey's inequality, $E(t)$, $B(t)$ are Lipschitz continuous on $\Omega$. Now pick any $x_\parallel \in \mathbb R^2$, and $0< x_3 < 1$, from the fundamental theorem of calculus, we have \[ \begin{split} E^\ell (t,x_\parallel, 0 ) = & E^\ell(t,x_\parallel, x_3) -\int_0^{x_3} \partial_{3} E^\ell(t,x_\parallel, y ) dy, \\ B^\ell(t,x_\parallel, 0 ) = & B^\ell(t,x_\parallel, x_3) -\int_0^{x_3} \partial_{3} B^\ell(t,x_\parallel, y ) dy. \end{split} \] Therefore \[ \begin{split}
& \| E^\ell(t,\cdot , 0 ) \|_{L^\infty(\partial \Omega ) } \le \| E^\ell(t) \|_{L^\infty(\Omega ) } + x_3 \| \partial_{3} E^\ell(t ) \|_{L^\infty(\Omega ) } ,
\\ & \| B^\ell(t,\cdot, 0 ) \|_{L^\infty(\partial \Omega ) } \le \| B^\ell(t) \|_{L^\infty(\Omega ) } + x_3 \| \partial_{3} B^\ell(t ) \|_{L^\infty(\Omega ) } , \end{split} \] for any $x_3 > 0 $. Thus \[
\| E^\ell(t,\cdot , 0 ) \|_{L^\infty(\partial \Omega ) } \le \| E^\ell(t) \|_{L^\infty(\Omega ) } < \infty, \ \| B^\ell(t,\cdot, 0 ) \|_{L^\infty(\partial \Omega ) } \le \| B^\ell(t) \|_{L^\infty(\Omega ) } < \infty, \] and we conclude \eqref{EBpO}. \end{proof}
Next, we prove the strong convergence of the sequence $f^\ell$.
\begin{lemma} \label{fEBsollemmain} Suppose $f_0, g$ satisfies \eqref{f0bdd}, \eqref{inflowdata}, $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{E0B0bdd}. There exists functions $(f,E,B)$ with $ \langle v \rangle^{4 +\delta } f(t,x,v) \in L^\infty( (0, T) ; L^\infty( \bar \Omega \times \mathbb R^3 ) ) $, and $(E,B) \in L^\infty((0,T) ; L^\infty( \Omega) \cap L^\infty( \partial \Omega ) )$, such that as $\ell \to \infty$, \begin{equation} \label{EnBnconvergein}
\sup_{0 \le t \le T} \left( \| E^\ell (t) - E(t) \|_{L^\infty( \Omega)} + \| E^\ell (t) - E(t) \|_{L^\infty( \partial \Omega)} + \| B^\ell (t) - B(t) \|_{L^\infty( \Omega)} + \| B^\ell (t) - B(t) \|_{L^\infty( \partial \Omega)} \right) \to 0, \end{equation} and \begin{equation} \label{fnconvergein}
\sup_{0 \le t \le T} \| \langle v \rangle^{4 +\ell } f^\ell(t) - \langle v \rangle^{4 +\delta } f (t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} \to 0.
\end{equation} Moreover, $(f,E,B)$ is a (weak) solution of the system \eqref{VMfrakF1}-\eqref{rhoJ1}, and \eqref{inflow}.
\end{lemma} \begin{proof}
Let $m > n \ge 1$. Note that $f^m - f^n $ satisfies $(f^m - f^n ) |_{t = 0 } = 0 $ and \[
(f^m- f^n )|_{\gamma_-} = 0. \]
The equation for $f^m - f^n $ is \[ \partial_t(f^m- f^n ) + \hat v \cdot \nabla_x (f^m - f^n ) + \mathfrak F^{m-1} \cdot \nabla_v (f^{m} - f^n ) = - ( \mathfrak F^{m-1} - \mathfrak F^{n-1} ) \cdot \nabla_v f^n. \] Thus, for any $(t,x,v) \in (0,T) \times \bar \Omega \times \mathbb R^3$, using \eqref{Vsvin}, we get \[ \begin{split}
& | \langle v \rangle^{4 + \delta } (f^m - f^n)(t,x,v) |
\\ \le & C_1 \int_{\max \{ {t_{\mathbf{b}}}^{m-1} , 0 \} }^t | \langle V^{m-1}(s) \rangle^{4 + \delta } ( \mathfrak F^{m-1} - \mathfrak F^{n-1} ) \cdot \nabla_v f^n )(s, X^{m-1}(s), V^{m-1}(s) ) | ds
\end{split} \] where $C_{1} = (1+ T (M_2+g) )^{4 + \delta}$. Then from \eqref{flElBldsqbdin}, \begin{equation} \label{fellnmiterate1in} \begin{split}
\| \langle v \rangle^{4 + \delta} f^m(t) - \langle v \rangle^{4 + \delta} f^n)(t) \|_\infty \le & C_1 \left( \sup_{\ell} \sup_{0 \le s \le t } \| \langle v \rangle^{4 + \delta} \nabla_v f^\ell(s) \|_\infty \right) \int_0^t \| \mathfrak F^{m-1} (s) - \mathfrak F^{n-1}(s) \|_\infty ds
\\ \le & C_2 \int_0^t \| \mathfrak F^{m-1} (s) - \mathfrak F^{n-1}(s) \|_\infty ds, \end{split} \end{equation} where $C_2 = C_1M_3$.
Now,
from \eqref{fellseqin} and using the same argument as Lemma \ref{EBlinflemma} with \eqref{fellnmiterate1in}, we have \begin{equation} \label{iterate2in} \begin{split}
\| \mathfrak F^{m-1} (s) - \mathfrak F^{n-1}(s) \|_\infty \le & \| E^{n-1}(s) - E^{m-1}(s) \|_\infty + \| B^{n-1}(s) - B^{m-1}(s) \|_\infty
\\ \le & C \left( \sup_{0 \le s' \le s } \| \langle v \rangle^{4 + \delta } (f^{n-1} - f^{m-1} )(s' ) \|_\infty + \int_0^s \| \mathfrak F^{m-2} (s') - \mathfrak F^{n-2 }(s') \|_\infty ds' \right)
\\ \le & C \int_0^s \| \mathfrak F^{m-2} (s') - \mathfrak F^{n-2}(s') \|_\infty ds'
\\ \le & C \int_0^s \left( \| E^{m-2}(s') - E^{n-2}(s') \|_\infty + \| B^{m-2}(s') - B^{n-2}(s') \|_\infty \right) ds'. \end{split} \end{equation} Iteration of \eqref{iterate2in} and using \eqref{fellboundin} yields \[ \begin{split}
& \| E^{m}(t) - E^{n}(t) \|_\infty + \| B^{m}(t) - B^{n}(t) \|_\infty
\\ \le & C^2 \int_0^t \int_0^s \left( \| E^{m-2}(s') - E^{n-2}(s') \|_\infty + \| B^{m-2}(s') - B^{n-2}(s') \|_\infty \right) ds' ds
\\ = & C^2 \int_0^t \tau \left( \| E^{m-2}(\tau) - E^{n-2}(\tau) \|_\infty + \| B^{m-2}(\tau) - B^{n-2}(\tau) \|_\infty \right) d\tau
\\ \le & C^l \int_0^t \frac{ \tau^{l-1}}{ (l-1)!} \left( \| E^{m-i}(\tau) - E^{n-k}(\tau) \|_\infty + \| B^{m-k}(\tau) - B^{n-k}(\tau) \|_\infty \right) d\tau
\\ \le &M_2 \frac{C^l t^l}{l!}. \end{split} \]
Thus the sequences $E^\ell$, $B^\ell$ are Cauchy in $L^\infty((0,T) \times \Omega ) $, moreover, from Lemma \ref{EBelltrace}, $E^\ell, B^\ell \in L^\infty([0,T] \times \partial \Omega )$. Therefore, there exists functions $E,B \in L^\infty((0,T) ; L^\infty( \Omega) \cap L^\infty( \partial \Omega ) )$, such that \begin{equation} \label{EnBncovin} E^\ell \to E, B^\ell \to B \text{ in } L^\infty((0,T) \times \Omega ) \cap L^\infty((0,T) \times \partial \Omega ) . \end{equation}
This proves \eqref{EnBnconvergein}. Also, from \eqref{fellnmiterate1in}, \eqref{iterate2in}, \[
\| \langle v\rangle^{4 + \delta} f^m(t) - \langle v \rangle^{4 + \delta} f^n)(t) \|_{L^\infty((0,T) \times \bar \Omega ) } \le M_2 \frac{C^{l-1} t^{l-1}}{(l-2)!}, \] therefore we get \eqref{fnconvergein}.
Now, take any $\phi(t,x,v) \in C_c^\infty( [0,T) \times \bar \Omega \times \mathbb R^3$ with $\text{supp } \phi \subset \{ [0, T) \times \bar \Omega \times \mathbb R^3 \} \setminus \{ (0 \times \gamma ) \cup (0,T) \times \gamma_0 \} $, from \eqref{fellseqin}, we have \begin{equation} \label{weakfellVMin} \begin{split} & \int_{\Omega \times \mathbb R^3 } f_0 \phi (0) dv dt + \int_0^T \int_{\Omega \times \mathbb R^3} f^\ell \left( \partial_t \phi + \hat v \cdot \nabla_x \phi + \mathfrak F^{\ell-1} \cdot \nabla_v \phi \right) dv dx dt \\ = & \int_0^T \int_{\gamma_+} \phi f^\ell \hat v_3 dv dS_x + \int_0^T \int_{\gamma_-} \phi g \hat v_3 dv dS_x. \end{split} \end{equation} Because of the strong convergence \eqref{EnBnconvergein}, \eqref{fnconvergein}, we have that as $\ell \to \infty$, each term in \eqref{weakfellVMin} goes to the corresponding terms with $f^\ell$ replaced by $f$ and $\mathfrak F^\ell$ replaced by $\mathfrak F$. Therefore we conclude that $(f,E,B)$ satisfy \eqref{weakf}.
Next, from Proposition \ref{Eiform} and Proposition \ref{Biform}, we have that $E^\ell$ and $B^\ell$ are (weak) solutions to the wave equations with the initial data, boundary condition and forcing term in \eqref{E12solin}-\eqref{B12solin}, with $\rho, J$ changed to $\rho^\ell ,J^\ell$. Then from \eqref{flElBldsqbdin} and Lemma \ref{wavetoMax}, we have \begin{equation} \label{Maxwellell} \begin{split} \partial_t E^\ell & = \nabla_x \times B^\ell - 4 \pi J^\ell, \, \nabla_x \cdot E^\ell = 4\pi \rho^\ell, \\ \partial_t B^\ell & = - \nabla_x \times E^\ell, \, \nabla_x \cdot B^\ell = 0, \end{split} \end{equation} with \begin{equation} \label{EellBellDin} E^\ell_1 = E^\ell_2 = 0, B^\ell_3 = 0, \ \text{ on } \partial \Omega. \end{equation} Clearly, \eqref{nablaEBweak} is satisfied. Now, for any test functions $\Psi(t,x) \in C_c^\infty([0,T) \times \bar \Omega ; \mathbb R^3 ), \ \Phi(t,x) \in C_c^\infty([0,T) \times \Omega ; \mathbb R^3 )$, from \eqref{Maxwellell} and \eqref{EellBellDin}, we have \begin{equation} \label{EBellweak1} \int_0^T \int_\Omega E^\ell \cdot \partial_t \Psi dx dt - \int_\Omega \Psi(0,x) \cdot E_0 dx = - \int_0^T \int_\Omega (\nabla_x \times \Psi) \cdot B^\ell dx dt + 4\pi \int_0^T \int_\Omega \Psi \cdot J^\ell dx dt, \end{equation} and \begin{equation} \label{EBellweak2} \int_0^T \int_\Omega B^\ell \cdot \partial_t \Phi dx dt + \int_\Omega \Phi(0,x) \cdot B_0 dx = \int_0^T \int_\Omega (\nabla_x \times \Phi) \cdot E^\ell dx dt, \end{equation}
Then from the strong convergence \eqref{fnconvergein}, \eqref{EnBnconvergein}, we can pass $\ell \to \infty$ and deduce that each term in \eqref{EBellweak1} and \eqref{EBellweak2} converges to the corresponding term with $E^\ell$, $B^\ell$, $J^\ell$ replace by $E$, $B$, and $J$ respectively. Therefore, $(f,E,B)$ satisfy \eqref{Maxweak1} and \eqref{Maxweak2}.
So we conclude that $(f,E,B)$ is a (weak) solution of the RVM system \eqref{VMfrakF1}-\eqref{rhoJ1} with inflow BC \eqref{inflow}. \end{proof}
In the next lemma, we consider the regularity of the solution.
\begin{lemma} \label{fEBregin} Let $\alpha(t,x,v)$ be defined as in \eqref{alphadef}. The solution $(f,E,B)$ obtained in Lemma \ref{fEBsollemma} satisfies \begin{equation} \label{pfbdlimitin}
\| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha^{} \partial_{x_3 } f(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_v f(t) \|_\infty < \infty, \end{equation} and \begin{equation} \label{pEBbdlimitin}
\| \partial_t E(t) \|_\infty + \| \partial_t B(t) \|_\infty + \| \nabla_{x } E(t) \|_\infty +\| \nabla_{x } B(t) \|_\infty < \infty. \end{equation} \end{lemma} \begin{proof} From the $L^\infty$ strong convergence \eqref{EnBnconvergein}, and the uniform-in-$\ell$ bound \eqref{flElBldsqbdin}, we can pass the limit up to subsequence if necessary and get the weak$-*$ convergence \begin{equation} \label{dEnBncovin} \partial_t E^\ell \overset{\ast}{\rightharpoonup} \partial_t E , \ \nabla_x E^\ell \overset{\ast}{\rightharpoonup} \nabla_x E, \ \partial_t B^\ell \overset{\ast}{\rightharpoonup} \partial_t B, \ \nabla_x B^\ell \overset{\ast}{\rightharpoonup} \nabla_x B \text{ in } L^\infty((0,T) \times \Omega ), \end{equation} and \begin{equation} \label{dfnconvergein} \langle v \rangle ^{4 + \delta} \nabla_{x_\parallel } f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{4 + \delta}\nabla_{x_\parallel } f, \ \langle v \rangle ^{4 + \delta} \nabla_{v } f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{4 + \delta}\nabla_{v } f \text{ in } L^\infty((0,T) \times \Omega \times \mathbb R^3 ). \end{equation} We also claim \begin{equation} \label{ap3fellcovin} \langle v \rangle ^{5 + \delta} \alpha^{\ell-1} \partial_{x_3} f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f \text{ in } L^\infty((0,T) \times \Omega \times \mathbb R^3 ). \end{equation} For any test function $\phi \in C^\infty_c((0,T) \times \Omega \times \mathbb R^3 ) $, we have \begin{eqnarray} & & \notag \int_0^t \iint_{\Omega \times \mathbb R^3} ( \langle v \rangle^{5 + \delta} \alpha^{\ell-1} \partial_{x_3} f^\ell - \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f ) \phi dv dx dt \\ \label{alphanpfn1in} = & &- \int_0^t \iint_{\Omega \times \mathbb R^3} ( \langle v \rangle^{5 + \delta} \alpha^{\ell-1} f^\ell - \langle v \rangle^{5 + \delta} \alpha f ) \partial_{x_3} \phi dv dx dt \\ \label{alphanpfn2in} & & - \int_0^t \iint_{\Omega \times \mathbb R^3} ( \langle v \rangle^{5 + \delta} \partial_{x_3} \alpha^{\ell-1} f^\ell - \langle v \rangle^{5 + \delta} \partial_{x_3} \alpha^{\ell-1} f ) \phi dv dx dt \\ \label{alphanpfn3in} & &- \int_0^t \iint_{\Omega \times \mathbb R^3} ( \langle v \rangle^{5 + \delta} \partial_{x_3} \alpha^{\ell-1} f - \langle v \rangle^{5 + \delta} \partial_{x_3} \alpha f ) \phi dv dx dt. \end{eqnarray} From \eqref{alphan} and \eqref{EnBnconvergein} we have \begin{equation} \label{alphancovin}
\| \alpha^\ell - \alpha \|_{L^\infty((0,T) \times \Omega \times \mathbb R^3)} \to 0 \text{ as } \ell \to \infty. \end{equation} Thus, together with \eqref{fnconvergein}, we have $\eqref{alphanpfn1in} \to 0$ as $n \to \infty$. Next, note that \[
\partial_{x_3} \alpha^\ell = \frac{x_3 - \left( E^\ell_3(t,x_\parallel, 0 ) + E_e + (\hat v \times B^\ell)_3(t,x_\parallel, 0 ) - g \right) \frac{1}{\langle v \rangle} }{\alpha^\ell}. \] For $(t,x,v) \in$ supp $\phi$, $x_3 > 0$ for some $c > 0$. Thus, $\frac{1}{\alpha^\ell(t,x,v) } < \frac{1}{c}$, so \[
| \partial_{x_3} \alpha^\ell \mathbf 1_{\text{supp}( \phi ) }(t,x,v) | < \frac{ C M_2}{c} \] From \eqref{fnconvergein}, this yields \[
| \eqref{alphanpfn2in} | \le \frac{ C M_2}{c} \int_0^t \iint_{\Omega \times \mathbb R^3} | \langle v \rangle^{5 + \delta} (f^\ell - f ) \phi | dv dx dt \to 0. \] For \eqref{alphanpfn3in}, since \[ \begin{split} & \partial_{x_3} \alpha^\ell - \partial_{x_3} \alpha \\ = & \frac{x_3 - \left( E^\ell_3(t,x_\parallel, 0 ) + E_e + (\hat v \times B^\ell)_3(t,x_\parallel, 0 ) - g \right) \frac{1}{\langle v \rangle} }{\alpha^\ell} - \frac{x_3 - \left( E_3(t,x_\parallel, 0 ) + E_e + (\hat v \times B)_3(t,x_\parallel, 0 ) - g \right) \frac{1}{\langle v \rangle} }{\alpha} \\ = & \frac{ - \left( (E^\ell-E)_3(t,x_\parallel, 0 ) + (\hat v \times( B^\ell - B))_3(t,x_\parallel, 0 ) \right) \frac{1}{\langle v \rangle} }{\alpha^\ell} + \\ & + \left( x_3 - \left( E_3(t,x_\parallel, 0 ) + E_e + (\hat v \times B)_3(t,x_\parallel, 0 ) - g \right) \right) \frac{1}{\langle v \rangle}\frac{ \alpha - \alpha^\ell }{\alpha^\ell \alpha}. \end{split} \] Again, for $(t,x,v) \in$ supp $\phi$, $x_3 > 0$ for some $c > 0$. Thus $\frac{1}{\alpha^\ell(t,x,v)} < \frac{1}{c}$, $\frac{1}{\alpha(t,x,v)} < \frac{1}{c}$. So from \eqref{EnBnconvergein}, \eqref{alphancovin}, we have \begin{equation} \label{px3alphacov3} \begin{split}
& \| (\partial_{x_3} \alpha^\ell - \partial_{x_3} \alpha ) \mathbf 1_{\text{supp} (\phi) }(t,x,v) \|_{L^\infty((0,T) \times \Omega \times \mathbb R^3)}
\\ & \le \frac{1}{c} \sup_{0 \le t \le T} \left( \| E(t) - E^\ell(t) \|_\infty + \| B(t) - B^\ell(t) \|_\infty \right) + \frac{CM_2}{c^2} \| \alpha^\ell - \alpha \|_{L^\infty((0,T) \times \Omega \times \mathbb R^3)} \to 0 \text{ as } \ell \to \infty. \end{split} \end{equation} Thus, we have $\eqref{alphanpfn3in} \to 0$, and this gives \eqref{ap3fellcovin}.
Therefore, from using the weak lower semi-continuity of the weak-$*$ convergence \eqref{dEnBncovin}, \eqref{dfnconvergein}, \eqref{ap3fellcovin}, and the uniform-in-$\ell$ bound \eqref{flElBldsqbdin}, we conclude \eqref{pfbdlimitin}, \eqref{pEBbdlimitin}.
\end{proof}
Next, we prove the uniqueness of the solutions of the RVM system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{inflow}.
\begin{lemma} \label{VMuniqlemmain} Suppose $(f,E_f, B_f)$ and $(g, E_g, B_g)$ are solutions to the VM system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{inflow} with $f(0) = g(0)$, $E_f(0) = E_g(0)$, $B_f(0) = B_g(0)$, and that \[ E_f, B_f, E_g, B_g \in W^{1,\infty}((0,T) \times \Omega ), \ \nabla_x \rho_{f}, \nabla_x J_f, \partial_t J_f , \nabla_x \rho_{g}, \nabla_x J_g, \partial_t J_g \in L^\infty((0,T); L_{\text{loc}}^p(\Omega)) \text{ for some } p>1. \] And \begin{equation} \label{dvfgbdin}
\sup_{0 < t < T} \| \langle v \rangle^{5+ \delta} \nabla_v f(t) \|_\infty <\infty, \sup_{0 < t < T} \| \langle v \rangle^{5+ \delta} \nabla_v g(t) \|_\infty <\infty. \end{equation} Then $f = g, E_f = E_g, B_f = B_g$. \end{lemma} \begin{proof}
The difference function $f-g $ satisfies \begin{equation} \label{fminusgeqin} \begin{split} (\partial_t + \hat v \cdot \nabla_x + \mathfrak F_f \cdot \nabla_v)(f-g) = (\mathfrak F_g - \mathfrak F_f ) \cdot \nabla_v g,
\\ (f-g)(0) = 0, \, (f- g )|_{\gamma_- } = 0, \end{split} \end{equation} where \[ \mathfrak F_f = E_f + E_{\text{ext}} + \hat v \times ( B_f + B_{\text{ext}}) - g \mathbf e_3 , \, \mathfrak F_g = E_g + E_{\text{ext}} + \hat v \times ( B_g + B_{\text{ext}}) - g \mathbf e_3, \] so \begin{equation} \label{mathfrakFfgin} \mathfrak F_g - \mathfrak F_f = E_f - E_g + \hat v \times (B_f - B_g ). \end{equation} From Lemma \ref{Maxtowave} we have $E_{f,1} - E_{g,1} , E_{f,2} - E_{g,2}, B_{f,3} - B_{g,3}$ solve the wave equation with the Dirichlet boundary condition \eqref{waveD} in the sense of \eqref{waveD_weak} with \begin{align}
u_0 = 0, \ u_1 = 0 , \ G = -4\pi \partial_{x_i} (\rho_f - \rho_g) - 4 \pi \partial_t (J_{f,i} - J_{g,i} ), \ g = 0 , \ \ \text{for} \ E_{f,i} - E_{g,i}, i =1,2, \label{E12solin} \\
u_0 = 0, \ u_1 = 0, \ G = 4 \pi (\nabla_x \times (J_f -J_g) )_3, \ g = 0, \ \ \text{for} \ B_{f,3}- B_{g,3}, \label{B3solin}
\end{align} respectively. And $E_{f,3} - E_{g,3}, B_{f,1} - B_{g,1}, B_{f,2}- B_{g,2}$ solve the wave equation with the Neumann boundary condition \eqref{waveNeu} in the sense of \eqref{waveinner} \text{ with } \begin{align}
u_0 = 0, \ u_1 = 0 , \ G = -4\pi \partial_{x_3} ( \rho_f - \rho_g) - 4 \pi \partial_t (J_{f,3} - J_{g,3} ) , \ g = - 4\pi (\rho_f - \rho_g), \ \ \text{for} \ E_{f,3} - E_{g,3}, \label{E3solin} \\
u_0 = 0, \ u_1 = 0, \ G = 4 \pi (\nabla_x \times (J_f - J_g) )_i, \ g = (-1)^{i+1} 4 \pi (J_{f,{\underline i}} - J_{g, \underline i } ), \ \ \text{for} \ B_{f,i} - B_{j,i}, \ i=1,2, \label{B12solin}
\end{align} respectively. Therefore, from Lemma \ref{wavesol} and Lemma \ref{wavesolD}, we know that $E_f - E_g$ and $B_f - B_g$ would have the form of \begin{equation} \label{EBdiffformin} \begin{split} & E_f - E_g = \eqref{Eesttat0pos} + \dots + \eqref{Eest3bdrycontri}, \ B_f -B_g = \eqref{Besttat0pos} + \dots + \eqref{Bestbdrycontri}, \\ & \text{ with } E_0, B_0 \text{ changes to } 0, \text{ and } f \text{ changes to } f -g. \end{split} \end{equation}
Now consider the characteristics \[ \begin{split} \dot X_f(s;t,x,v) = & \hat V_f(s;t,x,v) , \\ \dot V_f(s;t,x,v) = & \mathfrak F_f(s, X_f(s;t,x,v), V_f(s;t,x,v) ) . \end{split} \] Then from \eqref{fminusgeqin}, same as \eqref{fnmiterate2in}, we obtain \begin{equation} \label{fnmiterate2final} \begin{split}
& | \langle v \rangle^{4 + \delta } (f - g)(t,x,v) |
\\ \le & C_{1} \int_{0 }^{t } | \langle V_f(s) \rangle^{4 + \delta } ( \mathfrak F_g - \mathfrak F_g ) \cdot \nabla_v f^{} )(s, X_f^{}(s), V_f^{}(s) ) | ds
\end{split} \end{equation} So using \eqref{flElBldsqbdin}, we have
\begin{equation} \label{fgdiffrepin}
\sup_{ 0 \le s \le t } \| \langle v \rangle^{4 + \delta} (f-g)(s) \|_\infty \le C \int^t_0 \| (\mathfrak F_g - \mathfrak F_f )(s) \|_\infty \|\langle v \rangle^{4 + \delta} \nabla_v g (s) \|_\infty ds. \end{equation} Now, from \eqref{EBdiffformin} and the estimate in Lemma \ref{EBlinflemma}, we have \begin{equation} \label{FgFfdiffin} \begin{split}
\| (\mathfrak F_g - \mathfrak F_f )(s) \|_\infty \le & \| (E_f - E_g )(s) \|_\infty + \| (B_f - B_g )(s) \|_\infty
\\ \le & C \sup_{0 \le s' \le s } \| \langle v \rangle^{4 + \delta} (f-g )(s' ) \|_\infty, \end{split}
\end{equation} and from the assumption \eqref{dvfgbdin}, $\sup_{0 \le s \le t } \| \langle v \rangle^{4+\delta} \nabla_v g (s) \|_\infty < C$. Therefore from \eqref{fgdiffrepin} and \eqref{FgFfdiffin}, we have \begin{equation}
\sup_{0 \le s \le t } \| \langle v \rangle^{4 +\delta} (f-g)(s) \|_\infty \le C' \int^t_{0 } \sup_{0 \le s' \le s } \| \langle v \rangle^{4 +\delta}(f-g )(s' ) \|_\infty ds. \end{equation} Therefore from Gronwall \[
\sup_{0 \le s' \le t } \| \langle v \rangle^{4 +\delta} (f-g)(s') \|_\infty \le e^{C't} \| \langle v \rangle^{4 +\delta} (f-g)(0) \|_\infty = 0. \] Therefore we conclude that the solutions to \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{inflow}, is unique. \end{proof}
\begin{proof}[proof of Theorem \ref{main1}] Using the sequence $f^\ell, E^\ell , B^\ell$ constructed in \eqref{fellseqin}, \eqref{ElBlin}, we have from Lemma \ref{fEBsollemmain} that the limit $(f,E,B)$ is a solution to the VM system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{inflow}. This proves the existence. From Lemma \ref{fEBregin}, we have the regularity estimate \eqref{inflowfreg}, \eqref{inflowEBreg}. And from Lemma \ref{VMuniqlemmain}, we conclude the uniqueness. \end{proof}
\section{Diffuse BC} \label{DiffuseSec}
In \eqref{diffuseBC}, we denote $ \mu = \frac{1}{ (2 \pi )^{3/2} } e^{-\frac{|v|^2}{2} } $. And let the constant $c_{\mu}$ be such that $c_{\mu} \int_{v_3 > 0 } \hat v_{3} \mu (v) dv = 1$.
We first prove an a priori estimate for diffuse BC. \begin{proposition} \label{diffuseprop} Let $(f,E,B)$ be a solution of \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{diffuseBC}. Suppose the fields satisfies \eqref{signcondition},
and \begin{equation} \label{nablaEBass}
\sup_{0 \le t \le T} \left(\| \nabla_{x} E(t) \|_\infty + \| \nabla_{x} B(t) \|_\infty \right) < \infty. \end{equation} Assume that for $\delta > 0$, $ \langle v \rangle^{4 + \delta} \nabla_{x_\parallel} f, \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f, \langle v \rangle^{5 + \delta} \nabla_{v} f \in L^\infty((0,T) \times \Omega \times \mathbb R^3 ) $,
then for $0 < T \ll 1$, there exists a $C> 0$ such that \begin{equation} \label{diffusedfbd} \begin{split}
& \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta} \nabla_{x_\parallel} f(t) \|_\infty + \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f(t) \|_\infty + \| \langle v \rangle^{5+\delta} \nabla_{v} f(t) \|_\infty \right)
\\ &+ \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta} \nabla_{x_\parallel} f(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta} \nabla_{v} f(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right)
\\ < & C \left( \| \langle v \rangle ^{5 + \delta } \nabla_{x_\parallel} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha \partial_{x_3} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta } \nabla_v f_0 \|_\infty \right). \end{split} \end{equation} \end{proposition}
\begin{proof} For any $(t,x,v) \in (0,T) \times \Omega \times \mathbb R^3$, from \eqref{nablaxfest} and \eqref{nablavfest}, we have \begin{equation} \begin{split}
& | \partial_{x_i } f(t,x,v) |
\\ \lesssim & \mathbf 1_{t_{\mathbf{b}} > t } \{ | \nabla_x f_0( X(0), V(0)) | | \partial_{x_i} X(0) | + | \nabla_v f_0(X(0), V(0))| | \partial_{x_i} V(0) | \}
\\ & + \mathbf 1_{t_{\mathbf{b}} < t } \left( | \partial_t f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \frac{\delta_{i3} + t_{\mathbf{b}} }{ \hat {v_{\mathbf{b}}}_{,3} } + |\nabla_{x_\parallel } f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \frac{\delta_{i3} + t_{\mathbf{b}} }{ \hat {v_{\mathbf{b}}}_{,3} } + |\nabla_{v } f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \left( \frac{\delta_{i3} + t_{\mathbf{b}} }{ \hat {v_{\mathbf{b}}}_{,3} } + \langle v \rangle \right) \right), \end{split} \end{equation} and \begin{equation} \begin{split}
& | \partial_{v_i } f(t,x,v) |
\\ \lesssim & \mathbf 1_{t_{\mathbf{b}} > t } \{ | \nabla_x f_0( X(0), V(0)) | | \partial_{v_i} X(0) | + | \nabla_v f_0(X(0), V(0))| | \partial_{v_i} V(0) | \}
\\ & + \mathbf 1_{t_{\mathbf{b}} < t } \left( | \partial_t f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \frac{ t_{\mathbf{b}} }{ \hat {v_{\mathbf{b}}}_{,3} } + |\nabla_{x_\parallel } f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \frac{ t_{\mathbf{b}} }{ \hat {v_{\mathbf{b}}}_{,3} } + |\nabla_{v } f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) | \left( \frac{ t_{\mathbf{b}} }{ \hat {v_{\mathbf{b}}}_{,3} } + \langle v \rangle \right) \right) \end{split} \end{equation} Now, using the boundary condition \eqref{diffuseBC} and equation \eqref{VMfrakF1}, we have \[ \begin{split} \nabla_{x_\parallel } f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) = & c_\mu \mu (v_{\mathbf{b}}) \int_{u_3 < 0 } - \nabla_{x_\parallel } f(t-t_{\mathbf{b}}, x_{\mathbf{b}} , u ) \hat u _3 du, \\ \nabla_{v } f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) = & c_\mu v_{\mathbf{b}} \mu (v_{\mathbf{b}}) \int_{u_3 < 0 } f(t-t_{\mathbf{b}}, x_{\mathbf{b}} , u ) \hat u _3 du, \\ \partial_t f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, v_{\mathbf{b}} ) = & c_\mu \mu(v_{\mathbf{b}}) \int_{u_3 < 0 } - \partial_t f(t -t_{\mathbf{b}},x_{\mathbf{b}} , u ) \hat u _3 du
\\ = & c_\mu \mu(v_{\mathbf{b}}) \int_{u_3 < 0 } \left( \hat u \cdot \nabla_x f(t -t_{\mathbf{b}},x_{\mathbf{b}} , u ) + \mathfrak F \cdot \nabla_v f (t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) \right) \hat u _3 du
\\ = & c_\mu \mu(v_{\mathbf{b}}) \int_{u_3 < 0 } \hat u \cdot \nabla_x f(t -t_{\mathbf{b}},x_{\mathbf{b}} , u ) \hat u_3 + f (t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) \mathfrak F \cdot \nabla_u( \hat u _3 ) du . \end{split} \] Therefore, for $i = 1,2$, \begin{equation} \label{dxparallelfest1} \begin{split}
& | \partial_{x_i } f(t,x,v) |
\\ \lesssim & \mathbf 1_{t_{\mathbf{b}} > t } \{ | \nabla_x f_0( X(0), V(0)) | | \partial_{x_i} X(0) | + | \nabla_v f_0(X(0), V(0))| | \partial_{x_i} V(0) | \}
\\ & + \mathbf 1_{t_{\mathbf{b}} < t } \left( c_\mu \mu(v_{\mathbf{b}}) \langle v \rangle^2 \int_{u_3 < 0 } \left( | \nabla_x f (t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) | \hat u_3 + f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) \right) du \right)
\\ \lesssim & \mathbf 1_{t_{\mathbf{b}} > t } \{ | \nabla_{x_\parallel} f_0( X(0), V(0)) | |\partial_{x_i } X_\parallel (0 ) | + | \partial_{x_3} f_0( X(0), V(0)) | |\nabla_{x_\parallel} X_3 (0 ) | + | \nabla_v f_0(X(0), V(0))| |\partial_{x_i } V(0 ) | \}
\\ & + \mathbf 1_{t_{\mathbf{b}} < t } \left( c_\mu \mu(v_{\mathbf{b}}) \langle v \rangle^2 \int_{u_3 < 0 } \left( | \nabla_x f (t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) | \hat u_3 + f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) \right) du \right) , \end{split} \end{equation} where we've used \eqref{tbbdvb}. And for $i =3$, we have \begin{equation} \label{dx3fest1} \begin{split}
& | \partial_{x_3 } f(t,x,v) |
\\ \lesssim & \mathbf 1_{t_{\mathbf{b}} > t } \{ | \nabla_{x_\parallel} f_0( X(0), V(0)) | |\partial_{x_3 } X_\parallel (0 ) | + | \partial_{x_3} f_0( X(0), V(0)) | |\nabla_{x_3} X_3 (0 ) | + | \nabla_v f_0(X(0), V(0))| |\partial_{x_i } V(0 ) | \}
\\ & + \mathbf 1_{t_{\mathbf{b}} < t } \left( c_\mu \frac{1}{ \hat {v_{\mathbf{b}}}_{,3} } \mu(v_{\mathbf{b}}) \langle v \rangle^2 \int_{u_3 < 0 } \left( | \nabla_x f (t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) | \hat u_3 + f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) \right) du \right). \end{split} \end{equation} Also, \begin{equation} \label{dvfest1} \begin{split}
& | \partial_{v_i } f(t,x,v) |
\\ \lesssim & \mathbf 1_{t_{\mathbf{b}} > t } \{ | \nabla_{x_\parallel} f_0( X(0), V(0)) | |\partial_{v_i } X_\parallel (0 ) | + | \partial_{x_3} f_0( X(0), V(0)) | |\nabla_{v_i} X_3 (0 ) | + | \nabla_v f_0(X(0), V(0))| |\partial_{v_i } V(0 ) | \}
\\ & + \mathbf 1_{t_{\mathbf{b}} < t } \left( c_\mu \mu(v_{\mathbf{b}}) |v_{\mathbf{b}}| \langle v \rangle^2 \int_{u_3 < 0 } \left( | \nabla_x f (t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) | \hat u_3 + f(t-t_{\mathbf{b}}, x_{\mathbf{b}}, u ) \right) du \right) . \end{split} \end{equation}
Let $(x,v) \notin \gamma_0$ and $(t^0,x^0,v^0) = (t,x,v)$. For the characteristic \begin{equation} \label{diffusecycles0} \begin{split} \frac{d}{ds}X(s;t,x,v) &= \hat V(s;t,x,v),\\ \frac{d}{ds} V(s;t,x,v) &= \mathfrak F (s, X(s;t,x,v),V(s;t,x,v) ), \end{split} \end{equation} we define the stochastic (diffuse) cycles as \begin{equation} \label{diffusecycles1} \begin{split} & t^1 = t - t_{\mathbf{b}}(t,x,v), \, x^1 = x_{\mathbf{b}}(t,x,v) = X(t - t_{\mathbf{b}}(t,x,v);t,x,v), \\ & v_b^0 = V(t - t_{\mathbf{b}}(t,x,v);t,x,v) = v_{\mathbf{b}}(t,x,v), \end{split} \end{equation}
and $v^1 \in \mathbb R^3$ with $n(x^1) \cdot v^1 > 0$. For $l\ge1$, define \begin{equation} \begin{split} & t^{l+1} = t^l - t_{\mathbf{b}}(t^l,x^l,v^l), x^{l + 1 } = x_{\mathbf{b}}(t^l,x^l,v^l), \\ & v_b^l = v_{\mathbf{b}}(t^l,x^l,v^l), \end{split} \end{equation} and $v^{l+1} \in \mathbb R^3 \text{ with } n(x^{l+1}) \cdot v^{l+1} > 0$. Also, define \begin{equation} \label{diffusecycles2} X^l(s) = X(s;t^l,x^l,v^l), \, V^l(s) = V(s;t^l,x^l,v^l), \end{equation}
so $X(s) = X^0(s), V(s) = V^0(s)$.
Expanding $\nabla_x f(t^1, x^1 , v^1 ) + f(t^1,x^1,v^1)$ in \eqref{dx3fest1} again, we get for $i=1,2$, \[ \begin{split}
& | \partial_{x_i } f(t,x,v) |
\\ \lesssim & \mathbf 1_{t^1 < 0 } \{ | \nabla_x f_0( X(0), V(0)) | |\partial_{x_i } X(0 ) | + | \nabla_v f_0(X(0), V(0))| |\partial_{x_i } V(0 ) | \}
\\ & + \mathbf 1_{ t^2 < 0 < t^1 } \bigg\{ c_\mu \mu(v_{\mathbf{b}}) \langle v \rangle^2 \int_{v^1_3 < 0 } \big( \left( | \nabla_x f (0, X^1(0) , V^1(0) ) + \langle v^1 \rangle | \nabla_v f (0, X^1(0) , V^1(0) ) | \right) | \hat v^1_3 \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + f(0 , X^1(0) , V^1(0) ) \big) d v^1 \bigg \}
\\ & + \mathbf 1_{ t^2 > 0 } \left( c_\mu \mu(v_{\mathbf{b}}) \langle v \rangle^2 \int_{v^1_3 < 0 } \left( c_\mu \mu(v_{\mathbf{b}}^1) \langle v_1 \rangle^2 \frac{\hat v^1_3 }{\hat v_{\mathbf{b}}_{,3}^1} \int_{v^2_3 < 0 } | \left( \nabla_{x_\parallel} f(t^2,x^2, v^2 ) | \hat v^2_3 + f(t^2, x^2, v^2 ) \right) dv^2 \right) dv^1 \right), \end{split} \] keep doing the expansion we get for $\ell > 1$, \begin{equation} \label{dxparadiffusef2} \begin{split}
& | \nabla_{x_\parallel} f(t,x,v) |
\\ \lesssim & \mathbf 1_{t^1 < 0 } \{ | \nabla_{x_\parallel} f_0( X(0), V(0)) | |\nabla_{x_\parallel} X_\parallel (0 ) | + | \partial_{x_3} f_0( X(0), V(0)) | |\nabla_{x_\parallel} X_3 (0 ) | + | \nabla_v f_0(X(0), V(0))| |\nabla_{x_\parallel} V(0 ) | \}
\\ &+ \mu (v_{\mathbf{b}}) \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \sum_{i=1}^{l-1} \textbf{1}_{\{t^{i+1} < 0 < t^i \}} \bigg( \left( | \nabla_x f^{} (0,X^{i}(0), V^{i}(0)) | + \langle v^i \rangle | \nabla_v f^{} (0,X^{i}(0), V^{i}(0)) | \right) \hat v^i_3 \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + f(0 , X^i(0) , V^i(0) ) \bigg) \, d \Sigma_{i}^{l-1}
\\ & + \mu (v_{\mathbf{b}}) \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \textbf{1}_{\{t^{l} > 0 \}} \int_{\mathcal V_l} \left( | \nabla_{x_\parallel} f^{} (t^l,x^l, v^l ) | \hat v^l_3 + f(t^l, x^l, v^l ) \right) d v^l d \Sigma_{l-1}^{l-1}, \end{split} \end{equation} where $\mathcal V_j = \{ v^j \in \mathbb R^3: v^J_3 < 0 \}$, and \[ \begin{split}
d \Sigma_i^{l -1 } = & \{\prod_{j=i+1}^{l-1} \mu(v^j) c_\mu | \hat v_3^j | dv^j \}
\{\prod_{j=1}^{i-1} c_\mu \mu(v_{\mathbf{b}}^j ) \langle v^j \rangle^2 \frac{\hat v^J_3 }{\hat v_{\mathbf{b}}_{,3}^j} d v^j\}, \end{split} \]
where $c_\mu$ is the constant that $c_\mu \int_{\mathbb R^3 } \mu(v^j) | \hat v_3^j | dv^j = 1$. Similarly, we get \begin{equation} \begin{split}
& | \partial_{x_3 } f(t,x,v) |
\\ \lesssim & \mathbf 1_{t^1 < 0 } \{ | \nabla_{x_\parallel} f_0( X(0), V(0)) | |\partial_{x_3 } X_\parallel (0 ) | + | \partial_{x_3} f_0( X(0), V(0)) | | \partial_{x_3} X_3 (0 ) | + | \nabla_v f_0(X(0), V(0))| |\partial_{x_3 } V(0 ) | \}
\\ & + \frac{\mu (v_{\mathbf{b}})}{ \hat {v_{\mathbf{b}}}_{,3} } \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \sum_{i=1}^{l-1} \textbf{1}_{\{t^{i+1} < 0 < t^i \}} \bigg( \left( | \nabla_x f^{} (0,X^{i}(0), V^{i}(0)) | + \langle v^i \rangle | \nabla_v f^{} (0,X^{i}(0), V^{i}(0)) | \right) \hat v^i_3 \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + f(0 , X^i(0) , V^i(0) ) \bigg) \, d \Sigma_{i}^{l-1}
\\ & + \frac{\mu (v_{\mathbf{b}})}{ \hat {v_{\mathbf{b}}}_{,3} } \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \textbf{1}_{\{t^{l} > 0 \}} \int_{\mathcal V_l} \left( | \nabla_x f^{} (t^l,x^l, v^\ell ) | \hat v^\ell_3 + f(t^\ell, x^\ell, v^\ell ) \right) d v^l d \Sigma_{l-1}^{l-1}. \end{split} \end{equation} And \begin{equation} \label{dvdiffusef2} \begin{split}
& | \nabla_{v} f(t,x,v) |
\\ \lesssim & \mathbf 1_{t^1 < 0 } \{ | \nabla_{x_\parallel} f_0( X(0), V(0)) | |\nabla_{v} X_\parallel (0 ) | + | \partial_{x_3} f_0( X(0), V(0)) | |\nabla_{v} X_3 (0 ) | + | \nabla_v f_0(X(0), V(0))| |\nabla_{v} V(0 ) | \}
\\ &+ \mu (v_{\mathbf{b}}) |v_{\mathbf{b}}| \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \sum_{i=1}^{l-1} \textbf{1}_{\{t^{i+1} < 0 < t^i \}} \bigg( \left( | \nabla_x f^{} (0,X^{i}(0), V^{i}(0)) | + \langle v^i \rangle | \nabla_v f^{} (0,X^{i}(0), V^{i}(0)) | \right) \hat v^i_3 \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + f(0 , X^i(0) , V^i(0) ) \bigg) \, d \Sigma_{i}^{l-1}
\\ & + \mu (v_{\mathbf{b}}) |v_{\mathbf{b}}| \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \textbf{1}_{\{t^{l} > 0 \}} \int_{\mathcal V_l} \left( | \nabla_x f^{} (t^l,x^l, v^l ) | \hat v^l_3 + f(t^l, x^l, v^l ) \right) d v^l d \Sigma_{l-1}^{l-1}, \end{split} \end{equation}
Next, we claim that there exists $l_0 \gg 1$ such that for $l \ge l_0$, we have \begin{equation} \label{trajexpansionendterm} \int_{\prod_{j=1}^{l-1} \mathcal V_j} \textbf{1}_{\{t^{l}(t,x,v,v^1,...,v^{l-1} ) >0 \}} \, d \, \Sigma_{l-1}^{l-1} \lesssim \left( \frac{1}{2} \right)^{l}. \end{equation} Since \[
|v_{\mathbf{b}}^j|^2 \lesssim |v^j |^2 + (t^j - t^{j+1} ) ^2 ( \| E \|_\infty^2 + \| B \|_\infty^2 ), \, \langle v_{\mathbf{b}}^j \rangle \lesssim \langle v^j \rangle + (t^j - t^{j+1} ) ( \| E \|_\infty + \| B \|_\infty ) , \] and using \eqref{alphaest}, we have for some fixed constant $C_{0 } > 0$, \[ \begin{split}
\, d \, \Sigma_{l-1}^{l-1}
\le (C_{0 })^l \prod_{j=1}^{l-1} \sqrt{ \mu (v_j) } \langle v^j \rangle^2 dv^j. \end{split} \] Choose a sufficiently small $\delta = \delta (C_0) > 0$. Define \[ \mathcal V_j^\delta = \{ v^j \in \mathcal V_j: v^J_3 \ge \delta
\}, \] where we have $\int_{\mathcal V_j \setminus \mathcal V_j^\delta} C_0 \sqrt{ \mu (v_j) } \langle v^j \rangle^2 dv^j \lesssim \delta$.
On the other hand if $v^j \in \mathcal V_j^\delta$, we have from \eqref{Xiformula} \[ \begin{split}
| (t^j - t^{j+1} ) \hat{v}_3^j | = & | \int^{t^j }_{ t^{j+1} }\int^{t^j}_s \hat{\mathfrak F } _{3}(\tau, X^j (\tau), V^j(\tau) ) \mathrm{d} \tau \mathrm{d} s | \\ \le & \int^{t^j }_{ t^{j+1} }\int^{t^j}_s \frac{2 g }{ \langle V^j(\tau ) \rangle } d\tau ds \le \frac{(t^j - t^{j+1} )^2 g }{ \min_{ 0 \le \tau \le T} \langle V^j(\tau) \rangle }. \end{split} \] Thus \[
|t^j - t^{j+1} | \ge \frac{ v^J_3 }{g} \frac{ \min_{ 0 \le \tau \le T} \langle V^j(\tau) \rangle}{ \max_{ 0 \le \tau \le T} \langle V^j(\tau) \rangle } \gtrsim v^J_3 \ge \delta. \] Now if $t^l \ge 0$ then there are at most $ \left[ \frac{ C_\Omega }{ \delta } \right] + 1$ numbers of $v^m \in \mathcal V^\delta_m$ for $1 \le m \le l-1$. Equivalently there are at least $l -2 - \left[ \frac{ C_\Omega }{ \delta } \right]$ numbers of $v^{m_i} \in \mathcal V_{m_i} \setminus \mathcal V_{m_i }^\delta$. Therefore we have: \begin{equation} \label{tailtermsmall} \begin{split} \int_{\prod_{j=1}^{l-1} \mathcal V_j} & \textbf{1}_{\{t^{l}(t,x,v,v^1,...,v^{l-1} ) >0 \}} \, d \, \Sigma_{l-1}^{l-1} \\ \le & \sum_{m = 1}^{ \left[ \frac{ C_\Omega }{ \delta } \right] +1} \int_{ \left\{ \parbox{15em}{there are exactly $m$ of $v^{m_i} \in \mathcal V_{m_i}^\delta $ and $l -1 -m $ of $v^{m_i} \in \mathcal V_{m_i } \setminus \mathcal V^\delta_{m_i}$ } \right\} } \prod_{j =1}^{l-1} C_0 \sqrt{ \mu (v_j) } \langle v^j \rangle^2 d v^j \\ \le & \sum_{m = 1}^{ \left[ \frac{ C_\Omega }{ \delta } \right]+1} {l-1 \choose m} \left\{ \int_{\mathcal V} C_0 \sqrt{ \mu (v) } \langle v \rangle^2 dv \right\}^m \left\{ \int_{\mathcal V \setminus \mathcal V^\delta} C_0 \sqrt{ \mu (v) } \langle v \rangle^2 dv \right\}^{l-1-m} \\ \le & \left( \left[ \frac{ C_\Omega }{ \delta} \right] + 1 \right) ( l -1)^{ \left[ \frac{ C_\Omega }{ \delta } \right] + 1} ( \delta)^{l - 2 - \left[ \frac{ C_\Omega }{ \delta } \right] } \left\{ \int_{\mathcal V} C_0 \sqrt{ \mu (v) } \langle v \rangle^2 dv \right\}^{ \left[ \frac{ C_\Omega }{ \delta } \right]+1} \\ \le & C \delta^{l/2} \le C (\frac{1}{2})^{l}, \end{split} \end{equation} if $l \gg 1$, say $ l = 2 \left( \left[ \frac{ C_\Omega }{ \delta } \right] +1 \right) ^2$.
Therefore, from \eqref{tbdV0}, \eqref{pxviXVest}, \eqref{pxipviXV}, \eqref{dxparadiffusef2},\eqref{trajexpansionendterm}, and \eqref{tailtermsmall} we have \begin{equation} \label{dxparadiffusef3} \begin{split}
& \langle v \rangle^{4 +\delta } | \nabla_{x_\parallel} f(t,x,v) |
\\ \lesssim & \langle v \rangle^{4 +\delta } \{ | \nabla_{x_\parallel} f_0( X(0), V(0)) | + t | \partial_{x_3} f_0( X(0), V(0)) | + \langle v \rangle | \nabla_v f_0(X(0), V(0))| \}
\\ &+ \langle v \rangle^{4 +\delta } \mu (v_{\mathbf{b}}) \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \sum_{i=1}^{l-1} \textbf{1}_{\{t^{i+1} < 0 < t^i \}} \bigg( \left( | \nabla_x f^{} (0,X^{i}(0), V^{i}(0)) | + \langle v^i \rangle | \nabla_v f^{} (0,X^{i}(0), V^{i}(0)) | \right) \hat v^i_3 \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + f(0 , X^i(0) , V^i(0) ) \bigg) \, d \Sigma_{i}^{l-1}
\\ & + \langle v \rangle^{4 +\delta } \mu (v_{\mathbf{b}}) \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \textbf{1}_{\{t^{l} > 0 \}} \int_{\mathcal V_l} \left( | \nabla_{x_\parallel} f^{} (t^l,x^l, v^l ) | \hat v^l_3 + f(t^l, x^l, v^l ) \right) d v^l d \Sigma_{l-1}^{l-1}
\\ \lesssim & C_l \left( \| (1 +|v|^{4 + \delta } ) \nabla_{x_\parallel} f_0 \|_\infty + \| (1 +|v|^{4 + \delta } ) \alpha \partial_{x_3} f_0 \|_\infty + \| (1 +|v|^{5 + \delta } ) \nabla_{v} f_0 \|_\infty \right)
\\ & + C \left( \frac{1}{2} \right)^l \sup_{0 \le t \le T } \| \langle v \rangle^{4+\delta} \nabla_{x_\parallel} f(t) \|_\infty , \end{split} \end{equation} and similarly, \begin{equation} \label{dx3diffusef3} \begin{split}
& \langle v \rangle^{5 +\delta } | \alpha \partial_{x_3} f(t,x,v) |
\\ \lesssim & \langle v \rangle^{5 +\delta } \alpha(t,x,v) \{ | \nabla_{x_\parallel} f_0( X(0), V(0)) | + | \partial_{x_3} f_0( X(0), V(0)) | + \langle v \rangle | \nabla_v f_0(X(0), V(0))| \}
\\ &+ \langle v \rangle^{5 +\delta } \frac{\alpha(t,x,v) \mu (v_{\mathbf{b}})}{ \hat {v_{\mathbf{b}}}_{,3} } \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \sum_{i=1}^{l-1} \textbf{1}_{\{t^{i+1} < 0 < t^i \}} \bigg( \left( | \nabla_x f^{} (0,X^{i}(0), V^{i}(0)) | + \langle v^i \rangle | \nabla_v f^{} (0,X^{i}(0), V^{i}(0)) | \right) \hat v^i_3 \\ & \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad + f(0 , X^i(0) , V^i(0) ) \bigg) \, d \Sigma_{i}^{l-1}
\\ & + \langle v \rangle^{5 +\delta } \frac{\alpha(t,x,v) \mu (v_{\mathbf{b}})}{ \hat {v_{\mathbf{b}}}_{,3} } \langle v \rangle^2 \int_{\prod_{j=1}^{l-1} \mathcal V_j} \textbf{1}_{\{t^{l} > 0 \}} \int_{\mathcal V_l} \left( | \nabla_x f^{} (t^l,x^l, v^l ) | \hat v^l_3 + f(t^l, x^l, v^l ) \right) d v^l d \Sigma_{l-1}^{l-1}
\\ \lesssim & C_l \left( \| (1 +|v|^{5 + \delta } ) \nabla_{x_\parallel} f_0 \|_\infty + \| (1 +|v|^{5 + \delta } ) \alpha \partial_{x_3} f_0 \|_\infty + \| (1 +|v|^{5 + \delta } ) \nabla_{v} f_0 \|_\infty \right)
\\ & + C \left( \frac{1}{2} \right)^l \sup_{0 \le t \le T } \| \langle v \rangle^{5+\delta} \alpha \nabla_x f(t) \|_\infty , \end{split} \end{equation} and \begin{equation} \label{dvdiffusef4} \begin{split}
& \langle v \rangle^{5 +\delta } | \nabla_{v} f(t,x,v) |
\\ \lesssim & C_l \left( \| (1 +|v|^{4 + \delta } ) \nabla_{x_\parallel} f_0 \|_\infty + \| (1 +|v|^{4 + \delta } ) \alpha \partial_{x_3} f_0 \|_\infty + \| (1 +|v|^{5 + \delta } ) \nabla_{v} f_0 \|_\infty \right)
\\ & + C \left( \frac{1}{2} \right)^l \sup_{0 \le t \le T } \| \langle v \rangle^{5+\delta} \alpha \nabla_x f(t) \|_\infty , \end{split} \end{equation} where we've used \eqref{alphaest}. Adding \eqref{dxparadiffusef2}, \eqref{dx3diffusef3}, and \eqref{dvdiffusef4} and choosing $l \gg 1$, we get for a large $C > 0$, \begin{equation} \label{diffusefinal} \begin{split}
& \sup_{ 0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta}\nabla_{x_\parallel } f(t) \|_\infty + \| \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f(t) \|_\infty + \| \langle v \rangle^{5 + \delta} \nabla_v f(t) \|_\infty \right)
\\ < & C \left( \| \langle v \rangle^{5 + \delta} \nabla_{x_\parallel} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta} \alpha \partial_{x_3} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta} \nabla_{v} f_0 \|_\infty \right). \end{split} \end{equation}
Next, using the same argument in Lemma \ref{tracepf}, we obtain
\begin{equation} \label{xparavfbdp} \begin{split}
& \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f (t) \|_{L^\infty(\gamma \setminus \gamma_0) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel} f (t) \|_\infty,
\\ & \sup_{0 \le t \le T} \| \langle v \rangle^{5 + \delta } \nabla_{v} f (t) \|_{L^\infty(\gamma \setminus \gamma_0) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{5 + \delta } \nabla_{v} f (t) \|_\infty,
\\ & \sup_{0 \le t \le T} \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t,x,v) \|_{L^\infty(\gamma \setminus \gamma_0 ) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{5+\delta} \alpha \partial_{x_3} f (t,x,v) \|_\infty. \end{split} \end{equation}
Together \eqref{diffusefinal}, we conclude \eqref{diffusedfbd}.
\end{proof}
In order to construct a solution to the system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{diffuseBC}, we define a sequence of functions: \[ f^0(t,x,v) = f_0(x,v), \ E^0(t,x) = E_0(t,x), \ B^0(t,x) = B_0(x). \] For $\ell \ge 1$, let $f^\ell$ be the solution of \begin{equation} \label{fellseq} \begin{split} \partial_t f^\ell + \hat v \cdot \nabla_x f^\ell + \mathfrak F^{\ell-1} \cdot \nabla_v f^\ell = & 0, \text{ where } \mathfrak F^{\ell-1} = E^{\ell-1} + E_{\text{ext}} + \hat v \times ( B^{\ell-1} + B_{\text{ext}} ) - g \mathbf e_3, \\f^\ell(0,x,v) = & f_0(x,v) ,
\\ f^\ell(t,x,v) |_{\gamma_-} = & c_\mu \mu(v) \int_{u_3 < 0 } - f^{\ell-1} (t,x, u ) \hat u _3 du. \end{split} \end{equation} Let $\rho^\ell = \int_{\mathbb R^3 } f^\ell dv, j^\ell = \int_{\mathbb R^3 } \hat v f^\ell dv$. Let \begin{equation} \label{ElBl} E^\ell = \eqref{Eesttat0pos} + \dots + \eqref{Eest3bdrycontri}, \ B^\ell = \eqref{Besttat0pos} + \dots + \eqref{Bestbdrycontri}, \text{ with } f \text{ changes to } f^\ell. \end{equation} And let \begin{equation} \label{Fell} \mathfrak F^\ell = E^\ell + E_{\text{ext}} - \hat v \times ( B^\ell + B_{\text{ext}} ) - g \mathbf e_3. \end{equation} We prove several uniform-in-$\ell$ bounds for the sequence before passing the limit.
\begin{lemma} Suppose $f_0$ satisfies \eqref{f0bdd}, $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{E0B0bdd}, then there exits $M_1, M_2$, and $c_0$, such that for $0 < T \ll 1 $, \begin{equation} \label{fellbound} \begin{split}
\sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } f^\ell(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} \right) < & M_1, \
\\ \sup_\ell \sup_{0 \le t \le T} \left( \| E^\ell (t) \|_\infty + \| B^\ell (t) \|_\infty \right) + |B_e| + E_e + g < & M_2, \\ \inf_{\ell} \inf_{ t ,x_\parallel} \left( g - E_e - E^\ell_3(t,x_\parallel, 0 ) - (\hat v \times B^\ell)_3(t,x_\parallel, 0 ) \right) > & c_0. \end{split} \end{equation} \end{lemma} \begin{proof} Let $\ell \ge 1$. By induction hypothesis we assume that \begin{equation} \label{inductfEB} \begin{split}
\sup_{ 0 \le i \le \ell } \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta} f^{\ell-i}(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} \right) < & M_1,
\\ \sup_{ 0 \le i \le \ell } \sup_{0 \le t \le T} \left( \| E^{\ell-i} (t) \|_\infty + \| B^{\ell-i} (t) \|_\infty \right) + |B_e| + E_e + g< & M_2. \end{split} \end{equation}
Let the characteristics $(X^\ell, V^\ell)$ be defined as in \eqref{XV_ell}. We define the stochastic cycles:
\begin{equation}
\begin{split}\label{cycle}
t^{\ell}_1 (t,x,v)&:=
\sup\{ s<t:
X^\ell(s;t,x,v) \in \partial\Omega
\}
,\\
x^\ell_1 (t,x,v ) &:= X^\ell (t^{\ell}_1 (t,x,v);t,x,v)
,\\
t^{\ell-1}_2 (t,x,v, v_1) &:= \sup\{ s<t^\ell_1:
X^{\ell-1}(s;t^{\ell}_1 (t,x,v),x^{\ell}_1 (t,x,v),v_1) \in \partial\Omega
\}
,\\
x^{\ell-1}_2 (t,x,v, v_1 ) &:= X^{\ell-1} (t^{\ell-1}_2 (t,x,v,v_1);t^\ell_1(t,x,v),x^\ell_1(t,x,v),v_1)
,\\
\end{split}
\end{equation}
and inductively
\begin{equation}
\begin{split}\label{cycle_ellin}
& t^{\ell-(k-1)}_k (t,x,v, v_1, \cdots, v_{k-1}) \\
&:= \sup\big\{ s<t^{\ell-(k-2)}_{k-1}
:
X^{\ell-1}(s;t_{k-1}^{\ell - (k-2)} , x_{k-1}^{\ell - (k-2)} ,v_{k-1}) \in \partial\Omega
\big\},\\
& x_k^{\ell - (k-1)} (t,x,v, v_1, \cdots, v_{k-1})\\
&:= X^{\ell- (k-2)} (t_k^{\ell- (k-1)}; t_{k-1}^{\ell- (k-2)},x_{k-1}^{\ell- (k-2)} , v_{k-1})
.
\end{split}
\end{equation}
Here,
\begin{equation} \begin{split}\notag
t^{\ell-(i-1)}_{i } &:= t^{\ell-(i-1)}_{i }
(t,x,v,v_1, \cdots, v_{i-1}),\\
x^{\ell-(i-1)}_{i } &:= x^{\ell-(i-1)}_{i }
(t,x,v,v_1, \cdots, v_{i-1}).\end{split}\end{equation}
First, we note that for any $t^{\ell-i}_{i+1} \le s < t^{\ell-(i-1)}_{i}$, since \[ V^{\ell - i} (s; t^{\ell-(i-1)}_{i}, x^{\ell-(i-1)}_{i }, v_i ) = v_i - \int_{s}^{ t^{\ell-(i-1)}_{i} } \mathfrak F^{\ell - i} (\tau, X^{\ell-i}(\tau), V^{\ell-i}(\tau) ) d\tau, \] from \eqref{inductfEB}, we have \[
|v_i | - (t^{\ell-(i-1)}_{i} - t^{\ell-i}_{i+1})M_2 \le | V^{\ell - i} (s; t^{\ell-(i-1)}_{i}, x^{\ell-(i-1)}_{i }, v_i ) | \le |v_i | + (t^{\ell-(i-1)}_{i} - t^{\ell-i}_{i+1})M_2. \] Thus \begin{equation} \label{Vsv} \left( 1 + (t^{\ell-(i-1)}_{i} - t^{\ell-i}_{i+1})M_2 \right)^{-1} \langle v \rangle \le \langle V^{\ell - i} (s; t^{\ell-(i-1)}_{i}, x^{\ell-(i-1)}_{i }, v_i ) \rangle \le \left( 1 + (t^{\ell-(i-1)}_{i} - t^{\ell-i}_{i+1})M_2 \right) \langle v \rangle. \end{equation}
From \eqref{fellseq} we have for any $(t,x,v) \in (0,T) \times \bar \Omega \times \mathbb R^3$, \begin{equation} \begin{split} f^{\ell+1}(t,x,v) = & \mathbf 1_{t^\ell_1 \le 0} f^{\ell+1}( 0, X^\ell(0), V^\ell(0)) + \mathbf 1_{t_1^\ell \ge 0 } f^{\ell+1}(t_1^\ell , X^\ell(t_1^\ell; t,x,v), V^\ell(t_1^\ell;t,x,v) ) \\ = & \mathbf 1_{t^\ell_1 \le 0} f^{\ell+1}( 0, X^\ell(0), V^\ell(0)) - \mathbf 1_{t_1^\ell \ge 0 } c_\mu \mu(V^\ell(t^\ell_{1}) ) \int_{v_{1,3} < 0 } f^\ell (t^\ell_1, x^\ell_1, v_1 ) \hat{v}_{1,3} \mathrm{d} v_1. \end{split} \end{equation} And \eqref{Vsv} gives \begin{equation} \begin{split}
& \langle v \rangle^{4 + \delta } |f^{\ell+1}(t,x,v) |
\\ \le & \mathbf 1_{t^\ell_1 \le 0} (1+ T (M_2+g))^{4 + \delta} | \langle V^\ell(0) \rangle^{4 + \delta } f^{\ell+1}( 0, X^\ell(0), V^\ell(0)) |
\\ & + \mathbf 1_{t_1^\ell \ge 0 } c_\mu (1+ T (M_2+g))^{4 + \delta} | \langle V^\ell(t^\ell_{1}) \rangle^{4 + \delta} \mu(V^\ell(t^\ell_{1}) ) \int_{v_{1,3} < 0 } \langle v_1 \rangle^{4 +\delta} f^\ell (t^\ell_1, x^\ell_1, v_1 ) \frac{\hat{v}_{1,3}}{\langle v_1 \rangle^{4 + \delta } } \mathrm{d} v_1 | . \end{split} \end{equation}
Then inductively, we obtain \begin{equation} \label{vfinftyinduc} \begin{split}
& \langle v \rangle^{4 + \delta } | f^{\ell+1}(t,x,v) | \le \mathbf 1_{t^\ell_1 \le 0} (1+t(M_2+g))^{4 + \delta} | | \langle V^\ell(0) \rangle^{4 + \delta } f^{\ell+1}( 0, X^\ell(0), V^\ell(0)) |
\\ &+ (1+ T (M_2+g))^{4 + \delta} \int_{\prod_{j=1}^{k-1} \mathcal V_j} \sum_{i=1}^{k-1} \textbf{1}_{\{t^{\ell-i}_{i+1} \le 0 < t^{\ell-(i-1)}_{i} \}} | \langle V^{\ell - i } (0; v_i) \rangle^{4 + \delta } f^{\ell - (i-1)}(0 , X^{\ell-i}(0; v_i) , V^{\ell-i}(0;v_i) ) | \, d \Sigma_{i}^{k-1}
\\ & + (1+ T (M_2+g))^{4 + \delta} \int_{\prod_{j=1}^{k-1} \mathcal V_j} \textbf{1}_{\{t_k^{\ell - (k-1)} > 0 \}} \int_{\mathcal V_k} | f^{\ell - k } ( t_k^{\ell - (k-1) }, x_k^{\ell-(k-1) }, v_{k} ) | d v_k d \Sigma_{k-1}^{k-1}, \end{split} \end{equation} where \[ \begin{split} X^{\ell-i }(0;v_i) = & X^{\ell-i}(0; t^{\ell - (i-1) }_i, x^{\ell-(i-1) }_i, v_i), \\ V^{\ell-i }(0;v_i) = & V^{\ell-i}(0; t^{\ell - (i-1) }_i, x^{\ell-(i-1) }_i, v_i), \end{split} \] $\mathcal V_j = \{ v_j \in \mathbb R^3: v_{j,3} < 0 \}$, and \begin{equation} \label{Nujsigmaprod} \begin{split}
d \Sigma_i^{k -1 } = & \{\prod_{j=1}^{i-1} c_\mu (1+ T (M_2+g) )^{4 + \delta} \mu(V^{\ell-(j-1)}(t^{\ell- (j-1)}_{j } ) ) \frac{\hat v_{j,3} \langle V^{\ell-(j-1)}(t^{\ell- (j-1)}_{j } ) \rangle^{4 + \delta} }{ {\hat V}^{\ell-(j-1)}_3(t^{\ell- (j-1)}_{j } ) \langle v_j \rangle^{4 + \delta} } d v_j\} \{\prod_{j=i+1}^{k-1} \mu(v_j) c_\mu | \hat v_{j,3} | dv_j \}.
\end{split} \end{equation} From the same argument as in \eqref{trajexpansionendterm}-\eqref{tailtermsmall}, we get there exists $k_0 \gg 1$ such that for $k \ge k_0$, \begin{equation} \label{1over2k}
\int_{\prod_{j=1}^{k-1} \mathcal V_j} \textbf{1}_{\{t_k^{\ell - (k-1)} > 0 \}} d \Sigma_{k-1}^{k-1} \le \left( \frac{1}{2} \right)^{k }. \end{equation} Thus, from \eqref{vfinftyinduc}, \eqref{1over2k}, we have \begin{equation} \begin{split}
& \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } f^{\ell+1}(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)}
\\ & \le k (1 + TM_2)^{4 + \delta } \| \langle v \rangle^{4 + \delta } f_0 \|_\infty + (1 + T(M_2+g))^{4 + \delta } \left( \frac{1}{2} \right)^{k } \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta} f^{\ell-i}(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)}. \end{split} \end{equation} By choosing $M_1 \gg 1$ and then $ T \ll 1$, we get \begin{equation} \label{fell1final}
\sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } f^{\ell+1} (t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} < M_1. \end{equation} Now from \eqref{ElBl} and \eqref{E0B0g}, using the same argument as \eqref{BEellinftyestinflow1}--\eqref{BEellinftyestinflow4}, we get \begin{equation} \label{EBlifinal}
\sup_{0 \le t \le T} \| E^{\ell+1} (t) \|_\infty + \sup_{0 \le t \le T} \| B^{\ell+1} (t) \|_\infty + |B_e| +E_e + g < M_2, \end{equation} and \begin{equation}
\inf_{ t ,x_\parallel} \left( g - E_e - E^{\ell+1}_3(t,x_\parallel, 0 ) - (\hat v \times B^{\ell+1})_3(t,x_\parallel, 0 ) \right) > c_0. \end{equation} Thus we conclude \eqref{fellbound} by induction. \end{proof}
Next, we consider the derivative of the sequences. Define $\alpha^{\ell} $ as in \eqref{alphan}.
We have the following estimate. \begin{lemma} Suppose $f_0$ satisfies \eqref{f0bdd}, $E_0$, $B_0$ satisfy \eqref{E0B0bdd}, then there exits $M_3, M_4$ such that for $0 < T \ll 1 $, \begin{equation} \label{flElBldsqbd} \begin{split}
&\sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_\infty \right)
\\& + \sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right) < M_3 ,
\\ & \sup_\ell \sup_{0 \le t \le T} \left( \| \partial_t E^\ell(t) \|_\infty + \| \partial_t B^\ell(t) \|_\infty + \| \nabla_{x } E^\ell(t) \|_\infty +\| \nabla_{x } B^\ell(t) \|_\infty \right) < M_4. \end{split} \end{equation} \end{lemma} \begin{proof} The proof is essentially the same as the proof of Proposition \ref{diffuseprop}. The only difference is that instead of using the stochastic cycles \eqref{diffusecycles0}-\eqref{diffusecycles2} that flows under fixed $E(t,x), B(t,x)$, we use the \eqref{cycle}-\eqref{cycle_ellin} that flows with a different $E^\ell(t,x), B^\ell(t,x)$ after each bounce.
From the uniform estimate \eqref{fellbound}, and from the velocity lemma (Lemma \ref{vlemma}), we have for some $C>0$,
\begin{equation} e^{-C|t-s| } \alpha^\ell(t,x,v) \le \alpha^\ell(s, X^\ell(s;t,x,v) , V^\ell(s;t,x,v) ) \le e^{C|t-s| } \alpha^\ell(t,x,v), \ \text{ for all } \ell. \end{equation} Therefore, following the same proof of Proposition \ref{diffuseprop} we get \begin{equation} \label{} \begin{split}
& \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 +\delta } \nabla_{x_\parallel} f^{\ell+1} (t) \|_\infty + \| \langle v \rangle^{5 +\delta } \alpha^\ell \partial_{x_3} f ^{\ell+1} (t) \|_\infty + \| \langle v \rangle^{5 +\delta } \nabla_{v} f^{\ell+1} (t) \|_\infty \right)
\\ + & \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right)
\\ \le & C_k \left( \| \langle v \rangle^{4 +\delta } \nabla_{x_\parallel} f_0 \|_\infty + \| \| \langle v \rangle^{4 +\delta } \alpha \partial_{x_3} f_0 \|_\infty + \| \| \langle v \rangle^{5 +\delta } \nabla_{v} f_0 \|_\infty \right)
\\ & + C \left( \frac{1}{2} \right)^k \sup_{0 \le i \le \ell } \bigg( \sup_{0 \le t \le T } \left( \| \langle v\rangle^{4 +\delta} \nabla_{x_\parallel} f^i(t) \|_\infty + \| \langle v\rangle^{5 +\delta} \alpha \partial_{x_3} f^i(t) \|_\infty + \| \langle v \rangle^{5 +\delta } \nabla_{v} f^i(t) \|_\infty \right)
\\ & + \sup_{0 \le i \le \ell} \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^i(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{i-1} \partial_{x_3 } f^i(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{4 + \delta } \nabla_v f^i(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right) \bigg). \end{split} \end{equation} Thus, by choosing $k \gg 1 $ and $M_3 \gg 1$, we conclude \begin{equation} \label{dfelluni} \begin{split}
& \sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_\infty \right) \\& + \sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right)< M_3. \end{split} \end{equation} From this, we use the same argument to get \eqref{dxEBfinal} in the proof of Lemma \ref{EBW1inftylemma} and obtain \[ \begin{split}
\sup_{0 \le t \le T} & \left( \| \partial_t E^{\ell+1}(t) \|_\infty + \| \partial_t B^{\ell+1}(t) \|_\infty + \| \nabla_{x } E^{\ell+1}(t) \|_\infty +\| \nabla_{x } B^{\ell+1}(t) \|_\infty \right)
\\ \le & TC \sup_{1 \le i \le \ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{i}(t) \|_\infty + \| \partial_t B^{i}(t) \|_\infty + \| \nabla_{x } E^{i}(t) \|_\infty +\| \nabla_{x } B^{i}(t) \|_\infty \right)
\\ & C \left( \| E_0 \|_{C^2 } + \| B_0 \|_{C_2} \right) + C \sup_{0 \le t \le T} \left( \| \langle v \rangle ^{4 + \delta } \nabla_{x_\parallel } f^{\ell +1}(t) \|_\infty + \| \langle v \rangle^{\ell + \delta } \alpha^{n} \partial_{x_3 } f^{\ell +1}(t) \|_\infty \right)
\\ & + C \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right)
\\ & + C \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } f^{\ell+1}(t) \|_\infty + \| E^{\ell +1} (t) \|_\infty + \| B^{\ell +1} (t) \|_\infty \right). \end{split} \] From \eqref{fellbound} and \eqref{dfelluni}, this gives \[ \begin{split}
\sup_{\ell} \sup_{0 \le t \le T} & \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right)
\\ \le & TC \sup_{ \ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right)
\\ & + C \left( \| E_0 \|_{C^2 } + \| B_0 \|_{C_2} \right)+ C(M_1 +M_2 +M_3). \end{split} \] Therefore, by choosing $M_4 \gg 1$ and $T \ll 1$, we get \begin{equation}
\sup_{\ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right) < M_4. \end{equation} Together with \eqref{dfelluni}, we conclude \eqref{flElBldsqbd}. \end{proof}
Note that from \eqref{flElBldsqbd}, using the argument in Lemma \ref{EBelltrace} we have $E^\ell |_{\partial \Omega}, B^\ell |_{\partial \Omega} \in L^\infty((0,T) \times \partial \Omega ) $ for all $\ell$. Next, we prove the strong convergence of the sequence $f^\ell$.
\begin{lemma} \label{fEBsollemma} Suppose $f_0$ satisfies \eqref{f0bdd}, $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{E0B0bdd}. There exists functions $(f,E,B)$ with $ \langle v \rangle^{4 +\delta } f(t,x,v) \in L^\infty( (0, T) ; L^\infty( \bar \Omega \times \mathbb R^3 ) ) $, and $(E,B) \in L^\infty((0,T) ; L^\infty( \Omega) \cap L^\infty( \partial \Omega ) )$, such that as $\ell \to \infty$, \begin{equation} \label{EnBnconverge}
\sup_{0 \le t \le T} \left( \| E^\ell (t) - E(t) \|_{L^\infty( \Omega)} + \| E^\ell (t) - E(t) \|_{L^\infty( \partial \Omega)} + \| B^\ell (t) - B(t) \|_{L^\infty( \Omega)} + \| B^\ell (t) - B(t) \|_{L^\infty( \partial \Omega)} \right) \to 0, \end{equation} and \begin{equation} \label{fnconverge}
\sup_{0 \le t \le T} \| \langle v \rangle^{4 +\ell } f^\ell(t) - \langle v \rangle^{4 +\delta } f (t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} \to 0.
\end{equation} Moreover, $(f,E,B)$ is a (weak) solution of the system \eqref{VMfrakF1}--\eqref{rhoJ1}, and \eqref{diffuseBC}.
\end{lemma} \begin{proof}
Let $m > n \ge 1$. Note that $f^m - f^n $ satisfies $(f^m - f^n ) |_{t = 0 } = 0 $ and \[
(f^m- f^n )|_{\gamma_-} = c_\mu \mu \int_{\gamma_+} (f^{m-1} - f^{n-1} )(t,x,u ) \hat u_3 du. \]
The equation for $f^m - f^n $ is \[ \partial_t(f^m- f^n ) + \hat v \cdot \nabla_x (f^m - f^n ) + \mathfrak F^{m-1} \cdot \nabla_v (f^{m} - f^n ) = - ( \mathfrak F^{m-1} - \mathfrak F^{n-1} ) \cdot \nabla_v f^n. \] Thus, for any $(t,x,v) \in (0,T) \times \bar \Omega \times \mathbb R^3$, using \eqref{Vsv}, we get \[ \begin{split}
& | \langle v \rangle^{4 + \delta } (f^m - f^n)(t,x,v) |
\\ \le & C_1 \int_{\max \{ t^{m-1}_1 , 0 \} }^t | \langle V^{m-1}(s) \rangle^{4 + \delta } ( \mathfrak F^{m-1} - \mathfrak F^{n-1} ) \cdot \nabla_v f^n )(s, X^{m-1}(s), V^{m-1}(s) ) | ds
\\ & + \mathbf 1_{t^{m-1}_1 > 0 } C_{1} c_\mu V^{m-1}(t_1^m) \mu (V^m(t_1^m ) ) \int_{v_{1,3} < 0 } |(f^{m-1} - f^{n-1} )(t_1^{m-1}, x_1^{m-1} , v_1 ) \hat v_{1,3} | dv_1, \end{split} \] where $C_{1} = (1+ T (M_2+g) )^{4 + \delta}$ . Doing this inductively, we obtain \begin{equation} \label{fnmiterate2} \begin{split}
& | \langle v \rangle^{4 + \delta } (f^m - f^n)(t,x,v) | \\ \le & C_{1} \int_{\prod_{j=1}^{k-1} \mathcal V_j} \sum_{ i =1}^{k-1} \int_{\max \{ t^{m-i}_{i} , 0 \} }^{t^{m -(i-1) }_{ i-1} } \mathbf{1}_{ \{ t^{m -i }_i \le 0 < t^{m -(i-1) }_{ i-1} \} }
\\ & \quad \quad \quad \quad \times | \langle V^{m-i}(s) \rangle^{4 + \delta } ( \mathfrak F^{m-i} - \mathfrak F^{n-i} ) \cdot \nabla_v f^{n-(i-1)} )(s, X^{m-i}(s), V^{m-i}(s) ) | ds d \Sigma_{i}^{k-1}
\\ & + C_1 \int_{\prod_{j=1}^{k-1} \mathcal V_j} \textbf{1}_{\{t_{k-1}^{m - (k-1)} > 0 \}} \int_{\mathcal V_k} | (f^{m - k } - f^{n-k} )( t_k^{m - (k-1) }, x_k^{m-(k-1) }, v_{k} ) | d v_k d \Sigma_{k-1}^{k-1}.
\end{split} \end{equation} Where, $\mathcal V_j$ and $\Sigma_{i}^{k-1}$ are in \eqref{Nujsigmaprod}. Then from \eqref{1over2k} and \eqref{flElBldsqbd}, by fixing $k \gg 1$,we get \begin{equation} \label{fellnmiterate1} \begin{split}
\| \langle v \rangle^{4 + \delta} f^m(t) - \langle v \rangle^{4 + \delta} f^n)(t) \|_\infty \le & C_k C_1 \left( \sup_{\ell} \sup_{0 \le s \le t } \| \langle v \rangle^{4 + \delta} \nabla_v f^\ell(s) \|_\infty \right) \int_0^t \sup_{1 \le i \le k} \| \mathfrak F^{m-i} (s) - \mathfrak F^{n-i}(s) \|_\infty ds
\\ \le & C_2 \int_0^t \sup_{1 \le i \le k} \| \mathfrak F^{m-i} (s) - \mathfrak F^{n-i}(s) \|_\infty ds, \end{split} \end{equation} where $C_2 = C_k C_1M_3$.
Now,
from \eqref{Fell} and using the same argument as Lemma \ref{EBlinflemma} with \eqref{fellnmiterate1}, we have \begin{equation} \label{iterate2} \begin{split}
\| \mathfrak F^{m-i} (s) - \mathfrak F^{n-i}(s) \|_\infty \le & \| E^{n-i}(s) - E^{m-i}(s) \|_\infty + \| B^{n-i}(s) - B^{m-i}(s) \|_\infty
\\ \le & C \left( \sup_{0 \le s' \le s } \| \langle v \rangle^{4 + \delta } (f^{n-i} - f^{m-i} )(s' ) \|_\infty + \int_0^s \| \mathfrak F^{m-i-1} (s') - \mathfrak F^{n-i-1 }(s') \|_\infty ds' \right)
\\ \le & C \int_0^s \sup_{1 \le i_1 \le k } \| \mathfrak F^{m-i - i_1} (s') - \mathfrak F^{n-i - i_1}(s') \|_\infty ds'
\\ \le & C \int_0^s \sup_{1 \le i \le 2k} \left( \| E^{m-i}(s') - E^{n-i}(s') \|_\infty + \| B^{m-i}(s') - B^{n-i}(s') \|_\infty \right) ds'. \end{split} \end{equation} Iteration of \eqref{iterate2} and using \eqref{fellbound} yields \[ \begin{split}
& \| E^{m}(t) - E^{n}(t) \|_\infty + \| B^{m}(t) - B^{n}(t) \|_\infty
\\ \le & C^2 \int_0^t \int_0^s \sup_{1 \le i \le 2k} \left( \| E^{m-i}(s') - E^{n-i}(s') \|_\infty + \| B^{m-i}(s') - B^{n-i}(s') \|_\infty \right) ds' ds
\\ = & C^2 \int_0^t \tau \sup_{1 \le i \le 2k } \left( \| E^{m-i}(\tau) - E^{n-i}(\tau) \|_\infty + \| B^{m-i}(\tau) - B^{n-i}(\tau) \|_\infty \right) d\tau
\\ \le & C^l \int_0^t \frac{ \tau^{l-1}}{ (l-1)!} \sup_{1 \le i \le lk} \left( \| E^{m-i}(\tau) - E^{n-i}(\tau) \|_\infty + \| B^{m-i}(\tau) - B^{n-i}(\tau) \|_\infty \right) d\tau
\\ \le &M_2 \frac{C^l t^l}{l!}. \end{split} \]
Thus the sequences $E^\ell$, $B^\ell$ are Cauchy in $L^\infty((0,T) \times \Omega ) $, moreover, from Lemma \ref{EBelltrace}, $E^\ell, B^\ell \in L^\infty([0,T] \times \partial \Omega )$. Therefore, there exists functions $E,B \in L^\infty((0,T) ; L^\infty( \Omega) \cap L^\infty( \partial \Omega ) )$, such that \begin{equation} \label{EnBncov} E^\ell \to E, B^\ell \to B \text{ in } L^\infty((0,T) \times \Omega ) \cap L^\infty((0,T) \times \partial \Omega ) . \end{equation}
This proves \eqref{EnBnconverge}. Also, from \eqref{fellnmiterate1}, \eqref{iterate2}, \[
\| \langle v\rangle^{4 + \delta} f^m(t) - \langle v \rangle^{4 + \delta} f^n)(t) \|_{L^\infty((0,T) \times \bar \Omega ) } \le M_2 \frac{C^{l-1} t^{l-1}}{(l-2)!}, \] therefore we get \eqref{fnconverge}.
Now, take any $\phi(t,x,v) \in C_c^\infty( [0,T) \times \bar \Omega \times \mathbb R^3$ with $\text{supp } \phi \subset \{ [0, T) \times \bar \Omega \times \mathbb R^3 \} \setminus \{ (0 \times \gamma ) \cup (0,T) \times \gamma_0 \} $, from \eqref{fellseq}, we have \begin{equation} \label{weakfellVM} \begin{split} & \int_{\Omega \times \mathbb R^3 } f_0 \phi (0) dv dt + \int_0^T \int_{\Omega \times \mathbb R^3} f^\ell \left( \partial_t \phi + \hat v \cdot \nabla_x \phi + \mathfrak F^{\ell-1} \cdot \nabla_v \phi \right) dv dx dt \\ = & \int_0^T \int_{\gamma_+} \phi f^\ell \hat v_3 dv dS_x + \int_0^T \int_{\gamma_+ } \left( - c_\mu \int_{u_3 > 0 } \mu(u) \phi (t,x,u )\hat u_3 du \right) \hat v_3 f^\ell \, dv dS_x. \end{split} \end{equation} Because of the strong convergence \eqref{EnBnconverge}, \eqref{fnconverge}, we have that as $\ell \to \infty$, each term in \eqref{weakfellVM} goes to the corresponding terms with $f^\ell$ replaced by $f$ and $\mathfrak F^\ell$ replaced by $\mathfrak F$. Therefore we conclude that $(f,E,B)$ satisfy \eqref{weakf}.
Next, using the same argument as in \eqref{Maxwellell}-\eqref{EBellweak2}, we get $(f,E,B)$ satisfy \eqref{Maxweak1} and \eqref{Maxweak2}.
Therefore, we conclude that $(f,E,B)$ is a (weak) solution of the RVM system \eqref{VMfrakF1}--\eqref{rhoJ1} with diffuse BC \eqref{diffuseBC}. \end{proof}
In the next lemma, we consider the regularity of the solution.
\begin{lemma} \label{fEBreg} Let $\alpha(t,x,v)$ be defined as in \eqref{alphadef}. The solution $(f,E,B)$ obtained in Lemma \ref{fEBsollemma} satisfies \begin{equation} \label{pfbdlimit}
\| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f(t) \|_\infty + \| \langle v \rangle^{5 + \delta } \alpha^{} \partial_{x_3 } f(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_v f(t) \|_\infty < \infty, \end{equation} and \begin{equation} \label{pEBbdlimit}
\| \partial_t E(t) \|_\infty + \| \partial_t B(t) \|_\infty + \| \nabla_{x } E(t) \|_\infty +\| \nabla_{x } B(t) \|_\infty < \infty. \end{equation} \end{lemma} \begin{proof} From the $L^\infty$ strong convergence \eqref{EnBnconverge}, and the uniform-in-$\ell$ bound \eqref{flElBldsqbd}, we can pass the limit up to subsequence if necessary and get the weak$-*$ convergence \begin{equation} \label{dEnBncov} \partial_t E^\ell \overset{\ast}{\rightharpoonup} \partial_t E , \ \nabla_x E^\ell \overset{\ast}{\rightharpoonup} \nabla_x E, \ \partial_t B^\ell \overset{\ast}{\rightharpoonup} \partial_t B, \ \nabla_x B^\ell \overset{\ast}{\rightharpoonup} \nabla_x B \text{ in } L^\infty((0,T) \times \Omega ), \end{equation} and \begin{equation} \label{dfnconverge} \langle v \rangle ^{4 + \delta} \nabla_{x_\parallel } f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{4 + \delta}\nabla_{x_\parallel } f, \ \langle v \rangle ^{4 + \delta} \nabla_{v } f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{4 + \delta}\nabla_{v } f \text{ in } L^\infty((0,T) \times \Omega \times \mathbb R^3 ). \end{equation} Then using the same argument as in \eqref{ap3fellcovin}--\eqref{px3alphacov3}, we also have \begin{equation} \label{ap3fellcov} \langle v \rangle ^{5 + \delta} \alpha^{\ell-1} \partial_{x_3} f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{4 + \delta} \alpha \partial_{x_3} f \text{ in } L^\infty((0,T) \times \Omega \times \mathbb R^3 ). \end{equation}
Therefore, from using the weak lower semi-continuity of the weak-$*$ convergence \eqref{dEnBncov}, \eqref{dfnconverge}, \eqref{ap3fellcov}, and the uniform-in-$\ell$ bound \eqref{flElBldsqbd}, we conclude \eqref{pfbdlimit}, \eqref{pEBbdlimit}.
\end{proof}
Next, we prove the uniqueness of the solutions of the RVM system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{diffuseBC}.
\begin{lemma} \label{VMuniqlemma} Suppose $(f,E_f, B_f)$ and $(g, E_g, B_g)$ are solutions to the VM system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{diffuseBC} with $f(0) = g(0)$, $E_f(0) = E_g(0)$, $B_f(0) = B_g(0)$, and that \[ E_f, B_f, E_g, B_g \in W^{1,\infty}((0,T) \times \Omega ), \ \nabla_x \rho_{f}, \nabla_x J_f, \partial_t J_f , \nabla_x \rho_{g}, \nabla_x J_g, \partial_t J_g \in L^\infty((0,T); L_{\text{loc}}^p(\Omega)) \text{ for some } p>1. \] And \begin{equation} \label{dvfgbd}
\sup_{0 < t < T} \| \langle v \rangle^{5+ \delta} \nabla_v f(t) \|_\infty <\infty, \sup_{0 < t < T} \| \langle v \rangle^{5+ \delta} \nabla_v g(t) \|_\infty <\infty. \end{equation} Then $f = g, E_f = E_g, B_f = B_g$. \end{lemma} \begin{proof}
The difference function $f-g $ satisfies \begin{equation} \label{fminusgeq} \begin{split} (\partial_t + \hat v \cdot \nabla_x + \mathfrak F_f \cdot \nabla_v)(f-g) = (\mathfrak F_g - \mathfrak F_f ) \cdot \nabla_v g
\\ (f-g)(0) = 0, \, (f- g )|_{\gamma_- } = c_\mu \mu(v) \int_{u_3 < 0 } - (f - g) (t,x, u ) \hat u _3 du, \end{split} \end{equation} where \[ \mathfrak F_f = E_f + E_{\text{ext}} + \hat v \times ( B_f + B_{\text{ext}}) - g \mathbf e_3 , \, \mathfrak F_g = E_g +E_{\text{ext}} + \hat v \times ( B_g + B_{\text{ext} } ) - g \mathbf e_3, \] so \begin{equation} \label{mathfrakFfg} \mathfrak F_g - \mathfrak F_f = E_f - E_g + \hat v \times (B_f - B_g ). \end{equation} From Lemma \ref{Maxtowave} we have $E_{f,1} - E_{g,1} , E_{f,2} - E_{g,2}, B_{f,3} - B_{g,3}$ solve the wave equation with the Dirichlet boundary condition \eqref{waveD} in the sense of \eqref{waveD_weak} with \begin{align}
u_0 = 0, \ u_1 = 0 , \ G = -4\pi \partial_{x_i} (\rho_f - \rho_g) - 4 \pi \partial_t (J_{f,i} - J_{g,i} ), \ g = 0 , \ \ \text{for} \ E_{f,i} - E_{g,i}, i =1,2, \label{E12sol_A} \\
u_0 = 0, \ u_1 = 0, \ G = 4 \pi (\nabla_x \times (J_f -J_g) )_3, \ g = 0, \ \ \text{for} \ B_{f,3}- B_{g,3}, \label{B3sol_A}
\end{align} respectively. And $E_{f,3} - E_{g,3}, B_{f,1} - B_{g,1}, B_{f,2}- B_{g,2}$ solve the wave equation with the Neumann boundary condition \eqref{waveNeu} in the sense of \eqref{waveinner} \text{ with } \begin{align}
u_0 = 0, \ u_1 = 0 , \ G = -4\pi \partial_{x_3} ( \rho_f - \rho_g) - 4 \pi \partial_t (J_{f,3} - J_{g,3} ) , \ g = - 4\pi (\rho_f - \rho_g), \ \ \text{for} \ E_{f,3} - E_{g,3}, \label{E3sol_A} \\
u_0 = 0, \ u_1 = 0, \ G = 4 \pi (\nabla_x \times (J_f - J_g) )_i, \ g = (-1)^{i+1} 4 \pi (J_{f,{\underline i}} - J_{g, \underline i } ), \ \ \text{for} \ B_{f,i} - B_{j,i}, \ i=1,2, \label{B12sol_A}
\end{align} respectively. Therefore, from Lemma \ref{wavesol} and Lemma \ref{wavesolD}, we know that $E_f - E_g$ and $B_f - B_g$ would have the form of \begin{equation} \label{EBdiffform} \begin{split} & E_f - E_g = \eqref{Eesttat0pos} + \dots + \eqref{Eest3bdrycontri}, \ B_f -B_g = \eqref{Besttat0pos} + \dots + \eqref{Bestbdrycontri}, \\ & \text{ with } E_0, B_0 \text{ changes to } 0, \text{ and } f \text{ changes to } f -g. \end{split} \end{equation}
Now consider the characteristics \[ \begin{split} \dot X_f(s;t,x,v) = & \hat V_f(s;t,x,v) , \\ \dot V_f(s;t,x,v) = & \mathfrak F_f(s, X_f(s;t,x,v), V_f(s;t,x,v) ) . \end{split} \] Then from \eqref{fminusgeq}, same as \eqref{fnmiterate2}, we obtain \begin{equation} \label{fnmiterate2final} \begin{split}
& | \langle v \rangle^{4 + \delta } (f - g)(t,x,v) | \\ \le & C_{1} \int_{\prod_{j=1}^{k-1} \mathcal V_j} \sum_{ i =1}^{k-1} \int_{\max \{ t^{}_{i} , 0 \} }^{t^{ }_{ i-1} } \mathbf{1}_{ \{ t^{}_i \le 0 < t^{ }_{ i-1} \} }
\\ & \quad \quad \quad \quad \times | \langle V_f(s) \rangle^{4 + \delta } ( \mathfrak F_g - \mathfrak F_f ) \cdot \nabla_v f^{} )(s, X_f^{}(s), V_f^{}(s) ) | ds d \Sigma_{i}^{k-1}
\\ & + C_1 \int_{\prod_{j=1}^{k-1} \mathcal V_j} \textbf{1}_{\{t_{}^{m - (k-1)} > 0 \}} \int_{\mathcal V_k} | (f - g )( t_k^{}, x_k^{ }, v_{k} ) | d v_k d \Sigma_{k-1}^{k-1}.
\end{split} \end{equation} So using \eqref{tailtermsmall} and \eqref{1over2k}, we have
\begin{equation} \label{fgdiffrep}
\sup_{ 0 \le s \le t } \| \langle v \rangle^{5 + \delta} (f-g)(s) \|_\infty \le C \int^t_0 \| (\mathfrak F_g - \mathfrak F_f )(s) \|_\infty \|\langle v \rangle^{5 + \delta} \nabla_v g (s) \|_\infty ds. \end{equation} Now, from \eqref{EBdiffform} and the estimate in Lemma \ref{EBlinflemma}, we have \begin{equation} \label{FgFfdiff} \begin{split}
\| (\mathfrak F_g - \mathfrak F_f )(s) \|_\infty \le & \| (E_f - E_g )(s) \|_\infty + \| (B_f - B_g )(s) \|_\infty
\\ \le & C \sup_{0 \le s' \le s } \| \langle v \rangle^{5 + \delta} (f-g )(s' ) \|_\infty, \end{split}
\end{equation} and from the assumption \eqref{dvfgbd}, $\sup_{0 \le s \le t } \|(1 + |v |^{5 + \delta } ) \nabla_v g (s) \|_\infty < C$. Therefore from \eqref{fgdiffrep} and \eqref{FgFfdiff}, we have \begin{equation}
\sup_{0 \le s \le t } \| \langle v \rangle^{5 +\delta} (f-g)(s) \|_\infty \le C' \int^t_{0 } \sup_{0 \le s' \le s } \| \langle v \rangle^{5 +\delta}(f-g )(s' ) \|_\infty ds. \end{equation} Therefore from Gronwall \[
\sup_{0 \le s' \le t } \| \langle v \rangle^{5 +\delta} (f-g)(s') \|_\infty \le e^{C't} \| \langle v \rangle^{5 +\delta} (f-g)(0) \|_\infty = 0. \] Therefore we conclude that the solutions to \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{diffuseBC} is unique. \end{proof}
\begin{proof}[proof of Theorem \ref{main2}] Using the sequence $f^\ell, E^\ell , B^\ell$ constructed in \eqref{fellseq}, \eqref{ElBl}, we have from Lemma \ref{fEBsollemma} that the limit $(f,E,B)$ is a solution to the VM system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{diffuseBC}. This proves the existence. From Lemma \ref{fEBreg}, we have the regularity estimate \eqref{inflowfreg}, \eqref{inflowEBreg}. And from Lemma \ref{VMuniqlemma}, we conclude the uniqueness. \end{proof}
\section{Specular BC} \label{chapspec} In this section we consider the solution $f$ of the Vlasov-Maxwell system \eqref{VMfrakF1} satisfies the specular reflection boundary condition \eqref{spec}. We have the following a priori estimate for $f$. \begin{proposition} \label{specBCprop} Let $(f,E,B)$ be a solution of \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{spec}. Suppose the fields satisfies \eqref{gbig10}, and \begin{equation} \label{pEBassspec}
\sup_{0 \le t \le T} \left(\| \nabla_{x} E(t) \|_\infty + \| \nabla_{x} B(t) \|_\infty \right) < \infty. \end{equation} Assume that for some $\delta > 0$, and some $C > 0 $ such that \[ \begin{split}
\| \langle v \rangle^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha \langle v \rangle } } } \nabla_{x} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha \langle v \rangle } } } \nabla_v f_0 \|_\infty < \infty, \end{split} \] then there exists a $0 < T \ll 1$ small enough such that \begin{equation} \label{specdfbd} \begin{split}
& \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4+\delta} \nabla_{x} f(t) \|_\infty + \| \langle v \rangle^{4+\delta} \nabla_{v} f(t) \|_\infty \right)
\\ & + \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4+\delta} \nabla_{x} f(t) \|_{L^\infty(\gamma \setminus \gamma_0 ) } + \| \langle v \rangle^{4+\delta} \nabla_{v} f(t) \|_{L^\infty(\gamma \setminus \gamma_0 ) } \right) < \infty. \end{split} \end{equation} \end{proposition}
Let $(t,x,v) \in (0,T) \times \bar \Omega \times \mathbb R^3$. Recall the definition of $t_{\mathbf{b}}(t,x,v), x_{\mathbf{b}}(t,x,v), v_{\mathbf{b}}(t,x,v)$ in \eqref{tb}. Now let $(t^{0}, x^{0}, v^{0}) = (t,x,v).$ We define the specular cycles, for $\ell\geq 0,$ \[ (t^{\ell+1}, x^{\ell+1},v^{\ell+1}) = (t^{\ell}-t_{\mathbf{b}}(t^\ell, x^{\ell}, v^{\ell}), x_{\mathbf{b}}(t^\ell,x^{\ell},v^{\ell}), v_{\mathbf b}(t^\ell, x^{\ell}, v^{\ell}) - 2 v_{\mathbf b , 3}(t^\ell, x^{\ell}, v^{\ell}) \mathbf e_3 ). \] And we define the generalized characteristics for the specular BC as \begin{equation} \label{cycles} \begin{split} X_{\mathbf{cl}}(s;t,x,v) \ = \ \sum_{\ell} \mathbf{1}_{[t^{\ell+1},t^{ \ell})}(s) X(s;t^\ell, x^\ell , v^\ell ), \ \ V_{\mathbf{cl}}(s;t,x,v) \ = \ \sum_{\ell} \mathbf{1}_{[t^{\ell+1},t^{ \ell})}(s) V(s;t^\ell, x^\ell , v^\ell ). \end{split} \end{equation}
The key to prove Proposition \ref{specBCprop} is the following estimate for the derivative of the characteristics under the specular reflection.
\begin{lemma} \label{dXVcl} For any $(t,x,v) \in (0,T) \times \bar \Omega \times \mathbb R^3$, and $0 \le s \le t $, let $\partial_{\mathbf e } \in \{ \nabla_x , \nabla_v \} $, then for some $C_1 \gg 1$, we have \begin{equation}\label{lemma_Dxv} \begin{split}
| \partial_{\mathbf e } X_{\mathbf{cl}}(s;t,x,v)| & \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } } ,
\\ | \partial_{\mathbf e } V_{\mathbf{cl}}(s;t,x,v)| & \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } }. \end{split} \end{equation} \end{lemma}
\begin{proof} We need to estimate along the bounces: \begin{equation}\label{chain} \begin{split} & \frac{\partial ( X_{\mathbf{cl}}(s;t,x,v),V_{\mathbf{cl}}(s;t,x,v))}{\partial (x,v)}\\
= &\underbrace{\frac{\partial ( X_{\mathbf{cl}}(s), V_{\mathbf{cl}}(s))}{\partial (t^{\ell_{*}}, {x}_{\parallel_{\ell_{*}}}^{\ell_{*}}, {v}_{3_{\ell_{*}}}^{\ell_{*}}, {v}_{\parallel_{\ell_{*}}}^{\ell_{*}})} }_{\text{from the last bounce to the }s-\text{plane}} \times \underbrace{\prod_{\ell=1}^{\ell_*} \frac{\partial (t^{\ell+1}, {x}_{\parallel}^{\ell+1}, {v}_{3 }^{\ell+1},{v}_{\parallel }^{\ell+1})}{\partial (t^{\ell}, {x}_{\parallel }^{\ell}, {v}_{3 }^{\ell},{v}_{\parallel }^{\ell})} }_{\text{ intermediate groups}}
\times \underbrace{ \frac{\partial (t^{1}, {x}_{\parallel_{1}}^{1}, {v}_{3_{1}}^{1}, {v}_{\parallel_{1}}^{1})}{\partial (x,v)}}_{\text{from the } t-\text{plane to the first bounce} }. \end{split} \end{equation}
We first find out the matrix of derivatives in the intermediate groups from the $\ell$-th bounce to the $(\ell + 1)$-th bounce: \begin{equation} \label{Jell1matrix} \begin{split} J^{\ell+1}_{\ell}&:= \frac{\partial (t^{\ell+1}, {x}_{\parallel}^{\ell+1}, {v}_{3 }^{\ell+1},{v}_{\parallel }^{\ell+1})}{\partial (t^{\ell}, {x}_{\parallel }^{\ell}, {v}_{3 }^{\ell},{v}_{\parallel }^{\ell})}. \end{split} \end{equation} We have \begin{equation} \label{ellellplus1} (t^\ell - t^{\ell +1 } ) \hat v_3^\ell = \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \hat{\mathfrak F}_3(\tau ) d\tau ds. \end{equation} Taking $\frac{\partial}{\partial t^\ell } \eqref{ellellplus1} $ gives \[ \begin{split} (1 - \frac{\partial t^{\ell+1} }{\partial t^\ell } ) \hat v_3^\ell = - \frac{\partial t^{\ell+1} }{\partial t^\ell } \int_{t^{\ell+1}}^{t^\ell } \hat{\mathfrak F}_3(\tau ) d\tau + \int_{t^{\ell +1}}^{t^\ell } \hat{\mathfrak F}_3( t^\ell ) ds + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } d\tau ds, \end{split} \] so \[ \begin{split} - \hat v_3^{\ell +1 } \frac{\partial t^{\ell+1} }{\partial t^\ell } = & - ( \hat v_3^\ell - \int_{t^{\ell +1}}^{t^\ell } \hat{\mathfrak F}_3( t^\ell ) ds ) + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } d\tau ds \\ = & - \hat v_3^{\ell+1} + \int_{t^{\ell+1}}^{t^\ell } ( \hat{\mathfrak F}_3( t^\ell ) - \hat{\mathfrak F}_3( s ) ) ds + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } d\tau ds \\ = & - \hat v_3^{\ell+1} + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } \right) d\tau ds , \end{split} \] thus \begin{equation} \label{ptltlplus1} \frac{\partial t^{\ell+1} }{\partial t^\ell } = 1 - \frac{1}{ \hat v_3^{\ell +1 } } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } \right) d\tau ds. \end{equation} Taking $\frac{\partial}{\partial t^{\ell } } $ derivative to \begin{equation} \label{xell1rep} x_{\parallel}^{\ell + 1 } = x_\parallel^\ell - (t^\ell - t^{\ell +1 } ) \hat v_\parallel^\ell + \int_{t^{\ell + 1 }}^{t^\ell } \int_s^{t^\ell } \hat{ \mathfrak F}_\parallel (\tau ) d\tau ds, \end{equation} we get \begin{equation} \begin{split} \frac{ \partial x_\parallel^{\ell+1}}{\partial t^\ell } = & - ( 1 - \frac{\partial t^{\ell +1}}{\partial t^\ell } ) \hat v_\parallel^\ell - \frac{\partial t^{\ell +1}}{\partial t^\ell } \int_{t^{\ell+1}}^{t^\ell } \hat {\mathfrak F}_\parallel(\tau) d\tau + \int_{t^{\ell+1}}^{t^\ell } \hat {\mathfrak F}_\parallel(t^\ell) d\tau + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_\parallel (\tau )}{ \partial_{t^\ell } } d\tau ds \\ & = - \hat v_\parallel^\ell + \frac{\partial t^{\ell +1}}{\partial t^\ell } \hat v_\parallel^{\ell + 1 } + \int_{t^{\ell+1}}^{t^\ell } \hat {\mathfrak F}_\parallel(t^\ell) d\tau + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_\parallel (\tau )}{ \partial_{t^\ell } } d\tau ds \\ & = - \hat v_\parallel^\ell + \hat v_\parallel^{\ell + 1 } - \frac{\hat v_\parallel^{\ell+1} }{ \hat v_3^{\ell +1 } } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } \right) d\tau ds + \int_{t^{\ell+1}}^{t^\ell } \hat {\mathfrak F}_\parallel(t^\ell) d\tau + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_\parallel (\tau )}{ \partial_{t^\ell } } d\tau ds \\ & = \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_\parallel (\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_\parallel (\tau )}{ \partial_{t^\ell } } \right) d\tau ds - \frac{\hat v_\parallel^{\ell+1} }{ \hat v_3^{\ell +1 } } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } \right) d\tau ds. \end{split} \end{equation} And taking $\frac{\partial}{\partial t^{\ell } } $ derivative to \begin{equation} \label{vell1rep} v_\parallel^{\ell+1} = v_\parallel^\ell - \int_{t^{\ell+1}}^{t^\ell} \mathfrak F_\parallel(s) ds, \end{equation} we get \begin{equation} \begin{split} \frac{ \partial v_\parallel^{\ell+1}}{\partial t^\ell } = & - \mathfrak F_\parallel(t^\ell) + \frac{\partial t^{\ell +1}}{\partial t^\ell } \mathfrak F_\parallel(t^{\ell+1}) - \int_{t^{\ell+1}}^{t^\ell} \frac{ \partial \mathfrak F_\parallel(s) }{\partial t^\ell } ds \\ = & - \mathfrak F_\parallel(t^\ell) + F_\parallel(t^{\ell+1}) - \frac{ \mathfrak F_\parallel(t^{\ell+1} ) }{ \hat v_3^{\ell +1 } } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } \right) d\tau ds - \int_{t^{\ell+1}}^{t^\ell} \frac{ \partial \mathfrak F_\parallel(s) }{\partial t^\ell } ds \\ = & - \frac{ \mathfrak F_\parallel(t^{\ell+1} ) }{ \hat v_3^{\ell +1 } } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } \right) d\tau ds - \int_{t^{\ell+1}}^{t^\ell } \left( \frac{ \partial {\mathfrak F}_\parallel (s )}{ \partial_{s} } + \frac{ \partial {\mathfrak F}_\parallel (s )}{ \partial_{t^\ell } } \right) ds. \end{split} \end{equation} Similarly, taking taking $\frac{\partial}{\partial t^{\ell } } $ derivative to \begin{equation} \label{v3ell1rep} v_3^{\ell+1} = -v_3^\ell - \int_{t^{\ell+1}}^{t^\ell} \mathfrak F_3(s) ds, \end{equation} we get \begin{equation} \label{v3tlderi} \begin{split} \frac{ \partial v_3^{\ell+1}}{\partial t^\ell } = & - \mathfrak F_3(t^\ell) + \frac{\partial t^{\ell +1}}{\partial t^\ell } \mathfrak F_3(t^{\ell+1}) - \int_{t^{\ell+1}}^{t^\ell} \frac{ \partial \mathfrak F_3(s) }{\partial t^\ell } ds \\ = & - \mathfrak F_3(t^\ell) + F_3(t^{\ell+1}) - \frac{ \mathfrak F_3(t^{\ell+1} ) }{ \hat v_3^{\ell +1 } } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } \right) d\tau ds - \int_{t^{\ell+1}}^{t^\ell} \frac{ \partial \mathfrak F_3(s) }{\partial t^\ell } ds \\ = & - \frac{ \mathfrak F_3(t^{\ell+1} ) }{ \hat v_3^{\ell +1 } } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \left( \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } \right) d\tau ds - \int_{t^{\ell+1}}^{t^\ell } \left( \frac{ \partial {\mathfrak F}_3 (s )}{ \partial_{s} } + \frac{ \partial {\mathfrak F}_3 (s )}{ \partial_{t^\ell } } \right) ds. \end{split} \end{equation}
Now let's calculate the matrix $J_\ell^{\ell+1}$ in \eqref{Jell1matrix}. Taking $\partial_{\mathbf e } \in \{ \partial_{x_\parallel^\ell } , \partial_{v_3^\ell }, \partial_{v_\parallel^\ell } \}$ derivatives to \eqref{ellellplus1} we get \begin{equation} \label{partialtell1} \begin{split} \frac{ \partial t^{\ell+1}}{\partial x_\parallel^\ell } = & - \frac{1}{\hat v_3^{\ell +1} } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3 (\tau ) }{\partial_{x_\parallel^\ell } }d\tau ds, \\ \frac{ \partial t^{\ell+1}}{\partial v_3^\ell } = & \frac{t^\ell - t^{\ell +1 } }{\hat v_3^{\ell +1 } } \frac{ \partial \hat v_3^\ell }{\partial v_3^\ell } + \frac{1}{\hat v_3^{\ell +1} } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3 (\tau ) }{\partial_{v_3^\ell } }d\tau ds \\ = & \frac{(t^\ell - t^{\ell +1 } )}{\hat v_3^{\ell +1 } } \frac{ ( 1 - (\hat v_3^\ell )^2 )}{\langle v^\ell \rangle } + \frac{1}{\hat v_3^{\ell +1} } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3 (\tau ) }{\partial_{v_3^\ell } }d\tau ds, \\ \frac{ \partial t^{\ell+1}}{\partial v_\parallel^\ell } = & - \frac{1}{\hat v_3^{\ell +1} } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3 (\tau ) }{\partial_{v_\parallel^\ell } }d\tau ds. \end{split} \end{equation}
Taking $\partial_{\mathbf e } \in \{ \partial_{x_\parallel^\ell } , \partial_{v_3^\ell }, \partial_{v_\parallel^\ell } \}$ derivatives to \eqref{xell1rep} we get \begin{equation} \label{partialxell1} \begin{split} \frac{ \partial x_\parallel^{\ell+1}}{\partial {x_\parallel } ^\ell } = & \textbf{Id}_{2,2} + \frac{ \partial t^{\ell+1}}{\partial x_\parallel^\ell } \hat v_\parallel^{\ell +1 } + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_\parallel (\tau ) }{\partial_{x_\parallel^\ell } }d\tau ds, \\ \frac{ \partial x_\parallel^{\ell+1}}{\partial {v_3 } ^\ell } = & \frac{ \partial t^{\ell+1}}{\partial v_3^\ell } \hat v_\parallel^{\ell +1 } + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_\parallel (\tau ) }{\partial_{v_3^\ell } }d\tau ds, \\ \frac{ \partial x_\parallel^{\ell+1}}{\partial {v_\parallel } ^\ell } = & \frac{ \partial t^{\ell+1}}{\partial v_\parallel^\ell } \hat v_\parallel^{\ell +1 } - (t^\ell - t^{\ell +1 } ) \frac{ \partial \hat v_\parallel^\ell }{\partial v_\parallel^\ell } + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_\parallel (\tau ) }{\partial_{v_\parallel^\ell } }d\tau ds \\ = & \frac{ \partial t^{\ell+1}}{\partial v_\parallel^\ell } \hat v_\parallel^{\ell +1 } - (t^\ell - t^{\ell +1 } ) \frac{1 - (\hat v_\parallel^\ell ) ^2 }{\langle v^\ell \rangle } + \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_\parallel (\tau ) }{\partial_{v_\parallel^\ell } }d\tau ds . \end{split} \end{equation} Taking $\partial_{\mathbf e } \in \{ \partial_{x_\parallel^\ell } , \partial_{v_3^\ell }, \partial_{v_\parallel^\ell } \}$ derivatives to \eqref{vell1rep} we get \begin{equation} \label{partialvell1} \begin{split} \frac{ \partial v_\parallel^{\ell+1}}{\partial {x_\parallel } ^\ell } = & \frac{ \partial t^{\ell+1}}{\partial x_\parallel^\ell } \mathfrak F_\parallel (t^{\ell +1 } ) - \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_\parallel (s ) }{\partial_{x_\parallel^\ell } } ds, \\ \frac{ \partial v_\parallel^{\ell+1}}{\partial {v_3 } ^\ell } = & \frac{ \partial t^{\ell+1}}{\partial v_3^\ell } \mathfrak F_\parallel (t^{\ell +1 } ) - \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_\parallel (s ) }{\partial_{v_3^\ell } } ds, \\ \frac{ \partial v_\parallel^{\ell+1}}{\partial {v_\parallel } ^\ell } = & \textbf{Id}_{2,2} + \frac{ \partial t^{\ell+1}}{\partial v_\parallel^\ell } \mathfrak F_\parallel (t^{\ell +1 } ) - \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_\parallel (s ) }{\partial_{v_\parallel^\ell } } ds. \end{split} \end{equation} And finally, taking $\partial_{\mathbf e } \in \{ \partial_{x_\parallel^\ell } , \partial_{v_3^\ell }, \partial_{v_\parallel^\ell } \}$ derivatives to \eqref{v3ell1rep} we get \begin{equation} \label{partialv3ell1} \begin{split} \frac{ \partial v_3^{\ell+1}}{\partial {x_\parallel } ^\ell } = & \frac{ \partial t^{\ell+1}}{\partial x_\parallel^\ell } \mathfrak F_3 (t^{\ell +1 } ) - \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_3 (s ) }{\partial_{x_\parallel^\ell } } ds, \\ \frac{ \partial v_3^{\ell+1}}{\partial {v_3 } ^\ell } = & -1 - \frac{ \partial t^{\ell+1}}{\partial v_3^\ell } \mathfrak F_3 (t^{\ell +1 } ) + \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_3 (s ) }{\partial_{v_3^\ell } } ds, \\ \frac{ \partial v_3^{\ell+1}}{\partial {v_\parallel } ^\ell } = & \frac{ \partial t^{\ell+1}}{\partial v_\parallel^\ell } \mathfrak F_3 (t^{\ell +1 } ) - \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_3 (s ) }{\partial_{v_\parallel^\ell } } ds. \end{split}
\end{equation} For the estimate, from \eqref{ptltlplus1} and that $| \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{\tau} } + \frac{ \partial \hat{\mathfrak F}_3(\tau )}{ \partial_{t^\ell } } | \lesssim \frac{1}{ \langle v \rangle } $, we obtain \begin{equation} \begin{split}
| \frac{\partial t^{\ell+1} }{\partial t^\ell } | & \le 1 + M \frac{ |t^\ell - t^{\ell +1} |^2 }{ \hat v_3^{\ell +1 } \langle v \rangle },
\\ | \frac{\partial x_\parallel^{\ell+1} }{\partial t^\ell } | & \le M \frac{ |t^\ell - t^{\ell +1} |^2 }{ \hat v_3^{\ell +1 } \langle v \rangle },
\\ | \frac{\partial v_\parallel^{\ell+1} }{\partial t^\ell } | & \le M( \frac{ |t^\ell - t^{\ell +1} |^2 }{ \hat v_3^{\ell +1 } \langle v \rangle } + \frac{|t^\ell - t^{\ell +1 } | }{\langle v \rangle } ),
\\ | \frac{\partial v_3^{\ell+1} }{\partial t^\ell } | & \le M( \frac{ |t^\ell - t^{\ell +1} |^2 }{ \hat v_3^{\ell +1 } \langle v \rangle } + \frac{ |t^\ell - t^{\ell +1 } | }{\langle v \rangle} ). \end{split} \end{equation}
Next, we estimate $\frac{ \partial \hat{\mathfrak F} (\tau ) }{\partial_{x_\parallel^\ell } } $, $\frac{ \partial \hat{\mathfrak F} (\tau ) }{\partial_{v^\ell } }$, $\frac{ \partial {\mathfrak F} (\tau ) }{\partial_{x_\parallel^\ell } } $, $\frac{ \partial {\mathfrak F} (\tau ) }{\partial_{v^\ell } }$. Since \[ \begin{split}
\frac{ \partial \hat{\mathfrak F} (\tau ) }{\partial_{x_\parallel^\ell } } = \nabla_{x_\parallel} \hat{\mathfrak F}(\tau ) \cdot \partial_{x_\parallel^\ell } X_\parallel (\tau) + \partial_{x_3} \hat{\mathfrak F}(\tau ) \cdot \partial_{x_\parallel^\ell } X_3 (\tau) + \nabla_v \hat{\mathfrak F}(\tau ) \cdot \partial_{x_\parallel^\ell } V (\tau). \end{split} \] From \eqref{pxviXVest}, \eqref{pxiXj}, \eqref{nablaxhatF}, \eqref{nablavhatF}, we have \begin{equation} \notag \begin{split}
| \frac{ \partial \hat{\mathfrak F} (\tau ) }{\partial_{x_\parallel^\ell } }| \lesssim \frac{1}{\langle V(\tau ) \rangle^2 } |t^\ell - t^{\ell + 1 } | + \frac{1}{ \langle V(\tau ) \rangle }. \end{split} \end{equation} And, \begin{equation} \notag \begin{split}
| \frac{ \partial \hat{\mathfrak F} (\tau ) }{\partial_{v^\ell } } | = | \nabla_{x} \hat{\mathfrak F}(\tau ) \cdot \partial_{v^\ell } X (\tau) + \nabla_v \hat{\mathfrak F}(\tau ) \cdot \partial_{v^\ell } V (\tau) | \lesssim \frac{|t^\ell - t^{\ell +1} | }{ \langle V(\tau ) \rangle^2 } + \frac{1}{ \langle V(\tau ) \rangle^2 }. \end{split} \end{equation} Similarly, from \eqref{pxviXVest}, \eqref{pxiXj}, \eqref{nablaxfrakF} and \eqref{nablavfrakF}, \[ \begin{split}
| \frac{ \partial {\mathfrak F} (\tau ) }{\partial_{x_\parallel^\ell } } | = & | \nabla_{x_\parallel} {\mathfrak F}(\tau ) \cdot \partial_{x_\parallel^\ell } X_\parallel (\tau) + \partial_{x_3} {\mathfrak F}(\tau ) \cdot \partial_{x_\parallel^\ell } X_3 (\tau) + \nabla_v {\mathfrak F}(\tau ) \cdot \partial_{x_\parallel^\ell } V (\tau)| \lesssim \frac{1}{ \langle V(\tau) \rangle } |t^\ell - t^{\ell + 1 } | + 1,
\\ | \frac{ \partial {\mathfrak F} (\tau ) }{\partial_{v^\ell } } | = & | \nabla_{x} {\mathfrak F}(\tau ) \cdot \partial_{v^\ell } X (\tau) + \nabla_v {\mathfrak F}(\tau ) \cdot \partial_{v^\ell } V (\tau) | \lesssim \frac{|t^\ell - t^{\ell +1} | }{ \langle V(\tau ) \rangle } + \frac{1}{ \langle V(\tau ) \rangle }. \end{split} \] Thus, we have \begin{equation} \label{phatFpxvellest} \begin{split}
\int_{t^{\ell+1}}^{t^\ell } \int_s^{t^\ell } | \frac{ \partial \hat{\mathfrak F} (\tau ) }{\partial_{x_\parallel^\ell } }| d\tau ds \lesssim & \frac{ |t^\ell - t^{\ell +1 } |^2}{\langle v \rangle}, \ \ \int_{t^{\ell+1}}^{t^\ell } \int_s^{t^\ell } | \frac{ \partial \hat{\mathfrak F} (\tau ) }{\partial_{v^\ell } }| d\tau ds \lesssim \frac{ |t^\ell - t^{\ell +1 } |^2}{\langle v \rangle^2},
\\ \int_{t^{\ell+1}}^{t^\ell } | \frac{ \partial {\mathfrak F} (\tau ) }{\partial_{x_\parallel^\ell } }| ds \lesssim & |t^\ell - t^{\ell +1 } | , \ \ \int_{t^{\ell+1}}^{t^\ell } | \frac{ \partial {\mathfrak F} (s ) }{\partial_{v^\ell } }| ds \lesssim \frac{ |t^\ell - t^{\ell +1 } | }{\langle v \rangle} . \end{split} \end{equation}
Thus, from \eqref{partialtell1} and \eqref{phatFpxvellest} , we obtain \begin{equation} \begin{split}
| \frac{ \partial t^{\ell+1}}{\partial x_\parallel^\ell } | \le & M \frac{ |t^\ell - t^{\ell+1} | ^2}{ \hat v_3^{\ell +1 } \langle v \rangle },
\\ | \frac{ \partial t^{\ell+1}}{\partial v_3^\ell } | \le & M \frac{ |t^\ell - t^{\ell+1} |}{ \hat v_3^{\ell +1 } \langle v \rangle },
\\ | \frac{ \partial t^{\ell+1}}{\partial v_\parallel ^\ell } | \le & M \frac{ |t^\ell - t^{\ell+1} |^2}{ \hat v_3^{\ell +1 } \langle v \rangle^2 }. \end{split} \end{equation} From \eqref{partialxell1} and \eqref{phatFpxvellest}, \begin{equation} \begin{split}
| \frac{ \partial x_\parallel^{\ell+1}}{\partial x_\parallel^\ell } | \le & 1 + M \frac{ |t^\ell - t^{\ell+1} | ^2}{ \hat v_3^{\ell +1 } \langle v \rangle },
\\ | \frac{ \partial x_\parallel^{\ell+1}}{\partial v_3^\ell } | \le & M \frac{ |t^\ell - t^{\ell+1} |}{ \hat v_3^{\ell +1 } \langle v \rangle },
\\ | \frac{ \partial x_\parallel^{\ell+1}}{\partial v_\parallel ^\ell } | \le & M \left( \frac{ |t^\ell - t^{\ell+1} |^2}{ \hat v_3^{\ell +1 } \langle v \rangle^2 } + \frac{|t^\ell - t^{\ell +1} | }{\langle v \rangle } \right) . \end{split} \end{equation} From \eqref{partialvell1} and \eqref{phatFpxvellest}, \begin{equation} \begin{split}
| \frac{ \partial v_\parallel^{\ell+1}}{\partial x_\parallel^\ell } | \le & M \left( \frac{ |t^\ell - t^{\ell+1} | ^2}{ \hat v_3^{\ell +1 } \langle v \rangle } + |t^\ell - t^{\ell +1} | \right) ,
\\ | \frac{ \partial v_\parallel^{\ell+1}}{\partial v_3^\ell } | \le & M \left( \frac{ |t^\ell - t^{\ell+1} |}{ \hat v_3^{\ell +1 } \langle v \rangle } + \frac{ |t^\ell - t^{\ell +1} | }{\langle v \rangle} \right),
\\ | \frac{ \partial v_\parallel^{\ell+1}}{\partial v_\parallel ^\ell } | \le & 1 + M \left( \frac{ |t^\ell - t^{\ell+1} |^2}{ \hat v_3^{\ell +1 } \langle v \rangle^2 } + \frac{ | t^\ell - t^{\ell+1} |}{\langle v \rangle} \right) . \end{split} \end{equation} From \eqref{partialv3ell1} and \eqref{phatFpxvellest}, \begin{equation} \begin{split}
| \frac{ \partial v_3^{\ell+1}}{\partial x_\parallel^\ell } | \le & M \left( \frac{ |t^\ell - t^{\ell+1} | ^2}{ \hat v_3^{\ell +1 } \langle v \rangle } + |t^\ell - t^{\ell +1} | \right) ,
\\ | \frac{ \partial v_3^{\ell+1}}{\partial v_\parallel ^\ell } | \le & M \left( \frac{ |t^\ell - t^{\ell+1} |^2}{ \hat v_3^{\ell +1 } \langle v \rangle^2 } + \frac{ | t^\ell - t^{\ell+1} |}{\langle v \rangle} \right) . \end{split}
\end{equation} Now we estimate $| \frac{ \partial v_3^{\ell+1}}{\partial v_3^\ell } | $. Notice that since \begin{equation} \label{v3elliint} \begin{split} 0 = x_3^{\ell} = & x_3^{\ell+1} + \int_{t^{\ell +1}}^{t^\ell } \hat V(s ) ds \\ = & (t^\ell - t^{\ell+1} ) \hat v_3^{\ell+1} + \int_{t^{\ell +1}}^{t^\ell } \int_{t^{\ell+1}}^s \hat {\mathfrak F}_3 (\tau) d\tau ds \\ = & (t^\ell - t^{\ell+1} ) \hat v_3^{\ell+1} + \frac{(t^\ell - t^{\ell+1} )^2}{2} \hat {\mathfrak F}_3 (t^{\ell+1} ) + \int_{t^{\ell +1}}^{t^\ell } \int_{t^{\ell+1}}^s \int_{t^{\ell+1}}^\tau \frac{d}{d \tau' } \hat {\mathfrak F}_3 (\tau') d\tau' d\tau ds, \end{split} \end{equation} and from \eqref{nablaxhatF}, \eqref{nablavhatF}, \[
| \frac{d}{d \tau' } \hat {\mathfrak F}_3 (\tau') | \le \frac{M}{\langle v \rangle } \]
where $M$ depends on $\| \nabla_x E \|_\infty$, $\| \nabla_x B \|_\infty$, and $g$. Thus
\begin{equation} 1 + \frac{(t^\ell - t^{\ell+1} )}{2 \hat v_3^{\ell+1} } \hat {\mathfrak F}_3 (t^{\ell+1} ) = M \frac{ |t^\ell - t^{\ell+1} |^2}{\hat v_3^{\ell+1} \langle v \rangle } . \end{equation} From \eqref{hatF}, \[
\hat {\mathfrak F}_3 (t^{\ell+1} ) = \frac{ 1 - |\hat v_3^{\ell+1} |^2 }{\langle v^{\ell+1} \rangle } \mathfrak F_3(t^{\ell+1} ) + O_{\mathfrak F} (1) \frac{ \hat v_3^{\ell+1} }{\langle v^{\ell+1} \rangle }. \] Therefore, \begin{equation} \label{vell3d1}
| 1 + \frac{(t^\ell - t^{\ell+1} )}{2 \hat v_3^{\ell+1} } \frac{ 1 - |\hat v_3^{\ell+1} |^2 }{\langle v^{\ell+1} \rangle } \mathfrak F_3(t^{\ell+1} ) | \le M \left( \frac{ |t^\ell - t^{\ell+1} |^2}{\hat v_3^{\ell+1} \langle v \rangle } + |t^\ell - t^{\ell+1} | \right). \end{equation} Also, notice that \begin{equation} \label{vell3d2}
| \frac{ ( 1 - (\hat v_3^\ell )^2 )}{\langle v^\ell \rangle } - \frac{ ( 1 - (\hat v_3^{\ell+1} )^2 )}{\langle v^{\ell+1} \rangle } | = | \int_{t^{\ell+1}}^{t^\ell} \frac{d}{ds} \left( \frac{ ( 1 - (\hat V^\ell_3(s) )^2 )}{\langle V^\ell(s) \rangle } \right) ds | \le M \frac{ |t^\ell - t^{\ell+1} |}{ \langle v \rangle ^2 }. \end{equation} Therefore from \eqref{phatFpxvellest}, \eqref{phatFpxvellest}, \eqref{vell3d1}, and \eqref{vell3d2}, we have \begin{equation} \begin{split} \frac{ \partial v_3^{\ell+1}}{\partial {v_3 } ^\ell } = & -1 - \frac{ \partial t^{\ell+1}}{\partial v_3^\ell } \mathfrak F_3 (t^{\ell +1 } ) + \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_3 (s ) }{\partial_{v_3^\ell } } ds \\ = & -1 - \left( \frac{(t^\ell - t^{\ell +1 } )}{\hat v_3^{\ell +1 } } \frac{ ( 1 - (\hat v_3^\ell )^2 )}{\langle v^\ell \rangle } + \frac{1}{\hat v_3^{\ell +1} } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3 (\tau ) }{\partial_{v_3^\ell } }d\tau ds \right) \mathfrak F_3 (t^{\ell +1 } )+ \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_3 (s ) }{\partial_{v_3^\ell } } ds \\ = & - 1 - \frac{(t^\ell - t^{\ell +1 } )}{\hat v_3^{\ell +1 } } \frac{ ( 1 - (\hat v_3^{\ell+1} )^2 )}{\langle v^{\ell+1} \rangle } \mathfrak F_3 (t^{\ell +1 } ) + \frac{(t^\ell - t^{\ell +1 } )}{\hat v_3^{\ell +1 } } \left( \frac{ ( 1 - (\hat v_3^\ell )^2 )}{\langle v^\ell \rangle } - \frac{ ( 1 - (\hat v_3^{\ell+1} )^2 )}{\langle v^{\ell+1} \rangle } \right) \mathfrak F_3 (t^{\ell +1 } ) \\ &- \frac{ \mathfrak F_3 (t^{\ell +1 } )}{\hat v_3^{\ell +1} } \int_{t^{\ell +1}}^{t^\ell } \int_s^{t^\ell } \frac{ \partial \hat{\mathfrak F}_3 (\tau ) }{\partial_{v_3^\ell } }d\tau ds + \int_{t^{\ell +1}}^{t^\ell } \frac{ \partial {\mathfrak F}_3 (s ) }{\partial_{v_3^\ell } } ds
\\ = & -1 + 2 + M \left( \frac{ |t^\ell - t^{\ell+1} |^2}{\hat v_3^{\ell+1} \langle v \rangle^2 } + \frac{ |t^\ell - t^{\ell +1 } | }{\langle v \rangle} \right)
\\ = & 1 + M \left( \frac{ |t^\ell - t^{\ell+1} |^2}{\hat v_3^{\ell+1} \langle v \rangle^2 } + \frac{ |t^\ell - t^{\ell +1 } | }{\langle v \rangle} \right). \end{split} \end{equation} Finally, from \eqref{tbbdvb}, \eqref{alphaest}, \[
|t^\ell - t^{\ell+1} | \lesssim \hat v_3^{\ell+1} \langle v \rangle \lesssim \alpha(t,x,v) \langle v \rangle, \] and from \eqref{v3elliint}, \[
|t^\ell - t^{\ell+1} | \hat v_3^{\ell+1} = - \int_{t^{\ell +1}}^{t^\ell } \int_{t^{\ell+1}}^s \hat {\mathfrak F}_3 (\tau) d\tau ds \lesssim \frac{|t^\ell - t^{\ell+1}|^2 g }{\langle v \rangle } , \] so by \eqref{alphaest}, \[
\alpha(t,x,v) \langle v \rangle \lesssim \hat v_3^{\ell+1} \langle v \rangle \lesssim |t^\ell - t^{\ell+1} |. \]
Therefore, we have for $\ell \ge 1$, there exists $c, C >0$ depending on $T, \| E \|_\infty, \|B\|_\infty$, and $g$ such that \begin{equation} \label{tdiffalpha}
c \alpha(t,x,v) \langle v \rangle \le |t^\ell - t^{\ell+1} | \le C \alpha(t,x,v) \langle v \rangle. \end{equation} Put together the above estimates and using \eqref{tdiffalpha}, we have for some $M>0$, \begin{equation} \begin{split} J^{\ell+1}_{\ell} \le &
\left[\begin{array}{c|cc|c|cc}
1 + M \alpha \langle v \rangle & M \alpha \langle v \rangle & M \alpha \langle v \rangle & M & M \alpha \langle v \rangle & M \alpha \langle v \rangle
\\ \hline M \alpha \langle v \rangle & 1 + M \alpha \langle v \rangle & M \alpha \langle v \rangle & M & M \alpha \langle v \rangle & M \alpha \langle v \rangle
\\ M \alpha \langle v \rangle & M \alpha \langle v \rangle & 1+ M \alpha \langle v \rangle & M & M \alpha \langle v \rangle & M \alpha \langle v \rangle
\\ \hline M \alpha \langle v \rangle & M \alpha \langle v \rangle & M \alpha \langle v \rangle & 1 + M \alpha \langle v \rangle & M \alpha \langle v \rangle & M \alpha \langle v \rangle
\\ \hline M \alpha \langle v \rangle & M \alpha \langle v \rangle & M \alpha \langle v \rangle & M & 1 + M \alpha \langle v \rangle & M \alpha \langle v \rangle
\\ M \alpha \langle v \rangle & M \alpha \langle v \rangle & M \alpha \langle v \rangle & M & M \alpha \langle v \rangle & 1+ M \alpha \langle v \rangle
\end{array} \right]
\\ : = & J(\alpha \langle v \rangle) . \end{split} \end{equation} From diagonalization, we get \[ J(\alpha \langle v \rangle ) = \mathcal P \Lambda \mathcal P^{-1}, \] where \[ \Lambda = \text{diag} \left[ 1, 1, 1, 1, 1 + M \left( \sqrt{\alpha \langle v \rangle ( 4 \alpha \langle v \rangle + 5 ) } + 3 \alpha \langle v \rangle \right) , 1 + M \left( - \sqrt{\alpha \langle v \rangle ( 4 \alpha \langle v \rangle + 5 ) } + 3 \alpha \langle v \rangle \right) \right], \] and \[ \begin{split} \mathcal P = & \begin{bmatrix} -1 & -1 & -1 & -1 & 1 & 1 \\ 1 & 0 & 0 & 0 & 1 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & \sqrt{\alpha \langle v \rangle ( 4 \alpha \langle v \rangle + 5 ) } - 2 \alpha \langle v \rangle & - \sqrt{\alpha \langle v \rangle ( 4 \alpha \langle v \rangle + 5 ) } - 2 \alpha \langle v \rangle \\ 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 & 1 \end{bmatrix}, \\ \mathcal P^{-1} = & \begin{bmatrix} -\frac{1}{5} & \frac{4}{5} & -\frac{1}{5} & 0 & -\frac{1}{5} & -\frac{1}{5} \\ -\frac{1}{5} & -\frac{1}{5} & \frac{4}{5} & 0 & -\frac{1}{5} & -\frac{1}{5} \\ -\frac{1}{5} & -\frac{1}{5} & -\frac{1}{5} & 0 & \frac{4}{5} & -\frac{1}{5} \\ -\frac{1}{5} & -\frac{1}{5} & -\frac{1}{5} & 0 & -\frac{1}{5} & \frac{4}{5} \\ a & a & a & b & a & a \\ - a & -a & -a & -b & -a & -a \end{bmatrix}, \end{split} \] where \[ a = \frac{ 2 \alpha \sqrt{ \langle v \rangle} + \sqrt{ \alpha( 4 \alpha \langle v \rangle + 5 )} }{10 \sqrt{ \alpha( 4 \alpha \langle v \rangle + 5 )} } , \ b = \frac{1}{2 \sqrt{ \alpha \langle v \rangle (4 \alpha \langle v \rangle + 5 ) }}. \] Now from \eqref{tdiffalpha}, the number of bounces \begin{equation} \label{ellstarbd}
\ell^*(0;t,x,v) \le \frac{T}{c \alpha(t,x,v) \langle v \rangle }. \end{equation} Therefore, \begin{equation} \begin{split}
\prod_{\ell = 1 }^{\ell^*(0;t,x,v) } J^{\ell+1}_{\ell } & \le J^{\ell^*(0;t,x,v) } \le \widetilde{ \mathcal P } \widetilde{ \Lambda} ^{\ell^*} \widetilde{ \mathcal P^{-1} }
\le ( 1 + 2 M \sqrt{\alpha \langle v \rangle } )^{\frac{1}{c \alpha \langle v \rangle } } \widetilde{ \mathcal P } \widetilde{ \mathcal P^{-1} }, \end{split}
\end{equation} where we use the notation: for a matrix $A$, the entries of a matrix $\widetilde A$ are absolute values of the entries of $A$, i.e. $(\widetilde A)_{ij} = | (A)_{ij} |$. From \[ ( 1 + 2 M \sqrt{\alpha \langle v \rangle } )^{\frac{1}{c \alpha \langle v \rangle } } = \left( \left( 1 + 2 M \sqrt{\alpha \langle v \rangle } \right)^{\frac{1}{ 2 M \sqrt{ \alpha \langle v \rangle } } } \right)^{ \frac{2M}{c \sqrt{ \alpha \langle v \rangle } } } \le e^{ \frac{2M}{c \sqrt{ \alpha \langle v \rangle } } }, \] and that $ \left( \widetilde{ \mathcal P } \widetilde{ \mathcal P^{-1} } \right)_{ij} \le \frac{M}{\sqrt{\alpha \langle v \rangle } } $, we get \begin{equation} \label{midij}
\left( \prod_{\ell = 1 }^{\ell^*(0;t,x,v) } J^{\ell+1}_{\ell } \right)_{ij} \le e^{ \frac{2M}{c \sqrt{ \alpha \langle v \rangle } } } \left(\widetilde{ \mathcal P } \widetilde{ \mathcal P^{-1} } \right)_{ij} \le \frac{M}{\sqrt{\alpha \langle v \rangle } } e^{ \frac{2M}{c \sqrt{ \alpha \langle v \rangle } } } . \end{equation} Next, we estimate $ \frac{\partial ( X_{\mathbf{cl}}(s), V_{\mathbf{cl}}(s))}{\partial (t^{\ell_{*}}, {x}_\parallel^{\ell_{*}}, {v}_{3}^{\ell_{*}}, {v}_{\parallel}^{\ell_{*}})} $ and $ \frac{\partial (t^{1}, {x}_{\parallel}^{1}, {v}_{3}^{1}, {v}_{\parallel}^{1})}{\partial (x,v)}$. From \[ \begin{split} X(s;t^{\ell^*},x^{\ell^*},v^{\ell^*}) = & x^{\ell^*} - (t^{\ell^*}-s) \hat{v}^{\ell^*} + \int^{t^{\ell^*}}_s\int^{t^{\ell^*}}_\tau \hat{\mathfrak F } _{}(\tau^\prime, X (\tau^\prime), V(\tau^\prime) ) \mathrm{d} \tau^\prime \mathrm{d} \tau, \\ V(s;t^{\ell^*},x^{\ell^*},v^{\ell^*}) = & v^{\ell^*} - \int_s^{t^{\ell^*}} \mathfrak F (\tau, X(\tau) , V(\tau) ) \mathrm{d} \tau, \end{split} \] we have \[ \begin{split} \frac{\partial X(s) }{\partial t^{\ell^*} } = & - \hat{v}^{\ell^*} + \int^{t^{\ell^*}}_s \hat{\mathfrak F } ( t^{\ell^*} ) + \int^{t^{\ell^*}}_s\int^{t^{\ell^*}}_\tau \frac{\partial \hat{\mathfrak F } _{}(\tau^\prime )}{\partial t^{\ell^*} } \mathrm{d} \tau^\prime \mathrm{d} \tau \\ = & - \hat V(s) + \int^{t^{\ell^*}}_s\int^{t^{\ell^*}}_\tau \frac{\partial \hat{\mathfrak F } _{}(\tau^\prime )}{\partial t^{\ell^*} } \mathrm{d} \tau^\prime \mathrm{d} \tau \\ \frac{\partial V(s) }{\partial t^{\ell^*} } = & - \mathfrak F( t^{\ell^*} ) - \int_s^{t^{\ell^*}} \frac{ \partial \mathfrak F (\tau )}{\partial t^{\ell^*} } \mathrm{d} \tau. \end{split} \]
Therefore, $ | \frac{\partial X(s) }{\partial t^{\ell^*} } | \lesssim | \hat V(s) | \lesssim 1$, and $ \frac{\partial V(s) }{\partial t^{\ell^*} } \lesssim 1 $. Combine this with \eqref{pxviXVest}, we have \begin{equation} \label{lastij}
\frac{\partial ( X_{\mathbf{cl}}(s), V_{\mathbf{cl}}(s))}{\partial (t^{\ell_{*}}, {x}_\parallel^{\ell_{*}}, {v}_{3}^{\ell_{*}}, {v}_{\parallel}^{\ell_{*}})} \le
\begin{bmatrix}
M & M & \frac{|t^{\ell^*}-s| }{\langle v \rangle }
\\ M & |t^{\ell^*}-s| & M
\end{bmatrix}. \end{equation} Lastly, since $t^1 = t_{\mathbf{b}}(t,x,v)$, $(x^1_\parallel, 0 ) = x_{\mathbf{b}}(t,x,v)$, $(v^1_3, v^1_\parallel) = v_{\mathbf{b}}(t,x,v)$, from \eqref{pxitb} and \eqref{pvitb}, \[
| \partial_x t^1 | \lesssim \frac{1}{ \hat {v }_3^1} \lesssim \frac{1}{\alpha} , \ | \partial_v t^1 | \lesssim \frac{t^1 }{ \hat {v }_3^1 \langle v \rangle } \lesssim M. \] And thus from \eqref{pxbvb}, we have \begin{equation} \label{firstij}
\frac{\partial (t^{1}, {x}_{\parallel}^{1}, {v}_{3}^{1}, {v}_{\parallel}^{1})}{\partial (x,v)}
\le
\begin{bmatrix} \frac{M}{\alpha} & M \\ 1 + \frac{M}{\alpha} & M \\ \frac{M}{\alpha} & M
\end{bmatrix}. \end{equation} Finally, combining \eqref{midij}, \eqref{lastij}, and \eqref{firstij}, we get for some $C_1= C_1(M,c)$, \begin{equation}
\left| \left( \frac{\partial ( X_{\mathbf{cl}}(s;t,x,v),V_{\mathbf{cl}}(s;t,x,v))}{\partial (x,v)} \right)_{ij} \right| \le \frac{ 36 M^3}{\alpha^{3/2} \sqrt{ \langle v \rangle } } e^{ \frac{2M}{c \sqrt{ \alpha \langle v \rangle } }} \le C_1 \langle v \rangle e^{\frac{C_1}{\sqrt{ \alpha \langle v \rangle } } }. \end{equation} \end{proof}
Proposition \ref{specBCprop} comes as a consequence of the lemma. \begin{proof}[proof of Proposition \ref{specBCprop}] For any $(t,x,v) \in (0,T) \times \bar \Omega \times \mathbb R^3$, let $\partial_{\mathbf e } \in \{ \nabla_x , \nabla_v \} $, then \begin{equation} \label{pef} \begin{split} \partial_{\mathbf e } f(t,x,v ) & = \partial_{\mathbf e} (f(0, X_{\mathbf{cl}}(0;t,x,v) , V_{\mathbf{cl}}(0;t,x,v) ) \\ & = \nabla_x f_0 \cdot \partial_{\mathbf e } X_{\mathbf{cl}}(0;t,x,v) + \nabla_v f_0 \cdot \partial_{\mathbf e } V_{\mathbf{cl}}(0;t,x,v). \end{split} \end{equation} Now, from \eqref{alphaest} and \eqref{lemma_Dxv}, write $X(0) = X_{\mathbf{cl}}(0;t,x,v)$, and $V(0) = V_{\mathbf{cl}}(0;t,x,v)$, we have \begin{equation} \label{nablaxfest1} \begin{split}
& \langle v \rangle^{4+\delta} | \partial_{\mathbf e } f (t,x,v ) |
\\ \le & C_1 | \nabla_x f(0,X(0), V(0) ) | \langle v \rangle^{5+\delta} e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } } + C_1 | \nabla_v f(0,X(0), V(0) ) | \langle v \rangle^{5+\delta} e^{\frac{C_1}{\sqrt{ \alpha(t,x,v) \langle v \rangle } } }
\\ \le & C | \nabla_x f(0,X(0), V(0) ) \langle |V(0)\rangle^{5+\delta} e^{\frac{C}{\sqrt{ \alpha(0,X(0),V(0)) \langle V(0) \rangle } } }
\\ & + C | \nabla_v f(0,X(0), V(0) ) \langle V(0) \rangle ^{5+\delta} e^{\frac{C}{\sqrt{ \alpha(0,X(0),V(0)) \langle V(0) \rangle } } }
\\ \le & C \left( \| \langle v \rangle^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha \langle v \rangle } } } \nabla_{x} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha \langle v \rangle } } } \nabla_v f_0 \|_\infty \right) . \end{split} \end{equation}
Next, using the same argument in Lemma \ref{tracepf}, we obtain
\begin{equation} \label{xparavfbdpspec} \begin{split}
& \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{x} f (t) \|_{L^\infty(\gamma \setminus \gamma_0) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{x} f (t) \|_\infty,
\\ & \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{v} f (t) \|_{L^\infty(\gamma \setminus \gamma_0) } \le \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } \nabla_{v} f (t) \|_\infty.
\end{split} \end{equation}
Combining with \eqref{nablaxfest1}, we conclude \eqref{specdfbd}. \end{proof}
We consider the sequence of functions: \[ f^0(t,x,v) = f_0(x,v), \ E^0(t,x) = E_0(t,x), \ B^0(t,x) = B_0(x). \] For $\ell \ge 1$, let $f^\ell$ be the solution of \begin{equation} \label{fellseqspec} \begin{split} \partial_t f^\ell + \hat v \cdot \nabla_x f^\ell + \mathfrak F^{\ell-1} \cdot \nabla_v f^\ell = & 0, \text{ where } \mathfrak F^{\ell-1} = E^{\ell-1} + E_{\text{ext}} + \hat v \times (B^{\ell-1} + B_{\text{ext} } ) - g \mathbf e_3, \\f^\ell(0,x,v) = & f_0(x,v) ,
\\ f^\ell(t,x,v) |_{\gamma_-} = & f^{\ell-1}(t,x,v_\parallel, - v_3 ). \end{split} \end{equation} Let $\rho^\ell = \int_{\mathbb R^3 } f^\ell dv, J^\ell = \int_{\mathbb R^3 } \hat v f^\ell dv$. Let \begin{equation} \label{ElBlspec} E^\ell = \eqref{Eesttat0pos} + \dots + \eqref{Eest3bdrycontri}, \ B^\ell = \eqref{Besttat0pos} + \dots + \eqref{Bestbdrycontri}, \text{ with } f \text{ changes to } f^\ell. \end{equation} And let \begin{equation} \label{Fellspec} \mathfrak F^\ell = E^\ell + E_{\text{ext}} - \hat v \times (B^\ell + B_{\text{ext} } ) - g \mathbf e_3. \end{equation} We prove several uniform-in-$\ell$ bounds for the sequence before passing the limit.
\begin{lemma} \label{fEBellbdlemspec} Suppose $f_0$ satisfies \eqref{f0bdd}, $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{E0B0bdd}, then there exits $M_1, M_2$ such that for $0 < T \ll 1 $, \begin{equation} \label{fellboundspec} \begin{split}
\sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } f^\ell(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} \right) < & M_1, \
\\ \sup_\ell \sup_{0 \le t \le T} \left( \| E^\ell (t) \|_\infty + \| B^\ell (t) \|_\infty \right) + |B_e| + E_e + g < & M_2, \\ \inf_{\ell} \inf_{ t ,x_\parallel} \left( g - E_e - E^\ell_3(t,x_\parallel, 0 ) - (\hat v \times B^\ell)_3(t,x_\parallel, 0 ) \right) > & c_0. \end{split} \end{equation} \end{lemma} \begin{proof} By induction hypothesis we assume that \begin{equation} \label{inductfEBspec} \begin{split}
\sup_{ 0 \le i \le \ell } \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta} f^{i}(t) \|_{L^\infty(\bar \Omega \times \mathbb R^3)} \right) < & M_1,
\\ \sup_{ 0 \le i \le \ell } \sup_{0 \le t \le T} \left( \| E^{i} (t) \|_\infty + \| B^{i} (t) \|_\infty \right) + |B_e| + E_e + g < & M_2. \end{split} \end{equation}
Denote the characteristics $(X^\ell, V^\ell)$ which solves \begin{equation} \label{specXV_ell} \begin{split} \frac{d}{ds}X^\ell(s;t,x,v) &= \hat V^\ell(s;t,x,v),\\ \frac{d}{ds} V^\ell(s;t,x,v) &= \mathfrak F^\ell (s, X^\ell(s;t,x,v),V^\ell(s;t,x,v) ).\end{split} \end{equation} And define the specular cycles:
\begin{equation}
\begin{split}\label{speccycle}
t^{\ell}_1 (t,x,v)&:=
\sup\{ s \ge 0 :
X^\ell(\tau;t,x,v) \in \Omega \text{ for all } \tau \in (t-s, t )
\}
,\\
x^\ell_1 (t,x,v ) &:= X^\ell (t^{\ell}_1 (t,x,v);t,x,v), \\ v^{\ell}_1(t,x,v) & : = V^\ell(t^\ell_1(t,x,v);t,x,v) - 2 V_3^\ell(t^\ell_1(t,x,v);t,x,v) \mathbf e_3
\end{split}
\end{equation}
and inductively for $k \ge 2$,
\begin{equation}
\begin{split}\label{cycle_ell}
& t^{\ell-(k-1)}_k (t,x,v) \\
&:= \sup\big\{ s \ge 0
:
X^{\ell-(k-1)}(\tau;t_{k-1}^{\ell - (k-2)} , x_{k-1}^{\ell - (k-2)} ,v^{\ell - (k-2) }_{k-1}) \in \Omega \text{ for all } \tau \in (t-s, t )
\big\},\\
& x_k^{\ell - (k-1)} (t,x,v)\\
&:= X^{\ell- (k-1)} (t_k^{\ell- (k-1)}; t_{k-1}^{\ell- (k-2)},x_{k-1}^{\ell- (k-2)} , v^{\ell - (k-2) } _{k-1}),
\\ & v_k^{\ell - (k-1)} (t,x,v)\\
&:= V^{\ell- (k-1)} (t_k^{\ell- (k-1)}; t_{k-1}^{\ell- (k-2)},x_{k-1}^{\ell- (k-2)} , v^{\ell - (k-2) } _{k-1}) - 2 V_3^{\ell- (k-1)} (t_k^{\ell- (k-1)}; t_{k-1}^{\ell- (k-2)},x_{k-1}^{\ell- (k-2)} , v^{\ell - (k-2) } _{k-1}) \mathbf e_3 .
\end{split}
\end{equation}
And we define the generalized characteristics for the specular BC as \begin{equation} \label{speccycles} \begin{split} & X^\ell_{\mathbf{cl}}(s;t,x,v) \ = \ \mathbf 1_{[t^\ell_1(t,x,v) , t ) } (s) X^\ell(s;t,x,v) + \sum_{k \ge 1} \mathbf{1}_{[t^{\ell-k}_{k+1},t^{ \ell- (k-1)}_{k})}(s) X^{\ell- k}(s;t^{\ell - (k-1)}_{k}, x^{\ell - (k-1) }_{k} , v^{\ell-(k-1)}_{k} ), \\ & V^\ell_{\mathbf{cl}}(s;t,x,v) \ = \ \mathbf 1_{[t^\ell_1(t,x,v) , t ) } (s) V^\ell(s;t,x,v) + \sum_{k \ge 1} \mathbf{1}_{[t^{\ell-k}_{k+1},t^{ \ell- (k-1)}_{k})}(s) V^{\ell- k}(s;t^{\ell - (k-1)}_{k}, x^{\ell - (k-1) }_{k} , v^{\ell-(k-1)}_{k} ). \end{split} \end{equation}
From \eqref{fellseqspec} and \eqref{speccycle}, for any $(t,x,v) \in (0,T) \times \bar \Omega \times \mathbb R^3$, let $k$ be such that $ t^{\ell-k}_{k+1}(t,x,v) \le 0 < t^{\ell- (k-1)}_{k}(t,x,v) $, then we have \begin{equation} \label{fellspecto0} \begin{split} f^{\ell+1}(t,x,v)
= & f^{\ell- (k-1) } \left(0, X^{\ell- k }_{}\left(0; t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) , V^{\ell-k}_{}\left(0;t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) \right) \\ = & f_0 \left( X^{\ell- k }_{}\left(0; t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) , V^{\ell-k}_{}\left(0;t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) \right). \end{split} \end{equation} Thus \begin{equation} \label{specv4f1}
\langle v \rangle^{4 + \delta } f^{\ell+1}(t,x,v) \le \frac{ \langle v \rangle ^{4 + \delta}}{ \left( V^{\ell-k}_{}\left(0;t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) \right)^{4 + \delta} } \| \langle v \rangle^{4 + \delta} f_0 \|_\infty. \end{equation} Now, since \[ \begin{split}
&| v - V^{\ell-k}_{}\left(0;t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) |
\\ & = \left| v - v^{\ell}_1 + \sum_{i=1}^{k-1} \left( v^{\ell - (i-1)}_i - v^{\ell - i}_{i+1} \right) + v^{\ell - ( k-1)}_k - V^{\ell-k}_{}\left(0;t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) \right|
\\ & \le \int_{t^\ell_1}^t \| \mathfrak F^\ell (s) \|_\infty ds + \sum_{i=1}^{k-1} \int_{t^{\ell - i }_{i+1}}^{t^{\ell -(i-1) }_i } \| \mathfrak F^{\ell - i } (s) \|_\infty ds + \int_{0}^{t^{\ell - (k-1) }_k } \| \mathfrak F^{\ell - k } (s) \|_\infty ds
\\ & \le \int_0^t \sup_{0 \le i \le l } \| \mathfrak F^i(s) \|_\infty ds. \end{split} \] From \eqref{inductfEBspec} we have \[
|v| \le \left| V^{\ell-k}_{}\left(0;t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) \right| + t M_2, \] and this yields \begin{equation} \label{vfracVspec} \begin{split} \frac{ \langle v \rangle }{ \langle V^{\ell-k}_{}\left(0;t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}\right) \rangle } < 1 + t M_2 . \end{split} \end{equation} Thus \eqref{specv4f1} gives \begin{equation} \label{fell1finalspec}
\sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta } f^{\ell+1}(t ) \|_{L^\infty(\bar \Omega \times \mathbb R^3 ) } < (1 + T (g+M_2) ) \| \langle v \rangle^{4 + \delta} f_0 \|_\infty < M_1, \end{equation} for $T$ small enough. Now from \eqref{ElBlspec} and \eqref{E0B0g}, using the same argument as \eqref{BEellinftyestinflow1}--\eqref{BEellinftyestinflow4}, we get \begin{equation} \label{EBlifinal}
\sup_{0 \le t \le T} ( \| E^{\ell+1} (t) \|_\infty + \| B^{\ell+1} (t) \|_\infty) + |B_e| + E_e + g < M_2, \end{equation} and \begin{equation}
\inf_{ t ,x_\parallel} \left( g - E_e - E^{\ell+1}_3(t,x_\parallel, 0 ) - (\hat v \times B^{\ell+1})_3(t,x_\parallel, 0 ) \right) > c_0. \end{equation} Thus we conclude \eqref{fellbound} by induction.
\end{proof}
Next, we define $\alpha^\ell(t,x,v)$ in the same way as in \eqref{alphan}. Then we have
\begin{lemma} Suppose $f_0$ satisfies \eqref{f0spec}, $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{E0B0bdd}, then there exits $M_3, M_4$ such that for $0 < T \ll 1 $, \begin{equation} \label{flElBldsqbdspec} \begin{split}
&\sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x } f^\ell(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_\infty \right)
\\& + \sup_\ell \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{4 + \delta } \nabla_v f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right) < M_3 ,
\\ & \sup_\ell \sup_{0 \le t \le T} \left( \| \partial_t E^\ell(t) \|_\infty + \| \partial_t B^\ell(t) \|_\infty + \| \nabla_{x } E^\ell(t) \|_\infty +\| \nabla_{x } B^\ell(t) \|_\infty \right) < M_4. \end{split} \end{equation} And with the $M_2$ as in Lemma \ref{fEBellbdlemspec}, \begin{equation} \label{EBellpObd}
\sup_\ell \sup_{0 \le t \le T} \left( \| E^\ell (t,x ) \|_{L^\infty( \partial \Omega ) } + \| B^\ell (t,x ) \|_{L^\infty( \partial \Omega ) } \right) < M_2. \end{equation} \end{lemma}
\begin{proof} Let $\partial_{\mathbf e } \in \{ \nabla_x, \nabla_v \}$. From \eqref{fellspecto0} we have \begin{equation} \label{pefell1spec} \begin{split} \partial_{\mathbf e } f^{\ell+1}(t,x,v) = & \nabla_x f_0 \cdot \partial_{\mathbf e } X^{\ell- k }_{}\left(0; t^{\ell- (k-1)}_{k}(t,x,v), x^{\ell- (k-1)}_{k}(t,x,v), v^{\ell- (k-1)}_{k}(t,x,v) \right) \\ & + \nabla_v f_0 \cdot \partial_{\mathbf e } V^{\ell- k }_{}\left(0; t^{\ell- (k-1)}_{k}(t,x,v), x^{\ell- (k-1)}_{k}(t,x,v), v^{\ell- (k-1)}_{k}(t,x,v) \right) \end{split} \end{equation} Then from \eqref{fellboundspec}, we can use essentially the same argument as the proof of Lemma \ref{dXVcl} to get \begin{equation} \label{dXellkVellkspec} \begin{split}
& \left| \left( \frac{\partial ( X_{}^{\ell - k}(0; t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k} ),V_{}^{\ell - k }(0; t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k})}{\partial (x,v)} \right)_{ij} \right| \\ & \le C_1 \langle V^{\ell -k }(0) \rangle \exp \left({\frac{C_1}{\sqrt{ \alpha^{\ell - k }(0, X^{\ell - k}(0), V^{\ell -k }(0) ) \langle V^{\ell -k }(0) \rangle } } } \right), \end{split} \end{equation} where \[ X^{\ell - k}(0) = X^{\ell - k}(0; t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k} ), \ V^{\ell -k }(0) = V_{}^{\ell - k }(0; t^{\ell- (k-1)}_{k}, x^{\ell- (k-1)}_{k}, v^{\ell- (k-1)}_{k}), \] \[ t^{\ell- (k-1)}_{k} =t^{\ell- (k-1)}_{k} (t,x,v), \ x^{\ell- (k-1)}_{k} = x^{\ell- (k-1)}_{k} (t,x,v), \ v^{\ell- (k-1)}_{k} = v^{\ell- (k-1)}_{k} (t,x,v), \] and $C_1$ depends on $M_1$, $M_2$.
Therefore, from \eqref{pefell1spec} and \eqref{dXellkVellkspec}, \[
\langle v \rangle^{4 + \delta} \partial_{\mathbf e } f^{\ell+1}(t,x,v) \le C_1 \frac{\langle v \rangle^{4 + \delta}}{\langle V^{\ell -k }(0) \rangle^{4 + \delta} } \left( \| \langle v \rangle ^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha \langle v \rangle } } } \nabla_{x} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha \langle v \rangle } } } \nabla_v f_0 \|_\infty \right). \] And using \eqref{vfracVspec}, we conclude \begin{equation} \label{pefell1bdend}
\sup_\ell \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta} \partial_{\mathbf e } f^{\ell+1}(t) \|_\infty \le C \left( \| \langle v \rangle ^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha \langle v \rangle } } } \nabla_{x} f_0 \|_\infty + \| \langle v \rangle^{5 + \delta } e^{\frac{C}{\sqrt{ \alpha \langle v \rangle } } } \nabla_v f_0 \|_\infty \right) < M_3. \end{equation} Then, from the same argument as in Lemma \ref{tracepf}, we get \begin{equation} \label{pefell1pObdend} \begin{split}
\sup_\ell \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta} \partial_{\mathbf e } f^{\ell+1}(t) \|_{L^\infty(\gamma \setminus \gamma_0 ) } \le & \sup_\ell \sup_{0 \le t \le T} \| \langle v \rangle^{4 + \delta} \partial_{\mathbf e } f^{\ell+1}(t) \|_\infty
< M_3. \end{split} \end{equation}
From this, we use the same argument to get \eqref{dxEBfinal} in the proof of Lemma \ref{EBW1inftylemma} and obtain \[ \begin{split}
\sup_{0 \le t \le T} & \left( \| \partial_t E^{\ell+1}(t) \|_\infty + \| \partial_t B^{\ell+1}(t) \|_\infty + \| \nabla_{x } E^{\ell+1}(t) \|_\infty +\| \nabla_{x } B^{\ell+1}(t) \|_\infty \right)
\\ \le & TC \sup_{1 \le i \le \ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{i}(t) \|_\infty + \| \partial_t B^{i}(t) \|_\infty + \| \nabla_{x } E^{i}(t) \|_\infty +\| \nabla_{x } B^{i}(t) \|_\infty \right)
\\ & C \left( \| E_0 \|_{C^2 } + \| B_0 \|_{C_2} \right) + C \sup_{0 \le t \le T} \left( \| \langle v \rangle ^{4 + \delta } \nabla_{x_\parallel } f^{\ell +1}(t) \|_\infty + \| \langle v \rangle^{\ell + \delta } \alpha^{n} \partial_{x_3 } f^{\ell +1}(t) \|_\infty \right)
\\ & + C \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } \nabla_{x_\parallel } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} + \| \langle v \rangle^{5 + \delta } \alpha^{\ell-1} \partial_{x_3 } f^\ell(t) \|_{L^\infty(\gamma \setminus \gamma_0)} \right)
\\ & + C \sup_{0 \le t \le T} \left( \| \langle v \rangle^{4 + \delta } f^{\ell+1}(t) \|_\infty + \| E^{\ell +1} (t) \|_\infty + \| B^{\ell +1} (t) \|_\infty \right). \end{split} \] From \eqref{fellboundspec} and \eqref{pefell1bdend}, this gives \[ \begin{split}
\sup_{\ell} \sup_{0 \le t \le T} & \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right)
\\ \le & TC \sup_{ \ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right)
\\ & + C \left( \| E_0 \|_{C^2 } + \| B_0 \|_{C_2} \right)+ C(M_1 +M_2 +M_3). \end{split} \] Therefore, by choosing $M_4 \gg 1$ and $T \ll 1$, we get \begin{equation} \label{EBptnxbdspec}
\sup_{\ell} \sup_{0 \le t \le T} \left( \| \partial_t E^{\ell}(t) \|_\infty + \| \partial_t B^{\ell}(t) \|_\infty + \| \nabla_{x } E^{\ell}(t) \|_\infty +\| \nabla_{x } B^{\ell}(t) \|_\infty \right) < M_4. \end{equation} Together with \eqref{pefell1bdend} and \eqref{pefell1pObdend}, we conclude \eqref{flElBldsqbdspec}.
Now, from \eqref{EBptnxbdspec}, we use the same argument as in Lemma \ref{EBelltrace} to conclude that for any $0 < t <T$, \[
E^\ell(t,x ) |_{\partial \Omega } \in L^\infty( \partial \Omega ), \ B^\ell(t,x ) |_{\partial \Omega } \in L^\infty( \partial \Omega ), \] and \[
\sup_{0 \le t \le T} \left( \| E^\ell (t,x ) \|_{L^\infty( \partial \Omega ) } + \| B^\ell (t,x ) \|_{L^\infty( \partial \Omega ) } \right) < \sup_{0 \le t \le T} \left( \| E^\ell (t,x ) \|_{\infty } + \| B^\ell (t,x ) \|_{\infty } \right) < M_2. \] This proves \eqref{EBellpObd}. \end{proof}
Next, we prove a pointwise convergence result for $f^\ell$.
\begin{lemma} \label{specpointwise} Suppose $f_0$ satisfies \eqref{f0spec}, $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{E0B0bdd}. Then there exists a function $f$ such that $f^\ell \to f$ pointwise almost everywhere on $(0,T) \times ( \bar \Omega \times \mathbb R^3 \setminus \gamma_0 ) $. \end{lemma}
\begin{proof}
Fix any $(t,x,v) \in (0,T) \times ( \bar \Omega \times \mathbb R^3 \setminus \gamma_0 ) $, then it suffices to show that $\{ f^\ell (t,x,v) \}_{\ell=1}^\infty$ is a Cauchy sequence. Fix $\e > 0$, and let $n > m \ge N_0$. Note that $f^m - f^n $ satisfies $(f^m - f^n ) |_{t = 0 } = 0 $ and \[
(f^m- f^n )(t,x,v) |_{\gamma_-} = (f^{m-1} - f^{n-1} )(t,x,v_\parallel, - v_3) . \] The equation for $f^m - f^n $ is \begin{equation} \label{fmfneq} \partial_t(f^m- f^n ) + \hat v \cdot \nabla_x (f^m - f^n ) + \mathfrak F^{m-1} \cdot \nabla_v (f^{m} - f^n ) = - ( \mathfrak F^{m-1} - \mathfrak F^{n-1} ) \cdot \nabla_v f^n. \end{equation} Thus, for any $(t,x,v) \in (0,T) \times ( \bar \Omega \times \mathbb R^3 \setminus \gamma_0 )$, there is a $k \ge 0$ such that $ t^{m-k}_{k}(t,x,v) \le 0 < t^{m- (k-1)}_{k-1}(t,x,v) $, and we have from \eqref{fmfneq}, \begin{equation} \label{fnmiterate2spec} \begin{split}
& (f^m - f^n)(t,x,v)
\\ = & \int_{t^{m-1}_1}^t \left( - ( \mathfrak F^{m-1} - \mathfrak F^{n-1} ) \cdot \nabla_v f^n \right)(s, X^{m-1}(s) , V^{m-1}(s) ) ds
\\ & + \sum_{i=1}^{k-2} \int_{t^{m - (i +1)}_{i+1}}^{t^{m -i }_{i} } \left( - ( \mathfrak F^{m-1-i} - \mathfrak F^{n-1 - i} ) \cdot \nabla_v f^{n-i} \right) ( s, X^{m-1-i} (s), V^{m-1-i} (s)) ds
\\ & + \int_0^{ t^{m- (k-1)}_{k-1} } \left( - ( \mathfrak F^{m-1-(k-1)} - \mathfrak F^{n-1 - (k-1)} ) \cdot \nabla_v f^{n-(k-1)} \right) ( s, X^{m-1-(k-1)} (s), V^{m-1-(k-1)} (s)) ds. \end{split} \end{equation} Together with \eqref{vfracVspec}, this implies \begin{equation} \label{fmfnspecest1}
\langle v \rangle^{4 + \delta} | (f^m- f^n )(t,x,v) | \le C_1 \left( \sup_{\ell} \sup_{0 \le s \le t } \| \langle v \rangle^{4 + \delta} \nabla_v f^\ell(s) \|_\infty \right) \int_0^t \sup_{1 \le i \le k-1} \| \mathfrak F^{m-i} (s) - \mathfrak F^{n-i}(s) \|_\infty ds \end{equation} where $C_1 = (1+ T (M_2+g) )^{4 + \delta}$. Note that since $(x,v) \notin \gamma_0$, $\alpha(t,x,v ) > 0$. And from \eqref{tdiffalpha} and \eqref{ellstarbd}, we have \[ k \le \frac{T}{c \alpha(t,x,v) \langle v \rangle }, \] where $c $ depends on $M_2$ and $g$. Now, for some small $\delta' > 0$, we write \[ \begin{split}
\mathfrak F^{m-i} - \mathfrak F^{n-i} = & E_{f^{m-i} } - E_{f^{n-i} } + B_{f^{m-i} } - B_{f^{n-i} }
\\ = & E_{f^{m-i} -f^{n-i} }+ B_{f^{m-i} - f^{n-i} }
\\ = & E_{ \mathbf 1_{ \{ |v_3| > \delta' \}} (f^{m-i} -f^{n-i} ) } + E_{ \mathbf 1_{ \{ |v_3| \le \delta' \}} (f^{m-i} -f^{n-i} ) } + B_{ \mathbf 1_{ \{ |v_3| > \delta' \} } (f^{m-i} -f^{n-i} ) } + B_{ \mathbf 1_{ \{ |v_3| \le \delta' \}} (f^{m-i} -f^{n-i} ) },
\end{split}
\] where \begin{equation} \label{EBsplit} \begin{split}
& E_{ \mathbf 1_{ \{ |v_3| > \delta' \} } (f^{m-i} -f^{n-i} ) } = \eqref{Eesttat0pos} + \dots + \eqref{Eest3bdrycontri}, \ \text{ with } f \text{ changes to } \mathbf 1_{|v_3| > \delta'} (f^{m-i} -f^{n-i} ),
\\ & B_{ \mathbf 1_{ \{ |v_3| > \delta' \}} (f^{m-i} -f^{n-i} ) } = \eqref{Besttat0pos} + \dots + \eqref{Bestbdrycontri}, \ \text{ with } f \text{ changes to } \mathbf 1_{|v_3| > \delta'} (f^{m-i} -f^{n-i} ),
\\ & E_{ \mathbf 1_{ \{ |v_3| \le \delta' \} } (f^{m-i} -f^{n-i} ) } = \eqref{Eesttat0pos} + \dots + \eqref{Eest3bdrycontri}, \ \text{ with } f \text{ changes to } \mathbf 1_{|v_3| \le \delta'} (f^{m-i} -f^{n-i} ),
\\ & B_{ \mathbf 1_{ \{| v_3| \le \delta' \}} (f^{m-i} -f^{n-i} ) } = \eqref{Besttat0pos} + \dots + \eqref{Bestbdrycontri}, \ \text{ with } f \text{ changes to } \mathbf 1_{|v_3| \le \delta'} (f^{m-i} -f^{n-i} ). \end{split} \end{equation} Now, using the estimate in Lemma \ref{EBlinflemma} and that \[
\int_{|v_3 | < \delta' } \frac{1}{\langle v \rangle^{3 + \delta } } dv \le C ( \delta')^{\delta}, \] we have \begin{equation} \label{fmfnspecest2}
\| E_{ \mathbf 1_{ \{ |v_3| \le \delta' \} } (f^{m-i} -f^{n-i} ) } (s) \|_\infty + \| B_{ \mathbf 1_{ \{ |v_3| \le \delta' \}} (f^{m-i} -f^{n-i} ) } (s) \|_\infty \le C ( \delta')^{\delta} \sup_{0 \le s' \le s } \| \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s') \|_\infty. \end{equation} And \begin{equation} \label{fmfnspecest3} \begin{split}
& \| E_{ \mathbf 1_{ \{ |v_3| > \delta' \} } (f^{m-i} -f^{n-i} ) } (s) \|_\infty + \| B_{ \mathbf 1_{ \{ |v_3| > \delta' \}} (f^{m-i} -f^{n-i} ) } (s) \|_\infty
\\ \le & C \sup_{0 \le s' \le s } \| \mathbf 1_{ \{ |v_3| > \delta' \} } \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s') \|_\infty. \end{split} \end{equation} So from \eqref{fmfnspecest1}, \eqref{fmfnspecest2}, and \eqref{fmfnspecest3}, \begin{equation} \begin{split}
\langle v \rangle^{4 + \delta} | (f^m- f^n )(t,x,v) | \le C C_1 M_3 \int_0^t \left( \sup_{1 \le i \le k-1} \sup_{0 \le s' \le s } \| \mathbf 1_{ \{ |v_3| > \delta' \} } \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s') \|_\infty + M_1 ( \delta')^{\delta} \right) ds. \end{split}
\end{equation} Now, let $j$ be such that $t^{m-i-j}_{j}(s',x,v) \le 0 < t^{m - i - (j-1) }_{j -1} (s',x,v) $. Then if $|v_3 | > \delta'$, from \eqref{ellstarbd}, \[ j \le \frac{T}{c \alpha(s',x,v) \langle v \rangle } \le \frac{T}{c \delta ' } : = k'. \] So same as \eqref{fmfnspecest1}, this gives \[ \begin{split}
& \sup_{1 \le i \le k-1} \sup_{0 \le s' \le s } \| \mathbf 1_{ \{ |v_3| > \delta' \} } \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s') \|_\infty
\\ & \le C_1M_3 \int_0^s \sup_{1 \le i' \le k' -1} \sup_{1 \le i \le k-1} \| \mathfrak F^{m-i - i'} (s') - \mathfrak F^{n-i - i'}(s') \|_\infty ds'. \end{split} \] Using the same split \eqref{EBsplit}, like \eqref{fmfnspecest2} and \eqref{fmfnspecest3}, we thus get \begin{equation} \label{fmfnspecest4} \begin{split}
& \sup_{1 \le i \le k-1} \sup_{0 \le s' \le s } \| \mathbf 1_{ \{ |v_3| > \delta' \} } \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s') \|_\infty
\\ & \le C C_1M_3 \int_0^s \sup_{2 \le i \le k + k' } \sup_{0 \le s'' \le s'} \left( \| \mathbf 1_{ \{ |v_3| > \delta' \} } \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s'') \|_\infty + M_1 ( \delta')^{\delta} \right) ds'. \end{split} \end{equation} Plug \eqref{fmfnspecest4} into \eqref{fmfnspecest3} yields \[ \begin{split}
& \langle v \rangle^{4 + \delta} | (f^m- f^n )(t,x,v) |
\\ \le & (C C_1 M_3)^2 \int_0^t \int_0^s \sup_{2 \le i \le k + k' } \sup_{0 \le s'' \le s'} \| \mathbf 1_{ \{ |v_3| > \delta' \} } \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s'') \|_\infty ds' ds
\\ & + M_1 (\delta')^\delta \left( CC_1M_3 t + \frac{ (CC_1M_3 t)^2}{2} \right).
\end{split} \] Iteration of the above gives \[ \begin{split}
& \langle v \rangle^{4 + \delta} | (f^m- f^n )(t,x,v) |
\\ \le & (C C_1 M_3)^l \frac{t^l}{l!} \sup_{2 \le i \le k + l k' } \sup_{0 \le s \le t} \| \mathbf 1_{ \{ |v_3| > \delta' \} } \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s) \|_\infty + M_1 (\delta')^\delta \sum_{i =1}^l \frac{ (CC_1M_3 t)^i}{i!}.
\end{split} \] Now, by choosing $\delta'$ small enough we have \[
M_1 (\delta')^\delta \sum_{i =1}^l \frac{ (CC_1M_3 t)^i}{i!} < M_1 (\delta')^\delta e^{ CC_1M_3 t} < \frac{\e}{2}. \] And choosing $l$ large enough such that \[
(C C_1 M_3)^l \frac{t^l}{l!} \sup_{2 \le i \le k + l k' } \sup_{0 \le s \le t} \| \mathbf 1_{ \{ |v_3| > \delta' \} } \langle v \rangle^{4+\delta} (f^{m-i} -f^{n-i} ) (s) \|_\infty < \frac{\e}{2}. \] Finally choose $N_0 $ large enough such that $N_0 > k + l k'$, we get for $n,m > N_0$, \[
\langle v \rangle^{4 + \delta} | (f^m- f^n )(t,x,v) | < \e. \] Therefore the sequence $ \{ f^\ell(t,x,v) \}_{\ell=1}^\infty$ is Cauchy, and this proves the lemma.
\end{proof}
\begin{lemma} \label{fEBsollemmaspec} Suppose $f_0$ satisfies \eqref{f0spec}, $E_0$, $B_0$ satisfy \eqref{E0B0g}, \eqref{E0B0bdd}. Then for $0 < T \ll 1$, there exists functions $(f,E,B)$ with $ \langle v \rangle^{4 +\delta } f(t,x,v) \in L^\infty( (0, T) ; L^\infty( \bar \Omega \times \mathbb R^3 ) ) $, and $(E,B) \in L^\infty((0,T) ; L^\infty( \Omega) \cap L^\infty( \partial \Omega ) )$, such that
$(f,E,B)$ is a (weak) solution of the system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{spec}. Moreover, \begin{equation} \label{pfbdlimitspec}
\| \langle v \rangle^{4 + \delta } \nabla_{x } f(t) \|_\infty + \| \langle v \rangle^{4 + \delta } \nabla_v f(t) \|_\infty < \infty, \end{equation} and \begin{equation} \label{pEBbdlimitspec}
\| \partial_t E(t) \|_\infty + \| \partial_t B(t) \|_\infty + \| \nabla_{x } E(t) \|_\infty +\| \nabla_{x } B(t) \|_\infty < \infty. \end{equation}
\end{lemma}
\begin{proof} From the uniform-in-$\ell$ bound \eqref{fellboundspec}, we can pass the limit up to subsequence if necessary and get the weak$-*$ convergence \begin{equation} \label{fellweakcov}
\langle v \rangle^{4 + \delta } f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{4 + \delta } f \text{ in } L^\infty((0,T) \times \Omega \times \mathbb R^3 ) \cap L^\infty((0,T) \times \gamma), \end{equation} \begin{equation} \label{EBellweakcov} E^\ell \overset{\ast}{\rightharpoonup} E, \ B^\ell \overset{\ast}{\rightharpoonup} B \text{ in } L^\infty((0,T) \times \Omega ) \cap L^\infty((0,T) \times \partial \Omega ). \ \end{equation} for some $(f, E, B)$. Then from \eqref{flElBldsqbdspec} we also have \begin{equation} \label{dEnBncovspec} \partial_t E^\ell \overset{\ast}{\rightharpoonup} \partial_t E , \ \nabla_x E^\ell \overset{\ast}{\rightharpoonup} \nabla_x E, \ \partial_t B^\ell \overset{\ast}{\rightharpoonup} \partial_t B, \ \nabla_x B^\ell \overset{\ast}{\rightharpoonup} \nabla_x B \text{ in } L^\infty((0,T) \times \Omega ), \end{equation} and \begin{equation} \label{dfnconvergespec} \langle v \rangle ^{4 + \delta} \nabla_{x } f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{4 + \delta}\nabla_{x } f, \ \langle v \rangle ^{4 + \delta} \nabla_{v } f^\ell \overset{\ast}{\rightharpoonup} \langle v \rangle^{4 + \delta}\nabla_{v } f \text{ in } L^\infty((0,T) \times \Omega \times \mathbb R^3 ). \end{equation}
Now it left to show that $(f,E,B)$ is a solution to the system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{spec}. Take any $\phi(t,x,v) \in C_c^\infty( [0,T) \times \bar \Omega \times \mathbb R^3$ with $\text{supp } \phi \subset \{ [0, T) \times \bar \Omega \times \mathbb R^3 \} \setminus \{ \{ 0 \} \times \gamma \cup (0,T) \times \gamma_0 \} $, from \eqref{fellseqspec}, we have \begin{equation} \label{weakfellVMspec} \begin{split} & \int_{\Omega \times \mathbb R^3 } f_0 \phi (0) dv dt + \int_0^T \int_{\Omega \times \mathbb R^3} f^\ell ( \partial_t \phi + \hat v \cdot \nabla_x \phi + \mathfrak F^{\ell-1} \cdot \nabla_v \phi ) dv dx dt \\ = & \int_0^T \int_{\gamma_+} \phi f^\ell \hat v_3 dv dS_x + \int_0^T \int_{\gamma_+} \phi(t,x,v_\parallel, - v_3 ) f^\ell \hat v_3 \ dv dS_x. \end{split} \end{equation}
Because of \eqref{fellweakcov} and \eqref{EBellweakcov}, we have \begin{equation} \label{weakfellVMspec1} \begin{split}
& \int_0^T \int_{\Omega \times \mathbb R^3} f^\ell( \partial_t \phi + \hat v \cdot \nabla_x \phi ) dv dx dt + \int_0^T \int_{\gamma_+} \phi f^\ell \hat v_3 dv dS_x + \int_0^T \int_{\gamma_+} \phi(t,x,v_\parallel, - v_3 ) f^\ell_\pm \hat v_3 \ dv dS_x \\ \to & \int_0^T \int_{\Omega \times \mathbb R^3} f( \partial_t \phi + \hat v \cdot \nabla_x \phi ) dv dx dt + \int_0^T \int_{\gamma_+} \phi f \hat v_3 dv dS_x + \int_0^T \int_{\gamma_+} \phi(t,x,v_\parallel, - v_3 ) f \hat v_3 \ dv dS_x \end{split} \end{equation} as $\ell \to \infty$.
As for the term $\int_0^T \int_{\Omega \times \mathbb R^3 } f^\ell \mathfrak F^{\ell - 1 } \cdot \nabla_v \phi dv dx dt$, since \begin{equation} \label{weakfellVMspec2} \begin{split} & \int_0^T \int_{\Omega \times \mathbb R^3 } ( f^\ell \mathfrak F^{\ell - 1 } -f \mathfrak F) \cdot \nabla_v \phi dv dx dt \\ = & \int_0^T \int_{\Omega \times \mathbb R^3 } ( f^\ell - f ) \mathfrak F^{\ell - 1 } \cdot \nabla_v \phi dv dx dt + \int_0^T \int_{\Omega \times \mathbb R^3 } f ( \mathfrak F^{\ell - 1 } - \mathfrak F) \cdot \nabla_v \phi dv dx dt \end{split} \end{equation} From \eqref{EBellweakcov}, we have $ \int_0^T \int_{\Omega \times \mathbb R^3 } f ( \mathfrak F^{\ell - 1 } - \mathfrak F) \cdot \nabla_v \phi dv dx dt \to 0$ as $\ell \to \infty$.
Now, let $\text{supp} (\phi) = D$. From Lemma \ref{specpointwise}, $ \mathbf 1_D(t,x,v) f^\ell (t,x,v) $ converges to $\mathbf 1_D(t,x,v) f(t,x,v)$ pointwise almost everywhere. And from \eqref{fellboundspec}, $ | \mathbf 1_D(t,x,v) f^\ell (t,x,v) | \le \mathbf 1_D(t,x,v) M_1$. Therefore, from the dominated convergence theorem, we have \[
\int_0^T \int_{\Omega \times \mathbb R^3} | \mathbf 1_D ( f- f^\ell ) | dv dx dt \to 0 \text{ as } \ell \to \infty. \] Thus \begin{equation} \label{weakfellVMspec3} \begin{split} & \int_0^T \int_{\Omega \times \mathbb R^3} ( f- f^\ell ) \mathfrak F^{\ell-1} \cdot \nabla_v \phi dv dx dt
\\ \le & \sup_{0 \le t \le T } \| \mathfrak F^{\ell-1} (t) \|_\infty \sup_{0 \le t \le T } \| \nabla_v \phi (t) \|_\infty \int_0^T \int_{\Omega \times \mathbb R^3} | \mathbf 1_D ( f- f^\ell ) | dv dx dt \to 0 \text{ as } \ell \to \infty.
\end{split} \end{equation}
Put together \eqref{weakfellVMspec}--\eqref{weakfellVMspec3}, we deduce that $(f,E,B)$ satisfy \eqref{weakf}.
Next, using the same argument as in \eqref{Maxwellell}-\eqref{EBellweak2}, we get $(f,E,B)$ satisfy \eqref{Maxweak1} and \eqref{Maxweak2}. Therefore, we conclude that $(f,E,B)$ is a (weak) solution of the RVM system \eqref{VMfrakF1}--\eqref{rhoJ1}, with specular BC \eqref{spec}.
Finally, from using the weak lower semi-continuity of the weak-$*$ convergence \eqref{dEnBncovspec}, \eqref{dfnconvergespec}, and the uniform-in-$\ell$ bound \eqref{flElBldsqbdspec}, we conclude \eqref{pfbdlimitspec}, \eqref{pEBbdlimitspec}. \end{proof}
Lastly, we prove the uniqueness.
\begin{lemma} \label{VMuniqlemmaspec} Suppose $(f,E_f, B_f)$ and $(g, E_g, B_g)$ are solutions to the RVM system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{spec} with $f(0) = g(0)$, $E_f(0) = E_g(0)$, $B_f(0) = B_g(0)$, and that \[ E_{f}, B_f, E_g, B_g \in W^{1,\infty}((0,T) \times \Omega ), \ \nabla_x \rho_{f}, \nabla_x J_f, \partial_t J_f , \nabla_x \rho_{g}, \nabla_x J_g, \partial_t J_g \in L^\infty((0,T); L_{\text{loc}}^p(\Omega)) \text{ for some } p>1. \] And \begin{equation} \label{dvfgbdspec}
\sup_{0 < t < T} \| \langle v \rangle^{4+ \delta} \nabla_v f(t) \|_\infty <\infty, \sup_{0 < t < T} \| \langle v \rangle^{4+ \delta} \nabla_v g(t) \|_\infty <\infty. \end{equation} Then $f = g, E_f = E_g, B_f = B_g$. \end{lemma} \begin{proof}
The difference function $f-g $ satisfies \begin{equation} \label{fminusgeqspec} \begin{split} (\partial_t + \hat v \cdot \nabla_x + \mathfrak F_f \cdot \nabla_v)(f-g) = (\mathfrak F_g - \mathfrak F_f ) \cdot \nabla_v g
\\ (f-g)(0) = 0, \, (f- g )(t,x,v)|_{\gamma_- } = (f - g) (t,x, v_\parallel, -v_3 ), \end{split} \end{equation} where \[ \mathfrak F_f = E_f + E_{\text{ext}}+ \hat v \times ( B_f + B_{\text{ext}}) - g \mathbf e_3 , \, \mathfrak F_g = E_g + E_{\text{ext}}+ \hat v \times ( B_g + B_{\text{ext} } )- g \mathbf e_3, \] so \begin{equation} \label{mathfrakFfgspec} \mathfrak F_g - \mathfrak F_f = E_f - E_g + \hat v \times (B_f - B_g ). \end{equation} From Lemma \ref{Maxtowave} we have $E_{f,1} - E_{g,1} , E_{f,2} - E_{g,2}, B_{f,3} - B_{g,3}$ solve the wave equation with the Dirichlet boundary condition \eqref{waveD} in the sense of \eqref{waveD_weak} with \begin{align}
u_0 = 0, \ u_1 = 0 , \ G = -4\pi \partial_{x_i} (\rho_f - \rho_g) - 4 \pi \partial_t (J_{f,i} - J_{g,i} ), \ g = 0 , \ \ \text{for} \ E_{f,i} - E_{g,i}, i =1,2, \label{E12sol_B} \\
u_0 = 0, \ u_1 = 0, \ G = 4 \pi (\nabla_x \times (J_f -J_g) )_3, \ g = 0, \ \ \text{for} \ B_{f,3}- B_{g,3}, \label{B3sol_B}
\end{align} respectively. And $E_{f,3} - E_{g,3}, B_{f,1} - B_{g,1}, B_{f,2}- B_{g,2}$ solve the wave equation with the Neumann boundary condition \eqref{waveNeu} in the sense of \eqref{waveinner} \text{ with } \begin{align}
u_0 = 0, \ u_1 = 0 , \ G = -4\pi \partial_{x_3} ( \rho_f - \rho_g) - 4 \pi \partial_t (J_{f,3} - J_{g,3} ) , \ g = - 4\pi (\rho_f - \rho_g), \ \ \text{for} \ E_{f,3} - E_{g,3}, \label{E3sol_A} \\
u_0 = 0, \ u_1 = 0, \ G = 4 \pi (\nabla_x \times (J_f - J_g) )_i, \ g = (-1)^{i+1} 4 \pi (J_{f,{\underline i}} - J_{g, \underline i } ), \ \ \text{for} \ B_{f,i} - B_{j,i}, \ i=1,2, \label{B12sol_C}
\end{align} respectively. Therefore, from Lemma \ref{wavesol} and Lemma \ref{wavesolD}, we know that $E_f - E_g$ and $B_f - B_g$ would have the form of \begin{equation} \label{EBdiffformspec} \begin{split} & E_f - E_g = \eqref{Eesttat0pos} + \dots + \eqref{Eest3bdrycontri}, \ B_f -B_g = \eqref{Besttat0pos} + \dots + \eqref{Bestbdrycontri}, \\ & \text{ with } E_0, B_0 \text{ changes to } 0, \text{ and } f \text{ changes to } f -g. \end{split} \end{equation}
Now consider the characteristics \[ \begin{split} \dot X_f(s;t,x,v) = & \hat V_f(s;t,x,v) , \\ \dot V_f(s;t,x,v) = & \mathfrak F_f(s, X_f(s;t,x,v), V_f(s;t,x,v) ) . \end{split} \] Then from \eqref{fminusgeqspec}, same as \eqref{fnmiterate2spec}, we obtain \begin{equation} \label{fnmiterate2finalspec} \begin{split}
& (f - g)(t,x,v)
\\ = & \int_{t^{}_1}^t \left( ( \mathfrak F_g - \mathfrak F_f ) \cdot \nabla_v g \right)(s, \dot X(s) , \dot V(s) ) ds
\\ & + \sum_{i=1}^{k-2} \int_{t^{}_{i+1}}^{t^{}_{i} } \left( ( \mathfrak F_g^{} - \mathfrak F_f^{} ) \cdot \nabla_v g^{} \right) ( s, \cdot X^{} (s), \cdot V^{} (s)) ds
\\ & + \int_0^{ t^{}_{k-1} } \left( ( \mathfrak F_g^{} - \mathfrak F_f^{} ) \cdot \nabla_v g^{} \right) ( s, \cdot X^{} (s), \cdot V^{} (s)) ds. \end{split} \end{equation} So using \eqref{vfracVspec}, \eqref{dvfgbdspec}, we have
\begin{equation} \label{fgspecrep} \begin{split}
\sup_{ 0 \le s \le t } \| \langle v \rangle^{4 + \delta} (f-g)(s) \|_\infty \le & \sup_{0 \le t < T} \|\langle v \rangle^{4 + \delta} \nabla_v g (t) \|_\infty \int^t_0 \sup_{0 \le s' \le s } \| (\mathfrak F_g - \mathfrak F_f )(s') \|_\infty ds
\\ \le & C \int^t_0 \sup_{0 \le s' \le s } \| (\mathfrak F_g - \mathfrak F_f )(s') \|_\infty ds . \end{split} \end{equation} Now, from \eqref{EBdiffform} and the estimate in Lemma \ref{EBlinflemma}, we have \begin{equation} \label{FgFfdiff} \begin{split}
\sup_{0 \le s' \le s} \| (\mathfrak F_g - \mathfrak F_f )(s') \|_\infty \le & \sup_{0 \le s' \le s} \| (E_f - E_g )(s') \|_\infty + \sup_{0 \le s' \le s} \| (B_f - B_g )(s') \|_\infty
\\ \le & C \sup_{0 \le s' \le s } \| \langle v \rangle^{4 + \delta} (f-g )(s' ) \|_\infty, \end{split} \end{equation} Therefore from \eqref{fgspecrep} and \eqref{FgFfdiff}, we have \begin{equation}
\sup_{0 \le s \le t } \| \langle v \rangle^{4 +\delta} (f-g)(s) \|_\infty \le C' \int^t_{0 } \sup_{0 \le s' \le s } \| \langle v \rangle^{4 +\delta}(f-g )(s' ) \|_\infty ds. \end{equation} Therefore from Gronwall \[
\sup_{0 \le s' \le t } \| \langle v \rangle^{4 +\delta} (f-g)(s') \|_\infty \le e^{C't} \| \langle v \rangle^{4 +\delta} (f-g)(0) \|_\infty = 0. \] Therefore we conclude that the solutions to \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{spec} is unique. \end{proof}
We conclude the section by proving Theorem \ref{main3}.
\begin{proof}[proof of Theorem \ref{main3}] Using the sequence $f^\ell, E^\ell , B^\ell$ constructed in \eqref{fellseqspec}, \eqref{ElBlspec}, we have from Lemma \ref{VMuniqlemmaspec} that the limit $(f,E,B)$ is a solution to the RVM system \eqref{VMfrakF1}--\eqref{rhoJ1}, \eqref{diffuseBC}, and it satisfies the regularity estimate \eqref{specfbd}, \eqref{inflowEBreg}. And from Lemma \ref{VMuniqlemmaspec}, we conclude the uniqueness. \end{proof}
\appendix
\section{ }
In terms of energy. We have from \eqref{Maxwell}, we have \begin{equation} \label{ptEE} \int_{\Omega} ( \partial_t E \cdot E ) dx = \int_\Omega ( c(\nabla_x \times B ) \cdot E - 4 \pi J \cdot E ) dx, \end{equation} and \begin{equation} \label{ptBB} \int_{\Omega} ( \partial_t B \cdot B ) dx = \int_\Omega c( - \nabla_x \times E ) \cdot B dx, \end{equation} Adding \eqref{ptEE} and \eqref{ptBB} we have \begin{equation} \begin{split}
& \partial_t \int_\Omega \left(\frac{|E|^2}{2} + \frac{|B|^2}{2} \right) dx
\\ = & \int_{\Omega } c \big( (\partial_2 B_3 - \partial_3 B_2) E_1 - (\partial_1 B_3 - \partial_3 B_1) E_2 + ( \partial_1 B_2 - \partial_2 B_1) E_3
\\ & - (\partial_2 E_3 - \partial_3 E_2) B_1 + (\partial_1 E_3 - \partial_3 E_1) B_2 - ( \partial_1 E_2 - \partial_2 E_1) B_3 \big) dx - \int_\Omega (4 \pi J \cdot E )dx. \end{split} \end{equation} From integration by parts and the perfect conductor boundary condition \eqref{E12B3bc}, \[ \begin{split}
& \int_{\Omega } \big( (\partial_2 B_3 - \partial_3 B_2) E_1 - (\partial_1 B_3 - \partial_3 B_1) E_2 + ( \partial_1 B_2 - \partial_2 B_1) E_3
\\ & - (\partial_2 E_3 - \partial_3 E_2) B_1 + (\partial_1 E_3 - \partial_3 E_1) B_2 - ( \partial_1 E_2 - \partial_2 E_1) B_3 \big) dx
\\ = & \int_\Omega \left( - \partial_3 B_2 E_1 + \partial_3 B_1 E_2 + \partial_3 E_2 B_1 - \partial_3 E_1 B_2 \right) dx
\\ = & \int_{\partial \Omega } \left( - E_2 B_1 + E_1 B_2 \right) d x_\parallel = 0 \end{split} \] Therefore, \begin{equation} \label{pEEpBB}
\partial_t \int_\Omega \left(\frac{|E|^2}{2} + \frac{|B|^2}{2} \right) dx = - \int_\Omega (4 \pi J \cdot E )dx. \end{equation} On the other hand, define \begin{equation}
\langle v \rangle_\pm := \sqrt{ m_\pm^2 + |v|^2 / c^2 }. \end{equation} Note that $\hat v_\pm = \frac{v}{\langle v \rangle_\pm } $. Adding up the integration of $\langle v \rangle_\pm \times \eqref{VMfrakF}_\pm$ for both $f_+$ and $f_-$ gives \begin{equation} \label{vfint} \begin{split}
& \partial_t \int_{\Omega \times \mathbb R^3} (\langle v \rangle_+ f_+ + \langle v \rangle_- f_-) dvdx
\\ = & - \int_{\Omega \times \mathbb R^3} ( \langle v \rangle_+ \hat v_+ \cdot \nabla_x f_+ + \langle v \rangle_- \hat v_- \cdot \nabla_x f_-) dv dx - \int_{\Omega \times \mathbb R^3} \left( \langle v \rangle_+ \mathfrak F_+ \cdot \nabla_v f_+ + \langle v \rangle_- \mathfrak F_- \cdot \nabla_v f_- \right) dv dx
\\ = & \int_{\partial \Omega \times \mathbb R^3 } v_3 ( f_{+} + f_{-} ) dv dS_x + \int_{\Omega \times \mathbb R^3} \frac{1}{c^2} ( \hat v_+ \cdot \mathfrak F_+ f_+ + \hat v_- \cdot \mathfrak F_- f_- ) dv dx,
\end{split} \end{equation} where we've used $ \nabla_v \langle v \rangle_\pm = \frac{1}{c^2} \hat v_\pm $, and that $ \nabla_v \cdot \mathfrak F_\pm = 0 $. Now, since \[ \hat v_\pm \cdot \left( \hat v_\pm \times B \right) = 0, \] we have \begin{equation} \label{jFint} \begin{split}
& \frac{1}{c^2} \int_{\Omega \times \mathbb R^3} [ \hat v_+ \cdot \mathfrak F_+ f_+ + \hat v_- \cdot \mathfrak F_- f_- ] dv dx
\\ = & \frac{1}{c^2} \int_{\Omega \times \mathbb R^3 } (\hat v_+ ( e_+ (E + E_{\text{ext}}) - m_+ g \mathbf e_3 ) f_+ + \hat v_- ( e_- (E + E_{\text{ext}}) - m_- g \mathbf e_3 ) f_- ) dv dx
\\ = & \frac{1}{c^2} \left( \int_\Omega ( J \cdot ( E + E_{\text{ext}} )) dx - g \int_{\Omega \times \mathbb R^3} ( \hat v_{+,3} m_+ f_+ + \hat v_{-,3} m_- f_- ) dv dx \right). \end{split} \end{equation} Adding up \eqref{pEEpBB}, \eqref{vfint}, and \eqref{jFint}, we get \begin{equation} \begin{split}
& \frac{ \partial}{\partial t } \left( \int_\Omega \left(\frac{|E|^2}{2} + \frac{|B|^2}{2} \right) dx + 4 \pi c^2 \int_{\Omega \times \mathbb R^3} (\langle v \rangle_+ f_+ + \langle v \rangle_- f_-) dvdx \right)
\\ = & 4\pi c^2 \int_{\gamma_- } v_3 (f_{+} + f_{-} ) dv dS_x + 4\pi c^2 \int_{\gamma_+ } v_3 (f_{+} + f_{-} ) dv dS_x
\\ & + 4 \pi \int_{\Omega} ( J \cdot E_{\text{ext}} ) dx - 4 \pi g \int_{\Omega \times \mathbb R^3} ( \hat v_{+,3} m_+ f_+ + \hat v_{-,3} m_- f_- ) dv dx. \end{split} \end{equation} For the specular BC \eqref{spec}, \[ \int_{v_3 < 0} v_3 f_\pm dv = - \int_{v_3 > 0 } v_3 f_\pm dv \text{ for } x \in \partial \Omega. \] Thus a solution of the system \eqref{VMfrakF}-\eqref{rhoJ} with the specular BC \eqref{spec} has \begin{equation} \begin{split}
& \frac{ \partial}{\partial t } \left( \int_\Omega \left(\frac{|E|^2}{2} + \frac{|B|^2}{2} \right) dx + 4 \pi c^2 \int_{\Omega \times \mathbb R^3} (\langle v \rangle_+ f_+ + \langle v \rangle_- f_-) dvdx \right)
\\ = & 4 \pi E_e \int_{\Omega \times \mathbb R^3 }( \hat v_{+,3} e_+ f_+ + \hat v_{-,3} e_- f_- ) dv dx - 4 \pi g \int_{\Omega \times \mathbb R^3} ( \hat v_{+,3} m_+ f_+ + \hat v_{-,3} m_- f_- ) dv dx. \end{split} \end{equation} \hide
\section{}
We finish the section by demonstrating that $\partial_3 E_3$, $\partial_3 B_1$, $\partial_3 B_2$ have a trace at $\partial\Omega$: \begin{remark}
Suppose $\partial_t E, \nabla_x E, \partial_t B, \nabla_x B \in L^\infty((0,T) \times \Omega)$, and $ \nabla \rho, \partial_t J , \nabla_x J \in L^\infty((0,T); L_{\text{loc}}^p(\Omega))$ for $p>1$. Suppose
\begin{equation} \label{divE=rho}
\nabla_x \cdot E = 4 \pi \rho, \ \partial_t E = \nabla_x \times B - 4 \pi J \ \text{ in the sense of distribution } \mathcal{D}((0,T) \times \Omega),
\end{equation}
and $E_\parallel = (E_1, E_2)$ is a weak solution to
\begin{equation}
\begin{split}\label{pde:E_tan}
\partial_t^2 E_\parallel - \Delta E_\parallel, = - 4\pi \nabla_\parallel \rho -4 \pi \partial_{t} J_\parallel \ \ &\text{in } [0,T] \times \Omega\\
E_\parallel =0 \ \ &\text{on } \partial\Omega;
\end{split}
\end{equation}
$B_3$ is a weak solution to
\begin{equation}
\begin{split}\label{pde:B3}
\partial_t^2 B_3 - \Delta B_3 = 4 \pi \partial_1J_2 - 4\pi \partial_2 J_1 \ \ &\text{in } [0,T] \times \Omega\\
B_3 =0 \ \ &\text{on } \partial\Omega;
\end{split}
\end{equation}
Then
\begin{equation} \label{trace:E_3}
\partial_3 E_3 \ \text{has a trace and } \partial_3 E_3(t,x) = 4\pi \rho(t,x) \ \ \text{on} \ \ (0,T) \times \partial\Omega \ \text{ a.e.},
\end{equation}
and for $i=1,2$,
\begin{equation} \label{trace:B_tan}
\partial_3 B_i \ \text{has a trace and } \partial_3 B_i(t,x) = (-1)^{i+1} 4\pi J_{\underline i } (t,x) \ \ \text{on} \ \ (0,T) \times \partial\Omega \ \text{ a.e.}.
\end{equation}
Here, we have used a convenient notation:
\begin{equation} \label{def_under_i}
\underline i = \begin{cases} 2, \text{ if } i=1, \\ 1, \text{ if } i=2. \end{cases}
\end{equation} \end{remark} \begin{proof}
Step 1. Let $\phi (t) \in C([0,T]) \cap W^{1,1}(0,T)$ which is compactly supported in $(0,T)$, meaning that $\phi(0)= 0= \phi(T)$. We test $\phi (t)$ to \eqref{pde:E_tan}: this is possible since $E_\parallel$ might have a trace so the test function needs not vanish at the boundary $[0,T] \times \partial\Omega$. Then $\bar E_\parallel (x) = \int_0^T E_\parallel (s, x) \phi(s) \mathrm{d} s \in W^{1, \infty}(\Omega)$ solves
\begin{equation} \label{pde:barE_tan}
\begin{split}
- \Delta \bar E_\parallel = \int_0^T \partial_t E_\parallel (s,x) \partial_t \phi(s) \mathrm{d} s
-4 \pi \int_0^T \nabla_\parallel \rho \phi(s) \mathrm{d} s -4 \pi \int_0^T \partial_t J_\parallel(s,x ) \phi (s) \mathrm{d} s \ \ &\text{in } \Omega,\\
\bar{E}_\parallel =0 \ \ &\text{ on } \ \partial\Omega.
\end{split}
\end{equation}
Note that this is a Poisson equation of an $L^p(\Omega)$-source term with zero Dirichlet boundary condition. From the elliptic regularity theory, we have
\begin{equation}
\bar E_\parallel = \int^T_0 E_\parallel (s,x) \phi(s) \mathrm{d} s \in W_{\text{loc}}^{2,p}(\Omega)
\ \ \text{and } \ \ \nabla_\parallel \bar E_\parallel \in W_{\text{loc}}^{1,p}(\Omega).
\end{equation}
Then by the trace theorem $W_{\text{loc}}^{1,p}(\Omega) \rightarrow W_{\text{loc}}^{1-1/p,p}(\partial\Omega)$ (\cite{Leoni}, Theorem 18.27, Page 608), we conclude that
\begin{equation}
\nabla_\parallel \bar E_\parallel \ \text{has a trace and } \ \nabla_\parallel \bar E_\parallel \in W_{\text{loc}}^{1-1/p,p}(\partial\Omega).
\end{equation}
On the other hand, we note that $\nabla_\parallel \bar E_\parallel (x) = \int_0^T \nabla_\parallel E_\parallel (s, x) \phi(s) \mathrm{d} s$ a.e. in $\Omega$. Therefore we deduce that
\begin{equation}
\int_0^T \nabla_\parallel E_\parallel (s, x) \phi(s) \mathrm{d} s \ \text{has a trace and } \ \int_0^T \nabla_\parallel E_\parallel (s, x) \phi(s) \mathrm{d} s \in W_{\text{loc}}^{1-1/p,p}(\partial\Omega),
\end{equation}
for any $\phi (t) \in C([0,T]) \cap W^{1,1}(0,T)$ which is compactly supported in $(0,T)$. Since the dual of $W_0^{1,1}((0,T) )$ is $W^{-1,\infty}((0,T))$, we conclude that
\begin{equation} \label{trace:NE_p}
\nabla_\parallel E_\parallel
\in W^{-1, \infty} ((0,T) ; W_{\text{loc}}^{1,p} (\Omega))
\ \text{has a trace and } \ \nabla_\parallel E_\parallel \in W^{-1, \infty} ((0,T); W_{\text{loc}}^{1-1/p,p}(\partial\Omega)).
\end{equation}
On the other hand, since uniformly continuous function can be extended on the boundary, we regard $E$ to be continuous functions in $\bar \Omega$. For the continuous functions $E_\parallel |_{\partial\Omega}=0$, we derive that
\begin{equation} \label{bdry:N_pE}
\nabla_{\parallel} E_\parallel(x_1,x_2,0) =0 \ \ \text{for almost every } (x_1, x_2) \in \mathbb{R}^2.
\end{equation}
\hide
Since a distributional derivative $\nabla_x \cdot E$ equals $4 \pi \rho$ in $\Omega$, $\nabla_x \cdot E$ is also continuous function in $\bar \Omega$.
On the other hand, Therefore we conclude that a distributional derivative $\partial_3 E_3= \nabla_x \cdot E$ on $\partial\Omega$.
Moreover, since $E_\parallel=0$ in $\partial\Omega$ now we conclude that
\begin{equation} \label{bdry:N_pE}
\nabla_\parallel E_\parallel(t,\cdot ) =0 \ \text{ a.e. } \ x \in \Omega \ \text{ almost all } \ t \in (0,T).
\end{equation}\unhide
Step 2. From \eqref{divE=rho}, \eqref{trace:NE_p}, and $\nabla \rho \in L^\infty((0,T) ; L_{\text{loc}}^p(\Omega))$, we have
\begin{equation}
\partial_3 E_3 = - \partial_1 E_1 - \partial_2 E_2 + 4 \pi \rho\in W^{-1, \infty} ((0,T) ; W_{\text{loc}}^{1,p} (\Omega)).
\end{equation}
From the trace theorem $W^{-1,\infty} ((0,T) ; W_{\text{loc}}^{1,p} (\Omega)) \rightarrow W^{-1,\infty} ((0,T) ; W_{\text{loc}}^{1-1/p,p} (\partial\Omega))$ and \eqref{bdry:N_pE}, finally we prove that
\begin{equation} \label{trace:-1infty}
\partial_3 E_3 (t,x) - 4\pi \rho(t,x)=0 \ \ \text{ in } \ \ W^{-1,\infty} ((0,T) ; W_{\text{loc}}^{1-1/p,p} (\partial\Omega)).
\end{equation}
Note that $W^{-1,\infty} (0,T)$ should be considered as a subspace of $\mathcal{D}^\prime(0,T)$. For $g (t)\in W^{-1,\infty} (0,T)$, there exist $\psi_0(t) \in L^\infty$ and $\psi_1(t) \in L^\infty$ such that
\begin{equation}
\int^T_0g(s) \phi(s) \mathrm{d} s = \int^T_0\psi_0(s) \phi(s) \mathrm{d} s
- \int^T_0\psi_1(s)\partial_t \phi(s) \mathrm{d} s \ \ \text{for all } \phi \in W^{1,1}_0 (0,T)
. \end{equation}
Therefore \eqref{trace:-1infty} implies that $\partial_3 E_3 (t,\cdot) = 4\pi \rho(t,\cdot)$ for almost every $t \in (0,T)$. Therefore we conclude \eqref{trace:E_3}.
Step 3. Next, from \eqref{pde:B3} and using the same argument as in Step 1, we get
\begin{equation} \label{trace:B3_p}
\nabla_\parallel B_3
\in W^{-1, \infty} ((0,T) ; W_{\text{loc}}^{1,p} (\Omega))
\ \text{has a trace and } \ \nabla_\parallel B_3 \in W^{-1, \infty} ((0,T); L_{\text{loc}}^p(\partial\Omega)).
\end{equation}
And by regarding $B_3$ to be a continuous function in $\bar \Omega$ and $B_3 |_{\partial \Omega} = 0$, we have
\begin{equation} \label{bdry:N_B3}
\nabla_{\parallel} B_3 (x_1,x_2,0) =0 \ \ \text{for almost every } (x_1, x_2) \in \mathbb{R}^2.
\end{equation}
Now, for any $\phi (t) \in C([0,T]) \cap W^{1,1}(0,T)$ which is compactly supported in $(0,T)$,
\begin{equation}
\int_0^T \partial_t E_\parallel (s,x) \phi(s) \mathrm{d} s = \int_0^T E_\parallel (s,x) \partial_t \phi(s) \mathrm{d} s \in W^{1,\infty}(\Omega ).
\end{equation}
Thus
\[
\int_0^T \partial_t E_\parallel (s, x) \phi(s) \mathrm{d} s \ \text{has a trace and } \ \int_0^T \partial_t E_\parallel (s, x) \phi(s) \mathrm{d} s \in L^\infty(\partial\Omega),
\]
Again, using the dual of $W_0^{1,1}((0,T) )$ is $W^{-1,\infty}((0,T))$, we deduce that
\begin{equation} \label{trace:ptE}
\partial_t E_\parallel
\in W^{-1, \infty} ((0,T) ; W^{1,\infty} (\Omega))
\ \text{has a trace and } \ \partial_t E_\parallel \in W^{-1, \infty} ((0,T); L^{\infty}(\partial\Omega)).
\end{equation}
On the other hand, we regard $E$ to be continuous function in $\bar \Omega$ and $E_\parallel |_{\partial \Omega } = 0$, we derive that
\begin{equation} \label{ptEtan0}
\partial_t E_\parallel (x_1, x_2, 0 ) ) =0 \ \ \text{for almost every } (x_1, x_2) \in \mathbb{R}^2.
\end{equation}
From \eqref{divE=rho}, \eqref{trace:B3_p}, \eqref{trace:ptE}, and $\nabla j \in L^\infty((0,T) ; L_{\text{loc}}^p(\Omega) ) $, we have
\[
\begin{split}
\partial_3 B_1 = & \partial_t E + \partial_1 B_3 + 4 \pi J_2 \in W^{-1, \infty} ((0,T) ; W_{\text{loc}}^{1,p} (\Omega)),
\\ \partial_3 B_2 =& - \partial_t E + \partial_2 B_3 - 4 \pi J_1 \in W^{-1, \infty} ((0,T) ; W_{\text{loc}}^{1,p} (\Omega)).
\end{split}
\]
From the trace theorem $W^{-1,\infty} ((0,T) ; W_{\text{loc}}^{1,p} (\Omega)) \rightarrow W^{-1,\infty} ((0,T) ;L_{\text{loc}}^p (\partial\Omega))$ and \eqref{bdry:N_B3}, \eqref{ptEtan0}, finally we prove that
\begin{equation} \label{trace:-1infty1}
\partial_3 B_1 - 4 \pi J_2=0, \ \text{ and } \partial_3 B_2 + 4\pi J_1 = 0 \ \ \text{ in } \ \ W^{-1,\infty} ((0,T) ; L_{\text{loc}}^{p} (\partial\Omega)).
\end{equation}
This implies that $\partial_3 B_1 (t,\cdot) = 4\pi J_2(t,\cdot)$, and $\partial_3 B_2 (t,\cdot) = - 4\pi J_1(t,\cdot)$ for almost every $t \in (0,T)$. Therefore we conclude \eqref{trace:B_tan}.\end{proof}
\unhide
\end{document}
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arXiv
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Is this a misleading or undesirable implementation of a hash map?
I read a C++ implementation of a hash map here. https://www.geeksforgeeks.org/implementing-hash-table-open-addressing-linear-probing-cpp/
Let's say key k1 has a hash index of h. Suppose there's a collision with key k2 such that hash(k1) == hash(k2). Then won't the new hash index of k2 become h+1?
Key: k1 Index: h
Key: k2 Index: h+1
Suppose we introduce a third key k3 such that hash(k3) == h+1. Then when we insert k3 into the hash map, its hash index will become h+2.
Key: k1 Hash value: h Hash table index: h
Key: k2 Hash value: h Hash table index: h+1
Key: k3 Hash value: h+1 Hash table index: h+2
This can cause the hash indexes for keys to be shifted over 1 in the case of collisions (as seen above), and if collisions happen frequently then they could be shifted over more than once.
Is this a bad implementation of a hash map? From an aesthetic point of view I prefer the linked approach, where each hash node has a pointer to a next node, such that if multiple nodes have the same hash index, they are all part of the same linked list starting at that hash index.
In the linked approach, at least we have the assurance that a key will logically correspond to its hash index in the hash table, even if it's part of a linked list (in which case it doesn't physically correspond, but logically still does, since the head of the linked list is stored there).
Is the implementation of the hash map in GeeksForGeeks bad? Is the linked approach more logical and intuitive? What are your thoughts?
Note: The linked approach I refer to is simply to store a linked list at each hash index in the hash table, such that if multiple keys are hashed to that index, they are stored in this linked list.
data-structures hash-tables c++
ktm5124ktm5124
$\begingroup$ GeeksforGeeks is a very good question repository but it has some really ugly code and explanations. You should just try to look for trade-offs between different implementations. Hashmaps have a fixed size and shifting by 1 doesn't seem the most handsome solution. What if the hashmap is full, then no insertion would take place. $\endgroup$
– Navjot Singh
$\begingroup$ @NavjotWaraich Exactly! If the hash table is full you're in trouble. I wanted to confirm that's actually what is going on, since that was my takeaway too. $\endgroup$
– ktm5124
$\begingroup$ en.wikipedia.org/wiki/Hash_table#Open_addressing, en.wikipedia.org/wiki/Linear_probing, en.wikipedia.org/wiki/Hash_table#Separate_chaining $\endgroup$
– D.W. ♦
As others have indicated, this code example doesn't do resizing, and this is an important part of a "real" dynamic hash table. Whether or not it's "bad" depends on what it's trying to achieve as an example, but it's certainly not code that you should deploy.
Linear probing isn't necessarily a bad idea. After all, it has excellent locality. But since you asked about a specific scenario, let's explore that. The scenario is where you have two keys, k1 and k2, which have the same hash value h, and a third, k3, with the hash value h+1.
Most hash table designs rely on the premise that the hash function is "good". If you have a "good" hash function, this should be a difficult scenario to engineer on purpose, but it's still possible that it happens with high enough probability just by virtue of storing a lot of entries in the hash table.
Consider an open addressing hash table with $m$ slots and $n$ elements in it. Denote the load factor by $\lambda = \frac{n}{m}$. Then if the hash function is "good", and $m$ and $n$ are large enough, the probability that a given slot has $k$ elements that would "naturally" hash to that slot is given by the Poisson distribution:
$$P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!}$$
So the probability that a given slot has at least 2 elements is:
$$P_1 = 1 - P(0; \lambda) - P(1; \lambda)$$
And the probability that the very next slot has at least 1 element is:
$$P_2 = 1 - P(0; \lambda)$$
The probability of this scenario occurring to a given hash slot is:
$$P_1 P_2 = 1 - e^{-\lambda} (\lambda + 2) + e^{-2\lambda} (\lambda + 1)$$
...and so on. Multiply this by $m$ to give the expected number of times this scenario happens for any $m$ and $n$.
(Exercise: Suppose that you have $m$ pigeonholes and $n$ pigeons, and you assign pigeons to pigeonholes randomly and $m$ is large enough. Then the proportion of pigeonholes with $k$ pigeons assigned follows a Poisson distribution. Show that the expected number of "pigeon collisions" is $\frac{1}{2}$ when $n = \sqrt{m}$.)
Remember, though, that short probes aren't necessarily a problem. What you're trying to avoid is long probes. Specifically, you're trying to ensure that the maximum probe length, that is, the largest distance that an entry is from its "home", is small. This number gives the maximum "work" that you may have to do to look for an entry.
In open addressing, it is common practice to keep track of this maximum probe length, and update it when a new element is inserted. This number is used for (at least!) two purposes: to decide when to stop searching, and to decide when to resize the table.
PseudonymPseudonym
It's quite normal.
If your hash table is getting too full, you will get a high number of collisions - time to resize the hash table.
If your hash table is not very full, a high number of collisions is possible, but not very likely.
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CommonCrawl
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\begin{document}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{remark} \newtheorem{remark}[theorem]{Remark}
\theoremstyle{definition} \newtheorem{definition}[theorem]{Definition}
\theoremstyle{definition} \newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{example}[theorem]{Example}
\title[ Finite dimensional Hopf algebras over Kac-Paljutkin algebra $H_8$]{ Finite dimensional Hopf algebras over Kac-Paljutkin algebra $H_8$}
\author[Shi]{Yuxing Shi } \address{School of Mathematics and Information Science, Guangzhou University, Guangzhou 510006, P.R. China}\email{[email protected]}
\subjclass{Primary 17B37, 81R50; Secondary 17B35}
\thanks{
\textit{Keywords:} Nichols algebra; Hopf algebra; Gelfand-Kirillov dimension; Kac-Paljutkin algebra.
}
\begin{abstract} Let $H_8$ be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra \cite{MR0208401}. All simple Yetter-Drinfel'd modules over $H_8$ are given. As for simple objects and direct sums of two simple objects in ${}_{H_8}^{H_8}\mathcal{YD}$, we calculated dimensions for the corresponding Nichols algebras, except four semisimple cases which are generally difficult. Under the assumption that the four undetermined Nichols algebras are all infinite dimensional, we determine all the finite dimensional Nichols algebras over $H_8$. It turns out that the already known finite dimensional Nichols algebras are all diagonal type. In fact, they are Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$. By the way, we calculate Gelfand-Kirillov dimensions for some Nichols algebras. As an application, we obtain five families of new finite dimensional Hopf algebras over $H_8$ according to the lifting method. \end{abstract} \maketitle
\section{Introduction} Let $\mathbb{K}$ be an algebraically closed field of characteristic zero. The question of classification of all Hopf algebras over $\mathbb{K}$ of a given dimension up to isomorphism was posed by Kaplansky in 1975 \cite{Kaplansky1975MR0435126}. Some progress has been made but, in general, it is a difficult question for lack of standard methods. One breakthrough is the so-called \emph{Lifting Method} \cite{MR1659895} introduced by Andruskiewitsch and Schneider in 1998, under the assumption that the coradical is a Hopf subalgebra.
We describe the procedure for the lifting method briefly. Let $H$ be a Hopf algebra whose coradical $H_0$ is a Hopf subalgebra. It is well-known that the associated graded Hopf algebra of $H$ is isomorphic to $R\#H_0$ where $R = \oplus_{n\in \mathbb N_0}R(n)$ is a braided Hopf algebra in the category ${}_{H_0}^{H_0}\mathcal{YD}$ of Yetter-Drinfield modules over $H_0$. $\#$ stands for the Radford biproduct or \textit{bosonization} of $R$ with $H_0$. As explained in \cite{andruskiewitsch2001pointed}, to classify finite-dimensional Hopf algebras $H$ whose coradical is isomorphic to $H_0$ we have to deal with the following questions: \begin{enumerate}\renewcommand{\theenumi}{\alph{enumi}}\renewcommand{\labelenumi}{(\theenumi)}
\item\label{que:nichols-fd} Determine all Yetter-Drinfiel'd modules $V$ over $H_0$ such that the Nichols algebra $\mathfrak{B}(V)$ has finite dimension; find an efficient set of relations for $\mathfrak{B}(V)$. \item\label{que:nichols-R} If $R = \oplus_{n\in \mathbb N_0}R(n)$ is a finite-dimensional Hopf algebra in ${}_{H_0}^{H_0}\mathcal{YD}$ with $V = R(1)$, decide if $R \simeq \mathfrak{B}(V)$. Here $V = R(1)$ is a braided vector space called the \textit{infinitesimal braiding}. \item\label{que:lifting} Given $V$ as in \eqref{que:nichols-fd}, classify all $H$ such that $\mathrm{gr}\, H \simeq \mathfrak{B}(V)\#H_0$ (lifting). \end{enumerate}
The lifting method was extensively used in the classification of finite dimensional pointed Hopf algebras such as \cite{andruskiewitsch2005classification}, \cite{andruskiewitsch2010nichols}, \cite{MR2811166}\cite{MR2862142}, \cite{andruskiewitsch2011finite}, \cite{andruskiewitsch2010pointed}, \cite{MR3493214}, \cite{MR3395052} and so on. It is also effective to study finite-dimensional copointed Hopf algebras \cite{MR2863448}, \cite{MR3119229}, \cite{2016arXiv160806167F}. We note that there are very few classification results on finite-dimensional Hopf algebras whose coradical is a Hopf subalgebra but not a group algebra and the dual of a group algebra, two exceptions being \cite{MR2037722, 2016arXiv160503995A}.
Here we would like to initiate a project for the study of Hopf algebras whose coradicals are low-dimensional neither commutative nor cocommutative semisimple Hopf algebras by running procedures of the lifting method. One important step is to study the Nichols algebras over those low-dimensional semisimple Hopf algebras. Nichols algebras were studied first by Nichols \cite{nichols1978bialgebras} . These are connected graded braided Hopf algebras \cite{MR1907185} generated by primitive elements, and all primitive elements are of degree one. In the past decades, the study of Nichols algebras was mainly focused on group algebras and which were finite dimensional, for those Nichols algebras were essential ingredients of the classification of finite-dimensional pointed Hopf algebras. Under the assumption that the base field has characteristic $0$, the classification of finite-dimensional Nichols algebras over abelian groups was completely solved in \cite{MR2207786, heckenberger2009classification} by using Lie theoretic structures, and the result of the classification played an important role later in the significant work \cite{andruskiewitsch2005classification}. The problem of classifying finite-dimensional Nichols algebras over non-abelian groups is difficult in general for lack of systematic method, related works please refer to \cite{andruskiewitsch2010nichols}, \cite{freyre2007nichols},\cite{grana2011nichols}, \cite{MR2891215} \cite{MR2732989}, \cite{MR3276225}, \cite{MR3272075}, etc.
In this paper, we mainly focus on Kac-Paljutkin algebra $H_8$. The structure of our paper is as follows. In Section \ref{Preliminaries}, we recall the fundamental notions related to Yetter-Drinfel'd modules, Nichols algebras and Gelfand-Kirillov dimension. In section \ref{YDMH8}, we construct all the simple left Yetter-Drinfel'd modules over $H_8$ according to Radford's method. In section \ref{section:NicholsAlgebras}, we get all the possible finite dimensional Nichols algebras from Yetter-Drinfel'd modules over $H_8$ under the assumption that the four undetermined cases over semisimple modules for which are difficult for us at the moment. It turns out that all the already known finite dimensional Nichols algebras are Cartan types $A_1$, $A_2$, $A_2\times A_2$, $A_1\times \cdots \times A_1$, and $A_1\times \cdots \times A_1\times A_2$. Here is our first main result. \begin{mteor} \label{NicholsAlg:maintheorem}
Suppose
$$\dim\mathfrak{B}\left(M\langle(xy,x)\rangle \oplus W^{b_1,-1}\right)=\infty =\dim\mathfrak{B}\left(M\langle(y,xy)\rangle \oplus W^{b_1,-1}\right)$$ holds for $b_1=\pm 1$.
If $M\in {}_{H_8}^{H_8}\mathcal{YD}$ such that $\dim\mathfrak{B}(M)<\infty$, then $M$ is isomorphic either to one of the following Yetter-Drinfel'd modules
\begin{enumerate}
\item $\Omega_1(n_1,n_2,n_3,n_4)\triangleq
\bigoplus_{j=1}^4 M\left<1|V_1(b_j),g_j\right>^{\oplus n_j}$
with $\sum_{j=1}^4 n_j\geq 1$, $(b_1,g_1)=(\mathrm{i},x)$, $(b_2,g_2)=(-\mathrm{i},x)$, $(b_3,g_3)=(\mathrm{i},y)$, $(b_4,g_4)=(-\mathrm{i},y)$, the infinitesimal braiding is type $\underbrace{A_1\times \cdots \times A_1}_{n_1+n_2+n_3+n_4}$.
\item $\Omega_2(n_1,n_2)\triangleq M\langle\mathrm{i},x\rangle^{\oplus n_1}
\oplus M\langle -\mathrm{i},x\rangle^{\oplus n_2}
\oplus M\langle(xy,x)\rangle$
, $n_1+n_2\geq 0$, the infinitesimal braiding is type
$\underbrace{A_1\times \cdots \times A_1}_{n_1+n_2}\times\, A_2$.
\item $\Omega_3(n_1,n_2)\triangleq M\langle\mathrm{i},y\rangle^{\oplus n_1}
\oplus M\langle -\mathrm{i},y\rangle^{\oplus n_2}
\oplus M\langle(y,xy)\rangle$
, $n_1+n_2\geq 0$, the infinitesimal braiding is type
$\underbrace{A_1\times \cdots \times A_1}_{n_1+n_2}\times\, A_2$.
\item $\Omega_4(n_1,n_2)\triangleq M\langle\mathrm{i},x\rangle^{\oplus n_1}
\oplus M\langle\mathrm{i},y\rangle^{\oplus n_2}
\oplus W^{1,-1}$
, $n_1+n_2\geq 0$, the infinitesimal braiding is type
$\underbrace{A_1\times \cdots \times A_1}_{n_1+n_2}\times\, A_2$.
\item $\Omega_5(n_1,n_2)\triangleq M\langle -\mathrm{i},x\rangle^{\oplus n_1}
\oplus M\langle -\mathrm{i},y\rangle^{\oplus n_2}
\oplus W^{-1,-1}$
, $n_1+n_2\geq 0$, the infinitesimal braiding is type
$\underbrace{A_1\times \cdots \times A_1}_{n_1+n_2}\times\, A_2$.
\item $\Omega_6\triangleq M\langle(xy,x)\rangle
\oplus M\langle(y,xy)\rangle$, the infinitesimal braiding is type $A_2\times A_2$.
\item $\Omega_7\triangleq W^{1,-1}\oplus W^{-1,-1}$, the infinitesimal braiding is type $A_2\times A_2$.
\end{enumerate} \end{mteor} \begin{remark} We point out which of the Yetter-Drinfeld modules have a principal realization and which not, since the liftings are known when there is a principal realization and not otherwise \cite{AAGI}. Let $(h)$ and $(\delta_h)$ be a dual basis of $H_8$ and $H_8^{*}$, and $b\in\{\pm 1, \pm\mathrm{i}\}$, then $$\chi_b:=\delta_1+\delta_{xy}+b(\delta_x+\delta_y)+ b^2(\delta_{z}+\delta_{zxy})+b^3(\delta_{zx}+\delta_{zy})\in\mathrm{Alg}(H_8,\mathbb{K}).$$ $(g,\chi_b)$ is a \emph{YD-pair} \cite{MR3133699} and $\mathbb{K}_g^{\chi_b}\simeq M\langle b,g\rangle$. $M\langle(g_1,g_2)\rangle$ for $(g_1,g_2)\in \{(xy,x),(y,xy)\}$ and $W^{b_1,-1}$ for $b_1=\pm 1$ don't have a \emph{principal realization} \cite[Subsection 2.2]{AAGI}, since $\mathbb{K}_{g_1}^{\chi_{b_1}}\oplus \mathbb{K}_{g_2}^{\chi_{b_2}}$ is of type $A_1\times A_1$ for $(b_1,g_1), (b_2,g_2)\in\{(\pm \mathrm{i},x), (\pm \mathrm{i},y)\}$. So only $\Omega_1(n_1,n_2,n_3,n_4)$ has a principal realization. \end{remark}
In section \ref{section:HopfAlgebras}, according to the lifting method, we give a classification for finite-dimensional Hopf algebras over $H_8$ such that their infinitesimal braidings are isomorphic to those Yetter-Drinfel'd modules listed in Theorem \ref{NicholsAlg:maintheorem}. Here is the second main result.
\begin{mteor}\label{HopfAlgOverH8}
Suppose $H$ is a finite-dimensional Hopf algebra over $H_8$ such that its
infinitesimal braiding is isomorphic to one of the Yetter-Drinfel'd modules listed in Theorem \ref{NicholsAlg:maintheorem}, then $H$ is isomorphic either to
\begin{enumerate}
\item $\mathfrak{A}_1(n_1,n_2,n_3,n_4;I_1)$, see Definition \ref{definition:HopfAlgA_1};
\item $\mathfrak{B}[\Omega_2(n_1, n_2)]\# H_8$, see Proposition
\ref{HopfAlg:Omega2};
\item $\mathfrak{A}_4(n_1,n_2;I_4)$, see Definition \ref{Definition:HopfAlgA_4};
\item $\mathfrak{A}_6(\lambda)$, see Definition \ref{definition:A_6};
\item $\mathfrak{A}_7(I_7)$, see Definition \ref{Definition:HopfAlgA_7}.
\end{enumerate} \end{mteor} $\mathfrak{A}_1(n_1,n_2,n_3,n_4;I_1)$ comprises two nonisomorphic nonpointed self-dual Hopf algebras of dimension $16$ with coradical $H_8$ described in \cite{MR2037722} as special cases. Except $\mathfrak{B}[\Omega_2(n_1, n_2)]\# H_8$, the remainder four families of Hopf algebras contain non-trial lifting relations.
\section{Preliminaries}\label{Preliminaries}
\subsection{Conventions} Let $H$ be a Hopf algebra over $\mathbb{K}$, with antipode $S$. We will use Sweedler's notation $\Delta(h)=h_{(1)}\otimes h_{(2)}$ for the comultiplication \cite{montgomery1993hopf}. Let ${}_{H}^{H}\mathcal{YD}$ be the category of left \emph{Yetter-Drinfel'd modules} over $H$. That is to say that if $M$ is an object of
${}_{H}^{H}\mathcal{YD}$ if and only if there exists
an action $\cdot$ such that $(M,\cdot)$ is a left
$H$-module and a coaction $\rho$ such that $(M,\rho)$
is a left $H$-comodule, subject to the following compatibility condition:
\begin{equation}
\rho(h\cdot m)=h_{(1)}m_{(-1)}S
(h_{(3)})\otimes h_{(2)} \cdot m_{(0)}, \forall m\in M, h\in H,
\end{equation}
where $\rho(m)=m_{(-1)}\otimes m_{(0)}$. It is a braided monoidal category.
The braiding $c\in \mathrm{End}_\mathbb{K}(M\otimes M)$ of $M$ is defined by $ c(v\otimes w)=v_{(-1)} \cdot w\otimes v_{(0)}$, and the inverse braiding is defined by $c^{-1}(v\otimes w)=w_{(0)}\otimes \left(S^{-1}(w_{(-1)})\cdot v\right)$.
\begin{definition}\cite[Definition. 2.1]{andruskiewitsch2001pointed} \label{defNicholsalgebra} Let $H$ be a Hopf algebra and $V \in {}^H_H\mathcal{YD}$. A braided $\mathbb{N}$-graded Hopf algebra $R = \bigoplus_{n\geq 0} R(n) \in {}^H_H\mathcal{YD}$ is called the \textit{Nichols algebra} of $V$ if \begin{enumerate}
\item[(i)] $\mathbb{K}\simeq R(0)$, $V\simeq R(1) \in {}^H_H\mathcal{YD}$,
\item[(ii)] $R(1) = \mathcal{P}(R)
=\{r\in R~|~\Delta_{R}(r)=r\otimes 1 + 1\otimes r\}$.
\item[(iii)] $R$ is generated as an algebra by $R(1)$. \end{enumerate} In this case, $R$ is denoted by $\mathfrak{B}(V) = \bigoplus_{n\geq 0} \mathfrak{B}^{n}(V) $. \end{definition} \begin{remark} The Nichols algebra $\mathfrak{B}(V)$ is completely determined by the braiding. Let $\mathfrak{B}(M)$ denote the Nichols algebra generated by $M\in {}_{H}^{H}\mathcal{YD}$. More precisely, as proved in \cite{MR1396857} and noted in \cite{andruskiewitsch2001pointed}, $$\mathfrak{B}(M)=\mathrm{K}\oplus M\oplus\bigoplus\limits_{n=2}^\infty M^{\otimes n} / \ker\mathfrak{S}_n=T(M)/\ker\mathfrak{S},$$ where $\mathfrak{S}_{n,1}\in \mathrm{End}_\mathbb{K}\left(M^{\otimes (n+1)}\right)$, $\mathfrak{S}_{n}\in \mathrm{End}_\mathbb{K}\left(M^{\otimes n}\right)$, $$\mathfrak{S}_{n,1}\coloneqq\mathrm{id}+c_n+c_nc_{n-1}+\cdots+c_nc_{n-1}\cdots c_1=\mathrm{id}+c_n\mathfrak{S}_{n-1,1}$$ $$\mathfrak{S}_1\coloneqq\mathrm{id}, \quad \mathfrak{S}_2\coloneqq\mathrm{id}+c, \quad \mathfrak{S}_n\coloneqq(\mathfrak{S}_{n-1}\otimes \mathrm{id})\mathfrak{S}_{n-1,1}.$$ \end{remark}
\begin{lemma}\label{TensorNicholsAlg} (\cite[Theorem 2.2]{MR1779599}, \cite[Remark 1.4]{andruskiewitsch2016finite}) Let $M_1, M_2\in{}^H_H\mathcal{YD}$ be both finite dimensional and assume $c_{M_1,M_2}c_{M_2,M_1}=\mathrm{id}_{M_2\otimes M_1}$. Then $\mathfrak{B}(M_1\oplus M_2)\simeq \mathfrak{B}(M_1) \otimes \mathfrak{B}(M_2)$ as graded vector spaces and $\mathrm{GKdim}\,\mathfrak{B}(M_1\oplus M_2)= \mathrm{GKdim}\,\mathfrak{B}(M_1)+ \mathrm{GKdim}\,\mathfrak{B}(M_2)$. \end{lemma}
\begin{proposition}(\cite[Radford biproduct]{radford1985structure}) $H$ is a Hopf algebra. Let $A\in{}_H^H\mathcal{YD}$ be a braided Hopf algebra. Then $A\# H$ is a Hopf algebra. \begin{eqnarray} (a\# h)(a^\prime\# h^\prime)&=&\sum a(h_{(1)}\cdot a^\prime)\# h_{(2)}h^\prime,\quad a,a^\prime\in A, h,h^\prime\in H \\ \Delta(a\# h)&=&\sum\left[a_{(1)}\# (a_{(2)})_{(-1)}h_{(1)}\right]\otimes \left[(a_{(2)})_{(0)}\# h_{(2)}\right] \\ S(a\# h)&=&\sum \left(1\# S_H(h) S_H(a_{(-1)})\right) \left(S_A(a_{(0)})\# 1\right) \end{eqnarray} \end{proposition} The map $\iota: H \to A\#H$ given by $\iota(h) = 1\#h$ for all $h\in H$ is an injective Hopf algebra map, and the map $\pi: A\#H \to H$ given by $\pi(a\#h) = \varepsilon_{A}(a)h$ for all $a\in A$, $h\in H$ is a surjective Hopf algebra map such that $\pi \circ \iota = \mathrm{id}_{H} $. Moreover, it holds that $A= (A\#H)^{\mathrm{co}\, \pi}$.
Conversely, let $B$ be a Hopf algebra with bijective antipode and $\pi: B\to H$ a Hopf algebra epimorphism admitting a Hopf algebra section $\iota: H\to B$ such that $\pi\circ\iota =\mathrm{id}_{H}$. Then $A=B^{\mathrm{co}\,\pi}$ is a braided Hopf algebra in ${}^H_H\mathcal{YD}$ and $B\simeq A\# H$ as Hopf algebras.
\subsection{GK-dimension} Let $A$ be a finitely generated algebra over a field $\mathbb{K}$, and assume $a_1, \cdots , a_m$ generate $A$. Set $V$ to be the span of $a_1, \cdots, a_m$, and denote $V^n$ the span of all monomials in the $a_i$'s of length $n$. As $a_i$'s generate $A$ one has $A =\bigcup_{k=0}^\infty A_k $ where $A_k =\mathbb{K}+V +V^2+\cdots+V^k$. The function $d_V (n) = \dim A_n$ is the growth function of $A$. The \emph{Gelfand-Kirillov dimension} of a $\mathbb{K}$-algebra $A$ is
$\mathrm{GKdim}~A=\varlimsup\mathrm{log}_n~d_V(n)$.
$\mathrm{GKdim}~A$ does not depend on the choice of $V$. Suppose that $\mathrm{GKdim} ~A< \infty$. We say that a finite-dimensional subspace $V\subseteq A$ is \emph{$GK$-deterministic} if \begin{equation} \mathrm{GKdim} ~A = \lim_{n\rightarrow \infty} \log_n \dim \sum_{0\leq j\leq n} V^n. \end{equation} Clearly, if $V$ is a $GK$-deterministic subspace of $A$, then any finite-dimensional subspace of $A$ containing $V$ is $GK$-deterministic. Let $A$ and $B$ be two algebras. Then \begin{equation} \mathrm{GKdim}(A \otimes B) \leq \mathrm{GKdim}~ A + \mathrm{GKdim} ~B, \end{equation} but the equality does not hold in general. For instance, it does hold when A or B has a GK-deterministic subspace, see \cite[Proposition 3.11]{MR1721834}. The Gelfand-Kirillov dimension is a useful tool in Ring theory and Hopf algebraic theories. We shall not discuss in detail its importance but we refer the reader to \cite{MR1721834} as a basic reference and \cite{MR3490761,MR3061686,MR2661247,andruskiewitsch2016finite} for additional informations related with Hopf algebras.
\section{Simple Yetter-Drinfel'd modules of $H_8$}\label{YDMH8} Recall that the neither commutative nor cocommutative semisimple $8$-dimensional Hopf algebra $H_8$ in \cite{MR1357764} is constructed as an extension of $\mathbb{K}[C_2 \times C_2]$ by $\mathbb{K}[C_2]$. A basis for $H_8$ is given by $\{1, x, y, xy = yx, z, xz,yz, xyz\}$ with the relations $$x^{2} =y^{2}= 1,\quad z^{2} = \frac{1}{2}(1+x+y-xy), \quad xy = yx,\quad zx=yz,\quad zy=xz.$$ The coalgebra structure and the antipode are defined by $$ \Delta(x) = x \otimes x, \quad \Delta(y) = y \otimes y, \quad \varepsilon(x) = \varepsilon(y)=1, \quad S(x) = x, \quad S(y) = y,$$ $$ \Delta(z) =\frac{1}{2} (1 \otimes 1 + 1 \otimes x+ y \otimes 1- y \otimes x )(z\otimes z), \quad \varepsilon(z) = 1, \quad S(z) = z. $$
The automorphism group of $H_8$ is the Klein four-group \cite{MR2879228}. These automorphisms are given in Table \ref{autotableforH8}, which are going to be used in Corollary \ref{Isomorphism:B(V)H_8}. \begin{center} \begin{table}
\begin{tabular}{|c||c|c|c|c|} \hline & $1$ & $x$& $y$& $z$\\ \hline $\tau_1=\mathrm{id} $& $1$ & $x$& $y$& $z$\\ $\tau_2$& $1$ & $x$& $y$ & $xyz$\\ $\tau_3$& $1$ & $y$& $x$& $\frac{1}{2}\left(z+xz+yz-xyz\right)$\\ $\tau_4$& $1$ & $y$& $x$& $\frac{1}{2}\left(-z+xz+yz+xyz\right)$\\ \hline \end{tabular}
\caption{Automorphisms of $H_8$} \label{autotableforH8} \end{table} \end{center}
Denote a set of central orthogonal idempotents of $H_8$ as $$e_1=\frac{1}{8}(1+x)(1+y)(1+z),\quad e_2=\frac{1}{8}(1+x)(1+y)(1-z),$$ $$e_3=\frac{1}{8}(1-x)(1-y)(1+\mathrm{i}\, z),\quad e_4=\frac{1}{8}(1-x)(1-y)(1-\mathrm{i}\,z),$$ $$e_5=\frac{1-xy}{2}, \quad e_je_k=\delta_{jk},\quad j,k=1,\cdots, 5; \mathrm{i}=\sqrt{-1}.$$ And denote idempotents $e_5^\prime=\frac{1}{4}(1-xy)(1+z), e_5^{\prime\prime}=\frac{1}{4}(1-xy)(1-z)$, then \begin{eqnarray*} H_8 &=& H_8e_1\oplus H_8e_2\oplus H_8e_3\oplus H_8e_4\oplus H_8e_5\\ &=& H_8e_1\oplus H_8e_2\oplus H_8e_3\oplus H_8e_4\oplus(H_8e_5^{\prime}+ H_8e_5^{\prime\prime}), \end{eqnarray*} where $H_8e_5^{\prime}\simeq H_8e_5^{\prime\prime}$ as left $H_8$-module, via $e_5^{\prime}\mapsto xe_5^{\prime\prime}, xe_5^{\prime}\mapsto e_5^{\prime\prime}$. \begin{definition}
Denote $V_1(b)\coloneqq\mathbb{K}\{v|x\cdot v=b^2v, y\cdot v=b^2v, z\cdot v=bv, b\in \{\pm 1, \pm \mathrm{i}\}\}$, where $v$ is a vector. Let $V_2\simeq H_8e_5^{\prime}$ as left $H_8$-module, the actions of the generators are given by $$x\mapsto \begin{pmatrix}
0 & 1\\
1 & 0 \end{pmatrix}, \quad y\mapsto \begin{pmatrix}
0 & -1\\
-1 & 0 \end{pmatrix}, \quad z\mapsto \begin{pmatrix}
1 & 0\\
0 & -1 \end{pmatrix}.$$ \end{definition} \begin{proposition} All simple left modules of $H_8$ are classified by $V_1(b), V_2, b\in \{\pm 1, \pm \mathrm{i}\}.$ \end{proposition} \begin{remark} Thanks to the referee's reminder, the result was also obtained in \cite{MR1987013} under a different basis. \end{remark} In the remaining part of the article, $V_1(b)$ and $V_2$ always mean a simple left $H_8$-module.
\begin{lemma} (\cite[Proposition 2]{radford2003oriented}) \label{constructionLRYD} Let $H$ is a bialgebra over field $\mathbb{K}$ and suppose $S$ is the antipode of $H$. \begin{enumerate} \item\label{que:LRYDM} If $L\in{}_{H}\mathcal{M}$, then $L\otimes H\in{}_H\mathcal{YD}^{H}$, the module and comodule actions are given by \begin{equation*} h\cdot(\ell\otimes a)=h_{(2)}\cdot\ell\otimes h_{(3)}aS^{-1}(h_{(1)}),~~\rho(\ell\otimes h)=(\ell\otimes h_{(1)})\otimes h_{(2)}, \forall h, a\in H, \ell\in L. \end{equation*} Let $M\in {}_H \mathcal{YD}^H$ . \item Suppose that $L \in {}_H\mathcal{M}$ and $p: M\longrightarrow L$ is a map of left $H$-modules. Then the linear map $f :M \longrightarrow L \otimes H$ defined by $f (m) = p(m_{(0)})\otimes m_{(1)}$ for all $m\in M$ is a map of Yetter-Drinfel'd $H$-modules, where $L \otimes H$ has the structure described in part \eqref{que:LRYDM}. Furthermore $\ker f$ is the largest Yetter-Drinfel'd $H$-submodule, indeed the largest subcomodule, contained in $\ker p$. \item $M$ is isomorphic to a Yetter-Drinfel'd
submodule of some $L\otimes H$ described in above.
\end{enumerate} \end{lemma}
Similarly according to Radford's method, any simple left Yetter-Drinfel'd module over $H_8$ could be constructed by the submodule of tensor product of a left module $V$ of $H_8$ and $H_8$ itself, where the module and comodule structures are given by : \begin{eqnarray} h\cdot (\ell\boxtimes g)=(h_{(2)}\cdot \ell)\boxtimes h_{(1)}gS(h_{(3)}),\label{eq:action}\\ \rho(\ell\boxtimes h)=h_{(1)}\otimes (\ell\boxtimes h_{(2)}) , \forall h, g,\in H_8, \ell\in V. \label{eq:coaction} \end{eqnarray} Here we use $\boxtimes$ instead of $\otimes$ to avoid confusion by using too many symbals of the tensor product. We are going to construct all simple left Yetter-Drinfel'd modules over $H_8$ in this way. Keeping in mind that $H_8$ is semisimple, it's possibly being done. In fact, it is much easier than making use of the fact that ${}_{H}^{H}\mathcal{YD}\simeq {}_{\mathcal{D}\left(H_8^{cop}\right)}\mathcal{M}$. The following is a list of useful formulae for looking for simple objects of ${}_{H_8}^{H_8}\mathcal{YD}$.
\begin{lemma} \begin{eqnarray} (\mathrm{id}^{\otimes 2}\otimes S)\Delta^{(2)}(z)= \frac{1}{4}[& (1+y)z\otimes z\otimes z(1+x)+(1-y)z\otimes xz\otimes z(1+x)+\\ +&(1+y)z\otimes yz\otimes z(1-x)+(y-1)z\otimes xyz\otimes z(1-x)] \end{eqnarray} \begin{eqnarray} z_{(2)}\otimes z_{(1)}?S(z_{(3)})= \frac{1}{4}[z\otimes (1+y)z?z(1+x)+ xz\otimes(1-y)z?z(1+x)+\\ +yz\otimes(1+y)z?z(1-x)+xyz\otimes(1-y)z?z(x-1)]. \end{eqnarray}
\begin{eqnarray} z_{(2)}\otimes z_{(1)}S(z_{(3)})= \frac{1}{4}[z\otimes (1+x)(1+y)+ xz\otimes(1+x)(1-y)+\label{eq:1}\\ +yz\otimes(1-x)(1+y)+xyz\otimes(1-x)(1-y)],\\ z_{(2)}\otimes z_{(1)}xS(z_{(3)})= \frac{1}{4}[z\otimes (1+x)(1+y)+ xz\otimes(1+x)(y-1)+\label{eq:x}\\ +yz\otimes(1-x)(1+y)+xyz\otimes(x-1)(1-y)],\\ z_{(2)}\otimes z_{(1)}yS(z_{(3)})= \frac{1}{4}[z\otimes (1+x)(1+y)+ xz\otimes(1+x)(1-y)+\label{eq:y}\\ +yz\otimes(x-1)(1+y)+xyz\otimes(x-1)(1-y)],\\ z_{(2)}\otimes z_{(1)}xyS(z_{(3)})= \frac{1}{4}[z\otimes (1+x)(1+y)+ xz\otimes(1+x)(y-1)+\label{eq:xy}\\ +yz\otimes(x-1)(1+y)+xyz\otimes(1-x)(1-y)],\\ z_{(2)}\otimes z_{(1)}zS(z_{(3)})= \frac{1}{2}[z\otimes (1+y)z+xyz\otimes x(y-1)z],\label{eq:z}\\ z_{(2)}\otimes z_{(1)}xzS(z_{(3)})= \frac{1}{2}[z\otimes (1+y)z+xyz\otimes x(1-y)z],\label{eq:xz}\\ z_{(2)}\otimes z_{(1)}yzS(z_{(3)})= \frac{1}{2}[z\otimes x(1+y)z+xyz\otimes (y-1)z],\label{eq:yz}\\ z_{(2)}\otimes z_{(1)}xyzS(z_{(3)})= \frac{1}{2}[z\otimes x(1+y)z+xyz\otimes (1-y)z].\label{eq:xyz} \end{eqnarray} \end{lemma} \begin{definition}
Define $M\langle b,g\rangle\coloneqq\mathbb{K}\{v\boxtimes g|v\in V_1(b)\}$, where $b\in\{\pm 1, \pm \mathrm{i}\}$ and $g\in\{1,x,y,xy\}$. \end{definition} \begin{lemma} There are eight pairwise non-isomorphic one dimensional Yetter-Drinfel'd modules over $H_8$ as $M\langle b,g\rangle$ with $(b,g)\in\{(\pm1,1),(\pm1,xy),(\pm\mathrm{i},x),(\pm\mathrm{i},y)\}$. The actions and coactions are given by \begin{eqnarray} x\cdot (v\boxtimes g)=b^2(v\boxtimes g),\quad y\cdot (v\boxtimes g)=b^2(v\boxtimes g), \quad z\cdot (v\boxtimes g)=b(v\boxtimes g),\\ \rho(v\boxtimes g)=g\otimes (v\boxtimes g),\quad v\boxtimes g\in M\langle b,g\rangle, \quad v\in V_1(b). \end{eqnarray} \end{lemma} \begin{proof} Let $v\in V_1(b)$, then \begin{eqnarray} z\cdot (v\boxtimes 1)\xlongequal{\eqref{eq:1}} \frac{bv}{4}\boxtimes [1+x+b^2(1-x)][1+y+b^2(1-y)],\label{eq:1dimXgrouplike1}\\ z\cdot (v\boxtimes xy)\xlongequal{\eqref{eq:xy}} \frac{bv}{4}\boxtimes [1+x+b^2(x-1)][1+y+b^2(y-1)],\\ z\cdot (v\boxtimes x)\xlongequal{\eqref{eq:x}} \frac{bv}{4}\boxtimes [1+x+b^2(1-x)][1+y+b^2(y-1)],\\ z\cdot (v\boxtimes y)\xlongequal{\eqref{eq:y}} \frac{bv}{4}\boxtimes [1+x+b^2(x-1)][1+y+b^2(1-y)].\label{eq:1dimXgrouplike4} \end{eqnarray} so \begin{eqnarray} z\cdot (v\boxtimes 1)=bv\boxtimes 1,\quad z\cdot (v\boxtimes xy)=bv\boxtimes xy,\text{when} ~ b=\pm 1;\\ z\cdot (v\boxtimes x)=bv\boxtimes x,\quad z\cdot (v\boxtimes y)=bv\boxtimes y,\text{when} ~ b=\pm \mathrm{i}. \end{eqnarray} Now it's easy to see that $M\langle b,g\rangle$ defined in above is a one-dimensional Yetter-Drinfel'd modules by Radford's method and the eight one-dimensional Yetter-Drinfel'd modules are pairwise non-isomorphic by observations on their actions and coactions. \end{proof} \begin{definition} Let $(g_1, g_2)\in \{(1,y), (x,1),(xy,x),(y,xy)\}$ and denote three vector spaces as
$$M\langle(1,xy)\rangle\coloneqq\mathbb{K}\{v\boxtimes 1, v\boxtimes xy|v\in V_1(\mathrm{i})\},$$
$$M\langle(x,y)\rangle\coloneqq\mathbb{K}\{v\boxtimes x, v\boxtimes y|v\in V_1(1)\},$$
$$M\langle(g_1,g_2)\rangle\coloneqq\mathbb{K}\{(v_1+v_2)\boxtimes g_1, (v_1-v_2)\boxtimes g_2|v_1,v_2\in V_2\}.$$ \end{definition} \begin{lemma}\label{YDM2dim1} There are six pairwise non-isomorphic two-dimensional simple Yetter-Drinfel'd modules over $H_8$ as below, where the action and coaction are given by formulae (\ref{eq:action}) and (\ref{eq:coaction}). \begin{enumerate} \item $M\langle(1,xy)\rangle$, the actions of generators on $(v\boxtimes 1, v\boxtimes xy)$ are given by $$x\mapsto \begin{pmatrix}
-1 & 0\\
0 & -1 \end{pmatrix}, \quad y\mapsto \begin{pmatrix}
-1 & 0\\
0 & -1 \end{pmatrix}, \quad z\mapsto \begin{pmatrix}
0 & \mathrm{i}\\
\mathrm{i} & 0 \end{pmatrix}.$$ \item $M\langle(x,y)\rangle$, the actions of generators on $(v\boxtimes x, v\boxtimes y)$ are given by $$x\mapsto \begin{pmatrix}
1 & 0\\
0 & 1 \end{pmatrix}, \quad y\mapsto \begin{pmatrix}
1 & 0\\
0 & 1 \end{pmatrix}, \quad z\mapsto \begin{pmatrix}
0 & 1\\
1 & 0 \end{pmatrix}.$$ \item $M\langle(g_1,g_2)\rangle$, where $(g_1,g_2)\in \{(1,y), (x,1),(xy,x),(y,xy)\}$. the actions of generators on the row vector $((v_1+v_2)\boxtimes g_1, (v_1-v_2)\boxtimes g_2)$ are given by $$x\mapsto \begin{pmatrix}
1 & 0\\
0 & -1 \end{pmatrix}, \quad y\mapsto \begin{pmatrix}
-1 & 0\\
0 & 1 \end{pmatrix}, \quad z\mapsto \begin{pmatrix}
0 & 1\\
1 & 0 \end{pmatrix}.$$ \end{enumerate} \end{lemma} \begin{proof} For the coactions are easy to see, we can focus on their structures as left $H_8$-modules. Part (1) and (2) of the lemma can be checked by formulae (\ref{eq:1dimXgrouplike1}) to (\ref{eq:1dimXgrouplike4}). Let $v_1, v_2\in V_2$, then \begin{eqnarray} z\cdot (v_1\boxtimes 1)\xlongequal{\eqref{eq:1}}\frac{1}{2}[v_1\boxtimes (x+y)+v_2\boxtimes (x-y)],\\ z\cdot (v_2\boxtimes 1)\xlongequal{\eqref{eq:1}}\frac{1}{2}[v_1\boxtimes (-x+y)+v_2\boxtimes (-x-y)],\\ z\cdot (v_1\boxtimes xy)\xlongequal{\eqref{eq:xy}}\frac{1}{2}[v_1\boxtimes (x+y)+v_2\boxtimes (y-x)],\\ z\cdot (v_2\boxtimes xy)\xlongequal{\eqref{eq:xy}}\frac{1}{2}[v_1\boxtimes (x-y)+v_2\boxtimes (-x-y)],\\ z\cdot (v_1\boxtimes y)\xlongequal{\eqref{eq:y}}\frac{1}{2}[v_1\boxtimes (1+xy)+v_2\boxtimes (1-xy)],\\ z\cdot (v_2\boxtimes y)\xlongequal{\eqref{eq:y}}\frac{1}{2}[v_1\boxtimes (-1+xy)+v_2\boxtimes (-1-xy)],\\ z\cdot (v_1\boxtimes x)\xlongequal{\eqref{eq:x}}\frac{1}{2}[v_1\boxtimes (1+xy)+v_2\boxtimes (-1+xy)],\\ z\cdot (v_2\boxtimes x)\xlongequal{\eqref{eq:x}}\frac{1}{2}[v_1\boxtimes (1-xy)+v_2\boxtimes (-1-xy)]. \end{eqnarray} So we have $$z\cdot [(v_1+v_2)\boxtimes 1]=(v_1-v_2)\boxtimes y,\quad z\cdot [(v_1-v_2)\boxtimes y]=(v_1+v_2)\boxtimes 1,$$ $$z\cdot [(v_1+v_2)\boxtimes x]=(v_1-v_2)\boxtimes 1,\quad z\cdot [(v_1-v_2)\boxtimes 1]=(v_1+v_2)\boxtimes x,$$ $$z\cdot [(v_1+v_2)\boxtimes xy]=(v_1-v_2)\boxtimes x,\quad z\cdot [(v_1-v_2)\boxtimes x]=(v_1+v_2)\boxtimes xy,$$ $$z\cdot [(v_1+v_2)\boxtimes y]=(v_1-v_2)\boxtimes xy,\quad z\cdot [(v_1-v_2)\boxtimes xy]=(v_1+v_2)\boxtimes y.$$ Part (3) is immediate to check. The six two-dimensional Yetter-Drinfel'd modules are pairwise non-isomorphic since they are pairwise non-isomorphic as comodules. \end{proof}
\begin{lemma} \label{YDM2dim2} Let $b_1,b_2\in \{\pm 1\} $ and $v\in V_1(b_2)$ and denote \begin{equation} w^{b_1,b_2}_1\triangleq v\boxtimes (1+\mathrm{i} b_1 y)z,\quad w^{b_1,b_2}_2\triangleq v\boxtimes x(1-\mathrm{i} b_1 y)z. \end{equation} Then $W^{b_1,b_2}=\mathbb{K}w^{b_1,b_2}_1+\mathbb{K}w^{b_1,b_2}_2$ is a family of 4 pairwise non-isomorphic two dimensional simple Yetter-Drinfel'd modules over $H_8$ with the actions of generators on the row vector $(w^{b_1,b_2}_1, w^{b_1,b_2}_2)$ and coactions given by $$x\mapsto \begin{pmatrix}
0 & -\mathrm{i} b_1\\
\mathrm{i} b_1 & 0 \end{pmatrix}, \quad y\mapsto \begin{pmatrix}
0 & -\mathrm{i} b_1\\
\mathrm{i} b_1 & 0 \end{pmatrix}, \quad z\mapsto \begin{pmatrix}
\frac{(1-\mathrm{i} b_1)b_2}{2} & \frac{(1-\mathrm{i} b_1)b_2}{2}\\
\frac{-(1-\mathrm{i} b_1)b_2}{2} & \frac{(1-\mathrm{i} b_1)b_2}{2} \end{pmatrix},$$ $$\rho\left(w^{b_1,b_2}_1\right)=\frac{(1+y)z}{2}\otimes w^{b_1,b_2}_1 +\frac{(1-y)z}{2}\otimes w^{b_1,b_2}_2,$$ $$\rho\left(w^{b_1,b_2}_2\right)=\frac{x(1+y)z}{2}\otimes w^{b_1,b_2}_2 +\frac{x(1-y)z}{2}\otimes w^{b_1,b_2}_1.$$ \end{lemma}
\begin{proof}
It's straightforward by the definition of Yetter-drinfel'd module. When $b_2\neq b_2^\prime$, $W^{b_1,b_2}\nsimeq W^{b_1,b_2^\prime}$ since we will see that their braidings are different in Proposition \ref{NAlgdim1}. As explained in the following remark, $W^{b_1,b_2}$ has another basis $\{p_1,p_2\}$ with $p_1\in V_1(b_2)$ and $p_2\in V_1(-b_1b_2\mathrm{i})$. So $W^{b_1,b_2}\nsimeq W^{b_1^\prime,b_2}$ if $b_1\neq b_1^\prime$.
\end{proof}
\begin{remark}
\begin{enumerate}
\item Let
$M=\mathbb{K}\{v\boxtimes z, v\boxtimes xz, v\boxtimes yz, v\boxtimes xyz| v\in V_1(b)\}$, $b\in\{\pm 1\}$. $z$ acts on elements of $M$ as \begin{align*} z\cdot (v\boxtimes z)&\xlongequal{\eqref{eq:z}} \frac{bv}{2}\boxtimes (1-x+y+xy)z, &z\cdot (v\boxtimes xz)\xlongequal{\eqref{eq:xz}} \frac{bv}{2}\boxtimes (1+x+y-xy)z,\\ z\cdot (v\boxtimes yz)&\xlongequal{\eqref{eq:yz}} \frac{bv}{2}\boxtimes (-1+x+y+xy)z, &z\cdot (v\boxtimes xyz)\xlongequal{\eqref{eq:xyz}} \frac{bv}{2}\boxtimes (1+x-y+xy)z. \end{align*} Then $M\simeq W^{1,b}\oplus W^{-1,b}$ as Yetter-Drinfel'd modules over $H_8$.
\item Let $f_{jk}\triangleq\frac{1}{4}[1+(-1)^jx][1+(-1)^ky]$, $j,k=0,1$. Denote $$p_1=w_1^{b_1, b_2}+\mathrm{i} b_1 w_2^{b_1,b_2},\quad p_2=w_1^{b_1, b_2}-\mathrm{i} b_1 w_2^{b_1,b_2},$$ then $W^{b_1,b_2}=\mathbb{K}p_1+\mathbb{K}p_2$ with the actions of generators on the row vector $(p_1, p_2)$ and coactions given by $$x\mapsto \begin{pmatrix}
1 & 0\\
0 & -1 \end{pmatrix}, \quad y\mapsto \begin{pmatrix}
1 & 0\\
0 & -1 \end{pmatrix}, \quad z\mapsto \begin{pmatrix}
b_2 & 0\\
0 & -\mathrm{i} b_1b_2 \end{pmatrix},$$ $$\rho(p_1)=\left[f_{00}- \mathrm{i} b_1f_{11}\right]z\otimes p_1+\left[f_{10}+ \mathrm{i} b_1f_{01}\right]z\otimes p_2,$$ $$\rho(p_2)=\left[f_{00}+ \mathrm{i} b_1f_{11}\right]z\otimes p_2+ \left[f_{10}- \mathrm{i} b_1f_{01}\right]z\otimes p_1.$$ \end{enumerate} \end{remark}
According to \cite[Remark 2.14]{MR1357764}, $H_8$ is presented by generators $x,y,w$, where the expressions containing $z$ are replaced by \begin{eqnarray} w = \left(f_{00}+\sqrt{\mathrm{i}} f_{10} +\frac{1}{\sqrt{\mathrm{i}}}f_{01}+\mathrm{i} f_{11}\right)z, \quad
w^2=1, \\
wx = yw,\quad S(w)=\left(\frac{1+\mathrm{i}}{2}x+\frac{1-\mathrm{i}}{2}y\right)w,\\ \Delta(w)=\left(\frac{1}{2}(1+xy)\otimes 1+\frac{1+\mathrm{i}}{4}(1-xy)\otimes x+\frac{1-\mathrm{i}}{4}(1-xy)\otimes y\right)(w\otimes w). \end{eqnarray}
Let $a+1=\pm \sqrt{2}$, we define \begin{eqnarray*} w_1^{(1)}\triangleq(v_1+\mathrm{i} av_2)\boxtimes \frac{\mathrm{i}}{2}\left[(x+y)+\sqrt{\mathrm{i}}(x-y)\right]w +(av_1-\mathrm{i} v_2)\boxtimes \frac{1}{2}\left[(x+y)-\sqrt{\mathrm{i}}(x-y)\right]w,\\ w_2^{(1)}\triangleq(v_1+\mathrm{i} av_2)\boxtimes \frac{\mathrm{i}}{2}\left[(1+xy)+\sqrt{\mathrm{i}}(1-xy)\right]w -(av_1-\mathrm{i} v_2)\boxtimes \frac{1}{2}\left[(1+xy)-\sqrt{\mathrm{i}}(1-xy)\right]w,\\ w_1^{(2)}\triangleq(v_1-\mathrm{i} av_2)\boxtimes \frac{\mathrm{i}}{2}\left[(x+y)+\sqrt{\mathrm{i}}(x-y)\right]w +(av_1+\mathrm{i} v_2)\boxtimes \frac{1}{2}\left[(x+y)-\sqrt{\mathrm{i}}(x-y)\right]w,\\ w_2^{(2)}\triangleq(v_1-\mathrm{i} av_2)\boxtimes \frac{\mathrm{i}}{2}\left[(1+xy)+\sqrt{\mathrm{i}}(1-xy)\right]w -(av_1+\mathrm{i} v_2)\boxtimes \frac{1}{2}\left[(1+xy)-\sqrt{\mathrm{i}}(1-xy)\right]w. \end{eqnarray*} \begin{lemma} \label{YDM2dim3} Let $a+1=\pm \sqrt{2}$, there are 4 pairwise non-isomorphic simple Yetter-Drinfel'd modules $W_1^a$ and $W_2^a$ over $H_8$ as following \begin{enumerate} \item Let $W_1^a=\mathbb{K}w_1^{(1)}\oplus \mathbb{K}w_1^{(1)}$, then $W_1^a$ is a two dimensional simple Yetter-Drinfel'd module over $H_8$ with actions given by \[ \left\{ \begin{array}{rl} &x\cdot w_1^{(1)}=-w_1^{(1)} \\ &y\cdot w_1^{(1)}=w_1^{(1)} \\ &z\cdot w_1^{(1)}=\frac{1}{2}(1-\mathrm{i})(a+1)w_2^{(1)}\\ &w\cdot w_1^{(1)}=\frac{1}{2\sqrt{\mathrm{i}}}(1-\mathrm{i})(a+1)w_2^{(1)} \end{array} \right.\quad \left\{ \begin{array}{rl} &x\cdot w_2^{(1)}=w_2^{(1)} \\ &y\cdot w_2^{(1)}=-w_2^{(1)} \\ &z\cdot w_2^{(1)}=\frac{1}{2}(1+\mathrm{i})(a+1)w_1^{(1)}\\ &w\cdot w_2^{(1)}=\frac{\sqrt{\mathrm{i}}}{2}(1+\mathrm{i})(a+1)w_1^{(1)} \end{array} \right. \] and coactions given by \begin{align*} \rho\left(w_1^{(1)}\right)&=\frac{1}{2}(x+y)w\otimes w_1^{(1)}+\frac{\sqrt{\mathrm{i}}}{2} (x-y)w \otimes w_2^{(1)},\\ \rho\left(w_2^{(1)}\right)&=\frac{1}{2}(1+xy)w\otimes w_2^{(1)}+\frac{\sqrt{\mathrm{i}}}{2} (1-xy)w \otimes w_1^{(1)}. \end{align*} \item Let $W_2^a=\mathbb{K}w_1^{(2)}\oplus \mathbb{K}w_1^{(2)}$, then $W_2^a$ is a two dimensional simple Yetter-Drinfel'd module over $H_8$ with actions given by \[ \left\{ \begin{array}{rl} &x\cdot w_1^{(2)}=w_1^{(2)} \\ &y\cdot w_1^{(2)}=-w_1^{(2)} \\ &z\cdot w_1^{(2)}=\frac{1}{2}(1-\mathrm{i})(a+1)w_2^{(2)}\\ &w\cdot w_1^{(2)}=\frac{\sqrt{\mathrm{i}}}{2}(1-\mathrm{i})(a+1)w_2^{(2)} \end{array} \right.\quad \left\{ \begin{array}{rl} &x\cdot w_2^{(2)}=-w_2^{(2)} \\ &y\cdot w_2^{(2)}=w_2^{(2)} \\ &z\cdot w_2^{(2)}=\frac{1}{2}(1+\mathrm{i})(a+1)w_1^{(2)}\\ &w\cdot w_2^{(2)}=\frac{1}{2\sqrt{\mathrm{i}}}(1+\mathrm{i})(a+1)w_1^{(2)} \end{array} \right. \] and coactions given by \begin{align*} \rho\left(w_1^{(2)}\right)&=\frac{1}{2}(x+y)w\otimes w_1^{(2)}+\frac{\sqrt{\mathrm{i}}}{2} (x-y)w \otimes w_2^{(2)} ,\\ \rho\left(w_2^{(2)}\right)&=\frac{1}{2}(1+xy)w\otimes w_2^{(2)}+\frac{\sqrt{\mathrm{i}}}{2} (1-xy)w \otimes w_1^{(2)}. \end{align*} \end{enumerate} \end{lemma} \begin{proof} It's straightforward to check by the definition of Yetter-drinfel'd module. Actually $M\simeq \bigoplus_{a+1=\pm \sqrt{2}}\left(W^a_1 \oplus W^a_2\right)$ as Yetter-Drinfel'd modules over $H_8$, where
$M=\mathbb{K}\{v_j\boxtimes z, v_j\boxtimes xz, v_j\boxtimes yz, v_j\boxtimes xyz|v_j\in V_2, j=1,2\}$.
Since $\sqrt{\mathrm{i}}=\cos\frac{\pi}{4}+\mathrm{i} \sin\frac{\pi}{4}$, $\frac{1}{2}\sqrt{2}\sqrt{\mathrm{i}}(1-\mathrm{i})=1$. Denote $a+1=b \sqrt{2}$, $b=\pm 1$, $p_1^{(1)}=\sqrt{\mathrm{i}}w_1^{(1)}+w_2^{(1)}$, $p_2^{(1)}=-\sqrt{\mathrm{i}}w_1^{(1)}+w_2^{(1)}$, then $W^a_1=\mathbb{K}p_1^{(1)}+\mathbb{K}p_2^{(1)}$ with actions on the row vector $\left(p_1^{(1)}, p_2^{(1)}\right)$ given by $$x\mapsto \begin{pmatrix}
0 & 1\\
1 & 0 \end{pmatrix}, \quad y\mapsto \begin{pmatrix}
0 & -1\\
-1 & 0 \end{pmatrix}, \quad z\mapsto \begin{pmatrix}
b & 0\\
0 & -b \end{pmatrix}.$$ Let $p_1^{(2)}=w_1^{(2)}+\frac{1}{\sqrt{\mathrm{i}}}w_2^{(2)}$, $p_2^{(2)}=w_1^{(2)}-\frac{1}{\sqrt{\mathrm{i}}}w_2^{(2)}$, then $W^a_2=\mathbb{K}p_1^{(2)}+\mathbb{K}p_2^{(2)}$ with actions on the row vector $\left(p_1^{(2)}, p_2^{(2)}\right)$ given by $$x\mapsto \begin{pmatrix}
0 & 1\\
1 & 0 \end{pmatrix}, \quad y\mapsto \begin{pmatrix}
0 & -1\\
-1 & 0 \end{pmatrix}, \quad z\mapsto \begin{pmatrix}
b & 0\\
0 & -b \end{pmatrix}.$$ Now we can observe that $W_1^{-1+\sqrt{2}}$ is isomorphic to $W_1^{-1-\sqrt{2}}$ as modules (or comodules) under suitably chosen base, but they are not isomorphic as modules and comodules at the same time. So $W_1^{-1+\sqrt{2}}\nsimeq W_1^{-1-\sqrt{2}}$ as Yetter-Drinfel'd modules. For the same reason, we have $W_2^{-1+\sqrt{2}}\nsimeq W_2^{-1-\sqrt{2}}$ and $W_1^a\nsimeq W_2^a$. \end{proof}
Obviously, any module in Lemma \ref{YDM2dim1} is not isomorphic to any one of modules in Lemma \ref{YDM2dim2} and \ref{YDM2dim3} as comodules. As $H_8$-modules, $W^{b_1,b_2}\simeq V_1(b_2)\oplus V_1(-b_1b_2\mathrm{i})$, and $W_1^a\simeq W_2^a\simeq V_2$. So Yetter-drinfel'd modules in Lemma \ref{YDM2dim1}, \ref{YDM2dim2} and \ref{YDM2dim3} are pairwise non-isomorphic. Keeping in mind that $H_8$ is semisimple, now we are arriving at \begin{theorem} All the simple Yetter-Drinfel'd modules over $H_8$ are classified by \begin{itemize} \item 8 pairwise non-isomorphic simple Yetter-drinfel'd modules of one-dimension: $$M\langle b,g\rangle,\quad (b,g)\in\left\{(\pm 1,1),(\pm 1,xy), (\pm \mathrm{i},x), (\pm \mathrm{i},y)\right\}.$$ \item 14 pairwise non-isomorphic simple Yetter-drinfel'd modules of two-dimension: $$M\langle(1,xy)\rangle, ~M\langle(x,y)\rangle, M\langle(g_1,g_2)\rangle,
W^{b_1,b_2}, W_1^a, W_2^a, $$ where $(g_1,g_2)\in \{(1,y), (x,1),(xy,x),(y,xy)\}, b_1, b_2\in\{\pm 1\}, a+1=\pm\sqrt{2}.$
\end{itemize} \end{theorem} \begin{remark}
Jun Hu and Yinhuo Zhang investigated $\mathcal{D}(H)$-modules in \cite{MR2336009} and \cite{MR2352888} by using Radford's construction \cite{radford2003oriented}, especially they constructed all simple modules of $\mathcal{D}(H_8)$ under a different basis comparing with ours. \end{remark}
\section{Nichols algebras in ${}_{H_8}^{H_8}\mathcal{YD}$} \label{section:NicholsAlgebras} In this section, we try to determine all the finite-dimensional Nichols algebras generated by Yetter-Drinfel'd modules over $H_8$. As a byproduct, we calculate Gelfand-Kirillov dimensions for some Nichols algebras.
We begin by studying the Nichols algebras of simple Yetter-Drinfel'd modules. \begin{proposition}\label{NAlgdim1} Given a simple Yetter-Drinfel'd module $M$ over $H_8$, $\dim\mathfrak{B}(M)$ ($\mathrm{Gkdim}\,\mathfrak{B}(M)$ for some cases) is presented in Table \ref{dimNAlgSimpleYDM_H8}. Especially,
\begin{center} \begin{table}
\begin{tabular}{|c||c|c|c|} \hline
$M\in {}_{H_8}^{H_8}\mathcal{YD}$
& condition & $\dim\mathfrak{B}(M)$ & $\mathrm{GKdim}\,\mathfrak{B}(M)$\\ \hline
\multirow{2}{*} {$M\langle b,g\rangle$}& $(b,g)\in\{(\pm 1, 1),(\pm 1, xy)\}$ & $\infty$& $1$\\\cline{2-4}
&$(b,g)\in\{(\pm\mathrm{i}, x),(\pm\mathrm{i}, y)\}$ & $2$& $0$ \\\hline $M\langle(1,xy)\rangle$ & & $\infty$& $2$\\\hline $M\langle(x,y)\rangle$ & & $\infty$& $2$\\\hline
\multirow{2}{*}{$M\langle(g_1,g_2)\rangle$}& $(g_1,g_2)\in\{(1,y),(x,1)\}$ & $\infty$& $\infty$\\\cline{2-4} & $(g_1,g_2)\in\{(xy,x),(y,xy)\}$ & $8$& $0$\\\hline
\multirow{2}{*}{$W^{b_1,b_2}$} & $b_1=\pm 1$, $b_2=-1$ & $8$& $0$\\\cline{2-4} & $b_1=\pm 1$, $b_2=1$ & $\infty$& $\infty$\\\hline $W_1^a$, $W_2^a$& $a+1=\pm\sqrt{2}$ & $\infty$& \\ \hline \end{tabular}
\caption{Nichols algebras of simple Yetter-Drinfel'd modules over $H_8$} \label{dimNAlgSimpleYDM_H8} \end{table} \end{center} \begin{enumerate} \item \label{NicholsAlg:M(b,g)} $ \mathfrak{B}\left(M\langle b,g\rangle\right) = \left\{ \begin{array}{rl} \mathbb{K}[p], & \text{if } (b,g)\in\{(\pm 1,1),(\pm 1,xy)\},\\ \mathbb{K}[p]/(p^{2})= \bigwedge \mathbb{K}p, &\text{if } (b,g)\in\{(\pm \mathrm{i},x), (\pm \mathrm{i},y)\}.\\ \end{array} \right.$ \item \label{NicholsAlg:A_2} The both braidings of $M\langle(g_1,g_2)\rangle$ for $(g_1,g_2)\in \{(xy,x),(y,xy)\}$ and $W^{b_1,-1}$ for $b_1=\pm 1$ are Cartan type $A_2$, so their corresponding Nichols algebras are isomorphic to an algebra which is generated by $p_1$, $p_2$ satisfying relations $p_1p_2p_1p_2+p_2p_1p_2p_1=0$, $p_1^2=p_2^2=0.$ \end{enumerate} \end{proposition}
\begin{proof}
\begin{itemize}\renewcommand{$\diamond$}{$\diamond$} \item
Because $ c(p\otimes p) =g\cdot p\otimes p= \left\{ \begin{array}{rl} p\otimes p, & \text{if } (b,g)\in\{(\pm 1,1),(\pm 1,xy)\}\\ -p\otimes p, &\text{if } (b,g)\in\{(\pm \mathrm{i},x), (\pm \mathrm{i},y)\}\\ \end{array} \right. $ under the assumption that $M\langle b,g\rangle=\mathbb{K}p$, the part \eqref{NicholsAlg:M(b,g)} is obvious. \item As for the part \eqref{NicholsAlg:A_2}, we only give a proof for the case $W^{b_1,-1}$ for $b_1=\pm 1$. Let $p_1=w_1^{b_1,b_2}+\mathrm{i} b_1w_2^{b_1,b_2}$ and $p_2=w_1^{b_1,b_2}-\mathrm{i} b_1w_2^{b_1,b_2}$, then the braiding of $W^{b_1,b_2}$ is given by \begin{eqnarray*} c(p_1\otimes p_1)&=&b_2 p_1\otimes p_1, \quad c(p_2\otimes p_2)=b_2 p_2\otimes p_2, \\ c(p_1\otimes p_2)&=&-b_2 p_2\otimes p_1, ~~ c(p_2\otimes p_1)=b_2 p_1\otimes p_2. \end{eqnarray*} When $b_2=1$, $\mathrm{GKdim}\,\mathfrak{B}\left(W^{b_1,1}\right)=\infty$ according to \cite[Lemma 2.8]{andruskiewitsch2016finite}. When $b_2=-1$, the braiding is type $A_2$. As discussed in \cite{MR2136919}, the Nichols algebra $\mathfrak{B}\left(W^{b_1,-1}\right)$ is generated by $p_1$, $p_2$ with relations $p_1p_2p_1p_2+p_2p_1p_2p_1=0$, $p_1^2=p_2^2=0.$ So $\mathrm{dim}\left(\mathfrak{B}\left(W^{b_1,-1}\right)\right)=8$. \item
Let $p_1=v\boxtimes 1, p_2=v\boxtimes xy\in M\langle(1,xy)\rangle$, then
$c(p_j\otimes p_k)=p_k\otimes p_j$, where $j,k=1,2$.
If we view $M\langle(1,xy)\rangle=\mathbb{K}p_1\oplus \mathbb{K}p_2$ as braided vector spaces, then $\mathrm{GKdim}~\mathfrak{B}\left(M\langle(1,xy) \rangle\right)= \mathrm{GKdim}\,\mathfrak{B}(\mathbb{K}p_1)+ \mathrm{GKdim}\,\mathfrak{B}(\mathbb{K}p_2)=2$ by Lemma \ref{TensorNicholsAlg}. Similarly, $\mathrm{GKdim}\,\mathfrak{B}\left(M\langle(x,y)\rangle\right)=2$. \item Let $p_1=(v_1+v_2)\boxtimes 1$, $p_2=(v_1-v_2)\boxtimes y\in M\langle(1,y)\rangle$. The braiding is given by \begin{eqnarray*} c(p_1\otimes p_1)=p_1\otimes p_1,\quad c(p_1\otimes p_2)=p_2\otimes p_1,\\ c(p_2\otimes p_1)=-p_1\otimes p_2,\quad c(p_2\otimes p_2)=p_2\otimes p_2. \end{eqnarray*} By \cite[Lemma 2.8]{andruskiewitsch2016finite}, $\mathrm{GKdim}\mathfrak{B}\left(M\langle(1,y)\rangle\right)=\infty$. For the same reason, we obtain $\mathrm{GKdim}\mathfrak{B}\left(M\langle(x,1)\rangle\right)=\infty$. \item Let $\theta=\frac{1}{2}(\mathrm{i}-1)(a+1)$, then \begin{eqnarray*} c\left(w_1^{(1)}\otimes w_1^{(1)}\right)= -\theta w_2^{(1)}\otimes w_2^{(1)},\quad c\left(w_1^{(1)}\otimes w_2^{(1)}\right)= \theta w_1^{(1)}\otimes w_2^{(1)},\\ c\left(w_2^{(1)}\otimes w_1^{(1)}\right)= -\theta w_2^{(1)}\otimes w_1^{(1)},\quad c\left(w_2^{(1)}\otimes w_2^{(1)}\right)= \theta w_1^{(1)}\otimes w_1^{(1)},\\ c\left(\mathrm{i} w_1^{(1)}\otimes w_1^{(1)}+w_2^{(1)}\otimes w_2^{(1)}\right) =-\mathrm{i}\theta \left(\mathrm{i} w_1^{(1)}\otimes w_1^{(1)}+w_2^{(1)}\otimes w_2^{(1)}\right), \\ c\left(-\mathrm{i} w_1^{(1)}\otimes w_1^{(1)}+w_2^{(1)}\otimes w_2^{(1)}\right) =\mathrm{i}\theta \left(-\mathrm{i} w_1^{(1)}\otimes w_1^{(1)}+w_2^{(1)}\otimes w_2^{(1)}\right), \end{eqnarray*} By induction, \begin{eqnarray*} \mathfrak{S}_{2n-1,1}\left(\left(w_1^{(1)}\otimes w_2^{(1)}\right)^{\otimes n}\right)=\frac{(1+\theta)[1-(-\theta^2)^n]}{1+\theta^2}\left(\left(w_1^{(1)}\otimes w_2^{(1)}\right)^{\otimes n}\right),\\ \mathfrak{S}_{2n,1}\left(\left(w_1^{(1)}\otimes w_2^{(1)}\right)^{\otimes n}\otimes w_1^{(1)}\right)=\frac{1-\theta+(-1)^n\theta^{2n+1}(1+\theta)}{1+\theta^2}\left(\left(w_1^{(1)}\otimes w_2^{(1)}\right)^{\otimes n}\otimes w_1^{(1)}\right). \end{eqnarray*} It means that $\left(w_1^{(1)}\otimes w_2^{(1)}\right)^{\otimes n}$ is an eigenvector of $\mathfrak{S}_{2n-1}$ and $\left(w_1^{(1)}\otimes w_2^{(1)}\right)^{\otimes n}\otimes w_1^{(1)}$ is an eigenvector of $\mathfrak{S}_{2n}$ both with nonzero eigenvalue.
So $\dim\mathfrak{B}\left(W_1^a\right)=\infty$. And $\dim\mathfrak{B}\left(W_2^a\right)=\infty$ is similar to prove.
\end{itemize} \end{proof}
\begin{proposition}\label{NicholsAlg:TensorOfTwoSimpleObjects} \begin{enumerate} \item \label{NicholsAlg:Tensor1}$\mathfrak{B}\left[M\langle b,g\rangle\oplus M\left<b^\prime,g^\prime\right>\right]\simeq \mathfrak{B}\left(M\langle b,g\rangle\right) \otimes \mathfrak{B}\left( M\left<b^\prime,g^\prime\right>\right)$ for
$(b,g)$, $(b^\prime,g^\prime)$ $\in$ $\{(\pm 1,1)$, $(\pm 1,xy)$, $(\pm \mathrm{i},x)$, $(\pm \mathrm{i},y)\}$. \item When $(b,g)\in\{(\pm1,1),(\pm1,xy)\}$, then \begin{align*} \mathfrak{B}\left[M\langle b,g\rangle\oplus M\langle(1,xy)\rangle\right] &\simeq \mathfrak{B}\left(M\langle b,g\rangle\right)\otimes
\mathfrak{B}\left(M\langle(1,xy)\rangle\right),\\ \mathfrak{B}\left[M\langle b,g\rangle\oplus M\langle(x,y)\rangle\right] &\simeq \mathfrak{B}\left(M\langle b,g\rangle\right)\otimes
\mathfrak{B}\left(M\langle(x,y)\rangle\right). \end{align*} \item \label{NicholsAlg:Tensor3}$\mathfrak{B}\left[M\langle b,g\rangle\oplus M\langle(g_1,g_2)\rangle\right] \simeq \mathfrak{B}\left(M\langle b,g\rangle\right)\otimes
\mathfrak{B}\left(M\langle(g_1,g_2)\rangle\right)$ for the following cases
\begin{enumerate}
\item $(b,g)=(\pm\mathrm{i}, x)$, $(g_1,g_2)=(xy,x)$;
\item $(b,g)=(\pm\mathrm{i}, y)$, $(g_1,g_2)=(y,xy)$;
\item $(b,g)=(\pm1,1)$, $(g_1,g_2)\in$
$\{(xy, x),(y, xy)\}$.
\end{enumerate} \item \label{NicholsAlg:Tensor4}$\mathfrak{B}\left[M\langle b,g\rangle\oplus W^{b_1,-1}\right] \simeq \mathfrak{B}\left(M\langle b,g\rangle\right)\otimes
\mathfrak{B}\left(W^{b_1,-1}\right)$ for the following cases
\begin{enumerate}
\item $(b,g)\in\{(1, 1), (1,xy)\}$, $b_1=\pm 1$;
\item $(b,g)\in\{(\mathrm{i}, x), (\mathrm{i},y)\}$, $b_1= 1$;
\item $(b,g)\in\{(-\mathrm{i}, x), (-\mathrm{i},y)\}$, $b_1= -1$.
\end{enumerate} \item $\mathfrak{B}\left[M\langle(xy,x)\rangle \oplus M\langle(y,xy)\rangle\right] \simeq \mathfrak{B}\left(M\langle(xy,x)\rangle\right) \otimes \mathfrak{B}\left(M\langle(y,xy)\rangle\right)$. \item \label{NicholsAlg:Tensor6}$\mathfrak{B}\left(W^{1,-1}\oplus W^{-1,-1}\right) \simeq \mathfrak{B}\left(W^{1,-1}\right) \otimes \mathfrak{B}\left(W^{-1,-1}\right)$. \item \label{NicholsAlg:Tensor7}$\mathrm{GKdim}\,\mathfrak{B}\left[M\langle b,g\rangle\oplus M\langle(g_1,g_2)\rangle\right] =\infty$ for $(b,g)=(\pm\mathrm{i}, x)$, $(g_1,g_2)=(y,xy)$ or $(b,g)=(\pm\mathrm{i}, y)$, $(g_1,g_2)=(xy,x)$. \item \label{NicholsAlg:Tensor8}$\mathrm{GKdim}\,\mathfrak{B}\left[M\langle b,g\rangle\oplus W^{b_1,-1}\right] =\infty$ for $(b,g)\in\{(\mathrm{i}, x), (\mathrm{i},y)\}$, $b_1= -1$ or $(b,g)\in\{(-\mathrm{i}, x), (-\mathrm{i},y)\}$, $b_1= 1$. \item \label{NicholsAlg:Tensor9}$\mathrm{dim}~\mathfrak{B}\left(\left(M\langle(g_1,g_2)\rangle\right)^{\oplus 2}\right)=\infty$ for
$(g_1,g_2)\in \{(xy,x),(y,xy)\}$.
\item \label{NicholsAlg:Tensor10}$\dim \mathfrak{B}\left(W^{b_1,-1}\oplus W^{b_1,-1} \right)=\infty$ for $b_1=\pm 1$. \end{enumerate} \end{proposition} \begin{remark} According to the above two proposition, we calculate some Nichols algebras over direct sum of two simple objects of ${}_{H_8}^{H_8}\mathcal{YD}$ in Table \ref{dimNAlgTwoSimpleYDM_H8}. \end{remark}
\begin{center} \begin{table} \newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}}
\begin{tabular}{|c|c|c|c|} \hline
$M\in {}_{H_8}^{H_8}\mathcal{YD}$
& condition & $\dim\mathfrak{B}(M)$ & $\mathrm{GKdim}\mathfrak{B}(M)$\\ \hline
\multirow{5}{*} {\tabincell{c}{$M\langle b_1,g_1\rangle$\\ $\oplus M\langle b_2,g_2\rangle$}} &\tabincell{c}{$(b_1,g_1)$, $(b_2,g_2)\in$ \\ $\{(\pm 1, 1),(\pm 1, xy)\}$ }& $\infty$& $2$\\\cline{2-4} &\tabincell{c}{$(b_1,g_1)\in$$\{(\pm 1, 1),(\pm 1, xy)\}$ \\$(b_2,g_2)\in$ $\{(\pm\mathrm{i}, x),(\pm\mathrm{i}, y)\}$} & $\infty$& $1$\\\cline{2-4}
&\tabincell{c}{$(b_1,g_1)$, $(b_2,g_2)\in$ \\$\{(\pm\mathrm{i}, x),(\pm\mathrm{i}, y)\}$ }& $4$& $0$ \\\hline \tabincell{c}{$M\langle b,g\rangle$\\
$\oplus M\langle(1,xy)\rangle$} & $(b,g)\in$ $\{(\pm 1, 1),(\pm 1, xy)\}$ & $\infty$& $3$\\\hline
\tabincell{c}{$M\langle b,g\rangle$ \\ $\oplus M\langle(x,y)\rangle$} &$(b,g)\in$ $\{(\pm 1, 1),(\pm 1, xy)\}$& $\infty$& $3$\\\hline
\multirow{3}{*}{\tabincell{c}{$M\langle(g_1,g_2)\rangle$\\
$\oplus M\langle(g_1^\prime,g_2^\prime)\rangle$}}&
\tabincell{c}{$(g_1,g_2)=(g_1^\prime,g_2^\prime)=$\\ $(xy,x)$ or $(y,xy)$ }& $\infty$& $$\\\cline{2-4} &\tabincell{c}{ $(g_1,g_2)=(xy,x)$, \\$(g_1^\prime,g_2^\prime) =(y,xy)$} & $64$& $0$\\\hline
\multirow{2}{*}{$W^{b_1,-1}\oplus W^{b_1^\prime,-1}$} & $b_1=b_1^\prime=\pm 1$ & $\infty$& $$\\\cline{2-4} & $b_1=1$, $b_1^\prime=-1$ & $64$& $0$\\\hline $M\langle(g_1,g_2)\rangle\oplus W^{b_1,-1}$& \tabincell{c}{ $(g_1,g_2)=(xy,x)$ \\or $(y,xy)$, $b_1=\pm 1$ }& ? & \\ \hline
\multirow{6}{*} {\tabincell{c}{$M\langle b,g\rangle$\\ $\oplus M\langle(g_1,g_2)\rangle$}} &\tabincell{c}{$(b,g)=(\pm\mathrm{i}, x)$, $(g_1,g_2)=(xy,x)$ }& $16$& $0$\\\cline{2-4} &\tabincell{c}{$(b,g)=(\pm\mathrm{i}, x)$, $(g_1,g_2)=(y,xy)$ }& $\infty$& $$\\\cline{2-4} &\tabincell{c}{$(b,g)=(\pm\mathrm{i}, y)$, $(g_1,g_2)=(xy,x)$ }& $\infty$& $$\\\cline{2-4} &\tabincell{c}{$(b,g)=(\pm\mathrm{i}, y)$, $(g_1,g_2)=(y,xy)$ }& $16$& $0$\\\cline{2-4} &\tabincell{c}{$(b,g)\in$$\{(\pm 1, 1)\}$ \\$(g_1,g_2)\in$ $\{(xy, x),(y, xy)\}$} & $\infty$& $1$\\\hline
\multirow{6}{*} {\tabincell{c}{$M\langle b,g\rangle$ $\oplus W^{b_1,-1}$}} &\tabincell{c}{$(b,g)\in\{(1, 1), (1,xy)\}$, $b_1=\pm 1$ }& $\infty$& $1$\\\cline{2-4} &\tabincell{c}{$(b,g)\in\{(\mathrm{i}, x), (\mathrm{i},y)\}$, $b_1= 1$ }& $16$& $0$\\\cline{2-4} &\tabincell{c}{$(b,g)\in\{(\mathrm{i}, x), (\mathrm{i},y)\}$, $b_1= -1$ }& $\infty$& $$\\\cline{2-4} &\tabincell{c}{$(b,g)\in\{(-\mathrm{i}, x), (-\mathrm{i},y)\}$, $b_1= -1$ }& $16$& $0$\\\cline{2-4} &\tabincell{c}{$(b,g)\in\{(-\mathrm{i}, x), (-\mathrm{i},y)\}$, $b_1= 1$ }& $\infty$& $$\\\hline \end{tabular}
\caption{Nichols algebras over direct sum of two simple objects in ${}_{H_8}^{H_8}\mathcal{YD}$} \label{dimNAlgTwoSimpleYDM_H8} \end{table} \end{center}
\begin{proof} The part \eqref{NicholsAlg:Tensor1}-\eqref{NicholsAlg:Tensor6} are direct results of Lemma \ref{TensorNicholsAlg}. We only prove some cases as a byproduct in the following. \begin{itemize}\renewcommand{$\diamond$}{$\diamond$} \item Let $p_1=(v_1+v_2)\boxtimes g_1$, $p_2=(v_1-v_2)\boxtimes g_2\in M\langle(g_1,g_2)\rangle$, where $(g_1,g_2)\in\{(xy,x),(y,xy)\}$. Let $p=v\boxtimes g \in M\langle b,g\rangle$, then \begin{eqnarray*} c(p\otimes p_1) &=& \left\{ \begin{array}{rl} -p_1\otimes p, & \text{if }g\in\{y,xy\}\\ p_1\otimes p, &\text{if } g\in\{1,x\},\\ \end{array}\right. \quad c(p\otimes p_2) = \left\{ \begin{array}{rl} -p_2\otimes p, & \text{if }g\in\{x,xy\}\\ p_2\otimes p, &\text{if } g\in\{1,y\},\\ \end{array}\right. \\ c(p_1\otimes p) &=& \left\{ \begin{array}{rl} b^2p\otimes p_1, & \text{if } g_1=y\\ p\otimes p_1, &\text{if } g_1=xy,\\ \end{array}\right. \quad~~ c(p_2\otimes p) = \left\{ \begin{array}{rl} b^2p\otimes p_2, & \text{if }g_2=x\\ p\otimes p_2, &\text{if } g_2=xy.\\ \end{array}\right. \end{eqnarray*} \begin{itemize}\renewcommand{$\circ$}{$\circ$} \item When $(g_1,g_2)=(y,xy)$ and $(b,g)=(\pm\mathrm{i},x)$, then
\begin{align*}
c(p\otimes p_1)&=p_1\otimes p_1,
&c(p\otimes p_2)&=-p_2\otimes p,\\
c(p_1\otimes p)&=-p\otimes p_1,
&c(p_2\otimes p)&=p\otimes p_2.
\end{align*}
The generalized Dynkin diagram is given by Figure \ref{figureEightA1A2}. According to \cite{heckenberger2009classification},
$\dim \mathfrak{B}[M\langle\pm\mathrm{i},x\rangle
\oplus M\langle(y,xy)\rangle]=\infty$.
\begin{figure}\label{figureEightA1A2}
\end{figure} \item When $(g_1,g_2)=(xy,x)$ and $(b,g)=(\pm\mathrm{i},y)$, the generalized Dynkin diagram associated to the braiding is given by Figure \ref{figureEightA1A2}. According to \cite{heckenberger2009classification},
$\dim \mathfrak{B}[M\langle\pm\mathrm{i},y\rangle
\oplus M\langle(xy,x)\rangle]=\infty$. We finish the part \eqref{NicholsAlg:Tensor7}. \item As for cases listed in the part \eqref{NicholsAlg:Tensor6}, $\mathfrak{B}\left[M\langle b,g\rangle\oplus M\langle(g_1,g_2)\rangle\right] \simeq \mathfrak{B}\left(M\langle b,g\rangle\right)\otimes
\mathfrak{B}\left(M\langle(g_1,g_2)\rangle\right)$ by Lemma \ref{TensorNicholsAlg}. \end{itemize} \item Let $p=v\boxtimes g\in M\langle b,g\rangle$, where $(b,g)\in\{(\pm 1,1),(\pm 1,xy), (\pm \mathrm{i},x), (\pm \mathrm{i},y)\}$, then \begin{eqnarray*} c(p\otimes w_1^{b_1,-1}) &=& \left\{ \begin{array}{rl} w_1^{b_1,-1}\otimes p, & \text{if }~
(b,g)\in\{(\pm 1,1),(\pm 1,xy)\}\\ \mathrm{i} b_1 w_2^{b_1,-1}\otimes p, &\text{if }~ (b,g)\in\{(\pm\mathrm{i},x),(\pm \mathrm{i},y)\},\\ \end{array}\right. \\ c(p\otimes w_2^{b_1,-1}) &=& \left\{ \begin{array}{rl} w_2^{b_1,-1}\otimes p, & \text{if }~
(b,g)\in\{(\pm 1,1),(\pm 1,xy)\}\\ -\mathrm{i} b_1 w_1^{b_1,-1}\otimes p, &\text{if }~
(b,g)\in\{(\pm\mathrm{i},x),(\pm \mathrm{i},y)\},\\ \end{array}\right. \\ c(w_1^{b_1,-1}\otimes p) &=& \left\{ \begin{array}{rl} bp\otimes w_1^{b_1,-1}, & \text{if } ~
(b,g)\in\{(\pm 1,1),(\pm 1,xy)\}\\ bp\otimes w_2^{b_1,-1}, &\text{if } ~
(b,g)\in\{(\pm\mathrm{i},x),(\pm \mathrm{i},y)\},\\ \end{array}\right. \\ c(w_2^{b_1,-1}\otimes p) &=& \left\{ \begin{array}{rl} bp\otimes w_2^{b_1,-1}, & \text{if }~
(b,g)\in\{(\pm 1,1),(\pm 1,xy)\}\\ -bp\otimes w_1^{b_1,-1}, &\text{if }~
(b,g)\in\{(\pm\mathrm{i},x),(\pm \mathrm{i},y)\}.\\ \end{array}\right. \end{eqnarray*} \begin{itemize}\renewcommand{$\circ$}{$\circ$} \item In case $(b,g)\in\{(1,1),(1,xy)\}$, according to Lemma \ref{TensorNicholsAlg}, we have $$\mathfrak{B}\left(M\langle b,g\rangle\oplus W^{b_1, -1}\right)\simeq \mathfrak{B}\left(M\langle b,g\rangle\right)\otimes \mathfrak{B}\left( W^{b_1, -1}\right).$$ \item In case $(b,g)\in\{(\pm\mathrm{i},x),(\pm \mathrm{i},y)\}$, if $\mathrm{i} b_1b=-1$, according to Lemma \ref{TensorNicholsAlg}, we have $\mathfrak{B}\left(M\langle b,g\rangle\oplus W^{b_1, -1}\right)\simeq \mathfrak{B}\left(M\langle b,g\rangle\right)\otimes \mathfrak{B}\left( W^{b_1, -1}\right)$. If $\mathrm{i} b_1b=1$, the generalized Dynkin diagram associated to the braiding of $M\langle b,g\rangle$ $\oplus$ $W^{b_1, -1}$ is given by Figure \ref{figureEightA1A2}. Now we finish the part \eqref{NicholsAlg:Tensor4} and \eqref{NicholsAlg:Tensor8}. \end{itemize} \item As for $(g_1,g_2)\in \{(xy,x),(y,xy)\}$, $\mathrm{dim}~\mathfrak{B}\left(\left(M\langle(g_1,g_2)\rangle\right)^{\oplus 2}\right)=\infty$ by \cite{heckenberger2009classification}, since the generalized Dynkin diagram associated to the braiding is given by Figure \ref{figureEightM23}. \begin{figure}\label{figureEightM23}
\end{figure} \item As for $W^{b_1,-1}\oplus W^{b_1^\prime,-1}$ with $b_1$ and $b_1^\prime$ in $\{\pm 1\}$. Let $p_1=w_1^{b_1,-1}+\mathrm{i} b_1^\prime w_2^{b_1,-1}$, $p_2=w_1^{b_1,-1}-\mathrm{i} b_1^\prime w_2^{b_1,-1}$, $p_1^\prime=w_1^{b_1^\prime,-1}+\mathrm{i} b_1^\prime w_2^{b_1^\prime,-1}$ and $p_2^\prime=w_1^{b_1^\prime,-1}-\mathrm{i} b_1^\prime w_2^{b_1^\prime,-1}$, then \begin{align*} c(p_1\otimes p_1^\prime)&=- p_1^\prime\otimes p_1, &c(p_2\otimes p_2^\prime)&=- p_2^\prime\otimes p_2, \\ c(p_1\otimes p_2^\prime)&= p_2^\prime\otimes p_1, &c(p_2\otimes p_1^\prime)&=- p_1^\prime\otimes p_2. \end{align*} When $b_1=b_1^\prime$, the generalized Dynkin diagram associated to the braiding is given by Figure \ref{figureEightM23}. By \cite{heckenberger2009classification}, $\dim \mathfrak{B}\left(W^{b_1,-1}\oplus W^{b_1,-1} \right)=\infty$. This finish \eqref{NicholsAlg:Tensor10}.
When $b_1=-b_1^\prime$, then $p_2=w_1^{b_1,-1}+\mathrm{i} b_1 w_2^{b_1,-1}$, $p_1=w_1^{b_1,-1}-\mathrm{i} b_1 w_2^{b_1,-1}$, $p_2^\prime=w_1^{b_1^\prime,-1}+\mathrm{i} b_1 w_2^{b_1^\prime,-1}$, $p_1^\prime=w_1^{b_1^\prime,-1}-\mathrm{i} b_1 w_2^{b_1^\prime,-1}$, and \begin{align*} c(p_2^\prime\otimes p_2)&=- p_2\otimes p_2^\prime, &c(p_1^\prime\otimes p_1)&=- p_1\otimes p_1^\prime, \\ c(p_2^\prime\otimes p_1)&= p_1\otimes p_2^\prime, &c(p_1^\prime\otimes p_2)&=- p_2\otimes p_1^\prime. \end{align*}
By Lemma \ref{TensorNicholsAlg}, we have $\mathfrak{B}\left(W^{b_1,-1}\oplus W^{-b_1,-1}\right) \simeq \mathfrak{B}\left(W^{b_1,-1}\right) \otimes \mathfrak{B}\left(W^{-b_1,-1}\right)$. This finish \eqref{NicholsAlg:Tensor6}. \end{itemize} \end{proof}
\begin{conjecture}\label{conjecture} $\dim\mathfrak{B}\left(M\langle(xy,x)\rangle \oplus W^{b_1,-1}\right)=\infty =\dim\mathfrak{B}\left(M\langle(y,xy)\rangle \oplus W^{b_1,-1}\right)$ hold for $b_1=\pm 1$. \end{conjecture} \begin{remark} Let $p_1=(v_1+v_2)\boxtimes xy$, $p_2=(v_1-v_2)\boxtimes x$ $\in M\langle(xy,x)\rangle$, then \begin{align*} c\left(p_1\otimes w_1^{b_1,-1}\right)&=w_1^{b_1,-1}\otimes p_1, &c\left(p_1\otimes w_2^{b_1,-1}\right)&=w_2^{b_1,-1}\otimes p_1,\\ c\left(p_2\otimes w_1^{b_1,-1}\right)&=\mathrm{i} b_1 w_2^{b_1,-1}\otimes p_2, &c\left(p_2\otimes w_2^{b_1,-1}\right)&=-\mathrm{i} b_1w_1^{b_1,-1}\otimes p_2,\\ c\left(w_1^{b_1,-1}\otimes p_1\right)&=p_2\otimes w_1^{b_1,-1}, &c\left(w_1^{b_1,-1}\otimes p_2\right)&=p_1\otimes w_2^{b_1,-1},\\ c\left(w_2^{b_1,-1}\otimes p_1\right)&=-p_2\otimes w_2^{b_1,-1}, &c\left(w_2^{b_1,-1}\otimes p_2\right)&=p_1\otimes w_1^{b_1,-1}. \end{align*} Let $p_1^\prime=w_1^{b_1,-1}+\mathrm{i} b_1w_2^{b_1,-1}$ and $p_2^\prime=w_1^{b_1,-1}-\mathrm{i} b_1w_2^{b_1,-1}$, then \begin{align*} c(p_1\otimes p_1^\prime)&=p_1^\prime\otimes p_1, &c(p_1\otimes p_2^\prime)&=p_2^\prime\otimes p_1,\\ c(p_2\otimes p_1^\prime)&=p_1^\prime\otimes p_2, &c(p_2\otimes p_2^\prime)&=-p_2^\prime\otimes p_2,\\ c(p_1^\prime\otimes p_1)&=p_2\otimes p_2^\prime, &c(p_1^\prime\otimes p_2)&=\mathrm{i} b_1 p_1\otimes p_2^\prime,\\ c(p_2^\prime\otimes p_1)&=p_2\otimes p_1^\prime, &c(p_2^\prime\otimes p_2)&=-\mathrm{i} b_1 p_1\otimes p_1^\prime. \end{align*} \end{remark}
\subsection*{Proof of Theorem \ref{NicholsAlg:maintheorem}} Firstly, we recall the truth that for any submodule $
M_1\subset M_2\in{}_{H}^{H}\mathcal{YD}$, $\mathfrak{B}(M_1)
\subset \mathfrak{B}(M_2)$, $\dim \mathfrak{B}(M_2)=\infty$ if $\dim \mathfrak{B}(M_1)=\infty$. So the only possible $M\in{}_{H_8}^{H_8}\mathcal{YD}$ such that $\dim\mathfrak{B}(M)< \infty$ is in the list of Theorem \ref{NicholsAlg:maintheorem} according to Table \ref{dimNAlgSimpleYDM_H8}, Table \ref{dimNAlgTwoSimpleYDM_H8},
Proposition \ref{NAlgdim1} and
Proposition \ref{NicholsAlg:TensorOfTwoSimpleObjects}, under the assumption that Conjecture \ref{conjecture} is true. Now we only need to check that Nichols algebras $ \mathfrak{B}(M)$ for $M$ listed in Theorem \ref{NicholsAlg:maintheorem} is finite dimensional. In fact, $\Omega_1(n_1,n_2,n_3,n_4)$ is of Cartan type $\underbrace{A_1\times \cdots \times A_1}_{n_1+n_2+n_3+n_4}$;
$\Omega_k(n_1,n_2)$ for $k=2,3,4,5$ is of Cartan type
$\underbrace{A_1\times \cdots \times A_1}_{n_1+n_2}\times\, A_2$;
$\Omega_k$ for $k=6,7$ is of Cartan type $A_2\times A_2$.
\section{Hopf algebras over $H_8$} \label{section:HopfAlgebras} In this section, according to lifting method, we determine finite-dimensional Hopf algebra $H$ with coradical $H_8$ such that its infinitesimal braiding is isomorphic to a Yetter-Drinfel'd module $M$ listed in Theorem \ref{NicholsAlg:maintheorem}. We begin by proving $H$ is generated by elements of degree one in Theorem \ref{generatedByDegreeOne}. That is, $\mathrm{gr}\, H\simeq \mathfrak{B}(M)\#H_8$.
\begin{theorem}\label{generatedByDegreeOne} Let $H$ be a finite-dimensional Hopf algebra over $H_8$ such that its infinitesimal braiding is isomorphic to a Yetter-Drinfel'd module over $H_8$ which is in the list of Theorem \ref{NicholsAlg:maintheorem}. Then the diagram of $H$ is a Nichols algebra, and consequently $H$ is generated by the elements of degree one with respect to the coradical filtration. \end{theorem} \begin{proof} Since $\mathrm{gr}\, H \simeq R \#H_8$, with $R=\bigoplus_{n\geq 0} R(n)$ the diagram of $H$, we need to prove that $R$ is a Nichols algebra. Let $\mathcal{J}=\bigoplus_{n\geq 0} R(n)^*$ be the graded dual of $R$, then $\mathcal{J}$ is a graded Hopf algebra in ${}_{H_8} ^{H_8}\mathcal{YD}$ with $\mathcal{J}(0)=\mathbb{K}1$. According to \cite[Lemma 5.5]{MR1780094}, $R(1)= \mathcal{P}(R)$ if and only if $\mathcal{J}$ is generated as an algebra by $\mathcal{J}(1)$, that is, if $\mathcal{J}$ is itself a Nichols algebra.
Considering $\mathfrak{B}(M)\in {}_{H_8} ^{H_8}\mathcal{YD}$ for $M$ in the list of Theorem \ref{NicholsAlg:maintheorem}, since $\mathfrak{B}(M)=T(M)/\mathcal{I}$, in order to show $ \mathcal{P}(\mathcal{J})=\mathcal{J}(1)$, it is enough to prove that the relations that generate the ideal $\mathcal{I}$ hold in $\mathcal{J}$. This can be done by a case by case computation. We perform here three cases, and leave the rest to the reader.
Suppose $M=\Omega_1(n_1,n_2,n_3,n_4)$. A direct computation shows that the elements $r$ in $\mathcal{J}$ representing the quadratic relations are primitive and since the braiding is -flips, they satisfy that $c(r\otimes r)=r\otimes r$. As $\dim \mathcal{J} <\infty$, it must be $r=0$ in $\mathcal{J}$ and hence there exists a projective algebra map $\mathfrak{B}(M)$ $\rightarrow\mathcal{J}$, which implies that $ \mathcal{P}(\mathcal{J})=\mathcal{J}(1)$.
Suppose $M=\Omega_6$, then $M$ is generated by elements $p_1=(v_1+v_2)\boxtimes xy$, $p_2=(v_1-v_2)\boxtimes x$, $p_1^\prime=(v_1+v_2)\boxtimes y$, $p_2^\prime=(v_1-v_2)\boxtimes xy$ and the ideal defining the Nichols algebra is generated by the elements $p_1^2$, $p_2^2$, ${p_1^\prime}^2$, ${p_2^\prime}^2$, $p_1p_2p_1p_2+p_2p_1p_2p_1$, $p_1^\prime p_2^\prime p_1^\prime p_2^\prime+p_2^\prime p_1^\prime p_2^\prime p_1^\prime$, $p_1p_1^\prime+p_1^\prime p_1$, $p_1p_2^\prime+p_2^\prime p_1$, $p_2p_1^\prime-p_1^\prime p_2$, $p_2p_2^\prime+p_2^\prime p_2$.
We can check directly that all those generators of the defining ideal of $\mathfrak{B}(M)$ are primitive elements, or by \cite[Theorem 6]{MR2842083}. It's enough to show $c(r\otimes r)=r\otimes r$ for all
generators given in above for the defining ideal. Since
$\rho(p_1)=xy\otimes p_1$, $\rho(p_2)=x\otimes p_2$,
$\rho(p_1^\prime)=y\otimes p_1^\prime$, so $\rho(p_1^2)=1\otimes p_1^2$, $\rho(p_1p_2p_1p_2+p_2p_1p_2p_1)=1\otimes (p_1p_2p_1p_2+p_2p_1p_2p_1)$,
$\rho(p_1p_1^\prime+p_1^\prime p_1)=x\otimes (p_1p_1^\prime+p_1^\prime p_1)$. It's easy to see $c(r\otimes r)=r\otimes r$ holds for $r=p_1^2$, $p_1p_2p_1p_2+p_2p_1p_2p_1$ and $p_1p_1^\prime+p_1^\prime p_1$. We leave the rest to the reader.
Suppose $M=\Omega_4(n_1,n_2)$, then $M$ is generated by elements $p_1=w_1^{1 ,-1}+\mathrm{i} w_2^{1,-1}$, $p_2=w_1^{1 ,-1}-\mathrm{i} w_2^{1 ,-1}$, $\{X_j\}_{j=1,\cdots, n_1}$, $\{Y_k\}_{k=1,\cdots, n_2}$ with $\mathbb{K}X_j\simeq M\langle\mathrm{i},x\rangle$, $\mathbb{K}Y_k\simeq M\langle\mathrm{i},y\rangle$ and the ideal defining the Nichols algebra is generated by the elements $p_1^2$, $p_2^2$, $p_1p_2p_1p_2+p_2p_1p_2p_1$, $X_j^2$, $\{X_{j_1}X_{j_2}+ X_{j_2}X_{j_1}\}_{1\leq j_1< j_2\leq n_1}$, $Y_k^2$, $\{Y_{k_1}Y_{k_2}+Y_{k_2}Y_{k_1}\}_{1\leq k_1< k_2\leq n_2}$, $p_1Y_k-Y_k p_1$, $p_2Y_k+Y_k p_2$, $p_1X_j-X_jp_1$, $p_2X_j+X_jp_2$. We can check directly that all those generators of the defining ideal of $\mathfrak{B}(M)$ are primitive elements, or by \cite[Theorem 6]{MR2842083}. It's enough to show $c(r\otimes r)=r\otimes r$ for all
generators given in above for the defining ideal. Since
$\rho(p_1)=(f_{00}-\mathrm{i} f_{11})z\otimes p_1+ (f_{10}+\mathrm{i} f_{01})z\otimes p_2$, $\rho(p_2)=(f_{00}+\mathrm{i} f_{11})z\otimes p_2+ (f_{10}-\mathrm{i} f_{01})z\otimes p_1$,
$\rho(X_j)=x\otimes X_j$,
\begin{align*}
\rho(p_1p_2p_1p_2+p_2p_1p_2p_1)
&=[(f_{00}-\mathrm{i} f_{11})z(f_{00}+\mathrm{i} f_{11})z]^2\otimes p_1p_2p_1p_2+\\
&\quad+[(f_{00}+\mathrm{i} f_{11})z(f_{00}-\mathrm{i} f_{11})z]^2\otimes
p_2p_1p_2p_1+\\
&\quad +[(f_{10}+\mathrm{i} f_{01})z(f_{10}-\mathrm{i} f_{01})z]^2\otimes
p_2p_1p_2p_1+\\
&\quad +[(f_{10}-\mathrm{i} f_{01})z(f_{10}+\mathrm{i} f_{01})z]^2\otimes
p_1p_2p_1p_2\\
&=xy\otimes (p_1p_2p_1p_2+p_2p_1p_2p_1), \\
\rho(p_1X_j-X_jp_1)=(f_{00}+\mathrm{i} &f_{11})z\otimes
(p_1X_j-X_jp_1)+(f_{10}-\mathrm{i} f_{01})z\otimes (p_2X_j+X_jp_2).
\end{align*}
Because
\begin{align*}
(f_{10}-\mathrm{i} f_{01})z\cdot (p_1X_j-X_jp_1)
&=\frac{f_{10}-\mathrm{i} f_{01}}{2}\cdot \left[((1+y)z\cdot p_1)(z\cdot X_j)-((1-y)z\cdot X_j)(xz\cdot p_1)\right]\\
&=(-\mathrm{i})(f_{10}-\mathrm{i} f_{01})\cdot (p_1X_j-X_jp_1)=0,\\
(f_{00}+\mathrm{i} f_{11})z\cdot (p_1X_j-X_jp_1)
&=(-\mathrm{i})(f_{00}+\mathrm{i} f_{11})\cdot (p_1X_j-X_jp_1)=p_1X_j-X_jp_1,\\
xy\cdot (p_1p_2p_1p_2+p_2p_1p_2p_1)&=p_1p_2p_1p_2+p_2p_1p_2p_1,
\end{align*}
$c(r\otimes r)=r\otimes r$ holds for $r=p_1p_2p_1p_2+p_2p_1p_2p_1$ and $p_1X_j-X_jp_1$. We leave the rest to the reader. \end{proof}
\begin{lemma}\cite[Lemma 6.1]{MR1780094} Let $H$ be a Hopf algebra, $\psi: H\rightarrow H$ an automorphism of Hopf algebras, $V$, $W$ Yetter-Drinfel'd modules over $H$. \begin{enumerate} \item Let $V^\psi$ be the same space underlying $V$ but with action and coaction $$h\cdot_\psi v=\psi(h)\cdot v,\quad \rho^\psi (v) =\left(\psi^{-1}\otimes \mathrm{id}\right)\rho(v), \quad h\in H, v\in V.$$ Then $V^\psi$ is also a Yetter-Drinfel'd module over $H$. If $T: V\rightarrow W$ is a morphism in ${}_H^H\mathcal{YD}$, then $T^\psi: V^\psi\rightarrow W^\psi$ also is. Moreover, the braiding $c: V^\psi\otimes W^\psi\rightarrow W^\psi \otimes V^\psi$ coincides with the braiding $c: V\otimes W\rightarrow W\otimes V$. \item If $R$ is an algebra (resp., a coalgebra, a Hopf algebra) in ${}_H^H\mathcal{YD}$, then $R^\psi$ also is, with the same structural maps. \item Let $R$ be a Hopf algebra in ${}_H^H\mathcal{YD}$. Then the map $\Psi: R^\psi\# H\rightarrow R\#H$ given by $\Psi(r\#h)=r\#\psi(h)$ is an isomorphism of Hopf algebras. \end{enumerate} \end{lemma} \begin{corollary}\label{Isomorphism:B(V)H_8} \begin{enumerate} \item $\left[M\langle b\mathrm{i}, x\rangle\right]^{\tau_3}\simeq M\langle -b\mathrm{i}, y\rangle$, $b=\pm 1$. \item $\left[M\langle(xy, x)\rangle\right]^{\tau_3}\simeq M\langle(y, xy)\rangle$, $\left(W^{b_1,-1}\right)^{\tau_3}\simeq W^{-b_1,-1}$ with $b_1=\pm 1$. \item $\mathfrak{B}\left(\Omega_2(n_1,n_2)\right)\#H_8\simeq \mathfrak{B}\left(\Omega_3(n_2,n_1)\right)\#H_8$, $\mathfrak{B}\left(\Omega_4(n_1,n_2)\right)\#H_8\simeq \mathfrak{B}\left(\Omega_5(n_2,n_1)\right)\#H_8$. \end{enumerate} \end{corollary}
\begin{definition}\label{definition:HopfAlgA_1} For $n_1, n_2, n_3, n_4\in \mathbb{N}^{\geq 0}$ with $n_1+ n_2+ n_3+ n_4\geq 1$, and a set $I_1$ of parameters $\lambda_{j,s}$, $\mu_{j,t}$, $\zeta_{k,s}$, $\theta_{k,t}$ in $\mathbb{K}$ with $j=1,\cdots, n_1$, $k=1,\cdots, n_2$, $s=1,\cdots, n_3$, $t=1,\cdots, n_4$, denote by $\mathfrak{A}_1(n_1,n_2,n_3,n_4;I_1)$ the algebra generated by $x$, $y$, $z$, $\{X_j\}_{j=1,\cdots, n_1}$, $\{Y_k\}_{k=1,\cdots, n_2}$, $\{p_s\}_{s=1,\cdots, n_3}$, $\{q_t\}_{t=1,\cdots, n_4}$ satisfying the following relations: \begin{align} x^2=y^2=1,\quad z^2=\frac{1}{2}(1+x+y-xy), \\ xy=yx,\quad zx=yz,\quad zy=xz,\\
xX_j=-X_jx,\quad yX_j=-X_jy,\quad zX_j=\mathrm{i} X_jxz,\\
xY_k=-Y_kx,\quad yY_k=-Y_ky,\quad zY_k=-\mathrm{i} Y_kxz,\\
xp_s=-p_sx,\quad yp_s=-p_sy,\quad zp_s=\mathrm{i} p_sxz,\\ xq_t=-q_tx,\quad yq_t=-q_ty,\quad zq_t=-\mathrm{i} q_txz,\\ X_j^2=0,\quad Y_k^2=0,\quad p_s^2=0,\quad q_t^2=0,\label{ideal:directsumOneDim1}\\ X_{j_1}X_{j_2}+X_{j_2}X_{j_1}=0,\quad j_1, j_2\in\{1,\cdots, n_1\},\label{ideal:directsumOneDim2}\\ Y_{k_1}Y_{k_2}+Y_{k_2}Y_{k_1}=0,\quad k_1, k_2\in\{1,\cdots, n_2\}, \label{ideal:directsumOneDim3}\\ p_{s_1}p_{s_2}+p_{s_2}p_{s_1}=0,\quad s_1,s_2\in\{1,\cdots, n_3\}, \label{ideal:directsumOneDim4}\\ q_{t_1}q_{t_2}+q_{t_2}q_{t_1}=0,\quad t_1,t_2\in\{1,\cdots, n_4\}, \label{ideal:directsumOneDim5}\\ X_jY_k+Y_kX_j=0,\quad X_jp_s+p_sX_j=\lambda_{j,s}(1-xy), \label{ideal:directsumOneDim6}\\ X_jq_t+q_tX_j=\mu_{j,t}(1-xy),\quad Y_kp_s+p_sY_k=\zeta_{k,s}(1-xy), \label{ideal:directsumOneDim7}\\ Y_kq_t+q_tY_k=\theta_{k,t}(1-xy),\quad p_sq_t+q_tp_s=0. \label{ideal:directsumOneDim8} \end{align} It is a Hopf algebra with its structure determined by \begin{align} \Delta(X_j)=X_j\otimes 1+x\otimes X_j,\quad \Delta(Y_k)=Y_k\otimes 1+x\otimes Y_k,\\ \Delta(p_s)=p_s\otimes 1+y\otimes p_s,\quad \Delta(q_t)=q_t\otimes 1+y\otimes q_t,\\ \Delta(x)=x\otimes x,\quad \Delta(y)=y\otimes y, \quad \Delta(z)=\frac{1}{2}[(1+y)z\otimes z+(1-y)z\otimes xz]. \end{align} \end{definition} \begin{remark} \begin{enumerate} \item In fact, $\mathfrak{A}_1(n_1,n_2,n_3,n_4;I_1)\simeq \left[T\left( \Omega_1(n_1, n_2, n_3, n_4)\right)\#H_8\right]/{\mathcal{I}(I_1)}$,
where ${\mathcal{I}(I_1)}$ is a Hopf ideal generated by relations\eqref{ideal:directsumOneDim1}--\eqref{ideal:directsumOneDim8}. Especially, when parameters in $I_1$ are all equal to zero, then $\mathfrak{A}_1(n_1,n_2,n_3,n_4;I_1)\simeq \mathfrak{B}\left( \Omega_1(n_1, n_2, n_3, n_4)\right)\#H_8$. \item We can observe that any element of $\mathfrak{A}_1(n_1,n_2,n_3,n_4;I_1)$ can be expressed by a linear sum of $\{X_1^{\alpha_1}\cdots X_{n_1}^{\alpha_{n_1}} Y_1^{\beta_1}\cdots Y_{n_2}^{\beta_{n_2}} p_1^{\gamma_1}\cdots p_{n_3}^{\gamma_{n_3}} q_1^{\kappa_1}\cdots q_{n_4}^{\kappa_{n_4}}x^cy^dz^e\}$ for all parameteres $\alpha_1$, $\cdots$, $\alpha_{n_1}$, $\beta_1$, $\cdots$, $\beta_{n_2}$, $\gamma_1$, $\cdots$, $\gamma_{n_3}$, $\kappa_1$, $\cdots$, $\kappa_{n_4}$, $c$, $d$, $e$ in $\{0,1\}$. So $\mathfrak{A}_1(n_1,n_2,n_3,n_4;I_1)$ is finite dimensional. \end{enumerate} \end{remark} \begin{proposition}\label{HopfAlge:A_1} Let $H$ be a finite-dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $\Omega_1(n_1,n_2,n_3,n_4)$, then $H\simeq \mathfrak{A}_1(n_1,n_2,n_3,n_4;I_1)$. \end{proposition} \begin{proof} By Theorem \ref{generatedByDegreeOne}, we have $\mathrm{gr}\,H\simeq \mathfrak{B}(\Omega_1(n_1,n_2,n_3,n_4))\#H_8$. We can suppose $H$ is generated by elements $x$, $y$, $z$ in $H$ and \begin{align} X_j&=(v\boxtimes x)\#1\in M\langle\mathrm{i},x\rangle\#1,\quad j=1,\cdots, n_1,\\ Y_k&=(v\boxtimes x)\#1\in M\langle -\mathrm{i},x\rangle\#1,\quad k=1,\cdots, n_2,\\ p_s&=(v\boxtimes y)\#1\in M\langle\mathrm{i},y\rangle\#1,\quad s=1,\cdots, n_3,\\ q_t&=(v\boxtimes y)\#1\in M\langle -\mathrm{i},y\rangle\#1,\quad t=1,\cdots, n_4. \end{align} Then it's easy to check that formulae listed in Definition \ref{definition:HopfAlgA_1} except \eqref{ideal:directsumOneDim1}--\eqref{ideal:directsumOneDim8} hold in $H$ from the bosonization $\mathfrak{B}[\Omega(n_1,n_2,n_3,n_4)]\#H_8$. Let $p=(v\boxtimes g)\#1$ $\in [M\langle b,g\rangle]\#1$, $p^\prime=(v^\prime\boxtimes g^\prime)\#1\in [M\langle b^\prime,g^\prime\rangle]\#1$, then $\Delta(pp^\prime+p^\prime p)=(pp^\prime+p^\prime p) \otimes 1+gg^\prime\otimes (pp^\prime+p^\prime p)$, from which we obtain the lifting relations \eqref{ideal:directsumOneDim1}--\eqref{ideal:directsumOneDim8} are only possible for the given generators. In fact, those lifting relations \eqref{ideal:directsumOneDim1}--\eqref{ideal:directsumOneDim8} generate a Hopf ideal. \end{proof}
\begin{remark} \begin{enumerate} \item Suppose $H_1(b)$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $M\langle b,x\rangle$,
where $b=\pm \mathrm{i}$, then $H_1(b)\simeq \mathfrak{B}[M\langle b,x\rangle]\#H_8$. Denote $p=(v\boxtimes x)\# 1\in [M\langle b,x\rangle] \#1$, then $H_1(b)$ is generated by $H_8$, $p$, and with the following relations. $$p^2=0,\quad xp=-px,\quad yp=-py,\quad zp=bpxz,\quad \Delta(p)=p\otimes 1+x\otimes p.$$ \item Suppose $H_2(b)$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $M\langle b,y\rangle$, where $b=\pm \mathrm{i}$, then $H_2(b)\simeq \mathfrak{B}[M\langle b,y\rangle]\#H_8$. Denote $p=(v\boxtimes y)\# 1\in [M\langle b,y\rangle]\#1$, then $H_2(b)$ is generated by $H_8$, $p$, and with the following relations. $$p^2=0,\quad xp=-px,\quad yp=-py,\quad zp=bpxz,\quad \Delta(p)=p\otimes 1+y\otimes p.$$ \item $H_1(\pm \mathrm{i})$ are exactly the two nonisomorphic nonpointed self-dual Hopf algebras of dimension $16$ with coradical $H_8$ described by C{\u{a}}linescu, D{\u{a}}sc{\u{a}}lescu, Masuoka and Menini in \cite{MR2037722}. In fact, $H_1(b)\simeq H_2(-b)$ by Corollary \ref{Isomorphism:B(V)H_8}. \end{enumerate} \end{remark}
\begin{lemma}\label{HopfAlg:M2_3(y,xy)} Suppose $H$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $M\langle(y,xy)\rangle$, then $H\simeq \mathfrak{B}[M\langle(y,xy)\rangle]\# H_8$. Denote $p_1=[(v_1+v_2)\boxtimes y]\# 1\in M\langle(y,xy)\rangle]\#$ and $p_2=[(v_1-v_2)\boxtimes xy]\# 1\in M\langle(y,xy)\rangle]\#1$, then $H$ is generated by $x$, $y$, $z$, $p_1$, $p_2$, which satisfy the following relations. \begin{eqnarray} x^2=y^2=1,\quad z^2=\frac{1}{2}(1+x+y-xy), \quad xy=yx,\quad zx=yz,\quad zy=xz,\\
xp_1=p_1x,\quad yp_1=-p_1y,\quad zp_1=p_2 z,\\
xp_2=-p_2x,\quad
yp_2=p_2y,\quad zp_2= p_1 xz,\\
p_1^2=0,\quad p_2^2=0,\quad
p_1p_2p_1p_2+p_2p_1p_2p_1=0.
\end{eqnarray}
Its Hopf algebra structure is determined by
\begin{align}
\Delta(p_1)=p_1\otimes 1+y\otimes p_1,\quad \Delta(p_2)=p_2\otimes 1+xy\otimes p_2,\\
\Delta(x)=x\otimes x,\quad \Delta(y)=y\otimes y, \quad \Delta(z)=\frac{1}{2}[(1+y)z\otimes z+(1-y)z\otimes xz].
\end{align} \end{lemma} \begin{proof} By Theorem \ref{generatedByDegreeOne}, $\mathrm{gr}\,H\simeq \mathfrak{B}[M\langle(y,xy)\rangle]\# H_8$. It's straightforward to prove that $p_1^2$, $p_2^2$ and $p_1p_2p_1p_2+p_2p_1p_2p_1$ are primitive elements, so $H\simeq \mathrm{gr}\,H$. \end{proof}
\begin{lemma}\label{HopfAlg:M2_3(xy,x)} Suppose $H$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $M\langle(xy,x)\rangle$, then $H\simeq \mathfrak{B}[M\langle(xy,x)\rangle]\# H_8$. Let $p_1=[(v_1+v_2)\boxtimes xy]\# 1$, $p_2=[(v_1-v_2)\boxtimes x]\# 1$ be a basis of $M\langle(xy,x)\rangle]\#1$, then $H$ is generated by $x$, $y$, $z$, $p_1$, $p_2$, which satisfy the following relations. \begin{eqnarray} x^2=y^2=1,\quad z^2=\frac{1}{2}(1+x+y-xy), \quad xy=yx,\quad zx=yz,\quad zy=xz,\\
xp_1=p_1x,\quad yp_1=-p_1y,\quad zp_1=p_2 z,\\
xp_2=-p_2x, \quad
yp_2=p_2y,\quad zp_2= p_1 xz,\\
p_1^2=0,\quad p_2^2=0,\quad
p_1p_2p_1p_2+p_2p_1p_2p_1=0.
\end{eqnarray}
Its Hopf algebra structure is determined by
\begin{align}
\Delta(p_1)=p_1\otimes 1+xy\otimes p_1,\quad \Delta(p_2)=p_2\otimes 1+x\otimes p_2,\\
\Delta(x)=x\otimes x,\quad \Delta(y)=y\otimes y, \quad \Delta(z)=\frac{1}{2}[(1+y)z\otimes z+(1-y)z\otimes xz].
\end{align} \end{lemma} \begin{proof} By Theorem \ref{generatedByDegreeOne}, $\mathrm{gr}\,H\simeq \mathfrak{B}[M\langle(xy,x)\rangle]\# H_8$. It's straightforward to prove that $p_1^2$, $p_2^2$ and $p_1p_2p_1p_2+p_2p_1p_2p_1$ are primitive elements, so $H\simeq \mathrm{gr}\,H$. In fact, $\mathfrak{B}[M\langle(xy,x)\rangle]$$\# H_8$ is isomorphic to $\mathfrak{B}[M\langle(y,xy)\rangle]\# H_8$ by Corollary \ref{Isomorphism:B(V)H_8}. \end{proof}
\begin{proposition}\label{HopfAlg:Omega2} Suppose $H$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $\Omega_2(n_1, n_2)$, then $H\simeq \mathfrak{B}[\Omega_2(n_1, n_2)]\# H_8$. Denote \begin{align} p_1=[(v_1+v_2)\boxtimes xy]\# 1,~~ p_2=[(v_1-v_2)\boxtimes x]\# 1,\quad v_1, v_2\in V_2,\\ X_j=(v\boxtimes x)\#1, \quad v\in V_1(\mathrm{i}), \quad j=1,\cdots, n_1,\\ Y_k= (v^\prime\boxtimes x)\#1, \quad v^\prime\in V_1(-\mathrm{i}),\quad k=1,\cdots, n_2, \end{align} then $H$ is generated by $x$, $y$, $z$, $p_1$, $p_2$, $\{X_j\}_{j=1,\cdots, n_1}$, $\{Y_k\}_{k=1,\cdots, n_2}$ satisfying the following relations.
\begin{align} x^2=y^2=1,\quad z^2=\frac{1}{2}(1+x+y-xy), \\ xy=yx,\quad zx=yz,\quad zy=xz,\\
xp_1=p_1x,\quad yp_1=-p_1y,\quad zp_1=p_2 z,\\
xp_2=-p_2x, \quad
yp_2=p_2y,\quad zp_2= p_1 xz,\\
p_1^2=0,\quad p_2^2=0,\quad
p_1p_2p_1p_2+p_2p_1p_2p_1=0,\\
xX_j=-X_jx,\quad yX_j=-yX_j,\quad zX_j=\mathrm{i}
X_jxz,\\
xY_k=-Y_kx,\quad yY_k=-yY_k,\quad zY_k=-\mathrm{i}
Y_kxz,\\
X_{j_1}X_{j_2}+X_{j_2}X_{j_1}=0,\quad j_1,j_2\in\{1,\cdots,n_1\},\\
Y_{k_1}Y_{k_2}+Y_{k_2}Y_{k_1}=0,\quad k_1,k_2\in\{1,\cdots, n_2\},\\
X_j^2=0,\quad Y_k^2=0,\quad X_jY_k+Y_kX_j=0, \\
p_2X_j+X_jp_2=0,
\quad p_2Y_k+Y_kp_2=0,\label{ideal:xxM(2_3xy,x)1}\\
p_1X_j-X_jp_1=0, \quad
p_1Y_k-Y_kp_1=0.\label{ideal:xxM(2_3xy,x)2}
\end{align}
Its Hopf algebra structure is determined by
\begin{align}
\Delta(p_1)=p_1\otimes 1+xy\otimes p_1,\quad \Delta(p_2)=p_2\otimes 1+x\otimes p_2,\\
\Delta(X_j)=X_j\otimes 1+x\otimes X_j, \quad
\Delta(Y_k)=Y_k\otimes 1+x\otimes Y_k,\\
\Delta(x)=x\otimes x,\quad \Delta(y)=y\otimes y, \quad \Delta(z)=\frac{1}{2}[(1+y)z\otimes z+(1-y)z\otimes xz].
\end{align} \end{proposition} \begin{proof} By Theorem \ref{generatedByDegreeOne}, $\mathrm{gr}\,H\simeq \mathfrak{B}[\Omega_2(n_1,n_2)]\# H_8$. By Lemma \ref{HopfAlg:M2_3(xy,x)} and Proposition \ref{HopfAlge:A_1}, we only need to prove that the lifting relations \eqref{ideal:xxM(2_3xy,x)1} and \eqref{ideal:xxM(2_3xy,x)2} are only possible by the given generators, which can be obtained from the following formulae \begin{align*} x(p_1X_j-X_jp_1)&=-(p_1X_j-X_jp_1)x,\quad x(p_1Y_k-Y_kp_1)=-(p_1Y_k-Y_kp_1),\\ \Delta(p_1X_j-X_jp_1)&=(p_1X_j-X_jp_1)\otimes 1+y \otimes (p_1X_j-X_jp_1),\\ \Delta(p_1Y_k-Y_kp_1)&=(p_1Y_k-Y_kp_1)\otimes 1+y \otimes (p_1Y_k-Y_kp_1),\\ \Delta(p_2X_j+X_jp_2)&=(p_2X_j+X_jp_2)\otimes 1+1\otimes (p_2X_j+X_jp_2),\\ \Delta(p_2Y_k+Y_kp_2)&=(p_2Y_k+Y_kp_2)\otimes 1+1\otimes (p_2Y_k+Y_kp_2). \end{align*} So $H\simeq \mathrm{gr}\, H$. \end{proof}
\begin{definition}\label{definition:A_6} For $\lambda \in \mathbb{K}$, denote by $\mathfrak{A}_6(\lambda)$ the algebra generated by $x$, $y$, $z$, $p_1$, $p_2$, $q_1$, $q_2$ satisfying the following relations \begin{eqnarray} x^2=y^2=1,\quad z^2=\frac{1}{2}(1+x+y-xy), \\ xy=yx,\quad zx=yz,\quad zy=xz,\\
xp_1=p_1x,\quad yp_1=-p_1y,\quad zp_1=p_2 z,\\
xp_2=-p_2x,\quad
yp_2=p_2y,\quad zp_2= p_1 xz,\\
p_1^2=0,\quad p_2^2=0,\quad
p_1p_2p_1p_2+p_2p_1p_2p_1=0,
\label{ideal:TwoM2_3_1}\\
xq_1=q_1x,\quad yq_1=-q_1y,\quad zq_1=q_2 z,\\
xq_2=-q_2x, \quad
yq_2=q_2y,\quad zq_2= q_1 xz,\\
q_1^2=0,\quad q_2^2=0,\quad
q_1q_2q_1q_2+q_2q_1q_2q_1=0,\label{ideal:TwoM2_3_2}\\
p_1q_1+q_1p_1=\lambda(1-x),\quad p_2q_2+q_2p_2=\lambda(1-y), \label{ideal:TwoM2_3_3}\\
p_1q_2-q_2p_1=0,\quad p_2q_1+q_1p_2=0.\label{ideal:TwoM2_3_4}
\end{eqnarray}
It is a Hopf algebra with its structure determined by \begin{align} \Delta(x)=x\otimes x,\quad \Delta(y)=y\otimes y, \quad \Delta(z)=\frac{1}{2}[(1+y)z\otimes z+(1-y)z\otimes xz],\\ \Delta(p_1)=p_1\otimes 1+y\otimes p_1,\quad \Delta(p_2)=p_2\otimes 1+xy\otimes p_2,\\ \Delta(q_1)=q_1\otimes 1+xy\otimes q_1,\quad \Delta(q_2)=q_2\otimes 1+x\otimes q_2. \end{align} \end{definition} \begin{remark} In fact, $\mathfrak{A}_6(\lambda)\simeq [T(\Omega_6)\#H_8]/{\mathcal{I}(\lambda)}$, where ${\mathcal{I}(\lambda)}$ is a Hopf ideal generated by relations \eqref{ideal:TwoM2_3_1}, \eqref{ideal:TwoM2_3_2}, \eqref{ideal:TwoM2_3_3} and \eqref{ideal:TwoM2_3_4}. It's obvious that $\mathfrak{A}_6(\lambda)$ is finite-dimensional. \end{remark} \begin{proposition}\label{HopfAlg:A_6} Suppose $H$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $\Omega_6$, then $H\simeq \mathfrak{A}_6(\lambda)$. \end{proposition} \begin{proof} By Theorem \ref{generatedByDegreeOne}, $\mathrm{gr}\,H\simeq \mathfrak{B}(\Omega_6)\#H_8$. We can suppose that $H$ is generated by generators $x$, $y$, $z$ in $H_8$ and $p_1=[(v_1+v_2)\boxtimes y]\# 1$, $p_2=[(v_1-v_2)\boxtimes xy]\# 1$, $q_1=[(v_1+v_2)\boxtimes xy]\# 1$, $q_2=[(v_1-v_2)\boxtimes x]\# 1$ in $[M\langle(y,xy)\rangle\oplus M\langle(xy,x)\rangle]\#1$. It's easy to see that formulae above in Definition \ref{definition:A_6} except \eqref{ideal:TwoM2_3_3} and \eqref{ideal:TwoM2_3_4} hold in $H$ from the bosonization $\mathfrak{B}(\Omega_6)\#H_8$ and Lemma \ref{HopfAlg:M2_3(y,xy)}, \ref{HopfAlg:M2_3(xy,x)}. Since $\mathrm{gr}\,[T(\Omega_6)\#H_8]/{\mathcal{I}(\lambda)}\simeq \mathfrak{B}(\Omega_6)\#H_8$, it's enough to prove that \eqref{ideal:TwoM2_3_3} and \eqref{ideal:TwoM2_3_4} are the only possible lifting relations by the given generators.
Since $r=0$ in $\mathrm{gr}\,H$ for $r=p_1q_1+q_1p_1$, $p_2q_2+q_2p_2$,
$p_1q_2-q_2p_1$, $p_2q_1+q_1p_2$, we have $r\in H_8\oplus \mathbb{K}\left(\Omega_5\#1\right)$. It's only possible that $p_1q_1+q_1p_1=\lambda_1(1-x)$, $p_2q_2+q_2p_2=\lambda_2(1-y)$,
$p_1q_2-q_2p_1=\lambda_3(1-xy)$, $p_2q_1+q_1p_2=0$ for $\lambda_1$, $\lambda_2$ and $\lambda_3$ in $\mathbb{K}$, because \begin{align*} \Delta(p_1q_1+q_1p_1)=(p_1q_1+q_1p_1)\otimes 1+x\otimes (p_1q_1+q_1p_1),\\ \Delta(p_2q_2+q_2p_2)=(p_2q_2+q_2p_2)\otimes 1+ y\otimes (p_2q_2+q_2p_2),\\ \Delta(p_1q_2-q_2p_1)=(p_1q_2-q_2p_1)\otimes 1+xy \otimes (p_1q_2-q_2p_1),\\ \Delta(p_2q_1+q_1p_2)=(p_2q_1+q_1p_2)\otimes 1+ 1\otimes (p_2q_1+q_1p_2). \end{align*} Since $z(p_1q_1+q_1p_1)=(p_2q_2+q_2p_2)z$ and $z(p_1q_2-q_2p_1)=(p_2q_1+q_1p_2)xz$, we have $\lambda_1=\lambda_2$ and $\lambda_3=0$. \end{proof}
\begin{lemma}\label{HopfAlg:W(b_1,-1)} Suppose $H$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $W^{b_1,-1}$, where $b_1=\pm 1$.
Then there exist parameters $\lambda_1$ and $\lambda_2$ in $\mathbb{K}$ such that $H$ is generated by $x$, $y$, $z$, $p_1$, $p_2$, which satisfy the following relations. \begin{eqnarray} x^2=y^2=1,\quad z^2=\frac{1}{2}(1+x+y-xy), \\ xy=yx,\quad zx=yz,\quad zy=xz,\\ xp_1=p_1x,\quad yp_1=p_1y,\quad xp_2=-p_2x,\quad yp_2=-p_2y,\label{HopfAlg:W(b_1,-1)0}\\ zp_1=-p_1 z,\quad zp_2=\mathrm{i} b_1 p_2 xz,\label{HopfAlg:W(b_1,-1)1}\\ p_1^2=\lambda_1(1-xy), \quad p_2^2=\mathrm{i} b_1\lambda_1(1-xy),\label{ideal:W(b_1,-1)1}\\ p_1p_2p_1p_2+p_2p_1p_2p_1=\lambda_2(1-xy).\label{ideal:W(b_1,-1)2} \end{eqnarray} Its Hopf algebra structure is determined by \begin{align} \Delta(x)&=x\otimes x,\quad \Delta(y)=y\otimes y, \quad \Delta(z)=\frac{1}{2}[(1+y)z\otimes z+(1-y)z\otimes xz],\\ \Delta(p_1)&=\left[f_{00}- \mathrm{i} b_1f_{11}\right]z\otimes p_1+\left[f_{10}+ \mathrm{i} b_1f_{01}\right]z\otimes p_2+ p_1\otimes 1, \label{Delta:W(b_1-1)p1}\\ \Delta(p_2)&=\left[f_{00}+ \mathrm{i} b_1f_{11}\right]z\otimes p_2+ \left[f_{10}- \mathrm{i} b_1f_{01}\right]z\otimes p_1+p_2\otimes 1.\label{Delta:W(b_1-1)p2} \end{align} \end{lemma} \begin{remark} In fact, $H\simeq \left[T\left(W^{b_1,-1}\right)\#H_8\right]\Big/{\mathcal{I}(\lambda_1,\lambda_2)}$, where ${\mathcal{I}(\lambda_1,\lambda_2)}$ is a Hopf ideal generated by \eqref{ideal:W(b_1,-1)1} and \eqref{ideal:W(b_1,-1)2}. It's obvious that $H$ is finite dimensional. \end{remark} \begin{proof} By Theorem \ref{generatedByDegreeOne},
$\mathrm{gr}\,H\simeq \mathfrak{B}(W^{b_1,-1})\#H_8$. We can suppose $H$ is generated by $x$, $y$, $z$ and $p_1$, $p_2$
with $x$, $y$, $z$ in $H$ and $p_1=\left(w_1^{b_1,-1}+\mathrm{i} b_1w_2^{b_1,-1}\right)\#1$, $p_2=\left(w_1^{b_1,-1}-\mathrm{i} b_1w_2^{b_1,-1}\right)\#1$.
Formulae \eqref{HopfAlg:W(b_1,-1)0}, \eqref{HopfAlg:W(b_1,-1)1},
\eqref{Delta:W(b_1-1)p1} and \eqref{Delta:W(b_1-1)p2} hold in $H$
by a straightforward computation for the bosonization $\mathfrak{B}(W^{b_1,-1})\#H_8$. Since
\begin{align*} \Delta(p_1^2)
&=\frac{1}{2}(1+xy)\otimes p_1^2+ \frac{\mathrm{i} b_1}{2}(1-xy)\otimes p_2^2+ p_1^2\otimes 1,\\ \Delta(p_2^2) &=\frac{1}{2}(1+xy)\otimes p_2^2-\frac{\mathrm{i} b_1}{2}(1-xy)\otimes p_1^2+ p_2^2\otimes 1, \end{align*}
there must exist a parameter $\lambda_1\in\mathbb{K}$ such that $p_1^2=\lambda_1(1-xy)$ and $p_2^2=\mathrm{i} b_1 \lambda_1(1-xy)$. \begin{align*} \Delta(p_1 p_2)
&=\frac{1}{2}(x+y)\otimes p_1 p_2 +\frac{\mathrm{i} b_1}{2}(x-y)\otimes p_2 p_1+ p_1 p_2\otimes 1+ \\&\quad + p_2\left[f_{00}- \mathrm{i} b_1f_{11}\right]z\otimes p_1+
p_2\left[f_{10}+ \mathrm{i} b_1f_{01}\right]z\otimes p_2+ \\ &\quad + p_1\left[f_{00}+ \mathrm{i} b_1f_{11}\right]z\otimes p_2+
p_1\left[f_{10}- \mathrm{i} b_1f_{01}\right]z\otimes p_1, \end{align*}
\begin{align*} \Delta( p_2 p_1)
&=\frac{1}{2}(x+y)\otimes p_2 p_1+ \frac{\mathrm{i} b_1}{2}(y-x)\otimes p_1 p_2+ p_2 p_1\otimes 1+\\ &\quad + \left[f_{00}+ \mathrm{i} b_1f_{11}\right]zp_1\otimes p_2+
\left[f_{10}- \mathrm{i} b_1f_{01}\right]zp_1\otimes p_1+\\ & \quad +p_2\left[f_{00}- \mathrm{i} b_1f_{11}\right]z\otimes p_1+
p_2\left[f_{10}+ \mathrm{i} b_1f_{01}\right]z\otimes p_2. \end{align*} Denote $\Delta (p_1 p_2 )=B-A+E_1$ and $\Delta(p_2 p_1 )=B+A+E_2$, where \begin{align*} A&=\left[f_{00}+ \mathrm{i} b_1f_{11}\right]zp_1\otimes p_2+ \left[f_{10}-\mathrm{i} b_1f_{01}\right]zp_1\otimes p_1,\\ B&=p_2\left[f_{00}- \mathrm{i} b_1f_{11}\right]z\otimes p_1+ p_2\left[f_{10}+\mathrm{i} b_1f_{01}\right]z\otimes p_2,\\ E_2&=\frac{1}{2}(x+y)\otimes p_2 p_1+ \frac{\mathrm{i} b_1}{2}(y-x)\otimes p_1 p_2+ p_2 p_1\otimes 1,\\ E_1&=\frac{1}{2}(x+y)\otimes p_1 p_2 +\frac{\mathrm{i} b_1}{2}(x-y)\otimes p_2 p_1+ p_1 p_2\otimes 1. \end{align*}
We can obtain $A^2+B^2=0$, since \begin{align*} A^2
&=-\frac{1}{2}(x+y)p_1^2\otimes p_2^2 +\frac{\mathrm{i} b_1}{2}(1-xy)p_1^2\otimes p_1^2
=\mathrm{i} b_1\lambda_1^2(1-xy)\otimes (1-xy), \end{align*}
\begin{align*} B^2
&=- \frac{1}{2}(x+y) p_2^2 \otimes p_1^2+
\frac{\mathrm{i} b_1}{2}(1-xy) p_2^2 \otimes p_2^2
=-\mathrm{i} b_1\lambda_1^2(1-xy)\otimes (1-xy). \end{align*}
Keeping in mind that \begin{align*} p_1(p_1p_2+p_2p_1)&=(p_2p_1+p_1p_2)p_1,\quad p_2(p_1p_2+p_2p_1)=(p_2p_1+p_1p_2)p_2,\\ p_1(p_1p_2-p_2p_1)&=(p_2p_1-p_1p_2)p_1,\quad p_2(p_1p_2-p_2p_1)=(p_2p_1-p_1p_2)p_2,\\ (x+y)p_2(f_{00}&-\mathrm{i} b_1f_{11})z=-p_2(f_{00}-\mathrm{i} b_1f_{11})z(x+y),\\ (x-y)p_2(f_{00}&-\mathrm{i} b_1f_{11})z=p_2(f_{00}-\mathrm{i} b_1f_{11})z(x-y),\\ (x+y)p_2(f_{10}&+\mathrm{i} b_1f_{01})z=-p_2(f_{10}+\mathrm{i} b_1f_{01})z(x+y),\\ (x-y)p_2(f_{10}&+\mathrm{i} b_1f_{01})z=p_2(f_{10}+\mathrm{i} b_1f_{01})z(x-y),\\ (p_1p_2+p_2p_1)&p_2(f_{00}-\mathrm{i} b_1f_{11})z=-p_2(f_{00}-\mathrm{i} b_1f_{11})z(p_1p_2+p_2p_1),\\ (p_1p_2+p_2p_1)&p_2(f_{10}+\mathrm{i} b_1f_{01})z=-p_2(f_{10}+\mathrm{i} b_1f_{01})z(p_1p_2+p_2p_1), \end{align*} we deduce $B(E_1+E_2) +(E_1+E_2)B=0$. Similarly, we have $A(E_2-E_1)+(E_2-E_1)A=0$. \begin{align*} &\quad \Delta(p_1 p_2 p_1 p_2 +p_2 p_1 p_2 p_1) =(B-A+E_1)^2+(B+A+E_2)^2\\ &=2(A^2+B^2)+B(E_1+E_2) +(E_1+E_2)B+A(E_2-E_1)+(E_2-E_1)A+E_1^2+E_2^2\\ &=E_1^2+E_2^2=\left( \frac{1}{2}(x+y)\otimes p_1 p_2 +\frac{\mathrm{i} b_1}{2}(x-y)\otimes p_2 p_1+ p_1 p_2\otimes 1\right)^2+\\ &\quad +\left( \frac{1}{2}(x+y)\otimes p_2 p_1+ \frac{\mathrm{i} b_1}{2}(y-x)\otimes p_1 p_2+ p_2 p_1\otimes 1\right)^2\\ &=\frac{1}{2}(1+xy)\otimes \left(p_1 p_2\right)^2 -\frac{1}{2}(1-xy)\otimes \left(p_2 p_1\right)^2 +\left(p_1 p_2\right)^2\otimes 1+\\ &\quad+ \frac{1}{2}(1+xy)\otimes \left(p_2 p_1\right)^2 -\frac{1}{2}(1-xy)\otimes \left(p_1 p_2\right)^2 +\left(p_2 p_1\right)^2\otimes 1\\ &=xy\otimes \left[\left(p_1 p_2\right)^2+\left(p_2 p_1\right)^2\right] +\left[\left(p_1 p_2\right)^2+\left(p_2 p_1\right)^2\right]\otimes 1. \end{align*} So there exists a parameter $\lambda_2\in\mathbb{K}$ such that $p_1p_2p_1p_2+p_2p_1p_2p_1=\lambda_2(1-xy)$.
We have $H\simeq \left[T(W^{b_1,-1})\#H_8\right]\Big/{\mathcal{I}(\lambda_1,\lambda_2)}$, because $\mathrm{gr}\,\left\{\left[T(W^{b_1,-1})\#H_8\right]\Big/{\mathcal{I}(\lambda_1,\lambda_2)}\right\}\simeq \mathfrak{B}\left(W^{b_1,-1}\right)\#H_8$. \end{proof}
\begin{definition}\label{Definition:HopfAlgA_7}
For a set of parameters $I_7=\{\lambda_j\in \mathbb{K}|j=1,\cdots, 5\}$, denote by $\mathfrak{A}_7(I_7)$ the algebra generated by $x$, $y$, $z$, $p_1$, $p_2$, $q_1$, $q_2$ satisfying the following relations \begin{eqnarray} x^2=y^2=1,\quad z^2=\frac{1}{2}(1+x+y-xy), \\ xy=yx,\quad zx=yz,\quad zy=xz,\\ xp_1=p_1x,\quad yp_1=p_1y,\quad xp_2=-p_2x,\quad yp_2=-p_2y,\\ zp_1=-p_1 z,\quad zp_2=\mathrm{i} p_2 xz,\\ p_1^2=\lambda_1(1-xy), \quad p_2^2=\mathrm{i} \lambda_1(1-xy), \label{ideal:TwoW(pm 1,-1)1}\\ p_1p_2p_1p_2+p_2p_1p_2p_1=\lambda_2(1-xy), \label{ideal:TwoW(pm 1,-1)2}\\ xq_1=q_1x,\quad yq_1=q_1y,\quad xq_2=-q_2x,\quad yq_2=-q_2y,\\ zq_1=-q_1 z,\quad zq_2=-\mathrm{i} q_2 xz,\\ q_1^2=\lambda_3(1-xy), \quad q_2^2=-\mathrm{i} \lambda_3(1-xy), \label{ideal:TwoW(pm 1,-1)3}\\ q_1q_2q_1q_2+q_2q_1q_2q_1=\lambda_4(1-xy), \label{ideal:TwoW(pm 1,-1)4}\\ p_1q_2+q_2p_1=0,\quad p_2q_1+q_1p_2=0, \label{ideal:TwoW(pm 1,-1)5}\\ p_1q_1+q_1p_1=\lambda_5(x+y-2),\quad p_2q_2-q_2p_2=-\mathrm{i} \lambda_5(x-y). \label{ideal:TwoW(pm 1,-1)6} \end{eqnarray}
It is a Hopf algebra with its structure determined by
\begin{align}
\Delta(p_1)&=\left[f_{00}- \mathrm{i} f_{11}\right]z\otimes p_1+ \left[f_{10}+\mathrm{i} f_{01}\right]z\otimes p_2+ p_1\otimes 1,\\ \Delta(p_2)&=\left[f_{00}+ \mathrm{i} f_{11}\right]z\otimes p_2+ \left[f_{10}-\mathrm{i} f_{01}\right]z\otimes p_1+p_2\otimes 1,\\ \Delta(q_1)&=\left[f_{00}+\mathrm{i} f_{11}\right]z\otimes q_1+ \left[f_{10}-\mathrm{i} f_{01}\right]z\otimes q_2+ q_1\otimes 1,\\ \Delta(q_2)&=\left[f_{00}-\mathrm{i} f_{11}\right]z\otimes q_2+ \left[f_{10}+\mathrm{i} f_{01}\right]z\otimes q_1+q_2\otimes 1,\\ \Delta(x)&=x\otimes x,\quad \Delta(y)=y\otimes y, \quad \Delta(z)=\frac{1}{2}[(1+y)z\otimes z+(1-y)z\otimes xz]. \end{align} \end{definition} \begin{remark} In fact, $\mathfrak{A}_7(I_7)\simeq\left[T\left(W^{1,-1}\oplus W^{-1,-1}\right) \# H_8\right]\Big/{\mathcal{I}(I_7)}$, where $\mathcal{I}(I_7)$ is a Hopf ideal generated by relations \eqref{ideal:TwoW(pm 1,-1)1}, \eqref{ideal:TwoW(pm 1,-1)2}, \eqref{ideal:TwoW(pm 1,-1)3}, \eqref{ideal:TwoW(pm 1,-1)4}, \eqref{ideal:TwoW(pm 1,-1)5} and \eqref{ideal:TwoW(pm 1,-1)6}. Since \begin{align*} p_1p_2p_1p_2p_1&=p_1[\lambda_2(1-xy)-p_1p_2p_1p_2]= [\lambda_2p_1-\lambda_1p_2p_1p_2](1-xy),\\ p_2p_1p_2p_1p_2&=p_2[\lambda_2(1-xy)-p_2p_1p_2p_1]= [\lambda_2p_2-\lambda_1\mathrm{i} p_1p_2p_1](1-xy), \end{align*} $\dim\left<1, p_1, p_2\right><\infty$ for the subalgebra $\left<1, p_1, p_2\right>$ generated by $1$, $p_1$, $p_2$. Similarly, we have $\dim\left<1, q_1, q_2\right><\infty$. We can deduce that $$\dim A_7(I_7)=\dim\left<1, p_1, p_2\right> \dim\left<1, q_1, q_2\right> \dim H_8<\infty. $$ \end{remark} \begin{proposition}\label{HopfAlg:A_7} Suppose $H$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $\Omega_7$, then $H\simeq \mathfrak{A}_7(I_7)$. \end{proposition} \begin{proof} By Theorem \ref{generatedByDegreeOne}, we have $\mathrm{gr}\,H\simeq \mathfrak{B}(\Omega_7)\#H_8$. We can suppose $H$ is generated by $x$, $y$, $z$, $p_1$, $p_2$, $q_1$, $q_2$ with $x$, $y$, $z\in H_8$ and
\begin{align} p_1&=\left(w_1^{1,-1}+\mathrm{i} w_2^{1,-1}\right)\#1, &p_2&=\left(w_1^{1,-1}-\mathrm{i} w_2^{1,-1}\right)\#1,\\ q_1&=\left(w_1^{-1,-1}-\mathrm{i} w_2^{-1,-1}\right)\#1, &q_2&=\left(w_1^{-1,-1}+\mathrm{i} w_2^{-1,-1}\right)\#1. \end{align} By lemma \ref{HopfAlg:W(b_1,-1)}, we only need to prove that \eqref{ideal:TwoW(pm 1,-1)5} and \eqref{ideal:TwoW(pm 1,-1)6} hold in $H$. It's only possible for $p_1q_2+q_2p_1=0$, $p_2q_1+q_1p_2=0$, since $x(p_1q_2+q_2p_1)=-(p_1q_2+q_2p_1)x$, $ x(p_2q_1+q_1p_2)=-(p_2q_1+q_1p_2)x$, and \begin{align*} \Delta(p_1q_2+q_2p_1) &=\frac{1}{2}\left[(1+xy)+\mathrm{i} (1-xy)\right]\otimes (p_1q_2+q_2p_1)+ (p_1q_2+q_2p_1)\otimes 1,\\ \Delta(p_2q_1+q_1p_2) &=\frac{1}{2}\left[(1+xy)-\mathrm{i}(1-xy)\right]\otimes (p_2q_1+q_1p_2)+ (p_2q_1+q_1p_2)\otimes 1, \end{align*}
Similarly, we can get \eqref{ideal:TwoW(pm 1,-1)6}, since \begin{align*} z(p_1q_1+q_1p_1)&=(p_1q_1+q_1p_1)z, \quad z(p_2q_2-q_2p_2)=-(p_2q_2-q_2p_2)z,\\ \Delta(p_1q_1+q_1p_1)
&=\frac{x+y}{2} \otimes (p_1q_1+q_1p_1)+ (p_1q_1+q_1p_1)\otimes 1 +\frac{\mathrm{i}(x-y)}{2}\otimes (p_2q_2-
q_2p_2),\\
\Delta(p_2q_2-q_2p_2)
&=\frac{x+y}{2}\otimes (p_2q_2-q_2p_2)+(p_2q_2-q_2p_2)\otimes 1-\frac{\mathrm{i}(x-y)}{2}\otimes (p_1q_1+q_1p_1). \end{align*}
We have $H\simeq \mathfrak{A}_7(I_7)$, because $\mathrm{gr}\,\{\left[T\left(\Omega_7\right) \# H_8\right]\big/\mathcal{I}(I_7)\}\simeq \mathfrak{B}(\Omega_7)\#H_8$. \end{proof}
\begin{definition}\label{Definition:HopfAlgA_4}
For a set of parameters $I_4=\{\lambda_1, \lambda_2, \lambda_{j,k}\in \mathbb{K}|j=1,\cdots, n_1, k=1,\cdots, n_2\}$, denote by $\mathfrak{A}_4(n_1,n_2;I_4)$ the algebra generated by $x$, $y$, $z$, $p_1$, $p_2$,
$\{X_j\}_{j=1,\cdots, n_1}$, $\{Y_k\}_{k=1,\cdots, n_2}$ satisfying the following relations
\begin{align}
x^2=y^2=1,\quad z^2=\frac{1}{2}(1+x+y-xy), \label{formulae:A_4_1}\\ xy=yx,\quad zx=yz,\quad zy=xz,\\ xp_1=p_1x,\quad yp_1=p_1y,\quad xp_2=-p_2x,\quad yp_2=-p_2y,\label{FGK1}\\ zp_1=-p_1 z,\quad zp_2=\mathrm{i} p_2 xz,\\ p_1^2=\lambda_1(1-xy), \quad p_2^2=\mathrm{i} \lambda_1(1-xy), \label{ideal:xyW(b1,-1)1}\\ p_1p_2p_1p_2+p_2p_1p_2p_1=\lambda_2(1-xy),\label{ideal:xyW(b1,-1)2}\\
xX_j=-X_jx,\quad yX_j=-X_jy,\quad zX_j=\mathrm{i} X_j xz,\\
xY_k=-Y_kx,\quad yY_k=-Y_ky,\quad zY_k=\mathrm{i} Y_k xz,\\
X_{j_1}^2=0,\quad X_{j_1}X_{j_2}+X_{j_2}X_{j_1}=0,\quad j_1,j_2\in\{1,\cdots,n_1\},\label{ideal:xyW(b1,-1)3}\\
Y_{k_1}^2=0,\quad Y_{k_1}Y_{k_2}+Y_{k_2}Y_{k_1}=0,\quad k_1,k_2\in\{1,\cdots, n_2\},\label{ideal:xyW(b1,-1)4}\\
X_jY_k+Y_kX_j=\lambda_{j,k}(1-xy), \label{ideal:xyW(b1,-1)5}
\end{align}
\begin{align} p_1Y_k-Y_kp_1=0 , \quad p_2Y_k+Y_kp_2=0,\quad p_1X_j-X_jp_1=0, \quad p_2X_j+X_jp_2=0.\label{ideal:xyW(b1,-1)6} \end{align}
It is a Hopf algebra with its structure determined by
\begin{align}
\Delta(X_j)&=X_j\otimes 1+x\otimes X_j, \quad
\Delta(Y_k)=Y_k\otimes 1+y\otimes Y_k, \label{coproduct:A_4_1}\\ \Delta(p_1)&=(f_{00}-\mathrm{i} f_{11})z\otimes p_1+ (f_{10}+\mathrm{i} f_{01})z\otimes p_2+p_1\otimes 1,\\ \Delta(p_2)&=(f_{00}+\mathrm{i} f_{11})z\otimes p_2+ (f_{10}-\mathrm{i} f_{01})z\otimes p_1+p_2\otimes 1,\\ \Delta(x)&=x\otimes x,\quad \Delta(y)=y\otimes y, \quad \Delta(z)=\frac{1}{2}[(1+y)z\otimes z+(1-y)z\otimes xz]. \label{coproduct:A_4_2} \end{align} \end{definition} \begin{remark} We can observe that $$\dim \mathfrak{A}_4(n_1,n_2;I_4)=\dim\left<1,p_1,p_2\right> \dim\left<1, \{X_j\}_{j=1,\cdots, n_1} \right>\dim\left<1, \{Y_k\}_{k=1,\cdots, n_2}\right>\dim H_8<\infty$$ for subalgebra $\left<1,p_1,p_2\right>$ generated by $1$, $p_1$, $p_2$, subalgebra $\left<1, \{X_j\}_{j=1,\cdots, n_1} \right>$ generated by $1$, $\{X_j\}_{j=1,\cdots, n_1}$, and subalgebra $\left<1, \{Y_k\}_{k=1,\cdots, n_2}\right>$ generated by $1$, $\{Y_k\}_{k=1,\cdots, n_2}$.
In fact, $\mathfrak{A}_4(n_1,n_2;I_4)\simeq T[\Omega_4(n_1,n_2)]\#H_8/{\mathcal{I}(I_4)}$, where $\mathcal{I}(I_4)$ is a Hopf ideal genereated by relations
\eqref{ideal:xyW(b1,-1)1}, \eqref{ideal:xyW(b1,-1)2}, \eqref{ideal:xyW(b1,-1)3}, \eqref{ideal:xyW(b1,-1)4}, \eqref{ideal:xyW(b1,-1)5}, \eqref{ideal:xyW(b1,-1)6}. \end{remark}
\begin{proposition}\label{HopfAlg:Omega_4} Suppose $H$ is a finite dimensional Hopf algebra with coradical $H_8$ such that its infinitesimal braiding is isomorphic to $\Omega_4(n_1,n_2)$, then $H\simeq \mathfrak{A}_4(n_1,n_2; I_4)$. \end{proposition} \begin{proof} By Theorem \ref{generatedByDegreeOne}, we have $\mathrm{gr}\,H\simeq \mathfrak{B}[\Omega_4(n_1,n_2)]\#H_8$. We can suppose $H$ is generated by $x$, $y$, $z$, $p_1$, $p_2$, $\{X_j\}_{j=1,\cdots, n_1}$, $\{Y_k\}_{k=1,\cdots, n_2}$ with $x$, $y$, $z\in H_8$ and \begin{align} p_1=\left(w_1^{b_1,-1}+\mathrm{i} b_1w_2^{b_1,-1}\right)\#1, \quad p_2=\left(w_1^{b_1,-1}-\mathrm{i} b_1w_2^{b_1,-1}\right)\#1,\\ X_j=(v\boxtimes x)\#1, \quad j=1,\cdots, n_1,\\ Y_k= (v\boxtimes y)\#1, \quad v\in V_1(\mathrm{i}),\quad k=1,\cdots, n_2. \end{align} As similarly proved in Proposition \ref{HopfAlge:A_1} and Lemma \ref{HopfAlg:W(b_1,-1)}, formulae \eqref{formulae:A_4_1}--\eqref{ideal:xyW(b1,-1)5} and \eqref{coproduct:A_4_1}--\eqref{coproduct:A_4_2} hold in $H$. Since $r=0$ in $\mathrm{gr}\,H$ for $r=p_1Y_k-Y_kp_1$ and $p_2Y_k+Y_kp_2$, $r$ is an element of at most degree one. It's only possible for \begin{align*} p_1Y_k-Y_kp_1=-\mu_k\left(-f_{10}+\mathrm{i} b_1f_{01}\right)z , \quad p_2Y_k+Y_kp_2=-\mu_k\left(f_{00}-\mathrm{i} b_1f_{11}\right)z+ \mu_k 1, \end{align*} because of the following relations \begin{align*} x\left(p_1Y_k-Y_kp_1\right)&=-\left(p_1Y_k-Y_kp_1\right)x,\quad
z\left(p_1Y_k-Y_kp_1\right)=-b_1\mathrm{i} \left(p_1Y_k-Y_kp_1\right)xz,\\
x\left(p_2Y_k+Y_kp_2\right)&=\left(p_2Y_k+Y_kp_2\right)x,\quad
z\left(p_2Y_k+Y_kp_2\right)=\left(p_2Y_k+Y_kp_2\right)z,\\ \Delta(p_1Y_k-Y_kp_1) &=(p_1Y_k-Y_kp_1)\otimes 1+(f_{00}+\mathrm{i} b_1 f_{11})z\otimes (p_1Y_k-Y_kp_1)+\\ &\quad+ (-f_{10}+\mathrm{i} b_1 f_{01})z\otimes (p_2Y_k+Y_kp_2),\\ \Delta(p_2Y_k+Y_kp_2) &=(p_2Y_k+Y_kp_2)\otimes 1+(f_{00}-\mathrm{i} b_1 f_{11})z\otimes (p_2Y_k+Y_kp_2)-\\ &\quad-(f_{10}+
\mathrm{i} b_1 f_{01})z\otimes (p_1Y_k-Y_kp_1). \end{align*} Similarly, we get \begin{align*} p_1X_j-X_jp_1=-\mu_j^\prime\left(f_{10}-\mathrm{i} b_1f_{01}\right)z , \quad p_2X_j+X_jp_2=-\mu_j^\prime\left(f_{00}-\mathrm{i} b_1f_{11}\right)z+ \mu_j^\prime 1, \end{align*} from the following formulae \begin{align*} x\left(p_1X_j-X_jp_1\right)&=-\left(p_1X_j-X_jp_1\right)x, \quad z\left(p_1X_j-X_jp_1\right)=-b_1\mathrm{i} \left(p_1X_j-X_jp_1\right)xz,\\ x\left(p_2X_j+X_jp_2\right)&=\left(p_2X_j+X_jp_2\right)x,\quad z\left(p_2X_j+X_jp_2\right)=\left(p_2X_j+X_jp_2\right)z,\\ \Delta\left(p_1X_j-X_jp_1\right) &=(f_{00}+\mathrm{i} b_1 f_{11})z\otimes \left(p_1X_j-X_jp_1\right)+\left(p_1X_j-X_jp_1\right)\otimes 1+\\ &\quad+(f_{10}-\mathrm{i} b_1 f_{01})z\otimes \left(p_2X_j+X_jp_2\right),\\ \Delta\left(p_2X_j+X_jp_2\right) &=(f_{00}-\mathrm{i} b_1 f_{11})z\otimes \left(p_2X_j+X_jp_2\right)+ \left(p_2X_j+X_jp_2\right)\otimes 1+\\ &\quad+(f_{10}+\mathrm{i} b_1 f_{01})z\otimes \left(p_1X_j-X_jp_1\right). \end{align*} Since $X_jY_k+Y_kX_j=\lambda_{j,k}(1-xy)$, $p_1(X_jY_k+Y_kX_j)=(X_jY_k+Y_kX_j)p_1\Rightarrow \mu_j^\prime=\mu_k=0$. So \eqref{ideal:xyW(b1,-1)6} holds in $H$. We have $H\simeq \mathfrak{A}_4(n_1,n_2;I_4)$ because $ \mathrm{gr}\,\left\{T[\Omega_4(n_1,n_2)]\#H_8/{\mathcal{I}(I_4)}\right\}\simeq \mathfrak{B}[\Omega_4(n_1,n_2)]\#H_8$. \end{proof}
\subsection*{Proof of Theorem \ref{HopfAlgOverH8}} Let $M$ be one of Yetter-Drinfel'd modules listed in Theorem \ref{NicholsAlg:maintheorem}. We need to give a construction for any finite-dimensional Hopf algebra $H$ over $H_8$ up to isomorphism such that its infinitesimal braiding is isomorphic to $M$. By Theorem \ref{generatedByDegreeOne}, $\mathrm{gr}\,H\simeq \mathfrak{B}(M)\# H_8$. According to Corollary \ref{Isomorphism:B(V)H_8}, up to isomorphism, $\mathrm{gr}\,H\simeq \mathfrak{B}(M)\# H_8$ for $M=\Omega_1(n_1,n_2,n_3,n_4)$, $\Omega_2(n_1, n_2)$, $\Omega_4(n_1,n_2)$, $\Omega_6$, $\Omega_7$. Proposition \ref{HopfAlge:A_1}, \ref{HopfAlg:Omega2}, \ref{HopfAlg:Omega_4}, \ref{HopfAlg:A_6} and \ref{HopfAlg:A_7} finish the proof.
\end{document}
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arXiv
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Ultrasmall nanostructured drug based pH-sensitive liposome for effective treatment of drug-resistant tumor
Yongxia Zhai1,
Wei Liu2,
Kaixiang Zhang2,3,4,
Junjie Liu ORCID: orcid.org/0000-0002-9404-050X2,3,4,
Jinjin Shi2,3,4 &
Zhenzhong Zhang2,3,4
Cancer cells always develop ways to resist and evade chemotherapy. To overcome this obstacle, herein, we introduce a programmatic release drug delivery system that imparts avoiding drug efflux and nuclear transport in synchrony via a simple nanostructured drug strategy.
The programmatic liposome-based nanostructured drugs (LNSD) contained two modules: doxorubicin (DOX) loaded into tetrahedral DNA (TD, ~ 10 nm) to form small nanostructured DOX, and the nanostructured DOX was encapsulated into the pH-sensitive liposomes. In the in vitro and in vivo studies, LNSD shows multiple benefits for drug resistance tumor treatment: (1) not only enhanced the cellular DOX uptake, but also maintained DOX concentration in an optimum level in resistant tumor cells via nanostructure induced anti-efflux effect; (2) small nanostructured DOX efficiently entered into cell nuclear via size depended nuclear-transport for enhanced treatment; (3) improved the pharmacokinetics and biodistribution via reducing DOX leakage during circulation.
The system developed in this study has the potential to provide new therapies for drug-resistant tumor.
The emergence of multiple-drug resistance (MDR) is remaining a main obstacle for successful treatment of cancer [1]. Cancer cells often develop drug resistance and stop responding to chemotherapeutics after repeated sessions of chemotherapy [2]. MDR can be induced by various mechanisms, including decreased drug uptake, increased drug efflux, activation of detoxifying systems, activated of DNA repair mechanisms and evasion of drug-induced apoptosis [3,4,5]. In particular, MDR is typically mediated by the overexpression of a membrane transporter, P-glycoprotein (P-gp), actively increases the efflux of drugs from cancer cells [6, 7]. The efflux drugs reduced the therapeutic effect and cancer cells often develop drug resistance and stop responding to chemotherapeutics [8].
To reverse MDR, great effort has been devoted to developing specific drug delivery systems (DDS) [9, 10]. In the past 10 years, nanoscaled drug delivery systems such as liposome [11], solid lipid nanoparticles (SLN) [12], polymer micelles [13], mesoporous silica [14], carbon nanomaterial [15], and gold nanomaterial [16]. etc., which increased tumor selectivity and reduced toxicity have been receiving a lot of attentions. More importantly, the developed DDS could bring more drugs into the resistant tumor cells, therefore, significantly improved the antitumor efficacy [17]. Besides improving drug uptake via DDS, how to maintain the drug concentration in an optimum level in the resistant tumor cells is another main challenge for reversing MDR [18, 19]. To overcome the obstacle, the strategy of the super small nanoparticle was used. Firstly, compared with the free small molecule drugs, the small nanoparticle could not be efflux from drug resistant tumor cells, thus maintain the drug concentration [20]; Secondly, the super small nanoparticle could efficiently enter into the cell nuclear, and this is important for treatment. For example, Liang, etc. have developed a series of gold nanoparticles with different sizes, and they found the small gold nanoparticles could efficiently enter into the cell nuclear [21]. Therefore, in this study, the small nanoparticle was used for reversing MDR in tumor cells.
Recently, DNA nanotechnology has been widely investigated in different biomedical fields [22,23,24]. Tetrahedral DNA nanostructures (TDNs) have attracted a great deal of attention in biomedical fields due to the biological nature of DNA and convenient synthesis [25]. More importantly, TDNs have a super small size (~ 10 nm), and have a high drug loading efficacy [26]. TDNs have been considered a promising drug delivery system for cancer treatment [25,26,27]. In this study, TDNs were used as the secondary drug delivery vehicle.
The in vivo kinetic behavior of TDNs is a key point for tumor treatment [28], and a pharmacokinetic standpoint as nanoparticles less than 10 nm have been reported to be cleared by kidney, while larger nanoparticles have been reported to preferentially home into tumors through leaky tumor neovasculature as a result of the enhanced permeability and retention (EPR) effect [29, 30]. Liposomes are widely accepted as targeted delivery systems for antitumor drugs as demonstrated by the commercialization success of a number drug molecules [31]. pH-sensitive liposomes, usually PEGylated (pPSL), have been investigated to refine conventional liposomes in effective targeted extra- and intra-cellular delivery of anticancer drugs [32]. To improve the in vivo kinetic behavior of TDNs, pH-sensitive liposomes were used as the senior drug delivery vehicles.
In current study, we rationally designed a pH-sensitive liposome-based nanostructured DOX (LNSD) for drug resistance tumor treatment. As schemed in Fig. 1, DOX was loaded into TD to form the small nanostructured DOX (TD/DOX), and then the nanostructured DOX was encapsulated into the pH-sensitive liposomes to form LNSD. The prepared LNSD has multiple benefits for drug resistance tumor treatment: (1) not only enhanced the cellular DOX uptake, but also maintained DOX concentration in an optimum level in resistant tumor cells; (2) the small nanostructured DOX could enter into the cell nuclear for enhanced treatment; 3) accumulated in tumor site via EPR effect of liposomes. The enhanced antitumor efficacy and reversing DOX resistant effect of LNSD were investigated using MCF-7/ADR cells and DOX-resistant breast tumor models.
The preparation, anti-efflux and cell nuclear-transport effects of LNSD
Synthesis and characterization of LNSD
Tetrahedral DNA nanostructure was assembled with four 55-mer strands (Table 1) prepared with a high-yield, single-step synthesis originally reported by Turberfield et al. [33]. The successful preparation of TD was confirmed by the results of electrophoresis (Fig. 2a). Along with the self-assembly of S1 (55 bases), S1(55 bases) + S2(55 bases), S1(55 bases) + S2(55 bases) + S3(55 bases) and S1(55 bases) + S2(55 bases) + S3(55 bases) + S4(55 bases), the migration speed was gradually decline, and TD was composed of S1 + S2 + S3 + S4 (total 220 bases) (Fig. 2a). AFM results showed the size of TD was ~ 10 nm (Fig. 2b), and the average size of TD was 14.3 ± 1.6 nm confirmed by DLS (Fig. 2f). DOX loading was achieved by incubating TD with DOX for 12 h at room temperature. After DOX loading, the zeta potential of TD showed a significant decrease (from − 20.4 to − 12.8 mV, Fig. 2f). While after DOX loading, the results of AFM, electrophoresis and DLS showed no significant difference compared to TD (Fig. 2c, d and f), showing DOX loading did not influence the structure of TD. Finally, pH-sensitive liposomes were used to encapsulate the DOX loaded TD (TD/DOX) to obtain the liposome-based nanostructured DOX (LNSD). The pH-sensitive liposomes were prepared using a thin-film hydration method. The particle size distributions of liposome, TD@liposome, DOX@liposome and LNSD were shown in Additional file 1, and the average particle size of LNSD was ~ 147 nm (Fig. 2f). TEM images showed LNSD had a uniform size and a ball-like structure (Fig. 2e). The DOX encapsulation efficiency (EE) of LNSD was calculated to 29.6%. Interestingly, the DOX encapsulation efficiency of DOX@liposome was 16.7%, which was much lower than that of LNSD, suggesting that the strategy of nanostructured DOX could significantly increase the EE of liposomes (Additional file 2).
Table 1 DNA oligonucleotides used in TD
Synthesis and characterization of LNSDs. a Electrophoretic analysis of TD, a: S1 alone, b: S1 + S2, c: S1 + S2 + S3, d: S1 + S2 + S3 + S4 (TD), M: marker. b AFM image of TD. c Electrophoretic analysis of TD and TD/DOX, a: TD, b: TD/DOX, M: marker. d AFM image of TD/DOX. e TEM images of LNSD, insert: photo of LNSD nanosuspension. f Size and zeta potential of TD, TD/DOX, DOX@liposome and LNSD (n = 3). g TEM images of LNSD at pH 5.0 buffer for 8 h. h DOX release from DOX@liposome or LNSD at pH 5.0 and pH 7.4, respectively (n = 3)
The pH-sensitivity of LNSD was investigated by intuitive observation of the morphological change of LNSD in pH 5.0 buffer. According to the results of TEM (Fig. 2g), when LNSD was incubated in pH 5.0 buffer for 8 h, the shell of LNSD obviously ruptured, showing the high pH-sensitive ability of LNSD. Furthermore, the DOX release from DOX@liposome and LNSD were shown in Fig. 2h, in the case of DOX@liposome at pH 7.4, ~ 36.7% of DOX was released. While when the pH was declined to 5.0, the release of DOX significantly increased to ~ 89.6%, indicating the pH sensitive ability of the prepared liposomes. On the other hand, in the case of LNSD at 5.0 group, only ~ 25.1% of DOX was released after 24 h, much lower than that of DOX@liposome at pH 5.0 group (~ 89.6%), demonstrating most of DOX was not released from TD (Fig. 2h). More importantly, compared with DOX@liposome (free DOX strategy), LNSD (nanostructured DOX strategy) significantly decreased the unexpected DOX leakage (10.1% vs 36.7% after incubation with pH 7.4 buffer for 24 h, Fig. 2h), and the results were also confirmed by the photos of DOX@liposome and LNSD after centrifugation (Additional file 3).
Anti-efflux and cell nuclear distribution effects of LNSD in MCF-7/ADR cells
MCF-7/ADR cells were used as the model cells in the in vitro studies. Firstly, the biodistribution of TD@liposome was investigated, and the TD was labeled via FAM. The results were shown in Additional file 4. After incubation with FAM-TD@liposome for 4 h, a large amount of the green fluorescence (FAM) were observed in MCF-7/ADR cells, and most of the fluorescence were in the cytoplasm; While when the incubation time was prolonged to 12 h, a considerable part of green fluorescence were in the cell nuclear, indicating that TD could enter into the cell nuclear.
The intracellular stability of TD determined the distribution of the TD/DOX in the cells, a lot of previous studies had proved that TD could retain structural integrity in cells. [25, 30, 34]. Next, to investigate the intracellular stability of TD/DOX encapsulated in LNSD, FAM-TD/DOX@pH-sensitive liposome was first prepared, and after incubating with the MCF/ADR cells for 4 h, the results was shown in Additional file 5. According to the results, the green fluorescence of TD (FAM) and red fluorescence of DOX displayed well-overlapped, confirming that DOX was not released from DOX-loaded TD within cells, thus demonstrating the stability of DOX-loaded TD in MCF/ADR cells.
Next, we investigated the intracellular distribution of LNSD in MCF/ADR cells, and the results were shown in Fig. 3a and b. After incubation with DOX for 4 h, only weak red fluorescence (DOX) were found in MCF-7/ADR cells, while in the case of DOX@liposome and LNSD groups, more fluorescence were observed, indicating that LNSD or liposomes could carry DOX into the MCF-7/ADR cells. Interestingly, compared with DOX@liposome, more fluorescence was found in the cell nuclear (Fig. 3a). In the case of LNSD group, the co-localization ratio of DOX and DAPI (blue fluorescence) was significant higher than that of DOX@liposome (36.9% vs 28.3%, Fig. 3d). When the incubation time was prolonged to 12 h, almost no red fluorescence was detected in DOX group, indicating the efflux effect of MCF-7/ADR cells (Fig. 3b). The significant efflux effect was also found in the case of DOX@liposome group: compared with incubation for 4 h, the signals of red fluorescence in cells significantly decreased after another 8 h of incubation (Fig. 3b). However, in the case of LNSD group, and with the passage of incubation time, the red fluorescence of DOX did not show significant decrease (Fig. 3b).
Anti-efflux, cell nuclear distribution and proliferation inhibiting effects of LNSD in MCF-7/ADR cells. a CLSM images of MCF-7/ADR cells treated with DOX, DOX@liposome and LNSD for 4 h, scale bar: 10 μm. b CLSM images of MCF-7/ADR cells treated with DOX, DOX@liposome and LNSD for another 8 h, the total incubation time was 12 h, scale bar: 10 μm. c Intracellular pharmacokinetics of DOX, DOX@liposome and LNSD (n = 4). d Co-localization ratio of DOX (red FLR) and cell nuclear (blue FLR) in DOX, DOX@liposome or LNSD treated MCF-7/ADR cells (n = 20). e, f The cell proliferation inhibition rates of DOX, DOX@liposome and LNSD with different DOX concentrations for 24 h and 48 h (n = 6). Data presented are means ± SD. ***p < 0.01
The efflux effects in DOX, DOX@liposome and LNSD groups were also investigated via detecting the concentration of DOX in cells for different incubation times, and the results were shown in Fig. 3c. In the case of DOX group, the concentration of DOX in MCF-7/ADR cells was very low at every time point, while the concentration of DOX was much higher in DOX@liposome and LNSD groups. According to the results, after incubation for 4 h, the DOX concentration in cells decreased very fast in the case of DOX@liposome group, and after incubation for 12 h, the DOX concentration decreased to ~ 1.1 μg (Fig. 3c). On sharp contrast, the DOX concentration in cells decreased much slower in the case of LNSD group, and after incubation for 12 h, the DOX concentration was still much higher (~ 5.9 μg, Fig. 3c). The results showed the released DOX in cytoplasm could be discharged from the resistant cancer cells, while when DOX was encapsulated in nanoparticles (TD), the efflux effect of the resistant cancer cells was much weaker. More importantly, not like DOX and DOX@liposome groups, the co-localization ratios all showed significant decrease with the passage of incubation time, the co-localization ratio of LNSD showed a significant increase (from ~ 36.9% to ~ 66.4%, Fig. 3d). The above results showed not only the anti-efflux effect, but also the nuclear transport ability of the prepared LNSD.
The in vitro antitumor effect was highly correlated with the concentration of DOX in MCF-7/ADR cells, and the results were shown in Fig. 3e and f. As expect, the inhibition of LNSD was much higher than that of DOX or DOX@liposome after incubation for 24 or 48 h. When the cells were treated by LNSD (DOX concentration: 16 μg/mL) for 48 h, the inhibition was calculated to 71.5%, much higher than that of DOX (29.8%). More importantly, the inhibition of LNSD was also much higher than that of DOX@liposome (42.8%), indicating that the higher concentration of DOX in LNSD treated cells than that of DOX@liposome group. The toxicity of the blank carrier (TD@liposome) to MCF-7/ADR cells was also investigated, and the results were shown in Additional file 6, indicating that the blank carrier had a low toxicity to MCF-7/ADR cells.
In vivo studies of LNSD
The pharmacokinetics of DOX, TD/DOX, DOX@liposome and LNSD were shown in Fig. 4a, showing the decrease of DOX in LNSD group was slower than that of DOX, TD/DOX or DOX@liposome after administration. The circulation half-life of LNSD was 3.059 h, much longer than that of DOX (1.667 h), TD/DOX (1.342 h) or DOX@liposome (2.176 h) (Fig. 4b). The area under the curve (AUC) of LNSD (42.267 μg/mL*h) was about twenty times greater than that of DOX (2.215 μg/mL*h) and about eight times greater than that of TD/DOX (5.499 μg/mL*h). The AUC of LNSD was also greater than that of DOX@liposome (42.267 vs 32.044 μg/mL*h, Fig. 4c). The results of pharmacokinetics indicated that compared with the other groups, LNSD significantly increased the blood circulation time and the bioavailability of DOX.
Pharmacokinetics and biodistribution. a Mean concentration of DOX in plasma after intravenous administration of DOX, TD/DOX, DOX@liposome and LNSD (n = 3, with the same DOX dosage of 5 mg/kg). b Circulation half-lives of DOX, TD/DOX, DOX@liposome and LNSD obtained by fitting the circulation profile data (n = 3). c AUC of DOX, TD/DOX, DOX@liposome and LNSD after injection (n = 3). d To determine the DOX leakage of DOX@liposome and LNSD, the drug-containing NPs were incubated with plasma for 4 h and the level of DOX was determined in the plasma (n = 3). e Biodistribution of DOX, TD/DOX, DOX@liposome and LNSD over a span of 24 h after injection (n = 3). f Tumor-targeting efficacy of DOX, TD/DOX, DOX@liposome and LNSD over a span of 24 h after injection (n = 3). g In vivo optical images of tumor-bearing mice at different time after injection with IR783 loaded LNSD. Data presented are means ± SD. ***p < 0.01
The leakage of DOX from the NPs could influence the pharmacokinetic behavior in vivo, therefore, the leakages of DOX from DOX@liposome and LNSD in plasma were investigated (Fig. 4d). After incubation with plasma for 4 h, in the case of DOX@liposome group, the relative DOX in plasma was 16.7%, however, the DOX leakage in LNSD group significantly decreased (7.7%), indicating that LNSD with the nanostructured DOX strategy could significantly increase the blood circulation time of DOX via reducing the leakage of DOX in vivo.
Next, we investigated biodistribution of DOX in various organs (heart, liver, spleen, lung, kidney and tumor), and there were significant differences for the biodistribution of DOX in DOX, TD/DOX, DOX@liposome and LNSD treated tumor-bearing mice (Fig. 4e). After injection for 24 h, the level DOX in LNSD treated group was 5.433, about 5.4, 4.5 and 1.2-times higher than that of DOX (1.011), TD/DOX (1.220) and DOX@liposome (4.423). The tumor-targeting efficacy (TTE) of LNSD, DOX, TD/DOX or DOX@liposome was 16.3%, 8.02%, 8.32% or 14.52%, respectively (Fig. 4f), indicating that LNSD had the best tumor-targeting ability. Besides that, the tumor-targeting ability of LNSD was also confirmed by a real-time imaging system (Fig. 4g). According to the results, after injection of free IR783 and LNSD-IR783, whole body distribution were observed in tumor-bearing mice, a significant tumor site accumulation were observed in the case of LNSD-IR783, and the FLR signal increased with the extend of time. At the same time, no obvious fluorescence in the tumor site was observed after free IR783 treated group, because the unstable IR783 could be rapidly excreted from body. It is worth noting that the fluorescence intensity of LNSD in liver and spleen are also high, which may attribute to higher blood perfusion in these organs.
The MCF-7/ADR tumor-bearing mice were receiving different treatments for 15 days, and the results were monitored in terms of tumor volume change (Fig. 5a). In the case of control and TD@liposome (blank drug carrier) group, the MCF-7/ADR tumor-bearing mice were closely monitored for the continuous growth of tumor, which grew ~ 6 times larger on day 15 than it was initially, indicating that the blank drug carrier (TD@liposome) did not show any significant influence to the tumor growth. No significant difference of the relative tumor volume was shown between control group and DOX treated group, indicating the drug resistance property of MCF-7/ADR. Compared with DOX group (~ 5.40), the relative tumor volume of DOX@liposome significantly decreased to ~ 3.87, indicating the DOX delivery ability of the as-prepared liposome. More important, compared with DOX group (~ 5.40) and DOX@liposome group (~ 3.87), the relative tumor volume in LNSD showed a significantly decrease (2.04), demonstrating the tumor-targeting DOX delivery and reversing resistance abilities of LNSD. The tumor weights of different groups were shown in Fig. 5c, and the results were consistent with the relative tumor volume.
Antitumor activities of LNSD in MCF-7/ADR tumor-bearing mice. a Tumor volume changes of mice (n = 6) injected with LNSD or saline. b Body weight monitoring of the mice received different treatments. c Tumor weights of mice received the treatments measured on day 14. d Representative H&E staining images of tumor tissue received different treatments, scale bar: 100 μm. e Representative TUNEL staining images of tumor tissue received different treatments, scale bar: 100 μm. Data presented are means ± SD. *p < 0.05, ***p < 0.01
The weight loss of the tumor-bearing mice during different treatments was shown in Fig. 5b, according to the results, in the cases of control group and the blank drug carrier (TD@liposome) group, the tumor-bearing mice both gained the weight. In the cases of DOX@liposome group and LNSDs group, no significant weight loss was observed compared to the control group. While, in the case of DOX group, after treatment for 15 days, the weight of tumor-bearing mice significantly decrease, indicating the potential toxicity of DOX.
The therapeutic efficacy was also evaluated by the histological tissue images through H&E and TUNEL staining. As shown in Fig. 5d, severe tumor tissue damage was clearly observed in the LNSD-treated tumor compared with tumors in other groups, especially in DOX@liposome group. The results of apoptosis in different treatments were also shown in Fig. 5e, according to the results, a large number of apoptotic cells (dyed brown) were clearly observed in the case of LNSD-treated tumor, and the apoptotic cells were much more than that of saline, blank carrier, DOX and DOX@liposome-treated tumors, indicating that the therapeutic efficacy of LNSD.
The cell nuclear-targeting ability of LNSD in tumor-bearing nude mice was investigated after injection of DOX, DOX@liposome and LNSD for 24 h. As shown in Fig. 6a, in the case of DOX group, only little DOX signal (red fluorescence) was observed, indicating the poor tumor-targeting of DOX. Compared with DOX-treated group, the DOX signals of DOX@liposome and LNSD showed significantly increase, demonstrating the tumor-targeting abilities of liposome-based drug delivery. More importantly, in the cases of DOX and DOX@liposome groups, the DOX signals were mainly in cytoplasm (Fig. 6a), and the overlap degree of DOX and the cell nuclear (DAPI, blue fluorescence) were ~ 14.3% and ~ 37.2%, respectively (Fig. 6b). However, in the case of LNSD group, the co-localization ratio of DOX and DAPI was increased to ~ 74.1% (Fig. 6b), indicating that most of DOX were in the cell nuclear, further demonstrating the cell nuclear-targeting ability of TD/DOX, and the results were consistent with the in vitro studies.
In vivo anti-efflux, cell nuclear distribution and biosafety studies. a CLSM images of tumor tissues harvested from DOX, DOX@liposome or LNSD treated MCF-7/ADR tumor-bearing mice at 24 h post injection (red FLR: DOX; blue FLR: DAPI), scale bar: 100 μm. b Co-localization ratio of DOX (red FLR) and cell nuclear (blue FLR) in tumor tissues harvested from DOX, DOX@liposome or LNSD treated MCF-7/ADR tumor-bearing mice (n = 20). c Representative H&E staining images of liver tissue received saline or LNSD treatments, scale bar: 200 μm. d ALT, AST, BUN and CR in serum were detected in different treatments on day 14 (n = 5). Data presented are means ± SD. *p < 0.05***p < 0.01
In vivo toxicity is a great concern in the development of nanomedicine, and according to the results of biodistribution, the nanomedicine were mainly in liver after injection, therefore, the toxicity of LNSD in liver was evaluated by H&E staining (Fig. 6c), and no significant toxicity was observed via the pathological section. Furthermore, alanine aminotransferase (ALT), aspartate aminotransferase (AST), blood urea nitrogen (BUN), and creatinine (CR) which are highly reflected the physiological state of liver and kidney were also investigated, and no significant difference between LNSD-treated mice and the saline-treated mice (Fig. 6d).
The unexpected release of drug during circulation always lead to the unexpected side effects and low therapeutic efficiency [35]. So an ideal drug delivery system should avoid the drug leaks as possible. In this study, DOX was not directly loaded into a liposome, instead, DOX was firstly loaded into a tetrahedral DNA nanostructure (TD) to obtained a nanostructured DOX, and then the DOX-loaded TD (nanostructured DOX) was enveloped via liposome. The DOX release from LNSD was much lower than that of DOX@liposome (DOX was directly loaded into the liposome) after 24 h, (~ 9.4% vs ~ 36.7%). This is beneficial to the pharmacokinetics and biodistribution of DOX, therefore, reduced the side effects and enhanced the delivery efficacy of DOX [36].
The emergence of drug resistance is a main obstacle for successful treatment of cancer [37]. Generally, typical drug resistance is mediated by the overexpression of a membrane transporter, P-glycoprotein (P-gp), actively increases the efflux of drugs from cancer cells [38]. In this study, DOX was loaded into TD to obtain a (TD/DOX), when LNSD entered into the MCF/ADR cells via endocytosis, the nanostructured TD/DOX was released into the cell cytoplasm, unlike free DOX, which is efflux via P-gp rapidly, the nanostructured DOX could not be efflux from MCF/ADR cells even after 12 h of incubation. The nanostructured TD/DOX significantly increased the intracellular accumulation of DOX in MCF-7/ADR cells and maintained a relative high DOX concentration in an optimum level.
Furthermore, the size of the nanostructured TD/DOX was ~ 15 nm, and the diameter of the nuclear pore is ~ 30 nm [39], so due to the small size of the nanostructured DOX complex, the DOX complex could enter into the cell nuclear via nuclear pore, making the nanostructured DOX have the cell nuclear-targeting ability, and this was also proved both in the in vitro and in vivo studies. The therapeutic target of DOX is the tumor cell nuclear, the cell nuclear-targeting ability is very important for DOX delivery. In general, LNSD showed not only the anti-efflux effect, but also the nuclear transport ability, and both the abilities are crucial for reversal of tumor resistance to DOX. This reversal of tumor resistance of LNSD effect was substantiated by: 1) significantly increase the DOX concentration in MCF-7/ADR cells in vitro and increase the DOX levels in the MCF-7/ADR tumor-bearing mice when compared to DOX and DOX@liposome (DOX was directly loaded into the liposome); 2) significantly improved the DOX distribution in the nuclear of MCF-7/ADR cells in vitro and in vivo compared to DOX and DOX@liposome; 3) significantly enhanced the antitumor efficiency in MCF-7/ADR cells in vitro and in MCF-7/ADR tumor-bearing mice in vivo compared to DOX and DOX@liposome.
In total, a liposome-based nanostructured DOX (LNSD) was rational designed and investigated. In the in vitro and in vivo studies, LNSD could efficiently enter into the MCF-7/ADR cells and accumulated in the tumor tissue in MCF-7/ADR tumor-bearing mice. Furthermore, the DOX resistance of MCF-7/ADR cells could be reversed via the anti-efflux effect and the size-depended cell nuclear-targeting ability of the nanostructured DOX in LNSDs. Therefore, LNSD significantly improved the antitumor efficacy of DOX and the liposome-based DOX in drug-resistant breast tumor-bearing mice.
Doxorubicin (DOX, 20,170,511, purity >98%) was gotten from Beijing Yi-He Biotech Co. Ltd. All single-stranded DNAs were purchased from Takara Bio Co. (Dalian, China), and the specific sequences were displayed in Table 1. Phospholipids, 1,2-dioleoy-sn-glycero-3-phosphoethanolamine (DOPE), 1,2-dipalmitoyl-sn-glycero-3-phosphocholine (DPPC), 1,2-distearoyl-sn-glycero-3-phosphocholine (DSPC) and cholesteryl hemisuccinate (CHEMS) were purchased from Avanti Polar Lipids (Alabama, USA), and N-(carbonyl-methoxy-polyethylene-glycol-2000)-1,2-distearoyl-sn-glycero-3-phosphoethanolame (DSPE-mPEG2000) from Lipoid (Steinhausen, Switzerland). Cholesterol and calcein were obtained from Sigma-Aldrich Ltd (Auckland, New Zealand). PCR primer, loading Buffer, and Golden View was obtained from Beijing Ding Guo Chang Sheng Biotechnology Co. Ltd. Gel Extraction Kit, 2× Taq Master Mix, and 50× TAE was obtained from Beijing Com Win Biotech Co. Ltd. Sulforhodamine B (SRB), DMEM cell culture medium, penicillin, streptomycin, fetal bovine serum (FBS), and heparin sodium were bought from Gibco Invitrogen. DAPI, hematoxylin and eosin were supplied by Beyotime Biotechnology Co. Ltd. Other reagents were acquired from China National Medicine Corporation Ltd.
Synthesis of LNSD
Synthesis of tetrahedral DNA nanostructure (TD)
TDs were synthesized as previously reported [33]. In brief, the single-stranded DNA S1, S2, S3 and S4 were mixed in buffer (10 mM Tris–HCl, 50 mM MgCl2.6H2O, pH 8.0) at the same concentrations. The mixture solution was next quickly heated to 95 °C for 5 min, and then the mixture solution was cooled down to 4 °C for 0.5 min. The FAM marked TDs (TD-FAM) were prepared under the same conditions using FAM-S1, S2, S3 and S4.
Preparation of DOX-Loaded TD (TD/DOX)
DOX (500 μM) was added to the prepared TD solution (1 mM), and then the mixture was incubated for 12 h at room temperature with shaking (100 rpm) to make sure DOX was sufficiently intercalated into TD, after that, the mixture was centrifuged (10,000 rpm, 10 min) to remove unloaded DOX, and DOX-loaded TD (TD/DOX) was obtained.
pH-sensitive liposome wrapping
The blank pH-sensitive liposome (liposome), DOX loaded pH-sensitive liposome (DOX@liposome) and LNSD were prepared using a thin-film hydration method. In detail, for the blank pH-sensitive liposome, DOPE, DSPC, CHEMS, cholesterol and DSPE-PEG2000 (molar ratio 4:2:2:2:0.3) were dissolved in chloroform and methanol mixture (3:1 v/v, 2 mL) and dried using a rotary evaporator under vacuum. After that, 1.5 mL of water was added, and the mixture (emulsion) was sonicated at room temperature for 30 min. For DOX@liposome and LNSD, with the same steps of dying, after that, 1.5 mL of DOX solution (500 μM) and 1.5 mL of TD/DOX solution was added, and the mixture (emulsion) was sonicated at room temperature for 30 min. The liposome suspension was then extruded through 0.2 μm pore sized polycarbonate membrane filters (Whatman, UK) using a 10 mL LIPEXTM Extruder (Northern Lipids Inc., Burnaby, Canada).
The loaded DOX in DOX@liposome or LNSD was determined by lipid emulsification method. In brief, 2 mL of methanol was added to 0.2 mL of DOX@liposome or LNSD, after sonicated at room temperature for 1 h, the samples were centrifuged (10,000 rpm, 10 min), and then the amount of the loaded DOX was determined by high performance liquid chromatography (HPLC, 1100 Agilent, USA) with the following conditions: an Eclipse XDB-C18 column (150 mm × 4.6 mm, 5.0 μm); mobile phase sodium acetate solution (0.02 mol/L)/acetonitrile 80: 20; column temperature 40 °C; fluorescence detector with the excitation and emission wavelengths set at 475 nm and 560 nm, respectively; flow rate 1.0 mL/min; and injection volume 20 μL. The DOX encapsulation efficiency (EE) was calculated using the following formula:
$$EE=\frac{\mathrm{W}}{\mathrm{W}_{0}}\times 100\%$$
W was the weight of DOX loaded into liposomes. W0 was the weight of DOX@liposome or LNSD.
DLS (Zetasizer Nano ZS-90, Malvern, UK), TEM (Tecnai G2 20, FEI) and AFM (SPM-9700, Shimadzu, Japan) were used for characterizing zeta potential, particle size and morphology of LNSD and TD, respectively. Polyacrylamide gel electrophoresis (PAGE, 8%) was used to verify TD and TD/DOX.
DOX release profile
DOX@liposome (1 mL) and LNSD (1 mL) with the same DOX concentration (1 mg/mL) were sealed in the dialysis membranes (MW cutoff 12–14 KD, Spectrapor). The dialysis bags were incubated in 10 mL of PBS buffer (pH 7.4) and acetate buffer (pH 5.0), respectively. The released DOX was quantified by HPLC.
In vitro studies using MCF-7/ADR cells
MCF-7/ADR cells (human breast cancer cell line, P-gp highly expression) were cultured in normal DMEM culture medium with 10% fetal bovine serum (FBS) and 1% penicillin/streptomycin in 5% CO2 and 95% air at 37 °C in a humidified incubator.
Cellular distributions of TD-FAM@liposome
MCF-7/ADR cells were seeded at 5 × 104 cells per well in 6-well plates. When cells reached 70% confluence, they were treated with TD-FAM@liposome in the dark for 4 and 12 h. After staining with DAPI, the cells were washed with PBS for 3 times and imaged by a Confocal Microscopy (Zeiss, LSM 700, Germany).
Intracellular stability of TD/DOX
MCF-7/ADR cells were seeded at 5 × 104 cells per well in 6-well plates. When cells reached 70% confluence, they were treated with TD-FAM/DOX@liposome in the dark for 4 h. After 4 h of incubation, the cells were washed with PBS for 3 times and imaged by a Confocal Microscopy (Zeiss, LSM 700, Germany).
Cellular internalization studies
MCF-7/ADR cells were seeded at 5 × 104 cells per well in 6-well plates. When cells reached 70% confluence, they were treated with DOX, DOX@liposome and LNSD (with the same DOX concentration: 10 μg/mL) in the dark for 4 h. After staining with DAPI, the cells were washed with PBS for 3 times and imaged by a Confocal Microscopy. After imaging, the cells were incubated for another 8 h (the total incubation time was 12 h), and then the cells were imaged by a Confocal Microscopy again.
In vitro antitumor studies
MCF-7/ADR cells were plated in 96-well plates and then incubated for 24 h. After incubation, the cell medium was replaced with fresh culture medium containing DOX, DOX@liposome and LSND (with the same concentration of DOX) for 4 h, respectively, and then the medium was replaced with the fresh medium. After incubation for another 20 or 44 h (the total incubation time was 24 h or 48 h), standard SRB was carried out to determine the cell viabilities.
Detection of DOX in MCF-7/ADR cells
MCF-7/ADR cells were treated with DOX, DOX@liposome and LNSD (with the same DOX concentration 20 μg/mL). At the designated time points (2, 4, 6, 8, 10 and 12 h after incubation), the cells were washed thoroughly with PBS, followed by trypsinization and centrifugation at 3000 rpm for 5 min to harvest the cells as pellets. The cells were re-suspended in water, ultrasonicated and extracted by 1 mL of methanol. Finally, the amount of DOX internalized by MCF-7/ADR cells was measured by HPLC.
In vivo studies
Xenograft tumor mouse model
All animal experiments were performed under a protocol approved by Henan laboratory animal center. The MCF-7/ADR tumor models were generated by subcutaneous injection of 2 × 106 MCF-7/ADR cells in 0.1 mL saline into the right shoulder of nude mice (20–23 g, Shanghai Institutes for Biological Sciences, CAS). The mice were used when the tumor volume reached 60 to 100 mm3 (~ 10 days after tumor inoculation).
Pharmacokinetics and biodistribution
For pharmacokinetics: 0.2 mL of DOX (5 mg/kg), TD/DOX (DOX dosage: 5 mg/kg), DOX@liposome (DOX dosage: 5 mg/kg; liposome dosage: ~ 25 mg/kg) and LNSD (DOX dosage: 5 mg/kg; TD@liposome dosage: ~ 12 mg/kg) were intravenously injected into the tumor-free mice (3 mice per group). After injection for 0.25, 0.5, 1, 2, 4, 8, 12, and 24 h, 0.5 mL of blood was drawn from eyes of the tumor-free mice, centrifuged and then homogenized in methanol. DOX in blood samples were detected by HPLC.
For biodistribution: 0.2 mL of DOX (5 mg/kg), TD/DOX (DOX dosage: 5 mg/kg), DOX@liposome (DOX dosage: 5 mg/kg; liposome dosage: ~ 25 mg/kg) and LNSD (DOX dosage: 5 mg/kg; TD@liposome dosage: ~ 12 mg/kg) were intravenously injected into the tumor-bearing mice (3 mice per group). After injection for 24 h, the mice were killed to collect heart, liver, spleen, lung, kidney and tumor, weighed, and homogenized in buffer (methanol to saline ratio, 1:1). DOX in different tissues were determined by HPLC. Furthermore, a near-infrared dye (IR783, water-soluble) was used to mark LSND. A sample of 0.2 mL of IR783@liposome was intravenously injected into tumor-bearing mice, and the whole body fluorescence imaging was performed at 0, 0.5, 1, 6, 8, 10, 12 and 24 h after injection using a small animal imaging system (Xtreme, Bruke).
In vivo antitumor effect
MCF-7/ADR tumor-bearing mice were divided into 5 groups (six mice per group), minimizing the differences of weights and tumor sizes in each group. The mice were administered with (1) Saline (0.2 mL), (2) TD@liposome (0.2 mL, ~ 12 mg/kg), (3) DOX (5 mg/kg, 0.2 mL), (4) DOX@liposome (DOX: 5 mg/kg; liposome: ~ 25 mg/kg), (5) LNSD (DOX: 5 mg/kg; TD@liposome: ~ 12 mg/kg, 0.2 mL) were intravenously injected into mice via the tail vein every 2 days, respectively. The mice were observed daily for clinical symptoms and the tumor sizes were measured by a caliper every other day and calculated as the volume = (tumor length) × (tumor width)2/2. After treatment for 15 days, the mice were sacrificed to collect tumor for H&E and TUNEL staining.
Quantitative data are expressed as mean ± SD and analyzed by use of Student's t test. P values < 0.05 were considered statistically significant.
All data generated or analyzed during this study are included in this published article.
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Thanks to the modern analysis and computing center of Zhengzhou University for technical assistance.
The work is supported by grants from the National Natural Science Foundation of China (Nos. 81601597, U1704178 and 31900991), Innovation Talent Support Program of Henan Province (No. 19HASTIT006), Key Scientific Research Projects (Education Department of Henan Province, No. 17A350003), Postdoctoral Science Foundation of China (Nos. 2015M582210, 2018T110745 and 2017M622380) and Zhengzhou University Initiative Scientific Research Program (No. 32210809).
The Fifth Affiliated Hospital of Zhengzhou University, Kangfu Road, Zhengzhou, 450052, China
& Yongxia Zhai
School of Pharmaceutical Sciences, Zhengzhou University, Zhengzhou, China
Wei Liu
, Kaixiang Zhang
, Junjie Liu
, Jinjin Shi
& Zhenzhong Zhang
Collaborative Innovation Center of New Drug Research and Safety Evaluation, Zhengzhou, Henan, China
Kaixiang Zhang
Key Laboratory of Targeting Therapy and Diagnosis for Critical Diseases, Zhengzhou, Henan, China
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YL and JS performed the experiments, and with JL, KZ designed the experiments and drafted the manuscript. JL and ZZ contributed to the data analysis, manuscript preparation, and manuscript review and revision process. YZ and WL participated in the animal experiments. All authors read and approved the final manuscript.
Correspondence to Kaixiang Zhang or Junjie Liu or Jinjin Shi.
The study was approved by Ethics Committee of Zhengzhou University.
All the authors have approved the manuscript and agree with submission to your esteemed journal.
Additional file 1. Size distributions of nanopreparations.
Additional file 2. Encapsulation efficiency of DOX in nanopreparations.
Additional file 3. Photos of DOX@liposome and LNSD in PBS.
Additional file 4. Cellular distribution of FAM-TD@liposome.
Additional file 5. Co-localization ratio of DOX and TD in cells.
Additional file 6. Cell viability of MCF-7/ADR cells treated with different concentrations of blank carrier.
Li, Y., Zhai, Y., Liu, W. et al. Ultrasmall nanostructured drug based pH-sensitive liposome for effective treatment of drug-resistant tumor. J Nanobiotechnol 17, 117 (2019). https://doi.org/10.1186/s12951-019-0550-7
Reversing drug resistance
Drug efflux
Nuclear-transport
Nanostructured drug
Programmatic release
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CommonCrawl
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\begin{document}
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\def\tilde} \def\Ric{\text{\rm{Ric}}{\tilde} \def\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}{\text{\rm{Ric}}} \def\text{\rm{cut}}} \def\P{\mathbb P} \def\ifn{I_n(f^{\bigotimes n}){\text{\rm{cut}}} \def\P{\mathbb P} \def\ifn{I_n(f^{\bigotimes n})} \def\scr C} \def\aaa{\mathbf{r}} \def\r{r{\scr C} \def\aaa{\mathbf{r}} \def\r{r} \def\text{\rm{gap}}} \def\prr{\pi_{{\bf m},\varrho}} \def\r{\mathbf r{\text{\rm{gap}}} \def\prr{\pi_{{\bf m},\varrho}} \def\r{\mathbf r} \def\mathbb Z} \def\vrr{\varrho} \def\ll{\lambda{\mathbb Z} \def\vrr{\varrho} \def\ll{\lambda} \def\scr L}\def\Tt{\tt} \def\TT{\tt}\def\II{\mathbb I{\scr L}\def\Tt{\tilde} \def\Ric{\text{\rm{Ric}}} \def\TT{\tilde} \def\Ric{\text{\rm{Ric}}}\def\II{\mathbb I} \def{\rm in}}\def\Sect{{\rm Sect}} \def\H{\mathbb H{{\rm in}}\def\Sect{{\rm Sect}} \def\H{\mathbb H} \def\scr M}\def\Q{\mathbb Q} \def\texto{\text{o}} \def\LL{\Lambda{\scr M}\def\Q{\mathbb Q} \def\texto{\text{o}} \def\LL{\Lambda} \def{\rm Rank}} \def\B{\scr B} \def\i{{\rm i}} \def\HR{\hat{\R}^d{{\rm Rank}} \def\B{\scr B} \def{\rm in}}\def\Sect{{\rm Sect}} \def\H{\mathbb H{{\rm i}} \def\HR{\hat{\mathbb R} \def\ff{\frac} \def\ss{\sqrt}^d} \def\rightarrow}\def\l{\ell}\def\iint{\int{\rightarrow}\def\l{\ell}\def\iint{\int} \def\scr E}\def\Cut{{\rm Cut}{\scr E}\def\Cut{{\rm Cut}} \def\scr A} \def\Lip{{\rm Lip}{\scr A} \def\Lip{{\rm Lip}} \def\scr B}\def\Ent{{\rm Ent}{\scr B}\def\Ent{{\rm Ent}}
\title{{f Exponential Contraction in Wasserstein Distances for Diffusion Semigroups with Negative Curvature}
\begin{abstract} Let $P_t$ be the (Neumann) diffusion semigroup $P_t$ generated by a weighted Laplacian on a complete connected Riemannian manifold $M$ without boundary or with a convex boundary. It is well known that the Bakry-Emery curvature is bounded below by a positive constant $\ll>0$ if and only if $$W_p(\mu_1P_t, \mu_2P_t)\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} W_p (\mu_1,\mu_2),\ \ t\ge 0, p\ge 1 $$ holds for all probability measures $\mu_1$ and $\mu_2$ on $M$, where $W_p$ is the $L^p$ Wasserstein distance induced by the Riemannian distance. In this paper, we prove the exponential contraction $$W_p(\mu_1P_t, \mu_2P_t)\le c\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} W_p (\mu_1,\mu_2),\ \ p\ge 1, t\ge 0$$ for some constants $c,\ll>0$ for a class of diffusion semigroups with negative curvature where the constant $c$ is essentially larger than $1$. Similar results are derived for SDEs with multiplicative noise by using explicit conditions on the coefficients, which are new even for SDEs with additive noise. \end{abstract} \noindent
AMS subject Classification:\ 60J75, 47G20, 60G52. \\ \noindent
Keywords: Wasserstein distance, diffusion semigroup, Riemannian manifold, curvature condition, SDEs with multiplicative noise.
\vskip 2cm
\section{Introduction}
Let $M$ be a $d$-dimensional connected complete Riemannian manifold possibly with a convex boundary $\pp M$. Let $\rr$ be the Riemannian distance. Consider $L=\DD+Z$ for the Laplace-Beltrami operator $\DD$ and some $C^1$-vector field $Z$ such that the (reflecting) diffusion process generated by $L$ is non-explosive. Then the associated Markov semigroup $P_t$ is the (Neumann if $\pp M\ne\emptyset$) semigroup generated by $L$ on $M$. In particular, it is the case when the curvature of $L$ is bounded below; that is, \beq\label{C}\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z:= \text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}} -\nabla} \def\pp{\partial} \def\E{\mathbb E Z\ge K\end{equation} holds for some constant $K\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt$. Here and throughout the paper, we write $\scr T\ge h$ for a (not necessarily symmetric) $2$-tensor $\scr T$ and a function $h$ provided
$$\scr T (X, X) \ge h(x)|X|^2,\ \ X\in T_x M, x\in M.$$ There exist many inequalities on $P_t$ which are equivalent to the curvature condition \eqref{C}, see \cite{BGL, O, RS, WBook} for details. In particular, for any constant $K\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt$, the Wasserstein distance inequality \beq\label{W} W_p ( \mu_1P_t, \mu_2P_t)\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-Kt} W_p (\mu_1,\mu_2),\ \ t\ge 0, p\ge 1, \mu_1,\mu_2\in \scr P(M)\end{equation} is equivalent to the curvature condition \eqref{C}.
Here, $\scr P(M)$ is the class of all probability measures on $M$;
$W_p$ is the $L^p$-Warsserstein distance induced by $\rr$, i.e.,
$$W_p(\mu_1,\mu_2) :=\inf_{\pi\in \scr C(\mu_1,\mu_2)} \|\rr\|_{L^p(\pi)}, \ \ \mu_1,\mu_2\in \scr P(M), $$ where $\scr C (\mu_1,\mu_2)$ is the class of all couplings of $\mu_1$ and $\mu_2$; and for a Markov operator
$P$ on $\B_b(M)$ (i.e. $P$ is a positivity-preserving linear operator with $P1=1$), $$(\nu P)(A) := \nu(P 1_A),\ \ A\in \B(M), \nu\in \scr P(M),$$ where $\nu(f):=\int_M f\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\nu$ for $f\in L^1(\nu)$. In some references, $\nu P$ is also denoted by $P^*\nu$. In the sequel we will use $P_t^*$ to stand for the adjoint operator of $P_t$ in $L^2(\mu)$ for the invariant probability measure $\mu$, hence adopt the notation $\nu P$ rather than $P^*\nu$ to avoid confusion. When the curvature is positive (i.e. $K>0$), \eqref{W} implies the $W_p$-exponential contraction of $P_t$ for $p\ge 1.$
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In this paper, we aim to consider the case when \eqref{C} only holds for some negative constant $K,$ and to prove the exponential contraction \beq\label{W'}W_p (\mu_1P_t, \mu_2P_t)\le c\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} W_p (\mu_1,\mu_2),\ \ t\ge 0, p\ge 1, \mu_1,\mu_2\in \scr P(M)\end{equation} for some constants $c,\ll>0$. It is crucial that the exponential rate $\ll$ is independent of $p$. Due to the equivalence of \eqref{C} and \eqref{W}, in the negative curvature case it is essential that $c>1$.
According to \cite{WAnn}, even when $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ is unbounded below, i.e. $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ goes to $-\infty$ when $\rr_o:=\rr(o,\cdot)\rightarrow}\def\l{\ell}\def\iint{\int\infty$ for a fixed $o\in M$, there may exist the log-Sobolev inequality which implies the exponentially convergence of $P_t$ in entropy. This suggests that \eqref{W'} may also hold for a class of diffusion semigroups with negative curvature.
Recently, some efforts have been made in this direction for $M=\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$, see \cite{EB, EB2, LW}. More precisely, let $P_t$ be the diffusion semigroup for the solution to the following SDE on $\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$: $$\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D X_t = \ss 2\, \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t +b(X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t, $$where $B_t$ is the $d$-dimensional Brownian motion and $b: \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ is continuous. If there exist constants $K_1,K_2, r_0>0$ such that
\beq\label{EB} \<b(x)-b(y), x-y\>\le 1_{|x-y|\le r_0} (K_1+K_2) |x-y|^2 - K_2 |x-y|^2,\ \ x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d,\end{equation} then due to \cite{EB,EB2} we have
\beq\label{Eb2} W_1(\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t)\le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} |x-y|,\ \ x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d, t\ge 0\end{equation} for some constants $c,\ll>0$, where $\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x$ is the Dirac measure at point $x$. Indeed, \cite{EB,EB2} proved the $W_1$-exponential contraction with respect to a modified distance
$f(|x-y|)$ in place of $|x-y|$ as constructed in \cite{CW, CW2} for estimates of the spectral gap using the coupling by reflection. Under condition \eqref{EB} the modified distance is comparable with the usual one so that \eqref{Eb2} follows. As mentioned in \cite{EB2} that there is essential difficulty to prove \eqref{W'} for $p>1$ even for this flat case.
In Luo and Wang \cite{LW} the estimate \eqref{Eb2} was extended as
\beq\label{Eb3} W_p( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t)\le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t/p} (|x-y|+ |x-y|^{\ff 1 p}),\ \ x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d, t\ge 0, p\ge 1 \end{equation} for some constants $c,\ll>0$. Comparing with \eqref{W'} which is equivalent to
$$W_p(\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_y P_t)\le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} |x-y|,\ \ p\ge 1, x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d, t\ge 0$$ according to \cite{KK2} (see Proposition \ref{T3.1} below),
\eqref{Eb3} is less sharp for small $|x-y|$ and/or large $p$. It is open whether \eqref{EB}, or in the Riemannian setting that $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ is uniformly positive outside a compact domain, implies \eqref{W'} for some constants $c,\ll>0$.
\
As in \cite{KK, KK2}, we will consider the Warsserstein distances induced by Young functions in the class \begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} \scr N:= \Big\{\Phi\in C^1([0,\infty); [0,\infty)): &\ \Phi' \ \text{is\ nonnegative\ and \ increasing},\\
\Phi(0)=0, &\Phi(r)>0\ \text{for}\ r>0, \lim_{r\rightarrow}\def\l{\ell}\def\iint{\int\infty} \ff{\Phi(r)}r=\infty\Big\}.\end{align*} For any $\Phi\in \scr N$ and a measure $\nu$ on $M$, consider the gauge norm in $L^\Phi(\nu):$ $$\|f\|_{L^\Phi(\nu)}:= \inf\Big\{r>0: \nu\big(\Phi(r^{-1}|f|)\big)\le 1\Big\},\ \ \inf\emptyset:=\infty.$$
In particular, we have $\|f\|_{L^{\Phi_p} (\nu)}=\|f\|_{L^p(\nu)}$ for $\Phi_p(r):= r^p,\ p\in (1,\infty)$. This is the reason why we do not take $\Phi_p(r) =\ff 1 p r^p$ in the characterization of Legendre conjugates. We extend the notion $\Phi_p$ to $p= 1,\infty$ by letting $\Phi_1(r)=r, \Phi_\infty=\lim_{p\rightarrow}\def\l{\ell}\def\iint{\int\infty}\Phi_p$ and
$\|f\|_{L^{\Phi_p}(\nu)}=\|f\|_{L^p(\nu)}$ for all $p\in [1,\infty].$ Now, let $$W_\Phi(\mu_1,\mu_2)= \inf_{\pi\in \scr C(\mu_1,\mu_2)} \|\rr\|_{L^\Phi(\pi)},\ \
\Phi\in \bar{\scr N}:= \scr N\cup\{\Phi_1,\Phi_\infty\}.$$ In particular, $W_{\Phi_p}=W_p$ for $p\in [1,\infty].$ We aim to prove the exponential decay \beq\label{LL}W_\Phi( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t) \le c \Phi^{-1}(1)\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \rr(x,y),\ \ \ x,y\in M, t\ge 0, \Phi\in \bar{\scr N}\end{equation} when \eqref{C} only holds for a negative constant $K$, where $\Phi^{-1}$ is the inverse of $\Phi (\ne \Phi_\infty)$ and we set $\Phi^{-1}_\infty(1)=1$ by convention.
To extend condition \eqref{EB} to the Riemannian setting, consider the index
$$I(x,y)= \int_0^{\rr(x,y)} \sum_{i=1}^{d-1}\Big\{|\nabla} \def\pp{\partial} \def\E{\mathbb E_{\dot\gg}J_i|^2 -\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\scr R(\dot\gg, J_i) \dot\gg,J_i\>\Big\}
(\gg_s) \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ x,y\in M,$$ where $\rr$ is the Riemannian distance, $\scr R$ is the curvature tensor; $\gg: [0,\rr(x,y)]\rightarrow}\def\l{\ell}\def\iint{\int M$ is the minimal geodesic from $x$ to $y$ with unit speed;
$\{J_i\}_{i=1}^{d-1}$ are Jacobi fields along $\gg$ such that $$J_i(y)=P_{x, y}\,J_i(x), \quad i=1,\ldots,d-1 $$ holds for the parallel transform $P_{x,y}: T_x M\rightarrow}\def\l{\ell}\def\iint{\int T_y M$ along the geodesic $\gg$, and $\{\dot \gg(s), J_i(s): 1\le i\le d-1\}$ ($s=0,\rr(x,y)$) is an orthonormal basis of the tangent space (at points $x$ and $y$, respectively).
Note that when $(x,y)\in \Cut(M)$, i.e. $x$ is in the cut-locus of $y$, the minimal geodesic may be not unique. As a convention in the literature, all conditions on the index $I$ are given outside $\Cut(M)$. We now extend condition \eqref{EB} to the non-flat case as follows: for some constants $K_1,K_2>0$, \beq\label{EB'} \begin} \def\beq{\begin{equation}} \def\F{\scr F{split} I_Z(x,y)& := I(x,y)+ \<Z,\nabla} \def\pp{\partial} \def\E{\mathbb E \rr(\cdot, y)\>(x)+ \<Z,\nabla} \def\pp{\partial} \def\E{\mathbb E \rr(x,\cdot,)\>(y)\\
& \le \big\{(K_1+K_2)1_{\{\rr(x,y)\le r_0\}}- K_2\big\}\rr(x,y),\ \ x,y\in M.\end{split}\end{equation} In the flat case we have $I(x,y)=0$ and $\rr(x,y)=|x-y|$,
so that this condition reduces back to \eqref{EB}. Moreover, the curvature condition \eqref{C} is equivalent to
$$I_Z(x,y)\le -K\rr(x,y),\ \ x,y\in M,$$ so that \eqref{EB'} implies $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z\ge -(K_1+K_2).$
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In the next section, we state our main results and present examples. With condition \eqref{EB'} we first extend the main results of \cite{EB, LW} to the present Riemannian setting and give the exponential convergence of $P_t$ in $W_2$. Under the ultracontractivity and condition \eqref{C} for some $K<0$, our the second result ensures the desired inequality \eqref{LL}. Finally, we extend these results to SDEs with multiplicative noise by using explicit conditions on the coefficients. To prove these results, we make some preparations in Section 3. Complete proofs of the main results are addressed in Sections 4-6 respectively.
\section{Main Results and examples}
We first consider the Riemannian setting, then extend to SDEs with multiplicative noise by using explicit conditions on the coefficients instead of the less explicit curvature condition.
\subsection{The Riemannian setting}
We start with condition \eqref{EB'}. Besides the extension of \eqref{Eb3}, this condition also implies the hypercontractivity and the exponential convergence in $W_2$ for the semigroup $P_t$. For a measure $\mu$ and constants $p, q\ge 1$, let
$\|\cdot\|_{L^p(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^q(\mu)} $ stand for the operator norm form $L^p(\mu)$ to $L^q(\mu)$. Recall that $P_t$ is called hypercontractive if it has a unique invariant probability measure $\mu$ and $\|P_t\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^4(\mu)}=1$ holds for large $t>0$. By interpolation theorem, $\|P_t\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^4(\mu)}=1$ can be replaced by $\|P_t\|_{L^p(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^q(\mu)}=1$ for some $\infty >q>p>1.$
\begin} \def\beq{\begin{equation}} \def\F{\scr F{thm}\label{T1.1} Let $\eqref{EB'}$ hold for some constants $K_1, K_2$ and $ r_0> 0$. Then: \begin} \def\beq{\begin{equation}} \def\F{\scr F{enumerate} \item[$(1)$] There exist two constants $c,\ll>0$ such that for any $\Phi\in \bar{\scr N}$ and $x,y\in M$, \beq\label{EB1'} W_\Phi(\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t)\le \inf\Big\{r>0: \sup_{s\in (0, 1+ \rr(x,y)]} \ff{\Phi(r^{-1} s)}{s}\le \ff {\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{\ll t}}{c\rr(x,y)}\Big\},\ \ t\ge 0.\end{equation} In particular, $$W_p(\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_xP_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t)\le \{c\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t}\}^{\ff 1 p} (\rr(x,y)+\rr(x,y)^{\ff 1 p}),\ \ p\ge 1, t\ge 0, x,y\in M.$$ \item[$(2)$] $P_t$ has a unique invariant probability measure $\mu$ and the log-Sobolev inequality
\beq\label{LS} \mu(f^2\log f^2) \le C \mu(|\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2) +\mu(f^2)\log \mu(f^2),\ \ f\in C_b^1(M)\end{equation} holds for some constant $C>0$. Consequently, $P_t$ is hypercontractive. \item[$(3)$] There exist constants $c,\ll>0$ such that \beq\label{W2} W_2( \nu P_t, \mu)\le c\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} W_2(\nu, \mu),\ \ t\ge 0, \nu\in \scr P(M).\end{equation} \end{enumerate} \end{thm}
To illustrate this result, we present below a consequence with explicit curvature conditions in the spirit of \cite{WAnn}. These conditions allow $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ to be negative everywhere, for instance, when $-C_1\le \text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}\le -C_2$ and $C_2>-\nabla} \def\pp{\partial} \def\E{\mathbb E Z\ge \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho$ for some constants $C_1>C_2>\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho>0$.
As indicated in Introduction that \eqref{EB'} implies $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z\ge -(K_1+K_2),$ so in the following corollary we assume that $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ is bounded below.
\begin} \def\beq{\begin{equation}} \def\F{\scr F{cor}\label{C1.2} Assume that $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ is bounded below. Let $\rr_o=\rr(o,\cdot)$ for a fixed point $o\in M$. If there exist constants $\sigma} \def\ess{\text{\rm{ess}}>0$ and $\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho>\sigma} \def\ess{\text{\rm{ess}} (1+\ss 2) \ss{d-1}$ such that \beq\label{C1} -\nabla} \def\pp{\partial} \def\E{\mathbb E Z \ge -\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho \ {\rm and} \ \text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}\ge -\sigma} \def\ess{\text{\rm{ess}}^2\rr_o^2 \ {\rm outside\ a\ compact\ set},\end{equation} then all assertions in Theorem $\ref{T1.1}$ hold. \end{cor}
Next, we introduce sufficient conditions for \eqref{LL} which allow $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ to be negative. Due to technical reason, we will need the ultracontractivity of $P_t$, which is essentially stronger than the hypercontractivity. We call $P_t$ ultracontractive if $\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}<\infty$ for all $t>0.$ The ultracontractivity implies that $P_t$ has a density $p_t(x,y)$ with respect to $\mu$ (called heat kernel) and $$\|p_t\|_{L^\infty(\mu\times \mu)} = \|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}<\infty,\ \ t>0.$$ In references (see e.g. \cite{DS}), the ultracontractivity is also defined by $\|P_t\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}<\infty$ for $t>0$. When $P_t$ is symmetric in $L^2(\mu)$ we have
\beq\label{SY}\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}\le\|P_{t/2}\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}^2,\ \ t>0,\end{equation} so that these two definitions are equivalent. However, when $P_t$ is non-symmetric, the former might be stronger than the latter. The appearance of the ultracontractivity in our study is very nature: by Theorem \ref{T1.2}(1) we already have \eqref{LL} for $\Phi=\Phi_1$ (the weakest case), and by the ultracontractivity we are able to deduce the inequality from $\Phi_1$ to $\Phi_\infty$ (the strongest case). On the other hand, the result also indicates that \eqref{LL} implies the hypercontractivity of $P_t$.
\begin} \def\beq{\begin{equation}} \def\F{\scr F{thm}\label{T1.2} Assume that $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ is bounded below. \begin} \def\beq{\begin{equation}} \def\F{\scr F{enumerate} \item[$(1)$] If $P_t$ is ultracontractive, then there exist constants $c,\ll>0$ such that for any $\Phi\in \bar{\scr N} $, \beq\label{LL0} W_{\Phi}( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_xP_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t) \le \ff{c}{\Phi^{-1}(1)} \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \min\Big\{ \rr(x,y),\ G_\Phi(t) \Big\},\ \ t>0, x,y\in M \end{equation} holds for
$$G_\Phi(t):=\inf\Big\{r>0:\ (\mu\times\mu)\Big(\Phi\big(r^{-1}\rr\big)\Big)\le \|P_{t/2}\|^{-2}_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}\Big\}.$$ Consequently, for any $p\in [1, \infty], t\ge 0$ and $\mu_1,\mu_2\in \scr P(M)$,
\beq\label{LL1} W_p(\mu_1P_t, \mu_2P_t)\le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \min\Big\{W_p(\mu_1,\mu_2),\ \|\rr\|_{L^p(\mu\times \mu)}\|P_{t/2}\|^{\ff 2p}_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}\Big\}.\end{equation} \item[$(2)$] On the other hand, if there exist constants $c,\ll>0$ such that \beq\label{WIF} W_\infty(\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_xP_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_y P_t)\le c\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \rr(x,y),\ \ x,y\in M, t\ge 0,\end{equation} then the log-Sobolev inequality $\eqref{LS}$ holds for $c=\ff{2c^2}\ll$, so that $P_t$ is hypercontractive. \end{enumerate} \end{thm}
We note that in Theorem \ref{T1.2}(1) we have $\|\rr\|_{L^p(\mu\times \mu)}<\infty$ for $p\in [1,\infty)$. Indeed,
since $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ is bounded below, by \cite[Theorem 2.1]{RW} the ultracontractivity implies the super log-Sobolev inequality \eqref{SLS} below, so that due to Herbst we have $(\mu\times \mu)(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{r \rr^2})<\infty$ for all $r>0$ (see e.g. \cite{AMS}). Therefore, $G_\Phi(t)<\infty$ for $t>0$ and $\Phi\in \scr N$ satisfying $$\limsup_{r\rightarrow}\def\l{\ell}\def\iint{\int\infty}\ff{\log \Phi(r)}{r^2}<\infty.$$
In the symmetric case (i.e. $Z=\nabla} \def\pp{\partial} \def\E{\mathbb E V$ for some $V\in C^2(M)$), explicit sufficient conditions for the ultracontractivity have been introduced in \cite{WAnn} by using the dimension-free Harnack inequality in the sense of \cite{W97}. Together with a suitable exponential estimate on the diffusion process, this inequality implies $\|P_t\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}<\infty$ for $t>0$ and thus, $P_t$ is ultracontractive due to \eqref{SY}. The conditions can be formulated as \beq\label{C2} -\nabla} \def\pp{\partial} \def\E{\mathbb E Z\ge \Psi_1\circ\rr_o\ {\rm and}\ \text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}} \ge -\Psi_2\circ\rr_o\ \text{hold \ outside\ a\ compact\ subset\ of\ }M,\end{equation} where $\Psi_1, \Psi_2: (0,\infty)\rightarrow}\def\l{\ell}\def\iint{\int (0,\infty)$ are increasing functions such that \beq\label{4.3}\int_1^\infty \ff{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s}{\ss s\int_0^{\ss s}\Psi_1(u)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D u}<\infty,\ \ \lim_{r\rightarrow}\def\l{\ell}\def\iint{\int\infty}\min\Big\{\Psi_1(r), \ff{(\int_0^r \Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s)^2}{\Psi_1(r)}\Big\} =\infty,\end{equation} and for some constants $\theta\in (0, 1/(1+\ss 2))$ and $C>0,$ \beq\label{4.4} \ss{\Psi_2(r+t)(d-1)}\le \theta \int_0^r \Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s +\ff 1 2 \int_0^{t/2}\Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s+C,\ \ r,t\ge 0.\end{equation} When $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}$ is bounded below, \eqref{4.4} as well as the second inequality in \eqref{C2} hold for $\Psi_2$ being a large enough constant. In general, since $\int_0^r \Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \ge 2 \int_0^{r/2} \Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s$, \eqref{4.4} with $\theta=\ff 1 4<\ff 1 {1+\ss 2}$ follows from
\beq\label{4.4'}\begin} \def\beq{\begin{equation}} \def\F{\scr F{split} \ss{\Psi_2(r)(d-1)}&\le \ff 1 2\inf_{t\in [0,r]} \bigg\{ \int_0^{t/2} \Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s + \int_0^{(r-t)/2}\Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\bigg\}+C\\ &= \int_0^{r/4} \Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s+C,\ \ r\ge 0.\end{split}\end{equation}
Since \eqref{SY} fails for non-symmetric semigroups, we apply the inequality
$$\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}\le \|P_{t/2}\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^2(\mu)}\|P_{t/2}\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}$$ due to the semigroup property. So, to ensure the ultracontractivity, we need an additional condition implying $\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^2(\mu)}<\infty$ (see Corollary \ref{C1.3}(2) below).
To estimate $G_\Phi(t)$ in \eqref{LL0} using $\Psi_1$, we introduce $$\LL_1(r):= \ff 1 {\ss r} \int_0^{\ss r} \Psi_1(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ \ \
\LL_2(r):= \int_r^\infty \ff{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\ s}{\ss s\int_0^{\ss s}\Psi_1(u)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D u},\ \ \ r>0.$$ Obviously, the inverse function $\LL_2^{-1}$ exists on $(0,\infty)$, and since $\LL_1$ is increasing with $\LL_1(\infty)=\infty$, we have
$$\LL_1^{-1}(r):=\inf\{s\ge 0: \LL_1(s)\ge r\}<\infty,\ \ r\ge 0.$$
\begin} \def\beq{\begin{equation}} \def\F{\scr F{cor}\label{C1.3} Assume that $\eqref{4.3}$ and $\eqref{4.4}$ hold for some constants $\theta\in (0, 1/(1+\ss 2))$ and $C>0.$ \begin} \def\beq{\begin{equation}} \def\F{\scr F{enumerate} \item[$(1)$] If $P_t$ is symmetric, i.e. $Z=\nabla} \def\pp{\partial} \def\E{\mathbb E V$ for some $V\in C^2(M)$, then
there exist constants $c,\ll>0$ such that $\eqref{LL0}$ and $\eqref{LL1}$ hold for $$G_\Phi(t):=\inf\Big\{\ll>0:\ (\mu\times\mu)\big(\Phi (\ll^{-1}\rr )\big)\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{- ct^{-1}\{1+\LL_1^{-1}(c t^{-1})-\LL_2^{-1} ( c^{-1}t )\}}\Big\},\ \ t>0.$$
\item[$(2)$] If $P_t$ is non-symmetric but there exists continuous $h\in C([0,1]; [0,\infty)) $ with $h(r)>0$ for $r>0$ such that $\int_0^1 \ff{ h(r)} r\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r<\infty$ and $$H(\theta):=\int_0^1 \ff \theta {h(r)}\Big\{1+\LL_1^{-1} \big( \theta/ h(r)\big) + \LL_2^{-1} \big( h(r)/\theta\big)\Big\}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r<\infty,\ \ \theta>0,$$ then there exist constants $c,\ll>0$ such that $\eqref{LL0}$ holds for $$G_\Phi(t):=\inf\Big\{\ll>0:\ (\mu\times\mu)\big(\Phi (\ll^{-1}\rr )\big)\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{- ct^{-1}\{1+\LL_1^{-1}(c t^{-1})-\LL_2^{-1} ( c^{-1}t )\}-c H(ct^{-1})}\Big\}.$$
\end{enumerate} \end{cor}
To conclude this part, we
present a simple example to illustrate Corollary \ref{C1.3}.
\paragraph{Example 2.1.} Let $M$ have non-positive sectional curvatures and a pole $o\in M$. Let $Z= Z_0-\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho \nabla} \def\pp{\partial} \def\E{\mathbb E \rr_o^{2+\vv}$ outside a compact domain, where $\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho,\vv>0$ are constants and $Z_0$ is a $C^1$ vector field with
\beq\label{Z1}\limsup_{\rr_o\rightarrow}\def\l{\ell}\def\iint{\int\infty}\ff{|\nabla} \def\pp{\partial} \def\E{\mathbb E Z_0|}{\rr_o^\vv}<\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho(1+\vv)(2+\vv).\end{equation} Let $\Psi_2: (0,\infty)\rightarrow}\def\l{\ell}\def\iint{\int (0,\infty)$ be increasing such that \beq\label{Z2} \text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}} \ge - \Psi_2(\rr_o), \ \lim_{r\rightarrow}\def\l{\ell}\def\iint{\int\infty} \ff{\Psi_2(r)}{r^{2(1+\vv)}}=0. \end{equation} By \eqref{Z1}, \eqref{Z2} and the Hessian comparison theorem, we see that \eqref{C2}, \eqref{4.3} and \eqref{4.4'} hold with $\Psi_1(r)= c_1 r^\vv$ for some constant $c_1>0$. According to Corollary \ref{C1.3}, there exist constants $c,\ll>0$ such that for any $p\ge 1$,
$$W_p(\mu_1P_t, \mu_2P_t) \le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \min\Big\{W_p(\mu_1,\mu_2),\ \|\rr\|_{L_p(\mu\times\mu)}\exp\Big[\ff {c } {pt^{1+\ff 2\vv}}\Big] \Big\},\ \ t>0, \mu_1,\mu_2\in \scr P(M).$$
\subsection{SDEs with multiplicative noise}
Consider the following SDE on $\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$: \beq\label{SDE} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D X_t= b(X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t +\ss 2 \sigma} \def\ess{\text{\rm{ess}}(X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t,\end{equation} where $B_t$ is the $m$-dimensional Brownian motion, $b: \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ and $\sigma} \def\ess{\text{\rm{ess}}: \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d \rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\otimes\mathbb R} \def\ff{\frac} \def\ss{\sqrt^m $ (the space of $d\times m$-matrices) are locally Lipshitz such that
$$\|\sigma} \def\ess{\text{\rm{ess}}\|_{HS}^2(x) + \<b(x),x\>\le C(1+|x|^2),\ \ x\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$$ holds for some constant $C>0$, where and in the following, $\|\cdot\|_{HS}$ and $\|\cdot\|$ denote the Hilbert-Schmidt and the operator norms respectively. Then the SDE has a unique solution $\{X_t(x)\}_{t\ge 0}$ for every initial point $x\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$. Let $P_t$ be the associated Markov semigroup: $$P_t f(x):= \E[f(X_t(x))],\ \ t\ge 0, x\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d, f\in \B_b(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d).$$ We intend to investigate the $W_p$-exponential contraction for $p\in [1,\infty)$. As mentioned in Introduction that existing results only apply to $p=1$ and $\sigma} \def\ess{\text{\rm{ess}}=I$, and as mentioned in \cite{EB2,LW} that there is essential difficulty to prove \eqref{W'} for $p>1$ even for $\sigma} \def\ess{\text{\rm{ess}}=I$. So, the present study is non-trivial.
Corresponding to that \eqref{C} implies \eqref{W} in the Riemannian setting, we have the following assertion.
\begin} \def\beq{\begin{equation}} \def\F{\scr F{thm}\label{ST1} Let $p\in [1,\infty)$. If
\beq\label{DSS} \begin} \def\beq{\begin{equation}} \def\F{\scr F{split} & \ff{(p-2) |(\sigma} \def\ess{\text{\rm{ess}}(x)-\sigma} \def\ess{\text{\rm{ess}}(y))^*(x-y)|^2}{|x-y|^2}+ \|\sigma} \def\ess{\text{\rm{ess}}(x)-\sigma} \def\ess{\text{\rm{ess}}(y)\|_{HS}^2 +\<b(x)-b(y), x-y\>\\
&\le -K_p|x-y|^2,\ \ x\ne y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\end{split}\end{equation} holds for some constant $K_p\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt$, then $$ W_{p}(\mu_1P_t, \mu_2P_t)\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-K_p t} W_{p}(\mu_1,\mu_2),\ \ \ t\ge 0, \mu_1,\mu_2\in \scr P(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d).$$
\end{thm} Note that this result does apply to $p=\infty$ when $\sigma} \def\ess{\text{\rm{ess}}$ is non-constant. Next, as in the Riemannian case, we intend to prove the exponential contraction in $W_p$ when \eqref{DSS} only holds for some negative constant $K_p$. To this end, we need the SDE to be non-degenerate. The following result contains analogous assertions in Theorems \ref{T1.1} and \ref{T1.2}, where the first assertion extends \eqref{Eb2} to the multiplicative noise setting.
\begin} \def\beq{\begin{equation}} \def\F{\scr F{thm}\label{ST2} Assume that $ \sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^* \ge \ll_0^2 I$ for some constant $\ll_0>0$. \begin} \def\beq{\begin{equation}} \def\F{\scr F{enumerate} \item[$(1)$] If there exist constants $K_1,K_2,r_0>0$ such that $Z$ and $\sigma} \def\ess{\text{\rm{ess}}_0:= \ss{\sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^* -\ll_0^2 I}$ satisfy
\beq\label{DSS2} \begin} \def\beq{\begin{equation}} \def\F{\scr F{split} & \|\sigma} \def\ess{\text{\rm{ess}}_0(x)-\sigma} \def\ess{\text{\rm{ess}}_0(y)\|_{HS}^2-\ff{|(\sigma} \def\ess{\text{\rm{ess}}(x)-\sigma} \def\ess{\text{\rm{ess}}(y))^*(x-y)|^2}{|x-y|^2} +\<b(x)-b(y), x-y\>\\
&\le \big\{(K_1+K_2)1_{\{|x-y|\le r_0\}} - K_2\big\} |x-y|^2,\ \ x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d,\end{split}\end{equation} then there exist constants $c,\ll>0$ such that $$ W_1(\mu_1 P_t, \mu_2P_t)\le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} W_1(\mu_1,\mu_2),\ \ \ t\ge 0, \mu_1,\mu_2\in \scr P(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d).$$ \item[$(2)$] Let $P_t$ have a unique invariant probability measure $\mu$ such that the log-Sobolev inequality
\beq\label{LSA} \mu(f^2\log f^2)\le C\mu(|\sigma} \def\ess{\text{\rm{ess}}^*\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2),\ \ f\in C_b^1(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d), \mu(f^2)=1\end{equation} holds for some constant $C>0$.
If there exists a constant $K>0$ such that
\beq\label{DSS2'} \|\sigma} \def\ess{\text{\rm{ess}}(x)-\sigma} \def\ess{\text{\rm{ess}}(y)\|_{HS}^2 +\<b(x)-b(y), x-y\>
\le K |x-y|^2,\ \ x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d, \end{equation} then $\eqref{W2}$ holds for some constants $c,\ll>0$ and $M=\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$.
\item[$(3)$] Let $P_t$ be ultracontractive and let $\eqref{DSS2'}$ hold for some constant $K>0$. Then there exist a constant $\ll>0$ such that for any $p\in [1,\infty)$, condition $\eqref{DSS}$ implies $\eqref{LL1}$ for some constant $c=c(p)>0$, and all $t\ge 0, \mu_1,\mu_2\in \scr P(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$. \end{enumerate} \end{thm} According to \cite[Lemma 3.3]{PW}, we have
\beq\label{EP} \|\sigma} \def\ess{\text{\rm{ess}}_0(x)-\sigma} \def\ess{\text{\rm{ess}}_0(y)\|^2\le \ff 1 {4\ll_0} \|(\sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^*)(x)-(\sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^*)(y)\|_{HS}^2,\ \ x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d.\end{equation}
Combining this with $\|\cdot\|_{HS}^2 \le d \|\cdot\|^2$, we see that \eqref{DSS2} follows from the following more explicit condition:
\beq\label{DSS3} \begin} \def\beq{\begin{equation}} \def\F{\scr F{split} &\ff {d-1} {4\ll_0} \|(\sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^*)(x)-(\sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^*)(y)\|_{HS}^2 +\<b(x)-b(y), x-y\> \\
& \le \big\{(K_1+K_2)1_{\{|x-y|\le r_0\}} - K_2\big\} |x-y|^2,\ \ x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d. \end{split}\end{equation}
Note that conditions in Theorem \ref{ST1} and Theorem \ref{ST2}(1) are explicit. To illustrate Theorem \ref{ST2}(2)-(3), we present below sufficient conditions for the log-Sobolev inequality \eqref{LSA} and the ultracontractivity of $P_t$. For $a:=\sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^*$ and $(g_{ij})_{1\le i,j\le d}:= a^{-1}$, we introduce the Christoffel symbols $$\GG_{ij}^k:= \ff 1 2 \sum_{m=1}^d \big(\pp_i g_{mj}+\pp_j g_{im} - \pp_m g_{ij}\big)a_{km},\ \ 1\le i, j, k\le d,$$ and the matrix $\GG ab$: $$(\GG a b)_{ij}:= \sum_{k,l=1}^d \GG_{kl}^i a_{kj} b_k,\ \ 1\le i,j\le d.$$
\begin} \def\beq{\begin{equation}} \def\F{\scr F{prp}\label{PPN} Let $\sigma} \def\ess{\text{\rm{ess}}\in C_b^2(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\otimes \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$ such that $a:= \sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^*\ge \aa I$ for some constant $\aa>0$, and let $b\in C^1(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$ such that \beq\label{CC0} \ff 1 2 (\GG a b +\nabla} \def\pp{\partial} \def\E{\mathbb E_b a) - (\nabla} \def\pp{\partial} \def\E{\mathbb E b)a\ge -K_0 I \end{equation} for some constant $K_0$. If there exist constants $c_1,c_2>0$ and $\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho>1$ such that
\beq\label{UL} L |\cdot|^2 \le c_1 -c_2|\cdot|^{2\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho}, \end{equation}
then $P_t$ has a unique invariant probability measure $\mu$ and there exists a constant $c>0$ such that $$\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}\le \exp\Big[c+ct^{-\ff{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho}{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho-1}}\Big],\ \ t>0.$$ \end{prp}
We now introduce a simple example to illustrate Theorem \ref{ST2}.
\paragraph{Example 2.2.} Let $\sigma} \def\ess{\text{\rm{ess}}\in C_b^2(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\rightarrow}\def\l{\ell}\def\iint{\int \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d\otimes \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$ such that $a:= \sigma} \def\ess{\text{\rm{ess}}\sigma} \def\ess{\text{\rm{ess}}^*\ge \aa I$ for some constant $\aa>0$. Let $b(x)= -c_0 |x|^\theta x $ for large $|x|,$ where $c_0>0$ and $\theta>0$ are constants. Obviously, condition \eqref{DSS2'} holds. If
\beq\label{*D} \lim_{|x|\rightarrow}\def\l{\ell}\def\iint{\int\infty} |x|\cdot\|\nabla} \def\pp{\partial} \def\E{\mathbb E \sigma} \def\ess{\text{\rm{ess}}(x)\| =0,\end{equation} then \eqref{CC0} holds for some constant $K_0$. Moreover, it is easy to see that
$$L |\cdot|^2 \le c_1-c_2 |x|^{\theta+2},\ \ \ll>0, x\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$$ holds for some constants $c_1,c_2>0$. By Proposition \ref{PPN} and Theorem \ref{ST2}(3), for any $p\in [1,\infty)$, there exist constants $\ll,c>0$ such that $$W_p(\mu_1P_t, \mu_2P_t)\le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \min\Big\{W_p(\mu_1,\mu_2),\ \exp\Big[ c t^{-\ff{\theta+2}\theta}\Big] \Big\},\ \ t> 0, \mu_1,\mu_2\in \scr P(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d).$$
\section{Preparations}
This section includes some propositions which will be used to prove the results introduced in Section 2. We first recall a link between the Wasserstein distance and gradient estimates due to \cite{KK2}, then deduce the hyperboundedness and the exponential convergence in entropy from the log-Sobolev inequality for non-symmetric diffusion semigroups, and finally prove the exponential contraction in gradient for ultracontractive semigroups in a general framework including both diffusion and jump Markov semigroups.
\subsection{Wasserstein distance and gradient inequalities}
Let $(E,\rr)$ be a geodesic Polish space, i.e. it is a Polish space and for any two different points $x,y\in E$, there exists a continuous curve $\gg: [0,1]\rightarrow}\def\l{\ell}\def\iint{\int E$ such that $\gg_0=x, \gg_1=y$ and $\rr(\gg_s,\gg_t)=|s-t|\rr(x,y)$ for $s, t\in [0,1].$ Then for any $f\in \Lip_b(E)$, the class of bounded Lipschitz functions on $E$, the length of gradient
$$|\nabla} \def\pp{\partial} \def\E{\mathbb E f|(x):= \limsup_{\rr(x,y)\downarrow 0} \ff{|f(x)-f(y)|}{\rr(x,y)},\ \ x\in E$$ is measurable. Moreover, let $P(x,\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D y)$ be a Markov transition kernel and define the Markov operator $$Pf(x):= \int_E f(y)P(x,\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D y),\ \ x\in E, f\in \B_b(E).$$ For any $\Phi\in\bar{\scr N}\setminus\{\Phi_\infty\}$, consider the Young norm induced by $\Phi$ with respect to $P$
\beq\label{*NP}\|f\|_{L_*^\Phi(P)}(x) :=\sup\Big\{P(fg)(x): g\in \scr B_b(E), P\Phi(|g|)(x)\le 1\Big\},\ \ x\in E, f\in \B_b(E), \end{equation}
and set $\|f\|_{L_*^{\Phi_\infty}(P)}(x)= P|f|(x).$ Then $\|\cdot\|_{L_*^{\Phi_p}}=\|\cdot\|_{L^{\Phi_q}}$ for $p\in [1,\infty], q=\ff{p}{p-1}.$ The following result follows from \cite[Theorem 2.2, Remark 2 and Remark 3]{KK2}.
\begin} \def\beq{\begin{equation}} \def\F{\scr F{prp}[\cite{KK2}] \label{T3.1} For any constant $C>0$ and $\Phi\in \bar{\scr N}$, the following statements are equivalent to each other: \begin} \def\beq{\begin{equation}} \def\F{\scr F{enumerate} \item[$(1)$]
$ |\nabla} \def\pp{\partial} \def\E{\mathbb E P f| \le C \|\nabla} \def\pp{\partial} \def\E{\mathbb E f\|_{L_*^\Phi(P)} $ for $f\in \Lip_b(E).$ \item[$(2)$] $W_{\Phi}(\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_xP, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP) \le C \rr(x,y),\ \ x,y\in E.$ \end{enumerate} When $\Phi=\Phi_p$ for $p\in [1,\infty]$, they are also equivalent to \begin} \def\beq{\begin{equation}} \def\F{\scr F{enumerate} \item[$(3)$] $W_{p}(\mu_1P, \mu_2P) \le C W_{p} (\mu_1,\mu_2),\ \ \mu_1,\mu_2\in \scr P(E).$\end{enumerate} \end{prp}
\subsection{Hyperboundedness and exponential convergence in entropy}
When $P_t$ is symmetric, it is well known that the hyperbounddeness, exponential convergence in entropy and the log-Sobolev inequality are equivalent each other, see \cite{BGL, Wbook} and references within. In the non-symmetric case, the log-Sobolev inequality implies the former two properties if the generator $L$ and the symmetric part of the Dirichlet form $\scr E}\def\Cut{{\rm Cut}$ satisfy \beq\label{NC} \begin} \def\beq{\begin{equation}} \def\F{\scr F{split} &-\mu((1+\log f)Lf)\ge c_0\scr E}\def\Cut{{\rm Cut}\big(\ss f,\ss f\big) \ {\rm and}\\ & -\mu(f^{p-1} Lf)= \ff{c_0(p-1)}{p^2} \scr E}\def\Cut{{\rm Cut}( f^{\ff p 2}, f^{\ff p 2}),\ \ p>1, f\in \D\end{split}\end{equation} for some constant $c_0>0$ and a reasonable class $\D$ of non-negative bounded functions, which is stable under $P_t$ and dense in $L^p_+(\mu) :=\{f\in L^p(\mu): f\ge 0\}$ for any $p\ge 1$, see e.g. \cite{Gross}. In applications, it may be not easy to figure out the class $\D$ such that \eqref{NC} holds. But in general this condition can be replaced by the following approximation formula Lemma \ref{L0} in the spirit of \cite{RW04}.
Now, consider the (Neumann) semigroup $P_t$ generated by $L:= \DD+Z$ for a local bounded vector field $Z$ such that $P_t$ has a unique invariant probability measure $\mu$. Let $$\D_0=\big\{f\in C_0^\infty(M):\ f\ \text{satisfies\ the\ Neumann\ condition\ if\ } \pp M\ne\emptyset\big\}.$$ Then $(L,\D_0)$ is dissipative (thus, closable) in $L^1(\mu)$ with closure $(L,\D_1(L))$ generating $P_t$ in $L^1(\mu)$, see e.g. \cite{ST} and references within. Let $$\D= \{f\in \D_1(L)\cap L^\infty(\mu):\ f\ge 0\}. $$
\begin} \def\beq{\begin{equation}} \def\F{\scr F{lem}\label{L0} Let $f\in \D$ and $\psi \in C_b^\infty([{\rm ess}_\mu\inf f, \infty))$. There exists a sequence $\{f_n\}_{n\ge 1}\subset \D_0$ with $\inf f_n=\inf f$ such that $f_n\rightarrow}\def\l{\ell}\def\iint{\int f$ in $L^m(\mu)$ for any $m\ge 1$, $L f_n\rightarrow}\def\l{\ell}\def\iint{\int Lf$ in $L^1(\mu)$, and
$$\mu(\psi(f)Lf) = -\lim_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu(\psi'(f_n)|\nabla} \def\pp{\partial} \def\E{\mathbb E f_n|^2).$$\end{lem} \begin} \def\beq{\begin{equation}} \def\F{\scr F{proof} Since $f\in \D\subset \D_1(L)\cap L^\infty(\mu)$, there exists a uniformly bounded sequence $\{f_n\}_{n\ge 1}\subset \D_0$ such that $\inf f_n= {\rm ess}_\mu\inf f$ and $f_n\rightarrow}\def\l{\ell}\def\iint{\int f, L f_n\rightarrow}\def\l{\ell}\def\iint{\int Lf$ in $L^1(\mu)$. By the uniform boundedness, $f_n\rightarrow}\def\l{\ell}\def\iint{\int f$ in $L^m(\mu)$ for any $m\ge 1$. Since $\psi\in C_b^\infty([\inf f_n, \infty))$, $$g_n:= \int_{\inf f_n}^{f_n}\psi(s)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s\in \D_c:=\{g+c:\ c\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt, g\in \D_0\}\subset \D_1(L).$$ This implies $\mu(L g_n)=0$ since $\mu$ is $P_t$-invariant. So, by the dominated convergence theorem,
$$\mu(\psi(f)Lf)= \lim_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu(\psi(f_n)L f_n) = \lim_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu(L g_n - \psi'(f_n)|\nabla} \def\pp{\partial} \def\E{\mathbb E f_n|^2) = - \lim_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu(\psi'(f_n)|\nabla} \def\pp{\partial} \def\E{\mathbb E f_n|^2).$$ \end{proof}
\begin} \def\beq{\begin{equation}} \def\F{\scr F{prp}\label{PN} Let $Z$ be a locally bounded vector field such that the (Neumann) semigroup $P_t$ generated by $L:=\DD+Z$ has a unique invariant probability measure $\mu$. \begin} \def\beq{\begin{equation}} \def\F{\scr F{enumerate} \item[$(1)$] If the super log-Sobolev inequality
\beq\label{SLS} \mu(f^2\log f^2)\le r \mu(|\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2) +\bb(r),\ \ r>0,\ \ f\in C_b^1(M), \mu(f^2)=1. \end{equation} holds for some $\bb\in C((0,\infty);(0,\infty))$, then for any constants $q>p\ge 1$ and $\gg\in C((p,q); (0,\infty))$ such that $t:=\int_p^q \ff{\gg(r)}r\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r<\infty,$ there holds
$$\|P_t\|_{L^p(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^q(\mu)}\le \exp\bigg[\int_p^q\ff{\bb(4\gg(r)(1-r^{-1}))}{r^2}\,\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r\bigg].$$ \item[$(2)$] If the log-Sobolev inequality
\beq\label{LS} \mu(f^2\log f^2)\le C \mu(|\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2) + \mu(f^2)\log \mu(f^2),\ \ f\in C_b^1(M) \end{equation} holds for some constant $C>0$, then $$\mu((P_t g)\log P_t g)\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-4t/C} \mu(g\log g),\ \ g\in \B_b(M), g\ge 0, \mu(g)=1.$$ \end{enumerate} \end{prp}
\begin} \def\beq{\begin{equation}} \def\F{\scr F{proof} (1) According to Lemma \ref{L0}, for any $f\in \D$ and $p>1$, there exists $\{f_n\}_{n\ge 1}\subset \D_0$ such that $f_n\rightarrow}\def\l{\ell}\def\iint{\int f^{\ff p 2}$ in $L^m(\mu)$ for all $m\ge 1$, and
\beq\label{IT} -\mu(f^{p-1} Lf)= \ff{4(p-1)}{p^2} \limsup_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu(|\nabla} \def\pp{\partial} \def\E{\mathbb E f_n|^2).\end{equation} Applying \eqref{SLS} to $f_n$ and using \eqref{IT}, we obtain \begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} & p\mu(f^p\log f)=\lim_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu(f_n^2\log f_n^2)
\le r \liminf_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu(|\nabla} \def\pp{\partial} \def\E{\mathbb E f_n|^2)+\bb(r) \\ &\le \ff{rp^2}{4(p-1)}\Big(-\mu(f^{p-1} Lf)+\ff{4\bb(r)(p-1)}{rp^2}\Big),\ \ r>0.\end{align*} Set $c(p) =\ff{rp}{4(p-1)},$ we have $$\ff{4\bb(r)(p-1)}{r p^2}= \ff{\bb(4c(p)(1-p^{-1}))}{pc(p)},\ \ p>1,$$ so that the above inequality becomes $$\mu(f^p\log f) \le c(p) \Big(-\mu(f^{p-1} Lf) +\gg(p)\Big),\ \ p>1, f\in \D$$ for $\gg(p):= \ff{\bb(4c(p)(1-p^{-1}))}{pc(p)}.$ Noting that $\D$ is $P_t$-invariant (i.e. $P_t\D\subset \D$) and dense in $L_+^p(\mu)$ for any $p\ge 1$, the desired assertion follows from the proof of \cite[Corollary 3.13]{Gross}.
(2) It suffices to prove for $g\in \D$ with $\inf g>0.$ Applying Lemma \ref{L0} to $f= P_tg$ and $\psi(s)= 1+ \log s$, and using \eqref{LS}, we obtain
\begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} &\ff{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D}{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t} \mu( (P_t g)\log P_t g) = \mu((1+\log P_t g) L P_t g) = -4\lim_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu\big(\big|\nabla} \def\pp{\partial} \def\E{\mathbb E\ss f_n\big|^2\big)\\ & \le -\ff 4 C\liminf_{n\rightarrow}\def\l{\ell}\def\iint{\int\infty} \mu(f_n\log f_n) = -\ff 4 C \mu((P_t g)\log P_t g),\ \ t\ge 0.\end{align*} This implies the desired exponential estimate. \end{proof}
\subsection{Exponential contraction in gradient}
In this part, we consider a general framework including both diffusion and jump processes. Let $(E,\F,\mu)$ be a separable complete probability space, and let $P_t$ be a Markov semigroup on $L^2(\mu)$ with $\mu$ as invariant probability measure. Let $(L,\D(L))$ be the generator of $P_t$ in $L^2(\mu)$. We assume that there exists an algebra $\scr A\subset \D(L)$ such that \begin} \def\beq{\begin{equation}} \def\F{\scr F{enumerate} \item[{\rm (i)}] $1\in \scr A$, $\scr A$ is dense in $L^2(\mu)$ and the algebra induced by $$\D:=\{P_s f: s\ge 0, f\in \scr A\}$$ is contained in $\D(L)$. \item[{\rm (ii)}] $\GG(f,g):= \ff 1 2 (L(fg) - fLg -gLf)$ gives rise to a non-degenerate positive definite bilinear form on $\scr D\times\scr D$; i.e., for any $f\in \scr D$, $\GG(f,f)\ge 0$ and it equals to $0$ if and only if $f$ is constant. \end{enumerate} In particular, when $P_t$ is the (Neumann) semigroup generated by $L:=\DD+Z$ on $M$ with $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$ bounded below, the assumption holds for $$\scr A:= \{f+c: f\in C_0^\infty(M) \ {\rm satisfying \ the\ Neumann\ condition\ if} \ \pp M\ne \emptyset, c\in\mathbb R} \def\ff{\frac} \def\ss{\sqrt\}.$$ Under the above conditions, $$\scr E}\def\Cut{{\rm Cut}(f,g):=\mu(\GG(f,g)),\ \ f,g\in \scr A$$ is closable and the closure $(\scr E}\def\Cut{{\rm Cut}, \D(\scr E}\def\Cut{{\rm Cut}))$ is a conservative symmetric Dirichlet form. Although $P_t$ is not associated to $(\scr E}\def\Cut{{\rm Cut}, \D(\scr E}\def\Cut{{\rm Cut}))$ when it is non-symmetric, we have \beq\label{REL} \ff{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D }{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t} \mu((P_t f)^2) = - 2 \scr E}\def\Cut{{\rm Cut}(P_tf, P_t f),\ \ t\ge 0, f\in \scr D.\end{equation}
If $\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}<\infty,$ then $P_t$ has a heat kernel $p_t(x,y)$ with respect to $\mu$, i.e. $$P_t f=\int_E p_t(\cdot, y)f(y)\mu(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D y),\ \ f\in L^2(\mu),$$ and
$${\rm ess}_{\mu\times \mu}\sup p_t=\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}<\infty.$$
We consider the $``$gradient" length $|\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG f|= \ss{\GG(f,f)}$ induced by $\GG$. Note that for jump processes the length is non-local and thus essentially different from the usual gradient length. As shown below that estimates of $|\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_t|$ have a close link to functional inequalities of the associated Dirichlet form.
\begin} \def\beq{\begin{equation}} \def\F{\scr F{prp}\label{P2.1} Assume that there exist $t_1>0$ and $\eta \in C([0,\infty); (0,\infty))$ such that
\beq\label{AS} \|P_{t_1}\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)} <\infty,\ \ |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_t f|^2\le \eta(t) P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG f|^2,\ t\ge 0, f\in \scr D.\end{equation}
Then there exist constants $c,\ll, t_2>0$ such that for any $q\ge 1$ and $\eta_q\in C([0,\infty); (0,\infty))$, the gradient estimate \beq\label{GP} |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_t f|^2\le \eta_q(t) (P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG f|^{q})^{\ff 2 q},\ t\ge 0, f\in\D\end{equation} implies
\beq\label{GE} \|\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_tf \|_{L^\infty(\mu)}^2 \le \Big(c\sup_{[0,t_2]} \eta_q\Big) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} {\rm ess}_\mu\inf(P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG f|^{q})^{\ff 2 q}, \ \ t\ge t_2, f\in \D.\end{equation} \end{prp}
\begin} \def\beq{\begin{equation}} \def\F{\scr F{proof} (a) We first prove \beq\label{EP} \scr E}\def\Cut{{\rm Cut}(P_tf, P_tf) \le C \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \scr E}\def\Cut{{\rm Cut}(f,f),\ \ f\in \D, t\ge 0\end{equation} for some constants $C,\ll>0$. By the second inequality in \eqref{AS}, for any $t>0$ and $f\in \D$ we have
$$\ff{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D }{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s} P_s (P_{t-s}f)^2 = 2 P_s |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_{t-s} f|^2 \le 2 \eta(t-s) P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG f|^2,\ \ s\in [0,t].$$ Integrating both sides over $[0,t]$ leads to
$$ P_t f^2\le (P_t f)^2 + C(t)P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG f|^2,\ \ C(t):=2\int_0^t \eta(s) \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s,\ t>0. $$ Taking $t=t_1$ and noting that $\mu$ is the invariant probability measure of $P_t$, we obtain
\beq\label{DP} \mu(f^2)\le C(t_1) \scr E}\def\Cut{{\rm Cut}(f,f)+ \|P_{t_1}\|_{1\rightarrow}\def\l{\ell}\def\iint{\int\infty}^2 \mu(|f|)^2,\ \ f\in \D.\end{equation} Since $\D(\scr E}\def\Cut{{\rm Cut})$ is the closure of $\D$ under the $\scr E}\def\Cut{{\rm Cut}_1$-norm, this inequality also holds for $f\in \D(\scr E}\def\Cut{{\rm Cut}).$ By condition (ii), the symmetric Dirichlet form is irreducible. So, according to \cite[Corollary 1.2]{W14} the defective Poincar\'e inequality \eqref{DP} implies the Poincar\'e inequality \beq\label{P} \mu(f^2)\le \ff 1 {\ll} \scr E}\def\Cut{{\rm Cut}(f,f) +\mu(f)^2,\ \ f\in \D(\scr E}\def\Cut{{\rm Cut}) \end{equation} for some constant $\ll>0$. By \eqref{REL} and that $\D$ is dense in $L^2(\mu)$, the Poincar\'e inequality is equivalent to
\beq\label{EXP} \|P_t f-\mu(f)\|_2\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \|f-\mu(f)\|_2,\ \ t\ge 0, f\in L^2(\mu).\end{equation} On the other hand, by the second inequality in \eqref{AS}, for any $t>0$ and $f\in \D$ we have
$$\ff{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D }{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s} P_s (P_{t-s}f)^2 = 2 P_s |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_{t-s} f|^2 \ge \ff 2 {\eta(s)} |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_t f|^2,\ \ s\in [0,t].$$ So,
$$|\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_tf|^2 \le \ff {P_t f^2-(P_tf)^2} {2\int_0^t \eta(s)^{-1} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s},\ \ t>0, f\in \D.$$ Using $P_tf-\mu(f)$ to replace $f$ and integrating with respect to $\mu$, we obtain
$$\scr E}\def\Cut{{\rm Cut}(P_{2t}f, P_{2t}f) \le \ff {\|P_tf-\mu(f)\|_2^2} {2\int_0^t \eta(s)^{-1} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s},\ \ t>0, f\in \D.$$ Combining this with \eqref{EXP} and \eqref{P} we arrive at $$ \scr E}\def\Cut{{\rm Cut}(P_{t}f, P_{t}f) \le c_1 \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \scr E}\def\Cut{{\rm Cut}(f,f),\ \ t\ge 1, f\in \D$$ for some constant $c_1>0$; that is, \eqref{EP} holds for $t>1.$ Finally, \eqref{AS} implies \eqref{EP} for $t\in [0,1].$
(b) Next, we intend to find out a constant $t_0\ge t_1$ such that \beq\label{H} \ff 1 2 \le p_t \le 2,\ \ (\mu\times\mu){\text -a.e.}, t\ge t_0.\end{equation} Indeed, by \eqref{EXP} and the first inequality in \eqref{AS}, we obtain
\begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} &\bigg|\int_E (p_{t+2 t_1}(\cdot, y)-1)f(y)\mu(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D y)\bigg|= |P_{t_1}(P_{t+t_1}f-\mu(f))| \\
&\le c_0 \mu(|P_{t+t_1}f-\mu(f)|)
\le c_0 \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \|P_{t_1}f-\mu(f)\|_2\le c_0^2 \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \mu(|f|),\ \ \mu{\text -a.e.},\ t\ge 0,\end{align*} where $c_0:=\|P_{t_1}\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}.$ This implies the desired assertion for $t_0>0$ such that $c_0^2\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t_0} \le \ff 1 2$.
(c) Finally, combining \eqref{AS}, \eqref{H}, \eqref{EP} and \eqref{P}, we obtain
\begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} &\|\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_{t+2 t_0 }f\|_{L^\infty(\mu)}^2 \le c_1 \| P_{t_0} |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG P_{t+t_0}f|^2 \|_{L^\infty(\mu)}\le 2c_1 \scr E}\def\Cut{{\rm Cut}(P_{t+t_0}f, P_{t+t_0}f)\\ &\le c_2 \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \scr E}\def\Cut{{\rm Cut}(P_{t_0}f, P_{t_0}f)
\le c_2\eta_q(t_0) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \mu\big((P_{t_0} |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG f|^q)^{\ff 2 q}\big)\\
&\le c_3\eta_q(t_0) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} {\rm ess}_\mu\inf(P_{t+2t_0} |\nabla} \def\pp{\partial} \def\E{\mathbb E_\GG f|^q)^{\ff 2 q} \end{align*} for some constants $c_1, c_2, c_3 >0$. Then \eqref{GE} holds for $t_2= 2 t_0.$ \end{proof}
\section{Proof of Theorem \ref{T1.1} } The first assertion is a generalization of the main result in \cite{LW} where $M=\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ is considered. As in \cite{LW}, the key point of the proof is to construct a coupling by parallel transform for long distance but by reflection for short distance. The only difference is that we are working on a non-flat Riemannian manifold for which the curvature term appears in calculations. Since It\^o's formula of the distance process has been well developed for couplings by both parallel displacement and reflection, the proof is also straightforward.
The proofs of the other two assertions are based on the log-Sobolev inequality and the log-Harnack inequality derived in \cite{RW} and \cite{W10} respectively for bounded below $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z$.
\begin} \def\beq{\begin{equation}} \def\F{\scr F{proof}[Proof of Theorem \ref{T1.1}] (a) For two different points $x,y\in M$, let $P_{x,y}: T_xM \rightarrow}\def\l{\ell}\def\iint{\int T_yM$ be the parallel displacement along the minimal geodesic $\gg: [0,\rr(x,y)]\rightarrow}\def\l{\ell}\def\iint{\int M$ from $x$ to $y$, and let $M_{x,y}:= P_{x,y} - 2 \langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\cdot, \dot \gg_{0}\> \dot\gg_{\rr(x,y)}: T_xM\rightarrow}\def\l{\ell}\def\iint{\int T_y M$ be the mirror reflection. Both maps are smooth in $(x,y)$ outside the cut-locus $\Cut(M)$. According to \cite{K} and \cite{W94}, the appearance of the cut-locus and/or a convex boundary helps for the success of coupling, i.e. it makes the distance between two marginal processes smaller. So, for simplicity, we may and do assume that both the cut-locus and the boundary are empty, see \cite[Section 3]{ATW06} or \cite[Chapter 2]{Wbook} for details.
Now, let $X_t$ solve the SDE $$\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_I X_t = \ss 2 u_t \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t +Z(X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t,\ \ X_0=x,$$ where $\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_I$ denotes the It\^o differential introduced in \cite{E} on Riemannian manifolds, $B_t$ is the $d$-dimensional Brownian motion, and $u_t$ is the horizontal lift of $X_t$ to the frame bundle $O(M)$. Then $X_t$ is a diffusion process generated by $L$. To construct the
coupling by reflection for short distance and parallel displacement for long distance, we introduce a cut-off function $h\in C^1([0,\infty))$ which is decreasing such that $h(r)=1$ for $r\le r_0,$ $h(r)=0$ for $r\ge r_0+1$, and $\ss{1-h^2}$ is also in $C^1$, see e.g. \cite[(3.1)]{W15} for a concrete example. To construct the coupling in the above spirit, we split the noise into two parts, i.e. to replace $\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t$ by $h(\rr(X_t,Y_t))\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t' + \ss{1-h(\rr(X_t,Y_t))^2}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t''$ for two independent Brownian motions $B_t'$ and $B_t''$,
then make reflection for the $B_t'$ part and parallel displacement for the $B_t''$ part. More precisely, let $(X_t, Y_t)$ solve the following SDE on $M\times M$ for $(X_0,Y_0)=(x,y)$: \begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} &\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_I X_t = \ss 2 \Big(h(\rr(X_t,Y_t)) u_t \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t' + \ss{1-h(\rr(X_t,Y_t))^2}u_t \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t''\Big) +Z(X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t,\\ &\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D_I Y_t= \ss 2 \Big(h(\rr(X_t,Y_t))M_{X_t,Y_t} u_t \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t' + \ss{1-h(\rr(X_t,Y_t))^2}P_{X_t,Y_t}u_t \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t''\Big)+Z(Y_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t. \end{align*} Since the coefficients of the SDE are at least $C^1$ outside the diagonal $\{(z,z): z\in M\}$, it has a unique solution up to the coupling time $$T:=\inf\{t\ge 0: X_t=Y_t\}.$$ We then let $X_t=Y_t$ for $t\ge T$ as usual. By the second variational formula and the index lemma (see e.g. the proof of \cite[Lemma 2.3]{WAnn} and \cite[(2.4)]{W94}), the process $\rr_t:=\rr(X_t,Y_t)$ satisfies $$\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \rr_t\le 2\ss 2 h(\rr_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D b_t + I_Z(X_t,Y_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t,\ \ t\le T$$ for some one-dimensional Brownian motion $b_t$. Thus, by condition \eqref{EB'}, \beq\label{ITO} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \rr_t \le 2\ss 2 h(\rr_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D b_t +\big\{(K_1+K_2)1_{\{\rr_t\le r_0\}}-K_2\big\}\rr_t\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t,\ \ t\le T.\end{equation} Since $h(\rr_t)=0$ for $\rr_t\ge r_0+1$ while $\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D\rr_t<0$ when $\rr_t\ge r_0+1,$ this implies \beq\label{DD0}\rr_t\le (r_0+1)\lor\rr_0\le 1+r_0+\rr(x,y).\end{equation} On the other hand, since $h(\rr_t)=1$ for $\rr_t\le r_0$, as observed in \cite{LW} we have \beq\label{DD}\E \rr_t\le c\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t}\rr(x,y),\ \ t\ge 0\end{equation} for some constants $c,\ll>0$. Indeed, let $$\bar \rr_t= \vv \rr_t + 1- \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-N\rr_t},\ \ N=\ff {r_0} 2( K_1+K_2), \vv=N\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-Nr_0}.$$ Then $$\vv \rr_t\le \bar\rr_t\le (N+\vv)\rr_t,\ \ \ff{4N^2}{r(\vv\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{Nr}+N)}\ge K_1+K_2 \ \text{for}\ r\in (0,r_0],$$ so that
\eqref{ITO} and It\^o's formula lead to \begin} \def\beq{\begin{equation}} \def\F{\scr F{align*}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \bar\rr_t &\le 2\ss 2 (\vv+N\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-N\rr_t})h(\rr_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D b_t \\ &\qquad + (\vv+N\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-N\rr_t})\Big\{ (K_1+K_2)1_{\{\rr_t\le r_0\}}- K_2 -\ff {4N^2} {\rr_t (\vv\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{N\rr_t}+N}1_{\{\rr_t\le r_0\}}\Big\}\rr_t\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\\ &\le 2\ss 2 (\vv+N\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-N\rr_t})h(\rr_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D b_t -c_1\bar\rr_t\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t,\ \ t\le T\end{align*} for some constant $c_1$. This implies $\E \bar\rr_t\le \bar\rr_0\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-c_1 t}.$ Then \eqref{DD} holds for some constants $c,\ll>0$. Combining \eqref{DD0} with \eqref{DD} we arrive at $$\E \Phi(\rr_t/r) \le \sup_{s\in (0, 1+r_0+\rr_0]} \ff{\Phi(s/r)}{s} \E\rr_t \le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \rr(x,y) \sup_{s\in (0, 1+r_0+\rr_0]} \ff{\Phi(s/r)}{s}.$$ So,
\begin} \def\beq{\begin{equation}} \def\F{\scr F{align*}&W_{\Phi}(\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_xP_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t) \le \|\rr_t\|_{L^\Phi(\P)}= \inf\big\{r>0: \E \Phi(\rr_t/r)\le 1\big\}\\ &\le \inf\Big\{r>0: \sup_{s\in (0, 1+ \rr(x,y)]} \ff{\Phi(\ff s r)}{s}\le \ff {\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{\ll t}}{c\rr(x,y)}\Big\},\end{align*} which proves \eqref{EB1'}. Therefore, the proof of (1) is finished since the second inequality therein is a simple consequence of \eqref{EB1'}.
(b) According to the proofs of \cite[Proposition 3.1 and Theorem 1.1]{WAnn}, our conditions imply that $P_t$ is hyperbounded; that is, $\|P_t\|_{2\rightarrow}\def\l{\ell}\def\iint{\int 4}<\infty$ holds for some $t>0$. Since \eqref{EB'} implies $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z\ge -(K_1+K_2)$, by the hyperboundedness and \cite[Theorem 2.1]{RW}, we have the defective log-Sobolev inequality
$$\mu(f^2\log f^2)\le C_1 \mu(|\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2) + C_2,\ \ f\in C_b^1(M), \mu(f^2)=1$$ for some constants $C_1, C_2>0$. Since the symmetric Dirichlet form $\scr E}\def\Cut{{\rm Cut}(f,g):= \mu(\langle} \def\>{\rangle} \def\GG{\Gamma} \def\gg{\gamma\nabla} \def\pp{\partial} \def\E{\mathbb E f,\nabla} \def\pp{\partial} \def\E{\mathbb E g\>)$ with domain $H^{1,2}(\mu)$ is irreducible, according to \cite{W14} (see also \cite{M}), the log-Sobolev inequality \eqref{LS} holds for some constant $C>0$, so that (2) is proved.
(c) According to \cite[Theorem 1.10]{S} (see \cite{BGL0,W04,OV} for the case without boundary), the log-Sobolev inequality implies the Talagrand inequality \beq\label{TL}W_2(f\mu, \mu)^2\le \ff C 2 \mu(f\log f),\ \ f\ge 0, \mu(f)=1. \end{equation} Next, let $P_t^*$ be the adjoint of $P_t$ in $L^2(\mu)$. By Proposition \ref{PN} for $P_t^*$ in place of $P_t$, the log-Sobolev inequality implies \beq\label{EX0}\mu((P_t^*f)\log P_t^*f)\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-4t/C} \mu(f\log f),\ \ t\ge 0, f\ge 0, \mu(f)=1.\end{equation} Moreover, according to \cite[Theorem 1.1]{W10}, the curvature condition $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z\ge -( K_1+K_2)=:-K$ is equivalent to the log-Harnack inequality $$P_t (\log f )(x) \le \log P_tf (y) + \ff{K\rr(x, y)^2}{2(1-\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-2Kt})},\ \ t\ge 0, x,y\in M, 0\le f\in \B_b(M).$$
By \cite[Proposition 1.4.4(3)]{WBook}, this implies \beq\label{ET}\mu((P_t^* f)\log P_t^* f)\le \ff{K}{2(1-\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-2Kt})}W_2(f\mu,\mu)^2,\ \ f\ge 0, \mu(f)=1, t>0.\end{equation} Combining \eqref{TL}, \eqref{EX0} and \eqref{ET}, we obtain \beq\label{SL}\begin} \def\beq{\begin{equation}} \def\F{\scr F{split}& W_2((f\mu)P_{1+t}, \mu)^2= W_2((P_{1+t}^* f)\mu,\mu)^2\le \ff C 2 \mu((P_{1+t}^*f)\log P_{1+t}^*f)\\ &\le \ff C 2 \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-4t/C} \mu((P_1^* f)\log P_1^*f)
\le c_1\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-4t/C} W_2(f\mu, \mu)^2,\ \ t\ge 0, f\ge 0, \mu(f)=1 \end{split}\end{equation} for some constant $c_1>0$. Noting that $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z\ge -K$ implies $|\nabla} \def\pp{\partial} \def\E{\mathbb E P_t f|\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{Kt}P_t|\nabla} \def\pp{\partial} \def\E{\mathbb E f|$ (see e.g. \cite{W10}), by Proposition \ref{T3.1} we have $$W_2((f\mu)P_t,\mu)=W_2((f\mu) P_t, \mu P_t)\le c_2 W_2(f\mu,\mu),\ \ t\in [0,1], f\ge 0, \mu(f)=1.$$ Combining with \eqref{SL} yields $$W_2( (f\mu) P_t,\mu)\le c\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} W_2(f\mu,\mu),\ \ t\ge 0, f\ge 0, \mu(f)=1$$ for some constants $c,\ll>0$. Therefore, the proof of (3) is finished. \end{proof}
\section{Proof of Theorem \ref{T1.2} and Corollary \ref{C1.3}}
\begin} \def\beq{\begin{equation}} \def\F{\scr F{proof}[Proof of Theorem \ref{T1.2}] (1) Since $\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z\ge -K$ for some constant $K\ge 0$, we have (see e.g. \cite{W10})
$$|\nabla} \def\pp{\partial} \def\E{\mathbb E P_t f|\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{Kt}P_t|\nabla} \def\pp{\partial} \def\E{\mathbb E f|,\ \ f\in C_b^1(M).$$ Combining this with Proposition \ref{P2.1} for $q=1$ and noting that $P_t|\nabla} \def\pp{\partial} \def\E{\mathbb E f|$ is continuous, we obtain
$$|\nabla} \def\pp{\partial} \def\E{\mathbb E P_t f| \le c_0\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} P_t|\nabla} \def\pp{\partial} \def\E{\mathbb E f|,\ \ t\ge t_0, f\in C_b^1(M)$$ for some constants $c_0,\ll,t_0>0$. Obviously, \eqref{*NP} implies
$$\|\cdot\|_{L^1(P_t)}\le \ff{\|\cdot\|_{L_*^\Phi(P_t)}}{\Phi^{-1}(1)},\ \ \Phi\in \bar {\scr N}.$$ Then
$$|\nabla} \def\pp{\partial} \def\E{\mathbb E P_t f| \le \ff{c_0}{\Phi^{-1}(1)}\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \|\nabla} \def\pp{\partial} \def\E{\mathbb E f\|_{L_*^\Phi(P_t)},\ \ t\ge 0, \Phi\in \bar{\scr N}, f\in C_b^1(M).$$
According to Proposition \ref{T3.1}, this is equivalent to \beq\label{WW1} W_{\Phi}( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t)\le c_0 \Phi^{-1}(1) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \rr(x,y),\ \ t\ge 0, x,y\in M. \end{equation} On the other hand, noting that
$$\scr C( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_xP_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t)\ni \pi_t:=( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_t)\times ( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t) \le \|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}^2 (\mu\times \mu),$$ we obtain
$$W_{\Phi}( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_y P_t)\le \|\rr\|_{L^{\Phi}(\pi_t)} \le G_\Phi(2t),\ \ t>0. $$
Combining this with \eqref{WW1} and the semigroup property, we arrive at \begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} W_{\Phi}( \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_xP_t, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_yP_t) \le\ff{c_0}{\Phi^{-1}(1)}\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t/2} W_\Phi(\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_x P_{t/2}, \delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho_y P_{t/2})
\le \ff{c_0}{\Phi^{-1}(1)}\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t/2} G_\Phi(t).\end{align*} This together with \eqref{WW1} implies \eqref{LL0} for some constants $c,\ll>0.$ Moreover, \eqref{LL1} follows from \eqref{LL0} according to Proposition \ref{T3.1}.
(2) By Proposition \ref{T3.1}, \eqref{WIF} implies
$$|\nabla} \def\pp{\partial} \def\E{\mathbb E P_t f|\le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E f|,\ \ \ t\ge 0, f\in C_b^1(M).$$ Then using the standard semigroup calculation of Bakry-Emery, this implies
\begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} &P_t (f^2\log f^2) - (P_tf^2)\log P_t f^2 = \int_0^t \ff{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D}{\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s} P_s\big\{(P_{t-s} f^2)\log P_{t-s} f^2\big\}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \\
&= \int_0^t P_s\Big(\ff{|\nabla} \def\pp{\partial} \def\E{\mathbb E P_{t-s} f^2|^2}{P_{t-s} f^2}\Big) \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \le 4 c^2 \int_0^t \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-2\ll(t-s)} P_s \Big(\ff{(P_{t-s} \{f|\nabla} \def\pp{\partial} \def\E{\mathbb E f|\})^2}{P_{t-s} f^2}\Big)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s \\
&\le 4 c^2 \int_0^t \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{2\ll (t-s)} (P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s = \ff{2c^2 (1-\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-2\ll t})}{\ll} P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2,\ \ t\ge 0.\end{align*}
Since $\lim_{t\rightarrow}\def\l{\ell}\def\iint{\int\infty} P_t g= \mu(g)$ for $g\in \B_b(M)$ due to the ergodicity, by letting $t\rightarrow}\def\l{\ell}\def\iint{\int\infty$ we prove the log-Sobolev inequality for
\eqref{LS} for $C= \ff{2c^2}\ll.$ \end{proof}
\begin} \def\beq{\begin{equation}} \def\F{\scr F{proof}[Proof of Corollary \ref{C1.3}] We first observe that the proof of \cite[Theorem 4.2]{WAnn} works also for the non-symmetric case with $\nabla} \def\pp{\partial} \def\E{\mathbb E Z$ in place of $\Hess_V$, so that
\beq\label{ULT} \|P_t\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)} \le \exp\Big[c + \ff c t\Big(1+\LL_1^{-1}(ct^{-1})+ \LL_2^{-1}(c^{-1}t)\Big)\Big],\ \ t>0.\end{equation}
Since in the symmetric case we have $\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}\le \|P_{t/2}\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}^2$, the first assertion follows immediately from Theorem \ref{T1.2}.
As for the non-symmetric case, since $$\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}\le \|P_{t/2}\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^2(\mu)}\|P_{t/2}\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)},$$ by Theorem \ref{T1.2} and \eqref{ULT} it suffices to prove
\beq\label{UU} \|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^2(\mu)}\le c'+c'H(c't^{-1}),\ \ t>0\end{equation} for some constant $c'>0.$ According to \cite[Theorem 2.1]{RW}, \eqref{ULT} implies the super log-Sobolev inequality \eqref{SLS} for $$\bb(r):= c+ \ff c r\Big\{1+\LL_1^{-1}(cr^{-1})+ \LL_2^{-1}(c^{-1}r)\Big\},\ \ r>0$$ for some (possibly different) constant $c>0$. Then Proposition \ref{PN} with $p=1,q=2$ and $\gg(r):=\ff{trh(r-1)}{(r-1)\int_0^1 s^{-1}h(s) \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D s}$ implies \eqref{UU}.
\end{proof}
\section{Proofs of Theorems \ref{ST1}-\ref{ST2} and Proposition \ref{PPN}}
\begin} \def\beq{\begin{equation}} \def\F{\scr F{proof}[Proof of Theorems \ref{ST1}] Let $X_t(x)$ solve \eqref{SDE} with initial point $x$. By It\^o's formula and condition \eqref{DSS} we obtain
\begin} \def\beq{\begin{equation}} \def\F{\scr F{align*}&\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D |X_t(x)-X_t(y)|^p\\
&\le \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_t + p |X_t(x)-X_t(y)|^{p-2}\bigg\{\ff{(p-2)|(\sigma} \def\ess{\text{\rm{ess}}(X_t(x))-\sigma} \def\ess{\text{\rm{ess}}(X_t(y))^*(X_t(x)-X_t(y))|^2}{|X_t(x)-X_t(y)|^2}\\
&\quad +\|\sigma} \def\ess{\text{\rm{ess}}(X_t(x))-\sigma} \def\ess{\text{\rm{ess}}(X_t(y))\|^2_{HS} + 2 \<b(X_t(x)-b(X_t(y)), X_t(x)-X_t(y)\>\bigg\}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\\
&\le \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_t - p K_p|X_t(x)-X_t(y)|^p \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t\end{align*} for some martingale $M_t$. This implies
$$\E|X_t(x)-X_t(y)|^p\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-pK_p t}|x-y|^p,\ \ t\ge 0, x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d,$$ and thus,
\beq\label{LGR} \begin} \def\beq{\begin{equation}} \def\F{\scr F{split} |\nabla} \def\pp{\partial} \def\E{\mathbb E P_tf(x)| &\le \limsup_{y\rightarrow}\def\l{\ell}\def\iint{\int x} \E\Big(\ff{|f(X_t(x))-f(X_t(y))|}{|X_t(x)-X_t(y)|}\cdot \ff{|X_t(x)-X_t(y)|}{|x-y|}\Big)\\
&\le \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-K_pt} (P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E f|^{\ff p{p-1}})^{\ff{p-1}p}.\end{split}\end{equation} Then the desired assertion follows from Proposition \ref{T3.1}. \end{proof}
\begin} \def\beq{\begin{equation}} \def\F{\scr F{proof}[Proof of Theorem \ref{ST2}] (1) We reformulate \eqref{SDE} as \beq\label{SDE'} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D X_t= b(X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t +\ss 2\big(\sigma} \def\ess{\text{\rm{ess}}_0(X_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t' + \ll_0\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t''\big),\end{equation} where $B_t'$ and $B_t''$ are independent $d$-dimensional Brownian motions. For any $x\ne y$, let $X_t$ solve this SDE with $X_0=x$, and let $Y_t$ solve the following coupled SDE with $Y_0=y$:
$$ \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D Y_t= b(Y_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t +\ss 2 \,\sigma} \def\ess{\text{\rm{ess}}_0(Y_t)\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t' + \ll_0\ss 2\,\bigg(\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t''- 2 \ff{\<X_t-Y_t, \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t''\>(X_t-Y_t)}{|X_t-Y_t|^2}\bigg).$$ That is, under the flat metric we have made coupling by reflection for $B_t''$ and coupling by parallel displacement for $B_t'$. Obviously, the coupled SDE has a unique solution up to the coupling time $$T_{x,y}:= \inf\{t\ge 0: X_t=Y_t\}.$$ We set $Y_t=X_t$ for $t\ge T_{x,y}$ as usual. Then by \eqref{DSS2} and It\^o's formula, we obtain
\beq\label{MM} \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D |X_t-Y_t|\le \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_t + \big\{(K_1+K_2)1_{\{|X_t-Y_t|\le r_0\}} - K_2\big\} |X_t-Y_t|\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t,\ \ t\le T_{x,y}\end{equation} for
$$\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D M_t:= \ff{\ss 2\<2\ll_0 \text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t'' + (\sigma} \def\ess{\text{\rm{ess}}_0(X_t)-\sigma} \def\ess{\text{\rm{ess}}_0(Y_t))\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D B_t', X_t-Y_t\> }{|X_t-Y_t|}$$ being a martingale with \beq\label{MM'}\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D \<M\>_t\ge 8\ll_0^2\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D t.\end{equation} By repeating the argument leading to \eqref{DD}, it is easy see that \eqref{MM} and \eqref{MM'} imply
$$\E |X_t-Y_t|\le c\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} |x-y|,\ \ t\ge 0$$ for some constants $c,\ll>0$ independent of $x,y$. Therefore,
$$|\nabla} \def\pp{\partial} \def\E{\mathbb E P_t f|\le c \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} \|\nabla} \def\pp{\partial} \def\E{\mathbb E f\|_\infty,\ \ t\ge 0, f\in C_b^1(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d),$$ so that the first assertion follows from Proposition \ref{T3.1}.
(2) According to \cite[Theorem 1.1]{W11}, $a\ge \aa I$ and \eqref{DSS2'} imply the log-Harnack inequality
$$P_t (\log f )(x) \le \log P_tf (y) + \ff{c_1|x-y|^2}{1-\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-c_2t}},\ \ t\ge 0, x,y\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d, 0\le f\in \B_b(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$$ for some constants $c_1,c_2>0$. Combining this with the log-Sobolev inequality, we prove the second assertion as in (c) in the proof of Theorem \ref{T1.1}.
(3) According to the proof of Theorem \ref{ST1}, the condition \eqref{DSS} implies the gradient estimate \eqref{LGR}. Next, by Proposition \ref{P2.1}, the ultracontractivity and \eqref{LGR} imply
$$|\nabla} \def\pp{\partial} \def\E{\mathbb E P_t f|\le c(p) \text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{-\ll t} (P_t |\nabla} \def\pp{\partial} \def\E{\mathbb E f|^{\ff p{p-1}})^{\ff{p-1}p},\ \ t\ge 0, f\in C_b^1(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$$ for some $c(p)>0$ and $\ll>0$ independent of $p$. Then the proof if finished by Proposition \ref{T3.1}. \end{proof}
\begin} \def\beq{\begin{equation}} \def\F{\scr F{proof}[Proof of Proposition \ref{PPN}] We will apply results in \cite{RW} and \cite{WAD}. To this end, we introduce the Riemannian metric $$g(\pp_i,\pp_j)= g_{ij}:= (a^{-1})_{i,j},\ \ 1\ge i,j\le d,$$ and let $\DD^g, \nabla} \def\pp{\partial} \def\E{\mathbb E^g, \Hess^g$ be the corresponding Laplacian, gradient and Hessian tensor respectively. Then $L=\DD^g+Z$ for some $C^1$ vector field $Z$. We first verify the Bakry-Emery curvature condition \eqref{C} for some constant $K$. Using the Christoffel symbols, the intrinsic Hessian tensor induced by $g$ is formulated as $$\Hess_f^g(\pp_i,\pp_j)= \pp_{ij}^2 f - \sum_{k=1}^d \GG_{ij}^k\pp_k f.$$ So, for any $x\in \mathbb R} \def\ff{\frac} \def\ss{\sqrt^d$ and $f\in C^2(\mathbb R} \def\ff{\frac} \def\ss{\sqrt^d)$ with $\Hess^g_f(x)=0$, we have $$\pp_{ij}^2f(x)= \sum_{n=1}^d \GG_{ij}^n \pp_n f(x),\ \ 1\le i,j\le d.$$
Thus, by Bochner-Weitzenb\"ock formula and \eqref{CC0}, at point $x$ there holds
\begin} \def\beq{\begin{equation}} \def\F{\scr F{align*} &\text{\rm{Ric}}} \def\Hess{\text{\rm{Hess}}_Z(\nabla} \def\pp{\partial} \def\E{\mathbb E^g f,\nabla} \def\pp{\partial} \def\E{\mathbb E^g f) + K_0|\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2 = \ff 1 2 L\<a\nabla} \def\pp{\partial} \def\E{\mathbb E f, \nabla} \def\pp{\partial} \def\E{\mathbb E f\>- \<a\nabla} \def\pp{\partial} \def\E{\mathbb E f, \nabla} \def\pp{\partial} \def\E{\mathbb E Lf\>+K_0|\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2\\
&\ge \ff 1 2\sum_{i,j,k,l=1}^d a_{kl} \Big[(\pp^2_{kl} a_{ij}) (\pp_if)(\pp_jf)+ 2 a_{ij} (\pp_{ki}^2 f) (\pp_{il}^2f)
+2 (\pp_l a_{ij})\big\{(\pp^2_{ki}f)\pp_j f- (\pp_{ij}^2f)\pp_k f\big\}\Big] \\ & = \ff 1 2 \sum_{i,j,k,l=1}^da_{kl} \Big[ (\pp^2_{kl} a_{ij}) (\pp_if)(\pp_jf) +2 (\pp_l a_{ij})\sum_{n=1}^d (\pp_n f)\big\{\GG_{ki}^n \pp_i f- \GG_{ij}^n \pp_k f\big\}\Big] \\
&\ge - K_1|\nabla} \def\pp{\partial} \def\E{\mathbb E f|^2 \ge -\ff{K_1}\aa \<a\nabla} \def\pp{\partial} \def\E{\mathbb E f,\nabla} \def\pp{\partial} \def\E{\mathbb E f\>=-\ff{K_1}\aa g(\nabla} \def\pp{\partial} \def\E{\mathbb E^g f,\nabla} \def\pp{\partial} \def\E{\mathbb E^gf) \end{align*} for some constant $K_1$. Then \eqref{C} hold for some constant $K$.
Next, \eqref{UL} implies that $P_t$ has a unique invariant probability measure $\mu$
such that $\mu(\text{\rm{e}}} \def\ua{\underline a} \def\OO{\Omega} \def\oo{\omega^{c_2 |\cdot|^2})<\infty$ for some $c_2>\ff K {2\aa}$. By our assumption on $a$, the Riemannian distance $\rr $ induced by the metric $g$ is equivalent to the Euclidian metric:
\beq\label{EQ} \ff 1 {\|a\|_\infty} |\cdot|^2\le \rr_a^2(0,\cdot)\le\ff 1 \aa |\cdot|^2.\end{equation} Then we may repeat
the proof of \cite[Corollary 2.5]{RW} with $\gg(r)= c_2 r^{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho} $ and $\rr=|\cdot|$ to prove
\beq\label{*D5} \|P_t\|_{L^2(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)} \le \exp\Big[c_3 t^{-\ff{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho}{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho-1}}\Big],\ \ t>0\end{equation} for some constant $c_3>0.$ Combining this with the curvature condition \eqref{C}, we obtain from \cite[Theorem 2.1]{RW} for $p=2$ and $q=\infty$ that $$\mu(f^2\log f^2)\le r \scr E}\def\Cut{{\rm Cut}(f,f) + c_4 r^{-\ff{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho}{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho-1}},\ \ r\in (0,1), \mu(f^2)=1$$ holds for some constant $c_4>0$. Applying Proposition \ref{PN} below for $p=1, q=2$ and $\gg(r)= c_5 t (r-1)^{\ff{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho-1}{2\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho}-1}$ for constant $c_5>0$ such that $t= \int_1^2 \ff{\gg(r)}r\text{\rm{d}}} \def\bb{\beta} \def\aa{\alpha} \def\D{\scr D r$, we obtain
$$\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^2(\mu)}\le \exp\Big[c_6 t^{-\ff{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho}{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho-1}}\Big],\ \ t\in (0,1)$$ for some constant $c_6>0$. Combining this with \eqref{*D5} we arrive at
$$\|P_t\|_{L^1(\mu)\rightarrow}\def\l{\ell}\def\iint{\int L^\infty(\mu)}\le c_7\exp\Big[c_7 t^{-\ff{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho}{\delta} \def\DD{\Delta} \def\vv{\varepsilon} \def\rr{\rho-1}}\Big],\ \ t>0$$ for some constant $c_7>0$.
\end{proof}
\paragraph{Acknowledgement.} The author would like to thank Jian Wang for helpful comments.
\end{document}
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arXiv
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Asian Pacific Journal of Cancer Prevention
Pages.7561-7566
Asian Pacific Organization for Cancer Prevention (아시아태평양암예방학회)
Dietary Ziziphus jujuba Fruit Influence on Aberrant Crypt Formation and Blood Cells in Colitis-Associated Colorectal Cancer Mice
Periasamy, Srinivasan (Department of Environmental and Occupational Health, College of Medicine, National Cheng Kung University) ;
Liu, Chung-Teng (Department of Environmental and Occupational Health, College of Medicine, National Cheng Kung University) ;
Wu, Wang-Hung (Department of Environmental and Occupational Health, College of Medicine, National Cheng Kung University) ;
Chien, Se-Ping (Department of Food and Beverage Service, Tainan University of Technology) ;
Liu, Ming-Yie (Department of Environmental and Occupational Health, College of Medicine, National Cheng Kung University)
https://doi.org/10.7314/APJCP.2015.16.17.7561
Ziziphus jujuba (ZJ) fruit is rich in bioactive functional components such as polysaccharides, triterpenoid acid, flavonoids and oleamide. It has been commonly used in the treatment of various diseases including diabetes, digestive disorders, diarrhea, skin infections, liver and urinary diseases. However, its dietary effect on chemoprevention of colon cancer has never been studied. The present study was to evaluate the protective effects of dietary ZJ on colitis-associated colon carcinogenesis in azoxymethane (AOM)-dextran sodium sulphate (DSS)-treated mice. AOM was injected (10 mg/kg b.wt., i.p.) and three cycles of 2% DSS in drinking water for 7 days with 14 days of normal drinking water in-between was administered to induce colitis-associated colon cancer. ZJ fruit was supplemented in feed as 5 and 10%. Dietary ZJ significantly attenuated aberrant crypt foci (ACF) formation thereby decreasing the progression of hyperplasia to dysplasia. In addition, it significantly reduced circulating white blood cells, lymphocytes, neutrophils, monocytes, eosinophils, basophils and platelets compared to colon cancer mice. We conclude that ZJ supplementation delayed the progression of colon cancer from hyperplasia to dysplasia and ultimately adenocarcinoma and cancer. In addition, it decreased circulating tumor-related leucocytes, main regulators of cancer inflammation. Therefore, dietary consumption of ZJ fruit attenuated the formation of ACF and delayed the progression of colon cancer.
Ziziphus jujuba;colon cancer;azoxymethane;blood cells;aberrant crypt foci
Supported by : Taiwan Ministry of Sciences and Technology
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CommonCrawl
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\begin{document}
\setlength{\parindent}{0pt} \setlength{\parskip}{7pt}
\title[Derived Equivalences of Upper Triangular DGA's] {Derived Equivalences of Upper Triangular Differential Graded Algebras} \author{Daniel Maycock}
\maketitle \section{Introduction} The question of when two derived categories of rings are equivalent has been studied extensively. Morita theory answered the question of when two module categories of rings are equivalent and a version of Morita theory for derived categories was developed by Rickard in \cite{Rik} which made use of the concept of tilting modules. This approach was applied by Ladkani in \cite{Lad} to the situation of derived equivalences of upper triangular matrix rings. In this paper we extend the main results from \cite{Lad} to the more general case of upper triangular matrix differential graded algebras (henceforth referred to as DGAs). For this we will make extensive use of the tool of recollements and in particular the situation given by J\o rgensen in \cite{Jor}.
Section 2 sets out the notation used. We begin properly in section 3 by introducing the upper triangular matrix DGA $\Lambda$ which has the form $\begin{bmatrix}R&M\\0&S\end{bmatrix}$, where $R$ and $S$ are DGAs and $\leftidx{_R}{M}{_S}$ is a $R$-$S$-DG-bimodule. There are also the left-DG-modules $B=\begin{bmatrix}R\\0\end{bmatrix}$ and $C=\begin{bmatrix}M\\S\end{bmatrix}$ which we will use throughout the paper. Next, by proving some properties of $B$ and $C$, we are able to use the main result from \cite{Jor} to obtain the recollement $$\xymatrix@C5pc{D(R) \ar[r]^{i_*} & D(\Lambda) \ar@/^2pc/[l]^{i^!} \ar@/_2pc/[l]_{i^*} \ar[r]^{j^*} & D(S) \ar@/^2pc/[l]^{j_*} \ar@/_2pc/[l]_{j_!}}$$ where $i_*(_RR)\cong B$ and $j_!(_SS)\cong C$. We then conclude the section by presenting some useful results obtained from the recollement which we will need in the next section.
In section 4 we turn our attention to the main aim of the paper, generalizing the main theorem from Ladkani to DGA's. To do so we follow a similar idea as used in the proof of \cite[Theorem 4.5]{Lad}, by considering the DG-module $T=\Sigma i_*X\oplus j_*j^*\Lambda$ where $X$ is compact and $\langle X\rangle=D(R)$, where $\langle X\rangle$ denotes the smallest triangular subcategory containing $X$ which is closed under the taking of coproducts. We begin with a statement of Keller's theorem which we will require to prove the following theorem, our ``first attempt'' at generalising \cite[Theorem 4.5]{Lad}. \begin{Theorem*} Let $X$ be a DG $R$-module such that $_RX$ is compact and $\left\langle _RX\right\rangle=D(R)$. Let $_RM_S$ be compact as a DG-$R$-module. Let $T=\Sigma i_*X\oplus j_*j^*\Lambda$ with $\mathscr{E}=\operatorname{End}_\Lambda(P)$, where $P$ is a K-projective resolution of $T$. Then $\mathscr{E}$ is an DGA with $D(\Lambda)\simeq D(\mathscr{E}^{\operatorname{op}})$. \end{Theorem*} We then turn our attention to considering $P$, the K-projective resolution of $T$, and by doing so we are able to calculate its endomorphism DGA, which leads to our generalisation of the main theorem of Ladkani below. \begin{Theorem*} Let $X$ be a DG $R$-module such that $_RX$ is compact and $\left\langle _RX\right\rangle=D(R)$. Let $_RM_S$ be compact as an DG $R$-module and let $U$ and $V$ be K-projective resolutions of $X$ and $M$ respectively. Then for the upper triangular differential graded algebras $$\Lambda=\begin{bmatrix}R&M\\0&S\end{bmatrix}\textrm{ and } \tilde{\Lambda}=\begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}$$ we have that $D(\Lambda)\simeq D(\tilde{\Lambda})$. \end{Theorem*}
One specific advantage of considering the DGA case rather than the ring case is that with the DGA case we can do without a lot of constraints which are required in the ring case to ensure that the derived equivalence is between two triangular matrix rings.
Finally in section 5 we conclude with a look at some special cases. In the first we reconsider the original case in Ladkani, involving just rings and show that by making the same assumptions in our general theorem we obtain the same equivalence.
We then briefly consider what happens in the special case where $\leftidx{_R}{X}=\leftidx{_R}{R}$. In the final example, we require that our DGA's are over some field $k$ and that $R$ is self dual, that is, $\operatorname{Hom}_k(R,k)\cong R$ in the derived category of DG-R-modules. This gives us the following result. \begin{Corollary*} Let $R$ be a self dual finite dimensional DGA and $S$ be a DGA, both over a field $k$. Let $\leftidx{_R}{M}{_S}$ be compact as a DG-$R$-module. Then $$\Lambda=\begin{bmatrix}R&M\\0&S\end{bmatrix} \textrm{ and } \tilde{\Lambda}=\begin{bmatrix}S&DM\\0&R\end{bmatrix}$$ are derived equivalent. \end{Corollary*}
\section{Notation and terminology} In this we will fix the notation which we shall use throughout the paper; more details on DGA's and DG-modules can be found in \cite{Kel} and \cite{Fra}.
Throughout this paper we will make use of Differential Graded Algebras; these are always assumed to be over some commutative ground ring $k$ unless stated otherwise. For any graded object $r$ we will denote its degree by $|r|$. Note that we shall observe the Koszul sign convention so that whenever two graded elements of degrees $m$ and $n$ are interchanged we introduce a sign $(-1)^{mn}$.
For a DGA $R$ we can define the opposite DGA, denoted $R^{\operatorname{op}}$, this is the same as $R$ except that the product is given by $r.s=(-1)^{|r||s|}sr$ where . denotes multiplication in $R^{\operatorname{op}}$. We will often identify DG-right-$R$-modules with DG-left-$R^{\operatorname{op}}$-modules.
We shall often need to consider DG-modules with more than one DG-module structure, for instance a DG-left-$R$-right-$S$-module, denoted by $\leftidx{_R}{M}{_S}$. In these cases the different structures are required to be compatible, for the $\leftidx{_R}{M}{_S}$ case this means that the rule $(rm)s=r(ms)$ holds.
For a DGA $R$ we denote the category of all DG-left-$R$-modules by Mod$\:R$. We define the homotopy category of $R$, which we denote by $K(R)$, as the category consisting of all DG-left-$R$-modules whose morphisms are the morphisms of DG-modules mudololo homotopy. We define the derived category of $R$, denoted by $D(R)$, from $K(R)$ by formally inverting the quasi-isomorphisms. Both $K(R)$ and $D(R)$ are triangulated categories. A more detailed construction of the derived category and details of triangulated categories can be found in \cite{Har}.
Since we can identify DG-right-$R$-modules with DG-left-$R^{\operatorname{op}}$-modules we can also identify the derived category of DG-right-$R$-modules with $D(R^{\operatorname{op}})$.
\section{A Recollement Situation} We begin by defining the differential graded algebras and DG-modules which we will be using throughout the paper.
\begin{Definition} \label{UT DG def} Throughout this paper, let $R$ and $S$ be Differential Graded algebras with $\leftidx{_R}{M}{_S}$ a DG-bimodule which is quasi-isomorphic to $\leftidx{_R}{V}{_S}$ where $V$ is K-projective as a DG-left-$R$-module and let $\Lambda=\begin{bmatrix}R & M\\0 & S \end{bmatrix}$ denote the upper triangular matrix DGA with the differential $\partial^\Lambda\begin{bmatrix}r & m\\0 & s \end{bmatrix} =\begin{bmatrix}\partial^R r & \partial^M m\\0 & \partial^S s\end{bmatrix}$. \end{Definition}
\begin{Remark} In the case where the base ring $k$ is a field we always have that $\leftidx{_R}{M}{_S}$ is quasi-isomorphic to some $\leftidx{_R}{V}{_S}$ with $V$ K-projective as a DG-$R$-module. \end{Remark}
\begin{Definition} Let $e_R=\begin{bmatrix} 1 & 0\\0 & 0\end{bmatrix}$ and $e_S=\begin{bmatrix} 0 & 0\\0 & 1\end{bmatrix}$ and define the DG-left-$\Lambda$-modules $$B=\Lambda e_R=\begin{bmatrix} R\\0\end{bmatrix}\textrm{ and } C=\Lambda e_S=\begin{bmatrix} M\\S\end{bmatrix}$$ where $B$ has the differential $\partial\left(\begin{bmatrix}r\\0\end{bmatrix}\right)=\left(\begin{bmatrix}\partial^Rr\\0\end{bmatrix}\right)$ and $C$ has the differential $\partial\left(\begin{bmatrix}m\\s\end{bmatrix}\right)=\left(\begin{bmatrix}\partial^Mm\\\partial^Ss\end{bmatrix}\right)$. \end{Definition}
The first aim is to construct a recollement involving the objects we have defined above. To do this we shall use \cite[Theorem 3.3]{Jor} but before we can use this theorem we first need the following Lemmas involving the DG-modules $B$ and $C$.
\begin{Definition} For $X$ a full subcategory of a triangulated category $T$, we can define a full subcategory
$$X^\bot=\{Y\in T\:|\:\operatorname{Hom}_T(\Sigma^l X,Y)=0 \textrm{ for all } l\}$$ \end{Definition}
\begin{Lemma} $\Lambda\cong B\oplus C$ in $D(\Lambda)$ and hence both $B$ and $C$ are K-projective DG-left-$\Lambda$-modules which are compact in $D(\Lambda)$. \end{Lemma}
\begin{proof} Define $\Theta : \Lambda \rightarrow B\oplus C$ and $\Phi : B\oplus C \rightarrow \Lambda$ by $$\Theta\left(\begin{bmatrix}r & m\\0 & s \end{bmatrix}\right) = \left(\begin{bmatrix} r\\0\end{bmatrix},\begin{bmatrix} m\\s\end{bmatrix}\right)$$ and $$\Phi \left(\left(\begin{bmatrix} r\\0\end{bmatrix},\begin{bmatrix} m\\s\end{bmatrix}\right)\right) = \begin{bmatrix}r & m\\0 & s \end{bmatrix}.$$
It is obvious that $\Theta$ and$\Phi$ are inverses of each other and it is straightforward to check that they are homomorphisms of DG-modules. So we have that $\Lambda\cong B\oplus C$ as DG-$\Lambda$-modules and hence also in $D(\Lambda)$. \end{proof}
\begin{Lemma} $B\in C^\bot$ as DG-modules and hence in $D(\Lambda)$. \end{Lemma}
\begin{proof} Let $C \stackrel{f}{\rightarrow} B$ be a morphism of DG-modules. It suffices to show that $f=0$ and since $\begin{bmatrix}0\\1\end{bmatrix}$ generates $C$ we only need to show that $f\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=0$.
Let $f\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\begin{bmatrix}r\\0\end{bmatrix}$ for some $r\in R$. Then $$f\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=f\left(e_S.\begin{bmatrix}0\\1\end{bmatrix}\right)=e_Sf\left(\begin{bmatrix}0\\1\end{bmatrix}\right)= \begin{bmatrix}0&0\\0&1\end{bmatrix}\begin{bmatrix}r\\0\end{bmatrix}=0$$ as required, hence $f=0$ and so $B\in C^\bot$. \end{proof}
\begin{Lemma} $B^\bot\cap C^\bot =0$ in $D(\Lambda)$. \end{Lemma}
\begin{proof} Let $X\in B^\bot\cap C^\bot$, then $\operatorname{Hom}_{D(\Lambda)}(\Sigma^iB,X)=0$ and \\$\operatorname{Hom}_{D(\Lambda)}(\Sigma^iC,X)=0$ for each $i$. $$H^iX\cong H^i\operatorname{Hom}_\Lambda(\Lambda,X)\cong \operatorname{Hom}_{K(\Lambda)}(\Lambda,\Sigma^iX)$$ $$\cong \operatorname{Hom}_{D(\Lambda)}(\Lambda,\Sigma^iX) \cong \operatorname{Hom}_{D(\Lambda)}(B\oplus C,\Sigma^iX)$$ $$\cong \operatorname{Hom}_{D(\Lambda)}(B,\Sigma^iX)\oplus \operatorname{Hom}_{D(\Lambda)}(C,\Sigma^iX)$$ $$\cong 0\oplus 0=0$$ for all $i$. Hence we have that $X\cong 0$ in $D(\Lambda)$ and so $B^\bot\cap C^\bot =0$. \end{proof}
We now have shown that $B$ and $C$ satisfy the conditions required to apply \cite[Theorem 3.3]{Jor}. However before we do so we prove the following lemma about the endomorphism DGAs of $B$ and $C$.
\begin{Lemma} Let $\mathscr{F}=\operatorname{End}_\Lambda(B)$ and $\mathscr{G}=\operatorname{End}_\Lambda(C)$, then $\mathscr{F}^{\operatorname{op}}\cong R$ and $\mathscr{G}^{\operatorname{op}}\cong S$ as Differential Graded Algebras. \end{Lemma}
\begin{proof} Since $\begin{bmatrix}1\\0\end{bmatrix}$ is a generator of $B$ each element of $\operatorname{End}_\Lambda(B)$ depends entirely on where it sends $\begin{bmatrix}1\\0\end{bmatrix}$. For each $r\in R$ define the homomorphism $f_r$ as the element of $\mathscr{F}$ which sends $\begin{bmatrix}1\\0\end{bmatrix}$ to $\begin{bmatrix}r\\0\end{bmatrix}$.
We can now define $\phi:R^{\operatorname{op}} \rightarrow \mathscr{F}$ by $\phi(r)=f_r$. Since elements of $\mathscr{F}$ depend entirely on where they send $\begin{bmatrix}1\\0\end{bmatrix}$ this is obviously a bijection. It is also straightforward to show that $\phi$ is a homomorphism and so an isomorphism of DGAs.
Now let $g\in\mathscr{G}$. Since $C$ is generated by $\begin{bmatrix}0\\1\end{bmatrix}$ we know that $g$ depends entirely on where it sends $\begin{bmatrix}0\\1\end{bmatrix}$. Let $g\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\begin{bmatrix}m\\s\end{bmatrix}.$
However $g\left(e_S.\begin{bmatrix}0\\1\end{bmatrix}\right)=g\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=e_Sg\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=e_S\begin{bmatrix}m\\s\end{bmatrix} =\begin{bmatrix}0\\s\end{bmatrix}$, so $m=0$ and hence $g\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\begin{bmatrix}0\\s\end{bmatrix}$. So for each $s\in S$ we can define the homomorphism $g_s\in\mathscr{G}$ as the element of $\mathscr{G}$ which sends $\begin{bmatrix}0\\1\end{bmatrix}$ to $\begin{bmatrix}0\\s\end{bmatrix}.$
Hence we can define a map $\theta:S^{\operatorname{op}} \rightarrow \mathscr{G}$ sending $s\mapsto g_s$ which is easily shown to be an isomorphism and so $S^{\operatorname{op}}\cong\mathscr{G}$. \end{proof}
By taking the above lemmas together with \cite[Theorem 3.3]{Jor} we get the following recollement:
$$\xymatrix@C5pc{D(R) \ar[r]^{i_*} & D(\Lambda) \ar@/^2pc/[l]^{i^!} \ar@/_2pc/[l]_{i^*} \ar[r]^{j^*} & D(S) \ar@/^2pc/[l]^{j_*} \ar@/_2pc/[l]_{j_!}}.$$
Five of the functors are given by $$\begin{array}{ll} &j_!(\_)=\leftidx{_\Lambda}{C}{_S}\stackrel{L}{\otimes}_S\_,\\ i_*(\_)=\leftidx{_\Lambda}{B}{_R}\stackrel{L}{\otimes}_R\_,&j^*(\_)=\operatorname{RHom}_\Lambda(_{\Lambda}C_S,\_),\\ i^!(\_)=\operatorname{RHom}_\Lambda(_{\Lambda}B_R,\_),&j_*(\_)=\operatorname{RHom}_S(_SC^*_\Lambda,\_). \end{array}$$
Here $_SC^*_\Lambda =\operatorname{RHom}_\Lambda (_\Lambda C_S,\Lambda)$.
In particular, $i_*(R)\cong B$ and $j_!(S)\cong C.$
We shall now end this section with a number of results, involving the recollement we have constructed, which we will find to be of great use in the next section.
\begin{Remark} The functor $i_*(-)=\leftidx{_\Lambda}{B}{_R}\stackrel{L}{\otimes}_R\_$ sends a DG-$R$-module $X$ to the DG-$\Lambda$-module $\begin{bmatrix}X\\0\end{bmatrix}$. \end{Remark}
\begin{Proposition} \label{Lem: C* K-proj} For $_SC^*_\Lambda =\operatorname{RHom}_\Lambda (_\Lambda C_S,\Lambda)$ we have that: \begin{enumerate}
\item $C^*\cong _S\!\begin{bmatrix}0 & S\end{bmatrix}_\Lambda$ as DG-left-$S$-right-$\Lambda$-modules where $\begin{bmatrix}0&S\end{bmatrix}$ has the differential
$$\partial^{\left[\begin{smallmatrix}0 & S\end{smallmatrix}\right]}\left(\begin{bmatrix}0 & s\end{bmatrix}\right)
=\begin{bmatrix}0 & \partial^Ss \end{bmatrix}.$$
\item $\leftidx{_S}{C}{_\Lambda^*}$ is a K-projective object over both $S$ and $\Lambda$. \end{enumerate}
\end{Proposition}
\begin{proof} (i) First observe that $C$ is generated by $\begin{bmatrix} 0\\ 1\end{bmatrix}$ and that $$\operatorname{RHom}_\Lambda (_\Lambda C_S,\Lambda)\simeq \operatorname{Hom}_\Lambda (_\Lambda C_S,\Lambda)$$ since $C$ is K-projective over $\Lambda$.
Let $\theta\in\operatorname{Hom}_\Lambda (C,\Lambda)$ such that $\theta\left(\begin{bmatrix} 0\\ 1\end{bmatrix}\right)= \begin{bmatrix}r & m\\0 & s \end{bmatrix} \in \Lambda$.
However $$\theta\left(\begin{bmatrix} 0\\ 1\end{bmatrix}\right)= \theta\left(e_S.\begin{bmatrix} 0\\ 1\end{bmatrix}\right)=e_S.\theta\left(\begin{bmatrix} 0\\ 1\end{bmatrix}\right)=e_S\begin{bmatrix}r & m\\0 & s \end{bmatrix}=\begin{bmatrix}0 & 0\\0 & s \end{bmatrix}.$$
So $\theta\left(\begin{bmatrix} 0\\ 1\end{bmatrix}\right)= \begin{bmatrix}0 & 0\\0 & s \end{bmatrix}$.
So for every $s\in S$ we can define $\theta_s\in\operatorname{Hom}_\Lambda(C,\Lambda)$ to be the element which sends $\begin{bmatrix} 0\\ 1\end{bmatrix}$ to $\begin{bmatrix}0 & 0\\0 & s \end{bmatrix}$.
We can now use this to define an map $\Theta:\operatorname{Hom}_\Lambda(C,\Lambda)\rightarrow \begin{bmatrix} 0 & S \end{bmatrix}$ given by $\Theta(\theta_s)=\begin{bmatrix} 0 & s \end{bmatrix}$. This map is obviously a bijection and it is straightforward to check that it is an isomorphism of DG-left-S-right-$\Lambda$-modules.
(ii) To see that $C^*$ is K-projective over $S$ we observe that $C^*\cong S$ as $S$-modules. It remains to show now that $C^*$ is also K-projective over $\Lambda$. We do this by showing that $C^*\cong\begin{bmatrix} 0 & S \end{bmatrix}$ is a direct summand of $\Lambda$ as DG-right-$\Lambda$-modules.
First observe that $\begin{bmatrix} R & M \end{bmatrix}$ is a DG-right-$\Lambda$-module with the differential $\partial^ {\left[\begin{smallmatrix} r & m \end{smallmatrix}\right]}= \begin{bmatrix} \partial^Rr & \partial^Mm \end{bmatrix}$.
Now define $\Phi:\Lambda_\Lambda\rightarrow\begin{bmatrix} R & M \end{bmatrix}_\Lambda\oplus\begin{bmatrix} 0 & S \end{bmatrix}_\Lambda$ such that $$\Phi\left(\begin{bmatrix} r & m \\ 0 & s \end{bmatrix}\right)=\left(\begin{bmatrix} r & m \end{bmatrix},\begin{bmatrix} 0 & s \end{bmatrix}\right).$$
It is clear to see that $\Phi$ is bijective and a homomorphism of DG-modules. So $C^*_\Lambda\cong\begin{bmatrix} 0 & S \end{bmatrix}_\Lambda$ is a direct summand of $\Lambda_\Lambda$ and so is a K-projective DG-right-$\Lambda$-module.
\end{proof}
\begin{Lemma} \label{useful facts} In the set up of the recollement we have that: \begin{enumerate}
\item $j^*(\Lambda)\cong\leftidx{_S}{S}$ in $D(S)$,
\item $j_*(_SS)\cong \dfrac{\leftidx{_\Lambda}{C}}{\leftidx{_\Lambda}{\left[\begin{smallmatrix} M\\0\end{smallmatrix}\right]}{}}$ in $D(\Lambda)$. \end{enumerate} \end{Lemma} \begin{proof} (i) $j^*(\Lambda)=\operatorname{RHom}_\Lambda(\leftidx{_\Lambda}{C}{_S},\Lambda)=\leftidx{_S}{C}{^*}\cong\leftidx{_S}{S}$.
(ii) Since $C^*$ is a K-projective $S$-module we have that $$j_*(_SS)=\operatorname{RHom}_S(_SC^*_\Lambda,_SS)\cong \operatorname{Hom}_S(_SC^*_\Lambda,_SS)\cong \operatorname{Hom}_S(_S\begin{bmatrix} 0 & S \end{bmatrix}_\Lambda,_SS).$$
Now observe that $\leftidx{_S}{\begin{bmatrix} 0 & S \end{bmatrix}}$ is generated by $\begin{bmatrix} 0 & 1 \end{bmatrix}$ and for all $s\in S$ define $\phi_s \in \operatorname{Hom}_S(\leftidx{_S}{C}{^*_\Lambda},\leftidx{_S}{S})$ to be the element which sends $\begin{bmatrix} 0 & 1 \end{bmatrix}$ to $s$. We can now define the map $\Phi : \operatorname{Hom}_S(\leftidx{_S}{C}{^*_\Lambda},\leftidx{_S}{S}) \rightarrow \leftidx{_\Lambda}{C}/\leftidx{_\Lambda}{\left[\begin{smallmatrix} M\\0\end{smallmatrix}\right]}$ given by $\Phi(\phi_s)=\overline{\begin{bmatrix} 0\\s\end{bmatrix}}$, where $\overline{\begin{bmatrix} 0\\s\end{bmatrix}}$ denotes the element $\begin{bmatrix} 0\\s\end{bmatrix}+\begin{bmatrix}M\\0\end{bmatrix}$ in $\leftidx{_\Lambda}{C}/\leftidx{_\Lambda}{\left[\begin{smallmatrix} M\\0\end{smallmatrix}\right]}$. This is obviously a bijection and it is easy to show that it is a homomorphism of DG-$\Lambda$-modules. \end{proof}
\section{Derived Equivalences of Upper Triangular DGA's}
We are now almost in the position where we can make a start on what is the main aim of the paper, to obtain a generalised version of \cite[Theorem 4.5]{Lad} for upper triangular DGAs. In Theorem \ref{1st attempt}, which is the first major step towards our goal, we obtain a dervived equivalence between $D(\Lambda)$ and $D(\mathscr{E})$, where $\mathscr{E}$ is the endomorphism DGA of a K-projective resoultion of the DG-module $T=\Sigma i_*X\oplus j_*j^*\Lambda$. We then follow this up by constructing a K-projective resoultion for $T$ in proposition \ref{Structure of P} which in turn is followed by the structure of the endomorphism DGA $\mathscr{E}$ in Proposition \ref{structure of E}. The remainder of this section is involved in the details of computing an quasi-isomorphisms between the DGA $\mathscr{E}$ and the upper triangular matrix DGA $\tilde{\Lambda}=\begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}$ with the final result being Theorem \ref{Main}, the main result of the paper which gives us a derived equivalnence between the upper triangular matrix DGAs $\Lambda$ and $\tilde{\Lambda}$.
We start however with a statement of Keller's Theorem which we use in the proof of Theorem \ref{1st attempt}.
\begin{Theorem}[Keller's Theorem] \label{Keller's Theorem} Let $A$ be a DGA and let $N$ be a K-projective DG $A$-module which is compact in $D(A)$ such that $\langle N\rangle=D(A)$ and let $\mathscr{H}=\operatorname{End}_A(N,N)$. Then $D(A)\simeq D(\mathscr{H}^{\operatorname{op}})$. \end{Theorem}
\begin{proof} See \cite{Kel}. \end{proof}
We are now able to make our first attempt at generalising \cite[Theorem 4.5]{Lad} for DGAs. For this we follow a similar method by introducing a DG-$\Lambda$-module $T=\Sigma i_*X\oplus j_*j^*\Lambda$.
\begin{Theorem} \label{1st attempt} Let $X$ be a DG $R$-module such that $_RX$ is compact and $\left\langle _RX\right\rangle=D(R)$. Let $\leftidx{_R}{M}{_S}$ be compact as a DG-$R$-module. Let $\mathscr{E}=\operatorname{End}_\Lambda(P)$, where $P$ is a K-projective resolution of $T=\Sigma i_*X\oplus j_*j^*\Lambda$. Then $\mathscr{E}$ is an DGA with $D(\Lambda)\simeq D(\mathscr{E}^{\operatorname{op}})$. \end{Theorem}
\begin{proof} Our aim to to apply Kellers theorem. To do this we need to show that $T$ is compact and that $\left\langle T \right\rangle\cong D(\Lambda)$. We begin with the compactness of $T$.
Since $T$ is a direct sum it is sufficent to show that both its direct summands $i_*X$ and $j_*j^*\Lambda$ are compact.
To show that $i_*X$ is compact we first note that by adjointness $$\operatorname{Hom}_{D(\Lambda)}(i_*X,\coprod A_k)\simeq\operatorname{Hom}_{D(R)}(X,i^!(\coprod A_k))$$ and that $i^!(\coprod A_k)=\operatorname{RHom}_\Lambda(B,\coprod A_k)$. Also, since $B$ is compact it is not hard to show that $i^!(\coprod A_k)\cong\coprod i^!(A_k)$ in $D(\Lambda)$.
We therefore have that $$\operatorname{Hom}_{D(\Lambda)}(i_*X,\coprod A_k)\simeq\operatorname{Hom}_{D(R)}(X,i^!(\coprod A_k))$$ $$\cong\operatorname{Hom}_{D(R)}(X,\coprod i^! A_k)\cong\coprod\operatorname{Hom}_{D(R)}(X,i^! A_k)$$ $$\simeq \coprod\operatorname{Hom}_{D(\Lambda)}(i_*X,A_k).$$ So $i_*X$ is compact as required.
To show that $j_*j^*\Lambda$ is compact we observe from Lemma \ref{useful facts} that $j_*j^*\Lambda\cong \dfrac{C}{\left[\begin{smallmatrix} M\\0\end{smallmatrix}\right]}$. We know that $C$ is compact and since there is a distinguished triangle $\begin{bmatrix} M\\0\end{bmatrix}\rightarrow C\rightarrow \dfrac{C}{\left[\begin{smallmatrix} M\\0\end{smallmatrix}\right]}$ in $D(\Lambda)$ it is sufficent to show that $\begin{bmatrix} M\\0\end{bmatrix}$ is compact.
Since $\begin{bmatrix} M\\0\end{bmatrix}\cong i_*M=B\stackrel{L}{\otimes}_RM$ and both $B$ and $M$ are compact we have that $$\operatorname{Hom}_{D(\Lambda)}(B\stackrel{L}{\otimes}_R M,\coprod A_k)\cong H^0\operatorname{RHom}_\Lambda(B\stackrel{L}{\otimes}_R M,\coprod A_k)$$ $$\cong H^0\operatorname{RHom}_R(M,\operatorname{RHom}_\Lambda(B,\coprod A_k))\cong H^0\operatorname{RHom}_R(M,\coprod\operatorname{RHom}_\Lambda(B,A_k))$$ $$\cong \coprod H^0\operatorname{RHom}_R(M,\operatorname{RHom}_\Lambda(B,A_k))\cong \coprod H^0\operatorname{RHom}_\Lambda(B\stackrel{L}{\otimes}_R M,A_k)$$ $$\cong \coprod\operatorname{Hom}_{D(\Lambda)}(i_*M,A_k).$$ So $\begin{bmatrix} M\\0\end{bmatrix}$ is compact and since $C$ is also compact we have that $j_*j^*\Lambda\cong\dfrac{C}{\left[\begin{smallmatrix} M\\0\end{smallmatrix}\right]}$ is compact, and so $T=\Sigma i_*X\oplus j_*j^*\Lambda$ is compact.
It remains to show that $\left\langle T \right\rangle=D(\Lambda)$. For this it is sufficent to show that $\Lambda\in\left\langle T \right\rangle$.
Since $\Lambda\cong B\oplus C$ we only have to show that both $B$ and $C$ are in $\left\langle T\right\rangle$.
To show that $B$ is contained in $\left\langle T\right\rangle$ we first observe that the functor $i_*(-)$ respects the operations of taking distinguished triangles, set indexed coproducts, quotients and suspensions. This gives us that $i_*(\left\langle X\right\rangle)\subseteq\left\langle i_*(X)\right\rangle$ for all $X\in D(R)$. Hence $$B=i_*R\in\operatorname{Ess.\!Im}(i_*)=i_*(D(R))=i_*\left\langle X\right\rangle\subseteq\left\langle i_*X\right\rangle\subseteq\left\langle T\right\rangle.$$
To show that $C\in \langle T\rangle$ we first observe that $\dfrac{C}{\left[\begin{smallmatrix} M\\0\end{smallmatrix}\right]}\cong j_*j^*\Lambda\in\langle T\rangle$ so if we can show that $\begin{bmatrix} M \\0 \end{bmatrix}\in\langle T\rangle$ then $C$ is in $\langle T \rangle$. To show this we first observe that $\langle X\rangle=D(R)$ so $_RM$ can be built from $X$. Since $i_*$ preserves the possible constructions, we can build $\begin{bmatrix} M\\0\end{bmatrix}=i_*M$ from $i_*X\in\langle T\rangle$.
Hence we have that both $B$ and $C\in \langle T\rangle$ and therefore that $\Lambda\in\langle T\rangle$ so $\langle T\rangle=D(\Lambda)$.
We are now in a position to apply Keller's Therorem to get that $D(\Lambda)\simeq D(\mathscr{E}^{op})$. \end{proof}
Our aim now is to find a K-projective resolution of $T$ in the above theorem so that we can calculate $\mathscr{E}$. For this we first need the following lemmas.
\begin{Lemma} \label{K-projective fact} Let $U$ be a K-projective resolution of a DG-$R$-module $X$. Then $\begin{bmatrix} U\\0 \end{bmatrix}$ is a K-projective resolution of $\begin{bmatrix} X\\0 \end{bmatrix}$ over $\Lambda$. \end{Lemma}
\begin{proof} Let $J$ be an exact DG-$\Lambda$-module. Then $$\operatorname{Hom}_\Lambda\left(\begin{bmatrix} U\\0 \end{bmatrix},J\right)\cong\operatorname{Hom}_\Lambda(B\otimes_RU,J)\cong\operatorname{Hom}_R(U,\operatorname{Hom}_\Lambda(B,J)).$$ Since both $U$ and $B$ are K-projective we have that this is exact and hence $\begin{bmatrix} U\\0 \end{bmatrix}$ is K-projective. \end{proof}
\begin{Lemma} \label{K-proj mapping cones} Let $f:X\rightarrow Y$ be a morphism of K-projective DG-modules over some DGA $R$ and let $Z$ be the mapping cone of $f$. Then $Z$ is also K-projective. \end{Lemma}
\begin{Remark} From definition of $_RM_S$ we have a quasi-isomorphism $_RV_S \stackrel{f}{\rightarrow}\leftidx{_R}{M}{_S}$ where $V$ is K-projective over $R$. Also for the DG-$R$-module $_RX$ we can choose a quasi-isomorphism $\leftidx{_R}{U}\stackrel{g}{\rightarrow}\leftidx{_R}{X}$ where $U$ is a K-projective resolution. \end{Remark}
We can now prove the following proposition about the structure of $P$, a K-projective resolution of $T$.
\begin{Proposition} \label{Structure of P} Let $T=\Sigma i_*X\oplus j_*j^*\Lambda$ as defined in Theorem \ref{1st attempt}. Then $T$ has K-projective resolution $P=\Sigma\begin{bmatrix} U\\0 \end{bmatrix}\oplus W$ over $\Lambda$ where $W$ is the mapping cone of $\begin{bmatrix} V\\0 \end{bmatrix}\stackrel{\left[\begin{smallmatrix} f\\0 \end{smallmatrix}\right]}{\longrightarrow}\begin{bmatrix} M\\S \end{bmatrix}$. \end{Proposition}
\begin{proof} By lemma \ref{K-projective fact} we have that $\begin{bmatrix} U\\0 \end{bmatrix}$ is a K-projective resolution of $i_*X=\begin{bmatrix} X\\0 \end{bmatrix}$ and that $\begin{bmatrix} V\\0 \end{bmatrix}$ is a K-projective resolution of $\begin{bmatrix} M\\0 \end{bmatrix}$ over $\Lambda$.
We now wish to find a K-projective resolution of $j_*j^*\Lambda$. To do this we first recall that $j_*j^*\Lambda=\dfrac{C}{\left[\begin{smallmatrix}M\\0\end{smallmatrix}\right]}$.
We now consider the map $$\begin{bmatrix} V\\0 \end{bmatrix}\stackrel{\left[\begin{smallmatrix} f\\0 \end{smallmatrix}\right]}{\longrightarrow}\begin{bmatrix} M\\S \end{bmatrix}$$ of DG-$\Lambda$-modules. This embeds into the distinguished triangle $$\begin{bmatrix} V\\0 \end{bmatrix}\stackrel{\left[\begin{smallmatrix} f\\0 \end{smallmatrix}\right]}{\longrightarrow}\begin{bmatrix} M\\S \end{bmatrix}\longrightarrow W.$$
We can now use this to obtain the diagram $$\xymatrix{{\begin{bmatrix} M\\0 \end{bmatrix}} \ar[r] & C \ar[r] & {C/\left[\begin{smallmatrix} M\\0 \end{smallmatrix}\right]} \ar[r] & {}\\ {\begin{bmatrix} V\\0 \end{bmatrix}} \ar[u]^{\simeq} \ar[r] & C \ar@{=}[u] \ar[r] & W \ar@{.>}[u]^{\exists} \ar[r] & {}}$$ of distinguished triangles in $D(\Lambda)$ so there exists a quasi-isomorphism $W\rightarrow {C}/{\left[\begin{smallmatrix} M\\0 \end{smallmatrix}\right]}$.
By Lemma \ref{K-proj mapping cones} we also have that $W$ is K-projective and hence a K-projective resolution of $C/\left[\begin{smallmatrix} M\\0\end{smallmatrix}\right]\cong j_*j^*\Lambda$.
We now have K-projective resolutions for both direct summands of $T$ and hence $T$ has the K-projective resolution, $P=\Sigma\begin{bmatrix} U\\0\end{bmatrix}\oplus W$.
\end{proof}
Now that we have a K-projective resolution for $T$ in Theorem \ref{1st attempt} we can try to calculate the endomorphism DGA $\mathscr{E}=\operatorname{End}_\Lambda(T)$, but before we do so we need a few facts about $W$ as defined in the above propostion.
\begin{Remark} $W$ is the mapping cone of $\begin{bmatrix}V\\0\end{bmatrix}\stackrel{\left[\begin{smallmatrix} f\\0 \end{smallmatrix}\right]}{\longrightarrow}\begin{bmatrix} M\\S \end{bmatrix}$ so $$W^\natural=\begin{bmatrix}M\\S\end{bmatrix}^\natural\oplus\Sigma\begin{bmatrix}V\\0\end{bmatrix}^\natural$$ i.e any element $w\in W^\natural$ is of the form $w=\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}$. In addition $W$ is equipped with the diffential $$\partial^W=\begin{bmatrix}\partial^C&\left[\begin{smallmatrix}f&0\\0&0\end{smallmatrix}\right]\\[0.3cm]0&-\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}\end{bmatrix}.$$ and the quasi-ismorphism $W \rightarrow {C}/{\left[\begin{smallmatrix} M\\0 \end{smallmatrix}\right]}$ in the proof of proposition \ref{Structure of P} is give by $$\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\mapsto\overline{\begin{bmatrix}0\\s\end{bmatrix}}.$$ \end{Remark}
\begin{Lemma} \label{W quasi} $W$ is isomorphic in $K(\Lambda)$ to $\begin{bmatrix}Z\\S\end{bmatrix}$ where $Z$ is the mapping cone of $f$. Furthermore $Z$ is exact. \end{Lemma}
\begin{proof} Since $Z$ is the mapping cone of $f$ we have a distinguished triangle of the form: $$\xymatrix{V \ar[r]^f & M\ar[r]^h & Z \ar[r]^k & \Sigma V \ar[r] &{}}.$$
We can use this to construct a distinguished triangle $$\xymatrix{{\begin{bmatrix}V\\0\end{bmatrix}} \ar[r]^{\left[\begin{smallmatrix}f\\0\end{smallmatrix}\right]} & {\begin{bmatrix}M\\S\end{bmatrix}}\ar[r]^{\left[\begin{smallmatrix}h\\1\end{smallmatrix}\right]} & {\begin{bmatrix}Z\\S\end{bmatrix}} \ar[r]^{\left[\begin{smallmatrix}k\\0\end{smallmatrix}\right]} & \Sigma{\begin{bmatrix}V\\0\end{bmatrix}} \ar[r] &{}}.$$
Which in turn we can use to obtain the diagram of distinguished triangles: $$\xymatrix{{\begin{bmatrix}V\\0\end{bmatrix}} \ar[r]^{\left[\begin{smallmatrix}f\\0\end{smallmatrix}\right]} \ar@{=}[d] & {\begin{bmatrix}M\\S\end{bmatrix}}\ar[r]^{\left[\begin{smallmatrix}h\\1\end{smallmatrix}\right]} \ar@{=}[d]& {\begin{bmatrix}Z\\S\end{bmatrix}} \ar[r]^{\left[\begin{smallmatrix}k\\0\end{smallmatrix}\right]} \ar@{-->}[d]^{\exists} & \Sigma {\begin{bmatrix}V\\0\end{bmatrix}} \ar[r] \ar@{=}[d]& {} \\{\begin{bmatrix}V\\0\end{bmatrix}} \ar[r]^{\left[\begin{smallmatrix}f\\0\end{smallmatrix}\right]} & {\begin{bmatrix}M\\S\end{bmatrix}} \ar[r] & W \ar[r] & \Sigma{\begin{bmatrix}V\\0\end{bmatrix}} \ar[r] & {.}} $$
Hence we have that there exists an isomorphism $\begin{bmatrix}Z\\S\end{bmatrix}\rightarrow W$.
Finally since $Z$ is the mapping cone of a quasi-isomorphism it is exact. \end{proof}
\begin{Lemma} \label{Hom equiv} Let $A$ and $B$ be DG-$R$-modules. Then $$\operatorname{Hom}_R(A,B)\cong\operatorname{Hom}_\Lambda\!\left(\begin{bmatrix}A\\0\end{bmatrix},\begin{bmatrix}B\\0\end{bmatrix}\right)$$ as complexes of abelian groups.
Furthermore $$\operatorname{Hom}_R(A,A)\cong\operatorname{Hom}_\Lambda\!\left(\begin{bmatrix}A\\0\end{bmatrix},\begin{bmatrix}A\\0\end{bmatrix}\right)$$ as DGA's. \end{Lemma}
\begin{proof} Define $\Theta:\operatorname{Hom}_R(A,B)\rightarrow\operatorname{Hom}_\Lambda\!\left(\begin{bmatrix}A\\0\end{bmatrix},\begin{bmatrix}B\\0\end{bmatrix}\right)$ by $\Theta(\phi)=\begin{bmatrix}\phi&0\\0&0\end{bmatrix}$. It is easy to see that $\Theta$ is an isomorphism of complexes of abelian groups. In addition in the case $B=A$, $\Theta$ becomes an isomorpism of DGA's. \end{proof}
The following proposition give the structure of $\mathscr{E}$ which by Theorem \ref{1st attempt} is dervived equivalent to the upper triangular matix DGA $\Lambda$.
\begin{Proposition} \label{structure of E} In the setup of Theorem \ref{1st attempt}, $$\mathscr{E}\cong\begin{bmatrix} \operatorname{Hom}_R(U,U) & \operatorname{Hom}_\Lambda(W,\Sigma \left[\begin{smallmatrix} U \\0\end{smallmatrix}\right])\\ \operatorname{Hom}_\Lambda(\Sigma \left[\begin{smallmatrix} U \\0\end{smallmatrix}\right],W) & \operatorname{Hom}_\Lambda(W,W)\end{bmatrix}.$$ \end{Proposition}
\begin{proof} Since $P=\Sigma U\oplus W$ consists of a direct sum we have that $$\mathscr{E}=\operatorname{Hom}_\Lambda(P,P)=\begin{bmatrix} \operatorname{Hom}_\Lambda(\Sigma \left[\begin{smallmatrix} U \\0\end{smallmatrix}\right],\Sigma \left[\begin{smallmatrix} U \\0\end{smallmatrix}\right]) & \operatorname{Hom}_\Lambda(W,\Sigma \left[\begin{smallmatrix} U \\0\end{smallmatrix}\right])\\ \operatorname{Hom}_\Lambda(\Sigma \left[\begin{smallmatrix} U \\0\end{smallmatrix}\right],W) & \operatorname{Hom}_\Lambda(W,W)\end{bmatrix}.$$
Furthermore from Lemma \ref{Hom equiv} above we have that $$\operatorname{Hom}_\Lambda\!\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)\cong\operatorname{Hom}_\Lambda\!\left(\begin{bmatrix}U\\0\end{bmatrix},\begin{bmatrix}U\\0\end{bmatrix}\right)\cong\operatorname{Hom}_R(U,U).$$ \end{proof}
Our attention now is with obtaining a quasi-isomorphism between the entries of $\mathscr{E}$ and the corresponding entries $\tilde{\Lambda}=\begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}$ which will allow us to construct a isomorpism between the two DGAs.
\begin{Lemma} \label{Exactness} The complex of abelian groups $\operatorname{Hom}_\Lambda\!\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},W\right)$ is exact. \end{Lemma}
\begin{proof} For all $i$ we have that $$\operatorname{H}^i\operatorname{Hom}_\Lambda\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},W\right)\cong\operatorname{H}^0\operatorname{Hom}_\Lambda\left(\begin{bmatrix}U\\0\end{bmatrix},\Sigma^{i-1}W\right)$$ $$\cong\operatorname{Hom}_{K(\Lambda)}\left(\begin{bmatrix}U\\0\end{bmatrix},\Sigma^{i-1}W\right)\cong\operatorname{Hom}_{K(\Lambda)}\left(\begin{bmatrix}U\\0\end{bmatrix},\Sigma^{i-1}\begin{bmatrix}Z\\S\end{bmatrix}\right)$$
However for $\theta\in\operatorname{Hom}_\Lambda\left(\begin{bmatrix}U\\0\end{bmatrix},\Sigma^{i-1}\begin{bmatrix}Z\\S\end{bmatrix}\right)$ such that $\theta\left(\begin{bmatrix}u\\0\end{bmatrix}\right)=\begin{bmatrix}z\\s\end{bmatrix}$ for some $u\in U,z\in Z$ and $s\in S$, we have $$\begin{bmatrix}z\\s\end{bmatrix} =\theta\left(\begin{bmatrix} u \\0\end{bmatrix}\right) =\theta\left(\begin{bmatrix}1&0\\0&0\end{bmatrix}\begin{bmatrix}u\\0\end{bmatrix}\right) =\begin{bmatrix}1&0\\0&0\end{bmatrix}\theta\left(\begin{bmatrix}u\\0\end{bmatrix}\right)$$ $$=\begin{bmatrix}1&0\\0&0\end{bmatrix}\begin{bmatrix}z\\s\end{bmatrix} =\begin{bmatrix}z\\0\end{bmatrix}.$$
So $s=0$ and so $\theta\left(\begin{bmatrix}u\\0\end{bmatrix}\right)=\begin{bmatrix}z\\0\end{bmatrix}$.
Hence $\operatorname{Hom}_\Lambda\!\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},\Sigma^{i-1}\begin{bmatrix}Z\\S\end{bmatrix}\right)\cong\operatorname{Hom}_\Lambda\!\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},\Sigma^{i-1}\begin{bmatrix}Z\\0\end{bmatrix}\right)$ and by Lemma \ref{Hom equiv} this is isomorphic to $\operatorname{Hom}_R(\Sigma U,\Sigma^{i-1}Z)$.
Taking this together with $U$ being K-projective and the exactness of $Z$ gives us that $$\operatorname{Hom}_{K(\Lambda)}\left(\begin{bmatrix}U\\0\end{bmatrix},\Sigma^{i-1}\begin{bmatrix}Z\\S\end{bmatrix}\right)\cong\operatorname{Hom}_{K(\Lambda)}\left(U,\Sigma^{i-1}Z\right)$$ $$\cong\operatorname{Hom}_{D(\Lambda)}\left(U,\Sigma^{i-1}Z\right)\cong 0,$$
Hence $\operatorname{H}^i\operatorname{Hom}_\Lambda\!\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},W\right)\cong 0$ for all $i$ and so $\operatorname{Hom}_\Lambda\!\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},W\right)$ is exact. \end{proof}
\begin{Proposition} \label{quasi S^op} There is a quasi-isomorphism of DGAs $$\alpha:S^{\operatorname{op}}\rightarrow\operatorname{Hom}_\Lambda(W,W).$$ \end{Proposition}
\begin{proof}
Define $\alpha:S^{\operatorname{op}}\rightarrow\operatorname{Hom}_\Lambda(W,W)$ by $\alpha(\tilde{s})=\begin{bmatrix} g_{\tilde{s}}&0\\0&l_{\tilde{s}}\end{bmatrix}$ where $g_{\tilde{s}}\left(\begin{bmatrix} m\\s\end{bmatrix}\right)=(-1)^{|\tilde{s}||s|}\begin{bmatrix} m\tilde{s}\\s\tilde{s}\end{bmatrix}$ and $l_{\tilde{s}}\left(\begin{bmatrix} v\\0\end{bmatrix}\right)=(-1)^{|\tilde{s}|(|v|+1)}\begin{bmatrix} v\tilde{s}\\0\end{bmatrix}$.
We first want to show that $\alpha$ is a homomorphism of Differential Graded Algebras.
It is straightforward to check that $\alpha$ respects the operations of addition and multiplication. So it remains to check that $\alpha$ is compatible with the differential.
So $$\partial^{\operatorname{Hom}_\Lambda(W,W)}(\alpha(\tilde{s})) =\partial^{\operatorname{Hom}_\Lambda(W,W)}\left(\begin{bmatrix}g_{\tilde{s}}&0\\0&l_{\tilde{s}}\end{bmatrix}\right)$$
$$=\partial^W\begin{bmatrix}g_{\tilde{s}}&0\\0&l_{\tilde{s}}\end{bmatrix}-(-1)^{|\tilde{s}|}\begin{bmatrix}g_{\tilde{s}}&0\\0&l_{\tilde{s}}\end{bmatrix}\partial^W$$
$$=\begin{bmatrix}\partial^C&\left[\begin{smallmatrix}f&0\\0&0\end{smallmatrix}\right]\\[0.3cm]0&-\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}\end{bmatrix}\begin{bmatrix}g_{\tilde{s}}&0\\0&l_{\tilde{s}}\end{bmatrix}-(-1)^{|\tilde{s}|}\begin{bmatrix}g_{\tilde{s}}&0\\0&l_{\tilde{s}}\end{bmatrix}\begin{bmatrix}\partial^C&\left[\begin{smallmatrix}f&0\\0&0\end{smallmatrix}\right]\\[0.3cm]0&-\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}\end{bmatrix}$$
$$=\begin{bmatrix}\partial^Cg_{\tilde{s}}&\left[\begin{smallmatrix}f&0\\0&0\end{smallmatrix}\right]l_{\tilde{s}}\\[0.3cm]0&-\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}l_{\tilde{s}}\end{bmatrix}-(-1)^{|\tilde{s}|}\begin{bmatrix}g_{\tilde{s}}\partial^C&g_{\tilde{s}}\left[\begin{smallmatrix}f&0\\0&0\end{smallmatrix}\right]\\[0.3cm]0&-l_{\tilde{s}}\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}\end{bmatrix}$$
$$=\begin{bmatrix}\partial^Cg_{\tilde{s}}-(-1)^{|\tilde{s}|}g_{\tilde{s}}\partial^C&\left[\begin{smallmatrix}f&0\\0&0\end{smallmatrix}\right]l_{\tilde{s}}-(-1)^{|\tilde{s}|}g_{\tilde{s}}\left[\begin{smallmatrix}f&0\\0&0\end{smallmatrix}\right]\\[0.3cm]0&-\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}l_{\tilde{s}}+(-1)^{|\tilde{s}|}l_{\tilde{s}}\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}\end{bmatrix}.$$
Considering each of the terms in this matrix seperately, starting with the upper left term, we get:
$$\left(\partial^Cg_{\tilde{s}}-(-1)^{|\tilde{s}|}g_{\tilde{s}}\partial^C\right)\left(\begin{bmatrix}m\\s\end{bmatrix}\right)$$
$$=\partial^C\left((-1)^{|\tilde{s}||s|}\begin{bmatrix}m\tilde{s}\\s\tilde{s}\end{bmatrix}\right)-(-1)^{|\tilde{s}|}g_{\tilde{s}}\left(\begin{bmatrix}\partial^M(m)\\\partial^S(s)\end{bmatrix}\right)$$
$$=(-1)^{|\tilde{s}||s|}\begin{bmatrix}\partial^M(m\tilde{s})\\\partial^S(s\tilde{s})\end{bmatrix}-(-1)^{|\tilde{s}|}(-1)^{|\tilde{s}|(|s|-1)}\begin{bmatrix}\partial^M(m)\tilde{s}\\\partial^S(s)\tilde{s}\end{bmatrix}$$
$$=(-1)^{|\tilde{s}||s|}\begin{bmatrix}\partial^M(m)\tilde{s}+(-1)^{|s|}m\partial^S(\tilde{s})\\\partial^S(s)\tilde{s}+(-1)^{|s|}s\partial^S(\tilde{s})\end{bmatrix}-(-1)^{|\tilde{s}||s|}\begin{bmatrix}\partial^M(m)\tilde{s}\\\partial^S(s)\tilde{s}\end{bmatrix}$$
$$=(-1)^{|\tilde{s}||s|}\begin{bmatrix}(-1)^{|s|}m\partial^S(\tilde{s})\\(-1)^{|s|}s\partial^S(\tilde{s})\end{bmatrix}=(-1)^{|\partial^S(\tilde{s})||s|}\begin{bmatrix}m\partial^S(\tilde{s})\\s\partial^S(\tilde{s})\end{bmatrix}$$ $$=g_{\partial^S(\tilde{s})}\left(\begin{bmatrix}m\\s\end{bmatrix}\right).$$
So $\partial^Cg_{\tilde{s}}-(-1)^{|\tilde{s}|}g_{\tilde{s}}\partial^C=g_{\partial^S(\tilde{s})}$.
By a similar arguement we also have for the lower right entry that $-\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}l_{\tilde{s}}+(-1)^{|\tilde{s}|}l_{\tilde{s}}\partial^{\left[\begin{smallmatrix}V\\0\end{smallmatrix}\right]}=l_{\partial^S(\tilde{s})}$.
Finally for the upper right entry we have:
$$\left(\begin{bmatrix}f&0\\0&0\end{bmatrix}l_{\tilde{s}}-(-1)^{|\tilde{s}|}g_{\tilde{s}}\begin{bmatrix}f&0\\0&0\end{bmatrix}\right)\left(\begin{bmatrix}v\\0\end{bmatrix}\right)$$
$$=\begin{bmatrix}f&0\\0&0\end{bmatrix}\left((-1)^{|\tilde{s}|(|v|+1)}\begin{bmatrix}v\tilde{s}\\0\end{bmatrix}\right)-(-1)^{|\tilde{s}|}g_{\tilde{s}}\left(\begin{bmatrix}f(v)\\0\end{bmatrix}\right)$$
$$(-1)^{|\tilde{s}|(|v|+1)}\begin{bmatrix}f(v\tilde{s})\\0\end{bmatrix}-(-1)^{|\tilde{s}|}(-1)^{|\tilde{s}||v|}\begin{bmatrix}f(v)\tilde{s}\\0\end{bmatrix}$$
$$(-1)^{|\tilde{s}|(|v|+1)}\begin{bmatrix}f(v)\tilde{s}\\0\end{bmatrix}-(-1)^{|\tilde{s}|(|v|+1)}\begin{bmatrix}f(v)\tilde{s}\\0\end{bmatrix}=0.$$
Substituting these values back into the matrix gives us that: $$\partial^{\operatorname{Hom}_\Lambda(W,W)}(\alpha(\tilde{s}))=\begin{bmatrix}g_{\partial^S(\tilde{s})}&0\\0&l_{\partial^S(\tilde{s})}\end{bmatrix}=\alpha(\partial^S(\tilde{s})).$$
Hence we have that $\alpha$ is a homomorphism of Differential Graded Algebras. It remains to show that it is also a quasi-isomorphism.
Now let $\theta\in\operatorname{Hom}_\Lambda\!\left(W,C/\left[\begin{smallmatrix}M\\0\end{smallmatrix}\right]\right)$ such that $\theta\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)=\overline{\begin{bmatrix}0\\s\end{bmatrix}}$. This quasi-isomorphism gives us the homomorphism of complexes of abelian groups $$\operatorname{Hom}_\Lambda(W,\theta):\operatorname{Hom}_\Lambda(W,W)\rightarrow\operatorname{Hom}_\Lambda\left(W,C/\left[\begin{smallmatrix}M\\0\end{smallmatrix}\right]\right).$$
Define $\beta=\operatorname{Hom}_\Lambda(W,\theta)\circ\alpha:S^{\operatorname{op}}\rightarrow\operatorname{Hom}_\Lambda\left(W,C/\left[\begin{smallmatrix}M\\0\end{smallmatrix}\right]\right)$. Then $\beta$ is a homomorphism of complexes of abelian groups with $$\beta(\tilde{s})\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)
=\theta\left((-1)^{|s||\tilde{s}|}\begin{bmatrix}\left[\begin{smallmatrix}m\tilde{s}\\s\tilde{s}\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)
=(-1)^{|s||\tilde{s}|}\overline{\begin{bmatrix}0\\s\tilde{s}\end{bmatrix}}.$$
Now let $\psi\in \operatorname{Hom}_{\Lambda}\left(W,C/\left[\begin{smallmatrix}M\\0\end{smallmatrix}\right]\right)$. Then for any $m\in M, s\in S$ and $v\in V$ we have that $\psi\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right) =\overline{\begin{bmatrix}0\\\tilde{s}\end{bmatrix}}$ for some $\tilde{s}\in S$. Hence we have $$\psi\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right) =\overline{\begin{bmatrix}0\\\tilde{s}\end{bmatrix}} =\begin{bmatrix}0&0\\0&1\end{bmatrix}\overline{\begin{bmatrix}0\\\tilde{s}\end{bmatrix}} =\begin{bmatrix}0&0\\0&1\end{bmatrix}\psi\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)=\psi\left(\begin{bmatrix}\left[\begin{smallmatrix}0\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}0\\0\end{smallmatrix}\right]\end{bmatrix}\right).$$
Furthermore $$\psi\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right) =\psi\left(\begin{bmatrix}\left[\begin{smallmatrix}0\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}0\\0\end{smallmatrix}\right]\end{bmatrix}\right) =\begin{bmatrix}0&0\\0&s\end{bmatrix}\psi\left(\begin{bmatrix}\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\\\left[\begin{smallmatrix}0\\0\end{smallmatrix}\right]\end{bmatrix}\right).$$
So each element of $\operatorname{Hom}_{\Lambda}\left(W,C/\left[\begin{smallmatrix}M\\0\end{smallmatrix}\right]\right)$ depends entirely on where it sends $\begin{bmatrix}\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\\\left[\begin{smallmatrix}0\\0\end{smallmatrix}\right]\end{bmatrix}$.
We therefore have that for every $s\in S$ we have that $\beta(s)$ is the element of $\operatorname{Hom}_{\Lambda}\left(W,C/\left[\begin{smallmatrix}M\\0\end{smallmatrix}\right]\right)$ which sends $\begin{bmatrix}\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\\\left[\begin{smallmatrix}0\\0\end{smallmatrix}\right]\end{bmatrix}$ to $\overline{\begin{bmatrix}0\\s\end{bmatrix}}$.
Since elements of $\operatorname{Hom}_{\Lambda}\left(W,C/\left[\begin{smallmatrix}M\\0\end{smallmatrix}\right]\right)$ depend entirely on where they send $\begin{bmatrix}\left[\begin{smallmatrix}0\\1\end{smallmatrix}\right]\\\left[\begin{smallmatrix}0\\0\end{smallmatrix}\right]\end{bmatrix}$ and since $\overline{\left[\begin{matrix}0\\s\end{matrix}\right]}\neq\overline{\left[\begin{matrix}0\\s'\end{matrix}\right]}$ for all $s,s'\in S$ with $s\neq s'$ we have that $\beta$ is a bijection and so a isomorphism of complexes of abelian groups.
Furthermore since $W$ is K-projective and $\theta$ is a quasi-isomorphism we have that $\operatorname{Hom}_\Lambda(W,\theta)$ is a quasi-isomorphism and therefore since $\beta$ is an isomorphism we have that $\alpha$ must also be a quasi-isomorphism. \end{proof}
\begin{Lemma} \label{Hom iso2}
There exists a quasi-isomorphism $\Psi:\operatorname{Hom}_R(V,U)\rightarrow\operatorname{Hom}_\Lambda\left(W,\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)$ of complexes of abelian groups, such that $$\Psi(\theta)\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)=(-1)^{|\theta|}\begin{bmatrix}\theta(v)\\0\end{bmatrix}.$$ \end{Lemma}
\begin{proof} Consider the distinguished triangle $$\begin{bmatrix}V\\0\end{bmatrix}\stackrel{\left[\begin{smallmatrix}f\\0\end{smallmatrix}\right]}{\rightarrow}\begin{bmatrix}M\\S\end{bmatrix}\stackrel{\iota}{\rightarrow}W\stackrel{\pi}{\rightarrow}\Sigma\begin{bmatrix}V\\0\end{bmatrix}\rightarrow$$ in $K(\Lambda)$.
Since $W$ is the mapping cone of $\begin{bmatrix}f\\0\end{bmatrix}$ we have that $\pi$ is given by $\pi\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)=\begin{bmatrix}v\\0\end{bmatrix}$.
By applying the functor $\operatorname{Hom}_\Lambda\left(-,\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)$ we get a distinguished triangle
$$\leftarrow\operatorname{Hom}_\Lambda\left(\begin{bmatrix}M\\S\end{bmatrix},\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right) \leftarrow\operatorname{Hom}_\Lambda\left(W,\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)$$ $$\stackrel{\pi^*}{\leftarrow}\operatorname{Hom}_\Lambda\left(\Sigma\begin{bmatrix}V\\0\end{bmatrix},\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right) \leftarrow\operatorname{Hom}_\Lambda\left(\Sigma\begin{bmatrix}M\\S\end{bmatrix},\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)$$ in $K(\operatorname{Ab})$.
Now let $\theta\in\operatorname{Hom}_\Lambda\left(\begin{bmatrix}M\\S\end{bmatrix},\Sigma^i\begin{bmatrix}U\\0\end{bmatrix}\right)$. Then, since $\begin{bmatrix}M\\S\end{bmatrix}$ is generated by $\begin{bmatrix}0\\1\end{bmatrix}$ as a DG-$\Lambda$-module, we have that $\theta$ depends entirely upon where it sends $\begin{bmatrix}0\\1\end{bmatrix}$.
Let $\theta\left(\begin{bmatrix}0\\1\end{bmatrix}\right)=\begin{bmatrix}u\\0\end{bmatrix}$. Then
$$\begin{bmatrix}u\\0\end{bmatrix} =\theta\left(\begin{bmatrix}0\\1\end{bmatrix}\right) =\theta\left(\begin{bmatrix}0&0\\0&1\end{bmatrix}\begin{bmatrix}0\\1\end{bmatrix}\right) =\begin{bmatrix}0&0\\0&1\end{bmatrix}\theta\left(\begin{bmatrix}0\\1\end{bmatrix}\right) =\begin{bmatrix}0&0\\0&1\end{bmatrix}\begin{bmatrix}u\\0\end{bmatrix}=0,$$
so $\theta=0$ and hence $\operatorname{Hom}_\Lambda\left(\begin{bmatrix}M\\S\end{bmatrix},\Sigma^i\begin{bmatrix}U\\0\end{bmatrix}\right)=0$ for all $i$.
Hence the distinguished triangle above shows that $$\pi^*:\operatorname{Hom}_\Lambda\left(\Sigma\begin{bmatrix}V\\0\end{bmatrix},\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)\rightarrow\operatorname{Hom}_\Lambda\left(W,\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)$$ is a quasi-isomorphism.
We can now use this along with the suspension $\Sigma$ and the isomorphism $\Theta$ defined in the proof of Lemma \ref{Hom equiv} to obtain the diagram $$\operatorname{Hom}_R(V,U)\stackrel{\Theta}{\longrightarrow}\operatorname{Hom}_{\Lambda}\left(\begin{bmatrix}V\\0\end{bmatrix},\begin{bmatrix}U\\0\end{bmatrix}\right)\stackrel{\Sigma(-)}{\longrightarrow}$$ $$\operatorname{Hom}_{\Lambda}\left(\Sigma\begin{bmatrix}V\\0\end{bmatrix},\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)\stackrel{\pi^*}{\longrightarrow}\operatorname{Hom}_{\Lambda}\left(W,\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)$$
Since each of the maps in the diagram is a quasi-isomorphism we can use them to define the quasi-isomorphism $\Psi:\operatorname{Hom}_R(V,U)\rightarrow\operatorname{Hom}_\Lambda\!\left(W,\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)$ by the composition $$\Psi=\pi^*\circ\Sigma(-)\circ\Theta.$$
Finally for $\theta\in\operatorname{Hom}_R(V,U)$ we have that $$\Psi(\theta)=\pi^*\circ\Sigma\circ\Theta(\theta)$$ $$=\pi^*\Sigma\left(\begin{bmatrix}\theta&0\\0&0\end{bmatrix}\right)$$
$$=\pi^*\left((-1)^{|\theta|}\begin{bmatrix}\theta&0\\0&0\end{bmatrix}\right)$$
$$=(-1)^{|\theta|}\begin{bmatrix}\theta&0\\0&0\end{bmatrix}\circ\pi.$$
So for $\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\in W$, we have that $$\Psi(\theta)\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)
=(-1)^{|\theta|}\begin{bmatrix}\theta&0\\0&0\end{bmatrix}\circ\pi\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)$$
$$=(-1)^{|\theta|}\begin{bmatrix}\theta&0\\0&0\end{bmatrix}\left(\begin{bmatrix}v\\0\end{bmatrix}\right)
=(-1)^{|\theta|}\begin{bmatrix}\theta(v)\\0\end{bmatrix}.$$
\end{proof}
\begin{Remark} From the right DG-$S$-module structure on $V$ we have that $\operatorname{Hom}_R(V,U)$ is a left DG-$S$-module. In addition $\operatorname{Hom}_R(V,U)$ is a left DG-$\operatorname{Hom}_R(U,U)$-module. Hence we have that $\begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}$ is a DGA. \end{Remark}
We are now in a position to produce our main Theorem, a version of \cite[Theorem 4.5]{Lad} for DGAs.
\begin{Theorem} \label{Main} Let $X$ be a DG $R$-module such that $_RX$ is compact with $\left\langle _RX\right\rangle=D(R)$ and let $\leftidx{_R}{M}{_S}$ be compact as a DG-$R$-module. Then for the upper triangular differential graded algebras $$\Lambda=\begin{bmatrix}R&M\\0&S\end{bmatrix}\textrm{ and } \tilde{\Lambda}=\begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}$$ we have that $D(\Lambda)\simeq D(\tilde{\Lambda})$. \end{Theorem}
\begin{proof} From Theorem \ref{1st attempt} and lemma \ref{structure of E} we have that $D(\Lambda)\simeq D(\mathscr{E}^{\operatorname{op}})$ where $$\mathscr{E}=\begin{bmatrix}\operatorname{Hom}_R(U,U)&\operatorname{Hom}_\Lambda(W,\Sigma\left[\begin{smallmatrix}U\\0\end{smallmatrix}\right])\\\operatorname{Hom}_R(\Sigma \left[\begin{smallmatrix}U\\0\end{smallmatrix}\right],W)&\operatorname{Hom}_\Lambda(W,W)\end{bmatrix}.$$
We therefore only need to show that there is a quasi-isomorphism of DGA's from $\tilde{\Lambda}^{\operatorname{op}}=\begin{bmatrix}\operatorname{Hom}_R(U,U)&\operatorname{Hom}_R(V,U)\\0&S^{\operatorname{op}}\end{bmatrix}$ to $\mathscr{E}$.
From proposition \ref{quasi S^op} we have that there exists a quasi-isomorphism $\alpha:S^{\operatorname{op}}\rightarrow\operatorname{Hom}_\Lambda(W,W)$. Hence we can define the map $$\Phi:\begin{bmatrix}\operatorname{Hom}_R(U,U)&\operatorname{Hom}_R(V,U)\\0&S^{\operatorname{op}}\end{bmatrix} \rightarrow \begin{bmatrix}\operatorname{Hom}_R(U,U)&\operatorname{Hom}_\Lambda(W,\Sigma\left[\begin{smallmatrix}U\\0\end{smallmatrix}\right])\\\operatorname{Hom}_R(\Sigma \left[\begin{smallmatrix}U\\0\end{smallmatrix}\right],W)&\operatorname{Hom}_\Lambda(W,W)\end{bmatrix}$$
by $\Phi\left(\begin{bmatrix}\phi&\theta\\0&s\end{bmatrix}\right)=\left(\begin{bmatrix}\phi&(-1)^{|\theta|}\Psi(\theta)\\0&\alpha(s)\end{bmatrix}\right)$.
Here $\Psi:\operatorname{Hom}_R(V,U)\rightarrow\operatorname{Hom}_\Lambda\!\left(W,\Sigma\begin{bmatrix}U\\0\end{bmatrix}\right)$ is the quasi-isomorphism from Lemma \ref{Hom iso2}.
We now need to show that $\Phi$ is a morphism of DGA's.
Both addition and compatibility with the differential follow from the fact that $\alpha$ is a morphism of diffential graded algebras. So we only need to check multiplication:
Let . denote multiplication in $S^{\operatorname{op}}$. Let $\begin{bmatrix}\phi&\theta\\0&s\end{bmatrix}\in \tilde{\Lambda}_i$ and $\begin{bmatrix}\phi'&\theta'\\0&s'\end{bmatrix}\in \tilde{\Lambda}_j$; then we have $$\Phi\left(\begin{bmatrix}\phi&\theta\\0&s\end{bmatrix}\right)\Phi\left(\begin{bmatrix}\phi'&\theta'\\0&s'\end{bmatrix}\right) =\begin{bmatrix}\phi&(-1)^i\Psi(\theta)\\0&\alpha(s)\end{bmatrix}\begin{bmatrix}\phi'&(-1)^j\Psi(\theta')\\0&\alpha(s')\end{bmatrix}$$ $$=\begin{bmatrix}\phi\phi'&(-1)^j\phi\Psi(\theta')+(-1)^i\Psi(\theta)\alpha(s')\\0&\alpha(s)\alpha(s')\end{bmatrix}$$ $$=\begin{bmatrix}\phi\phi'&(-1)^j\phi\Psi(\theta')+(-1)^i\Psi(\theta)\alpha(s')\\0&\alpha(s.s')\end{bmatrix}$$ and $$\Phi\left(\begin{bmatrix}\phi&\theta\\0&s\end{bmatrix}\begin{bmatrix}\phi'&\theta'\\0&s'\end{bmatrix}\right) =\Phi\left(\begin{bmatrix}\phi\phi'&\phi\theta+\theta.s'\\0&s.s'\end{bmatrix}\right)$$ $$=\begin{bmatrix}\phi\phi'&(-1)^{(i+j)}\Psi(\phi\theta'+(-1)^{ij}s'\theta)\\0&\alpha(s.s')\end{bmatrix}.$$
However $$(-1)^{(i+j)}\Psi((\phi\theta'+(-1)^{ij}s'\theta))\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)$$ $$=(-1)^{(i+j)}\Psi(\phi\theta')\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)+(-1)^{(i+j)}(-1)^{ij}\Psi(s'\theta)\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)$$ $$=(-1)^{(i+j)}(-1)^{(i+j)}\begin{bmatrix}\phi\theta'(v)\\0\end{bmatrix}+(-1)^{(i+j)}(-1)^{ij}(-1)^{(i+j)}\left(\begin{bmatrix}(s'\theta)(v)\\0\end{bmatrix}\right)$$ $$=\phi\begin{bmatrix}\theta'(v)\\0\end{bmatrix}+(-1)^{ij}(-1)^{j(i+(i+1))}\begin{bmatrix}\theta(vs')\\0\end{bmatrix}$$ $$=\phi\begin{bmatrix}\theta'(v)\\0\end{bmatrix}+(-1)^{j(i+1)}\begin{bmatrix}\theta(vs')\\0\end{bmatrix}$$ $$=(-1)^j\phi\Psi(\theta')\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right)+(-1)^{j(i+1)}(-1)^i\Psi(\theta)\left(\begin{bmatrix}\left[\begin{smallmatrix}ms'\\ss'\end{smallmatrix}\right]\\\left[\begin{smallmatrix}vs'\\0\end{smallmatrix}\right]\end{bmatrix}\right)$$ $$=((-1)^j\phi\Psi(\theta')+(-1)^i\Psi(\theta)\alpha(s'))\left(\begin{bmatrix}\left[\begin{smallmatrix}m\\s\end{smallmatrix}\right]\\\left[\begin{smallmatrix}v\\0\end{smallmatrix}\right]\end{bmatrix}\right),$$
so $(-1)^j\phi\Psi(\theta')+(-1)^i\Psi(\theta)\alpha(s')=(-1)^{(i+j)}\Psi((\phi\theta'+(-1)^{ij}s'\theta))$ and therefore $$\Phi\left(\begin{bmatrix}\phi&\theta\\0&s\end{bmatrix}\right)\Phi\left(\begin{bmatrix}\phi'&\theta'\\0&s'\end{bmatrix}\right)=\Phi\left(\begin{bmatrix}\phi&\theta\\0&s\end{bmatrix}\begin{bmatrix}\phi'&\theta'\\0&s'\end{bmatrix}\right).$$
We therefore have that $\Phi$ is a morphism of Diffential Graded Algebras. Furthermore since from lemma \ref{Exactness} we have that $\operatorname{Hom}_\Lambda\!\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},W\right)$ is exact and so the map $0\rightarrow\operatorname{Hom}_\Lambda\!\left(\Sigma\begin{bmatrix}U\\0\end{bmatrix},W\right)$ is a quasi isomorphism.
Taking these together with the fact that $\alpha:S^{\operatorname{op}}\rightarrow\operatorname{Hom}_\Lambda(W,W)$ is a quasi-isomorphism we have that $\Phi$ is a quasi-isomorphism.
Hence $\mathscr{E}\simeq\tilde{\Lambda}^{\operatorname{op}}$ and so $$D(\Lambda)\simeq D(\mathscr{E}^{\operatorname{op}})\simeq D(\tilde{\Lambda}).$$
\end{proof}
\section{Examples}
We shall now conclude with some examples. In the first example we will show that by taking $R$ and $S$ to be $k$-algebras and making the same assumptions as in \cite{Lad}, we obtain what is in essence the same result.
\begin{Definition} An $R$-module $X$ is called rigid if $\operatorname{Ext}_R^i(X,X)=0$ for all $i\neq 0$. \end{Definition}
\begin{Theorem} Let $R$ and $S$ be rings and $\leftidx{_R}{M}{_S}$ a $R$-$S$-bimodule such that $\leftidx{_R}{M}$ is compact in $D(R)$ and when $R$ and $S$ are considered as DGAs then $\leftidx{_R}{M}{_S}$ as a DG-bimodule is quasi-isomorphic to $\leftidx{_R}{V}{_S}$ which is a K-projective DG-$R$-module. Let $\leftidx{_R}{X}$ be a compact and rigid $R$-module with $\langle X\rangle=D(R)$ and $\operatorname{Ext}_R^n(\leftidx{_R}{M},\leftidx{_R}{X})=0$ for all $n\neq 0$. Then the triangular matrix rings $$\Lambda=\begin{bmatrix}R&M\\0&S\end{bmatrix} \textrm{ and } \tilde{\Lambda}=\begin{bmatrix}S&\operatorname{Hom}_R(M,X)\\0&\operatorname{End}_R(X)^{\operatorname{op}}\end{bmatrix}$$ are derived equivalent. \end{Theorem} \begin{proof} By considering the rings $R$ and $S$ and modules $M$ and $X$ to be DGA's and DG-modules respectively we can apply Theorem \ref{Main} to get that the DGA's $$\begin{bmatrix}R&M\\0&S\end{bmatrix}\textrm{ and } \begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}$$ are derived equivalent, where $U$ is a K-projective resolution of $X$.
Since $\operatorname{Hom}_R(U,U)=\operatorname{RHom}_R(X,X)$ we have that $$H^i\operatorname{Hom}_R(U,U)=H^i\operatorname{RHom}_R(X,X)=\operatorname{Ext}_R^i(X,X)=0$$ for all $i\neq 0$ since $X$ is rigid and $$H^0\operatorname{Hom}_R(U,U)=H^0\operatorname{RHom}_R(X,X)=\operatorname{End}_R(X).$$ Similarly since $\operatorname{Hom}_R(V,U)=\operatorname{RHom}_R(M,X)$ we have that $$H^i\operatorname{Hom}_R(V,U)=H^i\operatorname{RHom}_R(M,X)=\operatorname{Ext}_R^i(M,X)=0$$ for all $i\neq 0$ and $$H^0\operatorname{Hom}_R(V,U)=H^0\operatorname{RHom}_R(M,X)=\operatorname{Hom}_R(M,X).$$
Hence we have that $H^i\begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}=0$ for all $i\neq 0$ and $$H^0\begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}=\begin{bmatrix}S&\operatorname{Hom}_R(M,X)\\0&\operatorname{End}_R(X)^{\operatorname{op}}\end{bmatrix}.$$
We therefore have that the matrix ring $\begin{bmatrix}S&\operatorname{Hom}_R(M,X)\\0&\operatorname{End}_R(X)^{\operatorname{op}}\end{bmatrix}$ is derived equivalent to the DGA $\begin{bmatrix}S&\operatorname{Hom}_R(V,U)\\0&\operatorname{Hom}_R(U,U)^{\operatorname{op}}\end{bmatrix}$ and so derived equivalent to the matrix ring $\begin{bmatrix}R&M\\0&S\end{bmatrix}$.
\end{proof}
Our next example considers the special case obtained when we take $\leftidx{_R}{X}=\leftidx{_R}{R}$.
\begin{Corollary} \label{X=R} Let $\leftidx{_R}{M}{_S}$ be compact as a DG-$R$-module. Then the triangular matrix DGAs $$\Lambda=\begin{bmatrix}R&M\\0&S\end{bmatrix} \textrm{ and } \tilde{\Lambda}=\begin{bmatrix}S&\operatorname{Hom}_R(V,R)\\0&R\end{bmatrix}$$ where $V$ is K-projective over $R$ and is quasi-isomorphic to $\leftidx{_R}{M}{_S}$, are derived equivalent. \end{Corollary}
For the next example we require the idea of the duality on $\operatorname{Mod} R$ which we define next.
\begin{Definition} Let $R$ be a finite dimensional DGA over a field $k$. Then we can define the duality on $\operatorname{Mod} R$ by $D:\operatorname{Mod} R\rightarrow\operatorname{Mod} R^{\operatorname{op}}$ where $D(-)=\operatorname{Hom}_k(-,k)$. \end{Definition}
The final example below considers the case where the DGA's $R$ and $S$ are over some field $k$ and $R$ is self dual in the sense of the above definition.
\begin{Theorem} Let $R$ be a finite dimensional and self dual in the sense that $DR\cong R$ in the derived category of DG-bi-$R$-modules and let $\leftidx{_R}{M}{_S}$ be compact as a DG-$R$-module. Then $$\Lambda=\begin{bmatrix}R&M\\0&S\end{bmatrix} \textrm{ and } \tilde{\Lambda}=\begin{bmatrix}S&DM\\0&R\end{bmatrix}$$ are derived equivalent. \end{Theorem} \begin{proof} From corollary \ref{X=R} we have that $$\begin{bmatrix}R&M\\0&S\end{bmatrix} \textrm{ and } \begin{bmatrix}S&\operatorname{Hom}_R(V,R)\\0&R\end{bmatrix}$$ are derived equivalent, where $\leftidx{_R}{V}{_S}$ is quasi-isomorphic to $\leftidx{_R}{M}{_S}$ and $\leftidx{_R}{V}$ is K-projective.
Since $R$ is self dual we have that $$\operatorname{Hom}_R(V,R)\cong\operatorname{Hom}_R(V,DR)=\operatorname{Hom}_R(V,\operatorname{Hom}_k(R,k))$$ $$\cong\operatorname{Hom}_k(R\otimes_R V,k)\cong\operatorname{Hom}_k(V,k)=DV.$$
Furthermore, applying the functor $D(-)$ to the quasi-isomorphism $V\rightarrow M$ gives us the quasi-isomorphim $DM\rightarrow DV$. This in turn allows us to define a quasi-isomorphism $\begin{bmatrix}S&DM\\0&R\end{bmatrix}\rightarrow\begin{bmatrix}S&DV\\0&R\end{bmatrix}$, so $$\begin{bmatrix}S&DM\\0&R\end{bmatrix} \textrm{ and }\begin{bmatrix}S&DV\\0&R\end{bmatrix}$$ are derived equivalent and hence $$\begin{bmatrix}R&M\\0&S\end{bmatrix} \textrm{ and }\begin{bmatrix}S&DM\\0&R\end{bmatrix}$$ are derived equivalent. \end{proof}
\end{document}
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arXiv
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Implementing a quattromodal freight hub: an approach for the city of Vienna
Karin Markvica ORCID: orcid.org/0000-0001-8083-73691,
Matthias Prandtstetter ORCID: orcid.org/0000-0003-1263-79841,
Jürgen Zajicek1,
Bernhard Heilmann1,
Gernot Lenz1,
Georg Hauger2,
Monika Wanjek2,
Claudia Berkowitsch2,
Sarah Pfoser3,
Oliver Schauer3,
Lisa-Maria Putz3,
Reinhold Schodl4 &
Sandra Eitler4
In terms of freight transportation it is essential to pick the most convenient mode(s) of transport (MOT). To get a more flexible system, one can assume that the number of transport options at each hub should be maximized. Therefore, it is investigated how a new hub concept comprising four MOT, a so-called "quattromodal freight hub", can be implemented in the existing transport system from a traffic planning, technological and organizational point of view.
The research incorporates screening existing literature and best practices, conducting stakeholder interviews as well as performing site visits at the best practice areas Hamburg (Germany) and Constanta (Romania). Furthermore, the implementation perspective for a quattromodal freight hub in the city of Vienna (Austria) is examined.
The research revealed the strengths, weaknesses, opportunities and threats of the concept from a theoretical and practical perspective. Four options to create a quattromodal freight hub in the city of Vienna could be identified taking the effects on the overall transport system capacity, arising costs and the legal framework into account.
Cost and efficiency related decision criteria are decisive for the implementation of a quattromodal freight hub. An implementation in the city of Vienna is attractive in terms of prestige and unique selling proposition for the region but at the same time requires further research in terms of legal aspects and impacts on the region.
In December 2008, the European Commission (EC) agreed upon a package of directives and targets for climate protection and energy ("20–20-20 targets"). One of the targets for the year 2020 addresses the reduction of greenhouse gas emissions by 20% compared to the year 2005 [1]. Other important steps in this direction were the Paris Agreement in 2015 [2] and the Katowice Climate Package 2018 [3] as a global agreement implementing the Paris Agreement. Since the transport sector is a key factor for the achievement of these objectives, the EC has set the objective to shift 50% of transports in the European Union over 300 km to environmentally friendly transports (such as rail and inland navigation) by 2050 to reduce CO2 emissions and further negative impacts on the environment [4].
In alignment with these goals the concept of the Physical Internet (PI) [5] emerged in the logistics sector. Here, the main idea is to enhance efficiency by vertical and especially horizontal collaboration and cooperation among all stakeholders and competitors along the supply chain. However, to be more responsive to user requests the PI idea contains also strategies related to (intermediate) warehouse placements [6]. One of the initiatives within the PI is the transition of the current (partly intermodal freight) transportation network towards a synchromodal transportation network [7] as outlined in the ALICE roadmaps towards the PI [8]. Synchromodality, also referred to as Transport-as-a-Service in some contexts, is the continued development of intermodal and multimodal transport systems, whereby the actors along the transport chain cooperate and goods can switch in real-time between the modes of transport (MOT) in the most flexible way adapted to the respective resources [9]. Only the most basic transport requirements, such as costs, duration or sustainability requirements are defined by the consignor in advance to optimize transport and utilize existing resources best possible [7]. Hence, synchromodality refers to a transport concept which aims at shifting traffic from road to environmentally friendly MOT, such as rail and navigation while keeping flexibility of road transports.
Relevant publications on multimodality in freight transport such as the "Green Paper on Sustainable Logistics" [10] and the "Green Paper on Sustainable Logistics in Urban Areas" [11] focus on offering more than one MOT for freight transport at a location but are nonetheless currently limited to bi- or trimodal transport incorporating road, rail and/or inland navigation only. However, air cargo is dealt with in terms of the Trans-European Airport network but apart from that left out in most papers on multimodal transportation. To be able to pick the most convenient MOT, variety at a transshipment point is a crucial factor.
As passenger numbers tend to increase and are expected to reach 7 billion persons by the year 2034 according to the International Air Transport Association [12], air traffic is going to expand which offers capacity for transport of goods. The air industry agreed on targets to reduce environmental damage [12]. Therefore, air cargo is expected to be more environmentally friendly in the future and hence more competitive compared with other MOT. As a further logical step, an integration of air cargo in the existing hub structure would be desirable. Taking traffic planning as well as organizational and technological matters into account is one step in the process. The main research question is therefore: "Is it possible to create something bigger, like a quattromodal freight hub, out of an existing infrastructure?"
Quattromodal transport offering four different MOT is rarely discussed even though some studies [13, 14] already deal with possible models and effects of a regional and/or organizational linkage of several MOT taking into account air cargo and giving practical examples. Furthermore, there are hardly any hubs which are characterized by an integration of air cargo into freight transport and already integrated in an optimal manner in the freight network in terms of transport technology and/or organization. The technical and/or organizational integration of air cargo in the freight transport network is currently taking place at the largest airports in Europe. Only a few European cities (e.g. Hamburg) are a quattromodal transport provider for freight transport so far.
Since the concept of quattromodality is only slightly targeted by research, an exhaustive literature review only revealed partially useful definitions mainly pointing out the opportunity to use different MOT at a location and to link them together [15]. For this research, quattromodal hubs were therefore defined as "logistics pivots where the four MOT road, rail, waterways and air cargo come together" [16]. Pipelines as transport mode as well as the distinctions between inland and maritime navigation and regular and broad gauge suggested by Kummer [15] are deliberately left out to make the definition more readable and easier to comprehend for non-specialists [17].
Due to areal constraints (limited space), the potential of such a quattromodal freight hub offering four different MOT has to be assessed in a regional context. Beyond that, its practical relevance for freight transportation has to be investigated to reveal ecological, economic and social effects deriving from a new hub concept and to legitimate additional costs that might derive from it. Therefore, the question "How to establish a quattromodal freight hub in a region with potential?" should be addressed.
Details on the methodology are given in Chapter 2. Information on the site visit locations and key findings from the expert interviews can be found in Chapter 3. Chapter 4 examines the implementation perspective for a quattromodal freight hub in City of Vienna (Austria) and deals with the impact on the overall transport network. Chapter 5 outlines the conclusions and the discussion of the findings and further research needed.
The concept of quattromodality is not only in an early development stage from a theoretical point of view, but also from a practical one. Because of the lacking practical implementation, there are no reliable empirical values, figures and facts available for existing approaches. To generate knowledge on potential offered by quattromodality for the City of Vienna, our research focused on knowledge carriers as well as best practice hubs.
The research revolving around the two main questions "Is it possible to create something bigger out of an existing infrastructure?" and "How to establish a quattromodal freight hub in a region with potential?" therefore concentrated on
Desk research on
quattromodality and quattromodal freight hubs
best practice examples in Europe and beyond (including telephone enquiry)
specifications (technical, economical etc.) of various transport modes
framework conditions for an implementation in Vienna
Qualitative and quantitative data collection via
site visits to best practice areas Hamburg (March 2016) and Constanta (May 2016)
interviews with experts at the best practice areas Hamburg and Constanta based on guiding questions (see Additional file 1) to diagnose the state of the implementation (March 2016)
interviews with (inter-)national experts based on a standardized interview guideline (see Additional file 2) to assess the potential for Vienna (December 2015 until May 2016)
Scenario assessment/forecasting techniques for the impact estimation of such systems on transport in Vienna
In terms of site visits, Hamburg, Liège and Constanta were identified as especially interesting not only due to their importance for freight logistics but also due to the range of freight related organizations located there (cf. Table 1). Since scheduling stakeholders for an interview failed for Liège, the site visits were performed in Hamburg and Constanta.
Table 1 Site visits
Employees from 23 organizations participated in the interviews (see Table 2) which were conducted between December 2015 and May 2016. Except for three interviews via phone, all interviews were face-to-face interviews. Nine interviews were conducted during site visits to Hamburg and Constanta based on an interview guideline (see Additional file 2), all other interviews were performed in Austria based on guiding questions (see Additional file 1).
Table 2 Stakeholderinterviews
The insights on the practical implementation from best practices site visits and the expert point of view on the potential arising from quattromodality not only resulted in a SWOT analysis which is explained in detail in [18] but also in a conceptualization of the implementation of a quattromodal freight hub for the City of Vienna. This conceptualization not only included practicability, technical aspects and costs but also the effect on the overall transport system.
Site visits and expert interviews
In this section, details on the site visit locations and results of the expert interviews in Hamburg, Constanta and Austria are given. A more detailed representation of the interview results can be found in [18].
Hamburg site visit
The city of Hamburg in Germany has an area of 755.1 sq. km [19] and around 1.83 mio. Inhabitants in 2018 [20]. In the metropolitan region of Hamburg, the logistics sector directly or indirectly accounted for around 250,000 jobs in 2010 [21].
Hamburg offers road, rail, navigation, aviation and pipeline. The airport occupies an area of 5.7 sq. km [19] and had 17.6 mio. Passengers and 74,948 tons air freight in 2017 [22]. Nevertheless, the port is decisive for the handling of goods. Road freight transport benefits enormously from the volume of shipping transported. According to the industry portrait "Freight traffic in Hamburg" (2010) only about a third of the containers handled in the port of Hamburg had their destination or origin in the metropolitan region [21]. Around half of the containerized goods were either processed there or were destined for further transport via distribution centers. Due to the comparatively short distances 80% to 90% of these collection and delivery journeys were handled via trucks [21].
Hamburg port stretches on an area of 72 sq. km [19]. Measured by container turnover, Hamburg port is the third largest port in Europe in 2017 with 8.82 mio. TEU [23]. The seaborne cargo handling at that time was 136.5 mio. Tons mainly concentrating on container (90.3 mio. tons) and grab cargo (23.5 mio. tons), followed by liquid cargo (13.7 mio. tons), agribulk (7.5 mio. tons) and break bulk (1.4 mio. tons) [23]. The modal split in container traffic was 56.6% truck, 41.4% rail and 2% barge. The port railway is the link between the transshipment terminals of the container ships and the European rail network with 46 tons cargo handling a year.
Constanta site visit
Constanta is located in south-east Romania and has a size of 58 sq. km with a population of around 300,000 inhabitants [24]. It is the only Romanian city served by road, rail, sea, river and air, and therefore among the most important cities of Romania due to its economic importance and activities in logistics [25]. At the same time, not all of these MOT are interconnected and therefore intermodality/multimodality is limited.
Constanta has the biggest port on the Black Sea which is one of the major European ports [25]. 413,253 containers and 670,536 TEUs were handled in 2017 [26]. The port offers access to the European railway network from every terminal through a 300 km railway system [26]. In 2017, a Memorandum of Cooperation with Romanian state rail cargo operator CFR Marfa and the Port of Constanta Authority was signed by the freight forwarder PKP Cargo and the Port of Gdansk Authority. This was done to enhance the logistics services on the railway corridor between the two ports Gdansk (Poland) and Constanta [27].
The Mihail Kogalniceanu International airport is in 25 km distance from the city and serves as the main airport [25] which is primarily used for passenger transport. The freight terminal was constructed a few years ago and has a capacity of 2500 cubic meters [28].
Results from expert interviews
To develop a baseline for the case study, the most important interview results are summarized. Certain aspects, pointed out by the experts, are emphasized since they influence a successful linkage of the existing infrastructure. Current shortcomings of the site visit locations are stated to give insights on the implementation potential in these regions.
Spatial, technological and organizational aspects
The stakeholder interviews with experts from Austria and neighboring countries showed that the term "quattromodal hub" is not entirely self-explanatory, i.e. it requires a clear definition.
The spatial proximity of the modes of transport is generally regarded as important, with respondents considering a quattromodal hub to be more spacious and not bundled to one location. In a previous question it was already pointed out that aviation takes up a lot of space and therefore requires spatial expansion. Two interviewees remarked that shipping and aviation are not compatible and therefore do not make sense at one location.
The uneven picture in terms of spatial distance is conclusive based on different perspectives: For logistics service providers, spatial distances are part of the business basis, while spatial bundling tends to bring advantages for infrastructure operators.
Only a few interviewees regard a technical coordination as little or not important. In this context, technological differences between the airport and container terminals as well as the low level of networking between the airport and the outside world are referred to. The other participants regard it as a very important, if not a basic prerequisite. In this context, reference is made to the importance of compatible containers and security aspects.
Two interviewees consider the organizational coordination to be of little importance, the other interviewees consider the organizational coordination to be important or very important. Here it is pointed out that this is in any case closely linked to technological coordination.
Potential assessment and SWOT
More than one interviewee emphasized the feasibility or practical relevance of a quattromodal hub. It was also mentioned that the definition should be as broad as possible. It was further mentioned that unused potentials of trimodal hubs should serve as a starting point and different regional understanding (thus also the role of pipelines) should be considered.
Regarding the SWOT (strengths, weaknesses, opportunities and threats) arising from a quattromodal freight hub, various aspects were mentioned by the experts. The technological differences between the airport and container terminals, carrier incompatibility, safety and IT aspects were mentioned as hindering factors at current freight hubs due to the lack in standardization. In case of the IT related aspects, the concerns were later on overcome by an expert in the data communication domain. Furthermore, the low level of transport network integration of airports was detected as a problem. The potential resistance of the population or residents and conflicts of interest between different stakeholders were pointed out as main weaknesses of the concept of quattromodality. Even though these concerns exist, more than two thirds of the experts interviewed ascribe a quattromodal freight hub efficiency potential in either ecological or economical regard. Most interview partners agreed with the USP a quattromodal freight hub offers and the potential for easier handling of the most appropriate MOT. A rather neutral attitude was shown towards the spatial proximity. This was reasoned by the low transport costs between MOT that distance is not a relevant factor in terms of connecting air cargo to other transport modes.
Implementation potential for site visits
On closer examination of best practice regions, the experts in Hamburg revealed that the combination of air cargo and navigation is currently used for certain categories of goods such as luxury items, repair parts, (economically) perishable and other time-sensitive goods as well as medicines and drugs pointing out the lack of a shared storage space at the site. Regarding the inclusion of air cargo, the transport volume was stressed as one limiting factor since a combination must be efficient and the cost-benefit-ratio has to support the decision. The interview partners in Constanta were less concerned about the concept itself and more focused on the lacking implementation perspective at the site due to various limitations such as the current military use of the airport. Therefore, they agreed with the value of a quattromodal freight hub as new business concept for the location but revealed that Constanta would not be able to manage such a freight hub soon.
Application for the City of Vienna
In Austria, the transport sector accounts for almost a quarter of greenhouse gas emissions [29], which can (apart from passenger transport) lead back to its importance as a logistic base. The high logistics competence of Austria is emphasized both by the "International Business Compass 2014" and the Hamburg Institute of International Economics. The World Economic Forum places the location among the top 10 with the quality of rank 6 for roads and rank 12 for railway [30]. For the year 2011, the Austrian motorway network density was 20.5 km per 1000 sq. km (compared to 16.3 km per 1000 sq. km in the EU average) and the Austrian railroad density with 58.3 km per 1000 sq. km (compared to 49.2 km per 1000 sq. km in the EU average) [31].
Compared to other European countries, the Austrian infrastructure is well developed. This also accounts for the City of Vienna where the port of Vienna functions as a trimodal hub covering rail, road and navigation and at approximately 9.5 km air-line distance the Vienna International Airport operates as a bimodal freight hub covering road and air cargo; not including rail as freight transport option due to a lack of freight rail tracks. Both hubs are located at the western bank of the Danube River with direct access in the case of the port of Vienna and a 3 km air-line distance for the Vienna International Airport, see also Fig. 1.
Existing transport connection between port of Vienna and Vienna International Airport [14]
To create in quattromodal freight hub in Vienna, the port of Vienna and the Vienna International Airport have to be connected to each other since there are no other locations providing adequate air cargo and inland navigation qualities. This requires a well elaborated connection between these two sites. Considering the geographic location and existing transport system, we identified four different options to connect the Vienna International Airport to the port of Vienna to operate freight transport:
Connection via road
Connection via rail
Connection via circulating cable car and inland navigation
Connection via freight-zeppelin
Among all four options, the connection via road between the port of Vienna and the Vienna International Airport is the only existing one. It includes A4 Ost Autobahn (East Highway), B228 Simmeringer Straße (Simmeringer Road) and B14 Klosterneuburger Straße (Klosterneuburger Road) (Fig. 1). The terminal at the port of Vienna as well as the cargo center at the Vienna International Airport have road access and are therefore well connected via the road network.
Since this is an existing transport connection, this solution does not involve additional costs apart from infrastructure maintenance costs in the case of increased traffic volume. However, a barrier of implementation could be the environmental aspect and the associated image for the City of Vienna on the one side and the current traffic levels at this highway section on the other side. To evaluate the connection between the two locations, an estimation of the effects of additional trucks on travel times on this route was assessed (see Chapter 4.5).
The Vienna International Airport has currently three terminals, an office park, a General Aviation Center/VIP Terminal and an air cargo center. To deal with the growing demand for international air traffic and to maintain the position of Vienna as an attractive and competitive passenger and freight hub within Europe, expansion blueprints for the Vienna International Airport were introduced. The airport management strives for a third runway which will be 3680 m long and 60 m wide and constructed until the year 2025 [32]. As a result of the citizens' participation process in order to achieve minimized noise pollutions, it shall run 2400 m south of the first runway (runway 29/11) and parallel to it which means that the B10 Bruckner Bundesstraße (Bruckner Federal Road) has to be relocated further to the south. Nowadays, no further changes to the existing infrastructure are known to public but the possibility is given due to major improvements. Therefore, we did not take the current location of the cargo center as a limiting factor for the rail concept.
In contrast to the surrounding companies, the Vienna International Airport mainly focusses on passenger transport and is not connected to the rail network via freight rail tracks. The port of Vienna on the other hand has freight rail tracks in use and therefore already enjoys better transportation connections. To connect the airport to the rail network it is necessary to dock on the freight rail tracks of the Petrochemie Danubia northwest of the Vienna International Airport. This not only requires the new installation of a 4 km section but also the relocation of the air cargo center between the current airfield and the planned extension (Fig. 2).
Additional tracks and air cargo relocation for the rail connection [14]
The existing rail network furthermore does not offer a straight connection between the port of Vienna and the Vienna International Airport. A connection without detours would require using the station Kaiserebersdorf as connection point between the tracks from the port of Vienna and the Vienna International Airport. The second option would be to accept a detour and use the loop via Kledering – Hauptbahnhof (Vienna Central Station) – Meidling – Donaulaendebahn – Kaiserebersdorf (see also Fig. 3) which would be the more cost-efficient solution since the installation of a skidway is more expensive than the additional travel time. Besides costs, the detour has the additional advantage that is offers a direct connection to the new freight yard Inzersdorf which went into full operation in the year 2017 and the skidway Kledering offering the routing of wagon groups in the direction of international destinations.
Overview about the planned and existing railway connections the airport
Apart from using the rail network most efficiently, the air cargo center has to be designed taking certain requirements into account. Since the freight rail track mark an end at the air cargo center, a terminal station would result. At least three tracks should be envisaged to transform it to a skidway of which two tracks are available for unloading and loading the trains. An essential part of the track system is that the design allows the arriving trains to enter the hall by means of the line locomotive. The line locomotive is consequently separated from the train and can use the loop around the hall to move from one end to the other. This does not require a lot of personnel since only a specially trained person is needed to interlock and perform the brake test.
Another alternative to connect the two freight hubs is based on the consideration that the hardly used Danube River is nearby the airport and would provide a direct connection to the port of Vienna without disturbance of the existing transport network. A hindering factor in this regard is the Danube-Auen nature reserve which is located north of the airport. A direct connection via road or tunnel is therefore not possible.
An alternative approach would be to use the air space and therefore a circulating cable car which connects the riverbank to the air cargo center in 1.5 km distance (Fig. 4). The pillars of the cable car do not lead to dissection of the landscape and do not require much space on such a short distance. As a transshipment area to load the goods on the barge, an area with agricultural use on the Danube River could be considered.
Cable car line for the connection with the riverbank [14]
Apart from the already introduced advantages, the cable car can be automated. With a minimum additional staff, it is a very efficient way to transport goods from one place to another [33]. To use a circulating cable car would mean that this infrastructure is only used for freight purposes and that the air cargo center does not have to be relocated. Furthermore, a simple implementation of an automatic reloading of the containers between the cable car and the barge can be arranged.
The main limitation factors to this concept are the arising investment costs for the construction of the cable car as well as the territory of the national park which is affected. It cannot be predicted whether such a connection between airport and river is feasible in terms of environmental law as well as sites to be crossed (owned by railway and/or road operator) and what additional arising costs may accompany it.
Despite the insolvency of the company Cargolifter in the year 2002, the concept of Zeppelins for transport purposes is not yet a thing of the past. Cargolux, a freight forwarder from Luxembourg, signed a strategic partnership with the aerospace company Aeros in the year 2015 to test the vertical removal of freight containers or large loads, which has been the main weakness of this means of transport [34].
An unconventional way to connect port and airport would be the introduction of a freight-airship [35, 36] which operates on its own tracks. The line could be for instance guided along the existing infrastructure (Fig. 1) using guiding ropes with a length of up to approximately 9.6 km and pillars of 20 m to 30 m height. Since standard solutions are currently not available on the market, the technical configuration can only rely on the solutions from the cable railway sector taking into account the unequal wind impact area of the freight-airship to the conventional gondolas which require to be picked up by special constructions in order to prevent the round rope from being pulled out.
The freight-airship would by far be the least space consuming and the quietest option to connect the port and the airport but there is no existing infrastructure and therefore costs including investment, ground rent and cost of operation arise. Additional requirements of owners of the sites to be crossed (especially the railway operator) should be taken into account since these are very likely to occur.
Impact on existing road network
As mentioned in Chapter 4.1, the road link is the only existing connection between the airport and the port. To assess the impact of additional truck traffic on the road network, the effect on the travel time was investigated for the section concerned on the A4 Ost Autobahn (East Highway) for each direction.
The targeted road segment between the highway junctions "ASt Flughafen Schwechat" and "ASt Simmering" has a length of approximately 11 km. In both directions a two-lane section between "ASt Simmering" and "ASt Schwechat" (link with the highway "S1") and a three-lane section between "ASt Schwechat" and "ASt Flughafen Schwechat" is available (Fig. 5). Thus, the capacity of the two-lane section limits how much additional vehicles can use the entire route.
Targeted road section on the A4 Ost Autobahn (East Highway)
In terms of methodological approach, Volume Delay Functions (VDF) were used which express the travel time or travel time loss as a function of the traffic volume and therefore require a capacity value of the road traffic system as an input variable [36].
The fact that the capacitator value decreases with increasing interval duration has been described as a constant, for example, by Keller et al. [37] and Ponzlet [38]. A more recent study explains that the probability of congestion increases with increasing interval duration [39]. The investigation of Keller et al. [37] shows that a fundamental diagram based on 5-min intervals in the area of free and partial traffic represents a fundamental diagram valid for stationary states. However, a linear calculation of the ascertained capacitator value C5 from 5-min intervals to the maximum traffic volume per hour would exceed the capacity, which is why Keller et al. [37] derive a factor for determining the capacity from their empirical investigations. For the calculation of capacity C60 therefore applies:
$$ {C}_{60}=0,84\ast {C}_5\ast 12 $$
To measure the influence of heavy traffic on traffic density, the traffic volume is calculated in passenger car units instead of vehicle traffic intensities.
$$ {q}_{PCE}={q}_{PKW}+{q}_{LKW}\ast {f}_{LKW} $$
q PCE Total traffic volume in passenger car units [passenger car units per hour]
q PKW Passenger car traffic density [passenger cars per hour]
q LKW Truck traffic density [trucks per hour]
f LKW Conversion factor of truck traffic in passenger car units [−]
Geistefeldt [40] describes the influence of a truck on the occupancy rate of a four-lane section with two to three passenger car units (equivalent fSV for the conversion of the heavy traffic volume in passenger car units with a heavy traffic share of 5–15%). Based on these findings, a factor of fLKW = 2 is assumed.
The speed at free traffic flow v0 is a constant cross section specific value but must not be equated with the legally permissible speed. In this study, v0 is assumed to be a 95% quantile of all driven current speeds at the cross section. A representative of the conical VDF is the hyperbolic function according to Akcelik [41]:
$$ {t}_{akt}={t}_0+0,25\bullet T\left(\frac{q_{PCE}}{C}-1+\sqrt{\left(\frac{q_{PCE}}{C}-1\right)2+\frac{8\bullet {J}_D\bullet \frac{q_{PCE}}{C}}{T\bullet C}}\right) $$
CCapacity [passenger car units per hour]
TFlow period (Time interval in which a certain traffic volume q prevails)
J D Delay parameter
The function parameters are estimated by a non-linear regression analysis using a Least Square Method.
The capacity was calculated with an Aerde function [42] for all available cross sections in both directions on the highway A4 between the junction "ASt Prater" (km 0.0) and the Airport "ASt Flughafen Schwechat" (roughly at km 12.5) in the period from January to June 2012. Night hours between 8 pm and 6 am were not considered [36]. The results on capacity and parameter of the VDF are clearly dependent on the cross section or the type of the cross section. From the results of the REFEREE project [36] two cross sections of the highway A4 were selected, which had comparable characteristics to the route under consideration.
The following figure (Fig. 6) shows a v-k-diagram of the two-lane cross section "MQ_A04_2_003.200" in the direction of travel to Vienna, Fig. 7 for the three-lane cross section "MQ_A04_1_011.597" in the direction of travel to the Vienna International Airport near the junction "ASt Flughafen Schwechat".
k-v-diagram and Volume Delay function of the two-lane highway
k-v-diagram and Volume Delay function of the three-lane highway
The parameters free speed v0, capacity C and delay parameter JD of the two-lane highway section (MQ_A04_2_003.200) were the basis for the calculation of the VDF. As a basic load of the line segment, a typical time series of one working day of the year 2013 was selected on the respective route for one direction of travel (Fig. 8).
Typical hydrograph of the highway A4
The calculated travel times for various additional loads (+ 100 to + 1000 trucks per hour) show that the considered routes on highway A4 could take up additional journeys of trucks at certain times of the day, but at other times the capacity is significantly exceeded which means an additional travel time up to 10 min per road kilometer (Figs. 9 and 10). Especially during 4 pm and 10 pm both routes are already heavily used and can only accommodate a few additional trucks which means less than 300 trucks per hour.
Travel time depending on additional truck traffic intensity for route 1
The results are rated for the measured values, in particular the hydrographs of the passenger car traffic intensity of the year 2013.
As a result of the travel time extensions determined, the attractiveness for the connection in motorized individual traffic (MIV) is reduced in the section between junction of the highway A4 at "ASt Simmeringer Haide" and "ASt Flughafen Schwechat".
According to a survey commissioned by the Planungsgemeinschaft Ost (planning association east), the relations between the districts of Vienna-region (east), Bruck/Leitha, Neusiedl and Eisenstadt as well as the neighboring countries of Slovakia and Hungary to and from Vienna are affected [43]. The reduction in the demand for transport on these relations which results from increased travel times is estimated with an elasticity approach. Several studies [44,45,46] indicate a mean elasticity with respect to the travel time in the MIV with − 0.4 which means that a travel time increase of 10% is estimated to result in a decrease of traffic demand of MIV by 4% for this transport connection. According to the predicted travel times (Figs. 9 and 10) on the envisaged sections of the highway A4, some significant declines in MIV traffic could be expected as a result of the increase in truck traffic.
Impact on existing rail network
The presented approach to create a quattromodal freight hub by connecting the port of Vienna and the Airport of Vienna via rail in Chapter 4.2 requires usage of the existing rail network. Therefore, it is limited by the network capacity. To assess the feasibility of such a connection, the underlying assumptions are disclosed in the following section.
The developed scenario is represented by a new freight center situated in the south of the runway 11/34 (east-west) and the existing railway connection to the airport train station. The freight center will be connected via new railway siding which should have its starting point west of the existing train station "Mannswörth Bahnhof" (Fig. 5). The length of this single-track connection will be about 4 km. It opens at the freight center to three loading tracks and a bypass track with a dead-end track of about 200 m length to replace the locomotives for the retour journey.
The logistics concept includes the handling of the air freight to containers in the freight center for the transport of these goods to the harbor or for the long-distance transport of complete wagonloads. Regarding to the number of 300 trucks (2 × 20″ or 1 × 40″ container) mentioned in Chapter 4.5 and based on the capacity of a typical container rail freight wagon type "Sgnss" (3 × 20″ containers or 1 × 40″ and 1 × 20″) it is expected that about 13 to14 short trains (300 m) will run in both directions each day.
The logistics concept foresees the transfer of the freight wagons to the central shunting yard in "Kledering" to shunt the wagons dedicated to the harbor and to other destinations. This approach has the advantage that the freight wagons can be separated and transferred to the harbor without shunting activities in the small train station of "Wien Kaiserebersdorf".
The current time table of the railway connection between the City of Vienna and the airport shows about 7 to 9 trains in each direction depending to the day time and some freight trains heading to the refinery at Schwechat. The operations capacity of the railway line between station "Mannswörth Bahnhof" and "Wien Kaiserebersdorf" shows enough potential to manage the additional trains in both directions.
Conclusions and outlook
This paper describes the concept of quattromodality from a theoretical and practical point of view, covering not only the definition and details on spatial, technological and organizational aspects as well as arising potentials but also offering implementation concepts for the City of Vienna.
To investigate the mentioned aspects of quattromodal freight hubs designating "logistic pivots where the four MOT road, rail, waterways and air cargo come together" [16]. stakeholder interviews and site visits were performed. In many discussions, it was often mentioned that the combination of air cargo and inland navigation is not meaningful. However, the main advantage of a quattromodal freight hub is not the ability to transship directly from plan to vessel (although this is also done for specific freight groups as outlined by the interviewees during the Hamburg site visit) but the savings potential due to synergies. E.g., instead of having to implement customs treatment at two locations, one combined treatment can be employed. To get one step further than assessing the concept itself and the current situation at two important logistic hubs, the spatial implementation for the City of Vienna was analyzed.
The investigation of quattromodal freight hubs has revealed the strengths and opportunities offered by combining four different MOT but also the weaknesses and risks that should be envisaged before implementing. The experts' opinions unfold cost and efficiency related decision criteria as most important and (not surprisingly) do not focus on environmental aspects too much. Apart from that, it is agreed with the positive aspect on the prestige of a region and hence on the USP of a site. This would indeed be beneficial for the City of Vienna which is already an important hub within Europe and could strengthen its position by offering this flexible and cost-efficient approach of selecting the most appropriate MOT out of the four offered ones.
A further investigation of an implementation in the City of Vienna revealed four different options to connect the port of Vienna (trimodal hub) with the Vienna International Airport (bimodal hub). Each of them offers certain advantages but also leads to challenges since the implementation does not happen on a brownfield but within a well elaborated transport network and a city structure characterized by high building density as well as strong legal regulations.
This also indicates the shortcoming in the research when it comes to legal aspects of an implementation of new infrastructure elements in the existing network. While the effect of a connection on the existing transport network was assessed, the legal implications could only be noted and not be solved on a theoretical basis so far. Since land holding might be involved and these legal questions cannot be answered generally.
Another factor that might be worth considering is the effect on the region by offering faster transport channels. More efficiency in freight transport does offer potential for additional loads and most probably connects the region much stronger. Although technological developments are already taking place and the already mentioned concepts emerge as promising alternatives (e.g. high-speed transport and efficient maritime links), its effect on network quality and thus on the flows of goods and persons and the affected regions (fusion of agglomerations) is currently not a major concern at the European level.
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We thankfully acknowledge the time and expertise provided by the stakeholders and experts during our site visits and interviews.
This work has been partially funded by the Austrian Federal Ministry for Transport, Innovation and Technology (bmvit) in the 'Mobilitaet der Zukunft' program under grant number 850339 ("Q4").
AIT Austrian Institute of Technology, Center for Mobility Systems, Giefinggasse 4, 1210, Vienna, Austria
Karin Markvica
, Matthias Prandtstetter
, Jürgen Zajicek
, Bernhard Heilmann
& Gernot Lenz
Vienna University of Technology, Augasse 2-6, 1090, Vienna, Austria
Georg Hauger
, Monika Wanjek
& Claudia Berkowitsch
University of Applied Sciences Upper Austria, LOGISTIKUM Steyr, Wehrgrabengasse 1, Steyr, Austria
Sarah Pfoser
, Oliver Schauer
& Lisa-Maria Putz
University of Applied Sciences BFI Vienna, Wohlmutstrasse 22, 1020, Vienna, Austria
Reinhold Schodl
& Sandra Eitler
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KM participated in the Hamburg site visit, carried out interviews with stakeholders and worked on the application for the city of Vienna. KM and MP are responsible for the conceptualization, original draft preparation, review and editing. MP carried out interviews with stakeholders and worked on the application for the city of Vienna as well as on the visualizations. JZ worked on technical details of the application for the city of Vienna as well as the validation. BH and GL measured the impact on the existing road network in Vienna. GH was responsible for the funding acquisition and supervision. He contributed to the methodology and gave feedback on the work performed. MW and CB worked on the project administration, participated in the Hamburg site visit, carried out interviews with stakeholders and did research on the gaps in knowledge with regard to the integration of the air freight to quattromodal freight hub. SP organized and participated in the Hamburg site visit, carried out part of the desk research and performed interviews with stakeholders. OS and LP carried out part of the desk research. RS and SE participated in the Constanta site visit, carried out interviews with stakeholders, performed the potential assessment and SWOT analysis and gave feedback to the other tasks performed. All authors read and approved the final manuscript.
Correspondence to Karin Markvica.
Questionnaire. (DOCX 18 kb)
Interview Guide. (DOCX 31 kb)
Markvica, K., Prandtstetter, M., Zajicek, J. et al. Implementing a quattromodal freight hub: an approach for the city of Vienna. Eur. Transp. Res. Rev. 11, 34 (2019) doi:10.1186/s12544-019-0367-3
Freight logistics
Quattromodality
Quattromodal freight hub
Vienna region
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CommonCrawl
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Prof Richard Bower
BA Physics, PhD.
+44 (0) 191 33 43526
[email protected]
https://www.dur.ac.uk/images/profiles/527/eagleprojectandBower.jpg
Professor in the Department of Physics ocw214 +44 (0) 191 33 43526
Member of the Centre for Extragalactic Astronomy
Member of the Institute for Computational Cosmology OCW214 +44 (0) 191 33 43526
Member of the Institute of Medieval and Early Modern Studies
Responsibilities within department
Organisation of the astronomical computingfor the Level 4 astronomy projects.
Post-graduate course in Statistical Methods in Astronomy
My reasearch time is shared between observational andtheoretical cosmology.In particular, I work on the formation and evolution of galaxiesand clusters of galaxies. I am fascinated by the existence ofgalaxies and the ways in which they formed. Why is the nightsky so full of stars? What has made the different types ofgalaxies so different from each other? I like to speculate aboutwhat it would be like to live in a galaxy cluster. Interested? metoo. Take this link to find out more about my research interestsand recent papers...
extragalactic astronomy and cosmology
Institute for Computational Cosmology
Ordered Universe
Available for media contact about:
Astronomy and Astrophysics: Cosmology: the formation of galaxies and the universe we see.
Tanner, B. K., Bower, R. G., McLeish, T. C. B. & Gasper, Giles E. M. (2016). Unity and Symmetry in the De Luce of Robert Grosseteste. In Robert Grosseteste and the Pursuit of Religious and Scientific learning in the Middle-Ages. Cunningham, Jack P. & Hocknull, M. Springer. 18: 3-20.
Schaller, M., Bower, R. G. & Theuns, T. (2013), On the use of particle based methods for cosmological hydrodynamical simulations, 8th International SPHERIC Workshop. Trondheim, Norway, Trondheim.
Mitchell, N.L., Bower, R.G., Theuns, T. & Vorobyov, E.I. (2012), Towards Understanding Simulated Feedback in AMR and SPH Codes and the Multi-Phase Nature of the ISM, in Capuzzo-Dolcetta, R., Limongi, M. & Tornambè, A. eds, Astronomical Society of the Pacific Conference Series 453: Advances in Computational Astrophysics: Methods, Tools, and Outcome. 19.
Scharf, C. A., Smail, I., Ivison, R. J. and Bower, R. G. (2004), X-ray Emission from a Proto-cluster at z = 3.8, Clusters of Galaxies: Probes of Cosmological Structure and Galaxy Evolution
Kodama, T., Smail, Ian, Nakata, F., Okamura, S. and & Bower, R. G. (2004), History of Mass Assembly and Star Formation in Galaxy Cluster, Studies of Galaxies in the Young Universe with New Generation Telescope 23-31.
Bacon, Roland, Bauer, Svend-Marian, Bower, Richard, Cabrit, Sylvie, Cappellari, Michele, Carollo, Marcella, Combes, Francoise, Davies, Roger L., Delabre, Bernard, Dekker, Hans, Devriendt, Julien, Djidel, Slimane, Duchateau, Michel, Dubois, Jean-Pierre, Emsellem, Eric, Ferruit, Pierre, Franx, Marijn, Gilmore, Gerard F., Guiderdoni, Bruno, Henault, Francois, Hubin, Norbert, Jungwiert, Bruno, Kelz, Andreas, Le Louarn, Miska, Lewis, Ian J., Lizon, Jean-Louis, McDermid, Richard, Morris, Simon L., Laux, Uwe, Le Fèvre, Olivier, Lantz, Blandine, Lilly, Simon, Lynn, James, Pasquini, Luca, Pecontal, Arlette, Pinet, Patrick, Popovic, Dan, Quirrenbach, Andreas, Reiss, Roland, Roth, Martin M., Steinmetz, Matthias, & Stuik, Remko, Wisotzki, Luc and de Zeeuw, P. Tim (2004), The second-generation VLT instrument MUSE: science drivers and instrument design, Ground-based Instrumentation for Astronomy. Edited by Alan F. M. Moorwood and Iye Masanori. Proceedings of the SPIE, Volume 5492, pp. 1145-1149 (2004) 5492: 1145-1149.
Fritz, A., Ziegler, B. L., Bower, R. G., Smail, I. and & Davies, R. L. (2004), Early-type Galaxies in the Cluster Abell 2390 at z = 0.23, Clusters of Galaxies: Probes of Cosmological Structure and Galaxy Evolution
Baldry, I. K., Balogh, M. L., Bower, R., Glazebrook, K., & and Nichol, R. C. (2004), Color bimodality: Implications for galaxy evolution, AIP Conf. Proc. 743: The New Cosmology: Conference on Strings and Cosmology 743: 106-119.
Gilbank, David G., Castander, F. J., Balogh, M. L. and & Bower, R. G. (2004), Wide-field spectroscopy of a galaxy cluster pair at z=0.4, IAU Colloq. 195: Outskirts of Galaxy Clusters: Intense Life in the Suburbs 95-97.
Balogh, M.L. & Bower, R.G. (2003), Galaxy Evolution: Internally or Externally Driven?, Revista Mexicana de Astronomia y Astrofisica Conference Series 17: 220-221.
Couch, W. J., Balogh, M., Bower, R. and Lewis, I. (2003), The Cluster `Sphere of Influence': Tracking Star Formation with Environment via Hα Surveys, ASP Conf. Ser. 289: The Proceedings of the IAU 8th Asian-Pacific Regional Meeting, Volume I 289: 235-.
Kodama, T., Smail, I., Nakata, F., Okamura, S. and Bower, & R. G. (2003), Large Scale Environmental Effects in Clusters of Galaxies, Astronomical Society of the Pacific Conference Series 301: 235-.
Castander, F. J., Balogh, M. L., Bernardi, M., Bower, R. G., Connolly, A. J., Gilbank, D. G., Gómez, P. L., Goto, T., Hopkins, A. M., Miller, C. J., Nichol, R. C., Schneider, D. P., Seth, R. and Zabludoff, A. & I. (2003), Galaxy Star Formation as a function of Environment, Revista Mexicana de Astronomia y Astrofisica Conference Series 16: 229-232.
Kodama, T., Smail, I., Nakata, F., Okamura, S. and Bower, & R. G. (2002), History of Mass Assembly and Star Formation in Clusters, ASP Conf. Ser. 268: Tracing Cosmic Evolution with Galaxy Clusters 268: 301-.
Quilis, V., Bower, R. & Balogh, M. (2002), Blowing Bubbles in the ICM, ASP Conf. Ser. 268: Tracing Cosmic Evolution with Galaxy Clusters 268: 253-.
Morris, Simon, Content, Robert, Sharples, Ray, Bower, Richard, Davies, Roger & Baugh, Carlton (2002), A Million Element Integral Field Unit (MEIFU), Scientific Drivers for ESO Future VLT/VLTI Instrumentation Proceedings of the ESO Workshop held in Garching, Germany, 11-15 June, 2001 ESO Workshop. Garching, Germany, Springer, 99-107.
Bower, R.G. (2002), Making the Connection: Feedback, X-rays and Galaxy Formation, ASP Conf. Ser. 268: Tracing Cosmic Evolution with Galaxy Clusters 268: 257-.
Bridle, S. L., Bower, R. G., Smail, I. and Kneib, J.-P. (2001), Weak lensing reconstructions of low X-ray luminosity clusters from HST, Clusters of Galaxies and the High Redshift Universe Observed in X-rays
Gilbank, D. G., Bower, R. G. and Castander, F. Javier (2001), Optical vs. X-ray Selection for Finding Clusters of Galaxies, ASP Conf. Ser. 240: Gas and Galaxy Evolution 240: 644-.
Ziegler, B. L., Fricke, K. J., Balogh, M. L., Bower, R. & G., Gaztelu, A., Smail, I. and Davies, R. L. (2001), Galaxy Transformation in Poor Clusters at z≈0.25, ASP Conf. Ser. 240: Gas and Galaxy Evolution 240: 619-.
Bower, R.G. (2001), Galaxy Transformation in the Cluster Environment, ASP Conf. Ser. 240: Gas and Galaxy Evolution 240: 613-.
Quilis, V., Moore, B. & Bower, R. (2001), The origin of SO galaxies in clusters, Highlights of Spanish astrophysics II 65-.
Moore, S. A. W., Lucey, J. R., Colless, M., Kuntschner, H., Bower, R. & Davies, R. L. (2000), The fundamental properties of early-type galaxies in the Coma Cluster, IAU Symposium 201.
Terlevich, Alejandro, Kuntschner, Harald and Bower, & Richard (2000), Stellar Populations and the Colour-Magnitude Relation in Coma, ASP Conf. Ser. 215: Cosmic Evolution and Galaxy Formation: Structure, Interactions, and Feedback 215: 222-.
Moore, Ben, Quilis, Vicent and Bower, Richard (2000), Dynamical Effects on Galaxies in Clusters, ASP Conf. Ser. 197: Dynamics of Galaxies: from the Early Universe to the Present 197: 363-.
Bell, E. F., Bower, R. G., de Jong, R. S., Rauscher, B. & J., Barnaby, D., Harper, D. A., Hereld, M. and Loewenstein, R. F. (1999), The star formation histories of Low Surface Brightness galaxies, ASP Conf. Ser. 170: The Low Surface Brightness Universe 170: 245-.
Kodama, T., Bell, E. F. and Bower, R. G. (1999), Identification of High Redshift Clusters Using Photometric Redshifts, ASP Conf. Ser. 191: Photometric Redshifts and the Detection of High Redshift Galaxies 191: 160-.
Knapp, G. R., Binette, L., Bower, R. G., Brinks, E., & Goudfrooij, P., Hau, G., Pogge, R. W. and Young, L. M. (1999), Panel Discussion: Star Formation in Early-Type Galaxies, ASP Conf. Ser. 163: Star Formation in Early Type Galaxies 163: 142-.
Bower, R. G., Terlevich, A., Kodama, T. and Caldwell, N. (1999), The Formation History of Early-Type Galaxies: an Observational Perspective, ASP Conf. Ser. 163: Star Formation in Early Type Galaxies 163: 211-.
Bower, R.G. & Kay, S.T. (1999), Cosmological Parameters from the X-Ray Evolution of Clusters, IAU Symp. 183: Cosmological Parameters and the Evolution of the Universe 183: 243-.
Terlevich, A. I., Bower, R. G., Caldwell, N. and Rose, J. & A. (1998), The star formation history of early type galaxies in the Coma cluster, Untangling Coma Berenices: A New Vision of an Old Cluster 111-.
Terlevich, A. I., Bower, R. G., Smail, I., Barger, A. J., & and Ellis, R. S. (1997), The X-Ray Structure of the Butcher-Oemler Clusters AC114 and AC118, The Hubble Space Telescope and the High Redshift Universe 227-.
Bower, R.G. (1997), The Influence of Environmental Effects on Galaxy Formation, The Evolution of the Universe: report of the Dahlem Workshop on the Evolution of the Universe 245-.
Guzmán, R., Lucey, J.R. & Bower, R.G. (1993), The Fundamental Properties of Giant Ellipticals, Structure, Dynamics and Chemical Evolution of Elliptical Galaxies 19-.
Lorrimer, S.J. & Bower, R.G. (1991), Clustering from a fresh perspective: correlations in the Press-Schechter formalism, Clusters and Superclusters of Galaxies 125-.
Bower, R., Lucey, J. R. and Ellis, R. S. (1991), Cosmological implications of the colour-magnitude relation, Clusters and Superclusters of Galaxies 11-.
Rose, J., Sharples, R. M., Ellis, R. S. and Bower, R. G. (1989), The stellar content of early-type galaxies in dense environments, NATO ASIC Proc. 264: The Epoch of Galaxy Formation 371-.
Bower, R. G., Ellis, R. S. & Efstathiou, G. F. (1988), Dynamic friction in the rich cluster A2029, NATO ASIC Proc. 229: Cooling Flows in Clusters and Galaxies 115-119.
Bower, R.G. (1990). The stellar populations of early-type galaxies in groups and clusters. Durham Univ.~(England). PhD.
Icaza-Lizaola, M, Bower, Richard G, Norberg, Peder, Cole, Shaun, Schaller, Matthieu & Egan, Stefan (2021). A sparse regression approach to modelling the relation between galaxy stellar masses and their host haloes. Monthly Notices of the Royal Astronomical Society 507(3): 4584.
Pearce, Francesca A, Kay, Scott T, Barnes, David J, Bahé, Yannick M & Bower, Richard G (2021). Redshift evolution of the hot intracluster gas metallicity in the C-EAGLE cluster simulations. Monthly Notices of the Royal Astronomical Society 507(2): 1606-1622.
Aylett-Bullock, J., Cuesta-Lazaro, C., Quera-Bofarull, A., Icaza-Lizaola, M., Sedgewick, A., Truong, H., Curran, A., Elliott, E. Caulfield, T., Fong, K., Vernon, I. Williams, J., Bower, R. & Krauss, F. (2021). JUNE: open-source individual-based epidemiology simulation. Royal Society Open Science 8(7).
Chan, T K, Theuns, Tom, Bower, Richard & Frenk, Carlos (2021). Smoothed particle radiation hydrodynamics: two-moment method with local Eddington tensor closure. Monthly Notices of the Royal Astronomical Society 505(4): 5784-5814.
Borrow, Josh, Schaller, Matthieu & Bower, Richard G (2021). Inconsistencies arising from the coupling of galaxy formation sub-grid models to pressure-smoothed particle hydrodynamics. Monthly Notices of the Royal Astronomical Society 505(2): 2316-2327.
Habouzit, Mélanie, Li, Yuan, Somerville, Rachel S, Genel, Shy, Pillepich, Annalisa, Volonteri, Marta, Davé, Romeel, Rosas-Guevara, Yetli, McAlpine, Stuart, Peirani, Sébastien, Hernquist, Lars, Anglés-Alcázar, Daniel, Reines, Amy, Bower, Richard, Dubois, Yohan, Nelson, Dylan, Pichon, Christophe & Vogelsberger, Mark (2021). Supermassive black holes in cosmological simulations I: MBH − M⋆ relation and black hole mass function. Monthly Notices of the Royal Astronomical Society 503(2): 1940-1975.
Borrow, Josh, Schaller, Matthieu, Bower, Richard G. & Schaye, Joop (2021). SPHENIX: Smoothed Particle Hydrodynamics for the next generation of galaxy formation simulations. Monthly Notices of the Royal Astronomical Society
Masini, Alberto, Hickox, Ryan C., Carroll, Christopher M., Aird, James, Alexander, David M., Assef, Roberto J., Bower, Richard, Brodwin, Mark, Brown, Michael J. I., Chatterjee, Suchetana, Chen, Chien-Ting J., Dey, Arjun, DiPompeo, Michael A., Duncan, Kenneth J., Eisenhardt, Peter R. M., Forman, William R., Gonzalez, Anthony H., Goulding, Andrew D., Hainline, Kevin N., Jannuzi, Buell T., Jones, Christine, Kochanek, Christopher S., Kraft, Ralph, Lee, Kyoung-Soo, Miller, Eric D., Mullaney, James, Myers, Adam D., Ptak, Andrew, Stanford, Adam, Stern, Daniel, Vikhlinin, Alexey, Wake, David A. & Murray, Stephen S. (2020). The Chandra Deep Wide-field Survey: A New Chandra Legacy Survey in the Boötes Field. I. X-Ray Point Source Catalog, Number Counts, and Multiwavelength Counterparts. The Astrophysical Journal Supplement Series 251(1): 2.
Jackson, Thomas M., Rosario, D. J., Alexander, D. M., Scholtz, J., McAlpine, Stuart & Bower, R. G. (2020). The Star-Formation Properties of the Observed and Simulated AGN Universe: BAT vs EAGLE. Monthly Notices of the Royal Astronomical Society 498(2): 2323-2338.
Mitchell, P. D. Schaye, J. & Bower, R. G. (2020). Galactic inflow and wind recycling rates in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 497(4): 4495-4516.
Stott, J. P. Bielby, R. M. Cullen, F. Burchett, J. N. Tejos, N. Fumagalli, M. Crain, R. A. Morris, S. L. Amos, N. Bower, R. G. & Prochaska, J. X. (2020). Quasar Sightline and Galaxy Evolution (QSAGE) survey - II. Galaxy overdensities around UV luminous quasars at z = 1-2. Monthly Notices of the Royal Astronomical Society 497(3): 3083-3096.
Crain, Robert A., Bower, Richard G., Schaye, Joop & Mitchell, Peter D. (2020). Galactic outflow rates in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 494(3): 3971-3997.
Ludlow, Aaron D, Schaye, Joop, Schaller, Matthieu & Bower, Richard (2020). Numerical convergence of hydrodynamical simulations of galaxy formation: the abundance and internal structure of galaxies and their cold dark matter haloes. Monthly Notices of the Royal Astronomical Society 493(2): 2926-2951.
Gillman, S., Tiley, A.L., Swinbank, A.M., Harrison, C.M., Smail, I., Dudzevičiūtė, U, Sharples, R.M., Cortese, L., Obreschkow, D., Bower, R.G., Theuns, T., Cirasuolo, M., Fisher, D.B., Glazebrook, K., Ibar, E., Mendel, J. T. & Sweet, S.M. (2020). From Peculiar Morphologies to Hubble-type Spirals: The relation between galaxy dynamics and morphology in star-forming galaxies at z\raisebox-0.5ex\textasciitilde1.5. Monthly notices of the Royal Astronomical Society 492(1): 1492-1512.
Salcido, Jaime Bower, Richard G. & Theuns, Tom (2020). How feedback shapes galaxies: an analytic model. Monthly Notices of the Royal Astronomical Society 491(4): 5083-5100.
Pearce, Francesca A, Kay, Scott T, Barnes, David J, Bower, Richard G & Schaller, Matthieu (2020). Hydrostatic mass estimates of massive galaxy clusters: a study with varying hydrodynamics flavours and non-thermal pressure support. Monthly Notices of the Royal Astronomical Society 491(2): 1622-1642.
Ludlow, Aaron D, Schaye, Joop & Bower, Richard (2019). Numerical convergence of simulations of galaxy formation: the abundance and internal structure of cold dark matter haloes. Monthly Notices of the Royal Astronomical Society 488(3): 3663-3684.
McAlpine, Stuart, Smail, Ian, Bower, Richard G, Swinbank, A M, Trayford, James W, Theuns, Tom, Baes, Maarten, Camps, Peter, Crain, Robert A & Schaye, Joop (2019). The nature of sub-millimeter and highly star-forming galaxies in the eagle simulation. Monthly Notices of the Royal Astronomical Society 488(2): 2440–2454.
Swinbank, A. M., Harrison, C. M., Tiley, A. L., Johnson, H. L., Smail, I., Stott, J. P., Best, P. N., Bower, R. G., Bureau, M., Bunker, A., Cirasuolo, M., Jarvis, M., Magdis, G. E., Sharples, R. M. & Sobral, D. (2019). The energetics of starburst-driven outflows at z ∼ 1 from KMOS. Monthly Notices of the Royal Astronomical Society 487(1): 381-393.
Cooke, E A, Smail, Ian, Stach, S M, Swinbank, A M, Bower, R G, Chen, Chian-Chou, Koyama, Y & Thomson, A P (2019). The submillimetre view of massive clusters at z ∼ 0.8–1.6. Monthly Notices of the Royal Astronomical Society 486(3): 3047-3058.
Sweet, Sarah M, Fisher, Deanne B, Savorgnan, Giulia, Glazebrook, Karl, Obreschkow, Danail, Gillman, Steven, Tiley, Alfred L, Lagos, Claudia D P, Wang, Liang, Swinbank, A Mark, Bower, Richard & Sharples, Ray M (2019). Angular momentum of z ∼ 1.5 galaxies and their local analogues with adaptive optics. Monthly Notices of the Royal Astronomical Society 485(4): 5700-5714.
Lovell, Mark R, Barnes, David, Bahé, Yannick, Schaye, Joop, Schaller, Matthieu, Theuns, Tom, Bose, Sownak, Crain, Robert A, Vecchia, Claudio dalla, Frenk, Carlos S, Hellwing, Wojciech, Kay, Scott T, Ludlow, Aaron D & Bower, Richard G (2019). The signal of decaying dark matter with hydrodynamical simulations. Monthly Notices of the Royal Astronomical Society 485(3): 4071-4089.
Bahé, Yannick M, Schaye, Joop, Barnes, David J, Dalla Vecchia, Claudio, Kay, Scott T, Bower, Richard G, Hoekstra, Henk, McGee, Sean L & Theuns, Tom (2019). Disruption of satellite galaxies in simulated groups and clusters: the roles of accretion time, baryons, and pre-processing. Monthly Notices of the Royal Astronomical Society 485(2): 2287-2311.
Thob, Adrien C.R., Crain, Robert A., McCarthy, Ian G., Schaller, Matthieu, Lagos, Claudia D.P., Schaye, Joop, Talens, Geert Jan J., James, Philip A., Theuns, Tom & Bower, Richard G. (2019). The relationship between the morphology and kinematics of galaxies and its dependence on dark matter halo structure in EAGLE. Monthly Notices of the Royal Astronomical Society 485(1): 972–987.
Rosas-Guevara, Yetli M, Bower, Richard G, McAlpine, Stuart, Bonoli, Silvia & Tissera, Patricia B (2019). The abundances and properties of Dual AGN and their host galaxies in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 483(2): 2712-2720.
Kulier, A. Padilla, N. Schaye, J. Crain, R. A. Schaller, M. Bower, R. G. Theuns, T. & Paillas, E. (2019). The evolution of the baryon fraction in halos as a cause of scatter in the galaxy stellar mass in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society 482(3): 3261-3273.
Tissera, P. B. Rosas-Guevara, Y. Bower, R. G. Crain, R. A. del P Lagos, C. Schaller, M. Schaye, J. & Theuns, T. (2019). The oxygen abundance gradients in the gas discs of galaxies in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society 482(2): 2208-2221.
Tiley, A. L. Bureau, M. Cortese, L. Harrison, C. M. Johnson, H. L. Stott, J. P. Swinbank, A. M. Smail, I. Sobral, D. Bunker, A. J. Glazebrook, K. Bower, R. G. Obreschkow D. Bryant, J. J. Jarvis, M. J. Bland-Hawthorn, J. Magdis, G. Medling, A. M. Sweet, S. M. Tonini, C. Turner, O. J. Sharples, R. M. Croom, S. M. Goodwin, M., Konstantopoulos, I. S., Lorente, N. P. F. Lawrence, J. S. Mould, J., Owers, M. S. & Richards, S. N. (2019). KROSS-SAMI: A Direct IFS Comparison of the Tully-Fisher Relation Across 8 Gyr Since z ≈ 1. Monthly Notices of the Royal Astronomical Society 482(2): 2166–2188.
McAlpine, Stuart, Bower, Richard G., Rosario, David J., Crain, Robert A., Schaye, Joop & Theuns, Tom (2018). The rapid growth phase of supermassive black holes. Monthly Notices of the Royal Astronomical Society 481(3): 3118-3128.
Clauwens B. Schaye J. Franx M. & Bower R. G. (2018). The three phases of galaxy formation. Monthly Notices of the Royal Astronomical Society 478(3): 3994-4009.
Salcido, J., Bower, R. G., Barnes, L. A., Lewis, G. F., Elahi, P. J., Theuns, T., Schaller, M., Crain, R. A. & Schaye, J. (2018). The impact of dark energy on galaxy formation. What does the future of our Universe hold? Monthly Notices of the Royal Astronomical Society 477(3): 3744-3759.
Barnes, L. A., Elahi, P. J., Salcido, J., Bower, R. G., Lewis, G. F., Theuns, T., Schaller, M., Crain, R. A. & Schaye, J. (2018). Galaxy Formation Efficiency and the Multiverse Explanation of the Cosmological Constant with EAGLE Simulations. Monthly Notices of the Royal Astronomical Society 477(3): 3727-3743.
Knebe, A., Pearce, F. R., Gonzalez-Perez, V., Thomas, P. A., Benson, A., Asquith, R., Blaizot, J., Bower, R., Carretero, J., Castander, F. J., Cattaneo, A., Cora, S. A., Croton, D. J., Cui, W., Cunnama, D., Devriendt, J. E., Elahi, P. J., Font, A., Fontanot, F., Gargiulo, I. D., Helly, J., Henriques, B., Lee, J., Mamon, G. A., Onions, J., Padilla, N. D., Power, C., Pujol, A., Ruiz, A. N., Srisawat, C., Stevens, A. R. H., Tollet, E., Vega-Martínez, C. A. & Yi, S. K. (2018). Cosmic CARNage I: on the calibration of galaxy formation models. Monthly Notices of the Royal Astronomical Society 475(3): 2936-2954.
Scholtz, J., Alexander, D. M., Harrison, C. M., Rosario, D. J., McAlpine, S., Mullaney, J. R., Stanley, F., Simpson, J., Theuns, T., Bower, R. G., Hickox, R C, Santini, P & Swinbank, A. M. (2018). Identifying the subtle signatures of feedback from distant AGN using ALMA observations and the EAGLE hydrodynamical simulations. Monthly Notices of the Royal Astronomical Society 475(1): 1288-1305.
Johnson, H. L., Harrison, C. M., Swinbank, A. M., Tiley, A. L., Stott, J. P., Bower, R. G., Smail, I., Bunker, A. J., Sobral, D., Turner, O. J., Best, P., Bureau, M., Cirasuolo, M., Jarvis, M. J., Magdis, G., Sharples, R. M., Bland-Hawthorn, J., Catinella, B., Cortese, L., Croom, S. M., Federrath, C., Glazebrook, K., Sweet, S. M., Bryant, J. J., Goodwin, M., Konstantopoulos, I. S., Lawrence, J. S., Medling, A. M., Owers, M. S. & Richards, S. (2018). The KMOS Redshift One Spectroscopic Survey (KROSS): the origin of disc turbulence in z ≈ 1 star-forming galaxies. Monthly Notices of the Royal Astronomical Society 474(4): 5076-5104.
Mitchell, Peter D., Lacey, Cedric G., Lagos, Claudia D. P., Frenk, Carlos S., Bower, Richard G., Cole, Shaun, Helly, John C., Schaller, Matthieu, Gonzalez-Perez, Violeta & Theuns, Tom (2018). Comparing galaxy formation in semi-analytic models and hydrodynamical simulations. Monthly Notices of the Royal Astronomical Society 474(1): 492-521.
Lagos, C. d. P., Stevens, A. R. H., Bower, R. G., Davis, T. A., Contreras, S., Padilla, N. D., Obreschkow, D., Croton, D., Trayford, J. W., Welker, C. & Theuns, T. (2018). Quantifying the impact of mergers on the angular momentum of simulated galaxies. Monthly Notices of the Royal Astronomical Society 473(4): 4956-4974.
Correa, C. A., Schaye, J., Wyithe, J. S. B., Duffy, A. R., Theuns, T., Crain, R. A. & Bower, R. G. (2018). The formation of hot gaseous haloes around galaxies. Monthly Notices of the Royal Astronomical Society 473(1): 538-559.
Tescari, E., Cortese, L., Power, C., Wyithe, J. S. B., Ho, I.-T., Crain, R. A., Bland-Hawthorn, J., Croom, S. M., Kewley, L. J., Schaye, J., Bower, R. G., Theuns, T., Schaller, M., Barnes, L., Brough, S., Bryant, J. J., Goodwin, M., Gunawardhana, M. L. P., Lawrence, J. S., Leslie, S. K., López-Sánchez, Á. R., Lorente, N. P. F., Medling, A. M., Richards, S. N., Sweet, S. M. & Tonini, C. (2018). The SAMI Galaxy Survey: understanding observations of large-scale outflows at low redshift with EAGLE simulations. Monthly Notices of the Royal Astronomical Society 473(1): 380-397.
Correa, Camila A., Schaye, Joop, Clauwens, Bart, Bower, Richard G., Crain, Robert A., Schaller, Matthieu, Theuns, Tom & Thob, Adrien C. R. (2017). The relation between galaxy morphology and colour in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society: Letters 472(1): L45-L49.
De Rossi, María Emilia, Bower, Richard G., Font, Andreea S., Schaye, Joop & Theuns, Tom (2017). Galaxy metallicity scaling relations in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 472(3): 3354-3377.
Jian, Hung-Yu, Lin, Lihwai, Lin, Kai-Yang, Foucaud, Sebastien, Chen, Chin-Wei, Chiueh, Tzihong, Bower, R. G., Cole, Shaun, Chen, Wen-Ping, Burgett, W. S., Draper, P. W., Flewelling, H., Huber, M. E., Kaiser, N., Kudritzki, R.-P., Magnier, E. A., Metcalfe, N., Wainscoat, R. J. & Waters, C. (2017). The Pan-STARRS1 Medium-deep Survey: Star Formation Quenching in Group and Cluster Environments. The Astrophysical Journal 845(1): 74.
Katsianis, A., Blanc, G., Lagos, C. P., Tejos, N., Bower, R. G., Alavi, A., Gonzalez, V., Theuns, T., Schaller, M. & Lopez, S. (2017). The evolution of the star formation rate function in the EAGLE simulations: a comparison with UV, IR and Hα observations from z ∼ 8 to z ∼ 0. Monthly Notices of the Royal Astronomical Society 472(1): 919-939.
Bahé, Y. M., Barnes, D. J., Dalla Vecchia, C., Kay, S. T., White, S. D. M., McCarthy, I. G., Schaye, J., Bower, R. G., Crain, R. A., Theuns, T., Jenkins, A., McGee, S. L., Schaller, M., Thomas, P. A. & Trayford, J. W. (2017). The Hydrangea simulations: galaxy formation in and around massive clusters. Monthly Notices of the Royal Astronomical Society 470(4): 4186-4208.
Balogh, M. L., Gilbank, D. G., Muzzin, A., Rudnick, G., Cooper, M. C., Lidman, C., Biviano, A., Demarco, R., McGee, S. L., Nantais, J. B., Noble, A., Old, L., Wilson, G., Yee, H. K. C., Bellhouse, C., Cerulo, P., Chan, J., Pintos-Castro, I., Simpson, R., van der Burg, R. F. J., Zaritsky, D., Ziparo, F., Alonso, M. V., Bower, R. G., De Lucia, G., Finoguenov, A., Lambas, D. G., Muriel, H., Parker, L. C., Rettura, A., Valotto, C. & Wetzel, A. (2017). Gemini Observations of Galaxies in Rich Early Environments (GOGREEN) I: survey description. Monthly Notices of the Royal Astronomical Society 470(4): 4168-4185.
Artale, M. Celeste, Pedrosa, Susana E., Trayford, James W., Theuns, Tom, Farrow, Daniel J., Norberg, Peder, Zehavi, Idit, Bower, Richard G. & Schaller, Matthieu (2017). Small-scale galaxy clustering in the eagle simulation. Monthly Notices of the Royal Astronomical Society 470(2): 1771-1787.
Trayford, J. W., Camps, P., Theuns, T., Baes, M., Bower, R. G., Crain, R. A., Gunawardhana, M. L. P., Schaller, M., Schaye, J. & Frenk, C. S. (2017). Optical colours and spectral indices of z = 0.1 eagle galaxies with the 3D dust radiative transfer code skirt. Monthly Notices of the Royal Astronomical Society 470(1): 771-799.
Pearson, R. J., Ponman, T. J., Norberg, P., Robotham, A. S. G., Babul, A., Bower, R. G., McCarthy, I. G., Brough, S., Driver, S. P. & Pimbblet, K. (2017). Galaxy And Mass Assembly: search for a population of high-entropy galaxy groups. Monthly Notices of the Royal Astronomical Society 469(3): 3489-3504.
Algorry, D. G., Navarro, J. F., Abadi, M. G., Sales, L. V., Bower, R. G., Crain, R. A., Dalla Vecchia, C., Frenk, C. S., Schaller, M., Schaye, J. & Theuns, T. (2017). Barred galaxies in the EAGLE cosmological hydrodynamical simulation. Monthly Notices of the Royal Astronomical Society 469(1): 1054-1064.
Ludlow, A. D., Benítez-Llambay, A., Schaller, M., Theuns, T., Frenk, C. S., Bower, R., Schaye, J., Crain, R. A., Navarro, J. F., Fattahi, A. & Oman, K. A. (2017). Mass-Discrepancy Acceleration Relation: A Natural Outcome of Galaxy Formation in Cold Dark Matter Halos. Physical Review Letters 118(16): 161103.
Pujol, A., Skibba, R. A., Gaztañaga, E., Benson, A., Blaizot, J., Bower, R., Carretero, J., Castander, F. J., Cattaneo, A., Cora, S. A., Croton, D. J., Cui, W., Cunnama, D., De Lucia, G., Devriendt, J. E., Elahi, P. J., Font, A., Fontanot, F., Garcia-Bellido, J., Gargiulo, I. D., Gonzalez-Perez, V., Helly, J., Henriques, B. M. B., Hirschmann, M., Knebe, A., Lee, J., Mamon, G. A., Monaco, P., Onions, J., Padilla, N. D., Pearce, F. R., Power, C., Somerville, R. S., Srisawat, C., Thomas, P. A., Tollet, E., Vega-Martínez, C. A. & Yi, S. K. (2017). nIFTy cosmology: the clustering consistency of galaxy formation models. Monthly Notices of the Royal Astronomical Society 469(1): 749-762.
McAlpine, S., Bower, R. G., Harrison, C. M., Crain, R. A., Schaller, M., Schaye, J. & Theuns, T. (2017). The link between galaxy and black hole growth in the eagle simulation. Monthly Notices of the Royal Astronomical Society 468(3): 3395-3407.
Sharma, M., Theuns, T., Frenk, C., Bower, R. G., Crain, R. A., Schaller, M. & Schaye, J. (2017). Winds of change: reionization by starburst galaxies. Monthly Notices of the Royal Astronomical Society 468(2): 2176-2188.
Swinbank, M., Harrison, C., Trayford, J., Schaller, M., Smail, I., Schaye, J., Theuns, T., Smit, R., Alexander, D., Bacon, R., Bower, R., Contini, T., Crain, R., de Breuck, C., Decarli, R., Epinat, B., Fumagalli, M., Furlong, M., Galametz, A., Johnson, H.L., Lagos, C., Richard, J., Vernet, J., Sharples, R., Sobral, D. & Stott, J.P. (2017). Angular momentum evolution of galaxies over the past 10 Gyr: a MUSE and KMOS dynamical survey of 400 star-forming galaxies from z = 0.3 to 1.7. Monthly Notices of the Royal Astronomical Society 467(3): 3140-3159.
van de Voort, F., Bahé, Y. M., Bower, R. G., Correa, C. A., Crain, R. A., Schaye, J. & Theuns, T. (2017). The environmental dependence of gas accretion onto galaxies: quenching satellites through starvation. Monthly Notices of the Royal Astronomical Society 466(3): 3460-3471.
Ramraj, R., Gilbank, D. G., Blyth, S.-L., Skelton, R. E., Glazebrook, K., Bower, R. G. & Balogh, M. L. (2017). HST Hα grism spectroscopy of ROLES: a flatter low-mass slope for the z ˜ 1 SSFR-mass relation. Monthly Notices of the Royal Astronomical Society 466(3): 3143-3160.
Rodrigues, Luiz Felippe S., Vernon, Ian & Bower, Richard G. (2017). Constraints on galaxy formation models from the galaxy stellar mass function and its evolution. Monthly Notices of the Royal Astronomical Society 466(2): 2418-2435.
Norris, M. A., Van de Ven, G., Schinnerer, E., Crain, R. A., Meidt, S., Groves, B., Bower, R. G., Furlong, M., Schaller, M., Schaye, J. & Theuns, T. (2016). Being WISE II: Reducing the Influence of Star formation History on the Mass-to-Light Ratio of Quiescent Galaxies. Astrophysical Journal 832(2): 198.
Matthee, J., Schaye, J., Crain, R. A., Schaller, M., Bower, R. & Theuns, T. (2017). The origin of scatter in the stellar mass-halo mass relation of central galaxies in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society 465(2): 2381-2396.
Furlong, M., Bower, R. G., Crain, R. A., Schaye, J., Theuns, T., Trayford, J. W., Qu, Y., Schaller, M., Berthet, M. & Helly, J. C. (2017). Size evolution of normal and compact galaxies in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society 465(1): 722-738.
Bower, R. G., Schaye, J., Frenk, C. S., Theuns, T., Schaller, M., Crain, R. A. & McAlpine, S. (2017). The dark nemesis of galaxy formation: why hot haloes trigger black hole growth and bring star formation to an end. Monthly Notices of the Royal Astronomical Society 465(1): 32-44.
Ferrero, I., Navarro, J. F., Abadi, M. G., Sales, L. V., Bower, R. G., Crain, R. A., Frenk, C. S., Schaller, M., Schaye, J. & Theuns, T. (2017). Size matters: abundance matching, galaxy sizes, and the Tully-Fisher relation in EAGLE. Monthly Notices of the Royal Astronomical Society 464(4): 4736-4746.
Crain, R. A., Bahé, Y. M., Lagos, C. d. P., Rahmati, A., Schaye, J., McCarthy, I. G., Marasco, A., Bower, R. G., Schaller, M., Theuns, T. & van der Hulst, T. (2017). The EAGLE simulations: atomic hydrogen associated with galaxies. Monthly Notices of the Royal Astronomical Society 464(4): 4204-4226.
Bahé, Y. M., Schaye, J., Crain, R. A., McCarthy, I. G., Bower, R. G., Theuns, T., McGee, S. L. & Trayford, J. W. (2017). The origin of the enhanced metallicity of satellite galaxies. Monthly Notices of the Royal Astronomical Society 464(1): 508-529.
Sales, L. V., Navarro, J. F., Oman, K., Fattahi, A., Ferrero, I., Abadi, M., Bower, R., Crain, R. A., Frenk, C. S., Sawala, T., Schaller, M., Schaye, J., Theuns, T. & White, S. D. M. (2017). The low-mass end of the baryonic Tully–Fisher relation. Monthly Notices of the Royal Astronomical Society 464(2): 2419-2428.
Qu, Y., Helly, J. C., Bower, R. G., Theuns, T., Crain, R. A., Frenk, C. S., Furlong, M., McAlpine, S., Schaller, M., Schaye, J. & White, S. D. M. (2017). A chronicle of galaxy mass assembly in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society 464(2): 1659-1675.
Salcido J. Bower R. G. Theuns T. McAlpine S. Schaller M. Crain R. A. Schaye J. & Regan J. (2016). Music from the heavens - gravitational waves from supermassive black hole mergers in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 463(1): 870-885.
Rosas-Guevara, Y., Bower, R.G., Schaye, J., McAlpine, S., Dalla Vecchia, C., Frenk, C.S., Schaller, M. & Theuns, T. (2016). Supermassive black holes in the EAGLE Universe. Revealing the observables of their growth. Monthly Notices of the Royal Astronomical Society 462(1): 190-205.
Marasco A. Crain R. A. Schaye J. Bahé Y. M. van der Hulst T. Theuns T. & Bower R. G. (2016). The environmental dependence of H I in galaxies in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 461(3): 2630-2649.
Guo, Q., Gonzalez-Perez, V., Guo, Q., Schaller, M., Furlong, M., Bower, R. G., Cole, S., Crain, R. A., Frenk, C. S., Helly, J. C., Lacey, C. G., Lagos, C. d. P., Mitchell, P., Schaye, J. & Theuns, T. (2016). Galaxies in the EAGLE hydrodynamical simulation and in the Durham and Munich semi-analytical models. Monthly Notices of the Royal Astronomical Society 461(4): 3457-3482.
Zavala, J., Frenk, C. S., Bower, R., Schaye, J., Theuns, T., Crain, R. A., Trayford, J. W., Schaller, M. & Furlong, M. (2016). The link between the assembly of the inner dark matter halo and the angular momentum evolution of galaxies in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society 460(4): 4466-4482.
Segers M. C. Schaye J. Bower R. G. Crain R. A. Schaller M. & Theuns T. (2016). The origin of the α-enhancement of massive galaxies. Monthly Notices of the Royal Astronomical Society Letters 461(1): L102-L106.
Oppenheimer, B. D., Crain, R. A., Schaye, J., Rahmati, A., Richings, A. J., Trayford, J. W., Tumlinson, J., Bower, R. G., Schaller, M. & Theuns, T. (2016). Bimodality of low-redshift circumgalactic O VI in non-equilibrium EAGLE zoom simulations. Monthly Notices of the Royal Astronomical Society 460(2): 2157-2179.
Johnson, H. L., Harrison, C. M., Swinbank, A. M., Bower, R. G., Smail, I., Koyama, Y. & Geach, J. E. (2016). The Spatially Resolved Dynamics of Dusty Starburst Galaxies in a z ~ 0.4 Cluster: Beginning the Transition from Spirals to S0s. Monthly Notices of the Royal Astronomical Society 460(1): 1059-1076.
Barber, C., Schaye, J., Bower, R. G., Crain, R. A., Schaller, M. & Theuns, T. (2016). The origin of compact galaxies with anomalously high black hole masses. Monthly Notices of the Royal Astronomical Society 460(1): 1147-1161.
Hellwing, W. A., Schaller, M., Frenk, C. S., Theuns, T., Schaye, J., Bower, R. G. & Crain, R. A. (2016). The effect of baryons on redshift space distortions and cosmic density and velocity fields in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society (Letters) 461(1): L11-L15.
Tiley, A. L., Stott, J. P., Swinbank, A. M., Bureau, M., Harrison, C. M., Bower, R., Johnson, H. L., Bunker, A. J., Jarvis, M. J., Magdis, G., Sharples, R., Smail, I., Sobral, D. & Best, P. (2016). The KMOS Redshift One Spectroscopic Survey (KROSS): the Tully–Fisher relation at z ∼ 1. Monthly Notices of the Royal Astronomical Society 460(1): 103-129.
Lagos, C. D. P., Theuns, T., Schaye, J., Furlong, M., Bower, R. G., Schaller, M., Crain, R. A., Trayford, J. W. & Matthee, J. (2016). The Fundamental Plane of star formation in galaxies revealed by the EAGLE hydrodynamical simulations. Monthly Notices of the Royal Astronomical Society 459(3): 2632-2650.
McAlpine, S., Helly, J. C., Schaller, M., Trayford, J. W., Qu, Y., Furlong, M., Bower, R. G., Crain, R. A., Schaye, J., Theuns, T., Dalla Vecchia, C., Frenk, C. S., McCarthy, I. G., Jenkins, A., Rosas-Guevara, Y., White, S. D. M., Baes, M., Camps, P. & Lemson, G. (2016). The EAGLE simulations of galaxy formation: Public release of halo and galaxy catalogues. Astronomy and Computing 15: 72-89.
Sharma, M., Theuns, T., Frenk, C., Bower, R., Crain, R., Schaller, M. & Schaye, J. (2016). The brighter galaxies reionised the Universe. Monthly Notices of the Royal Astronomical Society Letters 458(1): L94-L98.
Sawala, Till, Frenk, Carlos S., Fattahi, Azadeh, Navarro, Julio F., Bower, Richard G., Crain, Robert A., Dalla Vecchia, Claudio, Furlong, Michelle, Helly, John. C., Jenkins, Adrian, Oman, Kyle A., Schaller, Matthieu, Schaye, Joop, Theuns, Tom, Trayford, James & White, Simon D. M. (2016). The APOSTLE simulations: solutions to the Local Group's cosmic puzzles. Monthly Notices of the Royal Astronomical Society 457(2): 1931-1943.
Stott, J.P., Swinbank, A.M., Johnson, H.L., Tiley, A., Magdis, G., Bower, R., Bunker, A.J., Bureau, M., Harrison, C.M., Jarvis, M.J., Sharples, R., Smail, I., Sobral, D., Best, P. & Cirasuolo, M. (2016). The KMOS Redshift One Spectroscopic Survey (KROSS): dynamical properties, gas and dark matter fractions of typical z~1 star-forming galaxies. Monthly Notices of the Royal Astronomical Society 457(2): 1888-1904.
Magdis, G.E., Bureau, M., Stott, J.P., Tiley, A., Swinbank, A.M., Bower, R., Bunker, A.J., Jarvis, M., Johnson, H. & Sharples, R. (2016). KROSS: mapping the Hα emission across the star formation sequence at z ≈ 1. Monthly Notices of the Royal Astronomical Society 456(4): 4533-4541.
Balogh, M. L., McGee, S. L., Mok, A., Muzzin, A., van der Burg, R. F. J., Bower, R. G., Finoguenov, A., Hoekstra, H., Lidman, C., Mulchaey, J. S., Noble, A., Parker, L. C., Tanaka, M., Wilman, D. J., Webb, T., Wilson, G. & Yee, H. K. C. (2016). Evidence for a change in the dominant satellite galaxy quenching mechanism at z = 1. Monthly Notices of the Royal Astronomical Society 456(4): 4364-4376.
Segers, M. C., Crain, R. A., Schaye, J., Bower, R. G., Furlong, M., Schaller, M. & Theuns, T. (2016). Recycled stellar ejecta as fuel for star formation and implications for the origin of the galaxy mass-metallicity relation. Monthly Notices of the Royal Astronomical Society 456(2): 1235-1258.
Harrison, C. M., Alexander, D. M., Mullaney, J. R., Stott, J. P., Swinbank, A. M., Arumugam, V., Bauer, F. E., Bower, R. G., Bunker, A. J. & Sharples, R. M. (2016). The KMOS AGN Survey at High redshift (KASHz): the prevalence and drivers of ionised outflows in the host galaxies of X-ray AGN. Monthly Notices of the Royal Astronomical Society 456(2): 1195-1220.
Bahé, Y. M., Crain, R. A., Kauffmann, G., Bower, R. G., Schaye, J., Furlong, M., Lagos, C., Schaller, M., Trayford, J. W., Dalla Vecchia, C. & Theuns, T. (2016). The distribution of atomic hydrogen in EAGLE galaxies: morphologies, profiles, and H I holes. Monthly Notices of the Royal Astronomical Society 456(1): 1115-1136.
Sawala, Till, Frenk, Carlos S., Fattahi, Azadeh, Navarro, Julio F., Theuns, Tom, Bower, Richard G., Crain, Robert A., Furlong, Michelle, Jenkins, Adrian, Schaller, Matthieu & Schaye, Joop (2016). The chosen few: the low-mass haloes that host faint galaxies. Monthly Notices of the Royal Astronomical Society 456(1): 85-97.
Rosas-Guevara, Y. M., Bower, R. G., Schaye, J., Furlong, M., Frenk, C. S., Booth, C. M., Crain, R. A., Dalla Vecchia, C., Schaller, M. & Theuns, T. (2015). The impact of angular momentum on black hole accretion rates in simulations of galaxy formation. Monthly Notices of the Royal Astronomical Society 454(1): 1038-1057.
Trayford, J. W., Theuns, T., Bower, R. G., Schaye, J., Furlong, M., Schaller, M., Frenk, C. S., Crain, R. A., Dalla Vecchia, C. & McCarthy, I. G. (2015). Colours and luminosities of z = 0.1 galaxies in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society 452(3): 2879-2896.
Eke, V. R., Bower, K. E., Diserens, S., Ryder, M., Yeomans, P. E. L., Teodoro, L. F. A., Elphic, R. C., Feldman, W. C., Hermalyn, B., Lavelle, C. M. & Lawrence, D. J. (2015). The effect of craters on the lunar neutron flux. Journal of Geophysical Research: Planets 120(8): 1377-1395.
Knebe, A., Pearce, F. R., Thomas, P. A., Benson, A., Blaizot, J., Bower, R., Carretero, J., Castander, F. J., Cattaneo, A., Cora, S. A., Croton, D. J., Cui, W., Cunnama, D., De Lucia, G., Devriendt, J. E., Elahi, P. J., Font, A., Fontanot, F., Garcia-Bellido, J., Gargiulo, I. D., Gonzalez-Perez, V., Helly, J., Henriques, B., Hirschmann, M., Lee, J., Mamon, G. A., Monaco, P., Onions, J., Padilla, N. D., Power, C., Pujol, A., Skibba, R. A., Somerville, R. S., Srisawat, C., Vega-Martínez, C. A. & Yi, S. K. (2015). nIFTy cosmology: comparison of galaxy formation models. Monthly Notices of the Royal Astronomical Society 451(4): 4029-4059.
Schaller, Matthieu, Frenk, Carlos S., Bower, Richard G., Theuns, Tom, Jenkins, Adrian, Schaye, Joop, Crain, Robert A., Furlong, Michelle, Dalla Vecchia, Claudio & McCarthy, Ian G. (2015). Baryon effects on the internal structure of ΛCDM haloes in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 451(2): 1247-1267.
Crain, R. A., Schaye, J., Bower, R. G., Furlong, M., Schaller, M., Theuns, T., Dalla Vecchia, C., Frenk, C. S., McCarthy, I. G., Helly, J. C., Jenkins, A., Rosas-Guevara, Y. M., White, S. D. M. & Trayford, J. W. (2015). The EAGLE simulations of galaxy formation: calibration of subgrid physics and model variations. Monthly Notices of the Royal Astronomical Society 450(2): 1937-1961.
Sawala, T., Frenk, C.S., Fattahi, A., Navarro, J.F., Bower, R.G., Crain, R.A., Dalla Vecchia, C., Furlong, M., Jenkins, A., McCarthy, I.G., Qu, Y., Schaller, M., Schaye, J. & Theuns, T. (2015). Bent by baryons: the low-mass galaxy-halo relation. Monthly Notices of the Royal Astronomical Society 448(3): 2941-2947.
Lagos, C. d. P., Crain, R. A., Schaye, J., Furlong, M., Frenk, C. S., Bower, R. G., Schaller, M., Theuns, T., Trayford, J. W., Bahé, Y. M. & Dalla Vecchia, C. (2015). Molecular hydrogen abundances of galaxies in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 452(4): 3815-3837.
Robertson, A., Massey, R., Eke, V. & Bower, R. (2015). Self-interacting dark matter scattering rates through cosmic time. Monthly Notices of the Royal Astronomical Society 453(3): 2267-2276.
Livermore, R.C., Jones, T.A., Richard, J., Bower, R.G., Swinbank, A.M., Yuan, T.-T., Edge, A.C., Ellis, R.S., Kewley, L.J., Smail, I., Coppin, K.E.K. & Ebeling, H. (2015). Resolved spectroscopy of gravitationally lensed galaxies: global dynamics and star-forming clumps on ∼100 pc scales at 1 < z < 4. Monthly Notices of the Royal Astronomical Society 450(2): 1812-1835.
Rahmati, A., Schaye, J., Bower, R. G., Crain, R. A., Furlong, M., Schaller, M. & Theuns, T. (2015). The distribution of neutral hydrogen around high-redshift galaxies and quasars in the EAGLE simulation. Monthly Notices of the Royal Astronomical Society 452(2): 2034-2056.
Creasey, P., Theuns, T. & Bower, R.G. (2015). The metallicity of galactic winds. Monthly Notices of the Royal Astronomical Society 446(2): 2125-2143.
Schaller, Matthieu, Robertson, Andrew, Massey, Richard, Bower, Richard G. & Eke, Vincent R. (2015). The offsets between galaxies and their dark matter in Λ cold dark matter. Monthly Notices of the Royal Astronomical Society: Letters 453(1): L58-L62.
Schaller, Matthieu, Frenk, Carlos S., Bower, Richard G., Theuns, Tom, Trayford, James W., Crain, Robert A., Furlong, Michelle, Schaye, Joop, Dalla Vecchia, Claudio & McCarthy, Ian G. (2015). The effect of baryons on the inner density profiles of rich clusters. Monthly Notices of the Royal Astronomical Society 452(1): 343-355.
Furlong, M., Bower, R. G., Theuns, T., Schaye, J., Crain, R. A., Schaller, M., Dalla Vecchia, C., Frenk, C. S., McCarthy, I. G., Helly, J., Jenkins, A. & Rosas-Guevara, Y. M. (2015). Evolution of galaxy stellar masses and star formation rates in the EAGLE simulations. Monthly Notices of the Royal Astronomical Society 450(4): 4486-4504.
Oman, K. A., Navarro, J. F., Fattahi, A., Frenk, C. S., Sawala, T., White, S. D. M., Bower, R., Crain, R. A., Furlong, M., Schaller, M., Schaye, J. & Theuns, T. (2015). The unexpected diversity of dwarf galaxy rotation curves. Monthly Notices of the Royal Astronomical Society 452(4): 3650-3665.
Schaller, Matthieu, Dalla Vecchia, Claudio, Schaye, Joop, Bower, Richard G., Theuns, Tom, Crain, Robert A., Furlong, Michelle & McCarthy, Ian G. (2015). The EAGLE simulations of galaxy formation: the importance of the hydrodynamics scheme. Monthly Notices of the Royal Astronomical Society 454(3): 2277-2291.
Velliscig, M., Cacciato, M., Schaye, J., Hoekstra, H., Bower, R. G., Crain, R. A., van Daalen, M. P., Furlong, M., McCarthy, I. G., Schaller, M. & Theuns, T. (2015). Intrinsic alignments of galaxies in the EAGLE and cosmo-OWLS simulations. Monthly Notices of the Royal Astronomical Society 454(3): 3328-3340.
Velliscig, M., Cacciato, M., Schaye, J., Crain, R. A., Bower, R. G., van Daalen, M. P., Dalla Vecchia, C., Frenk, C. S., Furlong, M., McCarthy, I. G., Schaller, M. & Theuns, T. (2015). The alignment and shape of dark matter, stellar, and hot gas distributions in the EAGLE and cosmo-OWLS simulations. Monthly Notices of the Royal Astronomical Society 453(1): 721-738.
Schaye, Joop, Crain, Robert A., Bower, Richard G., Furlong, Michelle, Schaller, Matthieu, Theuns, Tom, Dalla Vecchia, Claudio, Frenk, Carlos S., McCarthy, I. G., Helly, John C., Jenkins, Adrian, Rosas-Guevara, Y. M., White, Simon D. M., Baes, Maarten, Booth, C. M., Camps, Peter, Navarro, Julio F., Qu, Yan, Rahmati, Alireza, Sawala, Till, Thomas, Peter A. & Trayford, James (2015). The EAGLE project: simulating the evolution and assembly of galaxies and their environments. Monthly Notices of the Royal Astronomical Society 446(1): 521-554.
Vernon, Ian, Goldstein, Michael & Bower, Richard (2014). Galaxy Formation: Bayesian History Matching for the Observable Universe. Statistical Science 29(1): 81-90.
McGee, S.L., Bower, R.G. & Balogh, M.L. (2014). Overconsumption, outflows and the quenching of satellite galaxies. Monthly Notices of Royal Astronomical Society Letters 442(1): L105-L109.
Mok, A., Balogh, M.L., McGee, S.L., Wilman, D.J., Finoguenov, A., Tanaka, M., Bower, R.G., Hou, A., Mulchaey, J.S. & Parker, L.C. (2014). Star formation and environmental quenching of GEEC2 group galaxies at z ∼ 1. Monthly Notices of Royal Astronomical Society 438(4): 3070-3085.
Bower, R.G., McLeish, T.C.B., Tanner, B.K., Smithson, H.E., Panti, C., Lewis, N. & Gasper, G.E.M. (2014). A medieval multiverse?: Mathematical modelling of the thirteenth century universe of Robert Grosseteste. Proceedings of the Royal Society A 470(2167): 40025.
Lin, L., Jian, H.-Y., Foucaud, S., Norberg, P., Bower, R.G., Cole, S., Arnalte-Mur, P., Chen, C.-W., Coupon, J., Hsieh, B.-C., Heinis, S., Phleps, S., Chen, W.-P., Lee, C.-H., Burgett, W., Chambers, K.C., Denneau, L., Draper, P., Flewelling, H., Hodapp, K.W., Huber, M.E., Kaiser, N., Kudritzki, R.-P., Magnier, E.A., Metcalfe, N., Price, P.A., Tonry, J.L., Wainscoat, R.J. & Waters, C. (2014). The Pan-STARRS1 Medium-Deep Survey: The Role of Galaxy Group Environment in the Star Formation Rate versus Stellar Mass Relation and Quiescent Fraction out to z ~ 0.8. The Astrophysical Journal 782(1): 33.
Stott, J.P., Sobral, D., Smail, I., Bower, R., Best, P.N. & Geach, J.E. (2013). The merger rates and sizes of galaxies across the peak epoch of star formation from the HiZELS survey. Monthly notices of the Royal Astronomical Society 430(2): 1158-1170.
Hou, A., Parker, L.C., Balogh, M.L., McGee, S.L., Wilman, D.J., Connelly, J.L., Harris, W.E., Mok, A., Mulchaey, J.S., Bower, R.G. & Finoguenov, A. (2013). Do group dynamics play a role in the evolution of member galaxies? Monthly Notices of Royal Astronomical Society 435(2): 1715-1726.
Stott, J.P., Sobral, D., Bower, R., Smail, I., Best, P.N., Matsuda, Y., Hayashi, M., Geach, J.E. & Kodama, T. (2013). A fundamental metallicity relation for galaxies at z = 0.84-1.47 from HiZELS. Monthly Notices of the Royal Astronomical Society 436(2): 1130-1141.
Mok, A., Balogh, M.L., McGee, S.L., Wilman, D.J., Finoguenov, A., Tanaka, M., Giodini, S., Bower, R.G., Connelly, J.L., Hou, A., Mulchaey, J.S. & Parker, L.C. (2013). Efficient satellite quenching at z∼1 from the GEEC2 spectroscopic survey of galaxy groups. Monthly Notices of the Royal Astronomical Society 431(2): 1090-1106.
Sobral, D., Swinbank, A.M., Stott, J.P., Matthee, J., Bower, R.G., Smail, I., Best, P., Geach, J.E. & Sharples, R.M. (2013). The dynamics of z = 0.8 Hα-selected star-forming galaxies from KMOS/CF-HiZELS. The Astrophysical Journal 779(2): 139.
Creasey, P., Theuns, T. & Bower, R.G. (2013). How supernova explosions power galactic winds. Monthly Notices of the Royal Astronomical Society 429(3): 1922-1948.
Merson, A.I., Baugh, C.M., Helly, J.C., Gonzalez-Perez, V., Cole, S., Bielby, R., Norberg, P., Frenk, C.S., Benson, A.J., Bower, R.G., Lacey, C.G. & Lagos, C. del P. (2013). Lightcone mock catalogues from semi-analytic models of galaxy formation - I. Construction and application to the BzK colour selection. Monthly Notices of the Royal Astronomical Society 429(1): 556-578.
Greene, C.R., Gilbank, D.G., Balogh, M.L., Glazebrook, K., Bower, R.G., Baldry, I.K., Hau, G.K.T., Li, I.H. & McCarthy, P. (2012). The slowly evolving role of environment in a spectroscopic survey of star formation in M∗ > 5 × 108 M⊙ galaxies since z ∼ 1. Monthly Notices of the Royal Astronomical Society 425(3): 1738-1752.
Hou, A., Parker, L.C., Wilman, D.J., McGee, S.L., Harris, W.E., Connelly, J.L., Balogh, M.L., Mulchaey, J.S. & Bower, R.G. (2012). Substructure in the most massive GEEC groups field-like populations in dynamically active groups. Monthly Notices of the Royal Astronomical Society 421(4): 3594-3611.
Connelly, J.L., Wilman, D.J., Finoguenov, A., Hou, A., Mulchaey, J.S., McGee, S.L., Balogh, M.L., Parker, L.C., Saglia, R., Henderson, R.D.E. & Bower, R.G. (2012). Exploring the Diversity of Groups at 0.1 \lt z \lt 0.8 with X-Ray and Optically Selected Samples. \apj 756: 139.
Murphy, D.N.A., Geach, J.E. & Bower, R.G. (2012). ORCA the overdense red-sequence cluster algorithm. Monthly Notices of the Royal Astronomical Society 420(3): 1861-1881.
Matsuda, Y., Smail, I., Geach, J.E., Best, P.N., Sobral, D., Tanaka, I., Nakata, F., Ohta, K., Kurk, J., Iwata, I., Bielby, R., Wardlow, J.L., Bower, R.G., Ivison, R.J., Kodama, T., Yamada, T., Mawatari, K. & Casali, M. (2012). H$\alpha$ emitters at z=2.23 (Matsuda+, 2011). VizieR Online Data Catalog 741: 62041.
Fanidakis, N., Baugh, C.M., Benson, A.J., Bower, R.G., Cole, S., Done, C., Frenk, C.S., Hickox, R.C., Lacey, C. & del P. Lagos, C. (2012). The evolution of active galactic nuclei across cosmic time what is downsizing? Monthly Notices of the Royal Astronomical Society 419(4): 2797-2820.
Stringer, M.J., Bower, R.G., Cole, S., Frenk, C.S. & Theuns, T. (2012). Principles of supernova-driven winds. Monthly Notices of the Royal Astronomical Society 423(2): 1596-1609.
Swinbank, A.M., Balogh, M.L., Bower, R.G., Zabludoff, A.I., Lucey, J.R., McGee, S.L., Miller, C.J. & Nichol, R.C. (2012). From star-forming spirals to passive spheroids: integral field spectroscopy of E+A galaxies. Monthly Notices of the Royal Astronomical Society 420(1): 672-683.
Bower, R.G., Benson, A.J. & Crain, R.A. (2012). What shapes the galaxy mass function ? exploring the roles of supernova-driven winds and active galactic nuclei. Monthly Notices of the Royal Astronomical Society 422(4): 2816-2840.
Livermore, R.C., Jones, T., Richard, J., Bower, R.G., Ellis, R.S., Swinbank, A.M., Rigby, J.R., Smail, I., Arribas, S., Rodriguez Zaurin, J., Colina, L., Ebeling, H. & Crain, R.A. (2012). Hubble Space Telescope Hα imaging of star-forming galaxies at z ≃ 1–1.5 evolution in the size and luminosity of giant H ii regions. Monthly notices of the Royal Astronomical Society 427(1): 688-702.
Livermore, R.C., Swinbank, A.M., Smail, I., Bower, R.G., Coppin, K.E.K., Crain, R.A., Edge, A.C., Geach, J.E. & Richard, J. (2012). Observational Limits on the Gas Mass of a z = 4.9 Galaxy. \apjl 758: L35.
Gilbank, D.G., Baldry, I.K., Balogh, M.L., Glazebrook, K. & Bower, R.G. (2011). Erratum: The local star formation rate density: assessing calibrations using [O II], H$\alpha$ and UV luminosities. Monthly Notices of the Royal Astronomical Society 412: 2111-2111.
Benson, A.J. & Bower, R. (2011). Accretion shocks and cold filaments in galaxy formation. Monthly Notices of the Royal Astronomical Society 410: 2653-2661.
Tyler, K.D., Rieke, G.H., Wilman, D.J., McGee, S.L., Bower, R.G., Bai, L., Mulchaey, J.S., Parker, L.C., Shi, Y. & Pierini, D. (2011). The Nature of Star Formation at 24 $\mu$m in the Group Environment at 0.3 \lt\~ z \lt\~ 0.55. \apj 738: 56.
Lagos, C.D.P. Lacey, C.G. Baugh, C.M. Bower, R.G. & Benson, A.J. (2011). On the impact of empirical and theoretical star formation laws on galaxy formation. Monthly Notices of the Royal Astronomical Society 416(2): 1566-1584.
Bower, R. (2011). Astronomy: Unlocking the secrets of the giant blobs. \nat 476: 288-289.
Fanidakis, N., Baugh, C.M., Benson, A.J., Bower, R.G., Cole, S., Done, C. & Frenk, C.S. (2011). Grand unification of AGN activity in the $\Lambda$CDM cosmology. Monthly Notices of the Royal Astronomical Society 410(1): 53-74.
Matsuda, Y., Smail, I., Geach, J.E., Best, P.N., Sobral, D., Tanaka, I., Nakata, F., Ohta, K., Kurk, J., Iwata, I., Bielby, R., Wardlow, J.L., Bower, R.G., Ivison, R.J., Kodama, T., Yamada, T., Mawatari, K. & Casali, M. (2011). An H$\alpha$ search for overdense regions at z = 2.23. Monthly Notices of the Royal Astronomical Society 416: 2041-2059.
Li, I.H., Glazebrook, K., Gilbank, D., Balogh, M., Bower, R., Baldry, I., Davies, G., Hau, G. & McCarthy, P. (2011). Dependence of star formation activity on stellar mass and environment from the Redshift One LDSS-3 Emission line Survey. Monthly Notices of the Royal Astronomical Society 411: 1869-1879.
Balogh, M.L., Mazzotta, P., Bower, R.G., Eke, V., Bourdin, H., Lu, T. & Theuns, T. (2011). The stellar and hot gas content of low-mass galaxy clusters. Monthly Notices of the Royal Astronomical Society 412: 947-959.
McCarthy, I.G., Schaye, J., Bower, R.G., Ponman, T.J., Booth, C.M., Dalla Vecchia, C. & Springel, V. (2011). Gas expulsion by quasar-driven winds as a solution to the overcooling problem in galaxy groups and clusters. Monthly Notices of the Royal Astronomical Society 412(3): 1965-1984.
McGee, S.L., Balogh, M.L., Wilman, D.J., Bower, R.G., Mulchaey, J.S., Parker, L.C. & Oemler, A. (2011). The Dawn of the Red: star formation histories of group galaxies over the past 5 billion years. Monthly Notices of the Royal Astronomical Society 413: 996-1012.
Balogh, M.L., McGee, S.L., Wilman, D.J., Finoguenov, A., Parker, L.C., Connelly, J.L., Mulchaey, J.S., Bower, R.G., Tanaka, M. & Giodini, S. (2011). Direct observational evidence for a large transient galaxy population in groups at 0.85 \lt z \lt 1. Monthly Notices of the Royal Astronomical Society 412: 2303-2317.
Creasey, P., Theuns, T., Bower, R.G. & Lacey, C.G. (2011). Numerical overcooling in shocks. Monthly Notices of the Royal Astronomical Society 415(4): 3706-3720.
Font, A.S., Benson, A.J., Bower, R.G., Frenk, C.S., Cooper, A., De Lucia, G., Helly, J.C., Helmi, A., Li, Y.-S., McCarthy, I.G., Navarro, J.F., Springel, V., Starkenburg, E., Wang, J. & White, S.D.M. (2011). The population of Milky Way satellites in the $\Lambda$ cold dark matter cosmology. Monthly Notices of the Royal Astronomical Society 417: 1260-1279.
Geach, J.E., Murphy, D.N.A. & Bower, R.G. (2011). 4098 galaxy clusters to z\tilde 0.6 in the Sloan Digital Sky Survey equatorial Stripe 82. Monthly Notices of the Royal Astronomical Society 413: 3059-3067.
Gilbank, D.G., Bower, R.G., Glazebrook, K., Balogh, M.L., Baldry, I.K., Davies, G.T., Hau, G.K.T., Li, I.H., McCarthy, P. & Sawicki, M. (2011). A spectroscopic measurement of galaxy formation time-scales with the Redshift One LDSS3 Emission line Survey. Monthly Notices of the Royal Astronomical Society 414: 304-320.
Lu, T., Gilbank, D.G., Balogh, M.L., Milkeraitis, M., Hoekstra, H., van Waerbeke, L., Wake, D.A., Edge, A.C. & Bower, R.G. (2010). Large-scale structure and dynamics of the most X-ray luminous galaxy cluster known - RX J1347-1145. Monthly Notices of the Royal Astronomical Society 403(4): 1787-1800.
Bower, R. G., Vernon, I., Goldstein, M., Benson, A. J., Lacey, C. G., Baugh, C. M., Cole, S. & Frenk, C. S. (2010). The Parameter Space of Galaxy Formation. Monthly Notices of the Royal Astronomical Society 407(4): 2017-2045.
Dariush, A.A., Raychaudhury, S., Ponman, T.J., Khosroshahi, H.G., Benson, A.J., Bower, R.G. & Pearce, F. (2010). The mass assembly of galaxy groups and the evolution of the magnitude gap. \mnras 405: 1873-1887.
Pratt, G.W., Arnaud, M., Piffaretti, R., Böhringer, H., Ponman, T.J., Croston, J.H., Voit, G.M., Borgani, S. & Bower, R.G. (2010). Gas entropy in a representative sample of nearby X-ray galaxy clusters (REXCESS): relationship to gas mass fraction. \aap 511: A85.
McCarthy, I.G., Schaye, J., Ponman, T.J., Bower, R.G., Booth, C.M., Dalla Vecchia, C., Crain, R.A., Springel, V., Theuns, T. & Wiersma, R.P.C. (2010). The case for AGN feedback in galaxy groups. Monthly Notices of the Royal Astronomical Society 406(2): 822-839.
Vernon, Ian., Goldstein, Michael. & Bower, Richard G. (2010). Galaxy Formation: a Bayesian Uncertainty Analysis. Bayesian Analysis 05(04): 619 - 670.
Weijmans, Anne-Marie, Bower, Richard G., Geach, James E., Swinbank, A. Mark, Wilman, R. J., de Zeeuw, P. T. & Morris, Simon L. (2010). Dissecting the Lyman α emission halo of LAB1. Monthly Notices of the Royal Astronomical Society 402(4): 2245-2252.
Pope, E.C.D., Babul, A., Pavlovski, G., Bower, R.G. & Dotter, A. (2010). Mass transport by buoyant bubbles in galaxy clusters. \mnras 406: 2023-2037.
Gilbank, D.G., Balogh, M.L., Glazebrook, K., Bower, R.G., Baldry, I.K., Davies, G.T., Hau, G.K.T., Li, I.H. & McCarthy, P. (2010). The Redshift One LDSS-3 Emission line Survey (ROLES): survey method and z \~ 1 mass-dependent star formation rate density. \mnras 405: 2419-2438.
Gilbank, D.G., Baldry, I.K., Balogh, M.L., Glazebrook, K. & Bower, R.G. (2010). The local star formation rate density: assessing calibrations using [OII], H and UV luminosities. \mnras 405: 2594-2614.
Benson, A.J. & Bower, R. (2010). Galaxy formation spanning cosmic history. \mnras 405: 1573-1623.
Wilman, D.J., Oemler, Jr., A., Mulchaey, J.S., McGee, S.L., Balogh, M.L. & Bower, R.G. (2009). Morphological Composition of z \~ 0.4 Groups: The Site of S0 Formation. \apj 692: 298-308.
McGee, S.L., Balogh, M.L., Bower, R.G., Font, A.S. & McCarthy, I.G. (2009). The accretion of galaxies into groups and clusters. \mnras 400: 937-950.
Cimatti, A., Robberto, M., Baugh, C., Beckwith, S.V.W., Content, R., Daddi, E., De Lucia, G., Garilli, B., Guzzo, L., Kauffmann, G., Lehnert, M., Maccagni, D., Martínez-Sansigre, A., Pasian, F., Reid, I.N., Rosati, P., Salvaterra, R., Stiavelli, M., Wang, Y., Zapatero Osorio, M., Balcells, M., Bersanelli, M., Bertoldi, F., Blaizot, J., Bottini, D., Bower, R., Bulgarelli, A., Burgasser, A., Burigana, C., Butler, R.C., Casertano, S., Ciardi, B., Cirasuolo, M., Clampin, M., Cole, S., Comastri, A., Cristiani, S., Cuby, J.-G., Cuttaia, F., de Rosa, A., Sanchez, A.D., di Capua, M., Dunlop, J., Fan, X., Ferrara, A., Finelli, F., Franceschini, A., Franx, M., Franzetti, P., Frenk, C., Gardner, J.P., Gianotti, F., Grange, R., Gruppioni, C., Gruppuso, A., Hammer, F., Hillenbrand, L., Jacobsen, A., Jarvis, M., Kennicutt, R., Kimble, R., Kriek, M., Kurk, J., Kneib, J.-P., Le Fevre, O., Macchetto, D., MacKenty, J., Madau, P., Magliocchetti, M., Maino, D., Mandolesi, N., Masetti, N., McLure, R., Mennella, A., Meyer, M., Mignoli, M., Mobasher, B., Molinari, E., Morgante, G., Morris, S., Nicastro, L., Oliva, E., Padovani, P., Palazzi, E., Paresce, F., Perez Garrido, A., Pian, E., Popa, L., Postman, M., Pozzetti, L., Rayner, J., Rebolo, R., Renzini, A., Röttgering, H., Schinnerer, E., Scodeggio, M., Saisse, M., Shanks, T., Shapley, A., Sharples, R., Shea, H., Silk, J., Smail, I., Spanó, P., Steinacker, J., Stringhetti, L., Szalay, A., Tresse, L., Trifoglio, M., Urry, M., Valenziano, L., Villa, F., Villo Perez, I., Walter, F., Ward, M., White, R., White, S., Wright, E., Wyse, R., Zamorani, G., Zacchei, A., Zeilinger, W.W. & Zerbi, F. (2009). SPACE: the spectroscopic all-sky cosmic explorer. Experimental Astronomy 23(1): 39-66.
Davies, G.T., Gilbank, D.G., Glazebrook, K., Bower, R.G., Baldry, I.K., Balogh, M.L., Hau, G.K.T., Li, I.H., McCarthy, P. & Savaglio, S. (2009). A spectroscopic measure of the star formation rate density in dwarf galaxies at z \~ 1. \mnras 395: L76-L80.
Rasmussen, J., Sommer-Larsen, J., Pedersen, K., Toft, S., Benson, A., Bower, R.G. & Grove, L.F. (2009). Hot Gas Halos Around Disk Galaxies: Confronting Cosmological Simulations with Observations. \apj 697: 79-93.
Gerssen, J., Wilman, D.J., Christensen, L., Bower, R.G. & Wild, V. (2009). Highly ionized gas on galaxy scales: mapping the interacting Seyfert galaxy LEDA 135736. \mnras 393: L45-L49.
Balogh, M.L., McGee, S.L., Wilman, D., Bower, R.G., Hau, G., Morris, S.L., Mulchaey, J.S., Oemler, A., Parker, L. & Gwyn, S. (2009). The colour of galaxies in distant groups. Monthly notices of the Royal Astronomical Society 398(2): 754-768.
Reed, DS, Bower, R, Frenk, CS, Jenkins, A & Theuns, T (2009). The clustering of the first galaxy haloes. Monthly Notices Of The Royal Astronomical Society 394(2): 624-632.
Geach, J.E., Alexander, D.M., Lehmer, B.D., Smail, I., Matsuda, Y., Chapman, S.C., Scharf, C.A., Ivison, R.J., Volonteri, M., Yamada, T., Blain, A.W., Bower, R.G., Bauer, F.E. & Basu-Zych, A. (2009). The Chandra Deep protocluster survey Lyα blobs are powered by heating, not cooling. The astrophysical journal 700(1): 1-9.
Swinbank, A.M., Webb, T.M., Richard, J., Bower, R.G., Ellis, R.S., Illingworth, G., Jones, T., Kriek, M., Smail, I., Stark, D.P. & van Dokkum, P. (2009). A spatially resolved map of the kinematics, star formation and stellar mass assembly in a star-forming galaxy at z = 4.9. \mnras 400: 1121-1131.
Finoguenov, A., Connelly, J.L., Parker, L.C., Wilman, D.J., Mulchaey, J.S., Saglia, R.P., Balogh, M.L., Bower, R.G. & McGee, S.L. (2009). The Roadmap for Unification in Galaxy Group Selection. I. A Search for Extended X-ray Emission in the CNOC2 Survey. \apj 704: 564-575.
Smail, I., Lehmer, B.D., Ivison, R.J., Alexander, D.M., Bower, R.G., Stevens, J.A., Geach, J.E., Scharf, C.A., Coppin, K.E.K. & van Breugel, W.J.M. (2009). A 100 kpc Inverse Compton X-Ray Halo Around 4C 60.07 at z = 3.79. \apjl 702: L114-L118.
Tanaka, M., Lidman, C., Bower, R.G., Demarco, R., Finoguenov, A., Kodama, T., Nakata, F. & Rosati, P. (2009). Star formation activities of galaxies in the large-scale structures at z = 1.2. \aap 507: 671-682.
Mitchell, N.L., McCarthy, I.G., Bower, R.G., Theuns, T. & Crain, R.A. (2009). On the origin of cores in simulated galaxy clusters. Monthly Notices of the Royal Astronomical Society 395(1): 180-196.
Balogh, M.L., McCarthy, I.G., Bower, R.G. & Eke, V.R. (2008). Testing cold dark matter with the hierarchical build-up of stellar light. \mnras 385: 1003-1014.
Font, A.S., Bower, R.G., McCarthy, I.G., Benson, A.J., Frenk, C.S., Helly, J.C., Lacey, C.G., Baugh, C.M. & Cole, S. (2008). The colours of satellite galaxies in groups and clusters. Monthly Notices of the Royal Astronomical Society 389(4): 1619-1629.
Croston, J.H., Pratt, G.W., Böhringer, H., Arnaud, M., Pointecouteau, E., Ponman, T.J., Sanderson, A.J.R., Temple, R.F., Bower, R.G. & Donahue, M. (2008). Galaxy-cluster gas-density distributions of the representative XMM-Newton cluster structure survey (REXCESS). \aap 487: 431-443.
Okamoto, T., Nemmen, R.S. & Bower, R.G. (2008). The impact of radio feedback from active galactic nuclei in cosmological simulations: formation of disc galaxies. Monthly Notices of the Royal Astronomical Society 385: 161-180.
McCarthy, I.G., Frenk, C.S., Font, A.S., Lacey, C.G., Bower, R.G., Mitchell, N.L., Balogh, M.L. & Theuns, T. (2008). Ram pressure stripping the hot gaseous haloes of galaxies in groups and clusters. Monthly Notices of the Royal Astronomical Society 383(2): 593-605.
McCarthy, I.G., Babul, A., Bower, R.G. & Balogh, M.L. (2008). Towards a holistic view of the heating and cooling of the intracluster medium. Monthly Notices of the Royal Astronomical Society 386(3): 1309-1331.
McGee, S.L., Balogh, M.L., Henderson, R.D.E., Wilman, D.J., Bower, R.G., Mulchaey, J.S., Oemler, Jr. & A. (2008). Evolution in the discs and bulges of group galaxies since z = 0.4. \mnras 387: 1605-1621.
Almeida, C., Baugh, C.M., Wake, D.A., Lacey, C.G., Benson, A.J., Bower, R.G. & Pimbblet, K. (2008). Luminous red galaxies in hierarchical cosmologies. Monthly notices of the Royal Astronomical Society 386(4): 2145-2160.
Hau, G.K.T., Bower, R.G., Kilborn, V., Forbes, D.A., Balogh, M.L. & Oosterloo, T. (2008). Is NGC 3108 transforming itself from an early- to late-type galaxy - an astronomical hermaphrodite? \mnras 385: 1965-1972.
Bower, R.G., McCarthy, I.G. & Benson, A.J. (2008). The flip side of galaxy formation: a combined model of galaxy formation and cluster heating. Monthly Notices of the Royal Astronomical Society 390(4): 1399-1410.
Wilman, D.J., Pierini, D., Tyler, K., McGee, S.L., Oemler, Jr., A., Morris, S.L., Balogh, M.L., Bower, R.G. & Mulchaey, J.S. (2008). Unveiling the Important Role of Groups in the Evolution of Massive Galaxies: Insights from an Infrared Passive Sequence at Intermediate Redshift. \apj 680: 1009-1021.
Swinbank, A.M., Bower, R.G., Smith, Graham P., Wilman, R.J., Smail, Ian, Ellis, R.S., Morris, S.L. & Kneib, J.-P. (2007). Resolved spectroscopy of a gravitationally lensed L* Lyman-break galaxy at z∼ 5. Monthly Notices of the Royal Astronomical Society 376(2): 479-491.
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Nemmen, RS, Bower, RG, Babul, A & Storchi-Bergmann, T (2007). Models for jet power in elliptical galaxies: a case for rapidly spinning black holes. Monthly Notices Of The Royal Astronomical Society 377(4): 1652-1662.
Bower, R (2007). Exploring superwinds through Ly-alpha emission from high redshift galaxies. New Astronomy Reviews 51(1-2): 190-193.
McCarthy, IG, Bower, RG, Balogh, ML, Voit, GM, Pearce, FR, Theuns, T, Babul, A, Lacey, CG & Frenk, CS (2007). Modelling shock heating in cluster mergers - I. Moving beyond the spherical accretion model. Monthly Notices Of The Royal Astronomical Society 376(2): 497-522.
Tanaka, M., Kodama, T., Kajisawa, M., Bower, R., Demarco, R., Finoguenov, A., Lidman, C. & Rosati, P. (2007). A deficit of faint red galaxies in the possible large-scale structures around the RDCS J1252.9−2927 cluster at z= 1.24. Monthly Notices of the Royal Astronomical Society 377(3): 1206-1214.
Bower, R. (2007). Exploring superwinds through Ly-$\alpha$ emission from high redshift galaxies. \nar 51: 190-193.
Balogh, ML, Wilman, D, Henderson, RDE, Bower, RG, Gilbank, D, Whitaker, R, Morris, SL, Hau, G, Mulchaey, JS, Oemler, A & Carlberg, RG (2007). The stellar mass content of distant galaxy groups. Monthly Notices Of The Royal Astronomical Society 374(3): 1169-1180.
McCarthy, IG, Bower, RG & Balogh, ML (2007). Revisiting the baryon fractions of galaxy clusters: a comparison with WMAP 3-yr results. Monthly Notices Of The Royal Astronomical Society 377(4): 1457-1463.
Reed, DS, Bower, R, Frenk, CS, Jenkins, A & Theuns, T (2007). The halo mass function from the dark ages through the present day. Monthly Notices Of The Royal Astronomical Society 374(1): 2-15.
Poggianti, BM, Von der Linden, A, De Lucia, G, Desai, V, Simard, L, Halliday, C, Aragon-Salamanca, A, Bower, R, Varela, J, Best, P, Clowe, DI, Dalcanton, J, Jablonka, P, Milvang-Jensen, B, Pello, R, Rudnick, G, Saglia, R, White, SDM & Zaritsky, D (2006). The evolution of the star formation activity in galaxies and its dependence on environment. Astrophysical Journal 642(1): 188-215.
Bunker, A, Smith, J, Parry, I, Ben Metcalf, R, Sharp, R, De Grijs, R, Bower, R & Swinbank, M (2006). Extragalactic integral field spectroscopy on the Gemini telescopes. New Astronomy Reviews 50(4-5): 422-425.
Trotta, R & Bower, R (2006). Surveying the dark side. Astronomy & Geophysics 47(4): 20-27.
Swinbank, A.M., Bower, R.G., Smith, G.P., Smail, I., Kneib, J.-P., Ellis, R.S., Stark, D.P. & Bunker, A.J. (2006). Galaxies under the cosmic microscope: resolved spectroscopy and new constraints on the z = 1 Tully-Fisher relation. Monthly Notices of Royal Astronomical Society 368: 1631-1645.
Kajisawa, M., Kodama, T., Tanaka, I., Yamada, T. & Bower, R. (2006). Protoclusters with evolved populations around radio galaxies at z \~ 2.5. Monthly Notices of Royal Astronomical Society 371: 577-582.
Baldry, IK, Balogh, ML, Bower, RG, Glazebrook, K, Nichol, RC, Bamford, SP & Budavari, T (2006). Galaxy bimodality versus stellar mass and environment. Monthly Notices Of The Royal Astronomical Society 373(2): 469-483.
Pedersen, K, Rasmussen, J, Sommer-Larsen, J, Toft, S, Benson, AJ & Bower, RG (2006). Discovery of a very extended X-ray halo around a quiescent spiral galaxy - The "missing link" of galaxy formation. New Astronomy 11(7): 465-470.
Bower, R. G. Benson, A. J. Malbon, R. K. Helly, J. C. Frenk, C. S. Baugh, C. M. Cole, S. & Lacey, C. G. (2006). Breaking the hierarchy of galaxy formation. Monthly Notices of the Royal Astronomical Society 370(2): 645-655.
Wilman, D. J., Balogh, M. L., Bower, R. G., Mulchaey, J. S., Oemler, A., Carlberg, R. G., Eke, V. R., Lewis, I., Morris, S. L. & Whitaker, R. J. (2005). Galaxy groups at 0.3 ⩽ z ⩽ 0.55 - II. Evolution to z ∼ 0. Monthly Notices of the Royal Astronomical Society 358(1): 88-100.
Wilman, D. J., Balogh, M. L., Bower, R. G., Mulchaey, J. S., Oemler, A., Carlberg, R. G., Morris, S. L. & Whitaker, R. J. (2005). Galaxy groups at 0.3 ⩽ z ⩽ 0.55 - I. Group properties. Monthly Notices of the Royal Astronomical Society 358(1): 71-87.
Swinbank, A.M., Smail, I., Bower, R.G., Borys, C., Chapman, S.C., Blain, A.W., Ivison, R.J., Howat, S.R., Keel, W.C. & Bunker, A.J. (2005). Optical and near-infrared integral field spectroscopy of the SCUBA galaxy N2 850.4. Monthly Notices of Royal Astronomical Society 359: 401-407.
Angulo, R. Baugh, C. M. Frenk, C. S. Bower, R. G. Jenkins, A. R. & Morris, S. L. (2005). Constraints on the dark energy equation of state from the imprint of baryons on the power spectrum of clusters. Monthly Notices of the Royal Astronomical Society 362(1): L25-L29.
Wilman, R. J., Gerssen, J., Bower, R. G., Morris, S. L., Bacon, R., de Zeeuw, P. T. & Davies, R. L. (2005). The discovery of a galaxy-wide superwind from a young massive galaxy at redshift z ≈ 3. Nature 436(7048): 227-229.
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Nakata, F., Bower, R.G., Balogh, M.L. & Wilman, D.J. (2005). The evolution of [O II] emission from cluster galaxies. Monthly Notices of Royal Astronomical Society 357(2): 679-686.
Swinbank, A. M., Balogh, M. L., Bower, R. G., Hau, G. K. T., Allington-Smith, J. R., Nichol, R. C. & Miller, C. J. (2005). Gemini Multi-Object Spectrograph Integral Field Spectroscopy of a Merging System with Enhanced Balmer Absorption. Astrophysical Journal 622(1): 260-266.
Fritz, A., Ziegler, B.L., Bower, R.G., Smail, I. & Davies, R.L. (2005). On the evolutionary status of early-type galaxies in clusters at z\~ 0.2 - I. The Fundamental Plane. Monthly Notices of Royal Astronomical Society 358: 233-255.
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Bower, R. G., Morris, S. L., Bacon, R., Wilman, R. J., Sullivan, M., Chapman, S., Davies, R. L., de Zeeuw, P. T. & Emsellem, E. (2004). Deep SAURON spectral imaging of the diffuse Lyman α halo LAB1 in SSA 22. Monthly Notices of the Royal Astronomical Society 351(1): 63-69.
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Bunker, A. J., Smith, J. K., Bower, R. G., Cambridge & CIRPASS Team and Durham/Gemini-Observatory GMOS Team (2002). Seeing the Universe in 3D: First Demonstration Science Results from CIRPASS and GMOS on Gemini Integral Field Spectrographs. Bulletin of the American Astronomical Society 34: 1131-.
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Supervision students
Mr Arnau Quera-Bofarull
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What is the length of the segment of the number line whose endpoints satisfy $|x-\sqrt[5]{16}|=3$?
We have $x-\sqrt[5]{16}=3$ or $x-\sqrt[5]{16}=-3$. Our two solutions are $x=\sqrt[5]{16}+3$ and $x=\sqrt[5]{16}-3$. These are the endpoints of the segment, and we need to find the length, so take the larger minus the smaller: $(\sqrt[5]{16}+3)-(\sqrt[5]{16}-3)=\boxed{6}$.
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Carrithers, J. et al., 2004. Concurrent Aerobic and Resistance Exercise Effects on Mixed, Myofibrillar, Mitochondrial, and Sarcoplasmic Protein Synthesis. Medicine & Science in Sports & Exercise, 36, p.S194.
Coker, R.H. et al., 2002. Prevention of Overt Hypoglycemia During Exercise Stimulation of Endogenous Glucose Production Independent of Hepatic Catecholamine Action and Changes in Pancreatic Hormone Concentration. Diabetes, 51, pp.1310–1318.
Honeal, K.P. et al., 2002. Synergistic Reduction in Low-Density Lipoprotein Cholesterol With Combined Hmg-Coa Reductase Inhibitor and Aerobic Exercise Therapy in Obese, Hypercholesterolemic Males. Medicine & Science in Sports & Exercise, 34, p.S50.
Koyama, Y. et al., 2002. Prior exercise and the response to insulin-induced hypoglycemia in the dog. American Journal of Physiology-Endocrinology And Metabolism, 282, pp.E1128–E1138.
Simonsen, L. et al., 2002. The effect of insulin and glucagon on splanchnic oxygen consumption. Liver, 22, pp.459–466.
Coker, R.H. et al., 2001. Stimulation of splanchnic glucose production during exercise in humans contains a glucagon-independent component. American Journal of Physiology-Endocrinology And Metabolism, 280, pp.E918–E927.
Crommett, A. et al., 2001. Excess post-exercise oxygen consumption following acute aerobic and resistance exercise in lean and obese women. Medicine & Science in Sports & Exercise, 33, p.S15.
Koyama, Y. et al., 2001. Role of carotid bodies in control of the neuroendocrine response to exercise. American Journal of Physiology-Endocrinology And Metabolism, 281, pp.E742–E748.
Coker, R.H. et al., 2000. Hepatic $\alpha$-and $\beta$-adrenergic receptors are not essential for the increase in Ra during exercise in diabetes. American Journal of Physiology-Endocrinology And Metabolism, 278, pp.E444–E451.
Koyama, Y. et al., 2000. Evidence that carotid bodies play an important role in glucoregulation in vivo. Diabetes, 49, pp.1434–1442.
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A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff
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Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
June 2019, 12(3): 507-549. doi: 10.3934/krm.2019021
Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation
Torsten Keßler , and Sergej Rjasanow
Saarland University, Department of Mathematics, P.O. Box 15 11 50, 66041 Saarbrücken, Germany
* Corresponding author: Torsten Keßler
Received December 2017 Revised July 2018 Published February 2019
Full Text(HTML)
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In this paper, we present an application of a Galerkin-Petrov method to the spatially one-dimensional Boltzmann equation. The three-dimensional velocity space is discretised by a spectral method. The space of the test functions is spanned by polynomials, which includes the collision invariants. This automatically insures the exact conservation of mass, momentum and energy. The resulting system of hyperbolic PDEs is solved with a finite volume method. We illustrate our method with two standard tests, namely the Fourier and the Sod shock tube problems. Our results are validated with the help of a stochastic particle method.
Keywords: Boltzmann equation, Galerkin-Petrov method, orthogonal polynomials, system of hyperbolic equations, finite volume method.
Mathematics Subject Classification: Primary: 82C40, 65N35; Secondary: 33C45, 35L04, 65M08.
Citation: Torsten Keßler, Sergej Rjasanow. Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation. Kinetic & Related Models, 2019, 12 (3) : 507-549. doi: 10.3934/krm.2019021
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Figure 1. Generalised spectrum of $ D $ with respect to $ M $ for $ K = 9 $, $ L = 9 $, i.e. $ n = 1000 $
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Figure 2. Approximation of the solution by a piecewise constant function. To fulfil boundary conditions, the dashed cells, called ghost cells, are added to the discretisation
Figure 3. Sketch of the one-dimensional Fourier problem. We seek the particle density function along the axis labeled by $ x $. $ T_l $, $ T_r $ are the temperatures of the walls, $ T_0 $ is the initial temperature of the gas
Figure 4. Course of the total mass for $ K = 3, L = 6 $, 256 spatial cells and $ {\rm{Kn}} = 0.1 $
Figure 5. Sketch of the initial situation of the shock tube problem. Two areas of same bulk velocities and temperatures but different densities are separated by diaphragm (dashed line), which is removed at $ t = 0 $
Figure 6. Contour plot of the final particle density function for $ {\rm{Kn}} = 0.25 $ and $ K = 3 $, $ L = 3 $ at $ x = 0.25 $ in the $ (v_1,v_2) $-plane
Figure 7. Comparison of different sets of basis functions for $ {\rm{Kn}} = 1.0 $. $ K $ is set to 3; $ L $ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $ N_x = 512 $ and $ \tau $ chosen as in equation (17). Figure 7a shows the density at the time $ t_f $, whereas in Figure 7b, the temperature at the time $ t_f $ is shown
Figure 8. Comparison of different sets of basis functions for $ {\rm{Kn}} = 1.0 $ near the walls. $ K $ is set to 3; $ L $ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $ N_x = 512 $ and $ \tau $ chosen as in equation (17). Figures 8a and 8b show the density at the time $ t_f $ near the left and the right wall, respectively. Figures 8c and 8d show the temperature at the time $ t_f $ near the left and the right wall, respectively
Figure 9. Comparison of different sets of basis functions for $ {\rm{Kn}} = 0.25 $. $ K $ is set to 3; $ L $ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $ N_x = 512 $ and $ \tau $ chosen as in equation (17). Figure 9a shows the density at the time $ t_f $, whereas in Figure 9b, the temperature at the time $ t_f $ is shown
Figure 10. Comparison of different sets of basis functions for ${\rm{Kn}} = 0.25$ near the walls. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figures 10a and 10b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 10c and 10d show the temperature at the time $t_f$ near the left and the right wall, respectively
Figure 11. Comparison of different sets of basis functions for $ {\rm{Kn}} = 0.025 $. $ K $ is set to 3; $ L $ the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $ N_x = 512 $ and $ \tau $ chosen as in equation (17). Figure 11a shows the density at the time $ t_f $, whereas in Figure 11b, the temperature at the time $ t_f $ is shown
Figure 12. Comparison of different sets of basis functions for ${\rm{Kn}} = 0.025$ near the walls. $K$ is set to 3; $L$ takes the values 3, 5 and 7. The stochastic curves are added for reference. The deterministic curves are obtained with $N_x = 512$ and $\tau$ chosen as in equation (17). Figures 12a and 12b show the density at the time $t_f$ near the left and the right wall, respectively. Figures 12c and 12d show the temperature at the time $t_f$ near the left and the right wall, respectively
Figure 13. Comparison of the final particle density function for $K = 3$, $L = 3$ with the stochastic particle density function at $x = 0.25$ for the Knudsen numbers $1.0$, $0.25$ and $0.025$
Figure 14. Numerical solution of the shock tube problem at $ t_f $ obtained with DSMC and the Galerkin–Petrov method for $ {\rm{Kn}} = 0.1 $. The exact solution of the Euler equations is shown by dashed lines
Figure 15. Numerical solution of the shock tube problem at $ t_f $ obtained with DSMC and the Galerkin–Petrov method for $ {\rm{Kn}} = 0.01 $. The exact solution of the Euler equations is shown by dashed lines
Figure 16. Numerical solution of the shock tube problem at $t_f$ obtained with DSMC and the Galerkin--Petrov method for ${\rm{Kn}} = 0.01$. The exact solution of the Euler equations is shown by dashed lines
Figure 17. Particle density functions for $ {\rm{Kn}} = 0.1 $ shortly after the diaphragm is removed
Figure 18. Particle density functions for ${\rm{Kn}} = 0.1$ when the shock discontinuity reaches the third evaluation point
Figure 19. Particle density functions for ${\rm{Kn}} = 0.1$ when the contact discontinuity reaches the third evaluation point at time $t_f$
Figure 20. Particle density functions for $ {\rm{Kn}} = 0.01 $ shortly after the diaphragm is removed
Figure 21. Particle density functions for ${\rm{Kn}} = 0.01$ when the shock discontinuity reaches the third evaluation point
Figure 22. Particle density functions for ${\rm{Kn}} = 0.01$ when the contact discontinuity reaches the third evaluation point at time $t_f$
Figure 23. Particle density functions for ${\rm{Kn}} = 0.001$ shortly after the diaphragm is removed
Figure 24. Particle density functions for ${\rm{Kn}} = 0.001$ when the shock discontinuity reaches the third evaluation point
Figure 25. Particle density functions for ${\rm{Kn}} = 0.001$ when the contact discontinuity reaches the third evaluation point at time $t_f$
Table 1. Number of basis and test functions for different choices of parameters. The set $ I_{K,L} $ is defined in equation (8)
$K$ $L$ $|{I_{K, L}}|$
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Table 2. Relative $ L^2 $-error for the mixture from equation (18) with parameters given in equation (19)
basis functions $L^2$-error
$K=1, L=2$ $5.666437\cdot 10^{-2}$
Table 3. Relative error of mass conservation for $ {\rm{Kn}} = 1 $
Spatial cells
$K=1, L=2$ $9.886570 \cdot 10^{-2}$ $9.875780 \cdot 10^{-2}$ $9.874720 \cdot 10^{-2}$ $9.874130 \cdot 10^{-2}$
Table 4. Relative error of mass conservation for $ {\rm{Kn}} = 0.1 $
Table 5. Relative error of mass conservation for $ {\rm{Kn}} = 0.01 $
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\begin{document}
\title{Solving systems of Boolean multivariate equations with quantum annealing}
\author{Sergi Ramos-Calderer} \affiliation{Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE.} \affiliation{Departament de F\'isica Qu\`antica i Astrof\'isica and Institut de Ci\`encies del Cosmos (ICCUB), Universitat de Barcelona, Mart\'i i Franqu\`es 1, 08028 Barcelona, Spain.}
\author{Carlos Bravo-Prieto} \affiliation{Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE.} \affiliation{Departament de F\'isica Qu\`antica i Astrof\'isica and Institut de Ci\`encies del Cosmos (ICCUB), Universitat de Barcelona, Mart\'i i Franqu\`es 1, 08028 Barcelona, Spain.}
\author{Ruge Lin} \affiliation{Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE.} \affiliation{Departament de F\'isica Qu\`antica i Astrof\'isica and Institut de Ci\`encies del Cosmos (ICCUB), Universitat de Barcelona, Mart\'i i Franqu\`es 1, 08028 Barcelona, Spain.}
\author{\\ Emanuele Bellini} \affiliation{Cryptography Research Centre, Technology Innovation Institute, Abu Dhabi, UAE.}
\author{Marc Manzano} \affiliation{Sandbox@Alphabet, Mountain View, CA, USA.} \affiliation{Electronics and Computing Department, Faculty of Engineering, Mondragon Unibertsitatea, Mondragon, Spain.}
\author{Najwa Aaraj} \affiliation{Cryptography Research Centre, Technology Innovation Institute, Abu Dhabi, UAE.}
\author{Jos\'e I. Latorre} \affiliation{Quantum Research Centre, Technology Innovation Institute, Abu Dhabi, UAE.} \affiliation{Departament de F\'isica Qu\`antica i Astrof\'isica and Institut de Ci\`encies del Cosmos (ICCUB), Universitat de Barcelona, Mart\'i i Franqu\`es 1, 08028 Barcelona, Spain.} \affiliation{Centre for Quantum Technologies, National University of Singapore, Singapore.}
\begin{abstract} Polynomial systems over the binary field have important applications, especially in symmetric and asymmetric cryptanalysis, multivariate-based post-quantum cryptography, coding theory, and computer algebra. In this work, we study the quantum annealing model for solving Boolean systems of multivariate equations of degree 2, usually referred to as the Multivariate Quadratic problem. We present different methodologies to embed the problem into a Hamiltonian that can be solved by available quantum annealing platforms. In particular, we provide three embedding options, and we highlight their differences in terms of quantum resources. Moreover, we design a machine-agnostic algorithm that adopts an iterative approach to better solve the problem Hamiltonian by repeatedly reducing the search space. Finally, we use D-Wave devices to successfully implement our methodologies on several instances of the Multivariate Quadratic problem. \end{abstract}
\maketitle
\section{Introduction}
Adiabatic quantum computation is a universal quantum computation scheme \cite{farhi2000quantum} where a quantum system is prepared in the ground state of an easy to prepare Hamiltonian and evolved towards a Hamiltonian that encodes the solution of a problem in its ground state. If the evolution is performed adiabatically, the quantum system still remains in its instantaneous ground state and the problem will be solved. Quantum annealers are special-purpose devices based upon the principles of adiabatic quantum computation that work with simpler Hamiltonains and more relaxed evolution times. However, these devices are believed to provide an edge when solving classical satisfiability problems by leveraging quantum phenomena and are easier to control and scale up for larger, real-life problems~\cite{finnila1994quantum, brooke1999quantum, johnson2011quantum, perdomo2012finding, mandra2016strengths, benedetti2017quantum, khoshaman2018quantum, benedetti2018quantum, ding2019towards, perdomo2019readiness, wilson2021quantum}. Interestingly, quantum annealers on the order of thousands of qubits are already commercially available by D-Wave~\cite{dwave}.
The quantum annealing approach to quantum computing is a research topic that can be relevant on many research problems in a variety of scientific fields. Motivated by this idea, we investigate the possibility of using D-Wave quantum annealer for a fundamental problem in computer science: solving systems of multivariate polynomial equations over the binary field. If all polynomials in the system are linear, then the system can be efficiently solved, for instance, by Gaussian elimination. The problem is easy also when the system is either underdetermined (much fewer equations than variables), overdetermined (much more equations than variables), or sparse (the number of terms is linear with respect to the number of variables). However, the problem is known to be NP-hard already for generic quadratic systems \cite{lewis1983computers}. Moreover, assuming the exponential time hypothesis
\cite{impagliazzo2001complexity}, there exists no sub-exponential time (worst-case) algorithm for this problem.
Polynomial systems over the binary field can be used to perform algebraic cryptanalysis \cite{bard2009algebraic} potentially against any cipher. Moreover, in the case of degree 2 polynomials, the problem, usually referred to as the Multivariate Quadratic ($\mathsf{MQ}$) problem, has important applications in post-quantum cryptography, since several post-quantum schemes exist basing their security on its difficulty to be solved \cite{rainbowround3nist, gemmsround3nist}. For these reasons, there is a spreading interest in the scientific community to find new algorithms to solve the $\mathsf{MQ}$ problem, both in the classical and quantum computation model. The former case has been extensively studied, see for example \cite{Bard2007,mou:tel-01110887,ullah2012,EDER2017719} for comprehensive surveys on the most effective algorithms. Regarding the latter, a detailed analysis of the required qubits and time for a Grover's algorithm approach is presented in Ref.~\cite{schwabe2016solving}. Building upon the previous work, the authors of Ref.~\cite{pring2018exploiting} demonstrate that by applying preprocessing the computational load on the quantum computer can be reduced and, in a generalization of the multi-target search for single targets, the efficiency of the basic quantum search oracle for the $\mathsf{MQ}$ problem over the binary field can be improved. In Ref.~\cite{faugere2017fast}, the authors present a quantum version of BooleanSolve \cite{bardet2013complexity}, which is currently the fastest asymptotic algorithm for classically solving systems of non-linear Boolean equations, that takes advantage of Grover’s quantum algorithm. Note that Ref.~\cite{bernstein2018asymptotically} also proposed a new Gr\"obner-based quantum algorithm for solving quadratic equations with a complexity comparable to QuantumBooleanSolve (we refer to Ref.~\cite{faugere2017fast} for further details). Finally, in Ref.~\cite{biasse2021framework}, the authors show how to reduce the use of quantum RAM and circuit complexity by delegating some precomputations to a classical computer.
Note that all the above quantum techniques have only considered the use of universal fault-tolerant quantum computers. Therefore, the aforementioned methods can not be implemented on current quantum computers without error correction. In contrast, the use of quantum annealers to solve systems of multivariate equations over a finite field is still unexplored, and in this work, we try to fill this gap.
In this paper, we explore how quantum annealing can be used for solving multivariate systems of quadratic equations over binary fields, namely, the $\mathsf{MQ}$ problem. We present different methodologies to translate the $\mathsf{MQ}$ problem into a Hamiltonian that can be solved by a quantum annealer. To support our proposal, we provide results obtained by running examples on D-Wave's Advantage quantum device \cite{dwave}, a commercially-available quantum computer. Our approach takes into consideration the decomposition of multi-qubit terms up to at most two-qubit interactions, which is a constraint of the underlying architecture of near-term quantum annealing devices.
Our work highlights the main obstacle of mapping Boolean systems of equations into Hamiltonian ground states, which is the fact that the inherent correlations in operations over the binary field have to be encoded into a real Hamiltonian, at the cost of an overhead. We present two alternative methods to circumvent the exponential overhead in quantum memory that the naive transformation would incur. Moreover, we also introduce a new iterative approach that aids quantum annealing devices in finding the ground state of the Hamiltonian by repeatedly shrinking the search space using information gained in previous executions of the system. This method allowed us to successfully solve small instances of the $\mathsf{MQ}$ problem using current D-Wave devices.
This paper is organized as follows. In Sec. \ref{sec:preliminaries}, we introduce the background required for this work. Next, in Sec. \ref{sec:formalizing} we present our methodologies to translate a given $\mathsf{MQ}$ problem into a Hamiltonian in the context of quantum annealing and analyze the needed resources. Finally, in Sec. \ref{sec:experiments} and Sec. \ref{sec:conclusions} we present the results of our experiments with D-Wave and the conclusions of this work, respectively.
\section{Preliminaries} \label{sec:preliminaries}
We denote by $\mathbb{F}_q$ the finite field with $q$ elements. $\mathbb{F}_q^n$ is the set of all vectors of length $n$, viewed as an $\mathbb{F}_q$-vector space. For compactness, we sometimes denote with $\Vec{x}$ the vector $(x_1, \dots, x_n)$.
The $\mathsf{MQ}$ problem is defined as follows. The input of the problem consists of $m$ quadratic polynomials $p_1(x_1, \dots, x_n), \dots, p_m(x_1, \dots, x_n) \in \mathbb{F}_q[x_1, \dots, x_n]$ in $n$ variables $x_1, \dots, x_n$ and coefficients in a finite field $\mathbb{F}_q$. The output of the problem is given by the set of $(a_1, \dots, a_n) \in \mathbb{F}_q^n$ for which $p_i(a_1, \dots, a_n) = 0$ for all $i = 1, \ldots, m$. The vector $(a_1, \dots, a_n)$ is called a \emph{solution} of the \emph{system of equations} \begin{equation}\label{equ:system} p_i(x_1, \dots, x_n) = 0, \quad i = 1,\dots,m . \end{equation} Three variants of the $\mathsf{MQ}$ problem can be defined. (1) The \emph{Decision} variant asks to determine if Eq.~\eqref{equ:system} has a solution. (2) The \emph{Search} variant asks to find a solution of Eq.~\eqref{equ:system}, if there is one. (3) The \emph{Exhaust} variant asks to find all solutions of Eq.~\eqref{equ:system}.
The decision variant of $\mathsf{MQ}$ is known to be an NP-complete problem \cite{fraenkel1979complexity}. It is easy to observe that solving the search variant also solves the decision one. On the other hand, by iteratively guessing each variable, it is possible to solve the search variant by solving the decision variant at most $n$ times.
For practical purposes, one is usually interested in the search variant, and sometimes in the exhaust variant. From now on, unless it is stated otherwise, we write $\mathsf{MQ}$ problem to refer to its search variant. Furthermore, we will focus on the Boolean case, i.e., where $q=2$. In this case, polynomials are called Boolean polynomials, and the corresponding unique map $f$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2$ is called a Boolean function.
It is common to refer to the Boolean polynomial as the Algebraic Normal Form (ANF) of $f$, which we indicate with $f^{(\mathbb{F})}$. In this work, we will also need another representation of $f$ called the Numerical Normal Form (NNF), which we indicate with $f^{(\mathbb{Z})}$. \begin{definition}\label{defNNF}
Let $f$ be a Boolean function on $\mathbb{F}_2^n$ taking values in the integer ring $\mathbb{Z}$.
We call the \emph{Numerical Normal Form (NNF)} of $f$
the following expression of $f$ as a polynomial:
$$
f(x_1,\ldots,x_n)
=
\sum_{u \in \mathbb{F}_2^n}\lambda_u (\prod_{i=1}^{n}x_i^{u_i})
=
\sum_{u \in \mathbb{F}_2^n}\lambda_{u}X^u\,,
$$
with $\lambda_{u} \in \mathbb{Z}$ and $u=(u_1,\ldots,u_n)$. \end{definition}
With abuse of notation and when clear from the context, we sometimes write $f = f^{(\mathbb{F})} = f^{(\mathbb{Z})}$, and we indicate with $+$ both the addition over $\mathbb{F}_2$ or over another field or ring. Given a Boolean function, its ANF and NNF are unique. It is also worth noting that, in general, if a Boolean function in ANF has about $k$ terms (i.e., nonzero coefficients), then its corresponding NNF will contain about $2^k$ terms (see Ref. \cite{bellini2018deterministic} for detailed proof). As described in this work, this significant increase of terms turns out to be the main obstacle when trying to solve Boolean polynomial systems using annealing evolution. We finally refer to \cite{carlet10boolean} for an exhaustive introduction to Boolean functions.
\begin{example} An example of a quadratic Boolean polynomial system with $n = 4$ variables and $m = 4$ equations is given below: {\footnotesize \begin{align*}
\begin{cases}
x_1 x_2 + x_1 x_3 + x_1 x_4 + x_1 + x_2 x_3 + x_2 x_4 + x_2 + x_3 x_4 + x_4 = 0 \\
x_1 x_2 + x_1 x_3 + x_2 + x_3 x_4 + x_3 = 0 \\
x_1 x_2 + x_1 x_3 + x_2 x_3 + x_2 + x_3 + x_4 = 0\\
x_1 x_3 + x_2 x_4 + x_4 + 1 = 0 \\
\end{cases} \end{align*} } The polynomials are given in Algebraic Normal Form and the only solutions of the system are the two binary vectors $(1, 0, 1, 0)$ and $(0, 0, 1, 1)$. For example, the Numerical Normal Form of $f^{(\mathbb{F})} = x_1 x_3 + x_2 x_4 + x_4 + 1$ (note that the addition is over $\mathbb{F}_2$) is given by $f^{(\mathbb{Z})} = -2 x_1 x_2 x_3 x_4 + 2 x_1 x_3 x_4 - x_1 x_3 + x_2 x_4 - x_4 + 1$ (note that the addition is over the integer ring $\mathbb{Z}$). \end{example}
\section{Formalizing the problem in a quantum annealer} \label{sec:formalizing}
In the early 2000s, Farhi et al. proposed a new universal quantum computation model based on the quantum adiabatic theorem \cite{farhi2000quantum, farhi2001quantum}. The so-called adiabatic quantum computation model was shown to be polynomially equivalent to the quantum gate-based model proposed by Deutsch in 1989 \cite{deutsch1989, aharonov2008}, and is one of the most promising models of quantum computing due to its natural robustness against errors~\cite{childs2001robustness}. The adiabatic theorem guarantees that if the Hamiltonian that dictates the energy of a quantum system is modified slowly enough, a quantum state will remain in its instantaneous ground state during the evolution~\cite{born1928, kato1950adiabatic}. This implies that we can encode the solution of a hard problem, the $\mathsf{MQ}$ problem in this case, into the ground state of a problem Hamiltonian $H_p$, and then, starting from an easy-to-prepare ground state of an initial Hamiltonian $H_0$, drive the system slowly to the problem Hamiltonian to then measure its solution.
A less restrictive, more hardware-friendly, technique to solve classical problems is quantum annealing. Quantum annealing also follows the evolution of a quantum Hamiltonian in order to find low-energy configurations of the system but does not demand adiabatic evolution and forgoes universality. Devices such as D-Wave are quantum annealers, and while not being universal quantum computers due to their limitations, are still useful for solving hard optimization problems \cite{kadowaki1998quantum, das2008}. Hard classical problems that can be codified to a problem Hamiltonian by only using the computational basis of the system, as is the case for the $\mathsf{MQ}$ problem tackled in this paper, are ideal for these available quantum annealers.
We want to solve the $\mathsf{MQ}$ problem defined in Sec. \ref{sec:preliminaries}, where we are given a set of $m$ quadratic polynomials $p_1(x_1,\ldots x_n),\ldots p_m(x_1, \ldots x_n)$ over the binary field and we are tasked with finding $\Vec{x}$ so that all $\Vec{p}$ are equal to zero, via quantum annealing. Therefore, we need to create a Hamiltonian with a ground state that encodes the solution to this problem.
\subsection{Direct embedding} \label{sec:direct}
A first direct approach is penalizing with positive energy each of the equations $p_i(\Vec{x})$ that is not fulfilled. The corresponding problem Hamiltonian can be constructed as \begin{equation}
H_p = \sum_{i=1}^mp_i(\Vec{x})\,, \label{eq:Hp} \end{equation} as it contributes with positive energy if the input bits for $p_i(\Vec{x})$ do not result in a zero solution.
Usually, the polynomials $p_i(\Vec{x})$ in Eq. \ref{eq:Hp} are given in ANF since bitwise operations are performed over the binary field $\mathbb{F}_2$. However, the quantum Hamiltonian we can encode into a quantum annealer device does not function with binary algebra, each positive term only adds more energy to the final state. Therefore, each polynomial $p_i(\Vec{x})$ has to be given in its NNF. This transformation can be obtained by recursively applying the change \begin{equation}
(x_i + x_j) \longrightarrow x_i + x_j - 2 x_i\cdot x_j \label{eq:ANFNNF} \end{equation} to the original ANF equations, where, with abuse of notation, the symbol $+$ on the left is the addition over $\mathbb{F}_2$, while the symbols $+$ and $-$ on the right are the regular addition and subtraction over the integer ring. This transformation introduces multi-qubit interaction terms (i.e., terms of degree greater than 1) that were not present in the ANF of $p_i(\Vec{x})$. In general, all combination of monomials present in the ANF of $p_i(\Vec{x})$ will appear in the NNF. Keep in mind that, in a binary field, it holds that $x^2=x$ (since the values of the variables $x_i$ are either $0$ or $1$), there are no powers in the monomials of the ANF or the NNF.
This transformation gives rise to a different issue, the chip architecture of currently available quantum annealers only allows for two-qubit interactions. Thus, to run a quantum annealing protocol on a real device, the interactions of the Hamiltonian have to be reduced. For a general many-body Hamiltonian, its interactions can be reduced to two-body using perturbation theory by adding ancilla qubits \cite{bravyi2008quantum, jordan2008perturbative, cao2015hamiltonian}. If all the problem Hamiltonian parts share the same basis, as is the case for a classical Hamiltonian such as ours, the reduction can be performed without perturbation theory \cite{biamonte2008non, babbush2013resource}. This reduction method yields a new Hamiltonian with a different energy spectrum but equal ground state and energy, therefore not altering the solution of the problem and is the one we follow for the direct embedding.
The method consists of exchanging a two-qubit interaction by an ancilla, reducing by one the order of the interaction. A penalty function is then introduced to the Hamiltonian that adds energy when the value of the ancilla is not equal to the product of the original two qubits. The penalty function can be written as \begin{equation}
s(x_i, x_j, x_{ij}) = 3x_{ij}+x_ix_j-2x_ix_{ij}-2x_jx_{ij}\,, \end{equation} where $x_{ij}$ is the label given to the ancillary qubit that is substituted. It can be seen that $s(x_i, x_j, x_{ij})=0$ if $x_ix_j=x_{ij}$ and $s(x_i, x_j, x_{ij})\geq 1$ otherwise. This keeps the ground state and energy unchanged.
Furthermore, a single ancilla $x_{ij}$ can be used for all terms in the Hamiltonian, where the term $x_ix_j$ appears. This is achieved by applying the substitution \begin{align}\label{eq:substitution}
\sum_K\alpha_{ijK}&x_ix_jx_K\longrightarrow\\\nonumber
&\sum_K\left(\alpha_{ijK}x_{ij}x_K+\left(1+\abs{\alpha_{ijK}}\right)s(x_i, x_j, x_{ij})\right)\,, \end{align} where the index $K$ is the product of multiple other variables in all terms where $x_ix_j$ is present. It can be shown that this transformation also yields a Hamiltonian with the same ground state \cite{babbush2013resource}.
When this procedure is used to reduce large multi-qubit terms, the resulting final Hamiltonian will have large coefficients. This introduces a problem for real-life implementation since the machine precision for coefficients of quantum annealing devices such as D-Wave's is limited. The quantum annealer developed by D-Wave scales the given coefficients between $[-1, 1]$ when introducing them to the machine, so small coefficients can vanish when translated into weights in the presence of other large parameters.
An alternative transformation is proposed in Ref. \cite{babbush2013resource} with the aim of reducing the precision needed for the control of the device. Introducing the term \begin{equation}
\delta_{ij}=\max\left(\sum_{K, \alpha_{ijK}>0}\alpha_{ijK}, \sum_{K, \alpha_{ijK}<0}-\alpha_{ijK}\right)\,, \end{equation} the substitution given in Eq. \eqref{eq:substitution} can be rewritten as \begin{equation}
\begin{split}
\sum_K\alpha_{ijK}x_ix_jx_K &\longrightarrow \\ &\sum_K\alpha_{ijK}x_{ij}x_K+\left(1+\delta_{ij}\right)s(x_i, x_j, x_{ij})\,,
\end{split} \label{eq:control} \end{equation} while still keeping the desired ground state. This reduces, but not completely solves, the precision problem.
If a given $n$-qubit Hamiltonian contains multi-qubit interactions involving all of its constituents, that is, an $n$-qubit Hamiltonian with up to $n$-body terms, one would require $2^{\frac{n+2}{2}}-2$ total qubits to reduce all possible combinations of qubit interactions to two-body terms for an even $n$ ($3\times 2^{\frac{n-1}{2}}-2$ for odd $n$). This can be achieved by dividing the total qubit register into two fully-connected graphs using ancillary variables and connecting both graphs with another ancillary qubit. Unfortunately, this will be the case for a general conversion from ANF to NNF due to the fact that an $n$ term sum in ANF will generally require \begin{equation}
\sum_{k=1}^n\binom{n}{k}=2^n-1 \end{equation} terms for the equivalent NNF equation. Therefore one would need an exponential amount of quantum resources, ancillary qubits in this case, to encode the ground state into a Hamiltonian following this first direct approach.
\subsection{Truncated embedding} \label{sec:truncated}
This problem can be circumvented by partitioning the original polynomials $p_i(\Vec{x})$ into smaller pieces with $k$-bounded length using ancillary variables. It is straightforward to see that a sum of $n_i$ monomials can be reduced to sums of up to $k$ terms by adding ancillas in the form \begin{equation}
\begin{split}
x_1+\ldots+x_{n_i}=0 \rightarrow& x_1+\ldots +x_{k-1}+a_1=0 \\
&a_1+x_k+\ldots +x_{2k-2}+a_2=0 \\
&\ldots \\
&a_l+x_{{n_i}-k+1}\ldots+x_{n_i} = 0\,,
\end{split} \end{equation} at the cost of expanding the number of equations to $\frac{n_i-2}{k-2}$ using $l=\frac{n_i-2}{k-2}-1=\frac{n_i-k}{k-2}$ ancilla variables we have labeled $a_i$. A similar technique can also be used when encoding the $\mathsf{MQ}$ problem to a SAT instance \cite{mq2sat}.
\begin{figure}
\caption{Number of logical qubits needed to embed an $\mathsf{MQ}$ problem into the ground state of a Hamiltonian for the direct and truncated approaches. It can be seen how the direct approach, while more resource-efficient in small systems, quickly outpaces the truncated approach. The inset shows the scaling for a larger number of variables. Note also that the optimal value for the cutoff variable is $k=4$.}
\label{fig:direct-vs-trunc}
\end{figure}
To have more precise control on the total number of ancillary qubits added to decompose the multi-qubit terms, we need to ensure that the parameter $k$ is also the maximum number of multi-body interactions. For that reason, we first introduce ancilla variables to substitute the two-qubit terms in the original ANF representation adding the required penalty functions. In the worst-case scenario, where all combinations of two-body interactions appear, we will need to add $\binom{n}{2}=n(n-1)/2$ ancillary variables. This, however, does not change the total number of monomials in the truncated system of equations.
The maximum number of qubits needed to represent an $\mathsf{MQ}$ problem with $m$ equations involving $n$ variables into a two-body Hamiltonian will then be \begin{equation}
\sum_{i=1}^m \left[\frac{n_i-2}{k-2}\left(2^{\frac{k+2}{2}}-2-k\right)+\frac{n_i-k}{k-2}\right]+\binom{n}{2}+n, \end{equation} where $n_i$ is the number of monomials in each $p_i(\Vec{x})$ of Eq.~\ref{equ:system} and $k$ (even) is the length of the partitions.
\begin{table*}[t!] \centering\renewcommand\cellalign{c} \setcellgapes{3pt}\makegapedcells
\begin{tabular}{|c|c|c|} \hline \textbf{Gate} & \,\,\,\textbf{Boolean operation}\,\,\, & \textbf{Penalty function} \\ \hline NOT & $z = \overline{x}$ & $2xz - x - z + 1$ \\ \hline Controlled-NOT & $z = x_c x_t$ & $\makecell{2x_c x_t - 2(x_c + x_t)z - 4(x_c + x_t)x_a + 4zx_a + x_c + x_t + z + 4x_a}$\\ \hline Toffoli & $z = x_{c1} x_{c2} x_t$ & $\makecell{-4x_{a1}x_{a2} + 4x_{a1}z - 4x_{a1}x_t - 2x_{a1}x_{c1} -\\ 2x_{a2}x_{c2} 2x_{a2}z + 2x_{a2}x_t + x_{c1}x_{c2} - 2x_t z + 4x_{a1} + 4x_{a2} + z + x_t}$\\
\hline\end{tabular} \caption{Summary of Boolean operations and their penalty function implementation in a quantum annealer as only constant, single and two-qubit monomials appear in $\mathsf{MQ}$ problems. The result is saved in the qubit corresponding to the variable $z$, while the variable $x$ corresponds to other qubits involved in the Boolean operation. The subscripts $c$, $t$, and $a$ correspond to \textit{control}, \textit{target}, and \textit{ancilla}, respectively.}
\label{table:operations} \end{table*}
The differences in scaling between the truncated approach with different values for $k$ and the direct embedding can be seen in Fig. \ref{fig:direct-vs-trunc}. While the direct approach is more efficient when the number of variables is small, its exponential scaling quickly makes it unfeasible when compared to the truncated approach. For the polynomial approach, the cutoff variable $k$ defines its scaling. We note that the scaling is optimal for $k=4$. Additionally, the precision issues raised in the substitution scheme will be significantly attenuated in the truncated embedding since now the non-locality of the ancillae is governed by $k$ and not the total number of variables.
A partition length of $k=4$ minimizes the total number of ancillae since the exponential term $2^{\frac{k+2}{2}}$ dominates and the truncation of the original equation needs half the parameters than $k=3$. Precisely, a $4$-term sum will only need $2$ extra ancillary variables to reduce it to up to $2$-body terms. Fixing the value for $k$ the total number of qubits needed reads \begin{equation}
\frac{n^2}{2}+\frac{n}{2}-4m+\frac{3}{2}\sum_{i=1}^mn_i. \end{equation} We encounter now a polynomial scaling with the number of parameters under the condition that the total number of terms in the system of equation scales reasonably with the number of variables.
In order to obtain a qubit scaling that only depends on the number of variables $n$, we can use average values for both $m$ and $n_i$. Generally, we will encounter as many equation as variables in the system, $m=n$, each with an average number of monomials given by the total possible combination of terms with two-body interactions, $n_i\sim \left(n+\binom{n}{2}\right)/2=(n+n^2)/4$. These two approximations yield the new scaling \begin{equation}
\frac{3}{8}n^3+\frac{7}{8}n^2-\frac{7}{8}n, \end{equation} a polynomial of degree $3$ in the number of variables of the problem.
\subsection{Penalty embedding} \label{sec:problem_penalty}
An alternative way to embed the ground state of the $\mathsf{MQ}$ problem is to model the equations in their ANF using logical quantum gates such as CNOT or Toffoli gates which natively act over the $\mathbb{F}_2$ field, and then reproduce that circuit as an adiabatic evolution using penalty functions. To be more precise, we model the $\mathsf{MQ}$ problem equations as Boolean operations on an output quantum register, that is, the actions of $+x_i$ and $+x_ix_j$ can be modeled to a CNOT and Toffoli gates targeting the output qubit and controlled by qubits $\{x_i\}$ and $\{x_i,\,x_j\}$ respectively. Then a Hamiltonian is constructed with a ground state that follows the correct gate-by-gate implementation of the resulting circuit.
This method of circuit-to-Hamiltonian encoding using penalty functions is reminiscent of Feynman's Hamiltonian clock \cite{feynman1985quantum}, where in order to create a Hamiltonian that faithfully represents the actions of a logical quantum circuit one would use an extra \textit{clock register} where the time step of each applied quantum gate is stored. In this implementation, an ancillary \textit{output} qubit register is added, which stores the result of the output qubit after each gate application.
The penalty functions needed to map the solution of an $\mathsf{MQ}$ problem into the ground state of a Hamiltonian are displayed in Tab. \ref{table:operations}. The \textit{output} ancilla qubit $z$ used in the penalty function of a given quantum gate will be used as the \textit{target} qubit $x_t$ in the penalty function of the immediately following gate. These penalty functions contribute with positive energy if the state of the qubits involved does not match the logical Boolean operation that they map. Additionally, the qubits used to initialize the \textit{output} ancilla register are penalized if they are in the $\ket{1}$ state as we assume an initial state of the output qubit of $\ket{0}$. The same thing is applied to the \textit{output} ancillae where the final result of applying each equation $p_i(\Vec{x})$ in Eq.~\ref{equ:system} is stored as we are interested in the solution where the output is zero.
It is straightforward to see that the quantum resources needed to apply this implementation are governed by the number of monomials present in the equations of a given $\mathsf{MQ}$ problem, as they will dictate the number of gates that are to be implemented. As discussed in Sec. \ref{sec:truncated} above, the average number of monomials appearing in a given problem will scale as $\order{n^3}$, and the ancilla overhead needed for the implementation of each CNOT or Toffoli gate, an extra $1$ or $2$ ancillae respectively, will not change the overall scaling. Therefore, up to the particularities of each implementation, both the truncated and the penalty function embedding will scale similarly, and in large system sizes outclass the direct embedding.
However, it is crucial to mention the number of physical qubits needed for the implementation when assessing the actual quantum resources. Due to chip architecture constraints, mapping a Hamiltonian into a real quantum annealing device will require an overhead to account for non-local interactions. The D-Wave API provides the automatic solver \textit{minorminer}~\cite{minorminer} to find a good embedding into their architecture. Highly non-local Hamiltonians will require a large number of physical qubits in order to represent each logical variable. Moreover, the amount of required physical qubits can change the scaling of a particular method, giving an edge to a more local embedding with more logical variables.
We show in Tab.~\ref{table:log-phys} the comparison between both the truncated and penalty embeddings in terms of physical and logical qubits required for their implementation into the Advantage D-Wave machine. For different instances up to 12 logical variables, we show the amount of required physical quantum resources for both the truncated and the penalty embedding. Each embedding has been averaged over $10$ instances in order to reduce the uncertainty due to the minimization method provided by D-Wave. We note that both the truncated and the penalty embedding scale in a similar manner when mapped into the physical qubits of a given chip architecture with a worse, albeit still polynomial, overall scaling. \begin{table}[t!] \centering\renewcommand\cellalign{c} \setcellgapes{3pt}\makegapedcells
\begin{tabular}{|c||c|c|c|c|c|} \hline Variables & 4 & 6 & 8 & 10 & 12 \\ \hline \hline Truncated (logical) & $30$ & $90$ & $231$ & $451$ & $718$ \\ \hline Truncated (physical) & $55.6$ & $223.0$ & $758.0$ & $1627.8$ & $2645.2$ \\ \hline \hline Penalty (logical) & $61$ & $150$ & $345$ & $645$ & $1005$ \\ \hline Penalty (physical) & $105.1$ & $309.4$ & $864.1$ & $1940.6$ & $3436.5$ \\ \hline\end{tabular} \caption{Number of logical and physical qubits needed to map the truncated and penalty embedding for different number of variables. Physical qubit values are averaged over 10 instances of the \textit{minorminer} algorithm provided by the D-Wave API \cite{minorminer}.} \label{table:log-phys} \end{table}
\iffalse \begin{figure}
\caption{Comparison between the logical and physical qubit requirements for the truncated and penalty embedding. It can be seen that both scale in a similar manner and that the overhead due to the transformation into physical qubits does not change the complexity of the embedding. The number of physical qubits averaged over 5 different embedding instances.}
\label{fig:phys-vs-log}
\end{figure} \fi
\section{Results} \label{sec:experiments}
In this section, we encode some reduced $\mathsf{MQ}$ problem instances using the methods presented in Sec. \ref{sec:formalizing} in order to be solved using D-Wave machines. We also propose an iterative method to aid in finding a singular correct solution in large, highly correlated, systems. For a detailed implementation, we refer to our code made available on Github \cite{nonlinear-code}.
So far we have presented several ways to encode the solution of an $\mathsf{MQ}$ problem into the ground state of a Hamiltonian that can be used for quantum annealing. The implementation of such a protocol on a real quantum device, however, will require adjusting to the specifics of the particular machine. The problem Hamiltonian assumes the implementation of all-to-all interaction. However, this is unrealistic because the superconducting chips for quantum annealing provided by D-Wave's Advantage device support a Pegasus chip architecture~\cite{dattani2019pegasus} and, therefore, the qubits need to be mapped accordingly to that restriction. The solution to this problem is the introduction of \textit{qubit chains}.
A logical qubit will be extended into a chain of qubits when mapped into the physical chip of the quantum device. This means that different physical qubits, which will represent the same variable, are bound together by an interaction term, a \textit{chain strength}, that penalizes members of the same qubit chain for being in different quantum states. We leave the mapping of the original variables to physical objects to the built-in compiler provided by the D-Wave library \cite{minorminer} and adjust the chain strength hyper-parameter in order to not overpower the variables of the problem while measuring as few broken chains as possible. This mapping will result in a more complex evolution, and consequently poorer results, especially for Hamiltonians with a large number of non-local qubit interactions.
We present the results of running the Hamiltonians proposed in the different embedding schemes. The uppermost graphs in Fig. \ref{fig:embedding_results} show the results of sampling the final state of a quantum annealing evolved under each corresponding Hamiltonian for the direct, penalty, and truncated embedding, respectively. We state for each case the number of logical and physical qubits the problem needed to be mapped to. We decided to focus on a similar number of physical qubits needed, therefore the direct embedding was able to reach a 9-variable problem while the truncated and penalty embedding are limited to 5 variables. We note that with a small number of variables, the annealing process does not yield the exact ground state that encodes the solution of the problem, the quantum state with zero energy. Longer annealing times, more precise control on the annealing schedule, or higher quality qubits are ways to improve the results. However, current quantum annealers might not have the capabilities of tuning those parameters to the required specifications of large problems. In order to achieve the ground state energy of the problem in a machine-agnostic way, we propose an iterative algorithm that closes in on a smaller, easier-to-solve, subspace where the ground state might be located.
\begin{figure*}
\caption{Samples below 100 energy after a 1000-sample execution on D-Wave's Advantage machine for the different $\mathsf{MQ}$ problem embeddings presented where the ground state was first observed. We have implemented problems with a similar number of physical qubits required in the first embedding of the problem. The direct embedding (left) encodes a 9-variable problem in 46 logical qubits that are mapped to 179 physical ones, the truncated (center) and penalty (right) embedding both encode a 5-variable problem with 67 logical qubits with 167 physical ones for the truncated and 114 logical qubits with 221 physical one for the penalty one. We show 4 different iterations in our iterative method to highlight how the energy approaches the ground state as the system gets smaller both in logical and physical qubits. The ground state energy of this problem is 0, depicted as a gray dotted line.}
\label{fig:embedding_results}
\end{figure*}
As detailed in Sec. \ref{sec:formalizing}, the proliferation of ancillary qubits in the different proposed embedding options appears when reducing the multi-qubit interactions into at most two-qubit interaction terms. This means that most of these added ancillae will represent products of other variables and will therefore have a stronger penalization than the original variables that they represent. That is, the wrong state of certain ancillary variables is penalized with a higher amount of energy than others. The following is a heuristic iterative method where we use that to our advantage.
After running an annealing protocol on a quantum device, if no quantum state with zero energy has been found, we may look at some of the low energy configurations of the obtained samples. If some qubits are found in the same result in all of the lowest energy states, we can assume that the Hamiltonian penalizes those variables more than the others. We can narrow the subsequent search space by substituting that variable in the original Hamiltonian by their, now known, preferred value. The more the search space is reduced, the easier it is for the quantum annealing device to find the lowest energy solution.
The amount of low-energy solutions to check for the same value of the ancilla variables is a hyper-parameter that can be optimized. On the one hand, if we set the value too low we might be excluding the ground state from the reduced search space by fixing ancillae to a wrong outcome. On the other hand, if we set it too high we might not find any variable that lays in the same output for all low energy configurations. More sampling at each iteration will also enhance our ability to fix ancillas but will impact the overall run time of the algorithm. The number of fixed parameters per run will depend on the problem, encoding and quality of the quantum device. We used a heuristic approach when tuning the number of low energy solutions checked. A method to check if the reduced subspace no longer contains the solution can be devised. If the lowest energy sample of the reduced Hamiltonian is lower than its equivalent value in the original Hamiltonian, adding back up the fixed variables of the original setting, then we have excluded the original ground state from the reduced subspace. The different rows in Fig. \ref{fig:embedding_results} show how this iterative approach indeed helps in finding the ground state of the problem. As more iterations go by and more ancillae are fixed, the system starts finding lower energy solutions until the ground state with zero energy is reached.
The first row in Fig. \ref{fig:embedding_results} is always the initial run of the algorithm. Then the following are some snapshots of the energy samples during the iterative algorithm, and the last row showcases the first iteration where a state with energy zero is reached. The direct embedding shows the result of having a highly non-linear Hamiltonian with large parameters. The initial runs are very far away from the ground state and it is not until the more volatile variables are fixed that the ground state can be found. The truncated and penalty embedding behave in a similar way to each other. It can be seen how after each iteration the median energy gets closer and closer to the ground energy until it is reached. We note that the penalty embedding in spite of requiring more qubits to embed the problem Hamiltonian, reaches the ground state with fewer iterations. This can be attributed to the lower coefficients that are needed to map the problem, making it more suited to an annealer machine such as the one provided by D-Wave.
\iffalse \begin{figure}
\caption{\textbf{9-bits (direct, 46 qubits, 179 physical qubits).} samples under 200 energy after 8 iterations of the algorithm after 1000 samples. The ground state energy of this problem is 0, depicted as a gray dotted line.}
\label{fig:direct-results-9}
\end{figure} \fi
\iffalse \begin{figure}
\caption{\textbf{5-bits (penalty, 114 qubits, 221 physical qubits)}. Samples under 50 energy after 4 iterations of the algorithm after 1000 samples. The ground state energy of this problem is 0, depicted as a gray dotted line.}
\label{fig:penalty-results-5}
\end{figure} \fi
\iffalse \begin{figure}
\caption{\textbf{4-bits (truncated, 30 qubits, 55 physical qubits)}. Samples under 50 energy after a single execution on D-Wave with 1000 samples. The ground state energy of this problem is 0, depicted as a gray dotted line.}
\label{fig:truncated-results-4}
\end{figure} \fi
\iffalse \begin{figure}
\caption{\textbf{5-bits (truncated, 67 qubits, 167 physical qubits)}. Samples under 100 energy after a single execution on D-Wave with 1000 samples. The ground state energy of this problem is 0, depicted as a gray dotted line.}
\label{fig:truncated-results-5}
\end{figure} \fi
\section{Conclusion} \label{sec:conclusions}
\iffalse TODO:\\ - mention that: the Fukuoka MQ Challenge website \cite{mqchallenge} reports the solution of a binary system of 37 equations and 74 variables (Type I challenge) and another binary system of 69 equations and 105 variables (Type IV challenge). How many qubits would we need to solve a similar challenge with dwave?
-------\fi
Our work is a first step towards demonstrating the efficiency of quantum annealing computations in solving the $\mathsf{MQ}$ problem, using practical experiments on the existing D-Wave quantum annealing platform. We show that we can construct a Hamiltonian with a ground state encoding the solution of the problem and subsequently find it using a quantum annealer. We propose different methods for the embedding of the problem into a Hamiltonian using a polynomial amount of quantum resources. As quantum technology advances, we foresee that the evolution of quantum annealing architectures (e.g., support to n-body interactions or larger coherence times) might provide a quantum advantage when solving such problems as the required numbers of ancilla qubits and required quantum control would decrease.
We have introduced an algorithm that simplifies the problems by fixing ancillary qubits that are easy to find for the quantum device in order to more reliably find the ground state of the more complex qubits with finer parameters. This method can help when dealing with large amounts of qubits in near-term devices and could be applied in problems beyond the scope of what is studied in this manuscript.
As an estimate of the quantum resources needed to solve state-of-the-art $\mathsf{MQ}$ problems, we refer to the Fukuoka MQ Challenge website \cite{mqchallenge} where the largest unsolved instances of $\mathsf{MQ}$ problems can be found. The Type I challenge problem instance of 37 equations and 74 variables would require an estimate of under 80000 logical qubits for its solution to be mapped into the ground state of a two-body Hamiltonian. For the Type IV challenge with 69 equations and 105 variables, one would require under 300000 logical qubits for the embedding. Quantum annealers are still far away from being able to tackle the problems at the edge of what is classically solvable, but quantum technologies are still emerging and new devices with more, and higher quality, qubits are being currently developed.
\section*{Acknowledgments} The authors would like to thank Andre Esser for his insights. M.M. is partially supported by the TRUSTIND project, under the grant agreement KK-2020/00054, from the Department of Economic Development and Infrastructures of the Basque Government. M.M. is also a member of the Intelligent Systems for Industrial Systems research group of Mondragon Unibertsitatea (IT1676-22), supported by the Department of Education, Universities and Research of the Basque Country.
\iffalse
\appendix \onecolumngrid \begin{center} \Large{Supplementary Material for \\ ``Adiabatic quantum computation for circuit codes''} \end{center}
\twocolumngrid \fi
\end{document}
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arXiv
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Descent along torsors
In mathematics, given a G-torsor X → Y and a stack F, the descent along torsors says there is a canonical equivalence between F(Y), the category of Y-points and F(X)G, the category of G-equivariant X-points.[1] It is a basic example of descent, since it says the "equivariant data" (which is an additional data) allows one to "descend" from X to Y.
When G is the Galois group of a finite Galois extension L/K, for the G-torsor $\operatorname {Spec} L\to \operatorname {Spec} K$, this generalizes classical Galois descent (cf. field of definition).
For example, one can take F to be the stack of quasi-coherent sheaves (in an appropriate topology). Then F(X)G consists of equivariant sheaves on X; thus, the descent in this case says that to give an equivariant sheaf on X is to give a sheaf on the quotient X/G.
Notes
1. Vistoli 2008, Theorem 4.46
References
• Vistoli, Angelo (September 2, 2008). "Notes on Grothendieck topologies, fibered categories and descent theory" (PDF).
• Algebraic Geometry I: Schemes. Springer Studium Mathematik - Master. 2020. doi:10.1007/978-3-658-30733-2. ISBN 978-3-658-30732-5. S2CID 124918611.
External links
• Stack of Tannakian categories? Galois descent?
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Wikipedia
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ChromTime: modeling spatio-temporal dynamics of chromatin marks
Petko Fiziev1,2,3 &
Jason Ernst ORCID: orcid.org/0000-0003-4026-78531,2,3,4,5,6
To model spatial changes of chromatin mark peaks over time we develop and apply ChromTime, a computational method that predicts peaks to be either expanding, contracting, or holding steady between time points. Predicted expanding and contracting peaks can mark regulatory regions associated with transcription factor binding and gene expression changes. Spatial dynamics of peaks provide information about gene expression changes beyond localized signal density changes. ChromTime detects asymmetric expansions and contractions, which for some marks associate with the direction of transcription. ChromTime facilitates the analysis of time course chromatin data in a range of biological systems.
Genome-wide mapping of histone modifications (HMs) and related chromatin marks using chromatin immunoprecipitation coupled with high-throughput sequencing (ChIP-seq) and DNA accessibility through assays for DNase I hypersensitivity (DNase-seq) or transposase-accessible chromatin (ATAC-seq) assays have emerged as a powerful approach to annotate genomes and study cell states [1,2,5]. Through the efforts of large consortia projects such as ENCODE [6], Roadmap Epigenomics [7], and BLUEPRINT [8] as well as individual labs [9,10,11], multiple different chromatin marks have been mapped across more than a hundred different cell and tissue types. These maps have yielded numerous insights into gene regulation and genetic and epigenetic association with disease [12,13,14,15,16].
While many mapping efforts have largely focused on single or unrelated cell and tissue types [3, 6], a growing number of biological processes have been studied with temporal epigenomic data using assays such as ChIP-seq, ATAC-seq, or DNase-seq over a time course, which map chromatin marks at consecutive stages during the particular biological process. Such datasets have been generated for a wide range of biological settings, including T-cell development [17], adipogenesis [18], hematopoiesis [19, 20], macrophage differentiation [21], neural differentiation [12], cardiac development [22, 23], somatic cell reprogramming [24,25,26,27], embryogenesis [28], and many others [7, 29,30,31,32,33,34,35,36,37]. The output of these experiments presents a unique opportunity to study the spatio-temporal changes of epigenetic peaks and associated regulatory elements. However, almost all computational methods designed or applied to epigenomic data have been developed based on single or multiple unrelated samples. For example, continuous regions of enrichments of single marks are detected by peak or domain calling methods [38,39,40,41,42]. In cases when multiple chromatin marks are mapped in the same cell type, methods such as ChromHMM [43] and Segway [44] can be used to produce genome-wide chromatin state annotations. In addition, methods have been developed for pairwise comparisons of ChIP-seq signal data by differential peak calling [45, 46].
In the context of time course chromatin data, only a few methods have been proposed that consider temporal dependencies between samples. One such method, TreeHMM [47], produces a chromatin state genome annotation similar to ChromHMM and Segway, while taking into account a tree-like structure that captures lineage relationships between the input cell types in order to potentially derive a more consistent annotation across samples. Another method, GATE [30], produces a genome annotation based on clustering fixed-length genomic loci that can be modeled with the same switch from one chromatin state to another over time.
One important limitation of methods for pairwise comparison or time course modeling of chromatin data is that they do not directly consider or model spatial changes in the genomic territory occupied by chromatin marks over time. Spatial properties of genomic peaks continuously marked by HMs have gained increasing attention as a potentially important characteristic of chromatin marks. For example, long peaks of H3K27ac have been associated with active cell type-specific locus control regions termed super-enhancers or stretch enhancers in a number of cell types [48, 49]. Also, the length of H3K4me3 peaks has been associated with transcriptional elongation and consistency of cell identity genes [50]. In the context of cancer, long H3K4me3 peaks have been linked to transcriptional elongation and enhancer activity at tumor suppressor genes and have been observed to be significantly shortened in tumor cells [51]. Long H3K4me3 domains have been implicated to mark loci involved in psychiatric disorders [52]. Expanded domains of H3K27me3 and H3K9me3 marks have been shown to be characteristic of terminally differentiated cells compared to stem cells [53]. These studies suggest that length of epigenetic peaks is a dynamic feature that can correlate with activity of putative functional elements regulating specific genes. Computational methods that do not explicitly reason about the spatial changes of chromatin marks have significant limitations for studying the dynamics of these properties because they are unable to detect some territorial changes that might be associated with redistribution of signal or identify asymmetric directional peak boundary movements.
In this work, we present ChromTime, a novel computational method for detection of expanding, contracting, and steady peaks, which can detect patterns of changes in the genomic territory occupied by chromatin mark peaks from time course sequencing data (Fig. 1a). We applied ChromTime to a diverse set of data from different developmental, differentiation, and reprogramming time courses (Table 1). Predicted expansions and contractions in general mark regulatory regions associated with changes in transcription factor (TF) binding or gene expression. ChromTime enables studying the directionality of spatial dynamics of chromatin mark peaks relative to other genomic features, which existing computational approaches do not directly address. Our results show that the direction of predicted expansions and contractions correlates with direction of transcription near transcription start sites (TSSs). ChromTime is a general method that can be used to analyze time course chromatin data from high-throughput sequencing assays such as from ChIP-seq, ATAC-seq, and DNase-seq for a wide range of biological systems to gain insights into the dynamics of gene regulation.
Overview of the ChromTime method. a Examples of H3K4me2 peaks with steady, expanding, and contracting boundary dynamics, shown from left to right, respectively, across five time points during mouse T-cell development [17]. Time points 1, 2, and 3 correspond to in vitro differentiated T-cell precursors (FLDN1, FLDN2a, and FLDN2b), whereas time points 4 and 5 correspond to in vivo purified thymocytes (ThyDN3 and ThyDP). Normalized ChIP-seq signal, MACS2 [38] peaks (black rectangles), and ChromTime output are shown for each time point. Peaks upstream of the Zfp148 gene are called steady by ChromTime despite fluctuations of MACS2 peak boundaries. In contrast, ChromTime calls a peak at the Skap1/GM11529 promoter to expand after time points 2 and 3. Conversely, ChromTime calls a peak upstream of the GPR141 gene to contract after time points 2, 3, and 4. b Overview of the ChromTime method. During the block-finding stage, input foreground and, optionally, control reads are used to determine blocks of signal enrichment. In the dynamics prediction stage, for each block, peak boundary positions are predicted at each time point and peak boundary dynamics are predicted at each pair of consecutive time points. c Predicting dynamics for one block. Boxes represent genomic bins at each time point. Foreground signal is depicted as blue bars for each bin whose height represents the number of reads mapped to the bin. ChromTime learns a probabilistic mixture model from the input data to partition each block at each time point into peak and background components. Bins in the peak component (orange) mark peaks of signal enrichment whereas those in the background component (white) mark flanking background signal. The movement of the boundaries on the left and the right side of peaks between consecutive time points is estimated by reasoning jointly about the input data from all time points
Table 1 Datasets used for analysis with ChromTime
Model for detecting expanding, contracting, and steady peaks from temporal chromatin data
We developed a computational method, ChromTime (https://github.com/ernstlab/ChromTime), designed for systematic detection of expansions, contractions, and steady peaks from time course chromatin data of a single chromatin mark ("Methods"; Fig. 1b). ChromTime takes as input a set of genomic coordinates of aligned sequencing reads from foreground experiments for a chromatin mark and, optionally, control experiments over the time course. The foreground experiments are data from a chromatin sequencing assay such as ChIP-seq, ATAC-seq, or DNase-seq performed at a series of time points. The method consists of two stages—block finding and dynamics prediction. During the block finding stage, ChromTime determines continuous genomic regions (blocks) that may contain peaks of foreground signal enrichment during the time course (Additional file 1: Figure S1A, B). To achieve this, ChromTime partitions the genome into fixed length bins and counts the number of foreground and control reads that map to each bin at each time point. Nearby bins that show significant enrichment are joined into continuous intervals, which subsequently are grouped into blocks if they overlap across time points. As a result, large portions of the genome that are likely to contain background noise at all time points are filtered out, so that peak boundary dynamics are determined within a subset of the genome potentially enriched for the chromatin mark.
During the dynamics prediction stage, for each block, ChromTime determines the most likely positions of the peak boundaries at each time point and whether the peak expands, contracts, or holds steady at each boundary between consecutive time points. The method uses a probabilistic mixture model to partition the signal within each block at each time point into background and peak components (Fig. 1c, Additional file 1: Figure S1C) by reasoning jointly about the data from all time points in the time course. The method assumes that central positions in blocks are more likely to be enriched for foreground reads and thus the peak component is flanked by the background components (Additional file 1: Figure S1D). The number of sequencing reads in bins from each component at each time point is modeled with different negative binomial distributions that can account for the local abundance of control reads. Furthermore, between any two consecutive time points the boundaries of the peaks are assumed to follow one of three possible dynamics: steady, expand, or contract. For steady dynamics, the peak boundaries are enforced to have the same genomic position. For expanding and contracting dynamics, the number of genomic bins that the peak boundaries move between the two time points is modeled with different negative binomial distributions which depend on the pair of time points and the corresponding dynamic. ChromTime models time points that have no bins in the peak component with zero length peaks. Thus, appearances of peaks, except at the first time point, are modeled as expansions from zero length peaks and the disappearances of peaks are modeled as contractions to zero length peaks. Each dynamic is also assumed to have a prior probability which captures information about its genome-wide frequency at each time point.
All model parameters are learned jointly from the whole time course. As a result, ChromTime can adapt to different boundary movements, dynamics frequencies, and noise levels across experiments and biological systems. The estimated parameters are used to make a prediction for each block for the most likely positions of the peak boundaries and the corresponding boundary dynamics that had generated the signal within the block. The final output contains predicted peak boundaries annotated and colored by their assigned dynamics, which can be used for downstream analysis with existing tools and visualized in genome browsers (Fig. 2, Additional file 1: Figure S2; https://github.com/ernstlab/ChromTime).
Sample output from ChromTime with contracting peaks. Genome browser screenshot with sample output of ChromTime for H3K4me2 from the T-cell development time course in mouse [17] with five time points at the Esam/Vsig2/Nrgn locus. Time points 1, 2, and 3 correspond to in vitro differentiated T-cell precursors (FLDN1, FLDN2a, and FLDN2b), whereas time points 4 and 5 correspond to in vivo purified thymocytes (ThyDN3 and ThyDP). The input ChIP-seq signal and MACS2 [38] peaks (black boxes under each signal track) are shown in the upper panel of the screenshot. The ChromTime-predicted peaks colored by their boundary dynamics for each block at each time point are shown in the bottom panel. The first peak in each block is colored in dark gray. Each subsequent peak is colored with respect to the predicted dynamic relative to its previous time point. Peaks with steady boundaries on both sides are shown in light gray, and those with at least one contracting boundary are shown in blue. Nearby peaks that touch boundaries are visualized as one peak by the genome browser. Not shown in the figure are expanding peaks, peaks at single time points, and peaks with opposite dynamics (EXPAND on the left and CONTRACT on the right, or vice versa), which would be colored in red, orange, and black, respectively. See Additional file 1: Figure S2 for examples of predicted expanding peaks
Reproducibility of ChromTime predictions and association with changes in gene expression, TF binding, and DNaseI hypersensitivity sites
To investigate the reproducibility of ChromTime predictions, we applied ChromTime separately to two biological replicate datasets for the H3K4me2 and H3K(9/14)ac marks in T-cell development in mouse [17] and confirmed, on average, strong enrichment for the same ChromTime annotations co-localizing across replicates (Additional file 1: Figure S3). We then applied the method to data from pooled replicates for the H3K4me2 mark from the mouse T-cell development study [17], to data for the H3K4me3 and H3K27ac marks from a study on stem cell reprogramming in human [24], to ATAC-seq data from a mouse stem cell reprogramming time course [27], and to a human fetal brain development time course that we constructed from DNase-seq datasets from Roadmap Epigenomics [7]. To investigate the biological relevance of ChromTime predictions, for blocks overlapping TSSs we examined changes in the corresponding gene expression. Peaks with predicted expanding and contracting boundaries that overlap annotated TSSs associated with increases and decreases, respectively, in gene expression (Fig. 3, Additional file 1: Figure S4). Additionally, for all chromatin marks we examined enrichments of TF binding sites across all blocks [6, 17, 27], and in the case of HMs, also enrichments of DNaseI hypersensitivity sites (DHSs) [7]. Predicted peaks with expanding and contracting boundaries were enriched for sites bound by important transcriptional regulators in each biological system in a cell type-specific manner. Expanding and contracting HM peaks were also enriched for cell type-specific DHSs. Furthermore, peaks with predicted steady boundaries showed enrichment for TF binding sites that are shared between the first and the last time point in the corresponding time courses, which mark potentially stable regulatory elements. Similar enrichments in the case of HM peaks were also seen for shared DHSs.
Changes in GATA3 binding and gene expression at predicted H3K4me2 dynamics in T-cell development. a Fold enrichments of cell type-specific and shared peaks of GATA3, which is a master regulator in T-cell development [17], are shown for three sets of blocks with predicted H3K4me2 peaks: 1) blocks with peaks present at all time points whose boundaries hold steady on both sides throughout the whole time course (T1-Tn Steady); 2) blocks with non-contracting peaks whose boundaries expand between at least one pair of consecutive time points and have a peak at the last time point (Tx-Tn Expand); and 3) blocks with non-expanding peaks whose boundaries contract between at least one pair of consecutive time points and have a peak at the first time point (T1-Tx Contract). The first column shows the percentage of bases out of all bases covered by peaks of the set. The last row shows the baseline percentage for each feature out of all bases covered by ChromTime peaks at any time point. Percentages are colored from 0 (white) to 100 (green). Fold enrichments in each column are colored from 1 (white) to the maximum value in the column (red). FLDN1 and ThyDP denote differentiated T-cell precursors and purified thymocytes, which are the first and the last time point, respectively. b Boundaries of predicted H3K4me2 peaks in blocks with at least one predicted non-zero length peak overlapping annotated TSSs were sorted in decreasing order by their posterior probability for EXPAND dynamic (left plots) and CONTRACT dynamic (right plots) at each pair of consecutive time points (Additional file 2: Supplementary methods). Gene expression differences between consecutive time points were calculated as the average difference across all genes with overlapping TSSs for each block. For each posterior rank (x-axis) the plot shows the cumulative average gene expression difference across all peak boundaries with equal or higher posterior probabilities (y-axis). Expanding boundaries associated with increase of gene expression and contracting boundaries associated with decrease of gene expression. Shaded regions correspond to 95% confidence intervals
Predicted spatial dynamics by ChromTime associate better with gene expression changes compared to boundary position changes of peaks called from individual time points in isolation
We next investigated whether ChromTime's approach for reasoning jointly about the whole time course increases power to detect associations with gene expression compared to considering boundary differences of peaks at consecutive time points called in isolation. Specifically, we analyzed gene expression changes of genes with TSSs overlapping ChromTime peaks in relation to posterior probabilities for expansions and contractions compared to boundary differences of peaks called with ChromTime from data from individual time points in isolation. We investigated this in the context of H3K4me2 peaks in mouse T-cell development [17] and for H3K4me3 peaks in stem cell reprogramming in human [24]. In most cases, ranking boundary changes of peaks in blocks with at least one non-zero length peak by their predicted ChromTime posterior probabilities for expansions and contractions associated, on average, with larger gene expression changes compared to ranking boundaries directly based on the change in the genomic positions of the boundaries of ChromTime peaks called at individual time points in isolation (Fig. 4, Additional file 1: Figure S5A). These results also held when using peaks from two different peak callers, MACS2 [38] and SICER [40], applied on data from individual time points (Additional file 1: Figure S5B, C).
ChromTime predictions associate better with expression changes than boundary movements of peaks called in isolation. a For H3K4me2 in mouse T-cell development [17] ChromTime was applied once with data from all time points (ChromTime ALL), and once with single time points in isolation (ChromTime SINGLE; Additional file 2: Supplementary methods). Time points 1, 2, and 3 correspond to T-cell precursors, 4 and 5 to purified thymocytes. Peaks called by both procedures overlapping annotated TSSs were analyzed for their relationship with gene expression changes. i Left: Comparison of agreement with expression for expansions when applying ChromTime ALL and ChromTime SINGLE for the change between time points 3 and 4. Peak boundaries were sorted in decreasing order of their EXPAND posterior probabilities from ChromTime ALL and compared to sorting them in decreasing order of the difference of peak boundary positions in ChromTime SINGLE peaks with positive differences in boundary positions indicating peaks expanding with time. Each boundary was also ranked by the average gene expression difference of TSSs overlapping its block in decreasing order with positive expression differences indicating gain with time. The cumulative average boundary rank of expression change (y-axis) is shown for the boundary change ranking for ChromTime ALL and ChromTime SINGLE (x-axis). Low Y-values indicate stronger association with expression changes. Black line shows expected average expression change rank. Shaded regions indicate 95% confidence intervals. Plots for other time points can be found in Additional file 1: Figure S5. Right: Analogous to left plots for contract posterior probabilities for ChromTime ALL, increasing order of the difference of boundary change positions for ChromTime SINGLE, and increasing order of expression changes. ii Differences between ChromTime ALL and ChromTime SINGLE values shown in i between time points 3 and 4 as well as for all other pairs of time points. Positive values correspond to boundary ranks for which ChromTime ALL posteriors better associate with gene expression changes than boundary movements of ChromTime SINGLE peaks. Black lines show expected difference of zero between random rankings. b As in a for H3K4me3 in human stem cell reprogramming [24]. Time points correspond to human inducible and immortalized fibroblast-like (hiF-T) cells, hiF-T at 5, 10, and 20 days after induction, and human induced pluripotent stem cells (hIPSC)
Spatial dynamics contain information about gene expression changes between consecutive time points not captured by corresponding pairwise signal density changes
We next investigated whether there is additional information in ChromTime predictions with respect to gene expression changes beyond what can be captured by pairwise signal density changes or by differential peak calls. For this analysis, we focused on H3K4me2 in mouse T-cell development [17] and H3K4me3 in human stem cell reprogramming [24]. For pairs of consecutive time points, we computed the change in signal density in the region starting at the left-most and ending at the right-most predicted peak boundary in the block (Additional file 2: Supplementary methods). We associated the signal density changes with gene expression changes at the nearest TSS within 50 kb of each block and computed the average gene expression change as a function of the signal density change within blocks (Fig. 5). We found that locations with the same signal density change can associate with significantly different average gene expression changes of proximal genes depending on the predicted ChromTime dynamics. Notably, bidirectional expansions, expansions occurring on both sides of a peak, associated for a range of signal density changes with greater average increase in gene expression than unidirectional expansions, those expansions occurring on one side but steady on the other, when controlling for the signal density change. These unidirectional expansions in turn associated for a range of signal density changes with greater expression change than steady regions, those regions with a steady call on both sides of a peak, when controlling for the signal density change. We observed a similar relationship for contractions and decrease of gene expression. These results were replicated also after substituting ChIP-seq signal density changes with differential peak scores from two differential peak calling methods, SICER [40] and MACS2 [38] (Additional file 1: Figure S6A, B). Therefore, ChromTime predictions can provide additional information about gene expression changes beyond what is contained in the corresponding signal density changes as measured directly or by utilizing differential peak-calling procedures.
Spatial dynamics can contain additional information about gene expression changes beyond signal density changes. Gene expression change is plotted as function of ChIP-seq signal density change after loess smoothing for each predicted ChromTime dynamic for a H3K4me2 dynamics in T-cell development in mouse [17] and b H3K4me3 dynamics in stem cell reprogramming in human [24] (Additional file 2: Supplementary methods). Peaks of each type of dynamics were pooled from all time points in each dataset for this analysis. Peaks with asymmetric dynamics E/S and S/E were pooled together in the "E-S" group. Similarly C/S and S/C peaks were pooled in the "C-S" group. The total number of peaks in each group is shown in parenthesis. In both systems, for a range of signal density changes, peaks with the same signal density change associated with different gene expression changes depending on the predicted spatial dynamic. Shaded regions represent 95% confidence intervals
Spatial dynamics are correlated between multiple chromatin marks
Previous studies have shown that the locations of different chromatin marks can be correlated [3, 54]. In this context, we tested whether multiple chromatin marks can also exhibit jointly the same type of spatio-temporal dynamics. For this purpose, we compared the genomic locations of predicted expansions, contractions, and steady peaks for different chromatin marks within the same time course. We focused on three previously published time courses—stem cell reprogramming in human [24], stem cell reprogramming in mouse [27], and adipogenesis in mouse [18]—where multiple chromatin marks were mapped (Fig. 6, Additional file 1: Figure S7). In all three datasets, we observed that predicted expansions co-localized preferentially for H3K4me2, H3K4me3, and H3K27ac and to a lesser extent for H3K4me1 and similarly for predicted contractions and steady peaks. In contrast, different predicted spatial dynamics for H3K36me3 and H3K27me3 tended to occupy distinct locations. In addition, in mouse reprogramming [27], ChromTime predicted dynamics of ATAC-seq, H3K4me2, H3K4me3, H3K27ac, H3K9ac, and, to a lesser extent, of H3K4me1 and H3K79me2 peaks co-localized preferentially (Additional file 1: Figure S7). These results suggest that spatial dynamics of chromatin marks are coordinated at least at a subset of genomic locations.
Spatial dynamics of multiple different chromatin marks co-localize within a time course. Hierarchical clustering with optimal leaf ordering [80] of the geometric average fold enrichments taken across all time points of the overlap of every pair of predicted spatial dynamics for mapped HMs in a mouse adipogenesis [18] and b human stem cell reprogramming [24]. At each pair of time points, "Expand" and "Contract" dynamics are defined as all peaks that are predicted as either unidirectional or bidirectional expansions and contractions, respectively, whereas "Steady" dynamics are defined as all peaks that have predicted steady boundaries at both sides. Peaks with "Expand" dynamic on one side and "Contract" dynamic on the other were excluded from this analysis. In both datasets, expansions, contractions, and steady peaks of H3K4me2, H3K4me3, and H3K27ac and, to a lesser extent, of H3K4me1 tend to cluster together within each of the three classes, whereas spatial dynamics of H3K27me3 and H3K36me3 peaks tend to occupy different locations. All enrichments were capped at 50 before clustering
Direction of expansions and contractions is correlated with direction of transcription
ChromTime can predict unidirectional expansions and contractions, which enables analysis of directionality of spatial dynamics of peaks, an aspect of chromatin regulation that has not been previously systematically explored. To investigate this, we applied ChromTime on data from 13 previously published studies from a variety of developmental, differentiation, and reprogramming processes (Table 1) for nine different HMs, including narrow and broad marks, and for Pol2, ATAC-seq, and DNase-seq. We observed that unidirectional expansions and contractions are predicted in most cases, on average, to be the majority of all expansions and contractions, respectively, at a given pair of consecutive time points (Additional file 1: Figure S8). One hypothesis for the prevalence of asymmetric boundary movements for the promoter-associated chromatin marks is that the direction of boundary movements is associated with the asymmetry of transcription initiation in promoter regions. To test this hypothesis, for each dataset we compared the prevalence of each class of unidirectional dynamics as a function of its distance to the nearest annotated TSS and the orientation of the corresponding gene (Fig. 7). Consistent with our hypothesis, for H3K4me3, H3K4me2, H3K(9/14)ac, H3K79me2, and Pol2, we found that unidirectional expansions that expand into the gene body (i.e., in the same direction as transcription) were substantially more frequently found in proximity of TSSs compared to unidirectional expansions in the opposite direction. Moreover, this difference was not observed for expansions that are distal from TSSs. Similarly, in most cases for these marks unidirectional contractions that contract towards the TSS of the nearest gene (i.e., in the opposite direction of transcription) were substantially more frequent compared to unidirectional contractions in the opposite direction in proximity of TSSs, whereas their frequencies at distal sites showed much smaller differences. HMs H3K27ac, H3K4me1, and H3K27me3 and ATAC-seq and DNase-seq exhibited the same trend, but to a lesser degree.
Direction of asymmetric dynamics correlates with direction of transcription. a i Left panel shows a schematic representation of unidirectional expansions that expand in the same direction as transcription and in the opposite direction of transcription. The adjacent plots show, for each mark, the average log2 ratio across all time points in each time course between the fraction of unidirectional expansions that expand in the same directions as transcription of the nearest gene and the fraction of unidirectional expansions that expand in the opposite direction of transcription of the nearest gene, separately for blocks that are within 1 kb of annotated TSSs and for more distal blocks. Positive values correspond to enrichment of unidirectional expansions in the same direction as transcription. For marks mapped in at least six time courses, a black line is plotted representing the average across all data sets and significant differences are denoted with asterisks based on a two-tailed Mann-Whitney test at a P value threshold of 0.05. ii Left panel shows analogous schematic for unidirectional contractions. Likewise, adjacent plots show, for each mark, the average log2 ratio between the fraction of unidirectional contractions that contract in the opposite direction of transcription of the nearest TSS and unidirectional contractions that contract in the same direction as transcription of the nearest TSS. b Left panel shows an example of unidirectional expansions between pairs of time points that expand in the same direction as transcription at the Hs6st1 gene of the H3K4me2 mark in the T-cell development dataset [17]. Right panel shows an example of unidirectional contractions in the opposite direction of transcription at the DNMT3B gene. Time points 1, 2, and 3 correspond to in vitro differentiated T-cell precursors, whereas time points 4 and 5 correspond to in vivo purified thymocytes. The predicted ChromTime peaks colored by their boundary dynamics are shown under the signal track for each time point
In this work, we presented ChromTime, a novel computational method for systematic detection of expanding, contracting, and steady peaks of chromatin marks from time course high-throughput sequencing data. ChromTime employs a probabilistic graphical model that directly models changes in the genomic territory occupied by single chromatin marks over time. This approach allowed us to directly encode our modeling assumptions about dependencies between variables in an interpretable and extendable framework.
We applied ChromTime on ChIP-seq data for broad and narrow HMs and for Pol2, and on ATAC-seq and DNase-seq data from a variety of developmental, differentiation, and reprogramming courses. Our results show that the method can identify sets of expanding and contracting peaks that are biologically relevant to the corresponding systems. In particular, expansions and contractions associate with up- and down-regulation of gene expression and differential TF binding, supporting the biological relevance of ChromTime predictions.
ChromTime gains power by both reasoning jointly about all time points in a time course and by explicitly modeling the peak boundary movements. Supporting this, in our analyses we observed that territorial changes identified by ChromTime had better agreement with gene expression changes compared to considering directly the boundary change of peaks called on data from individual time points in isolation. Additionally, we also observed for a range of cases that expanding and contracting peaks associated, on average, with greater change in gene expression compared to peaks with steady boundaries even after controlling for signal density changes. Some of the power that ChromTime gains from considering spatial information might be explained by its ability to differentiate territorial expansions or contractions, which can reflect changes in the number of TF binding sites in close vicinity, from changes in signal density within steady peak boundaries. Changes in signal density without territorial expansions or contractions might reflect a change in the proportion of cells with the chromatin mark without large changes in activity in any one cell. Additional power can come from the temporal and spatial information that allows the model to effectively smooth over noise in the data, thus enabling more biologically relevant inferences.
ChromTime enables novel analysis of directionality of spatial epigenetic dynamics. In this context, we found that asymmetric unidirectional expansions and contractions for several marks correlate strongly with direction of transcription in promoter proximal regions, which suggests that spatial dynamics at such locations may be related to actions of the transcriptional machinery. One possible explanation for the observed correlation between the direction of spatial dynamics of at least some HMs and transcription can be provided in part by previous studies that have shown that the Pol2 elongation machinery can recruit H3K4-methyltransferases, such as members of the SET [55] and MLL [56] families, at the promoters of genes. Our findings are consistent with such models where the Pol2 complex itself may be facilitating the attachment and removal of these marks [57].
The ChromTime software is also relatively efficient in terms of runtime, particularly when using its option to parallelize all computations during the parameter learning and prediction phases over multiple CPU cores. In our tests, processing ChIP-seq data for the H3K4me2 mark and control data from five time points in mouse T-cell development [17] took 3 h on a laptop computer using four CPU cores.
We applied ChromTime to a range of data types but found no single setting of the method options to be preferable in all cases ("Methods"). We thus created three modes with different default options: punctate mode used for ATAC-seq and DNase-seq, narrow mode used for ChIP-seq of narrow HMs, and broad mode used for ChIP-seq of broad HMs and Pol2. In principle, ChromTime can also be applied on ChIP-seq data of sequence-specific TFs in punctate mode. However, for these data, where binding can often be associated with a single point source such as individual instances of DNA sequence regulatory motifs, methods that predict the single point source across time points and the binding intensity associated with the source at each time point may be a more natural way to model the data.
Another limitation of the ChromTime method is that while the runtime of ChromTime still scales linearly with the number of time points, T, the number of observed combinations of dynamics can scale exponentially with T. This exponential growth can complicate downstream analyses that directly consider each combination of dynamics, as there will be 3T-1 possible sequences of dynamics at each side of a peak. Extensions of the ChromTime model could model the large number of combinations as being instances of a smaller number of more distinct dynamic patterns.
The increasing availability of time course chromatin data provides an opportunity to understand chromatin dynamics in many biological systems. To facilitate reaching this goal we developed ChromTime, which systematically detects expanding, contracting, and steady peaks, allowing extraction of additional information from these data. ChromTime gains power by both reasoning about data from all time points in the time course and by explicitly modeling movements of peak boundaries. We showed that ChromTime predictions associate with relevant genomic features such as changes in gene expression and TF binding. We demonstrated that territorial changes of peaks can contain additional information beyond signal density changes with respect to gene expression of proximal genes. ChromTime allows for novel analysis of directionality of spatial dynamics of chromatin marks. In this context, we showed for multiple chromatin marks that the direction of predicted asymmetric expansions and contractions of peaks strongly associates with direction of transcription in proximity of TSSs. ChromTime is generally applicable to modeling time courses of chromatin marks and thus should be a useful tool to gaining insights into dynamics of epigenetic gene regulation in a range of biological systems.
Overview of the ChromTime method
ChromTime takes as input a set of files in BED format with genomic coordinates of aligned sequencing reads from experiments for a single chromatin mark from a high-throughput sequencing experiment such as ChIP-seq, ATAC-seq, or DNase-seq over a time course and, optionally, from a set of control experiments. ChromTime consists of two stages (Fig. 1b, c):
Detecting genomic intervals (blocks) potentially containing regions of signal enrichment (peaks)
Learning a probabilistic mixture model for boundary dynamics of peaks within blocks throughout the time course and computing the most likely spatial dynamic and peak boundaries for each block throughout the whole time course
Detecting genomic blocks containing regions of signal enrichment
The aim of this stage is to determine approximately the genomic coordinates of regions with potential peaks of signal enrichment at any time point in the time course (Additional file 1: Figure S1A, B). The signal within these blocks will be used as input to build the mixture model in the next stage of ChromTime. ChromTime supports analysis of punctate, narrow, and broad marks in three different modes, which are defined by different default options. The method partitions the genome into non-overlapping bins of predefined length, BIN_SIZE (by default, 200 bp in narrow and punctate modes, 500 bp in broad mode) and counts for each bin and time point the number of sequencing reads whose alignment starting positions after shifting by a predefined number of bases (SHIFT, 100 bp in the direction of alignment by default) are within its boundaries. Next, each bin at each time point is tested for enrichment based on a Poisson background distribution at a predefined false discovery rate (FDR; 0.05 by default). The expected number of reads for a bin at position p and time point t, λt,p, in the Poisson test is computed conservatively as the maximum of:
If control reads are provided: for each window of size w = 1000 bp, 5000 bp, and 20,000 bp the average number of control reads in the window centered at the current bin, normalized by the ratio of total reads in the foreground and control experiments, that is:
$$ {\lambda}_{t,p,w}=\frac{\#\left[\mathrm{Total}\ \mathrm{Foreground}\ \mathrm{Reads}\right]}{\#\left[\mathrm{Total}\ \mathrm{Control}\ \mathrm{Reads}\right]}\frac{\mathrm{BIN}\_\mathrm{SIZE}}{w}{\mathrm{Ctrl}}_{t,p,w} $$
where Ctrlt,p,w is the total number of control reads in each window of size w around the bin at position p at time point t.
The average number of foreground reads per genomic bin.
One read.
Testing multiple different window sizes for the background is a strategy we adopted from the MACS2 peak caller [38].
Within each time point, consecutive bins that are significantly enriched are merged into continuous intervals. The intervals are further extended in both directions to include continuous stretches of bins where each bin is significant based on a Poisson background distribution at a weaker P value threshold (0.15 by default). Extended intervals within a predefined number of non-significant bins, MAX_GAP (3 bins by default), are further joined together. This joining strategy has been previously implemented by other peak callers for single datasets such as SICER [40]. Next, overlapping intervals across time points are grouped into blocks. To capture more of the potential background signal and to increase the likelihood that central bins within blocks contain higher foreground signal, the start and end positions of each block are extended additionally by a predefined number of bins, BLOCK_EXTEND (5 bins by default), upstream of the left-most coordinate and downstream of the right-most coordinate of the intervals in the block, respectively, or up to the middle point between the current block and its adjacent blocks if they are within BLOCK_EXTEND bins apart. Restricting BLOCK_EXTEND to a relatively limited number of bins helps to keep the running time of the method within reasonable bounds.
In narrow and punctate modes, blocks that contain multiple intervals at the same time point separated by gaps of non-significant bins longer than MAX_GAP are split into sub-blocks at each gap between those intervals. In particular, all gaps within a block are intersected across the time points that have gaps. For each gap intersection, the block is split at the position with the lowest average foreground signal across all time points. In broad mode, no such splitting is performed in order to avoid excessive peak fragmentation.
Probabilistic mixture model for boundary dynamics of peaks within blocks across the time course
The foreground and the expected signal within the blocks are used as input to build a probabilistic mixture model for the boundary dynamics of the peaks within blocks (Additional file 1: Figure S1C). One core assumption of the model is that each block contains at each time point exactly one peak, which can potentially have a length of zero bins. This implies that, at each time point, the bins within a block can be partitioned into three continuous intervals: left-flanking background, foreground peak, and right-flanking background. For the bin in block i, at time point t and position p, let Oi,t,p denote the random variable that models the number of observed foreground reads, and let oi,t,p denote the corresponding observed read counts. Let Vi,t,p denote the random variable for the label of the corresponding bin, which can either have the value PEAK or BACKGROUND. Let Xi,t,p denote a random variable for the vector of covariates for the corresponding bin, and xi,t,p their corresponding values. The distribution of Oi,t,p conditioned on Vi,t,p and Xi,t,p is modeled with different negative binomial distributions depending on the value of Vi,t,p and xi,t,p:
$$ P\left({O}_{i,t,p}={o}_{i,t,p}|{V}_{i,t,p}=\mathrm{PEAK},{X}_{i,t,p}={x}_{i,t,p}\right)=\mathrm{NB}\left({o}_{i,t,p};{\mu}_{\mathrm{PEAK},i,t,p},{\delta}_t\right)=\frac{\Gamma \left({o}_{i,t,p}+{\delta}_t\right)}{o_{i,t,p}!\Gamma \left({\delta}_t\right)}\times {\left(\frac{\delta_t}{\mu_{\mathrm{PEAK},i,t,p}+{\delta}_t}\right)}^{\delta_t}\times {\left(\frac{\mu_{\mathrm{PEAK},i,t,p}}{\mu_{\mathrm{PEAK},i,t,p}+{\delta}_t}\right)}^{o_{i,t,p}} $$
$$ P\left({O}_{i,t,p}={o}_{i,t,p}|{V}_{i,t,p}=\mathrm{BACKGROUND},{X}_{i,t,p}={x}_{i,t,p}\right)=\mathrm{NB}\left({o}_{i,t,p};{\mu}_{\mathrm{BACKGROUND},i,t,p},{\delta}_t\right)=\frac{\Gamma \left({o}_{i,t,p}+{\delta}_t\right)}{o_{i,t,p}!\Gamma \left({\delta}_t\right)}\times {\left(\frac{\delta_t}{\mu_{\mathrm{BACKGROUND},i,t,p}+{\delta}_t}\right)}^{\delta_t}\times {\left(\frac{\mu_{\mathrm{BACKGROUND},i,t,p}}{\mu_{\mathrm{BACKGROUND},i,t,p}+{\delta}_t}\right)}^{o_{i,t,p}} $$
where δt is the dispersion parameter. Similarly to negative binomial regression models [58], ChromTime models the mean of each component through the log link as a linear combination of a two-dimensional vector of covariates, xi, t,p = (1, log λi,t,p), which includes a constant term and the logarithm of the expected number of reads in the bin as computed in the previous section:
$$ {\mu}_{\mathrm{PEAK},i,t,p}=\exp \left[{\alpha}_t+{\gamma}_t\log {\lambda}_{i,t,p}\right] $$
$$ {\mu}_{\mathrm{BACKGROUND},i,t,p}=\exp \left[{\beta}_t+{\gamma}_t\log {\lambda}_{i,t,p}\right] $$
where αt, βt and γt are time point-specific scalar parameters. Negative binomial distributions have been successfully employed in a similar manner to capture the over-dispersion of sequencing reads in peak callers for single samples such as ZINBA [59]. Of note, however, ChromTime requires that the dispersion parameter δt and the coefficient γt are shared between the two components at each time point. The first requirement ensures that the distribution with the smaller mean value has higher probabilities compared to the distribution with the larger mean value for the lowest values of the support domain of the negative binomial distribution, and that the opposite holds for the largest values of the support domain (Additional file 2: Supplementary methods). Sharing the dispersion parameter here is analogous to sharing the variance parameter in Gaussian mixture models. The second requirement to share the γt parameter ensures that the control signal has equal importance in each component.
Formally, let Bi,L,t and Bi,R,t denote the random variables corresponding to the first and the last bin, respectively, in the peak partition at time t for block i relative to the beginning of the block, and let Ni be the length of the block. We then have 1 ≤ Bi,L,t ≤ Ni + 1 and 0 ≤ Bi,R,t ≤ Ni, with values of Bi,L,t = Ni + 1 and Bi,R,t = 0 corresponding to the special cases of starting a peak after all positions and ending a peak before all positions in a block, respectively. For Bi,L,t and Bi,R,t to denote valid interval boundaries, ChromTime also requires that Bi,L,t ≤ Bi,R,t + 1 at each time point. These constraints can be formally encoded by introducing one auxiliary binary variable for each time point in the model, Zi,t, such that:
$$ P\left({Z}_{i,t}=1|{B}_{i,L,t}=l,{B}_{i,R,t}=r\right)=\left\{\ \begin{array}{ll}1&\ \mathrm{if}\ 1\le l\le r+1\le {N}_i+1\\ {}0&\ \mathrm{otherwise}\end{array}\right. $$
and thus also
ChromTime treats all Zi,t variables as observed with values equal to 1 for all blocks and time points.
The conditional probability of the bin labels, Vi,t,p, given the peak boundaries, Bi,L,t and Bi,R,t, are defined to be:
$$ P\left({V}_{i,t,p}=\mathrm{PEAK}|{B}_{i,L,t}=l,{B}_{i,R,t}=r\right)=\left\{\ \begin{array}{ll}1&\ \mathrm{if}\ l\le p\le r\\ {}0&\ \mathrm{otherwise}\end{array}\right. $$
$$ P\left({V}_{i,t,p}=\mathrm{BACKGROUND}|{B}_{i,L,t}=l,{B}_{i,R,t}=r\right)=\left\{\ \begin{array}{ll}0&\ \mathrm{if}\ l\le p\le r\\ {}1&\ \mathrm{otherwise}\end{array}\right. $$
The probability of the observed read counts at time t, oi,t, and Zi,t = 1, conditioned on the values of the peak boundaries, Bi,L,t and Bi,R,t, and the covariates at time point t, xi,t, under the model is then:
$$ {\displaystyle \begin{array}{l}P\left({\boldsymbol{O}}_{i,t}={\boldsymbol{o}}_{i,t},{Z}_{i,t}=1|{B}_{i,L,t}=l,{B}_{i,R,t}=r,{\boldsymbol{X}}_{i,t}={\boldsymbol{x}}_{i,t}\right)\\ {}=P\left({Z}_{i,t}=1|{B}_{i,L,t}=l,{B}_{i,R,t}=r\right)\\ {}\times \prod \limits_{p=1}^{l-1}\mathrm{NB}\left({o}_{i,t,p};{\mu}_{\mathrm{BACKGROUND},i,t,p}=\exp\ \left[{\beta}_t+{\gamma}_t\log\ {\lambda}_{i,t,p}\right],{\delta}_t\right)\\ {}\times \prod \limits_{p=l}^r\mathrm{NB}\left({o}_{i,t,p};{\mu}_{\mathrm{PEAK},i,t,p}=\exp\ \left[{\alpha}_t+{\gamma}_t\log\ {\lambda}_{i,t,p}\right],{\delta}_t\right)\\ {}\times \prod \limits_{p=r+1}^{N_i}\mathrm{NB}\left({o}_{i,t,p};{\mu}_{\mathrm{BACKGROUND},i,t,p}=\exp\ \left[{\beta}_t+{\gamma}_t\log\ {\lambda}_{i,t,p}\right],{\delta}_t\right)\end{array}} $$
An important special case of the above formulation when Bi,L,t = Bi,R,t + 1 corresponds to modeling the whole signal at time point t as background, which enables ChromTime to accommodate time points that are all background by modeling them with zero length peaks. For this reason, ChromTime blocks internally have the same number of peak boundaries at all time points even if some time points are predicted as zero length peaks (i.e., all background). Boundaries of zero length peaks are treated by the model in the same way as boundaries of non-zero length peaks.
ChromTime assumes uniform prior probabilities for the left and the right end boundaries at the first time point:
$$ P\left({B}_{i,L,1}=l\right)= Unif\left(1,{N}_i+1\right) $$
$$ P\left({B}_{i,R,1}=r\right)= Unif\left(0,{N}_i\right) $$
where Unif(a, b) denotes the uniform distribution of integer numbers in the closed interval [a, b].
Let Di,s,t denote the dynamic between time points t and t + 1 on boundary side s, where s is one of L (left side) or R (right side). Between any two time points the ChromTime model allows for one of three possible dynamics at both the left and the right end boundaries of a peak: STEADY, EXPAND, or CONTRACT. To capture the change of boundary positions between consecutive time points t and t + 1 we define the quantities Ji,L,t = Bi,L,t − Bi,L,t + 1 and Ji,R,t = Bi,R,t + 1 − Bi,R,t corresponding to the left and right boundaries, respectively. Positive values of Ji,L,t and Ji,R,t indicate the number of bases a peak expanded, whereas negative values indicate the number of bases a peak contracted, and a value of 0 indicates that the peak held steady on the left and the right side, respectively. ChromTime models Ji,L,t and Ji,R,t with different probability distributions for each of the three dynamics. For STEADY dynamic, ChromTime uses the Kronecker delta function:
$$ P\left({J}_{i,s,t}|{D}_{i,s,t}=\mathrm{STEADY}\right)=\left\{\begin{array}{c}1, if\ {J}_{i,s,t}=0\\ {}0, otherwise\end{array}\right. $$
For expanding and contracting dynamics, ChromTime employs negative binomial distributions to model the number of genomic bins a peak boundary moves relative to the minimal movement of one bin required for peak expansions and contractions:
$$ P\left({J}_{i,s,t}=j|{D}_{i,s,t}=\mathrm{EXPAND}\right)= NB\left(j-1;{\mu}_{\mathrm{EXPAND},\mathrm{t}},{\delta}_{\mathrm{EXPAND},\mathrm{t}}\right) $$
$$ P\left({J}_{i,s,t}=j|{D}_{i,s,t}=\mathrm{CONTRACT}\right)=N\mathrm{B}\left(-j-1;{\mu}_{\mathrm{CONTRACT},\mathrm{t}},{\delta}_{\mathrm{CONTRACT},\mathrm{t}}\right) $$
Furthermore, each distribution is parametrized with a mean and dispersion parameter depending on the dynamic and the time point, t: μEXPAND,t, δEXPAND,t for expansions, and μCONTRACT,t, δCONTRACT,t for contractions. Of note, in negative binomial distributions the probabilities for negative integers are defined to be 0. Therefore, the above parametrization enforces that boundary movements of negative or zero length (i.e., contracting or steady, respectively) are impossible for expansions and that boundary movements of positive or zero length (i.e., expanding or steady) are impossible for contractions.
The ChromTime model additionally assumes that there is a prior probability to observe each dynamic between time points t and t + 1, P(Di,s,t = d) = πt,d, which is the same at each side (left and right). Users have the option to set a minimum prior probability (MIN_PRIOR) for the dynamics for all time points. This parameter can be used to avoid learning priors too close to zero, which in some cases can occur for more punctate marks where the short length of the peaks can cause the prior to become a dominant influence on the class assignment of the spatial dynamics. By default, MIN_PRIOR = 0 in narrow and broad modes and MIN_PRIOR = 0.05 in punctate mode.
For a time course with T time points we can express for block i the probability of a particular sequence of dynamics and boundary positions on the left side (dL and bL, respectively) and on the right side (dR and bR, respectively), and observing foreground counts oi and Zi = 1 conditioned on the values of the covariates, xi as:
$$ P\left({\boldsymbol{D}}_{i,L}={\boldsymbol{d}}_L,{\boldsymbol{B}}_{i,L}={\boldsymbol{b}}_L,{\boldsymbol{D}}_{i,R}={\boldsymbol{d}}_R,{\boldsymbol{B}}_{i,R}={\boldsymbol{b}}_R,{\boldsymbol{O}}_i={\boldsymbol{o}}_i,{\boldsymbol{Z}}_i=\mathbf{1}|{\boldsymbol{X}}_i={\boldsymbol{x}}_i\right)=P\left({B}_{i,L,1}={l}_1\right)\times P\left({B}_{i,R,1}={r}_1\right)\times P\left({\boldsymbol{O}}_{i,1}={\boldsymbol{o}}_{i,1},{Z}_{i,1}=1|{B}_{i,L,1}={l}_1,{B}_{i,R,1}={r}_1,{\boldsymbol{X}}_{i,1}={\boldsymbol{x}}_{i,1}\right)\times \prod \limits_{t=2}^T\left(P\left({\boldsymbol{O}}_{i,t}={\boldsymbol{o}}_{i,t},{Z}_{i,t}=1|{B}_{i,L,t}={l}_t,{B}_{i,R,t}={r}_t,{\boldsymbol{X}}_{i,t}={\boldsymbol{x}}_{i,t}\right)\times P\left({J}_{i,L,t-1}={l}_{t-1}-{l}_t|{D}_{i,L,t-1}={d}_{L,t-1}\right)\times P\left({D}_{i,L,t-1}={d}_{L,t-1}\right)\times P\left({J}_{i,R,t-1}={r}_t-{r}_{t-1}|{D}_{i,R,t-1}={d}_{R,t-1}\right)\times P\left({D}_{i,R,t-1}={d}_{R,t-1}\right)\right) $$
where Zi = 1 is used to denote Zi,t = 1 for all t, ds,t for t = 1,…,T-1 is the dynamic label for the tth pair of consecutive time points on the left or the right side (s = L or R), respectively. Also bL and bR are the vectors of T boundary positions containing lt and rt for t = 1,…,T, respectively.
The total probability of the signal in a block can be expressed as a sum over all possible sequences of dynamics and peak boundary positions that can generate the block across all time points. Thus, the probability of block i having observations oi and Zi = 1 given the covariates xi is:
$$ P\left({\boldsymbol{O}}_i={\boldsymbol{o}}_i,{\boldsymbol{Z}}_i=\mathbf{1}|{\boldsymbol{X}}_i={\boldsymbol{x}}_i\right)=\sum \limits_{{\boldsymbol{d}}_L,{\boldsymbol{b}}_L,{\boldsymbol{d}}_R,{\boldsymbol{b}}_R}P\left({\boldsymbol{D}}_{i,L}={\boldsymbol{d}}_L,{\boldsymbol{B}}_{i,L}={\boldsymbol{b}}_L,{\boldsymbol{D}}_{i,R}={\boldsymbol{d}}_R,{\boldsymbol{B}}_{i,R}={\boldsymbol{b}}_R,{\boldsymbol{O}}_i={\boldsymbol{o}}_i,{\boldsymbol{Z}}_i=\mathbf{1}|{\boldsymbol{X}}_i={\boldsymbol{x}}_i\right) $$
where dL and dR each iterate over all possible 3T-1 combinations of peak boundary dynamics, and bL and bR each iterate over all possible ways to place left and right end boundaries across all time points that are consistent with the requirements that 1 ≤ Bi,L,t ≤ Bi,R,t + 1 ≤ Ni + 1 at each time point.
Let o be the total set of observed read counts in all blocks in the data, x be the set of the corresponding two-dimensional vectors containing the constant term and the logarithm of the expected number of reads at each position and time point for each block, Z = 1 denotes all Zi = 1, and M be the total number of blocks. Then, the likelihood of all blocks conditioned on their covariates is:
$$ P\left(\boldsymbol{O}=\boldsymbol{o},\boldsymbol{Z}=\mathbf{1}|\boldsymbol{X}=\boldsymbol{x}\right)=\prod \limits_{i=1}^MP\left({\boldsymbol{O}}_i={\boldsymbol{o}}_i,{\boldsymbol{Z}}_i=\mathbf{1}|{\boldsymbol{X}}_i={\boldsymbol{x}}_i\right) $$
We note that the above formulation allows ChromTime to model the appearance of a peak, if it occurs after the first time point in the time course, as an expansion from a zero length peak at the previous time point. Similarly, the disappearance of a peak is modeled as a contraction to a zero length peak at the next time point.
Model optimization
The total set of parameters of the model consists of:
Prior probabilities of each dynamic d at each time point t: πt,d.
Parameters of the negative binomial distributions that model the PEAK and the BACKGROUND components at each time point: αt, βt, γt and δt.
Parameters of the negative binomial distributions that model the boundary movements in EXPAND and CONTRACT dynamics at each time point: μEXPAND,t, δEXPAND,t and μCONTRACT,t, δCONTRACT,t, respectively.
The optimal parameter values are attempted to be estimated by Expectation Maximization (EM). In particular, ChromTime attempts to optimize the conditional log-likelihood of the observed counts and Zi = 1 given the covariates (Additional file 2: Supplementary Methods):
$$ \sum \limits_{i=1}^M\log P\left({\boldsymbol{O}}_i={\boldsymbol{o}}_i,{\boldsymbol{Z}}_i=\mathbf{1}|{\boldsymbol{X}}_i={\boldsymbol{x}}_i\right) $$
Computing the most likely spatial dynamic and peak boundaries for each block across the whole time course
After the optimal values for all model parameters are estimated from the data, for each block the most likely positions of the peak boundaries at each time point are calculated. This procedure consists of two steps. First, ChromTime determines for each block all time points with significantly low probability of containing a false positive non-zero length peak. Second, conditioned on those time points, ChromTime computes the most likely assignment of the peak boundary variables at each side and each time point (Additional file 2: Supplementary methods).
ChromTime options used in this study
In this work, we applied ChromTime in narrow mode on all data for H3K4me2, H3K4me3, H3K27ac, and H3K(9,14)ac marks. We applied ChromTime in punctate mode on all ATAC-seq and DNase-seq data. No control reads were used for ATAC-seq and DNase-seq. In addition, foreground reads for ATAC-seq were shifted by 5 bp in the direction of alignment (SHIFT = 5), and for DNase-seq no shifting was applied (SHIFT = 0). We applied ChromTime in broad mode on all data for H3K79me2, Pol2, H3K4me1, H3K27me3, H3K9me3, and H3K36me3 marks. All other options were set to their default values.
Timing evaluation
The timing evaluation was conducted on a MacBook Pro laptop with 2.7GHz Intel Core i7 and 16 GB RAM using four CPU cores.
Analyses with external data
The procedures for analyses with external data are described in Additional file 2: Supplementary methods.
ATAC-seq:
Assay for transposase accessible chromatin coupled with high-throughput DNA sequencing
ChIP-seq:
Chromatin immunoprecipitation coupled with high-throughput DNA sequencing
DHS:
DNase I hypersensitive site
DNase-seq:
DNase I hypersensitivity assay followed by high-throughput DNA sequencing
Expectation maximization
FDR:
False discovery rate
HM:
Histone mark
TSS:
Transcription start site
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We are grateful to Constantinos Chronis, Kathrin Plath, and members of the Ernst lab for useful discussions.
This work was supported by the CIRM Training Grant TG2–01169, the Eli and Edythe Broad Center of Regenerative Medicine and Stem Cell Research at UCLA Training Program (P.F.); NIH grants R01ES024995, U01HG007912, DP1DA044371, an NSF CAREER Award #1254200 and an Alfred P. Sloan Fellowship (J.E.).
ChromTime software is released under GPL v3.0 and can be downloaded freely at: https://github.com/ernstlab/ChromTime
The version of the code used to perform all analyses in this manuscript is available at: https://doi.org/10.5281/zenodo.1219895 [60].
No new experimental datasets were generated within this study. We used the following publicly available datasets in applying or evaluating ChromTime:
ChIP-seq data for histone marks from adipogenesis time course in human and mouse (GEO GSE20752 [18]).
ChIP-seq data for histone marks and ATAC-seq data from a blood formation time course in mouse (GEO GSE60103 [19]).
ATAC-seq data from a blood formation time course in human. Data from all available healthy donors was pooled for each cell type from the hematopoietic tree (GEO GSE74912 [20]).
Aligned reads from DNase-seq experiments from three samples representing human fetal brain development from the Roadmap Epigenomics project (epigenome ids E003, E007, E082 [61,62,63]).
ChIP-seq data for histone marks and Pol2 from a cardiac development time course in mouse (GNomEx database accession number 44R [23, 64]).
ChIP-seq data for histone marks and Pol2 from a cardiac development time course in human (GEO GSE35583 [22]).
ChIP-seq data for histone marks from an embryogenesis time course in zebrafish (GEO GSE32483 [28]).
ChIP-seq data for histone marks from a macrophage differentiation time course in mouse (GEO GSE69101 [21]).
ChIP-seq data for histone marks from a neural differentiation time course in human (GEO GSE62193 [12]).
ChIP-seq data for histone marks from a stem cell reprogramming time course in human (replicate 1 for all marks and time points and pooled input DNA from all available time points as control, GEO GSE71033 [24]).
ChIP-seq data for histone marks and transcription factors, ATAC-seq data, and gene expression data from a stem cell reprogramming time course in mouse (GEO GSE90895 [27]).
ChIP-seq data for histone marks and Pol2 from a stem cell reprogramming time course in mouse (GEO GSE67520 [25]).
ChIP-seq data for histone marks and GATA3 transcription factor from a T-cell development time course in mouse (GEO GSE31235 [17]).
ChIP-seq peaks for OCT4 transcription factor in H1 human embryonic stem cells from the ENCODE project [6, 65].
ChIP-seq peaks for NANOG transcription factor in H1 human embryonic stem cells from the ENCODE project [6, 66].
ChIP-seq peaks for P300 in H1 human embryonic stem cells from the ENCODE project [6, 67].
ChIP-seq peaks for P300 in IMR90 cells were downloaded from ChIP-Atlas [68] at FDR 0.05 [69, 70].
ChIP-seq peaks for CEBP in H1 human embryonic stem cells from the ENCODE project [71].
ChIP-seq peaks for CEBP in IMR90 cells from the ENCODE project [72].
ChIP-seq peaks for Pol2 in H1 human embryonic stem cells from the ENCODE project [73].
ChIP-seq peaks for Pol2 in IMR90 cells from the ENCODE project [74].
ChIP-seq peaks for Rad21 in H1 human embryonic stem cells from the ENCODE project [75].
ChIP-seq peaks for Rad21 in IMR90 cells from the ENCODE project [76].
DNase-seq peaks for IMR90 cells from the Roadmap Epigenomics project (epigenome id E017 [77])
DNase-seq peaks for H1 human embryonic stem cells from the Roadmap Epigenomics project (epigenome id E003 [78])
Gene expression data from the Roadmap Epigenomics project (epigenome ids E003, E007, E082) [79].
Bioinformatics Interdepartmental Program, University of California, Los Angeles, CA, USA
Petko Fiziev & Jason Ernst
Department of Biological Chemistry, University of California, Los Angeles, CA, USA
Eli and Edythe Broad Center of Regenerative Medicine and Stem Cell Research at UCLA, Los Angeles, CA, USA
Computer Science Department, University of California, Los Angeles, CA, USA
Jason Ernst
Jonsson Comprehensive Cancer Center, University of California, Los Angeles, CA, USA
Molecular Biology Institute, University of California, Los Angeles, CA, USA
Petko Fiziev
JE conceived and supervised the project, designed the method, proposed analyses, and wrote the manuscript. PF designed the method, implemented the software, proposed analyses, performed all analyses, processed all data, and wrote the manuscript. Both authors read and approved the final manuscript.
Correspondence to Jason Ernst.
Additional figures supporting the main analyses. (PDF 8541 kb)
Further description of methods and analyses in this study. (PDF 711 kb)
Fiziev, P., Ernst, J. ChromTime: modeling spatio-temporal dynamics of chromatin marks. Genome Biol 19, 109 (2018). https://doi.org/10.1186/s13059-018-1485-2
Epigenomics
Time course
Spatial dynamics
Histone modifications
Chromatin marks
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CommonCrawl
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Tonelli–Hobson test
In mathematics, the Tonelli–Hobson test gives sufficient criteria for a function ƒ on R2 to be an integrable function. It is often used to establish that Fubini's theorem may be applied to ƒ. It is named for Leonida Tonelli and E. W. Hobson.
More precisely, the Tonelli–Hobson test states that if ƒ is a real-valued measurable function on R2, and either of the two iterated integrals
$\int _{\mathbb {R} }\left(\int _{\mathbb {R} }|f(x,y)|\,dx\right)\,dy$
or
$\int _{\mathbb {R} }\left(\int _{\mathbb {R} }|f(x,y)|\,dy\right)\,dx$
is finite, then ƒ is Lebesgue-integrable on R2.
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Wikipedia
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\begin{document}
\title{Percolation threshold for Brownian loop soup on metric graphs}
\author{Yinshan Chang\thanks{College of Mathematics, Sichuan University. Email: [email protected]}\and Hang Du\thanks{School of Mathematical Sciences, Peking University. Email: [email protected]}\and Xinyi Li\thanks{Beijing International Center for Mathematical Research, Peking University.\newline Email: [email protected]}}
\date{\today}
\maketitle \abstract{In this short note, we show that the critical threshold for the percolation of the Brownian loop soup on a large class of transient metric graphs (including quasi-transitive graphs such as $\mathbb{Z}^d$, $d\geq 3$) is $1/2$.} \section{Introduction and the main result} The model of Brownian loop soup was first introduced by Lawler and Werner in the seminal paper \cite{LawWer04} as a Poissonian collection of loops whose law is based on that of the Brownian motion. Its random walk analog, the random walk loop soup, was introduced by Lawler and Trujillo Ferreras in \cite{LawTru07}. Loop soups are intimately related to various objects of interest in probability and statistical physics, in particular via the isomorphism theorem discovered by Le Jan \cite{LeJ11} linking the loop soup of intensity\footnote{Note that in the early literature there is an inconsistency of a multiplicative factor of $2$ in the intensity parameter from the definition of loop soups; see e.g.\ \cite{Lupu16-2} for a detailed discussion on this issue.} $1/2$ to the Gaussian free field.
The percolation of loop soups was already considered by Lawler and Werner in \cite{LawWer04} and then by Sheffield and Werner in \cite{SheWer12} in the setting of the two-dimensional Brownian loop soup. The latter paper, among other results, identified the value of the critical intensity as $1/2$. In the discrete setting, Lupu in \cite{Lupu16-2} also identified the critical threshold in the case of the upper half plane. Subsequently, the works \cite{LJL13,ChS16,Chang17} considered loop percolation on $\mathbb{Z}^d$ for $d\geq 3$ and established various results regarding the phase transition in percolative properties.
In \cite{Lupu16}, Lupu considered the Brownian loop soup on the so-called ``metric graphs'' (also referred to as the ``cable system'' or ``cable graphs''), a notion that corresponds to the extension of discrete graphs to a continuous metric space in which each edge of the graph have a ``length'' and Markov chains are embedded in Brownian motions moving continuously along edges. This particular model interpolates between the discrete and the continuum, on which the power of the link to Gaussian free field is maximized, yielding exact formulas (in particular the two-point function from \cite{Lupu16}; see Proposition~ \ref{prop-two-point-function} below for more details) that allow the authors of \cite{DPR22} to conclude that the critical threshold for the percolation of Brownian loop soup on a large class of transient metric graphs (including the metric version of $\mathbb{Z}^d$, $d\geq3$ and regular trees) is greater or equal to $1/2$. It remained an open question whether this threshold is exactly equal to $1/2$. (In contrast, the critical threshold for the discrete loop percolation does not equal to $1/2$ in general, see e.g.\ \cite[Theorem~1.3]{ChS16} where it is shown that the threshold for the discrete loop percolation on $\mathbb{Z}^d$ tends to infinity as $d\to \infty$.)
In this short note, we give a positive answer to this question on a sufficiently general class of metric graphs by a simple application of the Russo's formula and the two-point function discovered in \cite{Lupu16}.
We now state our main result. Given a metric graph $G$, we denote by $\mathbb{P}_\alpha$ for the law of a Brownian loop soup on $G$ with intensity $\alpha>0$ and by $x_o\longleftrightarrow \infty$ the event that $x_o\in G$ is in an infinite cluster formed by the loop soup. \begin{thm}\label{thm:main}
For any quasi-transitive transient metric graph $G$ and any $x_o\in G$, it holds that
\begin{equation}
\mathbb{P}_\alpha[x_o\longleftrightarrow \infty]>0\iff \alpha>1/2\,.
\end{equation}
In other words, the critical threshold for the loop percolation on $G$ is $1/2$. \end{thm} \begin{rem} In fact, our proof works for more general metric graphs, see Remark \ref{rem:gen} for discussions on sufficient conditions for our result to hold. \end{rem}
{\bf Acknowledgments:} The authors acknowledge the support of National Key R\&D Program of China (No.\ 2021YFA1002700 and No.\ 2020YFA0712900). YC acknowledges the support of NSFC (No.\ 11701395). HD is partially supported by the elite undergraduate training program of School of Mathematical Science at Peking University. XL acknowledges the support of NSFC (No.\ 12071012).
\section{Preliminaries} In this section, we briefly introduce metric graphs as well as the associated Brownian loop soup and discuss a few classical preliminary facts that will be useful to the proof. For a more detailed introduction of metric graphs and related objects, see e.g.\ Section~2 of \cite{Lupu16}.
We start with metric graphs. \begin{defn}[Metric graphs] Let $G^{\rm skeleton}=(V,E,\lambda)$ be an unoriented finite or countably-infinite weighted connected graph of finite degrees such that $\lambda_{x,y}>0$ for any $x,y\in V$ and $w(x):=\sum_{y\sim x}\lambda_{x,y}<\infty$ for any $x\in V$. Then the metric graph\footnote{Although in the literature metric graphs are usually denoted with a tilde as in ``$\widetilde{G}$'' in contrast to its skeleton, in this note we do not follow this convention as we will be almost exclusively working on metric graphs.} $G$ associated with $G^{\rm skeleton}$ is the metric space where each edge $e\in E$ is regarded as an interval of length $(2\lambda_e)^{-1}$, referred to as the {\bf metric graph} $G$ with skeleton $G^{\rm skeleton}$. \end{defn} If in addition there is a finite subset $V_o\subset V$ such that for any $x\in V$ there is an automorphism of $G^{\text{skeleton}}$ mapping $x$ to some $x_o\in V_o$ (this automorphism should also preserve weights), we say that $G$ is {\bf quasi-transitive}. Examples of quasi-transitive metric graphs include the metric version of periodic lattices (in particular, $\mathbb{Z}^d$) and regular trees with a fixed set of possible choices of edge weights.
We now turn to the Brownian motion and Brownian loop soup on metric graphs.
Given a metric graph $G$, there is an associated canonical diffusion process, called the Brownian motion $(B^G_t)_{t\geq 0}$ on $G$. We write $(l_y)_{y\in G}$, $Q_\cdot$ and $E_\cdot$ for the corresponding local time process, the probability law and expectation respectively. If $G$ is transient, it is possible to define the Green's function $G(\cdot,\cdot):G\times G \to \mathbb{R}^+$ associated with $B^G$ as $$ G(x,y)= E_x[l_y],\, \forall x,y\in G, $$ i.e., the expected value of the local time at $y$ of a Brownian motion $B^G$ on $G$ starting from $x$. When $x,y\in V$, $G(x,y)$ coincides with the Green's function associated with the Markov jump process on $G^{\rm skeleton}$. For any open subset $K$ of $G$, it is also possible to define the killed Green's function $G_K(x,y)$ for $x,y\in K$, as the expected local time at $y$ before the Brownian motion started from $x$ exits $K$ for the first time.
\begin{defn}[The Brownian loop soup] Given a metric graph $G$, we can endow a $\sigma$-finite measure $\mu$ on the spaces of (rooted) loops on G (i.e.\ continuous paths $l:[0,T] \to G$ such that $l(0)=l(T)$) defined via $$ \mu=\int_{x\in G} \int_{0}^\infty Q_{x,x}^t p_t(x,x) \frac{dt}{t} dx $$ where we denote by $Q^t_{x,x}$ and $p_t(x,x)$ the bridge probability measure and transition kernel of $B^G$ from $x$ to $x$ of duration $t$ respectively. A Brownian loop soup with intensity $\alpha>0$ on the metric graph $G$, denoted by $\operatorname{BLS}_\alpha$ is a Poisson point process on the set of loops in $G$ with intensity $\alpha\mu$. \end{defn} A configuration $\omega$ of $\operatorname{BLS}_\alpha$, can be viewed as an open subset of the metric graph, which is the union of ranges of (random) loops in the point process. Denote $\mathbb{P}_\alpha=\mathbb{P}_\alpha^G$ for the probability measure of configurations sampled from $\operatorname{BLS}_\alpha$.
We now turn to loop percolation. For two Borel subsets $A,B$ of $G$ and a configuration $\omega\sim \operatorname{BLS}_\alpha$, we say $A {\longleftrightarrow} B$ if $A,B$ are connected by a path entirely lying in $\omega$. With slight abuse of notation we also write events like $x\longleftrightarrow \cdot$ as a shorthand for $\{x\}\longleftrightarrow \cdot$. For any Borel subset $K\subset G$, the event such that $A\cap K$ is connected to $B\cap K$ via clusters of loops of $\omega$ {\bf entirely} lying in $K$ is denoted by $A\stackrel{K}{\longleftrightarrow}B$. Given $x_o\in G$, write $\{x_o\longleftrightarrow \infty\}$ for the event that $x_o$ lies in an infinite cluster of the loop soup configuration $\omega$.
The following two-point function estimate is a paraphrase of Proposition 5.2 of \cite{Lupu16}. \begin{prop}[Killed two-point function]\label{prop-two-point-function}
For any open subset $K\subset G$ and $x,y\in K$, it holds that
\begin{equation}\label{eq-two-point-function}
\mathbb{P}_{1/2}[x\stackrel{K}{\longleftrightarrow}y]=\frac{2}{\pi}\arcsin\frac{G_K(x,y)}{\sqrt{G_K(x,x)G_K(y,y)}}\,.
\end{equation}
\end{prop}
For any $\alpha>1/2$, we note that a configuration $\omega\sim \operatorname{BLS}_\alpha$ has the same distribution of the superposition of $\omega_1$ and $\omega_2$, where $\omega_1,\omega_2$ are independent configurations sampled from $\mathbb{P}_{1/2}$ and $\mathbb{P}_{\alpha-1/2}$, respectively. By the quasi-transitivity of $G$, it is clear that $\omega_2$ stochastically dominates a ``Bernoulli 2-bond percolation model'' on $G$ where we independently open any 2-bond, i.e., pair of {\bf neighboring} edges with probability $p=p(\alpha-1/2,G)>0$. Motivated by this, we denote $\overline{\mathbb{P}}_{\varepsilon}$ as the law of superposing of two independent configurations $\omega_1\sim \mathbb{P}_{1/2}$ and $\omega^{\rm b}_\varepsilon$, that of a Bernoulli 2-bond percolation with parameter $\varepsilon\in [0,1)$, whose law we denote by $P_\varepsilon$.
Let $E^2(G)$ stand for the set of 2-bonds in $G$. For an increasing event $A$ and a configuration $\omega$, we say a 2-bond $ef\in E^2(G)$ is pivotal for $A$ in $\omega$, if $\omega\cup \{e,f\}\in A$ but $\omega\setminus \{e,f\}\notin A$. We have the following Russo's formula: \begin{prop}[Russo's formula]\label{prop-Russo}
For any $\varepsilon\ge 0$, and any increasing event $A$ which depends on a finite range of edges, it holds that as $\delta\downarrow 0$,
\begin{equation}
\overline{\mathbb{P}}_{\varepsilon+\delta}[A]-\overline{\mathbb{P}}_\varepsilon[A]=\delta\cdot\overline{\mathbb{E}}_\varepsilon\big[\#\{ef\in E^2(G):\;ef\mbox{ is pivotal for }A\}\big]+O(\delta^2).
\end{equation} \end{prop} We omit the proof as it follows from rather standard arguments.
Finally, we turn to the FKG inequality for the new measure $\overline{\mathbb{P}}_\varepsilon$, which is a direct consequence of the FKG inequality for general Poisson processes (see Lemma 2.1 of \cite{Jans84}) and the observation that when looking at percolative properties, one can effectively regard $\omega\sim \overline{\mathbb{P}}_\varepsilon$ as a Poisson process in ${\cal L}_G \cup E^2(G)$. \begin{prop}[FKG inequality]\label{prop:FKG} For $\varepsilon\in[0,1)$ and any increasing events $A,B$, $$ \overline{\mathbb{P}}_\varepsilon[A\cap B]\geq \overline{\mathbb{P}}_\varepsilon[A]\overline{\mathbb{P}}_\varepsilon[B]. $$ \end{prop}
\section{Proof of the main result} Without loss of generality we assume $x_o\in V$. Let $\operatorname{d}(\cdot,\cdot)$ be the {\bf discrete} graph metric on $G^{\rm skeleton}$. We write $$ B_n=\{x\in V:\operatorname{d}(x_o,x)\leq n\}\;\mbox{ and }\;\partial^{ \rm s} B_n=\{x\in V:\operatorname{d}(x_o,x)=n\}\,.$$ We then write $$ f_{n}(\varepsilon):=\overline{\mathbb{P}}_\varepsilon\big[x_o\longleftrightarrow \partial^{\rm s} B_n\big]. $$ The crux of the proof is the following differential inequality, which is remotely inspired by the argument in \cite{DCT16} \begin{prop}\label{prop:ODE}
There exist constants $c,C>0$ (depending on $G$ and $x_o$ only), such that for any sufficiently large $n$, it holds
\begin{equation}
f_{n,+}^{\prime}(\varepsilon)\ge c(1-Cf_n(\varepsilon))\,,\quad \forall\ 0\le \varepsilon<1\,,
\end{equation}
where $f_{n,+}'$ stands for the right derivative of $f_n$. \end{prop} \begin{proof} From Proposition~\ref{prop-Russo}, it suffices to show that there exists $c>0$ such that for any $\varepsilon \geq 0$, \[ \overline{\mathbb{E}}_\varepsilon\big[ \#\{ef\text{ is pivotal for }x_o\longleftrightarrow \partial^{\rm s} B_n\}\big]\ge c\big(1-Cf_n(\varepsilon)\big)\,. \] For any configuration $\omega$ sampled from $\overline{\mathbb{P}}_\varepsilon$, denote $\omega_{B_n}$ as its restriction in $B_n$. We consider the union of connected components of $\omega_{B_n}$ intersecting $\partial^{\rm s} B_n$, and denote its closure by $C_n=C_n(\omega)$. Define $K_n$ as the component of $B_n\setminus C_n$ containing $x_o$ (if $x_o\in C_n$, then set $K_n=\emptyset$). By exploring from $\partial^{\rm s} B_n$ inwards, we see that conditioned on the realization of $C_n$, the configuration in $K_n$ has the law of the superposition of the loop soup of intensity $1/2$ in $K_n$ and the Bernoulli 2-bond percolation of parameter $\varepsilon$ on $K_n^{\rm s}:=V(G)\cap K_n$, independent with the configurations in $C_n$.
Write $C_n^s:=V(G)\cap C_n$. Let $$ E(K_n,C_n)=\big\{ef=(x,y)(y,z)\in E^2(G)\,:\,x,y\in K_n^{\rm s},\;z\in C_n^s\big\} $$ be the boundary 2-bonds of $K_n$ and let $$ \partial_i K_n:=\big\{x\in K_n^s: {\rm d}(x,C^{\rm s}_n)=1\big\}\; \mbox{ and }\;\partial^2_i K_n=\big\{x\in K_n^s: {\rm d}(x,C^{\rm s}_n)=2\big\} $$
stand for the inner vertex boundary and 2-inner vertex boundary of $K_n$ respectively.
Write $$\mathcal A_n:=\{d(x_o,C_n^{\rm s})>2\}.$$ Note that $\mathcal A_n$ is measurable with respect to the realization of $C_n$.
We now make the following observation: conditioned on any realization of $C_n$ such that $\mathcal A_n$ holds, for any $ef=(x,y)(y,z)\in E(K_n,C_n)$, $ef$ is pivotal for $x_o\longleftrightarrow \partial^{\rm s} B_n$ if $x_o\stackrel{K_n}{\longleftrightarrow} x$. As a result we have
\begin{align}
&\ \overline{\mathbb{E}}_\varepsilon\big[\#\{ef\in E(K_n,C_n);\;ef\text{ is pivotal\ for }x_o\longleftrightarrow \partial^{\rm s} B_n\}\big]\nonumber\\
\ge&\ \overline{\mathbb{E}}_\varepsilon \bigg[\sum_{ef\in E(K_n,C_n)}\overline{\mathbb{P}}_\varepsilon\Big[ef \text{ is pivotal\ for }x_o\longleftrightarrow \partial^{\rm s} B_n]\Big| C_n\Big]\bigg]\nonumber\\
\ge&\ \overline{\mathbb{P}}_\varepsilon[\mathcal A_n]\cdot \overline{\mathbb{E}}_\varepsilon\bigg[ \sum_{e=(x,y)(y,z)\in E(K_n,C_n),\,x\in \partial^2_i K_n}\overline{\mathbb{P}}_\varepsilon\Big[x_o\stackrel{K_n}{\longleftrightarrow} x \Big | C_n\Big]\bigg| {\cal A}_n\bigg]\nonumber\\
\ge&\ \overline{\mathbb{P}}_\varepsilon[\mathcal A_n]\cdot \overline{\mathbb{E}}_\varepsilon\bigg[\sum_{x\in \partial^2_i K_n}\mathbb{P}_{1/2}\Big[x_o\stackrel{K_n}{\longleftrightarrow} x\Big | C_n\Big]\bigg| {\cal A}_n\bigg]\,.\label{eq-two-terms}
\end{align}
To estimate the first term in \eqref{eq-two-terms}, we define $\mathcal O(x_o)$ for the event that all edges incident to $x_o$ and graph neighbors of $x_o$ are covered by a single loop in the $1/2$-BLS. Also we note that $\mathcal A_n^c$ is equivalent to the event that one of the neighbors or 2-neighbors of $x_o$ in $V(G)$ is connected to $\partial^{\rm s} B_n$. Then it is clear that both $\mathcal A_n^c$ and $\mathcal O(x_o)$ are increasing, and $\mathcal A_n^c\cap \mathcal O(x_o)\subset \{x_o\longleftrightarrow \partial^{\rm s} B_n\}$. By FKG inequality (see Proposition \ref{prop:FKG}), we get \[ \overline{\mathbb{P}}_\varepsilon[\mathcal A_n]=1-\overline{\mathbb{P}}_\varepsilon[\mathcal A_n^c]\ge 1-\frac{\overline{\mathbb{P}}_\varepsilon[\mathcal A_n^c\cap \mathcal O(x_o)]}{\mathbb{P}_{1/2}[\mathcal O(x_o)]}\ge 1-C\overline{\mathbb{P}}_\varepsilon[x_o\longleftrightarrow \partial^{\rm s} B_n]=1-Cf_n(\varepsilon), \] where $C=C(G)=\sup_{x_o\in V(G)}\left(\mathbb{P}_{1/2}[\mathcal O(x_o)]\right)^{-1}$ is a positive constant depending on $G$ only. For the second term,
from \eqref{eq-two-point-function} we see that \begin{equation}\label{eq-est-1} \begin{aligned} \sum_{x\in \partial^2_i K_n}\mathbb{P}_{1/2}[x_o\stackrel{K_n}{\longleftrightarrow} x\mid C_n]=&\ \sum_{x\in \partial_i^2K_n}\frac{2}{\pi}\arcsin\frac{G_{K_n}(x_o,x)}{\sqrt{G_{K_n}(x_o,x_o)G_{K_n}(x,x)}}\\ \ge&\ c_1\sum_{x\in \partial^2_i K_n} \frac{G_{K_n}(x_o,x)}{\sqrt{G_K(x,x)}}\,, \end{aligned} \end{equation} where $c_1={2}\big(\pi\sqrt{G(x_o,x_o)}\big)^{-1}$ depends only on $G$ and $x_o$, and we have used the fact that $\arcsin x\ge x$ for $x\in (0,1]$ and $G_{K_n}(x_o,x_o)\le G(x_o,x_o)$ for any $K_n$. Furthermore, we have for each $x\in \partial_i^2 K_n$, \begin{equation}\label{eq-est-2}
\begin{aligned} \frac{G_{K_n}(x_o,x)}{\sqrt{G_{K_n}(x,x)}}=&\ \sqrt{G_{K_n}(x,x)}\cdot\frac{G_{K_n}(x_o,x)}{G_{K_n}(x,x)}\\ =&\ \sqrt{G_{K_n}(x,x)}\cdot Q_{x_o}[B^G\text{ hits }x\text{ before exiting }K_n]\,. \end{aligned} \end{equation}
In addition, $x\in \partial_i^2 K_n$ implies $G_{K_n}(x,x)\ge c_2$ for some constant $c_2>0$ depending only on $G$.
\begin{figure}
\caption{\small A possible realization of $K_n$, $C_n$, $\partial_i K_n$, $\partial^2_i K_n$ and an exemplary pivotal 2-bond. Note that in this figure $B_n$ and $C_n$ are only partially depicted.}
\end{figure}
Combining \eqref{eq-est-1} with \eqref{eq-est-2}, we see that the second term in \eqref{eq-two-terms} is bounded from below by \[ c_1\sqrt{c_2}\sum_{x\in \partial_i^2K_n}Q_{x_o}[B^G\text{ hits }x\text{ before leaving }K_n]\,. \] Note that under the event $\mathcal A_n$, the sum in the above formula is trivially bounded from below by $1$, since a path of the Brownian motion $B^G$ on $G$ starting from $x_o$ almost surely hits at least one vertex $x\in \partial_i^2 K_n$ before exiting $K_n$. Therefore, the proof is completed by taking $c=c_1\sqrt{c_2}$. \end{proof} It is worth noting that there is a difficulty in the argument above if we only consider $\partial_i K_n$, the interior graph boundary of $K_n$ (instead of its 2-inner boundary) in summing the two point functions. Indeed, for some $x\in \partial_i K_n$, it could happen that $x$ is very close to $C_n$ in the metric sense, and in this case one may not bound $G_{K_n}(x,x)$ from below by any universal constant. \begin{proof}[Proof of Theorem \ref{thm:main}] First, it is easy to see that quasi-transitive transient weighted graphs satisfy the ${\rm (Cap)}$ condition in \cite{DPR22}, hence by Corollary 3.6, ibid., $f(0)=0$, and the critical threshold for loop soup percolation on $G$ is greater or equal to $1/2$.
The main claim then follows from a standard and classical argument. It suffices to show that for any $\varepsilon>0$, $$f(\varepsilon):=\overline{\mathbb{P}}_\varepsilon[x_o\longleftrightarrow \infty]>0.$$ For any $\varepsilon\in(0,1)$, if $f(\varepsilon)\geq 1/(2C),$ where $C$ is the constant from Proposition \ref{prop:ODE}, then the claim naively follows; otherwise, one can find $N_0=N_0(\varepsilon)$ such that $f_n(\varepsilon)<1/(2C)$ for all $n\geq N_0$. By Proposition \ref{prop:ODE} and the monotonicity of $f_n(\cdot)$, uniformly for all $\delta\in(0,\varepsilon)$ and $n>N_0$, one has $f'_{n,+}(\delta)\geq c(1-Cf_n(\varepsilon))>c/2$. From this we conclude that $f_n(\varepsilon)>c\varepsilon/2$ for any $n\geq N_0$. Combining the two cases, one has $f(\varepsilon)>0$ for any $\varepsilon>0$. \end{proof} \begin{rem}\label{rem:gen} 1) Our proof does not essentially rely on the quasi-transitivity of $G$ except at places where we require a uniform constant depending on the graphs $G$. In fact, it applies to all transient weighted graphs satisfying the ${\rm (Cap)}$ condition from \cite{DPR22} with uniformly bounded degrees and edge weights uniformly bounded from above (but not necessarily from below -- since one may always break a ``long'' edge into shorter pieces, which produces a new skeleton graph but keep the metric space (of the metric graph) and the percolation threshold unchanged). \newline 2) Another direction of generalization is to take killing (which corresponds to massive loop soups and or loop soups on a metric graph with boundary) into consideration. If the killing is mild, i.e., $$ h_{\rm kill}(x) :=Q_x\big[B^G\mbox{ is killed }\big]<1,\;\forall x\in G, $$ (note that this is a necessary condition for {\rm (Cap)} condition from \cite{DPR22} to hold), then our arguments still work with little modification. It is then very natural to ask the following question: what is the sufficient and necessary condition one should pose on a metric graph for the critical threshold of loop percolation to be exactly $1/2$? \end{rem}
\end{document}
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arXiv
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\begin{document}
\title{Congruence Filter Pairs, Adjoints and Leibniz Hierarchy} \author{P. Arndt, H.L. Mariano, D.C. Pinto}
\maketitle
\begin{abstract}
We have introduced in \cite{AMP1} the notion of (finitary) filter pair as a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, given by presenting its lattice of theories as the image of a lattice homomorphism, with certain properties ensuring that the resulting logic is finitary and substitution invariant.
Every finitary, substitution invariant logic arises from a filter pair. Particular classes of logics can be characterized as arising from special classes of filter pairs. We consider so-called congruence filter pairs, i.e. filter pairs for which the domain of the lattice homomorphism is a lattice of congruences for some quasivariety. We show that the class of logics admitting a presentation by such a filter pair is exactly the class of logics having an algebraic semantics.
We study the properties of a certain Galois connection coming with such filter pairs. We give criteria for a congruence filter pair to present a logic in some classes of the Leibniz hierarchy by means of this Galois connection, and its interplay with the Leibniz operator.
As an application, we show a bridge theorem, stating that the amalgamation property implies the Craig interpolation property, for a certain class of logics including non-protoalgebraic logics.
\end{abstract}
\section*{Introduction}
The first main aim of this article is to introduce the notion of filter pair as a tool for creating and analyzing logics. A filter pair can be seen as a presentation of a logic, different in style from the usual syntactic presentations in terms of axioms and rules, or semantic presentations in terms of matrices. Roughly, a filter pair is a highly structured lattice homomorphism from some algebraic lattice into the power set of the formula algebra, whose image can be taken as the lattice of theories of a logic. The notion of filter pair can be developed into several different directions, some of which are sketched in Section \ref{SectionVista}.
In this article we concentrate on one direction, which constitutes its second main aim: to advance the general study of logics having an algebraic semantics. Recall that an algebraic semantics for a logic $l$ is given by a class of algebras ${\bf K}$ and a set of equations $\delta(x) = \epsilon(x)$ such that $$\Gamma \vdash_l \varphi \ \ \ \ \Leftrightarrow \ \ \ \ \{\delta(\gamma) = \epsilon(\gamma) \mid \gamma \in \Gamma\} \vDash_{{\bf K}} \delta(\varphi) = \epsilon(\varphi).$$ The notion was considered by Blok-Pigozzi in \cite{BP1}, but there mainly used as a stepping stone towards the notion of algebraizable logic. As Font writes \cite[p.114]{Fon}: ``The bare concept of algebraic semantics has not been extensively studied. [...] At the time of writing, \cite{BlokRebagliato} and \cite{Raftery} are probably the only papers that contain significant results about it.'' It turns out that logics having an algebraic semantics are exactly those which can be presented by a so-called equational filter pair - in the above picture of a filter pair as a highly structured lattice homomorphism, these are those homomorphisms whose domain is the lattice of congruences relative to some quasivariety ${\bf K}$. We find that such a presentation, and the additional structure that comes with it, can be employed to analyze logics with an algebraic semantics. We illustrate this by considering some criteria for being algebraizable or truth-equational and for satisfying the Craig interpolation property.
We now sketch the basic idea of the notion of filter pair:
Throughout the article the word \emph{logic} will mean a pair $(\Sigma, \vdash)$ where $\Sigma$ is a signature, i.e. a collection of connectives with finite arities, and $\vdash$ is a Tarskian consequence relation, i.e. an idempotent, increasing, monotonic, finitary and structural relation between subsets and elements of the set of formulas $\Fm{\Sigma}{X}$ built from $\Sigma$ and a set $X$ of variables.
It is well-known that every Tarskian logic gives rise to an algebraic lattice contained in the powerset $\wp(\Fm{\Sigma}{X})$, namely the lattice of theories. This lattice is closed under arbitrary intersections and directed unions.
Conversely an algebraic lattice $L \subseteq \wp(\Fm{\Sigma}{X})$ that is closed under arbitrary intersections and unions of increasing chains gives rise to a finitary closure operator (assigning to a subset $A \subseteq \Fm{\Sigma}{X}$ the intersection of all members of $L$ containing $A$). This closure operator need not be structural --- this is an extra requirement.
We observe that the structurality of the logic just defined is equivalent to the \emph{naturality} (in the sense of category theory) of the inclusion of the algebraic lattice into the power set of formulas with respect to endomorphisms of the formula algebra: Structurality means that the preimage under a substitution of a theory is a theory again or, equivalently, that the following diagram commutes:
$$\xymatrix{ \Fm{\Sigma}{X} \ar[d]_\sigma & L \ar@{^(->}[r]^-i & \wp(\Fm{\Sigma}{X}) \\
\Fm{\Sigma}{X} & L \ar[u]^{\sigma^{-1}|_{_L}} \ar@{^(->}[r]^-i & \wp(\Fm{\Sigma}{X}) \ar[u]_{\sigma^{-1}} }$$
Further, it is equivalent to demand this naturality for all $\Sigma$-algebras and homomorphisms instead of just the formula algebra.
We thus arrive at the definition of \emph{filter pair}, Def. \ref{DefinitionFilterPair}: A filter pair for the signature $\Sigma$ is a contravariant functor $G$ from $\Sigma$-algebras to algebraic lattices together with a natural transformation $i \colon G \to \wp(-)$ from $G$ to the functor taking an algebra to the power set of its underlying set, which preserves arbitrary infima and directed suprema.
The logic associated to a filter pair $(G, i)$ is simply the logic associated (in the above fashion) to the algebraic lattice given by the image $i(G(\Fm{\Sigma}{X})) \subseteq \wp(\Fm{\Sigma}{X})$.
In particular, it is clear that different filter pairs can give rise to the same logic, indeed this will happen precisely if the images of $i$ for the formula algebra are the same. A filter pair can thus be seen as a \emph{presentation} of a logic, and there can of course be different presentations of the same logic. We could have removed a bit of this ambiguity by requiring that $i$ be an inclusion, but it is one of the insights of this article that it is beneficial not to do this. Indeed this will give us greater flexibility for the choice of the functor $G$, and injectivity of $i$ can become a meaningful extra feature. Thus, for example, if $G$ is the functor associating to a $\Sigma$-structure the lattice of congruences relative to some quasivariety ${\bf K}$, i.e. when we have a so-called \emph{congruence filter pair}, then by Prop. \ref{2.3} the injectivity of $i$ means that the associated logic is algebraizable.
In this article we show how to recognize classes of logics, like algebraizable or truth-equational logics, through their presentations by filter pairs and how these presentations may permit to use algebraic methods even outside the realm of protoalgebraic logics, see the following overview for a sample of results.
Another theme that we take up is the theme of adjunctions arising in the context of filter pairs. It follows from the definition that the maps $i$ in a filter pair each have a left adjoint $\Xi$ (i.e. are part of a Galois connection). The closure operator obtained as the composition $i \circ \Xi$ is exactly the closure operator generating the theories of the logic. For congruence filter pairs we also have another operator back from theories to congruences, namely the Leibniz operator $\Omega$. The interplay of the three operators $i, \Xi, \Omega$ gives an interesting picture that one can observe for any logic with an algebraic semantics. As an illustration of the concrete relevance of these operators, we show in Section \ref{SectionCraigInterpolation} that whenever the maps $\Xi$ form a natural transformation (actually a weaker condition suffices), then the amalgamation property of a quasivariety implies the Craig interpolation property of a logic with an algebraic semantics in this quasivariety. While we have not yet been able to make this result bear fruits for the most common types of logics, we can already use it to cover a certain class of examples which includes non-protoalgebraic and non-truth-equational logics, showing that the results on amalgamation and interpolation can be extended beyond the known cases from the literature -- see Def. \ref{DefregularEquationsAndVarieties} and Cor. \ref{CorollaryAmalgamationInregularVarietiesImpliesInterpolation}.
{\bf Overview of the article:}
In section \ref{SectionPreliminaries} we fix terminology and gather some results characterizing different classes of logics, like algebraizable, truth-equational or protoalgebraic logics. No original results are contained in this section.
In section \ref{SectionFilterFunctors} we introduce the notion of filter pair (Def. \ref{DefinitionFilterPair}) and show how to get a logic from a filter pair (Prop. \ref{LogicsFromFilterPairs}). We show some additional structures that one obtains from a filter pair, for example a left adjoint to each map $i_A \ (A \in \Sigma\text{-Str})$ (Def. \ref{DefLeftAdjointOfAFilterPair}) whose composition with $i$ yields the closure operator of the logic (Prop. \ref{PropDescriptionOfConsequenceRelationByClosureOperator}) and a consequence relation on each $A \in \Sigma\text{-Str}$ (Prop. \ref{LogicsFromFilterPairs}). We establish the basic relations between these structures, e.g. that homomorphisms of $\Sigma$-structures induce translations between these consequence relations, which are conservative in the case of free algebras (Prop. \ref{PropHomomorphismsInduceTranslationsAndVariableInclusionsInduceConservativeTranslations}), that subsets in the image of $i$ are filters for the logic, and that in the case of free algebras all filters are in the image of $i$ (Prop. \ref{FreeAlgebraResult}).
In section \ref{SectionFilterPairsOverCo} we consider the special case of filter pairs where the functor $G$ is given by congruences relative to some class of algebras. In subsection 3.1 we consider the question what the properties of $G$ demanded in the definition of filter pair enforce on the class ${\bf K}$. We find that ${\bf K}$ must be closed under subalgebras (Prop. \ref{PropCoKIsAFunctor}) and products (Prop. \ref{PropKClosedUnderProducts}). This subsection may be skipped and ${\bf K}$ be assumed to be a quasivariety in the remainder of the article. In subsection 3.2 we introduce congruence filter pairs, and give a general construction of them, yielding the so-called equational filter pairs (Thm. \ref{TheoremLogicsFromEquations}). We give criteria when the associated logic is algebraizable (Prop. \ref{CriterionAlgebraizable}) or truth-equational (Cor. \ref{CriterionTruthEquational}, Prop. \ref{CriterionTruthEquationalPointedQuasiVar}).
In section \ref{filteradjunctionequationalfilterpairs} we show and analyze some of the extra structure coming with an equational filter pair, for example give more concrete descriptions of the left adjoint of the $i_A$ mentioned in the previous paragraph (Prop. \ref{PropFirstFormulaForLeftAdjointForEquationalFilterPairs}, Prop. \ref{PropOurLeftAdjointIsBlokPigozzisOmegaK}) and investigate its relation with the Leibniz operator (Prop. \ref{TheoremInclusionsXiIdLeibniz} and the following discussion). We conclude from these results that logics having an algebraic semantics are precisely those which admit a presentation by an equational filter pair (Thm. \ref{TeoEveryLogicWithAlgebraicSemanticsComesFromEqunlFilterPair}), and we get another criterion for the algebraizability of a logic (Thm. \ref{TheoremInclusionsXiIdLeibniz}).
In section \ref{SectionCraigInterpolation} we show how the language of filter pairs can be used to establish connections between the amalgamation property of a quasivariety and Craig interpolation of a logic with algebraic semantics in that quasivariety, Theorem \ref{TheoremMatrixAmalgamationImpliesCraigInterpolation}. We recover the fact that amalgamation implies interpolation for algebraizable logics (Cor. \ref{CorollaryForAlgebraizableLogicsAmalgamationImpliesCraigInterpolation}), but also cover some cases which include non-protoalgebraic logics (Lem. \ref{LemmaConditionForNaturalityOfXi}, Cor. \ref{CorollaryAmalgamationInregularVarietiesImpliesInterpolation}).
Finally, in section \ref{SectionVista} we discuss some of the other directions into which one can develop filter pairs.
\section{Preliminaries}\label{SectionPreliminaries}
\subsection{Ordered sets}
\begin{Df}[Galois connections]
Let $f \colon P \to Q$, $g \colon Q \to P$ be order preserving maps between partially ordered sets. Then $(f,g)$ is a Galois connection, where $f$ is called left adjoint to $g$ and $g$ is called right adjoint to $f$, if the following equivalence is satisfied: \[f(p) \leq q \Leftrightarrow p \leq g(q) \ \ \ \ \forall p \in P,\ q \in Q\] \end{Df}
A Galois connection (or adjunction) between posets, $(f,g)$, is uniquely determined by its left (respectively right) component, and induces an isomorphism between the subposets $\hat{P} \subseteq P$ and $\hat{Q} \subseteq Q$, where $\hat{P} := Fix(g \circ f)$ and $\hat{Q} := Fix(f \circ g)$.
\begin{Teo}\label{TheoremExistenceOfLeftAdjoint} {\em \cite[Thm. 3.6.9]{TaylorPracticalFoundations}} Let $g \colon Q \to P$ be a function between complete (and hence cocomplete) partially ordered sets that preserves arbitrary infima (resp. suprema). Then $g$ has a left adjoint (resp. right adjoint) $f \colon P \to Q$, given by $f(p)=inf\{q \in Q \, \mid \, p \leq g(q) \}$ (resp. $f(p) = sup\{q \in Q \, \mid \, p \geq g(q) \}$). \end{Teo}
We will be concerned with algebraic lattices.
\begin{Df}[algebraic lattices] \label{AL-df} \begin{enumerate} \item Let $L$ be a lattice. A element $a\in L$ is compact if for every (upward) directed subset $\{d_{i}\}$ of $L$ we have $a\leq \bigvee_{i}d_{i}\ \Leftrightarrow\ \exists i(a\leq d_{i})$. $L$ is said to be algebraic if it is a complete lattice such that every element is the join of the compact elements below it. \item We denote by $AL$ the category of all algebraic lattices and morphisms that preserve arbitrary meets and directed joins. In particular, these morphisms are monotonic functions that preserve the top elements of the lattices. \end{enumerate} \end{Df}
\begin{Obs}[sublattices of algebraic lattices] \label{AL-obs} Let $X$ be a set and denote $(P(X), \subseteq)$ the (algebraic) lattice of all subsets of $X$. Note that a subset ${\cal S} \subseteq P(X)$ that is closed under arbitrary intersections and directed unions determines an algebraic sublattice $({\cal S}, \subseteq) \hookrightarrow (P(X), \subseteq)$, where the compact elements coincide with the finitely generated elements of $({\cal S}, \subseteq)$. \end{Obs}
\subsection{Structures, logics, filters and matrices}
\begin{Df}[signatures, structures, formulas] A signature is a sequence of pairwise disjoint sets $\Sigma=(\Sigma_{n})_{n\in \mathbb{N}}$, where $\Sigma_{n}$ is thought of as a set of $n$-ary function symbols. A $\Sigma$-structure is a set with an interpretation of the function symbols as actual operations. A homomorphism of $\Sigma$-structures is a map respecting these operations. We denote by $\Sigma$-Str the category of $\Sigma$-structures and homomorphisms.
For a set $X$ denote by $\Fm{\Sigma}{X}$ (or $\Fm{}{X}$, if the signature is clear), the absolutely free $\Sigma$-structure, i.e. the set of $\Sigma$-formulas with variables from $X$. A homomorphism $\Fm{}{X} \to A$ with domain $\Fm{}{X}$ is also called a valuation. An endomorphism of $\Fm{\Sigma}{X}$ is also called a substitution. \end{Df}
In this work there will be naturally occurring consequence relations on non-free $\Sigma$-structures, hence we phrase the definition of logic slightly more generally than customary.
\begin{Df}[consequence relation, abstract logic, logic]\label{DefAbstractLogic} A Tarskian consequence relation on a $\Sigma$-structure $A$ is a relation $\vdash\subseteq\wp(A)\times A$, such that, for all sets $\Gamma,\Delta \subseteq A$ and all elements $\varphi,\psi \in A$ the following conditions are satisfied: \begin{itemize} \item[$\circ$]$\bf{Reflexivity}: $If $\varphi\in\Gamma,\ \Gamma\vdash\varphi$ \item[$\circ$]$\bf{Cut}: $If $\Gamma\vdash\varphi$ and for every $\psi\in\Gamma,\ \Delta\vdash\psi$, then $\Delta\vdash\varphi$ \item[$\circ$]$\bf{Monotonicity}: $If $\Gamma\subseteq\Delta$ and $\Gamma\vdash\varphi$, then $\Delta\vdash\varphi$ \item[$\circ$]$\bf{Finitarity}: $If $\Gamma\vdash\varphi$, then there is a finite subset $\Delta$ of $\Gamma$ such that $\Delta\vdash\varphi$. \item[$\circ$]$\bf{Structurality}: $If $\Gamma\vdash\varphi$ and $\sigma$ is an endomorphism of $A$, then $\sigma[\Gamma]\vdash\sigma(\varphi)$ \end{itemize}
An abstract logic (in the sense of \cite{BloomBrownSuszko}) is a pair $(A, \vdash)$, where $A$ is a $\Sigma$-structure and $\vdash$ is a Tarskian consequence relation on $A$.
A logic is a pair $(\Sigma,\vdash)$ where $\Sigma$ is a signature and $\vdash$ is a Tarskian consequence relation on $\Fm{\Sigma}{X}$. \end{Df}
A consequence determines, and is determined by, a closure operator on the power set $\wp(A)$, and we will switch freely between the two.
\begin{Df}[Filters, matrices] \label{filter-df} Let $l=(\Sigma,\vdash)$ be a logic. \begin{enumerate} \item Let $A\in \Sigma\text{-Str}$. A subset $F \subseteq A$ is an $l$-filter if for every $\Gamma\cup\{\varphi\}\subseteq \Fm{\Sigma}{X}$ such that $\Gamma\vdash \varphi$ and every valuation $v:\Fm{}{X}\to A$, if $v[\Gamma]\subseteq F$ then $v(\varphi)\in F$.
In particular, since $l$ is structural, when $A=\Fm{}{X}$, then the $l$-filters in $\Fm{}{X}$ coincide with the theories of $l$, i.e., the $\vdash$-closed subsets of $\Fm{}{X}$.
\item Let $A\in \Sigma\text{-Str}$ and $F \subseteq A$ be an $l$-filter. Then the pair $\langle M,F\rangle$ is then said to be a matrix model of $l$, or shortly, an $l$-matrix. \end{enumerate} \end{Df}
\begin{Obs}[Functoriality of filters]\label{RemarkFunctorialityOfFilters} Let $l=(\Sigma,\vdash)$ be a logic and $A$ be a $\Sigma$-structure. Denote by $(Fi_{l}(A), \subseteq) \overset{i_A}\hookrightarrow (P(A), \subseteq)$ the (sub)poset of all $l$-filters of $A$. Clearly $Fi_l(A)$ is closed under arbitrary intersections and, since $l$ is a finitary logic, $Fi_l(A)$ is closed under directed unions. Thus $(Fi_{l}(A), \subseteq)$ is an algebraic lattice where the compact elements are the finitely generated filters. Further, for a homomorphism of $\Sigma$-algebras $f \colon A \to A'$
and a filter $F' \in Fi_{l}(A')$, the preimage $f^{-1}(F')$ is a filter in $Fi_{l}(A)$. Defining $Fi_{l}(f) := f^{-1}$ (inverse image), then $Fi_{l}$ becomes a contravariant functor from the category $\Sigma\text{-Str}$ to the category $AL$. \end{Obs}
\subsection{Congruences and equational consequence}\label{SubsectionCongruencesAndEquationalConsequence}
A class ${\bf K} \subseteq \Sigma\text{-Str}$ is a quasivariety if it is axiomatizable by quasi-identities in the signature $\Sigma$, i.e., formulas of the form \[(p_{1}\approx q_{1}\wedge...\wedge p_{n}\approx q_{n})\to p\approx q.\] Quasivarieties are precisely the classes of $\Sigma$-structures that are closed under isomorphisms, subalgebras, products and directed colimits.
A ${\bf K}$-congruence on a $\Sigma$-structure is a $\Sigma$-congruence $\theta \subseteq A \times A$ such that $A/\theta\in \bf K$.
\begin{Obs}[functoriality of ${\bf K}$-congruences] Let ${\bf K}$ be a quasivariety. For a $\Sigma$-structure $A$ denote by $(Co_{\bf K}(A), \subseteq) \overset{i_A}\hookrightarrow (P(A \times A), \subseteq)$ the (sub)poset of all ${\bf K}$-congruences. As ${\bf K}$ is closed under isomorphisms, subalgebras and products, $Co_{\bf K}(A) \subseteq P(A \times A)$ is closed under arbitrary intersections and, since ${\bf K}$ is closed under directed colimits, $Co_{\bf K}(A) \subseteq P(A \times A)$ is closed under directed unions. Thus $(Co_{\bf K}(A), \subseteq)$ is an algebraic lattice where the compact elements are its finitely generated members. If $f\colon A \to A'$ is a homomorphism and $\theta' \in Co_{\bf K}(A')$ then, since ${\bf K}$ is closed under isomorphisms and subalgebras, $(f\times f)^{-1}(\theta') \in Co_{\bf K}(A)$. Defining $Co_{\bf K}(f) := (f \times f)^{-1}$, then $Co_{\bf K}$ becomes a contravariant functor from the category $\Sigma\text{-Str}$ to the category $AL$. \end{Obs}
Let $\Sigma$ be a signature and ${\bf K} \subseteq \Sigma\text{-Str}$ a class of algebras. Let $Eq:=\Fm{}{X}_{\{x\}}\times \Fm{}{X}_{\{x\}}$ be the set of pairs of formulas in at most the variable $x$. We think of such a pair of formulas $\langle \delta, \epsilon \rangle$ as an equation between the terms of both sides and write, following Blok-Pigozzi, $\delta(x)\approx\epsilon(x)$.
\begin{Df}[equational consequence]\label{DefEquationalConsequence} Given a class of algebras $\textbf{K}$ over the signature $\Sigma$, and a set of variables $X$, the associated equational consequence is the relation $\models_{\textbf{K}}^X$ between a set of equations $\Gamma$ and a single equation $\varphi\approx \psi$ over $\Sigma$ defined by: \[\Gamma\models_{\textbf{K}}^X\varphi\approx \psi\ i\!f\!\!f\ for\ every\ A\in\textbf{K}\ and\ every\ \Sigma-homomorphism \ \ h:\Fm{}{X}\to A,\] \[\ if \ h(\eta)=h(\nu)\ for\ all\ \eta\approx\nu\in \Gamma,\ then\ h(\varphi)=h(\psi).\] \end{Df}
In the context of amalgamation we will consider equational consequence for several sets of variables simultaneously. The following is the basic relation that we will need.
\begin{Lem}\label{LemmaInclusionOfVariablesGivesConservativeTranslationOfEquationalConsequences} Let $\textbf{K}$ be a class of algebras over the signature $\Sigma$, $X \subseteq Y$ sets of variables, and $\Gamma \cup\{\varphi\approx \psi\}$ sets of equations with variables in $X$. Then \[\Gamma\models_{\textbf{K}}^Y\varphi\approx \psi\ \ \ \Leftrightarrow\ \ \ \Gamma\models_{\textbf{K}}^X\varphi\approx \psi\] \end{Lem} \n{\bf Proof:\;\;} $\Leftarrow \colon$ Suppose $\Gamma\models_{\textbf{K}}^X\varphi\approx \psi$. Let $A\in\textbf{K}$ and $h\colon \Fm{}{Y}\to A$ be a $\Sigma$-homomorphism such that $h(\eta)=h(\nu)$ for all $\eta\approx\nu\in \Gamma$. We need to show that $h(\varphi)=h(\psi)$. For this, just precompose with the inclusion $i \colon \Fm{}{X}\hookrightarrow \Fm{}{Y}$. As $\Gamma \cup\{\varphi\approx \psi\} \subseteq \Fm{}{X}$, we have $(h \circ i)(\eta)=(h \circ i)(\nu)$ for all $\eta\approx\nu\in \Gamma$. From the assumption $\Gamma\models_{\textbf{K}}^X\varphi\approx \psi$ it follows that $h(\varphi)=(h \circ i)(\varphi)=(h \circ i)(\psi)=h(\psi)$.
$\Rightarrow \colon$ Suppose $\Gamma\models_{\textbf{K}}^Y\varphi\approx \psi$. Let $A\in\textbf{K}$ and $h\colon \Fm{}{X}\to A$ be a $\Sigma$-homomorphism such that $h(\eta)=h(\nu)$ for all $\eta\approx\nu\in \Gamma$. We need to show that $h(\varphi)=h(\psi)$.
Choose a left inverse $Y \to X$ to the inclusion $X \subseteq Y$. These two maps induce homomorphisms $i \colon \Fm{}{X}\hookrightarrow \Fm{}{Y}$ and $r \colon \Fm{}{Y}\to \Fm{}{X}$ such that $r \circ i = id_{\Fm{}{X}}$. The composition $h \circ r \colon \Fm{}{Y}\to\Fm{}{X}\to A$ is a homomorphism such that $(h \circ r)(\eta)=(h \circ r)(\nu)$ for all $\eta\approx\nu\in \Gamma$. As $\Gamma\models_{\textbf{K}}^Y\varphi\approx \psi$, we have that $(h \circ r)(\varphi)=(h \circ r)(\psi)$. Precomposing with $i$ yields $h(\varphi)=(h \circ r \circ i)(\varphi)=(h \circ r \circ i)(\psi)=h(\psi)$. \qed
It is shown in chapter 2 of \cite{BP1} that $\vDash_{\bf K}^X$ satisfies the axioms of a structural consequence relation, which is finitary if and only if $\vDash_{\bf K}^X = \vDash_{{\bf K}^Q}^X$, where ${\bf K}^Q$ denotes the quasivariety generated by ${\bf K}$. As usual, the consequence relation $\vDash_{\bf K}^X$ corresponds to a closure operator $Cn_{\bf K}^X$ on $\wp(Eq)$. A set of equations $\Gamma \subseteq Eq$ is called a ${\bf K}$-theory if is closed under ${\bf K}$-consequence, i.e. if $Cn_{\bf K}^X(\Gamma) = \Gamma$.
For the following Lemma keep in mind that for a set $\Gamma \subseteq Eq$ we have two view points: we can see $\Gamma$ as a set of equations, or as a set of pairs of formulas. From the latter view point it makes sense to ask whether a set is a congruence relation or not. We suppose all the formulas to be in $\Fm{}{X}$ for some set of variables $X$.
\begin{Lem}\label{LemmaEquationalClosureEqualsGeneratedCongruence}
For a set of equations $\Gamma=\{\langle \gamma^i_1, \gamma^i_2 \rangle \mid i \in I \} \subseteq Eq$ its closure $Cn_{\bf K}^X(\Gamma)$ is the congruence relation generated by $\Gamma$.
In particular a set $\Gamma \subseteq Eq$ is a ${\bf K}$-theory if and only if is a congruence relation. \end{Lem} \n{\bf Proof:\;\;} Denote the ${\bf K}$-consequence relation generated by $\Gamma$ by $\hat{\Gamma}$.
$Cn_{\bf K}^X(\Gamma) \supseteq \hat{\Gamma}$: Let $\langle \varphi(x), \psi(x) \rangle \in \hat{\Gamma}$. Note that $\hat{\Gamma}$ is the smallest congruence $\theta$ on $\Fm{}{X}$ such that $\bar{\gamma^i_1}=\bar{ \gamma^i_2}$ in $\Fm{}{X}/\theta$. Thus, every quotient $\Fm{}{X}/\theta$ of $\Fm{}{X}$ in which $\bar{\gamma^i_1(x)}=\bar{ \gamma^i_2(x)}\ (i \in I)$ holds will be itself a quotient of $\Fm{}{X}/\hat{\Gamma}$.
Now suppose that for some algebra $A \in {\bf K}$ and some $a\in A$ we have $(\gamma^i_1)^A(a)= (\gamma^i_2)^A(a)$. Then there is a homomorphism $h \colon \Fm{}{X} \to A$ sending $x$ to $a$ and the other variables to arbitrary elements. The image $Im\, h$ of this homomorphism is isomorphic to a quotient of $\Fm{}{X}$ and a subalgebra of $A$ in which $(\gamma^i_1)^A(a)= (\gamma^i_2)^A(a)$ holds. Thus $Im\, h$ is also isomorphic to a quotient of $\Fm{}{X}/\hat{\Gamma}$, and since $\langle \varphi(x), \psi(x) \rangle \in \hat{\Gamma}$, we have $\varphi^A(a)= \psi^A(a)$ in $Im\, h \subseteq A$. This shows $\langle \varphi(x), \psi(x) \rangle \in Cn_{\bf K}(\Gamma)$.
$Cn_{\bf K}^X(\Gamma) \subseteq \hat{\Gamma}$: Let $\langle \varphi(x), \psi(x) \rangle \in Cn_{\bf K}^X(\Gamma)$. Then since in $\Fm{}{X}/\hat{\Gamma}$ we have $\bar{\gamma^i_1(x)}=\bar{ \gamma^i_2(x)}\ (i \in I)$, we also have $\bar{\varphi(x)} = \bar{\psi(x)}$. Therefore $\langle \varphi(x), \psi(x) \rangle \in \hat{\Gamma}$. \qed
\subsection{Algebraicity of logics}
We summarize several ways in which a logic can be tied to, and its properties determined by, algebra.
\begin{Df}[algebraic semantics]
A logic $l=(\Sigma, \vdash_l)$ has an algebraic semantics in a class of algebras ${\bf K}$ given by a set of equations $\delta(x) \approx \epsilon(x)$ if the following equivalence holds: $$\Gamma \vdash_l \varphi \ \ \ \ \Leftrightarrow \ \ \ \ \{\delta(\gamma) \approx \epsilon(\gamma) \mid \gamma \in \Gamma\} \vDash_{{\bf K}} \delta(\varphi) \approx \epsilon(\varphi)$$ \end{Df}
In the following ``algebraizable logic'' will mean ``algebraizable'' in the sense of Blok-Pigozzi \cite{BP1}.
\begin{Df}[algebraizable] \label{BP-def} A logic $l=(\Sigma, \vdash_l)$ is algebraizable with equivalent algebraic semantics $\textbf{K}$ if \begin{enumerate} \item[(1)] $l$ has an algebraic semantics in ${\bf K}$ given by a set of equations $\delta(x) \approx \epsilon(x)$ \item[(2)]there is a finite set $\Delta_{j}(p,q),j=1,...,m$ of formulas in two variables such that for every set of equations $\Gamma$ and for every equation $\varphi\approx\psi$, we have that \[\Gamma\models_{\bf K}\varphi\approx\psi\ \Leftrightarrow\ \{\xi\Delta\eta:\xi\approx\eta\in\Gamma\}\vdash_l\varphi\Delta\psi\] \item[(3)]For every $\psi\in \Fm{}{X}$ we have that \[\psi\dashv \vdash_l\Delta(\tau(\psi)).\] \end{enumerate} \end{Df}
\begin{Df}[compatible congruence, Leibniz operator, reduced matrix]\label{DefCompatibleCongruence} Let $\Sigma$ be a signature, ${A}$ be a $\Sigma$-algebra and $F\subseteq A$.
(a) Let $\theta$ be a congruence in ${A}$. $\theta$ is said to be compatible with $F$ if, for all $a,b\in A$, if $a\in F$ and $\langle a,b\rangle\in\theta$ then $b\in F$. Given an algebra $A$ and a subset $F$ of its domain there always exists a greatest congruence of $A$ compatible with $F$ (Theorem 1.5, \cite{BP1}), denoted by $\Omega^{A}(F)$. The function $\Omega^{A}$ with domain the set of all subsets of $A$ is called the Leibniz operator on $A$.
(b) A matrix $(A,F)$ for a logic $l = (\Sigma, \vdash)$ is reduced when $\Omega^{A}(F) = Diag(A) = \{(a,b) \in A \times A : a=b\}$. We will denote $Matr^*_l$ the class of all reduced matrices for $l$; $Alg^{*}_l$ stands for the class of all $\Sigma$-algebras underlying to some reduced matrix. \end{Df}
\begin{Teo}(The Isomorphism Theorem, first version \cite[Thm 5.1]{BP1}).\label{IT2} \label{TheoremFirstIsoThmInclUniquenessOfTheQuasivariety}
Let $l$ be a logic and ${\bf K}$ a quasivariety. The following are equivalent. \begin{enumerate} \item $l$ is algebraizable with equivalent semantics ${\bf K}$ \item For every algebra $A$ the Leibniz operator $\Omega^{A} : Fi_{l}(A) \to Co_{{\bf K}}(A)$ is an isomorphism of algebraic lattices. \end{enumerate}
Moreover, the quasivariety ${\bf K}$ is unique for providing an equivalent semantics by \cite[Thm 2.15]{BP1}. \end{Teo}
\begin{Teo}(The Isomorphism Theorem, 2nd version \cite[Thm. 3.58]{Fon}).\label{IT1} \label{TheoremSecondIsoThmInclUniquenessOfTheQuasivariety}
Let $l$ be a logic and ${\bf K}$ be a quasivariety. The following conditions are equivalent: \begin{enumerate}[{(i)}] \item $l$ is algebraizable with equivalent algebraic semantics the class ${\bf K}$. \item For every algebra $A$ there is an isomorphism $\Phi^{A}$ between the algebraic lattices $Fi_{l}(A)$ and $Co_{{\bf K}}(A)$ that commutes with endomorphisms, i.e., for every $F\in Fi_{l}(A)$ and every $h\in End(A)$, $\Phi^{A}h^{-1}(F)=h^{-1}\Phi^{A}F$. \item There is an isomorphism of algebraic lattices $\Phi^{Fm} \colon \mathcal{T}h(l) \to Co_{{\bf K}}(Fm)$ that commutes with substitutions, i.e., for every $T\in \mathcal{T}h(l)$ and every $\sigma\in End(Fm)$, $\Phi\sigma^{-1}T=\sigma^{-1}\Phi T$. \end{enumerate} If the above conditions are satisfied, the isomorphism of (iii) is unique and given by the Leibniz operator $\Omega$. It can be expressed by via set $\Delta(p,q)$ of formulas in two variables as $\Omega(F)=\{\langle \varphi, \psi \rangle \mid \Delta(\varphi, \psi) \subseteq F \}$ and its inverse can be expressed via a set of equations $\delta(x)=\epsilon(x)$ as $\Omega^{-1}(\theta)=\{ \varphi \in Fm \mid \langle \delta(x),\epsilon(x)\rangle \subseteq \theta \}$.
For a more general algebra $A$, the same set of formulas, resp. set of equations, induce isomorphisms and there is at most one way to express an isomorphism by equations in this way (\cite[Exercise 3.40]{Fon}). \end{Teo}
The following corollary was the main motivation to define the congruence filter pairs in order to establish conditions to get algebraizable logics knowing previously a quasivariety of some similarity type $\Sigma$.
\begin{Cor}(The Isomorphism Theorem, 3rd version)
Let $l$ be a logic and ${\bf K}$ be a quasivariety. Then following conditions are equivalent: \begin{enumerate} \item $l$ is an algebraizable logic with equivalent algebraic semantics the class ${\bf K}$. \item There is a natural isomorphism between the functors $Fi_{l}$ and $Co_{{\bf K}}$. \end{enumerate} \end{Cor}
\n{\bf Proof:\;\;}
$''1\Rightarrow 2''$ Suppose that $l$ is an algebraizable logic. By theorem \ref{IT2} we have that for every $A\in\Sigma\text{-Str}$, the Leibniz operator $\Omega^{A}:Fi_{l}(A)\to Co_{K}(A)$ is a isomorphism. Let $\Omega=(\Omega^{A})_{A\in\Sigma\text{-Str}}$. We prove that $\Omega$ is a natural transformation. In other to do that, it is enough to prove that given $f\in Hom_{\Sigma\text{-Str}}(A,B)$, the following diagram commutes
\[ \xymatrix{ Fi_{l}(A)\ar[r]^{\Omega^{A}}&Co_{{\bf K}}(A)\\ Fi_{l}(B)\ar[u]^{f^{-1}}\ar[r]_{\Omega^{B}}&Co_{{\bf K}}(B)\ar[u]_{{(f \times f)}^{-1}} }\]
Let $F\in Fi_{l}(B)$. Firstly we prove that $f^{-1}(\Omega^{B}(F))$ is compatible with $f^{-1}(F)$. Let $(a,b)\in f^{-1}(\Omega^{B}(F))$ and suppose that $a\in f^{-1}(F)$. Then $(f(a),f(b))\in \Omega^{B}(F)$ and $f(a)\in F$. Therefore $f(b)\in F$, thus $b\in f^{-1}(F)$. Thus $(a,b)\in \Omega^{A}((f \times f)^{-1}(F))$.
Now let $(a,b)\in \Omega^{A}(f^{-1}(F))$, then by algebraizability of $l$, we have $\Delta^{A}(a,b)\in f^{-1}(F)$. Thus \linebreak $\Delta^{B}(f(a),f(b)) = f(\Delta^{A}(a,b))\subseteq F$. Therefore $(f(a),f(b))\in \Omega^{B}(F)$ and finally $(a,b)\in f^{-1}(\Omega^{B}(F))$. So $\Omega^{A}(f^{-1}(F))=f^{-1}(\Omega^{B}(F))$, which means that $\Omega$ is natural. \vspace*{0.3cm}
$''2\Rightarrow 1''$ Suppose that there is a natural isomorphism $\Phi:Fi_{l}\to Co_{{\bf K}}$. In particular we have that for every $A\in\Sigma\text{-Str}$, $\Phi^{A}:Fi_{l}(A)\to Co_{{\bf K}}(A)$ is a isomorphism and commutes with endomorphisms. By theorem \ref{IT1} we have that $l$ is an algebraizable logic. \qed \vspace*{0.3cm}
We consider now some special kinds of logics that are generalizations for algebraizable logics (see \cite{Fon}).
\begin{Df}[protoalgebraic, truth-equational]\label{SpecialLogics} Let $l=(\Sigma,\vdash)$ a logic: \begin{itemize} \item $l$ is a {protoalgebraic} logic if for any theory $T\in Th(l)$, \[ if \langle\varphi,\psi\rangle\in\Omega(T)\ then\ T,\varphi\vdash\psi\ and\ T,\psi\vdash\varphi.\]
\item A class of matrices $M$ has its filters equationally definable by a set of equations $\tau(p)$ if for every matrix $\langle A,F\rangle\in M$, for every $a\in A$, \[a\in F\ \ iff\ \ \delta^{A}(a) = \varepsilon^{A}(a),\ for\ every\ \delta\approx\varepsilon\in \tau(p).\] $l$ is a {truth-equational} logic if the class of reduced matrix $Matr^{*}_l$ has its filters equationally definable.
\item Let $K$ be a pointed class of algebra of a similarity type $\Sigma$. $l$ is the assertional logic of $K$ if, \[\Gamma\vdash\varphi \Leftrightarrow \{\gamma\approx T\ :\ \gamma\in\Gamma\}\models_{K}\varphi\approx
T\] \end{itemize} \end{Df}
Now we recall some characterizations of the classes of logics defined above.
\begin{Teo}\label{teocar} Let $l$ be a logic: \begin{itemize} \item $l$ is protoalgebraizable iff $\Omega$ is monotone on set of theories $Th(l)$. \item $l$ is equivalential iff $(\Omega^{A})_{A\in\Sigma\text{-Str}}$ commutes with homomorphism (categorial naturality) and $\Omega$ is monotone. \item $l$ is truth-equational iff there exists a set of equations $\tau(p)$ such that for every algebra $A$ and every $F\in Fi_{l}(A)$, \[F=\{a\in A;\ \tau^{A}(a)\subseteq\Omega^{A}(F)\}\] \end{itemize} \end{Teo}
The following diagram summarizes the relations between the different classes of logics mentioned in this section.
\[ \xymatrix@=0.4em{ &\text{algebraizable}\ar[dr]\ar[dl]&& \text{regularly weakly algebraizable} \ar[dl]\ar[rd]&\\ \text{equivalential}\ar[dr]&& \text{weakly algebraizable}\ar[dl]\ar[dr]&&\text{assertional}\ar[ld]\\ &\text{protoalgebraic} &&\text{truth-equational} \ar[dl]\\ && \text{has an algebraic semantics} & }\]
\section{Finitary Filter Pairs}\label{SectionFilterFunctors}
To motivate the notion of filter pair, we recall from Remark \ref{RemarkFunctorialityOfFilters} that, given a logic $l=(\Sigma, \vdash)$, one can associate to each $\Sigma$-structure $M$ its collection of $l$-filters $Fi_{l}(M)$ and that, together with taking inverse image, this constitutes a functor $Fi_l \colon \Sigma\text{-Str} \to AL$, from $\Sigma$-structures to algebraic lattices. Also recall from Remark \ref{RemarkFunctorialityOfFilters} that for $M\in \Sigma\text{-Str}$ the inclusion $i_{M} : (Fi_{l}(M), \subseteq) \hookrightarrow (\mathcal{P}(M),\subseteq)$ preserves arbitrary infima and directed suprema, i.e., it is a morphism in the category $AL$ (in particular preserves order).
Moreover, given a morphism $h:M\to N$ we have the following commutative diagram:
\[ \xymatrix{ M\ar[d]_{h}&Fi_{l}(M)\ar@{^(->}[r]^{i_{M}}&(\mathcal{P}(M),\subseteq)\\ N&Fi_{l}(N)\ar[u]^{h^{-1}}\ar@{^(->}[r]_{i_{N}}&(\mathcal{P}(N), \subseteq)\ar[u]_{h^{-1}}\\ } \]
This collection of data is the motivating example for the notion, and the name, of filter pair.
\begin{Df}\label{DefinitionFilterPair} Let $\Sigma$ be a signature. A {\bf filter pair} over $\Sigma$ is a pair $(G,i)$, consisting of a contravariant functor $G:\Sigma\text{-Str}\to AL$ and a collection of maps $i=(i_{M})_{M\in \Sigma\text{-Str}}$ such that for any $M\in\Sigma\text{-Str}$ the function $i_{M}:G(M)\to (\mathcal{P}(M),\subseteq)$ satisfies the following properties:
{\bf 1.} For any $M\in\Sigma\text{-Str}$, $i_{M}$ preserves arbitrary infima (in particular $i_{M}(\top)=M$) and directed suprema.
{\bf 2.} Given a morphism $h:M\to N$ the following diagram commutes:
\[ \xymatrix{ M\ar[d]_{h}&G(M)\ar[r]^{i^{G}_{M}}&(\mathcal{P}(M);\subseteq)\\ N&G(N)\ar[u]^{G(h)}\ar[r]_{i^{G}_{N}}&(\mathcal{P}(N);\subseteq)\ar[u]_{h^{-1}} } \] \end{Df}
\begin{Obs}
Condition {\bf 2.} says that $i$ is a natural transformation from $G$ to the functor $\wp\colon \Sigma\text{-Str}^{op}\to AL$ sending a $\Sigma$-structure to the power set of its underlying set and a homomorphism of $\Sigma$-structures to its associated inverse image function. \end{Obs}
From a filter pair we obtain a consequence relation on every $\Sigma$-structure, i.e. an abstract logic in the sense of Definition \ref{DefAbstractLogic}.
\begin{Prop}\label{LogicsFromFilterPairs} Let $(G,i)$ be a filter pair over the signature $\Sigma$ and $A$ a $\Sigma$-structure. Then there is a generalized logic $l_{G}^A=(A,\vdash_{G}^A)$, defined as follows:\\ Given $\Gamma\cup\{\varphi\}\subseteq A$, define \[\Gamma\vdash_{G}\varphi\ \ i\!f\!\!f\ \ for\ any\ a\in G(\Fm{}{X}),\ if\ \Gamma\subseteq i_{\Fm{}{X}}(a)\ then\ \varphi\in i_{\Fm{}{X}}(a).\] \end{Prop}
\n{\bf Proof:\;\;}
It is easy to see that $\vdash_{G}^A$ satisfies reflexivity, cut and monotonicity.
The structurality is a consequence of condition {\bf 2} (naturality). Indeed, let $\sigma\in hom(\Fm{}{X},\Fm{}{X})$ and $\Gamma\cup\{\varphi\}\subseteq \Fm{}{X}$ such that $\Gamma\vdash_{G}\varphi$. Consider $a\in G(\Fm{}{X})$ such that $\sigma[\Gamma]\subseteq i^{G}_{\Fm{}{X}}(a)$. This implies $\Gamma\subseteq\sigma^{-1}(i^{G}_{\Fm{}{X}}(a))$. By naturality we have $\sigma^{-1}(i^{G}_{\Fm{}{X}}(a))=i^{G}_{\Fm{}{X}}(G(\sigma)(a))$. Therefore $\varphi\in i^{G}_{\Fm{}{X}}(G(\sigma)(a))=\sigma^{-1}(i^{G}_{\Fm{}{X}}(a))$ and finally $\sigma(\varphi)\in i^{G}_{\Fm{}{X}}(a)$.
Now we are going to prove the finitarity. Let $\Gamma\cup\{\varphi\}\subseteq \Fm{}{X}$. Consider the set $S=\{\Gamma'\subseteq \Fm{}{X};\ \Gamma'\subseteq_{fin}\Gamma\}$. Notice that $S$ is a directed set. Suppose that for any $\Gamma'\in S,\ \Gamma'\nvdash_{G}\varphi$, hence there is $a\in G(\Fm{}{X})$ such that $\Gamma'\subseteq i_{\Fm{}{X}}(a)$ and $\varphi\not\in i_{\Fm{}{X}}(a)$. Denote by $a_{\Gamma'}=\wedge\{a\in G(\Fm{}{X});\ \Gamma'\subseteq i_{\Fm{}{X}}(a)\}$ . $i_{\Fm{}{X}}$ preserves inf, thus $\Gamma'\subseteq i_{\Fm{}{X}}(a_{\Gamma'})$ and $\varphi\not\in i_{\Fm{}{X}}(a_{\Gamma'})$. We obtain that the set $s=\{a_{\Gamma'};\ \Gamma'\in S\}$ is a directed set.
By the assumption $i_{\Fm{}{X}}$ preserves directed suprema, hence \[\Gamma=\cup S\subseteq \bigcup_{\Gamma'\in S}i_{\Fm{}{X}}(a_{\Gamma'})=i_{\Fm{}{X}}(\vee s).\] On the other hand $\varphi\not\in\bigcup_{\Gamma'\in S'}i_{\Fm{}{X}}(a_{\Gamma'})=i_{\Fm{}{X}}(\vee s)$. Therefore $\Gamma\nvdash_{G}\varphi.$\qed
\begin{Obs}
The main instance of interest of the construction of Prop. \ref{LogicsFromFilterPairs} is the case $A=\Fm{\Sigma}{X}$, the formula algebra over some set $X$ of variables. In this case one obtains a logic in the usual sense of the word, which we will denote by $(\Sigma, \vdash^X)$. Thus one can see a filter pair (together with a set $X$ of variables) as a \emph{presentation} of a logic, different in style from the usual presentations by axioms and rules or by matrices. It is clear from the definition that the logic defined in this way does not depend on the values of the filter pair at $\Sigma$-structures other than $\Fm{\Sigma}{X}$, and indeed it only depends on \emph{the image of the map} $G(\Fm{\Sigma}{X}) \to \wp(\Fm{\Sigma}{X})$, as this is exactly the collection of theories of the logic, by definition.
Thus, just as an algebraic structure can have many different presentations by generators and relations, a logic can have presentations by different filter pairs, each of which can be useful for different purposes. \end{Obs}
\begin{Ex}\label{ExampleTheFilterPairOfFilters} Given a Tarskian logic $l=(\Sigma,\vdash)$, by Remark \ref{RemarkFunctorialityOfFilters}, defining $Fi_{l}(A)$ to be the set of $l$-filters on a $\Sigma$-structure $A$ provides a functor $Fi_{l} : \Sigma\text{-Str}^{op}\to AL$, and hence a filter pair $(Fi_{l},i)$ where and $i$ is the inclusion of filters into all subsets. As filters on the formula algebra are exactly the theories, this shows that every logic admits a presentation by a filter pair. \end{Ex}
There is a direct description of the closure operator associated to the consequence relation of Prop. \ref{LogicsFromFilterPairs} through an application of the notion of adjunctions between posets. Since the natural transformation $i$ of a filter pair $(G,i)$ preserves infima, by Theorem \ref{TheoremExistenceOfLeftAdjoint} for each algebra $A$, the map $i_A$ has a (unique) left adjoint.
\begin{Df}\label{DefLeftAdjointOfAFilterPair}
Let $(G,i)$ be a filter pair. The left adjoint of $i_A$ will be denoted by $\Xi(i)_A$. \end{Df}
Note that in general there is no reason that the collection of left adjoints $\Xi(i)_A\ \ (A \in \Sigma\text{-Str})$ should form a natural transformation. It is Nevertheless a resource that comes for free with a filter pair and can be usefully employed to understand the properties of the associated logics.
\begin{Prop}\label{PropDescriptionOfConsequenceRelationByClosureOperator}
Let $(G,i^{G})$ be a filter pair and $A$ a $\Sigma$-structure. Then the closure operator on $A$ associated to the consequence relation $\vdash_{G}^A$ on $A$ of Prop. \ref{LogicsFromFilterPairs} is the one given by $i_A \circ \Xi(i)_A$. \end{Prop}
\n{\bf Proof:\;\;} Recall the definition of $\vdash_{G}^A$: \[D \vdash_{G}^A a\ \ i\!f\!\!f\ \ for\ any\ z\in G(A),\ if\ D\subseteq i_{A}(z)\ then\ a\in i_{A}(z).\]
Consider the set $Z_D:=\{z\in G(A) \mid D\subseteq i_{A}(z) \}$. We can rephrase the definition of $\vdash_G^A$ by saying \[D \vdash_{G}^A a\ \ i\!f\!\!f\ \ \forall z\in Z_D,\ a\in i_{A}(z)\]
Since $G(A)$ is complete, there exists an infimum, say $z'$, of $Z_D$. Since $i_A$ preserves arbitrary infima, we have $i_A(z')=\bigcap_{z \in Z_D} i_A(z)$. Since we have $D \subseteq i_A(z) \forall z \in Z_D$, we also have $D \subseteq \bigcap_{z \in Z_D} i_A(z) = i_A(z')$. Therefore $z' \in Z_D$, i.e. it is the minimal $z'$ for which $i_A(z)$ contains $D$. Since $i_A$ is order preserving, if $a \in i_A(z')$, then $a \in i_A(z)$ for every $z \in Z_D$ and hence $D \vdash_{G}^A a$. The opposite direction is also true, i.e. if $D \vdash_{G}^A a$, then $a \in i_A(z')$, just because $z' \in Z_D$.
Thus we can describe the consequence relation by $D \vdash_{G}^A a \ \text{ iff } \ a \in i_{A}(z')$.
By the formula for left adjoints from Theorem \ref{TheoremExistenceOfLeftAdjoint}, for a subset $D \subseteq A$ we have $\Xi(i)_A(D)=\bigwedge \{z \in G(A) \mid D \leq i_A(z) \} = z'$. Altogether we obtain \[D \vdash_{G}^A a\ \ \text{ iff } \ a \in i_{A}(\Xi(i)_A(D))\] This means exactly that the closure operator $i_A \circ \Xi(i)_A$ is the closure operator associated to the consequence relation $\vdash_G^A$. \qed
In the remainder of the section we will explore the relationship between filters of the logic $(\Sigma, \vdash^X)$ associated to a filter pair $(G,i)$, and sets in the image of $i_A$ for a $\Sigma$-structure $A$. We start by giving a name to the latter.
\begin{Df}\label{DefiFilter}
Let $(G,i)$ be a filter pair and $A$ a $\Sigma$-structure. An $i$-filter in $A$ is a subset $F \subseteq A$ such that for all $D \subseteq A$ the following implication holds: \[ ( D\subseteq F\text{ and }D \vdash_{G}^A a)\ \Rightarrow a\in F \] In other words, $i$-filters in $A$ are the theories for the abstract logic $(A,\vdash)$. \end{Df}
\begin{Lem}\label{Lemi-FiltersAreTheImageOfi} Let $(G,i^{G})$ be a filter pair and $A$ a $\Sigma$-structure. The image of $i_A$ in $\wp(A)$ consists exactly of the $i$-filters in $A$. \end{Lem} \n{\bf Proof:\;\;} The set $i-Fi(A)$ of Filters in a $\Sigma$-algebra $A$ is defined by the following condition: \[F \in i-Fi(A) \ \ \text{ iff } \ [ ( D\subseteq F\text{ and }D \vdash_{G}^A a)\ \Rightarrow a\in F]\] In view of the previous proposition this means \[F \in i-Fi(A) \ \ \text{ iff } \ [ (D\subseteq F\text{ and }a \in (i_A \circ \Xi(i)_A)(D)\ \Rightarrow a\in F]\] It is enough to check the condition on the right hand side on the maximal $D \subseteq F$, i.e. on $F$ itself, which leads to \[F \in i-Fi(A) \ \ \text{ iff } \ [ a \in (i_A \circ \Xi(i)_A)(F) \Rightarrow a\in F] \] Now since $i_A \circ \Xi(i)_A$ is a closure operator, this means exactly that $F$ is closed for this operator, i.e. $F=(i_A \circ \Xi(i)_A)(F)$, hence $F$ is in the image of $i_A$. Vice versa, it is clear from the properties of Galois adjunctions that sets in the image of $i_A$ are closed. \qed
The notions of $i$-filter for a filter pair and filter for a logic associated to that filter pair are, in general, different. The content of the next proposition is that one notion subsumes the other. It can be shown in relevant examples that this inclusion is strict.
\begin{Prop}\label{PropImageOfiConsistsOfFilters} Let $(G,i)$ be a filter pair, $X$ a set and $l=(\Sigma,\vdash^X)$ the associated logic with set of variables $X$. Then for any algebra $A$ the subsets in the image of $i_A$ are $l$-filters. \end{Prop} \n{\bf Proof:\;\;} Let $A$ be an algebra, $F=i_A(x) \subseteq A$ a subset in the image of $i$, $\Gamma \cup \{\varphi\} \subseteq \Fm{}{X}$ formulas with $\Gamma \vdash^X \varphi$ and $v \colon \Fm{}{X} \to A$ a homomorphism with $v(\Gamma) \subseteq F$. We need to show that $\varphi \in F$. For this consider the naturality square
\[ \xymatrix{ \Fm{}{X}\ar[d]_{v}&G(\Fm{}{X})\ar[r]^{i^{G}_{\Fm{}{X}}}&(\mathcal{P}(\Fm{}{X});\subseteq)\\ A&G(A)\ar[u]^{G(v)}\ar[r]_{i^{G}_{A}}&(\mathcal{P}(A);\subseteq)\ar[u]_{v^{-1}} } \]
We have $\Gamma \subseteq v^{-1}(F) = v^{-1}(i_A(x)) = i_{\Fm{}{X}}(G(v)(x))$,
So $v^{-1}(F)$, lying in the image of $i_{\Fm{}{X}}$, is a theory containing $\Gamma$. The assumption $\Gamma \vdash^X \varphi$ then implies $\varphi \in v^{-1}(F)$, i.e. $v(\varphi) \in F$. \qed
\begin{Prop}\label{PropiMatricesDefineTheSameLogicAsAllMatrices} Given a filter pair $(G,i)$ and a set $X$, the consequence relation $\vdash$ on $\Fm{}{X}$ defined by
\begin{center} $\Gamma\vdash\varphi$ iff for any $\Sigma$-structure $M$, for any $a\in G(M)$ and any valuation $v:\Fm{}{X}\to M$, if $v(\Gamma)\subseteq i_{M}(a)$ then $v(\varphi)\in i_{M}(a)$
\end{center} coincides with the consequence relation $\vdash^X$. \end{Prop} \n{\bf Proof:\;\;} Suppose $\Gamma\vdash\varphi$. Then, taking the identity as valuation, one has that if $\Gamma\subseteq i_{\Fm{}{X}}(a)$ then $\varphi\in i_{\Fm{}{X}}(a)$ for all $a \in G(\Fm{}{X})$. Since the sets in the image of $i_{\Fm{}{X}}$ are precisely the theories of the logic $l:=(\Sigma,\vdash^X)$, this shows $\Gamma \vdash^X \varphi$.
On the other hand suppose $\Gamma\vdash^X\varphi$. Then for any $l$-matrix $(M,F)$ and valuation $v:\Fm{}{X}\to M$ one has that if $v(\Gamma)\subseteq F$ then $v(\varphi)\in F$. Since by Proposition \ref{PropImageOfiConsistsOfFilters} all subsets in the image of $i_M$ are filters, this implies that the defining condition for $\Gamma\vdash\varphi$ is satisfied. \qed
\begin{Obs}\label{RemarkAlternativeDescriptionsOFLogicsFromFilterPairs} \begin{enumerate}[(a)] \item Call a matrix $(A,F)$ an $i$-matrix if the filter $F$ is in the image of $i_A$. Denoting the collection of all $i$-matrices by $i\text{-}Matr$, and the collection of \emph{all} matrices for the logic $l=(\Sigma, \vdash^X)$ by $l\text{-}Matr$, Proposition \ref{PropImageOfiConsistsOfFilters} can be subsumed as stating an inclusion $i\text{-}Matr \subseteq l\text{-}Matr$. Proposition \ref{PropiMatricesDefineTheSameLogicAsAllMatrices} then says that, although this inclusion can be strict, the two classes of matrices always define the same logic.
\item The inclusions of $i$-filters into all filters, established by Proposition \ref{PropImageOfiConsistsOfFilters} are also easily seen to form a natural transformation. Thus we can consider the natural transformation $i^{G}:G\Rightarrow Fi_{l}$. This exhibits $(Fi_l, i)$ as a weakly terminal filter pair among all filter pairs presenting the same logic $l$. The collection of filter pairs presenting a given fixed logic will be studied in a follow-up work. \end{enumerate} \end{Obs}
The following two propositions establish that, although the notions of $i$-filter and $l$-filter (where $l=(\Sigma,\vdash^X)$ is the logic with variables $X$) may differ for general algebras, this is not the case \emph{for an absolutely free algebra} $\Fm{}{Z}$ (with a set of generators $Z$ possibly different from $X$).
\begin{Prop}\label{2.19}\label{PropHomomorphismsInduceTranslationsAndVariableInclusionsInduceConservativeTranslations} Let $(G,i)$ be a filter pair over the signature $\Sigma$. \begin{enumerate}[(i)] \item For any homomorphism of $\Sigma$-structures $f\colon A\to B$ and $\Gamma\cup\{\varphi\}\subseteq A$: \[\Gamma\vdash^{A} \varphi\ \Rightarrow\ f[\Gamma]\vdash^{B} f(\varphi).\] \item For any injective map of sets $f:X\rightarrowtail Y$ and $\Gamma\cup\{\varphi\}\subseteq \Fm{}{X}$: \[\Gamma\vdash^{X}\varphi\ \Leftrightarrow\ f[\Gamma]\vdash^{Y} f(\varphi).\] \end{enumerate} \end{Prop}
\n{\bf Proof:\;\;} $\emph{(i)}$ Suppose $\Gamma\vdash^{A} \varphi$. Let $z\in G(B)$ such that $f[\Gamma]\subseteq i_{B}(z)$. Then $\Gamma\subseteq f^{-1}(i_{B}(z))=i_{A}(G(f)(z))$. Since $\Gamma\vdash^{A}\varphi$, we have that $\varphi\in i_{A}(G(f)(z))$. Therefore $f(\varphi)\in i_{B}(z)$. As $z$ was arbitrary we have $f[\Gamma]\vdash^{B} f(\varphi)$. \vspace*{0.3cm}
$\emph{(ii)}$ Let $f:X\to Y$ be injective. By $\emph{1}$ we have that $\Gamma\vdash^{X}\varphi\ \Rightarrow\ f[\Gamma]\vdash^{Y}f(\varphi)$. It remains to prove the converse. Let $z\in G(\Fm{}{X})$ such that $\Gamma\subseteq i^{G}_{X}(z)$. Since $f$ is injective there is a $g\colon Y\to X$ such that $g\circ f=Id_{X}$. Hence $g\circ f[\Gamma]=\Gamma$. Therefore $f[\Gamma]\subseteq g^{-1}(i_{X}(z))=i_{Y}(G(g)(z))$. Since $f[\Gamma]\vdash^{Y}f(\varphi)$, we have $f(\varphi)\in i^{G}_{Y}(G(g)(z))=g^{-1}(i^{G}_{X}(z))$. Therefore $\varphi=g(f(\varphi))\in i^{G}_{X}(z)$. \qed
\begin{Prop}\label{FreeAlgebraResult} Let $(G,i)$ be a filter pair and $l:=(\Fm{}{X}, \vdash^X)$ the associated logic with an infinite set of variables $X$. Let $Z$ be another set and $F\subseteq \Fm{}{Z}$ an $l$-filter. Then $F=i_Z(\Xi(F))$. In particular all filters in absolutely free algebras lie in the image of $i$. \end{Prop} \n{\bf Proof:\;\;} From the general properties of adjunctions we know $F \subseteq i_Z(\Xi(F))$. It remains to show $F \supseteq i_Z(\Xi(F))$. So let $\varphi \in i_Z(\Xi(F))$. By Prop. \ref{PropDescriptionOfConsequenceRelationByClosureOperator} this means $F \vdash^Z \varphi$.
By Prop. \ref{LogicsFromFilterPairs} the logic $(\Fm{}{Z}, \vdash^Z)$ is finitary, so there is a finite subset $F' \subseteq F$ such that $F' \vdash^Z \varphi$. Then there is a finite set $V \subseteq Z$ of variables occurring in the formulas of $F' \cup \{\varphi\}$. Choose a map $h \colon Z \to X$ which is injective on $V$ (this is possible since $X$ is infinite) and a map $v \colon X \to Z$ such that $v \circ h|_V = id_V$. We continue denoting the induced maps on formula algebras by $h$, resp. $v$. By construction we then have $v(h(F))=F$ and $v(h(\varphi))=\varphi$.
From $F' \vdash^Z \varphi$ and Prop. \ref{PropHomomorphismsInduceTranslationsAndVariableInclusionsInduceConservativeTranslations}(i) it follows that $h(F') \vdash^X h(\varphi)$, i.e. $h(F')$ implies $h(\varphi)$ in the logic $l$.
Now $v$ is a valuation such that $v(h(F'))=F'\subseteq F$ and $F$ was supposed to be an $l$-filter, so $\varphi = v(h(\varphi)) \in F$. \qed
\section{Congruence filter pairs }\label{SectionFilterPairsOverCo}
Motivated by the example of algebraizable logics, we introduce in this section a special class of filter pairs, namely the filter pairs whose functor part is given by $Co_{{\bf K}}: \Sigma\text{-Str}\to AL$, where ${\bf K}\subseteq \Sigma\text{-Str}$ is a class of $\Sigma$-structures, and for $M \in \Sigma\text{-Str}$ $Co_{{\bf K}}(M)$ denotes the ordered set of congruences whose associated quotient lies in ${\bf K}$. We will call such filter pairs \emph{congruence filter pairs}.
We begin by studying the question which classes ${\bf K}\subseteq \Sigma\text{-Str}$ are eligible for the formation of such a filter pair, i.e. when $Co_{{\bf K}}$ becomes a functor with values in algebraic lattices.
We will provide some examples of congruence filter pairs and present results relating these filter pairs with classes of logics in the Leibniz hierarchy.
\subsection{Functors of congruences}
To begin, we record some conditions on a class ${\bf K} \subseteq \Sigma\text{-Str}$ for obtaining a functor $Co_{\bf K} \colon \Sigma\text{-Str} \to Ord$, where $Ord$ denotes the category of partially ordered sets and order preserving maps.
\begin{Prop}\label{PropCoKIsAFunctor}
Let $\Sigma$ be a signature and ${\bf K} \subseteq \Sigma\text{-Str}$ a class of algebras that is closed under subalgebras and isomorphisms. Then there is a functor $Co_{\bf K} \colon \Sigma\text{-Str}^{op} \to Ord$ associating to a $\Sigma$-structure $A$ the set of congruences $\theta$ such that $A/\theta \in {\bf K}$ and to a homomorphism $f \colon A \to B$ the function $$Co_{\bf K}(f)\colon Co_{\bf K}(B) \ni \theta \mapsto (f \times f)^{-1}(\theta)=\{\langle a,a' \rangle \mid \langle f(a),f(a') \rangle \in \theta \} \in Co_{\bf K}(A)$$
Conversely, if ${\bf K} \subseteq \Sigma\text{-Str}$ is a class of algebras such that $Co_{\bf K} \colon \Sigma\text{-Str}^{op} \to Ord$ is a functor, which on morphisms is given by $Co_{\bf K}(f)=(f \times f)^{-1}$, then ${\bf K}$ is closed under substructures. \end{Prop} \n{\bf Proof:\;\;}
The set-theoretic preimage of a congruence is a congruence again: Indeed, congruences are exactly the set-theoretic preimages of the equality relation (Denoted $\Delta$) along homomorphisms. So, if $\theta \in Co_{\bf K}(B)$ is of the form $(g \times g)^{-1}(\Delta_C)$ for some $g \colon B \to C$, then $(f \times f)^{-1}(\theta)=((g \times g) \circ (f\times f))^{-1}(\Delta_C)$, and hence is a congruence again.
To see that the quotient by this congruence lies in ${\bf K}$, suppose that $\theta \in Co_{\bf K}(B)$. From the isomorphism theorems of universal algebra we obtain a commutative square \[ \xymatrix{ A \ar@{->>}[d] \ar[r]^{f}&B \ar@{->>}[d]\\ A/f^{-1}(\theta)\ar@{>->}[r]_{\bar{f}}& B/\theta } \] where the lower horizontal map is injective. So $A/f^{-1}(\theta)$ is isomorphic to a subalgebra of $B/\theta$, hence lies in ${\bf K}$ again.
Conversely, suppose that $Co_{\bf K}$ is a functor as described and let $B \in {\bf K}$ and $h \colon A \hookrightarrow B$ a substructure. Then $\Delta_B \subseteq B \times B$ is in $Co_{\bf K}(B)$. Hence so is $(h \times h)^{-1}(\Delta_B)=\Delta_A$, hence $A=A/\Delta_A \in {\bf K}$. \qed
The following result is essentially a result attributed to Malcev (see \cite{Art} Theorem 1.37, pp 869):
\begin{Prop}\label{PropKClosedUnderProducts}
Let $\Sigma$ be a signature and ${\bf K} \subseteq \Sigma\text{-Str}$ a class of algebras that is closed under subalgebras and isomorphisms and $A \in \Sigma\text{-Str}$.
Are equivalent:
(i) $Co_{\bf K}(A)$ has a minimal element;
(ii) ${\bf K}$ is
closed under products;
(iii) ${\bf K}$ is a reflective subcategory of $\Sigma\text{-Str}$.
\end{Prop}
\qed
\begin{Cor}\label{CorollaryUnderVopenkasPrincipleKMustBeAQuasivariety}
Assuming Vop{\v e}nka's principle, under the conditions of Prop. \ref{PropKClosedUnderProducts} the class ${\bf K}$ is a quasivariety. \end{Cor} \n{\bf Proof:\;\;} Assuming Vop{\v e}nka's principle, by the Corollary after Theorem 6.14 in \cite{AR}, a class of $\Sigma$-structures is a quasivariety if and only if it is closed under subalgebras and products. \qed
In view of Cor. \ref{CorollaryUnderVopenkasPrincipleKMustBeAQuasivariety} it seems possible that the question whether under the conditions of Prop. \ref{PropKClosedUnderProducts} ${\bf K}$ must be a quasivariety is undecidable. However, for our application to filter pairs we want $Co_{\bf K}$ to be taking values in algebraic lattices and can ask whether this condition forces ${\bf K}$ to be a quasivariety. We have not been able to settle this question and leave it as a question.
\begin{Que}
Let $\bf K$ be a class of $\Sigma$-structures that is closed under subalgebras and isomorphisms and suppose that the functor $Co_{\bf K}$ of Prop. \ref{PropCoKIsAFunctor} takes values in algebraic lattices. Is $\bf K$ then necessarily a quasivariety? \end{Que}
\begin{Obs}
One way of approaching the above question is to note that the proof of Proposition \ref{PropKClosedUnderProducts} shows that, in the situation in question ${\bf K}$ is a regular epi-reflective subcategory of $\Sigma\text{-Str}$, i.e. a reflexive subcategory for which the unit morphisms of the adjunction are regular epimorphisms. It is a consequence of this, as shown in the proof, that ${\bf K}$ is closed under subalgebras and products.
Quasivarieties are precisely the regular epi-reflective subcategories of $\Sigma\text{-Str}$ which are also closed under directed colimits (see e.g. \cite{AdamekRosickyVitale}, Cor. 10.24 together with the remarks on the next page). A quick proof of the relevant direction proceeds by noting that all we need for having a quasivariety is closure under ultraproducts, and ultraproducts are filtered colimits of pdiagrams of products.
The question can thus be reformulated as asking whether the additional assumption that $Co_{\bf K}$ takes values in algebraic lattices forces ${\bf K}$ to be closed under directed colimits, and hence to be a quasivariety.
To see the possible connection, note that the reflection functor is given by taking certain quotients. From this we obtain a reflection of ordered sets congruences, i.e. a left inverse $r \colon Co_{\Sigma\text{-Str}}(A) \to Co_{\bf K}(A)$ to the inclusion $Co_{\bf K}(A) \hookrightarrow Co_{\Sigma\text{-Str}}(A)$: given a congruence $\theta \in Co_{\Sigma\text{-Str}}(A)$, we define $r(\theta):=(p \times p)^{-1}(min_{A/\theta}) \in Co_{\bf K}(A)$ where $p$ denotes the projection $p \colon A \to A/\theta$.
By \cite[Prop 1.4.12]{Gorbunov}, a class $\mathbf{K}$ is a quasivariety iff $Co_{\mathbf{K}}(A) \subseteq Co(A)$ is an algebraic sublattice for every structure $A$. Thus the question is, whether the inclusion $Co_{\mathbf{K}}(A) \subseteq Co(A)$ preserves directed suprema (which could be used to show closure of $\bf K$ under directed colimits). A related question is whether the compact elements of $Co_{\bf K}(A)$ are exactly the finitely generated ${\bf K}$-congruences, i.e. the images under the lattice reflection of the compact elements of $Co_{\Sigma\text{-Str}}(A)$. \end{Obs}
\subsection{Congruence filter pairs}
\begin{Df}
Let $\Sigma$ be a signature and ${\bf K}$ a class of $\Sigma$-structures such that the association $A \mapsto Co_{\bf K}(A)$ is (the object part of) a functor from $\Sigma$-structures to algebraic lattices. A filter pair of the form $(Co_{\bf K}(A), i)$ is called congruence filter pair. \end{Df}
In all examples in this article ${\bf K}$ will actually be a quasivariety, and in this case the set $Co_{\bf K}(A)$ of congruences relative to ${\bf K}$, ordered by inclusion, is an algebraic lattice. The next proposition shows how in this case a presentation by a congruence filter pair is closely linked to the algebraizability of a logic.
\begin{Prop}\label{2.3}\label{CriterionAlgebraizable} Let $\Sigma$ be a signature and ${\bf K}\subseteq\Sigma\text{-Str}$ a quasivariety. If $(Co_{{\bf K}},i)$ is a filter pair, such that $i_{\Fm{}{X}}\colon Co_{\bf K}(\Fm{}{X}) \to \wp(\Fm{}{X})$ is injective, then the associated logic is algebraizable. \end{Prop}
\n{\bf Proof:\;\;} We know from Lemma \ref{Lemi-FiltersAreTheImageOfi} that $i^{{\bf K}}_{Fm}[Co_{{\bf K}}(Fm)]=Th(l_{{\bf K}})$. As $i^{{\bf K}}_{Fm}$ is injective, we have that $i^{{\bf K}}_{Fm}$ is bijective. Then $i^{{\bf K}}_{Fm}$ is an isomorphism. Now let $\sigma\in hom(Fm,Fm)$. As $i^{{\bf K}}$ is a natural transformation we have the following commutative diagram: \[ \xymatrix{ Fm\ar[d]_{\sigma}&Co_{{\bf K}}(Fm)\ar[r]^{i^{{\bf K}}_{Fm}}&(\mathcal{P}(Fm);\subseteq)\\ Fm&Co_{{\bf K}}(Fm)\ar[u]^{Co_{{\bf K}}(\sigma)}\ar[r]_{i^{{\bf K}}_{Fm}}&(\mathcal{P}(Fm);\subseteq)\ar[u]_{\sigma^{-1}} } \] Note that $\sigma^{-1}(T)\in Th(l_{{\bf K}})$ for any $T\in Th(l_{k})$. Therefore $i^{{\bf K}}_{Fm}$ is a isomorphism such that commutes with substitution. By the isomorphism theorem \ref{IT1}, $l_{{\bf K}}$ is an algebraizable logic.\qed
\begin{Obs}
The above proposition gives us an alternative proof of theorem 5.2 \cite{BR}. The condition of injectivity assumed here is exactly the condition assumed in loc. cit. to get algebraizability. \end{Obs}
The next theorem will provide us with an ample supply of examples of congruence filter pairs.
\begin{Teo}\label{equations}\label{TheoremLogicsFromEquations} Let $\Sigma$ be a signature, ${\bf K}\subseteq\Sigma\text{-Str}$ a quasivariety and $\tau$ a finite set of equations in at most one variable. The map $i^{{\bf K}}=(i^{{\bf K}}_{M})_{M\in\Sigma\text{-Str}}$ where: \[\begin{array}{rcl} i^{\tau}_{M}:Co_{{\bf K}}(M)&\to&(\mathcal{P}(M),\subseteq)\\ \theta&\mapsto&\{m\in M;\ \tau^{M}(m)\in\theta\} \end{array}\] is a natural transformation and for any $M\in \Sigma\text{-Str}$, $i^{\tau}_{M}$ preserves arbitrary infima and directed suprema, i.e., $(Co_{{\bf K}},i^{\tau})$ is a filter pair. \end{Teo}
\n{\bf Proof:\;\;} Let $f\in hom(M,N)$. Denote here $f(\tau^{M}(m))=\{\langle f(\varepsilon^{M}(m)),f(\delta^{M}(m))\rangle;\ \langle\varepsilon,\delta\rangle\in\tau\}$.
\[ \xymatrix{ M\ar[d]_{f}&Co_{{\bf K}}(M)\ar[r]^{i^{{\bf K}}_{M}}&(\mathcal{P}(M);\subseteq)\\ N&Co_{{\bf K}}(N)\ar[u]^{Co_{{\bf K}}(f)}\ar[r]_{i^{{\bf K}}_{N}}&(\mathcal{P}(N);\subseteq)\ar[u]_{f^{-1}} } \]
For $\theta\in Co_{{\bf K}}(N)$ we have
$$\begin{array}{rcl} f^{-1}(i^{{\bf K}}_{N}(\theta))&=&f^{-1}(\{n\in N;\ \tau^{N}(n)\subseteq\theta\})\\ &=&\{m\in M;\ \tau^{N}(f(m))\subseteq\theta\}\\ &=&\{m\in M;\ f(\tau^{M}(m))\subseteq\theta\}\\ &=&\{m\in M;\ \tau^{M}(m)\subseteq Co_{{\bf K}}(f)(\theta)\}\\ &=& i^{{\bf K}}_{M}(f^{-1}(\theta)) \end{array}$$
Thus $i$ is a natural transformation.
Consider a family of congruences $\theta_i \in Co_{{\bf K}}(M)\ \ (i \in I)$.
$$\begin{array}{rcl} i^{{\bf K}}_{M}(\bigcap_{i\in I}\theta_i)&=&\{m\in M;\ \tau^{M}(m)\subseteq\bigcap_{i\in I}\theta_i\}\\ &=&\{m\in M;\ \forall i \in I: \tau^{M}(m)\subseteq \theta_i\}\\ &=&\bigcap_{i\in I} \{m\in M;\ \tau^{M}(m)\subseteq \theta_i\} \end{array}$$
Thus $i_{M}$ preserves arbitrary infima.
Now let $U=\{\theta_{i};\ i\in I\}$ be an upwards directed set.
$$\begin{array}{rcl} i^{{\bf K}}_{M}(\bigvee U)&=&\{m\in M;\ \tau^{M}(m)\subseteq \bigvee U\}\\ &=&\{m\in M;\ \tau^{M}(m)\subseteq \bigcup_{i\in I}\theta_{i}\}\\
&=&\bigcup_{i\in I}\{m\in M;\ \tau^{M}(m)\subseteq\theta_{i}\} \end{array}$$ Here the last equality holds because $U$ is a directed set and the set of equations $\tau$ is finite, hence if each equation in $\tau(m)$ is contained in some $\theta_i$ there is a $\theta$ containing all those $\theta_i$ and hence all equations. Thus $i_{M}$ preserves directed suprema. \qed
\begin{Df}\label{DefinitionEquationalFilterPair} A filter pair $(G,i)$ arising as in Theorem \ref{TheoremLogicsFromEquations} will be called an equational filter pair. The transformation $i$ will be said to be given by the equations $\tau$. We will sometimes let the defining equations be part of the notation for an equational filter pair, as in $(Co_{\bf K}, i^\tau)$ or $(Co_{\bf K}, i^{\delta \approx \epsilon})$ \end{Df}
In Thm. \ref{TeoEveryLogicWithAlgebraicSemanticsComesFromEqunlFilterPair} below we will show that the logics admitting a presentation by a congruence filter pair, are exactly those admitting an algebraic semantics. For example, every algebraizable logic admits such a presentation, with the equations given by the algebraizig pair.
For concreteness, here is a typical example of a logic with a presentation by a congruence filter pair.
\begin{Ex}\label{ExampleImplicationlessFragmentOfIPC} Consider the signature $\Sigma = \{\wedge, \vee, \neg, \top, \bot \}$ and let ${\bf K}$ be the variety of pseudocomplemented distributive lattices. Using the single equation $\langle x, \top \rangle$
By \cite[Thm. 2.6]{BP1}, the logic associated to the filter pair $(Co_{{\bf K}},i)$ is the implicationless fragment of intuitionistic propositional logic $\mathbf{IPL^*}$. In \cite[Thm. 5.13]{BP1} it is shown that this logic is not protoalgebraic. \end{Ex}
One further, admittedly artifical, example of a logic that is neither protoalgebraic nor truth-equational but has an algebraic semantics:
\begin{Ex}\label{ExampleSuccessoralgebra} Consider the signature $\Sigma=(\Sigma)_{n\in\omega}$ which $\Sigma_{1}=\{s\}$ and $\Sigma_{n}=\emptyset$ for all $n\neq 1$. A $\Sigma$-algebra is simply a set with an endomorphism. The free $\Sigma$-algebra $F_{\Sigma}(X)$ on countably many generators $X=\{x_0, x_1, \ldots\}$ is isomorphic to $\mathbb{N}^X$, the disjoint union of countably many copies of the natural numbers, where $x_i \in X$ corresponds to $0$ in the $i$th copy, and the endomorphism $s$ acts as the successor function on each copy.
Consider the functor $Co:=Co_{\Sigma-Alg}$ and the map $i:Co \Rightarrow (\mathcal{P}(\ ),\subseteq)$ given as above by $\tau=\{\langle x,s(x)\rangle\}$. We denote the logic associated to this filter pair by $l_s$.
For a congruence $\theta$ on a $\Sigma$-algebra $A$ one has $i(\theta)=\{ a \in A \mid \bar{a}=s(\bar{a}) \text{ in } A/\theta \}$. Thus $i$ assigns to a congruence $\theta$ the set of elements of $A$ which become $s$-fixed points in $A/\theta$.
The logic $l_s$ is not protoalgebraic because it has no theorems: the theorems of $l_s$ are exactly the elements of $i(\theta_{min})$, where $\theta_{min}$ is the minimal congruence relation on $F_{\Sigma}(X)$. As $F_{\Sigma}(X)/\theta_{min} = F_{\Sigma}(X)$, this is exactly the set of $s$-fixed points in $F_{\Sigma}(X)$, which is empty. \end{Ex}
After having seen, in Prop. \ref{CriterionAlgebraizable}, a criterion for the algebraizability of a logic in terms of a presentation by an equational filter pair, in the next few items we discuss such a criterion for truth-equationality.
\begin{Prop}\label{truth}\label{CriterionTruthEquational} Let $(Co_{\bf K},i)$ be an equational filter pair as in Theorem \ref{TheoremLogicsFromEquations}. If $i_A$ is surjective onto the collection of filters for every algebra $A$, then the associated logic is truth-equational. \end{Prop} \n{\bf Proof:\;\;} By Definition \ref{SpecialLogics} a logic is truth-equational if and only if there is a set of equations defining the filters of all its reduced matrices. But if $i$ is surjective, then \emph{all} filters of all matrices are given by the equations defining $i$. \qed
The following example shows that the condition of $i$ being surjective onto filters is not always satisfied, and moreover, that logics associated to equational filter pairs need not be truth-equational.
\begin{Ex}\label{ExampleSuccessorLogicIsNotTruthEquational} Consider the logic $l_s$ of example \ref{ExampleSuccessoralgebra}. Take the $\Sigma$-algebra $A=\mathbb{N} \cup \{z\}$, where $z$ is an $s$-fixed point and $\mathbb{N}$ is endowed with the (fixed point-free) successor operation. Then $\{z\}$ is a filter on $A$ (belonging to the minimal congruence relation, whose quotient is just $A$ itself). Now $\Omega^A(\{z\})$ is the coarsest congruence relation which does not relate $z$ with any other element. This is the congruence relation identifying all the elements of $\mathbb{N}$ and leaving $z$ alone. Therefore $A/\Omega^A(\{z\})$ has two elements (both of which are $s$-fixed points). Note that $i_A (\Omega^A(\{z\}))$ has two elements and is not $\{z\}$, so that the natural candidate $\tau$ for a set of truth-equations does not work.
There is also no other set of equations defining this filter in the way required for truth-equationality: any equation satisfied by the element $\bar{z}$ in $A/\Omega^A(\{z\})$ is also satisfied by the other element, because the permutation of the two elements is an automorphism of the algebra $A/\Omega^A(\{z\})$. \end{Ex}
There is a natural and large class of examples where the condition of Prop. \ref{CriterionTruthEquational} is satisfied:
\begin{Prop}\label{PointedQuasiVarOmegaIsRetraction} Let ${\bf K}$ be a pointed quasivariety, i.e. a quasivariety over a signature with a constant symbol $0$. Consider the set of equations $\tau=\{\langle x,0\rangle\}$. Then $\Omega^{A}$ is a right inverse of $i^{\tau}_{A}$ for any $A\in\Sigma\text{-Str}$. In particular $i_A$ is surjective onto the collection of filters for every $\Sigma$-algebra $A$. \end{Prop} \n{\bf Proof:\;\;} Note that in the associated logic $l$ we have that $\vdash_l0$, $x,\varphi(x,\bar{z})\vdash_l\varphi(0,\bar{z})$ and $x,\varphi(0,\bar{z})\vdash_l\varphi(x,\bar{z})$ for any $\varphi(x,\bar{z})\in Fm$. Indeed, let $\theta\in Co_{{\bf K}}(Fm)$, then $\langle 0,0\rangle\in\theta$, thus $0\in i^{\tau}_{Fm}(\theta)$ and thus $\vdash_l 0$. Now let $\theta\in Co_{{\bf K}}(Fm)$ and suppose that $x,\varphi(x,\bar{z})\in i^{\tau}_{Fm}(\theta)$, then $\langle x,0\rangle\in\theta$ and $\langle \varphi(x,\bar{z}),0\rangle\in\theta$. Since $\theta$ is a congruence, we have that $\langle\varphi(x,\bar{z}),\varphi(0,\bar{z})\rangle\in\theta$. Therefore $\langle\varphi(0,\bar{z}),0\rangle\in\theta$, so $\varphi(0,\bar{z})\in i^{\tau}_{Fm}(\theta)$. Hence $x,\varphi(x,\bar{z})\vdash_l\varphi(0,\bar{z})$. The same proof can be used to prove that $x,\varphi(0,\bar{z})\vdash_l\varphi(x,\bar{z})$.
Now we are able to prove that for any $A\in\Sigma\text{-Str}$ and $F\in Fi_{l_{{\bf K}}}(A)$, $F=i^{\tau}_{A}(\Omega^{A}(F))$. Let $a\in F$ and $\varphi(x,\bar{z})\in Fm$. Let $\bar{c}\in A$ and suppose that $\varphi^{A}(a,\bar{c})\in F$. Since $x,\varphi(x,\bar{z})\vdash_l\varphi(0,\bar{z})$, we have that $\varphi^{A}(0,\bar{c})\in F$. Analogously we have that if $\varphi^{A}(0,\bar{c})\in F$, $\varphi^{A}(a,\bar{c})\in F$. Hence $\langle a,0\rangle\in\Omega^{A}(F)$. By definition of $i^{\tau}_{A}$ we have $a\in i^{\tau}_{A}(\Omega^{A}(F))$. Thus $F\subseteq i^{\tau}_{A}(\Omega^{A}(F))$. Let $a\in i^{\tau}_{A}(\Omega^{A}(F))$, then $\langle a,0\rangle\in\Omega^{A}(F)$. Since $\vdash_l 0$, we have $0\in F$, therefore $a\in F$. Hence $i^{\tau}_{A}(\Omega^{A}(F))\subseteq F$. \qed
With this we have arrived at the well-known fact that assertional logics are truth-equational:
\begin{Cor}\label{CriterionTruthEquationalPointedQuasiVar} Let ${\bf K}\subseteq\Sigma\text{-Str}$ be a pointed quasivariety. Then the logic associated to the equation $\tau=\{\langle x,0\rangle\}$ (as in Theorem \ref{TheoremLogicsFromEquations}) is truth-equational.
\end{Cor} \n{\bf Proof:\;\;} Follows from Prop. \ref{PointedQuasiVarOmegaIsRetraction} and Prop. \ref{truth}.\qed
\section{Filters and adjunctions for equational filter pairs}\label{filteradjunctionequationalfilterpairs}
This section provides a closer analysis of the left adjoint $\Xi_A$ of $i_A$ for equational filter pairs, its relation to the Leibniz operator and some consequences for the filters of the associated logics.
The main statements are Theorems \ref{TheoremInclusionsXiIdLeibniz} and \ref{ThmRegWeakAlgebraizableIffOmegaLeftAdjointOfI}.
\begin{Prop}\label{PropLogicsFromEquationalFilterPairsAreFilterWeakEquivalential}
For a logic presented by an equational filter pair $(Co_{\bf K}, i^{\tau})$, a $\Sigma$-structure $A$ and a congruence $\theta \in Co_{\bf K}(A)$, the filter $i^{\tau}_A(\theta)$ is compatible with the congruence relation $\theta$ (in the sense of Def. \ref{DefCompatibleCongruence}). \end{Prop} \begin{proof}
Let the filter pair be given by the equations $\tau = \langle \epsilon(x), \delta (x) \rangle$. Let $\theta \in Co_{\bf K}(A)$ and $\langle \varphi,\psi \rangle \in \theta$. Then we have $$\begin{array}{rcl}
\varphi \in i(\theta)& \Leftrightarrow & \langle \epsilon(\varphi), \delta(\varphi) \rangle \in \theta \text{ (by definition of $i(\theta)$)}\\ & \Leftrightarrow & \epsilon(\bar{\varphi})=\delta(\bar{\varphi}) \in A/\theta \\ & \Leftrightarrow & \epsilon(\bar{\psi})=\delta(\bar{\psi}) \in A/\theta \text{ (because $\bar{\varphi} = \bar{\psi} \in A/\theta$)} \\ & \Leftrightarrow & \psi \in i(\theta) \text{ (by definition of $i(\theta)$)} \\ \end{array}$$
\end{proof}
Next we give an explicit description of the operator $\Xi_A$ of Def. \ref{DefLeftAdjointOfAFilterPair} in the case of an equational filter pair.
\begin{Prop}\label{PropFirstFormulaForLeftAdjointForEquationalFilterPairs} Let $(Co_{\bf K},i)$ be an equational filter pair, with $i$ given by the equations $\delta(x)=\epsilon(x)$. Then for a $\Sigma$-algebra $A$ the left adjoint $\Xi_A$ to $i_A$ maps $S \subseteq A$ to the congruence relation generated by $\{\langle \delta(s), \epsilon(s) \rangle \mid s \in S\}$. \end{Prop} \n{\bf Proof:\;\;} This follows immediately from the general description of Theorem \ref{TheoremExistenceOfLeftAdjoint} of left adjoints between complete partially ordered sets:
\[ \begin{array}{rcl} \Xi_A(S) &=& \bigwedge \{ \theta \in Co_{\bf K}(A) \mid S \subseteq i_A(\theta) \} \\ &=& \bigwedge \{ \theta \in Co_{\bf K}(A) \mid S \subseteq \{a \in A \mid \delta(\bar{a})=\epsilon(\bar{a}) \text{ in } A/\theta \} \} \\ &=& \bigwedge \{ \theta \in Co_{\bf K}(A) \mid \{ \langle \delta(s), \epsilon(s)\rangle \mid s \in S \} \subseteq \theta\} \end{array} \] \qed
Recall from Def. \ref{DefEquationalConsequence} the notation $Cn_{\bf K}(R)$ for the set of equations that is implied in the quasivariety ${\bf K}$ by a set of equations $R$. Also recall from and Lemma \ref{LemmaEquationalClosureEqualsGeneratedCongruence} that, if one identifies equations with pairs of elements, $Cn_{\bf K}(R)$ is exactly the congruence relative ${\bf K}$ generated by the set of pairs $R$. Blok-Pigozzi, in \cite[p.28]{BP1}, on the formula algebra $\Fm{}{X}$ define the operator $$\Omega_{\bf K} \colon \wp(\Fm{}{X}) \to Co_{\bf K}(\Fm{}{X}), \ \ \ \ T \mapsto Cn_{\bf K}(\{ \delta(a)\approx\epsilon(a) \mid a \in T \}).$$ This makes sense on any algebra and gives us an alternative description of $\Xi_A$:
\begin{Cor}\label{PropOurLeftAdjointIsBlokPigozzisOmegaK} For an algebra $A$ and a subset $T \subseteq A$ one has $\Xi_A (T) = Cn_{\bf K}(\{ \delta(a)\approx\epsilon(a) \mid a \in T \})$. \end{Cor} \n{\bf Proof:\;\;} Follows immediately from Lemma \ref{LemmaEquationalClosureEqualsGeneratedCongruence} and Proposition \ref{PropFirstFormulaForLeftAdjointForEquationalFilterPairs}. \qed
With Corollary \ref{PropOurLeftAdjointIsBlokPigozzisOmegaK} we can now pin down the class of logics admitting a presentation by an equational filter pair.
\begin{Teo}\label{TeoEveryLogicWithAlgebraicSemanticsComesFromEqunlFilterPair} Let $\Sigma$ be a signature, $l$ a logic over $\Sigma$, ${\bf K}$ a class of $\Sigma$-algebras and ${\bf K}^Q$ the quasivariety generated by ${\bf K}$.
Then $l$ has an algebraic semantics in ${\bf K}$, if and only if it has a presentation by an equational filter pair $(Co_{{\bf K}^Q},i)$. \end{Teo} \n{\bf Proof:\;\;} Assume that $l$ has an algebraic semantics in ${\bf K}$ given by a set $\{\delta(x)=\epsilon(x)\}$ of equations over $\Sigma$.
Using Theorem \ref{TheoremLogicsFromEquations} we can define the filter pair $(Co_{{\bf K}^Q}, i^{\delta = \epsilon})$ and compare its associated logic with $l$. Let $\Xi_{\Fm{}{X}}$ be the left adjoint of $i_{\Fm{}{X}}$.
By \cite[Cor 2.3]{BP1} one can replace ${\bf K}$ by ${\bf K}^Q$, hence
$$\begin{array}{rclr} \Gamma \vdash_l \varphi & \Leftrightarrow & \{\delta(\gamma) = \epsilon(\gamma) \mid \gamma \in \Gamma\} \vDash_{{\bf K}} \delta(\varphi) = \epsilon(\varphi) & \\
& \Leftrightarrow & \{\delta(\gamma) = \epsilon(\gamma) \mid \gamma \in \Gamma\} \vDash_{{\bf K}^Q} \delta(\varphi) = \epsilon(\varphi)& \text{\cite[Cor 2.3]{BP1}} \\
& \Leftrightarrow & \delta(\varphi) = \epsilon(\varphi) \in Cn_{{\bf K}^Q}(\{\delta(\gamma) = \epsilon(\gamma) \mid \gamma \in \Gamma\})& \text{def. of $Cn_{{\bf K}^Q}$} \\
& \Leftrightarrow & \delta(\varphi) = \epsilon(\varphi) \in \Xi_{\Fm{}{X}}(\Gamma)& \text{Cor. \ref{PropOurLeftAdjointIsBlokPigozzisOmegaK}} \\
& \Leftrightarrow & \varphi \in i^{\delta = \epsilon}(\Xi_{\Fm{}{X}}(\Gamma))& \text{def. of $i^{\delta = \epsilon}$} \\
& \Leftrightarrow & \Gamma \vdash_{(Co_{{\bf K}^Q}, i^{\delta = \epsilon})} \varphi & \text{Prop. \ref{PropDescriptionOfConsequenceRelationByClosureOperator}} \end{array}$$
Vice versa it follows from the last four equivalences that a logic presented by an equational filter pair has an algebraic semantics. \qed
We dedicate the rest of the section to a study of the properties of the family of adjunctions $i^\tau, \Xi$. To begin, many of the results of Chapter 3 of \cite{BP1} on the operator $\Omega_{\bf K}$, there proven directly from the definition and only for the formula algebra $\Fm{}{X}$, follow now immediately from the fact that it is left adjoint to $i^\tau$.
\begin{Lem}\label{LemmaConsequencesOfAdjointnessForEquationalFilterPairs} Let $(Co_{\bf K}, i^{\tau})$ be an equational filter pair and $A$ a $\Sigma$-structure. \begin{enumerate}[(i)]
\item \emph{\cite[Lemma 3.3(i)]{BP1}} Write $C := i_A^\tau \circ \Xi_A$ for the closure operator of the abstract logic on $A$. Then for $\Gamma \subseteq A$ we have $\Xi_A(C(\Gamma)) = \Xi_A(\Gamma)$\footnote{Blok-Pigozzi's original statement follows together with Prop. \ref{PropDescriptionOfConsequenceRelationByClosureOperator}}.
\item \emph{\cite[Lemma 3.3(ii)]{BP1}} The map $\Xi_A$ preserves arbitrary suprema.
\item \emph{\cite[Lemma 3.3(iii)]{BP1}} The map $\Xi_A$ preserves unions of directed sets of theories of the abstract logic on $A$.
\item \emph{\cite[Lemma 3.4(i)]{BP1}} For a theory $T$ of the abstract logic on $A$ one has $(i^\tau_A \circ \Xi_A )(T) = T$
\item \emph{\cite[Lemma 3.4(ii)]{BP1}} For every $\theta \in Co_{\bf K}(A)$ one has $(\Xi_A \circ i^\tau)(\theta) \subseteq \theta$. Equality holds if and only if $\theta = \Xi_A(T)$ for some $A$-theory $T$.
\item \emph{\cite[Lemma 3.5(i)]{BP1}} $\Xi_A$ maps the lattice of $A$-theories isomorphically to a compact and join-complete sublattice of $Co_{\bf K}(A)$.
\end{enumerate} \end{Lem} \n{\bf Proof:\;\;} \begin{enumerate}[(i)]
\item This is just the triangle equality $\Xi_A \circ i^\tau \circ \Xi_A = \Xi_A$ coming from the adjointness.
\item The map $\Xi_A$ is a left adjoint, hence preserves arbitrary colimits.
\item Since both $\Xi_A$ and $i_A^\tau$ preserve directed suprema, the composite $i_A^\tau \circ \Xi_A$ preserves directed suprema in $\wp(A)$, i.e. directed unions, and takes values in the sublattice of theories. Thus directed suprema in the lattice of theories coincide with set-theoretic unions, hence this follows from part (ii).
\item By Lemma \ref{Lemi-FiltersAreTheImageOfi} theories of the abstract logic on $A$ (there called $i$-filters, see Def. \ref{DefiFilter}) are exactly the sets in the image of $i^\tau_A$. Hence the claim follows from the adjointness equality $i^\tau_A \circ \Xi_A \circ i^\tau = i^\tau$.
\item The inclusion is the counit of the adjunction, the condition for equality is clear from the adjunction properties.
\item Adjunctions induce isomorphisms between the sublattices occurring as the images. The map $\Xi_A \circ i^\tau$ is a coreflection onto the respective sublattice of $Co_{\bf K}(A)$ and coreflective sublattices are closed under suprema.
\end{enumerate} \qed
As mentioned earlier, the collection of adjoint maps $(\Xi_A)_{A \in \Sigma\textrm{-Str}}$ need not be a natural transformation with respect to preimage maps in general, and its behaviour with respect to homomorphisms of $\Sigma$-structures does not follow directly from adjointness. It is, however, of significance in the consideration of Section \ref{SectionCraigInterpolation}. The following lemma records the behaviour with respect to substitutions on the formula algebras.
\begin{Lem}\label{LemmaSubstitutionInvarianceForTheEquationalOperators} Let $\sigma \colon \Fm{}{X} \to \Fm{}{Y}$ be a homomorphism of free $\Sigma$-algebras. \begin{enumerate}[(i)]
\item\label{substitutionInvarianceKConsequence} For $\Gamma \subseteq \Fm{}{X} \times \Fm{}{X}$ we have $(\sigma \times \sigma)(Cn_{\bf K}^{\Fm{}{X}}(\Gamma)) \subseteq Cn_{\bf K}^{\Fm{}{Y}}(\sigma(\Gamma))$.
\item\label{equalityKConsequenceClosuresAndSubstitution} For $\Gamma \subseteq \Fm{}{X} \times \Fm{}{X}$ we have $Cn_{\bf K}^{\Fm{}{Y}}((\sigma \times \sigma)(Cn_{\bf K}^{\Fm{}{X}}(\Gamma))) = Cn_{\bf K}^{\Fm{}{Y}}(\sigma(\Gamma))$.
\item\label{substitutionInvarianceDirectImageXi} For any homomorphism of free $\Sigma$-algebras $\sigma \colon \Fm{}{X} \to \Fm{}{Y}$ and $T \subseteq \Fm{}{X}$ we have \linebreak $\Xi_{\Fm{}{Y}}(\sigma(T)) = Cn_{\bf K}^{\Fm{}{Y}}((\sigma\times \sigma)(\Xi_{\Fm{}{X}}(T)))$
\item\label{substitutionInvarianceInverseImageXi} If $\sigma \colon \Fm{}{X} \to \Fm{}{Y}$ is the homomorphism induced by an injective map of variables $X \hookrightarrow Y$ and $T \subseteq \Fm{}{Y}$, we have $\Xi_{\Fm{}{X}}(\sigma^{-1}(T)) \subseteq (\sigma\times \sigma)^{-1}(\Xi_{\Fm{}{Y}}(T))$
\end{enumerate}
\end{Lem}
\n{\bf Proof:\;\;} \begin{enumerate}[(i)]
\item Equational consequence satisfies, for any substitution $\sigma \colon \Fm{}{X} \to \Fm{}{Y}$, and set $\Gamma \cup \{\tau \} \subseteq \Fm{}{X}$ of equations, $\Gamma \vDash_{\bf K} \tau \Rightarrow \sigma\Gamma \vDash_{\bf K} \sigma \tau$. Hence we have $(\sigma \times \sigma)(Cn_{\bf K}^{\Fm{}{X}}(\Gamma)) = \{(\sigma \times \sigma)(\tau) \mid \Gamma \vDash_{\bf K} \tau \} \subseteq \{ \tau \mid (\sigma \times \sigma)(\Gamma) \vDash_{\bf K} \tau \} = Cn_{\bf K}^{\Fm{}{Y}}(\sigma(\Gamma))$.
\item Applying $Cn_{\bf K}^{\Fm{}{Y}}$ to the inclusion from (\ref{substitutionInvarianceKConsequence}) yields $$Cn_{\bf K}^{\Fm{}{Y}}((\sigma \times \sigma)(Cn_{\bf K}^{\Fm{}{X}}(\Gamma))) \subseteq Cn_{\bf K}^{\Fm{}{Y}} Cn_{\bf K}^{\Fm{}{Y}}(\sigma \times \sigma(\Gamma)) = Cn_{\bf K}^{\Fm{}{Y}}(\sigma \times \sigma(\Gamma)).$$
On the other hand $Cn_{\bf K}^{\Fm{}{X}}(\Gamma) \supseteq \Gamma$ implies $(\sigma \times \sigma)(Cn_{\bf K}^{\Fm{}{X}}(\Gamma)) \supseteq (\sigma \times \sigma)(\Gamma)$, which implies $Cn_{\bf K}^{\Fm{}{Y}}((\sigma \times \sigma)(Cn_{\bf K}^{\Fm{}{X}}(\Gamma))) \supseteq Cn_{\bf K}^{\Fm{}{Y}}((\sigma \times \sigma)(\Gamma)).$
\item See \cite[Lemma 3.6]{BP1} for the case $X=Y$. The general proof is no different.
\item In the special case we are considering now, $\sigma^{-1}$ (resp. $(\sigma \times \sigma)^{-1}$) is just restriction to formulas (resp. pairs of formulas) in $\Fm{}{X}$, i.e. $\sigma^{-1}(T) = T \cap \Fm{}{X}$.
Recall, from the text after Lemma \ref{LemmaInclusionOfVariablesGivesConservativeTranslationOfEquationalConsequences}, the notation $Cn_{\bf K}^X$ for the closure operator associated to the equational consequence relation $\vDash_{\bf K}^X$ of Def. \ref{DefEquationalConsequence}. Expressed in terms of these closure operators, Lemma \ref{LemmaInclusionOfVariablesGivesConservativeTranslationOfEquationalConsequences} says that $Cn_{\bf K}^Y(T \cap \Fm{}{X}^2) \cap \Fm{}{X}^2 = Cn_{\bf K}^X(T \cap \Fm{}{X}^2)$. Hence
$$Cn_{\bf K}^Y(T) \cap \Fm{}{X}^2 \supseteq Cn_{\bf K}^Y(T \cap \Fm{}{X}^2) \cap \Fm{}{X}^2 = Cn_{\bf K}^X(T \cap \Fm{}{X}^2)$$ which implies the claim. \end{enumerate} \qed
While in the usual treatments of abstract algebraic logic the Leibniz operator plays a prominent role, we haven't made any use of it so far. Even algebraizable logics can be characterized purely in terms of the natural transformations $i$ and $\Xi$ (namely if $\Xi(i(\theta))$ always holds, Thm. \ref{TheoremInclusionsXiIdLeibniz} a)$\Leftrightarrow$e) below).
We will now start considering the interplay of our operators with the Leibniz operator $\Omega$.
\begin{Teo}\label{TheoremInclusionsXiIdLeibniz} Let $(Co_{\bf K},i)$ be an equational filter pair, with $i$ given by equations $\delta(x)= \epsilon(x)$. Let $A$ be a $\Sigma$-algebra and $\theta \in Co_{\bf K}(A)$ a congruence on $A$. Then $\Xi_A(i(\theta)) \subseteq \theta \subseteq \Omega_A(i_A(\theta))$. Moreover the following are equivalent: \begin{enumerate}[(a)]
\item one of the inclusions is an equality for every $\theta \in Co_{\bf K}(\Fm{}{X})$ (where $X$ is any enumerable set)
\item both inclusions are equalities for all $\Sigma$-structures $A$ and every $\theta \in Co_{\bf K}(A)$
\item $\Xi_A = \Omega_A$ for all $\Sigma$-structures $A$, as maps from the lattice of filters to $Co_{\bf K}(A)$
\item $\Xi_{\Fm{}{X}} = \Omega_{\Fm{}{X}}$, as maps from the lattice of theories to $Co_{\bf K}(\Fm{}{X})$
\item the logic presented by $(Co_{\bf K},i)$ is algebraizable with $i_{\Fm{}{X}}$ being a substitution preserving lattice isomorphism.
\end{enumerate} \end{Teo} \n{\bf Proof:\;\;} The first inclusion has been noted in Lemma \ref{LemmaConsequencesOfAdjointnessForEquationalFilterPairs}(v).
For the second inclusion recall that by Prop. \ref{PropLogicsFromEquationalFilterPairsAreFilterWeakEquivalential} $\theta$ is compatible with $i(\theta)$. But $\Omega_A(i_A(\theta))$ is the coarsest congruence relation that is compatible with $i(\theta)$, so $\theta \subseteq \Omega_A(i_A(\theta))$.
For the equivalences we prove the cycle $(a) \Rightarrow (e) \Rightarrow (c) \Rightarrow (b) \Rightarrow (d) \Rightarrow (a)$.
$(a) \Rightarrow (e):$ If one of the inclusions $\theta \subseteq \Omega(i(\theta))$, $\theta \subseteq \Xi(i(\theta))$ is an equality, then $i$ has a left inverse and hence is injective. Since by definition of the associated logic, $i$ is also surjective onto the theories, we have a substitution preserving isomorphism between the lattice of congruences and the lattice of theories. By Thm. \ref{TheoremSecondIsoThmInclUniquenessOfTheQuasivariety} the associated logic is algebraizable.
$(e) \Rightarrow (c):$ By Thm. \ref{TheoremSecondIsoThmInclUniquenessOfTheQuasivariety} for any $\Sigma$-structure $A$ the equations defining $i_A$ induce isomorphisms, and the inverse is given by the Leibniz operator $\Omega_A$. Being an inverse of $i_A$, $\Omega_A$ is in particular a left adjoint. By uniqueness of adjoints, it follows that $\Xi_A = \Omega_A$.
$(c) \Rightarrow (b):$ Clear, using the two inclusions and the equality $\Xi_A = \Omega_A$.
$(b) \Rightarrow (d):$ By hypothesis $(b)$ the maps $\Xi_{\Fm{}{X}}$ and $\Omega_{\Fm{}{X}}$ coincide on the filters in the image of $i_{\Fm{}{X}}$. But by Prop. \ref{FreeAlgebraResult} all theories are in the image of $i_{\Fm{}{X}}$, hence $\Xi_{\Fm{}{X}} = \Omega_{\Fm{}{X}}$.
$(d) \Rightarrow (a):$ Clear, using the two inclusions and the equality $\Xi_{\Fm{}{X}} = \Omega_{\Fm{}{X}}$. \qed
\begin{Lem}\label{LemmaFiltersInTheImageOfiForReducedMatrices} Let $(Co_{\bf K},i)$ be an equational filter pair, with $i$ given by equations $\delta(x)= \epsilon(x)$. Let $\langle A, F \rangle$ be a reduced matrix for the associated logic with $F=i_A(\theta)$ in the image of $i_A$, for a congruence $\theta \in Co_{\bf K}(A)$. Then $F=\{a \in A \mid \delta(a)=\epsilon(a) \}$. \end{Lem} \n{\bf Proof:\;\;} The assumption that $\langle A, F \rangle$ is a reduced matrix means $\Omega_A(F)=\Delta$ (where $\Delta$ denotes the diagonal, i.e. the minimal congruence relation). We therefore have $\Omega_A(F)=\Delta \subseteq \Xi_A(F)$, and thus by the inclusions of Theorem \ref{TheoremInclusionsXiIdLeibniz} $\Delta=\Xi_A(F)=\theta$. We obtain \[ \begin{array}{rcl}
F = i_A(\theta) &=& \{ a \in A \mid \delta(\bar{a})=\epsilon(\bar{a})\text{ in } A/\theta \} \\ &=& \{ a \in A \mid \delta(a)=\epsilon(a) \text{ in } A \} \end{array} \] \qed
Our considerations about adjoints and filter pairs let us arrive at the following well-known fact about assertional logics.
\begin{Cor}\label{CorollaryDescriptionOfReducedMatricesForTruthEquationalFilterPairs} Let $(Co_{\bf K},i)$ be an equational filter pair, with $i$ given by equations $x=\top$. Let $\langle A, F \rangle$ be a reduced matrix for the associated logic. Then $F=\{ \top \}$. \end{Cor} \n{\bf Proof:\;\;} For a filter pair given by the equation $x=\top$ all filters are in the image of $i_A$ by Lemma \ref{PointedQuasiVarOmegaIsRetraction}. By Lemma \ref{LemmaFiltersInTheImageOfiForReducedMatrices} we have $F=\{a \in A \mid a=\top\}=\{ \top \}$. \qed
We know from Theorem \ref{TheoremInclusionsXiIdLeibniz} that if one of the inclusions $\Xi(i(\theta)) \subseteq \theta \subseteq \Omega(i(\theta))$ is actually an equality (and hence $i$ injective), then the filter pair in question presents an algebraizable logic. Applying $i$ to the first inclusion, by general properties of adjunctions, we obtain an equality $i(\Xi(i(\theta))) = i(\theta)$. In contrast we do not necessarily have the corresponding equality $i(\theta) = i(\Omega(i(\theta)))$ for the Leibniz operator.
\begin{Ex}
Consider again the logic $l_s$ of Examples \ref{ExampleSuccessoralgebra} and \ref{ExampleSuccessorLogicIsNotTruthEquational}. In Example \ref{ExampleSuccessorLogicIsNotTruthEquational} we considered the algebra $A:= \mathbb{N} \cup \{z\}$ with the successor operation on $\mathbb{N}$ and $z$ a fixed point for the unary operation $s$. As $\{z\}$ is the $s$-fixed point set of $A$, we have $\{z\} = i(\Delta_A)$ (where $\Delta_A$ denotes the minimal congruence relation on $A$). As noted in Example \ref{ExampleSuccessorLogicIsNotTruthEquational}, $\Omega_A(\{z\})$, being the coarsest congruence not relating elements of $\{z\}$ with elements outside $\{z\}$, is the congruence collapsing all elements of $\mathbb{N}$ to a single element and leaving $z$ alone. The quotient $A/\Omega_A(\{z\})$ consists of two elements, both of which adre $s$-fixed points. Hence $i (\Omega_A(\{z\}))$, the set of elements of $A$ which become $s$-fixed points in $A/\Omega_A(\{z\})$, is all of $A$. Altogether we have $i (\Omega_A(i(\Delta_A))) = A \supsetneq \{z\} = i(\Delta_A)$.
\end{Ex}
Continuing to consider the previous example we may apply $\Omega$ to the inequality $i (\Omega_{A}(i(\Delta_A))) = A \supsetneq \{z\} = i(\Delta_A)$ and obtain $\Omega(i (\Omega(i(\Delta_A)))) = \Omega(A) = A \times A \supsetneq \Omega(\{z\}) = \Omega(i(\Delta_A))$ where the strict inequality comes from the fact that the quotient of the left hand congruencce yields a one element set and the quotient of the right hand congruence yields a two element set. This shows that one also in general does not have equality in the inclusion $\Omega \circ i \circ \Omega \circ i \supseteq \Omega \circ i$. However, in all examples we considered the iterated application of $\Omega \circ i$ stabilizes at some point (in the above example this happens in the next step).
\begin{Que} Does iterated application of $\Omega \circ i$ always stabilize, i.e. is there for any congruence $\theta$ an $n \in \mathbb{N}$ such that $(\Omega \circ i)^{\circ n}(\theta) = (\Omega \circ i)^{\circ (n+1)}(\theta)$? If yes, is there a uniform bound on $n$ for all subsets $F$ of a fixed algebra $A$? \end{Que}
While the condition $\theta = \Omega(i(\theta))$ is equivalent to algebraizability, the weaker condition $i(\theta) = i(\Omega(i(\theta)))$ can, however, also hold for non-algebraizable logics: We have seen that for assertional logics we have $i(\Omega(F))=F$, hence $i = i\circ \Omega\circ i$.
Let $(Co_{\bf K},i)$ be an equational filter pair, with $i$ given by equations $x=\top$. Then we have $i = i\circ \Omega\circ i$.
\begin{Que}
Can one give a meaningful characterization of the class of equational filter pairs for which $i = i\circ \Omega\circ i$ holds? Is it precisely the class of filter pairs given by equations of the form $x=\top$? \end{Que}
The following discussion will shed some more light on this question.
We have seen that the operator $\Xi \circ i$ associates to a congruence $\theta$ the \emph{smallest} congruence in whose quotient the same elements get mapped to solutions of the equations $\tau$ set as in the quotient by $\theta$ -- in particular this smallest congruence exists. If $i$ has a right adjoint $R$, then the operator $R \circ i$ associates to a congruence $\theta$ the \emph{biggest} congruence which has the same $\tau$-solution set as $\theta$. This biggest congruence exists if and only if there is a right adjoint, but this need not be the case in general.
One may suggest the mental image that the Leibniz operator is trying to be a right adjoint of $i$, but may fail to be so. Indeed, the failure may lie in the fact that it is not even an order preserving map for non-protoalgebraic logics. For protoalgebraic logics, the Leibniz operator is not only order preserving, but also preserves arbitrary infima, and hence by Theorem \ref{TheoremExistenceOfLeftAdjoint} has a left adjoint. The description of the left adjoint offered by that same theorem resembles the description of $i$ in the case of an assertional logic, as we will see below.
\begin{Prop}
Let $l$ be a protoalgebraic logic. Then the Leibniz operator from the \emph{filters} to the congruences has a left adjoint given by $L(\theta) = \bigcap \{ F \ \mid \ \theta \textrm{ does not relate elements of $F$ with elements outside $F$} \}$. \end{Prop}
\n{\bf Proof:\;\;} Since it preserves arbitrary infima, $\Omega$ has a left adjoint $L$. By the adjunction formula Thm. \ref{TheoremExistenceOfLeftAdjoint} it is given by:
$$\begin{array}{rcl}
L(\theta) &=& \bigwedge \{ F \, \mid \, \theta \subseteq \Omega(F) \} \\
&=& \bigwedge \{ F \, \mid \, \theta \subseteq \vee \{\rho \mid F \textrm{ is a union of } \rho-\textrm{equivalence classes}\} \} \\
&=& \bigcap \{ F \ \mid \ \theta \textrm{ does not relate elements of $F$ with elements outside $F$} \} \end{array}$$
Arbitrary intersections of sets that are compatible with a given equivalence relation $\theta$ are compatible with $\theta$ again. Arbitrary intersections of filters are filters again, so replacing the infimum on the last step with an intersection is ok. \qed
We would like to relate this to the right adjoint $i$ occurring in a filter pair. Consider the case where $i$ is given by the equation $\langle x, \top \rangle$, then we have the formula $i(\theta) = \{a \ \mid \ a \theta \top\} = [\top]_{\theta}$. We have $i(\theta) \subseteq F$ for each filter $F$ such that $\theta \subseteq \Omega(F)$, thus $i(\theta) \subseteq L(\theta)$. The opposite inclusion, $L(\theta) \subseteq i(\theta)$, holds too, since $[\top]_{\theta}$ occurs among the filters over which we take the intersection in the formula for $L(\theta)$ (this inclusion is also a consequence of the adjunction, since $\theta \subseteq \Omega(i(\theta))$, because $\theta$ is compatible with $i(\theta)$). {\bf Thus $i=L$ !}
The hypothesis that our logic admits a presentation by a congruence filter pair with $i$ given by the equation $\langle x,\top\rangle$ is equivalent to asking that it is assertional. By \cite[6.125]{Fon}, a logic is assertional and protoalgebraic if and only if it is regularly weakly algebraizable.
Thus we arrive at the following result:
\begin{Teo}\label{ThmRegWeakAlgebraizableIffOmegaLeftAdjointOfI}
Let $l$ be the logic associated to a congruence filter pair $(Co_{\mathbf{K}},i)$, with $i$ given by the equation $\langle x,\top\rangle$. Then $l$ is regularly weakly algebraizable if and only $\Omega$ is a left adjoint of $i$. \end{Teo}
\n{\bf Proof:\;\;} One direction has been shown in the above discussion: If a logic is regularly weakly algebraizable, then it is protoalgebraic and we have seen above that then the left adjoint of $\Omega$ coincides with $i$.
In the other direction note that if $i$ is left adjoint to $\Omega$, then $\Omega$, being a right adjoint, preserves arbitrary infima and hence the logic is protoalgebraic. Assertionality follows directly from the filter pair being given by the equation $\langle x,\top\rangle$. \qed
Note that the equation $i = i\circ \Omega\circ i$, shown independently before, now follows immediately from the adjointness of $i$ and $\Omega$.
\begin{Lem}\label{protoalgebraic} $\Omega \colon Fi \to Co_K$ has a left adjoint if and only if the logic is protoalgebraic. \end{Lem} \n{\bf Proof:\;\;} $\Omega$ has a left adjoint if and only if it preserves arbitrary infima (by Thm. \ref{TheoremExistenceOfLeftAdjoint}) if and only if the logic is protoalgebraic, by Thm. 6.4 in \cite{Fon}. \qed
\begin{Prop}\label{PropProtoAlgAndReducedAlgsEqualKThenAlgebraizable} Let $l$ be the logic associated to a filter pair $(Co_K, i)$, with $i$ given by the equation $x=\top$. Suppose that $l$ is protoalgebraic and that the class of reduced algebras coincides with $K$. Then $l$ is algebraizable. \end{Prop} \n{\bf Proof:\;\;} We know already that $l$ is weakly regularly algebraizable (by a combination of the previous Lemma and Proposition). I remains to show that $l$ is also equivalential.
For a weakly algebraizable logic, by \cite[Thm. 6.117]{Fon} the Leibniz operator is a bijection $\Omega^A\colon Fi_l A \simeq Con_{Alg^{*}\,l}A$ from filters to congruences relative the class of reduced algebras for $l$. If this class coincides with $\mathbf{K}$, then we have that $\Omega$ is a bijection $\Omega^A\colon Fi_l A \cong Con_{\mathbf{K}}A$. Its adjoint $i$ is then also a bijection and hence the logic is algebraizable. Alternatively, since $i$ is the adjoint, and hence inverse, of $\Omega$, and $i$ is a natural transformation, $\Omega$ must be natural as well, which means that $l$ is equivalential by \cite[Cor. 6.69]{Fon}.\qed
Note that, in this case, {\bf K} is a quasivariety that is the equivalent algebraic semantics for $l$.
\begin{Ex}
Consider the filter pair $(Co_K, i^\tau)$ where {\bf K} is the variety of all groups, axiomatized with a constant symbol $e$ for the neutral element, and $i^\tau$ given by the equation $\tau=\langle x, e \rangle$.
The associated logic is protoalgebraic with implication connective $y^{-1}x$: This follows from the fact that if $y^{-1}x=e$ and $x^{-1}z=e$ then also $y^{-1}z = y^{-1}xx^{-1}z=ee = e$. Furthermore, every group has $\{e\}$ as a reduced filter, so every group is the underlying algebra of a reduced matrix. By Prop. \ref{PropProtoAlgAndReducedAlgsEqualKThenAlgebraizable} the associated logic is algebraizable.
A similar reasoning applies to the filter pair $(Co_K, i^\tau)$ with {\bf K} the variety of all rings, and $i^\tau$ given by the equation $\tau=\langle x, 0 \rangle$. \end{Ex}
On the other hand, classical logic $l_c$ has as algebraic semantics the quasivariety generated by the orthomodular lattice $O_6$ (see \cite[Ex. 4.79 ]{Fon}), which is not the class of boolean algebras. Thus the criterion of Prop. \ref{PropProtoAlgAndReducedAlgsEqualKThenAlgebraizable} is sufficient but not necessary for a logic to be algebraizable. This comes from the possibility of giving an algebraic semantics to an algebraizable logic that is not an equivalent algebraic semantics.
\section{Craig entailment interpolation property and filter pairs}\label{SectionCraigInterpolation}
In this section we present a correspondence between the Craig interpolation property for a logic associated to an equational filter pair, and the amalgamation property in the class ${\bf K}$ for whose congruences the filter pair is defined.
We recall the two central notions for this chapter.
\begin{Df} \begin{itemize}\label{DefinitionMatrixEmbeddingCraigEntailmentAmalgamationProperty}
\item A logic $l$ has the \emph{Craig entailment interpolation property} if for every set of formulas $\Gamma$, with variables $var(\Gamma)$ and every formula $\varphi$ with variables $var(\varphi)$, if $\Gamma\vdash \varphi$ then there is a set of formulas $\Gamma'$ with the variables in $var(\Gamma)\cap var(\varphi)$ such that $\Gamma\vdash \Gamma'$ and $\Gamma'\vdash \varphi$. \item We shall say that a class of algebras ${\bf K}$ has the \emph{amalgamation property} if given $A,B,C\in {\bf K}$ and injective homomorphisms $i_{B} \colon A \to B$, $i_{C} \colon A \to C$, there exist an algebra $D \in {\bf K}$ and injective homomorphisms $e_B:B\to D$, $e_C:C\to D$ such that $e_{B}\circ i_{B}=e_{C}\circ i_{C}$. \end{itemize} \end{Df}
We will employ the following auxiliary property which implies Craig interpolation.
\begin{Df}
[Def. 3.4 of \cite{CzP}] A logic $l$ has the \emph{flat theory amalgamation property} if for every two non-disjoint sets of variables $X$ and $Y$, and every $l$-filter $T$ of the formula algebra $\Fm{}{X}$ there is an $l$-filter $R$ of the formula algebra $\Fm{}{X\cup Y}$ such that $R\cap \Fm{}{X}=T$ and $R\cap \Fm{}{Y}=Fi_{l}[T\cap \Fm{}{X\cap Y}]=\bigcap\{T'\in Th(\Fm{}{Y});\ T\cap \Fm{}{X\cap Y}\subseteq T'\}$ \end{Df}
\begin{Lem}[\cite{CzP}, Thm. 3.5]\label{2.17} If a logic $l$ has the flat theory amalgamation property, it has the Craig entailment interpolation property. \end{Lem} \n{\bf Proof:\;\;} Let $\Gamma\cup\{\varphi\}$ set of formulas such that $\Gamma\vdash \varphi$, $X=var(\Gamma)$ and $Y=var(\varphi)$. Consider the $l$-filter $T$ of $\Fm{}{X}$ generated by $\Gamma$. By the flat theory amalgamation property, there is an l-filter $R$ of $\Fm{}{X\cup Y}$ such that $R\cap \Fm{}{X}=T$ and $R\cap \Fm{}{Y}=Fi_{l}[T\cap \Fm{}{X\cap Y}]$. Note that $\varphi\in R\cap \Fm{}{Y}$, and consider $\Gamma'=T\cap \Fm{}{X\cap Y}$. It is clear that $\Gamma\vdash \Gamma'$ and $\Gamma'\vdash\varphi$. \qed
We now introduce a property which ensures that from a situation in which one can ask for flat theory amalgamation one can pass to a situation in which one can apply amalgamation of algebras. It is the content of Theorem \ref{TheoremMatrixAmalgamationImpliesCraigInterpolation} below that whenever this property is satisfied, the amalgamation property for a quasivariety ${\bf K}$ implies Craig interpolation for the logic presented by a filter pair $(Co_{\bf K}, i)$.
\begin{Df}\label{DefTheoryLiftingProperty}
Let ${\bf K}$ be a class of algebras. We say that a congruence filter pair $(Co_{{\bf K}},i)$ has the \emph{theory lifting property}, if the following holds:
Given sets $X,Y$ such that $Z:=X \cap Y \neq \emptyset$ and a theory $T \subseteq \Fm{}{X}$ of the logic associated to $(Co_{{\bf K}},i)$, define $T'':=T \cap \Fm{}{Z}$ and $T' \subseteq \Fm{}{Y}$ to be the smallest $\Fm{}{Y}$-theory containing $T''$.
Then there exist $\theta_T \in Co_{\bf K}(\Fm{}{X})$, $\theta_{T''} \in Co_{\bf K}(\Fm{}{Z})$ and $\theta_{T'} \in Co_{\bf K}(\Fm{}{Y})$ such that \begin{enumerate}[(a)]
\item $T = i(\theta_{T})$, $T' = i(\theta_{T'})$ and $T'' = i(\theta_{T''})$
\item $\theta_{T} \cap \Fm{}{Z} = \theta_{T''} = \theta_{T'} \cap \Fm{}{Z} $
\end{enumerate}
\end{Df}
The following theorem is the justification for the notion we just introced.
\begin{Teo}\label{ida}\label{TheoremMatrixAmalgamationImpliesCraigInterpolation} Let $\Sigma$ be a signature, ${\bf K}\subseteq\Sigma\text{-Str}$ a class of algebras closed under subalgebras and $(Co_{{\bf K}},i)$ an equational filter pair having the theory lifting property. If ${\bf K}$ has the amalgamation property, then the logic associated to $(Co_{{\bf K}},i)$ has the Craig entailment interpolation property. \end{Teo}
\n{\bf Proof:\;\;} By Lemma \ref{2.17} it is enough to prove that $l$ has the flat theory amalgamation property. Let $X,Y$ be non disjoint sets and $T\in Fi_{l_{k}}(\Fm{}{X})$. Denote by $Z=X\cap Y$ and $W=X\cup Y$. Consider $T'=Fi^{Y}_{l_{{\bf K}}}(T\cap \Fm{}{Z})$. So $T'\cap \Fm{}{Z}=T\cap \Fm{}{Z}(=T'')$. Indeed, it is clear that $T\cap \Fm{}{Z}\subseteq T'\cap \Fm{}{Z}$. Suppose $\varphi\not\in T\cap \Fm{}{Z}$, hence $T\cap \Fm{}{Z}\not\vdash_{l_{{\bf K}}}\varphi$. Notice that $Z\subseteq Y$, by \ref{2.19} we have $T\cap \Fm{}{Z}\not\vdash^{Y}\varphi$, then $\varphi\not\in Fi_{l_{{\bf K}}}^{Y}(T\cap \Fm{}{Z})=T'$, proving that if $T'\cap \Fm{}{Z}\subseteq T\cap \Fm{}{Z}$.
From the assumed theory lifting property we obtain $\theta_T \in Co_{\bf K}(\Fm{}{X})$, $\theta_{T''} \in Co_{\bf K}(\Fm{}{Z})$ and $\theta_{T'} \in Co_{\bf K}(\Fm{}{Y})$ such that $T = i(\theta_{T})$, $T' = i(\theta_{T'})$ and $T'' = i(\theta_{T''})$ and $\theta_{T} \cap \Fm{}{Z} = \theta_{T''} = \theta_{T'} \cap \Fm{}{Z} $.
The inclusion $\theta_{T} \cap \Fm{}{Z} \supseteq \theta_{T''}$ tells us that the map $\Fm{}{Z} \hookrightarrow \Fm{}{X}$ induces a well-defined homomorphism $h_1 \colon \Fm{}{Z}/\theta_{T''} \to \Fm{}{X}/\theta_{T}$, and the opposite inclusion $\theta_{T} \cap \Fm{}{Z} \subseteq \theta_{T''}$ tells us that this homomorphism is injective. Likewise, from the equation $\theta_{T''} = \theta_{T'} \cap \Fm{}{Z}$ we obtain a well-defined and injective homomorphism $h_2 \colon \Fm{}{Z}/\theta_{T''} \to \Fm{}{Y}/\theta_{T'}$.
By hypothesis ${\bf K}$ has the amalgamation property, so we can complete these two maps to a commutative square of injective homomorphisms, for some algebra $\tilde{A} \in {\bf K}$. \[ \xymatrix{ \Fm{}{Z}/\theta_{T''} \ar@{>->}[d]_{h_1} \ar@{>->}[r]^{h_2} & \Fm{}{Y}/\theta_{T'} \ar@{>->}[d]^{g_2} \\
\Fm{}{X}/\theta_{T} \ar@{>->}[r]_{g_1} & \tilde{A} } \]
Since $F \colon Set \to \Sigma\text{-Str}$ is a left adjoint, we have that $\Fm{}{X\cup Y}$ is the pushout of $\Fm{}{X} \hookleftarrow \Fm{}{Z} \hookrightarrow \Fm{}{Y}$. Hence there is a natural homomorphism $\Fm{}{X\cup Y} \to \tilde{A}$. Its image is a subalgebra $A \subseteq \tilde{A}$, hence, by hypothesis, also belongs to ${\bf K}$. Thus $A \cong \Fm{}{X\cup Y}/\theta$ for some congruence $\theta \in Co_{\bf K}(\Fm{}{X\cup Y})$. We claim that the filter on $\Fm{}{X\cup Y}$ required for the flat theory amalgamation property is given by $R:=i_{\Fm{}{X\cup Y}}(\theta)$.
First note that the images of $g_1, g_2$ lie inside $A$, because $\text{Im}\, g_1 = \text{Im}\, (\Fm{}{X} \twoheadrightarrow \Fm{}{X}/\theta_T \to \tilde{A}) = \text{Im}\,(\Fm{}{X} \to \Fm{}{X\cup Y} \to \tilde{A})$ and likewise for $\text{Im}\, g_2$. We can thus in the above square replace $\tilde{A}$ by $A$.
Next, note that for $\varphi \in \Fm{}{X}$ we have $$\varphi \in T\ \ \Leftrightarrow\ \ \delta(\varphi)=\epsilon(\varphi) \in \Fm{}{X}/\theta_T \ \ \Leftrightarrow\ \ \delta(g_1(\varphi))=\epsilon(g_1(\varphi)) \in \Fm{}{X\cup Y}/\theta\ \ \Leftrightarrow\ \ g_1(\varphi) \in R.$$ Here the middle equivalence holds because of $\delta(\varphi)=\epsilon(\varphi)\ \ \Leftrightarrow \ \ g_1(\delta(\varphi))=g_1(\epsilon(\varphi))$ (where $\Leftarrow$ holds because of the injectivity of $g_1$) and $g_1(\delta(\varphi))=\delta(g_1(\varphi))$, $g_1(\epsilon(\varphi))=\epsilon(g_1(\varphi))$ (because $g_1$ is a homomorphism). Thus $R \cap \Fm{}{X} = T$. With exactly the same reasoning we obtain $R \cap \Fm{}{Y} = T'$. We have proved the flat amalgamation property, and thus the claim. \qed
In view of Theorem \ref{TheoremMatrixAmalgamationImpliesCraigInterpolation} we would like to know when an equational filter pair satisfies the theory lifting property. The following Lemma provides a sufficient criterion for this.
\begin{Lem}\label{LemmaTheoryLiftingHoldsIfXiIsNatural} Let $(Co_{{\bf K}},i)$ be a congruence filter pair for which the collection of left adjoints $(\Xi_A)_{A \in \Sigma\textrm{-Str}}$ is a natural transformation with respect to variable inclusions, i.e. such that for every inclusion of sets $Z \subseteq X$ the following diagram (in which $j_X \colon \Fm{}{Z} \hookrightarrow \Fm{}{X}$ denotes the induced map of formula algebras) commutes: \[ \xymatrix{ \Fm{}{Z}\ar[d]^{j_X} & Co_{{\bf K}}(\Fm{}{Z}) && Fi_{l}(\Fm{}{Z}) \ar[ll]^{\Xi_{\Fm{}{Z}}} \\ \Fm{}{X}&Co_{{\bf K}}(\Fm{}{X})\ar[u]^{Co_{\bf K}(j_{X})} && Fi_{l}(\Fm{}{X})\ar[u]_{j^{-1}_{X}} \ar[ll]^{\Xi_{\Fm{}{X}}} } \] Then $(Co_{{\bf K}},i)$ has the theory lifting property. \end{Lem} \n{\bf Proof:\;\;} Suppose we are given sets $X,Y$ such that $Z:=X \cap Y \neq \emptyset$ and a theory $T \subseteq \Fm{}{X}$ of the logic associated to $(Co_{{\bf K}},i)$, define $T'':=T \cap \Fm{}{Z}$ and $T' \subseteq \Fm{}{Y}$ to be the smallest $\Fm{}{Y}$-theory containing $T''$.
First note that $T'\cap \Fm{}{Z}=T\cap \Fm{}{Z}(=T'')$. Indeed, it is clear that $T\cap \Fm{}{Z}\subseteq T'\cap \Fm{}{Z}$. For the other inclusion suppose $\varphi\not\in T\cap \Fm{}{Z}$. Then $T\cap \Fm{}{Z}\not\vdash_{l_{{\bf K}}}\varphi$. As $Z\subseteq Y$, by Prop. \ref{2.19} (which says that the logic based on the set of variables $Y$ is a conservative extension of the logic based on the set of variables $Z$) we have $T\cap \Fm{}{Z}\not\vdash^{Y}\varphi$, hence $\varphi$ is not contained in the closure $T\cap \Fm{}{Z}$, i.e. in $T'$, proving that if $T'\cap \Fm{}{Z}\subseteq T\cap \Fm{}{Z}$.
Denote by $j_X$ (resp. $j_Y$) the map $\Fm{}{Z} \hookrightarrow \Fm{}{X}$ (resp. $\Fm{}{Z} \hookrightarrow \Fm{}{Y}$) induced by the inclusion $Z \subseteq X$ (resp. $Z \subseteq Y$). Since $i$ is a natural transformation we have the following commutative diagrams.
\[ \xymatrix{ \Fm{}{Z}\ar[d]^{j_X}&Co_{{\bf K}}(\Fm{}{Z})\ar[r]^{i^{{\bf K}}_{Z}}&Fi_{l}(\Fm{}{Z})&&\Fm{}{Z}\ar[d]^{j_Y}&Co_{{\bf K}}(\Fm{}{Z})\ar[r]^{i^{{\bf K}}_{Z}}&Fi_{l}(\Fm{}{Z})\\ \Fm{}{X}&Co_{{\bf K}}(\Fm{}{X})\ar[u]^{Co_{\bf K}(j_{X})}\ar[r]_{i^{{\bf K}}_{X}}&Fi_{l}(\Fm{}{X})\ar[u]_{j^{-1}_{X}}&&\Fm{}{Y}&Co_{{\bf K}}(\Fm{}{Y})\ar[u]^{Co_{\bf K}(j_{Y})}\ar[r]_{i^{{\bf K}}_{Y}}&Fi_{l}(\Fm{}{Y})\ar[u]_{j^{-1}_{Y}} } \]
Note that for any $\theta\in Co_{{\bf K}}(\Fm{}{X})$, we have $Co_{\bf K}(j_{X})(\theta)=\theta\cap \Fm{}{Z}^{2}$ (and the same for $j_{Y}$) and similarly for any $S \subseteq \Fm{}{X}$ we have $j_{X}^{-1}(S)=S\cap \Fm{}{Z}$.
We now construct the congruences of the claim and prove that they satisfy conditions (a) and (b).
By Prop. \ref{FreeAlgebraResult} (which says that for free algebras $i$ is surjective onto filters) there is a congruence $\theta \in Co_{{\bf K}}(\Fm{}{X})$ such that $i_{X}(\theta)=T$. We then know that there is also a minimal such congruence, namely $\theta_{T}:=\Xi_{\Fm{}{X}}(T)$. Define $\theta_{T''}:=\theta_{T}\cap \Fm{}{X}^{2}$.
By the naturality of $i$ we have $i_{\Fm{}{Z}}(\theta_{T''})=i_{\Fm{}{Z}}(\theta_{T}\cap \Fm{}{Z}^{2})=i_{\Fm{}{X}}(\theta_{T})\cap \Fm{}{Z}=T\cap \Fm{}{Z}=T''$. Thus we have constructed $\theta_T \in Co_{\bf K}(\Fm{}{X})$ and $\theta_{T''} \in Co_{\bf K}(\Fm{}{Z})$ and verified that they satisfy conditions (a) and (b) of the claim.
Since $j_{Y}$ is a split monomorphism (because it is induced by an injective map between sets of variables), we have that $Co_{\bf K}(j_{Y})$ is a split epimorphism, hence surjective. Therefore there exists a $\theta'\in Co_{{\bf K}}(\Fm{}{Y})$ such that $\theta'\cap \Fm{}{Z}^{2}=Co_{\bf K}(j_{Y})(\theta')=\theta_{T''}$. Again by naturality of $i$ we have that $T\cap \Fm{}{Z}=T''=i_{Z}(\theta_{T''})=i_{Z}(\theta'\cap \Fm{}{Z}^{2})=i_{Y}(\theta')\cap \Fm{}{Z}$. Thus $T\cap \Fm{}{Z}\subseteq i_{Y}(\theta')$. Since $T'$ is the smallest theory in $\Fm{}{Y}$ containing $T \cap \Fm{}{Z}$, we have $T'\subseteq i_{Y}(\theta')$.
By proposition \ref{FreeAlgebraResult} we have that there exists a $\theta\in Co_{{\bf K}}(\Fm{}{Y})$ such that $i_{Y}(\theta)=T'\subseteq i_{Y}(\theta')$. We now define $\theta_{T'}:=\theta\cap \theta'$. Then $i_{Y}(\theta_{T'})=i_{Y}(\theta)\cap i_{Y}(\theta')=T' \cap i_{Y}(\theta')=T'$, so $\theta_{T'}$ satisfies condition (a).
Next, note that $\theta_{T''}$ is the \emph{minimal} congruence which is mapped to $T''$ by $i_{\Fm{}{Z}}$. Indeed, by assumption $\Xi$ is a natural transformation with respect to homomorphisms induced by variable inclusions. Therefore $\theta_{T''} = \theta_{T}\cap \Fm{}{X}^{2} = Co_{\bf K}(j_{X})(\theta_{T}) = Co_{\bf K}(j_{X})(\Xi(T)) = \Xi(j^{-1}_{X}(T))=\Xi(T'')$.
Furthermore, we have that $\theta_{T'}\cap \Fm{}{Z}^{2}=(\theta\cap \theta')\cap \Fm{}{Z}^{2}=\theta\cap(\theta'\cap \Fm{}{Z}^{2})=\theta\cap \theta_{T''}$. Thus $\theta_{T'}\cap \Fm{}{Z}^{2}\subseteq \theta_{T''}$. We also have $i_Y(\theta_{T'}\cap \Fm{}{Z}^{2}) = i_Y(Co_{\bf K}(j_Y)(\theta_{T'})) = j_Y^{-1}(i_Y(\theta_{T'})) = j_Y^{-1}(T') = T' \cap \Fm{}{Z} = T''$, and since $\theta_{T''}$ was the \emph{minimal} congruence which is mapped to $T''$ by $i_Z$ this implies $\theta_{T''} \subseteq \theta_{T'}\cap \Fm{}{Z}^{2}$. Altogether we have $\theta_{T'}\cap \Fm{}{Z}^{2} = \theta_{T''}$ and thus $\theta_{T'}$ satisfies condition (b). \qed
\begin{Obs}
The naturality condition of Lemma \ref{LemmaTheoryLiftingHoldsIfXiIsNatural} can be spelled out as the requirement that \linebreak $\Xi_{\Fm{}{X}}(j_X^{-1}(T)) \subseteq (j_X \times j_X)^{-1}(\Xi_{\Fm{}{Y}}(T))$.
By Lemma \ref{LemmaSubstitutionInvarianceForTheEquationalOperators}(\ref{substitutionInvarianceInverseImageXi}), the inclusion $\Xi_{\Fm{}{X}}(\sigma^{-1}(T)) \subseteq (\sigma\times \sigma)^{-1}(\Xi_{\Fm{}{Y}}(T))$ always holds and one only needs to check the other inclusion. \end{Obs}
\begin{Cor}\label{CorollaryiInjectiveImpliesTheoryLifting}
Let $(Co_{{\bf K}},i)$ be a congruence filter pair with $i$ injective. Then $(Co_{{\bf K}},i)$ has the theory lifting property. \end{Cor} \n{\bf Proof:\;\;}
By Prop. \ref{CriterionAlgebraizable} the logic presented by $(Co_{{\bf K}},i)$ is algebraizable with $i$ being a substitution preserving lattice isomorphism. By Theorem \ref{TheoremSecondIsoThmInclUniquenessOfTheQuasivariety} the inverse is given by $\Omega$, which hence is a left adjoint. By uniqueness of adjoints we have $\Xi_A = \Omega_A$ for every $\Sigma$-structure $A$ (see also Theorem \ref{TheoremInclusionsXiIdLeibniz}). Since the logic associated to $(Co_{{\bf K}},i)$ is algebraizable, it is in particular equivalential and hence by \cite[Thm 6.68]{Fon} $\Omega$ is a natural transformation. Hence $\Xi = \Omega$ satisfies the condition of Lemma \ref{LemmaTheoryLiftingHoldsIfXiIsNatural}. \qed
We arrive at the following well-known fact about algebraizable logics.
\begin{Cor}\label{CorollaryForAlgebraizableLogicsAmalgamationImpliesCraigInterpolation}
Let $l$ be an algebraizable logic with equivalent semantics in a quasivariety ${\bf K}$. If ${\bf K}$ has the amalgamation property, then the $l$ has the Craig entailment property. \end{Cor} \n{\bf Proof:\;\;} Since the logic $l$ is algebraizable, it can be presented by an equational filter pair with $i$ injective, just consider the pair $(Co_{K}, i)$ such that K is the equivalent algebraic semantic for $l$ and $i_{A}$ is the inverse of $\Omega^{A}$ for any algebra $A$. Now one can apply Corollary \ref{CorollaryiInjectiveImpliesTheoryLifting} and Theorem \ref{TheoremMatrixAmalgamationImpliesCraigInterpolation}. \qed
Of course stronger statements than the preceding one are known, e.g. that amalgamation is equivalent to Craig interpolation for algebraizable logics, see \cite[Cor. 5.27]{CzP}. While we could obtain Corollary \ref{CorollaryForAlgebraizableLogicsAmalgamationImpliesCraigInterpolation} as a quick byproduct of Lemma \ref{LemmaTheoryLiftingHoldsIfXiIsNatural}, the aim of the latter is to pave the way for new cases.
We now give another criterion for the naturality condition of Lemma \ref{LemmaTheoryLiftingHoldsIfXiIsNatural} to be satisfied. For this recall from Section \ref{SubsectionCongruencesAndEquationalConsequence} the notation $Cn_{\bf K}^X (\Gamma)$ for the ${\bf K}$-congruence relation generated by a set of equations $\Gamma$ on the free algebra $\Fm{}{X}$ or, equivalently, for the closure operator for the equational consequence relation.
\begin{Lem}\label{LemmaConditionForNaturalityOfXi}
Let ${\bf K}$ be a quasivariety and $\langle \delta(\varphi), \epsilon(\varphi) \rangle$ an equation over its signature. Suppose that every inclusion $\sigma \colon \Fm{}{Z} \hookrightarrow \Fm{}{X}$ of free algebras coming from an inclusion of generators $Z \subseteq X$ induces a conservative extension of equational theories, i.e. that
for every congruence $\theta \in Co_{\bf K}(\Fm{}{X})$ we have
$$(Cn_{\bf K}^X\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{X} \}) \cap \Fm{}{Z}^2
= Cn_{\bf K}^Z(\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{Z} \}$$
Then the equational filter pair associated to the equation satisfies the naturality condition of Lemma \ref{LemmaTheoryLiftingHoldsIfXiIsNatural}, i.e. $(\sigma \times \sigma)^{-1}(\Xi_{\Fm{}{X}}(T)) = \Xi_{\Fm{}{Z}}(\sigma^{-1}T)$ \end{Lem} \n{\bf Proof:\;\;} First note that $\sigma^{-1}$ (resp. $(\sigma \times \sigma)^{-1}$) is just restriction to formulas (resp. pairs of formulas) in $\Fm{}{Z}$, i.e. $\sigma^{-1}(T) = T \cap \Fm{}{Z}$ and $(\sigma \times \sigma)^{-1}(\theta) = \theta \cap \Fm{}{Z}^2$. Next note that $\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{X} \} = \{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \mid \varphi \in i(\theta) \}$
As by Prop. \ref{FreeAlgebraResult} for free algebras filters are exactly the sets of formulas of the form $T=i(\theta)$ for some $\theta \in Co_{\bf K}(\Fm{}{X})$, we can rewrite the assumption of conservative extension of equational theories by saying that for every filter $T$ we have $$(Cn_{\bf K}^X\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \mid \varphi \in T \}) \cap \Fm{}{Z}^2 = Cn_{\bf K}^Z(\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \mid \varphi \in T \} \cap \Fm{}{Z}^2) \ \ \ \ \ (\ast \ast)$$
Now the claim follows from the series of equations $$\begin{array}{rclr} (\sigma \times \sigma)^{-1}(\Xi_{\Fm{}{X}}(T)) &=& \Xi_{\Fm{}{X}}(T) \cap \Fm{}{Z}^2 & \\
&=& (Cn_{\bf K}^X\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \mid \varphi \in T \}) \cap \Fm{}{Z}^2 & \text{(Cor. \ref{PropOurLeftAdjointIsBlokPigozzisOmegaK})}\\
&=& Cn_{\bf K}^Z(\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \mid \varphi \in T \} \cap \Fm{}{Z}^2) & (\ast \ast) \\
&=& Cn_{\bf K}^Z(\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \mid \varphi \in T \cap \Fm{}{Z}\} ) & \\
&=& \Xi_{\Fm{}{Z}}(T \cap \Fm{}{Z}) & \text{(Cor. \ref{PropOurLeftAdjointIsBlokPigozzisOmegaK})}\\
&=& \Xi_{\Fm{}{Z}}(\sigma^{-1}T) \end{array}$$ \qed
Note that the condition of Lemma \ref{LemmaConditionForNaturalityOfXi} is purely a condition on the quasivariety ${\bf K}$ and the equations, involving no logic or particularities of filter pairs.
\begin{Ex}\label{ExampleSuccessorLogicHasTheoryLiftingProperty} Consider again the logic $l_s$ of Example \ref{ExampleSuccessoralgebra} associated to the equational filter pair $(Co_{\bf K}, i^\tau)$ where $\Sigma$ is the signature with just one unary function symbol $s$, ${\bf K}$ is the class of all $\Sigma$-structures and $\tau$ is the single equation $\langle x, s(x) \rangle$. We will show that the conditions of Lemma \ref{LemmaConditionForNaturalityOfXi} are satisfied.
To this end note that the set $\{ \langle \varphi, s(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{X} \}$ occurring in this condition is the set of $\varphi \in \Fm{}{X}$ which are mapped to $s$-fixed points in $\Fm{}{X} / \theta$. The congruence relation generated by such a set is simply the congruence which identifies all sucessors of such a fixed point. Explicitly $\varphi$ and $\psi$ get identified if and only if they satisfy what one might call the successor-fixed point condition, namely that either $\varphi\textrm{ is an \emph{s}-fixed point in }\Fm{}{X}/\theta\textrm{ and }\psi=s^{\circ n}(\varphi)$ or $\psi\textrm{ is an \emph{s}-fixed point in }\Fm{}{X}/\theta\textrm{ and }\varphi=s^{\circ n}(\psi)$ for some $n \in \mathbb{N}$ (here $s^{\circ n}$ denotes the $n$-fold iteration of $s$). The left one of the two congruences occurring in the condition of Lemma \ref{LemmaConditionForNaturalityOfXi} then consists of those pairs $\langle \varphi, \psi \rangle$ which satisfy the said condition and whose variables lie in $Z$.
The corresponding description holds of course also for the right hand side; it is the set of pairs $\langle \varphi, \psi \rangle$ whose variables lie in $Z$ and which satisfy the successor-fixed point condition. Thus the two sets are equal and the hypotheses of Lemma \ref{LemmaConditionForNaturalityOfXi} are satisfied. \end{Ex}
We thus have a first example of a non-protoalgebraic logic for which we can infer the Craig entailment property from the amalgamation property:
\begin{Ex}\label{ExampleSucessorLogicHasMatrixAmalgamationProperty} Consider the logic $l_s$ of example \ref{ExampleSuccessoralgebra}. We continue using the notation of loc. cit., i.e. the signature $\Sigma$ is the one with exactly one unary operation and $i$ is the natural transformation from the filter pair defining that logic. The category of $\Sigma$-algebras (i.e. of sets with an endomorphism) is equivalent to the functor category $Set^\mathbb{N}$, where one sees the monoid $(\mathbb{N},+)$ as a category with one object. This is a functor category, thus pushouts are formed on the underlying sets. In particular, pushouts of monomorphisms are monomorphisms, because this is the case in the category of sets. This shows that $\Sigma$-Alg has the amalgamation property.
Example \ref{ExampleSuccessorLogicHasTheoryLiftingProperty} together with Lemma \ref{LemmaConditionForNaturalityOfXi} and Lemma \ref{LemmaTheoryLiftingHoldsIfXiIsNatural} implies that the filter pair we used to define $l_s$ has the theory lifting property.
Thus Theorem \ref{TheoremMatrixAmalgamationImpliesCraigInterpolation} applies and we can deduce that $l_s$ has the Craig entailment property. We have successfully applied algebraic reasoning to an (admittedly artificial) logic that is neither protoalgebraic nor truth-equational, see Examples \ref{ExampleSuccessoralgebra} and \ref{ExampleSuccessorLogicIsNotTruthEquational}. \end{Ex}
We conclude this section with determining a class of filter pairs for which the theory lifting property holds.
\begin{Df}\label{DefregularEquationsAndVarieties}
An equation of formulas $\langle \varphi, \psi \rangle \in \Fm{}{X}$ is called \emph{regular} if $\varphi, \psi $ contain exactly the same variables. A variety is called regular if it can be defined by regular equations. \end{Df}
\begin{Ex}
The variety of bounded semilattices is regular: it can be axiomatized over the signature $\Sigma=\{ \wedge, \top \}$ by the equations $x \wedge x = x$, $x \wedge y = y \wedge x$, $x \wedge (y \wedge z) = (x \wedge y) \wedge z$ and $x \wedge \top = x$.
Some varieties of bounded semilattices with a rudimentary implication operation are regular, e.g the one obtained by enhancing the above signature to $\Sigma'=\{ \to, \wedge, \top \}$ and adding the axioms $x \wedge (x \to y) = x \wedge y$ and $x \to (x \wedge y) = (x \to x) \wedge (x \to y)$. Further typical axioms like $x \to x = \top$ and $y \wedge (x \to y)$ are, however, not regular.
The variety of lattices is not regular. Its standard axiomatization contains the absorption law $x \wedge (y \vee x) = x$ which is not regular, in particular the free lattice over some set of generators satisfies that law. If there was an axiomatization by regular quasiequations then, by arguments similar to the ones in the proof of Prop. \ref{PropregularVarietiesHaveTheoryLiftingProperty} below, one could show that the free lattice does not satisfy this equation. \end{Ex}
\begin{Prop}\label{PropregularVarietiesHaveTheoryLiftingProperty} Let ${\bf K}$ be a regular variety. Then any equational filter pair $(Co_{\bf K}, i^\tau)$, with $\tau = \langle \delta(x), \epsilon(x) \rangle$ and $\delta(x)$ and $\epsilon(x)$ each having exactly the free variable $x$, has the theory lifting property. \end{Prop} \n{\bf Proof:\;\;}
We verify the hypothesis of Lemma \ref{LemmaConditionForNaturalityOfXi}, i.e. $$(Cn_{\bf K}^X\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{X} \}) \cap \Fm{}{Z}^2
= Cn_{\bf K}^Z(\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{Z} \}.$$
\noindent $\supseteq$: We have
$$\begin{array}{rclr} (Cn_{\bf K}^X\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{X} \}) \cap \Fm{}{Z}^2 &\supseteq & (Cn_{\bf K}^X\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{Z} \}) \cap \Fm{}{Z}^2 \\
&=& Cn_{\bf K}^Z(\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{Z} \}
\end{array}$$ where the first inclusion holds because of $\Fm{}{X} \supseteq \Fm{}{Z}$ and the second equality follows from Lemma \ref{LemmaInclusionOfVariablesGivesConservativeTranslationOfEquationalConsequences}.
\noindent $\subseteq$: Let $\langle \rho, \psi \rangle \in \Fm{}{Z}^2$ with $\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \} \vDash^X_{\bf K} \langle \rho, \psi \rangle$. Then by \cite[Thm 2.1]{Cze3} there exists a finite sequence $ \langle \alpha_1, \beta_1 \rangle, \ldots, \langle \alpha_n, \beta_n \rangle$ with $ \langle \alpha_n, \beta_n \rangle = \langle \rho, \psi \rangle$ and such that for every $i = 1, \ldots , n$ the equation $\langle \alpha_i, \beta_i \rangle$ satisfies one of the following: \begin{enumerate} \item one has $\alpha_i = \beta_i$.
\item one has $\langle \alpha_i, \beta_i \rangle = \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta$ for some $\varphi \in \Fm{}{X}$
\item there is a substitution mapping one of the defining equations for ${\bf K}$ to $\langle \alpha_i, \beta_i \rangle$.
\item the equation $\langle \alpha_i, \beta_i \rangle$ follows from $\{\langle \alpha_1, \beta_1 \rangle, \ldots, \langle \alpha_{i-1}, \beta_{i-1} \rangle\}$ by one of Birkhoff's rules $\langle \gamma_1, \gamma_2 \rangle \vDash \langle \gamma_2, \gamma_1 \rangle$ (symmetry), $\{ \langle \gamma_1, \gamma_2 \rangle, \langle \gamma_2, \gamma_3 \rangle \} \vDash \langle \gamma_1, \gamma_3 \rangle$ (transitivity) or \linebreak $\{ \langle \gamma_1, \gamma_1' \rangle, \ldots, \langle \gamma_k, \gamma_k' \rangle \} \vDash \langle f(\gamma_1, \ldots, \gamma_k), f(\gamma_1', \ldots, \gamma_k') \rangle$ for a $k$-ary function symbol $f$ (congruence). \end{enumerate} First, observe that by induction one can see that every equation $\langle \alpha_i, \beta_i \rangle$ is regular. Indeed, if $\langle \alpha_i, \beta_i \rangle$ is introduced by one of the first three rules, then it is regular because the image of a regular equation under a substitution is regular, and the equations $\langle x, x \rangle$, $\langle \delta(x), \epsilon(x) \rangle$ and the defining equations of ${\bf K}$ are regular, in the first case obviously so, in the second and third case by assumption on ${\bf K}$.
If $\langle \alpha_i, \beta_i \rangle$ is introduced by the rules of the last item, then observe that these transform regular equations, again into regular equations. Since by induction hypothesis $\{\langle \alpha_1, \beta_1 \rangle, \ldots, \langle \alpha_{i-1}, \beta_{i-1} \rangle\}$ are regular, so is $\langle \alpha_i, \beta_i \rangle$.
Now, again by induction on $n$, we show that one can derive $\langle \rho, \psi \rangle$ already from the hypotheses in $\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{Z} \}$, i.e. without using formulas in $\Fm{}{X} \setminus \Fm{}{Z}$. Indeed, if one arrives at the last step $ \langle \alpha_n, \beta_n \rangle = \langle \rho, \psi \rangle$ by application of one of the first three rules, this is clear.
If one arrives at the last equation by application of the symmetry rule $\langle \psi , \rho \rangle \vDash \langle \rho, \psi \rangle$, then the equation $\langle \psi , \rho \rangle$ also has variables in $Z$ and has a shorter proof, hence by induction hypothesis follows from $\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{Z} \}$.
If one arrives at the last equation by application of the congruence rule \linebreak $\{ \langle \gamma_1, \gamma_1' \rangle, \ldots, \langle \gamma_k, \gamma_k' \rangle \} \vDash \langle f(\gamma_1, \ldots, \gamma_k), f(\gamma_1', \ldots, \gamma_k') \rangle$ for a $k$-ary function symbol $f$, then, since the outcome only contains variables in $Z$, the equations $\langle \gamma_i, \gamma_i' \rangle$ also contain only variables from $Z$. By induction hypothesis all of these equations can be derived from $\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{Z} \}$.
Finally, if one arrives at the last equation by application of the transitivity rule $\{ \langle \rho, \gamma \rangle, \langle \gamma, \psi \rangle \} \vDash \langle \rho, \psi \rangle$, then, since $\rho$ and $\psi$ have variables in $Z$ and $\langle \rho, \gamma \rangle, \langle \gamma, \psi \rangle$ are regular, they also have variables in $Z$, hence by induction hypothesis these equations can be derived from $\{ \langle \delta(\varphi), \epsilon(\varphi) \rangle \in \theta \mid \varphi \in \Fm{}{Z} \}$. \qed
\begin{Cor}\label{CorollaryAmalgamationInregularVarietiesImpliesInterpolation}
Let ${\bf K}$ be a regular variety satisfying the amalgamation property.
Then any logic $l$ with an algebraic semantics in ${\bf K}$ given by a regular equation satisfies the Craig interpolation property. \end{Cor} \n{\bf Proof:\;\;} By Theorem \ref{TeoEveryLogicWithAlgebraicSemanticsComesFromEqunlFilterPair} the logic $l$ can be presented by an equational filter pair. By the construction of that filter pair, it will satisfy the hypotheses of Prop. \ref{PropregularVarietiesHaveTheoryLiftingProperty}, so that we have the theory lifting property. By Theorem \ref{TheoremMatrixAmalgamationImpliesCraigInterpolation} it follows that amalgamation implies Craig interpolation. \qed
We point out that Examples \ref{ExampleSuccessorLogicHasTheoryLiftingProperty}, \ref{ExampleSucessorLogicHasMatrixAmalgamationProperty} are an instance of Corollary \ref{CorollaryAmalgamationInregularVarietiesImpliesInterpolation}.
While the class of logics covered by Corollary \ref{CorollaryAmalgamationInregularVarietiesImpliesInterpolation} does not commonly show up in applications, the result nevertheless shows that systematic connections between the amalgamation property and the Craig interpolation property exist even beyond the classes of protoalgebraic or truth-equational logics. It seems worthwhile to look for further practical criteria for a filter pair to satisfy the theory lifting property, or to satisfy the condition on congruence generation appearing in Lemma \ref{LemmaConditionForNaturalityOfXi}.
\section{Final remarks and future works}\label{SectionVista}
We sketch a few directions into which the work on filter pairs may be taken. We begin with three sets of questions upon which we have touched in this work and which may be developed further.
\begin{Obs}\label{Tomasso Jansana } {\bf Adjoint operators and the Leibniz hierarchy}
With the sample of results of section \ref{filteradjunctionequationalfilterpairs}, the topic of adjunctions and filter pairs is far from exhausted. While we have considered the inclusions $\theta \subseteq \Omega(i(\theta))$, $i(\theta) \subseteq i(\Omega(i(\theta)))$ etc. and the meaning of equality holding in such an inclusion, one can, for example, also consider when equality holds for the opposite compositions $i(\Omega(F))=F$, $\Omega (i(\Omega(F)))=\Omega (F)$ and so on. Concrete descriptions of the operator $\Omega \circ i$ and its companions can be given for equational filter pairs, with assertional logics being the easiest case. Each of these conditions will define a class of logics, possibly extending the Leibniz hierarchy.
A natural further step will be to give corresponding treatments of the Suszko operator and work out the relation to the strong version of a logic.
Furthermore, in \cite{JM1}, \cite{JM2} there is an interesting approach to the Leibniz hierarchy: Roughly, Leibniz classes are taken to be classes defined by the condition of receiving a translation from a fixed class of logics. It is an interesting question whether the classes of logics defined via properties of the operators $i, \Omega, \Xi$ and their adjoints, are Leibniz classes in the sense of \cite{JM1}, \cite{JM2}. \end{Obs}
\begin{Obs}\label{Rem bridge-theorems} {\bf Bridge theorems through equational filter-pairs.} In Section \ref{SectionCraigInterpolation} we have started to look into results connecting amalgamation and Craig interpolation. One may try to refine these results, for example find new criteria for the theory lifting property to hold. The theory lifting does not need to occur through the operators $\Xi$ or $\Omega$; these two are just the extreme ends between which further operators might be found with the necessary naturality condition. Even concentrating on the operator $\Xi$ there is much room for further investigations. The question when the condition of Lemma \ref{LemmaConditionForNaturalityOfXi} holds is purely a matter of universal algebra, that is worth investigating. For particular quasivarieties and equations an analysis of the proofs in the associated equational logic, along the lines of the one carried out in the proof of Proposition \ref{PropregularVarietiesHaveTheoryLiftingProperty}, may lead to amalgamation results.
It will also be interesting to look for further results connecting algebraic properties with logical properties for congruence filter pairs, e.g. the Beth property, or being admissibly closed, which for an algebraizable logic can be expressed in terms of the category of matrices or the category found in \cite{MaPi2}. \end{Obs}
\begin{Obs}\label{Rem Logic by equations}{\bf Studying logics by variation of their defining equation.} A common instance of the situation of Theorem \ref{TheoremLogicsFromEquations} is when the signature $\Sigma$ contains connectives $\wedge, \vee, \top, \bot, \neg$ for conjunction, disjunction, truth, falsity and negation (and possibly others), ${\bf K}$ is a quasivariety of lattices with extra operations and the equation of the filter pair is simply given by $\langle x, \top \rangle$. This means that filters are sets of elements which are the equivalence class of the element $\top$ in some quotient.
Even if one is only interested in this logic, one may get insight by varying the equation. For example, maintaining the same variety ${\bf K}$, one has also the logics associated to the equations $\langle x, \neg x \rangle$, $\langle x \wedge \neg x, \bot \rangle$, $\langle x \vee \neg x, \top \rangle$ or $\langle x , \neg\neg x \rangle$. The theorems of these logics can be seen to be the ``contingent formulas'' (which are ``as true as they are false'', which e.g. makes sense in many-valued logic), the consistent formulas (which is a non-trivial property for paraconsistent logics), the exhaustive formulas (which, together with their negation exhaust all possible cases) or the classical formulas (e.g. considered in intuitionistic logic as the image of the double negation translation).
The consequence relation of these new logics can, roughly, be understood as $\Gamma \vdash_{cons} \varphi$ saying that if all $\gamma \in \Gamma$ are, say, consistent, then so is $\varphi$. The relationship to the initially given logic is ensured by taking the same quasivariety ${\bf K}$.
One can start exploring the new associated logics with the help of the congruence filter pair presentations. For example, suppose that one knows that the initial logic is algebraizable and enjoys the Craig interpolation property. Then one knows that ${\bf K}$ has the amalgamation property and can try to infer Craig interpolation for the new logics by the techniques of Section \ref{SectionCraigInterpolation}.
\end{Obs}
While the previous directions of further research are close to the contents of this article, there are also quite different directions into which the idea of filter pairs may be usefully developed.
\begin{Obs}\label{Rem new hierarchy for filter pairs} {\bf Further examples of filter pairs.}
In this work we have emphasized the class of congruence filter pairs. But there are other interesting classes. For example one can consider, instead of the lattice of (relative) congruences on algebras, the lattice of (relative) congruences for some subsignature (or more generally a morphism of signatures with target the signature of the given logic).
Let us consider the extreme example of the empty subsignature: Then the functor part of the filter pair will associate to an algebra the lattice of equivalence relations on its underlying set. In general a quotient by such an equivalence relation does not inherit an algebra structure. It \emph{does}, however, inherit a structure of \emph{multialgebra}, i.e. a gadget with many-valued operations corresponding to the connectives of the initially given signature. This could connect to Avron's non-deterministic semantics \cite{Avron} \cite{AvronZamanskyHandbook}, and maybe make it possible to relate metalogical properties to non-deterministic matrices in the style of algebraic logic.
Going further, one may take a subsignature and consider consequence relations only with respect to that subsignature. Repeating the considerations of the previous section, one could hope to obtain a partial Craig interpolation theorem, talking only about the fragment given by the subsignature.
Finally, allowing more general signature morphisms, and considering congruences for the pullbacks of algebras along these, one may obtain a framework for remote algebraization from the left -- the dual to the possible translation semantics of \cite{BCC1} \cite{BCC2}.
Also any functorial reflective sublattice of the lattice of all congruences can be considered, as long as the inclusion preserves directed suprema. The passage to a reflective sublattice will result in a stronger logic (to see this just consider the composition of the two adjunctions in question).
Of course, lattices do not need to consist of congruences at all. One possible direction, still staying on the side of algebra, is to let ${\bf K}$ be any reflexive subcategory of $\Sigma$-Str, not necessarily a quasivariety, and for a $\Sigma$-algebra $A$ consider a lattice of maps into certain members of ${\bf K}$ ordered by factorization. In a different direction, both the quotient and the subobject lattices in a locally finitely presentable category are algebraic lattices \cite[Thm. 5, Thm. 12]{PorstAlgLattices}. One might try to take the subobject lattices in a category of coalgebras and obtain a new point of view on coalgebraic semantics. \end{Obs}
\begin{Obs} {\bf Algebraic semantics in other categories than Sets.} One may adjust the notion of filter pair by changing the domain category from $\Sigma$-structures in the category of sets to $\Sigma$-structures in different categories. Not only will this extend the range of semantics that can be given to propositional logics. It should also render the statement of Prop. \ref{CorollaryAmalgamationInregularVarietiesImpliesInterpolation} far more useful: While regular varieties defined over the category of sets are scarce, it is precisely regularity that allows to interpret such varieties in arbitrary symmetric monoidal categories. For example, the variety of rings is not regular, but the variety of monoids is. Hence one can say what a monoid in a symmetric monoidal category is. Taking the category of abelian groups with tensor product, one recovers the category of rings. \end{Obs}
\begin{Obs}\label{Rem InfFilterPair+Category} {\bf On categories of infinitary logics and infinitary filter pairs.} Motivated by a question on the existence of natural extensions of a logic, posed by Cintula and Noguera in \cite{CN} and in the meantime solved by P{\v r}enosil \cite{Prenosil}, we have extended in \cite{AMP2} the notion of finitary filter pair to an infinitary version called $\kappa$-filter pair, where $\kappa$ is any regular cardinal. We show that a $\kappa$-filter pair gives rise to a logic of cardinality $\kappa$ and every logic of cardinality $\kappa$ comes from a $\kappa$-filter pair, extending the result for $\kappa = \omega$ contained in this work. Together with an analogue of Prop. \ref{PropHomomorphismsInduceTranslationsAndVariableInclusionsInduceConservativeTranslations} this yields a solution to the construction of natural extensions, which turns out to be identical to one of P{\v r}enosil's solutions, but obtained on a different route.
Due to the possible non-uniqueness of natural extensions, the ambiguity of different presentations of a single logic by different filter pairs is bigger than in the finitary case. This suggests a point of view of a filter pair as a logic together with a family of natural extensions to sets of variables of all arities.
The natural task arises to extend the results of this work to the infinitary case. This is carried out in \cite{AMP2} for the results of Section \ref{SectionFilterFunctors}. \end{Obs}
\begin{Obs}\label{Rem MetaLogic} {\bf On a representation theory of logics.} Taking adequate notions of morphisms, we show that the category of logics of cardinality $\kappa$ and translation morphisms is (isomorphic to) a full reflective subcategory of the category of $\kappa$-filter pairs: Thus this is can be see as a natural sequel of the work initiated in \cite{MaPi2}, where there is established a full correspondence of certain functors between categories of $\Sigma$-structures and translations between algebraizable logics.
The presentation of functorial encodings of morphisms of logics, will play a role in the long term project of on establishing a representation theory of logics, i.e. studying arbitrary logics through their translations into a class of "well-behaved logical objects", such as the class of algebraizable logics (\cite{MaPi1}, \cite{MaPi2}) or the class of filter pair matrices (\cite{AMP1}).
A possibly interesting use of these machinery is to establish local-global principles in the realm of Propositional Logic, in particular in the study of meta-logical properties of logical systems. A concrete goal is to establish sufficient conditions for a meta-logical property to be preserved under constructions such as products, filtered colimits, among others. \end{Obs}
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Topology seminar
Main seminar page
UMN Mathematics
Talks in Fall 2019
Weiyan Chen (University of Minnesota)
Topology and Arithmetic Statistics
Topology studies the shape of spaces. Arithmetic statistics studies the behavior of random algebraic objects such as integers and polynomials. I will talk about a circle of ideas connecting these two seemingly unrelated areas. To illuminate the connection, I will focus on three concrete examples: (1) the Burau representation of the braid groups, (2) analytic number theory for effective 0-cycles on a variety, and (3) cohomology of the space of multivariate irreducible polynomials. These projects are parts of a broader research program, with numerous contributions by topologists, algebraic geometers, and number theorists in the past decade, and lead to many future directions yet to be explored. (PS. This will be a rehearsal of a job talk accessible to the general audience. Any comments or suggestions are appreciated.)
Tyler Lawson
Affiliation: University of Minnesota
Cochain models for the unit group of a differential graded algebra
Abstract not available
Elden Elmanto (Harvard University)
Compactifying the étale topos
The speaker has long feared the technicalities and intricacies of equivariant stable homotopy theory. Fortunately, beginning with the work of Glasman, major simplification on the foundations of the subject has been made (cf. the work of Ayala-Mazel-Gee-Rozenblyum, Nikolaus-Scholze and the Barwick school). We offer another perspective (that the speaker has a chance of understanding) on equivariant stable homotopy theory, at least for the group C_2, via algebraic geometry. We view it as a way to remedy an infamous annoyance: the 2-étale cohomological dimension of the field of real numbers is infinite. We do this by identifying the genuine C_2-spectra with a category of motives based on Real algebraic geometry ala Scheiderer. This is joint work with Jay Shah.
Craig Westerland (University of Minnesota)
Second order terms in arithmetic statistics
The machinery of the Weil conjectures often allows us to relate the singular cohomology of the complex points of a scheme to the cardinality of its set of points over a finite field. When we apply these methods to a moduli scheme, we obtain an enumeration of the objects the moduli parameterizes. It's rare that we can actually fully compute the cohomology of these moduli spaces, but homological stability results often give a first order approximation to the homology.
In this talk, we'll explain how to obtain second order homological computations for a class of Hurwitz moduli spaces of branched covers; these yield second order terms in enumerating the moduli over finite fields. We may interpret these as second order terms in a function field analogue of the function which counts number fields, ordered by discriminant. Our second order terms match those of Taniguchi-Thorne/Bhargava-Shankar-Tsimerman in the cubic case, and give new predictions in other Galois settings.
This is joint (and ongoing) work with Berglund, Michel, and Tran.
Liam Keenan (University of Minnesota)
Descent properties of topological Hochschild homology
Algebraic K-theory is an extremely rich but notoriously difficult invariant to compute. In order to make calculations tractible, topological Hochschild homology and topological cyclic homology were introduced, along with the Dennis and cyclotomic trace maps. A natural question to consider is whether or not these invariants are sheaves for various topologies arising in algebraic geometry. In fact, it turns out that topological Hochschild homology is a sheaf for the fpqc topology on connective commutative ring spectra. In this talk, I plan to introduce the language necessary and sketch the argument of this result.
Sasha Voronov (University of Minnesota)
Mysterious Duality
"Mysterious Duality" was discovered by Iqbal, Neitzke, and Vafa in 2001. They noticed that toroidal compactifications of M-theory lead to the same series of combinatorial objects as del Pezzo surfaces do, along with numerous mysterious coincidences: both toroidal compactifications and del Pezzo surfaces give rise to the exceptional series E_k; the U-duality group corresponds to the Weyl group W(E_k), arising also as a group of automorphisms of the del Pezzo surface; a collection of various M- and D-branes corresponds to a set of divisors; the brane tension is related to the "area" of the corresponding divisor, etc. The mystery is that it is not at all clear where this duality comes from. In the talk, I will present another series of mathematical objects: certain versions of multiple loop spaces of the sphere S^4, which are, on the one hand, directly connected to M-theory and its combinatorics, and, on the hand, possess the same combinatorics as the del Pezzo surfaces. This is a report on an ongoing work with Hisham Sati.
Dongkwan Kim (University of Minnesota)
Homology class of Deligne-Lusztig varieties
Since first defined by Deligne and Lusztig, a Deligne-Lusztig variety has become unavoidable when studying the representation theory of finite groups of Lie type. This is a certain subvariety of the flag variety of the corresponding reductive group, and its cohomology groups are naturally endowed with the action such finite groups, which in turn gives a decomposition of irreducible representations called Lusztig series. In this talk, I will briefly discuss the background of Deligne-Lusztig theory and provide a formula to calculate the homology class of Deligne-Lusztig varieties in the Chow group of the flag variety. If time permits, I will also discuss their analogues.
Peter Webb (University of Minnesota)
Transfer in the homology and cohomology of categories
The cohomology of a category has many properties that extend those that are familiar when the category is a group. Second cohomology classifies equivalence classes of category extensions, first cohomology parametrizes conjugacy classes of splittings, first homology is the abelianization of the fundamental group, and second homology has a theory that extends that of the Schur multiplier. Defining restriction and corestriction maps on the homology of categories is problematic: most attempts to do this require induction and restriction functors to be adjoint on both sides, and this typically does not happen with categories. We describe an approach to defining these maps that includes all the situations where they can be defined in group cohomology, at least when the coefficient ring is a field. The approach uses bisets for categories, the construction by Bouc and Keller of a map on Hochschild homology associated to a bimodule, and the realization by Xu of category cohomology as a summand of Hochschild cohomology.
Cohomology of the space of complex irreducible polynomials in several variables
We will show that the space of complex irreducible polynomials of degree d in n variables satisfies two forms of homological stability: first, its cohomology stabilizes as d increases, and second, its compactly supported cohomology stabilizes as n increases. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.
Talks in Spring 2018
Nick Proudfoot (University of Oregon)
The Orlik-Terao algebra and the cohomology of configuration spaces
The Orlik-Terao algebra is the subalgebra of all rational functions in n variables generated by 1/(x_i - x_j). I will explain how to use topological techniques to understand this algebra as a graded representation of the symmetric group. I will also describe two different connections (one proven and one conjectural) between this algebra and the cohomology of configuration spaces.
Ben Knudsen (Harvard University)
Edge stabilization in the homology of graph braid groups
We discuss a novel type of stabilization map on the configuration spaces of a graph, which increases the number of particles occupying an edge. Through these maps, the homology of the configuration spaces forms a module over the polynomial ring generated by the edges of the graph, and we show that this module is finitely generated, implying eventual polynomial growth of Betti numbers over any field. Moreover, the action lifts to an action at the level of singular chains, which contains strictly more information; indeed, we show that this differential graded module is almost never formal over the ring of edges. These results, along with numerous calculations, arise from consideration of an explicit chain complex, which is a structured enhancement of a cellular model first considered by Swiatkowski. We arrive at this model through a local-to-global approach combining ideas from factorization homology and discrete Morse theory. This is joint work with Byung Hee An and Gabriel Drummond-Cole.
Dylan Wilson (University of Chicago)
HF_p is an equivariant Thom spectrum
Mahowald (and later many others) proved that HF_2 is a Thom spectrum for a bundle over the double loop space on the 3-sphere. This has had lots of applications, including (more recently) some nilpotence results for E_2-algebras and a calculation of THH of F_2. In this talk, we explain how to extend the result to the case of equivariant spectra over a cyclic group of prime power order. The case of HF_2 over C_2 is joint with Mark Behrens, and the general case is joint with Jeremy Hahn. If there's time, we'll discuss how this fits into a program of Hill-Hopkins-Ravenel for solving the 3-primary Kervaire invariant problem.
Jonathan Campbell (Vanderbilt University)
Topological Hochschild Homology and Characteristics
In this talk I'll review the definition of duality in categories and bicategories and how certain functors called "shadows", due to Kate Ponto, can be used to extract Euler characteristic-type invariants from this data. It turns out that topological Hochschild homology (which I'll define) is an example of such a shadow, and this can be used to relate classical invariants from fixed point theory (e.g. the Reidemeister trace) with the image of the cyclotomic trace in K-theory. Time permitting, I'll sketch how anything that one calls a "characteristic" should fit into this story. This is joint work with Kate Ponto.
Weiyan Chen (University of Minnesota, Twin Cities)
Topology of enumerative problems: inflection points on cubic curves
We will focus on one concrete example of a broader program to study certain topological questions arising from classical enumerative problems in algebraic geometry. For example, every smooth cubic plane curve naturally has 9 marked points on it, namely its 9 inflection points. We ask: are there other ways to choose n distinct points on a smooth cubic curve as the curve varies continuously in family? We will show that the 9 inflection points are almost universal among all those choices. The key is to understand the topology of complement of certain discriminant variety.
Kathryn Hess (École polytechnique fédérale de Lausanne)
Topological vistas in neuroscience (special colloquium)
I will describe results obtained in collaboration with the Blue Brain Project on the topological analysis of the structure and function of digitally reconstructed microcircuits of neurons in the rat cortex and outline our on-going work on topology and synaptic plasticity. The talk will include an overview of the Blue Brain Project and a brief introduction to the topological tools that we use. If time allows, I will also briefly sketch other collaborations with neuroscientists in which my group is involved.
Sasha Voronov (University of Minnesota, Twin Cities)
Quantum Deformation Theory
Classical deformation theory is based on the Classical Master Equation (CME), a.k.a. the Maurer-Cartan Equation: dS + 1/2 [S,S] = 0. Physicists have been using a quantized CME, called the Quantum Master Equation (QME), a.k.a. the Batalin-Vilkovisky (BV) Master Equation: dS + h \Delta S + 1/2 {S,S} = 0. The CME is defined in a differential graded (dg) Lie algebra, whereas the QME is defined in a space V[[h]] of formal power series or V((h)) of formal Laurent series with values in a dg BV algebra V. One can anticipate a generalization of classical deformation theory arising from the QME, quantum deformation theory. Quantum deformation functor and its representability will be discussed in the talk.
Kirsten Wickelgren (Georgia Tech)
Motivic Euler numbers and an arithmetic count of the lines on a cubic surface
A celebrated 19th century result of Cayley and Salmon is that a smooth cubic surface over the complex numbers contains exactly 27 lines. Over the real numbers, it is a lovely observation of Benedetti_Silhol, Finashin–Kharlamov and Okonek–Teleman that while the number of real lines depends on the surface, a certain signed count of lines is always 3. We extend this count to an arbitrary field k using an Euler number in A1-homotopy theory. The resulting count is valued in the Grothendieck-Witt group of non-degenerate symmetric bilinear forms. This is joint work with Jesse Kass.
Jesse Wolfson (UC Irvine)
The Theory of Resolvent Degree - After Hamilton, Hilbert, Segre and Brauer
Resolvent degree is an invariant of a branched cover which quantifies how "hard" is it to specify a point in the cover given a point under it in the base. Historically, this was applied to the branched cover P^n/S_{n-1} --> P^n/S_n, from the moduli of degree n polynomials with a specified root to the moduli of degree n polynomials. Classical enumerative problems and congruence subgroups provide two additional sources of branched covers to which this invariant applies. In ongoing joint work with Benson Farb, we develop the theory of resolvent degree as an extension of Buhler and Reichstein's theory of essential dimension. We apply this theory to systematize an array of classical results and to relate the complexity of seemingly different problems such as finding roots of polynomials, lines on cubic surfaces, and level structures on intermediate Jacobians.
Megan Maguire (University of Wisconsin -- Madison)
Cohomology algebras of configuration spaces
For a manifold X with finite-dimensional cohomology, we know that the cohomology algebra of each unordered configuration space of X is finitely generated, but can we say something stronger about its generators? More precisely, does there exist a D (depending only on X) so that the cohomology algebra of each unordered configuration space of X can be generated in degree at most D? We will answer this question in some cases.
Tasos Moulinos (University of Illinois, Chicago)
Derived Azumaya algebras and twisted K-theory
Topological K-theory of dg-categories is a localizing invariant of dg-categories over C taking values in the ∞-category of KU-modules. In this talk I describe a relative version of this construction; namely for X a quasi-compact, quasi-separated C-scheme I construct a functor valued in ShvSp(X(C)), the ∞-category of sheaves of spectra on X(C). For inputs of the form Perf(X,A) where A is an Azumaya algebra over X, I characterize the values of this functor in terms of the twisted topological K-theory of X(C). From this I deduce a certain decomposition, for X a finite CW-complex equipped with a bundle of projective spaces π : P → X, of KU(P) in terms of the twisted topological K-theory of X ; this is a topological analogue of a result of Quillen's on the algebraic K-theory of Severi-Brauer schemes.
Homotopy Lie algebroids and bialgebroids
Lie algebroids appear throughout geometry and mathematical physics and realize the idea of a family of Lie algebras parameterized by a smooth manifold. A well-known result of A. Vaintrob characterizes Lie algebroids and their morphisms in terms of homological vector fields on supermanifolds, which might be regarded as a version of derived geometry. This leads naturally to the notion of an L_infty-algebroid, which offers a way to think of a sheaf of L_infty-algebras over a smooth manifold. The situation with Lie bialgebroids and their morphisms is more complicated, as they combine covariant and contravriant features. We approach them in terms of odd symplectic dg-manifolds, building on the work of D. Roytenberg. We extend them to the homotopy Lie case and introduce the notions of an L_infty-bialgebroid and an L_infty-morphism between them. This is a joint work with Denis Bashkirov.
Jennifer Wilson (Stanford University)
Stability in the second homology of Torelli groups
In this talk I will describe stability results for the Torelli subgroups of mapping class groups of surfaces. Specifically, I will explain how to use tools from representation theory to establish patterns their second homology. These "representation stability" results are an application of advances in a general algebraic framework for studying sequences of group representations. This project is joint with Jeremy Miller and Peter Patzt.
David Roberts (University of Minnesota, Morris)
Monodromy for a large class of Hurwitz-Belyi maps
Hurwitz varieties covering configuration varieties are important objects in mathematics, with monodromy described via permutation representations of braid groups. In the case of the curves, the maps can be normalized to be Belyi maps, meaning that the base becomes just C-{0,1} and monodromy can be described by giving two permutations. The talk will present a uniform formula explicitly giving the monodromy of a large class of these Hurwitz-Belyi maps simultaneously. Many pictorial examples will be given, illustrating both the monodromy formula itself and related phenomena.
Tyler Lawson (University of Minnesota, Twin Cities)
Craig Westerland (University of Minnesota, Twin Cities)
Structure theorems for braided Hopf algebras
The Milnor-Moore and Poincaré-Birkhoff-Witt theorems are fundamental in studying the structure of Hopf algebras, particularly in characteristic zero. Central to these results is the fact that the space of primitives in a Hopf algebra is a Lie algebra. Part of the classification of pointed Hopf algebras involves a notion of ``braided Hopf algebras;" the primitives in these objects no longer form a Lie algebra.
I will present work in progress which will establish analogues of the Poincaré-Birkhoff-Witt and Milnor-Moore theorems in this setting. The main new tool is a braided analogue of the notion of a Lie algebra defined in terms of braided operads. This can be used to establish these results, and also presents an unexpected connection to profinite braid groups and related operads.
Saul Glasman (University of Minnesota, Twin Cities)
Symmetric power structures on algebraic K-theory
This is a talk about the lambda-operations and their relatives on algebraic K-theory, which we argue should be thought of as analogous to power operations in a homology theory. Classically constructed as operations on homotopy groups, we extend the lambda-operations to the space level by using the surprising observation that polynomial functors between additive categories give maps of K-theory spaces. Subsequently, we use the language of Lawvere theories to define and study a category of spectral lambda-rings. We end with some speculations concerning how these structures might relate to the trace map. This is joint work in progress with Barwick, Mathew and Nikolaus.
Braids, algebraic curves, and 0-cycles
We will see how ideas from topology can provide different perspectives on problems in arithmetic via the bridge of étale cohomology. I will present two concrete examples: (1) how braid groups and Burau representations (topology) could help one count points on algebraic curves over finite fields (arithmetic), and (2) how topology would lead to generalizations of the well-known analogy between integers and polynomials, and results about "analytic number theory for effective 0-cycles on varieties over finite fields".
Mike Hill (UCLA)
Evenness in equivariant topology
Motivated by work of Wilson, I will describe a version of ``evenness'' in equivariant homotopy theory. There are dual notions of even cells and even homotopy groups, and one can also ask for things which have both properties. I'll describe some of the ongoing work in this area, all of which is joint with Hopkins.
Sam Nariman (Northwestern University)
Friedlander-Milnor's problem for diffeomorphism groups
Let G be a finite dimensional Lie group and G^delta be the same group with the discrete topology. The natural homomorphism from G^delta to G induces a continuous map from BG^delta to BG. Milnor conjectured that this map induces a p-adic equivalence. In this talk, we discuss the same map for infinite dimensional Lie groups, in particular for diffeomorphism groups and symplectomorphisms. In these cases, we show that the map from BG^delta to BG induces split surjection on cohomology with finite coefficients in "the stable range". If time permits, I will discuss applications of these results in foliation theory, in particular, flat surface bundles.
Nitu Kitchloo (Johns Hopkins University)
Quantization of the Modular Functor and Equivariant Elliptic Cohomology
Let M be a compact G-space for a compact Lie group G. I will describe a procedure that can be seen as the categorical quantization of the category of parametrized positive energy representations of the loop group of G. This procedure is described in terms of dominant K-theory of the loop group parametrized over M. More concretely, I will construct a holomorphic sheaf over a universal elliptic curve with values in dominant K-theory of the loop space LM, and show that each stalk of this sheaf is a cohomological functor of M (thereby giving rise to an equivariant cohomology theory). I will also give compelling evidence that this theory is equivalent to equivariant elliptic cohomology of M as constructed by Grojnowski. I will assume very little background, and give a lot of motivation, but some general ideas of what a field theory is may be helpful.
Agnès Beaudry (University of Colorado)
The K(2)-local Picard group at p=2
The K(n)-local categories provide examples of interesting Picard groups. Their importance in chromatic homotopy theory is highlighted by the fact that the dualizing object for Brown-Commenetz duality comes from an invertible element. These groups have been computed at all primes when n=1 and all odd primes when n=2. Mahowald predicted that the torsion in the K(2)-local Picard group at the prime 2 would be very large compared to the torsion in the K(2)-local Picard groups at odd primes. In this talk, I will confirm this and explain our current understanding of the structure of this group.
Tyler Lawson (UMN)
Khovanov homotopy
The Jones polynomial is an invariant of a knot or link, and Khovanov homology is a categorical lift so that the coefficients of the Jones polynomial are ranks of Khovanov homology groups. I'll discuss a refinement of this to a homotopy type due to Lipshitz-Sarkar and Hu-Kriz-Kriz, and discuss ongoing work to refine this to an invariant for tangles.
Nikita Markarian (Dept Math, Higher School of Economics, Moscow)
Weyl n-algebras and the Swiss cheese operad
I will discuss the definition and applications of Weyl n-algebras and connected algebras over the Swiss cheese operad.
Aaron Royer (UCLA)
Affinization, Witt vectors and duality
The Hochschild--Kostant--Rosenberg theorem relates the Hochschild homology and Andre--Quillen homology of smooth commutative algebras. Using ideas from derived algebraic geometry, Ben-Zvi--Nadler proved an HKR theorem for rational E_\infty-rings sans smoothness hypotheses. Their argument ultimately relies on an equivalence between the algebra of rational cochains on the circle and a trivial square-zero extension. We will explain how the situation is more complicated away from characteristic zero, and why this is actually a good thing.
Jack Morava (Johns Hopkins University)
Topological Hochschild homology of (some) perfectoid fields
Beside its familiar topology, the field of complex numbers has complete p-adic topologies for all integral primes. Unlike the Archimedean topology, which has only the real numbers as a complete topological subfield, the p-adic topologies have many interesting (complete, totally ramified) subfields: in particular, the completions of maximal totally ramified extensions of locally compact number fields (such as the p-cyclotomic field obtained by adjoining all p-power roots of unity to the p-adic rationals).
Recent work of Scholze, Bhatt and others has transformed the study of algebraic geometry over such fields, while similarly recent work of Hesselholt and Madsen has transformed our understanding of their algebraic topology. This talk will focus on connections between Lubin-Tate (ie generalized cyclotomic field) theory and topological Hochschild homology, which seems to be a powerful tool for understanding perfectoid geometry.
Saul Glasman (UMN)
Goodwillie calculus and Mackey functors
This talk concerns a theorem stating that the theory of n-excisive functors from the category of spectra to a target stable oo-category E is equivalent to the theory of Mackey functors on a suitable indexing category - the category of finite sets of cardinality at most n with morphisms surjections - which has many of the salient properties of the orbit category of a finite group. We'll give motivation for this theorem, and give a broad sketch of the proof, which decomposes readily into a sequence of independent assertions. We'll then prove as many of these assertions as time allows.
An overview of equivariant stable homotopy theory and stable functor calculus
This will be an almost totally expository introduction to certain topics in stable homotopy theory. I'll attempt to motivate and introduce Elmendorf's theorem on G-spaces and the spectral Mackey functor approach to G-spectra. I'll then discuss Goodwillie's functor calculus for functors from spectra to spectra, which answers the question of what it means for such a functor to be "polynomial". If time permits, I'll start exploring the analogies between these two theories, a task which will be taken to its conclusion in next week's talk.
Akhil Mathew (Harvard University)
Artin induction for ring spectra and algebraic K-theory
A theorem of Mitchell states that the chromatic complexity of the algebraic K-theory spectrum of a discrete ring R is bounded by one, i.e., the Morava K-theory vanishes at heights at least two. We give an approach for bounding the chromatic complexity in the algebraic K-theory of ring spectra.
Let R be a ring spectrum. We say that R-based Artin induction holds for a family of groups if for every finite group G, the rationalized Grothendieck group of the category of perfect R-modules with G-action is induced from the given family. We show that Artin induction theorems can be used to bound chromatic complexity and give several examples in ring spectra, in line with the redshift philosophy of Rognes. This is joint work with Dustin Clausen, Niko Naumann, and Justin Noel.
Iterated algebraic K-theory and T-duality
We introduce a periodic form of the iterated algebraic K-theory of ku, the (connective) complex K-theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological T-duality for sphere bundles oriented with respect to this theory. This is joint work with John Lind and Hisham Sati.
Iterated algebraic K-theory and T-duality, II
We introduce a periodic form of the iterated algebraic K-theory of ku, the (connective) complex K-theory spectrum, as well as a natural twisting of this cohomology theory by higher gerbes. Furthermore, we prove a form of topological T-duality for sphere bundles oriented with respect to this theory. This is joint work with John Lind and Hisham Sati. I will continue from last week, and discuss T-duality.
Tyler Lawson (University of Minnesota)
Derived Morita theory
In this talk I'll discuss joint work with David Gepner on the study of Morita theory and Azumaya algebras in the derived context.
John Lind (University of Regensburg)
Duality in bicategories and the THH transfer
Associated to a fibration E --> B with homotopy finite fiber is a stable wrong way map LB --> LE of free loop spaces coming from the transfer map in THH. This transfer is defined under the same hypotheses as the Becker-Gottlieb transfer, but on different objects. I will use duality in bicategories to explain why the THH transfer contains the Becker-Gottieb transfer as a direct summand. The corresponding result for the A-theory transfer may then be deduced as a corollary. When the fibration is a smooth fiber bundle, the same methods give a three step description of the THH transfer in terms of explicit geometry over the free loop space. (Joint work with C. Malkiewich)
Shigeo Koshitani (Chiba University, Japan)
On endotrivial modules for dihedral defect groups
We will be discussing endotrivial modules for the group algebra kG of a finite group G over a field k of characteristic p > 0 (cf. [Koshitani-Lassueur, Manuscripta Math. 148 (2015), 265--282] and [Koshitani-Lassueur, J.Group Theory, 27 pages, in press]). Endotrivial modules go back to E.C. Dade [1978, Ann. of Math] and also J.L.Alperin. Endotrivial modules show up often and play a very important role in the representation theory of finite groups; for example an endotrivial module for kG induces a self stable equivalence of the module category. Our result is a kind of complement to [Carlson-Mazza-Thevanaz, European J. Math. 15 (2013), 157--177].
TriThang Tran (University of Oregon)
Topology Seminar - TBA
Søren Galatius (Stanford University and University of Copenhagen)
Martin Markl (Ordway visitor) (Institute of Mathematics of the Czech Academy)
Deformation theory IV
A mini-course on deformation theory based on L_\infty-algebras and the Maurer-Cartan equation, culminating with an outline of Kontsevich's deformation quantization proof.
Deformation theory I
Deformation theory III
Deformation theory II
Pedram Hekmati (IMPA, Brasil)
Topological T-duality and Hodgkin's theorem
A famous problem in topology, first solved by Hodgkin in 1967, is to determine the K-theory of compact simply-connected Lie groups. Hodgkin's original proof was extremely technical, motivating the discovery of a number of simpler proofs. In this talk I will present a new, surprisingly simple proof of Hodgkin's theorem using topological T-duality, an idea that originated in physics. This is based on joint work with David Baraglia.
Jesse Wolfson (University of Chicago)
Counting Problems and Homological Stability
In 1970, Arnold showed that the i'th homology of the space of un-ordered configurations of n points in the plane becomes independent of n for large n. A decade later, Segal extended Arnold's method to show that the i'th homology of the space of degree n holomorphic maps from P^1 to itself also becomes independent of n for large n, and, moreover, that both sequences of spaces have the same limiting homology. We explain how, using Weil's number field/function field dictionary, one might have predicted this topological coincidence from easily verifiable statements about specific counting problems. We then discuss ongoing joint work with Benson Farb and Melanie Wood in which we use other counting problems to predict and discover new instances of homological stability in the topology of complex manifolds.
Inna Zakharevich (University of Chicago)
The annihilator of the affine line
The Grothendieck ring of varieties is defined to be the free abelian group generated by k-varieties, modulo the relation that for any closed subvariety Y of a variety X, we impose the relation that [X] = [Y] + [X \ Y]; the ring structure is defined by [X][Y] = [X x Y]. Last December two longstanding questions about the Grothendieck ring of varieties were answered: 1. If two varieties X and Y are piecewise isomorphic then they are equal in the Grothendieck ring; does the converse hold? 2. Is the class of the affine line a zero divisor? Both questions were answered by Borisov, who constructed an element in the kernel of multiplication by the affine line; coincidentally, the proof also constructed two varieties whose classes in the Grothendieck ring are the same but which are not piecewise isomorphic. In this talk we will investigate these questions further by constructing a topological analog of the Grothendieck ring and analyzing its higher homotopy groups. Using this extra structure we will sketch a proof that Borisov's coincidence is not a coincidence at all: that any element in the annihilator of the Lefschetz motive can be represented by a difference of varieties which are equal in the Grothendieck ring but not piecewise isomorphic.
Andy Putman (Rice University)
Stability in the homology of congruence subgroups
I will explain how to use categorical tools arising from the Lannes-Schwartz Artinian conjecture (recently proved by myself and Sam, and independently by Sam-Snowden) to understand patterns in the cohomology of congruence subgroups of SL(n,Z). This is joint work with Steven Sam.
Alexander A. Voronov (University of Minnesota)
Categorification of Dijkgraaf-Witten theory
This is a new perspective on an old topic: Dijkgraaf-Witten theory, i.e., gauge theory with a finite gauge group, producing a TQFT, i.e., a functor from the category of cobordisms to that of vector spaces. I will recall earlier approaches to Dijkgraaf-Witten theory and then describe a categorification of it. This is based on a cap product between homology and cohomology with coefficients in Picard groupoids. This is a joint work with Amit Sharma.
Homotopy limits and colimits
Ordinary limits and colimits behave poorly from the point of view of homotopy equivalences, but they have generalizations which do. This will be an informal talk discussing these limits and colimits in algebraic topology, and in particular the impact of some of the "new" category theory from Lurie's "Higher Topos Theory".
Fox-Neuwirth cells, quantum shuffle algebras, and the homology of braid groups
The notion of a braided vector space V comes from the Hopf algebra community, and examples abound, from conjugacy classes in groups to braidings coming from Cartan matrices. From this definition, the tensor powers of V form a family of braid group representations. They also may be assembled into a non-commutative, non-cocommutative braided Hopf algebra called a quantum shuffle algebra. Our main result identifies the homology of the braid groups with these coefficients as the cohomology of this algebra. Using change of rings spectral sequences, we begin to get a handle on these homology groups. This is joint work with TriThang Tran.
Christian Haesemeyer (UCLA)
Algebraic invariants of singularities
I will discuss my work with Cortinas, Walker, Weibel on the K-theory and cyclic homology of singular algebraic varieties from the point of view of a concrete application to questions about the existence of vector bundles with certain properties.
The Blakers-Massey excision theorem
The Blakers-Massey excision theorem describes the homotopy groups of a union of two spaces through a range. I'll discuss a recent new proof of this result, with the unusual property that it was reverse-engineered by Rezk from a proof by Favonia in a computer proof assistant.
The Blakers-Massey excision theorem II
Nathan Perlmutter (U. Oregon)
Homological stability for the diffeomorphism groups of odd-dimensional manifolds
I will present a new homological stability result for the diffeomorphism groups of manifolds of dimension 2n+1 ≥ 9, with respect to forming the connected sum with copies of an arbitrary (n − 1)-connected, (2n + 1)-dimensional manifold that is stably parallelizable. This result is an odd dimensional analogue of a recent theorem of Galatius and Randal-Williams regarding the homological stability of the diffeomorphism groups of manifolds of dimension 2n ≥ 6, with respect to forming connected sums with S^n × S^n .
Martin Markl (Mathematical Institute of the Czech Academy)
Structures of string field theory via modular envelopes
Fabian Hebestreit (University of Notre Dame)
Twisted Spin cobordism and positive scalar curvature
After a brief exposition of twisted (co)homology and more generally parametrised homotopy theory, I want to explain the relevance of twisted Spin cobordism to the search for Riemannian metrics of positive scalar curvature and in particular the Gromov-Lawson-Rosenberg conjecture. I will then present a homotopical analysis of the underlying parametrised spectrum and, should time permit, report on ongoing joint work with Michael Joachim and Stephan Stolz aimed at proving an existence theorem for such metrics on manifolds whose universal cover admit a spin structure.
Dylan Wilson (Northwestern University)
Genera, Orientations, and tmf with level structure
We will review the work of Ando-Hopkins-Rezk on the Witten genus and show how their construction carries over to the case of topological modular forms with level structure. We also indicate how to improve genera on String manifolds to genera on Spin manifolds and construct the Ochanine genus.
Omar Ortiz (University of Western Ontario)
Moment graphs and the wonderful compactification
The wonderful compactification X of a semisimple group of adjoint type was first constructed by De Concini and Procesi in 1983 and has been extensively studied in algebraic geometry. The equivariant cohomology of X resembles in many ways that of the flag variety, allowing some Schubert calculus to be performed. This talk will be a review of these ideas and an indication of current work on a moment graph theory for X.
Motivic desuspension
Certain problems such as classifying manifolds up to cobordism are stable in the sense that they are solved in categories where it is possible to desuspend. Other problems, such as classifying algebraic vector bundles on schemes, require analogous unstable information. The EHP sequence in algebraic topology is a tool for turning stable information into unstable information. We will discuss the situation from algebraic topology, and present an EHP sequence in A^1 homotopy theory of schemes. This is joint work with Ben Williams.
John Terilla (Queens College, CUNY)
Homotopy Probability Theory with an application to Fluids
Homotopy probability theory is a theory in which the vector space of random variables is replaced with a chain complex. I'll discuss how to use homotopy algebra (rather than analysis) to extract meaningful expectations and correlations among random variables. Some natural examples will be introduced, including an application to fluid flow.
Alexander A. Voronov (University of Minnesota and IPMU)
Ambidexterity I: Dijkgraaf-Witten theory
I will continue the series of talks, started by talks on the Morava K-theory of Eilenberg-Mac Lane spaces by Tyler Lawson and Craig Westerland in the Spring 2014, aimed at ambidexterity after Jacob Lurie and Mike Hopkins, and go over the Dijkgraaf-Witten Topological Quantum Field Theory as a motivational example.
Ambidexterity II: DW theory and ambidextrous spaces
This is the second talk in a series, started by talks on the Morava K-theory of Eilenberg-Mac Lane spaces by Tyler Lawson and Craig Westerland in the Spring 2014, aimed at ambidexterity after Jacob Lurie and Mike Hopkins. I will show how Dijkgraaf-Witten Topological Quantum Field Theory leads to a phenomenon called ambidexterity and talk about ambidextrous maps and spaces.
Benjamin Ward (Simons Center for Geometry and Physics, Stony Brook University)
Maurer-Cartan elements and natural operations
The process of twisting by a Maurer-Cartan element in an operadic context produces cochain complexes computing, for example, the Chevalley-Eilenberg cohomology of a Lie algebra, the Hochschild cohomology of an A-infinity algebra, the singular cohomology of a topological space, the cyclic cohomology of a Frobenius algebra or the equivariant cohomology of an S^1 space. A byproduct of this perspective is the existence of certain families of cohomology and cochain operations, as I will discuss.
Configurations spaces and symmetric complements
In this talk, we discuss homological stability for spaces that are defined as the complements of the closures of certain strata in the n-fold symmetric product — named "symmetric complements". Symmetric complements are closely related to configuration spaces where homological stability was previously known to hold. Homological stability for symmetric complements, in particular, answers a conjecture by Vakil and Wood, which goes part way to understanding the relationship between homological stability, and the existence of certain limits in the Grothendieck ring of varieties.
The talk will proceed by providing a brief introduction to homological stability using configuration spaces as an example. We will then discuss how to leverage stability for configuration spaces to obtain stability for symmetric complements. This is joint work with A. Kupers and J. Miller.
Rune Haugseng (Max-Planck Institute for Mathematics)
The higher Morita category of E_n-algebras
I will discuss a construction of a higher category of E_n-algebras and iterated bimodules, generalizing the classical bicategory of algebras and bimodules. This leads to generalizations of the Picard and Brauer groups, which have been studied in stable homotopy theory as interesting invariants of ring spectra, and should also lead to an "algebraic" construction of factorization homology as an extended topological quantum field theory.
Torsion in the homology of random chain complexes and the Cohen-Lenstra distribution
The expected Betti numbers of "random" simplicial complexes have been studied for some time, beginning with remarkable properties of the Erdos-Renyi random graph. Very little is known about the expected torsion in the homology of a random simplicial complex X. Recently, Matt Kahle has conjectured that for each prime p, the p-torsion in H_*(X) might be Cohen-Lenstra distributed. Loosely, the Cohen-Lenstra distribution on finite abelian p-groups predicts that a given group arises with probability inversely proportional to the size of its automorphism group. In this talk, we'll give evidence for this conjecture by proving a version of it for a suitable notion of random chain complexes (rather than simplicial complexes), using somewhat standard techniques of random p-adic matrices. Additionally, we'll give some number-theoretic, category-theoretic, and measure-theoretic background on the Cohen-Lenstra distribution.
Tomer Schlank (Massachusetts Institute of Technology)
Stable étale obstruction to zero-cycles of degree 1
Given a field K and an algebraic variety X/K, a zero-cycle on X is a formal sum of points on X(\bar{K}) which is Galois equivariant . The zero-cycles with total degree 1 can be considered as a abelianized version of a K-point on X. When K is a number field Colliot-Thélène, conjectured that the famous Brauer-Manin obstruction is the only one for the existence of zero-cycles of degree 1 on a smooth projective variety. From the viewpoint of algebraic geometry, solutions to a Diophantine equation are just sections of a corresponding map of schemes X -> S, where schemes are usually considered as a certain type of "Spaces" . When considering sections of maps of spaces f:X->S in the realm of algebraic topology, Bousfield and Kan developed an Obstruction-Classification Theory using the cohomology of the S with coefficients in the homotopy groups of the fiber of f. By using étale homotopy it is possible to transform this obstruction theory to the realm of algebraic geometry. This allows reinterpretation of classical obstructions to existences of rational points (like the Brauer-Manin Obstruction) and to define new obstructions. One example is using stable étale homotopy theory to get a new obstruction for zero-cycles of degree 1. In a work in progress we hope to use this new obstruction to attack Colliot-Thélène's conjecture. This is a joint project with Y.Harpaz and V.Stojanoska.
I'll be discussing the calculation of the Morava K-theory of Eilenberg-Mac Lane spaces, as part of a discussion of the Hopkins-Lurie paper on ambidexterity in $K(n)$-local homotopy theory.
Yu Tsumura (Purdue University)
A 2-categorical extension of the Reshetikhin-Turaev theory
I will discuss a concrete construction of a 2-categorical extended topological field theory that extends the Reshetikhin-Turaev TQFT. An extended TQFT is defined to be a 2-functor from a 2-category of cobordisms with corners to the Kapranov-Voevodsky 2-vector spaces. As an intermediate 2-category between these 2-categories, a 2-category of special ribbon graphs is introduced.
Morava K-theory of Eilenberg-Mac Lane spaces
Robert Lipshitz (Columbia University)
Morse flows on products and Khovanov homotopy
We will start by recalling the definition of the Khovanov homology of a knot, and a refinement, a Khovanov homotopy type. We will then discuss the combinatorics of Morse flows on products of manifolds (this is surprisingly complicated), and a Kunneth theorem for the Khovanov homotopy type, and give an application to concordance. Along the way, we will present a number of as-yet unanswered questions, mostly of a homotopy-theoretic nature. This is joint work with Sucharit Sarkar.
Amit Patel (University of Minnesota (IMA))
Persistent Sheaves for Stratified Maps
Given a stratified map f : X -> Y, we may study the homology (relative homology) of its fibers by studying its Leray cosheaf (sheaf), one for each dimension. The Leray cosheaf assigns to each sufficiently small open set U of Y the ordinary homology of the pre-image f^{-1}(U). The Leray sheaf assigns to each sufficiently small open set U of Y the relative homology of the pair of spaces (X, X - f^{-1}(U)). For a stratified map g : X -> R to the real line, the indecomposable summands of the Leray cosheaf (sheaf) are in one-to-one correspondence with the appropriate (level-set zigzag) persistence diagram of g. It is therefore tempting to define the persistence diagram of an arbitrary stratified map f : X -> Y as the list of indecomposable summands associated to the Leray sheaf or Leray cosheaf of f. However, the Leray cosheaf (sheaf) is in general not stable to perturbations to f. For example, an arbitrarily small perturbation to f may introduce a hole (a point with a trivial co-stalk) well in the interior of the support of the Leray cosheaf (sheaf).
In this talk, I will present the persistent sheaf for any stratified map f : X -> M to an oriented manifold M. Roughly speaking, the persistent sheaf is a 2-functor that assigns to each sufficiently small open set U of M, a subgroup of the homology of the pre-image f^{-1}(U). I will give a rough construction of the persistent sheaf and show how it is stable to sufficiently small perturbations to f.
This is joint work with Robert MacPherson (IAS).
The Dieudonné module approach to the Morava K-theory of Eilenberg-MacLane spaces.
We'll introduce Dieudonné modules and p-divisible groups in the context of homology of H-spaces, and talk about the reformulation of Ravenel-Wilson's computation of the Morava K-theory of Eilenberg-MacLane spaces in this language.
The Dieudonné module approach to the Morava K-theory of Eilenberg-MacLane spaces, II
Continuing from last week, we will compute the Dieudonne module of K(n)_* K(Z/p^j, 1)), explain some duality results due to Buchstaber-Lazarev, and discuss the p-divisible groups associated to these Hopf algebras.
Donald Kahn (University of Minnesota)
Coincidence sets and Kervaire invariants, d'apres Koschorke and Randall
p-compact groups and étale homotopy types
Connected, compact Lie groups are quite rich in geometric structure. However, they have a classification in terms of the rather simple algebraic data of an integral reflection group, coming from the action of the Weyl group on the character lattice. From the point of view of homotopy theory, the geometric structure is largely invisible, and compact Lie groups are simply loop spaces with finite rank homology. Dwyer and Wilkerson introduced a p-complete analogue of this notion called "p-compact groups," and established some remarkable structure theorems for them. This culminated in a classification due to Andersen et al which looks remarkably similar to the classification of compact Lie groups, only employing p-adic reflection groups. Nonetheless, p-compact groups are decidedly lacking in geometry, and it is the purpose of this talk to try to equip them with some coming from the étale homotopy theory of algebraic groups.
Alexander Voronov (University of Minnesota)
Dijkgraaf-Witten theory using cohomology with coefficients in Picard groupoids
This is a report on work in progress with Amit Sharma in which we define cap product between homology and cohomology with coefficients in Picard groupoids and use it to construct Dijkgraaf-Witten theory, a gauge theory with a finite gauge group.
Cohomology operations and power operations
In the singular cohomology of spaces, one can define both cohomology operations and power operations, and the former turn out to be special instances of the latter. In different situations, however, these start to deviate from each other. I'll discuss some work in progress on understanding how these two structures interact with the specific goals of understanding Morava K-theory and E-theory.
Michael Mandell (Indiana University, Bloomington)
The homotopy theory of cyclotomic spectra
In joint work with Andrew Blumberg, we construct a category of cyclotomic spectra that is (something like) a closed model category and which has well-behaved mapping spectra. We show that topological cyclic homology (TC) is the corepresentable functor on this category given by maps out of the sphere spectrum, verifying a conjecture of Kaledin.
Pavle Blagojevic (Freie Universität Berlin and Mathematical Institute of the Serbian Academy)
On k-regular maps
The question about the existence of a continuous $k$-regular map from a topological space $X$ to an $N$-dimensional Euclidean space $R^N$, which would map any $k$ distinct points in $X$ to linearly independent vectors in $R^N$, was first considered by Borsuk in 1957. In this talk we present a proof of the following theorem, which extends results by F. Cohen and Handel 1978 (for $d=2$) and Chisholm 1979 (for $d$ power of 2): For integers $k$ and $d$ greater then zero, there is no $k$-regular map $R^d \to R^N$ for $N \lt d(k-a(k))+a(k)$, where $a(k)$ is the number of ones in the dyadic expansion of $k$.
Joint work with G. M. Ziegler and W. Lück.
F.R. Cohen (University of Rochester)
A leisurely excursion into polyhedral products
Polyhedral products are natural subspaces of a product space "indexed by a simplicial complex". They are natural extensions of "moment-angle complexes" of Buchstaber-Panov dating back to Poincaré, Coxeter, as well as many others.
Definitions, previous developments as well as a survey of applications of polyhedral products from toric manifolds, analogues of Stanley-Reisner ring of a simplicial complex to packing questions will be described. This talk is based on joint work with A. Bahri, M. Bendersky, S. Gitler, and the speaker.
Michael Hill (University of Virginia)
Equivariant Operadic Actions and the Transfer
Classically, loop spaces are algebras over little disks operads for some inner product space, and this is true equivariantly. Here, however, the indexing universe for spectra plays a role, as $G$-spectra on a universe $U$ have certain additional structure maps: transfers. In this talk, I'll describe how the transfer arises from the operadic structure directly. This is joint with Andrew Blumberg.
Gunnar Carlsson (Stanford University)
Derived completion and K-theory of fields
I will discuss a formulation of the descent problem for the algebraic $K$-theory of fields in terms of derived functors of completion applied to the complex representation ring of the absolute Galois group of the field, together with a discussion of how this conjecture can be proved.
Homotopy, schemes, and Weibel's road map
Daniel Schaeppi (University of Western Ontario)
A Tannakian characterization of categories of coherent sheaves
Classical Tannaka duality is a duality between groups and their categories of representations. The two basic questions it answers are the reconstruction problem (when can a group be reconstructed from its category of representations?) and the recognition problem (can we characterize categories of representations abstractly?).
I will outline how the notion of a Tannakian category can be weakened in order to solve the recognition problem for categories of coherent sheaves of algebraic stacks (if you are an algebraic geometer), respectively categories of comodules of Adams Hopf algebroids (if you are an algebraic topologist). As an application I will outline how this notion can be used to describe coherent sheaves on fiber products in terms of Deligne's tensor product of categories.
Andrew Blumberg (University of Texas, Austin)
The higher-categorical perspective on algebraic K-theory
This talk will give an overview of the recent work interpreting the formal properties of algebraic K-theory in the context of new formalisms for handling the category of homotopical categories.
Tarje Bargheer (Australian National University)
Stretching String Topology
Higher dimensional string topology generalizes the Chas-Sullivan product on the free loop space on M to the space of maps from higher-dimensional manifolds into M. I will present a geometric operad that describes structures on the space of maps from n-dimensional spheres into M, generalizing the Chas-Sullivan structure and shows they are $E_{n+1}$-algebras. I will use this introduction of the operad to show how the geometric construction of the operad allows us to stretch string topology in various directions, suggesting new ways of enriching string topology.
Secondary operations featured prominently in Adams' proof of the Hopf invariant one conjecture, which implies there are no division algebra structures on $\mathbb{R}^n$ when $n \neq 1,2,4,8$. I'll discuss how secondary products and secondary operations appear in different contexts in algebra and topology.
Secondary invariants from Dijkgraaf-Witten theory
As Tyler Lawson explained to us in his last seminar talk, secondary operations arise when a primary operation vanishes or two primary operations are equal. If you have a bordism between two closed manifolds, their homology classes will be equal and, by definition, their cobordism classes will be equal. A Topological Field Theory (TFT) creates a secondary invariant in this case: a linear map between certain vector spaces associated to each manifold. Dijkgraaf-Witten (DW) theory is an example of a TFT associated to a finite group. Recently, there has been a push towards describing an Extended TFT, and that generated deeper understanding of DW theory. I will discuss new insights on DW theory as per Jacob Lurie and others.
Forms of K-theory and elliptic cohomology
I'll discuss K-theory, and how its theory of Chern classes differs from that for ordinary cohomology. I'll then discuss "forms" of K-theory, which provide some twist on the multiplicative rules of ordinary K-theory, and how real K-theory is universal among those. Elliptic cohomology theories will then be introduced, along with the analogous properties a universal object should have.
Elliptic curves and elliptic cohomology theories
I'll be continuing last week's talk by introducing elliptic cohomology theories, and universal objects among these.
Erin Manlove (University of Minnesota)
Morava Stabilizer Groups
Morava stabilizer groups arise in the study of homotopy groups of spheres as objects that encode topological information. Though Morava stabilizer groups themselves remain relatively mysterious, we may study them by looking at their maximal finite subgroups. In my talk, I will give two characterizations of Morava stabilizer groups, describe the type of subgroups that are of interest, and discuss how we connect these subgroups to geometric objects by identifying them with automorphism groups of points in an algebraic stack.
Niles Johnson (Ohio State)
Obstruction theory for homotopical algebra maps
We take an obstruction-theoretic approach to the question of algebraic structure in homotopical settings. At its heart, this is an application of the Bousfield-Kan spectral sequence adapted for the action of a monad $T$ on a topological model category. This yields an obstruction theory for lifting maps from $T$-algebras in a homotopy category to the homotopy category of $T$-algebras. Maps in this latter category are required to commute with a coherent system of higher homotopies encoding the $T$-action. We give general conditions under which the $E_2$ page of the resulting spectral sequence may be identified with André-Quillen cohomology.
This talk will give an overview of the general theory, but focus mainly on the breadth of applications. These include categories of algebraic theories in the sense of Lawvere and categories of algebras over operads. We will describe specific examples in $G$-spaces, $G$-spectra, and rational $E_\infty$ algebras. Furthermore, we will outline calculations connected to rational unstable homotopy theory which distinguish between $E_\infty$ and $H_\infty$.
Eric Peterson (University of California, Berkeley)
Cotangent spaces of certain spectra
We outline a construction in derived algebraic geometry which produces elements of the $K(n)$-local Picard group. As an example, we produce a new model of a spectrum called the determinantal sphere, and as time permits we discuss newly indicated patterns in $K(n)$-local homotopy theory.
Robert Hank (University of Minnesota)
Massey products in $A_\infty$ algebras
During this talk, I'll develop a relationship between Massey products and multiplicative structure in such a way that the generalization of a Massey product from a strictly associative to an $A_\infty$ structure arises naturally.
William Abram (University of Michigan)
The equivariant complex cobordism ring of a finite abelian group
The calculation of the non-equivariant cobordism ring due to Milnor and Quillen was one of the great successes of algebraic topology. Equivariantly, Kriz described tom Dieck's stable equivariant complex cobordism ring $(MU_G)_*$, in the case $G=\mathbb Z/p$, as the pullback of a diagram of rings arising from the Tate diagram for $MU_{\mathbb Z/p}$. We extend this work to the case of $G$ a finite abelian group, where we describe $(MU_G)_*$ as the inverse limit over certain $G$-spectra $F(S)_*$ indexed over chains of subgroups of $G$. In the case $G=\mathbb Z/p^n$, we get a simple description of $(MU_G)_*$ as the n-fold pullback of a diagram of rings. In the general case, we are still able to compute the algebraic structure of $F(S)_*$ explicitly. I will discuss this computation in some detail.
Yifei Zhu (University of Minnesota)
Power operations in an elliptic cohomology theory
As a result of the geometry of vector bundles underlying K-theory, power operations in K-theory correspond to linear representations of finite symmetric groups. For elliptic cohomology theories, lacking analogous geometric interpretations, we can study their power operations via a correspondence to cyclic isogenies of elliptic curves, in hope of understanding their conjectural geometry. This approach is valid for Morava E-theories in general, including versions of K-theory and elliptic cohomology as special cases where computations can be carried out explicitly.
Schemes and homotopy after Fabian Morel
I will attempt to survey the works of Morel on his construction of the homotopy category of schemes and the relation to motivic homotopy with applications to to the Grothendieck-witt ring of quadratic forms and questions about when relations in classic homotopy are actually algebraic. This will have to be a sketch.
Nathaniel Stapleton (M.I.T.)
The E-theory of centralizers in symmetric groups
Using cohomology theories closely related to Morava E-theory, we will provide an algebro-geometric description of the cohomology of particular centralizers in symmetric groups in terms of the scheme classifying subgroups of a certain p-divisible group.
Older seminars
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CommonCrawl
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\begin{document}
\title[Ideals, supports and harmonic operators] {Ideals of the Fourier algebra, supports and harmonic operators}
\date{}
\author{M. Anoussis, A. Katavolos and I. G. Todorov}
\address{Department of Mathematics, University of the Aegean, Samos 83 200, Greece}
\email{[email protected]}
\address{Department of Mathematics, University of Athens, Athens 157 84, Greece}
\email{[email protected]}
\address{Pure Mathematics Research Centre, Queen's University Belfast, Belfast BT7 1NN, United Kingdom}
\email{[email protected]}
\keywords{Fourier algebra, masa-bimodule, invariant subspaces, harmonic operators}
\begin{abstract} We examine the common null spaces of families of Herz-Schur multipliers and apply our results to study jointly harmonic operators and their relation with jointly harmonic functionals. We show how an annihilation formula obtained in \cite{akt} can be used to give a short proof {as well as a generalisation} of a result of Neufang and Runde concerning harmonic operators with respect to a normalised positive definite function. We compare the two notions of support of an operator that have been studied in the literature and show how one can be expressed in terms of the other. \end{abstract}
\maketitle
\section{Introduction and Preliminaries}
In this paper we investigate, for a locally compact group $G$, the common null spaces of families of Herz-Schur multipliers (or completely bounded multipliers of the Fourier algebra $A(G)$) and their relation to ideals of $A(G)$.
This provides a new perspective for our previous results in \cite{akt} concerning (weak* closed) spaces of operators on $L^2(G)$ which are simultaneously invariant under all Schur multipliers and under {conjugation by the right regular representation} of $G$ on $L^2(G)$ ({\em jointly invariant} subspaces -- see below for precise definitions).
At the same time, it provides a new approach to, as well as an extension of, a result of Neufang and Runde \cite{neurun} concerning the space
$\widetilde{\cl H}_\sigma$ of operators which are `harmonic' with respect to a positive definite normalised function $\sigma:G\to\bb C$. The notion of $\sigma$-harmonic operators was introduced in \cite{neurun} as an extension of the notion of $\sigma$-harmonic functionals on $A(G)$ as defined and studied by Chu and Lau in \cite{chulau}. One of the main results of Neufang and Runde is that $\widetilde{\cl H}_\sigma$ is the von Neumann algebra on $L^2(G)$ generated by the algebra $\cl D$ of multiplication operators together with the space ${\cl H}_\sigma$ of harmonic functionals, considered as a subspace of the von Neumann algebra $\vn(G)$ of the group.
It will be seen that this result can be obtained as a consequence of the fact (see Corollary \ref{c_jho}) that, for any family $\Sigma$ of completely bounded multipliers of $A(G)$, the space $\widetilde{\cl H}_\Sigma$ of {\em jointly $\Sigma$-harmonic operators} can be obtained as the weak* closed $\cl D$-bimodule generated by the {\em jointly $\Sigma$-harmonic functionals} ${\cl H}_\Sigma$. In fact, the spaces $\widetilde{\cl H}_\Sigma$ belong to the class of jointly invariant subspaces of $\cl B(L^2(G))$ studied in \cite[Section 4]{akt}.
The space ${\cl H}_\Sigma$ is the annihilator in $\vn(G)$ of a certain ideal of $A(G)$. Now from any given closed ideal $J$ of the Fourier algebra $A(G)$, there are two `canonical' ways to arrive at a weak* closed $\cl D$-bimodule of $\cl B(L^2(G))$. One way is to consider its annihilator $J^\perp$ in $\vn(G)$ and then take the weak* closed $\cl D$-bimodule generated by $J^{\perp}$. We call this bimodule $\Bim(J^\perp)$. The other way is to take a suitable saturation $\Sat(J)$ of $J$ within the trace class operators on $L^2(G)$ (see Theorem \ref{th_satlcg}), and then form its annihilator. This gives a masa bimodule $(\Sat J)^{\perp}$ in $\cl B(L^2(G))$. In \cite{akt}, we proved that these
two procedures yield the same bimodule, that is, $\Bim(J^\perp) = (\Sat J)^{\perp}$. Our proof that $\widetilde{\cl H}_\Sigma=\Bim({\cl H}_\Sigma)$ rests on this equality.
The notion of {\em support}, $\suppG(T)$, of an element $T\in\vn(G)$ was introduced by Eymard in \cite{eymard} by considering $T$ as a linear functional on the function algebra $A(G)$; thus $\suppG(T)$ is a closed subset of $G$. This notion was extended by Neufang and Runde in \cite{neurun} to an arbitrary
$T\in\cl B(L^2(G))$ and used to describe harmonic operators. By considering joint supports, we show that this extended notion of $G$-support for an operator $T\in\cl B(L^2(G))$ coincides with the joint $G$-support of a family of elements of $\vn (G)$ naturally associated to $T$ (Proposition \ref{propsame2}).
On the other hand, the notion of support of an operator $T$ acting on $L^2(G)$ was first introduced by Arveson in \cite{arv} as a certain closed subset of $G \times G$. This notion was used in his study of what was later called operator synthesis. A different but related approach appears in \cite{eks}, where the notion of $\omega$-support, $\suppo(T)$, of $T$ was introduced and used to establish a bijective correspondence between reflexive masa-bimodules and $\omega$-closed subsets of $G\times G$.
We show that the joint $G$-support $\suppG(\cl A)$ of an arbitrary family
$\cl A\subseteq \cl B(L^2(G))$ can be fully described in terms of its joint $\omega$-support $\suppo(\cl A)$ (Theorem \ref{th_compsa}). The converse does not hold in general, as the $\omega$-support, being a subset of $G\times G$, contains in general more
information about an arbitrary operator than its $G$-support (see Remark \ref{last}); however, in case $\cl A$ is a (weak* closed) jointly invariant subspace, we show that its $\omega$-support can be recovered from its $G$-support (Theorem \ref{312}). We also show that, if a set $\Omega\subseteq G\times G$ is invariant under all maps $(s,t)\to (sr,tr), \, r\in G$, then $\Omega$ is marginally equivalent to an $\omega$-closed set if and only if it is marginally equivalent to a (topologically) closed set. This can fail for non-invariant sets (see for example \cite[p. 561]{eks}). {For a related result, see \cite[Proposition 7.3]{stt_clos}.}
\noindent\textbf{Preliminaries and Notation } Throughout, $G$ will denote a second countable locally compact group, equipped with left Haar measure. Denote by $\cl D\subseteq\cl{B}(L^2(G))$ the maximal abelian selfadjoint algebra (masa, for short) consisting of all multiplication operators $M_f:g\to fg$, where $f\in L^\infty(G)$. We write $\vn (G)$ for the von Neumann algebra $\{\lambda_s : s\in G\}''$ generated by the left regular representation $s\to \lambda_s$ of $G$ on $L^2(G)$ (here $(\lambda_sg)(t)=g(s\an t)$).
Every element of the predual of $\vn (G)$ is a vector functional, $\omega_{\xi,\eta}: T\to (T\xi,\eta)$, where $\xi,\eta\in L^2(G)$, and $\nor{\omega_{\xi,\eta}}$ is the infimum of the products
$\|\xi\|_2\|\eta\|_2$ over all such representations. This predual can be identified \cite{eymard} with the set $A(G)$ of all complex functions $u$ on $G$ of the form $s\to u(s)=\omega_{\xi,\eta}(\lambda_s)$. With the above norm and pointwise operations, $A(G)$ is a (commutative, regular, semi-simple) Banach algebra of continuous functions on $G$ vanishing at infinity, called the \emph{Fourier algebra} of $G$; its Gelfand spectrum can be identified with $G$ {\it via} point evaluations. The set $A_c(G)$ of compactly supported elements of $A(G)$ is dense in $A(G)$.
A function $\sigma:G\to\bb C$ is a {\em multiplier} of $A(G)$ if for all $u\in A(G)$ the pointwise product $\sigma u$ is again in $A(G)$. By duality, a multiplier $\sigma$ induces a bounded operator $T\to \sigma\cdot T$ on $\vn(G)$. We say $\sigma$ is {\em a completely bounded (or Herz-Schur) multiplier}, and write $\sigma\in M^{\cb}A(G)$, if the latter operator is completely bounded, that is, if there exists a constant $K$ such that $\nor{[\sigma\cdot T_{ij}]}\le K \nor{[T_{ij}]}$ for all $n\in\bb N$ and all $[T_{ij}]\in M_n(\vn (G))$ (the latter being the space of all $n$ by $n$ matrices with entries in $\vn (G)$). The least such constant is the \emph{cb norm} of $\sigma$. The space $M^{\cb}A(G)$ with pointwise operations and the cb norm is a Banach algebra into which $A(G)$ embeds contractively. For a subset $\Sigma\subseteq M^{\cb}A(G)$, we let $Z(\Sigma)=\{s\in G: \sigma(s)=0 \text{ for all } \sigma\in\Sigma\}$ be its \emph{zero set}.
A subset $\Omega\subseteq G\times G$ is called {\em marginally null} if there exists a null set (with respect to Haar measure) $X\subseteq G$ such that $\Omega\subseteq (X\times G)\cup(G\times X)$. Two sets $\Omega,\Omega'\subseteq G\times G$ are {\em marginally equivalent} if their symmetric difference is a marginally null set; we write $\Omega_1\cong \Omega_2$. A set $\Omega\subseteq G\times G$ is said to be {\em $\omega$-open} if it is marginally equivalent to a {\em countable} union of Borel rectangles $A\times B$; it is called {\em $\omega$-closed} when its complement is $\omega$-open.
Given any set $\Omega\subseteq G\times G$, we denote by $\frak M_{\max}(\Omega)$ the set of all $T\in\cl{B}(L^2(G))$ which are {\em supported} by $\Omega$ in the sense that $M_{\chi_ B}TM_{\chi_A}=0$ whenever $A\times B\subseteq G\times G$ is a Borel rectangle disjoint from $\Omega$ (we write $\chi_A$ for the characteristic function of a set $A$). Given any set $\cl U\subseteq \cl{B}(L^2(G))$ there exists a smallest, up to marginal equivalence, $\omega$-closed set $\Omega\subseteq G\times G$ supporting every element of $\cl U$, {\it i.e.} such that $\cl U\subseteq\frak M_{\max}(\Omega)$. This set is called {\em the $\omega$-support} of $\cl U$ and is denoted $\suppo(\cl U)$ \cite{eks}.
Two functions $h_1,h_2 : G\times G\to \bb{C}$ are said to be {\em marginally equivalent}, or equal {\em marginally almost everywhere (m.a.e.)}, if they differ on a marginally null set.
The predual of $\cl{B}(L^2(G))$ consists of all linear forms $\omega$ given by $\omega(T):= \sum\limits_{i=1}^{\infty} \sca{Tf_i, g_i}$ where $f_i, g_i\in L^2(G)$ and $\sum\limits_{i=1}^{\infty}\nor{f_i}_2\nor{g_i}_2<\infty$. Each such $\omega$ defines a trace class operator whose kernel is a function $h = h_\omega:G\times G\to\bb C$, unique up to marginal equivalence, given by $h(x,y)=\sum\limits_{i=1}^{\infty} f_i(x )\bar g_i(y)$. This series converges marginally almost everywhere on $G\times G$. We use the notation $\du{T}{h} :=\omega(T)$.
We write $T(G)$ for the Banach space of (marginal equivalence classes of) such functions, equipped with the norm of the predual of $\cl{B}(L^2(G))$.
Let $\frak{S}(G)$ be the multiplier algebra of $T(G)$; by definition, a measurable function $w : G\times G\rightarrow \bb{C}$ belongs to $\frak{S}(G)$ if the map $m_w: h\to wh$ leaves $T(G)$ invariant, that is, if $wh$ is marginally equivalent to a function from $T(G)$, for every $h\in T(G)$. Note that the operator $m_w$ is automatically bounded. The elements of $\frak{S}(G)$ are called \emph{(measurable) Schur multipliers}. By duality, every Schur multiplier induces a bounded operator $S_w$ on $\cl B(L^2(G))$, given by \[\du{S_w(T)}{h} = \du{T}{wh}, \ \ \ h\in T(G), \; T\in \cl B(L^2(G))\, .\] The operators of the form $S_w$, $w\in \frak{S}(G)$, are precisely the bounded weak* continuous $\cl D$-bimodule maps on $\cl B(L^2(G))$ (see \cite{haa}, \cite{sm}, \cite{pe} and \cite{kp}).
A weak* closed subspace $\cl U$ of $\cl B(L^2(G))$ is invariant under the maps $S_w$, $w\in \frak{S}(G)$, if and only if it is invariant under all left and right multiplications by elements of $\cl D$, {\it i.e.} if $M_fTM_g\in \cl U$ for all $f,g\in L^\infty(G)$ and all $T\in\cl U$, in other words, if it is a {\em $\cl D$-bimodule}. For any set $\cl T\subseteq \cl B(L^2(G))$ we denote by $\Bim\cl T$ the smallest weak* closed $\cl D$-bimodule containing $\cl T$; thus, $\mathrm{Bim}(\cl T)=\overline{[\mathfrak{S}(G)\cl T]}^{w^*}$.
We call a subspace $\cl U\subseteq \cl B(L^2(G))$ {\em invariant} if $\rho_rT\rho_r^*\in\cl A$ for all $T\in\cl A$ and all $r\in G$; here, $r\to \rho_r$ is the right regular representation of $G$ on $L^2(G)$. An invariant space, which is also a $\cl D$-bimodule, will be called a {\em jointly invariant space}.
It is not hard to see that, if $\cl A\subseteq \cl B(L^2(G))$, the smallest weak* closed jointly invariant space
containing $\cl A$ is the weak* closed linear span of $\{S_w(\rho_rT\rho_r^*): T\in\cl A, w\in \frak S(G), r\in G\}$.
For a complex function $u$ on $G$ we let $N(u):G\times G\to\bb C$ be the function given by $N(u)(s,t) = u(ts^{-1})$. For any subset $E$ of $G$, we write $E^*=\{(s,t)\in G\times G: ts^{-1}\in E\}$.
It is shown in \cite{bf} (see also \cite{j} and \cite{spronk}) that the map $u\rightarrow N(u)$ is an isometry from $M^{\cb}A(G)$ into $ \frak{S}(G)$ and that its range consists precisely of all {\em invariant} Schur multipliers, {\it i.e.} those $w\in \frak{S}(G)$ for which $w(sr,tr) = w(s,t)$ for every $r\in G$ and marginally almost all $s,t$. Note that the corresponding operators $S_{N(u)}$ are denoted $\hat\Theta(u)$ in \cite{neuruaspro}.
The following result from \cite{akt} is crucial for what follows.
\begin{theorem}\label{th_satlcg} Let $J\subseteq A(G)$ be a closed ideal and $\Sat(J)$ be the closed $L^\infty(G)$-bimodule of $T(G)$ generated by the set \[ \{N(u)\chi_{L\times L}: u \in J, L\ \text{compact, } \ L\subseteq G \}. \] Then $\Sat(J)^{\perp} = \Bim (J^{\perp})$. \end{theorem}
\section{Null spaces and harmonic operators}\label{s1}
Given a subset $\Sigma\subseteq M^{\cb}A(G)$, let
\[ \frak{N}(\Sigma) = \{T\in \vn(G) : \sigma\cdot T = 0, \ \mbox{ for all } \sigma\in \Sigma\} \] be the {\em common null set} of the operators on $\vn(G)$ of the form $T\to \sigma\cdot T$, with $\sigma\in \Sigma$. Letting \[\Sigma A \stackrel{def}{=} \overline{\sspp}(\Sigma A(G)) = \overline{\sspp\{ \sigma u : \sigma\in \Sigma, u\in A(G)\}},\] it is easy to verify that $\Sigma A$ is a closed ideal of $A(G)$ and that \begin{equation}\label{eq_prean} \frak{N}(\Sigma) = (\Sigma A)^\bot . \end{equation}
{\remark \label{remideal} The sets of the form $\Sigma A$ are precisely the closed ideals of $A(G)$ generated by their compactly supported elements.} \begin{proof} It is clear that, if $\Sigma\subseteq M^{\cb}A(G)$, the set $\{\sigma u: \sigma\in\Sigma, u\in A_c(G)\}$ consists of compactly supported elements and is dense in $\Sigma A$. Conversely, suppose that $J\subseteq A(G)$ is a closed ideal such that $J\cap A_c(G)$ is dense in $J$. For every $u\in J$ with compact support $K$, there exists $v\in A(G)$ which equals 1 on $K$ \cite[(3.2) Lemme]{eymard}, and so $u=uv\in JA$. Thus $J= \overline{J\cap A_c(G)}\subseteq JA\subseteq J$ and hence $J=JA$. \end{proof}
The following Proposition shows that it is sufficient to study sets of the form $\frak N(J)$ where $J$ is a closed ideal of $A(G)$.
\begin{proposition}\label{p_njan} For any subset $\Sigma$ of $M^{\cb}A(G)$, \[ \frak{N}(\Sigma)=\frak{N}(\Sigma A).\] \end{proposition} \proof If $\sigma\cdot T = 0$ for all $\sigma\in\Sigma$ then {\em a fortiori} $v\sigma\cdot T=0$, for all $v\in A(G)$ and all $\sigma\in \Sigma$. It follows that $w\cdot T=0$ for all $w\in \Sigma A$; thus $\frak{N}(\Sigma)\subseteq \frak{N}(\Sigma A)$.
Suppose conversely that $w\cdot T=0$ for all $w\in \Sigma A$ and fix $\sigma\in\Sigma$. Now $u\sigma\cdot T=0$ for all $u \in A(G)$, and so $ \du{\sigma\cdot T}{uv}=0$ when $u,v\in A(G)$. Since the products $uv$ form a dense subset of $A(G)$, we have $\sigma\cdot T=0$. Thus $\frak{N}(\Sigma)\supseteq \frak{N}(\Sigma A)$ since $\sigma\in\Sigma$ is arbitrary, and the proof is complete. \qed
It is not hard to see that $\lambda_s$ is in $\frak N(\Sigma)$ if and only if $s$ is in the zero set $Z(\Sigma)$ of $\Sigma$, and so $Z(\Sigma)$ coincides with the zero set of the ideal $J=\Sigma A$. Whether or not, for an ideal $J$, these unitaries suffice to generate $\frak N(J)$
depends on properties of the zero set.
For our purposes, a closed subset $E\subseteq G$ is a {\em set of synthesis} if there is a unique closed ideal $J$ of $A(G)$ with $Z(J)=E$. Note that this ideal is generated by its compactly supported elements \cite[Theorem 5.1.6]{kaniuth}.
\begin{lemma} \label{proto} Let $J\subseteq A(G)$ be a closed ideal. Suppose that its zero set $E=Z(J)$ is a set of synthesis. Then \[ \frak N(J)=J^\bot=\overline{\sspp\{\lambda_x:x\in E\}}^{w*} \] \end{lemma}
\proof Since $E$ is a set of synthesis, $J=JA$ by Remark \ref{remideal}; thus $J^\bot=(JA)^\bot=\frak N(J)$ by relation (\ref{eq_prean}). The other equality is essentially a reformulation of the fact that $E$ is a set of synthesis: a function $u\in A(G)$ is in $J$ if and only if it vanishes at every point of $E$, that is, if and only if it annihilates every $\lambda_s$ with $s\in E$ (since $\du{\lambda_s}{u}= u(s)$). \qed
A linear space $\cl U$ of bounded operators on a Hilbert space is called {\em a ternary ring of operators (TRO)}
if it satisfies $ST^*R\in\cl U$ whenever $S,T$ and $R$ are in $\cl U$. Note that a TRO containing the identity operator is automatically a selfadjoint algebra.
\begin{proposition}\label{deutero} Let $J\subseteq A(G)$ be a closed ideal. Suppose that its zero set $E=Z(J)$ is the coset of a closed subgroup of $G$. Then $\frak N(J)$ is a (weak-* closed) TRO. In particular, if $E$ is a closed subgroup then $\frak N(J)$ is a von Neumann subalgebra of $\vn (G)$. \end{proposition}
\proof We may write $E=Hg$ where $H$ is a closed subgroup and $g\in G$ (the proof for the case $E=gH$ is identical). Now $E$ is a translate of $H$ which is a set of synthesis by \cite{tatsuuma2} and hence $E$ is a set of synthesis. Thus Lemma \ref{proto} applies.
If $sg,tg,rg$ are in $E$ and $S=\lambda_{sg}, \, T=\lambda_{tg}$ and $R=\lambda_{rg}$, then $ST^*R=\lambda_{st\an rg}$ is also in $\frak N(J)$ because $st\an rg\in E$. Since $\frak N(J)$ is generated by $\{\lambda_x:x\in E\}$, it follows that $ST^*R\in\frak N(J)$ for any three elements $S,T,R$ of $\frak N(J)$. \qed
{\remark Special cases of the above result are proved by Chu and Lau in \cite{chulau} (see Propositions 3.2.10 and 3.3.9.)}
We now pass from $\vn(G)$ to $\cl B (L^2(G))$: The algebra $M^{\cb}A(G)$ acts on $\cl B(L^2(G))$ {\it via} the maps $S_{N(\sigma)},\, \sigma\in M^{\cb}A(G)$ (see \cite{bf} and \cite{j}), and this action is an extension of the action of $M^{\cb}A(G)$ on $\vn (G)$: when $T\in\vn (G)$ and $\sigma\in M^{\cb}A(G)$, we have $S_{N(\sigma)}(T)=\sigma\cdot T$. Hence, letting \[ \tilde{\frak N}(\Sigma) =
\{T\in\cl B(L^2(G)): S_{N(\sigma)}(T)=0 , \ \mbox{ for all } \sigma\in \Sigma\}, \] we have $\frak{N}(\Sigma)=\tilde{\frak N}(\Sigma)\cap \vn(G).$
The following is analogous to Proposition \ref{p_njan}; note, however, that the dualities are different.
\begin{proposition}\label{new} If $\Sigma\subseteq M^{\cb}A(G)$, \[ \tilde{\frak{N}}(\Sigma)=\tilde{\frak{N}}(\Sigma A).\] \end{proposition} \proof The inclusion $\tilde{\frak{N}}(\Sigma)\subseteq \tilde{\frak{N}}(\Sigma A)$ follows as in the proof of Proposition \ref{p_njan}. To prove that $\tilde{\frak N}(\Sigma A)\subseteq \tilde{\frak{N}}(\Sigma)$, let $T\in \tilde{\frak{N}}(\Sigma A)$; then $S_{N(v\sigma)}(T)=0$ for all $\sigma \in\Sigma$ and $v \in A(G)$. Thus, if $h\in T(G)$, \[\du{S_{N(\sigma)}(T)}{N(v) h} =\du{T}{N(\sigma v) h} = \du{S_{N(v\sigma )}(T)}{ h} = 0\, .\] Since the linear span of the set $\{N(v) h: v \in A(G), h \in T(G) \}$ is dense in $T(G)$, it follows that $S_{N(\sigma)}(T)=0$ and so $T \in \tilde{\frak{N}}(\Sigma)$. \qed
\begin{proposition}\label{prop2} For every closed ideal $J$ of $A(G),\quad \tilde{\frak N}(J)= \Bim(J^\bot) $. \end{proposition} \proof If $T\in\cl B(L^2(G)), h\in T(G)$ and $u\in A(G)$ then \[\langle S_{N(u)}(T),h\rangle = \langle T, N(u)h\rangle .\] By \cite[Proposition 3.1]{akt}, $\Sat(J)$ is the closed linear span of $\{N(u)h: u\in J , h\in T(G)\}$. We conclude that $T\in (\Sat(J))^\bot$ if and only if $S_{N(u)}(T) = 0$ for all $u\in J$, {\it i.e.} if and only if $T\in \tilde{\frak{N}}(J)$. By Theorem \ref{th_satlcg}, $(\Sat(J))^\bot=\Bim(J^\bot)$, and the proof is complete. $\qquad\Box$
\begin{theorem}\label{thbimn} For any subset $\Sigma$ of $M^{\cb}A(G)$, \[ \tilde{\frak N}(\Sigma)= \Bim(\frak{N}(\Sigma)).\] \end{theorem} \proof It follows from relation (\ref{eq_prean}) that $\Bim((\Sigma A)^\bot) = \Bim(\frak{N}(\Sigma))$. But $\Bim((\Sigma A)^\bot)=\tilde{\frak N}(\Sigma A)$ from Proposition \ref{prop2} and $\tilde{\frak N}(\Sigma A)=\tilde{\frak N}(\Sigma)$ from Proposition \ref{new}.
\qed
More can be said when the zero set $Z(\Sigma)$ is a subgroup (or a coset) of $G$.
\begin{lemma}\label{trito} Let $J\subseteq A(G)$ be a closed ideal. Suppose that its zero set $E=Z(J)$ is a set of synthesis. Then \begin{equation}\label{eq} \tilde{\frak N}(J) =\overline{\sspp\{M_g\lambda_x:x\in E,g\in L^\infty(G)\}}^{w*} \end{equation} \end{lemma} \proof By Theorem \ref{thbimn}, $\tilde{\frak N}(J) = \Bim(\frak N(J))$ and thus, by Lemma \ref{proto}, $\tilde{\frak N}(J)$ is the weak* closed linear span of the monomials of the form $M_f\lambda_sM_g$ where $f,g\in L^\infty(G)$ and $s\in E$. But, because of the commutation relation $\lambda_sM_g=M_{g_s}\lambda_s \ \ (\mbox{where } g_s(t)=g(s\an t))$, we may write $M_f\lambda_sM_g=M_\phi\lambda_s$ where $\phi=fg_s\in L^\infty(G)$.\qed
\begin{theorem}\label{tetarto} Let $J\subseteq A(G)$ be a closed ideal. Suppose that its zero set $E=Z(J)$ is the coset of a closed subgroup of $G$. Then $\tilde{\frak N}(J)$ is a (weak* closed) TRO. In particular if $E$ is a closed subgroup then $\tilde{\frak N}(J)$ is a von Neumann subalgebra of $\cl B(L^2(G))$ and \[ \tilde{\frak N}(J)=(\cl D\cup\frak N(J))''=(\cl D\cup\{\lambda_x:x\in E\})''. \] \end{theorem}
\proof As in the proof of Proposition \ref{deutero}, we may take $E=Hg$. By Lemma \ref{trito}, it suffices to check the TRO relation for monomials of the form $M_f\lambda_{sg}$; but, by the commutation relation, triple products $(M_f\lambda_{sg})(M_g\lambda_{tg})^*(M_h\lambda_{rg})$ of such monomials may be written in the form
$M_\phi\lambda_{st\an rg}$ and so belong to $\tilde{\frak N}(J)$ when $sg,tg$ and $rg$ are in the coset $E$. Finally, when $E$ is a closed subgroup, the last equalities follow from relation (\ref{eq}) and the bicommutant theorem. \qed
We next extend the notions of $\sigma$-harmonic functionals \cite{chulau} and operators \cite{neurun} to jointly harmonic functionals and operators:
\begin{definition}\label{d_jh} Let $\Sigma\subseteq M^{\cb}A(G)$. An element $T\in \cl \vn(G)$ will be called a \emph{$\Sigma$-harmonic functional } if
$\sigma\cdot T=T$ for all $\sigma\in\Sigma$. We write $\cl H_{\Sigma}$ for the set of all {$\Sigma$-harmonic} functionals.
An operator $T\in \cl B(L^2(G))$ will be called \emph{$\Sigma$-harmonic} if $S_{N(\sigma)}(T)=T$ for all $\sigma\in\Sigma$. We write $\widetilde{\cl H}_{\Sigma}$ for the set of all {$\Sigma$-harmonic} operators. \end{definition}
Explicitly, if $\Sigma'=\{\sigma -\mathbf 1:\sigma\in\Sigma\}$, \begin{align*} \cl H_{\Sigma} &= \{T\in \vn(G) : \sigma\cdot T=T \;\text{for all }\; \sigma\in\Sigma\} = \frak N(\Sigma') \\ \text{and }\quad \widetilde{\cl H}_{\Sigma} \ &= \{T\in \cl B(L^2(G)) : S_{N(\sigma)}(T)=T \;\text{for all }\; \sigma\in\Sigma\} = \tilde{\frak N}(\Sigma'). \end{align*} The following is an immediate consequence of Theorem \ref{thbimn}.
\begin{corollary}\label{c_jho} Let $\Sigma\subseteq M^{\cb}A(G)$. Then the weak* closed $\cl D$-bimodule $\Bim(\cl H_{\Sigma})$ generated by $\cl H_{\Sigma}$ coincides with $\widetilde{\cl H}_{\Sigma}$. \end{corollary}
Let $\sigma$ be a positive definite normalised function and $\Sigma = \{\sigma\}$. In \cite[Theorem 4.8]{neurun}, the authors prove, under some restrictions on $G$ or $\sigma$ (removed in \cite{kalantar}), that $\widetilde{\cl H}_{\Sigma}$ coincides with the von Neumann algebra $(\cl D\cup\cl H_{\Sigma})''$. We give a short proof of a more general result.
Denote by $P^1(G)$ the set of all positive definite normalised functions on $G$. Note that $P^1(G)\subseteq M^{\cb}A(G)$.
\begin{theorem} Let $\Sigma\subseteq P^1(G)$. The space $\widetilde{\cl H}_{\Sigma}$ is a von Neumann subalgebra of $\cl B(L^2(G))$,
and $\widetilde{\cl H}_{\Sigma}=(\cl D \cup\cl H_{\Sigma})''$. \end{theorem} \proof Note that $\cl H_{\Sigma}=\frak N(\Sigma')=\frak N(\Sigma' A)$ and $\widetilde{\cl H}_{\Sigma} = \tilde{\frak N}(\Sigma')= \tilde{\frak N}(\Sigma' A)$. Since $Z(\Sigma')$ is a closed subgroup \cite[Proposition 32.6]{hr2}, it is a set of spectral synthesis \cite{tatsuuma2}. Thus the result follows from Theorem \ref{tetarto}. \qed
{\remark It is worth pointing out that $\widetilde{\cl H}_{\Sigma}$ has an abelian commutant, since it contains a masa. In particular, it is a type I, and hence an injective, von Neumann algebra.}
In \cite[Theorem 4.3]{akt} it was shown that a weak* closed subspace $\cl U\subseteq \cl B(L^2(G))$ is jointly invariant if and only if it is of the form $\cl U = \Bim(J^{\perp})$ for a closed ideal $J\subseteq A(G)$. By Proposition \ref{prop2}, $\Bim(J^{\perp})=\tilde{\frak{N}}(J)$, giving another equivalent description. In fact, the ideal $J$ may be replaced by a subset of $M^{\cb}A(G)$:
\begin{proposition}\label{th_eqc} Let $\cl U\subseteq \cl B(L^2(G))$ be a weak* closed subspace. The following are equivalent:
(i) \ \ $\cl U$ is jointly invariant;
(ii) \ there exists a closed ideal $J\subseteq A(G)$ such that $\cl U = \tilde{\frak{N}}(J)$;
(iii) \ there exists a subset $\Sigma\subseteq M^{\cb}A(G)$ such that $\cl U = \tilde{\frak{N}}(\Sigma)$.
\end{proposition} \begin{proof}
We observed the implication (i)$\Rightarrow$(ii) above, and (ii)$\Rightarrow$(iii) is trivial.
Finally, (iii)$\Rightarrow$(i) follows from Theorem \ref{thbimn} and \cite[Theorem 4.3]{akt}. \end{proof}
{\remark It might also be observed that every weak* closed jointly invariant subspace $\cl U$ is of the form $\cl U = \widetilde{\cl H}_{\Sigma}$ for some $\Sigma\subseteq M^{\cb}A(G)$.}
We end this section with a discussion on the ideals of the form $\Sigma A$: If $J$ is a closed ideal of $A(G)$, then $J A\subseteq J$; thus, by (\ref{eq_prean}) and Proposition \ref{p_njan}, $J^\bot\subseteq \frak N(J)$ and therefore $\Bim (J^\bot)\subseteq\tilde{\frak N}(J)$, since $\tilde{\frak N}(J)$ is a $\cl D$-bimodule and contains ${\frak N}(J)$. The equality $J^\bot= \frak N(J)$ holds if and only if $J$ is generated by its compactly supported elements, equivalently if $J=JA$ (see Remark \ref{remideal}). Indeed, by Proposition \ref{p_njan} we have $\frak N(J)=\frak N(JA)= (JA)^\bot$ and so the equality $J^\bot= \frak N(J)$ is equivalent to $J^\bot= (JA)^\bot$. Interestingly, the inclusion $\Bim (J^\bot)\subseteq\tilde{\frak N}(J)$ is in fact always an equality (Proposition \ref{prop2}).
We do not know whether all closed ideals of $A(G)$ are of the form $\Sigma A$. They certainly are when $A(G)$ satisfies {\em Ditkin's condition at infinity} \cite[Remark 5.1.8 (2)]{kaniuth}, namely if every $u\in A(G)$ is the limit of a sequence $(uv_n)$, with $v_n\in A_c(G)$. Since $A_c(G)$ is dense in $A(G)$, this is equivalent to the condition that every $u\in A(G)$ belongs to the closed ideal $\overline{uA(G)}$.
This condition has been used before (see for example \cite{kl}). It certainly holds whenever $A(G)$ has a weak form of approximate identity; for instance, when $G$ has the approximation property (AP) of Haagerup and Kraus \cite{hk} and a fortiori when $G$ is amenable. It also holds for all discrete groups. See also the discussion in Remark 4.2 of \cite{lt} and the one following Corollary 4.7 of \cite{akt}.
\section{Annihilators and Supports}\label{s}
In this section, given a set $\cl A$ of operators on $L^2(G)$, we study the ideal of all $u\in A(G)$ which act trivially on $\cl A$; its zero set is the $G$-support of $\cl A$; we relate this to the $\omega$-support of $\cl A$ defined in \cite{eks}.
In \cite{eymard}, Eymard introduced, for $T\in\vn (G)$, the ideal $I_T$ of all $u\in A(G)$ satisfying $u\cdot T=0$. We generalise this by defining, for a subset $\cl A$ of $\cl B(L^2(G))$, \[ I_\cl{A} =\{u \in A(G): S_{N(u)}(\cl A)=\{0\}. \} \] It is easy to verify that $I_\cl{A}$ is a closed ideal of $A(G)$.
Let $\cl U(\cl A)$ be the smallest weak* closed jointly invariant subspace containing $\cl A$. We next prove that $\cl U(\cl A)$ coincides with the set $\tilde{\frak N}(I_\cl{A})$ of all $T\in\cl B(L^2(G))$ satisfying $S_{N(u)}(T)=0$ for all $u \in I_\cl{A}$.
\begin{proposition} \label{13} Let $\cl A\subseteq\cl B(L^2(G))$. If $\sigma\in M^{\cb}A(G)$ then $S_{N(\sigma)}(\cl A)=\{0\}$ if and only if $S_{N(\sigma)}(\cl U(\cl A))=\{0\}$. Thus, $I_\cl{A}=I_\cl{U(A)}$. \end{proposition} \proof Recall that $$\cl U(\cl A) = \overline{\sspp\{S_w(\rho_r T \rho_r^*) : T\in \cl A, w\in \frak{S}(G), r\in G\}}^{w^*}.$$ The statement now follows immediately from the facts that $S_{N(\sigma)}\circ S_w= S_w\circ S_{N(\sigma)}$ for all $w\in \frak S(G)$ and $S_{N(\sigma)}\circ {\rm Ad}_{\rho_r}= {\rm Ad}_{\rho_r}\circ S_{N(\sigma)}$ for all $r\in G$. The first commutation relation is obvious, and the second one can be seen as follows: Denoting by $\theta_r$ the predual of the map ${\rm Ad}_{\rho_r}$, for all $h\in T(G)$ we have $\theta_r(N(\sigma)h) = N(\sigma)\theta_r(h)$ since $N(\sigma)$ is right invariant and so \begin{align*} \du{S_{N(\sigma)}(\rho_rT\rho_r^*)}{h} &= \du{\rho_rT\rho_r^*}{N(\sigma)h}
= \du{T}{\theta_r(N(\sigma)h)} \\ & = \du{T}{N(\sigma)\theta_r(h)} = \du{S_{N(\sigma)}(T)}{\theta_r(h)} \\ &= \du{\rho_r(S_{N(\sigma)}(T))\rho_r^*}{h}. \end{align*} Thus $S_{N(\sigma)}(\rho_rT\rho_r^*)=\rho_r(S_{N(\sigma)}(T))\rho_r^*$. \qed
\begin{theorem} \label{prop16} Let $\cl A\subseteq\cl B(L^2(G))$. The bimodule $\tilde{\frak N}(I_\cl{A})$ coincides with the smallest weak* closed jointly invariant subspace $\cl U(\cl A)$ of $\cl B(L^2(G))$ containing $\cl A$. \end{theorem} \proof Since $\cl{U(A)}$ is weak* closed and jointly invariant, by \cite[Theorem 4.3]{akt} it equals $\Bim(J^\bot)$, where $J$ is the closed ideal of $A(G)$ given by \[J=\{u\in A(G): N(u)\chi_{L\times L} \in (\cl{U(A)})_\bot \;\text{for all compact $L\subseteq G$}\}.\]
We show that $J\subseteq I_\cl{A}$. Suppose $u\in J$; then, for all $w\in\frak S(G)$ and all $T\in\cl A$, since $S_w(T)$ is in $\cl{U(A)}$, by Theorem \ref{th_satlcg} it annihilates $ N(u)\chi_{L\times L}$ for every compact $L\subseteq G$. It follows that $$\du{S_{N(u)}(T)}{w\chi_{L\times L}} = \du{T}{N(u)w\chi_{L\times L}} =\du{S_w(T)}{N(u)\chi_{L\times L}} = 0$$ for all $w\in\frak S(G)$ and all compact $L\subseteq G$. Taking $w=f\otimes\bar g$ with $f,g\in L^\infty(G)$ supported in $L$, this yields \[ \sca{S_{N(u)}(T)f,g} = \du{S_{N(u)}(T)}{w\chi_{L\times L}} = 0 \] for all compactly supported $f,g\in L^\infty(G)$ and therefore $S_{N(u)}(T)=0$. Since this holds for all $T\in\cl A$, we have shown that $u\in I_\cl{A}$.
It follows that $\cl{U(A)}=\Bim(J^\perp)\supseteq \Bim(I_\cl{A}^\perp)$. But $\Bim(I_\cl{A}^\perp)=\tilde{\frak N}(I_\cl{A})$ by Proposition \ref{prop2}, and this space is clearly jointly invariant and weak* closed. Since it contains $\cl A$, it also contains $\cl{U(A)}$ and so \[\cl{U(A)}=\Bim(J^\perp)= \Bim(I_\cl{A}^\perp)=\tilde{\frak N}(I_\cl{A}). \qquad\Box\]
\noindent\textbf{Supports of functionals and operators} In \cite{neurun}, the authors generalise the notion of support of an element of
$\vn(G)$ introduced by Eymard \cite{eymard} by defining, for an arbitrary $T\in\cl B(L^2(G))$, \[ \suppG T := \{x\in G : u(x) = 0 \;\text{for all $u\in A(G)$ with }\; S_{N(u)}(T) = 0\}.\] Notice that $\suppG T$ coincides with the zero set of the ideal $I_T$ (see also \cite[Proposition 3.3]{neurun}). More generally, let us define the {\em $G$-support} of a subset $\cl A$ of $\cl B(L^2(G))$ by \[\suppG(\cl A) = Z(I_\cl{A}).\] When $\cl A\subseteq \vn(G)$, then $\suppG(\cl A)$ is just the support of $\cl A$ considered as a set of functionals on $A(G)$ as in \cite{eymard}.
The following is proved in \cite{neurun} under the assumption that $G$ has the approximation property of Haagerup and Kraus \cite{hk}:
\begin{proposition} Let $T\in\cl B(L^2(G))$. Then $\suppG(T)=\emptyset$ if and only if $T=0$. \end{proposition} \proof It is clear that the empty set is the $G$-support of the zero operator.
Conversely, suppose $\suppG(T)=\emptyset$, that is, $Z(I_T)=\emptyset$. This implies that $I_T=A(G)$ (see \cite[Corollary 3.38]{eymard}). Hence $S_{N(u)}(T)=0$ for all $u\in A(G)$, and so for all $h\in T(G)$ we have \[ \du{T}{N(u)h}= \du{S_{N(u)}(T)}{h}=0. \] Since the linear span of $\{N(u)h:u\in A(G), h\in T(G)\}$ is dense in $T(G)$, it follows that $T=0.$ \qed
\begin{proposition}\label{propsame} The $G$-support of a subset $\cl A\subseteq\cl B(L^2(G))$ is the same as
the $G$-support of the smallest weak* closed jointly invariant subspace $\cl{U(A})$ containing $\cl A$. \end{proposition} \proof Since $I_\cl{A}=I_{\cl{U(A})}$ (Proposition \ref{13}), this is immediate. \qed
The following proposition shows that the $G$-support of a subset $\cl A\subseteq\cl B(L^2(G))$ is in fact the support of a space of linear functionals on $A(G)$ (as used by Eymard): it can be obtained either by first forming the ideal $I_\cl{A}$ of all $u\in A(G)$ `annihilating' $\cl A$ (in the sense that $S_{N(u)}(\cl A)=\{0\}$) and then taking the support of the annihilator of $I_\cl{A}$ in $\vn(G)$; alternatively, it can be obtained by forming the smallest weak* closed jointly invariant subspace
$\cl{U(A})$ containing $\cl A$ and then considering the support of the set of all the functionals on $A(G)$ which are contained in $\cl{U(A})$.
\begin{proposition}\label{propsame2} The $G$-support of a subset $\cl A\subseteq\cl B(L^2(G))$ coincides with the supports of the following spaces of functionals on $A(G)$:
(i) \ the space $I_\cl{A}^\bot\subseteq\vn(G)$
(ii) the space $\cl{U(A})\cap\vn(G)=\frak N(I_\cl{A})$. \end{proposition} \proof By Proposition \ref{prop2} and Theorem \ref{prop16}, \[\cl{U(A})= \tilde{\frak N}(I_\cl{A})=\Bim( I_\cl{A}^\bot).\] Since the $\cl D$-bimodule $\Bim( I_\cl{A}^\bot)$ is jointly invariant, it coincides with $\cl U(I_\cl{A}^\bot)$. Thus $\cl{U(A})=\cl U(I_\cl{A}^\bot)$ and so Proposition \ref{propsame} gives $\suppG(\cl A)=\suppG( I_\cl{A}^\bot)$, proving part (i).
Note that $\cl U(\frak N(I_\cl{A}))=\Bim(\frak N(I_\cl{A}))=\tilde{\frak N}(I_\cl{A})$ and so $\cl U(\frak N(I_\cl{A}))=\cl{U(A})$. Thus by Proposition \ref{propsame}, $\frak N(I_\cl{A})$ and $\cl A$ have the same support. Since $\cl{U(A})\cap\vn(G)=\tilde{\frak N}(I_\cl{A})\cap\vn(G)=\frak N(I_\cl{A})$, part (ii) follows. \qed
We are now in a position to relate the $G$-support of a set of operators to their $\omega$-support as introduced in \cite{eks}.
\begin{theorem}\label{312} Let $\cl U\subseteq\cl B(L^2(G))$ be a weak* closed jointly invariant subspace. Then \begin{align*} \suppo(\cl U) &\cong (\suppG(\cl U))^*. \end{align*} In particular, the $\omega$-support of a jointly invariant subspace is marginally equivalent to a topologically closed set. \end{theorem} \proof Let $J = I_\cl{U}$. By definition, $\suppG(\cl U) = Z(J)$. By the proof of Theorem \ref{prop16}, $\cl U = \Bim(J^\bot)$, and hence, by Theorem \ref{th_satlcg}, $\cl U = (\Sat J)^\bot$. By \cite[Section 5]{akt}, $\suppo(\cl U)=\nul(\Sat J)=(Z(J))^*$, where $\nul (\Sat J)$ is the largest, up to marginal equivalence, $\omega$-closed subset $F$ of $G\times G$
such that $h|_F = 0$ for all $h\in\Sat J$ (see \cite{st1}). The proof is complete. \qed
\begin{corollary}\label{c_nss} Let $\Sigma\subseteq M^{\cb}A(G)$. Then \[ \suppo \tilde{\frak{N}}(\Sigma) \cong Z(\Sigma)^*. \] If $Z(\Sigma)$ satisfies spectral synthesis, then $\tilde{\frak{N}}(\Sigma) = \frak{M}_{\max}(Z(\Sigma)^*)$. \end{corollary} \begin{proof} From Theorem \ref{thbimn}, we know that $\tilde{\frak{N}}(\Sigma)=\Bim((\Sigma A)^\bot)=\tilde{\frak{N}}(\Sigma A)$ and so $\suppo \tilde{\frak{N}}(\Sigma) \cong Z(\Sigma A)^*$ by \cite[Section 5]{akt}. But $Z(\Sigma A)=Z(\Sigma)$ as can easily be verified (if $\sigma(t)\ne 0$ there exists $u\in A(G)$ so that $(\sigma u)(t)\ne 0$; the converse is trivial).
The last claim follows from the fact that, when $Z(\Sigma)$ satisfies spectral synthesis, there is a unique weak* closed $\cl D$ bimodule whose $\omega$-support is $Z(\Sigma)^*$ (see \cite[Theorem 4.11]{lt} or the proof of \cite[Theorem 5.5]{akt}).
\end{proof}
Note that when $\Sigma\subseteq P^1(G)$, the set $Z(\Sigma)$ satisfies spectral synthesis.
The following corollary is a direct consequence of Corollary \ref{c_nss}.
\begin{corollary}\label{corsyn} Let $\Sigma\subseteq M^{\cb}A(G)$ and $\Sigma'=\{\mathbf 1-\sigma:\sigma\in\Sigma\}$.
If $Z(\Sigma') $ is a set of spectral synthesis, then $\widetilde{\cl H}_{\Sigma} = \frak{M}_{\max}(Z(\Sigma')^*)$. \end{corollary}
\begin{corollary} Let $\Omega$ be a subset of $G\times G$ which is invariant under all maps $(s,t)\to (sr,tr), \, r\in G$.
Then $\Omega$ is marginally equivalent to an $\omega$-closed set if and only if it is marginally equivalent to a topologically closed set. \end{corollary} \proof A topologically closed set is of course $\omega$-closed. For the converse, let $\cl U=\frak{M}_{\max}(\Omega)$, so that $\Omega\cong\suppo(\cl U)$. Note that $\cl U$ is a weak* closed jointly invariant space. Indeed, since $\Omega$ is invariant, for every $T\in\cl U$ the operator $T_r=: \rho_rT\rho_r^*$ is supported in $\Omega$ and hence is in $\cl U$. Of course $\cl U$ is invariant under all Schur multipliers. By Theorem \ref{312}, $\suppo(\cl U)$ is marginally equivalent to a closed set. \qed
\begin{theorem}\label{th_compsa} Let $\cl A\subseteq \cl B(L^2(G))$. Then $\suppG(\cl A)$ is the smallest closed subset $E\subseteq G$ such that $E^*$ marginally contains $\suppo(\cl A)$. \end{theorem}
\proof {Let $\cl U=\cl{U(A})$ be the smallest jointly invariant weak* closed subspace containing $\cl A$. Let $Z=Z(I_\cl{A})$; by definition, $Z=\suppG\cl A$. But $\suppG\cl A=\suppG\cl U=Z$ (Proposition \ref{propsame}) and so $\suppo\cl{U}\cong Z^*$ by Theorem \ref{312}.
Thus $Z^*$ does marginally contain $\suppo(\cl A)$.
On the other hand, let $E\subseteq G$ be a closed set such that $E^*$ marginally contains $\suppo(\cl A)$. Thus any operator $T\in\cl A$ is supported in
$E^*$. But since $E^*$ is invariant, $\rho_rT\rho_r^*$ is also supported in $E^*,$ for every $r\in G$. Thus $\cl U$ is supported in $E^*$.
This means that $Z^*$ is marginally contained in $E^*$; that is, there is a null set $N\subseteq G$ such that $Z^*\setminus E^*\subseteq (N\times G)\cup (G\times N)$. We claim that $Z\subseteq E$. To see this, assume, by way of contradiction, that there exists $s\in Z\setminus E$. Then the `diagonal' $\{(r,sr):r\in G\}$ is a subset of $Z^*\setminus E^*\subseteq (N\times G)\cup (G\times N)$. It follows that for every $r\in G$, either $r\in N$ or $sr\in N$, which means that $r\in s\an N$. Hence $G\subseteq N\cup s\an N$, which is a null set. This contradiction shows that $Z\subseteq E$.
\qed
We note that for subsets $\cl S$ of $\vn(G)$ the relation $\suppo (\cl{S})\subseteq (\suppG(\cl S))^*$ is in \cite[Lemma 4.1]{lt}.
In \cite{neurun} the authors define, for a closed subset $Z$ of $G$, the set \[ \cl B_Z(L^2(G)) = \{T\in\cl B(L^2(G): \suppG(T)\subseteq Z\}. \]
\begin{corollary}\label{rem38} If $Z\subseteq G$ is closed, the set $\cl B_Z(L^2(G))$ consists of all $T\in\cl B(L^2(G))$
which are $\omega$-supported in $Z^*$; that is, $\cl B_Z(L^2(G))=\frak{M}_{\max}(Z^*)$. In particular, this space is a reflexive jointly invariant subspace. \end{corollary}
\proof If $T$ is $\omega$-supported in $Z^*$, then by Theorem \ref{th_compsa}, $\suppG(T)\subseteq Z$.
Conversely if $\suppG(T)\subseteq Z$ then $\suppG(\cl U(T))\subseteq Z$ by Proposition \ref{propsame}. But, by Theorem \ref{312}, $\suppo(\cl U(T)) \cong (\suppG(\cl U(T)))^*\subseteq Z^*$
and so $T$ is $\omega$-supported in $Z^*$. \qed
\begin{remark} \label{last} The $\omega$-support $\suppo(\cl A)$ of a set $\cl A$ of operators is more \lq sensitive' than $\suppG(\cl A)$ in that it encodes more information about $\cl A$. Indeed, $\suppG(\cl A)$ only depends on the (weak* closed) jointly invariant subspace generated by $\cl A$, while $\suppo(\cl A)$ depends on the (weak* closed) masa-bimodule
generated by $\cl A$. \end{remark}
{\example Let $G=\bb Z$ and $\cl A=\frak{M}_{\max}\{ (i,j):i+j \in\{0,1\}\}$. The $\omega$-support of $\cl A$ is of course the two-line set $\{ (i,j):i+j \in\{0,1\}\}$, while its $G$-support is $\bb Z$ which gives no information about $\cl A$.}
Indeed, if $E\subseteq\bb Z$ contains $\suppG(\cl A)$, then by Theorem \ref{th_compsa} $E^*=\{(n,m)\in\bb Z\times\bb Z:m-n\in E\}$ must contain $\{ (i,j):i+j \in\{0,1\}\}$. Thus for all $n\in\bb Z$, since $(-n,n)$ and $(-n,n+1)$ are in $\suppo(\cl A)$ we have $n-(-n)\in E$ and $n+1-(-n)\in E$; hence $\bb Z\subseteq E$.
\def$'$} \def\cprime{$'${$'$} \def$'$} \def\cprime{$'${$'$}
\end{document}
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arXiv
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Jyoti Dhar
Volume 39 Issue 5 November 1992 pp 541-545 Research Articles
Majorana neutrino transition magnetic moment in a variant of Zee model with horizontal symmetry
Jyoti Dhar S Dev
A SU(2)H symmetric variant of Zee model of lepton flavor violation is presented and is shown to lead to neutrino transition magnetic moment of the order required to explain the solar neutrino deficit and the possible anticorrelation of solar neutrino flux with sunspot activity via VVO mechanism. The use of horizontal symmetry leads to totally degenerate neutrino states which may be combined to form a ZKM Dirac neutrino with naturally small mass.
Volume 44 Issue 4 April 1995 pp 347-356
Two-loop Majorana neutrino mass and magnetic moment in a gauge model
Jyoti Dhar Umesh Pandey S Dev
Two-loop contributions to Majorana mass and transition magnetic moment in a gauge model not in conflict with decaying neutrino dark matter (DDM) hypothesis have been studied. Another variant of an earlier model [J Dhar and S Dev,Pramana — J. Phys.39 541 (1992)] consistent with the DDM hypothesis is proposed and is shown to lead to large enough neutrino magnetic moment and consistent with the phenomenological constraints on neutrino mass.
Volume 61 Issue 1 July 2003 pp 67-83
Resonant spin-flavor precession constraints on the neutrino parameters and the twisting structure of the solar magnetic fields from the solar neutrino data
S Dev Jyoti Dhar Sharma U C Pandey S P Sud B C Chauhan
Resonant spin-flavor precession (RSFP) scenario with twisting solar magnetic fields has been confronted with the solar neutrino data from various ongoing experiments. The anticorrelation apparent in the Homestake solar neutrino data has been taken seriously to constrain (Δm2,φ′) parameter space and the twisting profiles of the magnetic field in the convective zone of the Sun. The twisting profiles, thus derived, have been used to calculate the variation of the neutrino detection rates with the solar magnetic activity for the Homestake, Super-Kamiokande and the gallium experiments. It is found that the presence of twisting reduces the degree of anticorrelation in all the solar neutrino experiments. However, the anticorrelation in the Homestake experiment is expected to be more pronounced in this scenario. Moreover, the anticorrelation of the solar neutrino flux emerging from the southern solar hemisphere is expected to be stronger than that for the neutrinos emerging from the northern solar hemispheres.
Volume 82 Issue 6 June 2014 pp 1103-1117 Research Articles
First-principle study of nanostructures of functionalized graphene
Naveen Kumar Jyoti Dhar Sharma P K Ahluwalia
We present first-principle calculations of 2D nanostructures of graphene functionalized with hydrogen and fluorine, respectively, in chair conformation. The partial density of states, band structure, binding energy and transverse displacement of C atoms due to functionalization (buckling) have been calculated within the framework of density functional theory as implemented in the SIESTA code. The variation in band gap and binding energy per add atom have been plotted against the number of add atoms, as the number of add atoms are incremented one by one. In all, 37 nanostructures with 18C atoms, $3 \times 3 \times 1$ (i.e., the unit cell is repeated three times along 𝑥-axis and three times along 𝑦-axis) supercell, have been studied. The variation in C–C, C–H and C–F bond lengths and transverse displacement of C atoms (due to increase in add atoms) have been tabulated. A large amount of buckling is observed in the carbon lattice, 0.0053–0.7487 Å, due to hydrogenation and 0.0002–0.5379 Å, due to fluorination. As the number of add atoms (hydrogen or fluorine) is increased, a variation in the band gap is observed around the Fermi energy, resulting in change in behaviour of nanostructure from conductor to semiconductor/insulator. The binding energy per add atom increases with the increase in the number of add atoms. The nanostructures with 18C+18H and 18C+18F have maximum band gap of 4.98 eV and 3.64 eV, respectively, and binding energy per add atom –3.7562 eV and –3.3507 eV, respectively. Thus, these nanostructures are stable and are wide band-gap semiconductors, whereas the nanostructures with 18C+2H, 18C+4H, 18C+4F, 18C+8F, 18C+10F and 18C+10H atoms are small band-gap semiconductors with the band gap lying between 0.14 eV and 1.72 eV. Fluorine being more electronegative than hydrogen, the impact of electronegativity on band gap, binding energy and bond length is visible. It is also clear that it is possible to tune the electronic properties of functionalized graphene, which makes it a suitable material in microelectronics.
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CommonCrawl
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Let $a_n = 4n^3 + 6n^2 + 4n + 1.$ Find
\[a_8 + a_9 + a_{10} + \dots + a_{23}.\]
We see that $a_n = 4n^3 + 6n^2 + 4n + 1 = (n^4 + 4n^3 + 6n^2 + 4n + 1) - n^4 = (n + 1)^4 - n^4,$ so
\[a_8 + a_9 + a_{10} + \dots + a_{23} = (9^4 - 8^4) + (10^4 - 9^4) + (11^4 - 10^4) + \dots + (24^4 - 23^4) = 24^4 - 8^4 = \boxed{327680}.\]
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Math Dataset
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\begin{document}
\begin{frontmatter}
\title{Bernoulli and tail-dependence compatibility} \runtitle{Bernoulli and tail-dependence compatibility}
\begin{aug}
\author[A]{\fnms{Paul}~\snm{Embrechts}\thanksref{m1,T1}\ead[label=e1]{[email protected]}}, \author[B]{\fnms{Marius}~\snm{Hofert}\corref{}\thanksref{m2,T2}\ead[label=e2]{[email protected]}} \and \author[B]{\fnms{Ruodu}~\snm{Wang}\thanksref{m2,T2,T3}\ead[label=e3]{[email protected]}}
\thankstext{T1}{Supported by the Swiss Finance Institute.} \thankstext{T2}{Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC Discovery Grant numbers 5010 and 435844, resp.).} \thankstext{T3}{Supported by the Forschungsinstitut f\"ur Mathematik (FIM) at ETH Zurich.}
\runauthor{P. Embrechts, M. Hofert and R. Wang} \affiliation{ETH Zurich\thanksmark{m1} and University of Waterloo\thanksmark{m2}}
\address[A]{P. Embrechts\\ RiskLab\\ Department of Mathematics\\ \quad and Swiss Finance Institute\\ ETH Zurich\\ 8092 Zurich\\ Switzerland\\ \printead{e1}}
\address[B]{M. Hofert\\ R. Wang\\ Department of Statistics\\ \quad and Actuarial Science\\ University of Waterloo\\ 200 University Avenue West\\ Waterloo, Ontario N2L 3G1\\ Canada\\ \printead{e2}\\ \phantom{E-mail:\ }\printead*{e3}}
\end{aug}
\received{\smonth{1} \syear{2015}}
\begin{abstract} The tail-dependence compatibility problem is introduced. It raises the question whether a given $d\times d$-matrix of entries in the unit interval is the matrix of pairwise tail-dependence coefficients of a $d$-dimensional random vector. The problem is studied together with Bernoulli-compatible matrices, that is, matrices which are expectations of outer products of random vectors with Bernoulli margins. We show that a square matrix with diagonal entries being 1 is a tail-dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant. We introduce new copula models to construct tail-dependence matrices, including commonly used matrices in statistics. \end{abstract}
\begin{keyword}[class=AMS] \kwd{60E05} \kwd{62H99} \kwd{62H20} \kwd{62E15} \kwd{62H86} \end{keyword}
\begin{keyword} \kwd{Tail dependence} \kwd{Bernoulli random vectors} \kwd{compatibility} \kwd{matrices} \kwd{copulas} \kwd{insurance application} \end{keyword}
\end{frontmatter}
\section{Introduction} The problem of how to construct a bivariate random vector $(X_1,X_2)$ with log-normal marginals $X_1\sim\operatorname{LN}(0,1)$, $X_2\sim\operatorname{LN}(0,16)$ and correlation coefficient $\operatorname{Cor}(X_1,X_2)=0.5$ is well known in the history of dependence modeling, partially because of its relevance to risk management practice. The short answer is: There is no such model; see Embrechts et al. \cite{EMS02} who studied these kinds of problems in terms of copulas. Problems of this kind were brought to RiskLab at ETH Zurich by the insurance industry in the mid-1990s when dependence was thought of in terms of correlation (matrices). For further background on quantitative risk management, see McNeil et al. \cite{MFE15}. Now, almost 20 years later, copulas are a well established tool to quantify dependence in multivariate data and to construct new multivariate distributions. Their use has become standard within industry and regulation. Nevertheless, dependence is still summarized in terms of numbers [as opposed to (copula) functions], so-called \emph {measures of association}. Although there are various ways to compute such numbers in dimension $d>2$, measures of association are still most widely used in the bivariate case $d=2$. A~popular measure of association is tail dependence. It is important for applications in quantitative risk management as it measures the strength of dependence in either the lower-left or upper-right tail of the bivariate distribution, the regions quantitative risk management is mainly concerned with.
We were recently asked\setcounter{footnote}{3}\footnote{By Federico Degen (Head Risk Modeling and Quantification, Zurich Insurance Group) and Janusz Milek (Zurich Insurance Group).} the following question which is in the same spirit as the log-normal correlation problem if one replaces ``correlation'' by ``tail dependence''; see Section~\ref{sec:Lambda:1} for a definition.
\begin{quote} \textit{For which $\alpha\in[0,1]$ is the matrix}
\begin{equation} \Gamma_d(\alpha)=\pmatrix{ 1 & 0 & \cdots& 0 & \alpha\vspace*{2pt} \cr 0 & 1 & \cdots& 0 & \alpha\vspace*{2pt} \cr \vdots& \vdots& \ddots& \vdots& \vdots\vspace*{2pt} \cr 0 & 0 & \cdots& 1 & \alpha\vspace*{2pt} \cr \alpha& \alpha& \cdots& \alpha& 1 }\label{eq:Gammad} \end{equation}
\textit{a matrix of pairwise (either lower or upper) tail-dependence coefficients?} \end{quote}
Intrigued by this question, we more generally consider the following \emph{tail-dependence compatibility problem} in this paper:
\begin{quote} \textit{When is a given matrix in $[0,1]^{d\times d}$ the matrix of pairwise (either lower or upper) tail-dependence coefficients?} \end{quote}
In what follows, we call a matrix of pairwise tail-dependence coefficients a \emph{tail-dependence matrix}. The compatibility problems of tail-dependence coefficients were studied in \cite{J97}. In particular, when $d=3$, inequalities for the bivariate tail-dependence coefficients have been established; see Joe \cite{J97}, Theorem~3.14, as well as Joe \cite{J14}, Theorem~8.20. The sharpness of these inequalities is obtained in \cite{NJL09}. It is generally open to characterize the tail-dependence matrix compatibility for $d>3$.
Our aim in this paper is to give a full answer to the tail-dependence compatibility problem; see Section~\ref{sec:Lambda}. To this end, we introduce and study \emph{Bernoulli-compatible matrices} in Section~\ref {sec:bern}. As a main result, we show that a matrix with diagonal entries being 1 is a compatible tail-dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant. In Section~\ref{sec:model}, we provide probabilistic models for a large class of tail-dependence matrices, including commonly used matrices in statistics. Section~\ref{sec:con} concludes.
Throughout this paper, $d$ and $m$ are positive integers, and we consider an atomless probability space $(\Omega, \mathcal A, \mathbb{P})$ on which all random variables and random vectors are defined. Vectors are considered as column vectors. For two matrices $A,B$, $B\ge A$ and $B\le A$ are understood as component-wise inequalities. We let $A\circ B$ denote the Hadamard product, that is, the element-wise product of two matrices $A$ and $B$ of the same dimension. The $d\times d$ identity matrix is denoted by $I_d$. For a square matrix $A$, $\operatorname{diag}(A)$ represents a diagonal matrix with diagonal entries equal to those of $A$, and $A^\top$ is the transpose of $A$. We denote $\mathrm{I}_E$ the indicator function of an event (random or deterministic) $E\in\mathcal A$. $\mathbf{0}$ and $\mathbf{1}$ are vectors with all components being 0 and 1, respectively, as long as the dimension of the vectors is clear from the context.
\section{Bernoulli compatibility}\label{sec:bern} In this section, we introduce and study the \emph{Bernoulli-compatibility problem}. The results obtained in this section are the basis for the \emph{tail-dependence compatibility problem} treated in Section~\ref{sec:Lambda}; many of them are of independent interest, for example, for the simulation of sequences of Bernoulli random variables.
\subsection{Bernoulli-compatible matrices}
\begin{definition}[(Bernoulli vector, $\mathcal{V}_d$)] A \emph{Bernoulli vector} is a random vector $\mathbf{X}$ supported by $\{0,1\}^d$ for some $d\in\mathbb{N}$. The set of all $d$-Bernoulli vectors is denoted by $\mathcal{V}_d$. \end{definition}
Equivalently, $\mathbf{X}=(X_1,\ldots,X_d)$ is a Bernoulli vector if and only if $X_i\sim\mathrm{B}(1,p_i)$ for some $p_i\in[0,1]$, $i=1,\ldots,d$. Note that here we do not make any assumption about the dependence structure among the components of $\mathbf{X}$. Bernoulli vectors play an important role in credit risk analysis; see, for example, Bluhm and Overbeck \cite{BO06} and Bluhm et al. \cite{BOW02}, Section~2.1.
In this section, we investigate the following question which we refer to as the \emph{Bernoulli-compatibility problem}.
\begin{question}\label{Q:B:compatibility} Given a matrix $B\in[0,1]^{d\times d}$, can we find a Bernoulli vector $\mathbf{X}$ such that $B=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]$? \end{question}
For studying the Bernoulli-compatibility problem, we introduce the notion of Bernoulli-compatible matrices.
\begin{definition}[(Bernoulli-compatible matrix, $\mathcal{B}_d$)] A $d\times d$ matrix $B$ is a \emph{Bernoulli-compatible matrix}, if $B=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]$ for some $\mathbf{X}\in\mathcal{V}_d$. The set of all $d\times d$ Bernoulli-compatible matrices is denoted by $\mathcal{B}_d$. \end{definition}
Concerning covariance matrices, there is extensive research on the compatibility of covariance matrices of Bernoulli vectors in the realm of statistical simulation and time series analysis; see, for example, Chaganty and Joe \cite{CJ06}. It is known that, when $d\ge3$, the set of all compatible $d$-Bernoulli correlation matrices is strictly contained in the set of all correlation matrices. Note that $\mathbb{E}[\mathbf{X}\mathbf{X}^\top]=\operatorname{Cov}(\mathbf{X})+\mathbb{E}[\mathbf {X}]\mathbb{E}[\mathbf{X}]^\top$. Hence, Question~\ref{Q:B:compatibility} is closely related to the characterization of compatible Bernoulli covariance matrices.
Before we characterize the set $\mathcal{B}_d$ in Section~\ref{chara:Bcm}, and thus address Question~\ref{Q:B:compatibility}, we first collect some facts about elements of $\mathcal{B}_d$.
\begin{proposition}\label{prop:prelim} Let $B,B_1,B_2\in\mathcal{B}_d$. Then:
\begin{longlist}[(iii)]
\item[(i)]$B\in[0,1]^{d\times d}$.
\item[(ii)] $\max\{b_{ii}+b_{jj}-1,0\}\le b_{ij}\le \min\{b_{ii},b_{jj}\}$ for $i,j=1,\ldots,d$ and $B=(b_{ij})_{d\times d}$.
\item[(iii)]$t B_1+(1-t)B_2 \in\mathcal{B}_d$ for $t\in[0,1]$, that is, $\mathcal{B}_d$ is a convex set.
\item[(iv)] $B_1\circ B_2 \in\mathcal{B}_d$, that is, $\mathcal{B}_d$ is closed under the Hadamard product.
\item[(v)]$(0)_{d\times d}\in\mathcal{B}_d$ and $(1)_{d\times d}\in\mathcal{B}_d$.
\item[(vi)] For any $\mathbf p=(p_1,\ldots,p_d)\in[0,1]^d$, the matrix $B=(b_{ij})_{d\times d}\in\mathcal{B}_d$ where $b_{ij}=p_i p_j$ for $i\ne j$ and $b_{ii}=p_i$, $i,j=1,\ldots,d$. \end{longlist}
\end{proposition}
\begin{pf} Write $B_1=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]$ and $B_2=\mathbb{E}[\mathbf {Y}\mathbf{Y}^\top]$ for $\mathbf{X},\mathbf{Y}\in\mathcal{V}_d$, and $\mathbf{X}$ and $\mathbf{Y}$ are independent.
\begin{longlist}[(iii)]
\item[(i)] Clear.
\item[(ii)] This directly follows from the Fr\'echet--Hoeffding bounds; see McNeil et al. \cite{MFE15}, Remark~7.9.
\item[(iii)] Let $A\sim\mathrm{B}(1,t)$ be a Bernoulli random variable independent of $\mathbf{X},\mathbf{Y}$, and let $\mathbf{Z}=A\mathbf{X}+(1-A)\mathbf{Y}$. Then $\mathbf{Z}\in\mathcal{V} _d$, and $\mathbb{E}[\mathbf{Z}\mathbf{Z}^\top]=t \mathbb{E}[\mathbf{X}\mathbf{X}^\top]+(1-t)\mathbb{E}[\mathbf{Y}\mathbf{Y}^\top ]=t B_1+(1-t)B_2$. Hence, $t B_1+(1-t)B_2\in \mathcal{B}_d$.
\item[(iv)] Let $\mathbf p=(p_1,\ldots,p_d)$, $\mathbf q=(q_1,\ldots,q_d)\in\mathbb{R}^d$. Then
\begin{eqnarray*} (\mathbf p\circ\mathbf q) (\mathbf p\circ\mathbf q)^\top&=&(p_iq_i)_{d} (p_iq_i)_d^\top =(p_iq_ip_jq_j)_{d\times d}=(p_ip_j)_{d\times d} \circ(q_iq_j)_{d\times d}\\ &=&\bigl(\mathbf p\mathbf p^\top\bigr)\circ\bigl(\mathbf q\mathbf q^\top\bigr). \end{eqnarray*}
Let $\mathbf{Z}=\mathbf{X}\circ\mathbf{Y}$. It follows that $\mathbf{Z}\in\mathcal{V}_d$ and $\mathbb{E}[\mathbf{Z}\mathbf{Z}^\top]=\mathbb{E} [(\mathbf{X}\circ\mathbf{Y}) (\mathbf{X}\circ\mathbf{Y})^\top ]=\mathbb{E}[(\mathbf{X}\mathbf{X}^\top)\circ(\mathbf{Y}\mathbf{Y}^\top )]=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]\circ\mathbb{E}[\mathbf{Y}\mathbf {Y}^\top]=B_1\circ B_2$. Hence, $B_1\circ B_2 \in\mathcal{B}_d$.
\item[(v)] Consider $\mathbf{X}=\mathbf{0}\in\mathcal{V}_d$. Then $(0)_{d\times d}=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]\in\mathcal{B}_d$ and similarly for $(1)_{d\times d}$.
\item[(vi)] Consider $\mathbf{X}\in\mathcal{V}_d$ with independent components and $\mathbb{E}[\mathbf{X}]=\mathbf p$.\quad\qed \end{longlist}
\noqed\end{pf}
\subsection{Characterization of Bernoulli-compatible matrices}\label {chara:Bcm} We are now able to give a characterization of the set $\mathcal{B}_d$ of Bernoulli-compatible matrices and thus address Question~\ref {Q:B:compatibility}.
\begin{theorem}[(Characterization of $\mathcal{B}_d$)]\label{thm:bern} $\mathcal{B}_d$ has the following characterization:
\begin{equation}\label{bern-1} \mathcal{B}_d= \Biggl\{ \sum_{i=1}^{n} a_i \mathbf{p}_i\mathbf{p}_i^\top: \mathbf{p}_i\in\{0,1\}^d, a_i\ge0, i=1,\ldots,n, \sum_{i=1}^n a_i=1, n\in \mathbb{N} \Biggr\};\hspace*{-25pt} \end{equation}
that is, $\mathcal{B}_d$ is the convex hull of $\{\mathbf p\mathbf p^\top: \mathbf p\in\{ 0,1\}^d\}$. In particular, $\mathcal{B}_d$ is closed under convergence in the Euclidean norm. \end{theorem}
\begin{pf} Denote the right-hand side of \eqref{bern-1} by $\mathcal M$. For $B\in \mathcal{B}_d$, write $B=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]$ for some $\mathbf {X}\in\mathcal{V}_d$. It follows that
\[ B=\sum_{\mathbf p \in\{0,1\}^d}\mathbf p \mathbf p^\top\mathbb{P}(\mathbf{X}= \mathbf p)\in \mathcal M, \]
hence $\mathcal{B}_d \subseteq\mathcal M$. Let $\mathbf{X}=\mathbf{p}\in\{ 0,1\}^d$. Then $\mathbf{X}\in\mathcal{V}_d$ and $\mathbb{E}[\mathbf{X}\mathbf {X}^\top]=\mathbf p \mathbf p^\top\in\mathcal{B}_d$. By Proposition~\ref{prop:prelim}, $\mathcal{B}_d$ is a convex set which contains $\{\mathbf p\mathbf p^\top: \mathbf p\in\{0,1\}^d\}$, hence $\mathcal M\subseteq\mathcal{B}_d$. In summary, $\mathcal M=\mathcal{B}_d$. From \eqref {bern-1}, we can see that $\mathcal{B}_d$ is closed under convergence in the Euclidean norm. \end{pf}
A matrix $B$ is \emph{completely positive} if $B=AA^\top$ for some (not necessarily square) matrix $A\ge0$. Denote by $\mathcal{C}_d$ the set of completely positive matrices. It is known that $\mathcal{C}_d$ is the convex cone with extreme directions $\{\mathbf p\mathbf p^\top: \mathbf p\in[0,1]^d\}$; see, for example, R\"uschendorf \cite{R81} and Berman and Shaked-Monderer \cite{BS03}. We thus obtain the following result.
\begin{corollary}\label{cpm} Any Bernoulli-compatible matrix is completely positive. \end{corollary}
\begin{remark}\label{rem:Bd} One may wonder whether $B=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]$ is sufficient to determine the distribution of $\mathbf{X}$, that is, whether the decomposition
\begin{equation} B=\sum_{i=1}^{2^d} a_i \mathbf{p}_i\mathbf{p}_i^\top\label{op1} \end{equation}
is unique for distinct vectors $\mathbf{p}_i$ in $\{0,1\}^d$. While the decomposition is trivially unique for $d=2$, this is in general false for $d\ge3$, since there are $2^d-1$ parameters in \eqref{op1} and only $d(d+1)/2$ parameters in $B$. The following is an example for $d=3$. Let
\begin{eqnarray*} B&=&\frac{1}4\pmatrix{ 2 & 1 & 1 \vspace*{2pt} \cr 1 & 2 & 1 \vspace*{2pt} \cr 1 & 1 & 2 } \\ &=&\frac{1}4 \bigl( (1,1,1)^\top(1,1,1) + (1,0,0)^\top(1,0,0) + (0,1,0)^\top(0,1,0)\\ &&{} + (0,0,1)^\top(0,0,1) \bigr) \\ &=&\frac{1}4 \bigl( (1,1,0)^\top(1,1,0) + (1,0,1)^\top(1,0,1) + (0,1,1)^\top(0,1,1)\\ &&{} + (0,0,0)^\top(0,0,0) \bigr). \end{eqnarray*}
Thus, by combining the above two decompositions, $B\in\mathcal{B}_3$ has infinitely many different decompositions of the form \eqref{op1}. Note that, as in the case of completely positive matrices, it is generally difficult to find decompositions of form \eqref{op1} for a given matrix $B$. \end{remark}
\subsection{Convex cone generated by Bernoulli-compatible matrices} In this section, we study the convex cone generated by $\mathcal{B}_d$, denoted by $\mathcal{B}_d^*$:
\begin{equation} \mathcal{B}_d^*=\{aB: a\ge0, B\in\mathcal{B}_d\}.\label{bern-2} \end{equation}
The following proposition is implied by Proposition~\ref{prop:prelim} and Theorem~\ref{thm:bern}.
\begin{proposition}\label{prop:bern} $\mathcal{B}_d^*$ is the convex cone with extreme directions $\{\mathbf p\mathbf p^\top : \mathbf p\in\{0,1\}^d\}$. Moreover, $\mathcal{B}_d^*$ is a commutative semiring equipped with addition $(\mathcal{B}_d^*, +)$ and multiplication $(\mathcal{B}_d^*, \circ)$. \end{proposition}
It is obvious that $\mathcal{B}_d^*\subseteq\mathcal{C}_d$. One may wonder whether $\mathcal{B} _d^*$ is identical to $\mathcal{C}_d$, the set of completely positive matrices. As the following example shows, this is false in general for $d\ge2$.
\begin{example} Note that $B \in\mathcal{B}_d^*$ also satisfies Proposition~\ref{prop:prelim}, part~(ii). Now consider $\mathbf p=(p_1,\ldots,p_d)\in (0,1)^d$ with $p_i> p_j$ for some $i\neq j$. Clearly, $\mathbf p\mathbf p^\top\in\mathcal{C}_d$, but $p_ip_j> p_j^2=\min\{p_i^2,p_j^2\}$ contradicts Proposition~\ref{prop:prelim}, part~(ii), hence $\mathbf p\mathbf p^\top\notin\mathcal{B}_d^*$. \end{example}
For the following result, we need the notion of diagonally dominant matrices. A matrix $A\in\mathbb{R}^{d\times d}$ is called \emph{diagonally dominant} if, for all
$i=1,\ldots,d$, $\sum_{j\neq i}|a_{ij}|\le|a_{ii}|$.
\begin{proposition}\label{ddm} Let $\mathcal{D}_d$ be the set of nonnegative, diagonally dominant $d\times d$-matrices. Then $\mathcal{D}_d\subseteq\mathcal{B}_d^*$. \end{proposition}
\begin{pf} For $i,j=1,\ldots,d$, let $\mathbf p^{(ij)}=(p^{(ij)}_1,\ldots,p^{(ij)}_d)$ where $p^{(ij)}_k=\mathrm{I}_{\{k=i\}\cup\{k=j\}}$. It is straightforward to verify that the $(i,i)$-, $(i,j)$-, $(j,i)$- and $(j,j)$-entries of the matrix $M^{(ij)}=\mathbf p^{(ij)} (\mathbf p^{(ij)})^\top$ are 1, and the other entries are 0. For $D=(d_{ij})_{d\times d}\in\mathcal{D}_d$, let
\[ D^*=\bigl(d_{ij}^*\bigr)_{d\times d}=\sum _{i=1}^d\sum_{j=1, j\ne i}^d d_{ij}M^{(ij)}. \]
By Proposition~\ref{prop:bern}, $D^*\in\mathcal{B}_d^*$. It follows that $d_{ij}^*=d_{ij}$ for $i\ne j$ and $d_{ii}^*=\sum_{j=1,j\ne i}^d d_{ij}\le d_{ii}$. Therefore, $D=D^*+\sum_{i=1}^d (d_{ii}-d_{ii}^*)M^{(ii)}$, which, by Proposition~\ref{prop:bern}, is in $\mathcal{B}_d^*$. \end{pf}
For studying the tail-dependence compatibility problem in Section~\ref {sec:Lambda}, the subset
\[ \mathcal{B}^I_d=\bigl\{B: B\in\mathcal{B}^*_d, \operatorname{diag}(B)=I_d\bigr\} \]
of $\mathcal{B}_d^*$ is of interest. It is straightforward to see from Proposition~\ref{prop:prelim} and Theorem~\ref{thm:bern} that $\mathcal{B}^I_d$ is a convex set, closed under the Hadamard product and convergence in the Euclidean norm. These properties of $\mathcal{B}^I_d$ will be used later.
\section{Tail-dependence compatibility}\label{sec:Lambda}
\subsection{Tail-dependence matrices}\label{sec:Lambda:1} The notion of tail dependence captures (extreme) dependence in the lower-left or upper-right tails of a bivariate distribution. In what follows, we focus on lower-left tails; the problem for upper-right tails follows by a reflection around $(1/2,1/2)$, that is, studying the survival copula of the underlying copula.
\begin{definition}[(Tail-dependence coefficient)] The \emph{(lower) tail-dependence coefficient} of two continuous random variables $X_1\sim F_1$ and $X_2\sim F_2$ is defined by
\begin{equation} \lambda=\lim_{u\downarrow0}\frac{\mathbb{P}(F_1(X_1)\le u, F_2(X_2)\le u)}{u},\label{def:tail:dep} \end{equation}
given that the limit exists. \end{definition}
If we denote the copula of $(X_1,X_2)$ by $C$, then
\[ \lambda=\lim_{u\downarrow0}\frac{C(u,u)}{u}. \]
Clearly, $\lambda\in[0,1]$, and $\lambda$ only depends on the copula of $(X_1,X_2)$, not the marginal distributions. For virtually all copula models used in practice, the limit in \eqref{def:tail:dep} exists; for how to construct an example where $\lambda$ does not exist; see Kortschak and Albrecher \cite{kortschakalbrecher2009}.
\begin{definition}[(Tail-dependence matrix, $\mathcal{T}_d$)] Let $\mathbf{X}=(X_1,\ldots,X_d)$ be a random vector with continuous marginal distributions. The \emph{tail-dependence matrix} of $\mathbf{X}$ is $\Lambda=(\lambda_{ij})_{d\times d}$, where $\lambda_{ij}$ is the tail-dependence coefficient of $X_i$ and $X_j$, $i,j=1,\ldots,d$. We denote by $\mathcal{T}_d$ the set of all tail-dependence matrices. \end{definition}
The following proposition summarizes basic properties of tail-dependence matrices. Its proof is very similar to that of Proposition~\ref {prop:prelim} and is omitted here.
\begin{proposition} For any $\Lambda_1, \Lambda_2\in\mathcal{T}_d$, we have that:
\begin{longlist}[(iii)]
\item[(i)]$\Lambda_1=\Lambda_1^\top$.
\item[(ii)]$t \Lambda_1+(1-t)\Lambda_2 \in\mathcal{T}_d$ for $t\in[0,1]$, that is, $\mathcal{T}_d$ is a convex set.
\item[(iii)]$I_d\le\Lambda_1\le(1)_{d\times d}$ with $I_d\in\mathcal{T}_d$ and $(1)_{d\times d}\in\mathcal{T}_d$. \end{longlist}
\end{proposition}
As we will show next, $\mathcal{T}_d$ is also closed under the Hadamard product.
\begin{proposition}\label{prop:hadamard:prod} Let $k\in\mathbb{N}$ and $\Lambda_1,\ldots,\Lambda_k\in\mathcal{T}_d$. Then $\Lambda_1\circ\cdots\circ\Lambda_k\in\mathcal{T}_d$. \end{proposition}
\begin{pf} Note that it would be sufficient to show the result for $k=2$, but we provide a general construction for any $k$. For each $l=1,\ldots,k$, let $C_l$ be a $d$-dimensional copula with tail-dependence matrix $\Lambda_l$. Furthermore, let $g(u)=u^{1/k}$, $u\in[0,1]$. It follows from Liebscher \cite{L08} that $C(u_1,\ldots,u_d)=\prod_{l=1}^k C_l(g(u_1),\ldots,g(u_d))$ is a copula; note that
\begin{equation} \Bigl(g^{-1}\Bigl(\max_{1\le l\le k}\{U_{l1}\} \Bigr),\ldots,g^{-1}\Bigl(\max_{1\le l\le k}\{U_{ld} \}\Bigr) \Bigr)\sim C\label{eq:liebscher:cop} \end{equation}
for independent random vectors $(U_{l1},\ldots,U_{ld})\sim C_l$, $l=1,\ldots,k$. The $(i,j)$-entry $\lambda_{ij}$ of $\Lambda$ corresponding to $C$ is thus given by
\begin{eqnarray*} \lambda_{ij}&=&\lim_{u\downarrow0}\frac{\prod_{l=1}^k C_{l,ij}(g(u),g(u))}{u}=\lim _{u\downarrow 0}\prod_{l=1}^k \frac{C_{l,ij}(g(u),g(u))}{g(u)}\\ &=&\prod_{l=1}^k\lim _{u\downarrow 0}\frac{C_{l,ij}(g(u),g(u))}{g(u)} \\ &=&\prod_{l=1}^k\lim_{u\downarrow0} \frac{C_{l,ij}(u,u)}{u}=\prod_{l=1}^k \lambda_{l,ij}, \end{eqnarray*}
where $C_{l,ij}$ denotes the $(i,j)$-margin of $C_l$ and $\lambda_{l,ij}$ denotes the $(i,j)$th entry of $\Lambda_l$, $l=1,\ldots,k$. \end{pf}
\subsection{Characterization of tail-dependence matrices} In this section, we investigate the following question.
\begin{question}\label{Q:characterization} Given a $d\times d$ matrix $\Lambda\in[0,1]^{d\times d}$, is it a tail-dependence matrix?
\end{question}
The following theorem fully characterizes tail-dependence matrices, and thus provides a theoretical (but not necessarily practical) answer to Question~\ref{Q:characterization}.
\begin{theorem}[(Characterization of $\mathcal{T}_d$)]\label{thm:main:characterization} A square matrix with diagonal entries being 1 is a tail-dependence matrix if and only if it is a Bernoulli-compatible matrix multiplied by a constant. Equivalently, $\mathcal{T}_d=\mathcal{B}^I_d$. \end{theorem}
\begin{pf} We first show that $\mathcal{T}_d\subseteq\mathcal{B}^I_d$. For each $\Lambda=(\lambda_{ij})_{d\times d}\in\mathcal{T}_d$, suppose that $C$ is a copula with tail-dependence matrix $\Lambda$ and $\mathbf{U}=(U_1,\ldots ,U_n)\sim C$. Let $\mathbf{W}_u=(\mathrm{I}_{\{U_1\le u\}},\ldots,\mathrm{I}_{\{U_d\le u\}})$. By definition,
\[ \lambda_{ij}=\lim_{u\downarrow0}\frac{1}u\mathbb{E}[ \mathrm{I}_{\{U_i\le u\}}\mathrm{I} _{\{U_j\le u\}}] \]
and
\[ \Lambda=\lim_{u\downarrow0}\frac{1}u \mathbb{E}\bigl[ \mathbf{W}_u\mathbf {W}_u^\top\bigr]. \]
Since $\mathcal{B}^I_d$ is closed and $\mathbb{E}[\mathbf{W}_u\mathbf{W}_u^\top ]/u\in\mathcal{B}^I_d$, we have that $\Lambda\in\mathcal{B}^I_d$.
Now consider $\mathcal{B}^I_d\subseteq\mathcal{T}_d$. By definition of $\mathcal{B}^I_d$, each $B\in \mathcal{B}^I_d$ can be written as $B=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]/p$ for an $\mathbf{X}\in\mathcal{V}_d$ and $\mathbb{E}[\mathbf{X}]=(p,\ldots,p)\in(0,1]^d$. Let $U,V\sim\mathrm {U}[0,1]$, $U,V,\mathbf{X}$ be independent and
\begin{equation} \mathbf{Y}=\mathbf{X}pU+(\mathbf{1}-\mathbf{X}) \bigl(p+(1-p)V \bigr).\label{smodel} \end{equation}
We can verify that for $t\in[0,1]$ and $i=1,\ldots,d$,
\begin{eqnarray*} \mathbb{P}(Y_i\le t)&=&\mathbb{P}(X_i=1)\mathbb{P}(pU \le t)+ \mathbb{P}(X_i=0)\mathbb{P}\bigl(p+(1-p)V\le t\bigr) \\ &=&p\min\{t/p,1\} +(1-p) \max\bigl\{(t-p)/(1-p),0\bigr\}=t, \end{eqnarray*}
that is, $Y_1,\ldots,Y_d$ are $\mathrm{U}[0,1]$-distributed. Let $\lambda_{ij}$ be the tail-dependence coefficient of $Y_i$ and $Y_j$, $i,j=1,\ldots,d$. For $i,j=1,\ldots,d$ we obtain that
\begin{eqnarray*} \lambda_{ij}&=&\lim_{u\downarrow0}\frac{1}u \mathbb{P}(Y_i\le u, Y_j\le u)=\lim_{u\downarrow0} \frac{1}u\mathbb{P}(X_i=1, X_j=1)\mathbb{P}(pU\le u)\\ &=& \frac{1}p \mathbb{E}[X_iX_j]. \end{eqnarray*}
As a consequence, the tail-dependence matrix of $(Y_1,\ldots,Y_d)$ is $B$ and $B\in\mathcal{T}_d$. \end{pf}
It follows from Theorem~\ref{thm:main:characterization} and Proposition~\ref{prop:bern} that $\mathcal{T}_d$ is the ``1-diagonals'' cross-section of the convex cone with extreme directions $\{\mathbf p\mathbf p^\top: \mathbf p\in\{0,1\}^d\}$. Furthermore, the proof of Theorem~\ref{thm:main:characterization} is constructive. As we saw, for any $B\in\mathcal{B}^I_d$, $\mathbf{Y}$ defined by \eqref{smodel} has tail-dependence matrix $B$. This interesting construction will be applied in Section~\ref{sec:model} where we show that commonly applied matrices in statistics are tail-dependence matrices and where we derive the copula of $\mathbf{Y}$.
\begin{remark} From the fact that $\mathcal{T}_d=\mathcal{B}_d^I$ and $\mathcal{B}_d^I$ is closed under the Hadamard product [see Proposition~\ref{prop:prelim}, part~(iv)], Proposition~\ref{prop:hadamard:prod} directly follows. Note, however, that our proof of Proposition~\ref{prop:hadamard:prod} is constructive. Given tail-dependence matrices and corresponding copulas, we can construct a copula $C$ which has the Hadamard product of the tail-dependence matrices as corresponding tail-dependence matrix. If sampling of all involved copulas is feasible, we can sample $C$; see Figure~\ref{fig:liebscher} for examples.\footnote{All plots can be reproduced via the \textsf{R} package \texttt{copula} (version $\ge$ 0.999-13) by calling \texttt{demo(tail\_compatibility)}.}
\begin{figure}
\caption{Left-hand side: Scatter plot of 2000 samples from \protect\eqref{eq:liebscher:cop} for $C_1$ being a Clayton copula with parameter $\theta=4$ ($\lambda_1=2^{-1/4}\approx0.8409$) and $C_2$ being a $t_3$ copula with parameter $\rho=0.8$ [tail-dependence coefficient $\lambda_2=2t_4(-2/3)\approx0.5415$]. By Proposition~\protect\ref {prop:hadamard:prod}, the tail-dependence coefficient of \protect\eqref{eq:liebscher:cop} is thus $\lambda=\lambda_1\lambda_2=2^{3/4}t_4(-2/3)\approx0.4553$. Right-hand side: $C_1$ as before, but $C_2$ is a survival Marshall--Olkin copula with parameters $\alpha_1=2^{-3/4},\alpha_2=0.8$, so that $\lambda=\lambda_1\lambda_2=1/2$.}
\label{fig:liebscher}
\end{figure} \end{remark}
Theorem~\ref{thm:main:characterization} combined with Corollary~\ref{cpm} directly leads to the following result.
\begin{corollary} Every tail-dependence matrix is completely positive, and hence positive semi-definite. \end{corollary}
Furthermore, Theorem~\ref{thm:main:characterization} and Proposition~\ref{ddm} imply the following result.
\begin{corollary}\label{coro:ddm} Every diagonally dominant matrix with nonnegative entries and diagonal entries being $1$ is a tail-dependence matrix. \end{corollary}
Note that this result already yields the if-part of Proposition~\ref {prop:example} below.
\section{Compatible models for tail-dependence matrices} \label{sec:model}
\subsection{Widely known matrices} We now consider the following three types of matrices $\Lambda=(\lambda_{ij})_{d\times d}$ which are frequently applied in multivariate statistics and time series analysis and show that they are tail-dependence matrices.
\begin{longlist}[(a)]
\item[(a)] Equicorrelation matrix with parameter $\alpha \in[0,1]$: $\lambda_{ij}=\mathrm{I}_{\{i=j\}} +\alpha\mathrm{I}_{\{i\ne j\}}$, $ i,j=1,\ldots,d$.
\item[(b)] AR(1) matrix with parameter $\alpha\in[0,1]$:
$\lambda_{ij}= \alpha^{ |i-j| }$, $ i,j=1,\ldots,d$.
\item[(c)] MA(1) matrix with parameter $\alpha\in [0,1/2]$: $\lambda_{ij}=\mathrm{I}_{\{i=j\}}
+\alpha\mathrm{I}_{\{|i-j|=1\}}$, $ i,j=1,\ldots,d$. \end{longlist}
Chaganty and Joe \cite{CJ06} considered the compatibility of correlation matrices of Bernoulli vectors for the above three types of matrices and obtained necessary and sufficient conditions for the existence of compatible models for $d=3$. For the tail-dependence compatibility problem that we consider in this paper, the above three types of matrices are all compatible, and we are able to construct corresponding models for each case.
\begin{proposition} Let $\Lambda$ be the tail-dependence matrix of the $d$-dimension\-al random vector
\begin{equation} \mathbf{Y}=\mathbf{X}pU+(\mathbf{1}-\mathbf{X}) \bigl(p+(1-p)V \bigr),\label{smodel2} \end{equation}
where $U,V\sim\mathrm{U}[0,1]$, $\mathbf{X}\in\mathcal{V}_d$ and $U,V,\mathbf{X}$ are independent.
\begin{longlist}[(iii)]
\item[(i)] For $\alpha\in[0,1]$, if $\mathbf{X}$ has independent components and $\mathbb{E}[X_1]=\cdots=\mathbb{E}[X_d]=\alpha$, then $\Lambda$ is an equicorrelation matrix with parameter $\alpha$; that is, \textup{(a)} is a tail-dependence matrix.
\item[(ii)] For $\alpha\in[0,1]$, if $X_i=\prod_{j=i}^{i+d-1} Z_j$, $i=1,\ldots,d$, for independent $\mathrm{B}(1,\alpha)$ random variables $Z_1,\ldots,Z_{2d-1}$, then $\Lambda$ is an AR(1) matrix with parameter $\alpha$; that is, \textup{(b)} is a tail-dependence matrix.
\item[(iii)] For $\alpha\in[0,1/2]$, if $X_i=\mathrm{I}_{\{Z\in [(i-1)(1-\alpha), (i-1)(1-\alpha)+1]\}}$, $i=1,\ldots,d$, for $Z\sim\mathrm{U}[0,d]$, then $\Lambda$ is an MA(1) matrix with parameter $\alpha$; that is, \textup{(c)} is a tail-dependence matrix. \end{longlist}
\end{proposition}
\begin{pf} We have seen in the proof of Theorem~\ref{thm:main:characterization} that if $\mathbb{E}[X_1]=\cdots=\mathbb{E}[X_d]=p$, then $\mathbf{Y}$ defined through \eqref{smodel2} has tail-dependence matrix $\mathbb{E}[\mathbf{X}\mathbf{X}^\top]/p$. Write $\Lambda=(\lambda_{ij})_{d\times d}$ and note that $\lambda_{ii}=1$, $i=1,\ldots,d$, is always guaranteed.
\begin{longlist}[(iii)]
\item[(i)] For $i\neq j$, we have that $\mathbb{E}[X_iX_j]=\alpha^2$ and thus $\lambda_{ij}=\alpha^2/\alpha=\alpha$. This shows that $\Lambda$ is an equicorrelation matrix with parameter $\alpha$.
\item[(ii)] For $i<j$, we have that
\begin{eqnarray*} \mathbb{E}[X_iX_j]&=&\mathbb{E} \Biggl[ \prod _{k=i}^{i+d-1} Z_k \prod _{l=j}^{j+d-1} Z_l \Biggr]=\mathbb{E} \Biggl[ \prod _{k=i}^{j-1} Z_k \Biggr]\mathbb{E} \Biggl[ \prod_{k=j}^{i+d-1} Z_k \Biggr]\mathbb{E} \Biggl[ \prod_{k=i+d}^{j+d-1} Z_k \Biggr] \\ &=&\alpha^{j-i} \alpha^{i+d-j}\alpha^{j-i}= \alpha^{j-i+d} \end{eqnarray*}
and $\mathbb{E}[X_i]=\mathbb{E}[X_i^2]=\alpha^d$. Hence,
$\lambda_{ij}=\alpha^{j-i+d}/\alpha^{d}=\alpha^{j-i}$ for $i< j$. By symmetry, $\lambda_{ij}= \alpha^{ |i-j| }$ for $i\ne j$. Thus, $\Lambda$ is an AR(1) matrix with parameter $\alpha$.
\item[(iii)] For $i<j$, note that $2(1-\alpha)\ge1$, so
\begin{eqnarray*} \mathbb{E}[X_iX_j]&=&\mathbb{P}\bigl(Z\in\bigl[(j-1) (1-\alpha), (i-1) (1-\alpha)+1\bigr]\bigr) \\ &=&\mathrm{I}_{\{j=i+1\}} \mathbb{P}\bigl(Z\in\bigl[i(1-\alpha),(i-1) (1-\alpha)+1\bigr] \bigr)=\mathrm{I}_{\{ j=i+1\}}\frac{\alpha} d \end{eqnarray*}
and $\mathbb{E}[X_i]=\mathbb{E}[X_i^2]=\frac{1}d$. Hence, $\lambda_{ij}=\alpha \mathrm{I}_{\{j-i=1\}}$ for $i<j$. By symmetry, $\lambda_{ij}=
\alpha\mathrm{I}_{\{|i-j|=1\}}$ for $i\ne j$. Thus, $\Lambda$ is an MA(1) matrix with parameter $\alpha$.\quad\qed \end{longlist}
\noqed\end{pf}
\subsection{Advanced tail-dependence models} Theorem~\ref{thm:main:characterization} gives a characterization of tail-dependence matrices using Bernoulli-compatible matrices and \eqref{smodel} provides a compatible model $\mathbf{Y}$ for any tail-dependence matrix $\Lambda(=\mathbb{E}[\mathbf{X}\mathbf {X}^\top]/p)$.
It is generally not easy to check whether a given matrix is a Bernoulli-compatible matrix or a tail-dependence matrix; see also Remark~\ref{rem:Bd}. Therefore, we now study the following question.
\begin{question}\label{Q:model:for:Y} How can we construct a broader class of models with flexible dependence structures and desired tail-dependence matrices? \end{question}
To enrich our models, we bring random matrices with Bernoulli entries into play. For $d, m\in\mathbb{N}$, let
\[ \mathcal{V}_{d\times m}= \Biggl\{X=(X_{ij})_{d\times m}: \mathbb{P}\bigl(X\in \{0,1\} ^{d\times m}\bigr)=1, \sum_{j=1}^m X_{ij}\le1, i=1,\ldots,d \Biggr\}, \]
that is, $\mathcal{V}_{d\times m}$ is the set of $d\times m$ random matrices supported in $\{0,1\}^{d\times m}$ with each row being \emph{mutually exclusive}; see Dhaene and Denuit \cite{DD99}. Furthermore,
we introduce a transformation $\mathcal{L}$ on the set of square matrices, such that, for any $i,j=1,\ldots,d$, the $(i,j)$th element $\tilde{b}_{ij}$ of $\mathcal{L} (B)$ is given by
\begin{equation} \tilde{b}_{ij}=\cases{ b_{ij},&\quad$\mbox{if } i\neq j$, \vspace*{2pt} \cr 1,&\quad$\mbox{if } i=j$;} \end{equation}
that is, $\mathcal{L}$ adjusts the diagonal entries of a matrix to be 1, and preserves all the other entries. For a set $S$ of square matrices, we set $\mathcal{L}(S)=\{\mathcal{L}(B):B\in S\}$. We can now address Question~\ref{Q:model:for:Y}.
\begin{theorem}[(A class of flexible models)]\label{thm:main:tdm:Y} Let $\mathbf{U}\sim C^{\mathbf{U}}$ for an $m$-dimensional copula $C^{\mathbf{U}}$ with tail-dependence matrix $\Lambda$ and let $\mathbf{V}\sim C^{\mathbf{V}}$ for a $d$-dimensional copula $C^{\mathbf{V}}$ with tail-dependence matrix $I_d$. Furthermore, let $X \in\mathcal{V}_{d\times m}$ such that $X,\mathbf{U},\mathbf{V}$ are independent and let
\begin{equation} \mathbf{Y}=X\mathbf{U}+\mathbf{Z}\circ\mathbf{V},\label{eq:stoch:rep} \end{equation}
where $\mathbf{Z}=(Z_1,\ldots,Z_d)$ with $Z_i=1-\sum_{k=1}^mX_{ik}$, $i=1,\ldots,d$. Then $\mathbf{Y}$ has tail-dependence matrix $\Gamma=\mathcal{L}(\mathbb{E}[X\Lambda X^\top])$. \end{theorem}
\begin{pf} Write $X=(X_{ij})_{d\times m}$, $\mathbf{U}=(U_1,\ldots,U_m)$, $\mathbf{V}=(V_1,\ldots,V_d)$, $\Lambda=(\lambda_{ij})_{d\times d}$ and $\mathbf{Y}=(Y_1,\ldots,Y_d)$. Then, for all $i=1,\ldots,d$,
\begin{eqnarray*} Y_i=\sum_{k=1}^m X_{ik} U_k+Z_iV_i=\cases{ V_i,&\quad$\mbox{if } X_{ik}=0 \mbox{ for all } k=1,\ldots,m, \mbox{ so } Z_i=1$,\vspace*{2pt} \cr U_k,&\quad$\mbox{if } X_{ik}=1 \mbox{ for some } k=1,\ldots,m, \mbox{ so } Z_i=0$.} \end{eqnarray*}
Clearly, $\mathbf{Y}$ has $\mathrm{U}[0,1]$ margins. We now calculate the tail-dependence matrix $\Gamma=(\gamma_{ij})_{d\times d}$ of $Y$ for $i\neq j$. By our independence assumptions, we can derive the following results:
\begin{longlist}[(iii)]
\item[(i)]$\mathbb{P}(Y_i\le u, Y_j\le u, Z_i=1, Z_j=1)=\mathbb{P}(V_i\le u, V_j\le u, Z_i=1, Z_j=1)=C_{ij}^{\mathbf {V}}(u,u)\mathbb{P}(Z_i=1,Z_j=1)\le C_{ij}^{\mathbf{V}}(u,u)$, where $C_{ij}^{\mathbf{V}}$ denotes the $(i,j)$th margin of $C^{\mathbf{V}}$. As $\mathbf{V}$ has tail-dependence matrix $I_d$, we obtain that
\[ \lim_{u\downarrow0}\frac{1}u\mathbb{P}(Y_i\le u, Y_j\le u, Z_i=1, Z_j=1)=0. \]
\item[(ii)]$\mathbb{P}(Y_i\le u, Y_j\le u, Z_i=0, Z_j=1)=\sum_{k=1}^m\mathbb{P}(U_k\le u, V_j\le u, X_{ik}=1, Z_j=1)=\sum_{k=1}^m\mathbb{P}(U_k\le u)\mathbb{P}(V_j\le u)\mathbb{P} (X_{ik}=1,Z_j=1)\le u^2$, and thus
\[ \lim_{u\downarrow0}\frac{1}u\mathbb{P}(Y_i\le u, Y_j\le u, Z_i=0, Z_j=1)=0. \]
Similarly, we obtain that
\[ \lim_{u\downarrow0}\frac{1}u\mathbb{P}(Y_i\le u, Y_j\le u, Z_i=1, Z_j=0)=0. \]
\item[(iii)]$\mathbb{P}(Y_i\le u, Y_j\le u, Z_i=0, Z_j=0)=\sum_{k=1}^m\sum_{l=1}^m\mathbb{P}(U_k\le u, U_l\le u, X_{ik}=1, X_{jl}=1)=\sum_{k=1}^m\sum_{l=1}^mC_{kl}^{\mathbf{U}}(u,u)\mathbb{P}(X_{ik}=1, X_{jl}=1)= \sum_{k=1}^m\sum_{l=1}^mC_{kl}^{\mathbf{U}}(u,u)\*\mathbb{E} [X_{ik}X_{jl}]$ so that
\begin{eqnarray*} \lim_{u\downarrow0}\frac{1}u\mathbb{P}(Y_i\le u, Y_j\le u, Z_i=0, Z_j=0)&=&\sum _{k=1}^m\sum_{l=1}^m \lambda_{kl}\mathbb{E}[X_{ik}X_{jl}]\\ &=&\mathbb{E} \Biggl[ \sum _{k=1}^m\sum_{l=1}^mX_{ik} \lambda_{kl}X_{jl} \Biggr] \\ &= &\bigl(\mathbb{E}\bigl[X\Lambda X^\top\bigr] \bigr)_{ij}. \end{eqnarray*}
\end{longlist}
By the law of total probability, we thus obtain that
\begin{eqnarray*} \gamma_{ij}&=&\lim_{u\downarrow0}\frac{\mathbb{P}(Y_i\le u, Y_j\le u)}{u}=\lim _{u\downarrow0}\frac{\mathbb{P}(Y_i\le u, Y_j\le u, Z_i=0, Z_j=0)}{u} \\ &=& \bigl(\mathbb{E}\bigl[X\Lambda X^\top\bigr] \bigr)_{ij}. \end{eqnarray*}
This shows that $\mathbb{E}[X\Lambda X^\top]$ and $\Gamma$ agree on the off-diagonal entries. Since $\Gamma\in\mathcal{T}_d$ implies that $\operatorname{diag}(\Gamma)=I_d$, we conclude that $\mathcal{L}(\mathbb{E}[X\Lambda X^\top])=\Gamma$. \end{pf}
A special case of Theorem~\ref{thm:main:tdm:Y} reveals an essential difference between the transition rules of a tail-dependence matrix and a covariance matrix. Suppose that for $X\in\mathcal{V}_{d\times m}$, $\mathbb{E}[X]$ is a stochastic matrix (each row sums to 1), and $\mathbf{U}\sim C^{\mathbf{U}}$ for an $m$-dimensional copula $C^{\mathbf{U}}$ with tail-dependence matrix $\Lambda=(\lambda_{ij})_{d\times d}$. Now we have that $Z_i=0$, $i=1,\ldots,d$ in \eqref{eq:stoch:rep}. By Theorem~\ref{thm:main:tdm:Y}, the tail dependence matrix of $\mathbf {Y}=X\mathbf{U}$ is given by $\mathcal{L}(\mathbb{E}[X\Lambda X^\top])$. One can check the diagonal terms of the matrix $\Lambda^*=(\lambda^*_{ij})_{d\times d}=X\Lambda X^\top$ by
\[ \lambda^*_{ii}=\sum_{j=1}^m \sum_{k=1}^m X_{ik}\lambda _{kj}X_{ij}=\sum_{k=1}^m X_{ik}\lambda_{kk}=1,\qquad i=1,\ldots,m. \]
Hence, the tail-dependence matrix of $\mathbf{Y}$ is indeed $\mathbb{E} [X\Lambda X^\top]$.
\begin{remark} \label{transition} In summary:
\begin{longlist}[(ii)]
\item[(i)] If an $m$-vector $\mathbf{U}$ has covariance matrix $\Sigma$, then $X\mathbf{U}$ has covariance matrix $\mathbb{E}[X \Sigma X^\top]$ for any $d\times m$ random matrix $X$ independent of $\mathbf{U}$.
\item[(ii)] If an $m$-vector $\mathbf{U}$ has uniform $[0,1]$ margins and tail-dependence matrix $\Lambda$, then $X\mathbf{U}$ has tail-dependence matrix $\mathbb{E} [X\Lambda X^\top]$ for any $X\in\mathcal{V}_{d\times m}$ independent of $\mathbf{U}$ such that each row of $X$ sums to 1. \end{longlist}
It is noted that the transition property of tail-dependence matrices is more restricted than that of covariance matrices. \end{remark}
The following two propositions consider selected special cases of this construction which are more straightforward to apply.
\begin{proposition}\label{coro-b} For any $B \in\mathcal{B}_d$ and any $\Lambda\in\mathcal{T}_d$ we have that $\mathcal{L} (B\circ \Lambda)\in\mathcal{T}_d$. In particular, $\mathcal{L}(B) \in\mathcal{T}_d$, and hence $\mathcal{L}(\mathcal{B}_d)\subseteq\mathcal{T}_d$. \end{proposition}
\begin{pf} Write $B=(b_{ij})_{d\times d}=\mathbb{E}[\mathbf{W}\mathbf{W}^\top]$ for some $\mathbf{W}=(W_1,\ldots,W_d)\in \mathcal{V}_d$ and consider $X=\operatorname{diag}(\mathbf{W})\in\mathcal{V}_{d\times d}$. As in the proof of Theorem~\ref{thm:main:tdm:Y} (and with the same notation), it follows that for $i\ne j$, $\gamma_{ij}=\mathbb{E}[X_{ii}\lambda_{ij}X_{jj}]=\mathbb{E}[W_iW_j\lambda _{ij}]$. This shows that $\mathbb{E}[X\Lambda X^\top]=\mathbb{E}[\mathbf{W}\mathbf{W}^\top\circ \Lambda]$ and $B\circ\Lambda$ agree on off-diagonal entries. Thus, $\mathcal{L}(B\circ\Lambda)=\Gamma\in \mathcal{T}_d$. By taking $\Lambda=(1)_{d\times d}$, we obtain $\mathcal{L}(B)\in T_d$. \end{pf}
The following proposition states a relationship between substochastic matrices and tail-dependence matrices. To this end, let
\[ \mathcal{Q}_{d\times m}= \Biggl\{Q=(q_{ij})_{d\times m}: \sum _{j=1}^m q_{ij}\le 1, q_{ij} \ge0, i=1,\ldots,d, j=1,\ldots,m \Biggr\}, \]
that is, $\mathcal{Q}_{d\times m}$ is the set of $d\times m$ \emph{(row) substochastic matrices}; note that the expectation of a random matrix in $\mathcal V_{d\times m}$ is a substochastic matrix.
\begin{proposition} For any $Q\in\mathcal{Q}_{d\times m}$ and any $\Lambda\in\mathcal{T}_m$, we have that $\mathcal{L}(Q \Lambda Q^\top)\in\mathcal{T}_d$. In particular, $\mathcal{L}(QQ^\top) \in\mathcal{T}_d$ for all $Q\in \mathcal{Q}_{d\times m}$ and $\mathcal{L}(\mathbf p \mathbf p^\top) \in\mathcal{T}_d$ for all $\mathbf p\in[0,1]^d$. \end{proposition}
\begin{pf} Write $Q=(q_{ij})_{d\times m}$ and let $X_{ik}=\mathrm{I}_{\{Z_{i}\in [ \sum_{j=1}^{k-1}q_{ij}, \sum_{j=1}^{k}q_{ij})\}}$ for independent $Z_i\sim\mathrm{U}[0,1]$, $i=1,\ldots,d$, $k=1,\ldots,m$. It is straightforward to see that $\mathbb{E}[X]=Q$, $X\in\mathcal{V}_{d\times m}$ with independent rows, and $\sum_{k=1}^m X_{ik}\le1$ for $i=1,\ldots,d$, so $X\in\mathcal{V}_{d\times m}$. As in the proof of Theorem~\ref{thm:main:tdm:Y} (and with the same notation), it follows that for $i\ne j$,
\[ \gamma_{ij}= \sum_{l=1}^m\sum _{k=1}^m \mathbb{E}[X_{ik}] \mathbb{E}[X_{jl}]\lambda_{kl}= \sum_{l=1}^m \sum_{k=1}^m q_{ik}q_{jl} \lambda_{kl}. \]
This shows that $Q\Lambda Q^\top$ and $\Gamma$ agree on off-diagonal entries, so $\mathcal{L}(Q\Lambda Q^\top)=\Gamma\in\mathcal{T}_d$. By taking $\Lambda=I_d$, we obtain $\mathcal{L}(QQ^\top)\in T_d$. By taking $m=1$, we obtain $\mathcal{L}(\mathbf p \mathbf p^\top) \in\mathcal{T}_d$. \end{pf}
\subsection{Corresponding copula models} In this section, we derive the copulas of \eqref{smodel} and~\eqref{eq:stoch:rep} which are able to produce tail-dependence matrices $\mathbb{E}[\mathbf{X}\mathbf{X}^\top]/p$ and $\mathcal{L}(\mathbb{E}[X\Lambda X^\top])$ as stated in Theorems~\ref{thm:main:characterization} and \ref{thm:main:tdm:Y}, respectively. We first address the former.
\begin{proposition}[{[Copula of \eqref{smodel}]}] Let $\mathbf{X}\in\mathcal{V}_d$, $\mathbb{E}[\mathbf{X}]=(p,\ldots,p)\in(0,1]^d$. Furthermore, let $U,V\sim\mathrm{U}[0,1]$, $U,V,\mathbf{X}$ be independent and
\[ \mathbf{Y}=\mathbf{X}pU+(\mathbf{1}-\mathbf{X}) \bigl(p+(1-p)V\bigr). \]
Then the copula $C$ of $\mathbf{Y}$ at $\mathbf{u}=(u_1,\ldots,u_d)$ is given by
\[ C(\mathbf{u})=\sum_{\mathbf{i}\in\{0,1\}^d}\min \biggl\{ \frac{\min_{r:i_r=1}\{u_r\}}{p},1 \biggr\} \max \biggl\{\frac{\min_{r:i_r=0}\{ u_r\}-p}{1-p},0 \biggr\} \mathbb{P}( \mathbf{X}=\mathbf{i}), \]
with the convention $\min\varnothing=1$. \end{proposition}
\begin{pf} By the law of total probability and our independence assumptions,
\begin{eqnarray*} C(\mathbf{u})&=& \sum_{\mathbf{i}\in\{0,1\}^d}\mathbb{P}(\mathbf{Y}\le \mathbf{u}, \mathbf{X}=\mathbf{i}) \\[-2pt] &=& \sum_{\mathbf{i}\in\{0,1\}^d}\mathbb{P}\Bigl(pU\le\min_{r:i_r=1} \{u_r\}, p+(1-p)V\le\min_{r:i_r=0}\{u_r\}, \mathbf{X}=\mathbf{i}\Bigr) \\[-2pt] &= &\sum_{\mathbf{i}\in\{0,1\}^d}\mathbb{P} \biggl(U\le\frac{\min_{r:i_r=1}\{u_r\}}{p} \biggr) \mathbb{P} \biggl(V\le\frac{\min_{r:i_r=0}\{ u_r\}-p}{1-p} \biggr) \mathbb{P}(\mathbf{X}=\mathbf{i}); \end{eqnarray*}
the claim follows from the fact that $U,V\sim\mathrm{U}[0,1]$. \end{pf}
For deriving the copula of \eqref{eq:stoch:rep}, we need to introduce some notation; see also Example~\ref{ex:special:cases} below. In the following theorem, let $\mathrm{supp}(X)$ denote the support of~$X$. For a vector
$\mathbf{u}=(u_1,\ldots,u_d)\in[0,1]^d$ and a matrix $A=(A_{ij})_{d\times m}\in\mathrm{supp}(X)$, denote by $A_i$ the sum of the $i$th row of $A$, $i=1,\ldots,d$, and let $\mathbf{u}_A=(u_1\mathrm{I}_{\{A_1=0\}}+\mathrm{I}_{\{A_1=1\}},\ldots,u_d\mathrm{I}_{\{ A_d=0\}}+\mathrm{I}_{\{A_d=1\}})$, and $\mathbf{u}_A^*=(\min_{r:A_{r1}=1}\{u_r\},\ldots, \min_{r:A_{rm}=1}\{u_r\})$, where $\min\varnothing=1$.
\begin{proposition}[{[Copula of \eqref{eq:stoch:rep}]}] Suppose that the setup of Theorem~\ref{thm:main:tdm:Y} holds. Then the copula $C$ of $\mathbf{Y}$ in \eqref{eq:stoch:rep} is given by
\begin{equation} C(\mathbf{u})=\sum_{A\in\mathrm{supp}(X)} C^V( \mathbf{u}_A) C^U\bigl(\mathbf{u}_A^*\bigr) \mathbb{P}(X=A).\label{eq:cop:Y} \end{equation}
\end{proposition}
\begin{pf}
By the law of total probability, it suffices to verify that $\mathbb{P} (\mathbf{Y}\le
\mathbf{u} | X=A)= C^{\mathbf{V}}(\mathbf{u}_A)C^{\mathbf {U}}(\mathbf{u}_A^*)$. This can be seen from
\begin{eqnarray*}
&&\mathbb{P}(\mathbf{Y}\le\mathbf{u} | X=A) \\[-2pt] &&\qquad=\mathbb{P} \Biggl( \sum_{k=1}^m A_{jk}U_k+(1-A_j)V_j\le u_j, j=1,\ldots ,d \Biggr) \\[-2pt] &&\qquad=\mathbb{P}(U_k \mathrm{I}_{\{A_{jk}=1\}} \le u_j, V_j \mathrm{I}_{\{A_j=0\}}\le u_j, j=1,\ldots,d,~k=1,\ldots,m) \\[-2pt] &&\qquad=\mathbb{P} \Bigl(U_k \le\min_{r:A_{rk}=1}\{u_r\}, V_j\le u_j \mathrm{I}_{\{A_j=0\}}+\mathrm{I}_{\{A_j=1\}},\\[-2pt] &&\qquad\quad j=1,\ldots,d, k=1,\ldots,m \Bigr) \\[-2pt] &&\qquad=\mathbb{P} \Bigl(U_k \le\min_{r:A_{rk}=1}\{u_r \}, k=1,\ldots,m \Bigr) \mathbb{P} (V_j\le u_j \mathrm{I}_{\{A_j=0\}}+\mathrm{I}_{\{A_j=1\}},\\[-2pt] &&\qquad\quad j=1,\ldots,d ) \\[-2pt] &&\qquad=C^{\mathbf{U}}\bigl(\mathbf{u}_A^*\bigr) C^{\mathbf{V}}({ \mathbf{u}}_A).
\end{eqnarray*}
\upqed\end{pf}
As long as $C^{\mathbf{V}}$ has tail-dependence matrix $I_d$, the tail-dependence matrix of $\mathbf{Y}$ is not affected by the choice of $C^{\mathbf{V}}$. This theoretically provides more flexibility in choosing the body of the distribution of $\mathbf{Y}$ while attaining a specific tail-dependence matrix. Note, however, that this also depends on the choice of $X$; see the following example where we address special cases which allow for more insight into the rather abstract construction~\eqref{eq:cop:Y}.
\begin{example}\label{ex:special:cases}
1. For $m=1$, the copula $C$ in \eqref {eq:cop:Y} is given by
\begin{equation} C(\mathbf{u})=\sum_{\mathbf{A}\in\{0,1\}^d} C^{\mathbf{V}}(\mathbf {u}_{\mathbf{A}})C^{\mathbf{U}}\bigl(\mathbf{u}_{\mathbf{A}}^*\bigr)\mathbb{P} ( \mathbf{X}=\mathbf{A});\label{eq:case:m1} \end{equation}
note that $X,A$ in equation~(\ref{eq:cop:Y}) are indeed vectors in this case. For $d=2$, we obtain
\begin{eqnarray*} C(u_1,u_2)&=&M(u_1,u_2)\mathbb{P}\biggl( \mathbf{X}=\pmatrix{1\cr 1}\biggr)+C^{\mathbf {V}}(u_1,u_2)\mathbb{P}\biggl( \mathbf{X}=\pmatrix{0\cr 0}\biggr) \\ &&{}+\Pi(u_1,u_2)\mathbb{P}\biggl(\mathbf{X}=\pmatrix{1\cr 0} \mbox{ or }\mathbf{X}=\pmatrix{0\cr 1}\biggr), \end{eqnarray*}
and therefore a mixture of the Fr\'echet--Hoeffding upper bound $M(u_1,u_2)$ $=\min\{u_1,u_2\}$, the copula $C^{\mathbf{V}}$ and the independence copula $\Pi(u_1,u_2)=u_1u_2$. If $\mathbb{P}\bigl(\mathbf{X}=\bigl({0\atop 0}\bigr)\bigr)=0$ then $C$ is simply a mixture of $M$ and $\Pi$ and does not depend on $\mathbf {V}$ anymore.
Now consider the special case of \eqref{eq:case:m1} where $\mathbf {V}$ follows the $d$-dimensional independence copula $\Pi(\mathbf{u})=\prod_{i=1}^d u_i$ and $\mathbf{X}=(X_1,\ldots,X_{d-1},1)$ is such that at most one of $X_1,\ldots,X_{d-1}$ is 1 [each randomly with probability $0\le\alpha \le 1/(d-1)$ and all are simultaneously 0 with probability $1-(d-1)\alpha$]. Then, for all $\mathbf{u}\in[0,1]^d$, $C$ is given by
\begin{equation} C(\mathbf{u})
=\alpha\sum _{i=1}^{d-1} \Biggl(\min\{u_i,u_d \}\prod_{j=1, j\ne i}^{d-1}u_j \Biggr) + \bigl(1-(d-1)\alpha\bigr) \prod_{j=1}^{d}u_j.\label {ex:special:cases:FD} \end{equation}
This copula is a conditionally independent multivariate Fr\'echet copula studied in Yang et al. \cite{YQW09}. This example will be revisited in Section~\ref{sec:ex:qrm}; see also the left-hand side of Figure~\ref{fig:Gamma} below.
\begin{figure}
\caption{Scatter plots of 2000 samples from $\mathbf{Y}$ for $\mathbf {V}\sim\Pi$ and $\mathbf{U}$ following a bivariate ($m=2$) $t_3$ copula with Kendall's tau equal to 0.75 (top row) or a survival Marshall--Olkin copula with parameters $\alpha_1=0.25,\alpha_2=0.75$ (bottom row). For the plots on the left-hand side, the number of rows of $X$ with one 1 are randomly chosen among $\{0,1,2\ (=d)\}$, the corresponding rows and columns are then randomly selected among $\{1,2\ (=d)\}$ and $\{1,2\ (=m)\}$, respectively. For the plots on the right-hand side, $X$ is drawn from a multinomial distribution with probabilities 0.5 and 0.5 such that each row contains precisely one~1.}
\label{fig:Y:m2}
\end{figure}
2. For $m=2$, $d=2$, we obtain
\begin{eqnarray} \label{ex:C:d2:m2} C(u_1,u_2)&=&M(u_1,u_2) \mathbb{P}\biggl(X=\mat{1} {0} {1} {0} \mbox{ or } X=\mat {0} {1} {0} {1}\biggr) \nonumber \\ &&{}+C^{\mathbf{U}}(u_1,u_2)\mathbb{P}\biggl(X=\mat {1} {0} {0} {1}\biggr)+C^{\mathbf{U}}(u_2,u_1)\mathbb{P}\biggl(X=\mat{0} {1} {1} {0}\biggr) \nonumber \\[-8pt] \\[-8pt] \nonumber &&{}+C^{\mathbf{V}}(u_1,u_2)\mathbb{P}\biggl(X=\mat{0} {0} {0} {0}\biggr)\\ &&{}+\Pi (u_1,u_2)\mathbb{P}\biggl(X=\mat{0} {0} {1} {0} \mbox{ or } \mat{0} {0} {0} {1} \mbox { or } \mat{1} {0} {0} {0} \mbox{ or } \mat{0} {1} {0} {0}\biggr).\hspace*{-25pt}\nonumber \end{eqnarray}
Figure~\ref{fig:Y:m2} shows samples of size 2000 from \eqref {ex:C:d2:m2} for $\mathbf{V}\sim\Pi$ and two different choices of $\mathbf{U}$ (in different rows) and $X$ (in different columns). From Theorem~\ref{thm:main:tdm:Y}, we obtain that the off-diagonal entry $\gamma_{12}$ of the tail-dependence matrix $\Gamma $ of $\mathbf{Y}$ is given by
\[ \gamma_{12}=p_{(1,2)(1,1)}+p_{(1,2)(2,2)}+\lambda _{12}(p_{(1,2)(2,1)}+p_{(1,2)(1,2)}), \]
where $\lambda_{12}$ is the off-diagonal entry of the tail-dependence matrix $\Lambda$ of $\mathbf{U}$.
\end{example}
\subsection{An example from risk management practice}\label{sec:ex:qrm} Let us now come back to problem~\eqref{eq:Gammad} which motivated our research on tail-dependence matrices. From a practical point of view, the question is whether it is possible to find one financial position, which has tail-dependence coefficient $\alpha$ with each of $d-1$ tail-independent financial risks (assets). Such a construction can be interesting for risk management purposes, for example, in the context of hedging.\looseness=-1
Recall problem~\eqref{eq:Gammad}:
\begin{quote} \textit{For which $\alpha\in[0,1]$ is the matrix
\begin{eqnarray} \Gamma_d(\alpha)=\pmatrix{ 1 & 0 & \cdots& 0 & \alpha\vspace*{2pt} \cr 0 & 1 & \cdots& 0 & \alpha\vspace*{2pt} \cr \vdots& \vdots& \ddots& \vdots& \vdots\vspace*{2pt} \cr 0 & 0 & \cdots& 1 & \alpha\vspace*{2pt} \cr \alpha& \alpha& \cdots& \alpha& 1 } \label{eq:Gammadrec} \end{eqnarray}
a matrix of pairwise (either lower or upper) tail-dependence coefficients?} \end{quote}
Based on the Fr\'echet--Hoeffding bounds, it follows from Joe \cite{J97}, Theorem~3.14, that for $d=3$ (and thus also $d>3$), $\alpha$ has to be in $[0,1/2]$; however, this is not a sufficient condition for $\Gamma_d(\alpha)$ to be a tail-dependence matrix. The following proposition not only gives an answer to \eqref{eq:Gammadrec} by providing necessary and sufficient such conditions, but also provides, by its proof, a compatible model for $\Gamma_d(\alpha)$.
\begin{proposition}\label{prop:example} $\Gamma_d(\alpha) \in\mathcal{T}_d$ if and only if $0\le\alpha\le1/(d-1)$. \end{proposition}
\begin{pf} The if-part directly follows from Corollary~\ref{coro:ddm}. We provide a constructive proof based on Theorem~\ref{thm:main:tdm:Y}. Suppose that $0\le\alpha\le1/(d-1)$. Take a partition $\{\Omega_1,\ldots,\Omega_{d}\}$ of the sample space $\Omega$ with $\mathbb{P}(\Omega_i)=\alpha$, $i=1,\ldots,d-1$, and let $\mathbf{X}=(\mathrm{I}_{\Omega_1},\ldots,\mathrm{I}_{\Omega_{d-1}}, 1)\in\mathcal{V} _d$. It is straightforward to see that\looseness=-1
\[ \mathbb{E}\bigl[\mathbf{X}\mathbf{X}^\top\bigr]=\pmatrix{ \alpha& 0 & \cdots& 0 & \alpha\vspace*{2pt} \cr 0 & \alpha& \cdots& 0 & \alpha\vspace*{2pt} \cr \vdots& \vdots& \ddots& \vdots&\vdots \vspace*{2pt} \cr 0 & 0 & \cdots& \alpha& \alpha \vspace*{2pt} \cr \alpha& \alpha& \cdots& \alpha& 1 }. \]
By Proposition~\ref{coro-b}, $\Gamma_d(\alpha)=\mathcal{L}(\mathbb{E}[\mathbf {X}\mathbf{X}^\top])\in\mathcal{T}_d$.
For the only if part, suppose that $\Gamma_d(\alpha) \in\mathcal{T}_d$; thus $\alpha\ge0$. By Theorem~\ref{thm:main:characterization}, $\Gamma _d(\alpha)\in \mathcal{B}^I_d$. By the definition of $\mathcal{B}^I_d$, $\Gamma_d(\alpha)=B_d/p$ for some $p\in(0,1]$ and a Bernoulli-compatible matrix $B_d$. Therefore,
\[ p\Gamma_d(\alpha)=\pmatrix{ p & 0 & \cdots& 0 & p\alpha\vspace*{2pt} \cr 0 & p & \cdots& 0 & p\alpha\vspace*{2pt} \cr \vdots& \vdots& \ddots& \vdots& \vdots \vspace*{2pt} \cr 0 & 0 & \cdots& p & p\alpha\vspace*{2pt} \cr p\alpha& p \alpha& \cdots& p\alpha& p } \]
is a compatible Bernoulli matrix, so $p\Gamma_d(\alpha)\in\mathcal{B}_d$. Write $p\Gamma_d(\alpha)=\mathbb{E}[\mathbf{X}\mathbf{X}^\top]$ for some $\mathbf{X}=(X_1,\ldots,X_d)\in\mathcal{V}_d$. It follows that $\mathbb{P}(X_i=1)=p$ for $i=1,\ldots, d$, $\mathbb{P}(X_iX_j=1)=0$ for $i\ne j$, $i,j=1,\ldots,d-1$ and $\mathbb{P}(X_iX_d=1)=p\alpha$ for $i=1,\ldots,d-1$. Note that $\{X_iX_d=1\}$, $i=1,\ldots,d-1$, are almost surely disjoint since $\mathbb{P} (X_iX_j=1)=0$ for $i\ne j$, $i,j=1,\ldots,d-1$. As a consequence,
\[ p=\mathbb{P}(X_d=1)\ge\mathbb{P} \Biggl( \bigcup_{i=1}^{d-1} \{X_iX_d=1\} \Biggr)=\sum_{i=1}^{d-1} \mathbb{P}(X_iX_d=1)=(d-1)p\alpha, \]
and thus $(d-1)\alpha\le1$. \end{pf}
It follows from the proof of Theorem~\ref{thm:main:tdm:Y} that for $\alpha\in[0,1/(d-1)]$, a compatible copula model with tail-dependence matrix $\Gamma_d(\alpha)$ can be constructed as follows. Consider a partition $\{\Omega_1,\ldots,\Omega_{d}\}$ of the sample space $\Omega$ with $\mathbb{P}(\Omega_i)=\alpha$, $i=1,\ldots,d-1$, and let $\mathbf{X}=(X_1,\ldots,X_d)=(\mathrm{I}_{\Omega_1},\ldots,\mathrm{I}_{\Omega _{d-1}}, 1)\in\mathcal{V}_d$; note that $m=1$ here. Furthermore, let $\mathbf{V}$ be as in Theorem~\ref {thm:main:tdm:Y}, $U\sim\mathrm{U}[0,1]$ and $U,\mathbf{V},\mathbf{X}$ be independent. Then
\[ \mathbf{Y}=\bigl(UX_1+(1-X_1)V_1, \ldots,UX_{d-1}+(1-X_{d-1})V_{d-1},U\bigr) \]
has tail-dependence matrix $\Gamma_d(\alpha)$. Example~\ref {ex:special:cases}, part~1 provides the copula $C$ of $\mathbf{Y}$ in this case. It is also straightforward to verify from this copula that $\mathbf{Y}$ has tail-dependence matrix $\Gamma_d(\alpha)$.
Figure~\ref{fig:Gamma} displays pairs plots of 2000 realizations of $\mathbf{Y}$ for $\alpha=1/3$ and two different copulas for $\mathbf{V}$.
\begin{figure}
\caption{Pairs plot of 2000 samples from $\mathbf{Y}\sim C$ which produces the tail dependence matrix $\Gamma_4(1/3)$ as given by \protect\eqref {eq:Gammad}. On the left-hand side, $\mathbf{V}\sim\Pi$ [$\alpha$ determines how much weight is on the diagonal for pairs with one component being $Y_4$; see \protect\eqref{ex:special:cases:FD}] and on the right-hand side, $\mathbf{V}$ follows a Gauss copula with parameter chosen such that Kendall's tau equals 0.8.}
\label{fig:Gamma}
\end{figure}
\begin{remark} Note that $\Gamma_d(\alpha)$ is not positive semidefinite if and only if $\alpha>1/\sqrt{d-1}$. For $d<5$, element-wise nonnegative and positive semidefinite matrices are completely positive; see Berman and Shaked-Monderer \cite{BS03}, Theorem~2.4. Therefore, $\Gamma_3(2/3)$ is completely positive. However, it is not in $\mathcal{T}_3$. It indeed shows that the class of completely positive matrices with diagonal entries being 1 is strictly larger than $\mathcal{T}_d$. \end{remark}
\section{Conclusion and discussion}\label{sec:con} Inspired by the question whether a given matrix in $[0,1]^{d\times d}$ is the matrix of pairwise tail-dependence coefficients of a $d$-dimensional random vector, we introduced the tail-dependence compatibility problem. It turns out that this problem is closely related to the Bernoulli-compatibility problem which we also addressed in this paper and which asks when a given matrix in $[0,1]^{d\times d}$ is a Bernoulli-compatible matrix (see Question~\ref{Q:B:compatibility} and Theorem~\ref{thm:bern}). As a main finding, we characterized tail-dependence matrices as precisely those square matrices with diagonal entries being 1 which are Bernoulli-compatible matrices multiplied by a constant (see Question~\ref{Q:characterization} and Theorem~\ref{thm:main:characterization}). Furthermore, we presented and studied new models (see, e.g., Question~\ref{Q:model:for:Y} and Theorem~\ref{thm:main:tdm:Y}) which provide answers to several questions related to the tail-dependence compatibility problem.
The study of compatibility of tail-dependence matrices is mathematically different from that of covariances matrices. Through many technical arguments in this paper, the reader may have already realized that the tail-dependence matrix lacks a linear structure which is essential to covariance matrices based on tools from linear algebra. For instance, let $\mathbf{X}$ be a $d$-random vector with covariance matrix $\Sigma$ and tail-dependence matrix $\Lambda$, and $A$ be an $m\times d$ matrix. The covariance matrix of $A\mathbf{X}$ is simply given by $A\Sigma A^\top$; however, the tail-dependence matrix of $A\mathbf{X}$ is generally not explicit (see Remark~\ref{transition} for special cases). This lack of linearity can also help to understand why tail-dependence matrices are realized by models based on Bernoulli vectors as we have seen in this paper, in contrast to covariance matrices which are naturally realized by Gaussian (or generally, elliptical) random vectors. The latter have a linear structure, whereas Bernoulli vectors do not. It is not surprising that most classical techniques in linear algebra such as matrix decomposition, diagonalization, ranks, inverses and determinants are not very helpful for studying the compatibility problems we address in this paper.
Concerning future research, an interesting open question is how one can (theoretically or numerically) determine whether a given arbitrary nonnegative, square matrix is a tail-dependence or Bernoulli-compatible matrix. To the best of our knowledge there are no corresponding algorithms available. Another open question concerns the compatibility of other matrices of pairwise measures of association such as rank-correlation measures (e.g., Spearman's rho or Kendall's tau); see \cite{EMS02}, Section~6.2. Recently, \cite{FSS14} and \cite {SBS15} studied the concept of tail-dependence functions of stochastic processes. Similar results to some of our findings were found in the context of max-stable processes.
From a practitioner's point-of-view, it is important to point out limitations of using tail-depen\-dence matrices in quantitative risk management and other applications. One possible such limitation is the statistical estimation of tail-dependence matrices since, as limits, estimating tail dependence coefficients from data is nontrivial (and typically more complicated than estimation in the body of a bivariate distribution).
After presenting the results of our paper at the conferences ``Recent Developments in Dependence Modelling with Applications in Finance and Insurance---2nd Edition, Brussels, May 29, 2015'' and ``The 9th International Conference on Extreme Value Analysis, Ann Arbor, June 15--19, 2015,'' the references \cite{FSS14} and \cite{SBS15} were brought to our attention (see also Acknowledgments below). In these papers, a very related problem is treated, be it from a different, more theoretical angle, mainly based on the theory of max-stable and Tawn--Molchanov processes as well as results for convex-polytopes. For instance, our Theorem~\ref{thm:main:characterization} is similar to Theorem~6(c) in \cite{FSS14}.
\section*{Acknowledgments} We would like to thank Federico Degen for raising this interesting question concerning tail-dependence matrices and Janusz Milek for relevant discussions. We would also like to thank the Editor, two anonymous referees, Tiantian Mao, Don McLeish, Johan Segers, Kirstin Strokorb and Yuwei Zhao for valuable input on an early version of the paper.
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Hi I'm Ben and welcome to this Microsoft Office 365 QuickStart tutorial on working with SharePoint documents using the office online applications as part of Microsoft Office 365 you will usually have access to what is called the Microsoft Office online applications this gives you access to use versions of the Microsoft Office suite such as Word and Excel purely from within a web browser without needing the applications installed on your computer the online versions don't have all the functionality of the full versions however for most purposes they do provide all of the features you would need the office online applications can be super convenient for quick edits co-authoring and working on your documents when using someone else's computer to edit a document using word online first of all locate your file in your SharePoint document library using your web browser to do this open your web browser of choice I'm using Google Chrome and into the address of your SharePoint site this is usually in the format of your business name followed by a SharePoint calm for example my SharePoint site is grassroots IT SharePoint calm if prompted log in with your usual username and password once logged on to your SharePoint site navigate through to find the document that you're interested in you can find the document library in question here I'm going through - one called marketing in community into a folder in my case this is a folder called blog and in here we've got some documents that we can choose to work with I'm going to click on this first document here five signs you need a managed service provider when you click on the document name it will open in word online in a read-only mode you can then click on the edit document menu from up the top here and choose to either edit in word if you do have the full version of Microsoft Word installed on your computer or edit in word online we're going to select editing word online word online will then open the document in a full editable mode as you can see it looks just like the full version of Microsoft Word that you will be familiar with or without quite as many menus and options when working in word online your changes are automatically saved and when you have finished editing simply select file and then exit to return to your document library in this example we've used word online to open and then edit a document keep in mind that word online is only one of the tools available you also have access to Excel online and a number of the other Microsoft Office applications in an online format ready for you to use.
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What are the benefits of using LaTeX over MS Word, especially for a scientific researcher doing a lot of biology and mathematics?
As well as the general benefits others have mentioned I'll try and show you why LaTeX can be better from my own work.First off is general appearance. By default, LaTeX just looks nicer. It's easier to format professionally whereas MS Word takes a lot of effort to look good.This first report was done on Word and took a long time to get a title page that looked generally decent. Trying to put an image in and making things align well usually ends in disaster. In contrast…The one on LaTeX looks a lot more professional. It was easier to make and I never had to worry about alignment issues as you just say \centering or a similar command.I much prefer the two column format to my reports but do you know how difficult that is in Word? Trying to put an image in is a nightmare and will usually result in ruining the format of everything else. And it's almost impossible to change from two columns back to one.In LaTeX you just type in \begin{multicols}{2} or something similar and it works fine. The text also looks nicer and it's much easier to change the margins. You can also easily add in a nice contents page before that corresponds to all the names of the sections.Word does have a similar thing but it's really not up to standard that I just don't bother.Then there's references. I've tried references on Word before but it doesn't automatically order them, it doesn't have square brackets and there's not enough customability in their layout. I usually have to do them manually. Likewise for figure, table and equation numbers. The labeling system on LaTeX is sooooooo much nicer.The reference section at the end doesn't look as good on Word.It just looks basic and unprofessional.This looks a lot nicer.Finally we come to equations. On Word, something like this is a challenge.Thank God for copy and paste!In LateX this would be child's play. In fact something like this can be written out easily too:You also have much more control rather than having to depend on the options word gives you.There are other benefits too such as the Fix package that allows you to draw images such as Feynman diagrams in a vector format. These are much nicer than the shape tool on Word.So yeah, use LaTeX! You can also write out the awesome equations you learn on Quora!
How do you type fancy letters?
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Is LaTeX dead? If yes, what are some modern alternatives?
No. It is absolutely not dead. It still is the main tool for writing academic texts and it will be for a long time. But I would argue that [math]\LaTeX[/math] should die, let me explain why.First of all, let me differentiate between [math]\TeX[/math] and [math]\LaTeX[/math]. The first is a computational typography language/system developed by Don Knuth in the 70s. As such, [math]\TeX[/math] is the backend for [math]\LaTeX[/math], the typesetting language designed by Leslie Lamport in the 80s for easily producing documents with [math]\TeX[/math]. On my opinion [math]\TeX[/math] is beautiful and we have no reason to replace it. On other hand, [math]\LaTeX[/math] is a technology from another age which is not aligned with the best known practices of designing expert user-facing document production software. And no, I ain't comparing it with MS Word. I'm comparing it with other expert user-facing computer languages. 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This leads to a series of shortcomings I want to pinpoint:[math]\LaTeX[/math] mixes global formatting and document structure in the same language. Those are two very different tasks and LaTeX is sub-optimal for both. On a typical web environment for example, the first is done by CSS and the second by HTML (not that those are two particularly good languages, but separating those concerns is a good idea).The way you annotate formatting and document structure in [math]\LaTeX[/math] is intrusive and gets in the way of you actually reading the text while you type it. This is similar to what happens in HTML versus the seamless and natural annotations used by the Markdown standard or the Wikipedia markup standard. A typical file should be content centric, not annotation centric. The most prominent feature in the file should be the text itself, not the structure marks you use to format it. Note the difference between those options:Bold face formatting:\textbf{some bold text} ([math]\LaTeX[/math])bsome bold text/b (HTML)**some bold text** (Markdown)'''some bold text''' (Wikipedia)Creating a new section:\section{New Section} ([math]\LaTeX[/math])# New Section (Markdown)On [math]\LaTeX[/math] you have to decide the global structure of a document before hand (typically by choosing a documentclass. Once the document reaches a certain size, changing the documentclass is almost impossible. Once you separate formatting concerns, content structure concerns, and global presentation and rendering concerns, one should be able to deal with this almost transparently. When you're organizing the content structure of a document it shouldn't matter whether it's a book, a Physical Review Letters article, a tufte-latex handout, or whatever. 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Do you know an application that allows to easily create a document template, as it were in MS Word, and then get the LaTeX code?
I can offer a program in which you can type the formula as a string in the MS Word document or another text editor. The program analyzes the formula and generates a LaTeX string. This is a unique and very easy way to obtain a mathematical formula to LaTeX format. Among the unique features - validation of the formula with the issuance of diagnostic messages. Is it good to have the ability to use variable names that supports indexes, Greek characters, setting a range of values and change step, calculate and save result? That all are present. There are a lot of unique properties and it is easier try it to learn them yourself. Program has a good help and easy to use.Its name is "aneasycalc":The AnEasyCalc program by SteamAndWater labs
As a programmer, what tools do you use on a daily basis? (IDEs, code editors, etc.)
Sublime Text 3My baby. This is where I spend most of my time. I switched to Sublime from Notepad++ about 5 years ago, and since then I've tried most of the big competitors—Atom, Brackets, etc.—but none have been able to lure me away from Sublime. I'm continually impressed by how fast and smoothly it runs, even under heavy loads. The Command Palette and Package Control make it easy for me to find features I haven't yet memorized, or add new ones, and the plugins I've installed for FTP/SFTP transfer and git version control allow me to perform most basic tasks without leaving the app.I'm cross-platform, with a Windows PC and MacOS laptop, using Google Drive to keep my files synced. Sublime works (nearly) identically on both OSes, which means I can switch back and forth without hassle. I can save my project on Windows, open it in MacOS ten seconds later, and everything's right where I left it: same files, same folders, even the cursor is in the same position. I use and highly recommend the Material Theme plugin, which really polishes up the interface.Atom has gotten very close, and I could probably get by on it for a few days before I started seriously pining. There's a lot I like about it: the fully unified command menu, vastly superior management of profiles, preferences and plugins, much better git integration. But for me, Sublime still has a meaningful edge in performance and feature polish. Sublime never makes me wait, and very rarely gets in my way. Atom doesn't pass that test just yet. I also prefer the way Sublime handles projects and workspaces. Atom's "project = folder" approach is attractively simple, but doesn't cut it for some schemes.SourceTreeMy team uses Atlassian's BitBucket for version control, it's kind of a no-brainer to use their integrated client. And my eyes appreciate the new dark theme, even though it's still a bit on the hideous side.I've also tried the new hotness, GitKraken, and there's a lot I like about it. It's much prettier, the workflow is better conceived and more suited to modern development practices, , it feels more solid and stable in places where SourceTree tends to rattle a bit. Nonetheless, it's just not as capable (yet). SourceTree has better information density—I like being able to see the commit history and file diffs at the same time—and there are a lot of secondary and tertiary tasks, such as managing remotes, that GitKraken just doesn't do yet. The nice thing about git is that I can easily use them side-by-side on the same projects, so I'll be keeping an eye on GK as it develops. I can see it replacing SourceTree before long. But for now, SourceTree is the one that Gets The Job Done.PostmanIf you do any kind of work with REST APIs, I cannot recommend this guy enough. It's like Fiddler's "Compose" feature on steroids. They really thought of everything, and it's very clear that they use their own product. Almost every time I've thought "hey, you know what feature would be helpful here?" I've promptly found that it was not only included, but it was exactly where I'd expect it to be. Sometimes with a niche market you're stuck with a limited range of options, often there's one or two standards that everyone uses, and they get kind of complacent. So I have a lot of love for Postman, because they've really knocked it out of the park.Webkit InspectorA lot of people don't know about the surprisingly powerful developer tools that are built into their old vanilla web browser. Chrome and Safari both include Webkit's developer tools, which is super handy. I also use Firefox Developer Edition, and even Edge is (slowly) catching up. This obviously doesn't replace a full-fledged IDE, but for basic web development and maintenance on the scale my team works at, it's perfectly capable, and hence one less piece of software I need to install.WorkFlowyNot strictly a programming tool, but something I cannot live without. WorkFlowy is a deceptively simple concept: it's an infinite outliner. You can make bulleted outlines, and then zoom in on any individual item and work inside it. So any single bullet can become its own self-contained document, task list, outline, project hub, etc. It also has, of course, sharing and real-time collaboration. I've got my whole life in WorkFlowy, both personal and professional. It sits right next to my inbox with equal standing. It has virtually no learning curve—the concept is simple enough that you can very quickly get started and set it up however you want. If you're a hierarchal thinker like I am, I urge you to give it a try.KoalaKoala does one thing well: it watches a set of folders and automatically compiles your LESS, SASS/Compass, or CoffeeScript source files when it detects a change. In conjunction with Sublime's SFTP plugin, it makes my life a lot easier when I have to keep files in sync with a remote web server somewhere. All I have to do is save my changes, and both the compiling and syncing happen automatically, in the background. Since I tend to be a very iterative worker—make a tiny change, refresh the browser to see how it looks, go back to Sublime and make another tweak, rinse and repeat—I really value anything that cuts time, clicks and keystrokes out of my workflow.MacOSApple doesn't exactly need my endorsement, but I'd be remiss if I didn't acknowledge it as a major part of my toolbox. I started with Windows when I was five years old, I learned everything I know on PCs, and I never even used a Mac for an extended period of time until about two years ago. And, to be fair, it took that long to really get its hooks in me. It's only recently that I knew I was ready to switch to MacOS as my primary operating system. There are a lot of reasons, and a lot of them are a matter of aesthetics and personal preference, but there are a few that are relevant to this question:It has Apache, MySQL and PHP built-in. I have local mirrors of all my client sites right in my Documents folder, and I can edit them directly, without having to sync files by FTP or other means. I know you can do this just as easily with MAMP/XAMPP or a dozen other development bundles, but I like not having to pile too much third-party software onto my laptop.The Terminal has made me understand why so many of my developer friends actually like using the command line, which baffled me for so long. Windows' Command Prompt is a necessary evil, and PowerShell is a necessary improvement. Terminal is clean, friendly, a car that you'd actually want to drive.On a more fundamental level, MacOS is a Unix-based OS, which means it fits more naturally with the software, frameworks and (primarily Linux) server environments that I deal with as a web developer. The bash shell, SSH tunneling and keychain authentication, chmod file permissions, Ruby, Python—there are so many of these things that are so awkward and cumbersome on Windows, it feels like you're getting a room full of people who speak different languages to communicate with each other using charades and pictograms. On Mac/Linux they just freaking work.Google DriveGoogle doesn't need any help from me, either. But Drive is so fundamental to my digital life that it must be mentioned. I pay $2 a month—that's two dollars, not a typo—for 100 GB of storage, which is way more than I need, and it's enough to keep all of my stuff—all my documents, projects, photos, everything—synced and backed up in real time. I never have to worry about whether a file I need at home is on my work computer, or vice versa. More importantly, if my machine gets struck by lightning and vaporized in an instant, I won't lose more than the last few minutes of work. Whether it's Drive, Dropbox, OneDrive, iCloud, Box.com, or anything else, this kind of easy, seamless backup & sync is one of the delights of being a computer worker in 2017, and you really owe it to yourself to get one.Edited to remove some personal information from a screenshot.
How can I make my newly published novel "Going for Balm in Gilead" be known to people who are interested in immigration and migrant issues?
I'm very sorry, but User-12584786383630117302 is absolutely correct.You've been suckered. You might just as well have thrown all the money you spent on your book into a hole and set it on fire.Given how terrible the amazon page, the cover, and everything else is (and seriously, there are 11 year olds who could have done better) no one is going to take this book seriously, no matter what you do. In fact, I would venture to say that this vanity press did no editing, just did a rough conversion of your document to a .pdf file without bothering even to spellcheck, and sent it off to some cheap Chinese printing press.You cannot save this book. The only thing you can do is resign yourself to your losses start educating yourself on self-publishing and self-promotion, rewrite the book from scratch, hire a professional editor and book cover artist and self-publish it under another title.
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Determination of calcium by edta titration
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determination of calcium by edta titration To 0. spectrophotometric determination of calcium analytical. ) The given automated titration method can measure multi-parameters in one go. Nov 02, 2020 · The procedure for determination of water hardness was adapted from. The concentration of calcium ion in the blood is very important for the development and preservation of strong bones and teeth. EDTA is a versatile chelating agent. Both the total hardness and the individual calcium and magnesium hardnesses will be 20 hours ago · EDTA Titration: Calcium in Calcium Supplements Student Handout Purpose To determine the amount of calcium in a calcium supplement tablet by EDTA titration. A calcium ion selective electrode (ISE) is used as the titration indicator, which gives a potentiometric change when all the calcium has been complexed by the EDTA. Blank titration - determination of k and s. You will use EDTA complexometric titration to determine the hardness of a responsible for hardness, and total water hardness is defined as the sum of the. ICUMSA Method GS 8/2/3/4-9 (2000) Determination of Calcium in Sugar Products by EDTA Titration – Accepted Single Method, available as online version (2 One of the analytical technique to determine calcium oxide is complexometric titration with EDTA. Average values of 0. The student will use a 10-mL buret. 0 and the total calcium and magnesium is determined at pH 9. Volume Of Cacl 2 (V 1 ml) Burette Reading(ml) Volume of EDTA Initial Final Used(V 2 ml) 1 20 ml 0 2. In this case, an EDTA (ethylenediamine tetra-acetic acid) solution is the titrant, which will react or capture (also called chelate) the calcium ions in the water as shown in equation 1 below. determination of calcium in cinder block researchgate net. Determination of the Hardness of Tap Water 1. Once you know the volume of EDTA used for the titration, this value can be converted into milligrams of calcium carbonate or PPM (part per million) calcium carbonate, therefore determining the hardness of water. It turns red when it forms a complex with calcium, magnesium, or other metal ions. If EDTA is added as a titrant, the calcium and magnesium will be com-plexed, and when all of the magnesium and calcium has been complexed the solution turns from wine red to blue, marking the end point of the titration. pH. Mg 2+ + EDTA 4-→ MgEDTA 2-sample size. Vipulanandan 9 Nov 2017 Las soluciones de EDTA son valoradas comúnmente contra una solución estándar del ion Ca+2 que se prepara al disolver carbonato de calcio . Let the burette reading of EDTA be V 3 ml. EDTA Determination of Total Calcium and Determination of Individual Calcium and Magnesium. EDTA dissolved in water forms a colourless solution. In most water samples, calcium and magnesium are the chief contributors to water hardness. 5 by titration, using Solochrome BlackT as indicator, whilst a titrimetric macro method for calcium, in which Murexide is used as indicator" "there is some diagreement in the literature on the effect of phosphate in the determination of calcium in biological In-text: (Determination of Water Hardness By Complexometric Titration Class Notes, 2015) Your Bibliography: Homepages. With both types of method satisfactory accuracy and precision are attainable. In this titration standard EDTA solution is added to given sample containing metals using burette till the end point is achieved. The student will perform a complexometric titration using EDTA. The student will use the complexometric titration for analysis. 19 In brief, a 0. 01 M titrant and assuming 50 mL burette, aliquot taken for titration should contain about 0. While you are waiting for it to dry, prepare your EDTA solution, as in step 2. uring a titration with EDTA, the metal ion solution is in the flask and EDTA is added from the burette. 0 g of calcium carbonate standard and dry at 110°C for ~1 hr, or to constant weight. Ethylenediaminetetraacetic acid (EDTA) is used to complex the Co2+. Read about EDTA and its titration on pp. Dec 01, 1976 · Calcium is determined by titration with EGTA, calcium+magnesium+strontium by titration with EDTA and magnesium is obtained by difference. The reaction that takes place is the following: Ca 2+ + Y 4-<===> CaY 2-Before the equivalence point, the Ca 2+ concentration is nearly equal to the amount of unchelated (unreacted) calcium since the dissociation of the chelate is slight. Since most In a titration to determine the concentration of a metal ion, the added EDTA combines. Fabrics are boiled for 10-15 min with an excess of sodium-EDTA solution to extract Calcium Carbonate is a good primary standard as it has a relatively high molecular mass, is stable in air and water, and does not absorb water from the air, and has a high level of purity. Similarity of the relative standard deviation shows that both methods have similar degree of precision. Background Calcium is a mineral which is essential to the human body. The determination of these two elements by classical procedures (i. Digestion : see Method 2 under a. The method is based on the joint titration of calcium and magnesium with EDTA at pH ∼ 10 and the release of EDTA from the magnesium-EDTA calcium determination versene method comparison of magnesium determination methods as. A calcium ion selective electrode (ISE) is used as the titration indicator, which gives a potentiometric change when all the calcium and magnesium have been complexed by the EDTA. 140 ×10-3 M EDTA for titration. The sample is then titrated to the equivalence point using ethylenediaminetetraacetic acid (EDTA) titrant. very gradual change in color of the indicator over the course of the titration. 5 grams of sample add 10 cc's HCl and boil for several minutes. After dry ashing and treatment with concentrated nitric acid, calcium and magnesium were determined by Flame Atomic Absorption Spectrometry (FAAS) and complexometric titration with EDTA. DETERMINATION OF NICKEL BY DIRECT TITRATION WITH EDTA. water cations include calcium, magnesium, iron, zinc and the other polyvalent metal ions. 3 (if not, consult. The chelate, CaEDTA-2, is very stable so this reaction proceeds quantitatively from left to right. 00 mL aliquot of the solution required 24. If small amount of Eriochrome Balck is added to an aqueous solution containing Ca and Mg ions at pH 10 + 0. Instead of the NH 4 Cl /NH 4 OH buffer, we use a mixture of TRIS and acetylacetone. This reaction can be used to determine the amount of these minerals in a sample by a complexometric titration. , the well known Eriochrome. How many moles of CaCO3 were used? b. The titrations were carried Both magnesium and calcium can be easily determined by EDTA titration in the pH 10 against Eriochrome. SIONG, KHOR SWAN CHOO and SITI MIZURA SHAHID Division ofHuman Nutrition Institute for Medical Research 50588 Kuala Lumpur, Malaysia. It is. 2) Add 2. (iv) Repeat the titration till concordant readings are obtained. Hardness, Calcium, Oil and Gas (200,000 mg/L) 5 B. Calculate the hardness in terms of parts per million calcium carbonate. Lab 5 Copper Titration with EDTA YouTube. This titration gives the amount of calcium in the solution. Sep 10, 2017 · EDTA is insoluble in water at low pH because H4Y is predominant in that pH (less than 2). 06671 (standard deviation 0. Before coming to lab: • Record the analytical reaction EDTA Titration: Calcium in Calcium Supplements Student Handout Purpose To determine the amount of calcium in a calcium supplement tablet by EDTA titration. Calcein is a calcium-dependent fluorescent molecule. With the use of complexometric titration the total hardness of water sample was determined. Calcium analysis is a titration of a filtrate using EDTA reagent, plus a high pH buffer so that the magnesium ions are precipitated and only calcium ion is being analyzed. The simplest and most rapid method for use with ex- tracts of soils is the procedure of Cheng and Bray (3) in which calcium plus magnesium is determined in one. Prepare 500 mL of 0. Keywords: pitavastatin; calcium; EDTA; eriochrome black T; complexometric titration. After all the free calcium and magnesium ions are consumed, the EBT is replaced by EDTA from the unstable complex and liberates the free Erichrome Black-T. Background The presence in water of salts of calcium and magnesium is spoken of as Oct 29, 2020 · Calcium and Magnesium can affect muds in different ways, therefore, they may need to be analyzed separately. CaCO3 is a carbonate. The difference between the first and the second titration gives the magnesium content present in water. Synonym: (Ethylenedinitrilo)tetraacetic acid calcium disodium salt, EDTA calcium disodium salt, Edathamil Linear Formula: C 10 H 12 N 2 O 8 CaNa 2 Molecular Weight: 374. Other complexometric indicators are Eriochrome Black T for the titration of calcium and magnesium ions, and the chelating agent EDTA used to titrate metal ions in solution. Report the water hardness as ppm CaCO 3 This closer to the literature value of $0. Complexometric titration Wikipedia. Its usefulness arises because of its role as a chelating agent, i. 1. Determination of Calcium Oxide by Titration with a Chelating Ligand, Ethylenediamminetetraacetic Acid (EDTA) Ethylenediamminetetraacetic acid, more commonly known as EDTA, belongs to a class of synthetic compounds known as polyaminocarboxylic acids. EDTA Determination of Total Calcium and Determination of Individual Calcium and Magnesium Read about EDTA and its titration on pp. In this case it will be complexometric titration against EDTA (ethylenediamine tetraacetic acid). of aluminium and up to 6 mg. methodology for the determination of exchangeable calcium and magnesium in soil extracts by Ion Selective Calcium Electrode and Complexometric Titration (ISE-CT). Objective: The objective of this experiment is to practice the use of titration with the ethylenediaminetetraacetic acid in order to determine the quantitative properties of calcium supplements (mg Ca 2+ /tablet). Calcium and magnesium in the sample are titrated to the equivalence point using an ethylenediaminetetracetic acid (EDTA) titrant1,2,3. th. Determination Of Water Hardness By Complexometric Titration Class Notes. The EDTA complexes the Ca2+ or Mg2+ metal ion Ca2+ + (EDTA)4-⇄ Ca(EDTA)2-Note that the reaction has a 1:1 stoichiometry. One of the analytical technique to determine calcium oxide is complexometric titration wiIt isth EDTA. Determination of Zinc Oxide in Pharmaceutical Preparations by EDTA Titration: A Practical Class for a Quantitative Analysis Course. Endpoints in the titration are detected using indicators that change color when they complex with mineral ions. In fact 1. 800 M EDTA Titration Cartridge for each 0. The main constituent of the cement and clinker is calcium oxide (CaO) which is the major factor for cement quality. Dec 24, 2013 · This paper has been conducted based on the MSc thesis entitled: "Design of a simple direct colorimetric method for calcium determination in EDTA treated plasma" in which a novel method has been presented, patented on 29. Two reagents, calmagite and EDTA, are used to perform the determinations. The titration curve is displayed on the recording device. It is mainly required to study bone metabolism under in vivo conditions and helps in the staining of pit area under in vitro conditions. By EDTA titration: for the milk samples: 140. 10 M EDTA until a color change from red to blue occurs. This bulletin describes the determination of calcium, magnesium, and alkalinity in water by complexometric titration with EDTA as titrant. Simple Water and Complexes 4. If the sample solution initially contains also other metal ions, one should first remove or mask them, as EDTA react easily with most of the cations (with the exception of alkali metals). 01114 M. 8 mL min-1 and sample volume is 1. Determination of Permanent hardness Take 100 ml of sample hard water in 250 ml beaker. Calcium is determined by titration with 5 mM EDTA in 0. 5 Eriochrome Black-T(EBT) is the metal ion indicator used in the determination of hardness by complexometric titration with EDTA. Black T. Objective Apply the concept of complexometric titration in the determination of total hardness in drinking water 3. 310-318; 325-328; 853e in your text. Experimental: Preparation of EDTA Solution Titration of the test sample with EDTA using ammonium purpureate as an indicator is done to determine the calcium content of the sample. determination of calcium and magnesium in milk. After standardizing the EDTA, the average molarity was found to be 0. INVESTIGATIONS OF THE DETERMINATION OF CALCIUM IN THE. The determination of calcium in natural waters with EDTA has been described (Betz and No11 1950). 4 and titrated with EDTA in the presence of Patton and Reeder's indicator and sodium tartrate. ) which contained calcium carbonate (CaCO 3) as the active ingredient, with the intent of determining how much calcium ion the tablet contained by titration of samples from the tablet with EDTA. Separate determination of ethylenediaminetetraacetic acid and its calcium chelate in foods by colorimetry. org quantity QUANTITATIVE DETERMINATION OF TOTAL HARDNESS IN DRINKING WATER BY COMPLEXOMETRIC EDTA TITRATION In the below photo you can see that in its deprotonated form, Eriochrome Black T is blue. In the first titration the solution was titrated with 0. 2. The second titration to the first EQP with 0. The reaction that takes place is the following: \[Ca^{2+} + Y^{4-} \rightleftharpoons CaY^{2-}\] Calcium ions can be analyzed by titration with EDTA using an appropriate indicator. Black T was used as indicator. EDTA is then titrated with calcium carbonate to a permanent purple end point. EDTA is a polydentate ligand. icumsa. Journal of Chemical Education 2020 , 97 (2) , 522-527. Filter off and wash well with hot NH4Cl solution (2 parts water to 1 part NH4Cl). Are carbonates water soluble? Do egg shells dissolve easily in water ? Worked example: Determining solute concentration by acid-base titration · Titration of a strong acid where Ca is concentration of acid and Va is volume of acid Of these three ions, calcium can be quantified in a relatively simple and selective manner by complexometric titration (Belcher et al. 025 mol/L EDTA to the first equivalence point (EQP), from which the calcium content was calculated. Determination of Calcium in Foods by the Atomic Absorption Spectrophotom~tricand TitriInetric Methods TEE E. The results were compared to those obtained by conventional analytical techniques such as Complexometric Titration (CT), Flame Spectrometry Atomic Absorption (FAAS) and even Firstly, the total calcium content is determined, 0. equivalence point is reached when moles of a standard solution (titrant) equals moles of a solution of unknown concentration 10. The Determination of Calcium in Milk by EDTA Titration - Free download as Word Doc (. To 2 ml serum 18 ml water and 1% NaOH to pH 12 were added; Ca was estimated by titration with ethylenediamine tetra-acetate with Cal-Red as indicator. Both the total hardness and the individual calcium and magnesium hardnesses will be A simple complexometric method has been developed for the determination of magnesium directly and of calcium by difference at pH ∼ 10, using EDTA in a single aliquot of solution. The use of murexide as an indicator for complexometric determination of calcium, copper and nickel is well known [1]. 5% of the human body is made up of calcium, and not just the obvious uses such as bone and teeth formation but it is also a vital factor in many enzyme reactions, for example blood clotting. COMPLEXOMETRIC DETERMINATION OF CALCIUM WITH EDTA In this experiment you will determine the concentration of Ca+2 in an unknown solution by complexing it with an excess of EDTA. 0 Calculation and Reporting a. 30 ± 0. 00004) for Ca:Cl‰ and 0. Jul 31, 2017 · The titration is performed by adding a standard solution of EDTA to the sample containing the Ca. Eriochrome black T was used as the indicator at pH = 10 in both the titrations. The endpoint detection in complexometric titration can be done by two methods. This buffer solution allows separation between Calcium and Magnesium when a Calcium selective electrode is used as measuring electrode Standardization of EDTA will be performed first and then the determination of calcium concentration. 5. The salt was dissolved Complexometric titrations with EDTA have been reported for the analysis of nearly all metal ions. 00 ± 0. The calcium is precipitated as calcium oxalate CaC2O4. After sufficient EDTA has been added to complex all the magnesium and calcium, the solution will turn from wine red to blue. 1, the solution becomes wine red. Perhaps the most important feature of complexometric titrimetry, which has contributed to its popularity, is its speed. 0 mL in a volumetric flask. T. EDTA forms very stable, soluble, stoichiometric 1:1 complexes with almost all ions. notube. Key words: Calcium in foods, atomic absorption spectrophotometry, potassium permanganate titration. Adding the EDTA solution to the calcium, standard or unknown, is done by titration. 5 2 20 ml 2. 3 to 0. Reactions taking place aqueous solution containing calcium and magnesium ions at a pH of 10. pared by dilution as required. 26 Jan 2016 Find more similar flip PDFs like Calcium Analysis by EDTA Titration - CCRI. 27 Determination of Water Hardness using Complexometric titration You will use EDTA complexometric titration to determine the hardness of a sample of water brought from your home. Calcium and magnesium are easily measured by titration with the complexing agent ethylene-diaminetetraacetate (EDTA). @article{osti_4389108, title = {THE DETERMINATION OF FLUORIDE BY TITRATION WITH CALCIUM CHLORIDE}, author = {Belcher, R and Clark, S J}, abstractNote = {}, doi = {10. calcium ions changing colour from blue to pink/red in the process, but the dye–metal ion complex is less stable than the EDTA–metal ion complex. chemistry 321: quantitative analysis lab webnote edta titration for determination of calcium and magnesium before attempting this experiment, you may need to. Both the total hardness and the individual calcium and magnesium hardnesses will be measured. 04. 025 mol/L EDTA allowed the barium content and hence the sulfate content to be determined. , in CH 321. It is grouped into two parts, the potentiometric determination and the photometric determination. 25 for magnesium, for the water melon samples: 111. Since EDTA is insoluble in water, the disodium salt of EDTA is taken for this experiment. It was found out that the water hardness of Viva mineral water is classified as "hard" in terms of calcium and magnesium ions content that was expressed in terms of ppm CaCO3. 00 ml 10 mM NaCl instead of the bicarbonate sample. If EDTA is then added as a titrant, the calcium and magnesium will be complexed. The EDTA solution is standardized by titration of aliquots of the standard zinc solution. Determination of Calcium by Titration with EDTA CHEM 334 Quantitative Analysis Laboratory Determination of Calcium by EDTA Titration Introduction Complexometric titration is a form of volumetric analysis in which the formation of a colored complex is used to indicate the equivalence point of a titration. Calcium ions react with the Based on the fact of determination of Mn from a solution of pH 4. Jan 04, 2012 · Lab 2: Determination of Calcium in dietary supplement Tablets by EDTA Titration. Background. Nov 26, 2009 · Ethylenediaminetetraacetate (EDTA) complexes with numerous mineral ions, including calcium and magnesium. 00, using a NH 4Cl /NH 4OH buffer and a colorimetric detection of the equivalent point. Eriochrome Black T will be used as an indicator. Titrate the solutions with 0. 34 Titration curves illustrating how we can use the titrand's pH to control EDTA's selectivity. View EDTA Titration. The EDTA titration method gives good This standard calcium solution, of known concentration, is used to determine the exact concentration of the EDTA solution. Applicability This procedure is applicable to the determination of calcium or bone in meat and poultry products. Titration of another test sample with EDTA using Eriochrome Black T as indicator is done to determine the total calcium and magnesium content. Mar 19, 2006 · The amount of EDTA used in the tirtation is proportional to the amount of calcium and magnesium present. The part that containing the calcium and magnesium ions is reacted with an excess of EDTA. The purpose of this experiment is to determine the total hardness of water at the university's Chemistry Laboratory using a standardized titrant, ethylenediaminetetraacetic acid, by applying the principles of complexometric and displacement calcium, sodium hydroxide is added to a sample to raise the pH to 12 to 13. Ideally, pH should be ≥ 10. For 0. 1 1. Method 4: Sulphuric acid and peroxide digestion followed by flame photometry a. Ch 11 EDTA Titrations University of Windsor. Add 40 cc's NH4Cl solution and 15 cc's NH4OH and bring to a boil. Calcium is titrated in the presence of magnesium at pH 12. 010 M EDTA. 5 4 1. PROCEDURE Good planning says to do the digestion at the same time that the gravimetric samples are done. An internal indicator, for example The results obtained by ICP-AES are well consistent with the results determined by the traditional EDTA titration and weigh method, and suggest that the ICP-AES analysis is a simple, fast and accurate method for simultaneous determination of calcium and phosphorous in calcium phosphate based bioceramics. In this experiment a solution of EDTA will be standardize by titration against a standard solution made from calcium carbonate, CaCO3. Direct Titration. Although neither the EDTA titrant nor its calcium and magnesium complexes are col-ored, the end point of the titration can be visually detected by adding a metallochromic indicator to the water sample. j Record the final buret reading. Both magnesium and calcium can be easily determined by EDTA titration in the pH 10 against Eriochrome Black T. H2O by treating a hot hydrochloric acid solution with ammonium oxalate, and slowly neutralizing with aqueous ammonia solution. This application note also uses EDTA titration with a potentiometric determination of the equivalent This bulletin describes the determination of calcium, magnesium, and alkalinity in water by complexometric titration with EDTA as titrant. The amount of calcium ion in a dietary calcium supplement will be determined by titration with the standard ethylenediaminetetraacetic acid solution (EDTA). Determination of Calcium as. of manganese, 3 mg. Oxalate. Just like during determination The determination of total hardness of water can be made quickly and accurately by titration with ethylenediaminetetraacetic disodium salt (EDTA). 35-0. Complexometric Titration. ICP-AES in the determination of water hardness Annika Larson . Erlenmeyer flask 4. The "harder" the water (the more minerals are dissolved in it) the quicker mineral deposits build-up in water pipes and tanks that it runs through, requiring more neutralization reaction as in the acetic acid titration. 3) Titrate with standard 0. The precipitate is washed with dilute ammonium oxalate solution Weighed in one of the An indirect back-titration method for the determination of (Ca + Mg) in flameproofed cotton fabrics is developed. Aug 09, 2013 · Take 50. (EDTA) titrant. The use of the EDTA methods leads to greater speed and convenience but practice in recognizing the correct titration end-points is necessary. By gravimetric method Discussion. It derives its stability in part to calcium being connected to the EDTA by six bonds. 2012 in State Organization for Registration of Deeds and Properties with registration number of 74804. pdf from ENGLISH CO 102 at Rutgers University. Add 6 drops of indicator and 3 mL of buffer solution. 10 M Mg/EDTA solution and 4 to 5 drops of Erio T indicator. 0 mL of 0. calcium-analysis-by-edta-titration 1/1 Downloaded from www. The Complex solutions were formed by titration with the chelating agent EDTA. With increasing the pH, each hydrogen ion in the carboxyl groups of EDTA will start to dissociate. One of the most common methods for determination of endpoint owing to its simplicity, least cost and Feb 04, 2012 · Quantitative Determination of Total Hardness in Drinking Water by Complexometric EDTA Titration 1. Page 3. +2. … Nov 10, 2015 · EDTA, the titrant, complexes with Magnesium and Calcium ions, removing them from association with the indicator. A simple colorimetric method is described for the separate determination of ethylenediaminetetra acetic acid (free EDTA) and its calcium chelate (Ca-EDTA) in commercial foods. Determination of zinc in the presence of copper cadmium. A chelating agent is a substance whose molecules can form several bonds to a single metal ion. Calcium and Magnesium ion concentration determination with EDTA titration. 4) Indirect titration. reaction. International Journal of Advanced Research in. 2-0. If the sample solution initially contains also Hardness is defined as calcium and magnesium ion content. 45 millimoles of magnesium Apr 01, 2014 · The calcium in the water will be measured by performing a titration with EDTA. of phosphate in 50 ml. Determination of calcium by Standardized EDTASolution. 03020~\mathrm{L}$, which resulted in a solubility of $0. Experimental: Preparation of EDTA Solution Determination of Water Hardness using Complexometric titration You will use EDTA complexometric titration to determine the hardness of a sample of water brought from your home. It is similar to acid-base titration technique. EDTA titrant and a standard Zn 2+ solution are prepared. 01 M. The indicator is added and remains blue as all The classic method of determining calcium and other suitable cations is titration with a standardized solution of ethylenediaminetetraacetic acid (EDTA). St S, Moorhead, MN 56562 . Blood serum calcium was determined by both methods. 3/1 mol EDTA x MM CaCO. For the titration, the indicator is For the titration, the sample solution containing the calcium and magnesium ions is reacted with an excess of EDTA. Burette 2. At the Direct semi-automatic and automatic derivative potentiometric EDTA procedures for the determination of calcium and/or magnesium with a calcium-selective electrode are described. Related Applications: Determination of total hardness of water by complexometric titration of calcium and magnesium with EDTA at pH 10 using a Phototrode and Erio T color indicator. This titration must be completed in less than 5 minutes to minimize precipitation of calcium. EDTA has Water hardness can be readily determined by titration with the chelating agent EDTA. Sep 20, 2017 · The Determination of Calcium in Milk by EDTA Titration Description Introduction/Theory: Essential to the proper formation of teeth and bones calcium (Ca) is a mineral that can be found in cereals fruits and vegetables. EDTA is a chelating agent that binds to metals through four carboxylic acids. Calcein is used for the fluorometric determination of calcium and EDTA (ethylenediaminetetraacet ic acid) titration of calcium. PROCEDURE: Solutions needed: 1. The edta reagent can be used to measure the total quantity of dissolved Ca 2+ and Mg 2+ ions in a water sample. Then the water color change from wine red to blue that In the determination of water hardness, ethylene-diaminetetraacetic acid (EDTA) is used as the titrant that complexes Ca2+ and Mg2+ ions. For the titration, the indicator is added to the sample solution Total hardness is usually measured in one of two ways. Write the equations for the endpoint color change of Eriochrome Black T (EBT) with Mg+2 3. The presence in water of salts of calcium and magnesium is spoken of as . Standardization of EDTA will be performed first and then the determination of calcium concentration. •In this experiment, The determination of calcium in milk is based on a complexometric titration of calcium with an aqueous solution of the disodium salt of EDTA at high pH value (12). This method requires pH adjustment for good results. 0. EDTA or Ethylenediaminetetraacetic acid is commonly used as an indicator for complexometric titration because it can act as a ligand which can bind to , so that the Calcium is titrated first before the Magnesium. Aug 01, 2016 · Solutions of EDTA are prepared from the soluble disodium salt, Na 2 H 2 Y. 0 ml of 0. This means that the moles of EDTA consumed in the titration is exactly equal to the moles of {eq}Ca^{2+}{/eq Calcium can also be estimated with barium or strontium present in a 30-fold or a 20-fold molar excess, respectively. 035L x 1 mol CaCO. Abstract . 93 mL. pdf), Text File (. More than 95% of calcium in our body can be found in bones and teeth. EDTA itself is not very water soluble so the disodium salt is used, Na 2H 2C 10H 12N 2O 4. - In this experiment, The determination of calcium in milk is based on a complexometric titration of calcium with an aqueous solution of the disodium salt of EDTA at high pH . This dye-stuff tends to polymerize in strongly acidic solutions to a red brown product, and hence the indicator is generally used in EDTA titration with solutions having pH greater than 6. Hardness of water measures the sum of calcium and magnesium ions present in the water. ius. 0022 divided by milliliters EDTA used in standardization titration, B = milliliters EDTA solution used in titration, with Calcon as indicator, C = milligrams magnesium per mil-liliter EDTA solution, which is equal to 1. Bottle etc. 0-102. Practical to Determine total hardness of water sample in terms of Caco3 by EDTA Titration method using Eriochrome black T indicator diaminetetra-acetate (EDTA) for the semi-micro determination of calcium and magnesium in milk and milk diffusate are described. 02120 (standard deviation 0. If phosphates are present they must be removed by passing an aliquot through an ion exchange column before the final titration steps. In this experiment, you dissolved in HCl a commercial antacid tablet (Tums, Rolaids, etc. Because EDTA has four acidic protons, the formation of metal-ion/EDTA complexes is dependent upon the pH. determination of total calcium and magnesium ion new zea. 4 When the titration is complete, we adjust the titrand's pH to 9 and titrate the Ca 2+ with EDTA. 0216~\mathrm{M}$ than my second method. 12 Dec 2016 CHEM 249 Extra credit by Heydi Dutan and Gabriella Hetesy. EDTA Titration: Calcium in Calcium Supplements Student Handout Purpose To determine the amount of calcium in a calcium supplement tablet by EDTA titration. its ability to "sequester" metal ions such as Ca 2+ and Fe 3+ . In the remaining solution the sum of Ca and Mg is titrated by EDTA using Eriochrome Black T as indicator aluminum, manganese, calcium and magnesium mixture by EDTA titration The solution after determination of Fe is treated with EDTA in a slightly excess for calcium. EDTA is ethylene diamine tetraacetic acid or H 4C 10H 12N 2O 4. The correct result for this titration is 10 digits of the 0. a. In addition dissolved impurities such as Cu(II) or Fe(III) will interfere with the indicator color change to blue. 010 M Ca 2 + at a pH of 3 and a pH of 9 using 0. of iron, 1. Nov 17, 2009 · EDTA is an organic acid chemists use in many applications. For storage between lab periods, the tops of these columns should be stoppered and sealed with plastic wrap and the stopcocks turned off firmly. Thus the total hardness of a water sample can be estimated by titration with a standard solution of edta. Reaction taking place during titration is. A. by non- Calcium can be precipitated as carbonate or oxalate, although presence of oxalates may make end point detection difficult. ICUMSA Method GS 8/2/3/4-9 (2000) Determination of Calcium in Sugar Products by EDTA Titration - Accepted Single Method, available as online version (2 years) on www. The method Calcium and magnesium had been determined for some time in this laboratory by Jenness' method (2). The results are contained in Table II. EDTA , AgNO 3, Na 2 S 2 O 3,, Ce(SO 4) 2 Determination of pharmaceutical bases with perchloric An indirect back-titration method for the determination of (Ca + Mg) in flameproofed cotton fabrics is developed. standardizing NaOH solution : Zeolite Analysis! (stuck on ion calc. Calculate hardness Hardness (EDTA), as mg/L = × ×1,000 where A = mL of EDTA titrant used T = Titer of EDTA titrant, mg CaCO3 per mL of EDTA titrant S = mL of sample volume b. The indicator is added and remains blue as all the Ca2+ and Mg2+ ions present are complexed with the EDTA. Copper, barium, zinc, mercury, aluminum, lead, bismuth, chromium Statistical analysis of EDTA titration vs. Now you have your EDTA solution standardized and your standard EDTA solution should be ~0. The determination of water hardness or the concentrations of( 2+Ca and Mg2+ ions present in drinking water) is not a new phenomenon. The flow rate is 0. Water hardness can be readily determined by titration with the The amount of calcium may be determined by titration with standard ethylenediaminetetraacetic acid (EDTA). Molarity of EDTA (M) Determination of Calcium plus Magnesium Initial burette reading (mL) Final burette reading (mL) Volume of EDTA used for the titration (mL) Determination of Calcium Initial burette reading (mL) Final burette reading (mL) Volume of EDTA used for the titration (mL) From your data, calculate the following (Must show work) m mol of Ca^2+ present in the 50. 5 g of the rea-gent in 100 mL of ethanol (2 × 10 –2 mol ⋅L –1). Once the calcium carbonate standard has dried and cooled, weigh 0. 6 M HCl. SUMMARY The determination of calcium by rapid titration with disodium dihydro- complexometric titration problem on calcium with EDTA: Calcium Analysis by EDTA Complexometric Titration: Composition of a mixture of cations: I do not know how to figure out what to put in the blank spots or what I have put in is close to being correct. The Patton-Reeder Indicator (hereafter PR) is used as the indicator. 005510~\mathrm{M}$. 5 M NaOH, with murexide as indicator and monitoring at 510 nm. Visual Method. The EDTA we use is thus Na 2H 2Y. 4505 g sample of CaCO3 was dissolved in HCl and the resulting solution was diluted to 250. The calcium ISE overcomes problems typical of color indicators such as endpoints that are not clear and sharp, endpoint color changes that are Abstract: The flow system described incorporates two valves to select the indicator and EDTA reagent solution required for the determination of Ca and the sum of Ca and Mg. At any point in the titration we can calculate the value of pM (= -log[M n+]). Overview . 1958, Kim and. 215 divided by milliliters EDTA A Preparing a standard EDTA solution 1. Namely, the solution after determination of Mn is used for determination of Ca and Mg in alkaline medium as usual and the procedure is carried out as follows: Nov 26, 2009 · Ethylenediaminetetraacetate (EDTA) complexes with numerous mineral ions, including calcium and magnesium. Sample Complexometric Titration with EDTA B. For Mg, 2 ml serum, 18 ml water, 10 ml ammonia buffer, pH 10. edu/EDTASA WEBSITE If you wish, you can also compare the AAS method to the EDTA titration method for the determination of total hardness, based on your past experience with the ETDA method (e. Transfer 50 mL of tap water to four different Erlenmeyer flasks. A necessary requirement is that the ligand combines (complexes) quantitatively with a particular metal ion under the solution conditions. An EDTA titration method has been developed for the determination of calcium in small biological samples. EDTA Determination of Total Hardness and Determination of Individual Calcium and Magnesium. For the titration of Mg2+, one must buffer the solution to a pH of 10 so that complex formation will be quantitative. It is measured either by a titration using EDTA, or by a soap titration method. 1016/S0003-2670(00)87636-9}, journal = {Analytica Chimica Acta (Netherlands)}, number = , volume = Vol: 8, place = {Country unknown/Code not available}, year = {Sun Mar 01 00:00:00 Calcium Edta found in: Calcium Disodium EDTA, FCC, SIGMA Calcium lactobionate monohydrate 98. The stock calcium solution was standardized against ethylenediamine tetraactate e (EDTA) using murexide as metallochromic indicator for compleximetric titration. Also, the reaction with EDTA and a metal species typically goes to completion at a fast rate. Write the equation for the titration of Ca+2 with EDTA. This reagent is a weak acid that can lose Five procedures were compared for E. A back titration is carried out using a solution of magnesium chloride. Mg is estimated by titration of the supernatant fluid. edta titration of calciumii and 8. http:/genchem. This forms a complex with the excess EDTA molecules until the end-point, when all molarity of the EDTA. My question is if either of these methods are correct? determination of aluminum by back titration,EDTA titration,quantitatively determin,determination of aluminium by edta titration, complexometric titration of aluminium,aluminium edta complex which is the end point. EDTA After the Co2+ in your unknown is separated from Fe3+ it can be quantitatively measured by a complexometric titration. The main aim of this titration reaction is to determine the presence of calcium ions in a titrant by a method referred to as titrimetric. ABSTRAK 5. At the end of the titration shut off the pump and the recording device. Experiment 7 2. Pipette 3. Since you will be controlling the total moles of EDTA added in the first place, you will be able to Aug 09, 2013 · Take 50. EDTA titration with a potentio-metric determination of the equivalent point by means of a Calcium ion-selective electrode . When all the Mg+2 and Ca+2 are complexed with EDTA, the indicator will turn blue. tetraacetate ion (EDTA) for titration of calcium ion supports this suggestion. Precipitation is effected with an excess of standard calcium chloride and, after standing overnight, the unconsumed calcium ions are back-titrated with ethylenediamine tetra-acetic acid (EDTA) using Eriochrome Black T as indicator. 0% (EDTA titration), Calcium bromide hydrate ; >/=. 25 mL of an EDTA solution for titration to the Eriochrome Black T end point. The overall goal of this experiment was to compose a standardized EDTA solution to use as a titrant in a complexometric titration in order to identify the amount of calcium oxide, CaO, in an unknown sample. Erichrome Black-T and patton and reeder volume of EDTA consumed. The optimum reaction conditions and other analytical parameters were evaluated. EDTA TITRATION of CALCIUM in WATER Determination of Water Hardness Analytical Chain PRELAB. 5. At the Operate the pump to start the titration. B . The excess EDTA will be titrated with a standard solution of Zn+2. You can perform a Zinc determination using this titration method with methyl orange as your indicator. Add 3 drops of NH3 and boil until the NH3 is expelled. 1 M sodium hydroxide (pH 12-13) against murexide. </p> <p>There are multiple definitions of the different types of water hardness. Procedure- 1. Calmagite and eriochrome black T (EBT) are such indicators that Complex-Formation Titration Determination of Zinc using EDTA Zinc ion forms a stable water-soluble 1:1 complex with EDTA, which is the basis for the determination of Zn in this experiment. Figure 9. This compound (the ammonium salt of 5,5'-nitri- As a result, when the calcium ion–PR complex is titrated with EDTA the Ca2+ ions react to form a stronger complex with the EDTA. This method is similar to acid-base titrations, and involves adding the standard chelon solution to the metal ion solution till the end point is attained. EDTA Titration of CalciumII and MagnesiumII Calcium and magnesium ions are the primary contributors to "hardness" of water and they are important components of limestone. Calcium and Magnesium. Introduction The quantitative determination of many metal ions in solution can be achieved by titrating with a standard solution of a Lewis base (ligand). There are many ways to determine the concentrations of calcium and magnesium ions in tap water. Department of Chemistry, Concordia College, 901 8. Under the conditions described the determination of calcium in the presence of an equivalent amount of magnesium, using Cal‐red as indicator, could tolerate up to 2. Recently Elliott (4) presented a macromethod for the determination of serum calcium by this direct titra- tion method. Complexometric Titration with EDTA Chemistry 3200 Complexometric Titration with EDTA In this experiment you will use ethylenediaminetetraaectic acid (EDTA) to determine metals in aqueous solution by complexation titration. Background Calcium is an important element for our body. Fresh or dried material is extracted with 1 nitric acid and the extract is alkalinised to pH 13. In the EDTA titration, it is assumed that the total hardness is due to the presence mainly of calcium and magnesium ions. May 06, 2020 · Complexation Titration: Determination of Total Hardness of Water The use of EDTA as a complexometric titrant for metal ions has been investigated in the past . Test the pH of the solution using universal pH paper. 3. Determination of Calcium by Titration with EDTA In this investigation, a known concentration of EDTA is complexed to calcium ions to determine water sample hardness. Determination of Total Hardness in Water by Automatic Titration Introduction Total hardness due to calcium and magnesium in water is determined using the preprogrammed method, T7 Total Hardness. 01 M EDTA solution was prepared by weighing 4 g EDTA disodium salt into a 400-ml beaker. Fabrics are boiled for 10-15 min with an excess of sodium-EDTA solution to extract and to bind the ('a and Mg, present in fabrics, into their corresponding EDTA complexes. 5 g of a specimen is weighed and is put in a beaker, 20-30 ml of hydrochloric acid and 8-10 ml of a saturated boric acid solution are added, heating dissolution is performed for 20-40 minutes, and volume fixing is performed after filtering and cooling; titration is performed by using an Apr 10, 2013 · Determination of Calcium ions in milk using titration. Acting as a ligand that shows multiple coordination sites, EDTA forms very strong 1:1 See full list on titrations. EDTA can form four or six Miguel Garcia Chemistry 221 Section 801 February 13, 2017 Determination of Calcium in Dietary Supplement Tablets By EDTA Titration Objective: The purpose of the experiment was to determine the amount of calcium content of a dietary supplement and compare its results from the results of another method which is the ion-exchange chromatography. DETERMINATION OF CALCIUM WITH EDTA. Calcium is also of biological importance and is contained in teeth and bone. Introduction Aqueous calcium hydroxide, also known as lime water is used to verify the presence of carbon dioxide gas, (carbon dioxide reacts with the calcium hydroxide to produce calcium carbonate) this is achieved by bubbling the gas through the solution, if the solution turns cloudy then the precipitate calcium carbonate has formed, thus carbon dioxide is present. Complexometric titration (sometimes chelatometry) is a form of volumetric analysis in which the formation of a colored complex is used to indicate the end point of a titration. Determination of Calcium by Displacement Titration 1) Take your unknown sample into a 250 mL conical flask and add 50 mL distilled water. Though the determination of calcium and magnesium by complexometric titration with standard solutions of disodium dihydrogen tetraacetate, utilising Eriochrome Black T as indicator is widely accepted and quite adequately understood, it, like other complexometric titration methods, suffers from the limitations of having an indistinct end point (where a photometric titrator is needed Determination of Cobalt by Titration with EDTA I. The above-mentioned standard lays down a titration with EDTA at pH 10. The atomic absorption method and inductively coupled plasma method are accurate means of determining calcium. For the titration of the total magnesium and calcium by E. 1-mL addition of the standard solution. A 0. In such a titration, a metal ion is titrated with a ligand that readily forms a complex with the metal ion. The presence in water of salts of calcium and magnesium is spoken of as hardness. Principles: Figure 1 EDTA Download file to see previous pages In this experiment, the searchers are trying to perform a complex formation reaction for analytical purposes. Repeat the above titration procedure using 5. At pH 10, calcium and magnesium ions form colourless, water soluble, complexes with EDTA: calcium ion complexed by EDTA : CaEDTA 2-magnesium ion complexed by EDTA : MgEDTA 2-An indicator, known as a metal ion indicator, is required for the titration. Determination of Water Hardness using Complexometric titration You will use EDTA complexometric titration to determine the hardness of a sample of water brought from your home. The endpoint of a complexometric EDTA titration using either Calmagite or EBT as the indicator is detected as the colour changes from pink to blue. Apr 02, 2018 · Calcium, Magnesium and Total Hardness Determination by Potentiometric Titration Using TRIS Buffer Method Abstract #126 Scope and Application This method is useful for monitoring the calcium, magnesium and total hardness in water samples. Magnesium is then calculated from the difference between the calcium titration and one for total hardness. We herewith present to you our Titration handbook. Titration of Unknown Calcium Sample Prepare a clean beaker and ask your GA for 100 mL of unknown solution. Therefore, a buffer solution may be added to the titration chamber to maintain the pH. A sample of the waterbuffered at pH 10 is titrated with a standard solutionof EDTA. 0050 second titration only calcium reacts with EDTA solution. 0 0. 5 2. Determination of Total hardness Repeat the above titration method for sample hard water instead of standard hard water. The end point in the calcium determination is some- what 23 Jan 2009 This video demonstrates the titration of calcium with an EDTA titrant. lime salts determination by direct colorimetric titration. 00 mL aliquots of the diluted solution for titration, treating each as follows: Add about 2 mL of pH 10 buffer,1 mLof Mg/EDTA solution, and 3 to 4 drops of Eriochrome Blake T or Calmagite indicator. - Complexometric titration is a type of titration based on complex formation between the analyte and titrant. your TA). e. Either dry the EDTA for 2 hours at 80 o C or plan to standardize the EDTA using CaCO 3, 2. 01M Na 2 H 2 Y to a color change from red to blue. hardness. - 2. The determination of calcium and mag nesium specifically is usually performed as two titrations. Chelometric Titration : The Determination of Water Hardness and Water Filtration Things for the lab notebook and to be included in the lab report: 1. 1 (EDTA4-) (CaEDTA2-) Oct 13, 2020 · Freeman and Company European community classification of olive oil Aljalimetric. EDTA= titrant; EDTA is a powerful ligand that binds with metal ions to make a stable metal-EDTA complex when the pH of the solution is greater than 10 9. Apparatus 1. 3. edu. A copper Experiment 5: EDTA Determination of Calcium and Magnesium. EDTA Titration for Determination of. 0 mg. edta titration virtual lab Labs in C 6 will undertake the spectrophotometric HINT on the second titration, add the EDTA to with 1 mL of the endpoint of the first titration, then titrate dropwise and very slowly with swirling to the red to blue color change endpoint. ch on November 6, 2020 by guest [MOBI] Calcium Analysis By Edta Titration Getting the books calcium analysis by edta titration now is not type of inspiring means. It is a variation of Standard Method 2340C and ASTM D-1126. For example, calcium gluconate injection, calcium lactate tablets, and compound sodium lactate injection for the assay of calcium chloride calcium ions in the saturated solution can be calculated and the solubility of calcium sulfate can then be determined. txt) or read online for free. Direct semi-automatic and automatic derivative potentiometric EDTA procedures for the determination of calcium and/or magnesium with a calcium-selective electrode are described. 0 mL of pH 10 buffer, 10 mL of 0. A novel approach based on flow injection gradient titration is proposed for the simultaneous spectrophotometric determination of Ca and Mg based on parameters of a single signal. After titrating with EDTA Ca and Mg will be complex the solution turns from wine red to blue. EDTA is a hexaprotic acid, meaning that each of the amines nitrogen's and each of the acid's oxygen's can donate one electron pair. Next, the EDTA titration is used to test the effectiveness of an ion exchanger that removes Calcium and magnesium are separated by precipitation of Ca as its oxalate, which is centrifuged off, dissolved in perchloric acid and titrated with ethylenediamine tetra-acetate in the presence of ethanolamine-magnesium-ethylenediamine tetraacetic acid buffer. 2, and 15 ml 20% sodium tungstate were mixed; after boiling for 2 to 3 min, cooling and adding 10 ml buffer and 1% NaOH to pH 10, the mixture was titrated with Eriochrome black T as A comparative study was carried out for the determination of calcium and magnesium in 5 different brands of milk (Haleeb, Milk Pak, Olpers, Dairy Queen and Nurpur). NO. As mentioned previously, calconcarboxylic acid (or Patton-Reeder Indicator) is used for the determination of calcium ion concentration by complexometric titration. "why?" •Complexometric titration is a type of titration based on complex formation between the analyte and titrant. Second method I used the volume of the entire solution at the end of the titration, $0. of solution. The disodium salt of EDTA was used to determine the concentration of M 2+ metal ion impurities in hard water by a complexometirc titration. Ethylenediaminetetraacetate (EDTA) complexes with numerous mineral ions, including calcium and magnesium. Eqn. 7. The method uses EDTA (ethylenediaminetetraacetic acid) to form a complex with calcium (Ca 2+) ions. Sep 23, 2010 · In the first titration the solution was titrated with 0. The 0. Concentrations can be determined directly from the known mass of EDTA, however, for more accurate work, standardisation is accomplished by titrating against a solution made from the primary standard Ca(NO 3) 2. ; CAS Number: 154071-48-4; Synonym: This standard method is usually used to check the accuracy of the other methods [5], complexometry or the titration method of calcium with EDTA is a rapid and However, various acids and salt solutions are used in the investigation of the durability of concrete, and the adaptability of the EDTA titration method to determine in the titration of calcium with EDTA. . The estimation of hardness is based on complexometric titration. g. 2H 2 O. It is determined by several analytical techniques. Add methyl Water "hardness" refers to the amount calcium and magnesium dissolved in the water. 05 M EDTA. For the purpose of simplicity, Y will stand for C 10H 12N 2O 4. Calcium Carbonate was used as a primary standard in a control experiment to determine if back titration was a suitable technique. This means How can you Determination of Zinc (Zn+2) in solution determination of zinc ion,determination of zinc by edta titration, Determination of Zinc ion by Direct Titration using Eriochrome Black T as indicator,Complexometric Titration of Zn(II) with EDTA,determination of zinc by titration ,determination of zinc by titration method Direct Titration-It is the most convenient and simple method of complexometric titration using EDTA. Background The presence in water of salts of calcium and magnesium is spoken of as hardness. 5, a continuous determination of Fe, Mn, Ca by titration with EDTA becomes possible. It is grouped into two parts, Ethylenediaminetetraacetic acid (EDTA), shown on the right in its deprotonated form, is commonly used in a titration to determine the concentration of Ca2+ and Titrate with EDTA 0,01 N. Suitable conditions for the titration are achieved by the addition of a buffer solution of pH 10. 00 mL titration E815-17b Standard Test Method for Determination of Calcium Fluoride in Fluorspar by EDTA Complexometric Titrimetry calcium fluoride content~ complexometric titration~ fluorspar~ A new method for the determination of fluoride ions is described. aqueous solution containing calcium and magnesium ions at a pH of 10. If much more or less titrant was used, there can be a problem with user technique, reagents, apparatus or an interference. sodium concentration for the NaCl standards and derive the calibration equation for the two sets of second titration only calcium reacts with EDTA solution. After being bound by EDTA, metal ions remain in solution but exhibit diminished reactivity (Wikipedia, 2009). doc), PDF File (. Titrate [2] with standard 0. 50 ± 0. Before attempting this experiment, you may need to consult the section in the textbook dealing. i. diaminetetra-acetate (EDTA) for the semi-micro determination of calcium and magnesium in milk and milk diffusate are described. This is the end point of the titration. 00014) for Mg:Cl‰ were obtained for samples from tropical North Atlantic Ocean. Now, when this solution is titrated against EDTA, then the calcium and magnesium ions started to form a stable metal-EDTA complex. For example, consider the titration of 50. 26 for calcium and 140. The salts causing Determination of calcium in water by edta titration || Analysis of water || environmental chemistry || Determination of calcium in water || calcium analysis A = milligrams calcium per milli-liter EDTA solution, which is equal to 2. Calcein Used for the fluorometric determination of calcium and EDTA titration of calcium in the presence of magnesium. Methods include titrations as well as ion analysis using AS you will see, the binding constants are determined for the EDTA form, because this is the form Say we wish to titrate Ca with EDTA at pH 10, what is our. and are determined by EDTA titration at different. 8. Calcium is titrated separately with EDTA at pH 12 to 13, while magnesium is masked as magnesium hydroxide. D. Since you will be controlling the total moles of EDTA added in the first place, you will be able to A relatively simple procedure for the titration of calcium and magnesium in milk with EDTA. The application note describes a potentiometric and Photometric method for the titer determination of CaCl2 by different procedure i. As a result, when the calcium ion–PR complex is titrated with EDTA the Ca2+ ions react to form a stronger complex with the EDTA. Preparing the calcium carbonate primary standard. A 25. rutgers. This titration uses the metal ion indicator reaction Therefore, we can also determine how much CaCO3 is in them. (ethylenediaminetetraacetic acid). (v) Calculate Normality of EDTA required by sample using given formula below: N 1 V 1 = N 2 V 2 Determination of volume of EDTA S. Complexometric titrations are particularly useful for the determination of a mixture of different metal ions in solution. A buffer will be used. Since both EDTA and Ca2+ are colorless, it is necessary to use a special indicator to detect the end point of the titration. people rit edu. TITRATION Ca+2 + EDTA-4 ------> 2 Oct 2020 prevent additional dissolution. (a) Titration of 50. The EDTA solution can then be used to determine the hardness of an unknown water sample. 6 Sep 2009 Calcium can be determined by EDTA titration in solution of 0. The method which combined Ling's method for removal of A method was developed to determine the total calcium and magnesimn in diluted milk by direct titration with ethylenediamine tetraacetate (EDTA). Many water samples can be analyzed with one click of button. the goal - to prepare and standardize an EDTA solution in order to determine total hardness in an unknown water sample Terms in this set (20) Calcium; Magnesium titration of calcium oxalate was selected as the most universally accepted procedure. (See Gravimetric Nickel Determination) COMPLEXOMETRIC DETERMINATION OF CALCIUM WITH EDTA In this experiment you will determine the concentration of Ca+2 in an unknown solution by complexing it with an excess of EDTA. titration. A relatively simple procedure for the titration of calcium and magnesium in milk with EDTA. Let the burette reading of EDTA be V 2 ml. The titration was carried out both by direct titration with EDTA and back titration with Zn2+. Miguel Garcia Chemistry 221 Section 801 February 13, 2017 Determination of Calcium in Dietary Supplement Tablets By EDTA Titration Objective: The purpose of the experiment was to determine the amount of calcium content of a dietary supplement and compare its results from the results of another method which is the ion-exchange chromatography. For finding out the total hardness of water the total molarity of calcium and magnesium ions in the water. The stock color reagent was GBHA solution, prepared by dissolving 0. Titrate with standard EDTA, 25 mL of unknown solution after addition of 3 mL ammonium chloride Ethylenediaminetetraacetic acid (EDTA), shown on the right in its deprotonated form, is commonly used in a titration to determine the concentration of Ca 2+ and Mg 2+ ions in water because both ions form complexes with EDTA. 1 EDTA (ethylenediamine tetraacetic acid or its salts) is added to a sample containing calcium and magnesium ions after the pH of the solution is adjusted to 10 for the determi-nation of calcium and magnesium or from pH 12 to 13 for the determination of calcium alone. Calcium Analysis by EDTA Titration PRESTUDY 1. B. Hardness of water is determined by titrating with a standard solution of ethylene diamine tetra acetic acid (EDTA) which is a complexing agent. The EDTA initially complexes the calcium and then the magnesium. 24 for magnesium. The titration is performed by adding a standard solution of EDTA to the sample containing the Ca. 01 M EDTA until the color changes from wine red to blue. Hardness analysis typically involved the addition of an ammonium hydroxide buffer to […] Also, the reaction with EDTA and a metal species typically goes to completion at a fast rate. Chelating agents are multi-dentate ligands. 20 hours ago · EDTA Titration: Calcium in Calcium Supplements Student Handout Purpose To determine the amount of calcium in a calcium supplement tablet by EDTA titration. by using H2EDTA-Na2 heavy metals are present, determine calcium and magnesium by a non-EDTA method (see Sections 3500-Ca and 3500-Mg) and obtain hardness by calculation. 2015. Obtain ~ 0. In the second part of this experiment, the calcium ion concentration will be determined by an entirely different method. on p85 "at pH 10 to 10. Methods involving the photometric determination of Al2O3 with anthrazochrome, the complexometric back titration of CaO, and the direct complexometric titration of MgO have been developed. The amount of calcium in a solution can be determined using titration. 24 for calcium and 125. 1). OBJECTIVES: 1. ) (3) Tabulate and plot the emission intensity vs. Determination of total hardness of water by complexonometric titration (EDTA) File: Hardness_EDTA Quantity Unit Definition Θ °C Difference of the actual laboratory temperature from 20 degrees centigrade γ 1/°C The coefficient of volume expansion for water V EDTA_rep ml Value and the repeatability unceratinty component of the volume Jun 25, 2015 · PRINCIPLE :- EDTA form chelated soluble complex when added to a solution of Certain metal cations. The overall procedure to be used involves the standardization of an EDTA solution by titration with a known amount of calcium followed by using the calibrated solution to determine an unknown amount of calcium. Introduction. V2 = volume in ml of EDTA solution consumed in titration; VI = volume in ml of EDTA consumed in titration for calcium ~determination, in the same aliquot of solution of-sample; and M = mass in g of the sample taken for test. info chemistry 321: quantitative analysis lab webnote edta titration for determination of calcium and magnesium before attempting this experiment, you may need to ! 1! Determination!of!calcium!by!Standardized!EDTASolution!! Introduction!! The!classic!method!of!determining!calcium!andothersuitablecations!is titration!with!a The present analysis is concerned with the determination of Ca by the use of a complexometric titration of the type that is described above. It is vital for the development of bones and teeth. Hamano T, Mitsuhashi Y, Tanaka K, Matsuki Y, Oji Y, Okamoto S. determination of calcium by edta titration
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\begin{document}
\flushbottom \title{Minimum Copies of Schr\"{o}
\thispagestyle{empty}
\section*{\label{sec:level1}Introduction}
From fundamental tests of quantum mechanics \cite{EINsTEIN} to quantum teleportation, quantum key distribution, and quantum communications \cite{Lee,Tang,Vallone}, quantum entanglement has wide applications in different areas. Recently, a single photon has been recognized to teleport multiple degrees of freedom simultaneously, which includes spin and orbital angular momentum \cite{Wang}. The Greenberger-Horne-Zeilinger (GHZ) states created in experiment have been obtained by combining the momentum and polarization \cite{1,2,3,4,5}. Several experiments have been performed to validate multi-photon entanglement \cite{threephotonGHZ,1,2,3,5,39}. An indispensable tool is the entanglement witness for certification of entanglement in these related experiments \cite{43,GO,HR}. Generally, the expectation value of entanglement witness can be evaluated by fidelity \cite{37}. Precise estimation of fidelity requires many identical copies of the prepared state \cite{fidcopy}. On the other hand, the coincidence count rate of multi-photon entangled states decreases exponentially with a linear increase in the number of entanglement photons, which is generated by the phenomenon of a nonlinear process of parametric down-conversion in BBO \cite{PANRMP,8,Crosse,Ghosh,Parigi}. Hence, collecting sufficient copies of multi-photon entanglement state costs much longer time, such as, in eight-photon entanglement it takes $170$ hours to produce the sufficient copies of eight-photon Schr\"{o}dinger's cat (SC) state (See the label of Fig.3 of Ref.\cite{2}). Up to now, a study on ten-photon entanglement or more is inaccessible in experiment since the coincidence count rate of ten-photon entanglement state is lower than 9 counts per hour \cite{2}. It needs nearly three months to prepare sufficient copies, for example, 110, of the ten-photon SC state to certify entanglement according to the current technology. (See Appendix ``Preparation of Ten-photon SC state").
Discrimination of a quantum state by adaptive process is developed recently. Adaptive process is to split the conventional measurement into several pieces and to choose the current measurement based on the results of previous measurements. The standard of selection is chosen to be minimizing the probability of error \cite{pryde}. The probability is estimated by the known information. Generally the adaptive process is split into two steps. The first step is to get rough information. The second step is to rectify it and get a precise density matrix \cite{D.H.Mahler, thePaperOfGuoLaoshi}.
In this paper, an efficient method is developed to reduce the number of copies of an unknown state to certificate entanglement in the multi-photon experiment. Specifically, the conventional measurement is split into several steps. For each step, the least number and the optimal distribution of identical copies of the unknown state on different measurement settings are calculated by the proposed model. The measurement result from previous steps provides the value of parameters for future steps. In our model, the unknown state is supposed to be a pure SC state or SC state in the presence of noise. Since the entanglement validation of SC state is through fidelity \cite{43}, the optimization introduces fidelity as a criterion. When fidelity is greater than 0.5, the experimentally prepared state is certified to be entangled \cite{43}. The target of optimization is to search for the minimum number of copies of the unknown state that can confirm the error gap of fidelity belonging to a small area; therefore the fidelity interval can be estimated and the minimum number of copies of state is obtained.
\section*{\label{sec:level2} Minimum copies of multi-photon Schr\"{o}dinger's Cat state}
The experimental n-qubit SC state is denoted by a $2^n\times2^n$ density matrix $\rho_{exp}$. Its fidelity with the pure state $|SC\rangle$ is defined as \begin{eqnarray}
F_{exp}(|SC\rangle)=\langle SC|\rho_{exp}|SC\rangle = {\rm Tr}(\rho_{exp}|SC\rangle\langle SC|).\label{4} \end{eqnarray}
To calculate the $F_{exp}(|SC\rangle)$, Eq.(\ref{4}) can be written as \begin{eqnarray}
F_{exp}(|SC\rangle) ={\rm Tr}\left\{\rho_{exp}\left[\frac{1}{2}I-\left(\frac{1}{2}I-|SC\rangle\langle SC|\right)\right]\right\} .\label{FEXP} \end{eqnarray} Now setting entanglement witness operator $w$, \begin{eqnarray}
w=\frac{1}{2}I-|SC\rangle\langle SC|\label{6} \end{eqnarray} in Eq.(\ref{FEXP}), we arrive at \begin{eqnarray}
F_{exp}(|SC\rangle)={\rm Tr}\left[\rho_{exp}\left(\frac{1}{2}I-w\right)\right]=\frac{1}{2}-\langle w\rangle, \end{eqnarray}
where $\langle w\rangle$ is the expectation of entanglement witness \cite{Guhne2,37}. Hence, $F_{exp}(|SC\rangle)$ can be calculated by evaluating $\langle w\rangle$. In Eq.(\ref{6}), $|SC\rangle\langle SC|$ is decomposed into the form \begin{eqnarray}
&&|SC\rangle\langle SC|=\nonumber\\
&&\frac{1}{2}\left[(|H\rangle\langle H|)^{\otimes n}+(|V\rangle\langle V|)^{\otimes n}+\frac{1}{n}\sum_{k=1}^{n} (-1)^{k}M^{\otimes n}_{k\pi/n}\right],\label{DEsc} \end{eqnarray} where $M_{k\pi/n}=\cos(k\pi/n)\sigma_{x}+\sin(k\pi/n)\sigma_{y}$ \cite{5,37}. See Appendix ``Entanglement Witness" for more details.
The $n$-qubit SC state requires at least $n+1$ settings to calculate fidelity (see Observation 1 in \cite{37}). Based on Eq.(\ref{DEsc}), the standard deviation of fidelity is deduced, \begin{eqnarray} &&\Delta F_{exp}=\sqrt{\frac{1}{4}\frac{P_1(1-P_1)}{t_1}+\frac{1}{n^2}\sum_{j=2}^{n+1}\frac{P_j(1-P_j)}{t_j}}.\label{deltaF1} \end{eqnarray}
In Eq.(\ref{deltaF1}), $t_j$ is the total number of copies of n-qubit entanglement state that projected into the $j$th measurement setting, $j=1,2,\cdots, n+1.$ Its value equals the sum of accumulated n-fold coincidence counts in all different bases of the $j$th setting. Here accidental coincidence count is ignored since it is almost zero when $n$ is large. The $P_1$ is equal to the summation of two relative frequencies. One is the case that all qubits are projected into horizontal polarizations $(|H\rangle\langle H|)^{\otimes n}$, the other is the case that all qubits are projected into vertical polarizations $(|V\rangle\langle V|)^{\otimes n}$. Here, the meaning of relative frequency is the ratio between the number of copies of a state projected into a base and the number of copies of the state measured in all the bases belonging to this setting. Similarly, the $P_j$ is the linear combination of relative frequencies of different basis in the $j$th setting. It should be kept in mind that measurement setting means a group of complete basis that copies of a state are projected in the same period of time and relative frequencies being gained simultaneously. The details are shown in the appendix ``Standard Deviation of Fidelity".
We intend to apply fewer copies of the unknown state to estimate fidelity with same accuracy. Let $\epsilon_0$ denote the given upper bound of standard deviation of fidelity since the number of copies of a state required relies on it. Our objective is to use as few copies of the state as possible and, at the same time, to narrow down the fidelity to a small interval. Therefore, the following model is proposed: for $n$-qubit SC state, we have \begin{eqnarray} && {\rm Minimize}\ \sum_{j=1}^{n+1}t_j\ \nonumber\\ && {\rm subject\ to}\ \nonumber\\ &&\sqrt{\frac{1}{4}\frac{P_1(1-P_1)}{t_1}+\frac{1}{n^2}\sum_{j=2}^{n+1}\frac{P_j(1-P_j)}{t_j}}\leq \epsilon_0.\label{Optimizationproblem} \end{eqnarray}
It should be noted that $t_j$ obtained by solving Eq.(\ref{Optimizationproblem}) is sufficiently large, since the larger $t_j$ is, the higher the probability for the result of Eq.(\ref{Optimizationproblem}) to hold, which will be discussed in section of ``Characteristics of optimization of the successful probabilities". Based on numerical results, the solution of Eq.(\ref{Optimizationproblem}) has large $t_j$ in most cases and the probability for the above model is nearly $1$. Following is the analytical solution of Eq.(\ref{Optimizationproblem}) obtained. Let $\epsilon_0^2=\epsilon$, then
\begin{eqnarray} && t_1=\frac{\frac{1}{2}\sqrt{P_1(1-P_1)}\left[\frac{1}{2}\sqrt{P_1(1-P_1)}+\sum_{j=2}^{n+1}\frac{1}{n}\sqrt{P_j(1-P_j)}\right]}{\epsilon}, \nonumber\\ && t_j=\frac{\frac{1}{n}\sqrt{P_j(1-P_j)}\left[\frac{1}{2}\sqrt{P_1(1-P_1)}+\sum_{j=2}^{n+1}\frac{1}{n}\sqrt{P_j(1-P_j)}\right]}{\epsilon}. \label{optimalt} \end{eqnarray}
The process to get the Eq.(\ref{optimalt}) can be found in ``Theoretical derivation of minimum copies of multi-photon Schrodinger's Cat state"
\section*{Results} \subsection*{Direct estimation of fidelity for experimental eight-photon SC state and simulated ten-photon SC state}
The advantage of our method over the existing approaches can be demonstrated by the experiment of eight-photon entanglement. When sufficient copies of eight-photon SC state in the presence of noise ($\rho_{8photons}$) are projected into different settings in experiment, fidelity is calculated by total accumulated coincidence counts on different basis and then the eight-photon entanglement can be verified \cite{2}. In this section, our method is to change the number of copies of prepared SC state ($\rho_{8photons}$) measured in different settings. The results show that total copies of prepared SC state can be saved; while, fidelity precision remains same.
To apply Eq.(\ref{optimalt}) for certificating of eight-photon entanglement in our model, some parameters need to be given, such as $n=8$. Let the prepared eight-photon SC state in experiment be $\rho_{8photons}$, which is a SC state mixed with noise. In order to have optimized results compared with the experiment. The error bound of fidelity, $\epsilon_0$, is set at 0.016, which is exactly the same value as the one used in experiment. This number can be found in the second to last paragraph of Ref.\cite{2}. According to entanglement witness, an eight-photon SC state requires nine settings to determine fidelity uniquely, see equation (2) in Ref.\cite{2}. Let $|H\rangle$ represent horizontal polarization and $|V\rangle$ represent vertical polarization. And define $|+\rangle$$=$$(|H\rangle+ e^{i\theta}|V\rangle)/\sqrt{2}$, $|-\rangle$$=$$(|H\rangle- e^{i\theta}|V\rangle)/\sqrt{2}$. Then the nine measurement settings are defined as
\begin{eqnarray}
&& S_{8photons,1}=\{(|H\rangle\langle H|)^{\otimes8}, (|H\rangle\langle H|)^{\otimes7}(|V\rangle\langle V|),\cdots,(|V\rangle\langle V|)^{\otimes8} \}, \nonumber\\
&& S_{8photons,2}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes8}, (|+,\theta\rangle\langle +,\theta|)^{\otimes7}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&& (|-,\theta\rangle\langle -,\theta|)^{\otimes8}, \theta=0 \}, \nonumber\\
&& S_{8photons,3}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes8}, (|+,\theta\rangle\langle +,\theta|)^{\otimes7}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&&(|-,\theta\rangle\langle -,\theta|)^{\otimes8}, \theta=\pi/8 \}, \nonumber\\
&& S_{8photons,4}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes8}, (|+,\theta\rangle\langle +,\theta|)^{\otimes7}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&& (|-,\theta\rangle\langle -,\theta|)^{\otimes8}, \theta=2\pi/8 \}, \nonumber\\
&& S_{8photons,5}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes8}, (|+,\theta\rangle\langle +,\theta|)^{\otimes7}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&&(|-,\theta\rangle\langle -,\theta|)^{\otimes8}, \theta=3\pi/8 \}, \nonumber\\
&& S_{8photons,6}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes8}, (|+,\theta\rangle\langle +,\theta|)^{\otimes7}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&& (|-,\theta\rangle\langle -,\theta|)^{\otimes8}, \theta=4\pi/8 \}, \nonumber\\
&& S_{8photons,7}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes8}, (|+,\theta\rangle\langle +,\theta|)^{\otimes7}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&&(|-,\theta\rangle\langle -,\theta|)^{\otimes8}, \theta=5\pi/8 \}, \nonumber\\
&& S_{8photons,8}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes8}, (|+,\theta\rangle\langle +,\theta|)^{\otimes7}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&&(|-,\theta\rangle\langle -,\theta|)^{\otimes8}, \theta=6\pi/8 \}, \nonumber\\
&& S_{8photons,9}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes8}, (|+,\theta\rangle\langle +,\theta|)^{\otimes7}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&&(|-,\theta\rangle\langle -,\theta|)^{\otimes8}, \theta=7\pi/8 \}. \nonumber\\
\end{eqnarray}
Furthermore, 1305 copies of $\rho_{8photons}$ are prepared in total in experiment \cite{2}. Notice that 1305 is not directly given in the Ref.\cite{2}, but is used to draw the graphes, calculate fidelity in Ref.\cite{2} and provided by the author of that paper. The number can also be roughly calculated by the copies of $\rho_{8photons}$ per hour and the total hours spent. That is $9\times40+9\times25+9\times15\times7=1544$, in which the coincidence counting rate can be found in the 11th paragraph of Ref. \cite{2} and the hours spent for different settings can be found in the label of Figure 3 of Ref. \cite{2}. Accidental coincidence count is very small in eight-photon experiment, therefore it is neglected. A numerical test is performed. Firstly, a set of copies of a quantum state measured at various settings is defined as ``distribution of copies".
Three different distributions (experimentally applied distribution of copies of $\rho_{8photons}$, optimal distribution of copies of the state, which is obtained from Eq. (\ref{optimalt}), and uniformity distribution of copies of the state) are considered separately, and compared with each other. The number of copies of $\rho_{8photons}$ for each case is listed in Table \ref{tablecopy}. The first column is the tag of setting. The optimal distribution of the copies of $\rho_{8photons}$ calculated by Eq.(\ref{optimalt}) is listed in the last column. Obviously, the total number of copies of $\rho_{8photons}$ required is cut down to 1253, thus 52 copies of $\rho_{8photons}$ (about 5 percent) are saved compared with experiment. Since coincidence count rate is only nearly nine (8.88) per hour (in the 11th paragraph of Ref.\cite{2}), around 5.9 hours could be reduced in the experiment while the same precision of fidelity can be hold.
\begin{table}
\begin{tabular}{|c|c|c|c|}
\hline
Setting & Experiment & Uniformity &Optimization \\
\hline
$S_{8photons,1}$ & 352 & 145 & 415 \\
$S_{8photons,2}$ & 200&145 & 106 \\
$S_{8photons,3}$ &107& 145 & 103 \\
$S_{8photons,4}$ &100 &145 & 106\\
$S_{8photons,5}$ & 110 & 145 &103 \\
$S_{8photons,6}$ & 111 &145 & 108 \\
$S_{8photons,7}$ & 106&145 & 101 \\
$S_{8photons,8}$& 116 & 145 &108 \\
$S_{8photons,9}$ & 103 &145 & 103 \\
\hline
Summation & 1305& 1305 &1253\\
\hline \end{tabular} \caption{ Distribution of copies of $\rho_{8photons}$ under different case. The first row represents the number of copies of $\rho_{8photons}$ that is the summation of the number of accumulated coincidence counts projected into all the bases of the first measurement setting. The following eight rows represent the number of copies of $\rho_{8photons}$ that projected into the other eight settings corresponding to $\theta=0, \pi/8,\cdots,7\pi/8$. The last row is the total cost of copies of $\rho_{8photons}$ in the experiment of Ref.\cite{2}, uniform distribution and our optimization. The second column is the cost of number of copies of $\rho_{8photons}$ in the experiment for different setting. The third column represents uniform distribution of copies of $\rho_{8photons}$ in each setting. The last column represents the copies of $\rho_{8photons}$ obtained in the optimization.}\label{tablecopy} \end{table}
For each case, fidelity can be calculated from new relative frequencies obtained by simulating the experimental process in computer according to the real precise relative frequencies in different settings. In simulation, the real relative frequency is calculated according to Born's rule. It also requires density matrix to be known in this rule. Fortunately, the density matrix of $\rho_{8photons}$ can be obtained by experimental data and phaselift approach, which will be given in detail in ``optimization of multi-qubit experimental and simulated data via density matrices". Since summation of the real relative frequency of different bases in a same setting is equal to one, the interval between 0 and 1 is divided into $2^8$ sub-intervals and the range for each of the sub-interval is equal to the value of the corresponding relative frequency. And a random number between 0 and 1 is produced with the equal probability for each value between 0 and 1. And the interval it lies in is found and the number of event for this interval is added to one. After producing random numbers with the number of copies of $\rho_{8photons}$ for the setting, different interval gets a different number of event. Then relative frequencies can be calculated. After that simulated fidelity is obtained. We also divide the fidelity range from 0 to 1 into 50 equal portions. Event number is added to one when the calculated fidelity belongs to the corresponding interval. All three situations (experiment, optimization and uniformity) are all repeated for 550 times separately, which means 550 fidelities are calculated. The number of events per interval is accumulated and observed, as shown in Figure \ref{eightphoto1}. Figure \ref{eightphoto1}a shows that: when all 1305 copies of $\rho_{8photons}$ are applied, the experimental results give better estimation of fidelity than the uniform distribution since the height of the outline for uniform distribution on vertical axis direction is lower than the experimental one. The outlines for experiment and optimization are also described, which almost coincide with each other. However, optimization only costs 1253 copies of the $\rho_{8photons}$, which is smaller than the 1305 copies of $\rho_{8photons}$ required by the experiment, as shown in Figure \ref{eightphoto1}b. The Figure \ref{eightphoto1}c demonstrates that optimization is also better than the uniform distribution.
At present there is no way to create enough copies of ten-photon SC state to certify entanglement in experiment. Numerical test is produced to estimate fidelity based on a computer created density matrix $\rho_{10photons}$ that its fidelity with pure ten-photon SC state is 0.8414. It is carried out in the situation of uniform distribution in each setting (100 copies of ten-photon SC state for each setting) and optimization. The process is same as eight-photon entanglement. Both cases are all repeated for 100 times separately, then the distributions of fidelities are obtained, as shown in Figure \ref{densitymatriten}. It is observed that 22.45 \% copies of simulated ten-photon SC state $\rho_{10photons}$ can be saved according to Eq.(\ref{optimalt}) compared with uniform distribution on each setting.
Obviously, the optimization yields a better estimation of fidelity with limited copies of state available. The scheme given here is useful in certifying the multi-qubit entanglement state and can be generalized to any state by changing the form of constraint of Eq.(\ref{Optimizationproblem}).
\subsection*{Optimization of multi-qubit experimental and simulated data via density matrices}
In addition to the direct estimation of fidelity, we also estimate a density matrix first by phaselift \cite{luyiping}, and then calculate fidelity.
The model for calculation of the density matrix is constructed based upon the procedures given in Refs.\cite{James,Banaszek,21, 20, 31, 33, 35, 36}; the noise case is applied, which is in \cite{34} and \cite{luyiping},
\begin{eqnarray}
&& {\rm Minimize}\ \sum_{\mu,\nu}|{\rm Tr}(\rho M_{\mu,\nu})-f_{\mu,\nu}| \nonumber\\ && {\rm subject\ to}\ \ \rho\geq0, {\rm Tr}(\rho)=1,\label{optidensitymatirx} \end{eqnarray} where $\rho$ is density matrix, $M_{\mu,\nu}$ is positive operator valued measure (POVM) in the $\mu$-th bases of the $\nu$-th setting, $f_{\mu,\nu}$ is the relative frequency in the $\mu$-th bases of the $\nu$-th setting.
The quantum state tomography for three, four and eight-photon entanglement is conducted. When corresponding experimental frequencies $f_{\mu,\nu}$ are put in
Eq.(\ref{optidensitymatirx}), the density matrix is calculated out. Our objective is to use the least copies of an unknown state to obtain a density matrix close to the real one. The real density matrix $\rho_{exp}$ is approximately obtained with the use of a large number of copies of the state prepared in experiment. Then, $\rho_{exp}$ is applied to gain new frequencies according to Born's rule through the simulation of experiment process on computer. These frequencies are applied to obtain the density matrix $\rho_{re}$. Then many density matrices $\rho_{re}$ are obtained under different number of copies of the state and compared with the $\rho_{exp}$ achieved in the experiment.
Several examples are given below in the following. The density matrix of the three-photon SC state ($\rho_{exp_{-}3qubits}$) is obtained by using the experimental data and construction method summarized by Eq.(\ref{optidensitymatirx}) with Pauli measurement, as shown in Figure \ref{Figure5789}a. The two large elements on the diagonal of the density matrix $\rho_{exp_{-}3qubits}$ are equal to 0.50188 for $|HHH\rangle\langle HHH|$ and 0.38419 for $|VVV\rangle\langle VVV|$. And the real parts of two main elements on the anti-diagonal are both 0.37238 on $|HHH\rangle\langle VVV|$ and $|VVV\rangle\langle HHH|$. The imaginary parts are quite small, so are not drawn. The density matrix of four-photon SC state ($\rho_{exp_{-}4qubits}$) is also obtained by using experimental data and phaselift, as shown in Figure \ref{Figure5789}b. The result is obtained by Pauli measurement and Eq.(\ref{optidensitymatirx}). Two large elements on the diagonal of the density matrix equal to 0.50637 for $|HHHH\rangle\langle HHHH|$ and 0.36161 for $|VVVV\rangle\langle VVVV|$. The real parts of elements on the anti-diagonal are 0.35944 on $|HHHH\rangle\langle VVVV|$ and $|VVVV\rangle\langle HHHH|$. Since the $\rho_{exp_{-}3qubits}$ (Figure \ref{Figure5789}a) and $\rho_{exp_{-}4qubits}$ (Figure \ref{Figure5789}b) have very small noise and the purity is high, density matrix of three-qubit ($\rho_{3qubits}$) (Figure \ref{Figure5789}c, Figure \ref{Figure5789}d) with much more noise is created for the following simulation. Two large elements on the diagonal of the density matrix are equal to 0.3716 for $|HHH\rangle\langle HHH|$ and 0.3412 for $|VVV\rangle\langle VVV|$. And the real parts of two main elements on the anti-diagonal are both 0.3504 on $|HHH\rangle\langle VVV|$ and $|VVV\rangle\langle HHH|$. The corresponding imaginary part is nearly approach to zero, so is not shown. The density matrix of the eight-photon system ($\rho_{8photons}$) is also drawn in Figure \ref{densitymatrix}. Obviously, only the real part of elements in four corners of the density matrix are larger than 0.2; other elements are much less than it, which is the characteristic of SC state. Besides, the imaginary part is too small; therefore, it is not drawn. By the density matrix of Figure \ref{Figure5789}a, we reconstructed the $\rho_{exp_{-}3qubits}$ under different number of pauli measurement. The reconstructed density matrix is $\rho_est$. Figure \ref{threefreprob} exhibits fidelities and Mean Square Error (MSE) when a different number of POVM is applied. When sampling number of POVM achieves around 45, fidelity is in a stable value (around 0.7) and the corresponding MSE is near 0.
Similarly, the following four measurement settings are defined for three qubits measurement. \begin{eqnarray}
&& S_{3qubits,1}=\{(|H\rangle\langle H|)^{\otimes3}, (|H\rangle\langle H|)^{\otimes2}(|V\rangle\langle V|),\cdots,(|V\rangle\langle V|)^{\otimes3} \}, \nonumber\\
&& S_{3qubits,2}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes3}, (|+,\theta\rangle\langle +,\theta|)^{\otimes2}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&&(|-,\theta\rangle\langle -,\theta|)^{\otimes3}, \theta=0 \}, \nonumber\\
&& S_{3qubits,3}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes3}, (|+,\theta\rangle\langle +,\theta|)^{\otimes2}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&&(|-,\theta\rangle\langle -,\theta|)^{\otimes3}, \theta=\pi/3 \}, \nonumber\\
&& S_{3qubits,4}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes3}, (|+,\theta\rangle\langle +,\theta|)^{\otimes2}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&& (|-,\theta\rangle\langle -,\theta|)^{\otimes3}, \theta=2\pi/3 \}. \nonumber\\ \end{eqnarray} Two measurement settings shown in \cite{55}, which is to project the state into $S_{3qubits,1}$ and $S_{3qubits,2}$, can also be applied to certify entanglement. For the three-qubit SC state, the density matrix ($\rho_{3qubits}$) are measured in four settings or two settings, respectively, and then the fidelity is estimated, as shown in Figure \ref{optimizeRandoAverage}. In the figure, it shows the distribution of number of event of fidelity between a target pure SC state and estimated one when 10000 copies of three-photon SC state $\rho_{3qubits}$ are measured. The fidelity between $\rho_{3qubits}$ and pure three-qubit SC state is 0.7068. We use $Random$ to represent the distribution of $500_{-}1000_{-}5000_{-}3500$, which represents the number of copies of $\rho_{3qubits}$ prepared in four settings, such as 500 is prepared in the $S_{3qubits,1}$, 1000 is for the setting of $S_{3qubits,2}$, 5000 for $S_{3qubits,3}$, and 3500 for $S_{3qubits,4}$. Optimization means $3630_{-}2570_{-}2670_{-}1130$, in which 3630 copies of $\rho_{3qubits}$ is projected into the setting of $S_{3qubits,1}$, 2570 copies of $\rho_{3qubits}$ is projected into $S_{3qubits,2}$, 2670 is for the setting of $S_{3qubits,3}$, 1130 is for the setting of $S_{3qubits,4}$. $``Uniformity"$ means $2500_{-}2500_{-}2500_{-}2500$, which means all four settings are projected with the same number of copies of $\rho_{3qubits}$ (2500). ``Two setting" means $5000_{-}5000$, which represents the number of copy of $\rho_{3qubits}$ prepared in two settings, such as 5000 is for $S_{3qubits,1}$, and 5000 for $S_{3qubits,2}$. Both two-setting distribution and the optimized distribution of copies of $\rho_{3qubits}$ ($3630_{-}2570_{-}2670_{-}1130$) gives the best estimation of fidelity, while the randomized distribution ($500_{-}1000_{-}5000_{-}3500$) gives the worst estimation. Christ mentioned that a bias exists for fidelity estimation when the semi-definite constraint is added to the maximum likelihood approach, and this bias is based on density matrix \cite{christ}. Here phaselift is applied, and there is no obvious bias for fidelity estimation when the number of copies of $\rho_{3qubits}$ approaches 10000, as shown in Figure \ref{optimizeRandoAverage}. However, there is an obvious bias when the number of copies of $\rho_{3qubits}$ drops to 1000 and the number for the setting of $S_{3qubits,2}$ is switched into the setting of $S_{3qubits,4}$ in two settings case, as exhibited in Figure \ref{fifu1}.
Fidelity estimation is also compared and analyzed in different initial conditions, such as the number of copies of $\rho_{3qubits}$, as shown in Figure \ref{fi1}, which presents that 10000 copies of $\rho_{3qubits}$ provide much better estimation of fidelity than 200 copies of the state. Furthermore, optimization always gives better estimation of fidelity than uniform distribution of copies of $\rho_{3qubits}$.
\section*{Optimization of the number of copies via experimental feedback}
Let us note that $P_j$ in Eq.(\ref{optimalt}) should be known before calculating $t_j$, and there is no way to obtain the precise value of $P_j$ without knowing density matrix via Born's rule or without experimental measurement. In ``Direct estimation of fidelity for experimental eight-photon SC state and simulated ten-photon SC state", we prior estimate a density matrix based on the preparation of copies experimentally. In ``Optimization of multi-qubit experimental and simulated data via density matrices", we estimate a precise density matrix by phaselift. Here, we show to calculate them through the experiment itself. Take eight-photon SC state experiment as an example. In experiment, pure eight-photon SC state is the target state that needs to be prepared. It can be taken as a priori to approximately decide $P_j$, such as $P_1$ is a value near to ${\rm Tr}((|H\rangle\langle H|)^{\otimes 8} |SC\rangle\langle SC|)+{\rm Tr}((|V\rangle\langle V|)^{\otimes 8} |SC\rangle\langle SC|)$, $P_j$ is near to a value given by Eq.(\ref{p1pj1pj}), $j=2, 3, \cdots, n+1$. However, the experimentally prepared state is not pure $|SC\rangle$ and it takes too long time to precisely estimate $P_j$, an optimization procedure is proposed based upon the experiment. It divides the process of measurement into a few steps. Instead of measuring one setting for a prolonged time to obtain the frequencies within small error margins, and then continue to measure the next setting for the same time, and so on. We divided this total long time into several intervals, and changed the order of measurements. The order is to measure all the required settings one by one for a much shorter time, then based on the measurement results; the $P_j$ can be estimated roughly. After that, the extra number of copies of a quantum state that needs to be prepared and measured for each setting can be given by Eq.(\ref{optimalt}) by inputting the rough $P_j$. Afterwards, more copies of the quantum state are prepared and measured according to the $t_j$ given. Later, more precise frequencies can be obtained and this process can be repeated until the final precision for fidelity is reached. Figure \ref{celiangfangxiang} shows the measurement order when the process is conducted only twice. We simulated this process in computer, it only costs a very short time, as shown in Figure \ref{RunTimeVSCopyNumber}.
To be specific, the main process is as following. Firstly, we introduce a superscript to represent the number of steps in optimization. The superscript $l$ of a parameter represents the parameter applied in the $l$-th step, i.e. $\epsilon^{1}$ represents the value $\epsilon$ used in Eq.(\ref{optimalt}) for the first round of measurement. At the beginning of fidelity estimation, $\epsilon^{1}$ is set to a large number, such as 0.01, and all of $P_j$ are originally set to $P_j=P_j^1=1/2$, $j=1,2,3,\cdots,n+1$, ($P_j^1$ can also be chosen according to target pure state $|SC\rangle$, such as $P_1^1$ can be a value near ${\rm Tr}((|H\rangle\langle H|)^{\otimes n} |SC\rangle\langle SC|)+{\rm Tr}((|V\rangle\langle V|)^{\otimes n} |SC\rangle\langle SC|)$ so that the suitable solution $t_j^l$ can be obtained by solving Eq.(\ref{optimalt}). The experiment is performed according to the $t_j^1$ copies of the quantum state. When all $t_j^1$ copies of the prepared quantum state is projected into a measurement setting, $P_j^2$ can be obtained. After that, input $P_j^2$ instead of $P_j^1$, and have $\epsilon^1$ become smaller; consequently, the $t_j^2$ can be obtained. Then copies of the state with the number of $t_j^2-t_j^1$ are projected into the $j$th measurement setting in the second round of experiment, so on and so forth. Measurement is ended when $\epsilon^{iter}$ is sufficiently small, obvious, \begin{eqnarray} \epsilon^{1}>\epsilon^{2}>\epsilon^{3}>\cdots>\epsilon^{l}\cdots>\epsilon^{iter}. \end{eqnarray}
Extra time is needed for optimization; however it is much shorter compared with the time required for the preparation of copies of multi-photon entanglement state, as shown in Figure \ref{RunTimeVSCopyNumber}. The iteration makes the experiment to have more pauses between different settings during the measurement process. Mostly, switching settings cost more time, generally is about 3 or 4 times of the switching time in conventional measurement. Anyhow the time required for optimization is much shorter than that spent on the preparation of the copies of multi-photon entanglement state. Generally, switches and optimizations only cost less than two minutes, while the coincidence count rate of eight-photon entanglement state is too low that it costs several hours to produce just enough copies of the state for only one setting. Therefore, the time in calculation and switching time can be neglected compared to the preparation of copies of multi-photon entanglement state.
In the following, a specific example is given. Numerical simulation applies a four-qubit SC state mixed with gaussian noise. The density matrix is $\rho_{4qubits}$, which fulfills trace equal to one and semi-definite condition. The fidelity between $\rho_{4qubits}$ and pure SC state is 0.9374. In simulation, parameters are chosen as $\epsilon^{1}=0.01$, $\epsilon^{2}=0.001$, $\epsilon^{3}=0.0001$, $\epsilon^{4}=0.00001$. The five measurement settings are required and listed as follows: \begin{eqnarray}
&& S_{4qubits,1}=\{(|H\rangle\langle H|)^{\otimes4}, (|H\rangle\langle H|)^{\otimes3}(|V\rangle\langle V|),\cdots,(|V\rangle\langle V|)^{\otimes4} \}, \nonumber\\
&& S_{4qubits,2}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes4}, (|+,\theta\rangle\langle +,\theta|)^{\otimes3}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&& (|-,\theta\rangle\langle -,\theta|)^{\otimes4}, \theta=\pi/4 \}, \nonumber\\
&& S_{4qubits,3}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes4}, (|+,\theta\rangle\langle +,\theta|)^{\otimes3}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&& (|-,\theta\rangle\langle -,\theta|)^{\otimes4}, \theta=2\pi/4 \}, \nonumber\\
&& S_{4qubits,4}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes4}, (|+,\theta\rangle\langle +,\theta|)^{\otimes3}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&&(|-,\theta\rangle\langle -,\theta|)^{\otimes4}, \theta=3\pi/4 \}, \nonumber\\
&& S_{4qubits,5}=\{(|+,\theta\rangle\langle +,\theta|)^{\otimes4}, (|+,\theta\rangle\langle +,\theta|)^{\otimes3}(|-,\theta\rangle\langle -,\theta|),\cdots,\nonumber\\
&& (|-,\theta\rangle\langle -,\theta|)^{\otimes4}, \theta=\pi \} \nonumber\\
\end{eqnarray}
All the initial numbers of copies of $\rho_{4qubits}$ for each setting are set at 5. Other initial parameters for 5 measurement settings are set at: $P_1=1/2$, $P_2=1/2$, $P_3=1/2$, $P_4=1/2$, $P_5=1/2$, respectively.
In Figure \ref{copieoptimiz}, the $\rho_{4qubits}$ is taken as the test matrix. Its horizontal axis represents the number of iteration, which means the number of round of measurement. The corresponding point is the average number of extra copies of the state $\rho_{4qubits}$ that needs to be projected into each setting for the next round of measurement. The curve connects the number of required copies of $\rho_{4qubits}$ for each same setting. The error bar is one standard deviation, which is obtained by repeating the optimization program for 100 times. When the iteration ends, the $P_j$ is listed in Table \ref{lastpjandkj}.
\begin{table}[h] \begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$P_1$ &0.9339 \\ $P_2$&0.0316 \\ $P_3$& 0.9491 \\ $P_4$& 0.0325 \\ $P_5$& 0.9474 \\
\hline \end{tabular} \caption{\label{lastpjandkj} The final $P_j$ when iteration ends.} \end{center} \end{table}
We define the ratio between the $\epsilon^l$ applied in the current round of measurement and the $\epsilon^{l-1}$ applied in the previous round of measurement as the ratio of $\epsilon$, and let the ratio of $\epsilon$ for different round of measurement be equal to each other, that is $\epsilon^{2}/\epsilon^{1}=\epsilon^{3}/\epsilon^{2}=\cdots=\epsilon^{l}/\epsilon^{l-1}=\cdots$. The different values are tested to search for the best ratio that costs the least number of copies of a state. Figure \ref{copiep} shows the number of copies of state randomly created at different ratios of $\epsilon$ when it equals to a value between 0.05 and 0.9. It is observed that it requires most number of copies when the ratio is 1/2, it increases at a wave type before this value, and decreases after this value. Specifically, the following procedure is conducted. Initially, the number of copies of a randomly created state is an integer between $4$ to $7$ for each setting, $P_1$, $P_2$, $\cdots$, $P_5$ are all given a value randomly created between 0.25 and 0.75 and $\epsilon$ is $0.01$. Then, a fixed ratio of $\epsilon$, such as 0.1, is applied. It means $\epsilon$ is set to $0.001$ in the second round of measurement, $0.0001$ in the third round of measurement and so on. Eq.(\ref{optimalt}) is applied to calculate the needed copies ($t_j$) of the state for each round of measurement. Then $t_j$ is summed up to gain the total number of the copies of the state in current round of measurement ($\sum_jt_j$). The iteration stops when $\epsilon$ is smaller than 0.0003. The minimum number of copies of the state can be found by repeating the above steps by changing the ratio of $\epsilon$. It is found that the value near 0.15 for the ratio requires the least number of copies of a randomly created state, which is 178. The total iteration number for each setting for most created states is about 3 or 4 to achieve the 0.0003 of finial $\epsilon$. It is also noticed that the required number of copies is even less when the ratio of $\epsilon$ approaches 0.9. However, it is not suitable to apply the large value for the ratio of $\epsilon$ since too less copies of a state may lead to our model hold with a low probability, which will be discussed in the following section.
\section*{Discussion} \subsection*{Characteristics of optimization of the successful probabilities}\label{ivc}
It is noticed that the summation of number of copies of an unknown state projected into the same setting must be larger than a certain value, since a large value can confirm the model to hold with probability nearly one.
In the above sections, the minimum number of copies of an unknown state is obtained and its fidelity belongs to the interval with the certain high probability. Hoeffding's inequality is a mathematical way to describe the probability. It states that the sample average $\overline{X}=\sum X/t$ of $t$ independent, not essentially identical distributed, bounded random variables with $Prob[X_i\in[a_i,b_i]]=1$ for $i=1,2,\cdots,t$ satisfies \begin{eqnarray} && Prob\left[\overline{X}-E[\overline{X}]\leq-h\right]\nonumber\\ &&\leq exp[-2t^2h^2/\sum(b_i-a_i)^2] \end{eqnarray} for all $h>0$, where $X_i$ is a variable, $a_i$ is the lower bound, $b_i$ is the upper bound, $t$ is the number of samples, $E[\overline{X}]$ denoting the mean value of $\overline{X}$, $h$ is the definite value that equals to the maximum deviation from expectation \cite{Hoeffdingone, Hoeffdingtwo}.
Now this inequality is applied to multi-photon entanglement certificate experiment. The measurement response of a single copy of an unknown state is taken as the value of a single random variable. Since photon detectors can only give the feedback, $0$ or $1$, this leads to $b_i=1$ and $a_i=0$. $\overline{X}$ corresponds to a relative frequency, which is denoted to be $f_j$, $j$ is to distinguish different measurement settings. Hence the expectation $E[\overline{X}]$ corresponds to the probability $p_j$. The total copy of a state for the $j_{th}$ setting is represented by $t_j$ instead of $t$. By replacing all of them, we obtain \begin{eqnarray} && Prob\left[f_j-p_j\leq-h_j\right]\leq exp[-2t_jh_j^2]\label{probb}, \end{eqnarray} where $h_j$ is the deviation from true probability $p_j$. It means \begin{eqnarray} && Prob\left[f_j\in (p_j-h_j,p_j+h_j)\right]\geqslant 1-2exp[-2t_jh_j^2]. \end{eqnarray} Then \begin{eqnarray} && Prob\left[1-f_j\in (1-p_j-h_j,1-p_j+h_j)\right]\nonumber\\ && \geqslant 1-2exp[-2t_jh_j^2]. \end{eqnarray} Therefore, \begin{eqnarray} && Prob\left[f_j-(1-f_j)\in (2(p_j-h_j)-1,2(p_j+h_j)-1)\right]\nonumber\\ && \geqslant 1-2exp[-2t_jh_j^2]. \end{eqnarray} Let $f_j-(1-f_j)$ be $p_j$. Hence, \begin{eqnarray} && Prob\left[P_j\in (2(p_j-h_j)-1,2(p_j+h_j)-1)\right]\nonumber\\ &&\geqslant 1-2exp[-2t_jh_j^2],j=2,3,\cdots,n+1. \end{eqnarray} Let $2(p_j-h_j)-1$ be $P_j^-$ and $2(p_j+h_j)-1$ be $P_j^+$, and based on Eq. (\ref{k1j}), $k_j\in[k_j^-,k_j^+]$, where \begin{eqnarray} && k_1^+=Max{\left\{\frac{1}{4}P_1^+(1-P_1^+), \frac{1}{4}P_1^-(1-P_1^-)\right\}}, \end{eqnarray} \begin{eqnarray} && k_1^-=Min{\left\{\frac{1}{4}P_1^+(1-P_1^+), \frac{1}{4}P_1^-(1-P_1^-)\right\}}, \end{eqnarray} \begin{eqnarray} && k_j^+=Max{\left\{\frac{1}{n^2}P_j^+(1-P_j^+), \frac{1}{n^2}P_j^-(1-P_j^-)\right\}},\nonumber\\ && j=2, 3, \cdots, n+1, \end{eqnarray} \begin{eqnarray} && k_j^-=Min{\left\{\frac{1}{n^2}P_j^+(1-P_j^+), \frac{1}{n^2}P_j^-(1-P_j^-)\right\}}, \nonumber\\ &&j=2, 3, \cdots, n+1, \end{eqnarray} then from Eq.(\ref{optimalt}), one has \begin{eqnarray} && t_i^+=\frac{\sqrt{k_i^+}\left(\sum_{j=1}^{n+1}\sqrt{k_j^+}\right)}{\epsilon}, \end{eqnarray} \begin{eqnarray} && t_i^-=\frac{\sqrt{k_i^-}\left(\sum_{j=1}^{n+1}\sqrt{k_j^-}\right)}{\epsilon}. \end{eqnarray}
Therefore the $t_i$ $\in[t_i^-,t_i^+]$ in Eq.(\ref{optimalt}) when the holding probability of model Eq.(\ref{Optimizationproblem}) is considered.
Obviously, $h_j$ has the impact on $t_i^+$ and $t_i^-$. The larger $h_j$ is, the larger the gap between $t_i^+$ and $t_i^-$. Large $h_j$ and $t_i$ from Eq.(\ref{probb}) are needed to keep results with high probability. However, large $t_i$ costs too much experimental time. Large $h_j$ may introduce too large a gap between $t_i^+$ and $t_i^-$, which may then lead to the wrong number of copies of an unknown state. Therefore, it requires to choose suitable $h_j$ and $t_i$.
By comparing the optimization results with the experiment, it is found that only $986$ copies of $\rho_{8qubits}$ are used compared with the $1305$ copies of $\rho_{8qubits}$ in eight photon experiment, which specifies that $24$ percent copies of eight-photon SC state ($\rho_{8qubits}$) can be saved. Specifically, $h_j$ is chosen to be $0.2$ for all $j$. According to joint probability, $\prod_jp_j$ is calculated, in which $p_j$ is the successful probability for each setting. For eight photon measurement, $j=1,2,\cdots,9$, the final probability is 0.9972 for experiment after $1305$ copies of $\rho_{8qubits}$ are measured. We observed same probability is obtained when $110$ copies of $\rho_{8qubits}$ for each setting are used and all $h_j$s' are chosen as 0.2.
In the above analysis, we assume Hoeffding's inequality describes the probability precisely. In the following, numerical simulation is produced to confirm the above mathematical tool is true. The density matrix ($\rho_{8qubits}$) is calculated from experimental frequencies, and new relative frequencies are obtained under a certain number of copies of $\rho_{8qubits}$ in a random simulation of experimental process that gets the relative frequency by computer. $1-P_1$ is the summation of relative frequency that all the qubits projected into horizontal polarization and the relative frequency that all the qubits projected into vertical polarization. The real value of $1-P_1$ is $0.8068$ when the number of copy is sufficiently large. When failing probability is set less than $0.0001$, Figure \ref{probaibcopy} shows how $1-P_1$ behaves under different number of copy of $\rho_{8qubits}$. In the figure, red circle and blue triangle are drawn according to Hoeffding's inequality, $1-P_1$ can be estimated much more precisely with an increasement of prepared copies of the state $\rho_{8qubits}$. It is observed that all the numerical simulated points lie in the region between upper bound and lower bound. Therefore, Hoeffding's inequality can be applied to describe the $P_j$ in multi-photon entanglement.
\section*{Extension of the optimization of the number of copies of a state to quantum-state tomography}
The surprising thing brought to us by the optimization in the fidelity estimation, is to extend it to quantum state tomography. The optimization model for tomography is constructed as follows: Let $\rho_0$ be a $d\times d$ density matrix of real experimental created and $\rho$ be the estimated density matrix via limited copies of $\rho_0$. For $n$ qubit state tomography, we build \begin{eqnarray} &&{\rm Minimize}\ \sum_{\nu=1}^{n_s}T_{\nu}\ \nonumber\\
&&{\rm subject\ to}\ \ ||\rho-\rho_0||_F\leq \epsilon_0,\label{Optimizationtomography} \end{eqnarray} where $T_{\nu}$ represents the number of copies of $\rho_0$ of the $\nu$th setting, $\nu$ can distinguish different measurement settings, $n_s$ is total number of measurement setting. The solution of Eq.(\ref{Optimizationtomography}) is
See ``Theoretical derivation of minimum number of copies of a state in quantum-state tomography" for the details to get the solution of Eq.(\ref{Optimizationtomography})
\section*{Conclusions} We proposed an optimal approach that assists to find the minimum distribution of copies of a state that is sufficient to certify the entanglement of the state by fidelity. The main purpose is to facilitate an experiment to obtain better measurement strategy for fidelity estimations, for example, by changing the ratio of the number of copies of the state in different settings. To estimate fidelity directly from fewer copies of SC state (1253 copies), with optimized distribution, almost the same distribution of fidelity as the experimental one (1305) can be obtained. It not only saves time, about five percent of measurement time (6 hours) is saved, but also small error of fidelity. Additionally, the distribution on the number of copies of ten-photon SC state is also simulated, and 22.45\% of copies of the ten-photon SC state are saved, which further highlights the superiority of this scheme, and reveals that the optimized distribution of copies of a state in different settings gives better estimation of the fidelity than uniform distribution of copies of a state in all settings. Fidelity can also be estimated by the reconstructed density matrix. It is observed that the optimized distribution provides the best estimation of the true state, the uniform distribution provides a worse estimation, while randomized distribution provides the worst estimation. With the increase of the number of copies of the state the differences between different distributions (uniform distribution and optimized distribution) become much smaller. Besides the state with high similarity with SC state, this approach can also be extended into other states in parallel. And the scheme is extendable to tomography when the MSE between the estimated density matrix and real density matrix is limited to a fixed value.
\appendix
\section*{Preparation of ten-photon SC state}
From second paragraph of Reference \cite{2}, the count rate of eight-photon event is about $2.8\times10^{-5}Hz$. Accidental coincidence counts can be neglected for eight-fold entanglement. Therefore two-photon event count rate is $\sqrt[4]{2.8\times10^{-5}}Hz$. Detecting ten-photon entanglement requires totally five independent pairs of entangled photons to present at the same time, so the ten-photon coincidence event scales as $(\sqrt[4]{2.8\times10^{-5}\times3600})^5=0.0568$ per hour. For ten-photon entanglement, 11 measurement settings are required according to the entanglement witness of SC state. If only 10 copies of ten-photon SC state are prepared and measured in one setting, then 110 copies of SC state are required. Therefore, the corresponding time is $(110/0.0568) hours=1.9366\times10^3 hours=80.6917 days\approx3 months$.
\section*{Entanglement witness}
To calculate $F_{exp}$, each term in the decomposition of $|SC\rangle\langle SC|$ has to be measured to determine its expectation value. For an eight-qubit SC state, $n = 8$, the expectation values of all the terms on the right hand of Eq.(\ref{DEsc}) should be calculated. Specifically the total accumulated coincident counts on the i-th base is defined as $n_i$s such as $n_1$ copies of $\rho_{8photons}$ with all qubits are projected into horizontal polarization $|H\rangle$. $n_{256}$ copies of $\rho_{8photons}$ with all qubits are projected into vertical polarization $|V\rangle$. Relative frequencies on $(|H\rangle\langle H|)^{\otimes 8}$ or
$(|V\rangle\langle V|)^{\otimes 8}$ can be calculated by $n_1/(\sum_{i=1}^{2^8}n_i)$ or $n_{256}/(\sum_{i=1}^{2^8}n_i)$. To get the expectation value of the third term of Eq.(\ref{DEsc}), we have
\begin{eqnarray} &&\frac{1}{2}[{\rm Tr}\rho_{exp}\frac{1}{8}\sum_{k=1}^{n}(-1)^{k}M^{\otimes 8}_{k\pi/8}] \nonumber\\ &&=\frac{1}{16}[-{\rm Tr}\rho_{exp}M^{\otimes 8}_{\pi/8}+{\rm Tr}\rho_{exp}M^{\otimes 8}_{2\pi/8}-{\rm Tr}\rho_{exp}M^{\otimes 8}_{3\pi/8}\nonumber\\ &&\hspace{1.3cm}+{\rm Tr}\rho_{exp}M^{\otimes 8}_{4\pi/8}-{\rm Tr}\rho_{exp}M^{\otimes 8}_{5\pi/8}+{\rm Tr}\rho_{exp}M^{\otimes 8}_{6\pi/8}\nonumber\\ &&\hspace{1.3cm}-{\rm Tr}\rho_{exp}M^{\otimes 8}_{7\pi/8}+{\rm Tr}\rho_{exp}M^{\otimes 8}_{\pi}]\nonumber\\ &&=\frac{1}{16}\sum_{k=1}^{8}(-1)^{k}\langle M^{\otimes 8}_{k\pi/8}\rangle, \end{eqnarray} in which $\langle M^{\otimes 8}_{k\pi/8}\rangle$ represents the expectation of the operator $M^{\otimes 8}_{k\pi/8}$. The estimation of expectation value of the operator
$M^{\otimes 8}_{k\pi/8}=(|+,\theta\rangle\langle +,\theta|-|-,\theta\rangle\langle -,\theta|)^{\otimes 8}$
is equivalent all the expectations of various combinations of $|+,\theta\rangle\langle +,\theta|$ and $|-,\theta\rangle\langle -,\theta|$, due to \begin{eqnarray} M^{\otimes 8}_{k\pi/8}
&=&(|+,\theta\rangle\langle +,\theta|-|-,\theta\rangle\langle -,\theta|)^{\otimes 8}\nonumber\\
&=&(|+,\theta\rangle\langle +,\theta|)^{\otimes 8}-(|+,\theta\rangle\langle +,\theta|)^{\otimes 7}(|-,\theta\rangle\langle -,\theta|)\nonumber\\
&&+ .... +(|-,\theta\rangle\langle -,\theta|)^{\otimes 8}.\label{A2} \end{eqnarray} There are 256 terms in all for a fixed $\theta$.
When the number of copies of state that projected into different combinations of bases $|+,\theta\rangle$ and $|-,\theta\rangle$ are collected, the copy numbers corresponding to $M^{\otimes 8}_{k\pi/8}$ are calculated from Eq.(\ref{A2}). Thus, the $\langle M^{\otimes 8}_{k\pi/8}\rangle$ can be evaluated \cite{43,5}. From these measurements, the expectations of different terms appearing in the decomposition of the SC state entanglement witness are obtained.
\section*{Standard deviation of fidelity}
The error is calculated from Poisson distribution in Fig.2 and Fig.3 in Ref.\cite{5} for each term of Eq.(\ref{DEsc}).
Based on the experimental data of Ref.\cite{2}, the all eightfold coincidences are mainly projected into $(|H\rangle\langle H|)^{\otimes8}$ or $(|V\rangle\langle V|)^{\otimes8}$ in $S_{8photons,1}$ setting. When the state is projected into the setting of horizontal or vertical polarization, the copy of $\rho_{8qubits}$ projected into $(|H\rangle\langle H|)^{\otimes8}$ is 148, the number of copies of $\rho_{8qubits}$ is projected into $(|V\rangle\langle V|)^{\otimes8}$ is 136. The summation of total number is 68 for the case that eight qubits are projected into other bases in $S_{8photons,1}$. Therefore, the ratio $P_1$ between the number that projected into $(|H\rangle\langle H|)^{\otimes8}$ or $(|V\rangle\langle V|)^{\otimes8}$ and the total number is (148+136)/(136+148+68)=(148+136)/352=284/352=0.8068, the ratio ($1-P_1$) between the copy that some qubits are projected into the horizontal polarization $|H\rangle$ and some are projected into vertical polarization $|V\rangle$ and the total copy of $\rho_{8photons}$ for this setting is 1-0.8068=0.1932. The smaller value in $P_1$ and $1-P_1$ is defined as $\widetilde{P_1}$. Similarly, according to Eq.(\ref{A2}). A smaller value between $\langle M_{(j-2)\pi/8}^{\otimes8}\rangle$ and $1-\langle M_{(j-2)\pi/8}^{\otimes8}\rangle$ is chosen as $\widetilde{P_j}$, $j=2,3,\cdots,9$. Therefore, $\widetilde{P_2}=40/200=0.2$, $\widetilde{P_3}=20/107= 0.1869$, $\widetilde{P_4}=20/100=0.2$, $\widetilde{P_5}=21/110= 0.1909$, $\widetilde{P_6}=23/111= 0.2072$, $\widetilde{P_7}=19/106= 0.1792$, $\widetilde{P_8}=24/116= 0.2069$, $\widetilde{P_9}=20/103= 0.1942$, in which the largest ratio is 0.2072.
When the number of copy of state in the whole measurement time are large and the relative frequency that the copy of state projected into a base or several bases in a setting is close to 0, Poisson distribution can be approximated by binomial distribution (Page 291 of Ref.\cite{ProbabStatisc}). Notice that the Poisson distribution here is not for the entangled photons created in BBO in time scale but the distribution of number of copies of state on different measurement basis satisfied. The binomial distribution is a special case of the Poisson binomial distribution, which is a sum of $t_j$ independent non-identical Bernoulli trials \cite{9PoissonBino}. In our optimization model, binomial distribution is applied since the number of copy of state is quite large. Let $P_1$ represent the probability that the copy of state is projected into $|H\rangle\langle H|^{\otimes n}$ or $|V\rangle\langle V|^{\otimes n}$ in the setting of $S_{1}$, where $S_{1}=\{|H\rangle\langle H|^{\otimes n}, |H\rangle\langle H|^{\otimes n-1}|V\rangle\langle V|,\cdots,|V\rangle\langle V|^{\otimes n} \}$. And let $\overline{P_1}$ denote the ratio that the state collapses to other bases in $S_{1}$, hence $\overline{P_1}=1-P_1$. According to Fig.2 and Fig.3 of Ref.\cite{5}, a value in $P_1$ or $\overline{P_1}$ is close to $1$ and the other is close to $0$ when $t_1$ is much larger than 20. It satisfies the condition that Poisson binomial distribution can be approximately replaced by binomial distribution. Since the variance of the binomial distribution is $t_1\overline{P_1}(1-\overline{P_1})$ (Page 277 of Ref.\cite{ProbabStatisc}), then variance of number of events that SC state collapses to $|H\rangle^{\otimes n}$ or $|V\rangle^{\otimes n}$ is also the same value since $t_1\overline{P_1}(1-\overline{P_1})=t_1P_1(1-P_1)$. Therefore the standard deviation is $\sqrt{t_1P_1(1-P_1)}$. Besides, the $P_1$ is defined as the ratio between the number of copies of state detected on a $(|H\rangle\langle H|)^{\otimes n}$ or $(|V\rangle\langle V|)^{\otimes n}$ basis and the total copies of state in $S_1$. Therefore the standard deviation for the relative frequency is $\sqrt{t_1P_1(1-P_1)}/t_1$, which is equal to $\sqrt{P_1(1-P_1)/t_1}$.
From Eq.(\ref{4}) and Eq.(\ref{DEsc}),
\begin{eqnarray}
&&F_{exp}(|SC\rangle)\nonumber\\
&&= {\rm Tr}\left(\rho_{exp}|SC\rangle\langle SC|\right)\nonumber\\
&&= {\rm Tr}\left\{\rho_{exp}\left[\frac{1}{2}\left((|H\rangle\langle H|)^{\otimes n}+(|V\rangle\langle V|)^{\otimes n}+\frac{1}{n}\sum_{k=1}^{n} (-1)^{k}M^{\otimes n}_{k\pi/n}\right)\right]\right\}\nonumber\\
&&= {\rm Tr}\left\{\rho_{exp}\left[\frac{1}{2}\left((|H\rangle\langle H|)^{\otimes n}+(|V\rangle\langle V|)^{\otimes n}\right)\right]\right\}+{\rm Tr}\left\{\rho_{exp}\left[\frac{1}{2n}\sum_{k=1}^{n} (-1)^{k}M^{\otimes n}_{k\pi/n}\right]\right\}\nonumber\\
&&= {\rm Tr}\left\{\rho_{exp}\left[\frac{1}{2}\left(|H\rangle\langle H|\right)^{\otimes n}+\left(|V\rangle\langle V|\right)^{\otimes n}\right]\right\}+\sum_{k=1}^{n}{\rm Tr}\left\{\rho_{exp}\left[\frac{1}{2n}(-1)^{k}M^{\otimes n}_{k\pi/n}\right]\right\}\nonumber\\
&&= \frac{1}{2}{\rm Tr}\left\{\rho_{exp}\left[\left(|H\rangle\langle H|\right)^{\otimes n}+(|V\rangle\langle V|)^{\otimes n}\right]\right\}+\sum_{k=1}^{n}\frac{1}{2n}(-1)^{k}{\rm Tr}\left\{\rho_{exp}\left[M^{\otimes n}_{k\pi/n}\right]\right\}\nonumber\\
&&= \frac{1}{2}{\rm Tr}\left\{\rho_{exp}\left[\left(|H\rangle\langle H|\right)^{\otimes n}+(|V\rangle\langle V|)^{\otimes n}\right]\right\}\nonumber\\
&&+\sum_{k=1}^{n}\frac{1}{2n}(-1)^{k}{\rm Tr}\left\{\rho_{exp}\left[(|+,\theta\rangle\langle +,\theta|)^{\otimes n}-(|+,\theta\rangle\langle +,\theta|)^{\otimes n-1}(|-,\theta\rangle\langle -,\theta|)
+\cdots +(|-,\theta\rangle\langle -,\theta|)^{\otimes n}\right]\right\}\nonumber\\
&&= \frac{1}{2}{\rm Tr}\left\{\rho_{exp}\left[\left(|H\rangle\langle H|\right)^{\otimes n}+(|V\rangle\langle V|)^{\otimes n}\right]\right\}\nonumber\\
&&+\sum_{k=1}^{n}\frac{(-1)^{k}}{2n}{\rm Tr}\left\{\rho_{exp}\left[(|+,\theta\rangle\langle +,\theta|)^{\otimes n}+(|+,\theta\rangle\langle +,\theta|)^{\otimes n-2} \otimes(|-,\theta\rangle\langle -,\theta|)^{\otimes 2}
+\cdots+(|-,\theta\rangle\langle -,\theta|)^{\otimes n}\right]\right\}\nonumber\\
&&-\sum_{k=1}^{n}\frac{(-1)^{k}}{2n}{\rm Tr}\{\rho_{exp}[(|+,\theta\rangle\langle +,\theta|)^{\otimes n-1}(|-,\theta\rangle\langle -,\theta|)+(|+,\theta\rangle\langle +,\theta|)^{\otimes n-2}(|-,\theta\rangle\langle -,\theta|)\otimes(|+,\theta\rangle\langle +,\theta|)
+\cdots+ (|-,\theta\rangle\langle -,\theta|)^{\otimes n-1}\otimes(|+,\theta\rangle\langle +,\theta|)]\}.\label{FEXPDECOMPp} \end{eqnarray}
Define \begin{eqnarray}
&&P_1={\rm Tr}\left\{\rho_{exp}\left[\left(|H\rangle\langle H|\right)^{\otimes n}+(|V\rangle\langle V|)^{\otimes n}\right]\right\},\nonumber\\
&&P_j={\rm Tr}\left\{\rho_{exp}\left[(|+,\theta\rangle\langle +,\theta|)^{\otimes n}+(|+,\theta\rangle\langle +,\theta|)^{\otimes n-2} \otimes(|-,\theta\rangle\langle -,\theta|)^{\otimes 2}+\cdots+(|-,\theta\rangle\langle -,\theta|)^{\otimes n}\right]\right\},\nonumber\\
&&1-P_j={\rm Tr}\{\rho_{exp}[(|+,\theta\rangle\langle +,\theta|)^{\otimes n-1}(|-,\theta\rangle\langle -,\theta|)+(|+,\theta\rangle\langle +,\theta|)^{\otimes n-2}(|-,\theta\rangle\langle -,\theta|)\otimes(|+,\theta\rangle\langle +,\theta|)\nonumber\\
&&+\cdots+ (|-,\theta\rangle\langle -,\theta|)^{\otimes n-1}\otimes(|+,\theta\rangle\langle +,\theta|)]\}\}.\label{p1pj1pj} \end{eqnarray}
Then Eq.(\ref{FEXPDECOMPp}) can be rewritten as
\begin{eqnarray} && \frac{1}{2}P_1+\sum_{k=1}^{n}\frac{(-1)^k}{2n}P_{k+1}-\sum_{k=1}^{n}\frac{(-1)^k}{2n}(1-P_{k+1})\nonumber\\ &&=\frac{1}{2}P_1+2\sum_{k=1}^{n}\frac{(-1)^k}{2n}P_{k+1}-\sum_{k=1}^{n}\frac{(-1)^k}{2n}\nonumber\\ &&=\frac{1}{2}P_1+\sum_{j=2}^{n+1}\frac{(-1)^{j-1}}{n}P_{j}-\sum_{j=2}^{n+1}\frac{(-1)^{j-1}}{2n}. \label{FEXPDECOMP} \end{eqnarray}
Here Eq.(\ref{A2}) is applied when $n=8$ and $k+1$ is denoted as $j$ in last second step.
According to the previous analysis, the standard deviation of $P_j$ is \begin{eqnarray} \sqrt{\frac{P_{j}(1-P_{j})}{t_j}},\quad j=1, 2, \cdots, 9.\label{fHVP} \end{eqnarray}
Further considering the formula of combined standard uncertainty \cite{erroruncertainty}, the standard deviation of fidelity can be derived. We use $\Delta F_{exp}$ to represent it. Therefore, \begin{eqnarray} &&\Delta F_{exp}\nonumber\\ &&=\sqrt{\left[ \frac{\partial F_{exp}}{\partial P_{1}}\sqrt{\frac{P_1(1-P_1)}{t_1}} \right]^2+\sum_{j=2}^{n+1}\left[\frac{\partial F_{exp}}{\partial P_{j}}\sqrt{\frac{P_j(1-P_j)}{t_j}} \right]^2}\nonumber\\ &&=\sqrt{\frac{1}{4}\frac{P_1(1-P_1)}{t_1}+\frac{1}{n^2}\sum_{j=2}^{n+1}\frac{P_j(1-P_j)}{t_j}}.\label{deltaF} \end{eqnarray}
\section*{Theoretical derivation of Minimum copies of multi-photon Schr\"{o}dinger's Cat state}
Let \begin{eqnarray} && k_1=\frac{1}{4}P_1(1-P_1),\nonumber\\ && \ k_j=\frac{1}{n^2}P_j(1-P_j), j=2, 3, \cdots, n+1.\nonumber \\ && \epsilon=\epsilon_{0}^2, \label{k1j} \end{eqnarray}
then the optimization problem is equivalent to
\begin{eqnarray} && {\rm Minimize_{t_1,t_2,\cdots,t_{n+1}}}\ \sum_{j=1}^{n+1}t_j\ \nonumber\\ && {\rm subject\ to}\ \sum_{j=1}^{n+1}\frac{k_j}{t_j}\leq \epsilon, t_j>0,k_j\geq0, j=1,2,\cdots,n+1\label{optimizationp} \end{eqnarray}
where $\epsilon$ and $k_j$ are positive real constants, $n$ is a positive integer, $t_j$ is a variable and also a positive integer. In order to solve Eq.(\ref{optimizationp}) easily, all the variables, including the number of copies of state, $t_j$, are considered as a real. The optimized number of copies of state is then rounded off to the smallest integer greater than the final real $t_j$.
The solution of the optimization problem is assumed to satisfy $\sum_{j=1}^{n+1}\frac{k_j}{t_j}= \epsilon$. If the optimal solution is not on the boundary, it means $\sum_{j=1}^{n+1}\frac{k_j}{t_j} < \epsilon$. Appropriate reduction in the number of $t_j$ can be made, while the inequality ($\sum_{j=1}^{n+1}\frac{k_j}{t_j}\leq \epsilon$) is still satisfied. This is contradictory with the target function ``${\rm Minimize} \sum_{j=1}^{n+1}t_j$", therefore the optimal solution must exist on the bound.
The detailed process of how to find the analytical solution of Eq.(\ref{optimizationp}) will be shown below.
Now let \begin{eqnarray} && t_j=\frac{1}{x_j}. \label{tjtran} \end{eqnarray}
By substituting Eq.(\ref{tjtran}) into Eq.(\ref{optimizationp}), we obtain \begin{eqnarray} &&{\rm Minimize}\ \sum_{j=1}^{n+1}\frac{1}{x_j}\ \nonumber\\ &&{\rm subject\ to} \nonumber\\ && \sum k_jx_j=\epsilon. \label{Minxj} \end{eqnarray}
The Lagrange multiplier method is applied to solve the problem. Since the target is the minimization of $\sum_{j=1}^{n+1}\frac{1}{x_j}$, Eq.(\ref{Minxj}) leads to \begin{eqnarray} L=\sum \frac{1}{x_j}-\lambda(\sum k_jx_j-\epsilon). \end{eqnarray} To find its minimum, partial derivative for each $x_j$ is expressed as \begin{eqnarray} \frac{\partial L}{\partial x_j}=-\frac{1}{x_j^{2}}-k_j\lambda=0. \label{lagr0} \end{eqnarray}
From Eq.(\ref{lagr0}), the following equation is obtained, \begin{eqnarray} x_j=\sqrt{\frac{-1}{k_j\lambda}}. \label{lagr1} \end{eqnarray}
The constraint of Eq.(\ref{Minxj}) is \begin{eqnarray} \sum k_jx_j=\epsilon. \label{lagr2} \end{eqnarray}
From Eq.(\ref{lagr1}) and Eq.(\ref{lagr2}), $\lambda$ is given by \begin{eqnarray} \lambda=-\left(\frac{\sum\sqrt{k_j}}{\epsilon}\right)^2. \label{lagr3} \end{eqnarray}
By substituting Eq.(\ref{lagr3}) into Eq.(\ref{lagr1}), we arrive at
\begin{eqnarray} x_j=\frac{\epsilon}{\sqrt{k_j}(\sum_{j=1}^{n+1}\sqrt{k_j})}.\label{xjee} \end{eqnarray}
Rewriting Eq.(\ref{xjee}) using Eq.(\ref{tjtran}), we have
\begin{eqnarray} t_j=\frac{\sqrt{k_j}(\sum_{j=1}^{n+1}\sqrt{k_j})}{\epsilon}.\label{tjjjj} \end{eqnarray}
By substituting Eq.(\ref{k1j}) into Eq.(\ref{tjjjj}), Eq.(\ref{optimalt}) is obtained. The optimal results of Eq.(\ref{optimalt}) can be compared with experiment when the same coefficient $P_1$ and $P_j$s are substituted.
\section*{Theoretical derivation of minimum number of copies of a state in quantum-state tomography}
According to Born's rule, \begin{eqnarray} &&P_{\mu,\nu}={\rm Tr}(M_{\mu,\nu}\rho_0),\nonumber\\ &&f_{\mu,\nu}={\rm Tr}(M_{\mu,\nu}\rho),\nonumber\\ &&\mu=1,2,\cdots,d,\nu=1,2,\cdots,n_s,\label{pifi} \end{eqnarray} where $\mu$ distinguishes different measurement operators in the same setting, $d$ is the dimension of density matrix and $P_{\mu,\nu}$ is the probability when $\rho_0$ is measured by operator $M_{\mu,\nu}$. Namely, $\rho\in C^{d\times d}$, then \begin{eqnarray} &&\rho_0=\sum_{\mu,\nu}P_{\mu,\nu}M_{\mu,\nu},\nonumber\\ &&\rho=\sum_{\mu,\nu}f_{\mu,\nu}M_{\mu,\nu}. \end{eqnarray} Therefore one has \begin{eqnarray} &&\rho-\rho_0=\sum_{\mu,\nu}(f_{\mu,\nu}-P_{\mu,\nu})M_{\mu,\nu}. \end{eqnarray}
Consider $M_{\mu,\nu}$ is orthogonal to each other, then \begin{eqnarray}
&&||\rho-\rho_0||_F\nonumber\\ &&=\left[{\rm Tr}\left((\rho-\rho_0)(\rho-\rho_0)^*\right)\right]^{1/2}\nonumber\\ &&=\left[\sum_{\mu,\nu}(f_{\mu,\nu}-P_{\mu,\nu})^{2}{\rm Tr}(M_{\mu,\nu}M_{\mu,\nu}^*)\right]^{1/2}\nonumber\\ &&=\left[\sum_{\mu,\nu}\left(\sqrt{\frac{f_{\mu,\nu}(1-f_{\mu,\nu})}{T_{\mu,\nu}}}\right)^{2}{\rm Tr}(M_{\mu,\nu}M_{\mu,\nu}^*)\right]^{1/2}.\label{rhoF} \end{eqnarray}
In the last step of Eq.(\ref{rhoF}), standard deviation of binomial distribution is applied, see the first paragraph of ``Standard deviation of fidelity" in appendix for detail. When the measurement operator $M_{\mu,\nu}$ belongs to the same setting $\nu$, they have the identical number of copies $T_{\mu,\nu}$ of $\rho_0$, i.e.$T_{1,\nu}=T_{2,\nu}=\cdots=T_{d,\nu}$. We denote them to be $T_\nu$. The model of Eq. (\ref{Optimizationtomography}) is equivalent to
\begin{eqnarray} && {\rm Minimize}\ \sum_{\nu=1}^{n_s}T_{\nu}\ \nonumber\\ && {\rm subject\ to}\ \nonumber\\ &&\left[\sum_{\nu=1}^{n_s}\sum_{\mu=1}^{d}\left(\sqrt{\frac{f_{\mu,\nu}(1-f_{\mu,\nu})}{T_{\nu}}}\right)^{2}{\rm Tr}(M_{\mu,\nu}M_{\mu,\nu}^*)\right]^{1/2} \nonumber\\ &&\leq \epsilon_0.\label{Optimizationtomography2} \end{eqnarray}
It is easy to find that the target of Eq.(\ref{Optimizationtomography2}) is similar to Eq.(\ref{Optimizationproblem}) except the larger required number of settings and different coefficients.
Let $\sum_{\mu=1}^{d}[f_{\mu,\nu}(1-f_{\mu,\nu}){\rm Tr}(M_{\mu,\nu}M_{\mu,\nu}^*)]$ be $k_\nu$ and $\epsilon=\epsilon_0^2$, then the model has the following form \begin{eqnarray} && {\rm Minimize}\ \sum_{\nu=1}^{n_s}T_{\nu}\ \nonumber\\ && {\rm subject\ to}\ \sum_{\nu=1}^{n_s}\frac{k_\nu}{T_\nu}\leq \epsilon, T_\nu>0,k_\nu\geq0, \nu=1,2,\cdots,n_s.\label{optimizationp2} \end{eqnarray}
Obviously, it has a similar form to Eq.(\ref{optimizationp}); therefore the solution is the same as that of Eq.(\ref{optimalt}) \begin{eqnarray} T_\nu=\frac{\sqrt{k_\nu}(\sum_{\nu=1}^{n_s}\sqrt{k_j})}{\epsilon}.\label{optimalt2} \end{eqnarray}
If $M_{\mu,\nu}$ is non orthogonal with each other, then \begin{eqnarray}
&&||\rho-\rho_0||_F\nonumber\\ &&=\left\{{\rm Tr}\left[(\rho-\rho_0)(\rho-\rho_0)^*\right]\right\}^{1/2}\nonumber\\ &&=\left[\sum_{\mu,\nu,\mu^{'},\nu^{'}}(f_{\mu,\nu}-P_{\mu,\nu})(f_{\mu^{'},\nu^{'}}-P_{\mu^{'},\nu^{'}}){\rm Tr}(M_{\mu,\nu}M_{\mu^{'},\nu^{'}}^*)\right]^{\frac{1}{2}}\nonumber\\ &&=\left[\sum_{\mu,\nu,\mu^{'},\nu^{'}}\sqrt{\frac{f_{\mu,\nu}(1-f_{\mu,\nu})}{T_{\mu,\nu}}}\sqrt{\frac{f_{\mu^{'},\nu^{'}}(1-f_{\mu^{'},\nu^{'}})}{T_{\mu^{'},\nu^{'}}}} {\rm Tr}(M_{\mu,\nu}M_{\mu^{'},\nu^{'}}^*)\right]^{1/2}. \end{eqnarray} Applying the similar substitutions as used in the orthogonal case, we have
\begin{eqnarray} &&{\rm Minimize}\ \sum_{\nu=1}^{n_s}T_{\nu}\ \nonumber\\ &&{\rm subject\ to}\ \sum_{\nu,\nu^{'}=1}^{n_s}\frac{k_{\nu,\nu^{'}}}{\sqrt{T_{\nu}}\sqrt{T_{\nu^{'}}}}\leq \epsilon, \nonumber\\ &&T_{\nu}>0,T_{\nu^{'}}>0,k_{\nu,\nu^{'}}\geq0, \nonumber\\ &&\nu=1,2,\cdots,n_s, \nu^{'}=1,2,\cdots,n_s.\label{Optimizationtomography3} \end{eqnarray} Substitute $q_{\nu}=1/T_{\nu}$, the constraint in the optimization becomes \begin{eqnarray} &&\sum_{\nu,\nu^{'}=1}^{n_s}\frac{k_{\nu,\nu^{'}}}{\sqrt{T_{\nu}}\sqrt{T_{\nu^{'}}}}\nonumber\\ &&=\sum_{\nu,\nu^{'}=1}^{n_s}k_{\nu,\nu^{'}}\sqrt{q_{\nu}}\sqrt{q_{\nu^{'}}}\nonumber\\ &&\leq \sum_{\nu,\nu^{'}=1}^{n_s}k_{\nu,\nu^{'}}(q_{\nu}+q_{\nu^{'}})/2\nonumber\\ &&\leq \sum_{p=1}^{n_s}\left[\sum_{\nu^{'}=1}^{n_s}k_{p,\nu^{'}}+\sum_{\nu=1}^{n_s}k_{\nu,p}\right]q_{p}/2\nonumber\\ &&\leq \epsilon. \end{eqnarray} Therefore, the non orthogonal case has a similar result with the orthogonal one Eq.(\ref{optimizationp2}) except the coefficient is different.
\section*{Author contributions statement}
Y.L. constructed the model and performed the numerical simulations, Q.Z. supervised the research. All authors contributed to the preparation of this manuscript.
\section*{Additional information} \textbf{Competing financial interests}: The authors declare no competing financial interests.
\begin{figure}
\caption{a: The number of events versus fidelities for both experimental distribution and uniform distribution. b: The outlines for experiment and optimization, which almost coincide with each other. Optimization only costs 1253 copies of $\rho_{8photons}$, which is a smaller number than the 1305 copies of $\rho_{8photons}$ required by the experiment. c: The number of events versus fidelities for both optimization distribution and uniform distribution.}
\label{eightphoto1}
\end{figure}
\begin{figure}
\caption{ The distribution of events' number of fidelity of ten-photon entanglement state ($\rho_{10photons}$). The blue vertical line represents the fidelity between the simulated state $\rho_{10photons}$ and pure ten-photon SC state, which equals to 0.8414. Different lines are used to connect adjacent points. Black line represents the optimized fidelity distribution and red line represents uniform distribution. It is observed that optimized distribution has more events accumulated near the real fidelity at 0.8414 than uniform distribution.}
\label{densitymatriten}
\end{figure}
\begin{figure}\label{Figure5789}
\end{figure}
\begin{figure}
\caption{ The real part of experimental density matrix of eight-photon SC state ($\rho_{8photons}$). Different colors are applied to represent the values of the elements of density matrix.}
\label{densitymatrix}
\end{figure}
\begin{figure}\label{threefreprob}
\end{figure}
\begin{figure}
\caption{ The distribution of fidelities between the target pure SC state and the estimated states under different number of copy distribution on different settings. $``Random"$ means the number of event of fidelity when distribution of copy of $\rho_{3qubits}$ goes $``500_{-}1000_{-}5000_{-}3500"$, in which the $500$ copies of state $\rho_{3qubits}$ are projected into $S_{3qubits,1}$; $1000$ copies of $\rho_{3qubits}$ is projected into $S_{3qubits,2}$; $5000$ copies of $\rho_{3qubits}$ is projected into the basis of set of $S_{3qubits,3}$ and $3500$ copies of $\rho_{3qubits}$ is projected into $S_{3qubits,4}$. Similarly, ``Optimization" represents the event number of fidelity when distribution of copies of $\rho_{3qubits}$ is $``3630_{-}2570_{-}2670_{-}1130"$, in which $3630$ copies of state $\rho_{3qubits}$ are projected into $S_{3qubits,1}$; $2570$ copies of $\rho_{3qubits}$ is projected into $S_{3qubits,2}$; $2670$ copies of $\rho_{3qubits}$ is projected into $S_{3qubits,3}$ and 1130 copies of $\rho_{3qubits}$ is projected into $S_{3qubits,4}$. ``Uniformity" means the distribution is $``2500_{-}2500_{-}2500_{-}2500"$, which represents all is equal to $2500$ for the number of copies of state $\rho_{3qubits}$ that projected into $S_{3qubits,1}$, $S_{3qubits,2}$, $S_{3qubits,3}$ and $S_{3qubits,4}$. ``Two Setting" represents the both equals to 5000 for the copies of $\rho_{3qubits}$ that projected into $S_{3qubits,1}$ and $S_{3qubits,2}$. The range of fidelity is split into $250$ intervals on average between $0$ and $1$ to compare the event number. Different number of events that fidelity lies into a certain interval is gained, such as, if the calculated fidelity is $0.005$, then it belongs to the interval between $0.004$ and $0.008$, the number of events belong to this interval is added to $1$, and so forth. Fidelity is estimated for $550$ times in all four situations. Black squares represent the number of events accumulated in each interval for the case of $``500_{-}1000_{-}5000_{-}3500"$; Red circles represent for the case of $``3630_{-}2570_{-}2670_{-}1130"$; Blue triangles represent for the case of $``2500_{-}2500_{-}2500_{-}2500"$ and pink triangles represent for the case of $``5000_{-}5000"$. The fidelity between pure three-qubit SC state and $\rho_{3qubits}$ is $0.7068$. The points are connected by the lines with the same color of the points. It is observed that red circles and pink triangles have the most events near this value. Therefore both the ``Optimization" ($``3630_{-}2570_{-}2670_{-}1130"$) and ``Two setting" ($``5000_{-}5000"$), perform better for the estimation of fidelity. While the black squares for the random one gives the worst estimation.}
\label{optimizeRandoAverage}
\end{figure}
\begin{figure}\label{fifu1}
\end{figure}
\begin{figure}
\caption{ The distributions of number for events of fidelities between estimations and target pure SC state when the fidelities are obtained from the density matrices that are constructed by 200, 1000 and 10000 copies of $\rho_{3qubits}$. The black down triangle connected by black solid line represents the number of events of fidelities when they are calculated by optimization distribution of total 200 copies of $\rho_{3qubits}$ on all the four settings. Purple star connected by dashed line represents the number of events of fidelities when it is calculated by uniform distribution of 200 copies of $\rho_{3qubits}$ on all the four settings. Blue up triangle connected by solid line represents the number of events of fidelities when it is calculated by optimization distribution of 1000 copies of $\rho_{3qubits}$ on all the four settings. Magenta diamond connected by dashed line represents the number of events of fidelities when it is calculated by uniform distribution of 1000 copies of $\rho_{3qubits}$ on all the four settings. Olive square connected by solid line represents the number of events of fidelities when it is calculated by optimization distribution of 10000 copies of $\rho_{3qubits}$ on all the four settings. Red circle connected by short dashed line represents the number of events of fidelities when it is calculated by uniform distribution of 10000 copies of $\rho_{3qubits}$ on all the four settings. Optimization distribution of 200 copies of $\rho_{3qubits}$ is $``73_{-}51_{-}53_{-}23"$, which represents the distribution of fidelity when $73$ copies of state $\rho_{3qubits}$ are projected into $S_{3qubits,1}$; $51$ copies of $\rho_{3qubits}$ is projected into $S_{3qubits,2}$, $53$ copies of $\rho_{3qubits}$ is projected into $S_{3qubits,3}$ and $23$ copies of $\rho_{3qubits}$ is projected into the bases of $S_{3qubits,4}$. Similarly, uniform distribution ($``50_{-}50_{-}50_{-}50"$), ($``250_{-}250_{-}250_{-}250"$), ($``2500_{-}2500_{-}2500_{-}2500"$), optimization distribution ( $``363_{-}257_{-}267_{-}113"$) and ($``3630_{-}2570_{-}2670_{-}1130"$) all follow the same rule as $``73_{-}51_{-}53_{-}23"$. Namely, the first number is the number of copy of $\rho_{3qubits}$ that projected into the setting of $S_{3qubits,1}$; the second number is the number of copy of $\rho_{3qubits}$ that projected into the $S_{3qubits,2}$; the third number is the number of copy of $\rho_{3qubits}$ that projected into $S_{3qubits,3}$ and the last number is the copies of $\rho_{3qubits}$ that is projected into the bases of $S_{3qubits,4}$. 10000 copies of $\rho_{3qubits}$ give much smaller error or standard deviation of fidelity than 200. Optimization always gives more centralized estimation of fidelity than uniform distribution under the same number of copies of $\rho_{3qubits}$.
}
\label{fi1}
\end{figure}
\begin{figure}
\caption{ Comparison between traditional and optimization measurement of three-qubit state. The color of line segment represents the different setting. The length of line segment represents the time for the corresponding measurement. Traditional measurement order is to finish the measurement of each setting one by one, as shown by the Pastel yellow area. The optimization measurement is iterated twice as shown by the light blue area.}
\label{celiangfangxiang}
\end{figure}
\begin{figure}
\caption{ The required time of optimization for different number of copies of a state. The red circle represents running time of eight-photon optimization. Black square represents running time of four-qubit optimization. It is observed that the total time to calculate Eq.(\ref{optimalt}) and simulate the experiment is less than 100 seconds for both cases. Therefore compared with several hours spend to prepare copies of eight-photon state in experiment. It can be negelected.}
\label{RunTimeVSCopyNumber}
\end{figure}
\begin{figure}
\caption{ The change of optimization number of copy of $\rho_{4qubits}$ of fidelity estimation corresponding to different iteration numbers with different settings: $\pi/4$, $2\pi/4$, $3\pi/4$, and $\pi$. ``Setting for H/V" represents the number of copy of $\rho_{4qubits}$ projected into $S_{4qubits,1}$. ``Setting for $\theta=\pi/4$" represents for $S_{4qubits,2}$, ``Setting for $\theta=2\pi/4$" represents for $S_{4qubits,3}$, ``Setting for $\theta=3\pi/4$" represents for $S_{4qubits,4}$, ``Setting for $\theta=\pi$" represents for $S_{4qubits,5}$. Error bar represents one standard deviation.}
\label{copieoptimiz}
\end{figure}
\begin{figure}
\caption{ The required copies of a state for different ratio of $\epsilon$. For each ratio of $\epsilon$, numerical test is conducted for 10 times. Black square is used to represent the number of copies of the state. Error bar is mean standard deviation. It is observed that the number of copies of state rises at wave type when the ratio of $\epsilon$ increases but no larger than 1/2 and decreases with the ratio of $\epsilon$ when it is larger than 1/2.}
\label{copiep}
\end{figure}
\begin{figure}
\caption{ The change of $1-P_1$ in the setting of $S_1$ corresponding to different number of copies of eight-photon SC state($\rho_{_8qubits}$). Black square represents the numerical simulation, red circle represents the theoretical lower bound and upper triangle represents the theoretical upper bound. Numerical simulation is repeated for 10 times for each number of copies. It is observed that all the simulated points lie in the region that consists of point that is larger than the lower bound and smaller than the upper bound.}
\label{probaibcopy}
\end{figure}
\end{document}
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arXiv
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\begin{document}
\allowdisplaybreaks
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\ShortArticleName{Representations and Conjugacy Classes of Semisimple Quasitriangular Hopf Algebras}
\ArticleName{Representations and Conjugacy Classes \\ of Semisimple Quasitriangular Hopf Algebras}
\Author{Sebastian BURCIU}
\AuthorNameForHeading{S.~Burciu}
\Address{Institute of Mathematics ``Simion Stoilow'' of the Romanian Academy,\\ P.O.~Box 1-764, RO-014700, Bucharest, Romania} \Email{\href{[email protected]}{[email protected]}} \URLaddress{\url{http://www.imar.ro/~sburciu/}}
\ArticleDates{Received September 16, 2019, in final form April 27, 2020; Published online May 06, 2020}
\Abstract{In this paper we give two general formulae for the M\"uger centralizers in the category of representations of a semisimple quasitriangular Hopf algebra. The first formula is given in the terms of the Drinfeld map associated to the quasitriangular Hopf algebra. The second formula for the M\"uger centralizer is given in the terms of the conjugacy classes introduced by Cohen and Westreich in [\textit{J.~Algebra} \textbf{283} (2005), 42--62]. In the case of a factorizable Hopf algebra these formulae extend some particular cases obtained by the author in [\textit{Math.~Z.} \textbf{279} (2015), 227--240].}
\Keywords{quasi-triangular Hopf algebras; centralizers; braided fusion categories; normal coideal subalgebras}
\Classification{18D10; 16T05; 19D23}
\section{Introduction}
The notion of centralizer in a braided fusion category was introduced by M\"uger in \cite{proclond}. It was shown in \cite[Theorem~3.13]{dgno2} that the centralizer of a nondegenerate fusion subcategory of a~braided category is a categorical complement of the nondegenerate subcategory. This principle is the basis of many classification results of braided fusion categories, see for example papers \cite{DGNO, dgno2, eno-adv} and references therein.
Despite its importance, in the current literature there is no concrete known formula for the M\"uger centralizer of all fusion subcategories of a given fusion category. Only few particular cases are completely known in the literature. For instance, in the same paper~\cite{proclond}, M\"uger described the centralizer of all fusion subcategories of the category of finite-dimensional representations of a Drinfeld double of a finite abelian group. More generally, for the category of representations of a (twisted) Drinfeld double of an arbitrary finite group, not necessarily abelian, a similar formula was then given in~\cite{nnw}. For the braided center of Tambara--Yamagami categories, in~\cite{gnn}, the centralizer can be described by computing completely the $S$-matrix of the modular category. In~\cite{bcg} a different approach gave a partial formula for the centralizer of fusion subcategories of a~braided equivariantized fusion category.
Given a fusion subcategory $\cd$ of a braided fusion category~$\cc$, the notion of {\it M\"uger centralizer of~$\cd$} was introduced in~\cite{dgno2}. The centralizer~$\cd'$ is defined as the fusion subcategory $\cd'$ of~$\cc$ generated by all simple objects $X$ of $\cc$ satisfying \begin{gather*} c_{X, Y}c_{Y, X}=\mathrm{id}_{X\otimes Y}\end{gather*} for all objects $Y\in\mathcal O(\cd)$ (see also~\cite{proclond}). For a fusion category~$\cc$ as usually, we denote by $\mathcal O(\cc)$ the set of isomorphism classes of simple objects of~$\cc$.
If $(A,R)$ is a quasitriangular Hopf algebra then the category $\rep(A)$ of finite-dimensional $A$-modules is a braided category with the braiding given by \begin{gather*} c_{M, N}\colon \ M\otimes N\rightarrow N\otimes M,\qquad m\otimes n \mapsto R_{21}(n\otimes m)=R^{(2)}n\otimes R^{(1)}m, \end{gather*} for any two objects $M, N\in \rep(A)$.
Given a quasitriangular Hopf algebra $(A, R)$ one can also define the Drinfeld map \begin{gather*} \phi_R\colon \ A^* \rightarrow A, \qquad f \mapsto (f \otimes \id)(R_{21}R)=f(Q_{1})Q_{2}, \end{gather*} where $Q=R_{21}R$ is the monodromy matrix.
We prove the following theorem which gives a general description for the centralizer of any fusion subcategory of the category of representations of a quasitriangular Hopf algebra:
\begin{Theorem} \label{main1}Let $(A, R)$ be a semisimple quasitriangular Hopf algebra and $L$ be a left normal coideal subalgebras of $A$. Then \begin{gather*} \rep(A//{L})'= \rep(A//{M}),\qquad \text{where}\quad {M}=\phi_{R}((A//{L})^{*}). \end{gather*} \end{Theorem} We denote by $F_{0}, F_{1}, \dots, F_{r}$ the central primitive idempotents of the character ring~$C(A)$ where $F_{0}=t$ is the idempotent integral of~$A^{*}$. Following~\cite{CW2} one can define the conjugacy classes~$\cc^{j}$ of~$A$ as $\cc^{j}:=\Lam \leftharpoonup F_{j}A^{*}$, where $\Lam$ is an idempotent integral of $A$ and $a \leftharpoonup f=\langle f, a_{1}\rangle a_{2}$ for all $a \in A$ and $f \in A^{*}$. It is well known that these conjugacy classes are the simple $D(A)$-submodules of the induced $D(A)$-module $\Bbbk\uparrow^{D(A)}_{A}\simeq A$, see~\cite{zind}.
Let $(A, R)$ be a semisimple quasitriangular Hopf algebra and $V_{0}=\Bbbk, \dots ,V_{r}$ be a complete set of isomorphism classes of irreducible $A$-modules. Let also $\mathrm{Irr}(A)=\{\chi_{0}=\epsilon,\, \chi_{1}, \dots ,\chi_{r}\}$ be the set of irreducible characters afforded by these modules and $E_{i}\in \mathcal Z(A)$ be the associated central primitive idempotent of the irreducible character~$\chi_{i}$. Since the Drinfeld map $\phi_{R}\colon C(A)\rightarrow \mathcal Z(A) $ is an algebra map we may suppose that $\phi_{R}(F_{j})=\sum\limits_{i \in \mathcal A_{j}}E_{i}$ for some subset $\cA_j\subseteq \{0, \dots , r\}$. Since $\phi_{R}(1)=1$ we obtain a partition for the set of indices of all irreducible representations $\{0,1, \dots, r\}= \bigsqcup\limits_{j\in \mathcal J}\mathcal A_{j}$. For any $0\leq i\leq r$ we denoted by $m(i)$ the unique index $j \in \mathcal J$ such that $i \in \mathcal A_{j}$. Therefore in this way we obtain a unique function \begin{gather*} m\colon \ \{0, 1,\dots, r\}\rightarrow \mathcal J \end{gather*} with the property that $E_i\phi_R(F_{m(i)})\neq 0$ for all $i \in \{0, 1,\dots, r\}$.
Our second main result is the following: \begin{Theorem} \label{mgqtr}Suppose that $(A,R)$ is a semisimple quasitriangular Hopf algebra and $L$ is a left normal coideal subalgebra of~$A$. With the above notations one has \begin{gather*}
\mathcal O(\rep(A//L)')=\big\{\chi_{i}\,|\, \cc^{m(i)}\subseteq L\big\}. \end{gather*} \end{Theorem}
Recall that the quasitriangular Hopf algebra $(A,R)$ is called {\it factorizable} if the Drinfeld map $\phi_{R}\colon A^*\rightarrow A$ is an isomorphism of algebras. In this case, its restriction $\phi_R|_{C(A)}\colon C(A)\rightarrow \mathcal Z(A)$ is an isomorphism of algebras. For a factorizable semisimple Hopf algebra we can record the primitive central idempotents $F_j$ of $C(A)$ such that $F_{j}:=\phi_{R}^{-1}(E_{j})$ any $1\leq j \leq r$. With these notations, $m(i)=i$ for any $0\leq i\leq r$ and Theorem~\ref{mgqtr} implies the following: \begin{Corollary}\label{main2} Let $(A, R)$ be a semisimple factorizable Hopf algebra and~$L$ be a left normal coideal subalgebra of~$A$. Then with the above notations one has that \begin{gather*}
\mathcal O(\rep(A//L)')=\big\{\chi_{i}\,|\,\cc^{i}\subseteq L\big\}. \end{gather*} \end{Corollary}
Shortly, this paper is organized as follows. In Section~\ref{prelim} we recall the basic notions of Hopf algebras and fusion categories that are used throughout this paper. In this section we also prove a canonical decomposition of a left normal coideal subalgebra in terms of the decomposition of its integral, see equation~\eqref{decl3}. In Section~\ref{qtr} we recall the main properties of quasitriangular Hopf algebras and their associated Drinfeld maps. In Section~\ref{fmr} we prove Theorem~\ref{main1} and some consequences of it. In particular we apply Theorem~\ref{main1} to the adjoint subcategory of the category of representations of a factorizable Hopf algebra. In this way we obtain a relation, via the Drinfeld map, between the Hopf center and the first commutator of a factorizable Hopf algebra.
In Section~\ref{smr} we prove Theorem~\ref{mgqtr}. Some consequences of this result are also described. In Section~\ref{h8} we give an example, by considering the semisimple quasitriangular Hopf algebra~$H_8$ of dimension~$8$. Based on our results we are able to compute the function~$m$ in this case and therefore the centralizer of any fusion subcategory of~$\rep(H_8)$.
We work over an algebraically closed field~$\Bbbk$ of characteristic zero. The comultiplication and antipode of a Hopf algebra are denoted by~$\Delta$ and~$S$ respectively. We use Sweedler’s notation for comultiplication with the sigma symbol dropped. All the other Hopf algebra notations are those used in~\cite{montg}.
\section{Preliminaries}\label{prelim} Let $A$ be a finite-dimensional semisimple Hopf algebra over an algebraically closed field $\Bbbk$ of characteristic zero. Then~$A$ is also cosemisimple and $S^{2}=\id$~\cite{Lard}. The character ring $C(A):=\G_0(A)\otimes_{\mathbb Z}\Bbbk$ is a semisimple subalgebra of~$A^*$ and it has a vector space basis given by the set $\mathrm{Irr}(A)$ of irreducible characters of~$A$, see~\cite{Z}. Moreover, $C(A)=\mathrm{Cocom}(A^*)$, the space of cocommutative elements of~$A^*$. By duality, the character ring of $A^*$ is a semisimple subalgebra of $A$ and $C(A^*)=\mathrm{Cocom}(A)$. If $M$ is an $A$-representation with character $\chi$ then~$M^*$ is also an $A$-representation with character $\chi^*=\chi \circ S$. This induces an involution ``$\;^*\;$'': $C(A)\rightarrow C(A)$ on~$C(A)$.
Throughout of this paper we denote by $\Lam$ an idempotent integral of $A$ and by $t$ an idempotent integral of $A^{*}$. Moreover one has that $t(\Lam)=\frac{1}{\dim_{\Bbbk}(A)}$. Recall also~\cite{Lar} that \begin{gather*} \dim_{\Bbbk}(A)\Lam=\sum_{d \in \mathrm{Irr}(A^{*})}\epsilon(d)d \end{gather*} is the regular character of $A^{*}$. Dually \begin{gather*} \dim_{\Bbbk}(A)t=\sum_{\chi \in \mathrm{Irr}(A)}\chi(1)\chi \end{gather*} is the regular character of~$A$.
Recall that a {\it left} coideal subalgebra of~$A$ is a subalgebra $L\subseteq A$ with $\Delta(L)\subset A\otimes L$. Then~$L$ is called {\it left normal coideal subalgebra} if $L$ is closed under the left adjoint action of $A$, i.e., $a_{1}lS(a_{2})\in L$ for any $l \in L$ and any $a\in A$. Recall also from~\cite{gmj} that given a left coideal subalgebra~$L$ of~$A$ there is a unique element $\Lam_{L}\in L$ (called {\it integral}) such that $l\Lam_{L}=\epsilon(l)\Lam_{L}$ for all $l \in L$, see also~\cite{kopp}. Then the coideal subalgebra~$L$ is normal if and only if~$\Lam_{L}$ is central, i.e., $\Lam_L\in Z(A)$.
For any left normal coideal subalgebra $L$ of $A$ the augmentation ideal $AL^{+}$ is a Hopf ideal and it has the following form $AL^{+}=A(1-\Lambda_{L})=\mathrm{Ann}_{A}(\Lambda_{L})$. Thus one can define the Hopf quotient $A//L:=A/AL^+$. It is well known that any fusion subcategories of~$\rep(A)$ can be written as $\rep(A//L)$ for some Hopf quotient of~$A$. \begin{Remark}\label{regq} Since $A$ is free as left $L$-module \cite{sk} it follows that the map \begin{gather*} A\otimes_{L}\Bbbk\simeq A\Lam_{L},\qquad a\otimes_{L}1\mapsto a\Lam_{L} \end{gather*} is an isomorphism of $A$-modules. Moreover, by \cite[Proposition~3.11]{iop} it follows that the regular module of the quotient Hopf algebra~$A//L$ is isomorphic to the induced module~$A\otimes_{L}\Bbbk$. \end{Remark} \subsection{Duality between the character ring and the center} Let $A$ be a semisimple Hopf algebra over the ground field $\Bbbk$. Let us denote by $\mathrm{Irr}(A)$ the set of irreducible characters of $A$. We suppose that $\mathrm{Irr}(A)=\{\chi_{0}, \chi_{1}, \dots, \chi_{r}\}$. Without loss of generality we may suppose that $\chi_{0}=\epsilon$. Let also $E_{0}, E_{1}, \dots, E_{r}$ be the corresponding central primitive central idempotents in $A$. The evaluation form \begin{gather*}\label{evform} C(A)\otimes \mathcal Z(A)\rightarrow \Bbbk,\qquad \chi \otimes a\mapsto \chi(a) \end{gather*} is nondegenerate. A pair of dual bases for this form is given by $\big\{\chi_{i}, \frac{1}{n_{i}}E_{i}\big\}$ since $\big\langle \chi_{i}, \frac{1}{n_{j}}E_{j}\big\rangle =\delta_{i,j}$ for any $1 \leq i,j \leq r$.
\subsection{Another pair of dual basis in the commutative case} Let $A$ be a semisimple Hopf algebra with a commutative character ring. According to~\cite{CW2} in the case of a commutative ring~$C(A)$ there is another pair of dual bases corresponding to this nondegenerate form. This pair of dual bases is given in terms of the conjugacy class sums as defined in~\cite{CW2}.
The {\it conjugacy classes $\cc^{j}$} of $A$ are defined as $\cc^{j}=\Lam \leftharpoonup F_{j}A^{*}$, where $\Lam=\Lam_A$ is a two-sided idempotent integral of $A$ and $\{F_j\}_j$ is the (complete) set of central primitive idempotents of the semisimple algebra $C(A)$. This notion of conjugacy classes generalizes the usual notion of conjugacy classes in finite groups.
\begin{Example}Let $G$ be a finite group and $A=\Bbbk G$ be the associated group algebra. It is easy to see that the conjugacy classes as defined above coincide with the usual notion of conjugacy class in a group. Indeed, let ${\mathcal C}_0, {\mathcal C}_1, \dots ,\mathcal C_r$ be the usual conjugacy classes of~$G$. Then the set of central primitive idempotents of $C(\Bbbk G)$ can be described as $p_{j}=\sum\limits_{h \in \cc^j}p_h$ where $p_h\in \Bbbk^G$ is defined as $p_h(g)=\delta_{g,h}$. Since $\Lam=\frac{1}{|G|}\sum\limits_{g \in G}g$ it follows that \begin{gather*} \cc^j=\Lam\leftharpoonup p_j\Bbbk^G=\Bbbk [\mathcal C_j] \end{gather*} is the vector sub-space of $\Bbbk G$ generated by all group elements of~$\cc_j$. \end{Example}
Recall that the Fourier transform $\mathcal{F}\colon A^*\rightarrow A$ defined by $f\mapsto \Lam \leftharpoonup f$ for any $f \in A^*$ is a~$\Bbbk$-linear isomorphism. Since $A^*=\bigoplus\limits_{j=0}^rF_jA^*$ and $\cc^j=\mathcal{F}(F_jA^*)$ one has \begin{gather*} A=\bigoplus_{j=0}^{r}{\cc^{j}}. \end{gather*}
One can also define the corresponding {\it conjugacy class sums} \begin{gather*} {\bf C}_{j}=\Lam \leftharpoonup (\dim A)F_{j}. \end{gather*} Note that ${\bf C}_j\in \mathcal Z(A)$ and since $\dim_{\Bbbk}\mathcal Z(A)=\dim_{\Bbbk}C(A)$ it follows that ${\cc^{j}}\cap \mathcal Z(A)=\Bbbk {\bf C}_j$. Note also that $\dim_{\kk}(A)\Lam=\sum\limits_{j=0}^{r}{\bf C}_{j}$. Indeed, $\Lam=\Lam \leftharpoonup \epsilon=\Lam\leftharpoonup \sum\limits_{j=0}^rF_j= \frac{1}{\dim_{\kk}(A)}\sum\limits_{j=0}^r\mathbf{ C}_j$. \begin{Example}
If $A=\Bbbk G$ then \begin{gather*}
{\bf C}_{j}=\bigg(\frac{1}{|G|}\sum_{g \in G}g\bigg) \leftharpoonup |G|p_{j}=\sum_{g \in G}p_j(g)g=\sum_{h \in C_j}h \end{gather*} is the usual class sum of a conjugacy class $\mathcal C_j$. \end{Example} \begin{Remark}\label{genmz} By the class equation for semisimple Hopf algebras, see~\cite{leq}, one has that the value $n_{j}:=\frac{\dim_{\Bbbk}A^{*}}{\dim_{\Bbbk}(A^{*}F_{j})}$ is an integer. Moreover as in \cite[equation~(11)]{CW2} one can deduce that $F_{j}(\Lambda)=\frac{1}{n_{j}}$. \end{Remark}
\begin{Example}If $A=\Bbbk G$ then $n_j=\frac{|G|}{|C_j|}$ is the order of the centralizer of any group element $g_j \in \mathcal C_j$. \end{Example}
This implies that a second pair of dual bases for the form of equation~\eqref{evform} can be given by $ \big\{F_{i}, \frac{n_{i}}{\dim_{\Bbbk}(A)}{\bf C}_{i}\big\}$, see also \cite[equation~(17)]{CW2}. This can be written as $\big\langle F_{i}, \frac{n_{j}}{\dim_{\Bbbk}(A)}{\bf C}_j\big\rangle =\delta_{i,j}$.
Indeed, one has \begin{gather*} \left\langle F_{i}, \frac{n_{j}}{\dim_{\Bbbk}(A)}{\bf C}_j\right\rangle = \frac{n_{j}}{\dim_{\Bbbk}(A)}\langle F_i, \Lam\leftharpoonup (\dim_{\kk}(A))F_j\rangle
= {n_{j}}\langle F_jF_i, \Lam\rangle =\delta_{i,j}n_jF_j(\Lam)=\delta_{i,j}. \end{gather*}
\subsection{Decomposition of the integral} Let $L$ be a left normal coideal subalgebra of a semisimple Hopf algebra $A$ with a commutative character ring~$C(A)$. We shall use the notation $\lambda_{L}\in (A//L)^{*}$ for the idempotent integral of the Hopf algebra $(A//L)^{*}$. Clearly $\lambda_{L}\in C((A//L)^{*})\subset C(A^{*})$ and we may suppose that \begin{gather}\label{intldec} \lambda_L=\sum_{j \in \mtc{I}_L}F_j \end{gather} for some subset of indices $\mtc{I}_L\subseteq \{0,1, \dots, r\}$. Note that by \cite[Lemma~1.1]{CW10} $\widetilde{\Lam_{L}}:=\Lam \leftharpoonup \lambda_{L}$ is a~left integral for~$L$. It follows from above that \begin{gather*} \widetilde{\Lam_{L}}=\frac{1}{\dim_{\kk}(A)}\sum_{j \in \mtc{I}_L}\mathbf{ C}_j. \end{gather*} Then one has \begin{gather*} L=\Lam_{L}\leftharpoonup A^{*}=\bigoplus_{j \in \mathcal I_{L}}{\bf C}_j\leftharpoonup A^{*}=\bigoplus_{j \in \mathcal I_{L}}{\cc^{j}}. \end{gather*} Note also that \begin{gather*} \epsilon(\mathbf{ C}_j)= \dim_{\kk}(A) F_j(\Lam)=\frac{\dim_{\kk}(A)}{n_j}=\dim_{\kk}(F_jA^*)=\dim_{\kk}\big(\cc^j\big). \end{gather*} Thus \begin{gather*} \epsilon\big(\widetilde{\Lam_{L}}\big)=\frac{1}{\dim_{\kk}(A)}\bigg(\sum_{j \in \mtc J_L}\epsilon(\mathbf{ C}_j)\bigg)=\frac{1}{\dim_{\kk}(A)}\bigg(\sum_{j \in \mtc J_L}\dim_{\kk}\big(\cc^j\big)\bigg)=\frac{\dim_{\kk}(L)}{\dim_{\kk}(A)}. \end{gather*} It follows then that{\samepage \begin{gather}\label{decl3} \blam_L=\frac{\dim_{\kk}(A)}{\dim_{\kk}(L)}\widetilde{\Lam_{L}}=\frac{1}{\dim_{\kk}(L)}\sum_{j \in \mtc{I}_L}\mathbf{ C}_j \end{gather}
is a formula for the idempotent integral of~$L$.}
Define the functional $p_{{\cc^{j}}}\in A^{*} $ as the unique linear functional that coincides to~$\epsilon$ on~${\cc^{j}}$ and it is equal to zero on the other conjugacy classes $\mathcal C_{l}$ with $l \neq j$. The following lemma was proven in \cite[Theorem~5.13]{repalg}.
\begin{Lemma} \label{fct}Suppose that $A$ is a semisimple Hopf algebra with a commutative character ring $C(A)$. Let $\{F_j\}_{0\leq j\leq r}$ be a complete set of central primitive idempotents of~$C(A)$. Then $F_{j}=p_{{\cc^{j}}}$ for all $0\leq j\leq r$. \end{Lemma}
Equation \eqref{decl3} and the above lemma implies the following: \begin{Lemma}\label{charval} Let $L$ be a left normal coideal subalgebra of a semisimple Hopf algebra~$A$. With the above notations one has $j \in \mathcal I_{L} \iff F_{j}(\Lam_{L})\neq 0$. \end{Lemma}
Let $A$ be a semisimple Hopf algebra with a commutative character ring $C(A)$. Then $\{F_{j}\}$ form a $\Bbbk$-linear basis for $C(A)$ and for any character $\chi \in C(A)$ one can write $\chi=\sum\limits_{j=0}^{r}\alpha_{\chi,j}F_{j}$ with $\alpha_{\chi,j }\in \Bbbk$. Previous lemma implies the following result, see also \cite[Theorem~1.12]{CW-r}.
\begin{Proposition} \label{charl}Let $A$ be a semisimple Hopf algebra with a commutative character ring and $\chi \in C(A)$. Then one has $ \chi \in C(A//L) \iff \chi F_{j}=\chi(1)F_{j}$ for all $j \in \mathcal I_{L}.$ \end{Proposition}
\begin{proof} We may suppose that $\chi=\chi_{M}$ is the character of an $A$-module $M$. If $\chi \in C(A//L)$ then $\chi\lambda_{L}=\chi(1)\lambda_{L}$ and equation~\eqref{intldec} implies that $\chi F_{j}=\chi(1)F_{j}$, for any $j \in \mathcal J_{L}$. Conversely if $\chi F_{j}=\chi(1)F_{j}$ for all $j \in J_{L}$ then $\chi \lambda_{L}=\lambda_{L}\chi(1)$. Thus \begin{gather*} \chi(\Lam_{L})=\frac{\dim_{\kk}(A)}{\dim_{\kk}(L)}\chi\big(\widetilde{\Lam_{L}}\big)=\frac{\dim_{\kk}(A)}{\dim_{\kk}(L)}\chi(\lambda_{L}\rightharpoonup \Lam)=\frac{\dim_{\kk}(A)}{\dim_{\kk}(L)}\chi\lambda_{L}(\Lam)=\frac{\dim_{\kk}(A)}{\dim_{\kk}(L)}\chi(1)\lambda_{L}(\Lam).\! \end{gather*} On the other hand note that $\lambda_{L}(\Lam)=\epsilon\big(\widetilde{\blam_L}\big)=\frac{\dim_{\kk}(L)}{\dim_{\kk}(A)} $ by equation~\eqref{decl3}. Thus $\chi(\Lam_{L})=\chi(1)$ which shows that the restriction of the $A$-module $M$ to~$L$ is trivial. It follows that $M\in \rep(A//L)$. \end{proof} \begin{Example}If $A=\Bbbk G$ then a left normal coideal subalgebra of $A$ is of the form $L=\Bbbk N$ for some normal subgroup $N\unlhd G$. Then \begin{gather*}
\lambda_L=\sum_{C_j\subseteq N}p_j \qquad \text{and}\qquad \mathcal{I}_L=\{j\,|\, C_j\subseteq N\}. \end{gather*} \end{Example}
\subsection{Left kernels and Burnside formula}\label{lkern} Let $M$ be an $A$-module and let $\mathrm{LKer}_{ _A}(M)$ be the {\it left kernel of $M$}. Recall \cite{gmj} that $\mathrm{LKer}_{ _A}(M)$ is defined by \begin{gather*}
\mathrm{LKer}_{ _{A}}(M)=\{a \in A\,|\, a_1\otimes a_2m=a\otimes m,\,\text{for all}\, m\in M\}. \end{gather*}
Then by~\cite{gmj} it follows that $\lker_A(M)$ is the largest left coideal subalgebra of $A$ that acts trivially on~$M$. It is also a left normal coideal subalgebra. For example, if $A=\Bbbk G$ is the group algebra of a finite group $G$ and~$M$ is a $\Bbbk G$-module then $\lker_A(M)=\Bbbk \ker_G(M)$, where $\ker_G(M)=\{g \in G\,|\, g.m=m, \,\text{for all}\, m\in M\}$ is the usual kernel of~$M$.
Next theorem generalizes a well known result of Brauer in the representation theory of finite groups. \begin{Theorem}[{\cite[Theorem~4.2.1]{gmj}}]\label{charofim} Suppose that $M$ is a finite-dimensional module over a~semisimple Hopf algebra~$A$. Then \begin{gather*} \langle M\rangle =\mathrm{Rep}(A//\mathrm{LKer}_A(M)), \end{gather*} where $\langle M\rangle $ is the fusion subcategory of~$\mathrm{Rep}(A)$ generated by $M$. \end{Theorem}
This implies that for any left normal coideal subalgebra $L$ of $A$ one has that \begin{gather}\label{brint} \bigcap_{M\in \mathrm{Irr}(A//L)} \lker_{A}(M)=L. \end{gather}
The previous theorem also implies that any fusion subcategory of $\rep(A)$ is of the type $\rep(A//L)$ for some left normal coideal subalgebra~$L$ of~$A$. Moreover, for any $V\in \rep(A)$ one has
\begin{gather}\label{inq} M\in \rep(A//L)\iff \lker_{A}(M)\supseteq L. \end{gather}
\subsection{Quasitriangular and factorizable Hopf algebras}\label{qtr}
Recall that a Hopf algebra $A$ is called {\it quasitriangular} if $A$ admits an $R$-matrix, i.e., an element $R \in A\otimes A$ satisfying the following properties: \begin{enumerate}\itemsep=0pt \item[1)] $R \Delta(x)=\Delta^{\cop}(x)R $ for all $x \in A$, \item[2)] $(\Delta \otimes \id)(R)= R^{(1)} \otimes r^{(1)} \otimes R^{(2)}r^{(2)}$, \item[3)] $( \id \otimes \Delta)(R)=R^{(1)}r^{(1)} \otimes r^{(2)}\otimes R^{(2)}$, \item[4)] $( \id \otimes \epsilon)(R)=1=(\epsilon \otimes \id)(R)$. \end{enumerate} Here $R=r=R^{(1)}\otimes R^{(2)}=r^{(1)}\otimes r^{(2)}$. If $(A, R)$ is a quasitriangular Hopf algebra then the category of representations is a braided fusion category with the braiding given by \begin{gather*} c_{M, N}\colon \ M\otimes N\rightarrow N\otimes M,\qquad m\otimes n \mapsto R_{21}(n\otimes m)=R^{(2)}n\otimes R^{(1)}m \end{gather*} for any two left $A$-modules $M, N\in \rep(A)$ (see~\cite{Kas}). Recall that $R_{21}:= R^{(2)}\otimes R^{(1)}$. Denote $Q:=R_{21}R$. Then the monodromy of two objects is defined as \begin{gather*} c_{M,N}c_{N,M}\colon \ M\otimes N \rightarrow N\otimes M,\qquad (m\otimes n)\mapsto R^{(2)}R^{(1)}m\otimes R^{(2)}R^{(1)}n=Q(m\otimes n). \end{gather*}
A quasitriangular Hopf algebra $(A, R)$ is called {\it factorizable} if and only if the Drinfeld map \begin{gather*} \phi_R\colon \ A^* \rightarrow A, \qquad f \mapsto (f \otimes \id)(R_{21}R) \end{gather*}
is an isomorphism of vector spaces. In this situation, following \cite[Theorem~2.3]{schfact} $\phi_R$ maps the character ring~$C(A)$ onto the center $\mathcal Z(A)$ of~$A$ and the restriction $\phi_R|_{C(A)}$ is an isomorphism of algebras. \begin{Remark} \label{nat} By \cite[Lemma~4.1]{rqts} one has that $\phi_{R}(C)$ is a left normal coideal for any subcoalgebra $C$ of $A^{*}$. \end{Remark}
One can also define the map ${\;_R\phi}\colon A^*\rightarrow A$ by $ _{R}\phi(f)=(\id \otimes f)(Q)$ for all $f\in A^*$. Moreover by \cite[Theorem~2.1]{schfact} one has that ${}_R\phi(f\chi)={}_R\phi(f)\,{}_R\phi(\chi)$ for all $f \in A^*$ and $\chi \in C(A)$. Thus $_{R}\phi|_{C(A)}\colon C(A)\rightarrow \mathcal Z(A)$ is an isomorphism of $\Bbbk$-algebras.
\begin{Remark}\label{rphir} By \cite[Lemma~2.3]{rqts} one has that $S\circ_{R}\phi=\phi_{R}\circ s$ where $S$ and $s$ are the antipodes of $A$ and $A^{*}$ respectively. \end{Remark}
In the case of a factorizable Hopf algebra $_{R}\phi$ is also bijective and moreover by~\cite{CW5} the two maps $_{R}\phi$ and $\phi_R$ coincide on the character ring~$C(A)$.
\section{Proof of Theorem \ref{main1} on M\"uger centralizer}\label{fmr}
In this section we prove the first main theorem mentioned in the introduction. Given a fusion subcategory $\cd$ of a braided fusion category~$\cc$, recall that the M\"uger centralizer~$\cd'$ is defined as the fusion subcategory of~$\cc$ generated by all simple objects~$X$ of~$\cc$ satisfying \begin{gather*} c_{X, Y}c_{Y, X}=\mathrm{id}_{X\otimes Y}\end{gather*} for all objects $Y \in \mathcal O(\cd)$ (see also~\cite{proclond}). Recall that $\mathcal O(\cd)$ denotes the set of isomorphism classes of simple objects of~$\cd$.
Let $A$ be a semisimple quasitriangular Hopf algebra over $\Bbbk$ and $\cd=\rep(A//L)$ be a fusion subcategory of $\rep(A)$ where $L$ is a left normal coideal subalgebra of~$A$.
\begin{Lemma}\label{firstequiv} Let $(A, R)$ be a semisimple quasi\-trian\-gular Hopf algebra and $L$, $N$ be two left normal coideal subalgebras. Then the following assertions are equivalent: \begin{enumerate}\itemsep=0pt \item[$1)$] $\rep(A//N)\subseteq \rep(A//L)'$, \item[$2)$] the following equation holds in $A\otimes A$: \begin{gather}\label{cond1} Q(\Lam_{L}\otimes \Lam_{N})=\Lam_{L}\otimes \Lam_{N}, \end{gather} \item[$3)$] $N\supseteq \phi_R((A//L)^*)$. \end{enumerate}
\end{Lemma}
\begin{proof}$(1)\iff (2)$ It is well known that two fusion subcategories of $\rep(A)$ centralize each other if and only if their regular representations centralize. Thus one needs to show that the two regular representations of $A//L$ and $A//N$ centralize each other if and only if equation~\eqref{cond1} holds. On the other hand from the definition of the braiding in $\rep(A)$ the two representations centralize each other if and only if $Q=R^{(2)}r^{(1)}\otimes R^{(1)}r^{(2)}$ acts as identity on their tensor product $A//L\otimes A//N$. By Remark~\ref{regq} one has $\Bbbk\uparrow^{A}_{L}\otimes \Bbbk\uparrow^{A}_{N}=A\Lam_{L}\otimes A\Lam_{N}$. Since $\Lam_{L}$ and $\Lam_{N}$ are central elements of $A$ it is clear that~$Q$ acts as identity on this subspace of $A\otimes A$ if and only if equation~\eqref{cond1} holds.
$(2)\implies (3)$ By \cite[Lemma~1.1]{CW10} one has $(A//L)^*=\Lam_L\rightharpoonup A^*$. Therefore $\phi_{R}(\Lam_{{L}}\rightharpoonup f)=f(Q^{1}\Lam_{L})Q^{2}$ for any $f \in A^{*}$. From here, applying equation~\eqref{cond1} it follows \begin{gather}\label{condx} \phi_{R}(\Lam_{{L}}\rightharpoonup f)\Lam_{{N}}=f\big(Q^{1}\Lam_{{L}}\big)Q^{2}\Lam_{{N}}=f(\Lam_{{L}})\Lam_{{N}} .\end{gather} On the other hand note that \begin{gather}\label{inp} \epsilon_A(\phi_{R}(\Lam_{{L}}\rightharpoonup f))=f(\Lam_{{L}}). \end{gather} If $L':=\phi_{R}((A//{L})^{*})$ then equation~\eqref{inp} gives that $A(L')^{+} \subseteq AN^{+}$ and by \cite[Lemma~6.2]{iop} one has that $L'\subseteq {N}$.
$(3)\implies (2)$ In this situation equation~\eqref{condx} is satisfied for any $f \in A^*$ which shows that equation~\eqref{cond1} also holds.\end{proof} \begin{Remark} If $\mathcal B\subseteq \cd$ are fusion subcategories of a braided fusion category $\cc$ then clearly $\cd'\subseteq \mathcal B'$. In particular if $
\rep(A//N)\subseteq \rep(A//L)'$ then by centralizing once more one has that $\rep(A//L)\subseteq \rep(A//L)''\subseteq \rep(A//N)'$. Thus the three above conditions from the previous lemma are also equivalent to: \begin{enumerate}\itemsep=0pt \item[1)] $\rep(A//L)\subseteq \rep(A//N)'$, \item[2)] $Q(\Lam_{N}\otimes \Lam_{L})=(\Lam_{N}\otimes \Lam_{L})$, \item[3)] $L\supseteq \phi_R((A//N)^*)$. \end{enumerate} \end{Remark}
\subsection{Proof of Theorem \ref{main1}}
\begin{proof} Let $M:=\phi_R((A//L)^*)$ as in the statement of the theorem. By Remark~\ref{nat} it is well-known that~$M$ is also a left normal coideal subalgebra of $A$. Suppose also that $\rep(A//L)'=\rep(A//L^\circ)$ for some other left normal coideal subalgebra $L^\circ$ of $A$.
We need to prove that $L^\circ=M$. Note that for any left normal coideal subalgebra $N$ of $A$ one has $\rep(A//N)\subseteq \rep(A//L)'$ if and only if $\rep(A//N)\subseteq \rep(A//L^\circ)$, i.e $L^\circ\subseteq N$. Then the previous Lemma~\ref{firstequiv} shows that for any left normal coideal subalgebra~$N$ of~$A$ one has \begin{gather*} L^\circ\subseteq N\iff M\subseteq N. \end{gather*} In particular, for $N=L^\circ$ one obtains that $M\subseteq L^\circ$. For $N=M$ one obtains the other inclusion $L^\circ\subseteq M$. Thus $L^\circ=M$ and the proof is complete. \end{proof}
Theorem~\ref{main1} can be rewritten as following by using the notion of left kernel of an $A$-module, see Section~\ref{lkern}: \begin{Theorem}\label{putgh1} Let $(A, R)$ be a quasitriangular semisimple Hopf algebra and~$L$ be a left normal coideal subalgebra of~$A$. If $M\in \rep(A)$ then the following assertions are equivalent: \begin{enumerate}\itemsep=0pt \item[$1)$] $M\in \rep(A//L)'$, \item[$2)$] $\phi_{R}((A//L)^{*})\subseteq \lker_{A}(M)$. \end{enumerate} \end{Theorem} \begin{proof}One has that $\rep(A//L)'=\rep(A//L^\circ)$ where $L^\circ=\phi_{R}((A//L)^{*})$. Then by equation~\eqref{inq} one has $M\in \rep(A//L^\circ)\iff \lker_{A}(M)\supseteq L^\circ$. \end{proof}
\subsection{On the commutators and Hopf centre}
Given a fusion category~$\cc$ we denote by ${\cc_{\mtr{pt}}}$ the fusion subcategory generated by the invertible objects of~$\cc$. In the case $\cc=\rep(A)$ for a semisimple Hopf algebra we have that ${\cc_{\mtr{pt}}}$ is the full abelian subcategory generated by one-dimensional modules. It was shown in~\cite{iop} that $\rep(A)_{\rm pt}=\rep(A/I)$ where $I:=\{ab-ba\,|\, a, b\in A\}$ is the first commutator of the $\Bbbk$-algebra~$A$. Moreover the commutator ideal $[A, A]$ is a Hopf ideal and by Takeuchi's correspondence it corresponds to a left normal coideal subalgebra~$A'$. Thus $A(A')^+=I$ and \begin{gather*} \rep(A)_{\rm pt}=\rep(A//A'). \end{gather*} Moreover by \cite{iop} $A'$ is the smallest left normal coideal subalgebra~$L$ with the property that~$A//L$ is a commutative Hopf algebra. $A'$ is called the {\it the commutator} of~$A$. \begin{Example} If $A=\Bbbk G$ then $A'=\Bbbk G'$ where $G'$ is the first commutator of $G$, i.e., $G'=[G,\;G]$. \end{Example}
For a fusion category $\cc$, recall that the adjoint subcategory $\cc_{\rm ad}$ is defined as the smallest fusion subcategory generated by all objects of the type $X\otimes X^*$ with $X$ a~simple object of~$\cc$. If $\cc=\rep(A)$ is the category of representations of a semisimple Hopf algebra~$A$ then it is well-known that \begin{gather*} \rep(A)_{\rm ad}=\rep(A//K(A)), \end{gather*} where $K(A)$ is the Hopf centre of $A$, i.e., largest central Hopf subalgebra of~$A$.
Recall that a braided fusion subcategory $\cc$ is called {\it nondegenerate} if its M\"uger center is trivial, i.e., $\cc'=\mathrm{Vec}$. If $\cc$ is a nondegenerate braided fusion category then by \cite[Corollary~3.11] {DGNO} one has \begin{gather}\label{cadc} (\cc_{\rm ad})'=\cc_{\rm pt}. \end{gather} Since $\cd''=\cd$ for any fusion subcategory $\cd$ of a nondegenerate braided category $\cc$ we can also write that \begin{gather}\label{cadc2} (\cc_{\rm pt})'=\cc_{\rm ad}. \end{gather} \begin{Proposition} Let $A$ be a factorizable semisimple Hopf algebra. With the above notations, one has that \begin{gather*} \phi_{R}((A//K(A))^{*})=A'\qquad \text{and}\qquad \phi_R((A/I)^*)=K(A). \end{gather*} \end{Proposition}
\begin{proof} For a quasitriangular Hopf algebra $(A, R)$ it is well known that $\rep(A)$ is nondegenerate if and only if~$A$ is a factorizable Hopf algebra. In this case equation~\eqref{cadc} gives that
\begin{gather*}
\rep(A//K(A))'=\rep(A//A').
\end{gather*} Then Theorem \ref{main1} gives $\phi_R((A//K(A))^*)=A'$.
Similarly, equation~\eqref{cadc2} gives that
\begin{gather*}
\rep(A//A')'=\rep(A//K(A))
\end{gather*} and Theorem~\ref{main1} implies $K(A)=\phi_{R}((A//A')^{*})$ .\end{proof}
\section{Conjugacy classes and M\"uger centralizer}\label{smr} In this section we will prove Theorem~\ref{mgqtr}. Suppose that $(A,R)$ is a semisimple quasitriangular Hopf algebra. Let as above $V_{0}, V_{1},\dots ,V_{r}$ be a complete set of isomorphism classes of irreducible $A$-modules. Let also $\mathrm{Irr}(A)=\{\chi_{0}, \chi_{1}, \dots ,\chi_{r}\}$ be the set of irreducible characters afforded by these modules and $\{E_{0}, \dots ,E_r\}$ be their associated central primitive idempotents of~$A$. Without loss of generality we may suppose that $V_0=\Bbbk$ is the trivial $A$-module and therefore $\chi_0=\epsilon$ and $E_0=\Lam$.
Recall by \cite{EG}, that $\rep(A)$ is a ribbon category with the canonical ribbon element $v=u^{-1}$, where $u:=S\big(R^1\big)R^2$ is the Drinfeld element of $(A,R)$. With respect to the canonical ribbon structure given by this ribbon element, the $S$-matrix of $(A, R)$ has entries \begin{gather*} s_{ii'}:=\mathrm{tr}_{V_{i}\otimes V_{i'}}(Q)=(\chi_{i}\otimes \chi_{i'})(Q)=\langle \chi_{i'}, \phi_{R}(\chi_{i})\rangle. \end{gather*}
It follows from \cite{dgno2} that one has $|s_{ii'}|\leq \chi_{i}(1)\chi_{i'}(1)$ and $V_{i}, V_{i'}$ centralize each other if and only if $s_{ii'}=\chi_{i}(1)\chi_{i'}(1).$
The Drinfeld map $\phi_{R}\colon C(A)\rightarrow \mathcal Z(A) $ is an algebra map and we may suppose as in the introduction that \begin{gather*} \phi_{R}(F_{j})=\sum_{i \in \mathcal A_{j}}E_{i} \end{gather*} for some subset $\cA_j\subseteq \{0, \dots , r\}$.
Without loss of generality we may also suppose that $F_{0}=t$, the idempotent integral of~$A^{*}$. Then $\phi_R(F_{0})$ is the idempotent integral of ${ \Phi(A)}:=\phi_R(A^*)$ since $\phi_R(f)\phi_R(F_{0})= \phi_R(fF_{0})=f(1)\phi_R(F_{0})=\epsilon(\phi_R(f))\phi_R(F_{0})$ for any $f \in A^{*}$ and also $\epsilon(\phi_R(F_{0}))=F_0(1)=1$.
Note that the set $\mathcal A_{j}$ is empty if and only if $\phi_{R}(F_{j})=0$. Denote by $\mathcal J\subseteq \{0,1,\dots, r\}$ the set of all indices $j$ with $\mathcal A_{j}$ not a empty set. Since $\phi_{R}(1)=1$ we obtain in this way a partition for the set of indices of all irreducible representations $\{0,1, \dots, r\}=\bigsqcup\limits_{j\in \mathcal J}\mathcal A_{j}$.
For any index $0\leq i\leq r$ we denoted by $m(i)$ the unique index $j \in \mathcal J$ such that $i \in \mathcal A_{j}$. Therefore in this way we obtain a unique function \begin{gather*} m\colon \ \{0, 1,\dots, r\}\rightarrow \mathcal J \end{gather*} with the property that $E_i\phi_R(F_{m(i)})\neq 0$ for all $i \in \{0, 1,\dots, r\}$.
Recall from Section~\ref{lkern} the definition of the left kernel of an $A$-module. \begin{Lemma}Let $(A, R)$ be a quasitriangular Hopf algebra and $V_{i}$, $V_{i'}$ be two irreducible $A$-representations. Then, with the above notations the following assertions are equivalent: \begin{enumerate}\itemsep=0pt \item[$1)$] $V_{i}$ and $V_{i'}$ centralize each other in $\rep(A)$, \item[$2)$] $\chi_{i'}F_{m(i)}=\chi_{i'}(1)F_{m(i)}$, \item[$3)$] $\chi_{i}F_{m(i')}=\chi_{i}(1)F_{m(i')}$, \item[$4)$] $\cc^{m(i)}\subseteq \lker_{A}(V_{i'})$, \item[$5)$] $\cc^{m(i')}\subseteq \lker_{A}(V_{i})$. \end{enumerate} \end{Lemma}
\begin{proof}For any character $\chi \in C(A)$ write as above $\chi=\sum\limits_{j=0}^{r}\alpha_{\chi,j}F_{j}$. Then one has that \[ \phi_{R}(\chi)=\sum\limits_{j=0}^{r}\alpha_{\chi,j}\phi_{R}(F_{j})=\sum\limits_{j=0}^{r}\alpha_{\chi,j}\bigg(\sum\limits_{s \in \mathcal A_{j}}E_{s}\bigg). \] With these formulae note that \begin{gather*} s_{ii'}=\langle \chi_{i}, \phi_{R}(\chi_{i'})\rangle =\bigg\langle \chi_{i}, \sum_{j\in \mathcal J}\sum_{s\in \mathcal A_{j}}\alpha_{\chi_{i'}, j}E_{s}\bigg\rangle =\chi_{i}(1)\alpha_{\chi_{i'}, m(i)}, \end{gather*} where $m(i)$ as above, is the unique index $j \in J$ with $i \in \mathcal A_{j}$. Therefore we see that~$V_{i}$ centralize~$V_{i'}$ if and only if \begin{gather}\label{centralize} \alpha_{\chi_{i'}, m(i)}=\chi_{i'}(1) \ \iff \ \chi_{i'}F_{m(i)}=\chi_{i'}(1) \ \iff \ \cc^{m(i)}\subseteq \lker_{A}(V_{i'}). \end{gather}
The equivalence of assertions (2) and (4) follows from \cite[Theorem~3.6]{CW5}. The rest of the equivalences follow from the symmetry property of the centralizer. \end{proof}
\begin{Remark}The above lemma also shows that if $V_{i}$ centralizes $V_{i'}$ then $V_{i}$ centralize all $V_{i''}$ with $i''\in \mathcal A_{m(i')}$. \end{Remark} Next theorem is a generalization of Theorem~\ref{mgqtr}.
\subsection{Proof of Theorem \ref{mgqtr}}
\begin{proof}Using the previous lemma we have the following equalities: \begin{align*} \mathcal O(\rep(A//L)')& =\bigcap_{M\in \mathrm{Irr}(A//L)}\mathcal O(\langle M\rangle ')
=\bigcap_{M\in \mathrm{Irr}(A//L)}\big\{V_{i}\,|\, \cc^{m(i)}\subseteq \lker_{A}(M)\big\}\\ & =
\bigg\{V_{i}\,|\,\cc^{m(i)}\subseteq \bigcap_{M\in \mathrm{Irr}(A//L)} \lker_{A}(M)\bigg\}. \end{align*} On the other hand by equation~\eqref{brint} one has that \begin{gather*} \cap_{M\in \mathrm{Irr}(A//L)} \lker_{A}(M)=L \end{gather*}
and therefore $\mathcal O(\rep(A//L)')=\big\{\chi_{i}\,|\,\cc^{m(i)}\subseteq L\big\}$. \end{proof}
Another description of the simple objects of $\rep(A//L)'$ is given in the following: \begin{Proposition}\label{evlamqtr}If $(A, R)$ is a semisimple quasitriangular Hopf algebra then \begin{gather*}
\mathcal O(\rep(A//L)')=\{\chi_{i}\,|\, m(i) \in \mathcal I_{L}\}
=\{\chi_{i}\,|\, F_{m(i)}(\Lam_{L})\neq 0\}\end{gather*} for any left normal coideal subalgebra $L$ of $A$. \end{Proposition}
\begin{proof} By Theorem \ref{mgqtr} one has that $\mathcal O(\rep(A//L)')=\big\{\chi_{i}\,|\, \cc^{m(i)}\subseteq L\big\}$, i.e., $\mathcal O(\rep(A//L)')=\{\chi_{i}\,|\, m(i)\in \mathcal I_L \}$. On the other hand by Lemma~\ref{charval} one has that $\cc^{m(i)}\subseteq L$ if and only if $F_{m(i)}(\Lam_L)\neq 0$. \end{proof}
Corollary~\ref{putgh1} together with Theorem~\ref{mgqtr} gives the following: \begin{Theorem}\label{puttgh} Let $(A, R)$ be a quasitriangular semisimple Hopf algebra and~$L$ be a left normal coideal subalgebra of $A$. For an irreducible representation~$V_{i}$ of $A$ we have that the following assertions are equivalent: \begin{enumerate}\itemsep=0pt \item[$1)$] $V_{i}\in \mathcal O(\rep(A//L)')$, \item[$2)$] $\cc^{m(i)}\subseteq L $, \item[$3)$] $\phi_{R}((A//L)^{*})\subseteq \lker_{A}(V_{i})$. \end{enumerate} \end{Theorem}
\subsection[Description of ${ \Phi(A)}$]{Description of $\boldsymbol{{ \Phi(A)}}$}
As above denote by ${ \Phi(A)}:=\phi_R(A^*)$ the image of the Drinfeld map. \begin{Proposition}\label{mcenter} Suppose that $(A,R)$ is a quasitriangular Hopf algebra and $\phi_{R}(F_0) =\sum\limits_{i \in \mathcal A_{0}}E_{i}$ where $F_0=t$ is the idempotent integral of $A^{*}$. Then \begin{enumerate}\itemsep=0pt \item[$1)$] $\rep(A)'=\rep(A//{ \Phi(A)})$,
\item[$2)$] $\mathrm{Irr}(A//{ \Phi(A)})=\{\chi_{i}\,|\,i \in \mathcal A_{0}\}$, \item[$3)$] ${ \Phi(A)}=\bigoplus\limits_{j \in \mathcal J}{\cc^{j}}$. \end{enumerate} \end{Proposition} \begin{proof}By Theorem \ref{main1} one has that \begin{gather*} \rep(A)'=\rep(A//\Bbbk)'=\rep(A//\phi_R(A^*))=\rep(A//{ \Phi(A)}). \end{gather*} On the other hand, Theorem~\ref{mgqtr} gives the following equality \begin{gather*}
\mathcal O(\rep(A)')=\mathcal O(\rep(A//\Bbbk)')=\big\{\chi_{i}\,|\,\cc^{m(i)}\subseteq \Bbbk\big\}=\big\{\chi_{i}\,|\,\cc^{m(i)}= \Bbbk\big\}. \end{gather*}
It is easy to see that $\cc^{m(i)}= \Bbbk$ if and only if $m(i)=0$. Indeed, $\cc^{m(i)}=\mathcal F(F_{m(i)}A^*)= \Lam \leftharpoonup F_{m(i)}A^*$ and $\Bbbk=\mathcal F(F_{0}A^*)$. Since $\mathcal F$ is bijective the statement follows. Therefore $\mathcal O(\rep(A//{ \Phi(A)}))=\mathcal O(\rep(A)')=\{\chi_{i}\,|\,m(i)=0\}=\{\chi_{i}\,|\,i \in \mathcal A_{0}\}$.
For the last item note that $\mathcal O(\rep(A))=\mathcal O(\rep(A//{ \Phi(A)})')=\big\{\chi_{i}\,|\,\cc^{m(i)}\subseteq { \Phi(A)}\big\}$. This implies that ${ \Phi(A)}=\bigoplus_{j \in \mathcal J}{\cc^{j}}$. \end{proof}
\subsection{Proof of Corollary \ref{main2}} This is now a particular case of Theorem~\ref{mgqtr}.
\begin{proof} Note that if $A$ is factorizable then $\phi_R$ is bijective and every set $\mtc A_j$ is a~singleton. Moreover $\phi_R(F_0)=E_0$, the integral of~$A$ in this case. Then, without loss of generality, after a permutation of the indices, we may suppose $\phi_R(F_i)=E_i$ and therefore $m(i)=i$ for all $0\leq i \leq r$. Then the statement of Theorem~\ref{mgqtr} becomes Theorem~\ref{main2}. \end{proof}
For the rest of this subsection we suppose that $A$ is a semisimple factorizable Hopf algebra. As explained above without loss of generality we may also assume $\phi_R(F_i)=E_i$ and therefore that the function $m\colon \{0, 1,\dots, r\}\rightarrow \{0, 1,\dots, r\}$ is the identity map.
\begin{Proposition} Suppose that $(A,R)$ is a semisimple factorizable Hopf algebra. Then for any irreducible $A$-module $V_{i}$ one has that \begin{gather*}
\lker_{A}(V_{i})=\bigoplus_{\{i' \,|\, V_{i'}\;\text{centralize}\; V_i\}}{\cc^{i'}}. \end{gather*} \end{Proposition} \begin{proof} Since $\mathcal J=\{0, 1,\dots, r\}$ in this case, by equation~\eqref{centralize} and \cite[Theorem 3.6]{CW5} one has that \begin{gather*} \chi_{i}F_{i'}=\chi_{i}(1)F_{i'} \ \iff \ {\cc^{i'}}\subseteq \lker_{A}(V_{i}) \ \iff \ V_{i}\;\text{centralizes}\; V_{i'}. \end{gather*} It follows that in this case one has \begin{gather*}
\lker_{A}(V_{i})=\bigoplus_{\cc^{i'}\subseteq\lker_{A}(V_{i})}\cc^{i'}=\bigoplus_{\{i'\,|\, V_{i}\;\text{centralizes}\; V_{i'}\}}\cc^{i'}.\tag*{\qed} \end{gather*}\renewcommand{\qed}{} \end{proof}
\begin{Remark}\quad \begin{enumerate}\itemsep=0pt \item From the previous proposition, in the case of a semisimple factorizable Hopf algebra one can deduce that for any two irreducible characters $\chi_{i}$ and $\chi_{i'}$ one has $\mathcal C^{i'}\subseteq \lker_{A}(V_{i})\iff\mathcal C^{i}\subseteq \lker_{A}(V_{i'})$ \item Recall that in \cite[Theorem 1.4]{mathz} it is shown that \begin{gather*}
\mathcal O(\rep(A//K)')=\{\chi_{i}\,|\, F_{i}(\Lam_{K})\neq 0\} \end{gather*} for any normal Hopf subalgebra $K$ of a factorizable Hopf algebra~$A$. Note that Proposition~\ref{evlamqtr} generalizes the above result from normal Hopf subalgebras~$K$ to left normal coideal subalgebras $L$ of $A$. It also drops the factorizability assumption on~$A$. \end{enumerate} \end{Remark}
Define $C_{V_{i}}:=C_{\chi_{i}}\subset A^{*}$ as the subcoalgebra of $A^{*}$ generated by $\chi_{i}$. By \cite[Lemma~4.2(i)]{CW5}, in the factorizable case one has that $\phi_{R}(C_{V_{i}})=\mathcal C_{i}.$ for all $0\leq i \leq r$. \begin{Remark} Let $A$ be a semisimple factorizable Hopf algebra and~$L$ a left normal coideal subalgebra of~$A$. If $\cd:=\rep(A//L)$ then $(A//L)^{*}=\bigoplus\limits_{\chi_{j} \in \mathcal O(\cd)}C_{V_{j}}$ and $L^\circ=\phi_{R}((A//L)^{*})=\bigoplus\limits_{\chi_{j} \in \mathcal O(\cd)}\phi_{R}(C_{V_{j}})$, i.e., $L^\circ=\bigoplus\limits_{\chi_{j} \in \mathcal O(\cd)}\cc^{j}$. This gives another proof for Theorem~\ref{main2} in the case of a factorizable Hopf algebra since in this case $\cd''=\cd$. \end{Remark}
\section[Example $H_8$]{Example $\boldsymbol{H_8}$}\label{h8}
In this section we compute the centralizer of any fusion subcategory of the quasi-triangular Hopf algebra $H_8$, the unique semisimple non-trivial Hopf algebra of dimension~$8$. We note that the category of representations~$\rep(H_8)$ is a braided Tambara--Yamagami category and therefore~$\rep(H_8)\subset \rep(D(H_8))$. The $S$-matrix of the center of a Tambara--Yamagami was computed in~\cite{gnn}. Using this one can describe completely the centralizer of any fusion subcategory of~$\rep(H_8)$. However, we decided to include this example here to illustrate how Theorem~\ref{mgqtr} can be applied in a concrete example.
The eight-dimensional semisimple Hopf algebra (see \cite{kp-66, ma-6-8}) is generated by $\{x,y,z\}$ subject to the relations \begin{gather*} x^2=y^2=z^2=1,\qquad xz=zx,\qquad zy=yz, \qquad xyz=yx. \end{gather*} The comultiplication is given by \begin{gather}\label{d1}\Delta(x)=xe_0\otimes x+xe_1\otimes y, \qquad \Delta(y)=ye_1\otimes x+ye_0\otimes y,\qquad \Delta(z)=z\otimes z, \end{gather} and the counit by $\epsilon(x)=\epsilon(y)=\epsilon(z)=1.$ The antipode has the formulae \begin{gather*} S(x)=xe_0+ye_1,\qquad S(y)=xe_1+ye_0,\qquad S(z)=z. \end{gather*} Based on equation~\eqref{d1} one can compute that \begin{gather*} \Delta(xy)=xye_0\otimes xy+xye_1\otimes yx,\qquad \Delta(yx)=yxe_0\otimes yx+yxe_1\otimes xy. \end{gather*} It can also be checked that \begin{gather*} \Delta(xz)=xe_0\otimes xz-xe_1\otimes yz,\qquad \Delta(yz)=ye_0\otimes yz-ye_1\otimes xz. \end{gather*} Since $z$ is a central element in $H_8$, there are two central orthogonal idempotents: \begin{gather*} e_0=\frac{1}{2}(1+z),\qquad e_1=\frac{1}{2}(1-z). \end{gather*} \subsection[Dual basis of $H^*$]{Dual basis of $\boldsymbol{H^*}$}
$H_8$ has a $\Bbbk$-linear basis given by the set of elements $\{1, x, y, z, {xy}, {yx}, {xz}, {yz}\}$. We consider its linear dual basis on $H_8^*$ given by $\{p_1, p_x, p_y, p_z, p_{xy}, p_{yx}, p_{xz},p_{yz}\}$.
It is easy to see that the idempotent integrals of $H_8$ and $H_8^*$ are given by \begin{gather*} \Lam=\frac{e_0}{4}(1+x+y+xy)=\frac{1}{8}(1+x+y+xy+z+zx+zy+yx),\qquad \lambda=p_1. \end{gather*}
\subsection[$H_8^*$ representations]{$\boldsymbol{H_8^*}$ representations}
Since $H_8$ is a self dual Hopf algebra~\cite{alaoui} it has also four-1-dimensional representations given by the group like elements of $H_8$ and a~2-dimensional representation. One has that \begin{gather*} G(H_8)=\{1, g_1, g_2, z\}, \end{gather*} with $g_1=xy(e_0+ie_1)$, $g_2=xy(e_0-ie_1)$. It can be easily checked that $g_1g_2=z$, $zg_i=g_i$ and $g_i^2=1$. Moreover, the set of central grouplike elements of~$H_8$ is given by $\bar G(H_8)=\{1, z\}$.
\subsubsection{On the 2-dimensional comodule}
From equation~\eqref{d1} one can compute that \begin{alignat*}{3} & \Delta(xe_0)=xe_0\otimes xe_0+xe_1\otimes ye_1,\qquad && \Delta(xe_1)=xe_0\otimes xe_1+xe_1\otimes ye_0,& \\ & \Delta(ye_0)=ye_0\otimes ye_0+ye_1\otimes xe_1,\qquad && \Delta(ye_1)=ye_0\otimes ye_1+ye_1\otimes xe_0.& \end{alignat*} This shows that $W=\Bbbk w_1\oplus \Bbbk w_2$ is a left $H_8$-comodule with the comodule structure given by \begin{gather*} \rho(w_1)=xe_0\otimes w_1+xe_1\otimes w_2,\qquad \rho(w_2)=ye_1\otimes w_1+ye_0\otimes w_2. \end{gather*} Thus \begin{gather}\label{comsd} H_8=\Bbbk 1\oplus \Bbbk g_1 \oplus \Bbbk g_2\oplus \Bbbk z \oplus(\Bbbk xe_0\oplus \Bbbk ye_1 )\oplus (\Bbbk ye_0 \oplus \Bbbk xe_1) \end{gather} is a decomposition of $H_8$ into simple left $H_8$-comodules. We denote $M_0=\Bbbk 1$, $M_1=\Bbbk g_1$, $ M_2=\Bbbk g_2$, $M_3=\Bbbk z$ the four one-dimensional $H_8^*$-modules. Moreover, $M_4:= (\Bbbk xe_0\oplus \Bbbk ye_1 )$ is an irreducible $H_8^*$-module and $M_4\simeq M_5:=(\Bbbk ye_0 \oplus \Bbbk xe_1)$.
\subsection{Fourier transform} Based on the comultiplication formulae one can compute the Fourier transform \begin{gather*} \mathcal F\colon \ H_8^*\rightarrow H_8,\qquad f \mapsto \dim_{\kk} (H_8)f\rightharpoonup \Lam. \end{gather*} After some computations it follows that under $\mathcal F$ one has \begin{gather*} p_1 \mapsto 1, \qquad p_z\mapsto z,\;p_x\mapsto xe_0+ye_1, \qquad p_{xz}\mapsto xe_0-ye_1 \end{gather*} and \begin{gather*} p_y\mapsto ye_0+xe_1, \qquad p_{yz}\mapsto ye_0-xe_1,\qquad p_{xy}\mapsto yx, \qquad p_{yx}\mapsto xy. \end{gather*}
\subsection[Irreducible $H_8$-modules and their characters]{Irreducible $\boldsymbol{H_8}$-modules and their characters}
$H_8$ has four-1-dimensional modules $V_0$, $V_1$, $V_2$, $V_3$ and a 2-dimensional irreducible module~$V$. The action of the generators on these modules is given as follows.
For $V_0$, the trivial $H$-module the action is given by $xv=yv=zv=v$ and therefore the character is given by \begin{gather*} \chi_0=p_1+p_z+p_x+p_y+p_{xy}+p_{yx}+p_{xz}+p_{yz}. \end{gather*} For $V_1$ the action is given by $xv=-v$, $yv=v$, $zv=v.$ The character $\chi_1$ of $V_1$ has $\chi_1(1)=\chi_1(z)=\chi_1(y)=\chi_1(yz)=1$, $\chi_1(x)=\chi_1(yx)=\chi_1(xy)=\chi_1(xz)=-1$. Thus \begin{gather*} \chi_1=p_1+p_z-p_x+p_y-p_{xy}-p_{yx}-p_{xz}+p_{yz}. \end{gather*} For $V_2$ the action is given by $xv=-v$, $yv=-v$, $zv=v$ and the character is given by \begin{gather*} \chi_2=p_1+p_z-p_x-p_y+p_{xy}+p_{yx}-p_{xz}-p_{yz}. \end{gather*}
For $V_3$ one has $xv=v$, $yv=-v$, $zv=v$ and the character is given by \begin{gather*} \chi_3=p_1+p_z+p_x-p_y-p_{xy}-p_{yx}+p_{xz}-p_{yz}. \end{gather*} For $V_4=V$, the 2-dimensional simple module, if $V=\Bbbk v_1\oplus \Bbbk v_2$ then the action of generators given by \begin{gather*} xv_1=v_1,\qquad yv_1=v_2,\qquad zv_1=-v_1,\qquad xv_2=-v_2,\qquad yv_2=v_1,\qquad zv_2=-v_2. \end{gather*} The character $\chi_4$ of $V$ has the values $\chi_4(1)=2 \chi_4(z)=-2$, $\chi_4(x)=\chi_4(y)= \chi_4(yx)=\chi_4(xy)=\chi_4(xz)=\chi_4(zx)=0$. This gives that \begin{gather*} \chi_4=2p_1-2p_z. \end{gather*} \subsubsection{Multiplication of the characters}It can be easily checked that $\chi_4^2=\sum\limits_{i=0}^3\chi_i$, and $\chi_i\chi_4=\chi_4\chi_i=\chi_i$ for all $0\leq i \leq 3$. Moreover $\chi_i\chi_j=\chi_k$ if $\{i,j,k\}=\{1,2,3\}$.
\subsection{Central primitive idempotent of the character ring} Based on the above multiplication, the central primitive idempotents of $C(H_8)$ can be computed as follows \begin{gather*} F_0=\frac{1}{8}(\epsilon+\chi_1+\chi_2+\chi_3+2\chi_4)=p_1 \qquad \text{is the integral of $H_8^*$},\\ F_1=\frac{1}{4}(\epsilon+\chi_1-\chi_2-\chi_3)=p_y+p_{yz}, \qquad F_2=\frac{1}{4}(\epsilon-\chi_1+\chi_2-\chi_3)=p_{xy}+p_{yx}, \\ F_3=\frac{1}{4}(\epsilon-\chi_1-\chi_2+\chi_3)=p_x+p_{xz},\qquad F_4=\frac{1}{8}(\epsilon+\chi_1+\chi_2+\chi_3-2\chi_4)=p_z. \end{gather*} Note that $F_4$ is the central primitive idempotent of $H_8^*$ attached to the central grouplike element $z\in \bar G(H_8):=G(H_8)\cap \mathcal Z(H_8)$.
\subsection[Conjugacy class sums of $H_8$]{Conjugacy class sums of $\boldsymbol{H_8}$} Using the above formulae for the central primitive idempotents since $ {C}_i=8\big(\Lam\leftharpoonup F_i\big)$ it follows that \begin{gather*}
{C}_0=8(\Lam\leftharpoonup F_0)=1,\qquad {C}_1=8(\Lam\leftharpoonup F_1)=2ye_0,\\
{C}_2=8(\Lam\leftharpoonup F_2)=xy+yx=g_1+g_2, \end{gather*} and \begin{gather*}
{C}_3=8(\Lam\leftharpoonup F_3)=2xe_0,\qquad {C}_4=8(\Lam\leftharpoonup F_4)=z. \end{gather*}
\subsection{Description of the adjoint action} Using the antipode formulae one can see that the adjoint action of $H_8$ on itself can be given by \begin{gather*} x.a=xax,\qquad y.a=yay,\qquad z.a=a,\qquad \text{for all} \ a\in H_8. \end{gather*} \subsection[Conjugacy classes of $H_8$]{Conjugacy classes of $\boldsymbol{H_8}$}
Recall that the conjugacy classes are simple $D(H_8)$-modules~\cite{zind}. We rewrite the decomposition of $H_8$ into left $H_8$-comodules from equation~\eqref{comsd} as follows \begin{gather*} H_8=\Bbbk 1\oplus \Bbbk z\oplus (\Bbbk g_1\oplus \Bbbk g_2)\oplus (\Bbbk xe_0\oplus \Bbbk ye_1) \oplus (\Bbbk xe_1\oplus \Bbbk ye_0). \end{gather*} Moreover, by the above description of the left adjoint action it can be checked each of the above five subspaces is closed under the adjoint action of~$H_8$.
Clearly, the first two subspaces are irreducible $D(H_8)$-modules being one-dimensional. Since $ C_0\in \Bbbk 1$ and $ C_4=z\in \Bbbk z$ we deduce that \begin{gather*} \cc^0=\Bbbk 1,\qquad \cc^4=\Bbbk z. \end{gather*} It can be checked directly that the third $D(H_8)$-module is an irreducible $D(H_8)$-module since \begin{gather*} x.g_1=g_2,\qquad x.g_2=g_1,\qquad y.g_1=g_2,\qquad y.g_2=g_1. \end{gather*} Since $ C_2=g_1+g_2=xy+yx$ it follows that \begin{gather*} \cc^2=\Bbbk g_1\oplus \Bbbk g_2. \end{gather*}
Similarly one can check that the simple $H_8$-comodule $M_4=\Bbbk xe_0\oplus \Bbbk ye_1$ is an irreducible $D(H_8)$-module and since $ C_3=2xe_0=x+xz\in M_4$ we can say that \begin{gather*} \cc^3=\Bbbk xe_0\oplus \Bbbk ye_1. \end{gather*} By the same argument \begin{gather*} \cc^1=\Bbbk xe_1\oplus \Bbbk ye_0. \end{gather*}
\subsection[Presentation of the central primitive idempotents of $H_8$]{Presentation of the central primitive idempotents of $\boldsymbol{H_8}$}
The associated central primitive idempotents of $V_0$, $V_1$, $V_2$, $V_3$, $V$ can be computed as \begin{gather*} E_0=\frac{e_0}{4}(1+x+y+xy)=\Lam \end{gather*} is the central primitive idempotent of $\chi_0$, i.e., the idempotent integral~$\Lam_{H_8}$ of~$H_8$, \begin{gather*} E_1=\frac{e_0}{4}(1-x+y-xy) \end{gather*} is the central primitive idempotent of~$\chi_1$, \begin{gather*} E_2=\frac{e_0}{4}(1-x-y+xy) \end{gather*} is the central primitive idempotent of $\chi_2$, \begin{gather*} E_3=\frac{e_0}{4}(1+x-y-xy) \end{gather*} is the central primitive idempotent of $\chi_3$, \begin{gather*} E_4=e_1 \end{gather*} is the central primitive idempotent of $\chi_4$.
\subsection[The $R$-matrix and the Drinfeld map]{The $\boldsymbol{R}$-matrix and the Drinfeld map}
It is well-known that $H_8$ is a semisimple quasitriangular Hopf algebra~\cite{ma-6-8} with the $R$-matrix given by \begin{gather*} R=\frac{1}{2}(1\otimes 1+g_2\otimes 1+ 1\otimes g_1-g_2\otimes g_1). \end{gather*} It follows that \begin{gather*} Q=R_{21}R = \frac{1}{4}(1\otimes (1+z+g_1+g_2) + g_1\otimes (1+g_1-g_2-z)\\ \hphantom{Q=R_{21}R =}{}+ g_2\otimes (1-g_1+g_2-z)+z\otimes (1-g_1-g_2+z). \end{gather*}
\subsubsection[The Drinfeld map on $\Bbbk$-linear basis]{The Drinfeld map on $\boldsymbol{\Bbbk}$-linear basis} One can compute that the Drinfeld map $\phi_R\colon H_8^*\rightarrow H_8$, $f\mapsto f\big(Q^1\big)Q^2$ is given by \begin{gather*} p_1\mapsto \Lambda_{\Bbbk G}=\frac{1}{4}(1+g_1)(1+g_2)=e_{0,0},\qquad p_z\mapsto \frac{1}{4}(1-g_1)(1-g_2)= e_{11}, \\ p_{xy}\mapsto \frac{1}{4}(1+ig_1-ig_2-z),\qquad p_{yx}\mapsto \frac{1}{4}(1-ig_1+ig_2-z),\qquad p_x, p_y, p_{xz}, p_{yz}\mapsto 0. \end{gather*}
\subsubsection[The Drinfeld map given on the central primitive idempotents of the character ring]{The Drinfeld map given on the central primitive idempotents\\ of the character ring}
The following formulae hold for the Drinfeld map: \begin{gather*} \phi_R (F_0)=E_0+E_2,\qquad \phi_R (F_1)=\phi_R (F_3)=0,\qquad \phi_R (F_2)=E_4,\qquad \phi_R (F_4)=E_1+E_3. \end{gather*}
\subsubsection[The map $m$ and the subset $\mathcal J$]{The map $\boldsymbol{m}$ and the subset $\boldsymbol{\mathcal J}$}
It follows that in this case the subset $\mathcal J=\{0,2,4\}$ and the function $m\colon \{0,1,2,3,4\}\rightarrow \mathcal J$ is given by \begin{gather} \label{m} m(0)=m(2)=0, \qquad m(1)=m(3)=4, \qquad m(4)=2. \end{gather} Therefore $\phi_R(H^*_8)=\Bbbk G(H_8)$ and by Proposition \ref{mcenter} one has \begin{gather*}
\mathcal O(\rep(H_8)')=\{\chi_i\,|\, m(i)=0\}=\{\chi_0, \chi_2\}. \end{gather*} \subsection[Description of the centralizer based on the function $m$]{Description of the centralizer based on the function $\boldsymbol{m}$}
In this subsection we describe the centralizer of any fusion subcategory of $\rep(H_8)$. For the entire category, $\rep(H_8)$ this was done above.
There are two normal coideal subalgebras of $H_8$ which are not Hopf subalgebras, \begin{gather*} L_1=\Bbbk\langle 1, z, ye_0, xe_1\rangle =\lker_H(V_1) \end{gather*} and \begin{gather*} L_3=\Bbbk\langle 1, z, xe_0, ye_1\rangle =\lker_H(V_3). \end{gather*} One has \begin{gather*} L_1=\cc^0\oplus \cc^1\oplus \cc^4,\qquad L_2=\cc^0\oplus \cc^3\oplus \cc^4. \end{gather*} Theorem \ref{charofim} gives that $\rep(H_8//\lker_{H_8}(V_i))\simeq \langle V_i\rangle$, the fusion subcategory of $\rep(H_8)$ generated by $V_i$. Using Theorem \ref{mgqtr} it follows \begin{gather*}
\rep(A//L_1)'=\big\{\chi_i\,|\,\cc^{m(i)}\subseteq L_1\big\}=\{\chi_i\,|\,m(i)\in \{0,1,4\}\}. \end{gather*} By description of the function $m$ from equation~\eqref{m} it follows that \begin{gather*} \mathcal O(\langle V_1\rangle')=\{\chi_0, \chi_1, \chi_2, \chi_3\}. \end{gather*} Similarly, one has that \begin{gather*}
\mathcal O(\langle V_3\rangle')=\rep(A//L_3)'=\big\{\chi_i\,|\,\cc^{m(i)}\subseteq L_3\big\}=\{\chi_i\,|\, m(i)\in \{0,1,3\}\}. \end{gather*} By description of the function $m$ from equation \eqref{m} it follows that \begin{gather*} \mathcal O(\langle V_3\rangle ')=\{\chi_0, \chi_1, \chi_2, \chi_3\}. \end{gather*} Note that for the central linear character $\chi_2\in \mathcal Z(H_8^*)$ one has \begin{gather*} L_2:=\lker_H(V_2)=\Bbbk G(H_8)=\cc^0\oplus \cc^2\oplus \cc^4. \end{gather*} As above, using Theorem~\ref{mgqtr} it follows \begin{gather*}
\mathcal O(\langle V_2\rangle')=\{\chi_i\,|\,m(i)\in \{0,2,4\}\}=\mathcal O(\rep(H_8)). \end{gather*} On the other hand since $\langle V_4\rangle =\rep(H_8)$ Theorem~\ref{charofim} gives that $L_4=\lker_{H_8}(V)=\Bbbk=\cc^0.$ Thus, using again Theorem~\ref{mgqtr} it follows that \begin{gather*}
\mathcal O(\langle V_4\rangle')=\{\chi_i\,|\,m(i)=0\}=\{\chi_0, \chi_2\}=\mathcal O(\rep(H_8)'). \end{gather*} All the four one-dimensional centralize each other. $V$ centralizes only $\chi_2$.
\subsubsection{On the first commutator and adjoint subcategory} One has that the first commutator of $H_8$ is given by $H_8'=\Bbbk \langle 1, z\rangle$ and moreover, for this Hopf algebra ${\cc_{\mtr{ad}}}={\cc_{\mtr{pt}}}$. Thus in this case \begin{gather*} {\cc_{\mtr{ad}}}'=\langle V_1, V_2, V_3\rangle'=\bigcap_{i=0}^3 \langle V_i\rangle'={\cc_{\mtr{pt}}}. \end{gather*}
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Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity
Statistical stability for multi-substitution tiling spaces
October 2013, 33(10): 4595-4611. doi: 10.3934/dcds.2013.33.4595
Canard trajectories in 3D piecewise linear systems
Rafel Prohens 1, and Antonio E. Teruel 1,
Dep. Ciències Matemàtiques i Informàtica, Universitat de les Illes Balears, 07122 Palma de Mallorca, Illes Balears, Spain, Spain
Received July 2012 Revised January 2013 Published April 2013
We present some results on singularly perturbed piecewise linear systems, similar to those obtained by the Geometric Singular Perturbation Theory. Unlike the differentiable case, in the piecewise linear case we obtain the global expression of the slow manifold ${\mathcal S}_{\varepsilon}$. As a result, we characterize the existence of canard orbits in such systems. Finally, we apply the above theory to a specific case where we show numerical evidences of the existence of a canard cycle.
Keywords: canard solutions, piecewise linear differential systems, slow--fast system, Singular perturbation, slow manifold..
Mathematics Subject Classification: Primary: 34E15; Secondary: 34E17, 37G05, 34D1.
Citation: Rafel Prohens, Antonio E. Teruel. Canard trajectories in 3D piecewise linear systems. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4595-4611. doi: 10.3934/dcds.2013.33.4595
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Qunying Zhang, Zhigui Lin. Blowup, global fast and slow solutions to a parabolic system with double fronts free boundary. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 429-444. doi: 10.3934/dcdsb.2012.17.429
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Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123
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Rafel Prohens Antonio E. Teruel
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You are here: Home ∼ Weekly Papers on Quantum Foundations (35)
Weekly Papers on Quantum Foundations (35)
Published by editor on August 27, 2016
This is a list of this week's papers on quantum foundations published in various journals or uploaded to preprint servers such as arxiv.org and PhilSci Archive.
Operational General Relativity: Possibilistic, Probabilistic, and Quantum. (arXiv:1608.06940v1 [gr-qc])
hep-th updates on arXiv.org
on 2016-8-27 4:36am GMT
Authors: Lucien Hardy
In this paper we develop an operational formulation of General Relativity similar in spirit to existing operational formulations of Quantum Theory. To do this we introduce an operational space (or op-space) built out of scalar fields. A point in op-space corresponds to some nominated set of scalar fields taking some given values in coincidence. We assert that op-space is the space in which we observe the world. We introduce also a notion of agency (this corresponds to the ability to set knob settings just like in Operational Quantum Theory). The effects of agents' actions should only be felt to the future so we introduce also a time direction field. Agency and time direction can be understood as effective notions. We show how to formulate General Relativity as a possibilistic theory and as a probabilistic theory. In the possibilistic case we provide a compositional framework for calculating whether some operationally described situation is possible or not. In the probabilistic version we introduce probabilities and provide a compositional framework for calculating the probability of some operationally described situation. Finally we look at the quantum case. We review the operator tensor formulation of Quantum Theory and use it to set up an approach to Quantum Field Theory that is both operational and compositional. Then we consider strategies for solving the problem of Quantum Gravity. By referring only to operational quantities we are able to provide formulations for the possibilistic, probabilistic, and (the nascent) quantum cases that are manifestly invariant under diffeomorphisms.
Comment on "Non-representative Quantum Mechanical Weak Values". (arXiv:1608.07185v1 [quant-ph])
quant-ph updates on arXiv.org
Authors: Alon Ben Israel, L. Vaidman
Svensson [Found. Phys. \textbf{45}, 1645 (2015)] argued that the concept of the weak value of an observable of a pre- and post-selected quantum system cannot be applied when the expectation value of the observable in the initial state vanishes. Svensson's argument is analyzed and shown to be inconsistent using several examples.
Universal Decoherence under Gravity: A Perspective through the Equivalence Principle
PRL: General Physics: Statistical and Quantum Mechanics, Quantum Information, etc.
on 2016-8-24 2:00pm GMT
Author(s): Belinda H. Pang, Yanbei Chen, and Farid Ya. Khalili
Pikovski et al. [Nat. Phys. 11, 668 (2015)] show that a composite particle prepared in a pure initial quantum state and propagated in a uniform gravitational field undergoes a decoherence process at a rate determined by the gravitational acceleration. By assuming Einstein's equivalence principle to …
[Phys. Rev. Lett. 117, 090401] Published Wed Aug 24, 2016
Quantum Mechanics of a Photon. (arXiv:1608.06479v1 [quant-ph])
Authors: Hassan Babaei, Ali Mostafazadeh
A first quantized free photon is a complex massless vector field $A=(A^\mu)$ whose field strength satisfies Maxwell's equations in vacuum. We construct the Hilbert space $\mathscr{H}$ of the photon by endowing the vector space of the fields $A$ in the temporal-Coulomb gauge with a positive-definite and relativistically invariant inner product. We give an explicit expression for this inner product, identify the Hamiltonian for the photon with the generator of time translations in $\mathscr{H}$, determine the operators representing the momentum and the helicity of the photon, and introduce a chirality operator whose eigenfunctions correspond to fields having a definite sign of energy. We also construct a position operator for the photon whose components commute with each other and with the chirality and helicity operators. This allows for the construction of the localized states of the photon with a definite sign of energy and helicity. We derive an explicit formula for the latter and compute the corresponding electric and magnetic fields. These turn out to diverge not just at the point where the photon is localized but on a plane containing this point. We identify the axis normal to this plane with an associated helicity axis, and show that each choice of this axis determines a position basis and the corresponding position representation of the quantum mechanics of photon. In particular, we examine the position wave functions determined by such a position basis, elucidate their relationship with the Riemann-Silberstein and Landau-Peierls wave functions, and determine the probability density for the spatial localization of the photon.
Lieb-Robinson Bound and the Butterfly Effect in Quantum Field Theories
PRL Editors' Suggestions
Author(s): Daniel A. Roberts and Brian Swingle
A universal regime describing charge transport in holographic theories with particle-hole symmetry is theoretically predicted, suggesting a link between strongly coupled quantum field theories and quantum chaos.
[Phys. Rev. Lett. 117, 091602] Published Tue Aug 23, 2016
Decoherent histories approach to the cosmological measure problem. (arXiv:1608.05672v1 [quant-ph])
gr-qc updates on arXiv.org
Authors: Seth Lloyd
The method of decoherent histories allows probabilities to be assigned to sequences of quantum events in systems, such as the universe as a whole, where there is no external observer to make measurements. This paper applies the method of decoherent histories to address cosmological questions. Using a series of simple examples, beginning with the harmonic oscillator, we show that systems in a stationary state such as an energy eigenstate or thermal state can exhibit decoherent histories with non-trivial dynamics. We then examine decoherent histories in a universe that undergoes eternal inflation. Decoherent histories that assign probabilities to sequences of events in the vicinity of a timelike geodesic supply a natural cosmological measure. Under reasonable conditions, such sequences of events do not suffer from the presence of unlikely statistical fluctuations that mimic reality.
Posted in @all Weekly Papers
Article written by editor
active 17 hours ago
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Tetraview
A tetraview is an attempt to graph a complex function of a complex variable, by a method invented by Davide P. Cervone.
A graph of a real function of a real variable is the set of ordered pairs (x,y) such that y = f(x). This is the ordinary two-dimensional Cartesian graph studied in school algebra.
Every complex number has both a real part and an imaginary part, so one complex variable is two-dimensional and a pair of complex variables is four-dimensional. A tetraview is an attempt to give a picture of a four-dimensional object using a two-dimensional representation—either on a piece of paper or on a computer screen, showing a still picture consisting of five views, one in the center and one at each corner. This is roughly analogous to a picture of a three-dimensional object by giving a front view, a side view, and a view from above.
A picture of a three-dimensional object is a projection of that object from three dimensions into two dimensions. A tetraview is set of five projections, first from four dimensions into three dimensions, and then from three dimensions into two dimensions.
A complex function w = f(z), where z = a + bi and w = c + di are complex numbers, has a graph in four-space (four dimensional space) R4 consisting of all points (a, b, c, d) such that c + di = f(a + bi).
To construct a tetraview, we begin with the four points (1,0,0,0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1), which are vertices of a spherical tetrahedron on the unit three-sphere S3 in R4.
We project the four-dimensional graph onto the three-dimensional sphere along one of the four coordinate axes, and then give a two-dimensional picture of the resulting three-dimensional graph. This provides the four corner graph. The graph in the center is a similar picture "taken" from the point of view of the origin.
External links
• http://www.math.union.edu/~dpvc/professional/art/tetra-exp.html
• http://www.maa.org/cvm/1998/01/sbtd/article/tour/tetra-Z3/tetra-Z3.html
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\begin{document}
\title{f Galois Representations hanks{The work on this article was
partially supported by NSF Grant DMS-9702885.}
\thispagestyle{first} \setcounter{page}{449}
\begin{abstract}\vskip 3mm In the first part of this paper we try to explain to a general mathematical audience some of the remarkable web of conjectures linking representations of Galois groups with algebraic geometry, complex analysis and discrete subgroups of Lie groups. In the second part we briefly review some limited recent progress on these conjectures. \vskip 4.5mm
\noindent {\bf 2000 Mathematics Subject Classification:} 11F80.
\noindent {\bf Keywords and Phrases:} Galois representations, $L$-function, Automorphic forms. \end{abstract}
\vskip 12mm
\section*{Introduction}
\vskip-5mm \hspace{5mm}
The organisers requested a talk which would both be a colloquium style talk understandable to a wide spectrum of mathematicians and one which would survey the recent developments in the subject. I have found it hard to meet both desiderata, and have opted to concentrate on the former. Thus the first three sections of this paper contain a simple presentation of a web of deep conjectures connecting Galois representations to algebraic geometry, complex analysis and discrete subgroups of Lie groups. This will be of no interest to the specialist. My hope is that the result is not too banal and that it will give the non-specialist some idea of what motivates work in this area. I should stress that nothing I write here is original. In the final section I briefly review some of what is known about these conjectures and {\em very briefly} mention some of the available techniques. I also mention two questions which lie outside the topic we are discussing, but which would have important implications for it. Maybe someone can make progress on them?
Due to lack of space much of this article is too abbreviated. A somewhat expanded version is available on my website {\tt www.math.harvard.edu/\~{}rtaylor} and will hopefully be published elsewhere.
\Section{Galois representations}\label{s1}
\vskip-5mm \hspace{5mm}
We will let ${\mathbb{Q}}$ denote the field of rational numbers and $\overline{{\Q}}$ denote the field of algebraic numbers, the algebraic closure of ${\mathbb{Q}}$. We will also let $G_{\mathbb{Q}}$ denote the group of automorphisms of $\overline{{\Q}}$, that is ${\operatorname{Gal}\,}(\overline{{\Q}}/{\mathbb{Q}})$, the absolute Galois group of ${\mathbb{Q}}$. Although it is not the simplest it is arguably the most natural Galois group to study. An important technical point is that $G_{\mathbb{Q}}$ is naturally a (profinite) topological group, a basis of open neighbourhoods of the identity being given by the subgroups ${\operatorname{Gal}\,}(\overline{{\Q}}/K)$ as $K$ runs over subextensions of $\overline{{\Q}}/{\mathbb{Q}}$ which are finite over ${\mathbb{Q}}$.
To my mind the Galois theory of ${\mathbb{Q}}$ is most interesting when one looks not only at $G_{\mathbb{Q}}$ as an abstract (topological) group, but as a group with certain additional structures associated to the prime numbers. I will now briefly describe these structures.
For each prime number $p$ we may define an absolute value $| \,\,\,
|_p$ on ${\mathbb{Q}}$ by setting
\[ | \alpha |_p = p^{-r} \]
if $\alpha = p^r a/b$ with $a$ and $b$ integers coprime to $p$. If we complete ${\mathbb{Q}}$ with respect to this absolute value we obtain the field of $p$-adic numbers ${\mathbb{Q}}_p$, a totally disconnected, locally compact topological field. We will write $G_{{\mathbb{Q}}_p}$ for its absolute Galois group, ${\operatorname{Gal}\,}(\overline{{\Q}}_p/{\mathbb{Q}}_p)$. The absolute value $|\,\,\,|_p$ has a unique extension to an absolute value on $\overline{{\Q}}_p$ and $G_{{\mathbb{Q}}_p}$ is identified with the group of automorphisms of $\overline{{\Q}}_p$ which preserve
$|\,\,\,|_p$, or equivalently the group of continuous automorphisms of $\overline{{\Q}}_p$. For each embedding $\overline{{\Q}} \hookrightarrow \overline{{\Q}}_p$ we obtain a closed embedding $G_{{\mathbb{Q}}_p} \hookrightarrow G_{\mathbb{Q}}$ and as the embedding $\overline{{\Q}} \hookrightarrow \overline{{\Q}}_p$ varies we obtain a conjugacy class of closed embeddings $G_{{\mathbb{Q}}_p} \hookrightarrow G_{\mathbb{Q}}$. Slightly abusively, we shall consider $G_{{\mathbb{Q}}_p}$ a closed subgroup of $G_{\mathbb{Q}}$, suppressing the fact that the embedding is only determined up to conjugacy.
This can be compared with the situation `at infinity'. Let
$|\,\,\,|_\infty$ denote the usual Archimedean absolute value on
${\mathbb{Q}}$. The completion of ${\mathbb{Q}}$ with respect to $|\,\,\,|_\infty$ is the field of real numbers ${\mathbb{R}}$ and its algebraic closure is ${\mathbb{C}}$ the field of complex numbers. Each embedding $\overline{{\Q}} \hookrightarrow {\mathbb{C}}$ gives rise to a closed embedding \[ \{ 1, c\} = G_{\mathbb{R}} = {\operatorname{Gal}\,}({\mathbb{C}}/{\mathbb{R}}) \hookrightarrow G_{\mathbb{Q}}. \] As the embedding $\overline{{\Q}} \hookrightarrow {\mathbb{C}}$ varies one obtains a conjugacy class of elements $c \in G_{\mathbb{Q}}$ of order $2$, which we refer to as complex conjugations.
There are however many important differences between the case of finite places (i.e. primes) and the infinite place $| \,\,\,
|_\infty$. For instance $\overline{{\Q}}_p/{\mathbb{Q}}_p$ is an infinite extension and $\overline{{\Q}}_p$ is not complete. We will denote its completion by ${\mathbb{C}}_p$. The Galois group $G_{{\mathbb{Q}}_p}$ acts on ${\mathbb{C}}_p$ and is in fact the group of continuous automorphisms of ${\mathbb{C}}_p$.
The elements of ${\mathbb{Q}}_p$ (resp. $\overline{{\Q}}_p$) with absolute value less than or equal to $1$, form a closed subring ${\mathbb{Z}}_p$ (resp. ${\cal{O}}_{\overline{{\Q}}_p}$). These rings are local with maximal ideals $p{\mathbb{Z}}_p$ (resp. ${\mathfrak{m}}_{\overline{{\Q}}_p}$) consisting of the elements with absolute value strictly less than $1$. The field ${\cal{O}}_{\overline{{\Q}}_p}/ {\mathfrak{m}}_{\overline{{\Q}}_p}$ is an algebraic closure of the finite field with $p$ elements ${\mathbb{F}}_p = {\mathbb{Z}}_p/p{\mathbb{Z}}_p$, and we will denote it by $\overline{{\F}}_p$. Thus we obtain a continuous map \[ G_{{\mathbb{Q}}_p} \longrightarrow G_{{\mathbb{F}}_p} \] which is surjective. Its kernel is called the inertia subgroup of $G_{{\mathbb{Q}}_p}$ and is denoted by $I_{{\mathbb{Q}}_p}$. The group $G_{{\mathbb{F}}_p}$ is procyclic and has a canonical generator called the (geometric) Frobenius element and defined by \[ {\operatorname{Frob}}_p^{-1} (x) = x^p. \] In many circumstances it is technically convenient to replace $G_{{\mathbb{Q}}_p}$ by a dense subgroup $W_{{\mathbb{Q}}_p}$, which is referred to as the Weil group of ${\mathbb{Q}}_p$ and which is defined as the subgroup of $\sigma \in G_{{\mathbb{Q}}_p}$ such that $\sigma$ maps to \[ {\operatorname{Frob}}_p^{\mathbb{Z}} \subset G_{{\mathbb{F}}_p}. \] We endow $W_{{\mathbb{Q}}_p}$ with a topology by decreeing that $I_{{\mathbb{Q}}_p}$ with its usual topology should be an open subgroup of $W_{{\mathbb{Q}}_p}$.
We will take a moment to describe some of the finer structure of $I_{{\mathbb{Q}}_p}$ which we will need for technical purposes later. First of all there is a (not quite canonical) continuous surjection \[ I_{{\mathbb{Q}}_p} \rightarrow \hspace{-.86em} \rightarrow \prod_{l \neq p} {\mathbb{Z}}_l \] such that \[ t({\operatorname{Frob}}_p \sigma {\operatorname{Frob}}_p^{-1}) = p^{-1}t(\sigma) \] for all $\sigma \in I_{{\mathbb{Q}}_p}$. The kernel of $t$ is a pro-$p$-group called the wild inertia group. The fixed field $\overline{{\Q}}_p^{\ker t}$ is obtained by adjoining $\sqrt[n]{p}$ to $\overline{{\Q}}_p^{I_{{\mathbb{Q}}_p}}$ for all $n$ coprime to $p$ and \[ \sigma \sqrt[n]{p} = \zeta_n^{t(\sigma)} \sqrt[n]{p}, \] for some primitive $n^{th}$-root of unity $\zeta_n$ (independent of $\sigma$, but dependent on $t$).
In my opinion the most interesting question about $G_{\mathbb{Q}}$ is to describe it together with the distinguished subgroups $G_{\mathbb{R}}$, $G_{{\mathbb{Q}}_p}$, $I_{{\mathbb{Q}}_p}$ and the distinguished elements ${\operatorname{Frob}}_p \in G_{{\mathbb{Q}}_p}/ I_{{\mathbb{Q}}_p}$.
I want to focus here on attempts to describe $G_{\mathbb{Q}}$ via its representations. Perhaps the most obvious representations to consider are those representations \[ G_{\mathbb{Q}} \longrightarrow GL_n({\mathbb{C}}) \] with open kernel, and these so called Artin representations are already very interesting. However one obtains a richer theory if one considers representations \[ G_{\mathbb{Q}} \longrightarrow GL_n(\overline{{\Q}}_l) \] which are continuous with respect to the $l$-adic topology on $GL_n(\overline{{\Q}}_l)$. We refer to these as {\em $l$-adic representations}.
One justification for considering $l$-adic representations is that they arise naturally from geometry. Here are some examples of $l$-adic representations. \begin{enumerate} \item A choice of embeddings $\overline{{\Q}} \hookrightarrow {\mathbb{C}}$ and $\overline{{\Q}} \hookrightarrow \overline{{\Q}}_l$ establishes a bijection between isomorphism classes of Artin representations and isomorphism classes of $l$-adic representations with open kernel. Thus Artin representations are a special case of $l$-adic representations: those with finite image.
\item There is a unique character \[ \chi_l:G_{\mathbb{Q}} \longrightarrow {\mathbb{Z}}_l^\times \subset \overline{{\Q}}_l^\times \] such that \[ \sigma \zeta = \zeta^{\chi_l(\sigma)} \] for all $l$-power roots of unity $\zeta$. This is called the $l$-adic {\em cyclotomic character}.
\item If $X/{\mathbb{Q}}$ is a smooth projective variety (and we choose an embedding $\overline{{\Q}} \subset {\mathbb{C}}$) then the natural action of $G_{\mathbb{Q}}$ on the cohomology \[ H^i(X({\mathbb{C}}),\overline{{\Q}}_l) \cong H^i_{\operatorname{et}}(X \times_{\mathbb{Q}} \overline{{\Q}}, \overline{{\Q}}_l) \] is an $l$-adic representation. For instance if $E/{\mathbb{Q}}$ is an elliptic curve then we have the concrete description \[ H^1_{\operatorname{et}}(E \times_{\mathbb{Q}} \overline{{\Q}}, \overline{{\Q}}_l) \cong {\operatorname{Hom}\,}_{{\mathbb{Z}}_l}(\lim_{\leftarrow r} E[l^r](\overline{{\Q}}) , \overline{{\Q}}_l) \cong \overline{{\Q}}_l^2, \] where $E[l^r]$ denotes the $l^r$-torsion points on $E$.
We will write $H^i(X({\mathbb{C}}),\overline{{\Q}}_l(j))$ for the twist \[ H^i(X({\mathbb{C}}),\overline{{\Q}}_l) \otimes \chi_l^{j}. \] \end{enumerate}
Before discussing $l$-adic representations of $G_{\mathbb{Q}}$ further, let us take a moment to look at $l$-adic representations of $G_{{\mathbb{Q}}_p}$. The cases $l\neq p$ and $l=p$ are very different. Consider first the much easier case $l \neq p$. Here $l$-adic representations of $G_{{\mathbb{Q}}_p}$ are not much different from representations of $W_{{\mathbb{Q}}_p}$ with open kernel. More precisely define a {\em WD-representation} of $W_{{\mathbb{Q}}_p}$ over a field $E$ to be a pair \[ r:W_{{\mathbb{Q}}_p} \longrightarrow GL(V) \] and \[ N \in {\operatorname{End}\,}(V), \] where $V$ is a finite dimensional $E$-vector space, $r$ is a representation with open kernel and $N$ is a nilpotent endomorphism which satisfies \[ r(\phi)Nr(\phi^{-1}) = p^{-1}N \] for every lift $\phi \in W_{{\mathbb{Q}}_p}$ of ${\operatorname{Frob}}_p$. The key point here is that there is no reference to a topology on $E$, indeed no assumption that $E$ is a topological field. Given $r$ there are up to isomorphism only finitely many choices for the pair $(r,N)$ and these can be explicitly listed without difficulty. A WD-representation $(r,N)$ is called {\em unramified} if $N=0$ and $r(I_{{\mathbb{Q}}_p}) = \{ 1 \}$. It is called Frobenius semi-simple if $r$ is semi-simple. Any WD-representation $(r,N)$ has a canonical Frobenius semi-simplification $(r,N)^{\operatorname{ss}}$ (see \cite{tate:ntb}). In the case that $E=\overline{{\Q}}_l$ we call $(r,N)$ $l$-integral if all the eigenvalues of $r(\phi)$ have absolute value $1$. This is independent of the choice of Frobenius lift $\phi$.
If $l \neq p$, then there is an equivalence of categories between $l$-integral WD-representations of $W_{{\mathbb{Q}}_p}$ over $\overline{{\Q}}_l$ and $l$-adic representations of $G_{{\mathbb{Q}}_p}$. To describe it choose a Frobenius lift $\phi \in W_{{\mathbb{Q}}_p}$ and a surjection $t_l:I_{{\mathbb{Q}}_p} \rightarrow \hspace{-.86em} \rightarrow {\mathbb{Z}}_l$. Up to natural isomorphism the equivalence does not depend on these choices. We associate to an $l$-integral WD-representation $(r,N)$ the unique $l$-adic representation sending \[ \phi^n \sigma \longmapsto r(\phi^n \sigma) \exp(t_l(\sigma)N) \] for all $n \in {\mathbb{Z}}$ and $\sigma \in I_{{\mathbb{Q}}_p}$. The key point is Grothendieck's observation that for $l \neq p$ any $l$-adic representation of $G_{{\mathbb{Q}}_p}$ must be trivial on some open subgroup of the wild inertia group. We will write ${\operatorname{WD}}_p(R)$ for the WD-representation associated to an $l$-adic representation $R$. Note that ${\operatorname{WD}}_p(R)$ is unramified if and only if $R(I_p)=\{ 1\}$. In this case we call $R$ {\em unramified}.
The case $l=p$ is much more complicated because there are many more $p$-adic representations of $G_{{\mathbb{Q}}_p}$. These have been extensively studied by Fontaine and his co-workers. They single out certain $p$-adic representations which they call {\em de Rham} representations. I will not recall the somewhat involved definition here (see however \cite{derham} and \cite{semisimp}), but note that `most' $p$-adic representations of $G_{{\mathbb{Q}}_p}$ are not de Rham. To any de Rham representation $R$ of $G_{{\mathbb{Q}}_p}$ on a $\overline{{\Q}}_p$-vector space $V$ they associate the following. \begin{enumerate} \item A WD-representation ${\operatorname{WD}}_p(R)$ of $W_{{\mathbb{Q}}_p}$ over $\overline{{\Q}}_p$ (see \cite{berger} and \cite{fpr}). \item A multiset ${\operatorname{HT}}(R)$ of $\dim V$ integers, called the Hodge-Tate
numbers of $R$. The multiplicity of $i$ in ${\operatorname{HT}}(R)$ is \[ \dim_{\overline{{\Q}}_p} ( V \otimes_{{\mathbb{Q}}_p} {\mathbb{C}}_p(i))^{G_{{\mathbb{Q}}_p}}, \] where ${\mathbb{C}}_p(i)$ denotes ${\mathbb{C}}_p$ with $G_{{\mathbb{Q}}_p}$-action $\chi_p(\sigma)^i$ times the usual (Galois) action on ${\mathbb{C}}_p$. \end{enumerate}
We now return to the global situation (i.e. to the study of $G_{{\mathbb{Q}}}$). The $l$-adic representations of $G_{\mathbb{Q}}$ that arise `in nature', by which I mean `from geometry', have a number of very special properties which I will now list. Let $R:G_{\mathbb{Q}} \longrightarrow GL(V)$ be a subquotient of $H^i(X({\mathbb{C}}),\overline{{\Q}}_l(j))$ for some smooth projective variety $X/{\mathbb{Q}}$ and some integers $i \geq 0$ and $j$.
\begin{enumerate} \item (Grothendieck) The representation $R$ is unramified at all but finitely many primes $p$.
\item (Fontaine, Messing, Faltings, Kato, Tsuji, de Jong, see e.g. \cite{illu}, \cite{bert}) The representation $R$ is de Rham in the sense that its restriction to $G_{{\mathbb{Q}}_l}$ is de Rham.
\item (Deligne, \cite{deligne:w1}) The representation $R$ is {\em pure} of weight $w=i-2j$ in
the following sense. There is a finite set of primes $S$, such that
for $p \not\in S$, the representation $R$ is unramified at $p$ and for every eigenvalue $\alpha$ of $R({\operatorname{Frob}}_p)$ and every
embedding $\iota: \overline{{\Q}}_l \hookrightarrow {\mathbb{C}}$
\[ | \iota\alpha |_\infty^2 = p^w. \] In particular $\alpha$ is algebraic (i.e. $\alpha \in \overline{{\Q}}$). \end{enumerate}
An amazing conjecture of Fontaine and Mazur (see \cite{fontaine:arbeit} and \cite{fm}) asserts that any irreducible $l$-adic representation of $G_{\mathbb{Q}}$ satisfying the first two of these properties arises from geometry in the above sense and so in particular also satisfies the third property.
\begin{conjecture}[{\rm Fontaine-Mazur}]\label{cfm} Suppose that \[ R:G_{\mathbb{Q}} \longrightarrow GL(V) \]
is an irreducible $l$-adic representation which is unramified at all but finitely many primes and with $R|_{G_{{\mathbb{Q}}_l}}$ de Rham. Then there is a smooth projective variety $X/{\mathbb{Q}}$ and integers $i \geq 0$ and $j$ such that $V$ is a subquotient of $H^i(X({\mathbb{C}}),\overline{{\Q}}_l(j))$. In particular $R$ is pure of some weight $w\in {\mathbb{Z}}$. \end{conjecture}
We will discuss the evidence for this conjecture later. We will call an $l$-adic representation satisfying the conclusion of this conjecture {\em geometric}.
Algebraic geometers have formulated some very precise conjectures about the action of $G_{\mathbb{Q}}$ on the cohomology of varieties. We don't have the space here to discuss these in general, but we will formulate, in an as algebraic a way as possible, some of their conjectures.
\begin{conjecture}[{\rm Tate}]\label{cgeo} Suppose that $X/{\mathbb{Q}}$ is a smooth projective variety. Then there is a decomposition \[ H^i(X({\mathbb{C}}), \overline{{\Q}}) = \bigoplus_j M_j \] with the following properties. \begin{enumerate} \item For each prime $l$ and for each embedding $\iota:\overline{{\Q}} \hookrightarrow \overline{{\Q}}_l,$ $M_j \otimes_{\overline{{\Q}}, \iota} \overline{{\Q}}_l$ is an irreducible subrepresentation of $H^i(X({\mathbb{C}}),\overline{{\Q}}_l)$. \item For all indices $j$ and for all primes $p$ there is a WD-representation ${\operatorname{WD}}_p(M_j)$ of $W_{{\mathbb{Q}}_p}$ over $\overline{{\Q}}$ such that \[ {\operatorname{WD}}_p(M_j) \otimes_{\overline{{\Q}},\iota} \overline{{\Q}}_l \cong {\operatorname{WD}}_p(M_j \otimes_{\overline{{\Q}},\iota} \overline{{\Q}}_l) \] for all primes $l$ and all embeddings $\iota:\overline{{\Q}} \hookrightarrow \overline{{\Q}}_l$. \item There is a multiset of integers ${\operatorname{HT}}(M_j)$ such that \begin{enumerate} \item for all primes $l$ and all embeddings $\iota:\overline{{\Q}} \hookrightarrow \overline{{\Q}}_l$ \[ {\operatorname{HT}}(M_j \otimes_{\overline{{\Q}},\iota} \overline{{\Q}}_l) = HT(M_j)\] \item and for all $\iota:\overline{{\Q}} \hookrightarrow {\mathbb{C}}$ \[ \dim_{\mathbb{C}} ((M_j \otimes_{\overline{{\Q}},\iota} {\mathbb{C}}) \cap H^{a,i-a}(X({\mathbb{C}}),{\mathbb{C}})) \] is the multiplicity of $a$ in $HT(M_j)$. \end{enumerate} \end{enumerate} \end{conjecture}
If one considers the whole of $ H^i(X({\mathbb{C}}), \overline{{\Q}})$ rather than its pieces $M_j$, then part 2. is known to hold up to Frobenius semisimplification for all but finitely many $p$ and part 3. is known to hold (see \cite{illu}). It follows from a theorem of Faltings \cite{falt:mor} that the whole conjecture is true for $H^1$ of an abelian variety. The putative constituents $M_j$ are one incarnation of what people call `pure motives'.
If one believes conjectures \ref{cfm} and \ref{cgeo} then `geometric' $l$-adic representations should come in compatible families as $l$ varies. There are many ways to make precise the notion of such a compatible family. Here is one.
By a {\em weakly compatible system of $l$-adic representations} ${\cal{R}}=\{ R_{l,\iota} \}$ we shall mean a collection of semi-simple $l$-adic representations \[ R_{l,\iota}: G_{{\mathbb{Q}}} \longrightarrow GL(V \otimes_{\overline{{\Q}},\iota} \overline{{\Q}}_l), \] one for each pair $(l,\iota)$, where $l$ is a prime and $\iota:\overline{{\Q}} \hookrightarrow \overline{{\Q}}_{l}$, which satisfy the following conditions. \begin{itemize} \item There is a multiset of integers ${\operatorname{HT}}({\cal{R}})$ such that for each prime $l$ and each embedding $\iota:\overline{{\Q}} \hookrightarrow \overline{{\Q}}_l$ the restriction
$R_{l,\iota}|_{G_{{\mathbb{Q}}_l}}$ is de Rham and
${\operatorname{HT}}(R_{l,\iota}|_{G_{{\mathbb{Q}}_l}}) = {\operatorname{HT}}({\cal{R}})$. \item There is a finite set of primes $S$ such that if $p \not\in S$ then ${\operatorname{WD}}_p(R_{l,\iota})$ is unramified for all $l$ and $\iota$. \item For all but finitely many primes $p$ there is a Frobenius semi-simple WD-representation ${\operatorname{WD}}_p({\cal{R}})$ over $\overline{{\Q}}$ such that for all primes $l \neq p$ and for all $\iota$ we have \[ {\operatorname{WD}}_p(R_{l,\iota})^{\operatorname{ss}} \sim {\operatorname{WD}}_p({\cal{R}}). \] \end{itemize} We make the following subsidiary definitions. \begin{itemize}
\item We call ${\cal{R}}$ {\em strongly compatible} if the last condition
(the existence of ${\operatorname{WD}}_p({\cal{R}})$) holds for all primes $p$.
\item We call ${\cal{R}}$ {\em irreducible} if each $R_{l,\iota}$ is
irreducible.
\item We call ${\cal{R}}$ {\em pure} of weight $w \in {\mathbb{Z}}$, if for all but
finitely many $p$ and for all eigenvalues $\alpha$ of
$r_p({\operatorname{Frob}}_p)$, where ${\operatorname{WD}}_p({\cal{R}})=(r_p,N_p)$, we have
\[ |\iota \alpha|_\infty^2 = p^w \]
for all embeddings $\iota:\overline{{\Q}} \hookrightarrow {\mathbb{C}}$.
\item We call ${\cal{R}}$ {\em geometric} if there is a smooth projective variety
$X/{\mathbb{Q}}$ and integers $i \geq 0$ and $j$ and a subspace
\[ W \subset H^i(X({\mathbb{C}}),\overline{{\Q}}) \]
such that for all $l$ and $\iota$, $W \otimes_{\overline{{\Q}}, \iota} \overline{{\Q}}_l$ is
$G_{\mathbb{Q}}$ invariant and realises $R_{l,\iota}$. \end{itemize}
Conjectures \ref{cfm} and \ref{cgeo} lead one to make the following conjecture.
\begin{conjecture}\label{ccs}\begin{enumerate} \item If $R:G_{\mathbb{Q}} \rightarrow GL_n(\overline{{\Q}}_l)$ is a continuous semi-simple de Rham representation unramified at all but finitely many primes then $R$ is part of a weakly compatible system. \item Any weakly compatible system is strongly compatible. \item Any irreducible weakly compatible system ${\cal{R}}$ is geometric and pure of weight $(2/\dim {\cal{R}}) \sum_{h \in {\operatorname{HT}}({\cal{R}})} h$. \end{enumerate} \end{conjecture}
A famous theorem of Cebotarev asserts that if $K/{\mathbb{Q}}$ is any Galois extension in which all but finitely many primes are unramified (i.e. for all but finitely many primes $p$ the image of $I_{{\mathbb{Q}}_p}$ in ${\operatorname{Gal}\,}(K/{\mathbb{Q}})$ is trivial) then the Frobenius elements at unramified primes ${\operatorname{Frob}}_p \in {\operatorname{Gal}\,}(K/{\mathbb{Q}})$ are dense in ${\operatorname{Gal}\,}(K/{\mathbb{Q}})$. It follows that an irreducible weakly compatible system ${\cal{R}}$ is uniquely determined by ${\operatorname{WD}}_p({\cal{R}})$ for all but finitely many $p$ and hence by one $R_{l,\iota}$.
Conjectures \ref{cfm} and \ref{ccs} are known for one dimensional representations, in which case they have purely algebraic proofs based on class field theory (see \cite{serre:alr}). Otherwise only fragmentary cases have been proved, where amazingly the arguments are extremely indirect involving sophisticated analysis and geometry. We will come back to this later.
\Section{{\boldmath $L$}-functions}\label{s2}
\vskip-5mm \hspace{5mm}
$L$-functions are certain Dirichlet series \[ \sum_{n=1}^\infty a_n/n^s \] which play an important role in number theory. A full discussion of the role of $L$-functions in number theory is beyond the scope of this talk. The simplest example of an $L$-function is the Riemann zeta function \[ \zeta(s)=\sum_{n=1}^\infty 1/n^s. \] It converges to a holomorphic function in the half plane ${\operatorname{Re}\,} s > 1$ and in this region of convergence it can also be expressed as a convergent infinite product over the prime numbers \[ \zeta(s) = \prod_p (1-1/p^s)^{-1}. \] This is called an {\em Euler product} and the individual factors are called Euler factors. Lying deeper is the fact that $\zeta(s)$ has meromorphic continuation to the whole complex plane, with only one pole: a simple pole at $s=1$. Moreover if we set \[ Z(s)=\pi^{-s/2} \Gamma(s/2) \zeta(s) \] then $Z$ satisfies the functional equation \[ Z(1-s)=Z(s). \] Encoded in the Riemann zeta function is lots of deep arithmetic information. For instance the location of the zeros of $\zeta(s)$ is intimately connected with the distribution of prime numbers. Moreover its special values at negative integers (where it is only defined by analytic continuation) turn out to be rational numbers encoding deep arithmetic information about the cyclotomic fields ${\mathbb{Q}}(e^{2 \pi \sqrt{-1}/p})$.
Another celebrated example is the $L$-function of an elliptic curve $E$: \[ y^2=x^3+ax+b. \] In this case the $L$-function is defined as an Euler product (converging in ${\operatorname{Re}\,} s>3/2$) \[ L(E,s) = \prod_p L_p(E,p^{-s}), \] where $L_p(E,X)$ is a rational function, and for all but finitely many $p$ \[ L_p(E,X) = (1-a_p(E)X+pX^2)^{-1}, \] with $p-a_p(E)$ being the number of solutions to the congruence \[ y^2 \equiv x^3+ax+b \bmod p \] in ${\mathbb{F}}_p^2$. It has recently been proved \cite{bcdt} that $L(E,s)$ can be continued to an entire function, which satisfies a functional equation \[ (2 \pi)^{-s} \Gamma(s) L(E,s)= \pm N(E)^{1-s} (2 \pi)^{s-2} \Gamma(2-s) L(E,2-s), \] for some explicit positive integer $N(E)$. A remarkable conjecture of Birch and Swinnerton-Dyer \cite{bsd} predicts that $y^2=x^3+ax+b$ has infinitely many rational solutions if and only if $L(E,1)=0$. Again we point out that it is the behaviour of the $L$-function at a point where it is only defined by analytic continuation, which is governing the arithmetic of $E$. This conjecture has been proved (see \cite{koly}) when $L(E,s)$ has at most a simple zero at $s=1$.
One general setting in which one can define $L$-functions is $l$-adic representations. Let us look first at the local setting. If $(r,N)$ is a WD-representation of $W_{{\mathbb{Q}}_p}$ on an $E$-vector space $V$, where $E$ is an algebraically closed field of characteristic zero, we define a local L-factor
\[ L((r,N),X) = \det(1-X {\operatorname{Frob}}_p)|_{V^{I_{{\mathbb{Q}}_p},N=0}}^{-1} \in E(X). \] ($V^{I_{{\mathbb{Q}}_p},N=0}$ is the subspace of $V$ where $I_{{\mathbb{Q}}_p}$ acts trivially and $N=0$.) One can also associate to $(r,N)$ a conductor $f(r,N) \in {\mathbb{Z}}_{\geq 0}$, which measures how deeply into $I_{{\mathbb{Q}}_p}$ the WD-representation $(r,N)$ is nontrivial, and a local epsilon factor $\epsilon((r,N),\Psi_p) \in E$, which also depends on the choice of a non-trivial character $\Psi_p:{\mathbb{Q}}_p \rightarrow E^\times$ with open kernel. (See \cite{tate:ntb}.)
If $R:G_{\mathbb{Q}} \rightarrow GL(V)$ is an $l$-adic representation of $G_{\mathbb{Q}}$ which is de Rham at $l$ and pure of some weight $w \in {\mathbb{Z}}$, and if $\iota:\overline{{\Q}}_l \hookrightarrow {\mathbb{C}}$ we will define an $L$-function \[ L(\iota R,s) = \prod_p L(\iota {\operatorname{WD}}_p(R),p^{-s}), \] which will converge to a holomorphic function in ${\operatorname{Re}\,} s> 1+w/2$. For example \[ L(1,s) = \zeta(s) \] and if $E/{\mathbb{Q}}$ is an elliptic curve then \[ L(\iota H^1(E({\mathbb{C}}),\overline{{\Q}}_l), s) = L(E,s) \] (for any $\iota$). Note the useful formulae \[ L(\iota(R_1 \oplus R_2),s) = L(\iota R_1,s)L(\iota R_2,s) \,\,\,\,\,\,\,\,\,\, {\rm and } \,\,\,\,\,\,\,\,\,\, L(\iota (R \otimes \chi_l^r),s) = L(\iota R, s+r). \] Also note that $L(\iota R,s)$ determines $L({\operatorname{WD}}_p(R),X)$ for all $p$ and hence ${\operatorname{WD}}_p(R)^{\operatorname{ss}}$ for all but finitely many $p$. Hence by the Cebotarev density theorem $L(\iota R,s)$ determines $R$ (up to semisimplification).
Write $m^R_i$ for the multiplicity of an integer $i$ in ${\operatorname{HT}}(R)$ and, if $w/2 \in {\mathbb{Z}}$, define $m_{w/2,\pm}^R \in (1/2){\mathbb{Z}}$ by: \[ \begin{array}{rcl} m_{w/2,+}^R+m_{w/2,-}^{R} &=& m^R_{w/2} \\ \\ m_{w/2,+}^R-m_{w/2,-}^{R} &=& (-1)^{w/2}(\dim V^{c=1} - \dim V^{c=-1}). \end{array} \] Assume that $m^R_{w/2,\pm}$ are integers, i.e. that $m^R_{w/2} \equiv \dim V \bmod 2$. Then we can define a $\Gamma$-factor, $\Gamma(R,s)$, which is a product of functions $\pi^{-(s+a)/2}\Gamma((s+a)/2)$ as $a$ runs over a set of integers depending only on the numbers $m^R_i$ and $m^R_{w/2,\pm}$. We can also define an epsilon factor $\epsilon_\infty(R,\Psi_\infty) \in {\mathbb{C}}^\times$ which again only depends on $m^R_i$, $m^R_{w/2,\pm}$ and a non-trivial character $\Psi_\infty:{\mathbb{R}} \rightarrow {\mathbb{C}}^\times$. Set \[ \Lambda(\iota R,s)= \Gamma(R,s) L(\iota R, s) \] and \[ N(R) = \prod_p p^{f({\operatorname{WD}}_p(R))} \] (which makes sense as $f({\operatorname{WD}}_p(R))=0$ for all but finitely many $p$) and \[ \epsilon(\iota R) = \epsilon_{\infty}(R,e^{2 \pi \sqrt{-1} x}) \prod_p \iota \epsilon({\operatorname{WD}}_p(R), \Psi_p), \] where $\iota \Psi_p(x) = e^{-2 \pi \sqrt{-1} x}$.
The following conjecture is a combination of conjecture \ref{cfm} and conjectures which have become standard. \begin{conjecture}\label{cfe} Suppose that $R$ is an irreducible $l$-adic representation of $G_{\mathbb{Q}}$ which is de Rham and pure of weight $w \in {\mathbb{Z}}$. Then $m_p^R=m_{w-p}^R$ for all $p$, so that $m_{w/2} \equiv \dim V \bmod 2$. Moreover the following should hold. \begin{enumerate} \item $L(\iota R,s)$ extends to an entire function, except for a single simple pole if $R=\chi_l^{-w/2}$. \item $\Lambda(\iota R,s)$ is bounded in vertical strips $\sigma_0 \leq {\operatorname{Re}\,} s \leq \sigma_1$. \item $\Lambda(\iota R,s)= \epsilon(\iota R) N(R)^{-s} \Lambda(\iota R^\vee,1-s)$. \end{enumerate} \end{conjecture}
It is tempting to believe that something like properties 1., 2. and 3. should characterise those Euler products which arise from $l$-adic representations. We will discuss a more precise conjecture along these lines in the next section. Why Galois representations should be {\bf the} source of Euler products with good functional equations seems a complete mystery.
\Section{Automorphic forms}\label{s3}
\vskip-5mm \hspace{5mm}
Automorphic forms may be thought of as certain smooth functions on the quotient $GL_n({\mathbb{Z}})\backslash GL_n({\mathbb{R}})$. We need several preliminaries before we can make a precise definition.
Let ${\widehat{\Z}}$ denote the profinite completion of ${\mathbb{Z}}$, i.e. \[ {\widehat{\Z}} = \lim_{\leftarrow N} {\mathbb{Z}} / N{\mathbb{Z}} = \prod_p {\mathbb{Z}}_p, \] a topological ring. Also let ${\mathbb{A}}^\infty$ denote the topological ring of finite adeles \[ {\mathbb{A}}^\infty = {\widehat{\Z}} \otimes_{\mathbb{Z}} {\mathbb{Q}}, \] where ${\widehat{\Z}}$ is an open subring with its usual topology. As an abstract ring, ${\mathbb{A}}^\infty$ is the subring of $\prod_p {\mathbb{Q}}_p$ consisting of elements $(x_p)$ with $x_p \in {\mathbb{Z}}_p$ for all but finitely many $p$. However the topology is not the subspace topology. We define the topological ring of adeles to be the product \[ {\mathbb{A}} = {\mathbb{A}}^\infty \times {\mathbb{R}}. \] Note that ${\mathbb{Q}}$ embeds diagonally as a discrete subring of ${\mathbb{A}}$ with compact quotient \[ {\mathbb{Q}} \backslash {\mathbb{A}} = {\widehat{\Z}} \times {\mathbb{Z}} \backslash {\mathbb{R}}. \]
We will be interested in $GL_n({\mathbb{A}})$, the locally compact topological group of $n \times n$ invertible matrices with coefficients in ${\mathbb{A}}$. We remark that the topology on $GL_n({\mathbb{A}})$ is the subspace topology resulting from the closed embedding \[ \begin{array}{rcl} GL_n({\mathbb{A}}) &\hookrightarrow & M_n({\mathbb{A}}) \times M_n({\mathbb{A}}) \\
g & \mapsto & (g, g^{-1}). \end{array} \] $GL_n({\mathbb{Q}})$ is a discrete subgroup of $GL_n({\mathbb{A}})$ and the quotient $GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}})$ has finite volume. If $U \subset GL_n({\widehat{\Z}})$ is an open subgroup with $\det U = {\widehat{\Z}}^\times$, then \[ GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}) / U = (GL_n({\mathbb{Q}}) \cap U) \backslash GL_n({\mathbb{R}}). \] Note that $GL_n({\mathbb{Q}}) \cap U$ is a subgroup of $GL_n({\mathbb{Z}})$ of finite index. Most of the statements we make concerning $GL_n({\mathbb{A}})$ can be rephrased to involve only $GL_n({\mathbb{R}})$, but at the expense of making them much more cumbersome. To achieve brevity (and because it seems more natural) we have opted to use the language of adeles. We hope that this extra abstraction will not be too confusing for the novice.
Before continuing our introduction of automorphic forms let us digress to mention class field theory, which provides a concrete example of the presentational advantages of the adelic language. It also implies essentially all the conjectures we are considering in the case of one dimensional Galois representations. Indeed this article is about the search for a non-abelian analogue of class field theory. Class field theory gives a concrete description of the abelianisation (maximal continuous abelian quotient) $G_{\mathbb{Q}}^{\operatorname{ab}}$ of $G_{\mathbb{Q}}$ and $W_{{\mathbb{Q}}_p}^{\operatorname{ab}}$ of $W_{{\mathbb{Q}}_p}$. First the local theory asserts that there is an isomorphism \[ {\operatorname{Art}\,}_p: {\mathbb{Q}}_p^\times \iso W_{{\mathbb{Q}}_p}^{\operatorname{ab}} \] with various natural properties, including the facts that ${\operatorname{Art}\,}({\mathbb{Z}}_p^\times)$ is the image of the inertia group $I_{{\mathbb{Q}}_p}$ in $W_{{\mathbb{Q}}_p}^{\operatorname{ab}}$, and that the induced map \[ {\mathbb{Q}}_p^\times / {\mathbb{Z}}_p^\times \longrightarrow W_{{\mathbb{Q}}_p}^{\operatorname{ab}}/I_{{\mathbb{Q}}_p} \subset G_{{\mathbb{F}}_p} \] takes $p$ to the geometric Frobenius element ${\operatorname{Frob}}_p$. Secondly the global theory asserts that there is an isomorphism \[ {\operatorname{Art}\,}: {\mathbb{A}}^\times/{\mathbb{Q}}^\times {\mathbb{R}}^\times_{>0} \iso G_{\mathbb{Q}}^{\operatorname{ab}} \] such that the restriction of ${\operatorname{Art}\,}$ to ${\mathbb{Q}}_p^\times$ coincides with the composition of ${\operatorname{Art}\,}_p$ with the natural map $W_{{\mathbb{Q}}_p}^{\operatorname{ab}} \rightarrow G_{\mathbb{Q}}^{\operatorname{ab}}$. Thus ${\operatorname{Art}\,}$ is defined completely from a knowledge of the ${\operatorname{Art}\,}_p$ (and the fact that ${\operatorname{Art}\,}$ takes $-1 \in {\mathbb{R}}^\times$ to complex conjugation) and the reciprocity theorem of global class field theory can be thought of as a determination of the kernel of $\prod_p {\operatorname{Art}\,}_p$.
We now return to our (extended) definition of automorphic forms. For each partition $n=n_1+n_2$ let $N_{n_1,n_2}$ denote the subgroup of $GL_n$ consisting of matrices of the form \[ \left( \begin{array}{cc} I_{n_1} & * \\ 0 & I_{n_2} \end{array} \right) . \] Let $O(n) \subset GL_n({\mathbb{R}})$ denote the orthogonal subgroup. Let ${\mathfrak{z}}_n$ denote the centre of the universal enveloping of ${\mathfrak{gl}}_n$, the complexified Lie algebra of $GL_n({\mathbb{R}})$ (i.e. ${\mathfrak{gl}}_n=M_n({\mathbb{C}})$ with $[X,Y]=XY-YX$). Via the Harish-Chandra isomorphism (see for example \cite{hc}) we may identify homomorphisms ${\mathfrak{z}}_n \rightarrow {\mathbb{C}}$ with multisets of $n$ complex numbers. We will write $\chi_H$ for the homomorphism corresponding to a multiset $H$. Thus ${\mathfrak{z}}_n$ acts on the irreducible finite dimensional ${\mathfrak{gl}}_n$-module with highest weight $(a_1,...,a_n) \in {\mathbb{Z}}^n$ ($a_1 \geq ... \geq a_n$) by $\chi_{\{ a_1+(n-1)/2,...,a_n+(1-n)/2 \} }$.
Fix such a multiset $H$ of cardinality $n$. The space of cusp forms with infinitesimal character $H$, ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))$ is the space of smooth bounded functions \[ f: GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}) \longrightarrow {\mathbb{C}} \] satisfying the following conditions. \begin{enumerate} \item ($K$-finiteness) The translates of $f$ under $GL_n({\widehat{\Z}}) \times O(n)$ (where $O(n)$ denotes the orthogonal group) span a finite dimensional vector space; \item (Infinitesimal character $H$) If $z \in {\mathfrak{z}}_n$ then $zf=\chi_H(z) f$; \item (Cuspidality) For each partition $n=n_1+n_2$, \[ \int_{N_{n_1,n_2}({\mathbb{Q}}) \backslash N_{n_1,n_2}({\mathbb{A}})} f(ug) du =0. \] \end{enumerate} Note that if $U \subset GL_n({\widehat{\Z}})$ is an open subgroup with $\det U = {\widehat{\Z}}^\times$ then one may think of ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))^U$ as a space of functions on $(GL_n({\mathbb{Q}}) \cap U) \backslash GL_n({\mathbb{R}})$.
One would like to study ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))$ as a representation of $GL_n({\mathbb{A}})$, unfortunately it is not preserved by the action of $GL_n({\mathbb{R}})$ (because the $K$-finiteness condition depends on the choice of a maximal compact subgroup $O(n) \subset GL_n({\mathbb{R}})$). It does however have an action of $GL_n({\mathbb{A}}^\infty) \times O(n)$ and of ${\mathfrak{gl}}_n$, which is essentially as good. More precisely it is an admissible $GL_n({\mathbb{A}}^\infty) \times ({\mathfrak{gl}}_n,O(n))$-module in the sense of \cite{flath}. In fact it is a direct sum of irreducible, admissible $GL_n({\mathbb{A}}^\infty) \times ({\mathfrak{gl}}_n,O(n))$-modules each occurring with multiplicity one. We will (slightly abusively) refer to these irreducible constituents as cuspidal automorphic representations of $GL_n({\mathbb{A}})$ with infinitesimal character $H$.
${\cal{A}}^\circ_{\{ 0 \} }({\mathbb{Q}}^\times \backslash {\mathbb{A}}^\times)$ is just the space of locally constant functions on ${\mathbb{A}}^\times/{\mathbb{Q}}^\times {\mathbb{R}}^\times_{>0}$ and so cuspidal automorphic representations of $GL_1({\mathbb{A}})$ with infinitesimal character $\{ 0 \}$, are just the (finite order) complex valued characters of ${\mathbb{A}}^\times/{\mathbb{Q}}^\times {\mathbb{R}}^\times_{>0} \cong {\widehat{\Z}}^\times$, i.e. Dirichlet characters. ${\cal{A}}^\circ_{\{ s \} }({\mathbb{Q}}^\times \backslash {\mathbb{A}}^\times)$ is simply obtained from ${\cal{A}}^\circ_{\{ 0 \} }({\mathbb{Q}}^\times
\backslash {\mathbb{A}}^\times)$ by twisting by $|| \,\,\,||^s$, where $||\,\,\,||:
{\mathbb{A}}^\times/{\mathbb{Q}}^\times \rightarrow {\mathbb{R}}^\times_{>0}$ is the product of the absolute values $| \,\,\,|_x$. Thus in the case $n=1$ cuspidal automorphic representations are essentially Dirichlet characters.
The case $n=2$ is somewhat more representative. In this case we have ${\cal{A}}^\circ_{ \{ s,t \} }(GL_2({\mathbb{Q}}) \backslash GL_2({\mathbb{A}}))=(0)$ unless $s-t \in i{\mathbb{R}}$, $s-t \in {\mathbb{Z}}$ or $s-t \in (-1,1)$. It is conjectured that the third possibility can not arise unless $s=t$. Let us consider the case $s-t \in {\mathbb{Z}}_{>0}$ a little further. If $s-t \in {\mathbb{Z}}_{>0}$ then it turns out that the irreducible constituents of ${\cal{A}}^\circ_{ \{ s,t \} }(GL_2({\mathbb{Q}}) \backslash GL_2({\mathbb{A}}))$ are in bijection with the weight $1+s-t$ holomorphic cusp forms on the upper half plane, which are normalised newforms (see for example \cite{miy}). Thus in some sense cuspidal automorphic representations are are also generalisations of classical holomorphic normalised newforms.
Note that if $\psi$ is a character of ${\mathbb{A}}^\times/{\mathbb{Q}}^\times {\mathbb{R}}^\times_{>0}$ and if $\pi$ is a cuspidal automorphic representation of $GL_n({\mathbb{A}})$ with infinitesimal character $H$ then $\pi \otimes (\psi \circ \det)$ is also a cuspidal automorphic representation with infinitesimal character $H$ and the contragredient (dual) $\pi^*$ of $\pi$ is a cuspidal automorphic representation with infinitesimal character $-H=\{ -h:\,\, h \in H \}$.
One of the main questions in the theory of automorphic forms is to describe the irreducible constituents of ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))$. If we are to do this we first need some description of all irreducible admissible $GL_n({\mathbb{A}}^\infty) \times ({\mathfrak{gl}}_n,O(n))$-modules, and then we can try to say which occur in ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))$.
Just as a character $\psi: {\mathbb{A}}^\times \rightarrow {\mathbb{C}}^\times$ can be factored as \[ \psi = \psi_\infty \times \prod_p \psi_p \] where $\psi_p:{\mathbb{Q}}_p^\times \rightarrow {\mathbb{C}}^\times$ (resp. $\psi_\infty:{\mathbb{R}}^\times \rightarrow {\mathbb{C}}^\times$), so an irreducible, admissible $GL_n({\mathbb{A}}^\infty) \times ({\mathfrak{gl}}_n,O(n))$-module can be factorised as a restricted tensor product (see \cite{flath}) \[ \pi \cong {\bigotimes_x}' \pi_x, \] where $\pi_{\infty}$ is an irreducible, admissible $({\mathfrak{gl}}_n,O(n))$-module (see for example \cite{wall}), and each $\pi_p$ is an irreducible smooth (i.e. stabilisers of vectors are open in $GL_n({\mathbb{Q}}_p)$) representation of $GL_n({\mathbb{Q}}_p)$ with $\pi_p^{GL_n({\mathbb{Z}}_p)} \neq (0)$ for all but finitely many $p$. To the factors $\pi_x$ one can associate various invariants (see \cite{jac}). \begin{itemize}
\item A central character $\psi_x:{\mathbb{Q}}_x^\times \rightarrow {\mathbb{C}}^\times$.
\item L-factors $L(\pi_p,X) \in {\mathbb{C}}(X)$.
\item A $\Gamma$-factor $\Gamma(\pi_{\infty},s)$.
\item Conductors $f(\pi_p) \in {\mathbb{Z}}_{\geq 0}$.
\item For each non-trivial character $\Psi_x:{\mathbb{Q}}_x \rightarrow
{\mathbb{C}}^{\times}$ an epsilon factor $\epsilon(\pi_x,\Psi_x) \in
{\mathbb{C}}^{\times}$. \end{itemize} (We also remark that ${\mathfrak{z}}_n$ acts via a character ${\mathfrak{z}}_n \rightarrow {\mathbb{C}}$ on any irreducible, admissible $({\mathfrak{gl}}_n,O(n))$-module $\pi_\infty$. This character is called the infinitesimal character of $\pi_\infty$.) Now we may attach to $\pi$ \begin{itemize} \item a central character $\psi_\pi = \prod_x \psi_x: {\mathbb{A}}^\times \rightarrow {\mathbb{C}}^\times$; \item an $L$-function $L(\pi,s) = \prod_p L(\pi_p,p^{-s})$ (which may or may not converge); \item an extended $L$-function $\Lambda(\pi,s)=\Gamma(\pi_\infty,s)L(\pi,s)$; \item a conductor $N(\pi) = \prod_p p^{f(\pi_p)}$ (which makes sense because $f(\pi_p)=0$ when $\pi_p^{GL_n({\mathbb{Z}}_p)} \neq (0)$); \item and an epsilon constant $\epsilon(\pi) = \prod_x \epsilon(\pi_x,\Psi_x) \in {\mathbb{C}}^{\times}$ where $\prod_x \Psi_x: {\mathbb{A}}/{\mathbb{Q}} \rightarrow {\mathbb{C}}^\times$ is any non-trivial character. \end{itemize}
The following theorem and conjecture describe the (expected) relationship between automorphic forms and $L$-functions with Euler product and functional equation. We suppose $n>1$. A similar theorem to theorem \ref{gjt} is true for $n=1$, except that $L(\pi,s)$ may have one simple pole. In this case it was due to Dirichlet. Conjecture \ref{cct} becomes vacuous if $n=1$.
\begin{thm}[{\rm Godement-Jacquet, \cite{gj}}]\label{gjt} Suppose that $\pi$ is an irreducible \linebreak constituent of ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))$ with $n>1$. Then $L(\pi,s)$ converges to a holomorphic function in some right half complex plane ${\operatorname{Re}\,} s > \sigma$ and can be continued to a holomorphic function on the whole complex plane so that $\Lambda(\pi,s)$ is bounded in all vertical strips $\sigma_1 \geq {\operatorname{Re}\,} s \geq \sigma_2$. Moreover $\Lambda(\pi,s)$ satisfies the functional equation \[ \Lambda(\pi,s)=\epsilon(\pi) N(\pi)^{-s} \Lambda(\pi^*,1-s). \] \end{thm}
\begin{conj}[{\rm Cogdell-Piatetski-Shapiro, \cite{cps1}}]\label{cct} Suppose that $\pi$ is an irreducible, admissible $GL_n({\mathbb{A}}^\infty) \times ({\mathfrak{gl}}_n,O(n))$-module such that the central character of $\pi$ is trivial on ${\mathbb{Q}}^\times$ and such that $L(\pi,s)$ converges in some half plane. Suppose also that for all characters $\psi: {\mathbb{A}}^\times/{\mathbb{Q}}^\times {\mathbb{R}}^\times_{>0} \rightarrow {\mathbb{C}}^\times$ the $L$-function $\Lambda (\pi \otimes (\psi \circ \det),s)$ (which will then converge in some right half plane) can be continued to a holomorphic function on the entire complex plane, which is bounded in vertical strips and satisfies a functional equation \[ \Lambda(\pi \otimes (\psi \circ \det),s)=\epsilon(\pi \otimes (\psi \circ \det)) N(\pi \otimes (\psi \circ \det))^{-s} \Lambda(\pi^* \otimes (\psi^{-1} \circ \det),1-s). \] ($\Lambda (\pi^* \otimes (\psi^{-1} \circ \det),s)$ also automatically converges in some right half plane.) Then there is a partition $n=n_1+...+n_r$ and cuspidal automorphic representations $\pi_i$ of $GL_{n_i}({\mathbb{A}})$ such that \[ \Lambda(\pi,s) = \prod_{i=1}^r \Lambda(\pi_i,s). \] \end{conj}
This conjecture is known to be true for $n=2$ (\cite{weil}, \cite{jl}) and $n=3$ (\cite{jpss:gl3}). For $n>3$ a weaker form of this conjecture involving twisting by higher dimensional automorphic representations is known to hold (see \cite{cps1}, \cite{cps2}). These results are called `converse theorems'.
The reason for us introducing automorphic forms is because of a putative connection to Galois representations, which we will now discuss. But first let us discuss the local situation. It has recently been established (\cite{ht}, \cite{hen}, \cite{har:icm}) that there is a natural bijection, ${\operatorname{rec}}_p$, from irreducible smooth representations of $GL_n({\mathbb{Q}}_p)$ to $n$-dimensional Frobenius semi-simple WD-representations of $W_{{\mathbb{Q}}_p}$ over ${\mathbb{C}}$. The key point here is that the bijection should be natural. We will not describe here exactly what this means (instead we refer the reader to the introduction to \cite{ht}). It does satisfy the following. \begin{itemize} \item $\psi_{\pi} \circ {\operatorname{Art}\,}_p^{-1} = \det {\operatorname{rec}}_p (\pi)$, where $\psi_\pi$ is the central character of $\pi$. \item $L({\operatorname{rec}}_p(\pi),X)= L(\pi,X)$. \item $f({\operatorname{rec}}_p(\pi))=f(\pi)$. \item $\epsilon({\operatorname{rec}}_p(\pi), \Psi_p)=\epsilon(\pi,\Psi_p)$ for any non-trivial character $\Psi_p:{\mathbb{Q}}_p \rightarrow {\mathbb{C}}^\times$. \end{itemize} The existence of ${\operatorname{rec}}_p$ can be seen as a non-abelian generalisation of local class field theory, as in the case $n=1$ we have ${\operatorname{rec}}_p(\pi)=\pi \circ {\operatorname{Art}\,}_p^{-1}$.
Now suppose that $\iota:\overline{{\Q}}_l \rightarrow {\mathbb{C}}$ and that $R$ is a de Rham $l$-adic representation of $G_{\mathbb{Q}}$ which is unramified at all but finitely many primes. Using the local reciprocity map ${\operatorname{rec}}_p$, we can associate to $R$ an irreducible, admissible $GL_n({\mathbb{A}}^\infty) \times ({\mathfrak{gl}}_n,O(n))$-module \[ \pi(\iota R)= \pi_{\infty}(R) \otimes \prod_p {\operatorname{rec}}_p^{-1}(\iota {\operatorname{WD}}_p(R)), \] where $\pi_{\infty}(R)$ is a tempered irreducible, admissible $({\mathfrak{gl}}_n,O(n))$-module with infinitesimal character
${\operatorname{HT}}(R|_{G_{{\mathbb{Q}}_l}})$ and with $\Gamma(\pi_{\infty}(R),s)=\Gamma(R,s)$. The definition of $\pi_\infty(R)$ depends only on the numbers $m^R_i$ and $m^R_{w/2,\pm}$. Then we have the following conjectures.
\begin{conj}\label{cgfa} Suppose that $H$ is a multiset of $n$ {\em integers} and that $\pi$ is an irreducible constituent of ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))$. Identify $\overline{{\Q}} \subset {\mathbb{C}}$. Then each ${\operatorname{rec}}_p(\pi_p)$ can be defined over $\overline{{\Q}}$ and there is an irreducible geometric strongly compatible system of $l$-adic representations ${\cal{R}}$ such that ${\operatorname{HT}}({\cal{R}})=H$ and ${\operatorname{WD}}_p({\cal{R}})^{{\operatorname{ss}}}={\operatorname{rec}}_p (\pi_p)$ for all primes $p$. \end{conj}
\begin{conj}\label{cafg} Suppose that \[ R:G_{\mathbb{Q}} \longrightarrow GL(V) \]
is an irreducible $l$-adic representation which is unramified at all but finitely many primes and for which $R|_{G_{{\mathbb{Q}}_l}}$ is de Rham. Let $\iota: \overline{{\Q}}_l \rightarrow {\mathbb{C}}$. Then $\pi(\iota R)$ is a cuspidal automorphic representation of $GL_n({\mathbb{A}})$. \end{conj}
These conjectures are essentially due to Langlands \cite{lang:conj}, except we have used a precise formulation which follows Clozel \cite{claa} and we have incorporated conjecture \ref{cfm} into conjecture \ref{cafg}.
Conjecture \ref{cafg} is probably the more mysterious of the two, as only the case $n=1$ and fragmentary cases where $n=2$ are known. This will be discussed further in the next section. Note the similarity to the main theorem of global class field theory that $\prod_p {\operatorname{Art}\,}_p:{\mathbb{A}}^\times \rightarrow G_{\mathbb{Q}}^{\operatorname{ab}}$ has kernel ${\mathbb{Q}}^\times$.
The following theorem provides significant evidence for conjecture \ref{cgfa}.
\begin{thm}[{\rm \cite{kotjams}, \cite{clihes}, \cite{ht}}]\label{kt} Suppose that $H$ is multiset of $n$ {\em distinct} integers and that $\pi$ is an irreducible constituent of ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))$. Let $\iota:\overline{{\Q}}_l \hookrightarrow {\mathbb{C}}$. Suppose moreover that $\pi^* \cong \pi \otimes (\psi \circ \det)$ for some character $\psi:{\mathbb{A}}^\times/{\mathbb{Q}}^\times \rightarrow {\mathbb{C}}^\times$, and that either $n \leq 2$ or for some prime $p$ the representation $\pi_p$ is square integrable (i.e. ${\operatorname{rec}}_p(\pi_p)$ is indecomposable). Then there is a continuous representation \[ R_{l,\iota}: G_{\mathbb{Q}} \longrightarrow GL_n(\overline{{\Q}}_l) \] with the following properties. \begin{enumerate} \item $R_{l,\iota}$ is geometric and pure of weight $2/n \sum_{h \in H} h$.
\item $R_{l,\iota}|_{G_{{\mathbb{Q}}_l}}$ is de Rham and ${\operatorname{HT}}(R_{l,\iota}|_{G_{{\mathbb{Q}}_l}})= H$. \item For any prime $p \neq l$ there is a representation $r_p:W_{{\mathbb{Q}}_p} \rightarrow GL_n(\overline{{\Q}}_l)$ such that ${\operatorname{WD}}_p(R_{l,\iota})^{\operatorname{ss}}=(r_p,N_p)$ and ${\operatorname{rec}}_p(\pi_p)= (\iota r_p, N_p')$. \end{enumerate} \end{thm}
This was established by finding the desired $l$-adic representations in the cohomology of certain unitary group Shimura varieties. It seems not unreasonable to hope that similar techniques might allow one to improve many of the technical defects in the theorem. However Clozel has stressed that in the cases where $H$ does not have distinct elements or where $\pi^* \not\cong \pi \otimes(\psi \circ \det)$, there seems to be no prospect of finding the desired $l$-adic representations in the cohomology of Shimura varieties. It seems we need a new technique.
\Section{What do we know?} \label{s4}
\vskip-5mm \hspace{5mm}
Let us first summarise in a slightly less precise way the various conjectures we have made, in order to bring together the discussion so far. Fix an embedding $\overline{{\Q}} \hookrightarrow {\mathbb{C}}$ and let $H$ be a multiset of integers of cardinality $n>1$. Then the following sets should be in natural bijection. One way to make precise the meaning of `natural' is to require that two objects $M$ and $M'$ should correspond if the local L-factors $L_p(M,X)$ and $L_p(M',X)$ are equal for all but finitely many $p$. Note that in each case the factors $L_p(M,X)$ for all but finitely many $p$, completely determine $M$.
\begin{enumerate} \item[(AF)]\label{af} Irreducible constituents $\pi$ of ${\cal{A}}^\circ_H(GL_n({\mathbb{Q}}) \backslash GL_n({\mathbb{A}}))$.
\item[(LF)]\label{lf} Near equivalence classes of irreducible, admissible $GL_n({\mathbb{A}}^\infty) \times ({\mathfrak{gl}}_n,O(n))$-modules $\pi$ with the following properties. (We call two $GL_n({\mathbb{A}}^\infty) \times ({\mathfrak{gl}}_n,O(n))$-modules, $\pi$ and $\pi'$ nearly equivalent if $\pi_p \cong \pi_p'$ for all but finitely many $p$.) \begin{enumerate} \item $\pi_\infty$ has infinitesimal character $H$. \item The central character $\psi_\pi$ is trivial on ${\mathbb{Q}}^{\times} \subset {\mathbb{A}}^{\times}$. \item For all characters $\psi:{\mathbb{A}}^\times/{\mathbb{Q}}^\times {\mathbb{R}}^\times_{>0}$ the $L$-function $\Lambda(\pi \otimes (\psi \circ \det),s)$ converges in some right half plane, has holomorphic continuation to the entire complex plane so that it is bounded in vertical strips and satisfies the functional equation \[ \Lambda(\pi \otimes \psi,s)=\epsilon(\pi \otimes \psi) N(\pi \otimes \psi )^{-s} \Lambda(\pi^* \otimes \psi^{-1},1-s). \] \item (See \cite{js} for an explanation of this condition.) There is a finite set of primes $S$ containing all primes $p$ for which ${\operatorname{rec}}_p(\pi_p)$ is ramified, such that, writing $L(\pi_p,X) = \prod_{i=1}^n (1-\alpha_{p,i}X)^{-1}$ for $p \not\in S$, \[ \sum_{p\not\in S,i,j} \sum_{m=1}^{\infty} \alpha_{p,i}^m \alpha_{p,j}^{-m} /mp^{ms} + \log(s-1) \] is bounded as $s \rightarrow 1$ from the right. \end{enumerate} In this case $L_p(\pi,X)=L(\pi_p,X)$.
\begin{description} \item[(lR)]\label{lr} (Fix $\iota: \overline{{\Q}}_l \rightarrow {\mathbb{C}}$.) Irreducible $l$-adic representations \[ R:G_{\mathbb{Q}} \longrightarrow GL_n(\overline{{\Q}}_l) \] which are unramified at all but finitely many primes and for which
$R|_{G_{{\mathbb{Q}}_l}}$ is de Rham with ${\operatorname{HT}}(R|_{G_{{\mathbb{Q}}_l}})=H$. In this case $L_p(R,X)=\iota L({\operatorname{WD}}_p(R),X)$.
\item[(WCS)]\label{cs} Irreducible weakly compatible systems of $l$-adic representations ${\cal{R}}$ with \linebreak ${\operatorname{HT}}({\cal{R}})=H$. In this case $L_p({\cal{R}},X)=L_p({\operatorname{WD}}_p({\cal{R}}),X)$.
\item[(GCS)]\label{gcs} Irreducible geometric strongly compatible systems of $l$-adic representations ${\cal{R}}$ with ${\operatorname{HT}}({\cal{R}})=H$. In this case $L_p({\cal{R}},X)=L_p({\operatorname{WD}}_p({\cal{R}}),X)$. \end{description} \end{enumerate}
For $n=1$ we must drop the item $(LF)$, because it would need to be modified to allow $L(\pi \otimes (\psi \circ \det),s)$ to have a simple pole, while, in any case condition (LF) (b) would make the implication $(LF) \Longrightarrow (AF)$ trivial. This being said, in the case $n=1$ all the other four sets are known to be in natural bijection (see \cite{serre:alr}). This basically follows because global class field theory provides an isomorphism \[ {\operatorname{Art}\,}: {\mathbb{A}}^{\times}/{\mathbb{Q}}^{\times} {\mathbb{R}}^\times_{>0} \liso G_{\mathbb{Q}}^{\operatorname{ab}}. \]
I would again like to stress how different are these various sorts of objects and how surprising it is to me that there is any relation between them. Items (AF) and (LF) both concern representations of adele groups, but arising in rather different settings: either from the theory of discrete subgroups of Lie groups or from the theory of $L$-functions with functional equation. Items (lR) and (WCS) arise from Galois theory and item (GCS) arises from geometry.
So what do we know about the various relationships for $n>1$?
Not much. Trivially one has $(GCS) \Longrightarrow (WCS) \Longrightarrow (lR)$. The passage $(AF) \Longrightarrow (LF)$ is OK by theorem \ref{gjt}. As discussed in section \ref{s3} we have significant partial results in the directions $(LF) \Longrightarrow (AF)$ and $(AF) \Longrightarrow (GCS)$, but both seem to need new ideas. (Though I should stress that I am not really competent to discuss converse theorems.)
One way to establish the equivalence of all five items would be to complete the passages $(LF) \Longrightarrow (AF)$ and $(AF) \Longrightarrow (GCS)$ and to establish the passage $(lR) \Longrightarrow (AF)$. It is these implications which have received most study, though it should be pointed out that in the function field case the equivalence of the analogous objects was established by looking at the implications \[ (lR) \Longrightarrow (LF) \Longrightarrow (AF) \Longrightarrow (GCS). \] (See \cite{laff}. It is the use of techniques from Grothendieck's $l$-adic cohomology to prove the first of these implications which is most special to function fields.) However it is striking that in the case of number fields all known implications from items (lR), (WCS) or (GCS) to (LF) go via (AF).
For the rest of this article we will concentrate on what still seems to be the least understood problem: the passage from (lR) or (WCS) to (AF) or (LF). Although the results we have are rather limited one should not underestimate their power. Perhaps the most striking illustration of this is that the lifting theorems discussed in section \ref{s4.2} (combined with earlier work using base change and converse theorems) allowed Wiles \cite{wiles} to finally prove Fermat's last theorem.
The discussion in the rest of this paper will of necessity be somewhat more technical. In particular we will need to discuss automorphic forms, $l$-adic representations and so on over general number fields (i.e. fields finite over ${\mathbb{Q}}$). We will leave it to the reader's imagination exactly how such a generalisation is made. In this connection we should remark that if $L/K$ is a finite extension of number fields and if $R$ is a semi-simple de Rham $l$-adic representation of $G_L$ which is unramified at all but finitely many primes, then (see \cite{artin}) \[ L(R,s) = L({\operatorname{Ind}\,}_{G_L}^{G_K} R,s) \] (formally if the $L$-functions don't converge). In fact this is true Euler factor by Euler factor and similar results hold for conductors and $\epsilon$-factors (see \cite{tate:ntb}). This observation can be extremely useful.
\subsection{Cyclic base change}\label{s4.1}
\vskip-5mm \hspace{5mm}
Suppose that $G$ is a group, $H$ a normal subgroup such that $G/H$ is cyclic with generator of $\sigma$. It is an easy exercise that an irreducible representation $r$ of $H$ extends to a representation of $G$ if and only if $r^\sigma \cong r$ as representations of $H$. If one believes conjectures \ref{cgfa} and \ref{cafg}, one might expect that if $L/K$ is a cyclic Galois extension of number fields of prime order, if $\sigma$ generates ${\operatorname{Gal}\,}(L/K)$ and if $\pi$ is a cuspidal automorphic representation of
$GL_n({\mathbb{A}}_L)$ with $\pi \circ \sigma \cong \pi$, then there should be a cuspidal automorphic representation $\Pi$ of $GL_n({\mathbb{A}}_K)$, such that for all places $w$ of $L$ we have ${\operatorname{rec}}_w(\pi_w)={\operatorname{rec}}_{w|_K}(\Pi_{w|_K})|_{W_{L_w}}$. This is indeed the case. For $n=1$ we have $\pi=\Pi \circ {\mbox{\bf N}}_{L/K}$, Langlands \cite{lang:bc} proved it for $n=2$ using the trace formula and Arthur and Clozel \cite{ac} generalised his method to all $n$.
One drawback of this result is that if $v$ is a place of $K$ inert in $L$ then there is no complete recipe for $\Pi_v$ in terms of $\pi$. This can be surprisingly serious. It can however be alleviated, if we know how to associate irreducible $l$-adic representations to both $\Pi$ and $\pi$. Langlands used this to show that many two dimensional Artin representations (i.e. $l$-adic representations with finite image) were automorphic (i.e. associated to a cuspidal automorphic representation). In fact using additional results from the theory of $L$-functions, particularly the converse theorem for $GL_3$ (see section \ref{s4.4}), he and Tunnell (\cite{tunnell}) were able to establish the automorphy of all continuous two dimensional Artin representations with soluble image.
\subsection{Brauer's theorem}\label{s4.3}
\vskip-5mm \hspace{5mm}
The result I want to discuss is a result of Brauer \cite{brauer} about finite groups. \begin{thm}[{\rm Brauer}] Suppose that $r$ is a representation of a finite group $G$. Then there are soluble subgroups $H_i<G$, one dimensional representations $\psi_i$ of $H_i$ and integers $n_i$ such that as virtual representations of $G$ we have \[ r = \sum_i n_i {\operatorname{Ind}\,}_{H_i}^G \psi_i. \] \end{thm}
As Artin \cite{artin} had realised this theorem has the following immediate consequence. (Indeed Brauer proved his theorem in response to Artin's work.)
\begin{cor} Let $\iota:\overline{{\Q}}_l \rightarrow {\mathbb{C}}$. Suppose that \[ R:G_{\mathbb{Q}} \longrightarrow GL_n(\overline{{\Q}}_l) \] is an $l$-adic representation with {\em finite image}. Then the $L$-function $L(\iota R, s)$ has meromorphic continuation to the entire complex plane and satisfies the expected functional equation. \end{cor}
Artin's argument runs as follows. Let $G$ denote the image of $R$ and write \[ R = \sum_i n_i {\operatorname{Ind}\,}_{H_i}^G \psi_i \] as in Brauer's theorem. Let $L/{\mathbb{Q}}$ be the Galois extension with group $G$ cut out by $R$ and let $K_i=L^{H_i}$. Then one has equalities \[ \begin{array}{rcl} L(\iota R,s) &=& \prod_i L(\iota {\operatorname{Ind}\,}_{G_{K_i}}^{G_{\mathbb{Q}}} \psi_i,s)^{n_i} \\ &=& \prod_i L( \iota \psi_i,s)^{n_i}. \end{array} \] By class field theory for the fields $K_i$, the character $\psi_i$ is automorphic on $GL_1({\mathbb{A}}_{K_i})$ and so $L(\iota \psi_i,s)$ has holomorphic continuation to the entire complex plane (except possibly for one simple pole if $\psi_i=1$) and satisfies a functional equation. It follows that $L(\iota R,s)$ has meromorphic continuation to the entire complex plane and satisfies a functional equation. The problem with this method is that some of the integers $n_i$ will usually be negative so that one can only conclude the meromorphy of $L(\iota R,s)$, not its holomorphy.
\subsection{Converse theorems}\label{s4.4}
\vskip-5mm \hspace{5mm}
As Cogdell and Piatetski-Shapiro point out, conjecture \ref{cct} would have very important implications for Galois representations. For instance the cases $n=2$ and $3$ played a key role in the proof of the automorphy of two dimensional Artin representations (see \ref{s4.1}). Conjecture \ref{cct} combined with Brauer's theorem and a result of Jacquet and Shalika \cite{js} in fact implies that many (all? - certainly those with soluble or perfect image) Artin representations are automorphic. A similar argument shows that in many other cases, in order to check the automorphy of an $l$-adic representation of $G_{\mathbb{Q}}$, it suffices to do so after a finite base change. For instance one has the following result.
{\em Assume conjecture \ref{cct}. Let $\iota:\overline{{\Q}}_l \hookrightarrow {\mathbb{C}}$ and let $K/{\mathbb{Q}}$ be a finite, totally real Galois extension. Suppose that $\Pi$ is a cuspidal automorphic representation of $GL_n({\mathbb{A}}_K)$ with infinitesimal character corresponding to a multiset $H$ consisting of $n$ distinct integers. If $n>2$ also suppose that $\Pi_v$ is square integrable (i.e. ${\operatorname{rec}}_v(\Pi_v)$ is indecomposable) for some finite place $v$ of $K$. Let \[ R:G_{\mathbb{Q}} \longrightarrow GL_n(\overline{{\Q}}_l) \]
be an $l$-adic representation such that $R \sim R^* \otimes \psi$ for some character $\psi$ of $G_{\mathbb{Q}}$, and such that $R|_{G_{K}}$ is irreducible. Suppose finally that $R|_{G_K}$ and $\Pi$ are associated, in the sense that, for all but finitely many places $v$ of $K$, we have
\[ \iota L({\operatorname{WD}}_v(R|_{G_K}),X) = L(\Pi_v,X). \] Then there is a regular algebraic cuspidal automorphic representation $\pi$ of $GL_n({\mathbb{A}})$ associated to $R$ in the same sense.}
\subsection{Lifting theorems}\label{s4.2}
\vskip-5mm \hspace{5mm}
To describe this sort of theorem we first remark that if $R:G_{\mathbb{Q}} \rightarrow GL_n(\overline{{\Q}}_l)$ is continuous then after conjugating $R$ by some element of $GL_n(\overline{{\Q}}_l)$ we may assume that the image of $R$ is contained in $GL_n({\cal{O}}_{\overline{{\Q}}_l})$ and so reducing we obtain a continuous representation \[ \overline{{R}}: G_{\mathbb{Q}} \longrightarrow GL_n(\overline{{\F}}_l). \]
The lifting theorems I have in mind are results of the general form if $R$ and $R'$ are $l$-adic representations of $G_{\mathbb{Q}}$ with $R'$ automorphic and if $\overline{{R}}=\overline{{R}}'$ then $R$ is also automorphic. Very roughly speaking the technique (pioneered by Wiles \cite{wiles} and completed by the author and Wiles \cite{tw}) is to show that $R \bmod l^r$ arises from automorphic forms for all $r$ by induction on $r$. As $\ker(GL_n({\mathbb{Z}}/l^r{\mathbb{Z}}) \rightarrow \hspace{-.86em} \rightarrow GL_n({\mathbb{Z}}/l^{r-1} {\mathbb{Z}}))$ is an abelian group one is led to questions of class field theory and Galois cohomology.
I should stress that such theorems are presently available only in very limited situations. I do not have the space to describe the exact limitations, which are rather technical, but the sort of restrictions that are common are as follows. \begin{enumerate} \item If $R:G_{\mathbb{Q}} \rightarrow GL(V)$ then there should be a character $\mu:G_{\mathbb{Q}} \rightarrow GL_n(\overline{{\Q}}_l)$ and a non-degenerate bilinear form $(\,\,\, ,\,\,\, )$ on $V$ such that \begin{itemize} \item $(R(\sigma)v_1,R(\sigma)v_2) = \mu(\sigma)(v_1,v_2)$ and \item $(v_2,v_1) = \mu(c) (v_1,v_2)$. \end{itemize} (This seems to be essential for the method of \cite{tw}.)
\item $R$ should be de Rham with distinct Hodge-Tate numbers. (This again seems essential to the method of \cite{tw}, but see \cite{bt}.)
\item Either $R$ and $R'$ should be ordinary (i.e. their restrictions to $G_{{\mathbb{Q}}_l}$ should be contained in a Borel subgroup); or $R$ and $R'$ should be crytsalline (not just de Rham) at $l$ with the same Hodge-Tate numbers and $l$ should be large compared with the differences of elements of ${\operatorname{HT}}(R)$. (The problems here are connected with the need for an integral Fontaine theory, but they are not simply technical problems. There are some complicated results pushing back this restriction in isolated cases, see \cite{cdt}, \cite{bcdt}, \cite{savitt}, but so far our understanding is very limited.)
\item The image of $\overline{{R}}$ should not be too small (e.g. should be irreducible when restricted to ${\mathbb{Q}}(e^{2 \pi i/l})$), though in the case $n=2$ there is beautiful work of Skinner and Wiles (\cite{sw1} and \cite{sw2}) dispensing with this criterion, which this author has unfortunately not fully understood.
\end{enumerate}
In addition, all the published work is for the case $n=2$. However there is ongoing work of a number of people attempting to dispense with this assumption. Using a very important insight of Diamond \cite{diam}, the author, together with L.Clozel and M.Harris, has generalised to all $n$ the so called minimal case (originally treated in \cite{tw}) where $R$ is no more ramified than $\overline{{R}}$. One would hope to be able to deduce the non-minimal case from this, as Wiles did in \cite{wiles} for $n=2$. In this regard one should note the work of Skinner and Wiles \cite{swe} and the work of Mann \cite{russ}. However there seems to be one missing ingredient, the analogue of the ubiquitous Ihara lemma, see lemma 3.2 of \cite{ihara} (and also theorem 4.1 of \cite{rib2}). As this seems to be an important question, but one which lies in the theory of discrete subgroups of Lie groups, let us take the trouble to formulate it, in the hope that an expert may be able to prove it. It should be remarked that there are a number of possible formulations, which are not completely equivalent and any of which would seem to suffice. We choose to present one which has the virtue of being relatively simple to state.
\begin{conjecture} Suppose that $G/{\mathbb{Q}}$ is a unitary group which becomes an inner form of $GL_n$ over an imaginary quadratic field $E$. Suppose that $G({\mathbb{R}})$ is compact. Let $l$ be a prime which one may assume is large compared to $n$. Let $p_1$ and $p_2$ be distinct primes different from $l$ with $G({\mathbb{Q}}_{p_1}) \cong GL_n({\mathbb{Q}}_{p_1})$ and $G({\mathbb{Q}}_{p_2}) \cong GL_n({\mathbb{Q}}_{p_2})$. Let $U$ be an open compact subgroup of $G({\mathbb{A}}^{p_1,p_2})$ and consider the representation of $GL_n({\mathbb{Q}}_{p_1}) \times GL_n({\mathbb{Q}}_{p_2})$ on the space $C^\infty(G({\mathbb{Q}})\backslash G({\mathbb{A}}) / U, \overline{{\F}}_l)$ of locally constant $\overline{{\F}}_l$-valued functions on \[ G({\mathbb{Q}}) \backslash G({\mathbb{A}}) /U = (G({\mathbb{Q}}) \cap U) \backslash (GL_n({\mathbb{Q}}_{p_1}) \times GL_n({\mathbb{Q}}_{p_2}). \] (Note that $G({\mathbb{Q}}) \cap U$ is a discrete cocompact subgroup of $GL_n({\mathbb{Q}}_{p_1}) \times GL_n({\mathbb{Q}}_{p_2})$.) Suppose that $\pi_1 \otimes \pi_2$ is an irreducible {\em sub-representation} of $C^\infty(G({\mathbb{Q}})\backslash G({\mathbb{A}}) / U, \overline{{\F}}_l)$ with $\pi_1$ generic. Then $\pi_2$ is also generic. \end{conjecture}
The most serious problem with applying such lifting theorems to prove an $l$-adic representation $R$ is automorphic is the need to find some way to show that $\overline{{R}}$ is automorphic. The main success of lifting theorems to date, has been to show that if $E$ is an elliptic curve over the rationals then $H^1(E({\mathbb{E}}),\overline{{\Q}}_l)$ is automorphic, so that $E$ is a factor of the Jacobian of a modular curve and the $L$-function $L(E,s)$ is an entire function satisfying the expected functional equation (\cite{wiles}, \cite{tw},\cite{bcdt}). This was possible because $GL_2({\mathbb{Z}}_3)$ happens to be a pro-soluble group and there is a homomorphism $GL_2({\mathbb{F}}_3) \longrightarrow GL_2({\mathbb{Z}}_3)$ splitting the reduction map. The Artin representation \[ G_{\mathbb{Q}} \longrightarrow GL(H^1(E({\mathbb{C}}),{\mathbb{F}}_3)) \longrightarrow GL_2({\mathbb{Z}}_3) \] is automorphic by the Langlands-Tunnell theorem alluded to in section \ref{s4.1}.
\subsection{Other techniques?}
\vskip-5mm \hspace{5mm}
I would like to discuss one other technique which has been some help if $n=2$ and may be helpful more generally. We will restrict our attention here to the case $n=2$ and $\det R(c)=-1$. We have said that the principal problem with lifting theorems for proving an $l$-adic representation $R:G_{{\mathbb{Q}}} \rightarrow GL_2(\overline{{\Q}}_l)$ is automorphic is that one needs to know that $\overline{{R}}$ is automorphic. This seems to be a very hard problem. Nonetheless one can often show that $\overline{{R}}$ becomes automorphic over some Galois totally real field $K/{\mathbb{Q}}$. (Because $K$ is totally real, if $\overline{{R}} (G_{\mathbb{Q}})
\supset SL_2({\mathbb{F}}_l)$ and $l>3$ then $\overline{{R}} (G_K) \supset SL_2({\mathbb{F}}_l)$. So this `potential automorphy' is far from vacuous). The way one does this is to look for an abelian variety $A/K$ with multiplication by a number field $F$ with $[F:{\mathbb{Q}}]=\dim A$, and such that $\overline{{R}}$ is realised on $H^1(A({\mathbb{C}}),{\mathbb{F}}_l)[\lambda]$ for some prime $\lambda|l$, while for some prime $\lambda'|l' \neq l$ the image of $G_K$ on $H^1(A({\mathbb{C}}),{\mathbb{F}}_{l'})[\lambda']$ is soluble. One then argues that $H^1(A({\mathbb{C}}),{\mathbb{F}}_{l'})[\lambda']$ is automorphic, hence by a lifting theorem $H^1(A({\mathbb{C}}),{\mathbb{Q}}_{l'}) \otimes_{F_{l'}} F_{\lambda'}$ is automorphic, so that
(tautologically) $H^1(A({\mathbb{C}}),{\mathbb{F}}_l)[\lambda]$ is also automorphic, and hence, by another lifting theorem, $R|_{G_K}$ is automorphic. One needs $K$ to be totally real, as over general number fields there seems to be no hope of proving lifting theorems, or even of attaching $l$-adic representations to automorphic forms. In practice, because of various limitations in the lifting theorems one uses, one needs to impose some conditions on the behaviour of a few primes, like $l$, in $K$ and some other conditions on $A$. The problem of finding a suitable $A$ over a totally real field $K$, comes down to finding a $K$-point on a twisted Hilbert modular variety. This is possible because we are free to choose $K$, the only restriction being that $K$ is totally real and certain small primes (almost) split completely in $K$. To do this, one has the following relatively easy result.
\begin{prop}[{\rm \cite{bailly},\cite{pop}}] Suppose that $X/{\mathbb{Q}}$ is a smooth geometrically irreducible variety. Let $S$ be a finite set of places of ${\mathbb{Q}}$ and suppose that $X$ has a point over the completion of ${\mathbb{Q}}$ at each place in $S$. Let ${\mathbb{Q}}_S$ be the maximal extension of ${\mathbb{Q}}$ in which all places in $S$ split completely (e.g. ${\mathbb{Q}}_{\{ \infty \}}$ is the maximal totally real field). Then $X$ has a ${\mathbb{Q}}_S$-point. \end{prop}
In this regard it would have extremely important consequences if, in the previous proposition, one could replace ${\mathbb{Q}}_S$ by ${\mathbb{Q}}_S^{\operatorname{sol}}$, the maximal soluble extension of ${\mathbb{Q}}$ in which all places in $S$ split completely. I do not know if it is reasonable to expect this.
Using this method one can, for instance, prove the following result.
\begin{thm}[{\rm \cite{mero}}] \label{mer} Suppose that ${\cal{R}}$ is an irreducible weakly compatible system of two dimensional $l$-adic representations with ${\operatorname{HT}}({\cal{R}})=\{ n_1,n_2 \}$ where $n_1 \neq n_2$. Suppose also that $\det R_{l,\iota}(c)=-1$ for one (and hence for all) pairs $(l,\iota)$. Then there is a Galois totally real field $K/{\mathbb{Q}}$ and a cuspidal automorphic representation $\pi$ of $GL_2({\mathbb{A}}_K)$ such that \begin{itemize}
\item for all $v|\infty$, $\pi_v$ has infinitesimal character
$H$, and
\item for all $(l,\iota)$ and for all finite places $v {\mbox{$\not| $}} l$
of $K$ we have
\[ {\operatorname{rec}}_v(\pi_v) = {\operatorname{WD}}_v(R_{l,\iota}|_{G_K})^{\operatorname{ss}}. \] \end{itemize} In particular ${\cal{R}}$ is pure of weight $(n_1+n_2)/2$. Moreover ${\cal{R}}$ is strongly compatible and $L(\iota {\cal{R}},s)$ has meromorphic continuation to the entire complex plane and satisfies the expected functional equation. \end{thm}
The last sentence of this theorem results from the first part and Brauer's theorem. We remark that conjecture \ref{cct} would imply that this theorem could be improved to assert the automorphy of ${\cal{R}}$ over ${\mathbb{Q}}$.
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arXiv
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Letters in High Energy Physics
Vol 2 No 2 (2019) /
Search for more sensitive observables to charged scalar in \(B \rightarrow D^{(*)}\tau\nu_{\tau}\) decays.
Lobsang Dhargyal Harish-Chandra Research Institute
DOI: https://doi.org/10.31526/lhep.2.2019.108
Keywords: new R(D(*)) observables
It has been known that $B \rightarrow D^{(*)} \tau \nu_{\tau}$ are good observables in the search for the charged Higgs. The recent obervation of deviation from standard-model by almost 4$\sigma$ by Babar, Belle and LHCb in $R(D^{(*)})$ revived the interest in possible signal of presence of charged Higgs in these modes. But such a large deviation in the rates, where standard-model has tree level contribution, coming from a charged Higgs alone is highly unlikely. However these decay modes are good probes to search for small charged Higgs signal if we can construct sensitive observables in these modes. In this work we would like to propose four new observables which shows much more sensitivity to the presence of charged Higgs than the usual observables such as $A_{\lambda}^{D^{(*)}}$ and $A_{\theta}^{D^{(*)}}$. These four observable are (1) $\frac{1}{A_{\lambda}^{D}}$,\ (2) $Y_{1}(q^{2}) = \frac{A^{D}_{\theta}}{A^{D}_{\lambda}}$,\ (3) $Y_{2}(q^{2}) = \frac{d\Gamma(B \rightarrow D^{*}\tau\nu_{\tau})}{d\Gamma_{D}(\lambda_{\tau}=+1/2) - d\Gamma_{D}(\lambda_{\tau}=-1/2)}$ and (4) $Y_{3}(q^{2}) = (\frac{q^{2}}{m^{2}_{\tau}})(A^{D}_{\lambda} + 1)\frac{1}{A^{D}_{\lambda}}$.
Letters in High Energy Physics (LHEP) is an open access journal published by Andromeda Publishing and Education Services. The articles in LHEP are distributed according to the terms of the creative commons license CC-BY 4.0. Under the terms of this license, copyright is retained by the author while use, distribution and reproduction in any medium are permitted provided proper credit is given to original authors and sources.
By submitting an article for publication in LHEP, the submitting author asserts that:
1. The article presents original contributions by the author(s) which have not been published previously
in a peer-reviewed medium and are not subject to copyright protection.
2. The co-authors of the article, if any, as well as any institution whose approval is required,
agree to the publication of the article in LHEP.
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CommonCrawl
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\begin{definition}[Definition:Thousand]
A '''thousand''' is $1000$: $10$ to the power of $3$:
:$1000 = 10^3 = 10 \times 10 \times 10$
\end{definition}
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ProofWiki
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Predictive modeling in pediatric traumatic brain injury using machine learning
Shu-Ling Chong1,
Nan Liu2,3,
Sylvaine Barbier3 &
Marcus Eng Hock Ong2,4
Pediatric traumatic brain injury (TBI) constitutes a significant burden and diagnostic challenge in the emergency department (ED). While large North American research networks have derived clinical prediction rules for the head injured child, these may not be generalizable to practices in countries with traditionally low rates of computed tomography (CT). We aim to study predictors for moderate to severe TBI in our ED population aged < 16 years.
This was a retrospective case–control study based on data from a prospective surveillance head injury database. Cases were included if patients presented from 2006 to 2014, with moderate to severe TBI. Controls were age-matched head injured children from the registry, obtained in a 4 control: 1 case ratio. These children remained well on diagnosis and follow up. Demographics, history, and physical examination findings were analyzed and patients followed up for the clinical course and outcome measures of death and neurosurgical intervention. To predict moderate to severe TBI, we built a machine learning (ML) model and a multivariable logistic regression model and compared their performances by means of Receiver Operating Characteristic (ROC) analysis.
There were 39 cases and 156 age-matched controls. The following 4 predictors remained statistically significant after multivariable analysis: Involvement in road traffic accident, a history of loss of consciousness, vomiting and signs of base of skull fracture. The logistic regression model was created with these 4 variables while the ML model was built with 3 extra variables, namely the presence of seizure, confusion and clinical signs of skull fracture. At the optimal cutoff scores, the ML method improved upon the logistic regression method with respect to the area under the ROC curve (0.98 vs 0.93), sensitivity (94.9% vs 82.1%), specificity (97.4% vs 92.3%), PPV (90.2% vs 72.7%), and NPV (98.7% vs 95.4%).
In this study, we demonstrated the feasibility of using machine learning as a tool to predict moderate to severe TBI. If validated on a large scale, the ML method has the potential not only to guide discretionary use of CT, but also a more careful selection of head injured children who warrant closer monitoring in the hospital.
Head Injury remains an important cause of mortality and morbidity for children, worldwide. Injury-related deaths in the pediatric age group are mostly associated with head injury [1]. Emergency Departments (EDs) worldwide are seeing an increase in pediatric head injury attendance [2]. The admission rates for head injured children are also on the rise [3]. While the majority of these are mild, severe head injuries have potential for mortality and long-term neurological devastation. The prevalence of neurological disability among children and youths admitted for traumatic brain injury approximates 20% [4]. Compared to adults with head injury, children tend to present in a varied way. Younger children are unable to provide a clear history and may be difficult to examine. A matched retrospective cohort study performed to inform an evidence-based triage assessment showed that young age and injuries to the temporo-parietal region were more likely to be associated with significant closed head injury, as identified on computed tomography (CT) [5].
CT scans are frequently performed in the adult head injured population. In children however, the rapidly developing brain, when exposed to radiation, is at risk of developing malignancies [6,7]. When deciding on whether a CT is warranted in a young child, the physician has to weigh the need to promptly diagnose an intracranial injury against the radiation that the child will be exposed to. Locally, there is great reluctance to order unnecessary CT scans.
Clinical prediction rules [8-10] have been published by large North American research networks to guide the ED physician on when to order a CT scan for a head-injured child. The Pediatric Emergency Care Applied Research Network (PECARN) [7] rule specifically, has been reported to be of excellent performance [11]. However, prior to application, it has been encouraged that the question of generalizability and performance to the individual population be addressed [12]. The CT rate in the Singapore population has been maintained at a low level of under 2%, as opposed to the estimated 30-50% reported in the literature. This is because a large majority of our patients comprise of young children presenting with mild head injuries after falls, as well as the availability of inpatient observation in most cases.
While most of the published clinical rules [6-8] were derived with recursive partitioning [13], emerging computational methods like machine learning (ML) have potential in solving complex and challenging medical problems [14-17]. ML procedures are capable of discovering interaction, nonlinear, and high-order effects in the predictive variables [14], which are difficult to handle with conventional parametric regression methods. In this study, we aim to (1) select clinical predictors for moderate to severe traumatic brain injury (TBI) in children aged < 16 years, (2) derive a ML model and a logistic regression model (3) Compare the performance of both tools.
Study design and patient recruitment
This was a retrospective case–control study. Cases were included if patients presented during the period from 2006 to 2014, with moderate to severe TBI. Due to the very low event rate, a case–control design was chosen [18,19] instead of a cohort analysis.
Data was collected from KK Women's and Children's Hospital, Singapore, the main pediatric emergency department in Singapore, with an annual trauma attendance (of all severities) of about 28,000. The majority of head injuries that we see in the emergency department are mild. We defined cases as patients aged < 16 years who presented to the ED with a Glasgow coma scale (GCS) of ≤13 or those who presented with GCS 15 but deteriorated after admission, and were confirmed on CT scan to have a bleed or fracture, during the period January 2006 – June 2014. Controls were obtained from an ongoing prospective head injury database. Controls were age-matched, year for year, at a ratio of 4 controls: 1 case. This study was approved by the Singapore Health Services (SingHealth) Centralized Institutional Review Board with a waiver of patient consent.
We obtained the individual predictive variables based on those published in similar studies [6-8], as well as from departmental head injury protocols. We divided the collection of data into demographics, mechanism of injury, presenting symptoms and physical examination findings. Symptoms studied included seizures, confusion, loss of consciousness (and duration), difficult arousal, and vomiting. Caregivers of preverbal children were questioned for irritability while verbal children were questioned for headache and amnesia.
From the physical findings, data were documented on the GCS, altered mental status, presence of unequal pupils, signs of vault fractures and basal skull fractures, scalp hematoma, focal neurological signs and gait abnormalities. Basal skull fractures signs included: blood or cerebrospinal fluid from the nose or ears, bruising at the posterior auricular region, and periorbital bruises. Among young children with open fontanelles, the presence or absence of a tense fontanelle was documented.
Admitted patients were followed up and the need for neurosurgery or any resultant death was documented. Among the controls, a clinical research coordinator followed up discharged patients with a call 72 hours after ED attendance, to ask about any deterioration or attendance at another institution.
Patients with and without TBI were studied for differences in clinical characteristics, using Student t-tests for continuous variables and Chi-Square or Fisher Exact test for categorical variables. Continuous variables are expressed as mean and standard deviation and categorical variables as absolute numbers and percentages.
In the approach using a classical logistic regression, we used a two-step selection for the contributing factors. Univariable logistic regressions were performed on each of them and those achieving a p-value below 0.2 were selected. Then, we fitted a multivariable model, following a stepwise algorithm (p-value of entry = 0.1, p-value of removal = 0.05). The models' selection was based on the Akaike Criterion (AIC), the Bayesian Information Criteron (BIC) and log-likelihood, incorporating clinical knowledge. The predictive performance of the final model was reflected by the AUC, sensitivity, specificity, Positive and Negative Predictive Values (PPV and NPV). The data were analyzed using STATA v12 (Stata Corp, College Station, TX) and MATLAB R2009a (Mathworks, Natick, MA).
Predictive modeling with machine learning
The machine learning (ML) method [20] implemented for predictive modeling in this pediatric traumatic brain injury study was originally designed for the prediction of acute cardiac complications, with an ensemble learning-based risk assessment as the core of decision making. The rationale behind this ML method is that in most scenarios we often seek a second or more opinion before making final decisions. For example in choosing a proper treatment of a disease, people usually consult with more than one physician to reach a conclusion. In machine learning, this process of decision making is called ensemble learning where the decision is made by combining the outcomes of several individual classifiers (a classifier in machine learning is considered as a physician in the real-world).
Due to its flexibility in many application domains, the above ML method is readily adaptable to our study with minor changes. The architecture of the ML method used in this study is illustrated in Figure 1. Each ensemble classifier φ t where t = 1, 2, …, T and T is the number of individual classifiers in the decision ensemble. Ensemble learning methods [21,22] usually generate a predictive label rather than a score as the output. The ML method uses a simple and straightforward approach to convert the predictive decision into a risk score. Details are elaborated as follows.
The architecture of the machine learning (ML) method. Input x is the patient whose risk of abnormal CT scan is being evaluated. L t is the training set consisting of K samples (x k , y k ) where k = 1, 2, …, K and y k is the class label. By using the training data, a total of T individual classifiers φ t (x, L t ) are created to form the decision ensemble. Each individual classifier is built based on a subset of the training data. Then the prediction outcomes are combined by means of majority voting scheme to generate a final risk score for patient x.
Assume that we have a training dataset L t consisting of K samples (x k , y k ) where k = 1, 2, …, K and y k is the class label. Given a testing sample x, its label y can be predicted by a single classifier φ t (x, L t ) where the class label is either C0 or C1. Label C0 indicates that the patient is normal (negative outcome) while label C1 indicates that the patient has abnormal CT scan (positive outcome). As illustrated in Figure 1, we can derive T independent classifiers from training samples. The risk score on the testing sample x is calculated using
$$ R{S}_x=100\times \frac{{\displaystyle {\sum}_{t=1}^T{\varphi}_t\left(x,{L}_t\right)}}{T} $$
The advantage of the ML method is its ability to handle data imbalance, select suitable individual classifiers for decision ensemble creation and decision combination, such as for our dataset (i.e. positive samples are less than negative samples with a ratio of 1:4).
Instead of applying a sophisticated hybrid-sampling scheme [20] to create the decision ensemble, in this study we used a simplified under-sampling scheme. Given the minority set P and the majority set N, the under-sampling method [21] randomly samples a subset N t from N where |N t | < |N| and |N t | = |P|. Dataset P represents a set of samples with positive outcomes and N represents a set of samples with negative outcomes. The balanced dataset L t consists of both P and N t and is used for classification model derivation. We then estimate a risk score using Eq. (1).
In the ML method, neural network [23,24] was chosen as the individual classifier φ because of its reliable performance and efficiency. The individual classifier was single layer feed-forward neural network where extreme learning machine [25] was adopted as the training method. In implementing the ensemble learning and neural network-based risk scoring method, the ensemble size T was 100, and the number of hidden neurons was 30. The sigmoid function was chosen as the activation function in neural network training.
In our study, two sets of predictive variables were used to build the ML model. One set of variables was derived from logistic regression according to the statistical significance, while another set of variables were determined by physicians in terms of clinically relevance. Compared to traditional regression analysis, the ML method is flexible where the predictive variables used to build the model are not necessarily significant in statistical analysis. Furthermore, the ML method may be able to discover nonlinear correlations among all variables.
Thirty-nine cases of moderate to severe TBI children were analyzed, with a corresponding 156 age-matched controls. Table 1 shows the comparison of patient demographics and mechanism of injury, between both groups. Among the cases, 26 patients required neurosurgical intervention and 8 patients died. From the prospective database, our event rate was 0.5% and our CT rate was 1%. Among the controls in this study, 4 patients had a CT brain (2.6%). Retrospective application of the published rules [6-8] to the prospective database showed that they would indeed increase the CT rate in our population: CHALICE 24.0%, CATCH (for high risk only) 5.7%, CATCH (for high and medium risk) 20.1%, PECARN (for high risk in children < 2 years) 1.7%, PECARN (for high risk in children ≥ 2 years) 2.1%, PECARN (high and intermediate risk in children < 2 years) 14.0%, PECARN (high and intermediate risk in children ≥ 2 years) 24.6%.
Table 1 Patient demographics and mechanism of injury
Table 1 presents patient demographics. With regards to the primary mechanism of injury, 44% of the cases were involved in a road traffic accident as compared to only 2% in the controls (p < 0.001), while the majority of controls presented to the ED after falls. A similar trend was seen (although with small numbers) among children aged 2 years and under.
Table 2 describes the individual variables obtained from history and physical examination. Variables from history or physical evidence that described altered mental status – difficult arousal, confusion/disorientation and signs of altered mental status were each statistically significant. Besides altered mental status, the presence of signs of base of skull fracture, unequal pupils, and scalp hematoma were statistically significant. Among those with scalp hematomas, frontal hematomas appeared to be protective. Among infants with open fontanelles, the presence of a tense fontanelle was also statistically significant.
Table 2 Univariable analysis of variables from history and physical examination
On multivariable analysis (Table 3), the following four predictors showed an independent significant effect: mechanism of road traffic accident (OR: 19.62, p = 0.001), history of loss of consciousness (OR: 16.32, p < 0.001), vomiting (OR: 4.89, p = 0.006) and signs of base of skull fracture (OR: 13.94, p = 0.001). A ML model was created using three more variables, namely presence of seizure activity, confusion and clinical signs of skull fracture.
Table 3 Independent predictors for traumatic brain injury (univariable then multivariable logistic regressions)
Two receiver operating characteristic (ROC) curves shown in Figure 2 were drawn using both prediction models, from which ML method was observed to outperform logistic regression method. Detailed comparison results are presented in Table 4. In general, the ML method significantly improved upon the logistic regression method with respect to sensitivity (94.9% vs 82.1%) and PPV (90.2% vs 72.7%). The cutoff scores were chosen to give the best trade-off between sensitivity and specificity, where the optimal cutoff is determined by the point that is nearest to the upper-left corner in the ROC curve.
ROC curves produced by logistic regression and machine learning.
Table 4 Prediction results using receiver operating characteristic (ROC) analysis
Figure 3 illustrates the differences in predicted scores by the logistic regression method and the ML method in terms of frequency distribution. Figure 3(a) shows the results on TBI patients and Figure 3(b) presents the results on non-TBI patients. In non-TBI patients, both methods performed similarly with the ML prediction being slightly more accurate. In TBI patients, the ML method performed better at categorizing most of the TBI patients at high risk for moderate to severe injury. These matched the observations that the ML method achieved higher sensitivity and PPV than the logistic regression method.
Frequency distribution of the logistic regression method and the machine learning method in predicting pediatric TBI.
In current practice, 3 clinical decision rules (CDRs) have been widely referenced: CHALICE, PECARN and CATCH. PECARN and physician practice were demonstrated to be superior in identifying all clinically important traumatic brain injuries in a recent prospective observational study that compared these rules [11]. Specifically, apart from being derived and validated in a large population (n = 42412), the PECARN had a separate rule for preverbal children (<2 years old) [7]. The PECARN was intended as a rule-out tool, identifying low risk children who do not require the CT scan. The rate of CT in this study was 35.3%.
It has been previously noted that applicability of the clinical prediction rules may vary based on population characteristics, and before implementing them, their performance should first be assessed [10]. We identified a few differences in the Singapore population compared to that reported in the PECARN study. The mean age of children from our prospective database was 4.6 years, as compared to 7.1 years in the latter. Most of our head injured population comprised of young children presented after low mechanism falls. This likely accounted for the low event rate in our population – a unique characteristic that may hinder the direct use of the above clinical decision rules. Our center sees a low event rate of moderate to severe TBI (<1%) and a baseline CT rate of less than 2%. We found that the direct application of these rules to our population would in most cases, increase our CT rate, which would be undesirable. Hence, we sought to derive high risk clinical predictors from our population, and test their utility in our local setting.
The multivariable analysis revealed 4 independent predictors – road traffic accident as the mechanism of injury, a history of loss of consciousness, vomiting and signs of base of skull fracture. The presence of a change in conscious level and evidence of base of skull fracture were consistently reported in the 3 high performing CDRs. The presence of vomiting, on the other hand, was variable (reported in PECARN for children 2 years and older, as well as 3 or more discrete episodes of vomiting in CHALICE). Dayan et al., on the other hand, reported that the presence of isolated vomiting among children with a minor blunt head injury was unlikely to be associated with clinically important TBI [26].
We also investigated the utility of ML for predicting pediatric TBI. Compared with the logistic regression method, ML is more flexible in terms of predictor selection as it is able to discover nonlinear interactions among clinical variables [14]. As a result, the presence of seizure activity, confusion and clinical signs of skull fracture were combined with the above mentioned four variables used in regression method to build a ML predictive model. It is observed in Table 2 that both seizure activity and confusion are statistically significant, while the presence of clinical signs of skull fracture is not. Possible explanations on improved performance by adding in non-significant variables are that a complex neural network structure is capable of detecting nonlinear correlations among variables and associating them with the clinical outcome, i.e. TBI in our study. There is superiority of ML over logistic regression as shown in Figure 2 and Table 4 where at the optimal cutoff scores ML achieves much higher sensitivity and PPV. However, it is worth noting that all reported performance indicators have overlapping confidence intervals. Further investigation will be conducted to determine if the ML method is statistically superior to classic logistic regression method.
To the best of our knowledge, machine learning has yet been applied to predict pediatric TBI, although it received attention in various medical areas [14,15,20,27]. Amongst many machine learning methods, neural network has been widely implemented for predictive modeling and shows excellent prediction performance compared to logistic regression [28-31]. The ability of a neural network to model complex nonlinear relationships between independent and dependent variables [32] makes it a natural tool to predict moderate to severe TBI in our study. However the application of neural networks is limited by the lack of interpretability, more specifically, the difficulty in assessing the relative contribution of each variable to the predictive modeling [31]. In developing predictive models, it is usually recommended to consider both advantages and limitations of the approaches [32,33].
We believe that our findings may apply to populations with low event rates of moderate to severe traumatic brain injury, in which the majority of head injured patients attend after mild mechanisms of injury. We recognize the following limitations of the study: in our population, we see a very low rate of moderate to severe TBI, therefore a case–control method was chosen. Cases were obtained from retrospective recruitment spanning 8 years – During this period there may have been changes to ED practices and protocols within the department. Also, we acknowledge that exaggerated results can trigger premature adoption of diagnostic tests [34]. In order that physicians make accurate informed decisions about the care for individual patients, larger prospective studies are required in a new population to validate these findings. We chose to perform age-matching in this study, to aid the ED physician when faced with a head-injured child of known age. We recognize, however, that matching by age would affect the independence of the observations, and that age could be associated with the other cofactors. This was not explored in this analysis. Given continued accrual of patients with moderate to severe head injury in the prospective database, we aim to take into account this aspect in the analysis. Finally, we recognize that the ML model in our study was built partially from statistically significant variables from logistic regression, and therefore the incorporation of variables for the two methods was not similar. The ML method serves to build on the logistic regression method as an improved tool, rather than a replacement of logistic regression. With a larger database, we will be able to validate this model on a separate dataset.
In a population with a low event rate of moderate to severe TBI and a low CT rate, the following predictors were demonstrated to be significant in predicting moderate to severe TBI: road traffic accident as the mechanism of injury, a history of loss of consciousness, vomiting and signs of base of skull fracture. Moreover, seizure activity, confusion and clinical signs of skull fracture held predictive power in the diagnosis of pediatric TBI. In this study, we demonstrated the feasibility and the advantages of using machine learning as a tool to predict TBI. If validated on a large scale, the ML method has the potential not only to guide discretionary use of CT, but also a more careful selection of head injured children who warrant closer monitoring in the hospital.
AIG:
Akaike criterion
Area under the ROC curve
Bayesian information criterion
CATCH:
Canadian assessment of tomography for childhood head injury
CHALICE:
Children's head injury algorithm for the prediction of important clinical events
CDR:
Clinical decision rules
CT:
ED:
GCS:
ML:
PECARN:
Pediatric Emergency Care Applied Research Network
TBI:
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The ongoing prospective head injury surveillance is supported by the Paediatrics Academic Clinical Program (Paeds ACP) Young Researcher Pilot Grant, Singapore. This publication was supported in part by the SGH Centre Grant from the National Medical Research Council, Singapore (ref: NMRC/CG/016/2013). We would like to thank Ms Dianna Sri, Ms Lau Yew Ping and Ms Jasmine Feng for their contributions in data collection.
Department of Emergency Medicine, KK Women's and Children's Hospital, Singapore, Singapore
Shu-Ling Chong
Department of Emergency Medicine, Singapore General Hospital, Singapore, Singapore
Nan Liu & Marcus Eng Hock Ong
Centre for Quantitative Medicine, Duke-NUS Graduate Medical School, Singapore, Singapore
Nan Liu & Sylvaine Barbier
Health Services and Systems Research, Duke-NUS Graduate Medical School, Singapore, Singapore
Marcus Eng Hock Ong
Nan Liu
Sylvaine Barbier
Correspondence to Nan Liu.
SLC and NL planned and established the study, performed data collection and analysis, and drafted the manuscript. SB performed detailed statistical analysis and drafted the manuscript. MEHO planned the study, drafted the manuscript and reviewed critical revisions. All authors took part in manuscript writing and approved the final manuscript.
Chong, SL., Liu, N., Barbier, S. et al. Predictive modeling in pediatric traumatic brain injury using machine learning. BMC Med Res Methodol 15, 22 (2015). https://doi.org/10.1186/s12874-015-0015-0
Prediction rules
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CommonCrawl
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\begin{document}
\title{Test vectors for trilinear forms : the case of two principal series} \author{Mladen Dimitrov and Louise Nyssen \cr {\footnotesize
[email protected], [email protected]} } \date{\today}
\maketitle
\section{Introduction}
Let $F$ be a finite extension of $\mathbb Q_p$ with ring of integers ${\mathcal O}$ and uniformizing parameter $\pi$. Let $V_1$, $V_2$ and $V_3$ be three irreducible, admissible, infinite dimensional representations of $G={\rm GL}_2(F)$ of central characters $\omega_1$, $\omega_2$ and $\omega_3$ and conductors $n_1$, $n_2$ and $n_3$. Using the theory of Gelfand pairs, Diprenda Prasad proves in \cite{P} that the space of $G$-invariant linear forms on $V_1\otimes V_2 \otimes V_3$ has dimension at most one and gives a precise criterion for this dimension to be one, that we will now explain.
Let $D^*$ be the group of invertible elements of the unique quaternion division algebra $D$ over $F$. When $V_i$ is a discrete series representation of $G$, denote by $V'_i$ the irreducible representation of $D^*$ associated to $V_i$ by the Jacquet-Langlands correspondence. Again, by the theory of Gelfand pairs, the space of $D^*$-invariant linear forms on $V'_1\otimes V'_2 \otimes V'_3$ has dimension at most one.
A necessary condition for the existence on a non-zero $G$-invariant linear form on $V_1\otimes V_2\otimes V_3$ (resp. non-zero $D^*$-invariant linear form on $V'_1\otimes V'_2 \otimes V'_3$), that we will {\it always assume}, is that $$ \enspace \omega_1\omega_2\omega_3=1.$$
Let $\sigma_i$ be the two dimensional representations of the Weil-Deligne group of $F$ associated to $V_i$. The triple tensor product $\sigma_1 \otimes \sigma_2 \otimes\sigma_3$ is an eight dimensional symplectic representation of the Weil-Deligne group having a local root number $\varepsilon(\sigma_1 \otimes \sigma_2 \otimes \sigma_3)$ equal to $1$ or $-1$. When $\varepsilon(\sigma_1 \otimes \sigma_2 \otimes \sigma_3)=- 1$, one can prove that the $V_i$'s are all discrete series representations of $G$.
\begin{theo} (Prasad \cite[Theorem 1.4]{P}) If all the $V_i$'s are supercuspidal, assume that the residue characteristic of $F$ is not 2. Then \par $\centerdot$ $\varepsilon(\sigma_1 \otimes \sigma_2 \otimes \sigma_3)=1$ if, and only if, there exists a non-zero $G$-invariant linear form on $V_1\otimes V_2 \otimes V_3$ , and \par $\centerdot$ $\varepsilon(\sigma_1 \otimes \sigma_2 \otimes \sigma_3)=-1$ if, and only if, there exists a non-zero $D^*$ invariant linear form on $V'_1\otimes V'_2 \otimes V'_3$.\par \end{theo}
Given a non zero $G$-invariant linear form $\ell$ on $V_1\otimes V_2 \otimes V_3$, or a non-zero $D^*$-invariant linear form $\ell'$ on $V'_1\otimes V'_2 \otimes V'_3$, the goal is to find a vector in $V_1 \otimes V_2 \otimes V_3$ which is not in the kernel of $\ell$, or a vector in $V'_1\otimes V'_2 \otimes V'_3$ which is not in the kernel of $\ell'$. Such a vector is called a test vector. The following results of Prasad and Gross-Prasad show that new vectors can sometimes be used as test vectors. In what follows $v_i$ denotes a new vector in $V_i$ (see \S\ref{nv}).
\begin{theo}\label{vt-000} (Prasad \cite[Theorem 1.3]{P}) If all the $V_i$'s are unramified principal series, then $v_1 \otimes v_2 \otimes v_3$ is a test vector. \end{theo}
\begin{theo}\label{vt-111} (Gross and Prasad \cite[Proposition 6.3]{GP}) Suppose all the $V_i$'s are unramified twists of the Steinberg representation. \begin{itemize} \item If $\varepsilon(\sigma_1 \otimes \sigma_2 \otimes \sigma_3)=1$, then $v_1 \otimes v_2 \otimes v_3$ is a test vector. \item If $\varepsilon(\sigma_1 \otimes \sigma_2 \otimes \sigma_3)=-1$ and if $R$ is the the unique maximal order in $D$, then any vector belonging to the unique line in $V'_1\otimes V'_2 \otimes V'_3$ fixed by $R^* \times R^*\times R^*$ is a test vector. \end{itemize} \end{theo}
Actually, the proof by Gross and Prasad of the first statement of the above theorem contains another result :
\begin{theo}\label{vt-110} If two of the $V_i$'s are unramified twists of the Steinberg representation and the third one is an unramified principal series, then $v_1 \otimes v_2 \otimes v_3$ is a test vector. \end{theo}
However, as mentioned in \cite{GP}, new vectors are not always test vectors. Let $K={\rm GL} ({\mathcal O})$ be the maximal compact subgroup of $G$ and suppose that $V_1$ and $V_2$ are unramified, but $V_3$ is ramified. Since $v_1$ and $v_2$ are $K$-invariant and $\ell$ is $G$-equivariant, $v \mapsto \ell(v_1\otimes v_2\otimes v)$ defines a $K$-invariant linear form on $V_3$. Since $V_3$ is ramified, so is its contragredient, and therefore the above linear form has to vanish. In particular $\ell(v_1\otimes v_2\otimes v_3)=0.$
To go around this obstruction for new vectors to be test vectors, Gross and Prasad made the following suggestion : suppose that $V_3$ has conductor $n=n_3\geq 1$; since $V_3$ has unramified central character, its contragredient representation has non-zero invariant vectors by the $n$-th standard Iwahori subgroup $I_n=\begin{pmatrix} {\mathcal O}^\times & {\mathcal O} \\ \varpi^n{\mathcal O} & {\mathcal O}^\times\end{pmatrix}$ of $G$; put $\gamma=\begin{pmatrix}\pi^{-1} & 0
\\ 0 & 1\end{pmatrix}$ and let $v_1^*\in V_1$ be a non-zero vector on the line fixed by the maximal compact subgroup $\gamma^{n}K\gamma^{-n}$ of $G$; since $K\cap \gamma^{n}K\gamma^{-n}=I_n$, the linear form on $V_3$ given by $v \mapsto \ell(v_1^*\otimes v_2\otimes v)$ is not necessarily zero and there is still hope for $v_1^* \otimes v_2 \otimes v_3$ to be a test vector. This is the object of the following theorem
\begin{theo}\label{vt-00n} If $V_1$ and $V_2$ are unramified and $V_3$ has conductor $n_3$, then $v_1^* \otimes v_2 \otimes v_3$ is a test vector, where $v_1^*=\gamma^{n_3}\!\cdot\! v_1$. \end{theo} Theorem \ref{vt-00n} for $n_3=1$, together with Theorems \ref{vt-000}, \ref{vt-111} and \ref{vt-110}, completes the study of test vectors when the $V_i$'s have conductors $0$ or $1$ and unramified central characters.
Assume from now on that $V_1$ and $V_2$ are (ramified or unramified) principal series. Then for $i=1,2$ there exist quasi-characters $\mu_i$ and $\mu'_i$
of $F^\times$ such that $\mu'_i\mu_i^{-1}\neq |\cdot|^{\pm 1}$, and
$$V_i = {\rm Ind}_{B}^{G} \chi_i \text{ , with }
\chi_i \begin{pmatrix} a & b \cr 0 & d \cr \end{pmatrix}
= \mu_i(a)\mu'_i(d).$$
According to Theorem 1 there exists a non-zero $G$-invariant linear form $\ell$ on $V_1\otimes V_2 \otimes V_3$, so we are looking for a test vector in $V_1\otimes V_2 \otimes V_3$.
The following theorem is our main result.
\begin{theo}\label{vt-mkn} Suppose that $V_1$ and $V_2$ are principal series such that $\mu_1$ and $\mu'_2$ are unramified. Put $$x=\max(n_2-n_1,n_3-n_1) \qquad{\rm and}\qquad v_1^*=\gamma^{x}\!\cdot\! v_1.$$ Then $x\geq 0$ and, if $v_1^* \otimes v_2 \otimes v_3$ is {\it not} a test vector, then \begin{itemize} \item either $n_1=0$, $n_2=n_3>0$ and $\gamma^{n_2-1}\!\cdot\! v_1 \otimes v_2 \otimes v_3$ is a test vector, \item or $n_2=0$, $n_1=n_3>0$ and $v_1 \otimes \gamma\!\cdot\! v_2 \otimes v_3$ is a test vector, \item or $\widetilde{V_3}$ is a quotient of ${\rm Ind}_{B}^{G}(\chi_1 \chi_2 \delta^{\frac{1}{2}})$, $n_1+n_2=n_3$ and $ v_1 \otimes \gamma^{n_1}\!\cdot\! v_2 \otimes v_3$ is a test vector. \end{itemize} \end{theo} The assumptions of the theorem imply in particular that $V_1$ and $V_2$ have minimal conductor among their twists. If $V_1$ and $V_2$ are two arbitrary principal series, then one can always find characters $\eta_1$, $\eta_2$ and $\eta_3$ of $F^\times$ with $\eta_1 \eta_2 \eta_3 =1$, such that the above theorem applies to $(V_1 \otimes \eta_1) \otimes(V_2\otimes \eta_2 ) \otimes(V_3\otimes \eta_3)$. Nevertheless, we found also interesting to study the case when $\mu_1$ or $\mu'_2$ is ramified. Then we are able to show that certain new vectors are {\it not} test vectors, while {\it a priori} this cannot be seen by a direct argument (the obstruction of Gross and Prasad described above does not apply to this case). Put $m_1={\rm cond}(\mu'_1)$ and $m_2={\rm cond}(\mu'_2)$
\begin{theo}\label{no-vt} Suppose that $\mu_1$ or $\mu'_2$ is ramified. Let $x$, $y$ and $z$ be integers such that \begin{itemize} \item $x\geq m_1$, \item $y \geq m_2$, \item $x-n_3\geq z \geq y$, and \item $x-y\geq \max( n_1-m_1, n_2-m_2,1)$. \end{itemize} Put \begin{equation}\label{v12} \begin{split} v_1^*=\begin{cases} \gamma^{x-m_1}\!\cdot\! v_1 & \text{ , if } \mu'_1 \text{ is ramified,}\\ \gamma^{x}\!\cdot\! v_1-\beta_1 \gamma^{x-1}\!\cdot\! v_1 & \text{ , if } \mu'_1 \text{ is unramified.} \end{cases} \\ v_2^*=\begin{cases} \gamma^{y-m_2}\!\cdot\! v_2 & \text{ , if } \mu_2 \text{ is ramified.}\\ \gamma^{y-n_2}\!\cdot\! v_2-\alpha_2^{-1} \gamma^{y-n_2+1}\!\cdot\! v_2 & \text{ , if } \mu_2 \text{ is unramified.} \end{cases} \end{split} \end{equation} Then $$\ell(v_1^* \otimes v_2^* \otimes \gamma^{z} \!\cdot\! v_3)=0.$$ \end{theo}
We will prove theorems \ref{vt-mkn} and \ref{no-vt} by following the pattern of the proof of Theorem \ref{vt-000} in \cite{P}, with the necessary changes.
We believe that suitable generalization of the method of Gross and Prasad would give test vectors in the case where at least two of the $V_i$'s are special representations, as well as in the case where one is a special representation and one is a principal series.
On the other hand in order to find test vectors in the case where at least two of the $V_i$'s are supercuspidal, one should use different techniques, involving probably computations in Kirillov models.
The search for test vectors in our setting is motivated by subconvexity problems for $L$-functions of triple products of automorphic forms on ${\rm GL}(2)$. Roughly speaking, one wants to bound the value of the $L$-function along the critical line $\Re(z)=\frac{1}{2}$. In \cite{BR1} and \cite{BR2} Joseph Bernstein and Andre Reznikov establish a {\it subconvexity bound} when the {\it eigenvalue} attached to one of the representations varies. Philippe Michel and Akshay Venkatesh considered the case when the {\it level} of one representation varies. More details about subconvexity and those related techniques can be found in \cite{V} or \cite{MV}. Test vectors are key ingredients. Bernstein and Reznikov use an explicit test vector. Venkatesh uses a theoretical one, but explains that the bounds would be better with an explicit one (see \cite[\S 5]{V}).
There is an extension of Prasad's result in \cite{HS}, where Harris and Scholl prove that the dimension of the space of $G$-invariant linear forms on $V_1\otimes V_2 \otimes V_3$ is one when $V_1$, $V_2$ and $V_3$ are principal series representations (either irreducible or reducible, but with infinite dimensional irreducible subspace). They apply their result to the global setting to construct elements in the motivic cohomology of the product of two modular curves predicted by Beilinson.
\subsection*{Acknowledgments.} We would like to thank Philippe Michel for suggesting the study of this problem, and of course Benedict Gross and Diprenda Prasad for their articles full of inspiration. The second named author would like to thank also Paul Broussous and Nicolas Templier for many interesting discussions, and Wen-Ching Winnie Li for the opportunity to spend one semester at PennState University where the first draft of this paper was written.
\section{Background on induced admissible representations of ${\rm GL}(2)$.}
\subsection{About induced and contragredient representations.}\label{notations} Let $(\rho, W)$ be a smooth representation of a closed subgroup $H$ of $G$. Let $\Delta_H$ be the modular function on $H$. The induction of $\rho$ from $H$ to $G$, denoted ${\rm Ind}_{H}^{G} \rho $, is the space of functions $f$ from $G$ to $W$ satisfying the two following conditions :
(1) $\forall h \in H, \quad \forall g \in G, \quad f(hg)=\Delta_H(h)^{-\frac{1}{2}} \rho(h) f(g)$,
(2) there exists an open compact subgroup $K_f$ of $G$ such that $$\forall k \in K_f, \quad \forall g \in G, \quad f(gk)= f(g)$$
\noindent where $G$ acts by right translation as follows : $$\forall g, g' \in G, (g\cdot f)(g') = f(g'g).$$ With the additional condition that $f$ must be compactly supported modulo $H$, one gets the {\it compact} induction denoted by ${\rm ind}_{H}^{G}$. When $G/H$ is compact, there is no difference between ${\rm Ind}_{H}^{G}$ and ${\rm ind}_{H}^{G}$.
Let $B$ the Borel subgroup of upper triangular matrices in $G$, and let $T$ be the diagonal torus. The character $\Delta_T$ is trivial and we will use $\Delta_B=\delta^{-1} $ with $\delta \begin{pmatrix} a & b \cr 0 & d \cr \end{pmatrix} = \vert \frac{a}{d} \vert$ where $ \vert \enspace \vert$ is the normalised valuation of $F$. The quotient $B \backslash G$ is compact and can be identified with $\mathbb{P}^1(F)$.
For a smooth representation $V$ of $G$, the contragredient representation $\widetilde{V}$ is the space of smooth linear forms $l$ on $V$, where $G$ acts as follows : $$\forall g\in G, \qquad \forall v\in V, \qquad (g\cdot l)(v)= l(g^{-1}\cdot v).$$
We refer the reader to \cite{BZ} for more details about induced and contragredient representations.
\subsection{New vectors and ramification.}\label{nv} Let $V$ be an irreducible, admissible, infinite dimensional representation of $G$ with central character $\omega$. Then $\widetilde{V}\cong V\otimes \omega^{-1}$. To the descending chain of compact subgroups of $G$ $$ K=I_0 \supset I_1 \supset \cdots \supset I_n \supset I_{n+1} \cdots $$ one can associate an ascending chain of vector spaces $$V^{I_0,\omega}=V^K \text{ , and for all } n\geq 1, \quad
V^{I_{n},\omega}=\left\{v\in V \Big{|} \begin{pmatrix} a & b \\ c &
d\end{pmatrix}\!\cdot\! v=\omega(d)v \text{ , for all } \begin{pmatrix} a & b \\ c & d\end{pmatrix}\in I_n \right\}.$$ There exists a minimal $n$ such that the vector space $V^{I_{n},\omega}$ is non-zero. It is necessarily one dimensional and any non-zero vector in it is called a {\it new
vector} of $V$. The integer $n$ is the {\it conductor} of $V$. The representation $V$ is said to be {\it unramified} if $n=0$.
More information about new vectors can be found in \cite{C}.
\subsection{New vectors as functions on $G$.}\label{nv-functions}
Let $V$ be a principal series of $G$, with central character $\omega$, and conductor $n$. There exist quasi-characters $\mu$ and $\mu'$
of $F^\times$ such that $\mu'\mu^{-1}\neq |\cdot|^{\pm 1}$, and
$$V = {\rm Ind}_{B}^{G} (\chi) \qquad \text{with } \qquad
\chi \begin{pmatrix} a & * \cr 0 & d \cr \end{pmatrix}
= \mu(a)\mu'(d).$$
Then $\omega=\mu\mu'$ and $n = {\rm cond}(\mu)+{\rm cond}(\mu')$. A new vector $v$ in $V$ is a non-zero function from $G$ to $\mathbb C$ such that for all $b \in B$, $g \in G$ and $k= \begin{pmatrix} * & * \\ * & d \end{pmatrix} \in I_n$ $$v(bgk)=\chi(b)\delta(b)^\frac{1}{2} \omega(d)v(g).$$
Put $$\alpha^{-1}= \mu(\pi)|\pi|^{\frac{1}{2}} \qquad{\rm and} \qquad \beta^{-1}= \mu'(\pi)|\pi|^{-\frac{1}{2}}.$$
First, we assume that $V$ is unramified, and we normalise $v$ so that $v(1)=1$.
\begin{lemma}\label{calcul-NR} If $V$ is unramified then for all $r\in \mathbb N$, $$(\gamma^{r}\!\cdot\! v)(k)= \begin{cases} \beta^r & \text{ , if } k\in K\backslash I, \\ \alpha^s\beta^{r-s} & \text{ , if } k\in I_s\backslash I_{s+1} \text{ for } 1\leq s\leq r-1,\\ \alpha^{r} & \text{ , if } k\in I_r.\\ \end{cases}$$
Similarly, $$(\gamma^{r}\!\cdot\! v-\alpha^{-1}\gamma^{r+1}\!\cdot\! v)(k)= \begin{cases} \alpha^s\beta^{r-s}-\alpha^{s-1}\beta^{r+1-s} & \text{ , if } k\in I_s\backslash I_{s+1} \text{ for } 0\leq s\leq r,\\ 0 & \text{ , if } k\in I_{r+1}.\\ \end{cases} $$
Finally, for $r\geq 1$, $$(\gamma^{r}\!\cdot\! v-\beta\gamma^{r-1}\!\cdot\! v)(k)= \begin{cases} \alpha^r(1-\frac{\beta}{\alpha}) & \text{ , if } k\in I_r,\\ 0 & \text{ , if } k\in K \backslash I_{r}. \end{cases} $$ \end{lemma}
\noindent{\it Proof : } If $k\in I_r$, then $\gamma^{-r}k\gamma^{r}\in K$, so $$(\gamma^{r}\!\cdot\! v)(k)=\alpha^{r}v(\gamma^{-r}k\gamma^{r})= \alpha^{r}.$$
Suppose that $k =\begin{pmatrix}a & b \\ c & d\end{pmatrix}\in I_s\backslash I_{s+1}$ for some $0\leq s\leq r-1$ (recall that $I_0=K$). Then $\pi^{-s} c \in {{\mathcal O}}^{\times}$ and $$ (\gamma^{r}\!\cdot\! v)(k)=\alpha^{r}v\begin{pmatrix}a & \pi^{r}b \\ \pi^{-r}c & d\end{pmatrix} = \alpha^{r}v\begin{pmatrix} (ad-bc)\pi^{r-s} & a \\ 0 & \pi^{-r}c \end{pmatrix} = \alpha^s\beta^{r-s}.$$ The second part of the lemma follows by a direct computation. $
\Box $
For the rest of this section we assume that $V$ is ramified, that is $n \geq 1$. We put $$m={\rm cond}(\mu') \qquad {\rm so \quad that} \qquad n-m={\rm cond}(\mu).$$
By Casselman \cite[pp.305-306]{C} the restriction to $K$ of a new vector $v$ is supported by the double coset of $\begin{pmatrix} 1 & 0 \\ \pi^{m} & 1 \end{pmatrix}$ modulo $I_{n}$. In particular if $\mu'$ is unramified ($m=0$), then $v$ is supported by $$I_{n} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}I_{n} =I_{n} \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}I_{n} =K\backslash I.$$ If $1 \leq m \leq n-1$, then $v$ is supported by $$I_{n} \begin{pmatrix} 1 & 0 \\ \pi^m & 1 \end{pmatrix}I_{n} =I_{m} \backslash I_{m+1}.$$ If $\mu$ is unramified ($m=n$), then $v$ is supported by $I_{n}$. We normalise $v$ so that $$v\begin{pmatrix} 1 & 0 \\ \pi^{m} & 1 \end{pmatrix}=1.$$
\begin{lemma}\label{calcul-SP0} If $\mu$ and $\mu'$ are both ramified ($0<m<n$), then for all $r\in \mathbb N$ and $k\in K$, $$(\gamma^{r}\!\cdot\! v)(k)= \begin{cases}\alpha^r \mu\Bigl(\frac{\det{k}}{\pi^{-(m+r)}c}\Bigr)\mu'(d) &
\text{ , if } k=\begin{pmatrix} * & * \\ c & d \end{pmatrix}\in I_{m+r}\backslash I_{m+r+1}, \\ 0 & \text{ , otherwise}. \end{cases}$$ \end{lemma}
\noindent{\it Proof : } For $k=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in K$ we have $$\alpha^{-r}(\gamma^{r}\!\cdot\! v)(k)= v(\gamma^{-r}k\gamma^{r})= v\begin{pmatrix} a &\pi^r b \\ \pi^{-r}c & d \end{pmatrix}.$$
It is easy to check that for every $s\geq 1$, $$K\cap B\gamma^{r}I_s\gamma^{-r}=I_{s+r}.$$ It follows that $\gamma^{r}\!\cdot\! v$ has its support in $I_{m+r}\backslash I_{m+r+1}$. If $k\in I_{m+r}\backslash I_{m+r+1}$ then $c \in {\pi}^{m+r} {{\mathcal O}}^{\times}$, $d\in {{\mathcal O}}^{\times}$ and we have the following decomposition : \begin{equation}\label{decomposition} \begin{pmatrix} a &\pi^r b \\ \pi^{-r}c & d \end{pmatrix}= \begin{pmatrix} \det{k} & \pi^{-m}cb\\ 0 & \pi^{-m-r}cd \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \pi^{m} & 1 \end{pmatrix} \begin{pmatrix} d^{-1} & 0 \\ 0 & \pi^{m+r}c^{-1} \end{pmatrix}. \end{equation}
Hence $$\alpha^{-r}(\gamma^{r}\!\cdot\! v)(k)= \mu\Bigl(\det(k)\Bigr)\mu'(\pi^{-m-r}cd)(\mu\mu')( \pi^{m+r}c^{-1})= \mu\Bigl(\frac{\det(k)}{\pi^{-(m+r)}c}\Bigr)\mu'(d).$$ $
\Box $
Similarly we obtain :
\begin{lemma}\label{calcul-SP1} Suppose that $\mu$ is unramified and $\mu'$ is ramified. Then, for all $r\in \mathbb N$ and $k\in K$, $$ (\gamma^{r}\!\cdot\! v)(k)= \begin{cases}\alpha^r \mu'(d) & \text{ , if } k=\begin{pmatrix} * & * \\ * & d \end{pmatrix}\in I_{{n}+r}, \\ 0 & \text{ , otherwise}. \end{cases} $$ $$\Bigl( \gamma^{r}\!\cdot\! v-\alpha^{-1} \gamma^{r+1}\!\cdot\! v \Bigr)(k)= \begin{cases}\alpha^r \mu'(d) & \text{ , if } k=\begin{pmatrix} * & * \\ * & d \end{pmatrix}\in I_{n+r}\backslash I_{n+r+1}, \\ 0 & \text{ , otherwise}. \end{cases} $$ \end{lemma}
\begin{lemma}\label{calcul-SP2} Suppose that $\mu'$ is unramified and $\mu$ is ramified. Then for all $r\in \mathbb N$, $$(\gamma^{r}\!\cdot\! v)(k)= \begin{cases}\alpha^s\beta^{r-s}
\mu\left(\frac{\det(k)}{\pi^{-s}c}\right) & \text{ , if } k=\begin{pmatrix} * & * \\ c & * \end{pmatrix}\in I_s\backslash I_{s+1}\text{, with } 0\leq s\leq r, \\ 0 & \text{ , if } k\in I_{r+1}. \end{cases} $$
Moreover, if $r \geq 1$, then $$\Bigl(\gamma^{r}\!\cdot\! v-\beta \gamma^{r-1}\!\cdot\! v \Bigr)(k) = \begin{cases}\alpha^r
\mu\left(\frac{\det(k)}{\pi^{-r}c}\right) & \text{ , if } k=\begin{pmatrix} * & * \\ c & * \end{pmatrix}\in I_r\backslash I_{r+1}, \\ 0 & \text{ , otherwise}. \end{cases} $$ \end{lemma}
\noindent{\it Proof : } We follow the pattern of proof of lemma \ref{calcul-SP0}. The restriction of $\gamma^{r}\!\cdot\! v$ to $K$ is zero outside $$K\cap B\gamma^{r}(K\backslash I)\gamma^{-r}=K\backslash I_{r+1}.$$ For $ 0\leq s\leq r$ and $k=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in I_s\backslash I_{s+1}$ we use the following decomposition : \begin{equation}\label{decomposition2} \begin{pmatrix} a &\pi^r b \\ \pi^{-r}c & d \end{pmatrix}= \begin{pmatrix} -\frac{\det{k}}{\pi^{-r}c} & a+\frac{\det{k}}{\pi^{-r}c}\\ 0 & \pi^{-r}c \end{pmatrix} \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \begin{pmatrix} 1 & 1+\frac{d}{\pi^{-r}c} \\ 0 & -1 \end{pmatrix}. \end{equation} Since $d\in{\mathcal O}$ and $\pi^{r}c^{-1}\in{\mathcal O}$ we deduce that : $$\alpha^{-r}(\gamma^{r}\!\cdot\! v)(k)= \mu\Bigl(\frac{\det{k}}{\pi^{-r}c}\Bigr) \mu'(-\pi^{-r}c)\left\vert \pi^{r}c^{-1} \right\vert = \mu\Bigl(\frac{\det{k}}{\pi^{-s}c}\Bigr)\alpha^{s-r}\beta^{r-s}.$$ $
\Box $
As direct consequence of these lemmas we obtain
\begin{lemma}\label{support} Let $v_1^*$ and $v_2^*$ be as in Theorem \ref{no-vt}. Then the support of $v_1^*$ is $$\begin{cases} I_{x}\backslash I_{x+1} & \text{ , if } \mu_1 \text{ is ramified, }\\ I_{x} & \text{ , if } \mu_1 \text{ is unramified, } \end{cases}$$ and the support of $v_2^*$ is $$\begin{cases} I_{y}\backslash I_{y+1} & \text{ , if } \mu'_2 \text{ is ramified, }\\ K \backslash I_{y+1} & \text{ , if } \mu'_2 \text{ is unramified. } \end{cases}$$ \end{lemma}
\section{Going down Prasad's exact sequence.}\label{suites-exactes}
In this section we will explain how Prasad finds a non-zero $\ell\in {\rm Hom}_G ( V_1 \otimes V_2 \otimes V_3 , \mathbb C)$ in the case
where $V_1$ and $V_2$ are principal series representations.
\subsection{Prasad's exact sequence.} The space ${\rm Hom}_G ( V_1 \otimes V_2 \otimes V_3 , \mathbb C)$ is canonically isomorphic to ${\rm Hom}_G ( V_1 \otimes V_2 , \widetilde{V_3} )$, hence finding $\ell$ it is the same as finding a non-zero element $\Psi$ in it. We have $$V_1 \otimes V_2 = {\rm Res}_{G}\,{\rm Ind}_{B \times B}^{G \times G} \Bigl( \chi_1 \times \chi_2 \Bigr)$$ where the restriction is taken with respect to the diagonal embedding of $G$ in $G\times G$. The action of $G$ on $(B\times B) \backslash (G\times G) \cong \mathbb{P}^1(F)\times \mathbb{P}^1(F)$ has precisely two orbits.
The first is the diagonal $\Delta_{B \backslash G}$, which is closed and can be identified with $B \backslash G$. The second is its complement which is open and can be identified with $T \backslash G$ via the bijection :
$$ \begin{matrix} T \backslash G & \longrightarrow & \Bigl( B \backslash G \times B \backslash G \Bigr) \setminus \Delta_{B \backslash G} \cr
Tg & \longmapsto & \Bigl( Bg , B\begin{pmatrix} 0 & 1 \cr 1 & 0
\cr \end{pmatrix}g \Bigr)
\end{matrix}$$
Hence, there is a short exact sequence of $G$-modules : \begin{equation}\label{courtesuite} 0 \rightarrow {\rm ind}_{T}^{G}\Bigl( \chi_1\chi'_2 \Bigr) \xrightarrow{{\rm \bf ext}} V_1 \otimes V_2 \xrightarrow{{\rm \bf res}} {\rm Ind}_{B}^{G} \Bigl(\chi_1 \chi_2 \delta^{\frac{1}{2}} \Bigr)\rightarrow 0, \end{equation} where $ \chi'_2 \begin{pmatrix} a & b \cr 0 & d \cr \end{pmatrix}
= \mu'_2(a)\mu_2(d).$ The surjection ${\rm \bf res}$ is given by the restriction to the diagonal. The injection ${\rm \bf ext}$ takes a function $f \in {\rm ind}_{T}^{G}\Bigl( \chi_1\chi'_2 \Bigr)$ to a function $F \in {\rm Ind}_{B \times B}^{G \times G} \Bigl( \chi_1 \times \chi_2 \Bigr)$ vanishing on $\Delta_{B \backslash G}$, such that for all $g\in G$ $$F \Bigl( g, \begin{pmatrix} 0 & 1 \cr 1 & 0 \cr \end{pmatrix} g \Bigr) = f(g) \label{rel}.$$
Applying the functor ${\rm Hom}_G \Bigl( \bullet , \widetilde{V_3}
\Bigr) $ yields a long exact sequence :
\begin{multline}\label{longuesuite} 0 \rightarrow {\rm Hom}_G \Bigl( {\rm Ind}_{B}^{G} \Bigl(\chi_1 \chi_2 \delta^{\frac{1}{2}} \Bigr) , \widetilde{V_3} \Bigr)
\rightarrow {\rm Hom}_G \Bigl( V_1 \otimes V_2 , \widetilde{V_3} \Bigr)
\rightarrow {\rm Hom}_G \Bigl( {\rm ind}_{T}^{G}\Bigl( \chi_1\chi'_2 \Bigr), \widetilde{V_3} \Bigr) \\
\downarrow \hskip20mm\\
\cdots \leftarrow {\rm Ext}_G^1 \Bigl( {\rm Ind}_{B}^{G} \Bigl(\chi_1 \chi_2 \delta^{\frac{1}{2}} \Bigr) , \widetilde{V_3} \Bigr) \end{multline}
\subsection{The simple case.}\label{cas-simple}
The situation is easier if $V_3$ occurs in ${\rm Ind}_{B}^{G} (\chi_1^{-1} \chi_2^{-1} \delta^{-\frac{1}{2}})$. Then $\chi_1\chi_2$ does not factor through the determinant and there is a natural surjection $${\rm Ind}_{B}^{G} \Bigl(\chi_1 \chi_2 \delta^{\frac{1}{2}} \Bigr) \twoheadrightarrow \widetilde{V_3}.$$ This surjection is an isomorphism, unless there exists a quasi-character $\eta$ of $F^{\times}$ such that $\chi_1\chi_2\delta=\eta \circ \det$ in which case the kernel is a line generated by the function $\eta \circ {\rm det}$. From (\ref{courtesuite}) we obtain a surjective homomorphism $\Psi$ completing the following commutative diagram : \begin{equation}\label{diagramme} \begin{matrix}
V_1 \otimes V_2 & \xrightarrow{{\rm \bf res}} & {\rm Ind}_{B}^{G} \Bigl(\chi_1 \chi_2 \delta^{\frac{1}{2}} \Bigr) \cr
{\scriptstyle \Psi}\! \searrow & & \swarrow
\cr &\widetilde{V_3}&\cr \end{matrix} \end{equation}
Finding a test vector is then reduced to finding an element of $V_1 \otimes V_2$ whose image by ${\rm \bf res}$ is not zero (resp. not a multiple of $\eta \circ {\rm det}$), if $V_3$ is principal series (resp. special representation).
Following the notations of paragraph \ref{nv-functions} put, for $i=1$ and $i=2$
$$m_i={\rm cond}(\mu'_i) \qquad \alpha_i^{-1}= \mu_i(\pi)|\pi|^{\frac{1}{2}}\quad \text{and} \qquad
\beta_i^{-1}= \mu'_i(\pi)|\pi|^{-\frac{1}{2}}.$$
\subsubsection{Proof of theorem \ref{no-vt} in the simple case.}
To prove theorem \ref{no-vt}, suppose that $\mu_1$ or $\mu'_2$ is ramified. By our assumptions $x>y$, hence $I_x\cap (K\backslash I_{y+1})=\varnothing$. Therefore the supports of $v_1^*$ and $v_2^*$ are disjoint and $${\rm \bf res}( v_1^* \otimes v_2^*)=0.$$
Using the diagram (\ref{diagramme}) we see that for any $v \in V_3$ : $$\ell( v_1^* \otimes v_2^* \otimes v ) = \Psi(v_1^* \otimes v_2^*)(v)=0.$$
In particular $\ell( v_1^* \otimes v_2^* \otimes \gamma^z\!\cdot\! v_3)=0$ which proves Theorem \ref{no-vt} in the simple case.
The rest of section \ref{cas-simple} will be devoted to the proof of Theorems \ref{vt-00n} and \ref{vt-mkn} in the simple case. Consequently, we will suppose that $\mu_1$ and $\mu'_2$ are unramified, that is $m_1-n_1=m_2=0$.
\subsubsection{Proof of Theorem \ref{vt-00n} in the simple case.}
Since $V_1$ and $V_2$ are unramified, by theorem \ref{vt-000} we may assume that $V_3$ is ramified. Then necessarily $$\widetilde{V_3}= \eta \otimes \rm St, $$ where $\rm St$ is the Steinberg representation and $\eta$ is an unramified character. Hence $ n_3=1$ and we will prove that $\gamma \cdot v_1\otimes v_2\otimes v_3$ is a test vector.
The function $$\left\{ \begin{matrix} G & \longrightarrow & \mathbb C
\\ g & \mapsto & \eta \Bigl({\rm det}(g)\Bigr)^{-1}{\rm \bf res}(\gamma \cdot v_1\otimes v_2)(g) \\ \end{matrix}\right.$$ is not constant, since according to lemma \ref{calcul-NR} $$ \eta \Bigl({\rm det}(1)\Bigr)^{-1}(\gamma \cdot v_1 \otimes v_2)(1) = v_1( \gamma) v_2(1)=\alpha_1 $$ and $$ \eta \Bigl({\rm det}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\Bigr)^{-1} (\gamma \cdot v_1 \otimes v_2 )\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \eta(-1) v_1\begin{pmatrix} 1 & 0 \\ 0 & \pi^{-1} \end{pmatrix} = \beta_1, $$ and $\alpha_1\neq \beta_1$ because $V_1$ is a principal series.
Hence $\Psi(\gamma \cdot v_1\otimes v_2)\neq 0$. Moreover, since $$\gamma K \gamma^{-1} \cap K =I$$ we deduce that $$\Psi(\gamma \cdot v_1\otimes v_2)\in \widetilde{V_3}^{I,{\omega_3}^{-1}}.$$ Hence $\Psi(\gamma \cdot v_1\otimes v_2)$ cannot vanish on the line ${V_3}^{I,{\omega_3}}$, which is generated by $v_3$, and therefore
$\gamma \cdot v_1\otimes v_2 \otimes v_3$ is a test vector.
This completes the proof of Theorem \ref{vt-00n} in the simple case.
\subsubsection{Proof of Theorem \ref{vt-mkn} in the simple case, when
$\widetilde{V_3}$ is a special representation.}
Assume now that $$\widetilde{V_3}= \eta \otimes \rm St, $$
where $\rm St$ is the Steinberg representation and $\eta$ is a character. Since $$\eta = \mu_1\mu_2|\cdot|=\mu'_1\mu'_2|\cdot|^{-1}$$ and $\mu_1$ and $\mu'_2$ are unramified, it follows that $\eta$ is unramified if, and only if, both $V_1$ and $V_2$ are unramified. Since this case was taken care of in the previous paragraph, we can assume for the rest of this paragraph that $\eta$ is ramified. Then $$n_1=n_2={\rm cond}(\eta)\geq 1 \qquad \text{and} \qquad n_3=2n_1=n_1+n_2.$$ We will prove that $v_1\otimes \gamma^{{n_1}} \cdot v_2\otimes v_3$ is a test vector.
The function $$\left\{ \begin{matrix} G & \longrightarrow & \mathbb C
\\ g & \mapsto & \eta \Bigl({\rm det}(g)\Bigr)^{-1}{\rm \bf res}(v_1\otimes \gamma^{{n_1}} \cdot v_2)(g) \\ \end{matrix}\right.$$ is not constant, since according to lemmas \ref{calcul-SP1} and \ref{calcul-SP2} $$ \eta \Bigl({\rm det}(1)\Bigr)^{-1}(v_1\otimes \gamma^{{n_1}} \cdot v_2)(1) = 0 $$ whereas $$ \eta \Bigl({\rm det}\begin{pmatrix} 1 & 0 \\ \pi^{n_1} & 1 \end{pmatrix}\Bigr)^{-1} (v_1\otimes \gamma^{{n_1}} \cdot v_2 )\begin{pmatrix} 1 & 0 \\ \pi^{n_1}
& 1 \end{pmatrix} = \alpha_2^{n_1} \neq 0. $$
Hence $\Psi(v_1\otimes \gamma^{{n_1}} \cdot v_2)\neq 0$. Moreover, since $$I_{n_1}\cap \gamma^{n_1}I_{n_2}\gamma^{-n_1}=I_{n_1+n_2}=I_{n_3}$$ we deduce that $$\Psi(v_1\otimes \gamma^{{n_1}} \cdot v_2)\in \widetilde{V_3}^{I_{n_3},{\omega_3}^{-1}}.$$ Hence $\Psi(v_1\otimes \gamma^{{n_1}} \cdot v_2)$ cannot vanish on the line ${V_3}^{I_{n_3},{\omega_3}}$, which is generated by $v_3$, and therefore
$v_1\otimes \gamma^{n_1} \cdot v_2\otimes v_3$ is a test vector.
\subsubsection{Proof of Theorem \ref{vt-mkn} in the simple case, when
$\widetilde{V_3}$ is a principal series.} \label{cas-simple2}
Finally, we consider the case where $\widetilde{V_3}$ is a principal series representation. Then $$ \widetilde{V_3} = {\rm Ind}_{B}^{G} \Bigl(\chi_1 \chi_2 \delta^{\frac{1}{2}} \Bigr)$$ and $$ n_3 = {\rm cond}(\mu_1\mu_2)+ {\rm cond}(\mu'_1\mu'_2)=n_2+n_1.$$ We will prove that $v_1\otimes \gamma^{n_1} \cdot v_2\otimes v_3$ is a test vector.
According to lemmas \ref{calcul-NR}, \ref{calcul-SP1} and \ref{calcul-SP2} we have $$ (v_1\otimes \gamma^{n_1} \cdot v_2) \begin{pmatrix}1 & 0 \cr
\pi^{n_1} & 1\cr\end{pmatrix} = \alpha_2^{n_1} \neq 0,$$ hence ${\rm \bf res}(v_1\otimes \gamma^{n_1} \cdot v_2)\neq 0$.
Therefore $\Psi(v_1\otimes \gamma^{n_1}v_2)\neq 0$. Moreover, since $$I_{n_1}\cap \gamma^{n_1}I_{n_2}\gamma^{-n_1}=I_{n_1+n_2}=I_{n_3}$$ we deduce that $$\Psi(v_1\otimes \gamma^{n_1}v_2)\in (\widetilde{V_3})^{I_{n_3},{\omega_3}^{-1}}.$$ Hence $\Psi(v_1\otimes \gamma^{n_1}v_2)$ cannot vanish on the line ${V_3}^{I_{n_3},{\omega_3}}$, which is generated by $v_3$. Thus $v_1\otimes \gamma^{n_1} \cdot v_2\otimes v_3$ is a test vector.
This completes the proof of Theorem \ref{vt-mkn} in the simple case.
\subsection{The other case.} The situation is more complicated if ${\rm Hom}_G ( {\rm Ind}_{B}^{G} (\chi_1 \chi_2 \delta^{\frac{1}{2}} ) , \widetilde{V_3} ) =0$. By
\cite[Corollary 5.9]{P} we have ${\rm Ext}_G^1 ( {\rm Ind}_{B}^{G}(\chi_1 \chi_2 \delta^{\frac{1}{2}} ) , \widetilde{V_3} )=0$, hence
the long exact sequence (\ref{longuesuite}) yields the following isomorphism : $${\rm Hom}_G \Bigl( V_1 \otimes V_2 , \widetilde{V_3} \Bigr) \simeq {\rm Hom}_G \Bigl( {\rm ind}_{T}^{G}( \chi_1\chi'_2 ), \widetilde{V_3} \Bigr).$$ Finally, by Frobenius reciprocity $${\rm Hom}_G \Bigl( {\rm ind}_{T}^{G}( \chi_1\chi'_2 ) , \widetilde{V_3} \Bigr) \simeq {\rm Hom}_T \Bigl( \chi_1\chi'_2 , \widetilde{V_{3\vert T}} \Bigr).$$ By \cite[ Lemmes 8-9]{W} the latter space is one dimensional, since the restriction of $\chi_1\chi'_2$ to the center equals $\omega_3^{-1}$ (recall that $\omega_1\omega_2\omega_3=1$). Thus, we have four canonically isomorphic lines with corresponding bases :
\begin{equation}\label{chaine} \begin{matrix} 0\neq\ell & \in & {\rm Hom}_G \Bigl( V_1 \otimes V_2 \otimes V_3 , \mathbb C \Bigr) \\
& & \downarrow \wr \\ 0\neq\Psi & \in & {\rm Hom}_G \Bigl( V_1 \otimes V_2 , \widetilde{V_3} \Bigr) \\ & & \downarrow \wr \\ 0\neq\Phi & \in & {\rm Hom}_G \Bigl( {\rm ind}_{T}^{G}(\chi_1\chi'_2), \widetilde{V_3} \Bigr) \\ & & \downarrow \wr \\ 0\neq\varphi & \in & {\rm Hom}_T \Bigl( \chi_1\chi'_2 , \widetilde{V_{3\vert T}} \Bigr) \\ \end{matrix} \end{equation} Observe that $\varphi$ can be seen as a linear form on $V_3$ satisfying : \begin{equation}\label{phi} \forall t \in T, \qquad \forall v \in V_3, \qquad \varphi(t\!\cdot\! v) = (\chi_1\chi'_2)(t)^{-1}\varphi(v). \end{equation}
\begin{lemma}\label{lemmeV3} $\varphi(v_3) \neq 0$ if, and only if, $\mu_1\mu'_2$ is unramified. \end{lemma}
\noindent{\it Proof : } Suppose $\varphi(v_3) \neq 0$. Since $v_3\in V_3$ is a new vector, for all $a,d\in{\mathcal O}^\times$ we have $$\begin{pmatrix} a & 0 \\ 0 &
d\end{pmatrix}\!\cdot\! v_3=\omega_3(d)v_3= (\mu_1\mu'_1\mu_2\mu'_2)(d)^{-1}v_3.$$ Comparing it with (\ref{phi}) forces $\mu_1\mu'_2$ to be unramified.
Conversely, assume that $\mu_1\mu'_2$ is unramified. Take any $v\in V_3$ such that $\varphi(v) \neq 0$. By smoothness $v$ is fixed by the principal congruence subgroup $\ker(K\rightarrow {\rm GL}_2({\mathcal O}/\pi^s))$, for some $s\geq 0$. Then $\varphi(\gamma^{s}\!\cdot\! v)=(\mu_1\mu'_2)(\pi^s)\varphi(v) \neq 0$ and $\gamma^{s}\!\cdot\! v$ is fixed by the congruence subgroup
$$I_{2s}^{1}:=\left\{k\in K \Big{|} k\equiv \begin{pmatrix} 1 & * \\ 0 &
1\end{pmatrix} \pmod{\pi^{2s}} \right\}.$$ By replacing $\gamma^{s}\!\cdot\! v $ by $v$ and $2s$ by $s$, we may assume that $v\in V_3^{I_{s}^{1}}$ for some $s\geq 0$. Since $I_{s}/I_{s}^{1}$ is a finite abelian group, $V_3^{I_{s}^{1}}$ decomposes as a direct sum of spaces indexed by the characters of $I_{s}/I_{s}^{1}$. Then $\varphi$ has to be non-zero on $V_3^{I_{s},\omega_3}$ (defined in paragraph \ref{nv}) since by (\ref{phi}), $\varphi$ vanishes on all other summands of $V_3^{I_{s}^{1}}$.
By Casselman \cite[Theorem 1]{C} the space $V_3^{I_{s},\omega_3}$ has dimension $n_3-s+1$ and has a basis $$\Bigl( \quad v_3\quad, \quad \gamma\!\cdot\! v_3\quad, \dots , \quad\gamma^{n_3-s}\!\cdot\! v_3 \quad\Bigr)$$ (recall that $n_3$ denotes the conductor of $V_3$). Again by (\ref{phi}), $\varphi(\gamma^{i}\!\cdot\! v_3)\neq 0$ for some $i$ is equivalent to $\varphi(v_3)\neq 0$. $
\Box$
Notice that, when $\mu_1\mu'_2$ and $\mu'_1\mu_2$ are both unramified, the claim follows from the first case in \cite[Proposition 2.6]{GP} applied to the split torus $T$ of $G$.
\section{Going up Prasad's exact sequence.}
In this section we take as a starting point lemma \ref{lemmeV3} and follow the isomorphisms (\ref{chaine}).
\subsection{From $\varphi$ to $\Phi$.}\label{phi-f}
Let $x$, $y$ and $z$ be integers such that $$x-n_3\geq z \geq y \geq 0 \qquad {\rm and} \qquad x-y \geq 1.$$ For the proof of Theorem \ref{vt-mkn} we will take $$x=\max(n_1,n_3)\geq 1 \qquad {\rm and} \qquad y=z=0.$$ Given a quasi-character $\mu$ of $F^\times$ define : $${\mathcal O}^{\mu}=\begin{cases}{\mathcal O} & \text{ , if } \mu \text{ is unramified, } \\ {\mathcal O}^\times & \text{ , if } \mu \text{ is ramified.} \end{cases}$$
Put $$ I_{f}=\begin{pmatrix} 1 & \pi^{-y}{\mathcal O}^{\mu'_2} \\
\pi^{x}{\mathcal O}^{\mu_1}& 1\end{pmatrix}, $$ and consider the unique function $f\in {\rm ind}_{T}^{G}( \chi_1\chi'_2 )$ which is zero outside the open compact subset $T I_f$ of $T\backslash G$ and such that for all $b_0\in \pi^{-y}{\mathcal O}^{\mu'_2}$ and $ c_0 \in \pi^{x}{\mathcal O}^{\mu_1}$ we have : \begin{equation}\label{f} f\begin{pmatrix} 1 & b_0 \\ c_0 & 1\end{pmatrix} = \begin{cases} \mu_1(\frac{\pi^{x}}{c_0})\mu'_2(b_0\pi^{y}) & \text{ , if } \mu_1 \text{ and } \mu'_2 \text{ are ramified ; } \\
\mu'_2(b_0\pi^{y}) & \text{ , if } \mu_1 \text{ is unramified } \text{ and } \mu'_2 \text{ is ramified ; } \\
\mu_1(\frac{\pi^{x}}{c_0}) & \text{ , if }\mu_1 \text{ is ramified } \text{ and } \mu'_2 \text{ is unramified ; } \\
1 & \text{ , if } \mu_1 \text{ and } \mu'_2 \text{ are unramified. } \\ \end{cases} \end{equation}
Since $x-n_3\geq z \geq y \geq 0$ and $x-y \geq 1$ we have $$ I_{f}\subset \gamma^{z}I_{n_3}^1\gamma^{-z}$$ and so every $k_0\in I_{f}$ fixes $\gamma^{z}\!\cdot\! v_3$.
By definition, the function $g \mapsto f(g)\varphi( g\gamma^{z}\!\cdot\! v_3)$ on $G$ factors through $T\backslash G$ and
$$\Bigl( \Phi(f) \Bigr)(\gamma^{z}\!\cdot\! v_3) = \int_{T \backslash G} \! f(g) \, \varphi( g\gamma^{z}\!\cdot\! v_3) dg =\varphi( \gamma^{z}\!\cdot\! v_3 ) \int_{ I_{f}} \! f(k_0) dk_0.$$
If we write $k_0=\begin{pmatrix} 1 & b_0 \\ c_0 & 1\end{pmatrix}\in I_f$, then by separating the variables $b_0$ and $c_0$ we obtain
$$ \int_{ I_{f}} \! f(k_0) dk_0=\begin{cases} |\pi|^{x-y} & \text{, if } \mu_1 \text{ and } \mu'_2 \text{ are unramified,} \\ 0 & \text{, otherwise.} \end{cases}$$
From this and from lemma \ref{lemmeV3} we deduce :
\begin{lemma}\label{phi-f-v_3} $\Phi(f) (\gamma^{z}\!\cdot\! v_3)\neq 0$ if, and only if,
$\mu_1$ and $\mu'_2$ are both unramified.
\end{lemma}
\subsection{From $\Phi$ to $\Psi$.}
Now, we are going to compute $F={\rm \bf ext}(f)$ as a function on $G\times G$. Recall that $F:G\times G\rightarrow \mathbb C$ is a function such that :
- for all $b_1,b_2\in B$, $g_1,g_2\in G$, $F(b_1g_1,b_2g_2)=\chi_1(b_1)\chi_2(b_2)\delta^{\frac{1}{2}}(b_1b_2)F(g_1,g_2)$,
- for all $g\in G$, $F(g,g)=0$ and $F(g,\begin{pmatrix} 0 & 1 \\ 1 &
0\end{pmatrix}g) = f(g)$.
\noindent Since $G=BK$, $F$ is uniquely determined by its restriction to $K\times K$. Following the notations of paragraph \ref{nv-functions} put
$$\alpha_i^{-1}= \mu_i(\pi)|\pi|^{\frac{1}{2}}\quad \text{and} \qquad
\beta_i^{-1}= \mu'_i(\pi)|\pi|^{-\frac{1}{2}}.$$
\begin{lemma} \label{FV} Suppose that $x-n_3\geq z \geq y \geq 0$ and $x-y \geq\max (n_1-m_1, n_2-m_2,1)$. Then for all $k_1 =\begin{pmatrix} * & * \\ c_1 & d_2 \end{pmatrix}$ and $k_2 =\begin{pmatrix} * & * \\ c_2 & d_2 \end{pmatrix}$ in $K$ we have $F(k_1,k_2)=0$ unless $$d_1c_2 \neq 0, \qquad \frac{c_1}{d_1}\in \pi^x{{\mathcal O}}^{\mu_1} \qquad {\rm and}\qquad \frac{d_2}{c_2}\in \pi^{-y}{{\mathcal O}}^{\mu'_2},$$ in which case, if we denote by $s$ the valuation of $c_2$, we have
$$F(k_1,k_2)=\begin{cases} \mu_1\left(\frac{\det(k_1)}{\pi^{-x}c_1}\right) \mu'_1(d_1) \mu_2\left(\frac{-\det(k_2)}{\pi^{-s}c_2}\right) \mu'_2(d_2) \left(\frac{\alpha_2}{\beta_2}\right)^s & \text{, if }\mu_1 \text{ and } \mu'_2 \text{ are ramified ; } \\ \mu'_1(d_1) \mu_2\left(\frac{-\det(k_2)}{\pi^{-s}c_2}\right) \mu'_2(d_2) \left(\frac{\alpha_2}{\beta_2}\right)^s & \text{, if } \mu_1 \text{ is unramified } \text{ and } \mu'_2 \text{ is ramified ; } \\ \mu_1\left(\frac{\det(k_1)}{\pi^{-x}c_1}\right) \mu'_1(d_1) \mu_2\left(\frac{-\det(k_2)}{\pi^{-s}c_2}\right) \left(\frac{\alpha_2}{\beta_2}\right)^s & \text{, if }\mu_1 \text{ is ramified } \text{ and } \mu'_2 \text{ is unramified ; } \\ \mu'_1(d_1) \mu_2\left(\frac{-\det(k_2)}{\pi^{-s}c_2}\right) \left(\frac{\alpha_2}{\beta_2}\right)^s & \text{, if } \mu_1 \text{ and } \mu'_2 \text{ are unramified. } \end{cases}$$ \end {lemma}
\noindent{\it Proof : } By definition $F(k_1,k_2)=0$ unless there exist $k_0=\begin{pmatrix} 1 & b_0 \\ c_0 & 1 \end{pmatrix} \in I_{f}$ such that $$k_1k_0^{-1}\in B \qquad \text{ and } \qquad k_2k_0^{-1}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\in B ,$$ in which case $$F(k_1,k_2)=\chi_1(k_1k_0^{-1})\chi_2 \Bigl( k_2k_0^{-1}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \Bigr) \delta^{\frac{1}{2}}\Bigl(k_1k_0^{-1}k_2k_0^{-1}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \Bigr) f(k_0).$$ From $k_1k_0^{-1}\in B$, we deduce that $c_1=c_0d_1$. From $k_2k_0^{-1}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\in B $ we deduce that $d_2=b_0c_2$. Hence $$d_1\in {\mathcal O}^\times, \qquad \frac{c_1}{d_1}\in \pi^x{{\mathcal O}}^{\mu_1}, \qquad c_2 \neq 0 \qquad \text{and} \qquad \frac{d_2}{c_2}\in \pi^{-y}{{\mathcal O}}^{\mu'_2}.$$ Moreover $$k_1k_0^{-1}=\begin{pmatrix}\frac{\det{k_1}}{d_1\det{k_0}} & * \\ 0 & d_1 \end{pmatrix} \text{ and } k_2k_0^{-1}\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} =\begin{pmatrix} \frac{-\det{k_2}}{c_2\det{k_0}} & * \\ 0 & c_2 \end{pmatrix}.$$ Since $x-y\geq n_1-m_1$, $x-y \geq n_2-m_2$ and $x-y \geq 1$ we have $$\mu_1(\det{k_0})=\mu_2(\det{k_0})=1.$$ Hence $$ F(k_1,k_2) = \mu_1(\frac{\det{k_1}}{d_1}) \mu'_1(d_1)\mu_2(\frac{-\det{k_2}}{c_2}) \mu'_2(c_2) \left\vert \frac{1}{c_2}\right\vert f\begin{pmatrix} 1 & \frac{d_2}{c_2}\\ \frac{c_1}{d_1} & 1 \end{pmatrix}.$$
From here and (\ref{f}) follows the desired formula for $F$.
Conversely, if $k_1$ and $k_2$ are such that
$\frac{c_1}{d_1}\in \pi^x{{\mathcal O}}^{\mu_1}$ and
$\frac{d_2}{c_2}\in \pi^{-y}{{\mathcal O}}^{\mu'_2}$ one can take $$k_0=\begin{pmatrix} 1 & d_2c_2^{-1} \\ c_1d_1^{-1} & 1 \end{pmatrix}.$$
$\Box$
\begin{rem} One can compute $F$ without the assumption $x-y \geq\max (n_1-m_1, n_2-m_2,1)$. However, $F$ needs not decompose as a product of functions of one variable as in the above lemma.
For example, if $x=n_3=0$ and $n_1=n_2$, then for all $k_1\in K$ and $k_2\in K$ $$F(k_1,k_2)=\begin{cases}
\omega_1(\frac{c_1d_2-d_1c_2}{\det k_2})& \text{ , if } d_1\in {\mathcal O}^\times,
\enspace c_2\in {\mathcal O}^\times \text{ and } c_1d_2\neq d_1c_2 \\ 0 & \text{ , otherwise}. \end{cases}$$ \end{rem}
\subsection{From $\Psi$ to $\ell$}
Now, we want to express $F \in V_1\otimes V_2$ in terms of the new vectors $v_1$ and $v_2$.
From now on we suppose that $x$, $y$ and $z$ are integers as in theorem \ref{no-vt}.
We may also suppose that $x\geq 1$, because otherwise $V_1$, $V_2$ and $V_3$ are all unramified and this case is covered in Theorem \ref{vt-000}. Observe also that if $y=0$, then $\mu'_2$ is unramified and therefore $ {\mathcal O}^{\mu'_2}={\mathcal O}$.
For $i=1,2$, since $k_i\in K$, both $c_i$ and $d_i$ are in ${\mathcal O}$, and one of them is in ${\mathcal O}^\times$. Hence \begin{itemize} \item $\frac{c_1}{d_1}\in \pi^x{{\mathcal O}}^\times$ if, and only if $k_1\in I_{x}\backslash I_{x+1}$, \item $\frac{c_1}{d_1}\in \pi^x{{\mathcal O}}$ if, and only if $k_1\in I_{x}$, \item $\frac{d_2}{c_2}\in \pi^{-y}{{\mathcal O}}^\times$ with $y\geq 1$ if, and only if $k_2\in I_{y}\backslash I_{y+1}$, \item $\frac{d_2}{c_2}\in \pi^{-y}{{\mathcal O}}$ with $y\geq 0$ if, and only if $k_2\in K \backslash I_{y+1}$. \end{itemize}
\begin{lemma} \label{calcul-F} With the notations of (\ref{v12}), $F$ is a non-zero multiple of $ v_1^* \otimes v_2^*$. \end{lemma}
{\it Proof : } Both $F$ and $ v_1^* \otimes v_2^*$ are elements in ${\rm Ind}_{B \times B}^{G \times G} \Bigl( \chi_1 \times \chi_2 \Bigr)$, hence it is enough to compare their restrictions to $K\times K$. By the above discussion together with lemmas \ref{FV} and \ref{support} the two restrictions are supported by $$\begin{cases} (I_{x}\backslash I_{x+1})\times (I_{y}\backslash I_{y+1}) & \text{ , if } \mu_1 \text{ and } \mu'_2 \text{ are ramified ; }\\ I_{x}\times (I_{y}\backslash I_{y+1}) & \text{ , if } \mu_1 \text{ is unramified } \text{ and } \mu'_2 \text{ is ramified ; }\\ (I_{x}\backslash I_{x+1})\times (K \backslash I_{y+1}) & \text{ , if } \mu_1 \text{ is ramified } \text{ and } \mu'_2 \text{ is unramified ; }\\ I_{x}\times (K \backslash I_{y+1}) & \text{ , if } \mu_1 \text{ and } \mu'_2 \text{ are unramified. } \end{cases}$$
There are $16$ different cases depending on whether each one among $\mu_1$, $\mu'_1$, $\mu_2$ and $\mu'_2$ is ramified or unramified. Since it is a straightforward verification from lemmas \ref{calcul-NR}, \ref{calcul-SP0}, \ref{calcul-SP1} and \ref{calcul-SP2}, in order to avoid repetitions or cumbersome notations, we will only give the final result : \begin{equation}\begin{split}
&F=\lambda_1\lambda_2\mu_2(-1) \alpha_1^{m_1-x}\alpha_2^{m_2}\beta_2^{-y}
(v_1^* \otimes v_2^*) \text{ , where }\\ &\lambda_i=\begin{cases} \Bigl(1-\frac{\beta_i}{\alpha_i}\Bigr)^{-1} & \text{ , if } V_i \text{ is unramified,}\\ 1 & \text{ , if } V_i \text{ is ramified. } \end{cases}
\end{split}\end{equation}
In all cases $F$ is a non-zero multiple of $ v_1^* \otimes v_2^*$.
$\square$
Since by definition $\ell(F\otimes\bullet)=\Psi(F)=\Phi(f)$, the above lemma together with lemma \ref{phi-f-v_3} imply theorem \ref{no-vt}.
\subsection{Proof of Theorems \ref{vt-00n} and \ref{vt-mkn}.}
We assume henceforth that $\mu_1$ and $\mu'_2$ are {\it both} unramified ($n_1-m_1=m_2=0$). We put $y=z=0$ and
$x=\max(n_1,n_3)\geq 1$. Since $\omega_1\omega_2\omega_3=1$, $\max(n_1,n_3)=\max(n_1,n_2,n_3)\geq 1$.
Then lemma \ref{phi-f-v_3} yields : \begin{equation} \ell(F\otimes v_3)=\Psi(F)(v_3)=\Phi(f)(v_3)\neq 0. \end{equation}
From this and lemma \ref{calcul-F} we deduce : \begin{lemma}\label{l-F-v_3} We have $\ell(v_1^*\otimes v_2^*\otimes v_3)\neq 0$ where \begin{equation*}\begin{split} &v_1^*=\begin{cases} \gamma^{x-n_1}\!\cdot\! v_1 & \text{ , if } \mu'_1 \text{ is ramified,}\\ \gamma^{x}\!\cdot\! v_1-\beta_1 \gamma^{x-1}\!\cdot\! v_1 & \text{ , if } \mu'_1 \text{ is unramified.} \end{cases} \\ &v_2^*=\begin{cases} v_2 & \text{ , if } \mu_2 \text{ is ramified.}\\ v_2-\alpha_2^{-1} \gamma\!\cdot\! v_2 & \text{ , if } \mu_2 \text{ is unramified.} \end{cases} \end{split}\end{equation*} \end{lemma}
\subsubsection{The case of two unramified representations.}
Suppose that $n_1=n_2=0$, so that $x=n_3$. Then lemma \ref{l-F-v_3} yields : $$\ell\Bigl((\gamma^{{n_3}}\!\cdot\! v_1-\beta_1\gamma^{n_3-1}\!\cdot\! v_1)\otimes (\gamma\!\cdot\! v_2-\alpha_2v_2)\otimes v_3\Bigr)\neq 0. $$
This expression can be simplified as follows. Consider for $m\geq 0$ the linear form : $$\psi_m(\bullet)=\ell(\gamma^{m}\!\cdot\! v_1\otimes v_2 \otimes\bullet)\in\widetilde{V_3}.$$
As observed in the introduction, $\psi_m$ is invariant by $\gamma^{m}K\gamma^{-m}\cap K =I_m $, hence vanishes if $m<n_3={\rm cond}(\widetilde{V_3})$. Therefore, for ${n_3}\geq 2$ : \begin{equation*}\begin{matrix} \ell\Bigl((\gamma^{{n_3}}\!\cdot\! v_1-\beta_1\gamma^{n_3-1}\!\cdot\! v_1)\otimes (\gamma\!\cdot\! v_2-\alpha_2v_2)\otimes v_3\Bigr)
\\ \hskip1cm = -\alpha_2\psi_{n_3}(v_3)+\beta_1\alpha_2\psi_{{n_3}-1}(v_3)+\psi_{{n_3}-1}(\gamma^{-1} \!\cdot\! v_3)-\beta_1\psi_{{n_3}-2}(\gamma^{-1} \!\cdot\! v_3)
\\ \hskip1cm = -\alpha_2\psi_{n_3}(v_3)
\\ \hskip1cm = -\alpha_2\ell(\gamma^{{n_3}}\!\cdot\! v_1\otimes v_2\otimes v_3) \neq 0.
\\ \end{matrix}\end{equation*}
If ${n_3} = 1$, only the two terms in the middle vanish and we obtain $$\alpha_2\ell(\gamma\!\cdot\! v_1\otimes v_2\otimes v_3) +\beta_1\ell(v_1\otimes \gamma\!\cdot\! v_2\otimes v_3)\neq 0.$$
Put $g=\begin{pmatrix}0 & 1 \cr \pi & 0\cr\end{pmatrix}$. Then $g \gamma = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\in K$ and $\gamma^{-1} g = \begin{pmatrix} 0 & \pi \\ \pi & 0 \end{pmatrix}\in \pi K$. Hence : $$\begin{matrix} \beta_1\ell(v_1\otimes \gamma\!\cdot\! v_2\otimes v_3)
& = & \beta_1\ell(\gamma\gamma^{-1} g\!\cdot\! v_1\otimes g\gamma\!\cdot\! v_2\otimes g\!\cdot\! v_3)
\\ & = & \beta_1\omega_1(\pi)\ell\bigl(\gamma\!\cdot\! v_1\otimes v_2\otimes g\!\cdot\! v_3\bigr)
\\ & = & \alpha_1^{-1}\ell(\gamma\!\cdot\! v_1\otimes v_2\otimes g\!\cdot\! v_3).
\\ \end{matrix}$$
Therefore $$\ell\Bigl(\gamma\!\cdot\! v_1\otimes v_2\otimes(g\!\cdot\! v_3+\alpha_1\alpha_2v_3)\Bigr)\neq 0,$$ in particular $$\Psi(\gamma\!\cdot\! v_1\otimes v_2)\neq 0.$$ By the same argument as in paragraph \ref{cas-simple2} we conclude that $$\ell(\gamma\!\cdot\! v_1\otimes v_2\otimes v_3)=\Psi(\gamma\!\cdot\! v_1\otimes v_2)(v_3)\neq 0.$$ Hence, if $n_3\geq 1$, $\gamma^{{n_3}}\!\cdot\! v_1\otimes v_2\otimes v_3$ is a test vector. This completes the proof of Theorem \ref{vt-00n}.
\subsubsection{The case of two ramified principal series.}
Suppose that $V_1$ and $V_2$ are both ramified ($m_1>0$, $n_1-m_1=0$, $m_2=0$, $n_2>0$) and put $n=x-n_1=\max(n_2-n_1,n_3-n_1)$. Then lemma \ref{l-F-v_3} yields : $$\ell(\gamma^{n}\!\cdot\! v_1 \otimes v_2 \otimes v_3)\neq 0,$$ hence $\gamma^{n}\!\cdot\! v_1 \otimes v_2 \otimes v_3$ is a test vector.
\subsubsection{The case where $V_1$ is unramified and $V_2$ is ramified.}
Suppose that $n_1=0$, but $n_2>0$. Then $x=n_3\geq n_2$ and lemma \ref{l-F-v_3} yields :
$$\ell\Bigl((\gamma^{{n_3}}\!\cdot\! v_1-\beta_1\gamma^{n_3-1}\!\cdot\! v_1)\otimes v_2\otimes v_3\Bigr)\neq 0.$$
If $n_2<{n_3}$, then $$\gamma^{n_3-1}K\gamma^{1-n_3}\cap I_{n_2}\supset I_{{n_3}-1},$$
and therefore $$\ell(\gamma^{n_3-1}\!\cdot\! v_1\otimes v_2\otimes \bullet)\in \widetilde{V_3}^{I_{{n_3}-1},\omega_3^{-1}}=\{0\}.$$ Hence $$\ell(\gamma^{{n_3}}\!\cdot\! v_1\otimes v_2\otimes v_3)\neq 0,$$ that is $\gamma^{{n_3}}\!\cdot\! v_1\otimes v_2\otimes v_3$ is a test vector.
If $n_2={n_3}$, the condition on the central character forces $V_3$ and $\omega_3$ to have the same conductor. Hence $V_3$ is also a principal series. In this case we do not see {\it a priori} a reason for either $\ell(\gamma^{{n_3}}\!\cdot\! v_1\otimes v_2\otimes v_3)$ or $\ell(\gamma^{n_3-1}\!\cdot\! v_1\otimes v_2\otimes v_3)$ to vanish. But we can notice that the two linear forms $$\ell(\gamma^{{n_3}}\!\cdot\! v_1\otimes v_2\otimes \bullet) \quad {\rm and} \quad \ell(\gamma^{n_3-1}\!\cdot\! v_1\otimes v_2\otimes \bullet) $$ belong both to the new line $\widetilde{V_3}^{I_n,\omega_3^{-1}}$ of $\widetilde{V_3}$, hence they are proportionals.
\subsubsection{The case where $V_1$ is ramified and $V_2$ is unramified.}
Suppose that $n_1>0$ and $n_2=0$. Then $x=n_3\geq n_1$ and lemma \ref{l-F-v_3} yields : $$\ell\Bigl(\gamma^{n_3-n_1}\!\cdot\! v_1 \otimes (\gamma\!\cdot\! v_2-\alpha_2 v_2)\otimes v_3\Bigr)\neq 0. $$
If $n_1<{n_3}$, then $$\ell(\gamma^{n_3-n_1-1}\!\cdot\! v_1\otimes v_2\otimes \bullet)\in \widetilde{V_3}^{I_{{n_3}-1},\omega_3^{-1}}=\{0\}.$$ Then $$\ell(\gamma^{n_3-n_1}\!\cdot\! v_1 \otimes \gamma\!\cdot\! v_2\otimes v_3)= \ell(\gamma^{n_3-n_1-1}\!\cdot\! v_1 \otimes v_2\otimes\gamma^{-1} \!\cdot\! v_3)=0.$$ Hence $$\ell(\gamma^{n_3-n_1}\!\cdot\! v_1 \otimes v_2\otimes v_3)\neq 0,$$ that is $\gamma^{n_3-n_1}\!\cdot\! v_1 \otimes v_2\otimes v_3$ is a test vector.
If $n_1={n_3}$, the condition on the central character forces $V_3$ to be also a principal series. In this case we do not see {\it a priori} a reason for either $\ell(v_1 \otimes v_2\otimes v_3)$ or $\ell(v_1 \otimes \gamma\!\cdot\! v_2\otimes v_3)$ to vanish. But we can once again notice that the two linear forms $$\ell(v_1 \otimes v_2\otimes \bullet) \quad {\rm and} \quad \ell(v_1 \otimes \gamma\!\cdot\! v_2\otimes \bullet)$$ belong to the line generated by a new vector in $\widetilde{V_3}$, hence are proportionals.
The proof of Theorem \ref{vt-mkn} is now complete.
\end{document}
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arXiv
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Legendre's theorem on spherical triangles
In geometry, Legendre's theorem on spherical triangles, named after Adrien-Marie Legendre, is stated as follows:
Let ABC be a spherical triangle on the unit sphere with small sides a, b, c. Let A'B'C' be the planar triangle with the same sides. Then the angles of the spherical triangle exceed the corresponding angles of the planar triangle by approximately one third of the spherical excess (the spherical excess is the amount by which the sum of the three angles exceeds π).
The theorem was very important in simplifying the heavy numerical work in calculating the results of traditional (pre-GPS and pre-computer) geodetic surveys from about 1800 until the middle of the twentieth century.
The theorem was stated by Legendre (1787) who provided a proof (1798) in a supplement to the report of the measurement of the French meridional arc used in the definition of the metre (Delambre 1798) harv error: no target: CITEREFDelambre1798 (help). Legendre does not claim that he was the originator of the theorem despite the attribution to him. Tropfke (1903) maintains that the method was in common use by surveyors at the time and may have been used as early as 1740 by La Condamine for the calculation of the Peruvian meridional arc.
Girard's theorem states that the spherical excess of a triangle, E, is equal to its area, Δ, and therefore Legendre's theorem may be written as
${\begin{aligned}A-A'\;\approx \;B-B'\;\approx \;C-C'\;\approx \;{\frac {1}{3}}E\;=\;{\frac {1}{3}}\Delta ,\qquad a,\;b,\;c\,\ll \,1.\end{aligned}}$
The excess, or area, of small triangles is very small. For example, consider an equilateral spherical triangle with sides of 60 km on a spherical Earth of radius 6371 km; the side corresponds to an angular distance of 60/6371=.0094, or approximately 10−2 radians (subtending an angle of 0.57° at the centre). The area of such a small triangle is well approximated by that of a planar equilateral triangle with the same sides: 1⁄2a2sin(π/3) = 0.0000433 radians corresponding to 8.9″.
When the sides of the triangles exceed 180 km, for which the excess is about 80″, the relations between the areas and the differences of the angles must be corrected by terms of fourth order in the sides, amounting to no more than 0.01″:
${\begin{aligned}\Delta &=\Delta '\left(1+{\frac {a^{2}+b^{2}+c^{2}}{24}}\right),\\A&=A'+{\frac {\Delta }{3}}+{\frac {\Delta }{180}}\left(-2a^{2}+b^{2}+c^{2}\right),\\B&=B'+{\frac {\Delta }{3}}+{\frac {\Delta }{180}}\left({\quad a^{2}-2b^{2}+c^{2}}\right),\\C&=C'+{\frac {\Delta }{3}}+{\frac {\Delta }{180}}\left({\quad a^{2}+b^{2}-2c^{2}}\right).\end{aligned}}$
(Δ′ is the area of the planar triangle.) This result was proved by Buzengeiger (1818)—an extended proof may be found in Osborne (2013) (Appendix D13). Other results are surveyed by Nádeník (2004).
The theorem may be extended to the ellipsoid if a, b, c are calculated by dividing the true lengths by the square root of the product of the principal radii of curvature (see Osborne (2013) Chapter 5) at the median latitude of the vertices (in place of a spherical radius). Gauss (1828, Art. 26–28) provided more exact formulae.
References
• Buzengeiger, Karl Heribert Ignatz (1818), "Vergleichung zweier kleiner Dreiecke von gleichen Seiten, wovon das eine sphärisch, das andere eben ist", Zeitschrift für Astronomie und verwandte Wissenschaften, 6: 264–270
• Clarke, Alexander Ross (1880), Geodesy, Clarendon Press. Republished at Forgotten Books. {{citation}}: External link in |postscript= (help)CS1 maint: postscript (link)
• Gauss, C. F. (1902) [1828]. General Investigations of Curved Surfaces of 1827 and 1825. Princeton Univ. Lib. English translation of Disquisitiones generales circa superficies curvas (Dieterich, Göttingen, 1828). {{cite book}}: External link in |postscript= (help)CS1 maint: postscript (link)
• Legendre, Adrien-Marie (1787), Mémoire sur les opérations trigonométriques, dont les résultats dépendant de la figure de la Terre, Article VI , p. 7 {{citation}}: External link in |others= (help)
• Legendre, Adrien-Marie (1798), Méthode pour déterminer la longueur exacte du quart du méridien d'après les observations faites pour la mesure de l'arc compris entre Dunkerque et Barcelone, pp. 12–14 (Note III )
• Nádeník, Zbynek (2004), Legendre theorem on spherical triangles (PDF), archived from the original (PDF) on 2014-01-16
• Osborne, Peter (2013), The Mercator Projections, archived from the original on 2013-09-24
• Tropfke, Johannes (1903), Geschichte der Elementar-Mathematik (Volume 2)., Verlag von Veit, p. 295
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Wikipedia
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Dynamic elementary mode modelling of non-steady state flux data
Abel Folch-Fortuny ORCID: orcid.org/0000-0001-6845-08071,2,
Bas Teusink3,
Huub C.J. Hoefsloot4,
Age K. Smilde4 &
Alberto Ferrer1
BMC Systems Biology volume 12, Article number: 71 (2018) Cite this article
A novel framework is proposed to analyse metabolic fluxes in non-steady state conditions, based on the new concept of dynamic elementary mode (dynEM): an elementary mode activated partially depending on the time point of the experiment.
Two methods are introduced here: dynamic elementary mode analysis (dynEMA) and dynamic elementary mode regression discriminant analysis (dynEMR-DA). The former is an extension of the recently proposed principal elementary mode analysis (PEMA) method from steady state to non-steady state scenarios. The latter is a discriminant model that permits to identify which dynEMs behave strongly different depending on the experimental conditions. Two case studies of Saccharomyces cerevisiae, with fluxes derived from simulated and real concentration data sets, are presented to highlight the benefits of this dynamic modelling.
This methodology permits to analyse metabolic fluxes at early stages with the aim of i) creating reduced dynamic models of flux data, ii) combining many experiments in a single biologically meaningful model, and iii) identifying the metabolic pathways that drive the organism from one state to another when changing the environmental conditions.
Data analysis methods are widely used in Systems Biology to interpret different kinds of data. In the field of fluxomics, principal component analysis (PCA) [1] models have been proposed to obtain a set of key pathways in metabolic networks, assuming steady state conditions [2, 3]. Basically, these key pathways are groups of correlated metabolic fluxes measured in different experiments. Multivariate curve resolution (MCR) [4] was afterwards proposed to obtain this set of metabolic pathways, exploiting the ability of MCR to include constraints in the algorithm, driving the model to a more biologically meaningful solution [5].
The drawback of PCA and MCR is that the components do not represent metabolic routes connecting substrates with end-products, but separate groups of concatenated reactions in the network. To enhance the interpretability of PCA and MCR, principal elementary mode analysis (PEMA) [6] was proposed to build a multivariate model using thermodynamically feasible pathways retrieved directly from the network. In the PEMA model, fluxes from different experiments are projected into the most representative set of elementary modes (EMs) from the metabolic network. The EMs are the simplest representations of pathways in the metabolic network. Basically, each EM connects substrates with end-products concatenating reactions.
In non-steady state conditions, the state of the network at a particular time point of the biological process is defined by the concentration of each metabolite in the cell, and metabolites may interact via one or more reactions. Each reaction is represented by an ordinary differential equation (ODE) relating chemical compounds. Since metabolic networks may have hundreds of reactions, it is hard to build kinetic models requiring kinetic parameters. When given the initial concentrations of metabolites and the full kinetic model (including the values for the kinetic parameters), the concentration of the metabolites along time can be simulated to produce a state transition path or trajectory, i.e. the succession of states adopted by the network over time [7]. Methodologies commonly applied when dealing with the aforementioned ODE systems, however using different data sources, are kinetic modelling [8], dynamic flux balance analysis (DFBA) [9], and a recently proposed approach combining time-resolved metabolomics and dynamic FBA (MetDFBA) [10], among others.
Once the kinetic model is built and the data is gathered, either simulated or (partially) measured, a comparison between experimental conditions can be performed to discover which groups of metabolites, reactions or pathways show differences between substrates, environment, etc. For this purpose, partial least squares regression discriminant analysis (PLS-DA) [11] can be used to find metabolites that are strongly related to a response variable (e.g. group of experiments) [12]. The problem with this approach is that no topological information is included in the multivariate model. The identified metabolites can be scattered in the network, not showing clear metabolic routes, as it happened in PCA with steady state data.
The Goeman's test was proposed in [13] to tackle the lack of topological information in the PLS-DA model. In that case, discrimination between experiments using metabolite concentrations was investigated using the set of pathways retrieved from the Kyoto Encyclopedia of Genes and Genomes (KEGG) database [14–16]. The aim was to find which pathways have a different activation pattern depending on the initial conditions of the experiment at particular time points. This model includes topological information, as metabolites are tested in groups of KEGG pathways, but these pathways sometimes do not connect directly substrates with end products, and the model is not built including all pathways and time points simultaneously.
To solve the aforementioned drawbacks of PLS-DA and the Goeman's global test, a novel framework is proposed to analyse non-steady state metabolite concentrations, based on an extension of the PEMA model. For this, we introduce the concept of dynamic EMs (dynEMs), i.e. EMs activated partially at each time point of the experiment. The dynEMs are used in a discriminant model to identify which metabolic routes have different activations depending on the initial conditions, i.e. which pathways discriminate between experimental conditions (as for example different substrate concentrations). As opposed to PLS-DA, dynEMR-DA integrates topological information to make the model more interpretable, as the set of candidates are drawn from the elementary mode matrix of the metabolic network; and, as opposed to Goeman's test, includes all metabolic routes connecting substrates with end-products and all time points of the experiment in the same discriminant model.
The MATLAB code for dynEMR-DA, related functions and example data are freely available in http://www.bdagroup.nl/content/Downloads/software/software.php, with instructions about how to use the method with own data. This way, practitioners are guided through the procedure, from the definition of the inputs, elementary mode matrix and concentration or flux data (either can be used), to the outputs, i.e. coefficients for the dynamic elementary modes to reconstruct the flux data. The N-way toolbox [17] and efmtool [18] for MATLAB are required to use dynEMR-DA code.
The structure of the article is as follows. In Methods, the metabolic models and data sets of S. cerevisiae are presented and the adaptation of the PEMA model from a steady to a non-steady state environment is introduced, describing dynEMA, dynEMR-DA and the validation scheme. In Results, the output of dynEMR-DA is analysed using simulated and real concentration data. Finally, some conclusions are drawn in the last section.
Metabolic networks
Two metabolic models of the well-known baker's yeast S. cerevisiae are used here to build the multivariate discriminant models (see Additional file 1 for a list of reactions). The first one was used in [19] to study the dynamics in glycolysis. The metabolic network (see Fig. 1a) has M=23 metabolites and K=18 reactions. This metabolic model has 26 elementary modes.
S. cerevisiae metabolic models. Model a), from [19], is used for the simulated study, and b), from [13], for the real case study
The second model was proposed in [10], and comprises M=12 metabolites and K=20 reactions, and describes the glycolysis and the tricarboxylic acid (TCA) cycle (see Fig. 1b). This second metabolic model has 13 elementary modes.
Two models are used in this article since the metabolites whose measurements were available in the real case study were not exactly the same as in the simulated model. Also, kinetic parameters were only available for the simulated case study. However, since both models are describing glycolysis in the same organism, the results are comparable.
Concentration data
The concentration data used in the first model (Fig. 1a) are simulated using COmplex PAthway SImulation (COPASI) software [20]. The initial concentrations of the metabolites match the measurements used in the original paper [19] (see Table 1). In this case, COPASI is used to simulate the concentrations from 0 to 60 s in 20 intervals of 3 s using a deterministic method (LSODA) [21]. The metabolic fluxes and the set of EMs are also obtained directly from COPASI.
Table 1 Initial concentrations in the simulated study. Experimental conditions taken from [19]
The aim in the simulated study consists of discriminating between scenarios using a high versus low initial concentration of glucose. 64 experiments are simulated using the data in Table 1, plus 20% noise, that is: c=(1+0.2ε)c0, where c is the concentration used in the analysis, c0 is the concentration given by COPASI and ε follows a Normal distribution with mean 0 and standard deviation 1. In the first 32 experiments the initial glucose concentration is set to 10mMol/l (plus noise), while in the last 32, this concentration is set to 2.5 mMol/l (also adding noise). These two values are indeed interesting, since they mimic the glucose concentrations used in the real case study (see paragraph below). The other common metabolites between metabolic models have comparable values in both concentration data sets. The set of EMs is obtained in this case using efmtool software [18].
In the real case, the concentrations of S. cerevisiae along 24 time points were obtained experimentally using liquid chromatography–mass spectrometry (LC-MS) [22, 23] at the Biotechnology Department of Delft University of Technology (The Netherlands), and were used afterwards in [13]. 12 different cultures are used in the present work (see Table 2). Regarding experiments 1 to 8, different initial glucose concentrations in aerobic conditions were used in these cultures: 10 mMol of glucose were used in the first 4 experiments and 2.3-2.5 mMol in experiments 5-8. Also, 4 more cultures, experiments 9 to 12, were performed using similar initial glucose concentrations as in experiments 5-8 but in anaerobic conditions (see Availability of data and materials section for more information on these data).
Table 2 Experiments used for the real case study. More details in Availability of data and materials section and in [13, 22, 23]
The aim in the real case study consists of discriminating between i) high and low glucose concentrations (i.e. experiments 1-4 vs 5-8), and ii) aerobic and anaerobic conditions (experiments 5-8 vs 9-12).
Scalar values are represented here as italic capital letters (e.g. N) and indices will appear as italic lower-case letters (e.g. j). Vectors are represented as bold lower-case letters (e.g. v). Data matrices are represented as bold capital letters (e.g. X). Superindex T denotes the transpose of a matrix. Observations or individuals within matrices are represented by rows, while variables are represented as columns. 3-dimensional arrays will be denoted as underlined bold capital letters (e.g. X). The mathematical operator × is used here to denote the size of the modes of a matrix (e.g. Y is a N×M matrix). No mathematical operator is used for products between scalars, vectors and matrices. Operator ∘ denotes the Hadamard element-wise product between vectors or matrices. Finally, operator ⊗ denotes the Kronecker tensor product between vectors or matrices, that is:
$$ \mathbf{X}\otimes \mathbf{Y}=\left[\begin{array}{cc} x_{11} & x_{12} \\ x_{21} & x_{22} \end{array}\right] \otimes \mathbf{Y}=\left[\begin{array}{cc} x_{11}\mathbf{Y} & x_{12}\mathbf{Y} \\ x_{21}\mathbf{Y} & x_{22}\mathbf{Y} \end{array}\right] $$
Squares and rectangles are used in figure drawings as a representation of matrices.
Dynamic elementary mode analysis (dynEMA)
Any steady state flux distribution x=(x1,…,x K ) can be decomposed as a positive linear combination of a set of E EMs [24]:
$$ \mathbf{x}=\sum\limits_{e=1}^{E} \lambda_{e}\mathbf{p}_{e} $$
where K is the number of fluxes (matching the number of reactions in the network), \(\phantom {\dot {i}\!}\mathbf {p}_{e}=(p_{e_{1}},\ldots,p_{e_{K}})\) is the eth EM, λ e is the positive weighting factor of the eth EM, and E is the number of EMs needed to reconstruct the flux distribution x. The set of E EMs is a subset of the complete set of Z EMs of the metabolic network.
Figure 2a shows an example of this modelling using a small network with M=5 metabolites and K=8 reactions. There are Z=3 EMs in the network: (1,1,1,1,0,0,0,0), (1,1,0,0,1,1,0,0) and (1,1,0,0,1,0,1,1). Let us assume that there is only flux on reactions 1 to 6. A linear combination of the first E=2 EMs will reconstruct the flux carried by the reactions in the system in Fig. 2b. In this case, all reactions in each EM are multiplied by the same value. The weighting factors correspond to the flux shown in the graphics beside reactions.
a Small metabolic network. b Steady state flux distribution. In b), the flux carried by each reaction is shown. Reactions 7-8 have no flux
When N flux distributions are considered, coming from different experiments or cultures, a PEMA model can be built:
$$ \mathbf{X}=\boldsymbol{\Lambda}\mathbf{P}^{\mathrm{T}}+\mathbf{F} $$
where X is the N×K flux data matrix, P is the K×E principal elementary mode (PEM) matrix, formed by a subset of E EMs; Λ is the N×E weighting matrix; and F is the N×K residual matrix. A schematic representation of a PEMA model is shown in Fig. 3.
Schematic representation of data matrices in the PEMA model
Non-steady state flux distributions cannot be decomposed as linear combinations of EMs, as in steady state. When the biological system has not reached yet the steady state, the system is not in equilibrium and fluxes can change over time. However, the EMs are indeed the simplest pathways along which the non-steady state fluxes have to flow, but not in a constant fashion. Thus, the EMs must be modified or adapted to fit this dynamical system. These are the so-called dynamic elementary modes (dynEMs).
To adapt an EM, there is not only a single coefficient multiplying the EM (Λ values in PEMA):
$$ \lambda_{e}\mathbf{p}_{e}=(\lambda_{e} p_{e_{1}},...,\lambda_{e} p_{e_{K}}) $$
but a different coefficient multiplying each reaction activated by the EM:
$$ \boldsymbol{\alpha}_{e_{j}}\circ\mathbf{p}_{e}=(\alpha_{e_{j,1}} p_{e_{1}},\ldots\alpha_{e_{j,K}} p_{e_{K}}) $$
where \(\boldsymbol {\alpha }_{e_{j}}\) includes the coefficients that adapt reactions 1 to K in the selected eth dynamic EM to reproduce the metabolic fluxes at time point j, and ∘ is the Hadamard element-wise product of matrices.
Thus, a single non-steady state flux distribution x at time point j can be decomposed as:
$$ \mathbf{x}_{j}=\sum\limits_{e=1}^{E} \boldsymbol{\alpha}_{e_{j}} \circ \mathbf{p}_{e} $$
Consider now a set of non-steady state flux distributions, which can be obtained from a single experiment measuring the concentration of the metabolites at J consecutive time points. Figure 4 shows an example of this scenario using the previous small network. Let us assume that there are fluxes only in reactions 1 to 4. In this case, only E=1 EM is needed. However, at each time point (j=1,…,4) the flux at each reaction (k=1,…,8) is different. High values are registered at the beginning of the experiment in the first reaction (Fig. 4a). Afterwards, the flux reaches all metabolites in the EM (Fig. 4b-c). Finally, the experiment reaches the steady state at the last time point (Fig. 4d), and all fluxes in the reactions are similar.
Small metabolic network with non-steady state fluxes from time point 1 to 4 (a) to d), respectively). Graphics show the flux carried by each reaction, which changes depending on the time point. The first subindex of the weighting factor \(\alpha _{e_{j,k}}\) indicates the EM E=1. The other two subindices indicate time point j=1,..,4 and reaction k=1,..,8
Considering non-steady state flux distributions along J time points, the set of active dynEMs can be obtained, in a PEMA/PCA-like fashion, from the new dynamic elementary mode analysis (dynEMA) model:
$$ \mathbf{X}=(\mathbf{I}_{J}\otimes\mathbf{1}^{\mathrm{T}}_{E}) \lbrack \mathbf{A}\circ(\mathbf{1}_{J}\otimes \mathbf{P}^{\mathrm{T}})\rbrack + \mathbf{F} $$
where A is the EJ×K coefficients matrix, I J is the J×J identify matrix, P is the K×E principal elementary mode (PEM) matrix, 1 E and 1 J represent column vectors of E and J ones respectively, F is the J×K residual matrix (containing the fluxes not explained by the set of dynamic elementary modes) and ⊗ is the Kronecker matrix product. In this case, X is a J×K data matrix representing the non-steady state fluxes from a single experiment along J time points; while in the PEMA model, X is a N×K matrix representing the steady state fluxes of N different experiments. Figure 5 shows a representation of dynEMA model.
Schematic representation of data matrices in the dynEMA model
The coefficients matrix A in the previous equation is, in fact, a E×K×J 3-way matrix unfolded reaction-wise, and each entry in the matrix \(\alpha _{e_{jk}}\) represents the coefficient multiplying reaction k of EM e to reconstruct the flux at time point j. Using this modelling it is possible to study the time evolution of a dynEM, i.e. how the dynEM is adapted or dynamically used along all measured time points for a given experimental condition.
This system of equations is solved similarly to PEMA. The candidates for first dynEM are selected from the complete K×ZEM matrix in a step-wise fashion. After selecting an EM, the coefficients multiplying it (thus creating the dynEM) are obtained solving Eq. 7 using non-negative least squares. Once all EMs are evaluated, the dynEM explaining most variance in data (as in PEMA) is classified as the first dynEM (1st column of PEM matrix P). Afterwards, this first dynEM is set, and the search for the second one starts, recalculating the coefficients in matrix A for both the first and the second dynEMs at each evaluation. In this way, the dynEMA model is built in a greedy way, explaining as much variance as possible at each step.
Regarding the number of dynEM extracted, this depends on the aim of the analysis, as explained in [6] with the PEMA model. For example, when the aim is to identify the main dynamic behaviour, one dynEM is enough. If the aim is to identify the main dynEM utilizing one particular section of the network, the model needs as many dynEMs as required to represent those reactions. Alternatively, one can extract as many dynEMs needed to reach certain percentage of explained variance (e.g. 95%).
The dynEMA model is useful to identify the dynEMs active in an experiment and how each dynEM is used in the culture at different time points of the experiment.
Dynamic elementary mode regression discriminant analysis (dynEMR-DA)
When the aim is to establish differences between environmental or experimental conditions, e.g. presence/absence of a compound or case/control studies, a discriminant model is needed. For this, dynamic elementary mode regression discriminant analysis (dynEMR-DA) is proposed here. This model focuses on finding which are the dynEMs with a strongly different time evolution or performance between conditions. In essence, dynEMR-DA is a two-step procedure. First, it projects the flux data into the space defined by each single dynEM. Then, fits a NPLS-DA [25] model with discriminant purposes.
To build a dynEMR-DA model, the set of different experiments are combined in a single \(\underline {\mathbf {X}}\) 3-way matrix (see Fig. 6). In X we consider N experiments, measuring K fluxes along J time points. Therefore, it is mandatory to have the same time points in all experiments.
dynEMR-DA procedure. XH and XL denote the flux data matrices of two different experimental conditions
The algorithm of dynEMR-DA has the following steps:
For each EM in the metabolic network (candidate to dynEM):
Unfold reaction-wise the N×K×JX matrix in Fig. 6 in a two-way JN×K matrix X.
Calculate the coefficients matrix A using the dynEMA model:
$$ \mathbf{X}=\left(\mathbf{I}_{JN}\otimes\mathbf{1}^{\mathrm{T}}_{E}\right) \left\lbrack \mathbf{A}\circ\left(\mathbf{1}_{JN}\otimes \mathbf{p}^{\mathrm{T}}\right)\right\rbrack + \mathbf{F} $$
where p denotes the candidate EM from step 1.
Reconstruct the flux data \(\hat {\mathbf {X}}\) using the dynEMA model:
$$ \hat{\mathbf{X}}=\left(\mathbf{I}_{JN}\otimes\mathbf{1}^{\mathrm{T}}_{E}\right) \left\lbrack \mathbf{A}\circ\left(\mathbf{1}_{JN}\otimes \mathbf{p}^{\mathrm{T}}\right)\right\rbrack $$
Fold the reconstructed data to build again a three-way data structure \(\underline {\hat {\mathbf {X}}}\)
Fit an NPLS-DA model between the reconstructed data and the y data, where y denotes the class of experiments (having 1s and 0s).
The dynEM whose NPLS-DA model explains most variance in y is classified as the first dynEM.
Check the predictions of NPLS-DA model. If the current model discriminates perfectly, stop. If not, set the first dynEM and repeat steps 1-3 to extract the second dynEM following the dynEMR-DA procedure.
NPLS-DA was proposed for studying N-dimensional data structures with discriminant purposes. NPLS is the natural extension of PLS to N-way structures, which tries to maximize the covariance between the \(\underline {\mathbf {X}}\) and Y data arrays. Y is denoted as y when one variable is predicted. NPLS-DA models in this paper have been computed using the N-way toolbox for MATLAB [17].
The dynEMR-DA algorithm can select many dynEMs until attaining a perfect discrimination. However, in practice, individual dynEMs are able to discriminate between two experimental conditions, so there is no need of considering two dynEMs simultaneously active to obtain a discriminant model. Moreover, some dynEMs are discriminating between initial conditions, but some of their reactions are not used at any time point of the experiment (so the flux does not flow through the metabolic pathway from the beginning to the end). These dynEMs do not represent actual metabolic pathways, so they should be removed when they are selected.
Triple cross-validation (3CV)
Proper validation of multivariate models is a subtle issue in Systems Biology. When enough data are available, single cross-validation procedures may lead to too optimistic models, especially when the aim is discrimination between classes. As commented in [26], when discriminant models, such as PLS-DA, are used on datasets with much more variables than samples, the models cannot be built as accurately as when there are more samples than variables. Then, the high number of variables can lead to chance discriminations, i.e. models that give good results because a variable had by chance lower values in all samples from one group. To avoid this sometimes spurious results, double cross validation (2CV) was proposed [26]. Using this procedure, a subset of the original data is used to model fitting, another subset to decide the complexity of the model (e.g. number of components of a multivariate model), and finally, a third subset is used for validation. This kind of models are especially useful for (N)PLS-DA model validation [26, 27].
In this work, though, we need an extra round of validation. dynEMR-DA models involve the projection, as first step, of the flux data into the space defined by each single dynEM. Afterwards, an NPLS-DA model is fitted, determining at the end which dynEMs are discriminating between groups. Therefore, we propose here a triple cross validation (3CV) scheme (see Fig. 7). This procedure consists of the following steps:
3CV procedure. 75% of the samples from both classes (red and blue) are used in the calibration, projection and test sets (25% in each). The remaining 25% of samples are used in validation set
Divide the data set in four groups: calibration, test, selection, and validation. The latter is left out of the analysis until the final external validation.
Fit a dynEMR-DA model using the calibration set, using a maximum of K components (as many as fluxes).
Project the test set, first to the corresponding dynEM, and then to each of the K NPLS-DA calibration models. At this point, the minimum number of components, A, needed to classify each experiment in its corresponding class, is selected.
Project the selection set into the previous dynEMR-DA model with A NPLS-DA components and evaluate the predictive power of each dynEM.
Steps 2-4 are repeated three times, changing the roles of the subsets. That is, the models are built using, in steps 2 to 4 respectively: calibration-test-selection, test-selection-calibration and selection-calibration-test sets.
The dynEMs with perfect classification rates using the selection set in the three rounds are used finally for validation, so the discrimination power of each dynEM is evaluated with completely external data. This prediction is performed substituting the selection group by these validation samples in the three models previously fitted.
A 2CV strategy is used for the NPLS-DA section of the dynEMR-DA models, but an extra validation round is needed to assess the performance of the selected dynEMs in terms of discrimination. Therefore, the 3CV procedure is built basically replacing the validation step, in the original 2CV, by the selection step, and performing the external validation in the last step.
Simulated flux data
The metabolic model of S. cerevisiae in Fig. 1a is used in this section to assess the performance of dynEMR-DA on simulated data. 64 experiments are simulated using COPASI, with the initial concentrations described in Methods (see Table 1). Thus, 32 experiments have a high initial concentration of glucose and 32 a low concentration. The fluxes derived from the concentration data, and also the set of EMs of the metabolic model, are also obtained using COPASI.
To validate the discriminant models, the 3CV scheme is used here, using the N-way Toolbox for MATLAB [17] to fit the NPLS-DA models. 8 experiments of each class selected at random (16 in total) are used for calibration. 16 more experiments are used to select the number of NPLS-DA components. And 16 more are used as selection samples. As described in Fig. 7, the first 3 subsets are used as calibration, test and selection sets, and then the roles change, i.e. test-selection-calibration and selection-calibration-test (steps 2-4 described in 3CV). Finally, 16 additional experiments are used as validation set.
When applying the dynEMR-DA procedure described in the previous section, only one dynEM (from the whole set of 26 EMs) is able to discriminate perfectly between both experimental conditions: dynEM 8. Finally, the remaining 16 cultures are used for the final validation of this dynEM (see Fig. 7). Again, all experiments are correctly classified in the dynEMR-DA model.
Figure 8a shows dynEM8. This mode covers the whole glycolytic pathway, starting from glucose (GLCo), producing all the intermediate products until reaching pyruvate (PYR), acetate (ACE) and finally ethanol (ETOH). The coefficients multiplying the EM are visualized in Fig. 8b-e. The first three time points (3, 6, and 9 s) reveal changes in the coefficients. Afterwards, changes are small. At 36 s, the system reaches the steady state, when fluxes do not change any more.
Simulated study. a dynEM8 depicted on the metabolic model. b-e dynEM8 coefficients at 3, 6, 9 and 36 s (first 3 times points and when the fluxes reach the steady state). Blue (red) lines show the mean of the coefficients for the high (low) glucose experiments
The differences between both experimental conditions can be seen in Fig. 8b-e (blue versus red bars). The usage of all reactions in the dynEM, i.e. the coefficients in A matrix, are higher in the high glucose concentration experiments than in the low glucose. This implies that these scenarios take advantage of the higher amount of glucose to carry more flux through the glycolysis until reaching ethanol.
It is worth mentioning that the system is close to steady state from the first time point. However, we used this set up to have a simulated case as close as possible to the real case, in order to find out i) whether there are differences between the initial concentrations of glucose, and ii) if the discriminant dynEM resembles the real case one(s) (see next section).
Real flux data
High vs low initial glucose concentrations
To assess the performance of dynEMR-DA in a real case study, a set of cultures of S. cerevisiae are used to discriminate between experiments using a high or a low initial glucose concentration. Unfortunately, the number of available cultures is low for this case study (4 in each class), so no 3CV, neither 2CV, is possible. Therefore, single CV is applied here: 3+3 experiments are used for dynEMR-DA model building and selection of NPLS-DA components, and the remaining 1+1 experiments are used for validation. This procedure is repeated 4 times, leaving out a couple of cultures each time.
The dynEMR-DA model has to be built using fluxes, not concentrations. Therefore, we computed the fluxes based on the changes in the concentrations between two consecutive time points solving an optimization problem (similarly as in [10]). Specifically, the objective function in this formulation makes the fluxes smooth along time (penalizing the sum of the differences between fluxes in consecutive time points) and small (penalizing the sum of squared fluxes), and the constraints force them to fulfil the stoichiometric equations.
In the actual data set, M=12 metabolites are measured in 24 time points within 2 min (1 measurement every 3 s). The metabolic network (see Fig. 1b) has K=20 reactions. Thus, the optimization problem to solve is:
$$ \left\{ \begin{array}{l} \min_{x_{jk}} {\sum\nolimits}_{j=1}^{22} {\sum\nolimits}_{k=1}^{20} (x_{j+1,k}-x_{j,k})^{2} + {\sum\nolimits}_{j=1}^{23} {\sum\nolimits}_{k=1}^{20} x_{j,k}^{2}\\ s.t. \quad \mathbf{S}\mathbf{X}^{\mathrm{T}}=\frac{d\mathbf{C}^{\mathrm{T}}}{dj}\\ \qquad \; \, \mathbf{X}\geq\mathbf{0}\\ \qquad \; \, \mathbf{X}_{0} \: \mathrm{initial \: solution}\\ \end{array} \right. $$
where X={x jk } is the 23×20 (time points × reactions) flux data matrix. The quadratic optimization problem needs an initial guess on X, i.e. X0. This guess is obtained solving \(\mathbf {S}\mathbf {X}_{0}^{\mathrm {T}}=\frac {d\mathbf {C}^{\mathrm {T}}}{dj}\) using non-negative least squares. Indices k and j denote flux number and time point, respectively, S denotes the 12×20 stoichiometric matrix (metabolites × reactions), and C is the 24×12 concentration matrix (time points × metabolites). It is worth noting that, since fluxes are computed based on the differences between concentrations at consecutive time points, there is one time point less in the flux data matrix (J=23) than in the concentration data (24).
The objective function used in the optimization problem resembles the MOMA function (minimize the squared difference of the reaction rates with steady state) used in [10], with the difference that we minimize the flux differences between consecutive time points.
In this case, only dynEM9 (from the set of 20 EMs) is able to discriminate the left out experiments. This dynEM can be visualised, jointly with the coefficients in matrix A, in Fig. 9. The differences between high and low glucose are also clear in this example. The usage of this dynEM is stronger in scenarios with a high initial glucose concentration than with a low concentration.
Real case study. a dynEM9 depicted on the metabolic model. b-e dynEM9 coefficients at 3, 6, 9 and 24 s (when system is close to steady state). Blue (red) lines show the coefficients for the high (low) glucose experiments
The results in this example follows the scheme described in Fig. 4. In both experiments (high and low), the fluxes are higher in the first steps of glycolysis (3, 6, and 9 s) and lower at the end. As time goes by, fluxes in the last part of the glycolysis increase. This shows that the flux data cannot be modelled in the same way at the first time points as when the culture reaches the steady state, therefore it necessitates to use of dynEMs to model non-steady state flux data, instead of applying a PEMA-based approach.
It is worth noting the similarity between the dynEM identified here and dynEM8 of the simulated case study. Both dynEMs are describing the same phenomena, the glycolysis until reaching pyruvate. They are not exactly the same because the metabolic models are different: acetate and ethanol were not measured in experimental conditions. However, when comparing the simulated and the actual data, the dynEM discriminating between experimental conditions is basically the same one.
Finally, it is difficult to assess when the system reaches the steady state in the real case study. In the simulated case, steady state was reached clearly at 36 s (since fluxes did not change anymore). In the real case, after 24 s (see Fig. 9) fluxes do not change significantly. However, since measurement error is present in the real case, it is difficult to asses whether the steady state was reached at 24 s or afterwards.
Aerobic vs anaerobic conditions
For the second real case study, four cultures performed in aerobic conditions versus four more in anaerobic conditions are compared. As in the previous example, fluxes are calculated from the real concentration data using the optimization framework (see Equation 10); also, a single cross validation procedure is applied here.
In this case study, dynEM8 is able to discriminate between both experimental conditions. The dynEM and the coefficients at 3, 6, 9 and 24 s (when system seems to reach steady state) can be visualized in Fig. 10. Again, the differences between both classes can be seen in the plots; the anaerobic experiments having higher coefficients. This behaviour has been outlined also in the literature [28–31]. To satisfy the redox balances, the flux is deviated from glycolysis to the production of glycerol (in our case, after reaction 4, flux is going through reactions 5 and 6). Glycerol is produced by reduction of the glycolytic intermediate dihydroxyacetone phosphate to glycerol 3-phosphate (g3p) followed by a dephosphorylation of g3p to glycerol. Despite glycerol does not appear explicitly in the network, because this metabolite was not measured in all original experiments, it is likely that the flux flowing through g3p produce glycerol at the end, as suggested in the literature.
Real case study. a dynEM8 depicted on the metabolic model. b-e dynEM8 coefficients at 3, 6, 9 and 24 s (when reaching steady state). Blue (red) lines show the coefficients for aerobic (anaerobic) experiments
Comparison to other state-of-the-art techniques
NPLS-DA
As in [6], it is worth to compare the approach of an elementary-mode based projection model to a classical projection method, which in this case, is NPLS-DA. To perform this comparison, the real case studies presented in the two previous subsections have been modelled using NPLS-DA algorithm.
Figure 11 shows the loadings of the fluxes using the high versus low initial glucose data. The model in this case has 3 components, explaining 92 and 95% of variance in flux and discriminant variables, respectively. This number of components corresponds to the most parsimonious model needed to correctly classify all experiments. Firstly, it is difficult to extract from the loading plots which fluxes are the most important for discrimination, as no clear threshold can be drawn in the plot. Secondly, even varying this hypothetical threshold, the significant fluxes (those with high absolute loading coefficient) represent disconnected reactions through the network and do not correspond to physical pathways, since no topological information is included in the model. The NPLS-DA loadings are the elementary modes in dynEMR-DA, therefore interpretation is more straightforward, as they represent real pathways.
NPLS-DA loading plots for the fluxes (high versus low intial glucose data)
Figure 12 shows the results for the aerobic versus anaerobic case study. Here, 6 components are needed, explaining 98 and 99% of variance in flux and discriminant variables, respectively. As in the high versus low initial glucose example, loading plots are very difficult to interpret.
NPLS-DA loading plots for the fluxes (aerobic versus anaerobic data)
The computation time with these case studies is 17 s (dynEMR-DA model) versus 0.5 s (NPLS-DA model). In the dynEMR-DA algorithm, as many NPLS-DA models as EMs (in this model, 13) are fitted to find the most discriminant one, therefore it is clear that one single NPLS-DA model will be faster than dynEMR-DA. However, the time needed to interpret the output of NPLS-DA is longer than the pathway-oriented result that dynEMR-DA provides.
dynEMR-DA, as opposed to NPLS-DA, can be strongly affected by the size of the EMs matrix. When having several hundreds of EMs, a pre-selection of EMs can be performed to speed up the analysis. One strategy would be to study the reactions that are active in all EMs and include only those EMs with different active reactions (i.e. coefficient different from zero). For example, if many elementary modes use the same reactions with the same directionality for the reversible ones, only one EM can be included in the set of EMs to test. Another possibility would be to use the set of extreme pathways of the network instead of the EMs [24].
Goeman's global test
The Goeman's global test was applied in [13] to find which KEGG pathways show differences between experimental conditions. The output in that case was a p-value indicating which pathways were different depending on the groups at discrete time points. Their results showed that glycolysis and TCA cycle were significant but not for all time points when comparing high versus low initial glucose. For the aerobic versus anaerobic case, both the glycolysis and TCA were significant for all time points.
This approach is not directly comparable to dynEMR-DA, as all pathways are tested simultaneously in dynEMR-DA, instead of individual pathway testing. No EM containing TCA was significant here, which can be also due to i) all time points are used simultaneously in dynEMR-DA, instead of discrete time point analysis (4 time points in [13]), and ii) the dynEMs containing TCA might not show differences between experimental conditions in the non-TCA section of the dynEM.
Finally, authors stated in the Goeman's test article [13] that a dynamic model would be more suitable for this type of data, which is what was pursued here.
The approach for dynamic elementary mode modelling proposed here permits decomposing non-steady state flux distributions into a set of active dynEMs. This way, dynEMA can be used to study the active dynEMs in an experiment, or a set of experiments, extending the PEMA model to a dynamic environment. For discrimination purposes, the main interest in this article, dynEMR-DA allows identifying which dynEMs have different patterns of activation depending on the culture initial conditions.
Actual and simulated concentration data of S. cerevisiae have been used here to evaluate dynEMR-DA. When changing the amount of glucose present in the experiment in both data sets, dynEMR-DA is able to identify that the dynEM flowing through the glycolytic pathway from glucose to pyruvate is discriminating between high and low initial glucose concentration experiments. Even considering two different metabolic models, for data availability reasons, the results of dynEMR-DA seem coherent between case studies. When analysing data from aerobic versus anaerobic conditions, dynEMR-DA indicates that the most discriminant dynEM drives the initial glucose concentration to the glycerol production. Previously published research confirms the results obtained using this new methodology.
The framework presented here will serve to create reduced dynamic models of flux data while preserving biological and thermodynamical meaning, as a tool to analyse non-steady state flux distributions in many experiments and to identify the hidden metabolic patterns that drive the organism from one state to another when changing the environmental conditions. dynEMA and dynEMR-DA have potential applications in bioprocess engineering to understand the small changes in cell metabolism at early stages of cultures.
2CV:
double cross-validation
triple cross-validation
COPASI:
complex pathway simulation
DFBA:
dynamic flux balance analysis
dynEM(s):
dynamic elementary mode(s)
dynEMA:
dynamic elementary mode analysis
dynEMR-DA:
dynamic elementary mode regression discriminant analysis
EM(s):
elementary mode(s)
FBA:
flux balance analysis
Kyoto Encyclopaedia of Genes and Genomes
LC-MS:
liquid chromatography–mass spectrometry
MCR:
multivariate curve resolution
MetDFBA:
time-resolved metabolomics and dynamic flux balance analysis
NPLS:
N-way partial least squares regression
NPLS-DA:
N-way partial least squares regression discriminant analysis
ODE:
ordinary differential equation
PEM(s):
principal elementary mode(s)
PEMA:
principal elementary mode analysis
PLS:
partial least squares regression
PLS-DA:
partial least squares regression discriminant analysis
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Authors would like to acknowledge Professor Henk A.L. Kiers (University of Groningen, The Netherlands), for his help during algorithm development, and the Biotechnology Department of Delft University of Technology (The Netherlands), for the real case study data sets.
This research work was partially supported by the Spanish Ministry of Economy and Competitiveness under the project DPI2014-55276-C5-1R.
The metabolic model of the simulated data can be retrieved from [19], with initial concentrations given in Table 1. The concetration data that support the findings of the real case study are available from the Biotechnology Department of Delft University of Technology (The Netherlands) but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are however available from the authors upon reasonable request and with permission of the Biotechnology Department of Delft University of Technology (The Netherlands). Additional information on the concentration data can be found in [22, 23].
Grupo de Ingeniería Estadística Multivariante, Departamento de Estadística e IO Aplicadas y Calidad, Universitat Politècnica de València, Valencia, Spain
Abel Folch-Fortuny & Alberto Ferrer
Genetics BioIT DBC Department, DSM Food Specialties, Delft, The Netherlands
Abel Folch-Fortuny
Systems Bioinformatics, Centre for Integrative Bioinformatics, Free University of Amsterdam, Amsterdam, The Netherlands
Bas Teusink
Biosystems Data Analysis, Swammerdam Institute for Life Sciences, University of Amsterdam, Amsterdam, The Netherlands
Huub C.J. Hoefsloot & Age K. Smilde
Huub C.J. Hoefsloot
Age K. Smilde
Alberto Ferrer
AF-F performed the analyses and wrote the manuscript. BT, HCJH and AKS conceived the study. AF-F, HCJH, AKS and AF developed the algorithms. BT, HCJH, AKS and AF and reviewed the manuscript. All authors read and approved the final manuscript.
Correspondence to Abel Folch-Fortuny.
Additional file 1
An additional file is provided with the detailed metabolic models. (PDF 105 kb)
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Folch-Fortuny, A., Teusink, B., Hoefsloot, H. et al. Dynamic elementary mode modelling of non-steady state flux data. BMC Syst Biol 12, 71 (2018). https://doi.org/10.1186/s12918-018-0589-3
Metabolic network
Elementary mode
Dynamic modelling
N-way
Cross validation
Methods, software and technology
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CommonCrawl
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\begin{document}
\publicationdetails{19}{2017}{2}{1}{3291} \title{A bijection between the set of nesting-similarity classes and L\&P matchings}
\begin{abstract} Matchings are frequently used to model RNA secondary structures; however, not all matchings can be realized as RNA motifs. One class of matchings, called the L \& P matchings, is the most restrictive model for RNA secondary structures in the Largest Hairpin Family (LHF). The L \& P matchings were enumerated in $2015$ by Jefferson, and they are equinumerous with the set of nesting-similarity classes of matchings, enumerated by Klazar. We provide a bijection between these two sets. This bijection preserves noncrossing matchings, and preserves the sequence obtained reading left to right of whether an edge begins or ends. \end{abstract}
Ribonucleic acid (RNA) is an essential molecule found in the cells of all living things. Usually, RNA is formed by a string of nucleotides which folds over on itself, creating secondary bonds. The structure of these secondary bonds is a topic of great interest and has been studied from a biological and mathematical perspective \cite{Condon, Jefferson}. Mathematically, RNA secondary structures can be modeled by considering each nucleotide as a vertex and bonds between nucleotides that are not part of the RNA backbone as edges. Each vertex is incident to at most one edge and thus the graph obtained is a matching.
We represent these matchings as $2n$ points drawn along a horizontal line (the backbone) and arcs drawn between pairs of points represent the nucleotide bonds. For simplicity, we assume all matchings are \emph{complete} (i.e., all vertices are incident to an edge), and therefore contain $n$ edges. RNA often has nucleotides that are not bonded, but these structures can be reconstructed by adding any number of isolated vertices to a complete matching. We notate the set of complete matchings with $n$ edges by $\mathcal{M}(n)$.
For a matching, $M$, we denote the set of edges $E(M)$ and label these edges with $[n]$ in increasing order from left to right by their left endpoints. For the edge labeled $i$, we write $i=(i_1,i_2)$ where $i$ represents the label of the edge and $i_1$, $i_2$ represent the position of the left and right endpoints of the edge, respectively. A pair of edges $i=(i_1,i_2)$ and $j=(j_1, j_2)$ are said to be \emph{nested} if $i_1<j_1<j_2<i_2$ and \emph{crossing} if $i_1<j_1<i_2<j_2$. For a matching $M \in \mathcal{M}(n)$, let $ne(M)$ and $cr(M)$ denote the number of pairs of nested edges and crossing edges in $M$, respectively.
Additionally, the edges $i=(i_1,i_2)$ and $j=(j_1, j_2)$ are said to form a \emph{hairpin} if $j_1=i_1+1$ and $j_2=i_2+1$ (Figure~\ref{hairpin}). A nested sequence of edges that can be drawn above the backbone is called a \emph{ladder} (Figure~\ref{ladder4}).
\begin{figure}
\caption{The left matching is a hairpin, the middle matching is a ladder of four edges, and the right matching is an example of an L \& P matching with edges labeled by left endpoint.}
\label{hairpin}
\label{ladder2}
\label{ladder4}
\label{LPMatching}
\label{ladder3}
\end{figure}
Matchings that contain no crossing pairs of edges are called \emph{noncrossing matchings}. The number of noncrossing matchings with $n$ edges is well-known to be counted by the $n$th Catalan number, $C_n=\frac{1}{n+1}\binom{2n}{n}$, and has been studied in several contexts, including pattern avoidance \cite{Bloom}.
For each matching, we can examine the vertices from left to right and list whether vertices are left or right endpoints of an edge. For example, for the matching in Figure \ref{example of matching}, this sequence is LLRLLRRRLR; we call this the \emph{LR-sequence} of a matching. LR-sequences are in bijection with Dyck paths, and each noncrossing matching has a distinct LR-sequence. For any $M \in \mathcal{M}(n)$, define $nc(M)$ to be the noncrossing matching with the same LR-sequence as $M$.
\begin{figure}
\caption{Two nesting-similar matchings. Each matching has LR-sequence LLRLLRRRLR and two pairs of nested edges.}
\label{nesting similarity}
\end{figure}
In order to model RNA structures that contain crossings, different families of matchings have been studied. In this paper, we focus on the family of L \& P matchings. The L \& P matchings were first rigorously defined by Condon et al.\ \cite{Condon} and this definition was later refined by Jefferson \cite{Jefferson}. Each matching in the L \& P family can be constructed inductively by starting from either a hairpin or a single edge, and either a) inflating an edge by a ladder, or b) inserting a noncrossing matching into an L \& P matching \cite{Jefferson}. For an example of an L \& P matching, see Figure~\ref{LPMatching}. The counting sequence for the number of L \& P matchings with $n$ edges begins 1, 3, 12, 51, 218, 926, 16323, 67866, 280746 and is given by the formula $2\cdot 4^{n-1}-\frac{3n-1}{2n+2}\binom{2n}{n}$ \cite{Jefferson}.
It was noted by Jefferson that L \& P matchings with $n$ edges, which we denote $\mathcal{LP}_n$, are equinumerous to the equivalence classes on matchings with $n$ edges determined by the nesting-similarity equivalence, $\sim_{ne}$, defined by Klazar \cite{Klazar}. We say that two matchings $M, N \in \mathcal{M}(n)$ are \emph{nesting-similar}, and write $M \sim_{ne} N$, if and only if $ne(M)=ne(N)$ and $M,N$ have the same LR-sequence. An example of this equivalence is shown in Figure \ref{nesting similarity}. Klazar showed that there are $2\cdot 4^{n-1}-\frac{3n-1}{2n+2}\binom{2n}{n}$ nesting-similarity equivalence classes for matchings in $\mathcal{M}(n)$; however, no explicit bijection was discovered connecting L \& P matchings to nesting-similarity classes. Our main result is to construct a bijective correspondence between these two structures.
Klazar \cite{Klazar} utilized a different, but equivalent, definition for nesting-similarity in his work. He examined the \emph{tree of matchings} where the vertex set is the infinite set of matchings, $\bigcup_{n=0}^{\infty} \mathcal{M}(n)$. Matchings are connected if one can be obtained from the other by inserting a new edge whose left endpoint occurs earliest. With this construction, two matchings are said to be nesting-similar if we can record the number of nestings for the children of each matching, which have one added edge, and obtain the same multiset. It is straightforward to show that this definition is equivalent to the definition above.
In his paper, Klazar shows a correspondence between his nesting-similarity matchings and tunnel pairs in Dyck paths. Our bijection between L \& P matchings and the set of nesting-similarity classes will be a composition of two bijections. The composed mapping will pass through the set of nestings in noncrossing matchings, which correspond to tunnel pairs in Dyck paths. Define $\mathcal{NCN}_n = \{(M,a,b) \mid M\in \mathcal{M}(n) \text{ noncrossing};a=b=0 \text{ or \allowbreak edges } a<b\text{ are nested} \allowbreak \text{ in }M\}$; so, $\mathcal{NCN}_n$ is the set of noncrossing matchings with a chosen nested pair of edges ($a=b=0$ indicates no nested pair has been indicated).
In Section~\ref{sec:LP}, we define a bijection between L \& P matchings and $\mathcal{NCN}_n$, and in Section~\ref{sec:nes}, we define a bijection between nesting-similarity classes and $\mathcal{NCN}_n$.Thus We show that both L \& P matchings and nesting-similarity classes are equinumerous to $\mathcal{NCN}_n$.
\section{L \& P matchings and noncrossing matchings} \label{sec:LP}
The process of constructing an L \& P matching implies that such a matching contains a crossing exactly if the matching can be built inductively from a hairpin. As a result, any L \& P matching that contains a crossing will have all crossings occur in an inflated hairpin. Given a matching, $M$, we will label edges by left endpoint, as in Figure~\ref{LPMatching}. In this Figure, edges 1, 2 crossing edges 4, 5 comprise an inflated hairpin. Below we provide a precise definition of this structure.
\begin{definition} A \emph{maximal inflated hairpin} in an L \& P matching is two sets of edges $A=\{a_1,a_2,\ldots,a_k\}$ and $B=\{b_1,b_2,\ldots,b_{\ell}\}$ such that \begin{enumerate} \item every pair of edges in $A$ and every pair of edges in $B$ is nested, \item for every $a_i \in A$, $a_i$ crosses every edge in $B$, and \item every crossing in $M$ occurs between edges in $A$ and $B$. \end{enumerate} We let $A$ be the set of edges with smaller labels (the left side of the inflated hairpin), and we say that $M$ contains the inflated hairpin $(A,B)$. \end{definition}
Any L \& P matching consists of a maximal inflated hairpin with noncrossing matchings inserted below the hairpin as in Figure \ref{fig:structure}. It is possible for the inflated hairpin to be empty, yielding a noncrossing matching. For example, the matching in Figure~\ref{example of matching} is L \& P: the first and third edges crossing the fourth edge form an inflated hairpin. However, the matching in Figure~\ref{nesting similarity b} is not L \& P, since the first four edges are all involved in crossings, but these four edges do not form an inflated hairpin.
Notice that every edge not in the inflated hairpin of an L \& P matching must begin and end between two vertices that are adjacent in the inflated hairpin, otherwise it would be part of an inflated hairpin itself. This fact will be used in our bijection.
\begin{figure}
\caption{The structure of an L \& P matching; Each $\star$ indicates a position where a noncrossing matching may be inserted. The edges $a_1, a_2,\ldots, a_k$ cross the edges $b_1,b_2,\ldots,b_{\ell}$ to form the inflated hairpin.}
\label{fig:structure}
\end{figure}
\begin{lemma} \label{lem:LP} Let $M \in \mathcal{LP}_n$ such that $M$ contains an inflated hairpin $(A, B)$. Then the right endpoints of the inflated hairpin appear in the order $(a_k,a_k-1,\ldots,a_1,b_{\ell},b_{\ell}-1,\ldots,b_1)$ and can be rearranged to $(b_{\ell},b_{\ell}-1,\ldots,b_1,a_k,a_k-1,\ldots,a_1)$ to construct $nc(M)$. Additionally, every pair of edges in $A \cup B$ are nested in $nc(M)$. \end{lemma}
\begin{proof} Since we defined $A$ to contain edges with smaller labels than $B$, the result that the right endpoints are in the order $(a_k,a_k-1,\ldots,a_1,b_{\ell},b_{\ell}-1,\ldots,b_1)$ follows immediately. Now consider rearranging the right endpoints such that they appear in the order $(b_{\ell},b_{\ell}-1,\ldots,b_1,a_k,a_k-1,\ldots,a_1)$. It is easy to see that every pair of edges in $A \cup B$ is now nested and that no additional crossings can be created by this rearrangement. Therefore the result is a noncrossing matching. Notice that, since noncrossing matchings are in direct bijection with LR-sequences, the resulting matching is in fact $nc(M)$. \end{proof}
For an example of this conversion of an L \& P matching with an inflated hairpin into a noncrossing matching, see Figure \ref{fig:LPMatching}. The results of Lemma \ref{lem:LP} will be useful as we define our bijection from $\mathcal{LP}_n$ to $\mathcal{NCN}_n$.
\begin{definition} Define $\phi: \mathcal{LP}_n \rightarrow \mathcal{NCN}_n$ where, \[\phi(M) = \begin{cases} (M,0,0), & \text{if $M$ noncrossing}, \\ (nc(M),\max(A),\max(B)), & \text{otherwise}. \end{cases}\] where $(A,B)$ is the inflated hairpin of $M$. \end{definition}
The results of Lemma \ref{lem:LP} imply that $\max(A), \max(B)$ are a nested pair of edges in $nc(M)$, and therefore $\phi$ is well-defined. All that remains is to show that $\phi$ is in fact a bijection.
\begin{theorem}\label{phi} The mapping $\phi: \mathcal{LP}_n \rightarrow \mathcal{NCN}_n$ is a bijection. \end{theorem}
\begin{proof} We will construct the inverse of $\phi$. First note that $\phi^{-1}(M,0,0)=M$.
Now consider some noncrossing matching $M$ with nested pair of edges $(a,b)$. Let $A=\{a_i \in E(M) \mid (a_i,a)\text{ are nested and } a_i\leq a\text{ such that }a_1<a_2<\cdots<a_k=a\}$ and $B=\{b_i \in E(M) \mid (b_i,b)\text{ are nested and } a<b_i\leq b\text{ such that }b_1<b_2<\cdots<b_{\ell}=b\}$. Then, since edges are labeled by left endpoints and $M$ is noncrossing, the right endpoints of $A \cup B$ appear in the order $(b_{\ell},b_{\ell}-1,\ldots,b_1,a_k,a_k-1,\ldots,a_1)$.
Let $M'$ be the resulting matching when the right endpoints of $A \cup B$ are reordered to appear as $(a_k,a_k-1,\ldots,a_1,b_{\ell},b_{\ell}-1,\ldots,b_1)$. It is straightforward to show that $M'$ is an L \& P matching with inflated hairpin $(A,B)$ where $(\max(A),\max(B))=(a,b)$. It follows that $\phi^{-1}(M,a,b)=M'$. Therefore $\phi$ is a bijection, as desired. \end{proof}
\begin{figure}
\caption{On the left is an L \& P matching with labeled edges. In this matching, the inflated hairpin contains edges 1, 2 crossing edges 4, 5. On the right is the corresponding noncrossing matching obtained by swapping the right endpoints of edges in the inflated hairpin.}
\label{fig:LPMatching}
\end{figure}
\section{Nesting-Similarity Classes and Noncrossing Matchings} \label{sec:nes}
Recall that $M \sim_{ne}N$ if and only if $ne(M)=ne(N)$ and $M,N$ have the same LR-sequence. Klazar showed that the nesting-similarity classes are equinumerous with tunnel pairs in Dyck Paths through the use of transpositions that swap the endpoints of nestings with minimal width. Using this map iteratively, for any LR-sequence, if $M$ is the corresponding noncrossing matching, then for every $i$ where $0\leq i \leq ne(M)$, Klazar proved that there exists some matching with the same LR-sequence and $i$ pairs of nested edges. Notice that the noncrossing matchings contain the maximum possible number of nestings for a particular LR-sequence, so Klazar's result encompasses all the nesting-similarity classes.
We define a bijection between nesting-similarity classes and the set $\mathcal{NCN}_n$ which explicitly defines a representative for each equivalence class. Let $nep(M)$ denote the list of nested pairs of edges in $M$ ordered lexicographically by prioritizing the second element. For example, $$nep(M)=\{(1,2), (1,3), (2,3), (1,4), (2,4), (1,5), (2,5), (4,5), (1,6), (2,6), (4,6), (1,7)\}$$ for the noncrossing matching of Figure~\ref{fig:LPMatching}. Let $rperm(M)$ be the order in which the right endpoints of edges appear. For example, $rperm(M)=3564271$. We define $rperm$ for all matchings, not necessarily noncrossing. However, when we have noncrossing matchings, rperm is useful for identifying nestings.
\begin{lemma} \label{lem:rperm} In a noncrossing matching $M$, edges $a,b$ are nested, with $a<b$, if and only if $b$ appears before $a$ in $rperm(M)$. \end{lemma}
This follows quickly from the method of labeling edges and the definition of nestings.
Consider a noncrossing matching $M$ with $k$ pairs of nested edges. Given some $i$ with $0 \leq i \leq k$, we define a process by which we rearrange the vertices in a noncrossing matching to obtain a matching with the same LR-sequence and $i$ pairs of nested edges. These matchings will be the representatives of the nesting-similarity classes.
In the definition below, given edges $a,b$ where $1 \leq a,b \leq n$ ($a \neq b$), let $(a,b).M$ denote the matching that results by swapping the left endpoints of edges $a$ and $b$.
\begin{definition}\label{def:swaps} Let $M$ be a noncrossing matching with $k$ pairs of nested edges, and $nep(M)=\{(a_{1},b_{1}),\newline(a_{2},b_{2}),\ldots,(a_k,b_k)\}$. Define the sequence of matchings $M_0, M_1, M_2, \ldots, M_k$ by $M_0=M$ and $M_i=(a_i,b_i).M_{i-1}$ for all $i \in [k]$. Additionally, let $lperm(M_i)$ denote the order in which the left endpoints of matching $M_i$ appear. \end{definition}
For an example of this definition, see Figure~\ref{fig:swaps}. It is clear that each $M_i$ has the same LR-sequence as $M$. Our goal is to additionally show that $ne(M_i)=k-i$. This would imply that the matchings in $\{M_0,M_1,\ldots,M_k \mid M_0 \text{ noncrossing}\}$ form a set of representatives of the $k+1$ distinct nesting-similarity classes for matchings with the same LR-sequence as $M$.
\begin{figure}
\caption{An example of the matching obtained by swapping left endpoints in a noncrossing matching, as in Definition \ref{def:swaps}. Note the $nep(M) = \{(1,2),(1,3),(1,4),(3,4)\}$, which defines the order in which left endpoints are swapped.}
\label{fig:swaps}
\end{figure}
\begin{lemma} \label{lem:adj} Let $M$ be a noncrossing matching, let $M_0, M_1,\ldots,M_k$ be the sequence of matchings as defined in Definition~\ref{def:swaps}, and let $nep(M)=\{(a_{1},b_{1}),(a_{2},b_{2}),\ldots,(a_k,b_k)\}$. Then for every $i\in[k]$, $a_{i+1}, b_{i+1}$ appear in order and are adjacent in $lperm(M_i)$. \end{lemma}
\begin{proof} We use proof by induction. The result clearly holds for $M_0$. Assume that $a_{j+1},b_{j+1}$ are adjacent and in order in $lperm(M_j)$ for $0 \leq j <i$. First notice that this assumption immediately implies that $a_{i+1},b_{i+1}$ appear in order in $lperm(M_i)$.
To obtain a contradiction, assume that $a_{i+1}, b_{i+1}$ are not adjacent in $lperm(M_i)$. So, there exists some $c$ such that $c$ appears between $a_{i+1},b_{i+1}$. Notice that nestings in $M_0$ that are lexicographically smaller than $(a_{i+1},b_{i+1})$ have had their left endpoints swapped to obtain $M_i$. This fact will allow us to obtain a contradiction. We will consider three cases based on the size of the label $c$.
First assume that $a_{i+1}<b_{i+1}<c$. This implies that $c$ and $b_{i+1}$ appear out of order in $lperm(M_i)$, and must have been swapped. However, $(b_{i+1},c)$ is lexicographically larger than $(a_{i+1},b_{i+1})$, which gives a contradiction.
Next assume that $c<a_{i+1}<b_{i+1}$. In this case, $c$ and $a_{i+1}$ appear out of order in $lperm(M_i)$. The inductive assumption implies that at some step, $(c,a_{i+1})$ must have been swapped and were a nesting in $M_0$. However, this would also imply that $(c,b_{i+1})$ is a nesting in $M_0$ that is lexicographically smaller than $(a_{i+1},b_{i+1})$; applying this swap would reverse the order of $c,b_{i+1}$ as well. So, $c$ could not appear between $a_{i+1}$ and $b_{i+1}$ and we have a contradiction.
Finally assume that $a_{i+1}<c<b_{i+1}$. We need to consider two further subcases dependent on $rperm(M_0)$. If $c$ appears before $b_{i+1}$ in $rperm(M_0)$, then $c$ and $b_{i+1}$ are not nested edges by Lemma~\ref{lem:rperm}. However, $a_{i+1}$ nests both $c$ and $b_{i+1}$ in this case. It follows that $(a_{i+1},c)$ is a lexicographically smaller nesting in $nep(M_0)$ and $c,a_{i+1}$ must have already been swapped. As a result, they cannot appear in order in $M_i$, giving a contradiction.
If instead $b_{i+1}$ appears before $c$ in $rperm(M_0)$, the condition on noncrossing edges in $M_0$ and the fact that $a_{i+1}$ and $b_{i+1}$ are nested implies that $a_{i+1}$ and $c$ are also nested. Again, this is a lexicographically smaller nesting, which implies that $a_{i+1}$ and $c$ must appear out of order in $M_i$, giving a contradiction. \end{proof}
\begin{lemma} \label{lem:nestings} Let $M$ be a noncrossing matching, let $M_0, M_1,\ldots,M_k$ be the sequence of matchings as defined in Definition~\ref{def:swaps}, and let $nep(M)=\{(a_{1},b_{1}),(a_{2},b_{2}),\ldots,(a_k,b_k)\}$. Then $nep(M_i)=\{(a_{i+1},b_{i+1}),\newline(a_{i+2},b_{i+2}),\ldots,(a_k,b_k)\}$; in particular, $ne(M_i)=k-i$. \end{lemma}
\begin{proof} We use proof by induction on $i$. It is straightforward to show that $nep(M_1)=\{(a_{2},b_{2}),\ldots \\,(a_k,b_k)\}$.
Now assume that $nep(M_{i-1})=\{(a_{i},b_{i}),\ldots,(a_k,b_k)\}$. By definition, we know that $(a_i,b_i)$ is no longer a nesting in $M_i=(a_i,b_i).M_{i-1}$. Additionally, since $(a_i,b_i)$ were adjacent in $lperm(M_{i-1})$ by Lemma~\ref{lem:adj}, we know that $(a_{i+1},b_{i+1}),(a_{i+2},b_{i+2}),\ldots,(a_k,b_k)$ are all still nestings in $M_i$ and that no additional nestings are created by swapping $(a_i,b_i)$. Therefore $nep(M_i)=\{(a_{i+1},b_{i+1}),\\ (a_{i+2},b_{i+2}),\ldots,(a_k,b_k)\}$. \end{proof}
It follows that the matchings we generate by the swaps in Definition~\ref{def:swaps} form a set of representatives of the $k+1$ distinct nesting-similarity classes for matchings with the same LR-sequence as $M$. Let $\mathcal{NS}_n$ be the set of representatives of the nesting-similarity classes; that is $\mathcal{NS}_n = \{N \in \mathcal{M}(n) \mid \text{$N=M_i$ for some noncrossing matching $M$ with at least $i$ nestings}\}$. We can now define the other half of our bijection.
\begin{definition} Define $\tau: \mathcal{NCN}_n \rightarrow \mathcal{NS}_n$ where, if $M$ is a noncrossing matching with $nep(M)=\{(a_{1},b_{1}),(a_{2},b_{2}),\ldots,(a_k,b_k)\}$, then $\tau(M,a_i,b_i)=M_i$ and $\tau(M,0,0)=M$. \end{definition}
So, $\tau$ will take the noncrossing matching $M$, which has the maximum number of nestings of any matching with the same LR-sequence, and perform a sequence of left vertex swaps. Each swap converts precisely one nesting pair of edges into a crossing pair of edges; these swaps continue until the associated pair of edges is no longer nested. For an example of the mapping $\tau$, see Figure~\ref{fig:tau}.
\begin{theorem} \label{tau} The mapping $\tau: \mathcal{NCN}_n \rightarrow \mathcal{NS}_n$ is a bijection. Additionally, if $M \in \mathcal{M}(n)$ is noncrossing, $\tau(M,0,0)=M$. \end{theorem}
\begin{proof} Given some $N \in \mathcal{NS}_n$, consider $M = nc(N)$. If $N=M$ (implying that $N$ is noncrossing), then $\tau^{-1}(N)=(N,0,0)$. Otherwise, by the definition of $\mathcal{NS}_n$, there exists some $i$ such that $N = M_i$. If $nep(M) =\{(a_{1},b_{1}),(a_{2},b_{2}),\ldots,(a_k,b_k)\}$, set $\tau^{-1}(N)=(M,a_i,b_i)$. \end{proof}
\begin{figure}
\caption{An example of the bijection $\tau$, mapping from the noncrossing matching on the left, with chosen nesting pair $(2,5)$ to the representative of the nesting-similarity class with 5 nestings and LR-sequence LLLRLLRLRRRLRR on the right.}
\label{fig:tau}
\end{figure}
Now, combining Theorems \ref{phi} and \ref{tau}, we immediately obtain our desired result.
\begin{theorem} The map $\sigma = \tau \circ \phi:\mathcal{LP}_n \rightarrow \mathcal{NS}_n$ is a bijection between L \& P matchings and nesting-similarity classes. \end{theorem}
An example of this composition can be seen in Figure~\ref{fig:WholeBijection}.
\begin{figure}
\caption{Above is the result of first applying $\phi$ to an L \& P matching to obtain a noncrossing matching with an indicated nesting pair. Then, we see the result of applying $\tau$ to the noncrossing matching with indicated nesting pair to obtain a representative of a nesting equivalence class. Composed, this is the mapping $\sigma$.}
\label{fig:WholeBijection}
\end{figure}
The images under the map $\sigma$ also form a set of representatives for the nesting similarity classes $\mathcal{NS}_n$. Although Klazar proved that the classes exist, representatives of those classes were not explicitly provided, and now we have done so.
\begin{corollary} Since the bijection $\sigma$ has an intermediate step at the noncrossing matching associated to a matching M, $\sigma$ has the following properties:
\begin{itemize} \item{If $M$ is a noncrossing matching, then $\sigma(M) = M$.} \item{The LR-sequence of M is the LR-sequence of $\sigma(M)$.} \end{itemize}
\end{corollary}
Other properties that are sometimes preserved in bijections between matchings, such as number of nestings or number of crossings, are not preserved by $\sigma$. However these statistics all fail to be equidistributed between the two sets $LP_n$ and $NS_n$, so no other bijection exists that preserves them.
In the larger context, we note that none of the other Largest Hairpin Family matchings (LHF, D\&P, R\&G, C\&C) have closed forms for their enumeration sequences. However given that our map $\sigma$ relates an L \& P matching to a noncrossing matching and a nesting edge pair, and the fact that all of these families are constructed inductively, we believe it may be possible to find a similar mapping involving a noncrossing matching and some other matching property. The authors hope to explore this possibility in future work.
The authors would like to thank the organizers of the Dagstuhl Seminar $16071$: Pattern Avoidance and Genome Sorting, which is where the problem was first presented, and where the authors had generous working time to explore the ideas in this paper. The authors also thank the referees for their helpful comments which improved the readability of the paper.
\end{document}
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arXiv
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Minimally invasive, patient specific, beat-by-beat estimation of left ventricular time varying elastance
Shaun Davidson ORCID: orcid.org/0000-0002-5868-86401,
Chris Pretty1,
Shun Kamoi1,
Joel Balmer1,
Thomas Desaive2 &
J. Geoffrey Chase1
BioMedical Engineering OnLine volume 16, Article number: 42 (2017) Cite this article
The aim of this paper was to establish a minimally invasive method for deriving the left ventricular time varying elastance (TVE) curve beat-by-beat, the monitoring of which's inter-beat evolution could add significant new data and insight to improve diagnosis and treatment. The method developed uses the clinically available inputs of aortic pressure, heart rate and baseline end-systolic volume (via echocardiography) to determine the outputs of left ventricular pressure, volume and dead space volume, and thus the TVE curve. This approach avoids directly assuming the shape of the TVE curve, allowing more effective capture of intra- and inter-patient variability.
The resulting TVE curve was experimentally validated against the TVE curve as derived from experimentally measured left ventricular pressure and volume in animal models, a data set encompassing 46,318 heartbeats across 5 Piétrain pigs. This simulated TVE curve was able to effectively approximate the measured TVE curve, with an overall median absolute error of 11.4% and overall median signed error of −2.5%.
The use of clinically available inputs means there is potential for real-time implementation of the method at the patient bedside. Thus the method could be used to provide additional, patient specific information on intra- and inter-beat variation in heart function.
Cardiovascular disease and dysfunction (CVD) are leading causes of Intensive Care Unit (ICU) admission and mortality worldwide. CVD was responsible for 31% of global deaths in 2013, and the estimated worldwide cost associated with CVD in 2010 was $863 billion USD, accounting for approximately 1.39% of gross world product in the same year [1]. Despite the need with aging populations for optimised cardiovascular care in the ICU, inadequate or incorrect diagnosis of cardiac disturbances resulting in increased length of stay, cost and mortality is an ongoing issue [2, 3]. Cardiac management in the ICU is often informed by measurements taken from catheters placed near the heart. However, despite the rich information available from such catheter waveforms, their use is not necessarily associated with improved clinical outcomes [4, 5]. Hence, improvement in the extraction of cardiac information from these catheter waveforms has the potential to yield value from readily available data that has potentially been under-utilised to date.
Time varying elastance (TVE) is an important means of expressing internal cardiac dynamics and function [6]. The TVE curve represents the active elastance changes in the ventricles that drive heart contraction, thus providing valuable intra-beat information about cardiac behaviour and energetics [6–8]. The TVE curve has a wide range of potential applications. It is often employed as an input in large cardiovascular models that seek to model the entire circulatory system [9–13], and also frequently used in the determining of important cardiac metrics, such as the end-systolic pressure–volume relation (ESPVR) [7].
There has been more limited investigation into the utility of the TVE curve in isolation. The area under the curve of the TVE curve is analogous to work done by the ventricle. It is thus a potentially significant indicator of patient condition. However, the typical normalisation of the TVE curve means only relative changes in work for a given inotropic state can be compared. The TVE curve relies on a combination of ventricular pressure and volume waveforms similar to pressure–volume (P–V) loops. It thus contains information concerning cardiac work [14, 15] and contractility [6, 16], both of which change in response to cardiac dysfunction. As such, it has been suggested that the shape of the ventricular elastance curve itself may have diagnostic use [17–19].
Unfortunately, the TVE curve can only be directly measured by placing catheters into the heart chambers, which is, understandably, not common practice. As such, the traditional approach to non-invasively generate a TVE curve is to fix it to a population based waveform [7, 11, 12]. Most existing work surrounding TVE curves is focused on using the TVE curve to estimate a clinical parameter, such as ejection fraction [20], and, most commonly, end-systolic elastance [7, 21, 22]. However, these studies are validated on correlations with the derived parameter, rather than on the shape and change in shape of the TVE curve itself. Thus, these approaches are not validated for use as part of a larger model of cardiac dynamics, or for direct use as a diagnostic aid.
Previous work specifically focused on experimentally generating a TVE curve and validating it based on its correlation with the analytically derived function showed promise, and noted changes in the TVE curves during pulmonary embolism and septic shock [17]. However, this work was limited by the availability of data for validation and the reliance on an assumed driver shape [17, 18]. Work has also been undertaken in modelling time varying ventricular elastance, split into active and passive components, in humans [19]. Active elastance was shown to compare well with other metrics of contractility, and the properties of these elastance curves were shown to change for different disease states. However, this method required highly invasive ventricular catheterisation and thus is not broadly implementable in a clinical environment. Hence, there is a significant gap created by the current clinical inability to directly measure or estimate the TVE curve every beat.
This paper presents a novel, minimally invasive method for deriving the TVE curve beat-by-beat. The method focuses on combining simple physiological assumptions with clinically available catheter waveforms to individually simulate the pressure and volume components that define the TVE curve, rather than generating the TVE curve itself. Importantly, this approach avoids the need to directly assume a shape for the TVE curve, and is thus better equipped to capture variations in this shape over time and condition, as well as the corresponding alterations in intra-beat cardiac behaviour.
Clinically feasible measurements mean the method has the potential for real-time implementation at the patient bedside, without requiring additional, invasive instrumentation. Such a TVE curve could be used to provide additional, patient specific information on intra-beat behaviour and inter-beat variation in the functioning of the heart. Finally, monitoring the driver's evolution over time could add significant new data and insight to improve diagnosis and treatment.
The TVE curve is defined:
$$e(t) = \frac{{P_{lv} \left( t \right)}}{{V_{lv} \left( t \right) - V_{d} }}$$
where P lv is the pressure in the left ventricle, V lv is the volume in the left ventricle and V d is the 'dead space' volume in the ventricle [6, 23]. Thus, the TVE curve is defined by two waveforms (P lv and V lv ) and one constant (V d ). These values can be measured directly, but doing so is not clinically feasible [24].
The proposed method approximates these waveforms (V lv , P lv ) and constant (V d ) using three inputs, as shown in Fig. 1. These inputs include continuously sampled aortic pressure waveforms (P ao ) and heart rate (HR), data which is typically available in a modern ICU. The final input required is baseline end-systolic (V es ) and end-diastolic (V ed ) Volume, obtained from a brief echocardiography reading, which is increasingly available in a clinical setting [25].
Flowchart of the proposed method. Roman numerals indicate different method regions
The overall goal of this method is to use the clinically available inputs P ao , HR and baseline V es and V ed to determine the outputs P lv , V d and V lv , and thus the TVE curve e(t) as set out in Fig. 1 and Eq. 1. This resulting TVE curve can be experimentally validated against the TVE curves derived from experimentally measured P lv and V lv in animal models. The availability of a nearby continuous pressure measurement (P ao ) forms an effective basis for the continuous approximation of P lv , and the availability of a baseline volume measurement (V es ) forms an effective basis for the approximation of baseline V d . However, there is no continuous volume measurement available for continuous approximation of Vlv, resulting in this process being considerably more involved. Thus, the shaded central Region III in Fig. 1 is considerably more complicated than Regions I and II.
While the overall method involves approximating two output waveforms (V lv , P lv ) from a single input waveform (P ao ), it's important to note that all three waveforms (P lv , V lv , P ao ) have distinct features. Different regions of behaviour are governed by different physiological phenomenon, and these waveforms have been extensively characterised [26]. As such, all three waveforms are heavily interconnected and information rich, making this task more reasonable than it might first appear.
Determining P lv from P ao (Region I, Fig. 1)
The left ventricle is situated directly upstream from the aorta, separated by the aortic valve. This valve is open during systole and closed during diastole. As such, if aortic valve resistance is neglected, P lv is equivalent to P ao , with a slight phase lag (δ) during the majority of systole (Section P. 1, Fig. 2). While aortic valve resistance is non-negligible in conditions such as aortic stenosis [26], valve dysfunction of any type is present in only 3.61% of CVD mortalities in the US [27]. Further, such conditions are typically chronic in nature, relatively easily diagnosed [28] and evolve slowly, while the method is designed to monitor short term changes in an ICU environment. As such, this assumption should not significantly affect the methods' applicability to the vast majority of the target cohort.
Simulating left ventricular pressure. Note P ao has been shifted left by δ to account for phase lag
During diastole the aortic valve is closed and little information is available from the aortic waveform about ventricular behaviour. However, the ventricle behaves in a largely passive manner in this region, meaning the TVE curve is typically near zero during diastole [26]. As such, a generic function consisting of two exponentials was used to approximate left ventricular pressure during diastole. In early diastole (Section P. 2, Fig. 2) an exponential decay to a fixed baseline pressure captures ventricular relaxation. In late diastole-early systole (Section P. 3, Fig. 2), an exponential increase captures the beginning of ventricular contraction [26].
While atrial contraction contributes significantly to late diastolic filling, ventricular elastance remains largely passive in late diastole [26]. Thus the TVE curve is typically at its baseline value until the beginning of ventricular contraction in early systole. While atrial function affects the magnitude of the driver function, which is normalised in this work, and may affect its shape, this effect is indirect as the driver function represents the impact of contraction in driving pulsatile blood from the heart to the arterial system. Further, with the increasing unpopularity of pulmonary artery catheters [29] none of the typically available instrumentation in an ICU provides a clear picture of atrial behaviour. As such, while the exponential in section P. 3 is broadly intended to capture ventricular filling, no specific atrial behaviour component is integrated into this model.
Using Fig. 2, the left ventricular pressure for the n th heartbeat is thus defined using P ao :
$$t_{1} = t\left( {\frac{{dP_{ao} }}{dt}_{max} } \right)_{n}$$
$$t_{2} = t\left( {\frac{{dP_{ao} }}{dt}_{min } } \right)_{n}$$
$$t_{4} = t\left( {\frac{{dP_{ao} }}{dt}_{max } } \right)_{n + 1}$$
$$t_{3} = 0.62t_{2} + 0.38t_{4}$$
(2d)
$$P_{lv} (t) = \left\{ {\begin{array}{*{20}l} {P_{ao} \left( {t_{1} + \delta < t < t_{2} + \delta } \right)} \hfill & {t_{1} < t < t_{2} } \hfill \\ {6 + \left( {P_{ao} \left( {t_{2} } \right) - 6} \right)e^{{ - 17.5\left( {t - t_{2} } \right)}} } \hfill & {t_{2} < t < t_{3} } \hfill \\ {P_{lv} \left( {t_{3} } \right) + \left( {P_{ao} \left( {t_{4} } \right) - P_{lv} \left( {t_{3} } \right)} \right)e^{{37.5\left( {t - t_{4} } \right)}} } \hfill & {t_{3} < t < t_{4} } \hfill \\ \end{array} } \right\}$$
where: \(\delta = 0.008{\text{s}}\)
Determining V d from baseline V es (Region II, Fig. 1)
A recent method has been developed to approximate V d from ventricular volume measurements [30]. This approach relies on linear regression of the Frank-Starling curve (SV–V ed ) and its end-systolic equivalent (SV–V es ) to the point where SV = 0 and 'the ventricle cannot develop any systolic pressure', the definition of V d [23]. This work also showed V d for a baseline, healthy pig be an approximately fixed percentage of V es [30]. Defining V d as a percentage of baseline V es allows approximation of baseline V d during the initial echocardiographic reading where measured V es is available. While V d has been shown to change with condition, there is no practical means of capturing this change short of additional echocardiography measurements. Thus, while intermittent measures are feasible, in this study V d is fixed at a baseline value.
$$V_{d} = 0.48 \times V_{es}$$
Determining V lv from P ao , HR and V d (Region III, Fig. 1)
Unlike pressure, little volume or flow information is readily available from the typical, clinically available instrumentation. Simulating V lv is thus more challenging. The shape of the ventricular volume waveform was approximated by a piecewise sine wave consisting of two sections: systole (Section V. 1, Fig. 3) and diastole (Section V. 2, Fig. 3), with a 90° phase shift at the beginning of systole. The underlying physiological behaviour might be better represented by a series of exponentials [26]. However, using sine waves achieves a similar result with considerably fewer variables involved.
Simulating left ventricular volume
Thus, six points per heartbeat (t 1, t 2, t 3 and (V ed ) n , (V es ) n , (V ed ) n+1 ) are required to define the ventricular volume waveform. The timing associated with systole start (t 1), systole end (t 2), and diastole end (t 3) are readily determined from the aortic pressure waveform (Fig. 3):
$$t_{1} = t\left( {P_{ao_{min}} } \right)_{n}$$
$$t_{2} = t\left( {P_{DN} } \right)_{n}$$
$$t_{3} = t\left( {P_{ao_{min}} } \right)_{n + 1}$$
Using existing work [31] SV can be approximated beat-to-beat using the aortic waveform. Thus, only one of V es or V ed is required, as SV can be used to convert between the two. The ESPVR allows for determination of V es , and is defined [23]:
$$P_{es} = E_{es} \times \left( {V_{es} - V_{0} } \right)$$
where E es is the end-systolic elastance and V 0 is the ventricular volume at zero pressure. Equation 6 can be rewritten:
$$P_{DN} = E_{es} \times \left( {V_{es} - V_{d} } \right)$$
where this change is justified by:
The pressures in the ventricle and aorta are roughly equivalent until the aortic valve closes, thus P DN is close to P es
V d and V 0 have similar, but distinct, physiological significance and values. The two are often used interchangeably [17, 18, 23]
Finally, it is necessary to account for E es , which changes in response to a number of factors including contractility [6], and loading conditions [32, 33]. Thus, Eq. 7 is modified:
$$P_{DN} = \left( {E_{c} \times HR^{3} } \right) \times \left( {V_{es} - V_{d} } \right)$$
here E es is defined as a function of HR and a coefficient (E C ), with a cubic selected as it provides the best compromise between simplicity and effective tracking for the data set presented here. In particular, the cardiovascular system responds to most changes in conditions in a number of ways, including changes in heart rate and elastance. As heart rate is easily measured, it provides an easy to obtain, if incomplete, indication of cardiovascular system response, which can be used to inform an approximated elastance [34]. Further supporting evidence is provided in the validation and discussion of results.
During the echocardiography calibration, measurements for P DN , HR and V es are available [35]. Thus, using Eq. 8, a constant value for E C can be defined, allowing approximation of E es and thus determination of V es on a beat-by-beat basis. The beat-to-beat ventricular volume can thus be determined:
$$V_{lv} (t) = \left\{ {\begin{array}{*{20}l} {\left( {V_{ed} } \right)_{n} + \left( {\left( {V_{es} } \right)_{n} - \left( {V_{ed} } \right)_{n} } \right)\sin \left( {\frac{{\pi \left( {t - t_{1} } \right)}}{{2\left( {t_{2} - t_{1} } \right)}}} \right)} \hfill & {t_{1} < t < t_{2} } \hfill \\ {\left( {V_{es} } \right)_{n} - \left( {\left( {V_{ed} } \right)_{n + 1} - \left( {V_{es} } \right)_{n} } \right)\left( {\frac{1}{2}\cos \left( {\frac{{\pi \left( {t - t_{2} } \right)}}{{\left( {t_{3} - t_{2} } \right)}}} \right) - \frac{1}{2}} \right)} \hfill & {t_{2} < t < t_{3} } \hfill \\ \end{array} } \right\}$$
$$V_{es} = \frac{{P_{DN} }}{{\left( {E_{c} \times HR^{3} } \right)}} + V_{d}$$
$$V_{ed} = V_{es} + SV$$
Summary of proposed method
The overall derivation of the TVE curve can be summarised:
Initially or intermittently
Calculate V d using Eq. 4 and baseline V es (Region I, Fig. 1)
Calculate E C using Eq. 8, P DN , HR and baseline V es (Region III, Fig. 1)
Every heartbeat
Simulate P lv using Eq. 2, Eq. 3 and P ao (Region II, Fig. 1)
Determine V es using Eq. 10, P DN , HR and E C (Region III, Fig. 1)
Determine SV using [31] and P ao (Region III, Fig. 1)
Determine V ed using Eq. 11, V es and SV (Region III, Fig. 1)
Simulate V lv using Eq. 5, Eq. 9, P ao , V es and V ed (Region III, Fig. 1)
Calculate and normalise the TVE curve e(t) using Eq. 1
Analysis and validation
The proposed method was validated on experimentally gathered data. A range of input (P ao ) and output (V lv , P lv ) waveforms were continuously measured via catheter. This data allowed validation of individual model assumptions through comparison with directly measured output waveforms, as well as validation of the overall method through comparisons between the TVE curve as calculated using simulated and directly measured V lv and P lv wave forms.
The data set encompasses 46,318 heartbeats across 5 Piétrain pigs. A diverse clinical protocol provides the ability to assess intra- and inter-subject variability across a large amount of invasive and non-invasive measurements. Together they enable rigorous assessment and validation of the method.
Experimental procedure
The experimental protocol was approved by the Ethics Commission for the Use of Animals at the University of Liège, Belgium. Five male, pure Piétrain pigs weighing between 18.5 and 29 kg were sedated, anaesthetised and mechanically ventilated (GE Engstrom CareStation) with a baseline positive end-expiratory pressure (PEEP) of 5 cmH2O (Fig. 4, Additional file 1: Table S1). Proximal aortic pressure was continually sampled using a pressure catheter (Transonic, NY, USA) with a sampling rate of 250 Hz. To provide direct measurements of P lv and V lv for validation, the heart was accessed via a median sternotomy, and an admittance pressure–volume catheter (Transonic, NY, USA) with a sampling rate of 250 Hz inserted into the left ventricle via an apical stab [36, 37].
The fully instrumented Piétrain Pig. The catheters for measuring P ao , P lv and V lv are positioned to the left of the image
To demonstrate a diverse range of cardiac states, several procedures were performed:
A single infusion of endotoxin (lipopolysaccharide from E. Coli, 0.5 mg/kg injected over 30 min) to induce septic shock. Septic shock drives a change in afterload conditions and is associated with a large variety of effects including an inflammatory response and capillary leakage that may lead to hypovolemia, decreased cardiac output, decreased ejection fraction and cardiac failure [38].
Several PEEP driven recruitment manoeuvres (RMs), both pre- and post-endotoxin infusion. RMs drive a change in preload conditions and are typically associated with a decrease in mean blood pressure and cardiac output [39].
One to four infusions of 500 mL saline solution over 30 min, pre- and post-endotoxin infusion, simulating fluid resuscitation therapy, a key component of hemodynamic resuscitation in patients with severe sepsis, which itself results in a change in circulatory volume [40].
Validation of significant model assumptions
Two major assumptions made in deriving this model are:
That V d can be expressed as a function of baseline V es , as in Eq. 4
That E es can be expressed as a function of HR, as in Eq. 8
Direct evaluation of the tracking of V es using different forms of ESPVR allows validation of both of these assumptions. In particular, 3 different methods of tracking V es were compared:
Fixed E es and neglected V 0 : The standard ESPVR (Eq. 6) with V 0 = 0 (a commonly used assumption [17, 18, 23])
Fixed E es and fixed V 0 : The standard ESPVR (Eq. 7) with V 0 = V d (allowing assessment of the validity of Eq. 4)
Dynamic E es and fixed V 0 : The ESPVR as used in the proposed method (Eq. 8) with V 0 = V d , and E es as a function of HR (allowing assessment of the validity of Eq. 8)
Validation of overall model
The overall method presented here is designed to simulate the TVE curve beat-by-beat, without requiring invasive instrumentation of the heart or real-time image-based monitoring, neither of which is clinically or ethically feasible in care. As such, validation of the method relies on comparison of the simulated TVE curve to the invasively measured, 'true' TVE curve, which is calculated using the catheter measured V lv and P lv waveforms for a single beat. This comparison is achieved by calculating the absolute and signed 'error area' between the measured and simulated TVE curve, according to Eqs. 12 and 13:
$$\varepsilon_{abs} = \frac{{\mathop \smallint \nolimits_{t = 0}^{1} \left| {e_{sim} (t) - e_{meas} (t)} \right|}}{{\mathop \smallint \nolimits_{t = 0}^{1} \left( {e_{meas} (t)} \right)}}$$
$$\varepsilon_{\text{sgn}} = \frac{{\mathop \smallint \nolimits_{t = 0}^{1} \left( {e_{sim} (t) - e_{meas} (t)} \right)}}{{\mathop \smallint \nolimits_{t = 0}^{1} \left( {e_{meas} (t)} \right)}}$$
where ε sim and ε meas are the simulated and measured TVE curves respectively, t is normalised time set to 1 for every heart beat to enable comparison over different beats, and ε abs and ε sgn and denote the absolute and signed errors respectively. An example TVE curve with an absolute error of 7.8% and bias of −3.4% is shown in Fig. 5, where shading denotes the error area.
Example TVE curve error. This simulated driver has an error of 7.8% and bias of −3.4%
Table 1 shows the percentage error associated with the 3 methods specified to track V es . Moving from neglected V 0 (Method 1) to fixed V 0 (Method 2), via Eq. 4, shows a modest reduction in median and 25th percentile errors (15.9–14.5% and 3.1–1.3%, respectively), but a highly significant reduction in 75th percentile error (43.2–25.5%). Introduction of dynamic E es (Method 3) as opposed to fixed E es (Method 2), via Eq. 8, shows a very significant reduction in median and 75th percentile errors (14.5–4.3% and 25.5–15.0%, respectively). These results support the validity and usefulness of Eqs. 4 and 8, their associated assumptions in the tracking of V es , and thus the use of Method 3 in the overall approximation of the TVE curve.
Table 1 Absolute percentage error associated with different methods of tracking Ves
Note the values quoted from the final row of Table 1 are an average of the 25th percentile, median and 75th percentile error for the pigs rather than an overall 25th percentile, median and 75th percentile error. The decision was made to represent population values in this way as the overall median is less representative than the average median due a differing number of heartbeats recorded for different pigs.
The comparative behaviour of these 3 methods is well illustrated in Fig. 6, which shows the tracking of V es for Pig 1. Method 1 agrees reasonably well with measured V es during normal behaviour, but diverges significantly from measured V es during recruitment manoeuvres and the onset of severe sepsis, where there are large vertical changes in V es . Method 2 accurately captures these large vertical changes in V es , reducing the large 75th percentile error in Method 1. Method 3 retains the tracking of large vertical changes in Method 2, and tracks normal behaviour more effectively by accounting for changes in E es , significantly reducing median error.
Tracking of V es . Comparison between the three methods in 2.2.2 for Pig 1
The overall purpose of this model is to track the shape of the TVE curve, and how this shape changes when circulatory behaviour changes. Table 2 shows the area under the curve errors, ϵ abs and ϵ sgn , for all 5 pigs. The median error is relatively small at 11.4%, suggesting the method is effective, and the interquartile range relatively narrow at 9.2–14.7%, suggesting the method is consistent. The bias is also small at −2.5%, with an interquartile range of −6.1 to 0.8%, suggesting only a slight method bias.
Table 2 TVE curve percentage area under the curve errors (ε abs and ε sgn ) associated with proposed method (identifying V es and V ed )
Table 3 shows, as a point of comparison, the area under the curve errors associated with the method if the directly measured values for V es and V ed are used. Using these measured values removes the majority of Region III, Fig. 1 (Eqs. 4 and 6–11) from the overall method, where a lot of relatively significant assumptions are made. The errors in Table 3 are very comparable to those in Table 2, with an overall modest reduction in average median error (11.4–10.2%) and bias (−2.5 to −2.2%), as expected when measuring V es and V ed directly and invasively rather than estimating them. Hence, the assumptions made had very little impact on error when removed.
Table 3 TVE curve percentage area under the curve error (ε abs and ε sgn ) associated with using measured V es and V ed values
Figure 6 shows a variety of measured and modelled TVE curve shapes for each of the 5 pigs, capturing both intra- and inter-subject variability. For each figure, Panel I shows a baseline waveform, Panel II a waveform during a RM and Panel III a waveform after endotoxin infusion has occurred and sepsis develops. The range of shapes and inter-pig variability indicated that assuming a generic population or cohort TVE curve is not particularly valid, and is further supported by the range of intra-pig variability evidenced. There are a variety of shoulder heights, relative gradient and maximum/minimum gradient timings that result in TVE curves with distinct shapes, variations which are well captured by the method. These TVE curves were also selected to demonstrate a range of error values similar to the interquartile error range for that pig (Table 2). Thus, these drivers are reasonably representative of the overall ability of the method to capture TVE curve shapes, despite different overall shapes for each pig and changes in shape as condition changes.
Validation of significant model assumptions (Table 1)
A direct evaluation of the effectiveness of three different ESPVR equations set out in 2.2.2 in tracking V es allowed validation of the assumptions that:
That V d can be derived from V es as in Eq. 4 (implemented in Method 2)
That E es can be expressed as a function of HR as in Eq. 8 (implemented in Method 3)
Method 1 serves as a control and uses a simplified ESPVR, assuming V 0 = 0 and E es is constant. This method tracks V es reasonably well, yielding an overall median error of 15.9% across all 5 pigs compared to V es as directly measured. The assumption that V 0 can be neglected is often used due to a combination of V 0 being difficult to measure directly, as it requires a significant artificial reduction in ventricular pressure [41], and V 0 typically being relatively small [42]. This assumption is largely supported by these results, as a 15.9% median error seems acceptable when weighed against the type of highly invasive and involved protocol traditionally required to determine V 0.
However, as shown in Fig. 6, this 15.5% median error fails to capture the extremely large inaccuracies associated with using Method 1 to track sudden changes in V es due, for example, to recruitment manoeuvres or the onset of severe sepsis. This failure to track sudden changes leads to a significantly larger 75th percentile error of 43.2% for Method 1. This high error when sudden changes occur is of concern in an ICU or cardiac surgery clinical scenario, where sudden changes and accurate, rapid determination of patient responses to these sudden changes is extremely important [26].
Method 2 introduces the assumption that V d can be derived from baseline V es , as in Eq. 4, and that V d can be used as a surrogate for V 0. There is a minor, but important physiological distinction between the two values: V 0 is the ventricular volume at 0 pressure, while V d is the volume at which the ventricle cannot develop any systolic pressure [23]. However, the purpose of both terms is to account for the subject-specific inactive volume within the ventricle, and the two values have been shown to be similar for a given subject [23].
Method 2 results in a notable decrease in 75th percentile error compared to Method 1 (43.2–25.5%), and is able to track sudden changes in V es significantly more effectively, as shown in Fig. 6. However, Method 2 yields only a modest reduction in median error compared to Method 1 (15.9–14.5%), as both methods fail to capture the dynamic nature of E es . Thus, large errors are reduced, but overall accuracy is not greatly improved. Regardless, these results provide support for the validity of Eq. 4, and the use of V d as a surrogate for V 0 here. The fact that these reductions in error are sustained over significant changes in cardiac output and ejection fraction as sepsis develops suggests that the absolute value of V d does not change significantly enough under such conditions to detract from method accuracy.
Method 3 further assumes that E es can be expressed as a function of HR. This assumption is unusual, but is supported by the results. The full method sees a further, significant reduction in 75th percentile error compared to Methods 1 and 2 (43.2% and 25.5–15.0%, Table 1), and, most importantly, a very large reduction in median error compared to both Methods 1 and 2 (15.9% and 14.5–4.3%, Table 1). This result suggests general tracking of trends in V es is being significantly improved, also supporting the validity of expressing E es as a function of HR. This behaviour can also be observed in Fig. 6. This result, combined with the minimal addition in method complexity required to include HR, which is very easy to measure, provides a strong case for the use of Method 3.
However, it is still important to note that the relationship between E es and HR expressed in Eq. 8 is a significant simplification of actual cardiac behaviour. The cardiac system uses a large variety of responses to maintain cardiac output. The cubic used to approximate E es changes as a function of HR attempts to mathematically approximate the sympathetic nature of some of these responses, but, inevitably, the relationship between HR and E es varies between subjects (accounted for by calibration), and as time and condition changes. While the cubic approximation of this relationship has been demonstrated to remain effective across the full progression of septic shock in the data presented here, further validation of this relationship, and especially the use of a cubic, across other conditions is desired.
Simulating the TVE curve using the proposed method resulted in a relatively low median absolute error area, ranging from 9.4 to 13.4% (Table 2) across all pigs. This narrow range of median absolute errors implies the method is able to consistently and effectively capture inter-subject variations in TVE curve behaviour, suggesting it is generalizable to other subjects. This inter-subject variability is demonstrated well by various baseline drivers shown in Fig. 7. All 5 pigs demonstrate relatively distinct baseline TVE curve shapes, showing a considerable level of inter-subject variability impossible to capture with a generic cohort or population based TVE curve that relies on a basic assumed shape [17, 18].
The simulated TVE curve also had a relatively narrow average interquartile error range of 9.2–14.7% (Table 2), suggesting the method is able to relatively consistently and effectively capture inter-subject variations in TVE curve behaviour. This intra-subject variability, and inter-quartile error range, is illustrated in Fig. 7. Accurate capture of relative changes and trends in patient behaviour is extremely important, as these changes are critical in assessing whether a patient is recovering, responding to treatment, or is in need of a change in treatment. Most pigs demonstrated notable changes and intra-pig variability in driver shape during recruitment manoeuvres and as sepsis developed. A couple of the more unusual driver shapes were not fully captured, for example Panel 2 of Pig 4 in Fig. 7, where the driver measured driver displayed two peaks and the simulated driver only one.
Example TVE curves for each pig. A range of error values and cardiac states are shown
The simulated TVE curve demonstrated consistently low signed area error, at −2.5% (IQR −6.1 to 0.8%), shown in Table 2. This result shows the method only slightly underestimating the TVE curve, supported further by 2 pigs having a slight positive bias and the other 3 a slight negative bias. This outcome further supports the ability of the TVE curve to accurately capture both intra- and inter-subject variability over time and condition.
In assessing the impact of assumptions on the TVE curve, a comparison of Table 2, using approximated V es and V ed , and Table 3, using measured V es and V ed , show very similar error values. For example, overall median error fell from 11.4 to 10.2% and overall bias from −2.5 to −2.2%, only a modest reduction in error. This implies that the body of assumptions and equations in Region III, Fig. 1 concerned with the approximating the V es and V ed for simulation of V lv do not result in a large increase in error compared to using measured V es and V ed . The assumptions made in areas of the method not involving simulating V es and V ed are relatively minimal, mostly involving using P ao to determine waveform timing. As such, it would seem that much of the error associated with this method is the result of the necessity of assuming equations for parts the two waveforms being reconstructed (P lv and V lv ), as shown in Figs. 2 and 3. Thus, the assumptions employed in Region III, Fig. 1 appear to function as intended, and further reduction in error would probably require increased method complexity or an increase in the clinically available data.
An important point to consider is the fact that the TVE curve is consistently normalised to a duration and amplitude of 1.0, as it is designed specifically as an indicator of how the heart is behaving relatively over the course of a beat, to be coupled with a lumped metric (E es ) indicating the overall strength of that heartbeat [6, 17, 18]. This normalisation does mean that some of the errors associated with the various assumptions and approximations made throughout the method are negated, and that indicators of absolute cardiac work and its changes, for example, are not able to be directly extracted from the TVE curve created. Fortunately, the driver does not exist alone, the intent is that the shape of the TVE curve, indicative of transient, relative cardiac behaviour, be used alongside other existing metrics, such as Cardiac Output or Stroke Volume [31, 43], indicative of lumped, absolute cardiac behaviour, to provide further diagnostic information.
There are study limitations that should be considered. First, all data presented is derived from a single protocol involving a single, but complex and varied [38], condition (sepsis). This data set encompasses several pigs, a full progression from healthy, baseline behaviour to cardiac failure and clinically standard ventilation and fluid interventions. Nevertheless, there is a much larger range of possible cardiac conditions, and further validation over several of these would be beneficial. For example, the method would benefit from validation on contractility altering drugs such as dobutamine [44], which may alter the behaviour of the elastance term in Eq. 8. However, the method already detects changes in haemodynamics, including those due to circulatory or cardiac muscle changes during sepsis in this study. Thus, the ability to detect changes due to inotropes should be similar to what is presented for the range of behaviours already observed in sepsis. Further, given inotrope infusions are determined and performed by a clinician, it would be possible to recalibrate the method directly after such an infusion to adjust to the new inotropic state if it proved necessary. Overall, the underlying physiology and data supporting the development of this method has been discussed in detail, and would be expected to generalise well to a wider range of conditions, as there are no intervention or condition specific assumptions made.
The method also requires validation on human subjects to ensure the methodology as presented here remains physiologically accurate, though the strong similarities between porcine and human physiology and the effectiveness of porcine models are well established [45, 46]. Equally, only an animal model, as used here, allows the direct validation against cardiac measured PV loops, which would not be possible in humans. Thus, only an animal trial allows this important first validation.
The method does require an initial calibration via echocardiography or similar means. Echocardiography equipment is increasingly available in modern ICUs [25]. Further, echocardiography is non-invasive and the calibration period required is relatively short, requiring approximately 10 heartbeats. However, the requirement of such a calibration still prevents the method from being fully implementable without modest additional clinical workload using normal ICU instrumentation.
The TVE curve is an important, but difficult to clinically measure, expression of internal cardiac dynamics that captures the heart's ability as a pump and can evolve over time, condition and patients. A novel, minimally invasive method for deriving the TVE curve beat-by-beat, by combining simple physiological assumptions with readily available catheter waveforms to individually simulate the components of the TVE curve, is proposed. This method was assessed across a cohort of 5 Piétrain pigs undergoing a progression from healthy behaviour to cardiac failure due to sepsis. The TVE curve generated by the method was shown to effectively track a directly measured function throughout the experiments, with low overall median absolute (11.4%) and signed (−2.5%) area under the curve errors. There is the potential for this method to provide real time, patient specific information on intra-beat behaviour and inter-beat variation in the heart, at the patient bedside, without requiring additional, invasive instrumentation.
CVD:
cardiovascular disease and dysfunction
ESPVR:
end-systolic pressure–volume relation
pressure–volume
PEEP:
positive end-expiratory pressure
RM:
recruitment manoeuvre
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SD was responsible for the conception and design of the study, analysis and interpretation of data, and drafting the manuscript; CP and JCG were responsible for the conception and design of the study and revising the manuscript; SK and JB were responsible for analysis and interpretation of data and revising the manuscript; AP and TD were responsible for the acquisition of data and revising the manuscript. All authors read and approved the final manuscript.
All experimental procedures and protocols used in this investigation were reviewed and approved by the Institutional Animal Care and Use Ethics Committee of the University of Liège, Belgium (Reference Number 14-1726). Their guidelines conform completely with the Guide for the Care and Use of Laboratory Animals published by the US National Institutes of Health (NIH Publication No. 85-23, revised 1996), as well as EU DIRECTIVE 2010/63/EU on the protection of animals used for scientific purposes.
This work was supported by the Engineering Technology-based Innovation in Medicine (eTIME) consortium grant [eTIME 318943]; the EU FP7 International Research Staff Exchange Scheme (IRSES) grant [#PIRSES-GA-2012-318943]; the Fonds de la Recherche Scientifique (F.R.S) grant; and the University of Canterbury 'Canterbury Scholarship' grant. These funding bodies had no role in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript.
Department of Mechanical Engineering, University of Canterbury, Christchurch, New Zealand
Shaun Davidson, Chris Pretty, Shun Kamoi, Joel Balmer & J. Geoffrey Chase
GIGA-Cardiovascular Sciences, University of Liège, Liège, Belgium
Antoine Pironet & Thomas Desaive
Shaun Davidson
Chris Pretty
Shun Kamoi
Joel Balmer
J. Geoffrey Chase
Correspondence to Shaun Davidson.
Additional file 1: Table S1. Supplementary Subject Specific Experimental Information.
Davidson, S., Pretty, C., Pironet, A. et al. Minimally invasive, patient specific, beat-by-beat estimation of left ventricular time varying elastance. BioMed Eng OnLine 16, 42 (2017). https://doi.org/10.1186/s12938-017-0338-7
Time varying elastance
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CommonCrawl
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Discrete uniform distribution
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen".
discrete uniform
Probability mass function
n = 5 where n = b − a + 1
Cumulative distribution function
Notation ${\mathcal {U}}\{a,b\}$ or $\mathrm {unif} \{a,b\}$
Parameters $a,b$ integers with $b\geq a$
$n=b-a+1$
Support $k\in \{a,a+1,\dots ,b-1,b\}$
PMF ${\frac {1}{n}}$
CDF ${\frac {\lfloor k\rfloor -a+1}{n}}$
Mean ${\frac {a+b}{2}}$
Median ${\frac {a+b}{2}}$
Mode N/A
Variance ${\frac {n^{2}-1}{12}}$
Skewness $0$
Ex. kurtosis $-{\frac {6(n^{2}+1)}{5(n^{2}-1)}}$
Entropy $\ln(n)$
MGF ${\frac {e^{at}-e^{(b+1)t}}{n(1-e^{t})}}$
CF ${\frac {e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}}$
PGF ${\frac {z^{a}-z^{b+1}}{n(1-z)}}$
A simple example of the discrete uniform distribution is throwing a fair die. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a uniform spanning tree is a spanning tree generated uniformly from the spanning trees of a given graph.
The discrete uniform distribution itself is inherently non-parametric. It is convenient, however, to represent its values generally by all integers in an interval [a,b], so that a and b become the main parameters of the distribution (often one simply considers the interval [1,n] with the single parameter n). With these conventions, the cumulative distribution function (CDF) of the discrete uniform distribution can be expressed, for any k ∈ [a,b], as
$F(k;a,b)={\frac {\lfloor k\rfloor -a+1}{b-a+1}}$
Estimation of maximum
Main article: German tank problem
This example is described by saying that a sample of k observations is obtained from a uniform distribution on the integers $1,2,\dotsc ,N$, with the problem being to estimate the unknown maximum N. This problem is commonly known as the German tank problem, following the application of maximum estimation to estimates of German tank production during World War II.
The uniformly minimum variance unbiased (UMVU) estimator for the maximum is given by
${\hat {N}}={\frac {k+1}{k}}m-1=m+{\frac {m}{k}}-1$
where m is the sample maximum and k is the sample size, sampling without replacement.[1] This can be seen as a very simple case of maximum spacing estimation.
This has a variance of[1]
${\frac {1}{k}}{\frac {(N-k)(N+1)}{(k+2)}}\approx {\frac {N^{2}}{k^{2}}}{\text{ for small samples }}k\ll N$
so a standard deviation of approximately ${\tfrac {N}{k}}$, the (population) average size of a gap between samples; compare ${\tfrac {m}{k}}$ above.
The sample maximum is the maximum likelihood estimator for the population maximum, but, as discussed above, it is biased.
If samples are not numbered but are recognizable or markable, one can instead estimate population size via the capture-recapture method.
Random permutation
See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation.
Properties
The family of uniform distributions over ranges of integers (with one or both bounds unknown) has a finite-dimensional sufficient statistic, namely the triple of the sample maximum, sample minimum, and sample size, but is not an exponential family of distributions, because the support varies with the parameters. For families whose support does not depend on the parameters, the Pitman–Koopman–Darmois theorem states that only exponential families have a sufficient statistic whose dimension is bounded as sample size increases. The uniform distribution is thus a simple example showing the limit of this theorem.
See also
• Dirac delta distribution
• Continuous uniform distribution
References
1. Johnson, Roger (1994), "Estimating the Size of a Population", Teaching Statistics, 16 (2 (Summer)): 50–52, CiteSeerX 10.1.1.385.5463, doi:10.1111/j.1467-9639.1994.tb00688.x
Probability distributions (list)
Discrete
univariate
with finite
support
• Benford
• Bernoulli
• beta-binomial
• binomial
• categorical
• hypergeometric
• negative
• Poisson binomial
• Rademacher
• soliton
• discrete uniform
• Zipf
• Zipf–Mandelbrot
with infinite
support
• beta negative binomial
• Borel
• Conway–Maxwell–Poisson
• discrete phase-type
• Delaporte
• extended negative binomial
• Flory–Schulz
• Gauss–Kuzmin
• geometric
• logarithmic
• mixed Poisson
• negative binomial
• Panjer
• parabolic fractal
• Poisson
• Skellam
• Yule–Simon
• zeta
Continuous
univariate
supported on a
bounded interval
• arcsine
• ARGUS
• Balding–Nichols
• Bates
• beta
• beta rectangular
• continuous Bernoulli
• Irwin–Hall
• Kumaraswamy
• logit-normal
• noncentral beta
• PERT
• raised cosine
• reciprocal
• triangular
• U-quadratic
• uniform
• Wigner semicircle
supported on a
semi-infinite
interval
• Benini
• Benktander 1st kind
• Benktander 2nd kind
• beta prime
• Burr
• chi
• chi-squared
• noncentral
• inverse
• scaled
• Dagum
• Davis
• Erlang
• hyper
• exponential
• hyperexponential
• hypoexponential
• logarithmic
• F
• noncentral
• folded normal
• Fréchet
• gamma
• generalized
• inverse
• gamma/Gompertz
• Gompertz
• shifted
• half-logistic
• half-normal
• Hotelling's T-squared
• inverse Gaussian
• generalized
• Kolmogorov
• Lévy
• log-Cauchy
• log-Laplace
• log-logistic
• log-normal
• log-t
• Lomax
• matrix-exponential
• Maxwell–Boltzmann
• Maxwell–Jüttner
• Mittag-Leffler
• Nakagami
• Pareto
• phase-type
• Poly-Weibull
• Rayleigh
• relativistic Breit–Wigner
• Rice
• truncated normal
• type-2 Gumbel
• Weibull
• discrete
• Wilks's lambda
supported
on the whole
real line
• Cauchy
• exponential power
• Fisher's z
• Kaniadakis κ-Gaussian
• Gaussian q
• generalized normal
• generalized hyperbolic
• geometric stable
• Gumbel
• Holtsmark
• hyperbolic secant
• Johnson's SU
• Landau
• Laplace
• asymmetric
• logistic
• noncentral t
• normal (Gaussian)
• normal-inverse Gaussian
• skew normal
• slash
• stable
• Student's t
• Tracy–Widom
• variance-gamma
• Voigt
with support
whose type varies
• generalized chi-squared
• generalized extreme value
• generalized Pareto
• Marchenko–Pastur
• Kaniadakis κ-exponential
• Kaniadakis κ-Gamma
• Kaniadakis κ-Weibull
• Kaniadakis κ-Logistic
• Kaniadakis κ-Erlang
• q-exponential
• q-Gaussian
• q-Weibull
• shifted log-logistic
• Tukey lambda
Mixed
univariate
continuous-
discrete
• Rectified Gaussian
Multivariate
(joint)
• Discrete:
• Ewens
• multinomial
• Dirichlet
• negative
• Continuous:
• Dirichlet
• generalized
• multivariate Laplace
• multivariate normal
• multivariate stable
• multivariate t
• normal-gamma
• inverse
• Matrix-valued:
• LKJ
• matrix normal
• matrix t
• matrix gamma
• inverse
• Wishart
• normal
• inverse
• normal-inverse
• complex
Directional
Univariate (circular) directional
Circular uniform
univariate von Mises
wrapped normal
wrapped Cauchy
wrapped exponential
wrapped asymmetric Laplace
wrapped Lévy
Bivariate (spherical)
Kent
Bivariate (toroidal)
bivariate von Mises
Multivariate
von Mises–Fisher
Bingham
Degenerate
and singular
Degenerate
Dirac delta function
Singular
Cantor
Families
• Circular
• compound Poisson
• elliptical
• exponential
• natural exponential
• location–scale
• maximum entropy
• mixture
• Pearson
• Tweedie
• wrapped
• Category
• Commons
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A Decade of Dual Pairing Vector Spaces
★Invited talk
Tatsuaki Okamoto
Efficient Attribute-Based Signatures for Unbounded Arithmetic Branching Programs Abstract
Pratish Datta Tatsuaki Okamoto Katsuyuki Takashima
This paper presents the first attribute-based signature (ABS) scheme in which the correspondence between signers and signatures is captured in an arithmetic model of computation. Specifically, we design a fully secure, i.e., adaptively unforgeable and perfectly signer-private ABS scheme for signing policies realizable by arithmetic branching programs (ABP), which are a quite expressive model of arithmetic computations. On a more positive note, the proposed scheme places no bound on the size and input length of the supported signing policy ABP's, and at the same time, supports the use of an input attribute for an arbitrary number of times inside a signing policy ABP, i.e., the so called unbounded multi-use of attributes. The size of our public parameters is constant with respect to the sizes of the signing attribute vectors and signing policies available in the system. The construction is built in (asymmetric) bilinear groups of prime order, and its unforgeability is derived in the standard model under (asymmetric version of) the well-studied decisional linear (DLIN) assumption coupled with the existence of standard collision resistant hash functions. Due to the use of the arithmetic model as opposed to the boolean one, our ABS scheme not only excels significantly over the existing state-of-the-art constructions in terms of concrete efficiency, but also achieves improved applicability in various practical scenarios. Our principal technical contributions are (a) extending and refining the techniques of Okamoto and Takashima [PKC 2011, PKC 2013], which were originally developed in the context of boolean span programs, to the arithmetic setting; and (b) innovating new ideas to allow unbounded multi-use of attributes inside ABP's, which themselves are of unbounded size and input length.
Decentralizing Inner-Product Functional Encryption Abstract
Michel Abdalla Fabrice Benhamouda Markulf Kohlweiss Hendrik Waldner
Multi-client functional encryption (MCFE) is a more flexible variant of functional encryption whose functional decryption involves multiple ciphertexts from different parties. Each party holds a different secret key and can independently and adaptively be corrupted by the adversary. We present two compilers for MCFE schemes for the inner-product functionality, both of which support encryption labels. Our first compiler transforms any scheme with a special key-derivation property into a decentralized scheme, as defined by Chotard et al. (ASIACRYPT 2018), thus allowing for a simple distributed way of generating functional decryption keys without a trusted party. Our second compiler allows to lift an unnatural restriction present in existing (decentralized) MCFE schemes, which requires the adversary to ask for a ciphertext from each party. We apply our compilers to the works of Abdalla et al. (CRYPTO 2018) and Chotard et al. (ASIACRYPT 2018) to obtain schemes with hitherto unachieved properties. From Abdalla et al., we obtain instantiations of DMCFE schemes in the standard model (from DDH, Paillier, or LWE) but without labels. From Chotard et al., we obtain a DMCFE scheme with labels still in the random oracle model, but without pairings.
Non-zero Inner Product Encryption Schemes from Various Assumptions: LWE, DDH and DCR Abstract
Shuichi Katsumata Shota Yamada
In non-zero inner product encryption (NIPE) schemes, ciphertexts and secret keys are associated with vectors and decryption is possible whenever the inner product of these vectors does not equal zero. So far, much effort on constructing bilinear map-based NIPE schemes have been made and this has lead to many efficient schemes. However, the constructions of NIPE schemes without bilinear maps are much less investigated. The only known other NIPE constructions are based on lattices, however, they are all highly inefficient due to the need of converting inner product operations into circuits or branching programs.To remedy our rather poor understanding regarding NIPE schemes without bilinear maps, we provide two methods for constructing NIPE schemes: a direct construction from lattices and a generic construction from schemes for inner products (LinFE). For our first direct construction, it highly departs from the traditional lattice-based constructions and we rely heavily on new tools concerning Gaussian measures over multi-dimensional lattices to prove security. For our second generic construction, using the recent constructions of LinFE schemes as building blocks, we obtain the first NIPE constructions based on the DDH and DCR assumptions. In particular, we obtain the first NIPE schemes without bilinear maps or lattices.
Efficient Invisible and Unlinkable Sanitizable Signatures Abstract
Xavier Bultel Pascal Lafourcade Russell W. F. Lai Giulio Malavolta Dominique Schröder Sri Aravinda Krishnan Thyagarajan
Sanitizable signatures allow designated parties (the sanitizers) to apply arbitrary modifications to some restricted parts of signed messages. A secure scheme should not only be unforgeable, but also protect privacy and hold both the signer and the sanitizer accountable. Two important security properties that are seemingly difficult to achieve simultaneously and efficiently are invisibility and unlinkability. While invisibility ensures that the admissible modifications are hidden from external parties, unlinkability says that sanitized signatures cannot be linked to their sources. Achieving both properties simultaneously is crucial for applications where sensitive personal data is signed with respect to data-dependent admissible modifications. The existence of an efficient construction achieving both properties was recently posed as an open question by Camenisch et al. (PKC'17). In this work, we propose a solution to this problem with a two-step construction. First, we construct (non-accountable) invisible and unlinkable sanitizable signatures from signatures on equivalence classes and other basic primitives. Second, we put forth a generic transformation using verifiable ring signatures to turn any non-accountable sanitizable signature into an accountable one while preserving all other properties. When instantiating in the generic group and random oracle model, the efficiency of our construction is comparable to that of prior constructions, while providing stronger security guarantees.
Function Private Predicate Encryption for Low Min-Entropy Predicates Abstract
Sikhar Patranabis Debdeep Mukhopadhyay Somindu C. Ramanna
In this work, we propose new constructions for zero inner-product encryption (ZIPE) and non-zero inner-product encryption (NIPE) from prime-order bilinear pairings, which are both attribute and function private in the public-key setting. Our ZIPE scheme is adaptively attribute private under the standard Matrix DDH assumption for unbounded collusions. It is additionally computationally function private under a min-entropy variant of the Matrix DDH assumption for predicates sampled from distributions with super-logarithmic min-entropy. Existing (statistically) function private ZIPE schemes due to Boneh et al. [Crypto'13, Asiacrypt'13] necessarily require predicate distributions with significantly larger min-entropy in the public-key setting.Our NIPE scheme is adaptively attribute private under the standard Matrix DDH assumption, albeit for bounded collusions. In addition, it achieves computational function privacy under a min-entropy variant of the Matrix DDH assumption for predicates sampled from distributions with super-logarithmic min-entropy. To the best of our knowledge, existing NIPE schemes from bilinear pairings were neither attribute private nor function private. Our constructions are inspired by the linear FE constructions of Agrawal et al. [Crypto'16] and the simulation secure ZIPE of Wee [TCC'17]. In our ZIPE scheme, we show a novel way of embedding two different hard problem instances in a single secret key - one for unbounded collusion-resistance and the other for function privacy. For NIPE, we introduce new techniques for simultaneously achieving attribute and function privacy. We further show that the two constructions naturally generalize to a wider class of predicate encryption schemes such as subspace membership, subspace non-membership and hidden-vector encryption.
Group Signatures with Selective Linkability Abstract
Lydia Garms Anja Lehmann
Group signatures allow members of a group to anonymously produce signatures on behalf of the group. They are an important building block for privacy-enhancing applications, e.g., enabling user data to be collected in authenticated form while preserving the user's privacy. The linkability between the signatures thereby plays a crucial role for balancing utility and privacy: knowing the correlation of events significantly increases the utility of the data but also severely harms the user's privacy. Therefore group signatures are unlinkable per default, but either support linking or identity escrow through a dedicated central party or offer user-controlled linkability. However, both approaches have significant limitations. The former relies on a fully trusted entity and reveals too much information, and the latter requires exact knowledge of the needed linkability at the moment when the signatures are created. However, often the exact purpose of the data might not be clear at the point of data collection. In fact, data collectors tend to gather large amounts of data at first, but will need linkability only for selected, small subsets of the data. We introduce a new type of group signature that provides a more flexible and privacy-friendly access to such selective linkability. When created, all signatures are fully unlinkable. Only when strictly needed or desired, should the required pieces be made linkable with the help of a central entity. For privacy, this linkability is established in an oblivious and non-transitive manner. We formally define the requirements for this new type of group signatures and provide an efficient instantiation that provably satisfies these requirements under discrete-logarithm based assumptions.
Let a Non-barking Watchdog Bite: Cliptographic Signatures with an Offline Watchdog Abstract
Sherman S. M. Chow Alexander Russell Qiang Tang Moti Yung Yongjun Zhao Hong-Sheng Zhou
We study how to construct secure digital signature schemes in the presence of kleptographic attacks. Our work utilizes an offline watchdog to clip the power of subversions via only one-time black-box testing of the implementation. Previous results essentially rely on an online watchdog which requires the collection of all communicating transcripts (or active re-randomization of messages).We first give a simple but generic construction, without random oracles, in the partial-subversion model in which key generation and signing algorithms can be subverted. Then, we give the first digital signature scheme in the complete-subversion model in which all cryptographic algorithms can be subverted. This construction is based on the full-domain hash. Along the way, we enhance the recent result of Russell et al. (CRYPTO 2018) about correcting a subverted random oracle.
Adaptively Single-Key Secure Constrained PRFs for $\mathrm {NC}^1$ Abstract
Nuttapong Attrapadung Takahiro Matsuda Ryo Nishimaki Shota Yamada Takashi Yamakawa
We present a construction of an adaptively single-key secure constrained PRF (CPRF) for $$\mathbf {NC}^1$$ assuming the existence of indistinguishability obfuscation (IO) and the subgroup hiding assumption over a (pairing-free) composite order group. This is the first construction of such a CPRF in the standard model without relying on a complexity leveraging argument.To achieve this, we first introduce the notion of partitionable CPRF, which is a CPRF accommodated with partitioning techniques and combine it with shadow copy techniques often used in the dual system encryption methodology. We present a construction of partitionable CPRF for $$\mathbf {NC}^1$$ based on IO and the subgroup hiding assumption over a (pairing-free) group. We finally prove that an adaptively single-key secure CPRF for $$\mathbf {NC}^1$$ can be obtained from a partitionable CPRF for $$\mathbf {NC}^1$$ and IO.
Obfuscating Simple Functionalities from Knowledge Assumptions Abstract
Ward Beullens Hoeteck Wee
This paper shows how to obfuscate several simple functionalities from a new Knowledge of OrthogonALity Assumption (KOALA) in cyclic groups which is shown to hold in the Generic Group Model. Specifically, we give simpler and stronger security proofs for obfuscation schemes for point functions, general-output point functions and pattern matching with wildcards. We also revisit the work of Bishop et al. (CRYPTO 2018) on obfuscating the pattern matching with wildcards functionality. We improve upon the construction and the analysis in several ways:attacks and stronger guarantees: We show that the construction achieves virtual black-box security for a simulator that runs in time roughly $$2^{n/2}$$, as well as distributional security for larger classes of distributions. We give attacks that show that our results are tight.weaker assumptions: We prove security under KOALA.better efficiency: We also provide a construction that outputs $$n+1$$ instead of 2n group elements. We obtain our results by first obfuscating a simpler "big subset functionality", for which we establish full virtual black-box security; this yields a simpler and more modular analysis for pattern matching. Finally, we extend our distinguishing attacks to a large class of simple linear-in-the-exponent schemes.
Zero-Knowledge Elementary Databases with More Expressive Queries Abstract
Benoît Libert Khoa Nguyen Benjamin Hong Meng Tan Huaxiong Wang
Zero-knowledge elementary databases (ZK-EDBs) are cryptographic schemes that allow a prover to commit to a set $$\mathsf {D}$$ of key-value pairs so as to be able to prove statements such as "x belongs to the support of $$\mathsf {D}$$ and $$\mathsf {D}(x)=y$$" or "x is not in the support of $$\mathsf {D}$$". Importantly, proofs should leak no information beyond the proven statement and even the size of $$\mathsf {D}$$ should remain private. Chase et al. (Eurocrypt'05) showed that ZK-EDBs are implied by a special flavor of non-interactive commitment, called mercurial commitment, which enables efficient instantiations based on standard number theoretic assumptions. On the other hand, the resulting ZK-EDBs are only known to support proofs for simple statements like (non-)membership and value assignments. In this paper, we show that mercurial commitments actually enable significantly richer queries. We show that, modulo an additional security property met by all known efficient constructions, they actually enable range queries over keys and values – even for ranges of super-polynomial size – as well as membership/non-membership queries over the space of values. Beyond that, we exploit the range queries to realize richer queries such as $$k$$-nearest neighbors and revealing the $$k$$ smallest or largest records within a given range. In addition, we provide a new realization of trapdoor mercurial commitment from standard lattice assumptions, thus obtaining the most expressive quantum-safe ZK-EDB construction so far.
Efficient Non-Interactive Zero-Knowledge Proofs in Cross-Domains Without Trusted Setup Abstract
Michael Backes Lucjan Hanzlik Amir Herzberg Aniket Kate Ivan Pryvalov
With the recent emergence of efficient zero-knowledge (ZK) proofs for general circuits, while efficient zero-knowledge proofs of algebraic statements have existed for decades, a natural challenge arose to combine algebraic and non-algebraic statements. Chase et al. (CRYPTO 2016) proposed an interactive ZK proof system for this cross-domain problem. As a use case they show that their system can be used to prove knowledge of a RSA/DSA signature on a message m with respect to a publicly known Pedersen commitment $$g^m h^r$$. One drawback of their system is that it requires interaction between the prover and the verifier. This is due to the interactive nature of garbled circuits, which are used in their construction. Subsequently, Agrawal et al. (CRYPTO 2018) proposed an efficient non-interactive ZK (NIZK) proof system for cross-domains based on SNARKs, which however require a trusted setup assumption.In this paper, we propose a NIZK proof system for cross-domains that requires no trusted setup and is efficient both for the prover and the verifier. Our system constitutes a combination of Schnorr based ZK proofs and ZK proofs for general circuits by Giacomelli et al. (USENIX 2016). The proof size and the running time of our system are comparable to the approach by Chase et al. Compared to Bulletproofs (SP 2018), a recent NIZK proofs system on committed inputs, our techniques achieve asymptotically better performance on prover and verifier, thus presenting a different trade-off between the proof size and the running time.
What About Bob? The Inadequacy of CPA Security for Proxy Reencryption Abstract
Aloni Cohen
In the simplest setting of proxy reencryption, there are three parties: Alice, Bob, and Polly (the proxy). Alice keeps some encrypted data that she can decrypt with a secret key known only to her. She wants to communicate the data to Bob, but not to Polly (nor anybody else). Using proxy reencryption, Alice can create a reencryption key that will enable Polly to reencrypt the data for Bob's use, but which will not help Polly learn anything about the data.There are two well-studied notions of security for proxy reencryption schemes: security under chosen-plaintext attacks (CPA) and security under chosen-ciphertext attacks (CCA). Both definitions aim to formalize the security that Alice enjoys against both Polly and Bob.In this work, we demonstrate that CPA security guarantees much less security against Bob than was previously understood. In particular, CPA security does not prevent Bob from learning Alice's secret key after receiving a single honestly reencrypted ciphertext. As a result, CPA security provides scant guarantees in common applications.We propose security under honest reencryption attacks (HRA), a strengthening of CPA security that better captures the goals of proxy reencryption. In applications, HRA security provides much more robust security. We identify a property of proxy reencryption schemes that suffices to amplify CPA security to HRA security and show that two existing proxy reencryption schemes are in fact HRA secure.
Sub-logarithmic Distributed Oblivious RAM with Small Block Size Abstract
Eyal Kushilevitz Tamer Mour
Oblivious RAM (ORAM) is a cryptographic primitive that allows a client to securely execute RAM programs over data that is stored in an untrusted server. Distributed Oblivious RAM is a variant of ORAM, where the data is stored in $$m>1$$ servers. Extensive research over the last few decades have succeeded to reduce the bandwidth overhead of ORAM schemes, both in the single-server and the multi-server setting, from $$O(\sqrt{N})$$ to O(1). However, all known protocols that achieve a sub-logarithmic overhead either require heavy server-side computation (e.g. homomorphic encryption), or a large block size of at least $$\varOmega (\log ^3 N)$$.In this paper, we present a family of distributed ORAM constructions that follow the hierarchical approach of Goldreich and Ostrovsky [17]. We enhance known techniques, and develop new ones, to take better advantage of the existence of multiple servers. By plugging efficient known hashing schemes in our constructions, we get the following results:1.For any number $$m\ge 2$$ of servers, we show an m-server ORAM scheme with $$O(\log N/\log \log N)$$ overhead, and block size $$\varOmega (\log ^2 N)$$. This scheme is private even against an $$(m-1)$$-server collusion.2.A three-server ORAM construction with $$O(\omega (1)\cdot \log N/\log \log N)$$ overhead and a block size almost logarithmic, i.e. $$\varOmega (\log ^{1+\epsilon }N)$$. We also investigate a model where the servers are allowed to perform a linear amount of light local computations, and show that constant overhead is achievable in this model, through a simple four-server ORAM protocol. From theoretical viewpoint, this is the first ORAM scheme with asymptotic constant overhead, and polylogarithmic block size, that does not use homomorphic encryption. Practically speaking, although we do not provide an implementation of the suggested construction, evidence from related work (e.g. [12]) confirms that despite the linear computational overhead, our construction is practical, in particular when applied to secure computation.
Collusion Resistant Broadcast and Trace from Positional Witness Encryption Abstract
Rishab Goyal Satyanarayana Vusirikala Brent Waters
An emerging trend is for researchers to identify cryptography primitives for which feasibility was first established under obfuscation and then move the realization to a different setting. In this work we explore a new such avenue—to move obfuscation-based cryptography to the assumption of (positional) witness encryption. Our goal is to develop techniques and tools, which we will dub "witness encryption friendly" primitives and use these to develop a methodology for building advanced cryptography from positional witness encryption.We take a bottom up approach and pursue our general agenda by attacking the specific problem of building collusion-resistant broadcast systems with tracing from positional witness encryption. We achieve a system where the size of ciphertexts, public key and private key are polynomial in the security parameter $$\lambda $$ and independent of the number of users N in the broadcast system. Currently, systems with such parameters are only known from indistinguishability obfuscation.
Shorter Quadratic QA-NIZK Proofs Abstract
Vanesa Daza Alonso González Zaira Pindado Carla Ràfols Javier Silva
Despite recent advances in the area of pairing-friendly Non-Interactive Zero-Knowledge proofs, there have not been many efficiency improvements in constructing arguments of satisfiability of quadratic (and larger degree) equations since the publication of the Groth-Sahai proof system (JoC'12). In this work, we address the problem of aggregating such proofs using techniques derived from the interactive setting and recent constructions of SNARKs. For certain types of quadratic equations, this problem was investigated before by González et al. (ASIACRYPT'15). Compared to their result, we reduce the proof size by approximately 50% and the common reference string from quadratic to linear, at the price of using less standard computational assumptions. A theoretical motivation for our work is to investigate how efficient NIZK proofs based on falsifiable assumptions can be. On the practical side, quadratic equations appear naturally in several cryptographic schemes like shuffle and range arguments.
Adaptively Secure Proxy Re-encryption Abstract
Georg Fuchsbauer Chethan Kamath Karen Klein Krzysztof Pietrzak
A proxy re-encryption (PRE) scheme is a public-key encryption scheme that allows the holder of a key pk to derive a re-encryption key for any other key $$pk'$$ . This re-encryption key lets anyone transform ciphertexts under pk into ciphertexts under $$pk'$$ without having to know the underlying message, while transformations from $$pk'$$ to pk should not be possible (unidirectional). Security is defined in a multi-user setting against an adversary that gets the users' public keys and can ask for re-encryption keys and can corrupt users by requesting their secret keys. Any ciphertext that the adversary cannot trivially decrypt given the obtained secret and re-encryption keys should be secure.All existing security proofs for PRE only show selective security, where the adversary must first declare the users it wants to corrupt. This can be lifted to more meaningful adaptive security by guessing the set of corrupted users among the n users, which loses a factor exponential in , rendering the result meaningless already for moderate .Jafargholi et al. (CRYPTO'17) proposed a framework that in some cases allows to give adaptive security proofs for schemes which were previously only known to be selectively secure, while avoiding the exponential loss that results from guessing the adaptive choices made by an adversary. We apply their framework to PREs that satisfy some natural additional properties. Concretely, we give a more fine-grained reduction for several unidirectional PREs, proving adaptive security at a much smaller loss. The loss depends on the graph of users whose edges represent the re-encryption keys queried by the adversary. For trees and chains the loss is quasi-polynomial in the size and for general graphs it is exponential in their depth and indegree (instead of their size as for previous reductions). Fortunately, trees and low-depth graphs cover many, if not most, interesting applications.Our results apply e.g. to the bilinear-map based PRE schemes by Ateniese et al. (NDSS'05 and CT-RSA'09), Gentry's FHE-based scheme (STOC'09) and the LWE-based scheme by Chandran et al. (PKC'14).
Break-glass Encryption Abstract
Alessandra Scafuro
"Break-glass" is a term used in IT healthcare systems to denote an emergency access to private information without having the credentials to do so.In this paper we introduce the concept of break-glass encryption for cloud storage, where the security of the ciphertexts – stored on a cloud – can be violated exactly once, for emergency circumstances, in a way that is detectable and without relying on a trusted party.Detectability is the crucial property here: if a cloud breaks glass without permission from the legitimate user, the latter should detect it and have a proof of such violation. However, if the break-glass procedure is invoked by the legitimate user, then semantic security must still hold and the cloud will learn nothing. Distinguishing that a break-glass is requested by the legitimate party is also challenging in absence of secrets.In this paper, we provide a formalization of break-glass encryption and a secure instantiation using hardware tokens. Our construction aims to be a feasibility result and is admittedly impractical. Whether hardware tokens are necessary to achieve this security notion and whether more practical solutions can be devised are interesting open questions.
Lossy Algebraic Filters with Short Tags Abstract
Benoît Libert Chen Qian
Lossy algebraic filters (LAFs) are function families where each function is parametrized by a tag, which determines if the function is injective or lossy. While initially introduced by Hofheinz (Eurocrypt 2013) as a technical tool to build encryption schemes with key-dependent message chosen-ciphertext (KDM-CCA) security, they also find applications in the design of robustly reusable fuzzy extractors. So far, the only known LAF family requires tags comprised of $$\varTheta (n^2)$$ group elements for functions with input space $$\mathbb {Z}_p^n$$, where p is the group order. In this paper, we describe a new LAF family where the tag size is only linear in n and prove it secure under simple assumptions in asymmetric bilinear groups. Our construction can be used as a drop-in replacement in all applications of the initial LAF system. In particular, it can shorten the ciphertexts of Hofheinz's KDM-CCA-secure public-key encryption scheme by 19 group elements. It also allows substantial space improvements in a recent fuzzy extractor proposed by Wen and Liu (Asiacrypt 2018). As a second contribution, we show how to modify our scheme so as to prove it (almost) tightly secure, meaning that security reductions are not affected by a concrete security loss proportional to the number of adversarial queries.
Short Discrete Log Proofs for FHE and Ring-LWE Ciphertexts Abstract
Rafael del Pino Vadim Lyubashevsky Gregor Seiler
In applications of fully-homomorphic encryption (FHE) that involve computation on encryptions produced by several users, it is important that each user proves that her input is indeed well-formed. This may simply mean that the inputs are valid FHE ciphertexts or, more generally, that the plaintexts m additionally satisfy $$f(m)=1$$ for some public function f. The most efficient FHE schemes are based on the hardness of the Ring-LWE problem and so a natural solution would be to use lattice-based zero-knowledge proofs for proving properties about the ciphertext. Such methods, however, require larger-than-necessary parameters and result in rather long proofs, especially when proving general relationships.In this paper, we show that one can get much shorter proofs (roughly 1.25 KB) by first creating a Pedersen commitment from the vector corresponding to the randomness and plaintext of the FHE ciphertext. To prove validity of the ciphertext, one can then prove that this commitment is indeed to the message and randomness and these values are in the correct range. Our protocol utilizes a connection between polynomial operations in the lattice scheme and inner product proofs for Pedersen commitments of Bünz et al. (S&P 2018). Furthermore, our proof of equality between the ciphertext and the commitment is very amenable to amortization – proving the equivalence of k ciphertext/commitment pairs only requires an additive factor of $$O(\log {k})$$ extra space than for one such proof. For proving additional properties of the plaintext(s), one can then directly use the logarithmic-space proofs of Bootle et al. (Eurocrypt 2016) and Bünz et al. (IEEE S&P 2018) for proving arbitrary relations of discrete log commitment.Our technique is not restricted to FHE ciphertexts and can be applied to proving many other relations that arise in lattice-based cryptography. For example, we can create very efficient verifiable encryption/decryption schemes with short proofs in which confidentiality is based on the hardness of Ring-LWE while the soundness is based on the discrete logarithm problem. While such proofs are not fully post-quantum, they are adequate in scenarios where secrecy needs to be future-proofed, but one only needs to be convinced of the validity of the proof in the pre-quantum era. We furthermore show that our zero-knowledge protocol can be easily modified to have the property that breaking soundness implies solving discrete log in a short amount of time. Since building quantum computers capable of solving discrete logarithm in seconds requires overcoming many more fundamental challenges, such proofs may even remain valid in the post-quantum era.
Generic Constructions of Robustly Reusable Fuzzy Extractor Abstract
Yunhua Wen Shengli Liu Dawu Gu
Robustly reusable Fuzzy Extractor (rrFE) considers reusability and robustness simultaneously. We present two approaches to the generic construction of rrFE. Both of approaches make use of a secure sketch and universal hash functions. The first approach also employs a special pseudo-random function (PRF), namely unique-input key-shift (ui-ks) secure PRF, and the second uses a key-shift secure auxiliary-input authenticated encryption (AIAE). The ui-ks security of PRF (resp. key-shift security of AIAE), together with the homomorphic properties of secure sketch and universal hash function, guarantees the reusability and robustness of rrFE. Meanwhile, we show two instantiations of the two approaches respectively. The first instantiation results in the first rrFE from the LWE assumption, while the second instantiation results in the first rrFE from the DDH assumption over non-pairing groups.
Publicly Verifiable Proofs from Blockchains Abstract
Alessandra Scafuro Luisa Siniscalchi Ivan Visconti
A proof system is publicly verifiable, if anyone, by looking at the transcript of the proof, can be convinced that the corresponding theorem is true. Public verifiability is important in many applications since it allows to compute a proof only once while convincing an unlimited number of verifiers.Popular interactive proof systems (e.g., $$\varSigma $$-protocols) protect the witness through various properties (e.g., witness indistinguishability (WI) and zero knowledge (ZK)) but typically they are not publicly verifiable since such proofs are convincing only for those verifiers who contributed to the transcripts of the proofs. The only known proof systems that are publicly verifiable rely on a non-interactive (NI) prover, through trust assumptions (e.g., NIZK in the CRS model), heuristic assumptions (e.g., NIZK in the random oracle model), specific number-theoretic assumptions on bilinear groups or relying on obfuscation assumptions (obtaining NIWI with no setups).In this work we construct publicly verifiable witness-indistinguishable proof systems from any $$\varSigma $$-protocol, based only on the existence of a very generic blockchain. The novelty of our approach is in enforcing a non-interactive verification (thus guaranteeing public verifiability) while allowing the prover to be interactive and talk to the blockchain (this allows us to circumvent the need of strong assumptions and setups). This opens interesting directions for the design of cryptographic protocols leveraging on blockchain technology.
Safety in Numbers: On the Need for Robust Diffie-Hellman Parameter Validation Abstract
Steven D. Galbraith Jake Massimo Kenneth G. Paterson
We consider the problem of constructing Diffie-Hellman (DH) parameters which pass standard approaches to parameter validation but for which the Discrete Logarithm Problem (DLP) is relatively easy to solve. We consider both the finite field setting and the elliptic curve setting.For finite fields, we show how to construct DH parameters (p, q, g) for the safe prime setting in which $$p=2q+1$$ is prime, q is relatively smooth but fools random-base Miller-Rabin primality testing with some reasonable probability, and g is of order q mod p. The construction involves modifying and combining known methods for obtaining Carmichael numbers. Concretely, we provide an example with 1024-bit p which passes OpenSSL's Diffie-Hellman validation procedure with probability $$2^{-24}$$ (for versions of OpenSSL prior to 1.1.0i). Here, the largest factor of q has 121 bits, meaning that the DLP can be solved with about $$2^{64}$$ effort using the Pohlig-Hellman algorithm. We go on to explain how this parameter set can be used to mount offline dictionary attacks against PAKE protocols. In the elliptic curve case, we use an algorithm of Bröker and Stevenhagen to construct an elliptic curve E over a finite field $${\mathbb {F}}_p$$ having a specified number of points n. We are able to select n of the form $$h\cdot q$$ such that h is a small co-factor, q is relatively smooth but fools random-base Miller-Rabin primality testing with some reasonable probability, and E has a point of order q. Concretely, we provide example curves at the 128-bit security level with $$h=1$$ , where q passes a single random-base Miller-Rabin primality test with probability 1/4 and where the elliptic curve DLP can be solved with about $$2^{44}$$ effort. Alternatively, we can pass the test with probability 1/8 and solve the elliptic curve DLP with about $$2^{35.5}$$ effort. These ECDH parameter sets lead to similar attacks on PAKE protocols relying on elliptic curves.Our work shows the importance of performing proper (EC)DH parameter validation in cryptographic implementations and/or the wisdom of relying on standardised parameter sets of known provenance.
Identity-Based Broadcast Encryption with Efficient Revocation Abstract
Aijun Ge Puwen Wei
Identity-based broadcast encryption (IBBE) is an effective method to protect the data security and privacy in multi-receiver scenarios, which can make broadcast encryption more practical. This paper further expands the study of scalable revocation methodology in the setting of IBBE, where a key authority releases a key update material periodically in such a way that only non-revoked users can update their decryption keys. Following the binary tree data structure approach, a concrete instantiation of revocable IBBE scheme is proposed using asymmetric pairings of prime order bilinear groups. Moreover, this scheme can withstand decryption key exposure, which is proven to be semi-adaptively secure under chosen plaintext attacks in the standard model by reduction to static complexity assumptions. In particular, the proposed scheme is very efficient both in terms of computation costs and communication bandwidth, as the ciphertext size is constant, regardless of the number of recipients. To demonstrate the practicality, it is further implemented in Charm, a framework for rapid prototyping of cryptographic primitives.
Hunting and Gathering – Verifiable Random Functions from Standard Assumptions with Short Proofs Abstract
Lisa Kohl
A verifiable random function (VRF) is a pseudorandom function, where outputs can be publicly verified. That is, given an output value together with a proof, one can check that the function was indeed correctly evaluated on the corresponding input. At the same time, the output of the function is computationally indistinguishable from random for all non-queried inputs.We present the first construction of a VRF which meets the following properties at once: It supports an exponential-sized input space, it achieves full adaptive security based on a non-interactive constant-size assumption and its proofs consist of only a logarithmic number of group elements for inputs of arbitrary polynomial length.Our construction can be instantiated in symmetric bilinear groups with security based on the decision linear assumption. We build on the work of Hofheinz and Jager (TCC 2016), who were the first to construct a verifiable random function with security based on a non-interactive constant-size assumption. Basically, their VRF is a matrix product in the exponent, where each matrix is chosen according to one bit of the input. In order to allow verification given a symmetric bilinear map, a proof consists of all intermediary results. This entails a proof size of $$\varOmega (L)$$ group elements, where L is the bit-length of the input.Our key technique, which we call hunting and gathering, allows us to break this barrier by rearranging the function, which – combined with the partitioning techniques of Bitansky (TCC 2017) – results in a proof size of $$\ell $$ group elements for arbitrary $$\ell \in \omega (1)$$.
Tightly Secure Hierarchical Identity-Based Encryption Abstract
Roman Langrehr Jiaxin Pan
We construct the first tightly secure hierarchical identity-based encryption (HIBE) scheme based on standard assumptions, which solves an open problem from Blazy, Kiltz, and Pan (CRYPTO 2014). At the core of our constructions is a novel randomization technique that enables us to randomize user secret keys for identities with flexible length.The security reductions of previous HIBEs lose at least a factor of $$ Q $$, which is the number of user secret key queries. Different to that, the security loss of our schemes is only dependent on the security parameter. Our schemes are adaptively secure based on the Matrix Diffie-Hellman assumption, which is a generalization of standard Diffie-Hellman assumptions such as $$k$$-Linear. We have two tightly secure constructions, one with constant ciphertext size, and the other with tighter security at the cost of linear ciphertext size. Among other things, our schemes imply the first tightly secure identity-based signature scheme by a variant of the Naor transformation.
Lattice-Based Revocable (Hierarchical) IBE with Decryption Key Exposure Resistance Abstract
Shuichi Katsumata Takahiro Matsuda Atsushi Takayasu
Revocable identity-based encryption (RIBE) is an extension of IBE that supports a key revocation mechanism, which is an indispensable feature for practical cryptographic schemes. Due to this extra feature, RIBE is often required to satisfy a strong security notion unique to the revocation setting called decryption key exposure resistance (DKER). Additionally, hierarchal IBE (HIBE) is another orthogonal extension of IBE that supports key delegation functionalities allowing for scalable deployments of cryptographic schemes. So far, R(H)IBE constructions with DKER are only known from bilinear maps, where all constructions rely heavily on the so-called key re-randomization property to achieve the DKER and/or hierarchal feature. Since lattice-based schemes seem to be inherently ill-fit with the key re-randomization property, no construction of lattice-based R(H)IBE schemes with DKER are known.In this paper, we propose the first lattice-based RHIBE scheme with DKER without relying on the key re-randomization property, departing from all the previously known methods. We start our work by providing a generic construction of RIBE schemes with DKER, which uses as building blocks any two-level standard HIBE scheme and (weak) RIBE scheme without DKER. Based on previous lattice-based RIBE constructions without DKER, our result implies the first lattice-based RIBE scheme with DKER. Then, building on top of our generic construction, we construct the first lattice-based RHIBE scheme with DKER, by further exploiting the algebraic structure of lattices. To this end, we prepare a new tool called the level conversion keys, which enables us to achieve the hierarchal feature without relying on the key re-randomization property.
Leakage-Resilient Identity-Based Encryption in Bounded Retrieval Model with Nearly Optimal Leakage-Ratio Abstract
Ryo Nishimaki Takashi Yamakawa
We propose new constructions of leakage-resilient public-key encryption (PKE) and identity-based encryption (IBE) schemes in the bounded retrieval model (BRM). In the BRM, adversaries are allowed to obtain at most $$\ell $$ -bit leakage from a secret key and we can increase $$\ell $$ only by increasing the size of secret keys without losing efficiency in any other performance measure. We call $$\ell /|\mathsf {sk}|$$ leakage-ratio where $$|\mathsf {sk}|$$ denotes a bit-length of a secret key. Several PKE/IBE schemes in the BRM are known. However, none of these constructions achieve a constant leakage-ratio under a standard assumption in the standard model. Our PKE/IBE schemes are the first schemes in the BRM that achieve leakage-ratio $$1-\epsilon $$ for any constant $$\epsilon >0$$ under standard assumptions in the standard model.As previous works, we use identity-based hash proof systems (IB-HPS) to construct IBE schemes in the BRM. It is known that a parameter for IB-HPS called the universality-ratio is translated into the leakage-ratio of the resulting IBE scheme in the BRM. We construct an IB-HPS with universality-ratio $$1-\epsilon $$ for any constant $$\epsilon >0$$ based on any inner-product predicate encryption (IPE) scheme with compact secret keys. Such IPE schemes exist under the d-linear, subgroup decision, learning with errors, or computational bilinear Diffie-Hellman assumptions. As a result, we obtain IBE schemes in the BRM with leakage-ratio $$1-\epsilon $$ under any of these assumptions. Our PKE schemes are immediately obtained from our IBE schemes.
Towards Non-Interactive Zero-Knowledge for NP from LWE Abstract
Ron D. Rothblum Adam Sealfon Katerina Sotiraki
Non-interactive zero-knowledge ( $$\mathsf {NIZK}$$ ) is a fundamental primitive that is widely used in the construction of cryptographic schemes and protocols. Despite this, general purpose constructions of $$\mathsf {NIZK}$$ proof systems are only known under a rather limited set of assumptions that are either number-theoretic (and can be broken by a quantum computer) or are not sufficiently well understood, such as obfuscation. Thus, a basic question that has drawn much attention is whether it is possible to construct general-purpose $$\mathsf {NIZK}$$ proof systems based on the learning with errors ( $$\mathsf {LWE}$$ ) assumption.Our main result is a reduction from constructing $$\mathsf {NIZK}$$ proof systems for all of $$\mathbf {NP}$$ based on $$\mathsf {LWE}$$ , to constructing a $$\mathsf {NIZK}$$ proof system for a particular computational problem on lattices, namely a decisional variant of the Bounded Distance Decoding ( $$\mathsf {BDD}$$ ) problem. That is, we show that assuming $$\mathsf {LWE}$$ , every language $$L \in \mathbf {NP}$$ has a $$\mathsf {NIZK}$$ proof system if (and only if) the decisional $$\mathsf {BDD}$$ problem has a $$\mathsf {NIZK}$$ proof system. This (almost) confirms a conjecture of Peikert and Vaikuntanathan (CRYPTO, 2008).To construct our $$\mathsf {NIZK}$$ proof system, we introduce a new notion that we call prover-assisted oblivious ciphertext sampling ( $$\mathsf {POCS}$$ ), which we believe to be of independent interest. This notion extends the idea of oblivious ciphertext sampling, which allows one to sample ciphertexts without knowing the underlying plaintext. Specifically, we augment the oblivious ciphertext sampler with access to an (untrusted) prover to help it accomplish this task. We show that the existence of encryption schemes with a $$\mathsf {POCS}$$ procedure, as well as some additional natural requirements, suffices for obtaining $$\mathsf {NIZK}$$ proofs for $$\mathbf {NP}$$ . We further show that such encryption schemes can be instantiated based on $$\mathsf {LWE}$$ , assuming the existence of a $$\mathsf {NIZK}$$ proof system for the decisional $$\mathsf {BDD}$$ problem.
Additively Homomorphic IBE from Higher Residuosity Abstract
Michael Clear Ciaran McGoldrick
We present an identity-Based encryption (IBE) scheme that is group homomorphic for addition modulo a "large" (i.e. superpolynomial) integer, the first such group homomorphic IBE. Our first result is the construction of an IBE scheme supporting homomorphic addition modulo a poly-sized prime e. Our construction builds upon the IBE scheme of Boneh, LaVigne and Sabin (BLS). BLS relies on a hash function that maps identities to $$e^{\text {th}}$$ residues. However there is no known way to securely instantiate such a function. Our construction extends BLS so that it can use a hash function that can be securely instantiated. We prove our scheme secure under the (slightly modified) $$e^{\text {th}}$$ residuosity assumption in the random oracle model and show that it supports a (modular) additive homomorphism. By using multiple instances of the scheme with distinct primes and leveraging the Chinese Remainder Theorem, we can support homomorphic addition modulo a "large" (i.e. superpolynomial) integer. We also show that our scheme for $$e > 2$$ is anonymous by additionally assuming the hardness of deciding solvability of a special system of multivariate polynomial equations. We provide a justification for this assumption by considering known attacks.
More Efficient Algorithms for the NTRU Key Generation Using the Field Norm Abstract
Thomas Pornin Thomas Prest
NTRU lattices [13] are a class of polynomial rings which allow for compact and efficient representations of the lattice basis, thereby offering very good performance characteristics for the asymmetric algorithms that use them. Signature algorithms based on NTRU lattices have fast signature generation and verification, and relatively small signatures, public keys and private keys.A few lattice-based cryptographic schemes entail, generally during the key generation, solving the NTRU equation: $$\begin{aligned} f G - g F = q \mod x^n + 1 \end{aligned}$$Here f and g are fixed, the goal is to compute solutions F and G to the equation, and all the polynomials are in $${\mathbb {Z}}[x]/(x^n + 1)$$. The existing methods for solving this equation are quite cumbersome: their time and space complexities are at least cubic and quadratic in the dimension n, and for typical parameters they therefore require several megabytes of RAM and take more than a second on a typical laptop, precluding onboard key generation in embedded systems such as smart cards.In this work, we present two new algorithms for solving the NTRU equation. Both algorithms make a repeated use of the field norm in tower of fields; it allows them to be faster and more compact than existing algorithms by factors $${\tilde{O}}(n)$$. For lattice-based schemes considered in practice, this reduces both the computation time and RAM usage by factors at least 100, making key pair generation within range of smart card abilities.
Upper and Lower Bounds for Continuous Non-Malleable Codes Abstract
Dana Dachman-Soled Mukul Kulkarni
Recently, Faust et al. (TCC'14) introduced the notion of continuous non-malleable codes (CNMC), which provides stronger security guarantees than standard non-malleable codes, by allowing an adversary to tamper with the codeword in a continuous way instead of one-time tampering. They also showed that CNMC with information theoretic security cannot be constructed in the 2-split-state tampering model, and presented a construction in the common reference string (CRS) model from collision-resistant hash functions and non-interactive zero-knowledge proofs.In this work, we ask if it is possible to construct CNMC from weaker assumptions. We answer this question by presenting lower as well as upper bounds. We show that it is impossible to construct 2-split-state CNMC, with no CRS, for one-bit messages from any falsifiable assumption, thus establishing the lower bound. We additionally provide an upper bound by constructing 2-split-state CNMC for one-bit messages, assuming only the existence of a family of injective one way functions. We note that in a recent work, Ostrovsky et al. (CRYPTO'18) considered the construction of a relaxed notion of 2-split-state CNMC from minimal assumptions.We also present a construction of 4-split-state CNMC for multi-bit messages in CRS model from the same assumptions. Additionally, we present definitions of the following new primitives: (1) One-to-one commitments, and (2) Continuous Non-Malleable Randomness Encoders, which may be of independent interest.
Efficiently Masking Binomial Sampling at Arbitrary Orders for Lattice-Based Crypto Abstract
Tobias Schneider Clara Paglialonga Tobias Oder Tim Güneysu
With the rising popularity of lattice-based cryptography, the Learning with Errors (LWE) problem has emerged as a fundamental core of numerous encryption and key exchange schemes. Many LWE-based schemes have in common that they require sampling from a discrete Gaussian distribution which comes with a number of challenges for the practical instantiation of those schemes. One of these is the inclusion of countermeasures against a physical side-channel adversary. While several works discuss the protection of samplers against timing leaks, only few publications explore resistance against other side-channels, e.g., power. The most recent example of a protected binomial sampler (as used in key encapsulation mechanisms to sufficiently approximate Gaussian distributions) from CHES 2018 is restricted to a first-order adversary and cannot be easily extended to higher protection orders.In this work, we present the first protected binomial sampler which provides provable security against a side-channel adversary at arbitrary orders. Our construction relies on a new conversion between Boolean and arithmetic (B2A) masking schemes for prime moduli which outperforms previous algorithms significantly for the relevant parameters, and is paired with a new masked bitsliced sampler allowing secure and efficient sampling even at larger protection orders. Since our proposed solution supports arbitrary moduli, it can be utilized in a large variety of lattice-based constructions, like NewHope, LIMA, Saber, Kyber, HILA5, or Ding Key Exchange.
Improved Security Evaluation Techniques for Imperfect Randomness from Arbitrary Distributions Abstract
Takahiro Matsuda Kenta Takahashi Takao Murakami Goichiro Hanaoka
Dodis and Yu (TCC 2013) studied how the security of cryptographic primitives that are secure in the "ideal" model in which the distribution of a randomness is the uniform distribution, is degraded when the ideal distribution of a randomness is switched to a "real-world" (possibly biased) distribution that has some lowerbound on its min-entropy or collision-entropy. However, in many constructions, their security is guaranteed only when a randomness is sampled from some non-uniform distribution (such as Gaussian in lattice-based cryptography), in which case we cannot directly apply the results by Dodis and Yu.In this paper, we generalize the results by Dodis and Yu using the Rényi divergence, and show how the security of a cryptographic primitive whose security is guaranteed when the ideal distribution of a randomness is a general (possibly non-uniform) distribution Q, is degraded when the distribution is switched to another (real-world) distribution R. More specifically, we derive two general inequalities regarding the Rényi divergence of R from Q and an adversary's advantage against the security of a cryptographic primitive. As applications of our results, we show (1) an improved reduction for switching the distributions of distinguishing problems with public samplability, which is simpler and much tighter than the reduction by Bai et al. (ASIACRYPT 2015), and (2) how the differential privacy of a mechanism is degraded when its randomness comes from not an ideal distribution Q but a real-world distribution R. Finally, we show methods for approximate-sampling from an arbitrary distribution Q with some guaranteed upperbound on the Rényi divergence (of the distribution R of our sampling methods from Q).
Decryption Failure Attacks on IND-CCA Secure Lattice-Based Schemes Abstract
Jan-Pieter D'Anvers Qian Guo Thomas Johansson Alexander Nilsson Frederik Vercauteren Ingrid Verbauwhede
In this paper we investigate the impact of decryption failures on the chosen-ciphertext security of lattice-based primitives. We discuss a generic framework for secret key recovery based on decryption failures and present an attack on the NIST Post-Quantum Proposal ss-ntru-pke. Our framework is split in three parts: First, we use a technique to increase the failure rate of lattice-based schemes called failure boosting. Based on this technique we investigate the minimal effort for an adversary to obtain a failure in three cases: when he has access to a quantum computer, when he mounts a multi-target attack or when he can only perform a limited number of oracle queries. Secondly, we examine the amount of information that an adversary can derive from failing ciphertexts. Finally, these techniques are combined in an overall analysis of the security of lattice based schemes under a decryption failure attack. We show that an attacker could significantly reduce the security of lattice based schemes that have a relatively high failure rate. However, for most of the NIST Post-Quantum Proposals, the number of required oracle queries is above practical limits. Furthermore, a new generic weak-key (multi-target) model on lattice-based schemes, which can be viewed as a variant of the previous framework, is proposed. This model further takes into consideration the weak-key phenomenon that a small fraction of keys can have much larger decoding error probability for ciphertexts with certain key-related properties. We apply this model and present an attack in detail on the NIST Post-Quantum Proposal – ss-ntru-pke – with complexity below the claimed security level.
On Tightly Secure Primitives in the Multi-instance Setting Abstract
Dennis Hofheinz Ngoc Khanh Nguyen
We initiate the study of general tight reductions in cryptography. There already exist a variety of works that offer tight reductions for a number of cryptographic tasks, ranging from encryption and signature schemes to proof systems. However, our work is the first to provide a universal definition of a tight reduction (for arbitrary primitives), along with several observations and results concerning primitives for which tight reductions have not been known.Technically, we start from the general notion of reductions due to Reingold, Trevisan, and Vadhan (TCC 2004), and equip it with a quantification of the respective reduction loss, and a canonical multi-instance extension to primitives. We then revisit several standard reductions whose tight security has not yet been considered. For instance, we revisit a generic construction of signature schemes from one-way functions, and show how to tighten the corresponding reduction by assuming collision-resistance from the used one-way function. We also obtain tightly secure pseudorandom generators (by using suitable rerandomisable hard-core predicates), and tightly secure lossy trapdoor functions.
Reducing the Key Size of McEliece Cryptosystem from Automorphism-induced Goppa Codes via Permutations Abstract
Zhe Li Chaoping Xing Sze Ling Yeo
In this paper, we propose a new general construction to reduce the public key size of McEliece cryptosystems constructed from automorphism-induced Goppa codes. In particular, we generalize the ideas of automorphism-induced Goppa codes by considering nontrivial subsets of automorphism groups to construct Goppa codes with a nice block structure. By considering additive and multiplicative automorphism subgroups, we provide explicit constructions to demonstrate our technique. We show that our technique can be applied to automorphism-induced Goppa codes based cryptosystems to further reduce their key sizes.
Key Encapsulation Mechanism with Explicit Rejection in the Quantum Random Oracle Model Abstract
Haodong Jiang Zhenfeng Zhang Zhi Ma
The recent post-quantum cryptography standardization project launched by NIST increased the interest in generic key encapsulation mechanism (KEM) constructions in the quantum random oracle (QROM). Based on a OW-CPA-secure public-key encryption (PKE), Hofheinz, Hövelmanns and Kiltz (TCC 2017) first presented two generic constructions of an IND-CCA-secure KEM with quartic security loss in the QROM, one with implicit rejection (a pseudorandom key is return for an invalid ciphertext) and the other with explicit rejection (an abort symbol is returned for an invalid ciphertext). Both are widely used in the NIST Round-1 KEM submissions and the ones with explicit rejection account for 40%. Recently, the security reductions have been improved to quadratic loss under a standard assumption, and be tight under a nonstandard assumption by Jiang et al. (Crypto 2018) and Saito, Xagawa and Yamakawa (Eurocrypt 2018). However, these improvements only apply to the KEM submissions with implicit rejection and the techniques do not seem to carry over to KEMs with explicit rejection.In this paper, we provide three generic constructions of an IND-CCA-secure KEM with explicit rejection, under the same assumptions and with the same tightness in the security reductions as the aforementioned KEM constructions with implicit rejection (Crypto 2018, Eurocrypt 2018). Specifically, we develop a novel approach to verify the validity of a ciphertext in the QROM and use it to extend the proof techniques for KEM constructions with implicit rejection (Crypto 2018, Eurocrypt 2018) to our KEM constructions with explicit rejection. Moreover, using an improved version of one-way to hiding lemma by Ambainis, Hamburg and Unruh (ePrint 2018/904), for two of our constructions, we present tighter reductions to the standard IND-CPA assumption. Our results directly apply to 9 KEM submissions with explicit rejection, and provide tighter reductions than previously known (TCC 2017).
Registration-Based Encryption from Standard Assumptions Abstract
Sanjam Garg Mohammad Hajiabadi Mohammad Mahmoody Ahmadreza Rahimi Sruthi Sekar
The notion of Registration-Based Encryption (RBE) was recently introduced by Garg, Hajiabadi, Mahmoody, and Rahimi [TCC'18] with the goal of removing the private-key generator (PKG) from IBE. Specifically, RBE allows encrypting to identities using a (compact) master public key, like how IBE is used, with the benefit that the PKG is substituted with a weaker entity called "key curator" who has no knowledge of any secret keys. Here individuals generate their secret keys on their own and then publicly register their identities and their corresponding public keys to the key curator. Finally, individuals obtain "rare" decryption-key updates from the key curator as the population grows. In their work, they gave a construction of RBE schemes based on the combination of indistinguishability obfuscation and somewhere statistically binding hash functions. However, they left open the problem of constructing RBE schemes based on standard assumptions.In this work, we resolve the above problem and construct RBE schemes based on standard assumptions (e.g., CDH or LWE). Furthermore, we show a new application of RBE in a novel context. In particular, we show that anonymous variants of RBE (which we also construct under standard assumptions) can be used for realizing abstracts forms of anonymous messaging tasks in simple scenarios in which the parties communicate by writing messages on a shared board in a synchronized way.
Factoring Products of Braids via Garside Normal Form Abstract
Simon-Philipp Merz Christophe Petit
Braid groups are infinite non-abelian groups naturally arising from geometric braids. For two decades they have been proposed for cryptographic use. In braid group cryptography public braids often contain secret braids as factors and it is hoped that rewriting the product of braid words hides individual factors. We provide experimental evidence that this is in general not the case and argue that under certain conditions parts of the Garside normal form of factors can be found in the Garside normal form of their product. This observation can be exploited to decompose products of braids of the form ABC when only B is known.Our decomposition algorithm yields a universal forgery attack on WalnutDSATM, which is one of the 20 proposed signature schemes that are being considered by NIST for standardization of quantum-resistant public-key cryptography. Our attack on WalnutDSATM can universally forge signatures within seconds for both the 128-bit and 256-bit security level, given one random message-signature pair. The attack worked on 99.8% and 100% of signatures for the 128-bit and 256-bit security levels in our experiments.Furthermore, we show that the decomposition algorithm can be used to solve instances of the conjugacy search problem and decomposition search problem in braid groups. These problems are at the heart of other cryptographic schemes based on braid groups.
Non-interactive Keyed-Verification Anonymous Credentials Abstract
Geoffroy Couteau Michael Reichle
Anonymous credential ($$\mathsf {AC}$$) schemes are protocols which allow for authentication of authorized users without compromising their privacy. Of particular interest are non-interactive anonymous credential ($$\mathsf {NIAC}$$) schemes, where the authentication process only requires the user to send a single message that still conceals its identity. Unfortunately, all known $$\mathsf {NIAC}$$ schemes in the standard model require pairing based cryptography, which limits them to a restricted set of specific assumptions and requires expensive pairing computations. The notion of keyed-verification anonymous credential ($$\mathsf {KVAC}$$) was introduced in (Chase et al., CCS'14) as an alternative to standard anonymous credential schemes allowing for more efficient instantiations; yet, making existing $$\mathsf {KVAC}$$ non-interactive either requires pairing-based cryptography, or the Fiat-Shamir heuristic.In this work, we construct the first non-interactive keyed-verification anonymous credential ($$\mathsf {NIKVAC}$$) system in the standard model, without pairings. Our scheme is efficient, attribute-based, supports multi-show unlinkability, and anonymity revocation. We achieve this by building upon a combination of algebraic $$\mathsf {MAC}$$ with the recent designated-verifier non-interactive zero-knowledge ($$\mathsf {DVNIZK}$$) proof of knowledge of (Couteau and Chaidos, Eurocrypt'18). Toward our goal of building $$\mathsf {NIKVAC}$$, we revisit the security analysis of a $$\mathsf {MAC}$$ scheme introduced in (Chase et al., CCS'14), strengthening its guarantees, and we introduce the notion of oblivious non-interactive zero-knowledge proof system, where the prover can generate non-interactive proofs for statements that he cannot check by himself, having only a part of the corresponding witness, and where the proof can be checked efficiently given the missing part of the witness. We provide an efficient construction of an oblivious $$\mathsf {DVNIZK}$$, building upon the specific properties of the $$\mathsf {DVNIZK}$$ proof system of (Couteau and Chaidos, Eurocrypt'18).
FE for Inner Products and Its Application to Decentralized ABE Abstract
Zhedong Wang Xiong Fan Feng-Hao Liu
In this work, we revisit the primitive functional encryption (FE) for inner products and show its application to decentralized attribute-based encryption (ABE). Particularly, we derive an FE for inner products that satisfies a stronger notion, and show how to use such an FE to construct decentralized ABE for the class $$\{0,1\}$$-$$\mathsf {LSSS} $$ against bounded collusions in the plain model. We formalize the FE notion and show how to achieve such an FE under the LWE or DDH assumption. Therefore, our resulting decentralized ABE can be constructed under the same standard assumptions, improving the prior construction by Lewko and Waters (Eurocrypt 2011). Finally, we also point out challenges to construct decentralized ABE for general functions by establishing a relation between such an ABE and witness encryption for general NP statements.
Shorter Ring Signatures from Standard Assumptions Abstract
Alonso González
Ring signatures, introduced by Rivest, Shamir and Tauman (ASIACRYPT 2001), allow to sign a message on behalf of a set of users while guaranteeing authenticity and anonymity. Groth and Kohlweiss (EUROCRYPT 2015) and Libert et al. (EUROCRYPT 2016) constructed schemes with signatures of size logarithmic in the number of users. An even shorter ring signature, of size independent from the number of users, was recently proposed by Malavolta and Schröder (ASIACRYPT 2017). However, all these short signatures are obtained relying on strong and controversial assumptions. Namely, the former schemes are both proven secure in the random oracle model while the later requires non-falsifiable assumptions.The most efficient construction under mild assumptions remains the construction of Chandran et al. (ICALP 2007) with a signature of size $$\varTheta (\sqrt{n})$$, where n is the number of users, and security is based on the Diffie-Hellman assumption in bilinear groups (the SXDH assumption in asymmetric bilinear groups).In this work we construct an asymptotically shorter ring signature from the hardness of the Diffie-Hellman assumption in bilinear groups. Each signature comprises $$\varTheta (\root 3 \of {n})$$ group elements, signing a message requires computing $$\varTheta (\root 3 \of {n})$$ exponentiations, and verifying a signature requires $$\varTheta (n^{2/3})$$ pairing operations. To the best of our knowledge, this is the first ring signature based on bilinear groups with $$o(\sqrt{n})$$ signatures and sublinear verification complexity.
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CommonCrawl
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\begin{definition}[Definition:Quasigroup/Right Quasigroup]
Let $\struct {S, \circ}$ be a magma.
$\struct {S, \circ}$ is a '''right quasigroup''' {{iff}}:
:for all $a \in S$, the right regular representation $\rho_a$ is a permutation on $S$.
That is:
:$\forall a, b \in S: \exists ! x \in S: x \circ a = b$
\end{definition}
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ProofWiki
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EURASIP Journal on Bioinformatics and Systems Biology
Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations
Amin Zollanvari1 &
Edward R. Dougherty2
EURASIP Journal on Bioinformatics and Systems Biology volume 2016, Article number: 2 (2016) Cite this article
In classification, prior knowledge is incorporated in a Bayesian framework by assuming that the feature-label distribution belongs to an uncertainty class of feature-label distributions governed by a prior distribution. A posterior distribution is then derived from the prior and the sample data. An optimal Bayesian classifier (OBC) minimizes the expected misclassification error relative to the posterior distribution. From an application perspective, prior construction is critical. The prior distribution is formed by mapping a set of mathematical relations among the features and labels, the prior knowledge, into a distribution governing the probability mass across the uncertainty class. In this paper, we consider prior knowledge in the form of stochastic differential equations (SDEs). We consider a vector SDE in integral form involving a drift vector and dispersion matrix. Having constructed the prior, we develop the optimal Bayesian classifier between two models and examine, via synthetic experiments, the effects of uncertainty in the drift vector and dispersion matrix. We apply the theory to a set of SDEs for the purpose of differentiating the evolutionary history between two species.
A purely data-driven classifier design with small samples encounters a fundamental conundrum: since the error rate of a classifier quantifies its predictive accuracy, the salient epistemic attribute of any classifier and re-sampling strategies such as cross-validation and bootstrap is generally very inaccurate on small samples due to excessive variance and lack of regression with the true error [1]. The inability to satisfactorily estimate the error with model-free methods with small samples implies that classifier error estimation is virtually impossible without the use of prior information. Prior knowledge can be incorporated in a Bayesian framework by assuming that the feature-label distribution belongs to an uncertainty class of feature-label distributions governed by a prior distribution [2, 3]. Given the latter, in conjunction with sample data, one can optimally estimate the error of any classifier, relative to the mean square error (MSE) between the true and estimated errors, where expectations are taken with respect to a posterior distribution derived from the prior distribution and the data [4, 5]. Hence, optimality is with respect to our prior knowledge and the data. Furthermore, one can derive an optimal classifier relative to the expected error of the classifier over the posterior distribution, this being called the optimal Bayesian classifier (OBC) [6, 7]. Closed-form solutions have been developed for multinomial and Gaussian models. In other situations, Markov Chain Monte Carlo (MCMC) methods can be used [8].
Having developed the statistical theory, one is confronted with an engineering problem: transform scientific knowledge given in some mathematical form into a prior distribution. Intuitively, given a set of mathematical relations among the features and labels, these relations constrain the uncertainty class of feature-label distributions that could potentially govern the classification and the relative strengths of the relations can be transformed so as to determine the probability mass of the prior distribution. For instance, in phenotype classification based on gene expression, genetic regulatory pathways constitute graphical prior knowledge and this prior knowledge can be employed to formulate a prior distribution governing the uncertainty class of feature-label distributions [9, 10]. Another genomic application involves using prior knowledge concerning RNA-seq data to form sequence-based classifiers [8].
From a general perspective, when using Bayesian methods, prior construction constitutes the highest hurdle. A half century ago, E. T. Jaynes remarked,
Bayesian methods, for all their advantages, will not be entirely satisfactory until we face the problem of finding the prior probability squarely [11].
The aim of this paper is to utilize prior knowledge in the form of stochastic differential equations (SDEs) to classify time-series data. Although we will confine ourselves to a Gaussian problem so that we can take advantage of existing closed-form OBC representations, one can envision further applications using MCMC methods. Hence, the approach taken in the present paper may lead to utilizing SDEs across a number of time-series classification problems, keeping in mind that SDEs play a major role in many disciplines including physics, biology, finance, and chemistry. Vector SDEs, our concern here, have various applications. Not only do they arise naturally in many systems with vector value states, but they also arise in many systems where the process is restricted to lie on certain manifolds [12].
In the stochastic setting, training data are collected over time processes. Given certain Gaussian assumptions, classification in the SDE setting takes the same form as ordinary classification in the Gaussian model and we can apply the optimal Bayesian classification theory once we have a prior distribution constructed in accordance with known stochastic equations. In this paper, we provide the mathematical framework to synthesize an OBC in the presence of prior knowledge induced in the form of SDEs governing the dynamics of the system. We consider a vector SDE in integral form involving a drift vector and dispersion matrix, develop the OBC between two models, and examine via synthetic experiments the effects of uncertainty in the drift vector and dispersion matrix.
We compare the performance of the OBC with quadratic discriminant analysis (QDA), a classical approach to building classifiers in the Gaussian model (see Additional file 2: Section I for definition of QDA). Such comparisons are useful because, even though the OBC is optimal given the uncertainty, its optimality is on average across the uncertainty class, so that its performance advantage varies for different feature-label distributions in the uncertainty class (and can be disadvantageous for some distributions, although these will have small probability mass in the posterior distribution). Comparison to QDA is instructive because, as we will explain in the next section, QDA is a sample-based approximation to the optimal classifier for the true feature-label distribution. In addition to synthetic experiments, we apply optimal Bayesian classification using a form of the Ornstein-Uhlenbeck process that has been employed for modeling the evolutionary change of species; specifically, we use a set of SDEs to construct a classifier to differentiate the evolutionary history between two species.
In a two-class classification, the population is characterized by a feature-label distribution F for a random pair (X,Y), where X is a vector of p features and Y is the binary label, 0 or 1, of the class containing X. The prior class probabilities are defined by c j =P(Y=j) and the class-conditional densities by p j (x)=p(x∣Y=j), for j=0,1. To avoid trivialities, we assume min{c 0,c 1}≠0. A classifier is a function ψ(X) assigning a binary label to each feature vector X. The error, ε[ψ], of ψ is the probability P(ψ(X)≠Y), which can be decomposed into ε=c 0 ε 0+c 1 ε 1, where ε j=P(ψ(X)=1−j|Y=j), for j=0,1. A classifier with minimum error among all classifiers is known as a Bayes classifier for F. The minimum error is called the Bayes error. Epistemologically, the error is the key issue since it quantifies the predictive capacity.
In practice, F is unknown and a classification rule ψ is used to design a classifier ψ n from a random sample S n ={(X 1,Y 1),(X 2,Y 2),…,(X n ,Y n )} of pairs drawn from F. If feature selection is involved, then it is part of the classification rule. Since the true classifier error ε[ ψ n ] depends on F, which is unknown, ε[ψ n ] is unknown. The true error must be estimated by an estimation rule, Ξ. Thus, the random sample S n yields a classifier ψ n =Ψ(S n ) and an error estimate \(\hat {\varepsilon } [\!\psi _{n}]=\Xi (S_{n})\) (see Additional file 2: Section II for more information).
When a large amount of data is available, the sample can be split into independent training and test sets, the classifier being designed on the training data and its error being estimated by the proportion of errors on the test data; however, when data are limited, the sample cannot be split without leaving too little data to design a good classifier. Hence, training and error estimation must take place on the same data set. As noted in Section 1, accurate error estimation is virtually impossible with small samples in the absence of distributional assumptions.
Optimal Bayesian classification
Distributional assumptions can be imposed by defining a prior distribution over an uncertainty class of feature-label distributions. This results in a Bayesian approach with the uncertainty class being given a prior distribution and the data being used to construct a posterior distribution.
Let Π 0 and Π 1 denote the class-0 and class-1 conditional distributions, respectively; let c be the probability of a point coming from Π 0 (the "mixing" probability); and let Π 0 and Π 1 be parameterized by θ 0 and θ 1, respectively. The overall model is parameterized by θ=(c,θ 0,θ 1) with prior distribution π(θ). Given a random sample, S n , a classifier ψ n is designed and we wish to minimize the MSE between its true error, ε, and an error estimate, \(\widehat {\varepsilon }\). The minimum mean square error (MMSE) error estimator is the expected true error, \(\widehat {\varepsilon }(\psi _{n},S_{n})=\mathrm {E}_{\theta }[\varepsilon (\psi _{n},\theta)|S_{n}]\). The expectation given the sample is over the posterior density of θ, denoted by π ∗(θ). Thus, we write the Bayesian MMSE error estimator as \(\widehat {\varepsilon }=\mathrm {E}_{\pi ^{\ast }}[\varepsilon ]\).
The Bayesian error estimate is not guaranteed to be the optimal error estimate for any particular feature-label distribution but optimal for a given sample, and assuming the parameterized model and prior probabilities, it is both optimal on average with respect to MSE and unbiased when averaged over all parameters and samples. To facilitate analytic representations, we assume c, θ 0, and θ 1 are all mutually independent prior to observing the data. Denote the marginal priors of c, θ 0, and θ 1 by π(c), π(θ 0), and π(θ 1), respectively, and suppose data are used to find each posterior, π ∗(c), π ∗(θ 0), and π ∗(θ 1), respectively. Independence is preserved, i.e., π ∗(c,θ 0,θ 1)=π ∗(c)π ∗(θ 0)π ∗(θ 1) [4].
If ψ n is a trained classifier given by ψ n (x)=0 if x∈R 0 and ψ n (x)=1 if x∈R 1, where R 0 and R 1 are measurable sets partitioning the sample space, then the Bayesian MMSE error estimator can be found from effective class-conditional densities, which are derived by taking the expectations of the individual class-conditional densities with respect to the posterior distribution,
$$ f\left(\mathbf{x}|y\right) =\int_{\mathbf{\Theta }_{y}}f_{\theta_{y}}\left(\mathbf{x}|y\right) \pi^{\ast }\left(\theta_{y}\right) d\theta_{y}. $$
Using these [6] (see Additional file 2: Section III for more information),
$$ {}\widehat{\varepsilon }\left(\psi_{n},S_{n}\right) \,=\,\mathrm{E}_{\pi^{\ast}}[\!c]\!\int_{R_{1}}\!f\!\left(\mathbf{x}|0\right) d\mathbf{x}+(1-\mathrm{E}_{\pi^{\ast }}[\!c]\!)\int_{R_{0}}f\!\left(\mathbf{x}|1\right) d\mathbf{x}. $$
In the context of a prior distribution, an optimal Bayesian classifier, ψ OBC, is any classifier satisfying
$$ \mathrm{E}_{\pi^{\ast }}\left[ \varepsilon (\psi_{\text{OBC}},\theta) \right] \leq \mathrm{E}_{\pi^{\ast }}\left[ \varepsilon (\psi,\theta) \right] $$
for all \(\psi \in \mathcal {C}\), where \(\mathcal {C}\) is an arbitrary family of classifiers. Under the Bayesian framework, this is equivalent to minimizing the probability of error,
$$ \begin{aligned} \mathrm{P}\left(\psi_{n}\left(\mathbf{X}\right) \neq Y|S_{n}\right)& =\mathrm{E}_{\pi^{\ast }}\left[ P\left(\psi_{n}\left(\mathbf{X}\right) \neq Y|\theta,S_{n}\right) \right]\\ & =\widehat{\varepsilon }\left(\psi_{n},S_{n}\right). \end{aligned} $$
If \(\mathcal {C}\) is the set of all classifiers with measurable decision regions (which we will assume), then an optimal Bayesian classifier, ψ OBC, satisfying (3) for all \(\psi \in \mathcal {C}\) exists and is given pointwise by [6]
$$ \psi_{\text{OBC}}\left(\mathbf{x}\right) =\left\{\! \begin{array}{ll} 0 & \text{if }\mathrm{E}_{\pi^{\ast }}[\!c]f\left(\mathbf{x}|0\right) \geq (1-\mathrm{E}_{\pi^{\ast }}[\!c])f\left(\mathbf{x}|1\right), \\ 1 & \text{otherwise}. \end{array} \right. $$
In many applications, especially in biomedicine, the sample S n is obtained by first deciding how many sample points will be taken from each class and then randomly sampling from each class separately, the resulting sample said to be "separately sampled." With separate sampling, the data cannot be used to generate a posterior distribution for c, so that c must be known. Stratified sampling is a special case of separate sampling in which the sample is drawn so that the proportion of sample points from class 0 is equal to c. In such a case, there is no posterior \(\mathrm {E}_{\pi ^{\ast }\phantom {\dot {i}\!}}[\!c]\) and \(\mathrm {E}_{\pi ^{\ast }\phantom {\dot {i}\!}}[\!c]\) is replaced by c in (5). We will utilize stratified sampling in our examples.
Binary classification of Gaussian processes
In this section, we frame the setting in which we are working and then define the problem of binary classification in the context of Gaussian processes. To begin with, a collection {X t :t∈T} of \(\mathbb {R}^{p}\)-valued random variables defined on a common probability space \((\Omega,\mathcal {F},P)\) indexed by a parameter \( t\in \mathbf {T}\subset \mathbb {R}\) (here assumed to be time) and \(\mathcal {F}\) being a σ-algebra of subsets of the sample space Ω (events) constitutes a stochastic process X with state space \(\mathbb {R} ^{p}\). Throughout this work, we consider \(\mathcal {F}\) as the σ-algebra of Borel subsets of \(\mathbb {R}^{p}\). A stochastic process X is adapted to an increasing family of σ-algebra \(\{\mathcal {F}_{t}:t\geq 0\}\) (a filtration) if for each t≥0, X t is \(\mathcal {F}_{t}\)-measurable.
We study classification in the context of multivariate Gaussian processes (see Additional file 2: Section IV for a review of literature pertaining to classification of stochastic processes). Consider the p-dimensional column random vectors \(\mathbf {X}_{t_{1}}\), \(\mathbf {X}_{t_{2}}\),...., \(\mathbf {X}_{t_{N}}\). A random process X is a multivariate Gaussian process if any finite-dimensional vector \(\left [\mathbf {X}_{t_{1}}^{T},\mathbf {X}_{t_{2}}^{T},...,\mathbf {X}_{t_{N}}^{T}\right ]^{T} \) possesses a multivariate normal distribution \(\mathcal {N}\left (\boldsymbol {\mu }_{\mathbf {t}_{N}},\boldsymbol {\Sigma }_{\mathbf {t}_{N}}\right)\), where
$$ \boldsymbol{\mu }_{\mathbf{t}_{N}}=\left[\boldsymbol{\mu }_{t_{1}}^{T}, \boldsymbol{\mu }_{t_{2}}^{T},...,\boldsymbol{\mu }_{t_{N}}^{T}\right]_{Np\times 1}^{T}, $$
with \(\boldsymbol {\mu }_{{t}_{i}\phantom {\dot {i}\!}}=E[\mathbf {X}_{t_{i}\phantom {\dot {i}\!}}]\), and \(\boldsymbol { \Sigma }_{\mathbf {t}_{N}\phantom {\dot {i}\!}}\) is the N p×N p covariance matrix dependent on t N =[t 1,t 2,...,t N ]T and structured as
$$ \begin{aligned} \boldsymbol{\Sigma }_{\mathbf{t}_{N}}\,=\, \left[ \begin{array}{cccc} \boldsymbol{\Sigma }_{{t}_{1},{t}_{1}} &\boldsymbol{\Sigma }_{t_{1},{t}_{2}} &... & \boldsymbol{\Sigma }_{t_{1},{t}_{N}} \\[1ex] \boldsymbol{\Sigma }_{{t}_{2},{t}_{1}} &\boldsymbol{\Sigma }_{t_{2},{t}_{2}} &... & \boldsymbol{\Sigma }_{t_{2},{t}_{N}} \\... &... &... &... \\ \boldsymbol{\Sigma }_{{t}_{N},{t}_{1}} & \boldsymbol{\Sigma }_{t_{N},{t}_{2}} &... &\boldsymbol{\Sigma }_{t_{N},{t}_{N}} \end{array} \right]_{Np\times Np}, \end{aligned} $$
$$ \boldsymbol{\Sigma }_{{t}_{i},{t}_{j}}=E\left[\left(\mathbf{X}_{t_{i}}-E(\mathbf{X} _{t_{i}}))(\mathbf{X}_{t_{j}}-E(\mathbf{X}_{t_{j}})^{T}\right)\right]. $$
We refer to t N as the observation time vector. For any fixed ω∈Ω, a sample path is a collection {X t (ω):t∈t}. We denote a realization of X at sample path ω and time vector t N by \( \mathbf {x}_{\mathbf {t}_{N}\phantom {\dot {i}\!}}(\omega)\).
We consider a general framework, referred to as binary classification of Gaussian processes (BCGP). Consider two independent multivariate Gaussian processes X 0 and X 1, where for any t N , X 0 and X 1 possess mean and covariance \(\boldsymbol {\mu }_{\mathbf {t}_{N}}^{0}\) and \(\boldsymbol {\Sigma }_{\mathbf {t}_{N}}^{0}\), and \(\boldsymbol {\mu }_{\mathbf {t}_{N}}^{1}\) and \(\boldsymbol {\Sigma }_{\mathbf {t}_{N}}^{1}\), respectively. For y=0,1, \( \boldsymbol {\mu }_{\mathbf {t}_{N}}^{y}\) is defined similarly to (6) with \(\boldsymbol {\mu }_{{t}_{i}}^{y}=E\left [\mathbf {X}_{t_{i}}^{y}\right ]\) and \(\boldsymbol {\Sigma }_{\mathbf {t}_{N}}^{y}\) is defined similarly to (7) with
$$ \boldsymbol{\Sigma}_{{t}_{i},{t}_{j}}^{y}=E\left[\left(\mathbf{X}_{t_{i}}^{y}-E(\mathbf{X}_{t_{i}}^{y})\right)\left(\mathbf{X}_{t_{j}}^{y}-E(\mathbf{X} _{t_{j}}^{y})\right)^{T}\right]. $$
Let \(\mathbf {S}_{\mathbf {t}_{N}}^{y}\) denote a set of n y sample paths from process X y at t N ,
$$ \mathbf{S}_{\mathbf{t}_{N}}^{y}=\left\{\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{1}),\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{2}),\ldots,\mathbf{x}_{ \mathbf{t}_{N}}^{y}(\omega_{n^{y}})\right\}. $$
((10))
We assume that t N is the same for both classes. Let \(\mathbf {X}_{\mathbf {t}_{N}}^{y}(\omega _{s})\) denote a future test sample path observed on the same observation time vector as the training sample paths, where y∈{0,1} indicates the label of the class-conditional process the sample path is coming from, either X 0 or X 1. Note that, as compared with the classical probabilistic definition of classification where the sample points are observations of p-dimension, here we define stochastic-process classification in connection with a set of sample paths, which can be considered as observations of Np dimension. A classification problem arises from the fact that the experimenter is blind to the class label of \(\mathbf {X}_{\mathbf {t}_{N}}^{y}(\omega _{s})\), i.e., to y, and desires a discriminant \(\psi _{\mathbf {t}_{N}\phantom {\dot {i}\!}}(.)\) such that
$$ y= \left\{ \begin{array}{ll} 0,\quad \text{if}\;\;\psi_{\mathbf{t}_{N}}\left(\mathbf{x}_{\mathbf{t} _{N}}^{y}(\omega_{s})\right)>0 \\ 1,\quad \text{otherwise}. \end{array} \right.\,. $$
Other types of classification could be defined. For example, one might be interested in classifying a test sample path \(\mathbf {x}_{\mathbf {t}_{N+M}}^{y}(\omega _{s})\) where the observation time vector of the test sample path is obtained by augmenting t N by another vector [t N+1,t N+2,...,t N+M ], where M is a positive integer. In this case, the time of observation for the future sample path is extended. Similarly, one may define problems where the future time of observation is shrunken to a subset of time points in t N or problems where the future observation time vector is a set of time points totally or partially different from time points in t N . Throughout this work, we are mainly concerned with solving the classification problem as defined in (11), which we refer to as the standard type, and we discuss the feasibility of solving other cases when possible.
General presentation of stochastic differential equations (SDEs)
To define SDEs, we consider a diffusion process, the most fundamental being the Wiener process. For a general definition of a q-dimensional Wiener process, see the Appendix. Let W={W t :t≥0} be a q-dimensional Wiener process. For each sample path and for 0≤t 0≤t≤T, we consider a vector SDE in the integral form as follows:
$$ \begin{aligned} \mathbf{X}_{t}(\omega)&=\mathbf{X}_{t_{0}}(\omega)+\int_{t_{0}}^{t}\mathbf{f }\left(s,\mathbf{X}_{s}(\omega)\right) ds\\ &\quad+\int_{t_{0}}^{t}\mathbf{G}\left(s,\mathbf{X}_{t}(\omega)\right) d\mathbf{W}_{s}(\omega), \end{aligned} $$
where \(\mathbf {f}:[\!0,T]\times \Omega \rightarrow \mathbb {R}^{p}\) (the p-dimensional drift vector) and \(\mathbf {G}:[\!0,T]\times \Omega \rightarrow \mathbb {R}^{p\times q}\) (the p×q dispersion matrix). The first integral in (12) is an ordinary Lebesgue integral, and throughout this work, we assume an Itô integration for the second integral. With slightly more work, the results can be extended to Stratonovich integration. Let \(\mathcal {L}\) be the σ-algebra of Lebesgue subsets of \(\mathbb {R}\). A function h(t,ω) defined on a probability space \((\Omega,\mathcal {F},P)\) belongs to \(\mathcal {L}_{T}^{\omega }\) if it is jointly \(\mathcal {L}\times \mathcal {F}\) measurable, h(t,.) is \(\mathcal {F}_{t}\)-measurable for each t∈[ 0,T], and with probability 1, \({\int _{0}^{T}}h(s,\omega)^{2}ds<\infty \). Let f i and g i,j denote the components of f and G, respectively. If we assume X 0(ω) is \(\mathcal {F}_{0}\)-measurable and if \(\sqrt {|f^{i}|}\in \mathcal {L}_{T}^{\omega }\) and \(g^{i,j}\in \mathcal {L}_{T}^{\omega }\), then each component of the p-dimensional process X t (ω) is \(\mathcal {F}_{t}\)-measurable [12]. The \(\mathcal {F}_{t}\)-measurability of X t (ω) along with the martingale property of W indicates "nonanticipativeness" of X t (ω) in general.
The integral Eq. (12) is commonly written in a symbolic form as
$$ \mathrm{d}\mathbf{X}_{t}=\mathbf{f}(t,\mathbf{X}_{t})\mathrm{d}t+\mathbf{G}(t, \mathbf{X}_{t})\mathrm{d}\mathbf{W}_{t}, $$
which is the representation of a vector SDE.
SDE prior knowledge in the BCGP model
Prior knowledge in the form of a set of stochastic differential equations constrains the possible behavior of the dynamical system to an uncertainty class. If such prior knowledge is available, then it can be used in the BCGP model to improve classification performance. The core underlying assumption of the BCGP model is that the data are generated from two Gaussian processes for which binary classification is desired. In this regard, we define valid prior knowledge (in the form of SDEs) as a set of SDEs with a unique solution that does not contradict the Gaussianity assumption of the dynamics of the model. For nonlinear f(t,X t ) and G(t,X t ) (w.r.t. to state X t ), the solution of SDE (13) is generally a non-Gaussian process. Fortunately, under a wide class of linear functions, the SDE solutions are Gaussian. To wit, the SDEs become valid prior knowledge for each class-conditional process defined in the BCGP model. Henceforth, we focus on this type of SDE.
For class label y=0,1, the linear classes of SDEs that we consider are defined by replacing
$$ \begin{aligned} &\mathbf{f}^{y}(t,\mathbf{X}_{t})=\mathbf{A}^{y}(t)\mathbf{X}_{t}^{y}+ \mathbf{a}^{y}(t), \\ & \mathbf{G}^{y}(t,\mathbf{X}_{t})=\mathbf{B}^{y}(t), \end{aligned} $$
in (13) with A y(t) (a p×p matrix), a y(t) (a p×1 vector), and B y(t) (a p×q matrix), these being measurable and bounded on [ t 0,T]. This results in
$$ \begin{aligned} \mathrm{d}\mathbf{X}_{t}^{y}=\!(\mathbf{A}^{y}(t)\mathbf{X}_{t}^{y}+ \mathbf{a}^{y}(t))\mathrm{d}t+\mathbf{B}^{y}(t)\mathrm{d}\mathbf{W}_{t}^{y}, \,\,\,\mathbf{X}_{t_{0}}^{y}(\omega)\!=c^{y}. \end{aligned} $$
This initial value problem has a unique solution that is a Gaussian stochastic process if and only if the initial conditions c y are constant or normally distributed (Theorem 8.2.10 [13]). Note that in this model, G y(t,X t ) (i.e. B y(t)) is independent of ω. Under this model, it can be shown that the mean (at a time index t i ) and the covariance matrix (at t i and t j ) of the Gaussian process \(\mathbf {X}_{t}^{y}\) are given by [13]
$$ \begin{aligned} \mathbf{m}_{t_{i}}^{y}=E\left[\!\mathbf{X}_{t_{i}}^{y}\right]=\boldsymbol{\Phi }^{y}(t_{i})\left(E[\!c^{y}]+\int_{t_{0}}^{t_{i}}\boldsymbol{\Phi }^{y}(s)^{-1}\mathbf{a}^{y}(s)\mathrm{d}s\right) \end{aligned} $$
$$ {\small{\begin{aligned} {}{\boldsymbol{\Psi }}_{t_{i},t_{j}}^{y}&=E\left[ \left(\mathbf{X}_{t_{i}}^{y}-E\left[\!\mathbf{X}_{t_{i}}^{y}\right]\right)\left(\mathbf{X}_{t_{j}}^{y}-E\left[\! \mathbf{X}_{t_{j}}^{y}\right]\right)^{T}\right] \\ & =\boldsymbol{\Phi }^{y}(t_{i})\left(E\left[\left(c^{y}-E[\!c^{y}]\right)\left(c^{y}-E[\!c^{y}]\right)^{T}\right]\right.\\ &\quad\left.+\int_{t_{0}}^{t_{i}}{\boldsymbol{ \Phi }^{y}(u)}^{-1}\mathbf{B}^{y}(u){\mathbf{B}^{y}(u)}^{T}\left({\boldsymbol{\Phi }^{y}(u)}^{-1}\right)^{T}\mathrm{d}u\right) {\boldsymbol{\Phi }^{y}(t_{j})}^{T}, \end{aligned}}} $$
where t 0≤t i ≤t j ≤T and Φ y(t i ) is the fundamental matrix of the deterministic equation
$$ \dot{\mathbf{X}}_{t}^{y}=\mathbf{A}^{y}(t)\mathbf{X}_{t}^{y}. $$
SDEs as perfect representatives for the dynamics of class-conditional processes
If the SDE model presented in (15) could perfectly represent the dynamics of the underlying stochastic processes of the BCGP model, then there would be no need for training sample paths. To see this, note that in this case \({\boldsymbol {\mu }}_{t}^{y}\) and \({\boldsymbol {\Sigma }} _{t_{i},t_{j}}^{y}\) defined in (6) and (7) are obtained by
$$ \begin{aligned} & {\boldsymbol{\mu }}_{\mathbf{t}_{N}}^{y}=\mathbf{m}_{\mathbf{t}_{N}}^{y} \\ & {\boldsymbol{\Sigma}}_{\mathbf{t}_{N}}^{y}={\boldsymbol{\Psi}}_{\mathbf{t}_{N}}^{y} \end{aligned}, $$
$$ {\mathbf{m}}_{\mathbf{t}_{N}}^{y}=\left[{\mathbf{m}_{t_{1}}^{y\,T}},{\mathbf{m} _{t_{2}}^{y\,T}},...,{\mathbf{m}_{t_{N}}^{y\,T}}\right]_{Np\times 1}^{T} $$
$$ \begin{aligned} \boldsymbol{\Psi }_{\mathbf{t}_{N}}^{y}\,=\, \left[ \begin{array}{cccc} \boldsymbol{\Psi }_{{t}_{1},{t}_{1}}^{y} &\boldsymbol{\Psi }_{t_{1},{t}_{2}}^{y} &... & \boldsymbol{\Psi }_{t_{1},{t}_{N}}^{y} \\[1ex] \boldsymbol{\Psi }_{{t}_{2},{t}_{1}}^{y} &\boldsymbol{\Psi }_{t_{2},{t}_{2}}^{y} &... & \boldsymbol{\Psi }_{t_{2},{t}_{N}}^{y} \\... &... &... &... \\ \boldsymbol{\Psi }_{{t}_{N},{t}_{1}}^{y} & \boldsymbol{\Psi }_{t_{N},{t}_{2}}^{y} &... &\boldsymbol{\Psi }_{t_{N},{t}_{N}}^{y}\end{array} \right]_{Np\times Np} \end{aligned}, $$
where \({\mathbf {m}_{t_{i}}^{y\,T}}\) and \(\boldsymbol {\Psi }_{{t}_{i},{t} _{j}}^{y}\) are obtained from (16) and (17), respectively. Therefore, one can obtain the exact (or at least approximately exact) values of the means and auto-covariances used to characterize the Gaussian processes involved in the BCGP model. To obtain \({\mathbf {m}_{t_{i}}^{y}}\) and \(\boldsymbol {\Psi }_{{t}_{i},{t}_{j}}^{y}\), two approaches can be taken. First, one may analytically solve (18) where possible and then use numerical methods to evaluate the integrations presented in (16) and (17). For example, if A y(t)=A y, i.e., being independent of t, the solution of (18) is given by a matrix exponential as
$$ \boldsymbol{\Phi }^{y}(t)=e^{\mathbf{A}^{y}(t-t_{0})}, $$
which can be used in (16) and (17). In general, where one may not be able to analytically solve (18), numerical methods such as the Euler-Maruyama scheme [14] can be used to directly solve for \( \mathbf {X}_{t}^{y}(\omega)\) and obtain
$$ {\small{\begin{aligned} \hat{\mathbf{m}}_{\mathbf{t}_{N}}^{y}&=\frac{1}{l^{y}}\sum_{i=1}^{l^{y}} \mathbf{x}_{\mathbf{t}_{N}}^{y,\text{SDE}}(\omega_{i}), \\ \hat{{\boldsymbol{\Psi}}}_{\mathbf{t}_{N}}^{y}&\,=\,\frac{1}{l^{y}-1}\!\sum_{i=1}^{l^{y}}\!\left(\mathbf{x}_{\mathbf{t}_{N}}^{y,\text{SDE}}(\omega_{i})\,-\,\bar{\mathbf{x}}_{\mathbf{t}_{N}}^{y,\text{SDE}}\right)\!\!\left(\!\mathbf{x}_{ \mathbf{t}_{N}}^{y,\text{SDE}}(\omega_{i})\,-\,\bar{\mathbf{x}}_{\mathbf{t}_{N}}^{y,\text{SDE}}\!\right)^{\!T}\!, \end{aligned}}} $$
where \(\mathbf {x}_{\mathbf {t}_{N}}^{y,\text {SDE}}(\omega _{i}),i=1,2,...,l^{y}\), are the generated sample paths obtained from solving SDEs. Since there is no restriction on generating an arbitrary number of sample paths from \(\mathbf {X}_{t}^{y}(\omega)\), one can take l y>>N p to have a positive definite \( \hat {{\boldsymbol {\Psi }}}_{\mathbf {t}_{N}}^{y}\) and, at the same time, obtain an accurate estimate of the actual values of \({\mathbf {m}}_{\mathbf {t}_{N}}^{y}\) and \({{\boldsymbol {\Psi }}}_{\mathbf {t}_{N}}^{y}\). In this approach, the knowledge of (16) and (17) is used in the existence of the limits \({\lim }_{l^{y}\rightarrow \infty }\,\hat {\mathbf {m}}_{\mathbf {t}_{N}}^{y}\) and \({\lim }_{l^{y}\rightarrow \infty }\,\hat {{\boldsymbol {\Psi }}}_{\mathbf {t}_{N}}^{y}\), i.e., justifies generating more sample paths as \({\lim }_{l^{y}\rightarrow \infty }\,\hat {\mathbf {m}}_{\mathbf {t}_{N}}^{y}={\mathbf {m}}_{\mathbf {t}_{N}}^{y}\) and \({\lim }_{l^{y}\rightarrow \infty }\,\hat {{\boldsymbol {\Psi }}}_{\mathbf {t}_{N}}^{y}={{\boldsymbol {\Psi }}}_{\mathbf {t}_{N}}^{y}\).
In any case, we can assume exact (approximately exact) values of \(\mathbf {m}_{t_{i}}^{0}\), \(\mathbf {m}_{t_{i}}^{1}\), \({\boldsymbol {\Psi }}_{t_{i},t_{j}}^{0}\), and \({\boldsymbol {\Psi }}_{t_{i},t_{j}}^{1}\) are available. The optimal discriminant in this case is obtained by using the conventional quadratic discriminant analysis (QDA), which is now defined by using the following statistic in (11):
$$ {\small{\begin{aligned} {}\psi_{\mathbf{t}_{N}}^{\text{QDA}}\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})\right)&\,=\,-\frac{1}{2}\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})\,-\,{\mathbf{m}}_{\mathbf{t}_{N}}^{0}\right)^{T}\!{{\boldsymbol{\Psi}}}_{\mathbf{t}_{N}}^{0\;-1}\!\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})-{\mathbf{m}}_{\mathbf{t}_{N}}^{0}\right) \\ & \quad+\frac{1}{2}\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})-{\mathbf{m}}_{\mathbf{t}_{N}}^{1}\right) \boldsymbol{\Psi }_{t_{N}}^{1\text{}-1}\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})\,-\,{\mathbf{m}}_{\mathbf{t}_{N}}^{1}\right)\\ &\quad+\frac{1}{2}\text{log}\frac{|\boldsymbol{\Psi}_{\mathbf{t}_{N}}^{1\;-1}|}{|\boldsymbol{\Psi }_{\mathbf{t}_{N}}^{0\;-1}|}-\log \frac{\alpha_{1}}{1-\alpha_{1}}. \end{aligned}}} $$
The use of (24) is justified by the fact that the BCGP classification reduces to differentiating independent observations of Np dimension generated from two multivariate Gaussian distributions. Therefore, taking the same set of machinery as in [15] results in (24). We restate that in this case where (19) holds, there is no need for utilizing the sample path measurements (training sample paths) in finding the discriminant (24). This is due to the fact that the statistical properties of a Gaussian process at t N are solely determined by \({\mathbf {m}}_{\mathbf {t}_{N}}^{y}\) and Ψ t N y and, as mentioned before, either closed-form solutions of these are available or they can be approximated element-wise with an arbitrary small error rate by generating a sufficiently large number of sample paths.
The optimal solution proposed in (24) is, in fact, a function of the observation time vector of future sample paths. Therefore, if a future sample point \(\mathbf {x}_{\mathbf {t}_{L}}^{y}(\omega _{s})\) is measured at an arbitrary time vector t L , which can be partially or totally different from t N , then the optimal discriminant \(\psi _{\mathbf {t}_{L}}\left (\mathbf {x}_{\mathbf {t}_{L}}^{y}(\omega _{s})\right)\) is obtained by determining the solution of SDEs at t L and replacing \({\mathbf {m}}_{\mathbf {t}_{N}}^{y}\) and \({\boldsymbol {\Psi }}_{\mathbf {t}_{N}}^{y}\) with \({\mathbf {m}}_{\mathbf {t}_{L}}^{y}\) and \({ \boldsymbol {\Psi }}_{\mathbf {t}_{L}}^{y}\), respectively, in (24).
SDEs as prior information for the dynamics of class-conditional processes
In practice, the SDEs usually do not provide complete description and are then viewed as prior knowledge concerning the underlying dynamics of the BCGP model. Since we assume that a Gaussian process governs both the dynamics of each class-conditional process (BCGP model in Section 3) and its corresponding set of SDEs (by using model (15)), incompleteness of the SDEs results from the fact that (19) does not necessarily hold. We make the following assumptions on the nature of the prior information to which the set of SDEs corresponding to each class give rise: (i) before observing the sample paths at an observation time vector, the SDEs characterize the only information that we have about the system and (ii) the statistical properties of all Gaussian processes that may generate the data are on average (over the parameter space) equivalent to the statistical properties determined from the SDEs. The latter statement will subsequently be formalized.
Assume that the parameters \(\boldsymbol {\mu }_{\mathbf {t}_{N}}^{y}\) and \(\boldsymbol {\Sigma }_{\mathbf {t}_{N}}^{y}\) defining the BCGP model constitute a realization of the random vector \(\mathbf {\theta }_{\mathbf {t}_{N}}^{y}=\left [\boldsymbol {\mu }_{\mathbf {t}_{N}}^{y},\boldsymbol {\Sigma }_{\mathbf {t}_{N}}^{y}\right ]\), where \(\mathbf {\theta }_{\mathbf {t}_{N}}^{y}\) has a prior distribution \(\pi (\mathbf {\theta }_{\mathbf {t}_{N}}^{y})\) parameterized by a set \(\left \{\breve {\mathbf {m}}_{\mathbf {t}_{N}}^{y},\breve {\boldsymbol {\Psi }}_{\mathbf {t}_{N}}^{y},\nu _{\mathbf {t}_{N}}^{y},\kappa _{\mathbf {t}_{N}}^{y}\right \}\) of hyperparameters. The quantities \(\nu _{\mathbf {t}_{N}}^{y}\) and \(\kappa _{\mathbf {t}_{N}}^{y}\) define our certainty about the prior knowledge (here, the set of SDEs presenting the dynamics of the model). If we take the conjugate priors for mean and covariance when the sampling is Gaussian, i.e., a normal-inverse-Wishart distribution (which depends on t N ), then
$$ {\small{\begin{aligned} {}\pi& \left(\mathbf{\theta}_{\mathbf{t}_{N}}^{y}\right)\propto |\boldsymbol{\Sigma }_{\mathbf{t}_{N}}^{y}|^{-(\kappa_{\mathbf{t}_{N}}^{y}+Np+1)/2}\text{exp}\left(-\frac{1}{2}\text{tr}\left(\breve{\boldsymbol{\Psi }}_{\mathbf{t}_{N}}^{y}\left(\boldsymbol{\Sigma }_{\mathbf{t}_{N}}^{y}\right)^{-1}\right)\right) \\ & \times\! |\boldsymbol{\Sigma }_{\mathbf{t}_{N}}^{y}|^{-1/2}\text{exp}\!\left(\!\!-\frac{\nu_{\mathbf{t}_{N}}^{y}}{2}\!\left(\boldsymbol{\mu }_{\mathbf{t}_{N}}^{y}\,-\,\mathbf{m}_{\mathbf{t}_{N}}^{y}\right)^{T}\!\left(\boldsymbol{\Sigma }_{\mathbf{t}_{N}}^{y}\right)^{-1}\!\left(\boldsymbol{\mu}_{\mathbf{t}_{N}}^{y}\,-\,\mathbf{m}_{\mathbf{t}_{N}}^{y}\right)\!\right)\!, \end{aligned}}} $$
with \(\boldsymbol {\mu }_{\mathbf {t}_{N}}^{y}\) and \(\boldsymbol {\Sigma }_{\mathbf {t}_{N}}^{y}\) defined in (6) and (7). Therefore, the above assumption (ii) on the nature of the prior information means that
$$ \begin{aligned} &\breve{\mathbf{m}}_{\mathbf{t}_{N}}^{y}=\mathbf{m}_{\mathbf{t}_{N}}^{y} \\ &\breve{{\boldsymbol{\Psi }}}_{\mathbf{t}_{N}}^{y}=\left(\kappa_{\mathbf{t}_{N}}^{y}-Np-1\right){\boldsymbol{\Psi }}_{\mathbf{t}_{N}}^{y}, \end{aligned} $$
with \(\mathbf {m}_{\mathbf {t}_{N}}^{y}\) defined by (16) and (20) and \({\boldsymbol {\Psi }}_{\mathbf {t}_{N}}^{y}\) defined by (17) and (21). To see (26), note that from (25) and independence of \(\boldsymbol {\mu }_{\mathbf {t}_{N}}^{y}\) and \( \boldsymbol {\Sigma }_{\mathbf {t}_{N}}^{y}\), we have \(E_{\pi }\left [\!\boldsymbol {\mu }_{\mathbf {t}_{N}}^{y}\right ]=\breve {\mathbf {m}}_{\mathbf {t}_{N}}^{y}\) and \( E_{\pi }\left [\!\boldsymbol {\Sigma }_{\mathbf {t}_{N}}^{y}\right ]=\frac {\breve {{\boldsymbol {\Psi }}}_{\mathbf {t}_{N}}^{y}}{\kappa _{\mathbf {t}_{N}}^{y}-Np-1} \) (the latter is the mean of an inverse-Wishart distribution). The more confident we are about an a priori set of SDEs that is supposed to represent the underlying stochastic processes at t N y, the larger we might choose the values of \(\nu _{\mathbf {t}_{N}}^{y}\) and \(\kappa _{\mathbf {t}_{N}}^{y}\) and the more concentrated become the priors of the mean and covariance about \(\mathbf {m}_{\mathbf {t}_{N}}^{y}\) and \({\boldsymbol {\Psi }}_{\mathbf {t}_{N}}^{y}\), respectively. To ensure a proper prior distribution, we assume \(\breve {\boldsymbol {\Psi }}_{\mathbf {t}_{N}}^{y}\) is positive definite, \(\kappa _{\mathbf {t}_{N}}^{y}>Np-1\), and \(\nu _{\mathbf {t}_{N}}^{y}>0\) for all t N (cf. p. 126 in [16], p. 178 in [17], and p. 427 in [3]).
Given the preceding framework for uncertainty in the BCGP model, the optimal Bayesian classification theory can be directly adapted. Specifically, the normal-inverse-Wishart distribution prior as defined in (25) and the independence of \(\mathbf {X}_{\mathbf {t}_{N}}^{y}(\omega _{s})\) from training sample paths resemble the same set of conditions as in [6], i.e., having a normal-inverse-Wishart distribution prior and independence of future data points from training data points. As a result, we can follow the same set of machinery to find the effective class-conditional distributions of the processes (similar to equation (64) in [6]) and from there obtain the optimal discriminant. Therefore, extending the dimensionality of the problem to Np and using the set of parameters \(\left \{\breve {\mathbf {m}}_{\mathbf {t}_{N}}^{y},\breve {\boldsymbol {\Psi }}_{\mathbf {t}_{N}}^{y},\nu _{\mathbf {t}_{N}}^{y},\kappa _{ \mathbf {t}_{N}}^{y}\right \}\) in the discriminant presented by Eq. (65) in [6] yields
$$ \begin{aligned} {}\psi_{\mathbf{t}_{N}}^{\text{OBC}}\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})\right)& =K\left(1+\frac{1}{k^{0}}\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})-{\mathbf{m}}_{\mathbf{t}_{N}}^{0\;\ast }\right)^{T}{\boldsymbol{\Pi}}_{\mathbf{t}_{N}}^{0\;-1}\right.\\ &\quad\times\left.\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})-{\mathbf{m}}_{\mathbf{t}_{N}}^{0\;\ast }\right)\vphantom{\frac{0}{0}}\right)^{k^{0}+Np} \\ &\quad-\left(1+\frac{1}{k^{1}}\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}\left(\omega_{s}\right)-{\mathbf{m}}_{\mathbf{t}_{N}}^{1\;\ast}\right)^{T}{\boldsymbol{\Pi}}_{\mathbf{t}_{N}}^{1\;-1}\right.\\ &\quad\times\left.\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{s})-{\mathbf{m}}_{\mathbf{t}_{N}}^{1\;\ast}\right)\vphantom{\frac{0}{0}}\right)^{k^{1}+Np}, \end{aligned} $$
$$ {{\begin{aligned} {}K\!=\left(\frac{\alpha_{1}}{1-\alpha_{0}}\right)^{2}\left(\frac{k^{0}}{k^{1}} \right)^{Np}\frac{|{\boldsymbol{\Pi }}_{\mathbf{t}_{N}}^{0}|}{|{\boldsymbol{\Pi}}_{\mathbf{t}_{N}}^{1}|}\left(\frac{\Gamma (k^{0}/2)\Gamma ((k^{1}+pN)/2)}{\Gamma (k^{1}/2)\Gamma ((k^{0}+pN)/2)}\right)^{2}, \end{aligned}}} $$
$$ {{\begin{aligned} &{\boldsymbol{\Pi}}_{\mathbf{t}_{N}}^{y}=\frac{\nu_{\mathbf{t}_{N}}^{y\;\ast}+1}{\left(\kappa_{\mathbf{t}_{N}}^{y\;\ast }-Np+1\right)\nu_{\mathbf{t}_{N}}^{y\;\ast }}{\boldsymbol{\Psi}}_{\mathbf{t}_{N}}^{y\;\ast },\\ &{\boldsymbol{\Psi}}_{\mathbf{t}_{N}}^{y\;\ast}\,=\,\breve{\boldsymbol{\!\Psi}}_{\mathbf{t}_{N}}^{y}\!\,+\,\!(n^{y}\!\,-\,\!1)\hat{\mathbf{\Sigma }}_{\mathbf{t}_{N}}^{y}\!\,+\,\!\frac{\nu_{\mathbf{t}_{N}}^{y}n^{y}}{\nu_{\mathbf{t}_{N}}^{y}+n^{y}}\!\left(\hat{\boldsymbol{\mu}}_{\mathbf{t}_{N}}^{y}\!\,-\,\breve{\mathbf{m}}_{\mathbf{t}_{N}}^{y}\right)\!\!\left(\hat{ \boldsymbol{\mu}}_{\mathbf{t}_{N}}^{y}\,-\,\breve{\mathbf{m}}_{\mathbf{t}_{N}}^{y}\right)^{T}\!, \\ &\nu_{\mathbf{t}_{N}}^{y\;\ast }=\nu_{\mathbf{t}_{N}}^{y}+n^{y},\quad \kappa_{\mathbf{t}_{N}}^{y\;\ast }=\kappa_{\mathbf{t}_{N}}^{y}+n^{y},\quad k^{y}=\kappa_{\mathbf{t}_{N}}^{y\;\ast }-Np+1, \\ &{\mathbf{m}}_{\mathbf{t}_{N}}^{y\;\ast }=\frac{\nu_{\mathbf{t}_{N}}^{y}\breve{\mathbf{m}}_{\mathbf{t}_{N}}^{y}+n^{y}\hat{ \boldsymbol{\mu}}_{\mathbf{t}_{N}}^{y}}{\nu_{\mathbf{t}_{N}}^{y}+n^{y}}, \end{aligned}}} $$
where \(\breve {\mathbf {m}}_{\mathbf {t}_{N}}^{y}\) and \(\breve {{\boldsymbol { \Psi }}}_{\mathbf {t}_{N}}^{y}\) are determined from (26), and \(\hat { \mathbf {\Sigma }}_{\mathbf {t}_{N}}^{y}\) and \(\hat {\boldsymbol {\mu }}_{\mathbf {t}_{N}}^{y}\) are the sample mean and sample covariance matrix obtained by using the sample path training sets \(\mathbf {S}_{\mathbf {t}_{N}}^{0}\) and \(\mathbf {S}_{\mathbf {t}_{N}}^{1}\) as follows:
$$ {{\begin{aligned} {} \hat{\boldsymbol{\mu}}_{\mathbf{t}_{N}}^{y}&=\frac{1}{n^{y}}\sum_{i=1}^{n^{y}} \mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{i}) \,,\\ {}\hat{\boldsymbol{\Sigma}}_{\mathbf{t}_{N}}^{y}&=\frac{1}{n^{y}-1}\sum_{i=1}^{n^{y}}\left(\mathbf{x}_{\mathbf{t}_{N}}^{y}(\omega_{i})\,-\,\hat{\boldsymbol{\mu}}_{\mathbf{t}_{N}}^{y}\right)\left(\mathbf{x}_{ \mathbf{t}_{N}}^{y}(\omega_{i})\,-\,\hat{\boldsymbol{\mu}}_{\mathbf{t}_{N}}^{y}\right)^{T}\,. \end{aligned}}} $$
As opposed to Section 4.1, where the discriminant can be applied to any future sample path with an arbitrary observation time vector, here, the discriminant depends on both the future and training observation time vectors. Thus, if the future observation time vector t L y contains only a set of time points t i where t i ∈t N y, one may easily apply the optimal discriminant. This is easily doable by reducing the dimensionality of the problem by considering the training sample paths only at t L y, i.e., by discarding the training sample points at those t N y not in t L y(denoted by t N y∖t L y). However, solving the case where t L y includes time points not included in t N y is more difficult and requires further study. In this case, although one is able to construct the class of prior knowledge for t L y (i.e., constructing \({\boldsymbol {\mu }} _{\mathbf {t}_{N}}^{y}\) and \(\boldsymbol {\Psi }_{\mathbf {t}_{N}}^{y}\)), the paucity of training sample paths at t L y∖t N y does not permit employing (27).
In this section, we analyze the effect of prior knowledge in the form of stochastic differential equations on the performance of the stochastic discriminant, \(\psi _{\mathbf {t}_{N}}^{OBC}\left (\mathbf {x}_{\mathbf {t} _{N}}^{y}(\omega _{s})\right)\), defined by (27)–(29). As the metric of performance, we take the true error averaged over the sampling space. The true error of a discriminant trained on an observation time vector t N , i.e., \(\psi _{\mathbf {t}_{N}\phantom {\dot {i}\!}}(.)\), is the probability of misclassification, which by considering (11) is defined as
$$ {{\begin{aligned} {}\epsilon_{\mathbf{t}_{N}}\!\,=\,\sum_{y=0}^{1}\alpha_{\mathbf{t}_{N}}^{y}P\!\left((-1)^{y}\psi_{\mathbf{t}_{N}}\!\left(\mathbf{X}_{\mathbf{t}_{N}}^{y}(\omega_{s})\right)\!>\!0\!\mid\!\mathbf{S}_{\mathbf{t}_{N}}^{0},\mathbf{S}_{\mathbf{t}_{N}}^{1}, \mathbf{X}_{\mathbf{t}_{N}}^{y}(\omega_{s})\!\in\! \mathbf{X}^{y}\right)\!, \end{aligned}}} $$
where X y denotes the class-conditional process that generates the future sample path \(\mathbf {X}_{\mathbf {t}_{N}}^{y}(\omega _{s})\) (we assume independence of future sample paths from training sample paths), \(\mathbf {S}_{\mathbf {t}_{N}}^{y}\) denotes the set of training sample paths from class y, and \(\alpha _{\mathbf {t}_{N}}^{y}\) is the mixing probability of the class-conditional process.
Recall that in this work, we consider a separate sampling scheme. With separate sampling in a classical binary classification problem where sample points are generated from two class-conditional densities, there is no sensible estimate of prior probabilities of classes from the sample [15]. In that case, either the ratio of the number of sample points in either class to the total sample size needs to reflect the corresponding prior probability of the class or the prior probabilities need to be known a priori; otherwise, classification rules or error estimation rules suffer performance degradation [15, 18, 19]. The same argument applies to this work in which we consider a binary classification of sample paths that are generated from two class-conditional processes under a separate sampling scheme. In this regard, we assume that the prior probability \(\alpha _{\mathbf {t}_{N}}^{y}\) is known a priori.
Taking expectation over the sample space, that is over the mixture of Gaussian processes with the means and covariance matrices defined by (16), (20), (17), and (21), yields
$$ {\small{\begin{aligned} {}E[\!\epsilon_{\mathbf{t}_{N}}]\,=\,\sum_{y=0}^{1}\alpha_{\mathbf{t}_{N}}^{y}P\!\left((-1)^{y}\psi_{\mathbf{t}_{N}}\!\left(\mathbf{X}_{\mathbf{t}_{N}}^{y}(\omega_{s})\right)\!>\!0\!\mid\!\! \,\mathbf{X}_{\mathbf{t}_{N}}^{y}(\omega_{s})\in\! \mathbf{X}^{y}\right). \end{aligned}}} $$
As benchmarks for evaluating the performance of \(\psi _{\mathbf {t}_{N}}^{\text {OBC}} \left (\mathbf {x}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\), we compare its performance to (1) the performance of the stochastic QDA, \(\psi _{\mathbf {t}_{N}}^{\text {QDA}}\left (\mathbf {X}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\), which is defined by (23) and (24), where l y =n y , with n y indicating the number of available sample paths, and (2) the performance of a Bayes classifier obtained by plugging (16), (17), (20), and (21), into (24).
Synthetic experiments
Experimental set-up
The following steps are used to set up the experiments:
To fix the ground-truth model governing the underlying dynamics of the data, we consider a set of three-dimensional SDEs (p=3) defined by (15) along with the following set of parameters:
$$ {\small{\begin{aligned} \mathbf{A}^{0}(t)&=\mathbf{A}^{1}(t)=[\!0.01,0.01,0.01]^{T},\\ \mathbf{a}^{0}(t)&=\mathbf{a}^{1}(t)=[\!0,0,0]^{T}, \\ &\mathbf{X}_{t_{0}}^{0}(\omega)=[\!0,0,0]^{T},\quad \mathbf{X}_{t_{0}}^{1}(\omega)=[\!0.25,0.25,0.25]^{T}, \\ &\mathbf{B}^{0}(t)=\mathbf{B}^{1}(t)=0.1\!\times \left\{ \begin{array}{ll} \sigma^{2}=1 &\text{diagonal elements} \\ \rho =0.4 & \text{otherwise} \end{array}\right.. \end{aligned}}} $$
The only difference between the SDEs describing X 0 and X 1 is in the constant initial conditions. Figure 1 presents a single sample path of these two three-dimensional processes for 0≤t≤100.
A single sample path taken from the two three-dimensional processes described by the set of parameters introduced in (33)
Use the ground-truth set of SDEs to generate a set of training sample paths, \(\mathbf {S}_{\mathbf {t}_{N}}^{y}\), of size n y for class y=0,1. We let n 0=n 1=n, where n∈, let the length of the observation time vector be N=20, and take [ t 1,t 2,...,t N ] such that t i −t i−1=1, i=2,…,20.
Use the ground-truth set of SDEs to generate a set of test sample paths, \(\mathbf {S}_{\mathbf {t}_{N}}^{y,\,\text {test}}\), of size n y, test=2,000 for class y=0,1, where n 0, test=n 1, test=n test.
Use \(\mathbf {S}_{\mathbf {t}_{N}}^{0}\cup \mathbf {S}_{\mathbf {t} _{N}}^{1}\) to train the stochastic QDA, \(\psi _{\mathbf {t}_{N}}^{\text {QDA}}\left (\mathbf {x}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\), which is defined by (23) and (24) with l y=n y. Apply the trained classifier to the set of test sample paths, \(\mathbf {S}_{\mathbf {t} _{N}}^{0,\,\text {test}}\cup \mathbf {S}_{\mathbf {t}_{N}}^{1,\,\text {test}}\), to determine the true error, \(\epsilon _{\mathbf {t}_{N}}^{\text {QDA}}\), which is defined by replacing \(\psi _{\mathbf {t}_{N}}\left (\mathbf {X}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\) with \(\psi _{\mathbf {t}_{N}}^{\text {QDA}}\left (\mathbf {X}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\) in (31). This procedure obtains an accurate estimate of true error.
Assume a set of SDEs obtained from prior knowledge (a priori SDEs). Let this a priori set of SDEs be presented by replacing A y(t), B y(t), A y(t), and \(\mathbf {X} _{t_{0}}^{y}(\omega)\) in (15) with \(\tilde {\mathbf {A}}^{y}(t)\), \(\tilde {\mathbf {B}}^{y}(t)\), \(\tilde {\mathbf {A}}^{y}(t)\), and \(\tilde {\mathbf { X}}_{t_{0}}^{y}(\omega)\), respectively. To examine the effects of deviations in the drift vector and dispersion matrix in the a priori set of SDEs from the ground-truth model introduced in (33), we assume
\(\tilde {\mathbf {A}}^{0}(t)={\mathbf {A}}^{0}(t)\), \(\tilde {\mathbf {B}} ^{0}(t)={\mathbf {B}}^{0}(t)\), \(\tilde {\mathbf {X}}^{0}_{t_{0}}(\omega)= \mathbf {X}_{t_{0}}^{0}(\omega)\), \(\tilde {\mathbf {X}}^{1}_{t_{0}}(\omega)= \mathbf {X}_{t_{0}}^{1}(\omega)\), \(\tilde {\mathbf {a}}^{0}(t)=\mathbf {a} ^{0}(t),\tilde {\mathbf {a}}^{1}(t)=\mathbf {a}^{1}(t)\).
To study the effect of shift in the drift vector, we take \(\tilde { \mathbf {A}}^{1}(t)={\mathbf {A}}^{1}(t)+[\!\Delta \mu,\Delta \mu,\Delta \mu ]^{T}\), where Δ μ=0,0.1,0.2,0.3. Here we assume \(\tilde {\mathbf {B}} ^{1}(t)={\mathbf {B}}^{1}(t)\).
To study the effect of shift in the dispersion matrix, we assume the off-diagonal elements of \(\tilde {\mathbf {B}}^{1}(t)\) are defined by replacing ρ with ρ d in (33), where ρ d −ρ=Δ ρ=0,0.03,0.06,0.1. Here we assume \(\tilde {\mathbf {A}} ^{1}(t)={\mathbf {A}}^{1}(t)\).
The hyperparameters defining our uncertainty about the specific choice of a priori SDEs (in fact, about the resultant prior distributions) are \(\nu _{\mathbf {t}_{N}}^{0}=\nu _{\mathbf {t}_{N}}^{1}=\kappa _{\mathbf {t} _{N}}^{0}=\kappa _{\mathbf {t}_{N}}^{1}=Np+\kappa \). The choice of N p+κ, κ=20,50,100,500, is made to have proper prior distributions (see Section 4.2).
Generate 2,000 sample paths from the a priori set of SDEs introduced in Step 5. These sample paths are used to calculate the hyperparameters \(\mathbf {m}_{\mathbf {t}_{N}}^{y}\) and \({\boldsymbol {\Psi }}_{ \mathbf {t}_{N}}^{y}\) being used in (26) (alternatively, one may solve (16), (17), (20), and (21) directly and use them in (26)).
Use \(\mathbf {m}_{\mathbf {t}_{N}}^{y}\) and \({\boldsymbol {\Psi }}_{ \mathbf {t}_{N}}^{y}\) obtained from Step 6 along with \(\mathbf {S}_{\mathbf {t} _{N}}^{0}\cup \mathbf {S}_{\mathbf {t}_{N}}^{1}\) to train \(\psi _{\mathbf {t} _{N}}^{\text {OBC}}\left (\mathbf {x}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\), which is defined in (27). Apply the trained classifier to the set of test sample paths, \(\mathbf {S}_{\mathbf {t}_{N}}^{0,\,\text {test}}\cup \mathbf {S}_{ \mathbf {t}_{N}}^{1,\,\text {test}}\), to determine the true error, \(\epsilon _{ \mathbf {t}_{N}}^{\text {OBC}}\), which is defined by replacing \(\psi _{\mathbf {t}_{N}} \left (\mathbf {X}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\) with \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}\left (\mathbf {x}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\) in (31).
Repeat Steps 2 through 7 a total of T=1,000 times to estimate \( E\left [\epsilon _{\mathbf {t}_{N}}^{\text {QDA}}\right ]\) and \(E\left [\epsilon _{\mathbf {t}_{N}}^{\text {OBC}}\right ] \).
Generate 2,000 sample paths from the ground-truth set of SDEs introduced in (33). Use these sample paths to train the stochastic QDA, \(\psi _{\mathbf {t}_{N}}^{\text {QDA}}\left (\mathbf {x}_{\mathbf {t} _{N}}^{y}(\omega _{s})\right)\), which is defined by (23) and (24) with l y=2,000. This provides an accurate estimate of the Bayes (optimal) classifier. Apply this classifier to \(\mathbf {S}_{\mathbf {t} _{N}}^{0,\,\text {test}}\cup \mathbf {S}_{\mathbf {t}_{N}}^{1,\,\text {test}}\) to obtain the Bayes error, which is a lower bound on the error of any classifier. Note that in our experiments obtaining the Bayes error is possible since we have complete knowledge of the underlying ground-truth models.
Figure 2 shows the effect of a shift in the drift vector from the ground-truth model via plots of the expected true error of \(\psi _{\mathbf {t}_{N}}^{\text {QDA}}(.)\) and \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\) as functions of the size of training sample paths and κ for y=0,1, \(\tilde {\mathbf {B}}^{y}(t)={\mathbf {B}}^{y}(t)\), \(\tilde {\mathbf {X}} ^{y}_{t_{0}}(\omega)=\mathbf {X}_{t_{0}}^{y}(\omega)\), \(\tilde {\mathbf {A}} ^{0}(t)={\mathbf {A}}^{0}(t)\), and \(\tilde {\mathbf {A}}^{1}(t)={\mathbf {A}} ^{1}(t)+[\!\Delta \mu,\Delta \mu,\Delta \mu ]^{T}\), where Δ μ=0,0.1,0.2,0.3. If the set of a priori SDEs is equivalent or close to the ground-truth model, e.g., Δ μ=0 or Δ μ=0.1, then \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\) outperforms \(\psi _{\mathbf {t}_{N}}^{\text {QDA}}(.)\) for a wide range of training sample sizes and κ. The more the prior distribution generated from the set of a priori SDEs is concentrated about the true underlying parameters of the model and the larger κ, the better is the performance achieved by using \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\).
Expected true error, \(E\left [\protect \epsilon ^{\text {QDA}}_{\mathbf {t}_{N}}\right ]\) and \(E\left [\protect \epsilon ^{\text {OBC}}_{\mathbf {t}_{N}}\right ]\), as a function of number of training sample paths in each class and various choices of Δ μ and κ. The dashed line shows the Bayes error
Figure 3 presents the effect of the discrepancy between the dispersion matrix of the ground-truth model and that of the a priori set of SDEs. Again, the closer the prior knowledge is to the ground-truth model and the larger κ, the better is the performance achieved by using \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\).
Expected true error, \(E\left [\epsilon ^{\text {QDA}}_{\mathbf {t}_{N}}\right ]\) and \(E\left [\epsilon ^{\text {OBC}}_{\mathbf {t}_{N}}\right ]\), as a function of number of training sample paths in each class and various choices of Δ ρ and κ. The dashed line shows the Bayes error
An experiment inspired by a model of the evolutionary process
In this section, we use a form of an Ornstein-Uhlenbeck process introduced in [20] for modeling the evolutionary change of species. This model has been recently employed by [21] to simulate quantitative trait data as a function of single nucleotide polymorphism (SNP) states. The model is presented by the following SDE:
$$ \mathrm{d}{X}_{t}^{y}=-\beta^{y}\left[{\!X_{t}^{y}}-\theta^{y}\right]\,\mathrm{d}t+\sigma^{y} \mathrm{d}{W}_{t}^{y},\quad {X_{0}^{y}}={X_{a}^{y}}, $$
where \({X}_{t}^{y}\) is the quantitative trait value in a species y, θ y is the primary target value of the trait, \({X_{a}^{y}}\) is the mean state in an ancestor a, and \({W}_{t}^{y}\) represents Brownian motion. The parameter β y is the rate of adaptation of species y to the target value—a low rate of adaptation means very slow evolution while a large β y practically indicates an instantaneous adaptation. The parameter σ y is an indicator of perturbation due to random selective factors such as random mutations and environmental fluctuations [20]. Similar to [21], we assume the value of the primary target is constant over the history of the species. Nevertheless, the model in (34) can be extended to include situations where the primary target can change over the evolutionary history of the species (see [20]).
Using the model of (34), we generate the evolutionary histories of a quantitative trait of two species, 0 and 1, over a time span of 30 million years with time steps of 1 million years. Similarly to [20, 21], to fix the ground-truth model that generates the data, we vary values of β y, take σ y=1, and assume θ 0=80 and θ 1=85. Furthermore, we assume both species have a common ancestor at the state \({X_{a}^{y}}=1\). Figure 4 presents 20 sample paths from each of these evolutionary processes for the case where β 0=β 1=β, β=0.1 (Fig. 4 a) and β=0.15 (Fig. 4 b). A larger β indicates a faster adaptation of species to the target value. The problem considered here is to use a set of a priori SDEs in constructing a classifier to differentiate the evolutionary history of an n-size population of species 0 from an n-size population of species 1, where n∈[ 60,140].
Multiple sample paths taken from the two one-dimensional evolutionary processes for two values of adaptation rate, (a): β=0.1; (b): β=0.15
The general protocol for evaluating the performance of ψ t N OBC(.) is similar to Section 5.1, except for replacing the ground-truth model (33) with (34) and using the following the step instead of Step 5:
Assume a set of SDEs obtained from prior knowledge (a priori SDEs). Let this a priori set of SDEs be presented by replacing β y, θ y, \({X_{a}^{y}}\), and σ y by \(\tilde {\beta } ^{y}\), \(\tilde {\theta }^{y}\), \(\tilde {X}_{a}^{y}\), and \(\tilde {\sigma }^{y}\), respectively, in (34). To examine the effect of deviation of the adaptation rate in the a priori set of SDEs from the ground-truth model, we let \( \tilde {\theta }^{y}={\theta }^{y}\), \(\tilde {\sigma }^{y}={\sigma }^{y}\), \( \tilde {X}_{a}^{y}={X}^{y}\), and \(\tilde {\beta }^{0}={\beta }^{0}\) and take \( \tilde {\beta }^{1}={\beta }^{1}+\Delta \beta \).
Figures 5 and 6 (β=0.1 and β=0.15, respectively) show the effect of a deviation from the true rate of adaptation to the target value by considering \(\tilde {\beta }^{1}={\beta } ^{1}+\Delta \beta,\) where Δ β=0, 0.02, 0.04, 0.06. They provide plots of the expected true error of \(\psi _{\mathbf {t}_{N}}^{\text {QDA}} (.)\) and \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\) as functions of the size of training sample paths and κ. In both figures, the closer the prior knowledge is to the ground-truth evolutionary models, the better is the performance achieved by using \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\). The performance deteriorates and eventually becomes worse than \(\psi _{ \mathbf {t}_{N}}^{\text {QDA}}(.)\) as the prior knowledge diverges from the ground-truth model and the certainty about the prior knowledge increases (a bad combination when utilizing prior knowledge). In addition, comparing Figs. 5 and 6 shows that the smaller is the true value of β and the more destructive is a fixed deviation of prior knowledge from the true β.
Expected true error, \(E\big [\protect \epsilon ^{\text {QDA}}_{\mathbf {t}_{N}}\big ]\) and \(E\left [\protect \epsilon ^{\text {OBC}}_{\mathbf {t}_{N}}\right ]\), as a function of number of training sample paths in each class and various choices of Δ β and κ and for β=0.1. The dashed line shows the Bayes error
Expected true error, \(E\big [\protect \epsilon _{\mathbf {t}_{N}}^{\text {QDA}}\big ]\) and \(E\left [\protect \epsilon _{\mathbf {t}_{N}}^{\text {OBC}}\right ]\), as a function of number of training sample paths in each class and various choices of Δ β and κ and for β =0.15. The dashed line shows the Bayes error
This paper provides the first instance in which prior knowledge in the form of SDEs is used to construct a prior distribution over an uncertainty class of feature-label distributions for the purpose of optimal classification. Given the ubiquity of small samples in biomedicine and other areas where sample data is expensive, time-consuming, limited by regulation, or simply unavailable, we have previously made the point that prior knowledge is the only avenue available. To achieve the mapping of SDE prior knowledge into a prior distribution, we have taken advantage of the form and Gaussianity of (12). This mapping is heavily dependent on the form of the SDEs, and one can expect widely varying mappings for different SDE settings.
In general, all parameters used in the a priori set of SDEs can affect the performance of \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\). These parameters include every element of the matrices \(\tilde {\mathbf {A}}^{y}(t)\) and \(\tilde {\mathbf {B}}^{y}(t)\) and all the elements of the vectors \(\tilde { \mathbf {a}}^{y}(t)\) and \(\tilde {\mathbf {X}^{y}}_{t_{0}}(\omega)\) used in the SDE's presentation in (15). For example, in the experiment of the evolutionary change of species considered in (34), a deviation from each of the parameters, namely \(\tilde {\beta }^{y}\), \(\tilde {\sigma } ^{y} \), \(\tilde {\theta }^{y}\), and \(\tilde {X}_{a}^{y},\) can affect the performance of \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\). Although simulation studies can elucidate the effects of deviation of prior knowledge from the ground-truth model (as done herein), it would be beneficial to analytically characterize the performance of \(\psi _{\mathbf {t}_{N}}^{\text {OBC}} (.)\) in terms of all the hyperparameters; however, this may be very difficult to accomplish. One possible approach may be to use an asymptotic Bayesian framework [22] to characterize the performance of \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}(.)\) in terms of sample size, dimensionality, and hyperparameters.
Recognizing that the construction of robust classifiers is simply a special case of optimal Bayesian classification where there are no sample data, so that the "posterior" is identical to the prior [7], the application of SDEs in this paper is at once applicable to optimal robust classification in a stochastic setting. Beyond that, one can consider the more general setting of optimal Bayesian robust filtering of random processes, where optimization across an uncertainty class of random processes, ideal and observed, is relative to process characteristics such as the auto- and cross-correlation functions [23]. Whereas in this paper we have considered using SDE prior knowledge to construct prior distributions governing uncertainty classes of feature-label distributions, it seems feasible to use SDE knowledge to construct prior distributions governing uncertainty classes of random-process characteristics in the case of optimal filtering. Of course, one must confront the increased abstraction presented by canonical representation of random processes [24, 25]; nevertheless, so long as one remains in the framework of second-order canonical expansions, it should be doable.
Definition of q-dimensional Wiener process
A one-dimensional Wiener process over [ 0,T] is a Gaussian process W={W t :t≥0} satisfying the following properties:
For 0≤t 1<t 2<T, \(W_{t_{2}}-W_{t_{1}}\) is distributed as \( \sqrt {t_{2}-t_{1}}N\left (0,\sigma ^{2}\right)\), where σ>0 (for the standard Wiener process, σ=1).
For 0≤t 1<t 2<t 3<t 4<T, \(W_{t_{4}}-W_{t_{3}}\) is independent of \(W_{t_{2}}-W_{t_{1}}\).
W 0=0 with probability 1.
The sample paths of W are almost surely continuous everywhere.
In general, a q-dimensional Wiener process is defined using the homogenous Markov process X t for t∈[t 0,T]. Let \(P(t_{1},x;t_{2}, B)=P(\mathbf {X}_{t_{2}\phantom {\dot {i}\!}} \in B|\mathbf {X}_{t_{1}\phantom {\dot {i}\!}}=x)\) denote the transition probabilities of a Markov process X t for t 1<t 2. For fixed values of t 1, x, and t 2, P(t 1,x;t 2,.) is a probability function (measure) on the σ-algebra \(\mathcal {B}\) of Borel subsets of the sample space R q. Intuitively, P(t 1,x;t 2,B) is the probability that the process be in the set \(B\in \mathcal {B}\) at time t 2 given it was in state x at time t 1. A Markov process is homogenous with respect to t if its transition probability P(t 1,x;t 2,B) is stationary. That is, for t 0<t 1<t 2<T and t 0<t 1+u<t 2+u<T, it satisfies
$$ P(t_{1}+u,x;t_{2}+u, B)=P(t_{1},x;t_{2}, B). $$
In this case P(t 1,x;t 2,B) is commonly denoted by P(t 2−t 1,x;B). A q-dimensional Wiener process is a q-dimensional homogenous Markov process defined on [0,∞) with stationary transition probability defined by a multivariate Gaussian distribution as follows:
$$ P(t,x;B)=\int_{B} \frac{1}{(2\pi t)^{d/2}}e^{-\frac{|y-x|^{2}}{2t}} dy. $$
Therefore, each dimension of a q-dimensional Wiener process is a one-dimensional Wiener process per se.
The computational complexity of the algorithm is determined by the computational cost of solving the set of SDEs from the Euler-Maruyama scheme (see Section 4.1) along with the computational cost of evaluating (27). The computational cost of the Euler-Maruyama scheme per sample path is inversely proportional to Δ t [26], where Δ t=T/N, with T and N being defined in Section 3. Thus, for l=l 0+l 1 sample paths, it is O(l/Δ t). In (27), the computational cost of evaluating \(\mathbf {x}_{\mathbf {t}_{N}}^{y}(\omega _{s})-{\mathbf {m}}_{\mathbf {t}_{N}}^{0\;\ast },\) with y=0,1, breaks down to a computation of \(\breve {\mathbf {m}}_{\mathbf {t}_{N}}^{y}\) and \(\hat {\boldsymbol {\mu }}_{\mathbf {t}_{N}}^{y}\), which are operations with computational costs of O(l y N p) and O(n y N p), respectively.
Computation of \({{\boldsymbol {\Pi }}_{\mathbf {t}_{N}}^{y\;-1}}\) in (27) by Gaussian elimination is an O(max{n y,N p}N 2 p 2)+O(l y N 2 p 2) operation (cf. section 3.7.2 in [27]). This will be further simplified because, in order to have a positive definite \(\breve {{ \boldsymbol {\Psi }}}_{\mathbf {t}_{N}}^{y}\), we assume we generate many sample paths by solving the set of SDEs such that l y>>N p (see Section 4.1), but since \({\boldsymbol {\Psi }}_{\mathbf {t}_{N}}^{y\;\ast }\) and \({{ \boldsymbol {\Pi }}_{\mathbf {t}_{N}}^{y\;-1}}\) defined in (29) become positive definite, we do not need to impose the condition of n y>N p. Having a realistic assumption on the number of available sample paths, we can assume l y>>n y, and therefore, the computation of \({{\boldsymbol {\Pi } }_{\mathbf {t}_{N}}^{y\;-1}}\) becomes an O(l y N 2 p 2) calculation. Furthermore, the product of \(\mathbf {x}_{\mathbf {t}_{N}}^{y}(\omega _{s})-{ \mathbf {m}}_{\mathbf {t}_{N}}^{0\;\ast }\) with \({{\boldsymbol {\Pi }}_{\mathbf { t}_{N}}^{y\;-1}}\) is an O(N 2 p 2) calculation. Altogether, by assuming 1/Δ t<(N p)2 and k 0+k 1+N p<(N p)2, the overall computational cost of \(\psi _{\mathbf {t}_{N}}^{\text {OBC}}\left (\mathbf {x}_{\mathbf {t}_{N}}^{y}(\omega _{s})\right)\) is O(max{l 0,l 1}N 2 p 2).
Using a similar approach, we see that the computational cost of QDA, which is solely constructed by using n 0+n 1 training sample paths from classes 0 and 1 (i.e., no prior knowledge) is O(max{n 0,n 1}N 2 p 2). We also note that for computing QDA we need to have min{n 0,n 1}>N p because, otherwise, the sample covariance matrices used in QDA are not invertible.
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Department of Electrical and Electronic Engineering, Nazarbayev University, Astana, 010000, Kazakhstan
Amin Zollanvari
The Center for Bioinformatics and Genomic Systems Engineering and the Department of Electrical and Computer Engineering, Texas A&M University, College Station, 77840, Texas
Edward R. Dougherty
Correspondence to Amin Zollanvari.
Additional file 1
Supplementary information. I. Definition of QDA in a classical setting. II. Error estimation accuracy. III. Bayesian MMSE error estimator. IV. Review of literature pertaining to classification of stochastic processes.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Zollanvari, A., Dougherty, E.R. Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations. J Bioinform Sys Biology 2016, 2 (2016). https://doi.org/10.1186/s13637-016-0036-y
DOI: https://doi.org/10.1186/s13637-016-0036-y
Stochastic differential equations
Optimal Bayesian classifier
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\begin{document}
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\title{Model theoretic-characterization of predicate intuitionistic formulas} \author{Grigory K. Olkhovikov\\ Department of Ontology and Cognition Theory\\ Ural Federal University\\ Fulbright Visiting Scholar at the Philosophy Dept,\\Stanford University \\Bldg 90, Stanford, CA, USA} \date{} \maketitle \begin{quote} {\bf Abstract.} Notions of asimulation and $k$-asimulation introduced in \cite{Ol} are extended onto the level of predicate logic. We then prove that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula iff it is invariant with respect to $k$-asimulations for some $k$, and then that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula over a class of intuitionistic models (intuitionistic models with constant domain) iff it is invariant with respect to asimulations between intuitionistic models (intuitionistic models with constant domain). \end{quote}
Van Benthem's well-known modal characterization theorem shows that expressive power of modal propositional logic as a fragment of first-order logic can be described via the notion of bisimulation invariance. Moreover, it is known that modal predicate logic, initially considered as an extension of first-order logic, can also be viewed as its fragment, although somewhat bigger than the fragment induced by propositional modal logic. Expressive power of modal predicate logic, from this vantage point, is described by the notion of world-object bisimulation which appears to be a rather direct combination of bisimulation and partial isomorphism (see, e.\,g. \cite[p. 124, Theorem 21]{vB}).
Although intuitionistic logic has been treated as a fragment of modal logic for quite a long while, results analogous to propositional and predicate version of Van Benthem's modal characterization theorem were not obtained for it until recently. In \cite{Ol} we filled this gap for intuitionistic propositional logic. In this paper we introduced the notion of asimulation and its parametrized version, $k$-asimulation, and showed that they can be used to characterize expressive power of intuitionistic propositional logic in much the same way bisimulation and $k$-bisimulation are used to characterize modal propositional logic. In this paper we do the same job for intuitionistic predicate logic without identity.
The layout of the paper is as follows. Starting from some notational conventions and preliminary remarks in section \ref{S:Prel}, we then define a predicate version of $k$-asimulation and move on to the proof of a `parametrized' version of model-theoretic characterization of intuitionistic predicate logic in section \ref{S:Param}. Then, in section \ref{S:Main}, we introduce the predicate version of asimulation and prove the full unparametrized counterpart to Theorem 21 of \cite{vB}. In section \ref{S:Rest} we discuss possibilities of restriction of the latter result to special subclasses of first-order models and the final sections contains some conclusions, and mentions possible directions of further research.
\section{Preliminaries}\label{S:Prel}
We take $\mathbb{N}$ to be the set of natural numbers \emph{without} $0$. A formula is a formula of classical predicate logic with identity whose predicate letters are in a vocabulary $\Sigma = \{\,R^2,E^2\,\} \cup \{\,P^n_m \mid n,m \in\mathbb{N}\,\}$, where the upper subscript denotes the arity of the letter, so $0$-ary predicate letters or propositional letters are not allowed. We refer to formulas with Greek letters distinct from $\alpha$ and $\beta$, and to sets of formulas with upper-case Greek letters distinct from $\Sigma$ and $\Theta$. We refer to variables with letters $w, x, y, z$, sometimes using primes or subscripts. If $\varphi$ is a formula, then we associate with it the following finite vocabulary $\Sigma_\varphi \subseteq \Sigma$ such that $\Sigma_\varphi = \{\,R^2, E^2\,\} \cup \{\,P^j_i \mid P^j_i \text{ occurs in }\varphi\,\}$. More generally, we refer with $\Theta$ to an arbitrary subset of $\Sigma$ such that $R^2, E^2 \in \Theta$. If $\psi$ is a formula and every predicate letter occurring in $\psi$ is in $\Theta$, then we call $\psi$ a $\Theta$-formula.
We refer to sequence $x_1,\dots, x_n$ of any objects as $\bar{x}_n$. We denote ordered pair of ordered $n$-tuple $(\bar{x}_n)$ and ordered $m$-tuple $(\bar{y}_m)$ by $(\bar{x}_n;\bar{y}_m)$. We identify ordered $1$-tuple with its only member. We denote the ordered $0$-tuple by $\Lambda$. If all free variables of a formula $\varphi$ (set of formulas $\Gamma$) are among $\bar{x}_n$, we write $\varphi(\bar{x}_n)$ ($\Gamma(\bar{x}_n)$).
For a binary relation $S$ and any objects $s, t$ we abbreviate the fact that $sSt \wedge tSs$ by $s\hat{S}t$.
We will denote models of classical predicate logic by letters $M$, $N$ or $\alpha, \beta$. We refer to the domain of a model $M$ by $D(M)$. For $n \geq 0$ by an $n$-ary evaluation $\Theta$-point we mean a sequence $(M, a, \bar{b}_n)$ such that $M$ is a $\Theta$-model and $(a,\bar{b}_n)$ is a sequence of elements of $D(M)$. If $(M, a, \bar{b}_n)$ is an $n$-ary evaluation point then we say that $\varphi(x, \bar{w}_n)$ is true at $(M, a, \bar{b}_n)$ and write $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$ iff for any variable assignment $f$ in $M$ such that $f(x) = a$, $f(w_i) = b_i$ for any $1 \leq i \leq n$ we have $M, f \models \varphi(x, \bar{w}_n)$. It follows from this convention that truth of a formula $\varphi(x, \bar{w}_n)$ at an $n$-ary evaluation point is to some extent independent of a choice of its free variables.
An intuitionistic formula is a formula of intuitionistic predicate logic without identity. Propositional (i.\,e. $0$-ary predicate) letters are allowed. We refer to intuitionistic formulas with letters $i, j, k$, possibly with primes or subscripts. Their variables are represented in the same way as in formulas. We assume a standard Kripke semantics for intuitionistic predicate logic where in a given world a predicate letter might be true only for some tuples of objects present in this world.
If $x$ is an individual variable in a first-order language, then by a standard $x$-translation of intuitionistic formulas into formulas we mean the following map $ST$ defined by induction on the complexity of the corresponding intuitionistic formula. First we assume some map of intuitionistic predicate letters into classical ones which correlates with each $n$-ary intuitionistic predicate letter $P$ an $(n + 1)$-ary classical predicate letter $P'$ distinct from $R^2, E^2$. We assume that this correlation is surjective, that is, that every predicate letter in $\Sigma$ distinct from $R^2, E^2$ is standard translation of an intuitionistic predicate letter. Then our induction goes as follows: \begin{align*} &ST(P(\bar{w}_n), x) = P'(x, \bar{w}_n);\\ &ST(\bot, x) = (x \neq x);\\ &ST(i(\bar{w}_n) \wedge j(\bar{w}_n), x) = ST(i(\bar{w}_n), x) \wedge ST(j(\bar{w}_n), x);\\ &ST(i(\bar{w}_n) \vee j(\bar{w}_n), x) = ST(i(\bar{w}_n), x) \vee ST(j(\bar{w}_n), x);\\ &ST(i(\bar{w}_n) \to j(\bar{w}_n), x) = \forall y(R(x, y) \to (ST(i(\bar{w}_n), y) \to ST(j(\bar{w}_n), y)));\\ &ST(\exists w'i(\bar{w}_n, w'), x) = \exists w'(E(x,w') \wedge ST(i(\bar{w}_n, w'), x));\\ &ST(\forall w'i(\bar{w}_n, w'), x) = \forall yw'((R(x, y) \wedge E(y, w')) \to ST(i(\bar{w}_n, w'), y)). \end{align*}
Standard conditions are imposed on the variables $x, y, \bar{w}_n, w'$.
By degree of a formula we mean the greatest number of nested quantifiers occurring in it. A degree of a formula $\varphi$ is denoted by $r(\varphi)$. Its formal definition by induction on the complexity of $\varphi$ goes as follows: \begin{align*} &r(\varphi) = 0 &&\text{for atomic $\varphi$}\\ &r(\neg\varphi) = r(\varphi)\\ &r(\varphi \circ \psi) = max(r(\varphi), r(\psi)) &&\text{for $\circ \in \{\,\wedge, \vee, \to\,\}$}\\ &r(Qx\varphi) = r(\varphi) + 1 &&\text{for $Q \in \{\,\forall, \exists\,\}$} \end{align*}
If $k \in \mathbb{N}$ and $\varphi(x, \bar{w}_n)$ is a $\Theta$-formula such that $r(\varphi) \leq k$, then $\varphi$ is a $(\Theta, (x, \bar{w}_n), k)$-formula.
\section{Characterization of intuitionistic predicate formulas via $k$-asimulations}\label{S:Param}
We begin with extending our previous notion of $k$-asimulation to cover the general case of predicate logic. \begin{definition}\label{D:k-asim} Let $(M, a, \bar{b}_n)$, $(N, c, \bar{d}_n)$ be two $n$-ary evaluation $\Theta$-points. A binary relation \[ A \subseteq \bigcup_{m \geq 1, l \geq 0}(((D(M)^m \times D(M)^l) \times (D(N)^m \times D(N)^l)) \cup ((D(N)^m \times D(N)^l) \times (D(M)^m \times D(M)^l))), \] is called $\langle (M, a, \bar{b}_n), (N, c, \bar{d}_n)\rangle_k$-asimulation iff $(a; \bar{b}_n)A(c; \bar{d}_n)$ and for any $\alpha, \beta \in \{\,M, N\,\}$, any $(\bar{a'}_m, a';\bar{b'}_l) \in D(\alpha)^{m+1} \times D(\alpha)^l$, $(\bar{c'}_m, c';\bar{d'}_l) \in D(\beta)^{m+1} \times D(\beta)^l$, whenever we have $(\bar{a'}_m, a';\bar{b'}_l)A(\bar{c'}_m, c';\bar{d'}_l)$, the following conditions hold:
\begin{align} &\forall P \in \Theta\setminus\{\,R^2,E^2\,\}(\alpha, a', \bar{b'}_l\models P(x, \bar{w}_l) \Rightarrow \beta, c', \bar{d'}_l\models P(x, \bar{w}_l))\label{E:c1}\\ &(m + l < n + k \wedge c'' \in D(\beta) \wedge c'R^\beta c'') \Rightarrow\notag\\ &\Rightarrow \exists a'' \in D(\alpha)(a'R^\alpha a'' \wedge (\bar{c'}_m, c', c'';\bar{d'}_l)\hat{A}(\bar{a'}_m, a', a'';\bar{b'}_l));\label{E:c2}\\ &(m + l < n + k \wedge b'' \in D(\alpha) \wedge E^\alpha(a', b'')) \Rightarrow\notag\\ &\Rightarrow \exists d'' \in D(\beta)(E^\beta(c',d'') \wedge (\bar{a'}_m, a';\bar{b'}_l, b'')A(\bar{c'}_m,c';\bar{d'}_l, d''));\label{E:c3}\\ &(m + l + 1 < n + k \wedge c'', d'' \in D(\beta) \wedge c'R^\beta c''\wedge E^\beta(c'', d'')) \Rightarrow\notag\\ &\Rightarrow \exists a'',b'' \in D(\alpha)(a'R^\alpha a''\wedge E^\alpha(a'',b'') \wedge (\bar{a'}_m, a', a'';\bar{b'}_l, b'')A(\bar{c'}_m,c', c'';\bar{d'}_l, d'')).\label{E:c4} \end{align} \end{definition}
\begin{lemma}\label{L:asim} Let $\varphi(x, \bar{w}_n) = ST(i(\bar{w}_n), x)$ for some intuitionistic formula $i(\bar{w}_n)$, and let $r(\varphi) = k$. Let $\Sigma_\varphi \subseteq \Theta$, let $(M, t,\bar{u}_s)$, $(N, t', \bar{u'}_s)$ be two $s$-ary evaluation $\Theta$-points, and let $A$ be an $\langle(M, t,\bar{u}_s), (N, t', \bar{u'}_s)\rangle_p$-asimulation. Then \begin{align*} &\forall\alpha, \beta \in \{\,M, N\,\}\forall(\bar{a}_m, a;\bar{b}_n) \in (D(\alpha)^{m+1} \times D(\alpha)^n)\forall(\bar{c}_m, c;\bar{d}_n)\in (D(\beta)^{m+1} \times D(\beta)^n)\\ &(((\bar{a}_m, a;\bar{b}_n)A(\bar{c}_m, c;\bar{d}_n) \wedge m + n + k \leq p + s \wedge \alpha, a, \bar{b}_n\models \varphi(x, \bar{w}_n) \Rightarrow \beta, c,\bar{d}_n \models \varphi(x, \bar{w}_n)). \end{align*} \end{lemma} \begin{proof} We proceed by induction on the complexity of $i$. In what follows we will abbreviate the induction hypothesis by IH.
\emph{Basis.} Let $i(\bar{w}_n) = P(\bar{w}_n)$. Then $\varphi(x,\bar{w}_n) = P'(x,\bar{w}_n)$ and we reason as follows: \begin{align} &(\bar{a}_m, a;\bar{b}_n)A(\bar{c}_m, c;\bar{d}_n)\label{E:0l1} &&\text{(premise)}\\ &\alpha, a, \bar{b}_n\models P'(x,\bar{w}_n)\label{E:0l1-1} &&\text{(premise)}\\ &P' \in \Theta\setminus\{\,R^2,E^2\,\}\label{E:0l1-2} &&\text{(by $\Sigma_\varphi \subseteq \Sigma'$)}\\ &\forall Q \in \Theta\setminus\{\,R^2,E^2\,\}(\alpha, a, \bar{b}_n\models Q(x, \bar{w}_n) \Rightarrow \beta, c, \bar{d}_n\models Q(x, \bar{w}_n))\label{E:0l2} &&\text{(from \eqref{E:0l1} by \eqref{E:c1})}\\ &\alpha, a, \bar{b}_n\models P'(x,\bar{w}_n) \Rightarrow \beta, b,\bar{d}_n \models P'(x,\bar{w}_n)\label{E:0l3} &&\text{(from \eqref{E:0l1-2} and \eqref{E:0l2})}\\ &\beta, c,\bar{d}_n\models P'(x,\bar{w}_n)\label{E:0l4} &&\text{(from \eqref{E:0l1-1} and \eqref{E:0l3})} \end{align}
The case $i = \bot$ is obvious.
\emph{Induction step.}
\emph{Case 1.} Let $i(\bar{w}_n) = j(\bar{w}_n) \wedge k(\bar{w}_n)$. Then $\varphi(x, \bar{w}_n) = ST(j(\bar{w}_n), x) \wedge ST(k(\bar{w}_n),x)$ and we reason as follows:
\begin{align} &(\bar{a}_m, a;\bar{b}_n)A(\bar{c}_m, c;\bar{d}_n)\label{E:1l1} &&\text{(premise)}\\ &\alpha, a, \bar{b}_n\models ST(j(\bar{w}_n), x) \wedge ST(k(\bar{w}_n),x)\label{E:1l2} &&\text{(premise)}\\ &m + n + r(ST(j(\bar{w}_n), x) \wedge ST(k(\bar{w}_n),x)) \leq p + s\label{E:1l3} &&\text{(premise)}\\ &r(ST(j(\bar{w}_n), x)) \leq r(ST(j(\bar{w}_n), x) \wedge ST(k(\bar{w}_n),x))\label{E:1l4} &&\text{(by df of $r$)}\\ &r(ST(k(\bar{w}_n), x)) \leq r(ST(j(\bar{w}_n), x) \wedge ST(k(\bar{w}_n),x))\label{E:1l5} &&\text{(by df of $r$)}\\ &\alpha, a, \bar{b}_n\models ST(j(\bar{w}_n), x)\label{E:1l6} &&\text{(from \eqref{E:1l2})}\\ &\alpha, a, \bar{b}_n\models ST(k(\bar{w}_n), x)\label{E:1l7} &&\text{(from \eqref{E:1l2})}\\ &m + n + r(ST(j(\bar{w}_n), x)) \leq p + s\label{E:1l8} &&\text{(from \eqref{E:1l3} and \eqref{E:1l4})}\\ &m + n + r(ST(k(\bar{w}_n), x)) \leq p + s\label{E:1l9} &&\text{(from \eqref{E:1l3} and \eqref{E:1l5})}\\ &\beta, c,\bar{d}_n \models ST(j(\bar{w}_n), x)\label{E:1l10} &&\text{(from \eqref{E:1l1}, \eqref{E:1l6} and \eqref{E:1l8} by IH)}\\ &\beta, c,\bar{d}_n \models ST(k(\bar{w}_n), x)\label{E:1l11} &&\text{(from \eqref{E:1l1}, \eqref{E:1l7} and \eqref{E:1l9} by IH)}\\ &\beta, c,\bar{d}_n \models ST(j(\bar{w}_n), x) \wedge ST(k(\bar{w}_n),x)\label{E:1l12} &&\text{(from \eqref{E:1l10} and \eqref{E:1l11})} \end{align}
\emph{Case 2.} Let $i(\bar{w}_n) = j(\bar{w}_n) \vee k(\bar{w}_n)$. Then $\varphi(x, \bar{w}_n) = ST(j(\bar{w}_n), x) \vee ST(k(\bar{w}_n),x)$ and we have then $\alpha, a, \bar{b}_n\models ST(j(\bar{w}_n), x) \vee ST(k(\bar{w}_n),x)$. Assume, without a loss of generality, that $\alpha, a, \bar{b}_n\models ST(j(\bar{w}_n), x)$. Then we reason as follows:
\begin{align} &\alpha, a,\bar{b}_n\models ST(j(\bar{w}_n), x)\label{E:2l0} &&\text{(premise)}\\ &(\bar{a}_m, a;\bar{b}_n)A(\bar{c}_m, c;\bar{d}_n)\label{E:2l1} &&\text{(premise)}\\ &m + n + r(ST(j(\bar{w}_n), x) \vee ST(k(\bar{w}_n),x)) \leq p + s\label{E:2l2} &&\text{(premise)}\\ &r(ST(j(\bar{w}_n), x)) \leq r(ST(j(\bar{w}_n), x) \vee ST(k(\bar{w}_n),x))\label{E:2l3} &&\text{(by df of $r$)}\\ &m + n + r(ST(j(\bar{w}_n), x)) \leq p + s\label{E:2l4} &&\text{(from \eqref{E:2l2} and \eqref{E:2l3})}\\ &\beta, c,\bar{d}_n \models ST(j(\bar{w}_n), x)\label{E:2l5} &&\text{(from \eqref{E:2l0}, \eqref{E:2l1} and \eqref{E:2l4} by IH)}\\ &\beta, c,\bar{d}_n \models ST(j(\bar{w}_n), x) \vee ST(k(\bar{w}_n),x)\label{E:2l6} &&\text{(from \eqref{E:2l5})} \end{align}
\emph{Case 3.} Let $i(\bar{w}_n) = j(\bar{w}_n) \to k(\bar{w}_n)$. Then \[ \varphi(x, \bar{w}_n) = \forall y(R(x, y) \to (ST(j(\bar{w}_n), y) \to ST(k(\bar{w}_n), y))). \] Let \[ \alpha, a,\bar{b}_n \models \forall y(R(x, y) \to (ST(j(\bar{w}_n), y) \to ST(k(\bar{w}_n), y))), \] and let \[ \beta, c,\bar{d}_n \models \exists y(R(x, y) \wedge (ST(j(\bar{w}_n), y) \wedge \neg ST(k(\bar{w}_n), y))). \] This means that we can choose a $c' \in D(\beta)$ such that $cR^\beta c'$ and $\beta, c', \bar{d}_n\models ST(j(\bar{w}_n), y) \wedge \neg ST(k(\bar{w}_n), y)$. We now reason as follows: \begin{align} &\beta, c', \bar{d}_n\models ST(j(\bar{w}_n), y) \wedge \neg ST(k(\bar{w}_n), y)\label{E:4l00} &&\text{(by choice of $c'$)}\\ &c' \in D(\beta) \wedge cR^\beta c'\label{E:4l0} &&\text{(by choice of $c'$)}\\ &(\bar{a}_m, a;\bar{b}_n)A(\bar{c}_m, c;\bar{d}_n)\label{E:4l1} &&\text{(premise)}\\ &m + n + r(\varphi(x, \bar{w}_n)) \leq p + s\label{E:4l2} &&\text{(premise)}\\ &r(\varphi(x, \bar{w}_n))) \geq 1 \label{E:4l3} &&\text{(by df of $r$)}\\ &m + n < p + s\label{E:4l4} &&\text{(from \eqref{E:4l2} and \eqref{E:4l3})}\\ &\exists a' \in D(\alpha)(aR^\alpha a' \wedge (\bar{c}_m, c,c';\bar{d}_n)\hat{A}(\bar{a}_m, a,a';\bar{b}_n))\label{E:4l5} &&\text{(from \eqref{E:4l0}, \eqref{E:4l1} and \eqref{E:4l4} by\eqref{E:c2})} \end{align} Now choose an $a'$ for which \eqref{E:4l5} is satisfied; we add the premises following from our choice of $a'$ and continue our reasoning as follows: \begin{align} &a' \in D(\alpha) \wedge aR^\alpha a'\label{E:4l6} &&\text{(by choice of $a'$)}\\ &(\bar{c}_m, c,c';\bar{d}_n)A(\bar{a}_m, a,a';\bar{b}_n)\label{E:4l7} &&\text{(by choice of $a'$)}\\ &(\bar{a}_m, a,a';\bar{b}_n)A(\bar{c}_m, c,c';\bar{d}_n)\label{E:4l7-1} &&\text{(by choice of $a'$)}\\ &r(ST(j(\bar{w}_n), y)) \leq r(\varphi(x,\bar{w}_n)) - 1\label{E:4l8} &&\text{(by df of $r$)}\\ &r(ST(k(\bar{w}_n), y)) \leq r(\varphi(x,\bar{w}_n)) - 1\label{E:4l8-1} &&\text{(by df of $r$)}\\ &m + 1 + n + r(ST(j(\bar{w}_n), y)) \leq p + s\label{E:4l9} &&\text{(from \eqref{E:4l2} and \eqref{E:4l8})}\\ &m + 1 + n + r(ST(k(\bar{w}_n), y)) \leq p + s\label{E:4l9-1} &&\text{(from \eqref{E:4l2} and \eqref{E:4l8-1})}\\ &\alpha, a',\bar{b}_n \models ST(j(\bar{w}_n), x)\label{E:4l10} &&\text{(from \eqref{E:4l00}, \eqref{E:4l7}, \eqref{E:4l9} by IH)}\\ &\alpha, a',\bar{b}_n \models \neg ST(k(\bar{w}_n), x)\label{E:4l10-1} &&\text{(from \eqref{E:4l00}, \eqref{E:4l7-1}, \eqref{E:4l9-1} by IH)}\\ &\alpha, a',\bar{b}_n \models ST(j(\bar{w}_n), y) \wedge \neg ST(k(\bar{w}_n), y)\label{E:4l10-2} &&\text{(from \eqref{E:4l10}, \eqref{E:4l10-1})}\\ &\alpha, a,\bar{b}_n \models \exists y(R(x, y) \wedge (ST(j(\bar{w}_n), y) \wedge \neg ST(k(\bar{w}_n), y)))\label{E:4l11} &&\text{(from \eqref{E:4l6} and \eqref{E:4l10-2})} \end{align} The last line contradicts our initial assumption that \[ \alpha, a,\bar{b}_n \models \forall y(R(x, y) \to (ST(j(\bar{w}_n), y) \to ST(k(\bar{w}_n), y))), \]
\emph{Case 4.} Let $i(\bar{w}_n) = \exists w' j(\bar{w}_n, w')$. Then \[ \varphi(x, \bar{w}_n) = \exists w'(E(x, w') \wedge ST(j(\bar{w}_n, w'), x)). \] Let $\alpha, a, \bar{b}_n \models \exists w'(E(x, w') \wedge ST(j(\bar{w}_n, w'), x))$. This means that we can choose a $b' \in D(\alpha)$ such that $aE^\alpha b'$ and $\alpha, a, \bar{b}_n,b' \models ST(j(\bar{w}_n, w'), x)$. We now reason as follows: \begin{align} &\alpha, a, \bar{b}_n,b' \models ST(j(\bar{w}_n, w'), x)\label{E:5l0} &&\text{(by choice of $b'$)}\\ &b' \in D(\alpha) \wedge E^\alpha(a, b')\label{E:5l1} &&\text{(by choice of $b'$)}\\ &(\bar{a}_m, a;\bar{b}_n)A(\bar{c}_m, c;\bar{d}_n)\label{E:5l2} &&\text{(premise)}\\ &m + n + r(\varphi(x, \bar{w}_n)) \leq p + s\label{E:5l3} &&\text{(premise)}\\ &r(\varphi(x, \bar{w}_n)) \geq 1 \label{E:5l4} &&\text{(by df of $r$)}\\ &m + n < p + s\label{E:5l5} &&\text{(from \eqref{E:5l3} and \eqref{E:5l4})}\\ &\exists d' \in D(\beta)(E^\beta(c,d') \wedge (\bar{a}_m, a; \bar{b}_n, b')A(\bar{c}_m,c;\bar{d}_n, d'))\label{E:5l6} &&\text{(from \eqref{E:5l1}, \eqref{E:5l2} and \eqref{E:5l5} by \eqref{E:c3})} \end{align} Now choose a $d'$ for which \eqref{E:5l6} is satisfied; we add the premises following from our choice of $d'$ and continue our reasoning as follows: \begin{align} &d' \in D(\beta) \wedge E^\beta(c,d')\label{E:5l7} &&\text{(by choice of $d'$)}\\ &(\bar{a}_m, a; \bar{b}_n, b')A(\bar{c}_m,c;\bar{d}_n, d')\label{E:5l8} &&\text{(by choice of $d'$)}\\ &r(ST(j(\bar{w}_n,w'), x)) = r(\varphi(x, \bar{w}_n)) - 1\label{E:5l9} &&\text{(by df of $r$)}\\ &m + n + 1 + r(ST(j(\bar{w}_n,w'), x)) \leq p + s\label{E:5l10} &&\text{(from \eqref{E:5l3} and \eqref{E:5l9})}\\ &\beta, c, \bar{d}_n, d' \models ST(j(\bar{w}_n,w'), x)\label{E:5l11} &&\text{(from \eqref{E:5l0}, \eqref{E:5l8}, \eqref{E:5l10} by IH)}\\ &\beta, c, \bar{d}_n \models \exists w'(E(x, w') \wedge ST(j(\bar{w}_n, w'), x))\label{E:5l12} &&\text{(from \eqref{E:5l7} and \eqref{E:5l11})} \end{align}
\emph{Case 5}. Let $i(\bar{w}_n) = \forall w'j(\bar{w}_n, w')$. Then \[ \varphi(x, \bar{w}_n) = \forall yw'((R(x,y) \wedge E(y, w')) \to ST(j(\bar{w}_n, w'), y)). \] Let \[ \alpha, a, \bar{b}_n \models \forall yw'((R(x,y) \wedge E(y, w')) \to ST(j(\bar{w}_n, w'), y)), \] and let \[ \beta, c, \bar{d}_n \models \exists yw'((R(x,y) \wedge E(y, w')) \wedge \neg ST(j(\bar{w}_n, w'), y)). \] The latter fact means that we can choose some $c',d' \in D(\beta)$ such that $cR^\beta c'$, $E^\beta(c',d')$, and $\beta, c', \bar{d}_n, d' \models \neg ST(j(\bar{w}_n, w'), y)$. We now reason as follows: \begin{align} &\beta, c', \bar{d}_n, d' \models \neg ST(j(\bar{w}_n, w'), y)\label{E:6l0} &&\text{(by choice of $c', d'$)}\\ &c' \in D(\beta) \wedge cR^\beta c'\label{E:6l1} &&\text{(by choice of $c'$)}\\ &d' \in D(\beta) \wedge E^\beta(c',d')\label{E:6l1-1} &&\text{(by choice of $c', d'$)}\\ &(\bar{a}_m, a;\bar{b}_n)A(\bar{c}_m, c;\bar{d}_n)\label{E:6l2} &&\text{(premise)}\\ &m + n + r(\varphi(x, \bar{w}_n)) \leq p + s\label{E:6l3} &&\text{(premise)}\\ &r(\varphi(x, \bar{w}_n))) \geq 2 \label{E:6l4} &&\text{(by df of $r$)}\\ &m + n + 1 < p + s\label{E:6l5} &&\text{(from \eqref{E:6l3} and \eqref{E:6l4})} \end{align} \begin{align} \exists a'b' \in D(aR^\alpha a'\wedge E^\alpha(a',b') \wedge (\bar{a}_m, a, a'; \bar{b}_n, b')A(\bar{c}_m,c, c';\bar{d}_n, d'))\label{E:6l6}\\ \text{(from \eqref{E:6l1}, \eqref{E:6l1-1}, \eqref{E:6l2} and \eqref{E:6l5} by\eqref{E:c4})}\notag \end{align}
Then choose $a',b' \in D(\alpha)$ for which \eqref{E:6l6} is satisfied. We add the premises following from our choice of $a', b'$ and continue our reasoning as follows:
\begin{align} &a' \in D(\alpha) \wedge aR^\alpha a'\label{E:6l7} &&\text{(by choice of $a'$)}\\ &b' \in D(\alpha) \wedge E^\alpha(a',b')\label{E:6l7-1} &&\text{(by choice of $a', b'$)}\\ &(\bar{a}_m, a, a'; \bar{b}_n, b')A(\bar{c}_m,c, c';\bar{d}_n, d')\label{E:6l8} &&\text{(by choice of $a', b'$)}\\ &r(\neg ST(j(\bar{w}_n, w'), y)) = r(\varphi(x, \bar{w}_n)) - 2\label{E:6l9} &&\text{(by df of $r$)}\\ &m + 1 + n + 1 + r(\neg ST(j(\bar{w}_n, w'), y)) \leq p + s\label{E:6l10} &&\text{(from \eqref{E:6l3} and \eqref{E:6l9})}\\ &\alpha, a', \bar{b}_n,b' \models \neg ST(j(\bar{w}_n, w'), y)\label{E:6l11} &&\text{(from \eqref{E:6l0}, \eqref{E:6l8}, \eqref{E:6l10} by IH)}\\ &\alpha, a, \bar{b}_n \models \exists yw'((R(x,y) \wedge E(y, w')) \wedge \neg ST(j(\bar{w}_n, w'), y))\label{E:6l12} &&\text{(from \eqref{E:6l7}, \eqref{E:6l7-1} and \eqref{E:6l11})} \end{align} The last line contradicts our initial assumption that \[ \alpha, a, \bar{b}_n \models \forall yw'((R(x,y) \wedge E(y, w')) \to ST(j(\bar{w}_n, w'), y)). \] \end{proof}
\begin{definition}\label{D:k-inv} A formula $\varphi(x, \bar{w}_n)$ is invariant with respect to $k$-asimulations iff for any $\Theta$ such that $\Sigma_\varphi \subseteq \Theta$, any two $n$-ary evaluation $\Theta$-points $(M, a, \bar{b}_n)$ and $(N, c, \bar{d}_n)$, if there exists a $\langle (M, a, \bar{b}_n), (N, c, \bar{d}_n)\rangle_k$-asimulation $A$ and $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$, then $N, c, \bar{d}_n \models \varphi(x, \bar{w}_n)$. \end{definition}
\begin{corollary}\label{L:c-k-inv} If $\varphi(x, \bar{w}_n)$ is a standard $x$-translation of an intuitionistic formula and $r(\varphi) = k$, then $\varphi(x, \bar{w}_n)$ is invariant with respect to $k$-asimulations. \end{corollary}
Corollary \ref{L:c-k-inv} immediately follows from Lemma \ref{L:asim} setting $\alpha = M$, $\beta = N$, $m = 0$, $p = k$, $t = a$, $\bar{u}_s = \bar{b}_n$, $t' = c$, $\bar{u'}_s = \bar{d}_n$.
Before we state and prove our main result, we need to mention a fact from classical model theory of first-order logic.
\begin{lemma}\label{L:fin} For any finite $\Theta$ and any natural $n, k$ there are, up to logical equivalence, only finitely many $(\Theta, (x, \bar{w}_n), k)$-formulas. \end{lemma}
This fact is proved as Lemma 3.4 in \cite[pp. 189--190]{EFT}.
\begin{definition}\label{D:conj} Let $\varphi(x, \bar{w}_n)$ be a formula. A conjunction of $(\Sigma_\varphi,(x, \bar{w}_n), k)$-formulas $\Psi(x, \bar{w}_n)$ is called a complete $(\varphi, (x, \bar{w}_n), k)$-conjunction iff (1) every conjunct in $\Psi(x)$ is a standard $x$-translation of an intuitionistic formula; and (2) there is an $n$-ary evaluation point $(M, a, \bar{b}_n)$ such that $M, a, \bar{b}_n \models \Psi(x, \bar{w}_n) \wedge \varphi(x, \bar{w}_n)$ and for any $(\Sigma_\varphi, (x, \bar{w}_n),k)$-formula $\psi(x, \bar{w}_n)$, if $\psi(x, \bar{w}_n)$ is a standard $x$-translation of an intuitionistic formula and $M, a, \bar{b}_n \models \psi(x, \bar{w}_n)$, then $\Psi(x, \bar{w}_n) \models\psi(x, \bar{w}_n)$. \end{definition}
\begin{lemma}\label{L:conj-ex} For any formula $\varphi(x, \bar{w}_n)$, any natural $k$, and any $n$-ary evaluation point $(M, a, \bar{b}_n)$ such that $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$ there is a complete $(\varphi,(x, \bar{w}_n),k)$-conjunction $\Psi(x, \bar{w}_n)$ such that $M, a, \bar{b}_n \models \Psi(x, \bar{w}_n) \wedge \varphi(x, \bar{w}_n)$. \end{lemma} \begin{proof} Let $\{\,\psi_1(x, \bar{w}_n)\ldots, \psi_n(x, \bar{w}_n),\ldots\,\}$ be the set of all $(\Sigma_\varphi, (x, \bar{w}_n), k)$-formulas that are standard $x$-translations of intuitionistic formulas true at $(M, a, \bar{b}_n)$. This set is non-empty since $ST(\bot \to \bot, x)$ will be true at $(M, a, \bar{b}_n)$. Due to Lemma \ref{L:fin}, we can choose in this set a non-empty finite subset $\{\,\psi_{i_1}(x, \bar{w}_n)\ldots, \psi_{i_n}(x, \bar{w}_n)\,\}$ such that any formula from the bigger set is logically equivalent to (and hence follows from) a formula in this subset. Therefore, every formula in the bigger set follows from $\psi_{i_1}(x, \bar{w}_n) \wedge\ldots, \wedge\psi_{i_n}(x, \bar{w}_n)$ and we also have $M, a, \bar{b}_n \models \psi_{i_1}(x, \bar{w}_n) \wedge\ldots, \wedge\psi_{i_n}(x, \bar{w}_n)$, therefore, $\psi_{i_1}(x, \bar{w}_n) \wedge\ldots, \wedge\psi_{i_n}(x, \bar{w}_n)$ is a complete $(\varphi, (x, \bar{w}_n), k)$-conjunction. \end{proof}
\begin{lemma}\label{L:conj-fin} For any formula $\varphi(x, \bar{w}_n)$ and any natural $k$ there are, up to logical equivalence, only finitely many complete $(\varphi,(x, \bar{w}_n), k)$-conjunctions. \end{lemma} \begin{proof} It suffices to observe that for any formula $\varphi(x, \bar{w}_n)$ and any natural $k$, a complete $(\varphi,(x, \bar{w}_n), k)$-conjunction is a $(\Sigma_\varphi, (x, \bar{w}_n), k)$-formula. Our lemma then follows from Lemma \ref{L:fin}. \end{proof}
In what follows we adopt the following notation for the fact that for any sequence $(x, \bar{w}_n)$ of variables all $(\Sigma_\varphi, (x, \bar{w}_n), k)$-formulas that are standard translations of intuitionistic formulas true at $(M, a, \bar{b}_n)$, are also true at $(N, c, \bar{d}_n)$: \[ (M, a, \bar{b}_n) \leq_{\varphi, n, k} (N, c, \bar{d}_n). \]
\begin{theorem}\label{L:t1} Let $r(\varphi(x, \bar{w}_n)) = k$ and let $\varphi(x, \bar{w}_n)$ be invariant with respect to $k$-asimulations. Then $\varphi(x, \bar{w}_n)$ is equivalent to a standard $x$-translation of an intuitionistic formula. \end{theorem} \begin{proof} We may assume that both $\varphi(x)$ and $\neg\varphi(x)$ are satisfiable, since both $\bot$ and $\top$ are obviously invariant with respect to $k$-asimulations and we have, for example, the following valid formulas:
\begin{align*} &\bot \leftrightarrow ST(\bot,x), \top \leftrightarrow ST(\bot \to \bot, x). \end{align*}
We may also assume that there are two complete $(\varphi, (x, \bar{w}_n), k + 2)$-conjunctions $\Psi(x, \bar{w}_n), \Psi'(x, \bar{w}_n)$ such that $\Psi'(x, \bar{w}_n)\models\Psi(x, \bar{w}_n)$, and both formulas $\Psi(x, \bar{w}_n) \wedge \varphi(x, \bar{w}_n)$ and $\Psi'(x, \bar{w}_n) \wedge \neg\varphi(x, \bar{w}_n)$ are satisfiable.
For suppose otherwise. Then take the set of all complete $(\varphi, (x, \bar{w}_n),k + 2)$-conjunctions $\Psi(x, \bar{w}_n)$ such that the formula $\Psi(x, \bar{w}_n) \wedge \varphi(x, \bar{w}_n)$ is satisfiable. This set is non-empty, because $\varphi(x, \bar{w}_n)$ is satisfiable, and by Lemma \ref{L:conj-ex}, it can be satisfied only together with some complete $(\varphi, (x, \bar{w}_n),k + 2)$-conjunction. Now, using Lemma \ref{L:conj-fin}, choose in it a finite non-empty subset $\{\,\Psi_{i_1}(x, \bar{w}_n)\ldots, \Psi_{i_n}(x, \bar{w}_n)\,\}$ such that any complete $(\varphi, (x, \bar{w}_n),k + 2)$-conjunction is equivalent to an element of this subset. We can show that $\varphi(x, \bar{w}_n)$ is logically equivalent to $\Psi_{i_1}(x, \bar{w}_n)\vee\ldots \vee\Psi_{i_n}(x, \bar{w}_n)$. In fact, if $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$ then, by Lemma \ref{L:conj-ex}, at least one complete $(\varphi, (x, \bar{w}_n),k + 2)$-conjunction is true at $(M, a, \bar{b}_n)$ and therefore, its equivalent in $\{\,\Psi_{i_1}(x, \bar{w}_n)\ldots, \Psi_{i_n}(x, \bar{w}_n)\,\}$ is also true at $(M, a, \bar{b}_n)$, and so, finally we have $M, a, \bar{b}_n \models \Psi_{i_1}(x, \bar{w}_n)\vee\ldots, \vee\Psi_{i_n}(x, \bar{w}_n)$. In the other direction, if $M, a, \bar{b}_n \models \Psi_{i_1}(x, \bar{w}_n)\vee\ldots, \vee\Psi_{i_n}(x, \bar{w}_n)$ then for some $1 \leq j \leq n$ we have $M, a, \bar{b}_n \models \Psi_{i_j}(x, \bar{w}_n)$. Then, since $\Psi_{i_j}(x, \bar{w}_n)\models\Psi_{i_j}(x, \bar{w}_n)$ and since by the choice of $\Psi_{i_j}(x, \bar{w}_n)$ the formula $\Psi_{i_j}(x, \bar{w}_n) \wedge \varphi(x, \bar{w}_n)$ is satisfiable, so, by our assumption, the formula $\Psi_{i_j}(x, \bar{w}_n) \wedge \neg\varphi(x, \bar{w}_n)$ must be unsatisfiable, and hence $\varphi(x, \bar{w}_n)$ must follow from $\Psi_{i_j}(x, \bar{w}_n)$. But in this case we will have $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$ as well. So $\varphi(x, \bar{w}_n)$ is logically equivalent to $\Psi_{i_1}(x, \bar{w}_n)\vee\ldots, \vee\Psi_{i_n}(x, \bar{w}_n)$ but the latter formula, being a disjunction of conjunctions of standard $x$-translations of intuitionistic formulas is itself a standard $x$-translation of an intuitionistic formula and so we are done.
If, on the other hand, one can take two complete $(\varphi, (x, \bar{w}_n),k + 2)$-conjunctions $\Psi(x, \bar{w}_n), \Psi'(x, \bar{w}_n)$ such that $\Psi'(x, \bar{w}_n)\models\Psi(x, \bar{w}_n)$, and formulas $\Psi(x, \bar{w}_n) \wedge \varphi(x, \bar{w}_n)$ and $\Psi'(x, \bar{w}_n) \wedge \neg\varphi(x, \bar{w}_n)$ are satisfiable, we reason as follows.
Take any $n$-ary evaluation $\Sigma_\varphi$-point $(M, a, \bar{b}_n)$ such that both $M, a, \bar{b}_n \models \Psi(x, \bar{w}_n) \wedge \varphi(x, \bar{w}_n)$ and for any $(\Sigma_\varphi, (x, \bar{w}_n),k)$-formula $\psi(x, \bar{w}_n)$, if $\psi(x, \bar{w}_n)$ is a standard $x$-translation of an intuitionistic formula and $M, a, \bar{b}_n \models \psi(x, \bar{w}_n)$, then $\Psi(x, \bar{w}_n) \models\psi(x, \bar{w}_n)$. Then take any $n$-ary evaluation $\Sigma_\varphi$-point $(N, c, \bar{d}_n)$ such that $N, c, \bar{d}_n \models \Psi'(x, \bar{w}_n) \wedge \neg\varphi(x, \bar{w}_n)$.
We can construct a $\langle (M, a, \bar{b}_n), (N, c, \bar{d}_n)\rangle_k$-asimulation and thus obtain a contradiction in the following way.
Let $\alpha, \beta \in \{\,M, N\,\}$ and let $(\bar{a'}_m, a', \bar{b'}_l) \in (D(\alpha)^{m+1} \times D(\alpha)^l)$ and $(\bar{c'}_m,c'; \bar{d'}_l) \in (D(\beta)^{m+1} \times D(\beta)^n)$. Then $(\bar{a'}_m, a'; \bar{b'}_l)A(\bar{c'}_m,c'; \bar{d'}_l)$ iff \[ m + l\leq n + k \wedge (\alpha, a', \bar{b'}_l) \leq_{\varphi, l, n + k + 2 - m - l} (\beta, c', \bar{d'}_l). \]
By the choice of $\Psi(x, \bar{w}_n), \Psi'(x, \bar{w}_n)$ and the independence of truth at an $n$-ary evaluation point from the choice of free variables in a formula we obviously have $(a; \bar{b}_n)A(c; \bar{d}_n)$. It remains to verify conditions \eqref{E:c1}--\eqref{E:c4} of Definition \ref{D:k-asim}.
\emph{Verification of \eqref{E:c1}}. Since the degree of any atomic formula is $0$, and the above condition implies that $ n + k + 2 - m - l \geq 2$, it is evident that for any $(\bar{a'}_m, a'; \bar{b'}_l)A(\bar{c'}_m,c'; \bar{d'}_l)$ and any predicate letter $P\in\Sigma_\varphi\setminus\{\,R^2,E^2\,\}$ we have $\alpha, a', \bar{b'}_l \models P(x, \bar{w}_l) \Rightarrow \beta, c', \bar{d'}_l \models P(x, \bar{w}_l)$.
\emph{Verification of \eqref{E:c2}}. Assume then that for some $(\bar{a'}_m, a'; \bar{b'}_l)A(\bar{c'}_m,c'; \bar{d'}_l)$ such that $m + l < n + k$ there exists a $c'' \in D(\beta)$ such that $c'R^\beta c''$. In this case we will also have $m + 1 + l \leq n + k$.
Then consider the following two sets: \begin{align*} &\Gamma = \{\,ST(i(\bar{w}_l), x) \mid ST(i(\bar{w}_l), x)\text{ is a $(\Sigma_\varphi, (x, \bar{w}_l), n + k + 1 - m - l)$-formula, }\beta, c'', \bar{d'}_l \models ST(i(\bar{w}_l), x)\,\};\\ &\Delta = \{\,ST(i(\bar{w}_l), x) \mid ST(i(\bar{w}_l), x)\text{ is a $(\Sigma_\varphi, (x, \bar{w}_l), n + k + 1 - m - l)$-formula, }\beta, c'', \bar{d'}_l \models \neg ST(i(\bar{w}_l), x)\,\}. \end{align*} These sets are non-empty, since by our assumption we have $n + k + 1 - m - l\geq 1$. Therefore, as we have $r(ST(\bot, x)) = 0$ and $r(ST(\bot \to \bot, x)) = 1$, we will also have $ST(\bot, x) \in \Delta$ and $ST(\bot \to \bot, x) \in \Gamma$. Then, according to our Lemma \ref{L:fin}, there are finite non-empty sets of logical equivalents for both $\Gamma$ and $\Delta$. Choosing these finite sets, we in fact choose some finite $\{\,ST(i_1(\bar{w}_l),x)\ldots ST(i_t(\bar{w}_l), x)\,\} \subseteq \Gamma$, $\{\,ST(j_1(\bar{w}_l),x)\ldots ST(j_u(\bar{w}_l), x)\,\} \subseteq \Delta$ such that \begin{align*} &\forall \psi(x,\bar{w}_l) \in \Gamma(ST(i_1(\bar{w}_l),x)\wedge\ldots \wedge ST(i_t(\bar{w}_l), x) \models \psi(x,\bar{w}_l));\\ &\forall \chi(x,\bar{w}_l) \in \Delta(\chi(x,\bar{w}_l)\models ST(j_1(\bar{w}_l),x)\vee\ldots \vee ST(j_u(\bar{w}_l), x)). \end{align*} But then we obtain that the formula \[ ST((i_1(\bar{w}_l)\wedge\ldots \wedge i_t(\bar{w}_l)) \to (j_1(\bar{w}_l)\vee\ldots \vee j_u(\bar{w}_l)), x) \] is false at $(\beta, c',\bar{d'}_l)$. In fact, $c''$ disproves this implication for $(\beta, c',\bar{d'}_l)$. But every formula both in $\{\,ST(i_1(\bar{w}_l),x)\ldots ST(i_t(\bar{w}_l), x)\,\}$ and $\{\,ST(j_1(\bar{w}_l),x)\ldots ST(j_u(\bar{w}_l), x)\,\}$ is, by their choice, a $(\Sigma_\varphi, x, n + k + 1 - m - l)$-formula, and so standard translation of the implication under consideration must be a $(\Sigma_\varphi,x,n + k + 2 - m - l)$-formula. Note, further, that by $(\bar{a'}_m, a'; \bar{b'}_l)A(\bar{c'}_m,c'; \bar{d'}_l)$ we must have \[ (\alpha, a', \bar{b'}_l) \leq_{\varphi, l, n + k + 2 - m - l} (\beta, c', \bar{d'}_l), \] and therefore this implication must be false at $(\alpha, a',\bar{b'}_l)$ as well. But then take any $a''$ such that $a'R^\alpha a''$ and $(\alpha, a'', \bar{b'}_l)$ verifies the conjunction in the antecedent of the formula but falsifies its consequent. We must conclude then, by the choice of $\{\,ST(i_1(\bar{w}_l),x)\ldots ST(i_t(\bar{w}_l), x)\,\}$, that $\alpha, a'',\bar{b'}_l \models \Gamma$ and so, by the definition of $A$, and given that $m + l + 1 \leq n + k$, that $(\bar{c'}_m,c', c'';\bar{d'}_l)A(\bar{a'}_m,a', a'';\bar{b'}_l)$. Since, in addition, $(\alpha, a'', \bar{b'}_l)$ disproves every formula from $\{\,ST(j_1(\bar{w}_l),x)\ldots ST(j_u(\bar{w}_l), x)\,\}$ then by the choice of this set we must conclude that every $(\Sigma_\varphi,x,n + k + 1 - m - l)$-formula that is a standard $x$-translation of an intuitionistic formula false at $(\beta, c', \bar{d'}_l)$ is also false at $(\alpha, a'', \bar{b'}_l)$. But then, again by the definition of $A$, and given the fact that $m + l + 1 \leq n + k$, we must also have $(\bar{a'}_m,a', a'';\bar{b'}_l)A(\bar{c'}_m,c', c'';\bar{d'}_l)$, and so condition \eqref{E:c2} holds.
\emph{Verification of \eqref{E:c3}}. Assume then that for some $(\bar{a'}_m, a'; \bar{b'}_l)A(\bar{c'}_m,c'; \bar{d'}_l)$ such that $m + l < n + k$ there exists a $b'' \in D(\alpha)$ such that $E^\alpha(a', b'')$. In this case we will also have $m + l + 1 \leq n + k$.
Then consider the following set: \begin{align*} &\Gamma = \{\,ST(i(\bar{w}_l,w'), x) \mid ST(i, x)\text{ is a $(\Sigma_\varphi, (x, \bar{w}_l,w'), n + k + 1 - m - l)$-formula, }\beta, a', \bar{b'}_l,b'' \models ST(i(\bar{w}_l,w'), x)\,\}. \end{align*} This set is non-empty, since by our assumption we have $n + k + 1 - m - l\geq 1$. Therefore, as we have $r(ST(\bot \to \bot, x)) = 1$, we will also have $ST(\bot \to \bot, x) \in \Gamma$. Then, according to our Lemma \ref{L:fin}, there is a finite non-empty set of logical equivalents for $\Gamma$. Choosing this finite set, we in fact choose some finite $\{\,ST(i_1(\bar{w}_l,w'),x)\ldots ST(i_t(\bar{w}_l,w'), x)\,\} \subseteq \Gamma$ such that \begin{align*} &\forall \psi(x,\bar{w}_l,w') \in \Gamma(ST(i_1(\bar{w}_l,w'),x)\wedge\ldots \wedge ST(i_t(\bar{w}_l,w'), x) \models \psi(x,\bar{w}_l,w')). \end{align*} But then we obtain that the formula \[ ST(\exists w'(i_1(\bar{w}_l,w')\wedge\ldots \wedge i_t(\bar{w}_l,w')), x) \] is true at $(\alpha, a',\bar{b'}_l)$. Moreover, every formula in $\{\,ST(i_1(\bar{w}_l,w'),x)\ldots ST(i_t(\bar{w}_l,w'), x)\,\}$ is, by their choice, a $(\Sigma_\varphi, x, n + k + 1 - m - l)$-formula, and so standard translation of the quantified conjunction under consideration must be a $(\Sigma_\varphi,x,n + k + 2 - m - l)$-formula. Since we have, by $(\bar{a'}_m, a'; \bar{b'}_l)A(\bar{c'}_m,c'; \bar{d'}_l)$, that \[ (\alpha, a', \bar{b'}_l) \leq_{\varphi, l, n + k + 2 - m - l} (\beta, c', \bar{d'}_l), \] then the formula in question must be true at $(\beta, c',\bar{d'}_l)$ as well. But then take any $d''$ such that $E^\beta(c',d'')$ and $(\beta, c', \bar{d'}_l,d'')$ verifies a standard translation of the conjunction after the existential quantifier. We must conclude then, by the choice of $\{\,ST(i_1(\bar{w}_l,w'),x)\ldots ST(i_t(\bar{w}_l,w'), x)\,\}$, that $\beta, c',\bar{d'}_l,d'' \models \Gamma$ and so, by the definition of $A$, and given that $m + l + 1 \leq n + k$, that $(\bar{a'}_m,a';\bar{b'}_l,b'')A(\bar{c'}_m,c';\bar{d'}_l,d'')$.
\emph{Verification of \eqref{E:c4}}. Assume then that for some $(\bar{a'}_m, a', \bar{b'}_l)A(\bar{c'}_m,c', \bar{d'}_l)$ such that $m + l + 1 < n + k$ there exist some $c'', d'' \in D(\beta)$ such that $c'R^\beta c'' \wedge E^\beta(c'', d'')$, but there are no $a'',b'' \in D(\alpha)$ such that $a'R^\alpha a'' \wedge E^\alpha(a'', b'')$ and $(\bar{a'}_m, a',a''; \bar{b'}_l, b'')A(\bar{c'}_m,c',c'';\bar{d'}_l, d'')$. In this case we will have $m + 1 + l + 1 \leq n + k$.
Then consider the following set: \begin{align*} &\Delta = \{\,ST(i(\bar{w}_l,w'), x) \mid ST(i(\bar{w}_l,w'), x)\text{ is a $(\Sigma_\varphi, (x, \bar{w}_l,w'), n + k - m - l)$-formula, }\beta, c'', \bar{d'}_l,d'' \models \neg ST(i(\bar{w}_l,w'), x)\,\}. \end{align*} This set is non-empty, since by our assumption we have $n + k - m - l\geq 0$. Therefore, as we have $r(ST(\bot, x)) = 0$, we will also have $ST(\bot, x) \in \Delta$. Then, according to our Lemma \ref{L:fin}, there is a finite non-empty set of logical equivalents for $\Delta$. Choosing this finite set, we in fact choose some finite $\{\,ST(j_1(\bar{w}_l,w'),x)\ldots ST(j_u(\bar{w}_l,w'), x)\,\} \subseteq \Delta$ such that \begin{align*} &\forall \chi(x,\bar{w}_l,w') \in \Delta(\chi(x,\bar{w}_l,w')\models ST(j_1(\bar{w}_l,w'),x)\vee\ldots \vee ST(j_u(\bar{w}_l,w'), x)). \end{align*} But then we obtain that the formula \[ ST(\forall w'(j_1(\bar{w}_l,w')\vee\ldots \vee j_u(\bar{w}_l,w')), x) \] is false at $(\beta, c',\bar{d'}_l)$. In fact, $c'',d''$ jointly disprove standard translation of this universally quantified disjunction for $(\beta, c',\bar{d'}_l)$. Further, every formula in $\{\,ST(j_1(\bar{w}_l),x)\ldots ST(j_u(\bar{w}_l), x)\,\}$ is, by their choice, a $(\Sigma_\varphi, x, n + k - m - l)$-formula, and so standard translation of the universally quantified disjunction under consideration must be a $(\Sigma_\varphi,x,n + k + 2 - m - l)$-formula. Since we have, by $(\bar{a'}_m, a'; \bar{b'}_l)A(\bar{c'}_m,c'; \bar{d'}_l)$, that \[ (\alpha, a', \bar{b'}_l) \leq_{\varphi, l, n + k + 2 - m - l} (\beta, c', \bar{d'}_l), \] then the formula in question must be false at $(\alpha, a',\bar{b'}_l)$ as well. But then take any $a'',b''$ for which we have $a'R^\alpha a''$ and $E^\alpha(a'',b'')$ such that $(\alpha,a'', \bar{b'}_l,b'')$ falsifies standard translation of the disjunction after the quantifier. We must conclude, by the choice of $\{\,ST(j_1(\bar{w}_l,w'),x)\ldots ST(j_u(\bar{w}_l,w'), x)\,\}$,
that every $(\Sigma_\varphi,x,n + k - m - l)$-formula that is a standard $x$-translation of an intuitionistic formula false at $(\beta, c'',\bar{d'}_l,d'')$ is also false at $(\alpha, a'',\bar{b'}_l,b'')$. But then, again by the definition of $A$, and given the fact that $m + 1 + l + 1 \leq n + k$, we must also have $(\bar{a'}_m,a', a'';\bar{b'}_l,b'')A(\bar{c'}_m,c', c'';\bar{d'}_l,d'')$, so condition \eqref{E:c4} is satisfied. \end{proof}
\begin{theorem}\label{L:param} A formula $\varphi(x, \bar{w}_n)$ is equivalent to a standard $x$-translation of an intuitionistic formula iff there exists $k \in \mathbb{N}$ such that $\varphi(x, \bar{w}_n)$ is invariant with respect to $k$-asimulations. \end{theorem} \begin{proof} Let $\varphi(x, \bar{w}_n)$ be equivalent to $ST(i(\bar{w}_n),x)$. Then by Corollary \ref{L:c-k-inv}, $ST(i(\bar{w}_n),x)$ is invariant with respect to $r(ST(i(\bar{w}_n),x))$-asimulations, and, therefore, so is $\varphi(x, \bar{w}_n)$. In the other direction, let $\varphi(x, \bar{w}_n)$ be invariant with respect to $k$-asimulations for some $k$. If $k \leq r(\varphi(x, \bar{w}_n))$, then every $r(\varphi(x, \bar{w}_n))$-asimulation is a $k$-asimulation, therefore, $\varphi(x, \bar{w}_n)$ is invariant with respect to $r(\varphi(x, \bar{w}_n))$-asimulations, and so, by Theorem \ref{L:t1}, $\varphi(x, \bar{w}_n)$ is logically equivalent to a standard $x$-translation of an intuitionistic formula. If, on the other hand, $r(\varphi(x, \bar{w}_n)) < k$, then set $l = k - r(\varphi(x, \bar{w}_n))$ and consider a sequence $\bar{y}_l$ of variables not occurring in $\varphi(x, \bar{w}_n)$. Formula $\forall\bar{y}_l\varphi(x, \bar{w}_n)$ is logically equivalent to $\varphi(x, \bar{w}_n)$, hence $\forall\bar{y}_l\varphi(x, \bar{w}_n)$ is invariant with respect to $k$-asimulations as well. But we have $r(\forall\bar{y}_l\varphi(x, \bar{w}_n)) = k$, so, by Theorem \ref{L:t1}, $\forall\bar{y}_l\varphi(x, \bar{w}_n)$ is logically equivalent to a standard $x$-translation of an intuitionistic formula. Hence $\varphi(x, \bar{w}_n)$ is equivalent to this translation, too. \end{proof}
\section{The main result}\label{S:Main}
We begin by introducing a somewhat simpler, unparametrized version of asimulation: \begin{definition}\label{D:asim} Let $(M,a, \bar{b}_n)$, $(N,c,\bar{d}_n)$ be two $n$-ary evaluation $\Theta$-points. A binary relation \[ A \subseteq \bigcup_{n \geq 0}(((D(M)\times D(M)^n) \times (D(N) \times D(N)^n)) \cup ((D(N)\times D(N)^n) \times (D(M) \times D(M)^n))), \] is called $\langle (M,a, \bar{b}_n), (N,c,\bar{d}_n)\rangle$-asimulation iff $(a; \bar{b}_n)A(c; \bar{d}_n)$ and for any $\alpha, \beta \in \{\,M, N\,\}$, any $(a';\bar{b'}_l) \in D(\alpha) \times D(\alpha)^l$, $(c';\bar{d'}_l) \in D(\beta) \times D(\beta)^l$, whenever we have $(a';\bar{b'}_l)A(c';\bar{d'}_l)$, the following conditions hold:
\begin{align} &\forall P \in \Theta\setminus\{\,R^2,E^2\,\}(\alpha, a', \bar{b'}_l\models P(x, \bar{w}_l) \Rightarrow \beta, c', \bar{d'}_l\models P(x, \bar{w}_l))\label{E:c11}\\ &(c'' \in D(\beta) \wedge c'R^\beta c'') \Rightarrow\notag\\ &\Rightarrow \exists a'' \in D(\alpha)(a'R^\alpha a'' \wedge (c'';\bar{d'}_l)\hat{A}(a'';\bar{b'}_l));\label{E:c22}\\ &(b'' \in D(\alpha) \wedge E^\alpha(a', b'')) \Rightarrow\notag\\ &\Rightarrow \exists d'' \in D(\beta)(E^\beta(c',d'') \wedge (a';\bar{b'}_l, b'')A(c';\bar{d'}_l, d''));\label{E:c33}\\ &(c'', d'' \in D(\beta) \wedge c'R^\beta c''\wedge E^\beta(c'', d'')) \Rightarrow\notag\\ &\Rightarrow \exists a'',b'' \in D(\alpha)(a'R^\alpha a''\wedge E^\alpha(a'',b'') \wedge (a'';\bar{b'}_l, b'')A(c'';\bar{d'}_l, d'')).\label{E:c44} \end{align} \end{definition} \begin{lemma}\label{L:k-asim1} Let $A$ be an $\langle (M,a, \bar{b}_n), (N,c,\bar{d}_n)\rangle$-asimulation, and let \[ A' = \{\,\langle(\bar{a'}_m, a';\bar{b'}_l),(\bar{c'}_m, c';\bar{d'}_l)\rangle\mid (a';\bar{b'}_l)A(c';\bar{d'}_l)\,\}. \] Then $A'$ is an $\langle (M,a, \bar{b}_n), (N,c,\bar{d}_n)\rangle_k$-asimulation for any $k \in \mathbb{N}$. \end{lemma} \begin{proof} We obviously have $(a; \bar{b}_n)A'(c; \bar{d}_n)$, and since for any $\alpha, \beta \in \{\,M, N\,\}$, and any $(\bar{a'}_m, a';\bar{b'}_l)$ in $D(\alpha)^{m+1}\times D(\alpha)^l$, $(\bar{c'}_m, c';\bar{d'}_l)$ in $D(\beta)^{m+1}\times D(\beta)^l$ such that $(\bar{a'}_m, a';\bar{b'}_l)A'(\bar{c'}_m, c';\bar{d'}_l)$ we have $(a';\bar{b'}_l)A(c';\bar{d'}_l)$, condition \eqref{E:c1} for $A'$ follows from the fulfilment of condition \eqref{E:c11} for $A$. So it remains to verify that the other three conditions hold for $A'$ for every $k$.
\emph{Condition \eqref{E:c2}}: If $(\bar{a'}_m, a';\bar{b'}_l)A'(\bar{c'}_m, c';\bar{d'}_l)$ then $(a';\bar{b'}_l)A(c';\bar{d'}_l)$, and if, further, $c'' \in D(\beta)$ and $c'R^\beta c''$ then by condition \eqref{E:c22} we can choose $a'' \in D(\alpha)$ such that $a'R^\alpha a''$, and $(c'';\bar{d'}_l)\hat{A}(a'';\bar{b'}_l)$. But then, by definition of $A'$ we will also have $(\bar{c'}_m, c',c'';\bar{d'}_l)\hat{A'}(\bar{a'}_m, a',a'';\bar{b'}_l)$.
\emph{Condition \eqref{E:c3}}: If $(\bar{a'}_m, a';\bar{b'}_l)A'(\bar{c'}_m, c';\bar{d'}_l)$ then $(a';\bar{b'}_l)A(c';\bar{d'}_l)$, and if, further, $b'' \in D(\alpha)$ and $E^\alpha(a',b'')$ then by condition \eqref{E:c33} we can choose $d'' \in D(\beta)$ such that $E^\beta(c',d'')$, and $(a';\bar{b'}_l,b'')A(c';\bar{d'}_l,d'')$. But then, by definition of $A'$ we will also have $(\bar{a'}_m, a';\bar{b'}_l,b'')A'(\bar{c'}_m, c';\bar{d'}_l,d'')$.
\emph{Condition \eqref{E:c4}}: If $(\bar{a'}_m, a';\bar{b'}_l)A'(\bar{c'}_m, c';\bar{d'}_l)$ then $(a';\bar{b'}_l)A(c';\bar{d'}_l)$, and if, further, $c'',d'' \in D(\beta)$, $c'R^\beta c''$ and $E^\beta(c'',d'')$ then by condition \eqref{E:c44} we can choose $a'',b'' \in D(\alpha)$ such that $a'R^\beta a''$, $E^\alpha(a'',b'')$, and $(a'';\bar{b'}_l,b'')A(c'';\bar{d'}_l,d'')$. But then, by definition of $A'$ we will also have $(\bar{a'}_m, a',a'';\bar{b'}_l,b'')A'(\bar{c'}_m, c',c'';\bar{d'}_l,d'')$. \end{proof} \begin{definition}\label{D:inv} A formula $\varphi(x, \bar{w}_n)$ is invariant with respect to asimulations iff for any $\Theta$ such that $\Sigma_\varphi \subseteq \Theta$, any $n$-ary evaluation $\Theta$-points $(M,a, \bar{b}_n)$ and $(N,c,\bar{d}_n)$, if there exists an $\langle (M,a, \bar{b}_n), (N,c,\bar{d}_n)\rangle$-asimulation $A$ and $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$, then $N, c, \bar{d}_n \models \varphi(x, \bar{w}_n)$. \end{definition} \begin{corollary}\label{L:c-inv} If $\varphi(x, \bar{w}_n)$ is equivalent to a standard $x$-translation of an intuitionistic formula, then $\varphi(x, \bar{w}_n)$ is invariant with respect to asimulations. \end{corollary} \begin{proof} Let $\varphi(x, \bar{w}_n)$ be not invariant with respect to asimulations, and let $A$ be an $\langle (M,a, \bar{b}_n), (N,c,\bar{d}_n)\rangle$-asimulation such that $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$, but not $N, c,\bar{d}_n \models \varphi(x, \bar{w}_n)$. Let $A'$ be defined as in Lemma \ref{L:k-asim1}. Then by this Lemma $A'$ is an $\langle (M,a, \bar{b}_n), (N,c,\bar{d}_n)\rangle_k$-asimulation for any $k \in \mathbb{N}$. Hence, by Theorem \ref{L:param}, $\varphi(x, \bar{w}_n)$ cannot be equivalent to a standard $x$-translation of an intuitionistic formula. \end{proof}
To proceed further, we need to introduce some notions and results from classical model theory. For a model $M$ and $\bar{a}_n \in D(M)$ let $[M, \bar{a}_n]$ be the extension of $M$ with $\bar{a}_n$ as new individual constants denoting themselves. It is easy to see that there is a simple relation between truth of a formula at a $\Theta$-evaluation point and truth of its substitution instance in an extension of the above-mentioned kind; namely, for any $\Theta$-model $M$, every $\Theta$-formula $\varphi(\bar{y}_n,\bar{w}_m)$ and any $\bar{a}_n,\bar{b}_m \in D(M)$ it holds that:
\[ [M, \bar{a}_n], \bar{b}_m \models \varphi(\bar{a}_n,\bar{w}_m) \Leftrightarrow M, \bar{a}_n, \bar{b}_m \models \varphi(\bar{y}_n,\bar{w}_m). \]
We will call a theory of $M$ (and write $Th(M)$) the set of all first-order sentences true at $M$. We will call an $n$-type of $M$ a set of formulas $\Gamma(\bar{w}_n)$ consistent with $Th(M)$.
\begin{definition} Let $M$ be a $\Theta$-model. $M$ is $\omega$-saturated iff for all $k \in \mathbb{N}$ and for all $\bar{a}_n \in D(M)$, every $k$-type $\Gamma(\bar{w}_k)$ of $[M, \bar{a}_n]$ is satisfiable in $[M, \bar{a}_n]$. \end{definition}
Definition of $\omega$-saturation normally requires satisfiability of $1$-types only. However, our modification is equivalent to the more familiar version: see e.g. \cite[Lemma 4.31, p. 73]{Doe}.
It is known that every model can be elementarily extended to an $\omega$-saturated model; in other words, the following lemma holds:
\begin{lemma}\label{L:ext} Let $M$ be a $\Theta$-model. Then there is an $\omega$-saturated extension $N$ of $M$ such that for all $\bar{a}_n \in D(M)$ and every $\Theta$-formula $\varphi(\bar{w}_n)$: \[ M, \bar{a}_n \models \varphi(\bar{w}_n) \Leftrightarrow N, \bar{a}_n \models \varphi(\bar{w}_n). \] \end{lemma} The latter lemma is a trivial corollary of e.g. \cite[Lemma 5.1.14, p. 216]{ChK}.
In what follows we adopt the following notation for the fact that for any $x$ all $\Theta$-formulas that are standard $x$-translations of intuitionistic formulas true at $(M,a, \bar{b}_n)$, are also true at $(N,c, \bar{d}_n)$: \[ (M,a, \bar{b}_n) \leq_{\Theta} (N,c, \bar{d}_n). \]
\begin{lemma}\label{L:sat} Let $\Theta \subseteq \Sigma$, let $M$, $N$ be $\omega$-saturated $\Theta$-models and let $(M,a, \bar{b}_n) \leq_{\Theta} (N,c,\bar{d}_n)$. Then relation $A$ such that for any $\alpha, \beta \in \{\,M, N\,\}$, any $(a';\bar{b'}_l) \in D(\alpha) \times D(\alpha)^l$, $(c';\bar{d'}_l) \in D(\beta) \times D(\beta)^l$ \[ (a';\bar{b'}_l)A(c';\bar{d'}_l) \Leftrightarrow (\alpha,a', \bar{b'}_l) \leq_{\Theta} (\beta,c', \bar{d'}_l) \] is an $\langle (M,a, \bar{b}_n),(N,c,\bar{d}_n)\rangle$-asimulation.\footnote{This definition of $A$ makes sense only when $D(M) \cap D(N) = \varnothing$. However, the latter can always be assumed without a loss of generality.} \end{lemma}
\begin{proof} Throughout this proof every formula mentioned is supposed to be a $\Theta$-formula. It is obvious that $(a; \bar{b}_n)A(c; \bar{d}_n)$, and since for any predicate letter $P$ distinct from $R^2, E^2$ and variables $x,\bar{w}_n$ formula $P(x,\bar{w}_n)$ is a standard $x$-translation of an atomic intuitionistic formula, condition \eqref{E:c11} is trivially satisfied for $A$.
To verify \emph{condition \eqref{E:c22}}, choose any $\alpha, \beta \in \{\,M, N\,\}$, any $(a';\bar{b'}_l) \in D(\alpha) \times D(\alpha)^l$, $(c';\bar{d'}_l) \in D(\beta) \times D(\beta)^l$ such that $(\alpha, a',\bar{b'}_l)\leq_{\Theta}(\beta,c',\bar{d'}_l)$ and choose any $c'' \in D(\beta)$ for which we have $c'R^\beta c''$.
Then choose any variables $x, \bar{w}_n$ and consider the following two sets: \begin{align*} &\Gamma = \{\,i(\bar{w}_l) \mid \beta, c'',\bar{d'}_l \models ST(i(\bar{w}_l), x)\,\};\\ &\Delta = \{\,i(\bar{w}_l) \mid \beta, c'',\bar{d'}_l \models \neg ST(i(\bar{w}_l), x)\,\}. \end{align*} We have by the choice of $\Gamma$, $\Delta$ that for every finite $\Gamma' \subseteq \Gamma$ and $\Delta' \subseteq \Delta$ the formula $ST(\bigwedge(\Gamma') \to \bigvee(\Delta'), x)$ is disproved by $c''$ for $(\beta, c',\bar{d'}_l)$. So, by our premise that $(\alpha, a',\bar{b'}_l)\leq_{\Theta}(\beta,c',\bar{d'}_l)$, the standard translation of every such implication must be false at $(\alpha, a',\bar{b'}_l)$ as well. This means that every finite subset of the set \[ \{\,R(a', x)\,\} \cup \{\,ST(i(\bar{b'}_l),x)\mid i(\bar{w}_l)\in\Gamma\,\} \cup \{\,\neg ST(i(\bar{b'}_l),x)\mid i(\bar{w}_l)\in\Delta\,\} \]
is satisfiable at $[\alpha, a',\bar{b'}_l]$. (We set $\Delta' = \{\,ST(\bot, x)\,\}$ if the finite set in question has an empty intersection with $\Delta$ and $\Gamma' = \{\,ST(\bot \to \bot, x)\,\}$ if it has an empty intersection with $\Gamma$.) Therefore, by compactness of first-order logic, this set is consistent with $Th([\alpha, a',\bar{b'}_l])$ and, by $\omega$-saturation of both $M$ and $N$ it must be satisfied in $[\alpha, a',\bar{b'}_l]$ by some $a'' \in D(\alpha)$. So for any such $a''$ we will have $a'R^\alpha a''$ and, moreover \[ \alpha, a'',\bar{b'}_l \models \{\,ST(i(\bar{w}_l),x)\mid i(\bar{w}_l)\in\Gamma\,\} \cup \{\,\neg ST(i(\bar{w}_l),x)\mid i(\bar{w}_l)\in\Delta\,\}. \] Thus, by choice of $\Gamma$ and $\Delta$ plus independence of truth at a pointed model from the choice of free variables in a formula we will have both $(\alpha, a'',\bar{b'}_l)\leq_{\Theta}(\beta,c'',\bar{d'}_l)$ and $(\beta,c'',\bar{d'}_l)\leq_{\Theta}(\alpha, a'',\bar{b'}_l)$ and condition \eqref{E:c22} is verified.
To verify \emph{condition \eqref{E:c33}}, choose any $\alpha, \beta \in \{\,M, N\,\}$, any $(a';\bar{b'}_l) \in D(\alpha) \times D(\alpha)^l$, $(c';\bar{d'}_l) \in D(\beta) \times D(\beta)^l$ such that $(\alpha, a',\bar{b'}_l)\leq_{\Theta}(\beta,c',\bar{d'}_l)$ and choose any $b'' \in D(\alpha)$ for which we have $E^\alpha(a',b'')$.
Then choose any variables $x, \bar{w}_n,w'$ and consider the following set: \begin{align*} &\Gamma = \{\,i(\bar{w}_l,w') \mid\alpha, a',\bar{b'}_l,b'' \models ST(i(\bar{w}_l,w'), x)\,\}. \end{align*} We have by the choice of $\Gamma$ that for every finite $\Gamma' \subseteq \Gamma$ the formula $ST(\exists w'\bigwedge(\Gamma'), x)$ is verified by $b''$ for $(\alpha, a',\bar{b'}_l)$. So, by our premise that $(\alpha, a',\bar{b'}_l)\leq_{\Theta}(\beta,c',\bar{d'}_l)$, the standard translation of every such quantified conjunction must be true at $(\beta, c',\bar{d'}_l)$ as well. This means that every finite subset of the set \[ \{\,E(c', w')\,\} \cup \{\,ST(i(\bar{d'}_l,w'),c')\mid i(\bar{w}_l,w')\in\Gamma\,\} \]
is satisfiable at $[\beta, c',\bar{d'}_l]$. Therefore, by compactness of first-order logic, this set is consistent with $Th([\beta, c',\bar{d'}_l])$ and, by $\omega$-saturation of both $M$ and $N$, it must be satisfied in $[\beta, c',\bar{d'}_l]$ by some $d'' \in D(\beta)$. So for any such $d''$ we will have $E^\beta(c',d'')$ and, moreover \[ \beta, c',\bar{d'}_l,d'' \models \{\,ST(i(\bar{w}_l,w'),x)\mid i(\bar{w}_l,w')\in\Gamma\,\}. \] Thus, by choice of $\Gamma$ plus independence of truth at a pointed model from the choice of free variables in a formula we will have $(\alpha, a',\bar{b'}_l,b'')\leq_{\Theta}(\beta,c',\bar{d'}_l,d'')$
and condition \eqref{E:c33} is verified.
To verify \emph{condition \eqref{E:c44}}, choose any $\alpha, \beta \in \{\,M, N\,\}$, any $(a';\bar{b'}_l) \in D(\alpha) \times D(\alpha)^l$, $(c';\bar{d'}_l) \in D(\beta) \times D(\beta)^l$ such that $(\alpha, a',\bar{b'}_l)\leq_{\Theta}(\beta,c',\bar{d'}_l)$ and choose any $c'',d'' \in D(\beta)$ for which we have $c'R^\beta c''$ and $E^\beta(c'',d'')$.
Then choose any variables $x, \bar{w}_n, w'$ and consider the following set: \begin{align*} &\Delta = \{\,i(\bar{w}_l,w') \mid\beta, c'',\bar{d'}_l,d'' \models \neg ST(i(\bar{w}_l,w'), x)\,\}. \end{align*} We have by the choice of $\Delta$ that for every finite $\Delta' \subseteq \Delta$ the formula $ST(\forall w'\bigvee(\Delta'), x)$ is disproved by $c'',d''$ for $(\beta, c',\bar{d'}_l)$. So, by our premise that $(a';\bar{b'}_l)\leq_{\Theta}(c';\bar{d'}_l)$, the standard translation of every such quantified disjunction must be false at $(\alpha, a',\bar{b'}_l)$ as well. This means that every finite subset of the set \[ \{\,R(a', x), E(x,w')\,\} \cup \{\,\neg ST(i(\bar{b'}_l,w'),x)\mid i(\bar{w}_l,w')\in\Delta\,\} \]
is satisfiable at $[\alpha, a',\bar{b'}_l]$. Therefore, by compactness of first-order logic, this set is consistent with $Th([\alpha, a',\bar{b'}_l])$ and, by $\omega$-saturation of both $M$ and $N$, it must be satisfied in $[\alpha, a',\bar{b'}_l]$ by some $a'',b'' \in D(\alpha)$. So for any such $a''$ and $b''$ we will have $a'R^\alpha a''$, $E^\alpha(a'',b'')$ and, moreover \[ \alpha, a'',\bar{b'}_l,b'' \models \{\,\neg ST(i(\bar{w}_l,w'),x)\mid i(\bar{w}_l,w')\in\Delta\,\}. \] Thus, by choice of $\Delta$ plus independence of truth at a pointed model from the choice of free variables in a formula we will have $(\alpha, a'',\bar{b'}_l,b'')\leq_{\Theta}(\beta,c'',\bar{d'}_l,d'')$ and condition \eqref{E:c44} is verified. \end{proof}
We are prepared now to state and prove our main result.
\begin{theorem}\label{L:main} Let $\varphi(x, \bar{w}_n)$ be invariant with respect to asimulations. Then $\varphi(x, \bar{w}_n)$ is equivalent to a standard $x$-translation of an intuitionistic formula. \end{theorem} \begin{proof} We may assume that $\varphi(x, \bar{w}_n)$ is satisfiable, for $\bot$ is clearly invariant with respect to asimulations and $\bot \leftrightarrow ST(\bot, x)$ is a valid formula. In what follows we will write $IC(\varphi(x, \bar{w}_n))$ for the set of $\Sigma_\varphi$-formulas in variables $x, \bar{w}_n$ that are standard $x$-translations of intuitionistic formulas following from $\varphi(x, \bar{w}_n))$. For any $n$-ary evaluation $\Sigma_\varphi$-point $(M, a, \bar{b}_n)$ we will denote the set of $\Sigma_\varphi$-formulas in variables $x, \bar{w}_n$ that are standard $x$-translations of intuitionistic formulas true at $(M, a, \bar{b}_n)$, or \emph{intuitionistic $\Sigma_\varphi$-theory} of $(M, a, \bar{b}_n)$ by $IT_\varphi(M, a, \bar{b}_n)$. It is obvious that for any $n$-ary evaluation $\Sigma_\varphi$-points $(M, a, \bar{b}_n)$ and $(N, c, \bar{d}_n)$ we will have $(M, a, \bar{b}_n) \leq_{\Sigma_\varphi}(N, c, \bar{d}_n)$ if and only if $IT_\varphi(M, a, \bar{b}_n) \subseteq IT_\varphi(N, c, \bar{d}_n)$.
Our strategy will be to show that $IC(\varphi(x, \bar{w}_n)) \models \varphi(x, \bar{w}_n)$. Once this is done we will apply compactness of first-order logic and conclude that $\varphi(x, \bar{w}_n)$ is equivalent to a finite conjunction of standard $x$-translations of intuitionistic formulas and hence to a standard $x$-translation of the corresponding intuitionistic conjunction.
To show this, take any $n$-ary evaluation $\Sigma_\varphi$-point $(M, a, \bar{b}_n)$ such that $M, a, \bar{b}_n \models IC(\varphi(x, \bar{w}_n))$. Such a model exists, because $\varphi(x, \bar{w}_n)$ is satisfiable and $IC(\varphi(x, \bar{w}_n))$ will be satisfied in any pointed model satisfying $\varphi(x, \bar{w}_n)$. Then we can also choose an $n$-ary evaluation $\Sigma_\varphi$-point $(N, c, \bar{d}_n)$ such that $N, c, \bar{d}_n \models \varphi(x, \bar{w}_n)$ and $IT_\varphi(N, c, \bar{d}_n) \subseteq IT_\varphi(M, a, \bar{b}_n)$.
For suppose otherwise. Then for any $n$-ary evaluation $\Sigma_\varphi$-point $(N, c, \bar{d}_n)$ such that $N, c, \bar{d}_n \models \varphi(x, \bar{w}_n)$ we can choose an intuitionistic formula $i_{(N, c, \bar{d}_n)}(\bar{w}_n)$ such that $ST(i_{(N, c, \bar{d}_n)}(\bar{w}_n), x)$ is a $\Sigma_\varphi$-formula true at $(N, c, \bar{d}_n)$ but not at $(M, a, \bar{b}_n)$. Then consider the set \[ S = \{\,\varphi(x, \bar{w}_n)\,\} \cup \{\,\neg ST(i_{(N, c, \bar{d}_n)}(\bar{w}_n), x)\mid N, c, \bar{d}_n \models \varphi(x, \bar{w}_n)\,\} \] Let $\{\,\varphi(x, \bar{w}_n), \neg ST(i_1(\bar{w}_n), x)\ldots , \neg ST(i_u(\bar{w}_n), x)\,\}$ be a finite subset of this set. If this set is unsatisfiable, then we must have $\varphi(x) \models ST(i_1(\bar{w}_n), x)\vee\ldots \vee ST(i_u(\bar{w}_n), x)$, but then we will also have $(ST(i_1(\bar{w}_n), x)\vee\ldots \vee ST(i_u(\bar{w}_n), x)) \in IC(\varphi(x, \bar{w}_n)) \subseteq IT_\varphi(M, a, \bar{b}_n)$, and hence $(ST(i_{(N_1, b_1)}, x)\vee\ldots \vee ST(i_{(N_u, b_u)}, x))$ will be true at $(M, a, \bar{b}_n)$. But then at least one of $ST(i_1(\bar{w}_n), x)\ldots ,ST(i_u(\bar{w}_n), x)$ must also be true at $(M, a, \bar{b}_n)$, which contradicts the choice of these formulas. Therefore, every finite subset of $S$ is satisfiable, and by compactness $S$ itself is satisfiable as well. But then take any pointed $\Sigma_\varphi$-model $(N',c', \bar{d'}_n)$ of $S$ and this will be a model for which we will have both $N', c', \bar{d'}_n \models ST(i_{(N',c', \bar{d'}_n)}(\bar{w}_n), x)$ by choice of $i_{(N',c', \bar{d'}_n)}$ and $N',c', \bar{d'}_n \models \neg ST(i_{(N',c', \bar{d'}_n)}(\bar{w}_n), x)$ by the satisfaction of $S$, a contradiction.
Therefore, we will assume in the following that $(M, a, \bar{b}_n)$, $(N,c, \bar{d}_n)$ are $n$-ary evaluation $\Sigma_\varphi$-points, $M, a, \bar{b}_n \models IC(\varphi(x, \bar{w}_n))$, $N,c, \bar{d}_n \models \varphi(x, \bar{w}_n)$, and $IT_\varphi(N,c, \bar{d}_n) \subseteq IT_\varphi(M, a, \bar{b}_n)$. Then, according to Lemma \ref{L:ext}, consider $\omega$-saturated elementary extensions $M'$, $N'$ of $M$ and $N$, respectively. We have: \begin{align} &M, a, \bar{b}_n \models \varphi(x, \bar{w}_n) \Leftrightarrow M', a, \bar{b}_n \models \varphi(x,\bar{w}_n)\label{E:m1}\\ &N', c, \bar{d}_n \models \varphi(x,\bar{w}_n)\label{E:m2} \end{align} Also since $M'$, $N'$ are elementarily equivalent to $M$, $N$ we have \[ IT_\varphi(N',c, \bar{d}_n) = IT_\varphi(N,c, \bar{d}_n) \subseteq IT_\varphi(M, a, \bar{b}_n) = IT_\varphi(M', a, \bar{b}_n). \] But then we have $(N',c, \bar{d}_n) \leq_{\Sigma_\varphi} (M',a, \bar{b}_n)$, and, by $\omega$-saturation of $M'$, $N'$, relation $A$ as defined in Lemma \ref{L:sat} is an $\langle(N',c, \bar{d}_n),(M',a, \bar{b}_n)\rangle$-asimulation. But then by \eqref{E:m2} and asimulation invariance of $\varphi(x,\bar{w}_n)$ we get $M', a, \bar{b}_n \models \varphi(x,\bar{w}_n)$, and further, by \eqref{E:m1} we conclude that $M, a, \bar{b}_n \models \varphi(x,\bar{w}_n)$. Therefore, $\varphi(x,\bar{w}_n)$ in fact follows from $IC(\varphi(x,\bar{w}_n))$. \end{proof}
The following theorem is an immediate consequence of Corollary \ref{L:c-inv} and Theorem \ref{L:main}: \begin{theorem}\label{L:final} A formula $\varphi(x,\bar{w}_n)$ is invariant with respect to asimulations iff it is equivalent to a standard $x$-translation of an intuitionistic formula. \end{theorem}
\section{Criteria for first-order definable classes}\label{S:Rest}
Theorem \ref{L:final} stated above establishes a criterion for the equivalence of first-order formula to a standard translation of intuitionistic formula on arbitrary first-order models. But one may have a special interest in a proper subclass $K$ of the class of first-order models viewing the models which are not in this subclass as irrelevant, non-intended etc. In this case one may be interested in the criterion for equivalence of a given first-order formula to a standard translation of an intuitionistic predicate formula \emph{over} this particular subclass. It turns out that if some parts of this subclass are first-order axiomatizable then only a slight modification of our general criterion is necessary to solve this problem.
To tighten up on terminology, we introduce the following definitions: \begin{definition}\label{D:k} Let $K$ be a class of models. Then: \begin{enumerate} \item $K(\Theta) = \{\,M \in K\mid K\text{ is a $\Theta$-model}\,\}$; \item $K(\Theta)$ is first-order axiomatizable iff there is a set $Ax$ of $\Theta$-sentences, such that a $\Theta$-model $M$ is in $K$ iff $M \models Ax$; \item A set $\Gamma$ of $\Theta$-formulas is $K$-satisfiable iff $\Gamma$ is satisfied by some model in $K$; \item A $\Theta$-formula $\varphi$ $K$-follows from $\Gamma$ $(\Gamma \models_K \varphi)$ iff $\Gamma \cup \{\,\varphi\,\}$ is $K$-unsatisfiable; \item $\Theta$-formulas $\varphi$ and $\psi$ are $K$-equivalent iff $\varphi \models_K \psi$ and $\psi \models_K \varphi$. \end{enumerate} \end{definition} It is clear that for any class $K$, such that $Ax$ first-order axiomatizes $K(\Theta)$, any set $\Gamma$ of $\Theta$-formulas and any $\Theta$-formula $\varphi$, $\Gamma$ is $K$-satisfiable iff $\Gamma \cup Ax$ is satisfiable, and $\Gamma \models_K \varphi$ iff $\Gamma \cup Ax \models \varphi$. \begin{definition}\label{D:int-inv} A formula $\varphi(x, \bar{w}_n)$ is $K$-invariant with respect to asimulations iff for any $\Theta$ such that $\Sigma_\varphi \subseteq \Theta$, any $n$-ary evaluation $\Theta$-points $(M, a, \bar{b}_n)$ and $(N,c, \bar{d}_n)$, if $M, N \in K$, there exists an $\langle (M, a, \bar{b}_n),(N,c, \bar{d}_n)\rangle$-asimulation $A$, and $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$, then $N, c, \bar{d}_n \models \varphi(x, \bar{w}_n)$. \end{definition} Now for the criterion of $K$-equivalence: \begin{theorem}\label{L:int-main} Let $K$ be a class of first-order models such that $K(\Theta)$ is first-order axiomatizablefor any finite $\Theta$, and let $\varphi(x, \bar{w}_n)$ be $K$-invariant with respect to asimulations. Then $\varphi(x, \bar{w}_n)$ is $K$-equivalent to a standard $x$-translation of an intuitionistic formula. \end{theorem}
\begin{proof} Let $Ax_\varphi$ be the set of first-order sentences that axiomatizes $K(\Sigma_\varphi)$. We may assume that $\varphi(x, \bar{w}_n)$ is $K(\Sigma_\varphi)$-satisfiable, otherwise $\varphi(x, \bar{w}_n)$ is $K$-equivalent to $ST(\bot, x)$ and we are done. In what follows we will write $KC(\varphi(x, \bar{w}_n))$ for the set of $\Sigma_\varphi$-formulas in variables $x, \bar{w}_n$ that are standard $x$-translations of intuitionistic formulas $K$-following from $\varphi(x,\bar{w}_n)$.
Our strategy will be to show that $KC(\varphi(x,\bar{w}_n)) \models_K \varphi(x,\bar{w}_n)$. Once this is done we will conclude that \[ Ax_\varphi \cup KC(\varphi(x,\bar{w}_n)) \models \varphi(x,\bar{w}_n). \] Then we apply compactness of first-order logic and conclude that $\varphi(x,\bar{w}_n)$ is equivalent to a finite conjunction $\psi_1(x,\bar{w}_n)\wedge\ldots \wedge\psi_m(x,\bar{w}_n)$ of formulas from this set. But it follows then that $\varphi(x,\bar{w}_n)$ is $K$-equivalent to the conjunction of the set $KC(\varphi(x)) \cap \{\,\psi_1(x,\bar{w}_n)\ldots, \psi_m(x,\bar{w}_n)\,\}$. In fact, by our choice of $KC(\varphi(x,\bar{w}_n))$ we have \[ \varphi(x,\bar{w}_n) \models_K \bigwedge(KC(\varphi(x,\bar{w}_n)) \cap \{\,\psi_1(x,\bar{w}_n)\ldots, \psi_m(x,\bar{w}_n)\,\}), \] And by our choice of $\psi_1(x,\bar{w}_n)\ldots, \psi_m(x,,\bar{w}_n)$ we have
\[ Ax_\varphi \cup (KC(\varphi(x,\bar{w}_n)) \cap \{\,\psi_1(x,\bar{w}_n)\ldots, \psi_m(x,\bar{w}_n)\,\}) \models \varphi(x,\bar{w}_n) \] and hence \[ KC(\varphi(x,\bar{w}_n)) \cap \{\,\psi_1(x,\bar{w}_n)\ldots, \psi_m(x,,\bar{w}_n)\,\} \models_K \varphi(x,\bar{w}_n). \]
To show that $KC(\varphi(x,\bar{w}_n)) \models_K \varphi(x,\bar{w}_n)$, take any $n$-ary evaluation $\Sigma_\varphi$-point $(M, a, \bar{b}_n)$ such that $M \in K$ and $M, a, \bar{b}_n \models KC(\varphi(x,\bar{w}_n))$. Such a model exists, because $\varphi(x,\bar{w}_n)$ is $K(\Sigma_\varphi)$-satisfiable and $KC(\varphi(x,\bar{w}_n))$ will be $K$-satisfied in any $n$-ary evaluation $\Sigma_\varphi$-point satisfying $\varphi(x,\bar{w}_n)$. Then we can also choose an $n$-ary evaluation $\Sigma_\varphi$-point $(N, c, \bar{d}_n)$ such that $N \in K$ and $N, c, \bar{d}_n \models \varphi(x,\bar{w}_n)$ and $IT_\varphi(N, c, \bar{d}_n) \subseteq IT_\varphi(M, a, \bar{b}_n)$.
For suppose otherwise. Then for any $\Sigma_\varphi$-model $N \in K$ and any $n$-ary evaluation $\Sigma_\varphi$-point $(N, c, \bar{d}_n)$ such that $N, c, \bar{d}_n \models \varphi(x,\bar{w}_n)$ we can choose an intuitionistic formula $i_{(N, c, \bar{d}_n)}(\bar{w}_n)$ such that $ST(i_{(N, c, \bar{d}_n)}(\bar{w}_n), x)$ is a $\Sigma_\varphi$-formula true at $(N, c, \bar{d}_n)$ but not at $(M, a, \bar{b}_n)$. Then consider the set \[ S = \{\,\varphi(x,\bar{w}_n)\,\} \cup \{\,\neg ST(i_{(N,c, \bar{d}_n)}(\bar{w}_n), x)\mid N \in K \wedge N, c, \bar{d}_n \models \varphi(x,\bar{w}_n)\,\} \] Let $\{\,\varphi(x,\bar{w}_n), \neg ST(i_1(\bar{w}_n), x)\ldots , \neg ST(i_u(\bar{w}_n), x)\,\}$ be a finite subset of this set. If this set is $K$-unsatisfiable, then we must have \[ \varphi(x,\bar{w}_n) \models_K ST(i_1(\bar{w}_n), x)\vee\ldots \vee ST(i_u(\bar{w}_n), x), \] but then we will also have \[ (ST(i_1(\bar{w}_n), x)\vee\ldots \vee ST(i_u(\bar{w}_n), x)) \in KC(\varphi(x,\bar{w}_n)) \subseteq IT_\varphi(M, a, \bar{b}_n), \] and hence $(ST(i_1(\bar{w}_n), x)\vee\ldots \vee ST(i_u(\bar{w}_n), x))$ will be true at $(M, a, \bar{b}_n)$. But then at least one of $ST(i_1(\bar{w}_n), x)\ldots ,ST(i_u(\bar{w}_n), x)$ must also be true at $(M, a, \bar{b}_n)$, which contradicts the choice of these formulas. Therefore, every finite subset of $S$ is $K$-satisfiable. But then every finite subset of the set $S \cup Ax_\varphi$ is satisfiable as well. By compactness of first-order logic $S \cup Ax_\varphi$ is satisfiable, hence $S$ is satisfiable over $K$.
But then take any $n$-ary evaluation $\Sigma_\varphi$-point $(N',c', \bar{d'}_n)$ satisfying $S$ such that $N' \in K$ and this will be an evaluation point for which we will have both $N', c', \bar{d'}_n \models ST(i_{(N',c', \bar{d'}_n)}(\bar{w}_n), x)$ by choice of $i_{(N',c', \bar{d'}_n)}$ and $N',c', \bar{d'}_n \models \neg ST(i_{(N',c', \bar{d'}_n)}(\bar{w}_n), x)$ by the satisfaction of $S$, a contradiction.
Therefore, for any given $n$-ary evaluation $\Sigma_\varphi$-point $(M, a, \bar{b}_n)$ satisfying $ KC(\varphi(x,\bar{w}_n))$ such that $M \in K$ we can choose an $n$-ary evaluation $\Sigma_\varphi$-point $(N,c, \bar{d}_n)$ such that $N \in K$, $N, c, \bar{d}_n\models \varphi(x, \bar{w}_n)$ and $IT_\varphi(N, c, \bar{d}_n) \subseteq IT_\varphi(M,a, \bar{b}_n)$. Then, reasoning exactly as in the proof of Theorem \ref{L:main}, we conclude that $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$. Therefore, $\varphi(x, \bar{w}_n)$ in fact $K$-follows from $KC(\varphi(x, \bar{w}_n))$. \end{proof}
\begin{theorem}\label{L:int-final} Let $K$ be a class of first-order models such that for any finite $\Theta$ the class $K(\Theta)$ is first-order axiomatizable. Then a formula $\varphi(x, \bar{w}_n)$ is $K$-invariant with respect to asimulations iff it is $K$-equivalent to a standard $x$-translation of an intuitionistic formula. \end{theorem} \begin{proof} From left to right our theorem follows from Theorem \ref{L:int-main}. In the other direction, assume that $\varphi(x, \bar{w}_n)$ is $K$-equivalent to $ST(i(\bar{w}_n),x)$ and assume that for some $\Theta$ such that $\Sigma_\varphi \subseteq \Theta$, some $n$-ary evaluation $\Theta$-points $(M, a, \bar{b}_n)$ and $(N, c, \bar{d}_n)$ such that $M,N \in K$, and some $\langle (M, a, \bar{b}_n),(N, c, \bar{d}_n)\rangle$-asimulation $A$ we have $M, a, \bar{b}_n \models \varphi(x, \bar{w}_n)$. Then, by Corollary \ref{L:c-inv} we have $N, c, \bar{d}_n \models ST(i(\bar{w}_n),x)$, but since $ST(i(\bar{w}_n),x)$ is $K$-equivalent to $\varphi(x, \bar{w}_n)$ and $N$ is in $K$, we also have $N, c, \bar{d}_n \models\varphi(x, \bar{w}_n)$. Therefore, $\varphi(x, \bar{w}_n)$ is $K$-invariant with respect to asimulations. \end{proof}
One obvious instantiation for $K$ would be the class of all \emph{intuitionistic} models which are normally viewed as intended models for intuitionistic predicate logic within the framework of Kripke semantics. A first-order axiomatization for $K(\Theta)$ would be $RT \cup Mon \cup ER \cup Type$, where: \begin{align*} &RT = \{\,\forall yR(y,y), \forall yzw((R(y,z) \wedge R(z,w)) \to R(y,w))\,\};\\ &Mon = \{\,\forall yz\bar{w}_n((P(y, \bar{w}_n) \wedge R(y,z)) \to P(z, \bar{w}_n))\mid P \in \Theta \setminus \{\,R\,\}\,\};\\ &ER = \{\,\forall x(\exists yE(x,y) \leftrightarrow \neg\exists yE(y,x)), \forall xy(R(x,y) \to \exists zw(E(x,z) \wedge E(y,w)))\,\};\\ &Type = \{\,\forall y\bar{z}_n(P(y,\bar{z}_n) \to \bigwedge^n_{i=1}(E(y,z_i))\mid P \in \Theta \setminus \{\,R\,\}\,\}. \end{align*}
Another instantiation for $K$ might be, e.g. the class of \emph{intuitionistic models with constant domains}. In this case, if $R^2, E^2 \in \Theta$, a first-order axiomatization for $K(\Theta)$ is given by $RT \cup Mon \cup ER \cup Type \cup \{\,CD\,\}$, where \[ CD = \forall x(\exists yE(y,x) \to \forall yE(y,x)). \]
Thus our Theorem \ref{L:int-final} yields, among others, a simple equivalence criterion for these two particular classes of models.
\section{Conclusion and further research}\label{S:final}
Theorems \ref{L:param}, \ref{L:final}, and \ref{L:int-final} proved above show that the general idea of asimulation for intuitionistic propositional logic is a faithful analogue of the idea of world-object bisimulation for modal predicate logic in many important respects. However, in the predicate case differences from the corresponding notion of bisimulation are much more conspicuous than in the propositional case. Thus, if we introduced `asimulation games' corresponding to the propositional version of asimulation defined in \cite{Ol} (the main difference from propositional case being the absence of conditions \eqref{E:c33} and \eqref{E:c44}) then, given the strength of condition \eqref{E:c22} we would have these games indistinguishable from bisimulation games on the segment beginning from the first move of Duplicator. Every link between worlds established by this player would have to be symmetrical and the asymmetry of asimulation would be important only for the intial pair of worlds.\footnote{This asymmetry would also possibly lead to exclusion of some successors of the left world of the link from the domain of the bisimulation game to follow.} This does not hold in the predicate case. Here, depending on the strategy chosen by Spoiler, the whole game might be played with the asymmetrical links between sequences of world and objects; also asymmetry can be reinstated after the players reach the first symmetrical link in the game, and the direction of asymmetry can be switched by moves of the players. All these features show that specific features of intuitionistic logic can be actualized within the setting of quantifiers and predicates only, while on the propositional level one can find but mere rudiments and traces of them.
One interesting further question lying beyond the scope of the present paper is the status of the proofs presented above from the viewpoint of intuitionistic philosophy. It is well-known that $\omega$-saturated models whose existence is guaranteed by Lemma \ref{L:ext} might turn out to be uncountable. Hence our proof might be viewed by a hardcore intuitionist as having no sense at all. As it happens, there is a way to give another proof of our main result that looks more favorable to an intuitionistic eye. This proof uses countable models only and employs the notion of recursive saturation instead of saturation \emph{simpliciter}. However, this variant of proof is also a little bit less clear and more indirect, so we postpone its publication to another occasion.
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arXiv
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Revisiting the 1986 computer classic Number Munchers!
Belgin gets hooked on a classic maths game…in 16 bits! Here's her review…
©MECC 1990, reproduced for the purpose of review.
by Belgin Seymenoglu. Published on 11 December 2020.
Ready to play Number Munchers. Image: ©MECC 1990, reproduced for the purpose of review.
If you were a child in the eighties or nineties, you might have seen the educational game Number Munchers on your school PC. It was originally released by MECC in 1986, and was re-released several times (for MS-DOS, Apple, and more). Nearly three decades later, Number Munchers received a Readers' Choice Award in 2005 from Tech and Learning.
Believe it or not, I didn't play it as a kid—rather I just watched a classmate play the 1990s version on a Macintosh. It wasn't until two decades later (read: last winter) when I had a go at playing it. I couldn't find the Macintosh version of the game, but I did come across the older MS-DOS version, so I played that.
Yum, yum!
The controls are quite straightforward—just use the arrow keys to move your green muncher around, and the space bar when it's time to eat a number. Granted, most games I have played are for the PC, so I find keyboard controls easy to use.
Your green guy is sitting in a 5 by 6 grid, and each square on the grid contains a number. You get points by eating numbers that satisfy the rule given on the top of the screen. Meanwhile, if you eat a wrong number, you lose one life. The game ends when you run out of lives. Example rules include:
Multiples of 5: eat 5, 10, 15, etc
Factors of 14: only eat 1, 2, 7 and 14
Prime numbers: eat primes
Equals 6: you get expressions such as $6\times 1$, $3 + 0$, and need to pick the ones that equal 6
Less than 12: eat only the numbers 1–11
There's even a challenge mode that lets you mix and match the rules! Moreover, there are lots of difficulty levels to pick from. There are 11 levels in total; they start at 'third grade easy' (that's year 4 for Brits like me), and go all the way up to 'seventh grade easy/advanced', and finally eighth grade and above.
Number Munchers features five fearsome foes to fight or flee. Image: ©MECC 1990, reproduced for the purpose of review.
You will also want to avoid the Troggles—they are the monsters who want to eat your little muncher! It's another surefire way to lose a life. When I first saw the game as a child, I didn't notice that there were five types of Troggles, each coming in different colours and walking in specific patterns. I also forgot that when a Troggle walks over a square, it leaves a new number behind. If that's not challenging enough, things start to get more frantic in later levels. More Troggles will turn up on the same board, and they'll move faster, so you'd better be quick on your feet or have picked an easy maths mode! You're also more likely to see what happens when Troggles meet: one eats the other, then the surviving Troggle continues walking as if nothing happened.
The Troggles at it again in this cutscene. Image: ©MECC 1990, reproduced for the purpose of review.
When you've eaten all the numbers on the board that fit the rule, you get to move on to the next level! Also, every three or four levels you get treated to a funny cutscene featuring the muncher and the Troggles! In most of the cutscenes, the Troggles try to capture the muncher, only for the plan to backfire, so the muncher gets the last laugh! You can even hear the muncher a little jingle, as if they were singing "Nyah-nyah-nyah-nyah-nyah-nyah!" Apparently there are at least five more fun cutscenes out there. No, not all of them feature Troggles. Sorry Troggle fans!
My favourite mode
As a schoolgirl I watched my classmate play the level where you only eat prime numbers, and the moment he lost a life. No—he did not get eaten! The disaster was what he ate…the number 1. The game then said that 1 is not prime, but didn't explain why.
Late breaking news from Number Munchers: 1 is not prime! Image: ©MECC 1990, reproduced for the purpose of review.
Then the teacher's assistant was watching too. When the muncher lost a life, she turned to me and asked, "Why do you think the number 1 is not prime?". How was I supposed to know? I was only just starting to learn what a prime number is! I was aware that a prime is divisible only by 1 and itself, but didn't realise that these two divisors should be distinct. It only dawned on me years later, but I'd already moved into secondary school by then!
This is why the prime numbers round became my favourite level in the game. It showed me a something I didn't realise until then, and made me go "ooh". And now I'm older, I'm having no difficulties with the prime level…as long as there are no three-digit numbers!
Number Munchers is definitely one of those maths games that can be enjoyed by people of (almost) all ages. Just make sure you didn't pick the hardest difficulty setting! I did that, and I instantly regretted it—I found myself struggling to figure out which of the three-digit numbers I got were multiples of 19! It didn't help that I initially misread the question, and thought I was supposed to avoid said multiples! An easy way to throw a life away. And as if I didn't have enough to do already, I had to keep dodging the Troggles to make sure I didn't eaten! Unsurprisingly I gave up, and switched to an easier setting.
We do not recommend starting with this mode. Image: ©MECC 1990, reproduced for the purpose of review.
If you're after graphics, I recommend the 90s Apple version—the creatures are prettier in there (especially your little green muncher). The graphics on the DOS version are not as great, but the gameplay's the same and the Troggles still look quite nice in that version. If you want to try the game yourself, the original version is available to the public on the Internet Archive, all for free. Better still, no emulator is required. What's not to like?
Believe it or not, this is not the only maths-themed game in the Munchers series—there's another game called Fraction Munchers! It features fractions instead of whole numbers, but I've never seen it! If you've been lucky enough to have played that game, why not send your review of Fraction Munchers to Chalkdust? It might just become an online article in here, too!
Belgin Seymenoglu
Belgin is a data scientist having got her PhD in population genetics. When not working, you can usually find Belgin either playing the piano or playing Math Blaster. She is pictured here standing next to her copy of Zeeman's catastrophe machine.
www.ucl.ac.uk/~zcahge7/ + More articles by Belgin
Maths trumps review
A mathematically-themed version of the classic card game, with several new features
Significant figures: Sir Christopher Zeeman
Biography of Sir Christopher Zeeman
Math Blaster
Belgin plays a classic mathsy game from her childhood...in 16-bit graphics! Here's her review...
MathsJam 2017
Chalkdust descends upon the UK's largest pop maths gathering and tells you what you missed
Creating hot ice
We created hot ice from scratch, a solution that remains liquid even below its freezing point!
Cutting my birthday cake
How to make the most slices from just a few cuts of cake
← Flo-maps fractograms: the game
Christmas puzzle #1: Christmas tree sudoku →
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CommonCrawl
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Complex Numbers Birth Trigonometry!
Measuring Angles of Rotation in a Dimensionless Way
Recall that we have interpreted complex multiplication in the context of rotating points in a plane representing complex numbers about the origin. Such rotations may be some part of a full rotation, a full rotation or more, or even rotations in the opposite direction (when the associated angles are negative). Exactly how much we rotate has so far been described in terms of degrees.
Hipparchus of Rhodes
Most will be familiar with measuring angles in degrees, but what many may not know is that this unit of measurement traces back to the second century BC, when the Greek astronomer and mathematician Hipparchos of Rhodes began applying geometry to Babylonian astronomy (likely due to the amazing collection of astronomical data the Babylonians had compiled -- i.e., 800 years worth of nightly written records).
Despite the existence at the time of Euclid's Elements (c. 300 BC), a text which largely set the standard for learning geometry -- that work never provided a unit of measurement for angles besides the right angle. Hipparchos thus borrowed the Babylonian division of the ecliptic -- a great circle on the celestial sphere representing the sun's apparent path during the year, so called because lunar and solar eclipses can occur only when the moon crosses it.
The ancient Babylonians had divided this circular path into 12 sections called beru (interestingly, with names that in Greek translate to Gemini, Cancer, Leo, etc.). Then, they divided each of these sections into 30 equal subsections -- allowing the position of the sun to be described in one of $12 \cdot 30 = 360$ ways. Thus, the notion of a degree -- one $360^{th}$ of a full rotation -- was born. Why $12$ sections and $30$ subdivision you ask? Likely this was due to the fact that the calendar Babylonians used at the time was based on lunar cycles (cycles of the moon). Each such cycle lasted approximately 30 days (i.e., sometimes 29, othertimes 30), with $12$ lunar cycles occurring with each year.
All that fascinating history aside, there are better things we can use to compare the sizes of angles than the number of days in a lunar cycle!
As a starting place, consider the length of the path that some complex $z$ creates as it moves from $1$ counter-clockwise around the unit circle. Recall that one complete rotation about $0$ (the origin) should result in a path $2\pi$ times as long as the radius. But how long is that, actually -- are we measuring in inches? ..in feet?
Rather than try to pick some arbitrary dimension for the measure for these lengths (which is actually not unlike the somewhat arbitrary Babylonian division of a full rotation into 360 degrees), what if we instead define the angle measure by the ratio of the length of the path $z$ takes, and the length of the radius? Note that as both would presumably be measured in the same dimension, allowing these to cancel -- leaving a "dimensionless" radian measure of the angle in question.
For clarity (especially when using both degrees and radians), we sometimes write angles measures expressed in radians as $\theta$ rad, but given that radians are actually dimensionless, we will more often omit the "rad" part.
Measuring angles in this way equates a full rotation of $360^{\circ}$ to a radian measure of $2\pi$, and all other angles to their proportional equivalents (for example, half of $360^{\circ}$ is $180^{\circ}$ and half of $2\pi$ is $\pi$, so $180^{\circ}$ and $\pi$ are equal angle measures. Rotations of some common radian measures are shown below:
Given that a radian measure of $\pi$ is the same as $180^{\circ}$ and dividing both sides by $180$, we quickly see that $1^{\circ} = \frac{\pi}{180}$, we can then use this to easily convert the degree measure of any angle to radians and the radian measure of any angle to degrees. Some example conversions are shown below:
Convert to radians: $20^{\circ}$, $30^{\circ}$, and $-60^{\circ}$ (by multiplying by $\frac{\pi}{180^{\circ}}$):
$20^{\circ} = 20 \left( \frac{\pi}{180^{\circ}} \right) = \frac{\pi}{9}$
$-60^{\circ} = -60 \left( \frac{\pi}{180^{\circ}} \right) = -\frac{\pi}{3}$
Convert the given radian measures to degrees: $\frac{7\pi}{6}$ rad, $-\frac{\pi}{12}$ rad, and $0.76$ rad
$\frac{7\pi}{6} = \frac{7\pi}{6} \cdot \left( \frac{180^{\circ}}{\pi} \right) = 210^{\circ}$
$-\frac{\pi}{12} = -\frac{\pi}{12} \cdot \left( \frac{180^{\circ}}{\pi} \right) = -15^{\circ}$
$0.76 = 0.76 \cdot \left( \frac{180^{\circ}}{\pi} \right) = \left( \frac{136.8}{\pi} \right)^{\circ} \approx 48.54^{\circ}$
Precisely because they do not rely on any (arbitrarily) chosen dimension, the use of radians to measure angles will greatly simplify many things in both calculus and other areas of mathematics.
The Cosine and Sine Functions
We saw when we introduced complex numbers previously that multiplication of one complex number by another involves both rotation about the origin and scaling the distance from the same. Of course, if the complex numbers involved were both of unit magnitude, then only rotation was involved.
With this in mind, let us consider again the unit circle of complex values with unit magnitude. In particular, let us consider the real and imaginary parts of the $z$ on that unit circle where $\theta = arg(z)$ is known. We should recognize that knowing $\theta$ fixes exactly where this $z$ must be located in the complex plane. To see this, imagine one starts at $0$ (the origin) and then moves exactly one unit distance in a direction corresponding to $\theta$.
The real part of such a $z$ and its imaginary coefficient (and consequently the $x$ and $y$ coordinates of the corresponding point in the related coordinate plane -- also called the Cartesian plane, named after French mathematician Rene Descartes.) are also then fixed once we know $\theta$. In this way, we can think of these $x$ and $y$ coordinates as functions of $\theta$. To attach specific names to these functions, let us make the following definitions:
For any $\theta$, let $z = x + iy$ be the unique complex value on the unit circle (i.e., $|z| = 1$) with argument $\theta$. Then, define functions cosine and sine, denoted* $\cos \theta$ and $\sin \theta $ respectively, so that: $$\cos \theta = x = Re(z) \quad \textrm{ and } \quad \sin \theta = y = Im(z)$$
*Note that -- just like log functions -- we traditionally omit the parentheses around the input when doing so doesn't cause confusion.
To visualize what these functions do, consider the diagram drawn on the left below. Of course, we can also (and more frequently do) draw such things instead on the related coordinate plane, as shown on the below right.
You may be curious about why we have included the shaded blue triangle in the diagram. This triangle -- formed by drawing a vertical line from the point on the unit circle to the real axis (or $x$-axis on the right) is called a reference triangle, and serves as a way to quickly compute sine and cosine values for certain commonly encountered angles. We will have more to say about it shortly.
For now, let us simply say that the cosine and sine functions form the basis of the area of mathematics called trigonometry, a word which traces back to the Greek words trigonan which means "triangle" and metron, meaning "to measure". In many ways, this reference triangle can be thought of as the triangle to which the word "trigonometry" owes its origin!
Some Basic Properties of the Sine and Cosine Functions
Whenever we develop a new function, there are always questions we should ask -- things like what's the domain? ..what's the image/range? ..what interesting properties does it have? ..etc. The below address some of these. Given the above equivalent ways to interpret the cosine and sine functions -- as related to the complex plane or the Cartesian plane -- let us opt for the latter in what we have to say below. That is to say, let us interpret $\cos(\theta)$ for the moment as an $x$-coordinate and $\sin \theta$ as a $y$-coordinate:
Since the inputs to both the cosine and sine functions are angles measures corresponding to rotations (which can be negative when the rotation involved is clockwise), the domain of these two functions both agree and equal the set of all real numbers, $\mathbb{R}$.
Since the related complex value $z$ for any $\theta$ lies on the unit circle, it must be the case that both the $x$ and $y$ values taken on by the cosine and sine functions respectively, must in the interval $[-1,1]$, which then forms the image/range of these two functions. That is to say: $$-1 \le \cos \theta \le 1 \quad \textrm{ and } \quad -1 \le \sin \theta \le 1 \quad \textrm{ for all } \theta \in \mathbb{R}$$
Note that the reference triangle will always be a right triangle with base measuring $a = |\cos \theta|$ and height $b = |\sin \theta|$, with a hypotenuse always $1$ unit long (i.e., the radius of the unit circle). As such, the Pythagorean theorem applies -- giving us the famed Pythagorean Identity: $$\cos^2 \theta + \sin^2 \theta = 1 \quad \textrm{ for all } \theta \in \mathbb{R}$$
Since co-terminal angles (i.e., angles whose measures differ by either $360^{\circ} n$ or equivalently $2\pi n$ radians for some integer $n$) have the same coordinates, the following must be true: $$\left.\begin{array}{rcl} \cos(\theta \pm 2\pi n) &=& \cos \theta\\ \sin(\theta \pm 2\pi n) &=& \sin \theta \end{array}\right\} \quad \textrm{ for all } \theta = 1,2,3,\ldots$$
The quadrant in which some complex $z$ falls can be used to determine the signs of the corresponding values of $\cos \theta$ and $\sin \theta$, in accordance with the following diagram:
Exploiting Familiar Triangles and Symmetries
For some commonly-encountered angles of $\theta$, the values of $\cos \theta$ and $\sin \theta$ are easily found using geometry. Using these values in conjunction with the symmetries seen in the unit circle, the values corresponding to other commonly encountered angles can also be easily found. We explore this idea below:
Sine and Cosine Values for Integer Multiples of $\frac{\pi}{2}$
Note that if the related $\theta = \frac{n \pi}{2}$ where $n$ is an integer, the related point on the unit circle must be on either the $x$ or $y$ axis. Consequently, there are only four points of interest, with cosine and sine values as shown in the table and image below:
$$\begin{array}{|c|c|c|} t & \cos t & \sin t\\\hline 0 & 1 & 0 \\\hline \frac{\pi}{2} & 0 & 1 \\\hline \pi & -1 & 0 \\\hline \frac{3\pi}{2} & 0 & -1\\\hline \end{array}$$
Of course, similar results are obtained for angles co-terminal to those above, as seen in the examples below: $$\cos \left( -\frac{3\pi}{2} \right) = 0, \quad \sin(5\pi) = 0, \quad \cos(-6\pi) = 1, \quad \sin \left( -\frac{5\pi}{2} \right) = -1$$
Sine and Cosine Values Related to $30^{\circ}$, $45^{\circ}$, and $60^{\circ}$
When the angles $\theta$ is $30^{\circ}$, $45^{\circ}$, or $60^{\circ}$ (or equivalently in radians: $\theta = \frac{\pi}{6}$, $\frac{\pi}{4}$, or $\frac{\pi}{3}$), we can appeal to the corresponding reference triangle and a little bit of geometry to find the exact values of $\cos \theta$ and $\sin \theta$.
Let us first consider the reference triangle for $45^{\circ} = \frac{\pi}{4}$:
Consider a $45^{\circ}$ angle, as shown in the triangle below. Recall for a reference triangle, the hypotenuse length is always $1$ as it is a radius of the unit circle. Since the sum of angles in a triangle is $180^{\circ}$, the unmarked angle must also measure $45^{\circ}$. With two angles of the same measure, geometry tells us that the triangle is isosceles, with lengths $x$ and $y$ equal.
Then, the Pythagorean Theorem gives us:
$$\begin{array}{rcll} x^2 + y^2 &=& 1 & \scriptsize{\textrm{since } x = y, \textrm{we make a substitution}}\\ x^2 + x^2 &=& 1 & \scriptsize{\textrm{now we solve for } x}\\ 2x^2 &=& 1 &\\ x &=& \frac{\sqrt{2}}{2} &\scriptsize{\textrm{keeping only the positive solution, as } x \textrm{ represents a length}}\\ y &=& \frac{\sqrt{2}}{2} &\scriptsize{\textrm{again, recalling } x = y}\\ \end{array}$$
Thus, we can immediately deduce: $$\cos 45^{\circ} = \frac{\sqrt{2}}{2} \quad \textrm{ and } \quad \sin 45^{\circ} = \frac{\sqrt{2}}{2}$$
We may similarly use geometry to find the values of the sine and cosine functions when applied to angles measuring $30^{\circ}$ and $60^{\circ}$.
This time we start with a $30^{\circ}-60^{\circ}-90^{\circ}$ right triangle whose hypotenuse is again one unit in length, as shown on the right.
Starting with the reference triangle for $\theta = 30^{\circ}$, we construct a congruent $30^{\circ}-60^{\circ}-90^{\circ}$ triangle directly below this one. In this way, we form a larger triangle whose angle measures are all $60^{\circ}$, as shown.
Thus, this larger triangle must then be equilateral. This means that all side lengths must be $1$ -- forcing the value of $y$ to be half that (i.e., $y = \frac{1}{2}$).
Again appealing to the Pythagorean theorem, we now have:
$$\begin{array}{rcll} x^2 + y^2 &=& 1 & \scriptsize{\textrm{then we substitute } y = \frac{1}{2}}\\ x^2 + \left(\frac{1}{2}\right)^2 &=& 1 & \scriptsize{\textrm{from here we can solve for } x}\\ 4x^2 + 1 &=& 4 &\\ x &=& \frac{\sqrt{3}}{2} & \overset{\normalsize{\textrm{again keeping only the positive value}}}{\scriptsize{\textrm{ of } x \textrm{ as it represents a length}}}\\ \end{array}$$
This immediately gives us
$$\cos 30^{\circ} = \frac{\sqrt{3}}{2} \quad \textrm{ and } \quad \sin 30^{\circ} = \frac{1}{2}$$
Armed with the cosine and sine values just argued and their radian equivalents (as shown below), we can find many more cosine and sine values by taking advantage of the various symmetries seen in the unit circle:
$$\begin{array}{cc} \cos \frac{\pi}{6} = \cos 30^{\circ} = \frac{\sqrt{3}}{2} \quad & \quad \sin \frac{\pi}{6} = \sin 30^{\circ} = \frac{1}{2}\\\\\hline \\\cos \frac{\pi}{4} = \cos 45^{\circ} = \frac{\sqrt{2}}{2} \quad & \quad \sin \frac{\pi}{4} = \sin 45^{\circ} = \frac{\sqrt{2}}{2} \end{array}$$
For example, consider $\theta = 60^{\circ} = \frac{\pi}{3}$. Noting that the triangle whose red side points at $\frac{\pi}{3}$ must be a reflection of the other triangle across the $y=x$ line and thus congruent to it, we must have both triangles congruent to the "$30^{\circ} - 60^{\circ} - 90^{\circ}$" triangle previously discussed. As such:
$$\textstyle{\cos \frac{\pi}{3} = \cos 60^{\circ} = \frac{1}{2} \quad \textrm{ and } \quad \sin \frac{\pi}{3} = \sin 60^{\circ} = \frac{\sqrt{3}}{2}}$$
Indeed, any angle whose reference triangle has a hypotenuse that forms either a $30^{\circ}$ angle with one of the axes will have its long side of length $\frac{\sqrt{3}}{2}$ and its short side of length $\frac{1}{2}$. Using this fact and the previously mentioned signs attached to the $x$ and $y$ coordinates in each quadrant, the cosine and sine values of any such angles will be immediate.
Similarly, for any reference triangle whose hypotenuse forms a $45^{\circ}$ angle with an axis, the sides will both have length $\frac{\sqrt{2}}{2}$. Again, combining these side lengths with what we know about the signs attached to the $x$ and $y$ coordinates in each quadrant, the cosine and sine values for these angles are also quickly determined.
All this allows us to considerably expand our list of known sine and cosine values, as the graphic below suggests.
On Names and (Other) Trigonometric Functions
One might naturally wonder why the functions discussed above are named sine and cosine. The answer is interesting, in that it provides an example of how some mathematical ideas that originated in India were picked up first by the Muslims and then finally spread to Europe. The origin of sine traces back to the Sanscrit word jiya, which means "bowstring".
It is not hard to see why -- consider the images below. If the circle on the right is a unit circle, note how the sine value associated with $\theta = m\angle BOC$ is the half the length of the (undrawn) "bowstring", segment $BC$. Coincidently, in the same image we can even see the "bow" as arc $\stackrel{\mbox{$\frown$}}{BAC}$, and the arrow $OA$ notched at $O$ to the (drawn) bowstring formed by the union of radii $OB$ and $OC$!
In Arabic, however the bowstring is called jiba, although vowels are not always written in Arabic, and thus someone like $12^{th}$-century Gherardo of Cremona, who was translating an Arabic text on geometry, would have seen simply the Arabic equivalent to the letters "jb". As a consequence, Gherado failed to translate this word correctly, thinking it was another (i.e., jaib) -- a word that means "curve, fold, or hollow". In Latin, the equivalent for "curve" is sinus, and from there one can more easily see the final evolution into sine upon one more translation into English.
With regard to the origins of the word cosine, consider the following diagrams:
Clearly, $\theta_1$ and $\theta_2$ are complementary angles since their measures add to $90^{\circ}$ (or equivalently, their radian measures add to $\frac{\pi}{2}$).
Note that the two triangles containing angles marked in blue must be congruent as they are reflections of each other across the line $y=x$. The same can be said of the triangles containing angles marked in red. As such, we can see there is a relation between the sine and cosine of an angle $\theta_1$ and its complement $\theta_2$. Namely, $$\cos \theta_1 = \sin \theta_2 \quad \textrm{ and } \quad \sin \theta_1 = \cos \theta_2$$ Given complement to any angle $\theta$ (in radians) is $\frac{\pi}{2} - \theta$, we can equivalently say for any $\theta$ that
$$\textstyle{\cos \theta = \sin (\frac{\pi}{2} - \theta) \quad \textrm{ and } \quad \sin \theta = \cos (\frac{\pi}{2} - \theta)}$$
In this way, we see that the cosine of an angle is the sine of its complement. This important relationship is actually captured in the name of the cosine function. In Medieval Latin this function was expressed as complementi sinus (note the use of the word sinus discussed earlier). Around 1620, English mathematician Edmund Gunter abbreviated this with co.sinus -- which ultimately was contracted to the "cosine" we use today.
Recall, we defined the cosine and sine functions of an angle $\theta$ as the $x$ and $y$ coordinates of the point $(a,b)$ on the unit circle with $\arg(a+b) = \theta$. However, this is not the only route we could have taken to relate $\theta$, a point on the unit circle, and a unique pair of $x$ and $y$ values.
Rather than focusing on the coordinates of the point itself, perhaps we draw instead a tangent to the point in question and notice the $x$ and $y$ values where this tangent cuts through the two axes, as seen in the picture below at the points labeled $x$ and $y$.
We call the function that gives the $x$ coordinate where this tangent cuts through the $x$-axis the secant function, denoted by $\sec \theta$, noting that the Latin word secare means "to cut" into pieces, not unlike the modern word "section" (when used as a verb).
In a related way, we call the function that gives the $y$-coordinate where this tangent cuts through the $y$ axis the cosecant function, denoting this by $\csc \theta$, given that the relationship between the secant of an angle and its complement is similar to that of the sine of an angle and its complement. To see this, again note that under a reflection of the entire image above over the line $y=x$, the terminal side of the angle $\theta$ (i.e., the one labeled with unit length) would move to the same for the complement of $\theta$, and the red segment would fall on the $x$-axis, as the blue segment is now.
$$\textstyle{\sec \theta = \csc (\frac{\pi}{2} - \theta) \quad \textrm{ and } \quad \csc \theta = \sec (\frac{\pi}{2} - \theta)}$$
Alternatively, we can see this relation between complements upon discovering the relationship between the secant and cosine functions. Notice that one can easily argue $\triangle BZO$ above is similar to $\triangle ZPO$, upon which we have the proportion $\sec \theta = \frac{1}{c}$.
Recognizing the convenience of picking an angle in the first quadrant, which keeps $c = \cos \theta$ and $s = \sin \theta$ (note $c$ and $s$ are distances and thus always positive, but $\cos \theta$ and $\sin \theta$ are coordinates which can sometimes be negative.), we see that in this context the secant is simply the reciprocal of the cosine function.
In the same manner, using the similarity of $\triangle OZR$ and $\triangle ZPO$, we can argue $\csc \theta = \frac{1}{s}$. Nicely, the argument above easily extends to other quadrants, making this relationship true in general:
$$\sec \theta = \frac{1}{\cos \theta} \quad \textrm{ and } \quad \csc \theta = \frac{1}{\sin \theta}$$
Of course, drawing the tangent segment $\overline{RB}$ to the point $Z$ makes us wonder something. How long are the two segments $\overline{ZR}$ and $\overline{ZB}$ (shown below in orange and magenta, respectively) that comprise it?
Let us focus on $\overline{ZB}$ first. Note that $\triangle BZO$ must be similar to $\triangle ZPO$, so $ZB = \frac{s}{c}$, suggesting (at least for this quadrant) that $ZB$ is the quotient of the related sine and cosine values.
In other quadrants, we note that the quotient of the sine and cosine values can sometimes be negative (as they are both coordinates), but the segments produced there always have positive distance. Still, in these other quadrants similar arguments establish that the magnitudes of these two things always agree.
Using a similar argument involving $\triangle RZO$ and $\triangle ZPO$ establishes the length $ZR$ and the magnitude of $\cos \theta$ divided by $\sin \theta$ (i.e., the reciprocal of $ZB$).
As the lengths we are finding are both on the tangent line to the unit circle at the point corresponding to $\theta$, let us call the functions of $\theta$ that produce them the tangent and cotangent functions, denoted $\tan \theta$ and $\cot \theta$, defining these in the following way:
$$\tan \theta = \frac{\sin \theta}{\cos \theta} \quad \textrm{ and } \quad \cot \theta = \frac{\cos \theta}{\sin \theta}$$
The typical "co-function" relationship seen for other trigonometric function pairs (i.e., sine and cosine, secant and cosecant) holds for the two functions above as well. Again, think of reflecting the entire image above over the line $y=x$ to see this.
$$\textstyle{\tan \theta = \cot (\frac{\pi}{2} - \theta) \quad \textrm{ and } \quad \cot \theta = \tan (\frac{\pi}{2} - \theta)}$$
Domain, Image/Range, Graphs, and "Inverses" for the Six Trigonometric Functions
Of course, with new functions introduced, we will want to know as much as we can about them. We need to ask all the standard questions:
"What is the domain?" (assuming this is some subset of the reals, $\mathbb{R}$)
"What is the related image/range?"
"What does its graph look like?"
"Does it have an inverse? ..and if not -- does it have any 'pieces' that are invertible?"
We aim to address all of these in this section. Let us address these in their natural pairs:
Properties of Sine and Cosine
First, as we can rotate a point $(x,y)$ about the origin on the unit circle as much or as little as we desire -- and in either a positive direction (i.e., counter-clockwise) or negative direction (clockwise) -- the coordinates $x = \cos \theta$ and $y=\sin \theta$ will always be defined. Hence, the domain of both the sine and cosine functions will be the set of all reals, $\mathbb{R}$.
As for the image/range, notice that the aforementioned point $(x,y)$ is contrained to fall on the unit circle. As such, neither coordinate may get larger than $1$ in magnitude. This combined with the fact that any horizontal line drawn whose distance from the $x$-axis does not exceed $1$ intersects the unit circle at least once (much more often twice), we can see the image/range of both sine and cosine will be the interval $[-1,1]$.
Recalling that multiple rotations around the unit circle can land one in the same position multiple times throughout that rotation. As such, the $x$ and $y$ coordinates -- and thus, the outputs of the $\cos \theta$ and $\sin \theta$ functions -- will be repeated whenever the related angles are co-terminal. Seeing the same ouputs for different input angles $\theta$ means the sine and cosine functions will not be invertible.
More generally, the lack of an inverse is a property shared by every periodic function, with these defined to be a function with some non-zero $p$ called its period where $f(x) = f(x+p)$ all $x$ in its domain. Going further, noting that sine and cosine are clearly periodic, and given how the other four trigonometric functions are defined in terms of these two, all of the trigonometric functions will be periodic, and hence none will be invertible!
Still, as we have done in the past to other non-invertible functions, we can often restrict their domain to produce a related function that is invertible. For example, $f(x) = x^2$ becomes invertible if we restrict its domain to all real $x$ where $x \ge 0$. We try of course, when we restrict the domain, to ensure that every output attained by the original (non-invertible) function is also attained by the function with the restricted domain. There are often many ways to do this -- especially for periodic functions. The below shows the standard way this is accomplished for the sine function.
Note, the inputs above are given as radian measures, and we have switched to using an input variable of $x$ inside parentheses instead of a $\theta$ without parentheses so that we draw both graphs on the same axes. Note that (traditionally) the use of $\theta$ as a variable is highly suggestive that its associated value should be interpreted as an angle. Contrast this to the fact that the input to $sin^{-1}(x)$ is always a $y$-coordinate -- definitely not an angle! As such, we really should avoid using $\theta$ as an input variable for $sin^{-1}$. However, the use of the variable $x$ (and $y$ and $z$) traditionally only suggests their associated values are real numbers -- which both the measure of an angle in radians and the value of a sine certainly would both be!
Note that for the purpose of creating a related inverse function, we have chosen to restrict the domain to $[\frac{\pi}{2},\frac{\pi}{2}]$. This is useful on a couple of fronts. Not only do we produce an invertible function (shown in solid blue), but we also produce one that has the exact same range as the sine function with an unrestricted domain.
Additionally, while there are several intervals that do both of these things (e.g., $[\pi/2,3\pi/2]$, $[-3\pi/2,-5\pi/2]$, etc.), the interval $[-\pi/2,\pi/2]$ minimizes the magnitudes of the angles involved.
We call the inverse to the restricted-domain version of the sine function (drawn above in red) the arcsine function, denoting it by either by $\arcsin x$, or $\sin^{-1} x$. The function was likely named as was as it gives as output the measure of an "arc" (or angle) that has a given sine value.
This function behaves in many ways (but not all) as an inverse to the unrestricted domain version of $\sin x$.
Like most inverse function pairs, the domain of $\sin^{-1} x$ is identical to the range of the $\sin x$, both being $[-1,1]$. However, the range of $\sin^{-1} x$ is only $[\pi/2,\pi/2]$, a small subset of the domain of the $\sin x$.
For any $x$ in the domain of $\sin^{-1}$, we have $\sin(\sin^{-1} x) = x$. However, $\sin^{-1} (\sin x) = x$ is true only if $x \in [-\pi/2,\pi/2]$ and false otherwise!
As always, notice the symmetry across the line $y=x$ between a function and its inverse (here, the un-named solid blue function with a restricted domain and the solid red arcsine function).
In a similar way, we can create a new invertible function that is identical to the cosine function, except with a restricted domain of $[0,\pi]$, to create a "pseudo-inverse" to the cosine function we call the arccosine function. The choice of the restricted domain here keeps the inputs as small as possible (and positive), which means the outputs of the related inverse function, called the arccosine function will also be as small as possible and positive. We denote the arccosine function by either $\arccos(x)$ or $\cos^{-1}(x)$:
Properties of Tangent and Cotangent
Addressing first the domain of these two functions, recall again that $$\tan \theta = \frac{\sin \theta}{\cos \theta} \quad \textrm{ and } \quad \csc \theta = \frac{\cos \theta}{\sin \theta}$$ The quotients seen above open the door to some possible domain issues, should the denominators be zero. Indeed, noting that $\cos = 0$ (i.e., one touches the $y$-axis) with every half-circle rotation from $\frac{pi}{2}$ (i.e., $90^{\circ}$), we must exclude all $\theta = \frac{\pi}{2} + n\pi$ where $n$ is an integer, from the domain of the tangent function. That is to say, the domain of the tangent is $$\{x \in \mathbb{R} \ | \ x \neq \frac{\pi}{2} + n\pi, \, \textrm{ whenever } \, n \in \mathbb{Z}\}$$ Similarly, since $\sin \theta = 0$ (i.e., one touches the $x$-axis) with every half-circle rotation from $0$, we will need to exclude all integer multiples of $\pi$ from the domain of the cotangent function. Equivalently, the domain of the cotangent function is given by $$\{x \in \mathbb{R} \ | \ x \neq n\pi, \, \textrm{ whenever } \, n \in \mathbb{Z}\}$$
Regarding the image/range for each of these functions, recall the image shared earlier showing the magnitudes of these functions as segment lengths:
By letting $\theta$ approach $\pi/2$ (i.e., $90^{\circ}$) the purple segment with length equal to the magnitude of the tangent function grows without bound. If $\theta$ instead approaches $0$ this purple length can surely be made as small as desired (including zero). Had we constructed a similar image in the second quadrant, where $\tan \theta = \frac{\sin \theta}{\cos \theta}$ is negative, we would see a similar span of magnitudes possible. Combining these, the value of $\tan(x)$ ranges over all (positive or negative) real values, $\mathbb{R}$. Finding the range of the cotangent function works similarly, and is again all reals, $\mathbb{R}$.
As we see the magnitude of $\tan(x)$ growing without bound (either in a positive or negative way) as $x$ approaches each odd multiple of $\frac{\pi}{2}$, we can expect to see some vertical asymptotes in its graph. In the image below, we can see these asymptotes drawn as vertical, dashed blue lines (they are not part of the function!)
Switching discussion to the graphs and invertibility of these two functions, note that -- as mentioned before -- their periodic nature prohibits them from having inverses. However, we are free to construct for each a related domain-restricted function with an identical image/range that is invertible.
Traditionally, we restrict the domain of the tangent function to $(-\frac{\pi}{2},\frac{\pi}{2}$) so that graph can be drawn as a single continuous curve. Note in particular that the endpoints at $\pm\frac{\pi}{2}$ are not included in the restricted domain here, unlike the corresponding function used to produce $\arcsin(x)$. We call the inverse to this restricted-domain version of $\tan(x)$ the arctangent function, denoting it by $\arctan(x)$ or $\tan^{-1}(x)$.
Interestingly, the choice for which "piece" of the cotangent function to invert to produce the arccotangent function is not universally agreed upon. We choose to define it to be $(0,π)$ so that the resulting arccotangent function, denoted $\textrm{arccot}(x)$ or $\cot^{-1}(x)$, is continuous and defined everywhere, and behaves in a manner more consistent with that seen in the related $\tan^{−1}(x)$ function.
Properties of Secant and Cosecant
Finally, let us consider $\sec(x)$ and $\csc(x)$.
With regard to their domains, we can see in formulas describing these functions in terms of sine and cosine, we again have domain issues to worry about. $$\sec \theta = \frac{1}{\cos \theta} \quad \textrm{ and } \quad \csc \theta = \frac{1}{\sin \theta}$$ The denominators force these functions to have the same domains as the tangent and cotangent functions, respectively, with the domain of the secant function being $$\{x \in \mathbb{R} \ | \ x \neq \frac{\pi}{2} + n\pi, \, \textrm{ whenever } \, n \in \mathbb{Z}\}$$ and the domain of the cosecant function being $$\{x \in \mathbb{R} \ | \ x \neq n\pi, \textrm{ whenever } \, n \in \mathbb{Z}\}$$ Since both functions are reciprocals of a function with image/range $[-1,1]$ (all real values whose magnitude is on or less than $1$) the $\sec(x)$ and $\csc(x)$ functions must have a range of all real values whose magnitude exceed $1$! That is to say, the range of both secant and cosecant is $$\{x \in \mathbb{R} \ | \ |x| \ge 1\} = (-\infty,-1] \cup [1,\infty)$$
As another shared trait with the tangent and cotangent functions, note that being reciprocals of the cosine and sine function, as either $\cos(x)$ or $\sin(x)$ gets closer and closer to zero, the magnitude of either the $sec(x)$ or $\sin(x)$ respectively, again grows without bound. Thus, we can again expect vertical asymptotes in the graphs of the secant and cosecant funtions. Below, the blue dashed vertical lines represent these asymptotes -- these vertical lines should once again not be considered as points on the function!.
With regard to the graph of their associated "arc" functions, there is also (sadly) no universal agreement how to restrict the domain of each to create an invertible function. We choose here to define the arcsecant and arccosecant functions by choosing domain restrictions of $[0,\pi]$ and $[-\frac{\pi}{2},0) \cup (0,\frac{\pi}{2}]$, denoting the "arc" functions thus created in the typical ways:
Some Useful Identities (Some Resulting From Complex Numbers!)
An identity is an equation whose left and right sides are equal for all values of the variables in their respective implicit domains. Some trigonometric identities (i.e., identities involving trigonometric functions) that prove useful in a great many contexts are given below, with a discussion of why each must hold. While all of these identities can be proven without appealing to complex numbers, some are proven far more easily with them!
The Pythagorean Identities:
$$\begin{array}{c} \cos^2 \theta + \sin^2 \theta = 1\\ 1 + \tan^2 \theta = \sec^2 \theta\\ 1 + \cot^2 \theta = \csc^2 \theta \end{array}$$
We have already discussed the first in the list above -- which is an immediate result of how we measure the magnitude of $|z|$ for some $z = \cos(\theta) + i \sin(\theta)$ on the unit circle (i.e., when $|z| = 1$). Recall, $|z| = Re(z)^2 + Im(z)^2$
The other two pythagorean identities listed are quick consequences of this relationship. The first being the result of dividing both sides by $\cos^2 \theta$, and the second being the result of dividing both sides by $\sin^2 \theta$ instead, as the calculations below reveal:
$$\begin{array}{c|c} \displaystyle{\frac{\cos^2 \theta}{\cos^2 \theta} + \frac{\sin^2 \theta}{\cos^2 \theta} = \frac{1}{\cos^2 \theta}} \quad & \quad \displaystyle{\frac{\cos^2 \theta}{\sin^2 \theta} + \frac{\sin^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta}}\\\\ 1 + \tan^2 \theta = \sec^2 \theta & 1 + \cot^2 \theta = \csc^2 \theta \end{array}$$
The Even/Odd Identities
$$\begin{array}{rcl} \cos (-\theta) &=& \phantom{-}\cos \theta\\ \sin (-\theta) &=& -\sin \theta\\ \tan (-\theta) &=& -\tan \theta \\ \end{array}$$
Examining the next picture immediately reveals the three "even/odd function identities" given above. (You do of course remember how we defined "even" and "odd" functions when we introduced functions of the form $(x) = x^n$, right?) To see how these three results can be deduced, consider the image below. Note the right triangles, sharing a common acute angle and hypotenuse length, must consequently be congruent. As such, the vertical sides of each are the same length.
The common length of the vertical sides tell us the associated $y$-coordinates for the red and blue angles have opposite signs, but the same magnitude, which in turn implies $\sin (-\theta) = -\sin \theta$.
Meanwhile, the horizontal side common to both triangles tells us the associated $x$-coordinates for these angles are identical, and thus $\cos (-\theta) = \cos \theta$.
Knowing the values of the sine and cosine for $-\theta$, we can find the tangent for this angle as well,
$$\tan (-\theta) = \frac{\sin (-\theta)}{\cos (-\theta)} = \frac{-\sin \theta}{\cos \theta} = -\tan \theta$$
The Complementary Angle Identities
$$\begin{array}{rcl} \cos(\pi/2 - \theta) &=& \sin \theta\\ \sin(\pi/2 - \theta) &=& \cos \theta\\ \tan(\pi/2 - \theta) &=& \cot \theta \end{array}$$
Why these hold has already been discussed. They are included here only as we should now start thinking of these as tools to argue more interesting results.
The Sum and Difference Identities
$$\begin{array}{c} \cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta\\ \sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta\\\\ \tan(\alpha \pm \beta) = \displaystyle{\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}}\\ \end{array}$$
These three identities are the great benefactors of the complex numbers!
Let $z_{\theta}$ denote the complex value on the unit circle with $\arg{z} = \theta$. Then, $$\begin{array}{rcl} z_{\alpha \pm \beta} &=& z_{\alpha} \cdot z_{\pm \beta}\\ &=& (\cos \alpha + i \sin \alpha )(\cos \beta + i \sin(\pm \beta))\\ &=& (\cos \alpha + i \sin \alpha )(\cos \beta \pm i \sin \beta ) \quad {\scriptstyle {\textrm{taking advantage of the even/odd property for $\sin(x)$}}}\\ &=& \left[\cos \alpha \cos\beta \mp \sin \alpha \sin \beta \right] + \left[\sin \alpha \cos \beta + \cos \alpha \sin \beta \right] \cdot i \end{array}$$ Note the last line above results from simply expanding the product immediately before, replacing $i^2$ with $-1$, and collecting the remaining multiples of $i$ together.
Now recall that $z_{\alpha \pm \beta}$ is on the unit circle, so its real part and imaginary coefficient give the cosine and sine values of its argument, $\alpha \pm \beta$, respectively. Thus, the first and second sum and difference identities are immediately established! $\require{cancel}$
All that remains is to prove the related result for the tangent. Fortunately, once the above identities are established, this one is trivial:
$$\begin{array}{rcll} \tan (\alpha \pm \beta) &=& \displaystyle{\frac{\sin \alpha \cos \beta \pm \cos \alpha \sin \beta}{\cos \alpha \cos \beta \mp \sin \alpha \sin \beta}}\\\\ &=& \frac {\displaystyle{\frac{\sin \alpha \cancel{\cos \beta}}{\cos \alpha \cancel{\cos \beta}} \pm \frac{\cancel{\cos \alpha} \sin \beta}{\cancel{\cos \alpha} \cos \beta}}} {\displaystyle{\frac{\cancel{\cos \alpha \cos \beta}}{\cancel{\cos \alpha \cos \beta}} \mp \frac{\sin \alpha \sin \beta}{\cos \alpha \cos \beta}}} & \overset{\normalsize{\textrm{after dividing by } \cos \alpha \cos \beta}}{\scriptsize{\textrm{to introduce tangents}}}\\\\ &=& \displaystyle{\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}} \end{array}$$
The Double and Triple Angle Identities
$$\begin{array}{c} \cos 2\theta &=& \cos^2 \theta - \sin^2 \theta\\ \sin 2 \theta &=& 2 \sin \theta \cos \theta\\\\ \cos 3\theta &=& \cos^3 - 3\cos \theta \sin^2 \theta\\ \sin 3\theta &=& 3\cos^2 \theta \sin \theta - \sin^3 \theta \end{array}$$
Directly applying the sum formula to $\cos (\theta + \theta)$ and $\sin (\theta + \theta)$ would quickly give us formulas for $\cos 2\theta$ and $\sin 2\theta$, but we instead again appeal to complex numbers as doing so provides an efficient way to establish more general results.
To see this, consider the below -- noting how the special product rules for the squares and cubes of binomials play a significant role. As a reminder of these special product rules, recall that $(a+b)^2 = a^2 + 2ab + b^2$ and $(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$.
Similar to how we began to prove the sum and difference formulas, let $z_{\theta}$ denote the complex value on the unit circle with $\arg{z} = \theta$. Then, $$\begin{array}{rcl} z_{2\theta} &=& z_{\theta}^2\\ &=& (\cos \theta + i \sin \theta)^2\\ &=& \cos^2 \theta + 2i\sin \theta \cos \theta + i^2 \sin^2 \theta \quad {\scriptstyle {\textrm{by the special product rule for $(a+b)^2$}}}\\ &=& (\cos^2 \theta - \sin^2 \theta) + (2\sin \theta \cos \theta) \cdot i \quad {\scriptstyle {\textrm{recalling that $i^2 = -1$ and then collecting terms}}} \end{array}$$ As argued before for the sum and difference formulas, remember that $z_{2\theta}$ is on the unit circle, so its real part and imaginary coefficient give the values of $\cos 2\theta$ and $\sin 2\theta$ seen in the box above.
To find the triple angle formula we proceed similarly, noting: $$\begin{array}{rcl} z_{3\theta} &=& z_{\theta}^3\\ &=& (\cos \theta + i \sin \theta)^3\\ &=& \cos^3 \theta + 3(\cos^2 \theta)(i\sin \theta) + 3(\cos \theta)(i^2 \sin^2 \theta) + i^3 \sin^3 \theta \, \, {\scriptstyle {\textrm{by the special product rule for $(a+b)^3$}}}\\ &=& (\cos^3 \theta - 3\cos \theta \sin^2 \theta) + (3\cos^2 \theta \sin \theta - \sin^3 \theta) \cdot i \quad {\scriptstyle {\textrm{recalling that $i^2 = -1$, and therefore $i^3 = -i$}}} \end{array}$$ Again recalling $z_{3\theta}$ is on the unit circle, its real part and imaginary coefficient must give the values for $\cos 3\theta$ and $\sin 3\theta$ also seen in the box above.
Alternate Forms for the Double Angle Identities
$$\begin{array}{rcl|rcl} \cos 2\theta &=& 2\cos^2 \theta - 1 \phantom{abc} & \phantom{abc} \cos 3\theta &=& 4\cos^3 \theta - 3\cos \theta\\ &=& 1 - 2\sin^2 \theta & \sin 3\theta &=& 3\sin \theta - 4\sin^3 \theta\\ \end{array}$$
The Pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$ provides an easy way to find an alternate form for any expression involving either $\cos^2 \theta$ or $\sin^2 \theta$, as we may solve it for these two expressions to find $$\cos^2 \theta = 1 - \sin^2 \theta \quad \textrm{ and } \quad \sin^2 \theta = 1 - \cos^2 \theta$$ Note that when we apply these to the double-angle identity for cosine, we can produce replacements for $\cos 2\theta$ that involve only a single trigonometric function. To see this, consider the following. $$\begin{array}{rcl} \cos 2\theta &=& \cos^2 \theta - \sin^2 \theta\\ &=& \cos^2 \theta - (1 - \cos^2 \theta)\\ &=& 2\cos^2 \theta - 1 \end{array}$$
$$\begin{array}{rcl} \cos 2\theta &=& \cos^2 \theta - \sin^2 \theta\\ &=& (1 - \sin^2 \theta) - \sin^2 \theta\\ &=& 1 - 2 \sin^2 \theta \end{array}$$
We may do something similar with the triple angle formulas, as seen below: $$\begin{array}{rcl} \cos 3\theta &=& \cos^3 - 3\cos \theta \sin^2 \theta\\ &=& \cos^3 \theta - 3\cos \theta (1-\cos^2 \theta)\\ &=& \cos^3 \theta - 3\cos \theta + 3\cos^3 \theta\\ &=& 4\cos^3 \theta - 3\cos \theta \end{array}$$ and $$\begin{array}{rcl} \sin 3\theta &=& 3\cos^2 \theta \sin \theta - \sin^3 \theta\\ &=& 3(1-\sin^2 \theta) \sin \theta - \sin^3 \theta\\ &=& 3\sin \theta - 3\sin^3 \theta - \sin^3 \theta\\ &=& 3\sin \theta - 4\sin^3 \theta \end{array}$$
The Half-Angle Identities
Lastly, if we take these two alternate forms for $\cos 2\theta$ and solve for $\sin^2 \theta$ and $\cos^2 \theta$, respectively -- we produce the half angle identities,
$$\begin{array}{rcl} \cos^2 \theta &=& \displaystyle{\frac{1+\cos 2\theta}{2}}\\\\ \sin^2 \theta &=& \displaystyle{\frac{1-\cos 2\theta}{2}}\\ \end{array}$$
Knowing the half angle identities in the above form will be the most useful for applications in calculus. That said, why these identities are called the "half angle" identities is made more clear upon making a substitution of $x = 2\theta$ and then taking a square root:
$$\begin{array}{c} \left| \cos \left( \frac{x}{2} \right) \right| &=& \displaystyle{\sqrt{\frac{1+\cos x}{2}}}\\\\ \left| \sin \left( \frac{x}{2} \right) \right| &=& \displaystyle{\sqrt{\frac{1-\cos x}{2}}} \end{array}$$
In this way, we can discover the magnitudes of the sine and cosine for half an angle if we know the cosine of the full angle. As for whether the sine and cosine are positive or negative, this is most easily managed by determining in which quadrant the half angle lies.
Proving Other Trigonometric Identities
To prove that a trigonometric equation is an identity, one typically starts by trying to show that either one side of the proposed equality can be transformed into the other, or that both sides can be transformed into the same expression.
In other words, suppose $A$ and $B$ are some trigonometric expressions and we are trying to determine if $A=B$.
We hope that both expressions will simplify to some common form $C$, as if we can show the following:
$$\begin{array}{rclcrcl} A &=& A_1 \quad && \quad B &=& B_1\\ &=& A_2 && &=& B_2\\ &=& \cdots & \textrm{and} & &=& B_3\\ &=& A_n && &=& B_4\\ &=& C && &=& \cdots\\ & & && &=& B_m\\ & & && &=& C \end{array}$$
then we will know
$$A = A_1 = A_2 = \cdots = A_n = C = B_m = \cdots = B_2 = B_1 = B$$
and thus,
$$A = B$$
That is our general "plan of attack" -- although, we might get lucky and the sequence of $A_1, A_2, \cdots$ will terminate in $B$, or the sequence $B_1, B_2, \cdots$ will terminate in $A$, which then shortens our argument a bit.
There are some basic strategies to help us get to that common form $C$ as efficiently as possible:
Only manipulate one side of the proposed identity at a time. Start by attempting to simplify the more complicated side first, as which steps one should take will likely be more obvious for this side.
First "trigonometrically simplify" the side in question.
One should try rewriting all of trigonometric functions involved in terms of sines and cosines, unless there is a compelling reason not to do this.
$$\begin{array}{rcl} \tan \theta + 2\csc \theta &=& \displaystyle{\left( \frac{\sin \theta}{\cos \theta} \right) + 2\left( \frac{1}{\sin \theta} \right)}\\ &=& \cdots \end{array}$$
Likewise, if you see trigonometric functions involving more than one angle measure, try to use known identities to rewrite things so that only a single angle measure is involved.
$$\begin{array}{rcl} \displaystyle{\frac{\cos 2\theta + \sin \theta}{\sin 2\theta + \sin(-\theta)}} &=& \displaystyle{\frac{ (\cos^2 \theta - \sin^2 \theta) + \sin \theta}{2\sin \theta \cos \theta - \sin \theta}}\\ &=& \cdots \end{array}$$
If one side can be written in terms involving (perhaps multiple occurrences of) a single trigonometric function of a single angle measure, doing so may help. In particular, this often helps in cases where one side of the proposed identity is already in this form, and the other side consists of a mixture of trigonometric functions.
$$\begin{array}{rcll} \displaystyle{\frac{\cos 2\theta}{\cos(\pi/2-\theta)}} &=& \displaystyle{\frac{1 - 2\sin^2 \theta}{\sin \theta}} & \overset{\normalsize{\textrm{assuming we were trying to get}}}{\scriptsize{\textrm{things in terms of only sine functions}}}\\ &=& \cdots \end{array}$$
As the previous examples attest, trigonometrically simplifying an expression sometimes makes it algebraically more cumbersome. This can actually be a good thing -- as it gives us a direction to proceed. Specifically -- after trigonometrically simplifying a side of the proposed identity -- one can next focus on "algebraically simplifying" it.
Complex fractions (i.e., fractions with fractions in either the numerator or denominator) should be collapsed
$$\begin{array}{rcll} \frac{\displaystyle{\frac{\sin \theta}{\cos^2 \theta}}}{\displaystyle{1 + \frac{1}{\cos \theta}}} &=& \displaystyle{\frac{\sin \theta}{\cos^2 \theta + \cos \theta}} & \scriptsize{\textrm{ after multiplying by } \displaystyle{\frac{\cos^2 \theta}{\cos^2 \theta}}}\\ &=& \cdots \end{array}$$
When fractional expressions appear in a sum or difference, these terms should be combined into a single fraction, finding common denominators as necessary. The resulting fractional expression may, upon factoring, admit a common factor that can be cancelled -- or may be simplified in some other fashion.
Example$\require{cancel}$
$$\begin{array}{rcll} \displaystyle{\frac{1}{\sin^2 \theta} + \frac{1}{\cos^2 \theta} + \frac{\sin \theta \cos^2 \theta - 1}{\sin^2 \theta \cos^2 \theta}} &=& \displaystyle{\frac{\cos^2 \theta + \sin^2 \theta - 1 + \sin \theta \cos^2 \theta}{\sin^2 \theta \cos^2 \theta}}\\\\ &=& \displaystyle{\frac{1 -1 + \sin \theta \cos^2 \theta}{\sin^2 \theta \cos^2 \theta}}\\\\ &=& \displaystyle{\frac{\sin \theta \cos^2 \theta}{\sin^2 \theta \cos^2 \theta}}\\\\ &=& \displaystyle{\frac{1}{\sin \theta}}\\\\ &=& \cdots \end{array}$$
Try to manipulate the side in question into the same "form" as the other side. For example, suppose one is attempting to simplify the left side of a proposed identity and the right side is a product. Then one should attempt to factor the left side, so that it is also expressed as a product.
Also, when the left and right sides get to the point where there is a partial "match", one should leave the matched parts alone from that point forward, and only manipulate the parts that still don't look like one another.
Show $\sin \theta \cos^2 \theta - \sin \theta = -\sin^3 \theta$ is an identity.
Starting with the left side, we have...
$$\begin{array}{rcll} \sin \theta \cos^2 \theta - \sin \theta &=& \sin \theta (\cos^2 \theta - 1) & \overset{\normalsize{\textrm{notice the right side, } -\sin^3 \theta, \textrm{ is a product, so}}}{\scriptsize{\textrm{we factor the left side to have the same form}}}\\\\ &=& \cdots\\\\ &=& \sin \theta (-\sin^2 \theta) & \overset{\normalsize{\textrm{working backwards from the sought expression below}}}{\scriptsize{\textrm{we expose a "match" of } \sin \theta \textrm{ with the above}}}\\\\ &=& -\sin^3 \theta\\\\ \end{array}$$
To fill in the missing steps above, we just need to "massage" $(\cos^2 \theta - 1)$ into $(-\sin^2 \theta)$. This of course is immediate, given the Pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$.
Showing an Equation is Not an Identity
It may be the case that in the course of trying to prove a given equation is an identity, one begins to suspect that it is not.
In such situations, one should test whether the equation's left and right sides are actually equal by plugging in some values for the variables it contains. Remember, one only needs a single counter-example to prove an equation is not an identity.
If however, one tests a particular value (or set of values) and the left and right sides of the given equation agree in value, that particular test is inconclusive -- and a decision must be made whether to continue the search for a counter-example and test additional values, or to return to trying to prove the given equation is an identity.
To have the best chance of selecting values that will show a given equation is not an identity, one should keep the following in mind:
While picking angle measures that are integer multiples of $\pi/2$ will lend itself to easy evaluation of the expressions involved in the equation, such values will often fail to reveal an equation is not an identity due to the fact that either the sine or cosine for these angle measures will be zero.
The following is clearly not an identity: $\sin \theta = \cos \theta + \sin \theta$. However, if one tests this with either $\theta = \pi/2$ or $\theta = 3\pi/2$, the results will be inconclusive, as the left and right sides will have the same value (i.e., $1$ in the first case, $-1$ in the second).
A similar problem presents itself when testing a proposed identity with angle measures that are odd integer multiples of $\pi/4$, but for a different reason. For these angle measures, recall the sine and cosine values are either identical or differ only in sign. This too can create inconclusive results for an equation that is not actually an identity.
The following is clearly not an identity: $2\sin \theta= \cos \theta + \sin \theta$. However, if one tests this with either $\theta = \pi/4$ or $\theta = 5\pi/4$, the results will be inconclusive, as both sides evaluate to $\sqrt{2}$.
Testing a proposed identity with angle measures that are integer multiples of $\pi/6$ or $\pi/3$ (when reduced) can be good first choices, as the exact values of the trigonometric functions are easy to find, and the problems seen above don't occur.
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CommonCrawl
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Comprehensive analysis of the differences between left- and right-side colorectal cancer and respective prognostic prediction
Mengye Niu1,2 na1,
Chengyang Chen1,2 na1,
Xian Gao1,2,
Yi Guo1,
Bingzhou Zhang1,2,
Xin Wang1,2,3,
Shihao Chen1,2,
Xupeng Niu1,2,
Chao Zhang1,2,
Like Li1,2,
Zhongxin Li1,
Zengren Zhao1 &
Xia Jiang1,2
BMC Gastroenterology volume 22, Article number: 482 (2022) Cite this article
Previous studies have reported that the tumor heterogeneity and complex oncogenic mechanisms of proximal and distal colon cancer (CRC) are divergent. Therefore, we aim to analyze the differences between left-sided CRC (L_cancer) and right-sided CRC (R_cancer), as well as constructing respective nomograms.
We enrolled 335 colon cancer patients (146 L_cancer patients and 189 R_cancer patients) from The Cancer Genome Atlas (TCGA) data sets, and 102 pairs of color cancer tissue and adjacent normal tissue (51 L_cancer patients and 51 R_cancer patients) from our hospital. Firstly, we analyzed the differences between the L_cancer patients and R_cancer patients, and then established the L_cancer and R_cancer prognostic models using LASSO Cox.
R_cancer patients had lower survival than L_cancer patients. R_cancer patients had higher ESTIMATE and immune scores and lower tumor purity. These patterns of expression of immune checkpoint-related genes and TMB level were higher in R_cancer than in L_cancer patients. Finally, we using Lasso Cox regression analyses established a prognostic model for L_cancer patients and a prognostic model for R_cancer patients. The AUC values of the risk score for OS in L_cancer were 0.862 in the training set and 0.914 in the testing set, while those in R_cancer were 0.835 in the training set and 0.857 in the testing set. The AUC values in fivefold cross-validation were between 0.727 and 0.978, proving that the two prognostic models have great stability. The nomogram of L_cancer included prognostic genes, age, pathological M, pathological stage, and gender, the AUC values of which were 0.800 in the training set and 0.905 in the testing set. Meanwhile, the nomogram of R_cancer comprised prognostic genes, pathological N, pathological T, and age, the AUC values of which were 0.836 in the training set and 0.850 in the testing set. In the R_cancer patients, high-risk patients had a lower proportion of 'B cells memory', 'Dendritic cells resting', immune score, ESTIMATE score, immune checkpoint-related genes, and HLA-family genes, and a higher proportion of 'T cells follicular helper', 'Dendritic cells activated', and 'Mast cells activated'.
We found significant differences between L_cancer and R_cancer patients and established a clinical predictive nomogram for L_cancer patients and a nomogram for R_cancer patients. Additionally, R_cancer patients in low-risk groups may be more beneficial from immunotherapy.
Colon cancer (CRC) is one of the most common cancers and cause of cancer death globally, seriously endangering the health of patients [1]. In recent years, there has been a growing body of evidence demonstrating that the primary tumor location of CRC is an important prognostic factor, owing to distinct biological features [2,3,4]. Despite the fact that the primary tumor site is not generally considered in CRC management, left-sided colon cancers (L_cancer) and right-sided colon cancers (R_cancer) exhibit different clinical and biological characteristics [5]. A meta-analysis of 66 studies with more than 1.4 million patients with a median follow-up of 65 months revealed that the tumor side had a significant prognostic impact on overall survival, with a 20% percent longer life expectancy, independent of stage, race, adjuvant chemotherapy, year of study, number of participants, and quality of included studies. [6]. The differences in colon cancer by its location have been identified through extensive research, including survival, tumor microenvironment, methylation profile, microbiota, gene expression, and epigenetic changes. [2, 3, 6,7,8]. In addition, the tumor location also influences the outcome of adjuvant chemotherapy, palliative therapy, or targeted therapy. Therefore, it is of special significance to classify CRC by its location.
Nomograms are widely used for prognosis in CRC patients. However, few previous studies have separately built predictive models to predict patient prognosis with respect to location. In this study, we separately build predictive models for L_cancer and R_cancer, identifying potential prognostic biomarkers for left and right CRC. Age, sex, histological classification, and so forth, are also important factors that can influence clinical outcomes and can improve the accuracy of models. Therefore, we also aimed to analyze the differences between L_cancer and R_cancer and construct respective nomograms for L_cancer and R_cancer, containing prognostic gene signatures and clinical prognostic factors, which are expected to allow for more accurate predictions in the prognosis of CRC, facilitating accurate diagnosis and treatment.
The transcriptome data, somatic mutation data, and clinical information of CRC patients were downloaded from The Cancer Genome Atlas (TCGA, https://portal.gdc.cancer.gov/), which includes transcriptome data for 332 CRC patients (146 L_cancer patients and 189 R_cancer patients) and somatic mutation data for 329 CRC patients (142 L_cancer patients and 187 R_cancer patients).
L_cancer patients were divided into L_cancer training and L_cancer internal validation sets at a ratio of 7:3. The L_cancer external validation set contained those who operated in our hospital, including 51 L_cancer patients.
R_cancer patients were also divided into R_cancer training set and R_cancer internal validation sets at a ratio of 7:3. The R_cancer external validation set contained those who operated in our hospital, including 51 R_cancer patients.
A total of 102 pairs of colon cancer and adjacent normal control samples were stored at − 80 °C. Patients were followed up by telephone interviews. As of the final data cutoff, December 30, 2021, the median duration of follow-up in the study was 4.5 years and the criterion to proceed with the final OS analysis was met.
The term "R_cancer" refers to any (histologically confirmed) adenocarcinoma arising from the caecum, ascending colon, or hepatic flexure. Any tumor that arises in the splenic flexure, descending colon or sigmoid colon was referred to as L_cancer.
Survival analysis
Using Kaplan–Meier survival analysis, we evaluated the differences in survival between patients with different clinicopathological characteristics, between high-risk and low-risk groups and between the L_cancer and R_cancer groups in the data sets mentioned above. The 'survival' package in R was used to perform a two‐sided log‐rank test and univariate and multivariate Cox regression analyses [9].
Differential gene analysis and functional annotation
By using the "edgeR" package in R, we identified differentially expressed genes (DEGs) between L_cancer and R_cancer, L_cancer and L_normal, R_cancer and R_normal based on differential expression analysis. To screen for DEGs, |log2 FC (fold-change)|> 1 and P < 0.05 were set as thresholds. To investigate the possible biological processes, cellular components, and molecular functions of DEGs, GO enrichment and KEGG pathway analyses were performed by using the R software package "clusterProfiler" [10,11,12].
Gene set variation analysis (GSVA)
By using the "GSVA" package in R, we evaluated the t-scores and assigned pathway activity conditions to L_cancer and R_cancer patients to reveal pathway enrichment. The "limma" package in R was also used to show differences in pathway activation between L_cancer and R_cancer patients [13,14,15].
The proportion of immune cell infiltration and the calculation of tumor purity
In each cancer sample, the relative proportions of 22 immune cell types were calculated using the CIBERSORT software [16]. A file called "LM22.txt", containing 547 gene signatures (https://cibersort.stanford.edu/download.php), is also needed in R. ESTIMATE was used to calculate immune, stromal, and ESTIMATE scores, as well as tumor purity, based on Yoshihara et al. [17].
Profiles of tumor mutation burden (TMB) and correlation analysis
The TMB was defined as: TMB = (total count of variants)/(the whole length of exons). In a waterfall plot, the mutation profiles of two groups were compared using the maftools package [18]. Afterward, the difference in mutation frequencies between the two groups was measured with the chi-square test. TMB was derived for each patient, calculated using Pearson correlation analysis with estimated P-values.
LASSO cox regression analysis
LASSO Cox regression analysis with the R package glmnet was then used to identify hub genes associated with the prognosis of L_cancer or R_cancer, and a Risk Score was calculated for each sample using the screened hub genes following the following formula [19]:
$$Riskscore = \sum\limits_{i = 1}^{N} {\left( {Expi \times Coef} \right)}$$
where N represents the number of signature genes, Expi is the gene expression levels, and Coef is the estimated regression coefficient value from the Cox proportional-hazards analysis. Based on this optimal cutoff value, the R survival package "survminer" was used to divide patient groups into Low- and High-Risk groups. Moreover, model predictive power was evaluated by calculating the AUC of 1-, 3-, 5-, 7-year, and all time-dependent ROC curves, using the "survivalROC" package.
Building and validating a predictive nomogram
To construct the nomograms, we used univariate and multivariate Cox regression analyses. Forest plots were used to display the P-value, HR, and 95% CI for each variable, using R's 'forest plot' package. Based on independent prognostic factors, the nomograms were generated in R using the rms, nomogramEx, and ggDCA packages. In the next step, Using calibration curves, we determined whether the predicted survival outcome matched the actual outcome. Moreover, training set decision curve analysis (DCA) and internal validation set DCA, which is a statistical method for assessing and comparing predictive models, was used to determine the clinical suitability of our established nomograms.
RNA isolation and quantitative reverse transcription PCR assay
For total RNA isolation, the TRIzol reagent by Invitrogen was used, and for complementary DNA synthesis, the PrimeScript RT reagent kit by Takara was used. RT-PCR was carried out using SYBR Premix Ex Taq I. GAPDH served as an internal control. Relative RNA abundances were calculated by using the standard 2-ΔCt method.
A two-sided significance level of 0.05 was used to determine statistical significance in all analyses using R software (version 3.6.3). All significance levels were two-sided.
Differences between L_cancer and R_cancer patients
Differences in demographic characteristics between L_cancer and R_cancer patients
An overview of the steps is presented as a flow chart in Fig. 1. The demographic characteristics of patients are summarized in Table 1. The L_cancer patients found a significant difference between R_cancer patients regarding age, stage N, and survival rate (P < 0.05). It is noteworthy that we observed lower survival after R_cancer versus L_cancer (Fig. 2A).
The flow diagram shows that: 1 the difference between L_cancer and R_cancer; 2 Nomograms were established to predict the prognosis of L_cancer and R_cancer, respectively. (L_cancer, left-side colon cancer; R_cancer, right-side colon cancer)
Table 1 Demographic and clinical characteristics of patients
Differentially expressed genes and functional annotation between L_cancer and R_cancer patients. A Survival rates difference between L_cancer patients and R_cancer patients. B Volcano plot for differentially expressed genes (DEGs) of L_cancer patients and R_cancer patients. C Heatmap plot for top 40 DEGs of the two groups. D, E GO enrichment analysis and KEGG analysis of the up-regulated DEGs in D L_cancer and E R_cancer. F Heatmap demonstrated the top 10 different gene set enrichment analysis (GSVA) pathways of the two groups
Moreover, there is no difference between the training set and the verification set except T stage. The difference in the T stage may due to the poor stage of patients from our hospital, but it does not affect the internal validation.
Differential expressed genes and functional annotation between L_cancer and R_cancer patients
By comparing the transcriptome data, we identified 540 significantly up-regulated DEGs in the L_cancer group and 1507 significantly up-regulated DEGs in the R_cancer group (Fig. 2B). The heatmap was shown the top 40 DEGs with the greatest variation (Fig. 2C).
Further, we applied the DEGs for functional enrichment analysis. L_cancer up-regulated DEGs were enriched in 38 GO terms and 3 KEGG pathways (FDR < 0.5, Fig. 2D), while R_cancer up-regulated DEGs were enriched in 129 GO terms and 2 KEGG pathways (FDR < 0.5, Fig. 2E).
In addition, GSVA revealed that MIS vs. MSS, 20Q11 anplicon chr20q11, chr20q13, reactome digestion of dietary lipids, DNA methylation involved in gamete generation and so on were different in L_cancer and R_cancer patients (|log2FC|> 0.2, all P < 0.05; Fig. 2F).
Differential immune microenvironment between L_cancer and R_cancer patients
By comparing the immune microenvironments between L_cancer and R_cancer patients, significant differences were observed between the two groups with regard to immune infiltration components.
In the R_cancer patients, the proportions of 'T cell CD8', 'T cells CD4 naïve', 'T cells follicular helper', 'Mast cells resting' were significantly higher and 'B cells memory', 'macrophages M0' were lower than in L_cancer patients (Wilcoxon test, all P < 0.05; Fig. 3A).
Differential immune microenvironment between L_cancer and R_cancer patients. A The comparison of immune infiltration levels between L_cancer and R_cancer patients, based on CIBERSORT. B The Stromal Score difference, Immune Score difference, ESTIMATE Score difference, and tumor purity difference between L_cancer and R_cancer patients. C The immune checkpoint-related gene expression levels in L_cancer and R_cancer patients. D HLA-related gene expression level in L_cancer and R_cancer patients. Notes: ns P > 0.05, *P < 0.05, **P < 0.01, ***P < 0.001
Comparing the Stromal score, ESTIMATE score, immune score, and tumor purity of L_cancer and R_cancer patients, we found that the R_cancer patients had a lower tumor purity and higher ESTIMATE and immune scores (Wilcoxon test, P < 0.05; Fig. 3B) than L_cancer patients.
We also analyzed the immune checkpoint-related genes (PD-1, PD-L1, CTLA4, CD86, LAG3, HAVCR2, TIGIT) and HLA family-related genes levels, which are considered biomarkers for predicting the efficacy of immunotherapy, between L_cancer and R_cancer patients and found that the expression levels of immune checkpoint-related genes and HLA family-related genes were significantly higher in R_cancer patients (Wilcoxon test, all P < 0.05; Fig. 3C, D).
Differential TMB landscape between L_cancer and R_cancer
The mutation prevalence varied dramatically within CRC in different locations. The mutation frequency in R_cancer patients was relatively higher than that in L_cancer patients (Fig. 4A). Moreover, the L_cancer and R_cancer groups contained different mutant genes. Waterfall plots (Fig. 4B, C) show the first 30 gene mutation rates in each location. A major discrepancy can be seen, as TP53 presented a higher mutation rate in L_cancer (L_cancer, 68%; R_cancer, 48%), while PIK3CA (L_cancer, 18%; R_cancer, 33%) and KRAS (L_cancer, 36%; R_cancer, 46%) showed higher yield mutation rates in R_cancer.
Differential TMB landscape between L_cancer and R_cancer patients. A The tumor mutation burden difference between L_cancer and R_cancer patients. B, C waterfall lot demonstrated the top 30 frequently mutated genes in B L_cancer and C R_cancer patients. D, E The mutation of microsatellite instability(MSI)-related genes in L_cancer and R_cancer patients. Notes: ***P < 0.001
We analyzed microsatellite instability (MSI)-related genes' mutation in each group, which showed that the L_cancer patient had MSI (Fig. 4D, E).
Identifying DEGs and functional annotation in tumor and normal patients
By comparing the transcriptome data of L_cancer and L_normal groups, we identified 4788 up-regulated DEGs and 4062 down-regulated DEGs (Fig. 5A). The top 20 up-regulated and down-regulated genes were displayed by heatmap (Fig. 5C). Further, we analyzed these DEGs between L_cancer and L_normal groups for functional enrichment analysis. This evaluation revealed the enrichment of 1139 GO terms and 65 KEGG pathways (FDR < 0.05). We chose to show the top 10 GO terms and 15 KEGG pathways in Fig. 5E, G.
Identifying DEGs and Functional Annotation in Tumor and Normal Patients. A Volcano plot for DEGs between L_cancer and L_normal patients. B Volcano plot for DEGs between R_cancer and R_narmal patients. C Heatmap of the top 40 DEGs between L_cancer and L_normal patients. D Heatmap of the top 40 DEGs between R_cancer and R_narmal patients. E GO enrichment analysis of the DEG between L_cancer and L_normal patients. F GO enrichment analysis of the DEG between R_cancer and R_narmal patients. G Top 15 KEGG analysis of the DEG between L_cancer and L_normal patients. H Top 15 KEGG analysis of the DEG between R_cancer and R_narmal patients
Likewise, the DEGs between R_cancer and R_normal identified 6261 up-regulated DEGs and 4501 down-regulated DEGs (Fig. 5B). The top 20 up-regulated and down-regulated genes were displayed by heatmap (Fig. 5D). These DEGs between R_cancer and R_normal groups be analyzed for functional enrichment analysis. A total of 1072 GO terms and 61 KEGG pathways had been enriched (FDR < 0.05). We chose to show the top 10 GO terms and 15 KEGG pathways in Fig. 5F, H.
Construction of prognostic gene model
To identify prognosis-related genes, we first screened genes using the Kaplan–Meier method in DEGs with P < 0.05, in order to screen survival-related DEGs as candidate genes affecting prognosis. Then, to avoid model overfitting, we performed a multivariate Cox regression analysis with the LASSO penalty algorithm to solve the multi-collinearity problem. Finally, we obtained 10 genes associated with the prognosis of L_cancer patients and 10 genes associated with the prognosis of R_cancer patients. These genes have a significant impact on the survival of patients (Additional file 1: Fig. S1).
The L_cancer patient prognosis features and risk score were calculated as: KNG1 × 0.621 + CYP11A1 × 0.600 + SMPD1 × 1.370 + DAND5 × 0.859 + NKPD1 × 0.721 + RP11-59D5_B.2 × 0.568 + CTD-2184C24.2 × 0.514 + RP11-680F8.3 × 0.517 − RP11-51F16.9 × 0.731 + CTD-2012K14.8 × 0.765 (Fig. 6A, B). The cutoff of risk score is 7.801, which had a great impact on OS (Fig. 6C). Scores lower than 7.801 have been defined as low-risk L_cancer patients, while scores higher than 7.801 have been defined as high-risk L_cancer patients. The AUC values of the risk score in the training set for 1-year, 3-year, 5-year, 7-year, and all-time OS were 0.554, 0.582, 0.593, 0.597, and 0.862, respectively (Fig. 6D).
Construction and validation of the prognostic model in L_cancer group. A LASSO coefficient profiles of DEGs. B Selection of the optimal parameter (lambda) in the LASSO model. C Differences in overall survival between high-risk and low-risk groups based on the risk scores in L_cancer patients. D Time-dependent ROC curves in the training set at 1-year, 2-year, 3-year, 5-year, 7-year and all-year in L_cancer patients. E Time-dependent ROC curves in the testing set at 1-year, 2-year, 3-year, 5-year, 7-year and all-year in L_cancer patients. F The ROC curves of Five-fold cross-validation in L_cancer patients. G–L Comparison of survival rates of high-risk and low-risk groups in different clinical subtypes in L_cancer patients. Survival analysis of different clinical characteristics including G Age < 65, H Age ≥ 65, I Female, J Male, K Stage I-II, L Stage III-IV
The R_cancer prognosis features and risk score were calculated as: MOCS1 × 1.100 − PTGS2 × 0.722 + PLEKHA8P1 × 0.409 − ZC3H12C × 0.571 + LPO × 0.575 + METTL11B × 0.294 + RP11-278A23.1 × 0.508 + RP11-452K12.7 × 0.405 − RP11-742B18.1 × 0.360 + RP11-626H12.2 × 0.787 (Fig. 7A, B). The cutoff of risk score is 11.981, which had a great impact on OS (Fig. 7C). Scores lower than 1.981 have been defined as low-risk R_cancer patients, while scores higher than 1.981 have been defined as high-risk R_cancer patients. The AUC values of the risk score in the training set for 1-year, 3-year, 5-year, 7-year, and all-time OS were 0.557, 0.610, 0.626, 0.692, and 0.835, respectively (Fig. 7D).
Construction and validation of the prognostic model in R_cancer group. A LASSO coefficient profiles of DEGs. B Selection of the optimal parameter (lambda) in the LASSO model. C Differences in overall survival between high-risk and low-risk groups based on the risk scores in R_cancer patients. D Time-dependent ROC curves in the train set at 1-year, 2-year, 3-year, 5-year, 7-year, and all-year in R_cancer patients. E Time-dependent ROC curves in the test set at 1-year, 2-year, 3-year, 5-year, 7-year, and all-year in R_cancer patients. F The ROC curves of Five-fold cross-validation in R_cancer patients. G–L Comparison of survival rates of high-risk and low-risk groups in different clinical subtypes in R_cancer patients. Survival analysis of different clinical characteristics including G Age < 65, H Age ≥ 65, I Female, J Male, K Stage I-II, L Stage III-IV
Internal validation of the prognosis genes model and stratified analysis by clinical factors
The efficacy of the prognostic signature was validated using a testing set of TCGA patients. Five-fold cross-validation was used to assess the stability of the model.
Among the L_cancer patients, the area under the curve (AUC) values of risk scores predicted in the testing set for 1-year, 3-year, 5-year, 7-year, and all-time OS were 0.597, 0.696, 0.722, 0.723, and 0.914, respectively (Fig. 6E). The AUC values of fivefold cross-validation were 0.860, 0.792, 0.908, 0.854, and 0.978, respectively, and the integrated AUC value was 0.863 (Fig. 6F). The results revealed that the AUC values of fivefold cross-validation were high and similar, indicating that the model had good predictability and stability. Based on the obtained sample clinical characteristics, patients were stratified into age < 65 years and age ≥ 65 years sub-groups (Fig. 6G, H), female and male sub-groups (Fig. 6I, J), and pathological tumor Stage I/II and Stage III/IV sub-groups (Fig. 6K, L). The overall survival analysis was performed in each sub-group, based on the level of risk score, and all results showed statistical differences.
Likewise, in R_cancer patients, the AUC values of risk scores predicted in the test set for 1-year, 3-year, 5-year, 7-year, and all-time OS were 0.679, 0.725, 0.771, 0.801, and 0.857, respectively (Fig. 7E). The AUC values of fivefold cross-validation were 0.838, 0.727, 0.796, 0.793, and 0.826, respectively, and the integrated AUC value was 0.792 (Fig. 7F). The results revealed that the AUC values of fivefold cross-validation were high and similar, indicating the model had good predictability and stability. Patients were also stratified into age < 65 years and age ≥ 65 years sub-groups (Fig. 7G, H), female and male sub-groups (Fig. 7I, J), and pathological tumor Stage I/II and Stage III/IV sub-groups (Fig. 7K, L). Overall survival analysis was also performed in each sub-group, based on the level of risk score, and all the results showed statistical differences.
Incorporating clinical factors to develop individualized nomograms
Clinical characteristics, including Age, Gender, T, N, M, Stage, and risk score, were utilized to perform univariate analyses in the training sets of L_cancer (Fig. 8A) and R_cancer (Fig. 9A), respectively. After statistical adjustment for other variables with multivariate Cox regression analysis, we found that the Risk, pathological M, pathological stage, gender, and age were the only six independent prognostic factors that could be used to predict the survival rate in L_cancer (Fig. 8B), while the Risk, pathological N, pathological T, and age were the only four independent prognostic factors that could be used to predict the survival rate in R_cancer. (Fig. 9B). L_cancer patients' nomogram (Fig. 8C) and R_cancer patients' nomogram (Fig. 9C) were developed using the above prognostic features, with the total points calculated by adding the points of individual prognostic features.
Validation of the nomogram in predicting the overall survival of L_cancer in the TCGA dataset. A, B Univariate and multivariate Cox regression analysis of L_cancer prognostic signatures and clinical characteristics. C Developed incorporating clinical factors nomogram of L_cancer patients. D Calibration curve of the nomogram in the train set and test set of L_cancer patients. E Decision curve analysis of the nomogram in the train set and test set of L_cancer patients. F Time-dependent ROC curves in the train set at 1-year, 2-year, 3-year, 5-year, 7-year, and all-year in L_cancer patients. G Time-dependent ROC curves in the test set at 1-year, 2-year, 3-year, 5-year, 7-year, and all-year in L_cancer patients
Validation of the nomogram in predicting overall survival of R_cancer in the TCGA dataset. A, B Univariate and multivariate Cox regression analysis of R_cancer prognostic signatures and clinical characteristics. C Developed incorporating clinical factors nomogram of R_cancer patients. D Calibration curve of the nomogram in the train set and test set of R_cancer patients. E Decision curve analysis of the nomogram in the train set and test set of R_cancer patients. F Time-dependent ROC curves in the train set at 1-year, 2-year, 3-year, 5-year, 7-year, and all-year in R_cancer patients. G Time-dependent ROC curves in the test set at 1-year, 2-year, 3-year, 5-year, 7-year, and all-year in R_cancer patients
Predictive performance of the established nomogram
Among L_cancer patients, the calibration curve and decision curve analysis for predicting median survival time OS in the training and testing sets indicated that the nomogram-predicted survival similarly corresponded with actual survival outcomes (Fig. 8D, E). The AUC of the nomogram was 0.8 in the training set and 0.905 in the testing set (Fig. 8F, G).
In R_cancer patients, the calibration curve and decision curve analysis for predicting median survival time OS in the training and testing sets indicated that the nomogram-predicted survival similarly corresponded with actual survival outcomes (Fig. 9D, E). The AUC of the nomogram was 0.836 in the training set and 0.850 in the testing set. (Fig. 9F, G).
External validation of the prognosis signature by qRT-PCR
The obtained results were further validated by qRT-PCR, as shown in Fig. 10.
A–E qRT-PCR validation of the expression of DAND5, SMPD1, KNG1, NKPD1, and CYP11A1 in 50 pairs of L_cancer patients. F–J qRT-PCR validation of the expression of ZC3h12c, LPO, METTL11B, PTGS2, and MOCS1 in 50 pairs of R_cancer patients. Notes: *P < 0.05, **P < 0.01, ***P < 0.001
In 51 pairs of L_cancer patients, compared with adjacent cancer tissues, the expression of DAND5, SMPD1, KNG1, NKPD1, and CYP11A1 were found to be down-regulated in cancer tissues (two-tailed paired t-test; all P < 0.05, Fig. 10A–E).
Moreover, in 51 pairs of R_cancer patients, compared with adjacent cancer tissues, the expression of LPO, METTL11B, and PTGS2 were found to be up-regulated, and ZC3H12C and MOCS1 were down-regulated in cancer tissues (two-tailed paired t-test; all P < 0.05, Fig. 10F–J).
Differences in the immune microenvironment, TMB landscape, immune checkpoint-related genes, and HLA-family genes level between high- and low-risk patients
Based on the difference in the immune microenvironment and TMB landscape between left and right CRC, we next analyzed the difference in these aspects between high- and low-risk patients based on prognostic gene models.
In the R_cancer patients, high-risk patients had a lower proportion of 'B cells memory', 'Dendritic cells resting', immune score, ESTIMATE score, immune checkpoint-related genes, and HLA-family genes, and a higher proportion of 'T cells follicular helper', 'Dendritic cells activated', and 'Mast cells activated' (Wilcoxon test, P < 0.05; Fig. 11A–E). These results indicate that R_cancer patients in high- and low-risk groups may have different responses to immunotherapy, and immunotherapy in R_cancer low-risk patients may be more beneficial.
A The comparison of immune infiltration levels between high-risk and low-risk groups in R_cancer patients, based on CIBERSORT. B The Stromal Score difference, Immune Score difference, ESTIMATE Score difference, and tumor purity difference between high-risk and low-risk groups in R_cancer patients. C The immune checkpoint-related gene expression levels in high-risk and low-risk groups in R_cancer patients. D The tumor mutation burden difference between high-risk and low-risk groups in R_cancer patients. E HLA-related gene expression level between high-risk and low-risk groups in R_cancer patients. Notes: ns P > 0.05, * P < 0.05, ** P < 0.01, *** P < 0.001
In the L_ancer patients, there was no difference in these indicators between high- and low-risk patients (Additional file 2: Fig. S2A–E).
Correlation of hub gene and risk score with immune-related score and genes
Correlation analyses were carried out for risk scores and hub genes with immune-related scores and genes. As we can see, in R_cancer patients, R_cancer risk score was strongly correlated with immune-related scores and genes (Fig. 12). In particular, it has a significant negative correlation with immune checkpoint-related genes, Stromal score, immune score, and ESTIMATE score and a positive correlation with tumor purity. These results prove that R_cancer patients with R_cancer low-risk score may benefit more from immunotherapy. In addition, the R_cancer risk score was positively associated with the content of 'B cells memory', 'T cells CD4 naïve', 'T cells regulatory Tregs', 'Macrophages M0', and 'Dendritic cells resting' and negatively associated with the content of 'T cells follicular helper', 'Dendritic cells activated', 'Mast cells activated' and 'Neutrophils'. In L_cancer patients, L_cancer risk score was no correlation with immune-related scores and genes (Additional file 3: Fig. S3).
Show the correlation of R_cancer RiskScore and R_cancer hub genes expression with immune infiltration level in R_cancer patients
CRC has a heterogeneous tumor composition and complex oncogenic mechanisms. The development of individualized treatment strategies and the evaluation of patient prognoses based on tumor location are crucial. This study is the first to separately build predictive models for L_cancer and R_cancer, to the best of our knowledge. We presented two nomograms for CRC classified with respect to both tumor side and location based on prognostic gene signatures and clinical prognostic factors can be used to distinguish high-risk from low-risk patients effectively. The L_cancer nomogram includes prognostic genes (KNG1, CYP11A1, SMPD1, DAND5, NKPD1, RP11-59D5_B.2, CTD-2184C24.2, RP11-680F8.3, RP11-51F16.9, CTD-2012K14.8), pathological N, pathological T, and age, which can be used to predict the survival rate; meanwhile, the R_cancer nomogram comprises prognostic genes (MOCS1, PTGS2, PLEKHA8P1, ZC3H12C, LPO, METTL11B, RP11-278A23.1, RP11-452K12.7, RP11-742B18.1, RP11-626H12.2), age, pathological M, pathological T, pathological stage, and gender, which can also be used to predict the survival rate.
Numerous studies have confirmed that the right- and left-sided colons are distinct due to their embryological origins. The right-side colon originate from the midgut, whereas the left-side colon originate from the hindgut. In this study, we confirmed that there exist significant differences in the TMB and immune microenvironment between right- and left-sided CRC patients. Furthermore, right-sided CRC tend to have worse prognosis than left-sided CRC patients. The difference between right- and left-sided CRC patients' survival rates is might be caused by the higher frequency of mutations in addition to changes in the tumor microenvironment associated with tumor purity. According to recent research, mutation prevalence differs depending on side and location. RAS mutations declined from 70% in patients with right-sided CRC to 43% in those with left-sided CRC, while the number of BRAFV600 mutations increased from 10 to 22% between the same locations. Sigmoid and rectal tumors with left-sided mutations were more likely to harbor TP53 mutations than PIK3CA, BRAF, or CTNNB1 mutations [3]. Consistent with our results, in left-sided tumors, TP53 (L_cancer: 68%, R_cancer: 48%) showed a higher mutation rate; meanwhile, in right-sided tumors, PIK3CA (L_cancer: 18%, R_cancer: 33%) and KRAS (L_cancer: 36%, R_cancer: 46%) showed higher yield mutation rates. The results in our study align well with a recent report by Marshall et.al., who also demonstrated significant differences between L_cancer and R_cancer in mutation patterns.
The tumor microenvironment (TME) refers to the physical environment around a tumor, including the immune cells, neurons, blood vessels, extracellular matrix, and other cellular functions related to tumor progression and therapy effects. We also confirmed that the immune microenvironment affects the prognosis of patients with CRC. Aggressively growing tumors create a highly immunosuppressive TME that depletes antitumor responses and promotes tumor progression [19, 20].
Based on the Estimation of STromal and Immune cells in MAlignant Tumor tissues using Expression data approach, immune score and tumor purity can reveal information about the tumor's immune status. Low immune scores and high tumor purity have been associated with better prognoses in several studies [21,22,23]. Based on this, we examined the differences in tumor immune microenvironment between right- and left-sided CRC patients. In our study, L_cancer patients not only had poor prognosis but also had high ESTIMATE and immune scores, as well as low tumor purity. Thus, we further analyzed the effect of high- or low-risk on immune infiltration in patients in both L_cancer and R_cancer models. We found that, in the R_cancer model, high-risk patients had lower immune and ESTIMATE scores and higher tumor purity than low-risk patients. However, there was no difference between high- and low-risk in the L_cancer model with respect to immune infiltration. Besides, in the R_cancer model, high-risk patients were significantly different from low-risk patients in terms of immune infiltrating cell types, such as memory B-cells, dendritic, T follicular helper cells and mast cell activation. Nevertheless, in the L_cancer model, the high- and low-risk patients showed no difference. These results may be related to our different models for L_cancer and R_cancer. The findings of some studies were in line with our study, where low tumor purity result in poor prognosis in glioma and CRC [21, 22]. Additionally, the proportions of CD8 T-cells and T follicular helper cells were significantly higher in the R_cancer group, while M0 macrophages had higher infiltration in L_cancer groups. A recent single-cell RNA-Seq study between right- and left-sided CRC patients discussed the difference in single-cell transcriptomes between the two groups, which was in line with our findings. In summary, there has been increasing awareness of the body's ability to fight tumors through various types of cells cytokines, and chemokines. Immune cells, especially, play a critical role in this. Immunotherapy has become increasingly popular as a treatment option for cancer patients with refractory malignant tumors, which can benefit significantly from immune checkpoint inhibitors. To determine whether immunotherapy is effective, TMB, TME, and immune checkpoint levels are considered as biomarkers [23,24,25]. A previous study has demonstrated that, in CRC patients, the prognostic impact of PD-L1 and PD-1 expression varies according to the primary tumor site. Moreover, the presence level of PD-L1 is an independent prognostic factor for right-side tumors [26]. This finding was in line with our study, which demonstrated that there were significant differences in PD-1, PD-L1, and CTLA4 expression between right- and left-sided CRC patients.
Given this, this study independently assessed the effect of the tumor microenvironment in L_cancer and R_cancer of high- and low-risk patients from two aspects (TMB and immune microenvironment), leading us to speculate that R_cancer—especially low-risk R_cancer—patients may benefit more from immunotherapy [27, 28]. Validation is needed, but these results could be clinically significant as they indicate that tumor location is important to consider in therapeutic decisions, including eligibility for immunotherapy.
The hub genes in the signature have previously been shown to be potential biomarkers. Relevant research has reported that PTGS2-driven inflammatory responses can induce tumor expression of microRNA-21, which can increase the level of the inflammatory mediator prostaglandin E2 (PGE2) by down-regulating PGE2-metabolizing enzymes, contributing to colorectal cancer development [28,29,30,31,32]. PLEKHA8P1 expression has been associated with the development and progression of many malignancies in humans, such as CRC and renal cancer [33]; moreover, research has shown that its dysregulated expression affects 5-Fluorouracil-induced chemoresistance in the human hepatocellular carcinoma cell line FT3-7 [34]. Prior studies found ZC3H12A has links with immune homeostasis and post-transcriptional regulation which can stimulate tumor progression in lung and colon cancer [35,36,37]. LPO can collaborate with activated Wnt signaling to induce intestinal neoplasia [38]. METTL11B expression has been associated with poor prognosis in colorectal cancer and is higher in cancer tissues than in neighboring normal tissues [39]. NKPD1 has been predicted to be linked with the de novo synthesis of sphingolipids [40]. Increased DAND5 level is an independent risk factor for both colorectal and breast cancers and the prediction of poor prognoses [41, 42]. SMPD1 encodes lysosomal acid sphingomyelinase, which converts sphingomyelin to ceramide. Prior studies have found that the functional inhibition of acid sphingomyelinase contributes to tumor cell death by overactivation of hypoxia stress-response pathways [43]. Another study has shown that down-regulation of SMPD1 is linked with resistance to chemotherapy regimens including 5-Fluorouracil [44]. Studies have shown CYP11A1, which can hydroxylate the side-chain of vitamin D3 at carbons 17, 20, 22, and 23, are related to susceptibility to breast cancer [45, 46]. KNG1 can regulate the expressions of VEGF, cyclinD1, ki67, and caspase-3/9, exerting anti-angiogenic properties and inhibiting the proliferation of endothelial cells. Over-expression of KNG1 can inhibit the activity of PI3K/Akt, decrease tumor growth, and promote apoptosis [47]. On the contrary, other researchers have found that KNG1 expression was significantly increased in colorectal cancer lesions [48]. At present, there has been no reported association between MOSC1, RP11-278A23.1, RP11-452K12.7, RP11-742B18.1, RP11-626H12.2 RP11-59D5_B.2, CTD-2184C24.2, RP11-680F8.3, RP11-51F16.9, CTD-2012K14.8, and cancer. In the end, RT-qPCR was performed to verify the results from the bioinformatic analyses of LCC and RCC. We revealed that the prognostic gene expression results were consistent with the outcomes of our survival analysis, indicating that our results are reproducible and reliable. In addition, this further confirmed that these key genes are related to the occurrence and development of colon cancer.
This study had some limitations. The signatures and nomograms constructed in this study using vast datasets from TCGA and our patient database were robust, but the study was still a retrospective one. Second, we explored the TMB and immune microenvironment landscape between right- and left-sided CRC patients and between patients in different risk groups; however, the study lacked experimental verification. Third, as previously noted, obtaining risk scores requires knowledge of ten genes expressed in tumor tissues, thereby increasing the difficulty of applying the nomograms. It appears that many molecular diagnostic or prognostic models have the same problem. Researchers and clinicians need to figure out how to simplify the application of these models in clinical settings. In the future, molecular detection technology may solve this dilemma. The constructed nomograms may be used routinely.
We found significant differences between L_cancer and R_cancer patients, including clinical features, transcriptome, TMB, immune microenvironment landscape, suggesting that colon cancer can be classified and analyzed into different clinical types with respect to their differences in anatomical location and gene expression, thus aiding in the early diagnosis and prognosis of colon cancer. We established two clinical predictive nomograms in combination with clinical features to provide a basis for the personalized and precise treatment of L_cancer and R_cancer. These hub genes may become promising biomarkers for the diagnosis, treatment, and prognosis of colon cancer. Moreover, The findings support previous studies suggesting that proximal and distal CRC can be classified differently in terms of epidemiology, pathology, and genetics.
The datasets used and/or analyzed during the current study are available from TCGA repository: https://portalgdccancer.gov.
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No additional contribution to the manuscript as well as specific funding is to be acknowledged by all of the Authors.
This work was supported by Medical Scientific Research Foundation of Hebei Province, China (H2020206374, H2021206306, 8220160183), and the Talent Project of Hebei, China (LS202001).
Mengye Niu and Chengyang Chen contributed equally to this work and should be considered co-first authors
Department of General Surgery, The First Hospital of Hebei Medical University, No. 89 Donggang Street, Yuhua District, Shijiazhuang, Hebei, China
Mengye Niu, Chengyang Chen, Xian Gao, Yi Guo, Bingzhou Zhang, Xin Wang, Shihao Chen, Xupeng Niu, Chao Zhang, Like Li, Zhongxin Li, Zengren Zhao & Xia Jiang
Hebei Key Laboratory of Colorectal Cancer Precision, The First Hospital of Hebei Medical University, Shijiazhuang, China
Mengye Niu, Chengyang Chen, Xian Gao, Bingzhou Zhang, Xin Wang, Shihao Chen, Xupeng Niu, Chao Zhang, Like Li & Xia Jiang
Department of Pathology, The First Hospital of Hbei Medical University, Shijiazhuang, China
Xin Wang
Mengye Niu
Chengyang Chen
Xian Gao
Yi Guo
Bingzhou Zhang
Shihao Chen
Xupeng Niu
Chao Zhang
Like Li
Zhongxin Li
Zengren Zhao
Xia Jiang
Conceptualization and design, MYN, and CYC; Methodology, MYN, and CYC; Clinical investigation, XPN, LKL, CZ; Data Curation, XG, YG, BZZ, ZXL, XW, SHC; Writing—Original Draft Preparation, CYC; Writing—Review and Editing, MYN, and CYC; Supervision, ZRZ, ZXL and XJ; Funding Acquisition, ZRZ, ZXL and XJ; All authors have read and agreed to the published version of the manuscript.
Correspondence to Zengren Zhao or Xia Jiang.
The study was conducted in accordance with the Declaration of Helsinki and approved by The First Hospital of Hebei Medical University (Protocol Code: HBYDYY2018003005). Written informed consent was obtained from all subjects. All the experiment protocol for involving human data was in accordance with the guidelines of national/international/institutional or Declaration of Helsinki in the manuscript.
None of the authors has any conflict of interest to disclose.
Additional file 1: Fig. S1.
(A) Kaplan-Meier survival analysis of ten hub genes (KNG1, CYP11A1, SMPD1, DAND5, NKPD1, RP11-59D5_B.2, CTD-2184C24.2, RP11-680F8.3, RP11-51F16.9, CTD-2012K14.8) in L_cancer patients between high-expression and low-expression groups. (B) Kaplan-Meier survival analysis of ten hub genes (MOCS1, PTGS2, PLEKHA8P1, ZC3H12C, LPO, METTL11B, RP11-278A23.1, RP11-452K12.7, RP11-742B18.1, RP11-626H12.2) in R_cancer patients between high-expression and low-expression groups.
(A) The comparison of immune infiltration levels between high-risk and low-risk groups in L_cancer patients, based on CIBERSORT. (B) The Stromal Score difference, Immune Score difference, ESTIMATE Score difference, and tumor purity difference between high-risk and low-risk groups in L_cancer patients. (C) The immune checkpoint-related gene expression levels in high-risk and low-risk groups in L_cancer patients. (D) The tumor mutation burden difference between high-risk and low-risk groups in L_cancer patients. (E) HLA-related gene expression level between high-risk and low-risk groups in L_cancer patients. (Notes: ns P>0.05).
Show the correlation of L_cancer RiskScore and L_cancer hub genes expression with immune infiltration level in L_cancer patients.
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Niu, M., Chen, C., Gao, X. et al. Comprehensive analysis of the differences between left- and right-side colorectal cancer and respective prognostic prediction. BMC Gastroenterol 22, 482 (2022). https://doi.org/10.1186/s12876-022-02585-3
Left-sided colon cancer
Right-sided colon cancer
Nomogram
Immune microenvironment
Tumor mutation burden
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Seismic hazard assessment of Koyna region, Peninsular India: using geospatial approach
S. M. S. P. Dev ORCID: orcid.org/0000-0001-8868-546X1 &
R. Nagarajan1
Earthquake-prone regions from the stable continental regions (like Peninsular India) warrant total seismic hazard estimation from possible sources — reservoir induced or tectonic. The Koyna region falls in the so-called stable continental region, though previously considered as aseismic because of its location in the Peninsular India, which was argued to be the continental Shield region; however, it is seismically active ever since the December 10, 1967, strong earthquake (M6.5) which killed around 200 people and damaged properties. The event was initially attributed to reservoir-induced seismicity. The present study aims at mapping the spatial distribution of hazard intensity in terms of peak ground acceleration and damage potential expressed as Modified Mercalli Intensity (MMI) levels owing to the scenario earthquakes (M4.0 to M6.9) from the linear sources and background seismicity in the Koyna region. For the seismic source model, geospatial integration approach is used. To delineate the source boundaries we perform spatial analysis of seismicity attributes and assigning weights based on the geological information. In addition, spatial buffering of thematic layers of faults and lineaments is applied to truncate the spatial uncertainties. Thematic attribute layers are integrated over the SRTM-DEM-derived geomorphic features and spatial seismicity pattern to define the possible linear seismic source zones. We employ ArcGIS® to perform geospatial analysis and to integrate thematic spatial information finally to prepare the seismic hazard and damage potential zones for the Koyna region.
We present the probabilistically estimated seismic hazard and damage potential estimates for the Koyna region. Spatial mapping of the hazard parameters is performed on Geographic Information System (GIS) platform. Along with the historical and instrumental earthquake data, we incorporate updated and relocated seismic events from circa 1618 to 2017 covering ~ 400 years period, for analyzing the mean annual rates of seismicity and earthquake recurrence intervals. The anticipated recurrence intervals of earthquakes ~ M5.5 is around 100 ± 10 years. The estimated peak ground acceleration (PGA) for 10% probability being exceeded in 50 years (i.e. 500 years of return period) is greater than 21% g is anticipated for the area located (~30 km radius) between the Koyna and Warna Reservoirs.
The hazard estimates for the 500 years return period earthquake scenarios (M4.0 to M6.9) are based on the bedrock level estimation. Estimated PGA-values are grouped into three categories that are relatively defined as high, moderate, and low hazard zones, respectively. Slightly damaging intensity (MMI-VII level) expected in 50 years has ~ 40% probability and PGA greater than 21% g for the area which is located between Koyna and Warna Reservoirs, which is identified as high hazard zone.
It is evident from the recent devastating earthquakes that in many parts of the world that the existing seismic hazard maps frequently failed in making the predictions of expected seismic ground motions levels. This inadequacy can be attributed to imprecise and surmised identification/mapping of seismically active faults, inadequate paleoseismic data, and uncertainty in the fault slip rates estimations and other seismicity related information. Improvement of risk estimation methods and vulnerability assessment schemes is warranted whenever cutting-edge development of earthquake science and recent or updated earthquake data is available. Because reliable risk estimates are essential since it offers the primary information for risk management and mitigating agencies. The present study is carried out, by applying recently occurred earthquake events and the relocated seismicity in the Koyna seismic zone in Peninsular India, to map the damage potential probabilistically. We assume segmented linear seismic sources delineated based on geologic, morphotectonic characteristics and fault-slip rates.
Besides, primary seismic hazards (i.e. bedrock level ground motion) the amount of damage inflicted by earthquake events can also be attributed to heterogeneous geological material characteristics since unconsolidated surface deposits tend to amplify in the seismic ground motions. Earthquake scenarios represents a potential future earthquake by assuming a particular magnitude, location, and fault-rupture geometry and estimating ground shaking using a variety of approaches for the regions where the seismic sources are of tectonically active. In addition to the understanding of regional tectonic set-up, it is also important infer the dynamics of Reservoir Induced Seismicity (RIS) for a total seismic hazard assessment of a given region. RIS occurs only when the pre-existing stress, geologic, and hydrologic conditions at the site are suitable (Simpson, 1976). Seismicity in the Koyna region in Peninsular India, was initially attributed to the RIS, is still not well understood. Predominant geology of is basalt, about 1500 m to 1700 m thick of Deccan Traps basalt flow layers is reported from Koyna region (Kaila et al., 1981; Sarma et al., 2004).
India has witnessed a number of devastating earthquakes in the past five decades. According to a joint World Bank–UN study, around 200 million people living in India will be exposed to seismic risk by 2050 (WB–UN, 2010). Virtually, India's 50% land is declared as seismically-prone. Considering the seismic risk and economic losses, seismic microzonation and hazard assessment at regular interval should entail as a part of disaster management processes for mitigating the hazard. In India, seismic microzonation studies on major cities that are vulnerable to the earthquake hazards have been carried out by Mohanty et al., 2007 (Delhi), kanth and Iyengar, 2006 (Mumbai), Sitharam and Anbazhagan, 2007 (Bangalore), Nath et al., 2008 (Guwahati), Ganapathy 2011 (Chennai), Mahajan et al., 2007 (Dehradun), Rao et al., 2011 (Jabalpur), and Pal et al., 2008 (Sikkim) on regional scales (MoES, 2011). GIS based application for estimating the seismic risk assessment of urban areas is developed and demonstrated for Mumbai city by Sinha et al., (2008). Available probabilistic seismic hazard assessments for peninsular India is performed based on the regional tectonic-setup by assuming one or two seismic source models on national scale (Jaiswal and Sihna, 2007; Nath and Thingbaijam, 2012).
The data gaps have been moderated in these exercises. At the same time, towns situated in proximity to urban areas are being developed to decongest the cities. Therefore, there is a need to prepare hazard assessment and mitigation plan for intended/ongoing areas of development. The earthquakes occurring in this part of stable continental region (SCR) emphasize the need for seismic hazard assessment and damage potential zonation for non-urban areas incorporating updated seismicity and spatial uncertainties of seismic sources.
GIS technology enables the storage of point and polygon information as geo-referenced spatial database and allows for further upgrading of new information. Geo-referenced information could be retrieved, analysed and results could be presented on geospatial mode for data dissemination for disaster management. This study demonstrate the effective use of limited (spatial and quantitative) information (seismological, geological and geomorphological) to a more accurate spatial variation of the seismic hazard that can be modelled, which will provide an improved basis for seismic (micro) hazard zonation.
The objective of this study is to assess the seismic hazard and demarcate the damage potential zones by incorporating the updated seismic events on geospatial mode for the Koyna region.
The steps that are followed in this study are:
Collection of earthquake, active tectonic, geological, geophysical and topographic data from different sources;
Creation of geospatial information database, from available thematic maps, by digitizing and geo-referencing the theme polygons, point information, and extrapolate on space using the ArcGIS®;
Analyze the spatial distribution of seismicity and buffer the lineaments/faults indicating potential spatial extension for defining the source zones;
Determining the return periods for potentially damaging earthquakes based on the complied earthquake catalogue;
By assuming many scenarios and identifiable seismic source zones calculation of probabilities of event parameters;
Computing the mean annual probability for expected return periods using hazard integral equation for 10Km X 10Km grid points covering the entire study area and its surroundings;
Zoning of probabilistically estimated seismic intensity using ArcGIS® for seismic risk analysis and mitigation strategy planning.
Data used in this study are — earthquake catalog information of ~400 years period to determine earthquake rates, high resolution geology maps, local soil maps supported by field-based data in establishing subsurface information of the site conditions, Landsat-7 ETM+ satellite images (NASA Landsat Program, 2003) for maintaining the uniform spatial continuity and demarcation of lineaments/faults and SRTM data for deciphering landforms, elevation, natural slope classes.
Peninsular India
Peninsular India is considered as one of the largest Precambrian Shield areas of the world. This Shield area was described as the stable land mass associated with low or no seismicity. However, earthquakes events such as 2001 M7.7 (Bhuj), 1993 M6.3 (Killari), 1997 M5.7 (Jabalpur), and 1967 M6.5 (Koyna), stressed the need for the revisiting of the stable continent belief as manifested by intra-plate earthquakes. It is believed that, the bending of the Indian plate is due to the collision of the Indian plate with Tibet plate that generated the elastic stress build-up within the tectonic pockets. This results in consequent slip along the faults and cause seismic activity in the Peninsular India. Northwest striking faults under the Deccan Traps are believed to exist in this region (Chandra, 1977). Thermal springs are found to occur either near the contact of two geological units or along prominent tectonic units in the Peninsular Shield. The stored stresses are released by frequent micro to moderate earthquakes through the thermal spring areas located in the West Coast of India, whereas in the rest of the Peninsula the stress release is through less frequently occurring moderate earthquakes (Chadha, 1992).
Koyna-Warna region
Koyna-Warna region (Fig. 1) is part of Deccan Volcanic Province covered by basaltic lava flows of Cretaceous-Tertiary age (~65 Ma).The basement is occupied by migmatitic-gneiss that is normally found at mid crustal levels. The focal depth of the events occurs at depths of around 6–9 Km below the surface. The maximum thickness of the Deccan Traps is around 1500–1700 m and source of the any seismic events need to be within the Deccan Traps basaltic pile. Fault-plane solutions and allied geophysical modelling have yielded results that are not consistent with each other. Earthquakes do not elucidate an entire fault-zone structure in the near surface; it is difficult to assess the overall tectonic structure from seismicity alone (Catchings et al., 2015). The ductile basaltic layer conceals the surface expression of the underlain seismogenic faults if any. Fractures and cooling joints are seen on the surface. The reported seismicity near Koyna and Warna reservoirs is on the eastern side of the escarpment from the focal depth of 7–9 Km below the surface (Figs. 3 and 6). Moho geometry in the Koyna-Warna region varies from 37 Km to 42 Km, with a slight dip towards SE. The average velocities in Koyna are higher than in Warna indicating a fractured setting in the latter. The seismicity event distribution in Koyna region is along the NNE-SSW trending fault zone coincides with a low velocity zone between two competent zones with a very high velocity > 4.0 Km/s. It is believed that the diffusion process and not the reservoir load effect, is the dominating mechanism in triggering earthquakes in Koyna.
Location of Koyna region and the Peninsular India
Earthquakes are occurring in a 20 × 30 Km area around Koyna since the impoundment of Shivajisagar Reservoir (Koyna Dam) in 1962, including the largest triggered earthquake of M6.5 on December 10, 1967, and 21 earthquakes of > M5.0 occurred since 1962 (Figs. 7 and 1). These events were further enhanced by the impoundment of the nearby Warna Reservoir in 1993 (Gupta, 2011). The earthquake of April 25, 1997, was located close to Koyna and February 11, 1998 earthquake was close to the Warna Reservoir. The micro-earthquake activity close to the source region was found to increase before the occurrence of the main shocks and continue up to a month before reaching a background level (Ramana et al., 2007).
The western margin of India has a nearly straight NNW-SSE trending coastline that gradually rises in a step-like pattern and all of a sudden a drastic change in elevation could be observed. It is an elevated rift-flank generated by the lithospheric mechanics of continental rifting phenomenon. There is a lack of evidence of block-faulting as the Western Ghats escarpment is well-adjusted to rock type and crustal structure (Fig. 3).
Geomorphic information
Geomorphologic features are grouped into hilly uplands and Western Ghats Escarpment (WGE) that rise to an elevation greater than 1390 m a.s.l. The WGE divides the eastern Deccan Plateau; where the lowest elevation (around Karad and Kolhapur) is about 550 m a.s.l. The Deccan Plateau has the step-like topographic of alternating steep and flat slopes (Fig. 2). Their elevation range groups are: <300 m, 550–600 m; 800–900 m; 950–1000 m a.s.l. The elevation of Konkan Coastal Belt (KCB) is less than 250 m a.s.l.; extends about 30–40 km further westwards till the shoreline of the Arabian Sea. It has steep-sided, flat-topped hill ranges, interspersed with flat-bottomed valleys (Fig. 3).
Field photo showing basalt flow layers
Spatial distribution of elevation in and around Koyna region (100 Km radius)
SRTM data having three arc-second (~90 m) ground resolutions is used to produce the spatial distribution of elevation and slope. They are grouped into <40 m, 40 to 360 m, 360 to 660 m, 660 to 960 m and >960 m a.m.s.l. classes (Fig. 3). These class intervals roughly indicate laterite, colluvium and black soil deposits, and deeply weathered basalts/highland laterite.
Regional geomorphic features such as valleys and steep slope hilly landforms — forming high relief terrain can alter the direction of propagating seismic waves. Amount of natural slope contributes to the angle of reflection and diffraction of seismic waves and can also influence the site response parameters. Steep terrain slope tends to focus the reflected seismic waves at the slope crest, whereas gentle slopes scatter the diffracted seismic wave (Ashford et al., 1997). The spatial distributions of natural slopes (Fig. 4) are grouped into <3o, 3o-10o, 10o-20o, 20o-30o and >30o. It approximately designates the domination of soil, soil and weathered rock/laterite, laterite and fresh/hard rock and hard rock in the sub-surface (~30 m down to the engineering bedrock) that affects the shear wave velocity.
Spatial distribution of natural slope derived from the SRTM data
Rock and soil types
There are more than 11 volcanic flow basalt layers and they consist of massive and vesicular basalt, amygdaloidal basalt, agglomerate, tuff breccia and red bole. The individual flow varies between 45 m to 61 m thickness. Red bole layer up to a thickness of 2 m is found persistently along the contact between the individual flows. Lateritization is a process of low-temperature weathering associated with mineralogical breakdown and the degree of weathering diminishes with depth. Laterites that are found capping the elevated basalt mesas of Western Ghats is termed as high-level laterite (900–1500 m a.s.l.) and semi-continuous belt lying west of Western Ghats Escarpment as low-level laterite (30 to 100 m a.s.l.).
Coarse-textured shallow reddish soils occur on the foothill slopes, deep soils are found along river banks/valleys. Besides, low-level laterite deposits, coastal alluvial soils are found along the Konkan Coast it consists of deep sandy loam soil deposits.
Surface cover features
False colour composites (bands 5, 3, 1) of Landsat-7 ETM+ of November 14, 1999 and November 25, 2000, having 30 m spatial resolution are used for extracting the lineaments and land cover features — water bodies (WB), natural vegetation (F), floodplain (FP), settlement (R) etc. (Fig. 5).
False colour composite of the Landsat-7 ETM+ image 20th October 1999 showing the land cover features
Structural information
Cooling of lava flow causes shrinkage and results in formation of cooling joints. Long, linear/gently curvilinear joints that extending continuously across several metres and occur parallel to each other with a uniform spacing are found in the Deccan Traps of the Koyna area. Fracture planes are occurring sub-parallel to each other. Fracture zone is a narrow linear zone along which several close-spaced. Shear zone is a linear zone along which the fracturing of the basalt is very close-spaced (Kale et al., 2016). Due to deeply weathered bedrock, soil cover, thick shrub vegetation, the areal extension and intensity of fracturing cannot be determined (Kale et al., 2014). Considerable regional fracture lineaments in the study area are N-S (strong geomorphic and geophysical expression), NW-SE (strong geomorphic and geophysical expression), NE-SW (moderate Geomorphic and geophysical expression) and E-W (not clear). Majority of reported events are associated with NW-SE and NE-SW trending lineaments/faults. Figure 6 shows the linear features that represent fractures/faults demarcated and the reported seismic events and lineaments and concealed faults. In addition to NNW-SSE, N100E to N300E lineaments are also present.
Seismic events, concealed faults and lineaments
The potential activity of faults are grouped as — active faults — characterized by current activity (movement or event) /associated with surface-rupturing earthquakes in the past 11,000 years; capable faults having ability for movement; and potential capable of being or becoming active. It is associated with surface-rupturing earthquakes occurred in the Quaternary age (~1.6 m years). The said classification is used for identifying the seismic source zones and to define the potential faults, for estimating fault-to-site-distance probabilities, based on the published literature. Talwani (1997) inferred the Koyna-Warna fault system as Patan Fault — trending NE-SW by a dip of about 45° and NW-SE trending fractures extending from near-surface to hypocentral depths. Apparent, NNE-SSW and NNW-SSE striking seismicity patterns attributed to normal faults have been reported from the Koyna region and Warna region, respectively (Talwani, 1997; Srinagesh and Sharma, 2005).
Five linear seismic sources zones (NEZ, SEZ, WSZ, WCF, and WF), defined based on the focal mechanism parameters and fault slip rates (Dixit, 2014; Catchings, 2015) are employed in hazard assessments (Fig. 7).
Seismic source zones of Koyna region (compiled after: Dixit, 2014; Catchings, 2015). NEZ – North escarpment zone, SEZ – South escarpment zone, WCF – West coast fault, WF – Warna fault, WSZ – Warna seismic zone (Fig. 7)
Hazard estimation
Estimation of the tendency of seismicity is derived from the time series analysis of historical and instrumental earthquake catalogue and other local sources. Source characterization is derived from (i) five linear source model — epicentre, fault orientation (or bearing), and type of fault, and (ii) an area source model — epicentre locations and hypocentral depths, for considering the background seismicity. Assessment of the probability of occurrence of various hazard levels, within the next 50 years, from the potential sources is computed from the probabilistic seismic hazard assessment (PSHA) hazard-integral Eq. (2) by assuming that the rate of earthquake occurrence in time is governed by the Poisson law.
PSHA integrates the probabilities of all possible magnitudes (m), all fault-to-site distances (r), rate of seismicity (α) of all the seismic sources, and the ground motion attenuation to produce hazard curves in terms of different level of ground motion and an associated annual frequency of being exceeded for 500 years (10% exceedance in 50 years), 2500 years (2% exceedance in 50 years) of return periods respectively. The attenuation relationship for the stable continental region (Toro et al., 2002) is employed for predicting PGA in terms of acceleration due to gravity is
$$ \ln Y={C}_1+{C}_2\left(M-6\right)+{C}_{3\kern0.5em }{\left(M-6\right)}^2-{C}_{4\kern0.5em }\ln {R}_M-\left({C}_5-{C}_4\right)\max \left[\ln \left(\frac{R_M}{100}\right),0\right]-{C}_{4\kern0.5em }{R}_M+{\varepsilon}_e+{\varepsilon}_a $$
(where) \( \mathrm{RM}=\sqrt{R^2-{C^2}_{7\kern0.5em }} \)
In the above Eq. (1), lnY = PGA, M = magnitude, R = hypocentral distance, εe = epistemic uncertainty, and εa = aleatory uncertainty, C1 to C5 are regression coefficients.
The mean annual rate of PGA level for 10% probability being exceeded in 50 years is calculated by means of the hazard integral Eq. (2) (McGuire, 2004).
$$ {E}_i\left(Y\ge {y}_0\right)={\alpha}_i\underset{R}{\int}\underset{M}{\int }{f}_R(r){f}_M(m)P\left[Y\left(m,r\right)\ge {y}_0\left|m,r\right.\right] dmdr $$
where Y = PGA, y 0 = target PGA, m = magnitude, r = fault-to-site distance (~200 km radius), α = seismicity rate (0.01 for >M5.5 and 0.0025 for > M6.5), Ei (Y > y0) = annual probability of being exceeded for target PGA levels.
The annual rates of PGA-values (10% exceedance probability) are calculated for around 300 point locations which make up the grid of 10 Km X 10 Km covering entire study area and its surroundings. Extractions of characterized parameters are on grid cell basis. Characterized parameters are used to describe the grids such as number of faults, their intersection and maximum and minimum events and their numbers. These grids characterized parameters can be extracted automatically through spatial interpolation. Using ArcGIS® estimated PGA-values gridded data are interpolated by means of geostatistical Kriging method to obtain the spatial distribution of hazard levels.
Figure 8 shows the distribution of hazard levels expressed as peak ground acceleration (PGA) for 10% probability of exceedance in 50 years, contoured from high (>21% g) to low (<16% g) for rock site conditions. Earthquake ground shaking varies from place to place and it depends on the properties of the rocks and weathered or soil column that earthquake waves travel through.
Spatial distribution of estimated PGA (% g) as hazard levels and MMI –VII as damage potentials in 50 years
In order to map the severity of ground shaking, Modified Mercalli Intensity (MMI) is estimated from the mean annual rate of PGA using the PGA – MMI relationship (equation-3) given by Shebastari and Yamazaki (2001):
$$ \ln\ \left(\mathrm{PGA}\right)=0.2545\ \mathrm{MMI}+0.2977 $$
For damage potential estimates, MMI levels are obtained for various return periods (Fig. 8).
In 50 years, the estimate of 10% probability of exceedance reveal that the mean ground acceleration is greater than 0.21 g in the Koyna-Warna region and it is shown as the red zone, whereas PGA ranging from 0.18 g to 0.19 g is expected to experience by the towns located within 50Km radius are: Patan, Satara, Karad, and Chiplun shown in orange to yellow shades. The low accelerations (<0.16 g to 0.17 g) are found to be expected in Kolhapur, Mahabaleshwar, Rantagiri, Khed and their proximities (Fig. 8).
Slightly damageable ground motions are expected from the 100 ± 10 years return period earthquakes (~M5.5). For rock sites conditions, estimated total hazard attributable to ~500 years return period earthquakes is greater than 0.21 g, however at surface level PGA is expected to amplify due to the unconsolidated material which make up soil-regolith formation in the Koyna-Warna region. Figure 8 shows that the expected intensity levels (MMI-VII or lesser) in 50 years expressed as probability (%) for the earthquake scenarios. Damaging intensity (MMI-VII level) expected in 50 years has ~ 40% probability for the area is located between Koyna and Warna Reservoirs.
Seismic risk and damage potential can modify with the occurrence of fresh events. These events throw a new light on the dynamics of seismic sources in the stable continental regions, therefore the necessity of installing earthquake monitoring networks is warranted, which is neglected previously due to its aseismic label, particularly in the other parts of Peninsular India in addition to Koyna.
In this paper, probabilistic seismic hazard is estimated for the 500 years return period earthquake scenarios (M4.0 to M6.9) for bedrock level conditions in the Koyna seismic zone. For hazard estimation, seismic events are collated till early 2017 and total hazard from the five linear sources are presented as spatial distribution of PGA and hazard potentials. The following derived conclusions would give an opportunity for reinforcement options if needed.
The estimated PGA for 10% probability being exceeded in 50 years (i.e. for 500 years return period) ranging from less than 0.16 g to greater than 0.21 g for Koyna-Warna area (radius of 50Km area).
The hazard estimates can the grouped into three relative categories for bedrock level accelerations greater than 0.21 g, 0.19 g, less than 0.16 g defining high, moderate, and low hazard levels, respectively.
Slightly damaging intensity (MMI-VII level) expected in 50 years has ~ 40% probability for the area which has PGA >21% g that is located between Koyna and Warna Reservoirs. There are surface transport alignment with more than 3 m to 5 m of cutting, tunnels and high bridges located in the study area; therefore seismic site response based microzonation is indispensible.
GIS:
MMI:
Modified Mercalli Intensity
PGA:
Peak Ground Acceleration
PSHA:
Probabilistic Seismic Hazard Analysis
RIS:
Reservoir Induced Seismicity
SCR:
Stable Continental Region
SRTM:
Shuttle Radar Topography Mission
WGE:
Western Ghats Escarpment
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We thank Dr. Kiran Kumar Thingbaijam, Postdoctoral Fellow (King Abdullah University of Science and Technology, Thuwal, Saudi Arabia) for assistance with the seismicity modeling, and Omkar Shinde (Civil Engineer, Siddhivinayak Constructions) for assisting in the field-work. We thank "anonymous" reviewers for their suggestions. We are also grateful to Prof. Fawu Wang (Editor-in-Chief, Geoenvironmental Disasters) for his comments on the manuscript. Any persisting errors are our own and should not tarnish the reputations of these reputed persons.
Availability of data and material
Centre of Studies in Resources Engineering (CSRE), Indian Institute of Technology Bombay, Mumbai, 400 076, India
S. M. S. P. Dev
& R. Nagarajan
Search for S. M. S. P. Dev in:
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Both the authors have equal contributions in preparing the manuscript. Both authors read and approved the final version of the manuscript.
Correspondence to S. M. S. P. Dev.
The authors declares that they have no competing interests.
Dev, S.M.S.P., Nagarajan, R. Seismic hazard assessment of Koyna region, Peninsular India: using geospatial approach. Geoenviron Disasters 4, 27 (2017) doi:10.1186/s40677-017-0092-y
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CommonCrawl
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Traditional knowledge and cultural importance of Borassus aethiopum Mart. in Benin: interacting effects of socio-demographic attributes and multi-scale abundance
Kolawolé Valère Salako ORCID: orcid.org/0000-0002-7817-36871,
Francisco Moreira2,3,
Rodrigue Castro Gbedomon1,
Frédéric Tovissodé1,
Achille Ephrem Assogbadjo1,4 &
Romain Lucas Glèlè Kakaï1
Eliciting factors affecting distribution of traditional knowledge (TK) and cultural importance of plant resources is central in ethnobiology. Socio-demographic attributes and ecological apparency hypothesis (EAH) have been widely documented as drivers of TK distribution, but their synergistic effect is poorly documented. Here, we focused on Borassus aethiopum, a socio-economic important agroforestry palm in Africa, analyzing relationships between the number of use-reports and cultural importance on one hand, and informant socio-demographic attributes (age category and gender) on the other hand, considering the EAH at multi-scale contexts. Our hypothesis is that effects of socio-demographic attributes on use-reports and cultural importance are shaped by both local (village level) and regional (chorological region level) apparency of study species. We expected so because distribution of knowledge on a resource in a community correlates to the versatility in the resource utilization but also connections among communities within a region.
Nine hundred ninety-two face-to-face individual semi-structured interviews were conducted in six villages of low versus high local abundance of B. aethiopum spanning three chorological regions (humid, sub-humid and semi-arid) also underlying a gradient of increasing distribution and abundance of B. aethiopum. Number of use-reports and score of importance of uses of B. aethiopum were recorded in six use-categories including medicine, food, handcraft, construction, firewood, and ceremonies and rituals. Data were analyzed using Poisson and ordered logistic models.
Informants listed 121 uses for B. aethiopum: medicine (66 uses), handcraft (16 uses), food (16 uses), construction (12 uses), firewood (6 uses), and ceremonies and rituals (5 uses); but food use was the most culturally important use (2.45 ± 0.03), followed by construction (0.61 ± 0.03), medicinal (0.57 ± 0.03) and handcraft (0.56 ± 0.03), firewood (0.29 ± 0.02), and ceremonies and rituals (0.03 ± 0.01). Food use was the most important for women who were specialized in hypocotyls and fruits collection for commercialization. Men valued more the species for handcrafting, construction, and medicine. The number of use-reports was significantly dependent on age category and gender, and differences between age categories (young, adult, and old) in particular were dependent upon local and regional apparency. In particular, discrepancies among age categories were higher in areas of low abundance and distribution, which may be linked to different speed in the process of knowledge acquisition. In areas of low abundance, the species past abundance was also found instrumental in understanding current knowledge distribution.
Findings suggest that studies aiming at understanding relationship between current TK and cultural importance of a resource on one hand and socio-demographic attributes on the other hand should consider the resource current local and regional apparency but further its local past abundance. The study also confirms that B. aethiopum is a socio-economic important species in Benin.
The overall trend of biodiversity loss [1] and the need to develop effective strategies for its conservation has led to emergence of several paradigms and principles of conservation. One of them, the principle of "conservation through use or trade" has been proposed as a key mechanism to provide incentives for the conservation of species and habitats by turning them into sources of income [2, 3]. The main idea is that conservation is more successful and livelihoods are improved when social and community beliefs and rights are understood and addressed in conservation programs [4]. This has resulted in increasing interests over local communities which in turn have led to a growing interest in the traditional knowledge (TK) they have on their environment [4, 5]. Ethnobotany, which aims at documenting interactions between humans and plants, has therefore become a core subject of conservation biology [6]. The need to better understand factors that determine spatial and intergenerational variation in TK has emerged, and quantitative tools are increasingly being developed to cope with the related issues [6].
Previous studies have shown that knowledge on the use of plant resources and its actual practices are a compounded effect of socio-demographic attributes, including people's gender and age [4, 7, 8] with women and older people tending to have greater knowledge [8, 9]. Historical gender divisions of space and labor in households and societies [10], and increasing knowledge accumulation through time [6], have been often used respectively to explain such patterns.
In addition, the most highly available plants are more likely to be encountered, hence subject to greater experiment and, consequently, broader use and greater local importance [11]. This is referred as the ecological apparency hypothesis (EAH). Although this hypothesis has received mixed support in the literature [12, 13], partly because human culture is more complex to be influenced only by appearance [14], it is still central in ethnobotany [14].
The EAH has been often tested in studies dealing with the compared use of multiple species [15, 16] and rarely used for the study of a single species. In the latter case, the EAH predicts that people living in a landscape of high visibility (abundance) of a species have more knowledge of its uses than people in a landscape of low visibility. However, ecological apparency can be assessed at either local (e.g., village) or larger (e.g., chorological region) scale [17]. At local scale, ecological apparency hypothesis is expected to be a consequence of the direct contact between people and the used resource. However, at the regional scale, the amount of knowledge does not only result from the direct visibility of the species locally but also knowledge exchange between people of different communities/villages trough connectivity and networks. Whether and how local and global visibility of a species affect distribution of knowledge on its uses against age and gender has been poorly investigated. In an area of high abundance of a given resource, we might expect no or little difference of TK between age categories and to some extent between genders because the resource is so common that everyone knows and likely uses it. In contrast, in an area of low abundance, great discrepancies are predicted between both age categories and genders. Understanding such relationship is crucial for conservation biology in general and ethnobiology in particular. For example, this understanding would clarify on whether for documenting TK on a given taxon, sampling area of higher abundance is always better than area of lower abundance. In addition, it will also provide better insights in drivers of knowledge distribution in local communities.
There are several evidences that patterns of plant selection and use by local people, as well as the importance of plants, are driven by complex interactions of biophysical, social, cultural, cultual, political, and economic contexts [14, 18], including resource availability [19]; informant age, gender, and ethnic affiliation [20]; urbanization and informant education level [10, 21]; market importance, nutritive value, and number of complementary uses of species [22]; informant social network [23]; and taboos [24]. As such, TK on use of plants and their importance cannot be simplified to only resource abundance and socio-demographic attributes such as age and gender. The overarching goal of this study was rather to examine whether and how both local and regional apparency of a resource mediate gender- and age-related distribution of traditional knowledge on its uses as well as its cultural importance. To our knowledge, this issue has so far been little explored.
We used the case study of the palm species Borassus aethiopum Mart. in Benin, West-Africa. B. aethiopum is one of the most important wild palm species in West Africa [25]. Although the overall IUCN threat category of B. aethiopum is least concern (LC), the species suffers from local over-exploitation for palm wine and several other threats (http://www.iucnredlist.org/details/195913/0), hence considered threatened across its distribution range [26,27,28,29]. In Benin, B. aethiopum is found in traditional agroforestry systems and natural forests [30] but faces a serious threat of regeneration, and its populations are aging [26]. Based on IUCN Categories and Criteria of threat, B. aethiopum was categorized as "vulnerable" in Benin because of human disturbances on its populations [26].
Although several ethnobotanical reports exist on the species over its distribution range [28, 31,32,33,34,35], they are often narrowed to a locality or a small part of a country [35] and rarely used quantitative approaches to decipher complex interactions between humans and the species. Yet, such understanding could provide important baseline information for its sustainable use and management.
The main research question addressed here is whether (and how) local and regional abundance of B. aethiopum influences the relationships between TK and cultural importance on one hand and socio-demographic attributes (gender and age) on the other hand? We predicted that (1) TK and cultural importance of B. aethiopum increases with increased local and regional apparency, (2) TK and cultural importance of B. aethiopum are influenced by gender and age, and (3) both local and regional apparency influence the relationships between TK and cultural importance of B. aethiopum and socio-demographic attributes, with greater discrepancies between age and gender where the species is less apparent. The study also sought to document the countrywide uses and importance of B. aethiopum in Benin.
The study was conducted in Benin (6° 25′ N–12° 30′ and 0° 45′ E–4° E), West Africa (Fig. 1) [36]. Benin is characterized by three contrasting chorological regions [37], northwards: the Guinean region (humid climate), the Sudan-Guinean region (sub-humid climate), and the Sudanian region (semi-arid climate) regions [38]. The country's native vegetation has suffered severe degradation as a result of various intense anthropogenic activities. The vegetation becomes dominated by woodland and savannah northwards. The resident population in 2013 was nearly 10 million inhabitants, unequally distributed across the territory [39]. The population is mainly young (more than 40% are < 15 years old) and slightly female-biased (51.2%) [39]. The local economy is agriculture-based [39]. More than half the population (53.9%) live with less than 1 US dollars per day [40]. The average size of farmland per farmer is 1.7 ha, and more than 1/3 had less than 1 ha [41]. Regardless of socio-cultural groups, women have no or limited access to land [41].
Map of the republic of Benin showing the three chorological regions and the study villages
Study species
The African fan palm, B. aethiopum (Fig. 2), belongs to family Arecaceae, subfamily Coryphoideae [42]. It is widespread and common across sub-Saharan Africa where it is a well-known and conspicuous component of savannas. B. aethiopum has a large and straight stem to 25 m tall and may reach 80-cm diameter. The fruits are massive, ovoid, and orange at maturity (Fig. 2). The mesocarp is pulpy and fragrant with many longitudinal fibers [25]. In Benin, B. aethiopum is found in all three chorological regions but with great differences in regard with distribution and abundance [30]. Overall, B. aethiopum becomes common with higher abundance northwards, but in all three regions, there exist areas of local low and high abundance [30].
A female tree of B. aethiopum showing its fruits (white circle) in Northern Benin. Credits to Salako et al. [30]
Sampling and data collection
In Benin, B. aethiopum has a wider distribution and larger abundance (trees ha−1) in the semi-arid region (24.54 ± 1.51) than in the sub-humid region (16.24 ± 1.79), and than in the humid region (7.06 ± 0.56) [30]. However, these chorological region patterns hide strong local differences within each region, where it is possible to identify areas with low and high local abundance. Based on previous studies [30], we therefore identified villages with high (> 20 adult trees ha−1) and low (up to 10 adult trees ha−1) local abundance of B. aethiopum. In each chorological region, we randomly selected one village with high abundance and one with low abundance (Fig. 1). Abundance at chorological regions was assumed as an indicator of the regional apparency, while abundance at village level represented local apparency. Following [43], three age categories were considered (age < 30 years for young; 30 ≤ age < 60 years for adults; age ≥ 60 years for old persons). Gender (men and women) was also considered. Primarily, it was planned to select in each village 30 informants for each combination of gender and age category, making a total of 1080 informants. But because of insufficient informants in some villages, 992 informants participated to the work (Additional file 1). Data were collected using face-to-face individual interviews based on a questionnaire (Additional file 2). The goal of the study in terms of gathering knowledge on B. aethiopum for a PhD research (this study was part of the PhD research of the first author on the conservation biology of B. aethiopum in Benin) was first explained to village authorities and next to each of the informant as to have their consent for participation before starting the interviews. Only individuals that consented to participate in the study were considered. Interviews were conducted with assistance of a local translator when necessary. The questionnaire comprised three main sections. The first was related to informant's socio-demographic information (age, gender, and ethnic affiliations). The second consisted of a free-listing of the use-reports of B. aethiopum, defined as each specific use mentioned by the informant per plant part in the sense of [44]. For example if an informant mentioned use of fruits to cure malaria and the use of leaves to also cure malaria separately, we considered them as two separate use-reports. Use-reports were arranged per plant part in six use-categories adapted from [45]: food, handcraft, construction, firewood, medicine, and ceremonies and rituals. The use-report does not distinguish between "knowledge" and "real use", as various potential uses may be known but real use may be different [46]. To account for that, the third section of the questionnaire focused on the real use of the species by asking the informant to score the six use-categories based on the importance of their actual uses. The score varied from 3 ("high use") to 0 ("not used") with score 1 for "medium use" and score 1 for "low use" [7]. Informants were additionally asked to rank each plant part based on the importance of their actual uses. Each interview lasted between 2 h and 2 h 30 min.
Knowledge on the uses of B. aethiopum
First, for each use-report of B. aethiopum, the relative frequency of citation defined as how often a use-report was mentioned was calculated using the fidelity level (FL) [47]. Only significant use-report (with FL > 5%) were reported here.
$$ \mathrm{FL}\left(\%\right)=100\times x/n $$
where x is the number of informants who mentioned a specific use and n is the total number of informants.
Knowledge on uses of B. aethiopum was measured using the relative use-value (UV) [7] which is a modified version of the use-value method introduced by [11]. This modified version of UV allows capturing of all the known uses by an individual within and between use-categories [7]:
$$ \mathrm{UV}=\sum \limits_{\mathrm{uc}=1}^{n_{\mathrm{uc}}}{\mathrm{UV}}_{\mathrm{uc}}=\sum \limits_{\mathrm{uc}=1}^{n_{\mathrm{uc}}}\sum \limits_{i=1}^n{\mathrm{UR}}_{\mathrm{uc},i}/n $$
where URuc, i is the number of use-report mentioned by informant i in for a given use-category uc. In our dataset, URuc, i varied from 0 to maximum (URuc,i) = 6, meaning that the maximum number of use-report mentioned by an informant in a use-category was 6. UVuc is the use-value for a given use-category uc which is the mean of URuc, i for that use category; nuc is the number of use-categories in the study (nuc = 6); n is the number of informants.
UV stands as a mean of URuc,i and could vary from 0 to n × nuc × 6 (in case all informants mentioned all use-categories and that all informants cited a number of use-report equals to 6 in each use-category). Because URuc,i is a count data, a generalized linear model (GLM) with Poisson error distribution [48] was used to assess variation of UV (response variable) with respect to region, local abundance, age category, and gender of informants (predictors). All predictors were categorical with respectively three (humid, sub-humid, semi-arid), two (high, low), three (young, adult, old), and two (women, men) levels. Interaction terms in the model included (i) interaction of region with age category on one hand and gender on the other hand, (ii) interaction of local abundance with age category on one hand and gender on the other hand, and (iii) interaction of age category with gender. Non-significant terms were sequentially removed from the model. Likelihood ratio test was used to assess the goodness of fit of the final model. The deviance-based pseudo-R2 was also computed to assess the explanatory quality of the final model.
Cultural importance of B. aethiopum
The cultural importance of B. aethiopum was assessed using the importance index (IP) adapted from [49]:
$$ \mathrm{IP}=\sum \limits_{\mathrm{uc}=1}^{n_{\mathrm{uc}}}{\mathrm{IP}}_{\mathrm{uc}}=\sum \limits_{\mathrm{uc}=1}^{n_{\mathrm{uc}}}\sum \limits_{i=1}^n{\mathrm{S}}_{i,\mathrm{uc}}/n $$
Si,uc is the score of importance attributed by informant i (i = 1,…, n) for the use-category uc; nuc is the number of use-category (nuc = 6). IP is the overall importance value of B. aethiopum and IPuc, the importance value of the use-category uc of B. aethiopum.
Values of IP and IPuc vary from 0 to 18 (in case all informants scored all six use-categories as "high use") and 0 to 3 (in the case all informants scored the use-category uc as "high use") respectively, with higher values indicating higher cultural importance. Ordered logistic models were used to model the effect of age category, gender, local abundance, and region on the variation in the use importance score of B. aethiopum. Backward elimination as described in [50] was used to select the most parsimonious models.
All statistical analyses were performed in the R software v.3.3.2 [51]. Package fmsb [52] was used to compute Naglekerke's pseudo-R2. Ordered logistic models were run using the function clm2 within the package "ordinal" [53].
Diversity of uses
Countrywide, 121 different use-reports of B. aethiopum were recorded as follows: medicine (66), handcraft (16), food (16), construction (12), firewood (6), and ceremonies and rituals (5). However, only 28 of all those use-reports were found to be significant (Fidelity Level > 5%) (Table 1). The significant use-reports included 9 food uses (human), 6 construction uses, 5 medicinal uses, 5 handcraft uses, 2 firewood uses, and 1 ceremonies and rituals' use indicating that consensus was high on food uses than the other uses, medicinal uses being the use-category where the least consensus was observed (only 5 significant uses out of 66 uses) (Table 1).
Table 1 Significant use-report of B. aethiopum per plant part and use category: processing method, forms of use, purpose of use and fidelity level (FL) per chorological region (Hu = Humid, Sub-hu = Sub-humid, Sem = Semi-arid) with illustrations on Fig. 3. Only uses with FL ≥ 5% in at least one region are displayed
The number of significant use-reports per plant part was higher for fruits (6 uses) and leaves (6 uses). Stem was involved in four uses while seeds, hypocotyles, and petioles were each involved in three significant use-reports. Roots had only two significant use-reports (Table 1).
Uses of plant parts of B. aethiopum (see Fig. 3) varied across regions. Fruits and leaves had multiple forms of use, some uses being reported by more than 50% of the informants. These two plant parts in addition to hypocotyl were involved in uses with the highest fidelity level, up to 86% in some regions. Seed uses were relatively not common, with fidelity level less than 15%. Use of root was mentioned only in the humid and sub-humid regions where it is used either to treat malaria or to strengthen children (Table 1). Processing methods were also not similar across regions (Table 1). For example, ripe fruits are toasted (Fig. 3b, c) only in the semi-arid region, while boiled (Fig. 3d) in the humid region before consumption (Table 1). In the semi-arid region, fruits are often boiled in association with cereals (either maize or millet or sorghum) (see Fig. 3e). The hypocotyles were consumed either boiled (Fig. 3g–i) (in all regions) or toasted (only in the semi-arid region) (Table 1). The use of fruits to discard shrews and snakes from homestead or chicken coop was essentially reported in the sub-humid region. There was no mention of this use in the semi-arid region. Many handcraft products (e.g., fan, mat, hat, sponge; see Fig. 3l–q) are made from leaves and petioles of B. aethiopum. It is worth noticing that other plant parts including bark and flowers of male and female trees were involved in medicinal and ceremonies and rituals uses that were not significant (FL < 5%), hence not presented here.
Illustrations of some use-reports of B. aethiopum in Benin. a Fruits with removed flesh. b, c Toasted fruits. d, e Boiling the ripen fruits in water with corn (maize, rice or millet). f Almond after germination. g Freshly harvested hypocotyls. h Boiled and packaged hypocotyls ready for sale. i Boiled hypocotyls not cut yet. j Fresh hypocotyls cut and put in palm alcohol. k Solid potash from incinerated seed hulls. l An old man making sieve. m Samples of sieves. n Fans made from leaves. o Sponge made from petioles. p Battledore from petioles. q Gate from petioles. r Implement made of leaves for ceremonies in Berba region. s A farmer logging a male tree. t Stem used in construction. u Canoe made from stem. v Seat made from stem at public places. w Fruits sowed on farm for hypocotyls production. x Fruits sowed at home for hypocotyls production. y Petioles stored for firewood. z Soap "koto" made from seeds hull. Credits to Salako et al. [30]
Based on the ranking of plant parts with respect to their importance for informants, fruit was ranked first and was followed by hypocotyl (Fig. 4) due to their food uses and commercial value. These two plant parts were the most sold on the local market either in rural or urban areas mainly by women and children. Fruits are collected from the wild. Hypocotyles are either collected from the wild or harvested from fruits sown on farm or at homestead (Fig. 3w–x). Apart from fruits and hypocotyles, the following plant parts were successively leaves, petioles, and stem (Fig. 4).
Average actual use rank of plant parts of B. aethiopum
Use-value of B. aethiopum: effect of region, local abundance, gender, age category, and their interactions
Main effect of local and regional abundance, gender, and age category on overall and by use-category UV
There was a significant (p < 0.05) relationship between overall knowledge of B. aethiopum uses and age category, gender, local abundance and region, either as main effect or in a significant interaction term (Table 2). Informants in the drier regions have more knowledge on the species use. Informants in the humid region reported less uses (mean ± standard error; 3.21 ± 0.08) than in the sub-humid (5.32 ± 0.14) and semi-arid (4.07 ± 0.11) regions. The younger the interviewee is, the lesser he has knowledge on the species uses. Young informants reported less uses (3.80 ± 0.11) than adults and old informants who reported similar number of uses, 4.36 ± 0.11 and 4.24 ± 0.14, respectively. Men (4.31 ± 0.10) were more knowledgeable than women (3.98 ± 0.10). Informants from areas with high local abundance reported more uses (4.19 ± 0.09) than informants from areas with low local abundance (4.09 ± 0.10).
Table 2 Predictors in the best Poisson and ordered logistic models showing the relationship between socio-demographic attributes (gender and age category), regional (Region) and local abundance, with use-value and importance index, respectively for B. aethiopum (−, non-significant term)
Knowledge on the uses of B. aethiopum varied greatly across use-categories (Fig. 5). Irrespective of the examined factors, knowledge was higher for food use (1.87 ± 0.04) followed successively by handcraft (0.72 ± 0.03), construction (0.60 ± 0.03), medicinal (0.41 ± 0.02), firewood (0.36 ± 0.02), and ceremonies and rituals uses (0.18 ± 0.01) (Fig. 5). Knowledge on food use was higher in the semi-arid region than in the sub-humid and humid regions that had similar knowledge (Fig. 5a). Knowledge on medicinal, construction, and handcraft uses were higher in the sub-humid region than in the other regions (Fig. 5a). Knowledge on food use was higher in areas with low local abundance while higher in areas with high local abundance for handcraft and ceremonies and rituals uses (Fig. 5b). Regarding gender, men always reported more knowledge than women irrespective of the use-category except for firewood use where the UV was higher for women (Fig. 5c). With respect to age category, knowledge was always lower for young than adults and old informants who had similar knowledge, except that for the food uses, the differences were relatively narrower than for the other use-categories (Fig. 5d).
Radar chart showing the main effect of region (a), local abundance (b), gender (c), and age category (d) on the UV of B. aethiopum across use-categories
Interacting effect of region and local abundance with socio-demographic attributes
Among socio-demographic attributes, only age category was involved in significant interactions with local abundance (Table 2). Young, adult, and old informants had similar knowledge in areas with high local abundance while greater differences were observed in areas with low local abundance, in particular between young and both adult and old informants who had similar knowledge (Fig. 6).
Interacting effect of age category and local abundance on the UV of B. aethiopum
Main effect of region, local abundance, gender, and age category on the overall and by use-category importance
There was a significant (p < 0.05) relationship between overall importance (IP) of B. aethiopum and region, local abundance, gender and age category; either as main effect or in interacting effect (Table 2). B. aethiopum was more important (higher IP) in the sub-humid region (mean ± standard error; 5.65 ± 0.08) than in the semi-arid (4.54 ± 0.08) and humid (3.51 ± 0.13) regions; slightly more important for men (4.52 ± 0.10) than for women (4.50 ± 0.09); more important in areas of high local abundance (4.55 ± 0.08) than in area of low local abundance (4.46 ± 0.10); and more important for adults (4.71 ± 0.10) and young (4.44 ± 0.12) than old informants (4.34 ± 0.12).
The IP varied greatly among use-categories and for each of the examined predictors (Fig. 7). Irrespective of the predictors, food use was the most important (2.45 ± 0.03) followed by construction (0.61 ± 0.03), handcraft (0.57 ± 0.03), and medicinal (0.56 ± 0.03) uses and then firewood (0.29 ± 0.02) and ceremonies and rituals uses (0.03 ± 0.01) (Fig. 7). Considering region, food use was more important in the semi-arid and sub-humid regions than in the humid region (Fig. 7a). Medicinal use was more important in the sub-humid region while the firewood use was roughly not important there (Fig. 7a). Considering local abundance, food and medicinal uses were more important in areas with high local abundance than in areas with low local abundance (Fig. 7b). Handcraft and construction uses were more important in areas with low local abundance than in areas with high local abundance (Fig. 7b). With respect to gender, food and firewood uses were more important for women than men while handcraft use was more important for men than women (Fig. 7c). Regarding age categories, food use was more important for young and adult than old informants. Contrary to food use, medicinal and construction uses were more important for adults and old informants than young (Fig. 7d).
Radar chart showing the main effect of region (a), local abundance (b), gender (c), and age category (d) on the IP of B. aethiopum across use-categories
There was no significant interaction involving local abundance and either socio-demographic attributes. However, significant effect was observed for the interaction of region and age category for the overall importance of B. aethiopum (p = 0.036; Table 2). B. aethiopum was more important for young and adults in the sub-humid and semi-arid regions while less important for young in the humid region (Fig. 8a). There was also significant interaction of gender and age categories (p = 0.047; Table 2). Accordingly, B. aethiopum was relatively more important for young women than young men, similarly important for adult men and women while relatively more important for old men than old women (Fig. 8b).
Interacting effect of region and age category (a) and gender and age category (b) on the IP of B. aethiopum
This study reported on the relationships between traditional knowledge (TK) and cultural importance (CI) of B. aethiopum on one hand, and informant socio-demographic attributes (age categories and gender) on the other hand, considering the ecological apparency hypothesis (EAH) at both local and regional scale. It was found that TK and CI of B. aethiopum varied greatly across use-categories: first, food use and then successively handcraft, construction, medicinal, firewood, and ceremonies and rituals uses (1). Significant difference was also observed among age categories, younger informants reporting less uses than adults and old informants who reported similar number of uses (2). Men reported more uses than women (3). Local abundance had significant effect on TK and CI: informants in areas of higher abundance reported more uses and higher score than informants in areas of lower abundance (4). Also, regional abundance determined TK and CI, region of low abundance (humid region) reporting less uses and lower score than region of larger abundance (sub-humid and semi-arid regions) (5). It was also found that local abundance influences relationships between TK (respectively CI) and age categories: young, adult, and old informants having similar knowledge in areas of high local abundance while greater differences were observed in areas of low local abundance in particular between young and both adult and old who had similar knowledge (6).
Interacting effect of age categories, gender, local abundance, and region as drivers of TK
Our data support the general trend that TK depends on age categories and gender [4, 54, 55]. Young informants often reported less uses than adults and old informants, hence congruent with the assumption that TK is a time-dependent process of learning [6]. Therefore, informants from older age category, having spent a longer time with their natural environment, would normally have more knowledge than informants from younger age category [4]. Although women are reputed to have more contacts with NTFPs than men [56], the fact that men often have more knowledge than women in our case study may be because women are often specialized in some uses of a species, most often food and to some extent medicinal uses while men uses expand to other uses (e.g., construction, ceremonies and rituals). This utilization pattern of resources is often dictated by differences in activities and roles of men and women within households [54] and sometimes to cultural taboos or prohibitions [7].
The EAH has multiple implications in conservation biology. For example, a positive relationship between species visibility and their use imply gradual elimination of the more apparent species by predatory collection through the constant pressure for domestic/commercial use [17]. The finding that TK was globally higher in areas of higher apparency (either local or regional) supports the ecological apparency hypothesis. Previous studies using abundance as a quantitative predictor of UV have come to similar conclusions [57, 58]. However, others found either no link or at best a weak relationship [15, 16]. Therefore, the EAH would not always explain pattern of uses of species, and other aspects such as socio-cultural, economic, and political aspects should be accounted for [14] and may explain the failure of our model to capture about 60% of the variation in TK. Indeed, human decision processes are complex and cannot be reduced to only ecological or economic considerations; other factors such as social and cultural are important in the processes involved in choosing and harvesting a plant [59].
Informants in drier regions reported more uses than informants in the wettest region which is congruent with the EAH prediction at the regional scale following the regional pattern of B. aethiopum across the country: B. aethiopum abundance and distribution is proportional to dryness, increasing northwards (from the wettest to the driest region) [30]. In Benin, previous studies have shown that diversity of wild edible plant species and plant in home gardens declines towards the semi-arid region [60,61,62], suggesting a likely higher intensity of use on a narrow number of species when the climate becomes drier which may in turn results in more knowledge on the resource at least as food use is concerned. At a local scale, the EAH also overall proved true: the higher the local abundance, the higher the knowledge informants have on the species. However, looking at patterns within each region (not statistically tested because of lack of replicates), this seems to not be the case in the semi-arid region where informants in the village of low local abundance reported up to two times more uses than informants in the village of high local abundance (low abundance: 5.50 ± 0.14 use-reports, high abundance: 2.89 ± 0.09 use-reports), compared to the sub-humid (low abundance: 3.70 ± 0.21 use-reports, high abundance: 6.49 ± 0.13 use-reports) and humid regions (low abundance: 3.24 ± 0.14 use-reports, high abundance: 3.19 ± 0.10 use-reports). As such, differences in TK due local apparency could be related to regional apparency. This pattern suggests two hypotheses. First, in the village of low local abundance in the semi-arid region, B. aethiopum was abundant in a recent past and likely has undergone rapid decline as reported by informants in that village (Additional file 3). This thus raises the importance of understanding the species past abundance in understanding current patterns of knowledge people have on them. Second, in the village of high abundance, the spread of uses of B. aethiopum is narrowed because other species fulfill the role of B. aethiopum observed in the village of low local abundance as suggested by the diversification hypothesis. The diversification hypothesis considers that the presence of multiple species in an environment amplifies the spectrum of alternatives [63] for subsistence, health, and livelihoods in general, thus reducing the pressure (use) on a single species, hence knowledge of its use. Apart from the regional apparency, one additional factor that may explain between regions differences for a given local abundance (e.g., considering villages of high abundance) is differences in ethnic affiliations [61, 64, 65]. Different ethnic groups have often different life style, beliefs, and perceptions of their environment that translate in different knowledge of resources of their environment [64] as previously reported for many other species in Benin [65,66,67].
This study also provides empirical evidence that pattern of knowledge distribution across age categories depends on local and regional apparency of the studied resources as we predicted. The most common (locally or regionally) a species is, the most likely knowledge on its utilization is similar across age categories. In contrast, the less common a species is, the relatively greater is the gap of knowledge among age categories, especially between young on one hand and adult and old people on the other hand. While knowledge acquisition as time-dependent process [6] is straightforward in explaining the often lower knowledge of younger informants as compared to older informants, local or regional apparency may provide interesting potential explanation of the magnitude of the gap of knowledge among age categories. Our proposition is that higher local or regional apparency seems to speed the process of knowledge acquisition resulting in similar knowledge among age categories. At the opposite side, lower local or regional apparency slow the process of knowledge acquisition and therefore result in greater discrepancies among age categories, in particular between younger and older informants. Additional studies on others species would be needed to clarify this proposition.
Understanding the cultural importance of plant resource is crucial for an informed management [7]. Differences in form of uses across regions (see Table 1) are mostly linked to cultural differences due to different ethnic affiliations. For example, Gourmantché people from the semi-arid region toast the fruit before consumption. Such use was not mentioned in the other regions.
Most culturally important uses of B. aethiopum in study regions were successively food, handcraft, construction, and medicine. These use-categories are also the most known and important for palm species in Latin America [68, 69]. Irrespective of regions, local abundance, gender, and age categories, food use was the most culturally important, clearly indicating that B. aethiopum is primarily a food palm species in Benin, particularly in the sub-humid and semi-arid region. This corroborates Assogbadjo et al. [60] who reported B. aethiopum as a priority wild edible tree species in these two regions.
The finding that food uses was more important for women than men confirms the previous hypothesis of women specialization in the food use category but also stress on the relationship between patterns of plant uses and activities/roles in African households [54]: women are responsible for kitchen and most often are the sellers of food products in markets. This is further confirmed by the greater value of importance index for use-categories handcraft, construction and medicinal for men than women since culturally, men are often responsible for constructions and health care of the household members [70].
The fact that fruits and hypocotyles were the most important plant parts added to its greater commercial value are an additional evidence that B. aethiopum is a food palm species. Moreover, the greater commercial value of these two plant parts for women than men confirms B. aethiopum as a "women-palm" species (Fig. 9) as also reported in Brong Ahafo region in Ghana [35]. However, this great cultural and commercial value of fruits and hypocotyls if not well controlled may reduce regeneration potential in natural stands of B. aethiopum, threatening its population rejuvenation as reported for other species, e.g., Pentadesma butyracea [71].
An old women and children on the way back to home from collection of B. aethiopum fruits in northern Benin. Credits to Salako et al. [30]
Leaves were the second plant part with the highest number of significant use-report and the third most important plant part. They were often harvested from saplings and juveniles because adults are often taller (up to 20 m, see) and it is not easy to climb [30]. As reported in other studies [25, 72], stipe of B. aethiopum is a much appreciated material for construction and likely explain why it was so culturally important in the study regions. For example, roughly all houses in the village Loumbou-loumbou are made of B. aethiopum stipe (Salako, field observations; Fig. 3t). Gamba Begounou, an 83-year-old women, said "B. aethiopum is the only one good tree for house construction here. It is very resistant and can lives more than two hundred years".
Medicinal use of B. aethiopum was also culturally important for surveyed informants and aligned with [35] who reported its medicinal use in 14 out of 28 African countries where it occurs and as the third most used palm species in traditional medicines in Africa. However, in spite of having four times more uses than food use-category (66 versus 16), only 5 were significant, suggesting a lack of consensus on most of the medicinal uses. This is likely due to the fact that the medicinal uses were mostly ethnic-specific. The common use of fruits against malaria also consensually reported in this study has been recently confirmed by pharmacological prospects [35].
Surprisingly, the use of sap commonly reported for B. aethiopum in other countries (e.g., Cote d'Ivoire, Senegal and Guinea; see [28, 32, 33]) was not mentioned in our study. In the humid region, this may be due to the fact that people reputed for palm wine extraction preferred Elaeis guineensis which is the ancestral source of palm wine in Benin [73]. In the sub-humid and semi-arid regions where E. guineensis is roughly absent, this may be due to (but not limited to) the fact that palm wine extraction is not their habit and that they do not have such knowledge. However, the use of fruits to discard shrews and snakes appears as a "new" reported use for B. aethiopum. This property may have potential for biological control of pests and hence required phytochemical screening prospection.
Dearth of information on traditional knowledge and cultural importance of species has been implicated for their non-sustainable utilization. This study confirms that traditional knowledge is closely linked to gender and age but provides additional evidence that this relationship is further influenced by local and regional apparency of the resource: greater discrepancies between younger and older informants in areas of lower apparency. We propose that this is linked to the speed of knowledge acquisition which we postulate is lower in areas of lower apparency. Therefore, study reporting on knowledge distribution among age categories should account for the local and regional availability of the study resources in explaining the observed patterns and further shade this by the past abundance of the resource in the study environment. Additional studies on others species are needed to clarify this proposition. This study also showed the paramount local importance of B. aethiopum in Benin in particular for people of the sub-humid and semi-arid regions, providing them with fundamental good and services (food, medicine and materials for house construction) with a high potential to generate cash income for women. B. aethiopum is therefore a particularly important tree species which deserves more attention than it is currently given. From a management perspective, women should be trained for good practices of fruits and hypocotyls collection to avoid overutilization. From a domestication perspective, further studies should with priority focus on fruits and hypocotyls. As a first step, traditional classification will provide good insights, and because of their specialization, women could provide valuable knowledge. Women also should be of particularly interest when selecting "plus trees" for desired traits in fruits and hypocotyls.
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The authors are very grateful to the local people for kindly sharing their precious knowledge.
This research was supported by the International Foundation for Science, Stockholm, Sweden, through a research grant to VKS (no. D/5448–1). Additional funding was obtained from a PhD research fellow of the University of Abomey-Calavi under the project "WILD-PALM" also attributed to VKS. FM was funded by FCT (IF/01053/2015).
The datasets used and/or analyzed in the current study are available from the corresponding author on reasonable request.
Laboratoire de Biomathématiques et d'Estimation Forestières, Faculté des Sciences Agronomiques, Université d'Abomey-Calavi, 04 BP 1525, Cotonou, Bénin
Kolawolé Valère Salako
, Rodrigue Castro Gbedomon
, Frédéric Tovissodé
, Achille Ephrem Assogbadjo
& Romain Lucas Glèlè Kakaï
REN Biodiversity Chair, CIBIO/InBIO – Centro de Investigação em Biodiversidade e Recursos Genéticos, Universidade do Porto, Campus Agrário de Vairão, 4485-601, Vairão, Portugal
CEABN/InBIO – Centro de Ecologia Aplicada "Professor Baeta Neves", Instituto Superior de Agronomia, Universidade de Lisboa, Tapada da Ajuda, 1349-017, Lisbon, Portugal
Laboratoire d'Ecologie Appliquée, Faculté des Sciences Agronomiques, Université d'Abomey-Calavi, 03 BP 1974, Cotonou, Bénin
Achille Ephrem Assogbadjo
Search for Kolawolé Valère Salako in:
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SVK conceived the work with advices from AAE, GKR, and MF. SVK collected the data with assistance of RCG. SVK and TF processed the data with contribution of RCG. SVK drafted the manuscript with contribution of MF and RCG. All authors read and approved the final manuscript.
Correspondence to Kolawolé Valère Salako.
Individual consent to participate in the study was obtained prior to implementing the questionnaire. Only individuals that consented to participate in the study were considered.
Written informed consent for publication was obtained for photographs related to individual persons in Figs. 3 and 9. A copy of each consent form is available for review by the Editor of this journal.
Socio-demographic attributes (ethnic group, age category and gender) of informants and local names of B. aethiopum. (DOCX 13 kb)
Questionnaire for assessing use-value and cultural importance of B. aethiopum. (DOCX 15 kb)
Local perception on the dynamic of B. aethiopum. (DOCX 28 kb)
Salako, K.V., Moreira, F., Gbedomon, R.C. et al. Traditional knowledge and cultural importance of Borassus aethiopum Mart. in Benin: interacting effects of socio-demographic attributes and multi-scale abundance. J Ethnobiology Ethnomedicine 14, 36 (2018) doi:10.1186/s13002-018-0233-8
Received: 03 December 2017
Knowledge distribution
Wild palm
Borassus aethiopum Mart.
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CommonCrawl
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\begin{definition}[Definition:Complete Set of Events]
Let $I$ be an indexing set.
Let $\family {A_i}_{i \mathop \in I}$ be a family of events in a probability space indexed by $I$.
$\family {A_i}_{i \mathop \in I}$ is a '''complete set of events''' {{iff}}:
:$\ds \map \Pr {\bigcup_{i \mathop \in I} A_i} = 1$
\end{definition}
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ProofWiki
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Journal Home About Issues in Progress Current Issue All Issues Early Posting Feature Issues
Issue 22,
pp. 6414-6421
•https://doi.org/10.1364/AO.431490
Estimation of nominal ocular hazard distance and nominal ocular dazzle distance for multibeam laser radiation
Jaroslaw Mlynczak, Krzysztof Kopczynski, Miron Kaliszewski, and Maksymilian Wlodarski
Jaroslaw Mlynczak,* Krzysztof Kopczynski, Miron Kaliszewski, and Maksymilian Wlodarski
Military University of Technology, Institute of Optoelectronics, Gen. S. Kaliskiego 2, 00-908 Warszawa, Poland
*Corresponding author: [email protected]
Jaroslaw Mlynczak https://orcid.org/0000-0002-0823-9302
J Mlynczak
K Kopczynski
M Kaliszewski
M Wlodarski
Jaroslaw Mlynczak, Krzysztof Kopczynski, Miron Kaliszewski, and Maksymilian Wlodarski, "Estimation of nominal ocular hazard distance and nominal ocular dazzle distance for multibeam laser radiation," Appl. Opt. 60, 6414-6421 (2021)
Nominal ocular dazzle distance (NODD)
Craig A. Williamson, et al.
Appl. Opt. 54(7) 1564-1572 (2015)
Wavelength and ambient luminance dependence of laser eye dazzle
Appl. Opt. 56(29) 8135-8147 (2017)
Impact of windscreen scatter on laser eye dazzle
Opt. Express 26(21) 27033-27057 (2018)
Table of Contents Category
Lasers, Optical Amplifiers, and Laser Optics
Green lasers
Laser beams
Laser sources
Original Manuscript: May 14, 2021
Revised Manuscript: June 18, 2021
Manuscript Accepted: June 24, 2021
DERIVING OF NOHD FOR MULTIBEAM RADIATION
DERIVING OF NODD FOR MULTIBEAM RADIATION
ANALYSIS OF NOHD AND NODD FOR THREE RADIATION BEAMS
ANALYSIS OF NOHD AND NODD FOR COMMERCIAL RADIATION SOURCES WITH POTENTIAL APPLICATION TO LASER DAZZLERS
References and links
This paper presents a method of estimation of the nominal ocular hazard distance (NOHD) and the nominal ocular dazzle distance (NODD) for multibeam laser radiation. For the analysis, laser beams propagating in the same optical path (overlapping) but with different wavelength, power, and divergences in two perpendicular planes were assumed. To the authors' best knowledge, such a comprehensive analysis of multiple beams, considering the above parameters, is being presented for the first time. The dazzling possibilities described thus far assumed a single beam of radiation with a circular cross-section. This article also presents the calculation results of the NOHD and the NODD values for three laser beams with wavelengths in the red, green, and blue radiation spectrum with assumed parameters. Similar calculations were also made for a commercial laser source with potential use for laser dazzling. The presented analysis did not take into account the attenuation of radiation by the atmosphere. Moreover, the study provides recommendations on how to design effective, but safe, multiwavelength laser dazzlers.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Jaroslaw Mlynczak, Krzysztof Kopczynski, Miron Kaliszewski, and Maksymilian Wlodarski, "Estimation of nominal ocular hazard distance and nominal ocular dazzle distance for multibeam laser radiation: publisher's note," Appl. Opt. 60, 6849-6849 (2021)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-60-23-6849
Death of the enemy should be a last resort in military or preventive actions carried out by the military or other state services. If possible, the opponent should be overpowered and kept alive. One of the ways to neutralize an opponent in this way is to induce a glare effect on him, which disables his ability to carry out his tasks effectively. Thus, dazzling allows one to disrupt the opponent's activity and, as a result, overpower him while keeping him alive. This makes it possible to minimize fatalities, protect the innocent, and limit collateral damage [1]. There are already available many commercial laser dazzlers mainly emitting radiation at the green wavelength. However, there are also devices emitting violet, blue, or red wavelengths that can be used as laser dazzlers [2]. The available dazzlers can be handheld or vehicle-mountable and can be used not only to dazzle people but also birds to scare them away from airport areas [3].
When analyzing the possibilities of dazzling caused by laser radiation, one cannot forget about the potential damage to the human eye. There is an increasing amount of news information and scientific articles about the reckless use of laser radiation emitted mainly by laser pointers, which are becoming increasingly available on the market. Currently, the so-called laser pointers emitting green, red, or blue radiation with a power of up to several W can be easily purchased using the Internet [4]. Such devices are especially dangerous when in the hands of children who, unknowingly, may hurt themselves and others. An extensive review of the literature on this subject is presented in [5]. Even toys for children that emit laser radiation do not necessarily meet the safety requirements [6]. The increasing number of incidents of illuminating airplanes with lasers and cases of damage to the eyesight of pilots should also be mentioned [7]. Such situations are particularly dangerous because they may lead to air crashes and death. The potential for eye damage has been already defined in the relevant standards [8,9].
One of the basic parameters of a dazzler is the distance range, within which it can produce a glare effect. In the case of a divergent laser beam (the most common situation), this range extends from the radiation source to the nominal ocular dazzle distance (NODD) [10]. In addition to the dazzling range, one should also bear in mind the distance range where the eye damage may occur, from the radiation source to the nominal ocular hazard distance (NOHD). The knowledge of the two mentioned parameters allows potentially safe use of a dazzler, which can cause glare without damaging the human eye. Their calculation for a single beam has already been described in the literature [11].
It is relatively easy to counteract dazzling caused by one expected wavelength. This can be done, for example, by using appropriate attenuating filters. Such filters suppress the radiation responsible for the glare effect and at the same time allows the remaining radiation in the visible range to pass. This residual radiation must be sufficient to maintain an adequate level of vision so that certain activities can still be performed. There are commercially available laser protection spectacles that can attenuate the green, blue, and near-infrared radiation [12]; however, they still transmit some radiation mainly in the red wavelength range. Thus, the chance to dazzle an adversary who wears them still exist. Moreover, to limit the possibilities of counteracting glare, several radiation beams can be used in different wavelength ranges. By using three beams in the red, green, and blue band of radiation, potential filters to suppress this radiation would also significantly reduce the level of vision, effectively preventing a person using such filters from performing properly. Therefore, it becomes reasonable to use many beams of radiation for dazzling. Thus, there is a need to develop a simple model for the analysis of multibeam radiation with different wavelengths in the band of visible light. The paper presents a simple way to determine the NODD and the NOHD for such radiation. Laser beams with homogeneous and constant power distribution in the cross-section of a beam, the so-called "top-hat," propagating in the same optical path, with different power, different wavelengths, and different divergences in two perpendicular planes were considered for the analysis. The presented analysis did not take into account the attenuation of radiation by the atmosphere.
2. DERIVING OF NOHD FOR MULTIBEAM RADIATION
The nominal ocular hazard distance (NOHD) is defined as the distance from a radiation source at which eye exposure to laser radiation equals the maximum permissible exposure (MPE). The MPE has been defined in the relevant standards [8,9] based on research and case studies, for various ranges of wavelength and exposure times. Thus, determination of the NOHD for a single laser beam with a defined wavelength is a simple task. For the given beam power $P$, the divergence $\emptyset$ (in the case of a beam with circular cross-section), and the beam width at the output aperture $w$, it can be expressed using the following equation:
(1)$${\rm NOHD} = \frac{{\sqrt {\frac{{4P}}{{\pi {\rm MPE}}}} - w}}{{tg\emptyset}}.$$
The problem becomes more complex when dealing with several laser beams with different wavelengths, power, and elliptical cross-sections. If the radiation beams belong to different wavelength ranges but affect the same tissue, they should be treated as additive. A typical example can be radiation at a wavelength of 1064 and 532 nm. Both wavelengths are transmitted to the interior of the eye and focused onto the retina, which is affected due to the absorption. If the beams do not affect the same tissue, they should be treated independently. Here, a typical example can be radiation with a wavelength of 532 and 1500 nm. The former affects the retina; the latter affects the cornea because it is absorbed by it and does not reach the retina.
In the case of radiation used for dazzling, it should be in the wavelength range that is visible to the human eye, i.e., 400–700 nm. Due to the fact that radiation in this range affects the same tissue, specifically the retina of the eye, it should be treated additively.
In the case of the additivity of the beams, each of the beams contribute to a specific effect, to an extent that can be defined as the ratio of the exposure level to the level necessary for the given effect to occur for a given wavelength. Therefore, to induce this effect, the sum of the ratios defined in this way should be equal to 1. In the case of the effect of eye damage, the sum of the ratios of the exposure level El to the MPE for individual beams should be equal to 1. For $n$ beams with different wavelengths, this can be expressed by the equation
(2)$$\frac{{{{{\rm El}}_1}}}{{{{{\rm MPE}}_1}}} + \frac{{{{{\rm El}}_2}}}{{{{{\rm MPE}}_2}}} + \ldots + \frac{{{{{\rm El}}_n}}}{{{{{\rm MPE}}_n}}} = 1.$$
The exposure level should be expressed in the same units as the MPE, which is expressed in ${\rm J}/{{\rm cm}^2}$ or ${\rm W}/{{\rm cm}^2}$ in accordance with the relevant standards [8,9]. Therefore, knowing the power or energy of the emitted radiation, it becomes necessary to determine the cross-sectional area of the laser beam, at a distance from the radiation source where the exposure takes place. For beams with an elliptical cross-section, it can be done based on their divergence in two perpendicular planes. The diagram of three beams with an elliptical cross-section is shown in Fig. 1.
Fig. 1. Diagram of three beams with different elliptical cross-section.
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The area of the cross-section of a single beam $S$ at exposure distance $d$ can be expressed by the following equation:
(3)$$\!\!\!S = \frac{\pi}{4}ab = \frac{\pi}{4}\left({2d \cdot tg\frac{{{\phi _x}}}{2} + {w_x}} \!\right) \cdot \left({2d \cdot tg\frac{{{\phi _y}}}{2} + {w_y}}\! \right),\!$$
where $a$ is the beam diameter in the $x$ plane, $b$ is the beam diameter in the $y$ plane, $d$ is the distance from the output aperture to the exposure place, ${\phi _x}$ is the angle of divergence in the $x$ plane, ${\phi _y}$ is the angle of divergence in the $y$ plane, ${w_x}$ is the beam diameter at the output aperture in the $x$ plane, and ${w_y}$ is the beam diameter at the output aperture in the $y$ plane.
In most cases, the beam diameter at the exit aperture is small compared with the diameter of the beam at the exposure site, so it can be neglected. Such neglecting of the beam diameter at the exit aperture increases the NOHD, which is advantageous from a safety point of view. Thus, the Eq. (3) can be expressed as
(4)$$S \cong \pi {d^2} \cdot tg\frac{{{\phi _x}}}{2} \cdot tg\frac{{{\phi _y}}}{2}.$$
When the power of the emitted radiation is known for each beam, Eq. (2) can be written as follows:
(5)$$\frac{{\frac{{{P_1}}}{{\pi {d^2} \cdot tg\frac{{{\phi _{x1}}}}{2} \cdot tg\frac{{{\phi _{y1}}}}{2}}}}}{{{{{\rm MPE}}_1}}} + \frac{{\frac{{{P_2}}}{{\pi {d^2} \cdot tg\frac{{{\phi _{x2}}}}{2} \cdot tg\frac{{{\phi _{y2}}}}{2}}}}}{{{{{\rm MPE}}_2}}} + \ldots + \frac{{\frac{{{P_n}}}{{\pi {d^2} \cdot tg\frac{{{\phi _{{xn}}}}}{2} \cdot tg\frac{{{\phi _{{yn}}}}}{2}}}}}{{{{{\rm MPE}}_n}}} \cong 1.$$
It is tempting to assume that the exposure time of potential adversaries to visible radiation is limited to natural aversion to too intense radiation. It is usually related to eyelid closure time, which is estimated for approximately 0.25 s. For such exposure time, the MPE value for all wavelengths in the range of 400–700 nm is the same and is equal to ${25.46}\;{{\rm W/m}^2}$. However, for laser dazzlers, applied in a hostile engagement scenario, a more appropriate assumption is persistent viewing when the person being dazzled intentionally looks at the dazzle beam for a relatively long time. It is therefore appropriate to use the MPE value that relates to exposures from 10 to 30,000 s. For this exposure time, according to the standard [8], the MPE is wavelength dependent.
Thus, Eq. (5) can be written as follows:
(6)$$\frac{1}{{\pi {d^2}}}\left({\frac{{{P_1}}}{{{{{\rm MPE}}_1}tg\frac{{{\phi _{x1}}}}{2} \cdot tg\frac{{{\phi _{y1}}}}{2}}} + \frac{{{P_2}}}{{{{{\rm MPE}}_2}tg\frac{{{\phi _{x2}}}}{2} \cdot tg\frac{{{\phi _{y2}}}}{2}}} + \ldots + \frac{{{P_n}}}{{{{{\rm MPE}}_n}tg\frac{{{\phi _{{xn}}}}}{2} \cdot tg\frac{{{\phi _{{yn}}}}}{2}}}} \right) \cong 1.$$
Knowing that $d$ is identical to the NOHD, Eq. (6) can be transformed to
(7)$${\rm NOHD} \cong \sqrt {\frac{1}{\pi}\left({\frac{{{P_1}}}{{{{{\rm MPE}}_1}tg\frac{{{\phi _{x1}}}}{2} \cdot tg\frac{{{\phi _{y1}}}}{2}}} + \frac{{{P_2}}}{{{{{\rm MPE}}_2}tg\frac{{{\phi _{x2}}}}{2} \cdot tg\frac{{{\phi _{y2}}}}{2}}} + \ldots + \frac{{{P_n}}}{{{{{\rm MPE}}_n}tg\frac{{{\phi _{{xn}}}}}{2} \cdot tg\frac{{{\phi _{{yn}}}}}{2}}}} \right)} .$$
For small divergence angles, which for most laser sources are up to several milliradians, the $tg\phi = \phi$ approximation can be introduced; then, Eq. (7) can be simplified to
(8)$$\begin{split}&{\rm NOHD} \cong 2 \\&\cdot\sqrt {\frac{1}{\pi}\left(\!{\frac{{{P_1}}}{{{{{\rm MPE}}_1}{\phi _{x1}}{\phi _{y1}}}} + \frac{{{P_2}}}{{{{{\rm MPE}}_2}{\phi _{x2}}{\phi _{y2}}}} + \ldots + \frac{{{P_n}}}{{{{{\rm MPE}}_n}{\phi _{{xn}}}{\phi _{{yn}}}}}} \!\right)} .\end{split}$$
After calculating the constant values and rounding, we obtain
(9)$$\begin{split}&{\rm NOHD} \cong 1.128 \\& \cdot\sqrt {\frac{{{P_1}}}{{{{{\rm MPE}}_1}{\phi _{x1}}{\phi _{y1}}}} + \frac{{{P_2}}}{{{{{\rm MPE}}_2}{\phi _{x2}}{\phi _{y2}}}} + \ldots + \frac{{{P_n}}}{{{{{\rm MPE}}_n}{\phi _{{xn}}}{\phi _{{yn}}}}}} .\end{split}$$
In order to determine the NOHD expressed in meters, the power of individual beams, expressed in Watts, and their divergence in two perpendicular planes, expressed in radians, should be inserted into the above equation. Further, the values of the MPE should be expressed in ${\rm W} / {\rm m}^2$.
According to the standard [8] for exposure durations of 10 s or longer, the additive photochemical effects (400 to 600 nm) and the additive thermal effects (400 to 1400 nm) shall be assessed independently and the most restrictive value used. However, the laser sources used for dazzling are usually characterized by angular subtense below 1.5 mrad; thus, factor C6 can be assumed equal to 1, and the values of the MPE defined in Table A1 in the standard [8] should be used. In this table, two exposure time ranges from 10s to ${{10}^2}\;{\rm s}$ and ${{10}^2}\;{\rm s}$ to ${{3\cdot 10}^4}\;{\rm s}$ are defined. The second one is the most restrictive for all wavelengths in the range of 400–700 nm for photochemical as well as for thermal effects, and the values of the MPE for this exposure time should be used. Moreover, for this exposure time, there are no different values of the MPE for thermal and photochemical effects.
3. DERIVING OF NODD FOR MULTIBEAM RADIATION
The nominal ocular dazzle distance (NODD) is defined as the distance from the radiation source at which eye exposure to laser radiation is equal to the maximum dazzle exposure (MDE). Here, the additivity of spectral sensitivity of the eye for visible radiation can be assumed again; however, it has not been firmly proven for laser exposure [13,14]. Thus, Eq. (2) can be used to find the NODD, but the MPE should be replaced by the MDE. Equation (5) will then take the form
(10)$$\frac{{\frac{{{P_1}}}{{\pi {d^2} \cdot tg\frac{{{\phi _{x1}}}}{2} \cdot tg\frac{{{\phi _{y1}}}}{2}}}}}{{{{{\rm MDE}}_1}}} + \frac{{\frac{{{P_2}}}{{\pi {d^2} \cdot tg\frac{{{\phi _{x2}}}}{2} \cdot tg\frac{{{\phi _{y2}}}}{2}}}}}{{{{{\rm MDE}}_2}}} + \ldots + \frac{{\frac{{{P_n}}}{{\pi {d^2} \cdot tg\frac{{{\phi _{{xn}}}}}{2} \cdot tg\frac{{{\phi _{{yn}}}}}{2}}}}}{{{{{\rm MDE}}_n}}} \cong 1.$$
Standard ${\rm MDE}_S$ values, for maximum eye sensitivity, for four different dazzle levels (the angular size of the visual field that is obscured within an observer's field of vision: 2°, 10°, 20°, 40°) and for three ambient luminance levels (night ${0.1}\;{{\rm cd/m}^2}$, dusk ${10}\;{{\rm cd/m}^2}$, day ${1000}\;{{\rm cd/m}^2}$) were proposed for the first time by Williamson and McLin [11]. The dazzle levels are schematically shown in Fig. 2, where the area seen by the observer is marked with blue, and the area obscured by the dazzle effect is marked with green.
The standard ${\rm MDE}_S$ values were determined based on a number of experiments and extensive analysis, taking into account such parameters as the angular subtense of the object, the contrast of the object, the observer's age, and the eye pigmentation [11]. The values of the above parameters, adopted for the determination of the ${\rm MDE}_S$, are presented in Table 1, and the ${\rm MDE}_S$ values are presented in Table 2.
The above ${\rm MDE}_S$ values were defined for the maximum eye sensitivity, which is for the eye's photopic sensitivity ${\rm V}(\lambda)$ equal to 1 (for a wavelength of 556.1 nm) [15]. To determine the MDE for other wavelengths, the value from Table 2 should be divided by the appropriate ${\rm V}(\lambda)$ value in accordance with the expression
(11)$${\rm MDE} = \frac{{{{\rm MDE}_{\rm S}}}}{{\rm V}({\lambda} )}.$$
The ${\rm V}(\lambda)$ values for several basic wavelengths, which are used in calculations presented later in this paper, are given in Table 3.
Fig. 2. Four dazzle levels.
Table 1. Parameters Adopted for the Determination of the ${\rm MDE}_S$ [11]
View Table | View all tables in this article
Table 2. ${\rm MDE}_S$ Values for Different Dazzle and Ambient Light Levels [11]
Table 3. ${\rm V}(\lambda)$ Values for Several Basic Wavelengths [15]
Taking the above considerations into account and assuming small angles of divergence of the beams and identifying $d$ with the NODD, Eq. (10) can be transformed as follows:
(12)$${\rm NODD} \cong 2 \sqrt {\frac{1}{\pi}\left({\frac{{{P_1}}}{{{{{\rm MDE}}_1}{\phi _{x1}}{\phi _{y1}}}} + \frac{{{P_2}}}{{{{{\rm MDE}}_2}{\phi _{x2}}{\phi _{y2}}}} + \ldots + \frac{{{P_n}}}{{{{{\rm MDE}}_n}{\phi _{{xn}}}{\phi _{{yn}}}}}} \right)} .$$
(13)$$\begin{split}&{\rm NODD} \cong 1.128 \\&\cdot \sqrt {\frac{{{P_1}}}{{{{{\rm MDE}}_1}{\phi _{x1}}{\phi _{y1}}}} + \frac{{{P_2}}}{{{{{\rm MDE}}_2}{\phi _{x2}}{\phi _{y2}}}} + \ldots + \frac{{{P_n}}}{{{{{\rm MDE}}_n}{\phi _{{xn}}}{\phi _{{yn}}}}}} .\end{split}$$
In order to determine the NODD expressed in meters, the power of individual beams, expressed in Watts, their divergence in two perpendicular planes, expressed in radians, and MDE expressed in ${\rm W}/{{\rm m}^2}$ should be substituted to the above equation. It is worth noting that the values of the ${{\rm MDE}_1},\;{{\rm MDE}_2}, \ldots ,\;{{\rm MDE}_n}$ should be considered for the same ambient light levels and dazzle levels.
4. ANALYSIS OF NOHD AND NODD FOR THREE RADIATION BEAMS
In order to analyze the effect of the additivity of laser beams on the values of the NOHD and the NODD, calculations were performed for three hypothetical radiation sources, with beam parameters presented in Table 4. Wavelengths were selected based on the availability in the market, of simple and small radiation sources such as laser diodes. For all three sources, the same power and same divergence in both perpendicular planes, 1 W and 2 mrad, respectively, were assumed. In Table 4, the values of the MPE for exposure time longer than 100 s according to the standard [8] are also shown.
Using Eq. (8), the NOHD values for the above sources were determined, which were approximately 564 m for the blue source and 178 m for the green and red sources; for all sources simultaneously (taking into account additivity), it was approximately 618 m.
Using Eq. (12), the calculations of the NODD were made. The results of the calculations for four dazzle levels and three ambient light levels are presented graphically in Figs. 3–6. The highest values of the NODD, determined under the same ambient light levels and dazzle levels, were obtained for the green beam. This is due to the highest value of the eye's photopic sensitivity (${\rm V}(\lambda) = {0.7181}$) compared with the other beams. The maximum NODD value of over 151 km was obtained for the night conditions and the dazzle level of 2°. The smallest NODD values were obtained for the blue beam for which the ${\rm V}(\lambda)$ coefficient equals only 0.0574. For the dazzle level of 40° and day conditions, the dazzle effect may appear only at distance up to 8 m from the source of this radiation.
The analysis also shows that the highest NODD values are obtained for night conditions and, depending on the beam and the dazzle level, range from approximately 1.7 km for the blue beam and 40° dazzle level to over 151 km for the green beam and 2° dazzle level. In turn, the lowest NODD values are obtained for day conditions, and it ranges from approximately 8 m for the blue beam and 40° dazzle level to approximately 756 m for the green beam and 2° dazzle level.
Table 4. Parameters of Beams Emitted by Hypothetical Sources and Their MPE
Fig. 3. NODD values for 445 nm beam.
Fig. 6. NODD values for all beams.
It should also be noted that there are some distance ranges for specific ambient light levels and dazzle levels, where the NOHD is greater than the NODD. It covers the distances where the eye can be damaged without dazzle effect. In this case, the exposure level is high enough to damage the retina of the eye but does not yet cause a dazzle effect due to the low contrast to ambient light and the assumed high dazzle level. These ranges obviously appear for day conditions and high dazzle levels. For the assumed beam parameters, such a situation occurs for day conditions and all dazzle levels for the blue beam and 10°, 20°, and 40° dazzle levels for the green and red beams. For dusk conditions, it occurs again for the blue beam at 10°, 20°, and 40° dazzle levels and for the red beam at 40° dazzle level.
If three beams are analyzed simultaneously and their additivity is taken into account, all the NODD values are increased. The maximum NODD≅177 km, similarly to a single beam, is obtained for night conditions and a dazzle level of 2°. The minimum NODD value of approximately 32 m, occurs again for day conditions and the dazzle level of 40°. The ranges where the NOHD≅618 m is greater than the NODD occurs for day conditions and the dazzle levels of 10°, 20°, and 40° as well as for dusk conditions and dazzle levels of 20° and 40°.
It is also worth noting that the presented analysis did not take into account the attenuation of radiation by the atmosphere; therefore, the NOHD and the NODD values will be lower in real conditions.
The analysis also included calculations of the NOHD and ODD values in function of the power of the emitted radiation for a wavelength of 520 nm and divergence in both perpendicular planes of 2 mrad. Due to the common availability of laser pointers with power up to several mW, on the market, the analysis was limited to this range. Figure 7 shows the results of the NOHD calculations, while Figs. 8–10 show the results of the NODD calculations for power up to 10 mW. The NOHD value for the power of 10 mW was approximately 18 m. The NODD values for 10 mW power are presented in Table 5.
Fig. 7. NOHD values in the function of beam power for a wavelength of 520 nm and a divergence of 2 mrad.
Fig. 8. NODD values in the function of beam power for a wavelength of 520 nm, divergence of 2 mrad, and night conditions.
Fig. 9. NODD values in the function of beam power for a wavelength of 520 nm, divergence of 2 mrad, and dusk conditions.
Fig. 10. NODD values in the function of beam power for a wavelength of 520 nm, divergence of 2 mrad, and day conditions.
The analysis shows that the beam of radiation with a wavelength of 520 nm, a divergence of 2 mrad, and a power of 10 mW can damage the eye up to a distance of approximately 18 m, while the distance at which the dazzle effect may occur varies widely depends on the ambient light conditions and the dazzle level that need to be achieved. The highest NODD values are obtained for night conditions, which reach over 15 km for the dazzle level of 2° and decrease to approximately 617 m for the dazzle level of 40°. In turn, the lowest NODD values are obtained for day conditions, which is approx. 76 m for the dazzle level of 2°, but it drops to about 3 m for the dazzle level of 40°. As can be seen from the presented analysis, despite the relatively low beam power, it is possible to cause dazzling at relatively large distances under appropriate conditions. Due to the lower contrast in day conditions, these distances are much smaller. It should also be noted that for day conditions and the dazzle levels of 10°, 20°, and 40°, there are distance ranges at which the eye can be injured without causing the dazzle effect. For the dazzle level of 40°, dazzling will occur only at a distance of up to approximately 3 m, while eye damage may occur at a distance of up to approximately 18 m.
Table 5. NODD Values for a Laser Beam of 10 mW Power, 520 nm Wavelength, and 2 mrad Divergence
For the 638 nm wavelengths and the same divergencies, the NOHD value in function of the beam power will have the same values as for the 520 nm wavelength due to the same MPE value. However, for the 445 nm wavelength, the values of the NOHD will be much higher, reaching approximately 56 m at 10 mW beam power. The NODD values for 638 and 445 nm wavelengths will be correspondingly lower due to the dependence of the MDE on the eye's photopic sensitivity ${\rm V}(\lambda)$.
Assuming the additivity of three beams with a wavelength of 445, 520, and 638 nm and 2 mrad divergence and 10 mW power of each beam, the NOHD value will be approximately 62 m. This is more than a threefold increase in the distance up to which eye can be damaged compared with a single beam at 520 nm. In contrast, the NODD values will increase slightly due to the relatively low ${\rm V}(\lambda)$ values for 445 and 638 nm. Figure 11 shows the dependence of the NODD on the power of emitted radiation, assuming the same power for each beam for night conditions and different dazzle levels. For the remaining ambient light levels, the NODD values will be much lower, as presented for a single beam of 520 nm wavelength. Results of the calculations of the NODD value for the maximum power of 10 mW are presented in Table 6.
Fig. 11. NODD values in the function of power for three beams with a wavelength of 445, 520, and 638 nm; divergence of 2 mrad; and night conditions.
Table 6. NODD Values for Three Beams with a Wavelength of 445, 520, and 638 nm, Power of 10 mW and Divergence of 2 mrad
It can be seen from the analysis that the additivity of the beams of different wavelengths causes a slight increase in the distance range of eye damage compared with the blue beam. Also, the increase in the range of the distance at which the dazzle effect can occur is relatively small and becomes smaller the farther the wavelengths are away from 556.1 nm.
5. ANALYSIS OF NOHD AND NODD FOR COMMERCIAL RADIATION SOURCES WITH POTENTIAL APPLICATION TO LASER DAZZLERS
The above analysis shows that the parameters influencing the NOHD and the NODD are the power of the beams and their divergence. Moreover, they are affected by the wavelength due to different tabularized values of the MPE and the dependence of the MDE on the eye's photopic sensitivity ${\rm V}(\lambda)$. The beam divergence can always be adjusted and changed using appropriately designed optical systems. On the other hand, the radiation power and the wavelength are usually characteristic for a given source. Therefore, when selecting potential sources of radiation for use in a dazzler, besides a simple and rigid construction, small dimensions and weight, these last two parameters are decisive. There are many light sources available on the market, but laser diodes deserve special attention due to their simple construction, small dimensions and weight, and the ability to generate relatively high powers, reaching several watts. Examples include laser diodes produced by Nichia (generating blue and green radiation) [16] and Ushio (generating red radiation) [17]. One can also find laser modules that are based on laser diodes, whereby applying appropriate optical systems, much smaller beam divergence can be obtained [18]. There are also ready-to-use modules emitting radiation at three wavelengths of different colors [18]. The parameters of such exemplary module 4 W-RGB are presented in Table 7.
Table 7. Parameters of Laser Beams Generated by the Exemplary Laser Module [18]
Table 8. NOHD Values for the Analyzed Laser Module
Using Eqs. (8) and (12), analysis of the values of the NOHD and the NODD was carried out for the laser module. Table 8 presents the NOHD values for each source separately, taking into account the additivity of the beams emitted by the module.
The results presented in Table 8 show that the laser module can damage the eye at distances up to 887, 198, and 208 m for the blue, green, and red beams, respectively. The additivity of the beams causes slight increases in the NOHD value compared with the blue wavelength beam, due to the low value of the power of the other two beams (red and green), as well as a tenfold smaller value of the MPE.
Table 9 presents the results of the NODD calculated in meters for the analyzed laser module. The results show that applying commercial laser modules allows one to develop a sufficient dazzling device that can cause a dazzle effect at long distances under defined ambient light and dazzle levels. It should also be noted that, for high ambient light and dazzle levels, there are distance ranges where eye damage may occur without the dazzle effect.
Table 9. Results of the NODD Calculations in Meters for the Analysed Laser Module
The paper shows the method of the NOHD and the NODD determination for multiple laser beams. The above equations allow simple calculation of these values on the basis of the MPE and the MDE for specific ambient light and dazzle levels, taking into account the additivity of the beams. The proposed calculation method may turn out to be particularly desirable in the current situation when more radiation sources using multiple laser beams become available. The analysis also showed that, for high ambient light and dazzle levels, there are distance ranges where the NOHD values are greater than the NODD values. In these ranges, the eye may be damaged, while the dazzle effect does not occur. Such information is particularly important in order to ensure the safety of those being dazzled. Thus, the presented results can be used as a reference for designers developing devices for applications related to dazzling.
From the point of view of safety, for exposures lasting more than 100 s, the most dangerous wavelengths are those in the 400–450 nm range due to the low MPE value of ${1}\;{{\rm W/m}^2}$. Wavelengths in the 450–500 nm range are characterized by the different MPE values varying from ${1}\;{{\rm W/m}^2}$ to ${10}\;{{\rm W/m}^2}$. However, the safest wavelengths are those in the 500–700 nm range due to the highest MPE value of ${10}\;{{\rm W/m}^2}$. For dazzling, wavelengths in the vicinity of 556 nm are the most effective. The dazzle efficiency decreases gradually moving away from this wavelength in both the longer and shorter wavelength directions. Thus, it can be concluded that the most desirable wavelengths, from the point of view of safety and dazzle efficiency, are those in the vicinity of 556 nm. However, in order to ensure the widest possible spectral range for dazzling while ensuring safety and high dazzle efficiency, one could conclude that the most appropriate wavelength range is 510–610 nm where the ${\rm V}(\lambda)$ is still higher than 0.5. These are the colors between cyan (almost blue) and orange. Applying wavelengths from this range to multibeam dazzlers can provide the maximum increase of the NODD with the minimum increase of the NOHD. Unfortunately, the availability of such lasers is much more limited than in the case of the exemplary lasers described in the paper.
When using commercial lasers, such as those described here, in multiwavelength dazzlers, the NODD is mainly determined by the green beam and the NOHD by the blue beam. Thus, combining the red and blue beams to the green beam only slightly increases the NODD and significantly increases the NOHD. Nevertheless, in justified cases, it may be rational to use such multiwavelength dazzlers, especially in night conditions and low dazzle levels. The reason may be the need to cover the wider spectrum of visible radiation because of the possible use of laser filters by the adversary.
Regarding different ambient light levels, the best results are definitely obtained for night conditions when effective dazzling and appropriate safety can be ensured for almost the whole spectrum of visible radiation. However, for daytime conditions, effective dazzling and safety can only be achieved for the lowest dazzle level and for the limited spectrum of visible radiation.
It should also be noted that the standard MDEs values have been estimated on the basis of the specific angular subtense and contrast of the object, the observer's age, and the eye pigmentation; therefore, they may vary depending on the individual, applications, and intended dazzle effect.
The analysis was limited to the ideal conditions where no laser beam attenuation in the atmosphere was considered. One can assume that the NOHD and NODD values will be lower under real conditions. The introduction of such a supplement may be the subject of further work on the problem of dazzling people by laser radiation.
The results presented in this paper concern only laser sources that generate continuous wave (cw) radiation. However, the calculations of the NOHD can be expanded to single-pulse or repetitively pulsed sources by applying appropriate values of the MPE for the right exposure time according to the relevant standards. In case of the NODD, the situation is not so simple. The values of MDE were determined for cw radiation. Thus, to expand the presented calculations to single-pulse or repetitively pulsed sources, the new values of MDE should be determined, which has not been done thus far. Expansion of the calculation to single-pulse or repetitively pulsed sources should be the subject of future work.
Moreover, it would be useful to explore the dazzle additivity assumption experimentally, i.e., exploring whether the human perception of multiple lasers does actually equal the effect calculated by simply adding MDE values, as was done in this paper.
Narodowe Centrum Badań i Rozwoju (DOB-1-6/1/PS/2014).
No data were generated or analyzed in the presented research.
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2. https://www.beamq.com/.
3. https://www.jetlasers.org/.
4. J. Marshall, J. B. O'Hagan, and J. R. Tyrer, "Eye hazards of laser 'pointers' in perspective," Br. J. Ophthalmol. 100, 583–584 (2016). [CrossRef]
5. J. E. Neffendorf, G. D. Hildebrand, and S. M. Downes, "Handheld laser devices and laser-induced retinopathy (LIR) in children: an overview of the literature," Eye 33, 1203–1214 (2019). [CrossRef]
6. J. Mlynczak, "Laser toys fail to comply with safety standards–case study based on laser product classification," Adv. Opt. Technol. (to be published).
7. D. B. Gosling, J. B. O'Hagan, and F. M. Quhill, "Blue laser induced retinal injury in a commercial pilot at 1300 ft," Aerosp. Med. Human Perform. 87, 69–70 (2016). [CrossRef]
8. "Safety of laser products—part 1: equipment classification and requirements," IEC 60825-1:2014 (International Electrotechnical Commission, 2014).
9. "American National standard for safe use of lasers," ANSI Z136.1-2014 (American National Standards Institute, 2014).
10. C. A. Williamson and L. N. McLin, "Nominal ocular dazzle distance (NODD)," Appl. Opt. 54, 1564–1572 (2015). [CrossRef]
11. C. A. Williamson and L. N. McLin, "Determination of a laser eye dazzle safety framework," J. Laser Appl. 30, 032010 (2018). [CrossRef]
12. https://ownthenight.com/.
13. P. Lennie, J. Pokorny, and V. C. Smith, "Luminance," J. Opt. Soc. Ame. A 10, 1283–1293 (1993). [CrossRef]
14. H. Cai and T. Chung, "Evaluating discomfort glare from non-uniform electric light sources," Light. Res. Technol. 45, 267–294 (2012). [CrossRef]
15. "CVRL Database, CVRL functions, Luminous efficiency functions, 2-deg functions," http://www.cvrl.org/.
16. http://www.nichia.co.jp/.
17. http://www.ushio-optosemi.com/.
18. https://optlasers.com/.
Article Order
"Joint non-lethal weapons program, non-lethal optical distracters fact sheet," 2016, https://jnlwp.defense.gov/Portals/50/Documents/Press_Room/Fact_Sheets/NL_Optical_Distracters_Fact_Sheet_May_2016.pdf .
https://www.beamq.com/ .
https://www.jetlasers.org/ .
J. Marshall, J. B. O'Hagan, and J. R. Tyrer, "Eye hazards of laser 'pointers' in perspective," Br. J. Ophthalmol. 100, 583–584 (2016).
[Crossref]
J. E. Neffendorf, G. D. Hildebrand, and S. M. Downes, "Handheld laser devices and laser-induced retinopathy (LIR) in children: an overview of the literature," Eye 33, 1203–1214 (2019).
J. Mlynczak, "Laser toys fail to comply with safety standards–case study based on laser product classification," Adv. Opt. Technol. (to be published).
D. B. Gosling, J. B. O'Hagan, and F. M. Quhill, "Blue laser induced retinal injury in a commercial pilot at 1300 ft," Aerosp. Med. Human Perform. 87, 69–70 (2016).
"Safety of laser products—part 1: equipment classification and requirements," IEC 60825-1:2014 (International Electrotechnical Commission, 2014).
"American National standard for safe use of lasers," ANSI Z136.1-2014 (American National Standards Institute, 2014).
C. A. Williamson and L. N. McLin, "Nominal ocular dazzle distance (NODD)," Appl. Opt. 54, 1564–1572 (2015).
C. A. Williamson and L. N. McLin, "Determination of a laser eye dazzle safety framework," J. Laser Appl. 30, 032010 (2018).
https://ownthenight.com/ .
P. Lennie, J. Pokorny, and V. C. Smith, "Luminance," J. Opt. Soc. Ame. A 10, 1283–1293 (1993).
H. Cai and T. Chung, "Evaluating discomfort glare from non-uniform electric light sources," Light. Res. Technol. 45, 267–294 (2012).
"CVRL Database, CVRL functions, Luminous efficiency functions, 2-deg functions," http://www.cvrl.org/ .
http://www.nichia.co.jp/ .
http://www.ushio-optosemi.com/ .
https://optlasers.com/ .
Cai, H.
Chung, T.
Downes, S. M.
Gosling, D. B.
Hildebrand, G. D.
Lennie, P.
Marshall, J.
McLin, L. N.
Mlynczak, J.
Neffendorf, J. E.
O'Hagan, J. B.
Pokorny, J.
Quhill, F. M.
Smith, V. C.
Tyrer, J. R.
Williamson, C. A.
Aerosp. Med. Human Perform. (1)
Appl. Opt. (1)
Br. J. Ophthalmol. (1)
J. Laser Appl. (1)
J. Opt. Soc. Ame. A (1)
Light. Res. Technol. (1)
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\begin{document}
\title{f Regularity theory for the dissipative\ solutions of the MHD equations} \begin{scriptsize} \abstract{We study here a new generalization of Caffarelli, Kohn and Nirenberg's partial regularity theory for weak solutions of the MHD equations. Indeed, in this framework some hypotheses on the pressure $P$ are usually asked (for example $P\in L^q_tL^1_x$ with $q>1$) and then local H\"older regularity, in time and space variables, for weak solutions can be obtained over small neighborhoods. By introducing the notion of dissipative solutions, we weaken the hypothesis on the pressure (we will only assume that $P\in \mathcal{D}'$) and we will obtain H\"older regularity in the space variable for weak solutions.}\\
\noindent\textbf{Keywords:} MHD equations; dissipative solutions; partial regularity theory; Morrey spaces.\\ \textbf{MSC2020:} 76W05; 76D03; 35Q35. \end{scriptsize}
\section{Introduction} The main purpose of this work is to study the partial regularity theory for the incompressible 3D MagnetoHydroDynamic (MHD) equations which are given by the following system: \begin{equation}\label{EquationMHDoriginal} \begin{cases} \partial_{t}\vU=\Delta \vU -(\vU\cdot\vn)\vU+(\vB\cdot\vn)\vB-\vn P+\vF,\quad div(\vU) = div(\vF)=0,\\[3mm] \partial_{t}\vB=\Delta \vB -(\vU\cdot\vn)\vB+(\vB\cdot\vn)\vU+\vG,\quad div(\vB)=div(\vG)=0,\\[3mm] \vU(0,x)=\vU_{0}(x), \; div(\vU_0)=0 \mbox{ and } \vB(0,x)=\vB_{0}(x),\; div(\vB_0)=0, \qquad x\in \mathbb{R}^3, \end{cases} \end{equation} where $\vU, \vB:[0,T]\times \R\longrightarrow \R$ are two divergence-free vector fields which represent the velocity and the magnetic field, respectively, and the scalar function $P:[0,T]\times \R\longrightarrow \mathbb{R}$ stands for the pressure. The initial data $\vU_0, \vB_0: \R\longrightarrow \R$ and the external forces $\vF, \vG:[0,T]\times \R\longrightarrow \R$ are given. We will say that $(\vU, P, \vB)$ is a solution of the MHD equations if it solves the system (\ref{EquationMHDoriginal}) in some particular sense to be precised later on.\\
In the recent articles \cite{ChCHJ1} and \cite{ChCHJ2} we have studied two different regularity theories for the system (\ref{EquationMHDoriginal}) using parabolic Morrey spaces as a common framework. The aim of the current work is to mix in a very specific manner these two previous results to obtain a new class of solutions of (\ref{EquationMHDoriginal}) for which we can deduce more general regularity results.\\
Indeed, in the first article \cite{ChCHJ1} we obtained a \emph{local} regularity result for the MHD equations following some ideas of O'Leary \cite{OLeary} (which were originally stated for the classical 3D Navier-Stokes equations): if we assume that $\mathds{1}_{\Omega}\vU$ and $\mathds{1}_{\Omega}\vB$ belong to some suitable parabolic Morrey spaces $\mathcal{M}^{p,q}_{t,x}$ (see Section \ref{Section_Preliminaries} below for a definition of these functional spaces), where $\Omega$ is a bounded subset of $]0, +\infty[\times \mathbb{R}^3$ of the form \begin{equation}\label{DefConjuntoOmega} \Omega=]a,b[\times B(x_{0},r), \quad \mbox{with} \quad 0<a<b<+\infty, x_{0}\in \R \mbox{ and } 0<r<+\infty. \end{equation} Then it is possible to obtain a gain of regularity in the space variable even though the pressure is a general object (\emph{i.e.} $P\in \mathcal{D}'$). The result obtained in this framework is the following one:
\begin{Theoreme}[Local Regularity, \cite{ChCHJ1}]\label{Teo_SerrinMHD} Let $\vU_{0}, \vB_{0}:\R\longrightarrow \R$ such that $\vU_{0}, \vB_{0}\in L^{2}(\R)$ and $div(\vU_{0})=div(\vB_{0})=0$ be two initial data and consider two external forces $\vF, \vG:[0,+\infty[\times\R\longrightarrow \R$ such that $\vF, \vG\in L^{2}([0,+\infty[, \dot{H}^{1}(\R))$. Assume that $\Omega$ is a bouned set of the form (\ref{DefConjuntoOmega}), that $P\in \mathcal{D}'(\Omega)$ and that $\vU, \vB:[0,+\infty[\times\R\longrightarrow \R$ are two vector fields that belong to the space \begin{equation}\label{UBSolutionFaiblesMHD} L^{\infty}(]a,b[, L^{2}(B(x_{0},r)))\cap L^{2}(]a,b[, \dot{H}^{1}(B(x_{0},r))), \end{equation} such that they satisfy the MHD equations (\ref{EquationMHDoriginal}) over the set $\Omega$.\\
\noindent If moreover we have the following local hypotheses \begin{equation}\label{LocalHypo1} \begin{cases} \mathds{1}_{\Omega}\vU\in\mathcal{M}_{t,x}^{p_{0},q_{0}}(\mathbb{R}{\tiny \times}\R) & \mbox{ with } 2<p_{0}\leq q_{0}, 5<q_{0}<+\infty\\[3mm] \mathds{1}_{\Omega}\vB\in\mathcal{M}_{t,x}^{p_{1},q_{1}}(\mathbb{R}\times\R) &\mbox{ with } 2<p_{1}\leq q_{1}, 5<q_{1}<+\infty,
\end{cases} \end{equation} and $p_1 \leq p_0$, $q_1 \leq q_0$, then, for all $\alpha, \beta$ such that $a<\alpha<\beta<b$ and for all $\rho$ such that $0<\rho<r$, we have \begin{equation}\label{ConclusionSerrinMHD} \vU\in L^{q_0}(]\alpha,\beta[, L^{q_0}(B(x_{0},\rho)))\quad \mbox{and}\quad \vB\in L^{q_1}(]\alpha,\beta[, L^{q_1}(B(x_{0},\rho))). \end{equation} \end{Theoreme} Following Serrin's theory \cite{Serrin1} (see also Section 13.2 of \cite{PGLR1}) it can be shown that the conclusion of this theorem implies that \begin{equation}\label{ConclusionSerrinMHD1} \vU\in L^{\infty}(]\alpha',\beta'[, L^{\infty}(B(x_{0},\rho')))\quad \mbox{and}\quad \vB\in L^{\infty}(]\alpha',\beta'[, L^{\infty}(B(x_{0},\rho'))), \end{equation} for some $\alpha\leq \alpha'<\beta'\leq \beta$ and $0<\rho'\leq \rho$. \begin{Remarque}\label{Rem_RegulariteSerrin} The points $(t,x)\in [0,+\infty[\times \mathbb{R}^3$ such that the previous condition (\ref{ConclusionSerrinMHD1}) holds for $\vU(t,x)$ and $\vB(t,x)$ are called \emph{regular points} as we can prove that in this case the regularity (in the space variable) of the solution $\vU$ and $\vB$ is actually driven by the regularity of the external forces $\vF$ and $\vG$: in the framework of Theorem \ref{Teo_SerrinMHD} we obtain that $\vU, \vB\in L^\infty_tH^2_x\cap L^2_t H^3_x$ over a subset of $\Omega$. See Theorem 13.1 p. 397 of \cite{PGLR1} for a proof of this fact in the setting of the classical Navier-Stokes equations which can be easily extended to the MHD equations. \end{Remarque} These ideas are of course reminiscent from the work of Serrin \cite{Serrin1} (stated for the classical Navier-Stokes equations) which first considered the condition $\vU\in (L^p_tL^q_x)_{loc}$ with $\tfrac{2}{p}+\tfrac{3}{q}<1$ (the case $\tfrac{2}{p}+\tfrac{3}{q}=1$ was studied by Struwe \cite{Struwe} and Takahashi \cite{Takahashi}). One important point of all these results is related to Serrin's counter-example: as we do not impose any particular information on the pressure $P$, we may lose regularity in the time variable since the temporal regularity is closely linked to the pressure (see Section 13.1 of \cite{PGLR1} for this particular point) and thus only space regularity can be obtained. Note also that the Morrey space assumption (\ref{LocalHypo1}) or the $(L^p_tL^q_x)_{loc}$ control with $\tfrac{2}{p}+\tfrac{3}{q}\leq 1$ are quite strong hypotheses that can not be deduced from a general weak Leray-type solution of (\ref{EquationMHDoriginal}).
Let us now mention that in the study of the local regularity theory for the MHD equations, Wu \cite{Wu1} generalized Ladyzhenskaya-Prodi-Serrin-type criteria by assuming conditions on both the velocity $\vU$ and the magnetic field $\vB$. He and Xin \cite{He1} gave a Serrin-type regularity condition which only including the integrability condition on the velocity $\vU$. More precisely, they shown that weak solutions $(\vU, \vB)$ are smooth if $\vU \in (L^p_tL^q_x)_{loc}$ with $\tfrac{2}{p}+\tfrac{3}{q}\leq 1$ and $q >3$, which reveals that the velocity field plays a more dominant role than the magnetic field does on the regularity of solutions to the MHD equations. Chen, Miao and Zhang \cite{ChenMZ} then proved that $(\vU, \vB)$ is regular under a refined Serrin-type regularity criterion in the framework of Besov spaces with negative index in terms of the velocity only. Later on, Kang and Lee \cite{KangLee} present a new interior regularity criteria for suitable weak solutions to the MHD equations by supposing that some scaled norm of the velocity is small and some scaled norm of the magnetic field is bounded. Their result was improved by Wang and Zhang in \cite{WangZhang} by removing the bounded assumptions on the magnetic field. \\
In the second article \cite{ChCHJ2} we studied the so-called \emph{partial} regularity theory for the MHD equations (\ref{EquationMHDoriginal}) using the same parabolic Morrey spaces setting as above. This setting was initially applied for the Navier-Stokes equations by Kukavica in \cite{Kukavica} and \cite{Kukavica08} in order to generalize the Caffarelli, Kohn and Nirenberg regularity criterion \cite{CKN}. The main idea of this theory relies in the use of the of ``suitable weak solutions" (first developed by Scheffer for the Navier-Stokes equations in \cite{Scheffer} and \cite{Scheffer1}) which is a class of weak Leray-type solution $(\vU, P, \vB)$ such that the distribution $\mu$ given by the expression \begin{eqnarray}
\mu&=&-\partial_t(|\vU|^2 + |\vB|^2 )+ \Delta (|\vU|^2 + |\vB|^2 ) - 2 (|\vn \otimes \vU|^2 + |\vn \otimes \vB|^2 ) - 2 div (P \vU) \label{Formula_MesureMHD}\\
& &- div \left( (|\vU|^2 + |\vB|^2) \vU \right) + 2 div ((\vU \cdot \vB) \vB ) + 2 (\vF \cdot \vU + \vG \cdot\vB),\notag \end{eqnarray} is a non-negative locally finite measure on $\Omega$, where $\Omega\subset ]0,+\infty[\times \mathbb{R}^3$ is a bounded domain. Of course, in the formula above some conditions on the pressure $P$ must be imposed (usually $P\in L^q_tL^1_x$ with $q>1$) in order to make the quantity $div(P \vU)$ meaningful. Assuming moreover a particular behavior of the quantities $\vn\otimes \vU$ and $\vn\otimes \vB$ over a small neighborhood of points, then it is possible to obtain a gain of regularity on both variables, space and time. More precisely we have:
\begin{Theoreme}[Partial Regularity, \cite{ChCHJ2}]\label{Teo_CKNMHD} Let $\Omega$ be a bounded domain of the form given in (\ref{DefConjuntoOmega}). Let $(\vU, P, \vB)$ be a weak solution on $\Omega$ of the MHD equations \eqref{EquationMHDoriginal}. Assume that \begin{itemize} \item[1)] The vector $(\vU, \vB, P, \vF, \vG)$ satisfies the conditions $$\vU, \vB \in L^{\infty}_t L^{2}_x \cap L^{2}_t \dot{H}^{1}_x(\Omega),\qquad P \in L^{q_{0}}_{t,x}(\Omega) \mbox{ with } 1<q_0\leq \tfrac{3}{2},\qquad \vF, \vG \in L^{\frac{10}{7}}_{t,x}(\Omega).$$ \item[2)] The solution $(\vU, P, \vB)$ is suitable in the sense that the distribution (\ref{Formula_MesureMHD}) is a non-negative locally finite measure on $\Omega$. \item[3)] We have the following local information on $\vF$ and $\vG$: $\mathds{1}_{\Omega}\vF \in \mathcal{M}_{t,x}^{\frac{10}{7}, \tau_{a}}$ and $ \mathds{1}_{\Omega} \vG \in \mathcal{M}_{t,x}^{\frac{10}{7}, \tau_{b}}$ for some $\tau_{a}, \tau_b>\frac{5}{2-\alpha}$ with $0<\alpha<\frac{1}{3}$. \end{itemize} There exists a positive constant $\epsilon^{*}$ which depends only on $\tau_{a}$ and $\tau_b$ such that, if for some $\left(t_{0}, x_{0}\right) \in \Omega$, we have \begin{equation}\label{HypothesePetitesseGrad}
\limsup _{r \rightarrow 0} \frac{1}{r} \iint_{]t_{0}-r^{2}, t_{0}+r^{2}[ \times B\left(x_{0}, r\right)}|\vn \otimes \vU|^{2} + |\vn \otimes \vB|^{2} dxds <\epsilon^{*}, \end{equation} then $(\vU, \vB)$ is H\"older regular of exponent $\alpha$ (in the time variable and in the space variable) in a neighborhood of $\left(t_{0}, x_{0}\right)$ for some small $\alpha$ in the interval $0<\alpha<\frac{1}{3}$. \end{Theoreme}
It is worth noting here that considerable efforts have been made to weaken as much as possible the hypotheses on the pressure. Indeed, in the original paper of Caffarelli, Kohn and Nirenberg \cite{CKN} which deals with the Navier-Stokes equations, the authors assumed that $P \in (L^{\frac{5}{4}}_{t,x})_{loc}$. Later on, based on a better estimate on the pressure due to Sohr and von Wahl \cite{Sohrvon}, Lin proposed a simplified proof in \cite{Lin} and assumed that $P \in (L^{\frac{3}{2}}_{t,x})_{loc}$. Afterwards, under a more natural condition on the pressure, Seregin and Sverak \cite{SereginSve} showed the regularity of suitable solutions to the Navier-Stokes equations. Vasseur \cite{Vasseur} gave a proof relying on a method introduced by De Giorgi for elliptic equations to show that $P \in (L^{q_0}_{t}L^{1}_{x})_{loc}$ with $q_0 >1$ is actually enough.
Note also that in the setting of the MHD equations, He and Xin \cite{He2} studied uniform gradient estimates and extended the work of Caffarelli, Kohn and Nirenberg \cite{CKN} to the partial regularity theory of suitable weak solutions to the MHD equations under no condition for the magnetic field $\vB$. Recently, relying on the De Giorgi iteration developed by Vasseur for the Navier-Stokes eqautions, Jiu and Wang \cite{Jiu_Wang14} gave an alternative proof of the work of He and Xin \cite{He2} to show that one dimensional parabolic Hausdorff measure of the possible singular points is zero. In \cite{CaoWU}, Cao and Wu provided a regularity criteria suggesting that the directional derivative of the pressure $P$ is bounded in some Lebesgue spaces. We remark that all these results hold under some integrability conditions on the pressure.\\
In this article we are going one step further in the treatment of the pressure and following the ideas of \cite{CML} we will prove that a pressure $P\in \mathcal{D}'$ can be considered to obtain a more general version of Theorem \ref{Teo_CKNMHD}. Indeed, although the \emph{local} and \emph{partial} regularity theories are quite different in spirit, it is possible to combine them to obtain some new results on the regularity of the solutions of the MHD equations (\ref{EquationMHDoriginal}). To be more precise, we will use the local regularity theory given in Theorem \ref{Teo_SerrinMHD} (which does not imposes conditions on the pressure $P$) in order to generalize the partial regularity theory stated in Theorem \ref{Teo_CKNMHD}: even though $P$ is a distribution, we can give a sense to the expression (\ref{Formula_MesureMHD}) above and with some additional mild assumptions we will be able to obtain a gain of regularity. \\
Remark that the use of the local regularity theory will not be direct and we need to perform two main steps to make this theory useful: first we need to introduce and study harmonic corrections of the solutions (\emph{i.e.} new variables that differ up to harmonic functions to the original ones) and then we need to link the properties of these new variables to the old ones in a very particular manner to exploit the information carried out by (\ref{Formula_MesureMHD}), this second step will be achieved with a detailed study of the energy inequality satisfied by the Leray-type solutions of (\ref{EquationMHDoriginal}). Let us also stress here that, among other deep results -that will be highlighted in due time- it is the common framework of parabolic Morrey spaces developed in \cite{ChCHJ1} and \cite{ChCHJ2} that allows us to connect these two regularity theories.\\
Our main result which introduces a new notion of solutions for the MHD equations and generalizes the partial regularity theory is the following:
\begin{Theoreme}[Regularity for Dissipative solutions]\label{Theorem_main_original} Let $\Omega$ be a bounded domain of the form given in (\ref{DefConjuntoOmega}) and $(\vU, P, \vB)$ be a weak solution on $\Omega$ of the MHD equations \eqref{EquationMHDoriginal}. Assume that \begin{itemize} \item[1)] we have that $(\vU, \vB, P, \vF, \vG)$ satisfies the conditions: \begin{equation}\label{Hipotheses_Func} \vU, \vB \in L^{\infty}_t L^{2}_x \cap L^{2}_t \dot{H}^{1}_x(\Omega), \quad \vF, \vG \in L^{2}_{t} H^1_x(\Omega), \quad P \in \mathcal{D}'(\Omega); \end{equation} \item[2)] the solution $(\vU, P, \vB)$ is dissipative, \emph{i.e.}, the quantity \begin{eqnarray}
M&=&-\partial_t(|\vU|^2 + |\vB|^2 )+ \Delta (|\vU|^2 + |\vB|^2 ) - 2 (|\vn \otimes \vU|^2 + |\vn \otimes \vB|^2 - 2 \langle div (P \vU) \rangle \label{Formula_MesureDissipativeMHD}\\
& &- div \left( (|\vU|^2 + |\vB|^2) \vU \right) + 2 div ((\vU \cdot \vB) \vB ) + 2 (\vF \cdot \vU + \vG \cdot\vB),\notag \end{eqnarray} is well-defined as a distribution and is a locally finite non-negative measure on $\Omega$; \item[3)] there exists a positive constant $\epsilon^{*}$ such that for some $\left(t_{0}, x_{0}\right) \in \Omega$, we have \begin{equation}\label{Hypo_PetitesseGradMHD}
\limsup _{r \rightarrow 0} \frac{1}{r} \iint_{]t_{0}-r^{2}, t_{0}+r^{2}[ \times B\left(x_{0}, r\right)}|\vn \otimes \vU|^{2} + |\vn \otimes \vB|^{2} dxds <\epsilon^{*}, \end{equation} \end{itemize} then $(\vU, \vB)$ is locally bounded: for every parabolic ball $Q$ which is compactly supported in a small neighborhood of the point $(t_0, x_0)$, we have $(\vU, \vB) \in L^\infty_tL^\infty_x (Q)$. And thus, following Remark \ref{Rem_RegulariteSerrin} we obtain that $\vU, \vB\in L^\infty_t\dot H^2_x\cap L^2_t\dot H^3_x$ over a small neighborhood of the point $(t_0, x_0)$. \end{Theoreme}
Some important remarks are in order here. We first note that the assumption on $\vU, \vB$ and $\vF, \vG$ stated in the first point of the theorem are classical and can be obtained for any weak Leray-type solution of (\ref{EquationMHDoriginal}), however we only assume here that $P$ is a general distribution and this fact implies that we must define in a very specific way the quantity $M$ given in (\ref{Formula_MesureDissipativeMHD}): indeed we need to define the expression $\langle div(P \vU)\rangle$ as a very particular limit to make it meaningful (see formula \eqref{def_limlim} below). This general point of view about the pressure constitutes one of the main novelties of this article since it introduces a new class of solutions for the MHD equations, called \emph{dissipative solutions}, and it allows us to generalize all the previous results related to the partial regularity theory of such equations. Let us mention that these type of dissipative solutions were studied for the classical Navier-Stokes equations in \cite{CML1} and \cite{CML}. To end these preliminaries remarks, note also that the condition (\ref{Hypo_PetitesseGradMHD}) is rather classical in the setting of the partial regularity theory (see Hypothesis (\ref{HypothesePetitesseGrad}) in Theorem \ref{Teo_CKNMHD}). Moreover, in the framework of the previous theorem, if we denote by $\Sigma_0$ the set of points for which we have the following behavior
$$\limsup _{r \rightarrow 0} \frac{1}{r} \iint_{]t_{0}-r^{2}, t_{0}+r^{2}[ \times B\left(x_{0}, r\right)}|\vn \otimes \vU|^{2} + |\vn \otimes \vB|^{2} dx ds\geq \varepsilon^*,$$ then it can be shown, by a standard Vitali covering lemma, that the parabolic Hausdorff measure of the set $\Sigma_0$ is null, which means that this set of ``irregular points'' is actually very small (see Section 13.10 of the book \cite{PGLR1}). \\
\textbf{Outline of the paper.} In Section \ref{Section_Preliminaries} we fix some notation and we explain the main strategy that will be used in order to prove Theorem \ref{Theorem_main_original}: we will follow some steps and each one of the remaining sections will be devoted to the proof of these steps. Some technical lemmas are stated in Appendices.
\section{Notation and strategy of the proof}\label{Section_Preliminaries} We start this section by recalling some notation and useful facts. First we recall the notion of parabolic H\"older and Morrey spaces and for this we consider the homogeneous space $(\mathbb{R}\times \R, d, \mu)$ where $d$ is the parabolic quasi-distance given by \begin{equation*}
d\big((t,x), (s,y)\big)=|t-s|^{\frac{1}{2}}+|x-y|, \end{equation*} and where $\mu$ is the usual Lebesgue measure $d\mu=dtdx$. Note that the homogeneous dimension is now $Q=5$. More details about the homogeneous spaces can be found in the books \cite{Folland}, \cite{PGLR1}, \cite{Triebel}. Associated to this distance, we define homogeneous (parabolic) H\"older spaces $\dot{\mathcal{C}}^\alpha(\mathbb{R}\times \R, \R)$ where $\alpha\in ]0,1[$ by the following condition: \begin{equation}\label{Holderparabolic}
\|\vphi\|_{\dot{\mathcal{C}}^\alpha}=\underset{(t,x)\neq (s,y)}{\sup}\frac{|\vphi(t,x)-\vphi(s,y)|}{\left(|t-s|^{\frac{1}{2}}+|x-y|\right)^\alpha}<+\infty, \end{equation}
and this formula studies H\"older regularity in both time and space variables. Now, for $1< p\leq q<+\infty$, parabolic Morrey spaces $\mathcal{M}_{t,x}^{p,q}$ are defined as the set of measurable functions $\vphi:\mathbb{R}\times\R\longrightarrow \R$ that belong to the space $(L^p_{t,x})_{loc}$ such that $\|\vphi\|_{M_{t,x}^{p,q}}<+\infty$ where \begin{equation}\label{DefMorreyparabolico}
\|\vphi\|_{\mathcal{M}_{t,x}^{p,q}}=\underset{x_{0}\in \R, t_{0}\in \mathbb{R}, r>0}{\sup}\left(\frac{1}{r^{5(1-\frac{p}{q})}}\int_{|t-t_{0}|<r^{2}}\int_{B(x_{0},r)}|\vphi(t,x)|^{p}dxdt\right)^{\frac{1}{p}}. \end{equation} These spaces are generalization of usual Lebesgue spaces, note in particular that we have $\mathcal{M}_{t,x}^{p,p}=L_{t,x}^p$. We refer the readers to the book \cite{Triebel1} for a general theory concerning the Morrey spaces and Hölder continuity.\\
To continue, we will introduce a change of variables that will allow us to work with a more symmetric expression of the equations (\ref{EquationMHDoriginal}). Indeed, following Elsasser \cite{elsasser1950}, we define \begin{equation}\label{Def_ChangeVariable} \vu= \vU + \vB,\quad \vb= \vU-\vB, \quad \vf= \vF + \vG\quad \mbox{and} \quad\vg = \vF - \vG, \end{equation} and then the original system \eqref{EquationMHDoriginal} becomes \begin{equation}\label{EquationMHD} \begin{cases} \partial_{t}\vu=\Delta \vu -(\vb\cdot\vn)\vu-\vn P+\vf,\quad div (\vu) = div (\vf)=0,\\[3mm] \partial_{t}\vb=\Delta \vb -(\vu\cdot\vn)\vb- \vn P +\vg,\quad div (\vb)=div (\vg)=0,\\[3mm] \vu(0,x)=\vu_{0}(x), div(\vu_0)=0,\quad \vb(0,x)=\vb_{0}(x), div(\vb_0)=0, \end{cases} \end{equation} where, since $div(\vu)=div(\vb)=0$, we have that $P$ satisfies the equation \begin{equation}\label{EquationPression} \Delta P= - \sum^3_{i,j= 1} \partial_i \partial_j (u_i b_j), \end{equation} and from this equation, we remark that the pressure $P$ is only determined by the couple $(\vu, \vb)$. \\
Now, let $\Omega$ be a bounded subset of the form given in (\ref{DefConjuntoOmega}), we say that the couple $(\vu, \vb)\in L^\infty_tL^2_x\cap L^2_t\dot{H}^1_x(\Omega)$ satisfies the MHD equations (\ref{EquationMHD}) in the weak sense if for all $\vphi, \vpphi \in \mathcal{D}(\Omega)$ such that $div(\vphi)=div(\vpphi)=0$, we have $$ \begin{cases}
\langle\partial_{t}\vu-\Delta \vu +(\vb\cdot\vn)\vu-\vf|\vphi \rangle_{\mathcal{D}'\times \mathcal{D}}=0,\\[3mm]
\langle\partial_{t}\vb-\Delta \vb +(\vu\cdot\vn)\vb-\vg|\vpphi \rangle_{\mathcal{D}'\times \mathcal{D}}=0, \end{cases} $$ note that if $(\vu, \vb)$ are solutions of the previous system, then due to the expression (\ref{EquationPression}) there exists a pressure $P$ such that (\ref{EquationMHD}) is fulfilled in $\mathcal{D}'$. We will work from now on with the following set of hypotheses for the functions $\vu, \vb, \vf,\vg$ and $P$: \begin{equation}\label{Hypotheses_Travail} \begin{split} &\vu, \vb \in L^{\infty}_t L^{2}_x \cap L^{2}_t \dot{H}^{1}_x(\Omega), \quad \vf, \vg \in L^{2}_{t} H^1_x(\Omega), \quad P \in \mathcal{D}'(\Omega),\qquad \\[4mm]
&\limsup _{r \rightarrow 0} \frac{1}{r} \iint_{]t_{0}-r^{2}, t_{0}+r^{2}[ \times B\left(x_{0}, r\right)}|\vn \otimes \vu|^{2} + |\vn \otimes \vb|^{2} dxds <\epsilon^{*},\qquad (t_0, x_0)\in \Omega, \end{split} \end{equation} which can easily be deduced from (\ref{Def_ChangeVariable}), (\ref{Hipotheses_Func}) and (\ref{Hypo_PetitesseGradMHD}) and where the point $(t_0, x_0)\in \Omega$ will be fixed from now on. In addition, for the Elsasser form MHD equations \eqref{EquationMHD}, the corresponding dissipative distribution turns out to be \begin{eqnarray}
\lambda&=&-\partial_t(|\vu|^2 + |\vb|^2 )+ \Delta (|\vu|^2 + |\vb|^2 ) - 2 (|\vn \otimes \vu|^2 + |\vn \otimes \vb|^2 ) - \langle div \big(P (\vu + \vb)\big)\rangle \label{Formula_MesureDissipativeMHD1} \\
& & - div ( |\vu|^2 \vb + |\vb|^2 \vu ) + 2 (\vf \cdot \vu + \vg \cdot\vb) \notag, \end{eqnarray} let us remark that the quantity above includes less nonlinear terms than \eqref{Formula_MesureDissipativeMHD} in Theorem \ref{Theorem_main_original} due to the symmetric property of the Elsasser formulation.\\
Once our framework is clear, we will explain now the strategy that will be implemented for proving Theorem \ref{Theorem_main_original}:
\begin{itemize} \item{\bf Step 1} Using as a starting point the system (\ref{EquationMHD}), the variables $\vu, \vb, P$ and the initial data $\vf, \vg$ (every term under the hypotheses (\ref{Hypotheses_Travail}) above), we will derive from $\vu$ and $\vb$ two new variables, $\vv$ and $\vh$ respectively, that will be called \emph{harmonic corrections} of $\vu$ and $\vb$ as they differ (locally) from the original variables up to harmonic functions. Section \ref{Section_newvariable} will be devoted to a detailed study of these harmonic corrections since they are one key ingredient of our method: we will first show how the hypotheses on $\vu$ and $\vb$ are transmitted to the new variables $\vv$ and $\vh$ and then we will investigate the PDEs (which are of the same shape of the MHD equations) satisfied by these variables. We will see then that the global environment of the functions $\vv$ and $\vh$ is essentially better than the variables $\vu$ and $\vb$.
\item{\bf Step 2} Next, we carry out a precise study of the energy inequalities satisfied by the variables $(\vu, P, \vb)$ and by the harmonic corrections $\vv$ and $\vh$. This step is crucial since it will allow us first to define in a very specific manner the distribution \textcolor{red}{(\ref{Formula_MesureDissipativeMHD1})} and then to transfer some information from the original system to the new one satisfied by $\vv$ and $\vh$. All these computations will be performed in Section \ref{Section_link}. \item{\bf Step 3} The two previous steps allows us to obtain a better framework for the harmonic corrections $\vv$ and $\vh$. In this step we will see how to apply the usual partial regularity theory (as developed in \cite{ChCHJ2}) in order to obtain an actual gain of regularity for these variables $\vv$ and $\vh$. Note that at this step, regularity in time and space can still be obtained for the harmonic corrections $\vv$ and $\vh$. \item{\bf Step 4} This fourth step is devoted to deduce from the previous step a gain of regularity in the original variables $\vu$ and $\vb$. \item{\bf Step 5} In this step, we will recover the regularity on the variables $\vU$ and $\vB$ and then Theorem \ref{Theorem_main_original} will be completely proven. This will be done by using the local partial regularity obtained in \cite{ChCHJ1}. It is worth to remark here that by applying the local regularity Theorem \ref{Teo_SerrinMHD} we may loose some information in the time variable and we will only be able to obtain a gain regularity for the space variable. \end{itemize} The steps 3, 4 and 5 will be studied in Sections \ref{subsection_vh}, \ref{Sec_GainReguubb} and \ref{Sec_RegUB} respectively.
\section{The harmonic corrections - Step 1}\label{Section_newvariable} From the functions $\vu$ and $\vb$ we are going to derive two new variables $\vv$ and $\vh$ that locally (\emph{i.e.} over a small neighborhood of a point $(t_0,x_0)$ fixed in (\ref{Hypotheses_Travail}) above) differ from $\vu$ and $\vb$ only up to harmonic functions, but before we need to introduce some specific neighborhoods of $(t_0, x_0)$ that will fix our framework: for a radius $0<\rho$ small enough, let $Q_{\rho}$ be a bounded subset of $\Omega$ of the form \begin{equation}\label{Def_ParaBallde} Q_{\rho}:=]t_0-\rho^2,t_0+\rho^2[\times B(x_0,\rho). \end{equation} where $ B(x_0, \rho)$ is an open ball in $\R$ at center $x_0 \in \R$. When the context is clear, for usual (euclidean) balls, we will write $B_\rho$ instead of $B(x,\rho)$. We will also need a smaller subset $\Qro$ of the form (\ref{Def_ParaBallde}) above where $0< \rho_0 < \rho$. \\
Let us now construct a cut-off function $ \psi:\mathbb{R}\times \mathbb{R}^3\longrightarrow \mathbb{R}$ such that $\psi\in \mathcal{C}^{\infty}_0(\mathbb{R}\times \mathbb{R}^3, \mathbb{R})$ and \begin{equation}\label{cut_off}
supp(\psi)\subset \Qr \quad \text{and} \quad \psi\equiv 1 \mbox{ on } \Qrone. \end{equation} with $0<\rho_0 < \rho_1 < \rho$. We can now define the harmonic corrections $\vv$ and $\vh$. \begin{Definition} Assume that $\vu$ and $\vb$ satisfy the general hypotheses (\ref{Hypotheses_Travail}) and consider the localizing function $\psi$ given in (\ref{cut_off}). Then we define the variables $\vv$ and $\vh$ by: \begin{equation}\label{new_variable} \vv := - \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vu ), \quad \vh := - \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vb ). \end{equation} \end{Definition} Note that by the localizing properties of $\psi$ (in particular $\psi \equiv 1$ on $\Qro\subset Q_{\rho_1}$), by the vector identity $\vn \wedge ( \vn \wedge \vu ) = \vn (div \vu) - \Delta \vu$ and by the divergence free conditions for $\vu$ and $\vb$, we have the (local) identities \begin{equation}\label{Identite_Harmoniques1} \Delta \vv = \Delta \vu, \quad \text{and}\quad \Delta \vh = \Delta \vb, \quad \text{over} \quad \Qro, \end{equation} and these identities explain the denomination of \emph{harmonic corrections} given to the variables $\vv$ and $\vh$.\\
We state now some elementary facts on $\vv$ and $\vh$ that can be easily deduced from the general hypotheses on $\vu$ and $\vb$. \begin{Proposition}\label{Proposition_vh} Assume that $\vu, \vb \in L^{\infty}_t L^{2}_x \cap L^{2}_t \dot{H}^{1}_x(\Omega)$, then the functions $\vv, \vh$ defined by the formula \eqref{new_variable} satisfy the following two facts: \begin{itemize} \item[1)] the functions $\vv$ and $\vh$ are divergence free: $div(\vv) =0$ and $div(\vh) =0 $; \item[2)] we have $\vv, \vh \in L^{\infty}_t L^{2}_x \cap L^{2}_t H^{1}_x(\Qro)$. \end{itemize} \end{Proposition} {\bf Proof.} For the first point we recall that the divergence of a curl is always null, then by definition of the functions $\vv$ and $\vh$ given in (\ref{new_variable}) the first point follows easily.\\
For the second point we will only show that $\vv \in L^{\infty}_t L^{2}_x (\Qro)$ and $\vv \in L^{2}_t H^{1}_x(\Qro)$ and we will omit the details for $\vh$ as the arguments follow the same lines. Thus, using the vector identity $ \psi \vn \wedge \vu=\vn \wedge (\psi \vu ) - \vn \psi \wedge \vu$ and (\ref{new_variable}) we can write \begin{equation}\label{Reecriture_vv1} \vv = - \frac{1}{\Delta} \vn \wedge \big(\vn \wedge (\psi \vu )\big) + \frac{1}{\Delta} \vn \wedge \big(\vn \psi \wedge \vu\big), \end{equation} and taking the $L^2_x$-norm of $\vv$ on the ball $\Bro$ we obtain that \begin{eqnarray*}
\|\vv(t,\cdot)\|_{L^2_x (\Bro)}
&\leq& \left\| \frac{1}{\Delta} \vn \wedge \big(\vn \wedge (\psi \vu )\big)(t,\cdot) \right\|_{L^2_x (\Bro)} + \left\| \frac{1}{\Delta} \vn \wedge \big(\vn \psi \wedge \vu\big)(t,\cdot)\right\|_{L^2_x (\Bro)} \\
& \leq &\left\| \frac{1}{\Delta} \vn \wedge \big(\vn \wedge (\psi \vu )\big)(t,\cdot) \right\|_{L^2_x (\R)} + \left\| \frac{1}{\Delta} \vn \wedge \big(\vn \psi \wedge \vu\big)(t,\cdot)\right\|_{L^2_x (\R)} \\
& \leq &C\left(\|\psi \vu (t,\cdot)\|_{L^2_x (\R)} + \|\vn \psi \wedge \vu(t,\cdot) \|_{L^{\frac{6}{5}}_x (\R)}\right), \end{eqnarray*} where in the last estimate we used the Hardy–Littlewood–Sobolev inequality $\norm{(-\Delta)^{-\frac{1}{2}} f}_{L^2 (\R)} \leq C \norm{f}_{L^{\frac{6}{5}}(\R)}$. Now by the support and regularity properties of $\psi$ (see (\ref{cut_off})) and by Hölder's inequality we obtain
$$\|\vv(t,\cdot)\|_{L^2_x (\Bro)}
\leq C \left(\|\psi (t,\cdot)\|_{L^\infty_x (\Br)} + \|\vn \psi(t,\cdot)\|_{L^{3}_x (\Br)} \right) \|\vu (t,\cdot)\|_{L^{2}_x (\Br)}. $$ Then, taking the $L^\infty$ norm in the time variable, we get that $$ \norm{\vv}_{L^\infty_tL^2_x (\Qro)}
\leq C \left(\|\psi \|_{L^\infty_{t, x} (\Qr)} + \|\vn \psi\|_{L^\infty_t L^{3}_x (\Qr)} \right) \|\vu \|_{L^\infty_t L^{2}_x (\Omega)}\leq C_{\psi} \|\vu \|_{L^\infty_t L^{2}_x (\Omega)}<+\infty, $$ and we can conclude that $\vv \in L^\infty_tL^2_x (\Qro)$.
Now, using again the identity (\ref{Reecriture_vv1}) for $\vv$ and taking the $H^1_x$-norm we have \begin{eqnarray*}
\|\vv(t, \cdot)\|_{ H^1_x (\Bro)}&\leq &\left\| \frac{1}{\Delta} \vn \wedge \big(\vn \wedge (\psi \vu )\big) (t, \cdot)\right\|_{ H^1_x (\R)} + \left\| \frac{1}{\Delta} \vn \wedge \big(\vn \psi \wedge \vu\big)(t, \cdot)\right\|_{ H^1_x (\R)}\\
&\leq & C\|\psi \vu(t, \cdot)\|_{ H^1_x (\R)}+ \left\| \frac{1}{\Delta} \vn \wedge \big(\vn \psi \wedge \vu\big)(t, \cdot)\right\|_{L^2_x (\R)} \\
& &+ \left\|\vn \otimes \left(\frac{1}{\Delta} \vn \wedge \big(\vn \psi \wedge \vu\big)\right)(t, \cdot)\right\|_{L^2_x (\R)} \\
&\leq & C\|(\psi \vu)(t, \cdot)\|_{ H^1_x (\R)}+ C\| \vn \psi \wedge \vu(t, \cdot) \|_{L^{\frac{6}{5}}_x (\R)} + C\|\vn \psi \wedge \vu (t, \cdot)\|_{L^2_x (\R)}, \end{eqnarray*} where we used in the last line the same Hardy–Littlewood–Sobolev inequality. Now by the support properties of the localizing function $\psi$ and by H\"older's inequality we obtain \begin{eqnarray*}
\|\vv(t, \cdot)\|_{ H^1_x (\Bro)}&\leq &C\left(\|\psi(t,\cdot)\|_{L^\infty_x}+\|\vn \psi(t,\cdot)\|_{L^\infty_x}\right)\|\vu (t, \cdot)\|_{H^1_x (\Br)}\\
&+& C\left(\|\vn \psi(t, \cdot) \|_{L^3_x} + \|\vn \psi(t, \cdot)\|_{L^{\infty}_x} \right) \|\vu(t, \cdot) \|_{L^{2}_x (\Br)}, \end{eqnarray*} thus, taking the $L^2_t$-norm in the time variable and observing that since $\vu \in L^\infty_t L^2_x\cap L^2_t \dot{H}^1_x (\Omega)$ one trivially gets $\vu \in L^2_t H^1_x (\Omega)$, we have
$$\|\vv\|_{L^2_t H^1_x (\Qro)} \leq C_{\psi} \|\vu \|_{L^2_t H^1_x (\Omega)}<+\infty,$$ and this concludes the proof of Proposition \ref{Proposition_vh}.
$\blacksquare$\\
This first proposition is rather straightforward and we need to study in more detail the properties of the new variables $\vv$ and $\vh$ with respect to the original functions $\vu$ and $\vb$. To this end, we define the following quantities \begin{equation}\label{Def_DifferenceHarmonique} \vbe := \vu - \vv \qquad \mbox{and}\qquad \vga := \vb- \vh, \end{equation} we can see from the identity (\ref{Identite_Harmoniques1}) that $\vu$ is equal to $\vv$ on $\Qro$ up to the harmonic corrector $\vbe$ and thus we have $\Delta \vbe=0$ over $\Qro$ and we also have $\Delta \vga=0$ over $\Qro$. \begin{Remarque}\label{remark_bega}The restriction on the smaller set $\Qro$ is not essential as long as we are only interested in proving the previous Proposition \ref{Proposition_vh} and following the same ideas of Proposition \ref{Proposition_vh} we can show that $\vv, \vh \in L^{\infty}_t L^{2}_x \cap L^{2}_t H^{1}_x(\Qr)$, from which we deduce the following estimates:
$$\|\vbe\|_{L^\infty_tL^2_x (\Qr)} \leq C_{\psi} \|\vu \|_{L^\infty_tL^2_x (\Omega)}, \quad \|\vbe\|_{L^2_t H^1_x (\Qr)} \leq C_{\rho, \psi} \|\vu \|_{L^2_t H^1_x (\Omega)}$$ and
$$\|\vga\|_{L^\infty_tL^2_x (\Qr)} \leq C_{\psi} \|\vb \|_{L^\infty_tL^2_x (\Omega)}, \quad \|\vga\|_{L^2_t H^1_x (\Qr)} \leq C_{\rho, \psi} \|\vb \|_{L^2_t H^1_x (\Omega)}. $$ Note that this restriction over the set $\Qro$ will be crucial in the following Proposition \ref{Lemma_correctors}. \end{Remarque} The next proposition states a stronger estimate for $\vbe$ and $\vga$. \begin{Proposition}\label{Lemma_correctors} Under the general hypotheses (\ref{Hypotheses_Travail}) over $\vu$ and $\vb$ and using the variables $\vv$ and $\vh$ given in (\ref{new_variable}),
then the functions $\vbe $ and $\vga$ defined in (\ref{Def_DifferenceHarmonique}) satisfy $\vbe, \vga \in L^\infty_t Lip_x (\Qro)$. \end{Proposition} This is a very useful result as it gives information on the differences between the original variables $(\vu, \vb)$ and the harmonic corrections $(\vv, \vh)$.\\
\noindent{\bf Proof.} Using expression (\ref{Reecriture_vv1}) we can write $$\vbe = \vu -\vv = \vu + \frac{1}{\Delta} \vn \wedge \left( \vn \wedge (\psi \vu ) - \vn \psi \wedge \vu \right),$$ since we have the identity $\vn \wedge \vn \wedge (\psi \vu)=\vn(\vu \cdot \vn \psi) - \Delta (\psi \vu)$, as $\vu$ is divergence free, we can write $$\vbe = \vu+\frac{1}{\Delta} \left( \vn (\vu \cdot \vn \psi)\right)-\psi \vu - \frac{1}{\Delta} \vn \wedge (\vn \psi \wedge \vu ).$$ Now, since $\psi \equiv 1$ on $\Qro$ we obtain over the set $\Qro$ the identity \begin{equation}\label{betauv} \vbe = \frac{1}{\Delta} \left( \vn (\vu \cdot \vn \psi)\right) - \frac{1}{\Delta} \vn \wedge (\vn \psi \wedge \vu). \end{equation}
Thus, for all $(t,x)\in\Qro$, for any $i=1,2,3$ and denoting by $K$ the convolution kernel associated to the operator $\frac{1}{\Delta}$ (namely $\frac{1}{|x|}$), we have for each term of the formula (\ref{betauv}) above \begin{equation*}
\left|\partial_{x_i}\left( \frac{1}{\Delta} \big( \vn (\vu \cdot \vn \psi)\big) \right)(t,x)\right|= \left| \partial_{x_i} K * \big( \vn (\vu \cdot \vn \psi)\big) (t,x) \right|
\leq \sum_{j, k=1}^3 \left|\partial_{x_i} \partial_{x_k} K * \big(u_j \partial_j \psi \big) (t,x) \right| \end{equation*} and \begin{equation*}
\left| \partial_{x_i}\left( \frac{1}{\Delta} \vn \wedge (\vn \psi \wedge \vu)\right) (t, x)\right|
= \left| \partial_{x_i}K * \left( \vn \wedge (\vn \psi \wedge \vu ) \right) (t, x)\right|
\leq \sum_{j,k,\ell=1}^{3} \left|\partial_{x_i} \partial_{x_k} K * ( \partial_j \psi u_\ell) (t, x)\right|. \end{equation*} Now we recall that by construction (see (\ref{cut_off})) we have $supp(\vn \psi) \subset B^c_{\rho_1}$ with $\rho_0 < \rho_1<\rho$, thus for $(t,x) \in \Qro$, we have for $i=1,2,3$, \begin{eqnarray}
|\partial_{x_i} \vbe(t,x)|
& \leq &\sum_{j, k=1}^3 \left|\partial_{x_i} \partial_{x_k} K * \big(u_j \partial_j \psi \big) (t,x) \right| + \sum_{j, k, \ell=1}^{3} \left|\partial_{x_i} \partial_{x_k} K * ( \partial_j \psi u_\ell) (t, x)\right|\notag \\
&\leq &C\int_{\{|x-y| > \rho_1 - \rho_0 , y \in \Br\}} \frac{1}{|x-y|^3} |\vn \psi (t,y)| |\vu (t,y)| \,dy\notag \\ &\leq &C\norm{\vn \psi (t, \cdot)}_{L^2 (\Br)} \norm{\vu (t, \cdot)}_{L^2 (\Br)},\label{betau_Lip} \end{eqnarray} which implies that $\vbe \in L^\infty_t Lip_x (\Qro)$ by the properties of the localizing function $\psi$ and the fact that $\vu \in L^\infty_tL^2_x (\Omega)$. The estimate for $\vga \in L^\infty_t Lip_x (\Qro)$ can be deduced in a similar manner. This ends the proof of Proposition \ref{Lemma_correctors}.
$\blacksquare$ \begin{Remarque}\label{remark_bega1} Note that the conclusion of the Proposition \ref{Lemma_correctors} above can be obtained, \emph{mutatis mutandis}, in any proper subset of $\Qr$. \end{Remarque}
Once we have obtained some preliminary information on the local behavior of the functions $\vv$ and $\vh$, we will study now the PDEs satisfied by them. In this sense we have the following proposition: \begin{Proposition}\label{Proposition_newPressureForce} Over the set $\Qro$ of the form (\ref{Def_ParaBallde}), the functions $\vv$ and $\vh$ defined by (\ref{new_variable}) satisfy the system: \begin{equation}\label{EquationMHD_companion} \begin{cases} \partial_{t}\vv=\Delta \vv -(\vh\cdot\vn)\vv-\vn q+\vk,\\[3mm] \partial_{t}\vh=\Delta \vh -(\vv\cdot\vn)\vh- \vn r +\vl. \end{cases} \end{equation} where the functions $q, r, \vk$ and $\vl$ satisfy the following properties \begin{enumerate} \item[1)] $q \in L^{\frac{3}{2}}_{t,x} (\Qro)$ and $r \in L^{\frac{3}{2}}_{t,x}(\Qro)$, \item[2)] $div(\vk)=0$, $div(\vl)=0$ and $\vk -\vk_0 \in L^{2}_{t,x} (\Qro)$ and $\vl - \vl_0 \in L^{2}_{t,x}(\Qro)$, where \begin{equation}\label{Def_k0_l0} \vk_0=- \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vf )\qquad \mbox{and}\qquad \vl_0:= - \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vg ), \end{equation} are the harmonic corrections associated to the external forces $\vf$ and $\vg$. \end{enumerate} \end{Proposition} The system (\ref{EquationMHD_companion}) satisfied by the couple $(\vv, \vh)$ is very similar to the problem (\ref{EquationMHD}) satisfied by the couple $(\vu, \vb)$, however there are many deep differences between these two equations. Indeed: \begin{itemize} \item[$\bullet$] The main feature of the equations (\ref{EquationMHD_companion}) relies in the fact that the terms $\vn q$ and $\vn r$, which can be considered as ``modified pressures'', \emph{do not} depend on the initial pressure $P$. This fact is related to the definition of the new variables $\vv$ and $\vh$ given in (\ref{new_variable}): the first operation made over $\vu$ and $\vb$ is given the Curl operator $\vn \wedge$ that annihilates all the gradients. \item[$\bullet$] The ``modified pressures'' $q$ and $r$ in (\ref{EquationMHD_companion}) are obtained in a very particular way by using vector calculus identities and the main point here is that we can deduce some information on these objects (namely $q,r\in L^{\frac{3}{2}}_{t,x} $). \item[$\bullet$] Although equations (\ref{EquationMHD_companion}) and (\ref{EquationMHD}) are very similar, the information available for them is completely different: in (\ref{EquationMHD_companion}) we do have some control over the ``modified pressures'' while in (\ref{EquationMHD}) the pressure is only a general distribution. \end{itemize} {\bf Proof.} We begin by considering $\pat \vv$ over the set $\Qro$. Since $\pat \psi= 0$ over $\Qro$ and using the equation satisfied by $\vu$ (see (\ref{EquationMHD})) we have \begin{eqnarray} \pat \vv&=& \pat \left( - \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vu )\right)= - \frac{1}{\Delta} \vn \wedge (\psi \;\pat(\vn \wedge \vu ))\notag\\ &=&- \frac{1}{\Delta} \vn \wedge \left(\psi \left(\Delta (\vn \wedge \vu) -\vn \wedge ( (\vb\cdot\vn)\vu )+ \vn \wedge \vf \right) \right)\notag \\ & = &- \frac{1}{\Delta} \vn \wedge (\psi \Delta (\vn \wedge \vu)) + \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge ( (\vb\cdot\vn)\vu ) ) - \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vf)\notag\\ &:= &I_1 + I_2 + \vk_0.\label{equationdet_timevv} \end{eqnarray} Doing the same procedure for $\pat \vh$, we get that \begin{eqnarray} \pat \vh&=& - \frac{1}{\Delta} \vn \wedge (\psi \Delta (\vn \wedge \vb)) + \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge ( (\vu \cdot\vn)\vb ) ) - \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vg)\notag\\ &:= &I_3 + I_4 + \vl_0. \label{equationdet_timevh} \end{eqnarray} Note that $div(\vk_0)=div(\vl_0)=0$ since the divergence of a curl is always null.\\
The equations (\ref{equationdet_timevv}) and (\ref{equationdet_timevh}) are still far away from the wished result (\ref{EquationMHD_companion}). In the following lines we will display some vectorial identities in order to force the terms $I_1, I_2$ and $I_3, I_4$ to be as in (\ref{EquationMHD_companion}). This process will of course bring up additional terms and some of them will be classified as \emph{pressures} an other as \emph{external forces}. Since the terms $I_1, I_2$ and $I_3,I_4$ are symmetric, we will only detail the computations for $I_1, I_2$.\\ \begin{itemize} \item For the first term $I_1=- \frac{1}{\Delta} \vn \wedge (\psi \Delta (\vn \wedge \vu))$ given in (\ref{equationdet_timevv}), we use the classical identity $$\psi \Delta (\vn \wedge \vu) = \Delta(\psi (\vn \wedge \vu))+(\Delta \psi) (\vn \wedge \vu) -2\displaystyle{\sum_{i=1}^{3}\partial_{i}\left((\partial_{i}\psi) (\vn \wedge \vu)\right)},$$ and we obviously get that \begin{eqnarray*} -\frac{1}{\Delta} \vn \wedge (\psi \Delta (\vn \wedge \vu) ) &= &-\frac{1}{\Delta} \vn \wedge \left(\Delta(\psi \vn \wedge \vu)+(\Delta \psi)(\vn \wedge \vu) -2\displaystyle{\sum_{i=1}^{3}\partial_{i}\left((\partial_{i}\psi)(\vn \wedge \vu)\right)}\right)\\ &=& \Delta \left(-\frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vu)\right)-\frac{1}{\Delta} \vn \wedge \left((\Delta \psi)(\vn \wedge \vu) \right)\\ & &+2\frac{1}{\Delta} \vn \wedge \left(\sum_{i=1}^{3}\partial_{i}\left((\partial_{i}\psi)(\vn \wedge \vu)\right)\right) \end{eqnarray*} Recalling that $\vv=-\frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vu)$ (see expression (\ref{new_variable})) we can write $$I_1=\Delta \vv-\frac{1}{\Delta} \vn \wedge \left((\Delta \psi)(\vn \wedge \vu) \right)+2\frac{1}{\Delta} \vn \wedge \left(\sum_{i=1}^{3}\partial_{i}\left((\partial_{i}\psi)(\vn \wedge \vu)\right)\right).$$ Now, applying the vector calculus identity \begin{equation}\label{identiteVectorielle} \phi \vn \wedge \vec{Y} = \vn \wedge (\phi \vec{Y}) - (\vn \phi) \wedge \vec{Y}, \end{equation} to the last two terms of the right-hand side of the expression of $I_1$ given above, we get that \begin{eqnarray*} I_1&= &\Delta \vv-\frac{1}{\Delta} \vn \wedge\left(\vn \wedge((\Delta \psi) \vu)-\vn(\Delta \psi) \wedge \vu\right)\\ & &+2 \frac{1}{\Delta} \vn \wedge\left(\sum_{i=1}^{3} \partial_{i}\left(\vn \wedge\left(\left(\partial_{i} \psi\right) \vu\right) - (\vn(\partial_{i} \psi)) \wedge \vu \right)\right), \end{eqnarray*} which can be rewritten as \begin{equation}\label{first_term_I1} I_1 =\Delta \vv + \vk_1, \end{equation} with \begin{eqnarray} k_1&=&-\frac{1}{\Delta} \vn \wedge\left(\vn \wedge((\Delta \psi) \vu)-\vn(\Delta \psi) \wedge \vu\right)\notag\\ & &+2 \frac{1}{\Delta} \vn \wedge\left(\sum_{i=1}^{3} \partial_{i}\left(\vn \wedge\left(\left(\partial_{i} \psi\right) \vu\right) - (\vn(\partial_{i} \psi)) \wedge \vu \right)\right).\label{expression_first_force} \end{eqnarray} Remark in particular that we have $div(\vk_1)=0$ as the divergence of a curl is always null.\\
\item We now treat the second term $I_2= \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge ( (\vb\cdot\vn)\vu ))$ of (\ref{equationdet_timevv}). Using again the vectorial identity (\ref{identiteVectorielle}), we can write \begin{eqnarray*} I_2& =& \frac{1}{\Delta} \vn \wedge \left(\vn \wedge (\psi ( (\vb\cdot\vn)\vu )) - (\vn \psi ) \wedge ( (\vb\cdot\vn)\vu ) \right)\\ & = &\frac{1}{\Delta} \vn \wedge \vn \wedge (\psi ( (\vb\cdot\vn)\vu )) - \frac{1}{\Delta} \vn \wedge \left((\vn \psi ) \wedge ( (\vb\cdot\vn)\vu ) \right)\\ &= &\frac{1}{\Delta} \left( \vn div (\psi ( (\vb\cdot\vn)\vu )) \right) - \psi (\vb\cdot\vn)\vu - \frac{1}{\Delta} \vn \wedge \left((\vn \psi ) \wedge ( (\vb\cdot\vn)\vu ) \right) \end{eqnarray*} where in the last line we used the identity $\vn \wedge \vn \wedge \vY = \vn div(\vY) - \Delta \vY$. Let us now define $q_1$ and $\vk_2$ by the expressions \begin{equation}\label{expression_k2} q_1 = - \frac{1}{\Delta} div (\psi ( (\vb\cdot\vn)\vu )) \qquad \mbox{and}\qquad \vk_2 = - \frac{1}{\Delta} \vn \wedge \left((\vn \psi ) \wedge ( (\vb\cdot\vn)\vu ) \right), \end{equation} we can thus write: \begin{equation}\label{second_term_I2} I_2 = -\vn q_1 - \psi (\vb\cdot\vn)\vu + \vk_2, \end{equation} note here that we clearly have $div(\vk_2)= 0$ since again the divergence of a curl is always null.\\
We need now to treat the term $\psi (\vb\cdot\vn)\vu$ which is present in (\ref{second_term_I2}). Recalling the definitions of the functions $\vbe =\vu - \vv$ and $\vga= \vb- \vh$ given in (\ref{Def_DifferenceHarmonique}) and by the fact that $\psi \equiv 1$ on the set $\Qro$, then we have $$\psi \vu = \vv + \vbe, \quad \psi \vb = \vh + \vga \quad \text{on} \quad \Qro,$$ which implies that, on $\Qro$, we have the identities \begin{eqnarray*} \psi (\vb\cdot\vn)\vu= ((\psi \vb ) \cdot\vn )( \psi \vu ) = [(\vh + \vga) \cdot\vn](\vv + \vbe) = (\vh \cdot\vn ) \vv + \vA, \end{eqnarray*} where $ \vA := (\vh \cdot\vn)\vbe + (\vga \cdot\vn)\vv + (\vga \cdot\vn)\vbe $. Observe that on $\Qro$, we have $ \vA = \psi \vA $ and we can rewrite the quantity $\psi \vA$ in the following manner $$\psi \vA := \vn q_2 - \vk_3,$$ with $$q_2 = \frac{1}{\Delta} div (\psi \vA) \quad \text{and}\quad \vk_3 = -\psi \vA + \vn \frac{1}{\Delta} div (\psi \vA),$$ we then have \begin{equation}\label{Decomposition1} \psi (\vb\cdot\vn)\vu = (\vh \cdot\vn ) \vv + \vn q_2 - \vk_3. \end{equation} Notice also that $div(\vk_3)=0$, indeed $$div(\vk_3)=-div(\psi \vA) + div(\vn \frac{1}{\Delta} div (\psi \vA)))=-div(\psi \vA)+div(\psi \vA)=0.$$ Thus, substituting the expression (\ref{Decomposition1}) above into \eqref{second_term_I2} we obtain the following formula for the term $I_2$: \begin{equation}\label{second_term_I2_2} I_2 = -\vn q_1 - (\vh \cdot\vn ) \vv - \vn q_2 + \vk_3 + \vk_2.\\[5mm] \end{equation} \end{itemize} Now using the expression \eqref{first_term_I1} for $I_1$ and \eqref{second_term_I2_2} for $I_2$ and coming back to \eqref{equationdet_timevv} we finally obtain \begin{equation*} \pat \vv = \Delta \vv - (\vh \cdot\vn ) \vv -\vn (q_1+ q_2) + \vk_0 + \vk_1 + \vk_2 + \vk_3 . \end{equation*} Performing the same computations for the right-hand side of \eqref{equationdet_timevh}, we get a very similar equation for $\vh$ : \begin{equation*} \pat \vh = \Delta \vh - (\vv \cdot\vn ) \vh -\vn (r_1+ r_2) + \vl_0 + \vl_1 + \vl_2 + \vl_3 \end{equation*} where $$r_1 = - \frac{1}{\Delta} div \left(\psi [ (\vu\cdot\vn)\vb ]\right), \quad r_2 = \frac{1}{\Delta} div [\psi \left((\vv \cdot\vn)\vga + (\vbe \cdot\vn)\vh + (\vbe \cdot\vn)\vga \right) ],$$ and $$\vl_{1}=-\frac{1}{\Delta} \vn \wedge[\vn \wedge(\Delta \psi \vb)-\vn(\Delta \psi) \wedge \vb]+2 \frac{1}{\Delta} \vn \wedge\left[\sum_{j=1}^{3} \partial_{j}\left(\vn \wedge((\partial_{j} \psi) \vb ) - (\vn(\partial_{j} \psi)) \wedge \vb \right)\right],$$ $$\vl_2 = - \frac{1}{\Delta} \vn \wedge \left((\vn \psi ) \wedge ( (\vu\cdot\vn)\vb ) \right), \quad \vl_3 = \vn r_2 - \psi \left((\vv \cdot\vn)\vga + (\vbe \cdot\vn)\vh + (\vbe \cdot\vn)\vga\right).\\[5mm]$$ Up to now, we have obtained the following equations for the new couple $(\vv, \vh)$ over the set $\Qro$: \begin{equation*} \begin{cases} \partial_{t}\vv=\Delta \vv -(\vh\cdot\vn)\vv-\vn q+\vk,\\[3mm] \partial_{t}\vh=\Delta \vh -(\vv\cdot\vn)\vh- \vn r +\vl, \end{cases} \end{equation*} with $q= q_1 + q_2$, $r= r_1 + r_2$, $\vk= \vk_0 + \vk_1 + \vk_2 + \vk_3$ and $\vl= \vl_0 + \vl_1 + \vl_2 + \vl_3$ and where $div(\vk)=div(\vl)=0$.\\[5mm]
Once we have obtained the previous system for $\vv$ and $\vh$, we need to show that we have the two points of the Proposition \ref{Proposition_newPressureForce}, \emph{i.e.} that $q, r \in L^{\frac 32}_{t,x}(\Qro)$ and $\vk -\vk_0 , \vl - \vl_0 \in L^{2}_{t,x}(\Qro)$. Let us start with the first point of Proposition \ref{Proposition_newPressureForce}, but we will only detail the estimates for $q$ and $\vk$, as the corresponding estimates for $r$ and $\vl$ follow the same lines.
\begin{itemize} \item[1)] Let us prove here that $q \in L^{\frac{3}{2}}_{t,x} (\Qro)$. Recall that $q=q_1+q_2$ with $$q_1 = - \frac{1}{\Delta} div (\psi ( (\vb\cdot\vn)\vu )) \quad \text{and} \quad q_2 = \frac{1}{\Delta} div [\psi \left((\vh \cdot\vn)\vbe + (\vga \cdot\vn)\vv + (\vga \cdot\vn)\vbe\right)],$$ and we will study these two terms separately.\\
We start taking the $L^{\frac{3}{2}}_{x} (\Bro)$-norm of $q_1$, and we apply Hölder's inequality, the Hardy-Littlewood-Sobolev inequality $\norm{(-\Delta)^{-\frac{1}{2}} f}_{L^{\frac{9}{5}} (\R)} \leq C \norm{f}_{L^{\frac{9}{8}}(\R)}$ and we use the support properties of the function $\psi$ (recall (\ref{cut_off})) to obtain: \begin{eqnarray}
\norm{q_{1} (t, \cdot)}_{L^{\frac{3}{2}}_x (\Bro)}&\leq &C_{\rho_0} \|-\frac{1}{\Delta} div (\psi ( (\vb\cdot\vn)\vu )) (t, \cdot) \|_{L^{\frac{9}{5}}_x (\Bro)} \leq C_{\rho_0} \|-\frac{1}{\Delta} div (\psi ( (\vb\cdot\vn)\vu )) (t, \cdot) \|_{L^{\frac{9}{5}}_x (\mathbb{R}^3)}\notag \\
&\leq &C_{\rho_0} \|\psi ( (\vb\cdot\vn)\vu ) (t, \cdot) \|_{L^{\frac{9}{8}}_x (\mathbb{R}^3)} =C_{\rho_0} \|\psi ( (\vb\cdot\vn)\vu ) (t, \cdot) \|_{L^{\frac{9}{8}}_x (\Br)}\notag \\
&\leq &C_{\rho_0}\|\psi(t, \cdot)\|_{L^{\infty}_x (\Br)} \|\vb (t, \cdot) \|_{L^{\frac{18}{7}}_x (\Br)} \|\vu (t, \cdot) \|_{H^{1}_x (\Br)}.\label{estimate_q1} \end{eqnarray}
By interpolation we have the estimate $\|\vb (t, \cdot)\|_{L^{\frac{18}{7}}_x (\Br)} \leq \|\vb (t, \cdot) \|^{\frac{2}{3}}_{L^{2}_x (\Br)} \|\vb (t, \cdot) \|^{\frac{1}{3}}_{L^{6}_x (\Br)} $ and using the classical Sobolev embedding $H^1_x \hookrightarrow L^6_x$, we thus have \begin{equation*} \begin{aligned} \norm{q_{1} (t, \cdot)}_{L^{\frac{3}{2}}_x (\Bro)}
\leq C_{\rho_0} \|\psi(t, \cdot)\|_{L^{\infty}_x (\Br)}\|\vb (t, \cdot) \|^{\frac{2}{3}}_{L^{2}_x (\Br)} \|\vb (t, \cdot) \|^{\frac{1}{3}}_{H^{1}_x (\Br)} \|\vu (t, \cdot) \|_{H^{1}_x (\Br)}. \end{aligned} \end{equation*} Now, taking the $L^{\frac{3}{2}}$-norm in the time variable and using Hölder's inequality, we finally obtain \begin{equation}\label{estimate_q11}
\norm{q_{1}}_{L^{\frac{3}{2}}_{t,x} (\Qro)}\leq C_{\rho_0, \psi} \|\vb \|^{\frac{2}{3}}_{L^\infty_t L^{2}_x (\Qr)} \|\vb \|^{\frac{1}{3}}_{ L^2_t H^{1}_x (\Qr)}
\|\vu \|_{L^2_t H^{1}_x (\Qr)}<+\infty. \end{equation}
We study now the term $q_2$. We first take the $L^{\frac{3}{2}}_{x} (\Bro)$-norm of $q_2$ and following the same ideas displayed in the estimate \eqref{estimate_q1}, we obtain \begin{eqnarray} \norm{q_{2} (t, \cdot)}_{L^{\frac{3}{2}}_x (\Bro)}
&\leq& C_{\rho_0} \left\|\frac{1}{\Delta} div [\psi \left((\vh \cdot\vn)\vbe +
(\vga \cdot\vn)\vv +(\vga \cdot\vn)\vbe\right)] (t, \cdot) \right\|_{L^{\frac{9}{5}}_x (\Bro)} \notag\\
& \leq &C_{\rho_0} \left\|\psi \left((\vh \cdot\vn)\vbe + (\vga \cdot\vn)\vv + (\vga \cdot\vn)\vbe\right) (t, \cdot) \right\|_{L^{\frac{9}{8}}_x (\Br)} \notag\\
&\leq &C_{\rho_0}\|\psi(t, \cdot)\|_{L^\infty_x(\Br)}\left(\|\vh (t, \cdot)\|_{L^{\frac{18}{7}}_x (\Br)} \|\vbe (t, \cdot) \|_{H^{1}_x (\Br)}\right.\notag\\
& &\qquad\left. +\|\vga (t, \cdot) \|_{L^{\frac{18}{7}}_x (\Br)} \big(\|\vv (t, \cdot) \|_{H^{1}_x (\Br)} + \|\vbe (t, \cdot)\|_{H^{1}_x (\Br)} \big)\right).\label{estimate_q2} \end{eqnarray}
Note now that by interpolation and Sobolev embedding, we have $\|\vh \|_{L^{\frac{18}{7}}_x (\Br)}
\leq \|\vh\|^{\frac{2}{3}}_{L^{2}_x (\Br)} \|\vh \|^{\frac{1}{3}}_{H^{1}_x (\Br)} $ and using Remark \ref{remark_bega} we can write
$$\|\vga (t, \cdot) \|_{L^{\frac{18}{7}}_x (\Br)} \leq \|\vga (t, \cdot)\|^{\frac{2}{3}}_{L^{2}_x (\Br)} \|\vga (t, \cdot) \|^{\frac{1}{3}}_{H^{1}_x (\Br)}
\leq \|\vb (t, \cdot) \|^{\frac{2}{3}}_{L^{2}_x (\Br)} \|\vb (t, \cdot)\|^{\frac{1}{3}}_{H^{1}_x (\Br)}. $$
Since we also have $\norm{\vbe}_{H^1_x (\Br)} \leq C_{\rho, \psi} \|\vu \|_{H^1_x (\Br)}$, substituting all these three estimates into \eqref{estimate_q2}, one gets that \begin{equation*} \begin{aligned} \norm{q_{2} (t, \cdot)}_{L^{\frac{3}{2}}_x (\Bro)} \leq & C_{\rho_0, \psi}
\|\vh (t, \cdot)\|^{\frac{2}{3}}_{L^{2}_x (\Br)} \|\vh (t, \cdot) \|^{\frac{1}{3}}_{H^{1}_x (\Br)} \|\vu (t, \cdot) \|_{H^{1}_x (\Br)} \\
&+ C_{\rho_0, \psi} \left(\|\vb (t, \cdot) \|^{\frac{2}{3}}_{L^{2}_x (\Br)} \|\vb (t, \cdot) \|^{\frac{1}{3}}_{H^{1}_x (\Br)} \left(\|\vv (t, \cdot) \|_{H^{1}_x (\Br)} + \|\vu (t, \cdot) \|_{H^{1}_x (\Br)}\right)\right). \end{aligned} \end{equation*} Now, taking $L^{\frac{3}{2}}$-norm in the time variable, by H\"older's inequality, and recalling that by Remark \ref{remark_bega} we have $\vv, \vh \in L^{\infty}_t L^{2}_x \cap L^{2}_t H^{1}_x(\Qr)$, we can conclude that \begin{eqnarray} \norm{q_{2}}_{L^{\frac{3}{2}}_{t,x} (\Qro)}
&\leq & C_{\rho_0, \psi} \|\vh \|^{\frac{2}{3}}_{L^\infty_t L^{2}_x (\Qr)} \|\vh \|^{\frac{1}{3}}_{ L^2_t H^{1}_x (\Qr)} \|\vu \|_{L^2_t H^{1}_x (\Qr)} \notag\\
& & + C_{\rho_0, \psi} \|\vb \|^{\frac{2}{3}}_{L^\infty_t L^{2}_x (\Qr)} \|\vb \|^{\frac{1}{3}}_{ L^2_t H^{1}_x (\Qr)} \left(\|\vv \|_{L^2_t H^{1}_x (\Qr)} + \|\vu \|_{L^2_t H^{1}_x (\Qr)}\right)<+\infty. \label{estimate_q22} \end{eqnarray} Since $q=q_1+q_2$, by the previous estimates \eqref{estimate_q11} and \eqref{estimate_q22}, we finally obtain that $q \in L^{\frac{3}{2}}_{t,x} (\Qro)$.\\
\item[2)] We study now the second point of Proposition \ref{Proposition_newPressureForce}. As mentionned above (precisely, the line below the system of $(\vv, \vh))$, we know that $div(\vk)=0$, so it only remains to prove that $\vk-\vk_0=\vk_1+\vk_2+\vk_3\in L^{2}_{t,x} (\Qro)$ and we will study the terms $\vk_1, \vk_2$ and $\vk_3$ separately.\\ \begin{itemize} \item[$\bullet$]Let us now recall the expression of $\vk_1$ given in \eqref{expression_first_force}: \begin{equation*} \vk_{1}=-\frac{1}{\Delta} \vn \wedge[\vn \wedge(\Delta \psi \vu)-\vn(\Delta \psi) \wedge \vu] +2 \frac{1}{\Delta} \vn \wedge\left[\sum_{i=1}^{3} \partial_{i}\left(\vn \wedge\left(\left(\partial_{i} \psi\right) \vu\right) - (\vn(\partial_{i} \psi)) \wedge \vu \right)\right]. \end{equation*} Taking the $L^2_x$-norm of $\vk_1$ over $\Bro$, by the boundedness of Riesz transforms in Lebesgue spaces and by the Hardy-Littlewood-Sobolev inequalities, we obtain that \begin{eqnarray*}
\|\vk_{1} (t, \cdot)\|_{L^2_x (\Bro)}&\leq & \left\|\frac{1}{\Delta} \vn \wedge\vn \wedge(\Delta \psi\,\vu)(t, \cdot)\right\|_{L^2_x (\mathbb{R}^3)}+\left\|\frac{1}{\Delta} \vn \wedge(\vn(\Delta \psi) \wedge \vu) (t, \cdot)\right\|_{L^2_x (\mathbb{R}^3)}\\
& &+2\sum_{i=1}^{3}\left\| \frac{1}{\Delta} \vn \wedge \partial_{i}\left(\vn \wedge\left(\left(\partial_{i} \psi\right) \vu\right) - (\vn(\partial_{i} \psi)) \wedge \vu \right)(t, \cdot)\right\|_{L^2_x (\mathbb{R}^3)}\\
&\leq & C\left\|(\Delta \psi\, \vu)(t, \cdot)\right\|_{L^2_x (\mathbb{R}^3)}+C\left\|(\vn(\Delta \psi) \wedge \vu)(t, \cdot) \right\|_{L^{\frac65}_x (\mathbb{R}^3)}\\
& &+C\sum_{i=1}^{3}\left\|\left(\vn \wedge\left(\left(\partial_{i} \psi\right) \vu\right) - (\vn(\partial_{i} \psi)) \wedge \vu \right)(t, \cdot)\right\|_{L^2_x (\mathbb{R}^3)}. \end{eqnarray*} By the localization property of $\psi$ (see (\ref{cut_off})) and by H\"older's inequality we can write \begin{eqnarray*}
\|\vk_{1} (t, \cdot)\|_{L^2_x (\Bro)}&\leq &C\|\Delta \psi(t, \cdot)\|_{L^\infty_x (\Br)}\left\|\vu(t, \cdot)\right\|_{L^2_x (\Br)}+C\|\vn(\Delta \psi)(t, \cdot)\|_{L^{3}_x (\Br)} \|\vu(t, \cdot) \|_{L^{2}_x (\Br)}\\
&&+C\|\Delta \psi(t, \cdot)\|_{L^\infty_x (\Br)}\|\vu(t, \cdot)\|_{L^2_x (\Br)}+C\|\vn \psi(t, \cdot)\|_{L^\infty_x (\Br)}\|\vu(t, \cdot)\|_{H^1_x (\Br)}, \end{eqnarray*} thus, taking the $L^2$-norm in the time variable we obtain \begin{equation}\label{estimate_k1} \norm{\vk_{1}}_{L^2_{t, x} (\Qro)}\leq C_\psi \norm{\vu}_{L^2_t H^1_x (\Br)}<+\infty. \end{equation} \item[$\bullet$]For $\vk_2$, recalling its expression given in formula \eqref{expression_k2} and taking its $L^2_x$-norm on $\Bro$, we have by H\"older's inequality \begin{eqnarray*}
\norm{\vk_2 (t, \cdot)}_{L^2_x (\Bro)}&=& \left\|\frac{1}{\Delta} \vn \wedge \left((\vn \psi ) \wedge ( (\vb\cdot\vn)\vu ) \right) (t, \cdot) \right\|_{L^2_x (\Bro)} \\
&\leq &C_{\rho_0} \left \|\frac{1}{\Delta} \vn \wedge \left((\vn \psi ) \wedge ( (\vb\cdot\vn)\vu ) \right) (t, \cdot) \right\|_{L^\infty_x (\Bro)} \end{eqnarray*} We will use now some of the arguments displayed in the proof of Lemma \ref{Lemma_correctors}. Indeed, since $\vn \psi \subset B^c_{\rho_1}$ with $\rho_1 > \rho_0$, for $(t,x ) \in \Qro$ we have \begin{equation*} \begin{split}
\left| \frac{1}{\Delta} \vn \wedge \left((\vn \psi ) \wedge ( (\vb\cdot\vn)\vu ) \right)(t,x) \right|
&\leq C\int_{\{|x-y| > \rho_1 - \rho_0 , y \in \Br\}} \frac{1}{|x-y|^2} |\vn \psi (t,y)| |(\vb \cdot \vn )\vu (t,y)| \,dy \\ &\leq C_{\rho_0, \rho_1} \norm{\vn \psi (t, \cdot)}_{L^\infty_x (\Br)} \norm{\vb (t, \cdot)}_{L^2_x (\Br)} \norm{\vu (t, \cdot)}_{H^1_x (\Br)}. \end{split} \end{equation*} Now, taking the $L^2$-norm in the time variable of the previous inequality we obtain that \begin{equation}\label{estimate_k2} \norm{\vk_2}_{L^2_{t, x} (\Qro)} \leq C_{\rho_0, \rho_1, \psi} \norm{\vb}_{L^\infty_t L^2_x (\Qr)} \norm{\vu}_{L^2_t H^1_x (\Qr)}<+\infty. \end{equation}
\item[$\bullet$]Recall now the expression of $\vk_3$: \begin{eqnarray} \vk_3& = &- \psi \left((\vh \cdot\vn)\vbe + (\vga \cdot\vn)\vv + (\vga \cdot\vn)\vbe\right) +\vn \left(\frac{1}{\Delta} div [\psi ((\vh \cdot\vn)\vbe + (\vga \cdot\vn)\vv + (\vga \cdot\vn)\vbe ) ]\right) \notag \\ & = &- \psi \vA + \vn \frac{1}{\Delta} div (\psi \vA)\label{estimate_k31} \end{eqnarray} where we denoted $\vA : = (\vh \cdot\vn)\vbe + (\vga \cdot\vn)\vv + (\vga \cdot\vn)\vbe$.
We treat the first term in the right-hand side of \eqref{estimate_k31}. Since $\psi \equiv 1$ on the set $\Qro$, we have the following estimates in the space variable: \begin{eqnarray*}
\norm{\psi \vA(t, \cdot)}_{L^2_x (\Bro)}&=& \norm{\vA (t, \cdot)}_{L^2_x (\Bro)} = \|\big((\vh \cdot\vn)\vbe + (\vga \cdot\vn)\vv +
(\vga \cdot\vn)\vbe \big)(t, \cdot)\|_{L^2_x (\Bro)} \\
&\leq & \|\vh (t, \cdot)\|_{L^2_x (\Bro)} \| \vn \vbe (t, \cdot) \|_{L^\infty_x (\Bro)} + \|\vga (t, \cdot)\|_{L^\infty_x (\Bro)} \| \vn \vv (t, \cdot)\|_{L^2_x (\Bro)} \\
& & + \|\vga (t, \cdot)\|_{L^2_x (\Bro)} \| \vn \vbe (t, \cdot) \|_{L^\infty_x (\Bro)}. \end{eqnarray*} Then, taking the $L^2$-norm in the time variable and using Hölder's inequality we obtain \begin{eqnarray*} \norm{\psi \vA}_{L^2_{t, x} (\Qro)}
& \leq &\|\vh \|_{L^2_{t,x} (\Qro)} \| \vbe \|_{L^\infty_t W^{1, \infty}_x (\Qro)}
+ \|\vga \|_{L^\infty_{t,x} (\Qro)} \| \vv \|_{L^2_t H^1_x (\Qro)} \\
&& + \|\vga \|_{L^2_{t,x} (\Qro)} \|\vbe \|_{L^\infty_t W^{1, \infty}_x (\Qro)}. \end{eqnarray*} Now we recall that from Proposition \ref{Lemma_correctors} and Proposition \ref{Proposition_vh}, we have $\vbe, \vga \in L^\infty_t Lip_x (\Qro)$ and $\vv, \vh \in L^{\infty}_t L^{2}_x \cap L^{2}_t H^{1}_x(\Qro)$ and these informations allow us to conclude that \begin{equation}\label{estimate_k32} \norm{\psi \vA}_{L^2_{t, x} (\Qro)} < + \infty. \end{equation} It remains to study the second term in the right-hand side of \eqref{estimate_k31}. For this, we introduce now a new cut-off function $\Phi \in \mathcal{D} (\mathbb{R}\times \R)$ such that $$supp(\Phi)\subset Q_{\rho_2} \quad \text{and} \quad \Phi\equiv 1 \mbox{ on } Q_{\rho_3},$$ where $0< \rho_0 < \rho_3 < \rho_2 < \rho_1 < \rho$ (recall the general definition of parabolic balls given in (\ref{Def_ParaBallde})). Since $supp(\psi)\subset \Qr$ and $\psi\equiv 1 \mbox{ on } \Qrone$ by (\ref{cut_off}), we see that $\psi \Phi \equiv 1$ on $Q_{\rho_3}$ and $$\ \text{supp} (\psi \Phi) \subset Q_{\rho_2}, \quad \text{and} \quad \text{supp} (\psi (1-\Phi)) \subset Q_{\rho} \backslash Q_{\rho_3}.$$ With this cut-off function $\Phi$, we decompose the second term in the right-hand side of \eqref{estimate_k31} as \begin{equation}\label{estimate_k33} \vn \frac{1}{\Delta} div (\psi \vA) = \vn \frac{1}{\Delta} div (\psi \Phi \vA) + \vn \frac{1}{\Delta} div (\psi (1-\Phi) \vA) : = I_1 + I_2, \end{equation} and we treat the two terms separately. For $I_1$, taking its $L^2_{t,x}$-norm on $\Qro$ and applying the same strategy used for $\psi \vA$, we have \begin{eqnarray}
\left\|\vn \frac{1}{\Delta} div (\psi \Phi \vA)\right\|_{L^2_{t,x} (\Qro)} &\leq &C\norm{ \psi \Phi \vA }_{L^2_{t,x} (Q_{\rho_2})}
= C\|\psi \Phi \big((\vh \cdot\vn)\vbe +(\vga \cdot\vn)\vv +
(\vga \cdot\vn)\vbe \big)\|_{L^2_{t,x} (Q_{\rho_2})} \notag \\
&\leq & \; C \left( \|\vh \|_{L^2_{t,x} (Q_{\rho_2})} \| \vbe \|_{L^\infty_t W^{1, \infty}_x (Q_{\rho_2})} + \|\vga \|_{L^\infty_{t,x} (Q_{\rho_2})} \| \vv \|_{L^2_t H^1_x (Q_{\rho_2})} \right.\notag\\
&& \left. + \|\vga \|_{L^2_{t,x} (Q_{\rho_2})} \|\vbe \|_{L^\infty_t W^{1, \infty}_x (Q_{\rho_2})}\right) < + \infty,\label{estimate_k34} \end{eqnarray} where we used, following Remark \ref{remark_bega1}, the fact that we have $\vbe, \vga \in L^\infty_t Lip_x (Q_{\rho_2})$.\\
The estimate of the term $I_2$ is more subtle and different arguments shall be used: indeed taking the $L^2$-norm in the space variable, we obtain \begin{equation*}
\left\|\vn \frac{1}{\Delta} div (\psi (1-\Phi) \vA) (t, \cdot) \right\|_{L^2_{x} (\Bro)} \leq C_{\rho_0} \norm{\vn \frac{1}{\Delta} div (\psi (1-\Phi) \vA) (t, \cdot) }_{L^\infty_{x} (\Bro)}. \end{equation*} Then, by the support property of $\psi (1- \Phi)$ for any point $(t,x) \in \Qro$ we have
$$ \left|\vn \frac{1}{\Delta} div (\psi (1-\Phi) \vA) (t,x)\right| \leq \; C\int_{\{|x-y| > \rho_3 - \rho_0 , y \in \Br\}} \frac{1}{|x-y|^3} |\psi (1-\Phi) (t,y)| |\vA(t,y)| \,dy,$$ thus, by a careful study of the support properties of each term we can deduce the following estimate \begin{eqnarray*}
\left|\vn \frac{1}{\Delta} div (\psi (1-\Phi) \vA) (t,x)\right|&\leq &C_{\rho_0, \rho_3} \norm{\psi (1-\Phi) (t, \cdot)}_{L^\infty_x (\Br)} \|\vA\|_{L^2_x (\Br)} \\ &\leq & \; C_{\rho_0, \rho_3} \norm{\psi (1-\Phi) (t, \cdot)}_{L^\infty_x (\Br)} \left(\norm{\vh (t, \cdot)}_{L^2_x (\Br)} \norm{\vbe (t, \cdot)}_{H^1_x (\Br)} \right.\\ & &\left. + \norm{\vga (t, \cdot)}_{L^2_x (\Br)} \big( \norm{\vv (t, \cdot)}_{H^1_x (\Br)} + \norm{\vbe (t, \cdot)}_{H^1_x (\Br)} \big) \right) \end{eqnarray*} where we used Hölder's inequality for each term of $\vA= (\vh \cdot\vn)\vbe + (\vga \cdot\vn)\vv + (\vga \cdot\vn)\vbe$. We can thus write \begin{eqnarray*}
\left\|\vn \frac{1}{\Delta} div (\psi (1-\Phi) \vA) \right\|_{L^2_{t, x} (\Qro)}\notag &\leq & \; C_{\rho_0, \rho_3, \psi, \Phi} \left(\norm{\vh}_{L^\infty_t L^2_x (\Qr)} \norm{\vbe }_{L^2_t H^1_x (\Qr)}\right. \notag\\ & + &\left.\norm{\vga }_{L^\infty_t L^2_x (\Qr)} \big( \norm{\vv }_{L^2_t H^1_x (\Qr)} + \norm{\vbe}_{L^2_t H^1_x (\Qr)} \big) \right), \end{eqnarray*}
since by Remark \ref{remark_bega} we have the inequalities $\norm{\vbe}_{L^2_t H^1_x (\Qr)} \leq C_{\rho, \psi} \|\vu \|_{L^2_t H^1_x (\Qr)}$ and
$\norm{\vga}_{L^\infty_tL^2_x (\Qr)} \leq C_{\rho, \psi} \|\vb \|_{L^\infty_tL^2_x (\Qr)}$, then we obtain \begin{eqnarray}
\left\|\vn \frac{1}{\Delta} div (\psi (1-\Phi) \vA) \right\|_{L^2_{t, x} (\Qro)}&\leq & C_{\rho, \rho_0, \rho_3, \psi, \Phi} \left(\norm{\vh}_{L^\infty_t L^2_x (\Qr)} \norm{\vu }_{L^2_t H^1_x (\Qr)}\right. \notag\\ &+&\left. \norm{\vb }_{L^\infty_t L^2_x (\Qr)} \big( \norm{\vv }_{L^2_t H^1_x (\Qr)} + \norm{\vu}_{L^2_t H^1_x (\Qr)} \big) \right)<+\infty.\label{estimate_k35} \end{eqnarray} With estimates \eqref{estimate_k34} and \eqref{estimate_k35} we obtain the term $\vn \frac{1}{\Delta} div (\psi \vA)$ given in (\ref{estimate_k33}) belongs to $L^{2}_{t,x} (\Qro)$ and with the estimate \eqref{estimate_k32} we obtain by (\ref{estimate_k31}) that $\vk_3\in L^{2}_{t,x} (\Qro)$.\\ \end{itemize} \noindent With this previous information on $\vk_3$, and gathering together estimates \eqref{estimate_k1} and \eqref{estimate_k2}, we can finally conclude that $$\vk -\vk_0 \in L^{2}_{t,x} (\Qro).$$ \end{itemize} This completes the proof of Proposition \ref{Proposition_newPressureForce}.
$\blacksquare$
\section{Link of original system and the companion equations - Step 2}\label{Section_link} In this section, we aim to define the quantity \eqref{Formula_MesureDissipativeMHD1} by proving the following proposition. \begin{Proposition}\label{Prop_limlim} Assume that $(\vu, P, \vb)$ is a weak solution on $Q_{\rho_0}$ of the MHD equations \eqref{EquationMHD} and that $(\vu, \vb, P, \vf, \vg)$ satisfies the conditions \eqref{Hypotheses_Travail}.\\
\noindent Let $\theta \in \mathcal{D} (\mathbb{R})$ and $\varphi \in \mathcal{D} (\R)$ be two functions such that $\displaystyle{\int_{\mathbb{R}}}\theta(t) dt=1$, $\text{supp} (\theta) \subset (-1, 1)$ and $\displaystyle{\int_{\R} \varphi(x) dx=1}$, $\text{supp} (\varphi) \subset B(0, 1)$. We define the auxiliary function \begin{equation}\label{Mollifier} \phi_{\alpha, \ep} = \theta_{\alpha} (t) \varphi_{\ep} (x), \end{equation} with the standard mollifiers $\theta_{\alpha}= \frac{1}{\alpha} \theta(\frac{t}{\alpha})$, $\alpha>0$, and $ \varphi_{\ep}= \frac{1}{\ep^3} \varphi(\frac{x}{\ep})$, $\ep > 0$. Let $\widetilde{Q}_{\rho_0}$ be a subset of $Q_{\rho_0}$, then for $\alpha , \ep$ small enough, the distributions $\vu * \phi_{\alpha, \ep} $, $ \vb * \phi_{\alpha, \ep}$ and $P * \phi_{\alpha, \ep}$ are well-defined on $\widetilde{Q}_{\rho_0}$. Moreover, the limit $$\lim_{\ep \to 0} \lim_{\alpha \to 0} div \left( P * \phi_{\alpha, \ep} (\vu * \phi_{\alpha, \ep} + \vb * \phi_{\alpha, \ep}) \right), $$ exists in $\mathcal{D}'(\widetilde{Q}_{\rho_0})$ and does not depend on the functions $\theta$ and $\varphi$. \end{Proposition} Assuming this proposition is true, we can introduce the notation \begin{eqnarray}\label{def_limlim} \langle div \big(P (\vu + \vb)\big)\rangle := \lim_{\ep \to 0} \lim_{\alpha \to 0} div \left( P * \phi_{\alpha, \ep} (\vu * \phi_{\alpha, \ep} + \vb * \phi_{\alpha, \ep}) \right), \end{eqnarray} and thus, it is not hard to see that, within our framework, the quantity \begin{eqnarray*}
\lambda&=&-\partial_t(|\vu|^2 + |\vb|^2 )+ \Delta (|\vu|^2 + |\vb|^2 ) - 2 (|\vn \otimes \vu|^2 + |\vn \otimes \vb|^2 ) - div \left( |\vu|^2 \vb + |\vb|^2 \vu \right)\\ & & - \langle div \big(P (\vu + \vb)\big)\rangle + 2 (\vf \cdot \vu + \vg \cdot\vb), \end{eqnarray*} is a well-defined distribution which will lead us the following concept. \begin{Definition}[Dissipative solutions]\label{Def_dissi} Let $(\vu, P, \vb)$ be a weak solution over some subset $\Omega$ of the form (\ref{DefConjuntoOmega}) of equations (\ref{EquationMHD}) that satisfy \eqref{Hypotheses_Travail}. We will say that $(\vu, P, \vb)$ is a \emph{dissipative} solution if the distribution $\lambda$ given above is a non-negative locally finite measure on $\Omega$. \end{Definition} We will see later on how to exploit this definition in order to prove the main results of this article. Let us prove now the proposition above.\\
\noindent{\bf Proof of Proposition \ref{Prop_limlim}.} We begin by regularizing the MHD equations \eqref{EquationMHD} as follows: denoting $$\vu_{\alpha, \ep} = \vu * \phi_{\alpha, \ep},\; \vb_{\alpha, \ep} = \vb * \phi_{\alpha, \ep},\; P_{\alpha, \ep} = P * \phi_{\alpha, \ep},\; \vf_{\alpha, \ep} = \vf * \phi_{\alpha, \ep} \quad\mbox{and }\quad \vg_{\alpha, \ep} = \vg * \phi_{\alpha, \ep},$$ where $\phi_{\alpha, \ep}$ is the function given in (\ref{Mollifier}). Thus one has \begin{eqnarray*}
\partial_t(|\vu_{\alpha, \ep}|^2 + |\vb_{\alpha, \ep}|^2) &= &2 \vu_{\alpha, \ep} \cdot \pat \vu_{\alpha, \ep} + 2 \vb_{\alpha, \ep} \cdot \pat \vb_{\alpha, \ep}\\ &=&2 \vu_{\alpha, \ep} \cdot \left(\Delta \vu -(\vb\cdot\vn)\vu-\vn P+\vf \right) * \phi_{\alpha, \ep} + 2 \vb_{\alpha, \ep} \cdot \left(\Delta \vb -(\vu\cdot\vn)\vb-\vn P+\vg \right)* \phi_{\alpha, \ep}, \end{eqnarray*} which implies \begin{equation}\label{regular_alep_ub} \begin{aligned}
\partial_t(|\vu_{\alpha, \ep}|^2 + |\vb_{\alpha, \ep}|^2 )
&= \Delta (|\vu_{\alpha, \ep}|^2 + |\vb_{\alpha, \ep}|^2 )
- 2 (|\vn \otimes \vu_{\alpha, \ep}|^2 + |\vn \otimes \vb_{\alpha, \ep}|^2 ) - 2 div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep}) \right)\\ & \hspace{-0.5cm} - 2 \vu_{\alpha, \ep} \cdot \left( (\vb \cdot \vn \vu ) * \varphi_{\alpha, \ep} \right) - 2 \vb_{\alpha, \ep} \cdot \left( (\vu \cdot \vn \vb ) * \varphi_{\alpha, \ep} \right) + 2 (\vu_{\alpha, \ep} \cdot \vf_{\alpha, \ep} + \vb_{\alpha, \ep} \cdot \vg_{\alpha, \ep}), \end{aligned} \end{equation} where we used divergence free conditions when dealing with the terms involving the pressure: indeed, we have $((\vn P) * \phi_{\alpha, \ep}) \cdot \vu_{\alpha, \ep}= div \left( (P * \phi_{\alpha, \ep}) \vu_{\alpha, \ep}\right) $ and $((\vn p) * \phi_{\alpha, \ep}) \cdot \vu_{\alpha, \ep}= div \left( (P * \phi_{\alpha, \ep}) \vb_{\alpha, \ep} \right) $.\\
By the same procedure used above in (\ref{regular_alep_ub}), we obtain the following regularized equation for the companion system \eqref{EquationMHD_companion}: \begin{equation}\label{regular_alep_vh} \begin{aligned}
\partial_t(|\vv_{\alpha, \ep}|^2 + |\vh_{\alpha, \ep}|^2 )
&= \Delta (|\vv_{\alpha, \ep}|^2 + |\vh_{\alpha, \ep}|^2 )
- 2 (|\vn \otimes \vv_{\alpha, \ep}|^2 + |\vn \otimes \vh_{\alpha, \ep}|^2 ) - 2 div \left( q_{\alpha,\ep} \vv_{\alpha, \ep} + r_{\alpha,\ep} \vh_{\alpha, \ep}\right)\\ & \hspace{-0.5cm} - 2 \vv_{\alpha, \ep} \cdot \left( (\vh \cdot \vn \vv ) * \varphi_{\alpha, \ep} \right) - 2 \vh_{\alpha, \ep} \cdot \left( (\vv \cdot \vn \vh ) * \varphi_{\alpha, \ep} \right) + 2 (\vv_{\alpha, \ep} \cdot \vk_{\alpha, \ep} + \vh_{\alpha, \ep} \cdot \vl_{\alpha, \ep}). \end{aligned} \end{equation} Now, we need some convergence lemmas to help us to pass to the limit $\alpha \to 0$. For the sake of simplicity, we use the notations $$\vu_{\ep} = \vu * \varphi_{\ep},\quad \vb_{ \ep} = \vb * \varphi_{ \ep}, \quad \vf_{ \ep} = \vf * \varphi_{ \ep},\quad \vg_{ \ep} = \vg * \varphi_{ \ep},$$ to indicate that only a mollification in the space variable is considered (see (\ref{Mollifier})). \begin{Lemme}\label{lemme_limit_alpha} We have the following strong convergences on the subset $Q_{\rho_0/4}$: \begin{itemize} \item $\vu_{\alpha, \ep} \underset{\alpha \rightarrow 0}{\longrightarrow} \vu_{\ep}$ and $\vb_{\alpha, \ep} \underset{\alpha \rightarrow 0}{\longrightarrow} \vb_{\ep}$ in $L_{t,x}^{2}(Q_{\rho_0/4}) \bigcap L_{t}^{2} H_{x}^{1}(Q_{\rho_0/4})$, \item $\vec{\nabla} \otimes \vu_{\alpha, \ep} \underset{\alpha \rightarrow 0}{\longrightarrow} \vn \otimes \vu_{\ep}$ and $\vn \otimes \vb_{\alpha, \ep} \underset{\alpha \rightarrow 0}{\longrightarrow} \vec{\nabla} \otimes \vb_{\ep}$ in $L_{t,x}^{2}(Q_{\rho_0/4})$, \item $(\vu \otimes \vb) * \phi_{\alpha, \ep} \underset{\alpha \rightarrow 0}{\longrightarrow}(\vu \otimes \vb) * \varphi_{\ep}$ and $(\vb \otimes \vu) * \phi_{\alpha, \ep} \underset{\alpha \rightarrow 0}{\longrightarrow}(\vb \otimes \vu) * \varphi_{\ep}$ in $L_{t,x}^{2} (Q_{\rho_0/4})$, \item $\vf_{\alpha, \ep} \underset{\alpha \rightarrow 0}{\longrightarrow} \vf_{\ep}$ and $\vg_{\alpha, \ep} \underset{\alpha \rightarrow 0}{\longrightarrow} \vg_{\ep}$ in $L_{t,x}^{2} (Q_{\rho_0/4})$. \end{itemize} \end{Lemme} We omit the proof here, since one can easily check that these convergences hold by the definition of functions $\phi_{\alpha, \ep}$ and $\varphi_\ep$ given in (\ref{Mollifier}) and by the properties of $\vu, \vb, \vf, \vg$.\\
Note that a completelty similar result can be obtained for the variables $\vv, \vh$ and the forces $\vk, \vl$, which allows us to pass to the limit $\alpha \to 0$ for the companion equations. \begin{Remarque}\label{RemarqueStrongCV} Observe however, that by the first point of Proposition \ref{Proposition_newPressureForce}, we have an additional strong convergence for the terms $q$ and $r$, indeed: \begin{equation}\label{conver_alpha} \lim\limits_{\alpha \to 0} q * \phi_{\alpha, \ep} = q * \varphi_{\ep}, \quad \lim\limits_{\alpha \to 0} r * \phi_{\alpha, \ep} = r * \varphi_{\ep}, \quad \text{in} \quad L_{t,x}^{\frac{3}{2}} (Q_{\rho_0/4}). \end{equation} \end{Remarque} Now, with the help of Lemma \ref{lemme_limit_alpha}, we are able to pass to the limit $\alpha \to 0$ for all the terms in equality \eqref{regular_alep_ub}, except for the term involving pressure. Namely, \begin{equation}\label{regular_ep_ub} \begin{aligned}
\partial_t(|\vu_{\ep}|^2 + |\vb_{\ep}|^2 )
&= \Delta (|\vu_{\ep}|^2 + |\vb_{\ep}|^2 )
- 2 (|\vn \otimes \vu_{\ep}|^2 + |\vn \otimes \vb_{\ep}|^2 ) - 2 \vu_{ \ep} \cdot \left( (\vb \cdot \vn \vu ) * \varphi_{\ep} \right) \\ & \hspace{0.5cm} - 2 \vb_{\ep} \cdot \left( (\vu \cdot \vn \vb ) * \varphi_{\ep} \right) - 2 \lim_{\alpha \to 0} div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep}) \right) + 2 (\vu_{ \ep} \cdot \vf_\ep+ \vb_{\ep} \cdot \vg_\ep). \end{aligned} \end{equation} We can also pass to the limit $\alpha \to 0$ for every term involving in \eqref{regular_alep_vh}, \begin{equation}\label{regular_ep_vh} \begin{aligned}
\partial_t(|\vv_{\ep}|^2 + |\vh_{\ep}|^2 )
&= \Delta (|\vv_{\ep}|^2 + |\vh_{\ep}|^2 )
- 2 (|\vn \otimes \vv_{\ep}|^2 + |\vn \otimes \vh_{\ep}|^2 ) - 2 \vv_{ \ep} \cdot \left( (\vh \cdot \vn \vv ) * \varphi_{\ep} \right) \\ & \hspace{0.5cm}
- 2 \vh_{\ep} \cdot \left( (\vv \cdot \vn \vh ) * \varphi_{\ep} \right) - 2div \left( (q * \varphi_{\ep}) \vv_{\ep} + (r * \varphi_{\ep}) \vh_{\ep} \right) + 2 (\vv_{\ep} \cdot \vk_\ep+ \vh_{\ep} \cdot \vl_\ep) \end{aligned} \end{equation} Notice that, since we just assume that $P \in \mathcal{D}'$, the limit term $\lim\limits_{\alpha \to 0} div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep}) \right)$ involving the pressure remains in (\ref{regular_ep_ub}). Remark also that thanks to the additional strong convergence \eqref{conver_alpha}, there is no such term in \eqref{regular_ep_vh}.\\
We study now the limit when $\ep \to 0$ with the following lemma. \begin{Lemme}\label{lemme_limit_ep} We have the following strong convergence: \begin{itemize} \item $\vu_{\ep} \underset{\ep \rightarrow 0}{\longrightarrow} \vu$ and $\vb_{\ep} \underset{\ep \rightarrow 0}{\longrightarrow} \vb$ in $L_{t,x}^{2} (Q_{\rho_0/4}) \bigcap L_{t}^{2} \dot H_{x}^{1}(Q_{\rho_0/4})$, \item $\vec{\nabla} \otimes \vu_{\ep} \underset{\ep \rightarrow 0}{\longrightarrow} \vn \otimes \vu$ and $\vn \otimes \vb_{\ep} \underset{\ep \rightarrow 0}{\longrightarrow} \vec{\nabla} \otimes \vb$ in $L_{t,x}^{2} (Q_{\rho_0/4})$, \item $\vf_{\ep} \underset{\ep \rightarrow 0}{\longrightarrow} \vf$, $\vg_{\ep} \underset{\ep \rightarrow 0}{\longrightarrow} \vg$ in $L_{t,x}^{2} (Q_{\rho_0/4})$. \end{itemize} \end{Lemme} The proof of this lemma is again straightforward and we omit the details.\\
As pointed out by the Remark \ref{RemarqueStrongCV} and by the first point of Proposition \ref{Proposition_newPressureForce}, for new pressure terms $q$ and $r$, we have the following strong convergence:
\begin{equation}\label{conver_ep}
\lim\limits_{\ep \to 0} q * \varphi_{\ep} = q, \quad
\lim\limits_{\ep \to 0} r * \varphi_{\ep} = r, \quad \text{in} \quad L_{t,x}^{\frac{3}{2}} (Q_{\rho_0/4})
\end{equation} which allows us to pass to the limit for the term involving $q$ and $r$ in equality \eqref{regular_ep_vh}.\\
In order to deal with the general limit when $\ep,\alpha\to 0$, we introduce the following two distributions: \begin{equation}\label{Def_MuEspNuEsp} \begin{split} \mu_\ep &= 2 \vu_{ \ep} \cdot \left( (\vb \cdot \vn \vu ) * \varphi_{\ep} \right) + 2 \vb_{\ep} \cdot \left( (\vu \cdot \vn \vb ) * \varphi_{\ep} \right)
- div (|\vu|^2 \vb + |\vb|^2 \vu)\\ &and\\ \eta_\ep &= 2 \vv_{ \ep} \cdot \left( (\vh \cdot \vn \vv ) * \varphi_{\ep} \right) + 2 \vh_{\ep} \cdot \left( (\vv \cdot \vn \vh ) * \varphi_{\ep} \right)
- div (|\vv|^2 \vh + |\vh|^2 \vv). \end{split} \end{equation} With these two quantities at hand, we pass to the limit $\ep \to 0$ for the equalities \eqref{regular_ep_ub} and \eqref{regular_ep_vh} to get \begin{equation}\label{regular_ub} \begin{aligned}
\partial_t(|\vu|^2 + |\vb|^2 )
&= \Delta (|\vu|^2 + |\vb|^2 )
- 2 (|\vn \otimes \vu|^2 + |\vn \otimes \vb|^2 ) - div (|\vu|^2 \vb + |\vb|^2 \vu) \\ & \hspace{0.5cm} - 2 \lim_{\ep \to 0} \lim_{\alpha \to 0} div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep}) \right) - \lim_{\ep \to 0} \mu_\ep
+ 2 (\vu \cdot \vf+ \vb \cdot \vg), \end{aligned} \end{equation} and \begin{equation}\label{regular_vh} \begin{aligned}
\partial_t(|\vv|^2 + |\vh|^2 )
&= \Delta (|\vv|^2 + |\vh|^2 )
- 2 (|\vn \otimes \vv|^2 + |\vn \otimes \vh|^2 ) - div (|\vv|^2 \vh + |\vh|^2 \vv) \\ & \hspace{0.5cm} -2 div \left( q \vv + r \vh \right) - \lim_{\ep \to 0} \eta_\ep + 2 (\vv \cdot \vk+ \vh \cdot \vl) , \end{aligned} \end{equation}
Recall that we aim to give a sense to the quantity $$ \lim_{\ep \to 0} \lim_{\alpha \to 0} div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep}) \right). $$
In order to achieve this goal, we shall need the following proposition of $\mu_\ep, \eta_\ep$, which links essentially the original MHD system and the companion system. \begin{Proposition}\label{lemme_limit} We have the following convergence in $\mathcal{D}' (Q_{\rho_0/4})$ for $\mu_\ep$ and $\eta_\ep$: \begin{equation}\label{limit_muetq} \lim_{\ep \to 0} \mu_\ep - \eta_\ep = 0. \end{equation} \end{Proposition} Before giving the detailed proof of Proposition \ref{lemme_limit}, let us first assume this lemma is proven and let us continue the proof of Proposition \ref{Prop_limlim}.\\
Without loss of generality, we set $\widetilde{Q}_{\rho_0} := Q_{\rho_0/4}$, then $\vu * \phi_{\alpha, \ep} $, $ \vb * \phi_{\alpha, \ep}$ and $P * \phi_{\alpha, \ep}$ are well-defined on $\widetilde{Q}_{\rho_0}$. Indeed, if we choose $\ep < \frac{\rho_0}{2}$, then for any point $(t,x) \in Q_{\rho_0/4}$, we have $Q_{t,x,\ep} = ]t-\ep^2, t+\ep^2[ \times B(x, \ep)\subset Q_{\rho}$.\\
Next, we rewrite equalities \eqref{regular_ub} and \eqref{regular_vh} in the following manner: \begin{eqnarray*}
\lim_{\ep \to 0} \mu_\ep +2 \lim_{\ep \to 0} \lim_{\alpha \to 0} div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep})\right)&=& -\partial_t(|\vu|^2 + |\vb|^2 ) + \Delta (|\vu|^2 + |\vb|^2 )\\
& &- 2 (|\vn \otimes \vu|^2 + |\vn \otimes \vb|^2 )- div (|\vu|^2 \vb + |\vb|^2 \vu)+ 2 (\vu \cdot \vf+ \vb \cdot \vg), \end{eqnarray*} and \begin{eqnarray*}
- \lim_{\ep \to 0} \eta_\ep &=& \partial_t(|\vv|^2 + |\vh|^2 )- \Delta (|\vv|^2 + |\vh|^2 )+ 2 (|\vn \otimes \vv|^2 + |\vn \otimes \vh|^2 ) + div (|\vv|^2 \vh + |\vh|^2 \vv) \\ &&+2 div \left( q \vv + r \vh \right) -2 (\vv \cdot \vk+ \vh \cdot \vl). \end{eqnarray*} Thus summing these two terms we can write \begin{eqnarray*}
\lim_{\ep \to 0} (\mu_\ep-\eta_\ep) +2 \lim_{\ep \to 0} \lim_{\alpha \to 0} div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep})\right)&=& -\partial_t(|\vu|^2 + |\vb|^2 ) + \Delta (|\vu|^2 + |\vb|^2 )\\
& &- 2 (|\vn \otimes \vu|^2 + |\vn \otimes \vb|^2 )- div (|\vu|^2 \vb + |\vb|^2 \vu)\\
&&+ 2 (\vu \cdot \vf+ \vb \cdot \vg)+\partial_t(|\vv|^2 + |\vh|^2 )- \Delta (|\vv|^2 + |\vh|^2 ) \\
&&+ 2 (|\vn \otimes \vv|^2 + |\vn \otimes \vh|^2 ) + div (|\vv|^2 \vh + |\vh|^2 \vv)\\ &&+2 div \left( q \vv + r \vh \right) -2 (\vv \cdot \vk+ \vh \cdot \vl). \end{eqnarray*} Now we use the fact stated in formula \eqref{limit_muetq}, which is the conclusion of Proposition \ref{lemme_limit}, and we obtain the following identity \begin{eqnarray*} 2 \lim_{\ep \to 0} \lim_{\alpha \to 0} div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep})\right)
&=& -\partial_t(|\vu|^2 + |\vb|^2 ) + \Delta (|\vu|^2 + |\vb|^2 )\\
& &- 2 (|\vn \otimes \vu|^2 + |\vn \otimes \vb|^2 )- div (|\vu|^2 \vb + |\vb|^2 \vu)\\
&&+ 2 (\vu \cdot \vf+ \vb \cdot \vg)+\partial_t(|\vv|^2 + |\vh|^2 )- \Delta (|\vv|^2 + |\vh|^2 ) \\
&&+ 2 (|\vn \otimes \vv|^2 + |\vn \otimes \vh|^2 ) + div (|\vv|^2 \vh + |\vh|^2 \vv)\\ &&+2 div \left( q \vv + r \vh \right) -2 (\vv \cdot \vk+ \vh \cdot \vl),\end{eqnarray*} which shows the existence of the limit $ \lim\limits_{\ep \to 0} \lim\limits_{\alpha \to 0} div \left( P_{\alpha,\ep} (\vu_{\alpha, \ep} + \vb_{\alpha, \ep})\right)$ in $\mathcal{D}'(Q_{\rho_0/4})$ since all the terms present in the right-hand side of the previous formula are well defined and this ends the proof of Proposition \ref{Prop_limlim}.
$\blacksquare$\\
The rest of this section is devoted to the proof of Proposition \ref{lemme_limit} which mainly relies on an idea of Duchon and Robert in \cite{DuchonRobert} which studied some deep properties of energy dissipation for weak solutions in the Navier-Stokes equations. Let us remark that a similar result for the MHD equations has been given in \cite{GaoTan}, however, for the sake of completeness, we state and prove the corresponding results in the Elsasser formulation of MHD system \eqref{EquationMHD}. \\
\noindent{\bf Proof of Proposition \ref{lemme_limit}.} In order to simply the notations, we introduce the following operator, which denotes the difference of the translation in space variable of a function $\vX : \mathbb{R} \times \R \longrightarrow \R$ and the function $\vX$ itself: $$ \delta_{y}[\vX ](t, x) : = \vX (t, x-y) - \vX (t,x). $$ We will need the following general result: \begin{Lemme}\label{prop_NRST} Let $\vX, \vY, \vZ \in L^{3}_{t,x} (Q_{\rho_0})$ such that $div(\vX)=div(\vY)=div( \vZ)=0$. Let $\theta \in \mathcal{D}(\R)$ be a smooth function on $\R$ such that $\displaystyle{\int_{\R}} \theta(x)dx =1$ and $supp(\theta) \subset B(0,1)$. We define $\theta_\ep (x)= \frac{1}{\ep^3} \theta (\frac{x}{\ep})$ with $\ep > 0$. Then we define the following four quantities for all $(t,x) \in Q_{\rho_0/4} \subset Q_{\rho_0}$: \begin{equation}\label{four_quantities} \begin{aligned} &N_{\ep} (\vX, \vY, \vZ) (t, x)=(\vY * \theta_{\ep}) \cdot\left([(\vX \cdot \vn ) \vZ] * \theta_{\ep}\right) + (\vZ * \theta_{\ep}) \cdot\left([(\vX \cdot \vn ) \vY] * \theta_{\ep}\right)-div((\vY \cdot \vZ) \vX)\\ &R_{\ep} (\vX, \vY, \vZ) (t, x)=\int_{\R} \left(\vn \theta_{\ep}(y) \cdot \delta_{y}[\vX ](t, x) \right) \left(\vY(x-y)-\vZ(x)\right) \cdot \left(\vZ (x-y) - \vY (x)\right)d y\\ &S_{\ep} (\vX, \vY, \vZ) (t, x)= \int_{\R} \left( \vn \theta_{\ep}(y) \cdot \delta_{y}[\vX ](t, x) \right) \left(\delta_{y}[\vY ](t, x)\right) \cdot\left((\vZ * \theta_\ep)(x)-\vY(x)\right) d y\\ &T_{\ep} (\vX, \vY, \vZ) (t, x)= \int_{\R} \left(\vn \theta_{\ep}(y) \cdot \delta_{y}[\vX ](t, x)\right) \left(\delta_{y}[\vZ ](t, x) \right) \cdot\left((\vY * \theta_\ep)(x)-\vZ(x)\right) d y \end{aligned} \end{equation} with $0<\ep< \frac{\rho_0}{2}$. Then, we have the limit $$\lim_{\ep \to 0} (N_{\ep}+R_{\ep}-S_{\ep}-T_{\ep}) (\vX, \vY, \vZ)= 0.$$ \end{Lemme} The proof of Lemma \ref{prop_NRST} is technical and can be found in the Appendix below. The previous lemma will not be enough for our purposes and the following result gives a more precise limit of the previous quantities $(S_{\ep} + T_{\ep} - R_{\ep}) (\vX, \vY, \vZ)$ under an additional condition on $\vX, \vY, \vZ$. \begin{Lemme}\label{lemma_Tep} Let $\vX, \vY, \vZ \in L_{t,x}^3 (\Qro)$ with $div(\vX)=div(\vY)=div( \vZ)=0$. Assume that at least one of them belongs to $L_t^\infty Lip_x (\Qro)$, then we have $$\lim_{\ep \to 0}(S_{\ep} + T_{\ep} - R_{\ep}) (\vX, \vY, \vZ) = 0 \quad \text{in} \quad L^1_{t,x} (Q_{\rho_0/4}),$$ where $S_{\ep}, T_{\ep}, R_{\ep}$ are the same as in Lemma \ref{prop_NRST}. \end{Lemme} Let us postpone again the proof of Lemma \ref{lemma_Tep} to the Appendix.\\
Now, with these two lemmas at hand we can continue the proof of Proposition \ref{lemme_limit}. Indeed, we start noting that the quantites $\mu_\ep$ and $\eta_\ep$ given in (\ref{Def_MuEspNuEsp}) can be rewritten with the help of the quantity $N_\ep (\vX, \vY, \vZ)$ defined in Lemma \ref{prop_NRST}. To be more precise, we have \begin{equation*} \begin{aligned} \mu_\ep = N_{\ep} (\vu, \vb, \vb) + N_{\ep} (\vb, \vu, \vu) \quad \mbox{and}\quad \eta_\ep = N_{\ep} (\vv, \vh, \vh) + N_{\ep} (\vh, \vv, \vv). \end{aligned} \end{equation*} Thus, proving the limit \eqref{limit_muetq} turns out to prove that \begin{equation}\label{limit_ofNep} \begin{aligned} & \lim_{\ep \to 0} N_{\ep} (\vu - \vv, \vb, \vb) + N_{\ep} (\vv, \vb-\vh, \vb) - N_{\ep} (\vv, \vh, \vb-\vh) \\ &\hspace{0.5cm}+ N_{\ep} (\vb- \vh, \vu, \vu) + N_{\ep} (\vh, \vu-\vv, \vu) - N_{\ep} (\vh, \vv, \vu-\vv) =0. \end{aligned} \end{equation} Since $\vu, \vv, \vb, \vh \in L^\infty_t L^2_x \cap L_{t}^2H_{x}^1 (\Qr)$, we obtain easily that $\vu, \vv, \vb, \vh \in L^3_{t,x}(\Omega)$ by interpolation. By Lemma \ref{prop_NRST}, we know that $ \lim\limits_{\ep \to 0} N_{\ep} (\vX, \vY, \vZ) = \lim\limits_{\ep \to 0} (S_{\ep} + T_{\ep} -R_{\ep} ) (\vX, \vY, \vZ) $ for any $\vX, \vY, \vZ \in L^3_{t,x}(\Qr)$, so we may replace all the $N_\ep$ by $(S_{\ep} + T_{\ep} -R_{\ep} )$ in the formula above. We have moreover that $\vu-\vv, \vb - \vh \in L^\infty Lip_x (\Qro)$ from Proposition \ref{Lemma_correctors}, so applying Lemma \ref{lemma_Tep} to each term on the left-hand side of \eqref{limit_ofNep} we can conclude that $$\lim_{\ep \to 0} \mu_\ep - \eta_\ep = 0,$$ and this completes the proof of Lemma \ref{lemme_limit}
$\blacksquare$\\
\section{Gain of regularity for new variables $\vv$ and $\vh$ - Step 3}\label{subsection_vh} In this section, we will obtain a regularity result for the new variables $\vv$ and $\vh$ (which satisfy the system (\ref{EquationMHD_companion})) by using the partial regularity theory obtained in \cite{ChCHJ2} and stated in Theorem \ref{Teo_CKNMHD} above. To achieve this task, we only need to check the hypotheses \emph{1)-3)} of this theorem. Indeed, from Proposition \ref{Proposition_vh} and Proposition \ref{Proposition_newPressureForce}, we have already shown that $\vv, \vh \in L^{\infty}_t L^{2}_x \cap L^{2}_t H^{1}_x(\Qro)$, $q, r \in L^{\frac{3}{2}}_{t,x} (\Qro) $, $div(\vk) =0$, $div(\vl) =0$ and $\vk -\vk_0 \in L^{2}_{t,x} (\Qro)$ and $\vl - \vl_0 \in L^{2}_{t,x}(\Qro)$. It still remains to show the suitability of $(\vv, \vh)$, the local information of $\vk$ and $\vl$ and the small gradient condition \eqref{HypothesePetitesseGrad} on $\vv$ and $\vh$.
\begin{itemize} \item \emph{Suitability of $(\vv, \vh)$}. Let us remark that, from the identity \eqref{regular_ub} and by the dissipativity assumption over $(\vu, \vb)$, we get that the quantity \begin{eqnarray*}
\lambda&=&-\partial_t(|\vu|^2 + |\vb|^2 )+ \Delta (|\vu|^2 + |\vb|^2 ) - 2 (|\vn \otimes \vu|^2 + |\vn \otimes \vb|^2 )- div ( |\vu|^2 \vb + |\vb|^2 \vu )\\ & & - 2 \langle div \big(P (\vu + \vb)\big)\rangle + 2 (\vf \cdot \vu + \vg \cdot\vb) = \lim\limits_{\ep \to 0} \mu_\ep \end{eqnarray*} is well-defined as a distribution and moreover we have $\lambda \geq 0$ on $\Qr$. As in addition we have (by Lemma \ref{lemme_limit}) that $\lim\limits_{\ep \to 0} \mu_\ep = \lim\limits_{\ep \to 0} \eta_\ep$, it is easy to find that the quantity \begin{equation*} \begin{aligned} \lim\limits_{\ep \to 0} \eta_\ep
&= -\partial_t(|\vv|^2 + |\vh|^2 )
+ \Delta (|\vv|^2 + |\vh|^2 )
- 2 (|\vn \otimes \vv|^2 + |\vn \otimes \vh|^2 ) - div (|\vv|^2 \vh + |\vh|^2 \vv) \\ & \hspace{0.5cm} -2 div ( q \vv + r \vh ) + 2 (\vv \cdot \vk+ \vh \cdot \vl), \end{aligned} \end{equation*} is a non-negative locally finite Borel measure on $\Qro$, which implies immediately the suitability of $(\vv, \vh)$. \item \emph{Local information of $\vk$ and $\vl$}. First, we recall the definition of $ \vk_0$ and $ \vl_0$: \begin{equation*} \begin{split} \vk_0 = - \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vf ) \quad \text{and} \quad \vl_0 = - \frac{1}{\Delta} \vn \wedge (\psi \vn \wedge \vg ), \end{split} \end{equation*} and using the vector calculus identities $ \psi \vn \wedge \vf=\vn \wedge (\psi \vf) - \vn \psi \wedge \vf$ and $\vn \wedge \vn \wedge (\psi \vf)=\vn(\vf \cdot \vn \psi) - \Delta (\psi \vf)$, we get that \begin{eqnarray*} \vk_0=- \frac{1}{\Delta} \vn \wedge \left(\vn \wedge (\psi \vf) - \vn \psi \wedge \vf \right) = - \frac{1}{\Delta} \left( \vn (\vf \cdot \vn \psi)\right) + \psi \vf + \frac{1}{\Delta} \vn \wedge (\vn \psi \wedge \vf) =\vf + \vbe_{\vf}, \end{eqnarray*} where $\vbe_{\vf}:= - \frac{1}{\Delta} \left( \vn (\vf \cdot \vn \psi)\right) + \frac{1}{\Delta} \vn \wedge (\vn \psi \wedge \vf)$. Doing the same computation for $\vl_0$, we get the decomposition \begin{equation*} \begin{split} \vl_0 = \vg + \vga_{\vg} \quad \text{with} \quad \vga_{\vg} := - \frac{1}{\Delta} \left( \vn (\vg \cdot \vn \psi)\right) + \frac{1}{\Delta} \vn \wedge (\vn \psi \wedge \vg). \end{split} \end{equation*} By the hypotheses \eqref{Formula_MesureDissipativeMHD1}, we have $\vf, \vg \in L^2_{t}H^1_x (\Qr)$ for the forces $\vf, \vg$ in the MHD system \eqref{EquationMHD}. From the proof and the result of Proposition \ref{Lemma_correctors}, we can deduce that $\vbe_{\vf}, \vga_{\vf}$ belong to $ L^2_t Lip_x (\Qro)$. Indeed, using the similar computation as in \eqref{betau_Lip}, we have \begin{equation*} \begin{split}
&\|\vbe_{\vf}\|_{L^2_t H^1_x(\Qro)}
\leq C_{\rho_0} \|\vbe_{\vf}\|_{L^2_t \text{Lip}_x(\Qro)}
\leq C_{\rho_0} \|\vf\|_{L^2_{t,x}(\Qr)}
\leq C_{\rho_0} \|\vf\|_{L^2_t H^1_x(\Qr)}\\
&\|\vga_{\vg}\|_{L^2_t H^1_x(\Qro)}
\leq C_{\rho_0} \|\vga_{\vg}\|_{L^2_t \text{Lip}_x(\Qro)}
\leq C_{\rho_0} \|\vg\|_{L^2_{t,x}(\Qr)}
\leq C_{\rho_0} \|\vg\|_{L^2_t H^1_x(\Qr)}. \end{split} \end{equation*} Thus, $\vk_0, \vl_0 \in L^2_t H^1_x(\Qro) \subset L^2_{t,x} (\Qro)$ and it results in $$\vk, \vl \in L^{2}_{t,x} (\Qro) \quad \text{and} \quad \mathds{1}_{\Qro} \vk, \mathds{1}_{\Qro}\vl \in L^2_{t,x} \subset \mathcal{M}_{t,x}^{\frac{10}{7}, 2}.$$ \item \emph{Small gradient condition on $\vv$ and $\vh$}. As we know from Proposition \ref{Lemma_correctors} that $\vbe = \vu -\vv$ and $\vga= \vb -\vh$ belong to $L^\infty_t Lip_x (\Qro)$, thus for any $0<r< \rho_0$, we have \begin{equation*} \begin{split}
\frac{1}{r} \iint_{Q_r}|\vn \otimes \vbe \,(s, y)|^{2} + |\vn \otimes \vga \,(s, y)|^{2} dyds \leq C r^4 \left(\|\vbe\|^2_{L^\infty_t Lip_x (\Qro)} + \|\vga\|^2_{L^\infty_t Lip_x (\Qro)} \right) \end{split} \end{equation*} which results in \begin{equation*} \begin{split} \limsup _{r \rightarrow 0}
\frac{1}{r} \iint_{{Q_r}}|\vn \otimes \vu \,(s, y)|^{2} + |\vn \otimes \vb \,(s, y)|^{2} dyds
= \limsup _{r \rightarrow 0} \frac{1}{r} \iint_{{Q_r}}|\vn \otimes \vv \,(s, y)|^{2} + |\vn \otimes \vh \,(s, y)|^{2} dyds, \end{split} \end{equation*} so the small gradient condition \eqref{HypothesePetitesseGrad} is essentially equal to the following condition: there exists a positive constant $\epsilon^{*}$ which depends only on $\tau_{0} >5$ such that for some $\left(t_{0}, x_{0}\right) \in \Qro$, we have \begin{equation}\label{smallgradub}
\limsup _{r \rightarrow 0} \frac{1}{r} \iint_{\left]t_{0}-r^{2}, t_{0}+r^{2}\right[ \times B(x_0, r)}|\vn \otimes \vu \,(s, y)|^{2} + |\vn \otimes \vb \,(s, y)|^{2} dyds<\epsilon^{*}. \end{equation} \end{itemize} We have now verified all the assumption of Theorem \ref{Teo_CKNMHD} and we can thus conclude a gain of information for the variables $\vv$ and $\vh$: the couple $(\vv, \vh)$ is H\"older regular (in the time variable and in the space variable) in a neighborhood of $\left(t_{0}, x_{0}\right)$ and from this fact we can easily deduce that \begin{equation}\label{gain_reg_vh} \mathds{1}_{Q_{R_0}} \vv \in \mathcal{M}_{t,x}^{3, \tau_{0}}\qquad \mbox{and}\qquad \mathds{1}_{Q_{R_0}} \vh \in \mathcal{M}_{t,x}^{3, \tau_{0}}, \qquad \tau_0 > 5. \end{equation}
\section{Gain of regularity for the variables $\vu$ and $\vb$ - Step 4}\label{Sec_GainReguubb} This section is devoted to the proof of the following statement in which we have a gain of regularity in the original variables $\vu$ and $\vb$. \begin{Theoreme}\label{Theorem_main} Let $\Omega$ be a bounded domain of $\mathbb{R} \times \R$ and $(\vu, P, \vb)$ be a weak solution on $\Omega$ of the MHD equations \eqref{EquationMHD}. Assume that \begin{itemize} \item[1)] $(\vu, \vb, P, \vf, \vg)$ satisfies the conditions : $$\vu, \vb \in L^{\infty}_t L^{2}_x \cap L^{2}_t \dot{H}^{1}_x(\Omega), \quad \vf, \vg \in L^{2}_{t} H^1_x(\Omega), \quad P \in \mathcal{D}'(\Omega);$$ \item[2)] $(\vu, P, \vb)$ is dissipative in the sense of Definition \ref{Def_dissi}; \item[3)] there exists a positive constant $\epsilon^{*}$ which depends only on $\tau_{0} >5$ such that for some $\left(t_{0}, x_{0}\right) \in \Omega$, we have \begin{equation}\label{smallgradub}
\limsup _{r \rightarrow 0} \frac{1}{r} \iint_{\left]t_{0}-r^{2}, t_{0}+r^{2}\right[ \times B(x_0, r)}|\vn \otimes \vu \,(s, y)|^{2} + |\vn \otimes \vb \,(s, y)|^{2} dyds<\epsilon^{*}, \end{equation} \end{itemize} then there exists a (parabolic) neighborhood $Q_{R_0}$ of $(t_{0}, x_{0})$ such that \begin{eqnarray}\label{gain_reg_ub} \mathds{1}_{Q_{R_0}} \vec{u} \in \mathcal{M}_{t,x}^{3, \tau_{0}}, \quad \mathds{1}_{Q_{R_0}} \vb \in \mathcal{M}_{t,x}^{3, \tau_{0}}. \end{eqnarray} \end{Theoreme} \textbf{Proof.} Based on the arguments in the Section \ref{subsection_vh} and the conclusion of Proposition \ref{Proposition_vh}, it remains to show that the local information \eqref{gain_reg_vh} on $(\vv, \vh)$ implies the local information \eqref{gain_reg_ub} on $(\vu, \vb)$.
Since $\tau_0 >3$, by the (generic) local property of Morrey spaces $\|\mathds{1}_{Q_{R_0}}\vf\|_{\mathcal{M}_{t,x}^{3, \tau_0}} \leq C\|\mathds{1}_{Q_{R_0}}\vf\|_{\mathcal{M}_{t,x}^{\tau_0, \tau_0}} = C\|\vf\|_{L_{t,x}^{ \tau_0}(Q_{R_0})}$ and by the identity $\vu=\vv+\vbe$ (recall (\ref{Def_DifferenceHarmonique})) we obtain that \begin{equation*} \begin{split}
\|\mathds{1}_{Q_{R_0}}\vu\|_{\mathcal{M}_{t,x}^{3, \tau_0}}
\leq \|\mathds{1}_{Q_{R_0}}\vv\|_{\mathcal{M}_{t,x}^{3, \tau_0}}+ \|\mathds{1}_{Q_{R_0}}\vbe\|_{\mathcal{M}_{t,x}^{3, \tau_0}}
\leq \|\mathds{1}_{Q_{R_0}}\vv\|_{\mathcal{M}_{t,x}^{3, \tau_0}}+ \|\vbe\|_{L_{t,x}^{\infty}(Q_{R_0})} < +\infty, \end{split} \end{equation*} where we used the embedding $L_{t,x}^{\infty}(Q_{R_0}) \subset L_{t,x}^{\tau_0}(Q_{R_0})$. By exactly the same computations, we thus have $\mathds{1}_{Q_{R_0}} \vb \in \mathcal{M}_{t,x}^{3, \tau_{0}}$ and this ends the proof of Theorem \ref{Theorem_main}.
$\blacksquare$
\section{Regularity of the variables $\vU$ and $\vB$ - Step 5}\label{Sec_RegUB} This section is devoted to end the proof of Theorem \ref{Theorem_main_original} by making use of Theorem \ref{Theorem_main} and the local regularity result obtained in \cite{ChCHJ1} (see Theorem \ref{Teo_SerrinMHD}).\\
\noindent\textbf{Proof of Theorem \ref{Theorem_main_original}.} First, since we have the relationships $\vu= \vU + \vB$, $\vb= \vU-\vB$, $\vf= \vF + \vG$ and $\vg = \vF - \vG$, one can check that hypothesis \emph{2)} in Theorem \ref{Theorem_main_original} implies that the quantity $\lambda$ \begin{eqnarray*}
\lambda&=&-\partial_t(|\vu|^2 + |\vb|^2 )+ \Delta (|\vu|^2 + |\vb|^2 ) - 2 (|\vn \otimes \vu|^2 + |\vn \otimes \vb|^2 )- div ( |\vu|^2 \vb + |\vb|^2 \vu )\\ & & - 2 \langle div \big(P (\vu + \vb)\big)\rangle + 2 (\vf \cdot \vu + \vg \cdot\vb) \end{eqnarray*} is well-defined as a distribution and is a locally finite non-negative measure on $\Omega$, so that $(\vu, \vb)$ is dissipative solution of equations (\ref{EquationMHD}), i.e., point $2)$ in Theorem \ref{Theorem_main}. Moreover, the hypothesis \emph{3)} in Theorem \ref{Theorem_main_original} implies the point $3)$ in Theorem \ref{Theorem_main}. Indeed, \begin{equation*} \begin{split} &\limsup _{r \rightarrow 0} \frac{1}{r}
\iint_{\left]t_{0}-r^{2}, t_{0}+r^{2}\right[ \times B(x_0, r)}|\vn \otimes \vu \,(s, y)|^{2} + |\vn \otimes \vb \,(s, y)|^{2} dyds\\
= & \limsup _{r \rightarrow 0} \frac{1}{r} \iint_{\left]t_{0}-r^{2}, t_{0}+r^{2}\right[ \times B(x_0, r)}|\vn \otimes (\vU + \vB) \,(s, y)|^{2} + |\vn \otimes (\vU - \vB) \,(s, y)|^{2} dyds < \epsilon^*. \end{split} \end{equation*} Applying Theorem \ref{Theorem_main}, we get local information \eqref{gain_reg_ub} on $(\vu, \vb)$, which is indeed equal to \begin{eqnarray}\label{hy_localubUB}
\mathds{1}_{Q_{R_0}}\vU, \mathds{1}_{Q_{R_0}}\vB\in \mathcal{M}_{t,x}^{3,\tau_{0}}(\mathbb{R}\times\R)\quad \text{with} \quad \tau_0 >5. \end{eqnarray} Observe now that $(\vU, \vB)$ satisfies the local hypotheses \eqref{LocalHypo1} in Theorem \ref{Teo_SerrinMHD}. We conclude the proof by applying the Theorem \ref{Teo_SerrinMHD} and the arguments stated in the formulas \eqref{ConclusionSerrinMHD}-\eqref{ConclusionSerrinMHD1}.
$\blacksquare$
\section*{Appendix}
{\bf Proof of Lemma \ref{prop_NRST}.} For simplicity of the argument, we denote by $\vX_{\ep} := \vX * \theta_\ep$, $\vY_{\ep}: = \vY * \theta_\ep$ and $\vZ_{\ep} := \vZ * \theta_\ep$. We first remark that, as $div(\vX) = 0$ and $\theta$ is a smooth function on $\R$ with compact support, then for any point $(t,x)$ on the cylinder $Q_{\rho_0/4}$, we have the identity $\displaystyle{\int_{\R}} \vn \theta_{\ep}(y) \cdot\left( \vX (t, x-y) - \vX (t,x) \right) dy = 0$, so that $$\int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x) \right) (\vY \cdot \vZ )(t, x) d y = 0,$$ which allows us to rewrite $ R_{\ep} (\vX, \vY, \vZ)$ in the following ways : \begin{equation}\label{formular_R} \begin{aligned} R_{\ep} (\vX, \vY, \vZ)(t,x) & = \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) (\vY \cdot \vZ )(x-y)d y \\ &\hspace{0.5cm} - \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \left(\vY(x-y) \cdot \vY (x) + \vZ(x-y) \cdot \vZ (x)\right) d y \\ & = div \left(\theta_\ep * [(\vY \cdot \vZ) \vX] \right) - div \left([\theta_\ep * (\vY \cdot \vZ)] \vX \right)\\ &\hspace{0.5cm} - \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \left(\vY(x-y) \cdot \vY (x) + \vZ(x-y) \cdot \vZ (x)\right) d y, \end{aligned} \end{equation} where we used the identities \begin{equation*} \begin{aligned} & \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \vX ( x-y)\right) (\vY \cdot \vZ )(x-y)d y = div \left(\theta_\ep * [(\vY \cdot \vZ) \vX]\right), \\ & \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \vX ( x)\right) (\vY \cdot \vZ )(x-y)d y = div \left([\theta_\ep * (\vY \cdot \vZ)] \vX \right). \end{aligned} \end{equation*} In the same manner, for $S_\ep$ and $T_\ep$, we have \begin{equation}\label{formular_S} \begin{aligned} S_{\ep}(\vX, \vY, \vZ) (t,x) &=\int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \left(\delta_{y}[\vY ](t, x) \right) \cdot\left(\vZ_{\ep}(x)-\vY(x)\right) d y\\ &= \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \left(\vY(x-y)\right) \cdot\left(\vZ_{\ep}(x)-\vY(x)\right) d y\\ & = \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \vX ( x-y)\right) \left(\vY(x-y)\cdot\vZ_{\ep}(x)\right) d y \\ & \hspace{0.5cm} - \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \vX (x)\right) \left(\vY(x-y) \cdot\vZ_{\ep}(x)\right) d y\\ & \hspace{0.5cm} - \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \left(\vY(x-y)\cdot\vY(x)\right) d y\\ & = \vZ_\ep \cdot \left([(\vX \cdot \vn)\vY ]* \theta_\ep\right) - \vZ_\ep \cdot \left((\vX \cdot \vn)\vY_\ep\right) \\ & \hspace{0.5cm} - \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \left(\vY(x-y)\cdot\vY(x)\right) d y, \end{aligned} \end{equation} and \begin{equation}\label{formular_T} \begin{aligned} T_{\ep} & (\vX, \vY, \vZ) (t,x) = \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \left(\delta_{y}[\vZ ](t, x) \right) \cdot\left(\vY_{\ep}(x)-\vZ(x)\right) d y\\ & = \vY_\ep \cdot \left([(\vX \cdot \vn)\vZ ]* \theta_\ep\right)- \vY_\ep \cdot \left((\vX \cdot \vn)\vZ_\ep\right) - \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \left(\vZ(x-y)\cdot\vZ(x)\right) d y. \end{aligned} \end{equation} Gathering the expressions \eqref{formular_R}, \eqref{formular_S} and \eqref{formular_T} together, we have \begin{equation*} \begin{aligned} (R_{\ep} - S_{\ep} - T_{\ep}) (\vX, \vY, \vZ) & = div \left(\theta_\ep * [(\vY \cdot \vZ) \vX] \right) - div \left([\theta_\ep * (\vY \cdot \vZ)] \vX \right)- \vZ_\ep \cdot \left([(\vX \cdot \vn)\vY ]* \theta_\ep\right) \\ & \hspace{0.5cm} - \vY_\ep \cdot \left([(\vX \cdot \vn)\vZ ]* \theta_\ep\right) + \vZ_\ep \cdot \left((\vX \cdot \vn)\vY_\ep\right) + \vY_\ep \cdot \left((\vX \cdot \vn)\vZ_\ep\right)\\ & = div \left(\theta_\ep * [(\vY \cdot \vZ) \vX] - [\theta_\ep * (\vY \cdot \vZ)] \vX \right)\\ & \hspace{0.5cm}- \vZ_\ep \cdot \left([(\vX \cdot \vn)\vY ]* \theta_\ep\right) - \vY_\ep \cdot \left([(\vX \cdot \vn)\vZ ]* \theta_\ep\right) +div \left((\vY_\ep \cdot \vZ_\ep) \vX\right), \end{aligned} \end{equation*} where in the last line we used the identity $\vZ_\ep \cdot \left((\vX \cdot \vn)\vY_\ep\right) + \vY_\ep \cdot \left((\vX \cdot \vn)\vZ_\ep\right) = div \left((\vY_\ep \cdot \vZ_\ep) \vX\right)$ due to divergence free assumption on $\vX$. Recall now the expression of $N_{\ep} (\vX, \vY, \vZ)$ given in (\ref{four_quantities}): $$N_{\ep} (\vX, \vY, \vZ)=\vY_{\ep} \cdot\left([(\vX \cdot \vn ) \vZ] * \theta_{\ep}\right) + \vZ_{\ep} \cdot\left([(\vX \cdot \vn ) \vY] * \theta_{\ep}\right)-div((\vY \cdot \vZ) \vX),$$ we thus have \begin{equation*} \begin{aligned} (R_{\ep} - S_{\ep} - T_{\ep} + N_\ep) (\vX, \vY, \vZ) & = div \left(\theta_\ep * [(\vY \cdot \vZ) \vX] - [\theta_\ep * (\vY \cdot \vZ)] \vX \right) + div \left((\vY_\ep \cdot \vZ_\ep) \vX -(\vY \cdot \vZ) \vX \right). \end{aligned} \end{equation*} Since $\vX, \vY, \vZ \in L^{3}_{t,x} (Q_{\rho_0})$, we have $\lim\limits_{\ep \to 0} \left( \theta_\ep * [(\vY \cdot \vZ) \vX] - [\theta_\ep * (\vY \cdot \vZ)] \vX \right) = 0$ in $L^1_{t,x} (Q_{\rho_0/4} )$, hence, $$ \lim\limits_{\ep \to 0} div \left(\theta_\ep * [(\vY \cdot \vZ) \vX] - [\theta_\ep * (\vY \cdot \vZ)] \vX \right) = 0 \quad \text{in} \quad \mathcal{D}'(Q_{\rho_0/4}).$$ Moreover, we have limit $\lim\limits_{\ep \to 0} div \left((\vY_\ep \cdot \vZ_\ep) \vX -(\vY \cdot \vZ) \vX \right) = 0$ in $\mathcal{D}'( Q_{\rho_0/4} )$, which allows us to conclude that $$\lim_{\ep \to 0} (N_{\ep} + R_{\ep}- S_{\ep} - T_{\ep}) (\vX, \vY, \vZ)= 0.$$ The lemma \ref{prop_NRST} is now proved.
$\blacksquare$\\
\noindent{\bf Proof of Lemma \ref{lemma_Tep}.} By the definitions of $S_{\ep},T_{\ep}, R_{\ep}$ given in (\ref{four_quantities}), we can rewrite the operator $(S_{\ep} + T_{\ep} - R_{\ep}) (\vX, \vY, \vZ)$ in the following manner: \begin{equation*} \begin{aligned} &(S_{\ep} + T_{\ep} - R_{\ep}) (\vX, \vY, \vZ)(t,x) \\ & = \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \delta_{y}[\vY ](t, x) d y \cdot \left((\vZ * \theta_\ep)(t, x) - \vZ(t, x) + \vZ(t, x)-\vY(t, x)\right) \\ &\hspace{0.5cm} + \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right)\delta_{y}[\vZ ](t, x) d y \cdot \left( (\vY * \theta_\ep)(t, x) - \vY(t, x) + \vY(t, x)-\vZ(t, x)\right)\\ &\hspace{0.5cm} - \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right)\left(\delta_{y}[\vY ](t, x) + \vY(t, x)-\vZ(t, x)\right) \cdot \left(\delta_{y}[\vZ](t, x) + \vZ (t, x) - \vY (t, x)\right)d y\\ &= \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right) \delta_{y}[\vY ](t, x) d y \cdot \left((\vZ * \theta_\ep)(t, x) - \vZ(t, x)\right) \\ &\hspace{0.5cm} + \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right)\delta_{y}[\vZ ](t, x) d y \cdot \left( (\vY * \theta_\ep)(t, x) - \vY(t, x)\right)\\ &\hspace{0.5cm} - \int_{\R} \left(\vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x)\right)\left(\delta_{y}[\vY ](t, x)\right) \cdot \left(\delta_{y}[\vZ](t, x)\right)dy, \end{aligned} \end{equation*} where in the last line we used the identity $\displaystyle{\int_{\R}} \vn \theta_{\ep}(y)\cdot \delta_{y}[\vX ](t, x) d y = 0$ due to divergence free condition of $\vX$. Next, recall the definition of the mollifier
$\theta_\ep = \frac{1}{\ep^3} \theta (\frac{x}{\ep})$, we have $\displaystyle{\int_{\R}}\theta_\ep dx= 1$, $\text{supp} (\theta_\ep) \subset B(0, \ep)$ and $|\vn \theta_{\ep}(y)| \leq \frac{C}{\ep^4}$, hence, \begin{equation}\label{bigformula_Tep} \begin{aligned} \vert(S_{\ep} + T_{\ep} - R_{\ep}) (\vX, \vY, \vZ)(t,x) \vert
&\leq C \frac{1}{\ep^4} \int_{|y| < \ep} \left|\delta_{y}[\vX](t, x)\right|\,\left|\delta_{y}[\vY ](t, x)\right|\, d y \left|(\vZ * \theta_\ep)(t, x)-\vZ(t, x)\right|\\
&\hspace{0.5cm} + C \frac{1}{\ep^4} \int_{|y| < \ep} \left|\delta_{y}[\vX](t, x)\right|\, \left|\delta_{y}[\vZ ](t, x)\right| d y \, \left|(\vY * \theta_\ep)(t, x)-\vY(t, x)\right|\\
&\hspace{0.5cm} + C \frac{1}{\ep^4} \int_{|y| < \ep} \left| \delta_{y}[\vX](t, x)\right| \,\left|\delta_{y}[\vY ](t, x)\right| \, \left|\delta_{y}[\vZ](t, x)\right| d y\\
& \leq C \frac{1}{\ep^7} \int_{|y| < \ep} \left|\delta_{y}[\vX](t, x)\right|\,\left|\delta_{y}[\vY ](t, x)\right|\, d y \times \int_{|z| < \ep} \left| \delta_{z}[\vZ ](t, x)\right| d z\\
&\hspace{0.5cm} + C \frac{1}{\ep^7} \int_{|y| < \ep} \left|\delta_{y}[\vX](t, x)\right|\, \left|\delta_{y}[\vZ ](t, x)\right| d y \times \int_{|z| < \ep} \left| \delta_{z}[\vY](t, x)\right| d z\\
&\hspace{0.5cm} + C \frac{1}{\ep^4} \int_{|y| < \ep} \left| \delta_{y}[\vX](t, x)\right| \,\left|\delta_{y}[\vY ](t, x)\right| \, \left|\delta_{y}[\vZ](t, x)\right| d y \end{aligned} \end{equation} where we used the following estimates
$$ \left|(\vZ * \theta_\ep)(t, x)-\vZ(t, x)\right| = \left| \int_{\R} \theta_\ep (z) \left(\vZ(t, x-z) -\vZ(t, x) \right) d z \right| \leq C \frac{1}{\ep^3} \int_{|z| < \ep} \left| \delta_{z}[\vZ ](t, x)\right| d z,$$
$$\left|(\vY * \theta_\ep)(t, x)-\vY(t, x)\right| = \left| \int_{\R} \theta_\ep (z) \left(\vY(t, x-z) -\vY(t, x)\right) d z \right| \leq C \frac{1}{\ep^3} \int_{|z| < \ep} \left| \delta_{z}[\vY ](t, x)\right| d z.$$ Now, using Hölder's inequality to the right-hand side of \eqref{bigformula_Tep}, we have \begin{equation*} \begin{aligned} \vert(S_{\ep} + T_{\ep} - R_{\ep}) (\vX, \vY, \vZ)(t,x) \vert
& \leq C \frac{1}{\ep^4} \|\delta_{\cdot}[\vX](t, x)\|_{L_y^3(B(0,\ep))} \|\delta_{\cdot}[\vY](t, x)\|_{L_y^3(B(0,\ep))} \|\delta_{\cdot}[\vZ](t, x)\|_{L_y^3(B(0,\ep))} . \end{aligned} \end{equation*} Let us turn to prove the limit in $L^1_{t,x} (Q_{\rho_0/4})$ norm. Taking the $L^1$ norm of $(S_{\ep} + T_{\ep} - R_{\ep})(\vX, \vY, \vZ)(t,x)$ in time and in space variable over the cylinder $Q_{\rho_0/4}$ and applying Hölder's inequality we obtain \begin{eqnarray} &&\int_{Q_{\rho_0/4}} \vert (S_{\ep} + T_{\ep} - R_{\ep})(\vX, \vY, \vZ)(t,x) \vert d t d x\notag\\ &\leq & \; C \frac{1}{\ep^4}
\int_{Q_{\rho_0/4}} \|\delta_{\cdot}[\vX](t, x)\|_{L_y^3(B(0,\ep))} \|\delta_{\cdot}[\vY](t, x)\|_{L_y^3(B(0,\ep))} \|\delta_{\cdot}[\vZ](t, x)\|_{L_y^3(B(0,\ep))} d t d x\notag\\ &\leq & \;C \frac{1}{\ep^4} \left(\int_{Q_{\rho_0/4}}
\|\delta_{\cdot}[\vX](t, x)\|^3_{L_y^3(B(0,\ep))} \right)^{\frac{1}{3}}
\left(\int_{Q_{\rho_0/4}} \|\delta_{\cdot}[\vY](t, x)\|^3_{L_y^3(B(0,\ep))} \right)^{\frac{1}{3}}\notag\\
&& \times \left(\int_{Q_{\rho_0/4}} \|\delta_{\cdot}[\vZ](t, x)\|^3_{L_y^3(B(0,\ep))}\right)^{\frac{1}{3}}.\label{integ_Tep} \end{eqnarray} Recalling that $\vX, \vY, \vZ \in L_{t,x}^3 (\Qro)$ and at least one of them belongs to $L_t^\infty Lip_x (\Qro)$, so we may suppose for instance that $\vX \in L_t^\infty Lip_x (\Qro) \cap L_{t,x}^3 (\Qro)$ and $\vY, \vZ \in L_{t,x}^3 (\Qro)$. For the term involving $\vX$ on the right-hand side of \eqref{integ_Tep}, by a change of variables, we get \begin{equation*} \begin{aligned} \int_{Q_{\rho_0/4}}
\|\delta_{\cdot}[\vX](t, x)\|^3_{L_y^3(B(0,\ep))} d t d x
= \int_{Q_{\rho_0/4}} \int_{|y|<\ep} |\vX (t, x-y) - \vX (t,x) |^3 d y d t d x
\leq C \ep^6 \| \vn \otimes \vX \|^3_{L^\infty_{t,x}} |Q_{\rho_0/4}|, \end{aligned} \end{equation*} and thus we have \begin{equation}\label{firstX}
\frac{1}{\ep^2}\left(\int_{Q_{\rho_0/4}} \|\delta_{\cdot}[\vX](t, x)\|^3_{L_y^3(B(0,\ep))} d t d x\right)^{\frac{1}{3}}
\leq C \| \vn \otimes \vX \|_{L^\infty_{t,x}} |Q_{\rho_0/4}|^{\frac{1}{3}}. \end{equation} Moreover, for $\vY, \vZ \in L_{t,x}^3 (\Qro)$, by a change of variables, we have \begin{equation}\label{lasttwoYZ}
\frac{1}{\ep} \left(\int_{Q_{\rho_0/4}} \|\delta_{\cdot}[\vY](t, x)\|^3_{L_y^3(B(0,\ep))} \right)^{\frac{1}{3}} \to 0, \quad \frac{1}{\ep} \left(\int_{Q_{\rho_0/4}} \|\delta_{\cdot}[\vZ](t, x)\|^3_{L_y^3(B(0,\ep))}\right)^{\frac{1}{3}} \to 0, \quad \text{as} \quad \ep \to 0. \end{equation} Note that the convergence holds here by the dominated convergence theorem. Substituting the estimates \eqref{firstX} and \eqref{lasttwoYZ} into \eqref{integ_Tep}, we conclude that $$ \lim_{\ep \to 0}(S_{\ep} + T_{\ep} - R_{\ep})(\vX, \vY, \vZ) = 0 \quad \text{in} \quad L^1_{t,x} (Q_{\rho_0/4}). $$ The Lemma \ref{lemma_Tep} is now proved.
$\blacksquare$\\
\noindent{\bf Acknowledgments:} J. \textsc{He} is supported by the \emph{Sophie Germain} Post-doc program of the \emph{Fondation Math\'ematique Jacques Hadamard}.
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Tim Alberdingk Thijm View Jun 22, 2020 · 15 mins read
David Walker View Jun 22, 2020 · 15 mins read
Part 1: SIMPLE Models and Simulation
One of the keys to success of any verification effort is identifying a class of models capable of representing the important elements of the system under consideration. Such models inevitably elide information—the real world is too complicated to represent it in its entirety. However, a good model represents enough of the real world to be useful. When it comes to software reliability, such models need to represent the causes of important bugs or vulnerabilities.
For instance, relational algebra has been an effective model for database query languages for decades. It represents database tables as relations, which, mathematically speaking, are just sets of tuples, and is capable of defining a variety of standard database operations such as filters, joins and maps. It has been indispensable for helping database implementers define and prove correct complex query transformations and optimizations. Of course, there are infinitely many things that the (standard) relational algebra models will not capture about databases: the concrete syntax of user queries, the cost of computing filters or joins, or the semantics of the underlying storage system and its potential for failure, to name just a few.
The networking community, like the databases community, has its fair share of models as well. In this post, we will introduce a simple class of models that capture the essence of distributed routing protocols like eBGP, OSPF, RIP and ISIS. They describe the flow of routing messages between devices and allow us to analyze the routes (i.e., paths) computed by the network control plane, without getting bogged down by the bells and whistles of real-world protocols. The bells and whistles are important for practical reasons but have little bearing on a broad class of verification questions of interest such as: Does the network control plane compute a route from server A to server B? Might Google accidentally try to act as transport for traffic destined for Japan, possibly shutting off access to the internet there? Will Level 3 leak routes that should be kept internal, disrupting internet service across the US? By developing the capability to analyze these simple models, we can identify a range of potentially devastating errors in the configuration of modern networks.
The models we will discuss here are derived from the pioneering research of Griffin et al. [1] and Sobrinho [2]. Griffin identified the fact that network protocols like BGP are solving a kind of "stable paths problem". In doing so, he developed a solid framework for thinking rigorously about such protocols and analyzing their basic properties, including convergence and determinacy. Sobrinho generalized these ideas and formalized them from an algebraic perspective. More recently, we developed NV [3, 4], a language and system that allows users to define and verify such models using powerful logical tools.
SIMPLE: An Idealized Control Plane Protocol
While there are many distributed control plane protocols, they share a common structure. That common structure allows us to define a wide range of protocols in the same way, and makes it possible to implement generic analysis tools that can be reused to find bugs in any of them.
To illustrate the basic pieces of a model, let's take a look at a concrete example—a made-up protocol we will call SIMPLE. In SIMPLE, each router in the network either knows a route to the destination or it doesn't. We use the symbol None to represent "no route." Each known route is a triple (pref, length, next). The first component, pref, is a numeric preference for the route. The higher the number, the more desirable the route. In general, the reader can assume the protocol uses 32-bit numbers that range from $0$ to $2^{32}-1$, but the details do not matter. The second component is the length of the path to the destination. The third component identifies the router that serves as the next hop along the journey to the destination. For simplicity, SIMPLE is designed to route to a single destination, so we don't need to include the name of the destination (i.e., its IP address) in the routing messages. We'll discuss more elaborate models for multi-destination routing later in this series of articles. To summarize, we have now defined the first component of a control plane model, the set of messages M, that are used to disseminate routing information amongst routers.
Below is an example of a network running SIMPLE in an initial state, before the process of computing routes to the destination has begun. This network has 5 routers, named $\mathtt{R_0} - \mathtt{R_4}$. Each router is annotated with an initial route. $\mathtt{R_0}$ is the destination router—its initial route has a preference of 100, a length of 0 to the destination (it is the destination), and a dummy next-hop of 0. The other routers have None as their initial route.
In general, the current state X of one of our network models is a function from routers to their current route. The initial state init, is an example of such a state. In this case, we define the init state for this network as follows:
$\mathtt{\mathbf{init}_{R}} = \mathtt{(100, 0, 0)}~~~,~\mathrm{if}~\mathtt{R} = \mathtt{R_0}$
$\mathtt{\mathbf{init}_{R}} = \mathtt{\mathit{None}}~~~,~\mathrm{otherwise}$
The routing process for SIMPLE, and other protocols we model, operates by transmitting messages around the network, and updating the network state as we go. Eventually (we hope), the process stabilizes and no more state changes occur.
Informally, SIMPLE operates as follows. Each router with a valid route can choose to export the route to one or more of its neighbors. When a message traverses an edge in the graph, it is transformed. The length of the route will increase by one, the next hop field will change, and the importer will change the preference field, making the route more or less desirable. We call the function that executes those transformations the trans function. Since the transformation may vary from one edge of the graph to the next, when we care to be specific, we write $\mathbf{\mathtt{trans}}_{\mathtt{e}}$ for the transformation function to be applied across the edge e.
When a router receives one or more routes from its neighbors, it compares those routes with its initial route and chooses the most desirable route available. More specifically, it prefers the route with the highest preference value. If there is a tie, it will prefer the route with the shortest path length. If there is still a tie, it prefers the route through the lowest numbered neighbor (e.g., all other things being equal, router $\mathtt{R_3}$ prefers routes it receives from $\mathtt{R_1}$ over those it receives from $\mathtt{R_2}$). In general, when multiple routes are available at a router, the router will use information from those routes to compute its most desired route. We call the function that executes that computation the merge function, and often write $\mathtt{R_1~+~R_2}$ to denote the merge of two routes, $\mathtt{r_1}$ and $\mathtt{r_2}$. In principle, the merge function can differ at every router. If we care to distinguish between the merge functions of particular routers, we will write merge$_R$, or $\mathtt{R_1}~\mathtt{+_R}~\mathtt{R_2}$, for the merge at router R, but to keep the notation light, we will often assume the merge functions are the same across all nodes and elide the subscript.
Choices such as which router forwards routes to which other routers, and how to change a field like the preference value are typically part of the configuration of a protocol. In contrast, the notion of "most desirable route" (i.e., the merge function) is usually a fixed part of the protocol. The configurations are typically defined by network operators—they are what allows a protocol to be customized to the needs of a particular network. Router vendors like Cisco and Juniper have complex, proprietary languages for configuring their devices, and in large networks, such configurations can be hundreds or thousands of lines of code per router, and hundreds of thousands or millions of lines in total for a large data center network. However, when it comes to network verification, it does not really matter whether a route processing function is defined by a configuration or is an inherent, unchangeable part of the protocol. Hence, our models incorporate both and do not distinguish between fixed and configurable parts.
In the following picture, we add a little bit of configuration information to our network. In particular, we will assume that when $\mathtt{R_3}$ imports a message from $\mathtt{R_1}$ it changes the preference value to 0, making it less desirable than routes imported from $\mathtt{R_2}$. We can also configure the network so that certain links drop messages. For simplicity, we will assume links propagate messages from right to left, and drop all messages (i.e., producing None) from left to right.
Protocol Simulation
To determine which paths each router in the network chooses to use, we can simulate execution of the control plane protocol on the given network. Simulation updates the network state, one step at a time until a "stable state" is found and no more updates are required. The following (non-deterministic) algorithm implements the simulation process.
Generic Simulation Algorithm
Let X, the current state of the simulation, be the initial state init.
Select any router R:
Let $\mathtt{R_1} \ldots \mathtt{R_k}$ be the neighbors of R
Let $\mathtt{e_1} \ldots \mathtt{e_k}$ be the edges that connect those neighbors to R
Update X(R) as follows (leaving other components of X unchanged):
$\mathtt{X}(\mathtt{R}) := \mathbf{\mathtt{trans}}_{\mathtt{e1}}(\mathtt{X}(\mathtt{R_1}))~\mathtt{+_R}~\ldots~\mathtt{+_R}~\mathbf{\mathtt{init}}(\mathtt{R})$
Repeat step 2 until there are no further changes to be made.
i.e., until $\mathtt{X}(\mathtt{R}) = \mathbf{\mathtt{trans}}_{\mathtt{e1}}(\mathtt{X}(\mathtt{R_1}))~\mathtt{+_R}~\ldots~\mathtt{+_R}~\mathbf{\mathtt{init}}(\mathtt{R})$for all routers R
In our SIMPLE network, a possible first step in simulation could select router R1 and consider its neighbors, $\mathtt{R_3}$ and $\mathtt{R_0}$. The current chosen routes for those routers are None and (100, 0, 0) respectively. The initial route at $\mathtt{R_1}$ is None. Hence, the new route chosen by $\mathtt{R_1}$ after this step in simulation will be:
$\mathtt{\mathbf{trans}_{01}}(100,0,0) ~\mathtt{+}$
$\mathtt{\mathbf{trans}_{31}}(\mathtt{None})~\mathtt{+}$
$\mathtt{None}$
which is equal to the following (we add 1 to the length of the known route and leave the other fields unchanged):
$(100, 1, 0) ~\mathtt{+}~ \mathit{None} ~\mathtt{+}~ \mathit{None}$
which is equal to
$(100, 1, 0)$
as the SIMPLE protocol always prefers some route over None. The following picture diagrams the process. The green route is the route that emerges for $\mathtt{R_1}$ after a single step of simulation:
Cleaning up the picture, our network now looks like this:
Next, it may be $\mathtt{R_3}$'s turn to act. As mentioned above, when $\mathtt{R_3}$ imports its message from $\mathtt{R_1}$, it downgrades the preference to 0. However, since $\mathtt{R_3}$ only receives None (no route yet) from $\mathtt{R_2}$, it prefers the low-preference route from $\mathtt{R_1}$ over no route at all. After $\mathtt{R_3}$, $\mathtt{R_4}$ may take a turn, leaving the network in the following state after those two steps:
And then the simulation might choose to update $\mathtt{R_2}$:
And now you will notice that all routers have a valid route (none of the routers have None as their route). Still, the simulation is not complete. If router $\mathtt{R_3}$ goes again at this point, it will compute the new route (100, 2, 2), which differs from its current route of (0, 2, 1). It does so because SIMPLE prefers routes with higher preference value. As a result, $\mathtt{R_3}$ prefers the route it receives from $\mathtt{R_2}$ (with preference 100) rather than the route it received earlier from $\mathtt{R_1}$ (with preference 0). $\mathtt{R_3}$'s next hop is now 2 instead of 1.
And then finally, we need to consider $\mathtt{R_4}$ again. When $\mathtt{R_4}$ pulls routes from all its neighbors now, we arrive in the following state.
Model Solutions
At this point, we can examine all routers R and check to see whether the current chosen route equals $\mathbf{\mathtt{trans}}_1(\mathtt{r_1}) \mathtt{+_R} \ldots \mathtt{+_R} \mathbf{\mathtt{trans}}_k(\mathtt{r_k}) \mathtt{+_R} \mathbf{\mathtt{init}}(\mathtt{R})$ where $\mathtt{r_1}$ through $\mathtt{r_k}$ are the current routes of R's neighbors. It turns out they all do.
(Aside: Recall that we specified that edges, when traversed from left to right, drop all routes—that is, they convert any route from left to right into None—and so the route with higher local preference of 100 at $\mathtt{R_3}$ does not propagate backwards to $\mathtt{R_1}$. If we did not block this route, the system would not yet be stable.)
So we've reached a stable state of the system, which we also call a solution to the routing system. More formally, a solution L is any state of a network that is globally stable in the sense that all routers R satisfy the following equation:
$\mathbf{\mathtt{L}}(\mathtt{R}) = \mathbf{\mathtt{trans}}_{\mathtt{e1}}(\mathbf{\mathtt{L}}(\mathtt{R_1})) \mathtt{+_R} \ldots \mathtt{+_R}~\mathbf{\mathtt{init}}(\mathtt{R})$
$\mathtt{R_1}, \ldots, \mathtt{R_k}$ are the neighbors of R
$\mathtt{e_1}, \ldots, \mathtt{e_k}$ are the edges connecting those neighbors to R
Once we have a solution, we can examine its properties. For instance, given a solution L, we could ask whether L(R) = None for any router R, indicating that that router receives no route from the destination and hence cannot reach it. We could also ask what the length of the longest path from any node to the destination is. By using the next-hop fields of routes, we can also reconstruct the path that any router uses to reach the destination. If we were concerned with a property of that path, such as whether it traverses a particular waypoint (like $\mathtt{R_2}$ or $\mathtt{R_1}$) then we could deduce that as well.
Routing is the process of computing paths to a given destination or a collection of destinations. There are many different distributed routing protocols that engage in this process; they go by names such as BGP, OSPF, ISIS and others.
It turns out these protocols have a lot of structure in common. We can see that commonality by defining a class of models with the components (G, M, trans, merge, init):
G — A graph representing the topology of the network (with vertices V and edges E).
M — The set M of messages, also called routes, used by the protocol.
trans : E → M → M — The function that propagates, transforms or discards messages/routes as they travel across each edge in the network.
merge : V → M → M → M — The function that combines incoming information from neighboring routers, usually selecting one (or more) most preferred route for traffic traveling to the given destination.
init : V → M — The initial routes (init) at each node in the graph, which are used in the absence of receiving additional information about where or how to route traffic.
Given these components, we have enough information to simulate a model and find its solution, or stable state. And once we have a solution in hand, we can analyze it to determine what sorts of properties it has. For instance, we can determine whether a given router can reach a particular destination, or whether a computed route passes through a particular waypoint. Such properties can help us uncover important bugs in network configurations before they are deployed.
There's a lot more to learn about this topic, and in future blog posts, we will explore some of them. Can a system have more than one solution? How can we tell? Are there algorithms for finding them all? What happens when there are failures? Can we reason about quantitative properties like congestion? How do we construct models of real protocols like BGP, OSPF and their interactions? How does one actually implement network simulation and verification tools based on these models?
[1] The stable paths problem and interdomain routing. T. G. Griffin, F. B. Shepherd, G. Wilfong. IEEE/ACM Transactions on Networking 10(2). April 2002. https://ieeexplore.ieee.org/document/993304
[2] An algebraic theory of dynamic routing. J. L. Sobrinho. IEEE/ACM Transactions on Networking 13(5). Oct 2005. https://ieeexplore.ieee.org/document/1528502
[4] NV: Tools for modeling and analyzing network configurations. June 2020. https://github.com/NetworkVerification https://github.com/NetworkVerification/nv
[5] NV: An Intermediate Language for Verification of Network Control Planes. Nick Giannarakis, Devon Loehr, Ryan Beckett and David Walker. ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI). June 2020. https://www.cs.princeton.edu/~dpw/papers/nv-pldi20.pdf
Author Ryan Beckett
Ryan Beckett is a Computer Scientist in the Mobility and Networking research group at Microsoft. He holds a PhD from Princeton in Computer Science as well as a B.S. in Computer Science and a B.A. in Mathematics from the University of Virginia.
Author Nick Giannarakis Follow
Nick is a PhD candidate at Princeton University, where he is currently advised by Prof. David Walker. He received a master's degree in Computer Science from ENS Cachan, and a diploma of Electrical and Computer Engineering from National Technical University of Athens.
Author Aarti Gupta
Aarti Gupta is a Professor of Computer Science at Princeton University. Before joining the department, she worked at NEC Labs America where she led a team in investigating new techniques for formal verification of software and hardware systems, contributing both to their foundations and to successful industrial deployment. Professor Gupta received her Ph.D. in computer science from Carnegie Mellon University in 1994 after earning a master's degree in computer engineering from Rensselaer Polytechnic Institute and a bachelor's in electrical engineering from the Indian Institute of Technology in New Delhi.
Author Devon Loehr
Devon is a PhD candidate at Princeton University, where he is currently advised by Prof. David Walker. Devon graduated from Swarthmore College in 2018 with High Honors, with majors in Mathematics and Computer Science.
Author Ratul Mahajan Follow
Ratul Mahajan is an Associate Professor at the University of Washington (Paul G. Allen School of Computer Science) and a Co-founder and CEO of Intentionet, a company that is enabling safe, rapid evolution of computer networks using formal analysis and high-level design approaches. Prior to that, he was a Principal Researcher at Microsoft Research. He got his PhD at the University of Washington and B.Tech at Indian Institute of Technology, Delhi, both in Computer Science and Engineering.
Author Tim Alberdingk Thijm
Tim is a PhD candidate at Princeton University, where he is currently advised by Prof. Aarti Gupta. Tim completed his BSc at the University of Toronto in 2018 in Computer Science and English.
Author David Walker
David Walker is a Professor of Computer Science at Princeton University. He received his doctoral and master's degrees in computer science from Cornell, and his bachelor's from Queen's University in Kingston, Ontario. During sabbaticals from Princeton, he has served as a visiting researcher at Microsoft Research in Redmond (2008) and in Cambridge (2009), and as Associate Visiting Faculty at the University of Pennsylvania (2015-2016).
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Feature selection and dimension reduction for single-cell RNA-Seq based on a multinomial model
F. William Townes ORCID: orcid.org/0000-0003-0320-67871,2,
Stephanie C. Hicks3,
Martin J. Aryee1,4,5,6 &
Rafael A. Irizarry1,7
A Correction to this article was published on 22 July 2020
Single-cell RNA-Seq (scRNA-Seq) profiles gene expression of individual cells. Recent scRNA-Seq datasets have incorporated unique molecular identifiers (UMIs). Using negative controls, we show UMI counts follow multinomial sampling with no zero inflation. Current normalization procedures such as log of counts per million and feature selection by highly variable genes produce false variability in dimension reduction. We propose simple multinomial methods, including generalized principal component analysis (GLM-PCA) for non-normal distributions, and feature selection using deviance. These methods outperform the current practice in a downstream clustering assessment using ground truth datasets.
Single-cell RNA-Seq (scRNA-Seq) is a powerful tool for profiling gene expression patterns in individual cells, facilitating a variety of analyses such as identification of novel cell types [1, 2]. In a typical protocol, single cells are isolated in liquid droplets, and messenger RNA (mRNA) is captured from each cell, converted to cDNA by reverse transcriptase (RT), then amplified using polymerase chain reaction (PCR) [3–5]. Finally, fragments are sequenced, and expression of a gene in a cell is quantified by the number of sequencing reads that mapped to that gene [6]. A crucial difference between scRNA-Seq and traditional bulk RNA-Seq is the low quantity of mRNA isolated from individual cells, which requires a larger number of PCR cycles to produce enough material for sequencing (bulk RNA-Seq comingles thousands of cells per sample). For example, the popular 10x Genomics protocol uses 14 cycles [5]. Thus, many of the reads counted in scRNA-Seq are duplicates of a single mRNA molecule in the original cell [7]. Full-length protocols such as SMART-Seq2 [8] analyze these read counts directly, and several methods have been developed to facilitate this [9]. However, in many experiments, it is desirable to analyze larger numbers of cells than possible with full-length protocols, and isoform-level inference may be unnecessary. Under such conditions, it is advantagous to include unique molecular identifiers (UMIs) which enable computational removal of PCR duplicates [10, 11], producing UMI counts. Although a zero UMI count is equivalent to a zero read count, nonzero read counts are larger than their corresponding UMI counts. In general, all scRNA-Seq data contain large numbers of zero counts (often >90% of the data). Here, we focus on the analysis of scRNA-Seq data with UMI counts.
Starting from raw counts, a scRNA-Seq data analysis typically includes normalization, feature selection, and dimension reduction steps. Normalization seeks to adjust for differences in experimental conditions between samples (individual cells), so that these do not confound true biological differences. For example, the efficiency of mRNA capture and RT is variable between samples (technical variation), causing different cells to have different total UMI counts, even if the number of molecules in the original cells is identical. Feature selection refers to excluding uninformative genes such as those which exhibit no meaningful biological variation across samples. Since scRNA-Seq experiments usually examine cells within a single tissue, only a small fraction of genes are expected to be informative since many genes are biologically variable only across different tissues. Dimension reduction aims to embed each cell's high-dimensional expression profile into a low-dimensional representation to facilitate visualization and clustering.
While a plethora of methods [5, 12–15] have been developed for each of these steps, here, we describe what is considered to be the standard pipeline [15]. First, raw counts are normalized by scaling of sample-specific size factors, followed by log transformation, which attempts to reduce skewness. Next, feature selection involves identifying the top 500–2000 genes by computing either their coefficient of variation (highly variable genes [16, 17]) or average expression level (highly expressed genes) across all cells [15]. Alternatively, highly dropout genes may be retained [18]. Principal component analysis (PCA) [19] is the most popular dimension reduction method (see for example tutorials for Seurat [17] and Cell Ranger [5]). PCA compresses each cell's 2000-dimensional expression profile into, say, a 10-dimensional vector of principal component coordinates or latent factors. Prior to PCA, data are usually centered and scaled so that each gene has mean 0 and standard deviation 1 (z-score transformation). Finally, a clustering algorithm can be applied to group cells with similar representations in the low-dimensional PCA space.
Despite the appealing simplicity of this standard pipeline, the characteristics of scRNA-Seq UMI counts present difficulties at each stage. Many normalization schemes derived from bulk RNA-Seq cannot compute size factors stably in the presence of large numbers of zeros [20]. A numerically stable and popular method is to set the size factor for each cell as the total counts divided by 106 (counts per million, CPM). Note that CPM does not alter zeros, which dominate scRNA-Seq data. Log transformation is not possible for exact zeros, so it is common practice to add a small pseudocount such as 1 to all normalized counts prior to taking the log. The choice of pseudocount is arbitrary and can introduce subtle biases in the transformed data [21]. For a statistical interpretation of the pseudocount, see the "Methods" section. Similarly, the use of highly variable genes for feature selection is somewhat arbitrary since the observed variability will depend on the pseudocount: pseudocounts close to zero arbitrarily increase the variance of genes with zero counts. Finally, PCA implicitly relies on Euclidean geometry, which may not be appropriate for highly sparse, discrete, and skewed data, even after normalizations and transformations [22].
Widely used methods for the analysis of scRNA-Seq lack statistically rigorous justification based on a plausible data generating a mechanism for UMI counts. Instead, it appears many of the techniques have been borrowed from the data analysis pipelines developed for read counts, especially those based on bulk RNA-Seq [23]. For example, models based on the lognormal distribution cannot account for exact zeros, motivating the development of zero-inflated lognormal models for scRNA-Seq read counts [24–27]. Alternatively, ZINB-WAVE uses a zero-inflated negative binomial model for dimension reduction of read counts [28]. However, as shown below, the sampling distribution of UMI counts is not zero inflated [29] and differs markedly from read counts, so application of read count models to UMI counts needs either theoretical or empirical justification.
We present a unifying statistical foundation for scRNA-Seq with UMI counts based on the multinomial distribution. The multinomial model adequately describes negative control data, and there is no need to model zero inflation. We show the mechanism by which PCA on log-normalized UMI counts can lead to distorted low-dimensional factors and false discoveries. We identify the source of the frequently observed and undesirable fact that the fraction of zeros reported in each cell drives the first principal component in most experiments [30]. To remove these distortions, we propose the use of GLM-PCA, a generalization of PCA to exponential family likelihoods [31]. GLM-PCA operates on raw counts, avoiding the pitfalls of normalization. We also demonstrate that applying PCA to deviance or Pearson residuals provides a useful and fast approximation to GLM-PCA. We provide a closed-form deviance statistic as a feature selection method. We systematically compare the performance of all combinations of methods using ground truth datasets and assessment procedures from [15]. We conclude by suggesting best practices.
We used 9 public UMI count datasets to benchmark our methods (Table 1). The first dataset was a highly controlled experiment specifically designed to understand the technical variability. No actual cells were used to generate this dataset. Instead, each droplet received the same ratio of 92 synthetic spike-in RNA molecules from External RNA Controls Consortium (ERCC). We refer to this dataset as the technical replicates negative control as there is no biological variability whatsoever, and in principle, each expression profile should be the same.
Table 1 Single cell RNA-Seq datasets used
The second and third datasets contained cells from homogeneous populations purified using fluorescence-activated cell sorting (FACS). We refer to these datasets as biological replicates negative controls. Because these cells were all the same type, we did not expect to observe any significant differences in unsupervised analysis. The 10 × Zheng monocytes data had low total UMI counts, while the SMARTer Tung data had high counts.
The fourth and fifth datasets were created by [15]. The authors allocated FACS-purified peripheral blood mononuclear cells (PBMCs) from 10 × data [5] equally into four (Zheng 4eq dataset) and eight (Zheng 8eq dataset) clusters, respectively. In these positive control datasets, the cluster identity of all cells was assigned independently of gene expression (using FACS), so they served as the ground truth labels.
The sixth and seventh datasets contained a wider variety of cell types. However, the cluster identities were determined computationally by the original authors' unsupervised analyses and could not serve as a ground truth. The 10 × Haber intestinal dataset had low total UMI counts, while the CEL-Seq2 Muraro pancreas dataset had high counts.
The final Zheng dataset consisted of a larger number of unsorted PBMCs and was used to compare computational speed of different dimension reduction algorithms. We refer to it as the PBMC 68K dataset.
UMI count distribution differs from reads
To illustrate the marked difference between UMI count distributions and read count distributions, we created histograms from individual genes and spike-ins of the negative control data. Here, the UMI counts are the computationally de-duplicated versions of the read counts; both measurements are from the same experiment, so no differences are due to technical or biological variation. The results suggest that while read counts appear zero-inflated and multimodal, UMI counts follow a discrete distribution with no zero inflation (Additional file 1: Figure S1). The apparent zero inflation in read counts is a result of PCR duplicates.
Multinomial sampling distribution for UMI counts
Consider a single cell i containing ti total mRNA transcripts. Let ni be the total number of UMIs for the same cell. When the cell is processed by a scRNA-Seq protocol, it is lysed, then some fraction of the transcripts are captured by beads within the droplets. A series of complex biochemical reactions occur, including attachment of barcodes and UMIs, and reverse transcription of the captured mRNA to a cDNA molecule. Finally, the cDNA is sequenced, and PCR duplicates are removed to generate the UMI counts [5]. In each of these stages, some fraction of the molecules from the previous stage are lost [5, 7, 32]. In particular, reverse transcriptase is an inefficient and error-prone enzyme [35]. Therefore, the number of UMI counts representing the cell is much less than the number of transcripts in the original cell (ni≪ti). Specifically, ni typically ranges from 1000−10,000 while ti is estimated to be approximately 200,000 for a typical mammalian cell [36]. Furthermore, which molecules are selected and successfully become UMIs is a random process. Let xij be the true number of mRNA transcripts of gene j in cell i, and yij be the UMI count for the same gene and cell. We define the relative abundance πij as the true number of mRNA transcripts represented by gene j in cell i divided by the total number of mRNA transcripts in cell i. Relative abundance is given by πij=xij/ti where total transcripts \(t_{i}=\sum _{j} x_{ij}\). Since ni≪ti, there is a "competition to be counted" [37]; genes with large relative abundance πij in the original cell are more likely to have nonzero UMI counts, but genes with small relative abundances may be observed with UMI counts of exact zeros. The UMI counts yij are a multinomial sample of the true biological counts xij, containing only relative information about expression patterns in the cell [37, 38].
The multinomial distribution can be approximated by independent Poisson distributions and overdispersed (Dirichlet) multinomials by independent negative binomial distributions. These approximations are useful for computational tractability. Details are provided in the "Methods" section.
The multinomial model makes two predictions which we verified using negative control data. First, the fraction of zeros in a sample (cell or droplet) is inversely related to the total number of UMIs in that sample. Second, the probability of an endogenous gene or ERCC spike-in having zero counts is a decreasing function of its mean expression (equations provided in the "Methods" section). Both of these predictions were validated by the negative control data (Fig. 1). In particular, the empirical probability of a gene being zero across droplets was well calibrated to the theoretical prediction based on the multinomial model. This also demonstrates that UMI counts are not zero inflated, consistent with [29].
Multinomial model adequately characterizes sampling distributions of technical and biological replicates negative control data. a Fraction of zeros is plotted against the total number of UMI in each droplet for the technical replicates. b As a but for cells in the biological replicates (monocytes). c After down-sampling replicates to 10,000 UMIs per droplet to remove variability due to the differences in sequencing depth, the fraction of zeros is computed for each gene and plotted against the log of expression across all samples for the technical replicates data. The solid curve is theoretical probability of observing a zero as a function of the expected counts derived from the multinomial model (blue) and its Poisson approximation (green). d As c but for the biological replicates (monocytes) dataset and after down-sampling to 575 UMIs per cell. Here, we also add the theoretical probability derived from a negative binomial model (red)
To further validate the multinomial model, we assessed goodness-of-fit of seven possible null distributions to both the Tung and Zheng monocytes negative control datasets (Additional file 1: Figure S2). When applied to UMI counts, the multinomial, Dirichlet-multinomial, and Poisson (as approximation to multinomial) distributions fit best. When applied to read counts, the zero-inflated lognormal was the best fitting distribution followed by the Dirichlet-multinomial.
These results are consistent with [39], which also found that the relationship between average expression and zero probability follows the theoretical curve predicted by a Poisson model using negative control data processed with Indrop [4] and Dropseq [3] protocols. These are droplet protocols with typically low counts. It has been argued that the Poisson model is insufficient to describe the sampling distribution of genes with high counts and the negative binomial model is more appropriate [11]. The Tung dataset contained high counts, and we nevertheless found the Poisson gave a better fit than the negative binomial. However, the difference was not dramatic, so our results do not preclude the negative binomial as a reasonable sampling distribution for UMI counts. Taken together, these results suggest our data-generating mechanism is an accurate model of technical noise in real data.
Normalization and log transformation distorts UMI data
Standard scRNA-Seq analysis involves normalizing raw counts using size factors, applying a log transformation with a pseudocount, and then centering and scaling each gene before dimension reduction. The most popular normalization is counts per million (CPM). The CPM are defined as (yij/ni)×106 (i.e., the size factor is ni/106). This is equivalent to the maximum likelihood estimator (MLE) for relative abundance \(\hat {\pi }_{ij}\) multiplied by 106. The log-CPM are then \(\log _{2}(c+\hat {\pi }_{ij}10^{6}) = \log _{2}(\tilde {\pi }_{ij})+C\), where \(\tilde {\pi }_{ij}\) is a maximum a posteriori estimator (MAP) for πij (mathematical justification and interpretation of this approach provided in the "Methods" section). The additive constant C is irrelevant if data are centered for each gene after log transformation, as is common practice. Thus, normalization of raw counts is equivalent to using MLEs or MAP estimators of the relative abundances.
Log transformation of MLEs is not possible for UMI counts due to exact zeros, while log transformation of MAP estimators of πij systematically distorts the differences between zero and nonzero UMI counts, depending on the arbitrary pseudocount c (derivations provided in the "Methods" section). To illustrate this phenomenon, we examined the distribution of an illustrative gene before and after the log transform with varying normalizations using the biological replicates negative control data (Fig. 2). Consistent with our theoretical predictions, this artificially caused the distribution to appear zero inflated and exaggerated differences between cells based on whether the count was zero or nonzero.
Example of how current approaches to normalization and transformation artificially distort differences between zero and nonzero counts. a UMI count distribution for gene ENSG00000114391 in the monocytes biological replicates negative control dataset. b Counts per million (CPM) distribution for the exact same count data. c Distribution of log2(1+CPM) values for the exact same count data
Focusing on the entire negative control datasets, we applied PCA to log-CPM values. We observed a strong correlation (r=0.8 for technical and r=0.98 for monocytes biological replicates) between the first principal component (PC) and the fraction of zeros, consistent with [30]. Application of PCA to CPM values without log transform reduced this correlation to r=0.1 for technical and r=0.7 for monocytes biological replicates. Additionally, the first PC of log-CPM correlated with the log of total UMI, which is consistent with the multinomial model (Fig. 3). Note that in datasets with strong biological variability, the nuisance variation from zero fraction and total counts could appear in secondary PCs rather than the first PC, but it would still confound downstream analyses. Based on these results, the log transformation is not necessary and in fact detrimental for the analysis of UMI counts. The benefits of avoiding normalization by instead directly modeling raw counts have been demonstrated in the context of differential expression [40]. Where normalization is unavoidable, we propose the use of approximate multinomial deviance residuals (defined in the "Residuals and z-scores" section) instead of log-transformed CPM.
Current approaches to normalization and transformation induce variability in the fraction of zeros across cells to become the largest source of variability which in turn biases clustering algorithms to produce false-positive results based on distorted latent factors. a First principal component (PC) from the technical replicates dataset plotted against fraction of zeros for each cell. A red to blue color scale represents total UMIs per cell. b As a but for the monocytes biological replicates data. c Using the technical replicates, we applied t-distributed stochastic neighbor embedding (tSNE) with perplexity 30 to the top 50 PCs computed from log-CPM. The first 2 tSNE dimensions are shown with a blue to red color scale representing the fraction of zeros. d As c but for the biological replicates data. Here, we do not expect to find differences, yet we see distorted latent factors being driven by the total UMIs. PCA was applied to 5000 random genes
Zero inflation is an artifact of log normalization
To see how normalization and log transformation introduce the appearance of zero inflation, consider the following example. Let yij be the observed UMI counts following a multinomial distribution with size ni for each cell and relative abundance πj for each gene, constant across cells. Focusing on a single gene j, yij follows a binomial distribution with parameters ni and pj. Assume πj=10−4 and the ni range from 1000−3000, which is consistent with the biological replicates negative control data (Fig. 1 and Additional file 1: Figure S1). Under this assumption, we expect to see about 74–90% zeros, 22–30% ones, and less than 4% values above one. However, notice that after normalization to CPM and log transformation, all the zeros remain log2(1+0)=0, yet the ones turn into values ranging from log2(1+1/3000×106)= log2(334)≈8.4 to log2(1001)≈10. The few values that are 2 will have values ranging from log2(668)≈9.4 to log2(2001)≈11. The large, artificial gap between zero and nonzero values makes the log-normalized data appear zero-inflated (Fig. 2). The variability in CPM values across cells is almost completely driven by the variability in ni. Indeed, it shows up as the primary source of variation in PCA plots (Fig. 3).
Generalized PCA for dimension reduction of sparse counts
While PCA is a popular dimension reduction method, it is implicitly based on Euclidean distance, which corresponds to maximizing a Gaussian likelihood. Since UMI counts are not normally distributed, even when normalized and log transformed, this distance metric is inappropriate [41], causing PCA to produce distorted latent factors (Fig. 3). We propose the use of PCA for generalized linear models (GLMs) [31] or GLM-PCA as a more appropriate alternative. The GLM-PCA framework allows for a wide variety of likelihoods suitable for data types such as counts and binary values. While the multinomial likelihood is ideal for modeling technical variability in scRNA-Seq UMI counts (Fig. 1), in many cases, there may be excess biological variability present as well. For example, if we wish to capture variability due to clusters of different cell types in a dimension reduction, we may wish to exclude biological variability due to cell cycle. Biological variability not accounted for by the sampling distribution may be accomodated by using a Dirichlet-multinomial likelihood, which is overdispersed relative to the multinomial. In practice, both the multinomial and Dirichlet-multinomial are computationally intractable and may be approximated by the Poisson and negative binomial likelihoods, respectively (detailed derivations provided in the "Methods" section). We implemented both negative binomial and Poisson GLM-PCA, but we focused primarily on the latter in our assessments for simplicity of exposition. Intuitively, using Poisson instead of negative binomial implies, we assume the biological variability is captured by the factor model and the unwanted biological variability is small relative to the sampling variability. Our implementation also allows the user to adjust for gene-specific or cell-specific covariates (such as batch labels) as part of the overall model.
We ran Poisson GLM-PCA on the technical and biological (monocytes) replicates negative control datasets and found it removed the spurious correlation between the first dimension and the total UMIs and fraction of zeros (Fig. 4). To examine GLM-PCA as a visualization tool, we ran Poisson and negative binomial GLM-PCA along with competing methods on the 2 ground truth datasets (Additional file 1: Figure S3). For the Zheng 4eq dataset, we directly reduced to 2 dimensions. For the Zheng 8eq dataset, we reduced to 15 dimensions then applied UMAP [42]. While all methods effectively separated T cells from other PBMCs, GLM-PCA methods also separated memory and naive cytotoxic cells from the other subtypes of T cells. This separation was not visible with PCA on log-CPM. Computational speed is discussed in the "Computational efficiency of multinomial models" section.
GLM-PCA dimension reduction is not affected by unwanted fraction of zeros variability and avoids false-positive results. a First GLM-PCA dimension (analogous to the first principal component) plotted against the fraction of zeros for the technical replicates with colors representing the total UMIs. b As a but using monocytes biological replicates. c Using the technical replicates, we applied t-distributed stochastic neighbor embedding (tSNE) with perplexity 30 to the top 50 GLM-PCA dimensions. The first 2 tSNE dimensions are shown with a blue to red color scale representing the fraction of zeros. d As c but for the biological replicates data. GLM-PCA using the Poisson approximation to the multinomial was applied to the same 5000 random genes as in Fig. 3
Deviance residuals provide fast approximation to GLM-PCA
One disadvantage of GLM-PCA is it depends on an iterative algorithm to obtain estimates for the latent factors and is at least ten times slower than PCA. We therefore propose a fast approximation to GLM-PCA. When using PCA a common first step is to center and scale the data for each gene as z-scores. This is equivalent to the following procedure. First, specify a null model of constant gene expression across cells, assuming a normal distribution. Next, find the MLEs of its parameters for each gene (the mean and variance). Finally, compute the residuals of the model as the z-scores (derivation provided in the "Methods" section). The fact that scRNA-Seq data are skewed, discrete, and possessing many zeros suggests the normality assumption may be inappropriate. Further, using z-scores does not account for variability in total UMIs across cells. Instead, we propose to replace the normal null model with a multinomial null model as a better match to the data-generating mechanism. The analogs to z-scores under this model are called deviance and Pearson residuals. Mathematical formulae are presented in the "Methods" section. The use of multinomial residuals enables a fast transformation similar to z-scores that avoids difficulties of normalization and log transformation by directly modeling counts. Additionally, this framework allows straightforward adjustment for covariates such as cell cycle signatures or batch labels. In an illustrative simulation (details in the "Residuals and z-scores" section), residual approximations to GLM-PCA lost accuracy in the presence of strong batch effects, but still outperformed the traditional PCA (Additional file 1: Figure S4). Systematic comparisons on ground truth data are provided in the "Multinomial models improve unsupervised clustering" section.
Computational efficiency of multinomial models
We measured time to convergence for reduction to two latent dimensions of GLM-PCA, ZINB-WAVE, PCA on log-CPM, PCA on deviance residuals, and PCA on Pearson residuals. Using the top 600 informative genes, we subsampled the PBMC 68K dataset to 680, 6800, and 68,000 cells. All methods scaled approximately linearly with increasing the numbers of cells, but GLM-PCA was 23–63 times faster than ZINB-WAVE across sample sizes (Additional file 1: Figure S5). Specifically, GLM-PCA processed 68,000 cells in less than 7 min. The deviance and Pearson residuals methods exhibited speeds comparable to PCA: 9–26 times faster than GLM-PCA. We also timed dimension reduction of the 8eq dataset (3994 cells) from 1500 informative genes to 10 latent dimensions. PCA (with either log-CPM, deviance, or Pearson residuals) took 7 s, GLM-PCA took 4.7 min, and ZINB-WAVE took 86.6 min.
Feature selection using deviance
Feature selection, or identification of informative genes, may be accomplished by ranking genes using the deviance, which quantifies how well each gene fits a null model of constant expression across cells. Unlike the competing highly variable or highly expressed genes methods, which are sensitive to normalization, ranking genes by deviance operates on raw UMI counts. An approximate multinomial deviance statistic can be computed in closed form (formula provided in the "Methods" section).
We compared gene ranks for all three feature selection methods (deviance, highly expressed, and highly variable genes) on the 8eq dataset (Table 1). We found a strong concordance between highly deviant genes and highly expressed genes (Spearman's rank correlation r=0.9987), while highly variable genes correlated weakly with both high expression (r=0.3835) and deviance (r=0.3738).
Choosing informative genes by high expression alone would be ineffective if a gene had high but constant expression across cells. To ensure the deviance criterion did not identify such genes, we created a simulation with three types of genes: lowly expressed, high but constantly expressed, and high and variably expressed. Deviance preferentially selected high and variably expressed genes while filtering by highly expressed genes identified the constantly expressed genes before the variably expressed (Additional file 1: Figure S6, Table S1). Furthermore, an examination of the top 1000 genes by each criteria on the Muraro dataset showed that deviance did not identify the same set of genes as highly expressed genes (Additional file 1: Figure S7, Table S2). Empirically, deviance seems to select genes that are both highly expressed and highly variable, which provides a rigorous justification for a common practice.
Multinomial models improve unsupervised clustering
Dimension reduction with GLM-PCA or its fast multinomial residuals approximation improved clustering performance over competing methods (Fig. 5a, Additional file 1: Figure S8a). Feature selection by multinomial deviance was superior to highly variable genes (Fig. 5b).
Dimension reduction with GLM-PCA and feature selection using deviance improves Seurat clustering performance. Each column represents a different ground truth dataset from [15]. a Comparison of dimension reduction methods based on the top 1500 informative genes identified by approximate multinomial deviance. The Poisson approximation to the multinomial was used for GLM-PCA. Dev. resid. PCA, PCA on approximate multinomial deviance residuals. b Comparison of feature selection methods. The top 1500 genes identified by deviance and highly variable genes were passed to 2 different dimension reduction methods: GLM-PCA and PCA on log-transformed CPM. Only the results with the number of clusters within 25% of the true number are presented
Using the two ground truth datasets described under the "Datasets" section, we systematically compared the clustering performance of all combinations of previously described methods for normalization, feature selection, and dimension reduction. In addition, we compared against ZINB-WAVE since it also avoids requiring the user to pre-process and normalize the UMI count data (e.g., log transformation of CPM) and accounts for varying total UMIs across cells [28]. After obtaining latent factors, we used Seurat's Louvain implementation and k-means to infer clusters, and compared these to the known cell identities using adjusted Rand index (ARI, [43]). This quantified accuracy. We assessed cluster separation using the silhouette coefficient. We varied the number of latent dimensions and number of clusters to assess robustness. Where possible, we used the same combinations of hyperparameters as [15] to facilitate comparisons to their extensive benchmarking (details are provided in the "Methods" section).
We compared the Seurat clustering performance of GLM-PCA (with Poisson approximation to multinomial) to running PCA on deviance residuals, which adhere more closely to the normal distribution than log-CPM. We found both of these approximate multinomial methods gave similar results on the 4eq dataset and outperformed PCA on log-CPM z-scores. However, GLM-PCA outperformed the residuals method on the 8eq dataset. Also, performance on ZINB-WAVE factors degraded when the number of latent dimensions increased from 10 to 30, whereas GLM-PCA and its fast approximation with deviance residuals were robust to this change (Fig. 5a). GLM-PCA and its residual approximations produced better cluster separation than PCA or ZINB-WAVE, even in scenarios where all methods had similar accuracy (Additional file 1: Figure S8a). The performance of Pearson residuals was similar to that of deviance residuals (Additional file 1: Figure S9, S10).
Focusing on feature selection methods, deviance had higher accuracy than highly variable genes across both datasets and across dimension reduction methods (Fig. 5b). Filtering by highly expressed genes led to similar clustering performance as deviance (Additional file 1: Figure S9), because both criteria identified strongly overlapping gene lists for these data. The combination of feature selection with deviance and dimension reduction with GLM-PCA also improved clustering performance when k-means was used in place of Seurat (Additional file 1: Figure S11). A complete table of results is publicly available (see the "Availability of data and materials" section).
Finally, we examined the clustering performance of competing dimension reduction methods on two public datasets with more complex subtypes (Table 1). The 10 × Haber dataset [33] was annotated with 12 types of enteroendocrine cells from the intestine. The CEL-Seq2 Muraro dataset [34] was annotated with 9 types of pancreatic cells. Since these cluster labels were computationally derived, they did not constitute a ground truth comparison. Nevertheless, GLM-PCA had the closest concordance with the original authors' annotation in both datasets (Additional file 1: Tables S3, S4).
We have outlined a statistical framework for analysis of scRNA-Seq data with UMI counts based on a multinomial model, providing effective and simple to compute methods for feature selection and dimension reduction. We found that UMI count distributions differ dramatically from read counts, are well-described by a multinomial distribution, and are not zero inflated. Log transformation of normalized UMI counts is detrimental, because it artificially exaggerates the differences between zeros and all other values. For feature selection, or identification of informative genes, deviance is a more effective criterion than highly variable genes. Dimension reduction via GLM-PCA, or its fast approximation using residuals from a multinomial model, leads to better clustering performance than PCA on z-scores of log-CPM.
Although our methods were inspired by scRNA-Seq UMI counts, they may be useful for a wider array of data sources. Any high dimensional, sparse dataset where samples contain only relative information in the form of counts may conceivably be modeled by the multinomial distribution. Under such scenarios, our methods are likely to be more effective than applying log transformations and standard PCA. A possible example is microbiome data.
We have not addressed major topics in the scRNA-Seq literature such as pseudotime inference [44], differential expression [45], and spatial analysis [46]. However, the statistical ideas outlined here can also be used to improve methods in these more specialized types of analyses.
Our results have focused on (generalized) linear models for simplicity of exposition. Recently, several promising nonlinear dimension reductions for scRNA-Seq have been proposed. The variational autoencoder (VAE, a type of neural network) method scVI [47] utilizes a negative binomial likelihood in the decoder, while the encoder relies on log-normalized input data for numerical stability. The Gaussian process method tGPLVM [48] models log-transformed counts. In both cases, we suggest replacing log-transformed values with deviance residuals to improve performance. Nonlinear dimension reduction methods may also depend on feature selection to reduce memory consumption and speed computation; here, our deviance method may be utilized as an alternative to high variability for screening informative genes.
Multinomial model for scRNA-Seq
Let yij be the observed UMI counts for cell or droplet i and gene or spike-in j. Let \(n_{i}=\sum _{j} y_{ij}\) be the total UMIs in the sample, and πij be the unknown true relative abundance of gene j in cell i. The random vector \(\vec {y}_{i} = (y_{i1},\ldots,y_{iJ})^{\top }\) with constraint \(\sum _{j} y_{ij}=n_{i}\) follows a multinomial distribution with densit function:
$$f(\vec{y}_{i}) = \binom{n_{i}}{y_{i1},\ldots,y_{iJ}} \prod_{j} \pi_{ij}^{y_{ij}} $$
Focusing on a single gene j at a time, the marginal distribution of yij is binomial with parameters ni and πij. The marginal mean is E[yij]=niπij=μij, the marginal variance is \(\text {var}[y_{ij}] = n_{i} \pi _{ij}(1-\pi _{ij}) = \mu _{ij}-\frac {1}{n_{i}}\mu _{ij}^{2}\), and the marginal probability of a zero count is \((1-\pi _{ij})^{n_{i}} = \left (1-\frac {\mu _{ij}}{n_{i}}\right)^{n_{i}}\). The correlation between two genes j,k is:
$$\text{cor}[y_{ij},y_{ik}] = \frac{-\sqrt{\pi_{ij}\pi_{ik}}}{\sqrt{(1-\pi_{ij})(1-\pi_{ik})}} $$
The correlation is induced by the sum to ni constraint. As an extreme example, if there are only two genes (J=2), increasing the count of the first gene automatically reduces the count of the second gene since they must add up to ni under multinomial sampling. This means when J=2, there is a perfect anti-correlation between the gene counts which has nothing to do with biology. More generally, when either J or ni is small, gene counts will be negatively correlated independent of biological gene-gene correlations, and it is not possible to analyze the data on a gene-by-gene basis (for example, by ranking and filtering genes for feature selection). Rather, comparisons are only possible between pairwise ratios of gene expression values [49]. Yet, this type of analysis is difficult to interpret and computationally expensive for large numbers of genes (i.e., in high dimensions). Fortunately, under certain assumptions, more tractable approximations may be substituted for the true multinomial distribution.
First, note that if correlation is ignored, the multinomial may be approximated by J-independent binomial distributions. Intuitively, this approximation will be reasonable if all πij are very small, which is likely to be satisfied for scRNA-Seq if the number of genes J is large, and no single gene constitutes the majority of mRNAs in the cell. If ni is large and πij is small, each binomial distribution can be further approximated by a Poisson with mean niπij. Alternatively, the multinomial can be constructed by drawing J-independent Poisson random variables and conditioning on their sum. If J and ni are large, the difference between the conditional, multinomial distribution, and the independent Poissons becomes negligible. Since in practice ni is large, the Poisson approximation to the multinomial may be reasonable [50–53].
The multinomial model does not account for biological variability. As a result, an overdispersed version of the multinomial model may be necessary. This can be accommodated with the Dirichlet-multinomial distribution. Let \(\vec {y}_{i}\) be distributed as a multinomial conditional on the relative abundance parameter vector \(\vec {\pi }_{i}=(\pi _{i1},\ldots,\pi _{iJ})^{\top }\). If \(\vec {\pi }_{i}\) is itself a random variable with symmetric Dirichlet distribution having shape parameter α, the marginal distribution of \(\vec {y}_{i}\) is Dirichlet-multinomial. This distribution can itself be approximated by independent negative binomials. First, note that a symmetric Dirichlet random vector can be constructed by drawing J-independent gamma variates with shape parameter α and dividing by their sum. Suppose (as above) we approximate the conditional multinomial distribution of \(\vec {y}_{i}\) such that yij follows an approximate Poisson distribution with mean niπij. Let λij be a collection of non-negative random variables such that \(\pi _{ij}=\frac {\lambda _{ij}}{\sum _{j} \lambda _{ij}}\). We require that \(\vec {\pi }_{i}\) follows a symmetric Dirichlet, which is accomplished by having λij follow independent gamma distributions with shape α and mean ni/J. This implies \(\sum _{j} \lambda _{ij}\) follows a gamma with shape Jα and mean ni. As J→∞, this distribution converges to a point mass at ni, so for large J (satisfied by scRNA-Seq), \(\sum _{j} \lambda _{ij}\approx n_{i}\). This implies that yij approximately follows a conditional Poisson distribution with mean λij, where λij is itself a gamma random variable with mean ni/J and shape α. If we then integrate out λij we obtain the marginal distribution of yij as negative binomial with shape α and mean ni/J. Hence a negative binomial model for count data may be regarded as an approximation to an overdispersed Dirichlet-multinomial model.
Parameter estimation with multinomial models (and their binomial or Poisson approximations) is straightforward. First, suppose we observe replicate samples \(\vec {y}_{i}\), i=1,…,I from the same underlying population of molecules, where the relative abundance of gene j is πj. This is a null model because it assumes each gene has a constant expected expression level, and there is no biological variation across samples. Regardless of whether one assumes a multinomial, binomial, or Poisson model, the maximum likelihood estimator (MLE) of πj is \(\hat {\pi }_{j} = \frac {\sum _{i} y_{ij}}{\sum _{i} n_{i}}\) where ni is the total count of sample i. In the more realistic case that relative abundances πij of genes vary across samples, the MLE is \(\hat {\pi }_{ij}=\frac {y_{ij}}{n_{i}}\).
An alternative to the MLE is the maximum a posteriori (MAP) estimator. Suppose a symmetric Dirichlet prior with concentration parameter αi is combined with the multinomial likelihood for cell i. The MAP estimator for πij is given by:
$$\tilde{\pi}_{ij}=\frac{\alpha_{i}+y_{ij}}{J\alpha_{i}+n_{i}} = w_{i}\frac{1}{J}+(1-w_{i})\hat{\pi}_{ij} $$
where wi=Jαi/(Jαi+ni), showing that the MAP is a weighted average of the prior mean that all genes are equally expressed (1/J) and the MLE (\(\hat {\pi }_{ij}\)). Compared to the MLE, the MAP biases the estimate toward the prior where all genes have the same expression. Larger values of αi introduce more bias, while αi→0 leads to the MLE. If αi>0, the smallest possible value of \(\tilde {\pi }_{ij}\) is αi/(Jαi+ni) rather than zero for the MLE. When there are many zeros in the data, MAP can stabilize relative abundance estimates at the cost of introducing bias.
Mathematics of distortion from log-normalizing UMIs
Suppose the true counts in cell i are given by xij for genes j=1,…,J. Some of these may be zero, if a gene is not turned on in the cell. Knowing xij is equivalent to knowing the total number of transcripts \(t_{i}=\sum _{j} x_{ij}\) and the relative proportions of each gene πij, since xij=tiπij. The total number of UMI counts \(n_{i}=\sum _{j} y_{ij}\) does not estimate ti. However, under multinomial sampling, the UMI relative abundances \(\hat {\pi }_{ij}=\frac {y_{ij}}{n_{i}}\) are MLEs for the true proportions πij. Note that it is possible that \(\hat {\pi }_{ij}=0\) even though πij>0. Because \(\sum _{j} \hat {\pi }_{ij}=1\) regardless of ni, the use of multinomial MLEs is equivalent to the widespread practice of normalizing each cell by the total counts. Furthermore, the use of size factors si=ni/m leads to \(\hat {\pi }_{ij} \times m\) (if m=106, this is CPM).
Traditional bulk RNA-Seq experiments measured gene expression in read counts of many cells per sample rather than UMI counts of single cells. Gene counts from bulk RNA-Seq could thus range over several orders of magnitude. To facilitate comparison of these large numbers, many bulk RNA-Seq methods have relied on a logarithm transformation. This enables interpretation of differences in normalized counts as fold changes on a relative scale. Also, for count data, the variance of each gene is a function of its mean, and log transformation can help to prevent highly expressed outlier genes from overwhelming downstream analyses. Prior to the use of UMIs, scRNA-Seq experiments also produced read counts with wide ranging values, and a log transform was again employed. However, with single cell data, more than 90% of the genes might be observed as exact zeros, and log(0)=−∞ which is not useful for data analysis. UMI data also contain large numbers of zeros, but do not contain very large counts since PCR duplicates have been removed. Nevertheless, log transformation has been commonly used with UMI data as well.
The current standard is to transform the UMI counts as \(\log _{2}(c+\hat {\pi }_{ij} \times m)\) where c is a pseudocount to avoid taking the log of zero, and typically c=1. As before, m is some constant such as 106 for CPM (see also [54] for an alternative). Finally, the data are centered and scaled so that the mean of each gene across cells is 0, and the standard deviation is 1. This standardization of the data causes any subsequent computation of distances or dimension reduction to be invariant to constant additive or multiplicative scaling. For example, under Manhattan distance, d(x+c,y+c)=|x+c−(y+c)|=|x−y|=d(x,y). In particular, using size factors such as CPM instead of relative abundances leads to a rescaling of the pseudocount, and use of any pseudocount is equivalent to replacing the MLE with the MAP estimator. Let k=c/m and αi=kni. Then, the weight term in the MAP formula becomes wi=Jk/(1+Jk)=w which is constant across all cells i. Furthermore Jk=w/(1−w), showing that:
$${}{\begin{aligned} \log_{2}(c+\hat{\pi}_{ij} \times m) &= \log_{2}(k+\hat{\pi}_{ij}) + \log_{2}(m)\\ &= \log_{2}\left(\frac{w}{1-w}\frac{1}{J}+\hat{\pi}_{ij}\right)+\log_{2}(m)\\ &= \log_{2}\left(w\frac{1}{J}+(1-w)\hat{\pi}_{ij}\right)-\log_{2}(1-w)+\log_{2}(m)\\ &= \log_{2}(\tilde{\pi}_{ij})+C \end{aligned}} $$
Where C is a global constant that does not vary across cells or genes. For illustration, if c=1 and m=106, this is equivalent to assuming a prior where all genes are equally expressed and for cell i, a weight of w=J/(106+J) is given to the prior relative to the MLE. Since the number of genes J is on the order of 104, we have w≈.01. The prior sample size for cell i is Jαi=10−6Jni≈.01×ni where ni is the data sample size. The standard transformation is therefore equivalent to using a weak prior to obtain a MAP estimate of the relative abundances, then log transforming before dimension reduction.
In most scRNA-Seq datasets, the total number of UMIs ni for some cells may be significantly less than the constant m. For these cells, the size factors si=ni/m are less than 1. Therefore, after normalization (dividing by size factor), the counts are scaled up to match the target size of m. Due to the discreteness of counts, this introduces a bias after log transformation, if the pseudocount is small (or equivalently, if m is large). For example, let c=1 and m=106 (CPM). If ni=104 for a particular cell, we have si=.01. A raw count of yij=1 for this cell is normalized to 1/.01=100 and transformed to log2(1+100)=6.7. For this cell, on the log scale, there cannot be any values between 0 and 6.7 because fractional UMI counts cannot be observed and log2(1+0)=0. Small pseudocounts and small size factors combined with log transform arbitrarily exaggerate the difference between a zero count and a small nonzero count. As previously shown, this scenario is equivalent to using MAP estimation of πij with a weak prior. To combat this distortion, one may attempt to strengthen the prior to regularize \(\tilde {\pi }_{ij}\) estimation at the cost of additional bias, as advocated by [21]. An extreme case occurs when c=1 and m=1. Here, the prior sample size is Jni, so almost all the weight is on the prior. The transform is then \(\log _{2}(1+\hat {\pi }_{ij})\). But this function is approximately linear on the domain \(0\leq \hat {\pi }_{ij}\leq 1\). After centering and scaling, a linear transformation is vacuous.
To summarize, log transformation with a weak prior (small size factor, such as CPM) introduces strong artificial distortion between zeros and nonzeros, while log tranformation with a strong prior (large size factor) is roughly equivalent to not log transforming the data.
Generalized PCA
PCA minimizes the mean squared error (MSE) between the data and a low-rank representation, or embedding. Let yij be the raw counts and zij be the normalized and transformed version of yij such as centered and scaled log-CPM (z-scores). The PCA objective function is:
$$\min_{u,v} \sum_{i,j}(z_{ij}-\vec{u}_{i}'\vec{v}_{j})^{2} $$
where \(\vec {u}_{i},\vec {v}_{j}\in \mathbb {R}^{L}\) for i=1,…,I, j=1,…,J. The \(\vec {u}_{i}\) are called factors or principal components, and the \(\vec {v}_{j}\) are called loadings. The number of latent dimensions L controls the complexity of the model. Minimization of the MSE is equivalent to minimizing the Euclidean distance metric between the embedding and the data. It is also equivalent to maximizing the likelihood of a Gaussian model:
$$z_{ij}\sim\mathcal{N}\left(\vec{u}_{i}'\vec{v}_{j},\sigma^{2}\right) $$
If we replace the Gaussian model with a Poisson, which approximates the multinomial, we can directly model the UMI counts as:
$$y_{ij}\sim \text{Poi}\left(n_{i}\exp\{\vec{u}_{i}'\vec{v}_{j}\}\right) $$
or alternatively, in the case of overdispersion, we may approximate the Dirichlet-multinomial using a negative binomial likelihood:
$$y_{ij}\sim NB\left(n_{i}\exp\{\vec{u}_{i}'\vec{v}_{j}\};~\phi_{j}\right) $$
We define the linear predictor as \(\eta _{ij} = \log n_{i} + \vec {u}_{i}'\vec {v}_{j}\). It is clear that the mean \(\mu _{ij}=e^{\eta }_{ij}\) appears in both the Poisson and negative binomial model statements, showing that the latent factors interact with the data only through the mean. We may then estimate \(\vec {u}_{i}\) and \(\vec {v}_{j}\) (and ϕj) by maximizing the likelihood (in practice, adding a small L2 penalty to large parameter values improves numerical stability). A link function must be used since \(\vec {u}_{i}\) and \(\vec {v}_{j}\) are real valued whereas the mean of a Poisson or negative binomial must be positive. The total UMIs ni term is used as an offset since no normalization has taken place; alternative size factors si such as those from scran [20] could be used in place of ni. If the first element of each \(\vec {u}_{i}\) is constrained to equal 1, this induces a gene-specific intercept term in the first position of each \(\vec {v}_{j}\), which is analogous to centering. Otherwise, the model is very similar to that of PCA; it is simply optimizing a different objective function. Unfortunately, MLEs for \(\vec {u}_{i}\) and \(\vec {v}_{j}\) cannot be expressed in closed form, so an iterative Fisher scoring procedure is necessary. We refer to this model as GLM-PCA [55]. Just as PCA minimizes MSE, GLM-PCA minimizes a generalization of MSE called the deviance [56]. While generalized PCA was originally proposed by [31] (see also [57] and [58]), our implementation is novel in that it allows for intercept terms, offsets, overdispersion, and non-canonical link functions. We also use a blockwise update for optimization which we found to be more numerically stable than that of [31]; we iterate over latent dimensions l rather than rows or columns. This technique is inspired by non-negative matrix factorization algorithms such as hierarchical alternating least squares and rank-one residue iteration, see [59] for a review.
As an illustration, consider GLM-PCA with the Poisson approximation to a multinomial likelihood. The objective function to be minimized is simply the overall deviance:
$$\begin{array}{*{20}l} D &= \sum_{i,j} y_{ij}\log\left(\frac{y_{ij}}{\mu_{ij}}\right)-(y_{ij}-\mu_{ij})\\ \log\mu_{ij} &= \eta_{ij} = \log s_{i} + \vec{u}_{i}'\vec{v}_{j} = \log s_{i} + v_{j1} + \sum_{l=2}^{L} u_{il}v_{jl} \end{array} $$
where si is a fixed size factor such as the total number of UMIs (ni). The optimization proceeds by taking derivatives with respect to the unknown parameters: vj1 is a gene-specific intercept term, and the remaining uil and vjl are the latent factors.
The GLM-PCA method is most concordant to the data-generating mechanism since all aspects of the pipeline are integrated into a coherent model rather than being dealt with through sequential normalizations and transformations. The interpretation of the \(\vec {u}_{i}\) and \(\vec {v}_{j}\) vectors is the same as in PCA. For example, suppose we set the number of latent dimensions to 2 (i.e., L=3 to account for the intercept). We can plot ui2 on the horizontal axis and ui3 on the vertical axis for each cell i to visualize the relationships between cells such as gradients or clusters. In this way, the \(\vec {u}_{i}\) and \(\vec {v}_{j}\) capture biological variability such as differentially expressed genes.
Residuals and z-scores
Just as mean squared error can be computed by taking the sum of squared residuals under a Gaussian likelihood, the deviance is equal to the sum of squared deviance residuals [56]. Since deviance residuals are not well-defined for the multinomial distribution, we adopt the binomial approximation. The deviance residual for gene j in cell i is given by:
$${}r^{(d)}_{ij}=\text{sign}(y_{ij}-\hat{\mu}_{ij})\sqrt{2y_{ij}\log\frac{y_{ij}}{\hat{\mu}_{ij}} + 2(n_{i}-y_{ij})\log\frac{n_{i}-y_{ij}}{n_{i}-\hat{\mu}_{ij}}} $$
where under the null model of constant gene expression across cells, \(\hat {\mu }_{ij}=n_{i}\hat {\pi }_{j}\). The deviance residuals are the result of regressing away this null model. An alternative to deviance residuals is the Pearson residual, which is simply the difference in observed and expected values scaled by an estimate of the standard deviation. For the binomial, this is:
$$r^{(p)}_{ij}=\frac{y_{ij}-\hat{\mu}_{ij}}{\sqrt{\hat{\mu}_{ij}-\frac{1}{n_{i}}\hat{\mu}_{ij}^{2}}} $$
According to the theory of generalized linear models (GLM), both types of residuals follow approximately a normal distribution with mean zero if the null model is correct [56]. Deviance residuals tend to be more symmetric than Pearson residuals. In practice, the residuals may not have mean exactly equal to zero, and may be standardized by scaling their gene-specific standard deviation just as in the Gaussian case. Recently, Pearson residuals based on a negative binomial null model have also been independently proposed as the sctransform method [60].
The z-score is simply the Pearson residual where we replace the multinomial likelihood with a Gaussian (normal) likelihood and use normalized values instead of raw UMI counts. Let qij be the normalized (possibly log-transformed) expression of gene j in cell i without centering and scaling. The null model is that the expression of the gene is constant across all cells:
$$q_{ij}\sim\mathcal{N}\left(\mu_{j},~\sigma^{2}_{j}\right) $$
The MLEs are \(\hat {\mu }_{j} = \frac {1}{I}\sum _{i} q_{ij}\), \(\hat {\sigma }^{2}_{j} = \frac {1}{I}\sum _{i} (q_{ij}-\hat {\mu }_{j})^{2}\), and the z-scores equal the Pearson residuals \(z_{ij}=(q_{ij}-\hat {\mu }_{j})/\hat {\sigma }_{j}\).
We compared the accuracy of the residuals approximations by simulating 150 cells in 3 clusters of 50 cells each with 5000 genes, of which 500 were differentially expressed across clusters (informative genes). We also created 2 batches, batch 1 with total counts of 1000 and batch 2 with total counts of 2000. Each cluster had an equal number of cells in the 2 batches. We then ran GLM-PCA on the raw counts, PCA on log2(1+CPM), PCA on deviance residuals, and PCA on Pearson residuals with L=2 dimensions.
Genes with constant expression across cells are not informative. Such genes may be described by the multinomial null model where πij=πj. Goodness of fit to a multinomial distribution can be quantified using deviance, which is twice the difference in log-likelihoods comparing a saturated model to a fitted model. The multinomial deviance is a joint deviance across all genes,and for this reason is not helpful for screening informative genes. Instead, one may use the binomial deviance as an approximation:
$$D_{j} = 2\sum_{i}\left[y_{ij}\log\frac{y_{ij}}{n_{i}\hat{\pi}_{j}} + (n_{i}-y_{ij})\log\frac{(n_{i}-y_{ij})}{n_{i}(1-\hat{\pi}_{j})}\right] $$
A large deviance value indicates the model in question provides a poor fit. Those genes with biological variation across cells will be poorly fit by the null model and will have the largest deviances. By ranking genes according to their deviances, one may thus obtain highly deviant genes as an alternative to highly variable or highly expressed genes.
Systematic comparison of methods
We considered combinations of the following methods and parameter settings, following [15]. Italics indicate methods proposed in this manuscript. Feature selection: highly expressed genes, highly variable genes, and highly deviant genes. We did not compare against highly dropout genes because [15] found this method to have poor downstream clustering performance for UMI counts and it is not as widely used in the literature. The numbers of genes are 60, 300, 1500. Normalization, transformation, and dimension reduction: PCA on log-CPM z-scores, ZINB-WAVE [28], PCA on deviance residuals, PCA on Pearson residuals, and GLM-PCA. The numbers of latent dimensions are 10 and 30. Clustering algorithms are k-means [61] and Seurat [17]. The number of clusters is all values from 2 to 10, inclusive. Seurat resolutions are 0.05, 0.1, 0.2, 0.5, 0.8, 1, 1.2, 1.5, and 2.
All methods and assessments described in this manuscript are publicly available at https://github.com/willtownes/scrna2019 [62]. GLM-PCA is available as an R package from CRAN (https://cran.r-project.org/web/packages/glmpca/index.html). The source code is licensed under LGPL-3.
All datasets used in the study were obtained from public sources (Table 1). The three Zheng datasets (ERCCs, monocytes, and 68K PBMCs) [5] were downloaded from https://support.10xgenomics.com/single-cell-gene-expression/datasets. The Duo datasets were obtained through the bioconductor package DuoClustering2018 [15]. The remaining three datasets had GEO accession numbers GSE77288 (Tung) [32], GSE92332 (Haber) [33], and GSE85241 (Muraro) [34].
An amendment to this paper has been published and can be accessed via the original article.
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The authors thank Keegan Korthauer, Jeff Miller, Linglin Huang, Alejandro Reyes, Yered Pita-Juarez, Mike Love, Ziyi Li, and Kelly Street for the valuable suggestions.
The review history is available as Additional file 2.
FWT was supported by NIH grant T32CA009337, SCH was supported by NIH grant R00HG009007, MJA was supported by an MGH Pathology Department startup fund, and RAI was supported by Chan-Zuckerberg Initiative grant CZI 2018-183142 and NIH grants R01HG005220, R01GM083084, and P41HG004059.
Department of Biostatistics, Harvard University, Cambridge, MA, USA
F. William Townes, Martin J. Aryee & Rafael A. Irizarry
Present Address: Department of Computer Science, Princeton University, Princeton, NJ, USA
F. William Townes
Department of Biostatistics, Johns Hopkins University, Baltimore, MD, USA
Stephanie C. Hicks
Molecular Pathology Unit, Massachusetts General Hospital, Charlestown, MA, USA
Martin J. Aryee
Center for Cancer Research, Massachusetts General Hospital, Charlestown, MA, USA
Department of Pathology, Harvard Medical School, Boston, MA, USA
Department of Data Sciences, Dana-Farber Cancer Institute, Boston, MA, USA
Rafael A. Irizarry
SCH, MJA, and RAI identified the problem. FWT proposed, derived, and implemented the GLM-PCA model, its fast approximation using residuals, and feature selection using deviance. SCH, MJA, and RAI provided guidance on refining the methods and evaluation strategies. FWT and RAI wrote the draft manuscript, and revisions were suggested by SCH and MJA. All authors approved the final manuscript.
Correspondence to Rafael A. Irizarry.
Additional file 1
Contains supplementary figures S1–S11 and tables S1–S4.
Review history.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Townes, F.W., Hicks, S.C., Aryee, M.J. et al. Feature selection and dimension reduction for single-cell RNA-Seq based on a multinomial model. Genome Biol 20, 295 (2019). https://doi.org/10.1186/s13059-019-1861-6
Dimension reduction
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The functional inequality for the mixed quermassintegral
Fangwei Chen1,
Jianbo Fang1,
Miao Luo2 &
Congli Yang ORCID: orcid.org/0000-0001-9576-08352
In this paper, the functional Quermassintegrals of a log-concave function in \(\mathbb{R}^{n}\) are discussed. The functional inequality for the ith mixed Quermassintegral is established. Moreover, as a special case, a weaker log-Quermassintegral inequality in \(\mathbb{R}^{n}\) is obtained.
Let \(\mathcal{K}^{n}\) be the set of convex bodies (compact convex subsets with nonempty interiors) in \(\mathbb{R}^{n}\), the fundamental Brunn–Minkowski inequality for convex bodies states that, for \(K, L \in \mathcal{K}^{n}\), the volume of the bodies and of their Minkowski sum \(K+L=\{x+y:x\in K,\text{ and }y\in L\}\) are given by
$$\begin{aligned} V (K+L)^{\frac{1}{n}}\geq V(K)^{\frac{1}{n}}+V(L)^{\frac{1}{n}}, \end{aligned}$$
with equality if and only if K and L are homothetic, namely agreeing up to a translation and a dilation. The Brunn–Minkowski inequality exposes the crucial logarithmic concavity of the volume in \(\mathcal{K}^{n}\), because it has an equivalent formulation as
$$\begin{aligned} V \bigl((1-t)K+t L \bigr)\geq V(K)^{1-t}V(L)^{t}, \end{aligned}$$
for \(t\in (0,1)\). See for example [18, 19, 29] for more about the Brunn–Minkowski inequality. Another important geometric inequality related to the convex bodies K and L is the mixed Quermassintegral inequality,
$$\begin{aligned} W_{i}(K,L)^{n-i}\geq W_{i}(K)^{n-i-1}W_{i}(L),\quad 0\leq i< n-1, \end{aligned}$$
with equality if and only if K and L are homothetic. Here \(W_{i}(K)\) \((i=0, 1, \ldots, n)\) are the Quermassintegrals of K, which are defined by letting \(W_{0}(K)=V_{n}(K)\), the volume of K; \(W_{n}(K)=\omega _{n}\), the volume of the unit ball \(B^{n}_{2}\) in \(\mathbb{R}^{n}\) and, for general \(i=1, 2,\ldots, n-1\),
$$\begin{aligned} W_{n-i}(K)=\frac{\omega _{n}}{\omega _{i}} \int _{\mathcal{G}_{i,n}}\mathrm{vol}_{i}(K |_{\xi _{i}})\,d\mu (\xi _{i}), \end{aligned}$$
where the \(\mathcal{G}_{i,n}\) is the Grassmannian manifold of i-dimensional linear subspaces of \(\mathbb{R}^{n}\), \(d\mu (\xi _{i})\) is the normalized Haar measure on \(\mathcal{G}_{i,n}\), \(K|_{\xi _{i}}\) denotes the orthogonal projection of K onto the i-dimensional subspaces \(\xi _{i}\), and \(\mathrm{vol}_{i}\) is the i-dimensional volume on space \(\xi _{i}\).
In the 1960s, the Minkowski addition was extended to the \(L^{p}\) \((p\geq 1)\) Minkowski sum by defining \(h^{p}_{K+_{p}t \cdot L}=h^{p}_{K}+t h^{p}_{L}\). The extension of the mixed Quermassintegrals to the \(L^{p}\) mixed Quermassintegrals is due to Lutwak [24]. The inequalities for the \(L^{p}\) mixed Quermassintegrals are established by Lutwak. Let \(K, L\in \mathcal{K}^{n}\) with origin in their interiors, \(0\leq i< n-i\) and \(p>1\), then
$$\begin{aligned} W_{p,i}(K,L)^{n-i}\geq W_{i}(K)^{n-i-p}W_{i}(L)^{p}, \end{aligned}$$
with equality if and only if K and L are dilates. Here the \(L^{p}\) mixed Quermassintegrals are defined by
$$\begin{aligned} W_{p,i}(K, L):=\frac{p}{n-i}\lim_{t \rightarrow 0^{+}} \frac{W_{i}(K+_{p}t \cdot L)-W_{i}(L)}{t}, \end{aligned}$$
for \(i=0, 1,\ldots, n-1\). In particular, for \(p=1\) in (0.4), it becomes \(W_{i}(K,L)\), and \(W_{p,0}(K,L)\) is denoted by \(V_{p}(K,L)\), which is called the \(L_{p}\) mixed volume of K and L.
Motivated by the analogy properties between the log-concave functions and the volume of convex bodies in \(\mathcal{K}^{n}\), the interest in studying the log-concave functions has been considerably increasing. For example, the functional Blaschke–Santaló inequality for even log-concave function is discussed by Ball in [6, 7]; for the general case see [8, 17, 21, 28]. The mean width for a log-concave function is introduced by Klartag, Milman and Rotem (see [22, 26, 27]). The affine isoperimetric inequality for the log-concave function is proved by Artstein-Avidan, Klartag, Schütt and Werner [5]. The John ellipsoid for log-concave function has been establish by Gutiérrez, Merino Jiménez and Villa [2], the LYZ ellipsoid for log-concave function is established by Fang and Zhou [16]. See [1, 4, 9, 12–14, 23] for more about the pertinent results.
Let \(f=e^{-u}\), \(g=e^{-v}\) be log-concave functions, \(\alpha, \beta >0\), the "sum" and "scalar multiplication" of log-concave functions are defined as
$$\begin{aligned} \alpha \cdot f\oplus \beta \cdot g:=e^{-w},\quad \text{where } w^{*}=\alpha u^{*}+\beta v^{*}, \end{aligned}$$
here \(w^{*}\) denotes as usual the Fenchel conjugate. The total mass integral \(J(f)\) of f is defined as \(J(f)=\int _{\mathbb{R}^{n}}f(x)\,dx\). In [15], the quantity \(\delta J (f,g)\), which is called the first variation of J at f along g, is defined by Colesanti and Fragalà,
$$ \delta J(f,g)=\lim_{t\rightarrow 0^{+}} \frac{J(f\oplus t\cdot g)-J(f)}{t}. $$
The authors show that the functional form of Minkowski's first inequality is
$$\begin{aligned} \delta J(f,g)\geq J(f)\bigl[\log J(g)+n\bigr]+\mathrm{Ent}(f), \end{aligned}$$
where \(\mathrm{Ent}(f)\) is the entropy of f defined by \(\mathrm{Ent}(f)=\int _{\mathbb{R}^{n}}f\log f\, dx-J(f)\log J(f)\). We have inequality in (0.5) if and only if there exist \(x_{0}\in \mathbb{R}^{n}\) such that \(g(x)=f(x-x_{0})\).
Inspired by Ref. [15] of Colesanti and Fragalà, in this paper, we define the ith functional Quermassintegrals \(W_{i}(f)\) as the i-dimensional average total mass of f,
$$ W_{i}(f):=\frac{\omega _{n}}{\omega _{n-i}} \int _{\mathcal{G}_{n-i,n}}J_{n-i}(f)\,d \mu (\xi _{n-i}), $$
where \(J_{i}(f)\) denotes the i-dimensional total mass of f defined in (3.1), \(\mathcal{G}_{i,n}\) is the Grassmannian manifold of \(\mathbb{R}^{n}\) and \(d\mu (\xi _{n-i})\) is the normalized measure on \(\mathcal{G}_{i,n}\). The first variation of \(W_{i}\) at f along g is defined by (see Definition 3.3)
$$\begin{aligned} W_{i}(f,g)=\lim_{t\rightarrow 0^{+}} \frac{W_{i}(f\oplus t\cdot g)-W_{i}(f)}{t}. \end{aligned}$$
\(W_{i}(f,g)\) is a natural extension of the mixed Quermassintegrals of convex bodies in \(\mathbb{R}^{n}\), we call it the ith functional mixed Quermassintegral. In fact, if one takes \(f=\chi _{K}\), and \(dom(f)=K\in \mathbb{R}^{n}\), then \(W_{i}(\chi _{K})\) turns out to be \(W_{i}(K)\), and \(W_{i}(\chi _{K}, \chi _{L})\) equals \(W_{i}(K,L)\). In this paper, our main result is to show the inequality for the ith functional mixed Quermassintegrals. Let \(\mathcal{A}'\) denote the integrable functions in \(\mathbb{R}^{n}\).
Let f and g are integrable functions on \(\mathcal{A}'\), then
$$\begin{aligned} W_{i}(f,g)\geq W_{i}(f) \biggl[1+ \frac{1}{n-i}\log \frac{W_{i}(g)}{W_{i}(f)} \biggr]+\frac{1}{n-i}W_{i}(f \log f), \end{aligned}$$
with equality if and only if there exists \(x_{0}\in \mathbb{R}^{n}\) such that \(g(x)=f(x-x_{0})\), for all \(x\in \mathbb{R}^{n}\).
The paper is organized as follows, in Sect. 1, we introduce some notations about the log-concave function. In Sect. 2, the projection of log-concave function is discussed. In Sect. 3, we turn our attention to the functional inequalities involving \(W_{i}(f,g)\), we prove the ith functional mixed Quermassintegral inequality. Specially, the weaker log-Quermassintegral inequality for convex bodies is obtained as a corollary.
Let \(u:\Omega \rightarrow (-\infty,+\infty ]\) be a convex function, that is, \(u ((1-t)x+ty )\leq (1-t)u(x)+tu(y)\) for \(t\in (0,1)\), here \(\Omega =\{x\in \mathbb{R}^{n}:u(x)\in \mathbb{R}\}\) is the domain of u. The convexity of u implies that Ω is a convex set in \(\mathbb{R}^{n}\). We say that u is proper if \(\Omega \neq \emptyset \), and u is of class \(\mathcal{C}^{2}_{+}\) if it is twice differentiable on \(\operatorname{int} (\Omega ) \), with a positive definite Hessian matrix. Let
$$\begin{aligned} \mathcal{L}={}& \Bigl\{ u:\Omega \rightarrow (-\infty,+\infty ]: \text{ u is convex, low semi-continuous} \\ &{}\text{and } \lim_{\|x\|\rightarrow +\infty }u(x)=+\infty \Bigr\} . \end{aligned}$$
The Fenchel conjugate of \(u\in \mathcal{L}\) is defined by
$$\begin{aligned} u^{*}(y)=\sup_{x\in \Omega } \bigl\{ \langle x,y\rangle -u(x) \bigr\} . \end{aligned}$$
It is obvious that \(u(x)+u^{*}(y)\geq \langle x,y\rangle \) for all \(x, y\in \Omega \), and there is an equality if and only if \(x\in \Omega \), and y is the subdifferential of u at x, which means
$$\begin{aligned} u^{*}\bigl(\nabla u(x)\bigr)+u(x)=\bigl\langle x, \nabla u(x)\bigr\rangle . \end{aligned}$$
Moreover, if u is a lower semi-continuous convex function, then also \(u^{*}\) is a lower semi-continuous convex function, and \(u^{**}=u\). The infimal convolution of functions u and v is defined by
$$\begin{aligned} u\Box v(x)=\inf_{y\in \Omega } \bigl\{ u(x-y)+v(y) \bigr\} . \end{aligned}$$
The right scalar multiplication by a nonnegative real number α is given by
$$\begin{aligned} (u\alpha ) (x):= \textstyle\begin{cases} \alpha u (\frac{x}{\alpha } ) & \text{if }\alpha >0; \\ I_{\{0\}} & \text{if }\alpha =0. \end{cases}\displaystyle \end{aligned}$$
The following propositions below gather some elementary properties of the Fenchel conjugate and the infimal convolution of u and v, which can be found in [15, 25].
Let \(u\in \mathcal{L}\), then there exist constants a and b, with \(a>0\), such that for \(x\in \Omega \)
$$\begin{aligned} u(x)\geq a \Vert x \Vert +b. \end{aligned}$$
Moreover, \(u^{*}\) is proper and satisfies \(u^{*}(y)>-\infty \), \(\forall y\in \Omega \).
Let \(u, v:\Omega \rightarrow (-\infty,+\infty ]\) are convex functions. Then:
\((u\Box v )^{*}=u^{*}+v^{*}\);
\((u\alpha )^{*}=\alpha u^{*}, \alpha >0\);
\(\operatorname{dom}(u\Box v)=\operatorname{dom}(u)+\operatorname{dom}(v)\);
we have \(u^{*}(0)=-\inf (u)\), in particular if u is proper, then \(u^{*}(y)>-\infty \); \(\inf (u)>-\infty \) implies \(u^{*}\) is proper.
Let \(u: \Omega \rightarrow (-\infty,+\infty ]\) be a closed convex function, and set \(\mathcal{C}:=\operatorname{int} (\Omega )\), \(\mathcal{C}^{*}:=\operatorname{int} (\operatorname{dom}(u^{*}))\). Then \((\mathcal{C},u)\) is a convex function of Legendre type if and only if \((C^{*}, u^{*})\) is. In this case \((\mathcal{C}^{*},u^{*})\) is the Legendre conjugate of \((\mathcal{C},u)\) (and conversely). Moreover, \(\nabla u:=\mathcal{C}\rightarrow \mathcal{C}^{*}\) is a continuous bijection, and the inverse map of ∇u is precisely \(\nabla u^{*}\).
A function \(f: \mathbb{R}^{n}\rightarrow (-\infty, +\infty ]\) is called log-concave if, for \(x, y\in \mathbb{R}^{n}\) and \(0< t<1\), we have \(f ((1-t)x+ty )\geq f^{1-t}(x)f^{t}(y)\). If f is a strictly positive log-concave function on \(\mathbb{R}^{n}\), then there exists a convex function \(u:\Omega \rightarrow (-\infty,+\infty ]\) such that \(f=e^{-u}\). The log-concave function is closely related to the convex geometry of \(\mathbb{R}^{n}\). An example of a log-concave function is the characteristic function \(\chi _{K}\) of a convex body K in \(\mathbb{R}^{n}\), which is defined by
$$\begin{aligned} \chi _{K}(x)=e^{-I_{K}(x)}= \textstyle\begin{cases} 1 & \text{if } x\in K; \\ 0 & \text{if }x\notin K, \end{cases}\displaystyle \end{aligned}$$
where \(I_{K}\) is a lower semi-continuous convex function, and the indicator function of K is
$$\begin{aligned} I_{K}(x)= \textstyle\begin{cases} 0 & \text{if } x\in K; \\ \infty & \text{if } x\notin K. \end{cases}\displaystyle \end{aligned}$$
Let us generalize f to the domain of \(\mathbb{R}^{n}\) by
$$\begin{aligned} \overline{f}=\textstyle\begin{cases} f ,& x\in \Omega ; \\ 0, & x\in \mathbb{R}^{n}/\Omega . \end{cases}\displaystyle \end{aligned}$$
In the later sections, we also use f to denote f having been extended to \(\mathbb{R}^{n}\), let \(\mathcal{A}= \{f:\mathbb{R}^{n}\rightarrow (0,+\infty ]: f=e^{-u}, u\in \mathcal{L} \}\) be the subclass of f.
Let \(f, g\in \mathcal{A}\), and \(\alpha, \beta \geq 0\). The sum and multiplication of f and g are defined by \(\alpha \cdot f\oplus \beta \cdot g=e^{-[(u\alpha )\Box (v\beta )]}\). That means
$$\begin{aligned} (\alpha \cdot f\oplus \beta \cdot g ) (x)=\sup _{y\in \mathbb{R}^{n}}f \biggl(\frac{x-y}{\alpha } \biggr)^{\alpha }g \biggl(\frac{y}{\beta } \biggr)^{\beta }. \end{aligned}$$
In particularly, when \(\alpha =0\) and \(\beta >0\), we have \((\alpha \cdot f\oplus \beta \cdot g)(x)=g(\frac{x}{\beta })^{\beta }\); when \(\alpha >0\) and \(\beta =0\), then \((\alpha \cdot f\oplus \beta \cdot g)(x)=f(\frac{x}{\alpha })^{\alpha }\); finally, when \(\alpha =\beta =0\), we have \((\alpha \cdot f\oplus \beta \cdot g )=I_{\{0\}}\).
Proposition 1.1 grants that \(\mathcal{L}\) is closed under the operations of infimal convolution and right scalar multiplication.
Let u and v belong both to the same class \(\mathcal{L}\), and \(\alpha, \beta \geq 0\). Then \(u\alpha \Box v\beta \) belongs to the same class as u and v.
Let \(f\in \mathcal{A}\), the support function of \(f=e^{-u}\) is defined by
$$\begin{aligned} h_{f}(x)=\bigl(-\log f(x)\bigr)^{*}=u^{*}(x), \end{aligned}$$
here the \(u^{*}\) is the Legendre transform of u. The definition of \(h_{f}\) is a proper generalization of the support function \(h_{K}\), in fact, one can easily check \(h_{\chi _{K}}=h_{K}\) (see [3, 26]). Specifically, the function \(h:\mathcal{A}\rightarrow \mathcal{L}\) has the following properties [27]:
h is a bijective map from \(\mathcal{A}\rightarrow \mathcal{L}\).
h is order preserving: \(f\leq g\) if and only if \(h_{f}\leq h_{g}\).
h is additive: for every \(f, g\in \mathcal{A}\) we have \(h_{f\oplus g}=h_{f}+h_{g}\).
The following proposition shows that \(h_{f}\) is \(GL(n)\) covariant.
Let \(f\in \mathcal{A}\), and \(A\in GL(n)\). Then, for \(x\in \mathbb{R}^{n}\),
$$\begin{aligned} h_{f\circ A}(x)=h_{f}\bigl(A^{-t}x\bigr). \end{aligned}$$
Let \(u, v\in \mathcal{L}\), denote \(u_{t}=u\Box vt\) \((t>0)\), and \(f_{t}=e^{-u_{t}}\). The following lemmas describe the monotonicity and convergence of \(u_{t}\) and \(f_{t}\), respectively.
Let \(f, g\in \mathcal{A}\). For \(t>0\), set \(u_{t}=u\square (vt)\), \(f_{t}=e^{-u_{t}}\), and assume that \(v(0)=0\). Then, for every fixed \(x\in \mathbb{R}^{n}\), \(u_{t}(x)\) and \(f_{t}(x)\) are, respectively, pointwise decreasing and increasing with respect to t; in particular
$$\begin{aligned} u_{1}(x)\leq u_{t}(x)\leq u(x) \quad\textit{and}\quad f(x)\leq f_{t}(x) \leq f_{1}(x) \quad\forall x\in \mathbb{R}^{n}\ \forall t\in [0, 1]. \end{aligned}$$
Let u and v belong both to the same class \(\mathcal{L}\) and, for any \(t>0\), set \(u_{t}:=u\Box (vt)\), assume that \(v(0)=0\). Then
\(\forall x\in \Omega \), \(\lim_{t\rightarrow 0^{+}}u_{t}(x)=u(x)\);
\(\forall E\subset \subset \Omega \), \(\lim_{t\rightarrow 0^{+}}\nabla u_{t}(x)=\nabla u\) uniformly on E.
Let u and v belong both to the same class \(\mathcal{L}\), for any \(t>0\), let \(u_{t}:=u\Box (vt)\). Then \(\forall x\in \operatorname{int}(\Omega _{t})\), and \(\forall t>0\),
$$\begin{aligned} \frac{d}{dt} \bigl(u_{t}(x) \bigr)=-\psi \bigl(\nabla u_{t}(x)\bigr), \end{aligned}$$
where \(\psi:=v^{*}\).
Projection of functions onto linear subspace
Let \(\mathcal{G}_{i,n}\) \((0\leq i\leq n)\) be the Grassmannian manifold of i-dimensional linear subspace of \(\mathbb{R}^{n}\). The elements of \(\mathcal{G}_{i,n}\) will usually be denoted by \(\xi _{i}\) and \(\xi ^{\perp }_{i}\) stands for the orthogonal complement of \(\xi _{i}\) which is a \((n-i)\)-dimensional subspace of \(\mathbb{R}^{n}\). Let \(\xi _{i}\in \mathcal{G}_{i,n}\) and \(f:\mathbb{R}^{n}\rightarrow \mathbb{R}\). The projection of f onto \(\xi _{i}\) is defined by (see [20, 22])
$$\begin{aligned} f|_{\xi _{i}}(x):=\max \bigl\{ f(y): y\in x+{\xi _{i}}^{\perp }\bigr\} ,\quad \forall x\in \Omega |_{\xi _{i}}. \end{aligned}$$
Here \(\xi _{i}^{\perp }\) is the orthogonal complement of \(\xi _{i}\) in \(\mathbb{R}^{n}\), \(\Omega |_{\xi _{i}}\) is the projection of Ω onto \(\xi _{i}\). By the definition of the log-concave function \(f=e^{-u}\), for every \(x\in \Omega |_{\xi _{i}}\), one can rewrite (2.1) as
$$\begin{aligned} f|_{\xi _{i}}(x)=\exp \bigl\{ \max \bigl\{ -u(y):y\in x+\xi _{i}^{\perp }\bigr\} \bigr\} =e^{-u|_{\xi _{i}}}(x). \end{aligned}$$
As regards the "sum" and "multiplication" of f, we say that the projection keeps the structure on \(\mathbb{R}^{n}\). In other words, we have the following proposition.
Let \(f, g\in \mathcal{A}\), \(\xi _{i}\in \mathcal{G}_{i,n}\) and \(\alpha, \beta \geq 0\). Then
$$\begin{aligned} (\alpha \cdot f\oplus \beta \cdot g)|_{\xi _{i}}=\alpha \cdot f|_{ \xi _{i}}\oplus \beta \cdot g|_{\xi _{i}}. \end{aligned}$$
Let \(f, g\in \mathcal{A}\), set \(x_{1}, x_{2}, x\in \xi _{i}\) such that \(x=\alpha x_{1}+\beta x_{2}\). Then, by the definition of the projection, we have
$$\begin{aligned} (\alpha \cdot f\oplus \beta \cdot g)|_{\xi _{i}}(x)&\geq (\alpha \cdot f \oplus \beta \cdot g) \bigl(\alpha x_{1}+\beta x_{2}+\xi _{i}^{\perp }\bigr) \\ &\geq f\bigl(x_{1}+\xi _{i}^{\perp } \bigr)^{\alpha }g\bigl(x_{2}+\xi _{i}^{\perp } \bigr)^{\beta }. \end{aligned}$$
Taking the supremum of the second right hand inequality over all \(\xi _{i}^{\perp }\) we obtain \((\alpha \cdot f\oplus \beta \cdot g)|_{\xi _{i}}\geq \alpha \cdot f|_{ \xi _{i}}\oplus \beta \cdot g|_{\xi _{i}}\). On the other hand, for \(x\in \xi _{i}\), any \(x_{1}, x_{2}\in \xi _{i}\) such \(x_{1}+x_{2}=x\) then
$$\begin{aligned} (\alpha \cdot f|_{\xi _{i}}\oplus \beta \cdot g|_{\xi _{i}} ) (x)&= \sup _{x_{1}+x_{2}=x} (f|_{\xi _{i}})^{\alpha }\biggl( \frac{x_{1}}{\alpha }\biggr) (g|_{ \xi _{i}})^{\beta }\biggl( \frac{x_{2}}{\beta }\biggr) \\ &= \sup_{x_{1}+x_{2}=x} \biggl\{ \max \biggl\{ f^{\alpha }\biggl( \frac{x_{1}}{\alpha }+\xi _{i}^{\perp }\biggr) \biggr\} \max \biggl\{ g^{\beta }\biggl( \frac{x_{2}}{\beta }+\xi _{i}^{\perp } \biggr) \biggr\} \biggr\} \\ &\geq \sup_{x_{1}+x_{2}=x} \biggl\{ \max \biggl(f^{\alpha }\biggl( \frac{x_{1}}{\alpha }+\xi _{i}^{\perp }\biggr)g^{\beta } \biggl(\frac{x_{2}}{\beta }+ \xi _{i}^{\perp }\biggr) \biggr) \biggr\} \\ &=\max \bigl\{ (\alpha \cdot f\oplus \beta \cdot g ) \bigl( x_{1}+x_{2} +\xi _{i}^{\perp }\bigr) \bigr\} \\ &=(\alpha \cdot f\oplus \beta \cdot g)|_{\xi _{i}}(x). \end{aligned}$$
Since \(f, g\geq 0\), the inequality \(\max \{f\cdot g\}\leq \max \{f\}\cdot \max \{g\}\) holds. So we complete the proof. □
Let \(\xi _{i}\in \mathcal{G}_{i,n}\), f and g are functions on \(\mathbb{R}^{n}\), such that \(f(x)\leq g(x)\) holds. Then
$$ f|_{\xi _{i}}\leq g|_{\xi _{i}} $$
holds for any \(x\in \xi _{i}\).
For \(y\in x+\xi _{i}^{\perp }\), since \(f(y)\leq g(y)\), then \(f(y)\leq \max \{g(y): y\in x+\xi _{i}^{\perp }\}\). So \(\max \{f(y):y\in x+L^{\perp }_{i}\}\leq \max \{g(y):y\in x+\xi _{i}^{\perp }\}\), by the definition of the projection, we complete the proof. □
For the convergence of f we have the following.
Let \(\{f_{i}\}\) be functions such that \(\lim_{n\rightarrow \infty }f_{n}=f_{0}\). Let \(\xi _{i}\in \mathcal{G}_{i,n}\), then \(\lim_{n\rightarrow \infty }(f_{n}|_{\xi _{i}})=f_{0}|_{\xi _{i}}\).
Since \(\lim_{n\rightarrow \infty }f_{n}=f_{0}\), it means that, for \(\forall \epsilon >0\), there exists \(N_{0}\), \(\forall n>N_{0}\), such that \(f_{0}-\epsilon \leq f_{n}\leq f_{0}+\epsilon \). By the monotonicity of the projection, we have \(f_{0}|_{\xi _{i}}-\epsilon \leq f_{n}|_{\xi _{i}}\leq f_{0}|_{\xi _{i}}+ \epsilon \). Hence each \(\{f_{n}|_{\xi _{i}}\}\) has a convergent subsequence, we denote it also by \(\{f_{n}|_{\xi _{i}}\}\), converging to some \(f'_{0}|_{\xi _{i}}\). Then, for \(x\in \xi _{i}\), we have
$$\begin{aligned} f_{0}|_{\xi _{i}}(x)-\epsilon \leq f'_{0}|_{\xi _{i}}(x)= \lim_{n \rightarrow \infty }(f_{n}|_{\xi _{i}}) (x)&\leq f_{0}|_{\xi _{i}}(x)+ \epsilon. \end{aligned}$$
By the arbitrariness of ϵ we have \(f'_{0}|_{\xi _{i}}=f_{0}|_{\xi _{i}}\), so we complete the proof. □
Combining with Proposition 2.3 and Proposition 1.7, it is easy to obtain the following proposition.
Let u and v belong both to the same class \(\mathcal{L}\), \(\Omega \in \mathbb{R}^{n}\) be the domain of u, for any \(t>0\), set \(u_{t}=u\Box (v t)\). Assume that \(v(0)=0\) and \(\xi _{i}\in \mathcal{G}_{i,n}\), then
\(\forall x\in \Omega |_{\xi _{i}}\), \(\lim_{t\rightarrow 0^{+}}u_{t}|_{\xi _{i}}(x)=u|_{\xi _{i}}(x)\),
\(\forall x\in \operatorname{int}(\Omega |_{\xi _{i}}), \lim_{t \rightarrow 0^{+}}\nabla u_{t}|_{\xi _{i}}=\nabla u|_{\xi _{i}}\).
Now let us introduce some facts about the functions \(u_{t}=u\Box (vt)\) with respect to the parameter t.
Let \(\xi _{i}\in \mathcal{G}_{i,n}\), u and v belong both to the same class \(\mathcal{L}\), \(u_{t}:=u\Box (vt)\) (\(t>0\)) and \(\Omega _{t}\) be the domain of \(u_{t}\). Then, for \(x\in {\Omega _{t}}|_{\xi _{i}}\),
$$\begin{aligned} \frac{d}{dt} (u_{t}|_{\xi _{i}} ) (x)=-\psi \bigl(\nabla ( u_{t}|_{ \xi _{i}} ) (x) \bigr), \end{aligned}$$
where \(\psi:=v^{*}|_{\xi _{i}}\).
Set \(D_{t}:={\Omega _{t}}|_{\xi _{i}}\subset \xi _{i}\), for fixed \(x\in \operatorname{int}(D_{t})\), the map \(t\rightarrow \nabla (u_{t}|_{\xi _{i}} )(x)\) is differentiable on \((0,+\infty )\). Indeed, by the definition of Fenchel conjugate and the definition of projection u, it is easy to see that \((u|_{\xi _{i}})^{*}=u^{*}|_{\xi _{i}}\) and \((u\Box ut)|_{\xi _{i}}=u|_{\xi _{i}}\Box ut|_{\xi _{i}}\) hold. Lemma 1.4 and the property of the projection grant the differentiability. Set \(\varphi:=u^{*}|_{\xi _{i}}\) and \(\psi:=v^{*}|_{\xi _{i}}\), and \(\varphi _{t}=\varphi +t\psi \), then \(\varphi _{t}\) belongs to the class \(\mathcal{C}^{2}_{+}\) on \(\xi _{i}\). Then \(\nabla ^{2} \varphi _{t}=\nabla ^{2}\varphi +t\nabla ^{2} \psi \) is nonsingular on \(\xi _{i}\). So the equation
$$\begin{aligned} \nabla \varphi (y)+t\nabla \psi (y)-x=0 \end{aligned}$$
locally defines a map \(y=y(x,t)\) which is of class \(\mathcal{C}^{1}\). By Proposition 1.3, \(\nabla (u_{t}|_{\xi _{i}})\) is the inverse map of \(\nabla \varphi _{t}\), that is, \(\nabla \varphi _{t}(\nabla (u_{t}|_{\xi _{i}}(x))=x\), which means that, for every \(x\in \operatorname{int}(D_{t})\) and every \(t>0\), \(t\rightarrow \nabla (u_{t}|_{\xi _{i}})\) is differentiable. Using Eq. (1.2) again, we have
$$\begin{aligned} u_{t}|_{\xi _{i}}(x)= \bigl\langle x, \nabla (u_{t}|_{\xi _{i}}) (x) \bigr\rangle -\varphi _{t} \bigl(\nabla (u_{t}|_{\xi _{i}}) (x) \bigr), \quad\forall x\in \operatorname{int}(D_{t}). \end{aligned}$$
Moreover, note that \(\varphi _{t}=\varphi +t\psi \) and we have
$$\begin{aligned} u_{t}|_{\xi _{i}}(x)&= \bigl\langle x, \nabla (u_{t}|_{\xi _{i}}) (x) \bigr\rangle -\varphi \bigl(\nabla (u_{t}|_{\xi _{i}} ) (x) \bigr)-t \psi \bigl(\nabla ( u_{t}|_{\xi _{i}} ) (x) \bigr) \\ &=u_{t}|_{\xi _{i}} \bigl(\nabla ( u_{t}|_{\xi _{i}} ) (x) \bigr)-t \psi \bigl(\nabla ( u_{t}|_{\xi _{i}} ) (x) \bigr). \end{aligned}$$
Taking the differential of the above formally we obtain
$$\begin{aligned} \frac{d}{dt} (u_{t}|_{\xi _{i}} ) (x)=-\psi \bigl(\nabla ( u_{t}|_{ \xi _{i}} ) (x) \bigr). \end{aligned}$$
Then we complete the proof. □
Inequality for functional mixed quermassintegral
A function \(f\in \mathcal{A}\) is non-degenerate and integrable if and only if \(\lim_{\|x\|\rightarrow +\infty }\frac{u(x)}{\|x\|}=+\infty \). Then, let \(\mathcal{L}'= \{u\in \mathcal{L}: u\in \mathcal{C}^{2}_{+}(\mathbb{R}^{n}), \lim_{\| x\|\rightarrow +\infty }\frac{u(x)}{\|x\|}=+\infty \}\), and \(\mathcal{A}'= \{f:\mathbb{R}^{n}\rightarrow (0,+\infty ]: f=e^{-u}, u\in \mathcal{L}' \}\).
Let \(f\in \mathcal{A}'\), \(\xi _{i}\in \mathcal{G}_{i,n}\) \((i=1, 2,\ldots,n)\), and \(x\in \Omega |_{\xi _{i}}\). The ith total mass of f is defined as
$$\begin{aligned} J_{i}(f):= \int _{\xi _{i}}f|_{\xi _{i}}({x})\,dx, \end{aligned}$$
where \(f|_{\xi _{i}}\) is the projection of f onto \(\xi _{i}\) defined by (2.1), dx is the i-dimensional volume element in \(\xi _{i}\).
(1) The definition of the \(J_{i}(f)\) follows the i-dimensional volume of the projection of a convex body. If \(i=0\), we define \(J_{0}(f):=\omega _{n}\), the volume of the unit ball in \(\mathbb{R}^{n}\), for completeness.
(2) When one takes \(f=\chi _{K}\), the characteristic function of a convex body K, one has \(J_{i}(f)=V_{i}(K)\), the i-dimensional volume in \(\xi _{i}\).
Let \(f\in \mathcal{A}'\). Set \(\xi _{i}\in \mathcal{G}_{i,n}\) be a linear subspace and, for any \(x\in \Omega |_{\xi _{i}}\), the ith functional Quermassintegrals of f (or the i-dimensional mean projection mass of f) is defined as
$$\begin{aligned} W_{n-i}(f):=\frac{\omega _{n}}{\omega _{i}} \int _{\mathcal{G}_{i,n}}J_{i}(f)\,d \mu (\xi _{i}),\quad i=1,2,\ldots, n, \end{aligned}$$
where \(J_{i}(f)\) is the ith total mass of f defined by (3.1), \(d\mu (\xi _{i})\) is the normalized Haar measure on \(\mathcal{G}_{i,n}\).
(1) The definition of the \(W_{i}(f)\) follows the definition of the ith Quermassintegral \(W_{i}(K)\), that is, the ith mean total mass of f on \(\mathcal{G}_{i,n}\). Also in the recent paper of Bobkov, Colesanti and Fragala [10], the authors give the same definition by defining the Quermassintegral of the support set for the quasi-concave functions.
(2) When i equals n in (3.2), we have \(W_{0}(f)=\int _{\mathbb{R}^{n}}f(x)\,dx=J(f)\).
(3) From the definition of the Quermassintegrals \(W_{i}(f)\), the following properties are obtained (see also [10]):
Positivity. \(0\leq W_{i}(f)\leq +\infty \).
Monotonicity. \(W_{i}(f)\leq W_{i}(g)\), if \(f\leq g\).
Generally speaking, the \(W_{i}(f)\) has no homogeneity under dilations. That is, \(W_{i}(\lambda \cdot f)=\lambda ^{n-i}W_{i}(f^{\lambda })\), where \(\lambda \cdot f(x)=\lambda f(x/\lambda ), \lambda >0\).
Let f, \(g\in \mathcal{A}'\), ⊕ and ⋅ denote the operations of "sum" and "multiplication" in \(\mathcal{A}'\), \(W_{i}(f)\) be the ith Quermassintegrals of f. Whenever the following limit exists:
$$\begin{aligned} W_{i}(f,g)=\frac{1}{(n-i)}\lim_{t\rightarrow 0^{+}} \frac{W_{i}(f\oplus t\cdot g)-W_{i}(f)}{t},\quad i=0,1, \ldots, n-1, \end{aligned}$$
we denote it by \(W_{i}(f,g)\), and call it the first variation of \(W_{i}\) at f along g, or the ith functional mixed Quermassintegrals of f and g.
Let \(f=\chi _{K}\) and \(g=\chi _{L}\), with K, \(L\in \mathcal{K}^{n}\). In this case \(W_{i}(f\oplus t\cdot g)=W_{i}(K+t L)\), then \(W_{i}(f,g)=W_{i}(K, L)\).
The following is devoted to proving that \(W_{i}(f,g)\) exists under the fairly weak hypothesis. First, we prove that the first variation of i-dimensional total mass of f is translation invariant.
Let \(\xi _{i}\in \mathcal{G}_{i,n}\), \(f=e^{-u}\) and \(g=e^{-v}\) are integrable log-concave functions in \(\mathcal{A}'\). Let \(c=\inf u|_{\xi _{i}}=:u(0)\), \(d=\inf v|_{\xi _{i}}:=v(0)\), and set \(\widetilde{u}_{i}(x)=u|_{\xi _{i}}(x)-c\), \(\widetilde{v}_{i}(x)=v|_{\xi _{i}}(x)-d\), \(\widetilde{\varphi }_{i}(y)= (\widetilde{u}_{i})^{*}(y)\), \(\widetilde{\psi }_{i}(y)=(\widetilde{v}_{i})^{*}(y)\), and \(\widetilde{f}_{i}=e^{-\widetilde{u}_{i}}, \widetilde{g}_{i}=e^{- \widetilde{v}_{i}}\), \(\widetilde{f}_{t}|_{i}=\widetilde{f}\oplus t\cdot \widetilde{g}\). Then, if
$$ \lim_{t\rightarrow 0^{+}} \frac{J_{i}(\widetilde{f}_{t})-J_{i}(\widetilde{f})}{t}= \int _{\xi _{i}} \widetilde{\psi }_{i} \,d\mu _{i}(\widetilde{f}), $$
then we have
$$\begin{aligned} \lim_{t\rightarrow 0^{+}}\frac{J_{i}(f_{t})-J_{i}(f)}{t}= \int _{\xi _{i}} \psi _{i} \,d\mu _{i}(f). \end{aligned}$$
By the construction, we have \(\widetilde{u}_{i}(0)=0, \widetilde{v}_{i}(0)=0\), and \(\widetilde{v}_{i}\geq 0, \widetilde{\varphi }_{i}\geq 0, \widetilde{\psi }_{i}\geq 0\). Further, we have \(\widetilde{\psi }_{i}(y)=\psi _{i}(y)+d\), and \(\widetilde{f}_{i}=e^{c}f_{i}\). Then we have
$$\begin{aligned} \lim_{t\rightarrow 0^{+}} \frac{J_{i}(\widetilde{f}_{t})-J_{i}(\widetilde{f})}{t}= \int _{\xi _{i}} \widetilde{\psi }_{i} \,d\mu _{i}(\widetilde{f})=e^{c} \int _{\xi _{i}} \psi _{i} \,d\mu _{i}(f)+d e^{c} \int _{\xi _{i}} \,d\mu _{i}( f). \end{aligned}$$
On the other hand, since \(f_{i}\oplus t\cdot g_{i}=e^{-(c+dt)}(\widetilde{f}_{i}\oplus t\cdot \widetilde{g}_{i})\), we have \(J_{i}(f\oplus t\cdot g)=e^{-(c+dt)}J_{i}(\widetilde{f}_{i}\oplus t \cdot \widetilde{g}_{i})\). By derivation of both sides of the above formula, we obtain
$$\begin{aligned} &\lim_{t\rightarrow 0^{+}}\frac{J_{i}(f\oplus t\cdot g)-J_{i}(f)}{t} \\ &\quad=-d e^{-c}\lim_{t\rightarrow 0^{+}}J_{i}( \widetilde{f}_{i}\oplus t \widetilde{g}_{i}) \,dx+e^{-c}\lim_{t\rightarrow 0^{+}} \biggl[ \frac{J_{i}(\widetilde{f}_{t})-J_{i}(\widetilde{f})}{t} \biggr] \\ &\quad=-d e^{-c}J_{i}(\widetilde{f}_{i})+ \int _{\xi _{i}} \psi _{i} \,d\mu _{i}(f)+d \int _{\xi _{i}} \,d\mu _{i}( f) \\ &\quad= \int _{\xi _{i}}\psi _{i} \,d\mu _{i}(f). \end{aligned}$$
So we complete the proof. □
Let \(f, g\in \mathcal{A}'\), with \(-\infty \leq \inf (\log g)\leq +\infty \), and \(W_{i}(f)>0\). Then \(W_{j}(f,g)\) is differentiable at f along g, and
$$\begin{aligned} W_{j}(f,g)\in [-k,+\infty ], \end{aligned}$$
where \(k=\max \{d,0\}W_{i}(f)\).
Let \(\xi _{i}\in \mathcal{G}_{i,n}\), since \(u|_{\xi _{i}}:=-\log (f|_{\xi _{i}})=-(\log f)|_{\xi _{i}}\) and \(v|_{\xi _{i}}:=-\log (g|_{\xi _{i}})=- (\log f)|_{\xi _{i}}\). By the definition of \(f_{t}\) and the Proposition 2.1 we obtain
$$ f_{t}|_{\xi _{i}}=(f\oplus t\cdot g)|_{\xi _{i}}=f|_{\xi _{i}} \oplus t \cdot g|_{\xi _{i}}. $$
Notice that \(v|_{\xi _{i}}(0)=v(0)\), set \(d:=v(0)\), \(\widetilde{v}|_{\xi _{i}}(x):=v|_{\xi _{i}}(x)-d\), \(\widetilde{g}|_{\xi _{i}}(x):=e^{-\widetilde{v}|_{\xi _{i}}(x)}\), \(\widetilde{f}_{t}|_{\xi _{i}}:=f|_{\xi _{i}}\oplus t\cdot \widetilde{g}|_{\xi _{i}}\). Without loss of generality, we may assume \(\inf (v)=v(0)\). Lemma 1.6 says that, for every \(x\in \xi _{i}\),
$$ f|_{\xi _{i}}\leq \widetilde{f}_{t}|_{\xi _{i}}\leq \widetilde{f}_{1}|_{ \xi _{i}},\quad \forall x\in \mathbb{R}^{n}, \forall t\in [0,1]. $$
Then there exists \(\widetilde{f}|_{\xi _{i}}(x):=\lim_{t\rightarrow 0^{+}} \widetilde{f}_{t}|_{\xi _{i}}(x)\), moreover, \(\widetilde{f}|_{\xi _{i}}(x)\geq f|_{\xi _{i}}(x)\) and \(\widetilde{f}_{t}|_{\xi _{i}}\) is pointwise decreasing as \(t\rightarrow 0^{+}\). By Lemma 1.1 and Proposition 1.4, one shows that
$$ f|_{\xi _{i}}\oplus t\cdot \widetilde{g}|_{\xi _{i}}\in A',\quad \forall t\in [0,1]. $$
Then \(J_{i}(f)\leq J_{i}(\widetilde{f}_{t})\leq J_{i}(\widetilde{f}_{1})\), and, \(-\infty \leq J_{i}(f), J_{i}(\widetilde{f}_{1})<\infty \). Hence, by the monotonicity and convergence, we have \(\lim_{t\rightarrow 0^{+}}W_{i}( \widetilde{f}_{t})=W_{i}( \widetilde{f})\). In fact, by definition we have \(\widetilde{f}_{t}|_{\xi _{i}}(x)=e^{-\inf \{u|_{\xi _{i}}(x-y)+tv|_{ \xi _{i}}(\frac{y}{t})\}}\), and
$$ -\inf \biggl\{ u|_{\xi _{i}}(x-y)+tv|_{\xi _{i}}\biggl(\frac{y}{t} \biggr)\biggr\} \leq - \inf u|_{ \xi _{i}}(x-y)-t\inf v|_{\xi _{i}}\biggl( \frac{y}{t}\biggr). $$
Note that \(-\infty \leq \inf (v|_{\xi _{i}})\leq +\infty \), then \(- \inf u|_{\xi _{i}}(x-y)-t\inf v|_{\xi _{i}}(\frac{y}{t})\) is a continuous function of variable t, then
$$\begin{aligned} \widetilde{f}|_{\xi _{i}}(x):=\lim_{t\rightarrow 0^{+}} \widetilde{f}_{t}|_{ \xi _{i}}(x)=f|_{\xi _{i}}(x). \end{aligned}$$
Moreover, \(W_{i}(\widetilde{f}_{t})\) is a continuous function of t \((t\in [0,1])\), then \(\lim_{t\rightarrow 0^{+}}W_{i}(\widetilde{f}_{t})=W_{i}(f)\). Since \(f_{t}|_{\xi _{i}}=e^{-dt}\widetilde{f}|_{\xi _{i}}(x)\), we have
$$\begin{aligned} \frac{ W_{i}(f_{t})-W_{i}(f)}{t}=W_{i}(f)\frac{e^{-dt}-1}{t}+e^{-dt} \frac{W_{i}(\widetilde{f}_{t})-W_{i}(f)}{t}. \end{aligned}$$
Since \(\widetilde{f}_{t}|_{\xi _{i}}\geq f|_{\xi _{i}}\), we have the following two cases:
$$ \exists t_{0}>0: W_{i}(\widetilde{f}_{t_{0}})=W_{i}(f)\quad \text{or}\quad W_{i}(\widetilde{f}_{t})=W_{i}(f)\quad \forall t>0. $$
For the first case, since \(W_{i}(\widetilde{f}_{t})\) is a monotone increasing function of t, \(W_{i}(\widetilde{f}_{t})=W_{i}(f)\) for every \(t\in [0,t_{0}]\). Hence we have
$$\begin{aligned} \lim_{t\rightarrow 0^{+}}\frac{W_{i}(f_{t})-W_{i}(f)}{t}&=-dW_{i}(f), \end{aligned}$$
the statement of the theorem holds true.
In the latter case, since \(\widetilde{f}_{t}|_{\xi _{i}}\) is an increasing nonnegative function, \(\log (W_{i}(\widetilde{f}_{t}))\) is an increasing concave function of t. Then
$$ \exists \frac{\log (W_{i}(\widetilde{f}_{t}))-\log (W_{i}( f))}{t}\in [0,+ \infty ]. $$
On the other hand,
$$\begin{aligned} \log ' \bigl(W_{i}(\widetilde{f}_{t}) \bigr)\big|_{t=0}=\lim_{t \rightarrow 0^{+}} \frac{\log (W_{i}(\widetilde{f}_{t}))-\log (W_{i}( f))}{W_{i}(\widetilde{f}_{t})-W_{i}(f)}= \frac{1}{W_{i} (f)}. \end{aligned}$$
$$\begin{aligned} \lim_{t\rightarrow 0^{+}} \frac{W_{i}(\widetilde{f}_{t})-W_{i}(f)}{\log (W_{i}(\widetilde{f}_{t}))-\log (W_{i}( f))}=W_{i}(f)>0. \end{aligned}$$
From the above we infer that
$$\begin{aligned} \exists \lim_{t\rightarrow 0^{+}} \frac{W_{i}(\widetilde{f}_{t})-W_{i}(f)}{t}\in [0,+\infty ]. \end{aligned}$$
Combining the above formulas we obtain
$$\begin{aligned} \lim_{t\rightarrow 0^{+}}\frac{W_{i}(f_{t})-W_{i}(f)}{t}\in \bigl[-\max \{d,0 \}W_{i}(f),+\infty \bigr]. \end{aligned}$$
Let \(f\in \mathcal{A}'\), then
$$\begin{aligned} W_{i}(f,f)=W_{i}(f)+\frac{1}{(n-i)}W_{i}(f \log f). \end{aligned}$$
Since \(f\in \mathcal{A}'\), we have \(f|_{\xi _{i}}\in \mathcal{A}'\). \(u\Box ut=u(1+t)\), then \(u\Box ut|_{\xi _{i}}=u(1+t)|_{\xi _{i}}\). So
$$\begin{aligned} \frac{J_{i}(f\oplus t\cdot f)-J_{i}(f)}{t}&=\frac{1}{t} \biggl[(1+t)^{i} \int _{\xi _{i}}e^{-(1+t)u|_{\xi _{i}}}\,dx- \int _{\xi _{i}}e^{-u|_{ \xi _{i}}}\,dx \biggr] \\ &= \biggl[\frac{(1+t)^{i}-1}{t} \biggr] \int _{\xi _{i}}e^{-(1+t)u|_{\xi _{i}}}\,dx+ \int _{\xi _{i}}e^{-u|_{\xi _{i}}} \biggl( \frac{e^{-tu|_{\xi _{i}}}-1}{t} \biggr) \,dx. \end{aligned}$$
Now taking the limit when \(t\rightarrow 0^{+}\), we obtain
$$\begin{aligned} \lim_{t\rightarrow 0^{+}}\frac{J_{i}(f\oplus t\cdot f)-J_{i}(f)}{t}=iJ_{i}(f)+ \int _{\xi _{i}}f|_{\xi _{i}}\log f|_{\xi _{i}}\,dx. \end{aligned}$$
$$\begin{aligned} &\lim_{t\rightarrow 0^{+}}\frac{W_{i}(f\oplus t\cdot f)-W_{i}(f)}{t}\\ &\quad= \frac{\omega _{n}}{\omega _{n-i}} \int _{\mathcal{G}_{n-i,n}}\lim_{t \rightarrow 0^{+}}\frac{J_{n-i}(f\oplus t\cdot f)-J_{n-i}(f)}{t}\,d\mu ( \xi _{n-i}) \\ &\quad=\frac{\omega _{n}}{\omega _{n-i}} \int _{\mathcal{G}_{n-i,n}} \biggl[(n-i)J_{n-i}(f)+ \int _{\xi _{n-i}}f|_{\xi _{n-i}}\log f|_{\xi _{n-i}}\,dx \biggr] \,d\mu ( \xi _{n-i}) \\ &\quad=(n-i)W_{i}(f)+\frac{\omega _{n}}{\omega _{n-i}} \int _{\mathcal{G}_{n-i,n}} \int _{\xi _{n-i}}f|_{\xi _{n-i}}\log f|_{\xi _{n-i}}\,dx\,d\mu ( \xi _{n-i}) \\ &\quad=(n-i)W_{i}(f)+\frac{\omega _{n}}{\omega _{n-i}} \int _{\mathcal{G}_{n-i,n}} \int _{\xi _{n-i}} (f\log f )|_{\xi _{n-i}}\,dx\,d\mu (\xi _{n-i}) \\ &\quad=(n-i)W_{i}(f)+W_{i}(f\log f). \end{aligned}$$
Here we use the \((f\log f)|_{\xi _{i}}=f|_{\xi _{i}}\log f|_{\xi _{i}}\), due to the f and logf being increasing nonnegative functions. Then by the definition we obtain \(W_{i}(f,f)=W_{i}(f)+\frac{1}{(n-i)}W_{i}(f\log f)\). Then we complete the proof. □
The following lemma is useful in proving Minkowski's first inequality for Quermassintegrals.
Let \(f, g\in \mathcal{A}'\), and \(0< t<1\). Then
$$\begin{aligned} \lim_{t\rightarrow 0^{+}} \frac{W_{i} ((1-t)\cdot f\oplus t\cdot g )-W_{i}(f)}{t}=(n-i) \bigl[W_{i}(f,g)-W_{i}(f,f) \bigr]. \end{aligned}$$
First by Lemma 3.4, without loss of generality, we may assume that the function \(v=-\log g\) satisfies the condition \(v(0)=0\). For \(t\in (0,1)\), letting \(s(t)=\frac{t}{1-t}\), by (1.9) we obtain \((1-t)\cdot f\oplus t\cdot g=(1-t)\cdot (f\oplus s(t)\cdot g )\). Let \(f_{s(t)}=f\oplus s(t)\cdot g\), then we have
$$\begin{aligned} \frac{W_{i} ((1-t)\cdot f\oplus t\cdot g )-W_{i}(f)}{t}={}& \frac{W_{i} ((1-t)\cdot f_{s(t)} )-W_{i} (f_{s(t)} )}{t} \\ &{}+\frac{W_{i} (f_{s(t)} )-W_{i} (f )}{t}. \end{aligned}$$
Concerning the first term of the right hand side (3.13), by Lemma 1.6 we know that the function \(f_{s(t)}(x)\) converges decreasingly to some pointwise limit \(f(x)\) as \(t\rightarrow 0^{+}\), since \(s(t)\rightarrow 0^{+}\) as \(t\rightarrow 0^{+}\). In fact, we have \(\lim_{t\rightarrow 0^{+}}f_{s(t)}(x)=\lim_{t'\rightarrow 0}f_{t'}(x)=f(x)\). Then we obtain
$$\begin{aligned} \lim_{t\rightarrow 0^{+}} \frac{W_{i} ((1-t)\cdot f_{s(t)} )-W_{i} (f_{s(t)} )}{t}&= \lim _{t\rightarrow 0^{+}} \frac{W_{i} ((1-t)\cdot f )-W_{i} (f )}{t} \\ &=-(n-i)W_{i}(f, f). \end{aligned}$$
Concerning the second term, we have
$$\begin{aligned} \lim_{t\rightarrow 0^{+}} \frac{W_{i} (f_{s(t)} )-W_{i} (f )}{t}&=\lim _{t \rightarrow 0^{+}} \frac{W_{i} (f\oplus s(t)\cdot g )-W_{i} (f )}{t} \\ &=\lim_{t\rightarrow 0^{+}} \frac{W_{i} (f\oplus s(t)\cdot g )-W_{i} (f )}{s(t)} \cdot \frac{s(t)}{t} \\ &=(n-i)W_{i}{(f,g)}. \end{aligned}$$
Then one can show the conclusion by combining with (3.14) and (3.15). □
Now we give the proof of Theorem 0.1.
Proof of Theorem 0.1
Let \(0\leq t\leq 1\), we construct a function
$$\begin{aligned} \Psi (t)=\log \bigl(W_{i} \bigl((1-t)\cdot f\oplus t\cdot g \bigr) \bigr). \end{aligned}$$
In fact, for \(f, g, h\in \mathcal{A}'\) and \(0\leq t\leq 1\),
$$\begin{aligned} h|_{\xi _{i}}(z)&= \bigl((1-t)\cdot f|_{\xi _{i}}\oplus t\cdot g|_{\xi _{i}} \bigr) (z) \\ &=\sup \bigl\{ f|_{\xi _{i}}(x)^{1-t}g|_{\xi _{i}}(y)^{t}: (1-t)x+ty=z \bigr\} \\ &\geq \bigl\{ f|_{\xi _{i}}(x)^{1-t}g|_{\xi _{i}}(y)^{t}: (1-t)x+ty=z \bigr\} . \end{aligned}$$
By the Prékopa–Leindler inequality, for \(0\leq t\leq 1\), we have
$$\begin{aligned} \int _{\xi _{i}}h|_{\xi _{i}}\,dz\geq \biggl( \int _{\xi _{i}}f|_{\xi _{i}}(x)\,dx \biggr)^{1-t} \biggl( \int _{\xi _{i}}g|_{\xi _{i}}(y)\,dy \biggr)^{t}. \end{aligned}$$
That means that
$$\begin{aligned} J_{i}(h)\geq J_{i}(f)^{1-t}J_{i}(g)^{t}. \end{aligned}$$
Taking the integral of both sides of (3.16) on \(\mathcal{G}_{i,n}\) with measure \(\mu (\xi _{i})\), by the Prékopa–Leindler inequality once again, we obtain
$$\begin{aligned} W_{i} \bigl((1-t)\cdot f\oplus t\cdot g \bigr)\geq W_{i}(f)^{1-t}W_{i}(g)^{t}. \end{aligned}$$
Since \(\Psi (t):=\log (W_{i} ((1-t)\cdot f \oplus t\cdot g ) )\), we conclude that \(\Psi (t)\) is concave on \([0,1]\). Then
$$\begin{aligned} \frac{\Psi (t)-\Psi (0)}{t}\geq \Psi (1)-\Psi (0), \quad\forall t\in [0,1]. \end{aligned}$$
It means that \(\Psi (t)'|_{t=0}\geq \Psi (1)-\Psi (0)\).
By Lemma 3.7, we have
$$\begin{aligned} \Psi (t)'|_{t=0}= \frac{W_{i} ((1-t)\cdot f\oplus t\cdot g )'}{W_{i}((1-t)\cdot f\oplus t\cdot g)} \bigg|_{t=0}= \frac{(n-i) [W_{i}(f,g)-W_{i}(f,f) ]}{W_{i}(f)}. \end{aligned}$$
On the other hand, note that \(\Psi (1)-\Psi (0)=\log (W_{i}(g) )-\log ( W_{i}(f) )\). Therefore, we obtain
$$\begin{aligned} \frac{(n-i) [W_{i}(f,g)-W_{i}(f,f) ]}{W_{i}(f)}\geq \log \bigl(W_{i}(g) \bigr)-\log \bigl( W_{i}(f) \bigr). \end{aligned}$$
Then, combining with formula (3.10), we obtain
$$\begin{aligned} W_{i}(f,g)&\geq \frac{1}{n-i}W_{i}(f) \bigl[\log (W_{i}(g)-\log W_{i}(f) \bigr]+W_{i}(f,f) \\ &=W_{i}(f) \biggl[1+\frac{1}{n-i}\log \frac{W_{i}(g)}{W_{i}(f)} \biggr]+ \frac{1}{n-i}W_{i}(f\log f). \end{aligned}$$
Concerning the equality case, first, assume that \(g(x)=f(x-x_{0})\), by (3.10) and the invariance of the integral by translation of coordinates, we know that (0.6) holds with equality. On the other hand, if (0.6) holds with equality, by inspection of the above proof, one may see that the inequalities (3.16), (3.17) and (3.18) must hold as equalities. Moreover, whenever inequalities (3.16) and (3.17) hold with equality sign, then (3.18) automatic holds with equality. This entails that the Prékopa–Leindler inequality holds as an equality, therefore f and g must agree up to a translation. □
The inequality (0.6) is called the functional Brunn–Minkwoski first inequality for ith mixed Quremassintegrals or functional mixed Quermassintegral inequality. In the following we will give some special case of (0.6).
In fact, we take \(f=\chi _{K}\) and \(g=\chi _{L}\), with \(K, L\in \mathcal{K}^{n}\). In this case \(\chi _{K}\oplus t\cdot \chi _{L}=\chi _{K+tL}\), \(J_{i}(\chi _{K})=V_{i}(K)\), here \(V_{i}\) denotes the i-dimensional volume in \(\xi _{i}\), \(W_{i}(\chi _{K})=W_{i}(K)\), and \(W_{i}(\chi _{K}, \chi _{L})=W_{i}(K,L)\). Moreover, by (1.6) and (1.7) we have, for any x,
$$ f(x)\log f(x)=-e^{-I_{K}(x)}I_{K}(x)\equiv 0. $$
Then (0.6) turns out to be
$$\begin{aligned} W_{i}(K, L)&\geq W_{i}(K) \biggl[1+\frac{1}{n-i}\log \frac{ W_{i}(L)}{W_{i}(K)} \biggr] \\ &=W_{i}(K)+\frac{1}{n-i}W_{i}(K)\log \frac{W_{i}(L)}{W_{i}(K)}. \end{aligned}$$
We can rewrite the above formula (3.19) equivalently as the following:
$$\begin{aligned} \frac{W_{i}(K, L)-W_{i}(K)}{W_{i}(K)}\geq \frac{1}{n-i}\log \frac{W_{i}(L)}{W_{i}(K)}. \end{aligned}$$
By defining the i-cone volume probability measure \(\overline{V_{i}}_{K}\) similar to the \(\overline{V}_{K}\) defined in [11] by Böröczky,
$$\begin{aligned} d{V_{i}}_{K}=\frac{1}{n}h_{K}d{S_{i}}_{K}, \end{aligned}$$
where \(d{S_{i}}_{K}\) is the ith Borel measue on \(S^{n-1}\). The normalized i-cone volume probability measure \(\overline{V_{i}}_{K}\) is defined as
$$\begin{aligned} d\overline{V_{i}}_{K}=\frac{1}{W_{i}(K)}d{V_{i}}_{K}. \end{aligned}$$
Then the normalized i-mixed Quermassintegrals \(\overline{W}_{i}(K,L)\) can be expressed as
$$\begin{aligned} \overline{W}_{i}(K,L)=\frac{W_{i}(K, L)}{W_{i}(K)}= \int _{S^{n-1}} \frac{h_{L}}{h_{K}} \,d\overline{V_{i}}_{K}. \end{aligned}$$
Moreover, by the integral representation of \(W_{i}(K)\), we have
$$\begin{aligned} W_{i}(K)=\frac{1}{n} \int _{S^{n-1}}h_{K}\,d{S_{i}}_{K}= \int _{S^{n-1}}\,d{V_{i}}_{K}. \end{aligned}$$
Then Eq. (3.20) reads
$$\begin{aligned} \int _{S^{n-1}} \biggl(\frac{h_{L}}{h_{K}}-1 \biggr) \,d \overline{V_{i}}_{K} \geq \frac{1}{n-i}\log \frac{W_{i}(L)}{W_{i}(K)}. \end{aligned}$$
We call (3.22) the weaker of the ith log Quermassintegral inequality. In fact,
$$\begin{aligned} \frac{h_{L}}{h_{K}}-1\geq \log \frac{h_{L}}{h_{K}}, \end{aligned}$$
for all \(u\in S^{n-1}\), and the equality holds if and only if \(\frac{h_{L}}{h_{K}}=1\), that is, \(K=L\). For \(i=0\) and \(n=2\), since \(d{\overline{V}_{0}}_{K}=d\overline{V}_{K}\), the cone volume probability measure of K, then by (3.23) and (3.22) we obtain
$$\begin{aligned} \int _{S^{1}} \biggl(\frac{h_{L}}{h_{K}}-1 \biggr) \,d \overline{V}_{K} \geq \int _{S^{1}}\log \frac{h_{L}}{h_{K}}\,d\overline{V}_{K} \geq \frac{1}{2}\log \frac{V(L)}{V(K)}. \end{aligned}$$
So we have the following corollary.
Let \(K, L\in \mathcal{K}^{n}\), \(W_{i}(K)\) denotes the ith Quermassintegral of K, \({\overline{V}_{i}}_{K}\) be the normalized i-cone volume probability measure. Then
$$\begin{aligned} \int _{S^{n-1}} \biggl(\frac{h_{L}}{h_{K}}-1 \biggr) \,d{ \overline{V}_{i}}_{K} \geq \frac{1}{n-i}\log \frac{W_{i}(L)}{W_{i}(K)}. \end{aligned}$$
When \(h_{K}=h_{L}\), equality holds.
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Haddad, J., Jiménez, C., Montenegro, M.: Asymmetric Blaschke–Santaló functional inequalities. J. Funct. Anal. (2019). https://doi.org/10.1016/j.jfa.2019.108319
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Rotem, L.: Support functions and mean width for α-concave functions. Adv. Math. 243, 168–186 (2013)
Rotem, L.: A sharp Blaschke–Santaló inequality for α-concave functions. Geom. Dedic. 172, 217–228 (2014)
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The authors would like to strongly thank the anonymous referee for the very valuable comments and helpful suggestions that directly lead to improve the original manuscript.
The work is supported in part by CNSF (Grant No. 11561012, 11861024), Guizhou Foundation for Science and Technology (Grant No. [2019] 1055, [2019]1228), Science and technology top talent support program of Guizhou Eduction Department (Grant No. [2017]069).
School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, Guizhou, China
Fangwei Chen & Jianbo Fang
School of Mathematical Sciences, Guizhou Normal University, Guiyang, Guizhou, China
Miao Luo & Congli Yang
Fangwei Chen
Jianbo Fang
Miao Luo
Congli Yang
All authors jointly worked on the results, and they read and approved the final manuscript.
Correspondence to Congli Yang.
Chen, F., Fang, J., Luo, M. et al. The functional inequality for the mixed quermassintegral. J Inequal Appl 2020, 253 (2020). https://doi.org/10.1186/s13660-020-02521-7
Log-concave function
Quermassintegral
Mixed Quermassintegral inequality
Minkowski inequality
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CommonCrawl
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Charlotte Scott
Charlotte Angas Scott (8 June 1858 – 10 November 1931)[1] was a British mathematician who made her career in the United States and was influential in the development of American mathematics, including the mathematical education of women. Scott played an important role in Cambridge changing the rules for its famous Mathematical Tripos exam.
Charlotte Scott
Charlotte Angas Scott, from the 1910 yearbook of Bryn Mawr College
Born
Charlotte Angas Scott
(1858-06-08)8 June 1858
Lincoln, Lincolnshire, England
Died10 November 1931(1931-11-10) (aged 73)
Cambridge, Cambridgeshire, England
Scientific career
FieldsMathematics
InstitutionsCambridge University
Bryn Mawr College
Doctoral advisorArthur Cayley
Doctoral studentsLouise Cummings
Ada Maddison
Virginia Ragsdale
Emilie Martin
Mary Gertrude Haseman
Early life
She was the second of seven children to Caleb Scott, a minister of the Congregational Church, and Eliza Exley Scott.[2] Educated at Girton College, Cambridge from 1876 to 1880 on a scholarship, she was then a Resident Lecturer in Mathematics there until 1884. In 1885 she became one of the first British women to receive a doctorate,[3] and the first British woman to receive a doctorate in mathematics, which she received from the University of London.[4] She did her graduate research under Arthur Cayley at Cambridge University, but since Cambridge did not begin issuing degrees to women until 1948, Scott received her BSc (1882) and D.Sc. (1885) from the University of London[2] through external examinations.
Passing the Tripos
In 1880, Scott obtained special permission to take the Cambridge Mathematical Tripos Exam, as women were not normally allowed to sit for the exam. She came eighth on the Tripos of all students taking them, but due to her sex, the title of "eighth wrangler," a high honour, went officially to a male student.[2]
At the ceremony, however, after the seventh wrangler had been announced, all the students in the audience shouted her name.
The man read out the names and when he came to 'eighth,' before he could say the name, all the undergraduates called out 'Scott of Girton,' and cheered tremendously, shouting her name over and over again with tremendous cheers and waving of hats.
— contemporary report, "Charlotte Angas Scott (1858–1931)" in Women of Mathematics: A Biobibliographic Sourcebook[2]
Because she could not attend the award ceremony, Scott celebrated her accomplishment at Girton College where there were cheers and clapping at dinner, a special evening ceremony where the students sang "See the Conquering Hero Comes", received an ode written by a staff member, and was crowned with laurels.[2]
After this incident women were allowed to formally take the exam and their exam scores listed, although separately from the men's and thus not included in the rankings. Women obtaining the necessary score also received a special certificate instead of the BA degree with honours. In 1922, James Harkness remarked that Scott's achievement marked "the turning point in England from the theoretical feminism of Mill and others to the practical education and political advances of the present time".[2]
Work
Moving to the United States in 1885, she became one of eight founding faculty and Associate Professor of Mathematics at Bryn Mawr College, and Professor from 1888 to 1917. She was the first mathematician at Bryn Mawr College and the first department head.[3] During this period she directed the PhD theses of some pioneering women mathematicians. Of the nine other women to earn doctorates in mathematics in the nineteenth century, three studied with Scott.[2]
Her mathematical speciality was the study of specific algebraic curves of degree higher than two.[5] Her book An Introductory Account of Certain Modern Ideas and Methods in Plane Analytical Geometry was published in 1894 and reprinted thirty years later. Scott was one of the first English language textbook writers to be "perfectly aware" of the "distinction between a general principle and a particular example". She played an important role in the transition to twentieth century custom of abstract mathematical proofs.[2]
In 1891 she became the first woman to join the New York Mathematical Society, later called the American Mathematical Society.[6] She served as the first woman on the first Council of the American Mathematical Society in 1894, and received an acclaimed review from the Society in 1896.[3] She is also credited with being the author of the first mathematical research paper written in the US that was widely recognised in Europe, "A Proof of Noether's Fundamental Theorem" (Mathematische Annalen, Vol. 52 (1899)).[3][7] She was one of only four women to attend the inaugural International Congress of Mathematicians in Zurich in 1897; the other three were Iginia Massarini, Vera von Schiff, and Charlotte Wedell.[8] In 1906 Scott served as vice-president of the American Mathematical Society.
Women in mathematics
Scott maintained the view that personal conservatism was a requirement to promote women's educational and political equality. She disapproved of smoking and makeup, however she did bob her hair before moving to Bryn Mawr (short hair being controversial even in the 1920s). This view was also held by the early Girton College community, because unaccompanied women in Cambridge could be thrown into Spinning House, a special prison for prostitutes and suspected prostitutes.[2]
She was a staunch supporter of rigour in women's classes, writing in a letter to Bryn Mawr President M. Carey Thomas:
I am most disturbed and disappointed at present to find you taking the position that intellectual pursuits must be "watered down" to make them suitable for women, and that a lower standard must be adopted at a woman's college than in a man's. I do not expect any of the other members of the faculty to feel this way about it; they, like (nearly) all men that I have known, doubtless take an attitude of toleration, half amused and half kindly, on the whole question; for even where men are willing to help in women's education, it is with an inward reserve of condescension.
— Charlotte Scott, Scott Papers[2]
The word "nearly" is written in small lettering above the handwritten letter.[2]
Later life
Scott and Grace Andrews were the only two women listed in the first edition of American Men of Science, which appeared in 1906.[9]
Also in 1906, Scott developed an acute case of rheumatoid arthritis, which along with her increasing deafness, interrupted her work. Under the advice of a doctor to get outside exercise, Scott began gardening and developed a new strain of chrysanthemum. She retired in 1924, but stayed an extra year in Bryn Mawr to help her eighth doctoral student complete her dissertation before she returned to and settled in Cambridge.[2]
She died on 10 November 1931 and is buried at the Parish of the Ascension Burial Ground in Cambridge, in her cousin Eliza Nevin's grave.[10]
Now walk to the door of the chapel and look at grave 4C52 which is a curb in the second row in the second row on your right. There is a scroll on this grave of ELIZA NEVIN to CHARLOTTE ANGUS SCOTT, who entered Girton College in 1876 and became a Wrangler in the Mathematical Tripos in 1880.
— L.J. Slater, A Walk Around The Ascension Parish Burial Ground
Recognition
In 2016 the Council of the University of Cambridge approved the use of Scott's name to mark a physical feature within the North West Cambridge Development.[11]
Publications
• Scott, Charlotte A. (1894). An Introductory Account of Certain Modern Ideas and Methods in Plane Analytical Geometry. Macmillan.
Citations
1. "Scott, Charlotte Angas". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/38823. (Subscription or UK public library membership required.)
2. Patricia Clark Kenschaft (1987). "Charlotte Angas Scott (1858–1931)" in Women of Mathematics: A Biobibliographic Sourcebook. New York: Greenwood Press. pp. 193–203. ISBN 0-313-24849-4.
3. Chaplin, Stephanie (1997). "Biographies of Women Mathematicians: Charlotte Angas Scott". Agnes Scott College. Retrieved 22 October 2012.
4. "Charlotte Angas Scott". mathwomen.agnesscott.org.
5. Kenschaft, Patricia (1977). "Charlotte Angas Scott" (PDF). AWM Newsletter. 7 (6): 11–12.
6. Oakes, Elizabeth (2007). Encyclopedia of World Scientists, Revised Edition. Infobase Publishing. p. 655. ISBN 9781438118826.
7. Scott, Charlotte Angas (1 December 1899). "A proof of Noether's fundamental theorem". Mathematische Annalen. 52 (4): 593–597. doi:10.1007/BF01453778. S2CID 120672039.
8. Curbera, Guillermo (2009), Mathematicians of the World, Unite!: The International Congress of Mathematicians—A Human Endeavor, CRC Press, p. 16, ISBN 9781439865125
9. Bailey, Martha J. (1994). American Women in Science:A Biographical Dictionary. ABC-CLIO, Inc. ISBN 0-87436-740-9.
10. Goldie, Dr. Mark (2009). A Guide to Churchill College, Cambridge. pp. 62–63.
11. Administrator (29 January 2015). "Street Naming". www.nwcambridge.co.uk. Retrieved 8 March 2017.
References
• Green, Dr. Judy (29 June 2000). "How many women mathematicians can you name?" (PDF). Summer Undergraduate Mathematical Sciences Research Institute / Miami University. Archived from the original (PDF) on 20 February 2012. Retrieved 21 October 2012.
• Girton College Register 1869–1946, University Press, Cambridge, 1948
• Eaton, Shelby L. (21 August 1997). "Women in Mathematics in the United States: 1866–1900". Shelby L. Eaton. Archived from the original on 16 March 2012. Retrieved 21 October 2012.
External links
• Charlotte Scott at Find a Grave
• Charlotte Scott at the Mathematics Genealogy Project
• "Charlotte Agnas Scott" written by Isabel Maddison
• Digital Copy of "Charlotte Angas Scott (1858–1931)" in Women of Mathematics: A Biobibliographic Sourcebook by Patricia Clark Kenschaft
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MSC 2010: Numerical Analysis
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20 results in 65Yxx
JASMIN-based Two-dimensional Adaptive Combined Preconditioner for Radiation Diffusion Equations in Inertial Fusion Research
MSC 2010: Computer aspects of numerical algorithms
MSC 2010: Partial differential equations, boundary value problems
MSC 2010: Numerical linear algebra
Xiaoqiang Yue, Xiaowen Xu, Shi Shu
Journal: East Asian Journal on Applied Mathematics / Volume 7 / Issue 3 / August 2017
We present a JASMIN-based two-dimensional parallel implementation of an adaptive combined preconditioner for the solution of linear problems arising in the finite volume discretisation of one-group and multi-group radiation diffusion equations. We first propose the attribute of patch-correlation for cells of a two-dimensional monolayer piecewise rectangular structured grid without any suspensions based on the patch hierarchy of JASMIN, classify and reorder these cells via their attributes, and derive the conversion of cell-permutations. Using two cell-permutations, we then construct some parallel incomplete LU factorisation and substitution algorithms, to provide our parallel -GMRES solver with the help of the default BoomerAMG in the HYPRE library. Numerical results demonstrate that our proposed parallel incomplete LU preconditioner (ILU) is of higher efficiency than the counterpart in the Euclid library, and that the proposed parallel -GMRES solver is more robust and more efficient than the default BoomerAMG-GMRES solver.
GPU-Accelerated LOBPCG Method with Inexact Null-Space Filtering for Solving Generalized Eigenvalue Problems in Computational Electromagnetics Analysis with Higher-Order FEM
MSC 2010: Numerical methods
MSC 2010: Algorithms - Computer Science
A. Dziekonski, M. Rewienski, P. Sypek, A. Lamecki, M. Mrozowski
Journal: Communications in Computational Physics / Volume 22 / Issue 4 / October 2017
Published online by Cambridge University Press: 28 July 2017, pp. 997-1014
Print publication: October 2017
This paper presents a GPU-accelerated implementation of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method with an inexact nullspace filtering approach to find eigenvalues in electromagnetics analysis with higher-order FEM. The performance of the proposed approach is verified using the Kepler (Tesla K40c) graphics accelerator, and is compared to the performance of the implementation based on functions from the Intel MKL on the Intel Xeon (E5-2680 v3, 12 threads) central processing unit (CPU) executed in parallel mode. Compared to the CPU reference implementation based on the Intel MKL functions, the proposed GPU-based LOBPCG method with inexact nullspace filtering allowed us to achieve up to 2.9-fold acceleration.
Exponential Time Differencing Gauge Method for Incompressible Viscous Flows
MSC 2010: Incompressible viscous fluids
MSC 2010: Partial differential equations, initial value and time-dependent initial-boundary value problems
Lili Ju, Zhu Wang
Journal: Communications in Computational Physics / Volume 22 / Issue 2 / August 2017
In this paper, we study an exponential time differencing method for solving the gauge system of incompressible viscous flows governed by Stokes or Navier-Stokes equations. The momentum equation is decoupled from the kinematic equation at a discrete level and is then solved by exponential time stepping multistep schemes in our approach. We analyze the stability of the proposed method and rigorously prove that the first order exponential time differencing scheme is unconditionally stable for the Stokes problem. We also present a compact representation of the algorithm for problems on rectangular domains, which makes FFT-based solvers available for the resulting fully discretized system. Various numerical experiments in two and three dimensional spaces are carried out to demonstrate the accuracy and stability of the proposed method.
Numerical Effects of the Gaussian Recursive Filters in Solving Linear Systems in the 3Dvar Case Study
MSC 2010: Error analysis and interval analysis
MSC 2010: Multivariate analysis
Salvatore Cuomo, Ardelio Galletti, Giulio Giunta, Livia Marcellino
Journal: Numerical Mathematics: Theory, Methods and Applications / Volume 10 / Issue 3 / August 2017
In many applications, the Gaussian convolution is approximately computed by means of recursive filters, with a significant improvement of computational efficiency. We are interested in theoretical and numerical issues related to such an use of recursive filters in a three-dimensional variational data assimilation (3Dvar) scheme as it appears in the software OceanVar. In that context, the main numerical problem consists in solving large linear systems with high efficiency, so that an iterative solver, namely the conjugate gradient method, is equipped with a recursive filter in order to compute matrix-vector multiplications that in fact are Gaussian convolutions. Here we present an error analysis that gives effective bounds for the perturbation on the solution of such linear systems, when is computed by means of recursive filters. We first prove that such a solution can be seen as the exact solution of a perturbed linear system. Then we study the related perturbation on the solution and we demonstrate that it can be bounded in terms of the difference between the two linear operators associated to the Gaussian convolution and the recursive filter, respectively. Moreover, we show through numerical experiments that the error on the solution, which exhibits a kind of edge effect, i.e. most of the error is localized in the first and last few entries of the computed solution, is due to the structure of the difference of the two linear operators.
Modulus-based Synchronous Multisplitting Iteration Methods for an Implicit Complementarity Problem
MSC 2010: Nonlinear algebraic or transcendental equations
Chen-Liang Li, Jun-Tao Hong
Journal: East Asian Journal on Applied Mathematics / Volume 7 / Issue 2 / May 2017
Print publication: May 2017
We construct modulus-based synchronous multisplitting iteration methods to solve a large implicit complementarity problem on parallel multiprocessor systems, and prove their convergence. Numerical results confirm our theoretical analysis and show that these new methods are efficient.
Designing Several Types of Oscillation-Less and High-Resolution Hybrid Schemes on Block-Structured Grids
Zhenhua Jiang, Chao Yan, Jian Yu, Boxi Lin
Journal: Communications in Computational Physics / Volume 21 / Issue 5 / May 2017
Published online by Cambridge University Press: 27 March 2017, pp. 1376-1407
An idea of designing oscillation-less and high-resolution hybrid schemes is proposed and several types of hybrid schemes based on this idea are presented on block-structured grids. The general framework, for designing various types of hybrid schemes, is established using a Multi-dimensional Optimal Order Detection (MOOD) method proposed by Clain, Diot and Loubère [1]. The methodology utilizes low dissipation or dispersion but less robust schemes to update the solution and then implements robust and high resolution schemes to deal with problematic situations. A wide range of computational methods including central scheme, MUSCL scheme, linear upwind scheme and Weighted Essentially Non Oscillatory (WENO) scheme have been applied in the current hybrid schemes framework. Detailed numerical studies on classical test cases for the Euler system are performed, addressing the issues of the resolution and non-oscillatory property around the discontinuities.
Simulation of Maxwell's Equations on GPU Using a High-Order Error-Minimized Scheme
MSC 2010: Equations of mathematical physics and other areas of application
Tony W. H. Sheu, S. Z. Wang, J. H. Li, Matthew R. Smith
Journal: Communications in Computational Physics / Volume 21 / Issue 4 / April 2017
Print publication: April 2017
In this study an explicit Finite Difference Method (FDM) based scheme is developed to solve the Maxwell's equations in time domain for a lossless medium. This manuscript focuses on two unique aspects – the three dimensional time-accurate discretization of the hyperbolic system of Maxwell equations in three-point non-staggered grid stencil and it's application to parallel computing through the use of Graphics Processing Units (GPU). The proposed temporal scheme is symplectic, thus permitting conservation of all Hamiltonians in the Maxwell equation. Moreover, to enable accurate predictions over large time frames, a phase velocity preserving scheme is developed for treatment of the spatial derivative terms. As a result, the chosen time increment and grid spacing can be optimally coupled. An additional theoretical investigation into this pairing is also shown. Finally, the application of the proposed scheme to parallel computing using one Nvidia K20 Tesla GPU card is demonstrated. For the benchmarks performed, the parallel speedup when compared to a single core of an Intel i7-4820K CPU is approximately 190x.
Efficient and Stable Exponential Runge-Kutta Methods for Parabolic Equations
Liyong Zhu
Journal: Advances in Applied Mathematics and Mechanics / Volume 9 / Issue 1 / February 2017
Published online by Cambridge University Press: 11 October 2016, pp. 157-172
Print publication: February 2017
In this paper we develop explicit fast exponential Runge-Kutta methods for the numerical solutions of a class of parabolic equations. By incorporating the linear splitting technique into the explicit exponential Runge-Kutta schemes, we are able to greatly improve the numerical stability. The proposed numerical methods could be fast implemented through use of decompositions of compact spatial difference operators on a regular mesh together with discrete fast Fourier transform techniques. The exponential Runge-Kutta schemes are easy to be adopted in adaptive temporal approximations with variable time step sizes, as well as applied to stiff nonlinearity and boundary conditions of different types. Linear stabilities of the proposed schemes and their comparison with other schemes are presented. We also numerically demonstrate accuracy, stability and robustness of the proposed method through some typical model problems.
Finite Element Algorithm for Dynamic Thermoelasticity Coupling Problems and Application to Transient Response of Structure with Strong Aerothermodynamic Environment
MSC 2010: Thermodynamics and heat transfer
MSC 2010: Hypersonic flows
Zhihui Li, Qiang Ma, Junzhi Cui
Journal: Communications in Computational Physics / Volume 20 / Issue 3 / September 2016
As an exploratory study for structural deformation and thermodynamic response induced by spacecraft reentry aerodynamic force and thermal environment, a finite element algorithm is presented on the basis of the classic Fourier heat conductive law to simulate the dynamic thermoelasticity coupling performance of the material. The Newmark method and Crank-Nicolson scheme are utilized to discretize the dynamic thermoelasticity equation and heat conductive equation in the time domain, respectively, and the unconditionally stable implicit algorithm is constructed. Four types of finite-element computing schemes are devised and discussed to solve the thermodynamic coupling equation, all of which are implemented and compared in the computational examples including the one-dimensional transient heat conduction in considering and not considering the vibration, the transient heat flow for the infinite cylinder, and the dynamic coupling thermoelasticity around re-entry flat plate from hypersonic aerothermodynamic environment. The computational results show that the transient responses of temperature and displacement field generate lag phenomenon in case of considering the deformation effect on temperature field. Propagation, rebounding, attenuation and stabilized phenomena of elastic wave are also observed by the finite-element calculation of thermodynamic coupling problem considering vibration and damping, and the oscillation of the temperature field is simultaneously induced. As a result, the computational method and its application research platform have been founded to solve the transient thermodynamic coupling response problem of the structure in strong aerodynamic heating and force environment. By comparing various coupling calculations, it is demonstrated that the present algorithm could give a correct and reliable description of transient thermodynamic responses of structure, the rationality of the sequentially coupling method in engineering calculation is discussed, and the bending deformation mechanism produced by the thermodynamic coupling response from windward and leeward sides of flying body is revealed, which lays the foundation in developing the numerical method to solve material internal temperature distribution, structural deformation, and thermal damage induced by spacecraft dynamic thermoelasticity coupling response under uncontrolled reentry aerothermodynamic condition.
Parallel Solution of Linear Systems
Sidi-Mahmoud Kaber, Amine Loumi, Philippe Parnaudeau
Computational scientists generally seek more accurate results in shorter times, and to achieve this a knowledge of evolving programming paradigms and hardware is important. In particular, optimising solvers for linear systems is a major challenge in scientific computation, and numerical algorithms must be modified or new ones created to fully use the parallel architecture of new computers. Parallel space discretisation solvers for Partial Differential Equations (PDE) such as Domain Decomposition Methods (DDM) are efficient and well documented. At first glance, parallelisation seems to be inconsistent with inherently sequential time evolution, but parallelisation is not limited to space directions. In this article, we present a new and simple method for time parallelisation, based on partial fraction decomposition of the inverse of some special matrices. We discuss its application to the heat equation and some limitations, in associated numerical experiments.
Reduced Basis Approaches in Time-Dependent Non-Coercive Settings for Modelling the Movement of Nuclear Reactor Control Rods
MSC 2010: Numerical methods in calculus of variations and optimal control
Alberto Sartori, Antonio Cammi, Lelio Luzzi, Gianluigi Rozza
Journal: Communications in Computational Physics / Volume 20 / Issue 1 / July 2016
Published online by Cambridge University Press: 22 June 2016, pp. 23-59
In this work, two approaches, based on the certified Reduced Basis method, have been developed for simulating the movement of nuclear reactor control rods, in time-dependent non-coercive settings featuring a 3D geometrical framework. In particular, in a first approach, a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod. In the second approach, a "staircase" strategy has been adopted for simulating the movement of all the three rods featured by the nuclear reactor chosen as case study. The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion, which, in the present case, is a set of ten coupled parametrized parabolic equations (two energy groups for the neutron flux, and eight for the precursors). Both the reduced order models, developed according to the two approaches, provided a very good accuracy compared with high-fidelity results, assumed as "truth" solutions. At the same time, the computational speed-up in the Online phase, with respect to the fine "truth" finite element discretization, achievable by both the proposed approaches is at least of three orders of magnitude, allowing a real-time simulation of the rod movement and control.
Simulation of Flow in Multi-Scale Porous Media Using the Lattice Boltzmann Method on Quadtree Grids
MSC 2010: Flows in porous media; filtration; seepage
Lei Zhang, Qinjun Kang, Li Chen, Jun Yao
Published online by Cambridge University Press: 12 April 2016, pp. 998-1014
The unified lattice Boltzmann model is extended to the quadtree grids for simulation of fluid flow through porous media. The unified lattice Boltzmann model is capable of simulating flow in porous media at various scales or in systems where multiple length scales coexist. The quadtree grid is able to provide a high-resolution approximation to complex geometries, with great flexibility to control local grid density. The combination of the unified lattice Boltzmann model and the quadtree grids results in an efficient numerical model for calculating permeability of multi-scale porous media. The model is used for permeability calculation for three systems, including a fractured system used in a previous study, a Voronoi tessellation system, and a computationally-generated pore structure of fractured shale. The results are compared with those obtained using the conventional lattice Boltzmann model or the unified lattice Boltzmann model on rectangular or uniform square grid. It is shown that the proposed model is an accurate and efficient tool for flow simulation in multi-scale porous media. In addition, for the fractured shale, the contribution of flow in matrix and fractures to the overall permeability of the fractured shale is studied systematically.
Automated Parallel and Body-Fitted Mesh Generation in Finite Element Simulation of Macromolecular Systems
MSC 2010: Physiological, cellular and medical topics
Yan Xie, Tiantian Liu, Bin Tu, Benzhuo Lu, Linbo Zhang
Journal: Communications in Computational Physics / Volume 19 / Issue 3 / March 2016
Print publication: March 2016
Mesh generation is a bottleneck for finite element simulations of biomolecules. A robust and efficient approach, based on the immersed boundary method proposed in [8], has been developed and implemented to generate large-scale mesh body-fitted to molecular shape for general parallel finite element simulations. The molecular Gaussian surface is adopted to represent the molecular surface, and is finally approximated by piecewise planes via the tool phgSurfaceCut in PHG [43], which is improved and can reliably handle complicated molecular surfaces, through mesh refinement steps. A coarse background mesh is imported first and then is distributed into each process using a mesh partitioning algorithm such as space filling curve [5] or METIS [22]. A bisection method is used for the mesh refinements according to the molecular PDB or PQR file which describes the biomolecular region. After mesh refinements, the mesh is optionally repartitioned and redistributed for load balancing. For finite element simulations, the modification of region mark and boundary types is done in parallel. Our parallel mesh generation method has been successfully applied to a sphere cavity model, a DNA fragment, a gramicidin A channel and a huge Dengue virus system. The results of numerical experiments show good parallel efficiency. Computations of electrostatic potential and solvation energy also validate the method. Moreover, the meshing process and adaptive finite element computation can be integrated as one PHG project to avoid the mesh importing and exporting costs, and improve the convenience of application as well.
High Order Numerical Methods for the Dynamic SGS Model of Turbulent Flows with Shocks
D. V. Kotov, H. C. Yee, A. A. Wray, A. Hadjadj, B. Sjögreen
Journal: Communications in Computational Physics / Volume 19 / Issue 2 / February 2016
Simulation of turbulent flows with shocks employing subgrid-scale (SGS) filtering may encounter a loss of accuracy in the vicinity of a shock. This paper addresses the accuracy improvement of LES of turbulent flows in two ways: (a) from the SGS model standpoint and (b) from the numerical method improvement standpoint. In an internal report, Kotov et al. ( "High Order Numerical Methods for large eddy simulation (LES) of Turbulent Flows with Shocks", CTR Tech Brief, Oct. 2014, Stanford University), we performed a preliminary comparative study of different approaches to reduce the loss of accuracy within the framework of the dynamic Germano SGS model. The high order low dissipative method of Yee & Sjögreen (2009) using local flow sensors to control the amount of numerical dissipation where needed is used for the LES simulation. The considered improved dynamics model approaches include applying the one-sided SGS test filter of Sagaut & Germano (2005) and/or disabling the SGS terms at the shock location. For Mach 1.5 and 3 canonical shock-turbulence interaction problems, both of these approaches show a similar accuracy improvement to that of the full use of the SGS terms. The present study focuses on a five levels of grid refinement study to obtain the reference direct numerical simulation (DNS) solution for additional LES SGS comparison and approaches. One of the numerical accuracy improvements included here applies Harten's subcell resolution procedure to locate and sharpen the shock, and uses a one-sided test filter at the grid points adjacent to the exact shock location.
A parallel root-finding algorithm
M. J. P. Nijmeijer
Journal: LMS Journal of Computation and Mathematics / Volume 18 / Issue 1 / 2015
Published online by Cambridge University Press: 01 December 2015, pp. 713-729
We present a parallel algorithm to calculate a numerical approximation of a single, isolated root ${\it\alpha}$ of a function $f:\mathbb{R}\rightarrow \mathbb{R}$ which is sufficiently regular at and around ${\it\alpha}$ . The algorithm is derivative free and performs one function evaluation on each processor per iteration. It requires at least three processors and can be scaled up to any number of these. The order with which the generated sequence of approximants converges to ${\it\alpha}$ is equal to $(n+\sqrt{n^{2}+4})/2$ for $n+1$ processors with $n\geqslant 2$ . This assumes that particular combinations of the derivatives of $f$ do not vanish at ${\it\alpha}$ .
Computing the Smallest Eigenvalue of Large Ill-Conditioned Hankel Matrices
MSC 2010: Nontrigonometric harmonic analysis
Niall Emmart, Yang Chen, Charles C. Weems
This paper presents a parallel algorithm for finding the smallest eigenvalue of a family of Hankel matrices that are ill-conditioned. Such matrices arise in random matrix theory and require the use of extremely high precision arithmetic. Surprisingly, we find that a group of commonly-used approaches that are designed for high efficiency are actually less efficient than a direct approach for this class of matrices. We then develop a parallel implementation of the algorithm that takes into account the unusually high cost of individual arithmetic operations. Our approach combines message passing and shared memory, achieving near-perfect scalability and high tolerance for network latency. We are thus able to find solutions for much larger matrices than previously possible, with the potential for extending this work to systems with greater levels of parallelism. The contributions of this work are in three areas: determination that a direct algorithm based on the secant method is more effective when extreme fixed-point precision is required than are the algorithms more typically used in parallel floating-point computations; the particular mix of optimizations required for extreme precision large matrix operations on a modern multi-core cluster, and the numerical results themselves.
A Wavelet-Based Almost-Sure Uniform Approximation of Fractional Brownian Motion with a Parallel Algorithm
MSC 2010: Numerical methods in Fourier analysis
MSC 2010: Stochastic processes
Dawei Hong, Shushuang Man, Jean-Camille Birget, Desmond S. Lun
Journal: Journal of Applied Probability / Volume 51 / Issue 1 / March 2014
Published online by Cambridge University Press: 30 January 2018, pp. 1-18
We construct a wavelet-based almost-sure uniform approximation of fractional Brownian motion (FBM) (B t (H))_t∈[0,1] of Hurst index H ∈ (0, 1). Our results show that, by Haar wavelets which merely have one vanishing moment, an almost-sure uniform expansion of FBM for H ∈ (0, 1) can be established. The convergence rate of our approximation is derived. We also describe a parallel algorithm that generates sample paths of an FBM efficiently.
AN IMPROVED SECOND-ORDER NUMERICAL METHOD FOR THE GENERALIZED BURGERS–FISHER EQUATION
MSC 2010: Parabolic equations and systems
A. G. BRATSOS
Journal: The ANZIAM Journal / Volume 54 / Issue 3 / January 2013
Print publication: January 2013
A second-order in time finite-difference scheme using a modified predictor–corrector method is proposed for the numerical solution of the generalized Burgers–Fisher equation. The method introduced, which, in contrast to the classical predictor–corrector method is direct and uses updated values for the evaluation of the components of the unknown vector, is also analysed for stability. Its efficiency is tested for a single-kink wave by comparing experimental results with others selected from the available literature. Moreover, comparisons with the classical method and relevant analogous modified methods are given. Finally, the behaviour and physical meaning of the two-kink wave arising from the collision of two single-kink waves are examined.
AN ALGORITHM FOR FINDING ALL ZEROS OF VECTOR FUNCTIONS
IBRAHEEM ALOLYAN
Journal: Bulletin of the Australian Mathematical Society / Volume 77 / Issue 3 / June 2008
Computing a zero of a continuous function is an old and extensively researched problem in numerical computation. In this paper, we present an efficient subdivision algorithm for finding all real roots of a function in multiple variables. This algorithm is based on a simple computationally verifiable necessity test for the existence of a root in any compact set. Both theoretical analysis and numerical simulations demonstrate that the algorithm is very efficient and reliable. Convergence is shown and numerical examples are presented.
Stochastic Analysis of 'Simultaneous Merge–Sort'
MSC 2010: Limit theorems
M. Cramer
Journal: Advances in Applied Probability / Volume 29 / Issue 3 / September 1997
The asymptotic behaviour of the recursion is investigated; Yk describes the number of comparisons which have to be carried out to merge two sorted subsequences of length 2k –1 and Mk can be interpreted as the number of comparisons of 'Simultaneous Merge–Sort'. The challenging problem in the analysis of the above recursion lies in the fact that it contains a maximum as well as a sum. This demands different ideal properties for the metric in the contraction method. By use of the weighted Kolmogorov metric it is shown that an exponential normalization provides the recursion's convergence. Furthermore, one can show that any sequence of linear normalizations of Mk must converge towards a constant if it converges in distribution at all.
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CommonCrawl
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Cycle 2 (2010) Accepted Guest Observer Programs
PHOTOMETRY OF AN ECLIPSING SYSTEM WITH A WHITE DWARF COMPONENT, THE ONLY ONE KNOWN IN THE KEPLER FOV
Roi Alonso
Observatoire Astronomique de l'Universite de Geneve
GO20043
+data
We plan to continue our research on the only system of a white dwarf with a M star eclipsing component that is accesible to Kepler FOV. Extending the observations through Cycle2 will allow us to 1) improve the precision on the orbital parameters, 2) study the anual evolution of the flare activity on the M companion and its dependance with the orbital phase, 3) study the evolution of magnetic active regions on any of the components, 4) improve the precision on the expected detection of a secondary eclipse, 5) gain valuable data on the O-C residuals of the 1040 eclipses/year that Kepler is able to obtain, that might allow the detection of small stellar companions and probably substellar, and 6) search for pulsations of the WD component.
SIMULTANEOUS OPTICAL AND RADIO MONITORING OF NEARBY GALAXY NUCLEI WITH KEPLER AND THE ALLEN TELESCOPE ARRAY
Geoffrey Bower
We propose to obtain simultaneous optical and radio light curves for the nuclear regions of a sample of 117 nearby galaxies, including some which are known to host AGN. These light curves will be valuable for constraining physical conditions in the galaxies studied, including the origin of AGN activity and of radio-loudness, black hole accretion mechanisms, and the presence of supernovae, X-ray binaries, or other variables in the nuclear regions of galaxies. Optical and radio data will probe a range of energies and size scales. The Allen Telescope Array (ATA) is a radio telescope designed for fast surveying of large areas of sky, with a particular emphasis on transient and variable sources. With a 5 square degree field of view at 20 cm, a sensitivity of ~10 mJy in a one minute observation, and the ability to observe simultaneously in two 100 MHz bands anywhere in the 0.5 - 10 GHz range, the ATA is opening new regions of parameter space in cadence, sensitivity, and area covered. Several ongoing surveys are in progress, including a survey of a ~10 square degree field in Cygnus which is observed for ~8 hours every few days. This survey is designed to study transient emission from objects such as supernovae and gamma ray bursts, as well as variable sources such as AGN and flare stars. Since we have coverage of the full 10 square degree ATA field, and not just the objects we propose to observe with Kepler, we will also be able to supply radio light curves to the Kepler team for any other objects (e.g., brown dwarfs, flare stars, etc.) in our radio data which show interesting characteristics such as variability. We select a sample of galaxies (including known AGN), most of which are associated with the low-redshift rich galaxy cluster Abell 2319, within the Kepler and ATA Cygnus Survey fields of view. The ATA will observe this field with a rolling cadence, allowing us to explore variability on many timescales, from minutes to years. Tools have been developed for imaging, catalog extraction, and light curve generation which will allow us to easily compare variability in the radio with variability at optical wavelengths from Kepler.
STARSPOT EVOLUTION ON ACTIVE LATE-TYPE STARS IN THE KEPLER FIELD - CYCLE 2
University Of Colorado, Boulder
Starspots on late-type stars are a direct manifestation of the photospheric emergence of strong dynamo-generated magnetic fields. We propose to extend our Cycle 1 project of 30 minute cadence Kepler photometry, in which we are investigating how activity phenomena such as the growth, migration, and decay of starspots, differential rotation, activity cycles,and flaring operate on single and binary stars with a wide range of mass (and hence convection zone depth), with the expectation that such investigations will stimulate and enable theoretical studies of magnetic flux generation and transport processes in the extreme regime of fast rotation that any successful theory must be able to address. Our sample of 186 active stars was selected based on GALEX FUV and NUV imaging of the Kepler field. For accurately measuring the longitudes of active regions, spot filling-factor maps will be obtained from the Kepler photometry using light-curve inversion methods. Time-series analysis, using both Fourier and wavelet techniques, are used to obtain accurate rotation periods. After which the phased light-curves are processed with our existing inversion codes using both the Occamian approach and the Maximum Entropy method. A full suite of supporting high resolution optical spectroscopic observations will be obtained using the Hobby-Eberly, Keck, and Apache Point Observatory telescopes to accurately determine the stellar parameters, including effective temperature, surface gravity, and projected rotational velocity, and to identify which stars are spectroscopic or eclipsing binaries and measure their radial velocity curves. For many targets Doppler imaging, both conventional and magnetic, will be pursued.
A SEARCH FOR ASTEROIDS ORBITING WHITE DWARFS
Rosanne Di Stefano
Smithsonian Institution/Smithsonian Astrophysical Observatory
Do white dwarfs host asteroid systems? Because asteroids are fossils of planet formation, the answer to this question has implications for our understanding of the earliest stages of planetary systems. In addition, because white dwarfs are remnants of stars like our Sun, the discovery of asteroids orbiting them may teach us about the evolution and survival of planetary systems. NASA's launch of Kepler provides us with the first tool capable of helping to answer this question. We propose that Kepler observe two of the brightest white dwarfs in its field in 1-minute cadence mode with the goal of searching for transits by asteroids in orbit around them. This scientific investigation makes full use of NASA's newest mission, and in fact would not be possible without it. To detect the passage of 100-km class objects against the disk of a white dwarf requires Kepler's unique photometric sensitivity and continuous monitoring. The analysis we propose to conduct will open a new field of endeavor that can help achieve NASA's science goals for the study of the origin and evolution of planetary systems.
LENSING IN THE KEPLER FIELD
Lensing events occur regularly in the Kepler field. The Kepler mission therefore provides a unique and scientifically important opportunity to monitor lensing light curves. The unprecedented photometric sensitivity combined with 30-minute cadence over a period of months can be utilized to accomplish important goals These include (1) high-precision verification of the form of the lensing light curve predicted by general relativity, and (2) high-precision tests for a variety of system parameters, including lens mass and multiplicity, source multiplicity, parallax, and source size. Detectable lensing is most likely to be caused by nearby high-proper-motion masses,i.e., mesolenses. The Einstein rings of nearby masses are large enough that astrometric effects, as well as photometric variations may be detectable. A large set of high-proper-motion stars is already likely to be observed by Kepler during the coming year. We propose to analyze Kepler data from all of these. In addition, we propose that a new set of high-proper-motion stars be observed during cycle 2. The newly selected stars are those with the highest probability of producing an event. In order to conduct the analysis we propose, we will develop tools that allow the community of scientists who will use Kepler data to check for evidence of lensing events in the entire data set.
HIGH METALLICITY IN CATACLYSMIC VARIABLES AND THE RISE OF SUPERNOVAE
Peter Garnavich
We propose to study two of the three known cataclysmic variables (CV) in the old, metal-rich open cluster NGC 6791. Both appear to be dwarf novae, although, ground observations have been sparse. The instability of the accretion disk producing the dwarf nova outbursts may depend on the metal abundance of the accreting gas. Photometric properties, such as cycle time and outburst amplitude, maybe enhanced in the unique NGC 6791 environment. We also propose to monitor 93 bright galaxies at z<0.05 in the Kepler field to obtain the early light curve of a supernova. SDSS-II studies of type Ia light curves show that the rise-times are shorter than previously thought and the shape of the early rise provides insight into the explosion mechanism.
AN EXTENDED SEARCH FOR CIRCUMBINARY COMPANIONS OF INTERMEDIATE-MASS ECLIPSING BINARY STARS
Douglas Gies
There is abundant evidence that stellar companions are more commonplace among the more massive stars, and it is possible that massive star formation processes lead directly to binary and multiple systems as a repository of the angular momentum of the natal cloud. In particular, the formation of a close binary may require the presence of a distant third star to carry the bulk of the angular momentum. Our goal in this proposal is to search for evidence of companions surrounding close eclipsing pairs of intermediate mass, B, A, and F-type stars. Since these close binaries have periods of a few days, the search will focus on dynamically stable outer companions with orbital periods in the range 0.3 to 26 months. We will use precise light curves from Kepler of 40 binaries to measure accurate eclipse timings, and we will search for companions by investigating periodic variations in the times of minima caused by the light travel time across the orbital displacement of the close binary. In favorable situations, we will be able to detect the presence of objects as small as brown dwarf stars and/or massive planets. This work will establish the occurrence of low mass companions among intermediate mass stars. We began this program in Cycle 1 and here we seek an extension through Cycle 2 to double the size of the sample and to search for longer period companions.
TRANSITION IN VARIABLE STARS: FROM SOLAR-TYPE TO GAMMA DORADUS-TYPE
Joyce Guzik
Los Alamos National Laboratory/University of California
The Main Sequence solar-type pulsators are characterized by acoustic oscillation modes excited by turbulent granular convection in the upper convective boundary layer. As the stellar mass increases the convection zone shrinks, the scale and intensity of the turbulent motions increases, providing more energy for excitation of acoustic modes. When the stellar mass reaches about 1.6 solar masses (the gamma Doradus class) the upper convection zone consists of two very thin layers corresponding to H and He ionization, and in addition to the acoustic (p) modes the stars show strong internal gravity (g) modes The thin convection zone is often considered insignificant for the stellar dynamics and variability. However, recent 3D radiative hydrodynamics simulations reveal supersonic granular-type convection of the scale significantly larger than the solar granulation, and strong overshooting plumes penetrating into the stable radiative zone. These plumes may contribute to the excitation of the g-modes or hybrid modes with p- and g-characteristics. Pulsations of these types despite substantial efforts have not been observed on Sun. The goal of this proposal is investigate the physics of the interaction between the turbulent convection and oscillations along the Main Sequence, from the solar-type stars to more massive gamma Doradus stars. This interaction will be investigated by comparing the convective and oscillation spectra with the numerical simulation models. The numerical simulations, specifically developed at NASA Ames and Stanford Center for Turbulence Research, will provide a critical theoretical support for interpretation of the observed variability of these stars. This type of turbulent convection cannot be correctly described by the traditional mixing-length models. The proposed investigation will include a series of interesting questions about the role of turbulent surface and subsurface motions in the stellar variability and magnetism, e.g. how the supergranulation pattern changes in this transition, what is the effect of this transition on the local dynamo, formation of magnetic structures and atmospheric heating. The gamma Doradus stars show an increase in UV radiation but the mechanism of this is unclear. The Kepler short-cadence data and the realistic numerical simulations carried out in conjunction with project offer a unique opportunity to investigate the physics of the transition in turbulent convection and oscillations, and also potential role of magnetic fields. This will provide an important insight for the understanding of these and other types of variable stars.
A SEARCH FOR HYBRID GAMMA DORADUS/DELTA SCUTI PULSATING VARIABLES--IMPROVING THE STATISTICS
The delta Scuti and gamma Doradus pulsating variables are main-sequence (core hydrogen-burning) stars with masses somewhat larger than the sun (1.2 to 2.5 solar masses). The lower-mass gamma Dor stars are pulsating in nonradial gravity modes with periods of near one day, whereas the delta Scuti stars are radial and nonradial p-mode (acoustic mode) pulsators with periods of order two hours. Because of the near one-day periods of gamma Dor stars, it is very difficult to discover these stars and determine their pulsation frequency spectrum from ground-based photometry due to the 1 cycle/day alias, whereas Kepler observations have been able to detect them easily with one or two quarters of monitoring. Hybrid gamma Dor/delta Sct stars are among the most interesting targets for asteroseismology because the two types of modes (pressure and gravity) probe different regions of the star and are sensitive to the details of the two different driving mechanisms. Because the pulsations are driven by two different, and somewhat mutually exclusive, mechanisms, hybrid stars exhibiting both types of pulsations are expected to exist only in a small overlapping region of temperature-luminosity space in the Hertzsprung-Russell diagram. Before the advent of the Kepler and CoRoT missions, only four hybrid gamma Dor/delta Scuti pulsators had been discovered. Now the first analysis by the Kepler Asteroseismic Consortium (KASC) of 234 targets showing pulsations of either type has revealed hybrid behavior in essentially all of them! (Grighacene et al., ApJL in press, astro-ph 1001.0747) The existence and properties of these hybrids raise a number of questions: Why are hybrids much more common than predicted by theory? Why do some hybrid stars show frequencies in the gap predicted by theory between the gamma Dor and delta Sct frequency range? Are unknown pulsation mechanisms at work ? We hope to answer these questions by improving the statistics on the occurrence and properties of hybrids among the gamma Dor and delta Sct stars using Kepler observations. We propose to supplement the KASC search by observing an additional 187 stars from the Kepler Guest Observer Input Catalog that lie in or near the gamma Dor and delta Sct instability strips. Note that we are only requesting long-cadence data, and we are proposing to observe stars that have not yet been observed by Kepler. It is imperative that we not miss this opportunity to observe these stars with Kepler, as it is nearly impossible to discover and monitor the gamma Dor stars with pulsation periods of order one day from the ground. Analysis of Kepler data to date has shown that the long cadence data is also capable of detecting delta Sct frequencies that are more than 1 hour; the properties of the short frequencies can be refined later by short-cadence data later, or by ground-based photometry. We also will perform follow-up observations of the promising hybrid candidates with ground-based spectroscopic observations using the New Mexico State University 1 meter or 3.5 m Apache Peak telescopes to obtain accurate effective temperatures and surface gravities, constrain rotation rates, detect abundance peculiarities, and rule out binarity or star spots as a cause of periodicities. A larger survey of these stars with the high-precision photometry provided by Kepler is essential to help resolve the mysteries surrounding the theoretical model predictions and to realize the potential for asteroseismology of these stars.
MEASURING THE MASSES AND RADII OF THE LOWER MAIN SEQUENCE II: IDENTIFICATION OF NEW ECLIPSING M-DWARFS
Thomas Harrison
We propose to continue to use Kepler to search for new low-mass main-sequence eclipsing binaries and characterize intrinsic M dwarf activity. Recent studies of eclipsing low-mass stars, (which allow the determination of individual masses and radii to better than 1%), have shown that the radii of late-type dwarfs are consistently 10% larger than predicted by stellar models. The cause for this might be enhanced magnetic activity due to their binarity, and thus artificially enhanced rotation rates. If so, such an effect should diminish with increasing semi-major axis and thus period. Unfortunately, only a single known system has a period > 3 days, and thus this hypothesis cannot be tested. Additional eclipsing low-mass dwarfs, especially with long periods, are needed. Kepler is ideally suited to find these long-period systems, whereas ground-based surveys are cadence and/or magnitude limited. We present an optimal sample of 1,200 currently unobserved M dwarfs to monitor for eclipsing systems. We will use NMSU resources at Apache Point Observatory to obtain follow-up photometry and spectroscopy to determine the fundamental parameters of the components in each system in conjunction with the Kepler data. Additionally, we propose to study low-mass star rotation periods, flare rates, and spot cycles for all stars which turn out not to be binaries. In relation to the Kepler Mission and broader impacts, the knowledge of how the radii of low-mass stars depend on their intrinsic properties is critical to accurately determining the radii of transiting planets around such stars. As well, characterizing M dwarf variability at the mmag level is needed to understand how this variability affects planetary transit signatures over time in low-mass systems.
GROUND-BASED FOLLOW-UP AND LIGHT CURVE MODELING OF ECLIPSING BINARIES TO DETERMINE LIMB DARKENING EFFECTS FOR THE KEPLER BANDPASS
We will use NMSU facilities to obtain UBVRI light curves of Algols in the Kepler field of view to ascertain the limb darkening for the broad Kepler bandpass. As we show below, limb darkening strongly affects parameters extracted from exoplanet transits. The Kepler bandpass is very broad, and therefore the derived, mean limb darkening cannot be easily predicted. This is especially true given the fact that limb darkening for normal stars has been shown to be in error by +/-10 - 20%! We have a current program to derive the limb darkening effects for a sample of Algols with a large range of spectral types. To extrapolate our results to the Kepler bandpass we request observations of 15 Algols in the Kepler field-of-view, and support to observe these Algols using NMSU facilities. In this way we can combine our ongoing program on limb darkening measures for long period Algols, with one specifically tied to Algols observed with Kepler, to quantify the limb darkening in the Kepler bandpass.
UNLOCKING NEW DISCOVERY SPACE IN THE MICRO- AND MEGA-FLARE REGIMES ON LOW MASS STARS
Suzanne Hawley
Low mass stars with strong magnetic fields often exhibit energetic outbursts known as flares. Flares are believed to be caused by magnetic reconnection events in the coronae of these magnetically active stars. They occur on timescales of seconds to days, and span more than eight orders of magnitude in emitted energy. We propose to use the NASA Kepler satellite to monitor six low mass M dwarf stars at short (1 minute) cadence for two months each. The sample includes both early and late type M dwarfs, with both very active and relatively inactive magnetic fields. These data will improve the time sampling and duration of flare monitoring observations on this class of stars by more than a factor of 10, providing significantly improved sensitivity to both micro-flares (low energy) and mega-flares (high energy). These data will enable us to (a) sample equal energy flares across the M spectral sequence and therefore test the hypothesis that later type stars produce flares at a much higher rate but lower average energy compared to earlier type stars; (b) characterize the morphology of flare light curves, which are used to constrain the origin of flare emission; (c) provide the first investigation of the correlation between flare rates and underlying starspot coverage; and (d) determine the flaring properties of (relatively) inactive M dwarfs. A robust understanding of flares, including the morphology of their light curves and the rate at which flares of different energy occur is central to the understanding of the magnetic properties of cool stars. In addition, characterizing flare rates, energies and light curves is important for the interpretation of transient signals in surveys such as LSST and Pan-STARRS, and to predict the radiation environment of the habitable zones of exoplanets.
INVESTIGATING THE ORIGIN OF LSP VARIATIONS
Kenneth Hinkle
National Optical Astronomy Observatory
Long secondary period (LSP) variables are so named because they are late type giants with both long period variation and shorter period pulsation. While approximately 25 - 30% of all pulsating AGB stars show LSP behavior there is no known physical cause for the longer period. LSP variables are the only form of stellar variability that is not understood. However, LSP variables are known to obey a period-luminosity (P-L) relation. This limits the possible causes to two causes: binarity and pulsation. Strong arguments can be made against both binarity and radial pulsation. The remaining possibility is non-radial pulsation. While the long period mode fits this violates current interior models. We propose to use Kepler high precision photometry to look for higher order non-radial pulsation modes. Fourier analysis of the light curve should readily identify these modes. If found the techniques of asteroseismology will be applied. In the absence of non-radial pulsations, we will explore the detailed long term light curve to see if it agrees to high precision with models of ellipsoidal variations. Either the binary or the pulsation models allow interesting outcomes. The binary model involves near-planet sized companions with orbits evolved into a very specific configuration. The pulsation model is forbidden by present stellar interior models and will drive now understanding of stellar interior structure.
ABSOLUTE CALIBRATION OF KEPLER USING WHITE DWARFS
Jay Holberg
We propose to use newly discovered white dwarfs together with Cycle 1 white dwarfs in the Kepler Field to help establish the absolute flux calibration of the Kepler observed magnitudes. The technique employed uses synthetic photometry and the procedures described in Holberg & Bergeron (2006, AJ, 132, 1221), to place Kepler photometry on the Hubble Space Telescope photometric scale. This proposal will be of direct use to astronomers seeking to relate Kepler photometry to familiar astrophysical photometric scales.
FLUX EMERGENCE AND DIFFERENTIAL ROTATION TIMESCALES ON F-M TYPE STARS
Gaitee Hussain
We propose to study differential rotation and flux emergence timescales on 100+ stars, ranging in spectral type from late-F to M in the open cluster, NGC 6866. This cluster is young enough to contain a mixture of both slow and fast rotators. Theories suggest that rapidly rotating active low-mass stars have a different dynamo mechanism compared to slow rotators. We will characterize the activity levels of stars covering a wide range of spectral type and rotation rate. We will exploit the high precision of Kepler to measure surface flows and flux emergence timescales on stars at a range of activity levels and spectral types in order to gain further insight into angular momentum evolution and stellar magnetic activity. We will use custom software and techniques to match chromospheric and surface activity. Our findings will inform and test flux emergence models currently being developed for cool stars.
CHARACTERIZING THE TOP OF THE RED GIANT BRANCH USING KEPLER DROP-LIST STARS
Jason Jackiewicz
We propose to conduct a study of the internal properties of fifty-five stars located near the top of the red giant branch. All of these stars are currently on the Kepler drop list. Program stars have effective temperatures and surface gravities of less than 3600K and log(g) = 1.0 respectively. Project goals are 1) to quantify the range of pulsation spectra found in upper red giant branch stars, 2) to use state-of-the art FAMIAS software to determine the values of several key global and interior properties, and 3) to determine the decay rate (if any) of the observed pulsation modes. Parameters to be measured include masses, ages, metal contents, convective overshoot parameters, hydrogen contents, radii, surface rotations, and rotation profile. Many of the pulsation modes in red giants are thought to be unstable. Observational studies support this assertion, however, the measured decay times range from days to weeks, and even longer time frames are allowed. Since little is known about the long term stability of the oscillations in these stars, and the necessity of removing the longer term pulsations from our light curves, a full year of data is requested.
PHOTOMETRY OF A VARIABLE HOT SUBDWARF STAR IN NGC 6791
Steven Kawaler
Iowa State State University
We propose a one year observation of the unique hot blue star B4 in NGC 6791, one of only a handful of subdwarf B (sdB) stars known to exist in an old open cluster, and the only cluster sdB known to show photometric variability caused by binarity. The goal of these observations are twofold - we expect to observe nonradial pulsations in this star, and plan to study longer period variations caused by its binarity. The primary goal is to confirm our expectation that B4 should show nonradial pulsations, since its temperature and gravity place it within the instability region for g-mode sdB pulsators, where pulsations are seen in about 75% of the stars (Green et al. 2003). The discovery of a pulsator in a well-studied open cluster of known age and metallicity would provide new and unique probes of the pulsation mechanism for the pulsating sdB stars. Because of the faintness of the star, the time scale of the variations (periods of approximately 45 to 90 minutes) and the expected small amplitude of the pulsations, Kepler is the only instrument able to measure these oscillations to the degree of precision needed for asteroseismic analysis. Our secondary goal is based on the fact that this star is already known to be a low-amplitude (2%-9%) variable with a period of 0.8 (or 0.4) days, The proposed observations will provide a high signal-to-noise light curve for analysis of the binary system. From photometry alone, we will be able to constrain the orbital properties of the binary, and the mass and radius of the companion. Subdwarf B (sdB) stars belong to a class of stars that represent the post-helium core flash evolution of low mass stars. They lie at the extreme blue end of the horizontal branch (Teff ~ 25,000 - 35,000K), and are the remnant cores of stars that have experienced the core helium flash while on the RGB. They have extremely thin (and inert) hydrogen shells surrounding a core undergoing helium fusion. The mechanism(s) that produce these stars is/are currently unknown, though leading scenarios include mass transfer in a binary system. Single-star mechanisms have also been proposed and remain viable given the limitations of observables in these stars. Asteroseismic probes of this star, coupled with the additional constraints of cluster membership and the properties of the binary system, should provide important clues about the formation mechanism of the extremely hot subdwarf stars. Because this star is relatively faint (V=17.88, Kepler magnitude 18.27), published ground-based data are insufficient to establish the nature of the known variability or determine the properties of the binary system. Furthermore, ground-based data are insufficient to detect the shorter period variability expected for any pulsations. Only with an extended, uninterrupted time series can we answer these questions, and the Kepler spacecraft is the only instrument capable of providing the needed data. If it shows pulsation, B4 will be a uniquely valuable star - a nonradially pulsating star, in a close binary system, within a cluster. The binary nature will allow mass and perhaps radius determination, the presence in a cluster secures knowledge of its distance, age, and metallicity, and with these constraints the asteroseismology will be tightly constrained.
KEPLER'S DETAILED VIEW ON THE PULSATIONS OF THE RR LYRAE PROTOTYPE AND BLAZHKO STAR RR LYR
Katrien Kolenberg
Though the RR Lyrae stars have been studied for over a century now, several aspects of their pulsations remain ununderstood. An intriguing subclass consists of the stars showing the Blazhko effect, with light curves that are modulated on time scales of typically tens to hundreds of days. Despite numerous studies, the origin of these long-term cycles remains a mystery. Moreover, in several RR Lyrae stars glitches and short-term irregularities in the light curves have been observed. This phenomenon has never been studied in detail. RR Lyr, the eponym and prototype of the RR Lyrae stars, is one of the best studied stars of its class. It is also a well-known Blazhko star with a modulation period of about 39 days (Kolenberg et al. 2006). In photometry of the star spanning over a century, both the pulsation and the Blazhko cycle have shown variations that are too fast to be of an evolutionary nature. On top of this, short-term irregularities have also been reported in RR Lyr. The 33.5 days of photometry of RR Lyr gathered during Kepler's first roll showed the potential of the unprecedented accuracy of Kepler data. On the basis of these preliminary data we already detected previously unseen frequencies (Kolenberg et al. 2010). The nature of the newly detected frequencies and their connection to the Blazhko effect, as well as the small irregularities in the pulsation of RR Lyr, can only be investigated with short cadence data. This would be the first time such a study is undertaken, and no other instrument can explore these previously unseen aspects of the star's pulsation. We propose to observe RR Lyr with Kepler in short cadence during more than two complete modulation cycles (90 days). By observing RR Lyr itself, we will be able to study variations in the Blazhko cycle, and the nature of the additional observed frequencies and their stability. These observations will be a milestone in gaining a better understanding of the pulsations of RR Lyrae stars in general.A better understanding of the Blazhko effect and other deviations from strictly regular pulsation will improve RR Lyrae stars as distance scale calibrators and tracers of galactic history.
DETECTING THE RELATIVISTIC BEAMING EFFECT IN ECLIPSING BINARIES WITH KEPLER
Tsevi Mazeh
We propose to observe a set of known eclipsing binaries in the Kepler field, in order to detect a small periodic intensity modulation with the binary period, due to relativistic effect, never observed so far. The intensity modulation depends on the radial velocity of the two stars, and therefore can be used as photometric radial-velocity measurements, allowing to determine or at least constrain the binary masses.We expect the amplitude of the effect to be of the order of 100 ppm or more. We can detect this effect with 5 sigma significance for stars with non-periodic stellar jitter of 1000 ppm. We apply now for a modest set of eclipsing binaries, so we can establish the ability of Kepler to perform this novel kind of observations.
THE STRUCTURE AND GLOBAL PROPERTIES OF RED GIANT CLUMP STARS
Bernard McNamara
We propose to conduct a targeted study by using Kepler to measure the pulsation properties of 128 red clump stars over the one year period of cycle 2. Since the program stars were selected from the Kepler drop list, they are known to be highly variable. Stars in the red clump are the metal-rich counterparts to the horizontal branch stars. Using the tools of asteroseismology and Kepler light curves, the masses, radii, temperatures, and ages of these stars will be determined. Several interior giant star properties will also be measured. These include: composition gradients, core sizes, and the convective overshoot parameter. A secondary goal is to use Kepler light curves to quantify the pulsation lifetimes. Giant star oscillations are expected to be stochastically excited and then damped, but the damping time frame is disputed. Suggestions range from a few days to several weeks, but it could be much longer.
A CALIBRATION STUDY OF VARIABLE STARS IN THE KEPLER FIELD: CYCLE 2
Kenneth Mighell
We propose to do a calibration study of variable stars in the Kepler Field which will be enable us to produce enhanced data products that will support and extend the broad science goals of the Kepler mission. Our primary objective is to produce proper flux-calibrated astronomical-grade light curves for individual stars that will complement the detrended light curves produced by the Kepler data pipeline. Relying upon the planned calibration efforts of the Kepler Science Team, we plan to produce nearly time-continuous light curves which extend the planned current monthly time base differential light curves to at least a quarterly basis and possibly a time base covering the entire 3.5 year lifetime of the Kepler primary mission. These light curves will have a Y axis value of "Flux" (in ergs/sec) instead of "Relative Flux" (in electrons / cadence) as given in the standard Kepler detrended light curves that are delivered by the Multimission Archive at STScI. This extended time base capability will support Kepler mission efforts to characterize the nature of the host stars of detected planetary candidates; in particular we will be able to gain better insight to the nature of brightness fluctuations over days to months which might be caused by chromospheric activity.
HIGH-PRECISION KEPLER MONITORING OF ACTIVE GALACTIC NUCLEI
Richard Mushotzky
We propose to monitor 20 of the brightest AGN in the Kepler field (V = 11.0-18.7) to obtain the first AGN light curves that uniformly cover time scales of hours to months. Most AGN show significant optical variability on these time scales, which is connected to emission from the accretion disk and thus provides one of the few ways of to study the physics of accretion in these objects. For the 10^6 to 10^9 solar mass black holes thought to power most AGN, one expects time scales ranging from the light-crossing times of minutes to weeks to the thermal time scales of order months to years. Previous optical monitoring was unable to access the critical short time scales due to diurnal and weather-related interruptions and poor photometric repeatability. These uninterrupted, high-precision light curves will yield the first AGN optical power spectral density functions (PSDs) of comparable quality to those obtained in the X-rays. This will allow us to determine the overall shape of optical PSDs, and if they are like X-ray PSDs, we will be able to measure slopes to 0.02-0.1 and detect breaks indicative of a characteristic variability time scale indicative of light-crossing or dynamical time scales in the accretion disk. Based on what is known about the optical variability characteristics of AGN our simulations show that Kepler will represent a breakthrough in this area allowing the determination of precision PDSs for several 10s of AGN. This is directly connected to one of NASA key goals in astrophysics, understanding the nature of black holes and active galaxies.
ASTROMETRY OF STARS & GALAXIES IN THE KEPLER FIELD
Robert Olling
We propose to observe three distinct sets of objects: 1) ninety-one (91) stars that are likely to be within 100 parsec from the Sun, and which have not previously been identified as such [the SN sample], 2) nine (9) K Giant Stars in Kepler's field of view that are part of the SIM-Lite Grid-star Catalog [SGC], and 3) four hundred and sixty three (463) small, nucleated galaxies that we will use to define an Absolute Astrometric Reference system [AAR] and to determine astrometric accuracy. Our science and technical goals for these target groups are as follows. The SN sample will improve the census of stars in the solar neighborhood. Because of their relative proximity, these systems are well-suited for a Kepler-based astrometric search for stellar and sub-stellar companions. Through a study of their positions and motions, we expect to be able to find Brown Dwarfs and long period planets that can be added to the target lists for future missions in NASA's Exoplanet Exploration Program. The SN sample is unique in that it allows both astrometric and RV studies so that masses can be determined unambiguously. The SGC K-giant stars were selected by the SIM program on the presumption that they would not have measurable astrometric wobble. We will evaluate the astrometric and photometric stability of these systems for suitability as astrometric standards for the SIM-Lite mission. This study will be particularly relevant for NASA if the Astro2010 Decadal Committee gives SIM-Lite the go-ahead in NASA's Exoplanet Exploration Program. The AAR sample comprises galaxies as identified in the 2MASS extended source catalog with diameters not exceeding 10 arcsec. We select about ten such small galaxies for each of our stellar targets. As was done for the Lick Proper Motion programs, we will use these slightly extended sources as absolute astrometric standards. We will use methods developed by the VLBA astrometry community to assess the absolute astrometric errors. We will also monitor these galaxies for variability as might occur from low-level AGN activity. We will use the state-of-the art ePSF astrometric methodology as developed by Anderson and collaborators for undersampled point-sources as well as for slightly extended sources.
HIGH-PRECISION MASSES AND RADII OF LOW-MASS ECLIPSING BINARY STARS
Jerome Orosz
We propose to obtain Kepler light curves of 7 long-period low-mass eclipsing binary (EB) targets. By making high-precision observations during the eclipses of these binaries we aim to resolve the long standing discrepancy between the theoretical and observational mass-radius relations at the bottom of the main-sequence, namely that the observed radii of low-mass stars are up to 15% larger than predicted by structure models. It has been suggested that this discrepancy may be related to strong stellar magnetic fields, which are not properly accounted for in current theoretical models. All previously well-characterized low-mass main-sequence EBs have periods of a few days or less, and their components are therefore expected to be rotating rapidly as a result of tidal synchronization, thus generating strong magnetic fields. We hypothesize that the stars in the binaries with longer orbital periods, which are expected to have weaker magnetic fields, will better match the assumptions of theoretical stellar models. By employing Kepler's high-precision photometry we will be able to determine the radius of both components to within a fraction of percent, which thus far has not been done for any low-mass binary with periods longer than a few days.
KEPLER OBSERVATIONS OF MASS TRANSFER ACTIVITY IN DIRECT-IMPACT ALGOL-TYPE INTERACTING BINARY SYSTEMS
Geraldine Peters
We propose a combination of high and low cadence Kepler observations of seven direct-impact Algol-type binaries in the Kepler fields to study the physics of mass accretion in these interacting systems. Included are the identification of a hot accretion spot at the site of the gas stream impact and a determination of its size and longitude, a search for accretion- induced photospheric oscillations, and a search for micro-flaring that might result from variable shocks due to a clumpy gas stream. Since a splash from a direct impact and the radiative energy from hot spots can precipitate systemic mass loss, their existence influences the evolution of close binaries. We expect that a hot spot and micro-flaring will be visible only on the trailing hemisphere of the system. Oscillations should be global, but perhaps of an irregular nature on hemisphere experiencing the impact. Although we have a general understanding of how Algol systems are formed and their evolutionary state, little is known about the details of the mass accretion. We will investigate both short and long-term variability over many orbital cycles to identify unique light curve structure that will provide insight into the physics of mass transfer. Since observing time on the GALEX spacecraft has been approved for two of the systems, UZ Lyr and BR Cyg, we have the opportunity to acquire simultaneous UV and Kepler photometry that will aid in the modeling of mass transfer activity. The Kepler photometry will be analyzed with the latest version of the Wilson-Devinney light curve analysis program. The residual light will be analyzed using standard Fourier techniques. Frequencies found in the residuals will be interpreted with the aid of current asteroseismology software. The project addresses NASA's Strategic Subgoal 3D, Discover the origin, structure, evolution, and destiny of the universe, and search for Earth-like planets, as it will advance our understanding of the evolution of early-type close binary stars.
ECLIPSING BINARIES IN THE OLD OPEN CLUSTER NGC 6791
Ruth Peterson
Astrophysical Advances
We propose 73 photometrically-selected targets with V < 16.6 within 12' of the center of the old, metal-rich open cluster NGC 6791 for Kepler 30-min sequence observations. The goal is to detect eclipsing binaries suitable for determining the masses of the components, through future observations of radial velocities with large ground-based telescopes, and possibly of orbits with SIM. Our targets are giants and subgiants, not main-sequence stars, in order to reduce confusion in the Kepler field and to provide feasible targets for spectroscopy. Towards the center of the cluster, the high stellar densities dramatically increase crowding and cause binaries to be more readily perturbed. Consequently we are including many targets in the outer regions of the cluster, those which fall on the cluster color-magnitude and color-color diagrams defined by the inner members. We need a large target sample to isolate favorable binaries, as some stars will be non-members, only half of the members will be in binaries, many of these will have merged, and only a few of those remaining are useful. Suitable binary systems must not be triple, and should include a giant and a main-sequence turnoff star so that both components can be detected spectroscopically. The components must not have previously exchanged or lost mass. Binary periods must be nearly a year to a few years, so the orientation must be nearly edge-on and the eccentricity will be finite. We expect the proposed observations to yield at least two non-interacting binaries from which both component masses can be obtained. For such binaries, eclipse depths of 10% over a day or more are expected, and are readily apparent from applying standard filters to the pipeline light curves. Radial-velocity curves will be based on echelle spectroscopy analyzed with IRAF, as we have done in our decade-long survey of the brightest NGC 6791 giants with the Lick Hamilton echelle. The effective temperature, gravity, and metallicity of each of the stellar components will be found from theoretical spectral calculations, which now match such strong-lined stars reasonably well thanks to an updated list of line parameters. This work should stringently constrain comparisons of observed color-magnitude diagrams to produce meaningful cluster parameters. Such constraints would have major significance for the validation or refinement of stellar evolutionary tracks at high metallicity, and the derivation of age and metallicity from broadband colors of both individual stars and integrated spectra of old elliptical galaxies, for which NGC 6791 is a critical template.
ULTRA-HIGH PRECISION PHOTOMETRY OF OVERCONTACT BINARY STARS: FUNDAMENTAL PROPERTIES AND EVOLUTION
Andrej Prsa
This proposal focuses on the fundamental properties of overcontact binary stars -- short-period systems where the two main sequence components share a common envelope. Our understanding of formation, evolution and physical properties of overcontact binaries is incomplete, mostly due to the limited data accuracy. Kepler will alleviate this problem, allowing us not only to advance, but to essentially resolve the standing issues that have persisted in the field of close binaries for over 40 years. These are: 1) the formation of overcontact binaries. The two competing theories attribute the tightening of close binary orbits and subsequent coalescence to either a steady angular momentum loss due to tidal and rotational friction, or to interactions with the third body. Kepler's uninterrupted observations will establish the current angular momentum loss, which will enable us to turn back time and compute whether steady angular momentum loss could feasibly cause coalescence; 2) the dominant energy transport mechanism in overcontact binary envelopes. The current standing theory asserts that thermodynamic equilibrium is sustained by the so-called Thermal Relaxation Oscillation (TRO) cycle. In essence, one component overflows its Roche lobe, causing mass transfer on the other component. The transferred mass veils the component completely, blocking the flux, converting it to thermal energy and causing the increase in the radius of the veiled component. Once that component grows over its Roche lobe, the process is reversed. Lately, however, this model has been theoretically shaken by showing that the Coriolis force would cause veiling only in the equatorial regions of the star, thus enabling it to keep radiating energy through the polar regions. Kepler's photometric data accuracy will allow us to directly observe veiling: if only a band covers the star, there will be a discrete jump in its disk brightness at the band boundary, an effect routinely modeled in the field of eclipsing binaries. If such a jump is found, it will have proven that the TRO hypothesis cannot adequately describe the mechanism that sustains the thermodynamic equilibrium; 3) many overcontact binaries show evidence of geometric contact but not thermal contact. The data accuracy so far inhibited our ability to correlate the two, but with the promise that Kepler brings, physical parameters of overcontact binaries will be determined to a sufficient accuracy via modeling to formulate this correlation; 4) since most overcontact binaries show signs of chromospheric activity, we will be able to directly probe differential rotation and limb darkening of severely distorted stars; lastly, 5) we invested significant effort to reformulate the theoretical model backbone so that it withstands Kepler's data accuracy. Our model builds on the Roche constricted three body hypothesis, where stars are considered point sources, surrounded by a massless envelope. This approximation proved adequate for ground-based observations, but Kepler will put the extent of applicability of this model to the test. Any deviation will have strong implications on the eclipsing binary modeling in general. Our study will be based on a carefully selected sample of 50 overcontact binary stars in the Kepler field of different variation amplitudes and orbital periods. Two of those exhibit total eclipses, making them ideal astrophysical laboratories for the focused, in-depth study. For these we propose short cadence observations (half a year each, occupying a single short cadence channel overall), and for the remaining program stars we propose the long cadence mode. Analyzing this sample will not only answer the listed questions, it will also yield physical and geometrical properties of these stars of unprecedented accuracy in a uniform way. Our research team has extensive experience (both theoretical and observational) in eclipsing binary stars and we feel well positioned to conduct this research successfully.
UNDERSTANDING M-TYPE STARS AS EXOPLANET HOSTS: CHARACTERIZING VARIABILITY AT SHORT TIMESCALES
Ignasi Ribas
Institut de Ciencies de l'Espai
Exoplanet research has experienced an exponential growth over the past few years. This is both because of the impressive discoveries made recently and also because of the inherent appeal of the topic. One of the future challenges is the discovery and subsequent characterization of habitable exoplanets. Intensive efforts are being put into advancing towards this goal. A possible shortcut to find a potentially habitable planet is to carry out the searches around low-mass stars (M dwarfs). M-type exoplanet hosts have two major advantages: (1) Because of the lower stellar mass and luminosity, habitable planets are closer in, have shorter orbital periods and hence induce higher amplitude reflex motions on the star; and (2) because of the smaller radius, a transit of a terrestrial planet has a depth of a few per cent and therefore it is suitable for discovery and follow up even from the ground. The search for exoplanets around M-type stars is blooming with new experiments (like the MEarth transit search) and projects for the near future. A critical element to the advancement of this field is to attain a detailed characterization of the targets. This is chiefly because of the inherent stellar activity that affects M-type stars causing both photometric and radial velocity jitter. Such jitter is related to the overall light modulation induced by starspots and to the time variability of their position and properties. Surprisingly, the activity patterns of M-type stars are largely unknown at the required level for exoplanet investigations. We will utilize the Kepler data, with its exquisite precision and time coverage, to obtain the power spectrum of the variability of some 10 bright M-type stars in the Kepler field over timescales from minutes to months. We will investigate the photometric variations to understand the variability patterns (including starspots and other activity-related phenomena) and define the best strategy to mitigate their effect in photometric or spectroscopic transit searches from the ground. In addition, we will collect data that will be central to the missing overall characterization of M-type stars as potential exoplanet hosts.
DYNAMO PARAMETERS IN YOUNG SUNS: A KEPLER STUDY OF DIFFERENTIAL ROTATION AND ACTIVE REGION DECAY RATES IN NGC 6811 (1 GYR) AND NGC 6819 (2.5 GYR)
Steven Saar
The formation and evolution of a magnetic dynamo is an integral part of the evolution of low-mass stars and the basis for a wide variety of observable phenomena in such stars. Yet despite decades of observational and theoretical study, we do not have a predictive dynamo model even for the best studied case - the Sun. A limiting factor has been the difficulty of making observations that can properly constrain key physical properties, such as differential rotation and turbulent diffusivity, important to understand and model the stellar dynamo. We propose to take advantage of Kepler's superb photometry to measure differential rotation as well as the growth and decay rates of surface active regions (a proxy for diffusivity) for 235 known members of the open clusters NGC 6811 (1 Gyr) and NGC 6819 (2.5 Gyr). The proposed study will more than triple the existing differential rotation measurements for dwarf stars over a wide range of masses, with the added advantages of having fixed metallicity and well determined ages. This work will also yield the first extensive survey of the growth and decay rates in homogeneous samples of dwarfs. We will also explore the frequency of magnetic grand minima at younger ages. Our measurements will add important new constraints on magnetic dynamos in stars, permitting better, more physically realistic models.
AGE-SENSITIVE DETACHED ECLIPSING BINARIES IN OPEN STAR CLUSTERS NGC 6791 AND NGC 6819
Eric Sandquist
Age is difficult to measure extremely precisely for stars other than the Sun. In the field being observed by Kepler, the stars of the open clusters NGC 6791 and NGC 6819 are the ones that can be most precisely age-dated. However, different methods provide ages that differ significantly. We propose an effort to bring methods of stellar age determination into agreement through the use of Kepler data for these star clusters. Here we focus on the use of masses and sizes measured from weakly-interacting eclipsing binary star systems in the clusters. Massive stars run out of hydrogen fuel at their centers before less massive ones, and start to change rapidly in size - for such rapidly evolving stars, measurements of both mass and radius that are precise to 1% can lead to ages precise to 10% and better. Further, mass and radius measurements are conceptually simple to derive from observations, and avoid complicating effects like uncertainties in distance and reddening. High precision age measurements from this and other methods will make these star clusters important testbeds for models of stars and stellar populations in galaxies.
BRIGHTNESS VARIATIONS FROM THE ACCRETION DISK IN CH CYGNI
Jennifer Sokoloski
The aim of the proposed Kepler program is to determine whether the accretion disks around white dwarfs are fundamentally similar to those around neutron stars and black holes. We will accomplish this goal by generating the first power spectrum of optical brightness fluctuations from an accreting white dwarf to span a large enough range of frequencies to reveal all the features typically seen in the power spectra of X-ray fluctuations from X-ray binaries. Since this broad power spectrum will need to cover time scales ranging from less than a minute to months with an unprecedented level of sensitivity, we will combine long-cadence observations with the Kepler satellite with fast optical photometry from ground-based telescopes. As the accretion disks around white dwarfs primarily emit in the optical, whereas the accretion disks around neutron stars and black holes primarily emit in the X-rays, we will determine the degree to which white-dwarf and X-ray binary disks are similar by comparing the optical power spectrum of a specially selected accreting white dwarf --- CH Cygni--- to the well-studied X-ray power spectra of neutron-star and black-hole X-ray binaries. The accreting white dwarf in the symbiotic binary CH Cygni is ideal for this study because it has a long enough orbital period that there will be no confusion between brightness variations that are due to the behavior of the accretion disk and those that are due to the orbit of the binary. The power spectra of X-ray binaries (in particular low-mass X-ray binaries, or LMXBs) display a characteristic set of features. These features include quasi-periodic oscillations and broad components that can be fitted by Lorentzian functions. As the compact objects in LMXBs have weak or non-existent magnetic fields, these features in the power spectra are not due to magnetic accretion. Instead, they are thought to be related to the accretion disk itself, with possible connections to the dynamical, thermal, and viscous time scales at the inner edge of the disk. They have properties and relationships that hold whether the accreting compact object is a neutron star or stellar-mass black hole, and recent observations suggest that supermassive black holes and white dwarfs might also produce the same pattern of variations. If our Kepler observations confirm that the accretion onto an object for which general relativity is not needed to describe the trajectories of matter and radiation near its surface, such as a white dwarf, has the same variability properties as accretion onto a relativistic object such as a black hole or neutron star, there will be several major implications. The more than two decades of research on LMXB variability will become relevant to accreting white dwarfs, and the myriad studies of disks around white dwarfs in cataclysmic binaries will become relevant to LMXBs. Moreover, models for the X-ray variations from LMXBs that invoke general relativity, as the most popular models do, and the possibility of using the features in LMXB power spectra to probe strong gravity, will be called into question. Observing CH Cygni with Kepler is technically challenging because the source is very bright. However, we have made plans to use a custom aperture to overcome these difficulties. Tackling these technical challenges is worth the effort because achieving our science goal will have broad implications, and Kepler is the only instrument that can obtain a long enough continuous light curve with high enough sensitivity to accomplish this goal.
VARIABILITY OF AN AVERAGE SYMBIOTIC BINARY - STHA 169
We propose to determine whether or not the symbiotic star StHA 169 contains a hidden accretion disk by studying its sub-mmag-level flickering properties that only Kepler has the sensitivity to detect. Symbiotic stars are wide binaries in which a white dwarf (WD) accretes from a red-giant companion. It has proved difficult to determine exactly how mass is lost by the red giant (Roche-lobe overflow, spherical wind, or "focused wind"?) and how it is accreted by the white dwarf (via an accretion disk or direct impact?). Most cataclysmic variables (CVs, in which the mass donor is a Roche-lobe filling main-sequence star), on the other hand, are known to accrete via a disk. A variety of observational techniques (including eclipse mapping and Doppler tomography) have not only confirmed the presence of the disk beyond a shadow of a doubt in non-magnetic CVs, but also revealed the detailed physics of the disk. For symbiotic stars, ground-based fast optical photometry has revealed stochastic brightness variations (termed "flickering") at tens of percent level in a few objects. This phenomenon, which is routinely observed in CVs, is a well-known signature of the accretion disk; the power spectrum of this disk flickering has a characteristic steep powerlaw shape at high frequencies. However, this successful determination that accretion proceeds via an accretion disk has been limited to a small subset of symbiotic stars. In the vast majority, no flickering is seen with a typical upper limit of about a few tenth of a percent. Does this mean that the majority of symbiotics do not actually contain disks, contrary to simple expectations ? Another possibility is that symbiotic stars do contain disks, but that the amplitude of disk flickering is reduced due to the presence of some other constant source of light. In other words, the disks are hidden. We can distinguish between these possibilities by using the phenomenal sensitivity of Kepler to detect mmag-level and even sub-mmag-level flickering in an ordinary symbiotic star. Here we propose a one-month fast-cadence Kepler observation of an ordinary symbiotic star, StHA 169 (the only such system in the Kepler field-of-view), to detect and characterize its flickering. The Kepler sensitivity is sufficiently high that a non-detection flickering with a steep powerlaw power spectrum will imply that a disk does not exist in this system, and perhaps in many other ordinary symbiotic stars. Since symbiotic stars are known to be the progenitors of at least some type Ia supernovae, understanding how they accrete could also shed light on the generation of cosmologically important supernovae.
CATACLYSMIC VARIABLES IN THE KEPLER FIELD
Martin Still
NASA Ames Research Center
Cataclysmic variables provide the cleanest available natural laboratories to investigate the physical behaviour of accretion disks. The timing capabilities and sensitivity of Kepler are well matched to the timescales and amplitude of accretion disk variability in these sources. This combination provides an unprecedented opportunity to test and refine the paradigms of stellar accretion with high-precision, uniform data containing no diurnal or seasonal gaps. We propose a multi-faceted observational and modeling program that puts our current understanding of accretion disks to the test and has the potential to measure the spatial structure of model-dependent disk parameters. Kepler observations of cataclysmic variables will impact profoundly our understanding of accretion disk dynamics and the nature of astrophysical viscosity. The proposed observations will provide an outstanding astrophysical legacy for the Kepler mission.
CHANDRA-SELECTED X-RAY SOURCES IN THE KEPLER FIELD
The Kepler mission has a finite lifetime. *If* there is no mission extension in 2012, there will be only three Guest Observer cycles before the spacecraft is switched off. We expect the Kepler archive to provide a rich heritage but the onus is upon the community to choose Kepler targets now that maximize the impact of Kepler in the future. There are many ways to attack target selection, but the one we propose here is to add new Kepler targets to the observing list that have been X-ray selected. Based upon the ROSAT all-sky survey, the Kepler field contains thousands of X-ray sources. The majority of these have an undetermined nature but experience suggests that the sample is comprised mostly of magnetically active stars, accreting stars and background quasars. All such sources would be premium targets for an instrument with Kepler's strengths - uniform cadence, long uninterrupted data sequences and high photometric precision. We propose a conservative study in cycle 2 of the best-localized, unidentified X-ray sources from the Chandra Source Catalog, with the potential goal of expanding the survey greatly in cycle 3.
TEMPORAL ANALYSIS OF RV TAURI AND SEMI-REGULAR VARIABLES USING KEPLER
Donald Walter
RV Tauri stars are luminous, supergiant variables with periods of pulsation that are sometimes predictable and sometimes not. Their lightcurves show alternating deep and shallow minima with a primary period of variability in the range of 30-150 days while their spectra vary across several spectral types. Semi-regular (SR) variables show some periodicity, but are even less regular than RV Tauri stars. RV Tauri and other SR variables occupy the region of the HR Diagram between the Cepheid instability strip to the left and the long period Mira types to the right. The evolutionary status of these objects is uncertain and an adequate explanation of the changes in their spectra and light curves is lacking. The presence of a number of RV Tauri stars in the Local Group of galaxies and their potential use in distance calculations adds cosmological significance to better understanding their luminosities and other characteristics. Studies to date are constrained by the limitations of ground-based data from AAVSO and the literature (e.g. Pollard et. al., 1996, MNRAS, 279, 949). Our own, modeling efforts (Cash et. al, 2009, AIP Conference Proceedings, CP1170, 146) include curve-fitting of the AAVSO data using Fourier and other methods to determine the periods of pulsation in the light curves and to examine the stability of the calculated periods. We propose to use Kepler to observe approximately 15 of these objects in its field of view through several of the stellar phase cycles over a time span of 12 months. Using Kepler's long cadence exposures of 30 minutes will provide unprecedented temporal detail and photometric precision. In order to provide insight into the underlying physical processes of these stars, we will combine the Kepler photometry with our modeling techniques and ~800 high signal-to-noise, archival spectra we have taken at the Coude-Feed telescope at Kitt Peak National Observatory over the past decade. This proposed research is relevant to the stated objective of the solicitation for the acquisition and analysis of new data that uses the high-precision photometry of Kepler for asteroseismology and other variability studies of Galactic sources. This in turn fits NASA's mission to pioneer the future in scientific discovery, in particular the Astrophysics Division's Focus Area for Stars that includes understanding how stars form and evolve. The NASA Strategic Plan and Goals for 2006-2016 include Sub-goal 3D to which this proposal is relevant "Discover the origin, structure, evolution, and destiny of the universe, and search for Earth-like planets."
UNDERSTANDING BLAZAR VARIABILITY THROUGH KEPLER
Ann Wehrle
Space Science Institute
We propose to monitor four flat spectrum radio quasars (blazars) and one powerful radio galaxy, Cygnus A, to search for variability on timescales comparable to the light crossing time of the accretion disk around the central supermassive black hole and the base of the relativistic jet. We want to see if some optical variability in quasars is due to a bright feature in the accretion disk as it approaches the last stable orbit, or if it is due to inhomogeneities in the jet, possibly in a helical structure. When the quasars are in quiescent, faint states, a quasi-periodic light curve indicates an accretion disk origin, and provides a dynamical means of measuring a lower limit to the mass of the supermassive black hole which may be compared to those derived by other methods, such as the shape of X-ray iron K$\alpha$ lines and stellar velocity dispersions. When the quasars are in bright states, then long-lived quasi-periodic oscillations (QPOs) are very probably from helical features in the jets, but if several different short-lived QPOs are seen in one quasar, then the emission is probably coming from turbulence behind a shock. If during a faint state, instead of QPOs, we detect aperiodic variations, including high and low breaks in the power spectrum density (PSD), then we may obtain the physical scales of the inner and outer edges of accretion disks and hence the BH mass. Aperiodic variations during a high state, with breaks in the PSD, could yield the smallest and largest physical scales corresponding to light travel times, modulo the Doppler factor, in the relativistic jet. Kepler is ideally suited to the necessary measurements by delivering highly stable photometry continuously on timescales from minutes to days.
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doi: 10.3934/mcrf.2020052
Optimal control of a non-smooth quasilinear elliptic equation
Christian Clason , , Vu Huu Nhu and Arnd Rösch
Faculty of Mathematics, University Duisburg-Essen, Thea-Leymann-Strasse 9, 45127 Essen, Germany
* Corresponding author: Christian Clason.
Received October 2018 Revised December 2018 Published December 2020
Fund Project: This work was supported by the DFG under the grants CL 487/2-1 and RO 2462/6-1, both within the priority programme SPP 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization".
This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regularization scheme, we derive C- and strong stationarity conditions. Under the additional assumption that the nonlinearity is a $ PC^1 $ function with countably many points of nondifferentiability, we show that both conditions are equivalent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples.
Keywords: Optimal control, non-smooth optimization, optimality system, quasilinear elliptic equation.
Mathematics Subject Classification: 49K20, 49J52, 49M15.
Citation: Christian Clason, Vu Huu Nhu, Arnd Rösch. Optimal control of a non-smooth quasilinear elliptic equation. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020052
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Figure 1. constructed exact solution for $ \alpha = 10^{-7} $, $ \beta = 0.85 $
Table 1. numerical results. number of Newton iterations and relative errors for state $\bar y$ and adjoint $\bar w$ in dependence of $n_h$, $\alpha$, and $\beta$
$n_h$ $\alpha$ $\beta$ $\frac{\| y_h - \bar y\|_{H^1_0(\Omega)}}{\|\bar y\|_{H^1_0(\Omega)}}$ $\frac{\| w_h - \bar w\|_{H^1_0(\Omega)}}{\|\bar w\|_{H^1_0(\Omega)}}$ #SSN $\|y_d \|_{L^\infty(\Omega)} $
100 1·10−6 0.8 3.27·10−3 2.92·10−2 2 2.07·10−4
1000 1·10−6 0.8 3.36·10−4 3.24·10−3 3 2.07·10−4
(A) dependence on $n_h$
800 1·10−8 0.8 2.32·10−5 2.19·10−3 25 2.03·10−4
(B) dependence on $\alpha$
(C) dependence on $\beta$
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Christian Clason Vu Huu Nhu Arnd Rösch
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CommonCrawl
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\begin{document}
\begin{frontmatter}
\title{Crack occurrence in bodies with gradient polyconvex energies}
\author{Martin Kru\v{z}\'{\i}k*, Paolo Maria Mariano**, Domenico Mucci***}
\address{*Czech Academy of Sciences, Institute of Information Theory and Automation\\
Pod Vod\'{a}renskou v\u{e}\v{z}\'{\i} 4,
CZ-182 00 Prague 8,
Czechia\\e-mail: [email protected]}
\address{**DICEA, Universit\`{a} di Firenze\\via Santa Marta 3, I-50139 Firenze, Italy\\e-mail: [email protected], [email protected]}
\address{***DSMFI, Universit\`{a} di Parma\\ Parco Area delle Scienze 53/A, I-43134 Parma, Italy \\ e-mail: [email protected]}
\begin{abstract}
Energy minimality selects among possible configurations of a continuous body with and without cracks those compatible with assigned boundary conditions of Dirichlet-type. Crack paths are described in terms of curvature varifolds so that we consider both \textquotedblleft phase" (cracked or non-cracked) and crack orientation. The energy considered is gradient polyconvex: it accounts for relative variations of second-neighbor surfaces and pressure-confinement effects. We prove existence of minimizers for such an energy. They are pairs of deformations and varifolds. The former ones are taken to be $SBV$ maps satisfying an impenetrability condition. Their jump set is constrained to be in the varifold support.
\end{abstract}
\begin{keyword}
Fracture, Varifolds, Ground States, Shells, Microstructures, Calculus of Variations
\end{keyword}
\end{frontmatter}
\section{Introduction} Deformation-induced material effects involving interactions beyond those of first-neighbor-type can be accounted for by considering, among the fields defining states, higher-order deformation gradients. In short, we can say that these effects emerge from \emph{latent} microstructures, intending those which do not strictly require to be represented by independent (observable) variables accounting for small-spatial-scale degrees of freedom. Rather they are such that \textquoteleft though its effects are felt in the balance equations, all relevant quantities can be expressed in terms of geometric quantities pertaining to apparent placements' \cite[p. 49]{C85}. A classical example is the one of Korteweg's fluid: the presence of menisci in capillarity phenomena implies curvature influence on the overall motion; it is (say) measured by second gradients \cite{K1} (see also \cite{DS} for pertinent generalizations). In solids length scale effects appear to be non-negligible for sufficiently small test specimens in various geometries and loading programs; in particular, when plasticity occurs in poly-crystalline materials, such effects are associated with grain size and accumulation of both randomly stored and geometrically necessary dislocations \cite{FMAH94}, \cite{FH}, \cite{Gud}.
These higher-order effects influence possible nucleation and growth of cracks because the corresponding hyperstresses enter the expression of Hamilton-Eshelby's configurational stress \cite{M17}. Here we refer to this kind of influence. We look at energy minimization and consider a variational description of crack nucleation in a body with second-gradient energy dependence. We do not refer to higher order theories in abstract sense (see \cite{DS} for a general setting, \cite{C85} for a physical explanations in terms of microstructural effects, \cite{M17} for a generalization of \cite{DS} to higher-order complex bodies), rather we consider a specific energy, in which we account for the gradient of surface variations and confinement effects due to the spatial variation of volumetric strain. Specifically, the energy we consider reads as
\begin{equation} \begin{aligned} {\mathcal{F}}(y,V;\mathcal{B}):=&\int_{\mathcal{B}} \hat W\bigl( {\Greekmath 0272} y(x),{\Greekmath 0272}[\mathrm{cof}{\Greekmath 0272} y(x)],{\Greekmath 0272}[\det {\Greekmath 0272} y(x)]\bigr)\,dx\\ &+\bar{a}\m_V(\mathcal{B}) +\int_{{\mathcal{G}}_2({\mathcal{B}})}a_{1}\Vert A\Vert^{\ol p}\,dV+a_{2}\Vert\pa V \Vert\,, \end{aligned} \end{equation}
with $\mathcal{B}$ a fit region in the three-dimensional real space, $\bar{a}$, $a_{1}$, and $a_{2}$ positive constants, $y:\mathcal{B}\longrightarrow \tilde{\mathbb{R}}^{3}$, a special bounded variation map, a deformation that preserves the local orientation and is such that its jump set is contained in the support over $\mathcal{B}$ of a two-dimensional varifold $V$ with boundary $\partial V$ and generalized curvature tensor $A$. Such a support is a $2$-rectifiable subset of $\mathcal{B}$ with measure $\m_V(\mathcal{B})$. We identify such a set with a possible crack path, and the terms
$$\bar{a}\m_V(\mathcal{B}) +\int_{{\mathcal{G}}_2({\mathcal{B}})}a_{1}\Vert A\Vert^{\ol p}\,dV+a_{2}\Vert\pa V \Vert$$
represent a modification of the traditional Griffith energy \cite{Gri20}, which is just $\bar{a}\m_V(\mathcal{B})$ (i.e., it is proportional to the crack lateral surface area), so they have a configurational nature. The energy density $\hat{W}$ is assumed to be gradient polyconvex, according to the definition introduced in reference \cite{BKS}.
We presume that a minimality requirement for ${\mathcal{F}}(y,V;\mathcal{B})$ selects among cracked and free-of-crack configurations. To this aim we prove an existence theorem for such minima under Dirichlet-type boundary conditions; minimizing deformations satisfy also a condition allowing contact of distant body boundary pieces but avoiding self-penetration. This is the main result of this paper.
\section{Physical insight}
\subsection{Energy depending on ${\Greekmath 0272}[\mathrm{cof}(\cdot)]$: a significant case}
The choice of allowing a dependence of the energy density $\hat{W}$ on ${\Greekmath 0272}[\mathrm{cof}{\Greekmath 0272} y]$ has physical ground: we consider an effect due to relative variations of neighboring surfaces. Such a situation occurs, for example, in gradient plasticity. We do not tackle directly its analysis here, but in this section we explain just its geometric reasons.
In periodic and quasi-periodic crystals, plastic strain emerges from dislocation motion through the lattice \cite{Phill01}, such phenomenon includes meta-dislocations and their approximants in quasi-periodic lattices \cite{FH11}, \cite{M19}. In polycristalline materials, dislocations cluster at granular interstices obstructing or favoring the re-organization of matter. In amorphous materials other microstructural rearrangements determining plastic (irreversible) strain occur. Examples are creation of voids, entanglement and disentanglement of polymers.
At macroscopic scale, the one of large wavelength approximation, a traditional way to account indirectly for the cooperative effects of irreversible microscopic mutations is to accept a multiplicative decomposition of the deformation gradient, commonly indicated by $F$, into so-called \textquotedblleft elastic", $F^{e}$, and \textquotedblleft plastic", $F^{p}$ factors \cite{Kr60}, \cite{L69}, namely $F=F^{e}F^{p}$, which we commonly name the \emph{ Kr\"{o}ner-Lee decomposition}. The plastic factor $F^{p}$ describes rearrangements of matter at a low scale, while $F^{e}$ accounts for macroscopic strain and rotation.
In such a view, the plastic factor $F^{p}$ indicates through its time-variation just how much (locally) the material goes far from thermodynamic equilibrium transiting from an energetic well to another, along a path in which the matter rearranges irreversibly. In the presence of quasi-periodic atomic arrangements, as in quasicrystals, such a viewpoint requires extension to the phason field gradient \cite{LRT}, \cite{M06}.
Here, we restrict the view to cases in which just $F$ and its decomposition play a significant role: they include periodic crystals, polycrystals, even amorphous materials like cement or polymeric bodies, when we neglect at a first glance direct representation of the material microstructure in terms of appropriate morphological descriptors to be involved in Laundau-type descriptions coupled with strain.
With $\mathcal{B}$ a reference configuration for the body under scrutiny, at every its point $x$, the plastic factor $F^{p}$ maps the tangent space of $\mathcal{B}$ at $x$ into a linear space not otherwise specified, except assigning a metric $g_{\mathfrak{L}}$ to it---indicate such a space by $\mathfrak{L}_{F^{p}}$. Then, $F^{e}$ transforms such a space into the tangent space of the deformed configuration.
In general, the plastic factor $F^{p}$ allows us to describe an incompatible strain, so its curl does not vanish, i.e., $\mathrm{curl}F^{p}\neq 0$, unless we consider just a single crystal in which irrecoverable strain emerges from slips along crystalline planes. So, the condition $\mathrm{curl}F^{p}\neq 0$, which may hold notwithstanding $\mathrm{curl}F= 0$, does not allow us to sew up one another the linear spaces $\mathfrak{L}_{F^{p}}$, varying $x\in\mathcal{B}$, so we cannot reconstruct an intermediate configuration, with the exception of a single crystal behaving as a deck of cards, parts of which can move along slip planes. Of course, $\mathrm{curl}F^{p}= 0$ when $F^{e}$ reduces to the identity.
In modeling elastic-perfectly-plastic materials in large strain regime, we usually assume that the free energy density ${\Greekmath 0120}$ has a functional dependence on state variables of the type ${\Greekmath 0120}:=\tilde{{\Greekmath 0120}}(x,F,F^{p})$. Further assumptions are listed below.
\begin{itemize}
\item \emph{Plastic indifference}, which is invariance under changes in the reference shape, leaving unaltered the material structure (\emph{material isomorphims}); formally it reads as
\begin{equation*}
\tilde{{\Greekmath 0120}}(x,F,F^{p})=\tilde{{\Greekmath 0120}}(x,FG,F^{p}G),
\end{equation*}
for any orientation preserving unimodular second rank tensor $G$ mapping at every $x$ the tangent space $T_{x}\mathcal{B}$ of $\mathcal{B}$ at $x$ onto itself (the requirement $\mathrm{det}G=1$ ensures mass conservation along changes in reference configuration).
\item \emph{Objectivity}: invariance with respect to the action of $SO(3)$ on the physical space; it formally reads
\begin{equation*}
\tilde{{\Greekmath 0120}}(x,F,F^{p})=\tilde{{\Greekmath 0120}}(x,QF,F^{p}),
\end{equation*}
for any $Q\in SO(3)$. \end{itemize}
Plastic indifference implies $\tilde{{\Greekmath 0120}}(x,F,F^{p})=\hat{{\Greekmath 0120}}(x,F^{e})$. Then, objectivity requires $\hat{{\Greekmath 0120}}(x,F^{e})=\hat{{\Greekmath 0120}}(x,\tilde{C}^{e})$, with $\tilde{C}^{e}$ the right Cauchy-Green tensor $\tilde{C}^{e}=F^{e\mathrm{T}}F^{e}$, where $\tilde{C}^{e}=g_{\mathfrak{L}}^{-1}C^{e}$, with $C^{e}:=F^{e\ast}\tilde{g}F^{e}$ the pull-back in $\mathfrak{L}_{F^{p}}$ of the metric in $\mathcal{B}_{a}$ (the asterisk denotes formal adjoint, which coincides with the transpose when the metrics involved are flat). However, plastic indifference implies also $\tilde{{\Greekmath 0120}}(x,F,F^{p})=\hat{{\Greekmath 0120}}(x,F^{e}, \bar{g})$, where $ \bar{g}:=F^{p-\ast}gF^{p-1}$ is at each $x$ push-forward of the material metric $g$ onto the pertinent intermediate space $\mathfrak{L}_{F^{p}}$ through $F^{p}$. Since by the action of $G$ over the reference space $g$ becomes $G^{\ast}gG$, we get $\bar{g}=F^{p-\ast}gF^{p-1}\overset{G}\longrightarrow(F^{p-\ast}G^{-\ast})G^{\ast}gG(G^{-1}F^{p-1})=\bar{g}$.
To account for second-neighborhood effects, we commonly accept the free energy density to be like $\hat{{\Greekmath 0120}}(x,F^{e}, D_{{\Greekmath 010B}}F^{e})$ or $\hat{{\Greekmath 0120}}(x,F^{e}, \bar{g}, D_{{\Greekmath 010B}}F^{e})$, with ${\Greekmath 010B}$ indicating that the derivative is computed with respect to coordinates over $\mathfrak{L}_{F^{p}(x)}$.
We claim here that \emph{this choice---i.e., the presence of $D_{{\Greekmath 010B}}F^{e}$ in the list of state variables---is related to the possibility of assigning energy to oriented area variations of neighboring staking faults when $\mathrm{det}F^{p}=1$}.
To prove the statement, first consider that the second-rank minors of $F^{p}$, collected in $\mathrm{cof} F^{p}$, govern at each point $x$ the variations of oriented areas from the reference shape to the linear intermediate space associated with the same point. Neighboring staking faults determine such variations in the microstructural arrangements collected in what we call plastic flows. Since $\mathrm{det}F^{p}>0$, linear algebra tells us that $\mathrm{cof} F^{p}=(\mathrm{det}F^{p})F^{p-\ast}$, where $-\ast$ indicates adjoint of $F^{p-1}$.
Consequently, assigning energy to area variations due to first-neighbor staking faults, we may take a structure for the free energy as
\begin{equation*} {\Greekmath 0120}:=\tilde{{\Greekmath 0120}}(x,F,F^{p},^{\urcorner}D\mathrm{cof} F^{p}), \end{equation*}
where $D$ indicates the spatial derivative with respect to $x$, and the apex $^{\urcorner}$ indicates minor left adjoint operation of the first two indexes of a third order tensor (it corresponds to the minor left transposition when the metric is flat or the first two tensor components are both covariant or contravariant). At least in the case of volume-preserving crystal slips over planes ($\mathrm{det}F^{p}=1$), we have $\mathrm{cof} F^{p}=F^{p-\ast}$, whence we can write in operational form $D\mathrm{cof} F^{p}=F^{p-\ast}\otimes D$ so that $^{\urcorner}D\mathrm{cof} F^{p}=F^{p-1}\otimes D$. Under the action of $G$, describing a change in the reference shape, as above, we have $F^{p-\ast}\otimes D\overset{G}\longrightarrow ((GF^{p-1})^{\ast}\otimes D)G$. Consequently, for volume-preserving plastic flows, the requirement of \emph{plastic invariance} reads
\begin{equation*} \tilde{{\Greekmath 0120}}(x,F,F^{p},F^{p-1}\otimes D)=\tilde{{\Greekmath 0120}}(x,FG,F^{p}G,((G^{-1}F^{p-1}))\otimes DG) \end{equation*}
for any choice of $G$ with $\mathrm{det}G=1$. The condition implies
\begin{equation*} \begin{aligned} \tilde{{\Greekmath 0120}}(x,F,F^{p},F^{p-\ast}\otimes D)&=\tilde{{\Greekmath 0120}}(x,FF^{p-1}, \bar{g},((FF^{p-1})\otimes D)F^{p-1})\\ &=\tilde{{\Greekmath 0120}}(x,FF^{p-1}, \bar{g},(DF^{e})F^{p-1})=\hat{{\Greekmath 0120}}(x,F^{e}, \bar{g}, D_{{\Greekmath 010B}}F^{e}), \end{aligned} \end{equation*}
which concludes the proof.
Alternatively, if we choose \begin{equation*} {\Greekmath 0120}:=\tilde{{\Greekmath 0120}}(x,F,F^{p},D\mathrm{cof} F^{p}), \end{equation*}
with the same argument as above we get
\begin{equation*} \tilde{{\Greekmath 0120}}(x,F,F^{p},D\mathrm{cof} F^{p})=\hat{{\Greekmath 0120}}(x,F^{e}, \bar{g}, D_{{\Greekmath 010B}}F^{e\ast}). \end{equation*}
In our analysis here the density $\hat{W}$ is less intricate than $\tilde{{\Greekmath 0120}}(x,F,F^{p},D\mathrm{cof} F^{p})$, however, the analysis of its structure indicates a fruitful path for dealing with more complex situations.
Finally, from now on we just assume flat metrics so that we write ${\Greekmath 0272}$ instead of $D$, which appears to indicate the weak derivative of special bounded variation functions, a measure indeed. Also, we refer just to $F$ and do not consider the plasticity setting depicted by the multiplicative decomposition. Despite this, our choice of considering the gradient of $\mathrm{cof}F$ among the entries of $\tilde{W}$ is intended as an indicator of relative surface variation effects. Also, as already mentioned, the dependence of $\tilde{W}$ on ${\Greekmath 0272}\mathrm{det}F$ is a way of accounting for confinement effects due to non-homogeneous volume variations (see \cite{BMM} for a pertinent analysis in small strain regime).
Explanations a part are necessary for justifying the representation of cracks in terms of varifold, which are special vector-valued measures.
\subsection{Cracks in terms of varifolds}
Take a reference configuration $\mathcal{B}$ of a body that can be cracked, and a set of its infinitely many copies differing one another just by a possible crack path, each a $\mathcal{H}^{2}$-rectifiable set. In this reference picture, each crack path can be considered fictitious, i.e., the projection over $\mathcal{B}$ of the real crack occurring in the deformed shape; in other words, it can be considered as a shadow over a wall. Assigned boundary conditions, a question can be whether a crack may occur so that the deformed configuration is in one-to-one correspondence with at least one of the infinitely many reference configurations just depicted.
We may imagine of giving an answer by taking an expression of the energy including both bulk and crack components, asking its minimality as a criterion of selecting among configurations with or without cracks. This is what has been proposed in reference \cite{FM98} taking Griffith's energy \cite{Gri20} as the appropriate functional. This minimality criterion is also a first step to approximate a cracking process \cite{FM98}. To this aim we may select a finite partition of the time interval presuming to go from the state at instant $k$ to the one at $k+1$ by minimizing the energy. In principle, the subsequent step should be computing the limit as partition interval goes to zero. This path rests on De Giorgi's notion of minimizing movements \cite{DeG93}.
In the minimum problem, deformation and crack paths are the unknowns. A non-trivial difficulty emerges: in three dimensions we cannot control minimizing sequences of surfaces. A way of overcoming the difficulty is to consider as unknown just the deformation taken, however, in the space of those special functions with bounded variations, which are orientation preserving. We give their formal definition in the next section. Here, we just need to know that they admit a jump set with non-zero $\mathcal{H}^{2}$ measure. Once found minima of such a type, we identify the crack path with the deformation jump set \cite{DMT002}. Although such a view is source of nontrivial analytical problems and pertinent results \cite{DMT002}, it does not cover cases in which portions of the crack margins are in contact but material bonds across them are broken. To account for these phenomena, we need to recover the original proposal in reference \cite{FM98}, taking once again separately deformations and crack paths. However, the problem of controlling minimizing sequences of surfaces or more irregular crack paths reappears. A way of overcoming it is to select minimizing sequences with bounded curvature because this restriction would avoid surface blow up. This is the idea leading to the representation of cracks in terms of varifolds.
Take $x\in\mathcal{B}$, the question to be considered is not only whether $x$ belongs to a potential crack path or not but also, in the affirmative case, what is the tangent (even in approximate sense) of the crack there, among all planes $\Pi$ crossing $x$. Each pair $(x,\Pi)$ can be viewed as a typical point of a fiber bundle $\mathcal{G}_{k}(\mathcal{B})$, $k=1,2$, with natural projector ${\Greekmath 0119}:\mathcal{G}_{k}(\mathcal{B})\longrightarrow\mathcal{B}$ and typical fiber ${\Greekmath 0119}^{-1}(x)=\mathcal{G}_{k,3}$ the Grassmanian of 2D-planes or straight lines associated with $\mathcal{B}$. A $k$-varifold over $\mathcal{B}$ is a non-negative Radon measure $V$ over the bundle $\mathcal{G}_{k}(\mathcal{B})$ \cite{Alm65}, \cite{All72}, \cite{All75}, \cite{Mant96}. For the sake of simplicity, here we consider just $\mathcal{G}_{2}(\mathcal{B})$, avoiding one-dimensional crack in a $3D$-body. The generalization to include $1D$ cracks is straightforward. Itself, $V$ has a projection ${\Greekmath 0119}_\#V$ over $\mathcal{B}$, which is a Radon measure over $\mathcal{B}$, indicated for short by ${\Greekmath 0116}_{V}$. Specifically, we may consider varifolds supported by $\mathcal{H}^{2}$-rectifiable subsets of $\mathcal{B}$, i.e., by potential crack paths. We look at those varifolds admitting a certain notion of generalized curvature (its formal definition is in the next section) and parametrize through them the set of infinitely many reference configurations described above. Rather than sequences of cracks, we consider sequences of varifolds. The choice allows us to avoid the problem of controlling sequences of surfaces but forces us to include the varifold and its curvature in the energy, leading (at least in the simplest case) to a variant of Griffith's energy augmented by
$$\int_{{\mathcal{G}}_2({\mathcal{B}})}a_{1}\Vert A\Vert^{\ol p}\,dV+a_{2}\Vert\pa V \Vert$$
with respect to the traditional term just proportional to the surface crack area, namely $\bar{a}\m_V(\mathcal{B})$. Such a view point has been introduced first in references \cite{GMMM10} and \cite{M10} (see also \cite{GMM10}).
The discussion in this section justifies a choice of a energy functional like $\mathcal{F}(y,V;\mathcal{B})$, indicated above, which we analyze in the next sections.
\section{Background analytical material}
\subsection{Some notation} For $G:{\mathbb{R}}^n\to{\mathbb{R}}^N$ a linear map, where $n\geq 2$ and $N\geq 1$, we indicate also by $ G=(G^j_i)$, ${j=1,\ldots,N}$, $i=1,\ldots n $, the $(N\tim n)$-matrix representing $G$ once we have assigned bases $(e_1,\ldots,e_n)$ and $(\e_1,\ldots,\e_N)$ in ${\mathbb{R}}^n$ and ${\mathbb{R}}^N$, respectively.
For any ordered multi-indices $\a$ in $\{1,\ldots,n\}$ and
$\b$ in $\{1,\ldots,N\}$ with length $|\a|=n-k$ and $|\b|=k$, we denote by $G^{\b}_{\ol\a}$ the $(k\tim k)$-submatrix of $G$ with rows $\b=(\b_1,\ldots,\b_k)$ and columns $\ol\a=(\ol\a_1,\ldots,\ol\a_k)$, where $\ol\a$ is the element which complements $\a$ in $\{1,\ldots,n\}$, and $0\leq k\leq\ol n:=\min\{n,N\}$.
We also denote by
$$M^\b_{\ol\a}(G):= \det G^{\b}_{\ol\a}$$
the determinant of $G^{\b}_{\ol\a}$\,, and set $M^0_{0}(G):=1$. Also, the Jacobian $|M(G)|$ of the graph map $x\mapsto (Id\join G)(x):=(x,G(x))$ from ${\mathbb{R}}^n$ into ${\mathbb{R}}^n\tim{\mathbb{R}}^N$ satisfies
\begin{equation}\label{MG}
|M(G)|^2:=\sum_{|\a|+|\b|=n}M^\b_{\ol\a}(G)^2. \end{equation}
\subsection{Currents carried by approximately differentiable maps}
Let $\O\sb{\mathbb{R}}^n$ be a bounded domain, with ${\mathcal{L}}^n$ the pertinent Lebesgue measure. For $u:\O\to {\mathbb{R}}^N$ an ${\mathcal{L}}^n$-a.e. approximately differentiable map, we denote by ${\Greekmath 0272} u(x)\in{\mathbb{R}}^{N\tim n}$ its approximate gradient at a.e. $x\in\O$. The map $u$ has a \emph{Lusin representative} on the subset $\widetilde \O$ of Lebesgue points pertaining to both $u$ and ${\Greekmath 0272} u$. Also, we have ${\mathcal{L}}^n(\O\sm\widetilde\O)=0$.
In this setting, we write $u\in{\mathcal{A}}^1(\O,{\mathbb{R}}^N)$ if
\begin{itemize}
\item ${\Greekmath 0272} u\in L^1(\O,{\mathbb{M}}^{3\times 2})$ and
\item $M^\b_{\ol\a}({\Greekmath 0272} u)\in L^1(\O)$ for any ordered multi-indices $\a$ and
$\b$ with $|\a|+|\b|=n$. \end{itemize}
\par The {\em graph} ${\mathcal{G}}_ u$ of a map $u\in{\mathcal{A}}^1(\O,{\mathbb{R}}^N) $ is defined by
\begin{equation*} {\mathcal{G}}_ u:=\Set{ (x,y) \in \O\times {\mathbb{R}}^N \mid x\in \widetilde\O\,,\ y=\widetilde{u}(x)}, \end{equation*}
where $\widetilde{u}(x) $ is the Lebesgue value of $u$. It turns out that ${\mathcal{G}}_u$ is a countably $n$-rectifiable set of $\O\tim{\mathbb{R}}^N$, with ${\mathcal{H}}^n({\mathcal{G}}_u)<\infty$. The approximate tangent $n$-plane at $(x,\widetilde u(x))$ is generated by the vectors ${\mathbf{t}}_i(x)=(e_i,\pa_i u(x))\in{\mathbb{R}}^{n+N}$, for $i=1,\ldots,n$, where the partial derivatives are the column vectors of the gradient matrix ${\Greekmath 0272} u$, and we take ${\Greekmath 0272} u(x)$ as the Lebesgue value of ${\Greekmath 0272} u$ at $x\in\widetilde\O$.
The unit $n$-vector
$$ {\Greekmath 0118}(x):=\frac{{\mathbf{t}}_1(x)\wedge {\mathbf{t}}_2(x)\wedge\cdots\wedge {\mathbf{t}}_n(x)}{|{\mathbf{t}}_1(x)\wedge {\mathbf{t}}_2(x)\wedge\cdots\wedge {\mathbf{t}}_n(x)|} $$
provides an orientation to the graph ${\mathcal{G}}_u$.
\par For ${\mathcal{D}}^k(\O\tim{\mathbb{R}}^N)$ \emph{the vector space of compactly supported smooth $k$-forms in} $\O\tim{\mathbb{R}}^N$, and ${\mathcal{H}}^k$ the $k$-dimensional Hausdorff measure, one defines the current $G_u$ carried by the graph of $u$ through the integration of $n$-form on ${\mathcal{G}}_u$, namely
$$ \langle G_u,\o \rangle:=\int_{{\mathcal{G}}_u}\langle \o,{\Greekmath 0118}\rangle\,d{\mathcal{H}}^n\,,\qquad \o\in {\mathcal{D}}^n(\O\tim{\mathbb{R}}^N),$$
where $\langle ,\rangle$ indicates the duality pairing. Consequently, by definition $G_{u}$ is an element of the (strong) dual of the space ${\mathcal{D}}^n(\O\tim{\mathbb{R}}^N)$. Write ${\mathcal{D}}_n(\O\tim{\mathbb{R}}^N)$ for such a dual space. Any element of it is called a \emph{current}.
By writing $U$ for a open set in $\mathbb{R}^{n+N}$, we define \emph{mass} of $T\in{\mathcal{D}}_k(U)$ the number
$${\mathbf{M}}(T):=\sup\{\langle T,\o\rangle\mid \o\in{\mathcal{D}}^k(U)\,,\,\,\Vert\o\Vert\leq 1\} $$
and call a {\em boundary} of $T$ the $(k-1)$-current $\partial T$ defined by
$$\langle \pa T,\y\rangle := \langle T,d\y \rangle, \qquad \y\in {\mathcal{D}}^{k-1}(U),$$
where $d\y$ is the differential of $\y$.
A {\em weak convergence} $T_h\rightharpoonup T$ in the sense of currents in ${\mathcal{D}}_k(U)$ is defined through the formula
$$ \lim_{h\to\ii}\langle T_{h},\o \rangle = \langle T,\o \rangle\qquad\fa\,\o\in {\mathcal{D}}^k(U)\,.$$
If $T_h\rightharpoonup T$, by lower semicontinuity we also have $${\mathbf{M}}(T)\leq\liminf_{h\to\ii}{\mathbf{M}}(T_h)\,.$$
With these notions in mind, we say that $G_u$ is an \emph{integer multiplicity} (in short i.m.) \emph{rectifiable current} in ${\mathcal{R}}_n(\O\tim{\mathbb{R}}^N)$, with finite mass ${\mathbf{M}}(G_u)$ equal to the area ${\mathcal{H}}^n({\mathcal{G}}_u)$ of the $u$-graph.
According to \eqref{MG}, since the Jacobian $|M({\Greekmath 0272} u)|$ of the graph map $x\mapsto (Id\join u)(x)=(x,u(x))$ is equal to
$|{\mathbf{t}}_1(x)\wedge {\mathbf{t}}_2(x)\wedge\cdots\wedge {\mathbf{t}}_n(x)|$, by the area formula
$$ \langle G_u,\o \rangle= \int_\O (Id\join u)^\#\o=\int_\O \langle\o(x,u(x)),M({\Greekmath 0272} u(x))\rangle\, dx $$ for any $\o\in {\mathcal{D}}^n(\O\tim{\mathbb{R}}^N)$, so that
$$ {\mathbf{M}}(G_u)={\mathcal{H}}^n({\mathcal{G}}_u)=\int_\O|M({\Greekmath 0272} u)|\,dx < \infty\ . $$
\par If $u$ is of class $C^2$, the Stokes theorem implies
$$ \langle \pa G_u,\y\rangle=\langle G_u,d\y\rangle=\int_{{\mathcal{G}}_u}d\y=\int_{\partial {\mathcal{G}}_u}\y=0 $$
for every $\y\in {\mathcal{D}}^{n-1}(\O\times {\mathbb{R}}^N)$, i.e., the null-boundary condition
\begin{equation}\label{bdryzero} (\pa G_u)\pri\O\tim{\mathbb{R}}^N=0\,. \end{equation}
Such a property \eqref{bdryzero} holds true also for Sobolev maps $u\in W^{1,\ol n}(\O,{\mathbb{R}}^N)$, by approximation. However, in general, the boundary $\pa G_u$ does not vanish and may not have finite mass in $\O\tim{\mathbb{R}}^N$. On the other hand, if $\pa G_u$ has finite mass, the boundary rectifiability theorem states that $\pa G_u$ is an i.m. rectifiable current in ${\mathcal{R}}_{n-1}(\O\tim{\mathbb{R}}^N)$.
An extended treatment of currents is in the two-volume treatise \cite{GMS98}.
\subsection{Weak convergence of minors} Let $\{u_h\}$ be a sequence in ${\mathcal{A}}^1(\O,{\mathbb{R}}^N)$.
\par Take $N=1$, i.e., consider real-valued maps $u$. Suppose also to have in hands sequences $\{u_h\}$ and $\{{\Greekmath 0272} u_h\}$ such that $u_h\to u$ strongly in $L^1(\O)$ and ${\Greekmath 0272} u_h\rightharpoonup v$ weakly in $L^1(\O,{\mathbb{R}}^n)$, where $u\in L^1(\O)$ is an a.e. approximately differentiable map and $v\in L^1(\O,{\mathbb{R}}^n)$. In general, \emph{we cannot conclude} that $v={\Greekmath 0272} u$ a.e. in $\O$. The question has a positive answer provided that $\{u_h\}$ is a sequence in $W^{1,1}(\O)$.
Notice that, when $N=1$, the membership of a function $u\in{\mathcal{A}}^1(\O,{\mathbb{R}})$ to the Sobolev space $W^{1,1}(\O)$ is equivalent to the null-boundary condition \eqref{bdryzero}.
\par When $N\geq 2$, assume that $u_h\to u$ strongly in $L^1(\O,{\mathbb{R}}^N)$, with $u$ some a.e. approximately differentiable $ L^1(\O,{\mathbb{R}}^N)$ map. Presume also that $M^\b_{\overline\a}({\Greekmath 0272} u_h)\rightharpoonup v^\b_{\ol\a}$ weakly in $L^1(\O)$, with $v^\b_{\ol\a}\in L^1(\O)$, for every multi-indices $\a$ and $\b$, with $\vert\a\vert+\vert\b\vert=n$.
A \emph{sufficient condition} ensuring that $v^\b_{\ol\a}=M^\b_{\ol\a}({\Greekmath 0272} u)$ a.e. is again the validity of equation \eqref{bdryzero} for each $u_h$.
\par We can weaken such a condition by requiring a mass control on $G_{u_h}$ boundaries of the type
\begin{equation}\label{MbdGuk} \sup_h{\mathbf{M}}((\pa G_{u_h})\pri\O\tim{\mathbb{R}}^N)<\ii\,, \end{equation}
as stated by Federer-Fleming's closure theorem \cite{FF}, which refers to sequences of graphs $G_{u_h}$ which have equi-bounded masses, $\sup_h{\mathbf{M}}(G_{u_h})<\ii$ and satisfy the condition \eqref{MbdGuk} \cite[Vol.~I, Sec.~3.3.2]{GMS98}.
\begin{theorem}\label{TclosG}{\bf (Closure theorem).} Let
$\{u_h\}$ be a sequence in ${\mathcal{A}}^1(\O,{\mathbb{R}}^N)$ such that
$u_h\to u$ strongly in $L^1(\O,{\mathbb{R}}^N)$ to an a.e. approximately differentiable map $u\in
L^1(\O,{\mathbb{R}}^N)$. For any multi-indices $\a$ and $\b$ with $\vert\a\vert+\vert\b\vert=n$, assume
$$M^\b_{\ol\a}({\Greekmath 0272} u_h)\rightharpoonup v^\b_{\ol\a}\qquad{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{weakly in }}L^1(\O), $$
with $v^\b_{\ol\a}\in L^1(\O)$.
If the bound $\eqref{MbdGuk}$ holds, the inclusion $u\in{\mathcal{A}}^1(\O,{\mathbb{R}}^N)$ holds
and, for every $\a$ and $\b$,
\begin{equation}\label{vabe} v^\b_{\ol\a}(x)=M^\b_{\ol\a}({\Greekmath 0272}
u(x))\qquad {\mathcal{L}}^n{\mx{-a.e in }}\O\,. \end{equation}
Moreover, we find $G_{u_h}\rightharpoonup G_u$ weakly in ${\mathcal{D}}_n(\O\tim{\mathbb{R}}^N)$, and also
$$\begin{array}{rl}{\mathbf{M}}(G_u) \leq &\liminf\limits_{h\to\ii}{\mathbf{M}}(G_{u_h})<\ii \\
{\mathbf{M}}((\pa G_{u})\pri\O\tim{\mathbb{R}}^N)\leq &\liminf\limits_{h\to \ii}{\mathbf{M}}((\pa G_{u_h})\pri\O\tim{\mathbb{R}}^N)<\ii\,. \end{array}$$
\end{theorem}
\subsection{Special functions of bounded variation} A summable function $u\in L^1(\O)$ is said to be \emph{of bounded variation} if the distributional derivative $Du$ is a finite measure in $\O$.
Such a function $u$ is approximately differentiable ${\mathcal{L}}^n$-a.e. in $\O$. Its approximate gradient ${\Greekmath 0272} u$ agrees with the Radon-Nikodym derivative density of $Du$ with respect to ${\mathcal{L}}^n$.
Then, the decomposition $Du={\Greekmath 0272} u\,{\mathcal{L}}^n+D^su$ holds true, where the component $D^su$ is singular with respect to ${\mathcal{L}}^n$.
Also, the \emph{jump set} $S(u)$ of $u$ is a countably $(n-1)$-rectifiable subset of $\O$ that agrees ${\mathcal{H}}^{n-1}$-essentially with the complement of $u$ Lebesgue's set.
If, in addition, the singular component $D^su$ is concentrated on the jump set $S(u)$, we say that $u$ is a {\em special function of bounded variation}, and write in short $u\in SBV(\O)$.
\par A vector valued function $u:\O\to{\mathbb{R}}^N$ belongs to the class $SBV(\O,{\mathbb{R}}^N)$ if all its components $u^j$ are in $SBV(\O)$. In this case, $Du={\Greekmath 0272} u\,{\mathcal{L}}^n+D^su$, where the approximate gradient ${\Greekmath 0272} u$ belongs to $L^1(\O,{\mathbb{R}}^{N\tim n} )$, and the jump set $S(u)$ is defined component-wise as in the scalar case, so that $D^su=(u^+-u^-)\otimes\n{\mathcal{H}}^{n-1}\pri S(u)$, where $\n$ is an unit normal to $S(u)$ and $u^\pm$ are the one-sided limits at $x\in S(u)$. Therefore, for each Borel set $B\subset\Omega$ we get
$$ |Du|(B)=\int_B|{\Greekmath 0272} u|\,dx+\int_{ B\cap S(u)} |u^+ -u^- | \, d{\mathcal{H}}^{n-1}\,. $$
\par Compactness and lower semicontinuity results hold in $SBV$. The treatise \cite{AFP00} offers an accurate analysis of $SBV$ landscape.
Here, we just recall that the compactness theorem in \cite{A2} relies on a generalization of the following characterization of $SBV$ functions with ${\mathcal{H}}^{n-1}$-rectifiable jump sets.
According to reference \cite{ABG}, we denote by ${{\mathcal{T}}}(\O\tim{\mathbb{R}})$ the class of $C^1$-functions \,$\vf(x,y)$\, such that \,$\vert\vf\vert+\vert D\vf\vert$\, is bounded and the support of $\vf$\, is contained in $K\tim{\mathbb{R}}$ for some compact set $K\sb\O$.
\begin{proposition}\label{PABG} Take $u\in BV(\O)$. Then, $u\in SBV(\O)$, with
${\mathcal{H}}^{n-1}(S(u))<\ii$, if and only if for every $i=1,\ldots,n$
there exists a Radon measure $\m_i$ in $\O\tim{\mathbb{R}}$ such
that
$$
\int_\O\Bigl(\frac{\pa\vf}{\pa
x_i}(x,u(x))+ \frac{\pa\vf}{\pa
y}(x,u(x))\,\pa_i{u}(x)\Bigr)\,dx=
\int_{\O\tim{\mathbb{R}}}\vf\,d\m_i $$
for any \,$\vf\in{{\mathcal{T}}}(\O\tim{\mathbb{R}})$. In this case, we have
$$\m_i= -(Id\join u^+)_\#(\n_i{\mathcal{H}}^{n-1}\pri
S(u))+(Id\join u^-)_\#(\n_i{\mathcal{H}}^{n-1}\pri S(u))\,. $$ \end{proposition}
\par As a consequence, we infer that if a sequence $\{u_h\}\in{\mathcal{A}}^1(\O,{\mathbb{R}}^N)$ satisfies
$$\sup_h\Bigl( \Vert u_h\Vert_\ii+\int_\O|M({\Greekmath 0272} u_h)|^p\,dx\Bigl)<\ii \,,\qquad p>1$$
and the boundary mass bound \eqref{MbdGuk}, the inclusion $\{u_h\}\in SBV(\O,{\mathbb{R}}^N)$ and the $SBV$ compactness theorem hold. In fact, by Proposition~\ref{PABG} we get
$${\mathcal{H}}^{n-1}\pri S(u_h)\leq {\Greekmath 0119}_\#|\pa G_{u_h}|(B)\qquad\fa\,h $$
where ${\Greekmath 0119}:\O\tim{\mathbb{R}}^N\to\O$ is the projection onto the first $n$ coordinates, and $|\cdot|$ the total variation, so that ${\Greekmath 0119}_\#|\pa G_u|(B)=|\pa G_u|(B\tim{\mathbb{R}}^N)$ for each Borel set $B\sb\O$.
\subsection{Generalized functions of bounded variation}
When the bound $\sup_h\Vert u_h\Vert_\ii<\ii$ fails, the SBV compactness theorem cannot be applied. This happens, e.g., if $u_h={\Greekmath 0272} y_h$ for some sequence $\{y_h\}\sb W^{1,p}(\O)$. When such sequences play a role in the problems analyzed, we find it convenient to call upon {\em generalized special functions of bounded variation}, the class of which is commonly denoted by $GSBV$.
To define them, first write $SBV_{\textrm{loc}}(\O)$ for functions $v:\O\to{\mathbb{R}}$ such that $v_{\vert K}\in SBV(K)$ for every compact set $K\subset\O$.
\begin{definition} A function $u:\O\to{\mathbb{R}}^N$ belongs to the class $GSBV(\O,{\mathbb{R}}^N)$ if ${\Greekmath 011E}\circ u\in SBV_{\textrm{loc}}(\O)$ for every ${\Greekmath 011E}\in C^1({\mathbb{R}}^N)$ with the support of ${\Greekmath 0272}{\Greekmath 011E}$ compact. \end{definition}
The following compactness theorem holds.
\begin{theorem}\label{TGSBV} Let $\{u_h\}\sb GSBV(\O,{\mathbb{R}}^N)$ be such that
$$ \sup_h \Bigl( \int_\O \bigl(|u_h|^p+|{\Greekmath 0272} u_h|^p\bigr)\,dx +{\mathcal{H}}^{n-1}(S_{u_h})\Bigr)<\ii$$
for some real exponent $p>1$. Then, there exists a function $u\in GSBV(\O,{\mathbb{R}}^N)$ and a (not relabeled) subsequence of $\{u_h\}$ such that $u_h\to u$ in $L^p(\O,{\mathbb{R}}^N)$, ${\Greekmath 0272} u_h\rightharpoonup {\Greekmath 0272} u$ weakly in $L^p(\O,{\mathbb{R}}^{N\tim n})$, and ${\mathcal{H}}^{n-1}\pri S({u_h})$ weakly converges in $\O$ to a measure $\m$ greater than ${\mathcal{H}}^{n-1}\pri S(u)$. \end{theorem}
\subsection{Curvature varifolds with boundary} We now turn to the physical dimension $n=3$ and denote by ${\mathcal{B}}$ a connected bounded domain in ${\mathbb{R}}^3$ with surface-like boundary that can be oriented by the outward unit normal to within a finite number of corners and edges. In this setting, we take the deformation as a map $y:\mathcal{B}\longrightarrow\tilde{\mathbb{R}^{3}}$, where $\tilde{\mathbb{R}^{3}}$ is a isomorphic copy of $\mathbb{R}^{3}$, the isomorphism given by the identification. Such a distinction is necessary for example when we consider changes in observers (which are frames on the entire ambient space) leaving invariant the reference configuration, which is $\mathcal{B}$ in this case.
\begin{definition} A general 2-varifold in ${\mathcal{B}}$ is a non-negative Radon measure on the trivial bundle ${\mathcal{G}}_2({\mathcal{B}}):={\mathcal{B}}\times {\mathcal{G}}_{2,3}$, where ${\mathcal{G}}_{2,3}$ is the Grassmanian manifold of $2$-planes $\Pi$ through the origin in ${\mathbb{R}}^{3}$. \end{definition}
\par If $\fk{C}$ is a 2-rectifiable subset of ${\mathcal{B}}$, for ${\mathcal{H}}^2\pri\fk{C}$ a.e. $x\in {\mathcal{B}}$ there exists the approximate tangent $2$-space $T_{x}\fk{C}$ to $\fk{C}$ at $x$. We thus denote by $\Pi(x)$ the $3\tim 3$ matrix that identifies the orthogonal projection of ${\mathbb{R}}^3$ onto $T_{x}\fk{C}$ and define
\begin{equation}\label{rem} V_{\fk{C},\t}(\vf) :=\int_{{\mathcal{G}}_{2}({\mathcal{B}})}\vf(x,\Pi) \, dV_{\fk{C},\t}(x,\Pi):= \int_{\fk{C}}\t(x) \vf(x,\Pi(x)) \, d{\mathcal{H}}^2(x) \end{equation}
for any $\vf \in C_{c}^{0}({\mathcal{G}}_2({\mathcal{B}}))$, where $\t\in L^1(\fk{C},{\mathcal{H}}^2) $ is a nonnegative density function. If $\t$ is integer valued, then $V=V_{\fk{C},{\Greekmath 0112} }$ is said to be the {\em integer rectifiable varifold} associated with $(\fk{C},\t,{\mathcal{H}}^2)$.
\par The {\em weight measure} of $V$ is the Radon measure in ${\mathcal{B}}$ given by $\m_V:={\Greekmath 0119}_\# V$, where ${\Greekmath 0119}:{\mathcal{G}}_2({\mathcal{B}})\to{\mathcal{B}}$ is the canonical projection. Then, we have $\m_V=\t\,{\mathcal{H}}^2\pri\fk{C}$ and call
$$\Vert V\Vert:=V({\mathcal{G}}_2({\mathcal{B}}))=\m_V({\mathcal{B}})=\int_{\fk{C}}\t\, d{\mathcal{H}}^2\,$$
a \emph{mass} of $V$.
\begin{definition}\label{Dvarif} An integer rectifiable
$2$-varifold $V=V_{\fk{C},{\Greekmath 0112} }$ is called a \emph
{curvature $2$-varifold with boundary} if
there exist a function
$A\in L^{1}({\mathcal{G}}_2({\mathcal{B}}),{\mathbb{R}}^{3*}\otimes{\mathbb{R}}^3\otimes
{\mathbb{R}}^{3*})$, $A=(A^{\ell i}_j)$, and a ${\mathbb{R}}^3$-valued measure $\pa V$ in ${\mathcal{G}}_2({\mathcal{B}})$ with finite mass $\Vert\pa V\Vert$,
such that
\begin{equation*}
\int_{{\mathcal{G}}_2({\mathcal{B}})}(\Pi D_x\vf + A D_{\Pi} \vf + \vf \RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }^{t}\mathrm{tr}\,(A I)) \, dV(x,\Pi)
= - \int_{{\mathcal{G}}_{2}({\mathcal{B}})} \vf \, d\pa V(x,\Pi)
\end{equation*}
for every $\vf\in C_{c}^{\ii}({\mathcal{G}}_2({\mathcal{B}}))$.
Moreover, for some real exponent $\ol p>1$, the subclass of curvature $2$-varifolds with boundary
such that $|A|\in L^{\ol p}({\mathcal{G}}_2({\mathcal{B}}))$ is indicated by
$CV^{\ol p}_2({\mathcal{B}})$. \end{definition}
\par Varifolds in $CV^{\ol p}_2({\mathcal{B}})$ have generalized curvature in $L^{\ol p}$ \cite{Mant96}. Therefore, Allard's compactness theorem applies (see \cite{All72}, \cite{All75}, but also \cite{Alm65}):
\begin{theorem}\label{Tcomp1}
For $1<{\ol p}<\ii$, let $\set{V^{(h)}}\sb CV^{\ol p}_2({\mathcal{B}})$ be a sequence of
curvature $2$-varifolds $V^{(h)}=V_{\fk{C}_{h},{\Greekmath 0112}_{h}}$ with
boundary. The corresponding curvatures and boundaries are
indicated by $A^{(h)}$ and $\pa
V^{(h)}$, respectively. Assume that there exists a real constant $c>0$ such that for
every $h$
\begin{equation*}
\m_{V^{(h)}}({\mathcal{B}}) +
\Vert\pa V^{(h)}\Vert + \int_{{\mathcal{G}}_{2}({\mathcal{B}})} |A^{(h)}|^{\ol p}\, dV^{(h)}\le c.
\end{equation*}
Then, there exists a (not relabeled) subsequence of $\set{V^{(h)}}$
and a $2$-varifold $V=V_{\fk{C},{\Greekmath 0112}}\in CV^{\ol p}_2({\mathcal{B}})$, with
curvature $A$ and boundary $\pa V$, such that
\begin{equation*}
V^{(h)} \rhu V,\quad A^{(h)}\,dV^{(h)}\rhu A\,dV,
\qquad \pa V^{(h)} \rhu \pa V,
\end{equation*}
in the sense of measures. Moreover, for any convex and lower semicontinuous function
$f:{\mathbb{R}}^{3*}\otimes{\mathbb{R}}^3\otimes{\mathbb{R}}^{3*}\to [0,+\ii]$, we get
\begin{equation*}
\int_{{\mathcal{G}}_2({\mathcal{B}})} f(A)\, dV \le \liminf_{h\to\ii}
\int_{{\mathcal{G}}_2({\mathcal{B}})}f( A^{(h)}) \, dV^{(h)}.
\end{equation*} \end{theorem}
\subsection{Gradient polyconvexity} According to references \cite{BKS,KPS,KR}, we take a continuous function
$$\hat W:{\mathbb{R}}^{3\tim 3}\tim{\mathbb{R}}^{3\tim 3\tim 3}\tim{\mathbb{R}}^3\to(-\infty,+\infty], $$
and we set $\hat W=\hat W(G,\D_1,\D_2)$.
We assume also existence of four real exponents $p,q,r,s$ satisfying the inequalities
\begin{equation}\label{exp} p>2\,,\quad q\geq{p\over p-1}\,,\quad r>1\,,\quad s>0 \end{equation}
and a positive real constant $c$ such that for every $(G,\D_1,\D_2)\in{\mathbb{R}}^{3\tim 3}\tim{\mathbb{R}}^{3\tim 3\tim 3}\tim{\mathbb{R}}^3$ the following estimates holds:
$$ \hat W(G,\D_1,\D_2)\geq c\,\bigl( |G|^p+|\mathrm{cof}G|^q+(\det G)^r+(\det G)^{-s}+|\D_1|^q+|\D_2|^r\bigr) $$
if $\det G>0$, and $\hat W(G,\D_1,\D_2)=+\infty$ if $\det G\leq 0$.
\begin{definition}\label{Dgradpoly} With ${\mathcal{B}}\subset{\mathbb{R}}^3$ the domain already described, consider the functional
$$ J(F;\mathcal{B}):=\int_{\mathcal{B}} \hat W\bigl( F(x),{\Greekmath 0272}[\mathrm{cof}F(x)],{\Greekmath 0272}[\det F(x)]\bigr)\,dx $$
defined on the class of integrable functions $F:{\mathcal{B}}\to{\mathbb{R}}^{3\tim 3}$ for which the approximate derivatives ${\Greekmath 0272}[\mathrm{cof}F(x)]$, ${\Greekmath 0272}[\det F(x)]$ exist for ${\mathcal{L}}^3$-a.e. $x\in{\mathcal{B}}$ and are both integrable functions in ${\mathcal{B}}$. Then, $J(F;\mathcal{B})$ is called {\em gradient polyconvex} if the integrand $\hat W(G,\cdot,\cdot)$ is convex in
${\mathbb{R}}^{3\tim 3\tim 3}\tim{\mathbb{R}}^3$ for every $G\in {\mathbb{R}}^{3\tim 3}$. \end{definition}
\par To assign the Dirichlet condition, we assume that $\G_0\cup\G_1$ is an ${\mathcal{H}}^2$-measurable partition of the ${\mathcal{B}}$ boundary such that ${\mathcal{H}}^2(\G_0)>0$. For some given measurable function $y_0:\G_0\to{\mathbb{R}}$, we consider the class $$ \begin{array}{r} \hat{{\mathcal{A}}}_{p,q,r,s}:=\{ y\in W^{1,p}({\mathcal{B}},{\mathbb{R}}^3) \mid \mathrm{cof}{\Greekmath 0272} y \in W^{1,q}({\mathcal{B}},{\mathbb{R}}^{3\tim 3})\,,\,\, \det{\Greekmath 0272} y\in W^{1,r}({\mathcal{B}})\,, \quad\quad \\
\det{\Greekmath 0272} y>0\,\,{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{a.e. in }}{\mathcal{B}}\,,\,\, (\det{\Greekmath 0272} y)^{-1}\in L^s({\mathcal{B}})\,,\,\, y=y_0 \,\,{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{on }}\G_0 \}\,, \end{array} $$
where $p,q,r,s$ satisfy the inequalities (\ref{exp}).
\par The following existence result has been proven in reference \cite{BKS} (see also \cite{KPS}).
\begin{theorem}\label{Tgradpoly} Under the previous assumptions, if the class $\hat{{\mathcal{A}}}_{p,q,r,s}$ is non-empty and
$ \inf\{ J({\Greekmath 0272} y;\mathcal{B})\mid y\in \hat{{\mathcal{A}}}_{p,q,r,s}\}<\infty $,
the functional $y\mapsto J({\Greekmath 0272} y;\mathcal{B})$ attains a minimum in ${\mathcal{A}}_{p,q,r,s}$. \end{theorem}
\section{Gradient polyconvex bodies with fractures}
We now look at an energy modified by the introduction of a varifold, through which we parametrize possible fractured configurations with respect to the reference one. Specifically, we consider a curvature varifold with boundary: $V\in CV^{\ol p}_2({\mathcal{B}})$. The choice implies a fracture energy modified with respect to the Griffith one. In fact, the latter is just proportional to the crack area, which implies considering material bonds of spring-like type. The additional presence in our case of the generalized curvature tensor implies, instead, considering beam-like material bonds for which bending effects play a role. In a certain sense, the energy we propose is a regularization of the Griffith one, since we require that the coefficient in front of the curvature tensor square modulus does not vanish.
In this setting, we look for minimizing deformations that are bounded and may admit a jump set contained in the varifold support. We cannot assume the deformation $y$ to be a Sobolev map. More generally we require $y\in SBV({\mathcal{B}},{\mathbb{R}}^3)$. The main issue in proving existence is recovering the weak convergence of minors. To achieve it we look at the approximate gradient and exploit Federer-Fleming's closure theorem as in Theorem~\ref{TclosG}. On the other hand, since some properties as the bound $\Vert\mathrm{cof}{\Greekmath 0272} y\Vert_\ii<\ii$ fails to hold, we assume $\mathrm{cof}{\Greekmath 0272} y$ to be in the class $GSBV$, with jump set controlled by the varifold support. In this way we recover the weak continuity of the approximate gradients ${\Greekmath 0272}[\mathrm{cof}{\Greekmath 0272} y_h]$ along minimizing sequences.
Our existence result below could be generalized to the case in which the crack path is described by a stratified family of varifolds in the sense introduced in references \cite{GMMM10} and \cite{M10} (see also \cite{GMM10}). This choice had been made we could have assigned additional curvature-type energy to the crack tip, taking possibly into account energy concentrations at tip corners, when the tip is not smooth. Also, we could describe the formation of defects with codimension 2 in front of the crack tip, specifically dislocations nucleating in front of the tip (see \cite{GMM10}). However, for the sake of simplicity, we restrict ourselves to the choice of a single varifold, avoiding to foresee an additional tip energy and also corner energies.
Consequently, we consider the energy functional
$$ {\mathcal{F}}(y,V;\mathcal{B}):=J({\Greekmath 0272} y;\mathcal{B})+ {\mathcal{E}}(V;\mathcal{B})\,, $$
where $F\mapsto J(F;\mathcal{B})$ is the functional in Definition~\ref{Dgradpoly}, and
$$ {\mathcal{E}}(V;\mathcal{B}):=\bar{a}\m_V(\mathcal{B})+\int_{{\mathcal{G}}_2({\mathcal{B}})}a_{1}\Vert A\Vert^{\ol p}\,dV+a_{2}\Vert\pa V \Vert\,, $$
with $\bar{a}$, $a_{1}$, and $a_{2}$ positive constants.
The couples deformation-varifold are in the class ${\mathcal{A}}_{\ol p,p,q,r,s,K,C}$ defined below.
\begin{definition} Let $\ol p>1$ and $p,q,r,s$ real exponents satisfying
\eqref{exp}, let $K, C$ be two positive constants, and let $y_0:\G_0\to{\mathbb{R}}$ be a given measurable function, where $\G_0\cup\G_1$ is an ${\mathcal{H}}^2$-measurable partition of the boundary of ${\mathcal{B}}$.
We say that a couple $(y,V)$ belongs to the class ${\mathcal{A}}_{\ol p,p,q,r,s,K,C}$ if the following properties hold:
\begin{enumerate}
\item $V=V_{\fk{C},{\Greekmath 0112} }$ is a curvature $2$-varifold with boundary in $CV^{\ol p}_2({\mathcal{B}})$;
\item $y\in{\mathcal{A}}^1({\mathcal{B}},{\mathbb{R}}^3)$, with $\Vert y\Vert_\ii\leq K$ and $y=y_0$ on $\G_0$;
\item ${\Greekmath 0119}_\#|\pa G_y|\leq C \cdot {\Greekmath 0116}_V$;
\item the approximate gradient ${\Greekmath 0272} y\in L^p({\mathcal{B}},{\mathbb{R}}^{3\tim 3})$, $\mathrm{cof}{\Greekmath 0272} y \in L^{q}({\mathcal{B}},{\mathbb{R}}^{3\tim 3})$, and $\det{\Greekmath 0272} y\in L^r({\mathcal{B}})$;
\item $\det{\Greekmath 0272} y>0$ {\RIfM@\expandafter\text@\else\expandafter\mbox\fi{a.e. in}} ${\mathcal{B}}$, and $(\det{\Greekmath 0272} y)^{-1}\in L^s({\mathcal{B}})$;
\item $\mathrm{cof}{\Greekmath 0272} y \in GSBV({\mathcal{B}},{\mathbb{R}}^{3\tim 3})$, with $|{\Greekmath 0272}[\mathrm{cof}{\Greekmath 0272} y]|\in L^q(\O)$;
\item $\det{\Greekmath 0272} y \in GSBV({\mathcal{B}},{\mathbb{R}})$, with ${\Greekmath 0272}[\det{\Greekmath 0272} y]\in L^r(\O)$;
\item ${\mathcal{H}}^{n-1}\pri S(\mathrm{cof}{\Greekmath 0272} y)\leq \m_V$ and ${\mathcal{H}}^{n-1}\pri S(\det{\Greekmath 0272} y)\leq \m_V$.
\end{enumerate}
\end{definition}
\par Assumptions (2) and (3) imply $y\in SBV({\mathcal{B}},{\mathbb{R}}^3)$, with jump set contained in the varifold support, namely ${\mathcal{H}}^{n-1}\pri S(y)\leq \m_V$.
Moreover, if $y\in \hat{{\mathcal{A}}}_{p,q,r,s}$, the graph current $G_y$ has null boundary $(\pa G_y)\pri{\mathcal{B}}\tim{\mathbb{R}}^3=0$, see \cite[Vol.~I, Sec.~3.2.4]{GMS98}.
Therefore, taking $V=0$, i.e., in the absence of fractures, it turns out that the couple $(y,0)$ belongs to the class ${\mathcal{A}}_{\ol p,p,q,r,s,K,C}({\mathcal{B}})$, provided that $\Vert y\Vert_\ii\leq K$, independently from the choice of $\overline p$ and $C$.
\begin{theorem}\label{Tgradpolyfrac} Under previous assumptions, if the class ${\mathcal{A}}:={\mathcal{A}}_{\ol p,p,q,r,s,K,C}$ of admissible couples $(y,V)$ is non-empty and
$ \inf\{ {\mathcal{F}}(y,V;\mathcal{B})\mid (y,V)\in {\mathcal{A}}\}<\infty $, the functional $(y,V)\mapsto {\mathcal{F}}(y,V;\mathcal{B})$ attains a minimum in ${\mathcal{A}}$. \end{theorem}
\begin{proof} Let $\{(y_h,V^{(h)})\}$ be a minimizing sequence in ${\mathcal{A}}$.
By Theorem~\ref{Tcomp1}, since $\sup_h{\mathcal{E}}(V^{(h)};\mathcal{B})<\ii$ we can find a (not relabeled) subsequence of $\set{V^{(h)}}$ and a $2$-varifold $V=V_{\fk{C},{\Greekmath 0112}}\in CV^{\ol p}_2({\mathcal{B}})$, with
curvature $A$ and boundary $\pa V$, such that
$V^{(h)} \rhu V$, $A^{(h)}\,dV^{(h)}\rhu A\,dV$, and $\pa V^{(h)} \rhu \pa V$ in the sense of measures, so that by lower semicontinuity
$$ {\mathcal{E}}(V;\mathcal{B})\leq \liminf_{h\to\ii}{\mathcal{E}}(V^{(h)};\mathcal{B})<\ii\,. $$
\par The domain ${\mathcal{B}}$ being bounded, in terms of a (not relabeled) subsequence $\{y_h\}\sb {\mathcal{A}}^1({\mathcal{B}},{\mathbb{R}}^3)$,
we find an a.e. approximately differentiable map $y\in
L^1({\mathcal{B}},{\mathbb{R}}^3)$ such that $y_h\to y$ {\mx{strongly in }}$L^1({\mathcal{B}},{\mathbb{R}}^3)$ and
for any multi-indices $\a$ and $\b$, with $\vert\a\vert+\vert\b\vert=3$, functions $v^\b_{\ol\a}\in L^1({\mathcal{B}})$
such that
$$M^\b_{\ol\a}({\Greekmath 0272} y_h(x))\rightharpoonup v^\b_{\ol\a}(x)\qquad{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{weakly in }}L^1({\mathcal{B}})\,. $$
Moreover, we get the bound $ \sup_h{\mathbf{M}}(G_{y_h})<\ii$ on the mass of the i.m. rectifiable currents $G_{y_h}$ in ${\mathcal{R}}_3({\mathcal{B}}\tim{\mathbb{R}}^3)$ carried by the $y_h$ graphs, whereas the inequalities ${\Greekmath 0119}_\#|\pa G_{y_h}|\leq C \cdot {\Greekmath 0116}_{ V^{(h)} }$ imply the bound $\sup_h{\mathbf{M}}((\pa G_{y_h})\pri{\mathcal{B}}\tim{\mathbb{R}}^3)<\ii$ on the boundary current masses.
Therefore, Theorem~\ref{TclosG} yields
$y\in{\mathcal{A}}^1({\mathcal{B}},{\mathbb{R}}^3)$ and $v^\b_{\ol\a}(x)=M^\b_{\ol\a}({\Greekmath 0272}
y(x))$ a.e in ${\mathcal{B}}$, for every $\a$ and $\b$, whereas $G_{y_h}\rightharpoonup G_y$ weakly in ${\mathcal{D}}_3({\mathcal{B}}\tim{\mathbb{R}}^3)$; the current $G_y$ is i.m. rectifiable in
${\mathcal{R}}_3({\mathcal{B}}\tim{\mathbb{R}}^3)$, and the inequality ${\Greekmath 0119}_\#|\pa G_{y}|\leq C\cdot\m_{V}$ holds true.
\par By taking into account that ${\mathcal{H}}^{n-1}\pri S(y_h)\leq \m_{V^{(h)}}$ and $\sup_h\Vert y_h\Vert_\ii\leq K$, the compactness theorem in $SBV$ applies to the sequence $\{y_h\}\sb SBV({\mathcal{B}},{\mathbb{R}}^3)$, yielding the convergence $Dy_h\rhu Dy$ as measures, whereas ${\mathcal{H}}^{n-1}\pri S(y)\leq \m_{V}$ and $\Vert y\Vert_\ii\leq K$, by lower semicontinuity, and clearly $y=y_0$ on $\G_0$.
\par By using the uniform bound
$$ \sup_h \int_{\mathcal{B}}\bigl(|{\Greekmath 0272} y_h|^p+|\mathrm{cof}{\Greekmath 0272} y_h|^q+|\det {\Greekmath 0272} y_h|^r\bigr)dx<\ii \,,$$
which follows from the lower bound imposed on the density $\hat W$ of the functional $F\mapsto J(F;\mathcal{B})$, we get
${\Greekmath 0272} y_h\rhu {\Greekmath 0272} y$ in $L^p({\mathcal{B}},{\mathbb{R}}^{3\tim 3})$, $\mathrm{cof}{\Greekmath 0272} y_h\rhu \mathrm{cof}{\Greekmath 0272} y$ in $L^q({\mathcal{B}},{\mathbb{R}}^{3\tim 3})$, and $\det{\Greekmath 0272} y_h\rhu \det{\Greekmath 0272} y$ in $L^r({\mathcal{B}})$.
\par Also, the inequalities ${\mathcal{H}}^{n-1}\pri S(\mathrm{cof}{\Greekmath 0272} y_h)\leq \m_{V^{(h)}}$ and the lower bound on $\hat W$ imply that the sequence $\{\mathrm{cof}{\Greekmath 0272} y_h\}\sb GSBV({\mathcal{B}},{\mathbb{R}}^{3\tim 3})$ satisfies the inequality
$$ \sup_h \Bigl( \int_{\mathcal{B}} \bigl(|\mathrm{cof}{\Greekmath 0272} y_h|^q+|{\Greekmath 0272}[\mathrm{cof}{\Greekmath 0272} y_h]|^q\bigr)\,dx +{\mathcal{H}}^{n-1}(S({\mathrm{cof}{\Greekmath 0272} y_h}))\Bigr) <\ii\,. $$
Therefore, by Theorem~\ref{TGSBV} we infer that
\begin{itemize}
\item $\mathrm{cof}{\Greekmath 0272} y\in GSBV({\mathcal{B}},{\mathbb{R}}^{3\tim 3})\,,$
\item $\mathrm{cof}{\Greekmath 0272} y_h\to\mathrm{cof}{\Greekmath 0272} y$ in $L^q({\mathcal{B}},{\mathbb{R}}^{3\tim 3})\,,$
\item ${\Greekmath 0272}[\mathrm{cof}{\Greekmath 0272} y_h]\rhu {\Greekmath 0272}[\mathrm{cof}{\Greekmath 0272} y]$ weakly in $L^q({\mathcal{B}},{\mathbb{R}}^{3\tim 3\tim 3})$, and
\item ${\mathcal{H}}^{n-1}\pri S(\mathrm{cof}{\Greekmath 0272} y)\leq \m_V$.
\end{itemize}
Similarly, the inequalities ${\mathcal{H}}^{n-1}\pri S(\det{\Greekmath 0272} y_h)\leq \m_{V^{(h)}}$ and the lower bound on $\hat W$ imply that the sequence
$\{\det{\Greekmath 0272} y_h\}\sb GSBV({\mathcal{B}})$ satisfies the inequality
$$ \sup_h \Bigl( \int_{\mathcal{B}} \bigl(|\det{\Greekmath 0272} y_h|^r+|{\Greekmath 0272}[\det{\Greekmath 0272} y_h]|^r\bigr)\,dx +{\mathcal{H}}^{n-1}(S({\det{\Greekmath 0272} y_h})) \Bigr) <\ii\,, $$
so that Theorem~\ref{TGSBV} entails that
\begin{itemize}
\item $\det {\Greekmath 0272} y\in GSBV({\mathcal{B}})\,,$
\item $\det{\Greekmath 0272} y_h\to\det{\Greekmath 0272} y$ in $L^r({\mathcal{B}})\,,$
\item ${\Greekmath 0272}[\det{\Greekmath 0272} y_h]\rhu {\Greekmath 0272}[\det{\Greekmath 0272} y]$ weakly in $L^r({\mathcal{B}},{\mathbb{R}}^{3})$, and
\item ${\mathcal{H}}^{n-1}\pri S(\det{\Greekmath 0272} y)\leq \m_V$.
\end{itemize}
Arguing as in the proof of Theorem~\ref{Tgradpoly}, reported in reference \cite{KPS}, we obtain $\det{\Greekmath 0272} y>0$ a.e. in ${\mathcal{B}}$, and
$(\det{\Greekmath 0272} y)^{-1}\in L^s({\mathcal{B}})$, whence we get $(y,V)\in{\mathcal{A}}={\mathcal{A}}_{\ol p,p,q,r,s,K,C}$.
Finally, on account of the previous convergences, the gradient polyconvexity assumption implies the lower semicontinuity inequality
$$ J({\Greekmath 0272} y;\mathcal{B})\leq\liminf_{h\to\ii}J({\Greekmath 0272} y_h;\mathcal{B}). $$
Then,
$$ {\mathcal{F}}(y,V)\leq\liminf_{h\to\ii}{\mathcal{F}}(y_h,V^{(h)}), $$
which is the last step in the proof. \end{proof}
\subsection{By avoiding self-penetration}
The restriction imposed to $\det {\Greekmath 0272} y(x)$ ensures that the deformation locally preserves orientation. However, we have also to allow possible self-contact between distant portions of the boundary preventing at the same time self-penetration of the matter. To this aim, in 1987 P. Ciarlet and J. Ne\v{c}as proposed the introduction of an additional constraint, namely
$$ \int_{{\mathcal{B}}'} \det {\Greekmath 0272} y(x)\, dx \le {\mathcal{L}}^3(\widetilde y(\widetilde{\mathcal{B}}')) $$
for any sub-domain ${\mathcal{B}}'$ of ${\mathcal{B}}$, where $\widetilde {\mathcal{B}}'$ is intersection of ${\mathcal{B}}'$ with the domain $\widetilde {\mathcal{B}}$ of Lebesgue's representative $\widetilde y$ of $y$ \cite{CN}.
We adopt here a weaker constraint, introduced in 1989 by M. Giaquinta, G. Modica, and J. Sou\v{c}ek \cite{GMS-Arma} (see also \cite[Vol.~II,~Sec. 2.3.2]{GMS98}). It reads
$$ \int_{{\mathcal{B}}} f(x,u(x))\,\det {\Greekmath 0272} y(x)\, dx \le \int_{{\mathbb{R}}^3}\sup_{x\in{\mathcal{B}}} f(x,y)\,dy\,, $$
for every compactly supported smooth function $f:{\mathcal{B}}\tim{\mathbb{R}}^3\to[0,+\ii)$.
We thus denote by $\widetilde{\mathcal{A}}_{\ol p,p,q,r,s,K,C}$ the set of couples $(y,V)\in {\mathcal{A}}_{\ol p,p,q,r,s,K,C}$ such that the deformation map $y$ satisfies the previous inequality.
Since that constraint is preserved by the weak convergence as currents $G_{y_h}\rhu G_y$ along minimizing sequences, arguing as in Theorem~\ref{Tgradpolyfrac} we readily obtain the following existence result.
\begin{corollary}\label{Cgradpolyfrac} Under the previous assumptions, if the class $\widetilde{\mathcal{A}}:=\widetilde{\mathcal{A}}_{\ol p,p,q,r,s,K,C}$ of admissible couples $(y,V)$ is non-empty and
$ \inf\{ {\mathcal{F}}(y,V)\mid (y,V)\in \widetilde{\mathcal{A}}\}<\infty $,
then the minimum of the functional $(y,V)\mapsto {\mathcal{F}}(y,V)$ is attained in $\widetilde{\mathcal{A}}$. \end{corollary}
\ \ \
\textbf{Acknowledgements}. This work has been developed within the activities of the research group in \textquotedblleft Theoretical Mechanics\textquotedblright\ of the \textquotedblleft Centro di Ricerca Matematica Ennio De Giorgi\textquotedblright\ of the Scuola Normale Superiore in Pisa. PMM wishes to thank the Czech Academy of Sciences for hosting him in Prague during February 2020 as a visiting professor. We acknowledge also the support of GA\v{C}R-FWF project 19-29646L (to MK), GNFM-INDAM (to PMM), and GNAMPA-INDAM (to DM).
\end{document}
\subsection{Vector valued BV functions} Let $u:\O\to{\mathbb{R}}^N$ be a function in $BV(\O,{\mathbb{R}}^N)$, i.e., $u=(u^1,\ldots u^N)$ with all components $u^j\in BV(\O)$.
This means that for every $j$ there exists a vector valued measure $Du^j=(D_1u^j,\ldots,D_nu^j)$ such that all the $D_iu^j$'s are signed Radon measures and for any vector field $\vf=(\vf^1,\ldots,\vf^n)\in C^\ii_c(\O,{\mathbb{R}}^n)$
$$\langle Du^j,\vf \rangle:=\int_\O \vf\,dDu^j=-\int_\O u\,\mathrm{div}\vf\,dx=:- \langle u,\mathrm{div} \vf \rangle\,, $$
where $\mathrm{div}\vf:=\sum\nolimits_{i=1}^nD_i\vf^i$. If $u\in BV(\O,{\mathbb{R}}^N)$, one decomposes $$Du^j=D^a u^j+D^s u^j $$ into its absolute continuous and singular parts. It turns out that $u$ is approximately differentiable at almost every point $x\in\O$, with $$D^a u^j={\Greekmath 0272} u^j\,{\mathcal{L}}^n $$ ${\Greekmath 0272} u^j$ being the approximate gradient of $u^j$.
\par The {\em Jump set} of $u$ is the countably ${\mathcal{H}}^{n-1}$-rectifiable set $J_u$ in $\O$ given by the union of the complements of the Lebesgue sets of the $u^j$'s. More precisely, $$x_0\notin J_u\Longleftrightarrow\exists\, y\in{\mathbb{R}}^N\quad{\RIfM@\expandafter\text@\else\expandafter\mbox\fi{s.t. }}\lim_{\r\to 0}{1\over\r^n}\int\nolimits_{B^n_\r(x_0)} \vert u(x)-y\vert\,dx=0\,. $$
Let $\n=\n_u(x)$ be a unit vector in ${\mathbb{R}}^n$ orthogonal to $J_u$ at ${\mathcal{H}}^{n-1}$-a.e. point $x\in J_u$. Let $u^{\pm}(x)$ denote the one-sided approximate limits of $u$ on $J_u$, so that for ${\mathcal{H}}^{n-1}$-a.e. point $x\in J_u$
$$\lim_{\r\to 0^+}\r^{-n}\int_{B^\pm_\r(x)}|u(x)-u^\pm(x)|\,dx=0\,, $$ where $B^\pm_\r(x):=\{y\in B^n_\r(x):\pm \langle y-x,\n(x)\rangle\geq 0\}$. Notice that a change of sign of $\n$ induces a permutation of $u^+$ and $u^-$ and that only for scalar functions there is a canonical choice of the sign of $\n$ which ensures that $u^+(x)>u^-(x)$.
\par The distributional derivative of $u$ turns out to be decomposed into the sum of a ``gradient" measure, of a ``jump" measure, concentrated on a set that is $\s$-finite with respect to the ${\mathcal{H}}^{n-1}$-measure, and of a ``Cantor-type" measure. More precisely, $$ Du=D^au+D^Ju+D^Cu\,,$$ where $D^au$, $D^Ju$, and $D^Cu$ are mutually singular, $$D^au={\Greekmath 0272} u\,{\mathcal{L}}^n\,,\qquad D^Ju=(u^+(x)-u^-(x))\otimes\n(x)\,{\mathcal{H}}^{n-1}\pri J_u\,, $$
${\Greekmath 0272} u$ being the approximate gradient of $u$, compare \cite{AFP} \cite{GMSl1}.
\subsubsection{SBV functions} Free discontinuity problems are characterized by a competition between volume energies, concentrated on $n$-dimensional sets, and surface energies, concentrated on $(n-1)$-dimensional sets. Probably, the best-known example is the Mumford-Shah functional \cite{MSh}. To study in a systematic way such a problem, E.~De Giorgi and L.~Ambrosio introduced in \cite{DGA} the space of {\em special functions of bounded variation}, $SBV$-functions, that are $BV$-functions $u$ whose derivatives have null Cantor part.
The compactness and lower semicontinuity results in $SBV$, see \cite{Am} \cite{A2} \cite{AM}, lead to an existence theory for the weak formulation of certain free discontinuity problems, compare \cite{AFP}. We recall that the proof of the compactness theorem in \cite{A2} relies on a generalization of the following characterization of $SBV$-functions with ${\mathcal{H}}^{n-1}$-rectifiable jump sets. Following \cite{ABG}, we denote by ${{\mathcal{T}}}(\O\tim{\mathbb{R}})$ the class of $C^1$-functions \,$\vf(x,y)$\, such that \,$\vert\vf\vert+\vert D\vf\vert$\, is bounded and the support of $\vf$\, is contained in $K\tim{\mathbb{R}}$ for some compact set $K\sb\O$.
\begin{proposition} Let $u\in BV(\O)$. Then $u\in SBV(\O)$ with finite jump, ${\mathcal{H}}^{n-1}(J_u)<\ii$, if and only if for every $i=1,\ldots,n$ there exists a Radon measure $\m_i$ in $\O\tim{\mathbb{R}}$ such that $$ \int_\O\left(\frac{\pa\vf}{\pa
x_i}(x,u(x))+ \frac{\pa\vf}{\pa
y}(x,u(x)){\Greekmath 0272}_i{u}(x)\right)\,dx= \int_{\O\tim{\mathbb{R}}}\vf\,d\m_i $$ for any \,$\vf\in{{\mathcal{T}}}(\O\tim{\mathbb{R}})$. In this case, moreover, we have:
$$\m_i= -(Id\join u^+)_\#(\n_i{\mathcal{H}}^{n-1}\pri J_u)+(Id\join u^-)_\#(\n_i{\mathcal{H}}^{n-1}\pri J_u)\,. $$ \end{proposition}
\end{document}
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arXiv
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After the dry, algebraic discussion of the previous section it is a relief to finally be able to compute some variances.
We say that the variance of the sum is the sum of all the variances and all the covariances.
If $X_1, X_2 \ldots , X_n$ are independent, then all the covariance terms in the formula above are 0.
When the random variables are i.i.d., this simplifies even further.
Let $X_1, X_2, \ldots, X_n$ be i.i.d., each with mean $\mu$ and $SD$ $\sigma$. You can think of $X_1, X_2, \ldots, X_n$ as draws at random with replacement from a population, or the results of independent replications of the same experiment.
This implies that as the sample size $n$ increases, the distribution of the sum $S_n$ shifts to the right and is more spread out.
Here is one of the most important applications of these results.
Here is the distribution of $X$. You can see that there is almost no probability outside the range $E(X) \pm 3SD(X)$.
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CommonCrawl
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\begin{document}
\date{}
\maketitle
\begin{abstract} \noindent In this paper, we study the estimates of resolvents $ R(\lambda,\mathcal{L}_{\va})=(\mathcal{L}_{\va}-\lambda I)^{-1} $, where $$ \mathcal{L}_{\va}=-\operatorname{div}(A(x/\va)\nabla) $$ is a family of second elliptic operators with symmetric, periodic and oscillating coefficients defined on a bounded domain $ \om $ with $ \va>0 $. For $ 1<p<\infty $, we will establish uniform $ L^p\to L^p $, $ L^p\to W_0^{1,p} $, $ W^{-1,p}\to L^p $ and $ W^{-1,p}\to W_0^{1,p} $ estimates by using the real variable method. Meanwhile, we use Green functions for operators $ \mathcal{L}_{\va}-\lambda I $ to study the asymptotic behavior of $ R(\lambda,\mathcal{L}_{\va}) $ and obtain convergence estimates in $ L^p\to L^p $, $ L^p\to W_0^{1,p} $ norm.\\ \textbf{Keywords:} Homogenization; Uniform estimates; Resolvents; Convergence. \end{abstract}
\section{Introduction and main results} \noindent The main purpose of this paper is to study estimates of resolvents for a family of elliptic operators with rapidly oscillating and symmetric coefficients. Precisely speaking, we consider the resolvents of operators \begin{align} \mathcal{L}_{\va}=-\operatorname{div}(A(x/\va)\nabla)=-\frac{\pa}{\pa x_i}\left\{a_{ij}^{\al\beta}\left(\frac{x}{\va}\right)\frac{\pa}{\pa x_j}\right\},\quad\va>0,\label{ho} \end{align} where $ 1\leq i,j\leq d $ and $ 1\leq\al,\beta\leq m $. Here, $ d\geq 2 $ denotes the dimension of Euclidean space and $ m\geq 1 $ is the number of equations in the system. The summation convention for repeated indices is used throughout the paper. In discussion, we will always assume that the measurable matrix-valued functions $ A(y)=(a_{ij}^{\al\beta}(y)):\mathbb{R}^d\to\mathbb{R}^{m^2\times d^2} $ satisfy the symmetry condition \begin{align} a_{ij}^{\al\beta}(y)=a_{ji}^{\beta\al}(y)\text{ for any }1\leq i,j\leq d,1\leq\al,\beta\leq m\text{ and } y\in\mathbb{R}^d,\label{sy} \end{align} the uniform ellipticity condition \begin{align}
\mu|\xi|^2\leq a_{ij}^{\al\beta}(y)\xi_{i}^{\al}\xi_{j}^{\beta}\leq \mu^{-1}|\xi|^2\text{ for any }y\in\mathbb{R}^d \text{ and }\xi=(\xi_{i}^{\al})\in\mathbb{R}^{m\times d},\label{el} \end{align} where $ \mu>0 $ is a positive constant and the periodicity condition \begin{align} A(y+z)=A(y)\text{ for any }y\in\mathbb{R}^d\text{ and }z\in\mathbb{Z}^d.\label{pe} \end{align} To ensure $ L^p $, $ W^{1,p} $ and Lipschitz estimates of operators $ \mathcal{L}_{\va} $, in some situations, we need more smoothness conditions on the coefficients matrix $ A $, i.e. the Hölder regularity of $ A $, \begin{align}
|A(x)-A(y)|\leq \tau|x-y|^{\nu}\text{ for any }x,y\in\mathbb{R}^d,\label{Hol} \end{align} where $ \tau>0,\nu\in(0,1) $ and the $ \VMO $ condition (vanishing mean oscillation condition), \begin{align}
\sup_{x\in\mathbb{R}^d,0<\rho<t}\Xint-_{B(x,\rho)}\left|A(y)-\Xint-_{B(x,\rho)}A\right|dy\leq\omega(t)\text{ and } \lim_{t\to 0}\omega(t)=0,\label{VMO} \end{align} where $ \omega(t) $ is a continuous nondecreasing function. To make the notation simpler, we denote that
$ A\in\VMO(\mathbb{R}^d) $ if $ A $ satisfies the $ \VMO $ condition $ \eqref{VMO} $.
Assume that $ A(y) =(a_{ij}^{\al\beta}(y)) $, the coefficient matrix of $ \mathcal{L}_{\va}=-\operatorname{div}(A(x/\va)\nabla) $, satisfies $ \eqref{el} $ and $ \eqref{pe} $. Let $ \chi_{j}^{\beta}(y)=(\chi_{j}^{\al\beta}(y)) $ denote the matrix of correctors for $ \mathcal{L}_1=-\operatorname{div}(A(x)\nabla) $ in $ \mathbb{R}^d $, where $ \chi_j^{\beta}(y)=(\chi_j^{1\beta}(y),...,\chi_j^{m\beta}(y))\in H_{\operatorname{per}}^1(Y;\mathbb{R}^m) $ is defined by the following cell problem \begin{align} \left\{\begin{aligned} &-\operatorname{div}(A(x)\nabla\chi_{j}^{\beta})=\operatorname{div}(A(x)\nabla P_{j}^{\beta})\text{ in }[0,1)^d,\\ &\chi_{j}^{\beta}\text{ is periodic with respect to }\mathbb{Z}^d\text{ and }\int_{Y}\chi_{j}^{\beta}dy=0, \end{aligned}\right.\label{correctors} \end{align} where $ 1\leq j\leq d $, $ 1\leq\beta\leq m $, $ Y=[0,1)^d\cong \mathbb{R}^d/\mathbb{Z}^d $ and $ P_{j}^{\beta}=(P_{j}^{\al\beta}(x))=(x_j\delta^{\al\beta}) $ with $ \delta^{\al\beta}=1 $ if $ \al=\beta $, $ \delta^{\al\beta}=0 $ otherwise. The homogenized operator is defined by $ \mathcal{L}_{0}=-\operatorname{div}(\widehat{A}\nabla) $, where the coefficients $ \widehat{A}=(\widehat{a}_{ij}^{\al\beta}) $ are given by \begin{align} \widehat{a}_{ij}^{\al\beta}=\int_{Y}\left[a_{ij}^{\al\beta}(y)+a_{ik}^{\al\gamma}(y)\frac{\pa}{\pa y_k}\chi_{j}^{\gamma\beta}(y)\right]dy.\label{hoco} \end{align} It is easy to show that if $ A $ satisfies $ \eqref{sy} $, then $ \widehat{A} $ is also symmetric, that is \begin{align} \widehat{a}_{ij}^{\al\beta}=\widehat{a}_{ji}^{\beta\al}\text{ for any }1\leq i,j\leq d\text{ and } 1\leq\al,\beta\leq m.\label{syl0} \end{align} Moreover, if $ A $ satisfies $ \eqref{el} $, then there exists $ \mu_1>0 $ depending only on $ \mu $, such that \begin{align}
\mu_1|\xi|^2\leq \widehat{a}_{ij}^{\al\beta}\xi_{i}^{\al}\xi_{j}^{\beta}\leq \mu_1^{-1}|\xi|^2\text{ for any }\xi=(\xi_{i}^{\al})\in\mathbb{R}^{m\times d}.\label{ell0} \end{align} For the sake of simplicity, we set $ \min(\mu,\mu_1) $ as new $ \mu $ in the rest of the paper. Let \begin{align} b_{ij}^{\al\beta}(y)=\widehat{a}_{ij}^{\al\beta}-a_{ij}^{\al\beta}(y)-a_{ik}^{\al\gamma}(y)\frac{\partial}{\pa y_k}\chi_{j}^{\gamma\beta}(y),\label{Flux coefficients} \end{align} where $ 1\leq i,j\leq d $ and $ 1\leq\al,\beta\leq m $. It is easy to see that $ \int_{Y}b_{ij}^{\al\beta}(y)dy=0 $ by $ \eqref{hoco} $. Then in view of $ \eqref{correctors} $ and $ \eqref{hoco} $, there exists the flux corrector $ (F_{kij}^{\al\beta}(y))_{1\leq \al\leq m}\in H_{\operatorname{per}}^1(Y;\mathbb{R}^m) $ such that \begin{align} b_{ij}^{\al\beta}(y)=\frac{\pa}{\partial y_k}\left\{F_{kij}^{\al\beta}(y)\right\}\quad\text{and}\quad F_{kij}^{\al\beta}(y)=-F_{ikj}^{\al\beta}(y)\label{Flux correctors} \end{align} for any $ 1\leq i,j,k\leq d $ and $ 1\leq\al,\beta\leq m $. Moreover, $ (F_{kij}^{\al\beta})\in L^{\infty}(Y) $ if $ (\chi_{j}^{\al\beta}) $ is Hölder continuous. For details about the proof of this fact, one can refer to Chaper 2 of \cite{Shen2}.
Let $ A $ satisfy $ \eqref{sy} $, $ \eqref{el} $ and $ \eqref{pe} $. For $ F\in L^2(\om;\mathbb{C}^m) $ and $ \va\geq 0 $, we can define a linear operator $ T_{\va}:L^2(\om;\mathbb{C}^m)\to H_0^1(\om;\mathbb{C}^m)\subset L^2(\om;\mathbb{C}^m) $ by $ T_{\va}(F)=u_{\va}\in H_0^1(\om;\mathbb{C}^m) $ such that $ \mathcal{L}_{\va}(u_{\va})=F $ in $ \om $ and $ u_{\va}=0 $ on $ \pa\om $. Using standard arguments in \cite{Kenig3}, operators $ T_{\va} $ with $ \va\geq 0 $ are positive and self-adjoint. Therefore, applying the spectrum theory, it is natural to consider properties of resolvents $ R(\lambda,\mathcal{L}_{\va})=(\mathcal{L}_{\va}-\lambda I)^{-1} $ with $ \lambda\in\mathbb{C}\backslash(0,\infty) $. To simplify notations, we will use $ R(\lambda,\mathcal{L}) $ to denote the resolvent of the elliptic operator $ \mathcal{L} $ in the rest of the paper. In order to better characterize resolvents of operators, for $ \lambda=|\lambda|e^{i\theta}\in\mathbb{C}\backslash(0,\infty) $, we define a constant $ c(\lambda,\theta) $ by \begin{align} c(\lambda,\theta)=\left\{\begin{matrix} 1&\text{if}&\theta\in[\pi/2,3\pi/2]\text{ or }\lambda=0\\
|\sin\theta|^{-1}&\text{if}&\theta\in(0,\pi/2)\cup(3\pi/2,2\pi)\text{ and }|\lambda|>0. \end{matrix}\right.\label{ac} \end{align} To present our results more conveniently and study the estimates of resolvents, we need to introduce the matrix of Dirichlet correctors $ \Phi_{\va}(x)=(\Phi_{\va,j}^{\beta}(x))_{1\leq j\leq d,1\leq \beta\leq m} $ in $ \om $, defined by \begin{align} \mathcal{L}_{\va}(\Phi_{\va,j}^{\beta}(x))=0\text{ in }\om\quad\text{and}\quad \Phi_{\va,j}^{\beta}(x)=P_j^{\beta}(x)\text{ on }\partial\om.\label{Dirichlet correctors} \end{align} The Dirichlet correctors were first introduced in \cite{Av1} to study the uniform Lipschitz estimates of homogenization problems. It is known that if $ A $ satisfies $ \eqref{el} $, $ \eqref{pe} $, $ \eqref{Hol} $ and $ \om $ is a bounded $ C^{1,\eta} $ $ \eta\in(0,1) $ domain in $ \mathbb{R}^d $ with $ d\geq 2 $, then \begin{align}
\|\Phi_{\va,j}^{\beta}-P_j^{\beta}\|_{L^{\infty}(\om)}\leq C\va\quad\text{and}\quad \|\nabla\Phi_{\va}\|_{L^{\infty}(\om)}\leq C,\label{Estimate for Dirichlet correctors} \end{align} where $ C $ depends only on $ \mu,d,m,\tau,\nu,\eta $ and $ \om $. The following are the main results of this paper. For the sake of simplicity, we will denote $ \diam(\om) $, the diameter of the bounded domain $ \om $ in $ \mathbb{R}^d $ with $ d\geq 2 $ by $ R_0 $ throughout this paper.
\begin{thm}[$ L^2 $ and $ H_0^1 $ convergence of resolvents]\label{Approximation 1}
Suppose that $ d\geq 2 $, $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. Let $ \om $ be a bounded $ C^{1,1} $ domain in $ \mathbb{R}^d $ and $ \lambda=|\lambda|e^{i\theta}\in\mathbb{C}\backslash(0,\infty) $. For $ \va\geq 0 $ and $ F\in L^2(\om;\mathbb{C}^m) $, let $ u_{\va,\lambda}\in H_0^1(\om;\mathbb{C}^m) $ be the unique solution of the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $. Then \begin{align}
\|u_{\va,\lambda}-u_{0,\lambda}\|_{L^2(\om)}&\leq C\va c^2(\lambda,\theta)(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|F\|_{L^2(\om)},\label{Convergence rate 11}\\
\|u_{\va,\lambda}-u_{0,\lambda}-(\Phi_{\va,j}^{\beta}-P_{j}^{\beta})\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}\|_{H_0^1(\om)}&\leq C\va c^2(\lambda,\theta)\|F\|_{L^2(\om)},\label{Convergence rate 1} \end{align} where $ C $ depends only on $ \mu,d,m,\tau,\nu,\om $ and $ \Phi_{\va} $ is given by $ \eqref{Dirichlet correctors} $. In operator forms, \begin{align}
\|R(\lambda,\mathcal{L}_{\va})-R(\lambda,\mathcal{L}_{0})\|_{L^2(\om)\to L^2(\om)}&\leq C\va c^2(\lambda,\theta)(R_0^{-2}+|\lambda|)^{-\frac{1}{2}},\label{Operator estimate 11}\\
\|R(\lambda,\mathcal{L}_{\va})-R(\lambda,\mathcal{L}_{0})-K_{\va}(\lambda)\|_{L^2(\om)\to H_0^1(\om)}&\leq C\va c^2(\lambda,\theta),\label{Operator estimate 1} \end{align} where $ K_{\va}(\lambda) $ are the operator correctors given by the formula \begin{align} K_{\va}(\lambda)=\{K_{\va}^{\al}(\lambda)\}_{1\leq\al\leq m}=\{(\Phi_{\va,j}^{\al\beta}(x)-P_{j}^{\al\beta}(x))\pa_{x_j}R(\lambda,\mathcal{L}_{0})^{\beta}\}_{1\leq\al\leq m}.\label{Operator corrector} \end{align} \end{thm}
Theorem \ref{Approximation 1} is actually a quantitative result of the homogenization theory for the operator $ \mathcal{L}_{\va}-\lambda I $. In fact, if $ \om $ is a bounded Lipschitz domain in $ \mathbb{R}^d $ with $ d\geq 2 $ and $ A $ satisfies $ \eqref{el},\eqref{pe} $, it can be shown that $ u_{\va,\lambda}\to u_{0,\lambda} $ weakly in $ H_0^1(\om;\mathbb{C}^m) $ and strongly in $ L^2(\om;\mathbb{C}^m) $. For more details about the homogenization problems of elliptic systems, one can refer to \cite{Bensou} and \cite{Shen2}.
In \cite{Su1}, assuming that $ \om $ is a bounded and $ C^{1,1} $ domain, the convergence rates of such resolvents are established, that is, for $ \lambda\in\mathbb{C}\backslash[0,\infty) $ and $ 0\leq\va<1 $, \begin{align}
\|R(\lambda,\mathcal{L}_{\va})-R(\lambda,\mathcal{L}_{0})\|_{L^2(\om)\to L^2(\om)}&\leq Cc^2(\lambda,\theta)(\va|\lambda|^{-\frac{1}{2}}+\va^2),\label{su1l2}\\
\|R(\lambda,\mathcal{L}_{\va})-R(\lambda,\mathcal{L}_{0})-K_0(\va,\lambda)\|_{L^2(\om)\to H^1(\om)}&\leq Cc^2(\lambda,\theta)\va^{\frac{1}{2}},\label{su1l22} \end{align}
where $ K_0(\va,\lambda) $ is some corrector of the homogenization problem and $ C $ depends only on $ \mu,d,m $ and $ \om $. These estimates are obtained by applying the results when $ \om=\mathbb{R}^d $ in \cite{Birman1}, \cite{Birman2} and some extension theorems. By using such approximation estimates for $ R(\lambda,\mathcal{L}_{\va}) $, \cite{Meshkova1} and \cite{Meshkova2} gave the convergence rates for homogenization problems of parabolic and hyperbolic systems in $ L^2 $ and $ H_0^1 $ space. What is new for Theorem \ref{Approximation 1} is that we use different operator corectors and obtain a more brief proof under higher regularity assumptions of $ A $. The main method is developed in the proof of Theorem 2.4 in \cite{Kenig3}, which is used to deal with the case that $ \lambda=0 $. In Theorem 1.5 of \cite{Xu1}, the author generalized convergence results in \cite{Kenig3} to elliptic operators with lower order terms. Such results are somewhat similar to this paper, but this does not mean that the conclusions of this paper can be trivially covered. To some extent, Theorem \ref{Approximation 1} is a generalization of Theorem 2.4 in \cite{Kenig3} and Theorem 1.5 in \cite{Xu1}. The difference between this and Theorem 1.5 in \cite{Xu1} is that the constants on the right hand side of $ \eqref{Convergence rate 11} $ do not depend on the module of $ \lambda $, i.e. $ |\lambda| $. Moreover, we remark that these estimates in Theorem \ref{Approximation 1} can also be applied to evolution systems and obtain similar results given in \cite{Lin1}, \cite{Meshkova1} and \cite{Meshkova2}.
Given a sectorial domain \begin{align}
\Sigma_{\theta_0}=\left\{\lambda=|\lambda|e^{i\theta}\in\mathbb{C}:|\lambda|>0,|\arg\theta|>\pi-\theta_0\right\},\label{td} \end{align} where $ \theta_0\in(0,\frac{\pi}{2}) $, we can obtain the following estimates.
\begin{thm}[$ L^p $ and $ W_0^{1,p} $ estimates of resolvents]\label{Lp estimates of resolventsf} Suppose that $ \va\geq 0 $ and $ d\geq 2 $. Let $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ \om $ be a bounded $ C^1 $ domain in $ \mathbb{R}^d $. Assume that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{VMO} $. Then for any $ 1<p<\infty $, $ F\in L^p(\om;\mathbb{C}^m) $ and $ f\in L^p(\om;\mathbb{C}^{m\times d}) $, there exists a unique $ u_{\va,\lambda}\in W_0^{1,p}(\om;\mathbb{C}^m) $ such that $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F+\operatorname{div}(f) $ in $ \om $, $ u_{\va,\lambda}=0 $ on $ \pa\om $ and satisfies the uniform estimates \begin{align}
\|u_{\va,\lambda}\|_{L^p(\om)}&\leq C_{p,\theta_0}(R_0^{-2}+|\lambda|)^{-1}\|F\|_{L^p(\om)}+C_{p,\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|f\|_{L^p(\om)},\label{Lpestiamesu3}\\
\|\nabla u_{\va,\lambda}\|_{L^p(\om)}&\leq C_{p,\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|F\|_{L^p(\om)}+C_{p,\theta_0}\|f\|_{L^p(\om)},\label{LpW1pestiamesu3} \end{align} where $ C_{p,\theta_0} $ depends only on $ \mu,d,m,\omega(t),p,\theta_0 $ and $ \om $. In operator forms, \begin{align}
\|R(\lambda,\mathcal{L}_{\va})\|_{L^p(\om)\to L^p(\om)}&\leq C_{p,\theta_0}(R_0^{-2}+|\lambda|)^{-1},\label{reLpLp}\\
\|R(\lambda,\mathcal{L}_{\va})\|_{W^{-1,p}(\om)\to L^p(\om)}&\leq C_{p,\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}},\label{reW-1pLp}\\
\|R(\lambda,\mathcal{L}_{\va})\|_{L^p(\om)\to W_0^{1,p}(\om)}&\leq C_{p,\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}},\label{reLpW1p}\\
\|R(\lambda,\mathcal{L}_{\va})\|_{W^{-1,p}(\om)\to W_0^{1,p}(\om)}&\leq C_{p,\theta_0},\label{reW-1pW1p} \end{align} where $ W^{-1,p}(\om;\mathbb{C}^m)\triangleq(W_0^{1,p'}(\om;\mathbb{C}^m))^* $ with $ p'=\frac{p}{p-1} $ being the conjugate number of $ p $. \end{thm}
The $ L^p\to L^p $ estimates of resolvents for elliptic operators are widely studied and have abundant materials. We will list some results for non-homogenization problems. For $ m=1 $ and $ \BMO $ coefficients, see \cite{Kang1}; for constant coefficients and Dirichlet boundary conditions, see \cite{Shen1}; for constant coefficients and Neumann boundary conditions, see \cite{Wei2}; and for variable coefficients and Lipschitz domains, see \cite{Wei1}. It is noteworthy that in \cite{Wei1}, the authors derived the $ L^p\to L^p $ estimate of resolvents $ R(\lambda,\mathcal{L}_{1}) $ without any regularity assumptions on $ A $ with $ p $ being closed to $ 2 $ when $ \om $ is a bounded Lipschitz domain. These results can be used for homogenization problems. In this point of view, Theorem \ref{Lp estimates of resolventsf} gives stronger results under more assumptions of $ A $ and $ \om $. We see that the estimates established in Theorem \ref{Lp estimates of resolventsf} are sharp in view of the estimates for $ p=2 $, which will be given later.
The main tool for the proof of Theorem \ref{Lp estimates of resolventsf} is real variable method, which was original used in \cite{Caf1} to deal with $ W^{1,p} $ estimates of elliptic equations. We remark that for the operator $ \mathcal{L}_{\va}-\lambda I $, the use of real variable method is slightly different from the case for $ \mathcal{L}_{\va} $.
Besides the uniform $ W^{-1,p},L^p\to L^p,W_0^{1,p} $ estimates of resolvents, one may also concern about the convergence rates in $ L^p\to L^p $ or $ L^p\to W_0^{1,p} $ norm with $ 1<p<\infty $. For these topics, we have the following theorems.
\begin{thm}[$ L^p\to L^p $ convergence rates of resolvents]\label{Lpconres} Suppose that $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $, $ \eqref{Hol} $. Let $ \om $ be a bounded $ C^{1,1} $ domain in $ \mathbb{R}^d $. For $ \va\geq 0 $ and $ F\in L^p(\om;\mathbb{C}^m) $, let $ u_{\va,\lambda}\in W_0^{1,p}(\om;\mathbb{C}^m) $ be the unique solution of the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $. Then for any $ 1<p<\infty $, \begin{align}
\|u_{\va,\lambda}-u_{0,\lambda}\|_{L^p(\om)}&\leq C_{p,\theta_0}\va (R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|F\|_{L^p(\om)},\label{Convergence rate Lp} \end{align} where $ C_{p,\theta_0} $ depends only on $ \mu,d,m,\tau,\nu,p $ and $ \om $. In operator forms, \begin{align}
\|R(\lambda,\mathcal{L}_{\va})-R(\lambda,\mathcal{L}_{0})\|_{L^p(\om)\to L^p(\om)}&\leq C_{p,\theta_0}\va (R_0^{-2}+|\lambda|)^{-\frac{1}{2}}.\label{**-} \end{align} \end{thm}
\begin{thm}[$ L^p\to W_0^{1,p} $ convergence rates of resolvents]\label{LpW1pconreso} Suppose that $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $, $ \eqref{Hol} $. Let $ \om $ be a bounded $ C^{2,1} $ domain in $ \mathbb{R}^d $. For $ \va\geq 0 $ and $ F\in L^p(\om;\mathbb{C}^m) $, let $ u_{\va,\lambda}\in W_0^{1,p}(\om;\mathbb{C}^m) $ be the unique solution of the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $. Then for any $ 1<p<\infty $, \begin{align}
\|u_{\va,\lambda}-u_{0,\lambda}-(\Phi_{\va,j}^{\beta}-P_j^{\beta})\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}\|_{W_0^{1,p}(\om)}&\leq C_{p,\theta_0}\va\left\{\ln[\va^{-1}R_0+2]\right\}^{4|\frac{1}{2}-\frac{1}{p}|}\|F\|_{L^p(\om)},\label{Convergence rate LpW1p} \end{align} where $ C_{p,\theta_0} $ depends only on $ \mu,d,m,\tau,\nu,p $ and $ \om $. In operator forms, \begin{align}
\|R(\lambda,\mathcal{L}_{\va})-R(\lambda,\mathcal{L}_{0})-K_{\va}(\lambda)\|_{L^p(\om)\to W_0^{1,p}(\om)}&\leq C_{p,\theta_0}\va\left\{\ln[\va^{-1}R_0+2]\right\}^{4|\frac{1}{2}-\frac{1}{p}|},\label{***-} \end{align} where $ K_{\va}(\lambda) $ is defined by $ \eqref{Operator corrector} $. \end{thm}
For the operator $ \mathcal{L}_{\va} $, the estimates of $ L^p $ and $ W_0^{1,p} $ convergence rates are established in \cite{Kenig4}. These estimates are established by obtaining the convergence rates of Green functions of operators $ \mathcal{L}_{\va} $. The existences and pointwise estimates of Green functions for operators $ \mathcal{L}_{\va} $ are well-known and one can refer to \cite{Hofmann1} and \cite{Taylor} for details. However, for operators $ \mathcal{L}_{\va}-\lambda I $ with $ \lambda\in\Sigma_{\theta_0} $ and $ \theta_0\in(0,\frac{\pi}{2}) $, the constructions and estimates of Green functions are still unknown. We will construct the Green functions for operators $ \mathcal{L}_{\va}-\lambda I $ in this paper and establish the uniform estimates of them. We point out that the constructions of Green functions for operators $ \mathcal{L}_{\va}-\lambda I $ with $ d\geq 3 $ are similar to which of the Green functions of elliptic operators with lower order terms in \cite{Xu1}. There are some differences between these two. The first is that the problems considered in this paper are about complex valued functions. This makes calculations a little more complicated. The second difference is that the impact of the parameter $ \lambda $ should be calculated explicitly. On the other hand, for $ d=2 $, we need to employ arguments in \cite{Dong2}, \cite{Dong1} and \cite{Taylor}. The constructions of Green functions for $ d\geq 3 $ and $ d=2 $ are rather different and to deal with the case $ d=2 $, we need some properties of $ \BMO $ space and Hardy space. Moreover, proofs of uniform estimates on two dimensional Green functions are much more difficult than that of the case $ d\geq 3 $. These estimates are new and are the most important innovations of this paper. What is essential is that all the uniform regularity estimates of Green functions in this paper are scaling invariant (see \cite{Xu2} for such estimates of fundamental solutions related to generalized elliptic operators with lower order terms).
After constructing Green functions for the operator $ \mathcal{L}_{\va}-\lambda I $, we will use almost the same methods in \cite{Kenig4} to prove Theorem \ref{Lpconres}-\ref{LpW1pconreso}. In view of the convergence rates with $ p=2 $, we can also see that estimates $ \eqref{**-} $ and $ \eqref{***-} $ are sharp.
The rest of this paper is organized as follows. In Section \ref{Preliminaries}, we will give some basic ingredients in the proof, including the $ L^2 $ regularity theory, Caccioppoli's inequality and some properties about $ \BMO $, Hardy space. In Section \ref{W1pand Lipschitz estimates for}, we will establish the $ W^{1,p} $, Hölder and Lipschitz estimates for the operator $ \mathcal{L}_{\va}-\lambda I $. In Section \ref{Lp estimates of resolventssection}, we will first prove Theorem \ref{Lp estimates of resolventsf} by using real variable methods. Then we will construct Green functions $ G_{\va,\lambda}(x,y) $ for operators $ \mathcal{L}_{\va}-\lambda I $ and obtain the regularity estimates for them. In Section \ref{Estimates of convergence of resolvents}, firstly, we will use standard methods given in \cite{Kenig3} to prove Theorem \ref{Approximation 1}. Next, we will calculate the convergence rates of Green functions for $ (\mathcal{L}_{\va}-\lambda I) $. Using these tools, we can prove Theorem \ref{Lpconres}-\ref{LpW1pconreso}.
\section{Preliminaries}\label{Preliminaries} \subsection{Energy estimates and Caccioppoli's inequality}
A notable observation gives that for any $ \xi=\xi^{(1)}+i\xi^{(2)}\in \mathbb{C}^{m\times d} $, where $ \xi^{(1)},\xi^{(2)}\in\mathbb{R}^{m\times d} $, \begin{align} a_{kj}^{\al\beta}(y)\xi_{k}^{\al}\overline{\xi_{j}^{\beta}}&=a_{kj}^{\al\beta}(y)\left(\xi_{k}^{(1)\al}+i\xi_{k}^{(2)\al}\right)\left(\xi_{j}^{(1)\beta}-i\xi_{j}^{(2)\beta}\right)=a_{kj}^{\al\beta}(y)\xi_{k}^{(1)\al}\xi_{j}^{(1)\beta}+a_{kj}^{\al\beta}(y)\xi_{k}^{(2)\al}\xi_{j}^{(2)\beta}.\nonumber \end{align} This, together with $ \eqref{el} $, gives the ellipticity condition for $ A $ with complex variables, i.e. \begin{align}
2\mu|\xi|^2\leq a_{ij}^{\al\beta}(y)\xi_{i}^{\al}\overline{\xi_{j}^{\beta}}\leq 2\mu^{-1}|\xi|^2\text{ for any } y\in\mathbb{R}^d\text{ and }\xi=(\xi_i^{\al})\in\mathbb{C}^{m\times d}.\label{ellcom} \end{align} Similarly, in view of $ \eqref{syl0} $ and $ \eqref{ell0} $, we can also infer that \begin{align}
2\mu|\xi|^2\leq \widehat{a}_{ij}^{\al\beta}\xi_{i}^{\al}\overline{\xi_{j}^{\beta}}\leq 2\mu^{-1}|\xi|^2\text{ for any } y\in\mathbb{R}^d\text{ and }\xi=(\xi_i^{\al})\in\mathbb{C}^{m\times d}.\label{ellcoml0} \end{align}
Moreover, for $ d\geq 2 $, let $ \om $ be a bounded domain in $ \mathbb{R}^d $, $ \va\geq 0 $, $ \lambda=|\lambda|e^{i\theta}\in\mathbb{C}\backslash(0,\infty) $ and $ A $ satisfy $ \eqref{sy},\eqref{el} $. Define a bilinear form $ B_{\va,\lambda,\om}[\cdot,\cdot]:H_0^1(\om;\mathbb{C}^m)\times H_0^1(\om;\mathbb{C}^m)\to\mathbb{C} $ by \begin{align} B_{\va,\lambda,\om}[u,v]=\int_{\om}A_{\va}(x)\nabla u(x)\overline{\nabla v(x)}dx-\lambda\int_{\om}u(x)\overline{v(x)}dx,\label{bilinearBva} \end{align} where $ A_{\va}(x)=A(x/\va) $ for $ \va>0 $ and $ A_0(x)=\widehat{A} $. For $ \lambda\in\mathbb{C}\backslash[0,\infty) $ and $ u\in H_0^1(\om;\mathbb{C}^m) $, \begin{align}
B_{\va,\lambda,\om}[u,u]=\int_{\om}A_{\va}(x)\nabla u(x)\overline{\nabla u(x)}dx-\lambda\int_{\om}|u(x)|^2dx.\label{buu} \end{align} If $ \operatorname{Re}(\lambda)\geq 0 $, we can take imaginary parts of both sides of $ \eqref{buu} $ and obtain that \begin{align}
|B_{\va,\lambda,\om}[u,u]|\geq|\operatorname{Im}(\lambda)|\int_{\om}|u(x)|^2dx\Rightarrow \|u\|_{L^2(\om)}^2\leq c(\lambda,\theta)|\lambda|^{-1}|B_{\va,\lambda,\om}[u,u]|.\label{buuim} \end{align} If $ \operatorname{Re}(\lambda)<0 $, we can take real parts of both sides of $ \eqref{buu} $ and get that \begin{align}
|B_{\va,\lambda,\om}[u,u]|\geq|\operatorname{Re}(\lambda)|\int_{\om}|u(x)|^2dx.\label{buure} \end{align} Adding $ \eqref{buure} $ by $ \eqref{buuim} $, it can be easily shown that \begin{align}
\|u\|_{L^2(\om)}^2\leq Cc(\lambda,\theta)|\lambda|^{-1}|B_{\va,\lambda,\om}[u,u]|\text{ for any }\lambda\in\mathbb{C}\backslash[0,\infty).\label{ubuuboun} \end{align} This, together with $ \eqref{ellcom} $, $ \eqref{ellcoml0} $ and $ \eqref{buu} $, implies that \begin{align}
\mu\|\nabla u\|_{L^2(\om)}^2\leq |B_{\va,\lambda,\om}[u,u]|+|\lambda|\|u\|_{L^2(\om)}^2\leq 2c(\lambda,\theta)|B_{\va,\lambda,\om}[u,u]|.\label{laxmil} \end{align} If $ \lambda=0 $, it is obvious that $ \eqref{laxmil} $ is still true. By using Poincaré's inequality, then \begin{align}
\|u\|_{L^2(\om)}^2\leq CR_0^2\|\nabla u\|_{L^2(\om)}^2\leq Cc(\lambda,\theta)R_0^2|B_{\va,\lambda,\om}[u,u]|,\nonumber \end{align} which, together with $ \eqref{ubuuboun} $, implies that for any $ \lambda\in\mathbb{C}\backslash(0,\infty) $, \begin{align}
\|u\|_{L^2(\om)}^2\leq Cc(\lambda,\theta)(R_0^{-2}+|\lambda|)^{-1}|B_{\va,\lambda,\om}[u,u]|,\label{ABCDE} \end{align} where $ C $ depends only on $ \mu,d,m $ and $ \om $. Another important fact is that if $ A $ satisfies $ \eqref{sy} $, the adjoint operator of $ \mathcal{L}_{\va}-\lambda I $ is $ \mathcal{L}_{\va}-\overline{\lambda} I $. Indeed, \begin{align} \langle(\mathcal{L}_{\va}-\lambda I)(u),v\rangle_{H^{-1}(\om)\times H_0^1(\om)}&=B_{\va,\lambda,\om}[u,v]=\overline{B_{\va,\overline{\lambda},\om}[v,u]}=\langle u,(\mathcal{L}_{\va}-\overline{\lambda} I)(v)\rangle_{H_0^1(\om)\times H^{-1}(\om)}.\nonumber \end{align} In view of $ \eqref{laxmil} $ and well-known Lax-Milgram theorem, it is easy to show the following theorem for existence of solutions corresponding to operators $ \mathcal{L}_{\va}-\lambda I $. \begin{thm}\label{existencethm} Let $ \om $ be a bounded $ C^1 $ domain in $ \mathbb{R}^d $ with $ d\geq 2 $. Assume that $ A $ satisfies $ \eqref{sy} $ and $ \eqref{el} $. Then for any $ \va\geq 0 $, $ F\in H^{-1}(\om;\mathbb{C}^m) $, there is a unique weak solution $ u_{\va,\lambda}\in H_0^1(\om;\mathbb{C}^m) $ satisfying the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $. That is, for any $ \varphi\in H_0^1(\om;\mathbb{C}^m) $, \begin{align} B_{\va,\lambda,\om}[u_{\va,\lambda},\varphi]=\langle F,\varphi\rangle_{H^{-1}(\om)\times H_0^1(\om)}.\nonumber \end{align} \end{thm}
\begin{lem}\label{l2reses}
Suppose that $ d\geq 2 $ and $ \lambda=|\lambda|e^{i\theta}\in\mathbb{C}\backslash(0,\infty) $. Let $ \om $ be a bounded $ C^{1} $ domain in $ \mathbb{R}^d $. Assume that $ A $ satisfies $ \eqref{sy} $ and $ \eqref{el} $. Then for any $ \va\geq 0 $, $ F\in L^2(\om;\mathbb{C}^m) $ and $ f\in L^2(\om;\mathbb{C}^{m\times d}) $, there exists a unique weak solution $ u_{\va,\lambda}\in H_0^1(\om;\mathbb{C}^m) $ of the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F+\operatorname{div}(f) $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $, satisfying the uniform energy estimates \be \begin{aligned}
\|u_{\va,\lambda}\|_{L^2(\om)}&\leq Cc(\lambda,\theta)\left\{(R_0^{-2}+|\lambda|)^{-1}\|F\|_{L^2(\om)}+(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|f\|_{L^2(\om)}\right\},\\
\|\nabla u_{\va,\lambda}\|_{L^2(\om)}&\leq Cc(\lambda,\theta)\left\{(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|F\|_{L^2(\om)}+\|f\|_{L^2(\om)}\right\}, \end{aligned}\label{L2uva} \ee where $ C $ depends only on $ \mu,d,m $ and $ \om $. Moreover, if $ \om $ is a bounded $ C^{1,1} $ domain, $ \va=0 $ and $ f\equiv 0 $, then \begin{align}
\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}\leq Cc(\lambda,\theta)\|F\|_{L^2(\om)}.\label{L2n2u0} \end{align} In operator forms, \begin{align}
\|R(\lambda,\mathcal{L}_{\va})\|_{L^2(\om)\to L^2(\om)}&\leq Cc(\lambda,\theta)(R_0^{-2}+|\lambda|)^{-1},\label{L2o1}\\
\max\left\{\|R(\lambda,\mathcal{L}_{\va})\|_{H^{-1}(\om)\to L^2(\om)},\|R(\lambda,\mathcal{L}_{\va})\|_{L^2(\om)\to H_0^1(\om)}\right\}&\leq Cc(\lambda,\theta)(R_0^{-2}+|\lambda|)^{-\frac{1}{2}},\label{L2o2}\\
\max\left\{\|R(\lambda,\mathcal{L}_{0})\|_{L^2(\om)\to H^2(\om)},\|R(\lambda,\mathcal{L}_{\va})\|_{H^{-1}(\om)\to H_0^1(\om)}\right\}&\leq Cc(\lambda,\theta).\label{L2o4} \end{align} \end{lem} \begin{proof} The existence is ensured by Theorem \ref{existencethm}, we only need to show $ \eqref{L2uva} $ and $ \eqref{L2n2u0} $. Firstly, we can choose $ u_{\va,\lambda}^{(1)} $ and $ u_{\va,\lambda}^{(2)} $ in $ H_0^1(\om;\mathbb{C}^m) $ such that \begin{align} \left\{\begin{matrix} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda}^{(1)})=F&\text{in}&\om,\\ u_{\va,\lambda}^{(1)}=0&\text{on}&\pa\om, \end{matrix}\right.\quad\text{and}\quad \left\{\begin{matrix} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda}^{(2)})=\operatorname{div}(f)&\text{in}&\om,\\ u_{\va,\lambda}^{(2)}=0&\text{on}&\pa\om. \end{matrix}\right.\nonumber \end{align} The next thing is to estimate $ u_{\va,\lambda}^{(1)} $ and $ u_{\va,\lambda}^{(2)} $ respectively. For $ u_{\va,\lambda}^{(1)} $, we need to prove that \begin{align}
\|\nabla u_{\va,\lambda}^{(1)}\|_{L^2(\om)}+(R_0^{-2}+|\lambda|)^{\frac{1}{2}}\|u_{\va,\lambda}^{(1)}\|_{L^2(\om)}&\leq Cc(\lambda,\theta)(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|F\|_{L^2(\om)}.\label{ww71} \end{align} Choosing $ u_{\va,\lambda}^{(1)} $ as the test function, it can be obtained that \begin{align} B_{\va,\lambda,\om}[u_{\va,\lambda}^{(1)},u_{\va,\lambda}^{(1)}]=\int_{\om}F(x)\overline{u_{\va,\lambda}^{(1)}(x)}dx=(F,u_{\va,\lambda}^{(1)})_{L^2(\om)\times L^2(\om)}.\label{wef1} \end{align} In view of $ \eqref{laxmil} $ and $ \eqref{ABCDE} $, we have \begin{align}
\|u_{\va,\lambda}^{(1)}\|_{L^2(\om)}^2&\leq Cc(\lambda,\theta)(R_0^{-2}+|\lambda|)^{-1}\|u_{\va,\lambda}^{(1)}\|_{L^2(\om)}\|F\|_{L^2(\om)},\nonumber\\
\mu\|\nabla u_{\va,\lambda}^{(1)}\|_{L^2(\om)}^2&\leq Cc(\lambda,\theta)\|u_{\va,\lambda}^{(1)}\|_{L^2(\om)}\|F\|_{L^2(\om)},\nonumber \end{align} which complete the proof of $ \eqref{ww71} $. For $ u_{\va,\lambda}^{(2)} $, it suffices to show that \begin{align}
\|\nabla u_{\va,\lambda}^{(2)}\|_{L^2(\om)}+(R_0^{-2}+|\lambda|)^{\frac{1}{2}}\|u_{\va,\lambda}^{(2)}\|_{L^2(\om)}&\leq Cc(\lambda,\theta)\|f\|_{L^2(\om)}.\label{flam-1} \end{align} Choosing $ u_{\va,\lambda}^{(2)} $ as the test function, then \begin{align} B_{\va,\lambda,\om}[u_{\va,\lambda}^{(2)},u_{\va,\lambda}^{(2)}]=-\int_{\om}f(x)\overline{\nabla u_{\va,\lambda}^{(2)}(x)}dx=-(f,\nabla u_{\va,\lambda}^{(2)})_{L^2(\om)\times L^2(\om)}.\label{wef2} \end{align} Again, by applying $ \eqref{laxmil} $ and $ \eqref{ABCDE} $, it is easy to get that \begin{align}
\|u_{\va,\lambda}^{(2)}\|_{L^2(\om)}^2&\leq Cc(\lambda,\theta)(R_0^{-2}+|\lambda|)^{-1}\|\nabla u_{\va,\lambda}^{(2)}\|_{L^2(\om)}\|f\|_{L^2(\om)},\nonumber\\
\mu\|\nabla u_{\va,\lambda}^{(2)}\|_{L^2(\om)}^2&\leq Cc(\lambda,\theta)\|\nabla u_{\va,\lambda}^{(2)}\|_{L^2(\om)}\|f\|_{L^2(\om)},\nonumber \end{align} which implies $ \eqref{flam-1} $. At last, for the proof of $ \eqref{L2n2u0} $, since $ \om $ is $ C^{1,1} $, one can deduce that \begin{align}
\|\nabla^2u_{0,\lambda}\|_{L^2(\om)}\leq C\left\{\|F\|_{L^2(\om)}+|\lambda|\|u_{0,\lambda}\|_{L^2(\om)}\right\}\leq Cc(\lambda,\theta)\|F\|_{L^2(\om)},\nonumber \end{align} where, for the first inequality, we have used standard $ H^2 $ estimates for elliptic systems with constant coefficients and for the second inequality, we have used $ \eqref{L2uva} $. \end{proof}
\begin{rem} For $ d\geq 3 $ and $ \lambda\in\mathbb{C}\backslash(0,\infty) $, using Hölder's inequality and Sobolev embedding theorem $ H_0^1(\om)\subset L^{\frac{2d}{d-2}}(\om) $, we have \begin{align}
|(F,u_{\va,\lambda}^{(1)})_{L^2(\om)\times L^2(\om)}|\leq\|F\|_{L^{\frac{2d}{d+2}}(\om)}\|u_{\va,\lambda}^{(1)}\|_{L^{\frac{2d}{d-2}}(\om)}\leq \|F\|_{L^{\frac{2d}{d+2}}(\om)}\|\nabla u_{\va,\lambda}^{(1)}\|_{L^2(\om)}.\nonumber \end{align} By using the same arguments in the proof of Theorem \ref{l2reses}, it is easy to find that for any $ \va\geq 0 $, \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^2(\om)}+(R_0^{-2}+|\lambda|)^{\frac{1}{2}}\|u_{\va,\lambda}\|_{L^2(\om)}\leq Cc(\lambda,\theta)\left\{\|F\|_{L^{\frac{2d}{d+2}}(\om)}+\|f\|_{L^{2}(\om)}\right\},\label{u2dd+2} \end{align} where $ C $ depends only on $ \mu,d,m $ and $ \om $. Similarly, If $ d=2 $, in view of the Sobolev embedding $ H_0^1(\om)\subset L^{q'}(\om) $ with $ q'=\frac{q}{q-1}>1 $, it can be easily seen that for any $ q>1 $, \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^2(\om)}+(R_0^{-2}+|\lambda|)^{\frac{1}{2}}\|u_{\va,\lambda}\|_{L^2(\om)}\leq Cc(\lambda,\theta)\left\{R_0^{2(1-\frac{1}{q})}\|F\|_{L^{q}(\om)}+\|f\|_{L^{2}(\om)}\right\},\label{uq} \end{align} where $ C $ depends only on $ \mu,m,q $ and $ \om $. \end{rem}
Next, we will establish the Caccioppoli's inequality for the operator $ \mathcal{L}_{\va}-\lambda I $. In some point of view, Caccioppoli's inequality can be seen as the localization of $ \eqref{u2dd+2} $. In this paper, we need to obtain Caccioppoli's inequality that is scaling invariant. This means that $ \lambda $ cannot be regarded as a constant and its influence on the constants of this inequality should be calculated explicitly. The scaling invariant Caccioppoli's inequality plays a vital role in the proofs of other scaling invariant inequalities. The idea comes from \cite{Shen1,Xu2} and we combine the methods in both of them to complete the proof for the sake of completeness. To simplify the notations, we can define \begin{align} \om(x_0,R)=\om\cap B(x_0,R)\quad\text{and}\quad \Delta(x_0,R)=\pa\om\cap B(x_0,R)\label{bn} \end{align} for any $ 0<R<R_0 $ and $ x_0\in\om $. Sometimes we will use $ \om_r $ and $ \Delta_r $ to denote $ \om(x_0,r) $ and $ \Delta(x_0,r) $ if no confusion would be made.
\begin{lem}[Caccioppoli's inequality]\label{Caccio} Let $ \va\geq 0 $, $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ \om $ be a bounded $ C^1 $ domain in $ \mathbb{R}^d $. Assume that $ A $ satisfies $ \eqref{sy} $ and $ \eqref{el} $. Suppose that $ 0<R<R_0 $, $ x_0\in\om $, $ f\in L^2(\om(x_0,2R);\mathbb{C}^m) $ and $ F\in L^{q}(\om(x_0,2R);\mathbb{C}^m) $, where $ q=\frac{2d}{d+2} $ if $ d\geq 3 $ and $ q>1 $ if $ d=2 $. If $ \Delta(x_0,2R)\neq\emptyset $, assume that $ u_{\va,\lambda}\in H^1(\om(x_0,2R),\mathbb{C}^m) $ is the weak solution of the boundary problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F+\operatorname{div}(f)\text{ in }\om(x_0,2R)\quad\text{and}\quad u_{\va,\lambda}=0\text{ on }\Delta(x_0,2R).\nonumber \end{align} If $ \Delta(x_0,2R)=\emptyset $, assume that $ u_{\va,\lambda}\in H^1(B(x_0,2R);\mathbb{C}^m) $ is the weak solution of the interior problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F+\operatorname{div}(f)\text{ in } B(x_0,2R).\nonumber \end{align} Then for any $ k\in\mathbb{N}_+ $, \be \begin{aligned}
\left(\Xint-_{\om(x_0,R)}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}}\left(\Xint-_{\om(x_0,2R)}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\\
&\quad+\frac{C_{k,\theta_0}R}{(1+|\lambda|R^2)^{\frac{1}{2}}}\left\{R\left(\Xint-_{\om(x_0,2R)}|F|^q\right)^{\frac{1}{q}}+\left(\Xint-_{\om(x_0,2R)}|f|^2\right)^{\frac{1}{2}}\right\}, \end{aligned}\label{Cau1} \ee \be \begin{aligned}
\left(\Xint-_{\om(x_0,R)}|\nabla u_{\va,\lambda}|^2\right)^{\frac{1}{2}}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}R}\left(\Xint-_{\om(x_0,2R)}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\\
&\quad+C_{k,\theta_0}\left\{R\left(\Xint-_{\om(x_0,2R)}|F|^q\right)^{\frac{1}{q}}+\left(\Xint-_{\om(x_0,2R)}|f|^2\right)^{\frac{1}{2}}\right\}, \end{aligned}\label{Cau2} \ee where $ C_{k,\theta_0} $ depends only on $ \mu,d,m,q,k,\theta_0,\om $. \end{lem}
\begin{proof}
We will only prove the case for $ d\geq 3 $, since the case $ d=2 $ follows from almost the same methods by substituting the Sobolev embedding theorem $ H_0^1\subset L^{\frac{2d}{d-2}} $ with $ H_0^1\subset L^{q'} $, $ 1<q<\infty $. Firstly, we can assume that $ R=1 $ and $ x_0=0 $ and for general case, $ \eqref{Cau1} $ and $ \eqref{Cau2} $ follows by rescaling and translation. For $ \lambda =0 $, the results are obvious by standard Caccioppoli's inequality for the operator $ \mathcal{L}_{\va} $. Then we can assume that $ \lambda\neq 0 $. Choose $ \varphi\in C_0^{\infty}(B(0,2);\mathbb{R}) $ such that $ \varphi\equiv 1 $ for $ x\in B(0,1) $, $ \varphi\equiv 0 $ for $ x\in B(0,\frac{3}{2})^c $, $ |\nabla\varphi|\leq C $ and $ 0\leq\varphi\leq 1 $. We can set $ \psi=\varphi^2u_{\va,\lambda} $ as the test function. By applying the definition of weak solutions, we have \begin{align}
\int_{\om_2}A_{\va}\nabla u_{\va,\lambda}\overline{\nabla(\varphi^2u_{\va,\lambda})}-\lambda \int_{\om_2}\varphi^2|u_{\va,\lambda}|^2=\int_{\om_2}F\overline{\varphi^2u_{\va,\lambda}}-\int_{\om_2}f\overline{\nabla(\varphi^2u_{\va,\lambda})},\label{ww1} \end{align} where $ A_{\va}=A(x/\va) $ if $ \va>0 $ and $ A_{0}=\widehat{A} $. In the following calculations, we use $ C_{\theta_0} $ to denote a constant depending on $ \theta_0 $ and $ C_{k,\theta_0} $ a constant depending on $ \theta_0 $ and $ k $. If $ \operatorname{Re}\lambda\geq 0 $, we can take the imaginary parts of both sides of $ \eqref{ww1} $, then \be \begin{aligned}
&|\operatorname{Im}\lambda|\int_{\om_2}\varphi^2|u_{\va,\lambda}|^2\\
&\quad\quad\leq C\left\{\int_{\om_2}\varphi|\nabla u_{\va,\lambda}||\nabla\varphi||u_{\va,\lambda}|+\int_{\om_2}\varphi^2|F||u_{\va,\lambda}|+\int_{\om_2}\varphi|f|\left(|\nabla\varphi||u_{\va,\lambda}|+\varphi|\nabla u_{\va,\lambda}|\right)\right\}. \end{aligned}\label{ww42}
\ee Noticing that $ |\lambda|=c(\lambda,\theta)|\operatorname{Im}\lambda| $, it actually means that \begin{align}
\int_{\om_2}\varphi^2|u_{\va,\lambda}|^2&\leq \frac{C_{\theta_{0}}}{|\lambda|}\left\{\int_{\om_2}|\nabla\varphi|^2|u_{\va,\lambda}|^2+\delta\int_{\om_2}\varphi^2|\nabla u_{\va,\lambda}|^2dx+\int_{\om_2}|\varphi F||\varphi u_{\va,\lambda}|+\int_{\om_2}\varphi^2|f|^2\right\},\nonumber \end{align} where we have used Cauchy's inequality and the following inequality \begin{align} ab\leq \delta a^2+\frac{1}{4\delta}b^2\text{ for any }\delta>0\text{ and }a,b>0,\label{inte} \end{align} with $ \delta>0 $ being sufficiently small. By using Hölder's inequality, it is easy to see that \be \begin{aligned}
\int_{\om_2}\varphi^2|u_{\va,\lambda}|^2&\leq \frac{C_{\theta_{0}}}{|\lambda|}\left\{\int_{\om_2}|\nabla\varphi|^2|u_{\va,\lambda}|^2+\delta\int_{\om_2}\varphi^2|\nabla u_{\va,\lambda}|^2\right.\\
&\quad\left.+\left(\int_{\om_2}|\varphi F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}\left(\int_{\om_2}|\varphi u_{\va,\lambda}|^{\frac{2d}{d-2}}\right)^{\frac{d-2}{2d}}+\int_{\om_2}\varphi^2|f|^2\right\}. \end{aligned}\label{ww6} \ee If $ \operatorname{Re}\lambda<0 $, we can take real parts of both sides of $ \eqref{ww1} $ and conclude that \be \begin{aligned}
&|\operatorname{Re}\lambda|\int_{\om_2}\varphi^2|u_{\va,\lambda}|^2\\
&\quad\quad\leq C\left\{\int_{\om_2}\varphi|\nabla u_{\va,\lambda}||\nabla\varphi||u_{\va,\lambda}|+\int_{\om_2}\varphi^2|F||u_{\va,\lambda}|+\int_{\om_2}\varphi|f|\left(|\nabla\varphi||u_{\va,\lambda}|+\varphi|\nabla u_{\va,\lambda}|\right)\right\}. \end{aligned}\label{ww43} \ee By adding $ \eqref{ww42} $ to $ \eqref{ww43} $, we can also derive $ \eqref{ww6} $ for the case $ \operatorname{Re}\lambda<0 $. Then $ \eqref{ww6} $ is true for any $ \lambda\in\Sigma_{\theta_0} $. Owing to $ H_0^1(\om_2)\subset L^{\frac{2d}{d-2}}(\om_2) $ and $ \eqref{inte} $, it can be seen that \begin{align}
&\left(\int_{\om_2}|\varphi F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}\left(\int_{\om_2}|\varphi u_{\va,\lambda}|^{\frac{2d}{d-2}}\right)^{\frac{d-2}{2d}}\leq C\left(\int_{\om_2}|\varphi F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{d}}+\delta\left(\int_{\om_2}|\varphi u_{\va,\lambda}|^{\frac{2d}{d-2}}\right)^{\frac{d-2}{d}}\nonumber\\
&\quad\quad\quad\quad\quad\quad\leq C\left(\int_{\om_2}|\varphi F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{d}}+\delta\left(\int_{\om_2}(|\nabla\varphi u_{\va,\lambda}|+|\varphi\nabla u_{\va,\lambda}|)^{2}\right)^{\frac{1}{2}}.\nonumber \end{align} This, together with $ \eqref{ww6} $ and the properties of $ \varphi $, leads to \begin{align}
\left(\int_{\om_1}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\leq \frac{C_{\theta_0}}{|\lambda|^{\frac{1}{2}}}\left\{\left(\int_{\om_2}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}+\left(\int_{\om_2}|F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}+\left(\int_{\om_2}|f|^2\right)^{\frac{1}{2}}\right\}.\label{ww5} \end{align} Combining $ \eqref{ww1} $ and $ \eqref{ww5} $, we have \begin{align}
\left(\int_{\om_1}|\nabla u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\leq C_{\theta_0}\left\{\left(\int_{\om_2}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}+\left(\int_{\om_2}|F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}+\left(\int_{\om_2}|f|^2\right)^{\frac{1}{2}}\right\}.\label{ww4} \end{align}
In view of the obvious fact that $ \|u_{\va,\lambda}\|_{L^2(\om_1)}\leq C\|u_{\va,\lambda}\|_{L^2(\om_2)} $, we can deduce from $ \eqref{ww5} $ that for any $ \lambda\in\Sigma_{\theta_0} $, \begin{align}
\left(\int_{\om_1}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\leq \frac{C_{\theta_0}}{(1+|\lambda|)^{\frac{1}{2}}}\left\{\left(\int_{\om_2}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}+\left(\int_{\om_2}|F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}+\left(\int_{\om_2}|f|^2\right)^{\frac{1}{2}}\right\}.\nonumber \end{align} This, together with $ \eqref{ww4} $, gives the proof by repeating the procedure for $ 2k $ times. \end{proof}
\begin{rem}
Choose $ \varphi\in C_0^{\infty}(B(0,2R);\mathbb{R}) $ such that $ \varphi\equiv 1 $ for $ x\in B(x_0,R) $, $ \varphi\equiv 0 $ for $ x\in B(x_0,\frac{3}{2}R)^c $, $ |\nabla\varphi|\leq C/R $ and $ 0\leq\varphi\leq 1 $. According to $ \eqref{ww42} $, $ \eqref{inte} $ and $ \eqref{ww43} $, it can be easily seen that, if $ \lambda\neq 0 $, $ f\equiv 0 $ and $ F\equiv 0 $, then \begin{align}
\int_{\om(x_0,2R)}\varphi^2|u_{\va,\lambda}|^2&\leq\frac{C_{\theta_0}}{|\lambda|}\int_{\om(x_0,2R)}\varphi|\nabla u_{\va,\lambda}||\nabla\varphi||u_{\va,\lambda}|\nonumber\\
&\leq \frac{C_{\theta_0}}{|\lambda|^2}\int_{\om(x_0,2R)}|\nabla\varphi|^2|\nabla u_{\va,\lambda}|^2+\frac{1}{2}\int_{\om(x_0,2R)}\varphi^2|u_{\va,\lambda}|^2,\nonumber \end{align} which, together with the properties of $ \varphi $, implies that \begin{align}
\left(\Xint-_{\om(x_0,R)}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\leq\frac{C_{\theta_0}}{|\lambda|}\left(\Xint-_{\om(x_0,2R)}|\nabla\varphi|^2|\nabla u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\leq \frac{C_{\theta_0}}{|\lambda|R}\left(\Xint-_{\om(x_0,2R)}|\nabla u_{\va,\lambda}|^2\right)^{\frac{1}{2}}.\label{fanxiangzy} \end{align} \end{rem}
\subsection{$ \operatorname{BMO} $ and Hardy space} To deal with the Green functions with $ d=2 $, we will need some basic knowledge on real analysis, mainly about $ \BMO $ space and Hardy space. \begin{defn}[$ \operatorname{BMO} $ space and atom functions]\label{BMO atom} Let $ d\geq 2 $. Assume that $ x_0\in\mathbb{R}^d $, $ r>0 $ and $ \om $ is a bounded domain in $ \mathbb{R}^d $. Denote $ \om(x_0,r)=\om\cap B(x_0,r) $ as before. $ \operatorname{BMO}(\om;\mathbb{C}^m) $ is a space containing all measurable, $ \mathbb{C}^m $-valued functions such that \begin{align}
\left\|u\right\|_{\BMO(\om)}=\sup\left\{\Xint-_{\om(x_0,r)}|u-u_{x_0,r}|:x_0\in\overline{\om},r>0\right\}<\infty,\label{BMOmo} \end{align} where \begin{align} u_{x_0,r}:=\left\{\begin{array}{ccc} 0 & \text{ if } & r\geq \delta(x_0)=\operatorname{dist}(x_0,\pa\om), \\ \Xint-_{\om(x_0,r)}u & \text{ if } & r< \delta(x_0)=\operatorname{dist}(x_0,\pa\om). \end{array}\right.\label{ww35} \end{align} We call a bounded measurable function $ a(x) $ as an atom function in $ \om $ if $ \operatorname{supp}(a)\subset\om(x_0,r) $ with $ x_0\in\overline{\om} $, $ 0<r<R_0 $ and \begin{align}
\left\|a\right\|_{L^{\infty}(\om)}\leq\frac{1}{|\om(x_0,r)|},\quad a_{x_0,r}=0. \nonumber \end{align} \end{defn}
\begin{defn}[Hardy space] Let $ \om $ be a $ C^1 $ domain in $ \mathbb{R}^d $ with $ d\geq 2 $. A function $ f $ is an element in the Hardy space $ \mathcal{H}^1(\om;\mathbb{C}^m) $, if there exist a sequence of atoms $ \left\{a_i\right\}_{i=1}^{\infty} $ and a sequence of complex numbers $ \left\{\eta_{i}\right\}_{i=1}^{\infty}\subset l^1(\mathbb{C}) $, such that $ f(x)=\sum_{i=1}^{\infty}\eta_ia_i(x) $. We define the norm in this space by \begin{align}
\left\|f\right\|_{\mathcal{H}^1(\om)}=\inf\left\{\sum_{i=1}^{\infty}|\eta_i|:f(x)=\sum_{i=1}^{\infty}\eta_ia_i(x)\right\}.\nonumber \end{align} We notice the expression \begin{align} \sup\left\{\int_{\om}a(y)u(y)dy:a=a(x)\text{ is an atom in }\om\right\},\nonumber \end{align} gives the equivalent norm of $ \operatorname{BMO}(\om) $. This is because that $ \operatorname{BMO}(\om) $ is the dual space of $ \mathcal{H}^1 $. \end{defn}
\section{Uniform regularity estimates}\label{W1pand Lipschitz estimates for} It is well-known that for the homogenization problem with real vector values, the $ W^{1,p} $ estimates for $ u_{\va} $ are established (see \cite{Av1} and \cite{Shen3}). By simple observations, we can obtain similar results for the problem with complex vector values.
\begin{lem}[$ W^{1,p} $ estimates for the operator $ \mathcal{L}_{\va} $] For $\va\geq 0 $ and $ d\geq 2 $, let $ 2<p<\infty $ and $ \om $ be a bounded $ C^1 $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{VMO} $. Assume that $ f=(f_{i}^{\al})\in L^p(\om;\mathbb{C}^{m\times d}) $ and $ F\in L^q(\om;\mathbb{C}^m) $ with $ q=\frac{pd}{p+d} $. Then the weak solution of $ \mathcal{L}_{\va}(u_{\va})=F+\operatorname{div}(f) $ in $ \om $ and $ u_{\va}=0 $ on $ \pa\om $ satisfies the uniform estimate \begin{align}
\|\nabla u_{\va}\|_{L^p(\om)}\leq C\left\{\|f\|_{L^p(\om)}+\|F\|_{L^q(\om)}\right\},\label{W1pom} \end{align} where $ C $ depends only on $ \mu,d,m,p,q,\omega(t) $ and $ \om $. \end{lem} \begin{proof} For the case that $ f\in L^p(\om;\mathbb{R}^{m\times d}) $ and $ F\in L^q(\om;\mathbb{R}^m) $, the proof is trivial. For $ f\in L^p(\om;\mathbb{C}^{m\times d}) $ and $ F\in L^q(\om;\mathbb{C}^m) $, we can write that $ f=g+ih $ and $ F=H+iG $ with $ g,h\in L^p(\om;\mathbb{R}^{m\times d}) $, $ G,H\in L^q(\om;\mathbb{R}^m) $ and use the $ W^{1,p} $ estimates for the case of real functions. \end{proof}
\begin{thm}[Localization of $ W^{1,p} $ estimates for the operator $ \mathcal{L}_{\va}-\lambda I $]\label{LoW1p} For $ \va\geq 0 $ and $ d\geq 2 $, let $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, $ \om $ be a bounded $ C^1 $ domain in $ \mathbb{R}^d $ and $ 2<p<\infty $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{VMO} $. Assume that $ x_0\in\om $, $ 0<R<R_0 $, $ f\in L^p(\om(x_0,2R);\mathbb{C}^{m\times d}) $, $ F\in L^q(\om(x_0,2R);\mathbb{C}^m)$ and $ q=\frac{pd}{p+d} $. If $ \Delta(x_0,2R)\neq\emptyset $, assume that $ u_{\va,\lambda}\in H^1(\om(x_0,2R);\mathbb{C}^m) $ is the weak solution of the boundary problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F+\operatorname{div}(f)\text{ in }\om(x_0,2R)\quad\text{and}\quad u_{\va,\lambda}=0\text{ on }\Delta(x_0,2R).\nonumber \end{align} If $ \Delta(x_0,2R)=\emptyset $, assume that $ u_{\va,\lambda}\in H^1(B(x_0,2R);\mathbb{C}^m) $ is the weak solution of the interior problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F+\operatorname{div}(f)\text{ in } B(x_0,2R).\nonumber \end{align} Then there exists $ n\in\mathbb{N}_+ $, a constant integer depending only on $ d $, such that for any $ k\in\mathbb{N}_+ $, \be \begin{aligned}
\left(\Xint-_{\om(x_0,R)}|\nabla u_{\va,\lambda}|^p\right)^{\frac{1}{p}}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}R}\left(\Xint-_{\om(x_0,2R)}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\\
&\quad+C_{k,\theta_0}(1+|\lambda|R^2)^{n}\left\{R\left(\Xint-_{\om(x_0,2R)}|F|^q\right)^{\frac{1}{q}}+\left(\Xint-_{\om(x_0,2R)}|f|^p\right)^{\frac{1}{p}}\right\}, \end{aligned}\label{loW1pnu} \ee \be \begin{aligned}
\left(\Xint-_{\om(x_0,R)}|u_{\va,\lambda}|^p\right)^{\frac{1}{p}}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}}\left(\Xint-_{\om(x_0,2R)}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\\
&\quad+C_{k,\theta_0}(1+|\lambda|R^2)^{n}\left\{R^2\left(\Xint-_{\om(x_0,2R)}|F|^q\right)^{\frac{1}{q}}+R\left(\Xint-_{\om(x_0,2R)}|f|^p\right)^{\frac{1}{p}}\right\}, \end{aligned}\label{loW1pu} \ee where $ C_{k,\theta_0} $ depends only on $ \mu,d,m,k,\theta_0,p,\omega(t) $ and $\om $. \end{thm} \begin{proof}
By rescaling and translation, it can be assumed that $ R=1 $ and $ x_0=0 $. We can choose $ \varphi\in C_0^{\infty}(B(0,2);\mathbb{R}) $ such that $ 0\leq\varphi\leq 1 $, $ \varphi\equiv 1 $ in $ B(0,1) $, $ \varphi\equiv 0 $ in $ B(0,\frac{3}{2})^c $ and $ |\nabla\varphi|\leq C $. Then by setting $ A_{\va}$ as $ A(x/\va) $ if $ \va>0 $ and $ \widehat{A} $ if $ \va=0 $, we have \begin{align} \mathcal{L}_{\va}(\varphi u_{\va,\lambda})=F_{\va}+\operatorname{div}(f_{\va})\text{ in }\om_2\quad\text{and}\quad \varphi u_{\va,\lambda}=0\text{ on } \pa(\om_2),\nonumber \end{align} where $ F_{\va},f_{\va} $ are defined by \begin{align} F_{\va}=\lambda \varphi u_{\va,\lambda}+\varphi F-f\nabla\varphi-A_{\va}\nabla u_{\va,\lambda}\nabla\varphi\quad\text{and}\quad f_{\va}=\varphi f-A_{\va}\nabla \varphi u_{\va,\lambda}.\nonumber \end{align} Then owing to $ \eqref{W1pom} $ and Hölder's inequality, it follows that \begin{align}
&\|\nabla(\varphi u_{\va,\lambda})\|_{L^p(\om_{3/2})}\leq C\left\{\|F_{\va}\|_{L^q(\om_{3/2})}+\|f_{\va}\|_{L^p(\om_{3/2})}\right\}\nonumber\\
&\quad\quad\leq C\left\{|\lambda|\|u_{\va,\lambda}\|_{L^q(\om_{3/2})}+\|F\|_{L^q(\om_{3/2})}+\|u_{\va,\lambda}\|_{L^p(\om_{3/2})}+\|\nabla u_{\va,\lambda}\|_{L^q(\om_{3/2})}+\|f\|_{L^p(\om_{3/2})}\right\},\nonumber \end{align} where $ q=\frac{pd}{p+d} $. By using Hölder's inequality again, this implies that \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^p(\om_1)}&\leq C\left\{(1+|\lambda|)\|u_{\va,\lambda}\|_{L^p(\om_{3/2})}+\|F\|_{L^q(\om_{3/2})}+\|\nabla u_{\va,\lambda}\|_{L^q(\om_{3/2})}+\|f\|_{L^p(\om_{3/2})}\right\}\nonumber\\
&\leq C\left\{(1+|\lambda|)\left(\|\nabla u_{\va,\lambda}\|_{L^q(\om_2)}+\|u_{\va,\lambda}\|_{L^2(\om_2)}\right)+\|F\|_{L^q(\om_2)}+\|f\|_{L^p(\om_2)}\right\}.\nonumber \end{align} Here, we have used the inequality that for any $ 2<p<\infty $ and $ u\in W^{1,q}(\om;\mathbb{C}^m) $ with $ q=\frac{pd}{p+d} $, \begin{align}
\|u\|_{L^p(\om)}\leq C\|u\|_{W^{1,q}(\om)}\leq C\left\{\|\nabla u\|_{L^q(\om)}+\|u\|_{L^2(\om)}\right\},\label{aux2} \end{align} where $ C $ depends only on $ p,d,m,\om $. One can refer to \cite{RA} for details about $ \eqref{aux2} $. By iterating for finite times (depending only on $ d $), we can get $ n=n(d)\in\mathbb{N}_+ $, such that \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^p(\om_1)}\leq C(1+|\lambda|)^{n}\|u_{\va,\lambda}\|_{L^2(\om_2)}+C(1+|\lambda|)^{n-1}\left\{\|F\|_{L^q(\om_2)}+\|f\|_{L^p(\om_2)}\right\}.\nonumber \end{align} In view of $ \eqref{Cau1} $, this yields that for any $ k\in\mathbb{N}_+ $, \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^p(\om_1)}&\leq \frac{C_{k,\theta_0}(1+|\lambda|)^{n}}{(1+|\lambda|)^{k+n}}\|u_{\va,\lambda}\|_{L^2(\om_2)}+C_{k,\theta_0}(1+|\lambda|)^{n}\left\{\|F\|_{L^q(\om_2)}+\|f\|_{L^p(\om_2)}\right\}\nonumber\\
&\leq C_{k,\theta_0}(1+|\lambda|)^{-k}\|u_{\va,\lambda}\|_{L^2(\om_2)}+C_{k,\theta_0}(1+|\lambda|)^{n}\left\{\|F\|_{L^q(\om_2)}+\|f\|_{L^p(\om_2)}\right\}.\nonumber \end{align} This inequality implies $ \eqref{loW1pnu} $. On the other hand, $ \eqref{loW1pu} $ follows from Poincaré's inequality and the same arguments of localization. \end{proof}
By using Sobolev embedding theorem and convex arguments (see \cite{Shen4}), we can use $ W^{1,p} $ estimates to obtain the Hölder and $ L^{\infty} $ estimates as follows. The proofs are trivial and for more details, one can refer to \cite{Xu1}.
\begin{cor}[Localization of Hölder and $ L^{\infty} $ estimates for the operator $ \mathcal{L}_{\va}-\lambda I $] For $ \va\geq 0 $ and $ d\geq 2 $, let $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ \om $ be a bounded $ C^1 $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{VMO} $. Assume that $ x_0\in \om $, $ 0<R<R_0 $, $ f\in L^p(\om(x_0,2R);\mathbb{C}^{m\times d})$, $ F\in L^q(\om(x_0,2R);\mathbb{C}^m)$ and $ q=\frac{pd}{p+d} $. If $ \Delta(x_0,2R)\neq\emptyset $, assume that $ u_{\va,\lambda}\in H^1(\om(x_0,2R);\mathbb{C}^m) $ is the weak solution of the boundary problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F+\operatorname{div}(f)\text{ in }\om(x_0,2R)\quad\text{and}\quad u_{\va,\lambda}=0\text{ on }\Delta(x_0,2R).\nonumber \end{align} If $ \Delta(x_0,2R)=\emptyset $, assume that $ u_{\va,\lambda}\in H^1(B(x_0,2R);\mathbb{C}^m) $ is the weak solution of the interior problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F+\operatorname{div}(f)\text{ in } B(x_0,2R).\nonumber \end{align} Then there exists $ n\in\mathbb{N}_+ $, a constant integer depending only on $ d $, such that for $ \gamma=1-\frac{d}{p} $, any $ k\in\mathbb{N}_+ $ and $ 0<s<\infty $, \be \begin{aligned}
\left[u_{\va,\lambda}\right]_{C^{0,\gamma}(\om(x_0,R))}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}R^{\gamma}}\left(\Xint-_{\om(x_0,2R)}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\\
&\quad+\frac{C_{k,\theta_0}(1+|\lambda|R^2)^{n}R}{R^{\gamma}}\left\{R\left(\Xint-_{\om(x_0,2R)}|F|^q\right)^{\frac{1}{q}}+\left(\Xint-_{\om(x_0,2R)}|f|^p\right)^{\frac{1}{p}}\right\}, \end{aligned}\label{Holder} \ee \be \begin{aligned}
\|u_{\va,\lambda}\|_{L^{\infty}(\om(x_0,R))}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}}\left(\Xint-_{\om(x_0,2R)}|u_{\va,\lambda}|^s\right)^{\frac{1}{s}}\\
&\quad+C_{k,\theta_0}(1+|\lambda|R^2)^{n}\left\{R^{2}\left(\Xint-_{\om(x_0,2R)}|F|^q\right)^{\frac{1}{q}}+R\left(\Xint-_{\om(x_0,2R)}|f|^p\right)^{\frac{1}{p}}\right\}, \end{aligned}\label{Linfty} \ee where $ C_{k,\theta_0} $ depends only on $ \mu,d,m,k,\theta_0,p,s,\omega(t) $ and $ \om $. \end{cor}
\begin{thm}[Localization of Lipschitz estimates for the operator $ \mathcal{L}_{\va} $]\label{Lipschitz estimate localization} For $ \va\geq 0 $ and $ d\geq 2 $, let $ \om $ be a bounded $ C^{1,\eta} $ domain in $ \mathbb{R}^d $ with $ 0<\eta<1 $. Suppose that $ A $ satisfies $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. Assume that $ x_0\in\om $, $ 0<R<R_0 $ and $ F\in L^p(\om(x_0,2R);\mathbb{C}^m) $ with $ p>d $. If $ \Delta(x_0,2R)\neq\emptyset $, assume that $ u_{\va}\in H^1(\om(x_0,2R);\mathbb{C}^m) $ is the weak solution of the boundary problem \begin{align} \mathcal{L}_{\va}(u_{\va})=F\text{ in }\om(x_0,2R)\quad\text{and}\quad u_{\va}=0\text{ on }\Delta(x_0,2R).\nonumber \end{align} If $ \Delta(x_0,2R)=\emptyset $, assume that $ u_{\va}\in H^1(B(x_0,2R);\mathbb{C}^m) $ is the weak solution of the interior problem \begin{align} \mathcal{L}_{\va}(u_{\va})=F\text{ in } B(x_0,2R).\nonumber \end{align} then \begin{align}
\|\nabla u_{\va}\|_{L^{\infty}(\om(x_0,R))}\leq \frac{C}{R}\left\{\left(\Xint-_{\om(x_0,2R)}|u_{\va}|^2\right)^{\frac{1}{2}}+R^2\left(\Xint-_{\om(x_0,2R)}|F|^p\right)^{\frac{1}{p}}\right\},\label{LipforL} \end{align} where $ C $ depends only on $ \mu,d,m,\tau,\nu,\eta $ and $ \om $. \end{thm} \begin{proof} See \cite{Av2} or Theorem 3.1.1 and Theorem 4.5.1 of \cite{Shen2}. \end{proof}
\begin{thm}[Localization of Lipschitz estimates for the operator $ \mathcal{L}_{\va}-\lambda I $] For $ \va\geq 0 $ and $ d\geq 2 $, Let $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ \om $ be a bounded $ C^{1,\eta} $ domain in $ \mathbb{R}^d $ with $ 0<\eta<1 $. Suppose that $ A $ satisfies $ \eqref{sy} $,$ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. Assume that $ x_0\in\om $, $ 0<R<R_0 $ and $ F\in L^p(\om(x_0,2R);\mathbb{C}^m) $ with $ p>d $. If $ \Delta(x_0,2R)\neq\emptyset $, assume that $ u_{\va,\lambda}\in H^1(\om(x_0,2R);\mathbb{C}^m) $ is the weak solution of the boundary problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F\text{ in }\om(x_0,2R)\quad\text{and}\quad u_{\va,\lambda}=0\text{ on }\Delta(x_0,2R).\nonumber \end{align} If $ \Delta(x_0,2R)=\emptyset $, assume that $ u_{\va,\lambda}\in H^1(B(x_0,2R);\mathbb{C}^m) $ is the weak solution of the interior problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F\text{ in }B(x_0,2R).\nonumber \end{align} Then there exists $ n\in\mathbb{N}_+ $, a constant integer depending only on $ d $, such that for any $ k\in\mathbb{N}_+ $, \be \begin{aligned}
\|\nabla u_{\va,\lambda}\|_{L^{\infty}(\om(x_0,R))}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR}\left(\Xint-_{\om(x_0,2R)}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\\
&\quad+C_{k,\theta_0}(1+|\lambda|R^2)^{n}R\left(\Xint-_{\om(x_0,2R)}|F|^p\right)^{\frac{1}{p}}, \end{aligned}\label{Lipesu} \ee where $ C_{k,\theta_0} $ depends only on $ \mu,d,m,k,\theta_0,\tau,\nu,\eta $ and $ \om $. \end{thm} \begin{proof}
In view of $ \eqref{Lipschitz estimate localization} $, we can infer that \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^{\infty}(\om_R)}&\leq \frac{C}{R}\left\{\left(\Xint-_{\om_{3/2R}}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}+R^2\left(\Xint-_{\om_{3/2R}}|F|^p\right)^{\frac{1}{p}}+|\lambda|R^2\left(\Xint-_{\om_{3/2R}}|u_{\va,\lambda}|^p\right)^{\frac{1}{p}}\right\}.\nonumber \end{align} By applying $ \eqref{loW1pu} $ to $ u_{\va,\lambda} $ for index $ p $, this yields that for any $ k\in\mathbb{N}_+ $, \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^{\infty}(\om_R)}&\leq \frac{C}{R}\left(\Xint-_{\om_{3/2R}}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}+CR\left(\Xint-_{\om_{3/2R}}|F|^p\right)^{\frac{1}{p}}\nonumber\\
&\quad+\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k-1}}\left\{\left(\Xint-_{\om_{2R}}|u_{\va,\lambda}|^p\right)^{\frac{1}{2}}+(1+|\lambda|R^2)^{k+n}R^2\left(\Xint-_{\om_{2R}}|F|^p\right)^{\frac{1}{p}}\right\}.\nonumber \end{align} This, together with $ \eqref{Cau1} $, implies $ \eqref{Lipesu} $. \end{proof}
\section{Green functions of operators}\label{Lp estimates of resolventssection}
\subsection{Proof of Theorem \ref{Lp estimates of resolventsf} and relevant estimates}
To begin with, we will use the well-known real variable method to prove Theorem \ref{Lp estimates of resolventsf}. Unlike what had been done in the proof of $ W^{1,p} $ estimates for $ \mathcal{L}_{\va} $, the operators $ \mathcal{L}_{\va}-\lambda I $ do not have the homogeneous property, that is, if $ \lambda\neq 0 $ and $ 0\neq c\in\mathbb{C}^m $ is a constant vector, $ (\mathcal{L}_{\va}-\lambda I)(c)\neq 0 $. For this reason, we need to make some adjustments for the original method employed on $ \mathcal{L}_{\va} $. For simplicity, we use $ B=B(x,r) $ to denote a ball in $ \mathbb{R}^d $ ($ d\geq 2 $) and $ tB=B(x,tr) $ ($ t\in \mathbb{R}_+ $) to denote balls with center $ x $ and radius $ tr $ if no confusion would be caused.
\begin{thm}[Real variable method]\label{Real variable method}
Let $ q>2 $ and $ \om $ be a bounded Lipschitz domain in $ \mathbb{R}^d $ with $ d\geq 2 $. Let $ F\in L^{2}(\om;\mathbb{C}^m) $ and $ f\in L^{p}(\om;\mathbb{C}^{m\times d}) $ for some $ 2<p<q $. Suppose that for each ball $ B $ with the property that $ |B|\leq c_{0}|\om| $ and either $ 4B\subset\om $ or $ B $ is centered on $ \pa\om $, there exist two measurable functions $ F_{B} $ and $ R_{B} $ on $ \om\cap 2B $, such that $ |F|\leq|F_{B}|+|R_{B}|$ on $ \om\cap 2B $, \begin{align}
&\left(\Xint-_{\om\cap 2B}|R_{B}|^{q}\right)^{\frac{1}{q}} \leq N_{1}\left\{\left(\Xint-_{\om \cap 4B}|F|^{2}\right)^{\frac{1}{2}}+\sup _{4 B_{0} \supset B^{\prime} \supset B}\left(\Xint-_{\om \cap B^{\prime}}|f|^{2}\right)^{\frac{1}{2}}\right\},\label{rvc1}\\
&\left(\Xint-_{\om \cap 2 B}|F_{B}|^{2}\right)^{\frac{1}{2}} \leq N_{2} \sup _{4 B_{0} \supset B^{\prime} \supset B}\left(\Xint-_{\om \cap B^{\prime}}|f|^{2}\right)^{\frac{1}{2}}+\eta\left(\Xint-_{\om \cap 4B}|F|^{2}\right)^{\frac{1}{2}},\label{rvc2} \end{align} where $ N_{1}, N_{2}>0 $ and $ 0<c_{0}<1 $. Then there exists $ \eta_{0}>0 $, depending only on $ N_{1},N_{2},c_{0} $, $ p,q $ and the Lipschitz character of $ \om $, with the property that if $ 0 \leq \eta<\eta_{0} $, then $ F\in L^{p}(\om;\mathbb{C}^m) $ and \begin{align}
\left(\Xint-_{\om}|F|^{p}\right)^{\frac{1}{p}} \leq C\left\{\left(\Xint-_{\om}|F|^{2}\right)^{\frac{1}{2}}+\left(\Xint-_{\om}|f|^{p}\right)^{\frac{1}{p}}\right\},\label{rer} \end{align} where $ C $ depends at most on $ N_{1},N_{2},c_{0}, p,q $ and the Lipschitz character of $ \om $. \end{thm} \begin{proof} It is proved for the case that $ F\in L^2(\om;\mathbb{R}^m) $ and $ f\in L^p(\om;\mathbb{R}^m) $ in Theorem 3.2.6 of \cite{Shen2}. It is easy to generalize it to the complex case through almost the same arguments. \end{proof}
\begin{proof}[Proof of Theorem \ref{Lp estimates of resolventsf}] Firstly, we assume that $ \lambda\neq 0 $, since the case $ \lambda=0 $ is trivial in view of the results for $ W^{1,p} $ estimates of $ \mathcal{L}_{\va} $. If $ p=2 $, the results are given by Lemma \ref{l2reses}. For $ 2<p<\infty $, choose $ q=p+1 $. Consider functions \begin{align}
&H=|u_{\va,\lambda}|,\,\,h=(R_0^{-2}+|\lambda|)^{-1}|F|+(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}|f|,\,\,G=|\nabla u_{\va,\lambda}|\nonumber\\
&\quad\quad\text{and }g=(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}|F|+|f|.\nonumber \end{align}
Now we will apply Theorem \ref{Real variable method} to complete the proof. For each ball $ B $ with the property that $ |B|\leq \frac{1}{100}|\om| $ and $ 4B\subset \om $, we write $ u_{\va,\lambda}=u_{\va,\lambda,1}+u_{\va,\lambda,2} $ in $ 2B $, where $ u_{\va,\lambda,1}\in H_0^1(4B;\mathbb{C}^m) $ is the weak solution of $ \mathcal{L}_{\va}(u_{\va,\lambda,1})-\lambda u_{\va,\lambda,1}=F+\operatorname{div}(f) $ in $ 4B $ and $ u_{\va,\lambda,1}=0 $ on $ \pa(4B) $. Let \begin{align}
H_{B}=|u_{\va,\lambda,1}|,\,\, R_{B}=|u_{\va,\lambda,2}|,\,\, G_{B}=|\nabla u_{\va,\lambda,1}|\text{ and }T_{B}=|\nabla u_{\va,\lambda,2}|.\nonumber \end{align}
Obviously, $ |H|\leq H_{B}+R_{B} $ and $ |G|\leq G_B+T_B $ in $ 2B $. It follows from Lemma \ref{l2reses} that \be \begin{aligned}
\left(\Xint-_{4B}|H_{B}|^2\right)^{\frac{1}{2}}&\leq \frac{C_{\theta_0}}{r^{-2}+|\lambda|}\left(\Xint-_{4B}|F|^2\right)^{\frac{1}{2}}+\frac{C_{\theta_0}}{(r^{-2}+|\lambda|)^{\frac{1}{2}}}\left(\Xint-_{4B}|f|^2\right)^{\frac{1}{2}}\\
&\leq C_{\theta_0}\left\{\Xint-_{4B}\left(\frac{|F|+(R_0^{-2}+|\lambda|)^{\frac{1}{2}}|f|}{R_0^{-2}+|\lambda|}\right)^2\right\}^{\frac{1}{2}}\leq C_{\theta_0}\left(\Xint-_{4B}|h|^2\right)^{\frac{1}{2}} \end{aligned}\label{HBestimates} \ee and \be \begin{aligned}
\left(\Xint-_{4B}|G_{B}|^2\right)^{\frac{1}{2}}&\leq \frac{C_{\theta_0}}{(r^{-2}+|\lambda|)^{\frac{1}{2}}}\left(\Xint-_{4B}|F|^2\right)^{\frac{1}{2}}+C_{\theta_0}\left(\Xint-_{4B}|f|^2\right)^{\frac{1}{2}}\\
&\leq C_{\theta_0}\left\{\Xint-_{4B}\left(\frac{|F|}{(R_0^{-2}+|\lambda|)^{\frac{1}{2}}}+|f|\right)^2\right\}^{\frac{1}{2}}\leq C_{\theta_0}\left(\Xint-_{4B}|g|^2\right)^{\frac{1}{2}}, \end{aligned}\label{GBestimates} \ee where $ r $ is the radius of $ B $. Moreover, we note that $ u_{\va,\lambda,2}\in H^1(4B;\mathbb{C}^m) $ and $ \mathcal{L}_{\va}(u_{\va,\lambda,2})-\lambda u_{\va,\lambda,2}=0 $ in $ 4B $. Owing to $ \eqref{loW1pu} $, \eqref{HBestimates} and Lemma \ref{l2reses}, we can obtain that \begin{align}
\left(\Xint-_{2B}|R_B|^q\right)^{\frac{1}{q}}&\leq C_{\theta_0}\left(\Xint-_{4B}|u_{\va,\lambda,2}|^2\right)^{\frac{1}{2}}\leq C_{\theta_0}\left\{\left(\Xint-_{4B}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}+\left(\Xint-_{4B}|u_{\va,\lambda,1}|^2\right)^{\frac{1}{2}}\right\}\nonumber\\
&\leq C_{\theta_0}\left\{\left(\Xint-_{4B}|H|^2\right)^{\frac{1}{2}}+\left(\Xint-_{4B}|h|^2\right)^{\frac{1}{2}}\right\}.\nonumber \end{align} To estimate $ T_B $, we first note that \begin{align} (\mathcal{L}_{\va}-\lambda I)\left(u_{\va,\lambda,2}-\Xint-_{3B}u_{\va,\lambda,2}\right)=\lambda\left(\Xint-_{3B}u_{\va,\lambda,2}\right)\text{ in }3B.\nonumber \end{align} Then it is easy to get that \begin{align}
\left(\Xint-_{2B}|T_B|^q\right)^{\frac{1}{q}}&\leq \frac{C_{\theta_0}}{r}\left(\Xint-_{3B}\left|u_{\va,\lambda,2}-\Xint-_{3B}u_{\va,\lambda,2}\right|^2\right)^{\frac{1}{2}}+C_{\theta_0}(1+|\lambda|r^2)^n|\lambda|r\left(\Xint-_{3B}|u_{\va,\lambda,2}|\right)\nonumber\\
&\leq C_{\theta_0}\left\{\left(\Xint-_{3B}|\nabla u_{\va,\lambda,2}|^2\right)^{\frac{1}{2}}+|\lambda|r\left(\Xint-_{7/2 B}|u_{\va,\lambda,2}|^2\right)^{\frac{1}{2}}\right\}\leq C_{\theta_0}\left(\Xint-_{4B}|\nabla u_{\va,\lambda,2}|^2\right)^{\frac{1}{2}}\nonumber\\
&\leq C_{\theta_0}\left\{\left(\Xint-_{4B}|\nabla u_{\va,\lambda,1}|^2\right)^{\frac{1}{2}}+\left(\Xint-_{4B}|\nabla u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\right\}\leq C_{\theta_0}\left\{\left(\Xint-_{4B}|G|^2\right)^{\frac{1}{2}}+\left(\Xint-_{4B}|g|^2\right)^{\frac{1}{2}}\right\},\nonumber \end{align} where for the second inequality, we have used $ \eqref{Cau1} $ with $ k=n $, Poincaré's inequality, Hölder's inequality and for the third inequality, we have used $ \eqref{fanxiangzy} $.
For ball $ B=B(x,r) $ such that it is centered at $ \pa\om $ with $ 0<r<\frac{1}{16}R_0 $, write $ u_{\va,\lambda}=u_{\va,\lambda,3}+u_{\va,\lambda,4} $ in $ 4B\cap\om $, where $ u_{\va,\lambda,3}\in H_0^1(4B;\mathbb{C}^m) $ is the weak solution of $ \mathcal{L}_{\va}(u_{\va,\lambda,3})-\lambda u_{\va,\lambda,3}=F+\operatorname{div}(f) $ in $ 4B\cap\om $ and $ u_{\va,\lambda,3}=0 $ on $ \pa(4B\cap\om) $. Let \begin{align}
H_{B}=|u_{\va,\lambda,3}|,\,\, R_{B}=|u_{\va,\lambda,4}|,\,\, G_{B}=|\nabla u_{\va,\lambda,3}|\text{ and }T_{B}=|\nabla u_{\va,\lambda,4}|.\nonumber \end{align}
By using almost the same arguments, we can obtain $ |H|\leq H_{B}+R_{B} $, $ |G|\leq G_B+T_B $ in $ 2B\cap\om $, \begin{align}
\left(\Xint-_{2B\cap\om}|R_B|^q\right)^{\frac{1}{q}}&\leq C_{\theta_0}\left(\Xint-_{4B\cap\om}|H|^2\right)^{\frac{1}{2}}+C_{\theta_0}\left(\Xint-_{4B\cap\om}|h|^2\right)^{\frac{1}{2}},\nonumber\\
\left(\Xint-_{4B\cap\om}|G_{B}|^2\right)^{\frac{1}{2}}&\leq C_{\theta_0}\left(\Xint-_{4B\cap\om}|g|^2\right)^{\frac{1}{2}}\text{ and }
\left(\Xint-_{4B\cap\om}|H_{B}|^2\right)^{\frac{1}{2}}\leq C_{\theta_0}\left(\Xint-_{4B\cap\om}|h|^2\right)^{\frac{1}{2}}.\nonumber \end{align} Moreover, since $ u_{\va,\lambda,4}=u_{\va,\lambda}-u_{\va,\lambda,3}=0 $ on $ \pa\om\cap 4B $, then in view of Poincaré's inequality and $ \eqref{loW1pnu} $, we have \begin{align}
\left(\Xint-_{2B\cap\om}|T_B|^q\right)^{\frac{1}{q}}&\leq \frac{C_{\theta_0}}{r}\left(\Xint-_{4B\cap\om}|u_{\va,\lambda,4}|^2\right)^{\frac{1}{2}}\leq C_{\theta_0}\left(\Xint-_{4B\cap\om}|\nabla u_{\va,\lambda,4}|^2\right)^{\frac{1}{2}}\nonumber\\
&\leq C_{\theta_0}\left\{\left(\Xint-_{4B\cap\om}|\nabla u_{\va,\lambda,3}|^2\right)^{\frac{1}{2}}+\left(\Xint-_{4B\cap\om}|\nabla u_{\va,\lambda}|^2\right)^{\frac{1}{2}}\right\}\nonumber\\
&\leq C_{\theta_0}\left\{\left(\Xint-_{4B\cap\om}|G|^2\right)^{\frac{1}{2}}+\left(\Xint-_{4B\cap\om}|g|^2\right)^{\frac{1}{2}}\right\}.\nonumber \end{align} Then by using Theorem \ref{Real variable method}, we have, for any $ 2<p<\infty $, \begin{align}
\left(\Xint-_{\om}|u_{\va,\lambda}|^{p}\right)^{\frac{1}{p}}&\leq C_{\theta_0}\left\{\left(\Xint-_{\om}|u_{\va,\lambda}|^{2}\right)^{\frac{1}{2}}+\left(\Xint-_{\om}|h|^{p}\right)^{\frac{1}{p}}\right\},\nonumber\\
\left(\Xint-_{\om}|\nabla u_{\va,\lambda}|^{p}\right)^{\frac{1}{p}}&\leq C_{\theta_0}\left\{\left(\Xint-_{\om}|\nabla u_{\va,\lambda} |^{2}\right)^{\frac{1}{2}}+\left(\Xint-_{\om}|g|^{p}\right)^{\frac{1}{p}}\right\}.\nonumber \end{align} In view of Lemma \ref{l2reses}, definitions of $ g,h $ and Hölder's inequality, we can complete the proof for the case $ 2<p<\infty $. For $ 1<p<2 $, the results follow form duality arguments. Using the same definitions of $ u_{\va,\lambda}^{(1)} $ and $ u_{\va,\lambda}^{(2)} $ in the proof of Lemma \ref{l2reses}, we only need to show that for any $ 1<p<2 $, \begin{align}
\|\nabla u_{\va,\lambda}^{(1)}\|_{L^p(\om)}+(R_0^{-2}+|\lambda|)^{\frac{1}{2}}\|u_{\va,\lambda}^{(1)}\|_{L^p(\om)}&\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|F\|_{L^p(\om)},\nonumber\\
\|\nabla u_{\va,\lambda}^{(2)}\|_{L^p(\om)}+(R_0^{-2}+|\lambda|)^{\frac{1}{2}}\|u_{\va,\lambda}^{(2)}\|_{L^p(\om)}&\leq C_{\theta_0}\|f\|_{L^p(\om)}.\nonumber \end{align} For $ F_1\in L^{p'}(\om;\mathbb{C}^m) $ and $ f_1\in L^{p'}(\om;\mathbb{C}^{m\times d}) $ with $ p'=\frac{p}{p-1} $ being the conjugate number of $ p $, let $ v_{\va,\lambda}^{(1)} $ and $ v_{\va,\lambda}^{(2)} $ be solutions of Dirichlet problems: \begin{align} \left\{\begin{matrix} (\mathcal{L}_{\va}-\overline{\lambda} I)(v_{\va,\lambda}^{(1)})=F_1&\text{in}&\om,\\ v_{\va,\lambda}^{(1)}=0&\text{on}&\pa\om, \end{matrix}\right.\quad\text{and}\quad \left\{\begin{matrix} (\mathcal{L}_{\va}-\overline{\lambda} I)(v_{\va,\lambda}^{(2)})=\operatorname{div}(f_1)&\text{in}&\om,\\ v_{\va,\lambda}^{(2)}=0&\text{on}&\pa\om. \end{matrix}\right.\nonumber \end{align} Then it follows from direct calculations and the definition of $ B_{\va,\lambda,\om}[\cdot,\cdot] $ that \begin{align} \int_{\om}F_1\overline{u_{\va,\lambda}^{(1)}}dx&=B_{\va,\overline{\lambda},\om}[v_{\va,\lambda}^{(1)},u_{\va,\lambda}^{(1)}]=\overline{B_{\va,\lambda,\om}[u_{\va,\lambda}^{(1)},v_{\va,\lambda}^{(1)}]}=\int_{\om}\overline{F}v_{\va,\lambda}^{(1)}dx,\nonumber\\ -\int_{\om}f_1\overline{\nabla u_{\va,\lambda}^{(1)}}dx&=B_{\va,\overline{\lambda},\om}[v_{\va,\lambda}^{(2)},u_{\va,\lambda}^{(1)}]=\overline{B_{\va,\lambda,\om}[u_{\va,\lambda}^{(1)},v_{\va,\lambda}^{(2)}]}=\int_{\om}\overline{F}v_{\va,\lambda}^{(2)}dx,\nonumber\\ \int_{\om}F_1\overline{u_{\va,\lambda}^{(2)}}dx&=B_{\va,\overline{\lambda},\om}[v_{\va,\lambda}^{(1)},u_{\va,\lambda}^{(2)}]=\overline{B_{\va,\lambda,\om}[u_{\va,\lambda}^{(2)},v_{\va,\lambda}^{(1)}]}=-\int_{\om}\overline{f}\nabla v_{\va,\lambda}^{(1)}dx,\nonumber\\ -\int_{\om}f_1\overline{\nabla u_{\va,\lambda}^{(2)}}dx&=B_{\va,\overline{\lambda},\om}[v_{\va,\lambda}^{(2)},u_{\va,\lambda}^{(2)}]=\overline{B_{\va,\lambda,\om}[u_{\va,\lambda}^{(2)},v_{\va,\lambda}^{(2)}]}=-\int_{\om}\overline{f}\nabla v_{\va,\lambda}^{(2)}dx.\nonumber \end{align} These, together with the results for $ 2<p<\infty $, imply that \begin{align}
\left|\int_{\om}F_1\overline{u_{\va,\lambda}^{(1)}}dx\right|&\leq \|F\|_{L^p(\om)}\|v_{\va,\lambda}^{(1)}\|_{L^{p'}(\om)}\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-1}\|F\|_{L^p(\om)}\|F_1\|_{L^{p'}(\om)},\nonumber\\
\left|\int_{\om}f_1\overline{\nabla u_{\va,\lambda}^{(1)}}dx\right|&\leq\|F\|_{L^p(\om)}\|v_{\va,\lambda}^{(2)}\|_{L^{p'}(\om)}\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|F\|_{L^p(\om)}\|f_1\|_{L^{p'}(\om)},\nonumber\\
\left|\int_{\om}F_1\overline{u_{\va,\lambda}^{(2)}}dx\right|&\leq \|f\|_{L^p(\om)}\|\nabla v_{\va,\lambda}^{(1)}\|_{L^{p'}(\om)}\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|f\|_{L^p(\om)}\|F_1\|_{L^{p'}(\om)},\nonumber\\
\left|\int_{\om}f_1\overline{\nabla u_{\va,\lambda}^{(2)}}dx\right|&\leq \|f\|_{L^p(\om)}\|\nabla v_{\va,\lambda}^{(2)}\|_{L^{p'}(\om)}\leq C_{\theta_0}\|f\|_{L^p(\om)}\|f_1\|_{L^{p'}(\om)},\nonumber \end{align} which give the proof. \end{proof}
\begin{cor} Assume that $ \om $ is a bounded $ C^{1,1} $ domain in $ \mathbb{R}^d $ with $ d\geq 2 $ and $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in (0,\frac{\pi}{2}) $. If $ u_{0,\lambda}\in W^{2,p}(\om;\mathbb{C}^m) $ is the unique weak solution for the Dirichlet problem \begin{align} (\mathcal{L}_{0}-\lambda I)(u_{0,\lambda})=F\text{ in }\om\quad\text{and}\quad u_{0,\lambda}=0\text{ on }\pa\om\nonumber \end{align} with $ 1<p<\infty $ and $ F\in L^p(\om;\mathbb{C}^m) $. Then \begin{align}
\|\nabla^2u_{0,\lambda}\|_{L^p(\om)}\leq C_{\theta_0}\|F\|_{L^p(\om)},\label{W2p} \end{align} where $ C_{\theta_0} $ depends only on $ \mu,d,m,p,\theta_0 $ and $ \om $. In operator forms, \begin{align}
\|R(\lambda,\mathcal{L}_0)\|_{L^p(\om)\to W^{2,p}(\om)}\leq C_{\theta_0}.\label{W2pop} \end{align} \end{cor} \begin{proof} The results follow from the $ W^{2,p} $ estimates for $ \mathcal{L}_0 $, i.e. \begin{align}
\|\nabla u_{0,\lambda}\|_{L^p(\om)}\leq C\left\{|\lambda|\|u_{0,\lambda}\|_{L^p(\om)}+\|F\|_{L^p(\om)}\right\}\nonumber \end{align}
and $ \|u_{0,\lambda}\|_{L^p(\om)}\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-1}\|F\|_{L^p(\om)} $, which is given by Theorem \ref{Lp estimates of resolventsf}. \end{proof}
\begin{lem} Assume that $ \om $ is $ C^{1,1} $ domain in $ \mathbb{R}^d $ with $ d\geq 2 $, $ x_0\in\om $ and $ 0<R<R_0 $. If $ \Delta(x_0,2R)\neq\emptyset $, assume that $ u_{0,\lambda}\in H^2(\om(x_0,2R);\mathbb{C}^m) $ is the weak solution of the boundary problem \begin{align} (\mathcal{L}_0-\lambda I)(u_{0,\lambda})=0\text{ in }\om(x_0,2R)\quad\text{and}\quad u_{0,\lambda}=0\text{ on }\Delta(x_0,2R),\nonumber \end{align} if $ \Delta(x_0,2R)=\emptyset $, assume that $ u_{0,\lambda}\in H^2(B(x_0,2R);\mathbb{C}^m) $ is the weak solution of the interior problem \begin{align} (\mathcal{L}_0-\lambda I)(u_{0,\lambda})=0\text{ in }B(x_0,2R),\nonumber \end{align} with $ x_0\in\om $, $ 0<R<R_0 $ and $ 1<p<\infty $. Then there exists $ n\in\mathbb{N}_+ $, a constant depending only on $ d $, such that for any $ k\in\mathbb{N}_+ $, \be \begin{aligned}
\left(\Xint-_{\om(x_0,R)}|\nabla^2u_{0,\lambda}|^p\right)^{\frac{1}{p}}&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR^2}\left(\Xint-_{\om(x_0,2R)}|u_{0,\lambda}|^2\right)^{\frac{1}{2}}. \end{aligned}\label{W2p local} \ee If we further assume that $ \om $ is a bounded $ C^{2,1} $ domain and $ \rho\in(0,1) $, then \begin{align}
\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om(x_0,R))}&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR^{2}}\left(\Xint-_{\om(x_0,2R)}|u_{0,\lambda}|^2\right)^{\frac{1}{2}},\label{ww80}\\
\left[\nabla^2u_{0,\lambda}\right]_{C^{0,\rho}(\om(x_0,R))}&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR^{2+{\rho}}}\left(\Xint-_{\om(x_0,2R)}|u_{0,\lambda}|^2\right)^{\frac{1}{2}},\label{ww81} \end{align} where $ C $ depends on $ \mu,d,m,k,\theta_0,\rho $ and $ \om $. \end{lem} \begin{proof} By applying $ C^{2,\rho} $ estimates for the constant elliptic system (see Theorem 5.23 in \cite{MG}), $ \eqref{Cau1} $, $ \eqref{Cau2} $, $ \eqref{W2p} $ and Theorem \ref{Lp estimates of resolventsf}. \end{proof}
\begin{rem}\label{remimp} Under conditions that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $, $ \eqref{VMO} $, $ \va\geq 0 $ and $ \om $ is a bounded $ C^1 $ domain in $ \mathbb{R}^d $ with $ d=2 $, if $ u_{\va,\lambda} $ is the unique solution for the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $, then in view of $ \eqref{W1pom} $ and Theorem \ref{Lp estimates of resolventsf}, we have \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^p(\om)}\leq C\left\{|\lambda|\|u_{\va,\lambda}\|_{L^{\frac{2p}{p+2}}(\om)}+\|F\|_{L^{\frac{2p}{p+2}}(\om)}\right\}\leq C_{\theta_0}\|F\|_{L^{\frac{2p}{p+2}}(\om)},\label{impoestim} \end{align} for $ 2<p<\infty $. If $ v_{\va,\lambda} $ is the unique solution for the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(v_{\va,\lambda})=\operatorname{div}(f) $ in $ \om $ and $ v_{\va,\lambda}=0 $ on $ \pa\om $, it is not hard to see that \begin{align}
\|v_{\va,\lambda}\|_{L^{\frac{2p}{p-2}}(\om)}\leq C_{\theta_0}\|f\|_{L^{\frac{p}{p-1}}(\om)},\quad \text{for any }2<p<\infty,\label{impoestim2} \end{align} Indeed, for any $ F\in C_0^{\infty}(\om;\mathbb{C}^m) $, we can choose $ w_{\va,\lambda} $ such that $ (\mathcal{L}_{\va}-\overline{\lambda} I)(w_{\va,\lambda})=F $ in $ \om $ and $ w_{\va,\lambda}=0 $ on $ \pa\om $. Then it can be easily seen by duality that \begin{align} \int_{\om}F\overline{v_{\va,\lambda}}dx&=B_{\va,\overline{\lambda},\om}[w_{\va,\lambda},v_{\va,\lambda}]=\overline{B_{\va,\lambda,\om}[v_{\va,\lambda},w_{\va,\lambda}]}=-\int_{\om}\overline{f}\nabla w_{\va,\lambda}dx\nonumber \end{align} In view of Hölder's inequality and $ \eqref{impoestim} $, it can be got that \begin{align}
\left|\int_{\om}F\overline{v_{\va,\lambda}}dx\right|\leq \|f\|_{L^{\frac{p}{p-1}}(\om)}\|\nabla w_{\va,\lambda}\|_{L^p(\om)}\leq C_{\theta_0}\|f\|_{L^{\frac{p}{p-1}}(\om)}\|F\|_{L^{\frac{2p}{p+2}}(\om)},\nonumber \end{align} which directly implies $ \eqref{impoestim2} $. Moreover, for $ F\in \dot{W}^{-1,p'}(\om) $, where $ \dot{W}^{-1,p'}(\om) $ denotes the dual space for the homogeneous Sobolev space $ \dot{W}_0^{1,p}(\om) $ and $ p' $ is the conjugate number of $ p $, we have \begin{align}
\left\|u_{\va,\lambda}\right\|_{L^{\frac{2p}{p-2}}(\om)}\leq C_{\theta_0}\left\|F\right\|_{\dot{W}^{-1,p'}(\om)}. \label{Dual estimates} \end{align} This conclusion is also proved by using the duality arguments. Actually, for all $ g\in C_0^1(\om;\mathbb{C}^m) $ we can choose $ v_{\va,\lambda} $ such that $ (\mathcal{L}_{\va}-\overline{\lambda} I)(v_{\va,\lambda})=g $ in $ \om $ and $ v_{\va,\lambda}=0 $ on $ \pa\om $. Thus, \begin{align} \int_{\om}g\overline{u_{\va,\lambda}}dx=B_{\va,\overline{\lambda},\om}[v_{\va,\lambda},u_{\va,\lambda}]=\overline{B_{\va,\lambda,\om}[u_{\va,\lambda},v_{\va,\lambda}]}=\overline{\langle F,v_{\va,\lambda}\rangle_{\dot{W}^{-1,p'}(\om)\times\dot{W}_0^{1,p'}(\om)}}. \nonumber \end{align} By using Hölder's inequality and $ \eqref{impoestim} $, this shows that \begin{align}
\left|\int_{\om}g\overline{u_{\va,\lambda}}dx\right|&=\left|\langle F,v_{\va,\lambda}\rangle_{\dot{W}^{-1,p'}(\om)\times\dot{W}_0^{1,p'}(\om)}\right|\leq \left\|F\right\|_{\dot{W}^{-1,p'}(\om)}\left\|v_{\va,\lambda}\right\|_{\dot{W}_0^{1,p}(\om)}\nonumber\\
&\leq\left\|F\right\|_{\dot{W}^{-1,p'}(\om)}\left\|\nabla v_{\va,\lambda}\right\|_{L^p(\om)}\leq C_{\theta_0}\left\|F\right\|_{\dot{W}^{-1,p'}(\om)}\left\|g\right\|_{L^{\frac{2p}{p+2}}(\om)},\nonumber \end{align} which gives the proof of $ \eqref{Dual estimates} $. \end{rem}
\begin{lem} Assume that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $, $ \eqref{VMO} $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ \om $ is a $ C^{1} $ bounded domain in $ \mathbb{R}^2 $. Let $ a=a(x) $ be an atom function in $ \om $. If $ u_{\va,\lambda} $ is the unique weak solution for the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=a $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $, then there exists a constant $ C_{\theta_0} $ depending only on $ \mu,\omega(t),m,\theta_0 $ and $ \om $ such that \begin{align}
\left\|u_{\va,\lambda}\right\|_{L^{\infty}(\om)}\leq C_{\theta_0}.\label{Linftyatom} \end{align} \end{lem} \begin{proof} For atom function $ a=a(\cdot) $, assume that \begin{align}
\operatorname{supp}(a)\subset\om(x_0,\rho)\text{ and }\left\|a\right\|_{L^{\infty}(\om)}\leq\frac{1}{|\om(x_0,\rho)|}\nonumber \end{align} with $ x_0\in\om $ and $ 0<\rho< R_0 $. Fix $ z\in\om $, we can choose $ 2<p<\infty $. Then \begin{align}
|u(z)| &\leq |u(z)-u_{z,\rho}|+|u_{z,\rho}|\leq C\rho^{1-\frac{2}{p}}[u]_{C^{0,1-\frac{2}{p}}(\om(z,\rho))}+|u_{z,\rho}|\nonumber\\
&\leq C\left\{\rho^{1-\frac{2}{p}}\left\|\nabla u_{\va,\lambda}\right\|_{L^p(\om)}+\rho^{\frac{2}{p}-1}\left\|u_{\va,\lambda}\right\|_{L^{\frac{2p}{p-2}}(\om)}\right\},\nonumber \end{align} due to Morrey's inequality and Hölder's inequality. Using $ \eqref{impoestim} $ and $ \eqref{Dual estimates} $, we have \begin{align}
\left\|\nabla u_{\va,\lambda}\right\|_{L^p(\om)}&\leq C_{\theta_0}\left\|a\right\|_{L^{\frac{2p}{p+2}}(\om)}\leq C_{\theta_0}\rho^{\frac{2}{p}-1}\text{ and } \left\|u_{\va,\lambda}\right\|_{L^{\frac{2p}{p-2}}(\om)}\leq C_{\theta_0}\left\|a\right\|_{\dot{W}^{-1,p'}(\om)}.\nonumber \end{align}
We claim that $ \left\|a\right\|_{\dot{W}^{-1,p'}(\om)}\leq \rho^{1-\frac{2}{p}} $. It is because that for all $ v\in \dot{W}_0^{1,p} (\om;\mathbb{R}^m) $, \begin{align}
\left|\int_{\om}a^{\alpha}(y)v^{\alpha}(y)dy\right|&\leq\left|\int_{\om}a^{\alpha}(y)(v^{\al}(y)-v_{x_0,\rho}^{\al})dy\right|\leq \|a\|_{L^1(\om)}\|v-v_{x_0,\rho}\|_{L^{\infty}(\om(x_0,\rho))}\nonumber\\
&\leq C\rho^{1-\frac{2}{p}}[v]_{C^{0,1-\frac{2}{p}}(\om(x_0,\rho))}\leq C\rho^{1-\frac{2}{p}}\left\|\nabla v\right\|_{L^p(\om)},\nonumber \end{align} where we have used Morrey's theory again for the last inequality. This completes the proof. \end{proof} \begin{rem} The key point of $ \eqref{Linftyatom} $ is that $ C_{\theta_0} $ does not depend on the module of $ \lambda $. Such estimates are extremely important in the constructions of Green functions with $ d=2 $. If $ C_{\theta_0} $ depends on the module of $ \lambda $, we will not be able to obtain related estimates similar to the case that $ d\geq 3 $. \end{rem}
\subsection{Green functions with dimension no less than three}
\begin{thm}[Green functions of $ \mathcal{L}_{\va}-\lambda I $ with $ d\geq 3 $]\label{Green3} For $ \va\geq 0 $ and $ d\geq 3 $, let $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ \om $ be a bounded $ C^1 $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{VMO} $. Then there exists a unique Green function, $ G_{\va,\lambda}=(G_{\va,\lambda}^{\al\beta}(\cdot,\cdot)):\om\times\om\to \mathbb{C}^{m^2}\cup\left\{\infty\right\} $ with $ 1\leq\al,\beta\leq m $, such that $ G_{\va,\lambda}(\cdot,y) \in H^1(\om\backslash B(y,r);\mathbb{C}^{m^2})\cap W_0^{1,s}(\om;\mathbb{C}^{m^2}) $ for any $ s\in [1,\frac{d}{d-1}) $, $ y\in\om $ and $ 0<r<R_0 $. $ G_{\va,\lambda} $ satisfies \begin{align} B_{\va,\lambda,\om}[G_{\va,\lambda}^{\gamma}(\cdot,y),\phi(\cdot)]=\phi^{\gamma}(y),\label{defnG} \end{align} for any $ 1\leq \gamma\leq m $, $ \phi\in W_0^{1,p}(\om;\mathbb{C}^m) $ with $ p>d $. Particularly, if $ F\in L^q(\om;\mathbb{C}^m) $ with $ q>d/2 $, \begin{align} u_{\va,\lambda}(x)=\int_{\om}G_{\va,\lambda}(x,y)\overline{F(y)}dy,\label{repre} \end{align} satisfies the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $. Moreover, let $ G_{\va,\overline{\lambda}}(x,y) $ be the Green function of the operator $ \mathcal{L}_{\va}-\overline{\lambda} I $, then \begin{align} G_{\va,\lambda}(x,y)=[\overline{G_{\va,\overline{\lambda}}(y,x)}]^T,\label{duality} \end{align} which means that $ G_{\va,\lambda}^{\al\beta}(x,y)=\overline{G_{\va,\overline{\lambda}}^{\beta\al}(y,x)} $ for any $ 1\leq\al,\beta\leq m $, $ x,y\in\om $ and $ x\neq y $. For any $ \sigma_1,\sigma_2\in(0,1) $ and $ k\in\mathbb{N}_+ $, the following estimates \begin{align}
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d-2}}\min\left\{1,\frac{[\delta(x)]^{\sigma_1}}{|x-y|^{\sigma_1}},\frac{[\delta(y)]^{\sigma_2}}{|x-y|^{\sigma_2}}\right\}\label{Greene} \end{align}
hold for any $ x,y\in\om $ and $ x\neq y$, where $ \delta(x)=\dist(x,\pa\om) $ denotes the distance from $ x $ to the boundary of $ \om $ and $ C_{k,\theta_0} $ depends only on $ \mu,d,m,k,\theta_0,\omega(t),\sigma_1,\sigma_2 $ and $ \om $. \end{thm} \begin{proof} For $ \lambda=0 $, the results follow from constructions of Green functions of standard elliptic operators $ \mathcal{L}_{\va} $. Then we only need to consider the case that $ \lambda\neq 0 $. First of all, let $ I(u)=\Xint-_{\om(x,\rho)}u^{\gamma}(\cdot) $, then $ I\in H^{-1}(\om;\mathbb{C}^m) $. Precisely speaking, for any $ u\in H_0^1(\om;\mathbb{C}^m) $, we can easily get by using Sobolev embedding theorem $ L^{\frac{2d}{d-2}}(\om)\subset H_0^1(\om) $ that \begin{align}
|I(u)|\leq C|\om(x,\rho)|^{-\frac{d-2}{2d}}\|u\|_{L^{\frac{2d}{d-2}}(\om)}\leq C|\om(x,\rho)|^{-\frac{d-2}{2d}}\|u\|_{H_0^1(\om)}\leq C\rho^{-\frac{d-2}{2}}\|u\|_{H_0^1(\om)}.\label{I u} \end{align} Consider the approximating Green function $ G_{\rho,\va,\lambda}(\cdot,y)=(G_{\rho,\va,\lambda}^{\al\beta}(\cdot,y))\in H_0^1(\om;\mathbb{C}^{m^2}) $ with $ \rho>0 $ and $ 1\leq\al,\beta\leq m $, such that \begin{align} B_{\va,\lambda,\om}[G_{\rho,\va,\lambda}^{\gamma}(\cdot,y),u(\cdot)]=\Xint-_{\om(y,\rho)}u^{\gamma}(\cdot)\text{ for any }u\in H_0^1(\om;\mathbb{C}^m),\label{apf1} \end{align} where $ 1\leq\gamma\leq m $. We see that the existence of $ G_{\rho,\va,\lambda}^{\gamma}(\cdot,y) $ is a direct consequence of Theorem \ref{existencethm}. Choose $ G_{\rho,\va,\lambda}^{\beta}(\cdot,y) $ $ (1\leq\beta\leq m) $ itself as the test function, we have \begin{align} B_{\va,\lambda,\om}[G_{\rho,\va,\lambda}^{\gamma}(\cdot,y),G_{\rho,\va,\lambda}^{\beta}(\cdot,y)]=I(G_{\rho,\va,\lambda}^{\gamma\beta}(\cdot,y)).\label{ww9} \end{align} In view of $ \eqref{ubuuboun} $, $ \eqref{laxmil} $ and $ \eqref{I u} $, it is not hard to see that \begin{align}
\|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}^2&\leq C_{\theta_0}|\lambda|^{-1}\rho^{-\frac{d-2}{2}}\|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)},\nonumber\\
\|\nabla_1G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}^2&\leq C_{\theta_0}\rho^{-\frac{d-2}{2}}\|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)},\nonumber \end{align} where $ \nabla_i $ ($ i=1,2 $) denote the derivatives for the first or second variable of Green functions and we will use these notations throughout this paper. Then \begin{align}
\|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}\leq C_{\theta_0}\rho^{-\frac{d-2}{2}}\text{ and } \|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}\leq C_{\theta_0}|\lambda|^{-\frac{1}{2}}\rho^{-\frac{d-2}{2}}.\label{ww11} \end{align} On the other hand, using Poincaré's inequality, we can deduce that \begin{align}
\|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}\leq CR_0\|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}\leq C_{\theta_0}R_0\rho^{-\frac{d-2}{2}}.\nonumber \end{align} This, combined with $ \eqref{ww11} $ and the case of $ \lambda=0 $, gives the following estimate \begin{align}
\|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\rho^{-\frac{d-2}{2}},\label{ww27} \end{align} for any $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ and $ 1\leq\gamma\leq m $. Let $ F\in C_0^{\infty}(\om;\mathbb{C}^m) $, consider the Dirichlet problem $ (\mathcal{L}_{\va}-\overline{\lambda}I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $. If we choose $ G_{\rho,\va,\lambda}^{\gamma}(\cdot,y) $ as the test function, it follows from simple calculations that \begin{align} &\int_{\om}\overline{G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)}F(\cdot)=B_{\va,\overline{\lambda},\om}[u_{\va,\lambda}(\cdot),G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)]=\overline{B_{\va,\lambda,\om}[G_{\rho,\va,\lambda}^{\gamma}(\cdot,y),u_{\va,\lambda}(\cdot)]}=\Xint-_{\om(y,\rho)}\overline{u_{\va,\lambda}^{\gamma}(\cdot)}.\nonumber \end{align} Suppose that $ \operatorname{supp}F\subset B(y,R)\subset\om $, by using $ \eqref{Linfty} $, it can be got that \be \begin{aligned}
&\left|\int_{\om}\overline{G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)}F(\cdot)\right|\leq \|u_{\va,\lambda}\|_{L^{\infty}(\om(y,\rho))}\leq\|u_{\va,\lambda}\|_{L^{\infty}(B(y,\frac{R}{4}))}\\
&\quad\quad\quad\quad\leq C_{\theta_0}\left(\Xint-_{B(y,\frac{R}{2})}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}+(1+|\lambda|R^2)^{n}R^{2}\left(\Xint-_{B(y,\frac{R}{2})}|F|^q\right)^{\frac{1}{q}} \end{aligned}\label{ww15}
\ee for any $ \rho<\frac{R}{4} $ and $ q>\frac{d}{2} $. In view of $ \eqref{ww15} $, we need to estimate $ \left(\Xint-_{B(y,\frac{R}{2})}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}} $. Owing to $ \eqref{u2dd+2} $, we can obtain without difficulty that \begin{align}
\|\nabla u_{\va,\lambda}\|_{L^2(\om)}\leq C_{\theta_0}\|F\|_{L^{\frac{2d}{d+2}}(\om)}\text{ and }\|u_{\va,\lambda}\|_{L^2(\om)}\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}\|F\|_{L^{\frac{2d}{d+2}}(\om)}.\nonumber \end{align} These, together with Sobolev embedding theorem that $ H_0^1(\om)\subset L^{\frac{2d}{d-2}}(\om) $, imply that \be \begin{aligned}
\left(\Xint-_{B(y,\frac{R}{2})}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}&\leq\left(\Xint-_{B(y,\frac{R}{2})}|u_{\va,\lambda}|^{\frac{2d}{d-2}}\right)^{\frac{d-2}{2d}}\leq CR^{1-\frac{d}{2}}\left(\int_{\om}|u_{\va,\lambda}|^{\frac{2d}{d-2}}\right)^{\frac{d-2}{2d}}\\
&\leq CR^{1-\frac{d}{2}}\left(\int_{\om}|\nabla u_{\va,\lambda}|^{2}\right)^{\frac{1}{2}}\leq C_{\theta_0}R^{1-\frac{d}{2}}\left(\int_{\om}|F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}\\
&\leq C_{\theta_0}R^{1-\frac{d}{2}}\left(\int_{B(y,R)}|F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}=C_{\theta_0}R^2\left(\Xint-_{B(y,R)}|F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}. \end{aligned}\label{l2uvlab} \ee Since $ d\geq 3 $, we have $ q>\frac{d}{2}>\frac{2d}{d+2} $. Then by using Hölder's inequality and $ \eqref{l2uvlab} $, it is not hard to see that \begin{align}
\left(\Xint-_{B(y,\frac{R}{2})}|u_{\va,\lambda}|^2\right)^{\frac{1}{2}}&\leq C_{\theta_0}R^2\left(\Xint-_{B(y,R)}|F|^{\frac{2d}{d+2}}\right)^{\frac{d+2}{2d}}\leq C_{\theta_0}R^2\left(\Xint-_{B(y,R)}|F|^{q}\right)^{\frac{1}{q}}.\label{ww14} \end{align} This, together with $ \eqref{ww15} $, gives that \begin{align}
\left|\int_{\om}\overline{G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)}F(\cdot)\right|&\leq C_{\theta_0}(1+|\lambda|R^2)^{n}R^{2}\left(\Xint-_{B(y,R)}|F|^{q}\right)^{\frac{1}{q}}.\label{ww16} \end{align} By using duality arguments, we can get that \begin{align}
&\left(\Xint-_{B(y,R)}|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^s\right)^{\frac{1}{s}}\leq C_{\theta_0}(1+|\lambda|R^2)^{n}R^{2-d},\label{ww17} \end{align}
for any $ \rho<\frac{R}{4} $, $ R\leq\delta(x) $ and $ s\in (1,\frac{d}{d-2}) $. For $ x,y\in\om $ such that $ x\neq y $, set $ r=|x-y| $. If $ r\leq\frac{1}{2}\delta(y) $, choosing $ p\in(1,\frac{d}{d-2}) $, we have $ (\mathcal{L}_{\va}-\lambda I)(G_{\rho,\va,\lambda}^{\gamma}(\cdot,y))=0 $ in $ B(x,\frac{r}{2}) $ for any $ \rho<\frac{1}{4}r $. Then by using $ \eqref{Linfty} $ and $ \eqref{ww17} $, it is easy to show that for any $ s\in (1,\frac{d}{d-2}) $, \be \begin{aligned}
|G_{\rho,\va,\lambda}^{\gamma}(x,y)|&\leq \frac{C_{\theta_0}}{(1+|\lambda|r^2)^{n}}\left(\Xint-_{B(x,\frac{r}{2})}|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^s\right)^{\frac{1}{s}}\\
&\leq\frac{C_{\theta_0}(1+|\lambda|r^2)^{n}r^{2-d}}{(1+|\lambda|r^2)^{n}}\leq \frac{C_{\theta_0}}{|x-y|^{d-2}}, \end{aligned}\label{ww18}
\ee where we have chosen $ k=n\in\mathbb{N}_+ $. Assume that $ R<\frac{1}{4}\delta(y) $, then for any $ \rho<\frac{R}{4} $, we have $ (\mathcal{L}_{\va}-\lambda I)(G_{\rho,\va,\lambda}^{\gamma}(\cdot,y))=0 $ in $ \om\backslash B(x,R) $. Choose $ \varphi\in C_0^1(\om;\mathbb{R}) $, such that $ 0\leq\varphi\leq 1 $, $ \varphi\equiv 1 $ in $ \om\backslash B(y,2R) $, $ \varphi\equiv 0 $ in $ B(y,R) $ and $ |\nabla\varphi|\leq\frac{C}{R} $. Set $ u(z)=\varphi^2G_{\rho,\va,\lambda}^{\gamma}(z,y) $ as the test function, then it is not hard to obtain that \be \begin{aligned}
&\int_{\om}\varphi^2A_{\va}(\cdot)\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\overline{\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)}-\lambda\int_{\om}\varphi^2|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^2\\ &\quad\quad\quad\quad=-\int_{\om}2\varphi A_{\va}(\cdot)\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\nabla\varphi\overline{G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)}, \end{aligned}\label{ww19} \ee where $ A_{\va}(x)=A(x/\va) $ if $ \va>0 $ and $ A_{0}(x)=\widehat{A} $. For $ \operatorname{Re}\lambda\geq 0 $, we can take the imaginary parts of both sides of $ \eqref{ww19} $ and get that \be \begin{aligned}
\|\varphi G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}^2&\leq \frac{C_{\theta_0}}{|\lambda|}\int_{\om}|\nabla\varphi(\cdot)||G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)||\varphi(\cdot)||\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|\\
&\leq \frac{C_{\theta_0}}{|\lambda|}\left\{\|\nabla\varphi G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}^2+\delta\|\varphi \nabla_1G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}^2\right\}, \end{aligned}\label{ww20} \ee where $ \delta>0 $ is sufficiently small. For $ \operatorname{Re}\lambda<0 $, we can take the real parts of both sides of $ \eqref{ww19} $ and obtain that $ \eqref{ww20} $ is also true. Then $ \eqref{ww20} $ is true for any $ \lambda\in\Sigma_{\theta_0} $. Owing to $ \eqref{ww19} $, $ \eqref{ww20} $ and properties of $ \varphi $, it is obvious that \be \begin{aligned}
\|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om\backslash B(y,2R))}^2&\leq C_{\theta_0}\|\nabla\varphi G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}^2\\
&\leq \frac{C_{\theta_0}}{R^2}\|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(B(y,2R)\backslash B(y,R))}^2\\
&\leq \frac{C_{\theta_0}}{R^2}\int_{B(y,2R)\backslash B(y,R)}\frac{1}{|z-y|^{2d-4}}dz\leq C_{\theta_0}R^{2-d}. \end{aligned}\label{ww21} \ee On the other hand, if $ \rho>\frac{R}{4} $, according to $ \eqref{ww11} $, it can be obtained that \begin{align}
\int_{\om\backslash B(y,2R)}|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^2\leq \int_{\om}|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^2\leq C_{\theta_0}R^{2-d}.\label{ww22} \end{align} $ \eqref{ww21} $, together with $ \eqref{ww22} $, yields that \begin{align}
\int_{\om\backslash B(y,2R)}|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^2\leq C_{\theta_0}R^{2-d},\label{ww23} \end{align}
for any $ \rho>0 $ and $ R<\frac{1}{4}\delta(y) $. Let $ \psi\in C_0^1(\om;\mathbb{R}) $ such that $ 0\leq\psi\leq 1 $, $ \psi\equiv 0 $ in $ B(y,2R) $, $ \psi\equiv 1 $ in $ \om\backslash B(y,\frac{5}{2}R) $ and $ |\nabla\psi|\leq\frac{C}{R} $. With the help of the definition of $ \psi $ and Sobolev embedding theorem, we can deduce that for any $ \rho<\frac{R}{4} $ and $ R<\frac{1}{4}\delta(y) $, \be \begin{aligned}
\|\psi G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^{\frac{2d}{d-2}}(\om)}^{\frac{2d}{d-2}}&\leq C\|\nabla(\psi G_{\rho,\va,\lambda}^{\gamma}(\cdot,y))\|_{L^{2}(\om)}^{\frac{2d}{d-2}}\\
&\leq C\|\nabla \psi G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}^{\frac{2d}{d-2}}+C\|\psi\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\|_{L^2(\om)}^{\frac{2d}{d-2}}\leq C_{\theta_0}R^{-d}, \end{aligned}\label{psiGvala} \ee where for the third inequality, we have used $ \eqref{ww18} $ and $ \eqref{ww23} $. Combining $ \eqref{psiGvala} $ and properties of $ \psi $, it can be easily inferred that \begin{align}
\int_{\om\backslash B(y,\frac{5}{2}R)}| G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^{\frac{2d}{d-2}}\leq C_{\theta_0}R^{-d}\text{ for any }\rho<\frac{R}{4}\text{ and } R<\frac{1}{4}\delta(y).\nonumber \end{align} For the case that $ \rho\geq \frac{R}{4} $, due to $ \eqref{ww11} $, one can find that \begin{align}
\int_{\om\backslash B(y,\frac{5}{2}R)}| G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^{\frac{2d}{d-2}}\leq \left(\int_{\om}|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^{2}\right)^{\frac{d}{d-2}}\leq C_{\theta_0}R^{-d}.\nonumber \end{align} We now address ourselves to the uniform estimates of $ G_{\rho,\va,\lambda}^{\gamma}(\cdot,y) $ and $ \nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y) $ with respect to parameter $ \rho>0 $. In the case of $ t>(\frac{\delta(y)}{4})^{1-d} $, we can choose $ R=t^{-\frac{1}{d-1}}<\frac{1}{4}\delta(y) $ and obtain that for any $ \rho>0 $, \be \begin{aligned}
&\left|\left\{x\in\om:|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|>t\right\}\right|\leq CR^d+t^{-2}\int_{\om\backslash B(y,2R)}|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^2\\ &\quad\quad\quad\quad\quad\quad\leq CR^d+C_{\theta_0}t^{-2}R^{2-d}\leq Ct^{-\frac{d}{d-1}}+C_{\theta_0}t^{-2}t^{\frac{d-2}{d-1}}\leq C_{\theta_0}t^{-\frac{d}{d-1}}. \end{aligned}\label{ww31} \ee When $ t>(\frac{\delta(y)}{4})^{2-d} $, similarly, we can choose $ R=t^{-\frac{1}{d-2}} $ and obtain that for any $ \rho>0 $, \be \begin{aligned}
&\left|\left\{x\in\om:|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|>t\right\}\right|\leq CR^d+t^{-\frac{2d}{d-2}}\int_{\om\backslash B(y,R)}|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^{\frac{2d}{d-2}}\\ &\quad\quad\quad\quad\quad\quad\leq CR^d+C_{\theta_0}t^{-\frac{2d}{d-2}}R^{-d}\leq Ct^{-\frac{d}{d-2}}+C_{\theta_0}t^{-\frac{2d}{d-2}}t^{\frac{d}{d-2}}\leq C_{\theta_0}t^{-\frac{d}{d-2}}. \end{aligned}\label{ww32} \ee Then in view of $ \eqref{ww31} $ and $ \eqref{ww32} $, it can be shown by simple calculations that \begin{align}
\int_{\om}|G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^s&\leq C[\delta(y)]^{s(2-d)}+C_{\theta_0}\int_{(\delta(y)/4)^{2-d}}t^{s-1}t^{-\frac{d}{d-2}}dt\nonumber\\ &\leq C_{\theta_0}\left\{[\delta(y)]^{s(2-d)}+[\delta(y)]^{s(2-d)+d}\right\},\nonumber \end{align} for any $ s\in [1,\frac{d}{d-2}) $ and \begin{align}
\int_{\om}|\nabla_1 G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)|^s\leq C_{\theta_0}\left\{[\delta(y)]^{s(1-d)}+[\delta(y)]^{s(1-d)+d}\right\},\nonumber \end{align} for any $ s\in [1,\frac{d}{d-1}) $. From the uniform estimates above, it follows that there exists a subsequence of $ \{G_{\rho_n,\va,\lambda}^{\gamma}(\cdot,y)\}_{n=1}^{\infty} $ and $ G_{\va,\lambda}^{\gamma}(\cdot,y) $ such that for any $ s\in (1,\frac{d}{d-1}) $ and $ 1\leq\gamma\leq m $, \begin{align} G_{\rho_n,\va,\lambda}^{\gamma}(\cdot,y)\rightharpoonup G_{\va,\lambda}^{\gamma}(\cdot,y)\text{ weakly in }W_0^{1,s}(\om;\mathbb{C}^m)\text{ as }n\to\infty.\label{ww33} \end{align} Hence, we have, for any $ \phi\in W_0^1(\om;\mathbb{C}^m) $ with $ p>d $ \begin{align} B_{\va,\lambda,\om}[G_{\va,\lambda}^{\gamma}(\cdot,y),\phi(\cdot)]=\lim_{n\to\infty}B_{\va,\lambda,\om}[G_{\rho_n,\va,\lambda}^{\gamma}(\cdot,y),\phi(\cdot)]=\lim_{n\to\infty}\Xint-_{\om(y,{\rho_n})}\phi^{\gamma}(\cdot)=\phi^{\gamma}(y),\nonumber \end{align} where we have used the definition of the approximating Green matrix. We note that there exists a weak solution $ u_{\va,\lambda}\in W_0^{1,p}(\om;\mathbb{C}^m) $ satisfying $ (\mathcal{L}_{\va}-\overline{\lambda} I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $ for any $ F\in L^{\frac{p}{2}}(\om;\mathbb{C}^m) $ with $ p>d $ ($ \frac{pd}{p+d}<\frac{p}{2} $). Thus we can obtain that \begin{align} u_{\va,\lambda}^{\gamma}(y)&=B_{\va,\lambda,\om}[G_{\va,\lambda}^{\gamma}(\cdot,y),u_{\va,\lambda}(\cdot)]=\overline{B_{\va,\overline{\lambda},\om}[u_{\va,\lambda}(\cdot),G_{\va,\lambda}^{\gamma}(\cdot,y)]}=\int_{\om}G_{\va,\lambda}^{\gamma}(\cdot,y)\overline{F(\cdot)}.\nonumber \end{align} We now verify the uniqueness. If $ \widetilde{G}_{\va,\lambda}^{\gamma}(\cdot,y) $ is another Green matrix, we can also derive by representation formula that $ u_{\va,\lambda}^{\gamma}(y)=\int_{\om}\widetilde{G}_{\va,\lambda}^{\gamma}(\cdot,y)\overline{F(\cdot)} $. It follows from the uniqueness of the weak solution that \begin{align} \int_{\om}[\widetilde{G}_{\va,\lambda}^{\gamma}(\cdot,y)-G_{\va,\lambda}^{\gamma}(\cdot,y)]\overline{F(\cdot)}=0\text{ for any }F\in L^{\frac{p}{2}}(\om;\mathbb{C}^m).\nonumber \end{align} Then $ \widetilde{G}_{\va,\lambda}^{\gamma}(\cdot,y)=G_{\va,\lambda}^{\gamma}(\cdot,y) $ a.e. in $ \om $ due to the arbitrariness of $ F $. Next, let $ G_{\tau,\va,\overline{\lambda}}(\cdot,x) $ denote the approximating for Green functions of the operator $ \mathcal{L}_{\va}-\overline{\lambda}I $, which satisfy \begin{align} B_{\va,\overline{\lambda},\om}[G_{\tau,\va,\overline{\lambda}}^{\xi}(\cdot,x),u(\cdot)]=\Xint-_{\om(x,\tau)}u^{\xi}(\cdot)\text{ for any }u\in H_0^1(\om;\mathbb{C}^m).\label{ww34} \end{align} By the same argument, we can derive the existence and uniqueness of $ G_{\va,\overline{\lambda}}^{\xi}(\cdot,x) $. Thus for any $ \tau,\rho>0 $ and $ 1\leq\gamma,\xi\leq m $, it is obvious to see that \begin{align} \Xint-_{\om(y,\rho)}G_{\tau,\va,\overline{\lambda}}^{\gamma\xi}(\cdot,x)&=B_{\va,\lambda,\om}[G_{\rho,\va,\lambda}^{\gamma}(\cdot,y),G_{\va,\overline{\lambda}}^{\xi}(\cdot,x)]\nonumber\\ &=\overline{B_{\va,\overline{\lambda},\om}[G_{\va,\overline{\lambda}}^{\xi}(\cdot,x),G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)]}=\Xint-_{\om(x,\tau)}\overline{G_{\rho,\va,\lambda}^{\xi\gamma}(\cdot,y)}.\nonumber \end{align} Note that $ (\mathcal{L}_{\va}-\overline{\lambda} I)(G_{\tau,\va,\overline{\lambda}}^{\xi}(\cdot,x))=0 $ in $ \om\backslash B(x,\tau) $ and $ (\mathcal{L}_{\va}-\lambda I)(G_{\rho,\va,\lambda}^{\gamma}(\cdot,y))=0 $ in $ \om\backslash B(y,\rho) $. In view of the $ W^{1,p} $ estimates, $ G_{\tau,\va,\overline{\lambda}}^{\xi}(\cdot,x) $ and $ G_{\rho,\va,\lambda}^{\gamma}(\cdot,y) $ are locally Hölder continuous. Therefore, by taking $ \tau,\rho\to\infty $, we have $ \overline{G_{\va,\lambda}^{\xi\gamma}(x,y)}=G_{\va,\overline{\lambda}}^{\gamma\xi}(y,x) $, which implies that $ G_{\va,\lambda}(x,y)=[\overline{G_{\va,\overline{\lambda}}(y,x)}]^T $ for any $ x,y\in\om $ such that $ x\neq y $. This gives the proof of $ \eqref{duality} $.
Finally, we will prove $ \eqref{Greene} $. Let $ r=|x-y| $ and $ F\in C_0^{\infty}(\om(x,{\frac{r}{3}});\mathbb{C}^m) $. Assume that $ u_{\va,\lambda} $ is the solution of $ (\mathcal{L}_{\va}-\overline{\lambda}I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $. Then $ u_{\va,\lambda}(y)=\int_{\om}\overline{F(\cdot)}G_{\va,\lambda}(\cdot,y) $. Since $ (\mathcal{L}_{\va}-\overline{\lambda} I)(u_{\va,\lambda})=0 $ in $ \om\backslash \om(x,\frac{r}{3}) $ and $ \om(y,\frac{r}{3})\subset\om\backslash \om(x,\frac{r}{3}) $, it follows from $ \eqref{u2dd+2} $ and $ \eqref{Linfty} $ that for any $ k\in\mathbb{N}_+ $, \begin{align}
|u_{\va,\lambda}(y)|&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k}\left(\Xint-_{\om(y,\frac{r}{3})}|u_{\va,\lambda}|^{2}\right)^{\frac{1}{2}}\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k}r^{1-\frac{d}{2}}\|u_{\va,\lambda}\|_{L^{\frac{2d}{d-2}}(\om(y,\frac{r}{3}))}\nonumber\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k}r^{1-\frac{d}{2}}\|u_{\va,\lambda}\|_{L^{\frac{2d}{d-2}}(\om)}\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k}r^{1-\frac{d}{2}}\|\nabla u_{\va,\lambda}\|_{L^{2}(\om)}\nonumber\\
&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^{k}}r^{1-\frac{d}{2}}\|F\|_{L^{\frac{2d}{d+2}}(\om)}=\frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^{k}}r^{1-\frac{d}{2}} \|F\|_{L^{\frac{2d}{d+2}}(\om(x,\frac{r}{3}))}\nonumber\\
&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^{k}}r^{2-\frac{d}{2}} \|F\|_{L^{2}(\om(x,\frac{r}{3}))}.\nonumber \end{align} In view of duality arguments, we can infer that for any $ k\in\mathbb{N}_+ $, \begin{align}
\left(\Xint-_{\om(x,\frac{r}{3})}|G_{\va,\lambda}(\cdot,y)|^2\right)^{\frac{1}{2}}\leq\frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^{k}}r^{2-d}.\label{ww56} \end{align} Note that $ (\mathcal{L}_{\va}-\lambda I)(G_{\va,\lambda}^{\gamma}(\cdot,y))=0 $ in $ \om\backslash B(y,r) $ for any $ r>0 $. So in the case of $ \frac{r}{6}\leq \delta(x) $, it follows from $ \eqref{Linfty} $ and $ \eqref{ww56} $ that for any $ k\in\mathbb{N}_+ $, \begin{align}
|G_{\va,\lambda}(x,y)|\leq C_{\theta_0}\left(\Xint-_{\om(x,\frac{r}{3})}|G_{\va,\lambda}(\cdot,y)|^{2}\right)^{\frac{1}{2}}\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}}\frac{1}{|x-y|^{d-2}}.\nonumber \end{align} When $ \frac{r}{6}>\delta(x) $, on the other hand, in view of $ \eqref{Holder} $ and $ \eqref{ww56} $, for any $ \sigma_1\in (0,1) $ and $ k\in\mathbb{N}_+ $, we have, \begin{align}
|G_{\va,\lambda}(x,y)|&=|G_{\va,\lambda}(x,y)-G_{\va,\lambda}(\overline{x},y)|\leq [G_{\va,\lambda}(\cdot,y)]_{C^{0,\sigma_1}(\om(x,\frac{r}{6}))}|x-\overline{x}|^{\sigma_1}\nonumber\\
&\leq C_{\theta_0}\left(\frac{|x-\overline{x}|}{r}\right)^{\sigma_1}\left(\Xint-_{\om(x,\frac{r}{3})}|G_{\va,\lambda}(\cdot,y)|^{2}\right)^{\frac{1}{2}}\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}}\frac{[\delta(x)]^{\sigma_1}}{|x-y|^{d-2+\sigma_1}},\nonumber \end{align}
where $ \overline{x}\in\pa\om $ is the point such that $ |x-\overline{x}|=\delta(x) $. It is easy to complete the proof of $ \eqref{Greene} $ by considering the same estimates for $ G_{\va,\overline{\lambda}}(x,y) $ and using property $ \eqref{duality} $. \end{proof}
\subsection{Two dimensional Green functions}
\begin{thm}[Green functions of $ \mathcal{L}_{\va}-\lambda I $ with $ d=2 $]\label{Green2} Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $, $ \eqref{VMO} $, $ \va\geq 0 $ and $ \om $ is a bounded $ C^{1} $ domain in $ \mathbb{R}^2 $. If $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, then there exists a unique Green function $ G_{\va,\lambda}=(G_{\va,\lambda}^{\al\beta}(\cdot,\cdot)):\om\times\om\to \mathbb{C}^{m^2}\cup\left\{\infty\right\} $ with $ 1\leq\al,\beta\leq m $, such that \be \begin{aligned}
G_{\va,\lambda}(\cdot,y)\in \operatorname{BMO}(\om;\mathbb{C}^{m^2}) \text{ i.e. } \left\|G_{\va,\lambda}(\cdot,y)\right\|_{\BMO(\om)}\leq C_{\theta_0}\text{ uniformly for }y\in\om. \end{aligned}\label{BMO estimates} \ee Moreover, for all $ u_{\va,\lambda} $ being the weak solution for the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $, where $ F\in L^p(\om;\mathbb{C}^m) $, $ (p>1) $, we have \begin{align} u_{\va,\lambda}(x)=\int_{\om}G_{\va,\lambda}(x,y)\overline{F(y)}dy.\label{Representation formula} \end{align} Furthermore, for Green functions $ G_{\va,\overline{\lambda}}(x,y) $ corresponding to the operators of $ \mathcal{L}_{\va}-\overline{\lambda} I $, we have $ G_{\va,\lambda}(x,y)=\overline{[G_{\va,\overline{\lambda}}(y,x)]}^T $. For all $ \sigma_1,\sigma_2,\sigma\in (0,1) $ and $ k\in\mathbb{N}_+ $, Green functions satisfy pointwise estimates \be
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^k}\left(\frac{R_0}{|x-y|}\right)^{\sigma}\text{ for any }x,y\in\om,\label{preliminary} \ee \be
|G_{\va,\lambda}(x,y)|\leq C_{k,\theta_0}\left\{1+|\lambda|^{\frac{\sigma}{2}}R_0^{\sigma}+\ln\left(\frac{R_0}{|x-y|}\right)\right\}\text{ for any }x,y\in\om,\label{Pointwise estimates for Green functions 4} \ee \begin{align}
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}[\delta(x)]^{\sigma_1}}{(1+|\lambda||x-y|^2)^k|x-y|^{\sigma_1}} &\text{ if } \delta(x)<\frac{1}{4}|x-y|,\label{Pointwise estimates for Green functions 1}\\
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}[\delta(y)]^{\sigma_2}}{(1+|\lambda||x-y|^2)^k|x-y|^{\sigma_2}} & \text{ if } \delta(y)<\frac{1}{4}|x-y|,\label{Pointwise estimates for Green functions 2}\\
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}[\delta(x)]^{\sigma_1}[\delta(y)]^{\sigma_2}}{(1+|\lambda||x-y|^2)^k|x-y|^{\sigma_1+\sigma_2}}&\text{ if } \min\left\{\delta(x),\delta(y)\right\}<\frac{1}{4}|x-y|,\label{Pointwise estimates for Green functions 3} \end{align} where $ \delta(x)=\operatorname{dist}(x,\pa\om) $ denotes the distance from $ x $ to the boundary of $ \om $, $ x\neq y $ and $ C_{k,\theta_0} $ are constants depending only on $ \sigma_1,\sigma_2,\sigma,\mu,\omega(t),k,\theta_0,m $ and $ \om $. \end{thm}
\begin{proof} For any $ y\in\om $ and $ \rho>0 $, in view of Theorem \ref{existencethm}, there exists a matrix-valued function $ G_{\rho,\va,\lambda}(\cdot,y)=(G_{\rho,\va,\lambda}^{\alpha\beta}(\cdot,y)):\om\to\mathbb{C}^{m^2} $, such that for any $ 1\leq\gamma\leq m $, $ G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)\in H_0^1(\om;\mathbb{C}^m) $ satisfies \begin{align} B_{\va,\lambda,\om}[G_{\rho,\va,\lambda}^{\gamma}(\cdot,y),u(\cdot)]=\Xint-_{\om(y,\rho)}u^{\gamma}(\cdot)\text{ for any }u\in H_0^1(\om;\mathbb{C}^m).\nonumber \end{align} For any atom function in $ \om $ denoted by $ a=a(\cdot) $, s.t. \begin{align}
\operatorname{supp}(a)\subset\om(y,\rho)\text{ and }\left\|a\right\|_{L^{\infty}(\om)}\leq\frac{1}{|\om(y,\rho)|},\nonumber \end{align} we can get $ v_{\va,\lambda}\in H_0^1(\om;\mathbb{R}^m) $ such that $ (\mathcal{L}_{\va}-\overline{\lambda} I)(v_{\va,\lambda})=a $ in $ \om $ and $ v_{\va,\lambda}=0 $ on $ \pa\om $. Then \begin{align} \Xint-_{\om(y,\rho)}v_{\va,\lambda}^{\gamma}(\cdot)&=B_{\va,\lambda,\om}[G_{\rho,\va,\lambda}^{\gamma}(\cdot,y),v_{\va,\lambda}(\cdot)]=\overline{B_{\va,\overline{\lambda},\om}[v_{\va,\lambda}(\cdot),G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)]}=\int_{\om}G_{\rho,\va,\lambda}^{\alpha\gamma}(\cdot,y)\overline{a^{\alpha}(\cdot)}.\nonumber \end{align} This, together with $ \eqref{Linftyatom} $, implies the following estimate \begin{align}
\left|\int_{\om}G_{\rho,\va,\lambda}(\cdot,y)\overline{a(\cdot)}\right|\leq \left|\Xint-_{\om(y,\rho)}v_{\va,\lambda}(\cdot)\right|\leq\|v_{\va,\lambda}\|_{L^{\infty}(\om(y,\rho))} \leq\|v_{\va,\lambda}\|_{L^{\infty}(\om)}\leq C_{\theta_0},\nonumber \end{align} where $ C_{\theta_0} $ depends only on $ \mu,\omega(t),\theta_0,m $ and $ \om $. According to the fact that $ \mathcal{H}^1 $ is the dual space of $ \operatorname{BMO} $ space, we can derive that $ G_{\rho,\va,\lambda}(\cdot,y) $ has a uniform boundedness $ C_{\theta_0} $ in $ \operatorname{BMO} $ space, where $ C_{\theta_0} $ depends only on $ \mu,\omega(t),m,\theta_0 $ and $ \om $. In view of Banach-Alaoglu theorem, we have, for all $ y\in\om $, there exists a sequence $ \{\rho_j\} $ such that $ \rho_j\to 0 $ when $ j\to\infty $ and functions $ G_{\va,\lambda}^{\alpha\beta}(\cdot,y)\in \operatorname{BMO}(\om) $ such that $ G_{\rho_j,\va,\lambda}^{\alpha\beta}(\cdot,y) $ converge to $ G_{\va,\lambda}^{\alpha\beta}(\cdot,y) $ in the space $ \operatorname{BMO}(\om) $ with the sense of the weak*-topology. For $ F\in L^q(\om;\mathbb{R}^m) $ where $ q>1 $, we can choose $ 1<q_1<q $, $ p=\frac{2q_1}{2-q_1}>2 $ and $ u_{\va,\lambda}\in W_0^{1,p}(\om;\mathbb{R}^m) $, such that $ (\mathcal{L}_{\va}-\overline{\lambda}I)(u_{\va,\lambda})=F $ and $ u_{\va,\lambda}=0 $ on $ \pa\om $. Then it can be obtained that \be \begin{aligned} \Xint-_{\om(y,\rho)}u_{\va,\lambda}^{\gamma}(\cdot)&=B_{\va,\lambda,\om}[G_{\rho,\va,\lambda}^{\gamma}(\cdot,y),u_{\va,\lambda}(\cdot)]\\ &=\overline{B_{\va,\overline{\lambda},\om}[u_{\va,\lambda}(\cdot),G_{\rho,\va,\lambda}^{\gamma}(\cdot,y)]}=\int_{\om}G_{\rho,\va,\lambda}^{\al\gamma}(\cdot,y)\overline{F^{\al}(\cdot)}. \end{aligned}\label{ww181} \ee In view of Remark \ref{remimp}, Poincaré's inequality and Hölder's inequality, we have \begin{align}
\left\|u_{\va,\lambda}\right\|_{W^{1,p}(\om)}\leq C\|\nabla u_{\va,\lambda}\|_{L^p(\om)}\leq C_{\theta_0}\left\|F\right\|_{L^{q_1}(\om)}\leq C_{\theta_0}\left\|F\right\|_{L^q(\om)},\nonumber \end{align} where $ C_{\theta_0} $ is a constant depending only on $ \mu,\omega(t),\theta_0,m,p,q $ and $ \om $. Using Sobolev embedding theorem, we see that $ u_{\va,\lambda} $ is continuous. Letting $ \rho_j\to 0 $, the left hand side of $ \eqref{ww181} $ converges to $ u_{\va,\lambda}(y) $. On the other hand, according to the embedding theorem $ L^p\subset \mathcal{H}^1 $, one can obtain that \begin{align} u_{\va,\lambda}^{\gamma}(y)=\lim_{j\to\infty}\Xint-_{\om(y,\rho_j)}u_{\va,\lambda}^{\gamma}(\cdot)=\lim_{j\to\infty}\int_{\om}G_{\rho_j,\va,\lambda}^{\al\gamma}(\cdot,y)\overline{F^{\al}(\cdot)}=\int_{\om}G_{\va,\lambda}^{\al\gamma}(\cdot,y)\overline{F^{\al}(\cdot)}.\label{ww191} \end{align} Since the representation formula, uniqueness of Green functions and duality property follow from almost the same arguments for the case $ d\geq 3 $, here we will not repeat the proofs of them.
Finally, we will prove the pointwise estimates of Green functions. Namely, we will prove $ \eqref{preliminary} $-$ \eqref{Pointwise estimates for Green functions 3} $. Letting $ x_0,y_0\in\om $ with $ x_0\neq y_0 $ and assuming that $ \delta(x_0)<\frac{1}{2}|x_0-y_0|=\frac{1}{2}r $, we have $ \om(x_0,\frac{1}{2}r)\subset\om\backslash\left\{y_0\right\} $. According to the definition of $ G_{\va,\lambda}^{\gamma}(\cdot,y_0) $, we have $ (\mathcal{L}_{\va}-\lambda I)(G_{\va,\lambda}^{\gamma}(\cdot,y_0))=0 $ in $ \om(x_0,\frac{1}{2}r) $ and $ G_{\va,\lambda}^{\gamma}(\cdot,y_0)=0 $ on $ \pa\om\cap B(x_0,\frac{1}{2}r) $. Then by $\eqref{Linfty} $, we can obtain that for any $ k\in\mathbb{N}_+ $, \begin{align}
|G_{\va,\lambda}(x_0,y_0)|\leq C\left\|G_{\va,\lambda}(\cdot,y_0)\right\|_{L^{\infty}(\om(x_0,\frac{r}{4}))}\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k}\Xint-_{\om(x_0,\frac{1}{2}r)}|G_{\va,\lambda}(\cdot,y_0)|.\nonumber \end{align} According to the definition of $ \BMO(\om) $ and the observation that $ G_{\va,\lambda}(\cdot,y_0)_{x_0,\frac{r}{2}}=0 $ by $ \eqref{ww35} $, it is easy to see that for any $ k\in\mathbb{N}_+ $, \begin{align}
|G_{\va,\lambda}(x_0,y_0)|&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k}\Xint-_{\om(x_0,\frac{r}{2})}|G_{\va,\lambda}(z,y_0)-G_{\va,\lambda}(z,y_0)_{x_0,\frac{r}{2}}|dz\nonumber\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k}\left\|G_{\va,\lambda}(\cdot,y_0)\right\|_{\BMO(\om)}\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k},\nonumber \end{align}
where $ C_{k,\theta_0} $ depends only on $ \mu,\omega(t),k,\theta_0,m $ and $ \om $. With the help of estimates above, we can find that if $ \delta(x)<\frac{1}{2}|x-y| $, then \begin{align}
|G_{\va,\lambda}(x,y)|\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^k}\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^k}.\label{Boundedness estimates} \end{align}
Assume that $ \delta(x_0)<\frac{1}{4}|x_0-y_0|=\frac{1}{4}r $ and $ z_0\in\pa\om $ is chosen such that $ |x_0-z_0|=\delta(x_0) $, we note that $ (\mathcal{L}_{\va}-\lambda I)(G_{\va,\lambda}^{\gamma}(\cdot,y_0))=0 $ in $ \om(x_0,\frac{r}{2}) $ and $ G_{\va,\lambda}^{\gamma}(\cdot,y_0)=0 $ on $ \pa\om\cap B(z_0,\frac{r}{2}) $. In view of the localized boundary H\"{o}lder estimates $ \eqref{Holder} $, then for all $ \sigma_1\in (0,1) $, there is \be \begin{aligned}
|G_{\va,\lambda}^{\gamma}(x_0,y_0)|&= |G_{\va,\lambda}^{\gamma}(x_0,y_0)-G_{\va,\lambda}^{\gamma}(z_0,y_0)|\\
&\leq |x_0-z_0|^{\sigma_1}[G_{\va,\lambda}^{\gamma}(\cdot,y_0)]_{C^{0,\sigma_1}(\om(z_0,\frac{3r}{8}))}\\
&\leq C_{\theta_0}\left(\frac{[\delta(x_0)]}{r}\right)^{\sigma_1}\left(\Xint-_{\om(z_0,\frac{r}{2})}|G_{\va,\lambda}^{\gamma}(\cdot,y_0)|^2\right)^{\frac{1}{2}}. \end{aligned}\label{Gvalamcthe} \ee Simple obervations give that for all $ x\in \om(z_0,\frac{r}{2}) $, we have $ \delta(x)<\frac{1}{2}r $. Then due to $ \eqref{Boundedness estimates} $ and $ \eqref{Gvalamcthe} $ \begin{align}
|G_{\va,\lambda}(x_0,y_0)|\leq\frac{C_{k,\theta_0}[\delta(x_0)]^{\sigma_1}}{(1+|\lambda||x_0-y_0|^2)^k|x_0-y_0|^{\sigma_1}}.\nonumber \end{align}
This proves $ \eqref{Pointwise estimates for Green functions 1} $. On the other hand, $ \eqref{Pointwise estimates for Green functions 2} $ can be obtained naturally by considering the same estimates for Green functions of $ \mathcal{L}_{\va}-\overline{\lambda}I $. In addition, we can see from the above proof that $ \frac{1}{4} $ is not an essential constant in $ \eqref{Pointwise estimates for Green functions 1} $-$ \eqref{Pointwise estimates for Green functions 2} $. That is, $ \eqref{Pointwise estimates for Green functions 1} $-$ \eqref{Pointwise estimates for Green functions 2} $ are also true if we change $ \frac{1}{4} $ to any constants $ C_0 $ such that $ 0<C_0<\frac{1}{2} $ by using almost the same arguments. Next, let us prove $ \eqref{Pointwise estimates for Green functions 3} $. We can assume that $ \delta(x_0)<\frac{1}{4}|x_0-y_0| $ and $ \delta(y_0)<\frac{1}{4}|x_0-y_0| $. Otherwise, $ \eqref{Pointwise estimates for Green functions 3} $ follows from $ \eqref{Pointwise estimates for Green functions 1} $-$ \eqref{Pointwise estimates for Green functions 2} $ directly. By using almost the same methods, we have, for any $ 1\leq\gamma\leq m $, \begin{align}
|G_{\va,\lambda}^{\gamma}(x_0,y_0)|&\leq C_{\theta_0}\left(\frac{[\delta(x_0)]}{r}\right)^{\sigma_1}\left(\Xint-_{\om(z_0,\frac{7r}{16})}|G_{\va,\lambda}^{\gamma}(\cdot,y_0)|^2\right)^{\frac{1}{2}}.\label{ww10000} \end{align}
For all $ y\in B(z_0,\frac{7r}{16})\cap\om $, it is esay to find by triangular inequality that $ |y-y_0|\geq |x_0-y_0|-|x_0-y|\geq\frac{9}{16}r $. Meanwhile, it is not hard to obtain that \begin{align}
|G_{\va,\lambda}^{\gamma}(z,y_0)|\leq\frac{C_{k,\theta_0}[\delta(y_0)]^{\sigma_2}}{(1+|\lambda||z-y_0|^2)^k|z-y_0|^{\sigma_2}}\text{ for any }k\in\mathbb{N}_+\text{ and }z\in \om\left(z_0,\frac{7r}{16}\right).\nonumber \end{align} This, together with $ \eqref{ww10000} $, gives the proof of $ \eqref{Pointwise estimates for Green functions 3} $. Indeed, for any $ k\in\mathbb{N}_+ $ and $ \sigma_1,\sigma_2\in(0,1) $, \begin{align}
|G_{\va,\lambda}^{\gamma}(x_0,y_0)|&\leq C_{k,\theta_0}\left(\frac{[\delta(x_0)]}{r}\right)^{\sigma_1}\left(\Xint-_{B(z_0,\frac{7r}{16})\cap\om}\frac{[\delta(y_0)]^{2\sigma_2}}{(1+|\lambda||y-y_0|^2)^{2k}|y-y_0|^{2\sigma_2}}dy\right)^{\frac{1}{2}}\nonumber\\
&\leq\frac{C_{k,\theta_0}[\delta(x_0)]^{\sigma_1}[\delta(y_0)]^{\sigma_2}}{(1+|\lambda||x_0-y_0|^2)^{k}|x_0-y_0|^{\sigma_1+\sigma_2}}.\nonumber \end{align}
At last, if $ \delta(x_0)\geq\frac{1}{4}|x_0-y_0| $ and $ \delta(y_0)\geq\frac{1}{4}|x_0-y_0| $, we choose $ F\in C_0^{\infty}(\om(x_0,\frac{r}{4});\mathbb{C}^m) $. Obviously, here, $ \om(x_0,\frac{r}{4})=B(x_0,\frac{r}{4}) $. Let $ w_{\va,\lambda}\in H_0^1(\om;\mathbb{C}^m) $ satisfy $ (\mathcal{L}_{\va}-\overline{\lambda} I)(w_{\va,\lambda})=F $ in $ \om $ and $ w_{\va,\lambda}=0 $ on $ \pa\om $. In view of the representation theorem, we have that $ w_{\va,\lambda}(y)=\int_{\om}\overline{G_{\va,\lambda}(\cdot,y)}F(\cdot) $. Since $ F\equiv 0 $ in $ \om\backslash\om(x_0,\frac{r}{4}) $, then $ (\mathcal{L}_{\va}-\overline{\lambda} I)(w_{\va,\lambda})=0 $ in $ \om\backslash\om(x_0,\frac{r}{4}) $. For all $ p>2 $ and $ 1<q<2 $, by using $ \eqref{Linfty} $ and Sobolev embedding theorem $ H_0^1(\om)\subset L^p(\om) $ with $ p>1 $, it is not hard to get that \begin{align}
|w_{\va,\lambda}(y_0)|&\leq C_{\theta_0}\left(\Xint-_{\om(y_0,\frac{r}{4})}|w_{\va,\lambda}|^2\right)^{\frac{1}{2}}\leq C_{\theta_0}\left(\Xint-_{\om(y_0,\frac{r}{4})}|w_{\va,\lambda}|^p\right)^{\frac{1}{p}}=C_{\theta_0}R_0^{\frac{2}{p}}r^{-\frac{2}{p}}\left(\int_{\om}|\nabla w_{\va,\lambda}|^2\right)^{\frac{1}{2}}\nonumber\\
&\leq C_{\theta_0}R_0^{\frac{2}{p}+2-\frac{2}{q}}r^{-\frac{2}{p}}\left(\int_{\om}|F|^q\right)^{\frac{1}{q}}\leq C_{\theta_0}R_0^{\frac{2}{p}+2-\frac{2}{q}}r^{-\frac{2}{p}+\frac{2}{q}-1}\left(\int_{\om(x_0,\frac{r}{4})}|F|^2\right)^{\frac{1}{2}},\nonumber \end{align} where for the third inequality, we have used $ \eqref{uq} $. This implies that \begin{align}
\left|\int_{\om(x_0,\frac{r}{4})}G_{\va,\lambda}(\cdot,y_0)\overline{F(\cdot)}\right|\leq C_{\theta_0}r\left(\frac{R_0}{r}\right)^{\frac{2}{p}-\frac{2}{q}+2}\left(\int_{\om(x_0,\frac{r}{4})}|F|^2\right)^{\frac{1}{2}}.\label{ww211} \end{align} Owing to duality arguments, $ \eqref{ww211} $ implies that for all $ p>2 $ and $ 1<q<2 $ \begin{align}
\left(\Xint-_{\om(x_0,\frac{r}{4})}|G_{\va,\lambda}(\cdot,y_0)|^2\right)^{\frac{1}{2}}\leq C_{\theta_0}\left(\frac{R_0}{r}\right)^{\frac{2}{p}-\frac{2}{q}+2}.\label{ww2222} \end{align} For any $ \sigma\in(0,1) $, we can choose special $ p,q $ such that $ -\frac{2}{p}+\frac{2}{q}-2=-\sigma $. Then $ \eqref{ww2222} $, together with $ \eqref{Linfty} $, gives the proof of $ \eqref{preliminary} $. Of course, there is still a certain distance between this and $ \eqref{Pointwise estimates for Green functions 4} $, so we need to make more precise estimates.
For $ x_0,y_0\in\om $, letting $ r_1=\frac{1}{2}|x_0-y_0| $, it can be seen that if $ \delta(x_0)<\frac{1}{2}|x_0-y_0| $, then by $ \eqref{Boundedness estimates} $, we have $ |G_{\va,\lambda}(x_0,y_0)|\leq C_{\theta_0} $. If $ \delta(x_0)\geq\frac{1}{2}|x_0-y_0| $, we need to consider the sequence of subsets of $ \om $ denoted as $ \om_j=\om(x_0,2^jr_1) $ with $ j=0,1,...,N $ such that $ 2^Nr_1\geq \delta(x_0) $ and $ 2^{N-1}r_1<\delta(x_0) $. Obviously, we can bound $ N $ by the inequality \begin{align}
N\leq C\left\{1+\ln\left(\frac{ R_0}{|x_0-y_0|}\right)\right\}.\label{lnN} \end{align} According to the fact that $ G_{\va,\lambda}(\cdot,y_0)\in \operatorname{BMO}(\om) $, we have, if $ 1\leq j\leq N-2 $, then \be \begin{aligned}
&|G_{\va,\lambda}(\cdot,y_0)_{x_0,2^{j}r_1}-G_{\va,\lambda}(\cdot,y_0)_{x_0,2^{j+1}r_1}|\\
&\quad\quad=\left|\Xint-_{\om(x_0,2^jr_1)}G_{\va,\lambda}(\cdot,y_0)-\Xint-_{\om(x_0,2^{j+1}r_1)}G_{\va,\lambda}(\cdot,y_0)\right|\\
&\quad\quad\leq C\Xint-_{\om(x_0,2^{j+1}r_1)}|G_{\va,\lambda}(x,y_0)-G_{\va,\lambda}(\cdot,y_0)_{x_0,2^{j+1}r_1}|dx\\
&\quad\quad\leq C\left\|G_{\va,\lambda}(\cdot,y_0)\right\|_{\BMO(\om)}\leq C_{\theta_0}. \end{aligned}\label{Average difference estimates} \ee With the choice of $ N $, we can get that \be \begin{aligned}
\Xint-_{\om_{N-1}}|G_{\va,\lambda}(\cdot,y_0)|&\leq C\Xint-_{\om_{N}}|G_{\va,\lambda}(\cdot,y_0)|\leq C\left\|G_{\va,\lambda}(\cdot,y_0)\right\|_{\BMO(\om)}\leq C_{\theta_0}. \end{aligned}\label{Large scale integral} \ee In the view of $ \eqref{Linfty} $, if $ p>2 $ and $ q=\frac{2p}{p+2} $, then for any $ k\in\mathbb{N}_+ $, \begin{align}
|G_{\va,\lambda}(x_0,y_0)-G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|r_1^2)^k}\Xint-_{B(x_0,r_1)}|G_{\va,\lambda}(x,y_0)-G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|dx \nonumber\\
&\quad+C_{k,\theta_0}(1+|\lambda|r_1^2)^{n}|\lambda|r_1^2|G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|\label{The second important estimates}\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r_1^2)^k}+C_{k,\theta_0}(1+|\lambda|r_1^2)^{n}|\lambda|r_1^2|G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|,\nonumber \end{align} where the second inequality in $ \eqref{The second important estimates} $ is derived from $ \eqref{Average difference estimates} $ and the fact that \begin{align}
\Xint-_{B(x_0,r_1)}|G_{\va,\lambda}(x,y_0)-G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|dx\leq \left\|G_{\va,\lambda}(\cdot,y_0)\right\|_{\BMO(\om)}\leq C_{\theta_0}.\nonumber \end{align} In view of $ \eqref{Average difference estimates} $, $\eqref{Large scale integral} $ and $ \eqref{The second important estimates} $, it is not hard to see that \be \begin{aligned}
|G_{\va,\lambda}(x_0,y_0)|&\leq C\sum_{j=-1}^{N-2}\left|\Xint-_{\om_j}G_{\va,\lambda}(\cdot,y_0)-\Xint-_{\om_{j+1}}G_{\va,\lambda}(\cdot,y_0)\right|+\Xint-_{\om_{N-1}}|G_{\va,\lambda}(\cdot,y_0)|\\
&\leq C_{\theta_0}\left(1+\ln\left(\frac{ R_0}{|x_0-y_0|}\right)\right)+C|G_{\va,\lambda}(x_0,y_0)-G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|\\
&\leq C_{\theta_0}\left(1+\ln\left(\frac{ R_0}{|x_0-y_0|}\right)\right)+C_{\theta_0}(1+|\lambda|r_1^2)^{n}|\lambda|r_1^2|G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|, \end{aligned}\label{lnzuiy} \ee where we denote $ \Xint-_{\om_{-1}}G_{\va,\lambda}(\cdot,y_0)=G_{\va,\lambda}(x_0,y_0) $ and $ C_{\theta_0} $ depends only on $ \mu,\omega(t),\theta_0,m $ and $ \om $. Setting $ \sigma\in (0,1) $ and using $ \eqref{preliminary} $, then for any $ k\in\mathbb{N}_+ $, \begin{align}
|G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|&= \left|\Xint-_{\om_0}G_{\va,\lambda}(\cdot,y_0)\right|\leq\Xint-_{\om_0}|G_{\va,\lambda}(\cdot,y_0)|\nonumber\\
&\leq \Xint-_{B(x_0,r_1)}\frac{C_{k,\theta_0}R_0^{\sigma}}{(1+|\lambda||x-y_0|^2)^k|x-y_0|^{\sigma}}dx\leq\frac{C_{k,\theta_0}R_0^{\sigma}}{(1+|\lambda||x_0-y_0|^2)^k|x_0-y_0|^{\sigma}}.\nonumber \end{align} Choosing $ k\in\mathbb{N}_+ $ such that $ k>n $, we have \begin{align}
C_{\theta_0}(1+|\lambda|r_1^2)^{n}|\lambda|r_1^2|G_{\va,\lambda}(\cdot,y_0)_{x_0,r_1}|\leq C_{\theta_0} |\lambda|^{\frac{\sigma}{2}}R_0^{\sigma}.\nonumber \end{align} This, together with $ \eqref{lnzuiy} $, completes the proof. \end{proof}
\subsection{More estimates of Green functions}
\begin{thm}\label{lipgg} For $ \va\geq 0 $ and $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, let $ \om $ be a bounded $ C^{1,\eta} $ $ (0<\eta<1) $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. Then for any $ k\in\mathbb{N}_+ $, the Green functions of $ (\mathcal{L}_{\va}-\lambda I) $ satisfy the uniform pointwise estimates \begin{align}
|\nabla_1 G_{\va,\lambda}(x,y)|+|\nabla_2 G_{\va,\lambda}(x,y)|&\leq \frac{C_{k,\theta_0} }{(1+|\lambda||x-y|^2)^{k}|x-y|^{d-1}},\label{Green Lipschitz 11}\\
|\nabla_1G_{\va,\lambda}(x,y)|&\leq \frac{C_{k,\theta_0} \delta(y)}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d}},\label{Green Lipschitz 12}\\
|\nabla_2G_{\va,\lambda}(x,y)|&\leq \frac{C_{k,\theta_0}\delta(x)}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d}},\label{Green Lipschitz 13}\\
|\nabla_1\nabla_2G_{\va,\lambda}(x,y)|&\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d}},\label{Green Lipschitz 14} \end{align} for $ x,y\in\om $ and $ x\neq y $, where $ C_{k,\theta_0} $ depends only on $ \mu,d,m,\tau,\nu,k,\theta_0,\eta $ and $ \om $. \end{thm} \begin{rem} The Lipschitz estimates of Green functions for operators $ \mathcal{L}_{\va}-\lambda I $ given above are sharp. These are of great significance in this paper. Further proofs rely heavily on these estimates. In fact, we point out that after proving these estimates, the proofs of theorems on convergence rates for Green functions are standard and the reason why the standard proofs can be carried out later is that we calculated the impact of $ \lambda $ in Lipschitz estimates to the best. \end{rem} \begin{rem} The proof of Theorem \ref{lipgg} is quite different from which of Lipschitz estimates of Green functions for the operator $ \mathcal{L}_{\va} $, especially when $ d=2 $. Recall that in \cite{Shen2}, the case $ d=2 $ was discussed by using the fact that $ \mathcal{L}_{\va}(c)=0 $ for any constant vector $ c\in\mathbb{C}^m $. However operator $ \mathcal{L}_{\va}-\lambda I $ do not have such property. In this point of view, we need to modify the proofs in \cite{Shen2} and use some technical calculations here. \end{rem} \begin{proof}[Proof of Theorem \ref{lipgg}]
Firstly, we consider the case $ d\geq 3 $. Under the condition that $ A $ is Hölder continuous, we can improve the estimates $ \eqref{Greene} $ to the case that $ \sigma_1=\sigma_2=1 $. Setting $ R=|x-y| $ and applying $ \eqref{Greene} $, it is apparent that if $ \frac{R}{6}\leq \delta(x) $, then \begin{align}
|G_{\va,\lambda}(x,y)|\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}}\frac{1}{|x-y|^{d-2}},\nonumber \end{align}
for any $ k\in\mathbb{N}_+ $. If $ \frac{R}{6}>\delta(x) $, we can choose $ \overline{x}\in\pa\om $ such that $ |x-\overline{x}|=\delta(x) $. Then it follows from $ \eqref{Lipesu} $ that for any $ k\in\mathbb{N}_+ $, \begin{align}
|G_{\va,\lambda}(x,y)|&=|G_{\va,\lambda}(x,y)-G_{\va,\lambda}(\overline{x},y)|\leq|x-\overline{x}|\|\nabla_1G_{\va,\lambda}(\cdot,y)\|_{L^{\infty}(\om(x,\frac{R}{6}))}\nonumber\\
&\leq\frac{C_{k,\theta_0}\delta(x)}{(1+|\lambda|R^2)^kR}\left(\Xint-_{\om(x,\frac{R}{3})}|G_{\va,\lambda}(\cdot,y)|^2\right)^{\frac{1}{2}}.\nonumber \end{align} This, together with $ \eqref{ww56} $, implies that for any $ k\in\mathbb{N}_+ $, \begin{align}
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}\delta(x)}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d-1}}.\nonumber \end{align} By applying the same arguments on $ G_{\va,\overline{\lambda}}(x,y) $ and using $ \eqref{duality} $, we have \begin{align}
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d-2}}\min\left\{1,\frac{\delta(x)}{|x-y|},\frac{\delta(y)}{|x-y|}\right\}.\label{ww57} \end{align} Noticing $ (\mathcal{L}_{\va}-\lambda I)(G_{\va,\lambda}^{\gamma}(\cdot,y))=0 $ for any $ 1\leq\gamma\leq m $ in $ \om\backslash\{y\} $, it follows from $ \eqref{Lipesu} $ and $ \eqref{ww56} $ that \be \begin{aligned}
|\nabla_1G_{\va,\lambda}(x,y)|&\leq \frac{C_{\theta_0}}{R}\left(\Xint-_{\om(x,\frac{1}{2}R)}|G_{\va,\lambda}(\cdot,y)|^{2}\right)^{\frac{1}{2}}\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}R^{d-1}}\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d-1}}. \end{aligned}\label{weds} \ee for any $ k\in\mathbb{N}_+ $. Again, in view of $ \eqref{duality} $, we can see that $ \eqref{Green Lipschitz 11} $ is true. Similarly, if we use the estimate \begin{align}
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}\delta(y)}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d-1}}\nonumber \end{align} for the second inequality of $ \eqref{weds} $, it is easy to show that \begin{align}
|\nabla_1G_{\va,\lambda}(x,y)|\leq \frac{C_{k,\theta_0}\delta(y)}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d}},\label{ww70} \end{align} which gives the proof of $ \eqref{Green Lipschitz 12} $. Similarly, $ \eqref{Green Lipschitz 13} $ can be derived by using $ \eqref{duality} $ and $ \eqref{ww70} $. Now, we only need to show $ \eqref{Green Lipschitz 14} $. Obviously, for any $ 1\leq\gamma\leq m $, \begin{align} (\mathcal{L}_{\va}-\lambda I)(\nabla_2G_{\va,\lambda}^{\gamma}(\cdot,y))=0\quad\text{in}\quad\om\backslash\{y\}. \nonumber \end{align} For any $ k_1,k_2\in\mathbb{N}_+ $, $ k=k_1+k_2 $, using $ \eqref{Green Lipschitz 11} $, we have \begin{align}
|\nabla_1\nabla_2G_{\va,\lambda}(x,y)|&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k_2}R}\left(\Xint-_{\om(x,\frac{1}{2}R)}|\nabla_2G_{\va,\lambda}(\cdot,y)|^{2}\right)^{\frac{1}{2}}\nonumber\\
&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k_2}R}\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k_1}R^{d-1}}\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d}},\nonumber \end{align}
which completes the proof for the case $ d\geq 3 $. Next, we consider the case $ d=2 $. Set $ r=|x_0-y_0| $ with $ x_0\neq y_0 $ and $ x_0,y_0\in\om $. For $ F\in C_0^{\infty}(\om(x_0,\frac{r}{4});\mathbb{C}^m) $, let $ v_{\va,\lambda} $ satisfy $ (\mathcal{L}_{\va}-\overline{\lambda} I)(v_{\va,\lambda})=F $ in $ \om $ and $ v_{\va,\lambda}=0 $ on $ \pa\om $. Then $ v_{\va,\lambda}(y)=\int_{\om}\overline{G_{\va,\lambda}(\cdot,y)}F(\cdot) $ due to representation formula $ \eqref{Representation formula} $. Since $ F\equiv 0 $ in $ \om\backslash\om(x_0,\frac{r}{4}) $, we have, $ (\mathcal{L}_{\va}-\overline{\lambda} I)(v_{\va,\lambda})=0 $ in $ \om\backslash\om(x_0,\frac{r}{4}) $. Owing to $ \eqref{Linfty} $, $ \eqref{impoestim} $, Poincaré's inequality and Hölder's inequality, we can get, for any $ p>2 $, \begin{align}
|v_{\va,\lambda}(y_0)|&\leq C_{\theta_0}\left(\Xint-_{\om(y_0,\frac{r}{4})}|v_{\va,\lambda}|^2\right)^{\frac{1}{2}}\leq C_{\theta_0}\left(\Xint-_{\om(y_0,\frac{r}{4})}|v_{\va,\lambda}|^p\right)^{\frac{1}{p}}=C_{\theta_0}\delta(y_0)r^{-\frac{2}{p}}\left(\int_{\om}|\nabla v_{\va,\lambda}|^p\right)^{\frac{1}{p}}\nonumber\\
&\leq C_{\theta_0}\delta(y_0)r^{-\frac{2}{p}}\|F\|_{L^{\frac{2p}{p+2}}(\om)}\leq C_{\theta_0}\delta(y_0)r^{-\frac{2}{p}}\|F\|_{L^{\frac{2p}{p+2}}(\om(x_0,\frac{r}{4}))} \leq C_{\theta_0}\delta(y_0)\|F\|_{L^{2}(\om(x_0,\frac{r}{4}))}.\nonumber \end{align} By duality arguments, this implies that \begin{align}
\left(\Xint-_{\om(x_0,\frac{r}{4})}|G_{\va,\lambda}(\cdot,y_0)|^2\right)^{\frac{1}{2}}\leq \frac{C_{\theta_0}\delta(y_0)}{r}\leq \frac{C_{\theta_0}\delta(y_0)}{|x_0-y_0|}.\label{ww22122} \end{align} According to the Lipschitz estimates $ \eqref{Lipesu} $ and $ \eqref{ww22122} $, then for any $ k\in\mathbb{N}_+ $, \begin{align}
|\nabla_1G_{\va,\lambda}(x_0,y_0)|&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^kr}\left(\Xint-_{\om(x_0,\frac{r}{4})}|G_{\va,\lambda}(\cdot,y_0)|^2\right)^{\frac{1}{2}}\leq \frac{C_{k,\theta_0}\delta(y_0)}{(1+|\lambda||x_0-y_0|^2)^k|x_0-y_0|^2}.\nonumber \end{align} Then $ \eqref{Green Lipschitz 12} $ is proved and $ \eqref{Green Lipschitz 13} $ follows directly by considering same estimates of $ G_{\va,\overline{\lambda}}(x,y) $. Moreover, for $ v_{\va,\lambda} $, we have $ (\mathcal{L}_{\va}-\overline{\lambda} I)(v_{\va,\lambda}-(v_{\va,\lambda})_{y_0,\frac{r}{4}})=\overline{\lambda} (v_{\va,\lambda})_{y_0,\frac{r}{4}} $ in $ \om $. By using $ \eqref{L2uva} $, $ \eqref{Lipesu} $, $ \eqref{impoestim} $, Hölder's inequality and Poincaré's inequality, it can be obtained that \begin{align}
|\nabla v_{\va,\lambda}(y_0)|&\leq \frac{C_{\theta_0}}{r}\left(\Xint-_{\om(y_0,\frac{r}{4})}|v_{\va,\lambda}-(v_{\va,\lambda})_{y_0,\frac{r}{4}}|^2\right)^{\frac{1}{2}}+C_{\theta_0}(1+|\lambda|r^2)^n|\lambda|r|(v_{\va,\lambda})_{y_0,\frac{r}{4}}|\nonumber\\
&\leq C_{\theta_0}\left(\Xint-_{\om(y_0,\frac{r}{4})}|\nabla v_{\va,\lambda}|^2\right)^{\frac{1}{2}}+C_{\theta_0}(1+|\lambda|r^2)^n|\lambda|r\left(\Xint-_{\om(y_0,\frac{r}{4})}|v_{\va,\lambda}|^2\right)^{\frac{1}{2}}\nonumber\\
&=C_{\theta_0}r^{-\frac{2}{p}}\left(\int_{\om}|\nabla v_{\va,\lambda}|^p\right)^{\frac{1}{p}}+C_{\theta_0}(1+|\lambda|r^2)^n\|F\|_{L^{2}(\om(x_0,\frac{r}{4}))}\nonumber\\
&\leq C_{\theta_0}r^{-\frac{2}{p}}\|F\|_{L^{\frac{2p}{p+2}}(\om(x_0,\frac{r}{4}))}+C_{\theta_0}(1+|\lambda|r^2)^n\|F\|_{L^{2}(\om(x_0,\frac{r}{4}))}\nonumber\\
&\leq C_{\theta_0}(1+|\lambda|r^2)^n\|F\|_{L^{2}(\om(x_0,\frac{r}{4}))},\nonumber \end{align} when $ 2<p<\infty $. In view of the definition of $ v_{\va,\lambda} $, we can get that \begin{align}
\left|\int_{\om(x_0,\frac{r}{4})}\nabla_2G_{\va,\lambda}(\cdot,y_0)\overline{F(\cdot)}\right|\leq C_{\theta_0}(1+|\lambda|r^2)^n\|F\|_{L^{2}(\om(x_0,\frac{r}{4}))}.\label{ww2112222} \end{align} Again, by duality arguments, it can be inferred that \begin{align}
\left(\Xint-_{\om(x_0,\frac{r}{4})}|\nabla_2G_{\va,\lambda}(\cdot,y_0)|^2\right)^{\frac{1}{2}}\leq \frac{C_{\theta_0}(1+|\lambda|r^2)^n}{r}.\label{ww22122667} \end{align} Similarly, with the help of Lipschitz estimates $ \eqref{Lipesu} $ and $ \eqref{ww22122667} $, for any $ k\in\mathbb{N}_+ $, \begin{align}
|\nabla_1\nabla_2G_{\va,\lambda}(x_0,y_0)|&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|r^2)^{k+n}r}\left(\Xint-_{\om(x_0,\frac{r}{4})}|\nabla_2G_{\va,\lambda}(z,y_0)|^2dz\right)^{\frac{1}{2}}\nonumber\\
&\leq \frac{C_{k,\theta_0}(1+|\lambda|r^2)^n}{(1+|\lambda|r^2)^{k+n}r^2}\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_0|^2)^k|x_0-y_0|^2}.\nonumber \end{align}
This completes the proof of $ \eqref{Green Lipschitz 14} $. Then we only need to show $ \eqref{Green Lipschitz 11} $. To begin with, we note that if $ |y-z|<\frac{1}{2}|x-y| $, then \be \begin{aligned}
|\nabla_1(G_{\va,\lambda}(x,y)-G_{\va,\lambda}(x,z))|&\leq\|\nabla_1\nabla_2G_{\va,\lambda}(x,\cdot)\|_{L^{\infty}(\om(y,\frac{|x-y|}{2}))}|y-z|\\
&\leq \frac{C_{k,\theta_0}|y-z|}{(1+|\lambda||x-y|^2)^k|x-y|^2}\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^k|x-y|}, \end{aligned}\label{lianggexiangjian}
\ee for any $ k\in\mathbb{N}_+ $. If $ \frac{1}{4}|x_0-y_0|\leq\delta(y_0) $, we get that \begin{align}
|x-x_0|\leq \frac{1}{2}|x-y_0|\text{ for any }x\in B\left(x_0,\frac{1}{4}|x_0-y_0|\right).\label{ww39} \end{align} At first, there exists a point $ \overline{y}\in\partial\Omega $ (see Figure 1), such that \begin{align}
(x_0-\overline{y})//(x_0-y_0)\quad\text{and}\quad (x_0-\overline{y})\cdot(x_0-y_0)>0.\nonumber \end{align}
Then according to the fact that $ \frac{1}{4}|x_0-y_0|\leq\delta(y_0)\leq |y_0-\overline{y}| $, there always exists a positive integer $ N\in\mathbb{N}_+ $ and a sequence of points $ \left\{y_j\right\}_{j=1}^{N}\subset\om $ such that \begin{align} &y_{j}=x_0+\frac{5}{4}(y_{j-1}-x_0),\ j=1,\ldots, N,\nonumber\\
&\frac{1}{4}|x_0-y_{N-1}|\leq |y_{N-1}-\overline{y}|\quad\text{and}\quad \frac{1}{4}|x_0-y_{N}|>|y_{N}-\overline{y}|.\nonumber \end{align} Moreover, in view of the definition of $ \left\{y_j\right\}_{j=1}^{N} $, one can obtain that \begin{equation}
|x_0-y_{j}|=\frac{5}{4}|x_0-y_{j-1}|=\ldots=\left(\frac{5}{4}\right)^j|x_0-y_0|,\ \ j=1,\ldots, N. \end{equation}
\centerline{ \begin{tikzpicture} \filldraw[color=black, fill=gray!50] (-5,0) circle (0.5); \filldraw[color=black, fill=black](-5,0) circle (0.05); \filldraw[color=black, fill=black](-3,0) circle (0.05); \filldraw[color=black, fill=black](-2.5,0) circle (0.05); \filldraw[color=black, fill=black](-1.875,0) circle (0.05); \filldraw[color=black, fill=black](-1.093,0) circle (0.05); \filldraw[color=black, fill=black](5.125,0) circle (0.05); \filldraw[color=black, fill=black](8,0) circle (0.05); \node[color=black] at (-5,0.2) {$ x_0 $}; \node[color=black] at (-3,0.2) {$ y_0 $}; \node[color=black] at (-2.5,0.2) {$ y_1 $}; \node[color=black] at (-1.875,0.2) {$ y_2 $}; \node[color=black] at (-1.093,0.2) {$ y_3 $}; \node[color=black] at (5.125,0.2) {$ y_N $}; \node[color=black] at (8.2,0.2) {$ \overline{y} $}; \draw [color=black, thick][-](-5,0) -- (8,0); \draw [color=black, thick][dotted](3,0.2) -- (4.06,0.2); \draw [color=black, thick][dashed](8,1.5) -- (8,-0.5); \node[color=black] at (8.4,0.9) {$ \partial\Omega $}; \end{tikzpicture} } \centerline{Figure 1.} \noindent By applying $ \eqref{lianggexiangjian} $, we can obtain that for any $ k\in\mathbb{N}_+ $, \begin{align}
|\nabla_1 (G_{\va,\lambda}(x_0,y_0)-G_{\va,\lambda}(x_0,y_1))|\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_0|^2)^k|x_0-y_0|}.\label{ww431} \end{align}
Similarly, since $ |y_{j+1}-y_{j}|=\frac{1}{4}|x_0-y_j|<\frac{1}{2}|x_0-y_j| $, we have \begin{equation}
|\nabla_1 (G_{\va,\lambda}(x_0,y_j)-G_{\va,\lambda}(x_0,y_{j+1}))|\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_j|^2)^k|x_0-y_j|},\label{ww43-i} \end{equation} for any $ j=1,\ldots,N-1 $. Finally, owing to the simple observation that \be
\frac{1}{4}|x_0-y_N|>|y_N-\overline{y}|\geq\delta(y_N), \nonumber \ee it can be inferred that \begin{equation} \begin{aligned}
|\nabla_1 G_{\va,\lambda}(x_0,y_N)|&\leq\frac{C_{k,\theta_0}\delta(y_N)}{(1+|\lambda||x_0-y_N|^2)^k|x_0-y_N|^2}\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_N|^2)^k|x_0-y_N|},\\ \end{aligned}\label{ww43-N} \end{equation} where we have used $ \eqref{Green Lipschitz 12} $. From \eqref{ww431}-\eqref{ww43-N}, we can obtain that \begin{align}
|\nabla_1 G_{\va,\lambda}(x_0,y_0)|&\leq \sum_{j=0}^N\frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_j|^2)^k|x_0-y_j|}\nonumber\\
&\leq \sum_{j=0}^N\left(\frac{4}{5}\right)^j\frac{C_{k,\theta_0}}{(1+|\lambda|\left(\frac{5}{4}\right)^{2j}|x_0-y_0|^2)^k|x_0-y_0|}\nonumber\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_0|^2)^k|x_0-y_0|},\nonumber \end{align}
for any $ k\in\mathbb{N}_+ $ and $ \frac{1}{4}|x_0-y_0|\leq\delta(y_0) $. This, together with $ \eqref{Green Lipschitz 12} $ and $ \eqref{duality} $, gives $ \eqref{Green Lipschitz 11} $. \end{proof} \begin{rem} Under the same assumptions of Theorem \ref{Green2}, if we further assume that $ A $ satisfies $ \eqref{Hol} $, we can improve $ \eqref{Pointwise estimates for Green functions 4} $ to the sharp estimate. That is, for any $ k\in\mathbb{N}_+ $, \begin{align}
|G_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^k}\left\{1+\ln\left(\frac{R_0}{|x-y|}\right)\right\},\label{sharppointwise} \end{align}
where $ C_{k,\theta_0} $ depends only on $ \mu,m,\tau,\nu,k,\theta_0,\eta $ and $ \om $. The proof is similar to Theorem \ref{lipgg}. Like what we have done in $ \eqref{lianggexiangjian} $, one can find that if $ |y-z|<\frac{1}{2}|x-y| $, then \be \begin{aligned}
|G_{\va,\lambda}(x,y)-G_{\va,\lambda}(x,z)|&\leq\|\nabla_2G_{\va,\lambda}(x,\cdot)\|_{L^{\infty}(\om(y,\frac{|x-y|}{2}))}|y-z|\\
&\leq \frac{C_{k,\theta_0}|y-z|}{(1+|\lambda||x-y|^2)^k|x-y|}\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^k}, \end{aligned}\label{lianggexiangjian2} \ee where we have used $ \eqref{Green Lipschitz 11} $. Then using almost the same arguments, we can obtain that \begin{align}
|G_{\va,\lambda}(x_0,y_j)-G_{\va,\lambda}(x_0,y_{j+1})|\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_0|^2)^k},\label{ww431111} \end{align} for any $ j=0,1,2,\ldots,N-1 $. Also, it can be obtained that \begin{equation} \begin{aligned}
|G_{\va,\lambda}(x_0,y_N)|&\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_N|^2)^k},\\ \end{aligned}\label{ww43-N111} \end{equation} for any $ k\in\mathbb{N}_+ $, where we have used $ \eqref{Boundedness estimates} $. Noticing that the boundedness of $ N $, $ \eqref{lnN} $, is also true in this case, it follows from direct computations that \begin{align}
|G_{\va,\lambda}(x_0,y_0)|&\leq \sum_{j=0}^N\frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_j|^2)^k}\leq \frac{C_{k,\theta_0}}{(1+|\lambda||x_0-y_0|^2)^k}\left\{1+\ln\left(\frac{R_0}{|x_0-y_0|}\right)\right\},\nonumber \end{align} which completes the proof. \end{rem} \begin{lem} For $ \va\geq 0 $ and $ d\geq 2 $, let $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $ and $ \om $ be a bounded $ C^1 $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{VMO} $. Then for any $ x\in\om $, $ \sigma\in(0,1) $ and $ k\in\mathbb{N}_+ $, \begin{align}
\int_{\om}|G_{\va,\lambda}(x,y)|dy&\leq
C_{\theta_0}(R_0^{-2}+|\lambda|)^{-1}\cdot\left\{\begin{matrix} 1&\text{if}&d\geq 3,\\
(1+|\lambda|R_0^{2})^{\sigma}&\text{if}&d=2, \end{matrix}\right.\label{*}\\
\int_{\om}|\nabla_2G_{\va,\lambda}(x,y)|dy&\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}},\label{**} \end{align} where $ C_{\theta_0} $ depends only on $ \mu,d,m,\sigma,\omega(t),\theta_0 $ and $ \om $. If we further assume that $ A $ satisfies $ \eqref{Hol} $ and $ \om $ is a $ C^{1,\eta} $ domain with $ 0<\eta<1 $, then \begin{align}
\int_{\om}|\nabla_1G_{\va,\lambda}(x,y)|dy&\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}},\label{***} \end{align} where $ C $ depends only on $ \mu,d,m,\tau,\nu,\theta_0,\eta $ and $ \om $. \end{lem} \begin{proof} By using $ \eqref{Greene} $ with $ k=2 $, it is easy to infer that \be \begin{aligned}
\int_{\om}|G_{\va,\lambda}(x,y)|dy&\leq \int_{B(x,R_0)}\frac{C_{\theta_0}}{(1+|\lambda||x-y|^2)^{2}}\frac{1}{|x-y|^{d-2}}dy=\int_{0}^{R_0}\frac{C_{\theta_0}\rho}{(1+|\lambda|\rho^2)^{2}}d\rho\\
&=-\left.\frac{C_{\theta_0}}{|\lambda|(1+|\lambda|\rho^2)}\right|_0^{R_0}=C_{\theta_0}\left\{\frac{1}{|\lambda|}-\frac{1}{|\lambda|(1+|\lambda|R_0^2)}\right\}\leq \frac{C_{\theta_0}}{|\lambda|}. \end{aligned}\label{ww25} \ee On the other hand, it can be easily seen that, \begin{align}
\int_{\om}|G_{\va,\lambda}(x,y)|dy&\leq \int_{B(x,R_0)}\frac{C_{\theta_0}}{(1+|\lambda||x-y|^2)^{2}}\frac{1}{|x-y|^{d-2}}dy\leq\int_{0}^{R_0}C_{\theta_0}\rho d\rho\leq C_{\theta_0}R_0^2.\nonumber \end{align} This, together with $ \eqref{ww25} $, implies $ \eqref{*} $ with $ d\geq 3 $. For $ d=2 $, owing to $ \eqref{preliminary} $, we can get that, for any $ \sigma\in(0,1) $, \begin{align}
\int_{\om}|G_{\va,\lambda}(x,y)|dy&\leq\sum_{j=0}^{\infty}\int_{\om(x,2^{-j}R_0)\backslash\om(x,2^{-j-1}R_0)}\frac{C_{\theta_0}R_0^{\sigma}}{(1+|\lambda||x-y|^2)^2|x-y|^{\sigma}}dy\nonumber\\
&\leq\sum_{j=0}^{\infty}\frac{C_{\theta_0}(2^{-j}R_0)^2R_0^{\sigma}}{(1+|\lambda|(2^{-j}R_0)^2)^{1-\sigma}(2^{-j}R_0)^{\sigma}}\leq C_{\theta_0}(1+|\lambda|R_0^{2})^{\sigma}(R_0^{-2}+|\lambda|)^{-1}.\nonumber \end{align} For $ R>0 $, consider the annulus $ \om(x,2R)\backslash \om(x,R) $. One can use small $ d $ dimensional balls, whose radius are $ \frac{5}{8}R $ and centers are on $ \pa B(x,\frac{3}{2}R)\cap\om $, to cover the annulus $ \om(x,2R)\backslash \om(x,R) $. We denote these small balls as $ \left\{B(x_i,\frac{5}{8}R)\right\}_{i=1}^{N} $. Obviously, $ N\leq C\frac{\pi((2R)^d-R^d)}{\pi(\frac{5}{8} R)^d}\leq C $, where $ C $ is a constant, independent of $ R $. Then by using Hölder's inequality, we have \begin{align}
\int_{\om(x,2R)\backslash \om(x,R)}|\nabla_2 G_{\va,\lambda}(x,y)|dy&\leq C\left(\int_{\om(x,2R)\backslash \om(x,R)}|\nabla_2 G_{\va,\lambda}(x,y)|^2dy\right)^{\frac{1}{2}}R^{\frac{d}{2}}\nonumber\\
&\leq C\sum_{i=1}^{N}\left(\int_{\om(x_i,\frac{5}{8} R)}|\nabla_2 G_{\va,\lambda}(x,y)|^2dy\right)^{\frac{1}{2}}R^{\frac{d}{2}}.\nonumber \end{align} In view of Caccioppoli's inequality $ \eqref{Cau2} $ and $ \eqref{Greene} $, if $ d\geq 3 $, then for any $ k\in\mathbb{N}_+ $, \begin{align}
\left(\int_{\om(x_i,\frac{5}{8} R)}|\nabla_2 G_{\va,\lambda}(x,y)|^2dy\right)^{\frac{1}{2}}&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR}\left(\int_{\om(x_i,\frac{2}{3}R)}|G_{\va,\lambda}(x,y)|^2dy\right)^{\frac{1}{2}}\nonumber\\
&\leq\frac{C_{k,\theta_0}R^{\frac{d}{2}}}{(1+|\lambda|R^2)^kR^{d-1}}\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR^{\frac{d}{2}-1}}.\nonumber \end{align} This implies that \begin{align}
\int_{\om(x,2R)\backslash \om(x,R)}|\nabla_2 G_{\va,\lambda}(x,y)|dy&\leq \frac{C_{k,\theta_0}R}{(1+|\lambda|R^2)^k}.\nonumber \end{align} Thus \begin{align}
\int_{\om}|\nabla_2 G_{\va,\lambda}(x,y)|dy&\leq\sum_{i=0}^{\infty}\int_{\om(x,2^{-j}R_0)\backslash \om(x,2^{-j-1}R_0)}|\nabla_2 G_{\va,\lambda}(x,y)|dy\leq\sum_{i=0}^{\infty}\frac{C_{\theta_0}(2^{-j}R_0)}{(1+|\lambda|(2^{-j}R_0)^2)^{\frac{3}{2}}}\nonumber\\
&\approx\int_{B(0,2R_0)}\frac{1}{(1+|\lambda||x|^2)^{\frac{3}{2}}}\frac{1}{|x|^{d-1}}dy\leq C(R_0^{-2}+|\lambda|)^{-\frac{1}{2}},\nonumber \end{align} which gives the proof of $ \eqref{**} $ with $ d\geq 3 $. If $ d=2 $, to show $ \eqref{**} $, we have \be \begin{aligned}
\int_{\om(x,2R)\backslash \om(x,R)}|\nabla_2 G_{\va,\lambda}(x,y)|dy&\leq C\left(\int_{\om(x,2R)\backslash \om(x,R)}|\nabla_2 G_{\va,\lambda}(x,y)|^2dy\right)^{\frac{1}{2}}R\\
&\leq C\sum_{i=1}^{N}\left(\int_{\om(x_i,\frac{5}{8} R)}|\nabla_2 G_{\va,\lambda}(x,y)|^2dy\right)^{\frac{1}{2}}R. \end{aligned}\label{suibian} \ee Choose $ f\in C_{0}^{\infty}(\om(x_i,\frac{5}{8}R);\mathbb{C}^{m\times d}) $, $ w_{\va,\lambda} $ such that $ (\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda})=\operatorname{div}(f) $ in $ \om $ and $ w_{\va,\lambda}=0 $ in $ \pa\om $. Then $ w_{\va,\lambda}(z)=\int_{\om}\nabla_2G_{\va,\lambda}(z,\cdot)\overline{f(\cdot)} $ due to $ \eqref{Representation formula} $. In view of $ \eqref{Linfty} $, $ \eqref{impoestim2} $, Hölder's inequality, the fact that $ (\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda})=0 $ in $ \om\backslash\om(x_i,\frac{5}{8}R) $ and $ \om(x,\frac{1}{20}R)\subset\om\backslash\om(x_i,\frac{5}{8}R) $, it is not hard to deduce that for any $ 2<p<\infty $ and $ k\in\mathbb{N}_+ $, \begin{align}
|w_{\va,\lambda}(x)|&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\left(\Xint-_{\om(x,\frac{1}{20}R)}|w_{\va,\lambda}|^2\right)^{\frac{1}{2}}\leq \frac{C_{k,\theta_0}R^{-1+\frac{2}{p}}}{(1+|\lambda|R^2)^k}\|w_{\va,\lambda}\|_{L^{\frac{2p}{p-2}}(\om)}\nonumber\\
&\leq \frac{C_{k,\theta_0}R^{-1+\frac{2}{p}}}{(1+|\lambda|R^2)^k}\|f\|_{L^{\frac{p}{p-1}}(\om)}\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\|f\|_{L^{2}(\om(x_i,\frac{5}{8}R))}.\nonumber \end{align} Then by using the representation of $ w_{\va,\lambda} $ given above, we have \begin{align}
\left|\int_{\om(x_i,\frac{5}{8}R)}\nabla_2G_{\va,\lambda}(x,y)\overline{f(y)}dy\right|\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\|f\|_{L^{2}(\om(x_i,\frac{5}{8}R))}.\nonumber \end{align} By duality arguments, we can infer that \begin{align}
\left(\int_{\om(x_i,\frac{5}{8}R)}|\nabla_2G_{\va,\lambda}(x,y)|^2dy\right)^{\frac{1}{2}}\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}.\nonumber \end{align} With the help of $ \eqref{suibian} $, it can be seen that for any $ k\in\mathbb{N}_+ $, \begin{align}
\int_{\om(x,2R)\backslash \om(x,R)}|\nabla_2 G_{\va,\lambda}(x,y)|dy&\leq \frac{C_{k,\theta_0} R}{(1+|\lambda|R^2)^k}.\label{prepa} \end{align} Then according to the calculations with $ d\geq 3 $, \begin{align}
\int_{\om}|\nabla_2 G_{\va,\lambda}(x,y)|dy&\leq\sum_{i=0}^{\infty}\int_{\om(x,2^{-j}R_0)\backslash \om(x,2^{-j-1}R_0)}|\nabla_2 G_{\va,\lambda}(x,y)|dy\nonumber\\
&\leq\sum_{i=0}^{\infty}\frac{C_{\theta_0} 2^{-j}R_0}{(1+|\lambda|(2^{-j}R_0)^2)^{\frac{3}{2}}}\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}.\nonumber \end{align} For the proof of $ \eqref{***} $, we simply note that by $ \eqref{Green Lipschitz 11} $ with $ k=2 $, \begin{align}
\int_{\om}|\nabla_1G_{\va,\lambda}(x,y)|dy&\leq\int_{\om}\frac{C_{\theta_0}}{(1+|\lambda||x-y|^2)^2|x-y|^{d-1}}dy\leq C_{\theta_0}(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}.\nonumber \end{align} Then we can complete the proof. \end{proof}
\section{Convergence estimates of resolvents}\label{Estimates of convergence of resolvents}
\subsection{Convergence of Green functions}
Now we turn to estimate $ |G_{\va,\lambda}(x,y)-G_{0,\lambda}(x,y)| $, where $ G_{\va,\lambda} $ and $ G_{0,\lambda} $ are Green functions for operators $ \mathcal{L}_{\va}-\lambda I $ and $ \mathcal{L}_0-\lambda I $ respectively. Such estimates are essential in the proof of Theorem \ref{Lpconres} and \ref{LpW1pconreso}.
\begin{thm}[Convergence of Green functions I]\label{Convergence of Green's functions} For $ \va\geq 0 $, $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, let $ \om $ be a bounded $ C^{1,1} $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. Then for any $ k\in\mathbb{N}_+ $ and $ x,y\in\om $ with $ x\neq y $, \begin{align}
|G_{\va,\lambda}(x,y)-G_{0,\lambda}(x,y)|\leq \frac{C_{k,\theta_0}\va }{(1+|\lambda||x-y|^2)^{k}|x-y|^{d-1}},\label{Convergence of Green's functions formula} \end{align} where $ C_{k,\theta_0} $ depends only on $ \mu,d,m,\nu,\tau,k,\theta_0 $ and $ \om $. \end{thm}
For continuous function $ u $ in $ \om $, the nontangential maximal function is defined by \begin{align}
(u)^*(y)=\sup\left\{|u(x)|:x\in\om\text{ and }|x-y|<C_0\delta(x)\right\}\label{nontangential maximal function} \end{align} for $ y\in\pa\om $, where $ C_0=C_0(\om)>1 $ is sufficiently large depending on $ \om $.
\begin{thm}[Nontangential-maximal-function estimates] For $ \va\geq 0 $ and $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, let $ \om $ be a bounded $ C^{1,\eta} $ domain in $ \mathbb{R}^d $ with $ 0<\eta<1 $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $, $ \eqref{Hol} $ and $ 1<p\leq\infty $. For $ g\in L^p(\pa\om;\mathbb{C}^m) $, let $ u_{\va,\lambda} $ be the unique solution to the Dirichlet problem $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=0 $ in $ \om $ and $ u_{\va,\lambda}=g $ on $ \pa\om $ with the property $ (u_{\va,\lambda})^*\in L^p(\pa\om) $. Then \begin{align}
\|(u_{\va,\lambda})^*\|_{L^p(\pa\om)}\leq C_{\theta_0}\|g\|_{L^p(\pa\om)}.\label{Lp for nontan} \end{align} where $ C_{\theta_0} $ depends only on $ \mu,d,m,\nu,\tau,\theta_0,\eta $ and $ \om $. In the case that $ p=\infty $, $ \eqref{Lp for nontan} $ implies, \begin{align}
\|u_{\va,\lambda}\|_{L^{\infty}(\om)}\leq C_{\theta_0}\|g\|_{L^{\infty}(\pa\om)}.\label{maximal principle} \end{align} \end{thm} \begin{proof} By using the definition of Green functions, we can represent $ u_{\va,\lambda} $ by the formula \begin{align} u_{\va,\lambda}(x)=\int_{\pa\om}P_{\va,\lambda}(x,y)\overline{g(y)}dS(y),\nonumber \end{align} where the Poisson kernel $ P_{\va,\lambda}(x,y)=(P_{\va,\lambda}^{\al\beta}(x,y)) $ for $ \mathcal{L}_{\va}-\lambda I $ in $ \om $ is given by \begin{align} P_{\va,\lambda}^{\al\beta}(x,y)=-n_i(y)a_{ji}^{\gamma\beta}(y/\va)\frac{\pa }{\pa y_j} \{G_{\va,\lambda}^{\al\gamma}(x,y)\}\nonumber \end{align} for $ x\in\om $, $ y\in\pa\om $ and $ n(x)=(n_1(x),n_2(x),...,n_d(x)) $ denotes the outward unit normal to $ \pa\om $. In view of the estimate $ \eqref{Green Lipschitz 13} $, we have, for any $ k\in\mathbb{N}_+ $, \begin{align}
|P_{\va,\lambda}(x,y)|\leq \frac{C_{k,\theta_0}\delta(x)}{(1+|\lambda||x-y|^2)^{k}|x-y|^d}\leq \frac{C_{\theta_0}\delta(x)}{|x-y|^d}.\nonumber \end{align} Hence, we can obtain that \begin{align}
|u_{\va,\lambda}(x)|\leq \int_{\pa\om}\frac{C_{\theta_0}\delta(x)}{|x-y|^d}|g(y)|dS(y)\text{ for any }x\in\om.\nonumber \end{align} The rest of the proof is standard, which can be obtained in Theorem 4.6.5 of \cite{Shen2}. \end{proof}
\begin{lem} For $ \va\geq 0 $, $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, let $ \om $ be a bounded $ C^{1,1} $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. If $ \Delta(x_0,3R)\neq\emptyset $, assume that $ u_{\va,\lambda}\in H^1(\om(x_0,3R);\mathbb{C}^m) $ is a solution of the boundary problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=0\text{ in }\om(x_0,3R)\quad\text{and}\quad u_{\va,\lambda}=f\text{ on } \Delta(x_0,3R),\nonumber \end{align}
with $ \|f\|_{L^{\infty}(\om)}<\infty $. If $ \Delta(x_0,3R)=\emptyset $, assume that $ u_{\va,\lambda}\in H^1(B(x_0,3R);\mathbb{C}^m) $ is the weak solution of the interior problem \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=0\text{ in }B(x_0,3R).\nonumber \end{align} Then for any $ k\in\mathbb{N}_+ $, \begin{align}
\|u_{\va,\lambda}\|_{L^{\infty}(\om(x_0,R))}\leq C_{k,\theta_0}\|f\|_{L^{\infty}(\Delta(x_0,3R))}+\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}}\Xint-_{\om(x_0,{3R})}|u_{\va,\lambda}|,\label{ww65} \end{align} where $ C_{k,\theta_0} $ depends only on $ \mu,d,m,\theta_0,k,\nu,\tau $ and $ \om $. \end{lem} \begin{proof} We only need to show the case that $ \Delta(x_0,3R)\neq\emptyset $ since the other follows directly from $ \eqref{Linfty} $. If $ f\equiv 0 $, the estimate is also a consequence of $ \eqref{Linfty} $. To treat the general case, let $ v_{\va,\lambda} $ be the solution to $ (\mathcal{L}_{\va}-\lambda I)(v_{\va,\lambda})=0 $ in $ \widetilde{\om} $ with the Dirichlet condition $ v_{\va,\lambda}=f $ on $ \pa\widetilde{\om}\cap\pa\om $ and $ v_{\va,\lambda}=0 $ on $ \pa\widetilde{\om}\backslash\pa\om $, where $ \widetilde{\om} $ is a $ C^{1,1} $ domain such that $ \om(x_0,2R)\subset\widetilde{\om}\subset \om(x_0,3R) $. By the maximum principle $ \eqref{maximal principle} $, we have, \begin{align}
\|v_{\va,\lambda}\|_{L^{\infty}(\widetilde{\om})}\leq C_{\theta_0}\|v_{\va,\lambda}\|_{L^{\infty}(\pa\widetilde{\om})}\leq C_{\theta_0}\|f\|_{L^{\infty}(\Delta(x_0,3R))}.\nonumber \end{align} This, together with the fact that for any $ k\in\mathbb{N}_+ $, \begin{align}
\|u_{\va,\lambda}-v_{\va,\lambda}\|_{L^{\infty}(\om_R)}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}}\Xint-_{\om_{2R}}|u_{\va,\lambda}-v_{\va,\lambda}|\nonumber\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^{k}}\Xint-_{\om_{3R}}|u_{\va,\lambda}|+C_{k,\theta_0}\|f\|_{L^{\infty}(\Delta_{3R})},\nonumber \end{align} gives the results. \end{proof}
\begin{lem}\label{ww74} For $ \va\geq 0 $ and $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, let $ \om $ be a bounded $ C^{1,1} $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. Let $ u_{\va,\lambda}\in H^1(\om(x_0,4R);\mathbb{C}^m) $ and $ u_{0,\lambda}\in W^{2,p}(\om(x_0,4R);\mathbb{C}^m) $ for some $ d<p<\infty $. If $ \Delta(x_0,4R)\neq\emptyset $, assume that \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=(\mathcal{L}_{0}-\lambda I)(u_{0,\lambda})\text{ in }\om(x_0,4R)\quad\text{and}\quad u_{\va,\lambda}=u_{0,\lambda}\text{ on }\Delta(x_0,4R),\nonumber \end{align} and if $ \Delta(x_0,4R)=\emptyset $, assume that \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=(\mathcal{L}_{0}-\lambda I)(u_{0,\lambda})\text{ in }B(x_0,4R).\nonumber \end{align} Then for any $ k\in\mathbb{N}_+ $, \be \begin{aligned}
\|u_{\va,\lambda}-u_{0,\lambda}\|_{L^{\infty}(\om_R)}&\leq
\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\Xint-_{\om_{4R}}|u_{\va,\lambda}-u_{0,\lambda}|\\
&\quad+C_{k,\theta_0}\va\left\{ R^{1-\frac{d}{p}}\|\nabla^2u_{0,\lambda}\|_{L^p(\om_{4R})}+(1+|\lambda|R^2)\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\right\}, \end{aligned}\label{67} \ee where $ C_{k,\theta_0} $ depends on $ \mu,d,m,\nu,\tau,p,k,\theta_0 $ and $ \om $. \end{lem} \begin{proof} Firstly, we consider the case that $ \Delta(x_0,3R)\neq\emptyset $. Choose a domain $ \widetilde{\om} $, which is $ C^{1,1} $ such that $ \om(x_0,3R)\subset\widetilde{\om}\subset \om(x_0,4R) $. Consider \begin{align} w_{\va,\lambda}(x)=u_{\va,\lambda}(x)-u_{0,\lambda}(x)-\va\chi_{j}^{\beta}(x/\va)\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}=w_{\va,\lambda}^{(1)}(x)+w_{\va,\lambda}^{(2)}(x)\quad\text{in }\widetilde{\om},\nonumber \end{align} where $ w_{\va,\lambda}^{(1)} $ and $ w_{\va,\lambda}^{(2)} $ are weak solutions for the Drichlet problems \begin{align} (\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda}^{(1)})&=(\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda})\text{ in }\widetilde{\om}\quad\text{and}\quad w_{\va,\lambda}^{(1)}\in H_0^1(\widetilde{\om};\mathbb{C}^m),\label{ww63}\\ (\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda}^{(2)})&=0\text{ in }\widetilde{\om}\quad\text{and}\quad w_{\va,\lambda}^{(2)}=w_{\va,\lambda}\quad\text{on }\pa\widetilde{\om}.\label{ww64} \end{align}
Since $ w_{\va,\lambda}^{(2)}=w_{\va,\lambda}=-\va\chi(x/\va)\nabla u_{0,\lambda} $ on $ \Delta(x_0,3R) $ and $ \|\chi\|_{L^{\infty}(\om)}\leq C $, it follows from $ \eqref{ww65} $ that for any $ k\in\mathbb{N}_+ $, \begin{align}
\|w_{\va,\lambda}^{(2)}\|_{L^{\infty}(\om_R)}&\leq C_{k,\theta_0}\va \|\nabla u_{0,\lambda}\|_{L^{\infty}(\Delta_{3R})}+\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\Xint-_{\om_{3R}}|w_{\va,\lambda}^{(2)}|\nonumber\\
&\leq C_{k,\theta_0}\va\|\nabla u_{0,\lambda}\|_{L^{\infty}(\Delta_{3R})}+\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\Xint-_{\om_{3R}}|w_{\va,\lambda}^{(1)}|+\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\Xint-_{\om_{3R}}|w_{\va,\lambda}|\nonumber\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\left\{\Xint-_{\om_{3R}}|u_{\va,\lambda}-u_{0,\lambda}|+\|w_{\va,\lambda}^{(1)}\|_{L^{\infty}(\om_{3R})}\right\}+C_{k,\theta_0}\va \|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{3R})}.\nonumber \end{align} By using definitions of $ w_{\va,\lambda} $, $ w_{\va,\lambda}^{(1)} $ and $ w_{\va,\lambda}^{(2)} $, this gives \be \begin{aligned}
\|u_{\va,\lambda}-u_{0,\lambda}\|_{L^{\infty}(\om_R)}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\Xint-_{\om_{3R}}|u_{\va,\lambda}-u_{0,\lambda}|\\
&\quad+C_{k,\theta_0}\left\{\|w_{\va,\lambda}^{(1)}\|_{L^{\infty}(\om_{3R})}+\va\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{3R})}\right\}. \end{aligned}\label{ww66} \ee To estimate $ w_{\va,\lambda}^{(1)} $ on $ \om(x_0,3R) $, we use the representation formula to obtain that \begin{align} w_{\va,\lambda}^{(1)}(x)=\int_{\widetilde{\om}}\widetilde{G}_{\va,\lambda}(x,y)\overline{(\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda})}dy,\nonumber \end{align} where $ \widetilde{G}_{\va,\lambda}(x,y) $ denotes the matrix of the Green function for $ \mathcal{L}_{\va}-\lambda I $ in $ \widetilde{\om} $. Note that \be \begin{aligned} [(\mathcal{L}_{\va}-\lambda I) w_{\va,\lambda})]^{\al}&=-\va\frac{\pa}{\pa x_i}\left\{F_{kij}^{\al\beta}(x/\va)\frac{\pa^2u_{0,\lambda}^{\beta}}{\pa x_j\pa x_k}\right\}+\va\lambda\chi_{j}^{\al\beta}(x/\va)\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}\\ &\quad+\va\frac{\pa}{\pa x_i}\left\{a_{ij}^{\al\delta}(x/\va)\chi_k^{\delta\beta}(x/\va)\frac{\pa^2u_{0,\lambda}^{\beta}}{\pa x_j\pa x_k}\right\}. \end{aligned}\label{ww69} \ee Then by using the representation formula of Green functions $ \eqref{repre} $ and $ \eqref{Representation formula} $, \begin{align} w_{\va,\lambda}^{(1)\gamma}(x)&=\va\int_{\widetilde{\om}}\frac{\pa}{\pa y_i}\{\widetilde{G}_{\va,\lambda}^{\gamma\al}(x,y)\}\overline{\left[F_{kij}^{\al\beta}(y/\va)-a_{ij}^{\al\delta}(y/\va)\chi_k^{\delta\beta}(y/\va)\right]\frac{\pa^2 u_{0,\lambda}^{\beta}}{\pa y_j\pa y_k}}dy\nonumber\\ &\quad\quad+\va\overline{\lambda}\int_{\widetilde{\om}}\widetilde{G}_{\va,\lambda}^{\gamma\al}(x,y)\overline{\chi_{j}^{\al\beta}(y/\va)\frac{\pa u_{0,\lambda}^{\beta}}{\pa y_j}}dy.\nonumber \end{align}
Since $ \|F_{kij}\|_{L^{\infty}(\om)},\|\chi_j\|_{L^{\infty}(\om)}\leq C $ and $ p>d $, we have \begin{align}
\left|\va\lambda\int_{\widetilde{\om}}\widetilde{G}_{\va,\lambda}^{\gamma\al}(x,y)\overline{\chi_{j}^{\al\beta}(y/\va)\frac{\pa u_{0,\lambda}^{\beta}}{\pa y_j}}dy\right|&\leq C_{\theta_0}\va|\lambda|\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\int_{\widetilde{\om}}|\widetilde{G}_{\va,\lambda}^{\gamma\al}(x,y)|dy\nonumber\\
&\leq C_{\theta_0}\va |\lambda|R^2\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})},\nonumber \end{align} where we have used $ \eqref{*} $ for the second inequality. Then \begin{align}
|w_{\va,\lambda}^{(1)}(x)|&\leq C\va\int_{\widetilde{\om}}|\nabla_2\widetilde{G}_{\va,\lambda}(x,y)||\nabla^2u_{0,\lambda}(y)|dy+C_{\theta_0}\va |\lambda|R^2\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\nonumber\\
&\leq C\va\|\nabla^2u_{0,\lambda}\|_{L^p(\om_{4R})}\left(\int_{\widetilde{\om}}|\nabla_2\widetilde{G}_{\va,\lambda}(x,y)|^{p'}dy\right)^{\frac{1}{p'}}+C_{\theta_0}\va |\lambda|R^2\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}.\nonumber \end{align}
In view of $ \eqref{Green Lipschitz 11} $ with $ k=0 $, we have $ \left(\int_{\widetilde{\om}}|\nabla_2\widetilde{G}_{\va,\lambda}(x,y)|^{p'}dy\right)^{\frac{1}{p'}}\leq C_{\theta_0}R^{1-\frac{d}{p}} $ and then \begin{align}
|w_{\va,\lambda}^{(1)}(x)|\leq C_{\theta_0}\va\left\{R^{1-\frac{d}{p}}\|\nabla^2u_{0,\lambda}\|_{L^p(\om_{4R})}+|\lambda|R^2\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\right\}.\nonumber \end{align} This, together with $ \eqref{ww66} $, gives the proof for the case that $ \Delta(x_0,3R)\neq\emptyset $. If $ \Delta(x_0,3R)=\emptyset $, we can choose $ w_{\va,\lambda}^{(1)}$ and $ w_{\va,\lambda}^{(2)} $ by \begin{align} (\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda}^{(1)})&=(\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda})\text{ in }B\left(x_0,3R\right)\quad\text{and}\quad w_{\va,\lambda}^{(1)}\in H_0^1\left(B\left(x_0,3R\right);\mathbb{C}^m\right),\nonumber\\ (\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda}^{(2)})&=0\text{ in }B\left(x_0,3R\right)\quad\text{and}\quad w_{\va,\lambda}^{(2)}=w_{\va,\lambda}\text{ on }\pa\left(B\left(x_0,3R\right)\right).\nonumber \end{align} By using $ \eqref{Linfty} $, it can be seen that for any $ k\in\mathbb{N}_+ $, \begin{align}
\|w_{\va,\lambda}^{(2)}\|_{L^{\infty}(B_R)}&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\Xint-_{B_{3R}}|w_{\va,\lambda}^{(2)}|\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\left\{\Xint-_{B_{3R}}|w_{\va,\lambda}^{(1)}|+\Xint-_{B_{3R}}|w_{\va,\lambda}|\right\}\nonumber\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\left\{\Xint-_{B_{3R}}|u_{\va,\lambda}-u_{0,\lambda}|+\|w_{\va,\lambda}^{(1)}\|_{L^{\infty}(B_{3R})}+\va \|\nabla u_{0,\lambda}\|_{L^{\infty}(B_{3R})}\right\}.\nonumber \end{align} Then $ \eqref{ww66} $ is still true. Choosing $ \widetilde{\om}=B(x_0,3R) $ and using almost the same arguments for the case that $ \Delta(x_0,3R)\neq\emptyset $, we can complete the proof. \end{proof}
\begin{proof}[Proof of Theorem \ref{Convergence of Green's functions}]
Fix $ x_0,y_0\in\om $ and $ R=\frac{|x_0-y_0|}{16}>0 $. For $ F\in C_0^{\infty}(\om(y_0,R);\mathbb{C}^m) $, let \begin{align} u_{\va,\lambda}(x)=\int_{\om}G_{\va,\lambda}(x,y)\overline{F(y)}dy\quad\text{and}\quad u_{0,\lambda}(x)=\int_{\om}G_{0,\lambda}(x,y)\overline{F(y)}dy.\nonumber \end{align} Then $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=(\mathcal{L}_0-\lambda I)(u_{0,\lambda})=F $ in $ \om $ and $ u_{\va,\lambda}=u_{0,\lambda}=0 $ on $ \pa\om $. For any $ x\in\om $ and $ p>d $, using $ \eqref{Green Lipschitz 11} $ with $ k=0 $, it follows by Hölder's inequality that \begin{align}
|\nabla u_{0,\lambda}(x)|&\leq\left|\int_{\om}\nabla_1 G_{0,\lambda}(x,y)\overline{F(y)}dy\right|\leq C\|F\|_{L^p(\om(y_0,R))}\left(\int_{\om(y_0,R)}|\nabla_1G_{0,\lambda}(x,y)|^{p'}dy\right)^{\frac{1}{p'}}\nonumber\\
&\leq C_{\theta_0}\left(\int_{\om(y_0,R)}\frac{1}{|x-y|^{(d-1)p'}}dy\right)^{\frac{1}{p'}}\|F\|_{L^p(\om(y_0,R))}\leq C_{\theta_0}R^{1-\frac{d}{p}}\|F\|_{L^p(\om(y_0,R))}.\nonumber \end{align} To this end, we can obtain the $ L^{\infty} $ estimates of $ \nabla u_{0,\lambda} $, that is, \begin{align}
\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om)}\leq C_{\theta_0}R^{1-\frac{d}{p}}\|F\|_{L^p(\om(y_0,R))}.\label{nabla infty} \end{align}
To estimate $ \|u_{\va,\lambda}-u_{0,\lambda}\|_{L^{\infty}(\om(y_0,R))} $, set $ w_{\va,\lambda} $ by \begin{align} w_{\va,\lambda}(x)=u_{\va,\lambda}(x)-u_{0,\lambda}(x)-\va\chi_{j}^{\beta}(x/\va)\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}=v_{\va,\lambda}(x)+z_{\va,\lambda}(x),\nonumber \end{align} where $ v_{\va,\lambda}\in H_0^1(\om;\mathbb{C}^m) $ and $ (\mathcal{L}_{\va}-\lambda I)(v_{\va,\lambda})=(\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda}) $ in $ \om $. By using $ \eqref{L2uva} $, $ \eqref{W2p} $ and $ \eqref{ww69} $ with $ p=2 $, we can obtain \begin{align}
\|\nabla v_{\va,\lambda}\|_{L^2(\om)}&\leq C_{\theta_0}\va\|\nabla^2u_{0,\lambda}\|_{L^2(\om)}+C_{\theta_0}\va|\lambda|^{\frac{1}{2}}\|\nabla u_{0,\lambda}\|_{L^2(\om)}\nonumber\\
&\leq C_{\theta_0}\va\|F\|_{L^2(\om)}\leq C_{\theta_0}\va\|F\|_{L^2(\om(y_0,R))}.\nonumber \end{align} By Hölder's and Sobolev's inequalities, this implies that if $ d\geq 3 $, we have \be \begin{aligned}
\|v_{\va,\lambda}\|_{L^2(\om(y_0,R))}&\leq CR\|v_{\va,\lambda}\|_{L^{\frac{2d}{d-2}}(\om(y_0,R))}\leq CR\|\nabla v_{\va,\lambda}\|_{L^2(\om)}\\
&\leq C_{\theta_0}\va R\|F\|_{L^2(\om(y_0,R))}\leq C_{\theta_0}\va R^{1+\frac{d}{2}-\frac{d}{p}}\|F\|_{L^p(\om(y_0,R))}, \end{aligned}\label{zhongjianchanwu} \ee where $ p>d $. If $ d=2 $, one can use the following estimate \begin{align}
\|v_{\va,\lambda}\|_{L^2(\om(x_0,R))}\leq C\va R\|F\|_{L^2(\om(y_0,R))}.\nonumber \end{align} in place of $ \eqref{zhongjianchanwu} $. Indeed, for any $ 2<q<\infty $, by using $ \eqref{ww69} $, Hölder's inequality and Sobolev embdeding theorem that $ W_0^{1,\frac{2q}{q+2}}(\om)\subset L^q(\om) $, it can be got that \be \begin{aligned}
\|v_{\va,\lambda}\|_{L^2(\om(y_0,R))}&\leq CR^{1-\frac{2}{q}}\|v_{\va,\lambda}\|_{L^q(\om(y_0,R))}\leq CR^{1-\frac{2}{q}}\|v_{\va,\lambda}\|_{L^q(\om)}\leq CR^{1-\frac{2}{q}}\|\nabla v_{\va,\lambda}\|_{L^{\frac{2q}{q+2}}(\om)}\\
&\leq C_{\theta_0}\va R^{1-\frac{2}{q}}\left\{\|\nabla^2u_{0,\lambda}\|_{L^{\frac{2q}{q+2}}(\om)}+|\lambda|^{\frac{1}{2}}\|\nabla u_{0,\lambda}\|_{L^{\frac{2q}{q+2}}(\om)}\right\}\\
&\leq C_{\theta_0}\va R^{1-\frac{2}{q}}\|F\|_{L^{\frac{2q}{q+2}}(\om(y_0,R))}\leq C_{\theta_0}R\|F\|_{L^{2}(\om(y_0,R))}, \end{aligned}\label{1111} \ee where for the forth and fifth inequalities, we have used Theorem \ref{Lp estimates of resolventsf} and $ \eqref{W2p} $. Since $ (\mathcal{L}_{\va}-\lambda I)(z_{\va,\lambda})=0 $ in $ \om $ and $ z_{\va,\lambda}=w_{\va,\lambda} $ on $ \pa\om $, by maximum principle $ \eqref{maximal principle} $, \begin{align}
\|z_{\va,\lambda}\|_{L^{\infty}(\om)}\leq C_{\theta_0} \|z_{\va,\lambda}\|_{L^{\infty}(\pa\om)}\leq C_{\theta_0}\va \|\nabla u_{0,\lambda}\|_{L^{\infty}(\pa\om)}.\nonumber \end{align} In view of $ \eqref{nabla infty} $, we obtain \begin{align}
\|u_{\va,\lambda}-u_{0,\lambda}\|_{L^2(\om(x_0,R))}&\leq \|w_{\va,\lambda}\|_{L^2(\om(x_0,R))}+C\va R^{\frac{d}{2}}\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om)}\nonumber\\
&\leq\|v_{\va,\lambda}\|_{L^2(\om(x_0,R))}+\|z_{\va,\lambda}\|_{L^2(\om(x_0,R))}+C\va R^{\frac{d}{2}}\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om)}\nonumber\\
&\leq\|v_{\va,\lambda}\|_{L^2(\om(x_0,R))}+C_{\theta_0}\va R^{\frac{d}{2}}\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om)}\nonumber\\
&\leq C_{\theta_0}\va R^{1+\frac{d}{2}-\frac{d}{p}}\|F\|_{L^p(\om(y_0,R))},\nonumber \end{align} where $ p>d $ and $ d\geq 3 $. Similarly, if $ d=2 $, we have, for $ p>2 $, \begin{align}
\|u_{\va,\lambda}-u_{0,\lambda}\|_{L^2(\om(x_0,R))}&\leq\|v_{\va,\lambda}\|_{L^2(\om(x_0,R))}+\|z_{\va,\lambda}\|_{L^2(\om(x_0,R))}+C\va R\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om)}\nonumber\\
&\leq C_{\theta_0}\va R^{2-\frac{2}{p}}\|F\|_{L^p(\om(y_0,R))}.\nonumber \end{align} This, together with Lemma \ref{ww74} and $ \eqref{W2p} $, gives \begin{align}
|u_{\va,\lambda}(x_0)-u_{0,\lambda}(x_0)|\leq C_{\theta_0}\va R^{1-\frac{d}{p}}\|F\|_{L^p(\om(y_0,R))}.\nonumber \end{align} Then it follows by duality arguments that \begin{align}
\left(\int_{\om(y_0,R)}|G_{\va,\lambda}(x_0,y)-G_{0,\lambda}(x_0,y)|^{p'}dy\right)^{\frac{1}{p'}}\leq C_{\theta_0}\va R^{1-\frac{d}{p}}\text{ for any }p>d.\label{ww76} \end{align} Finally, since $ (\mathcal{L}_{\va}-\overline{\lambda} I)(G_{\va,\lambda}^{\gamma}(x_0,\cdot))=(\mathcal{L}_0-\overline{\lambda} I)(G_{0,\lambda}^{\gamma}(x_0,\cdot))=0 $ in $ \om(y_0,R) $ for any $ 1\leq \gamma\leq m $, we may invoke Lemma \ref{ww74} again to conclude that for any $ k\in\mathbb{N}_+ $, \be \begin{aligned}
|G_{\va,\lambda}(x_0,y_0)-G_{0,\lambda}(x_0,y_0)|&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^k}\Xint-_{\om(y_0,R)}|G_{\va,\lambda}(x_0,y)-G_{0,\lambda}(x_0,y)|dy\\
&\quad+C_{k,\theta_0}\va R^{1-\frac{d}{p}}\|\nabla_2^2 G_{0,\lambda}(x_0,\cdot)\|_{L^p(\om(y_0,R))}\\
&\quad+C_{k,\theta_0}\va(1+|\lambda|R^2)\|\nabla_2G_{0,\lambda}(x_0,\cdot)\|_{L^{\infty}(\om(y_0,R))}. \end{aligned}\label{ww75} \ee Firstly for the first term of $ \eqref{ww75} $, in view of $ \eqref{ww76}, $ we can obtain that it is bounded by \be
C_{k,\theta_0}\va(1+|\lambda|R^2)^{-k}R^{1-d}.\label{zjbddx} \ee For the third term of $ \eqref{ww75} $, using $ \eqref{Green Lipschitz 11} $, it is also bounded by $ \eqref{zjbddx} $. For the second term, if $ d\geq 3 $, to obtain the same boundedness, we can use the $ W^{2,p} $ estimates for $ \mathcal{L}_0-\lambda I $, $ \eqref{W2p local} $, that is, \begin{align}
\left(\Xint-_{\om(y_0,R)}|\nabla_2^2G_{0,\lambda}(x_0,y)|^pdy\right)^{\frac{1}{p}}\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR^2}\left(\Xint-_{\om(y_0,2R)}|G_{0,\lambda}(x_0,y)|^2dy\right)^{\frac{1}{2}}.\label{GreenW2p} \end{align} This, together with $ \eqref{Greene} $, completes the proof for the case that $ d\geq 3 $. If $ d=2 $, we divide the proof into two cases. If $ \Delta(y_0,3R)\neq\emptyset $, then it follows directly from $ \eqref{Boundedness estimates} $ and $ \eqref{GreenW2p} $. If $ \Delta(y_0,3R)=\emptyset $, we can obtain the same result by using the interior Lipschitz estimate for the function $ \nabla_2G_{0,\lambda}(x_0,y) $, \begin{align}
|\nabla_2^2G_{0,\lambda}(x_0,y_0)|\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR}\left(\Xint-_{B(y_0,2R)}|\nabla_2G_{0,\lambda}(x_0,y)|^2dy\right)^{\frac{1}{2}} \end{align} and $ \eqref{Green Lipschitz 11} $. \end{proof}
\begin{thm}[Convergence of Green functions II]\label{Convergence of Green's functions 2t} For $ \va>0 $ and $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, let $ \om $ be a bounded $ C^{2,1} $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. Then for any $ k\in\mathbb{N}_+ $, $ 1\leq\al\leq m $ and $ x,y\in\om $ with $ x\neq y $, \be \begin{aligned}
\left|\frac{\pa }{\pa x_i}\{G_{\va,\lambda}^{\al\beta}(x,y)\}-\frac{\pa}{\pa x_i}\{\Phi_{\va,j}^{\al\beta}(x)\}\frac{\pa}{\pa x_j}\{G_{0,\lambda}^{\beta\gamma}(x,y)\}\right|\leq\frac{C_{k,\theta_0}\va \ln[\va^{-1}|x-y|+2]}{(1+|\lambda||x-y|^2)^{k}|x-y|^d}, \end{aligned}\label{Convergence of Green's functions 2} \ee where $ C_{k,\theta_0} $ depends only on $ \mu,d,m,\nu,\tau,k,\theta_0 $ and $ \om $. \end{thm}
\begin{lem} For $ \va>0 $ and $ d\geq 2 $, $ \lambda\in\Sigma_{\theta_0}\cup\{0\} $ with $ \theta_0\in(0,\frac{\pi}{2}) $, let $ \om $ be a bounded $ C^{2,1} $ domain in $ \mathbb{R}^d $. Suppose that $ A $ satisfies $ \eqref{sy} $, $ \eqref{el} $, $ \eqref{pe} $ and $ \eqref{Hol} $. Assume that $ u_{\va,\lambda}\in H^1(\om(x_0,4R);\mathbb{C}^m) $ and $ u_{0,\lambda}\in C^{2,\rho}(\om(x_0,4R);\mathbb{C}^m) $ for some $ 0<\rho<1 $. If $ \Delta(x_0,4R)\neq\emptyset $, assume that \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=(\mathcal{L}_0-\lambda I)(u_{0,\lambda})\text{ in } \om(x_0,4R)\quad\text{and}\quad u_{\va,\lambda}=u_{0,\lambda}\text{ on }\Delta(x_0,4R).\nonumber \end{align} If $ \Delta(x_0,4R)=\emptyset $, assume that \begin{align} (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=(\mathcal{L}_0-\lambda I)(u_{0,\lambda})\text{ in } B(x_0,4R).\nonumber \end{align} Then for any $ 0<\va<r $, $ 1\leq\al\leq m $ and $ k\in\mathbb{N}_+ $, \begin{align}
\|\frac{\pa u_{\va,\lambda}^{\al}}{\pa x_i}-\frac{\pa \Phi_{\va,j}^{\al\beta}}{\pa x_i}\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}\|_{L^{\infty}(\om_R)}&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR}\Xint-_{\om_{4R}}|u_{\va,\lambda}-u_{0,\lambda}|\nonumber\\
&\quad+C_{k,\theta_0}\va \left\{ (1+|\lambda|R^2)R^{-1}\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\right.\label{ww79}\\
&\quad+\ln[\va^{-1}R+2]\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}+\left. R^{\rho}[\nabla^2 u_{0,\lambda}]_{C^{0,\rho}(\om_{4R})}\right\},\nonumber \end{align} where $ C_{k,\theta_0} $ depends on $ \mu,d,m,k,\theta_0,\nu,\tau,p,\rho $ and $ \om $. \end{lem} \begin{proof} We only prove the case that $ \Delta(x_0,3R)\neq\emptyset $ and the other is similar in view of the proof of Lemma \ref{ww74}. We start by choosing a $ C^{2,1} $ domain $ \widetilde{\om} $ such that $ \om(x_0,3R)\subset\widetilde{\om}\subset \om(x_0,4R) $. Let \begin{align} w_{\va,\lambda}(x)=u_{\va,\lambda}(x)-u_{0,\lambda}(x)-[\Phi_{\va,j}^{\beta}(x)-P_j^{\beta}(x)]\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}.\nonumber \end{align} Simple calculations imply that \be \begin{aligned} \left\{(\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda})\right\}^{\alpha}&=-\va\frac{\pa}{\pa x_i}\left\{ F_{jik}^{\al\gamma}(x/\varepsilon)\frac{\pa^2u_{0,\lambda}^{\gamma}}{\pa x_j\pa x_k}\right\}\\ &\quad+a_{ij}^{\al\beta}(x/\va)\frac{\pa}{\pa x_j}[\Phi_{\va,k}^{\beta\gamma}(x)-x_k\delta^{\beta\gamma}-\va\chi_k^{\beta\gamma}(x/\va)]\frac{\pa^2 u_{0,\lambda}^{\gamma}}{\pa x_i\pa x_k}\\ &\quad+\frac{\pa}{\pa x_i}\left\{a_{ij}^{\al\beta}(x/\va)[\Phi_{\va,k}^{\beta\gamma}(x)-x_k\delta^{\beta\gamma}]\frac{\pa^2u_{0,\lambda}^{\gamma}}{\pa x_j\pa x_k}\right\}\\ &\quad+\lambda[\Phi_{\va,k}^{\al\beta}(x)-x_k\delta^{\al\beta}]\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_k} \end{aligned}\label{Equality 1} \ee Note that $ w_{\va,\lambda}=0 $ on $ \Delta(x_0,4R) $. Write $ w_{\va,\lambda}=v_{\va,\lambda}+z_{\va,\lambda} $ in $ \widetilde{\om} $, where $ v_{\va,\lambda}\in H_0^1(\widetilde{\om};\mathbb{C}^m) $ and $ (\mathcal{L}_{\va}-\lambda I)(v_{\va,\lambda})=(\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda}) $ in $ \widetilde{\om} $. Since $ (\mathcal{L}_{\va}-\lambda I)(z_{\va,\lambda})=0 $ in $ \widetilde{\om} $ and $ z_{\va,\lambda}=w_{\va,\lambda}=0 $ on $ \Delta(x_0,3R) $, it follows from the boundary Lipschitz estimate $ \eqref{Linfty} $ and $ \eqref{Lipesu} $ that for any $ k\in\mathbb{N}_+ $, \begin{align}
\|\nabla z_{\va,\lambda}\|_{L^{\infty}(\om_R)}&\leq \frac{C_{\theta_0}}{R}\left(\Xint-_{\om_{3/2R}}|z_{\va,\lambda}|^{2}\right)^{\frac{1}{2}}\leq \frac{C_{\theta_0}}{R}\|z_{\va,\lambda}\|_{L^{\infty}(\om_{3/2R})}\nonumber\\
&\leq \frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR}\Xint-_{\om_{2R}}|z_{\va,\lambda}|\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR}\Xint-_{\om_{2R}}|w_{\va,\lambda}|+\frac{C_{k,\theta_0}}{R}\|v_{\va,\lambda}\|_{L^{\infty}(\om_{2R})}\nonumber\\
&\leq\frac{C_{k,\theta_0}}{(1+|\lambda|R^2)^kR}\Xint-_{\om_{2R}}|u_{\va,\lambda}-u_{0,\lambda}|+\frac{C_{k,\theta_0}}{R}\left\{\|v_{\va,\lambda}\|_{L^{\infty}(\om_{2R})}+\va\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{2R})}\right\},\nonumber \end{align} where we have used $ \eqref{Estimate for Dirichlet correctors} $. This implies that \begin{align}
&\|\nabla w_{\va,\lambda}\|_{L^{\infty}(\om_R)}\leq\|\nabla v_{\va,\lambda}\|_{L^{\infty}(\om_R)}+\|\nabla z_{\va,\lambda}\|_{L^{\infty}(\om_R)}\nonumber\\
&\quad\quad\leq\frac{C_{k\theta_0}}{(1+|\lambda|R^2)^kR}\Xint-_{\om_{2R}}|u_{\va,\lambda}-u_{0,\lambda}|+\frac{C_{k,\theta_0}}{R}\left\{R\|\nabla v_{\va,\lambda}\|_{L^{\infty}(\om_{2R})}+\va\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{2R})}\right\},\nonumber \end{align}
where we have used $ \|v_{\va,\lambda}\|_{L^{\infty}(\om_{2R})}\leq CR\|\nabla v_{\va,\lambda}\|_{L^{\infty}(\om_{2R})} $ since $ v_{\va,\lambda}=0 $ on $ \pa\widetilde{\om} $. Owing to \begin{align} \frac{\pa w_{\va,\lambda}^{\al}}{\pa x_i}=\frac{\pa u_{\va,\lambda}^{\al}}{\pa x_i}-\frac{\pa \Phi_{\va,j}^{\al\beta}}{\pa x_i}\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}-[\Phi_{\va,j}^{\al\beta}(x)-x_j\delta^{\al\beta}]\frac{\pa^2u_{0,\lambda}^{\beta}}{\pa x_i\pa x_j},\nonumber \end{align} we can obtain that for any $ 1\leq\al\leq m $ and $ k\in\mathbb{N}_+ $, \be \begin{aligned}
&\|\frac{\pa u_{\va,\lambda}^{\al}}{\pa x_i}-\frac{\pa \Phi_{\va,j}^{\al\beta}}{\pa x_i}\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}\|_{L^{\infty}(\om_R)}\leq \|\nabla w_{\va,\lambda}\|_{L^{\infty}(\om_R)}+\|[\Phi_{\va}-P]\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_R)}\\
&\quad\quad\leq \|\nabla w_{\va,\lambda}\|_{L^{\infty}(\om_R)}+C\va\|\nabla^2 u_{0,\lambda}\|_{L^{\infty}(\om_R)}\\
&\quad\quad\leq\frac{C_{k\theta_0}}{(1+|\lambda|R^2)^kR}\Xint-_{\om_{2R}}|u_{\va,\lambda}-u_{0,\lambda}|\\
&\quad\quad\quad\quad+C_{k,\theta_0}\left\{\va R^{-1}\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{2R})}
+\|\nabla v_{\va,\lambda}\|_{L^{\infty}(\om_{2R})}+\va\|\nabla^2 u_{0,\lambda}\|_{L^{\infty}(\om_{2R})}\right\}. \end{aligned}\label{ww78} \ee It remains to estimate $ \nabla v_{\va,\lambda} $ in $ \om_{2R} $. To this end, we use the representation formula \begin{align} v_{\va,\lambda}(x)=\int_{\widetilde{\om}}\widetilde{G}_{\va,\lambda}(x,y)\overline{(\mathcal{L}_{\va}-\lambda I)(w_{\va,\lambda})(y)}dy,\nonumber \end{align} where $ \widetilde{G}_{\va,\lambda}(x,y) $ is the Green function for $ \mathcal{L}_{\va}-\lambda I $ in the $ C^{2,1} $ domain $ \widetilde{\om} $. Let \begin{align} f_i^{\al}(x)=-\va F_{jik}^{\al\gamma}(x/\varepsilon)\frac{\pa^2u_{0,\lambda}^{\gamma}}{\pa x_j\pa x_k}+a_{ij}^{\al\beta}(x/\va)[\Phi_{\va,k}^{\beta\gamma}(x)-x_k\delta^{\beta\gamma}]\frac{\pa^2u_{0,\lambda}^{\gamma}}{\pa x_j\pa x_k}.\nonumber \end{align} In view of $ \eqref{Equality 1} $, we can obtain \begin{align} v_{\va,\lambda}(x)&=-\int_{\widetilde{\om}}\frac{\pa}{\pa y_i}\{\widetilde{G}_{\va,\lambda}(x,y)\}\overline{[f_i(y)-f_i(x)]}dy\nonumber\\ &\quad\quad+\int_{\widetilde{\om}}\widetilde{G}_{\va,\lambda}(x,y)\overline{a_{ij}(y/\va)\frac{\pa}{\pa y_i}[\Phi_{\va,k}(y)-P_k(y)-\va\chi_k(y/\va)]\frac{\pa^2 u_{0,\lambda}}{\pa y_i\pa y_k}}dy\nonumber\\ &\quad\quad+\overline{\lambda}\int_{\widetilde{\om}}\widetilde{G}_{\va,\lambda}(x,y)\overline{[\Phi_{\va,k}(y)-P_k(y)]\frac{\pa u_{0,\lambda}}{\pa y_k}}dy.\nonumber \end{align} It follows that \be \begin{aligned}
|\nabla v_{\va,\lambda}(x)|&\leq\int_{\widetilde{\om}}|\nabla_1\nabla_2\widetilde{G}_{\va,\lambda}(x,y)||f(y)-f(x)|dy\\
&\quad+C\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\int_{\widetilde{\om}}|\nabla_1\widetilde{G}_{\va,\lambda}(x,y)||\nabla[\Phi_{\va}(y)-P(y)-\va\chi(y/\va)]|dy\\
&\quad+C|\lambda|\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\int_{\widetilde{\om}}|\widetilde{G}_{\va,\lambda}(x,y)||\Phi_{\va}(y)-P(y)|dy. \end{aligned}\label{ww77} \ee In view of the Lipshcitz estimates of Green functions, $ \eqref{Green Lipschitz 14} $, we have \begin{align}
|\nabla_1\nabla_2\widetilde{G}_{\va,\lambda}(x,y)|\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}|x-y|^d}\text{ for any }k\in\mathbb{N}_+.\label{yyd} \end{align} Meanwhile, simple calculations and $ \eqref{Estimate for Dirichlet correctors} $ give the $ L^{\infty} $ and Hölder estimates of $ f $, that is \begin{align}
\|f\|_{L^{\infty}(\om_{4R})}&\leq C\va\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})},\nonumber\\
|f(x)-f(y)|&\leq C|x-y|^{\rho}\left\{\va^{1-\rho}\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}+\va[\nabla^2u_{0,\lambda}]_{C^{0,\rho}(\om_{4R})}\right\}.\nonumber \end{align} Choosing $ k=0 $ in $ \eqref{yyd} $, it can be obtained that \begin{align}
&\int_{\widetilde{\om}}|\nabla_1\nabla_2\widetilde{G}_{\va,\lambda}(x,y)||f(y)-f(x)|dy\nonumber\\
&\quad\quad\leq C_{\theta_0}\va\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\int_{\widetilde{\om}\backslash B(x,\va)}\frac{dy}{|x-y|^{d}}\nonumber\\
&\quad\quad\quad+C_{\theta_0} \left\{\va^{1-\rho}\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}+\va[\nabla^2u_{0,\lambda}]_{C^{0,\rho}(\om_{4R})}\right\}\int_{\widetilde{\om}\cap B(x,\va)}\frac{dy}{|x-y|^{d-\rho}}\nonumber\\
&\quad\quad\leq C_{\theta_0} \left\{\va\ln[\va^{-1}R+2]\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}+C\va^{1+\rho}[\nabla^2u_{0,\lambda}]_{C^{0,\rho}(\om_{4R})}\right\}.\nonumber \end{align} Finally, using the estimates \begin{align}
|\nabla_1\widetilde{G}_{\va,\lambda}(x,y)|&\leq\frac{C_{k,\theta_0}}{(1+|\lambda||x-y|^2)^{k}|x-y|^{d-1}}\min\left\{1,\frac{\dist(y,\pa\widetilde{\om})}{|x-y|}\right\}\text{ for any }k\in\mathbb{N}_+,\nonumber \end{align} as well as the observation that for any $ 1\leq j\leq d $ and $ 1\leq \beta\leq m $, \begin{align}
\left|\nabla\left\{\Phi_{\va,j}^{\beta}(x)-P_{j}^{\beta}(x)-\va \chi_{j}^{\beta}(x/\va)\right\}\right| \leq C \min \left\{1,\va[\operatorname{dist}(x,\pa\widetilde{\Omega})]^{-1}\right\},\nonumber \end{align} we can bound the second term in the right hand side of $ \eqref{ww77} $ by \begin{align}
&C_{\theta_0}\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\left\{\va\int_{\widetilde{\om}\backslash B(x,\va)}\frac{dy}{|x-y|^{d}}+\int_{\widetilde{\om}\cap B(x,\va)}\frac{dy}{|x-y|^{d-1}}\right\}\nonumber\\
&\quad\quad\leq C_{\theta_0}\va \ln[\va^{-1}R+2]\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}.\nonumber \end{align} For the third term of the right hand side of $ \eqref{ww77} $, by rescaling and $ \eqref{*} $, we can obtain that this term is bounded by \begin{align}
C_{\theta_0}\va (1+|\lambda|R^2)R^{-1}\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}.\nonumber \end{align} As a result, we have proved that \begin{align}
\|\nabla v_{\va,\lambda}\|_{L^{\infty}(\om_{3R})}&\leq C_{\theta_0}\va (1+|\lambda|R^2) R^{-1}\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\nonumber\\
&\quad\quad+C_{\theta_0} \left\{\va\ln[\va^{-1}R+2]\|\nabla^2u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}+\va R^{\rho}[\nabla^2u_{0,\lambda}]_{C^{0,\rho}(\om_{4R})}\right\}.\nonumber \end{align} This, together with $ \eqref{ww78} $, completes the proof. \end{proof}
\begin{proof}[Proof of Theorem \ref{Convergence of Green's functions 2t}] Fix $ x_0,y_0\in\om $ and $ R=\frac{|x_0-y_0|}{16} $. We may assume that $ 0<\va<r $, since the case $ \va\geq r $ is trivial and follows directly from the size estimates of $ |\nabla_1G_{\va,\lambda}(x,y)| $, $ |\nabla_2G_{\va,\lambda}(x,y)| $ (see $ \eqref{Green Lipschitz 11} $) and $ \eqref{Estimate for Dirichlet correctors} $. For any $ 1\leq\gamma\leq m $, let $ u_{\va,\lambda}(x)=G_{\va,\lambda}^{\gamma}(x,y_0) $ and $ u_{0,\lambda}(x)=G_{0,\lambda}^{\gamma}(x,y_0) $. Observe that $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=(\mathcal{L}_0-\lambda I)(u_{0,\lambda})=0 $ in $ \om(x_0,4R) $ and $ u_{\va,\lambda}=u_{0,\lambda}=0 $ on $ \Delta(x_0,4R) $ (if $ \Delta(x_0,4R)\neq\emptyset $). By Theorem \ref{Convergence of Green's functions}, it can be obtained, \begin{align}
\|u_{\va,\lambda}-u_{0,\lambda}\|_{L^{\infty}(\om(x_0,4R))}\leq C_{\theta_0}\va R^{1-d}.\nonumber \end{align} Also, since $ \om $ is $ C^{2,1} $, we have, for any $ k\in\mathbb{N}_+ $, \begin{align}
\|\nabla u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}&\leq \frac{C_{k,\theta_0} }{(1+|\lambda|R^2)^kR^{d-1}},\nonumber\\
\|\nabla^2 u_{0,\lambda}\|_{L^{\infty}(\om_{4R})}\leq \frac{C_{k,\theta_0} }{(1+|\lambda|R^2)^kR^d}\quad&\text{and}\quad\|\nabla^2 u_{0,\lambda}\|_{C^{0,\rho}(\om_{4R})}\leq \frac{C_{k,\theta_0} }{(1+|\lambda|R^2)^kR^{d+\rho}}.\nonumber \end{align} Here, we have used $ \eqref{ww80} $, $ \eqref{ww81} $, $ \eqref{Green Lipschitz 11} $ and arguments in the proof of Theorem \ref{Convergence of Green's functions}. Hence, in view of $ \eqref{ww79} $, we can complete the proof. \end{proof}
\subsection{Proof of Theorem \ref{Approximation 1}, Theorem \ref{Lpconres} and \ref{LpW1pconreso}} \begin{proof}[Proof of Theorem \ref{Approximation 1}] For $ u_{\va,\lambda}=R(\lambda,\mathcal{L}_{\va})F $ and $ u_{0,\lambda}=R(\lambda,\mathcal{L}_{0})F $, let \begin{align} w_{\va,\lambda}(x)=u_{\va,\lambda}(x)-u_{0,\lambda}(x)-[\Phi_{\va,j}^{\beta}(x)-P_{j}^{\beta}(x)]\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}.\nonumber \end{align} Using the equality $ \eqref{Equality 1} $ and choosing $ w_{\va,\lambda} $ as the test function, we can obtain that \begin{align}
B_{\va,\lambda,\om}[w_{\va,\lambda},w_{\va,\lambda}]=\int_{\om}A(x/\va)\nabla w_{\va,\lambda}\overline{\nabla w_{\va,\lambda}}dx-\lambda\int_{\om}|w_{\va,\lambda}|^2dx=J_{\va,\lambda,\om}[u_{0,\lambda},w_{\va,\lambda}],\quad\va>0,\label{Equation 1} \end{align} where the bilinear form $ J_{\va,\lambda,\om}[\cdot,\cdot]:H_0^1(\om;\mathbb{C}^m)\times H_0^1(\om;\mathbb{C}^m)\to\mathbb{C} $ is defined by \be \begin{aligned} J_{\va,\lambda,\om}[u,v]&=-\int_{\om}\va F_{jik}^{\al\gamma}(x/\varepsilon)\frac{\pa^2 u^{\gamma}}{\pa x_j\pa x_k}\overline{\frac{\pa v^{\al}}{\pa x_i}}dx+\lambda\int_{\om}[\Phi_{\va,k}^{\al\beta}(x)-x_k\delta^{\al\beta}]\frac{\pa u^{\beta}}{\pa x_k}\overline{v^{\al}}dx\\ &\quad\quad-\int_{\om}a_{ij}^{\al\beta}(x/\va)[\Phi_{\va,k}^{\beta\gamma}(x)-x_k\delta^{\beta\gamma}]\frac{\pa^2 u^{\beta}}{\pa x_j\pa x_k}\overline{\frac{\pa v^{\al}}{\pa x_i}}dx\\ &\quad\quad+\int_{\om}a_{ij}^{\al\beta}(x/\va)\frac{\pa}{\pa x_j}[\Phi_{\va,k}^{\beta\gamma}(x)-x_k\delta^{\beta\gamma}-\va\chi_k^{\beta\gamma}(x/\va)]\frac{\pa^2 u^{\gamma}}{\pa x_i\pa x_k}\overline{v^{\al}}dx. \end{aligned}\label{J linear} \ee Taking $ u=u_{0,\lambda} $ and $ v=w_{\va,\lambda} $ and using $ \eqref{Estimate for Dirichlet correctors} $, it can be inferred that \begin{align}
|J_{\va,\lambda,\om}[u_{0,\lambda},w_{\va,\lambda}]|&\leq C\va \int_{\om}|\nabla^2u_{0,\lambda}||\nabla w_{\va,\lambda}|dx+C\va|\lambda|\int_{\om}|\nabla u_{0,\lambda}||w_{\va,\lambda}|dx\nonumber\\
&\quad+C\int_{\om}|\nabla[\Phi_{\va}(x)-P(x)-\va\chi(x/\va)]||\nabla^2u_{0,\lambda}||w_{\va,\lambda}|dx\nonumber\\
&\leq C\va|\lambda|\|\nabla u_{0,\lambda}\|_{L^2(\om)}\|w_{\va,\lambda}\|_{L^2(\om)}+C\va\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}\nonumber\\
&\quad+C\|\nabla[\Phi_{\va}(\cdot)-P(\cdot)-\va\chi(\cdot/\va)]w_{\va,\lambda}\|_{L^2(\om)}\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}.\nonumber \end{align}
To estimate $ |J_{\va,\lambda,\om}[u_{0,\lambda},w_{\va,\lambda}]| $, we first claim that for any $ 1\leq j\leq d $ and $ 1\leq \beta\leq m $, \begin{align}
\|\nabla[\Phi_{\va,j}^{\beta}(\cdot)-P_j^{\beta}(\cdot)-\va\chi_j^{\beta}(\cdot/\va)]w_{\va,\lambda}\|_{L^2(\om)}\leq C\va\|\nabla w_{\va,\lambda}\|_{L^2(\om)}.\label{Claim 1} \end{align} To see $ \eqref{Claim 1} $, we fix $ 1\leq \beta_0\leq m $, $ 1\leq j_0\leq d $ and let \begin{align} h_{\va}(x)=\Phi_{\va,j_0}^{\beta_0}(x)-P_{j_0}^{\beta_0}(x)-\va\chi_{j_0}^{\beta_0}(x/\va),\text{ where }x\in\om.\nonumber \end{align} Note that $ h_{\va}\in H^1(\Omega;\mathbb{C}^m)\cap L^{\infty}(\Omega;\mathbb{C}^m) $ and $ \mathcal{L}_{\va}(h_{\va})=0 $ in $ \om $. It follows that \begin{align}
\mu\int_{\om}|\nabla h_{\va}|^2|w_{\va,\lambda}|^2&\leq\int_{\om}a_{ij}^{\al\beta}(x/\va)\frac{\pa h_{\va}^{\beta}}{\pa x_j}\overline{\frac{\pa h_{\va}^{\al}}{\pa x_i}}|w_{\va,\lambda}|^2dx\nonumber\\ &=-\int_{\om}\overline{h_{\va}^{\al}}a_{ij}^{\al\beta}(x/\va)\frac{\pa h_{\va}^{\beta}}{\pa x_j}\overline{\frac{\pa w_{\va,\lambda}^{\gamma}}{\pa x_i}}w_{\va,\lambda}^{\gamma}dx-\int_{\om}\overline {h_{\va}^{\al}}a_{ij}^{\al\beta}(x/\va)\frac{\pa h_{\va}^{\beta}}{\pa x_j}\frac{\pa w_{\va,\lambda}^{\gamma}}{\pa x_i}\overline{w_{\va,\lambda}^{\gamma}}dx,\nonumber \end{align} where we have used integration by parts. Hence \begin{align}
\int_{\om}|\nabla h_{\va}|^2|w_{\va,\lambda}|^2dx\leq C\int_{\om}|h_{\va}||\nabla h_{\va}||\nabla w_{\va,\lambda}||w_{\va,\lambda}|dx,\nonumber \end{align}
where $ C $ depends on $ d,m $ and $ \mu $. This directly implies the claim by noticing that $ \|h_{\va}\|_{L^{\infty}(\om)}\leq C\va $ and using the inequality \begin{align}
\int_{\om}|\nabla h_{\va}|^2|w_{\va,\lambda}|^2dx\leq \frac{1}{2}\int_{\om}|\nabla h_{\va}|^2|w_{\va,\lambda}|^2dx+C\int_{\om}| h_{\va}|^2|\nabla w_{\va,\lambda}|^2dx,\nonumber \end{align} where we have used the $ \eqref{inte} $. Then the claim implies that \begin{align}
|J_{\va,\lambda,\om}[u_{0,\lambda},w_{\va,\lambda}]|\leq C\va|\lambda|\|\nabla u_{0,\lambda}\|_{L^2(\om)}\|w_{\va,\lambda}\|_{L^2(\om)}+C\va\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}.\label{Estimate of J} \end{align} In view of $ \eqref{ABCDE} $, $ \eqref{inte} $ and $ \eqref{Equation 1} $, it can be easily shown that \begin{align}
\|w_{\va,\lambda}\|_{L^2(\om)}^2&\leq \frac{Cc(\lambda,\theta)}{R_0^{-2}+|\lambda|}|B_{\va,\lambda,\om}[w_{\va,\lambda},w_{\va,\lambda}]|\leq \frac{Cc(\lambda,\theta)}{R_0^{-2}+|\lambda|}|J_{\va,\lambda,\om}[u_{0,\lambda},w_{\va,\lambda}]|\nonumber\\
&\leq \frac{C\va c(\lambda,\theta)}{R_0^{-2}+|\lambda|}\left\{|\lambda|\|\nabla u_{0,\lambda}\|_{L^2(\om)}\|w_{\va,\lambda}\|_{L^2(\om)}+\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}\right\}\nonumber\\
&\leq C\va^2 c^2(\lambda,\theta)\|\nabla u_{0,\lambda}\|_{L^2(\om)}^2+\frac{1}{2}\|w_{\va,\lambda}\|_{L^2(\om)}^2+\frac{C\va c(\lambda,\theta)}{R_0^{-2}+|\lambda|}\|\nabla^2u_{0,\lambda}\|_{L^2(\om)}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}.\nonumber \end{align} Then it is not hard to obtain the $ L^2 $ estimate of $ w_{\va,\lambda} $, that is, \begin{align}
\|w_{\va,\lambda}\|_{L^2(\om)}^2\leq C\va^2 c^2(\lambda,\theta)\|\nabla u_{0,\lambda}\|_{L^2(\om)}^2+\frac{C\va c(\lambda,\theta)}{R_0^{-2}+|\lambda|}\|\nabla^2u_{0,\lambda}\|_{L^2(\om)}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}.\label{wvadiyi} \end{align} Similarly, owing to $ \eqref{laxmil} $, it can be obtained without difficulty that \begin{align}
\|\nabla w_{\va,\lambda}\|_{L^2(\om)}^2&\leq Cc(\lambda,\theta)|B_{\va,\lambda,\om}[w_{\va,\lambda},w_{\va,\lambda}]|\leq Cc(\lambda,\theta)|J_{\va,\lambda,\om}[u_{0,\lambda},w_{\va,\lambda}]|\nonumber\\
&\leq C\va c(\lambda,\theta)\left\{|\lambda|\|\nabla u_{0,\lambda}\|_{L^2(\om)}\|w_{\va,\lambda}\|_{L^2(\om)}+\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}\right\}\nonumber\\
&\leq C\va^2 c^2(\lambda,\theta)|\lambda|\|\nabla u_{0,\lambda}\|_{L^2(\om)}^2+|\lambda|\|w_{\va,\lambda}\|_{L^2(\om)}^2+C\va c(\lambda,\theta)\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}.\nonumber \end{align} This, together with $ \eqref{inte} $ and $ \eqref{wvadiyi} $, gives that \begin{align}
\|\nabla w_{\va,\lambda}\|_{L^2(\om)}^2&\leq C\va^2c^2(\lambda,\theta)|\lambda|\|\nabla u_{0,\lambda}\|_{L^2(\om)}^2+C\va c(\lambda,\theta)\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}\nonumber\\
&\leq C\va^2c^2(\lambda,\theta)|\lambda|\|\nabla u_{0,\lambda}\|_{L^2(\om)}^2+C\va^2 c^2(\lambda,\theta)\|\nabla^2 u_{0,\lambda}\|_{L^2(\om)}^2+\frac{1}{2}\|\nabla w_{\va,\lambda}\|_{L^2(\om)}^2.\nonumber \end{align}
In view of $ \eqref{L2uva}$ and $ \eqref{L2n2u0} $, we can estimate $ \|\nabla u_{0,\lambda}\|_{L^2(\om)} $ and $ \|\nabla^2u_{0,\lambda}\|_{L^2(\om)} $. Then \begin{align}
\|\nabla w_{\va,\lambda}\|_{L^2(\om)}\leq C\va c^2(\lambda,\theta)\|F\|_{L^2(\om)},\nonumber \end{align} which completes the proof of $ \eqref{Convergence rate 1} $. This, together with $\eqref{wvadiyi} $, shows $ \eqref{Convergence rate 11} $. \end{proof}
\begin{proof}[Proof of Theorem \ref{Lpconres}] In view of the representation formula $ \eqref{repre} $ and $ \eqref{Convergence of Green's functions formula} $, it can be seen that \begin{align}
\|R(\lambda,\mathcal{L}_{\va})-R(\lambda,\mathcal{L}_{0})\|_{L^{\infty}(\om)\to L^{\infty}(\om)}&\leq C_{\theta_0}\va(R_0^{-2}+|\lambda|)^{-\frac{1}{2}},\nonumber\\
\|R(\lambda,\mathcal{L}_{\va})-R(\lambda,\mathcal{L}_{0})\|_{L^{1}(\om)\to L^{1}(\om)}&\leq C_{\theta_0}\va(R_0^{-2}+|\lambda|)^{-\frac{1}{2}}.\nonumber \end{align} These, together with $ \eqref{Operator estimate 11} $ and the M. Riesz interpolation theorem, give $ \eqref{**-} $. \end{proof}
\begin{proof}[Proof of Theorem \ref{LpW1pconreso}] It follows directly from $ \eqref{Convergence of Green's functions 2} $ and the arguments in Theorem 6.5.2 in \cite{Shen2}. We give the proof here for the sake of completeness. Let $ u_{\va,\lambda},u_{0,\lambda}\in H_0^1(\om;\mathbb{C}^m) $ such that $ (\mathcal{L}_{\va}-\lambda I)(u_{\va,\lambda})=F $ and $ (\mathcal{L}_{0}-\lambda I)(u_{0,\lambda})=F $ with $ F\in L^p(\om;\mathbb{C}^m) $. By using $ \eqref{Estimate for Dirichlet correctors} $ and $ \eqref{W2p} $, we only need to show that for any $ 1\leq p\leq \infty $ and $ 1\leq\al\leq m $, \begin{align}
\|\frac{\pa u_{\va,\lambda}^{\al}}{\pa x_i}-\frac{\pa\Phi_{\va,j}^{\al\beta}}{\pa x_i}\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}\|_{L^p(\om)}\leq C_{\theta_0}\va\{\ln[\va^{-1}R_0+2]\}^{4|\frac{1}{2}-\frac{1}{p}|}\|F\|_{L^p(\om)}.\nonumber \end{align} In view of Theorem \ref{Convergence of Green's functions 2t}, we can obtain that \begin{align}
\left|\frac{\pa u_{\va,\lambda}^{\al}}{\pa x_i}-\frac{\pa\Phi_{\va,j}^{\al\beta}}{\pa x_i}\frac{\pa u_{0,\lambda}^{\beta}}{\pa x_j}\right|\leq C_{\theta_0}\int_{\om}K_{\va}(x,y)|F(y)|dy,\nonumber \end{align} where the kernel $ K_{\va} $ is defined by \begin{align} K_{\va}(x,y)=\left\{\begin{matrix}
\va|x-y|^{-d}\ln[\va^{-1}|x-y|+2]&\text{if}&|x-y|\geq\va,\\
|x-y|^{1-d}&\text{if}&|x-y|<\va. \end{matrix}\right.\nonumber \end{align} Direct computations imply that \begin{align} \sup_{x\in\om}\int_{\om}K_{\va}(x,y)dy+\sup_{y\in\om}\int_{\om}K_{\va}(x,y)dy\leq C_{\theta_0}\{\ln[\va^{-1}R_0+2]\}^2.\label{Kva1} \end{align} This gives $ \eqref{Convergence rate LpW1p} $ for cases $ p=1 $ and $ \infty $. Thus, by the M. Riesz interpolation theorem, the result is a direct consequence of the case $ p=2 $, which is given by $ \eqref{Convergence rate 1} $. \end{proof}
\section*{Acknowledgments} I am grateful to Professor Jun Geng of Lanzhou University for warm guidance on the topics of the homogenization theory for elliptic systems. I am also grateful to Professor Zhifei Zhang of Peking University for some important inspirations on the homogenization theory. I sincerely thank the anonymous reviewers for their constructive revision suggestions.
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