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\begin{document}
\date{\today} \author{Kathrin Bringmann} \address{Mathematical Institute\\University of Cologne\\ Weyertal 86-90 \\ 50931 Cologne \\Germany} \email{[email protected]} \author{Larry Rolen} \address{Mathematical Institute\\University of Cologne\\ Weyertal 86-90 \\ 50931 Cologne \\Germany} \email{[email protected]} \thanks{The research of the first author was supported by the Alfried Krupp Prize for Young University Teachers of the Krupp foundation and the research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER. The second author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative.}
\begin{abstract} In analogy with the classical theory of Eichler integrals for integral weight modular forms, Lawrence and Zagier considered examples of Eichler integrals of certain half-integral weight modular forms. These served as early prototypes of a new type of object, which Zagier later called a quantum modular form. Since then, a number of others have studied similar examples. Here we develop the theory in a general context, giving rise to a well-defined class of quantum modular forms. Since elements of this class show up frequently in examples of combinatorial and number theoretical interest, we propose the study of the general properties of this space of quantum modular forms. We conclude by raising fundamental questions concerning this space of objects which merit further study.
\end{abstract} \title[Eichler integrals and quantum modular forms]{Half-integral weight Eichler integrals and quantum modular forms} \dedicatory{To Winnie Li, who has been a great inspiration, on the occasion of her birthday} \maketitle
\section{Introduction and statement of results}
It is well-known that the derivative of a modular form is typically not a modular form. However, thanks to an identity of Bol \cite{Bol}, there exists a canonical differential operator $D^{k-1}\colon M_{2-k}^!\rightarrow M_k^!$ for $k\in\mathbb N$, where $D:=\frac1{2\pi i}\frac{\partial}{\partial\tau}$ and $M_{\ell}^!$ denotes the space of weight $\ell$ weakly holomorphic modular forms. Motivated by this, Eichler \cite{Eichler} considered the formal $(k-1)$-st antiderivative of a cusp form. Specifically, if $f(\tau)=\sum_{n\geq1}a_f(n)q^n$ (throughout $q:=e^{2\pi i \tau}$ with $\tau\in\mathbb H$) is a cusp form on $\operatorname{SL}_2(\mathbb Z)$, then we define the holomorphic Eichler integral by $\widetilde{f}(\tau):=\sum_{n\geq1}a_f(n) n^{1-k}q^n.$
Denoting by $|_{\ell}$ the Petersson slash operator in weight $\ell$ (defined in Section 2.1), we easily see from the remarks above that $D^{k-1}(\widetilde f\vert_{2-k}(1-S))=0$ (where $S:=(\begin{smallmatrix}0&-1\\ 1&0\end{smallmatrix}))$. Hence, $\widetilde f\vert_{2-k}(1-S)$ is a polynomial of degree at most $k-2$, called the period polynomial of $f$. Eichler integrals play a fundamental role in the study of integral weight modular forms. For example, as further elaborated on by Shimura \cite{Shimura} and Manin \cite{Manin}, the theory of Eichler integrals provides deep insights into the theory of elliptic curves and critical values of $L$-functions. For an interesting discussion of the general cohomology theory, see also \cite{Bruggeman}.
Since half-integral weight modular forms (see Section 2.1 for the definition) also encode deep arithmetic information, it is natural to ask what the analogous Eichler-Shimura theory is for half-integral weight. Here, the situation is more complicated. In particular, the operator $D^{k-1}$ no longer makes sense. In fact, it is a central problem in the theory of harmonic Maass forms to construct a suitable operator which plays a similar role as $D^{k-1}$ in half-integral weight (see, for example, \cite{FractionalDeriv}). The first pioneering examples of half-integral weight Eichler integrals were considered in connection with WRT invariants of 3-manifolds by Lawrence and Zagier \cite{Lawrence-Zagier}. Although the proof is more difficult for general weight modular forms, they formally considered the same Eichler integral as defined above for certain weight $3/2$ theta functions $\vartheta$. Even though it is impossible for $\widetilde{\vartheta}\vert_{\frac12}(1-\gamma)$ (where $\gamma$ is any element of the congruence subgroup of $\vartheta$) to be a polynomial of degree $k-2$ as in the integral weight case, they give a nice characterization of its modular properties as one approaches cusps. In fact, $\widetilde{\vartheta}$ can be extended to ${\mathbb Q}$, and the resulting function $\widetilde{\vartheta}\vert_{\frac12}(1-\gamma)$ becomes real-analytic on $\mathbb R\backslash\{\gamma^{-1}\infty\}$. This provides one of the first examples of the burgeoning new theory of quantum modular forms, laid out by Zagier \cite{ZagierQMF}, which we review in Section 2.2. Essentially, a quantum modular form of weight $k$ is a complex-valued function $f$ on ${\mathbb Q}$ whose modular obstructions, or cocycles, $f\vert_k(1-\gamma)$ are ``nicer'' than the original function in some analytic way. For example, $f$ is usually only well-defined on ${\mathbb Q}$, whereas $f\vert_k(1-\gamma)$ typically extends to an open set of $\mathbb R$ and is differentiable, smooth, etc.
Since \cite{Lawrence-Zagier}, there has been an explosion of research aimed at constructing examples of quantum modular forms related to non-holomorphic Eichler integrals, see for example \cite{Kimport,Bringmann-Creutzig-Rolen,Bringmann-Folsom-Rhoades,Folsom-Ono-Rhoades,ZagierVass}. For instance, quantum modular forms are closely tied to surprising identities relating the combinatorial generating functions counting ranks, cranks, and unimodal sequences \cite{Folsom-Ono-Rhoades}, and to the general theory of negative index Jacobi forms and Kac-Wakimoto characters \cite{Bringmann-Creutzig-Rolen}. In this paper, we elucidate the general picture in arbitrary half-integral weight. Although the previous proofs depended on the modular forms considered being theta functions, we show that a similar phenomenon is true in general, along the way constructing large families of quantum modular forms. Our main result is the following, where a more detailed definition of quantum modular forms and ``nice properties'' is given in Section 2.2. \begin{theorem}\label{mainthm} If $f\in S_k(N)$ with $k\in\frac12+\mathbb N_0$ and $N\in4\mathbb N$, then $\widetilde f$ is a quantum modular form of weight $2-k$. In this instance, the ``nice'' property of the cocycle is that for every $\gamma\in\Gamma_0(N)$, $\widetilde f\vert_{2-k}(1-\gamma)$ is real-analytic except at $\gamma^{-1}\infty$. \end{theorem}
\noindent {\it Two remarks.}
\noindent 1) In fact, the proof of Proposition 2.1 shows that $\widetilde f$ satisfies a much stricter condition of quantum modularity and, in the language of Zagier, is a strong quantum modular form (cf. Section 2.2).
\noindent 2) The quantum modular forms in Theorem \ref{mainthm} were recently described in a different guise using the theory of mock theta functions (see Theorem 1.5 and Lemma 3.1 of \cite{RhoadesChoiLim}). However, we give a different proof of their quantum modularity, as the techniques here are of general interest.
Although the definition of quantum modular forms by Zagier is (intentionally) vague, the evolution of the literature indicates that it is now worthwhile to split up the types of quantum modular forms which naturally arise into various categories. Theorem 1.1 shows that it makes sense to consider the vector space of Eichler integrals of cusp forms in $S_{k}(N)$ as an interesting space of quantum modular forms, and we propose this new area of study in a series of questions at the end of the paper. In addition to the results of Theorem \ref{mainthm}, we give explicit formulas for the resulting quantum modular forms. Recall that for a half-integral weight cusp form $f(\tau)=\sum_{n\geq1} a_f(n)q^n$, its $L$-function is defined for $\operatorname{Re} (s)\gg0$ by $L_f(s):=\sum_{n\geq1}\frac{a_f(n)}{n^s}.$ More generally, consider the twisted $L$-function defined for $\operatorname{Re} (s)\gg0$ and $\frac dc\in{\mathbb Q}$ (throughout we assume that fractions are expressed in lowest terms) by $L_f(\zeta_c^d;s):=\sum_{n\geq1}\frac{a_f(n)\zeta_c^{dn}}{n^s},$ where $\zeta_{a}^b:=e(\frac ab)$ with $e(x):=e^{2\pi ix}$, and define the function $Q_f\colon{\mathbb Q}\rightarrow\mathbb C$ by \[Q_f\left(\frac dc\right):=L_f\left(\zeta_c^d;k-1\right).\] We show that $Q_f$ is a quantum modular form.
\begin{corollary} Assume the notation of Theorem \ref{mainthm}. Then $Q_f$ is a quantum modular form on $\Gamma_0(N)$, with $Q_f\vert_{2-k}(1-\gamma)$ real-analytic except at $\gamma^{-1}\infty$ for all $\gamma\in\Gamma_0(N)$. \end{corollary} \begin{remark} We will see in Section 2.4 that $L_f(\zeta_c^d;s)$ has an analytic continuation to $\mathbb C$, so that $Q_f$ is well-defined. \end{remark}
The paper is organized as follows. In Section 2, we review the definitions of half-integral weight modular forms and quantum modular forms, recall an auxiliary non-holomorphic Eichler integral considered in \cite{Lawrence-Zagier}, and reduce the statement of Theorem \ref{mainthm} to a certain claim about asymptotic expansions. We conclude Section 2 by giving several useful facts about $L$-functions needed for the proof of Theorem \ref{mainthm}. In Section 3, we complete the proof of Theorem \ref{mainthm} and Corollary 1.2. We conclude in Section 4 with a list of further questions raised by this paper.
\section{Preliminaries} \subsection{Half-integral weight modular forms} Here we review some standard definitions and facts concerning modular forms. Firstly, for $k\in\frac12\mathbb Z$, recall that the Petersson slash operator is defined for functions $f\colon\mathbb H\rightarrow\mathbb C$ and $\gamma=(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix})\in\operatorname{SL}_2(\mathbb Z)$ (with $\gamma\in\Gamma_0(4)$ if $k\in\frac12+\mathbb Z$) by \[f\vert_{k}\gamma(\tau):=\begin{cases}(c\tau+d)^{-k}f\left(\frac{a\tau+b}{c\tau+d}\right)&\text{ for }k\in\mathbb Z,\\ \varepsilon_d^{2k}\left(\frac cd\right)(c\tau+d)^{-k}f\left(\frac{a\tau+b}{c\tau+d}\right)&\text{ for }k\in\frac12+\mathbb Z,\end{cases}\] where $(\frac{\cdot}{\cdot})$ denotes the Jacobi symbol and for odd $d$, \[\varepsilon_d:=\begin{cases}1&\text{ if }d\equiv1\pmod4,\\ i&\text{ if }d\equiv 3\pmod4.\end{cases}\] We require the following congruence subgroups of $\operatorname{SL}_2(\mathbb Z)$:
\begin{equation*}\begin{aligned}\Gamma_0(N)&:=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\operatorname{SL}_2(\mathbb Z)\colon N|c\right\},\\ \Gamma_1(N)&:=\left\{\begin{pmatrix}a&b\\ c&d\end{pmatrix}\in\Gamma_0(N)\colon a\equiv d\equiv 1\pmod N\right\}.\end{aligned}\end{equation*} We then make the following definition. \begin{defn} Let $k\in\frac12+\mathbb N_0$, $N\in4\mathbb N$, and $\chi$ be a Dirichlet character modulo $N$. Then a holomorphic function $f\colon\mathbb H\rightarrow\mathbb C$ is a cusp form of weight $k$ on $\Gamma_0(N)$ with Nebentypus $\chi$ if the following conditions hold: \begin{enumerate} \item For all $\gamma=(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix})\in\Gamma_0(N)$, we have $f\vert_{k}\gamma=\chi(d)f.$ \item As $\tau$ approaches any cusp of $\Gamma_0(N)$, $f$ decays exponentially fast. \end{enumerate}
\end{defn} We denote the space of cusp forms of weight $k$ on $\Gamma_0(N)$ and Nebentypus $\chi$ by $S_k(N,\chi)$. If $\chi$ is trivial, then we also use the notation $S_k(N)$. One may analogously define a space of cusp forms $S_k(\Gamma_1(N))$. We have the following well-known decomposition \begin{equation}\label{decompgamma1}S_k(\Gamma_1(N))=\bigoplus_{\chi} S_k(N,\chi),\end{equation} where $\chi$ runs over all even Dirichlet characters modulo $N$. Thus, in the study of modular forms on $\Gamma_1(N)$, it is often sufficient to consider modular forms on $\Gamma_0(N)$ with Nebentypus.
\subsection{Quantum modular forms} In this subsection, we recall some definitions and examples of quantum modular forms. Following Zagier, we make the following definition.
\begin{defn} A function $f\colon \mathbb{Q}\rightarrow\mathbb C$ is a quantum modular form of weight $k$ on a congruence subgroup $\Gamma$ if, for all $\gamma\in\Gamma$, the cocycle
$r_{\gamma}:=f|_k(1-\gamma)$ extends to an open subset of\, $\mathbb R$ and is analytically ``nice''. Here ``nice'' could mean continuous, smooth, real-analytic, etc. We say that $f$ is a strong quantum modular form if, in addition, $f$ has formal power series attached to each rational number which also have near-modularity properties (see \cite{ZagierQMF} for more details).
\end{defn}
\begin{remark}
All of the quantum modular forms occurring in this paper have cocycles defined on $\mathbb R$ which are real-analytic except at one point.
\end{remark}
One of the most striking examples of a quantum modular form is given by Kontsevich's ``strange function'' $F(q)$, as studied by Zagier in \cite{ZagierVass}, which is given by
\begin{equation}\label{kontstrange}F(q):=\sum_{n\geq0}(q;q)_n,\end{equation} where $(a;q)_n:=\prod_{j=0}^{n-1}(1-aq^j)$ denotes the usual $q$-Pochhammer symbol. This function is strange as it does not converge on any open subset of $\mathbb C$, but converges as a finite sum for $q$ any root of unity. Zagier's study of $F$ depends on the ``sum of tails'' identity \begin{equation} \label{sumoftails} \displaystyle\sum_{n\geq0}\left(\eta(\tau)-q^{\frac1{24}}\left(q;q\right)_n\right)=\eta(\tau)D\left(\tau\right)+\sqrt 6\widetilde{\eta}(\tau), \end{equation} \noindent where $\eta(\tau):=q^{1/24}(q;q)_{\infty}$ and $D(\tau):=-\frac12+\sum_{n\geq1}\frac{q^n}{1-q^n}.$ The key observation of Zagier is that in (\ref{sumoftails}), the functions $\eta(\tau)$ and $\eta(\tau)D(\tau)$ vanish of infinite order as $\tau\rightarrow\frac hk$, so at a root of unity $\xi$, $F(\xi)$ is essentially the limiting value of the Eichler integral of $\eta$, which he showed has quantum modular properties \cite{ZagierVass}.
\subsection{Non-holomorphic Eichler integrals}
We now suppose that $f\in S_k(N)$ for $N\in4\mathbb N$ and $k\in\frac12+\mathbb N_0$. The main idea in studying the modularity properties of half-integral weight Eichler integrals, due to Lawrence and Zagier, is to introduce the non-holomorphic Eichler integral
\[f^*(\tau):=\frac{(-2\pi i)^{k-1}}{\Gamma(k-1)}\int_{\overline{\tau}}^{i\infty}f(w)(w-\tau)^{k-2}dw,\] defined for $\tau\in\mathbb H^-:=\{u+iv\in\mathbb C\colon v<0\}.$ The point is that $\widetilde f$ has a $q$-series expansion (e.g., if $f$ is a theta function, then $\widetilde f$ is a partial theta function), while $f^*$ satisfies a nice transformation law. These transformation properties of $f^*$ transfer over to $\widetilde f$ near the real axis. For concreteness, we make the following definition.
\begin{defn} Let $f(\tau)$ and $g(\tau)$ be defined for $\tau\in\mathbb H$ and $\tau\in\mathbb H^-$, respectively. We say that the asymptotic expansions of $f$ and $g$ agree at a rational number $\frac dc$ if there exist $\beta(n)$ such that as $t\rightarrow0^+$,
\[f\left(\frac dc+\frac{it}{2\pi}\right)\sim\sum_{n\geq0}\beta(n)t^n,\quad\quad g\left(\frac dc-\frac{it}{2\pi}\right)\sim\sum_{n\geq0}\beta(n)(-t)^n.\] \end{defn} We now look at the transformation properties of $f^*$. We easily compute for
$\gamma=\(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\)\in\Gamma_0(N)$ and $\tau\in\mathbb H^-$ that
\begin{equation}\label{nonholtransform}f^*(\tau)-\chi_{-4}(d)f^*\vert_{2-k}\gamma(\tau)= \frac{(-2\pi i)^{k-1}}{\Gamma(k-1)}\int_{-\frac dc}^{i\infty}f(w)(w-\tau)^{k-2}dw,\end{equation} where $\chi_{-4}$ is the Dirichlet character defined by $\chi_{-4}(n):=(\frac{-4}n)$. This is the key transformation property giving rise to quantum modularity in Theorem \ref{mainthm}. The connection between $\widetilde f$ and $f^*$ is given by the following proposition, whose proof we defer to Section 3 (similar results were explored in less generality in e.g. \cite{Hikami,Lawrence-Zagier,ZagierVass}).
\begin{proposition}\label{fooprop} Assuming the notation of Theorem \ref{mainthm}, the asymptotic expansions of $\widetilde f$ and $f^*$ agree at any $\frac dc\in{\mathbb Q}$. \end{proposition}
Thus, by Proposition \ref{fooprop}, $\widetilde f$ inherits the same transformation properties as $f^*$ as one approaches the real line. \subsection{Properties of $L$-functions} In this subsection, we recall some basic properties of modular $L$-functions needed for the proof of Theorem \ref{mainthm}. Firstly, we require the following lemma. \begin{lemma}\label{vanishing} Let $f\in S_k(N)$, where $k\in\frac12+\mathbb N_0$, $N\in4\mathbb N$, and $\frac dc\in{\mathbb Q}$. Then $L_f(\zeta_c^d;s)$ has an analytic continuation to $\mathbb C$ and $L_f(\zeta_c^d;-m)=0$ for $m\in\mathbb N_0$.
\end{lemma}
\begin{proof} We first determine the modularity properties of $f_{\frac dc}(\tau):=\sum_{n\geq1}a_f(n)\zeta_c^{dn}q^n$. Note that \[f_{\frac dc}(\tau)=\sum_{j=0}^{c-1}\zeta_c^{dj}\sum_{n\equiv j\pmod c}a_f(n)q^n.\] It is well-known that $\sum_{n\equiv j\pmod c}a_f(n)q^n\in S_k(\Gamma_1(N c^2)),$ and hence $f_{\frac dc}\in S_k(\Gamma_1(N c^2)).$ By the decomposition (\ref{decompgamma1}), we see that $f_{\frac dc}$ can be written as a finite sum $f_{\frac dc}=\sum_{j=0}^{N_0}f_{\frac dc,j}$ where $f_{\frac dc,j}\in S_k(Nc^2,\chi_j)$ and $\chi_j$ is a Dirichlet character modulo $Nc^2$. Clearly $L_f(\zeta_c^d;s)=L_{f_{\frac dc}}(s)$, so it suffices to show that the lemma holds for $L_g(s)$ where $g\in S_{k}(M,\chi)$ with $M\in4\mathbb N$ and $\chi$ is a Dirichlet character modulo $M$. For such a $g$, the proof of the analytic continuation and vanishing condition of $L_g$ is essentially classical, due to Hecke (see Theorem 14.7 of \cite{Iwaniec-Kowalski} and Satz 4 of \cite{Petersson}). However, since the multiplier is different in half-integral weight, for completeness we prove it directly.
For this, we first recall the action of the Fricke involution, given by \begin{equation}\label{WNDefn}g\vert_kW_N(\tau):=\left(-i\sqrt N\tau\right)^{-k}g\left(-\frac1{N\tau}\right).\end{equation} It is well-known that $g\vert_kW_N\in S_k(M,\chi(\frac{N}{\cdot}))$ (see Section 3 of \cite{Bruinier}). We now consider the completed $L$-function \[\Lambda_g(s):=\int_0^{\infty}g\left(\frac{iv}{\sqrt N}\right)v^{s-1}dv.\] By a simple calculation, one sees that the completed $L$-function factors as \begin{equation}\label{completedsplitting}\Lambda_g(s)=\left(\frac{\sqrt{N}}{2\pi}\right)^s\Gamma(s)L_g(s).\end{equation} The key property of $\Lambda_g(s)$ is its functional equation. To determine it, we begin by splitting $\Lambda_g(s)$ into two pieces as \[\Lambda_g(s)=\int_0^{1}g\left(\frac{iv}{\sqrt N}\right)v^{s-1}dv+\int_1^{\infty}g\left(\frac{iv}{\sqrt N}\right)v^{s-1}dv,\] and make a change of variables in the first integral to obtain \[\Lambda_g(s)=\int_1^{\infty}g\left(\frac{i}{v\sqrt N}\right)v^{-s-1}dv+\int_1^{\infty}g\left(\frac{iv}{\sqrt N}\right)v^{s-1}dv.\] Inserting (\ref{WNDefn}), we find \begin{equation}\label{completedint}\Lambda_g(s)=\int_1^{\infty}g\vert_kW_N\left(\frac{iv}{\sqrt N}\right)v^{k-s-1}dv+\int_1^{\infty}g\left(\frac{iv}{\sqrt N}\right)v^{s-1}dv.\end{equation} Since both $g$ and $g\vert_kW_N$ are cusp forms, and hence have rapid decay as $v\rightarrow\infty$, (\ref{completedint}) immediately gives an analytic continuation of $\Lambda_g(s)$ to $\mathbb C$. The analytic continuation of $L_g(s)$ now follows immediately from (\ref{completedsplitting}) and the fact that $1/\Gamma(s)$ has no poles. The integral representation (\ref{completedint}) also directly implies the functional equation, namely \begin{equation}\label{functionaleqn}\Lambda_g(s)=\Lambda_{g\vert_kW_N}(k-s).\end{equation} Now the claims follow, as the Gamma factor $\Gamma(s)$ in (\ref{completedsplitting}) forces $L_g(s)$ to have zeros at non-positive integers, as $\Gamma(s)$ has a pole at these locations whereas the right hand side of (\ref{functionaleqn}) does not. \end{proof} Besides the analytic continuation of our $L$-functions, we require a growth estimate as $\vert\operatorname{Im}(s)\vert\rightarrow\infty$. For our purposes, the following basic lemma suffices. Since the proof is a standard application of the functional equation and the Phragm\'en-Lindel\"of principle, we omit the proof. \begin{lemma} \label{Lfunctiongrowth}
For fixed $x\in\mathbb R$, $L_{f}(\zeta_c^d;x+it)$ grows at most polynomially in $t$ as $|t|\rightarrow\infty$. \end{lemma}
\section{Proof of Theorem \ref{mainthm} and Corollary 1.2}
In this section, we show Proposition \ref{fooprop}, use it to prove Theorem \ref{mainthm}, and then deduce Corollary 1.2. To prove Proposition 2.1, we compute the asymptotic expansions of $\widetilde f$ and $f^*$ separately to show that they agree. This is done using the Mellin transform, defined by $\mathcal M(f)(s):=\int_0^{\infty}f(t)t^{s-1}dt$, which provides a fundamental connection between the asymptotic expansion of one function and the poles of another. Specifically, we require the following result for computing asymptotic expansions, which is a special case of Theorem 4 (i) of \cite{Flajolet}. \begin{lemma} \label{AsympExpExists} Let $F(x)$ be continuous on $(0,\infty)$ with Mellin transform $\mathcal M(F)(s)$ converging on a right half-plane $\operatorname{Re}(s)>\alpha$. Assume that $\mathcal M(F)(s)$ can be meromorphically continued to the half-plane $\operatorname{Re}(s)>\beta$, where $\beta<\alpha$, with a finite number of poles $a_0,a_1,\ldots, a_M$ in that half-plane, each simple with residue $\alpha_j$. Further assume that $\mathcal M(F)(s)$ is analytic on the line $\operatorname{Re}(s)=\beta$ and that in the right half-plane $\operatorname{Re}(s)>\beta$, the estimate
\[\mathcal M(F)(s)=O\left(|s|^{-r}\right)\]
holds as $|s|\rightarrow\infty$ for some $r>1$. Then we have the asymptotic expansion \[F(x)\sim\sum_{j=0}^M\alpha_jx^{-a_j}+O\left(x^{-\beta}\right).\]
\end{lemma} For later use, we also recall that for a general function $f$ with $f(x)=O(x^{\alpha}) \text{ as }x\rightarrow0^+$ and $f(x)=O(x^{\beta})\text{ as }x\rightarrow\infty,$ $\mathcal M(f)(s)$ converges in the strip $-\alpha\leq\operatorname{Re}(s)\leq-\beta$.
In order to apply Lemma \ref{AsympExpExists} to the function $\widetilde f(\frac dc+\frac{it}{2\pi})$, we first compute its Mellin transform by integrating termwise
\begin{equation}\label{neednow}\begin{aligned}\mathcal{M}\left(\widetilde f\left(\frac dc+\frac{it}{2\pi}\right)\right)\left(s\right)&=\int_0^{\infty}\sum_{n\geq1}n^{1-k}\zeta_c^{nd}a_f(n)e^{-nt}t^{s-1}dt &=\Gamma(s)L_f\left(\zeta_c^d;k-1+s\right).\end{aligned}\end{equation} We next show that this Mellin transform satisfies the conditions of Lemma \ref{AsympExpExists}. Firstly, note, using the right-hand side of (\ref{neednow}), that $\mathcal{M}(\widetilde f(\frac dc+\frac{it}{2\pi}))(s)$ is convergent on some right half-plane, and by Lemma \ref{vanishing}, it has an analytic continuation to $\mathbb C$ with poles only at non-positive integers (coming from the Gamma factor). To estimate the growth of the Mellin transform in vertical strips, we first recall Stirling's estimate (see 5.11.9 of \cite{Nist}):
\begin{equation}\label{Stirling}\left\vert\Gamma(x+iy)\right\vert=\sqrt{2\pi}|y|^{x-\frac12}e^{-\frac{\pi|y|}2}\ \text{ as }|y|\rightarrow\infty.\end{equation} By Lemma 2.3 and (\ref{Stirling}), the Mellin transform is thus of rapid decay for fixed $\operatorname{Re}(s)$ as $\vert\operatorname{Im}(s)\vert\rightarrow\infty$. Hence, letting $\beta\rightarrow -\infty$ in Lemma \ref{AsympExpExists}, we directly find \begin{equation}\label{HolEichAsympFinal}\widetilde f\left(\frac dc+\frac{it}{2\pi}\right)\sim\sum_{n\geq0}\frac{(-1)^n}{n!}L\left(\zeta_c^d;k-1-n\right)t^n\text{ as }t\rightarrow0^+,\end{equation}
We now turn to computing the asymptotic expansion of $f^*(\frac dc-\frac{it}{2\pi})$ as $t\rightarrow0^+$. We begin by expanding $f^*$. By a change of variables, we have for $n>0$ and $\tau=u+iv$ with $v<0$
\[\int_{\overline{\tau}}^{i \infty}e^{2\pi i nw}(w-\tau)^{k-2}dw=i^{k-1}(2\pi n)^{1-k}e^{2\pi i n \tau}\Gamma(k-1,4\pi n |v|),\] where $\Gamma(s,x):=\int_x^{\infty}t^{s-1}e^{-t}dt$ denotes the incomplete Gamma function. Thus, integrating term-by-term, we obtain
\begin{equation}\label{fstarexp}f^*(\tau)=\frac1{\Gamma(k-1)}\sum_{n\geq1}a_f(n) n^{1-k}e^{2\pi i n\tau}\Gamma(k-1,4\pi n|v|).\end{equation} Taking the Mellin transform of the right hand side of (\ref{fstarexp}), we obtain
\begin{equation}\label{mellinnonholeich}\begin{aligned}\mathcal{M}\left(f^*\left(\frac dc-\frac{it}{2\pi}\right)\right)(s)&=\frac1{\Gamma(k-1)}L_f\left(\zeta_c^d;k-1+s\right)\mathcal{M}\left(e^t\Gamma(k-1,2t)\right)(s).\end{aligned}\end{equation}
In order to use Lemma \ref{AsympExpExists} to compute the asymptotic expansion of $f^*$, we first determine the location and residues of the poles of $\mathcal{M}(f^*(\frac dc-\frac{it}{2\pi}))(s)$ using the representation on the right-hand side of (\ref{mellinnonholeich}).
\begin{lemma}\label{crap} Assuming the notation above, the following are true: \begin{enumerate} \item The function $\mathcal{M}(f^*(\frac dc-\frac{it}{2\pi}))(s)$ has a simple pole at $s=-n \in-\mathbb N_0$ with residue \[\frac{ L_f\left(\zeta_c^d;k-1-n\right)}{n!}.\] \item The function $\mathcal{M}(f^*(\frac dc-\frac{it}{2\pi}))(s)$ is holomorphic for $s\not\in-\mathbb N_0$. \end{enumerate} \end{lemma} \begin{proof} (1) We begin by rewriting the Mellin transform $\mathcal M(e^t\Gamma(k-1,2t))(s)$ in a more convenient form. Namely, we claim that for $\text{Re}(s)>0$ and $\text{Re}(k+s-1)>0$, we have \begin{equation}\label{bleh}\mathcal M\left(e^t\Gamma\left(k-1,2t\right)\right)(s)=\beta\left(\frac12;s,2-k-s\right)\Gamma\left(k-1+s\right),\end{equation} where, for $\operatorname{Re}(a)>0$, $\operatorname{Re}(b)>0$, and $0\leq z\leq 1$, \begin{equation*}\beta(z;a,b):=\int_0^zt^{a-1}(1-t)^{b-1}\mathrm dt\end{equation*}
is the incomplete beta function. To see (\ref{bleh}), define for $|z|<1$ the Gaussian hypergeometric series \begin{equation}\label{2F1Defn} _2F_1\left(a,b;c;z\right):=\sum_{n\geq0}\frac{(a)_n(b)_nz^n}{(c)_nn!},\end{equation} where $(x)_n:=\prod_{j=0}^{n-1}(x+j)$ is the Pochhammer symbol (note that $ _2F_1$ is sometimes denoted by $F$ in \cite{Nist}). Now use (8.14.6) of \cite{Nist}, which states that for $\text{Re}(r)>-1,\text{ Re}(a+b)>0,$ and $\text{ Re}(a)>0,$ \[\int_0^{\infty}x^{a-1}e^{-rx}\Gamma(b,x) dx=\frac{\Gamma(a+b)}{a(1+r)^{a+b}} \text{ }_2F_1\left(1,a+b;1+a;\frac r{1+r}\right).\] Choosing $b=k-1$, $a=s$, and $r=-\frac12$, we obtain, after a change of variables, \begin{equation}\label{labelfoo}\mathcal M\left(e^t\Gamma(k-1,2t)\right)(s)=2^{k-1}\frac{\Gamma(s+k-1)}{s}\text{}_2F_1(1,s+k-1;1+s;-1).\end{equation} Here we remark that the specialization of $ _2F_1$ in (\ref{labelfoo}) is defined by analytic continuation, as discussed in Section 15.2 of \cite{Nist}. We then use (8.17.9) of \cite{Nist} which states that \begin{equation}\label{beta2f1}\beta(x;a,b)=\frac{x^a(1-x)^{b-1}}{a}\text{}_2F_1\left(1,1-b;a+1;\frac{x}{x-1}\right).\end{equation} Plugging in $x=\frac12$, $a=s$, and $b=2-k-s$ and using (\ref{labelfoo}) gives (\ref{bleh}).
To determine the poles and residues of the incomplete beta function, we make use of the beta function $\beta(a,b):=\beta(1;a,b)$ and the regularized beta function $I(z;a,b):=\frac{\beta(z;a,b)}{\beta(a,b)}.$ Using the standard fact that \begin{equation}\label{betafactor}\beta(a,b)=\frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)},\end{equation} along with the Euler reflection formula for $\Gamma(s)$, we obtain
\begin{equation}\label{mellinrewrite}\begin{aligned} \beta\left(\frac12;s,2-k-s\right)\Gamma\left(k-1+s\right)=
\Gamma(s)\frac{\pi I\left(\frac12;s,2-k-s\right)}{\Gamma(2-k)\sin\left(\pi(k-1+s)\right)}.\end{aligned}\end{equation} Combining (\ref{mellinnonholeich}), (\ref{bleh}), and (\ref{mellinrewrite}) and again using Euler's reflection formula, we find that \begin{equation*}\mathcal{M}\left(f^*\left(\frac dc-\frac{it}{2\pi}\right)\right)(s)= L_f\left(\zeta_c^d;k-1+s\right) \Gamma(s)\frac{(-1)^{k+\frac12} I\left(\frac12;s,2-k-s\right)}{\sin\left(\pi(k-1+s)\right)}.\end{equation*} We next specialize to $s=-n$. Note that $\sin\left(\pi(k-1-n)\right)=(-1)^{k+\frac12+n}$. Using Lemma 2.2 again, Lemma 3.1 (1) thus follows immediately once we have established that \begin{equation}\label{foobar}I\left(\frac12;-n,2-k+n\right)=1.\end{equation}
To see (\ref{foobar}), first note the following two identities, stated in (26.5.2) and (26.5.15) of \cite{AS}, respectively: \begin{equation}\label{foo1}I(x;a,b)=1-I(1-x;b,a),\end{equation} \begin{equation} \label{recurrencebetareg} I(x;a,b)=\frac{\Gamma(a+b)}{\Gamma(a+1)\Gamma(b)}x^a(1-x)^{b-1}+I(x;a+1,b-1).\end{equation} For $n\in\mathbb N_0$, (\ref{recurrencebetareg}) implies that \begin{equation}\label{anotherrecurr}I\(\frac12;2-k+n,-n\)=I\(\frac12;2-k,0\),\end{equation}
since the factor $\frac{\Gamma(2-k)}{\Gamma(2-k+1)\Gamma(n)}$
is zero. Using (\ref{recurrencebetareg}) again implies that \begin{equation}\label{lastrecurr}I\(\frac12;2-k,0\)=I\(\frac12;1-k,1\)-2^{k-1}.\end{equation} The right-hand side of (\ref{lastrecurr}) may now be evaluated directly. For this, we note that
$ _2F_1(1,0;2-k;-1)=1$, as all but the first term in (\ref{2F1Defn}) vanish, and hence, by (\ref{beta2f1}), $\beta(\frac12;1-k,1)=\frac{2^{k-1}}{1-k}.$ Although this specialization is on the border of the region of convergence $|z|<1$ of $ _2F_1$, by Abel's Lemma the value of the analytic continuation to a point on the boundary of convergence is the value at that point, assuming it exists. Now note that by (\ref{betafactor}), $\beta(1-k,1)=(1-k)^{-1}$, and hence by the definition of $I$, we have that $I(\frac12;1-k,1)=2^{k-1}$. Thus, by (\ref{anotherrecurr}) and (\ref{lastrecurr}), we have $I(\frac12;2-k+n,-n)=0$. But then (\ref{foo1}) gives
\begin{equation*}I\left(\frac12;-n,2-k+n\right)=1-I\left(\frac12;2-k+n,-n\right) =1,\end{equation*} proving (\ref{foobar}).
\noindent (2) Before proving that $\mathcal{M}(f^*(\frac dc-\frac{it}{2\pi}))(s)$ is holomorphic for $s\not\in-\mathbb N_0$, we first analyze where potential poles could arise. To do this, note that using (\ref{mellinnonholeich}), (\ref{labelfoo}), and the standard Pfaff transformation formula \begin{equation}\label{Pfaff} _2F_1(a,b,c;z)=(1-z)^{-b}\, _2F_1\(b,c-a;c;\frac z{z-1}\)\end{equation}
together gives \begin{equation}\label{simpler}\mathcal{M}\(f^*\(\frac dc-\frac{it}{2\pi}\)\)(s)=\frac{1}{2^s\Gamma(k-1)}L_f\(\zeta_c^d;k-1+s\)\Gamma(s+k-1)\frac{ _2F_1\(s+k-1,s;1+s;\frac12\)}s.\end{equation}
Hence, using the fact that the only possible poles of $ _2F_1(a,b;c;z)$ for $|z|<1$ occur if $c\in-\mathbb N_0$, together with Lemma 2.2, we see that $\mathcal M(f^*(\frac dc-\frac{it}{2\pi}))(s)$ can only possibly have poles for $s\in-\mathbb N_0$ or for $s\in\frac12+\mathbb Z$ with $s\leq 1-k$.
Thus, to complete the proof of (2) we just need to prove that there are no poles for $s\in\frac12+\mathbb Z$ with $s\leq 1-k$. We thus fix $s$ to be such a half-integer. Now it suffices by (\ref{simpler}) to show that $L_f\(\zeta_c^d;k-1+s\)=0$, as the only potential pole arises from a simple pole in the function $\Gamma(s+k-1)$. However, by the assumption on $s$, we note that $k-1+s\in-\mathbb N_0$, so that the desired vanishing condition is stated in Lemma 2.2. \end{proof} We are now ready to provide the asymptotic expansion of $f^*$, which, by comparison with (\ref{HolEichAsympFinal}), proves Proposition 2.1. One may easily show, using (\ref{beta2f1}), (\ref{Pfaff}), and (\ref{Stirling}), that $\beta(\frac12;s,2-k-s)$ is bounded in vertical strips for $\vert\operatorname{Im}(s)\vert$ large enough. By this fact, together with (\ref{mellinnonholeich}), (\ref{bleh}), Lemma \ref{Lfunctiongrowth}, and (\ref{Stirling}), we find that $\mathcal{M}(f^*(\frac dc-\frac{it}{2\pi}))(s)$ is of rapid decay in vertical strips as $\vert\operatorname{Im}(s)\vert\rightarrow\infty$. Hence, we can directly plug Lemma \ref{crap} into Lemma \ref{AsympExpExists} to find that \begin{equation}\label{nonholfinalexp} f^*\left(\frac dc-\frac{it}{2\pi}\right)\sim\sum_{n\geq0}\frac{1}{n!}L\left(\zeta_c^d;k-1-n\right)t^n\text{ as }t\rightarrow0^+,\end{equation} which establishes Proposition 2.1.
We now have all of the pieces needed to prove Theorem \ref{mainthm}. \begin{proof}[Proof of Theorem \ref{mainthm}] By (\ref{nonholfinalexp}), we find that $f^*(\frac dc-\frac{it}{2\pi})$ has a well-defined limit as $t\rightarrow0^+$, namely, $\lim_{t\rightarrow0^+}f^*(\frac dc-\frac{it}{2\pi})=L(\zeta_c^d;k-1).$ By (\ref{nonholtransform}), we see that $f^*$ is a quantum modular form, as for $x\in\mathbb R$, the cocycle $\int_{-\frac dc}^{i\infty}f(w)(w-x)^{k-2}dw$ is real-analytic on $\mathbb R\setminus\{-\frac dc\}$. By (\ref{HolEichAsympFinal}), we see that $\widetilde f$ also extends to a function on ${\mathbb Q}$ with the same values as $f^*$, and hence is a quantum modular form. The claim that it is a strong quantum modular form follows from Proposition \ref{fooprop}. \end{proof} The proof of Corollary 1.2 is now straightforward. \begin{proof}[Proof of Corollary 1.2] As in the proof of Theorem \ref{mainthm}, we see by (\ref{HolEichAsympFinal}) that $\widetilde f$ is a quantum modular form, and that its values at the rational point $\frac dc$ is $L_f(\zeta_c^d;k-1)$. \end{proof}
\section{Questions and outlook} Theorem 1.1 and Corollary 1.2 provide a canonical family of quantum modular forms, which gives a different perspective on the quantum modular forms defined in \cite{RhoadesChoiLim}. As discussed in Sections 1 and 2.2, this family includes many examples in the literature, such as Kontsevich's strange function (\ref{kontstrange}) \cite{ZagierVass}, quantum modular forms arising from specializations of the crank and rank generating functions \cite{Folsom-Ono-Rhoades}, and decomposition formulas for Kac-Wakimoto characters \cite{Bringmann-Creutzig-Rolen}. Due to the number of connections with combinatorics and number theory, we propose that studying these quantum modular forms in more detail is worthwhile. Thus, we make the following definition. \begin{defn} We say that a quantum modular form is an Eichler quantum modular form if its values on ${\mathbb Q}$ are equal to the radial limits from inside the unit disk of $\widetilde f$ for some cusp form $f$. \end{defn} The results presented here leave many open questions about the structure of these quantum modular forms. We conclude by giving a few natural questions that merit further investigation. \begin{questions} \begin{enumerate} \item If we replace the cusp form $f$ by a holomorphic modular form which is not cuspidal, then an inspection of the proof of Theorem \ref{mainthm} shows that the same calculations formally hold, but the Eichler integral has a mild singularity at cusps at which $f$ is not cuspidal. Is there a natural (and nontrivial) way to subtract a canonical holomorphic function which gives an associated quantum modular form? \item If $f$ is a unary theta function, then $\widetilde f$ is a partial theta function. Can one add a non-holomorphic completion term to the partial theta function which corrects the modularity transformations on the upper half plane as well as on ${\mathbb Q}$? More generally, for any cusp form $f$, are the modularity transformations proven in Theorem \ref{mainthm} a consequence of a more general non-holomorphic completion on the upper or lower half plane? \item What are the arithmetic properties of Eichler quantum modular forms? For example, inspired by the congruences proven in \cite{Andrews}, is there a general theory of congruences for the coefficients of $\widetilde{f}(1-q)$ (note the slight abuse of notation here) or for the coefficients of the asymptotic expansion of $\widetilde f$ as $\tau$ approaches a given root of unity? In particular, if $f$ is a theta function, can one show in a uniform manner whether or not there are infinitely many such congruences, or does there exist a ``Sturm-type'' theorem which gives a finite condition to verify congruences? \item What does the Shimura correspondence tell us about the structure of half-integral weight Eichler quantum modular forms, given that the theory of integral weight Eichler integrals is much simpler?
\end{enumerate} \end{questions}
\end{document} | arXiv |
Zimní škola z abstraktní analýzy
Sekce analýza
Hlavní stránka Zvaní Registrace Informace Cestovní informace Účastníci Historie
Gilles Godefroy, Institut de Mathématiques de Jussieu - Paris Rive Gauche, Paris, France
Series of 3 lectures: Lipschitz isomorphisms and uniform isomorphisms between Banach spaces
Non-linear geometry of Banach spaces is a relatively new but rapidly expanding field of research. One of its goals is to determine how much of the linear structure of a Banach spaces can be recovered from a knowledge of its metric or its uniform structure. In other words, if two Banach spaces are Lipschitz-isomorphic, or merely uniformly isomorphic, are they linearly isomorphic? Or at least, do they share some significant isomorphic properties? We will devote three lectures to this field, which will be organized as follows.
1. The free space of a metric space.
A simple and canonical construction permits to associate to every metric space $M$ a Banach space $\mathcal{F}(M)$ in such a way that Lipschitz maps between metric spaces become linear maps between their corresponding free spaces. We will display this construction and see how to use the free spaces. In particular, they will provide canonical examples of Lipschitz isomorphic (non separable!) Banach spaces which are not linearly isomorphic.
2. The Gorelik principle.
The Gorelik principle is a topological result which roughly asserts that a Lipschitz isomorphism from $X$ to $Y$ maps a large enough ball of any subspace of finite codimension of $X$ to a subset of $Y$ of «compact codimension». When properly used, this principle provides a substitute to the fact that Lipschitz isomorphisms cannot be transposed to isomorphisms between the dual spaces. We will see how to use it for showing that a space which is Lipschitz-isomorphic to $c_0$ is actually linearly isomorphic to $c_0$.
3. Approximation properties.
A proper use of free spaces show that Grothendieck's bounded approximation property (BAP) is stable under Lipschitz-Isomorphisms. We will further study the interplay between BAP and extension of Lispchitz maps. For instance, a canonical approach permits to relate the existence of a Banach spaces failing BAP and the existence of Banach-space valued Lipschitz functions on a subset of a metric space $M$ which do not admit Lipschitz extensions to $M$.
In all these lectures, related open questions will be recalled and commented. An effort will be made to avoid technicalities.
José Orihuela, Universidad de Murcia, Spain
Series of 3 lectures: Compactness, Optimality and Applications
One of the most important achievements in optimization in Banach space theory is the James' weak compactness theorem. It says that a weakly closed subset $A$ of a Banach space $E$ is weakly compact if, and only if, every linear form $x^*\in E^*$ attains its supremum over $A$ at some point of $A$. We propose a tour around it with three lectures in the natural framework of variational analysis. Of course, we shall bring related open questions in every one of them. We will concentrate on recent extensions of James' theorem. Among them we shall study the following one:
Theorem 1
Let $A$ be a closed, convex, bounded and not weakly compact subset of a Banach space $E$. Let us fix a convex and weakly compact subset $D$ of $E$, a functional $z^*_0 \in E^*$ and $\epsilon>0$. Then there is a linear form $x^*_0\in B_{p_W}(z^*_0, \epsilon)$, i.e. $$|x^*_0(d)-z^*_0(d)|<\epsilon$$ for all $d\in D$, which does not attain its supremum on $A$. Moreover, if $z^*_0(A)<0$ the same can be provided for the former non attaining linear form : $x^*_0(A)<0$ (one sided James' theorem).
In the first lecture we shall concentrate in the case of Banach spaces with $w^*$-sequentially compact dual unit ball. We shall present one-sided versions of the well known results by Bishop and Phelps, Simons, Fonf and Lindenstrauss, which play their job and go back to ideas of a joint work with B. Cascales and A. P'erez. (2017)
In the second lecture we shall provide techniques for a proof of Theorem 1 in arbitrary Banach spaces. Our approach comes from Ru'iz Gal'an and Simons and it goes back to the Pryce' undetermined function technique. We will show the strong connection of James theorem with variational principles and optimization theory. In order to do it, we will study unbounded versions of the former results. The first case should be the epigraph of a weakly lower semicontinuous function $$\alpha: E\longrightarrow (-\infty, +\infty],$$ where we shall see that $\partial \alpha(E)=E^*$ if, and only if, the level sets $\{\alpha \leq c\}$ are weakly compact (the Fenchel conjugate $\alpha^*$ should be finite for the "if" implication), which goes back to ideas of Ru'iz Gal'an, Simons, Calvert and Fitzpatrick we have collected in a joint work with M. Ru'iz Gal'an (2012). Moors deserves special mention here since he has recently obtained a closely related variational principle too.
In the third lecture we shall present some new applications. We will see that reflexive spaces are the natural frame to develop variational analysis and we will show a robust representation theorem for a {\it risk measure} $\rho: \LL^\infty\longrightarrow \R$ in natural dual pairs appearing in financial mathematics, both applications are based in a joint work with M. Ru'iz Gal'an. We shall study the Mackey topology $\tau(\LL^\infty, \LL^1)$ and a new characterization for risk measures $\rho$ verifying the Lebesgue dominated convergence theorem, as the expectated value does. The proof of Theorem \ref{theo} together with these applications have been obtained in joint work with F. Delbaen and T. Pennanen ( preprint 2018). We shall finish our programme with $\sigma(E^*,E)$ versions of the discussed results. Indeed, we shall look for conditions that provide $\sigma(E^*,E)$-closedness of norm closed convex (not necessarely bounded) subsets of a dual Banach space $E^*$. One-sided versions of classical Godefroy' results will be presented and new applications considered. Some of them comes from the same joint work with B. Cascales and A. P'erez (2017).
Todor Tsankov, Université Paris, France
Series of 3 lectures: Analysis on automorphism groups of countable structures
The classical theory of harmonic analysis and dynamical systems is usually restricted to actions of locally compact groups but in recent years, some of the theory has been extended to more general Polish groups. In this series of talks, I will concentrate on closed subgroups of S_infty, which is perhaps the best understood class of non-locally compact groups in this context, and in the study of which model-theoretic tools play an important role. I will explain some aspects of the theory concerning the unitary representations of some of those groups and their measure-preserving actions.
Anush Tserunyan, University of Illinois at Urbana-Champaign, USA
Series of 2-3 lectures: Countable Borel equivalence relations
An equivalence relation $E$ on a Polish space $X$ (e.g., $X := \mathbb{R}$, $L^1(\mathbb{R})$, $\mathbb{N}^\mathbb{N}$) is \emph{Borel} if it is a Borel subset of $X^2$, and it is \emph{countable} if each $E$-class is countable. Countable Borel equivalence relations (CBERs) naturally arise as orbit equivalence relations of Borel actions of countable groups (e.g., $\mathbb{Q} \curvearrowright \mathbb{R}$ by translation). From another, rather combinatorial, angle, a CBER $E$ can always be viewed as the connectedness relation of a locally countable Borel graph (e.g., take $E$ as the set of edges). These connections between equivalence relations, group actions, and graphs create an extremely fruitful interplay between descriptive set theory, ergodic theory, measured group theory, probability, descriptive graph combinatorics, and geometric group theory.
These lectures will feature this interplay. On one hand, we will learn some tools from each of the aforementioned subjects to analyze the structure of CBERs. On the other hand, we will utilize the basic theory of CBERs to prove some well-known results in those subjects | CommonCrawl |
The Annals of Mathematical Statistics
Ann. Math. Statist.
Volume 40, Number 4 (1969), 1499-1502.
Uniform Consistency of Some Estimates of a Density Function
D. S. Moore and E. G. Henrichon
More by D. S. Moore
More by E. G. Henrichon
Let $X_1, \cdots, X_n$ be independent random variables identically distributed with absolutely continuous distribution function $F$ and density function $f$. Loftsgaarden and Quesenberry [3] propose a consistent nonparametric point estimator $\hat{f}_n(z)$ of $f(z)$ which is quite easy to compute in practice. In this note we introduce a step-function approximation $f_n^\ast$ to $\hat{f}_n$, and show that both $\hat{f}_n$ and $\hat{f}_n^\ast$ converge uniformly (in probability) to $f$, assuming that $f$ is positive and uniformly continuous in $(-\infty, \infty)$. For more general $f$, uniform convergence over any compact interval where $f$ is positive and continuous follows. Uniform convergence is useful for estimation of the mode of $f$, for it follows from our theorem (see [4], section 3) that a mode of either $\hat{f}_n$ or $f_n^\ast$ is a consistent estimator of the mode of $f$. The mode of $f_n^\ast$ is particularly tractable; it is applied in [2] to some problems in pattern recognition. From the point of view of mode estimation, we thus obtain two new estimates which are similar in conception to those proposed by some previous authors. Let $k(n)$ be an appropriate sequence of numbers in each case. Chernoff [1] estimates the mode as the center of the interval of length $2k(n)$ containing the most observations. Venter [5] estimates the mode as the center (or endpoint) of the shortest interval containing $k(n)$ observations. The estimate based on $\hat{f}_n$ is that $z$ such that the distance from $z$ to the $k(n)$th closest observation is least. Finally, the estimate from $f_n^\ast$ is that observation such that the distance from it to the $k(n)$th closest observation is least.
Ann. Math. Statist., Volume 40, Number 4 (1969), 1499-1502.
https://projecteuclid.org/euclid.aoms/1177697524
doi:10.1214/aoms/1177697524
links.jstor.org
Moore, D. S.; Henrichon, E. G. Uniform Consistency of Some Estimates of a Density Function. Ann. Math. Statist. 40 (1969), no. 4, 1499--1502. doi:10.1214/aoms/1177697524. https://projecteuclid.org/euclid.aoms/1177697524
Correction: D. S. Moore, E. G. Henrichon. Correction to "Uniform Consistency of Some Estimates of a Density Function". Ann. Math. Statist., Vol. 41, Iss. 3 (1970),1126--1127.
Project Euclid: euclid.aoms/1177696997
The Institute of Mathematical Statistics
A Note on the Estimation of the Mode
Wegman, Edward J., The Annals of Mathematical Statistics, 1971
Consistency Properties of Nearest Neighbor Density Function Estimators
Moore, David S. and Yackel, James W., The Annals of Statistics, 1977
Consistency in Nonparametric Estimation of the Mode
Sager, Thomas W., The Annals of Statistics, 1975
Asymptotic Normality of Statistics Based on the Convex Minorants of Empirical Distribution Functions
Groeneboom, Piet and Pyke, Ronald, The Annals of Probability, 1983
A New Nonparametric Estimator of the Center of a Symmetric Distribution
Schuster, E. F. and Narvarte, J. A., The Annals of Statistics, 1973
Estimation of a Multivariate Mode
On the Last Time and the Number of Times an Estimator is More than $\varepsilon$ From its Target Value
Hjort, Nils Lid and Fenstad, Grete, The Annals of Statistics, 1992
Estimation of Probability Density by an Orthogonal Series
Schwartz, Stuart C., The Annals of Mathematical Statistics, 1967
Fredholm Determinant of a Positive Definite Kernel of a Special Type and Its Application
Sukhatme, Shashikala, The Annals of Mathematical Statistics, 1972
Interpolating Spline Methods for Density Estimation I. Equi-Spaced Knots
Wahba, Grace, The Annals of Statistics, 1975
euclid.aoms/1177697524 | CommonCrawl |
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Uncountable admissibles. I. Forcing
Author: Sy D. Friedman
Journal: Trans. Amer. Math. Soc. 270 (1982), 61-73
MSC: Primary 03D60; Secondary 03C70, 03E45
DOI: https://doi.org/10.1090/S0002-9947-1982-0642330-8
MathSciNet review: 642330
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Abstract | References | Similar Articles | Additional Information
Abstract: Assume $V = L$. Let $\kappa$ be a regular cardinal and for $X \subseteq \kappa$ let $\alpha (X)$ denote the least ordinal $\alpha$ such that ${L_\alpha }[X]$ is admissible. In this paper we characterize those ordinals of the form $\alpha (X)$ using forcing and fine structure of $L$ techniques. This generalizes a theorem of Sacks which deals with the case $\kappa = \omega$.
References [Enhancements On Off] (What's this?)
R. David, Some applications of Jensen's coding theorem (to appear).
Keith J. Devlin, Aspects of constructibility, Lecture Notes in Mathematics, Vol. 354, Springer-Verlag, Berlin-New York, 1973. MR 0376351
Sy D. Friedman, Steel forcing and Barwise compactness, Ann. Math. Logic 22 (1982), no. 1, 31–46. MR 661476, DOI https://doi.org/10.1016/0003-4843%2882%2990014-6
Sy D. Friedman, Uncountable admissibles. I. Forcing, Trans. Amer. Math. Soc. 270 (1982), no. 1, 61–73. MR 642330, DOI https://doi.org/10.1090/S0002-9947-1982-0642330-8
Harvey Friedman, One hundred and two problems in mathematical logic, J. Symbolic Logic 40 (1975), 113–129. MR 369018, DOI https://doi.org/10.2307/2271891
R. O. Gandy and G. E. Sacks, A minimal hyperdegree, Fund. Math. 61 (1967), 215–223. MR 225653, DOI https://doi.org/10.4064/fm-61-2-215-223
Leo Harrington, Contributions to recursion theory in higher types, Ph.D. Thesis, MIT. R. Jensen, Forcing over admissible sets, Notes by K. Devlin, unpublished.
R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229–308; erratum, ibid. 4 (1972), 443. With a section by Jack Silver. MR 309729, DOI https://doi.org/10.1016/0003-4843%2872%2990001-0
---, Coding the universe by a real, unpublished manuscript.
R. B. Jensen and R. M. Solovay, Some applications of almost disjoint sets, Mathematical Logic and Foundations of Set Theory (Proc. Internat. Colloq., Jerusalem, 1968) North-Holland, Amsterdam, 1970, pp. 84–104. MR 0289291
Gerald E. Sacks, The $1$-section of a type $n$ object, Generalized recursion theory (Proc. Sympos., Univ. Oslo, Oslo, 1972), North-Holland, Amsterdam, 1974, pp. 81–93. Studies in Logic and the Foundations of Math., Vol. 79. MR 0398811
Gerald E. Sacks, Countable admissible ordinals and hyperdegrees, Advances in Math. 20 (1976), no. 2, 213–262. MR 429523, DOI https://doi.org/10.1016/0001-8708%2876%2990187-0
John R. Steel, Forcing with tagged trees, Ann. Math. Logic 15 (1978), no. 1, 55–74. MR 511943, DOI https://doi.org/10.1016/0003-4843%2878%2990026-8
R. David, Some applications of Jensen's coding theorem (to appear). Keith J. Devlin, Aspects of constructibility, Lecture Notes in Math., vol. 354, Springer-Verlag, Berlin and New York. Sy D. Friedman, Steel forcing and Barwise compactness, Ann. Math. Logic (to appear). ---, Uncountable admissibles II: Compactness, Israel J. Math. (to appear). Harvey M. Friedman, One hundred and two problems in mathematical logic, J. Symbolic Logic 4. R. Gandy and G. Sacks, A minimal hyperdegree, Fund. Math. 61. Leo Harrington, Contributions to recursion theory in higher types, Ph.D. Thesis, MIT. R. Jensen, Forcing over admissible sets, Notes by K. Devlin, unpublished. ---, The fine structure of the constructible hierarchy, Ann. Math. Logic 4. ---, Coding the universe by a real, unpublished manuscript. R. Jensen and R. Solovay, Some applications of almost disjoint sets, Math. Logic and Foundations of Set Theory, (Bar-Hillel, Editor), North-Holland, Amsterdam. G. Sacks, The $1$-section of a type $n$ object, Generalized Recursion Theory, (Fenstad-Hinman, Editors), North-Holland, Amsterdam. ---, Countable admissible ordinals and hyperdegrees, Adv. in Math. 20. J. Steel, Forcing with tagged trees, Ann. Math. Logic 15.
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Supporting line
In geometry, a supporting line L of a curve C in the plane is a line that contains a point of C, but does not separate any two points of C.[1] In other words, C lies completely in one of the two closed half-planes defined by L and has at least one point on L.
Properties
There can be many supporting lines for a curve at a given point. When a tangent exists at a given point, then it is the unique supporting line at this point, if it does not separate the curve.
Generalizations
The notion of supporting line is also discussed for planar shapes. In this case a supporting line may be defined as a line which has common points with the boundary of the shape, but not with its interior.[2]
The notion of a supporting line to a planar curve or convex shape can be generalized to n dimension as a supporting hyperplane.
Critical support lines
If two bounded connected planar shapes have disjoint convex hulls that are separated by a positive distance, then they necessarily have exactly four common lines of support, the bitangents of the two convex hulls. Two of these lines of support separate the two shapes, and are called critical support lines.[2] Without the assumption of convexity, there may be more or fewer than four lines of support, even if the shapes themselves are disjoint. For instance, if one shape is an annulus that contains the other, then there are no common lines of support, while if each of two shapes consists of a pair of small disks at opposite corners of a square then there may be as many as 16 common lines of support.
References
1. "The geometry of geodesics", Herbert Busemann, p. 158
2. "Encyclopedia of Distances", by Michel M. Deza, Elena Deza, p. 179
| Wikipedia |
Sergey Kislitsyn
Sergey S. Kislitsyn, (Russian: Серге́й Серге́евич Кисли́цын) is a Russian mathematician, specializing in combinatorics and coding theory.
Kislitsyn was born January 5, 1935, in Ivanovo, Soviet Union. He received his M.S. in mathematics from Leningrad State University in 1957. From 1962 until 1970 he worked at Yekaterinburg branch of the Steklov Institute of Mathematics (Krasovsky Institute of Mathematics and Mechanics). He defended his Ph.D. thesis in 1964 and continued working as a lecturer at Krasnoyarsk State University.[1]
Kislitsyn is known for posing the 1/3–2/3 conjecture for linear extensions of finite posets, which he published in 1968.[2] The conjecture is established in several special cases but open in full generality.[3][4]
References
1. "Кислицын Сергей Сергеевич". Биобиблиографический Указатель Научных Трудов Сотрудников Института Математики и Механики УрО РАН до 1975 г [Bio-Bibliographic Index of Scientific Works of Employees of the Institute of Mathematics and Mechanics UB RAS before 1975] (PDF) (in Russian). Vol. 1. Yekaterinburg, Russia: Ural Branch of the Russian Academy of Sciences. 2010. pp. 87–89.
2. Kislitsyn, S. S. (1968). "A finite partially ordered set and its corresponding set of permutations". Mathematical Notes. 4 (5): 798–801. doi:10.1007/BF01111312. S2CID 120228193.
3. Olson, Emily J.; Sagan, Bruce E. (2018). "On the 1/3–2/3 conjecture". Order. 35 (3): 581–596. doi:10.1007/s11083-017-9450-3. MR 3861401. S2CID 52965439.
4. Brightwell, Graham (1999-04-28). "Balanced pairs in partial orders". Discrete Mathematics. 201 (1): 25–52. doi:10.1016/S0012-365X(98)00311-2. ISSN 0012-365X.
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Critical hydraulic gradients for seepage-induced failure of landslide dams
Austin Chukwueloka-Udechukwu Okeke1 &
Fawu Wang1
Geoenvironmental Disasters volume 3, Article number: 9 (2016) Cite this article
Landslide dams formed by rock avalanche processes usually fail by seepage erosion. This has been related to the complex sedimentological characteristics of rock avalanche dams which are mostly dominated by fragmented and pulverized materials. This paper presents a comprehensive experimental programme which evaluates the critical hydraulic and geometrical conditions for seepage-induced failure of landslide dams. The experiments were conducted in a flume tank specifically designed to monitor time-dependent transient changes in pore-water pressures within the unsaturated dam materials under steady-state seepage. Dam models of different geometries were built with either mixed or homogeneous materials. Two critical hydraulic gradients corresponding to the onset of seepage erosion initiation and collapse of the dam crest were determined for different upstream inflow rates, antecedent moisture contents, compactive efforts, grain size ranges, and dam geometries.
Two major types of dam failure were identified: Type I and Type II. These were further subdivided into minor failure processes which include exfiltration, sapping, downstream toe bifurcation, and undermining of the downstream face. The critical hydraulic gradients for seepage erosion initiation varied from 0.042 to 0.147. Experiments conducted with the mixed materials indicate that the critical hydraulic gradients for collapse of the dam crest increased with an increase in uniformity coefficient.
The deformation behaviour of the dams was significantly influenced by particle density, pore geometry, hydraulic conductivity, and the amount of gravel and pebbles present in the materials. The results indicate that the critical seepage velocity for failure of the dams decreased with an increase in downstream slope angle, but increased with an increase in pore geometry, dam height, dam crest width, upstream inflow rate, and antecedent moisture content.
Landslide dams and other natural river blockages such as moraine dams and glacier-ice dams are formed in narrow valleys bordered by oversteepened slopes. Active geological processes in these settings such as erosion and weathering often lead to the availability of highly fractured and hydrothermally altered bedrock which constitute source materials for hillslope processes and landslide dam formation (Costa and Schuster 1988; Clague and Evans 1994; Korup et al. 2010). These potentially dangerous natural phenomena occur mostly in seismically-active regions where high orographic precipitations on rugged mountain terrain associated with frequent earthquakes and snowmelt contribute to several geological processes that lead to mass wasting and river-damming landslides (Korup and Tweed 2007; Allen et al. 2011; Evans et al. 2011; Crosta et al. 2013). Failure of landslide dams could trigger the sudden release of stored water masses from lakes created by these damming events. This consequently produces catastrophic outburst floods and debris flows that inundate the downstream areas, causing loss of lives and infrastructural damage (O'Connor and Costa 2004; Bonnard 2011; Plaza et al. 2011). For example, the worst recorded case of landslide dam disaster occurred during the 1786 Kangding-Luding earthquake in Sichuan Province, southwest China (Dai et al. 2005). The earthquake triggered a huge landslide which dammed the Dadu River but failed ten days later and generated a catastrophic outburst flood that drowned more than 100,000 people. Similarly, Chai et al. (2000) presented a comprehensive account of the catastrophic failure of three landslide dams (Dahaizi, Xiaohaizi, and Deixi), triggered by the August 1933, Ms 7.5 earthquake in Diexi town, Sichuan Province, China. These landslide dams failed two months later, triggering catastrophic outburst floods that traveled more than 250 km downstream, and claimed about 2,423 lives. Therefore, timely evaluation of landslide dams is important for prevention of catastrophic dam failures and mitigation of disasters caused by downstream flooding of the released water masses.
Seepage erosion is one of the undermining factors affecting the stability and long-term performance of landslide dams and embankment dams. Many civil engineering and geoenvironmental studies have defined subsurface erosion processes by several terms such as piping, heave or blowout, seepage erosion, tunneling or jugging, internal erosion and sapping or spring sapping (Zasłavsky and Kassiff 1965; Jones 1981; Higgins 1982, 1984; Hutchinson 1982; Hagerty 1991; Wörman 1993; Terzaghi et al. 1996). However, a few researchers have made clear distinctions between the different processes involved in soil destabilization caused by seepage and piping (Jones 1981; Bryan and Yair 1982; Dunne 1990). The role of seepage in increasing positive pore-water pressure and causing apparent reduction of matric suction (u a -u w ) in unsaturated soils has been documented in the literature (Fredlund et al. 1978; Lam et al. 1987; Fredlund et al. 2012). Generally, landslide dams, stream banks and soil slopes are composed of unconsolidated materials which exist in unsaturated conditions. The stability of landslide dams in unsaturated conditions depends on the presence of matric suction which increases the shear strength of the soil τ, as described by the equation proposed by Fredlund et al. (1978):
$$ \tau =c\hbox{'}+\left({\sigma}_n-{u}_a\right) \tan \varphi \hbox{'}+\left({u}_a-{u}_w\right) \tan {\varphi}^b $$
where c' = effective cohesion of the soil, (σ n -u a ) = net normal stress on the failure plane, ϕ' = effective friction angle with respect to the net normal stress, (u a -u w ) = matric suction, ϕ b = angle that denotes the rate of increase in shear strength relative to matric suction. Transient changes from unsaturated to saturated conditions under steady-state seepage initiate high hydraulic gradients that accentuate subsequent reduction of apparent cohesion of the soil. This, in turn, increases seepage forces that accelerate soil mobilization, exfiltration and downstream entrainment of the eroded soil particles, as described by the equation:
$$ {F}_s={\gamma}_wi $$
where F s = seepage force per unit volume, i = hydraulic gradient, γ w = unit weight of water. Detailed research on seepage erosion processes in unsaturated soils and the effects of pore-water pressure on the stability of soil slopes have been carried out by Hutchinson (1982), Iverson and Major (1986), Howard and McLane (1988), Fredlund (1995), Skempton and Brogan (1994), Crosta and Prisco (1999), Rinaldi and Casagli (1999), Dapporto et al. (2001), Lobkovsky et al. (2004), Wilson et al. (2007), Fox et al. (2007), Cancienne et al. (2008), and Pagano et al. (2010).
The concept of hydraulic criteria for assessing the likelihood of initiation of internal erosion in soils is based on the hydraulic load acting on a soil particle which must exceed the drag forces of the seeping water. This is related to the critical hydraulic gradient i c , defined as the hydraulic gradient at which the effective stress of the soil becomes negligible. Apparently, a large number of theoretical and experimental approaches have been used to obtain critical hydraulic gradients in embankment dams, levees, dykes and other water-retaining structures. For example, Terzaghi (1943) obtained i c value of 1 for upward directed seepage flow as described by the following equation:
$$ {i}_c=\frac{\gamma \hbox{'}}{\gamma_w} $$
where γ' = submerged unit weight of soil, and γ w = unit weight of water. However, Skempton and Brogan (1994) observed selective erosion of fines in internally unstable cohesionless soils for upward flow conditions at critical hydraulic gradients (i c = 0.2 ~ 0.34) lower than that obtained from Terzaghi's classical approach. Similarly, Den Adel et al. (1988) carried out tests for horizontal seepage flow and obtained critical hydraulic gradient values of 0.16 to 0.17 and 0.7 for unstable and stable soils, respectively. Ahlinhan and Achmus (2010) performed experiments with unstable soils for upward and horizontal seepage flows and obtained critical hydraulic gradient values of 0.18 to 0.23. Ke and Takahashi (2012) obtained critical hydraulic gradients of 0.21 to 0.25 for internal erosion with binary mixtures of silica sands under one-dimensional upward seepage flow.
Whilst a lot of research has been done on critical hydraulic gradients for internal erosion, problems still exist in defining and ascribing limit values of hydraulic gradients for seepage erosion. For instance, Samani and Willardson (1981) proposed the hydraulic failure gradient i f , defined as the hydraulic gradient at which the shear strength of a confined soil is reduced by the drag forces of the seeping water. Wan and Fell (2004) introduced i start and i boil to represent critical hydraulic gradients for the onset of internal erosion and boiling, respectively. However, the conventional one-dimensional upward seepage tests can only be used to determine the hydraulic criteria for seepage erosion in granular materials with the exclusion of other factors such as dam geometry (dam height, dam crest width, upstream and downstream slope angles), and rate of inflow into the upstream reservoir. Hence, elaborate evaluation of the influence of these geometrical and hydraulic factors on seepage processes in landslide dams would require carrying out flume experiments where the characteristic deformation behaviour of the dam models would allow for accurate determination of the limit values of these hydraulic parameters.
Brief review of seepage erosion in soils
Comprehensive research on seepage erosion and piping mechanisms in landslide dams (Meyer et al. 1994; Davies and McSaveney 2011; Wang et al. 2013; Okeke and Wang 2016; Wang et al. in press), levees and earth embankments (Richards and Reddy 2007), hillslopes (Ghiassian and Ghareh 2008), and stream banks (Fox and Wilson 2010), have all been completed. Variations in experimental results and opinions are strictly based on the design and method of experiment adopted, coupled with size and scale effects arising from the nature of material tested.
Seepage erosion involves the detachment and entrainment of finer soil particles through a porous medium under a hydraulic gradient caused by the seeping water (Cedergren 1977). The various processes involved in seepage erosion mechanisms in hillslopes and landslide dams have been identified. For example, sapping, as defined by Hagerty (1991) involves exfiltration over a broad area on a sloping surface such that large lenticular cavities appear as a result of concentrated seepage which removes soil particles at the exit point and increases the diameter of the evolving channel over time. Iverson and Major (1986) derived a generalized analytical method for the evaluation of seepage forces considering static liquefaction and Coulomb failure under steady uniform seepage in any direction within a hillslope. They observed that slope destabilization occurred as a result of seepage force vector, which represents a body force that corresponds to the hydraulic gradient potential. They concluded that slope stability will invariably occur when the direction of the seepage flow is such that λ = 90°-ϕ, whereas the existence of a vertically upward seepage component results in Coulomb failure at similar conditions required for static liquefaction, especially when the slope angle is more or less equal to φ. Howard (1988) used flume experiments and numerical simulations to evaluate sapping processes and sapping zone morphology in homogeneous, isotropic sand mixtures. His experiments identified three distinct zones at the sapping face: mass wasting zone, sapping zone and fluvial transport zone, whereas numerical simulations performed by Howard and McLane (1988) revealed that the rate of mass wasting at the sapping face is dependent on the rate of sediment transport through the fluvial transport zone.
Perzlmaier et al. (2007) presented an overview of empirically-derived critical hydraulic gradients for initiation of backward erosion in a range of soil types based on field experience in several dams and levees (Table 1). Richards and Reddy (2010) evaluated piping potential in earth structures using a modified triaxial system, referred to as the true triaxial piping test apparatus (TTPTA). This apparatus was designed for controlling confining stresses and determining critical hydraulic gradients and critical velocities required for initiation of internal erosion. Their tests found that the critical hydraulic gradient and the critical seepage velocity for internal erosion in uniform fine-grained quartz sand varied from 1.8 × 10−3 to 2.4 × 10−3 and 8.1 × 10−3 to 1.1 × 10−3 m/s, respectively. They concluded that the critical seepage velocity is an essential parameter for evaluation of piping potentials in non-cohesive soils. Moffat et al. (2011) used a rigid wall permeameter to study internal erosion susceptibility in widely graded cohesionless soils by imposing a unidirectional flow in either upward or downward directions such that a constant average hydraulic gradient was maintained across the specimen. They found that suffusion occurred by 'episodic migration' of the finer fraction when the imposed average hydraulic gradient was increased. Chang and Zhang (2012) determined the critical hydraulic gradients for internal erosion under complex stress states using a computer-controlled triaxial testing apparatus which allowed for independent control of hydraulic gradient and stress states. They found that under isotropic stress states, the initiation hydraulic gradient i start increased with an increase in effective mean stress. They further observed that under the same confining stress, the initiation gradients obtained under compression stress states were higher than those obtained under extension stress states. These findings may have cleared up some of the ambiguities associated with critical hydraulic gradients determined under one-dimensional seepage tests as noted by Fell and Fry (2013), due to the inability of the conventional method to monitor stress states of soils.
Table 1 Comparison of empirically-derived critical average gradients i c for initiation of backward erosion and piping in different soil types (Perzlmaier et al. 2007)
However, despite the wealth of research done so far, not much has been reported on the influence of geometrical and hydraulic conditions for seepage erosion development in landslide dams. This paper presents a comprehensive experimental programme conducted to investigate transient pore-water pressure variations and the critical hydraulic gradients for seepage-induced failure of landslide dams. A series of experiments were conducted in a flume tank modified to accurately determine the limit values of hydraulic gradients at the various stages of the dam failure process. This is in contrast to the conventional one-dimensional upward directed seepage tests performed in a modified triaxial chamber. The main objectives of this research are summarized as follows: (1) to determine the critical hydraulic gradients required for initiation i ini and failure i f of landslide dams under different geometrical and hydraulic conditions, as well as the critical seepage velocities for erosion and debris flow mobilization; (2) to investigate the effects of pore-water pressure during seepage processes and its role in initiating seepage erosion and dam failure; and (3) to identify the various failure mechanisms of landslide dams under steady-state seepage conditions.
The experiments were conducted in a rectangular flume tank 2 m long, 0.45 m wide and 0.45 m high. The flume tank was made of 5 mm-thick acrylic sheets (plexiglass) of high transparency which enables visual observation of wetting front propagation, deformation and failure mechanism of the dam models. The flume was tilted to make a bed slope of ψ = 5°. The downstream end of the flume was equipped with two 4 cm-diameter holes for outflow of fluidized sediments. The water entering the upstream reservoir was provided by a rubber hose attached to a water tap while discharge into the upstream reservoir was controlled by a flowmeter connected to the drainage hose. The generation and dissipation of pore-water pressures during the experiments were monitored with three pore-water pressure sensors, hereafter referred to as p1, p2, and p3, with rated capacity of 50 kPa each (Fig. 1a). The sensors were fixed underneath the center of the flume bed through three 10 mm-diameter holes drilled on a horizontal line at the center of the flume bed. The sensors were separated by horizontal distances of 0.1 m and 0.103 m, respectively. Each of the pore-water pressure sensors was equipped with an L-shaped manometer attached to the outer wall of the flume to ensure an equal balance between the fluid pressure and atmospheric pressure. Transient variation in upstream reservoir level was monitored with a water level probe positioned near the toe of the upstream slope. Deformations and settlements caused by seepage and pore-water pressure buildup were monitored with two 0.1 m-range CMOS multi-function analog laser displacement sensors attached to a wooden overboard (Fig. 1b). The two sensors, hereafter referred to as H d1 and H d2, were separated by a distance of 0.04 m.
a Experimental setup. H d Laser displacement sensors; Ups Upstream water level probe; p1, p2, and p3 Pore-water pressure sensors. b Side view of the flume tank before the commencement of an experiment
Soil characteristics
A series of experiments were conducted using different soils and testing conditions. Table 2 shows a summary of all the experiments conducted under different testing conditions while the results of the critical pore-water pressures and critical seepage velocities obtained from the tests are summarized in Table 3. Uniform commercial silica sand no. 8 was used to build the dam models, except in Exp 1 to 3 where the dam models were composed of different proportions of silica sand nos. 5 and 8, including industrial pebbles and gravel, hereafter referred to as sandfill dam (SD), gravelly dam I (GV-I), and gravelly dam II (GV-II), respectively. The grain size distribution curves of all the materials used are shown in Fig. 2. The mechanical and hydraulic characteristics of the materials used in the experiments are summarized in Table 4. Silica sand nos. 5 and 8 are generally composed of subangular to angular grains with dry repose angles of 32 and 35°, respectively. Constant-head permeability tests and other soil property tests were carried out on the soils based on the physical conditions (bulk density and antecedent moisture content) used in building the dam models in accordance with standards of the Japanese Geotechnical Society (JGS).
Table 2 Summary of all the experiments at different testing conditions
Table 3 Summary of results of critical pore-water pressures and critical seepage velocities obtained from the tests
Grain size distribution curves of the dam materials. GV-I Gravelly dam I, GV-II Gravelly dam II, SD Sandfill dam, SS-8 Silica sand no. 8
Table 4 Mechanical and hydraulic characteristics of the materials used in the experiments
Landslide dam model construction and experimental procedure
Landslide dam models of different geometries were built approximately 0.4 m downslope from the upstream water inlet (Fig. 3a). Effort was made in building the dam models so as to simulate naturally existing landslide dam prototypes. Mechanically mixed soils were placed in the flume tank in equal lifts using the moist tamping method. Initially, oven-dried soils were mixed with a known volume of water and then compacted to obtain the desired moisture content and bulk density. All the experiments were conducted with an antecedent moisture content of 5 %, except in Exp 8 to 15 where the antecedent moisture content was varied from 5 to 20 %. The geometrical characteristics of the dam models are shown in Fig. 3b. The dam height H d and the dam crest width D crw were varied from 0.15 to 0.3 m and 0.1 to 0.25 m, respectively. The angles α and β representing the upstream and downstream slope angles were varied from 35 to 40° and 30 to 60°, respectively.
a Plan view of the flume tank indicating the position of the dam model and monitoring sensors. b Schematic diagram of the dam geometry (not to scale)
Seven different series of experiments, all summing up to 27 runs of tests, were carried out, each with intent to assess transient pore-water pressure variations and the critical hydraulic gradients for seepage erosion initiation and dam failure under steady-state seepage. The main experiments were conducted after carrying out a series of initial tests which were mostly done to check sensor reliability, result validation, test repeatability and selection of appropriate mixtures of materials. However, the results of experiments conducted on dams built with dam crest width D crw of 0.1 and 0.15 m are excluded in this paper due to some challenges posed by the monitoring sensors. The initial conditions set for all the tests assumed that the upstream reservoir was empty. Filling of the upstream reservoir was carried out with a rubber hose attached to a water tap, and connected to a manually-operated flowmeter. A steady-state seepage through the dam models was achieved by ensuring that the upstream reservoir level remained constant at approximately two-thirds of the dam height. Real-time data was acquired by connecting all the sensors to a standard high-speed monitoring and recording workstation comprised of two synchronized universal recorders (PCD-330B-F) and a laptop computer. Sampling frequency was set at 50 Hz for all the tests.
At the beginning of each experiment, discharge into the upstream reservoir was set at the desired value using a manually-operated flowmeter. The discharge was maintained until the upstream reservoir level equaled two-thirds of the dam height. Afterward, an equilibrium hydraulic head was established by ensuring that the upstream reservoir level remained constant prior to the collapse of the dam crest. The change from unsaturated to saturated state began during the filling of the upstream reservoir. Consequently, loss of matric suction due to positive pore-water pressure buildup under steady-state seepage, as observed from sensor p3 (Fig. 4), marked the onset of static liquefaction and exfiltration of water from the downstream toe, which further led to debris flow mobilization and dam failure.
Schematic diagram for the determination of hydraulic gradients
Determination of critical hydraulic gradients
Variations in hydraulic gradients (i 1 and i 2) through the dam models were determined from pore-water pressure values obtained from the experiments. Darcy (1856) in Fredlund et al. (2012) postulated that the rate of water flow through a soil mass was equal to the hydraulic gradient, as described by the equation:
$$ {v}_w=-{k}_w\frac{\partial {h}_w}{\partial z} $$
where v w = flow rate of water (m3/s), k w = coefficient of permeability with respect to the water phase (m/s), ∂h w /∂z = hydraulic gradient in the z-direction. Hydraulic heads, h 1, h 2, and h 3 at three different locations within the dam models were computed from pore-water pressure values using the following equation (Fig. 4):
$$ h=\frac{u_w}{\gamma_w{ \cos}^2\psi } $$
where u w = pore-water pressure (kPa), γ w = unit weight of water (kN/m3), ψ = flume bed slope angle (degree). Therefore, hydraulic gradient i 1 between sensors p1 and p2 was determined as described by the equation below:
$$ {i}_1\approx \frac{-\left[\left({h}_2+{h}_{02}\right)-\left({h}_1+{h}_{01}\right)\right]}{\raisebox{1ex}{${L}_1$}\!\left/ \!\raisebox{-1ex}{$ \cos \psi $}\right.} $$
Similarly, the hydraulic gradient, i 2 between sensors p2 and p3 was determined as follows:
where h 01, h 02 and h 03 represent corresponding vertical distances between the flat firm base and the slope bed, whereas L 1 and L 2 are horizontal distances between p1 and p2, and p2 and p3, respectively. Two limit values of hydraulic gradients, corresponding to the onset of initiation of seepage erosion i ini and collapse of the dam crest i f , were determined based on results obtained from the initial tests.
General characteristics of the experiments
Two characteristic types of dam failure (Type I and Type II) were observed during the experiments and were found to depend on the geometry and hydromechanical characteristics of the dam materials. These were further subdivided into several interrelated failure processes which included wetting front propagation, downstream slope saturation, exfiltration, sapping/seepage-face erosion, toe bifurcation, undermining and progressive sloughing of the downstream face, and late-stage overtopping.
Type I involves failures which could be related to static liquefaction of the soil mass under steady-state seepage that reduced the apparent cohesion of the soil and led to debris flow mobilization. This type of failure was primarily initiated by sapping erosion which occurred as a result of steady exfiltration of water from the downstream toe; which by extension, triggered gradual undercutting and debuttressing of the downstream slope as the mobilized mass 'flowed' downstream, thus lowering the dam height (Fig. 5a). Dam failure occurred by overtopping as the upstream reservoir level reached the tip of the partially saturated dam material, eroding the entire crest to form a wide breach channel. This type of failure was characteristic of experiments conducted with low upstream inflow rates, low compactive effort (e o = 1.76), high downstream slope angle (β ≥ 40°), and dam crest width greater than 0.15 m.
Typical failure mechanisms of the dams. a Type I - Upslope propagation of wetting front, exfiltration, sapping and sloughing of the fluidized soil mass. b Type II - Downslope propagation of wetting front, bifurcation, and undermining of the slope toe
Type II involves failures triggered by downslope propagation of the wetting front and subsequent mobilization of the fluidized material at the upper part of the downstream face. This failure mechanism was characterized by downstream toe bifurcation and abrupt collapse of a large flank of the slope due to intense saturation which originated from the dam crest and progressed towards the downstream toe (Fig. 5b). Dam failure occurred by the formation of a hydraulic crack aligned perpendicular to the downstream face due to the reduction of the effective stress of the soil. This type of failure occurred mostly in dams of low downstream slope angle (β ≥ 40°), high shear strength of the soil relative to the shear stress of the seeping water, and high compactive effort (e o = 1.21).
Influence of dam composition
Three types of materials (SD, GV-I, and GV-II) were used to investigate transient changes in pore-water pressures and variations in hydraulic gradients under steady-state seepage through the dam models (Exp 1 ~ 3; Table 2 and 3). The dam models were built to obtain initial void ratios of 1.41, 0.71 and 0.84 for SD, GV-I, and GV-II materials, respectively. The resulting trends of pore-water pressures within the dam models indicate gross anisotropy and heterogeneity in dams composed of GV-I and GV-II, whereas the low critical pore-water pressures obtained in the dam built with homogeneous SD material demonstrates the liquefaction potential of cohesionless and isotropic sands (Fig. 6). The failure mechanism of the SD dam was basically characteristic of the Type I failure pattern. Enlargement of the sapping zone was characterized by occasional mass failures which were enhanced by a decrease in the effective stress of the soil as the energy of the exfiltrating water increased. In contrast, GV-I material showed Type II failure mechanism, whereas the failure mechanism of GV-II material evolved from Type II to Type I (Fig. 7). Critical pore-water pressure values (p crit-1) determined at p1, which correspond to the onset of failure of the dams were 1.30, 1.64 and 1.45 kPa for SD, GV-I, and GV-II, respectively. The observed trends of pore-water pressures within the dams were found to be inversely proportional to the initial void ratio e o (Table 3), and directly proportional to the coefficient of uniformity C u of the dam materials (Table 4). This could be potentially caused by capillary rise within the materials which depends on the grain size distribution and bulk density of the constituent soil mass that, in itself, affected the porosity of the soil. Thus, the stability and deformation characteristics of the dams increased as the grain size distribution changed from poorly to well graded. Similarly, the critical hydraulic gradients for seepage erosion initiation i ini , increased with a decrease in pore size, while the critical hydraulic gradient for collapse of the dam crest i f , was influenced by the grain size distribution. The effect of grain size distribution on the development of seepage in the dams was evidenced by the variations in seepage velocity as the dynamics of the seeping water changed from laminar flow to turbulent flow (Table 3; Additional file 1: Video S1). The fact that the longevity of the dam built with GV-II material (v crit-2, 1.21 × 10−6 m/s) was higher than those built with SD and GV-I materials (v crit-2, 5.68 × 10−6 and 5.39 × 10−6 m/s, respectively) demonstrates that other physical parameters such as particle density, hydraulic conductivity and gravel content affect seepage development in landslide dams and soil slopes (Kokusho and Fujikura 2008).
Time-dependent transient changes in pore-water pressures and trends of hydraulic gradients in dams built with (a) Sandfill dam (b) Gravelly dam I and (c) Gravelly dam II
Images of seepage-induced failure of dams built with (a) Gravelly dam I and (b) Gravelly dam II
Rate of inflow into the upstream reservoir
Exp 4~7 were conducted to evaluate the influence of inflow rate Q in into the upstream reservoir. The dam models were built with uniform geometrical and physical characteristics (Table 2). Figure 8 shows the variations in pore-water pressures through the dams at steady-state inflow rates of 1.67 × 10−5 m3/s, 5 × 10−5 m3/s, 1 × 10−4 m3/s, and 1.67 × 10−4 m3/s. The filling rate of the upstream reservoir initiated seepage processes that changed the dynamics of the pore-water pressures. The critical hydraulic gradients for initiation of seepage erosion (i ini-1 and i ini-2) varied from 0.067 to 0.122. A low p crit-1 value of 1.52 kPa was determined in the experiment conducted with Q in of 1.67 × 10−4 m3/s, relative to Q in of 5 × 10−5 m3/s (p crit-1 = 1.65 kPa) and 1 × 10−4 m3/s (p crit-1 = 1.68 kPa) (Table 3). This could be attributed to a rapid increase in the hydraulic head which initiated high seepage gradients that reduced the effective stress of the soil, leading to differential settlement, hydraulic cracking, and lowering of the dam crest. Thus, the rate of reduction of the shear strength of the soil due to a decrease in matric suction depends on the rate of inflow into the upstream reservoir Q in and the rate of propagation of the wetting front. Trends of hydraulic gradients through the dams indicate that i f1 decreased with an increase in Q in , whereas i f2 increased with an increase in Q in , suggesting a corresponding increase in seepage velocity between sensors p1 and p2 (Table 2; Fig. 13 in Appendix 1). Critical seepage velocities determined from the tests show that v crit-2 increased from 7.39 × 10−7 m/s for Q in of 1.67 × 10−5 m3/s to 1.01 × 10−6 m/s for Q in of 1.67 × 10−4 m3/s. Exfiltration, sapping and undercutting of the downstream toe, characteristic of Type I failure mechanism, occurred at low inflow rates as a result of low seepage processes that led to liquefaction and collapse of the dam crest (Exp 4 and 5). In contrast, hydraulic cracking, downstream face saturation, and toe bifurcation characteristic of Type II failure mechanism, occurred in experiments conducted with high inflow rates (Exp 6 and 7). The experimental results demonstrate that the stability and time of collapse of the dam crest T b decreased with an increase in inflow rate into the upstream reservoir. This was evidenced by the characteristic failure mechanism of the dam models which evolved from Type I to Type II with a corresponding increase in Q in (Table 3).
Transient variations in pore-water pressures in experiments conducted with upstream inflow rates of (a) 1.67 × 10−5 m3/s (b) 5 × 10−5 m3/s (c) 1 × 10−4 m3/s (d) 1.67 × 10−4 m3/s
Influence of material condition
Soil wetting is a major cause of shear strength reduction and volume change in unsaturated soils and is a common occurring factor in collapsible soils and expansive soils. Exp 8~11 were conducted to assess the influence of antecedent moisture content w on the deformation behaviour of landslide dams under steady-state seepage. Antecedent moisture contents of the soils were increased by 5 % during soil preparation and dam model construction. Figure 9 shows the resulting trends of hydraulic gradients through the dams. A linear relationship was observed between the antecedent moisture content and the rate of deformation and collapse of the dam models (Fredlund 1999). It is noteworthy to mention that the critical hydraulic gradients (i f1 and i f2) coincided with the onset of dam deformation and crest settlement. Measured critical hydraulic gradients for seepage erosion initiation varied from 0.053 to 0.118, while the critical hydraulic gradient for failure of the dams increased with an increase in antecedent moisture content. Similarly, the reduction of capillary forces due to an increase in soil moisture content caused the critical seepage velocity to decrease from 1.31 × 10−6 m/s for w = 5 % to 9.52 × 10−7 m/s for w = 20 %. The failure mechanism of the dams evolved from Type II to Type I as antecedent moisture content increased through the dams. The rate of exfiltration and sapping erosion at the downstream toe increased from low saturated soils to highly saturated soils. This was attributed to the reduction of matric suction caused by wetting resulting in high void ratios that accentuated the abrupt collapse of the dams.
Trends of hydraulic gradients in dams built with an e o of 1.76 and antecedent moisture contents of (a) 5 % (b) 10 % (c) 15 % (d) 20 %
Figure 10 shows trends of hydraulic gradients and the failure mechanism of dam models built with the same antecedent moisture contents (5, 10, 15 and 20 %), but packed at a higher compactive effort, e o = 1.21 (Exp 12~15). The characteristic trends displayed by the hydraulic gradients, as well as the low critical seepage velocities determined from the experiments indicate, that the initial void ratio e o of the soil affected the failure mechanism of the dams. It may be important to note that i f1 and i f2 increased with an increase in antecedent moisture content, thus suggesting that the dynamics of the seeping water were mainly characteristic of a laminar flow. The stability of the dam models increased as antecedent moisture content decreased from 20 to 5 %, as observed from T b and p crit-3, and thus indicates the effect of pore-water pressures in reducing the effective stress of the soil (Tables 2 and 3). This effect can be related to the influence of matric suction on the liquefaction potential and shear strength reduction in partially saturated soils (Simon and Collison 2001; Okamura and Soga 2006). Comparison between Exp 8 ~ 11 and Exp 12 ~ 15 shows that the deformation and collapse mechanism of the dam models were more pronounced in dams with an e o of 1.76 (Exp 8 ~ 11) than in those with an e o of 1.21 (Exp 12 ~ 15) (Figs. 16 and 17 in Appendix 2). Similarly, a comparison between the critical hydraulic gradients measurements in Exp 8 ~ 11 and Exp 12 ~ 15 shows that the critical hydraulic gradient decreased with a decrease in initial void ratio. The observed trends of wetting front propagation and the transient changes in pore-water pressures suggest that seepage flow through the dam materials was not essentially controlled by matric suction but by a hydraulic head gradient (Fredlund and Rahardjo 1993).
Influence of dam geometry
The geometry of landslide dams is one of the major factors contributing to seepage erosion and slope instability. The two major factors that control the critical hydraulic gradient for instability in soil slopes are the downstream slope angles β and the gradient of the soil layer ψ (Iverson and Major 1986; Budhu and Gobin 1996). Basically, the internal friction angle of a dry cohesionless soil, at zero external pressure, is equal to the maximum stable slope angle of the soil. However, the soil mass collapses to a lower slope angle if steady-state seepage occurs. A series of experiments were conducted to evaluate the effects of downstream slope angle β on the critical hydraulic gradients for failure of landslide dams (Exp 16~19). The downstream slope angles were varied from 30 to 60°. A close examination of the results indicates that the stability of the dams increased as the downstream slope angle decreased from 60 to 30° (Table 2). The time of collapse of the dam crest increased from β = 60° (T b , 900 s) to β = 30° (T b , 2300 s). Similarly, i f1 increased with an increase in β, whereas i f2 decreased with an increase in β (Fig. 14 in Appendix 1). Also, the critical seepage velocity decreased with an increase in β, indicating high failure potentials in dams of high downstream slope angles (Table 3). The variations in pore-water pressures and the failure mechanism of the dams are shown in Fig. 18 (Appendix 2). The failure mechanism of the dams built with β in the range of 30 to 40° was initiated by the bifurcation of the downstream toe (Type II), whereas exfiltration, sapping and undermining of the downstream toe were characteristic of dams with β in the range of 41 to 60° (Type I). Budhu and Gobin 1996 remarked that for a soil which has ϕ of 30°, the exit hydraulic gradient at the slope face increases from 1 (when λ = β) to a limit value of sin β (when λ = 90°).
The influence of dam height on the stability and longevity of landslide dams under steady-state seepage was evaluated in dams built with different dam heights H d , ranging from 0.15 m to 0.3 m (Tables 2 and 3). The experiments were conducted at a constant upstream inflow rate of 1.2 × 10−4 m3/s (Exp 20~23). A positive correlation was observed between the critical hydraulic gradients for dam failure (i f1 and i f2) and the dam height. The values of i f1 and i f2 increased from 1.17 and 0.55 for H d = 0.15 m, to 1.35 and 0.85 for H d = 0.30 m (Table 2; Fig. 15 in Appendix 1). Critical pore-water pressure values correlating with the onset of failure of the dams increased from 1.13 kPa (H d = 0.15 m) to 1.72 kPa (H d = 0.30 m) (Fig. 19 in Appendix 2). The results show that at constant α and β, the stability of the dams increased with a decrease in dam height H d . This was further evidenced by the failure mechanism of the dams which evolved from Type I for H d = 0.15 m to Type II for H d = 0.30 m. The results indicate that the height of landslide dams is an important parameter for assessing the stability of natural river blockages.
Exp 24~25 were conducted to evaluate the influence of dam crest width D crw on the failure mechanism of landslide dams. A steady-state seepage was maintained at a constant upstream inflow of 1.67 × 10−4 m3/s. The results of transient variations in pore-water pressures and the corresponding trends of hydraulic gradients in the dams built with D crw of 0.20 m and 0.25 m (Exp 24 and 25) are shown in Fig. 11. The critical hydraulic gradients for seepage erosion initiation (i ini-1 and i ini-2), varied from 0.081 to 0.118. Exfiltration, sapping and debuttressing of the downstream toe, characteristic of Type I failure pattern, were the major failure mechanisms of the dams (Fig. 12). The rate of propagation of the wetting front through the dams was strongly influenced by D crw /H d . High D crw /H d resulted in high values of i f1, i f2, and v crit . The continual propagation of wetting front through the dams resulted in a gradual reduction of the effective stress of the soil, and subsequent mobilization of the liquefied mass which travelled downstream with an initial speed of 1.2 × 10−5 m/s. The episodic occurrence of hydraulic cracks and undermining and sloughing of the fluidized slope mass continued until the dam breached by overtopping. The results demonstrate that at constant hydraulic and geometrical conditions (H d , α and β), i f1 and i f2, as well as v crit , increased with an increase in D crw , indicating that the critical seepage velocity and the critical hydraulic gradient for seepage erosion in landslide dams are influenced by dam crest width D crw and D crw /H d .
Evolution of pore-water pressures and hydraulic gradients in dams built with dam crest widths of (a) 0.20 m (b) 0.25 m
Exfiltration, sapping and downstream toe debuttressing under steady-state seepage in dams built with dam crest widths of (a) 0.20 m (b) 0.25 m
An extensive experimental programme was carried out to investigate the effects of transient variations in pore-water pressures and the critical hydraulic gradients for seepage-induced failure of landslide dams using a flume tank specifically designed for accurate determination of these hydraulic parameters. A steady-state seepage was maintained by ensuring that the upstream reservoir level remained constant prior to the collapse of the dam crest. Limit values of hydraulic gradients and seepage velocities were determined for different hydromechanical and geometrical conditions. Based on the experimental results, the following conclusions can be drawn:
Sapping was the most dominant mechanism of slope destabilization observed in all the experiments. Other significant interrelated failure processes of the dam models included wetting front propagation, downstream face saturation, exfiltration, hydraulic cracking, toe bifurcation, downstream slope undercutting, sloughing and late-stage overtopping.
Two characteristic types of failure, which depend on the geometrical and hydromechanical properties of the dams were observed: Type I and Type II. Type I commonly occurred in dams built with low compactive effort (e o = 1.76), high downstream slope angle (β ≥ 40°), crest width greater than 0.15 m, and moisture content lower than 15 %. This type of failure was initiated by exfiltration, sapping, and upslope propagation of the wetting front towards the dry upper region of the dam crest. Type I failure mechanism shares similar characteristics to the three distinct zones of slope deformation triggered by sapping, which are: fluvial, sapping and undermining zones, as reported by Howard and McLane (1988). In contrast, Type II was found in dams of low downstream slope angle (β ≥ 40°), dam height greater than 0.25 m, high upstream inflow rates and high compactive effort (e o = 1.21). Failure in these dams was triggered by downslope propagation of the wetting front, bifurcation of the damp lowermost part of the downstream toe, sapping erosion and sloughing of the fluidized slope material.
The build-up of positive pore-water pressure under steady-state seepage and its effects on the apparent cohesion of the soil were evaluated for different upstream inflow rates and antecedent moisture contents. The results indicated that the stability and longevity of the dam models increased with a decrease in upstream inflow rate and antecedent moisture content. Thus, demonstrating the significance of pore geometry, particle density, gradation, and hydraulic conductivity of materials forming landslide dams in the development of seepage processes.
In all the experiments, the critical hydraulic gradients for seepage erosion initiation (i ini-1 and i ini-2) ranged from 0.042 to 0.147. The critical hydraulic gradient for collapse of the dam crest i f was strongly influenced by several factors, such as the initial void ratio (compactive effort), antecedent moisture content, particle density, grain size distribution, inflow rate into the upstream reservoir and the geometrical characteristics of the dams.
In the dams built with mixed materials, i f1 and i f2 increased with an increase in uniformity coefficient. The critical hydraulic gradient for collapse of the dam crest i f increased with an increase in inflow rate into the upstream reservoir (filling rate). Similarly, i f1 and i f2 were controlled by the combined effects of antecedent moisture content and porosity of the soil. At low void ratios, i f1 decreased with an increase in antecedent moisture content, whereas i f2 increased as antecedent moisture content increased through the dams. However, at high void ratios, under the same antecedent moisture contents, i f1 and i f2 increased with an increase in antecedent moisture content, suggesting seepage flow dynamics typical of laminar flows.
Furthermore, both i f1 and i f2 increased with an increase in H d and D crw , whereas i f1 increased with an increase in β, and i f2 decreased as β increased. This indicates that the critical hydraulic gradient for dam failure for near-horizontal flow (Ψ = 5°), depends on β.
These experiments demonstrate that seepage mechanisms in landslide dams comprised of unsaturated homogeneous and isotropic cohesionless materials are influenced by the hydraulic properties of the materials, as well as the geometrical characteristics of the dams.
The textural characteristics of the materials used in these experiments are typical of landslide dams formed by rock avalanche processes where fragmentation and pulverization of the rock materials cause seepage processes to develop in the upper blocky carapace layer.
However, further research should be done considering a wide range of sediment sizes and the addition of commercially available kaolinite clay to evaluate the mechanism of shear strength reduction under steady-state seepage. It is believed that performing unsaturated seepage analysis and limit equilibrium analysis, with regards to the results and conditions set for these experiments, could give further insights into the critical conditions for stability of landslide dams under steady-state seepage.
C c = coefficient of curvature
C u = coefficient of uniformity
D crw (m) = dam crest width
D 50 (mm) = median grain size
e o = initial void ratio
F s (kN/m3) = seepage force per unit volume
H d (m) = height of the dam
i 1 = hydraulic gradient (between sensors p1 and p2)
i ini-1 = critical hydraulic gradient for seepage erosion initiation (between sensors p1 and p2)
i f1 = critical hydraulic gradient for collapse of the dam crest (between sensors p1 and p2)
K (m/s) = coefficient of permeability
p crit-1 (kPa) = critical pore-water pressure for collapse of the dam crest at p1
Q in (m3/s) = inflow rate into the upstream reservoir
T b (s) = time of collapse of the dam crest
u w (kPa) = pore-water pressure
V crit-1 (m/s) = critical seepage velocity determined at p1
w (%) = antecedent moisture content
α (degree) = upstream slope angle
β (degree) = downstream slope angle
γ' (kN/m3) = submerged unit weight of soil
w (kN/m3) = unit weight of water
λ = seepage direction
\( \rho \) dry (Mg/m3) = dry bulk density
ϕ (degree) = internal friction angle
ψ (degree) = flume bed slope angle
Ahlinhan, M.F, and M. Achmus. 2010. Experimental investigation of critical hydraulic gradients for unstable soils. In Proceedings of the Fifth International Conference on Scour and Erosion, San Francisco, California, ed. S.E. Burns, S.K. Bhatia, C.M.C. Avila, and B.E. Hunt, 599–608. doi:10.1061/41147(392)58.
Allen, S.K., S.C. Cox, and I.F. Owens. 2011. Rock avalanches and other landslides in the central Southern Alps of New Zealand: a regional study considering possible climate change impacts. Landslides 8(1): 33–48.
Bligh, W.G. 1910. Dams, barrages and weirs on porous foundations. Engineering News 64(26): 708–710.
Bonnard, C. 2011. Technical and human aspects of historic rockslide-dammed lakes and landslide dam breaches. In Natural and artificial rockslide dams, ed. S.G. Evans, R.L. Hermanns, A. Strom, and G. Scarascia-Mugnozza, 101–122. Berlin Heidelberg: Springer.
Bryan, R.B., and A. Yair. 1982. Badland Geomorphology and Piping, 408. Norwich: Geobooks.
Budhu, M., and R. Gobin. 1996. Slope instability from ground-water seepage. Journal of hydraulic Engineering 122(7):415–417.
Cancienne, R.M., G.A. Fox, and A. Simon. 2008. Influence of seepage undercutting on the stability of root-reinforced streambanks. Earth Surface Processes and Landforms 33(11): 1769–1786.
Cedergren, H.R. 1977. Seepage, drainage, and flow nets (Vol. 16), 1–534. New York: Wiley.
Chai, H.J., H.C. Liu, Z.Y. Zhang, and Z.W. Xu. 2000. The distribution, causes and effects of damming landslides in China. Journal of Chengdu University of Technology 27: 302–307.
Chang, D.S., and L.M. Zhang. 2012. Critical hydraulic gradients of internal erosion under complex stress states. Journal of Geotechnical and Geoenvironmental Engineering 139(9): 1454–1467.
Chugaev, R.R. 1962. Gründungsumriss von Wasserbauwerken (in Russian).. Moskau – Leningrad.
Clague, J.J., and S.G. Evans. 1994. Formation and failure of natural dams in the Canadian Cordillera, Geological Survey of Canada Bulletin, 464.
Costa, J.E., and R.L. Schuster. 1988. The formation and failure of natural dams. Geological Society of America Bulletin 100(7): 1054–1068.
Crosta, G., and C.D. Prisco. 1999. On slope instability induced by seepage erosion. Canadian Geotechnical Journal 36(6): 1056–1073.
Crosta, G.B., P. Frattini, and F. Agliardi. 2013. Deep seated gravitational slope deformations in the European Alps. Tectonophysics 605: 13–33.
Dai, F.C., C.F. Lee, J.H. Deng, and L.G. Tham. 2005. The 1786 earthquake-triggered landslide dam and subsequent dam-break flood on the Dadu River, southwestern China. Geomorphology 65(3): 205–221.
Dapporto, S., M. Rinaldi, and N. Casagli. 2001. Failure mechanisms and pore water pressure conditions: analysis of a riverbank along the Arno River (Central Italy). Engineering Geology 61(4): 221–242.
Darcy, H. 1856. Histoire des Foundataines Publique de Dijon, 590–594. Paris: Dalmont.
Davies, T.R., and M.J. McSaveney. 2011. Rock-avalanche size and runout–implications for landslide dams. In Natural and Artificial Rockslide Dams, ed. S.G. Evans, R.L. Hermanns, A. Strom, and G. Scarascia-Mugnozza, 441–462. Berlin Heidelberg: Springer.
Den Adel, H., K.J. Bakker, and M. Klein Breteler. 1988. Internal Stability of Minestone. In Proceedings of the International Symposium on Modelling Soil–Water–Structure Interaction, International Association for Hydraulic Research (IAHR), Netherlands, ed. P.A. Kolkman, J. Lindenberg, and K.W. Pilarczyk, 225–231. Rotterdam: Balkema.
Dunne, T. 1990. Hydrology, mechanics, and geomorphic implications of erosion by subsurface flow. In Groundwater Geomorphology: The Role of Subsurface Water in Earth-Surface Processes and Landforms, Geol. Soc. Am. Spec. Pap. 252, ed. C.G. Higgins and D.R. Coates, 1–28.
Evans, S.G., K.B. Delaney, R.L. Hermanns, A. Strom, and G. Scarascia-Mugnozza. 2011. The formation and behaviour of natural and artificial rockslide dams; implications for engineering performance and hazard management. In Natural and artificial rockslide dams, ed. S.G. Evans, R.L. Hermanns, A. Strom, and G. Scarascia-Mugnozza, 1–75. Berlin Heidelberg: Springer.
Fell, R., and J.J. Fry. 2013. State of the art on the likelihood of internal erosion of dams and levees by means of testing. In Erosion in Geomechanics Applied to Dams and Levees, Chapter 1, ed. S. Bonelli, 1–99. London: ISTE-Wiley.
Fox, G.A., and G.V. Wilson. 2010. The role of subsurface flow in hillslope and stream bank erosion: a review. Soil Science Society of America Journal 74(3): 717–733.
Fox, G.A., G.V. Wilson, A. Simon, E.J. Langendoen, O. Akay, and J.W. Fuchs. 2007. Measuring streambank erosion due to groundwater seepage: correlation to bank pore water pressure, precipitation and stream stage. Earth Surface Processes and Landforms 32(10): 1558–1573.
Fredlund, D.G. 1995. The stability of slopes with negative pore-water pressures. In The Ian Boyd Donald Symposium on Modern Developments in Geomechanics, vol. 3168, ed. C.M. Haberfield, 99–116. Clayton: Monash University, Department of Civil Engineering.
Fredlund, D.G. 1999. The scope of unsaturated soil mechanics: an overview. In The Emergence of Unsaturated Soil Mechanics: Fredlund Volume, ed. A.W. Clifton, G.W. Wilson, and S.L. Barbour, 140–156. Ottawa: NRC Research Press.
Fredlund, D.G., N.R. Morgenstern, and R.A. Widger. 1978. The shear strength of unsaturated soils. Canadian Geotechnical Journal 15(3): 313–321.
Fredlund, D.G., H. Rahardjo. 1993. Soil mechanics for unsaturated soils. John Wiley & Sons. New York.
Fredlund, D.G., H. Rahardjo, and M.D. Fredlund. 2012. Unsaturated soil mechanics in engineering practice, 926. New York: Wiley.
Ghiassian, H., and S. Ghareh. 2008. Stability of sandy slopes under seepage conditions. Landslides 5(4): 397–406.
Hagerty, D.J. 1991. Piping/sapping erosion: 1 Basic considerations. Journal of Hydraulic Engineering 117: 991–1008.
Higgins, C.G. 1982. Drainage systems developed by sapping on Earth and Mars. Geology 10(3): 147–152.
Higgins, C.G. 1984. Piping and sapping; development of landforms by groundwater outflow, 18–58. Boston: Allen & Unwin.
Howard, A.D. 1988. Groundwater sapping experiments and modeling. Sapping Features of the Colorado Plateau: A Comparative Planetary Geology Field Guide 491: 71–83.
Howard, A.D., and C.F. McLane. 1988. Erosion of cohesionless sediment by groundwater seepage. Water Resources Research 24(10): 1659–1674.
Hutchinson, J.N. 1982. Damage to slopes produced by seepage erosion in sands. In Landslides and mudflows, 250–265. Moscow: Centre of International Projects, GKNT.
Iverson, R.M., and J.J. Major. 1986. Groundwater seepage vectors and the potential for hillslope failure and debris flow mobilization. Water Resources Research 22(11): 1543–1548.
Jones, J.A.A. 1981. The nature of soil piping: A review of research. In Brit. Geomorphol. Res. Group Res. Monogr. Serie 3. Norwich: Geobooks.
Ke, L., and A. Takahashi. 2012. Influence of internal erosion on deformation and strength of gap-graded non-cohesive soil. In Proceedings of the Sixth International Conference on Scour and Erosion, Paris, 847–854.
Kokusho, T., and Y. Fujikura. 2008. Effect of particle gradation on seepage failure in granular soils. In 4th Int'l Conf. on Scour and Erosion, Tokyo, Japan, 497–504.
Korup, O., and F. Tweed. 2007. Ice, moraine, and landslide dams in mountainous terrain. Quaternary Science Reviews 26(25): 3406–3422.
Korup, O., A.L. Densmore, and F. Schlunegger. 2010. The role of landslides in mountain range evolution. Geomorphology 120(1): 77–90.
Lam, L., D.G. Fredlund, and S.L. Barbour. 1987. Transient seepage model for saturated-unsaturated soil systems: a geotechnical engineering approach. Canadian Geotechnical Journal 24(4): 565–580.
Lane, E.W. 1935. Security from under-seepage masonry dams on earth foundations. Transactions of the American Society of Agricultural Engineers 60(4): 929–966.
Lobkovsky, A.E., B. Jensen, A. Kudrolli, and D.H. Rothman. 2004. Threshold phenomena in erosion driven by subsurface flow. Journal of Geophysical Research 109: F04010. doi:10.1029/2004JF000172.
Meyer, W., R.L. Schuster, and M.A. Sabol. 1994. Potential for seepage erosion of landslide dam. Journal of Geotechnical Engineering 120(7): 1211–1229.
Moffat, R., R.J. Fannin, and S.J. Garner. 2011. Spatial and temporal progression of internal erosion in cohesionless soil. Canadian Geotechnical Journal 48(3): 399–412.
Müller-Kirchenbauer, H., M. Rankl, and C. Schlötzer. 1993. Mechanism for regressive erosion beneath dams and barrages. In Proceedings of the First International Conference on Filters in Geotechnical and Hydraulic Engineering, ed. J. Brauns, M. Heibaum, and U. Schuler, 369–376. Rotterdam: Balkema.
O'Connor, J.E., and J.E. Costa. 2004. The world's largest floods, past and present: their causes and magnitudes. In U.S Geological Survey Circular, 1254, 13.
Okamura, M., and Y. Soga. 2006. Effects of pore fluid compressibility on liquefaction resistance of partially saturated sand. Soils and Foundations 46(5): 695–700.
Okeke, A.C.U., and F. Wang. 2016. Hydromechanical constraints on piping failure of landslide dams: an experimental investigation. Geoenvironmental Disasters 3(1):1–17.
Pagano, L., E. Fontanella, S. Sica, and A. Desideri. 2010. Pore water pressure measurements in the interpretation of the hydraulic behaviour of two earth dams. Soils and Foundations 50(2): 295–307.
Perzlmaier, S., P. Muckenthaler, and A.R. Koelewijn. 2007. Hydraulic criteria for internal erosion in cohesionless soil. Assessment of risk of internal erosion of water retaining structures: dams, dykes and levees. In Intermediate Report of the European Working Group of ICOLD, 30–44. Munich: Technical University of Munich.
Plaza, G., O. Zevallos, and É. Cadier. 2011. La Josefina Landslide Dam and Its Catastrophic Breaching in the Andean Region of Ecuador. In Natural and artificial rockslide dams, ed. S.G. Evans, R.L. Hermanns, A. Strom, and G. Scarascia-Mugnozza, 389–406. Berlin Heidelberg: Springer.
Richards, K.S., and K.R. Reddy. 2007. Critical appraisal of piping phenomena in earth dams. Bulletin of Engineering Geology and the Environment 66(4): 381–402.
Richards, K.S., and K.R. Reddy. 2010. True triaxial piping test apparatus for evaluation of piping potential in earth structures. Journal of ASTM Geotech Test 33(1): 83–95.
Rinaldi, M., and N. Casagli. 1999. Stability of streambanks formed in partially saturated soils and effects of negative pore water pressures: the Sieve River (Italy). Geomorphology 26(4): 253–277.
Samani, Z.A., and L.S. Willardson. 1981. Soil hydraulic stability in a subsurface drainage system. Transactions of the American Society of Agricultural Engineers 24(3): 666–669.
Simon, A., and A.J. Collison. 2001. Pore‐water pressure effects on the detachment of cohesive streambeds: seepage forces and matric suction. Earth Surface Processes and Landforms 26(13): 1421–1442.
Skempton, A.W., and J.M. Brogan. 1994. Experiments on piping in sandy gravels. Geotechnique 44(3): 449–460.
Terzaghi, K. 1943. Theoretical soil mechanics, 1–510. New York: Wiley.
Terzaghi, K., R.B. Peck, and G. Mesri. 1996. Soil Mechanics in Engineering Practice, 3rd ed. New York: Wiley.
Wan, C.F., and R. Fell. 2004. Experimental investigation of internal instability of soils in embankment dams and their foundations, NICIV Report No. R429. Sydney: University of South Wales.
Wang, G., R. Huang, T. Kamai, and F. Zhang. 2013. The internal structure of a rockslide dam induced by the 2008 Wenchuan (M w 7.9) earthquake, China. Engineering Geology 156:28–36.
Wang, F.W., Y. Kuwada, A.C. Okeke, M. Hoshimoto, T. Kogure, and T. Sakai. in press. Comprehensive study on failure prediction of landslide dams by piping. 12th Proc. International Symposium on Landslides, Napoli, Italy.
Weijers, J.B.A., and J.B. Sellmeijer. 1993. A new model to deal with the piping mechanism. In Proceedings of the First International Conference on Filters in Geotechnical and Hydraulic Engineering, ed. J. Brauns, M. Heibaum, and U. Schuler, 345–355. Rotterdam: Balkema.
Wilson, G.V., R.K. Periketi, G.A. Fox, S.M. Dabney, F.D. Shields, and R.F. Cullum. 2007. Soil properties controlling seepage erosion contributions to streambank failure. Earth Surface Processes and Landforms 32(3): 447–459.
Wörman, A. 1993. Seepage-induced mass wasting in coarse soil slopes. Journal of Hydraulic Engineering 119(10): 1155–1168.
Zasłavsky, D., and G. Kassiff. 1965. Theoretical formulation of piping mechanism in cohesive soils. Geotechnique 15(3): 305–316.
This investigation was financially supported by JSPS KAKENHI Grant Number A-2424106 for landslide dam failure prediction. Dr Solomon Obialo Onwuka (University of Nigeria, Nsukka) is gratefully acknowledged for his valuable comments and suggestions. The authors would like to thank the anonymous reviewers for reviewing the draft version of the manuscript.
Department of Geoscience, Graduate School of Science and Engineering, Shimane University, 1060 Nishikawatsu-cho, Matsue, Shimane, 690-8504, Japan
Austin Chukwueloka-Udechukwu Okeke & Fawu Wang
Austin Chukwueloka-Udechukwu Okeke
Fawu Wang
Correspondence to Austin Chukwueloka-Udechukwu Okeke.
FW acquired the laboratory materials used in the research. ACO designed and conducted the experiments. FW supervised the research and made suggestions on the initial method adopted for the experiments. ACO analyzed the experimental results and wrote the first draft of the manuscript. All authors read and approved the final manuscript.
Failure mechanism of Sandfill Dam (Experiment 1). (MP4 187388 kb)
Trends of hydraulic gradients in experiments carried out with upstream inflow rates of (a) 1.67 × 10−5 m3/s (b) 5 × 10−5 m3/s (c) 1 × 10−4 m3/s (d) 1.67 × 10−4 m3/s
Trends of hydraulic gradients in dams built with downstream slope angles of (a) 30° (b) 40° (c) 50° (d) 60°
Trends of hydraulic gradients in dams built with dam heights of (a) 0.15 m (b) 0.20 m (c) 0.25 m (d) 0.30 m
Evolution of pore-water pressures in dams built with an e o of 1.76 and antecedent moisture contents of (a) 5 % (b) 10 % (c) 15 % (d) 20 %
Variations in pore-water pressures in dams built with downstream slope angles of (a) 30° (b) 40° (c) 50° (d) 60°
Transient changes in pore-water pressures in dams built with dam heights of (a) 0.15 m (b) 0.20 m (c) 0.25 m (d) 0.30 m
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Okeke, A.CU., Wang, F. Critical hydraulic gradients for seepage-induced failure of landslide dams. Geoenviron Disasters 3, 9 (2016). https://doi.org/10.1186/s40677-016-0043-z
Sapping
Hydraulic gradient
Critical seepage velocity
Wetting front propagation
Downstream slope saturation
Landslide Dams | CommonCrawl |
Evaluate $\left(2^{\left(1\frac{1}{4}\right)}\right)^{\frac{2}{5}} \cdot \left(4^{\left(3\frac{1}{8}\right)}\right)^{\frac{2}{25}}$.
In addition to knowing how to use mixed numbers, to solve this problem one also must recall two of the basic properties of exponents: \[a^b \cdot a^c = a^{b+c}\] and \[\left(l^m\right)^n = l^{m \cdot n}.\] Keeping these properties in mind, we can proceed with simplification \begin{align*}
\left(2^{\left(1\frac{1}{4}\right)}\right)^{\frac{2}{5}} \cdot \left(4^{\left(3\frac{1}{8}\right)}\right)^{\frac{2}{25}}
&= \left(2^{\frac{5}{4}}\right)^{\frac{2}{5}} \cdot \left(4^{\frac{25}{8}}\right)^{\frac{2}{25}}\\
&= 2^{\frac{5}{4} \cdot \frac{2}{5}} \cdot 4^{\frac{25}{8} \cdot \frac{2}{25}}\\
&= 2^{\frac{2}{4}} \cdot (2^2)^{\frac{2}{8}}\\
&= 2^{\frac{1}{2}} \cdot 2^{2 \cdot \frac{1}{4}}\\
&= 2^{\frac{1}{2}} \cdot 2^{\frac{1}{2}}\\
&= 2^{(\frac{1}{2} + \frac{1}{2})}\\
&= \boxed{2}.
\end{align*} | Math Dataset |
\begin{document}
\maketitle
\begin{abstract} Using unbounded Hilbert space representations basic results on the transition probability of positive linear functionals $f$ and $g$ on a unital $*$-algebra are obtained. The main assumption is the essential self-adjointness of GNS representations $\pi_f$ and $\pi_g$. Applications to functionals given by density matrices and by integrals and to vector functionals on the Weyl algebra are given. \end{abstract}
\textbf{AMS Subject Classification (2000)}.
46L50, 47L60; 81P68.\\
\textbf{Key words:} transition probability, non-commutative probability, unbounded representations
\section {Introduction} Let $f$ and $g$ be
states on a unital $*$-algebra $A$. Suppose that these states are realized as vectors states of a common $*$-representation $\pi$ of $A$ on a Hilbert space with unit vectors $\varphi$ and $\psi$, respectively, that is, $f(a)=\langle \pi(a)\varphi,\varphi\rangle$ and $g(a)=\langle \pi(a)\psi,\psi\rangle$ for $a\in A$. In quantum physics the number $|\langle \varphi,\psi\rangle|^2$ is then interpreted as the transition probability from $f$ to $g$ in these vector states. The (abstract) {\it transition probability} $P_A(f,g)$ is defined as the supremum of values $|\langle \varphi,\psi\rangle|^2$, where the supremum is taken over all realizations of $f$ and $g$ as vectors states in some common $*$-representation of $A$. This definition was introduced by A. Uhlmann \cite{uhlmann76}. The square root $\sqrt{P_A(f,g)}$ is often called fidelity in the literature \cite{alberti03}, \cite{josza}.
The transition probability is related to other important topics such as the Bures distance \cite{bures}, Sakai's non-commutative Radon-Nikodym theorem \cite{araki72} and the geometric mean of Pusz and Woronowicz \cite{pw}. It was extensively studied in the finite dimensional case (see e.g. the monograph \cite{bengtson06}) and a number of results have been derived for $C^*$-algebras and von Neumann algebras (see e.g. \cite{araki72}, \cite{arraggio82}, \cite{alberti83},\cite{alberti92}, \cite{albuhl00}, \cite{alberti03}, \cite{yama}).
The aim of the present paper is to study the transition probability $P_A(f,g)$ for positive linear functionals $f$ and $g$ on a general unital $*$-algebra $A$. Then, in contrast to the case of $C^*$-algebras, the corresponding $*$-representations of $A$ act by unbounded operators in general and a number of technical problems of
{\it unbounded} representation theory
on Hilbert space come up. Dealing with these difficulties in a proper way is a main purpose of this paper. In section \ref{unbrep} we therefore collect all basic definitions and facts on unbounded Hilbert space representations that will be used throughout this paper.
In section \ref{mainres} we state and prove our main theorems about the transition probability $P_A(f,g)$ for a general $*$-algebras. The crucial assumption for these results is the essential self-adjointness of the GNS representations $\pi_f$ and $\pi_g$. This means that we restrict ourselves to a class of "nice" functionals. In contrast we do not restrict the $*$-representation $\pi$ where the functionals $f$ and $g$ are realized as vector functionals. (In some results it is assumed that $\pi$ is closed or biclosed, but this is no restiction of generality, since any $*$-representation has a closed or biclosed extension.)
In section \ref{appl} we apply Theorem \ref{sumofgnspifg1} from section \ref{mainres} to generalize two standard formulas (\ref{fsfttrans0}) and (\ref{commutativecase}) for the transition probability to the unbounded case; these formulas concern trace functionals $f(\cdot)={\rm Tr}\, \rho(\cdot) t$\, and functionals on commutative $*$-algebras given by integrals. A simple counter-example based on the Hamburger moment problem shows that these formulas can fail if the assumption of essential self-adjointness of GNS representations is omitted. In section \ref{vectorweyl} we determine the transition probability of positive functionals on the Weyl algebra given by certain functions from $C_0^\infty(\mathbb{R})$. In this case both GNS representations $\pi_f$ and $\pi_g$ are {\it not} essentially self-adjoint and the corresponding formula for $P_A(f,g)$ is in general different from the standard formula (\ref{fsfttrans0}).
Throughout this paper we suppose that $A$ is a complex unital $*$-algebra. The involution of $A$ is denoted by $a\to a^+$ and the unit element of $A$ by $1$. Let ${\mathcal{P}}(A)$ be the set of all positive linear functionals on $A$. Recall that a linear functional $f$ on $A$ is called positive if $f(a^+a)\geq 0$ for all $a\in A$. Let $\sum A^2$ be the set of all finite sum of squares $a^+a$, where $a\in A$. All notions and facts on von Neumann algebras and on unbounded operators used in this paper can be found in \cite{kr} and \cite{schmu12}, respectively. \section{Basics on Unbounded Representations}\label{unbrep} Proofs of all unproven facts stated in this section and more details can be found in the author's monograph \cite{schmu90}. Proposition \ref{adrepbasic2} below is a new result that might of interest in itself.
Let $({\mathcal{D}},\langle\cdot,\cdot\rangle)$ be a unitary space and $({\mathcal{H}},\langle\cdot,\cdot\rangle)$ the Hilbert space completion of $({\mathcal{D}},\langle\cdot,\cdot\rangle)$. We denote by $L({\mathcal{D}})$ the algebra of all linear operators\, $a:{\mathcal{D}}\to {\mathcal{D}}$, by $I_{\mathcal{D}}$ the identity map of ${\mathcal{D}}$ and by ${\bf B}({\mathcal{H}})$ the $*$-algebra of all bounded linear operators on ${\mathcal{H}}$. \begin{thd} A\, {\em representation}\, of $A$ on ${\mathcal{D}}$ is an algebra homomorphism $\pi$ of $A$ into the algebra $L({\mathcal{D}})$ such that $ \pi(1)=I_{\mathcal{D}}$ and $\pi(a)$ is a closable operator on ${\mathcal{H}}$ for $a\in A$. We then write ${\mathcal{D}}(\pi):={\mathcal{D}}$ and ${\mathcal{H}}(\pi):={\mathcal{H}}$.
A\, {\em $*$-representation}\, $\pi$ of $A$ on ${\mathcal{D}}$ is a representation $ \pi$ satisfying \begin{align}\label{defadjointre1} \langle \pi(a)\varphi,\psi\rangle=\langle \varphi,\pi(a^+)\psi\rangle~~~~{\rm for}~~~a\in A,~\varphi, \psi\in {\mathcal{D}}(\pi). \end{align} \end{thd} Let\, $\pi$\, be a representation of $A$. Then \begin{align}\label{defadjointre} {\mathcal{D}}(\pi^\ast):=\cap_{a\in A}\, {\mathcal{D}}(\pi(a)^\ast)~~~{\rm and}~~~
\pi^\ast(a):=\pi(a^+)^\ast\lceil{\mathcal{D}}(\pi^\ast) ~~~{\rm for}~~~~a\in A, \end{align}
defines a representation $\pi^\ast$ of\, $A$ on ${\mathcal{D}}(\pi^\ast)$, called the\, {\it adjoint representation}\, to $\pi$. Clearly, $\pi$ is a $*$-representation if and only if\, $\pi\subseteq \pi^*$.
If\, $\pi$ is a $\ast$-representation of $A$, then \begin{align}
{\mathcal{D}}(\, \overline{\pi}\, )&:=\cap_{a\in A}\, {\mathcal{D}}(\, \overline{\pi(a)}\, )~~~{\rm and}~~~~ \overline{\pi}(a):=\overline{\pi(a)} \lceil{\mathcal{D}}(\, \overline{\pi}\, ) ,~~a\in A,\label{defclosurere}\\
{\mathcal{D}}(\pi^{**})&:=\cap_{a\in A}\, {\mathcal{D}}(\pi^*(a)^\ast)~~~{\rm and}~~~
\pi^{**}(a):=\pi^*(a^+)^\ast\lceil{\mathcal{D}}(\pi^{**}) ,~~a\in A,\label{defbiadjointre} \end{align}
are $\ast$-representations $\pi^*$ and $\pi^{**}$\, of\, $A$, called the\, {\it closure}\, resp. the\, {\it biclosure} of\, $\pi$. Then \begin{align*} \pi\subseteq \overline{\pi}\subseteq \pi^{**}\subseteq \pi^\ast. \end{align*} If $\pi$ is a $*$-representation, then ${\mathcal{H}}(\pi)={\mathcal{H}}(\pi^*)$. But for a representation $\pi$ it may happen that the domain ${\mathcal{D}}(\pi^\ast)$ is {\it not} dense in ${\mathcal{H}}(\pi)$, that is, ${\mathcal{H}}(\pi^*)\neq {\mathcal{H}}(\pi)$. \begin{thp}\label{adrepbasic2} Let\, $\pi$\, and\, $\rho$\, be representations of a $*$-algebra $A$ such that\, $\rho\subseteq \pi$. Then: \begin{itemize} \item[\rm (i)]~ $P_{{\mathcal{H}}(\rho)}\pi^*(a)\subseteq \rho^*(a)P_{{\mathcal{H}}(\rho)}$, where $P_{{\mathcal{H}}(\rho)}$\, is the projection of\, ${\mathcal{H}}(\pi)$ onto ${\mathcal{H}}(\rho)$. \item[\rm (ii)]~ If\, ${\mathcal{H}}(\rho)={\mathcal{H}}(\pi)$,\, then\, $\pi^*\subseteq \rho^*$. \item[\rm (iii)]~~ $\rho^{**}\subseteq \pi^{**}$. \end{itemize} \end{thp}
\begin{proof} (i): Let $P$ denote the projection $P_{{\mathcal{H}}(\rho)}$ and fix $\psi\in {\mathcal{D}}(\pi^*)$. Let $\varphi\in {\mathcal{D}}(\rho)$ and $a\in A$. Using the assumption $\rho\subseteq \pi$ we obtain \begin{align*} \langle & \rho(a^+)\varphi ,P\psi\rangle =\langle P\rho(a^+)\varphi,\psi\rangle =\langle \rho(a^+)\varphi,\psi\rangle =\langle \pi(a^+)\varphi,\psi\rangle\\&= \langle \varphi,\pi(a^+)^*\psi\rangle= \langle \varphi,\pi^*(a)\psi\rangle =\langle P\varphi,\pi^*(a)\psi\rangle=\langle \varphi,P\pi^*(a)\psi\rangle.
\end{align*} From this equality it follows that\, $P\psi \in {\mathcal{D}}(\rho(a^+)^*)$\, and\, $\rho(a^+)^*P\psi=P\pi^*(a)\psi$. Hence\, $\psi\in \cap_{b\in A}\, {\mathcal{D}}(\rho(b)^*)={\mathcal{D}}(\rho^*)$\, and\, $\rho^*(a)P\psi=P\pi^*(a)\psi$. This proves that $P\pi^*(a)\subseteq \rho^*(a)P.$
(ii) follows at once from (i), since $P=I$ by the assumption ${\mathcal{H}}(\rho)={\mathcal{H}}(\pi)$.
(iii):
Let\, $\xi\in {\mathcal{D}}(\rho^{**})$ and $\psi\in {\mathcal{D}}(\pi^{*})$. Since\, ${\mathcal{H}}(\rho^{**})\subseteq {\mathcal{H}}(\rho^*)\subseteq {\mathcal{H}}(\rho)$\, by definition, $P\xi=\xi$. By (i),\,
$P\psi \in {\mathcal{D}}(\rho^*)$\, and\,
$\rho^*(a)P\psi=P\pi^*(a)\psi$. Therefore, we derive
\begin{align*}
\langle & \pi^*(a)\psi ,\xi\rangle =
\langle \pi^*(a)\psi ,P\xi\rangle =\langle P\pi^*(a)\psi,\xi\rangle\\& =\langle \rho^*(a)P\psi,\xi\rangle =\langle \psi,P\rho^{*}(a)^*\xi\rangle= \langle \psi,\rho^{**}(a^+)\xi\rangle
\end{align*} for $a\in A$. Hence $\xi \in {\mathcal{D}}(\pi^*(a)^*)$\, and\, $\pi^*(a)^*\xi= \rho^{**}(a^+)\xi$\, for $a\in A$. This implies that\, $\xi\in{\mathcal{D}}(\pi^{**})$\, and\, $\pi^{**}(a^+)\xi=\pi^*(a)^*\xi= \rho^{**}(a^+)\xi$. Thus we have proved that\, $\rho^{**}\subseteq \pi^{**}$. \end{proof}
\begin{thd} A $\ast$-representation\, $\pi$\, of a $\ast$-algebra $A$\, is called\\ -- {\rm closed}\, if $\pi=\overline{\pi}$,\, or equivalently, if\, ${\mathcal{D}}(\pi)={\mathcal{D}}(\, \overline{\pi}\, )$,\\ -- {\rm biclosed}\, if $\pi=\pi^{**}$,\, or equivalently, if\, ${\mathcal{D}}(\pi)={\mathcal{D}}(\pi^{**} )$,\\ -- {\rm self-adjoint}\, if $\pi=\pi^\ast$,\, or equivalently, if\, ${\mathcal{D}}(\pi)={\mathcal{D}}(\pi^\ast)$,\\ -- {\rm essentially self-adjoint}\, if\, $\pi^*$ is self-adjoint, that is, if\, $\pi^*=\pi^{**}$,\, or equivalently, if\, ${\mathcal{D}}(\pi^{**})={\mathcal{D}}(\pi^{*})$. \end{thd} Remark. It should be emphasized that the preceding definition of {\it essential self-adjointness} is different form the definition given in \cite{schmu90}. In \cite[Definition 8.1.10]{schmu90}, a $*$-representation was called essentially self-adjoint if\, $\overline{\pi}$\, is self-adjoint, that is, if\, $\overline{\pi}=\pi^*$.
Let $\pi$ be a $*$-representation. Then the $*$-representations $\overline{\pi}$ and $\pi^{**}$ are closed, $\pi^{**}$ is biclosed and $(\overline{\pi})^*=\pi^*$. It may happen that $\overline{\pi}\neq \pi^{**}$, so that $\overline{\pi}$ is closed, but not biclosed. The locally convex topology on ${\mathcal{D}}(\pi)$ defined by the family of seminorms $\{\|\cdot\|_a:=\|\pi(a)\cdot\|; a\in A\}$ is called the {\it graph topology} and denoted by ${\mathfrak{t}}_{\pi(A)}$. Then the $*$-representation $\pi$ is closed if and only if the locally convex space ${\mathcal{D}}(\pi)[{\mathfrak{t}}_{\pi(A)}]$ is complete.
\begin{thp}\label{technicalself} If\, $\pi_1$ is a self-adjoint $\ast$-subrepresentation of a $*$-representation $\pi$ of $A$, then there exists a $*$-representation $\pi_2$ of $A$ on the Hilbert space ${\mathcal{H}}(\pi)\ominus {\mathcal{H}}(\pi_1)$ such that $\pi=\pi_1\oplus \pi_2$. \end{thp} \begin{proof} \cite[Corollary 8.3.3]{schmu90}. \end{proof} For a $*$-representation of $A$ we define two {\it commutants} \begin{align*} \pi(A)^\prime_s&=\{ T\in {\bf B}({\mathcal{H}}(\pi)): T\varphi\in {\mathcal{D}}(\pi),~~ T\pi(a)\varphi=\pi(a)T\varphi ~~~~ {\rm for} ~~~ a\in A, \varphi\in {\mathcal{D}}(\pi)\},\\ \pi(A)^\prime_{ss}&=\{ T\in {\bf B}({\mathcal{H}}(\pi)): T\, \overline{\pi(a)}\subseteq \overline{\pi(a)}\, T,~~T^*\, \overline{\pi(a)}\subseteq \overline{\pi(a)}\, T^*\, \}. \end{align*} The symmetrized commutant\, $\pi(A)^\prime_{ss}$ is always a von Neumann algebra. If $\pi$ is closed, then \begin{align}\label{strongcom} \pi(A)_{ss}^\prime=\pi(A)_s^\prime\cap(\pi(A)_s^\prime)^\ast. \end{align}
If $\pi_1$ and $\pi_2$ are representations of $A$, the {\it interwining space} $I(\pi_1,\pi_2)$ consists of all bounded linear operators $T$ of ${\mathcal{H}}(\pi_1)$ into ${\mathcal{H}}(\pi_2)$ satisfying \begin{align}\label{definterwiner} T\varphi \in {\mathcal{D}}(\pi_2)\quad {\rm and}\quad T\pi_1(a)\varphi=\pi_2(a)T\varphi \quad {\rm for}\quad a\in A, \varphi \in {\mathcal{D}}(\pi_1). \end{align}
The $*$-representation $\pi_f$ in the following proposition is called the {\it GNS representation} associated with the positive linear functional $f$.
\begin{thp}\label{gnsprop} Suppose that $f\in {\mathcal{P}}(A)$. Then there exists a $*$-representation $\pi_f$ with algebraically cyclic vector $\varphi_f$, that is, ${\mathcal{D}}(\pi_f)=\pi_f(A)\varphi_f$, such that \begin{align*} f(a)=\langle \pi_f(a)\varphi_f,\varphi_f\rangle, ~~~ a\in A. \end{align*} If $ \pi$ is another $*$-representation of $A$ with algebraically cyclic vector $\varphi$ such that $f(a)=\langle \pi(a)\varphi,\varphi\rangle$ for all $ a\in A$, then there exists a unitary operator $U$ of ${\mathcal{H}}(\pi)$ onto ${\mathcal{H}}(\pi_f)$ such that $U{\mathcal{D}}(\pi)={\mathcal{D}}(\pi_f)$ and $\pi_f(a)=U^*\pi(a)U$ for $a\in A$. \end{thp} \begin{proof} \cite[Theorem 8.6.4]{schmu90}. \end{proof} We study some of the preceding notions by a simple but instructive example. \begin{thex}\label{hambmp} ({\it One-dimensional Hamburger moment problem})\\ Let $A$ by the polynomial $*$-algebra $\mathbb{C}[x]$ with involution determined by $x^+:=x$. We denote by $M(\mathbb{R})$ the set of positive Borel measures $\mu$ such that $p(x)\in L^1(\mathbb{R},\mu)$ for all $p\in \mathbb{C}[x]$. The number $s_n=\int x^n d\mu(x)$ is the $n$-th moment and the sequence $s(\mu)=(s_n)_{n\in \mathbb{N}_0}$ is called the moment sequence of a measure $\mu\in M(\mathbb{R})$. The moment sequence $s(\mu)$, or likewise the measure $\mu$, is called {\it determinate}, if the moment sequence $s(\mu)$ determines the measure $\mu$ uniquely, that is, if $s(\mu)=s(\nu)$ for some $\nu\in M(\mathbb{R})$ implies that $\nu=\mu$.
For $\mu\in M(\mathbb{R})$ we define a $*$-representation $\pi_\mu$ of $A=\mathbb{C}[x]$ by\, $\pi_\mu(p)q =p\cdot q$\, for $p\in A$ and $q\in {\mathcal{D}}(\pi_\mu):=\mathbb{C}[x]$ on the Hilbert space ${\mathcal{H}}(\pi_\mu):=L^2(\mathbb{R},\mu)$. Put $f_\mu(p)=\int p(x)d\mu(x)$ for $p\in\mathbb{C}[x]$. Obviously, the vector $1\in {\mathcal{D}}(\pi_\mu):=\mathbb{C}[x]$ is algebraically cyclic for $\pi_\mu$. Therefore, since $f_\mu(p)=\langle \pi_\mu(p)1,1\rangle$ for $p\in \mathbb{C}[x]$, $\pi_\mu$ is (unitarily equivalent to) the GNS representation $\pi_{f_\mu}$ of the positive linear functional $f_\mu$ on $A=\mathbb{C}[x]$.
\noindent{\bf Statement:}\, {\it The $*$-representation $\pi_\mu$ is essentially self-adjoint if and only if the moment sequence $s(\mu)$ is determinate.} \begin{proof} By a well-known result on the Hamburger moment problem (see e.g. \cite[Theorem 16.11]{schmu12}), the moment sequence $s(\mu)$ is determinate if and only if the operator $\pi_\mu(x)$ is essentially sel-adjoint. By \cite[Proposition 8.1(v)]{schmu90}, the latter holds if and only if the $*$-representation $(\pi_\mu)^*$ is self-adjoint, that is, if $\pi_\mu$ is essentially self-adjoint. \end{proof}
By \cite[Proposition 8.1(vii)]{schmu90}, the closure $\overline{\pi}_\mu$ of the $*$-representation $\pi_\mu$ is self-adjoint if and only if all powers of the operator $\pi_\mu(x)$ are essentially self-adjoint. This is a rather strong condition. It is fulfilled (for instance) if $1$ is an analytic vector for the symmetric operator $\pi_\mu(x)$, that is, if there exists a constant $M>0$ such that
$$\|\pi_\mu(x)^n 1\|= s_{2n}^{1/2}\leq M^n n!\quad {\rm for}\quad n\in \mathbb{N}.$$ From the theory of moment problems it is well-known that there are examples of measures $\mu\in M(\mathbb{R})$ for which $\pi_\mu(x)$ is essentially self-adjoint, but $\pi_\mu(x^2)$ is not. In this case $\pi_\mu$ is essentially self-adjoint (which means that $(\pi_\mu)^*$ is self-adjoint), but the closure\, $\overline{\pi}_\mu$ of\, $\pi_\mu$ is {\it not} self-adjoint. \end{thex}
\section{Main Results on Transition Probabilities}\label{mainres}
Let ${\rm Rep} A$ denote the family of all $*$-representations of $A$. Given $\pi\in {\rm Rep} A$ and $f\in{\mathcal{P}}( A)$, let $S(\pi,f)$ be the set of all representing vectors for the functional $f$ in ${\mathcal{D}}(\pi)$, that is, $S(\pi,f)$ is the set of vectors $\varphi\in {\mathcal{D}}(\pi)$ such that $f(a)=\langle \pi(a)\varphi,\varphi\rangle$ for\, $a\in A$. Note that $S(\pi,f)$ may be empty, but by Proposition \ref{gnsprop} for each $f\in {\mathcal{P}}(A)$ there exists a $*$-representation $\pi$ of $A$ for which $S(\pi,f)$ is not empty. If $f$ is a state, that is, if $f(1)=1$, then all vectors $\varphi\in S(\pi,f)$ are unit vectors.
\begin{thd} For $f,g \in{\mathcal{P}}(A)$ the {\rm transition probability}\, $P_A(f,g)$\, of\, $f$ and $g$ is defined by \begin{align}\label{transtionprob}
P_A(f,g)= \sup_{\pi \in {\rm Rep} A}~ \sup_{\varphi\in {\mathcal{S}}(\pi,f),\psi\in {\mathcal{S}}(\pi,g)}~|\langle \varphi,\psi\rangle|^2. \end{align} \end{thd} If $A$ is a unital $*$-subalgebra of $B$ and $f,g\in {\mathcal{P}}(B)$, then it is obvious that \begin{align}\label{pafgpbfg} P_B(f,g)\leq {\mathcal{P}}_A(f\lceil A,g\lceil A), \end{align} because the restriction of any $*$-representation of $B$ is a $*$-representation of $A$.
Let ${\mathcal{G}}(f,g)$ denote the set of all linear functionals on $A$ satisfying \begin{align}\label{defigset}
|F(b^+ a)|^2\leq f(a^+a)g(b^+b)\quad {\rm for}\quad a,b\in A.
\end{align}
Any vector $\varphi\in S(\pi,f)$ is called an {\it amplitude} of $f$ in the representation $\pi$ and any linear functional of ${\mathcal{G}}(f,g)$ is called a {\it transition form} from $f$ to $g$. If $\varphi\in {\mathcal{S}}(\pi,f)$ and $\psi\in {\mathcal{S}}(\pi,g)$, then the functional $F_{\varphi,\psi}$ defined by \begin{align}\label{transvarphipsi} F_{\varphi,\psi}(a):=\langle\pi(a)\varphi, \psi\rangle,\quad a\in A, \end{align} is a transition form from $f$ to $g$. Indeed, for\, $a,b\in A$ we have \begin{align*}
|F_{\varphi,\psi}(b^+ a)|^2&=|\langle\pi(b^+a)\varphi, \psi\rangle|^2=|\langle \pi(a)\varphi,\pi(b)\psi\rangle|^2\\ &\leq \|\pi(a)\varphi \|^2\|\pi(b)\psi\|^2 = f(a^+a)g(b^+b) \end{align*} which proves that $F_{\varphi,\psi}\in {\mathcal{G}}(f,g)$. By Theorem \ref{chartransprob} below, {\it each} functional $F\in {\mathcal{G}}(f,g)$ arises in this manner.
The number\, $|F_{\varphi,\psi}(1)|^2=|\langle\varphi, \psi\rangle|^2$\, is called the {\it transition probability} of the amplitudes $\varphi$ and $\psi$ and by definition the transition probability\, $P_A(f,g)$\, is the supremum of all such transition amplitudes.
The following description of the transition probability was proved by P.M. Alberti for $C^*$-algebras \cite{alberti83} and by A. Uhlmann for general $*$-algebras \cite{uhlmann85}. \begin{tht}\label{chartransprob} Suppose that $f, \in {\mathcal{P}}(A)$. Then \begin{align}\label{cxhartranspobality}
P_A(f ,g)= \sup_{F\in {\mathcal{G}}(f,g)}~|F(1)|^2. \end{align} There exist a $*$-representation $\pi$ of $A$ and vectors $\varphi \in S(\pi,f)$ and $\psi \in S(\pi,g)$
such that \begin{align}\label{extemepsi12}
P_A(f ,g)=|\langle \varphi,\psi\rangle|^2. \end{align} \end{tht}
Next we express the transition forms of ${\mathcal{G}}(f,g)$ and hence the transition probability in terms of intertwiners of the corresponding GNS representations. This provides a powerful tool for computing the transition probability. Recall $\pi_f$ denotes the GNS representation of $A$ associated with $f\in {\mathcal{P}}(A)$ and $\varphi_f$ is the corresponding algebraically cyclic vector.
\begin{thp}\label{proponetooneft} Suppose that $f,g\in{\mathcal{P}}(A)$. Then there a one-to-one correspondence between the sets\, ${\mathcal{G}}(f,g)$\, and\, $I(\pi_f,(\pi_g)^*)$\, given by \begin{align}\label{onetoonegfd} F(b^+a)= \langle T\pi_f(a)\varphi_f,\pi_g(b)\varphi_g\rangle\quad {\rm for} \quad a,b\in A, \end{align} where\, $F\in {\mathcal{G}}(f,g)$\, and\, $T\in I(\pi_f,(\pi_g)^*)$.\, In particular,\, $F(1)=\langle T\varphi_f,\varphi_g\rangle$. \end{thp} \begin{proof} Let $F\in {\mathcal{G}}(f,g)$. Then $$
|F(b^+a)|^2\leq f(a^*a)g(b^*b)=\|\pi_f(a)\varphi_f\|^2\|\pi_g(b)\varphi_g\|^2 \quad {\rm for}\quad a,b\in A. $$
Hence there exists a bounded linear operator $T$ of ${\mathcal{H}}(\overline{\pi}_g)$ into ${\mathcal{H}}(\overline{\pi}_f)$ such that\, $\|T\|\leq 1$\, and (\ref{onetoonegfd}) holds. Let $a,b,c\in A$. Using\, (\ref{onetoonegfd})\, we obtain \begin{align*}
\langle T \pi_f(a)\varphi_f,\pi_g(c^+)\pi_g(b)\varphi_g\rangle&=F((c^+b)^+a)=F(b^+(ca))\\&=\langle T\pi_f(c)\pi_f(a)\varphi_f,\pi_g(b)\varphi_g \rangle. \end{align*} Hence\, $T\pi_f(b)\varphi_f\in {\mathcal{D}}(\pi_g(c)^*)$ and\, $\pi_g(c)^*T\pi_f(a)\varphi_f= T\pi_f(c^+)\pi_f(a)\varphi_f$.\, Because $c\in A$ was arbitrary,\, $T\pi_f(a)\varphi_f \in {\mathcal{D}}((\pi_g)^*)$. Then $$(\pi_g)^*(c^+)T\pi_f(a)\varphi_f= T\pi_f(c^+)\pi_f(a)\varphi_f\quad {\rm for}\quad a\in A,$$ which means that $T\in I(\pi_f,(\pi_g)^*)$.
Conversely, let $T\in I(\pi_f,(\pi_g)^*)$ and $\|T\|\leq 1$. Define $F(a)= \langle T\pi_f(a)\varphi_f,\varphi_g\rangle$\, for\, $a\in A$. It is straightforward to check that\, (\ref{onetoonegfd})\, holds and hence\, (\ref{defigset}), that is, $F\in {\mathcal{G}}(f,g)$.
Clearly, by\, (\ref{onetoonegfd}), $F=0$ is equivalent to $T=0$. Thus we have a one-to-one correspondence between functionals $F$ and operators $T$. \end{proof}
Combining Theorem \ref{chartransprob} and Proposition \ref{proponetooneft} and using the formula $F(1)=\langle T\varphi_f,\varphi_g\rangle$ we obtain \begin{thc} For any $f, g \in {\mathcal{P}}(A)$ we have \begin{align}\label{intertwtnull}
P_A(f,g)=\sup_{T\in I(\pi_f,(\pi_g)^*),\, \|T\|\leq 1}~ |\langle T \varphi_f,\varphi_g\rangle|^2. \end{align} \end{thc} If the GNS representations of $f$ and $g$ are essentially self-adjoint, a number of stronger results can be obtained. \begin{tht}\label{sumofgnspifg} Suppose that $f$ and $g$ are positive linear functionals on $A$ such that their GNS representations $\pi_f$ and $\pi_g$ are essentially self-adjoint. Let $\pi$ be a biclosed $\ast$-representation of $A$ such that the sets $S(\pi,f)$ and $S(\pi,g)$ are not empty. Fix vectors $\varphi \in S(\pi,f)$ and $\psi \in S(\pi,g)$. Then \begin{align}\label{proptrastcss}
P(f,g)=\sup_{T\in \pi(A)^\prime_{ss}, \|T\|\leq 1} ~|\langle T \varphi,\psi\rangle|^2. \end{align} \end{tht}
\begin{proof} Let $T\in \pi(A)^\prime_{ss}$ and $ \|T\|\leq 1$. Similarly, as in the proof of Proposition \ref{proponetooneft}, we define $F(a)= \langle T\pi(a)\varphi,\psi\rangle$, $a\in A$. Since $\pi(A)^\prime_{ss} \subseteq\pi(A)^\prime_{s}$, we obtain \begin{align*}
|F(b^+a)|^2&=|\langle T\pi(b^+a)\varphi,\psi\rangle|^2=|\langle \pi(b^+)T\pi(a)\varphi,\psi\rangle|^2\\&=|\langle T\pi(a)\varphi,\pi(b)\psi\rangle|^2\leq \|\pi (a)\varphi\|^2~\|\pi(b)\psi\|^2=f(a^+a)g(b^+b) \end{align*}
for $a,b\in A$, that is, $F\in {\mathcal{G}}(f,g)$. Clearly, we have $\langle T\varphi,\psi\rangle=F(1)$. Let $\rho_f$ and $\rho_g$ denote the restrictions $\pi \lceil \pi(A)\varphi$ and $\pi \lceil \pi(A)\psi,$ respectively. Since $\rho_f\subseteq \pi$ and $\rho_g\subseteq \pi$ and $\pi$ is biclosed, it follows from Proposition \ref{adrepbasic2}(iii) that $(\rho_f)^{**}\subseteq \pi^{**}=\pi$ and $(\rho_g)^{**}\subseteq \pi^{**}=\pi$. Since $\varphi \in S(\pi,f)$ and $\psi \in S(\pi,g)$, the representations $\rho_f$ and $\rho_g$ are unitarily equivalent to the GNS representations $\pi_f$ and $\pi_g$, respectively. For notational simplicity we identify $\rho_f$ with $\pi_f$ and $\rho_g$ with $\pi_g$. Since $\rho_f$ and $\rho_g$ are essentially self-adjoint by assumption, $(\rho_f)^{**}$ and $(\rho_g)^{**}$ are self-adjoint. Therefore, by Proposition \ref{technicalself}, there are subrepresentations $\rho_1$ and $\rho_2$ of $\pi$ such that $\pi=(\rho_f)^{**}\oplus \rho_1$ and $\pi=(\rho_g)^{**}\oplus \rho_2$.
Conversely, suppose that $F\in {\mathcal{G}}(f,g)$. By Proposition \ref{proponetooneft}, there is an intertwiner $T_0\in I(\rho_f,(\rho_g)^*)\cong I(\pi_f,(\pi_g)^*))$\, such that\, $\|T_0\|\leq 1$\, and (\ref{onetoonegfd})\, holds with $T$ replaced by $T_0$. Define $T:{\mathcal{H}}(\rho_f) \oplus {\mathcal{H}}(\rho_1)\to {\mathcal{H}}(\rho_g) \oplus {\mathcal{H}}(\rho_2)$ by $T(\xi_f,\xi_1)=(T_0\xi_f,0)$. Clearly, $T^*$ acts by $T^*(\eta_g,\eta_2)=(T_0^*\eta_g,0)$. Since $(\rho_g)^{**}=(\rho_g)^*$ and $(\rho_f)^{**}=(\rho_f)^*$ by assumption and\, $T_0\in I(\rho_f,(\rho_g)^*)$, it follows from Proposition 8.2.3,(iii) and (iv), in \cite{schmu90} that \begin{align*} T_0&\in I((\rho_f)^{**},(\rho_g)^*)=I((\rho_f)^{**},(\rho_g)^{**}),\\ T_0^* &\in I((\rho_g)^{**},(\rho_f)^*)=I((\rho_g)^{**},(\rho_f)^{**}). \end{align*}
From these relations we easily derive that the operators $T$ and $T^*$ are in $\pi(A)^\prime_{s}$, so that $T\in \pi(A)^\prime_{ss}$ by (\ref{strongcom}). Then we have $\|T\|=\|T_0\|\leq 1$ and
$F(1)=\langle T_0 \varphi,\psi\rangle=\langle T \varphi,\psi \rangle$. Togther with the first paragraph of this proof we have shown that the supremum over the operators $T\in \pi(A)^\prime_{ss}$, $ \|T\|\leq 1$, is equal to the supremum over the functionals $F\in {\mathcal{G}}(f,g)$. Since the latter is equal to $P_A(f,g)$ by Theorem \ref{chartransprob}, this proves (\ref{proptrastcss}). \end{proof}
Remark. A slight modification of the preceding proof shows the following: If we assume that the closures $\overline{\pi}_f$ and $\overline{\pi}_g$ of the GNS representations $\pi_f$ and $\pi_g$ are self-adjoint, then the assertion of Theorem \ref{sumofgnspifg} remains valid if it is only assumed that $\pi$ is {\it closed} rather than biclosed. A similar remark applies also for the subsequent applications of Theorem \ref{sumofgnspifg} given below.
Theorem \ref{sumofgnspifg} says that (in the case of essentially self-adjoint GNS representations $\pi_f$ and $\pi_g$) the transition probability $P(f,g)$ is given by formula (\ref{proptrastcss}) in {\it any} fixed biclosed $*$-representation $\pi$ for which the sets $S(\pi,f)$ and $S(\pi,g)$ are not empty and for {\it arbitrary} fixed vectors $\varphi \in S(\pi,f)$ and $\psi \in S(\pi,g)$. In particular, we may take $\pi:=(\pi_f)^{**}\oplus (\pi_g)^{**}$,\, $\varphi:=\varphi_f$,\, and $\psi:=\varphi_g.$
\begin{tht}\label{sumofgnspifg1} Suppose that $f, g\in {\mathcal{P}}(A)$ and the GNS representations $\pi_f$ and $\pi_g$ are essentially self-adjoint. Suppose that $\pi$ is a biclosed $\ast$-representation of $A$ and there exist vectors
$\varphi \in S(\pi,f)$ and $\psi \in S(\pi,g)$. Let $F_\varphi$ and $F_\psi$ denote the vector functionals on the von Neumann algebra\, ${\mathcal{M}}:=(\pi(A)^\prime_{ss})^\prime$ given by $F_\varphi (x)=\langle x\varphi,\varphi\rangle $ and $F_\psi (x)=\langle x\psi,\psi\rangle $,\, $x\in {\mathcal{M}}$.
Then we have
\begin{align}\label{proptrastcss1}
P_A(f,g)=P_{{\mathcal{M}}}\, (F_\varphi,F_\psi) .
\end{align}
Further, there exist vectors $\varphi^\prime\in S(\pi,f)$ and $\psi^\prime\in S(\pi,g)$ such that
$\langle x \varphi^\prime,\varphi^\prime\rangle =\langle x \varphi,\varphi\rangle$ and $\langle x \psi^\prime,\psi^\prime\rangle =\langle x \psi,\psi\rangle$ for $x\in {\mathcal{M}}$ and
\begin{align}\label{attaiendtp}
P_A(f,g)=|\langle\varphi^\prime,\psi^\prime\rangle|^2.
\end{align}
\end{tht}
\begin{proof} Since $\pi(A)^\prime_{ss}$ is a von Neumann algebra, we have $T\in \pi(A)^\prime_{ss}$ if and only if\, $T\in (\pi(A)^\prime_{ss})^{\prime\prime}={\mathcal{M}}^\prime $ . Therefore, applying formula (\ref{proptrastcss}) to the $*$-representation $\pi$ of $A$ and to the identity representation of the von Neumann algebra ${\mathcal{M}}$, it follows that the supremum of\, $|\langle T \varphi,\psi\rangle|^2$\, over all operators\, $T\in \pi(A)^\prime_{ss}={\mathcal{M}}^\prime$,\, $ \|T\|\leq 1$, \, is equal to $P_A(f,g)$ and also to $P_{{\mathcal{M}}}\, (F_\varphi,F_\psi)$. This yields the equality (\ref{proptrastcss1}).
Now we prove the existence of vectors $\varphi^\prime$ and $\psi^\prime$ having the desired properties. In order to do so we go into the details of the proof of \cite[Appendix 7]{alberti92}. Besides we use some facts from von Neumann algebra theory \cite{kr}. We define a normal linear functional on the von Neumann algebra ${\mathcal{M}}^\prime$ by $h(\cdot)=\langle \cdot \, \varphi,\psi\rangle$. Let $h=R_u|h|$ be the polar decomposition of $h$, where $u$ is a partial isometry from ${\mathcal{M}}^\prime$. Then we have $|h|=R_{u^*}h$ and hence $\|h\|=\|\, |h|\, \|= |h|(1)=h(u^*)=\langle u^*\varphi,\psi\rangle$. Therefore, we obtain \begin{align}\label{hnorm}
P_{{\mathcal{M}}}\, (F_\varphi,F_\psi)=\sup_{T\in {\mathcal{M}}^\prime, \|T\|\leq 1} ~|\langle T \varphi,\psi\rangle|^2= \|h\|^2=\langle u^*\varphi,\psi\rangle^2, \end{align} where the first equality follows formula (\ref{proptrastcss}) applied to the von Neumann algebra ${\mathcal{M}}$. In the proof of \cite[Appendix 7]{alberti92} it was shown that there exist partial isometries $v,w\in {\mathcal{M}}^\prime $ satisfying \begin{align}\label{uvw} &\langle u^*\varphi,\psi\rangle=\langle v^*w\varphi,\psi\rangle,\\ & w^*w\geq p(\varphi), \, v^*v\geq p(\psi),\label{vwvarphi} \end{align} where $p(\varphi)$ and $p(\psi)$ are the projections of ${\mathcal{M}}^\prime$ onto the closures of ${\mathcal{M}} \varphi$ and ${\mathcal{M}} \psi$, respectively. Set $\varphi^\prime:=w \varphi$ and $\psi^\prime:=v \psi$. Comparing (\ref{uvw}) with (\ref{hnorm}) and (\ref{proptrastcss1}) we obtain (\ref{attaiendtp}).
From (\ref{vwvarphi}) it follows that $\langle x \varphi^\prime,\varphi^\prime\rangle =\langle x \varphi,\varphi\rangle$ and $\langle x \psi^\prime,\psi^\prime\rangle =\langle x \psi,\psi\rangle$ for $x\in {\mathcal{M}}$ and that $w^*w\varphi=\varphi$ and $v^*v\psi=\psi$. Since $w,w^*\in {\mathcal{M}}^\prime=\pi(a)^\prime_{ss}$ and $\pi$ is closed, we have $w,w^*\in \pi(a)^\prime_{s}$ by (\ref{strongcom}. Therefore, $w$ and $w^*$ leave the domain ${\mathcal{D}}(\pi)$ invariant, so that $\varphi^\prime=w \varphi\in {\mathcal{D}}(\pi)$ and $\psi^\prime=v \psi\in {\mathcal{D}}(\pi)$. For $a\in A$ we derive \begin{align*} \langle\pi(a)\varphi^\prime,\varphi^\prime\rangle&= \langle\pi(a)x\varphi,w\varphi\rangle= \langle w^* \pi(a)w\varphi,\varphi\rangle\\ &= \langle \pi(a)w^*w\varphi,\varphi\rangle=\langle \pi(a)\varphi,\varphi\rangle =f(a). \end{align*} That is, $\varphi^\prime\in S(\pi,f)$. Similarly, $\psi^\prime\in S(\pi,g)$. \end{proof} Theorem \ref{proptrastcss1} allows us to reduce the computation of the transition probability of the functionals $f$ and $g$ on $A$ to that of the vector functionals $F_\varphi$ and $F_\psi$ of the von Neumann algebra\, ${\mathcal{M}}=(\pi(A)^\prime_{ss})^\prime$. In the next section we will apply this result in two important situations.
The following theorem generalizes a classical result of A. Uhlmann \cite{uhlmann76} to the unbounded case. \begin{tht} Let $f, g\in {\mathcal{P}}(A)$ be such that the GNS representations $\pi_f$ and $\pi_g$ are essentially self-adjoint. Suppose that there exist a positive linear functional $h$ on $A$ and elements $b, c\in A$ such that $f(a)=h(b^+ab)$ and $g(a)=h(c^+ac)$ for $a\in A$. Assume that\, $c^+b\in \sum A^2$. Then\, $$P_A(f,g)=h(c^+b)^2.$$ \end{tht} \begin{proof} Recall that $\pi_h$ is the GNS representation of $h$ with algebraically cyclic vector $\varphi_h$. By the assumptions $f(\cdot)=h(b^+\cdot b)$ and $g(\cdot)=h(c^+\cdot c)$ we have $\pi_h(b)\varphi_h\in S(\pi_h,f)$ and $\pi_h(c)\varphi_h\in S(\pi_h,g)$. Therefore, $$h(c^+b)^2=\langle \pi_h(b)\varphi_h,\pi_h(c)\varphi \rangle^2 \leq P_A(f,g).$$
To prove the converse inequality we want to apply Theorem \ref{sumofgnspifg} to the biclosed representation\, $\pi:=(\pi_h)^{**}$. Suppose that\, $T\in \pi(A)^\prime_{ss}$ and $\|T\|\leq 1$. Set $R:=\pi(c^+b)$. Since\, $c^+b \in\sum\, A^2$\, by assumption, $R$ is a positive, hence symmetric, operator. Since\, $\pi:=(\pi_h)^{**}$\, is closed, we have\, $T\in \pi(A)^\prime_{s}$. Using these facts and the Cauchy-Schwarz inequality we derive \begin{align}\label{tvarphibc1}
|\langle T\pi_h(b)\varphi_h ,\pi_h(c)\varphi_h\rangle|^2 &=|\langle T\pi(b)\varphi_h,\pi(c)\varphi_h\rangle|^2
= |\langle \pi(b)T\varphi_h,\pi(c)\varphi_h\rangle|^2\nonumber\\&=| \langle RT\varphi_h,\varphi_h\rangle|^2 \leq \langle RT\varphi_h,T\varphi_h\rangle \langle R\varphi_h,\varphi_h\rangle\nonumber \\& =\langle RT\varphi_h,T\varphi_h\rangle h(c^+b). \end{align} Since $T\in \pi(A)^\prime_{ss}$, we have $TR\subseteq RT$. There exists a positive self-adjoint extension $\tilde{R}$ of $R$ on ${\mathcal{H}}(\pi)$ such that\, $T\tilde{R}\subseteq \tilde{R}T$ \cite[Exercise 14.14]{schmu12}. The latter implies that\, $T\tilde{R}^{1/2}\subseteq \tilde{R}^{1/2}T$\, and hence \begin{align}\label{tvarphibc2} \langle RT\varphi_h,T\varphi_h\rangle &=\langle \tilde{R}T\varphi_h,T\varphi_h\rangle=\langle \tilde{R}^{1/2}T\varphi_h,\tilde{R}^{1/2}T\varphi_h\rangle \nonumber \\ &=\langle T \tilde{R}^{1/2}\varphi_h,T\tilde{R}^{1/2}\varphi_h\rangle\nonumber \leq \langle \tilde{R}^{1/2}\varphi_h,\tilde{R}^{1/2}\varphi_h\rangle\nonumber \\&= \langle \tilde{R} \varphi_h,\varphi_h\rangle=\langle R \varphi_h,\varphi_h\rangle=\langle \pi_h(c^+b) \varphi_h,\varphi_h\rangle=h(c^+b) \end{align}
Inserting (\ref{tvarphibc2}) into (\ref{tvarphibc1}) we get $$|\langle T\pi_h(b)\varphi_h,\pi_h(c)\varphi_h\rangle|^2\leq h(c^+b).$$ Hence $P_A(f,g)\leq h(c^+b)$\, by Theorem \ref{sumofgnspifg}. \end{proof} Remarks. 1. The assumption $c^+b\in \sum A^2$ was only needed to ensure that the operator\, $R=\pi(c^+b)\equiv(\pi_h)^{**}(c^+b)$\, is {\it positive}. Clearly, this is satisfied if\, $F(c^+b)\geq 0$\, for all positive linear functionals $F$ on $A$.
2. If the closures of the GNS representations $\pi_f$ and $\pi_g$ are self-adjoint, we can set\, $\pi:=\overline{\pi}_h$ in the preceding proof and it suffices to assume that\, $h(a^+c^+ba)\geq 0$\, for all $a\in A$ instead of\, $c^+b\in \sum A^2.$ \section{Two Applications}\label{appl}
To formulate our first application we begin with some preliminaries.
Let $\rho$ be a closed $*$-representation of $A$. We denote by ${\bf B}_1(\rho(A))_+$ the set of positive trace class operators on ${\mathcal{H}}(\rho)$ such that $t{\mathcal{H}}(\rho)\subseteq {\mathcal{D}}(\rho)$ and the closure of $\rho(a)t\rho(b)$ is trace class for all $a,b\in A$.
Now let $t\in{\bf B}_1(\rho(A))_+$. We define a positive linear functional $f_t$ by $$ f_t(a):={\rm Tr}\, \rho(a)t,\quad a\in A, $$ where ${\rm Tr}$ always denotes the trace on the Hilbert space ${\mathcal{H}}(\rho)$. Note that $f_t(a)\geq 0$ if $\rho(a)\geq 0$ (that is, $\langle \rho(a)\varphi,\varphi\rangle \geq 0$ for all $\varphi\in {\mathcal{D}}(\rho)$).
In unbounded representation theory a large class of positive linear functionals is of the form $f_t$. We illustrate this by restating the following theorem proved in \cite{schmu78}. Recall that a {\it Frechet--Montel space} is a complete metrizable locally convex space such that each bounded sequence has a convergent subsequence.
\begin{tht}\label{tracereptheorem} Let $f$ be a linear functional on $A$ and let $\rho$ be a closed $*$-representation of $A$. Suppose that the locally convex space\, ${\mathcal{D}}(\rho)[{\mathfrak{t}}_{\rho(A)}]$\, is a Frechet-Montel space and $f(a)\geq 0$ whenever $\rho(a)\geq 0$ for $a\in A$. Then there exists
an operator\, $t\in {\bf B}(\rho(A))_+$\, such that\, $f=f_t$, that is, $f(a)={\rm Tr}~ \rho(a)t$\, for $a\in {\mathcal{A}}$. \end{tht}
Further, let ${\mathcal{M}}$ be a type $I$ factor acting on the Hilbert space ${\mathcal{H}}(\rho)$ and let ${\rm tr_{\mathcal{M}}}$ denote its canonical trace. Since in particular $t$ is of trace class, $F_t(x)={\rm Tr}\, x t$, $x\in {\mathcal{M}}$, defines a positive normal linear functional $F_t$ on ${\mathcal{M}}$. Hence there exists a unique positive element $\hat{t}\in {\mathcal{M}}$ such that\, ${\rm tr}_{\mathcal{M}}\, (\, \hat{t}\, )<\infty$ and \begin{align}\label{that} F_t(x)\equiv {\rm Tr}\, x t={\rm tr}_{\mathcal{M}}\, x\hat{t}\quad {\rm for}\quad x\in {\mathcal{M}}. \end{align} The element $\hat{t}$ can be obtained as follows. Since ${\mathcal{M}}$ is a type $I$ factor, there exist Hilbert spaces ${\mathcal{H}}_0$ and ${\mathcal{H}}_1$ such that, up to unitary equvalence, ${\mathcal{H}}(\pi)={\mathcal{H}}_0\otimes {\mathcal{H}}_1$\, and\, ${\mathcal{M}}={\bf B}({\mathcal{H}}_0)\otimes \mathbb{C}\cdot I_{{\mathcal{H}}_1}$. The canonical trace of ${\mathcal{M}}$ is then given by\, ${\rm tr}_{\mathcal{M}} (y\otimes \lambda\cdot I_{{\mathcal{H}}_1}):= {\rm Tr}_1\,\lambda y$, where\, ${\rm Tr}_1$ denotes the trace on the Hilbert space ${\mathcal{H}}_1$. Now $\tilde{F}_t(y):= F_t(y\otimes I_{{\mathcal{H}}_1})$, $y\in{\bf B}({\mathcal{H}}_0)$, defines a positive normal linear functional $\tilde{F}_t$ on ${\bf B}({\mathcal{H}}_0)$. Hence there exists a unique positive trace class operator $\tilde{t}$ on the Hilbert space ${\mathcal{H}}_0$ such $\tilde{F}_t(y)={\rm Tr}\, y\tilde{t}$\, for $y\in{\bf B}({\mathcal{H}}_0)$. Set $\hat{t}:=\tilde{t}\otimes I_{{\mathcal{H}}_1}$. Then we have ${\rm tr}_{\mathcal{M}}\, \hat{t}={\rm Tr}_1\, t< \infty$ and (\ref{that}) holds by construction.
Note that ${\rm tr}_{\mathcal{M}}= {\rm Tr}$\, and\, $t=\hat{t}$\, if\, ${\mathcal{M}}={\bf B}({\mathcal{H}}(\rho))$.
\begin{tht}\label{tracegen} Let\, $\rho$\, be a closed $*$-representation of $A$ such that the von Neumann algebra ${\mathcal{M}}:=(\rho(A)^\prime_{ss})^\prime$ is a type $I$ factor. For $s,t\in {\bf B}(\rho(A))_+$, let $f_s, f_t $ denote the positive linear functionals on ${\mathcal{A}}$ defined by \begin{align*} f_s(a)={\rm Tr}\, \rho(a)s, \quad f_t(a)={\rm Tr}\, \rho(a)t \quad {\rm for}\quad a\in {\mathcal{A}}. \end{align*} Suppose that the GNS representations $\pi_{f_s}$ and $\pi_{f_t}$ are essentially self-adjoint. Then \begin{align}\label{fsfttrans0}
P_A(f_s,f_t)= ({\rm tr}_{\mathcal{M}}\,|\hat{t}^{1/2}\hat{s}^{1/2}|)^2= (\,{\rm tr}_{\mathcal{M}}\, (\hat{s}^{1/2}\, \hat{t}\, \hat{s}^{1/2})^{1/2}\, )^2. \end{align} \end{tht} \begin{proof} Let $\rho_\infty$ be the orthogonal sum $\oplus_{n=0}^\infty~ \rho$ on\, ${\mathcal{H}}_\infty=\oplus_{n=0}^\infty~ {\mathcal{H}}(\rho)$.\, Since $\rho$ is biclosed, so is the $*$-representation $\rho_\infty$ of $A$. We want to apply Theorem \ref{sumofgnspifg1}. First we will describe the GNS representations\, $\pi_{f_s}$\, and\, $\pi_{f_t}$\, as $*$-subrepresentations of\, $\rho_\infty$.
The result is well-known if ${\mathcal{H}}(\rho)$ is finite dimensional \cite{uhlmann76}, so we can assume that ${\mathcal{H}}(\rho)$ is infinite dimensional. Since $s\in{\bf B}_1(\rho(A))_+$, there are a sequence $(\lambda_n)_{n\in \mathbb{N}}$ of nonnegative numbers and an orthonormal sequence $(\varphi_n)_{n\in \mathbb{N}}$ of ${\mathcal{H}}(\rho)$ such that $\varphi_n\in {\mathcal{D}}(\rho)$ for $n\in \mathbb{N}$, $$s\varphi =\sum_n \langle \varphi, \varphi_n\rangle\lambda_n\varphi_n \quad{\rm for}\quad \varphi\in {\mathcal{H}}(\rho),$$ and $(\rho(a)\lambda_n^{1/2}\varphi_n)_{n\in \mathbb{N}}\in {\mathcal{H}}_\infty$ for all $a\in {\mathcal{A}}$. Further, for $a\in A$ we have \begin{align}\label{f_s}
f_s(a)=\sum_{n=1}^\infty\, \langle \rho(a) \varphi_n,\lambda_n\varphi_n\rangle.
\end{align} All these facts are contained in Propositions 5.1.9 and 5.1.12 in \cite{schmu90}. Hence $$\rho_\Phi(a)(\rho(b)\lambda_n^{1/2}\varphi_n):=(\rho(ab)\lambda_n^{1/2}\varphi_n),\quad a,b\in {\mathcal{A}},$$
defines a $\ast$-representation $\rho_\Phi$ of ${\mathcal{A}}$ on the domain $${\mathcal{D}}(\rho_\Phi):=\{(\rho(a)\lambda_n^{1/2}\varphi_n)_{n\in \mathbb{N}};a\in {\mathcal{A}}\}$$ with algebraically cyclic vector\, $\Phi:=(\lambda_n^{1/2}\varphi_n)_{n\in \mathbb{N}}$. From (\ref{f_s}) we derive $$ f_s(a) =\sum_{n=1}^\infty\, \langle \rho(a) \lambda_n^{1/2}\varphi_n,\lambda_n^{1/2}\varphi_n\rangle =\langle \rho_\Phi (a)\Phi,\Phi \rangle=:f_\Phi(a),~~a\in A , $$ that is, $f_s$ is equal to the vector functional $f_\Phi$ in the representaton $\rho_\Phi$. Therefore, by the uniqueness of the GNS representation, $\pi_{f_s}$ is unitarily equivalent to $\rho_\Phi$.
Likewise, the GNS representation $\pi_{f_t}$ is unitarily equivalent to the corresponding $*$-representation $\rho_\Psi$, where $t\varphi=\sum_n \langle \varphi ,\psi_n\rangle\mu_n\psi_n$
is a corresponding representation of\, the operator $t$\, and\, $\Psi:=(\mu_n^{1/2}\psi_n)_{n\in \mathbb{N}}$. Clearly, since\, $\rho_\Phi \subseteq \rho_\infty$\, and\, $\rho_\Phi \subseteq \rho_\infty$,\, we have $\Phi \in S(\rho_\infty, f_s)$ and $\Psi \in S(\rho_\infty, f_t)$.
Let ${\mathcal{M}}_\infty$ denote the von Neumann algebra $(\rho_\infty(A)^\prime_{ss})^\prime$. Then, by Theorem \ref{sumofgnspifg1}, we have \begin{align}\label{papm}P_A (f_s,f_t)\equiv P_A(f_\Phi,f_\Psi)=P_{{\mathcal{M}}_\infty}( F_\Phi, F_\Psi). \end{align}
Let $x\in {\bf B}({\mathcal{H}}_\infty)$. We write $x$ as a matrix\, $(x_{jk})_{j,k\in \mathbb{N}}$\, with entries\, $x_{jk}\in {\bf B}({\mathcal{H}}(\rho)))$. Clearly, $x$ belongs to in $\rho_\infty({\mathcal{A}})^\prime_{ss}$\, if and only if each entry $x_{jk}$ is in\, $\rho(A)^\prime_{ss}$. Further, it is easily verified that $x$ is in\, $(\rho_\infty({\mathcal{A}})^\prime_{ss})^\prime$\, if and only if there is a (uniquely determined) operator $x_0 \in(\rho(A)^\prime_{ss})^\prime$ such that $x_{jk}=\delta_{jk} x_0$ for all $j,k\in \mathbb{N}$. The map $\pi(x_0):=x$\, defines a $*$-isomorphism of von Neumann algebras ${\mathcal{M}}:=(\rho(A)^\prime_{ss})^\prime$ and ${\mathcal{M}}_\infty=(\rho_\infty({\mathcal{A}})^\prime_{ss})^\prime$, that is, $\pi$ is a $\ast$-representation of ${\mathcal{M}}$.
As above, we let $F_s$ and $F_t$ denote the normal functionals on ${\mathcal{M}}$ defined by $F_s(x):={\rm Tr}\, xs$ and $F_t(x):={\rm Tr}\, xt$, $x\in {\mathcal{M}}$. Repeating the preceding reasoning with $\rho$ and $A$ replaced by $\pi$ and ${\mathcal{M}}$, respectively, we obtain $F_s(\cdot ) = \langle \pi_0(\cdot) \Phi,\Phi \rangle\equiv F_\Phi(\cdot)$ and $F_t=F_\Psi$. Hence $P_{\mathcal{M}}(F_s,F_t)=P_{{\mathcal{M}}_\infty}(F_\Phi,F_\Psi)$, so that \begin{align}\label{pafsftm} P_A (f_s,f_t) =P_{\mathcal{M}}(F_s,F_t) \end{align} by (\ref{papm}). It is proved in \cite[Corollary 1]{alberti03} (see also \cite{uhlmann76}) that
$$P_{\mathcal{M}}(F_s,F_t)= ({\rm tr}_{\mathcal{M}}\,|\hat{t}^{1/2}\, \hat{s}^{1/2}|)^2.$$ Combined with (\ref{pafsftm}) this yields (\ref{fsfttrans}) and completes the proof. \end{proof} Let us remain the assumptions and the notations of Theorem \ref{tracegen}. In general, $P_A(f_s,f_t)$ is different from $({\rm Tr}\, ({s}^{1/2}\, {t}\, {s}^{1/2})^{1/2}\, )^2$ a simple examples show. Howover, if in addition $\rho$ is {\it irreducible} (that is, if $\rho(A)^\prime_{ss}=\mathbb{C}\cdot I$), then $s=\hat{s}$ and $t=\hat{t}$ as noted above and therefore by (\ref{fsfttrans0}) we have \begin{align}\label{trairr} P_A(f_s,f_t)= (\,{\rm Tr}\, (s^{1/2}\, t\, s^{1/2})^{1/2}\, )^2. \end{align}
We now apply the preceding theorem to an interesting example. \begin{thex}\label{schroding} ({\it Schr\"odinger representation of the Weyl algebra})\\ Let $A$ be the Weyl algebra, that is, $A$ is the unital $*$-algebra generated by two hermitian generators $p$ and $q$ satisfying $$pq-qp=-\mathrm{i} 1 ,$$ and let $\rho$ be the Schr\"odinger representation of $A$, that is, \begin{align}\label{schrorep}(\rho(q)\varphi )(x)=x\varphi (x), \quad (\rho(p)\varphi)(x)=-\mathrm{i} \frac{d}{dx}\varphi(x),\quad \varphi \in{\mathcal{D}}(\rho):={\mathcal{S}}(\mathbb{R}), \end{align}
on $L^2(\mathbb{R})$. Since $\rho$ is irreducible, $\rho_\infty(A)^\prime_{ss}=\mathbb{C} \cdot I$. Hence\, ${\mathcal{M}}={\bf B}({\mathcal{H}}(\rho))$\, and\, ${\rm tr}_{\mathcal{M}}={\rm Tr}$. Therefore, if $s,t\in {\bf B}(\pi(A))_+$ and the GNS representations $\pi_{f_s}$ and $\pi_{f_t}$ are essentially self-adjoint, it follows from Theorem \ref{tracegen} and formula (\ref{trairr}) that
\begin{align}\label{fsfttrans}
P_A(f_s,f_t)= ({\rm Tr}\,|t^{1/2}s^{1/2}|)^2= (\,{\rm Tr}\, (s^{1/2}ts^{1/2})^{1/2}\, )^2. \end{align}
Let us specialize this to the rank one case, that is, let $s=\varphi\otimes \varphi$ and $t=\psi\otimes \psi$ with $\varphi,\psi \in {\mathcal{D}}(\rho)$, so that $f_s(a)=\langle \rho(a)\varphi,\varphi\rangle$ and $f_t(a)=\langle \rho(a)\psi,\psi\rangle$ for $a\in A$. Then formula (\ref{fsfttrans}) yields \begin{align}\label{fsftvector}
P_A(f_s,f_t)= |\langle \varphi,\psi\rangle |^2. \end{align} Recall that (\ref{fsftvector}) holds under the assumption that the GNS representations $\pi_{f_s}$ and $\pi_{f_t}$ are essentially self-adjoint. We shall see in section\, \ref{vectorweyl} below that (\ref{fsftvector}) is no longer true if the latter assumption is omitted. \end{thex} Now we turn to the second main application. \begin{tht}\label{commtrans} Let $X$ be a locally compact topological Hausdorff space and let $A$ be a $*$-subalgebra of\, $C(X)$\, which contains the constant function $1$ and separates the points of $X$. Let $\mu$ be a positive regular Borel measure on $X$ such that $A\subseteq L^1(X,\mu)$ and let $\eta, \xi\in L^\infty(X,\mu)$ be nonnegative functions. Define positive linear functionals $f_\eta$ and $f_\xi$ on $A$ by \begin{align}\label{fetaxi} f_\eta(a)=\int_X a(x)\eta(x)\, d\mu(x),\quad f_\xi(a)=\int_X a(x)\xi(x)\, d\mu(x)\, \, {\rm for}\,\, a\in A. \end{align} Suppose that the GNS representations $\pi_{f_\eta}$ and $\pi_{f_\xi}$ are essentially self-adjoint. Then \begin{align}\label{commutativecase} P_{\mathcal{A}}(f_\eta,f_\xi)= \bigg(\,\int_X\, \eta(x)^{1/2}\xi(x)^{1/2}\, d\mu(x) \bigg)^2. \end{align} \end{tht} \begin{proof} We define a closed $*$-representation $\pi$ of the $*$-algebra $A$ on $L^2(X,\mu)$ by $\pi(a)\varphi=a\cdot \varphi$ for $a\in A$ and $\varphi$ in the domain $${\mathcal{D}}(\pi):=\{\varphi \in L^2(X,\mu):a\cdot \varphi\in L^2(X,\mu) ~{\rm for}~ a\in A\}.$$
First we prove that $\pi(A)^\prime_{ss}=L^\infty (X,\mu)$, where the functions of\, $L^\infty (X,\mu)$ act as multiplication operators on\, $L^2 (X,\mu).$ Let ${\mathfrak{A}}$ denote the $*$-subalgebra of $L^\infty (X,\mu)$ generated by the functions $(a\pm \mathrm{i})^{-1}$, where $a=a^+\in A$. Obviously, $L^\infty (X,\mu)\subseteq \pi(A)^\prime_{ss}$.
Conversely, let $x\in \pi(A)^\prime_{ss}$. It is straightforward to show that for any $a=a^+\in A$ the operator\, $\overline{\pi(a)}$\, is self-adjoint and hence equal to the (self-adjoint) multiplication operator by the function $a$. By definition $x$ commutes with\, $\overline{\pi(a)}$\,, hence with\, $(\overline{\pi(a)}\pm \mathrm{i} I)^{-1}=(a\pm \mathrm{i})^{-1}$, and therefore with the whole algebra ${\mathfrak{A}}$. The $*$-algebra $A$ separates the points of $X$, so does the $*$-algebra ${\mathfrak{A}}$. Therefore, from the Stone--Weierstrass theorem \cite[Corollary 8.2]{conway90}, applied to the one point compactification of $X$, it follows that ${\mathfrak{A}}$ is norm dense in $C_0(X)$. Hence $x$ commutes with $C_0(X)$ and so with its closure \, $L^\infty (X,\mu)$\, in the weak operator topology. Thus, $x\in L^\infty (X,\mu)^\prime$. Since $ L^\infty (X,\mu)^\prime= L^\infty (X,\mu)$, we have shown that $\pi(A)^\prime_{ss}=L^\infty (X,\mu)$. Therefore, ${\mathcal{M}}:=(\pi(A)^\prime_{ss})^\prime=L^\infty (X,\mu)$.
Let $F_\eta$ and $F_\xi$ denote the positive linear functionals on ${\mathcal{M}}$ defined by (\ref{fetaxi}) with $A$ replaced by ${\mathcal{M}}$.
For ${\mathcal{M}}=L^\infty (X,\mu)$ it is well-known (see e.g. formula (14) in \cite{alberti83}) that $P_{\mathcal{M}}(F_\eta,F_\xi)= (\,\int_X\, \eta(x)^{1/2}\xi(x)^{1/2}\, d\mu(x) \,)^2. $ Since $P_A(f_\eta,f_\xi)=P_{\mathcal{M}}(F_\eta,F_\xi)$ by Theorem \ref{sumofgnspifg1}, we obtain (\ref{commutativecase}).
\end{proof}
In the following two examples we reconsider the one dimensional Hamburger moment problem (see Example \ref{hambmp}) and we specialize the preceding theorem to the case where $X=\mathbb{R}$ and $A=\mathbb{C}[x]$. \begin{thex}\label{detmpform} {\it Determinate Hamburger moment problems}\\ Let $\mu_\eta$ and $\mu_\xi$ be the positive Borel measures on $\mathbb{R}$ defined by $d\mu_\eta=\eta d\mu$ and $d\mu_\xi =\xi d\mu$. Since $\mathbb{C}[x]\in L^1(\mathbb{R},\mu)$ and $\eta, \xi\in L^\infty(\mathbb{R},\mu)$, we have $\mu_\eta,\mu_\xi\in M(\mathbb{R})$. If both measures $\mu_\eta$ and $\mu_\xi$ are determinate, then the GNS representations $\pi_{f_{\mu_\eta}}$ and $\pi_{f_{\mu_\xi}}$ are essentially self-adjoint (as shown in Example \ref{hambmp}) and hence formula (\ref{commutativecase}) holds by Theorem \ref{commtrans}. \end{thex}
\begin{thex}\label{indetr} {\it {Indeterminate Hamburger moment problems}}\\ Suppose $\nu\in M(\mathbb{R})$ is an indeterminate measure such that $\nu(\mathbb{R})=1$.
Let $V_\nu$ denote the set of all positive Borel measures $\mu\in M(\mathbb{R})$ which have the same moments as $\nu$, that is, $\int x^n d\nu(x)=\int x^n d\mu(x)$ for all $n\in \mathbb{N}_0$. Since $\nu$ is indeterminate and $V_\nu$ is convex and weakly compact, there exists a measure $\mu\in V_\nu$ which is not an extreme point of $V_\nu$, that is, there are measures $\mu_1,\mu_2\in V_\nu$, $\mu_j\neq \mu$ for $j=1,2$, such that $\mu=\frac{1}{2}(\mu_1+\mu_2)$. Since $\mu_j(M)\leq 2\mu(M)$ for all measurable sets $M$ and $\mu_1+\mu_2=2\mu$, there exists functions $\eta, \xi \in L^\infty(\mathbb{R},\nu)$ satisfying \begin{align}\label{propetaxi}
\eta(x)+\xi(x)=2 ,~~ \|\xi\|_\infty \leq 2,~~ \|\eta\|_\infty \leq 2,~~ d\mu_1=\eta d\mu,~~ d\mu_2=\xi d\mu. \end{align}
Define $f(p)=\int p(x)d\mu(x)$ for $p\in \mathbb{C}[x]$. Since $\mu_1,\mu_2,\mu\in V_\nu$, the functionals $f_\eta$ and $f_\xi$ defined by (\ref{fetaxi}) are equal to $f$. Therefore, since $f(1)=\mu(\mathbb{R})=\nu(\mathbb{R})=1$, we have $P_A(f_\eta,f_\xi)=P_A(f,f)=1$.
Put $J:=(\,\int_X\, \eta(x)^{1/2}\xi(x)^{1/2}\, d\mu(x) )^2.$ From (\ref{propetaxi}) we obtain $\eta(x)\xi(x)=\eta(x)(2-\eta(x))\leq 1$ and hence $J\leq 1$, since $\mu(\mathbb{R})=1$. If $J$ would be equal to $1$, then $\eta(x)(2-\eta(x))=1$ $\mu$-a.e. on $\mathbb{R}$ which implies that $\eta(x)=1$\, $\mu$-a.e. on $\mathbb{R}$ by (\ref{propetaxi}). But then $\mu_1=\mu_2=\mu$ which contradicts the choice of measures $\mu_1$ and $\mu_2$. Thus we have proved that $J\neq 1=P_A(f_\eta,f_\xi)$, that is, formula (\ref{commutativecase}) does not hold in this case. \end{thex} The classical moment problem leads to a number of open problems concerning transition probabilities. We will state three of them.
Let $M(\mathbb{R}^d)$, $d\in \mathbb{N}$, denote the set of positive Borel measures $\mu$ on $\mathbb{R}^d$ such that all polynomials $p(x_1,\dots,x_d)\in \mathbb{C}[x_1,\dots,x_d]$ are $\mu$-integrable. For $\mu \in M(\mathbb{R}^d)$ we define a positive linear functional $g_\mu$ on the $*$-algebra $A:=\mathbb{C}[x_1,\dots,x_d]$ by $$g_\mu(p)=\int p\, d\mu,\quad p \in \mathbb{C}[x_1,\dots,x_d].$$ Then the main problem is the following:\\ {\it Problem 1: Given $\mu,\nu \in M(\mathbb{R}^d)$, what is\, $P_A(g_\mu,g_\nu)$ ?}
This seems to be a difficult problem and it is hard to expect a sufficiently complete answer. For $d=1$ Example \ref{detmpform} contains some answer under the assumption that both measures $\mu_\eta$ and $\mu_\xi$ are determinate. This suggests the following questions:\\ {\it Problem 2: What about the case when the measures $\mu_\eta$ and/or $\mu_\xi$ in Example \ref{detmpform} are not determinate?}\\ {\it Problem 3: Is formula (\ref{commutativecase})
still valid in the multi-dimensional case $d>1$ if $\mu_\eta$ and $\mu_\xi$ are determinate ?}
It can be shown that the answer to problem 3 is affirmative if all multiplication operators $\pi_\mu(x_j)$, $j=1,\dots,d,$ are essentially self-adjoint. The latter assumption is sufficient, but not neccessary for $\mu$ being determinate \cite{ps}.
In the multi-dimensional case determinacy turns out to be much more difficult than in the one-dimensional case, see e.g. \cite{ps}.
\section{Vector Functionals of the Schr\"odinger Representation}\label{vectorweyl} The crucial assumption for the results in preceding sections was the essential self-adjointness of GNS representations $\pi_f$ and $\pi_g$. In this section we consider the simplest situation where $\pi_f$ and $\pi_g$ are not essentially self-adjoint.
In this section $A$ denotes the Weyl algebra (see Example \ref{schroding}) and $\pi$ is the Schr\"odinger representation of $A$ given by (\ref{schrorep}). For $\eta \in {\mathcal{D}}(\pi)={\mathcal{S}}(\mathbb{R})$ let $f_\eta$ denote the positive linear functional $f_\eta$ on $A$ given by $$f_\eta(x)=\langle \pi(x)\eta,\eta\rangle,\quad x\in A.$$ Consider the following condition on the function $\eta$:
$(*)$ {\it There are finitely many mutually disjoint open intervals $J_l(\eta)=(\alpha_l,\beta_l)$, $l=1,\dots,r$, such that $\eta(t)\neq 0$ for $t\in J(\eta):=\cup_l\, J_l(\eta) $ and $\eta^{(n)}(t)=0$ for $t\in \mathbb{R}/ J(\eta)$ and all $n\in \mathbb{N}_0$.}
The main result of this section is the following theorem. \begin{tht}\label{weylstates} Suppose that $\varphi$ and $\psi$ are functions of $C_0^\infty(\mathbb{R})$ satisfying condition $(*)$. Then \begin{align}\label{vectorstates}
P_A(f_\varphi,f_\psi)= \bigg(\sum_{k,l} \bigg|\int_{{\mathcal{J}}_k(\varphi)\cap{\mathcal{J}}_l(\psi)}\, \varphi(x) \overline{\psi(x)}\, dx \bigg|\bigg)^2. \end{align} (If ~ ${\mathcal{J}}_k(\varphi)\cap{\mathcal{J}}_l(\psi)$~ is empty, the corresponding integral is set zero.) \end{tht}
Before we turn to the proof of the theorem let us discuss formula (\ref{vectorstates}) in two simple cases.
$\bullet$ If both sets ${\mathcal{J}}(\varphi)$ and ${\mathcal{J}}(\psi)$ consist of a single interval, then
$$P_A(f_\varphi,f_\psi)= \bigg|\int_\mathbb{R}\, \varphi(x) \overline{\psi(x)}\, dx \bigg|^2=|\langle \varphi,\psi\rangle|^2,$$ that is, in this case formula (\ref{fsftvector}) holds.
$\bullet$ Let $\varphi, \psi\in C_0^\infty(\mathbb{R}) $ be such that ${\mathcal{J}}(\varphi)={\mathcal{J}}(\psi)$, ${\mathcal{J}}_k(\varphi)={\mathcal{J}}_k(\psi)$ and $\varphi(x)=\epsilon_k\psi(x)$ on ${\mathcal{J}}_k(\varphi)$ for $k=1,\cdots,r$, where $\epsilon_k\in \{1,-1\}$. Then formula (\ref{vectorstates}) yields\, $P_A(f_\varphi,f_\psi)=\|\varphi\|^4$.\, It is easy to choose $\varphi\neq 0$ and the numbers $\epsilon_k$ such that $\langle \varphi,\psi\rangle=0$, so formula (\ref{fsftvector}) does not hold in this case.
The proof of Theorem \ref{weylstates} requires a number of technical preparations. The first aim is to desribe the closure $\overline{\pi}_{f_\eta}$ of the GNS representation $\pi_{f_\eta}$ for a function $ \eta\in C_0^\infty(\mathbb{R})$ satisfying condition $(*)$.
Let $\rho_\eta$ denote the restriction of $\pi$ to the dense domain \begin{align*} {\mathcal{D}}(\rho_\eta)= \{ \xi\in \bigoplus_{l=1}^r C^\infty((\alpha_l,\beta_l)): \xi^{(k)}(\alpha_l)=\xi^{(k)}(\beta_l)=0,\, k\in \mathbb{N}_0,\, l=1,\dots,r \} \end{align*} in the Hilbert space $L^2({\mathcal{J}}(\eta))$. The following lemma says that\, $\rho_\eta$\, is unitarily equivalent to $\overline{\pi}_{f_\eta}$.
\begin{thl}\label{gnseta} There is a unitary operator $U$ of\, ${\mathcal{H}}(\pi_{f_\eta})$\, onto $L^2({\mathcal{J}}(\eta))$ given by $U(\pi_{f_\eta}(a)\eta)= \rho_\eta(a)\eta$, $a\in A$, such that\, $\rho_\eta=U\overline{\pi}_{f_\eta} U^\ast$. \end{thl} \begin{proof} From the properties of GNS representations it follows easily that the unitary operator $U$ defined by $U(\pi_{f_\eta}(a)\eta)= \rho_\eta(a)\eta$, $a\in A$, provides unitary equivalences $ \tau_\eta=U\pi_{f_\eta} U^\ast$ and $\overline{\tau}_\eta=U\overline{\pi}_{f_\eta} U^\ast$, where $\tau_\eta$ denotes the restriction of $\pi$ to ${\mathcal{D}}(\rho_\eta)= \pi(A)\eta$. Clearly, $\tau_\eta\subseteq \rho_\eta$ and hence $\overline{\tau}_\eta\subseteq \rho_\eta$, since $\rho_\eta$ is obviously closed. To prove the statement it therefore suffices to show that $\rho_\eta$ is the closure of $\tau_\eta$, that is, $\pi(A)\eta$ is dense in ${\mathcal{D}}(\rho_\eta)$ in the graph topology of\, $\rho_\eta(A)$. For this the auxiliary Lemmas \ref{denselemma1} and \ref{denselemma2} proved below are essentially used.
Each element $a\in A$ is of a finite sum of terms $f(q)p^n$, where $n\in \mathbb{N}_0$ and $f\in \mathbb{C}[q]$. Since $ \eta\in C_0^\infty(\mathbb{R})$, the set\, ${\mathcal{J}}(\eta)$ and hence the operators $\pi_0(f(q))$ are bounded. Therefore, the graph topology ${\mathfrak{t}}_{\rho_\eta(A)}$ is generated by the seminorms $\|\rho_\eta(p)^n \cdot\|$, $n\in \mathbb{N}_0$, on $ {\mathcal{D}}(\rho_\eta)$. Let $\psi\in {\mathcal{D}}(\rho_\eta)$.
First assume that $\psi$ vanishes in some neighbourhoods of the end points $\alpha_l, \beta_l$. Then, by Lemma \ref{denselemma2}, for any $m\in\mathbb{N}$ there is sequence\, $(f_n)_{n\in \mathbb{N}}$\, of polynomials such that $$\lim_n~ \rho_\eta((\mathrm{i} p)^k)(\rho_\eta( f(q))\eta-\psi)=\lim_n~((f_n\eta)^{(k)} -\psi^{(k)})=0 $$ in $L^2({\mathcal{J}}(\eta))$ for $k=0,\dots,m$. This shows that $\psi$ is in the closure of $\rho_\eta(A)\eta$ with respect the graph topology of $\rho_\eta(A)$.
The case of a general function $\psi$ is reduced to the preceding case as follows. Suppose that $\varepsilon>0$ and~ $2\varepsilon <\min_l\, |\beta_l-\alpha_l|$. We define \begin{align*} \psi_\varepsilon(x)=\psi(x-\varepsilon+2\varepsilon(x-\alpha_l-\varepsilon)(\beta_l-\alpha_l-2\varepsilon)^{-1})\quad{\rm for}\quad x\in (\alpha_l,\beta_l) \end{align*} and $l=1,\dots,r$ and $\psi_\varepsilon(x)=0$ otherwise. Then $\psi_\varepsilon$ vanishes in some neighbourhoods of the end points\, $\alpha_l, \beta_l$, so $\psi_\varepsilon$ is in the closure of $\rho_\eta(A)\eta$ as shown in the preceding paragraph. Using the dominated Lebesgue convergence theorem it follows that $$\lim_{\varepsilon\to +0}\rho_\eta((\mathrm{i} p)^k)(\psi_\varepsilon-\psi)= \lim_{\varepsilon\to +0} (\psi_\varepsilon^{(k)}-\psi ^{(k)})=0$$ in\, $L^2({\mathcal{J}}(\eta))$ for $k\in \mathbb{N}_0$. Therefore, since $\psi_\varepsilon$ is in the closure of $\rho_\eta(A)\eta$, so is $\psi$. \end{proof} \begin{thl}\label{denselemma1} Suppose that\, $g\in C^{(k)}([\alpha,\beta])$,\, where $\alpha,\beta \in \mathbb{R}$ and $k\in \mathbb{N}$. Then there exists a sequence $(f_n)_{n\in \mathbb{N}}$ of polynomials such that\, $f^{(j)}_n(x)\Longrightarrow g^{(j)}(x)$\, uniformly on\, $[\alpha,\beta]$\, for $j=0,\dots,k$ as\, $n\to \infty$. \end{thl} \begin{proof} By the Weierstrass theorem there is a sequence $(h_n)_{n\in \mathbb{N}}$ of polynomials such that $h_n(x) \Longrightarrow g^{(k)}(x)$ uniformly on $[\alpha,\beta]$. Fix $\gamma\in [\alpha,\beta]$ and set\, $h_{n,k}:=h_n$. Then $$h_{n,k-1}(x):=g^{(k)}(\gamma)+ \int^x_\gamma h_{n,k}(s)\, ds \Longrightarrow g^{(k-1)}(x)=g^{(k)}(\gamma)+\int^x_\gamma g^{(k)}(s)\, ds. $$
Clearly, $(h_{n,k-1})_{n\in \mathbb{N}}$ is sequence of polynomials and we have $h_{n,k-1}^\prime(x)=h_{n,k}(x)$ on $[\alpha,\beta]$. Proceeding by induction we obtain sequences $(h_{n,k-j})_{n\in \mathbb{N}}$, $j=0,\cdots,k$, of polynomials such that\, $h_{n,k-j}(x)\Longrightarrow g^{(k-j)}(x)$\, and\, $h_{n,k-j}^\prime(x)=h_{n,k+1-j}(x)$\, on $[\alpha,\beta]$. Setting\, $f_n:=h_{n,0}$\, the sequence\, $(f_n)_{n\in \mathbb{N}}$\, has the desired properties. \end{proof}
\begin{thl}\label{denselemma2} Suppose that $\eta\in C_0^\infty (\mathbb{R})$ satisfies condition $(*)$. Let $\psi\in \bigoplus_{l=1}^r C^{(m)}_0((\alpha_l,\beta_l))$, where $m\in \mathbb{N}$. Then there exists a sequence $(f_n)_{n\in \mathbb{N}}$ of polynomials such that~ $\lim\nolimits_{n\to \infty}\,(f_n\eta)^{(k)}= \psi^{(k)}$\, in $L^2({\mathcal{J}}(\eta))$ for $k=0,\dots,m$. \end{thl} \begin{proof} By the assumption $\psi$ vanishes in some neighbourhoods of the end points $\alpha_l$ and $\beta_l$. Set $\psi(x)=0$ on $\mathbb{R}/ {\mathcal{J}}(\eta)$. Then, $\psi\eta^{-1}$ becomes a function\, of $ C^{(m)}([\alpha,\beta])$, where $\alpha:=\min_l \, \alpha_l$ and $\beta:=\max_l\, \beta_l$. Therefore, by Lemma \ref{denselemma1}, there exists a sequence $(f_n)_{n\in \mathbb{N}}$ of polynomials such that\, $f^{(j)}_n(x)\Longrightarrow (\psi \eta^{-1})^{(j)}(x) $\, for $j=0,\dots, m$\, uniformly on\, $[\alpha,\beta]$. Then \begin{align*} (f_n\eta)^{(k)}=\sum_{j=0}^k\, {k \choose j} f_n^{(j)}\eta^{(k-j)}\Longrightarrow \sum_{j=0}^k\, { k \choose j} (\psi \eta^{-1})^{(j)}\eta^{(k-j)} =\psi^{(k)} \end{align*} as\, $n\to \infty$\, uniformly on\, $[\alpha,\beta]$\, and hence in $L^2({\mathcal{J}}(\eta))$. \end{proof}
Now we are able to give the\\ {\it Proof of Thorem \ref{weylstates}:} Let us abbreviate $\pi_\varphi=\overline{\pi}_{f_\varphi}$ and $\pi_\psi=\overline{\pi}_{f_\psi}$. By Lemma \ref{gnseta} the closure $\pi_\psi=\overline{\pi}_{f_\psi}$ of the GNS representation $\pi_{f_\psi}$ is unitarily equivalent to the representation $\rho_\psi$. For notational simplicy we shall identify the representations $\pi_\psi$ and $\rho_\psi$ via the unitary $U$ defined in Lemma \ref{gnseta}. Using this description of $\pi_\psi\cong \rho_\psi$ it is straightforward to check that the domain ${\mathcal{D}}((\pi_\psi)^*)$ consists of all functions $g\in C^\infty({\mathcal{J}}(\psi))$ such that their restrictions to ${\mathcal{J}}_l(\psi)$ extend to functions of $C^\infty(\,\overline{{\mathcal{J}}_l(\psi)}\, )$ and $g(t)=0$ on $ \mathbb{R}/\, \overline{{\mathcal{J}}(\psi)}$. Further, we have $(\pi_\psi)^*(f(q))g =f \cdot g $ and $(\pi_\psi)^*(p)g =-\mathrm{i} g^\prime $ for $g \in {\mathcal{D}}((\pi_\psi)^*)$ and $f\in \mathbb{C}[q]$.
Suppose that $T\in I(\pi_\varphi, (\pi_\psi)^*)$ and $\|T\|\leq 1$. Set $\xi:=T\varphi$. By the intertwining property of $T$, for each polynomial $f$ we have \begin{align}\label{intfq} T(f\cdot\varphi)=T\pi_\varphi(f(q))\varphi =(\pi_\psi)^*(f(q))T\varphi=(\pi_\psi)^*(f(q))\xi=f\cdot \xi . \end{align}
Therefore, since $\|T\|\leq 1$, we obtain \begin{align}\label{tless}
\int_\alpha^\beta |f(x)|^2|\xi(x)|^2dx= \int_\alpha^\beta |T(f\cdot \varphi)(x)|^2dx \leq \int_\alpha^\beta |f(x)|^2|\varphi(x)|^2 dx \end{align} for all polynomials $f$ and hence for all functions $f\in C[\alpha,\beta]$ by the Weierstrass theorem. Hence\, (\ref{tless}) implies that \begin{align}\label{xivarphi}
|\xi(x)|\leq |\varphi(x)|\quad {\rm on}\quad [\alpha,\beta]. \end{align} Therefore, $\xi(x)=0$ if\, $x\in \mathbb{R}/\, {\mathcal{J}}(\varphi)$. Clearly, $\xi(x)=0$ if $x\in \mathbb{R}/\, \overline{{\mathcal{J}}(\psi)}$, since $\xi\in {\mathcal{D}}((\pi_\psi)^*)$. Since $\varphi$ satisfies condition $(*)$, the set $\{f\cdot \varphi:f\in \mathbb{C}[x]\}$ is dense in $L^2({\mathcal{J}}(\varphi))={\mathcal{H}}(\pi_\varphi)$. Therefore, it follows from (\ref{intfq}) that $T$ is equal to the multiplication operator by the bounded function $\xi \varphi^{-1}$. (Note that $\xi \varphi^{-1}$ is bounded by (\ref{xivarphi}).) In particular, we obtain
$$\varphi^\prime \cdot\xi \varphi^{-1} =T\varphi^\prime=T\pi_\varphi(\mathrm{i} p)\varphi =(\pi_\psi)^*(\mathrm{i} p)T\varphi= (\pi_\psi)^*(\mathrm{i} p)\xi= \xi^\prime$$
Thus,\, $\varphi^\prime(x)\xi(x)=\varphi(x) \xi^\prime(x)$\, which in turn implies that\, $(\frac{\xi}{\varphi})^\prime(x)=0$\, for all $x\in {\mathcal{J}}(\varphi)\cap{\mathcal{J}}(\psi)$. Hence $\frac{\xi}{\varphi}$ is constant, say $\xi(x)=\lambda \varphi(x)$ for some constant $\lambda \in \mathbb{C}$ on each connected component of ${\mathcal{J}}(\varphi)\cap{\mathcal{J}}(\psi)$. By (\ref{xivarphi}),\, $|\lambda|\leq 1$. The connected components of the open set ${\mathcal{J}}(\varphi)\cap{\mathcal{J}}(\psi)$ are precisely the intervals ${\mathcal{J}}_l(\varphi)\cap{\mathcal{J}}_k(\psi)$ provided the latter is not empty.
Conversely, suppose that for all indices $l,k$ such that\, ${\mathcal{J}}_l(\varphi)\cap {\mathcal{J}}_k(\psi)\neq \emptyset$\, a complex number $\lambda_{k,l}$, where\, $|\lambda_{k,l}|\leq 1$, is given. Set
$\xi(x)=\lambda_{kl} \varphi(x)$ for $x\in {\mathcal{J}}_l(\varphi)\cap{\mathcal{J}}_k(\psi)$ and $\xi(x)=0$ otherwise. From the description of the domain\, ${\mathcal{D}}((\pi_\psi)^*)$\, given in the first paragraph of this proof it follows that $\xi\in {\mathcal{D}}((\pi_\psi)^*)$. Define $T(\pi_\varphi(a)\varphi):=(\pi_\psi)^*(a)\xi$, $a\in A$. It is easily checked that $T$ extends by continuity to an operator $T$ of ${\mathcal{H}}(\pi_\varphi)=L^2({\mathcal{J}}(\varphi))$ into ${\mathcal{H}}((\pi_\psi)^*)=L^2({\mathcal{J}}(\psi))$ such that $T\in I(\pi_\varphi,( \pi_\psi)^*)$\, and $\|T\|\leq 1$. Since $T\varphi=\xi$, we have $$\langle T\varphi,\psi\rangle=\sum\nolimits_{k,l}~ \lambda_{k,l} \int_{{\mathcal{J}}_k(\varphi)\cap{\mathcal{J}}_l(\psi)}\, \varphi(x) \overline{\psi(x)}\, dx.$$
Therefore, the supremum of expressions\, $|\langle T\varphi,\psi\rangle|$ is obtained if we choose $\lambda_{k,l}$ such that the number
$\lambda_{k,l} \int_{{\mathcal{J}}_k(\varphi)\cap{\mathcal{J}}_l(\psi)}\, \varphi \overline{\psi}\, dx$\, is equal to its modulus\, $|\int_{{\mathcal{J}}_k(\varphi)\cap{\mathcal{J}}_k(\psi)}\, \varphi \overline{\psi}\, dx|$. This implies formula (\ref{vectorstates}).
$\Box$
\noindent{\bf Acknowledgements.} The author would like to thank P.M. Alberti for many fruitful discussions on transition probabilities.
\end{document} | arXiv |
Solve for: {\text{begin}array l x-y+z=-8 } 4x+2y-z=12-2x+4y-z=36\text{end}array .
Expression: $\left\{\begin{array} { l } x-y+z=-8 \\ 4x+2y-z=12 \\ -2x+4y-z=36\end{array} \right.$
Rewrite the system as two systems, each consisting of two equations
$\begin{array} { l }\left\{\begin{array} { l } x-y+z=-8 \\ 4x+2y-z=12\end{array} \right.,\\\left\{\begin{array} { l } x-y+z=-8 \\ -2x+4y-z=36\end{array} \right.\end{array}$
Solve the system of equations
$\begin{array} { l }5x+y=4,\\\left\{\begin{array} { l } x-y+z=-8 \\ -2x+4y-z=36\end{array} \right.\end{array}$
$\begin{array} { l }5x+y=4,\\-x+3y=28\end{array}$
Write as a system of equations
$\left\{\begin{array} { l } 5x+y=4 \\ -x+3y=28\end{array} \right.$
$\begin{array} { l }x=-1,\\y=9\end{array}$
Substitute the given values of $\begin{array} { l }x,& y\end{array}$ into the equation $x-y+z=-8$
$-1-9+z=-8$
Solve the equation for $z$
$z=2$
The possible solution of the system is the ordered triple $\left( x, y, z\right)$
$\left( x, y, z\right)=\left( -1, 9, 2\right)$
Check if the given ordered triple is a solution of the system of equations
$\left\{\begin{array} { l } -1-9+2=-8 \\ 4 \times \left( -1 \right)+2 \times 9-2=12 \\ -2 \times \left( -1 \right)+4 \times 9-2=36\end{array} \right.$
Simplify the equalities
$\left\{\begin{array} { l } -8=-8 \\ 12=12 \\ 36=36\end{array} \right.$
Since all of the equalities are true, the ordered triple is the solution of the system
Calculate: (4x+2)^2+4(4x+2)+4=0
Evaluate: -9.4-6.5
Evaluate: 3^x=81
Calculate: -2w * (w+9)=0
Evaluate: 4sqrt(3)sqrt(6)
Evaluate: \lim_{x arrow-1} ((x^3-x^2-5x-3)/((x+1)^2))
Solve for: 239/31
Evaluate: (x+4)/2 =(x+9)/3
Solve for: 6sqrt((7)/(5))-2sqrt((5)/(7))-3sqrt(140)
Solve for: 25^{x^2-9x}=3125^{4x}
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Exploring Math Practice Apps: Which Ones are Most Effective? | CommonCrawl |
\begin{definition}[Definition:Gradient Operator/Riemannian Manifold/Definition 2]
Let $\struct {M, g}$ be a Riemannian manifold equiped with a metric $g$.
Let $f \in \map {\CC^\infty} M : M \to \R$ be a smooth mapping on $M$.
The '''gradient''' of $f$ is the vector field obtained from the differential $\rd f$ obtained by raising an index:
:$\grad f := \paren {\rd f}^\sharp$
{{MissingLinks}}
\end{definition} | ProofWiki |
\begin{document}
\title{\noindent Local Unknottedness of Planar Lagrangians with Boundary} \maketitle \begin{center} Zi-Xuan Wang \par\end{center}
\begin{center} Uppsala University \par\end{center}
\noindent
{}
\section*{Abstract}
\noindent We show the smooth version of the nearby Lagrangian conjecture for the 2-dimensional pair of pants and the Hamiltonian version for the cylinder. In other words, for any closed exact Lagrangian submanifold of $T^{*}M$, there is a smooth or Hamiltonian isotopy, when $M$ is a pair of pants or a cylinder respectively, from it to the 0-section. For the cylinder we modify a result of G. Dimitroglou Rizell for certain Lagrangian tori to show that it gives the Hamiltonian isotopy for a Lagrangian cylinder. For the pair of pants, we first study some results from pseudo-holomorphic curve theory and the planar Lagrangian in $T^{*}\mathbb{R}^{2}$, then finally using a parameter construction to obtain a smooth isotopy for the pair of pants.
\noindent
{}
\section{Introduction}
\subsection{Background}
\noindent A smooth manifold $\left(W,\omega\right)$ is called a symplectic manifold if it is equipped with a closed and non-degenerate 2-form $\omega$. It has to be of even-dimension because of the non-degeneracy of $\omega$. For an n-dimensional smooth manifold $M$, its cotangent bundle $\left(T^{*}M,-d\theta\right)$, where $\theta=\sum_{i=1}^{n}p_{i}dq_{i}$ is the tautological 1-form, is naturally a symplectic manifold.
\noindent Lagrangian submanifolds are smooth half-dimensional submanifolds of symplectic manifolds on which the symplectic form vanishes. For a cotangent bundle $\left(T^{*}M,-d\theta\right)$, its section $\left(m,\alpha\right)$ is Lagrangian if the 1-form $\alpha$ satisfies $d\alpha=0$. Moreover, if $\alpha$ is exact, the section is called an exact Lagrangian. Lagrangian submanifolds have been shown to exhibit many rigidity phenomena since Gromov\textquoteright s pseudoholomorphic curve theory, such as the result by Eliashberg-Polterovich in \cite{key-10} numbered as Theorem 9 here \begin{thm*} \noindent Any flat at infinity Lagrangian embedding of $\mathbb{R}^{2}$ into the standard symplectic $\mathbb{R}^{4}$ is isotopic to the flat embedding via an ambient compactly supported smooth isotopy of $\mathbb{R}^{4}$. \end{thm*} \noindent In the smooth category, an isotopy is a family of diffeomorphisms $\rho_{t}$ such that $\rho_{0}=Id$. This family is generated by a family of vector fields $\left\{ X_{t}\right\} $ s.t. \[ \frac{d}{dt}\rho_{t}=X_{t}\left(\rho_{t}\right) \] If each $X_{t}$ is Hamiltonian, i.e. there exists a smooth family of functions $H_{t}:T^{*}M\rightarrow\mathbb{R}$, called Hamiltonian functions s.t. \[ \iota\left(X_{t}\right)\omega=dH_{t} \] this isotopy is called a Hamiltonian isotopy.
\noindent Eliashberg-Polterovich considered the case of a Lagrangian disc, and my work here concerns a generalization: the case of a Lagrangian pair of pants. The main theorem of this article can be stated as \begin{thm*} \noindent Any flat outside a compact set Lagrangian embedding of $\mathbb{R}^{2}-\left\{ 0,1\right\} $ into the standard symplectic cotangent bundle of the same manifold is isotopic to the flat embedding via an ambient compactly supported smooth isotopy of \[ T^{*}\left(\mathbb{R}^{2}-\left\{ 0,1\right\} \right)=\mathbb{R}^{2}\times\left(\mathbb{R}^{2}-\left\{ 0,1\right\} \right)=\mathbb{R}^{4}-\left(\mathbb{R}^{2}\times\left\{ 0\right\} \cup\mathbb{R}^{2}\times\left\{ 1\right\} \right) \] \end{thm*} \noindent This result is a direct consequence of Theorem 18, whose formulation is adapted to the strategy of the proof.
\noindent The same methods are expected to show that the analogous result holds for $T^{*}\left(\mathbb{R}^{2}-\left\{ p_{1},\cdots,p_{m}\right\} \right)$, i.e. the complement of m points. However, the special motivation behind studying pair of pants is that a closed surface of genus $g\geq2$ admits a pair of pants decomposition. If one can show, via stretching the neck, that an exact Lagrangian surface inside a cotangent bundle is isotopic to pieces in a pair of pants decomposition, where the pieces have standard behavior near their boundaries, then this plus the result for the pair of pants implies that any exact Lagrangian surface is smoothly isotopic to the zero section.
\noindent This can be seen as a strategy to partially prove the following conjecture from 1986 due to V.I. Arnol'd: \begin{conjecture} \noindent (The nearby Lagrangian conjecture) Let M be a closed manifold. Any closed exact Lagrangian submanifold of $T^{*}M$ is Hamiltonian isotopic to the 0-section. \end{conjecture}
\noindent By demanding that the exact Lagrangian agrees with $M=\mathbb{R}^{2}$ outside a compact subset, the situation is similar to that of a closed manifold. So do $M=\mathbb{R}^{n}$. So far, the conjecture has only been established in the cases when $M=\mathbb{R}^{1},S^{1},\mathbb{R}^{2},\mathbb{R}^{1}\times S^{1},\mathbb{T}^{2},S^{2}$. Conjecture 1 for $M=\mathbb{R}^{2}$ can be rephrased as ``local planar Lagrangians in $T^{*}\mathbb{R}^{2}$ are unknotted'', and our goal can be rephrased as ``local planar Lagrangians in $T^{*}(\mathbb{R}^{2}-\left\{ 0,1\right\} )$ are unknotted''. The result for $\mathbb{R}^{2}$ in the smooth category is proved by Eliashberg-Polterovich in \cite{key-10}, whose methods provide great enlightenment for our proof, and the Hamiltonian version is proved by the same authors in another article in \cite{key-7}. In this article, we also establish the nearby Lagrangian conjecture in the case of the cylinder $M=\mathbb{R}\times S^{1}$ by adapting the proof of the Hamiltonian classification result for Lagrangian tori in $T^{*}\mathbb{T}^{2}$ from \cite[Theorem B]{key-9}. \begin{thm*} \noindent (Theorem 17, the nearby Lagrangian conjecture for the cylinder) Let $L\subset T^{*}\mathbb{T}^{2}$ be an exact Lagrangian torus which coincides with the zero section above the subset $\theta_{2}\in\left[-\delta,\delta\right]$. Then $L$ is Hamiltonian isotopic to the zero section by a Hamiltonian isotopy which is supported in the same subset. \end{thm*}
\subsection{Organization of this paper}
\noindent Firstly in section 2, we will present and prove some preliminaries from Gromov's pseudo-holomorphic curve theory. Then in section 3, we will go through the proof of local smooth unknottedness of planar Lagrangians. In section 4, we will explain how Theorem B, which seems to be the torus version of nearby Lagrangian conjecture in \cite{key-9} gives an isotopy for the cylinder, and finally in section 5 prove the local unknottedness of planar Lagrangians with two punctures in the smooth category.
\section{Preliminaries from Pseudo-Holomorphic Curve Theory}
\subsection{Moser's trick}
\noindent In \cite{key-1}, J. Moser invented the following method which turns out to be useful in many cases: Consider two k-forms $\alpha_{1}$ and $\alpha_{0}$ on a smooth manifold $M$ and one wants to find a diffeomorphism \[ \varphi:M\rightarrow M,\varphi^{*}\alpha_{1}=\alpha_{0}. \] Moser's idea is to find a family of diffeomorphisms $\varphi_{t},0\leq t\leq1$ for a family of forms $\alpha_{t}$ connecting $\alpha_{1}$ and $\alpha_{0}$ s.t. \[ \varphi_{t}^{*}\alpha_{t}=\alpha_{0}. \] This is in fact \[ 0=\frac{d}{dt}\phi_{t}^{*}\alpha_{t}=\phi_{t}^{*}\left(\alpha_{t}^{\prime}+\mathcal{L}_{X_{t}}\alpha_{t}\right) \] Use Cartan's formula, \[ \alpha_{t}^{\prime}+d\iota_{X_{t}}\alpha_{t}+\iota_{X_{t}}d\alpha_{t}=0 \] In our case $\alpha_{t}=\omega_{t}$ is a symplectic form, so it's closed. The equation becomes \[ \omega_{t}^{\prime}+d\iota_{X_{t}}\omega_{t}=0. \]
\noindent As an application, we have a standard result \begin{thm} \noindent Let $\omega_{0}$ and $\omega_{1}$ be two symplectic forms on a compact manifold $M$ that belong to the same de Rham cohomology class, and $\omega_{t}:=\omega_{0}+td\beta=\left(1-t\right)\omega_{0}+t\omega_{1}$ is symplectic for each $t\in\left[0,1\right]$ Then there is a symplectomorphism $\phi:\left(M,\omega_{0}\right)\rightarrow\left(M,\omega_{1}\right)$. \end{thm}
\begin{proof} \noindent There exists a 1-form $\beta$ s.t. $\omega_{1}=\omega_{0}+d\beta$. Let $\omega_{t}:=\omega_{0}+td\beta=\left(1-t\right)\omega_{0}+t\omega_{1}$. Then the equation \[ \omega_{t}^{\prime}+d\iota_{X_{t}}\omega_{t}=0 \] becomes \[ L.H.S.=d(\beta+\iota_{X_{t}}\omega_{t})=0 \] which is solvable. \end{proof}
\subsection{Gromov's pseudo-holomorphic curve theory}
\subsubsection{Almost complex structure} \begin{defn} \noindent An almost complex structure $J\in End\left(TM\right)$ on a symplectic manifold $\left(M,\omega\right)$ is a smooth linear structure $J_{m}$ on each tangent space $T_{m}M$ which satisfies $J_{m}^{2}=-Id$. $J$ is $tamed$ by the symplectic form $\omega$ if \[ \omega\left(v,Jv\right)>0,\forall v\neq0. \] $J$ is compatible with $\omega$ if $\omega\left(v,Jv\right)$ is a Riemannian metric. A manifold that admits an almost complex structure is called an almost complex manifold. \end{defn}
\begin{rem*} \noindent Every symplectic manifold admits compatible almost complex structures. The space of tamed and compatible almost complex structures are denoted by \[ J^{tame}\left(M,\omega\right) \]
\[ J^{comp}\left(M,\omega\right) \] . \end{rem*} \begin{lem} \noindent (Gromov 1985, \cite{key-11}) The spaces $J^{tame}\left(M,\omega\right)$ and $J^{comp}\left(M,\omega\right)$ are both contractible. \end{lem}
\begin{proof} \noindent Sketch of proof: The key is to prove that for each tangent space the lemma holds. One can identify the latter with a convex and open subset of the space of metrics on a vector space, hence contractible. A similar but more complicated identification proves the former. \end{proof} \noindent A compatible almost complex structure on a manifold $M$ is a section over $M$ of the fibre bundle whose fibres on each point are compatible almost complex actions of that point. Sometimes it's hard to construct a compatible structure over the whole symplectic manifold, and the above Lemma gives us a way to firstly construct a section on a suitable submanifold of $M$, and then extend this section over the entire manifold $M$. This extension is ensured by the contractibility of fibres. We will apply this to $M=\mathbb{CP}^{2}$ in the next section.
\subsubsection{Positivity of intersection} \begin{defn} \noindent Let $\left(\Sigma,j\right)$ be a Riemann surface. A map from this Riemann surface to an almost complex manifold $u:\left(\Sigma,j\right)\rightarrow\left(M,J\right)$ is said to be $J-holomorphic$ (also called pseudo-holomorphic) if it satisfies the fully non-linear first order PDE \[ \bar{\partial}_{J}u=\frac{1}{2}\left(du+J\circ du\circ j\right)=0 \] of Cauchy-Riemann type. When $\Sigma=\mathbb{CP}^{1}$, $u$ is called a J-holomorphic sphere. \end{defn}
\noindent From the classical intersection theory, we have \begin{prop} \noindent (McDuff 1994, \cite{key-16}) Consider a connected holomorphic curve $u:\Sigma\rightarrow M$ and a holomorphic hypersurface $D\subset M$, i.e. the complex dimension of $D$ = the complex dimension of $M$ minus one, such that u is not contained inside D. Then:
\noindent $\bullet$ $u$ and $D$ intersect in a discrete subset;
\noindent $\bullet$ each geometric intersection point gives a positive contribution to the algebraic intersection number $\left[u\right]\cdot\left[D\right]\geq0$;
\noindent $\bullet$ if an intersection point moreover is not a transverse intersection (e.g. a tangency or an intersection of $D$ and a singular point of $u$), then that geometric point contributes at least +2. \end{prop}
\noindent This is of special importance to dim-4 manifolds, because under this occasion the hypersurface $D$ is also two-dimensional. We have \begin{thm} \noindent (McDuff 1994, \cite{key-16}) Positivity of intersection in dim 4: two closed distinct J -holomorphic curves $u$ and $u^{\prime}$ in an almost complex 4-manifold $(M,J)$ have only a finite number of intersection points. Each such point $x$ contributes a number $k_{x}\geq1$ to the algebraic intersection number $\left[u\right]\cdot\left[u^{\prime}\right]$;. Moreover, $k_{x}=1$ iff the curves $u$ and $u^{\prime}$ intersect transversally at $x$. \end{thm}
\subsubsection{Existence of a pseudo-holomorphic line passing through two points}
\noindent Recall that in algebraic geomrtry, there exists precisely one algebraic curve of degree one (i.e. homologus to $L\in H_{2}\left(\mathbb{CP}^{n}\right)=\mathbb{Z}\cdot L$) that passes through two given points $P_{1}\neq P_{2}\in\mathbb{CP}^{n}$: the complex line \[ \mathbb{CP}^{1}\rightarrow\mathbb{CP}^{n},\left[x_{1}:x_{2}\right]\mapsto x_{1}\cdot P_{1}+x_{2}\cdot P_{2} \] unique up to reparameterization.
\noindent For pseudo-holomorphic spheres, we have \begin{thm} \noindent (Gromov 1985, \cite{key-11}) There exists a unique up to reparameterization holomorphic curve of degree one (i.e. homologus to $L\in H_{2}\left(\mathbb{CP}^{n}\right)=\mathbb{Z}\cdot L$) that passes through two given points $P_{1}\neq P_{2}\in\mathbb{CP}^{n}$: the complex line \[ \mathbb{CP}^{1}\rightarrow\mathbb{CP}^{n},\left[x_{1}:x_{2}\right]\mapsto x_{1}\cdot P_{1}+x_{2}\cdot P_{2}. \] \end{thm}
\begin{proof} \noindent If there exists a curve \[ u:\left(\Sigma,j\right)\rightarrow\left(\mathbb{CP}^{n},J_{0}\right) \]
in class $\left[u\right]=L$ is not of the above form, then we can find a linear hyperplane (denoted by $H$) $\mathbb{CP}^{n-1}\subset\mathbb{CP}^{n}$ which is tangent to the curve at some point but not contain it. Positivity of intersection of the curve and the hyperplane implies that $H\cdot L\geq2$, because a tangency contributes at least 2 to the intersection number. However, $H\cdot\left[u\right]=1$, contradiction. \end{proof} \noindent This will be of great importance when n=2. We will discuss it later in section 3.
\section{Local Unknottedness of Planar Lagrangians without Boundary}
\noindent In this section we will elaborate Eliashberg-Polterovich's proof of the smooth version of the nearby Lagrangian conjecture of $\mathbb{R}^{2}$, which will provide great insight for our case about pair of pants. \begin{thm} \noindent (Eliashberg-Polterovich 1993, \cite{key-10}) Any flat at infinity Lagrangian embedding of $\mathbb{R}^{2}$ into the standard symplectic $\mathbb{R}^{4}$ is isotopic to the flat embedding via an ambient compactly supported smooth isotopy of $\mathbb{R}^{4}$. \end{thm}
\begin{rem} \noindent In fact, this isotopy can be made Hamiltonian by constructing a family of suitable hypersurfaces. See \cite{key-7}. \end{rem}
\noindent Consider the standard linear space $\mathbb{R}^{4}$ with coordinates $\left(p_{1},q_{1},p_{2},q_{2}\right)$ such that the symplectic form $\omega=dp_{1}\wedge dq_{1}+dp_{2}\wedge dq_{2}$. Let $l=\left\{ p_{1}=p_{2}=0\right\} $ be a Lagrangian plane and let $L\subset\mathbb{R}^{4}$ be a Lagrangian submanifold which is diffeomorphic to $\mathbb{R}^{2}$ and which coincides with $l$ outside a compact subset. We have to prove that $L$ is isotopic to $l$ via a compactly supported isotopy of $\mathbb{R}^{4}$. To do this, we will modify $\omega$ to make $l$ and $L$ symplectic at the same time, so that we can use compactification to make them projective lines and construct a smooth isotopy. \begin{lem} \noindent Consider the family of symplectic forms $\omega_{\varepsilon}=\omega+\varepsilon dq_{1}\land dq_{2}$, $\varepsilon>0$ small. There exists a compactly supported isotopy of $\mathbb{R}^{4}$ that coincides with a linear symplectic plane outside of a compact subset which takes $L$ to an $\omega_{\varepsilon}-$symplectic surface. \end{lem}
\begin{proof} \noindent On a closed tubular neighborhood $V$ of $L$, one can find a closed 2-form $\tau$ such that $\exists$ a compactly supported 1-form $\lambda$ on $V$, $d\lambda=\tau-dq_{1}\land dq_{2}$, which coincides with $dq_{1}\land dq_{2}$ outside a compact subset of $V$ and makes $L$ $\tau-$symplectic. Choose a bump function $\rho$ on $\mathbb{R}^{4}$ which vanishes outside $V$ and equals 1 near $L$. Set \[ \omega_{\varepsilon}^{\prime}=\omega+\varepsilon\left(dq_{1}\land dq_{2}+d\left(\rho\lambda\right)\right) \] then $L$ is $\omega_{\varepsilon}^{\prime}$-symplectic. The difference $\omega_{\varepsilon}^{\prime}-\omega_{\varepsilon}=\varepsilon d\left(\rho\lambda\right)$, which is 0 outside a compact subset of $\mathbb{R}^{4}$, and exact inside it. So one can use Moser's linear construction to conclude that $\omega_{\varepsilon}$ and $\omega_{\varepsilon}^{\prime}$ are isotopic. This isotopy is compactly supported and takes $L$ into an $\omega_{\varepsilon}-$symplectic surface, denoted by $L^{\prime}.$ \end{proof} \begin{lem} \noindent There exists a symplectic form on $\mathbb{R}^{4}$ which tames $J_{0}$, coincides with $\omega_{\varepsilon}$ on some given subset, and with a multiple of the Fubini-Study metric outside a compact subset. \end{lem}
\begin{proof} \noindent Take an interpolation between $\parallel z\parallel^{2}$ and $C\cdot log\left(1+\parallel z\parallel^{2}\right),$ for $C\gg0$ sufficiently large. Take $\frac{i}{2}\partial\overline{\partial}$ of this function. If the function is strictly convex as a function of the radius, then the obtained 2-form, which is an interpolation between $\omega_{\varepsilon}$ applied in the compact subset where the above isotopy takes place, and the Fubini-Study form \[ \omega_{FS}=C\cdot\frac{i}{2}\partial\overline{\partial}\left(log\left(1+\parallel z\parallel^{2}\right)\right), \] far away, is symplectic. Since $J_{0}$ is tamed by $\omega_{\varepsilon}$ and $\omega_{FS}$ at the same time, $J_{0}$ is tamed by the interpolated 2-form, too. Actually, recall that any symplectic form of the form $\frac{i}{2}\partial\overline{\partial}f$ for a smooth real function $f$ is a Kähler form with respect to the standard complex structure, and hence the standard complex structure is compatible with this symplectic form. \end{proof} \noindent After the above interpolation, the symplectic volume of $\mathbb{R}^{4}$ becomes finite, so that one can use a projective line $\Gamma_{\infty}$ to replace remote areas in $\mathbb{R}^{4}$ where the Fubini-Study form is applied, and make $\mathbb{R}^{4}$ into $\mathbb{CP}^{2}$. For convenience, we simply call the above process by ``compactifying $\mathbb{R}^{4}$ into $\mathbb{CP}^{2}$''. \begin{lem} \noindent One can compactify $\left(\mathbb{R}^{4},\omega_{\varepsilon}\right)$ into $\mathbb{CP}^{2}$ by adding a line $\Gamma_{\infty}$ at infinity, while making the symplectic plane $l=\left\{ \left(p_{1},p_{2}\right)=\left(0,0\right)\right\} $ a projectice line $\gamma_{0}$, $L$ an symplectic embedded sphere $\Sigma$, which is homologus to $\mathbb{CP}^{1}$ and intersects the infinity line $\Gamma_{\infty}$ at the same point $P$ as $\Gamma_{\infty}\cap\gamma_{0}$. \end{lem}
\begin{proof} \noindent Due to our choice of the bump function $\rho$ vanishing outside a tubular neibourhood $V$, $L^{\prime}$ agrees with $L$ at infinity (thus with the standard plane $l$ too) when applying the isotopy. This allows us to, as in Lemma 12, compactify $\left(\mathbb{R}^{4},\omega_{\varepsilon}\right)$ with coordinates \[ \left(\mathbb{R}^{4}=\left(p_{1},q_{1},p_{2},q_{2}\right),\omega_{\varepsilon}=\left(dp_{1}-\frac{\varepsilon}{2}dq_{2}\right)\wedge dq_{1}+\left(dp_{2}+\frac{\varepsilon}{2}dq_{1}\right)\wedge dq_{2}\right) \]
into $\mathbb{CP}^{2}$ with coordiantes $\left[z_{1},z_{2}\right]$, namely $z_{1}\sim\left(x_{1},y_{1}\right)\sim\left(p_{1}-\frac{\varepsilon}{2}q_{2},q_{1}\right),z_{2}\sim\left(x_{2},y_{2}\right)\sim\left(\frac{2}{\varepsilon}p_{2}+q_{1},\frac{\varepsilon}{2}q_{2}\right)$ and make use of the results of pseudo holomorphic curves in $\mathbb{CP}^{2}$. Choose an almost complex structure $j$ (which is also complex here in $\mathbb{R}^{4}$), such that $\omega_{\varepsilon}\left(\cdot,j\cdot\right)$ is a Euclidian metric. Take a large ball $B$ such that $L^{\prime}$ coinsides with $l$ outside it, and compactify the ball to $\mathbb{CP}^{2}$ by adding a line $\Gamma_{\infty}$ at infinity. Denote the compactifications of $\omega_{\varepsilon}$ and $j$ by $\Omega$ and $J_{0}$ respectively. After such a compactication the symplectic line $l$ corresponds to a projective line $\gamma_{0}=\left\{ z_{1}=iz_{2}\right\} $ and our knot $L$ is compactified to an symplectic embedded sphere $\Sigma$, which is homologus to $\mathbb{CP}^{1}$ and intersects $\Gamma_{\infty}$ at the same point $P$ as $\Gamma_{\infty}\cap\gamma_{0}$. See Figure 5,2.1. \end{proof} \noindent In order to prove the theorem, it is enough to show that: \begin{thm} \noindent $\Sigma$ is smoothly isotopic to the projective line $\gamma_{0}$ defined in Lemma 13 via an isotopy of $\mathbb{CP}^{2}$ which fixes $\Gamma_{\infty}$. Moreover, this isotopy can be taken to fix a neighborhood of $\Gamma_{\infty}$. \end{thm}
\begin{rem} \noindent Given a compatible $J$, a $J-$holomorphic line on $\mathbb{CP}^{2}$ is an embedded 2-sphere $C\subset\mathbb{CP}^{2}$ which is homologous to $\mathbb{CP}^{1}$ and whose tangent space $T_{x}C$ is $J-$invariant for all points $x\in C$. Under this definition, Theorem 8 is to say that for each compatible almost complex structure $J$ on $\mathbb{CP}^{2}$ and for each two distinct points $A,B\in\mathbb{CP}^{2}$ there exists a unique $J-holomorphic$ line which passes through $A$ and $B$. Moreover this line depends smoothly on $J,A,B$.
\noindent This implies that given a point $x_{0}\in\mathbb{CP}^{2}$, there is a pencil of lines based on this point, i.e. a set of pseudo-holomorphic lines that all intersect at $x_{0}$. This pencil of lines forms a foliation away from a point, where each leaf is determined by its tangency at the point $x_{0}$. \end{rem}
\begin{proof} \noindent \begin{figure}
\caption{Fig.2 in \cite{key-10}. The vertical line is $\Gamma_{\infty}$, horizonal lines are $\gamma_{0}$ and $\Sigma$. Gray line is the path $Q\left(t\right)$.}
\end{figure}
\noindent \LyXZeroWidthSpace{}
\noindent (See Figure 5,2.1) Choose a smooth path $Q\left(t\right),t\in\left[0,1\right]$ on $\mathbb{CP}^{2}-\Gamma_{\infty}$ such that $Q\left(0\right)\in\gamma_{0}$ and $Q\left(1\right)\in\Sigma$. There exists a smooth family of compatible almost complex structures $J\left(t\right),t\in\left[0,1\right]$ such that
\noindent $J\left(0\right)=J_{0}$
\noindent $\Sigma$ is $J\left(1\right)$-holomorphic
\noindent $\Gamma_{\infty}$ is $J\left(t\right)$-holomorphic for all t.
\noindent For the existence, one can first choose some appropriate almost complex structure along $\Gamma_{\infty}$, or say a section on the bundle of compatible actions over $\Gamma_{\infty}$. Since the space of compatible almost complex structures forms a contractible space, we can extend this section to $\mathbb{CP}^{2}$, which gives $J\left(t\right)$.
\noindent By Theorem 8, there exists a unique $J\left(t\right)-$holomorphic line, denoted by $C\left(t\right)$ passing through $P$ and $Q\left(t\right)$ changing smoothly w.r.t. parameter $t$. The deformation $\left\{ C\left(t\right),t\in\left[0,1\right]\right\} $ gives an isotopy between $\gamma_{0}$ and $\Sigma$.
\noindent Before we finally extend $\left\{ C\left(t\right),t\in\left[0,1\right]\right\} $ to an ambient isotopy of $\mathbb{CP}^{2}$ which preserves $\Gamma_{\infty}$, it remains to show that $C\left(t\right)$ intersects $\Gamma_{\infty}$ at the unique point $P$ transversally. If not,namely $C\left(t\right)$ intersect $\Gamma_{\infty}$ at more than one points(counted by multiplicity), by positivity of intersection in dim 4, each of them contributes positively to the intersection number $C\left(t\right)\cdot\Gamma_{\infty}$, so the number is bigger than one. However, $C\left(0\right)\cdot\Gamma_{\infty}=1$, contradicts the fact that the intersection number should remain unchanged for homologus lines.
\noindent Additionally, after a smooth deformation of $Q\left(t\right)$, one can require that for all $t\in\left[0,1\right]$, $Q\left(t\right)\cap\gamma_{0}$ not only at the point $x_{o}$, but also in a neighborhood $I\subset\gamma_{0}$ of $x_{0}$. Here it's important that $Q\left(t\right)$ is transverse to $\Gamma_{\infty}$. Then one can make the isotopy intersect $I\times\Gamma_{\infty}$ only at $I$, thus obtain a compactly supported isotopy. \end{proof}
\section{Local Unknottedness of One-punctured Planar Lagrangians}
\subsection{The Nearby Lagrangian conjecture for the cylinder}
\noindent We first explain the statement of \cite[Theorem B]{key-9} and show that this is actually equivalent to the nearby Lagrangian conjecture for the cylinder.
\noindent By cutting a torus $\mathbb{T}^{2}=\left(S^{1}\times S^{1}\right)$ parameterized by $\theta_{1}$ and $\theta_{2}$ between $\left\{ \theta_{2}=s\right\} $ and $\left\{ \theta_{2}=t\right\} $, we get two cylinders. This motivates us to realize that a Lagrangian cylinder inside the cotangent bundle of $T^{*}\left(S^{1}\times I\right)$ which is standard near the boundary can be extended to a Lagrangian torus in $T^{*}\left(S^{1}\times S^{1}\right)$ which is equal to the zero section in ${\theta_{2}\in\left[-\delta,\delta\right]}$ in the base. Namely, this isotopy of a Lagrangian torus keeps $S^{1}\times e^{i\left[-\delta,\delta\right]}$ fixed, so it is actually an isotopy of the rest part of a torus, which is a cylinder. We start by recalling the following result. \begin{thm} \noindent (\cite[Theorem B(2)]{key-9}, 2019) Suppose that $L\subset(T^{*}\mathbb{T}^{2},d\lambda_{\mathbb{T}^{2}})$ is an exact Lagrangian embedding. Then for any $\theta\in S^{1}$ consider the properly embedded Lagrangian disc with one interior point removed \[ \dot{D}_{\mathbf{P}^{0}}\left(\theta\right):=\left(S^{1}\times\left\{ \theta\right\} \right)\times\left(\left\{ p_{1}^{0}\right\} \times(-\infty,p_{2}^{0}]\right)\subset\mathbb{T}^{2}\times\mathbb{R}^{2}=T^{*}\mathbb{T}^{2}, \] \[ \mathbf{p}^{0}:=\left(p_{1}^{0},p_{2}^{0}\right) \] If it is the case that \[ L\cap\dot{D}_{\mathbf{p}^{0}}\left(e^{is}\right)=\partial\dot{D}_{\mathbf{p}^{0}}\left(e^{is}\right)=S^{1}\times\left\{ e^{is}\right\} \times\left\{ \mathbf{p}^{0}\right\} \] holds for all $\mid s\mid<\epsilon$, then the Hamiltonian isotopy can be assumed to be supported outside of the subset \[ \bigcup_{\mid s\mid<\delta}\dot{D}_{\mathbf{p}^{0}}\left(e^{is}\right):=S^{1}\times e^{i\left[-\delta,\delta\right]}\times\left\{ p_{1}^{0}\right\} \times(-\infty,p_{2}^{0}] \] for some $0<\delta<\epsilon$ sufficiently small (note that for symplectic action reasons, we may not be able to Hamiltonian isotope the Lagrangian to the constant section $\mathbb{T}^{2}\times\left\{ \mathbf{p}^{0}\right\} $). \end{thm}
\noindent
\noindent The above result can in particular be applied to the torus which is an extension of a Lagrangian cylinder in $T^{*}\left(S^{1}\times I\right)$ which is standard above the boundary. However, we need to strengthen it in the following manner: make sure that the Hamiltonian isotopy of the torus is fixed above the entire subset $\theta_{2}\in\left[-\delta,\delta\right]$.
\noindent We prove the following strengthening of the result, \begin{thm} \noindent (The nearby Lagrangian conjecture for the cylinder) Let $L\subset T^{*}\mathbb{T}^{2}$ be an exact Lagrangian torus which coincides with the zero section above the subset $\theta_{2}\in\left[-\delta,\delta\right]$. Then $L$ is Hamiltonian isotopic to the zero section by a Hamiltonian isotopy which is supported in the same subset. \end{thm}
\begin{proof} \noindent We follow exactly the same steps as the proof of theorem B in \cite[Section 9]{key-9}.
\noindent We prove by constructing a solid torus with core removed which is foliated by pseudo-holomorphic punctured discs.
\noindent The main step of the proof is to construct a proper embedding of a solid torus $\dot{\mathcal{T}}\subset T^{*}T^{2}$ with its core removed, which is foliated by pseudoholomorphic discs, and whose boundary is the Lagrangian $L$. For technical reasons this solid torus with core removed is constructed as a solid torus $\mathcal{T}\subset\mathbb{CP}^{1}\times\mathbb{CP}^{1}$ foliated by pseudoholomorphic discs, which gives rise to $\dot{\mathcal{T}}$ after removing the four holomorphic lines $\left(\left\{ 0,\infty\right\} \times\mathbb{CP}^{1}\right)\cup\left(\mathbb{CP}^{1}\times\left\{ 0,\infty\right\} \right)$. Recall that the complement of this divisor is a neighbourhood of the zero section of $T^{*}T^{2}$. In order to obtain the solid torus with the sought properties it is crucial that we have a family of (punctured) pseudoholomorphic discs that we can start with. Namely, one starts from the family of punctured pseudo-holomorphic discs \[ C\left(e^{is}\right)=S^{1}\times\left\{ e^{is}\right\} \times\left(-\infty,p_{1}^{0}\right)\times\left\{ p_{2}^{0}\right\} \subset T^{*}\mathbb{T}^{2}\setminus L \] with boundary on $L$ that exists by the assumption that $L$ is standard above the neighborhood $\theta_{2}\in\left(-\delta,\delta\right)$. We proceed to explain how the additional assumptions made in Theorem 16 here give us additional control over the solid torus, as compared to the assumptions in Theorem 15 above which was proven in \cite{key-9}. The goal is to use positivity of intersection to say that $\mathcal{\dot{T}}$ is standard above the region $\theta_{2}\in\left(-\delta,\delta\right)$.
\noindent In \cite[Theorem B(2)]{key-9}, the condition about the intersection with the Lagrangian disc $\dot{D}_{\mathbf{P}^{0}}\left(\theta\right)$, says that the Hamiltonian isotopy is ``one-sided'' fixed near some Lagrangian disc, in the sense that the subset $\cup_{\mid s\mid<\delta}\dot{D}_{\mathbf{P}^{0}}\left(e^{is}\right)$ contains only $p_{2}<p_{2}^{0}$, but not $p_{2}>p_{2}^{0}$ and $p_{2}=p_{2}^{0}$. As in Section 9.1 of \cite{key-9}, denote a smooth one-dimensional family of embedded symplectic cylinders by \[ C\left(e^{is}\right)=S^{1}\times\left\{ e^{is}\right\} \times\left(-\infty,p_{1}^{0}\right)\times\left\{ p_{2}^{0}\right\} \subset T^{*}\mathbb{T}^{2}\setminus L \] where $L$ is the Lagrangian torus, and for $p_{2}<p_{2}^{0}$, $p_{2}>p_{2}^{0}$, there are cylinders \[ C\left(p_{2},e^{is}\right)=S^{1}\times\left\{ e^{is}\right\} \times\mathbb{R}\times\left\{ p_{2}^{0}\right\} \subset T^{*}\mathbb{T}^{2}\setminus L. \] Together, we can find a well-defined compatible almost complex structure $J$ on $\left(T^{*}\mathbb{T}^{2}\setminus L,d\lambda_{\mathbb{T}^{2}}\right)$ which is cylindrical outside of a compact subset, and which agrees with $J_{cyl}$ in a neighborhood of the union of cylinders. In \cite{key-9}, only $p_{2}<p_{2}^{0}$ was considered, however, if we look at Figure 5,2.16 in \cite{key-9}, there is also space for cylinders corresponds to $p_{2}>p_{2}^{0}$ above the $p_{2}$ plane. This property is a direct consequence of the assumption that $L$ coincides with the zero section above $\theta_{2}\in\left[-\delta,\delta\right]$.
\noindent Let us consider the family of cyliders \[ C\left(e^{is}\right)\cup C\left(p_{2},e^{is}\right),\mid s\mid\leq\delta,p_{2}<p_{2}^{0},p_{2}>p_{2}^{0}. \]
\noindent
\noindent \LyXZeroWidthSpace{}
\noindent The argument in \cite[Section 9]{key-9} produces an embeded solid torus $\mathcal{\dot{T}}$ inside $T^{*}\mathbb{T}^{2}$ with core removed, whose boundary is $L$, and which is foliated by pseudo-holomorphic cylinders with boundary on $L$. Part of these cylinders are given by the explicitly constructed standard cylinders $C\left(e^{is}\right)$. After compactifying $T^{*}\mathbb{T}^{2}$ to $S^{2}\times S^{2}$, $\mathcal{\dot{T}}$ becomes a solid torus. The positivity of intersection argument in Lemma 9.8(2) of \cite{key-9} which shows that the solid torus is disjoint from $C\left(p_{2},e^{is}\right)$ also shows that the interior of these solid tori (which are foliated by pseudo-holomorphic curves) are disjoint from the cylinders $C\left(p_{2},e^{is}\right)$. In particular this solid torus coinsides with the union of standard cylinders $C\left(e^{is}\right)$ at origin but not in $p_{1}>0,p_{2}=0$ inside the subset $\theta_{2}\in\left[-\delta,\delta\right]$, because if so, either it is contained inside the domain, or it intersects the domain at a discrete subset. The former is impossible, by compactness, and the latter is also impossible, because it will also intersect the nearby cylinders $C\left(p_{2},e^{is}\right)$ by continuity. Hence $\mathcal{\dot{T}}$, and the isotopy, intersects \[ C\left(e^{is}\right)\cup C\left(p_{2},e^{is}\right),\mid s\mid\leq4\delta,p_{2}<p_{2}^{0},p_{2}>p_{2}^{0} \]
exactly at the origin(s) of the Figure, in other words, we get an isotopy of the cylinder which is supported in the cotangent bundle of the cylinder. \end{proof} \noindent
{}
\section{Local Unknottedness of Two-punctured Planes}
\noindent While the analog of Theorem 9 for one-punctured planes (i.e. cylinder) is shown in Section 4.1, namely any Lagrangian inside $T^{*}\left(\mathbb{R}^{2}-\left\{ 0\right\} \right)$ which coincides with the zero-section outside of a compact subset is Hamiltonian isotopic to the zero section, the analogous result for $T^{*}\left(\mathbb{R}^{2}-\left\{ 0,1\right\} \right)$ (i.e. pair of pants) is unknown. Rather than Hamiltonian isotopy, we will try to prove the smooth version in this section.
\noindent Observe that here ``compact'' requires the isotopy to be fixed not only near $\infty$, but also 0 and 1. So we add back 0-fibre and 1-fibre, use the same idea of modification of $\omega$ to make the Lagrangian symplectic and compactification as the non-punctured case to construct an isotopy fixed at infinity. However, more needs to be done to make this isotopy fixed near 0 and 1. As an analog of Lemma 11 and 12, we have \begin{lem} \noindent There exists a diffeomorphism of $T^{*}\mathbb{R}^{2}=\mathbb{R}^{4}$ which takes the zero section to the complex line $\gamma_{0}$, the Lagrangian $L$ to a symplectic surface which coincides with a complex line $\gamma_{0}$ outside a compact subset, while the fibres over $0$ and $1$ become mapped to two disjoint symplectic lines $\Gamma_{0}$ and $\Gamma_{1}$, respectively, that intersects $\gamma_{0}$ transversely and positively in a single point, each are complex near $\gamma_{0}$ and outside a compact subset. \end{lem}
\begin{proof} \noindent First, we perturb $L$ to a symplectic surface $\alpha$ equal to $\gamma_{0}$, (i.e. the zero section, which is a symplectic linear plane for the symplectic form $\omega_{\varepsilon}$) near fibres and outside a compact subset. We construct this perturbation by an application of Moser's trick as in Section 3, on a closed tubular neighborhood $V$ of $L$, one can find a closed 2-form $\tau$ such that there exists a 1-form $\lambda$ which is compactly supported on $V$ and vanishing near the fibres $\sigma_{0}=\left\{ q_{1}=q_{2}=0\right\} ,\sigma_{1}=\left\{ q_{1}=1,q_{2}=0\right\} $ with \[ d\lambda=\tau-dq_{1}\land dq_{2}, \] which coincides with $dq_{1}\land dq_{2}$ outside a compact subset of $V$ and makes $L$ $\tau-$symplectic. Choose a bump function $\rho$ on $\mathbb{R}^{4}$ which vanishes outside $V$, near the Lagrangian fibres $\sigma_{0}=\left\{ q_{1}=q_{2}=0\right\} ,\sigma_{1}=\left\{ q_{1}=1,q_{2}=0\right\} $, and equals 1 near the neighborhood where $L$ is Lagrangian. Set \[ \widetilde{\omega_{\varepsilon}^{\prime}}=\omega+\varepsilon\left(dq_{1}\land dq_{2}+d\left(\rho\lambda\right)\right). \] Then $L$ is $\widetilde{\omega_{\varepsilon}^{\prime}}$-symplectic. Use Moser's trick to deform $L$ to an $\omega_{\varepsilon}$-symplectic plane $\alpha$ by a smooth isotopy which is supported in the complement of $\sigma_{0}$, $\sigma_{1}$ defined above.
\noindent Second, we perturb Lagrangian fibres to symplectic planes. This can be done via a family of parallel 2-planes that intersect $\alpha$ transversely at two points. Namely, in the standard linear space $\mathbb{R}^{4}$ with coordinates $\left(p_{1},q_{1},p_{2},q_{2}\right)$ such that the symplectic form \[ \omega_{\varepsilon}=\left(dp_{1}-\frac{\varepsilon}{2}dq_{2}\right)\wedge dq_{1}+\left(dp_{2}+\frac{\varepsilon}{2}dq_{1}\right)\wedge dq_{2} \] Consider a family of parallel pair of 2-planes $\Gamma_{0}^{t}$, $\Gamma_{1}^{t}$: \[ \Gamma_{0}^{t}:=\left\{ \left(q_{1},q_{2}\right)^{T}=t\left(\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right)\cdot\left(p_{1},p_{2}\right)^{T}+\left(0,0\right)\right\} \] \[ \Gamma_{1}^{t}:=\left\{ \left(q_{1},q_{2}\right)^{T}=t\left(\begin{array}{cc} 0 & 1\\ -1 & 0 \end{array}\right)\cdot\left(p_{1},p_{2}\right)^{T}+\left(1,0\right)\right\} \] where $\Gamma_{0}^{0}$ is the Lagrangian fibre $\sigma_{0}=\left\{ q_{1}=q_{2}=0\right\} $, and $\Gamma_{1}^{0}=\sigma_{1}=\left\{ q_{1}=1,q_{2}=0\right\} .$ When $t=0$, $\Gamma_{0}^{0}$, $\Gamma_{1}^{0}$ are a pair of Lagrangian planes for the symplectic form $\omega_{\varepsilon}$, when $0<t\leq t_{0}$ for some $t_{0}>0$ small, $\Gamma_{0}^{t}$, $\Gamma_{1}^{t}$ are a pair of symplectic planes for the symplectic form $\omega_{\varepsilon}$. If needed, one can rescale the $p_{i}$ coordiantes to flatten the Lagrangian $L$ so that $\Gamma_{0}^{t}$, $\Gamma_{1}^{t}$ will not intersect $L$ at points other than our desired two transversal intersection points.
\noindent Third, map $\gamma_{0}$ to the complex line $\mathbb{C}\times0$ by a linear symplectomorphism. Meanwhile, denote the symplectic images of $\gamma_{0}$, $\Gamma_{0}^{t_{0}}$, $\Gamma_{1}^{t_{0}}$ by the same notations. One can obtain a smooth family $\Sigma_{s}$ of symplectic immersions, where $\Sigma_{0}=\gamma_{0}\cup\Gamma_{0}^{t_{0}}\cup\Gamma_{1}^{t_{0}}$ with fixed intersection points $pt_{0},pt_{1}$ on $\Gamma_{0}^{t_{0}}$, $\Gamma_{1}^{t_{0}}$ respectively.
\noindent Finally, use \cite[Proposition 4.9]{key-9} to obtain a new deformation $\widetilde{\Sigma_{t}}$ through symplectic immersions with exactly two transverse double points, such that $\widetilde{\Sigma_{0}}=\Sigma_{0}$ and that the deformation fixes $\gamma_{0}$ and the positions $pt_{0}$ and $pt_{1}$ of the double points and where the deformation has support near the double points and near $\infty$. After such a deformation we may assume that the sought properties are satisfied. \end{proof} \noindent Again as Lemma 12, take an interpolerated symplectic form and compactify $\mathbb{R}^{4}$ into $\left(\mathbb{CP}^{2},\Omega,J_{0}\right)$ by adding $\Gamma_{\infty}$. The complex lines $\Gamma_{0}$, $\Gamma_{1}$ from the previous proposition become symplectic spheres in the projective plane. Denote them again by $\Gamma_{0}$, $\Gamma_{1}$. Moreover, we may assume that this $J_{0}$ makes $\gamma_{0}$$,\Gamma_{0},\Gamma_{1},\Gamma_{\infty}$ simultaneously J-holomorphic. The symplectic plane $\alpha$ becomes an embedded symplectic sphere, denoted by $\alpha$ again, which intersects $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$ at the same points as $\gamma_{0}$ does. Denote the intersections by $\infty,$ $pt_{0}$, $pt_{1}$ respectively. See Figure 5,1. In the following we abuse notation and use $\Gamma_{0}$, $\Gamma_{1}$ in order to denote the compactifications of the symplectic planes to symplectic degree one spheres in $\mathbb{CP}^{2}.$
\noindent \begin{figure}
\caption{Vertical lines are $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$, they intersect at the same point $x_{0}$. The horizonal line is $\alpha$ which intersect $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$ at $\infty,$ $pt_{0}$, $pt_{1}$ respectively. }
\end{figure}
\noindent \LyXZeroWidthSpace{}
\noindent Note the following: $\Gamma_{i}$ are complex planes which are linear outside a compact subset and near $\mathbb{C}\times0.$ The key point to our proof is that we can interpolate through compatible almost complex structures that keep $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$ J-holomorphic, from one that makes $\alpha$ J-holomorphic, to one that makes $\gamma_{0}$ J-holomorphic. \begin{thm} \noindent There exists a smooth isotopy of $\mathbb{CP}^{2}$ that takes $\alpha$ to $\gamma_{0}$ which fixes the line $\Gamma_{\infty}$ pointwise, keeps the intersection point between this isotopy's image and $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$ at $\infty,$ $pt_{0}$, $pt_{1}$ respectively, and the image does not intersect $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$ at other points. Moreover, this isotopy can be taken to fix a neighborhood of $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$. \end{thm}
\begin{proof} \noindent By Lemma 18, we have an almost complex structure $J_{0}$ making $\gamma_{0},\Gamma_{0},\Gamma_{1},\Gamma_{\infty}$ simultaneously $J_{0}$-holomorphic. There is also an almost complex structure $J_{1}$ making $\alpha,\Gamma_{0},\Gamma_{1},\Gamma_{\infty}$ simultaneously $J_{1}$-holomorphic, again as argued in Theorem 14 by extending a section of a bundle with contractible fibres. Choose the path $J_{t}$ of almost complex structures for which $\Gamma_{0},\Gamma_{1},\Gamma_{\infty}$ remain $J_{t}$-holomorphic for all times t. Consider the unique $J_{t}$-holomorphic line, denoted by $\alpha_{t}$, passing through the two intersection points $\infty,0$. We have $\alpha_{0}=\gamma_{0}$, $\alpha_{1}=\alpha$ and the family $\left\{ \alpha_{t},0\leq t\leq1\right\} $ gives the smooth isotopy from $\alpha$ to $\gamma_{0}$. One can extend this isotopy to an isotopy of $\mathbb{CP}^{2}$ that fixes $\Gamma_{0},\Gamma_{1},\Gamma_{\infty}$, by \cite[Proposition 4.8]{key-9}. However, the problem is that $\alpha_{t}$ may not intersect $\Gamma_{1}$ at 1. See Figure 5,2.
\noindent \begin{figure}
\caption{The intersection points of $\alpha_{t}$ and $\Gamma_{\infty}$, $\Gamma_{0}$ is $\infty$ and $pt_{0}$ respectively, but the intersection of $\alpha_{t}$ and $\Gamma_{1}$may not be $pt_{1}$.}
\end{figure}
\noindent Use parameter $t$ to denote the above isotopy process $f\left(t,x\right),t\in\left[0,1\right],x\in\mathbb{CP}^{2}$ from $\alpha$ to $\gamma_{0}$, with Figure 5,2 as an intermediate status between them.
\noindent For every tamed $J$, $\mathbb{CP}^{2}-\left\{ x_{0}\right\} $ can be foliated by $J-$holomorphic spheres that all intersect at $x_{0}$. Especially, there is a unique $J_{t}$-holomorphic curve passing through $x_{0}$ and $1$, denoted by $\Gamma^{t}$. There is a smooth isotopy for each $t\in\left[0,1\right]$ that takes $\Gamma_{1}$ to $\Gamma^{t}$. Parameterize this isotopy by $s$. Together we have an isotopy \[ g\left(t,s,x\right),t\in\left[0,1\right],s\in\left[0,1\right],x\in\mathbb{CP}^{2} \] with the property that $g\left(0,s,x\right)=x$, $g\left(1,s,x\right)=x$. Take the top-path of the two-dimensional domain of $\left(s,t\right)$, we get the desired smooth isotopy. See Figure 5.3.
\noindent \begin{figure}
\caption{The two-dimendional domain of the parameter pair $\left(t,s\right)$.}
\end{figure}
\noindent Parameterize this path by $r$, finally we get a smooth isotopy from $\alpha$ to $\gamma_{0}$, which keeps the intersection points $\infty,$$\Gamma_{0}^{t_{0}}$, $\Gamma_{1}^{t_{0}}$. By the same intersection argument as in section 3, this isotopy intersect $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$ at no other points.
\noindent A standard topological argument like that in the end of Proposition 12 makes the smooth isotopy away from $\Gamma_{\infty}$, $\Gamma_{0}$, $\Gamma_{1}$. Moreover, one can use the method of \cite[Theorem 4.6]{key-9} to make the isotopy also symplectic. \end{proof}
\end{document} | arXiv |
Removal of heavy metals from tannery effluent using chitosan-g-poly(butyl acrylate)/bentonite nanocomposite as an adsorbent
R. Nithya1 &
P. N. Sudha1
In the present era, due to industrial revolution in the developing countries like India, the ground water system has been largely polluted. Tannery effluent is a major source of aquatic pollution, and a large number of tanneries are scattered all over India particularly in Tamilnadu, Uttar Pradesh and West Bengal. This work deals with the removal of heavy metals chromium and lead and the reduction of the important physicochemical parameters like total dissolved solid (TDS), total suspended solid (TSS), total solids (TS), biochemical oxygen demand (BOD), chemical oxygen demand (COD), total hardness, salinity, turbidity and electrical conductivity from the tannery wastewater by using chitosan-g-poly(butyl acrylate)/bentonite nanocomposite as an adsorbent. The batch system was used to conduct the biosorption experiments. The influence of different experimental parameters, such as contact time, pH and amount of adsorbent, was evaluated. The results showed that the prepared nanocomposite can be used efficiently for the treatment of tannery wastewater containing heavy metals.
Due to the rapid growth of tannery industries and pollution-provoking factories, it has become mandatory to realise the perilous reactions of toxic organic compounds and heavy metal ions in the water resources and take drastic steps to remove such pollutions from water to save mankind and environment. It was found that the pollutions caused by Lead and Chromium are found to be very crucial and need to be removed, failure of which would cause very baneful effects. Hexavalent chromium, due to the presence of mutagenic and carcinogenic properties, causes liver damage, pulmonary congestion, edema and skin irritation, resulting in ulcer formation (Raji and Anirudhan 1998). The World Health Organization (WHO) has recommended a maximum level of 0.05 mg/l of Cr (VI) in drinking water (Bhaumik et al. 2011) and 0.1 mg/l in surface water (Guidelines for Drinking-Water Quality 2006). The liver, kidney and reproductive system are affected severely by the toxic effect of lead, so also, the basic cellular processes and functions of the brain get negatively affected (Kushwaha et al. 2012). According to the WHO, the maximum permissible limit of lead for drinking water is 3–10 μg/l (Needleman 1999) and 0.1 mg/l for inland surface water.
Currently, biopolymers are industrially attractive because of their capacity to bind transition metal ions and are environmentally safe. One among the biopolymers is chitosan, a linear polysaccharide, composed of glucosamine and N-acetyl glucosamine units linked by β (1–4) glycosidic linkage. It has emerged as one of the low-cost adsorbents for the removal of heavy metals and dyes. Though it is affordable in terms of cost and is non toxic, it is strongly pH dependent and very much susceptible to biodegradation (Mi et al. 2002; Ravi Kumar 2000), thus limiting its practical use as adsorbent. Hence, chemical modification by grafting (Mishra et al. 2008; Srivastava et al. 2009) is one of the promising techniques to improve the physicochemical properties and stability of chitosan.
The syntheses of organic–inorganic polymeric materials have gained much attention recently for the removal of heavy metals from the wastewater because of their low cost, high mechanical resistance (Thambiannan et al. 2013; Junping et al. 2007) and effective adsorption of dyes (Mouzdahir et al. 2010). Bentonite is one of the most popular clay rocks with exceptional adsorption properties due to its large surface area (Schütz et al. 2013). It is a type of alumino silicate in the ratio of 2:1; its unit layer consists of one Al3+ octahedral sheet between two Si4+ tetrahedral sheets. However, negatively charged surface and large amount of exchangeable cations make natural bentonite highly hydrophilic, and its surface is covered with a layer of water, blocking, in part, its adsorption capability for organic pollutants. Moreover, it swells and form very stable colloidal suspension when it comes into contact with water, which causes the solid–liquid separation very difficult after adsorption (Wang et al. 2014). Hence, bentonite can be used as fillers in various polymer matrices to prepare nanocomposites because the functional groups of bentonite such as SiAOASi can interact strongly with functional groups in polymer matrices such as AOH, ANH2, ACOOH and N-acetyl glucosamine units (Darder et al. 2005). A different composite with chitosan has been done by several researchers (Vanamudan et al. 2014; Abdel Khalek et al. 2012; Thayyath Sreenivasan Anirudhan et al. 2010).
The objective of this work is to use the novel adsorbent, chitosan-grafted-poly(butyl acrylate)/bentonite nanocomposite (Cs-g-PBA/bent) which is so far not known as reported for the removal of heavy metal Cr(VI) and Pb(II) ions from the industrial effluent rather than other physicochemical parameters. But, the other physicochemical parameters are used to check the validity of Cs-g-PBA/bent nanocomposite for the removal of Cr(VI) and Pb(II) in a tannery industry wastewater.
Chitosan (CS) (degree of deacetylation = 95 % determined by 1H-NMR and molecular weight 13.45 × 104 Da) was purchased from India Sea Foods, Kerela, India. Ceric ammonium nitrate was purchased from Thomas Baker pvt Ltd. Acetic acid and glutaraldehyde was purchased from Sisco Research laboratories Pvt ltd. Bentonite was purchased from Aryem chemicals, Chennai.
Preparation of chitosan-g-poly(butyl acrylate)
A known amount of chitosan (0.5 g) was dissolved in a known volume (30 ml) of 2 % aqueous acetic acid. N-Butyl acrylate (0.6 ml dissolved in 20 ml ethanol) was added, and then, a freshly prepared solution of ceric ammonium nitrate (CAN) in 1 N nitric acid was added drop by drop with continuous stirring for 30 min. The reaction was stopped, and the product was precipitated using 2 N sodium hydroxide solution with vigorous stirring. The precipitate was washed with distilled water for several times and filtered.
Synthesis of chitosan-grafted poly (butyl acrylate)/bentonite (Cs-g-PBA/bent) nanocomposites
Bentonite powder (bent) was heated at 110 °C for 1 h to activate the surface. A known amount of bentonite (1 g) was immersed in distilled water to make a slurry. About 1 g of poly(butyl acrylate) grafted chitosan was dissolved in (5 %, v/v) acetic acid. This solution was then added to the slurry and the mixture was stirred for 30 min. Five millilitres of 25 % glutaraldehyde was then added and stirred vigorously for 5 min. The mixture was stirred and soaked in an ultrasonic bath for 30 min. The temperature of the dispersion was at room temperature (25 °C). This mixture was then washed with water and dried in oven at 50 °C to get Cs-g-PBA/bent powder. By analysing the powder with DLS method, the particle size was found to be 679.3 nm, and hence, this powder would be more suitable for sorption studies.
Batch adsorption experiments
The batch adsorption experiments were conducted in 250-ml conical flasks containing 100 ml of tannery wastewater using 1 g of Cs-g-PBA/bent nanocomposite adsorbent. The flasks were agitated in an orbit shaker at room temperature. Effect of initial pH was studied by varying solution pH from 2 to 8 at the sorbent dosage of 1 g/100 ml for 1 h contact time. The pH of wastewater was adjusted by using 0.1 N HCl or 0.1 N NaOH. The effect of contact time and adsorbent dose on the sorption capacity of sorbent was studied in the range 1–6 h and 1–6 g. After the period, the solution was filtered using Whatman filter paper no. 42 and the filtrate was analysed for physicochemical parameters and heavy metals chromium and lead. The adsorption removal percentage of various physicochemical parameters and heavy metals in a tannery industry wastewater was calculated by using the following formula:
$$ \%\mathrm{Removal}=\frac{\mathrm{initial}\kern0.5em \mathrm{concentration}-\mathrm{final}\kern0.5em \mathrm{concentration}}{\mathrm{initial}\kern0.5em \mathrm{concentration}}\times 100 $$
Collection of sample
The sample was collected from Ranipet (Ranipettai) a suburban town, which is a major industrial area located in Wallajah taluk, Vellore district, Tamil Nadu, at 79° 19′–79° 22′ E longitude and 12° 53′–12° 57′ N latitude and is situated 114 km west of Chennai. More than 200 tannery industrial units were located in and around this town. It is one of the biggest exporting centres of tanned leather in India and discharging their effluents on the open land and surrounding water bodies (Govil et al. 2004). This sample was collected in a plastic container and kept in the refrigerator at 4 °C, and the effluent was used for the experiment for the next day of sample collection.
Physicochemical study
The samples thus collected were analysed for total solids (TS), total dissolved solids (TDS), total suspended solids (TSS), biochemical oxygen demand (BOD), chemical oxygen demand (COD), total hardness, salinity, turbidity, electrical conductivity and presence of heavy metals like chromium and lead. The techniques and methods followed for analysis and interpretation of physicochemical parameters are those given by APHA (1995) and heavy metals by atomic absorption spectroscopy.
The mean values for each parameter of the wastewater revealed that most of them were beyond the standard provisional limit provided by ISI (1991), CPCB as shown in Table 1.
Table 1 Initial parameters of tannery wastewater
pH is the most important factor that affects the adsorption process. The effect of pH of tannery effluent on adsorption was investigated from pH 2–8 (Figs. 1 and 2). It was found that the reduction of physicochemical parameters and heavy metal ions were higher at low pH. This was due to the fact that low pH causes surface OH− groups to accept protons, facilitating ligand exchange since H2O is an easier ligand than OH− to get displaced from metal bonding sites, and hence, low pH promotes anion adsorption (Dessalew D. Alemayehu et al. 2012).
Effect of pH on reduction of physicochemical parameters
Effect of pH on removal of metal ions
In other words, at low pH, a large number of hydrogen ions can neutralise the oppositely charged surface and thus enhance the electrostatic attraction between the adsorbent and adsorbate overcoming electrostatic repulsion between them (Vasanthy and Lakshmana Perumalsamy 1995). Moreover from the literature, at low pH, the spontaneous reduction of Cr(VI) into Cr(III) also occurred due to high redox potential (1.3 V), and hence, chromium(VI) has been removed up to 97 % (Bhaumik et al. 2012; Karthik and Meenakshi 2014). Lead showed a minimal of 21 % removal efficiency, and this may be due to intervention of other ions which were also present in the effluent. The optimum pH 3 was selected for further experiments.
Effect of adsorbent dose
The effect of adsorbent dose was studied by varying dosage from 1–6 g with the optimum pH 3 (Figs. 3 and 4). With the increase of adsorbent dose, the reduction of physicochemical parameters and the removal of heavy metal ions increased up to 5 g. This was due to the increased available binding sites in the nanocomposite for the complexation of metal ions (Saravanan et al. 2013). Equilibrium was almost attained after 5 g of adsorbent dosage as a result of decrease in available sites in the adsorbent. The maximum removal percentage (%) obtained in 5 g of dose for all parameters is shown in Fig. 5. The highest removal efficiencies for different chemical parameters achieved were 78 % (BOD), 73 % (COD), 91 % (TSS), 39 % (TDS), 98 % (Cr), 99 % (total hardness), and 28 % (Pb), 40 % (electrical conductivity), 43 % (salinity), 45 % (sodium), 66 % (TS) and 98 % (turbidity).
Effect of adsorbent dose on reduction of physicochemical parameters
Effect of adsorbent dose on removal of metal ions
Effect of adsorbent dose on removal percentage (%) of some physicochemical parameters
Effect of time
The effect of contact time on the reduction of physicochemical parameters was determined by varying time from 1–6 h (Figs. 6 and 7) at pH 3. The dependency of removal efficiency of some physicochemical parameters on reaction time is shown in Fig. 8. The highest removal efficiencies for different chemical parameters achieved were 77 % (BOD), 78 % (COD), 93 % (TSS), 41 % (TDS), 98 % (Cr), 98 % (total hardness), and 42 % (Pb), 42 % (electrical conductivity), 44 % (salinity), 48 % (sodium), 67 % (TS) and 98 % (turbidity).
Effect of time on reduction of physicochemical parameters
Effect of time on removal of metal ions
Effect of time on removal percentage (%) of some physicochemical parameters
The reduction efficiency increased rapidly till 240 min. There was no significant change in equilibrium concentration after 240–360 min since the adsorption phase reached equilibrium. A faster initial removal rate was possibly due to the availability of sufficient vacant adsorbing sites in the adsorbent. Afterwards, the rate of removal decreased significantly due to availability of limited vacant adsorption sites (Sivakami et al. 2013).
Comparison of treated effluent with the raw effluent
The comparative study for some effluent quality parameters of the raw effluent and the treated effluent is presented in Table 2 where the treated effluent was mentioned at pH 3. The concentration of total chromium drops down from 1055 to 23.05 mg/l with a removal of 97.81 % of chromium, and hence, it is obvious that a chromium concentration has been reduced to a large extent by using Cs-g-PBA/bent nanocomposite. The concentration of lead was reduced from 0.43 to 0.34 mg/l with a removal percentage of 20.93 %. Electrical conductivity was also decreased to 39.19 % and cannot be reduced to a greater extent as observed in the case of chromium concentration. This might be due to some ionic species which do not get precipitated (Sabur et al. 2013). The initial value of COD was 4210 mg/l, and after treatment, it was decreased to 1320 and the percentage removed is 68.64 %. BOD was reduced from 1250 to 380 mg/l. TSS was removed to 91.16 % and TS to 65.97 %. Turbidity was removed from 453 to 18 NTU.
Table 2 Physical and chemical characteristics of the untreated and treated effluent
Though tannery is an indispensable industry for the economic and social growth of a country, the fact that the wastewater generated by these industries gets directly discharged into the nearby water body with insufficient treatment is alarming. Most of the physicochemical parameters investigated in this study showed that almost all the effluent characteristics were above the provisional discharge limit set by ISI. In the present study, experiments have been conducted for the removal of Cr(VI) and Pb(II) from the tannery industrial wastewater using Cs-g-PBA/bent nanocomposite as an adsorbent. To know the ability of Cs-g-PBA/bent, the experiments were conducted with varying adsorbent dosage, contact time and pH.
The results showed that the maximum removal percentage of Cr(VI) in the tannery industrial wastewater at an optimum adsorbent dosage of 5 g, contact time of 240 min and pH of 3 was 97.81 %. Also, the resultant maximum removal percentage of Cr(VI) in tannery industry wastewater with optimum processing parameters was verified with the other physicochemical parameters of BOD, COD, total hardness, TDS, TSS, TS, salinity, turbidity, electrical conductivity and sodium in the tannery industry wastewater. Further, it was understood that the various pollutants of tannery wastewater like hexavalent chromium, total hardness, TSS and turbidity were reduced to the maximum extent of satisfaction with a comparatively economical absorption process. As this process is promising, it can be concluded that chitosan-g-poly (butyl acrylate)/bentonite nanocomposite is more effective for removal of heavy metals and reduction of concentration of physicochemical parameters from the tannery effluent.
Abdel Khalek, M.A., Mahmoud, G.A., El-Kelesh, N.A. (2012). Synthesis and Characterization of Poly-Methacrylic Acid Grafted Chitosan-Bentonite Composite and its Application for Heavy Metals Recovery. Chemistry and Materials Research, 2(7), 1-12.
Alemayehu, D. D., Singh, S. K., & Tessema, D. A. (2012). Assessment of the adsorption capacities of fired clay soils from JIMMA (Ethiopia) for the removal of Cr (VI) from aqueous solution. Universal Journal of Environmental Research and Technology, 2(5), 411–420.
American Public Health Association (APHA). (1995). Standard methods for estimation of water and wastewater (19th ed.). Washington: American Water Works Association, Water environment Federation. application as electrochemical sensors.
Anirudhan, T. S., Rijith, S., & Tharun, A. R. (2010). Adsorptive removal of thorium(IV) from aqueous solutions using poly(methacrylic acid)-grafted chitosan/bentonite composite matrix: process design and equilibrium studies. Colloids and Surfaces A: Physicochem. Eng. Aspects, 368, 13–22.
Bhaumik, M., Maity, A., Srinivasu, V. V., & Onyango, M. S. (2011). Enhanced removal of Cr(VI) from aqueous solution using polypyrrole/Fe3O4 magnetic nanocomposite. Journal of Hazardous Materials, 190, 381–390.
Bhaumik, M., Maity, A., Srinivasu, V. V., & Onyango, M. S. (2012). Removal of hexavalent chromium from aqueous solution using polypyrrole–polyaniline nanofibers. Chemical Engineering Journal, 181–182, 323–333.
Central Pollution Control Board. (1986). Pollution Control Acts, Rules, and Notifications issued there under. Fourth edition (pp. 358–359). New Delhi: CPCB, Ministry of Environment and Forests. 897 pp.
Darder, M., Colilla, M., & Ruiz-Hitzky, E. (2005). Chitosan–clay nanocomposites: application as electrochemical sensors. Applied Clay Science, 28, 199–208.
Govil, P. K., Reddy, G. L. N., Krishna, A. K., Seshu, C. L. V., Satya Priya, V., & Sujatha, D. (2004). Inventorization of contaminated sites in India (NGRI technical report, pp. 54–66).
Guidelines for Drinking-Water Quality (2006) In (3rd Eds.). Geneva: World Health Organization
Indian Standard Institute (ISI). (1991). Drinking Water Specification'.
Junping, Z., Li, W., & Aiqin, W. (2007). Preparation and properties of chitosan-g-poly(acrylic acid)/montmorillonite superabsorbent nanocomposite via in situ intercalative polymerization. Industrial and Engineering Chemistry Research, 46, 2497–2502.
Karthik, R., & Meenakshi, S. (2014). Removal of hexavalent chromium ions from aqueous solution using chitosan/polypyrrole composite. Desalination and Water Treatment, 1–14.
Kushwaha, A.K., Gupta, N., Chattopadhyaya, M.C. (2012). Adsorption behavior of lead onto a new class of functionalized silica gel. Arabian Journal of Chemistry, xxx, xxx–xxx. DOI:10.1016/j.arabjc.2012.06.010
Mi, F. L., Tan, Y. C., Liang, H. F., & Sung, H. W. (2002). In vivo biocompatibility and degradability of a novel injectable-chitosan-based implant. Biomaterials, 23, 81–91.
Mishra, D. M., Tripathy, J., Srivastava, A., Mishra, M. M., & Behari, K. (2008). Graft copolymer (chitosan-g-N-vinyl formamide): synthesis and study of its properties like swelling, metal ion uptake and flocculation. Carbohydrate Polymers, 74, 632–639.
Mouzdahir, Y. E. I., Elmchaouri, A., Mahboub, R., Gil, A., & Korili, S. A. (2010). Equilibrium modelling for adsorption of methylene blue from aqueous solutions on activated clay minerals. Desalination, 250, 335–338.
Needleman, H.L. (1999). History of lead poisoning in the world. In: Proceedings of international conference on lead poisoning prevention and treatment: 8-10 Febrauary, 1999. (pp. 17-25). Bangalore, India: The George foundation.
Raji, C., & Anirudhan, T. S. (1998). Batch Cr(VI) removal by poly acrylamide-grafted sawdust: kinetics and thermodynamics. Water Research, 32, 3772.
Ravi Kumar, M. N. V. (2000). A review of chitin and chitosan applications. Reactive and Functional Polymers, 46, 1–27.
Sabur, M. A., Rahman, M. M., & Safiullah, S. (2013). Treatment of tannery effluent by locally available commercial grade lime. Journal of Scientific Research, 5(1), 143–150.
Saravanan, D., Gomathi, T., & Sudha, P. N. (2013). Sorption studies on heavy metal removal using chitin/bentonite biocomposite. International Journal of Biological Macromolecules, 53, 67–71.
Schütz, T., Dolinská, S., & Mockovčiaková, A. (2013). Characterization of bentonite modified by manganese oxides. Universal Journal of Geoscience, 1(2), 114–119.
Sivakami, M. S., Gomathi, T., Venkatesan, J., Jeong, H.-S., Kim, S.-K., & Sudha, P. N. (2013). Preparation and characterization of nano chitosan for treatment wastewaters. International Journal of Biological Macromolecules, 57, 204–212.
Srivastava, A., Mishra, D. M., Tripathy, J., & Behari, K. (2009). One pot synthesis of xanthan gum-g-N-vinyl-2-pyrrolidone and study of their metal ion sorption behavior and water swelling property. Journal of Applied Polymer Science, 111, 2872–2880.
Thambiannan, S., Rajendran, R., & Lima Rose, M. (2013). Clean—soil, air. Water, 41, 797–807.
Vanamudan, A., Bandwala, K., & Pamidimukkala, P. (2014). Adsorption property of rhodamine 6G onto chitosan-g-(N-vinylpyrrolidone)/montmorillonite composite. International Journal of Biological Macromolecules, 69, 506–513.
Vasanthy, M., & Lakshmana Perumalsamy, P. (1995). Removal of chromium using various adsorbents. Journal of Ecotoxicology and Environmental Monitoring, 3, 47–50.
Wang, X., Yang, L., Zhang, J., Wang, C., & Li, Q. (2014). Preparation and characterization of chitosan–poly(vinyl alcohol)/bentonite nanocomposites for adsorption of Hg(II) ions. Chemical Engineering Journal, 251, 404–412.
PG and Research Department of Chemistry, D.K.M College for Women, Vellore, Tamil Nadu, India
R. Nithya
& P. N. Sudha
Search for R. Nithya in:
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Correspondence to P. N. Sudha.
PNS and RN contributed to the conception and design of the study. RN carried out the experiments, analysed the data and drafted the manuscript, and PNS supervised the works. Both authors read and approved the final manuscript.
Nithya, R., Sudha, P.N. Removal of heavy metals from tannery effluent using chitosan-g-poly(butyl acrylate)/bentonite nanocomposite as an adsorbent. Text Cloth Sustain 2, 7 (2017) doi:10.1186/s40689-016-0018-1
Tannery effluent
Physicochemical parameters | CommonCrawl |
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What is known about the topological structure of spacetime?
General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions:
What topological restrictions do Einstein's equations put on the manifold? For instance, the existence of a Lorentz metric implies some topological things, like the Euler characteristic vanishing.
Are there any experiments being done or even any hypothetical experiments that can give information on the topology? E.g. is there a group of graduate students out there trying to contract loops to discover the fundamental group of the universe?
general-relativity spacetime universe differential-geometry topology
JamalS
EricEric
That's a great question! What you are asking about is one of the missing links between classical and quantum gravity.
On their own, the Einstein equations, $ G_{\mu\nu} = 8 \pi G T_{\mu\nu}$, are local field equations and do not contain any topological information. At the level of the action principle,
$$ S_{\mathrm{eh}} = \int_\mathcal{M} d^4 x \, \sqrt{-g} \, \mathbf{R} $$
the term we generally include is the Ricci scalar $ \mathbf{R} = \mathrm{Tr}[ R_{\mu\nu} ] $, which depends only on the first and second derivatives of the metric and is, again, a local quantity. So the action does not tell us about topology either, unless you're in two dimensions, where the Euler characteristic is given by the integral of the ricci scalar:
$$ \int d^2 x \, \mathcal{R} = \chi $$
(modulo some numerical factors). So gravity in 2 dimensions is entirely topological. This is in contrast to the 4D case where the Einstein-Hilbert action appears to contain no topological information.
This should cover your first question.
All is not lost, however. One can add topological degrees of freedom to 4D gravity by the addition of terms corresponding to various topological invariants (Chern-Simons, Nieh-Yan and Pontryagin). For instance, the Chern-Simons contribution to the action looks like:
$$ S_{cs} = \int d^4 x \frac{1}{2} \left(\epsilon_{ab} {}^{ij}R_{cdij}\right)R_{abcd} $$
Here is a very nice paper by Jackiw and Pi for the details of this construction.
There's plenty more to be said about topology and general relativity. Your question only scratches the surface. But there's a goldmine underneath ! I'll let someone else tackle your second question. Short answer is "yes".
$\begingroup$ Thanks for the answer. I do not see why EFEs cannot contain topological data since you need a global solution to them (you can solve it locally but they need to patch together to form a global metric). For example, if the EFEs implied something like positive scalar curvature then that would really limit the topology (being positive at a point is local, being positive everywhere is global). The adding of topological invariants looks very interesting-- I'll have to read more into it. $\endgroup$ – Eric Dec 10 '10 at 15:46
$\begingroup$ I get what you're trying to say. The EFE's should encode some sort of topological information aside from the addition of topological terms to the action. Or perhaps that is because we consider the EFE's to be fundamental, when the Ricci term and the other topological terms can arise from something more general such as $BF$ theory Reference which is a topological theory. Anyhow, if you like the answer could you accept it as the answer. Thanks :-) $\endgroup$ – user346 Dec 10 '10 at 19:43
$\begingroup$ @user346 "So gravity in 2 dimensions is entirely topological" Could you please expand on this in a less technical terms for me? $\endgroup$ – Leos Ondra May 2 '13 at 11:15
Just one additional point that I haven't seen mentioned above: if the space-time has non-trivial fundamental group, it won't be seen by an observer at infinity. This is the content of the Topological Censorship Theorem. The implication is that for an asymptotically flat space-time, any interesting topology will be hidden behind the event-horizon. The proof of the theorem is rather surprisingly simple: it is more or less a direct extension of Penrose's singularity theorem.
Friedman, J. L.; Schleich, K. & Witt, D. M. Topological censorship Phys. Rev. Lett., American Physical Society, 1993, 71, 1486-1489
Schleich, K. & Witt, D. M. Singularities from the Topology and Differentiable Structure of Asymptotically Flat Spacetimes http://arxiv.org/abs/1006.2890
Galloway, G. J.. On the topology of the domain of outer communication. Class. Quantum Grav. 12 No 10 (October 1995) L99 (3pp)
Willie WongWillie Wong
$\begingroup$ You are a mathematician, correct? So please explain things at the level of a physicist to me :-) My question is, how does this conclusion change is the spacetime is asymptotically deSitter or anti-deSitter? Also what is your view on the dodecahedral universe hypothesis? $\endgroup$ – user346 Dec 13 '10 at 23:41
$\begingroup$ @space_cadet: I don't know much about the dodecahedral universe hypothesis, but from what I know, isn't it an attempt to explain certain "features" of WMAP data? I don't think there's any a priori reason to rule it in or rule it out: only data will tell. As to topological censorship in dS or AdS spaces: the Penrose argument itself only uses the null energy condition, which is not affected by the cosmological constant. But the statement of topological censorship I think requires a time-like or null Scri to make sense. Indeed, in the AdS case, there is a 2001 paper by ... $\endgroup$ – Willie Wong Dec 16 '10 at 1:38
$\begingroup$ ... Galloway, Schleich, Witt, and Woolgar which shows that the same result (topological censorship) holds for asymptotically anti-de-sitter space-times. That is, defining the domain of outer communications as the intersection of the past and the future of Scri, they showed that for (n+1) dimensional (with n at least 3) asymptotically AdS space-times, the domain of outer communications is simply connected, in the sense that any time-like curve going from Scri to (the same connected piece of) Scri can be deformed continuously to a causal curve in Scri. $\endgroup$ – Willie Wong Dec 16 '10 at 1:43
$\begingroup$ Interesting answer, but you may be interested in this: link.springer.com/article/10.1134%2FS0202289313010064. $\endgroup$ – user3657 Mar 15 '13 at 5:37
I don't know the answer, but your intuition is right on -- the fact that the equations are local doesn't mean that there cannot be a constraint on the topology of a global solution. For example, in Euclidean signature, $R_{ij} = g_{ij}$ immediately implies that the scalar curvature is positive, which in turn leads to topological constraints. If the four-manifold is Einstein and complex, then it must be a del Pezzo surface (highly constrained). I don't know much about Lorentzian signature, but I know that the PDE's are a whole different beast. I have seen a few results about classification of possible holonomy groups of Lorentzian Einstein manifolds, but I don't know anything global (I actually don't know anything at all).
Eric ZaslowEric Zaslow
Einstein equations describe local structure of the space-time. They contain no global or topological information.
While I heard that some restricitons on the scale of topology can be derived from curvature of the Universe if the curvature is negative. (Something like "scale = integer multiple of 1/curvature".)
Well, if our space has non-trivial topology, then light rays will "wrap around" our universe multiple times and you'll be able to see the same (similar) copies of galaxies. I heard of people searching for such similarities without success.
Also nontrivial topology must result in some correlation in CMB -- no such correlations were found (yet?) either.
Sklivvz
KostyaKostya
$\begingroup$ What do you mean by scale of the topology? But Einstein's equations need to be solved globally so couldn't they put some restrictions on the topology? For example if Einstein's equations implied positive scalar curvature, then that would limit the possible manifolds. Also, With there not being any classification of even simply connected 4-manifolds, it seems likely there are nontrivial ones which wouldn't have the "wrap around" property of light rays. $\endgroup$ – Eric Dec 10 '10 at 15:35
$\begingroup$ Simplest example -- consider flat space-time. You can imagine it "wrapping", so when you travel distance L in one direction you will get to the same place. As far as I understand that would be called the 3D torus (in simplest case). The distance L is the scale of topology. It can be arbitrary -- Einstein equations do not impose any restrictions on it. $\endgroup$ – Kostya Dec 10 '10 at 17:03
$\begingroup$ Oh ok, so that would still be a geometric thing: scaling a cylinder doesn't change any topology. $\endgroup$ – Eric Dec 10 '10 at 17:55
$\begingroup$ @Kostya Can you list some papers where people attempt to model "Also nontrivial topology must result in some correlation in CMB ... "? $\endgroup$ – More Anonymous Nov 3 '18 at 13:58
$\begingroup$ @MoreAnonymous arxiv.org/pdf/astro-ph/0412569.pdf $\endgroup$ – Kostya Nov 3 '18 at 15:18
These are two independent questions, one mathematical, and one about observations.
What constraints do the Einstein equations imply about the global structure of space and/or spacetime? I don't know the general answer, my impression is that not as much as is known about Lorentzian manifolds as about Euclidean manifolds. Furthermore, there is no reason to suspect the space/spacetime is singularity-free (at the very least we know of many black holes in the universe), and I doubt much can be said about the global structure of any manifold if you allow for singularities.
About observational physics: the only observable I can think of that is sensitive to global structure is the low multipoles of the CMB, and every now and again there are papers on the subject, to explain anomalies in such multipoles (e.g. stories about football-shaped universe). Alas, cosmic variance limits how seriously you can take such observations and models aimed to explain them.
On the experiments and topology question, there is some work on the subject by Glenn Starkman et al. In their work, they search for structures in the CMB that would indicate some particular topology for the universe. There is a very nice lecture given in PI on the subject as well as other issues that have to do with CMB. To give you a spoiler on the lecture, they haven't found anything in large angle correlations.
VagelfordVagelford
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\begin{document}
\title{Quantum Correlations in NMR systems}
\author{T. S. Mahesh}
\email{[email protected]}
\author{C. S. Sudheer Kumar}
\email{[email protected]}
\author{Udaysinh T. Bhosale}
\email{[email protected]}
\affiliation{Department of Physics and NMR Research Center,\\
Indian Institute of Science Education and Research, Pune 411008, India}
\begin{abstract}
{ In conventional NMR experiments, the Zeeman energy gaps of the nuclear spin ensembles are much lower than their thermal energies, and accordingly exhibit tiny polarizations. Generally such low-purity quantum states are devoid of quantum entanglement. However, there exist certain nonclassical correlations which can be observed even in such systems. In this chapter, we discuss three such quantum correlations, namely, quantum contextuality, Leggett-Garg temporal correlations, and quantum discord. In each case, we provide a brief theoretical background and then describe some results from NMR experiments.
}
\end{abstract}
\maketitle
\begin{center} \textit{`Correlations cry out for explanation'} - J. S. Bell in \textit{Speakable and Unspeakable in Quantum Mechanics}, Cambridge university press (1989). \end{center}
\section{Introduction}
Quantum physics is known for many nonintuitive phenomena including certain classically forbidden correlations.
To study and understand these mysterious quantum correlations we require a suitable testbed. Nuclear Magnetic Resonance (NMR) \cite{NMR_Abragam_book,Levitt_Spindynabook}
of an ensemble of molecular nuclei in bulk liquids/solids form a convenient testbed even at room temperatures \cite{Corynmr_1st}.
The weakly perturbed nuclear spins in such systems can store quantum superpositions for long durations ranging from seconds to minutes. In addition, excellent unitary controls via radio-frequency pulses allow precise manipulations of spin-dynamics.
Even though one can not have local addressability of individual spins, and one works with the spin-ensemble as a whole, it is still possible to study many of the quantum correlations, namely contextuality, temporal correlation, discord etc. The ensemble measurements are often sufficient, since many of the quantum correlations can be evaluated via expectation values. However, at room temperature there is little entanglement in conventional NMR systems \cite{PhysRevLett.83.1054}. In fact, this makes NMR a good candidate for studying quantum correlations without entanglement.
\begin{figure}
\caption{Various correlations and corresponding bounds distinguishing quantum regime from classical.}
\label{overview}
\end{figure}
In the following sections we are going to review some NMR experiments investigating quantum contextuality, Leggett-Garg inequality, and quantum discord. For the sake of completeness, we have provided a brief theoretical background in each case.
\section{Quantum Contextuality} As the name suggests, outcome of a quantum measurement in general depends on the context i.e., measurement-setting, arrangement, situation, circumstance, etc. Quantum contextuality (QC) states that the outcome of a measurement depends not only on the system and the observable being measured, but also on the context of the measurement, i.e., on other compatible observables which are measured along with \cite{peres_context_1pg,quant_theory_peres,KS}. QC signifies a mysterious nonclassical correlation between measurement outcomes corresponding to distinct observables. One consequence of QC is violation of Bell's inequality \cite{Cabello_state_ind_context,Bell_vio_lopholefree_1pt3km}, which
has challenged the most cherished tenet of special theory of relativity, i.e., locality.
Peres explained quantum contextuality using a pair of electrons in a singlet state $(\ket{01}-\ket{10})/\sqrt{2}$ \cite{peres_context_1pg}. Suppose we measure a Pauli observable $\sigma_{i\alpha}$, where $\alpha \in \{x,y,z\}$, on the $i$th particle, and obtain an outcome $\alpha_i = \pm 1$. For the singlet state, the result of measuring $\sigma_{1x}\sigma_{2x}$ is $x_1x_2 = -1$ since $\expv{\sigma_{1x}\sigma_{2x}} = -1$. Similarly, $y_1y_2 = -1$ . However, if one measures $\sigma_{1x}\sigma_{2y}$ followed by
$\sigma_{1y}\sigma_{2x}$ one would obtain the outcome $x_1y_2y_1x_2 = -1$ since $\expv{\sigma_{1x}\sigma_{2y} \sigma_{1y}\sigma_{2x}} = \expv{\sigma_{1z}\sigma_{2z}} = -1$, which is in contradiction with $x_1x_2=y_1y_2=-1$.
Later Mermin \cite{Mermin} generalized quantum contextuality to a state-independent scenario. Consider a pair of spin-1/2 particles and a set of nine Pauli-observables arranged in the following fashion: \begin{eqnarray}
\begin{array}{|c|c|c||c|} \hline \sigma_{1z} & \sigma_{2z} & \sigma_{1z}\sigma_{2z} & +\mathbbm{1}\\ \hline \sigma_{2x} & \sigma_{1x} & \sigma_{1x}\sigma_{2x} & +\mathbbm{1} \\ \hline \sigma_{1z}\sigma_{2x} & \sigma_{1x}\sigma_{2z} & \sigma_{1y}\sigma_{2y} & +\mathbbm{1}\\ \hline \hline +\mathbbm{1} & +\mathbbm{1} & -\mathbbm{1} & \\ \hline
\end{array}.
\label{Merminsquare} \end{eqnarray} Here the last column (row) lists the product along the row (column). In this arrangement, all the operators along any row, or any column, mutually commute and therefore they can be measured sequentially or simultaneously without any mutual disturbance. Whatever may be the state of the spin-pair, if one measures the three consecutive observables along any row one would obtain the outcome $+1$, the only eigenvalue of $\mathbbm{1}$. Similarly, if one measures along first or second column one would obtain $+1$. On the other hand, choosing observables along the last column will lead to an outcome $-1$. However, no assignment of $\pm 1$ values to individual measurements of all the nine observables can satisfy the above joint-measurement outcomes, indicating that such noncontextual preassignments of measurement outcomes is incompatible with quantum physics.
\subsection{Contextuality studies using NMR systems} The first demonstration of contextuality in NMR systems was reported by Moussa \textit{et. al} \cite{Moussa}. Using a solid state NMR system, they evaluated the state independent inequality \cite{Cabello_state_ind_context} \begin{eqnarray} \beta = \expv{\pi_{r_1}}+\expv{\pi_{r_2}}+\expv{\pi_{r_3}}+ \expv{\pi_{c_1}}+\expv{\pi_{c_2}}-\expv{\pi_{c_3}} \le 4 \label{cabello} \end{eqnarray} where $\expv{\pi_{r_i}}$ are the expectation values obtained when all the observables along the $i$th row of matrix in \ref{Merminsquare} are measured. Similarly $\expv{\pi_{c_j}}$ is the expectation value for measurements along the $j$th column. Exploiting the state independent property, they initialized the system in the maximally mixed state and obtained the value $\beta = 5.2\pm 0.1$. While the result is in agreement with the quantum bound which is $\beta \le 6$, it strongly violates the inequality in \ref{cabello}.
Later, Xi Kong \textit{ et. al} demonstrated QC by a single three level system in a NV center setup \cite{spin_1_QC}. More recently, Dogra \textit{ et. al} demonstrated QC using a qutrit (spin-1) NMR system with a quadrupolar moment, oriented in a liquid crystalline environment. Using a set of 8 traceless observables (Gell-Mann matrices) and an inequality derived based on a noncontextual hidden variable (NCHV) model, they observed a clear violation of the NCHV inequality \cite{Contextuality_KDorai}.
\subsubsection*{Contextuality via psuedo spin mapping} Su \textit{et. al.} \cite{Cont_theory} have theoretically studied QC of eigenstates of one dimensional quantum harmonic oscillator ($1$D-QHO) by introducing two sets of pseudo-spin operators, \begin{eqnarray} {\bf \Gamma } = (\Gamma_x,\Gamma_y,\Gamma_z),~~ {\bf \Gamma '} = (\Gamma_x',\Gamma_y',\Gamma_z') \nonumber \end{eqnarray} with components, \begin{eqnarray} \Gamma_x = \sigma_x \otimes \mathbbm{1}_2, \Gamma_y = \sigma_z \otimes \sigma_y, \Gamma_z = -\sigma_y \otimes \sigma_y, \Gamma_x' = \sigma_x \otimes \sigma_z, \Gamma_y' = \mathbbm{1}_2 \otimes \sigma_y, \Gamma_z' = -\sigma_x \otimes \sigma_x, \label{Gammas defined} \end{eqnarray} where $ \mathbbm{1}_2 $ is $2\times 2$ identity matrix.
Defining the dichotomic unitary observables, \begin{eqnarray} A=\Gamma_x, ~~~ B=\Gamma_x' \cos \beta + \Gamma_z' \sin \beta, ~~~ C=\Gamma_z, ~~~ D=\Gamma_x'\cos \eta + \Gamma_z'\sin\eta, \label{A,B,C,D defined} \end{eqnarray} they setup Bell-Clauser-Horne-Shimony-Holt (Bell-CHSH) inequality \cite{quant_info_neilson_chuang}, \begin{eqnarray} \mathrm{\textbf{I}}= \expec{AB} + \expec{BC} + \expec{CD} - \expec{AD} \leq 2. \label{BellCHSHineqlty} \end{eqnarray} However, the quantum bound was shown to be $\mathrm{\textbf{I}}_Q \le 2\sqrt{2}$, clearly violating the above Bell-CHSH inequality and thus exhibiting QC of QHO.
Katiyar \textit{et. al.} carried out an NMR investigation of this inequality by mapping the QHO eigenstates to the spin-states of a 2-qubit system (with an additional ancilla qubit) \cite{QCQHOcssk}. Using the Moussa protocol \cite{Moussa} (described in the next section) to extract the joint-expectation values in the inequality \ref{BellCHSHineqlty}, they obtained $\textbf{I}_Q \approx 2.4 \pm 0.1$. Although decoherence limited the experimental value to below the quantum bound ($\textbf{I}_Q \le 2.82$), it is clearly above the classical bound ($\textbf{I} \le 2$) and therefore establishes QC of 1D-QHO.
Thus, we observe that even when a system is in a separable state, measuring nonlocal observables leads to violation of Bell-CHSH inequality \cite{Cabello_NCHV_ineqlty}.
\section{Temporal Correlations} Bell's inequalities (BI) are concerned with how two systems (each with a dimension of at least 2) are correlated over space, where as the Leggett-Garg inequality (LGI) is concerned with the correlation of a single system (with a dimension of at least 2), with itself at different time instants. While the former deals with context of the measurement, the latter deals with a temporal context.
LGI is based on the following two assumptions: \begin{itemize} \item[1.] Macroscopic realism (MR): A macroscopic system, with two or more macroscopically distinct states available to it, exists in one of these states at any given point of time. \item[2.] Noninvasive measurability (NM): It is possible to determine the state of the system with arbitrarily small perturbation to its future dynamics \cite{LG,LGI_review}. \end{itemize}
Although the original motivation of Leggett and Garg was to test the existence of quantumness even at a macroscopic level, most of the violations of LGI reported so far are on microscopic systems \cite{LGI_review}. The LGI violations in such systems were either due to invasive measurement or the system being in microscopic superpositions. Even though LGI violation in a macroscopic system such as a superconducting qubit has been reported \cite{LeggettGargPalacios2010}, the existence of macroscopically distinct states in such a system is not clear \cite{LGI_review}. Other experimental works on LGI include Nitrogen-Vacancy centers \cite{Waldherr2011,George2013}, photonic systems \cite{Dressel2011}, electron interferometers \cite{LeggettGargEmaryClive}, superconducting qubit
\cite{LeggettGargGroen2013}, and more recently in neutrino oscillations \cite{LeggettGargFormaggio}.
Recent theoretical extensions of LGI include its entropic formulation\cite{ELGI_UshaDevi}
and LGI in a large ensemble of qubits \cite{LeggettGargLambertNeill}.
The violation of the former was recently observed using NMR experiments by Katiyar {\it et. al.}\cite{ELGI_TSM}.
LGI is also studied for a system of qubits
coupled to a thermal environment \cite{LeggettGargLobejkoMarcin}. For more details reader can refer to the review
\cite{LGI_review}. LGI violation in a 3-level NMR system has also been reported recently
\cite{LeggettGargHemant2016}.
In the following we provide a brief theoretical as well as experimental review of LGI in the context of NMR.
\begin{figure}
\caption{Extreme values of TTCCs for a classical particle in a double-well potential for the cases of (a) three-time measurement and (b) four-time measurement. The left and right columns illustrate minimum and maximum values of $K_n$-strings respectively. (c) $K_n$ versus $n$ and $\omega \Delta t/\pi$ for a single qubit. The filled regions indicate LGI violations.}
\label{kstring}
\end{figure}
\subsection{Leggett-Garg string} Consider a system (the `target') evolving under some Hamiltonian. Let $\mathbbm{Q}$ be a dichotomic observable with eigenvalues $Q=\pm 1$, and let $Q(t_i)$ denotes the measurement outcome at time $t_i$. Repeating these measurements a large number of times we obtain the two-time correlation coefficient (TTCC) $C_{ij}$ for each pair: \begin{eqnarray} C_{ij} = \lim\limits_{N\rightarrow\infty}\frac{1}{N} \sum _{r=1}^{N}Q_{r} \left( t_i \right)\cdot Q_r \left(t_j \right)=\expec{Q\left( t_i \right)\cdot Q\left(t_j \right)}, \end{eqnarray} where $r$ is the trial number. Finally, the values of these coefficients are to be substituted in the \textit {n}-measurement LG string given by: \begin{eqnarray} K_{n} = C_{12}+ C_{23} + C_{34} +....+ C_{(n-1)n} - C_{1n}. \end{eqnarray} Each TTCC $C_{ij}$ is bounded by a maximum value of $+1$, corresponding to a perfect correlation, and a minimum value of $-1$, corresponding to a perfect anti-correlation. $C_{ij}=0$ indicates no correlation. Thus, the upper bound for $K_{n}$ consistent with \textit{macrorealism} comes out to be $(n-2)$, while the lower bound is $-n$ for $odd$ $n$, and $-(n-2)$ for $even$ $n$ (see Fig. \ref{kstring}(a) and (b)). With these considerations LGI reads $-n \le K_n \le (n-2) \;\; \mathrm{for \; odd} \;n \mathrm{,\; and} -(n-2) \le K_n \le (n-2) \;\; \mathrm{for \; even} \; n$.
In the following, we consider the case of a single qubit, namely a spin-1/2 nucleus precessing in an external static magnetic field.
\subsection{Violation of LGI with a single qubit} A spin-1/2 nucleus precessing in an external magnetic field along $z$-axis has the following Hamiltonian: $\frac{1}{2}\omega \sigma_z$, where $\omega$ is the Larmor frequency. Let $\sigma_x$ be the dichotomic observable \cite{LGITSM}. Starting from the definition of TTCCs, we obtain for an arbitrary initial state $\rho_0$ \cite{Brukner_multilevelLGI_Luders,tempCHSH_bound_fritz}, \begin{eqnarray} C_{ij}=\left \langle \sigma_x\left(t_i\right) \sigma_x\left(t_j\right) \right\rangle =\cos \left\{ \omega (t_j-t_i) \right\}. \label{Cij_theory} \end{eqnarray} Dividing the total duration from $t_{1}$ to $t_{n}$ into $(n - 1)$ parts each of length $\Delta t$, we can express the LG string consistent with equation \ref{Cij_theory} as \begin{eqnarray} K_{n}=(n-1)\cos\{\omega\Delta t\} - \cos\{(n-1)\omega\Delta t\}. \label{kn} \end{eqnarray}
Fig. \ref{kstring}(c) illustrates $K_n$ curves for $n=3$ to $8$ and for $\omega \Delta t \in [0,2\pi]$. The classical bounds in each case are shown by horizontal lines. As indicated by the filled areas, LGI is violated for each value of $n$ at specific regions of $\omega \Delta t$. Quantum bounds of $K_3$ are $-3$ and $+1.5$ and that for $K_4$ are $-2\sqrt{2}$ and $+2\sqrt{2}$, and so on. In the following we discuss an experimental protocol for evaluating the LG strings.
\subsection{Moussa protocol} As described before, one needs to extract TTCCs in a way as noninvasive as possible. One way to achieve this is by using an ancilla qubit and employing Moussa protocol (Fig. \ref{lginmr}). It involves preparing the ancilla in $\ket{+}$ state (an eigenstate of $\sigma_x$; or a pseudopure state $(1-\epsilon)\mathbbm{1}/2+\epsilon \proj{+}$) followed by a pair of CNOT gates separated by the delay $t_j-t_i$. Finally $\sigma_x$ observable of the ancilla qubit is measured in the form of transverse magnetization which reveals the corresponding TTCC \cite{Moussa}: \begin{equation} \expv{\sigma_x}_\mathrm{ancilla} = \mathrm{Tr}[\rho_s \sigma_x(t_i)\sigma_x(t_j)] = C_{ij}, \end{equation} where $\rho_s = \mathbbm{1}/2$ is the initial state of the system qubit.
\begin{figure}
\caption{Moussa circuits (left) to extract TTCCs for the three-measurement case and the experimental results (crosses in the right) of $C_{ij}$ and $K_3$ obtained with $^1$H (ancilla) and $^{13}$C (system) spins of chloroform (molecular structure shown in bottom-left). Both short-time and long-time behavior of $K_3$ are shown. Here smooth curves are drawn with with theoretical expression (Eq. \ref{Cij_theory}) along with an appropriate decay factor.
Parts of this figure are adapted from \cite{LGITSM}. }
\label{lginmr}
\end{figure}
The Moussa circuits are easy to implement using a two-qubit NMR system \cite{LGITSM,Souza_LGI}. Athalye \textit{et. al.} \cite{LGITSM} have used $^{13}$C and $^1$H spins of $^{13}$C-Chloroform as system and ancilla qubits respectively and found a clear violation of LGI by more than 10 standard deviations at short time scales. However, with longer time scales, the TTCCs decayed resulting in a gradual reduction in the violation, and ultimately satisfying the LGI bounds.
More recently, Knee \textit{et. al.} \cite{LGI_knee_noninvasive} have used ideal negative result measurements (INRM) to extract TTCCs noninvasively. The method involves two sets of experiments - one with CNOT and the other with anti-CNOT. In the former, the system qubit is unaltered if the ancilla (control-qubit) is in state $\ket{0}$, while in the latter, the system is unaltered if the ancilla is in state $\ket{1}$. Postselecting the subspaces wherein the system is unaltered is considered to be more noninvasive \cite{LGI_knee_noninvasive}. Using nuclear and electronic spins (in an ensemble of phosphorous donars in silicon) as system and ancilla, Knee \textit{et. al.} demonstrated LGI violation with INRM \cite{LGI_knee_noninvasive}.
\subsection{Entropic Leggett-Garg inequality (ELGI)} In 2013, Usha Devi {\it et. al.} \cite{ELGI_UshaDevi} have formulated the entropic Leggett-Garg inequality in which they place bounds on amount of information associated with a noninvasive measurement of a macroscopic system. The amount of information stored in a classical observable $\mathbbm{Q}(t_i)$ at time $t_i$ is given by the Shannon entropy, \begin{eqnarray} H(\mathbbm{Q}(t_i)) = -\sum_{Q(t_i)} P(Q(t_i)) \log_2 P(Q(t_i)), \end{eqnarray} where $P(Q(t_i))$ is the probability of the measurement outcome $Q(t_i)$ at time $t_i$. The conditional entropy $H(\mathbbm{Q}(t_j) \vert \mathbbm{Q}(t_i))$ is related to the joint-entropy \begin{eqnarray} H(\mathbbm{Q}(t_j),\mathbbm{Q}(t_i)) = -\sum_{Q(t_i),Q(t_j)} P(Q(t_i),Q(t_j)) \log_2 P(Q(t_i),Q(t_j)) \end{eqnarray} by Bayes' theorem, i.e., \begin{eqnarray} H(\mathbbm{Q}(t_j) \vert \mathbbm{Q}(t_i)) = H(\mathbbm{Q}(t_i),\mathbbm{Q}(t_j))-H(\mathbbm{Q}(t_i)). \end{eqnarray} For $n$ measurements performed at equal intervals $\Delta t$, we denote $h(\Delta t) = H(\mathbbm{Q}(\Delta t) \vert \mathbbm{Q}(0)) = H(\mathbbm{Q}(2\Delta t) \vert \mathbbm{Q}(\Delta t)) = \cdots$, and $h((n-1) \Delta t) = H(\mathbbm{Q}((n-1) \Delta t) \vert \mathbbm{Q}(0))$. By setting up a quantity called information deficit \begin{eqnarray} {\mathcal D}_n = \frac{(n-1)h(\Delta t) - h((n-1)\Delta t )}{\log_2(2s+1)}, \end{eqnarray} where $2s+1$ is the number of distinct states (where $s$ is spin number), Usha Devi \textit{et. al.} proved that ${\mathcal D}_n \ge 0$ for classical systems.
The experimental violation of ELGI was first demonstrated by Katiyar \textit{et. al.} \cite{ELGI_TSM} again using $^{13}$C-Choroform as the two-qubit register. The single-time probability $P(Q(t_i))$ and the joint probabilities $P(Q(t_i),Q(t_j))$ are extracted using the circuits shown in Fig. \ref{elgiresults}(a) and (b) respectively. Note that an ancilla spin is used to extract joint probabilities with the help of INRM procedure applied to the first measurement. The results displayed in Fig. \ref{elgiresults}(c), indicate a clear violation of ELGI by four standard deviations.
\begin{figure}
\caption{(a,b) The quantum circuits for extracting single-time and joint probabilities. Here $U_i^\dagger$ denotes the back-evolution of the system in the computational basis which is equivalent to having the dynamical observable $\mathbb{Q}(t_i)$. (c) Experimental information deficit (crosses with errorbars) compared to theoretical values (solid curve) for a spin-1/2 particle. The dashed line indicates the macrorealistic bound. Here $\theta = (n-1)\omega \Delta t$.
Parts of this figure are adapted from \cite{ELGI_TSM}.
}
\label{elgiresults}
\end{figure}
\section{Quantum Discord} In the early days of quantum information and quantum computation it was shown that entanglement is the key resource to perform various tasks \cite{Horodeckirpm}. However, it was later realized that quantum correlations beyond entanglement are also useful for quantum information processing \cite{Knill1998,Bennett1999,Niset2006,Horodecki2005}. It was shown theoretically \cite{PhysRevLett.83.1054,DavidMeyer2000} as well as experimentally \cite{Lanyon2008} that some tasks can be made efficient even with separable states, but with non-zero quantum correlations. Thus, quantifying the quantum correlation becomes important, and it can be achieved by using measures such as discord \cite{Ollivier,vedral} and geometric discord \cite{Dakic,Luo_geometric,PaoloGiorda2010}. For more details on the topic of quantum correlations one may refer to the reviews in \cite{ModiRMP,ModiOpenSyst,Zhang2012,Horodecki2013,Streltsov,AdessoReview}.
Discord has also been studied in the ground state of certain spin chains particularly close to quantum phase transitions \cite{Sarandy2013}. Signatures of chaos in the dynamics of quantum discord are found using the model of the quantum kicked top \cite{VaibhavMadhok2015}. Quantum critical behavior in the anisotropy $XY$ spin chain is studied using geometric discord \cite{Cheng2012}.
It is believed that discord is a resource behind the efficiency of the DQC-1 model \cite{Knill1998,DavidMeyer2000,Passante2009,Maziero2009,Animesh}. Quantum advantage with no entanglement but with non-zero quantum discord has been demonstrated in single-photon states \cite{Maldonado2016}. Quantum discord has also been estimated in optical systems using mixed states \cite{Lanyon2008} and in an anti ferromagnetic Heisenberg compound \cite{Mitra2015}.
Non-zero quantum discord in NMR systems has been observed by many researchers \cite{laflammeoct11,serraaug11,serradiscordquad,katiyarDiscord}. For various theoretical and experimental aspects of quantum discord and related measures reader can refer to review \cite{Celeri2011}. Investigations on the evolution of quantum discord under decoherence\cite{DiscordNMR_Oliveira} and under decoherence-suppression sequences \cite{katiyarDiscord} have also been reported. In the following we briefly describe some aspects related to discord and geometric discord.
\subsection{Discord and mutual information} \begin{figure}
\caption{Venn diagram representing total information $H(A,B)$,
individual informations $\left(H(A), H(B)\right)$, the conditional
information $\left( H(A|B), H(B|A)\right)$, and the mutual information
$I(A:B) = J(A:B)$ in classical information theory.}
\label{mutual}
\end{figure}
Mutual Information $I(A:B)$ is defined as the amount of information that is common to both the subsystems $A$ and $B$ of a bipartite system, and is given in terms of Shannon entropy \begin{equation} I(A:B)= H(A) + H(B) - H(A,B). \label{eq5} \end{equation}
It can be seen that mutual information is symmetric, i.e., $I(A:B) = I(B:A)$. Another classically equivalent expression based on Bayes rule can be obtained from Eq.~(\ref{eq5}) as follows: \begin{eqnarray}
J(A:B) = H(A) - H(A|B)
= H(A) - \sum_{i}p_{i}^{b}H(A|b=i) ~. \label{eq6} \end{eqnarray} These expressions can be intuitively understood using Fig.~\ref{mutual}.
In the quantum information theory, the von Neumann entropy gives the information content of a density matrix and is defined as \begin{equation} {\mathcal H}(\rho) = -\sum_{x}\lambda_{x}\log_{2}\lambda_{x}, \label{eq7} \end{equation} where $\lambda_{x}$'s are the eigenvalues of the density matrix $\rho$. Although the two expressions of mutual information given in Eqs.~(\ref{eq5}) and (\ref{eq6}) are equivalent in classical information theory this is not the case in quantum information theory. The reason for this difference is that the expression for mutual information given by Eq.~(\ref{eq6}) involves measurements and its value depends on the measurement outcomes. Measurements in quantum theory depends on the basis used and it changes the final state of the system. Henderson and Vedral \cite{vedral} have proved that the total classical correlation can be obtained as the maximum value of \begin{eqnarray}
{\mathcal J}(A:B) = {\mathcal H}(B) - {\mathcal H}(B|A)
= {\mathcal H}(B) - \sum_{i}p_{i}^{a}{\mathcal H}(B|a=i) ~, \label{eq8} \end{eqnarray} where the maximization is performed over all possible orthonormal measurement bases $\{\Pi_{i}^{a}\}$ for $A$. The quantum mutual information ${\mathcal I}(A:B)$ is defined in a way analogous to that of the classical mutual information, i.e., \begin{eqnarray} {\mathcal I}(A:B) = {\mathcal H}(A)+{\mathcal H}(B)-{\mathcal H}(A,B). \label{qmutualinfo} \end{eqnarray} Therefore, the non-classical correlations can be quantified as the difference \begin{equation}
D(B|A) = {\mathcal I}(A:B) - \max_{\{\Pi_{i}^{a}\}}{\mathcal J}(A:B). \label{disc} \end{equation} Ollivier and Zurek had called this difference as `\textit{discord}' \cite{Ollivier}. Zero-discord states or ``classical'' states are the ones in which the maximal amount of information about a subsystem can be obtained without disturbing its correlations with the rest of the system.
It should be noted that discord is not a symmetric function in general, i.e. $D(B|A)$ and $D(A|B)$ can differ. Datta \cite{animesh_nullity} has proved that a given state
$\rho_{AB}$ satisfies $D(B|A)=0$ if and only if there exists a complete set of orthonormal measurement operators on $A$ such that \begin{equation}
\rho_{AB} = \sum_{i}p_{i}^{a} \Pi_{i}^{a}\otimes\rho_{B|a=i}. \label{classical} \end{equation} When the first part of a general bipartite system is measured, the resulting density matrix is of the form given by Eq.~(\ref{classical}). Since the final state after measurements is a classical state, one can extract the classical correlations from it. Thus, for any quantum state and every orthonormal measurement basis, there exists a classically correlated state. Maximization of ${\mathcal J}(A:B)$ gives the maximum classical correlation that can be extracted from the system, and the remaining extra correlation is the quantum correlation.
\subsection{Evaluation of Discord} Given a density matrix $\rho_{AB}$, one can easily construct the reduced density matrices $\rho_{A}$ and $\rho_{B}$ of the individual subsystems. Then the total correlation ${\mathcal I}(A:B)$ can be found using the quantum mutual information Eq.(\ref{qmutualinfo}). Maximization of ${\mathcal J}(A:B)$ to evaluate discord is nontrivial. The brute force method is to maximize ${\mathcal J}(A:B)$ over as many orthonormal measurement bases as possible, taking into account all constraints and symmetries. Strictly speaking, this method gives a lower bound on ${\mathcal J}(A:B)$ since the maximization may not be perfect.
While a closed analytic formula for discord does not exist for a general quantum state, analytical results are available for certain special classes of states \cite{girolami}. For example, Chen \textit{et. al.} have described analytical evaluation of discord for two qubit $X$-states under specific circumstances \cite{chen,TingYu2007,ARPRau2009,ARPRau2010,Fanchini2010}. Luo has given an analytical formula for discord of the Bell-diagonal states which are a subset of the $X$-states \cite{luo}, and are defined as the states which are diagonal in the Bell basis \begin{equation} \vert \psi^{\pm}\rangle = \frac{1}{\sqrt{2}} ( \vert 01\rangle \pm \vert 10\rangle ) ~,~~ \vert \phi^{\pm}\rangle = \frac{1}{\sqrt{2}} ( \vert 00\rangle \pm \vert 11\rangle ). \end{equation} The generic structure of a Bell-diagonal state is $\rho_{BD} = \lambda_1 \vert \psi^-\rangle\langle \psi^- \vert + \lambda_2 \vert \phi^-\rangle\langle \phi^- \vert + \lambda_3 \vert \phi^+\rangle\langle \phi^+ \vert + \lambda_4 \vert \psi^+\rangle\langle \psi^+ \vert$. This state is separable iff it’s spectrum lies in $[0,1/2]$ \cite{RyszardHorodecki1996}.
Using only local unitary operations (so that the correlations remain unaltered), all Bell-diagonal states can be transformed to the form given by \begin{equation} \rho_{BD} = \frac{1}{4} \Big( \mathbbm{1} + \sum_{j=1}^{3}r_{j}\sigma_{j}\otimes\sigma_{j} \Big), \label{rhobd} \end{equation} where the real numbers $r_{j}$ are constrained such that all eigenvalues of
$\rho_{BD}$ remain in $[0,1]$. The symmetric form of $\rho_{BD}$ also implies that it has symmetric discord, i.e., $D_{BD}(B|A) = D_{BD}(A|B)$. Thus, the analytical formula for discord in this case is, using Eq.~(\ref{disc}), \begin{eqnarray}
{D}_{BD}(B|A) &=& 2 + \sum_{i=1}^{4}\lambda_{i}\log_2\lambda_{i} - \left(\frac{1-r}{2}\right)\log_{2}(1-r)
- \left(\frac{1+r}{2}\right)\log_{2}(1+r), \label{dbd} \end{eqnarray}
where $r=\max\{|r_{1}|,|r_{2}|,|r_{3}|\}$.
A special Bell-diagonal state, i.e., when $\lambda_1=(1+3\epsilon)/4$ and $\lambda_2=\lambda_3=\lambda_4=(1-\epsilon)/4$, is known as the Werner state \begin{eqnarray} \rho_{W}(\epsilon) = \frac{1-\epsilon}{4}\mathbbm{1} + \epsilon \vert\psi^- \rangle\langle \psi^- \vert. \label{werner} \end{eqnarray} It has entanglement iff $1/3 \leq \epsilon \leq 1$. In this case $r_j=-\epsilon$ for $j=1$, $2$, $3$ and $r=\epsilon$. The discord using Eq.~(\ref{dbd}) is then given by \begin{eqnarray} D_{W}(\epsilon) = \frac{1}{4} \log_2 \frac{(1-\epsilon)(1+3\epsilon)}{(1+\epsilon)^2}+ \frac{\epsilon}{4} \log_2 \frac{(1+3\epsilon)^3}{(1-\epsilon)(1+\epsilon)^2} = \frac{\epsilon^2}{\ln 2} + O(\epsilon^3). \label{dwerner} \end{eqnarray} This expression is plotted in Fig.~\ref{discordVspurity}.
\subsection{Geometric Discord} Geometric discord is a form of Discord that is relatively easier to compute \cite{Dakic,Luo_geometric}. In the following, we discuss the case of two-qubit geometric discord \cite{Dakic,Huang2016}. For every quantum state there exist a set of postmeasurement classical states ($\Omega_0$), and the geometric discord is defined as the distance between the quantum state ($\rho$) and the nearest classical state ($\chi$), \begin{equation}
D^{G}(B|A) = \min_{\chi \in \Omega_0}\|\rho-\chi\|^2, \end{equation} where
$\|\rho-\chi\|^2 = {\rm Tr}[(\rho-\chi)^2]$ is the Hilbert-Schmidt quadratic norm. Obviously, $D^{G}(B|A)$ is invariant under local unitary transformations. Explicit and tight lower bound on the geometric discord for an arbitrary $A_{m \times m} \otimes B_{n \times n}$ state of a bipartite quantum system is available \cite{Luo_geometric,Hassan}. Protocols to determine lower bounds on geometric discord without tomography have also been discovered recently \cite{Rana,Hassan}.
Following the formalism of Dakic \textit{et. al.} \cite{Dakic} analytical expression for the geometric discord for two-qubit states was obtained in \cite{RyszardHorodecki1996}. The two-qubit density matrix in the Bloch representation is \begin{equation} \rho = \frac{1}{4} \Big( \mathbbm{1} \otimes \mathbbm{1} + \sum_{i=1}^{3} x_{i}\sigma_{i}\otimes\mathbbm{1} + \sum_{i=1}^{3}y_{i} \mathbbm{1}\otimes\sigma_{i} + \sum_{i,j=1}^{3} T_{ij}\sigma_{i}\otimes\sigma_{j} \Big), \label{bloch} \end{equation} where $x_{i}$ and $y_{i}$ represent the Bloch vectors for the two qubits, and $T_{ij}={\rm Tr}[(\rho(\sigma_{i}\otimes\sigma_{j}))]$ are the components of the correlation matrix. The geometric discord for such a state is \begin{equation}
D^{G}(B|A) = \frac{1}{4}\left(\|x\|^2 + \|T\|^2 - \eta_{\rm max} \right), \label{dg} \end{equation}
where $\|T\|^2 = {\rm Tr}[T^\dagger T]$,\ and $\eta_{\rm max}$ is the largest eigenvalue of the matrix $\vec{x}\vec{x}^\dagger + TT^\dagger$. Explicit form of $\eta_{\rm max}$ and a remarkable tight lower bound on geometric discord are given in \cite{RyszardHorodecki1996}.
Using the transformed form of Bell-diagonal states as given in Eq.~(\ref{rhobd}) it can be seen that $x_i = y_i = 0$ and $T$ is a diagonal matrix with elements $T_{ii}=r_i$. Then the geometric discord is given as \begin{equation} D^{G}_{BD}= \frac{1}{4}\left(\sum_{i=1}^{3} r_i^2 - \mbox{max}(r_1^2,r_2^2,r_3^2) \right). \end{equation}
For the Werner state $r_i=-\epsilon$. Then $\|T\|^2 = 3\epsilon^2$ and all eigenvalues of $TT^\dagger$ are $\epsilon^2$, yielding \begin{equation} D^{G}_{W}(\epsilon) = \frac{1}{4}\left(3\epsilon^2-\epsilon^2 \right) = \frac{\epsilon^2}{2}. \end{equation} This expression is plotted versus the purity $\epsilon$ in Fig.~\ref{discordVspurity}. Comparison with Eq.~(\ref{dwerner}) reveals that discord and geometric discord are proportional for low-purity Werner states. Also, the numerical difference between $D_{W}(\epsilon)$ and $2 D^G_{W}(\epsilon)$ does not exceed 0.027 for all $\epsilon\in[0,1]$. An analytical formula for symmetric geometric discord for two-qubit systems is given in \cite{Shi2011} and geometric discord for qubit--qudit systems is given in \cite{Sai2012}.
\begin{figure}
\caption{Discord ($D_W$) and geometric discord ($D_W^G$) of Werner state as a function of its purity factor $\epsilon$. Typical ranges of purity and discord values for some spin-based architectures such as NMR, low-field ESR, and optically polarized electronic spin of nitrogen-vacancy center (NVC) are indicated.}
\label{discordVspurity}
\end{figure}
\subsection{NMR studies of quantum discord} Katiyar \textit{et. al.} \cite{katiyarDiscord} have studied discord and its evolution in certain NMR systems. After preparing the pseudopure state $\rho_0 = (1-\epsilon)\mathbbm{1}/2+\epsilon \proj{00}$ they applied the pulse sequence shown in Fig. \ref{DwCHCl3}(a). The initial state $\rho_0$ is transformed into a Werner state when $\theta$ is set to an odd integral multiple of $\pi/2$. Katiyar \textit{et. al.} measured quantum discord using extensive measurement method described earlier. Fig. \ref{DwCHCl3}(b) displays discord as a function of $\theta$. One can notice that discord is zero for the initial state $\rho_0$, grows with $\theta$ and reaches a maximum value at the Werner state. This experiment demonstrates the existence of small, but non-zero, nonclassical correlations in NMR systems even at room temperatures.
\begin{figure}
\caption{(a) NMR Pulse-sequence used by Katiyar \textit{et. al.} to prepare Werner state and measure discord and (b) experimental and simulated discord as a function of the nonlocal rotation $\theta$. Parts of this figure are adapted from \cite{katiyarDiscord}.}
\label{DwCHCl3}
\end{figure}
Maziero \textit{et. al.} studied the behavior of quantum discord under decoherence using an NMR testbed \cite{DiscordNMR_Oliveira}. They observed a sudden change in the behavior of classical and quantum correlations at a particular instant of time and found distinct time intervals where classical and quantum correlations are robust against decoherence. Yurishchev \cite{DiscordNMR_Yurishchev2014} has analytically and numerically studied NMR dynamics of quantum discord in gas molecules (with spin) confined in a closed nanopore. Kuznetsova and Zenchuk \cite{DiscordNMR_dimers_Kuznetsova20121029} have theoretically studied quantum discord in a pair of spin-1/2 particles (dimer) governed by the standard multiple quantum NMR Hamiltonian and shown the relation between discord and the intensity of the second-order multiple quantum coherence in NMR systems.
\section{Summary} In this chapter, we have briefly discussed three types of quantum correlations, namely quantum contextuality, Leggett-Garg temporal correlations, and quantum discord. In each case, we have surveyed a few NMR experiments.
Exploiting the state independent nature of quantum contextuality, Moussa \textit{et. al.} \cite{Moussa} demonstrated that even a content-less maximally mixed-state ($\mathbbm{1}/4$) violates certain noncontextual hidden variable inequalities when subjected to quantum measurements of certain observables. Similarly, the violation of Leggett-Garg inequalities can be observed even in a two-level quantum system (while quantum contextuality is exhibited by a quantum system with at least three levels). Hence, as demonstrated by Athalye \textit{et. al.} \cite{LGITSM} the violation of LGI is observable even in a spin-1/2 NMR system at room temperature. Moreover, Oliveira \textit{et. al.} \cite{DiscordNMR_Oliveira} and Katiyar \textit{et. al.} \cite{ELGI_TSM} showed the existence of nonzero discord in NMR systems.
NMR has wide-ranging applications from spectroscopy to imaging, and quantum information testbed is the latest of them. Although NMR offers excellent control operations and long coherence times, highly mixed nature of spin-ensembles at room temperatures allows only separable quantum states. In the absence of entanglement, does it have any resource for quantum information studies? This question was answered in terms of above nonclassical correlations.
\section*{Acknowledgments} TSM acknowledges support from DST/SJF/PSA-03/2012-13 and CSIR 03(1345)/16/EMR-II. UTB acknowledges support from DST-SERB-NPDF (File Number PDF/2015/000506).
\end{document} | arXiv |
\begin{document}
\maketitle
\begin{abstract} Extending results of Hargé and Hu for the Gaussian measure, we prove inequalities for the covariance ${\mathrm{{\rm Cov}}}_\mu(f,g)$ where $\mu$ is a general product probability measure on ${\mathbb{R}\ \!\!}^d$ and $f,g : {\mathbb{R}\ \!\!}^d \to {\mathbb{R}\ \!\!}$ satisfy some convexity or log-concavity assumptions, with possibly some symmetries. \end{abstract}
\maketitle
\section{Introduction} If $\mu$ is a probability measure on ${\mathbb{R}\ \!\!}^d$ and if $f,g \in L^2(d\mu)$ are two square integrable functions with respect to $\mu$, their covariance is defined by
\begin{align*} {\mathrm{{\rm Cov}}}_{\mu}(f,g)&= \int \left(f-\int f d\mu \right)\left(g-\int g d\mu\right) d \mu
\end{align*} and is a measure of the joint variability of the two functions. Here and in all the sequel, we make the assumptions that $f$ and $g$ have enough integrability and regularity conditions, so that all the written quantities are well defined.
Lying at the intersection of probability, analysis and geometry, covariance identities and inequalities provide a variety of tools. Without trying to be exhaustive, let us cite some of them: FKG inequalities (\cite{FKG}), (asymmetric) Brascamp-Lieb inequalities (\cite{otto-menz,carlen-cordero-lieb,abj}), Stein kernels (\cite{chatterjee:stein,nourdin-viens,ledoux-nourdin-peccati,Courtadeetal:19,Fathi:stein,saumard:wpi}), concentration inequalities (\cite{bobkov-gotze-houdre,HouPriv:02}, \cite[Section 5.5]{MR1849347}).
The proof techniques of these covariance identities and inequalities vary from semi-group techniques, other types of integration by parts, measure transportation or stochastic calculus. Gaussian measures offer a particularly fruitful framework in this perspective and in Theorem \ref{thm:harge} below, we recall famous covariance inequalities known for the standard Gaussian measure. The main point of this work is to discuss and extend partially these results beyond the Gaussian assumption, to the case of general product measures.
\begin{theo}\label{thm:harge}Let $\gamma$ be the standard Gaussian distribution on ${\mathbb{R}\ \!\!}^{d}$.
\begin{enumerate} \item \cite{hu-chaos,harge:04} Let $f$ and $g$ be two convex functions on ${\mathbb{R}\ \!\!}^{d}$, then
\begin{equation} {\mathrm{{\rm Cov}}}_\gamma(f,g) \geq {\mathrm{{\rm Cov}}}_\gamma (f,x)\cdot {\mathrm{{\rm Cov}}}_\gamma (g,x)\label{eq-harge-gaussienne} \end{equation} where $\cdot$ denotes the standard scalar product on ${\mathbb{R}\ \!\!}^{d}$. \item \cite{harge:04} Let $f$ be a log-concave function and $g$ be a convex function. Assume moreover that $f$ is orthogonal to the linear functions -- that is ${\mathrm{{\rm Cov}}}_\gamma (f,x)=0$ --, then \begin{equation} {\mathrm{{\rm Cov}}}_\gamma(f,g) \leq 0 \label{eq-harge-gaussienne-niv2}. \end{equation} \item \cite{Royen} Let $f$ and $g$ be some quasi-concave functions that are both even, then \begin{equation} {\mathrm{{\rm Cov}}}_\gamma(f,g) \geq 0 \label{eq-harge-gaussienne-niv3}. \end{equation}
\end{enumerate} \end{theo}
The first point of Theorem \ref{thm:harge} is due to Hu \cite{hu-chaos} and was recovered by Hargé \cite{harge:04}. Hu's proof is based on some Itô-Wiener chaos decomposition. This decomposition is based on the interpolation of the covariance by the standard heat semi-group.
Hargé's proof of the second point is based on optimal transport theory and Caffarelli's contraction theorem. Hargé obtained in fact an inequality when $f$ is not necessarily orthogonal to the linear functions, which by a limiting argument recovers (1).
Point (3) was proven by Royen \cite{Royen}. It is known as the Gaussian correlation inequality and was an open question during decades. We refer to \cite{latala-matlak:royen} and \cite{barthe:royen} for history of this result. Royen proved his result in its geometric form, for symmetric convex bodies, by approximation with finite intersections of symmetric slabs. The main ingredients are then an interpolation of some dependent and independent Gaussian measures through their covariance matrix and clever computations of the Laplace transform of multivariate Gamma distributions. Royen thus proves its result for some family of multivariate Gamma distributions. Subsequently, Eskenazis, Nayar and Tkocz \cite{Eskenazis} noticed that Theorem \ref{thm:harge}(3) still holds for product measures whose marginals are mixtures of centered Gaussian measures. In the Appendix, we also show that Theorem \ref{thm:harge}(2) is true in the latter situation.
\
The first main new results of this paper are devoted to dimension one. In dimension one, the covariance inequalities of Theorem \ref{thm:harge} are not limited to the Gaussian context but actually hold for \emph{any} probability measure on ${\mathbb{R}\ \!\!}$ having a finite variance.
\begin{theo}\label{thm:main-1d} Let $\mu$ be any probability measure on ${\mathbb{R}\ \!\!}$ admitting a second moment.
\begin{enumerate} \item For any convex functions $f$ and $g$, one has \begin{equation} {\mathrm{{\rm Var}}}(\mu)\, {\mathrm{{\rm Cov}}}_{\mu}(f,g)\geq{\mathrm{{\rm Cov}}}_{\mu}(f,x)\,{\mathrm{{\rm Cov}}}_{\mu}(g,x).\label{eq-main-1d} \end{equation}
\item Let $f$ be a log-concave function and $g$ be a convex function. Assume moreover that $f$ is orthogonal to the linear function $x$, then \begin{equation} {\mathrm{{\rm Cov}}}_\mu(f,g) \leq 0 \label{eq-main-1d-niv2}. \end{equation}
\item Let $f$ and $g$ be some quasi-concave functions that are both even, then \begin{equation} {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0 \label{eq-main-1d-niv3}. \end{equation} \end{enumerate} \end{theo}
The fact that Theorem \ref{thm:main-1d} holds for any probability measure whereas Theorem \ref{thm:harge} seems limited to the Gaussian setting is striking and rises the following natural question: what about general product measures? Before trying to answer this question, we shall introduce some notations and the hypotheses.
\
\textbf{Notations and hypotheses:} In all the sequel of the paper, we consider $\mu= \mu_1 \otimes \dots \otimes \mu_d$ to be a product measure on ${\mathbb{R}\ \!\!}^d$. Moreover for each $1 \leq k \leq d$, we denote by $a_k(x_k)$ a positive function on ${\mathbb{R}\ \!\!}$ and by $A_k$ its primitive, centered with respect to $\mu_k$ and we assume that $A_k \in L^2(\mu_k)$. When we apply the results with $a_k\equiv 1$, we thus implicitly assume that the measure $\mu_k$ admits a second moment. We assume that $f,g\in L^2(\mu)$. In order to apply the tensorization method and to exchange derivative and integral, we also assume that all the first and second partial derivatives of $f$ and $g$ are integrable with respect to $\mu$. \begin{remark} It is actually possible to weaken the regularity assumptions on the second order partial derivatives. It is indeed sufficient to assume that the first derivatives are monotonic on ${\mathbb{R}\ \!\!}^d$, at the price of standard approximation arguments. This is particularly transparent with the pure determinantal approach. But for clarity and simplicity, we prove the theorems by using the second order partial derivatives. \end{remark}
Arguably, the first basic idea to investigate general product measures is to use a tensorization argument. This allows us to obtain the following extension of Theorem \ref{thm:main-1d}(1) to the higher dimensional case.
\begin{theo}\label{thm:Hu-produit-A} Let $\mu$ be a product measure on ${\mathbb{R}\ \!\!}^d$.
Let $f$ and $g$ be two functions on ${\mathbb{R}\ \!\!}^d$ such that for each pair $ 1\leq i,j \leq d$, the signs of
\begin{equation}\label{eq:cond-l2-fg-mod} \partial_{j} \left(\frac{\partial_{i}f(x)}{a_{i}(x_{i})}\right) \textrm{ and } \partial_{j} \left(\frac{\partial_{i}g(x)}{a_{i}(x_{i})}\right) \end{equation} are constant on ${\mathbb{R}\ \!\!}^d$ and equal. Then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq \sum_{i=1}^{d}\frac{1}{{\mathrm{{\rm Var}}}_{\mu_i}(A_i)}\;{\mathrm{{\rm Cov}}}_{\mu}(f(x),A_{i}(x_i))\,{\mathrm{{\rm Cov}}}_{\mu}(g(x),A_{i}(x_i)). \]
\end{theo} Taking the functions $a_i\equiv 1$ gives the following corollary. \begin{coro}\label{cor:hu-produit} Let $\mu$ be a product measure on ${\mathbb{R}\ \!\!}^d$.
Let $f$ and $g$ be two functions on ${\mathbb{R}\ \!\!}^d$ such that for each couple $ 1\leq i,j \leq d$, the signs of \begin{equation}\label{eq:cond-l2-fg} \partial_{i,j} f(x) \textrm{ and } \partial_{i,j}g(x) \end{equation} are constant and equal. Then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq \sum_{i=1}^{d}\frac{1}{{\mathrm{{\rm Var}}}(\mu_i)}\;{\mathrm{{\rm Cov}}}_{\mu}(f(x),x_{i})\,{\mathrm{{\rm Cov}}}_{\mu}(g(x),x_{i}). \]
\end{coro}
A striking point is that Corollary \ref{cor:hu-produit} is not limited to the Gaussian setting, but holds for any product measure with marginals having a finite second moment. Particularizing to the Gaussian case, where $\mu=\gamma$, the conclusion of Corollary \ref{cor:hu-produit} is the same as in Theorem \ref{thm:harge}(1), but under different assumptions on the functions $f$ and $g$. Even if they coincide in dimension one, the two assumptions are different in higher dimensions and are not included one into another. The assumption of Corollary \ref{cor:hu-produit} seems less classical from a geometric point of view than the classical convexity assumption. Actually, such assumption on the sign of the second partial derivatives also appears in the context of Gaussian comparison theorems, see for instance \cite[Theorem 3.11]{LedTal:11}, that implies Slepian's lemma and Gordon's min-max theorem. Note finally that in the Gaussian setting, even if the statement of Corollary \ref{cor:hu-produit} seems to be new, its proof could be also deduced from the arguments developed in the proof of Hu \cite{hu-chaos}.
\begin{remark} The conditions stated in \eqref{eq:cond-l2-fg-mod} can also be written as the conjunction of Conditions \eqref{eq:cond-a-i} and \eqref{eq:cond-a-ij} below: for each $1\leq i \leq d$, the signs of \begin{equation}\label{eq:cond-a-i} \partial_{i} \left(\frac{\partial_{i}f(x)}{a_{i}(x_{i})}\right) \textrm{ and } \partial_{i} \left(\frac{\partial_{i}g(x)}{a_{i}(x_{i})}\right) \end{equation} are constant and equal and for each couple $1\leq i\neq j \leq d $, the signs of \begin{equation}\label{eq:cond-a-ij} \partial_{ij} f(x) \textrm{ and }\partial_{ij} g(x) \end{equation} are constant and equal. The condition described by Equation~\eqref{eq:cond-a-i} can be interpreted as follows: let $B_i : {\mathbb{R}\ \!\!}^d \to {\mathbb{R}\ \!\!}^d$ be the inverse bijection of \[ (x_1,\ldots,x_d) \mapsto (x_1,\ldots,A_i(x_i),\ldots,x_d),\] then Condition \eqref{eq:cond-a-i} means that the functions \[x_i \mapsto (f\circ B_i)(x_1,\ldots,x_d) \ , \ x_i \mapsto (g \circ B_i)(x_1,\ldots,x_d)\] are both convex or both concave. In the case where $a_i \equiv 1$ for all $ i=1,\dots, d$, and if moreover, all the signs in \eqref{eq:cond-a-i} are the same, then the functions $f$ and $g$ are both coordinatewise convex or both coordinatewise concave. \end{remark}
We now want to investigate what happens when the functions are assumed to satisfy some symmetries. As we shall see, the good notion that fits with the tensorization argument is quite strong and is the unconditionality of (at least) one function.
We recall that a function $f:{\mathbb{R}\ \!\!}^d \to {\mathbb{R}\ \!\!}$ is said to be \emph{unconditional} if it is symmetric with respect to each hyperplan of coordinates : for all $(x_1,\dots, x_d)\in {\mathbb{R}\ \!\!}^d$,
\[ f(x_1,\dots, x_d) = f ({\varepsilon\ \!\!}_1 x_1,\dots, {\varepsilon\ \!\!}_d x_d). \] holds for each choice of signs $( {\varepsilon\ \!\!}_1,\dots, {\varepsilon\ \!\!}_d ) \in \{-1,1\}^d$.
Theorem \ref{thm:Hu-tens-incond-A} below is a multi-dimensional extension of Theorem \ref{thm:main-1d}(1) with a symmetry assumption.
\begin{theo}\label{thm:Hu-tens-incond-A} Let $\mu= \mu_1 \otimes \dots \otimes \mu_d$ be a product measure on ${\mathbb{R}\ \!\!}^d$ and assume that for each $1\leq i \leq d$, the measure $\mu_i$ is even.
Let $f$ and $g$ be two functions on ${\mathbb{R}\ \!\!}^d$ such that for each $ 1\leq i \leq d$ and all $x\in {\mathbb{R}\ \!\!}^d$, the signs of \begin{equation}\label{eq:cond-l2-idem} \partial_{i}\left( \frac{\partial _i f(x)}{a_i(x_i)} \right) \textrm{ and \ } \partial_{i}\left( \frac{\partial _i g(x)}{a_i(x_i)} \right) \end{equation} are constant and equal. Assume moreover that one of the functions is unconditional. Then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \end{theo} The following corollary is directly obtained by setting again the functions $a_i$ to be identically equal to $1$.
\begin{coro}\label{cor:Hu-tens-incond} Let $\mu= \mu_1 \otimes \dots \otimes \mu_d$ be a product measure on ${\mathbb{R}\ \!\!}^d$ and assume that for $1\leq i \leq d$, the measures $\mu_i$ are even.
Let $f$ and $g$ be two functions on ${\mathbb{R}\ \!\!}^d$ such that for each $ 1\leq i \leq d$, the signs of \begin{equation}\label{eq:cond-l2-cor} \partial_{i,i}f(x) \textrm{ and \ }\partial_{i,i}g(x) \end{equation} are constant and equal. Assume moreover that one of the functions is unconditional. Then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \end{coro}
With these symmetries, the tensorization method also leads to the following extension of Theorem \ref{thm:main-1d}(2) and (3).
\begin{comment} \adp{Choix 1:} \begin{theo}\label{thm:Harge-Royen-tens} Let $\mu= \mu_1 \otimes \dots \otimes \mu_d$ be a product measure on ${\mathbb{R}\ \!\!}^d$ and assume that for $1\leq i \leq d$, the measures $\mu_i$ are even. \begin{enumerate} \item Assume in this specific point that the marginals $\mu_k$ are log-concave. \Hm{J'ai enlevé meaning that their potential $V_k$ is convex car on n'a pas parlé du potentiel pour le moment. } Let $f=e^{-\phi} $ be an unconditional positive log-concave function
and $g$ be a convex function on ${\mathbb{R}\ \!\!}^d$, then \Hm{Ca m'embête de ne pas donner le résultat optimal dans la première partie du thm 1.8. Ca ne m'embête pas pour le cas quasi-concave. Je n'ai pas encore défini \red{coordinatewise convex} Mais du coup, je ne vois pas comment rédiger cela au mieux.} \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \leq 0. \] \item Assume that $f$ and $g$ are both unconditional and quasi-concave. Then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \end{enumerate} \end{theo} \adp{ Actually, as it will be apparent form the proof, it is enough to suppose that $g$ is coordinate-wise convex in Theorem \ref{thm:Harge-Royen-tens}(1). This means that for any $(x_1,\dots,x_{i-1},x_{i+1},\dots x_d)\in {\mathbb{R}\ \!\!}^{d-1}$, the function \[ x_i\in {\mathbb{R}\ \!\!} \to g (x_1,\dots,x_{i-1},x_i,x_{i+1},\dots x_d) \] is convex. We will also prove the quasi-concave case of Theorem \ref{thm:Harge-Royen-tens}(2) under weaker hypothesis both on the functions and on the product measure. See Theorem \ref{thm:quasi-concave-prod} for the precise statement.} \end{comment}
\
\begin{theo}\label{thm:Harge-Royen-tens} Let $\mu= \mu_1 \otimes \dots \otimes \mu_d$ be a product measure on ${\mathbb{R}\ \!\!}^d$. \begin{enumerate} \item Assume that for $1\leq i\leq d$, the marginals $\mu_i$ are even and log-concave. Let $f=e^{-\phi} $ be an unconditional positive log-concave function
and $g$ be a coordinatewise convex function on ${\mathbb{R}\ \!\!}^d$, then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \leq 0. \] \item Assume that $f$ and $g$ are both unconditional and coordinatewise quasi-concave. Then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \end{enumerate} \end{theo}
\
A drawback of this tensorization approach is arguably that in Theorem \ref{thm:Hu-tens-incond-A} and Corollary \ref{cor:Hu-tens-incond}, we assume a strong symmetry property: the unconditionality of at least one function.
In order to require less symmetry assumptions, it is natural to try to use, instead of the tensorization argument, a more global approach. A first attempt would be to use the interpolation with the associated Markov semi-group and the covariance representation given in \eqref{eq:cov-rep-interpolation}. Actually, we shall provide a slightly different covariance representation, based on an argument of ``duplication'' of random variables (Lemma \ref{lem:var-prod}). The main reason for this choice is that the latter approach is much simpler than the semi-group approach and is also more effective.
See more comments in Section \ref{sec:comments}.
As expected, this approach allows us to reduce drastically the symmetries required on the functions, but at prize of considering some convexity type assumptions that are less common. Theorem \ref{thm:Harge-Royen-global} below provides an extension of Theorem \ref{thm:main-1d}(2) and (3).
\begin{theo}\label{thm:Harge-Royen-global} Let $\mu$ be a product measure on ${\mathbb{R}\ \!\!}^d$. \begin{enumerate} \item Let $f=e^{-\phi}$ and $g$ be two functions on ${\mathbb{R}\ \!\!}^d$ such that all $x\in {\mathbb{R}\ \!\!}^d$, \begin{equation}\label{eq:cond-ii-phi-g} \partial_{i} \left( \frac{\partial_i \phi(x)}{a_i(x_i)} \right) \leq 0 \textrm{ and } \partial_{i} \left( \frac{\partial_i g(x)}{a_i(x_i)} \right)\geq 0 \textrm{ for all } 1\leq i \leq d, \end{equation} and \begin{equation}\label{eq:cond-ij-phi-g} \partial_{i,j} \phi(x) \leq 0 \textrm{ and } \partial_{i,j }g(x)\geq 0 \textrm{ for all } 1\leq i\neq j \leq d. \end{equation} Assume moreover that $f$ is orthogonal to the functions $A_i(x_i)$ for all $1\leq i\leq d$, then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \item Let $f=e^{-\phi}$ and $g=e^{-\psi}$ be two functions on ${\mathbb{R}\ \!\!}^d$ such that for all $x\in {\mathbb{R}\ \!\!}^d$, \begin{equation}\label{eq:cond-ii-phi-psi} \partial_{i} \left( \frac{\partial_i \phi(x)}{a_i(x_i)} \right) \leq 0 \textrm{ and } \partial_{i} \left( \frac{\partial_i \psi(x)}{a_i(x_i)} \right)\leq 0 \textrm{ for all } 1\leq i \leq d, \end{equation} and \begin{equation}\label{eq:cond-ij-phi-psi} \partial_{i,j} \phi(x) \leq 0 \textrm{ and } \partial_{i,j }\psi(x)\leq 0 \textrm{ for all } 1\leq i\neq j \leq d. \end{equation} Assume moreover that the product measure $\mu$ is symmetric and the functions $a_i$ are even for $1\leq i \leq d$ and that also both $f$ and $g$ are even, then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \end{enumerate} \end{theo}
As before, the choice $a_k\equiv1$ is worth looking at and gives the following corollary. \begin{coro}\label{cor:Harge-Royen-global} Let $\mu$ be a product measure on ${\mathbb{R}\ \!\!}^d$. \begin{enumerate} \item Let $f=e^{-\phi}$ and $g$ be two functions on ${\mathbb{R}\ \!\!}^d$ such that for all $ 1\leq i,j \leq d$ and all $x\in {\mathbb{R}\ \!\!}^d$, \begin{equation}\label{eq:cond-l2-phi-g} \partial_{i,j} \phi(x) \leq 0 \textrm{ and } \partial_{i,j }g(x)\geq 0. \end{equation} Assume moreover that $f$ is orthogonal to the coordinate functions $x_i$ for all $1\leq i\leq d$, then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \item Let $f=e^{-\phi}$ and $g=e^{-\psi}$ be two functions on ${\mathbb{R}\ \!\!}^d$ such that for all $ 1\leq i,j \leq d$ and all $x\in {\mathbb{R}\ \!\!}^d$, \begin{equation}\label{eq:cond-l2-phi-psi} \partial_{i,j} \phi(x) \leq 0 \textrm{ and } \partial_{i,j }\psi (x)\leq 0. \end{equation} Assume moreover that the product measure $\mu$ is symmetric and that both $f$ and $g$ are even, then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \end{enumerate} \end{coro}
\
\textbf{Outline. \ }The paper is organized as follows. The case of the dimension one is investigated in Sections \ref{sec:Andreev-dim1}, \ref{sec:kernel-dim1} and \ref{sec:cheb}. In Sections \ref{sec:Andreev-dim1} and \ref{sec:kernel-dim1}, we produce two different proofs of Theorem \ref{thm:main-1d}. The first one, given in Section \ref{sec:Andreev-dim1}, is based on the use of determinants and the so-called Andreev's formula. The second one, detailed in Section \ref{sec:kernel-dim1}, is based on a covariance identity due to Hoeffding and the use on ${\mathbb{R}\ \!\!}^2$ of the classical FKG inequality. In Section \ref{sec:cheb}, we notice that more structure is actually present in dimension one: the kernel $k$ in Hoeffding's covariance identity is indeed \emph{totally positive} in the sense of Karlin \cite{karlin:book}. Consequently, determinantal covariance inequalities for general Chebyshev systems follow (see Theorem \ref{thm:cheb} for the precise statement). The latter inequalities are also recovered without using Hoeffding's covariance identity, through a direct approach with determinants and Andreev's formula.
The tensorization method and the proofs of Theorems \ref{thm:Hu-produit-A}, \ref{thm:Hu-tens-incond-A} and \ref{thm:Harge-Royen-tens} are given in Section \ref{sec:tens}, except for the proofs of Theorem \ref{thm:main-1d}(3) and Theorem \ref{thm:Harge-Royen-tens}(2), that pertain to the hypothesis of quasi-concavity and are detailed in Section \ref{sec:quasi-concave}. Indeed, the method for proving Theorem \ref{thm:main-1d}(3) in dimension one is very specific and independent from the rest of the paper. Theorem \ref{thm:Harge-Royen-tens}(2) is then obtained by tensorization.
In addition, a generalization of Hoeffding's covariance identity for product measures, obtained by a duplication argument, is provided in Section \ref{sec:global}. A second proof of Theorem \ref{thm:Hu-produit-A} and the proof of Theorem \ref{thm:Harge-Royen-global} are then given. As explained above, another natural generalization of Hoeffding's covariance identity would be given through the standard semi-group interpolation. Comments on the difficulty of using this covariance representation are provided in Section \ref{sec:comments}. Some possible examples of applications are given in Section \ref{sec:examples}. Finally, in the Appendix, we also prove that that Theorem \ref{thm:harge}(2) is true for product measures whose marginals are mixtures of centered Gaussian measures.
\section{A determinantal approach in dimension one} \label{sec:Andreev-dim1}
This section is devoted to a first proof of Theorem~\ref{thm:main-1d}(1) and (2). The proof is based on properties of determinants, in particular on the intertwining between determinants and the integral operator, a property known as Andreev's formula. Similar arguments will be used in Section~\ref{sec:cheb} in the more general framework of Chebyshev systems.
\subsection{Convexity and determinants}
\begin{defi} A pair of real-valued functions $(u,U)$ is said to satisfy Assumption ($\mathcal{C}$) if for any triple $(x_1,x_2,x_3) \in {\mathbb{R}\ \!\!}^3$ with $ x_1 < x_2 < x_3$, one has \begin{equation*}
D(x_1,x_2,x_3) = \left|\begin{array}{ccc} 1 & 1 & 1 \\ u(x_1) & u(x_2) & u(x_3) \\ U(x_1) & U(x_2) & U(x_3) \end{array} \right| \ge 0. \end{equation*} \end{defi}
In other terms, the couple $(u,U)$ satisfies Assumption ($\mathcal{C}$) if and only if the triple $(1,u,U)$ forms a Chebyshev system (see Definition~\ref{def:Cheby}). From elementary properties of determinants, it follows that:
\begin{prop} \label{prop:AssCsignD} Let $(u,U)$ be satisfying Assumption ($\mathcal{C}$) and $D : {\mathbb{R}\ \!\!}^3 \rightarrow {\mathbb{R}\ \!\!}$ be as defined above. Let $(x_1,x_2,x_3) \in {\mathbb{R}\ \!\!}^3$. Let $\sigma \in S_3$ be a permutation of $\{1,2,3\}$ such that $x_{\sigma(1)} \le x_{\sigma(2)} \le x_{\sigma(3)}$. Then \begin{equation*}
{\varepsilon\ \!\!}(\sigma) D(x_1,x_2,x_3) \ge 0 \end{equation*} \end{prop}
\begin{prop} \label{prop:ConvC} A function $f : {\mathbb{R}\ \!\!} \to {\mathbb{R}\ \!\!}$ is convex if and only if, for any $x\in {\mathbb{R}\ \!\!}$, the pair $(x,f(x))$ satisfies ($\mathcal{C}$). \end{prop}
\begin{proof} Take $(x_1,x_2,x_3) \in {\mathbb{R}\ \!\!}^3$ with $x_1 < x_2 < x_3$. Expanding the determinant $D(x_1,x_2,x_3)$ gives: \begin{equation*} D(x_1,x_2,x_3) = (x_2-x_1)(f(x_3)-f(x_2)) - (x_3-x_2)(f(x_2)-f(x_1)). \end{equation*} Dividing by the positive quantity $(x_3-x_2)(x_2-x_1)>0$, we obtain that $D(x_1,x_2,x_3) \ge 0$ if and only if \begin{equation*} \frac{f(x_3)-f(x_2)}{x_3-x_2} \ge \frac{f(x_2)-f(x_1)}{x_2-x_1}, \end{equation*} which is the slope inequality equivalent to convexity of $f$. \end{proof}
\begin{coro} If $U$ is an increasing bijection between ${\mathbb{R}\ \!\!}$ and some interval $I$, then $(1,u,U)$ satisfies ($\mathcal{C}$) if and only if $u \circ U^{-1}$ is concave on $I$. \end{coro}
\begin{proof} We notice that, for $x_1 < x_2 < x_3$ we have $$D(x_1,x_2,x_3) = D(U^{-1}(y_1),U^{-1}(y_2),U^{-1}(y_3))$$ for some triple $y_1<y_2<y_3 \in I$. Elementary properties of determinants then give: \begin{equation*}
D(x_1,x_2,x_3) = \left|\begin{array}{ccc} 1 & 1 & 1 \\ y_1 & y_2 & y_3 \\ (-u \circ U^{-1})(y_1) & (-u \circ U^{-1})(y_2) & (-u \circ U^{-1})(y_3) \end{array} \right| \end{equation*} and we conclude by applying Proposition~\ref{prop:ConvC}. \end{proof}
\
Let us now consider some positive function $f$. Let $F$ be a primitive of $f$. It is known (see \cite{bobkov:extremal}) that $f$ is log-concave if and only if $f \circ F^{-1}$ is concave. From the above, we deduce the following proposition.
\begin{prop} \label{prop:logconcC} A positive function $f$ is log-concave if and only if the pair $(f,F)$ satisfies ($\mathcal{C}$). \end{prop} \begin{comment} \textbf{Proof: } Let $(x_1,x_2,x_3) \in {\mathbb{R}\ \!\!}^3$ such that $x_1 \le x_2 \le x_3$. Then : \begin{eqnarray*}
D(x_1,x_2,x_3) &=& \left|\begin{array}{ccc} 1 & 1 & 1 \\ f(x_1) & f(x_2) & f(x_3) \\ F(x_1) & F(x_2) & F(x_3) \end{array} \right| \\ &=& (f(x_2)-f(x_1))(F(x_3)-F(x_2))-(f(x_3)-f(x_2))(F(x_2)-F(x_1)) \\ \end{eqnarray*} \Hm{\red{Attention, ici $f$ n'est pas forcément une densité de proba, cela change le domaine de définition de $F^{-1}$}}The argument is now taken from Bobkov (extremal properties 96) Since $f >0$, we can define the inverse $F^{-1}$, and due to Proposition 6.1, $f$ is log-concave if and only if $I(p)=f(F^{-1}(p))$ is concave on $(0,1)$.
Now $I$ is concave on $(0,1)$ if and only if if for all $0<p_1<p_2<p_3$, one has \[ I(p_2) \geq \frac{p_3-p_2}{p_3-p_1} I(p_1) + \frac{p_2-p_1}{p_3-p_1} I(p_3) \] or equivalently \[ (p_3-p_2) \left( I(p_2)- I(p_1) \right) \geq (p_2-p_1) \left( I(p_3)- I(p_2) \right). \]
Considering, the change of variable $x_1= F^{-1}(p_1),x_2= F^{-1}(p_2) $ and $x_3= F^{-1}(p_3)$, the last line reads exactly as \[ \left(F(x_3)-F(x_2) \right) \left(f(x_2)- f(x_1) \right) \geq \left(F(x_2)-F(x_1) \right) \left(f(x_3)- f(x_2) \right) \] and the result follows.
\end{comment}
\begin{comment} \blue{Enlever la suite de la preuve.} Then : \begin{eqnarray*}
D(x_1,x_2,x_3) &=& \left|\begin{array}{ccc} 1 & 1 & 1 \\ f(x_1) & f(x_2) & f(x_3) \\ F(x_1) & F(x_2) & F(x_3) \end{array} \right| \\ &=& (f(x_2)-f(x_1))(F(x_3)-F(x_2))-(f(x_3)-f(x_2))(F(x_2)-F(x_1)) \\ &=& (f(x_2)-f(x_1)) \int_{x_2}^{x_3} f(t) dt-(f(x_3)-f(x_2)) \int_{x_1}^{x_2} f(t) dt \\ &=& f(x_2) \int_{x_1}^{x_3} f(t) dt - \int_{x_1}^{x_2} f(x_3) f(t) dt - \int_{x_2}^{x_3} f(x_1) f(t) dt. \end{eqnarray*} Let $t$ be in the interval $[x_1,x_2]$. As $f$ is log-concave, we have, by the inequality from Definition~\ref{def:PF2} with $a_1=t+x_3-x-2$, $a_2 = x_3 \ge a_1$, $b_1=0$ and $b_2 = x_3-x_2 \ge b_1$: \begin{equation*}
0 \le \left|\begin{array}{cc} f(t+x_3-x_2) & f(t) \\ f(x_3) & f(x_2)\end{array} \right| = f(x_2) f(t+x_3-x_2)-f(t)f(x_3). \end{equation*} From this inequality (and the non-negativity of $f$) we deduce that: \begin{eqnarray*} \int_{x_1}^{x_2} f(x_3) f(t) dt & \le & f(x_2) \int_{x_1}^{x_2} f(t+x_3-x_2) dt \\ &=& f(x_2) \int_{x_1+x_3-x_2}^{x_3} f(t) dt. \end{eqnarray*} Let us now consider $t \in [x_2,x_3]$. Applying the log-concavity inequality with $a_1=x_2$, $a_2 = t \ge a_1$, $b_1=0$ and $b_2 = x_2-x_1 \ge b_1$ gives: \begin{equation*}
0 \le \left|\begin{array}{cc} f(x_2) & f(x_1) \\ f(t) & f(t+x_1-x_2)\end{array} \right| = f(x_2) f(t+x_1-x_2)-f(t)f(x_1), \end{equation*} and we deduce: \begin{eqnarray*} \int_{x_2}^{x_3} f(x_1) f(t) dt & \le & f(x_2) \int_{x_2}^{x_3} f(t-x_2+x_1) dt \\ &=& f(x_2) \int_{x_1}^{x_1+x_3-x_2} f(t) dt. \end{eqnarray*} We thus have: \begin{equation*} D(x_1,x_2,x_3) \ge f(x_2) \int_{x_1}^{x_3} f(t) dt-f(x_2) \int_{x_1+x_3-x_2}^{x_3} f(t) dt - f(x_2) \int_{x_1}^{x_1+x_3-x_2} f(t) dt = 0, \end{equation*} so the couple $(f,F)$ satisfies the assumption ($\mathcal{C}$). $\square$ \end{comment}
\subsection{An Andreev-type formula}
A key point in this approach is the following Andreev-type formula which exchanges expectation and determinants. \begin{prop} \label{prop:Andreev} Let $(f_i)_{1 \le i \le n}$ and $(g_i)_{1 \le i \le n}$ be two $n$-uples of functions in $L^2(\mu)$. We have: $$ \det\left(\mathbb{E}_\mu \left[f_i(X) g_j(X) \right]\right) = \frac{1}{n!} \mathbb{E}_{\mu \otimes \cdots \otimes \mu} \left[\det\left(f_i(X_j)\right)\det\left(g_i(X_j)\right)\right]. $$ \end{prop}
\begin{proof} An elementary formula for determinant asserts that: \begin{equation*} n! \det\left(\mathbb{E}_\mu \left[f_i(X) g_j(X) \right]\right) = \sum_{\sigma,\sigma' \in S_n} {\varepsilon\ \!\!}(\sigma) {\varepsilon\ \!\!}(\sigma') \prod_{i=1}^n \mathbb{E}_\mu \left[f_{\sigma(i)}(X) g_{\sigma'(i)}(X) \right]. \end{equation*} Fubini's theorem allows us to write: \begin{equation*} n! \det\left(\mathbb{E}_\mu \left[f_i(X) g_j(X) \right]\right) = \sum_{\sigma,\sigma' \in S_n} {\varepsilon\ \!\!}(\sigma) {\varepsilon\ \!\!}(\sigma') \mathbb{E}_{\mu \otimes \cdots \otimes \mu} \left[ \prod_{i=1}^n f_{\sigma(i)}(X_i) g_{\sigma'(i)}(X_i) \right]. \end{equation*} We thus have: \begin{eqnarray*} n! \det\left(\mathbb{E}_\mu \left[f_i(X) g_j(X) \right]\right) &=& \mathbb{E}_{\mu \otimes \cdots \otimes \mu} \left[ \sum_{\sigma,\sigma' \in S_n} {\varepsilon\ \!\!}(\sigma) {\varepsilon\ \!\!}(\sigma') f_{\sigma(i)}(X_i) g_{\sigma'(i)}(X_i) \right] \\ &=& \mathbb{E}_{\mu \otimes \cdots \otimes \mu} \left[ \left( \sum_{\sigma \in S_n} {\varepsilon\ \!\!}(\sigma) f_{\sigma(i)}(X_i) \right) \left( \sum_{\sigma' \in S_n} {\varepsilon\ \!\!}(\sigma') g_{\sigma'(i)}(X_i)\right) \right] \\ &=& \mathbb{E}_{\mu \otimes \cdots \otimes \mu} \left[ \det\left(f_j(X_i)\right)\det\left(g_j(X_i)\right) \right] \\ &=& \mathbb{E}_{\mu \otimes \cdots \otimes \mu} \left[ \det\left(f_i(X_j)\right)\det\left(g_i(X_j)\right) \right]. \end{eqnarray*} \end{proof} Note that with the particular choice $n=2$, $f_1=1$, $f_2=f$, $g_1=1$, $g_2=g$, Proposition~\ref{prop:Andreev} gives the so-called "Chebyshev's other inequality":
\begin{prop}[Chebyshev] \label{prop:Chebyshev1d} If $f,g \in L^2(\mu)$ are both non-increasing or both non-decreasing, then ${\mathrm{{\rm Cov}}}_\mu(f,g) \ge 0$. \end{prop}
\subsection{A first proof of Theorem~\ref{thm:main-1d}}
The first proof of Theorem~\ref{thm:main-1d} will be deduced from the following more general result.
\begin{theo} \label{th:CovaC} Let $(u,U)$ and $(v,V)$ be two pairs of functions satisfying Assumption ($\mathcal{C}$). Then \begin{equation*} {\mathrm{{\rm Cov}}}_\mu(U,V) {\mathrm{{\rm Cov}}}_\mu(u,v) \ge {\mathrm{{\rm Cov}}}_\mu(u,V) {\mathrm{{\rm Cov}}}_\mu(U,v). \end{equation*} \end{theo}
\begin{proof} We want to show that $D \ge 0$, where \begin{equation*}
D = \left|\begin{array}{cc} {\mathrm{{\rm Cov}}}_\mu(u,v) & {\mathrm{{\rm Cov}}}_\mu(u,V) \\ {\mathrm{{\rm Cov}}}_\mu(U,v) & {\mathrm{{\rm Cov}}}_\mu(U,V) \end{array}\right|. \end{equation*} But we also have: \begin{equation*}
D = \left|\begin{array}{ccc} 1 & \mathbb{E}_\mu[v] & \mathbb{E}_\mu[V] \\ \mathbb{E}_\mu[u] & \mathbb{E}_\mu[u v] & \mathbb{E}_\mu[u V] \\ \mathbb{E}_\mu[U] & \mathbb{E}_\mu[U v] & \mathbb{E}_\mu[U V]\end{array} \right|. \end{equation*} The latter equality can be proven by simply expanding the determinant. We now apply Proposition~\ref{prop:Andreev} with $f_1=1,f_2=u,f_3=U$ and $g_1=1,g_2=v,g_3=V$. This gives \begin{equation*}
D = \int_{{\mathbb{R}\ \!\!}^3} \left|\begin{array}{ccc} 1 & 1 & 1 \\ u(x_1) & u(x_2) & u(x_3) \\ U(x_1) & U(x_2) & U(x_3) \end{array} \right|\left|\begin{array}{ccc} 1 & 1 & 1 \\ v(x_1) & v(x_2) & v(x_3) \\ V(x_1) & V(x_2) & V(x_3) \end{array} \right| d\mu(x_1) d\mu(x_2) d\mu(x_3). \end{equation*} Let $(x_1,x_2,x_3) \in {\mathbb{R}\ \!\!}^3$ and $\sigma \in S_3$ be such that $x_{\sigma(1)} < x_{\sigma(2)} < x_{\sigma(3)}$. As both pairs $(u,U)$ and $(v,V)$ satisfy ($\mathcal{C}$), we apply Proposition~\ref{prop:AssCsignD} as follows, \begin{eqnarray*}
\left|\begin{array}{ccc} 1 & 1 & 1 \\ u(x_1) & u(x_2) & u(x_3) \\ U(x_1) & U(x_2) & U(x_3) \end{array} \right|\left|\begin{array}{ccc} 1 & 1 & 1 \\ v(x_1) & v(x_2) & v(x_3) \\ V(x_1) & V(x_2) & V(x_3) \end{array} \right| \\ = \left({\varepsilon\ \!\!}(\sigma) \left|\begin{array}{ccc} 1 & 1 & 1 \\ u(x_1) & u(x_2) & u(x_3) \\ U(x_1) & U(x_2) & U(x_3) \end{array} \right|\right) \left({\varepsilon\ \!\!}(\sigma)\left|\begin{array}{ccc} 1 & 1 & 1 \\ v(x_1) & v(x_2) & v(x_3) \\ V(x_1) & V(x_2) & V(x_3) \end{array} \right|\right) \ge 0, \end{eqnarray*} from which we deduce $D \ge 0$, as wanted. \end{proof}
As we prove now, points (1) and (2) of Theorem~\ref{thm:main-1d} are particular cases of Theorem~\ref{th:CovaC}, for a suitable choice for the pairs $(u,U)$ and $(v,V)$.
\begin{proof}[Proof of Theorem~\ref{thm:main-1d}(1) and (2)]
The first item is a direct consequence of Theorem~\ref{th:CovaC} and Proposition \ref{prop:ConvC} for the particular choice $u(x)=x$, $U(x)=f(x)$, $v(x)=x$ and $V(x) = g(x)$.
For the second item, we set $u(x) = f(x)$, $U(x) = \int_0^x f(t) dt$, $v(x)=x$, $V(x)=g(x)$. Propositions~\ref{prop:logconcC} and~\ref{prop:ConvC} show that the pairs $(u,U)$ and $(v,V)$ both satisfy Assumption ($\mathcal{C}$). By Theorem~\ref{th:CovaC}, we thus have: \begin{equation*} {\mathrm{{\rm Cov}}}_\mu(f,x) {\mathrm{{\rm Cov}}}_\mu(U,g) \ge {\mathrm{{\rm Cov}}}_\mu(f,g) {\mathrm{{\rm Cov}}}_\mu(U,x). \end{equation*} The orthogonality assumption gives ${\mathrm{{\rm Cov}}}_\mu(f,x)=0$. Moreover, as $f$ is non-negative, the function $U$ is non-decreasing. Proposition~\ref{prop:Chebyshev1d} then gives ${\mathrm{{\rm Cov}}}_\mu(U,x) \ge 0$, so that the inequlity ${\mathrm{{\rm Cov}}}_\mu(f,g)\le 0$ holds, as desired.
\begin{comment} For the third item \blue{à enlever?} we set $F(x) = \int_0^x f(t) dt$ and $G(x)=\int_0^x g(t) dt$. Both couples $(f,F)$ and $(g,G)$ satisfy assumption ($\mathcal{C}$), so we have: \begin{equation*} {\mathrm{{\rm Cov}}}_\mu(f,g) {\mathrm{{\rm Cov}}}_\mu(F,G) \ge {\mathrm{{\rm Cov}}}_\mu(f,G) {\mathrm{{\rm Cov}}}_\mu(F,g). \end{equation*} By Proposition~\ref{prop:Chebyshev1d}, the functions $F$ and $G$ being both non-decreasing, we have ${\mathrm{{\rm Cov}}}_\mu(F,G) \ge 0$. Moreover, $F$ and $G$ are odd functions, so in particular ${\mathrm{{\rm Cov}}}_\mu(f,G)=0$ and ${\mathrm{{\rm Cov}}}_\mu(F,g)=0$. We thus have ${\mathrm{{\rm Cov}}}_\mu(f,g) \ge 0$ as wanted. \end{comment}
\end{proof}
\section{The Hoeffding covariance identity approach in dimension one} \label{sec:kernel-dim1}
This section is devoted to a second proof of Theorem~\ref{thm:main-1d}(1) and (2). The proof will follow from the Hoeffding covariance identity and the use of the FKG inequality for a new probability measure on ${\mathbb{R}\ \!\!}^2$. A key point is that the kernel of the Hoeffding representation is \emph{totally positive}.
\subsection{Hoeffding's covariance identity} We start by recalling the following representation formula for the covariance, which is a consequence of a slightly more general covariance identity due to Hoeffding. See \cite{saumwellner2017efron} for more details about Hoeffding's covariance identity.
\begin{theo}\label{thm:covA} Let $\mu$ be a probability measure on ${\mathbb{R}\ \!\!}$ and denote by $F_{\mu}$ its cumulative distribution function, then for all functions $f$ and $g$ in $L^2(\mu)$ and absolutely continuous, one has \begin{equation} {\mathrm{{\rm Cov}}}_{\mu}(f,g)=\iint f'(x)k_\mu(x,y)g'(y)dxdy,\label{eq:cov-k} \end{equation} with \[ k_\mu(x,y)=F_{\mu}(x\wedge y)-F_{\mu}(x)F_{\mu}(y) \] and $x\wedge y=\min(x,y)$. \end{theo} For simplicity, when it is clear from context, we shall write $k=k_\mu$ in the sequel.
We now recall some properties of the kernel $k:{\mathbb{R}\ \!\!}^{2}\to[0,+\infty)$. Taking $f(\cdot)=\mathbf 1_{[x,\infty[}(\cdot)$ and $g(\cdot)=\mathbf 1_{[y,\infty[}(\cdot)$, one sees that the kernel $k$ is necessarily unique and can also be written \[ k(x,y)={\mathrm{{\rm Cov}}}_{\mu}\left(1_{\left\{ X\leq x\right\} },1_{\left\{ X\leq y\right\} }\right). \]
This kernel is non-negative, bounded, continuous if $\mu$ is assumed to be a continuous measure, but it is not differentiable on the line $y=x$.
Let us emphasize the fact that the kernel $k$ is \emph{totally positive} in the sense of Karlin \cite{karlin:book}. This result should be classical but we could not find a reference of it in the literature.
\begin{theo}\label{thm:totpos} For all $n\geq2$, $s_{1}\leq\dots\leq s_{n}\in{\mathbb{R}\ \!\!}$ and $t_{1}\leq\dots\leq t_{n}\in{\mathbb{R}\ \!\!}$, \[ \det\left(k(s_{i},t_{j})\right)_{1\leq i,j\leq n}\geq0. \] \end{theo}
\begin{proof} The proof follows from Theorem 3.1 in Karlin \cite{karlin:book}, or Theorem 4.2 in Pinkus \cite{pinkus:book}, by showing that the matrix $\left(k(s_{i},t_{j})\right)_{1\leq i,j\leq n}$ is a Green matrix. One can also directly use Corollary 3.1 in Karlin \cite{karlin:book} by writing \[ k(x,y)=\left\{ \begin{array}{l} \phi(x)\psi(y)\textrm{ if }x\geq y\\ \psi(x)\phi(y)\textrm{ if }x\leq y \end{array}\right. \] with $\phi(x)=F(x)$ non-decreasing and $\psi(y)=1-F(y)$ non-increasing. \end{proof}
In the case $n=2$, Theorem \ref{thm:totpos} provides the following inequality. \begin{coro} \label{cor:totpos2} For all $s_{1}\leq s_{2}$ and $t_{1}\leq t_{2}$, \begin{equation} k(s_{1},t_{1})k(s_{2},t_{2})\geq k(s_{1},t_{2})k(s_{2},t_{1}).\label{eq:totpos2} \end{equation} \end{coro}
The conclusion of Corollary \ref{cor:totpos2} is well known in the literature under different names. Inequality \eqref{eq:totpos2}, here in the case of ${\mathbb{R}\ \!\!}^{2}$, is sometimes referred to as the \emph{Holley condition} or the \emph{strong FKG condition}. The kernel $k$ is also called \emph{log-supermodular} or \emph{multivariate totally positive of order 2.} We shall also the need the following extension: \begin{coro}\label{cor:totpos} Let $a$ and $b$ be two positive functions on ${\mathbb{R}\ \!\!}$ and define the kernel $k_{a,b}$ on ${\mathbb{R}\ \!\!}^2$ by \[ k_{a,b}(x,y) = a(x) k(x,y) b(y). \] Then the kernel is totally positive and thus satisfies inequality \eqref{eq:totpos2}. \end{coro}
We recall now the classical result, due to Fortuin, Kasteleyn and Ginibre \cite{FKG}, which asserts that the Holley condition implies the FKG inequality. We first state the definition of the FKG inequality in ${\mathbb{R}\ \!\!}^d$.
\begin{defi}\label{def:FKG} Let $d \geq 1$. A function $f:{\mathbb{R}\ \!\!}^{d}\to{\mathbb{R}\ \!\!}$ is said to be \emph{coordinate increasing} if it is non-decreasing along each coordinate, that is if: \begin{equation} \textrm{for all }x,y\in{\mathbb{R}\ \!\!}^{d},\textrm{ satisfying }x_{i}\leq y_{i},1\leq i\leq d \textrm{ one has }f(x)\leq f(y).\label{eq:croissante} \end{equation} A probability measure $\nu$ on ${\mathbb{R}\ \!\!}^{d}$ is said to satisfy the \emph{FKG inequality} if for all functions $f$ and $g$ \emph{coordinate increasing}, one has: \begin{equation} {\mathrm{{\rm Cov}}}_{\nu}(f,g)\geq0.\label{e:def-FKG} \end{equation} \end{defi}
\begin{theo}\label{thm:FKG} Let $\nu$ be a probability measure ${\mathbb{R}\ \!\!}^{d}$ with density $k$ with respect to the Lebesgue measure. Assume that for all $x,y\in{\mathbb{R}\ \!\!}^{d}$, \begin{equation} k(x\wedge y)k(x\vee y)\geq k(x)k(y),\label{eq:k-FKG} \end{equation} where $x\wedge y=(\min(x_{1},y_{1}),\dots,\min(x_{d},y_{d}))$ and $x\vee y=(\max(x_{1},y_{1}),\dots,\max(x_{d},y_{d}))$. Then $\nu$ satisfies the FKG inequality. \end{theo}
\begin{remark}\label{rmk:bakry-michel} Writing $k=e^{H}$, the condition \eqref{eq:k-FKG} writes: \begin{equation}\label{eq:cond-Holley-H} H(x\wedge y)+H(x\vee y)\geq H(x)+H(y). \end{equation} In the case where $k$ is smooth - more precisely when $H$ is of class $\mathcal{C}^{2}$ here -, inequality \eqref{eq:cond-Holley-H} is equivalent to the following condition on the second order cross-derivatives of $H$: \[ \frac{\partial^{2}}{\partial x_{i}\partial x_{j}}H(x)\geq0\textrm{ for }1\leq i\neq j\leq d. \] In this case, Bakry and Michel \cite{bakry-michel} proved the slightly stronger result that the associated semi-group (see Section \ref{sec:comments}) preserves the class of coordinate increasing functions. Finally note that the kernel $k_{\mu}$ given in Theorem \ref{thm:covA} above, is not smooth on the diagonal of ${\mathbb{R}\ \!\!}^{2}$ and that $\partial_{x,y}^{2}\ln k_\mu(x,y)=0$ for all $x\neq y\in{\mathbb{R}\ \!\!}^{2}$. \end{remark}
\begin{remark} Condition \eqref{eq:k-FKG} is not equivalent to the FKG inequality. In the Gaussian setting, for a Gaussian vector with non-singular matrix covariance $\Gamma$ on ${\mathbb{R}\ \!\!}^{d}$, the Holley condition \eqref{eq:k-FKG} is equivalent to $(\Gamma^{-1})_{i,j}\leq0$ for $1\leq i\neq j\leq d$. But as proven by Pitt \cite{pitt:82} in the Gaussian setting, the FKG inequality is equivalent to $\Gamma_{i,j}\geq0$ (see also Tong \cite{tong:book}). The condition on the coefficient of $\Gamma^{-1}$ implies the one for $\Gamma$. But the converse does not hold. The example 4.3.2 in Tong \cite{tong:book} provides a covariance matrix for $d\geq3$ such that $\Gamma_{i,j}\geq0$ for all $1\leq i,j\leq d$ but not $(\Gamma^{-1})_{i,j}\leq0$ for all $1\leq i\neq j\leq d$. \end{remark}
\ \subsection{Hoeffding's formula as a relation between covariances}\label{subsec:nice-relation-dim1}
The main result of this section is Lemma \ref{thm:cov2} where we express the quantities appearing in Theorem \ref{thm:main-1d} as a covariance of the derivatives of the functions with respect to a new probability measure on ${\mathbb{R}\ \!\!}^2$. Let $\mu$ be a probability measure on ${\mathbb{R}\ \!\!}$ admitting a second moment. We recall that $k$ is the non-negative kernel: \[ k(x,y)=F_{\mu}(x\wedge y)-F_{\mu}(x)F_{\mu}(y). \] and that from Theorem \ref{thm:covA}, it satisfies \begin{equation} \iint k(x,y)dxdy={\mathrm{{\rm Var}}}_{\mu}(x)={\mathrm{{\rm Var}}}(\mu).\label{eq:varmu} \end{equation} By assumption, this last quantity is finite and we denote by $\mu^{(1)}$ the following probability measure on ${\mathbb{R}\ \!\!}^{2}$:
\[ d\mu^{(1)}(x,y)=\frac{k(x,y)}{\iint k(x',y')dx'dy'}dxdy \] In the case where $f$ and $g$ are some positive functions, we also denote, \[ d\mu_{f}^{(1)}(x,y)=\frac{f(x) k(x,y)}{\iint f(x') k(x',y')dx'dy'}dxdy \] and \[ d\mu_{f,g}^{(1)}(x,y)=\frac{f(x) k(x,y) g(y)}{\iint f(x') k(x',y')dx'dy'}dxdy \] The main result here is the following relation between the covariances of $\mu$ and $\mu^{(1)}$. It consists essentially in a rewriting of Hoeffding's covariance identity \eqref{eq:cov-k} and to highlight the slight difference, we call it ``Hoeffding's covariance relation''.
\begin{lem}[Hoeffding's covariance relation]\label{thm:cov2}
Let $\mu$ be a probability measure on ${\mathbb{R}\ \!\!}$ admitting a second moment, with ${\mathrm{{\rm Var}}}(\mu)>0$. Let $f,g:{\mathbb{R}\ \!\!}\to {\mathbb{R}\ \!\!}$ be some absolutely continuous functions that belong to $L^2(\mu)$.
\begin{enumerate} \item Then,
\begin{equation} \frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))}{{\mathrm{{\rm Var}}}(\mu)}-\frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),x)}{{\mathrm{{\rm Var}}}(\mu)}\,\frac{{\mathrm{{\rm Cov}}}_{\mu}(g(x),x)}{{\mathrm{{\rm Var}}}(\mu)}={\mathrm{{\rm Cov}}}_{\mu^{(1)}}(f'(x),g'(y)).\label{eq:cov2} \end{equation}
\item If moreover, $f=e^{-\phi}$ is positive: \begin{equation} \frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))}{Z_f}- \frac{1}{Z_f^2 }{\mathrm{{\rm Cov}}}_{\mu}(f(x),x)\,{\mathrm{{\rm Cov}}}_{\mu}(F(x),g(x))={\mathrm{{\rm Cov}}}_{\mu_f^{(1)}}(- \phi'(x),g'(y)).\label{eq:cov2b} \end{equation}
\item If moreover $g=e^{-\psi}$ is positive: \begin{equation} \frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))}{Z_{f,g}}- \frac{1}{Z_{f,g}^2 }{\mathrm{{\rm Cov}}}_{\mu}(f(x),G(x))\,{\mathrm{{\rm Cov}}}_{\mu}(F(x),g(x))={\mathrm{{\rm Cov}}}_{\mu_{f,g}^{(1)}}( \phi'(x),\psi'(y)).\label{eq:cov2c} \end{equation} \end{enumerate} Here $F$ and $G$ are primitives of $f$ and $g$ and \[ Z_f= \iint f(x)k(x,y) dxdy = {\mathrm{{\rm Cov}}}(F(x),x)>0, \] and \[ Z_{f,g}= \iint f(x)k(x,y)g(y) dxdy = {\mathrm{{\rm Cov}}}(F(x),G(x))>0. \] \end{lem}
Note that this approach is also linked to determinants in the sense that the left hand sides of the equalities \eqref{eq:cov2}, \eqref{eq:cov2b} \eqref{eq:cov2c} can be written as determinants. For example, formula \eqref{eq:cov2} can be written as \begin{equation}\label{formula-det} {\mathrm{{\rm Var}}}(\mu)^{2}{\mathrm{{\rm Cov}}}_{\mu^{(1)}}(f'(x),g'(y))=\det\begin{pmatrix}{\mathrm{{\rm Var}}}(\mu) & {\mathrm{{\rm Cov}}}_{\mu}(x,f(x))\\ {\mathrm{{\rm Cov}}}_{\mu}(x,g(x)) & {\mathrm{{\rm Cov}}}_{\mu}(f,g) \end{pmatrix}. \end{equation}
\
\begin{proof} Using several times the covariance representation of Theorem \ref{thm:covA}, one has: \begin{align*}
& \frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))}{{\mathrm{{\rm Var}}}(\mu)} = \iint f'(x)\frac{k(x,y)}{\iint k(x',y')dx'dy'}g'(y)dxdy\\ = & {\mathrm{{\rm Cov}}}_{\mu^{(1)}}(f'(x),g'(y))\\
& +\left(\iint f'(x)\frac{k(x,y)}{\iint k(x',y')dx'dy'}dxdy\right)\left(\iint g'(y)\frac{k(x,y)}{\iint k(x',y')dx'dy'}dxdy\right)\\ = & {\mathrm{{\rm Cov}}}_{\mu^{(1)}}(f'(x),g'(y))+\frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),x)}{{\mathrm{{\rm Var}}}(\mu)}\,\frac{{\mathrm{{\rm Cov}}}_{\mu}(g(x),x)}{{\mathrm{{\rm Var}}}(\mu)}. \end{align*} Similarly, if $f=e^{-\phi}$, \begin{align*}
& \frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))}{Z_f}
= \iint( - \phi'(x) ) \frac{ f(x) k(x,y)}{Z_f}g'(y)dxdy\\ = & {\mathrm{{\rm Cov}}}_{\mu_f^{(1)}}(- \phi'(x),g'(y))+ \frac{1}{Z_f^2 }\left(\iint f'(x)k(x,y)dxdy\right)\left(\iint f(x)k(x,y) g'(y)dxdy\right)\\ = & {\mathrm{{\rm Cov}}}_{\mu_f^{(1)}}(- \phi'(x),g'(y))+ \frac{1}{Z_f^2 }{\mathrm{{\rm Cov}}}_{\mu}(f(x),x)\,{\mathrm{{\rm Cov}}}_{\mu}(F(x),g(x)) \end{align*} and if moreover $g=e^{-\psi}$, \begin{align*}
& \frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))}{Z_{f,g}}
= \iint( - \phi'(x) ) \frac{ f(x) k(x,y)g(y) }{Z_{f,g}} ( -\psi'(y))dxdy\\ = & {\mathrm{{\rm Cov}}}_{\mu_{f,g}^{(1)}}( \phi'(x), \psi'(y))+ \frac{1}{Z_{f,g}^2 }\left(\iint f'(x)k(x,y) g(y) dxdy\right)\left(\iint f(x)k(x,y) g'(y)dxdy\right)\\ = & {\mathrm{{\rm Cov}}}_{\mu_{f,g}^{(1)}}(\phi'(x),\psi'(y))+ \frac{1}{Z_{f,g}^2 }{\mathrm{{\rm Cov}}}_{\mu}(f(x),G(x))\,{\mathrm{{\rm Cov}}}_{\mu}(F(x),g(x)). \end{align*} \end{proof}
\subsection{A second proof of Theorem \ref{thm:main-1d}}
\label{sec:proof1d} We are now ready to turn to the second proof of Theorem \ref{thm:main-1d}(1) and (2) pertaining to dimension one. The last ingredient will be the use of the FKG inequality.
\begin{proof}[Proofs of Theorem \ref{thm:main-1d}(1) and (2)] Let $\mu$ be any probability measure ${\mathbb{R}\ \!\!}$ admitting a second moment and let $f$ and $g$ be two convex functions on ${\mathbb{R}\ \!\!}$. Using the first covariance relation of Lemma \ref{thm:cov2}, it is equivalent to prove that: \[ {\mathrm{{\rm Cov}}}_{\mu^{(1)}}(f'(x),g'(y))\geq0. \] By Corollary \ref{cor:totpos2} and Theorem \ref{thm:FKG}, the probability measure $\mu^{(1)}$ on ${\mathbb{R}\ \!\!}^{2}$ satisfies the FKG inequality. Since $f$ and $g$ are convex, the functions $(x,y) \to f'(x)$ and $(x,y)\to g'(y)$ on ${\mathbb{R}\ \!\!}^2$ are in particular increasing along coordinates in ${\mathbb{R}\ \!\!}^{2}$, which implies the desired inequality.
As for the second item, let $f=e^{-\phi}$ be a log-concave function and $g$ be a convex function on ${\mathbb{R}\ \!\!}$ and such that $f$ is orthogonal to the linear function $x$. By the second covariance formula of Lemma \ref{thm:cov2}, and since ${\mathrm{{\rm Cov}}}_\mu(f(x),x)=0$, \[ {\mathrm{{\rm Cov}}}_\mu(f,g)= - {\mathrm{{\rm Cov}}}_{\mu^{(1)}_f} (\phi'(x), g'(y)). \] The second point follows similarly as above, since by Corollary \ref{cor:totpos} the kernel of the probability measure $\mu^{(1)}_f$ is also totally positive. \end{proof}
\begin{comment} \blue{To remove?} For the third point, let $f=e^{-\phi}$ and $g=e^{-\psi}$ be two log-concave and even functions and assume that the measure $\mu$ is symmetric. Then with $G$ the (centered) primitive of $g$, \[ {\mathrm{{\rm Cov}}}_\mu(f(x), G(x))= \int_{\mathbb{R}\ \!\!} f(x) G(x) d\mu(x) - \left(\int_{\mathbb{R}\ \!\!} f(x) d\mu(x) \right) \left(\int_{\mathbb{R}\ \!\!} G(x) d\mu(x) \right)=0 \] since $G$ is odd and by symmetry $\int_{\mathbb{R}\ \!\!} f(x) G(x) d\mu(x)= \int_{\mathbb{R}\ \!\!} G(x) d\mu(x)=0.$ The third covariance relation of Lemma \ref{thm:cov2} thus gives: \[ {\mathrm{{\rm Cov}}}_\mu(f,g)= {\mathrm{{\rm Cov}}}_{\mu^{(1)}_f} (\phi'(x), \psi'(y)) \geq 0 \] since as before by Corollary \ref{cor:totpos} the kernel of the probability measure $\mu^{(1)}_{f,g}$ is also totally positive.
\blue{make a comment on the last point: sans la symétrie de $\mu$,\\
le résultat reste vrai pour $f,g$ quasi-concave (ensemble de sur niveau convex) et paire\\
log-concave implique quasi concave\\ Prekopa: $f$ log-concave complètement sur $R^d$ implique $f_k$ (cf tensorisation) log-concave\\ qu'en est-il pour quasi-concave.}
\blue{Un autre résultat: Si $\mu$ mesure produit générale, $f,g$ log-concave paire, alors ${\mathrm{{\rm Cov}}}_\mu(f,g)\geq 0.$} \end{comment}
\section{More covariance inequalities in dimension one for Chebyshev systems} \label{sec:cheb}
In this section, we consider some generalizations in dimension one of Theorem \ref{thm:main-1d}, involving formulations through determinants for Chebyshev systems. As for Theorem \ref{thm:main-1d}, we provide two proofs, one based only through determinantal identities and the other taking advantage of the strong fact that the kernel $k_\mu$ is totally positive.
\subsection{Covariance inequalities for Chebyshev systems} Let us first define Chebyshev systems. \begin{defi} \label{def:Cheby} A $r$-uple of functions $(f_{1},\dots,f_{r})$ with $f_i : {\mathbb{R}\ \!\!} \to {\mathbb{R}\ \!\!}$ is said to form a \emph{Chebyshev system} (of order r) if for all $t_{1}\leq\dots\leq t_{r}\in{\mathbb{R}\ \!\!}$, \begin{equation} \det(f_{i}(t_{j}))_{1\leq i,j\leq r}\geq0.\label{eq:cheb-def} \end{equation} \end{defi}
The main result of this section is the following:
\begin{theo}\label{thm:cheb} Let $n\geq1$ and $f_{1},\dots,f_{n}:{\mathbb{R}\ \!\!}\to{\mathbb{R}\ \!\!}$ and $g_{1},\dots,g_{n}:{\mathbb{R}\ \!\!}\to{\mathbb{R}\ \!\!}$ be some functions such that
both the $(n+1)$-uples $(1,f_1,\ldots,f_n)$ and $(1,g_1,\ldots,g_n)$ form two Chebyshev systems.
Denote $F(x)=\begin{pmatrix}f_{1}(x)\\ \dots\\ f_{n}(x) \end{pmatrix}$ and $G(x)=\begin{pmatrix}g_{1}(x)\\ \dots\\ g_{n}(x) \end{pmatrix}$, then: \begin{equation} \det({\mathrm{{\rm Cov}}}(F,G))\geq0.\label{eq:cheb-concl} \end{equation} \end{theo}
\subsection{A first proof using the determinantal approach.} We produce here a proof of Theorem \ref{thm:cheb} which is based on the methods introduced in Section~\ref{sec:Andreev-dim1}.
We first claim that \begin{equation}\label{eq:egalite-det}
\det\left({\mathrm{{\rm Cov}}}_\mu(F ,G)\right)_{1 \le i,j \le n}= \det\left( \mathbb{E}_\mu\left[f_i g_j \right]_{0 \le i,j \le n}\right),
\end{equation} where we set $f_0(x) = 1$ and $g_0(x)=1$. Indeed, let $A$ be the matrix $\left(\mathbb{E}_\mu\left[f_i g_j \right]\right)_{0 \le i,j \le n}$. Let $C_0,\ldots,C_n$ denote the columns of the matrix $A$. Let us consider the matrix $B$ obtained by replacing, for every $1 \le j \le n$, the column $C_j$ by $C_j-\mathbb{E}[g_j] C_0$. As $B$ is obtained from $A$ only by elementary operations, they have the same determinant. Moreover, $B$ can be written in block form: \begin{equation*}
B=\begin{pmatrix} 1 & 0 \\ E_\mu(F) & {\mathrm{{\rm Cov}}}(F,G) \end{pmatrix}. \end{equation*} This proves that $\det(A) = \det\left({\mathrm{{\rm Cov}}}(F,G)\right)$ and \eqref{eq:egalite-det} follows.
\begin{proof}[Proof of Theorem \ref{thm:cheb}] By the equality \eqref{eq:egalite-det} and the Andreev-type formula of Proposition~\ref{prop:Andreev}, one has $$ \det\left({\mathrm{{\rm Cov}}}(F,G)\right) = \frac{1}{n!} \int_{{\mathbb{R}\ \!\!}^{n+1}} \det\left(f_i(x_j)\right)_{0 \le i,j \le n} \det\left(g_i(x_j)\right)_{0 \le i,j \le n} d\mu(x_0)\cdots d\mu(x_n). $$ Let us fix $(x_0,\ldots,x_n)\in {\mathbb{R}\ \!\!}^{n+1}$. There exists a permutation $\sigma \in S_n$ such that $x_{\sigma(0)} \le \cdots \le x_{\sigma(n)}$. As $(f_0,\ldots,f_n)$ and $(g_0,\ldots,g_n)$ are Chebyshev systems, one has $$ \varepsilon(\sigma) \det\left(f_i(x_j)\right)_{0 \le i,j \le n} \ge 0 \ , \ \varepsilon(\sigma) \det\left(g_i(x_j)\right)_{0 \le i,j \le n} \ge 0. $$ One then has $\det\left(f_i(x_j)\right)_{0 \le i,j \le n}\det\left(g_i(x_j)\right)_{0 \le i,j \le n} \ge 0$, so $$\det\left({\mathrm{{\rm Cov}}}(F,G)\right) \ge 0$$ and the result follows. \end{proof}
\subsection{A second proof with the Hoeffding covariance identity.} We turn now to the proof of the covariance inequality of Theorem \ref{thm:cheb} using Hoeffding's covariance identity \eqref{eq:cov-k}. The main point will be the use of a bivariate Andreev-type formula for bilinear integral operators. Another key point is to transfer the assumption on the functions to an assumption on their derivatives. \begin{prop} \label{prop:equiv-cheb}
Let $n\geq1$ and $f_{1},\dots,f_{n}:{\mathbb{R}\ \!\!}\to{\mathbb{R}\ \!\!}$
be some $\mathcal C^1$ functions. The following assertions are equivalent: \begin{enumerate}
\item The $(n+1)$-uple $(1,f_1,\ldots,f_n)$ forms a Chebyshev system.
\item The $n$-uple $(f_{1}',\dots,f_{n}')$ forms a Chebyshev system. \end{enumerate} \end{prop}
\begin{proof} Assume (1), let $x_0 <x_1 <\cdots <x_n$ and set $f_0 =1$. Then, replacing the $i$-th column $C_i$ by $C_i- C_{i-1}$ for $1\leq i \leq n$, yields \begin{align*}
0\leq \det(f_i (x_j) )_{0\leq i,j\leq n} &= \det(f_i (x_{j}) -f_i(x_{j-1}) )_{1\leq i,j\leq {n}}\\
&= \prod_{j=1}^n (x_{j}-x_{j-1}) \det \left( \frac{f_i (x_{j}) -f_i(x_{j-1})} {x_{j}-x_{j-1}} \right)_{1\leq i,j\leq {n}}. \end{align*} Letting successively $x_{j}$ tend to $x_{j-1}$ for $j=1,...,n$ gives \[ \det(f_i' (x_{j-1}))_{1\leq i,j\leq {n}} \geq 0 \] and (2) follows.
Now assume (2), for $x_0<x_1 <\cdots <x_n$. Since one has $f_0=1$, replacing the $i$-th column $C_i$ by $C_i- C_{i-1}$ for $1\leq i \leq n$, one gets \begin{align*} \det(f_i (x_j) )_{0\leq i,j\leq n} &= \det(f_i (x_j) -f_i(x_{j-1}) )_{1\leq i,j\leq n}\\ &= \det\left( \int_{x_{j-1}}^{x_j} f_i' (u_j) d u_j \right)_{1\leq i,j\leq n}\\ &= \int_{x_{n-1 }}^{x_{n}} \dots \int_{x_{1}}^{x_2} \int_{x_{0}}^{x_1} \det(f_i' (u_j))_{1\leq i,j\leq {n}} \; du_1 du_2 \dots du_n \end{align*} where the last line follows from an Andreev-type formula. Since $u_1 \leq u_2 \leq \cdots \leq u_n$, we have $\det(f_i' (u_j)) \geq 0$ by (2), and (1) follows. \end{proof}
\
The second key point is the following bivariate Andreev-type formula for bilinear kernel integral operators on ${\mathbb{R}\ \!\!}^{2n}$. It is stated here without a proof.
\begin{prop}\label{prop:bivar-andreev} With the same notation as in Theorem \ref{thm:cheb}, \begin{align*} \det({\mathrm{{\rm Cov}}}(F,G)) & =\det\left(\iint_{x,y\in{\mathbb{R}\ \!\!}}f'_{i}(x)k(x,y)g'_{j}(y)d\mu(x)d\mu(y)\right)_{1\leq i,j\leq n}\\
& =\iint_{\mathcal{D}}\det\left(f_{i}'(x_{j})\right)\det\left(k(x_{i},y_{j})\right)\det\left(g_{i}'(y_{j})\right)
dx_1 \dots dx_d dy_1 \dots dy_d \end{align*} where $\mathcal{D}$ is the domain of ${\mathbb{R}\ \!\!}^{2n}$ defined by \[ \mathcal{D}=\left\{ (x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})\in{\mathbb{R}\ \!\!}^{2n}\ :\ x_{1}<\cdots<x_{n}\ ,\ y_{1}<\cdots<y_{n}\right\} . \] \end{prop} A second proof of Theorem \ref{thm:cheb} is then immediate. \begin{proof}[Second proof of Theorem \ref{thm:cheb}] By hypothesis and since the kernel $k$ is totally positive, the three determinants in the integral on the second line in the equality of Proposition \ref{prop:bivar-andreev} are non-negative and the result follows. \end{proof}
\begin{comment} We first recall the classical result which follows from the change of variables formula: \Hm{ref et ecriture, j'ai vu une formule genre andreieff qulque part mais je ne sias plus trop où : un article de Pinkus :TP mais sans numero}
\begin{prop} \label{prop:intD} We introduce $\mathcal{D}$ as the domain of ${\mathbb{R}\ \!\!}^{2n}$ defined by \[ \mathcal{D}=\left\{ (x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})\in{\mathbb{R}\ \!\!}^{2n}\ :\ x_{1}<\cdots<x_{n}\ ,\ y_{1}<\cdots<y_{n}\right\} . \] Then for every function $\psi:{\mathbb{R}\ \!\!}^{2n}\rightarrow{\mathbb{R}\ \!\!}$ we have \[ \int_{{\mathbb{R}\ \!\!}^{2n}}\psi(x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})dXdY=\sum_{\sigma_{1},\sigma_{2}\in S_{n}}\int_{\mathcal{D}}\psi(x_{\sigma_{1}(1)},\ldots,x_{\sigma_{1}(n)},y_{\sigma_{2}(1)},\ldots,y_{\sigma_{2}(n)})dXdY. \] \end{prop} We now state a bivariate version of Andreev's formula :
\Hm{orthographe Andreiev, il faut choisir.} \begin{prop} \label{prop:andr} Let $(f_{1},\ldots,f_{n})$ and $(g_{1},\ldots,g_{n})$ be two $n$-uples of functions ${\mathbb{R}\ \!\!}\rightarrow{\mathbb{R}\ \!\!}$. Let $k:{\mathbb{R}\ \!\!}^{2}\rightarrow{\mathbb{R}\ \!\!}$ be another function. Then : \[ \det\left(\iint_{{\mathbb{R}\ \!\!}^{2}}f_{i}(x)k(x,y)g_{j}(y)dxdy\right)=\int_{\mathcal{D}}\det\left(f_{i}(x_{j})\right)\det\left(k(x_{i},y_{j})\right)\det\left(g_{i}(y_{j})\right)dx_{1}\cdots dx_{n}dy_{1}\cdots dy_{n}. \] \end{prop}
\begin{proof} We use the definition of determinant and Fubini's theorem to transform the left-hand side integral into \begin{eqnarray*} D & = & \det\left(\iint_{{\mathbb{R}\ \!\!}^{2}}f_{i}(x)K(x,y)g_{j}(y)dxdy\right)\\
& = & \sum_{\sigma\in S_{n}}{\varepsilon\ \!\!}(\sigma)\prod_{i=1}^{n}\iint_{{\mathbb{R}\ \!\!}^{2}}f_{i}(x_{i})k(x_{i},y_{i})g_{\sigma(i)}(y_{i})dx_{i}dy_{i}\\
& = & \iint_{{\mathbb{R}\ \!\!}^{2n}}\left(\prod_{i=1}^{n}f_{i}(x_{i})k(x_{i},y_{i})\right)\left(\sum_{\sigma\in S_{n}}{\varepsilon\ \!\!}(\sigma)\prod_{i=1}^{n}g_{\sigma(i)}(y_{i})\right)dXdY\\
& = & \iint_{{\mathbb{R}\ \!\!}^{2n}}\left(\prod_{i=1}^{n}f_{i}(x_{i})k(x_{i},y_{i})\right)D_{g}(y_{1},\ldots,y_{n})dXdY, \end{eqnarray*} where $D_{g}(y_{1},\ldots,y_{n})=\det(g_{i}(y_{j}))$ and where the shorter notation $dXdY$ stands for $dx_{1}\cdots dx_{n}dy_{1}\cdots dy_{n}$. Elementary properties of determinants allow us to write \[ D_{g}(y_{\sigma(1)},\ldots,y_{\sigma(n)})={\varepsilon\ \!\!}(\sigma)D_{g}(y_{1},\ldots,y_{n}). \] We then use Proposition~\ref{prop:intD} to write \begin{eqnarray*} D & = & \int_{\mathcal{D}}\sum_{\sigma_{2}\in S_{n}}\left(\sum_{\sigma_{1}\in S_{n}}\prod_{i=1}^{n}f_{i}(x_{\sigma_{1}(i)})K(x_{\sigma_{1}(i)},y_{\sigma_{2}(i)})\right)D_{g}(y_{\sigma_{2}(1)},\ldots,y_{\sigma_{2}(n)})dXdY\\
& = & \int_{\mathcal{D}}\sum_{\sigma_{2}\in S_{n}}\left(\sum_{\sigma_{1}\in S_{n}}\prod_{i=1}^{n}f_{i}(x_{\sigma_{1}(i)})k(x_{\sigma_{1}(i)},y_{\sigma_{2}(i)})\right){\varepsilon\ \!\!}(\sigma_{2})D_{g}(y_{1},\ldots,y_{n})dXdY\\
& = & \int_{\mathcal{D}}D_{g}(y_{1},\ldots,y_{n})\sum_{\sigma_{1}\in S_{n}}\prod_{i=1}^{n}f_{i}(x_{\sigma_{1}(i)})\left(\sum_{\sigma_{2}\in S_{n}}{\varepsilon\ \!\!}(\sigma_{2})k(x_{\sigma_{1}(i)},y_{\sigma_{2}(i)})\right)dXdY. \end{eqnarray*} We notice that \begin{eqnarray*} \sum_{\sigma_{2}\in S_{n}}{\varepsilon\ \!\!}(\sigma_{2})k(x_{\sigma_{1}(i)},y_{\sigma_{2}(i)}) & = & \det\left(k(x_{\sigma_{1}(i)},y_{j}\right)\\
& = & {\varepsilon\ \!\!}(\sigma_{1})\det(k(x_{i},y_{j})), \end{eqnarray*} so we have \begin{eqnarray*} D & = & \int_{\mathcal{D}}D_{g}(y_{1},\ldots,y_{n})\det(k(x_{i},y_{j}))\sum_{\sigma_{1}\in S_{n}}{\varepsilon\ \!\!}(\sigma_{1})\prod_{i=1}^{n}f_{i}(x_{\sigma_{1}(i)}dXdY\\
& = & \int_{\mathcal{D}}\det\left(f_{i}(x_{j})\right)\det\left(k(x_{i},y_{j})\right)\det\left(g_{i}(y_{j})\right)dXdY \end{eqnarray*} \end{proof}
We now turn to the proof of Theorem \ref{thm:cheb}.
\begin{proof}{[}Proof of Theorem \ref{thm:cheb}{]} By the covariance representation formula of Theorem \ref{thm:covA}, and the bivariate Andreiev formula of Proposition \ref{prop:andr}, one directly gets \begin{align*} \det({\mathrm{{\rm Cov}}}(F,G)) & =\det\left(\iint_{x,y\in{\mathbb{R}\ \!\!}}F'_{i}(x)k(x,y)G'_{j}(y)d\mu(x)\right)_{1\leq i,j\leq n}\\
& =\int_{\mathcal{D}}\det\left(f_{i}'(x_{j})\right)\det\left(k(x_{i},y_{j})\right)\det\left(g_{i}'(y_{j})\right)dXdY \end{align*} and the result follows since the three determinants are non-negative. \end{proof}
\blue{On enlève jusqu'ici.} \end{comment}
\subsection{Some applications.} We shall use Theorem \ref{thm:cheb} under the following particular form. \begin{coro}\label{cor:cheb} Let $n\geq1$ and $\phi_1,\dots,\phi_{n}:{\mathbb{R}\ \!\!}\to{\mathbb{R}\ \!\!}$ and $f,g:{\mathbb{R}\ \!\!}\to{\mathbb{R}\ \!\!}$ be some functions and denote $F(x)=\begin{pmatrix}\phi_{1}(x)\\ \dots\\ \phi_{n}(x)\\ f(x) \end{pmatrix}$ and $G(x)=\begin{pmatrix}\phi_{1}(x)\\ \dots\\ \phi_{n}(x)\\ g(x) \end{pmatrix}$. Assume that $(1,\phi_{1},\dots,\phi_{n},f)$ and $(1,\phi_{1},\dots,\phi_{n},g)$ form two Chebyshev systems, then: \begin{equation} \det({\mathrm{{\rm Cov}}}(F,G))\geq0.\label{eq:cheb-concl2} \end{equation} \end{coro}
It is well known that, if $f$ is smooth and if we choose more precisely $\phi_{1}(x)=x,\dots,\phi_{n}(x)=x^{n}$, the condition that $(1,\phi_{1},\dots,\phi_{n},f)$ forms a Chebyshev system is equivalent to $f^{(n+1)}(x) \geq 0,\textrm{ for all }x\in{\mathbb{R}\ \!\!}$. This is a well known generalization of Proposition \ref{prop:ConvC}, see \cite[Chapter 6 Example 4]{karlin:book}.
In the case where we consider $n=1$, we recover Theorem \ref{thm:main-1d}(1). Indeed, in dimension one, by Proposition \ref{prop:ConvC}, the convexity assumptions on $f$ and $g$ are equivalent to the fact that $(1,x,f)$ and $(1,x,g)$ both form a Chebyshev system.
\
We now describe the result for $n=2$ and $\phi_{1}(x)=x,\phi_{2}(x)=x^{2}$.
\begin{coro}\label{cor:n=00003D3} Let $\mu$ be probability measure on ${\mathbb{R}\ \!\!}$ admitting a fourth moment. Assume that $f^{(3)}(x)\geq0$ and $g^{(3)}(x)\geq0$ for all $x\in{\mathbb{R}\ \!\!}$, then \begin{align*} {\mathrm{{\rm Cov}}}_{\mu}(f,g)\left({\mathrm{{\rm Var}}}_{\mu}(x^{2}){\mathrm{{\rm Var}}}_{\mu}(x)-{\mathrm{{\rm Cov}}}_{\mu}(x,x^{2})^{2}\right)\\ \geq\begin{pmatrix}{\mathrm{{\rm Cov}}}_{\mu}(x,f) & {\mathrm{{\rm Cov}}}_{\mu}(x^{2},f)\end{pmatrix}\begin{pmatrix}{\mathrm{{\rm Var}}}_{\mu}(x^{2}) & -{\mathrm{{\rm Cov}}}_{\mu}(x,x^{2})\\ -{\mathrm{{\rm Cov}}}_{\mu}(x,x^{2}) & {\mathrm{{\rm Var}}}_{\mu}(x) \end{pmatrix}\begin{pmatrix}{\mathrm{{\rm Cov}}}_{\mu}(x,g)\\ {\mathrm{{\rm Cov}}}_{\mu}(x^{2},g) \end{pmatrix}. \end{align*}
If moreover $\int_{{\mathbb{R}\ \!\!}}xd\mu=\int_{{\mathbb{R}\ \!\!}}x^{3}d\mu=0$, the latter inequality writes \begin{align*} {\mathrm{{\rm Cov}}}(f,g)\geq\frac{1}{{\mathrm{{\rm Var}}}_{\mu}(x)}{\mathrm{{\rm Cov}}}(x,f)\cdot{\mathrm{{\rm Cov}}}(x,g)+\frac{1}{{\mathrm{{\rm Var}}}_{\mu}(x^{2})}{\mathrm{{\rm Cov}}}(x^{2},f)\cdot{\mathrm{{\rm Cov}}}(x^{2},g). \end{align*} \end{coro} \begin{comment}\begin{proof} The proof is a direct application of Corollary \ref{cor:cheb} with $F(x)=\begin{pmatrix}x\\ x^{2}\\ f(x) \end{pmatrix}$ and $G(x)=\begin{pmatrix}x\\ x^{2}\\ g(x) \end{pmatrix}$. The assumptions means exactly that $(1,x,x^2,f)$ and $(1,x,x^2,g)$ forms two Chebyshev systems. From Corollary \ref{cor:cheb}, we infer that $\det({\mathrm{{\rm Cov}}}(F,G))\geq0$ and the conclusion follows from a computation of $\det({\mathrm{{\rm Cov}}}(F,G))$. \end{proof} \end{comment}
\
\section{The tensorization method for product measures}\label{sec:tens} We investigate here product measures on ${\mathbb{R}\ \!\!}^d$ with $d\geq 2$, through the use of the tensorization argument. This method consists in decomposing the covariance for the product measure by the one-dimensional covariances of the marginals and then applying the covariance inequalities previously obtained in dimension one.
\subsection{The tensorization decomposition of the covariance} \begin{lem}\label{lem:usual-tens} Let $\mu=\mu_{1}\otimes\dots\otimes\mu_{d}$ be a product measure. For a function $f:{\mathbb{R}\ \!\!}^{d}\to{\mathbb{R}\ \!\!}$, set \[ f_{k}(x_{1},\dots x_{k})=\iint f(x_{1},\dots x_{d})d\mu_{k+1}(x_{k+1})\dots d\mu_{d}(x_{d}), \] for $1\leq k\leq d$, and set $f_{0}=\int fd\mu$. Then it holds,
\begin{equation}\label{eq:tensor}
{\mathrm{{\rm Cov}}}_{\mu}(f,g)=\sum_{k=1}^{d}\iint{\mathrm{{\rm Cov}}}_{\mu_{k}}(f_{k},g_{k})d\mu_{1}(x_{1})\dots d\mu_{k-1}(x_{k-1}). \end{equation} \end{lem}
In the above lemma, the function $f_k$ is the conditional expectation of $f$ knowing $(x_1,\dots,x_k)$ and ${\mathrm{{\rm Cov}}}_{\mu_{k}}(f_{k},g_{k})$ is the covariance with respect to the one-dimensional marginal $\mu_k$ of $f_k$ and $g_k$ ; that is the function depending on $(x_1,\dots, x_{k-1})$ given by: \[
{\mathrm{{\rm Cov}}}_{\mu_{k}}\big(x\mapsto f_{k}(x_1,\dots, x_{k-1}, x) , x\mapsto g_{k}(x_1,\dots, x_{k-1}, x)\big). \]
This decomposition of the covariance is well known and implies the famous tensorization property of the Poincar\'e inequality; see e.g. \cite{bakry-gentil-ledoux}. To be complete, \eqref{eq:tensor} is stated for the variance in \cite{ledoux-spin}, but it also applies to the covariance due to the following polarization identity: \[ 4 \, {\mathrm{{\rm Cov}}}(f,g) = {\mathrm{{\rm Var}}}(f+g) -{\mathrm{{\rm Var}}} (f-g). \]
\subsection{A weighted Hoeffding's covariance relation in dimension one.} For the sequel, we shall need a slight generalization of the covariance relation of Lemma~\ref{thm:cov2} in dimension one, that we describe now.
Let $\mu$ be a probability measure on ${\mathbb{R}\ \!\!}$. Let $a$ be a positive function on ${\mathbb{R}\ \!\!}$ and let $A$ be the (centered) primitive of $a$, $A=\int a dx+c$. We assume that $a$ is such that ${\mathrm{{\rm Var}}}_{\mu}(A)<+\infty$. With these notations and with also the same notations as in Section \ref{subsec:nice-relation-dim1}, we define \[ k_{a,a}(x,y):={a(x)}k(x,y){a(y)} \] and we set $Z_{a,a}=\iint k_{a,a}(x,y)dxdy.$ By Theorem \ref{thm:covA}, one gets $ Z_{a,a}={\mathrm{{\rm Var}}}_{\mu}(A). $ We will also consider the measure $\mu_{a,a}^{(1)}$, defined by \[ d\mu_{a,a}^{(1)}(x,y)=\frac{k_{a,a}(x,y)}{\iint k_{a,a}(x',y')dx'dy'}dxdy. \] Since the kernel $k$ is totally positive, by Corollary \ref{cor:totpos2}, one gets that the kernels $k_{a,a}$ are also totally positive.
The following lemma provides a generalization of Lemma \ref{thm:cov2}, which corresponds to the case $a\equiv1$.
\begin{lem}\label{thm:cov-mod-a}
Let $f,g:{\mathbb{R}\ \!\!}\to {\mathbb{R}\ \!\!}$ be some absolutely continuous functions that belong to $L^2(\mu)$.
\begin{enumerate}
\item Then, \begin{align}\label{weight_cov_dim1_conv_conv}
\nonumber & {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))\\
=&Z_{a,a}\left[{\mathrm{{\rm Cov}}}_{\mu_{a,a}^{(1)}}\left(\frac{f'(x)}{a(x)},\frac{g'(y)}{a(y)}\right)+\frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),A(x))}{Z_{a,a}}\,\frac{{\mathrm{{\rm Cov}}}_{\mu}(g(x),A(x))}{Z_{a,a}}\right]. \end{align}
\item If moreover $ f=e^{-\phi}$, then \begin{align}\label{weight_cov_dim1_logco_conv}
\nonumber & {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))\\
=&
Z_{af,a}\left[{\mathrm{{\rm Cov}}}_{\mu_{af,a}^{(1)}}\left(\frac{-\phi'(x)}{a(x)},\frac{g'(y)}{a(y)}\right)+\frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),A(x))}{Z_{af,a}}\,\frac{{\mathrm{{\rm Cov}}}_{\mu}(F_a(x),g(x))}{Z_{af,a}}\right]. \end{align} \item If moreover $g=e^{-\psi}$, then \begin{align} \label{weight_cov_dim1_logco_conv2} \nonumber & {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))\\ =&
Z_{af,ag}\left[{\mathrm{{\rm Cov}}}_{\mu_{af,ag}^{(1)}}\left(\frac{\phi'(x)}{a(x)},\frac{\psi'(y)}{a(y)}\right)+\frac{{\mathrm{{\rm Cov}}}_{\mu}(f(x),G_a(x))}{Z_{af,ag}}\,\frac{{\mathrm{{\rm Cov}}}_{\mu}(F_a(x),g(x))}{Z_{af,ag}}\right]. \end{align} \end{enumerate} Here $F_a$ and $G_a$ denotes respectively the primitives of $ af $ and $ag$ and the constants $Z_{af,a}$ and $Z_{af,ag}$ are defined as in Lemma \ref{thm:cov2}. \end{lem}
\subsection{Proofs of Theorems \ref{thm:Hu-produit-A}, \ref{thm:Hu-tens-incond-A} and \ref{thm:Harge-Royen-tens} }
We first consider the proof of Theorem \ref{thm:Hu-produit-A}. We then state a similar result in Theorem \ref{thm:tens-V}, but with slightly different assumptions.
\begin{proof}[Proof of Theorem \ref{thm:Hu-produit-A}] First, we assume that $f$ and $g$ are such that all the quantities in \eqref{eq:cond-a-i} and \eqref{eq:cond-a-ij} are non-negative. By the tensorization of the covariance \eqref{eq:tensor} and the first covariance relation of Lemma \ref{thm:cov-mod-a}, one has \begin{align} \nonumber & {\mathrm{{\rm Cov}}}_{\mu}(f,g)\\ = & \sum_{k=1}^{d}\frac{1}{Z_{k,a_k,a_{k}}}\iint{\mathrm{{\rm Cov}}}_{\mu_{k}}(f_{k},A_{k}(x_{k})){\mathrm{{\rm Cov}}}_{\mu_{k}}(g_{k},A_{k}(x_{k}))d\mu_{1}(x_{1})\dots d\mu_{k-1}(x_{k-1})\label{l1}\\
+& \sum_{k=1}^{d}Z_{k,a_k,a_{k}}\iint {\mathrm{{\rm Cov}}}_{\mu_{k,(a_k,a_{k})}^{(1)}}\left( \frac{\partial_{k}f_{k}(x_{k})}{a_{k}(x_{k})} ,\frac{\partial_{k}g_{k}(y_{k})}{a_{k}(y_{k})}\right)
d\mu_{1}(x_{1})\dots d\mu_{k-1}(x_{k-1}),\label{l2} \end{align} where more precisely \[ \partial_{k}f_{k}(x_{k}) =\partial_{k}f_{k}(x_{1},\dots,x_{k-1},x_{k}) \textrm{ and } \partial_{k}g_{k}(y_{k}) = \partial_{k}g_{k}(x_{1},\dots,x_{k-1},y_{k}).\]
We first prove that the sum in display \eqref{l2} is non-negative. The assumption \eqref{eq:cond-a-i} implies that
both $(x_k,y_k)\mapsto \frac{\partial_{k}f(x_{k})}{a_{k}(x_{k})}$ and $(x_k,y_k)\mapsto \frac{\partial_{k}g(y_{k})}{a_{k}(y_{k})}$ are coordinatewise increasing on ${\mathbb{R}\ \!\!}^2$ for every fixed $x_{1},\dots,x_{k-1}$. Since by Corollary \ref{cor:totpos}, the measure $\mu_{k,(a_k,a_{k})}^{(1)}$ satisfies the FKG criterion on ${\mathbb{R}\ \!\!}^{2}$, the term \eqref{l2} is non-negative. We now turn to the sum in display \eqref{l1}. With a similar notation for $g$, we set \begin{align*} F_{k,a_{k}}(x_{1},\dots x_{k-1})&={\mathrm{{\rm Cov}}}_{\mu_{k}}(f_{k},A_{k}(x_{k})).\\ \end{align*} Since the hypotheses allow to exchange derivation and integrals, one has
for $1\leq i \leq (k-1)$, \begin{align*} & \partial_i \, {\mathrm{{\rm Cov}}}_{\mu_{k}}(f_{k},A_{k}(x_{k}))\\ =& \iint_{x_k,y_k} \left( \iint_{x_{k+1}, \dots,x_d} \partial_{i,k} f(x) d\mu_{k+1} (x_{k+1})\dots d\mu_d(x_d) \right) k_{\mu_k}(x_k,y_k) a_k(y_k) d x_k dy_k. \end{align*} In particular, the assumptions \eqref{eq:cond-a-ij} give that the above integrands are non-negative. This implies that the functions $F_{k,a_{k}}$ and $G_{k,a_{k}}$ are both coordinate increasing on ${\mathbb{R}\ \!\!}^{k-1}$. By the standard FKG inequality for product measures, one gets that \begin{align*}
& \iint F_{k,a_{k}}\;G_{k,a_{k}}d\mu_{1}(x_{1})\dots d\mu_{k-1}(x_{k-1})\\ \geq & \iint F_{k,a_{k}}d\mu_{1}(x_{1})\dots d\mu_{k-1}(x_{k-1})\cdot\iint G_{k,a_{k}}d\mu_{1}(x_{1})\dots d\mu_{k-1}(x_{k-1})\\ = & \;{\mathrm{{\rm Cov}}}_{\mu}(f(x),A_{k}(x_{k}))\cdot{\mathrm{{\rm Cov}}}_{\mu}(g(x),A_{k}(x_{k})). \end{align*} Summing over the index $k$ ends the proof in the case of non-negative signs in assumptions \eqref{eq:cond-a-i} and \eqref{eq:cond-a-ij}. Finally, analyzing the above proof, one sees that it is still valid for general signs. Indeed, under the general case of assumption \eqref{eq:cond-a-i}, the functions $(x_k,y_k)\mapsto \frac{\partial_{k}f(x_{k})}{a_{k}(x_{k})}$ and $(x_k,y_k)\mapsto \frac{\partial_{k}g(y_{k})}{a_{k}(y_{k})}$ are either both coordinatewise increasing or both coordinatewise decreasing on ${\mathbb{R}\ \!\!}^2$ and thus have a non-negative covariance with respect to the measure $\mu_{k,(a_k,a_{k})}^{(1)}$. Secondly, the last argument relies on the FKG inequality for product measures. In this case, the FKG inequality is in fact also valid if the functions $F_{k,a_k}$ and $G_{k,a_k}$ are monotone along coordinates, with the same monotonicity along each coordinate. This is the case under the general assumption \eqref{eq:cond-a-ij} and the result follows.
\end{proof}
As announced in the beginning of this section, we also obtain with this tensorization approach, a similar result under slightly different conditions.
\begin{theo}\label{thm:tens-V} Let $\mu$ be a product measure on ${\mathbb{R}\ \!\!}^{d}$. Assume that the marginals $\mu_k$ are absolutely continuous with respect to the Lebesgue measure, with positive densities $e^{-V_k}$, for smooth potentials $V_k$. Let $f,g: {\mathbb{R}\ \!\!}^d \to {\mathbb{R}\ \!\!}$ and assume
that for each $1\leq k\leq d$, the signs of \begin{equation}\label{cond-V-i} \partial_{k}\left(\frac{\partial_{k}f(x)}{a_{k}(x_{k})}\right)\textrm{ and }\partial_{k}\left(\frac{\partial_{k}g(x)}{a_{k}(x_{k})}\right) \end{equation} are constant and equal and that for each $(j,k)$ with $1\leq j<k\leq d$, the signs of \begin{equation}\label{cond-V-ij} \partial_{j,k}\left(f(x)\frac{A_{k}(x_{k})}{V_{k}'(x_{k})}\right)\textrm{ and }\partial_{j,k}\left(g(x)\frac{A_{k}(x_{k})}{V_{k}'(x_{k})}\right) \end{equation} are also constant and equal. Assume furthermore the following technical assumption:
\begin{equation} \label{eq:HypoTech} \lim_{x_k \to \pm \infty} f_k(x_1,\ldots,x_k) \frac{A_k(x_k)}{V'_k(x_k)} e^{-V_k(x_k)} = 0, \end{equation} where $f_k$ is defined in Lemma~\ref{lem:usual-tens}. Then, it holds \[ {\mathrm{{\rm Cov}}}_{\mu}(f,g)\geq\sum_{k=1}^{d}\frac{1}{{\mathrm{{\rm Var}}}_{\mu_{k}}(A_{k})}{\mathrm{{\rm Cov}}}_{\mu}(f(x),A_{k}(x_{k}))\cdot{\mathrm{{\rm Cov}}}_{\mu}(g(x),A_{k}(x_{k})). \] \end{theo}
Note that since each $A_k$ vanishes exactly in one point, the hypothesis in \eqref{cond-V-ij} forces each $V_k$ to be unimodal (in the sense that $V_k$ has only one zero). In particular, if each potential $V_k$ is strictly convex, one can specialize the result to the case $A_k=V_k'$ or equivalently $a_k=V_k''$. With this specific choice, Theorems \ref{thm:tens-V} and \ref{thm:Hu-produit-A} coincide.
\begin{proof}[Proof of Theorem \ref{thm:tens-V}] The proof is similar to the one of Theorem \ref{thm:Hu-produit-A}. The only difference is that we use integration by parts to get a different representation of $F_{k,a_{k}}$. We obtain \[ F_{k,a_{k}}(x_{1},\dots x_{k-1})={\mathrm{{\rm Cov}}}_{\mu_{k}}\left(f_{k},V_{k}'(x_{k})\frac{A_{k}(x_{k})}{V_{k}'(x_{k})}\right)=\int\partial_{k}\left(f_{k}\frac{A_{k}(x_{k})}{V_{k}'(x_{k})}\right)d\mu_{k}(x_{k}). \] Notice that the bracket terms in the integration by parts are zero due to Assumption~\eqref{eq:HypoTech}. Furthermore, by Assumption \eqref{cond-V-ij}, the two functions $F_{k,a_k}$ and $G_{k,a_k}$ are coordinatewise monotone with the same kind of monotony along each coordinate. The result follows. \end{proof}
\begin{comment} \blue{ Corollaire et phrase à enlever!} \begin{coro}\label{cor:tens-V'} Let $\mu $ be a product measure on ${\mathbb{R}\ \!\!}^{d}$. With the above notation, assume that each potential $V_k$ for $1\leq k\leq d$ is strictly convex. Let $f,g \in L^2(\mu)$ and assume
that for each $1\leq k\leq d$, the signs of \begin{equation}\label{cond-V'-i} \partial_{k}\left(\frac{\partial_{k}f(x)}{V''_{k}(x_{k})}\right)\textrm{ and }\partial_{k}\left(\frac{\partial_{k}g(x)}{V''_{k}(x_{k})}\right) \end{equation} are constant and equal and that for each $(j,k)$ with $1\leq j<k\leq d$, the signs of \begin{equation}\label{cond-V-ij-cor} \partial_{j,k}f(x)\textrm{ and }\partial_{j,k}g(x) \end{equation} are also constant and equal.
\Hm{and satisfying the technical Assumption~\eqref{eq:HypoTech}} Then \red{Non, on retombe sur le thm d'avant et en plus avec $A_k $ et pas $x_k$} \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq \sum_{i=1}^d \frac{1}{{\mathrm{{\rm Var}}}(\mu_i)} {\mathrm{{\rm Cov}}}_\mu (f,x_i)\cdot {\mathrm{{\rm Cov}}}_\mu (g,x_i). \] \end{coro}
\blue{In the standard Gaussian case, Corollary \ref{cor:tens-V'} coincides with Corollary \ref{cor:hu-produit}. } \end{comment} \
We now add the symmetry assumptions and turn to the proofs of Theorem \ref{thm:Hu-tens-incond-A}. and \ref{thm:Harge-Royen-tens}.
\begin{proof}[Proof of Theorem \ref{thm:Hu-tens-incond-A}] We start by the same covariance formulae, given in \eqref{l1}, \eqref{l2}, as in the proof of Theorem \ref{thm:Hu-produit-A}. As previously, in view of assumption \eqref{eq:cond-l2-idem}, all terms in \eqref{l2} are non-negative. Now, we show that under the symmetry assumptions, all the terms in \eqref{l1} vanish. Indeed, since the product measure $\mu$ is symmetric, the function $f$ is unconditional and the function $a_k$ are even, we have that, for any $1\leq k \leq d$, and any $x_1,\dots x_{k-1}$, the functions \[ x_k \mapsto f_k(x_1,\dots x_{k-1},x_k) \] are even and that the primitive functions $A_k$ are odd. Hence, it holds that \[ {\mathrm{{\rm Cov}}}_{\mu_k}(f_k ,A_k(x_k)) =0 \textrm{ for each } 1\leq k \leq d \textrm{ and all } x \in {\mathbb{R}\ \!\!}^d \] and all the terms in \eqref{l1} vanish. The result follows. \end{proof}
\begin{proof}[Proof of Theorem \ref{thm:Harge-Royen-tens}(1)] With the same notations as above, writing $\phi_k=-\ln f_k$ and using the second point of Lemma \ref{thm:cov-mod-a}, one has \begin{align*}
& {\mathrm{{\rm Cov}}}_{\mu}(f,g)\\%
=&\sum_{k=1}^{d} \iint \frac{1}{Z_{k,f_k,1}} {\mathrm{{\rm Cov}}}_{\mu_{k}}(f_{k},x_{k})
\left(\iint f_k(x_k)k_{\mu_k}(x_k,y_k) \partial_k g_k(y_k)dx_kdy_k\right) d\mu_{1}\dots d\mu_{k-1} \\
+&\sum_{k=1}^{d}\iint Z_{k,f_k,1}{\mathrm{{\rm Cov}}}_{\mu_{k,(f_k, 1)}^{(1)}}\left( -\partial_{k}\phi_{k}(x_{k}) ,\partial_{k}g_{k}(y_{k})\right)d\mu_{1}\dots d\mu_{k-1}.
\end{align*}
Since $f$ is unconditional, for each fixed $(x_1,\dots,x_{k-1})$, the function $f_k$ is even, and thus \[ {\mathrm{{\rm Cov}}}_{\mu_{k}}(f_{k},x_{k})=0 \] and the terms of the sum in the right-hand side of the above equality vanish. Furthermore, as $f$ and the $\mu_i$ are log-concave, by stability through marginalization of log-concavity (Prékopa's theorem), the functions $f_k$ are log-concave, meaning that the functions $\phi_k$ are convex. Since by Corollary \ref{cor:totpos}, the measures $\mu_{k,(f_k, 1)}^{(1)}$ satisfy the FKG inequality on ${\mathbb{R}\ \!\!}^2$, one has for each fixed $(x_{1},\dots,x_{k-1})\in {\mathbb{R}\ \!\!}^{k-1}$,
\[ {\mathrm{{\rm Cov}}}_{\mu_{k,(f_k, 1)}^{(1)}}\left( -\partial_{k}\phi_{k}(x_{1},\dots,x_{k-1},x_{k}) ,\partial_{k}g_{k}(x_{1},\dots,x_{k-1},y_{k})\right)\leq 0 \] and the proof is complete. \end{proof} The proof of Theorem \ref{thm:Harge-Royen-tens}(2) is given in the next section.
One can note that we do not state a version of Theorem \ref{thm:Harge-Royen-tens} with the function $a_k$ or $A_k$. The reason is that we do not know a natural hypothesis on $f$ that would induce a sign for the quantities $\partial_k \left(\frac{\partial_k \phi_k}{a_k} \right)$.
\section{The quasi-concave case}\label{sec:quasi-concave} \begin{comment} \adp{A choisir?} This section is devoted to the proof of Theorem \ref{thm:main-1d}(3) and Theorem \ref{thm:Harge-Royen-tens}(2) in the quasi-concave case. Actually, as said before, Theorem \ref{thm:Harge-Royen-tens}(2) will be encompassed in Theorem \ref{thm:quasi-concave-prod} below. This quasi-concave case is very specific comparing to the rest of the paper. The result in dimension one is obtained through the layer-cake representation of the functions \eqref{eq:quasi-concave} and the result in dimension $d\geq 2$ is then obtained by tensorisation. Theorem \ref{thm:Harge-Royen-tens}(2) already appears in \cite{Schechtman}. As far as we know, Theorem \ref{thm:quasi-concave-prod} is a new generalisation. \end{comment}
This section is devoted to the proof of Theorem \ref{thm:main-1d}(3) and Theorem \ref{thm:Harge-Royen-tens}(2), related to the quasi-concave case. This assumption indeed requires different techniques than in the rest of the paper. The result in dimension one is obtained through the so-called layer-cake representation of the functions (see \eqref{eq:quasi-concave}) and the result in dimension $d\geq 2$ is then obtained by tensorisation. A similar result already appears in \cite{Schechtman}, but as far as we know, the statement of Theorem \ref{thm:Harge-Royen-tens}(2) is new.
\
Recall that a quasi-concave function $f$ on ${\mathbb{R}\ \!\!}^d$ is a real-valued function that satisfies, for any $x,y\in {\mathbb{R}\ \!\!}^d$ and any $\lambda \in [0,1]$, \[ f(\lambda x + (1-\lambda)y)\geq \min \left\{ f(x), f(y)\right\}. \]
An equivalent formulation of quasi-concavity consists in requiring that the upper level sets of the function are convex. In the following, we make use of a weaker notion than quasi-concavity, that we term ``coordinatewise quasi-concavity'': \begin{defi} A function $f:{\mathbb{R}\ \!\!}^d \to {\mathbb{R}\ \!\!}$ is said to be \emph{coordinatewise quasi-concave} if
for all $(x_1,...,x_{i-1},x_{i+1},...,x_d)\in {\mathbb{R}\ \!\!}^{d-1}$, the functions
\[
x_i \in {\mathbb{R}\ \!\!} \mapsto f(x_1,...,x_{i-1},x_i,x_{i+1},...,x_d) \in {\mathbb{R}\ \!\!}
\]
are quasi-concave. \end{defi}
Another characterization is thus that for any $\lambda \in [0,1]$, any $(x_1,...,x_{i-1},x_{i+1},...,x_d)\in {\mathbb{R}\ \!\!}^{d-1}$ and any $x,y\in {\mathbb{R}\ \!\!}$, \begin{align*}
& f(x_1,...,x_{i-1},\lambda x_i + (1-\lambda)y_i,x_{i+1},...,x_d)\\ & \geq \min \left\{ f(x_1,...,x_{i-1},x_i,x_{i+1},...,x_d), f(x_1,...,x_{i-1}, y_i,x_{i+1},...,x_d)\right\}. \end{align*}
The above definition and its characterization directly imply that quasi-concave functions are coordinatewise quasi-concave, but the converse is not true.
Note that the interpretation in terms of convex upper level sets does not hold anymore for the notion of coordinatewise quasi-concavity. But still, the upper level sets of a coordinatewise quasi-concave function are connected sets.
\begin{comment} \adp{Choix 2: A enlever!} In this section we prove Theorem \ref{thm:Harge-Royen-tens}(2) under a slightly weaker assumption: we only assume that the function are coordinatewise quasi-concave and do not assume any symmetry assumption on the product measure. \begin{theo}\label{thm:quasi-concave-prod} Let $\mu$ be a product measure. Assume that $f$ and $g$ are unconditional and coordinatewise quasi-concave. Then \[ {\mathrm{{\rm Cov}}}_\mu(f,g) \geq 0. \] \end{theo} \end{comment} We now turn to the proof of Theorem \ref{thm:main-1d} (3) in dimension one. \begin{proof}[Theorem \ref{thm:main-1d} (3)] Let $f$ and $g$ be two non-negative quasi-concave even function on ${\mathbb{R}\ \!\!}$. We write, for $x\in {\mathbb{R}\ \!\!}$, \begin{equation}\label{eq:quasi-concave}
f(x)= \int_0^\infty {\mathbf{1}}_{A_t}(x) dt \textrm{ and } g(x)= \int_0^\infty {\mathbf{1}}_{B_t}(x) dt \end{equation} where $A_t$ and $B_t$ for $t\geq 0$ are the the level sets of $f$ and $g$ defined by \[ A_t:= \{x\in {\mathbb{R}\ \!\!}, f(x)\geq t \} \textrm{ and } B_t:= \{x\in {\mathbb{R}\ \!\!}, f(x)\geq t\} . \] The key point is here that since $f$ and $g$ are quasi-concave and even, the sets $A_s$ and $B_t$ are symmetric intervals on ${\mathbb{R}\ \!\!}$ and therefore, for each $s,t \geq 0$, $A_s\subset B_t$ or $B_t \subset A_s$. Therefore by Fubini-Tonelli, one has \begin{align*}
\int f(x) g(x) d\mu(x) &= \int_x \int_{s=0}^\infty \int_{t=0}^\infty {\mathbf{1}}_{A_s}(x) {\mathbf{1}}_{B_t}(x) ds dt d\mu(x)\\
&= \int_{s=0}^\infty \int_{t=0}^\infty \mu (A_s \cap B_t) ds dt\\
&= \int_{s=0}^\infty \int_{t=0}^\infty \min( \mu (A_s), \mu(B_t)) ds dt\\
& \geq \int_{s=0}^\infty \int_{t=0}^\infty \mu (A_s) \mu (B_t) ds dt\\
&= \int f(x) d\mu(x) \; \int g(x) d\mu(x); \end{align*} which is precisely the desired inequality. \end{proof}
\
We now prove Theorem \ref{thm:Harge-Royen-tens}(2) by the tenzorisation method. The main argument is ensured by the following lemma, which states the stability of unconditional coordinatewise quasi-concavity by marginalization. Its proof can be found below. \begin{lem}\label{lem:marg_quasi_con} Consider an integer $d\geq 2$ and take $k\in\left\{1,...,d-1\right\}$. Assume that a function $f$ on ${\mathbb{R}\ \!\!}^d$ is unconditional and coordinatewise quasi-concave. Then the function \[ f_{k}(x_{1},\dots x_{k})=\int f(x_{1},\dots x_{d})d\mu_{k+1}(x_{k+1}) \dots d\mu_{d}(x_d) \] is coordinatewise quasi-concave and unconditional. \end{lem}
\begin{proof}[Proof of Theorem \ref{thm:Harge-Royen-tens}(2)] Let $f$ and $g$ be unconditional and coordinatewise quasi-concave functions. Let us first recall the standard tensorization formula: \[ {\mathrm{{\rm Cov}}}_{\mu}(f,g)=\sum_{k=1}^{d}\iint{\mathrm{{\rm Cov}}}_{\mu_{k}}(f_{k},g_{k})d\mu_{1}\dots d\mu_{k-1}. \] By Lemma \ref{lem:marg_quasi_con} above, for any $(x_1,...,x_{k-1})\in {\mathbb{R}\ \!\!}^{k-1}$ the functions $f_k(x_1,...,x_{k-1}, \cdot)$ and $g_k(x_1,...,x_{k-1}, \cdot)$ are even and quasi-concave on ${\mathbb{R}\ \!\!}$. By Theorem \ref{thm:main-1d} (3), one has ${\mathrm{{\rm Cov}}}_{\mu_{k}}(f_k,g_k) \geq 0$ and the result follows.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:marg_quasi_con}]
Unconditionality of $f_k$ directly follows from unconditionality of $f$. As for the coordinatewise quasi-concavity, we will make the reasoning for the first coordinate $x_1$ and the arguments readily extend to the other coordinates. Take a pair $(x_1,y_1)$ such that $|x_1|\leq|y_1|$. Assume without loss of generality that $y_1\geq0$ (otherwise replace it by $-y_1$). As $f$ is unconditional and coordinatewise quasi-concave, for any $y\in[-y_1,y_1]$ and for any $(x_2,...,x_d)\in {\mathbb{R}\ \!\!}^{d-1}$, we have \begin{align*}
f(y,x_2,\dots,x_d) & \geq \min \left\{ f(-y_1,x_2,...,x_d), f(y_1,x_2,...,x_d)\right\} \\
& = f(y_1,x_2,...,x_d) , \end{align*} where the latter equality follows from unconditionality of $f$. In particular, as $x_1 \in [-y_1,y_1]$, we have for any $(x_2,...,x_d)\in {\mathbb{R}\ \!\!}^{d-1}$, \[
f(x_1,x_2,...,x_d) \geq f(y_1,x_2,...,x_d). \] This gives, for any $\lambda \in [0,1]$, \begin{align*}
f_{k}(\lambda x_{1}+(1-\lambda)y_1,x_2,\dots x_{k})
& = \int f(\lambda x_{1}+(1-\lambda)y_1,x_2,\dots x_{d})d\mu_{k+1}\dots d\mu_{d}\\
& \geq \int f(y_1,x_2,\dots x_{d})d\mu_{k+1}\dots d\mu_{d}\\
& = \min \left\{ f_{k}(x_{1},x_2,\dots x_{k}), f_{k}(y_1,x_2,\dots x_{k}) \right\}. \end{align*}
By symmetry, the case $|y_1|\leq|x_1|$ follows, which finishes the proof. \end{proof}
\section{A global approach for product measures}\label{sec:global}
We provide in this section another proof of Theorem \ref{thm:Hu-produit-A} and we provide the proof of Theorem \ref{thm:Harge-Royen-global}. The first main ingredient that will be used is a generalization of Hoeffding's covariance identity \eqref{eq:cov-k} for product measures.
The two other ingredients are a generalization to product measures of the Hoeffding's covariance relations of Lemmas \ref{thm:cov2} and \ref{thm:cov-mod-a} and the use of FKG inequalities.
\subsection{Duplication and a generalization of Hoeffding's covariance identity} We first present in Lemma \ref{lem:var-prod} a duplication argument for the covariance of a product measure. Similar duplication representations are well known, see e.g. \cite{chatterjee:stein}.
We then deduce in Proposition \ref{prop:var-prod} a generalization of Hoeffding's covariance identity for product measures.
\begin{lem}\label{lem:var-prod}\ Let $\mu=\mu_1 \otimes \cdots \otimes \mu_d$ be a product measure on ${\mathbb{R}\ \!\!}^d$. Under suitable integrable conditions one has
\begin{equation}\label{eq:cov-prod} {\rm Cov}_\mu(f,g)= \frac{1}{2} \sum_{i=1}^d {\mathbb{E}\ \!\!} [\Delta_i f(X,X') \tilde \Delta_i g(X,X')], \end{equation} where $X$ and $X'$ are two independent random variables of law $\mu$, \[ \Delta_i f(X,X') = f(X_1,\dots,X_i , \dots , X_d) - f(X_1,\dots,X_i', \dots , X_d) \] and \[ \tilde \Delta_i g(X,X') = g(X_1,\dots, X_i ,X_{i+1}' \dots , X_d') - g(X_1,\dots,X_i', X_{i+1}' \dots , X_d'). \] \end{lem}
\begin{proof}[Proof of Lemma \ref{lem:var-prod}] Let $X'$ be an independent copy of $X$ with law $\mu$. By symmetrization and then the use of a telescopic sum, one has \begin{eqnarray*} {\mathrm{{\rm Cov}}}_\mu(f,g)&=& {\mathbb{E}\ \!\!}[f(X) (g(X) -g(X')) ]\\
&= & \sum_{i=1}^d {\mathbb{E}\ \!\!}[f(X) \tilde \Delta_i g(X,X') ]\\
&=& \sum_{i=1}^d {\mathbb{E}\ \!\!}\left[ U_i(X,X') \right], \end{eqnarray*} where we define $U_i(X,X')= f(X) \tilde \Delta_i g(X,X')$. Let us denote $(X,X')^{\{j\}}$ to be the random vector given by \[ (X,X')^{\{j\}}= \left(( X_1, \dots,X_{j-1}, X_j', X_{j+1}, \dots, X_d), ( X_1, \dots,X_{j-1}, X_j, X_{j+1}', \dots, X_d')\right) \]
We also write $(X,X')^{\{j\}}= \left(X^{\{j\}},X'^{\{j\}} \right)$ with the slight abuse of notation that $ X^{\{j\}}$ depends on $(X,X')$. Since $\mu$ is a product measure, for each $i$, $(X,X')^{\{i\}}$ is also of law $\mu\otimes \mu$ and thus \begin{eqnarray*} {\mathbb{E}\ \!\!}\left[ U_i (X,X') \right] &=& {\mathbb{E}\ \!\!}\left[ U_i \left((X,X')^{\{i\}}\right) \right]\\
&= & - {\mathbb{E}\ \!\!}[f(X^{\{i\}}) \tilde \Delta_i g (X,X') ]\\ \end{eqnarray*} since $\tilde \Delta_i g \left( (X,X')^{\{i\}}\right) = -\tilde \Delta_i g (X,X')$ and thus \[ {\mathbb{E}\ \!\!}\left[ U_i (X,X') \right] = \frac{1}{2} {\mathbb{E}\ \!\!}\left[ U_i (X,X') \right] + \frac{1}{2} {\mathbb{E}\ \!\!}\left[ U_i \left((X,X')^{\{i\}}\right) \right] = \frac{1}{2}{\mathbb{E}\ \!\!} [\Delta_i f(X,X') \tilde \Delta_i g(X,X')] \] and the result follows. \end{proof}
From the duplication argument, one obtains the following generalization to product measures of Hoeffding's covariance identity.
\begin{prop} \label{prop:var-prod}
Let $\mu=\mu_1\otimes \dots \otimes \mu_d$ be a product measure on ${\mathbb{R}\ \!\!}^d$. Let $f,g:{\mathbb{R}\ \!\!}^d \to {\mathbb{R}\ \!\!}$ be some coordinatewise absolutely continuous functions in $L^2(\mu)$, then \begin{equation}\label{eq:var-prod}
{\mathrm{{\rm Cov}}}_\mu(f,g) = \sum_{i=1}^d \iint_{ x,x' \in {\mathbb{R}\ \!\!}^d} \partial_i f(x) k_{\mu_i}(x_i,x_i') \partial_i g(\underline{x}_{i-1},\overline{x'}_{i})
dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i}) \end{equation} where for $x_i, x_i'\in {\mathbb{R}\ \!\!}$, $k_{\mu_i}$is the standard Hoeffding kernel for the marginal $\mu_i$: \[ k_{\mu_i}(x_i,x_i')= F_{\mu_i} (x_i\wedge x_i') -F_{\mu_i}(x_i) F_{\mu_i}(x_i') \] and for $x,x'\in {\mathbb{R}\ \!\!}^d$, $(\underline{x}_{i-1},\overline{x'}_{i})= (x_1,\dots,{x}_{i-1}, x_i', \dots,{x}'_{d})$, $x_{-i}=(x_1,\dots,x_{i-1}, x_{i+1},\dots x_d)$ and $d\mu(x_{-i})= d\mu_1(x_1) \dots d \mu_{i-1}(x_{i-1}) d \mu_{i+1}(x_{i+1})\dots d \mu_{d}(x_{d})$.
\end{prop} \begin{proof}[Proof of Proposition \ref{prop:var-prod}] We consider one term in the sum of the covariance formula of Lemma \ref{lem:var-prod}. We have \begin{align*} & {\mathbb{E}\ \!\!} [\Delta_i f(X,X') \tilde \Delta_i g(X,X')]\\ =
& \iint_{x,x' \in {\mathbb{R}\ \!\!}^d} \begin{pmatrix} f(\underline{x}_{i-1}, x_i, \overline{x}_{i+1})\\
- f ( \underline{x}_{i-1},x_i', \overline{x}_{i+1})
\end{pmatrix}
\begin{pmatrix} g(\underline{x}_{i-1}, x_i, \overline{x'}_{i+1})\\
- g( \underline{x}_{i-1},x_i', \overline{x'}_{i+1})
\end{pmatrix} d\mu(x)d\mu(x')\\ =& \iint_{x,x' \in {\mathbb{R}\ \!\!}^d} \iint_{s_i,t_i\in {\mathbb{R}\ \!\!}}
\partial_if(\underline{x}_{i-1}, s_i, \overline{x}_{i+1}) \partial_i g(\underline{x}_{i-1}, t_i, \overline{x'}_{i+1}) \\
& \hspace{3cm} \left( {\mathbf{1}}_{\{s_i\leq x_i\}} - {\mathbf{1}}_{\{s_i\leq x_i'\}} \right) \left( {\mathbf{1}}_{\{t_i\leq x_i\}} - {\mathbf{1}}_{\{t_i\leq x_i'\}} \right) ds_i dt_i d\mu(x)d\mu(x'). \end{align*} Furthermore, \begin{eqnarray*}
& &\iint_{x_i,x_i' \in {\mathbb{R}\ \!\!}} \left( {\mathbf{1}}_{\{s_i\leq x_i\}} - {\mathbf{1}}_{\{s_i\leq x_i'\}} \right) \left( {\mathbf{1}}_{\{t_i\leq x_i\}} - {\mathbf{1}}_{\{t_i\leq x_i'\}} \right) d\mu_i(x_i) d\mu_i(x_i')\\ &= & 2 \left( {\mathbb{P}\ \!\!}(X_i\geq \max(s_i,t_i)) - {\mathbb{P}\ \!\!}(X_i\geq s_i) {\mathbb{P}\ \!\!}(X_i\geq t_i )\right) \\ &= & 2 \left(F_{\mu_i} (s_i\wedge t_i) -F_{\mu_i}(s_i) F_{\mu_i}(t_i)\right)\\ &= & 2 k_{\mu_i}(s_i,t_i) \end{eqnarray*}
and the proof follows by Fubini theorem and by a change in the name of the letters in the integral. \end{proof}
We now study some symmetry properties of this covariance representation.
\begin{lem}\label{lem:kmu-sym} Assume $\mu_i$ is a symmetric one dimensional measure, then the kernel $k_{\mu_i}$ is even, that is \[k_{\mu_i}(-s_i,-t_i) = k_{\mu_i} (s_i,t_i).\] \end{lem}
\begin{proof} Without loss of generality assume that $s\leq t$, then $-t\leq -s$, and \begin{eqnarray*} k_{\mu_i} (-s,-t) &=& F_{\mu_i} (-t) - F_{\mu_i} (-s) F_{\mu_i} (-t) \\
&=& (1-F_{\mu_i} (t)) - (1- F_{\mu_i} (s)) (1 - F_{\mu_i} (t))\\
&= & F_{\mu_i} (s) - F_{\mu_i} (s) F_{\mu_i} (t)\\
&=& k_{\mu_i} (s,t). \end{eqnarray*} \end{proof} As a consequence, one obtains the following result. \begin{lem} \label{lem:even} Assume that $\mu$ is a symmetric product measure on ${\mathbb{R}\ \!\!}^d$. Let $f,g:{\mathbb{R}\ \!\!}^n \to {\mathbb{R}\ \!\!}$ be two even functions. Then, for any $1\leq i \leq d$, \[ \iint_{ x,x' \in {\mathbb{R}\ \!\!}^d} \partial_i f(x) k_{\mu_i}(x_i,x_i')
g(\underline{x}_{i-1},\overline{x'}_{i})
dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})=0. \] \end{lem} \begin{proof} The result follows from using the change of variables $(a,b)=(-x,-x')$ on ${\mathbb{R}\ \!\!}^{2d}$ and the fact that $\partial_i f $ is odd, $g$ is even and that the kernel $k_{\mu_i}$ is even. \end{proof} We also derive the following formulas, that will be instrumental in our proofs. \begin{lem}\label{lem:cov-prod-ei} Assume $\mu$ is a product measure on ${\mathbb{R}\ \!\!}^d$. For each $1\leq k \leq d$, let $a_k(x_k)$ be a positive function on ${\mathbb{R}\ \!\!}$ and let $A_k$ be a primitive, centered with respect to $\mu_k$. Let $f:{\mathbb{R}\ \!\!}^n \to {\mathbb{R}\ \!\!}$ be a coordinatewise absolutely continuous function. Then for any $1\leq i\leq d $, one has \[ \iint_{ x,x' \in {\mathbb{R}\ \!\!}^d} \partial_i f(x) k_{\mu_i}(x_i,x_i') a_i(x_i')dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})= {\mathrm{{\rm Cov}}}_\mu(f,A_i(x_i)). \] In particular, \[ \iint_{ x,x' \in {\mathbb{R}\ \!\!}^d} \partial_i f(x) k_{\mu_i}(x_i,x_i') dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})= {\mathrm{{\rm Cov}}}_\mu(f,x_i), \] where, by a slight abuse of notation, $x_i$ stands for the $i$th-coordinate function. It also holds
\[ \iint_{ x,x' \in {\mathbb{R}\ \!\!}^d} k_{\mu_i}(x_i,x_i') dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})= {\mathrm{{\rm Var}}}_\mu(x_i)={\mathrm{{\rm Var}}}(\mu_i). \] \end{lem}
\begin{proof} The proof is a direct application of Proposition \ref{prop:var-prod} with $g(x)=A_i(x_i)$, noticing that only one term in the sum is different from zero. \end{proof}
\subsection{Hoeffding's covariance relation for product measures} The main result here is Lemma \ref{lem:cov2-product} where a similar relation as in Lemma \ref{thm:cov2} is given for product measures.
Let $\mu=\mu_{1}\otimes\dots\otimes\mu_{d}$ be a product measure and write $\Gamma=\Gamma_{\mu}$ its covariance matrix. Since $\mu$ is a product measure, it is diagonal with $\Gamma_{i,i}={\mathrm{{\rm Var}}}_{\mu}(x_{i})={\mathrm{{\rm Cov}}}_{\mu}(x_{i},x_{i})$.
Since the kernels $k_{\mu_i}$ are non-negative, one can introduce the probability measures on ${\mathbb{R}\ \!\!}^{2d}$, defined for $1\leq i \leq d$ by
\[ d\mu_{(i)}^{(1)}(x,y)=\frac{1}{\Gamma_{i,i}} k_{\mu_i}(x_i,y_i) dx_i dy_i d\mu(x_{-i}) d\mu(y_{-i}). \]
If $f$ and $g$ are positive and integrable, we also introduce the following probability measures, \[ d\mu_{(i),f}^{(1)}(x,x')=\frac{1}{Z_{i,f}} f(x) k_{\mu_i}(x_i,x_i') dx_i dx'_i d\mu(x_{-i}) d\mu(x'_{-i}), \] with \[ Z_{i,f}= \iint_{x,x'} f(x)k_{\mu_i}(x_i,x'_i) dx_i dx'_i d\mu(x_{-i}) d\mu(x'_{-i}) \] and \[ d\mu_{(i),f,g}^{(1)}(x,x')=\frac{ 1}{Z_{i,f,g}} f(x) k_{\mu_i}(x_i,y_i)g( \underline{x}_{i-1}, \overline{x'}_{i}) dx_i dx'_i d\mu(x_{-i}) d\mu(x'_{-i}), \] with \[ Z_{i,f,g}= \iint_{x,x'} f(x) k_{\mu_i}(x_i,x'_i) g( \underline{x}_{i-1}, \overline{x'}_{i}) dx_i dx'_i d\mu(x_{-i}) d\mu(x'_{-i}). \]
The quantity $Z_{i,f}$ can still be written as a covariance with respect to $\mu$: $Z_{i,f}= {\mathrm{{\rm Cov}}}_\mu(F_i(x), x_i)$ where $F_i$ is a function such that $\partial_i F_i(x)= f(x)$. This is not anymore the case for $Z_{i,f,g}$.
In the case of a product measure $\mu$, Lemma \ref{thm:cov2} generalizes as follows.
\begin{lem}\label{lem:cov2-product} Let $f,g:{\mathbb{R}\ \!\!}^{d}\to{\mathbb{R}\ \!\!}$ be in $L^2(\mu)$ and coordinatewise absolutely continuous. \begin{enumerate}
\item Then, \begin{align*}
{\mathrm{{\rm Cov}}}_\mu(f,g) = &\sum_{i=1}^d {\Gamma_{i,i}} {\mathrm{{\rm Cov}}}_{\mu_{(i)}^{(1)}} (\partial_i f(x) ,\partial_i g( \underline{x}_{i-1}, \overline{x'}_{i}))\\
& +\sum_{i=1}^d
\frac{1}{\Gamma_{i,i}}
{\mathrm{{\rm Cov}}}_\mu(f(x),x_i) {\mathrm{{\rm Cov}}}_\mu(g(x),x_i).\\ \end{align*}
\item If moreover $f=e^{-\phi}$, then \begin{align*} {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x)) = &\sum_{i=1}^d Z_{i,f} {\mathrm{{\rm Cov}}}_{\mu_{(i),f}^{(1)}}(-\partial_{i} \phi(x),\partial_i g(\underline{x}_{i-1},\overline{x'}_{i}) )\\ &+\sum_{i=1}^d {\mathrm{{\rm Cov}}}_{\mu}(f(x),x_{i})\\
& \hspace{1.6cm} \times \left(\iint f(x) \frac{k_{\mu_i}^{(1)}(x_i,x_i')}{\Gamma_{i,i}} \partial_{i}g (\underline{x}_{i-1},\overline{x'}_{i})) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\right). \end{align*} In particular, if moreover $f$ is orthogonal to the linear functions $x_i$, $1\leq i\leq d$, \[ {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x)) = \sum_{i=1}^d Z_{i,f} {\mathrm{{\rm Cov}}}_{\mu_{(i),f}^{(1)}}(-\partial_{i} \phi(x),\partial_i g(\underline{x}_{i-1},\overline{x'}_{i}) ). \]
\item If $f=e^{-\phi}$ and $g=e^{-\psi}$,
\begin{align*} & {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x)) \\ = & \sum_{i=1}^d Z_{i,f,g} {\mathrm{{\rm Cov}}}_{\mu_{(i),f,g}^{(1)}}(\partial_{i} \phi(x),\partial_i \psi(\underline{x}_{i-1},\overline{x'}_{i}) )\\ &+ \sum_{i=1}^d Z_{i,f,g} \left(\iint\partial_{i}f(x)\frac{k_{\mu_i}^{(1)}(x_i,x_i')}{Z_{i,f,g} } g (\underline{x}_{i-1},\overline{x'}_{i})) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i}) \right)\\ & \hspace{1.4cm}\times \left(\iint f(x) \frac{k_{\mu_i}^{(1)}(x_i,x_i')}{Z_{i,f,g}} \partial_{i}g (\underline{x}_{i-1},\overline{x'}_{i})) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\right).\\ \end{align*} In particular, if the measure $\mu$ is symmetric and if both $f$ and $g$ are even, then \[ {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x)) = \sum_{i=1}^d Z_{i,f,g} {\mathrm{{\rm Cov}}}_{\mu_{(i),f,g}^{(1)}}(\partial_{i} \phi(x),\partial_i \psi(\underline{x}_{i-1},\overline{x'}_{i}) ). \] \end{enumerate} \end{lem}
In fact, we shall use in the sequel the following slight weighted generalization, similar to the one of Lemma \ref{thm:cov-mod-a}.
\begin{lem}\label{lem:cov2-product-mod} Let $f,g:{\mathbb{R}\ \!\!}^{d}\to{\mathbb{R}\ \!\!}$ be in $L^2(\mu)$ and coordinatewise absolutely continuous. \begin{enumerate}
\item Then, \begin{align*}
{\mathrm{{\rm Cov}}}_\mu(f,g) = &\sum_{i=1}^d {{\mathrm{{\rm Var}}}_{\mu_i}(A_i)} {\mathrm{{\rm Cov}}}_{\mu_{(i),a_i,a_i}^{(1)}} (\partial_i f(x) ,\partial_i g( \underline{x}_{i-1}, \overline{x'}_{i}))\\
& +\sum_{i=1}^d
\frac{1}{{\mathrm{{\rm Var}}}_{\mu_i}(A_i)}
{\mathrm{{\rm Cov}}}_\mu(f(x),A_i(x_i)) {\mathrm{{\rm Cov}}}_\mu(g(x),A_i(x_i)).\\ \end{align*} \item If moreover $f=e^{-\phi}$ and if $f$ is orthogonal to the functions $A_i(x_i)$, $1\leq i\leq d$, then
\[ {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x)) = \sum_{i=1}^d Z_{i,a_i f,a_i} {\mathrm{{\rm Cov}}}_{\mu_{(i),a_i f,a_i}^{(1)}}(-\partial_{i} \phi(x),\partial_i g(\underline{x}_{i-1},\overline{x'}_{i}) ). \] \item If moreover $f=e^{-\phi}$ and $g=e^{-\psi}$ and if the measure $\mu$ is symmetric, the function $a_k$ are even and both $f$ and $g$ are even, then
\[ {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x)) = \sum_{i=1}^d Z_{i,a_i f,a_i g} {\mathrm{{\rm Cov}}}_{\mu_{(i),a_i f,a_ig}^{(1)}}(\partial_{i} \phi(x),\partial_i \psi(\underline{x}_{i-1},\overline{x'}_{i}) ). \] \end{enumerate} \end{lem}
Since the other points are somehow similar, we only do the proof for the first item of Lemma \ref{lem:cov2-product}. \begin{proof}[Proof for the first item of Lemma \ref{lem:cov2-product}]
From Proposition \ref{prop:var-prod} and Lemma \ref{lem:cov-prod-ei}, one has \begin{align*}
& {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))\\ = &\sum_{i=1}^d \iint_{ x,x' \in {\mathbb{R}\ \!\!}^d} \partial_i f(x) k_{\mu_i}(x_i,x_i') \partial_i g(\underline{x}_{i-1},\overline{x'}_{i}) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\\ = & \sum_{i}\Gamma_{i,i} {\mathrm{{\rm Cov}}}_{\mu_{(i)}^{(1)}}(\partial_{i}f(x),\partial_i g(\underline{x}_{i-1},\overline{x'}_{i}) )\\ &+ \sum_{i}\Gamma_{i,i} \left(\iint\partial_{i}f(x)\frac{k_{\mu_i}^{(1)}(x_i,x_i')}{\Gamma_{i,i}} dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i}) \right)\\ & \hspace{1cm} \times \left(\iint\partial_{i}g (\underline{x}_{i-1},\overline{x'}_{i}) \frac{k_{\mu_i}^{(1)}(x_i,x_i')}{\Gamma_{i,i}} dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\right)\\ = & \sum_{i=1}^{d}\Gamma_{i,i}\;{\mathrm{{\rm Cov}}}_{\mu_{(i)}^{(1)}}(\partial_{i}f(x),\partial_{i}g(y))+\sum_{i=1}^{d}\frac{1}{\Gamma_{i,i}}\;{\mathrm{{\rm Cov}}}_{\mu}(f(x),x_{i})\,{\mathrm{{\rm Cov}}}_{\mu}(g(x),x_{i}). \end{align*}
\end{proof}
\begin{comment} Now if $f=e^{-\phi}$ \begin{align*}
& {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))\\ = &\sum_{i=1}^d Z_{i,f} \iint_{ x,x' \in {\mathbb{R}\ \!\!}^d}
- \partial_i \phi(x) \frac{f(x) k_{\mu_i}(x_i,x_i')}{Z_{i,f} } \partial_i g(\underline{x}_{i-1},\overline{x'}_{i}) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\\ = & \sum_{i} Z_{i,f} {\mathrm{{\rm Cov}}}_{\mu_{i,f}^{(1)}}(-\partial_{i} \phi(x),\partial_i g(\underline{x}_{i-1},\overline{x'}_{i}) )\\ &+ \sum_{i} Z_{i,f} \left(\iint\partial_{i}f(x)\frac{k_{\mu_i}^{(1)}(x_i,x_i')}{Z_{i,f} } dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i}) \right)\left(\iint f(x) \frac{k_{\mu_i}^{(1)}(x_i,x_i')}{\Gamma_{i,i}} \partial_{i}g (\underline{x}_{i-1},\overline{x'}_{i})) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\right)\\ = &\sum_{i=1}^d Z_{i,f} {\mathrm{{\rm Cov}}}_{\mu_{i,f}^{(1)}}(-\partial_{i} \phi(x),\partial_i g(\underline{x}_{i-1},\overline{x'}_{i}) )\\ &+\sum_{i=1}^d {\mathrm{{\rm Cov}}}_{\mu}(f(x),x_{i})\,\left(\iint f(x) \frac{k_{\mu_i}^{(1)}(x_i,x_i')}{\Gamma_{i,i}} \partial_{i}g (\underline{x}_{i-1},\overline{x'}_{i}) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\right)
\end{align*}
The result when $f$ is orthogonal to the coordinate functions $x_i$ directly follows.
Now if $f=e^{-\phi}$ and $g=e^{-\psi}$, \begin{align*}
& {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x))\\ = &\sum_{i=1}^n Z_{i,f,g} \iint_{ x,x' \in {\mathbb{R}\ \!\!}^n}
\partial_i \phi(x) \frac{f(x) k_{\mu_i}(x_i,x_i') g (\underline{x}_{i-1},\overline{x'}_{i}))}{Z_{i,f,g} } \partial_i \psi(\underline{x}_{i-1},\overline{x'}_{i}) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\\ = & \sum_{i} Z_{i,f,g} {\mathrm{{\rm Cov}}}_{\mu_{i,f,g}^{(1)}}(\partial_{i} \phi(x),\partial_i \psi(\underline{x}_{i-1},\overline{x'}_{i}) )\\ &+ \sum_{i} Z_{i,f,g} \left(\iint\partial_{i}f(x)\frac{k_{\mu_i}^{(1)}(x_i,x_i')}{Z_{i,f,g} } g (\underline{x}_{i-1},\overline{x'}_{i})) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i}) \right) \left(\iint f(x) \frac{k_{\mu_i}^{(1)}(x_i,x_i')}{Z_{i,f,g}} \partial_{i}g (\underline{x}_{i-1},\overline{x'}_{i}) dx_i dx_i' d\mu(x_{-i}) d\mu(x'_{-i})\right)\\ \end{align*}
The result when $\mu$ is symmetric and $f$ and $g$ are both even follows from Lemma \ref{lem:even}. \end{comment}
\subsection{Another proof of Theorem \ref{thm:Hu-produit-A} and a proof of Theorem \ref{thm:Harge-Royen-global}} Before we turn to the announced proofs, we highlight with the next statement that under our assumptions, the new probability measures on ${\mathbb{R}\ \!\!}^{2d}$ satisfy the Holley condition and thus the FKG inequality.
Recall that $\mu= \mu_1 \otimes \dots \otimes \mu_d$ is a product measure with marginals $\mu_k$, $k=1,\dots,d$, admitting densities, denoted by $\exp(-V_k)$, with respect to the Lebesgue measure. For some index $i\in \{1,\dots,d\}$ and for $f$ and $g$ some positive functions on ${\mathbb{R}\ \!\!}^d$, the kernel $k_{(i),f,g}$ is defined on ${\mathbb{R}\ \!\!}^{2d}$ by
\[ k_{(i),f,g}=f(x)k_{\mu_i}(x_i,y_i)g(x)\prod_{j\neq i}e^{-V_j(x_j)}\prod_{j\neq i}e^{-V_j(x'_j)}. \] The measure $\mu_{(i),f,g}^{(1)}$ has a density on ${\mathbb{R}\ \!\!}^{2d}$ equal to \[ d\mu_{(i),f,g}^{(1)}(x,x')=\frac{ 1}{Z_{i,f,g}} k_{(i),f,g}^{(1)} dx dx', \] with \[ Z_{i,f,g}= \iint_{x,x'} f(x) k_{\mu_i}(x_i,x'_i) g( \underline{x}_{i-1}, \overline{x'}_{i}) dx_i dx'_i d\mu(x_{-i}) d\mu(x'_{-i}). \]
\begin{prop}\label{prop:fkg-FKG} Let $\mu= \mu_1 \otimes \dots \otimes \mu_d$ be a product measure on ${\mathbb{R}\ \!\!}^d$ and grant the above notations. One has \begin{enumerate} \item For all $1\leq i \leq d$, the measures $\mu_{(i)}^{(1)}$ and $\mu_{(i),a_i,a_i}^{(1)}$ satisfy the Holley condition \eqref{eq:k-FKG}. Moreover, for any choice of signs $({\varepsilon\ \!\!}_1, \dots, {\varepsilon\ \!\!}_d)\in \{ +1,-1\} ^d$, the kernels $\tilde k_{(i)}$ and $\tilde k_{(i),a_i,a_i}$ defined by \[ \tilde k_{(i)} (x,x')= k_{(i)}^{(1)} ({\varepsilon\ \!\!} x, {\varepsilon\ \!\!} x') \textrm{ and } \tilde k_{(i),a_i,a_i} (x,x')= k_{(i),a_i,a_i} ({\varepsilon\ \!\!} x, {\varepsilon\ \!\!} x') \] with $k_{(i),a_i,a_i} $ the density - up to the constant factor ${\mathrm{{\rm Var}}}_{\mu_i}(A_i)$ - of the measure $\mu_{(i),a_i,a_i}^{(1)}$ with respect to the Lebesgue measure on ${\mathbb{R}\ \!\!}^{2d}$ and \[ ({\varepsilon\ \!\!} x, {\varepsilon\ \!\!} x')= ({\varepsilon\ \!\!}_1 x_1, \dots, {\varepsilon\ \!\!}_d x_d, {\varepsilon\ \!\!}_1 x'_1, \dots, {\varepsilon\ \!\!}_d x'_d), \] satisfy the Holley condition \eqref{eq:k-FKG}. \item Assume that $f=e^{-\phi}$ and that for all $1\leq i,j \leq d$ with $i\neq j$, \[ \partial_{i,j} \phi (x) \leq 0 \]
then for all $1\leq i \leq d$, the measures $\mu_{(i),f}^{(1)}$ and $\mu_{(i),a_if,a_i}^{(1)}$ satisfy the Holley condition \eqref{eq:k-FKG}.
\item Assume that $f=e^{-\phi}$ and $g=e^{-\psi}$ and that for all $1\leq i,j \leq d$ with $i\neq j$, \[ \partial_{i,j} \phi (x) \leq 0 \textrm{ and }\partial_{i,j} \psi (x) \leq 0 \] then for all $1\leq i \leq d$, the measures $\mu_{(i),f,g}^{(1)}$ and $\mu_{(i),a_i f ,a_i g}^{(1)}$ satisfiy the Holley condition \eqref{eq:k-FKG}. \end{enumerate} \end{prop} Note that in the latter proposition, the signs of the second-order cross derivatives for $\phi$ and $\psi$ should be both non-positive.
\begin{proof}
The logarithm $H_{(i),a_i,a_i}^{(1)}$ of the density of $\mu_{(i),a_i,a_i}^{(1)}$ with respect to the Lebesgue measure on ${\mathbb{R}\ \!\!}^{2d}$ is given by \[ H_{(i),a_i,a_i}^{(1)}(x,x')= \ln k_{\mu_i}(x_i,x_i') + \ln a_i(x_i)+ \ln a_i (x_i') - \sum_{j\neq i} V_j(x_j) - \sum_{j\neq i} V_j(x_j'). \] Since $k_{\mu_i}$ is a totally positive kernel on ${\mathbb{R}\ \!\!}^2$, it follows easily that $H_{(i),a_i,a_i}^{(1)}$ satisfies \eqref{eq:cond-Holley-H}.
Now for $({\varepsilon\ \!\!}_1, \dots, {\varepsilon\ \!\!}_d)\in \{ +1,-1\} ^d$ fixed, the logarithm $\tilde H_{(i),a_i,a_i}^{(1)}$ of the kernel $\tilde k_{(i),a_i,a_i}^{(1)}$ is given by:
\[ \tilde H_{(i),a_i,a_i}^{(1)}(x,x')= \ln k_{\mu_i}({\varepsilon\ \!\!}_i x_i,{\varepsilon\ \!\!}_i x_i') + \ln a_i({\varepsilon\ \!\!}_i x_i)+ \ln a_i ({\varepsilon\ \!\!}_i x_i') - \sum_{j\neq i} V_j({\varepsilon\ \!\!}_j x_j) - \sum_{j\neq i} V_j({\varepsilon\ \!\!}_j x_j'). \] Since the kernel $k_{\mu_i}({\varepsilon\ \!\!}_i x_i,{\varepsilon\ \!\!}_i x_i')$ is still totally positive on ${\mathbb{R}\ \!\!}^2$, the proof of the first point follows. We turn to the proof of the second point. The logarithm $H_{(i),a_i f, a_i}^{(1)}$ of the density of $\mu_{(i),a_i f,a_i}^{(1)}$ with respect to the Lebesgue measure on ${\mathbb{R}\ \!\!}^{2d}$ satisfies \[H_{(i),a_if, a_i}^{(1)}(x,x')= -\phi(x) + H_{(i),a_i , a_i}^{(1)}. \]
From assumption \eqref{eq:cond-l2-phi-g} and Remark \ref{rmk:bakry-michel}, the function $x\to -\phi(x)$ satisfies \eqref{eq:cond-Holley-H} on ${\mathbb{R}\ \!\!}^d$ and thus clearly the function $(x,x')\to -\phi(x)$ also satisfies \eqref{eq:cond-Holley-H} on ${\mathbb{R}\ \!\!}^{2d}$. Finally, by summation, Inequality \eqref{eq:cond-Holley-H} is also valid on ${\mathbb{R}\ \!\!}^{2d}$ for $H_{(i),a_if,a_i}^{(1)}$. The proof for the third point is similar and we omit the details. \end{proof}
We now provide another proof of Theorem \ref{thm:Hu-produit-A}.
\begin{proof}[Another proof of Theorem \ref{thm:Hu-produit-A}]
Let $f$ and $g$ be two functions on ${\mathbb{R}\ \!\!}^d$ satisfying \eqref{eq:cond-l2-fg-mod}.
We first assume that all the signs of the second derivatives in Assumption \ref{eq:cond-l2-fg-mod} are non-negative.
By Lemma \ref{lem:cov2-product-mod}, one has
\begin{align*}
& {\mathrm{{\rm Cov}}}_\mu(f,g) - \sum_{i=1}^d
\frac{1}{{\mathrm{{\rm Var}}}_{\mu_i}(A_i)}
{\mathrm{{\rm Cov}}}_\mu(f(x),A_i(x_i)) {\mathrm{{\rm Cov}}}_\mu(g(x),A_i(x_i))\\ =& \sum_{i=1}^d {{\mathrm{{\rm Var}}}_{\mu_i}(A_i)} {\mathrm{{\rm Cov}}}_{\mu_{(i),a_i,a_i}^{(1)}} \left( \frac{\partial_i f(x)}{a_i(x_i)} , \frac{\partial_i g( \underline{x}_{i-1}, \overline{x'}_{i})}{a_i(x_i')} \right).\\ \end{align*} Furthermore, by Proposition \ref{prop:fkg-FKG}(1), the measure $\mu_{(i),a_i,a_i}^{(1)}$, for $i\in \{1,\dots,d\}$, satisfies the Holley condition \eqref{eq:k-FKG}. By condition \eqref{eq:cond-l2-fg} both functions $(x,x')\to\partial_i f(x)/a_i(x_i)$ and $(x,x')\to\partial_i g( \underline{x}_{i-1}, \overline{x'}_{i})/a_i(x_i')$ are coordinate increasing on ${\mathbb{R}\ \!\!}^{2d}$ and thus, for each $1\leq i\leq d$, \[
{\mathrm{{\rm Cov}}}_{\mu_{(i),a_i,a_i}^{(1)}} \left( \frac{\partial_i f(x)}{a_i(x_i)} , \frac{\partial_i g( \underline{x}_{i-1}, \overline{x'}_{i})}{a_i(x_i')} \right) \geq 0. \] Summing these inequalities ends the proof in this specific case. In the general case, for any ${\varepsilon\ \!\!}= ({\varepsilon\ \!\!}_1, \dots, {\varepsilon\ \!\!}_d)\in \{ +1,-1\} ^d$, by the change of variable $(\tilde x, \tilde x')= ({\varepsilon\ \!\!} x,{\varepsilon\ \!\!} x')$, one has \[
{\mathrm{{\rm Cov}}}_{\mu_{(i),a_i,a_i}^{(1)}} \left( \frac{\partial_i f(x)}{a_i(x_i)} , \frac{\partial_i g( \underline{x}_{i-1}, \overline{x'}_{i})}{a_i(x_i')} \right) = {\mathrm{{\rm Cov}}}_{\tilde \mu_{(i),a_i,a_i}^{(1)}} \left( \frac{\partial_i f( {\varepsilon\ \!\!} x)}{a_i({\varepsilon\ \!\!}_i x_i)} , \frac{\partial_i g( \underline{ {\varepsilon\ \!\!} x}_{i-1}, \overline{ {\varepsilon\ \!\!} x'}_{i})}{a_i( {\varepsilon\ \!\!}_i x_i')} \right) \] and for each $1\leq i \leq d$, it is possible to find some vector ${\varepsilon\ \!\!}= ({\varepsilon\ \!\!}_1, \dots, {\varepsilon\ \!\!}_d)\in \{ +1,-1\} ^d$ such that $\frac{\partial_i f( {\varepsilon\ \!\!} x)}{a_i({\varepsilon\ \!\!}_i x_i)}$ and $ \frac{\partial_i g( \underline{ {\varepsilon\ \!\!} x}_{i-1}, \overline{ {\varepsilon\ \!\!} x'}_{i})}{a_i( {\varepsilon\ \!\!}_i x_i')} $ are both coordinate increasing. More precisely, it suffices to take ${\varepsilon\ \!\!}_j=\textrm{sign } \partial_j \left( \frac{\partial_i f}{a_i} \right)$. By Lemma \ref{lem:cov2-product-mod}(1), the measures $\tilde \mu_{(i),a_i,a_i}^{(1)}$ also satisfy the Holley condition and the result follows from the FKG inequality. \end{proof}
We turn now to the proof of Theorem \ref{cor:Harge-Royen-global}, where we add some symmetries.
\begin{proof}[Proof of Theorem \ref{cor:Harge-Royen-global}]
Let $f=e^{-\phi}$ and $g$ be two functions on ${\mathbb{R}\ \!\!}^d$ satisfying \eqref{eq:cond-ii-phi-g} and \eqref{eq:cond-ij-phi-g} and assume that $f$ is orthogonal to the functions $A_i$, $1\leq i\leq d$. By Lemma \ref{lem:cov2-product-mod}(2),
one has \[ {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x)) = \sum_{i=1}^d Z_{i,a_i f,a_i} {\mathrm{{\rm Cov}}}_{\mu_{(i),a_i f,a_i}^{(1)}}\left( -\frac{\partial_{i} \phi(x)}{a_i(x_i)},\frac{\partial_i g(\underline{x}_{i-1},\overline{x'}_{i})}{a_i(x_i')}\right) . \]
Now for each $i$, since $\phi$ satisfies \eqref{eq:cond-ij-phi-g}, by Proposition \ref{prop:fkg-FKG}(2) the measure $\mu_{(i),a_if,a_i}^{(1)}$ satisfies the Holley condition. Moreover adding condition \eqref{eq:cond-l2-phi-g} both functions $(x,x')\to \frac{\partial_{i} \phi(x)}{a_i(x_i)} $ and $(x,x')\to \frac{\partial_i g( \underline{x}_{i-1}, \overline{x'}_{i})}{a_i(x_i')}$ are coordinate increasing on ${\mathbb{R}\ \!\!}^{2d}$, and thus by the FKG inequality, for each $1\leq i\leq d$, one has: \[ {\mathrm{{\rm Cov}}}_{\mu_{(i),a_i f,a_i}^{(1)}}\left( -\frac{\partial_{i} \phi(x)}{a_i(x_i)},\frac{\partial_i g(\underline{x}_{i-1},\overline{x'}_{i})}{a_i(x_i')}\right) \geq 0. \]
Theorem \ref{cor:Harge-Royen-global}(1) thus follows. The proof of Theorem \ref{cor:Harge-Royen-global}(2) is similar, since by Lemma \ref{lem:cov2-product-mod}(3), the symmetry assumptions made on $f=e^{-\phi}$ and $g=e^{-\psi}$, give that \[ {\mathrm{{\rm Cov}}}_{\mu}(f(x),g(x)) = \sum_{i=1}^d Z_{i,a_i f,a_ig } {\mathrm{{\rm Cov}}}_{\mu_{(i),a_i f,a_i}^{(1)}}\left( \frac{\partial_{i} \phi(x)}{a_i(x_i)},\frac{\partial_i \psi(\underline{x}_{i-1},\overline{x'}_{i})}{a_i(x_i')}\right) . \] Finally, the assumptions \eqref{eq:cond-ii-phi-g} and \eqref{eq:cond-ij-phi-g} ensure that the measure $\mu_{i,f,g}^{(1)}$ satisfies the Holley condition and that the functions in the covariance are coordinate increasing, which gives the result. \end{proof}
\section{Comments on the standard semi-group interpolation} \label{sec:comments}
In this section, we explain what can be done using a standard covariance representation obtained by interpolation with the associated diffusion semi-group (see \eqref{eq:cov-rep-interpolation} below) and why we did not follow this natural approach, but rather used instead the covariance representation of Proposition \ref{prop:var-prod}.
We consider here a probability measure $\mu=e^{-V}dx$ with a smooth potential $V$. One can associate to it a diffusion semi-group with generator $L$ defined for $f$ smooth with compact support by \[Lf=\Delta f-\nabla V\cdot\nabla f.\] This diffusion operator is symmetric with respect to $\mu$: for $f,g\in\mathcal{C}_{c}^{\infty}({\mathbb{R}\ \!\!}^{d})$, \[ \int fLgd\mu=\int Lfgd\mu=-\int\nabla f\cdot\nabla gd\mu. \]
Under mild conditions on $V$, one we can define alternatively the semi-group associated to $L$ by the spectral theorem and functional calculus, or by a stochastic representation (see \cite{bakry-gentil-ledoux} for further details), \[ P_{t}f(x)=e^{tL}(f)(x)={\mathbb{E}\ \!\!}[f(X_{t}^{x})] \] for some Markov diffusion process $(X_{t}^{x})_{t\geq 0}$.
We assume moreover that the operator $ -{\mathcal{L}\ \!\!}+{\mathrm{{\rm Hess }}} V$, with ${\mathcal{L}\ \!\!}={\rm diag}(L,\dots,L)$ acting on gradients, is invertible. Note that this holds under some strong convexity of the potential $V$. In this situation, for $f,g:{\mathbb{R}\ \!\!}^{d}\to{\mathbb{R}\ \!\!}$ satisfying some integrability conditions on $f$ and $g$, one has \begin{equation} {\mathrm{{\rm Cov}}}_{\mu}(f,g)=\int_{{\mathbb{R}\ \!\!}^{d}}\nabla f(x)\cdot(-{\mathcal{L}\ \!\!}+{\mathrm{{\rm Hess }}} V)^{-1}\nabla g(x)d\mu(x)\label{eq:cov-L+V} \end{equation} and thus \[ {\mathrm{{\rm Cov}}}_{\mu}(f,g)=\iint_{{\mathbb{R}\ \!\!}^{d}\times{\mathbb{R}\ \!\!}^{d}}\nabla f(x)K(x,y)\nabla g(y)dxdy \] where $K$ is the matricial kernel (with respect to the Lebesgue measure) of the operator $(-{\mathcal{L}\ \!\!}+{\mathrm{{\rm Hess }}} V)^{-1}$. Moreover, the matricial kernel $K(x,y)$ admits the following stochastic Feynman-Kac representation, \begin{equation}
K(x,y)=e^{-V(x)}\int_{0}^{+\infty}{\mathbb{E}\ \!\!}[Y_{t,x}|X_{t}=y]p_{t}(x,y)e^{-V(y)}dy,\label{eq:K} \end{equation} where $p_{t}$ stands for the heat kernel associated to $P_{t}$ with respect to the measure $\mu$ and $Y_{t,x}$ is the matrix satisfying the following ordinary (random) differential equation, \begin{equation} \frac{d}{dt}Y_{t,x}=- Y_{t,x} {\mathrm{{\rm Hess }}} V(X_{t}^{x})\textrm{ for }t\geq0;\ Y_{0,x}=Id.\label{eq:FKY} \end{equation}
In the case of a product measure, we can write $V(x)=V_1(x_1)+\dots + V_d(x_d)$, for some real functions $V_k$. This gives the following generalization of Hoeffding's covariance identity, \begin{equation}\label{eq:cov-rep-interpolation} {\mathrm{{\rm Cov}}}_{\mu}(f,g)=\sum_{i=1}^{d}\iint_{x,y\in{\mathbb{R}\ \!\!}^{d}}\partial_{i}f(x)\, \kappa_{i}(x,y)\,\partial_{i}g(y)dxdy, \end{equation} where for each $1\leq i\leq d$ , $k_{i}:{\mathbb{R}\ \!\!}^{2d}\to{\mathbb{R}\ \!\!}_+$ is the kernel defined by \begin{equation}\label{eq:k1i}
\kappa_{i}(x,y)=\int_{t=0}^{\infty}{\mathbb{E}\ \!\!}\left[\exp\left(-\int_{0}^{t}V_{i}''(X_{s}^{x_{i},i})ds\right)|(X_{t}^{x_{i},i}=y_{i})\right]p_{t}(x,y) \, dt \, e^{-V(x)}e^{-V(y)}. \end{equation} This kernel also writes as \[ \kappa_{i}(x,y)=\int_{t=0}^{\infty} \kappa_{i,t} \, dt \] with \[
\kappa_{i,t}(x,y):= p_{t,i}^{V_i''}(x_i,y_i) \prod_{j=1, j\neq i}^{d}p_{t,j}(x_{j},y_{j}) e^{-V(x)}e^{-V(y)}, \] where $p_{t,j}$ is the kernel of the one dimensional diffusion semi-group with generator given by $L_j f(x_j):=f''(x_j) - V_j'(x_j) f'(x_j)$ and where $p_{t,i}^{V_i''}$ is the kernel of the one dimensional Schrödinger semi-group, with generator given by $L_i^{V_i''} f(x_i):=L_if(x_i) + V_i''(x_i) f(x_i).$ In the case of the standard Gaussian measure, one has $p_{t,i}^{V_i''}=e^{-t}p_{t,i}$.
We highlight that we do not know whether, in dimension $d\geq 2$, the probability measure with density proportional to $\kappa_{i}(x,y)$ satisfies the full FKG inequality on ${\mathbb{R}\ \!\!}^{2d}$. But one can also notice, that due to the \emph{coincidence formula}, diffusion kernels and Schrödinger kernels in dimension one are totally positive (see Karlin \cite{karlin:book}). As a consequence, the kernels $\kappa_{i,t}$ satisfy the Holley condition. And if slightly differently, one has \[ \kappa_{i}(x,y)=\int_{t=0}^{\infty} \kappa_{i,t} \, d\nu(t) \] for some probability measure $\nu$ on ${\mathbb{R}\ \!\!}_+$, one can use the following decomposition of the covariance,
\begin{equation}\label{eq:cov-mixture} {\mathrm{{\rm Cov}}}_{\mu_{\kappa_i}}(u,v)=\int_{0}^{\infty}{\mathrm{{\rm Cov}}}_{\mu_{\kappa_{i,t}}}(u,v)d\nu(t)+{\mathrm{{\rm Cov}}}_{\nu}\left(t\to\int u \, d\mu_{\kappa_{i,t}},t\to\int v \,d\mu_{\kappa_{i,t}}\right) \end{equation} and apply it with $u(x,y):=\partial_i f(x) $ and $v(x,y):=\partial_i g(y)$.
In view of proving Theorem \ref{thm:Hu-produit-A}, we were only able to pursue this approach when the marginals of $\mu_{\kappa_{i,t}}$ on ${\mathbb{R}\ \!\!}^d \times {\mathbb{R}\ \!\!}^d$ are constant for all $t>0$. In this situation,
if moreover, $(x,y)\to u(x)$ and $(x,y)\to v(y)$, the term related to ${\mathrm{{\rm Cov}}}_\nu$ appearing in the right-hand side of \ref{eq:cov-mixture} indeed vanishes and one obtains some \emph{partial FKG inequalities} for the measure $\mu_{\kappa_i}$ on ${\mathbb{R}\ \!\!}^{2d}$. Here, the terms ``partial'' means that it is applied only to coordinate increasing functions of the form $(x,y)\to u(x)$ and $(x,y)\to v(y)$.
This property that the marginals $\mu_{\kappa_{i,t}}$ are constant, holds for the standard Gaussian measure and this approach may be pursued, with a second order covariance representation, to recover Theorem \ref{thm:harge}(1) for the standard Gaussian measure. Let us give some details.
First, the first order representation \eqref{eq:cov-rep-interpolation} is well known for the standard Gaussian measure (see \cite{bobkov-gotze-houdre}). The measures $\gamma_{\kappa_{i,t}}$ are in fact independent of $i$, they are also Gaussian measures on ${\mathbb{R}\ \!\!}^{2d}$ and they have fixed marginals on ${\mathbb{R}\ \!\!}^d \times {\mathbb{R}\ \!\!}^d$. A second order covariance representation for $\gamma$ thus means a first order covariance of the new measure(s) $\gamma_{\kappa_{i,t}}$ similar to \eqref{eq:cov-rep-interpolation}.
It can be obtained either by a change of variable since $\gamma_{\kappa_{i,t}}$ is still a Gaussian measure or by solving explicitly the stochastic Feynman-Kac representation. This method is similar to the one of \cite{hu-chaos}, except that the latter approach specifically uses the fact that the Gaussian measure is the density at time $1$ of the classical heat semi-group, whereas instead we use here the Orstein-Uhlenbeck operator.
Finally, for general product measures, the ``constant marginal property'' also holds for the modified kernels $k_{\mu_i,(a_i,a_i)}$, with the choice $a_i(x_i)=\frac{1}{g_i'(x_i)}$ where $g_i$ is (if it exists) the first non-trivial eigenfunction associated to $L$. This leads to Theorem \ref{thm:Hu-produit-A}, but only for this specific choice. More importantly, this constant marginal property is valid for product measures under the symmetry assumptions of Theorem \ref{thm:Harge-Royen-global}(2) and this route may also be taken to provide another proof Theorem \ref{thm:Harge-Royen-global}(2).
\section{Examples}\label{sec:examples} In this final section, we provide a couple of examples of possible applications of our results.
First let $\mu$ be a product measure on ${\mathbb{R}\ \!\!}^d$ and for $\beta >0$ and consider the free energy, also known in the optimization community as the ``soft max'' function, \[ F_\beta(x):= \frac{1}{\beta} \ln\left( \sum_{i=1}^d e^{\beta x_i} \right). \] By setting $p_i:= \frac {e^{\beta x_i}} { \sum_j e^{\beta x_j} }$, it satisfies \[ \partial_i F_\beta = p_i \geq 0, \] \[ \partial_{ii} F_\beta(x)= \beta p_i(1-p_i)\geq 0 , \quad \partial_{ij} F_\beta(x)= -\beta p_i p_j \leq 0,\quad i\neq j. \] Thus, for any $\alpha,\beta>0$, Corollary \ref{cor:hu-produit} gives \begin{equation}\label{cov-soft-max} {\mathrm{{\rm Cov}}}_\mu(F_\alpha, F_\beta) \geq \sum_{i=1}^d \frac{1} {{\mathrm{{\rm Var}}}(\mu_i)} {\mathrm{{\rm Cov}}}(F_\alpha(x),x_i) {\mathrm{{\rm Cov}}}(F_\beta(x),x_i). \end{equation} Note that when $\alpha=\beta$, inequality \eqref{cov-soft-max} turns to the following Bessel inequality, \[
{\mathrm{{\rm Var}}}_\mu(F_\beta) \geq \sum_{i=1}^d \frac{1} {{\mathrm{{\rm Var}}}(\mu_i)} {\mathrm{{\rm Cov}}}(F_\beta(x),x_i)^2.
\]
\
We now turn to a second example. Let $\mu $ be a symmetric product measure on ${\mathbb{R}\ \!\!}^d$. Under some integrability condition, for $J\geq 0$, we consider the probability measure: \[ d\mu_J(x)= \frac{1}{Z_J} e^{J \sum_{i=1}^{d-1} x_i x_{i +1} } d\mu(x), \quad Z_J= \int_{{\mathbb{R}\ \!\!}^d}e^{J \sum_{i=1}^{d-1} x_i x_{i +1} } d\mu(x). \]
Let $\theta=(\theta_1,\dots,\theta_d)\in{\mathbb{R}\ \!\!}^d$ with $\theta_i \geq 0$, $1\leq i \leq d$, then by Corollary \ref{cor:Harge-Royen-global}, one has \begin{equation}
\int_{{\mathbb{R}\ \!\!}^d} \langle x,\theta \rangle^2 d \mu_J(x) \geq \int_{{\mathbb{R}\ \!\!}^d} \langle x,\theta \rangle^2 d \mu(x). \end{equation}
\section{Appendix} We consider here some product probability measures on ${\mathbb{R}\ \!\!}^d$ whose marginals are mixtures of centered Gaussian variables. This class of probability measures was investigated in \cite{Eskenazis}, where the authors prove that they satisfy \eqref{eq-harge-gaussienne-niv3} and provide interesting examples. Here we show that those measures also satisfy \eqref{eq-harge-gaussienne-niv2}.
We consider Gaussian mixtures of the form
\begin{equation}\label{def-mixture} \mu=\iint_{ \sigma\in (0,\infty)^d} \gamma_{\Gamma_\sigma} d\nu(\sigma),
\end{equation} where $\gamma_{\Gamma_\sigma}$ is the centered Gaussian random vector in ${\mathbb{R}\ \!\!}^d$ with covariance matrix $\Gamma_\sigma= diag(\sigma_1^2, \dots,\sigma_d^2)$ and where $\nu$ is also a product measure on $(0,\infty)^d$.
\begin{theo}\label{thm:mixture-Harge} Let $\mu$ be a product probability measure on ${\mathbb{R}\ \!\!}^d$, whose marginals are mixture of centered Gaussian variables. Then \eqref{eq-harge-gaussienne-niv2} holds. \end{theo}
The proof relies on the following Lemma, the key point of which being that no symmetry assumption is required in the convex situation. \begin{lem}\label{lem:coord-increase} The following points hold. \begin{enumerate}
\item Let $g$ be a convex function on ${\mathbb{R}\ \!\!}^d$, then the function \[ (\sigma_1, \dots,\sigma_d)\in (0,\infty)^d \to \int g(y) d\gamma_{\Gamma_\sigma}(y)
\] is coordinatewise increasing on $(0,\infty)^d$. \item Let $f$ be a quasi-concave and even function on ${\mathbb{R}\ \!\!}^d$, then the function \[ (\sigma_1, \dots,\sigma_d)\in (0,\infty)^d \to \int f(y) d\gamma_{\Gamma_\sigma}(y)
\] is coordinatewise decreasing on $(0,\infty)^d$. \end{enumerate} \end{lem}
\begin{proof}[Proof of Theorem \ref{thm:mixture-Harge}] Let $f$ be a log-concave and even function and let $g$ be a convex function. Using the decomposition of the covariance \eqref{eq:cov-mixture}, one has \begin{align*} {\mathrm{{\rm Cov}}}_\mu (f,g)&= \iint_{(0,\infty)^d} {\mathrm{{\rm Cov}}}_{\gamma_{\Gamma_\sigma}}(f,g) d\nu(\sigma) \\
& \quad + {\mathrm{{\rm Cov}}}_\nu \left( \sigma\in (0,\infty)^d \to \int f d\gamma_{\Gamma_\sigma} , \sigma\in (0,\infty)^d \to \int g d\gamma_{\Gamma_\sigma} \right). \end{align*} The rest of the proof consists in showing that the two terms in the right-hand side of the latter inequality are non-positive. Firstly, Hargé's result \eqref{eq-harge-gaussienne-niv2} also applies to any (centered) Gaussian distribution (see \cite{harge}) and thus ${\mathrm{{\rm Cov}}}_{\gamma_{\Gamma_\sigma}}(f,g)\leq 0$. Secondly, we use Lemma \ref{lem:coord-increase}, since $f$ is log-concave and even, it is also quasi-concave and even, and thus the two functions \[ \sigma\in (0,\infty)^d \to \int g d\gamma_{\Gamma_\sigma} \textrm{ and }\sigma\in (0,\infty)^d \to \int f d\gamma_{\Gamma_\sigma} \] are respectively coordinatewise increasing and coordinatewise decreasing on $(0,\infty)^2$. The measure $\nu$ being a product measure, by the FKG inequality for product measure, the term ${\mathrm{{\rm Cov}}}_\nu (\cdot,\cdot)$ is non-positive and the result follows. \end{proof} We turn now to the proof of Lemma \ref{lem:coord-increase}. \begin{proof}[Proof of Lemma \ref{lem:coord-increase}] Let $g$ be a convex function on ${\mathbb{R}\ \!\!}^d$. By a change of variable, one directly has \[ \int g(y) d\gamma_{\Gamma_\sigma}(y)= \int_{{\mathbb{R}\ \!\!}^d} g(\sigma_1 x_1, \dots,\sigma_d x_d) d\gamma(x), \]
where we recall that $\gamma$ is the standard Gaussian distribution. To prove the desired property, we compute for $1\leq l\leq d$,
\begin{align*} \frac{\partial}{\partial \sigma_l} \int_{{\mathbb{R}\ \!\!}^d} g(\sigma_1 x_1, \dots,\sigma_d x_d) d\gamma(x) &= \int_{{\mathbb{R}\ \!\!}^d} x_l \, \partial_l g(\sigma_1 x_1, \dots,\sigma_d x_d) d\gamma(x)\\ & = {\mathrm{{\rm Cov}}}_{\gamma} ( x_l, \partial_l g(\sigma_1 x_1, \dots,\sigma_d x_d)). \end{align*} Furthermore, by the covariance representation \eqref{eq:cov-rep-interpolation} for the standard Gaussian measure, one has \begin{align*} {\mathrm{{\rm Cov}}}_{\gamma} ( x_l, \partial_l g(\sigma_1 x_1, \dots,\sigma_d x_d)) = \iint_{x,y\in {\mathbb{R}\ \!\!}^d} \kappa(x,y) \sigma_l \partial_{ll} g(\sigma_1 y_1, \dots,\sigma_d y_d) dx dy \end{align*} and this quantity is non-negative since $g$ is convex and $\kappa(x,y)\geq 0$. The result follows.\\ For $f$ quasi-concave and even, we use the layer cake representation of $f$:
\[
f(x)= \int_0^\infty {\mathbf{1}}_{A_t}(x) dt \textrm{ and } A_t:= \{x\in {\mathbb{R}\ \!\!}, f(x)\geq t \}. \]
Here by assumption the $A_t$ are convex and even. Since by Fubini, \[ \int_{{\mathbb{R}\ \!\!}^d} f(\sigma_1 x_1, \dots,\sigma_d x_d) d\gamma(x) = \int_0 ^\infty \int_{{\mathbb{R}\ \!\!}^d} \mathbf 1_{A_t} (\sigma_1 x_1, \dots,\sigma_d x_d) d\gamma(x) dt. \] the result follows from \cite{Eskenazis} where the authors prove the following property: for each $t\geq 0$, \[ (\sigma_1, \dots,\sigma_d)\in (0,\infty)^d \to \int_{{\mathbb{R}\ \!\!}^d} \mathbf 1_{A_t} (\sigma_1 x_1, \dots,\sigma_d x_d) d\gamma(x) \] is coordinatewise decreasing. \end{proof}
\end{document} | arXiv |
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Some two thousand years ago in ancient Greece, philosophers Aristotle and Zeno asked some interesting thought-provoking questions, including a case of what are today known as Zeno's paradoxes. The most famous example is a race, known as Achilles and the tortoise. The setup is as follows:
Achilles and a tortoise are having a race, but (in the spirit of fairness) the tortoise is given a headstart. Then Achilles will definitely lose: he can never overtake the tortoise, since he must first reach the point where the tortoise started, so the tortoise must always hold a lead.
Since there are virtually infinitely many such points to be crossed, Achilles should not be able to reach the tortoise in finite time.
This argument is obviously flawed, and to see that we consider the point of view of the tortoise. From the tortoise's perspective, the problem is equivalent to just Achilles heading towards it at the speed equal to the difference between their speeds in the first version of the problem.
Since $\text{distance} = \text{speed}\times\text{time}$, we can say that after time $t$, Achilles has travelled a distance equal to $v_At$ and the tortoise $v_Tt$. The distance between them is
\begin{equation*}
D-v_At+v_Tt = D – (v_A-v_T)t,
\end{equation*}
and so Achilles catches the tortoise—ie the distance between them is $0$—when the time, $t$, is equal to $D/(v_A-v_T)$.
There is another way to see this problem that satisfies better the purpose of this article and directly tackles the problem Aristotle posed. To get to where the tortoise was at the start of the race, Achilles is going to travel the distance $D$ in time $t_1 = D/v_A$. By that time the tortoise will have travelled a distance equal to
D_1 = v_T t_1,
which is the new distance between them.
Travelling this distance will take Achilles time
t_2 = \frac{D_1}{v_A} = \left(\frac{v_T}{v_A}\right)t_1.
Then the tortoise will have travelled a distance
D_2 = v_Tt_2 = \left(\frac{v_T^2}{v_A}\right)t_1
and Achilles will cover this distance after time $t_3 = D_2/v_A = (v_T/v_A)^2t_1$.
Repeating this process $k$ times we notice that the distance between Achilles and the tortoise is
D_k &= v_T\left(\frac{v_T}{v_A}\right)^{k-1}t_1 \\ &= \left(\frac{v_T}{v_A}\right)^kD.
Summing up all these distances we get how far Achilles has to move before catching the tortoise: if we call this $D_A$ it's
D_A =
\lim_{n\to\infty}\sum_{k=0}^{n}D_k =\sum_{k=0}^{\infty}D_k = D \sum_{k=0}^{\infty}\left(\frac{v_T}{v_A}\right)^k.
This is probably the simplest example of an infinite convergent sum. In particular, this is the simplest example of a class of sums called geometric series, which are sums of the form
\sum_{k=0}^{n}a^k.
If $|a| < 1$, the sum tends to $(1-a)^{-1}$ as $n$ tends to $\infty$ and diverges otherwise, meaning that it either goes to $\pm\infty$, or a limiting value just doesn't exist. By 'doesn't exist', see for example what happens if $a=-1$: we get \begin{equation*} 1 - 1 + 1 - 1 + 1 - 1 + \cdots + (-1)^n \end{equation*} and the sum oscillates between $0$ (if $n$ is odd) and $1$ (if $n$ is even). In our case, $|a|=|v_T/v_A|<1$ so \begin{equation*} D_A = \frac{D}{1-v_T/v_A} = \frac{D v_A}{v_A-v_T}, \end{equation*} which, when divided by the speed of Achilles, $v_A$, gives exactly the time we found before. So Achilles and the tortoise will meet after Achilles has crossed a distance $D_A$ in time $t_A = D_A/v_A$. Thousands of years later, Leonhard Euler was thinking about evaluating the limit of \begin{equation*} \sum_{k=1}^{n}\frac{1}{k^2} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots + \frac{1}{n^2} \end{equation*} as $n\to\infty$, which is named as the Basel problem after Euler's hometown. This sum is convergent and it equals $\mathrm{\pi}^2/6$, as Euler ended up proving in 1734. He was one of the first people to study formal sums—which we will try to define shortly—and concretely develop the related theory. In his 1760 work De seriebus divergentibus he says
Whenever an infinite series is obtained as the development of some closed
expression, it may be used in mathematical operations as the equivalent of
that expression, even for values of the variable for which the series diverges.
So let's think about series which diverge. One way a series can diverge is simply by its terms getting bigger. One such example is the sum
\sum_{k=1}^{n}k = 1 + 2 + 3 + 4 + \cdots + n = \frac{n(n+1)}{2},
the limit of which when $n\to\infty$ is, of course, infinite.
But now let's think about the harmonic series,
\sum_{k=1}^{n}\frac{1}{k} = 1 + \frac12 + \frac13 + \frac14 + \cdots + \frac1n.
This time, although the terms themselves get smaller and smaller, the series still diverges as $n\to\infty$. But we can still describe the sum and its behaviour. It turns out that
\sum_{k=1}^{n}\frac1k = \ln(n)+\gamma + O(1/n),
where $\gamma$ is the Euler–Mascheroni constant, which approximately equals 0.5772, and $O(1/n)$ means 'something no greater than a constant times $1/n$'. You can see the sum and its approximation here:
Historically, the development of the seemingly unconventional theory of divergent sums has been debatable, with Abel, who at some point made contributions to the area, once describing them as shameful, calling them "an invention of the devil". Later contributions include works of Ramanujan and Hardy in the 20th century, about which more information can be found in the latter's book, Divergent Series.
More recently, a video on the YouTube channel Numberphile was published, and attempted to deduce the 'equality'
1+2+3+4+\cdots=-\frac{1}{12}.
This video sparked great controversy, and indicates one of the dangers of dealing with divergent sums. One culprit here is the Riemann zeta function, which is defined for $\operatorname{Re}(s)>1$ as
\zeta(s)=\sum_{k=1}^{\infty}\frac{1}{k^s}.
When functions are only defined on certain domains, it is sometimes possible to 'analytically continue' them outside of these original domains. Specifically at $-1$, doing so here gives $\zeta(-1)=-1/12$. The other culprit here is matrix summation—another method to give some value to divergent sums. By sheer (though neat) coincidence, these methods, such as the Cesáro summation method they use in the video, also give $-1/12$!
The main problem is this: at this point we no longer have an actual sum in the traditional sense.
Instead, we have a divergent sum which is formal, and by that, we mean that it is a symbol that denotes the addition of some quantities, regardless of whether it is convergent or not: it simply has the form of a sum.
These sums are not just naive mathematical inventions, instead, they show up in science and technology quite frequently and they can give us good approximations as they often emerge from standard manipulations, such as (as we'll see) integration by parts.
Applications in physics can be found in the areas of quantum field theory and quantum electrodynamics. In fact, formal series derived from perturbation theory can give very accurate measurements of physical phenomena like the Stark effect and the Zeeman effect, which characterise changes in the spectral lines of atoms under the influence of an external magnetic and electric field respectively.
In 1952, Freeman Dyson gave an interesting physical explanation of the divergence of formal series in quantum electrodynamics, explaining it via the stability of the physical system versus the spontaneous, explosive birth of particles in a scenario where the corresponding series that describes it is convergent. Essentially he argues that divergence is, in some sense, inherent in these types of systems otherwise we would have systems in pathological states. His paper from that year in Physical Review contains more information.
Euler's motivation
Sometimes, such assignments of formal sums to finite values (constants or functions) can be useful. The fact that they sometimes diverge does not make much difference in the end, if certain conditions are met.
An example that follows Euler's line of thought as described earlier emerges when trying to find an explicit formula for the function
\operatorname{Ei}(x):=\int_{-\infty}^{x}\frac{\mathrm{e}^t}{t}\, \mathrm{d} t,
for which repeated integration by parts yields
\operatorname{Ei}(x) = \int_{-\infty}^x\frac{\mathrm{e}^t}{t}\, \mathrm{d} t =& \left[\frac{\mathrm{e}^t}{t}\right]_{-\infty}^{x}-\int_{-\infty}^{x}\mathrm{e}^{t}\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{t}\right) \mathrm{d} t \\
=& \left[\frac{\mathrm{e}^x}{x} – 0\right] + \int_{-\infty}^{x}\frac{\mathrm{e}^{t}}{t^2} \mathrm{d} t \\
= \cdots =& \frac{\mathrm{e}^{x}}{x}\sum_{k=0}^{n-1}\frac{k!}{x^{k}}+O\left(\frac{\mathrm{e}^x}{x^{n+1}}\right),
where we've been able to say $\mathrm{e}^x/x\to 0$ on the second line as $x\to-\infty$. Dividing through by $\mathrm{e}^x$, this allows us to say
\mathrm{e}^{-x}\operatorname{Ei}(x) &= \sum_{k=0}^{n-1}\frac{k!}{x^{k+1}}+O\left(\frac{1}{x^{n+1}}\right)\\
&\sim\sum_{k=0}^{\infty}\frac{k!}{x^{k+1}}\text{ as }x\to\infty.
Now swap $x$ for $-1/x$ in this equation:
\mathrm{e}^{1/x}\operatorname{Ei}(-1/x)&\sim\sum_{k=0}^{\infty}k!(-x)^{k+1}\text{ as }x\to0\\
&=-x+x^2-2x^3+6x^4+\cdots.
As you can see below, this series now diverges as $x\to\infty$, but we still see convergence of the partial (truncated) sums as $x\to0$, even as we add more terms:
Euler noticed that $\mathrm{e}^{1/x}\operatorname{Ei}(-1/x)$, in its original integral form, solves the equation
x^2\frac{\mathrm{d}y}{\mathrm{d}x}+y=-x
(for $x\neq0$). Now here's the thing: the formal sum
\sum_{k=0}^{\infty}k!(-x)^{k+1},
to which $\mathrm{e}^{1/x}\operatorname{Ei}(-1/x)$ is asymptotic as $x\to0$, also (formally) 'solves' the same equation for any $x$.
This solution is not unique, and in fact, adding any constant multiple of $\mathrm{e}^{1/x}$ to $\mathrm{e}^{1/x}\operatorname{Ei}(-1/x)$ would still solve the equation; and the resulting solution would still be asymptotic to the same formal sum.
However, the coefficients of the powers of $x$ are unique. So there may be something in the formal sum that can give away the actual solution of the equation (which is often difficult to find via standard methods—unlike formal solutions that are easier to compute like the one above), at least up to some class of solutions and under certain conditions. In fact, this seems to actually be the case, at least for certain classes of formal sums—the ones that attain 'at most' factorial over power rate of divergence.
Solving a differential equation
To elaborate further, let's consider one more example, the differential equation
-\frac{\mathrm{d}y}{\mathrm{d}x}+y = \frac{1}{x}, \quad \text{where } y(x)\to 0 \text{ as }x \to \infty.
\label{fs1}
\tag{*}
Thinking about the boundary condition there, we could substitute in the (formal) sum of powers of $x$ which decay away as $x\to\infty$,
y(x) = \sum_{k=0}^\infty a_k x^{-k-1} = \frac{a_0}{x} + \frac{a_1}{x^2} + \frac{a_2}{x^3} + \cdots.
Doing so, we get
-\frac{\mathrm{d}}{\mathrm{d}x}\left[\sum_{k=0}^{\infty}a_kx^{-k-1}\right]+\sum_{k=0}^{\infty}a_kx^{-k-1} &= \frac{1}{x}
\\ \implies \sum_{k=0}^{\infty}(k+1)a_kx^{-k-2}+\sum_{k=0}^{\infty}a_kx^{-k-1} &=
\frac{1}{x} \\
\implies a_0x^{-1}+\sum_{k=0}^{\infty}\big[(k+1)a_k+a_{k+1}\big] x^{-k-2} &=\frac{1}{x}.
Then for our differential equation to be satisfied the coefficients have to satisfy
a_0=1 \qquad \text{and}
(k+1)a_k+a_{k+1}=0\implies a_{k+1} = -(k+1)a_k
which recursively means that $a_k=(-1)^kk!$ and our formal sum solution is
y(x) = \sum_{k=0}^{\infty}(-1)^k k!x^{-k-1}.
Now that we have a sum that solves the equation formally, we can obtain an actual solution assuming that it is asymptotic to the sum we found as $x\to\infty$ by using the repeated integration by parts result
\int_{0}^{\infty}\mathrm{e}^{-xs} s^k \,\mathrm{d} s = k! x^{-k-1} \text{ for }x>0,
which implies
y(x) &= \sum_{k=0}^{\infty}(-1)^{k}k!x^{-k-1} \\
&= \sum_{k=0}^{\infty}(-1)^k\int_{0}^{\infty}\mathrm{e}^{-xs}s^k \,\mathrm{d} s \\
&= \int_{0}^{\infty}\mathrm{e}^{-xs}\sum_{k=0}^{\infty}(-1)^ks^k \,\mathrm{d} s.
How is that helpful? Well, for $s:|s|<1$ we know that \begin{equation*} \sum_{k=0}^{\infty}(-1)^ks^{k} = 1-s+s^2+\cdots = \frac{1}{1+s}, \end{equation*} by the formula for geometric series for $a=-s$ from our discussion of Achilles and the tortoise. This is a nice function on the real line, having all the fine properties that we need in order to define \begin{equation*} y(x)=\int_{0}^{\infty}\frac{\mathrm{e}^{-xs}}{1+s}\,\mathrm{d}s, \end{equation*} which is the solution to our differential equation, \eqref{fs1}, we are looking for, and is also asymptotic to the formal sum $\sum_{k=0}^{\infty}(-1)^k k!x^{-k-1}$ as $x\to\infty$:
Notice that any linear combination of these formal sums will result from the same linear combination of the respective convergent (for $s:|s|<1$) series $1-s+s^2-s^3+\cdots$ inside the integral. In conclusion, it is possible to obtain a solution in closed form to a differential equation just by finding a formal power series to which the solution is asymptotic.
Not just reinventing the wheel
The aforementioned example is, of course, quite simple and trying to find a solution in the way we just described might look like we're reinventing the wheel using modern-era technology. However, the true potential of the method described above can be seen in nonlinear equations, to which we generally cannot find solutions in standard ways. In my own research I used formal sums to study an equation with applications in fluid mechanics.
In one of the first talks I gave about this topic, I remember noticing several of my peers tilting their heads in distrust when I mentioned that the emerging sums are divergent. This reaction was almost expected and for obvious reasons. It took an hour-long talk and several questions later to convince them that the mathematics involved is genuine.
Controversial as it may sound, at first sight, this concept is even more realistic than imaginary numbers, which are simply symbols with properties that we just accept and use. The idea is that, although imaginary, these numbers can demonstrably give us, when interpreted properly, very real results such as solutions to differential equations like
\frac{\mathrm{d}^2y}{\mathrm{d}x^2}+y=0.
The same is true for formal sums too.
Why do we assign actual numbers to formal sums in the first place? Because they are sometimes easier to work with and can lead to interesting results (such as solutions to differential equations) if interpreted properly. The underlying mechanisms should be well-defined mathematical processes and well-understood in order to avoid any serious mistakes when working with such sums. An example of erroneous use of such sums is Henri Poincaré's attempt to solve the three-body problem in order to win the King Oscar prize in 1889. He managed, however, in the next decade to spark the development of chaos theory. But that's for another time.
Nik Alexandrakis is a fourth-year PhD student at Lancaster University. Apart from differential equations, he is sometimes interested in environmental and animal rights activism. He hates writing but usually likes the outcome.
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Chalkdust is published by Chalkdust Magazine, UCL, Gower Street, London WC1E 6BT, United Kingdom. ISSN 2059-3805 (Print). ISSN 2059-3813 (Online). | CommonCrawl |
József Balogh (mathematician)
József Balogh is a Hungarian-American mathematician, specializing in graph theory and combinatorics.
József Balogh
BornDecember 1971 (age 51)
Hungary
Scientific career
FieldsMathematics
Combinatorics
Graph theory
InstitutionsUniversity of Illinois at Urbana–Champaign
Doctoral advisorBéla Bollobás
Doctoral students
• Wojciech Samotij
Websitesites.google.com/view/jozsefbaloghmath
Education and career
Balogh grew up in Mórahalom and attended secondary school in Szeged at Ságvári Endre Gyakorló Gimnázium (a special school for mathematics).[1] As a student, he won two silver medals (in 1989 and 1990) at the International Mathematical Olympiad. He studied at the University of Szeged (with one year TEMPUS grant at the University of Ghent), where he received his M.S, in mathematics in 1995 with advisor Péter Hajnal and thesis On the existence of MDS-cyclic codes. In 2001 Balogh received his doctorate from the University of Memphis with advisor Béla Bollobás and thesis Graph properties and Bootstrap percolation. As a postdoc Balogh was at AT&T Shannon Laboratories in Florham Park, New Jersey and for several months in 2002 at the Institute for Advanced Study. From 2002 to 2005 he was Zassenhaus Assistant Professor at Ohio State University. At the University of Illinois at Urbana–Champaign he was an assistant professor from 2005 to 2010 and an associate professor from 2010 to 2013 and is since 2013 a full professor. From 2009 to 2011 he was also an associate professor at University of California, San Diego.[2]
Balogh's research deals with extremal and probabilistic combinatorics (especially graph theory) and bootstrap percolation.[1] The latter models the spread of an infection on a d-dimensional grid, whereby nodes are infected in each time step in which at least r neighbors have already been infected. It is based on a randomly chosen starting structure and Bollobás, Balogh, Hugo Duminil-Copin and Robert Morris proved an asymptotic (for large grids) formula for the threshold probability that the whole grid is infected, depending on d and r. He had previously treated the three-dimensional case with r = 3 with Bollobás and Morris.
Recognition
In 2007 he received an NSF Career Grant.[2] In 2013/14 and 2020 he was a Simons Fellow, in 2013/14 Marie Curie Fellow. In 2016 he received the George Pólya Prize in combinatorics with Robert Morris and Wojciech Samotij.[3] In 2018 Balogh was an invited speaker at the International Congress of Mathematicians in Rio de Janeiro.
He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to extremal combinatorics, probability and additive number theory, and for graduate mentoring".[4]
Selected publications
• with Noga Alon, Peter Keevash, Benny Sudakov: The number of edge colorings with no monochromatic cliques, J. London Math. Soc., vol. 70, 2004, pp. 273–288. pdf
• with B. Bollobas, Robert Morris: Bootstrap percolation in three dimensions. Annals of Probability, vol. 37, 2009, pp. 1329–1380. Arxiv
• with Wojtek Samotij: The number of $K_{s,t}$-free graphs, J. Lond. Math. Soc., vol. 83, 2011, pp. 368–388, Abstract
• with John Lenz: Some Exact Ramsey-Turan Numbers, Bull. Lond. Math. Soc., vol. 44, 2012, pp. 1251–1258. Arxiv
• with Bela Bollobas, Hugo Duminil-Copin, R. Morris: The sharp threshold for bootstrap percolation in all dimensions, Trans. Amer. Math. Soc., vol. 364 2012, pp. 2667–2701. Arxiv
• with N. Alon, R. Morris, W. Samotij: A refinement of the Cameron-Erdös Conjecture, Proc. London Mathematical Society, vol. 108, 2014, pp. 44–72. Arxiv
• with Sarka Petrickova: The number of the maximal triangle-free graphs, Bull. London Math. Soc., vol. 46, 2014, pp. 1003–1006. Arxiv
• with Morris, Samotij: Independent sets in hypergraphs, J. AMS, vol. 28, 2015, pp. 669–709, Arxiv 2012
• with Hong Liu, Maryam Sharifzadeh, Andrew Treglown: The number of maximal sum-free subsets of integers, Proc. AMS, vol. 143, 2015, pp. 4713–4721, Arxiv 2014
• with J. Solymosi, On the number of points in general position in the plane, Discrete Analysis (2018), Paper No. 16, 20 pp.
• with R. Morris, W. Samotij, L. Warnke: The typical structure of sparse $K_{r+1}$-free graphs., Transactions AMS, 368 (2016) 6439–6485.Arxiv 2013
References
1. "28th Cumberland Conference on Combinatorics, Graph Theory & Computing Speaker, Jozsef Balogh (Plenary)". Interdisciplinary Mathematics Institute, University of South Carolina, Cumberland, SC, May 15–17, 2015.
2. "Curriculum Vitae, Jozsef Balogh" (PDF). Department of Mathematical Sciences, University of Illinois.
3. "SIAM: George Pólya Prize in Combinatorics". siam.org.
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Chinese Physics C> Just Accepted> Article
Energy staggering parameters in nuclear magnetic rotational bands
Wu-Ji Sun 1 ,
Jian Li 1,,
College of Physics, Jilin University, Changchun 130012, China
The systematics of energy staggering for the magnetic rotational bands with $ M1 $ and $ E2 $ transition properties strictly consistent with the features of good candidates of magnetic rotational bands in the $ A\sim80 $ , 110, 130 and 190 mass regions are presented. The regularities exhibited by these bands concerning the staggering parameter which increases with increasing spin are in agreement with the semiclassical description of shears mechanism. In addition, the abnormal behaviours in the backbend regions or close to band termination have also been discussed. Taking the magnetic dipole bands with same configuration in three $ N = 58 $ isotones, i.e., $ ^{103} {\rm Rh}$ , $ ^{105} {\rm Ag}$ , and $ ^{107} {\rm In}$ , as examples, the transition from chiral rotation to magnetic rotation with the proton number approaching $ Z = 50 $ is presented. Moreover, the self-consistent tilted axis cranking and principle axis cranking relativistic mean-field theories are applied to investigate the rotational mechanism in dipole band of $ ^{105} {\rm Ag}$ .
energy staggering ,
magnetic rotational band ,
shears mechanism ,
cranking relativistic mean-field theory
[1] S. Frauendorf, J. Meng, and J. Reif, in Proceedings of the Conference on Physics From Large-Ray Detector Arrays, Vol. II of Report LBL35687, edited by M. A. Deleplanque (University of California, Berkeley, 1994), p. 52
[2] R. M. Clark, R. Wadsworth, E. S. Paul et al, Phys. Lett. B, 275: 247 (1992) doi: 10.1016/0370-2693(92)91585-W
[3] A. Kuhnert, M. A. Stoyer, J. A. Becker et al, Phys. Rev. C, 46: 133 (1992) doi: 10.1103/PhysRevC.46.133
[4] G. Baldsiefen, H. Hüubel, D. Mehta et al, Phys. Lett. B, 275: 252 (1992) doi: 10.1016/0370-2693(92)91586-X
[5] S. Frauendorf, Nucl. Phys. A, 557: 259c (1993) doi: 10.1016/0375-9474(93)90546-A
[6] R. M. Clark, and A. O. Macchiavelli, Ann. Rev. Nucl. Part. Sci., 50: 1 (2000) doi: 10.1146/annurev.nucl.50.1.1
[7] S. Frauendorf, Rev. Mod. Phys., 73: 463 (2001) doi: 10.1103/RevModPhys.73.463
[8] H. Hüubel, Prog. Part. Nucl. Phys., 54: 1 (2005) doi: 10.1016/j.ppnp.2004.06.002
[9] J. Meng, J. Peng, S. Q. Zhang, and P. W. Zhao, Front. Phys., 8: 55 (2013) doi: 10.1007/s11467-013-0287-y
[10] J. Meng, S. Q. Zhang, and P. W. Zhao, in Relativistic Density Functional for Nuclear Structure, Vol. 10, edited by J. Meng, (World Scientific, Singapore, 2016)
[11] R. M. Clark, S. J. Asztalos, G. Baldsiefen et al, Phys. Rev. Lett., 78: 1868 (1997) doi: 10.1103/PhysRevLett.78.1868
[12] Amita, A. K. Jain, and B. Singh, At. Data Nucl. Data Tables, 74: 283 (2000) doi: 10.1006/adnd.2000.0831
[13] J. Li, C. Y. He, Y. Zheng et al, Phys. Rev. C, 88: 014317 (2013) doi: 10.1103/PhysRevC.88.014317
[14] Y. Zheng, J. Li, J. J. Liu et al, J. Phys. G, 42: 085108 (2015) doi: 10.1088/0954-3899/42/8/085108
[15] N. Kumar, S. Kumar, S. K. Mandal, S. Saha, J. Sethi, and R. Palit, Eur. Phys. J. A, 53: 25 (2017) doi: 10.1140/epja/i2017-12212-y
[16] B. Das, P. Datta, S. Chattopadhyay et al, Phys. Rev. C, 98: 014326 (2018) doi: 10.1103/PhysRevC.98.014326
[17] S. Rajbanshi, R. Raut, H. Pai et al, Phys. Rev. C, 98: 061304(R) (2018) doi: 10.1103/PhysRevC.98.061304
[18] C. J. Chiara, D. B. Fossan, V. P. Janzen et al, Phys. Rev. C, 64: 054314 (2001) doi: 10.1103/PhysRevC.64.054314
[19] T. Trivedi, R. Palit, J. Sethi et al, Phys. Rev. C, 85: 014327 (2012) doi: 10.1103/PhysRevC.85.014327
[20] A. Y. Deo, S. B. Patel, S. K. Tandel et al, Phys. Rev. C, 73: 034313 (2006) doi: 10.1103/PhysRevC.73.034313
[21] S. H. Yao, H. L. Ma, L. H. Zhu et al, Phys. Rev. C, 89: 014327 (2014) doi: 10.1103/PhysRevC.89.014327
[22] P. Datta, S. Roy, S. Pal et al, Phys. Rev. C, 78: 021306 (2008) doi: 10.1103/PhysRevC.78.021306
[23] D. Choudhury, R. Palit, P. Singh et al, Phys. Rev. C, 91: 014318 (2015) doi: 10.1103/PhysRevC.91.014318
[24] D. Negi, T. Trivedi, A. Dhal et al, Phys. Rev. C, 81: 054322 (2010) doi: 10.1103/PhysRevC.81.054322
[25] C. Y. He, X. Q. Li, L. H. Zhu et al, Phys. Rev. C, 83: 024309 (2011) doi: 10.1103/PhysRevC.83.024309
[26] S. Rajbanshi, S. Ali, A. Bisoi et al, Phys. Rev. C, 94: 044318 (2016) doi: 10.1103/PhysRevC.94.044318
[27] S. Rajbanshi, A. Bisoi, S. Nag et al, Phys. Rev. C, 89: 014315 (2014) doi: 10.1103/PhysRevC.89.014315
[28] C. B. Li, J. Li, X. G. Wu et al, Nucl. Phys. A, 892: 34 (2012) doi: 10.1016/j.nuclphysa.2012.07.013
[31] C. M. Petrache, Q. B. Chen, S. Guo et al, Phys. Rev. C, 94: 064309 (2016) doi: 10.1103/PhysRevC.94.064309
[32] T. Bhattacharjee, S. Chanda, S. Bhattacharyya et al, Nucl. Phys. A, 825: 16 (2009) doi: 10.1016/j.nuclphysa.2009.04.007
[33] S. Kaim, C. M. Petrache, A. Gargano et al, Phys. Rev. C, 91: 024318 (2015) doi: 10.1103/PhysRevC.91.024318
[34] N. T. Zhang, Y. H. Zhang, X. H. Zhou et al, Phys. Rev. C, 84: 057302 (2011) doi: 10.1103/PhysRevC.84.057302
[35] S. Y. Wang, D. P. Sun, B. Qi et al, Phys. Rev. C, 86: 064302 (2012)
[36] A. Y. Deo, R. Palit, Z. Naik et al, Phys. Rev. C, 79: 067304 (2009) doi: 10.1103/PhysRevC.79.067304
[37] R. Garg, S. Kumar, M. Saxena et al, Phys. Rev. C, 92: 054325 (2015) doi: 10.1103/PhysRevC.92.054325
[38] Y. Y. Cheng, S. Q. Zhang, X. Q. Li et al, Phys. Rev. C, 89: 054309 (2014)
[39] M. G. Procter, D. M. Cullen, C. Scholey et al, Phys. Rev. C, 81: 054320 (2010) doi: 10.1103/PhysRevC.81.054320
[40] K. Auranen, J. Uusitalo, S. Juutinen et al, Phys. Rev. C, 91: 024324 (2015) doi: 10.1103/PhysRevC.91.024324
[41] D. J. Hartley, E. P. Seyfried, W. Reviol et al, Phys. Rev. C, 78: 054319 (2008) doi: 10.1103/PhysRevC.78.054319
[42] H. J. Li, B. Cederwall, M. Doncel et al, Phys. Rev. C, 93: 034309 (2016) doi: 10.1103/PhysRevC.93.034309
[43] A. D. Ayangeakaa, S. Zhu, R. V. F. Janssens et al, Phys. Rev. C, 91: 044327 (2015) doi: 10.1103/PhysRevC.91.044327
[44] H. Pai, G. Mukherjee, S. Bhattacharyya et al, Phys. Rev. C, 85: 064313 (2012) doi: 10.1103/PhysRevC.85.064313
[45] T. Zerrouki, C. M. Petrache, R. Leguillon et al, Eur. Phys. J. A, 51: 50 (2015) doi: 10.1140/epja/i2015-15050-y
[46] A. Herzáň, S. Juutinen, K. Auranen et al, Phys. Rev. C, 96: 014301 (2017)
[47] D. Steppenbeck, R. V. F. Janssens, S. J. Freeman et al, Phys. Rev. C, 85: 044316 (2012) doi: 10.1103/PhysRevC.85.044316
[48] D. A. Torres, F. Cristancho, L.-L. Andersson et al, Phys. Rev. C, 78: 054318 (2008) doi: 10.1103/PhysRevC.78.054318
[49] D. Yuan, Y. Zheng, D. Zhou et al, Hyperfine Interact., 180: 49 (2008)
[50] P. Agarwal, S. Kumar, S. Singh et al, Phys. Rev. C, 76: 024321 (2007) doi: 10.1103/PhysRevC.76.024321
[51] A. Bohr and B. R. Mottelson, Nuclear Structure Volume II: Nuclear Deformation (W. A. Benjamin, Inc., 1975)
[52] Y. S. Chen, S. Frauendorf, and G. A. Leander, Phys. Rev. C, 28: 2437 (1983) doi: 10.1103/PhysRevC.28.2437
[53] Victor Velázquez, Jorge G. Hirsch, and Yang Sun, Nucl. Phys. A, 686: 129 (2001) doi: 10.1016/S0375-9474(00)00507-8
[54] Amita, A. K. Jain, A. Goel, and B. Singh, Pramana, 53: 463 (1999) doi: 10.1007/s12043-999-0016-9
[55] A. O. Macchiavelli, R. M. Clark, P. Fallon et al, Phys. Rev. C, 57: R1073 (1998) doi: 10.1103/PhysRevC.57.R1073
[56] P. Banerjee, S. Ganguly, M. K. Pradhan et al, Phys. Rev. C, 83: 024316 (2011) doi: 10.1103/PhysRevC.83.024316
[57] A. A. Pasternak, E. O. Lieder, R. M. Lieder et al, Eur. Phys. J. A, 37: 279 (2008) doi: 10.1140/epja/i2008-10631-5
[58] B. Das, N. Rather, P. Datta et al, Phys. Rev. C, 95: 051301 (2017) doi: 10.1103/PhysRevC.95.051301
[59] A. O. Macchiavelli, R. M. Clark, M. A. Deleplanque et al, Phys. Rev. C, 58: 3746 (1998) doi: 10.1103/PhysRevC.58.3746
[60] D. G. Jenkins, R. Wadsworth, J. A. Cameron et al, Phys. Rev. Lett., 83: 500 (1999) doi: 10.1103/PhysRevLett.83.500
[61] F. S. Stephens and R. S. Simon, Nucl. Phys. A, 183: 257 (1972) doi: 10.1016/0375-9474(72)90658-6
[62] M. Neffgen, G. Baldsiefen, S. Frauendorf et al, Nucl. Phys. A, 595: 499 (1995) doi: 10.1016/0375-9474(95)00375-4
[63] A. Afanasjev, D. Fossan, G. Lane, and I. Ragnarsson, Phys. Rep., 322: 1 (1999) doi: 10.1016/S0370-1573(99)00035-6
[64] S. Frauendorf and J. Meng, Nucl. Phys. A, 617: 131 (1997) doi: 10.1016/S0375-9474(97)00004-3
[65] I. Kuti, Q. B. Chen, J. Timár et al, Phys. Rev. Lett., 113: 032501 (2014) doi: 10.1103/PhysRevLett.113.032501
[66] D. Jerrestam, B. Fogelberg, R. A. Bark et al, Phys. Rev. C, 52: 2448 (1995) doi: 10.1103/PhysRevC.52.2448
[67] J. Timár, T. Koike, N. Pietralla et al, Phys. Rev. C, 76: 024307 (2007)
[68] S. Sihotra, Z. Naik, R. Palit et al, Eur. Phys. J. A, 43: 45 (2009)
[69] P. Datta, S. Chattopadhyay, P. Banerjee et al, Phys. Rev. C, 67: 014325 (2003) doi: 10.1103/PhysRevC.67.014325
[70] C. Y. He, L. H. Zhu, X. G. Wu et al, Phys. Rev. C, 81: 057301 (2010) doi: 10.1103/PhysRevC.81.057301
[71] P. Ring, Prog. Part. Nucl. Phys., 37: 193 (1996) doi: 10.1016/0146-6410(96)00054-3
[72] D. Vretenar, A. V. Afanasjev, G. A. Lalazissis, and P. Ring, Phys. Rep., 409: 101 (2005) doi: 10.1016/j.physrep.2004.10.001
[73] J. Meng, H. Toki, S. G. Zhou et al, Prog. Part. Nucl. Phys., 57: 470 (2006) doi: 10.1016/j.ppnp.2005.06.001
[74] W. Koepf, and P. Ring, Nucl. Phys. A, 493: 61 (1989) doi: 10.1016/0375-9474(89)90532-0
[75] H. Madokoro, J. Meng, M. Matsuzaki, and S. Yamaji, Phys. Rev. C, 62: 061301(R) (2000) doi: 10.1103/PhysRevC.62.061301
[76] P. W. Zhao, Phys. Lett. B, 773: 1 (2017) doi: 10.1016/j.physletb.2017.08.001
[77] P. W. Zhao, Y. K. Wang, and Q. B. Chen, Phys. Rev. C, 99: 054319 (2019) doi: 10.1103/PhysRevC.99.054319
[78] J. Peng, J. Meng, P. Ring, and S. Q. Zhang, Phys. Rev. C, 78: 024313 (2008) doi: 10.1103/PhysRevC.78.024313
[79] P. W. Zhao, S. Q. Zhang, J. Peng et al, Phys. Lett. B, 699: 181 (2011) doi: 10.1016/j.physletb.2011.03.068
[80] P. W. Zhao, J. Peng, H. Z. Liang, P. Ring, and J. Meng, Phys. Rev. C, 85: 054310 (2012) doi: 10.1103/PhysRevC.85.054310
[81] P. W. Zhao, J. Peng, H. Z. Liang, P. Ring, and J. Meng, Phys. Rev. Lett., 107: 122501 (2011) doi: 10.1103/PhysRevLett.107.122501
[82] X. W. Li, J. Li, J. B. Lu et al, Phys. Rev. C, 86: 057305 (2012) doi: 10.1103/PhysRevC.86.057305
[83] K. Y. Ma, J. B. Lu, D. Yang et al, Eur. Phys. J. A, 48: 82 (2012) doi: 10.1140/epja/i2012-12082-9
[84] P. Zhang, B. Qi, and S. Y. Wang, Phys. Rev. C, 89: 047302 (2014) doi: 10.1103/PhysRevC.89.047302
[85] J. Peng, and P. W. Zhao, Phys. Rev. C, 91: 044329 (2015) doi: 10.1103/PhysRevC.91.044329
[86] W. J. Sun, H. D. Xu, J. Li et al, Chin. Phys. C, 40: 084101 (2016) doi: 10.1088/1674-1137/40/8/084101
[87] P. W. Zhao, Z. P. Li, J. M. Yao, and J. Meng, Phys. Rev. C, 82: 054319 (2010) doi: 10.1103/PhysRevC.82.054319
[88] S. Frauendorf, and J. Meng, Z. Phys. A, 356: 263 (1996) doi: 10.1007/BF02769229
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Wu-Ji Sun 1,
Corresponding author: Jian Li, [email protected]
1. College of Physics, Jilin University, Changchun 130012, China
Abstract: The systematics of energy staggering for the magnetic rotational bands with $ M1 $ and $ E2 $ transition properties strictly consistent with the features of good candidates of magnetic rotational bands in the $ A\sim80 $ , 110, 130 and 190 mass regions are presented. The regularities exhibited by these bands concerning the staggering parameter which increases with increasing spin are in agreement with the semiclassical description of shears mechanism. In addition, the abnormal behaviours in the backbend regions or close to band termination have also been discussed. Taking the magnetic dipole bands with same configuration in three $ N = 58 $ isotones, i.e., $ ^{103} {\rm Rh}$ , $ ^{105} {\rm Ag}$ , and $ ^{107} {\rm In}$ , as examples, the transition from chiral rotation to magnetic rotation with the proton number approaching $ Z = 50 $ is presented. Moreover, the self-consistent tilted axis cranking and principle axis cranking relativistic mean-field theories are applied to investigate the rotational mechanism in dipole band of $ ^{105} {\rm Ag}$ .
Similar to rotational bands observed in molecules, many nuclei have an energy spectrum with a pronounced rotational character and the study of nuclear rotation has been at the forefront of nuclear structure for several decades. In particular, magnetic rotation (MR) [1], which is an exotic rotational phenomenon observed in weakly deformed or near-spherical nuclei and differs from conventional collective rotation in well-deformed nuclei, has been of great interest since the observation of cascades of magnetic dipole ($ M1 $) transitions in the region of neutron-deficient Pb isotopes in the 1990s [2–4].
The explanation of MR was given in terms of the shears mechanism [5]. In this interpretation, the magnetic dipole vector in the magnetic rotational bands arising from proton particles (holes) and neutron holes (particles) in high-j orbitals rotates around the total angular-momentum vector. Meanwhile, with increasing spin, the proton particles (holes) and neutron holes (particles) in the high-j orbitals align along the total angular momentum and this alignment reduces the perpendicular component of the magnetic dipole moment. As a result, a typical property of these bands is the decreasing of the $ B(M1) $ values with increasing spin. In all, experimental indicators for magnetic rotational bands can be summarized as follows [6–10]: 1) a $ \Delta I = 1 $ sequence of strong magnetic dipole ($ M1 $) transitions, corresponding to a reduced transition probability $ B(M1) \sim $ a few $ \mu_N^2 $, which decrease with increasing spin; 2) weak or absent quadrupole transitions, corresponding to a deformation parameter $ |\beta|\lesssim 0.15 $, which combined with strong $ M1 $ transitions results in large $ B(M1)/B(E2) $ ratios, $ \gtrsim 20\;\mu_N^2/(eb)^2 $; 3) a smooth variation in the $ \gamma $ transition energy with angular momentum; 4) a substantial moment of inertia, corresponding to the large ratio of the $ {\mathcal{J}}^{(2)}/B(E2) \gtrsim 100\;{\rm MeV}^{-1}(eb)^{-2} $, compared with the values in well-deformed [$ \sim 10\;{\rm MeV}^{-1}(eb)^{-2} $] or superdeformed [$ \sim 5\;{\rm MeV}^{-1}(eb)^{-2} $] rotational bands.
The first clear evidence of magnetic rotational bands was provided by the lifetime measurements for four $ M1 $-bands in $ ^{198,199} {\rm Pb}$ [11]. From then on, more and more magnetic rotational bands have been observed not only in the mass region of $ A\sim190 $, but also in the $ A\sim60 $, 80,110 and 130 regions. To date, more than 200 magnetic dipole bands spread over 110 nuclides have been observed, which have been summarized in the nuclear chart in Fig. 1 based on the review on the observed MR bands [12] and recent observations [13–50]. The blue squares in Fig. 1 represent nuclides with good candidate MR bands, in which $ M1 $ and $ E2 $ transition properties are strictly consistent with the features of MR, i.e., decreasing $ B(M1) $ values with increasing spin and $ B(M1)/B(E2)\gtrsim 20\;\mu_N^2/(eb)^2 $, including 50 bands spread over 39 nuclides. The red squares represent other MR candidate nuclides without lifetime measurements.
Figure 1. (Color online) The candidate nuclides with magnetic rotation observed in the nuclear chart. The blue squares represent nuclides with good candidate MR bands, in which M1 and E2 transition properties strictly consistent with the features of magnetic rotation. The red squares represent other candidates. The corresponding data are taken from Refs. [12–50].
Signature is the quantum number specifically appearing in a deformed intrinsic system and associated with symmetry under a rotation of $ 180^{\circ} $ about the axis of nuclear rotation. A rotational band with a $ \Delta I = 1 $ sequence could be divided into two branches classified by the signature quantum number [51–53]. Experimentally, energy staggering between alternate spin states, which is best visualized by the experimental quantity $S(I) = [E(I)- $$ E(I -1)]/2I $, could be observed in many rotational bands and is usually referred to as signature splitting. Theoretically, no signature splitting should be observed in ideal magnetic rotational bands due to the pure individual motion of nucleons (shears mechanism) and the tilted angle generally remains far from $ 90^{\circ} $ in tilted rotational mode of nuclei [5, 7]. However, it is noted that the experimental energy staggering could be found for some MR candidates as mentioned in Ref. [54].
In magnetic rotational bands, their energy spectra show rotational-like features with strong $ M1 $ transitions and it is noted in Ref. [6] that the rotational-like energy spectra (away from band crossings) approximately follow the pattern of $ E(I)-E_0 \varpropto (I-I_0)^2 $ such as in Pb isotopes [2–4], where $ E(I)-E_0 $ is the relative energy of a state with spin I at energy $ E(I) $ to the energy of the bandhead state, $ E_0 $. The rotational-like behaviour observed could be explained by a semiclassical analysis of the shears mechanism from a residual proton-neutron interaction [55]. Therefore in the MR bands, $ S(I) \varpropto 1-(2I_0+1)/2I $ are supposed to increase with increasing spin, and signature splitting, i.e., energy staggering, is not expected. In comparison, for conventional rotation in well-deformed nuclei, its energy spectra follow the pattern $ E(I)-E_0 \varpropto I(I+1) $ [51] based on the simple assumption of constant moment of inertia, and the $ S(I) $ values are constant at various spins.
Thus, it is interesting to systematically study energy staggering $ S(I) $ of magnetic rotational bands, especially for the ones with suitable electromagnetic transition properties. In the present work, systematic behaviours of $ S(I) $ in good candidates of MR bands would be investigated. In addition, taking the dipole bands with same configuration in $ ^{103} {\rm Rh}$, $ ^{105} {\rm Ag}$, and $ ^{107} {\rm In}$ as examples, the characteristics of $ S(I) $ in different rotational modes would be compared.
In Fig. 2, the experimental energy staggering $ S(I) $ for the good candidate MR bands in the mass $ A\sim80,110, 130 $, and $ 190 $ regions are shown, which include 50 bands spread out in 39 nuclides marked with blue squares in Fig. 1. In the 110, 130 and 190 mass regions, the MR bands in even-even, odd-odd and odd-A nuclei are presented respectively. The corresponding data are taken from the previous review [12] and recent observations [15–29, 56–58].
Figure 2. (Color online) The experimental energy staggering $ S(I)$ as a function of spin I for the good candidate MR bands in the mass number $ A\sim80$ (a), 110 (b), 130 (c) and 190 (d) regions. In the 110, 130 and 190 mass regions, MR bands in even-even, odd-odd and odd-A nuclei are presented separately. The filled symbols and the open symbols indicate positive parity bands and negative parity bands, respectively. The corresponding data are taken from the previous review Ref. [12] and recent observations [15–29, 56–58].
It could be seen from Fig. 2 that for almost all the MR candidates, the experimental energy staggering $ S(I) $ tend to increase with increasing spin except for the backbend regions (or band crossing), in agreement with the semi-classical formula for the rotational-like properties of MR, i.e. the deduced term $ S(I) \varpropto 1-(2I_0+1)/2I $, and different from the semi-classical formula for conventional rotation in well-deformed nuclei. For examples, for the MR bands of $ ^{79} {\rm Br}$, $ ^{81,83} {\rm Kr}$, $ ^{82,83,84} {\rm Rb}$ and $ ^{85} {\rm Sr}$ in the 80 mass region, $ S(I) $ clearly exhibits a increasing pattern as the spin increases. In Ref. [59], the semiclassical approach using two blades (particle and hole) interacting with an effective force is proposed and this simple scenario can account for the important features of the shears bands in neutron-deficient Pb nuclei and other mass regions. In fact, the moment of inertia for shears mechanism would decrease with the blades of the shears closing [7, 59]. The behaviour that kinematic moment of inertia $ {\mathcal{J}}^{(1)}\,( = 1/[2S(I)]) $ decreases with angular momentum is clearly demonstrated in Ref. [59], which further proves the increasing feature of $ S(I) $ with spin. However, $ S(I) $ for the MR band in $ ^{108} {\rm Sn}$ [60] shown in Fig. 2(b) is an exception, i.e., in the higher spin region of the band, $ S(I) $ begin to decrease while $ B(M1) $ still shows a decreasing tendency.
It could also be seen that the $ S(I) $ values in the backbend region show a decreasing trend with increasing spin for the MR bands in the 110, 130, and 190 mass regions. The backbend phenomenon is usually interpreted as the decoupling of a pair of particles from the rotating nuclear core and the subsequent rotational alignment of their angular momenta along the rotation axis [61]. The backbend would result in increasing moment of inertia caused by the shears opening due to the gradual alignment of the particles, and therefore $ S(I) $ shows the decreasing trend in the backbend regions of MR bands. For example, the $ S(I) $ of negative-parity MR band of $ ^{199} {\rm Pb}$ decreases from 11.5 to 5.3 $ {\rm keV}/\hbar $ in the backbend region [62] as shown in Fig. 2. Here, the configurations of the negative-parity MR band in $ ^{199} {\rm Pb}$ are $ \pi(h_{9/2}i_{13/2})_{K = 11^-}\otimes \nu i_{13/2}^{-1} $ before backbend ($ I\leqslant 17.5\hbar $) and $ \pi (h_{9/2}i_{13/2})_{K = 11^-}\otimes \nu i_{13/2}^{-3} $ after backbend ($ I\geqslant23.5\hbar $) respectively [62]. After the rotational alignment of two $ i_{13/2} $ neutron holes, the shears open up to $ 90^{\circ} $ coupling and a new shears band starts to build up.
In addition, it is easy to see in Fig. 2 that the $ S(I) $ values in higher mass regions are relatively smaller compared to that in lower mass regions. In the $ A\sim80 $ region, $ S(I) $ varies from 8 to 42 $ {\rm keV}/\hbar $ and it varies from 2 to 25 $ {\rm keV}/\hbar $ in the $ A\sim110 $ region. Then in the $ A\sim130 $ region, $ S(I) $ varies from 4 to 16 $ {\rm keV}/\hbar $ and finally it varies from 2 to 12 $ {\rm keV}/\hbar $ in the $ A\sim190 $ region. This behaviour is closely related to the mass dependence of the moment of inertia for these bands discussed in Ref. [55]. Moreover, $ S(I) $ is also related to the corresponding valence particle-hole configuration. For example, for the positive-parity band in $ ^{108,110,112} {\rm In}$ [18, 19] with the same configuration $ \pi g_{9/2}^{-1}\otimes \nu h^2_{11/2}(g_{7/2}/d_{5/2})^1 $, the $ S(I) $ values of these bands are quite similar as shown in Fig. 2(b). While for the two bands with different configurations in the same nuclide $ ^{108} {\rm In}$, there is a clear discrepancy in the $ S(I) $ values.
However, some MR bands in Fig. 2 show abnormal behaviours of $ S(I) $ for the band termination (high spin) region. For example, a sudden decline in $ S(I) $ happens at the highest spin ($ I = 23.5\hbar $) of MR band of $ ^{143} {\rm Eu} $ in Fig. 2(c), or $ S(I) $ decreases in the high-spin region ($ I>26.5\hbar $) of $ ^{193} {\rm Pb}$ in Fig. 2(d). Similar behaviours could also be seen in $ ^{104} {\rm Ag}$, $ ^{110,111,112} {\rm In}$, $ ^{106} {\rm Sn}$, $ ^{136} {\rm Ce}$, $ ^{139} {\rm Sm}$, $ ^{142} {\rm Gd}$ and $ ^{194,196,198} {\rm Pb}$. The decreasing tendency of $ S(I) $ in the high-spin region of the bands, in some cases a sudden decline at the highest spin, are mostly caused by the abrupt change of the configuration, which is similar to the pattern of $ S(I) $ in the backbend region. While for the MR band of $ ^{109} {\rm Ag}$ in Fig. 2(b), $ S(I) $ shows a staggering pattern in the high-spin region ($ I\geqslant15.5\hbar $), which could also be found in $ ^{79} {\rm Br}$, $ ^{81,83} {\rm Kr}$, $ ^{105,106,107} {\rm Ag}$ and $ ^{136} {\rm Ce}$ as shown in Fig. 2. In Ref. [63], it is noted that the competition and interaction with collective rotation happen in the high-spin region of the bands with certain configuration, as aligned states don't have the maximum spin.
In all the presented MR bands, the increasing tendency of energy staggering $ S(I) $ with increasing spin before and after backbend are observed. The systematic study of energy staggering parameter in MR bands shows a common behavior of $ S(I) $, implying that $ S(I) $ could be a potential indicator for MR. Therefore, it is necessary to test and compare with other exotic rotations, such as chiral rotation [64]. Taking the $ M1 $ bands in $ ^{103} {\rm Rh}$ [65], $ ^{105} {\rm Ag}$ [20, 66, 67], and $ ^{107} {\rm In}$ [24, 68] with the same configuration $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $ as examples, the extracted experimental $ S(I) $ values are shown and compared in Fig. 3.
Figure 3. (Color online) Experimental $ S(I)$ as a function of spin I for the negative parity $ M1$ bands of $ ^{103}{\rm Rh}$ (a), $ ^{105}{\rm Ag}$ (b), and $ ^{107}{\rm In}$ (c) with the same configuration $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1$. The corresponding data are taken from Refs. [20, 24, 65–68].
The two $ M1 $ bands of $ ^{103} {\rm Rh}$ have been proposed as chiral doublet bands [65]. It should be noted that multiple chiral doublet bands with the same configuration $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $ were observed in $ ^{103} {\rm Rh}$, and only the "yrast" chiral doublet bands are presented here. The $ S(I) $ values of those two chiral partner bands in Fig. 3(a) stay almost constant with a little staggering, and $ B(M1)/B(E2) $ ratios show typical values and behaviours of chiral doublet bands as suggested in Ref. [65].
For the $ M1 $ band of $ ^{107} {\rm In}$, the $ S(I) $ values increase with increasing spin before and after backbend. The behaviours of the experimental data in this band, including $ B(M1)/B(E2) $ ratios and $ B(M1) $ values, are consistent with ideal MR bands as suggested in Ref. [24].
For the $ M1 $ band of $ ^{105} {\rm Ag}$, the $ S(I) $ values of $ M1 $ band tend to increase with increasing spin in the lower spin region, which is consistent with the behaviour in ideal MR bands. In the higher spin region ($ I>15.5\hbar $), $ S(I) $ shows noticeable staggering. The staggering of $ S(I) $ is usually considered as a sign of the collective rotation, which indicates a competition between shears mechanism and collective motion.
The single-particle Routhians for the proton at the top of $ \pi g_{7/2} $ shell is close to the $ \pi g_{9/2} $ proton shell. Considering the contribution from the collective motion is increasing, there could be a mixing of the configuration $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $ and the configuration $ \pi g_{7/2}^{1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $ at high-spin states.
Based on the above statements, it could be concluded that the shears mechanism does not seem to dominate in the $ M1 $ bands of $ ^{103} {\rm Rh}$ and $ ^{105} {\rm Ag}$, and there is an obvious transition from chiral rotation to magnetic rotation in the $ A\sim110 $ region when the proton number is approaching Z = 50. Moreover, the quadrupole deformation in previous TAC calculations are 0.26, 0.19 and 0.12 respectively for $ ^{103} {\rm Rh}$ [65], $ ^{105} {\rm Ag}$ [69] and $ ^{107} {\rm In}$ [24] with the same configuration $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $, probably indicating that the deformation of nuclei becomes smaller with increasing Z number towards Z = 50. Similarly to Refs. [21, 70], the increasing importance of the shears mechanism and decreasing contribution of collective rotation with increasing proton number towards Z = 50 in the $ A\sim110 $ region also have been presented and discussed. Thus the systematic study of the different rotational modes in the $ A\sim110 $ region is an interesting question.
During the past few decades, relativistic mean-field (RMF) theory has been a great success in describing properties of nuclei and many nuclear phenomena [71–73]. The principle axis cranking relativistic mean-field (PAC-RMF) theory has been used to describe collective rotational motion in deformed nuclei [74]. Based on the RMF theory, the tilted axis cranking relativistic mean-field (TAC-RMF) theory has been developed for describing the nuclear magnetic and antimagnetic rotational modes [9, 10]. The cranking RMF model with arbitrary orientation of the rotational axis, i.e., three-dimensional cranking, has been developed and applied for the magnetic rotation in $ ^{84} {\rm Rb}$ [75]. Recently, the three-dimensional TAC-RMF theory with point-coupling interaction has been used to investigate multiple chirality in nuclear rotation [76, 77]. The two-dimensional cranking RMF theory based on the meson exchange [78] and the point-coupling interactions [79, 80] has also been established and applied successfully to describe magnetic rotation in $ A \sim 60$, 80, 130 and 190 regions [9, 10], and especially the 110 region [28, 80–86].
To further examine the rotational mechanism as well as the staggering patterns of $ S(I) $ for the bands in $ ^{105} {\rm Ag}$, the TAC-RMF calculations for the lower spin region of the dipole band in $ ^{105} {\rm Ag}$ with the configuration $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $ and the principle axis cranking RMF (PAC-RMF) calculations for the higher spin region of the band with the configuration $ \pi g_{7/2}^{1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $ in different signatures have been performed. The point-coupling interaction PC-PK1 [87] was used and pairing correlations were neglected. The Dirac equation for the nucleons is solved in a three-dimensional harmonic oscillator basis and a basis of 10 major oscillator shells is adopted. The calculated rotational excitation energies, total angular momenta and $ B(M1) $ values in comparison with the corresponding data [20, 66, 67] are shown in Fig. 4. The results of TAC-RMF calculations for lower spin region and PAC-RMF calculations for higher spin region in the $ M1 $ band are displayed respectively. The experimental rotational frequency can be extracted as in Ref. [88]: $ \hbar\omega_{\rm exp} = \frac{1}{2}[E_\gamma(I+1\rightarrow I) + E_\gamma(I\rightarrow I-1)] $.
Figure 4. (Color online) Rotational excitation energies (a) as a function of the total angular momentum, total angular momenta (b) and $ B(M1)$ values (c) as functions of the rotational frequency, for the configurations $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1$ in TAC-RMF calculations and $ \pi g_{7/2}^{1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1$ in PAC-RMF calculations. The corresponding data are taken from Refs. [20, 66, 67].
It could be seen in Fig. 4(a) that the calculated excitation energies of the band in both TAC-RMF and PAC-RMF calculations are both in good agreement with experimental data. In Fig. 4(b), for the lower spin region, the TAC-RMF results well reproduce the data, which further supports the significant contribution from shears mechanism. For the higher spin region, the PAC-RMF results are in reasonable agreement with the data indicating that the increasing contribution from the collective motion results from the intruder configuration $ \pi g_{7/2}^{1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $. Moreover, the experimental $ I-\omega $ plot are sandwiched between the corresponding plots for the configuration $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $ and $ \pi g_{7/2}^{1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $, further supporting the mixing of these two configurations at high-spin states.
In Fig. 4(c), the calculated $ B(M1) $ values are shown to be in agreement with the data, but do not decrease much with increasing rotational frequency, which corresponds to a small decline of the shears angle. As discussed in Ref. [78], the tilted angle $ \theta $ of the orientation of the angular velocity with respect to the principal axis of the density distribution are determined self-consistently in the TAC-RMF calculations. With the rotational frequency increasing from 0.18 to 0.50 MeV/$ \hbar $, the tilted angle of proton angular momentum $ \theta_{\pi} $ changes from $ 5^{\circ} $ to $ 10^{\circ} $, the tilted angle of neutron angular momentum $ \theta_\nu $ decreases from $ 81^{\circ} $ to $ 71^{\circ} $. The shears angle between these two blades decreases from $ 76^{\circ} $ to $ 61^{\circ} $, i.e., the proton and neutron angular momenta align toward each other with increasing rotational frequency, which exhibits a clear shears mechanism. The shears angle only decreases by a small amount due to the relatively high contribution from collective motion. In general, the TAC-RMF calculations support the MR interpretation for the lower spin region of the band.
In summary, 50 bands spread over 39 nuclides with $ M1 $ and $ E2 $ transition properties strictly consistent with the features of MR have been selected and a systematic study of energy staggering parameter $ S(I) $ in these bands has been performed. The present study shows that $ S(I) $ values increase with increasing spin for all the bands before and after backbends, which could be explained by the simple semiclassical description of shears mechanism. It could be treated as an indicator for MR and also needs more investigations. Moreover, the behaviours of $ S(I) $ in the backbend regions or close to band termination have been discussed. In addition, the $ M1 $ bands in three N = 58 isotones, i.e., $ ^{103} {\rm Rh}$, $ ^{105} {\rm Ag}$ and $ ^{107} {\rm In}$, with the same configuration $ \pi g_{9/2}^{-1}\otimes \nu h^1_{11/2}(g_{7/2}/d_{5/2})^1 $ are taken as examples to further examine the staggering behaviours of $ S(I) $ in different rotational modes. It is suggested that there is a transition from chiral rotation to magnetic rotation with the proton number approaching Z = 50, due to the competition between collective motion and shears mechanism. Furthermore, the TAC-RMF and PAC-RMF calculations have been performed, and the rotational modes in $ ^{105} {\rm Ag}$ is clearly shown.
The authors would like to thank Dr. P. W. Zhao and Y. K. Wang for helpful discussions and collaboration during the completion of this work. | CommonCrawl |
Conformal group
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
Several specific conformal groups are particularly important:
• The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V
$Q(Tx)=\lambda ^{2}Q(x)$
For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations.
• The conformal group of the sphere is generated by the inversions in circles. This group is also known as the Möbius group.
• In Euclidean space En, n > 2, the conformal group is generated by inversions in hyperspheres.
• In a pseudo-Euclidean space Ep,q, the conformal group is Conf(p, q) ≃ O(p + 1, q + 1) / Z2.[1]
All conformal groups are Lie groups.
Angle analysis
In Euclidean geometry one can expect the standard circular angle to be characteristic, but in pseudo-Euclidean space there is also the hyperbolic angle. In the study of special relativity the various frames of reference, for varying velocity with respect to a rest frame, are related by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities. Thus, they are conformal transformations with respect to the hyperbolic angle.
A method to generate an appropriate conformal group is to mimic the steps of the Möbius group as the conformal group of the ordinary complex plane. Pseudo-Euclidean geometry is supported by alternative complex planes where points are split-complex numbers or dual numbers. Just as the Möbius group requires the Riemann sphere, a compact space, for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping. Nevertheless, the conformal group in each case is given by linear fractional transformations on the appropriate plane.[2]
Mathematical definition
Given a (Pseudo-)Riemannian manifold $M$ with conformal class $[g]$, the conformal group ${\text{Conf}}(M)$ is the group of conformal maps from $M$ to itself.
More concretely, this is the group of angle-preserving smooth maps from $M$ to itself. However, when the signature of $[g]$ is not definite, the 'angle' is a hyper-angle which is potentially infinite.
For Pseudo-Euclidean space, the definition is slightly different.[3] ${\text{Conf}}(p,q)$ is the conformal group of the manifold arising from conformal compactification of the pseudo-Euclidean space $\mathbf {E} ^{p,q}$ (sometimes identified with $\mathbb {R} ^{p,q}$ after a choice of orthonormal basis). This conformal compactification can be defined using $S^{p}\times S^{q}$, considered as a submanifold of null points in $\mathbb {R} ^{p+1,q+1}$ by the inclusion $(\mathbf {x} ,\mathbf {t} )\mapsto X=(\mathbf {x} ,\mathbf {t} )$ (where $X$ is considered as a single spacetime vector). The conformal compactification is then $S^{p}\times S^{q}$ with 'antipodal points' identified. This happens by projectivising the space $\mathbb {R} ^{p+1,q+1}$. If $N^{p,q}$ is the conformal compactification, then ${\text{Conf}}(p,q):={\text{Conf}}(N^{p,q})$. In particular, this group includes inversion of $\mathbb {R} ^{p,q}$, which is not a map from $\mathbb {R} ^{p,q}$ to itself as it maps the origin to infinity, and maps infinity to the origin.
Conf(p,q)
For Pseudo-Euclidean space $\mathbb {R} ^{p,q}$, the Lie algebra of the conformal group is given by the basis $\{M_{\mu \nu },P_{\mu },K_{\mu },D\}$ with the following commutation relations:[4]
${\begin{aligned}&[D,K_{\mu }]=-iK_{\mu }\,,\\&[D,P_{\mu }]=iP_{\mu }\,,\\&[K_{\mu },P_{\nu }]=2i(\eta _{\mu \nu }D-M_{\mu \nu })\,,\\&[K_{\mu },M_{\nu \rho }]=i(\eta _{\mu \nu }K_{\rho }-\eta _{\mu \rho }K_{\nu })\,,\\&[P_{\rho },M_{\mu \nu }]=i(\eta _{\rho \mu }P_{\nu }-\eta _{\rho \nu }P_{\mu })\,,\\&[M_{\mu \nu },M_{\rho \sigma }]=i(\eta _{\nu \rho }M_{\mu \sigma }+\eta _{\mu \sigma }M_{\nu \rho }-\eta _{\mu \rho }M_{\nu \sigma }-\eta _{\nu \sigma }M_{\mu \rho })\,,\end{aligned}}$
and with all other brackets vanishing. Here $\eta _{\mu \nu }$ is the Minkowski metric.
In fact, this Lie algebra is isomorphic to the Lie algebra of the Lorentz group with one more space and one more time dimension, that is, ${\mathfrak {conf}}(p,q)\cong {\mathfrak {so}}(p+1,q+1)$. It can be easily checked that the dimensions agree. To exhibit an explicit isomorphism, define
${\begin{aligned}&J_{\mu \nu }=M_{\mu \nu }\,,\\&J_{+1,\mu }={\frac {1}{2}}(P_{\mu }-K_{\mu })\,,\\&J_{0,\mu }={\frac {1}{2}}(P_{\mu }+K_{\mu })\,,\\&J_{-1,0}=D.\end{aligned}}$
It can then be shown that the generators $J_{ab}$ with $a,b=-1,0,\cdots ,n=p+q$ obey the Lorentz algebra relations with metric ${\tilde {\eta }}_{ab}=\operatorname {diag} (-1,+1,-1,\cdots ,-1,+1,\cdots ,+1)$.
Conformal group in two spacetime dimensions
For two-dimensional Euclidean space or one-plus-one dimensional spacetime, the space of conformal symmetries is much larger. In physics it is sometimes said the conformal group is infinite-dimensional, but this is not quite correct as while the Lie algebra of local symmetries is infinite dimensional, these do not necessarily extend to a Lie group of well-defined global symmetries.
For spacetime dimension $n>2$, the local conformal symmetries all extend to global symmetries. For $n=2$ Euclidean space, after changing to a complex coordinate $z=x+iy$ local conformal symmetries are described by the infinite dimensional space of vector fields of the form
$l_{n}=-z^{n+1}\partial _{z}.$
Hence the local conformal symmetries of 2d Euclidean space is the infinite-dimensional Witt algebra.
Conformal group of spacetime
In 1908, Harry Bateman and Ebenezer Cunningham, two young researchers at University of Liverpool, broached the idea of a conformal group of spacetime[5][6][7] They argued that the kinematics groups are perforce conformal as they preserve the quadratic form of spacetime and are akin to orthogonal transformations, though with respect to an isotropic quadratic form. The liberties of an electromagnetic field are not confined to kinematic motions, but rather are required only to be locally proportional to a transformation preserving the quadratic form. Harry Bateman's paper in 1910 studied the Jacobian matrix of a transformation that preserves the light cone and showed it had the conformal property (proportional to a form preserver).[8] Bateman and Cunningham showed that this conformal group is "the largest group of transformations leaving Maxwell’s equations structurally invariant."[9] The conformal group of spacetime has been denoted C(1,3)[10]
Isaak Yaglom has contributed to the mathematics of spacetime conformal transformations in split-complex and dual numbers.[11] Since split-complex numbers and dual numbers form rings, not fields, the linear fractional transformations require a projective line over a ring to be bijective mappings.
It has been traditional since the work of Ludwik Silberstein in 1914 to use the ring of biquaternions to represent the Lorentz group. For the spacetime conformal group, it is sufficient to consider linear fractional transformations on the projective line over that ring. Elements of the spacetime conformal group were called spherical wave transformations by Bateman. The particulars of the spacetime quadratic form study have been absorbed into Lie sphere geometry.
Commenting on the continued interest shown in physical science, A. O. Barut wrote in 1985, "One of the prime reasons for the interest in the conformal group is that it is perhaps the most important of the larger groups containing the Poincaré group."[12]
See also
• Conformal map
• Conformal symmetry
References
1. Jayme Vaz, Jr.; Roldão da Rocha, Jr. (2016). An Introduction to Clifford Algebras and Spinors. Oxford University Press. p. 140. ISBN 9780191085789.
2. Tsurusaburo Takasu (1941) "Gemeinsame Behandlungsweise der elliptischen konformen, hyperbolischen konformen und parabolischen konformen Differentialgeometrie", 2, Proceedings of the Imperial Academy 17(8): 330–8, link from Project Euclid, MR14282
3. Schottenloher, Martin (2008). A Mathematical Introduction to Conformal Field Theory (PDF). Springer Science & Business Media. p. 23. ISBN 978-3540686255.
4. Di Francesco, Philippe; Mathieu, Pierre; Sénéchal, David (1997). Conformal field theory. New York: Springer. ISBN 9780387947853.
5. Bateman, Harry (1908). "The conformal transformations of a space of four dimensions and their applications to geometrical optics" . Proceedings of the London Mathematical Society. 7: 70–89. doi:10.1112/plms/s2-7.1.70.
6. Bateman, Harry (1910). "The Transformation of the Electrodynamical Equations" . Proceedings of the London Mathematical Society. 8: 223–264. doi:10.1112/plms/s2-8.1.223.
7. Cunningham, Ebenezer (1910). "The principle of Relativity in Electrodynamics and an Extension Thereof" . Proceedings of the London Mathematical Society. 8: 77–98. doi:10.1112/plms/s2-8.1.77.
8. Warwick, Andrew (2003). Masters of theory: Cambridge and the rise of mathematical physics. Chicago: University of Chicago Press. pp. 416–24. ISBN 0-226-87375-7.
9. Robert Gilmore (1994) [1974] Lie Groups, Lie Algebras and some of their Applications, page 349, Robert E. Krieger Publishing ISBN 0-89464-759-8 MR1275599
10. Boris Kosyakov (2007) Introduction to the Classical Theory of Particles and Fields, page 216, Springer books via Google Books
11. Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis, Springer, ISBN 0387-90332-1, MR520230
12. A. O. Barut & H.-D. Doebner (1985) Conformal groups and Related Symmetries: Physical Results and Mathematical Background, Lecture Notes in Physics #261 Springer books, see preface for quotation
Further reading
The Wikibook Associative Composition Algebra has a page on the topic of: Conformal spacetime transformations
• Kobayashi, S. (1972). Transformation Groups in Differential Geometry. Classics in Mathematics. Springer. ISBN 3-540-58659-8. OCLC 31374337.
• Sharpe, R.W. (1997), Differential Geometry: Cartan's Generalization of Klein's Erlangen Program, Springer-Verlag, New York, ISBN 0-387-94732-9.
• Peter Scherk (1960) "Some Concepts of Conformal Geometry", American Mathematical Monthly 67(1): 1−30 doi:10.2307/2308920
• Martin Schottenloher, The conformal group, chapter 2 of A mathematical introduction to conformal field theory, 2008 (pdf)
• page on conformal groups
| Wikipedia |
SN Applied Sciences
January 2020 , 2:107 | Cite as
Investigation of lithological control of iron enrichment in groundwater using geophysical techniques in Yenagoa, Southern Nigeria
K. S. Okiongbo
K. K. Oboshenure
A. R. C. Amakiri
First Online: 18 December 2019
Part of the following topical collections:
2. Earth and Environmental Sciences (general)
Hydrochemical analysis of water samples from Yenagoa in the Niger Delta shows widespread occurrence of iron (Fe) in the groundwater. The Fe concentration is more than 0.3 mg/L at many places, and the distribution is heterogeneous both vertically and horizontally. In order to identify the cause of the high heterogeneity, we carried out an integrated study consisting of hydrogeochemical, electrical resistivity sounding and induced polarization (IP) chargeability measurements at eleven sites and 2-D electrical resistivity profiling (at 2 sites). Data processing using inversion techniques resulted in 4-layered resistivity and chargeability—depth models. The results show that clean sand and gravel exhibit high resistivity but low chargeability and normalized chargeability values, whereas clay and sandy clay exhibit relatively low resistivity but high chargeability and normalized chargeability values. In sites where the aquifer is overlain by a thick clay layer, Fe concentration is high (Fe > 0.3 mg/L) in the groundwater and redox potential values range between 118 and 133 mV. We interpret that the low-permeability clay layer creates a relatively atmosphere-isolated state in the underlying aquifer, which is responsible for the reductive ambient subsurface groundwater environment. In sites where the aquifer is capped by silt, Fe concentration is low (< 0.3 mg/L) in the groundwater and redox potential values range between 115 and 164 mv indicating a mild oxidation environment. We interpret that the clay acts as a controlling factor to the Fe enrichment in the groundwater regime. Knowledge of the clay layer, which is identified in the present study, will be helpful in selecting suitable sites for boreholes.
Groundwater Electrical resistivity Induced polarization Normalized chargeability Yenagoa
In the last decade, several investigations were carried out in parts of the Niger Delta, Southern Nigeria, to determine the regulating processes and spatial distribution of Fe in the shallow alluvial aquifer [1, 2]. The results of these studies show that the concentration of Fe in the groundwater abstracted from boreholes is high. Aside, the distribution of Fe in the groundwater was reported to be extremely heterogeneous, both vertically and laterally, within a scale of tens of metres. The issue of high iron concentration (> 0.3 mg/L) in groundwater is a common problem, since over 90% of residents in most cities in the Niger Delta depend on water abstracted from shallow private boreholes. Presently, we remain remarkably ignorant of the cause of the high Fe heterogeneity in the groundwater.
Considering the above, it was pertinent to investigate the probable reasons responsible for the high spatial variation of Fe in the groundwater and provide a sustainable solution for mitigation. Most studies in this direction only concentrated on the geochemical aspects of Fe contamination and suggested various purification techniques for iron removal [3]. Designing various purification methods for iron removal is only temporary solution and suffers from many obvious problems such as waste disposal and maintenance and hence is not a sustainable solution. The strategic importance of groundwater in Yenagoa and the threats posed by excessive Fe concentration emphasize the significance of low-Fe groundwater sources and prompted the assessment of the hydrostratigraphy of the sedimentary sequences in the study area. In assessing the hydrostratigraphy of the sedimentary sequence, it was considered that investigating the alluvial aquifer over short distances ranging between tens to hundreds of metres and in different locations within the study area would perhaps help explain whether the variation in the groundwater Fe concentration was due to local variations in aquifer stratigraphy. Although borehole drilling could be one of the best ways to determine the lithological variation, this approach was considered tedious and would require drilling several boreholes of different depths. This will be time-consuming, laborious and cost-intensive. The use of non-invasive surface geophysical techniques is of great relevance in that reasonably factual subsurface information is obtained without any destruction to the environment within a relatively short time. Electrical resistivity and time domain induced polarization have shown a good complementarity in this regard. The geoelectrical method provides a wide range of variations in the subsurface electrical resistivity. The variations are often associated with water content and lithology; hence, it is one of the most powerful geophysical methods often used in providing solutions to hydrogeological problems [4, 5, 6]. Recently, induced polarization method which is based on the chargeability effect of the subsurface has proved to be of significant value in the investigation of lithological variability of unconsolidated sediments especially in the mapping of clay content. In this study, we explore the lithological control on Fe contamination using surface geophysical methods in Yenagoa and environs.
2 Description of the study area
2.1 Location, physiography and climate
Yenagoa is located within the Southern Nigeria sedimentary basin. It is the capital of Bayelsa State. The study area covers an area of about 50 km2 of Yenagoa, and its metropolis. Yenagoa is bounded by longitudes 006° 10′ 3.07″ and 00 6° 25′ 10.53″ East of the prime meridian and latitudes 04° 51′ 39.73″ and 05°.2′ 25.53″ North of the equator. Geographically, Yenagoa is within the coastal area of the Recent Niger Delta (Fig. 1) where the ground surface is relatively flat, sloping very gently seawards [7]. Its mean elevation is about 8 m above the mean sea level [8]. The study area has a tropical rain forest climate characterized by rainy season and dry season. The rainy season commences from April to October with a brief dry period in August. The dry season lasts between November and March. The mean annual rainfall is about 4500 mm [9] and about 85% of the mean annual rain falls during the wet season. The temperature varies between 25 and 32 °C. Fishing and farming are the main occupation of the people.
Map of the Niger Delta showing the study area
2.2 Geology and hydrogeology of the study area
The study area lies within the fresh water swamps, backswamps, deltaic plain, alluvium and meander belt geomorphic unit of the Niger Delta [9]). The Niger Delta is basically an alluvial plain and consists of the modern and Holocene delta top deposits. Grain-size profiles of the Holocene alluvial deposits consist of a fining-up sequence of sand capped by fine silts and clay indicating a fluvial environment of deposition [10]. The fine-grained silts and clay overlying the basal sandy sequence are often called the near surface aquitard. The near surface aquitard thickness varies between < 5 to about 12 m, and due to the varying clay, silt and fine sand content, [10] reported that the aquitard permeability is highly heterogeneous. The near surface aquitard becomes a confining unit if it is thick and impermeable, which prevents percolation of precipitation into the alluvial aquifer. Akpokodje [9] reported that groundwater flows from North to South in the region.
Three main subsurface lithostratigraphic units are reported in the Niger Delta [11]. From top to bottom, they are Benin, Agbada and Akata Formations. The Benin Formation which is fluvial in origin is the main aquifer. Groundwater occurs mainly under unconfined conditions in the Benin Formation. Abam [12] observed that the sediments of the Benin Formation were deposited during the Late Tertiary–Early Quaternary period and are about 2100 m thick. The sediments are lenticular and unconsolidated and consist of coarse- to medium-fine-grained sands with localized intercalations of clay/shale. Gravel and pebbles are minor components. Mbonu et al. [13] reported that the sands are moderately sorted and poorly cemented. The presence of thin clay beds creates discontinuities in the vertical and lateral continuity of the aquifer, resulting in the presence of local perched aquifers [10]. The aquifer is directly recharged through the infiltration of rain water. In the Niger Delta, the water table in many areas is close to the surface though subject to seasonal variations. The water table is about 3–4 m in the dry season [14], but rises considerably in the rainy season. Groundwater is the main source of drinking water for over 80% of the population in the study area.
3 Induced polarization (IP) method
In the electrical method of geophysical prospecting, current is injected using two current electrodes A and B. The passage of the electric current through the ground creates a potential difference (∆V) usually measured across a pair of potential electrodes C and D. If the inducing current is turned off, the difference of potential (∆V) does not immediately drop to zero, but decays slowly over a period of time. The recording of the decaying voltage gives a decay curve ∆Vip(t). In time domain surveys, the decay curve ∆Vip(t) is the object of study because it is characteristic of the medium in terms of initial magnitude, slope and relaxation time. The amplitude of ∆Vip is related to the polarizability of the earth materials [15]. This capacity to polarize is referred to as the IP response. The form of the primary wave and the IP decay is shown in Fig. 2. Induced polarization is due to two main sources: (1) membrane polarization and (2) electrode polarization. The presence of clay causes membrane polarization. The clay particles, which are negatively charged, attract positive ions from the electrolytes in the capillaries of the clay particles and thus behave as ion-selective membrane impeding their mobility through the capillaries.
Time domain IP discharge curve
Electrode polarization produces similar effect but occurs when metallic minerals are present. The flow of electrons through a metal is much faster than the flow of ions in the electrolyte, and hence, opposite charges accumulate on the surface of metallic grains that block the path of ionic flow through the pore fluid. In Fig. 2, the chargeability M is computed by integrating the signal Vip along the decay over n time windows, or gates. The chargeability of IP effect was measured by integrating the area under the IP decay curve according to the relation [16, 17] given below:
$$M = \frac{1}{{V_{o} }}\int\limits_{t1}^{{t_{2} }} {V(t)} {\text{d}}t$$
where Vo is the voltage measured before the current is turned off, t1 and t2 are the start and stop time intervals, respectively, and V(t) is the decaying voltage. The chargeability (M) is usually expressed in millisecond (msec) or milliVolt/Volt (mV/V).
4 Data acquisition and processing
Prior to the acquisition of geophysical data and drilling of boreholes, locations with contrasting dissolved Fe concentration (low and high) in the groundwater within Yenagoa and environs were selected for study. The selection of these locations was based on analysis of ten groundwater samples collected from the existing domestic boreholes spread over the study area. The depths of these boreholes range from 9 to 30 m. Geochemical analysis of the groundwater samples shows that groundwater from Tombia, Akenfa III, etc., (Fig. 3) exhibits Fe concentration within WHO acceptable limits (~ 0.02–0.3 mg/L), while groundwater samples from Amabolou, Azikoro, etc. (Fig. 3), have Fe concentrations greater than 0.3 mg/L. This analysis helped in planning the field layout of the geophysical profiles.
Map of study area showing VES-IP sounding, borehole locations and 2D traverses
4.1 Geoelectrical sounding and induced polarization
We acquired geoelectrical and IP data in eleven locations, i.e. five in low-Fe areas and six in high-Fe areas (Fig. 3). The electrical resistivity–IP soundings were carried out using the Schlumberger configuration. In general, soundings were carried out using the Abem Terrameter SAS 1000. Maximum current electrode separation AB/2 ranged between 100 and 150 m. In the Schlumberger configuration, current was injected into the ground through two outer electrodes A and B and the resulting voltage difference at two potential electrodes (C and D) was measured. An increase in the depth of current penetration is achieved by progressively increasing the electrode spacing. Field precautions observed to ensure good vertical electrical sounding (VES) data quality included firm grounding of the electrodes and checking for current leakage and creeps to avoid spurious measurements. During the survey, the resistance and chargeability were measured concurrently. These data were interpreted using IX1D (Interpex) software. The field resistivity data were converted to apparent resistivity (ρa) values and plotted against half-current spacing (AB/2) on log–log scale. Guided by the general trend of the field curves, partial curve smoothening of the field curves was made. The 1D inversion software (Interpex) takes advantage of least-squares optimization technique. The program iteratively compares the field data to a theoretical model curve. The starting model is modified or adjusted successively until the difference between the observation and the model output is reduced to a minimum. In constructing a model, we have used the principle that all maxima, minima and point of inflexion in a geoelectrical sounding curve indicate the existence of boundaries of different lithologies. Using this approach, the subsurface was divided into a number of horizontal layers of given thickness. The program iteratively changes the resistivities to obtain a best fit with the field data for the layer thicknesses chosen for the model. Due to the inherent problem of equivalence in geosounding data interpretation [18], lithological information from drilled boreholes was used to constrain all depth estimates in order to minimize the choice of equivalent models by fixing layer thicknesses and depths while allowing the resistivities to vary [19]. The resulting true resistivities represent the best average resistivity for the given layer and are shown in Tables 1 and 3, respectively.
Summary of VES-IP model results and their corresponding thicknesses at low-Fe areas
VES-IP No
Layer 1 (top soil)
Layer 2 (silty sand)
Layer 3 (sand)
Layer 4 (sandy clay)
(Ωm)
(ms)
(mS/m)
error (%)
VES-IP 1
ρ is bulk resistivity, η is chargeability, MN is normalized chargeability, and h is thickness
Because the chargeability represents a measure of polarization magnitude relative to conduction magnitude [17, 18] and thus is approximately linearly related to the bulk resistivity, we also calculated the normalized chargeability (MN), using the following expression:
$$MN = \frac{M}{\rho }\left( {mS/m} \right)$$
to separate the effects of conduction and polarization.Where M is the chargeability and ρ is the layer bulk resistivity.
4.2 Electrical resistivity imaging
We also acquired one each 2D electrical resistivity imaging profile in the low-Fe area and as well as in the high-Fe area using the Wenner array (Fig. 3). The 2D resistivity profile was acquired to supplement the vertical electrical sounding (VES) and IP sounding data. This is because the 2D resistivity imaging gives a clearer picture of the lateral and vertical variation of the subsurface geological sequences. The 2D resistivity imaging data were acquired manually using the Wenner configuration. Each 2D profile was 100 m in length. The electrode separation ranged between 5 and 30 m in an interval of 5 m, with a total of 21 electrode positions for each profile. Field measurements were taken using electrode spacing of 5.0 m at electrode positions 1, 2, 3 and 4 in each profile. Then, each electrode was moved a distance of 5.0 m (one unit electrode spacing), the active electrode positions being 2, 3, 4 and 5. This procedure was continued to the end of the profile with electrode positions for the last measurement being 18, 19, 20 and 21. The electrode spacing was then increased by 5.0 m, as mentioned above for measurements of next data level, such that the active positions were 1, 3, 5 and 7. The procedure was then repeated by moving each of the electrodes a distance 5.0 m (one unit electrode spacing) and maintaining the electrode spacing for the data level until the electrodes were at electrode positions 15, 17, 19 and 20. This procedure was continued until 6 data levels were observed, yielding a total of 63 data points in each of the profiles. RES2DINV computer code [19] was used in the inversion of the 2D data. The computer program takes advantage of the nonlinear optimization technique in which a 2D resistivity model of the subsurface is automatically determined for input apparent resistivity data [19, 20]. In this program, the subsurface is subdivided into a number of rectangular blocks based on the spread of the observed data. The 2D data were inverted using the least-squares inversion with standard least-squares constraint which minimizes the square difference between the observed and the calculated apparent resistivity values. The program displays the distribution of electrical properties in the form of 2D pseudo-section plot. 2D pseudo-section plot gives a simultaneous display of both horizontal and vertical variation of the subsurface resistivity and are useful for initial quality assessment [21]. In constructing a 2D pseudo-section plot, each measured value is put at the intersection of two 45o lines through centres of the quadripole. Each horizontal line is then associated with a specific value of n (inter electrode spacing) and gives a pseudo-depth of investigation. It is pertinent to note that the larger the n-values, the greater the depths of investigation [21].
4.3 Hydrogeochemical analysis
Eleven boreholes were drilled in the study area using rotary drilling method. Five boreholes were drilled in the low-Fe areas, while six were drilled in high-Fe areas. Each borehole was located close to a VES point. The locations of the boreholes in the low-Fe areas are Tombia, Kpansia (behind the market along the expressway), Igbogene, Akenfa III and Akenfa III (NNPC Road), while the high-Fe areas include Amabolou I, II and III, Biogbolo, Agbura and Azikoro (Fig. 3). The boreholes were developed, and groundwater samples were collected in clean 500-ml polyethylene bottles. Prior to sample collection, the boreholes were pumped continuously for about 10–20 minutes. In these samples, in situ measurements of temperature, redox potential (Eh) and pH were taken using precalibrated portable pH/ORP meter at the time of groundwater sampling. Major ions such as Na, K, Ca, Mg, Fe, HCO3, Cl and SO4 were determined including total dissolved solids (TDS) in the laboratory using standard procedures [22]. Major ions like sulphate (SO4−2) were determined by spectrophotometric turbidimetry. Using EDTA, calcium (Ca2+) and magnesium (Mg2+) were determined titrimetrically; chloride (Cl−) was determined by standard AgNO3 titration and bicarbonate (HCO3−) was determined using titration with HCl. Sodium (Na+) and potassium (K+) were measured using flame photometry; nitrate (NO3−) was determined by colourimetry with a UV–visible spectrophotometer (brucine method) while iron was measured using colourimeter with a UV–visible spectrophotometer at 520 nm. Subsequently, the groundwater composition was correlated with the colour characteristics of the sediments. Chemical composition of the groundwater at the low- and high-Fe areas as well as the redox potential values is shown in Tables 2, 4 and 5, respectively.
Chemical composition of the groundwater at low-Fe areas
Borehole No
Depth (m)
Na+
Ca2+
Mg2+
Cl−
HCO3
All parameters have been expressed as mg/L except pH, EC and Eh. The unit of EC is µS/cm, and that of Eh is mV
The drilled boreholes were lithologically logged and sampled at 3.0 m or more often when characteristics of the sediment changed based on their grain size and colour. Each sample was assigned to one of the three colours—grey, off-white and brown by virtual inspection of the sediments. The Fe concentration of about 3 g of the wet sediments was measured by AAS and after extraction with hydroxylamine hydrochloride (NH2OH.HCl) in 25% acetic acid, and filtering using a 0.45-µm cellulose acetic filter. The boreholes were screened at depth intervals of either oxidized brownish sand aquifers, off-white or greyish reduced sediments. The depth of these boreholes varied between 8 and 30 m. Fe concentration in the aquifer sediments is shown in Table 5.
In this study, resistivity soundings and profiling were carried out in locations with contrasting dissolved iron concentrations, low and high in the groundwater (Fig. 3). IP soundings were also carried out simultaneously to support the resistivity interpretation for investigating the lithological control of Fe enrichment in the groundwater.
Figure 4 shows resistivity and IP sounding curve types in locations where the dissolved Fe concentration is low, while Fig. 5 shows resistivity/IP curve, and a comparison between the I-D resistivity/IP model of VES-IP 1 (Table 1) with the lithological information obtained from the nearest borehole (B-6) in Tombia (Fig. 3). The I-D inversion results for the Schlumberger resistivity/IP soundings and results of the hydrochemical analysis from groundwater samples from boreholes near the resistivity/IP sounding points in the low-Fe areas are also presented in Tables 1 and 2, respectively. Correlation of the resistivity results of VES-IP1 with the lithological information from the nearest borehole (B-6) shows that the stratigraphic sequence consists of four layers (within the depth of investigation) in which the model resistivity of the third layer is higher than those of the upper and lower layers (Fig. 5). The resistivity curve obtained in this area is predominantly K-type (Fig. 4), and the stratigraphic sequence consists of top soil, silty sand, sand and a sandy clay (Fig. 5). The resistivity and thickness of the top soil vary between 19 and 43 Ωm and 0.5–1.1 m, but are 19–295 Ωm and 0.8–6.6 m in the silty sand layer underlying the top soil (Table 1). The resistivity and thickness of the sandy layer which serves as the aquifer vary between 219 and 2955 Ωm and 6.1–27 m, respectively, while the resistivity of the sandy clay layer ranges between 50 and 298 Ωm. The 2D resistivity cross section (Fig. 6) correlated well with the borehole information (Fig. 5) and shows the detailed variation of the subsurface lithological sequence. Figure 6 shows a clear trend of high apparent resistivity values at the top (values ranging between 200 and 680 Ωm), while at the bottom, the apparent resistivity values tend to decrease (values are lower than 100 Ωm). Correlation of the 2D resistivity cross section with the lithological information from the borehole located close to the profile line (Fig. 5) indicates that the top soil with an apparent resistivity range of 80–100 Ωm is underlain by a layer with an apparent resistivity range of 150–200 Ωm. The borehole data indicate that this layer is composed of silty sand. The apparent resistivity of the layer below the silty sand layer varies between 300 and 1000 Ωm. This layer serves as the aquifer and consists of sand. Below the aquifer is a layer with apparent resistivity range less than 100 Ωm. The lithological information from the borehole shows that this layer is a sandy clay layer.
Resistivity and IP sounding curves in the low-Fe areas
Resistivity–IP Sounding at Tombia Village (VES-IP 1, low-Fe area); a resistivity–IP model curves compared to field data; b model VES-IP results with borehole log and lithologic interpretation shown for comparison
Inverse model resistivity section at Tombia Village (low-groundwater Fe site). The vertical line indicates the location of borehole
The chargeability (M) and normalized chargeability (MN) of the top soil range between 0.79 and 2.47 ms and 0.014–0.073 mS/m but range between 1.3 and 3.8 ms and 0.009–0.186 mS/m, respectively, in the silty sand layer. In the sandy layer, the chargeability and normalized chargeability values are between 0.71 and 11.25 ms and 0.001–0.031 mS/m but range between 1.72 and 126.3 ms and 0.007–0.424 mS/m in the sandy clay layer. The chargeability of a given medium indicates polarizability of the medium. Thus, chargeability is related to the permittivity and resistivity of the subsurface materials as well as the porosity and moisture content in the subsurface media. The normalized chargeability (i.e. the ratio between chargeability and resistivity) has also been reported to be a good parameter for discriminating lithotypes [17, 18]. These authors suggested that clean sands have low chargeability and low normalized chargeability, while clay and clayey sands have high chargeability and normalized chargeability values.
A careful analysis of the model results (Figs. 5 and 8) shows a strong correlation between the resistivity and chargeability anomalies with high resistivity values corresponding to relatively low chargeability and normalized chargeability values (Table 1). Deposits of clean sand and gravel are easily distinguished by their high resistivity values from their surrounding clay and silt [23, 24, 25]. Small measurable IP effects are associated with clean sand and gravel deposits [26]. The model results show low chargeability and low normalized chargeability values in the second and third layers (i.e. silty sand and sand layers). This indicates that these layers are mainly composed of sandy materials and less disseminated clayey materials. In contrast, high chargeability and normalized chargeability values are observed in the sandy clay layer ranging between 1.72 and 126.3 ms and 0.007–0.424 mS/m relative to layers 2 and 3. This is consistent with the observation of Amaya et al.; Vonhala [26, 27] who suggested that strong IP effects are commonly observed in sediments containing clays disseminated on the surface of larger grains. Hence, the sandy clay layer displays large IP effects as a result of the presence of disseminated clay. The lithology suggested by the 2D resistivity interpretation (Fig. 6) correlated well with the observed subsurface materials obtained from ground-truthing in this location and also clearly delineate the basal sandy clay layer. Layer 3 serves as the aquifer in the area, and residents tap their water from this layer. Table 2 shows that the Fe concentrations obtained from the analysis of groundwater samples abstracted from boreholes in these areas are within WHO acceptable limits. Additionally, the measured redox potential values from the groundwater immediately after abstraction are relatively high, ranging between 115 and 164 mV, indicating a mild oxidation environment [28].
Figure 7 shows resistivity–IP sounding curves in locations that have high Fe concentration in the groundwater. For comparative study, IP sounding curves are presented along with resistivity sounding curves. Figure 8 shows field data, I-D inversion results for Schlumberger resistivity/IP sounding at Amabolou (VES-IP 6), a high-Fe-concentration area. The resulting I-D resistivity/IP model was compared to the borehole lithological information. Interpretation of the resistivity and IP sounding curves shows a large variation in resistivity and chargeability and hence normalized chargeability. Generally, resistivity, chargeability and normalized chargeability values vary between 5 and 2288 Ωm, 0.7–204 ms and 0.016–2.722 mS/m, respectively (Table 3). In Fig. 7, we observed that for values of AB/2 greater than 5 m, the resistivity and polarization curves have opposite slopes. The top soil is relatively dry and resistive and unpolarizable, but the lower layer, being wet, is lower in resistivity and high in polarizability. The low-resistivity anomaly and enhanced IP and normalized chargeability effect is attributed to the cationic exchange capacity due to the increase in clay volume [26].
Resistivity and IP sounding curves in the high-Fe areas
Resistivity–IP sounding at Amabolou Village (high-Fe area); a resistivity–IP model curves compared to field data; b model VES-IP results with borehole log and lithologic interpretation shown for comparison
Summary of VES-IP model results and their corresponding thicknesses at high-Fe areas
Layer 2 (clay)
RMS error(%)
VES-IP 10
Clay and clayey sand display large IP effect due to cationic exchange capacity [26, 27]. Table 3 shows that beside the top layer, the underlying layers, especially layers 2 and 3, have high chargeability and normalized chargeability values reflecting enhanced surface polarization caused by the presence of disseminated clay. Of particular interest with respect to the hydrogeology and the mobilization of Fe in the aquifer is the second layer, characterized by low resistivity (5–96 Ωm) and high chargeability and normalized chargeability values (0.67–13.9 ms and 0.039–2.242 mS/m). The correlation of VES-IP 6 interpretation results with the nearest lithological information (Fig. 8) shows that layer 2 is clay. The significant correlation of low resistivity and high chargeability affirms also that this layer is clay. The 2D resistivity cross section (Fig. 9) shows that underlying the top soil is a layer characterized by low apparent resistivity values in the depth range of about 2.5–7.8 m corroborating the results of the vertical electrical sounding (VES). The apparent resistivity values are generally less than 60 Ωm. The lithological information from the nearest borehole (Fig. 8) shows that this layer is a clay layer. Underlying the clay layer is a layer with slightly higher apparent resistivity values ranging 60–120 Ωm within the depth range of 8–13 m. The borehole data indicate that this layer is composed of fine sand in a matrix of finer sediments (clay) [29]. Below the sandy clay layer is the sandy layer with apparent resistivity range of about 120–220 Ωm. This clay layer (aquitard) acts as a confining layer. Analysis of groundwater samples abstracted from the sandy layer below the clay layer show elevated concentrations of dissolved Fe (Table 4). Redox potential values are relatively low and range between 116 and 133 mV, indicating a mild reducing environment. It is pertinent to mention that in the present study, analysis of the normalized chargeability results (Tables 1 and 3) did not give a clear difference of normalized chargeability values in layers with high clay content and coarse material in some layers contrary to the report of [26]. For instance, in VES-IP 7, 9 and 10 layer 2, results of the normalized chargeability did not show a clear trend of increasing values in this layer with high clay content and lower values when coarse material is the main soil content (VES-IP 9, 10 layer 4) and thus did not contribute significantly to the interpretation of the geological features in these layers. However, normalized chargeability values in these layers were used to complement the results of the chargeability.
Inverse model resistivity section at Azikoro Village (low-groundwater Fe site)
Chemical composition of the groundwater at high-Fe areas
SO42−
NO3−
All parameters have been expressed as mg/L except pH, EC and Eh. The unit of EC is µS/cm, and that of energy potential (Eh) is mV
5.1 Source of Fe and role of near surface clay layer on groundwater Fe distribution
Fe concentrations were extracted from aquifer sediments from both the low and high locations. The results show that Fe concentrations in the high-Fe aquifer sediments range between 0.26 and 0.90 mg/L but are between 0.5 and 0.82 mg/L in the low-Fe aquifer sediments. Table 5 lists the Fe concentrations in several sediment samples from borehole cores and indicates that aquifer sediments containing abundant Fe act as the supply source for the shallow groundwater. There is no significant overall difference in the amount of extractable Fe in the aquifer sediments. This is consistent with the studies of [27] who reported that paralic deposits often have plenty of Fe substances. The similar amount of extractable Fe in the aquifer sediments implies that Fe is present in the aquifer materials in sufficient amounts. Thus, transfer to the dissolved phase can cause a large increase in groundwater.
Fe concentration in the aquifer sediments
potential (mV)
Fe conc. (mg/L)
High-Fe locations
Low-Fe locations
We infer therefore that the spatial distribution of groundwater Fe is as a result of the variation of redox conditions in the host aquifer. This implies that a reductive ambient subsurface environment is favourable to Fe ions transferring from the aquifer matrix into the groundwater. Although the decomposition of organic matter in groundwater and soil can consume dissolved oxygen and thus create a reductive hydrochemistry, in this case this effect is assumed to be significantly small. We opine that a relatively atmosphere-isolated state in the aquifer is responsible for the stronger reducibility of groundwater in the high-Fe locations. The investigations of the sediment stratigraphy and lithology across the study area show a near surface aquitard composed of argillaceous materials (clay) widely occurs in the upper most subsurface sediments in the high-Fe locations. This near surface aquitard heterogeneity and variation in thickness lead to variation in vertical recharge, localized dilution and confinement, resulting in varying redox conditions in the aquifer affecting Fe release.
An integrated hydrogeophysical investigation consisting of electrical resistivity and induced polarization techniques in parts of the Niger Delta delineated a widespread clay layer characterized by low resistivity (5–96 Ωm) and high chargeability and normalized chargeability values (0.67–13.9 ms and 0.134–2.242 mS/m) overlying the aquifer in locations that show elevated dissolved Fe concentrations. The thickness of the clay layer varies, pinching out at some places. In locations where the clay layer pinches out, the Fe concentration is within WHO acceptable limits (< 0.3 mg/L), implying that the lithological set-up plays a significant role in understanding Fe enrichment in groundwater in the Niger Delta. The low-permeability clay layer acts a confining layer and thus helps in creating atmosphere-isolated state in the underlying aquifer which is responsible for the reductive ambient subsurface groundwater environment favourable to Fe ions transferring from the aquifer matrix into the groundwater. The knowledge of the clay layer will be very helpful in selecting suitable sites for the installation of boreholes.
We are grateful to the Post-graduate Geophysics students in the Department of Physics who assisted with the field work and Mr. Udofia for producing the maps.
We do not have conflict of interest in this paper.
Okiongbo KS, Douglas RK (2013) Hydrogeochemical analysis and evaluation of groundwater quality in Yenagoa city and environs, Southern Nigeria. Ife J Sci 15:209–222Google Scholar
Okiongbo KS, Douglas RK (2015) Evaluation of major factors influencing the geochemistry of groundwater using graphical and multivariate statistical methods in Yenagoa City, Southern Nigeria. J Appl Water Sci 5:27–37CrossRefGoogle Scholar
Ohimain EI, Angaye T, Okiongbo KS (2013) Removal of iron, coliforms and acidity from groundwater obtained from shallow aquifer using trickling Filter method. J Environ Sci Eng A 2:549–555Google Scholar
Niwas S, Singhal DC (1981) Estimation of aquifer transmissivity from Dar Zarrouk parameters in porous media. J Hydrol 50:393–399CrossRefGoogle Scholar
Dhakate R, Singh VS, Negi BC, Chandra S, Rao VA (2008) Geomorphological and geophysical approach for locating favourable groundwater zones in granitic terrain, Andhra Pradesh, India. J Environ Manag 88:1373–1383CrossRefGoogle Scholar
Okiongbo KS, Mebine P (2015) Estimation of aquifer hydraulic parameters from geoelectric method—A case study of Yenagoa and environs Southern Nigeria. Arab J Geosci 8:6085–6093CrossRefGoogle Scholar
Akpokodje EU, Etu-Efeotor JO (1987) The occurrence and Economic potential of clean sand deposits of the Niger Delta. J Afr Earth Sci 6:61–65Google Scholar
Akpokodje EU (1987) The engineering-geological characteristics and classification of the major superficial soils of the Niger Delta. Eng Geol 23:193–201CrossRefGoogle Scholar
Akpokodje EG (1986) A method of reducing the cement content of two stabilized Niger Delta soils. Q J Eng. Geol. Lond 19:359–363CrossRefGoogle Scholar
Amajor LC (1991) Aquifers in the Benin Formation (Miocene—Recent), Eastern Niger Delta, Nigeria. Lithostratigraphy, Hydraulics and water quality. Environ Geol Water Sci 17:85–101CrossRefGoogle Scholar
Short KC, Stauble AJ (1967) Outline of the geology of the Niger Delta. Bull AAPG 51:761–779Google Scholar
Abam TKS (1999) Dynamics and quality of water resources in the Niger Delta. Impacts of urban growth on surface water and groundwater quality. In: Proceedings of IUGG 99 symposium, vol 259. IAHS Publication, Birmigham, pp 429–437Google Scholar
Mbonu PDC, Ebeniro JO, Ofoegbu CO, Ekine AS (1991) Geoelectric sounding for the determination ofaquifer characteristics in parts of Umuahia area of Nigeria. Geophysics 56:84–291CrossRefGoogle Scholar
Ekine AS, Osobonye GT (1996) Surface geoelectric sounding for the determination of aquifer Characteristics in part of Bonny Local Government Area of Rivers State. Niger J Phys 85:93–99Google Scholar
Summer JS (1976) Principles of induced polarization for geophysical exploration. Elsevier, AmsterdamGoogle Scholar
Schon JP (1996) Physical properties of rocks: fundamentals and principles of petrophysics. Pergamon, New York, p 583Google Scholar
Slater LD, Lesmes D (2002) IP interpretation in environmental investigations. Geophysics 68:77–88CrossRefGoogle Scholar
Lesmes D, Frye KM (2001) Influence of pore fluid chemistry on the complex conductivity and induced polarization responses of Berea sandstone. J Geophys Res 160:4079–4090CrossRefGoogle Scholar
Loke MH, Baker RD (1996) Rapid least square inversion of apparent resistivity pseudosections by a quasi-Newton method. Geophys Prospect 44:131–152CrossRefGoogle Scholar
Griffiths DH, Barker RD (1993) Two-dimensional resistivity imaging and modelling in areas of complex geology. J Appl Geophys 29:211–226CrossRefGoogle Scholar
Samouelian A, Cousin I, Tabbagh A, Bruand A, Richard G (2005) Electrical resistivity survey in soil science: a review. Soil Tillage Res 83:173–193CrossRefGoogle Scholar
APHA (1998) Standard methods for the examination of water and wastewater, 19th edn. American Public Health Association, WashingtonGoogle Scholar
Baines D, Smith DG, Frose DG, Bauman P, Nimack G (2002) Electrical resistivity ground imaging [ERGI]: a new tool for mapping the lithology and geometry of channel-belts and valley-fills. Sedimentology 49:441–449CrossRefGoogle Scholar
Beresnev IA, Hruby CE, Davis CA (2002) The use of multi-electrode resistivity imaging in gravel prospecting. J Appl Geophys 49:245–254CrossRefGoogle Scholar
Magnusson MK, Fernlund JMR, Dahlin T (2010) Geoelectrical imaging in the interpretation of geological conditions affecting quarry operations. Bull Eng Geol Environ 69:465–486CrossRefGoogle Scholar
Amaya AG, Dahlin T, Barmen G, Rosberg JE (2016) Electrical resistivity tomography and induced polarization for mapping the subsurface of alluvial fans: a case study in Punata (Bolivia). Geosciences 6:1–13CrossRefGoogle Scholar
Vonhala H (1997) Mapping oil contaminated sand and till with spectral induced polarization (IP) method. Geophys Prospect 45:303–326CrossRefGoogle Scholar
Hasan MA, Ahmed KM, Sracek O, Bhattacharya P, von Brömssen M, Broms S, Fogelström J, Mazumder ML, Jacks G (2007) Arsenic in shallow groundwater of Bangladesh: investigations from three different physiographic settings. Hydrogeol J 15:1507–1522CrossRefGoogle Scholar
Okiongbo KS, Soronnadi-Ononiwu GC (2016) Characterizing aggregate deposits using electrical resistivity method: case history of sand search in the Niger Delta, Nigeria. J Earth Sci Geotech Eng 8:1–16Google Scholar
1.Geophysics Research Group, Department of PhysicsNiger Delta UniversityWilberforce IslandNigeria
2.Department of PhysicsRivers State UniversityPort-HarcourtNigeria
Okiongbo, K.S., Oboshenure, K.K. & Amakiri, A.R.C. SN Appl. Sci. (2020) 2: 107. https://doi.org/10.1007/s42452-019-1876-3
Received 29 May 2019
First Online 18 December 2019 | CommonCrawl |
\begin{document}
\title{On the vanishing of almost all primary components of the Shafarevich-Tate group of elliptic curves over the rationals}
\begin{abstract} The Shafarevich-Tate and Selmer groups arise in the context of Kummer theory for elliptic curves. The finiteness of the Shafarevich-Tate group of an elliptic curve $E$ over the field of rational numbers is included in the Birch and Swinnerton-Dyer conjectures, and is still an open question.
We present an overview of the Shafarevich-Tate and Selmer groups of an elliptic curve in the framework of Galois cohomology. Known results on the finiteness of the Shafarevich-Tate group are mentioned, including results of Coates and Wiles, Rubin, Gross and Zagier, and Kolyvagin.
We then prove the vanishing of the $\ell$-primary component of the Shafarevich-Tate group for almost all primes $\ell$, for any elliptic curve $E$ over the rationals without complex multiplication. \end{abstract}
\section{Introduction} \label{section:introduction}
\subsection{Statement of the problem} \label{subsection:statement}
We recall Kummer theory for elliptic curves \cite[pp. 331--332]{silverman2009} to introduce the Shafarevich-Tate groups and the related Selmer groups.
Let $K$ be an algebraic number field with absolute Galois group $\mathcal{G}=\Gal(\overline{K}/K)$, where $\overline{K}$ denotes the algebraic closure of $K$. Given an elliptic curve $E$ over $K$ and a prime number $\ell$, one has the short exact sequence (isogeny property) of $\mathcal{G}$-modules: \begin{equation} \label{eq:eq1final6} 0 \rightarrow E[\ell] \rightarrow E(\overline{K}) \xrightarrow{[\ell]} E(\overline{K}) \rightarrow 0, \end{equation} where $E[\ell]$ denotes the group of $\ell$-torsion points in $E(\overline{K})$. This yields, from the long exact sequence of Galois cohomology, the following exact sequence of Abelian groups: \begin{equation} \label{eq:eq2final6} 0 \rightarrow E(K)/[\ell](E(K)) \xrightarrow{\partial} \Homol^{1}(\mathcal{G},E[\ell]) \rightarrow \Homol^{1}(\mathcal{G},E(\overline{K}))[\ell] \rightarrow 0. \end{equation} Recall that Galois cohomology (see \cite{gruenberg1967} and \cite[pp. 333-335]{silverman2009}) is based on the profinite group structure of Galois groups (see \cite{gruenberg1967} and \cite[pp. 4-6]{neukirch1986}). In particular, $1$-cocycles are continuous maps with respect to the Krull topology on Galois groups \cite[p. 2]{neukirch1986} and the discrete topology on Galois modules. When the Galois group is finite, Galois cohomology coincides with group cohomology.
Given a place $v$ of $K$, let $K_v$ be the completion of $K$ at $v$, and denote $\mathcal{G}_v$ the absolute Galois group of $K_v$. Then, $\mathcal{G}_v$ may be viewed as a subgroup of $\mathcal{G}$ upon considering an embedding $\xi$ of $\overline{K}$ into $\overline{K}_v$. Moreover, the resulting embedding is continuous with respect to the Krull topology on Galois groups. One then has a commutative diagram: \begin{equation} \label{eq:eq3final6} \begin{CD} 0 @>>> E(K)/[\ell](E(K)) @>{\partial}>> \Homol^1(\mathcal{G},E[\ell]) @>>> \Homol^1(\mathcal{G},E(\overline{K}))[\ell] @>>> 0\\ @. @VVV @VV{Res^{\mathcal{G}}_{\mathcal{G}_v}}V @VV{Res^{\mathcal{G}}_{\mathcal{G}_v}}V \\ 0 @>>> E(K_v)/[\ell](E(K_v)) @>{\partial}>> \Homol^1(\mathcal{G}_v,E[\ell]) @>>> \Homol^1(\mathcal{G}_v,E(\overline{K}_v))[\ell] @>>> 0, \end{CD} \end{equation} where $Res^{\mathcal{G}}_{\mathcal{G}_v}$ denotes the restriction map of Galois cohomology \cite[pp. 331--332]{silverman2009}. At this point, recall that the Shafarevich-Tate group of $E$ over $K$ is defined as \cite[p. 332]{silverman2009}: \begin{equation} \label{eq:eq4final6} {\mbox{\textcyr{Sh}}}(E/K) = \Ker \Bigl \{ \Homol^1(\mathcal{G},E(\overline{K})) \rightarrow \oplus_v \Homol^1(\mathcal{G}_{v},E(\overline{K}_v)) \Bigr \}. \end{equation} Also, the $[\ell]$-Selmer group is defined as \cite[pp. 331--334]{silverman2009}: \begin{equation} \label{eq:eq5final6} S^{[\ell]}(E/K) =\Ker \Bigl \{ \Homol^1(\mathcal{G},E[\ell]) \rightarrow \oplus_{v} \Homol^1(\mathcal{G}_v,E(\overline{K}_v)) \Bigr \}. \end{equation} In both equations (\ref{eq:eq4final6}) and (\ref{eq:eq5final6}), $v$ covers the set at all places of $K$. Note that each homomorphism $\Homol^1(\mathcal{G},E[\ell]) \rightarrow \Homol^1(\mathcal{G}_v,E(\overline{K}_v))$ maps into the $\ell$-torsion group $\Homol^1(\mathcal{G}_v,E(\overline{K}_v))[\ell]$. The above commutative diagram then yields a short exact sequence of the form \cite[p. 333]{silverman2009}: \begin{equation} 0 \rightarrow E(K)/[\ell](E(K)) \xrightarrow{\partial} S^{[\ell]}(E/K) \rightarrow {\mbox{\textcyr{Sh}}}(E/K)[\ell] \rightarrow 0. \end{equation}
One can show that the Selmer group $S^{[\ell]}(E/K)$ is finite \cite[pp. 333--334]{silverman2009}, from which follows the finiteness of $E(K)/[\ell](E(K))$ (Weak Mordell-Weil Theorem). A descent argument based on the notion of height function then shows the following result \cite[Theorem 6.7, p. 239]{silverman2009}.
\begin{thm}[Mordell-Weil Theorem] \label{thm:Theorem1} For any elliptic curve over an algebraic number field $K$, the group $E(K)$ is finitely generated. \end{thm}
From the Mordell-Weil Theorem, one can define the rank of an elliptic curve: \begin{equation} \rank(E/K) = \rank_{\mathbb{Z}} E(K). \end{equation}
Given a prime number $\ell$, the Mordell-Weil Theorem implies that: \begin{equation} \label{eq:eq8final6} \dim_{\mathbb{F}_\ell} E(K)/[\ell](E(K)) = \rank(E/K) + \dim_{\mathbb{F}_\ell} E[\ell](K), \end{equation} where $E[\ell](K)$ is the group of $\ell$-torsion points of $E(K)$. Considering the special case where $K=\mathbb{Q}$, a Theorem of Mazur implies that $\# E_{tor}(\mathbb{Q})\leq 16$ \cite[Theorem 7.5, p. 242]{silverman2009}, so that $\dim_{\mathbb{F}_\ell} E[\ell](\mathbb{Q})=0$ for almost all prime numbers $\ell$. Then, $\rank(E/\mathbb{Q}) = \dim_{\mathbb{F}_\ell} E(\mathbb{Q})/[\ell](E(\mathbb{Q}))$.
Next, we recall the following results on the $L$-series attached to elliptic curves. Given an elliptic curve $E$ over a global field $K$, one defines the auxiliary function \cite[p. 450]{silverman2009}: \begin{equation} \Lambda(E,s) = N_E^{s/2} (2 \pi)^{-s} \Gamma(s) L(E,s), \end{equation} where $N_E$ denotes the conductor of $E/K$, $\Gamma$ denotes the Euler gamma function, and $L(E,s)$ is the $L$-series attached to $E$ \cite[pp. 449--452]{silverman2009}.
A construction of Eichler and Shimura associates to special functions, called modular forms \cite[Section C.12]{silverman2009}, a type of elliptic curves, called modular elliptic curves \cite[Section C.13]{silverman2009}, for which analytic continuation of the auxiliary function $\Lambda(E,s)$ to the entire complex plane can be demonstrated \cite[Section C.16]{silverman2009}. See \cite{murty1995} for a brief introduction.
Now, the Taniyama-Shimura-Weil conjecture states that any elliptic curve over $\mathbb{Q}$ is modular. This conjecture was proved in 1995 for semi-stable elliptic curves over $\mathbb{Q}$ \cite{wiles1995,taylor1995} and then, the proof was extended to cover all elliptic curves over $\mathbb{Q}$ \cite{breuil2001}. From there, one can conclude that $L(E,s)$ has an analytic continuation on the entire complex plane.
\begin{thm}[Wiles 1995, Taylor and Wiles 1995, Breuil et al. 2001] \label{thm:Theorem2} Modularity Theorem: Any elliptic curve $E$ over $\mathbb{Q}$ is modular. \end{thm}
Based on Theorem \ref{thm:Theorem2}, one can define the analytic rank of an elliptic curve over $\mathbb{Q}$: \begin{equation} \rank_{an}(E/\mathbb{Q}) = \ord_{s=1} L(E,s). \end{equation}
The Birch and Swinnerton-Dyer conjectures include the following statement \cite[Conjecture 16.5, part a), p. 452]{silverman2009}: \begin{itemize} \item[] BSD-1: $\rank(E/\mathbb{Q}) = \rank_{an}(E/\mathbb{Q})$, for any elliptic curve $E$ over $\mathbb{Q}$. \end{itemize}
The Birch and Swinnerton-Dyer conjectures assume also the following conjecture \cite[p. 341]{silverman2009} formulated independently by Shafarevich and Tate \cite{rubin2002}: \begin{itemize} \item[] S-T: the Shafarevich-Tate group ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})$ is finite, for any elliptic curve $E$ over $\mathbb{Q}$. \end{itemize} From S-T, one would have: \begin{equation} \rank(E/\mathbb{Q}) = \dim_{\mathbb{F}_\ell} E(\mathbb{Q})/[\ell](E(\mathbb{Q})) = \dim_{\mathbb{F}_\ell} S^{[\ell]}(E/\mathbb{Q}), \end{equation} for all but finitely many prime numbers $\ell$, based on Mazur's Theorem on torsion points.
Conjecture S-T is assumed in the second part of the BSD conjectures (BSD-2) on the value of the leading coefficient in the Taylor expansion of $L(E,s)$ at $s=1$. See \cite[Conjecture 16.5, part b), p. 452]{silverman2009}.
A result of Coates and Wiles \cite{coates1977} states that BSD-1 holds in the case of an elliptic curve over $\mathbb{Q}$ with complex multiplication (CM), if the analytic rank is equal to $0$. Rubin \cite{rubin1987} proved that conjecture S-T holds under the same conditions.
\begin{thm}[Coates and Wiles 1977, Rubin 1987] \label{thm:Theorem3} Let $E$ be an elliptic curve over $\mathbb{Q}$ with CM. Assume that $\rank_{an}(E/\mathbb{Q})= 0$. Then, conjectures BSD-1 and S-T hold. \end{thm}
Kolyvagin extended Rubin's result to any elliptic curve of analytic rank at most $1$ \cite{kolyvagin1988,kolyvagin1988b,kolyvagin1991}, building on a result of Gross and Zagier \cite{gross1986}; see also \cite[Theorem 1, p. 430]{kolyvagin1990}.
\begin{thm}[Gross and Zagier 1986, Kolyvagin 1988--1991] \label{thm:Theorem4} Let $E$ be an elliptic curve over $\mathbb{Q}$. Assume that $\rank_{an}(E/\mathbb{Q})\leq 1$. Then, conjectures BSD-1 and S-T hold. \end{thm}
The reader may consult \cite{rubin1999} and \cite{rubin2002} for further reading on the Birch and Swinnerton-Dyer conjectures and the rank of elliptic curves. In particular, the notions of Heegner points and Euler systems are explained. In this work, these notions
do not intervene.
Results on the finiteness of the Shafarevich-Tate group have been obtained by Kolyvagin and Logachev in the case of Abelian varieties \cite{logachev1989,logachev1991}.
The finiteness of ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})$ implies that its order is a perfect square, based on Cassels' pairing for elliptic curves \cite[p. 341]{silverman2009}. Poonen and Stoll have studied the Cassels-Tate pairing in the case of Abelian varieties \cite{poonen1999}.
\subsection{Main theorem of this work and consequences} \label{subsection:MainTheorems}
Firstly, as mentioned in \cite[Section 3]{poonen1999b}, the Shafarevich-Tate group is a torsion group, for the simple reason that it is a subgroup of the torsion group $\Homol^1(\mathcal{G},E(\overline{K}))$, having considered Galois cohomology.
The following proposition states equivalent formulations.
\begin{prop} \label{thm:Proposition1} Let $E$ be an elliptic curve over an algebraic number field $K$. Let $\ell$ be a prime number. Then, the following conditions are equivalent:
a) \begin{equation} {\mbox{\textcyr{Sh}}}(E/K)_{\ell} = 0, \end{equation} where ${\mbox{\textcyr{Sh}}}(E/K)_{\ell}$ denotes the $\ell$-primary component of ${\mbox{\textcyr{Sh}}}(E/K)$.
b) \begin{equation} {\mbox{\textcyr{Sh}}}(E/K)[\ell] = 0. \end{equation}
c) \begin{equation} \partial: E(K)/[\ell](E(K)) \xrightarrow{\approx} S^{[\ell]}(E/K). \end{equation}
d) \begin{equation} S^{[\ell]}(E/K) = \Ker\{ \Homol^{1}(\mathcal{G},E[\ell]) \rightarrow
\Homol^{1}(\mathcal{G},E(\overline{K})) \}. \end{equation} \end{prop}
\begin{proof} $a) \Rightarrow b)$. This is clear since ${\mbox{\textcyr{Sh}}}(E/K)[\ell] \subseteq {\mbox{\textcyr{Sh}}}(E/K)_{\ell}$.
$b) \Rightarrow a)$. Assume that ${\mbox{\textcyr{Sh}}}(E/K)[\ell]=0$. Let $A$ be any finite subgroup of ${\mbox{\textcyr{Sh}}}(E/K)_{\ell}$. Then, $A$ is of the form $\oplus_{i=1}^{n} \mathbb{Z}/\ell^{m_i}\mathbb{Z}$, where $m_i\in \mathbb{N}$. Now, one must have $n \leq \dim_{\mathbb{F}_\ell} {\mbox{\textcyr{Sh}}}(E/K)[\ell]$. Thus, $n=0$ and ${\mbox{\textcyr{Sh}}}(E/K)_{\ell}=0$.
$b) \Leftrightarrow c)$. This follows from the short exact sequence: \begin{equation} \label{eq:eq16final6} 0 \rightarrow E(K)/[\ell](E(K)) \xrightarrow{\partial} S^{[\ell]}(E/K) \rightarrow {\mbox{\textcyr{Sh}}}(E/K)[\ell] \rightarrow 0. \end{equation}
$c) \Leftrightarrow d)$. From the short exact sequence (\ref{eq:eq2final6}), one has an isomorphism: \begin{equation} \partial: E(K)/[\ell](E(K)) \xrightarrow{\approx} \Ker \Bigl \{ \Homol^{1}(\mathcal{G},E[\ell]) \rightarrow \Homol^{1}(\mathcal{G},E(\overline{K})) \Bigr \}. \end{equation} From the compositum of homomorphisms at any place $v$ of $K$: \begin{equation} \Homol^{1}(\mathcal{G},E[\ell]) \rightarrow \Homol^{1}(\mathcal{G},E(\overline{K})) \xrightarrow{Res^{\mathcal{G}}_{\mathcal{G}_v}} \Homol^{1}(\mathcal{G}_v,E(\overline{K}_v)), \end{equation} one obtains \begin{equation} \partial: E(K)/[\ell](E(K)) \xrightarrow{\approx} \Ker \Bigl \{ \Homol^{1}(\mathcal{G},E[\ell]) \rightarrow \Homol^{1}(\mathcal{G},E(\overline{K})) \Bigr \} \subseteq S^{[\ell]}(E/K). \end{equation} The equivalence is now clear. \end{proof}
{\bf Remark 1.} \label{remark1} Part d) of the proposition expresses a local-global principle: $f \in\Homol^{1}(\mathcal{G},E[\ell])$ splits in $E(\overline{K}_v)$ for all places $v$ if and only if it does in $E(\overline{K})$. The obstruction to this principle is thus ${\mbox{\textcyr{Sh}}}(E/K)_{\ell}$ in view of part a) of the proposition.\\
We now state the main results of this paper. As in \cite[IV-2.1]{serre1968}, we say that an elliptic curve has complex multiplication (CM), if it does over some finite extension $F_{\textrm{\tiny CM}}/\mathbb{Q}$; {\em i.e.}, $\End_{F_{\textrm{\tiny CM}}}(E)$ is an order in an imaginary quadratic field $K_{\textrm{\tiny CM}}$ \cite[Section 5]{rubin1999}.
Given a prime number $\ell$, one has a Galois representation $\rho_\ell: \mathcal{G} \rightarrow {\bf GL}_2(T_\ell)$, obtained by Galois action on the Tate module $T_\ell$ of $E$. This representation identifies $\Gal(L_{\infty}/\mathbb{Q})$ with $\rho_\ell(\mathcal{G})$, where $L_\infty$ denotes the field obtained by adjoining to $\mathbb{Q}$ the affine coordinates of all $\ell^n$-torsion points of $E$, with $n\geq 1$.
\begin{thm} \label{thm:Theorem5} Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM, and consider a Weierstrass equation of the form $y^2=x^3+Ax+B$,
with $A,B\in \mathbb{Z}$.
Let $\ell \not = 2,3,5,7,13$ be a prime number. Assume that: i) $\rho_\ell(\mathcal{G})$ is the full linear group ${\bf GL}_2(\mathbb{Z}_\ell)$; and ii) $\ell \nmid \Delta^\prime := 4A^3+27 B^2$.
Then, one has: \begin{equation} S^{[\ell]}(E/\mathbb{Q}) = \Ker \Bigl \{ \Homol^{1}(\mathcal{G},E[\ell]) \rightarrow
\Homol^{1}(\mathcal{G},E(\overline{\mathbb{Q}})) \Bigr \}. \end{equation} \end{thm}
Note that from Serre's Theorems \cite[Th\'eor\`eme 2, p. 294]{serre1972} and \cite[Th\'eor\`eme 4$^\prime$, p. 300]{serre1972}, it follows that the Galois group $\rho_\ell(\mathcal{G})$ is the full linear group for almost all primes, whenever $E$ has no CM. Furthermore, in the case $E$ is semi-stable ({\em i.e.}, with no additive reduction) without CM, Mazur's Theorem \cite[Theorem 4, p. 131]{mazur1978} implies that $\rho_\ell(\mathcal{G})$ is the full linear group for $\ell\geq 11$. It follows that ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{\ell}=0$, for any $\ell \geq 17$ not dividing $\Delta^\prime$, whenever $E$ is semi-stable.
Proposition \ref{thm:Proposition1} states that Theorem \ref{thm:Theorem5} implies the following consequences.
\begin{cor} \label{thm:Corollary1} Let $E$ be any elliptic curve over $\mathbb{Q}$ without CM. Then, for almost all prime numbers $\ell$, one has:
a) \begin{equation} {\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{\ell} = 0, \end{equation} where ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{\ell}$ denotes the $\ell$-primary component of ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})$;
b) \begin{equation} {\mbox{\textcyr{Sh}}}(E/\mathbb{Q})[\ell] = 0; \end{equation}
c) \begin{equation} \partial: E(\mathbb{Q})/[\ell](E(\mathbb{Q})) \xrightarrow{\approx} S^{[\ell]}(E/\mathbb{Q}). \end{equation} \end{cor}
Mazur's Theorem on torsion points \cite[Theorem 7.5, p. 242]{silverman2009} then implies the following result.
\begin{cor} \label{thm:Corollary2} Let $E$ be any elliptic curve over $\mathbb{Q}$ without CM. Then, for almost all prime numbers $\ell$, one has: \begin{equation} \rank(E/\mathbb{Q}) = \dim_{\mathbb{F}_\ell} S^{[\ell]}(E/\mathbb{Q}). \end{equation} \end{cor}
Since ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})[\ell]$ is finite for any prime $\ell$, as it is a quotient group of the finite group $S^{[\ell]}(E/\mathbb{Q})$, it follows that the $\ell$-primary component of ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})$ is of the form: \begin{equation} {\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{\ell} = \left ( \mathbb{Q}_\ell/\mathbb{Z}_\ell \right )^{n_\ell} \oplus T_\ell, \end{equation} where $n_\ell\geq 0$ and $T_\ell$ is a finite $\ell$-group \cite[Section 12]{poonen1999b}.
Thus, based on Theorem \ref{thm:Theorem5}, the only missing piece to proving that ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})$ is finite in the non-CM case, is a proof that ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})$ has no infinitely divisible element. See also \cite[p. 341]{silverman2009} on this issue.
Examples of CM elliptic curves of rank $2$ or $3$ with endomorphism ring $\mathbb{Z}[i]$ are studied in \cite{coates2010}. The statement of \cite[Theorems 1.2]{coates2010} assumes the condition $\ell \equiv 1 \mod 4$, and the very strong condition $\ell<30,000$ (and $\ell \not = 41$), in the case of a specific curve. In the case of \cite[Theorems 1.3]{coates2010}, the condition $\ell \equiv 1 \mod 4$ is also assumed, and the extra restriction that $\ell<30,000$ (except for finitely many exceptions), and the statement is valid for $5$ specific elliptic curves. In contrast, Theorem \ref{thm:Theorem5} is valid for any elliptic curve without CM, and all primes $\ell$, but finitely many. However, we have not succeeded in carrying out the strategy of our proof to the CM case, as of now.
In Section \ref{section:examples}, an example from \cite{penney1975} of an elliptic curve $E$ over the rationals without CM of rank at least $7$ is mentioned. Furthermore, we show that, in this example, $\ell=41$ is the smallest prime ({\em i.e.}, based on the conditions of Theorem \ref{thm:Theorem5}) for which Corollary \ref{thm:Corollary2} applies. Therefore, one can in principle find out the exact rank of $E/\mathbb{Q}$ from a computation of $\dim_{\mathbb{F}_{41}} S^{[41]}(E/\mathbb{Q})$.
It can be noticed that this example solves the open problem mentioned in \cite[Problem 2.16, p. 27]{stein2007} in the non-CM case.
An example that was communicated to us by Professor C. Wuthrich is also mentioned in Section \ref{section:examples}. This example shows that the condition $\rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{Z}_\ell)$ is not sufficient to conclude that ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_\ell=0$, if ever $\ell$ is one of the exceptional ones ({\em i.e.}, $2$, $3$, $5$, $7$, or $13$). This issue is crucial, in view of BSD-2.
We end this paper with a complement to Proposition \ref{thm:Proposition2} that clarifies its proof, but that is not needed as such for the proof of Theorem \ref{thm:Theorem5} that is presented here.
\section{Background on elliptic curves} \label{sec:background}
\subsection{Basic notions} \label{subsection:basicElliptic}
Let $K$ be a field and $E$ be an elliptic curve over $K$; {\em i.e.}, a smooth projective curve of genus $1$, together with a base point $O$. The elliptic curve admits a Weierstrass equation \cite[p. 42]{silverman2009}: \begin{equation} \label{eq:eq26final6} y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, \end{equation} with coefficients $a_i\in K$, $i=1,2,3,4,6$. One defines the quantities: \begin{equation} \begin{cases} b_2=a_1^2+4a_2;\\ b_4=2a_4+a_1a_3;\\ b_6=a_3^2+4a_6;\\ b_8=a_1^2a_6+4a_2a_6-a_1a_3a_4+a_2a_3^2-a_4^2. \end{cases} \end{equation} We also set: \begin{equation} \begin{cases} c_4=b_2^2-24b_4;\\ c_6=-b_2^3+36b_2b_4-216 b_6. \end{cases} \end{equation} Then, the discriminant $\Delta$ of $E$ corresponding to a given Weierstrass equation is equal to: \begin{equation} \Delta(E):=-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6, \end{equation} and its $j$-invariant (independent of the Weierstrass equation) is equal to: \begin{equation} j(E):=c_4^3/\Delta. \end{equation}
Given the cubic curve defined by a Weierstrass equation (\ref{eq:eq26final6}), there are three cases \cite[p. 45]{silverman2009}:
(1) The curve is non-singular if $\Delta\not =0$.
(2) The curve has a node if $\Delta=0$ and $c_4\not =0$.
(3) The curve has a cusp if $\Delta=0$ and $c_4=0$.
In cases (2) and (3), there is only one singular point. In case (1), the curve is an elliptic curve with base point $O=[0,1,0]$.
The elliptic curve has also Weierstrass equation $y^2=x^2-27 c_4 x -54 c_6$, if the characteristic of $K$ is different from $2$ and $3$ \cite[p. 43]{silverman2009}. Thus, it is of the form $y^2=x^3+Ax+B$. Two elliptic curves are isomorphic over $\overline{K}$ if and only if they have the same $j$-invariant \cite[p. 45]{silverman2009}. If $K$ has characteristic different from $2$ and $3$, the proof of that result \cite[pp. 46--47]{silverman2009} shows that an isomorphism holds over a base extension obtained by adjoining $(A/A^\prime)^{1/4}$ (case $j=1728$) or $(B/B^\prime)^{1/6}$ (case $j=0$) or $(A/A^\prime)^{1/4}=(B/B^\prime)^{1/6}$ (other cases) to $K$, where the two curves have equations $y^2=x^3+Ax+B$ and $y^2=x^3+A^\prime x+B^\prime$ over $K$, respectively. So, unless $j=0$ or $1728$, the base field extension has degree dividing $2$ (the g.c.d. of $4$ and $6$).
There is a group law defined on $E(K)$ that is a consequence of a special case of Bezout's Theorem, but that can also be defined explicitly. See \cite[Chapter II, Section \S 2]{silverman2009}.
Now, let $\ell$ be a prime number. If the characteristic of $K$ is different from $\ell$, then the group $E[\ell]=E[\ell](\overline{K})$ of $\ell$-torsion points of $E$ is isomorphic to $\mathbb{Z}/\ell\mathbb{Z} \oplus Z/\ell \mathbb{Z}$.
If $K$ has characteristic $\ell$, then $E[\ell]$ is isomorphic to $0$ or $\mathbb{Z}/\ell\mathbb{Z}$. See \cite[p. 86]{silverman2009}.
If $m$ is a positive integer coprime with the characteristic of $K$, then there is the Weil pairing $e_m: E[m] \times E[m] \rightarrow \mu_m$, which is bilinear, alternating, non-degenerate, Galois invariant, and compatible \cite[Proposition 8.1, p. 94]{silverman2009}. As a consequence, one deduces that $\mu_m \subset K$, if $E[m] \subset E(K)$, under the condition $\cha (K) \nmid m$ \cite[Corollary 8.1.1, p. 96]{silverman2009}.
Given an elliptic curve over a field $K$, one constructs its formal group $F$ as in \cite[pp. 115-120]{silverman2009}. If $K$ has characteristic $\ell$, multiplication by $\ell$ in $F$ (denoted $\ell[X]\in K[[X]]$) is either $0$ or else is of the form $g(X^{\ell^h})$, where $g^\prime(0)\not = 0$ \cite{kolyvagin1980}. In the latter case, $h$ is called the height of $F$.
Let $k$ be a finite field of characteristic $\ell$ and $\widetilde E$ be an elliptic curve over $k$. Then, either \cite[p. 144--145]{silverman2009}:
(1) The formal group of $\widetilde E$ has height $h=2$ and $\widetilde E[\ell]=0$ (the Hasse invariant is $0$, or the curve is supersingular);
or
(2) The formal group of $\widetilde E$ has height $1$ and $\widetilde E[\ell]=\mathbb{Z}/\ell\mathbb{Z}$ (the Hasse invariant is $1$, or the curve is ordinary).
The first case occurs if and only if $j(\widetilde E)\in \mathbb{F}_{\ell^2}$ and the map $[\ell]$ is purely inseparable.
\subsection{Elliptic curves over local fields} \label{subsection:localElliptic}
Let $K$ be a finite extension of ${\mathbb{Q}}_p$ and let $\overline{K}$ be its algebraic closure. Let $v$ be the discrete valuation of $K$. Given an elliptic curve $E$ over $K$, we consider its minimal Weierstrass equation \cite[pp. 185--187]{silverman2009}. That is a Weierstrass equation with coefficients in the integer ring $\mathcal{O}_v$ of $K$ with minimal value of $v(\Delta)$ among all such equations. Therefore, one can look at its reduction $\widetilde E$ modulo a uniformizer $\pi_v$ of $K$ \cite[p. 187--188]{silverman2009}, defined over the residue field $k_v$ of $K$. One says \cite[pp. 196--197]{silverman2009}:
1) $E$ has good reduction if $\widetilde E$ is non-singular ($v(\Delta)=0$).
2) $E$ has multiplicative reduction if $\widetilde E$ has a node ($v(\Delta)>0$ and $v(c_4)=0$).
3) $E$ has additive reduction if $\widetilde E$ has a cusp ($v(\Delta), v(c_4) >0$).
The set of non-singular points $\widetilde E_{ns}(\overline{k}_v)$ of the reduced curve forms a group \cite[p. 56]{silverman2009}. In the case of good reduction $\widetilde E_{ns}(\overline{k}_v)=\widetilde E(\overline{k}_v)$ is an elliptic curve defined over $k_v$. In the case of multiplicative reduction, $\widetilde E_{ns}(\overline{k}_v)\approx \overline{k}_v^*$. In the case of additive reduction, $\widetilde E_{ns}(\overline{k}_v)\approx \overline{k}_v^+$. See also \cite[Exercise 3.5, p. 105]{silverman2009}.
A sufficient condition for a Weierstrass equation to be minimal is that $v(\Delta)<12$ or that $v(c_4)<4$ \cite[Remark 1.1, p. 186]{silverman2009}. Therefore, in the case of good reduction ($v(\Delta)=0$) or multiplicative reduction ($v(c_4)=0$), a minimal Weierstrass equation remains minimal after base field extension \cite[Proposition 5.4.(b), p. 197]{silverman2009}. In the case of additive reduction, after a suitable finite base field extension (see below), the reduction turns either good or multiplicative. For an example of the former case, see \cite[Example 5.2, p. 196--197]{silverman2009}. For an example of the latter case, let $p$ be a prime number greater than $3$ and consider $E\::\: y^2=x^3+\sqrt[3]{p}x^2+p^2$ over $K=\mathbb{Q}_p(\sqrt[3]{p})$; then, over $K(\sqrt[2]{p})$, $E$ has Weierstrass equation $y^2=x^3+x^2+p$, as can be seen with the change of variable $y=\sqrt[2]{p}y^\prime$ and $x=\sqrt[3]{p}x^\prime$.
Next, recall that $E$ has good reduction after a base extension (potential good reduction) if and only if its $j$-invariant is an integer of $K$ \cite[p. 197]{silverman2009}. The proof of this result in the case $\cha(k_v)\not = 2$ \cite[p. 199]{silverman2009} relies on a Weierstrass equation in Legendre form $y^2=x(x-1)(x-\lambda)$, $\lambda\not = 0,1$, \cite[p. 49]{silverman2009}. Such an equation can be obtained after adjoining the roots of the cubic polynomial $x^3+(b_2/4) x^2+(b_4/2) x+b_6/4=(x-e_1)(x-e_2)(x-e_3)$ and then adjoining the square root of $e_2-e_1$. Thus, the base field extension $K^\prime/K$ can be taken of degree dividing $12$. If $\cha(k_v) =2$, the proof relies on a Weierstrass equation in Deuring normal form $y^2+\alpha x y + y=x^3$, $\alpha^3\not = 27$. Such an equation
is obtained after adjoining a root $\alpha$ of the polynomial $x^3(x^3-24)^3-(x^3-27)j(E)$, yielding a base field extension of degree dividing $3d^\prime$ with $1 \leq d^\prime \leq 4$, and then over an extra base field extension of degree $2$, $4$ or $6$ to obtain an isomorphism with the initial elliptic curve \cite[Proposition 1.3, p. 412, and p. 47]{silverman2009}. In all cases, the base field extension $K^\prime/K$ has degree $d$ divisible only by powers of $2$ and $3$.
If $\cha(k_v)\not =2$, consideration of a Weierstrass equation in Legendre form over a field extension $K^\prime$ of degree dividing $12$ shows that $E$ has either good or multiplicative reduction over $K^\prime$ \cite[p. 198]{silverman2009}. If $\cha(k_v) =2$, one considers a Weierstrass equation in Deuring normal form over a field extension $K^\prime/K$ of degree $d$ with only $2$ or $3$ as prime factors \cite[p. 413]{silverman2009}.
There is a well-defined reduction map $E(K)\rightarrow \widetilde E(k_v)$ \cite[p. 188]{silverman2009}. Let $E_0(K)$ denote the pre-image of $\widetilde E_{ns}(k_v)$ under the reduction map.
Then, there is an exact sequence \begin{equation} 0 \rightarrow E_1(K) \rightarrow E_0(K) \rightarrow \widetilde E_{ns}(k_v) \rightarrow 0, \end{equation}
where the second map is the reduction map, the first map is inclusion, and $E_1(K)$ consists of all points that reduce to the point $\widetilde O$ of $\widetilde E(k_v)$ \cite[pp. 187--188]{silverman2009}. From the above remark, in the case of good reduction or multiplicative reduction, the above sequence extends to a short exact sequence of Galois modules: \begin{equation} 0 \rightarrow E_1(\overline{K}) \rightarrow E_0(\overline{K}) \rightarrow \widetilde E_{ns}(\overline{k}_v) \rightarrow 0, \end{equation} where $E_{ns}(\overline{k}_v)\approx \overline{k}_v^*$ in the case of multiplicative reduction.
Moreover, there is an isomorphism \begin{equation} F_v({\mathcal M}_v)\approx E_1(K), \end{equation} where $F_v$ is the formal group of $E$ over $\mathcal{O}_v$ and ${\mathcal M}_v$ is the maximal ideal of $\mathcal{O}_{v}$ \cite[p. 191]{silverman2009}.
Now, let $\ell$ be a prime number (possibly different from $p$) and consider a finite extension $L/K$ with valuation $w$. Based on the above facts, we obtain an exact sequence of Abelian groups: \begin{equation} 0 \rightarrow W_1 \approx E_1[\ell] \rightarrow E_{0}[\ell] \rightarrow \widetilde E_{ns}[\ell]. \end{equation} Here, $W_1$ is the group of $\ell$-torsion points of $F_w({\mathcal M}_w)$, and $E_{1}[\ell]$, $E_{0}[\ell]$ and $\widetilde E_{ns}[\ell]$ are the groups of $\ell$ torsion points of $E_1(L)$, $E_0(L)$ and $\widetilde E_{ns}(k_w)$, respectively. Also, reduction is with respect to a minimal Weierstrass equation for $E$ over $L$ (not necessarily the same one as over $K$ in the case of additive reduction over $K$).
We also have an exact sequence of Abelian groups: \begin{equation} 0 \rightarrow E_{0}[\ell] \rightarrow E[\ell] \approx \mathbb{Z}/\ell\mathbb{Z} \times \mathbb{Z}/\ell\mathbb{Z} \rightarrow E[\ell]/E_{0}[\ell] \rightarrow 0. \end{equation} The Kodaira-N\'eron Theorem \cite[Theorem 6.1, p. 200]{silverman2009} states that the Abelian group $E[\ell]/E_{0}[\ell]$ has order at most $4$ except possibly in the case of split multiplicative reduction, in which case $E[\ell]/E_{0}[\ell]$ is cyclic of order $v(\Delta)=-v(j)$.
\begin{lem} \label{thm:Lemma1} Let $E$ be an elliptic curve defined over a local field $K$, with bad reduction. Let $\ell>3$ be a prime number different from the characteristic $p$ of the residue field $k_v$ of $K$. If $E$ has potential good reduction, then $E$ has good reduction over $L=K(E[\ell])$. If $E$ has potential multiplicative reduction, then $E$ has multiplicative reduction over $L$. \end{lem} \begin{proof} By way of contradiction, assume that $E$ has additive reduction over $L$. Then, there is a short exact sequence of Abelian groups: \begin{equation} 0 \rightarrow E_1(L) \rightarrow E_0(L) \rightarrow \widetilde E_{ns}(k_w)\approx k_w^+ \rightarrow 0, \end{equation} where $k_w$ denotes the residue field of $L$. This yields a short exact sequence: \begin{equation} 0 \rightarrow E_1[\ell] \rightarrow E_0[\ell] \rightarrow k_w^+[\ell], \end{equation} since $E_1[\ell],E_0[\ell] \subseteq E[\ell]$, as $L=K(E[\ell])$. But since $\ell\not = p$, it follows that $E_1[\ell]=0$ and that $k_w^+[\ell]=0$. Therefore, one obtains that $E_0[\ell]=0$. Now, let $o$ be the order of $E(L)/E_0(L)$. Then, an element $P$ of $E[\ell]$ satisfies both conditions $[\ell] P=O\in E_0(L)$ and $[o] P\in E_0(L)$. Therefore, since $(o,\ell)=1$, as $\ell>3$, one concludes that $P\in E_0[\ell]$. This means that $E[\ell]\subseteq E_0[\ell]=0$. However, this conclusion contradicts the fact that $E[\ell]\approx \mathbb{Z}/\ell \mathbb{Z} \oplus \mathbb{Z}/\ell \mathbb{Z}$,
as $L=K(E[\ell])$.
So, if $E$ has bad potentially good reduction ({\em i.e.}, potential good reduction), then $E$ must have good reduction over $L$. If $E$ has bad and no potential good reduction ({\em i.e.}, potential multiplicative reduction), then $E$ must have multiplicative reduction over $L$. \end{proof}
\begin{lem} \label{thm:Lemma2} Let $E$ be an elliptic curve defined over a local field $K$, with good reduction. Let $m$ be a positive integer coprime with
the characteristic $p$ of the residue field $k_v$ of $K$. Then, one has an isomorphism: \begin{equation} E[m](K) \xrightarrow{\approx} \widetilde E[m](k_v). \end{equation} \end{lem} \begin{proof} Firstly, there is a well-defined map $E[m](K) \rightarrow \widetilde E[m](k_v)$ obtained by restriction of the reduction map $E(K) \rightarrow \widetilde E(k_v)$. Next, this map is one-to-one, having assumed that $(m,p)=1$ and that $\widetilde E$ is non-singular \cite[Proposition 3.1, p. 192]{silverman2009}. Lastly, the Criterion of N\'eron-Ogg-Shafarevich implies that the extension $K(E[m])/K$ is unramified, having assumed good reduction and $(m,p)=1$ \cite[Theorem 7.1, p. 201]{silverman2009}. It follows that, for any $Q\in E[m](\overline{K})$, the degree of the extension $K(Q)/K$ is equal to its residue degree. Thus, a torsion point in $\widetilde E[m](k_v)$ can be lifted to a torsion point in $E[m](K)$. \end{proof}
\subsection{Elliptic curves over an algebraic number field} \label{subsection:globalEllipticK}
Let $E$ be an elliptic curve over an algebraic number field $K$. Then, $E$ admits a Weierstrass equation of the form $y^2=x^3+Ax+B$, with $A,B\in \mathcal{O}_K$, where $\mathcal{O}_K$ denotes the integer ring of $K$. Indeed, $E$ has a Weierstrass equation over $K$ of the form $y^2 = x^3 - 27 c_4 x - 54 c_6$, with $c_4,c_6\in K$ \cite[pp. 42--43]{silverman2009}. Writing $c_4=C_4/d$ and $c_6=C_6/d$, with $C_4,C_6,d\in \mathcal{O}_K$, one obtains the Weierstrass equation $y^2=x^3 - 27 C_4 d^3 x - 54 C_6 d^5$, upon replacing $(x,y)$ by $(x/d^2,y/d^3)$. Thus, $E$ has a Weierstrass equation of the form $y^2=x^3+Ax+B$, upon taking $A=- 27 C_4 d^3,B=- 54 C_6 d^5\in \mathcal{O}_K$.
The elliptic curve $E$ admits a global minimal Weierstrass equation with coefficients in the integer ring of the Hilbert class field of $K$ \cite[Corollary 8.3, p. 245]{silverman2009}.
We denote $\Sigma_E$ the set of places at which $E$ has bad reduction. The set $\Sigma_E$ is finite \cite[Remark 1.3, p. 211]{silverman2009}. We let $\Sigma_{E,add}$ ($\Sigma_{E,mult}$) denote the (finite) sets of places $v$ such that $E$ has additive (respectively, multiplicative) reduction at $v$. We denote $\Sigma_{E,p.g.}$ the set of places at which $E$ has potential good reduction and $\Sigma_{E,p.m.}$ the set of primes at which $E$ has potential multiplicative reduction. Thus, there is a decomposition of $\Sigma_E$ into a disjoint union $\Sigma_{E,p.g.} \ensuremath{\mathaccent\cdot\cup} \; \Sigma_{E,p.m.}$, with $\Sigma_{E,p.g.} \subseteq \Sigma_{E,add}$ and $\Sigma_{E,mult} \subseteq \Sigma_{E,p.m.}$ (both inclusions are a consequence of the other one). A place $v$ of $\Sigma_{E,add}$ is in $\Sigma_{E,p.g.}$ if and only if $v(j(E))\geq 0$. Here, the place $v$ is identified with the discrete valuation on the completion of $K$ at $v$, $K_{v}$, that maps $K_{v}^*$ onto $\mathbb{Z}$.
The following cases will be considered in Section \ref{subsection:SelmerL}:
Case A: $v \mid \ell$.
Case B: $v \nmid \ell$ and $v \not \in \Sigma_E$; $E$ has good reduction at $v$ and $\ell$ is not equal to the characteristic of the residue field of $K_{v}$.
Case C: $v \nmid \ell$ and $v \in \Sigma_E$, with potential good reduction of $E$ at $v$; {\em i.e.}, $v_0\in \Sigma_{E,p.g.}$. Then, $v \in \Sigma_{E,add}$ and $v(j(E))\geq 0$.
Case D: $v \nmid \ell$ and $v \in \Sigma_E$, with no potential good reduction of $E$ at $v$; {\em i.e.}, $v\in \Sigma_{E,p.m.}$. Then, $v \in \Sigma_{E,add} \cup \Sigma_{E,mult}$ and $v(j(E))< 0$.
In case A, $E$ has good reduction at $v\mid \ell$ for all but finitely many primes $\ell$. Then, $E$ has either supersingular or ordinary good reduction at $v \mid \ell$, according to whether the reduced elliptic curve $\widetilde E$ is supersingular or ordinary.
\subsection{Elliptic curves over $\mathbb{Q}$} \label{subsection:globalEllipticQ}
Let $E$ be an elliptic curve over $\mathbb{Q}$. Then, $E$ admits a global minimal Weierstrass equation (with coefficients in $\mathbb{Z}$) \cite[Corollary 8.3, p. 245]{silverman2009}. It is also convenient to consider a Weierstrass equation of the form $y^2=x^3+Ax+B$, with $A,B\in \mathbb{Z}$; for instance, see \cite[Corollary 7.2, p. 240]{silverman2009}.
One says \cite[IV-2.1]{serre1968} that $E/K$ has CM if for some finite extension $F_{\textrm{\tiny CM}}/K$, the endomorphism ring $\End_{F_{\textrm{\tiny CM}}}(E)$ is an order $\mathcal{O}$ of an imaginary quadratic extension $K_{\textrm{\tiny CM}}/\mathbb{Q}$ \cite[Section 5]{rubin1999}. One may assume that $F_{\textrm{\tiny CM}} \supseteq K_{\textrm{\tiny CM}}$. Indeed, if $F^\prime$ is a subfield of $F$, then $\End_{F^\prime}(E) \subseteq \End_{F}(E)$, so that one may replace $F_{\textrm{\tiny CM}}$ with the compositum $K_{\textrm{\tiny CM}} F_{\textrm{\tiny CM}}$, if necessary. In the case of a curve without CM, the endomorphism ring $\End_{\mathbb{C}}(E)$ is minimal; {\em i.e.}, it is isomorphic to $\mathbb{Z}$ \cite[Corollary 9.4, p. 102]{silverman2009}.
Let $E$ be an elliptic curve over $\mathbb{Q}$ with CM. Then, $\End_{F_{\textrm{\tiny CM}}}(E)$ is of the form $\mathcal{O} = \mathbb{Z} + c\, \mathcal{O}_{\textrm{\tiny CM}}$ over some finite base field extension $F_{\textrm{\tiny CM}}/K_{\textrm{\tiny CM}}$, where $\mathcal{O}_{\textrm{\tiny CM}}$ is the integer ring of the imaginary quadratic field $K_{\textrm{\tiny CM}}$, and $c = 1$, $2$, or $3$ \cite{serre1967b}.
Let $\ell$ be a prime number, and set $L=\mathbb{Q}(E[\ell])$. Based on \cite[Corollary 5.13]{rubin1999}, there exists an elliptic curve $E^\prime$ defined over $K_{\textrm{\tiny CM}}$, such that $\End_{K_{\textrm{\tiny CM}}}(E^\prime) = \mathcal{O}_{\textrm{\tiny CM}}$. From \cite[Proposition 5.3]{rubin1999}, one has $\End_{F_{\textrm{\tiny CM}}}(E^\prime)=\mathcal{O}_{\textrm{\tiny CM}}$ and $E[\ell]\xrightarrow{\approx} E^\prime[\ell]$ {\em as Galois modules} for any prime $\ell$ coprime with $c$, a condition satisfied if $\ell>3$. This relation is obtained from a short exact sequence based on an isogeny: \begin{equation} 0 \rightarrow E[c] \rightarrow E(\overline{K}_{\textrm{\tiny CM}}) \rightarrow E^\prime(\overline{K}_{\textrm{\tiny CM}}) \rightarrow 0, \end{equation} where $E[c]$ denotes the group of $c$-torsion points of $E$. In particular, one deduces the identities $L F_{\textrm{\tiny CM}} =F_{\textrm{\tiny CM}}(E[\ell]) = F_{\textrm{\tiny CM}}(E^\prime[\ell])$. Then, using \cite[Corollary 5.5]{rubin1999}, one obtains an embedding of groups: \begin{equation} \label{eq:eq40final6} \varphi: \Gal(L F_{\textrm{\tiny CM}}/F_{\textrm{\tiny CM}}) = \Gal(F_{\textrm{\tiny CM}}(E^\prime[\ell])/F_{\textrm{\tiny CM}}) \hookrightarrow \left ( \mathcal{O}_{\textrm{\tiny CM}}/(\ell) \right )^*, \end{equation}
From (\ref{eq:eq40final6}), it follows that $\ell \nmid \vert \Gal(\mathbb{Q}(E[\ell])/\mathbb{Q}) \vert$ in the CM case, unless possibly if $\ell$ ramifies in $K_{\textrm{\tiny CM}}$ or $\ell$ divides $[F_{\textrm{\tiny CM}}:\mathbb{Q}]$. In particular, $\Gal(\mathbb{Q}(E[\ell])/\mathbb{Q})$ is not the full linear group for almost all primes $\ell$, as $\vert {\bf GL}_2(\mathbb{F}_\ell) \vert$ is divisible by $\ell$.
In contrast, in the non-CM case, the Galois group $\Gal(\mathbb{Q}(E[\ell])/\mathbb{Q})$ is the full linear group for almost all primes $\ell$ \cite[Th\'eor\`eme 2, p. 294]{serre1972}; {\em i.e.}, the representation $\widetilde \rho_\ell:\mathcal{G}\rightarrow {\bf GL}_2(\mathbb{F}_\ell)$ obtained by Galois action on $\ell$-torsion points is surjective for almost all primes $\ell$. Moreover, sufficient conditions for the isomorphism $\widetilde \rho_\ell:\Gal(\mathbb{Q}(E[\ell])/\mathbb{Q}) \xrightarrow{\approx} {\bf GL}_2(\mathbb{F}_\ell)$ to hold at a specific prime $\ell$ are presented in Serre's work, together with several examples, in the case of semi-stable curves \cite[\S 5.4 and 5.5, p. 305--311]{serre1972}, as well as non semi-stable curves \cite[\S 5.6 to 5.10, p. 311--323]{serre1972}. See also \cite{mazur1978} for further results.
Note that from \cite[Proposition, IV-19]{serre1968} and \cite[Th\'eor\`eme 2, p. 294]{serre1972}, one has for almost all primes $\ell$, an isomorphism $\Gal(\mathbb{Q}(E[\ell^n])/\mathbb{Q}) \xrightarrow{\approx} {\bf GL}_2(\mathbb{Z}/\ell^n\mathbb{Z})$ induced by $\rho_\ell$, for any $n \geq 1$. Indeed, from \cite[p. IV-18]{serre1968}, the representation $\rho_\ell:\mathcal{G}\rightarrow {\bf GL}_2(\mathbb{Z}_\ell)$, obtained by Galois action on the Tate module $T_\ell$, composed with the determinant map yields the cyclotomic character $\psi_\ell$, whose image is $\mathbb{Z}_\ell^*$ (since the base field is $\mathbb{Q}$). Setting $X={\bf SL}_2(\mathbb{Z}_\ell) \cap \Image(\rho_\ell)$, one obtains a closed subgroup of ${\bf SL}_2(\mathbb{Z}_\ell)$. Then, assuming that the image of $X$ into ${\bf SL}_2(\mathbb{F}_\ell)$ is equal to ${\bf SL}_2(\mathbb{F}_\ell)$, one concludes that $\Image(\rho_\ell)={\bf SL}_2(\mathbb{Z}_\ell)$ whenever $\ell\geq 5$ \cite[Lemma 3, p. IV-23]{serre1968}. Altogether, one has: \begin{equation} \label{eq:eq41final7} \ell\geq 5 \textrm{ and } \Image(\widetilde \rho_\ell)={\bf GL}_2(\mathbb{F}_\ell) \Longrightarrow \Image(\rho_\ell)={\bf GL}_2(\mathbb{Z}_\ell). \end{equation} See \cite[pp. 299--301]{serre1972}.
We denote $\Sigma_E$ the set of primes at which $E$ has bad reduction. The set $\Sigma_E$ is finite \cite[Remark 1.3, p. 211]{silverman2009} and is non-empty \cite[Exerc. 8.15, p. 264]{silverman2009}. In the case of curves without CM, Serre proved that the set of primes $\ell$ at which $E$ has ordinary good reduction has density $1$ (c.f. \cite[Corollaire 1, p. 189]{serre1981}, using \cite[Exerc. 5.10, p. 154]{silverman2009}). See also \cite[Exerc. 5.11, p. 154]{silverman2009} for a weaker statement in the case of an arbitrary elliptic curve over $\mathbb{Q}$. On the other hand, Elkies proved that any elliptic curve over $\mathbb{Q}$ has infinitely many primes $\ell$ at which $E$ has supersingular good reduction \cite{elkies1987}.
If $p$ is a prime of $\mathbb{Z}$, $\ord_p$ denotes the valuation on $\mathbb{Q}_p$ such that $\ord_p(p)=1$.
\section{Liftings of points on reduced elliptic curves to points with coordinates in specific algebraic number fields} \label{section:liftingsPoints}
We collect in this section results on torsion points of elliptic curves that will be useful in the sequel.
\subsection{Torsion points over algebraic number fields} \label{subsection:torsionPointsK}
First of all, the following result on torsion points of elliptic curves over algebraic number fields expresses \cite[Theorem 7.1, p. 240]{silverman2009} in a context relevant to this work. Equation (\ref{eq:eq42final7}) follows from a result of Cassels; see \cite[Theorem 3.4, p. 193]{silverman2009}. Part b) is due to Lutz and Nagell independently in the case where $K=\mathbb{Q}$; see \cite[Corollary 7.2, p. 240]{silverman2009}.
\begin{lem} \label{thm:Lemma3} Let $E$ be an elliptic over an algebraic number field $K$, with Weierstrass equation of the form $y^2=x^3+Ax+B$, where $A,B\in \mathcal{O}_K$. Let $\ell$ be a prime number.
a) Assume that $\ell >3$. Then, any non-trivial $\ell^n$-torsion point $P$ of $E$ over $\overline{\mathbb{Q}}$, where $n\geq 1$, satisfies the conditions: \begin{eqnarray} \label{eq:eq42final7} && x(P), y(P) \in (\ell)^{-1};\\ &&\bigl ( \ell y(P) \bigr )^2 \mid \Delta^\prime \ell^5, \end{eqnarray} where $(x(P),y(P))$ are the affine coordinates of $P$, $\Delta^\prime=4 A^3+27 B^2$, so that $\Delta=-16 \Delta^\prime$ is the discriminant of the Weierstrass equation, and the divisibility condition holds in the integer ring $\mathcal{O}_n$ of $L_n:=K(E[\ell^n])$.
b) If $\ell$ is unramified in $K$, any non-trivial $\ell^n$-torsion point $P$ of $E$ over $K$, where $n\geq 1$, satisfies the conditions: \begin{eqnarray} && x(P), y(P) \in \mathcal{O}_K;\\ &&y(P)^2 \mid \Delta^\prime, \end{eqnarray} where the divisibility condition holds in the integer ring $\mathcal{O}_K$ of $K$. \end{lem}
\begin{proof} Part a). The proof follows closely \cite[pp. 240-241]{silverman2009}, but with some modifications.
From \cite[Theorem 7.1, p. 240]{silverman2009}, $x(P)$ and $y(P)$ are ${v}$-integral for any place $v \nmid \ell$ of $L_n$. Moreover, from that result, if $v \mid \ell$, then one has $v(y(P))\geq -3 v(\ell)/(\ell-1) > - v(\ell)$ and $v(x(P))\geq -2 v(\ell)/(\ell-1) > - v(\ell)$, having assumed that $\ell \geq 5$. Therefore, $x(P),y(P)\in (\ell)^{-1}$.
Next, from \cite[Sublemma 4.3, p. 222]{silverman2009}, one deduces the identity: \begin{equation} f(x(P)) \phi(x(P)) - g(x(P)) \psi(x(P)) = \Delta^\prime, \end{equation} where \begin{equation} \begin{cases} f(X) = 3 X^2 + 4 A;\\ \phi(X) = X^4 - 2 A X^2 - 8 B X + A^2;\\ g(X) = 3 X^3 - 5 AX - 27 B;\\ \psi(X) = X^3 + AX + B;\\ \Delta^\prime = 4 A^3 + 27 B^2. \end{cases} \end{equation} Note here that $-16 \Delta^\prime$ is the discriminant of the Weierstrass equation $y^2=x^3+Ax+B$ \cite[p. 45]{silverman2009}. But, one has the duplication identity, which can be deduced from \cite[p. 54]{silverman2009}, as $\ell\not=2$: \begin{equation} \label{eq:eq48final7} x([2]P) = \frac{\phi(x(P))}{4\psi(x(P))}. \end{equation} It follows that: \begin{equation} \label{eq:eq49final7} y(P)^2 \bigl [ 4 f(x(P)) x([2]P) - g(x(P)) \bigr ] = \Delta^\prime, \end{equation} since $y(P)^2=\psi(x(P))$. Now, multiplying by $\ell^5$, one obtains:
\begin{equation} \label{eq:eq50final7} \bigl ( \ell y(P) \bigr )^2 \ell^3 \bigl [ 4 f(x(P)) x([2]P) - g(x(P)) \bigr ] = \Delta^\prime \ell^5. \end{equation} But, $\ell^3 \bigl [ 4 f(x(P)) x([2]P) - g(x(P)) \bigr ]$ is an integral element of $L_n$ because both $\ell x(P)$ and $\ell x([2]P)$ are integral elements. This proves part a).
Part b). If $\ell$ is unramified in $K$, then $v(\ell)=1$. Thus, the above conditions $v(y(P))\geq -3 v(\ell)/(\ell-1) > - v(\ell)$ and $v(x(P))\geq -2 v(\ell)/(\ell-1) > - v(\ell)$ mean that $x(P),y(P)\in \mathcal{O}_K$. This yields:
\begin{equation} \bigl ( y(P) \bigr )^2 \bigl [ 4 f(x(P)) x([2]P) - g(x(P)) \bigr ] = \Delta^\prime, \end{equation} where $\bigl [ 4 f(x(P)) x([2]P) - g(x(P)) \bigr ] \in \mathcal{O}_K$. This proves part b). \end{proof}
\begin{lem} \label{thm:Lemma4} Let $E$ be an elliptic curve over an algebraic number field $K$. Given a prime number $\ell$ and a positive integer $n$, let $L_n$ denote the field extension over $K$ obtained by adjoining the affine coordinates of all $\ell^n$-torsion points of $E$.
a) Then, there is a group embedding: \begin{equation} 0 \rightarrow \Gal (L_n/K) \rightarrow {\bf GL}_2 \left ( \mathbb{Z}/\ell^n\mathbb{Z}\right). \end{equation}
b) The latter group has order dividing $\ell(\ell-1)^2(\ell+1)\ell^{4(n-1)}$. In particular, $[L_n:K]$ divides $\ell(\ell-1)^2(\ell+1)\ell^{4(n-1)}$.
c) Let $\ell_i$, $i=1,...,\nu$, be distinct prime numbers, and let $n_i$ be positive integers, $i=1,...,\nu$. Set $m=\prod_{i=1}^\nu \ell_i^{n_i}$. Let $K(E[m])$ be the field obtained by adjoining over $K$ the affine coordinates of all $m$-torsion points of $E$ (so, $L_n$ means $K(E[\ell^n])$). Then, there is a group embedding: \begin{equation} 0 \rightarrow \Gal (K(E[m])/K) \rightarrow \prod_{i=1}^{\nu} {\bf GL}_2 \left ( \mathbb{Z}/\ell_i^{n_i}\mathbb{Z}\right). \end{equation} In particular, $[K(E[m]):K]$ divides the integer: \begin{equation} \prod_{i=1}^{\nu} \ell_i(\ell_i-1)^2(\ell_i+1)\ell_i^{4(n_i-1)}. \end{equation} \end{lem}
\begin{proof} Part a). Since $\Gal (L_n/K)$ acts faithfully on the finite group of $\ell^n$-torsion points $E[\ell^n]$, one has a group embedding: \begin{equation} 0 \rightarrow \Gal (L_n/K) \rightarrow {\bf Aut} \left ( E[\ell^n] \right). \end{equation} Since $E[\ell^n]$ is isomorphic to $\mathbb{Z}/\ell^n\mathbb{Z} \oplus \mathbb{Z}/\ell^n\mathbb{Z}$ \cite[Corollary 6.4, p. 86]{silverman2009}, it follows that \begin{equation} {\bf Aut} \left ( E[\ell^n] \right) \xrightarrow{\approx} {\bf GL}_2 \left ( \mathbb{Z}/\ell^n\mathbb{Z}\right). \end{equation}
Part b). One has an exact sequence of groups: \begin{equation} 0 \rightarrow I + \ell {\bf Mat}_2 \left ( \mathbb{Z}/\ell^n\mathbb{Z}\right) \rightarrow {\bf GL}_2 \left ( \mathbb{Z}/\ell^n\mathbb{Z}\right) \xrightarrow{\pi_*} {\bf GL}_2 \left ( \mathbb{F}_\ell\right), \end{equation} where $I$ denotes the $2 \times 2$ identity matrix over $\mathbb{Z}/\ell^n\mathbb{Z}$, and the map $\pi_*$ is induced by the projection of rings $\pi: \mathbb{Z}/\ell^n\mathbb{Z} \rightarrow \mathbb{Z}/\ell\mathbb{Z}\approx \mathbb{F}_\ell$. But the rightmost factor ${\bf GL}_2 \left ( \mathbb{F}_\ell\right)$ has order $(\ell^2-1)(\ell^2-\ell)=\ell(\ell-1)^2(\ell+1)$, whereas the left most factor $I + \ell {\bf Mat}_2 \left ( \mathbb{Z}/\ell^n\mathbb{Z}\right)$ has order $\left ( \ell^{n-1} \right )^4$. Thus, the order of ${\bf GL}_2 \left ( \mathbb{Z}/\ell^n\mathbb{Z}\right)$ divides $\ell(\ell-1)^2(\ell+1)\ell^{4(n-1)}$. Lastly, part a) implies that $[L_n:K]$ divides the order of ${\bf GL}_2 \left ( \mathbb{Z}/\ell^n\mathbb{Z}\right)$.
Part c). Generalizing the proof of part a), one has a group embedding: \begin{equation} 0 \rightarrow \Gal (K(E[m])/K) \rightarrow {\bf Aut} \left ( E[m] \right) \xrightarrow{\approx} {\bf GL}_2 \left ( \mathbb{Z}/m\mathbb{Z}\right), \end{equation} since $\Gal (K(E[m])/K)$ acts faithfully on $E[m]$, which is isomorphic to $\mathbb{Z}/m\mathbb{Z} \oplus \mathbb{Z}/m\mathbb{Z}$ \cite[Corollary 6.4, p. 86]{silverman2009}. But then, the isomorphism of rings $\mathbb{Z}/m\mathbb{Z} \xrightarrow{\approx} \prod_{i=1}^{\nu} \mathbb{Z}/\ell_i^{n_i}\mathbb{Z}$ (from the Chinese Remainder Theorem) yields an isomorphism: \begin{equation} {\bf GL}_2 \left ( \mathbb{Z}/m\mathbb{Z}\right) \xrightarrow{\approx} \prod_{i=1}^{\nu} {\bf GL}_2 \left ( \mathbb{Z}/\ell_i^{n_i}\mathbb{Z}\right). \end{equation} Now, use part b) on each factor of the right-hand side of this equation. \end{proof}
\noindent {\bf Remark 2.} \label{remark2} Recall from \cite[\S 4.2, pp. 151--152]{serre1981} that the Galois group of the infinite extension $L_\infty/\mathbb{Q}$ obtained by adjoining over $\mathbb{Q}$ the affine coordinates of all $\ell^n$-torsion points of $E$, where $n\geq 1$, is an $\ell$-adic Lie group. Indeed, one has an embedding $\rho_\ell:\Gal(L_\infty/\mathbb{Q}) \hookrightarrow {\bf GL}_2(\mathbb{Z}_\ell)$ obtained by Galois action on the Tate module $\varprojlim E[\ell^n]$. Its dimension $N$ is at least $2$ and at most $4$, since $E[\ell^n] \approx \mathbb{Z}/\ell^n \mathbb{Z} \oplus \mathbb{Z}/\ell^n$ for all $n\geq 1$. Then, Lemma \ref{thm:Lemma4} shows that $[L_n:\mathbb{Q}]$ is of the form $b \ell^{nN}$, for some integer $b$ dividing $(\ell-1)^2(\ell+1)\ell^{\beta}$, with $\beta\geq 0$. See Appendix \ref{section:appendixA} for an expression of the different of the extension $L_n/\mathbb{Q}$ based on a theorem of Sen \cite{sen1972} that was conjectured by Serre \cite{serre1967c}. \\
\subsection{Multiplication by positive integers in elliptic curves} \label{subsection:Multiplication}
We consider an elliptic curve $E$ over a field $K$, with Weierstrass equation $y^2=x^3+Ax+B$.
We consider multiplication by a positive integer $n$ in $E(K)$, where $K$ is a field. For this purpose, we recall from \cite{lang1978,schoof1985} the polynomials over $\mathbb{Z}[A,B]$ (note that in \cite{schoof1985}, these polynomials are considered over a finite field): \begin{equation} \begin{cases} \Psi_{-1}(X,Y) = -1;\\ \Psi_{0}(X,Y) = 0;\\ \Psi_{1}(X,Y) = 1;\\ \Psi_{2}(X,Y) = 2Y;\\ \Psi_{3}(X,Y) = 3X^4+6 AX^2+12 B X - A^2;\\ \Psi_{4}(X,Y) = 4Y(X^6 + 5AX^4 + 20BX^3 - 5A^2 X^2 - 4AB X - 8 B^2 -A^3). \end{cases} \end{equation} Then, one has the recursion formulae for $n\geq 1$: \begin{equation} \begin{cases} \Psi_{2n}(X,Y) = \Psi_n(X,Y) \left ( \Psi_{n+2}(X,Y)\Psi_{n-1}^2(X,Y) - \Psi_{n-2}(X,Y)\Psi_{n+1}^2(X,Y) \right)/2Y;\\ \Psi_{2n+1}(X,Y) = \Psi_{n+2}(X,Y)\Psi_n^3(X,Y) - \Psi_{n+1}^3(X,Y)\Psi_{n-1}(X,Y). \end{cases} \end{equation}
As in \cite{schoof1985}, we denote $\Psi^\prime_n(X,Y)$ the polynomial obtained from $\Psi_n(X,Y)$ by replacing $Y^2$ with $X^3+AX+B$. Then, it turns out that: \begin{equation} \begin{cases} f_{n}(X) = \Psi^\prime_n(X,Y), \quad n \textrm { odd};\\ f_{n}(X) = \Psi^\prime_n(X,Y)/Y, \quad n \textrm { even}, \end{cases} \end{equation} are polynomials in $X$. From \cite[pp. 37--38]{lang1978}, $f_n(X) \in 2\mathbb{Z}[X]$ for $n$ even. Moreover, from \cite[p. 486]{schoof1985}, it follows that for any $n\geq 1$, one has: \begin{equation} \label{eq:eq63final7} \begin{cases} f_{n}(X) = c X^{(n^2-1)/2} + \cdot \cdot \cdot , \quad n \textrm { odd};\\ f_{n}(X) = c X^{(n^2-4)/2} + \cdot \cdot \cdot , \quad n \textrm { even}, \end{cases} \end{equation} for some element $c\not=0$ in $K$.
One can also show, with $P=(x,y)$, that: \begin{equation} [n] P = O \Longleftrightarrow f_n(x)=0, \end{equation} assuming that $[2]P\not = O$ \cite[Proposition (2.1)]{schoof1985}. One also has \cite[Chapter II]{lang1978}: \begin{equation} [n] P = O \Longleftrightarrow (\Psi^\prime_n(x,y))^2=0, \end{equation} for $P\not = O$.
Then, one has: \begin{equation} \label{eq:eq66final7} x([n]P) = x - \frac{\Psi^\prime_{n-1}(x,y)\Psi^\prime_{n+1}(x,y)}{\left (\Psi^\prime_{n}(x,y)\right)^2};\\ \end{equation} and \begin{equation} \label{eq:eq67final7} y([n]P) = \frac{\Psi^\prime_{n+2}(x,y)\left ( \Psi^\prime_{n-1}(x,y) \right)^2 - \Psi^\prime_{n-2}(x,y)\left (\Psi^\prime_{n+1}(x,y)\right)^2}{4y\left ( \Psi^\prime_{n}(x,y) \right)^3}, \end{equation} where $P=(x,y)$, assuming that $[n]P\not = O$; see \cite[Theorem 2.1, p. 38]{lang1978} and \cite[Proposition (2.2)]{schoof1985}.
The following result refines Eq. (\ref{eq:eq63final7}) ({\em i.e.}, \cite[p. 486]{schoof1985}) and \cite[Theorem 2.1, p. 38]{lang1978}.
\begin{lem} \label{thm:Lemma5} For any $n\geq 1$, one has: \begin{equation} \begin{cases} f_{n}(X) = n X^{(n^2-1)/2} + c X^{(n^2-1)/2-2} + \cdot \cdot \cdot , \quad n \textrm { odd};\\ f_{n}(X) = n X^{(n^2-4)/2} + c X^{(n^2-4)/2-2}+ \cdot \cdot \cdot , \quad n \textrm { even}. \end{cases} \end{equation} for some $c\in \mathbb{Z}[A,B]$ depending on $n$. \end{lem}
\begin{proof} The proof is by induction on $n\geq 1$.
The result is obviously true for $n=1,2$, since $f_1(X)=1$ and $f_2(X)$ is equal to $2$. It is also true for $n=3$, since $f_3(X)=3X^4+6 AX^2+12 B X - A^2$. For $n=4$, one has: \begin{equation} f_4(X) = 4 ( X^6 + 5AX^4 + 20BX^3 - 5A^2 X^2 - 4AB X - 8 B^2 -A^3), \end{equation} so that the result is true.
Assume by induction hypothesis that the result is true for any $1\leq n^\prime<n$, for some integer $n\geq 5$.
{\em Case 1}: $n=2m$, with $m\geq 4$ even. Then, one computes: \begin{eqnarray} f_n(X) &=& \Psi^\prime_{2m}(X,Y)/Y\nonumber\\ &=& \Psi^\prime_m(X,Y) \left ( \Psi^\prime_{m+2}(X,Y)(\Psi^\prime_{m-1}(X,Y))^2 - \Psi^\prime_{m-2}(X,Y)(\Psi^\prime_{m+1}(X,Y))^2 \right)/2Y^2\nonumber\\ &=& f_m(X)Y \left ( f_{m+2}(X)Yf_{m-1}^2(X) - f_{m-2}(X)Yf_{m+1}^2(X) \right)/2Y^2\nonumber\\ &=& f_m(X)\left ( f_{m+2}(X)f_{m-1}^2(X) - f_{m-2}(X)f_{m+1}^2(X) \right)/2. \end{eqnarray} We have: \begin{equation} \begin{cases} \deg (f_{m+2}(X)f_{m-1}^2(X)) = \frac{1}{2}((m+2)^2-4) + ((m-1)^2-1) = \frac{3}{2} m^2;\\ \lc (f_{m+2}(X)f_{m-1}^2(X)) = (m+2)(m-1)^2 = m^3 - 3m +2, \end{cases} \end{equation} and \begin{equation} \begin{cases} \deg (f_{m-2}(X)f_{m+1}^2(X)) = \frac{1}{2}((m-2)^2-4) + ((m+1)^2-1) = \frac{3}{2} m^2;\\ \lc (f_{m-2}(X)f_{m+1}^2(X)) = (m-2)(m+1)^2 = m^3 - 3m - 2, \end{cases} \end{equation} where $\lc(f(X))$ denotes here the leading coefficient of polynomial $f(X)$. We also let $\lc_{-}(f(X))$ denote the next coefficient. So, if $\deg(f(X))=d$, one has $f(X)=\lc(f(X))X^d + \lc_{-}(f(X)) X^{d-1}+ \cdot \cdot \cdot$. Thus, one has $\lc(f_{n}(X))=2m=n$, $\lc_{-}(f_n(X))=0$, and $\deg(f_n(X))=(n^2-4)/2$.
{\em Case 2}: $n=2m$, with $m\geq 3$ odd. Then, one computes: \begin{eqnarray} f_n(X) &=& \Psi^\prime_{2m}(X,Y)/Y\nonumber\\ &=& \Psi^\prime_m(X,Y) \left ( \Psi^\prime_{m+2}(X,Y)(\Psi^\prime_{m-1}(X,Y))^2 - \Psi^\prime_{m-2}(X,Y)(\Psi^\prime_{m+1}(X,Y))^2 \right)/2Y^2\nonumber\\ &=& f_m(X) \left ( f_{m+2}(X)f_{m-1}^2(X)Y^2 - f_{m-2}(X)f_{m+1}^2(X)Y^2 \right)/2Y^2\nonumber\\ &=& f_m(X)\left ( f_{m+2}(X)f_{m-1}^2(X) - f_{m-2}(X)f_{m+1}^2(X) \right)/2. \end{eqnarray} We have: \begin{equation} \deg (f_{m+2}(X)f_{m-1}^2(X)) = \frac{1}{2}((m+2)^2-1) + ((m-1)^2-4) = \frac{3}{2} (m^2 - 1), \end{equation} and \begin{equation} \deg (f_{m-2}(X)f_{m+1}^2(X)) = \frac{1}{2}((m-2)^2-1) + ((m+1)^2-4) = \frac{3}{2} (m^2-1), \end{equation} with leading coefficients as in Case 1. Thus, $\lc(f_{n}(X))=2m=n$, $\lc_{-}(f_n(X))=0$, and $\deg(f_n(X))=(n^2-4)/2$.
{\em Case 3}: $n=2m+1$, with $m\geq 2$ even. Then, one computes: \begin{eqnarray} f_n(X) &=& \Psi^\prime_{2m+1}(X,Y)\nonumber\\ &=& \Psi_{m+2}^\prime(X,Y)(\Psi^\prime_m(X,Y))^3 - (\Psi^\prime_{m+1}(X,Y))^3\Psi^\prime_{m-1}(X,Y)\nonumber\\ &=& f_{m+2}(X)Y(f_m(X)Y)^3 - f_{m+1}^3(X)f_{m-1}(X)\nonumber\\ &=& f_{m+2}(X)f^3_m(X)(X^3+AX+B)^2 - f_{m+1}^3(X)f_{m-1}(X). \end{eqnarray} We have: \begin{equation} \begin{cases} \deg (f_{m+2}(X)f^3_m(X)(X^3+AX+B)^2) = \frac{1}{2}((m+2)^2-4) + \frac{3}{2}(m^2-4) + 6\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad = 2m^2+2m;\\ \lc (f_{m+2}(X)f^3_m(X)(X^3+AX+B)^2) = (m+2)m^3 = m^4+2m^3, \end{cases} \end{equation} and \begin{equation} \begin{cases} \deg (f_{m+1}^3(X)f_{m-1}(X)) = \frac{3}{2}((m+1)^2-1) + \frac{1}{2}((m-1)^2-1) = 2m^2+2m;\\ \lc (f_{m+1}^3(X)f_{m-1}(X)) = (m+1)^3(m-1) = m^4 + 2m^3 -2m - 1. \end{cases} \end{equation} So, one has $\lc(f_{n}(X))=2m+1=n$, $\lc_{-}(f_n(X))=0$, and $\deg(f_n(X))=(n^2-1)/2$.
{\em Case 4}: $n=2m+1$, with $m\geq 3$ odd. Then, one computes: \begin{eqnarray} f_n(X) &=& \Psi^\prime_{2m+1}(X,Y)\nonumber\\ &=& \Psi_{m+2}^\prime(X,Y)(\Psi^\prime_m(X,Y))^3 - (\Psi^\prime_{m+1}(X,Y))^3\Psi^\prime_{m-1}(X,Y)\nonumber\\ &=& f_{m+2}(X)f^3_m(X) - (f_{m+1}(X) Y)^3(X)f_{m-1}(X)Y\nonumber\\ &=& f_{m+2}(X)f^3_m(X) - f_{m+1}^3(X)f_{m-1}(X)(X^3+AX+B)^2. \end{eqnarray} We have: \begin{equation} \deg (f_{m+2}(X)f^3_m(X) = \frac{1}{2}((m+2)^2-1) + \frac{3}{2}(m^2-1) = 2m^2+2m, \end{equation} and \begin{eqnarray} &&\deg (f_{m+1}^3(X)f_{m-1}(X)(X^3+AX+B)^2)\nonumber\\
&& = \frac{3}{2}((m+1)^2-4) + \frac{1}{2}((m-1)^2-4) + 6 = 2m^2+2m, \end{eqnarray} with same leading coefficients as in Case 3. So, one has $\lc(f_{n}(X))=2m+1=n$, $\lc_{-}(f_n(X))=0$, and $\deg(f_n(X))=(n^2-1)/2$. \end{proof}
We obtain the following refinement of \cite[Theorem 2.1, ii, p. 38]{lang1978}.
\begin{cor} \label{thm:Corollary3} Let $E$ be an elliptic curve with Weierstrass equation of the form $y^2=x^3+Ax+B$. For any $n\geq 1$, one has in $\mathbb{Z}[X,A,B]$: \begin{equation} (\Psi^\prime_{n}(X,Y))^2 = n^2 X^{n^2-1} + c X^{n^2-3} + \cdot \cdot \cdot. \end{equation} for some $c\in \mathbb{Z}[A,B]$ depending on $n$. \end{cor}
\begin{proof} For $n$ odd, one has directly from Lemma \ref{thm:Lemma5}: \begin{equation} (\Psi^\prime_{n}(X,Y))^2 = f_n^2(X) = n^2 X^{n^2-1} + c X^{n^2-3} + \cdot \cdot \cdot, \end{equation} For $n$ even, one has: \begin{eqnarray} (\Psi^\prime_{n}(X,Y))^2 &=& f_n^2(X) Y^2 = (n^2 X^{n^2-4} + c^\prime X^{n^2-6} + \cdot \cdot \cdot)(X^3+AX+B)\nonumber\\ &=& n^2 X^{n^2-1} + (c^\prime+n^2A) X^{n^2-3} + \cdot \cdot \cdot, \end{eqnarray} as was to be shown, taking $c=c^\prime + n^2 A$. \end{proof}
\begin{cor} \label{thm:Corollary4} For any prime $\ell>2$ and integer $n >1$, one has in $\mathbb{Z}[X,A,B]$: \begin{equation} \frac{(\Psi^\prime_{\ell^n}(X,Y))^2}{(\Psi^\prime_{\ell^{n-1}}(X,Y))^2} = \ell^2 X^{\ell^{2n-2}(\ell^2-1)} + c X^{\ell^{2n-2}(\ell^2-1)-2} + \cdot \cdot \cdot, \end{equation} for some $c\in \mathbb{Z}[A,B]$ depending on $\ell^n$. \end{cor}
\begin{proof} Any $\ell^{n-1}$-torsion point is also an $\ell^{n}$-torsion point. Therefore, the polynomial $(\Psi^\prime_{\ell^{n-1}}(X,Y))^2$ divides $(\Psi^\prime_{\ell^{n}}(X,Y))^2$. From \cite[Theorem 2.2-iii, p. 39]{lang1978}, the quotient of these two polynomials is actually in $\mathbb{Z}[X,A,B]$. One then computes directly from Corollary \ref{thm:Corollary3}: \begin{eqnarray} \frac{(\Psi^\prime_{\ell^n}(X,Y))^2}{(\Psi^\prime_{\ell^{n-1}}(X,Y))^2} &=& \frac{\ell^{2n} X^{\ell^{2n}-1} + c_1 X^{\ell^{2n}-3} + \cdot \cdot \cdot}{\ell^{2n-2} X^{\ell^{2n-2}-1} + c_2 X^{\ell^{2n-2}-3} + \cdot \cdot \cdot}\nonumber\\ &=& \ell^2 X^{\ell^{2n-2}(\ell^2-1)} + c X^{\ell^{2n-2}(\ell^2-1)-2} + \cdot \cdot \cdot, \end{eqnarray} as was to be shown. \end{proof}
\begin{cor} \label{thm:Corollary5} For any $m\geq 1$ and any element $\lambda$ in a field $K$ containing $A$ and $B$, one has in $K[X]$: \begin{eqnarray} \Phi_{m}(X,\lambda)&:=&(X-\lambda)(\Psi^\prime_{m}(X,Y))^2 - \Psi^\prime_{m-1}(X,Y)\Psi^\prime_{m+1}(X,Y)\nonumber\\ & = & X^{m^2} - \lambda m^2 X^{m^2-1} + \cdot \cdot \cdot. \end{eqnarray} \end{cor}
\begin{proof} For $m$ odd, one has: \begin{eqnarray} &&(X-\lambda)(\Psi^\prime_{m}(X,Y))^2 - \Psi^\prime_{m-1}(X,Y)\Psi^\prime_{m+1}(X,Y)\nonumber\\ && = (X-\lambda)f_{m}^2(X) - f_{m-1}(X)Yf_{m+1}(X)Y\nonumber\\
&& = (X-\lambda)f_{m}^2(X) - f_{m-1}(X)f_{m+1}(X)(X^3+AX+B), \end{eqnarray} which shows the result using Lemma \ref{thm:Lemma5}. For $m$ even, one has: \begin{eqnarray} &&(X-\lambda)(\Psi^\prime_{m}(X,Y))^2 - \Psi^\prime_{m-1}(X,Y)\Psi^\prime_{m+1}(X,Y)\nonumber\\ && = (X-\lambda)f_{m}^2(X)Y^2 - f_{m-1}(X)f_{m+1}(X)\nonumber\\
&& = (X-\lambda)f_{m}^2(X)(X^3+AX+B) - f_{m-1}(X)f_{m+1}(X), \end{eqnarray} which implies the result in that case. \end{proof}
\subsection{Torsion points over $\mathbb{Q}$ in the non-CM case} \label{subsection:torsionPointsQ}
In this section, we consider non-CM elliptic curves $E$ over $\mathbb{Q}$, with Weierstrass equation $y^2=x^3+Ax+B$, where $A,B\in \mathbb{Z}$.
Under these assumptions, the next results can be applied to almost all primes $\ell$, based on results of Serre. Namely, from \cite[Proposition, IV-19]{serre1968} and \cite[Th\'eor\`eme 2, p. 294]{serre1972}, one has for almost all prime numbers $\ell$, the isomorphism $\rho_\ell:\Gal(L_\infty/\mathbb{Q})\xrightarrow{\approx} {\bf GL}_2(\mathbb{Z}_\ell)$. The condition $\Gal(L_\infty/\mathbb{Q})\approx {\bf GL}_2(\mathbb{Z}_\ell)$, for a given $\ell$, is clearly equivalent to the condition $\Gal(L_n/\mathbb{Q})\approx {\bf GL}_2(\mathbb{Z}/\ell^n\mathbb{Z})$ for any $n\geq 1$. Moreover, the latter condition for a given $n>1$ implies the condition for all $1\leq n^\prime < n$.
\begin{cor} \label{thm:Corollary6} Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. Let $y^2=x^3+Ax+B$ be a Weierstrass equation for $E$, with $A,B\in\mathbb{Z}$. Given a prime number $\ell > 2$ and an integer $n\geq 1$, set $L_n=\mathbb{Q}(E[\ell^n])$ and $\tr_n=\tr_{L_n/\mathbb{Q}}$,
the trace map of $L_n$ over $\mathbb{Q}$. Assume that $\rho_\ell$ induces an isomorphism $\Gal(L_n/\mathbb{Q})\xrightarrow{\approx} {\bf GL}_2(\mathbb{Z}/\ell^n\mathbb{Z})$.
Then, one has: \begin{equation} \tr_{n}(x_n)=0, \end{equation} for any primitive $\ell^n$-torsion point $P_n$ of $E$, $x_n$ denoting its $x$-coordinate. \end{cor}
\begin{proof} From the assumption on $\ell$, all primitive $\ell^n$-torsion points of $E$ are conjugates. But there are $\ell^{2n} - \ell^{2(n-1)}$ of them, which yields $d=(\ell^{2n} - \ell^{2(n-1)})/2$ distinct $x$-coordinates. Now, the polynomial appearing in Corollary \ref{thm:Corollary4} is of the form $\left ( g_{\ell^n}(X) \right)^2$, where $g_{\ell^n}(X):=f_{\ell^n}(X)/f_{\ell^{n-1}}(X)$ has degree $d$. Thus, the polynomial $g_{\ell^n}(X)$ is the irreducible polynomial of $x_n$. As the coefficient of $X^{d-1}$ in this polynomial is equal to $0$ (using Corollary \ref{thm:Corollary4}), it follows that $\tr_{K/\mathbb{Q}} (x_n) =0$, where $K$ is the splitting field of $x_n$. Then, one computes: $\tr_n(x_n) = [L_n:K] \tr_{K/\mathbb{Q}} (x_n) =0$. \end{proof}
\begin{cor} \label{thm:Corollary7} Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM. Let $y^2=x^3+Ax+B$ be a Weierstrass equation for $E$, with $A,B\in\mathbb{Z}$. Given a prime number $\ell>2$ and an integer $n> 1$, set $L_n=\mathbb{Q}(E[\ell^n])$ and $\tr_{n,n-1}=\tr_{L_n/L_{n-1}}$, the trace map of $L_n$ over $L_{n-1}$. Assume that $\rho_\ell$ induces an isomorphism $\Gal(L_n/\mathbb{Q})\xrightarrow{\approx} {\bf GL}_2(\mathbb{Z}/\ell^n\mathbb{Z})$.
Then, one has $[L_n:L_{n-1}]=\ell^4$, and the following identity holds: \begin{equation} \frac{\tr_{n,n-1}(x_n)}{[L_n:L_{n-1}]}=x_{n-1}, \end{equation} for any primitive $\ell^n$-torsion point $P_n$ of $E$, $x_n$ and $x_{n-1}$ denoting the $x$-coordinate of $P_n$ and $[\ell]P_n$, respectively. In particular, $x_n - x_{n-1} \in \Ker \tr_{n,n-1}$. \end{cor}
\begin{proof} Given $n>1$, let $P_n$ be a primitive $\ell^n$-torsion point of $E$, and set $P_{n-1}:=[\ell]P_n$ and $x_{n-1}=x(P_{n-1})$. The roots of the polynomial $\Phi_{\ell}(X,x_{n-1})$ appearing in Corollary \ref{thm:Corollary5}, where we take $\lambda=x_{n-1}\in L_{n-1}$ and $m=\ell$, are the $x$-coordinates of the solutions $P$ to the equation $[\ell]P = \pm P_{n-1}$. Since $P$ and $-P$ have the same $x$-coordinate, we may restrict to the solutions of $[\ell]P=P_{n-1}$. There are $\ell^2$ solutions to this equation; namely $P=P_n+Q$, where $Q$ is an $\ell$-torsion point. This yields $\ell^2$ distinct $x$-coordinates, since $P_n+Q=\pm (P_n+Q^\prime)$ yields $Q=Q^\prime$, or else $[2]P_n=-Q-Q^\prime$, which is excluded since $\ell\not=2$ and $n>1$. Now, $\ell^2$ is the degree of the polynomial $\Phi_{\ell}(X,x_{n-1})$. From the assumption on the Galois group, it follows that the solutions to $[\ell]P=P_{n-1}$ are conjugates, and hence that the roots of $\Phi_{\ell}(X,x_{n-1})$ are conjugates. Therefore, $\Phi_{\ell}(X,x_{n-1})$ is the minimal polynomial of $x_n$ over $L_{n-1}$. We conclude that $\tr_{K/L_{n-1}}(x_n)=\ell^2 x_{n-1}$, where $K$ is the splitting field of $x_n$ over $L_{n-1}$. Thus, one obtains $\tr_{n,n-1}(x_n)=[L_n:K] \ell^2 x_{n-1}$. But then, $[L_n:L_{n-1}]=\ell^4$ from the assumption on the Galois group, whereas $[K:L_{n-1}]=\ell^2$ from above. Thus, $[L_n:K]=\ell^2$ and the result is shown. \end{proof}
In the next result, we consider a place $v$ of $\mathbb{Q}$, either $p$-adic or Archimedean, and pick the canonical norm $\vert \cdot \vert_v$ such that $\vert p \vert_v = p^{-1}$ if $v$ is the non-Archimedean place associated to a prime $p$, or $\vert \cdot \vert_v$ is the absolute value of a real number, if $v$ is the Archimedean place. See \cite[pp. 34--35]{lang1986}. One then has the product formula \cite[p. 99]{lang1986}: \begin{equation} \prod_{v} \vert x \vert_v = 1, \end{equation} for any rational number $x$, where the product covers all canonical places of $\mathbb{Q}$.
If $K$ is an algebraic field, we consider for each place $w$ lying above a place $v$ of $\mathbb{Q}$, the unique norm that extends $\vert \cdot \vert_v$. Namely, if $w$ is non-Archimedean, one defines: \begin{equation} \vert x \vert_w = p^{- w(x)/e_w}, \end{equation} where $w$ is viewed as the discrete valuation associated to the place $w$, $p$ is the prime number lying below $w$, and $e_w$ denotes the absolute ramification index of $p$ in $K_w$. If $w$ is Archimedean, $\vert \cdot \vert_w$ denotes the absolute value if $K_w=\mathbb{R}$, or the complex modulus if $K_w=\mathbb{C}$. See \cite[p. 35 and p. 99]{lang1986} for the alternative norm $\Vert x \Vert_w=\vert x \vert_w^{[K_w:\mathbb{Q}_v]}$ and the corresponding product formula, which is not used here.
We now introduce a topology on $E(\mathbb{C})$ as follows. Adapting \cite[Exerc. 7.6, pp. 203--204]{silverman2009}, we consider the Euclidean topology on $\mathbb{C}$ defined by the complex modulus $\vert \cdot \vert$. Then, we consider the product topology on the affine space $\mathbb{A}^2(\mathbb{C})$. Next, for each $0\leq i \leq 2$, there is an inclusion $\phi_i: \mathbb{A}^2(\mathbb{C}) \rightarrow \mathbb{P}^2(\mathbb{C})$ \cite[p. 9]{silverman2009}. This allows gluing together the product spaces $\phi_i(\mathbb{A}^2(\mathbb{C}))$, $i=0,1,2$. In this manner, we obtain a topology naturally defined on $\mathbb{P}^2(\mathbb{C})$, and hence on $E(\mathbb{C}) \hookrightarrow \mathbb{P}^2(\mathbb{C})$ based on a homogeneous equation $y^2z=x^3+Axz^2+Bz^3$ for $E$. Note that $E(\mathbb{C})$ is a Hausdorff space, so that a sequence of points in this topological space has at most one limit.
In the following result, we use the fact that the polynomial $X^3+AX+B$ has three distinct roots, since an elliptic curve is non-singular; equivalently, since its discriminant $\Delta$ does not vanish.
\begin{prop} \label{thm:Proposition2} Let $E$ be an elliptic curve over $\mathbb{Q}$, having Weierstrass equation $y^2=x^3+Ax+B$, with $A,B\in \mathbb{Z}$. Given a prime number $\ell$, set $L_n=\mathbb{Q}(E[\ell^n])$ and $\tr_{n}=\tr_{L_n/\mathbb{Q}}$.
a) Then, for all prime numbers $\ell >3$ not dividing $\Delta^\prime$, and for any integer $n \geq 1$, one has: \begin{equation} \label{eq:eq94final7} \Bigl \vert \frac{\tr_{n}(\alpha_n)}{[L_n:\mathbb{Q}]} \Bigr \vert_{v} \leq C_{*,v}, \end{equation} for any primitive $\ell^n$-torsion point $P_n$ of $E$, where $\alpha_n=\ell^3 \Delta^\prime/y^2(P_n)$, and any place $v\not=\ell$ of $\mathbb{Q}$, for some constant $C_{*,v}>0$ depending only on $A$ and $B$, the prime $\ell$ and the place $v$. Namely, one has explicitly: \begin{eqnarray} &&i):\quad C_{*,q} = \vert (\ell-1)^{-2} (\ell+1)^{-1} \vert_{q}, \textrm{ if } q\not=\ell;\\ &&ii):\quad C_{*,\infty} = \vert \Delta^\prime \vert \ell^3 \max \left ( 2, 1/\delta^{3} \right ), \end{eqnarray} where the constant $\delta>0$ is the minimal distance between $x$-coordinates $x_n$ and $e_1$ of any primitive $\ell^n$- and $2$-torsion points, respectively, of $E(\mathbb{C})$, such that $\vert x_n \vert < \sqrt{2(\vert A \vert + \vert B \vert)}$.
b) Let $E$ be a non-CM curve, and assume that $\rho_\ell$ induces an isomorphism $\Gal(\mathbb{Q}(E[\ell^n])/\mathbb{Q})\xrightarrow{\approx} {\bf GL}_2(\mathbb{Z}/\ell^n\mathbb{Z})$ for any $n\geq 1$, where $\ell>3$ is a prime number. Then, ${\tr_{n}(\alpha_n)}/{[L_n:\mathbb{Q}]}$ has only finitely many values for all $n\geq 1$ ($\ell$ being fixed). Moreover, (\ref{eq:eq94final7}) then also holds for the $\ell$-adic norm, taking: \begin{equation} iii):\quad C_{*,\ell} = \max_{n=1,2;\, \alpha_n} \Bigl \vert \frac{\tr_n(\alpha_n)}{[L_n:\mathbb{Q}]} \Bigr \vert_{\ell} < \infty.\\ \end{equation} Similar estimates also hold for the other norms. \end{prop}
\begin{proof} {\em Step 1.} Let us fix the prime $\ell > 3$, the integer $n\geq 1$, and the primitive $\ell^n$-torsion point $P_n$ with $x$ and $y$-coordinates $x_n=x(P_n)$ and $y_n=y(P_n)$, respectively. We set $\alpha_n=\ell^3 \Delta^\prime/y_n^2$. We assume that $\ell \nmid \Delta^\prime$.
{\em Step 2.} We consider first the case where $v$ is a non-Archimedean place of $\mathbb{Q}$ corresponding to a prime number $q\not = \ell$. Since $\alpha_n$ is a divisor of $\ell^5 \Delta^\prime$, it follows that $\tr_n (\alpha_n )$ is an integer. Therefore, one obtains: \begin{equation} \Bigl \vert \frac{\tr_{n}(\alpha_n)}{[L_n:\mathbb{Q}]} \Bigr \vert_{q} \leq \vert (\ell-1)^{-2} (\ell+1)^{-1} \vert_{q}, \end{equation} using Lemma \ref{thm:Lemma4}, which proves inequality i).
{\em Step 3.} Next, fix a place $w$ of $L_n$ lying above the Archimedean place $v=\infty$ of $\mathbb{Q}$; equivalently, $w$ corresponds to an embedding $\xi:\overline{\mathbb{Q}} \hookrightarrow \mathbb{C}$, with $\vert x \vert_{w} = \vert \xi(x) \vert$. We set $C_1:=\max \left ( 1, \sqrt{2(\vert A \vert_w + \vert B \vert_w )}\right ) = \sqrt{2(\vert A \vert + \vert B \vert)}$. Then, one has for $\vert x_n \vert_w \geq 1$: \begin{equation} \vert x_n^3+Ax_n+B \vert \geq \vert x \vert_w^3 - \vert A \vert_w \vert x_n \vert_w - \vert B \vert_w \geq \vert x_n \vert_w^3 - \left ( \vert A \vert_w + \vert B \vert_w \right ) \vert x_n \vert_w. \end{equation} Assuming also that $\vert x_n \vert_w \geq \sqrt{2(\vert A \vert_w + \vert B \vert_w )}$, one obtains $\vert x_n \vert_w^2/2 \geq (\vert A \vert_w + \vert B \vert_w )$, which yields: \begin{equation} \vert x_n \vert_w^3 - \left ( \vert A \vert_w + \vert B \vert_w \right ) \vert x_n \vert_w \geq \vert x_n \vert_w^3/2. \end{equation} Altogether, the lower bound $\vert x_n \vert_w \geq C_1$ implies the inequality: \begin{equation} \Bigl \vert \frac{\Delta^\prime \ell^3}{\left ( x_n^3+Ax_n+B \right )} \Bigr \vert_w \leq \frac{\vert \Delta^\prime \vert \ell^3}{\vert x_n \vert_w^3/2} \leq 2 \vert \Delta^\prime \vert \ell^3. \end{equation}
{\em Step 4.} We are left with the case where the norm $\vert x_n \vert_w$ corresponding to an Archimedean place $w$ of $L_n$, is within the bound $C_1$. Let $S$ denote the set of elements $x_n$ such that $\vert x_n \vert_w < C_1$. We then need a lower bound for $\vert x_n^3+Ax_n+B \vert_w$, whenever $x_n \in S$.
{\em Step 5.} Assume that the set $S$ of step 4 is finite.
Then, if any element $x_n$ of $S$ is equal to a root of $X^3+AX+B$, the point $(x_n,0)$ is a $2$-torsion point of $E(\overline{\mathbb{Q}})$, which contradicts the assumption that $x_n$ is the $x$-coordinate of a primitive $\ell^n$-torsion point, as $\ell \not =2$. Therefore, these elements $x_n$ are away from the roots of $X^3+AX+B$, say by a distance $\delta>0$, which implies that $1/(x_n^3+Ax_n+B)$ remains bounded on $S$. Therefore, one obtains: \begin{equation} \Bigl \vert \frac{\Delta^\prime \ell^3}{\left ( x_n^3+Ax_n+B \right )} \Bigr \vert_w \leq \vert \Delta^\prime \vert \ell^3/\delta^{3}, \end{equation} for all elements $x_n \in S$. In that case, one concludes that: \begin{equation} \label{eq:eq103final7} \Bigl \vert \frac{\tr_n(\alpha_n)}{[L_n:\mathbb{Q}]} \Bigr \vert \leq \vert \Delta^\prime \vert \ell^3 \max \left ( 2, 1/\delta^{3} \right ), \end{equation} making use of step 3.
{\em Step 6.} Next, we consider the case where the set $S$ of step 4 is infinite. At this point, we make use of the topology defined above on $E(\mathbb{C}) \hookrightarrow \mathbb{P}^2(\mathbb{C})$, and we identify $E(\overline{\mathbb{Q}})$ with its image under the embedding $\xi: E(\overline{\mathbb{Q}}) \hookrightarrow E(\mathbb{C})$.
As $S$ is bounded, there exists an accumulation point $x_{**} \in \mathbb{C}$ of $S$ such that $\vert x_{**} \vert \leq C_1$. Thus, $\lim_{k\rightarrow \infty} \xi(x_{n_k}) = x_{**}$ for some sequence $\{x_{n_k}\}$.
If the point $x_{**}$ is of the form $\xi(x_*)$, where $x_*:=e_1$ is a root of $X^3+AX+B=(X-e_1)(X-e_2)(X-e_3)$, then $P_*=(x_*,0)$ is a $2$-torsion point of $E(\overline{\mathbb{Q}})$. Since $\ell \not = 2$, one has $[\ell]P_*=P_*$. But, from (\ref{eq:eq66final7}), one has: \begin{equation} x([\ell]P_*) = x_* - \frac{\Psi_{\ell-1}^\prime (x_*,0) \Psi_{\ell+1}^\prime (x_*,0)}{\left ( \Psi_\ell^\prime (x_*,0)\right)^2}, \end{equation} knowing that the denominator $\left ( \Psi_\ell^\prime (x_*,0)\right)^2$ does not vanish, since $\ell \not = 2$. Thus, one concludes that $\Psi_{\ell-1}^\prime (x_*,0) \Psi_{\ell+1}^\prime (x_*,0) = 0$.
We also have: \begin{equation} x([\ell]P_n) = x_n - \frac{\Psi_{\ell-1}^\prime (x_n,y_n) \Psi_{\ell+1}^\prime (x_n,y_n)}{\left ( \Psi_\ell^\prime (x_n,y_n)\right)^2}, \end{equation} assuming that $n>1$. This yields: \begin{eqnarray} &&\vert \Psi_{\ell-1}^\prime (x_n,y_n) \Psi_{\ell+1}^\prime (x_n,y_n) \vert_w \nonumber\\ &=& \vert \Psi_{\ell-1}^\prime (x_n,y_n) \Psi_{\ell+1}^\prime (x_n,y_n) - \Psi_{\ell-1}^\prime (x_*,0) \Psi_{\ell+1}^\prime (x_*,0) \vert_w \nonumber\\ &\leq& C \vert x_n - x_* \vert_w, \end{eqnarray} for some constant $C>0$, since $\vert x_n \vert_w \leq C_1$. We thus conclude that: \begin{equation} \vert x_{n_k} - x([\ell]P_{n_k}) \vert_w = \Bigl \vert \frac{\Psi_{\ell-1}^\prime (x_n,y_n) \Psi_{\ell+1}^\prime (x_n,y_n)}{\left ( \Psi_\ell^\prime (x_n,y_n)\right)^2} \Bigr \vert_w \leq C^\prime \vert x_{n_k} - x_* \vert_w, \end{equation} for some constant $C^\prime>0$. In particular, one has $\lim_{k\rightarrow \infty} x([1 - \ell]P_{n_k}) = 0$. Therefore, for some infinite sequence $\{ P_{n_k}\}$ of primitive $\ell^{n_k}$-torsion points, one has both $\lim_{k\rightarrow \infty} P_{n_k} = P_*$, and $\lim_{k\rightarrow \infty} [1 - \ell]P_{n_k} = (0,\sqrt{B},1)$ or $\lim_{k\rightarrow \infty} [1 - \ell]P_{n_k} = (0,-\sqrt{B},1)$.
We now show that $\lim_{k\rightarrow \infty} [1 - \ell]P_{n_k} = O$, based on the assumption that $\lim_{k\rightarrow \infty} P_{n_k}$ is the $2$-torsion point $P_*$, which will yield a contradiction. For this purpose, we compute the $(x,y)$-coordinates of $Q_n := P_n [-] P_* = P_* [+] P_n$ \cite[pp. 53--54]{silverman2009}. Since $P_n \not = P_*$ and $a_1=a_2=a_3=0$, one has: \begin{equation} \begin{cases} P_* = (e_1,0); \quad P_{n_k} = (x_{n_k},y_{n_k});\\ \lambda_{n_k} = \frac{y_{n_k}}{(x_{n_k}-e_1)};\\ \nu_{n_k} = \frac{-y_{n_k} e_1}{(x_{n_k}-e_1)} = -\lambda_{n_k} e_1;\\ x(Q_{n_k}) = \lambda_{n_k}^2 - x_{n_k} - e_1;\\ y(Q_{n_k}) = - \lambda_{n_k} x(Q_{n_k}) - \nu_{n_k} = - \lambda_{n_k} \left ( \lambda_{n_k}^2 - x_{n_k} - 2 e_1 \right ). \end{cases} \end{equation} One develops: \begin{eqnarray} \lambda_{n_k}^2 &=& \frac{x_{n_k}^3+A x_{n_k} +B}{(x_{n_k}-e_1)^2} = \frac{(x_{n_k}-e_2)(x_{n_k}-e_3)}{(x_{n_k}-e_1)}, \end{eqnarray} where $e_2$ and $e_3$ are the two other roots of $X^3+AX+B$. Since $e_2,e_3\not = e_1$ by non-singularity of $E$, one obtains $\lim_{k \rightarrow \infty} \vert \lambda_{n_k} \vert_w = \infty$, and henceforth: \begin{equation} \begin{cases} \frac{x(Q_{n_k})}{y(Q_{n_k})} = - \frac{\lambda_{n_k}^2 - x_{n_k} - e_1}{\lambda_{n_k} \left ( \lambda_{n_k}^2 - x_{n_k} - 2 e_1 \right )} \rightarrow 0 \textrm{, as } {k \rightarrow \infty};\\ \frac{1}{y(Q_{n_k})} = - \frac{1}{\lambda_{n_k} \left ( \lambda_{n_k}^2 - x_{n_k} - 2 e_1 \right )} \rightarrow 0 \textrm{, as } {k \rightarrow \infty}. \end{cases} \end{equation} Thus, $\lim_{k\rightarrow \infty} Q_{n_k} = O$, as expected. Here, we have thus used the $(x,z)$-coordinates of $Q_{n_k}$.
Next, based on (\ref{eq:eq66final7}) and (\ref{eq:eq67final7}), one has: \begin{equation} \begin{cases} x([a]Q_{n_k}) = \frac{x(\Psi_{a}^\prime)^2 - \Psi_{a+1}^\prime\Psi_{a-1}^\prime}{(\Psi_{a}^\prime)^2};\\ y([a]Q_{n_k}) = \frac{\Psi_{a+2}^\prime (\Psi_{a-1}^\prime)^2 - \Psi_{a-2}^\prime (\Psi_{a+1}^\prime)^2}{ 4 y (\Psi_{a}^\prime)^3}, \end{cases} \end{equation} where $y$ stands for $y(Q_{n_k})$ and $a=\ell-1$, which is an even integer since $\ell\not =2$.
Now, one computes, based on Lemma \ref{thm:Lemma5}: \begin{eqnarray} \left ( \Psi_{a+2}^\prime (\Psi_{a-1}^\prime)^2 - \Psi_{a-2}^\prime (\Psi_{a+1}^\prime)^2 \right ) / (4 y) &=& \left ( f_{a+2}(f_{a-1})^2 - f_{a-2}(f_{a+1})^2 \right ) / 4\nonumber\\ &=& \left ( 4x^{3a^2/2} + \textrm{ terms of lower degree} \right) / 4.\nonumber\\ \end{eqnarray} One also has: \begin{eqnarray} (\Psi_{a}^\prime)^3 &=& (f_{a})^3 y^3\nonumber\\ &=& \left ( a^3 x^{3a^2/2 - 6} + \textrm{ terms of lower degree} \right ) y^3. \end{eqnarray} This yields: \begin{eqnarray} y([a]Q_{n_k}) &=& \frac{\left ( x^{3a^2/2} + \textrm{ terms of lower degree} \right)}{\left ( a^3 x^{3a^2/2 - 6} + \textrm{ terms of lower degree} \right ) y^3}\nonumber\\ &\sim & \frac{1}{a^3} \frac{x^6}{y^3}, \end{eqnarray} where $x$ stands for $x(Q_{n_k})$. It follows that: \begin{equation} \frac{1}{y([a]Q_{n_k})} \sim a^3 \frac{y^3}{x^6} \sim - a^3 \frac{\lambda_{n_k}^{9}}{\lambda_{n_k}^{12}}, \end{equation} which yields: \begin{equation} \lim_{k \rightarrow \infty} \frac{1}{y([a]Q_{n_k})} = 0, \end{equation} since $\lim_{k \rightarrow \infty} \vert \lambda_{n_k} \vert_w = \infty$.
One shows similarly that: \begin{equation} \lim_{k \rightarrow \infty} \frac{x([a]Q_{n_k})}{y([a]Q_{n_k})} = 0. \end{equation} Namely, one has from Lemma \ref{thm:Lemma5}: \begin{eqnarray} x (\Psi_{a}^\prime)^2 - \Psi_{a+1}^\prime \Psi_{a-1}^\prime &=& x (x^3+Ax+B) (f_{a})^2 - f_{a+1} f_{a-1}\nonumber\\ &=& \left ( x^{a^2} + \textrm{ terms of lower degree} \right).\nonumber\\ \end{eqnarray} One also has: \begin{eqnarray} (\Psi_{a}^\prime)^2 &=& (f_{a})^2 y^2\nonumber\\ &=& \left ( a^2 x^{a^2 - 4} + \textrm{ terms of lower degree} \right ) y^2. \end{eqnarray} This yields: \begin{eqnarray} x([a]Q_{n_k}) &=& \frac{\left ( x^{a^2} + \textrm{ terms of lower degree} \right)}{\left ( a^2 x^{a^2 - 4} + \textrm{ terms of lower degree} \right ) y^2}\nonumber\\ &\sim & \frac{1}{a^2} \frac{x^4}{y^2}, \end{eqnarray} so that: \begin{eqnarray} \frac{x([a]Q_{n_k})}{y([a]Q_{n_k})} &\sim& a \frac{y}{x^2} \sim - a \frac{\lambda_{n_k}^3}{\lambda_{n_k}^4}. \end{eqnarray}
One concludes that $\lim_{k \rightarrow \infty} [a]Q_{n_k} = O$; {\em i.e.}, $\lim_{k \rightarrow \infty} [a]P_{n_k} = O$, since $a$ is an even integer, as $\ell\not =2$, and $P_*$ is a $2$-torsion point. Henceforth, one obtains $\lim_{k \rightarrow \infty} [1-\ell]P_{n_k} = O$ because $[1-\ell]P_{n_k} = (x([\ell-1]P_{n_k}),-y([\ell-1]P_{n_k}))$ as $a_1=a_3=0$ \cite[p. 53]{silverman2009}. But on the other hand, from above, one has $\lim_{k \rightarrow \infty} [1-\ell]P_{n_k}=(0,\pm \sqrt{B}, 1) \not = O$, a contradiction.
Therefore, no root of $X^3+AX+B$ can be an accumulation point of the set $S$. This means that $\vert x_n - e_1 \vert_w \geq \delta$, for some constant $\delta>0$, for any $n\geq 1$ and any root $e_1$ of $X^3+AX+B$. This completes the proof of inequality ii), since (\ref{eq:eq103final7}) is then valid.
{\em Step 7.} Part a) being proved, we now show part b) under the assumptions on $E$ and $\ell$ stated in the proposition.
Let $F/\mathbb{Q}$ be the normal closure of the extension obtained by adjoining the roots $e_i$, $i=1,2,3$, of $X^3+AX+B$. Thus, $[F:\mathbb{Q}] \mid 6$. Consider $L_n^\prime = L_n F$, and let $\tr_{n,n-1}^\prime$ denote the trace map of the relative extension $L_{n}^\prime/L_{n-1}^\prime$. Note that $L_n$ and $L_{n-1}^\prime$ are linearly disjoint over $L_{n-1}$, since $[L_n:L_{n-1}]=\ell^4$ and $[L_{n-1}^\prime:L_{n-1}] \mid 6$, as $\ell>3$ by assumption.
Since $x_n$ is a root of the polynomial $f_{\ell^n}(X)$, whereas any root $e_i$ of $X^3+AX+B$ is not, as $\ell\not =2$, it follows that $x_n-e_i \not =0$. One then computes: \begin{eqnarray} \frac{1}{x_n^3+Ax_n+B} &=& \frac{1}{(x_n-e_1)(x_n-e_2)(x_n-e_3)}\nonumber\\ &=& \sum_{i=1}^{3}\frac{A_i}{(x_n-e_i)}, \end{eqnarray} where the coefficients $A_i=A_i(e_1,e_2,e_3)$, $i=1,2,3$, belong to $F$. Concretely, one has: \begin{equation} \begin{cases} A_1=- \frac{1}{(e_1-e_3)(e_2-e_1)};\\ A_2=- \frac{1}{(e_3-e_2)(e_3-e_1)};\\ A_3=- \frac{1}{(e_3-e_2)(e_1-e_3)}.\\ \end{cases} \end{equation} Thus, one obtains: \begin{eqnarray} \tr_{n,n-1}^\prime \left ( \alpha_n \right ) &=& \Delta^\prime \ell^3 \Bigl \{ \sum_{i=1}^{3} A_i\tr_{n,n-1}^\prime \Bigl ( \frac{1}{(x_n-e_i)} \Bigr )\Bigr \}. \end{eqnarray}
{\em Step 8.} We consider the polynomial of Corollary \ref{thm:Corollary5}, taking $m=\ell$ and $\lambda=x_{n-1}$: \begin{eqnarray} \Phi_{\ell}(X, x_{n-1}) &=& \left ( X - x_{n-1}\right ) f_{\ell}^2(X) - f_{\ell-1}(X) f_{\ell+1}(X) \left ( X^3 +AX +B \right )\nonumber\\ &:=& \sum_{j=0}^{\ell^2} a_j X^{j}. \end{eqnarray}
We observe that $\Phi_{\ell}(X + e_i, x_{n-1})$ is the minimal polynomial of $x_n-e_i$ over $L_{n-1}^\prime$. Furthermore, dividing $\Phi_{\ell}(X + e_i, x_{n-1})$ by $X^{\ell^2}$ and making the change of variable $Y:=1/X$, one obtains the polynomial: \begin{eqnarray} \sum_{j=0}^{\ell^2} a_j (X + e_i)^{j} X^{-j} X^{-(\ell^2-j)} &=& \sum_{j=0}^{\ell^2} a_j (1 + e_i Y)^{j} Y^{\ell^2-j}, \end{eqnarray} which is the minimal polynomial of $1/(x_n-e_i)$ over $L_{n-1}^\prime$.
We then obtain the trace $\tr_{n,n-1}^\prime$ of $1/(x_n-e_i)$ as the coefficient of the monomial $-Y^{\ell^2-1}$, which is: \begin{eqnarray} -\sum_{j=1}^{\ell^2} a_j j e_i^{j-1} &=& -\frac{d}{dX}\Phi_{\ell}^\prime(X,x_{n-1})\Bigr \vert_{X=e_i}. \end{eqnarray} But then, one computes: \begin{eqnarray} && \frac{d}{dX}\Phi_{\ell}^\prime(X,x_{n-1})\Bigr \vert_{X=e_i} \nonumber\\ && \quad = f_{\ell}^2(e_i) + (e_i-x_{n-1})2 f_{\ell}(e_i) \frac{d}{dX}f_{\ell}(X)\Bigr \vert_{X=e_i} - f_{\ell-1}(e_i) f_{\ell+1}(e_i) (3e_i^2+A), \nonumber\\ \end{eqnarray} since $e_i^3+Ae_i+B=0$. Therefore, one obtains: \begin{eqnarray} && \tr_{n,n-1}^\prime \left ( \alpha_n \right ) \nonumber\\ && \quad = - \Delta^\prime \ell^3 \Bigl \{ \sum_{i=1}^{3} A_i\Bigl ( f_{\ell}^2(e_i) + 2 e_i f_{\ell}(e_i) \frac{d}{dX}f_{\ell}(X)\Bigr \vert_{X=e_i} - f_{\ell-1}(e_i) f_{\ell+1}(e_i) (3e_i^2+A) \Bigr )\Bigr \} \nonumber\\ && \quad + \Delta^\prime \ell^3 \Bigl \{ \sum_{i=1}^{3} A_i\Bigl ( 2 x_{n-1}f_{\ell}(e_i) \frac{d}{dX}f_{\ell}(X)\Bigr \vert_{X=e_i} \Bigr )\Bigr \} \end{eqnarray}
Applying Corollary \ref{thm:Corollary7} iteratively, one then computes: \begin{eqnarray} && \frac{\tr_{L_n^\prime/L_1^\prime} \left ( \alpha_n \right )}{[L_{n}^\prime:L_1^\prime]} \nonumber\\ && \quad = - \frac{\Delta^\prime}{\ell} \Bigl \{ \sum_{i=1}^{3} A_i \Bigl ( f_{\ell}^2(e_i) + 2 e_i f_{\ell}(e_i) \frac{d}{dX}f_{\ell}(X)\Bigr \vert_{X=e_i} - f_{\ell-1}(e_i) f_{\ell+1}(e_i) (3e_i^2+A) \Bigr )\Bigr \} \nonumber\\ && \quad + \frac{\Delta^\prime}{\ell} \Bigl \{ \sum_{i=1}^{3} A_i 2 x_{1}f_{\ell}(e_i) \frac{d}{dX}f_{\ell}(X)\Bigr \vert_{X=e_i} \Bigr \}. \end{eqnarray}
Altogether, we have reached the identity: \begin{eqnarray} && \frac{\tr_{n} \left ( \alpha_n \right )}{[L_n:\mathbb{Q}]} = \frac{\tr_{n}^\prime \left ( \alpha_n \right )}{[L_n^\prime:\mathbb{Q}]} \nonumber\\ && \quad = - \frac{\Delta^\prime}{\ell [F:\mathbb{Q}]} \tr_{F/\mathbb{Q}} \Bigl \{ \sum_{i=1}^{3} A_i \Bigl ( f_{\ell}^2(e_i) + 2 e_i f_{\ell}(e_i) \frac{d}{dX}f_{\ell}(X)\Bigr \vert_{X=e_i} \Bigr )\Bigr \} \nonumber\\ && \quad + \frac{\Delta^\prime}{\ell [F:\mathbb{Q}]} \tr_{F/\mathbb{Q}} \Bigl \{ \sum_{i=1}^{3} A_i f_{\ell-1}(e_i) f_{\ell+1}(e_i) (3e_i^2+A) \Bigr \} \nonumber\\ && \quad + \frac{\Delta^\prime}{\ell [L_1^\prime:\mathbb{Q}]} \tr_{L_1^\prime/\mathbb{Q}} \Bigl \{ \sum_{i=1}^{3} A_i 2 x_{1}f_{\ell}(e_i) \frac{d}{dX}f_{\ell}(X)\Bigr \vert_{X=e_i} \Bigr \}. \end{eqnarray}
Therefore, the rational number ${\tr_{n} \left ( \alpha_n \right )}/{[L_n:\mathbb{Q}]}$ has only finitely many values for $n>1$, namely the values reached when taking $n=2$. As for $n=1$, there are only finitely many values for $\alpha_1$. This completes the proof of part b). \end{proof}
\subsection{Liftings of points on reduced elliptic curves} \label{subsection:liftingsReduced}
In this section, we continue to specialize results to the case where $E$ is an elliptic curve over $\mathbb{Q}$ without CM.
\begin{thm} \label{thm:Theorem6} Let $E$ be an elliptic over $\mathbb{Q}$ without CM, with Weierstrass equation $y^2=x^3+Ax+B$, where $A,B\in \mathbb{Z}$. Set $\Delta^\prime = 4 A^3 + 27 B^2$. Let $\ell>3$ be a fixed prime number that does not divide $\Delta^\prime$. Assume that the isomorphism $\rho_\ell:\Gal(L_\infty/\mathbb{Q})\xrightarrow{\approx} {\bf GL}_2(\mathbb{Z}_\ell)$ holds.
Let $p$ be a prime number such that $p \nmid \Delta^\prime \ell (\ell-1) (\ell+1)$. Let $\widetilde P$ be any non-trivial point of the $\ell$-component of the reduced curve $\widetilde E(\mathbb{F}_{p})$. Let the prime power $\ell^n$, with $n\geq 1$, be the order of the point $\widetilde P$.
Then, the point $\widetilde P$ can be lifted to a point $P$ of $E(\overline{\mathbb{Q}})$ with affine $y$-coordinate satisfying: \begin{equation} \label{eq:eq132final8} y(P) = \Bigl ( \frac{a}{b} \Bigr )^{1/2}, \end{equation} where $a$ and $b$ are integers such that $1\leq \vert a \vert, \vert b \vert \leq C$, for some constant $C$ independent of $n$, and where both $a$ and $b$ are coprime with $p$. \end{thm}
\begin{proof} {\em Step 1.} Let $\ell$ be a fixed prime number other than $2$ and $3$. Let $p\nmid \Delta^\prime \ell (\ell-1)(\ell+1)$ be a prime number. In particular, since $\ord_p(\Delta)=0$, the reduced curve $\widetilde E(\mathbb{F}_p)$ is non-singular.
Let $\widetilde P$ be a non-trivial point of the $\ell$-component of the reduced curve $\widetilde E(\mathbb{F}_{p})$. Let $(\bar{x},\bar{y})$ be the affine coordinates of $\widetilde P$ and set $\ell^n = \ord_{\widetilde E}(\widetilde P)$ with $n\geq 1$.
Let $\xi: \overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_p$ be a fixed embedding of fields. From Lemma \ref{thm:Lemma2} applied to $m=\ell^n$, $\widetilde P$ can be lifted to a point $P^\prime$ in $E(\mathbb{Q}_p)$, since $(\ell^n,p)=1$. Since $\xi$ induces an isomorphism $E[\ell^n](\overline{\mathbb{Q}}) \xrightarrow{\approx} E[\ell^n](\overline{\mathbb{Q}}_p )$ \cite[Corollary 6.4, part b), p. 86]{silverman2009}, it follows that there is an element $P_n \in E[\ell^n](\overline{\mathbb{Q}})$ such that $\xi(P_n) = P^\prime$.
{\em Step 2.} We denote $\mathbb{Q}(E[\ell^n])$ by $L_n$, and its integer ring by $\mathcal{O}_n$. From part a) of Lemma \ref{thm:Lemma3}, one has $\ell x(P_n), \ell y(P_n)\in \mathcal{O}_n$, and $\left ( \ell y(P_n) \right )^2 \alpha_n = \Delta^\prime \ell^5$, where $\alpha_n$ is given by (\ref{eq:eq50final7}): \begin{equation} \label{eq:eq133final8} \alpha_n = \ell^3 [ 4 f(x_n) x_n^\prime -g(x_n) ] = \ell^3 \left ( 12x_n^2 x_n^\prime + 16 A x_n^\prime - 3x_n^3 + 5A x_n + 27 B \right ), \end{equation} with $x_n=x(P_n)$ and $x_n^\prime=x([2]P_n)$. From step 1, there exists $y\in \mathbb{Z}$ such that: \begin{equation} y \equiv \ell y(P_n) \mod \mathfrak{P}, \end{equation} where $\mathfrak{P}\mid p$ is the prime ideal of $L_n$ that induces the embedding $\xi: \bigl ( L_n \bigr )_{\mathfrak{P}} \hookrightarrow \overline{\mathbb{Q}}_p$.
Taking $y\in \mathbb{Z}$ as above, we then have: \begin{eqnarray} \ell^2 y^2(P_n) \alpha_n = \Delta^\prime \ell^5 &\Rightarrow& y^2 \alpha_n = \Delta^\prime \ell^5 + \alpha_0 \nonumber\\ &\Rightarrow& y^2 \tr_n \left ( \alpha_n \right) = \Delta^\prime \ell^5 [L_n:\mathbb{Q}] + \tr_n \left ( \alpha_0 \right) \in \mathbb{Z}, \end{eqnarray} for some $\alpha_0 \in \mathfrak{P}$, where $\tr_n$ denotes the trace map from $L_n$ onto $\mathbb{Q}$ and $[L_n:\mathbb{Q}]$ denotes the degree of $L_n/\mathbb{Q}$.
Now, $\tr_n \left ( \alpha_0 \right) \in (p)$, since $\alpha_0 \in \mathfrak{P}$. Indeed, one has a commutative diagram: \begin{equation} \begin{CD} L_n \otimes_{\mathbb{Q}} \mathbb{Q}_p @>>> \prod_{\mathfrak{P}^\prime\mid p} \left ( L_n \right)_{\mathfrak{P}^\prime}\\ @VV{\tr_n}V @VV{\sum_{\mathfrak{P}^\prime\mid p} \tr_{\left ( L_n \right)_{\mathfrak{P}^\prime}/\mathbb{Q}_p}}V\\ \mathbb{Q}_p @= \mathbb{Q}_p.\\ \end{CD} \end{equation} Moreover, from \cite[Corollary 1, p. 142]{weil1974}, one has $\tr_{\left ( L_n \right)_{\mathfrak{P}^\prime}/\mathbb{Q}_p}(\alpha_0)\in (p)$, for each $\mathfrak{P}^\prime\mid p$.
This yields: \begin{equation} \label{eq:eq137final8} y^{2} \tr_n \left ( \alpha_n \right ) \equiv \Delta^\prime \ell^5 [L_n:\mathbb{Q}] \mod p. \end{equation} But $[L_n:\mathbb{Q}]$ divides $\ell(\ell-1)^2(\ell+1)\ell^{4(n-1)}$, as follows from Lemma \ref{thm:Lemma4} applied to $m=\ell^n$. Henceforth, having assumed that $p \nmid \Delta^\prime \ell(\ell-1)(\ell+1)$, one obtains: \begin{equation} \label{eq:eq138final8} y^{2} \equiv \frac{\Delta^\prime \ell^5}{\tr_n \left ( \alpha_n \right )/[L_n:\mathbb{Q}]} \mod p. \end{equation} Indeed, we see that $\tr_n \left ( \alpha_n \right ) \not \equiv 0 \mod p$, for otherwise we obtain the contradiction $\Delta^\prime \ell^5 [L_n:\mathbb{Q}] \equiv 0 \mod p$, using (\ref{eq:eq137final8}). A similar argument shows that $y\not \equiv 0 \mod p$.
{\em Step 3.} It then follows that: \begin{equation} y(P_n) \equiv y/\ell \mod \mathfrak{P}, \end{equation} and \begin{equation} y/\ell \equiv y(P) \mod \mathfrak{p}, \end{equation} where \begin{equation} \label{eq:eq141final8} y(P):=\left ( \frac{\Delta^\prime \ell^3}{\tr_n \left ( \alpha_n \right )/[L_n:\mathbb{Q}]} \right )^{1/2}, \end{equation} and $\mathfrak{p}$ is the prime ideal of $K:=\mathbb{Q}(y(P))$ that induces the embedding $\xi:K_{\mathfrak{p}} \hookrightarrow \overline{\mathbb{Q}}_p$.
{\em Step 4.} At this point, we consider the dependency of $\alpha_n$ on $n$. Under the assumptions stated in the theorem, Proposition \ref{thm:Proposition2} implies that: \begin{equation} \frac{\tr_n(\alpha_n)}{[L_n:\mathbb{Q}]} = \frac{b}{c}, \end{equation} where $c\not=0$ and $b$ are integers that are bounded independently of $n$, and which we may assume relatively prime. Moreover, we observe that $b$ has to be different from $0$ and actually coprime with $p$, since $\tr_n(\alpha_n) \not \equiv 0 \mod p$ from step 2. Furthermore, $c$ is coprime with $p$ since $[L_n:\mathbb{Q}]\not \equiv 0 \mod p$. The theorem follows using (\ref{eq:eq141final8}), upon setting $a=\Delta^\prime \ell^3 c$. \end{proof}
\noindent {\bf Remark 3.} \label{remark3} The lifting $P$ considered in Theorem \ref{thm:Theorem6} might not be a $\ell^n$-torsion point of $E$, but it projects down to a $\ell^n$-torsion point $\widetilde P$ of the reduced curve $\widetilde E(\mathbb{F}_p)$. \\
\noindent {\bf Remark 4.} \label{remark4} One observes that there are only finitely many admissible possibilities for the $y$-coordinate $y(P)=\Bigl ( \frac{a}{b} \Bigr )^{1/2}$ of the lifting $P$ considered in the statement of Theorem \ref{thm:Theorem6}. Note that both $a$ and $b$ are units of $\mathbb{Z}_p$ in the statement of Theorem \ref{thm:Theorem6}. If one chooses to express $b^{-1}$ as an integer modulo $p$, there might be infinitely many such resulting integers as $p$ varies. But this is unnecessary. The point here, is to have finitely many possibilities for $y^2(P)$, rather than expressing $y^2(P)$ as an integer modulo $p$. \\
To appreciate Theorem \ref{thm:Theorem6}, let us observe that, from \cite[Corollary 2]{kohel2000}, it follows directly that the reduced curve $\widetilde E(\mathbb{F}_p)$ of an elliptic curve $E$ over $\mathbb{Q}$ admits a set of generators $\widetilde P_i$ with $y$-coordinates satisfying the condition: \begin{equation} 0 \leq y(\widetilde P_i) \leq \lceil 20 (1 + \log p) p^{1/2} \rceil, \end{equation} upon taking the rational function $f=y$ of degree $3$ in this result. This implies in turn that any $\ell^n$-torsion point $\widetilde P$ of the reduced curve $\widetilde E$ can be lifted to a point $P$ of $E$ with $x$-coordinate belonging to the compositum of all field extensions over $\mathbb{Q}$ generated by the roots of cubic equations of the form \begin{equation} X^3+AX+B=y^2, \end{equation} for some $y \in \mathbb{Z}$ such that $0 \leq y \leq \lceil 20 (1 + \log p) p^{1/2} \rceil$. In particular, there are infinitely many such extensions to consider as prime $p$ varies.
On the other hand, Theorem \ref{thm:Theorem6} restricts to $\ell^n$-torsion points $\widetilde P$ of the reduced curve $\widetilde E$, and states that a lifting $P$ of $\widetilde P$ can be chosen so as to satisfy (\ref{eq:eq132final8}). Most importantly, there are only finitely many possibilities for the right-hand side of this equation.\\
\subsection{Properties of the specific algebraic number fields used for liftings} \label{subsection:liftingsField}
It will be convenient to define a field $K^\prime$ step by step as follows.
First, we consider the cyclotomic field: \begin{equation} \label{eq:eq145final8} K_1=\mathbb{Q}(\mu_{4})=\mathbb{Q}(\sqrt{-1}); \end{equation} then, the field obtained by adjoining radicals: \begin{equation} \label{eq:eq146final8} K_2: = K_1(p_1^{1/2},...,p_\nu^{1/2}), \end{equation} where $p_1,...,p_\nu$ are the $\nu$ distinct prime numbers other than $\ell$ that are bounded by the constant $C$ appearing in Theorem \ref{thm:Theorem6}; $C$ is a positive constant depending on $E$ and $\ell$. Next, we consider the field obtained by adjoining the remaining radical: \begin{equation} \label{eq:eq147final8} K_3: = K_2(\ell^{1/2}). \end{equation} Lastly, one adjoins over $K_3$ the roots $x_1,x_2,x_3$ of cubic equations of the form: \begin{equation} \label{eq:eq148final8} X^3+AX+B=y^2, \end{equation} as $y^{2}$ covers the set $\mathcal{U}$ of rational numbers of the form $\frac{a}{b}$, where $a$ and $b$ are integers satisfying $1 \leq \vert a \vert, \vert b \vert \leq C$. The resulting field extension is denoted $K_3^{(y)}$.
This yields the compositum of fields: \begin{equation} \label{eq:eq149final8} K^\prime: = \prod_{y^{2} \in \mathcal{U}} K_3^{(y)}. \end{equation}
\begin{prop} \label{thm:Proposition3} The extension $K^\prime/\mathbb{Q}$ defined in (\ref{eq:eq149final8}) is a normal extension that contains all affine coordinates of liftings $P$ appearing in Theorem \ref{thm:Theorem6}.
Moreover, the degree $[K^\prime:\mathbb{Q}]$ is of the form $2^{s} 3^{t}$, for some non-negative integers $s$ and $t$. In particular, $[K^\prime:\mathbb{Q}]$ is coprime with $\ell>3$. \end{prop}
\begin{proof} We proceed step by step as follows.
Firstly, the cyclotomic field $K_1:=\mathbb{Q}(\mu_{4})$ has degree $2$ over $\mathbb{Q}$.
Next, consider a Kummer extension of the form $K_1( p^{1/2} )/K_1$, where $p$ is the prime $\ell$ or one of the prime numbers bounded by $C$. This Kummer extension has relative degree dividing $2$. Moreover, since the prime $p$ is fixed under Galois action of $\Gal(K_1/\mathbb{Q})$, it follows that $K_1( p^{1/2} )$ is normal over $\mathbb{Q}$. Hence, both $K_2/\mathbb{Q}$ and $K_3/\mathbb{Q}$ are normal extensions.
It is clear that a Kummer extension of the form $K_1(y)/K_1$, where $y^2 = \frac{a}{b}$ with $1 \leq \vert a \vert, \vert b \vert \leq C$, is contained in $K_3$.
Next, fixing a rational number $y$ as above, one obtains a cubic equation: \begin{equation} X^3+AX+B = y^2, \end{equation} whose roots are in $K^\prime$, by construction. The relative normal closure of this equation over $K_3$ has relative degree dividing $6$.
Lastly, Galois action on the roots of such a cubic equation yields roots of another such cubic equation. Thus, $K^\prime$ is normal over $\mathbb{Q}$. \end{proof}
\begin{prop} \label{thm:Proposition4} Assume that $p \not \equiv 1 \mod \ell$. Then, the $\ell$-component of the reduced curve $\widetilde E(\mathbb{F}_p)$ is cyclic. \end{prop}
\begin{proof} If $\widetilde E(\mathbb{F}_p)_{\ell}$ is not cyclic, then it contains $\widetilde E[\ell]$, from which it follows that $\mu_\ell \subset \mathbb{F}_p$ \cite[Corollary 8.1.1, p. 96]{silverman2009} (consequence of the Weil pairing). Thus, one would have $\ell \mid (p-1)$, contrary to the assumption that $p \not \equiv 1 \mod \ell$. \end{proof}
The following results will be crucial in the proof of Lemma \ref{thm:Lemma9}.
\begin{prop} \label{thm:Proposition5} Let $\ell$ be a prime number. Let $n$ be a positive integer coprime with $\ell$. Define $K_1=\mathbb{Q}(\mu_n)$, and consider a field of the form \begin{equation} K_2 = K_1( p_1^{1/n},...,p_\nu^{1/n}), \end{equation} where $p_1,..,p_\nu$ are $\nu$ distinct prime numbers, each coprime with $\ell$. Then, $\ell$ is unramified in $K_2$. \end{prop}
\begin{proof} Firstly, $\ell$ is unramified in the cyclotomic field $K_1$ \cite[Theorem 2, p. 74]{lang1986}.
Next, let us consider the extension $K_{(p_1)}:=K_1(p_1^{1/n})$. This is a Kummer extension over $K_1$ of degree $d_1$ dividing $n$. Then, setting $y_1=p_1^{1/n}$, one deduces from Kummer theory that $y_1^{d_1}$ belongs to $K_1$. In particular, $f_1(X)=X^{d_1}-y_1^{d_1}$ is the minimal polynomial of $y_1$ over $K_1$. It follows that $\ell$ is coprime with the discriminant of the relative Kummer extension $K_{(p_1)}/K_1$, because the different of this extension divides the ideal generated by $f_1^\prime(y_1)=d_1y_1^{d_1-1}$ \cite[Corollary 2, p. 56]{serre1979}, as $y_1$ belongs to the integer ring of $K_1(y_1)$, and both $d_1$ and $y_1$ are coprime with $\ell$.
Proceeding by induction, one considers the Kummer extension $K_{(p_1,...,p_r,p_{r+1})}$ over $K_{(p_1,...,p_r)}$, defined as $K_{(p_1,...,p_r)}(p_{r+1}^{1/n})$, where $1\leq r < \nu$. The same argument as above shows that the relative different of this extension divides the ideal generated by $f_{r+1}(y_{r+1})=d_{r+1}y_{r+1}^{d_{r+1}-1}$, where $y_{r+1}=p_{r+1}^{1/n}$, $d_{r+1}$ is the degree of the relative extension, and $f_{r+1}(X)=X^{d_{r+1}} - y_{r+1}^{d_{r+1}}$ is the minimal polynomial of $y_{r+1}$ over $K_{(p_1,...,p_r)}$.
One concludes that $\ell$ is coprime with the different of the relative extension $K_2/K_1$, from the transitivity property \cite[Proposition 8, p. 51]{serre1979}. Altogether, it follows that $\ell$ is unramified in $K_2$, applying \cite[Proposition 6, p. 50]{serre1979} and \cite[Corollary 1, p. 53]{serre1979}. \end{proof}
\begin{cor} \label{thm:Corollary8} Let $\ell>2$ be a prime number. Let $K^\prime$ be the field defined in (\ref{eq:eq149final8}). Then,
a) $\ell$ is unramified in $K_2$, and hence, $\mathbb{Q}(\mu_\ell) \cap K_2 = \mathbb{Q}$;
b) $\mathbb{Q}(\mu_\ell) \cap K_3$ is equal to the unique quadratic subfield $K_0$ of $\mathbb{Q}(\mu_\ell)$ ({\em i.e.}, $\mathbb{Q}(\sqrt{\ell})$ if $\ell \equiv 1 \mod 4$, or $\mathbb{Q}(\sqrt{-\ell})$ if $\ell \equiv 2,3 \mod 4$);
c) the extension $(\mathbb{Q}(\mu_\ell) \cap K^\prime)/\mathbb{Q}$ has degree dividing $12$. \end{cor}
\begin{proof} Part a). The first statement follows from Proposition \ref{thm:Proposition5} applied to $n=4$ and the distinct prime factors $p_1,...,p_\nu$ other than $\ell$ that are bounded by $C$, having assumed that $\ell>2$.
Thus, the prime $\ell$ is unramified in $K_2 \subset K_1( p_1^{1/4},...,p_\nu^{1/4})$. See (\ref{eq:eq146final8}). On the other hand, $\ell$ is totally ramified in $\mathbb{Q}(\mu_\ell)$ \cite[Proposition 17, p. 78]{serre1979}. This yields the equality $\mathbb{Q}(\mu_\ell) \cap K_{2} = \mathbb{Q}$.
Part b). The extension $K_3/K_2$ has degree dividing $2$, and hence the same property holds true for $(\mathbb{Q}(\mu_\ell)\cap K_3)/(\mathbb{Q}(\mu_\ell)\cap K_2)$. But $\mathbb{Q}(\mu_\ell)\cap K_2 = \mathbb{Q}$ from part a). Furthermore, $\mathbb{Q}(\mu_\ell)\cap K_3$ contains both quadratic fields $\mathbb{Q}(\sqrt{\ell})$ and $\mathbb{Q}(\sqrt{-\ell})$. Thus, it contains the unique quadratic subfield of $\mathbb{Q}(\mu_\ell)$.
Part c). The field $K^\prime$ is obtained by adjoining to $K_3$ various roots of cubic equations. For each of these cubic equations, the splitting field has Galois group a quotient of the permutation group $\mathfrak{S}_3$, so that its Galois group has cardinality dividing $6$. Therefore, the normal extension $K^\prime/K_3$ has Galois group of exponent dividing $6$.
Now, we have an isomorphism of groups: \begin{equation} \Gal((\mathbb{Q}(\mu_\ell) \cap K^\prime) \cdot K_3/K_3) \xrightarrow{\approx} \Gal((\mathbb{Q}(\mu_\ell) \cap K^\prime)/(\mathbb{Q}(\mu_\ell) \cap K^\prime) \cap K_3). \end{equation} But since, $K_3 \subseteq K^\prime$, it follows that \begin{equation} \Gal((\mathbb{Q}(\mu_\ell) \cap K^\prime) \cdot K_3/K_3) \xrightarrow{\approx} \Gal((\mathbb{Q}(\mu_\ell) \cap K^\prime)/(\mathbb{Q}(\mu_\ell) \cap K_3)). \end{equation} Then, since $(\mathbb{Q}(\mu_\ell) \cap K^\prime) \cdot K_3 \subseteq K^\prime$, we obtain a surjective group homomorphism: \begin{equation} \Gal(K^\prime/K_3) \twoheadrightarrow \Gal((\mathbb{Q}(\mu_\ell) \cap K^\prime)/(\mathbb{Q}(\mu_\ell) \cap K_3)). \end{equation}
Therefore, $\Gal((\mathbb{Q}(\mu_\ell) \cap K^\prime)/(\mathbb{Q}(\mu_\ell) \cap K_3))$ is annihilated by $6$, and henceforth, $\Gal((\mathbb{Q}(\mu_\ell) \cap K^\prime)/\mathbb{Q})$ has exponent dividing $12$. Since $\Gal(\mathbb{Q}(\mu_\ell)/\mathbb{Q})$ is a cyclic group, we conclude that $(\mathbb{Q}(\mu_\ell) \cap K^\prime)/\mathbb{Q}$ has degree dividing $12$. \end{proof}
\section{Proof of the Main Theorem} \label{section:ProofMainTheorems}
Let $E$ be an elliptic curve over an algebraic number field $K$ and $\ell$ be a prime number. We denote the group of $\ell$-torsion points $E[\ell](\overline{\mathbb{Q}})$ by $E[\ell]$. We set $G=\Gal(L/K)$, where $L=K(E[\ell])$. The absolute Galois groups $\Gal(\overline{\mathbb{Q}}/K)$ and $\Gal(\overline{\mathbb{Q}}/L)$ are denoted $\mathcal{G}$ and $\mathcal{H}$, respectively.
For each places $v_0\mid p$ and $v\mid v_0$ of $K$ and $L$, respectively, where $p$ is a prime number, we set $\mathcal{G}_{v_0}=\Gal(\overline{\mathbb{Q}}_p/K_{v_0})$ and $\mathcal{H}_v=\Gal(\overline{\mathbb{Q}}_p/L_v)$. We consider the $[\ell]$-Selmer groups: \begin{eqnarray} S^{[\ell]}(E/L) &=&\Ker \Bigl \{ \Hom(\mathcal{H},E[\ell]) \xrightarrow{\oplus_v Res^{\mathcal{H}}_{\mathcal{H}_v}} \oplus_{v} \Homol^1(\mathcal{H}_v,E(\overline{\mathbb{Q}}_p)) \Bigr \};\\ S^{[\ell]}(E/K) &=& \Ker \Bigl \{ \Hom(\mathcal{G},E[\ell]) \xrightarrow{\oplus_{v_0} Res^{\mathcal{G}}_{\mathcal{G}_{v_0}}} \oplus_{v_0} \Homol^1(\mathcal{G}_{v_0},E(\overline{\mathbb{Q}}_p)) \Bigr \}. \end{eqnarray} Here, $v$ and $v_0$ cover all places of $L$ and $K$, respectively.
We are interested in computing the groups $S^{[\ell]}(E/K)$ and $S^{[\ell]}(E/L)$. Note that the $[\ell]$-Selmer groups are finite \cite[Theorem 4.2, part b), p. 333]{silverman2009} and, in fact, are finite vector spaces over $\mathbb{F}_\ell$. In Sections \ref{subsection:caseQ} and \ref{subsection:proofTheorem5}, we specialize to the case where $K=\mathbb{Q}$.
\subsection{Map from $S^{[\ell]}(E/K)$ into $S^{[\ell]}(E/L)$} \label{subsection:MapFrom}
We recall the exact inflation-restriction sequence of Galois cohomology.
\begin{lem}[Inflation-restriction exact sequence] \label{thm:Lemma6} Let $G$ be a (possibly infinite) profinite group with closed normal subgroup $N$. Let $A$ be a $G$-module. Then, there is an exact (inflation-restriction) sequence: \begin{equation} 0 \rightarrow \Homol^1(G/N,A^N) \xrightarrow{Inf^{G}_{G/N}} \Homol^1(G,A) \xrightarrow{Res^{G}_{N}} \Homol^1(N,A)^{G/N}, \end{equation} where $g\in G$ acts on a $1$-cocycle $f:N \rightarrow A$ as $(g\cdot f)(n)=\,^gf(g^{-1}n g)$ for $n\in N$. \end{lem}
\begin{proof} As in \cite[p. 420]{silverman2009}, one has an exact sequence: \begin{equation} 0 \rightarrow \Homol^1(G/N,A^N) \xrightarrow{Inf^{G}_{G/N}} \Homol^1(G,A) \xrightarrow{Res^{G}_{N}} \Homol^1(N,A). \end{equation} Then, one computes directly, assuming that $f$ is defined at $g$: \begin{eqnarray} \,^{g}f(g^{-1} n g) &=& \,^{g}\{f(g^{-1}) + \,^{g^{-1}}\left ( f(n) + \,^{n} f(g) \right )\}\nonumber\\ &=& \,^{g}f(g^{-1}) + f(n) + \,^{n} f(g) = -f(g) + f(n) + \,^{n} f(g)\nonumber\\ & =& f(n) + (n-1) f(g). \end{eqnarray} So, taking $g\in N$, one sees that the action of $G$ on $\Homol^1(N,A)$ factors through $G/N$. Moreover, if $f$ is defined on all of $G$, one deduces that $Res^{G}_{N}(f)\in \Homol^1(N,A)^{G/N}$. \end{proof}
\begin{prop} \label{thm:Proposition6} Let $\ell$ be a prime number and $E$ an elliptic curve over an algebraic number field $K$. Set $L=K(E[\ell])$. Then, one has a homomorphism of groups: \begin{equation} Res^{\mathcal{G}}_{\mathcal{H}}: S^{[\ell]}(E/K) \rightarrow S^{[\ell]}(E/L)^{\mathcal{G}/\mathcal{H}}, \end{equation} where $\mathcal{G}$ and $\mathcal{H}$ denote the absolute Galois groups of $K$ and $L$, respectively. \end{prop}
\begin{proof} From the inflation-restriction sequence, we obtain a commutative diagram for any places $v_0\mid p$ of $K$ and $v\mid v_0$ of $L$: \begin{equation} \begin{CD} \Homol^1(\mathcal{G},E[\ell]) @>{Res^{\mathcal{G}}_{\mathcal{H}}}>> \Homol^1(\mathcal{H},E[\ell])^{\mathcal{G}/\mathcal{H}}\\ @VV{Res^{\mathcal{G}}_{\mathcal{G}_{v_0}}}V @VV{Res^{\mathcal{H}}_{\mathcal{H}_v}}V \\ \Homol^1(\mathcal{G}_{v_0},E(\overline{\mathbb{Q}}_p))[\ell] @>{Res^{\mathcal{G}_{v_0}}_{\mathcal{H}_v}}>> \Homol^1(\mathcal{H}_v,E(\overline{\mathbb{Q}}_p))[\ell]. \end{CD} \end{equation} where $\mathcal{G}_{v_0}$ and $\mathcal{H}_v$ are the absolute Galois groups of $K_{v_0}$ and $L_v$, respectively. The result now follows from an easy diagram chasing. \end{proof}
\subsection{The group $S^{[\ell]}(E/L)$} \label{subsection:SelmerL}
In this section, $E$ is an elliptic curve over an algebraic number field $K$.
Motivated by Proposition \ref{thm:Proposition6}, we consider the $[\ell]$-Selmer group of $E$ over $L=K(E[\ell])$.
Now, by construction, the Galois group $\mathcal{H}$ acts trivially on $E[\ell]$, so that one has: \begin{equation} \Homol^1(\mathcal{H},E[\ell]) = \Hom(\mathcal{H},E[\ell])\approx \oplus_{i=1}^{2} \Hom(\mathcal{H},\mathbb{F}_\ell). \end{equation}
Moreover, $L$ contains necessarily $\mu_\ell$ because $\Lambda^2 T_\ell(E) \approx T_\ell(\mu)$, where $\mu$ denotes here the multiplicative group. See \cite[p. 99]{silverman2009}.
Thus, any non-trivial character $\chi$ in $\Hom(\mathcal{H},\mathbb{F}_\ell)$ factors through the Galois group of a Kummer extension of degree $\ell$. In particular, one has an isomorphism: \begin{equation} L^*/(L^*)^{\ell} \xrightarrow{\approx} \Hom(\mathcal{H},\mathbb{F}_\ell) = \Hom(\mathcal{H}/\mathcal{H}^\ell,\mathbb{F}_\ell). \end{equation} Since the group $L^*/(L^*)^{\ell}$ is infinite, we want to specify those group homomorphisms in $\Hom(\mathcal{H},E[\ell])$ that belong to the $[\ell]$-Selmer group of $E$ over $L$.
Thus, we consider a group homomorphism $\Psi \in \Hom(\mathcal{H},E[\ell])$ such that the following condition holds: \begin{equation} \label{eq:eq164final8} (*) \qquad Res^{\mathcal{H}}_{\mathcal{H}_v}(\Psi) \mapsto 0 \in \Homol^1(\mathcal{H}_v,E(\overline{\mathbb{Q}}_p)), \end{equation} for a given finite place $v\mid p$ of $L$.
Let $L^\prime=\overline{\mathbb{Q}}^{\Ker \Psi}$, $N=\Gal(L^\prime/L)$ and $N_{v^\prime}=N_v$ ($N$ is Abelian) be its decomposition group at $v^\prime\mid v$. Thus, $N$ can be viewed as a subgroup of $\mathbb{F}_\ell \oplus \mathbb{F}_\ell$. We have a commutative diagram with exact rows: \begin{equation} \begin{CD} 0 @>>> \Hom(N,E[\ell]) @>{Inf^{\mathcal{H}}_{N}}>> \Hom(\mathcal{H},E[\ell])\\ @. @VV{Res^{N}_{N_v}}V @VV{Res^{\mathcal{H}}_{\mathcal{H}_v}}V \\ 0 @>>> \Homol^1(N_v,E(L^\prime_{v^\prime})) @>{Inf^{\mathcal{H}_v}_{N_v}}>> \Homol^1(\mathcal{H}_v,E(\overline{\mathbb{Q}}_p)). \end{CD} \end{equation} Therefore, given $\Psi^\prime \in \Hom(N,E[\ell])$, $Res^{N}_{N_v}(\Psi^\prime)$ splits in $E(L^\prime_{v^\prime})$ if and only if $Res^{\mathcal{H}}_{\mathcal{H}_v} \circ Inf^{\mathcal{H}}_{N} (\Psi^\prime)$ does in $E(\overline{\mathbb{Q}}_p)$. Henceforth, condition (*) -- applied to $\Psi=Inf^{\mathcal{H}}_{N} (\Psi^\prime)$ -- is equivalent to: \begin{equation} \label{eq:eq166final8} (**) \qquad Res^{N}_{N_v}(\Psi^\prime) \mapsto 0 \in \Homol^1(N_v,E(L^\prime_{v^\prime})). \end{equation} In particular, if $N_v=0$, condition (**) holds trivially. Note also that $\Psi^\prime$ is not identically $0$ on any non-trivial subgroup of $N$.
\begin{prop} \label{thm:Proposition7} Let $\ell>3$ be a prime number.
a) In cases B and C, condition (**) holds at a place $v$ of $L$ if only if
$v$ is unramified in $L^\prime_{v^\prime}$ and $Res^{N}_{N_v}(\Psi^\prime)$ maps to $0\in \Homol^1(N_v,\widetilde E_v(k_{v^\prime}))$, where $k_{v^\prime}$ denotes the residue field of $L^\prime_{v^\prime}$.
b) In Case D, $v$ is at most tamely ramified in $L^\prime_{v^\prime}$. \end{prop}
\begin{proof} Part a). Case B: $v \mid v_0 \mid p \not = \ell$ and $v_0 \not \in \Sigma_E$. Assume that condition (**) holds. Let $I_v$ be the inertia subgroup of $L^\prime_{v^\prime}/L_v$. Then, there is an exact sequence: \begin{equation} 0 \rightarrow E_1(L^\prime_{v^\prime}) \rightarrow E(L^\prime_{v^\prime}) \rightarrow \widetilde E_v (k_{v^\prime}) \rightarrow 0, \end{equation} where $k_{v^\prime}$ denotes the residue field of $L^\prime_{v^\prime}$. Condition (**) then implies that $Res^{N}_{N_v}(\Psi^\prime)$ splits in $\widetilde E_v (k_{v^\prime})$. Since $I_v$ acts trivially on $\widetilde E_v (k_{v^\prime})$ and $E[\ell] \approx \widetilde E_v [\ell] \subset \widetilde E_v (k_{v^\prime})$, this implies that $\Psi^\prime$ is trivial on $I_v$. Therefore, $I_v=0$, which means that $v$ is unramified in $L^\prime_{v^\prime}$. Furthermore, we obviously have that $Res^{N}_{N_v}(\Psi^\prime)$ maps to $0\in \Homol^1(N_v,\widetilde E_v (k_{v^\prime}))$.
For the converse, it is sufficient to prove that \begin{equation} \Homol^1(N_v, E_1(L^\prime_{v^\prime}))=0, \end{equation} whenever $L^\prime_{v^\prime}/L_v$ is unramified of degree a power of $\ell$, for then one obtains a commutative diagram: \begin{equation} \begin{CD} @. \Hom(N,E[\ell]) @>\approx>> \Hom(N,\widetilde E_v[\ell])\\ @. @VV{Res^{N}_{N_v}}V @VV{Res^{N}_{N_v}}V \\ 0 @>>> \Homol^1(N_v,E(L^\prime_{v^\prime})) @>{reduction}>> \Homol^1(N_v,\widetilde E_v(k_{v^\prime})). \end{CD} \end{equation} Now, $E_1(L^\prime_{v^\prime})\approx F_v(\mathcal{M}_{v^\prime})$, where $F_v$ is the formal group of $E$ over $\mathcal{O}_{v}$, the integer ring of $L_v$, and $\mathcal{M}_{v^\prime}$ is the maximal ideal of the integer ring $\mathcal{O}_{v^\prime}$ of $L^\prime_{v^\prime}$. Moreover, from \cite[Proposition 6.3, p. 200]{silverman2009}, one has a short exact sequence: \begin{equation} 0 \rightarrow \pi_{v}^r \mathcal{O}_{v^\prime} \rightarrow F_v(\mathcal{M}_{v^\prime}) \rightarrow M \rightarrow 0 \end{equation} for some positive integer $r$, where $\pi_v$ is a uniformizer of $L_v$, and $M$ is a finite $\mathbb{Z}_p$-representation of $N_v$ that is annihilated by a power of $p$ (recall that $v\mid p\not = \ell$). Since $N_v$ has order $\ell$ (recall that $N_v$ is a cyclic subgroup of $\mathbb{F}_\ell \oplus \mathbb{F}_\ell$), it follows that $\Homol^1(N_v,M)=0$. Thus, it is sufficient to show that: \begin{equation} \Homol^1(N_v, \mathcal{O}_{v^\prime})=0, \end{equation} whenever $L^\prime_{v^\prime}/L_v$ is unramified. This result is well-known and can be proved by using the facts that $\Homol^1(N_v,k_{v^\prime})=0$ and that $\mathcal{O}_{v^\prime}$ is complete.
Case C: $v \mid v_0 \mid p \not=\ell$ and $p\in \Sigma_E$, with additive reduction of $E$ at $v_0$ and $v_0(j(E))\geq 0$. From Lemma \ref{thm:Lemma1} (by assumption, $\ell>3$), it follows that $E$ has good reduction over $L$. Therefore, the argument in case B applies.
Part b). Case D: This is clear since $\ell$ is coprime with the characteristic of the residue field of $L_v$. \end{proof}
At this point, we make use of class field theory. One may consult reference \cite{lang1986} for the classical approach, close to Takagi-Artin's treatment. One may also consult \cite{artin1967,tate1967} for global class field theory, and \cite{serre1967,serre1979} for local class field theory, with a treatment based extensively on homological algebra, including the notion of invariant class. A treatment based on the notion of henselian valuation with respect to a degree map can be found in \cite{neukirch1986}. Reference \cite{fesenko2002} should be consulted for a development of local class field theory, including explicit reciprocity laws, that does not rely on homological algebra. In this work, we found convenient to refer to \cite{lang1986,neukirch1986,serre1979}.
We now recall the notion of conductor of a finite Abelian extension of local fields $F/L_v$ \cite[p. 44]{neukirch1986}. Let $\pi_{v}$ be a uniformizer of $L_v$ and denote the prime ideal $(\pi_{v})$ by $\mathfrak{p}_v$. Set $U_{v}^{(0)}=U_{v}$ the group of units of the ring of integers $\mathcal{O}_v$ of $L_v$, and $U_{v}^{(n)}=1+\mathfrak{p}_v^n$ for $n\geq 1$. The conductor of $F/L_v$ is equal to $\mathfrak{f}_{v}=\mathfrak{p}_v^n$, where $n$ is the smallest integer such that $U_{v}^{(n)}\subset N_{F/L_v}(F^*)$.
Next, assume that $\mu_\ell\subset L_v^*$. Let $(\;\; ,\,L_v^{ab}/L_v): L_v^* \rightarrow \Gal(L_v^{ab}/L_v)$ be the local reciprocity map, where $L_v^{ab}$ denotes the maximal Abelian extension over $L_v$ \cite[pp. 168--171]{serre1979}. Let $\chi_a$ be the Kummer character associated to an element $a$ of $L_v^*$; {\em i.e.}, $\chi_a(\sigma)=\sigma(a^{1/\ell})/a^{1/\ell}$. Then, Hilbert's local symbol is defined as $(a,b)_{v,\ell}=\chi_a((b,\,L_v^{ab}/L_v))\in \mu_{\ell}$ \cite[p. 205--206]{serre1979}. One has: $(a,b)_{v,\ell}=1$ if and only if $b$ is a norm from the extension $L_v(a^{1/\ell})/L_v$ \cite[Proposition 4, p. 206]{serre1979}. It follows that the conductor of the Kummer extension $L_v(a^{1/\ell})/L_v$ is given by the smallest integer $n$ such that $(a,b)_{v,\ell}=1$ for all $b\in U_{v}^{(n)}$.
\begin{prop} \label{thm:Proposition8} Let $\ell>3$. Assume that condition (**) holds at a place $v$ of $L$. Let $\mathfrak{f}_v$ be the conductor of $L^\prime_{v^\prime}/L_v$. Then,
a) Case A: $\mathfrak{f}_v\mid \mathfrak{p}_v^{1+e_v\ell/(\ell-1)}$, where $e_v$ is the absolute ramification index of $L_v$.
b) Cases B and C: $\mathfrak{f}_v=1$.
c) Case D: $\mathfrak{f}_v\mid \mathfrak{p}_v$. \end{prop}
\begin{proof} Part a). Case A: The prime number $\ell$ is equal to the characteristic of the residue field of $L_v$. Moreover, one has $\mu_\ell \subset L_v^*$. It is sufficient to consider the case where $F^\prime/L_v$ is a cyclic sub-extension of $L^\prime_{v^\prime}/L_v$ of degree $\ell$. Indeed, $L^\prime_{v^\prime}/L_v$ is either the trivial extension, a cyclic extension of degree $\ell$ or the compositum of two cyclic extensions $F^\prime$ and $F''$ of degree $\ell$. In the first case, there is nothing to prove. In the third case, the inclusions $1+\mathfrak{p}_v^m \subset N_{F^\prime/L_v}((F^\prime)^*)$ and $1+\mathfrak{p}_v^n \subset N_{F''/L_v}((F'')^*)$ imply that $1+\mathfrak{p}_v^{\max(m,n)} \subset N_{F^\prime/L_v}((F^\prime)^*) \cap N_{F''/L_v}((F'')^*)=N_{F^\prime F''/L_v}((F^\prime F'')^*)$. Thus, we consider the case where $F^\prime/L_v$ is a Kummer extension of degree $\ell$, say $F^\prime=L_v(x^{1/\ell})$, that might be wildly ramified. From \cite[p. 186]{lang1986}, one has the inclusion $U_{v}^{(1+e_v\ell/(\ell-1))}\subset U_{v}^{\ell}$. But, one obviously has $(x,b)_{v,\ell}=1$ for all $b\in U_{v}^{\ell}$. Thus, $\mathfrak{f}_v\mid \mathfrak{p}_v^{1+e_v\ell/(\ell-1)}$.
Part b). Cases B and C: From local class field theory, one has $\mathfrak{f}_{v}=1$ if and only if $L^\prime_{v^\prime}/L_v$ is unramified \cite[Proposition (3.4), p. 44]{neukirch1986}, which holds from Proposition \ref{thm:Proposition7} (since $\ell>3$).
Part c). Case D: The prime number $\ell$ is coprime with the characteristic $p$ of the residue field of $L_v$. Moreover, one has $\mu_\ell \subset L_v^*$. As above, it is sufficient to consider the case where $F^\prime/L_v$ is a cyclic sub-extension of $L^\prime_{v^\prime}/L_v$ of degree $\ell$. Then, $F^\prime/L_v$ is tamely ramified and it is sufficient to consider the case where it is totally tamely ramified. Thus, we consider the case where $F^\prime/L_v$ is a Kummer extension of the form $L_v(\pi_{v}^{1/\ell})/L_v$ for some uniformizer $\pi_{v}$ of $L_v$ \cite[Proposition 12, p. 52]{lang1986}. In the case where $\ell$ is coprime with $p$, Hilbert's local symbol can be computed explicitly as in \cite[pp. 210--211]{serre1979}. Namely, let $(a)=(\pi_{v})^{\alpha}$ and $(b)=(\pi_{v})^{\beta}$. Set $c=(-1)^{\alpha \beta} a^{\beta}/b^{\alpha}$. Then, $(a,b)_{v,\ell}=\overline{c}^{(q-1)/\ell}$, where $\overline{c}$ is the image of $c$ in the residue field of $L_v$ and $q$ is the cardinality of the residue field. In our case, $a=\pi_{v}$, so that $\alpha=1$. Now, let $b\in U_{v}$, so that $\beta=0$. Then, $c=1/b$, so that $(a,b)_{v,\ell}=(\overline{b})^{-(q-1)/\ell}$. It follows that $(\pi_{v},b)_{v,\ell}=1$ for all $b\in U_{v}^{(1)}$ ({\em i.e.}, the group of units that map to $1$ in the residue field). On the other hand, the extension $L_v(\pi_v^{1/\ell})/L_v$ is ramified, so that $\mathfrak{f}_{v}\not = 1$. Therefore, in the totally tamely ramified case, one concludes that $\mathfrak{f}_{v}=\mathfrak{p}_v$. \end{proof}
Let $\mathfrak{m}$ be the cycle \cite[p. 97]{neukirch1986} defined as: \begin{equation} \label{eq:eq172final8} \mathfrak{m} = \prod_{v\mid \ell} \mathfrak{p}_v^{1+e_{v} \ell/(\ell-1)} \prod_{v\mid v_0 \in \Sigma_{E,p.m.}} \mathfrak{p}_v, \end{equation} where $e_{v}$ is the absolute ramification index of $L_v$. One considers the subgroup $I^{\mathfrak{m}}_L$ of the idele group $I_L$ \cite[p. 98]{neukirch1986}: \begin{equation} I^{\mathfrak{m}}_L = \prod_{v\mid \ell} U_v^{(1+e_{v} \ell/(\ell-1))} \times \prod_{v\mid v_0 \in \Sigma_{E,p.m.}} U_v^{(1)} \times \prod_{v \not \in T} U_v, \end{equation} where $T=\{v: v\mid \ell\} \cup \{v: v\mid v_0 \in \Sigma_{E,p.m.}\}$. Here, $\{v\}$ includes the set $S_\infty$ of infinite places of $L$. Since $\mu_{\ell}\subset L^*$, assuming that $\ell\not=2$, it follows that the infinite places of $L$ are all complex. In that case, one sets $U_v=\mathbb{C}^*$. From global class field theory \cite[Chapter IV, \S 7]{neukirch1986}, there exists a unique finite Abelian extension $L^{\mathfrak{m}}/L$ such that:
\begin{equation} (\;\; ,\,L^{\mathfrak{m}}/L)\::\: C_L/ C^{\mathfrak{m}}_L \xrightarrow{\approx} \Gal(L^{\mathfrak{m}}/L), \end{equation} where $C_L = \left (L^* \cdot I_L \right )/L^*$ and $C^{\mathfrak{m}}_L = \left (L^* \cdot I^{\mathfrak{m}}_L \right )/L^*$.
In the next result, we let $I_L^S$ denote $\prod_{v\in S} L_v^* \times \prod_{v\not \in S} U_v$ \cite[p. 76]{neukirch1986}. Now, let $T$ be any finite set of prime ideals of an algebraic number field $L$. Then, there exists a finite set of primes $S$, disjoint from $T$, such that the classes of the elements of $S$ generate the ideal class group of $L$ \cite[pp. 124--125]{lang1986}. Next, let $S$ be any finite set of primes of an algebraic number field $L$, such that: 1) $S$ includes the set $S_\infty$ of infinite places of $L$; 2) the classes of the elements of $S \setminus S_\infty$ generate the ideal class group of $L$. Then, $L^* \cdot I_L^S = L^* \cdot I_L$ \cite[pp. 77--78]{neukirch1986}.
Thus, given a finite set $T$ of non-Archimedean places, there exists a finite set of places $S \supseteq S_\infty$ disjoint from $T$ such that $L^* \cdot I_L^S = L^* \cdot I_L$.
\begin{lem} \label{thm:Lemma7} Let $E$ be an elliptic curve over $K$ and $\ell>2$ be a prime number. Set $L=K(E[\ell])$. Let $T=\{v: v\mid \ell\} \cup \{v: v\mid v_0 \in \Sigma_{E,p.m.}\}$. Let $S\supseteq S_\infty$ be a finite set of places of $L$, disjoint from $T$, such that $L^* \cdot I_L^S = L^* \cdot I_L$. Take $S$ sufficiently large, so that $S$ is closed under Galois action of $\Gal(L/K)$. Let $\mathcal{N}$ be the subgroup of $I_L$ defined as: \begin{equation} \mathcal{N} = \prod_{v\in T} U_v^\ell \times \prod_{v\in S} U_v \cdot (L_v^*)^\ell \times \prod_{v\not \in S \cup T} U_v. \end{equation}
Then, $\overline{\mathcal{N}} = \left ( L^* \cdot \mathcal{N} \right )/L^*$ is the class group of the maximal sub-extension $\widetilde L/L$ of $L^{\mathfrak{m}}/L$ whose Galois group is annihilated by $\ell$. In particular, $\Gal(\widetilde L/L)$ is a direct product of cyclic groups of order $\ell$ such that: \begin{equation} \Hom(\Gal(\widetilde L/L),E[\ell]) = \Hom(\Gal(L^{\mathfrak{m}}/L),E[\ell]). \end{equation} Moreover, $\widetilde L/K$ is a Galois extension. \end{lem}
\begin{proof} Let $\widetilde L/L$ be the maximal sub-extension of $L^{\mathfrak{m}}/L$ with Galois group annihilated by $\ell$. Then, one has: \begin{equation} \Hom(\Gal(\widetilde L/L),E[\ell]) = \Hom(\Gal(L^{\mathfrak{m}}/L),E[\ell]), \end{equation} and $\Gal(\widetilde L/L)$ is a direct product of cyclic groups of order $\ell$. We show that $\overline{\mathcal{N}}$ is its class group; {\em i.e.}, the class field $L_{\overline{\mathcal{N}}}$ is equal to $\widetilde L$.
First, we observe that $C_L/\overline{\mathcal{N}}\approx (L^* \cdot I_L )/(L^* \cdot \mathcal{N})$ is annihilated by $\ell$, since $L^* \cdot I_L = L^* \cdot I_L^S$ and $\left ( I_L^S \right)^{\ell} \subseteq \mathcal{N}$. Thus, $L_{\overline{\mathcal{N}}} \subseteq \widetilde L$.
Conversely, the Galois group $\Gal(\widetilde L/L)$ is the maximal quotient group of $\Gal(L^{\mathfrak{m}}/L) \approx C_L^{\mathfrak{m}} \approx (L^* \cdot I_L^S)/(L^* \cdot I_L^{\mathfrak{m}})$ that is annihilated by $\ell$. But $\prod_{v\in T} U_v^{\ell} \times \prod_{v \in S} \left ( L_v^*\right)^{\ell} \subset \left (I_L^S \right)^{\ell}$, and $\prod_{v \in S} U_v \times \prod_{v \not \in S \cup T} U_v \subset I_L^{\mathfrak{m}}$. Since both $\left (I_L^S \right)^{\ell}$ and $I_L^{\mathfrak{m}}$ map to $0$ under the natural projection $I_L \rightarrow C_L^{\mathfrak{m}}/\left ( C_L^{\mathfrak{m}} \right)^{\ell}$, $\overline{\mathcal{N}}$ is contained in the class group of $\widetilde L/L$, which means that $\widetilde L \subseteq L_{\overline{\mathcal{N}}}$.
Since $S$ is taken closed under Galois action of $\Gal(L/K)$, the same holds true for the class group $\overline{\mathcal{N}}$. It follows that $L_{\overline{\mathcal{N}}}$ is closed under any element of $\mathcal{G}=\Gal(\overline{K}/K)$. \end{proof}
Having assumed that $\ell>2$, the group $U_v \cdot (L_v^*)^\ell$ is actually equal to $U_v=L_v^*=\mathbb{C}^*$, for any $v\in S_\infty$.
Combining Propositions \ref{thm:Proposition7} and \ref{thm:Proposition8}, and Lemma \ref{thm:Lemma7}, we have reached the following result.
\begin{prop} \label{thm:Proposition9} Let $\ell>3$ be a prime number. Let $\overline{\mathcal{N}}$ be the class group defined in Lemma \ref{thm:Lemma7}. Let $\widetilde L = L_{\overline{\mathcal{N}}}$ be the corresponding class field. Set $\widetilde H=\Gal(\widetilde L/L)$. Then, one has: \begin{eqnarray} S^{[\ell]}(E/L) = Inf^{\mathcal{H}}_{\widetilde H} \Ker \Bigl \{ \Hom(\widetilde H,E[\ell]) \rightarrow \oplus_{w} \Homol^{1}(\widetilde H_w,E(\widetilde L_w)) \Bigr \}, \end{eqnarray} where $w$ covers all places of $\widetilde L$. \end{prop}
\subsection{Returning to the group $S^{[\ell]}(E/K)$} \label{subsection:Returning}
In this section, $E$ is an elliptic curve over an algebraic number field $K$.
Given a prime number $\ell>3$, we set $L=K(E[\ell])$. Let $\overline{\mathcal{N}}$ be the class group defined in Lemma \ref{thm:Lemma7}, and let $\widetilde L = L_{\overline{\mathcal{N}}}$ be the corresponding class field. We set $\widetilde G=\Gal(\widetilde L/K)$ and $\widetilde H=\Gal(\widetilde L/L)$.
Combining Propositions \ref{thm:Proposition6} and \ref{thm:Proposition9}, we have obtained the following result.
\begin{cor} \label{thm:Corollary9} Let $E$ be an elliptic curve over $K$. Let $\ell>3$ be a prime number. Then, one has: \begin{eqnarray} S^{[\ell]}(E/K) = Inf^{\mathcal{G}}_{\widetilde G} \Ker \Bigl \{ \Homol^1(\widetilde G,E[\ell]) \rightarrow \oplus_{w} \Homol^{1}(\widetilde G_{w},E(\widetilde L_w)) \Bigr \}, \end{eqnarray} where $w$ covers the places of $\widetilde L$ and $\widetilde G_{w}$ is the decomposition group of $w$ in $\widetilde L/K$. \end{cor}
\begin{proof} Let $w\mid v \mid v_0 \mid p$ be places of $\widetilde L$, $L$, $K$, and $\mathbb{Q}_p$, respectively.
We show the inclusion: \begin{equation} S^{[\ell]}(E/K) \subseteq Inf^{\mathcal{G}}_{\widetilde G} \Ker \Bigl \{ \Homol^1(\widetilde G,E[\ell]) \rightarrow \oplus_{w} \Homol^{1}(\widetilde G_{w},E(\widetilde L_w)) \Bigr \}. \end{equation} Let $f\in S^{[\ell]}(E/K)$. Then, $Res^{\mathcal{G}}_{\mathcal{H}}(f)=Inf^{\mathcal{H}}_{\widetilde H}(g)$, for some group homomorphism $g \in \Hom(\widetilde H,E[\ell])$ satisfying the property of Proposition \ref{thm:Proposition9}. Then, for $\sigma_1 \in \mathcal{G}$ and $\sigma_2 \in \Gal(\overline{\mathbb{Q}}/\widetilde L)$, one computes: \begin{equation} f(\sigma_1 \sigma_2)=f(\sigma_1)+\,^{\sigma_1} f(\sigma_2)=f(\sigma_1)+\,^{\sigma_1} g(\sigma_2) =f(\sigma_1). \end{equation} Thus, $f = Inf^{\mathcal{G}}_{\widetilde G}(\widetilde f)$, for some $\widetilde f \in \Homol^1(\widetilde G,E[\ell])$, and satisfies the stated property, as follows from the following diagram with exact bottom row: \begin{equation} \begin{CD} @. \Homol^1(\widetilde G,E[\ell]) @>{Inf^{\mathcal{G}}_{\widetilde G}}>> \Homol^1(\mathcal{G},E[\ell])\\ @. @VV{Res^{\widetilde G}_{\widetilde G_{w}}}V @VV{Res^{\mathcal{G}}_{\mathcal{G}_{v_0}}}V \\ 0 @>>> \Homol^1(\widetilde G_{w},E(\widetilde L_w))[\ell] @>{Inf^{\mathcal{G}_{v_0}}_{\widetilde G_{w}}}>> \Homol^1(\mathcal{G}_{v_0},E(\overline{\mathbb{Q}}_p))[\ell], \end{CD} \end{equation} where $\mathcal{G}_{v_0}$ denotes the absolute Galois group of $K_{v_0}$.
The other inclusion is clear. \end{proof}
\subsection{The group $S^{[\ell]}(E/\mathbb{Q})$} \label{subsection:caseQ}
We now specialize to the case where $E$ is an elliptic curve over $\mathbb{Q}$, of Weierstrass equation of the form $y^2=x^3+Ax+B$, where $A,B\in \mathbb{Z}$. Let $\Delta^\prime:=4A^3+27B^2$.
We consider a prime number $\ell>3$, and set $L=\mathbb{Q}(E[\ell])$. We let $\widetilde L$ denote the class field corresponding to the subgroup $\overline{\mathcal{N}}$ of $C_{L}$ as in Lemma \ref{thm:Lemma7}, when taking the base field $K=\mathbb{Q}$. We set $\widetilde H=\Gal(\widetilde L/L)$.
Next, we consider the field $K^\prime$ defined in (\ref{eq:eq149final8}), and we denote: \begin{equation} \label{eq:eq183final8} L^\prime: = L K^\prime. \end{equation} In addition to the field $\widetilde L$, we also consider $\widetilde L^\prime$ the class field corresponding to the subgroup $\overline{\mathcal{N}}$ of $C_{L^\prime}$ as in Lemma \ref{thm:Lemma7}, when taking the base field $K=K^\prime$.
The motivation for Theorem \ref{thm:Theorem6} and Proposition \ref{thm:Proposition3} was to reach the following result, which is useful for passing from $p$-adic rational points to algebraic ones.
\begin{prop} \label{thm:Proposition10} Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM, with Weierstrass equation of the form $y^2=x^3+Ax+B$, where $A,B\in \mathbb{Z}$. Set $\Delta^\prime=4A^3+27B^2$.
Let $\ell > 3$ be a prime number such that $\rho_\ell(\mathcal{G}) = {\bf GL}_2(\mathbb{Z}_\ell)$. Let $p$ be any prime number such that $p\nmid \Delta^\prime \ell (\ell-1) (\ell+1)$. Let $\xi$ be a fixed embedding of $\overline{\mathbb{Q}}$ into $\overline{\mathbb{Q}}_p$, inducing an embedding of Galois groups $\mathcal{G}_{p}=\Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p) \hookrightarrow \mathcal{G}=\Gal(\overline{\mathbb{Q}}/\mathbb{Q})$.
Then, one can write any point $P_0 \in E(\mathbb{Q}_{p})$ in the form: \begin{equation} P_0 = \xi(P) + [\ell] Q^\prime, \end{equation} for some point $P\in E(K^\prime)$, where $K^\prime$ is the field defined in (\ref{eq:eq149final8}), and $Q^\prime\in E(\mathbb{Q}_{p})$.
More precisely, let $\mathfrak{p}$ be the prime ideal of $K^\prime$ lying above $p$ such that $\xi$ induces $K^\prime_{\mathfrak{p}} \hookrightarrow \overline{\mathbb{Q}}_p$. Let $F$ be the subfield of $K^\prime$ fixed by the decomposition group $\Gal(K^\prime/\mathbb{Q})_{\mathfrak{p}}$. Then, in fact, one may take $P\in E(F)$. \end{prop}
\begin{proof} Since $\ord_p(\Delta)=0$, the reduced curve $\widetilde E$ of $E$ over the residue field $\mathbb{F}_{p}$ is non-singular. Let $\# \widetilde E(\mathbb{F}_{p}) = \ell^n m$, where $n\geq 0$ and $(m,\ell)=1$.
Let $P_0\in E(\mathbb{Q}_{p})$. Then, $[m]P_0$ projects to a point $\widetilde P$ of $\widetilde E[\ell^n]$. We first consider the non-trivial case where $n>0$. From Theorem \ref{thm:Theorem6}, there is a lifting $P^\prime$ of $\widetilde P$ with affine coordinates in the field $K^\prime$ that projects to $\widetilde P$. If $n=0$, then one may take $P^\prime=O$, so that $P^\prime\in E(\mathbb{Q})\subset K^\prime$ and $P^\prime$ is trivially a lifting of $[m]P_0 = O$.
Then, $[m]P_0 - \xi(P^\prime)\in E_1( K^\prime_{\mathfrak{p}})$, where $\mathfrak{p}$ is the maximal ideal of $K^\prime$ lying above $p$ that is compatible with the embedding $\xi$. Now, one has a commutative diagram: \begin{equation} \begin{CD} E(K^\prime) @>{\xi}>> E(K^\prime_{\mathfrak{p}})\\ @VV{\tr_{K^\prime/F}}V @VV{\tr_{K^\prime_{\mathfrak{p}}/\mathbb{Q}_{p}}}V\\ E(F) @>{\xi}>> E(\mathbb{Q}_{p}),\\ \end{CD} \end{equation} where $F$ is the fixed field of $K^\prime$ by the decomposition group $\Gal(K^\prime/\mathbb{Q})_{\mathfrak{p}}$. Here, we have used the fact that $F_{\mathfrak{p}^\prime}=\mathbb{Q}_p$, where $\mathfrak{p}^\prime$ is the prime ideal of $F$ lying below $\mathfrak{p}$, as well as \cite[Exerc. 1.12 b), p.16]{silverman2009}. This yields $[m]\tr_{K^\prime_{\mathfrak{p}}/\mathbb{Q}_{p}}(P_0) - \xi(\tr_{K^\prime/F}(P^\prime))\in E_1(\mathbb{Q}_{p})$. We set $P'':=\tr_{K^\prime/F}(P^\prime) \in E(F)$. Moreover, one has $\tr_{K^\prime_{\mathfrak{p}}/\mathbb{Q}_{p}}(P_0)=[m^\prime]P_0$, where $m^\prime = \vert \Gal(K^\prime/\mathbb{Q})_{\mathfrak{p}} \vert$ is coprime with $\ell$, based on Proposition \ref{thm:Proposition3}. We set $m'':=m \cdot m^\prime$.
Now, $E_1(\mathbb{Q}_{p})\approx F_{p}(\mathcal{M}_{p})$, where $F_{p}$ is the formal group of $E$ over $\mathbb{Q}_{p}$. From \cite{kolyvagin1980}, $F_{p}$ is necessarily a formal $\mathbb{Z}_p$-module. Since $\ell\in \mathbb{Z}_p^{*}$ by assumption, it follows that $[m'']P_0 - \xi(P'') = [\ell] Q'' \in [\ell](E_1(\mathbb{Q}_{p}))$, for some point $Q'' \in E_1(\mathbb{Q}_{p})$. Writing $1=m'' a+\ell b$, with $a,b\in\mathbb{Z}$, one deduces that \begin{eqnarray} P_0 &=& [a]([m'']P_0) + [\ell]([b]P_0) = [a](\xi(P'') + [\ell]Q'') + [\ell]([b]P_0)\nonumber\\ &=& \xi([a]P'') + [\ell] ([a]Q'' + [b]P_0), \end{eqnarray} where $P:=[a]P'' \in E(F)$ and $Q^\prime:=[a]Q'' + [b]P_0 \in E(\mathbb{Q}_{p})$. \end{proof}
Proposition \ref{thm:Proposition10} allows proving the following result.
\begin{prop} \label{thm:Proposition11} Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM, with Weierstrass equation $y^2 = x^3 + A x + B$, where $A,B\in \mathbb{Z}$. Set $\Delta^\prime=4A^3+27B^2$.
Let $\ell>3$ be a prime number such that $\rho_\ell(\mathcal{G}) = {\bf GL}_2(\mathbb{Z}_\ell)$. Let $p$ be a prime number such that $p\nmid \Delta^\prime \ell(\ell-1)(\ell+1)$. Consider $\widetilde L$, $K^\prime$, and $\widetilde L^\prime$ as above. Let $w^\prime$ be a place of $\widetilde L^\prime$ that lies above $p$, and consider the place $w$ of $\widetilde L$ that lies below $w^\prime$. Assume that the decomposition group $\widetilde H^\prime_{w^\prime}$, where $\widetilde H^\prime=\Gal(\widetilde L^\prime/L^\prime)$, maps onto the decomposition group $\widetilde H_w$, where $\widetilde H=\Gal(\widetilde L/L)$, under the natural projection $Res_{\widetilde L}^{\widetilde L^\prime}: \widetilde H^\prime \rightarrow \widetilde H$ (defined by restriction of automorphisms to $\widetilde L$).
Consider $\widetilde f$ in $\Ker \Bigl \{ \Homol^1(\widetilde G,E[\ell]) \xrightarrow{Res^{\widetilde G}_{\widetilde G_w}}
\Homol^{1}(\widetilde G_w,E(\widetilde L_w)) \Bigr \}$.
Then, there exists an element $\widetilde Q \in E(\overline{\mathbb{Q}})$ such that $[\ell]\widetilde Q \in E(K^\prime)$, and $\widetilde f(Res_{\widetilde L}^{\widetilde L^\prime}(\sigma)) = [\sigma-1] \widetilde Q$, for all $\sigma \in \widetilde H^\prime_{w^\prime}$. \end{prop}
\begin{proof} Firstly, consider a place $w^\prime$ of $\widetilde L^\prime$, as in the statement of the proposition. Let us fix an embedding $\xi:\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_p$ that yields the embedding $\bigl ( \widetilde L^\prime \bigr )_{w^\prime} \hookrightarrow \overline{\mathbb{Q}}_p$.
Let $\widetilde f:\widetilde G \rightarrow E[\ell]$ be a $1$-cocycle such that $\widetilde f(\sigma)=[\sigma-1] Q_0$ for all $\sigma\in \widetilde G_w$, for some $Q_0\in E(\widetilde L_w)$. Then, one has $P_0:=[\ell]Q_0 \in E(\widetilde L_w^{\widetilde G_w})=E(\mathbb{Q}_{p})$. Now, Proposition \ref{thm:Proposition10} applies to $P_0$, having assumed that $\ell > 3$, and $p\nmid \Delta^\prime \ell(\ell-1)(\ell+1)$. Thus, one may write $P_0$ in the form $\xi(P) + [\ell]Q^\prime$, where $P \in E(K^\prime)$, and $Q^\prime \in E(\mathbb{Q}_{p})$, based on Proposition \ref{thm:Proposition10}.
Let $Q \in E(\overline{\mathbb{Q}})$ such that $[\ell]Q=P$. The extension $L^\prime(Q)/L^\prime$ is an Abelian extension with Galois group embedded into $E[\ell]$, since $E[\ell] \subset L$. Indeed, the function $\Gal(L^\prime(Q)/L^\prime) \rightarrow E[\ell]$ that maps $\sigma \in \Gal(L^\prime (Q)/L^\prime)$ to $[\sigma -1]Q$ is a group homomorphism (it is a ``Kummer character'' of the elliptic curve). Moreover, it is injective. Furthermore, one computes in $E(\overline{\mathbb{Q}}_p)$: \begin{equation} [\ell](Q_0-\xi(Q) - Q^\prime)= P_0 - \xi(P) - [\ell] Q^\prime = O. \end{equation} Thus, one has $Q_0 = \xi(Q) + Q^\prime + \xi(Q'')$, with $Q'' \in E[\ell] \subset E(L)$. We set \begin{equation} \widetilde Q = Q + Q''. \end{equation}
Let us define $\widetilde g(\sigma):=[\sigma-1]\widetilde Q$, for $\sigma\in \mathcal{G}$. Let $w$ be the place of $\widetilde L$ lying below $w^\prime$, and consider the case where $\widetilde H_{w}=Res_{\widetilde L}^{\widetilde L^\prime}(\widetilde H^\prime_{w^\prime})$, as in the statement of the proposition. Let $\sigma \in \widetilde H^\prime_{w^\prime}$. One then computes: \begin{eqnarray} \xi \left ( \widetilde f(Res_{\widetilde L}^{\widetilde L^\prime}(\sigma)) \right ) &=& [\sigma-1] Q_0 = [\sigma-1](Q_0 - Q^\prime)\nonumber\\ &=& [\sigma -1](\xi(Q + Q'')) = [\sigma-1]\xi(\widetilde Q) = \xi ( [\sigma-1]\widetilde Q )\nonumber\\ &=& \xi \left ( \widetilde g(\sigma) \right ), \end{eqnarray} since $Q^\prime \in E(\mathbb{Q}_{p})$, and the embedding $\xi$ was chosen to be compatible with localization of $\widetilde L^\prime$ at $w^\prime$. Thus, one obtains $\widetilde f(Res_{\widetilde L}^{\widetilde L^\prime}(\sigma)) = g(\sigma)$ for any $\sigma \in \widetilde H^\prime_{w^\prime}$. \end{proof}
The hypotheses of Proposition \ref{thm:Proposition11} comprise a condition on the characteristic $p$ of the finite field intervening in the reduced curve, as well as condition $Res_{\widetilde L}^{\widetilde L^\prime}(\widetilde H_{w^\prime}^\prime)=\widetilde H_{w}$. Using Chebotarev's Density Theorem, one can show that these conditions can be met, in a form relevant to Proposition \ref{thm:Proposition12}. Namely, we have the following result.
\begin{lem} \label{thm:Lemma8} Let $E$ be an elliptic curve over $\mathbb{Q}$, of Weierstrass equation $y^2=x^3+Ax+B$, where $A,B\in \mathbb{Z}$. Let $\ell \not = 2,3,5,7,13$ be a prime number such that $\ell\nmid \Delta^\prime:=4A^3+27B^2$. Set $L=\mathbb{Q}(E[\ell])$, and let $d_0$ be a prime factor of $(\ell-1)/\gcd(\ell-1,12)$.
Let $K^\prime$ and $L^\prime$ be the fields defined in (\ref{eq:eq149final8}) and (\ref{eq:eq183final8}), respectively. Let $H_1$ be a subspace of $\widetilde H^\prime=\Gal(\widetilde L^\prime/L^\prime)$, of dimension $n_1$, that is a normal subgroup of $\Gal(\widetilde L^\prime/\left ( L^\prime \right )^{\sigma^\prime_0})$, for some element $\sigma^\prime_0$ of order $d_0$ in $\Gal(L^\prime/K^\prime)$.
Then, $H_1$ admits a decomposition $\oplus_{\nu=1}^{n_1} C_\nu$, where each $C_\nu$ is a cyclic group (of order $\ell$) that is closed under conjugation by elements of $\Gal(\widetilde L^\prime/\left ( L^\prime \right )^{\sigma^\prime_0})$. In particular, $\langle \sigma^\prime_0 \rangle$ acts on each group $C_\nu$ through some character $\chi_\nu$.
Then, in the case where $\chi_\nu$ is the trivial character, the group $C_\nu$ is equal to the decomposition group $\widetilde H^\prime_{w^\prime}$ of some place $w^\prime$ of $\widetilde L^\prime$ ({\em depending on} $C_\nu$) that lies above a prime $p$ satisfying the condition $p\nmid \Delta^\prime \ell (\ell-1)(\ell+1)$.
Moreover, assume that $Res_{\widetilde L}^{\widetilde L^\prime}(C_\nu)\not = 0$, where $Res_{\widetilde L}^{\widetilde L^\prime}$ denotes the natural projection $\widetilde H^\prime \rightarrow \widetilde H$, with $\widetilde H=\Gal(\widetilde L/L)$. Then, one has $Res_{\widetilde L}^{\widetilde L^\prime}(C_\nu)=\widetilde H_w$, where $w$ is the place of $\widetilde L$ lying below $w^\prime$. \end{lem}
\begin{proof} {\em Step 1.} We consider the fields $K^\prime$ and $L^\prime$ as in (\ref{eq:eq149final8}) and (\ref{eq:eq183final8}), respectively. We view the Galois group $\Gal(L^\prime/K^\prime)$ as the subgroup $\widetilde \rho_\ell(\Gal(L/(L\cap K^\prime)))$ of $\widetilde \rho_\ell(\Gal(L/\mathbb{Q})) < {\bf GL}_2(\mathbb{F}_\ell)$, under the Galois embedding $\widetilde \rho_\ell$.
We denote $Res_{L^\prime}^{\widetilde L^\prime}$ the projection of Galois groups $\Gal(\widetilde L^\prime/K^\prime) \rightarrow \Gal(L^\prime/K^\prime)$.
{\em Step 2.} Let $\sigma_0^\prime$ be an element of $\Gal(L^\prime/K^\prime)$ of order $d_0$. Thus, one has: i) $\ord(\sigma_0^\prime)=d_0$ is coprime with $\ell$.
Let then $\sigma_0''$ be any lifting of $\sigma_0^\prime$ in $\Gal(\widetilde L^\prime/K^\prime)$. Set $\widetilde \sigma_0^\prime:=(\sigma_0'')^{\ell}$. Then, $(\widetilde \sigma_0^\prime)^{d_0}=((\sigma_0'')^{d_0})^{\ell}=1$, since $(\sigma_0'')^{d_0} \in \widetilde H^\prime$, as $(\sigma_0^\prime)^{d_0}=1$, and $\widetilde H^\prime$ is a vector space over $\mathbb{F}_\ell$. Therefore, one has $d'':= \ord(\widetilde \sigma_0^\prime)\mid d_0$.
Moreover, the projection $Res_{L^\prime}^{\widetilde L^\prime}(\widetilde \sigma_0^\prime)\in \Gal(L^\prime/K^\prime)$ is equal to $(\sigma_0^\prime)^{\ell}$. Thus, one has $(\sigma_0^\prime)^{d''\ell}=1$, which combined with $(\sigma_0^\prime)^{d_0}=1$, yields $d_0 = \ord(\sigma_0^\prime) \mid \gcd(d''\ell,d_0)=d''$ (since $d''\mid d_0$ is coprime with $\ell$). Thus, one has ii) $\ord(\widetilde \sigma_0^\prime)=d_0$.
{\em Step 3.} Assume that $h^\prime$ is a non-trivial element of $\widetilde H^\prime$ that is in the centralizer of $\widetilde \sigma_0^\prime$. Consider the cyclic group $\langle h^\prime \widetilde \sigma_0^\prime \rangle$, where $\widetilde \sigma_0^\prime$ is as in step 2. Then, this cyclic group has order $d_0 \ell$, since $d_0$ is coprime with $\ell$. It follows that iii) $(h^\prime \widetilde \sigma_0^\prime)^{d_0}=(h^\prime)^{d_0}$ is a generator of the cyclic group $\langle h^\prime \rangle$.
{\em Step 4.} Assume now that $p$ is a prime that does not ramify in $\widetilde L^\prime$, and that $w^\prime \mid p$ is a place of $\widetilde L^\prime$ such that $\Frob_{\widetilde L^\prime/\mathbb{Q}}(w^\prime)=h^\prime \widetilde \sigma_0^\prime$, with $h^\prime$ as in step 3, and $\widetilde \sigma_0^\prime$ as in step 2.
Then, except for finitely many such primes, one may assume that $p\nmid \Delta^\prime \ell (\ell-1)(\ell+1)$.
{\em Step 5.} Since the element $Res_{L^\prime}^{\widetilde L^\prime}(\widetilde \sigma_0^\prime)$ considered in step 2 has order $d_0$, it follows that
a prime $p$ as in step 4 would have residue degree $f_{L^\prime/\mathbb{Q}}$ equal to $d_0$. Then, this means that any place $v^\prime$ of $L^\prime$ lying above $p$ would have Frobenius element equal to $\Frob_{\widetilde L^\prime/L^\prime}(w^\prime)=\Frob_{\widetilde L^\prime/\mathbb{Q}}(w^\prime)^{f_{L^\prime/\mathbb{Q}}} =(h^\prime)^{d_0}$, from property iii) of step 3. Therefore, $\Frob_{\widetilde L^\prime/L^\prime}(w^\prime)$ generates the cyclic group $\langle h^\prime \rangle$.
Therefore, all desired properties for $p$ would be met, provided $\sigma_0^\prime$ satisfies condition i) -- step 2 --, and $h^\prime$ is in the centralizer of $\widetilde \sigma_0^\prime$ -- step 3.
{\em Step 6.} Now, the cyclic group $\langle \sigma^\prime_0 \rangle < \Gal(L^\prime/K^\prime)$ acts by Galois conjugation on the $\mathbb{F}_\ell$-vector space $H_1$, which yields a representation $\langle \sigma^\prime_0 \rangle \hookrightarrow \Aut( H_1 )$. Since $\# \langle \sigma^\prime_0 \rangle =d_0 \mid (\ell-1)$, the eigenvalues of this representation belong to $\mathbb{F}_\ell^*$. Therefore, one obtains a decomposition of representations of the finite group $\langle \sigma^\prime_0 \rangle$ over $\mathbb{F}_\ell$: \begin{equation} H_1 = \oplus_{\nu=1}^{n_1} C_\nu, \end{equation} where $n_1:=\dim_{\mathbb{F}_\ell}( H_1 )$, and $C_\nu \approx \mathbb{F}_\ell(\chi_\nu)$, for some character $\chi_\nu$ of the group $\langle \sigma^\prime_0 \rangle$.
If ever $\chi_\nu$ is the trivial character, then the element $\sigma_0^\prime$ satisfies condition i), {\em and} $C_\nu$ is in the centralizer of $\widetilde \sigma_0^\prime$.
An application of Chebotarev's Density Theorem \cite{chebotarev1926} to $\widetilde L^\prime/\mathbb{Q}$ and the conjugacy class of $h^\prime \widetilde \sigma_0^\prime$, then yields $\Frob_{\widetilde L^\prime/\mathbb{Q}}(w^\prime_0) = \tau h^\prime \widetilde \sigma_0^\prime \tau^{-1}$, for some place $w^\prime_0$ of $\widetilde L^\prime$ and element $\tau$ of the Galois group $\Gal(\widetilde L^\prime/\mathbb{Q})$. But then, $\Frob_{\widetilde L^\prime/\mathbb{Q}}(\tau^{-1 }w^\prime_0)=h^\prime \widetilde \sigma_0^\prime$, so that one may take $w^\prime=\tau^{-1 }w^\prime_0$. Then, the prime $p$ lying below $w^\prime$ satisfies all the desired properties: $p$ is unramified in $\widetilde L^\prime$; $p\nmid \Delta^\prime \ell(\ell-1)(\ell+1)$; and $\Frob_{\widetilde L^\prime/L^\prime}(w^\prime)$ generates $C_\nu$.
{\em Step 7.} One has $\Frob_{\widetilde L/\mathbb{Q}}(w) = Res^{\widetilde L^\prime}_{\widetilde L} ( \Frob_{\widetilde L^\prime/\mathbb{Q}}(w^\prime) )$, where $w$ is the place of $\widetilde L$ lying below $w^\prime$. Let $v$ be the place of $L$ lying below $w$, and $v»^\prime$ be the place of $L^\prime$ lying below $w^\prime$. One computes: \begin{eqnarray} \Frob_{L/\mathbb{Q}}(v) &=& Res^{\widetilde L}_{L} ( \Frob_{\widetilde L/\mathbb{Q}}(w) )\nonumber\\
&=& Res^{\widetilde L}_{L} \circ Res^{\widetilde L^\prime}_{\widetilde L} ( \Frob_{\widetilde L^\prime/\mathbb{Q}}(w^\prime) )\nonumber\\ &=& Res^{L^\prime}_{L} \circ Res^{\widetilde L^\prime}_{L^\prime} ( \Frob_{\widetilde L^\prime/\mathbb{Q}}(w^\prime) )\nonumber\\ &=& Res^{L^\prime}_{L} ( \Frob_{L^\prime/\mathbb{Q}}(v^\prime) ), \end{eqnarray} which shows that $f_{L/\mathbb{Q}}=f_{L^\prime/\mathbb{Q}}$, as $\sigma_0=\Frob_{L/\mathbb{Q}}(v)$ has same order as $\sigma_0^\prime=\Frob_{L^\prime/\mathbb{Q}}(v^\prime)$ under the isomorphism $Res^{L^\prime}_{L}: \Gal(L^\prime/K^\prime) \xrightarrow{\approx} \Gal(L/(L \cap K^\prime))$. One then obtains: \begin{eqnarray} \Frob_{\widetilde L/L}(w) &=& \Frob_{\widetilde L/\mathbb{Q}}(w)^{f_{L/\mathbb{Q}}}\nonumber\\
&=& Res^{\widetilde L^\prime}_{\widetilde L} ( \Frob_{\widetilde L^\prime/\mathbb{Q}}(w^\prime) )^{f_{L/\mathbb{Q}}}\nonumber\\ &=& Res^{\widetilde L^\prime}_{\widetilde L} ( \Frob_{\widetilde L^\prime/\mathbb{Q}}(w^\prime)^{f_{L^\prime/\mathbb{Q}}} )\nonumber\\ &=& Res^{\widetilde L^\prime}_{\widetilde L} ( \Frob_{\widetilde L^\prime/L^\prime}(w^\prime) ), \end{eqnarray} since $f_{L/\mathbb{Q}}=f_{L^\prime/\mathbb{Q}}$. It then follows that $\Frob_{\widetilde L/L}(w)$ generates $Res_{\widetilde L}^{\widetilde L^\prime}(C_v)=\langle Res^{\widetilde L^\prime}_{\widetilde L} ( \Frob_{\widetilde L^\prime/L^\prime}(w^\prime) ) \rangle$. One then concludes that $Res_{\widetilde L}^{\widetilde L^\prime}(C_\nu)=\widetilde H_w$ since $p$ is unramified in $\widetilde L$, and $\widetilde H$ is an Abelian group of exponent $\ell$, which implies that any of its non-trivial cyclic subgroups has order $\ell$. \end{proof}
Combining Lemma \ref{thm:Lemma8} and Proposition \ref{thm:Proposition11}, we have reached the following result.
\begin{cor} \label{thm:Corollary10} Let $E$ be an elliptic curve over $\mathbb{Q}$, of Weierstrass equation $y^2=x^3+Ax+B$ without CM, where $A,B\in \mathbb{Z}$. Let $\ell \not = 2,3,5,7,13$ be a prime number such that i) $\rho_\ell(\mathcal{G}) = {\bf GL}_2(\mathbb{Z}_\ell)$; and ii) $\ell\nmid \Delta^\prime:=4A^3+27B^2$. Set $L=\mathbb{Q}(E[\ell])$, and let $d_0$ be a prime factor of $(\ell-1)/\gcd(\ell-1,12)$.
Let $K^\prime$ and $L^\prime$ be the fields defined in (\ref{eq:eq149final8}) and (\ref{eq:eq183final8}), respectively (depending on $A$, $B$ and $\ell$). Let $H_1$ be a subspace of $\widetilde H^\prime=\Gal(\widetilde L^\prime/L^\prime)$, of dimension $n_1$, that is a normal subgroup of $\Gal(\widetilde L^\prime/\left ( L^\prime \right )^{\sigma^\prime_0})$, for some element $\sigma^\prime_0$ of order $d_0$ in $\Gal(L^\prime/K^\prime)$.
Let $H_1 = \oplus_{\nu=1}^{n_1} C_\nu$ be the decomposition of representations of $\langle \sigma^\prime_0 \rangle$ over $\mathbb{F}_\ell$, as in Lemma \ref{thm:Lemma8}.
Let $f=Inf^{\mathcal{G}}_{\widetilde G}(\widetilde f)$ be a $1$-cocycle in the Selmer group $S^{[\ell]}(E/\mathbb{Q})$, where $\widetilde G=\Gal(\widetilde L/\mathbb{Q})$ and $\widetilde f\in \Homol^1(\widetilde G,E[\ell])$.
Then, in the case where $\chi_\nu$ is the trivial character and $Res_{\widetilde L}^{\widetilde L^\prime}(C_\nu)\not = 0$, one has $\widetilde f(Res_{\widetilde L}^{\widetilde L^\prime}(\sigma)) = [\sigma -1]\widetilde Q$, for any $\sigma\in C_\nu$, for some $\widetilde Q \in E(\overline{\mathbb{Q}})$ such that $[\ell]\widetilde Q \in E(K^\prime)$. The point $\widetilde Q$ depends on $\widetilde f$ and $C_\nu$. \end{cor}
\begin{proof} Let $E$ have Weierstrass equation $y^2=x^3+Ax+B$, where $A,B\in \mathbb{Z}$. Set $\Delta^\prime=4A^3+27B^2$. Assume that $C_\nu$ has conjugacy action defined by the trivial character of $\langle \sigma^\prime_0 \rangle$. Then, using Lemma \ref{thm:Lemma8}, one can take a place $w^\prime$ of $\widetilde L^\prime$ such that $\widetilde H^\prime_{w^\prime} = C_\nu$, where $w^\prime$ lies above a prime $p$ such that $p\nmid \Delta^\prime \ell (\ell-1)(\ell+1)$. The condition $Res_{\widetilde L}^{\widetilde L^\prime}(C_\nu)\not = 0$ implies that $\widetilde H_{w} = Res_{\widetilde L}^{\widetilde L^\prime}(C_\nu)$, where $w$ is the place of $\widetilde L$ lying below $w^\prime$. The corollary now follows from Proposition \ref{thm:Proposition11}. \end{proof}
\subsection{Proof of Theorem \ref{thm:Theorem5}} \label{subsection:proofTheorem5}
The following lemma will be crucial in the proof of the important intermediate result, Proposition \ref{thm:Proposition12}, and the proof of Theorem \ref{thm:Theorem5}.
\begin{lem} \label{thm:Lemma9} Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM, of Weierstrass equation $y^2=x^3+Ax+B$, with $A,B\in \mathbb{Z}$. Given a prime number $\ell \not = 2,3,5,7,13$, set $L=\mathbb{Q}(E[\ell])$. Assume that: i) $\Gal(L/\mathbb{Q}) \approx {\bf GL}_2(\mathbb{F}_\ell)$; and ii) $\ell\nmid \Delta^\prime:=4A^3+27B^2$. Let $K^\prime$ and $K_2$ be the fields defined in (\ref{eq:eq149final8}) and (\ref{eq:eq146final8}), respectively (depending on $A,B$ and $\ell$). Then, one has:
a) $\mathbb{Q}(\mu_\ell)/\mathbb{Q}$ is the maximal Abelian subextension of $L/\mathbb{Q}$, and $\Gal(L/\mathbb{Q}(\mu_\ell))$ corresponds to ${\bf SL}_2(\mathbb{F}_\ell)$ under $\widetilde \rho_\ell$;
b) $L \cap K_2 = \mathbb{Q}$, and $(L \cap K_3)/\mathbb{Q}$ has degree $2$;
c) $L \cap K^\prime \subseteq \mathbb{Q}(\mu_\ell)$;
d) $L \cap K^\prime = \mathbb{Q}(\mu_\ell) \cap K^\prime$ has degree over $\mathbb{Q}$ dividing $12$;
e) let $d_0$ be a prime divisor of $(\ell-1)/\gcd(\ell-1,12)$; then, there exists an element $\sigma_0\in \Gal(L/(L \cap K^\prime))$, of order $d_0$, such that $\widetilde \rho_\ell(\sigma_0)=\begin{pmatrix} \lambda_0 & 0\\0 & 1\end{pmatrix}$, with $\lambda_0\in \mathbb{F}_\ell^*$ (of order $d_0$); in particular, $\sigma_0$ acts on $X_{(\chi)}:=\langle \begin{pmatrix} 1\\0\end{pmatrix} \rangle$ through the character defined by $\chi(\sigma_0)=\lambda_0$, and acts on $X_{(1)}:=\langle \begin{pmatrix} 0\\1\end{pmatrix} \rangle$ through the trivial character;
f) the element $\tau \in \Gal(L/\mathbb{Q})$ defined by $\widetilde \rho_\ell(\tau)=\begin{pmatrix} 1 & 0\\1 & 1\end{pmatrix}$ belongs to the subgroup $\Gal(L/(L \cap K^\prime))$; in particular, $X_{(\chi)}$ is not stable under the action of $\tau$;
g) the element $\beta \in \Gal(L/\mathbb{Q})$ defined by $\widetilde \rho_\ell(\beta)=\begin{pmatrix} \lambda & 0\\0 & \lambda\end{pmatrix}$, where $\lambda$ has order $(\ell-1)/\gcd(\ell-1,12)$, belongs to the subgroup $\Gal(L/(L \cap K^\prime))$, and is non-trivial; in particular, $\beta$ is in the center of $\Gal(L/(L \cap K^\prime))$, and $\beta-1$ defines an automorphism of $E[\ell]$. \end{lem}
\begin{proof} {\em Step 1.} Firstly, we claim that the maximal Abelian sub-extension of $L/\mathbb{Q}$ is $\mathbb{Q}(\mu_\ell)$.
Indeed, $\mathbb{Q}(\mu_\ell)/\mathbb{Q}$ is an Abelian sub-extension of $L/\mathbb{Q}$. Let $\widetilde \rho_\ell$ be the Galois representation of $\Gal(L/\mathbb{Q})$ on $E[\ell]$. Having assumed that $\Gal(L/\mathbb{Q}) \approx {\bf GL}_2(\mathbb{F}_\ell)$, it follows that ${\bf SL}_2(\mathbb{F}_\ell) = \Ker \det (\widetilde \rho_\ell)$. But from the Weil pairing, one has $\det (\widetilde \rho_\ell)=\psi_\ell$, where $\psi_\ell$ is the cyclotomic character \cite[1.2, Example 2, pp. I-3-4]{serre1968}. Therefore, $\Gal(L/\mathbb{Q}(\mu_\ell))=\Ker \psi_\ell={\bf SL}_2(\mathbb{F}_\ell)$.
Now, let $K/\mathbb{Q}$ be an Abelian extension, with $L \supseteq K\supseteq \mathbb{Q}(\mu_\ell)$. Then, $\Gal(K/\mathbb{Q}(\mu_\ell))$ corresponds to an Abelian quotient of ${\bf SL}_2(\mathbb{F}_\ell)$. Since this special linear group is a perfect group for $\ell>3$ \cite[p. 61]{rose1994}, it follows that $\Gal(K/\mathbb{Q}(\mu_\ell))$ is trivial. Thus, $K$ is equal to $\mathbb{Q}(\mu_\ell)$, which completes the proof of part a).
{\em Step 2.} Let $K_1:=\mathbb{Q}(\mu_{4})$ as in equation (\ref{eq:eq145final8}). Also, as in equation (\ref{eq:eq146final8}), let $K_2=K_1(p_1^{1/2},...,p_\nu^{1/2})$, where $p_1,...,p_{\nu}$ are the distinct prime factors other than $\ell$ that are bounded by the constant $C$ of Theorem \ref{thm:Theorem6}.
Then, the extension $L \cap K_1$ is an Abelian sub-extension of $L/\mathbb{Q}$, and hence is contained in $\mathbb{Q}(\mu_\ell)$, from step 1. From part a) of Corollary \ref{thm:Corollary8}, it follows that $L \cap K_1 \subseteq \mathbb{Q}(\mu_\ell) \cap K_2=\mathbb{Q}$.
{\em Step 3.} Next, the extension $(L \cap K_2)/\mathbb{Q}$ is an Abelian sub-extension of $L$, since $K_2/K_1$ is Abelian, and $L \cap K_1 = \mathbb{Q}$ from step 2. From steps 1 and 2, one must have $L \cap K_2 \subseteq \mathbb{Q}(\mu_\ell) \cap K_2=\mathbb{Q}$, which proves the first statement of part b).
{\em Step 4.} Since the Kummer extension $K_3/K_2$ has degree dividing $2$, it follows that the extension $(L \cap K_3)/(L\cap K_2)$ has degree dividing $2$. But $L\cap K_2 = \mathbb{Q}$ from step 3. Thus, $(L\cap K_3)/\mathbb{Q}$ has degree $1$ or $2$. But $L\cap K_3$ contains the unique quadratic subextension of $\mathbb{Q}(\mu_\ell)$. This proves the second statement of part b).
{\em Step 5.} The field $K^\prime$ is obtained by adjoining to $K_3$ various roots of cubic equations, as in equation (\ref{eq:eq148final8}). Since $K^\prime/\mathbb{Q}$ is a normal extension of degree dividing a power of $6$, it follows that the extension $(L \cap K^\prime)/\mathbb{Q}$ has degree dividing a power of $6$.
Since both $K^\prime/\mathbb{Q}$ and $L/\mathbb{Q}$ are normal extensions, the extension $L \cap K^\prime$ is also normal over $\mathbb{Q}$. We consider the compositum $K'':=(L\cap K^\prime)\cdot \mathbb{Q}(\mu_\ell) \subseteq L$. Then, $K''$ is normal over $\mathbb{Q}$, and hence over $\mathbb{Q}(\mu_\ell)$. We set $N:=\Gal(L/K'')$, which is a normal subgroup of ${\bf SL}_2(\mathbb{F}_\ell)$, using part a). From \cite[Theorem 2, p. 62]{suprunenko1976}, there are only three cases if $\ell\geq 5$: $N=1$, $N=\{\pm I\}$, or $N={\bf SL}_2(\mathbb{F}_\ell)$, where $I$ denotes the $2\times 2$ matrix over $\mathbb{F}_\ell$. But the first two cases are ruled out, since the order of ${\bf SL}_2(\mathbb{F}_\ell)/N$ would then be divisible by $\ell$, whereas $K''/\mathbb{Q}(\mu_\ell)$ has degree dividing a power of $6$. It follows that $K''=\mathbb{Q}(\mu_\ell)$. This means that $L \cap K^\prime \subseteq \mathbb{Q}(\mu_\ell)$, which proves part c).
{\em Step 6.} From part c) of Corollary \ref{thm:Corollary8}, having assumed that $\ell >2$, the extension $(\mathbb{Q}(\mu_\ell) \cap K^\prime)/\mathbb{Q}$ has degree dividing $12$, which proves part d), making use of part c).
{\em Step 7.} Part e) is a consequence of part d) and the assumption that $\Gal(L/\mathbb{Q})$ is the full linear group. Indeed, consider $\sigma \in \Gal(L/\mathbb{Q})$ such that $\widetilde \rho_\ell(\sigma)=\begin{pmatrix} \lambda & 0\\0 & 1\end{pmatrix}$, where $\lambda$ is a generator of $\mathbb{F}_\ell^*$. Let $d_0$ be a divisor of $(\ell-1)/\gcd(\ell-1,12)$. Then, one has $\sigma_0:=\sigma^{(\ell-1)/d_0} \in \Gal(L/(L \cap K^\prime))$, using part d), since $\gcd(\ell-1,12)\mid (\ell-1)/d_0$. Then, the element $\sigma_0$ has order $d_0$, and $\widetilde \rho_\ell(\sigma_0)$ is of the form $\begin{pmatrix} \lambda_0 & 0\\0 & 1\end{pmatrix}$, where $\lambda_0$ has same order as $\sigma_0$.
{\em Step 8.} The element $\tau$ has order $\ell$, so that $\tau^{\ell} = 1 \in \Gal(L/(L \cap K^\prime))$. But, from part d), one has $\tau^{12} \in \Gal(L/(L \cap K^\prime))$. As $\gcd(\ell,12)=1$, one concludes that $\tau \in \Gal(L/(L \cap K^\prime))$, which proves part f).
{\em Step 9.} Let $\lambda_1$ have order $\ell-1$ in $\mathbb{F}_\ell^*$, and set $\beta_1=\begin{pmatrix} \lambda_1 & 0\\0 & \lambda_1\end{pmatrix}$. Then, the element $\beta:=\beta_1^{\gcd(\ell-1,12)}$ belongs to $\Gal(L/(L \cap K^\prime))$, using part d). Furthermore, since $\ell \not = 2,3,5,7,13$, the element $\beta$ is non-trivial, and it then follows that $\beta-1$ is an automorphism of $E[\ell]$. Lastly, $\widetilde \rho_\ell(\beta)$ is equal to $\begin{pmatrix} \lambda & 0\\0 & \lambda\end{pmatrix}$, where $\lambda:=\lambda_1^{\gcd(\ell-1,12)}$ has order $(\ell-1)/{\gcd(\ell-1,12)}$. This proves part g). \end{proof}
Lemma \ref{thm:Lemma9} and Corollary \ref{thm:Corollary10} play an important role in our proof of the following intermediate result.
\begin{prop} \label{thm:Proposition12} Let $E$ be an elliptic curve over $\mathbb{Q}$ without CM, of Weierstrass equation $y^2=x^3+Ax+B$, where $A,B\in \mathbb{Z}$. Let $\ell \not = 2,3,5,7,13$ be a prime number such that: i) $\rho_\ell(\mathcal{G}) = {\bf GL}_2(\mathbb{Z}_\ell)$; and ii) $\ell\nmid \Delta^\prime:=4 A^3 + 27 B^2$. Set $L=\mathbb{Q}(E[\ell])$, and let $K^\prime$ and $L^\prime$ be the fields defined in (\ref{eq:eq149final8}) and (\ref{eq:eq183final8}), respectively.
Then, for any element of $S^{[\ell]}(E/\mathbb{Q})$ represented by $1$-cocycle $f$, one has a decomposition of the form:
\begin{equation} Res^{\mathcal{G}}_{\mathcal{G}^\prime} (f) = g + Inf^{\mathcal{G}^\prime}_{G^\prime}(\widetilde h), \end{equation} where $\mathcal{G}^\prime:=\Gal(\overline{\mathbb{Q}}/K^\prime)$ and $G^\prime:=\Gal(L^\prime/K^\prime) \approx \Gal(L/(L \cap K^\prime))$, $g \in \Homol^1(\mathcal{G}^\prime,E(\overline{\mathbb{Q}}))$ splits in $E(\overline{\mathbb{Q}})$, and $\widetilde h\in \Homol^1(G^\prime,E[\ell])$. \end{prop}
\begin{proof} {\em Step 1.} From Corollary \ref{thm:Corollary9} (assuming that $\ell > 3$),
$f \in S^{[\ell]}(E/\mathbb{Q})$ is of the form $Inf^{\mathcal{G}}_{\widetilde G}(\widetilde f)$, for some $\widetilde f\in \Homol^{1}(\widetilde G,E[\ell])$, where $\widetilde L = L_{\overline{\mathcal{N}}}$ is the class field defined in Lemma \ref{thm:Lemma7} (with $K=\mathbb{Q}$) and $\widetilde G=\Gal(\widetilde L/\mathbb{Q})$.
{\em Step 2.} Let $y^2=x^3+Ax +B$ be a Weierstrass equation for $E$, with $A,B\in \mathbb{Z}$. We assume the non-CM case, with $\rho_\ell(\mathcal{G}) = {\bf GL}_2(\mathbb{Z}_\ell)$, $\ell\nmid \Delta^\prime:=4 A^3 + 27 B^2$, and $\ell \not = 2,3,5,7,13$. We consider the field $K^\prime$ defined in (\ref{eq:eq149final8}). From part d) of Lemma \ref{thm:Lemma9}, the extension $( L\cap K^\prime )/\mathbb{Q}$ has degree dividing $12$.
Having assumed that $\ell \not = 2,3,5,7,13$, let $d_0$ be a prime factor of $(\ell-1)/\gcd(\ell-1,12)$. Applying part e) of Lemma \ref{thm:Lemma9}, consider the element $\sigma_0^\prime \in \Gal(L^\prime/K^\prime)\approx \Gal(L/(L \cap K^\prime))$ defined by $\widetilde \rho_\ell(\sigma_0^\prime) = \begin{pmatrix} \lambda_0 & 0 \\ 0 & 1 \end{pmatrix}$, where $\lambda_0$ is an element of order $d_0$ in the multiplicative group $\mathbb{F}_\ell^{*}$. Let $X_{(\chi)} = \langle \begin{pmatrix} 1\\ 0 \end{pmatrix} \rangle$ be the one-dimensional $\mathbb{F}_\ell$-subspace of $E[\ell]$ on which the automorphism $\sigma^\prime_0 \in \Gal(L^\prime/K^\prime)$, acts through the character defined by $\chi(\sigma^\prime_0)=\lambda_0$. Let $X_{(1)} = \langle \begin{pmatrix} 0\\ 1 \end{pmatrix} \rangle$ be the one-dimensional $\mathbb{F}_\ell$-subspace of $E[\ell]$ on which $\sigma^\prime_0$ acts through the trivial character. One has the decomposition $E[\ell] = X_{(\chi)} \oplus X_{(1)}$.
{\em Step 3.} Given $\widetilde Q\in E(\overline{\mathbb{Q}})$ such that $[\ell]\widetilde Q=P\in E(K^\prime)$, the $1$-cocycle defined by $\widetilde g(\sigma):=[\sigma -1]\widetilde Q$ belongs to $S^{[\ell]}(E/K^\prime)$. Furthermore, by construction, any such element of this Selmer group splits in $E(\widetilde L^\prime)$. Let $S^{[\ell]}(E/K^\prime)_{split}$ be the subgroup of $S^{[\ell]}(E/K^\prime)$ consisting of such elements.
We define (with $L^\prime=L K^\prime$): \begin{equation} \begin{cases} \mathcal{G}^\prime = \Gal(\overline{\mathbb{Q}}/K^\prime);\qquad \mathcal{H}^\prime = \Gal(\overline{\mathbb{Q}}/L^\prime);\\ \widetilde G^\prime = \Gal(\widetilde L^\prime/K^\prime);\qquad \widetilde H^\prime = \Gal(\widetilde L^\prime/L^\prime);\\ G^\prime = \Gal(L^\prime/K^\prime). \end{cases} \end{equation} The $\mathbb{F}_\ell \langle \sigma^\prime_0 \rangle$-module $\widetilde H^\prime$ admits a decomposition: \begin{equation} \widetilde H^\prime = \oplus_{i=0}^{d_0-1} \widetilde H^\prime_{(\chi^i)}, \end{equation} where $\langle \sigma^\prime_0 \rangle$ acts on $\widetilde H^\prime_{(\chi^i)}$ through the character $\chi^i$ of the cyclic group $\langle \sigma^\prime_0 \rangle$, for $i=0,1,...,d_0-1$.
Let then $\mathcal{I}^\prime$ be the image of the map: \begin{equation} \begin{CD} \varrho^\prime: S^{[\ell]}(E/K^\prime)_{split} @>{Res^{\mathcal{G}^\prime}_{\mathcal{H}^\prime}}>> S^{[\ell]}(E/L^\prime)^{G^\prime}\\ @>>> \Hom(\widetilde H^\prime, E[\ell]/X_{(\chi)})^{\sigma_0^\prime}\\ @>{\approx}>> \Hom(\widetilde H^\prime/\bigl (\oplus_{i=1}^{d_0-1} \widetilde H^\prime_{(\chi^i)} \bigr ), E[\ell]/X_{(\chi)}),\\ \end{CD} \end{equation} where the first map is the one of Proposition \ref{thm:Proposition6} (applied to $K=K^\prime$), and the second map is induced by restriction of $G^\prime$ to $\langle \sigma_0^\prime \rangle$ and the projection $E[\ell] \rightarrow E[\ell]/X_{(\chi)}$. We have used the isomorphism $E[\ell]/X_{(\chi)}\approx X_{(1)}$ in the third map.
Then, from Pontryagin duality, one has an isomorphism: \begin{equation} \mathcal{I}^\prime \approx \Hom( \widetilde H^\prime/H_1, E[\ell]/X_{(\chi)}), \end{equation} for some closed subspace $H_1 \supseteq \oplus_{i=1}^{d_0-1} \widetilde H^\prime_{(\chi^i)}$ of $\widetilde H^\prime$.
{\em Step 4.} Since $X_{(\chi)}$ is stable under $\sigma_0^\prime$, it follows that $H_1$ is stable under conjugation by a lifting $\widetilde \sigma_0^\prime$ of $\sigma_0^\prime$ in $\Gal(\widetilde L^\prime/K^\prime)$. Indeed, with notation as above, the equality $[h - 1]\widetilde Q \in X_{(\chi)}$ for all $\widetilde Q$ such that $[\ell]\widetilde Q = \widetilde P \in E(K^\prime)$, implies that $[h - 1] (\widetilde \sigma_0^\prime)^{-1}(\widetilde Q ) \in X_{(\chi)}$ for all such point $\widetilde Q$, because $K^\prime$ is normal over $\mathbb{Q}$. But then, this implies that $[\widetilde \sigma_0^\prime h (\widetilde \sigma_0^\prime)^{-1} - 1]\widetilde Q = \widetilde \sigma_0^\prime [h - 1] (\widetilde \sigma_0^\prime)^{-1}(\widetilde Q ) \in \widetilde \sigma_0^\prime(X_{(\chi)})=X_{(\chi)}$ for all such point $\widetilde Q$. Therefore, $\widetilde \sigma_0^\prime h (\widetilde \sigma_0^\prime)^{-1} \in H_1$.
{\em Step 5.} We define: \begin{equation} \begin{cases} \mathcal{G} = \Gal(\overline{\mathbb{Q}}/\mathbb{Q});\qquad \mathcal{H} = \Gal(\overline{\mathbb{Q}}/L);\\ \widetilde G = \Gal(\widetilde L/\mathbb{Q});\qquad \widetilde H = \Gal(\widetilde L/L);\\ G = \Gal(L/\mathbb{Q}). \end{cases} \end{equation}
Let then $\mathcal{I}$ denote the image of the map \begin{equation} \begin{CD} \varrho: S^{[\ell]}(E/\mathbb{Q}) @>{Res^{\mathcal{G}}_{\mathcal{H}}}>> S^{[\ell]}(E/L)^{G}\\ @>>> \Hom(\widetilde H, E[\ell]/X_{(\chi)})^{\sigma_0}\\ @>{\left( Res_{\widetilde L}^{\widetilde L^\prime}\right )_*}>> \Hom(\widetilde H^\prime, E[\ell]/X_{(\chi)})^{\sigma_0^\prime}\\ @>{\approx}>> \Hom(\widetilde H^\prime/\bigl (\oplus_{i=1}^{d_0-1} \widetilde H^\prime_{(\chi^i)} \bigr ), E[\ell]/X_{(\chi)}),\\ \end{CD} \end{equation} where the first map is the one of Proposition \ref{thm:Proposition6} (applied to $K=\mathbb{Q}$), and $Res_{\widetilde L}^{\widetilde L^\prime}$ denotes the natural projection $\widetilde H^\prime \rightarrow \widetilde H$ of Galois groups. We have used the isomorphism $E[\ell]/X_{(\chi)}\approx X_{(1)}$ in the fourth map.
Then, from Pontryagin duality, one has an isomorphism: \begin{equation} \mathcal{I} \approx \Hom( \widetilde H^\prime/H_2, E[\ell]/X_{(\chi)}), \end{equation} for some closed subspace $H_2 \supseteq \oplus_{i=1}^{d_0-1} \widetilde H^\prime_{(\chi^i)}$ of $\widetilde H^\prime$.
{\em Step 6.} We claim that $\mathcal{I} \subseteq \mathcal{I}^\prime$, equivalently $H_1 \subseteq H_2$.
Based on step 4, the subgroup $H_1$ is normal in the Galois group $\Gal(\widetilde L^\prime/\left ( L^\prime \right )^{\sigma^\prime_0})$, where $\sigma^\prime_0$ is constructed in step 2. From Lemma \ref{thm:Lemma8}, one obtains a decomposition $H_1 = \oplus_{\nu=1}^{n_1} C_\nu$ of representations of $\langle \sigma^\prime_0 \rangle$ over $\mathbb{F}_\ell$.
If the one-dimensional representation $C_\nu$ maps to $0$ under the Galois projection $Res_{\widetilde L}^{\widetilde L^\prime}: \widetilde H^\prime \rightarrow \widetilde H$, then $C_\nu$ is contained in $H_2$, and there is nothing to prove.
If the one-dimensional representation $C_\nu$ is defined by the trivial character $\chi_\nu$ of $\langle \sigma^\prime_0 \rangle$, and $Res_{\widetilde L}^{\widetilde L^\prime}(C_\nu)\not = 0$, then Corollary \ref{thm:Corollary10} implies that $\widetilde f(Res_{\widetilde L}^{\widetilde L^\prime}(\sigma)) = [\sigma -1]\widetilde Q$, for any $\widetilde f \in S^{[\ell]}(E/\mathbb{Q})$ and any $\sigma\in C_\nu$, for some $\widetilde Q \in E(\overline{\mathbb{Q}})$ such that $[\ell]\widetilde Q \in E(K^\prime)$, depending on $\widetilde f$ and $C_\nu$. But then, the inclusion $C_\nu < H_1$ implies that $\widetilde f(Res_{\widetilde L}^{\widetilde L^\prime}(\sigma))=0$, for any $\widetilde f \in S^{[\ell]}(E/\mathbb{Q})$ and any $\sigma\in C_\nu$. This means that $C_\nu < H_2$.
If the one-dimensional representation $C_\nu$ is defined by a non-trivial character of $\langle \sigma^\prime_0 \rangle$, then $C_\nu < \oplus_{i=1}^{d_0-1} \widetilde H^\prime_{(\chi^i)}$. But from step 5, one has the inclusion $\oplus_{i=1}^{d_0-1} \widetilde H^\prime_{(\chi^i)} \subseteq H_2$, which means that $C_\nu < H_2$.
Altogether, we conclude that $H_1 \subseteq H_2$.
{\em Step 7.} From Step 6, we have $\mathcal{I} \subseteq \mathcal{I}^\prime$. So, let $f$ be in the Selmer group $S^{[\ell]}(E/\mathbb{Q})$. Then, $\varrho(f)$ satisfies: \begin{equation} \varrho(f)(\sigma) = \varrho^\prime (g)(\sigma) \mod X_{(\chi)}, \end{equation} valid for any $\sigma \in \mathcal{H}^\prime$, for some $1$-coboundary $g$ of the form: \begin{equation} \widetilde g(\sigma) : = [\sigma-1]\widetilde Q, \end{equation} where $\widetilde Q\in E(\overline{\mathbb{Q}})$ such that $[\ell]\widetilde Q=P\in E(K^\prime)$.
Define: \begin{equation} h:= Res^{\mathcal{G}}_{\mathcal{G}^\prime} (f) - g. \end{equation} Thus, $Res^{\mathcal{G}^\prime}_{\mathcal{H}^\prime}(h)$ maps to $0\in\Hom(\mathcal{H}^\prime,E[\ell]/X_{(\chi)})$. This means that $h$ maps the Galois group $\mathcal{H}^\prime$ into $X_{(\chi)}$.
{\em Step 8.} Now, from Proposition \ref{thm:Proposition6} applied to $K^\prime$, one has: \begin{equation} Res^{\mathcal{G}^\prime}_{\mathcal{H}^\prime} ( h ) \in \Hom(\mathcal{H}^\prime,E[\ell])^{G^\prime}. \end{equation} Therefore, for any $\sigma \in \mathcal{G}^\prime$ and $\tau\in \mathcal{H}^\prime$, one has: \begin{equation} \sigma(h(\tau)) = h(\sigma \tau \sigma^{-1}) \in X_{(\chi)}, \end{equation} since $\mathcal{H}^\prime$ is a normal subgroup of $\mathcal{G}^\prime$. Thus, one must have $h(\tau)=0$ for any $\tau\in \mathcal{H}^\prime$, since the one-dimensional subspace $X_{(\chi)}$ is not stable under Galois action of $G^\prime$. Indeed, one may take the element $\tau$ corresponding to the matrix $\begin{pmatrix} 1 & 0\\ 1 & 1 \end{pmatrix}$ of order $\ell$, as in part f) of Lemma \ref{thm:Lemma9}.
It follows that $Res^{\mathcal{G}}_{\mathcal{G}^\prime} (f) - g = 0$ on $\mathcal{H}^\prime$, so that \begin{equation} Res^{\mathcal{G}}_{\mathcal{G}^\prime} (f) - g = Inf^{\mathcal{G}^\prime}_{G^\prime}(\widetilde h), \end{equation} for some $\widetilde h \in \Homol^1(G^\prime, E[\ell])$, as was to be shown. \end{proof}
Equipped with Lemma \ref{thm:Lemma9} and Proposition \ref{thm:Proposition12}, we are now ready to prove Theorem \ref{thm:Theorem5}.
{\em Proof of Theorem \ref{thm:Theorem5}.} We show the inclusion: \begin{equation} \label{eq:eq207final8} S^{[\ell]}(E/\mathbb{Q}) \subseteq \Ker \Bigl \{ \Homol^{1}(\mathcal{G},E[\ell]) \rightarrow
\Homol^{1}(\mathcal{G},E(\overline{\mathbb{Q}})) \Bigr \}. \end{equation}
Let $\ell \not = 2,3,5,7,13$ be a prime number. Assuming that: i) $\rho_\ell(\mathcal{G}) = {\bf GL}_2(\mathbb{Z}_\ell)$, as well as the condition: ii) $\ell \nmid \Delta^\prime$, Proposition \ref{thm:Proposition12} applies. Set $L:=\mathbb{Q}(E[\ell])$ and $G:=\Gal(L/\mathbb{Q})$. Thus, for any $f \in S^{[\ell]}(E/\mathbb{Q})$, $Res^{\mathcal{G}}_{\mathcal{G}^\prime}(f)$ is of the form $g + Inf^{\mathcal{G}^\prime}_{G^\prime}(\widetilde h)$ on $\mathcal{G}^\prime$, for some $g \in \Homol^1(\mathcal{G}^\prime,E(\overline{\mathbb{Q}}))$ that splits in $E(\overline{\mathbb{Q}})$, and $\widetilde h\in \Homol^1(G^\prime,E[\ell])$.
Now, consider the element $\beta =\begin{pmatrix} \lambda & 0\\0 & \lambda \end{pmatrix}$, where $\lambda \in \mathbb{F}_\ell^*$ has order $(\ell-1)/\gcd(\ell-1,12)$. Then, from part g) of Lemma \ref{thm:Lemma9}, $\beta$ is in the center of $\Gal(L/(L \cap K^\prime)) \approx G^\prime$, and $\beta-1$ defines an automorphism of $E[\ell]$. Thus, from Sah's Theorem \cite[Theorem 5.1, p. 118]{lang1978}, one has: \begin{equation} \label{eq:eq208final8} \Homol^1(G^\prime,E[\ell])=0. \end{equation} See \cite[Proposition 19, p. 51]{bashmakov1972} in the context of Abelian varieties. Note that \cite[Theorem 11]{lawson2016} on vanishing of Galois cohomology groups defined on torsion points does not apply here, since $\mathbb{Q}(\mu_\ell) \cap K^\prime$ might be larger than $\mathbb{Q}$ ({\em c.f.} part d) of Lemma \ref{thm:Lemma9}). On the other hand, Coates' result \cite[Lemma 10, p. 179]{coates1970} does apply, since $G^\prime=\Gal(L^\prime/K^\prime)$ contains ${\bf SL}_2(\mathbb{F}_\ell)$, as follows from parts a) and d) of Lemma \ref{thm:Lemma9}.
Thus, one has for any $\sigma \in \mathcal{G}^\prime$: \begin{equation} f(\sigma) = [\sigma-1](\widetilde Q + R), \end{equation} where $\widetilde Q\in E(\overline{\mathbb{Q}})$ and $R \in E[\ell]$. Lastly, one has for any $\sigma \in \mathcal{G}$: \begin{equation} f(\sigma) - [\sigma-1](\widetilde Q + R ) = Inf^{\mathcal{G}}_{\Gal(K^\prime/\mathbb{Q})}(h^\prime), \end{equation} for some $h^\prime\in \Homol^{1}(\Gal(K^\prime/\mathbb{Q}),E(K^\prime))$. But then, from \cite[Corollary 10.2, p. 84]{brown1982}, $\Homol^{1}(\Gal(K^\prime/\mathbb{Q}),E(K^\prime))$ is annihilated by $\vert \Gal(K^\prime/\mathbb{Q}) \vert = n$. Therefore, one has for any $\sigma \in \mathcal{G}$: \begin{equation} [n] \left ( f(\sigma) - [\sigma-1](\widetilde Q + R ) \right ) = [\sigma-1]R^\prime, \end{equation} for some $R^\prime\in E(K^\prime)$. Thus, $[n]f$ maps to $0$ in $H^1(\mathcal{G},E(\overline{\mathbb{Q}}))$. Since, on the other hand, $f$ maps into $H^1(\mathcal{G},E(\overline{\mathbb{Q}}))[\ell]$, the inclusion (\ref{eq:eq207final8}) is proved, as $\gcd(n,\ell)=1$ by Proposition \ref{thm:Proposition3}.
The other inclusion is clear since, for any place $v_0$ of $\mathbb{Q}$, the homomorphism $Res^{\mathcal{G}}_{\mathcal{G}_{v_0}}: \Homol^{1}(\mathcal{G},E[\ell]) \rightarrow \Homol^{1}(\mathcal{G}_{v_0},E(\overline{\mathbb{Q}}_{v_0}))$ is the composition of homomorphisms $\Homol^{1}(\mathcal{G},E[\ell]) \rightarrow \Homol^{1}(\mathcal{G},E(\overline{\mathbb{Q}})) \xrightarrow{Res^{\mathcal{G}}_{\mathcal{G}_{v_0}}} \Homol^{1}(\mathcal{G}_{v_0}, E(\overline{\mathbb{Q}}_{v_0}))$.
$\square$
\section{Examples} \label{section:examples}
Recall that, given an elliptic curve $E$ over the rationals, there is, for each prime $\ell$, a Galois representation $\widetilde \rho_\ell: \mathcal{G} \rightarrow {\bf GL}_2(\mathbb{F}_\ell)$, where $\mathcal{G}$ is the absolute Galois group of $\mathbb{Q}$, through Galois action on the $\ell$-torsion points of $E$. Then, one has an isomorphism $\Gal(\mathbb{Q}(E[\ell])/\mathbb{Q}) \approx \widetilde \rho_\ell(\mathcal{G})$. In \cite{serre1972}, the notation is $\phi_\ell(G)$. \\
\noindent {\bf Example 1.} From \cite[Proposition 1.4, p. 45]{silverman2009}, to each $j\in \mathbb{Q}$ corresponds a unique class of elliptic curves over $\mathbb{Q}$ up to isomorphism
over $\overline{\mathbb{Q}}$. For each $j_0\in\mathbb{Q}$, the class of elliptic curves having $j_0$ as $j$-invariant is in one-to-one correspondence with $\mathbb{Q}^*/(\mathbb{Q}^*)^{n(j)}$, where $n(j)=2,4,6$ according to the cases $j\not=0,1738$, $j=1728$, or $j=0$, respectively \cite[Corollary 5.4.1, p. 343]{silverman2009}. From \cite[p. 427]{silverman2009}, there are exactly $13$ elliptic curves over $\mathbb{Q}$, up to isomorphism over $\overline{\mathbb{Q}}$, having CM. Then, for each corresponding $j$-invariant (see \cite[p. 295]{serre1967b} for a list), there are infinitely many elliptic curves over $\mathbb{Q}$ having CM ({\em i.e.}, over a finite base field extension). Moreover, except for these $13$ $j$-invariants, to any $j\in \mathbb{Q}$ corresponds a class of elliptic curves without CM. \\
\noindent {\bf Example 2.} Consulting \cite[Table 1]{rubin2002}, we consider the elliptic curve from \cite{penney1975}: \begin{equation} E\::\: y^2=x^3 + a x^2 + b x, \end{equation} with $a=1,692,602 = 2 \cdot 37 \cdot 89 \cdot 257$ and $b=-3 \cdot 5 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 37$. This equation corresponds to \cite[p. 42]{silverman2009}: \begin{equation} \begin{cases} a_1=a_3=a_6=0;\\ a_2=a; \quad a_4=b;\\ b_2=4a;\quad b_4=2b; \quad b_6=0;\\ b_8=-b^2;\\ c_4=16a^2-48b;\\ c_6=- 64a^3 + 36 \cdot 8 ab. \end{cases} \end{equation} One computes the discriminant: \begin{eqnarray} \Delta(E)&=&-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6=16a^2b^2-64b^3\nonumber\\ &=&16 b^2(a^2-4b), \end{eqnarray} and the $j$-invariant: \begin{equation} j(E)=c_4^3/\Delta=\frac{(16)^2(a^2-3b)^3}{b^2(a^2-4b)}. \end{equation} Furthermore, one has: \begin{equation} \begin{cases} \Delta(E)=2^8 \cdot 3^2 \cdot 5^2 \cdot 11^2 \cdot 13^2 \cdot 17^2 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2 \cdot 37^3 \cdot p^\prime;\\ j(E) = \frac{2^4 \cdot 7^3 \cdot 61^3 \cdot 347^3 \cdot (p'')^3} {3^2 \cdot 5^2 \cdot 11^2 \cdot 13^2 \cdot 17^2 \cdot 19^2 \cdot 23^2 \cdot 29^2 \cdot 31^2 \cdot p^\prime}, \end{cases} \end{equation} where $p^\prime$ and $p''$ are the prime numbers $8,420,798,017$ and $812,633$, respectively. In particular, $E$ is not semi-stable.
Thus, this elliptic curve has potential multiplicative reduction at $p_0=3$. Therefore, $E$ has no CM. Moreover, from \cite{penney1975}, $E$ has rank at least $7$ over $\mathbb{Q}$. Therefore, Theorem \ref{thm:Theorem4} does not apply, whereas Theorem \ref{thm:Theorem5} does.
Moreover, $p=7$ is the smallest prime number at which $E$ has good reduction. From \cite[Proposition 24, p. 314]{serre1972}, one concludes that $\widetilde \rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{F}_\ell)$, whenever $\ell \nmid \Delta(E)$, $\ell \nmid \ord_{p_0}(j(E))= -2$, and $\ell > (\sqrt{p}+1)^8$. This means that Theorem \ref{thm:Theorem5} applies to $E$ with $\ell > 31,210$ other than $p^\prime$.
Pushing further these computations, let us consider the corresponding Tate's curve \cite[Theorem 14.1, p. 445]{silverman2009} at $p_0=3$. One obtains an isomorphism $\phi: \overline{\mathbb{Q}}_{p_0}/\langle q \rangle \xrightarrow{\approx} E(\overline{\mathbb{Q}}_{p_0})$, as $\Gal(\overline{\mathbb{Q}}_{p_0}/K'')$-modules, for some unramified quadratic extension $K''/\mathbb{Q}_{p_0}$. Here, $q$ is defined in \cite[p. 444]{silverman2009}. One can easily check that $\ord_{p_0}(q)=2$, from the fact that $\ord_{p_0}(\Delta)=2$. Therefore, one obtains an isomorphism $\phi: (\langle q^{1/\ell}\rangle \times\mu_\ell)/\langle q \rangle \xrightarrow{\approx} E[\ell](\overline{\mathbb{Q}}_{p_0})$, as $\Gal(\overline{\mathbb{Q}}_{p_0}/K'')$-modules. This implies that $\widetilde \rho_\ell(\mathcal{G})$ has a cyclic subgroup of order $\ell$, for any prime $\ell\not = 2$.
Now, from \cite[Theorem 3]{mazur1978}, if $G=\widetilde \rho_\ell(\mathcal{G})\not = {\bf GL}_2(\mathbb{F}_\ell)$, then either $G$ is in the normalizer $N$ of a Cartan subgroup $C$, or $\ell$ is one of the exceptional primes $2,3,5,7,11,13,17,19,37,43,67,163$. But $C$ has index $2$ in $N$ \cite[\S 2.2, p. 279]{serre1972}, so that $N$ has order coprime with $\ell$. Therefore, one concludes that $\ell$ is one of the exceptional primes.
Furthermore, note that $E$ has potential good reduction at $2$, since $\ord_{2}(\Delta(E))=8$ and $\ord_{2}(j(E))=4$. It follows from \cite[a3), p. 312]{serre1972}, that the group $\Phi_{2}$ defined in \cite[pp. 311--312]{serre1972}, has cardinality $2$, $3$, $4$, $6$, $8$, or $24$. Recall that $\Phi_2$ is a quotient group of the inertia group $I_2$, and that it embeds into ${\bf GL}_2(\mathbb{F}_\ell)$, if $\ell \geq 5$ \cite[pp. 311--312]{serre1972}. Now, the Weierstrass equation $y^2=x^3+ax^2+bx$ is minimal over $\mathbb{Q}_2$ since $\ord_2(\Delta)=8<12$. From Section \ref{subsection:localElliptic}, the curve $E$ has good reduction over a finite extension $K^\prime/\mathbb{Q}_2$ of degree with only $2$ or $3$ as prime factors. Moreover, one can make a change of variable of the form $x=u^2x^\prime+r$ and $y=u^3y^\prime+u^2 s x^\prime+t$, where $u\in \mathcal{O}_{K^\prime}^*$, $r,s,t \in \mathcal{O}_{K^\prime}$ \cite[Proposition 1.3, part a), p. 186]{silverman2009}, and obtain the discriminant $u^{-12}\Delta$ \cite[Remark 1.1, p. 186]{silverman2009}. It follows that $v(u^{-12}\Delta)= - 12 v(u) + 8 v(2) = 0$, where $v$ is the discrete valuation of $K^\prime$. This in turn implies that $3 \mid v(2)$, so that the ramification index of the extension $K^\prime/\mathbb{Q}_2$ is divisible by $3$. One concludes that $\Phi_2$ has order $3$, $6$, or $24$ (see \cite[p. 312]{serre1972}).
Thus, for $\ell\not = 2$, if $\widetilde \rho_\ell(\mathcal{G})$ is not the full linear group, then it is contained in a Borel subgroup of the linear group. Indeed, \cite[Corollaire, p. 277]{serre1972} implies that $\widetilde \rho_\ell(\mathcal{G})$ contains a split Cartan semi-subgroup of the form $\begin{pmatrix} * & 0\\ 0 & 1 \end{pmatrix}$. Then, the condition $\ell \mid \# \widetilde \rho_\ell(\mathcal{G})$ for $\ell\not =2$ implies that $\widetilde \rho_\ell(\mathcal{G})$ is either the full linear group, or else is contained in a Borel subgroup. See \cite[Proposition 17 and remark a), p. 282]{serre1972}. But then, assuming $\ell \geq 5$, \cite[Proposition 23, part b), p. 313]{serre1972} implies that the divisor $3$ of $\vert \Phi_{2} \vert$ divides the order of $\left (\mathbb{Z}/2^n\mathbb{Z}\right)^*$, for some $n\geq 1$, which is not the case. So, actually, $\widetilde \rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{F}_\ell)$, for all $\ell\geq 5$. See \cite[5.7.1, p. 315]{serre1972} for this argument. It follows from (\ref{eq:eq41final7}) that $\rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{Z}_\ell)$, for all $\ell\geq 5$.
Next, we consider the Weierstrass equation of $E$: \begin{eqnarray} &&E : y^2 = x^3 + A x + B;\nonumber\\ && \; A = -27 c_4 = - 27 (16a^2-48b);\nonumber\\ && \; B = - 54 c_6 = -54 (- 64a^3 + 36 \cdot 8 ab);\nonumber\\ && \; \Delta^\prime(E) = - 2^{20} \cdot 3^{12} \cdot b^2 (a^2 - 4b). \end{eqnarray}
We conclude from Theorem \ref{thm:Theorem5} that ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{\ell}$ vanishes at all primes $\ell$, other than the ones in the set: \begin{equation} P=\{2,3,5,7,11,13,17,19,23,29,31,37,p^\prime\}. \end{equation} In particular, the smallest prime number for which Corollary \ref{thm:Corollary2} applies is $\ell=41$. Thus, one has: \begin{eqnarray} \rank (E/\mathbb{Q}) &=& \dim_{\mathbb{F}_{41}} S^{[41]}(E/\mathbb{Q}), \end{eqnarray} since ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})[41]=0$, from Theorem \ref{thm:Theorem5}, and $E[41](\mathbb{Q})=0$ from Mazur's Theorem on torsion points. \\
\noindent {\bf Remark 5.} \label{thm:Remark5} The previous example solves the open problem mentioned in \cite[Problem 2.16, p. 27]{stein2007} in the non-CM case. \\
\noindent {\bf Example 3.} The following example was communicated to us by Professor C. Wuthrich. The non-CM elliptic curve of rank $0$ defined by the cubic equation \cite{wuthrich1058e1}: \begin{eqnarray} &&E : y^2 + xy = x^3 - x^2 - 332,311 x - 73,733,731;\nonumber\\ && \; \Delta(E) = - 5,302,593,435,347,072 = - 2^7 \cdot 23^{10};\nonumber\\ && \; c_4 = 15,950,937 = 3 \cdot 19 \cdot 23^4;\nonumber\\ &&\; j(E) = - \frac{(3 \cdot 19 \cdot 23^4)^3}{2^7 \cdot 23^{10}} = - 2^{-7} \cdot 3^3 \cdot 19^3 \cdot 23^2, \end{eqnarray} has Shafarevich-Tate group of {\em analytic} order $25$, which is denoted as $\# {\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{an}$ $=25$. From \cite{miller2011}, one concludes that $\# {\mbox{\textcyr{Sh}}}(E/\mathbb{Q}) = 25$, as $E$ has conductor $N=1058<5000$, and rank $r\leq 1$.
Recall that $\# {\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{an}$ is based on BSD-2, and is computed as follows: \begin{equation} \# {\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{an} = \frac{\lim_{s\rightarrow 1} (s-1)^{-r}L(E,s) (\# E_{tor}(\mathbb{Q}))^{2}}{\Omega \, 2^r R \prod_p c_p}, \end{equation} where $L(E,s)$ denotes the $L$-series of $E$, $r$ is the rank of $E/\mathbb{Q}$, $\Omega$ is defined from the invariant differential, $R$ is the elliptic regulator of $E(\mathbb{Q})/E_{tor}(\mathbb{Q})$, and $c_p$ denotes $\# E(\mathbb{Q}_p)/E_0(\mathbb{Q}_p)$. See \cite[pp. 451--452]{silverman2009}. In the example, based on the information available on the Website \cite{wuthrich1058e1}, this expression simplifies to: \begin{equation} \# {\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{an} = \frac{L(E,1)}{\Omega} = 25, \end{equation} which is consistent with Cassels' result \cite[Theorem 4.14, p. 341]{silverman2009}.
Since $E$ has multiplicative reduction at $p_0=2$, it follows that $\widetilde \rho_\ell(\mathcal{G})$ has a cyclic subgroup of order $\ell$, for any prime $\ell \not = 7$. Indeed, the Tate's curve $E_q$, with $\ord_{p_0}(q)=7$, yields an isomorphism $\phi: (\langle q^{1/\ell}\rangle \times\mu_\ell)/\langle q \rangle \xrightarrow{\approx} E[\ell](\overline{\mathbb{Q}}_{p_0})$, as $\Gal(\overline{\mathbb{Q}}_{p_0}/K'')$-modules (where $K''$ is the unramified extension of degree $2$ over $\mathbb{Q}_{p_0}$, since $E$ has non-split multiplicative reduction at $p_0=2$).
Thus, unless $\ell$ is one of the exceptional primes $2,3,5,7,11,13,17,19,37,43,67$, or $163$, one has $\widetilde \rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{F}_\ell)$ \cite[Theorem 3]{mazur1978}.
Pushing further this example, observe that $E$ has potential good reduction at $23$. It follows from \cite[a1), p. 312]{serre1972}, that the group $\Phi_{23}$ has cardinality $6$, since $23 >3$ and $\ord_{23}(\Delta)=10$. Thus, for $\ell \not = 7$, if $\widetilde \rho_\ell(\mathcal{G})$ is not the full linear group, then it is contained in a Borel subgroup of the linear group. But then, assuming that $\ell \geq 5$, \cite[Proposition 23, part b), p. 313]{serre1972} implies that $6 = \vert \Phi_{23} \vert$ divides the order of $\left (\mathbb{Z}/23^n\mathbb{Z}\right)^*$, for some $n\geq 1$, which is not the case. So, actually, $\widetilde \rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{F}_\ell)$, for all $\ell \not = 2,3,7$, and hence, in particular for $\ell=5$. It follows from (\ref{eq:eq41final7}) that $\rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{Z}_\ell)$, for all $\ell$ except possibly $2$, $3$ and $7$. Actually, it is reported that $\widetilde \rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{F}_\ell)$ holds for any prime $\ell$ \cite{wuthrich1058e1}.
Next, we consider the Weierstrass equation of $E$: \begin{eqnarray} &&E : y^2 = x^3 + A x + B;\nonumber\\ && \; A = -430675299 = - 3^{4} \cdot 19 \cdot 23^{4};\nonumber\\ && \; B = - 3,443,997,030,498 = - 2 \cdot 3^6 \cdot 23^5 \cdot 367;\nonumber\\ && \; \Delta^\prime(E) = 2^{15} \cdot 3^{12} \cdot 23^{10}. \end{eqnarray} This yields the following Weierstrass equation, under the change of variable $(x,y)\mapsto (3^2 x^\prime,3^3 y^\prime)$: \begin{eqnarray} &&E : y^2 = x^3 + A_1 x + B_1;\nonumber\\ && \; A_1 = -5,316,979 = - 19 \cdot 23^{4};\nonumber\\ && \; B_1 = - 4724275762 = - 2 \cdot 23^5 \cdot 367;\nonumber\\ && \; \Delta_1^\prime(E) = 2^{15} \cdot 23^{10}. \end{eqnarray} Thus, one has to discard the prime $\ell=23 \mid \Delta_1^\prime(E)$ in addition to the exceptional primes $2,3,5,7,13$ (to avoid the exceptional condition $(\ell-1)/\gcd(\ell-1,12)=1$). So, although $\rho_\ell(\mathcal{G})={\bf GL}_2(\mathbb{Z}_\ell)$ at $\ell=5$ in this example, Theorem \ref{thm:Theorem5} does not predict the vanishing of ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{\ell}$ due to the exceptional condition $(\ell-1)\mid 12$.
Altogether, Theorem \ref{thm:Theorem5} predicts that ${\mbox{\textcyr{Sh}}}(E/\mathbb{Q})_{\ell}$ vanishes at any prime $\ell$ other than $2,3,5,7,13,23$. In particular, the conclusion is consistent with BSD-2 in this example ({\em i.e.}, $\# {\mbox{\textcyr{Sh}}}(E/\mathbb{Q})=25$). \\
\noindent {\bf Example 4.} Consider the elliptic curve presented in Example 3: \begin{eqnarray} &&E : y^2 = x^3 + A_1 x + B_1;\nonumber\\ && \; A_1 = -5,316,979 = - 19 \cdot 23^{4};\nonumber\\ && \; B_1 = - 4724275762 = - 2 \cdot 23^5 \cdot 367;\nonumber\\ && \; \Delta_1^\prime(E) = 2^{15} \cdot 23^{10}. \end{eqnarray} This curve has no rational points over $\mathbb{Q}$ \cite{wuthrich1058e1}.
Let us consider the prime $p=7$. The reduced curve has Weierstrass equation: \begin{eqnarray} &&\widetilde E_{p} : y^2 = x^3 + 4 x + 4, \end{eqnarray} since $y^2 = x^3 + A_1 x + B_1$ is a minimal Weierstrass equation for $E$ at $p$. The non-trivial points of the reduced curve modulo $p$ are: $(0,\pm 2)$, $(1,\pm 3)$, $(3,\pm 1)$, $(-3,0)$, $(-2,\pm 3)$. Thus, $\widetilde E(\mathbb{F}_p)$ has order $2 \cdot 5$. We take $\ell=5$, so that $\ell \nmid \Delta^\prime$. Since $p \nmid \Delta^\prime(\ell -1)(\ell +1)\ell$, it follows that Theorem \ref{thm:Theorem6} applies.
The non-trivial points that have order $\ell=5$ are the ones of the form $[2]P$ with $[2]P\not = O$: $(1,\pm 3)$, $(-2,\pm 3)$. One may take the generator $(1,3)$ of $\widetilde E(\mathbb{F}_p)_{\ell}$, so that $\frac{a}{b} = 9$ in Theorem \ref{thm:Theorem6}. We may choose the radical $y=\left ( \frac{a}{b} \right )^{1/2}=3$, in this simple situation. Let then $x$ be a root of $X^3+A_1X+B_1=3^2$ in $\overline{\mathbb{Q}}$ that maps to $1$ in the residue field $\mathbb{F}_p$, under an embedding $\xi:\mathbb{Q}(x) \rightarrow \overline{\mathbb{Q}}_p$ followed by projection into the residue field of $\mathbb{Q}(x)$ at some prime $\mathfrak{p}\mid p$. By Hensel's Lemma \cite[p. 43]{lang1986}, one has $\mathbb{Q}(x)_{\mathfrak{p}} = \mathbb{Q}_p$. Any point of $\widetilde E[\ell](\mathbb{F}_p)$ can be lifted to a point of $\mathbb{Q}(x)$, which is contained in the field $K^\prime$ defined in (\ref{eq:eq149final8}).
Then, for any point $P_0 \in E(\mathbb{Q}_p)$, the point $[m]P_0$, where $m=2$, projects to a point $\widetilde P \in \widetilde E(\mathbb{F}_p)$ that can be lifted to a point $P^\prime$ with affine coordinates in $K^\prime$ (in fact, in $\mathbb{Q}(x)$). Thus, $[m]P_0-\xi(P^\prime) \in E_1(\mathbb{Q}_p) \subset [\ell]E(\mathbb{Q}_p)$, so that $[m]P_0-\xi(P^\prime)=[\ell]Q''$, for some $Q''\in E(\mathbb{Q}_p)$. Since $3 m -\ell=1$, one obtains: $P_0=\xi([3]P^\prime) + [\ell]([3]Q'' - P_0)$, and we set $P=[3]P^\prime$ and $Q^\prime=[3]Q'' - P_0$.
Now, assume that $P_0 \not \in [\ell] E(\mathbb{Q}_p)$. Such a point exists since $\# \widetilde E(\mathbb{F}_p) = 2 \cdot \ell$ and $E(\mathbb{Q}_p)$ projects onto $\widetilde E(\mathbb{F}_p)$. Then, one must have $P\in E(\mathbb{Q}(x)) \setminus E(\mathbb{Q})$. Indeed, since $E(\mathbb{Q})=0$ in this example, the case $P\in E(\mathbb{Q})$ would imply that $P_0 \in [\ell]E(\mathbb{Q}_p)$. This issue was pointed out to us by Professor K. Rubin in an early draft of this paper. This motivated us to develop the results of Section \ref{section:liftingsPoints}. As the field $K^\prime$ is a finite extension over $\mathbb{Q}$, our approach in Section \ref{section:ProofMainTheorems} was then sufficient to prove Theorem \ref{thm:Theorem5}. \\
\appendix
\section{Ramification of the extension $L_\infty/\mathbb{Q}$} \label{section:appendixA}
The following result is a consequence of a theorem of Sen \cite{sen1972} that was conjectured by Serre \cite{serre1967c}.
\begin{prop} \label{thm:Proposition13} Let $E$ be an elliptic curve over the rationals. Let $\ell > 3$ be a prime number at which $E$ has good reduction. Let $L_n$ be the number field obtained by adjoining the affine coordinates of the $\ell^n$-torsion points of $E$. Then, the different $\mathfrak{D}_n$ of $L_n/\mathbb{Q}$ satisfies the estimate: \begin{equation} (\ell^{n} a) \subseteq \mathfrak{D}_n \subseteq (\ell^{n} a^{-1}) , \end{equation} for all $n\geq 1$, for some integer $a$. \end{prop}
\begin{proof} We consider the following four cases, in view of \cite[Proposition 10, p. 52]{serre1979}.
{\em Case A:} $p =\ell>3$ (and $p \not \in \Sigma_{E}$). Consider the Galois group $G$ of the infinite extension obtained by adjoining over $\mathbb{Q}$ the affine coordinates of all $\ell^n$-torsion points of $E$, where $n\geq 1$, as an $\ell$-adic Lie group. Let $\{ G_n \}$ be a Lie filtration on $G$. For instance, one may take $G_n:=\rho_\ell^{-1}(I+\ell^n{\bf Mat}_{2\times 2}(\mathbb{Z}_\ell))$. On the other hand, let $\{ G(n) \}$ denote the upper numbering filtration on the Galois group $G$ corresponding to an embedding $\overline{\mathbb{Q}} \hookrightarrow \overline{\mathbb{Q}}_\ell$. Sen's Theorem \cite[Theorem, p. 48]{sen1972} gives the estimate: \begin{equation} G(n e-c) \subseteq G_n \subseteq G(n e+c), \end{equation} valid for all $n$, for some constant $c$ (depending only on $G$, and hence on $E$ and $\ell$), where $e$ is the absolute ramification index of the ground field (so, $\mathbb{Q}_\ell$ here, and $e=1$).
Now, let $\mathfrak{P}$ denote a prime ideal of $L_n$ lying above $\ell$. Then, using \cite[Proposition 4, p. 64]{serre1979}, one has:
\begin{eqnarray} \val_{\mathfrak{P}} \left ( \mathfrak{D}_n \right ) &=& \sum_{u=0}^{u_0} \left ( \vert G[u] \vert - 1 \right), \end{eqnarray} where $G[u]$ denotes the lower numbering ramification groups, and $u_0$ is the largest integer $u$ such that $G[u]$ is non-trivial. One has:
\begin{eqnarray} \sum_{u=0}^{u_0} \vert G[u] \vert &=& \vert G[0] \vert \int_{u=0}^{u_0+1} \frac{1}{(G[0]:G[u])}\, du\nonumber\\ &=& \vert G[0] \vert \varphi(u_0+1), \end{eqnarray} where $\varphi$ denotes Herbrand's function \cite[p. 73]{serre1979}. From Sen's Theorem, one has $n-c \leq \varphi(u_0+1) \leq n+c$.
Moreover, from \cite[Exerc. 3 c), pp. 71--72]{serre1979}, one has $u_0\leq e_\mathfrak{P}/(\ell-1)$, and hence $u_0 + 1\leq e_\mathfrak{P}\ell/(\ell-1)$. Thus, we obtain:
\begin{equation} \prod_{\mathfrak{P}\mid \ell} \mathfrak{P}^{e_{\mathfrak{P}}(n+c)} \subseteq \prod_{\mathfrak{P}\mid \ell} \left ( \mathfrak{D}_n \right )_{\mathfrak{P}} \subseteq \prod_{\mathfrak{P}\mid \ell} \mathfrak{P}^{e_{\mathfrak{P}}(n-c-\ell/(\ell-1))}, \end{equation} that is:
\begin{equation} (\ell^{n+c} ) \subseteq \prod_{\mathfrak{P}\mid \ell} \left ( \mathfrak{D}_n \right )_{\mathfrak{P}} \subseteq (\ell^{n-c -\ell/(\ell-1)} ). \end{equation}
{\em Case B:} $p \not = \ell$ and $p\not \in \Sigma_{E}$. Then, from the Criterion of N\'eron-Ogg-Shafarevich, the extension $L_n/\mathbb{Q}$ is unramified. Hence, using \cite[Theorem 1, p. 53]{serre1979}, one obtains:
\begin{equation} \prod_{\mathfrak{P}\mid p} \left ( \mathfrak{D}_n \right )_{\mathfrak{P}} = (1), \end{equation} where $\mathfrak{P}$ stands for prime ideals of $L_n$.
{\em Case C:} $p \not = \ell$ and $p \in \Sigma_{E,p.g.}$. Then, from Lemma \ref{thm:Lemma1} (having assumed that $\ell > 3$), $E$ has good reduction over $L_1$. From the Criterion of N\'eron-Ogg-Shafarevich, the extension $L_n/L_1$ is unramified. Therefore, using \cite[Proposition 8, p. 51]{serre1979}, one has: \begin{equation} \prod_{\mathfrak{P}\mid p} \left ( \mathfrak{D}_n \right )_{\mathfrak{P}} =
\prod_{\mathfrak{p}\mid p} \left ( \mathfrak{D}_1 \right )_{\mathfrak{p}}, \end{equation} where $\mathfrak{p}$ denotes prime ideals of $L_1$.
{\em Case D:} $p \not = \ell$ and $p \in \Sigma_{E,p.m.}$. From Lemma \ref{thm:Lemma1} (having assumed that $\ell > 3$), $E$ has multiplicative reduction over $L_1$. Considering Tate's curves, the extension $L_n/L_1$ is at most tamely ramified. Therefore, one obtains: \begin{equation} \prod_{\mathfrak{p}\mid p} \left ( \mathfrak{D}_1 \right )_{\mathfrak{p}} p \subseteq \prod_{\mathfrak{p}\mid p} \left ( \mathfrak{D}_1 \right )_{\mathfrak{p}} \prod_{\mathfrak{P}\mid p} \mathfrak{P}^{e_{\mathfrak{P}}(L_n/L_1)-1} \subseteq \prod_{\mathfrak{P}\mid p} \left ( \mathfrak{D}_n \right )_{\mathfrak{P}} \subseteq (1), \end{equation} using \cite[Proposition 13, p. 58]{serre1979}. Here, $e_{\mathfrak{P}}(L_n/L_1)$ denotes the relative ramification index of $\mathfrak{P}$ in $L_n/L_1$. Since $e_{\mathfrak{P}}(L_n/L_1)$ is bounded by (in fact, divides) the ramification index $e_{\mathfrak{P}}$ of $\mathfrak{P}$ in $L_n/\mathbb{Q}$, the first inclusion holds, as $(p)=\prod_{\mathfrak{P}\mid p} \mathfrak{P}^{e_{\mathfrak{P}}}$. \end{proof}
\noindent {\bf Remark 6.} \label{remark6} Let $L_\infty$ be the infinite Galois extension obtained by adjoining the affine coordinates of all $\ell^n$-torsion points of $E$, with $n\geq 1$. Consider the cyclotomic fields $K_n=\mathbb{Q}(\mu_{\ell^n})$, for $n\geq 1$. From the Weil pairing, one has the inclusions $K_n \subset L_n$, for $n\geq 1$. From \cite[Theorem 3, p. 75]{lang1986}, one has: \begin{equation} \mathfrak{D}_{K_n/\mathbb{Q}} = (\ell^n \ell^{-1/(\ell-1)} ). \end{equation} This is consistent with the general results of Tate \cite[\S 3.1, pp. 170--172]{tate1967b}. Based on Sen's Theorem, one deduces that: \begin{eqnarray} (1) &\supseteq& \mathfrak{D}_{L_n/K_n} = \mathfrak{D}_{L_n/\mathbb{Q}} \, \mathfrak{D}_{K_n/\mathbb{Q}}^{-1} \supseteq (\ell^{n} a) (\ell^{-n} \ell^{1/(\ell-1)})\nonumber\\ &=& (a \ell^{1/(\ell-1)}) \supseteq (c), \end{eqnarray} for all $n\geq 1$, where $c=a \ell$. See also \cite[Remarque, p. 152]{serre1981} for a consequence of Sen's Theorem.
Now, consider $\alpha_n$ as in Proposition \ref{thm:Proposition2}, an element of the integer ring $\mathcal{O}_n$ of $L_n=\mathbb{Q}(E[\ell^n])$, with $n\geq 1$. Then, using Proposition \ref{thm:Proposition2} and Remark 2, one has: \begin{eqnarray} \Bigl \vert \frac{\tr_n(\alpha_n)}{[L_n:\mathbb{Q}]} \Bigr \vert_{\ell} &=& \Bigl \vert \frac{\tr_n(\alpha_n)}{b \ell^{nN}} \Bigr \vert_{\ell}\nonumber\\ &\sim& C_0 \ell^{n (N-1)} \Bigl \vert \frac{\tr_n(\alpha_n)}{\ell^{n}} \Bigr \vert_{\ell} \leq 1, \end{eqnarray} for some positive constant $C_0$. On the other hand, Proposition \ref{thm:Proposition13} implies that: \begin{equation} \vert a \vert_\ell \Bigl \vert \frac{\tr_n(\alpha_n)}{\ell^{n}} \Bigr \vert_{\ell} \leq \Bigl \vert \tr_n(\alpha_n \mathfrak{D}_{L_n/\mathbb{Q}}^{-1}) \Bigr \vert_{\ell} \leq 1. \end{equation} But $\lim_{n\rightarrow \infty} C_0 \ell^{n (N-1)} = \infty$, as $N\geq 2$, whereas $\bigl \vert a \bigr \vert_{\ell} < \infty$. Thus, the precise form of $\alpha_n$ in Proposition \ref{thm:Proposition2}, {\em i.e.}, $\alpha_n=\Delta^\prime \ell^3/y^2(P)$, is at stake in this proposition concerning the $\ell$-adic norm, in addition to the strong condition $\rho_{\ell}(\mathcal{G}) = {\bf GL}_2(\mathbb{Z}_\ell)$. For the Archimedean norm, one also needs the precise form of $\alpha_n$, but not the condition on $\rho_{\ell}(\mathcal{G})$. But for the other non-Archimedean norms, the mere fact that $\alpha_n$ is an integral element is sufficient, and this is the only trivial case. Thus, the result of Sen indicates that Proposition \ref{thm:Proposition2} is by no means elementary, as it seems. In particular, Serre's Theorems on the Galois group of $L_\infty/\mathbb{Q}$, in the case of non-CM elliptic curves over the rationals, played an essential role in our proof of Theorems \ref{thm:Theorem5} and \ref{thm:Theorem6}. \\
\subsection*{Acknowledgments} We are grateful to Professors Karl Rubin and Christian Wuthrich for their helpful comments on the first version of this article. Moreover, we acknowledge the helpful comments of Professor Rubin on drafts of the second version. The elliptic curve in Example 3 that was mentioned to us by Professor Wuthrich was very helpful in developing the second version of this work.
\end{document} | arXiv |
\begin{document}
\title[]{\large Generalized form of fixed-point theorems in generalized Banach algebra relative to the weak topology with an application}
\maketitle
\par\vskip0.1cm
\author\centerline{{ Aref Jeribi $^{(1)(i)}$, Najib Kaddachi $^{(1)(ii)}$ and Zahra Laouar$^{(1)(i)}$}}
\begin{center} \emph{$^{(1)}$ Department of Mathematics. University of Sfax. Faculty of Sciences of Sfax. \\ Soukra Road Km 3.5 B.P. 1171, 3000, Sfax, Tunisia}.\\
e-mail : \ $^{(i)}$ Aref.Jeribi$@$fss.rnu.tn\ \ \ \ \ $^{(ii)}$ najibkadachi$@$gmail.com
\end{center}
\vskip0.2cm {\footnotesize \noindent \parbox{5.5 in}{{\bf\small Abstract.} In this paper, a general hybrid fixed point theorem for the contractive mappings in generalized Banach spaces is proved via measure of weak non-compactness and it is further applied to fractional integral equations for proving the existence results for the solutions under mixed Lipschitz and weakly sequentially continuous conditions. Finally, an example is given to illustrate the result.
{\small \sloppy{}}}
\noindent \parbox{5.5 in}{{\bf\small Keywords:} Banach space, weakly compact, Measure of weak non-compactness, Fixed point theory, Integral equations.
{\small \sloppy{ }}}
\noindent{\bf Mathematics Subject Classification}: \sloppy{46E15, 46E40, 47H10, 37C25, 47N20.} }
\section{Introduction} Many nonlinear problems involve the study of nonlinear equations of the form
\begin{eqnarray} \label{abc} x=Ax \cdot Bx+Cx, \ \ x\in S \end{eqnarray} where $S$ is a nonempty, closed, and convex subset of a Banach algebra $X,$ see for example \cite{1,2,3,4,5} and the references therein. These studies were mainly based on the Schauder fixed point theorem, and properties of the operators $A, B$ and $C$ (cf. completely continuous, k-set contractive, condensing, and the potential tool of the axiomatic measure of noncompactness,$\ldots).$ Various attempts have been made in the literature to extend this study to a weak topology, see for example \cite{sofiane,so,wc}. In recent years, the authors Ben Amar et al. have established in \cite{sofiane} some fixed point theorems for the operator equation $(\ref{abc})$ in Banach algebra under the weak topology, their results were based on the class of weak sequential continuity, weakly compact and weakly condensing conditions. In other direction, the classical Banach contraction principle was extended for contractive maps on spaces endowed with vector-valued metrics by Perov \cite{{perov1},{perov2}}. Schauder's fixed point theorem has been extended from Banach spaces to generalized Banach spaces by Viorel \cite{Viorel} and Krasnosel'skii's fixed point theorem has been extended by Petru and ouahab in \cite{{Ouahab},{Petru}}. More recently, the authors Nieton et al. \cite{algebra} have also established some new variants of fixed-point theorems for operator equation $(\ref{abc})$ in vector-valued metrics endowed with an internal composition law $(\cdot).$ These studies were mainly based on the convexity and the closure of the bounded domain and properties of the operators $A, B$ and $C.$ Their analysis was carried out via arguments of strong topology.\\
\par This paper is centered around the following question: under which conditions on its entries, the operator equation $(\ref{abc})$ acting on a generalized Banach algebras with respect to the weak topology, has a solution?\\
Our main results are applied to investigate the existence of solutions for the following coupled system of quadratic integral equations of fractional order \begin{equation}\label{Sy3} x_i(t)= f_i(t,x_1(t),x_2(t))\cdot \displaystyle\int_{0}^{t}\displaystyle\frac{(t-s)^{\alpha_i-1}}{\Gamma(\alpha_i)}g_i(s,x_1(s),x_2(s))ds+\displaystyle\sum _{k=1}^{m}I^{\beta_{i}^{k}}h_{i}^{k}(t,x_1(t),x_2(t)),\ \ i=1,2$$ \end{equation}
where $k \in \{1,...,m\},$ $ t,\alpha_{i}\in (0,1),$ $I^{\beta_{i}^{k}}$ is the fractional Pettis integral of order $\beta_{i}^{k}>0,$ $\Gamma(\cdot)$ is the Gamma function and the functions $f_i,g_i,h_i^k$ are given functions, whereas $x_1=x_1(t)$ and $x_2=x_2(t)$ are unknown functions.\\
\par The present paper is arranged as follows. The next section, we give some preliminary results for future use. Moreover, we shall extend the result of B. Amar, Jeribi and Mnif [\cite{compact}, Theorem $2.5$]. In addition, we establish the fixed point theorem by using the concept of the measure of weak non compactness in generalized Banach space $($see Theorem \ref{condensing}$)$. Note our result $($Theorem \ref{condensing}$)$ improves and generalizes Theorem $3.2$ in \cite{ther3.2}. Apart from that we introduce a class of generalized Banach algebras satisfying certain sequential conditions called here the condition $(\mathcal{GP})$ $($see Definition \ref{defi} $ )$. In section $3,$ we present a collection of new fixed point theorems in generalized Banach algebras satisfying condition $(\mathcal{GP})$. Our results extend and improve well-known results in \cite{sofiane,so,algebra}. In the last section of this manuscript, we apply Theorem $\ref{3}$ to discuss the existence of solutions for a system of fractional integral equations $(\ref{Sy3})$ and an example is given to explain the applicability of the results.
\section{ Preliminaries and Results }
\noindent Let $(X,\|\cdot\|)$ be a generalized Banach space in the sense of Perov such that the vector-valued norm $\|\cdot\|:X \longrightarrow \mathbb{R}^n_+$ is given by
$$\|x\|=\left( \begin{array}{ccc}
\|x\|_1 \\
\vdots \\
\|x\|_n \\ \end{array} \right),\ \ x \in X $$
with $\|\cdot\|_i, i=1,...,n$ define $n$ norm on $X.$ We denote by $\theta$ the zero element of $X$ and ${B}_r =B(\theta,r) $ the closed ball centered at $\theta$ with radius $r \in \mathbb{R}^{n}_{+}.$ For more details, the reader may consult the monograph of John R. Graef, Johnny Henderson and Abdelghani Ouahab \cite{monograph}.\\ Let $ (Y_i,\tau_i)_{i \in I} $ be a family of topological spaces and let $f_i : X \longrightarrow Y_i,\ \ i \in I$ be a linear and continuous mappings. We define the smallest topology on $X$ such that all the mappings $f_i$ remain continuous with respect to this topology. Its basic open sets are of the form $ \displaystyle\cap_{i \in J}f_i^{-1}u_i, $ with $J$ a finite subset of $I$ and $u_i \in \tau_i$ for each $i.$ This topology is called the weak topology on $X$ generated by the $(f_i)_{i \in I} $ and we denote it by $\sigma(X,(f_i)_{i \in I} ).$ So that a sequence $(x_n)_n$ in $X$ converges to $x$ in $\sigma(X, (f_i)_{i \in I} )$ if and only if $(f_i(x_n))_n$ converges to $f_i ( x ),$ for all $i \in I.$ In generalized Banach spaces, weakly open subset, weakly closed subset and weak compactness, are similar to those for usual Banach spaces. We denote by $\mathcal{B}(X)$ the collection of all nonempty bounded subsets of X and $\mathcal{W}(X)$ is a subset of $\mathcal{B}(X)$ consisting of all weakly compact subset of $X.$ If $x,y \in \mathbb{R}^{n},$ $x=(x_1,...,x_n),\ \ y=(y_1,...,y_n)$ by $x\leq y$ we means $x_i \leq y_i$ for all $i=1,...,n.$
\begin{Definition}
A square matrix of real numbers $M$ is said to be convergent to zero if and only if its spectral radius $\rho(M) $ is strictly less than $1.$ In other words, $|\lambda|<1,$ for every $\lambda \in \mathbb{C}, $ with $det(M-\lambda I)=0,$ where $I$ denotes the unit matrix of $\mathcal{M}_{n \times n}(\mathbb{R}_+).$
$\diamondsuit$ \end{Definition}
\begin{Lemma}\label{lem}\cite{matric} Let $M $ The following assertions are equivalent:\\ \noindent{$(i)$} $M$ is a matrix convergent to zero,\\ \noindent{$(ii)$} $M^k \rightarrow 0 $ as $k \rightarrow \infty,$\\ \noindent {$ (iii)$} The matrix $(I-M)$ is nonsingular and $$(I-M)^{-1}= I+M+M^{2}+...+M^k+...,$$\\ \noindent {$(iv)$} The matrix $(I-M)$ is nonsingular and $(I-M)^{-1}$ has nonnegative elements.
$\diamondsuit$ \end{Lemma}
\begin{Definition}\cite{rus} Let $X$ be a generalized Banach space. An operator $ T:X\longrightarrow X$ is said to be contractive if there exists a matrix $M \in \mathcal{M}_{n \times n}(\mathbb{R}_+)$ convergent to zero such that
$$\|Tx-Ty\|\leq M\|x-y\|,\ \ \text{ for all } x, y \\ \text { in } X. $$ For $n = 1$, we recover the classical Banach's contraction fixed point result.
$\diamondsuit$ \end{Definition}
\begin{Definition}
An operator $T:X\longrightarrow X$ is called weakly sequentially continuous on $X,$ if for every sequence $(x_n)_{n \in \mathbb{N}}$ with
$x_n \rightharpoonup x \ \ \text{ we have }\ \ Tx_n \rightharpoonup Tx, $ here $\rightharpoonup$ denotes weak convergence.
$\diamondsuit$
\end{Definition}
\begin{Definition}
An operator $T:X\longrightarrow X$ is said to be weakly compact if $T(V)$ is relatively weakly compact for every bounded subset $ V\subseteq X. $
$\diamondsuit$ \end{Definition}
\begin{Theorem}\cite{Viorel,sch2}\label{C1} Let $X$ be a generalized Banach space, $S$ be a nonempty, compact and convex subset of $X$ and $A: S\longrightarrow S$ be a continuous operator. Then $A$ has at least a fixed point in $S.$
$\diamondsuit$ \end{Theorem}
\noindent Now, we will prove the following theorem. \begin{Theorem} \label{1} Let $S$ be a nonempty, closed and convex subset of a generalized Banach space $X.$ Let $T : S \longrightarrow S$ be a weakly sequentially continuous mapping. If $T(S)$ is relatively weakly compact, then $T$ has a fixed point.
$\diamondsuit$ \end{Theorem} \begin{pf}Let $M=\overline{co}(T(S))$ the closed convex hull of $T(S).$ Because $T(S)$ is relatively weakly compact, then $M$ is weakly compact convex subset of $X.$ On the other hand, $T(M)\subset T(S) \subset \overline{co}(T(S)=M $ ie : $T$ maps $M$ into itself. Since $T$ is weakly sequentially continuous on $M,$ we infer that $T$ is weakly continue on $M.$ (consider $X=(X,\sigma(X,(f_i)_{i \in {I}}))$ the space endowed with the weak topology and note that $T:M \longrightarrow M $ is continuous with M is compact). The use of Theorem \ref{C1}
achieves the proof \end{pf}
\begin{Definition} the measure of weak non-compactness of the generalized Banach space $X$ is the map $\mu: \mathcal{B}{(X)}\longrightarrow \mathbb{R}_{+}^{n}$ defined in the following way $$ \mu(S)= \inf \big\{ r \in \mathbb{R}^{n}_{+},\ \ \text{ there exists } K\in \mathcal{W}(X)\text{ such that } S \subseteq K+ {B}_r \big \},$$
for all $S \in \mathcal{B}(X).$
$\diamondsuit$ \end{Definition}
\begin{Lemma} Let $S_1$, $S_2$ be two elements of $\mathcal{B}(X).$ Then the functional $\mu$ has the properties:\\
\noindent $(i)$ $S_1 \subseteq S_2$ implies $\mu(S_1)\leq \mu(S_2),$\\
\noindent $(ii)$ $\mu(\overline{S_1^w})=\mu(S_1),$\\
\noindent $(iii)$ $\mu(S_1)=0,$ if and only if, $ \overline{S_1^w} \in \mathcal{W}{(X)},$ where $\overline{S_1^w}$ is the weak closure of $S_1,$\\ \noindent $(iv)$ $\mu(co(S_1))=\mu(S_1),$ where $co(S_1)$ is the convex hull of $S_1,$ \\ \noindent $(v)$ $\mu(S_1+S_2)\leq \mu(S_1)+\mu(S_2), \mu(\{a\}+S_1)\leq \mu(S_1),$\\ \noindent $(vi)$ $\mu(S_1 \cup S_2)= \max \{\mu(S_1),\mu(S_2)\},$\\
\noindent $(vii)$ $\mu(\lambda S_1)= |\lambda|\mu(S_1),$ for all $\lambda \in \mathbb{R}.$\\
$\diamondsuit$
\end{Lemma} \begin{pf} The statements $(i)$ and $(iv)$ - $(viii)$ are immediate consequences of the definition of $\mu$ . Let us prove $(ii)$. From the definition of $\mu$ there exists a subset $K_1 \in \mathcal{W}(X)$ and $r_1 \in \mathbb{R}^{n}_+\setminus\{0_{\mathbb{R}^{n}}\}$ such that $S_1 \subseteq K_1+ {B}(\theta,r_1+\mu(S_1)),$ then $ S_1 \subseteq \overline{co}{K_1}+ {B}(\theta,r_1+\mu(S)).$ By the Kerein-\u{S}mulian theorem $\overline{co}{K_1}$ is weakly compact. Since $ {B}(\theta,r_1+\mu(S_1))$ is weakly closed, we infer that
$ \overline{co}{K_1}+ {B}(\theta,r_1+\mu(S_1))$ is weakly closed. So $ \overline{S}_1^w \subseteq \overline{co}{K_1}+ {B}(\theta,r_1+\mu(S_1))$ implies that
$\mu(\overline{S}_1^w )\leq r_1+\mu(S_1) .$ Letting $r_1 \to 0_{\mathbb{R}^{n}}$ in the above inequality, we get $\mu(\overline{S}_1^w )\leq \mu(S_1) .$ The reverse inequality is obvious .
The proof of the "only if" part of $(iii).$ By the definition of $\mu,$ there exists a subset $K_1 \in \mathcal{W}(X)$ and $r_1 \in \mathbb{R}^{n}_+\setminus\{0_{\mathbb{R}^{n}}\}$ such that $$\overline{S}_1^w \subseteq \overline{K}_1^w+{B}(\theta, r_1). $$ Let $\{x_n\}_{n\in \mathbb{N}}$ be a sequence on $\overline{S}_1^w.$ So there existe sebsequences $\{y_n\}_{n \in \mathbb{N}}$ and $\{z_n\}_{n \in \mathbb{N}}$ of $\overline{K}_1^w$ and ${B}(\theta, r_1)$ respectively, such that $x_n=y_n+z_n.$
If $r_1 \longrightarrow 0_{\mathbb{R}^{n}},$ we deduce that $\{x_n\}$ has a weakly convergence subsequence, So $\overline{S}_1^w$ is weakly compact, while the "if" part is trivial.\\
\ \ \ \
\end{pf}
\noindent For $n = 1$, we recover the classical De Blasi measure of weak non-compactness.
\begin{Definition} Let $S$ be a nonempty subset of a generalized Banach space $X$ and $M \in \mathcal{M}_{n \times n}(\mathbb{R}_+) $ is a matrix convergent to zero. If $T$ maps $S$ into $X,$ we say that \\ \noindent $(i)$ $T$ is $M$-set contraction with respect to $ \mu$ if $T$ is bounded and for any bounded subset $V \text{ in } S$ and $\mu(T(V))\leq M\mu(V), $\\ \noindent $(ii)$ $T$ is condensing with $\mu$ if $T$ is bounded and $\mu(T(V))<\mu(V)$ for all bounded subsets $V$ of $S$ with $\mu(V)>0_{\mathbb{R}^{n}}.$
$\diamondsuit$ \begin{remark} If we assume that $T$ is $M$-set contraction, then $T$ is condensing. Indeed, let $V$ be a bounded subset on $S$ with $\mu(V)>0_{\mathbb{R}^{n}}.$ We claim that $M\mu(V)<\mu(V).$ If not we obtain $(I-M)\mu(V)\leq 0_{\mathbb{R}^{n}}.$ Since $(I-M)^{-1}$ has nonnegative elements , it follows that $\mu(V)=0_{\mathbb{R}^{n}}.$ which is a contradiction with $\mu(V)>0_{\mathbb{R}^{n}}.$
$\diamondsuit$ \end{remark}
\noindent By using the concept of a measure of weak noncompactness in vector-valued Banach spaces, we obtain the following fixed point theorem. \begin{Theorem} \label{condensing} Let $S$ be a nonempty closed convex subset of a generalized Banach space $X.$ Let $T:S\longrightarrow S$ be a weakly seqeuntially continuous operator and condensing with respect to $\mu .$ If $T(S)$ is bounded, then $T$ has a fixed point in $S.$
$\diamondsuit$ \end{Theorem} \begin{pf} let $x_0$ be fixed in $S$ and define the set $$\Sigma= \{K:\ \ x_0 \in K \subseteq S,\ \ K \text{ is closed, convex, bonded and } T(K) \subseteq K \}. $$ Clearly, $\Sigma \neq \emptyset$ since $\overline{co}(T(S)\cup \{x_0\})\subseteq S$ and we have $$T(\overline{co}(T(S)\cup \{x_0\})\subseteq T(S) \subseteq \overline{co}(T(S)\cup \{x_0\}). $$
Which shows that $\overline{co}(T(S)\cup \{x_0\})\in \Sigma.$
If we consider $M=\displaystyle\cap_{K \in \Sigma }K,$ then $x_0 \in M \subseteq S, M$ is also a closed convex subset and $T(M) \subseteq M.$ Therefore, we have that $ M \in \Sigma.$ We will prove that $M$ is weakly compact. Denoting by $M_0=\overline{co}(T(M)\cup{x_0})$, we have $ M_0 \subseteq M $, which implies that $T(M_0)\subseteq T(M) \subseteq M_0$. Therefore $M_0 \in \Sigma $, $M \subseteq M_0 $. Hence $M=M_0.$ Since $M$ is weakly closed, it suffices to show that $M$ is relatively weakly compact. If $\mu(M)>0_{\mathbb{R}^{n}},$ we obtain $$\mu(M)=\mu(\overline{co}(T(M)\cup \{x_0\}))< \mu(M) $$ which is a contradiction. Hence, $\mu(M)=0_{\mathbb{R}^{n}}$ and so $M$ is relatively weakly compact. Now, $T$ is weakly continuous on $M.$ Consider $X=(X,\sigma(X,(f_i)_{i \in {I}}))$ the space endowed with the weak topology. Hence, an application of Theorem \ref{C1} shows that $T$ has at least one fixed point in $M.$ \end{pf}
\end{Definition}
\begin{Definition}\cite{algebra} A generalized normed algebra $X$ is an algebra endowed with a norm satisfying the following property
$$\text{ for all } x, y \in X \ \ \|x.y\|\leq \|x\|\|y\|,$$ where $$\|x.y\|=\left( \begin{array}{ccc}
\|x.y\|_1 \\
\vdots \\
\|x.y\|_n \\ \end{array}
\right)$$ and $$\|x\|\|y\|=\left( \begin{array}{ccc}
\|x\|_1\|y\|_1 \\
\vdots \\
\|x\|_n\|y\|_n \\ \end{array} \right).$$ A complete generalized normed algebra is called a generalized Banach algebra.
$\diamondsuit$
\end{Definition}
\noindent Because the product of two sequentially weakly continuous functions in generalized Banach algaebra is not necessarily sequentially weakly continuous, we will introduce: \begin{Definition}\label{defi} we will say that the generalized Banach algebra $X$ satisfies conditions $(\mathcal{G P})$\ \ if
\vskip0.2cm\hskip0.2cm $(\mathcal{G P})\left\{
\begin{array}{ll}
\mbox{~For any sequences ~}\{x_n\} \mbox{~and~} \{y_n\} \mbox{~of~} X \mbox{~such that~} x_n\rightharpoonup x \mbox{~and~} y_n \rightharpoonup y,&\hbox{}\\ \mbox{~then~} x_n \cdot y_n \rightharpoonup x\cdot y; \mbox{~here~} \rightharpoonup \mbox{~denotes weak convergence~}.&\hbox{} \end{array}
\right.$
$\diamondsuit$ \end{Definition} \noindent If $X$ is Banach algebra, we recover the classical sequential condition $(\mathcal{P})$ [\cite{so},Definition $3.1$]
\begin{Lemma}\label{lemma} If $K$ and $ K'$ are weakly compact subsets of generalized Banach algebra $X$ satisfying the condition $(\mathcal{G P} ),$ then $K \cdot K'$ is weakly compact.
$\diamondsuit$ \end{Lemma}
\begin{pf} We will show that $K\cdot K'$ is weakly sequentially compact. For that, let $\{z_n\}_n$ be any sequence of $K \cdot K '.$ So, there exist sequences $\{x_n\}_n$ and $\{x'_n\}_n$ of $K$ and $K'$ respectively.
By hypothesis, there is a renamed subsequence $\{x_n\}_n$ such that $ x_n \rightharpoonup x \in K.$ Again, there is a renamed subsequence $\{x'_n\}_n$ of $K'$ such that $ x'_n \rightharpoonup x' \in K.$ This, together with condition $(\mathcal{GP})$ yields that $$z_n\rightharpoonup z= x\cdot x'.$$ This implies that $K \cdot K'$ is weakly sequentially compact. Hence, an application of the Eberlein-\u{S}mulian's theorem yields that $K \cdot K' $ is weakly compact.
\end{pf}
\section{ Existence solutions of Equation (\ref{abc})}
\noindent In this section, we are prepared to state our first fixed point theorems in generalized Banach algebra in order to found the existence of solutions for the operator equation $( \ref{abc} )$ under weak topology in the case where $A,B$ and $C$ are weakly sequentially continuous operators.
\begin{Theorem} \label{3.1} Let $S$ be a nonempty, closed, and convex subset of a generalized Banach algebra $X$ satisfying the sequential condition $(\mathcal{GP}).$ Assume that $A, C:X \longrightarrow X$ and $B:S \longrightarrow X$ are three weakly sequentially continuous operators satisfying the following conditions:\\ \noindent $(i)$ $A$ and $C$ are contractive with Lipschitz matrix $M_A$ and $M_C$ respectively,\\ \noindent $(ii)$ $A$ is regular on $X,$ $($i.e. $A$ maps $X$ into the set of all invertible elements of $X),$\\ \noindent $(iii)$ $B(S)$ is bounded,\\ \noindent $(iv)$ $x=Ax\cdot By+Cx,\ \ y \in S \Rightarrow x \in S,$ and \\ \noindent $(v)$ $\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}$ is weakly compact.\\
Then, the operator equation $($\ref{abc}$)$ has, at least, a solution in $S$ as soon as $\|B(S)\|M_A+M_C \in \mathcal{M}_{n \times n}(\mathbb{R}_+) $ is a matrix converging to zero.
$\diamondsuit$
\begin{remark}
If $M_A= (a_{ij})_{1\leq i,j\leq n}, \ \ M_C= (\overline{a}_{ij})_{1\leq i,j\leq n} $ and $\|B(S)\|=(b_j)_{1\leq j \leq n}$ then
$$\|B(S)\|M_A+M_C=\left( \begin{array}{ccc}
b_1a_{11}+\overline{a}_{11} & \ldots & b_na_{1n}+ \overline{a}_{1n} \\
\vdots & \ddots & \vdots \\
b_1a_{n1}+\overline{a}_{n1} &\ldots & b_na_{nn}+\overline{a}_{nn}, \\ \end{array} \right).$$
$\diamondsuit$ \end{remark}
\end{Theorem}
\noindent \textbf{ Proof of Theorem \ref{3.1}. } It is easy to check that the vector $x\in S$ is a solution for the operator equation $x=Ax\cdot Bx+Cx,$ if and only if $x$ is a fixed point for the inverse operator $T :=\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}B.$ The use of assumption $(i)$ as well as perov's theorem \cite{banach gen} leads to for each $y\in S,$ there is a unique $x_y\in X$ such that $x_y=Ty.$ Indeed, let $y\in S$ be arbitrary and let us define the mapping $\varphi_y: X \longrightarrow X$ by the formula $$\varphi_y(x)=Ax \cdot By+Cx.$$
Notice that $\varphi_y$ is contractive with a Lipschitz matrix $\|B(S)\|M_A+M_C.$ Applying Perov's theorem \cite{banach gen}, we obtain that $\varphi_y$ has a unique fixed point in $S,$ say $x_y.$ From assumption $(ii),$ it follows that the operator $\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}$ is well defined on $B(S).$\\
Let $\{x_n, \ n\in \mathbb{N} \}$ be a weakly convergent sequence of $S$ to a point $x\in S$ and let $y_n=Tx_n.$ Then the relation $y_n=Ay_n\cdot Bx_n+Cy_n$ holds and, therefore $\{y_n, \ n\in \mathbb{N} \}\subset S$ in view of assumption $(iv).$
Since $T(S)$ is relatively weakly compact, there is a subsequence $(x_{n_{k}})$ of $\{x_n, \ n\in \mathbb{N} \}$ such that $y_{n_{k}}=Tx_{n_{k}}\rightharpoonup y,$ { for some } $y\in X.$ The weak sequential closedness of $S$ gives $y\in S.$ Making use of the condition $(\mathcal{GP}),$ together with the assumptions on $A,B$ and $C,$ enables us to have $$y_{n_{k}}=Ay_{n_{k}}\cdot Bx_{n_{k}}+Cy_{n_{k}}\rightharpoonup Ay\cdot Bx+Cy.$$ This implies that $y=Ay\cdot Bx+Cy.$ Consequently $Tx_{n_{k}}\rightharpoonup Tx$ in light of assumption $(ii).$
\noindent Now a standard argument shows that $Tx_n\rightharpoonup Tx.$
Suppose the contrary, then there exists a weakly neighborhood $V^{w}$ of $Tx$ and a subsequence $(x_{n_j})_j$ of $\{x_{n}, \ n\in \mathbb{N} \}$ such that $Tx_{n_{j}} \notin V^{w},$ for all $j\geq 1.$ Arguing as before, we may extract a subsequence $(x_{n_{j_{k}}})_{k \in \mathbb{N}}$ of $\{x_{n_j}, \ j\in \mathbb{N} \}$ verifying $Tx_{n_{j_{k}}}\rightharpoonup y,$
which is a contradiction and consequently $T$ is weakly sequentially continuous. Hence, $T$ has, at least, one fixed point $x$ in $S$ in view of Theorem \ref{1}.
{Q.E.D.}
\noindent An interesting fixed point result of Theorem \ref{3.1} is
\begin{Corollary} Let $S$ be a nonempty, closed, and convex subset of a generalized Banach algebra $X$ satisfying the sequential condition $(\mathcal{GP}).$ Assume that $A, C:X \longrightarrow X$ and $B:S \longrightarrow X$ are three weakly sequentially continuous operators satisfying the following conditions:\\ \noindent $(i)$ $A$ and $C$ are contractive with Lipschitz matrix $M_A$ and $M_C$ respectively,\\ \noindent $(ii)$ $A$ is regular on $X,$\\
\noindent $(iii)$ $A(S),B(S)$ and $C(S)$ are relatively weakly compact,\\ \noindent $(iv)$ $x=Ax{\cdot}By+Cx,\ \ y \in S \Rightarrow x \in S. $\\
Then, the operator equation $($\ref{abc}$)$ has, at least, a solution in $S$ as soon as $\|B(S)\|M_A+M_C $ is a matrix converging to zero.
$\diamondsuit$
\end{Corollary}
\begin{Theorem}\label{3} Let $S$ be a nonempty, bounded, closed, and convex subset of a generalized Banach algebra $X$ satisfying the sequential condition $(\mathcal{GP}).$ Assume that $A, C:X \longrightarrow X$ and $B:S \longrightarrow X$ are three weakly sequentially continuous operators satisfying the following conditions:\\ \noindent $(i)$ $A$ and $C$ are contractive with Lipschitz matrix $M_A$ and $M_C$ respectively,\\ \noindent $(ii)$ $A$ is regular on $X$,\\ \noindent $(iii)$ $B$ is weakly compact, \\ \noindent $(iv)$ $x=Ax\cdot By+Cx,\ \ y \in S \Rightarrow x \in S. $\\
Then, the operator equation $($\ref{abc}$)$ has, at least, a solution in $S$ as soon as $\|B(S)\|M_A+M_C $ is a matrix converging to zero.
$\diamondsuit$
\end{Theorem}
\noindent Before proving the theorem, we need to establish two lemmas.
\begin{Lemma} \label{lemma3.2} Let $X$ be a generalized Banach algebra. If $S\in \mathcal{B}(X)$ and $K\in \mathcal{W}(X),$ then
$\mu \left(S \cdot K\right)\leq \|K\|\mu (S).$
$\diamondsuit$ \end{Lemma}
\begin{pf} Assume that $\|K\|>0_{\mathbb{R}^{n}}.$
By the definition of $\mu,$ there exists a subset $K'$ of $\mathcal{W}(X)$ and $ r \in {\mathbb{R}^{n}_+}\setminus \{0_{\mathbb{R}^{n}}\}$ such that
$$S\cdot K\subseteq K'\cdot K+{B}\left(\theta,\mu(S)+r \|K^{-1}\|\right)\cdot K, $$
where
$$ r \|K ^{-1}\|:=\left( \begin{array}{ccc}
\dfrac{r_1 }{\|K\|_1} \\
\vdots \\
\dfrac{r_ n}{\|K\|}_n \\ \end{array} \right).$$ So
$$S\cdot K\subseteq K'\cdot K+{B}\left(\theta, \| K\|\mu(S)+r \right).$$
Keeping in mind the subadditivity of the measure of weak noncompactness and using lemma \ref{lemma}, we get
$$\begin{array}{rcl}\displaystyle\mu(S\cdot K)&\leq&\displaystyle\mu\left(K\cdot K'\right) +\mu\left({B}\left(0,\| K\|\mu(S)+r \right) \right)\\\\&\leq& \|K\|\mu(S)+r.\end{array}$$
Since $\varepsilon$ is arbitrary, we deduce that $$\mu(S\cdot K)\leq \|K\|\mu(S).$$
\end{pf}
\begin{Lemma}\label{prop}Let $X$ be a generalized Banach algebra. Assume that
$T:X \longrightarrow X$ is weakly sequentially continuous. If $T$ is contractive with Lipschitz matrix $M \in \mathcal{M}_{n \times n}(\mathbb{R}_+),$ then {for any bounded subset } $S$ of $X,$ one has
$$\mu\left(T(S)\right)\leq M\mu\left(S\right).$$
$\diamondsuit$ \end{Lemma} \begin{pf}Let $S$ be a bounded subset of $X$ and $\varepsilon \in \mathbb{R}^{n}$ such that $\varepsilon> \mu(S)$ and let $M=(a_{ij})_{1\leq i,j \leq n}.$ By the definition of $\mu,$ we have
there exists $r\in \mathbb{R}^{n}$ and a weakly compact subset $K$ of $X$ such that $0<r<\varepsilon$ and $S \subseteq K+{B}(\theta,{r}).$ Let $y\in T\left(K+{B}(\theta,{r})\right),$ then there exists $x\in K+{B}(\theta,{r})$ such that $y=Tx.$ Since $x\in K+{B}(\theta,{r}),$ there are $k \in K$ and $b\in {B}\left(\theta,{r})\right)$ such that $x=k+b,$ and so
$$\|y-Tk\|_i=\|Tx-Tk\|_i\leq \sum_{j=1}^{n} a_{ij}\|x-k\|_j=\sum_{j=1}^{n} a_{ij}\|b\|_j\leq \sum_{j=1}^{n} a_{ij}r_j.$$ This means that
$$\|y-Tk\|\leq \left(
\begin{array}{c}
\displaystyle\sum_{j=1}^{n} a_{1j}r_j\\
\displaystyle\sum_{j=1}^{n} a_{2j}r_j\\
\vdots \\
\displaystyle\sum_{j=1}^{n} a_{nj}r_j\\
\end{array}
\right) =Mr.$$ That is, $y\in TK+{B}(\theta,{Mr})$ and consequently $TS\subset TK+{B}(\theta,{Mr}).$ Moreover, since $T$ is sequentially weakly continuous we have $\overline{TK}^{w}\in \mathcal{W}(X).$ Accordingly, $$\mu(TS)\leq Mr.$$ Letting $\varepsilon \to \mu(S)$ in the above inequality, we get $\mu(TS)\leq M\mu(S).$ \end{pf}
\noindent \textbf{ Proof of Theorem \ref{3}.} Following the same procedures as in the proof of Theorem \ref{3.1}, it can be proved that the inverse operator $T:=\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}B$ exists on $S.$ Now, we claim that $T(S)$ is a relatively compact subset of $X.$ If this is not the case, then $\mu(TS)>0_{\mathbb{R}^{n}}.$ Keeping in mind the subadditivity of the De Blasi's measure of weak noncompactness and using the equality: \begin{eqnarray}\label{dec}
\displaystyle
T=AT\cdot B+CT
\end{eqnarray} we obtain
\begin{align*} \mu(T(S)) &\leq \mu \left(AT(S)\cdot B(S)\right)+\mu \left(CT(S)\right). \end{align*} The use of Lemma \ref{lemma3.2} as well as Lemma \ref{prop} leads to
\begin{align*}
\mu(TS)&\leq \left( \|B(S)\|M_A+M_C\right) \mu(TS). \end{align*}
Since $\|B(S)\|M_A+M_C $ is a matrix converging to zero, we get a contradiction and consequently the claim is approved.
{Q.E.D.}
\noindent As easy consequences of Theorem \ref{3} we obtain the following result.
\begin{Corollary}
Let $S$ be a nonempty, bounded, closed and convex subset of a generalized Banach algebra $X$ satisfying the sequential condition $(\mathcal{GP}).$ Assume that $A,C:X\longrightarrow X$ and $B:S \longrightarrow X$ are three weakly sequentially continuous operators satisfying the following conditions:\\ \noindent $(i)$ $A$ is regular and is contractive with a Lipschitz matrix $M,$\\ \noindent $(ii)$ $B$ and $C$ are weakly compact, \\ \noindent $(iii)$ $\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}$ exists on $B(S),$\\ \noindent $(iv)$ $x=Ax\cdot By+Cx,\ \ y \in S \Rightarrow x \in S. $\\
Then, the operator equation $($\ref{abc}$)$ has, at least, one solution in $S$ as soon as $\|B(S)\|M$ is a matrix converging to zero.
$\diamondsuit$ \end{Corollary}
\noindent In the following result, we will consider that $A$ and $B$ are weakly compact operators.
\begin{Theorem}\label{rem} Let $S$ be a nonempty, bounded, closed and convex subset of a generalized Banach algebra $X$ satisfying the condition $(\mathcal{GP}).$ Assume that $A, B, C:S\longrightarrow X$ are three weakly sequentially continuous operators satisfying the following conditions:\\ \noindent{$(i)$} $A$ is regular,\\ \noindent{$(ii)$} $A$ and $B$ are weakly compact,\\ \noindent {$(iii)$} If $(I-C)x_n\rightharpoonup y,$ then there exists a weakly convergent subsequence of $(x_n)_n,$\\ \noindent{$(iv)$} $\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}$ exists on $B(S)$, and\\ \noindent{$(v)$} $x=Ax\cdot By+Cx \in S,\ \ y\in S$ $\Rightarrow$ $x \in S.$\\
Then, the operator equation $($\ref{abc}$)$ has, at least, one solution in $S.$
$\diamondsuit$ \end{Theorem} \begin{pf}
Let $\{y_n, \ n\in \mathbb{N} \}\subset T(S),$ there is a sequence $\{x_n, \ n\in \mathbb{N} \}\subset S$ such that $$y_n=Tx_n=\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}Bx_n.$$ Or equivalently, $y_n=Ay_n\cdot Bx_n+Cy_n.$ Taking into account the weak compactness of the weak closure of $A(S)$ and $B(S),$ we infer that
$$ATx_{n_{k}}\rightharpoonup x ~~~~~~~~~~~~~~~~\textrm{ and }~~~~~~~~~~~~~~~~ Bx_{n_{k_{j}}}\rightharpoonup x', \textrm{ for some } x,x'\in S,$$
where $(x_{n_{k}})$ is a subsequence of $\{x_n, \ n\in \mathbb{N} \}$ and $(x_{n_{k_{j}}})$ is a subsequence of $\{x_{n_{k}}, \ k>n \}.$ Using the condition $(\mathcal{GP}),$ we get
$$Ay_{n_{k_{j}}}\cdot Bx_{n_{k_{j}}}=(I-C)y_{n_{k_{j}}}\rightharpoonup x\cdot x' \ \ \text{ in } X.$$ Based on assumption $(iii),$ it follows that there exists a weakly convergent subsequence of $(y_{n_{k_{j}}})$ and consequently $T(S)$ is a relatively weakly compact subset of $X.$ The use of Theorem \ref{3.1} achieves the proof. \end{pf}
\begin{remark}If we assume that $C$ is sequentially weakly continuous and contractive with a Lipschitz Matrix $M,$ then $C$ satisfies the condition $(iii)$ of Theorem \ref{rem}. In fact, let $\{x_n, \ n\in \mathbb{N} \}$ be a sequence in $S$ such that $(I-C)x_n\rightharpoonup y,$ for some $y \in X.$ Based on the subadditivity of the De Blasi's measure of weak non-compactness it is shown that $$\beta(\{x_n, \ n\in \mathbb{N} \})\leq \beta(\{(I-C)x_n, \ n\in \mathbb{N} \})+\beta(\{Cx_n, \ n\in \mathbb{N} \}). $$ If we consider the weak compactness of the weak closure of $\{(I-C)x_n, \ n\in \mathbb{N} \}$ and deploy Lemma \ref{prop}, we get $$\beta(\{x_n, \ n\in \mathbb{N} \})\leq M\beta(\{x_n, \ n\in \mathbb{N} \}).$$ Since $M$ is a matrix converging to zero, then there is a weakly convergent subsequence of $\{x_n, \ n\in \mathbb{N} \}.$
$\diamondsuit$
\end{remark}
\noindent A consequence of Theorem \ref{rem} is
\begin{Corollary}Let $S$ be a nonempty, bounded, closed and convex subset of a generalized Banach algebra $X$ satisfying the condition $(\mathcal{GP}).$ Assume that $A,B,C:S\longrightarrow X$ are three weakly sequentially continuous operators satisfying the following conditions:\\ \noindent{$(i)$} $A$ is regular and $B$ is weakly compact,\\ \noindent {$(ii)$} If $\left(\displaystyle\dfrac{I-C}{A}\right) x_n\rightharpoonup y,$ then there exists a weakly convergent subsequence of $(x_n)_n,$\\ \noindent{$(iii)$} $\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}$ exists and $\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}B(S)\subseteq S.$\\
Then, the operator equation $($\ref{abc}$)$ has, at least, one solution in $S.$
$\diamondsuit$ \end{Corollary}
\noindent Let us study the case where the operators $B$ and $\left(I-\displaystyle\dfrac{I-C}{A}\right)$ are $M_1$-$\mu$-contraction and $M_2$-$\mu$-contraction respectively.
\begin{Theorem}\label{mesure} Let $S$ be a nonempty, bounded, closed and convex subset of a generalized Banach algebra $X.$ Assume that $A, C:X\longrightarrow X$ and $B:S\longrightarrow X$ are three operators satisfying the following conditions:\\ \noindent{$(i)$} $A$ is regular,\\ \noindent{$(ii)$} $B$ and $\left(I-\displaystyle\dfrac{I-C}{A}\right)$ are $M_1$-$\mu$-contraction and $M_2$-$\mu$-contraction respectively, \\
\noindent{$(iii)$} $B$ and $\left(\displaystyle\dfrac{I-C}{A}\right)$ are weakly sequentially continuous,\\
\noindent{$(iv)$} $\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}$ exists on $B(S),$ and\\ \noindent{$(v)$} $x=Ax\cdot By+Cx,\ \ y\in S$ $\Rightarrow$ $x \in S.$\\
Then, the operator equation $($\ref{abc}$)$ has, at least, one solution in $S$ as soon as $(I-M_2)^{-1}M_1 $ is a matrix converging to zero.
$\diamondsuit$ \end{Theorem} \begin{pf} It is easy to see that the operator $T:=\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}B:S\longrightarrow S$ is well defined. Let $\{x_n,\ \ n \in \mathbb{N}\}$ be a weakly convergent sequence of $S$ to a point $x\in S.$ Keeping in mind the weak sequential continuity of the operator $B$ and using the equality \begin{eqnarray}\label{mu} T=B+\left(I- \dfrac{I-C}{A}\right)T, \end{eqnarray}
we obtain $$\begin{array}{rcl}\mu \left(\{Tx_n, \ \ n \in \mathbb{N}\}\right)&\leq&\displaystyle\mu\left(Bx_n, \ \ n \in \mathbb{N}\right) +\mu\left( \left(I-\dfrac{I-C}{A}\right)(\{Tx_n,\ \ n \in \mathbb{N}\})\right)\\\\&\leq& M_2 \mu \left(\{Tx_n, \ \ n \in \mathbb{N}\}\right). \end{array}$$ This inequality means, in particular, that $\{Tx_n, \ \ n \in \mathbb{N}\}$ is relatively weakly compact. Consequently, there is a subsequence $(x_{n_k})_k$ of $\{Tx_n, \ \ n \in \mathbb{N}\}$ such that $Tx_{n_k}\rightharpoonup y,$ fore some $y \in S.$ Making use of equality $(\ref{mu}),$ together with the assumptions on $B$ and $\left(\frac{I-C}{A}\right)$; enables us to have $y = Tx.$
Now a standard argument shows that $Tx_n\rightharpoonup Tx.$
Suppose the contrary, then there exists a weakly neighborhood $V^{w}$ of $Tx$ and a subsequence $(x_{n_j})_j$ of $\{x_{n}, \ n\in \mathbb{N} \}$ such that $Tx_{n_{j}} \notin V^{w},$ for all $j\geq 1.$ Arguing as before, we may extract a subsequence $(x_{n_{j_{k}}})_{k \in \mathbb{N}}$ of $\{x_{n_j}, \ j\in \mathbb{N} \}$ verifying $Tx_{n_{j_{k}}}\rightharpoonup y,$ which is a contradiction and consequently $T$ is sequentially weakly continuous. Next, $T$ is $\mu$-condensing. In fact, let $V$ be a bounded subset of $S$ with $\mu(V) > 0.$ Using the subadditivity of the De Blasi's measure of weak noncompactness, we get $$\begin{array}{rcl} \mu\left(T(V) \right) &\leq& \mu \left(B(V)\right)+ \mu \left(\left(I-\dfrac{I-C}{A}\right)T(V) \right) \\\\&\leq& M_1\mu(V)+M_2\mu(T(V)). \end{array}$$ This implies that $$\mu\left(T(V) \right)\leq (I-M_2)^{-1}M_1\mu\left(V\right). $$ Hence, $T$ has, at least, one fixed point $x$ in $S$ in view of Theorem \ref{condensing} . \end{pf}
\noindent If we take $A = 1_X$ in the above result, where $1_X$ is the unit element of the generalized Banach algebra $X$, we obtain the following Corollary.
\begin{Corollary} Let $S$ be a nonempty, bounded, closed and convex subset of a generalized Banach algebra $X.$ Assume that $ C:X\longrightarrow X$ and $B:S\longrightarrow X$ are two sequentially weakly continuous operators satisfying the following conditions:\\ \noindent{$(i)$} $B$ and $C $ are $M_1$-$\mu$-contraction and $M_2$-$\mu$-contraction respectively,\\
\noindent{$(ii)$} $\left(I-C \right)^{-1}$ exists on $B(S),$\\ \noindent{$(iii)$} $x= By+Cx,\ \ y\in S$ $\Rightarrow$ $x \in S.$\\
Then, the operator equation $($\ref{abc}$)$ has, at least, one solution in $S$ as soon as $(I-M_2)^{-1}M_1 $ is a matrix converging to zero.
$\diamondsuit$ \end{Corollary} \begin{remark} Note that condition $(iii)$ in Theorem $\ref{mesure}$ may be replaced by "$A,B$ and $C$ are weakly sequentially continuous", but the generalized Banach algebra must satisfy condition $(\mathcal {GP}).$ Now, we can study the folllwing result.
$\diamondsuit$ \end{remark} \begin{Theorem} Let $S$ be a nonempty, bounded, closed, and convex subset of a generalized Banach algebra $X$ satisfying the sequential condition $(\mathcal{GP}).$ Assume that $A, C:X \longrightarrow X$ and $B:S \longrightarrow X$ are three weakly sequentially continuous operators satisfying the following conditions:\\ \noindent{$(i)$} $A$ is regular and weakly compact,\\ \noindent{$(ii)$} $B$ and $C $ are $M_1$-$\mu$-contraction and $M_2$-$\mu$-contraction respectively, \\
\noindent{$(iii)$} $\left(\displaystyle\dfrac{I-C}{A}\right)^{-1}$ exists on $B(S),$ and\\ \noindent{$(iv)$} $x=Ax\cdot By+Cx,\ \ y\in S$ $\Rightarrow$ $x \in S.$\\
Then, the operator equation $($\ref{abc}$)$ has, at least, one solution in $S$ as soon as $(I-M_2)^{-1}\|A(S)\|M_1 $ is a matrix converging to zero.
$\diamondsuit$
\end{Theorem}
\section{Integral Equations of Fractional Order}
Let $(X,\|\cdot\|)$ be a reflexive Banach and let ${C}(J, {X})$ be the Banach algebra of all
$X$-valued continuous functions defined on $J=[0,1]$, endowed with the norm $\|f\|_\infty = \displaystyle\sup_{t\in J}\|f(t)\|.$ We will use Theorem \ref{3} to examine the existence of solutions to the coupled system of quadratic integral equations of fractional order $(\ref{Sy3}).$
We need the following definition and proposition in the sequel. \begin{Definition} \cite{Pettis} Let $f:[0,T]\rightarrow X $ be a function. The fractional Pettis integral of the function $f$ of order $ \alpha \in \mathbb{R_+}$ is defined by $$ I^{\alpha}f(t)=\int_{0}^{t}\frac{(t-s)^{\alpha -1}}{\Gamma(\alpha)}f(s)ds, $$ where the sign $ "\int " $ denotes the Pettis integral. \end{Definition} \begin{Proposition}\cite{Pettis} If $f:[0,T]\rightarrow X $ is Riemann integrable on $[0,T],$ then $I^{\alpha}f$ exists on $[0,T]$ and fractional Pettis integral.
\end{Proposition}
\noindent Let us now introduce the following assumptions:\\ \noindent{($\mathcal{H}_0$)} The function $f_i:J\times X \times X \longrightarrow X, i=1,2 $ is such that:\\ \hspace*{20pt} {$($a$)$} The partial function $x\longrightarrow f_1(t,x,y)$ is regular on $X$ \\ \hspace*{20pt} {$($b$)$} The partial function $y\longrightarrow f_2(t,x,y)$ is regular on $X$ \\
\hspace*{20pt} {$($c$)$} The partial function $t \mapsto f_i(t,x,y) $ is continuous,\\ \hspace*{20pt} {$($d$)$} The partial function $(x,y) \mapsto f_i(t,x,y)$ is weakly sequentially continuous,\\ \hspace*{20pt} {$($e$)$} There is nonnegative real numbers $ a_{i1}$ and $a_{i2} ,$ $i=1,2$ such that \\
$$\|f_i(t,x,y)-f_i(t,\tilde {x},\tilde{y})\|\leq a_{i1}\|x-\tilde{x}\|+a_{i2}\|y-\tilde{y}\|.$$ \noindent{($\mathcal{H}_1$)} The function $g_i:J\times X \times X \longrightarrow X, i=1,2 $ is such that:\\ \hspace*{20pt} {$($a$)$}The partial function $t \mapsto g_i(t,x,y)$ is Riemann integrable,\\ \hspace*{20pt} {$($b$)$} The partial function $(x,y) \mapsto g_i(t,x,y)$ is weakly sequentially continuous.\\
\noindent{($\mathcal{H}_2$)} The function $h_{i}^{k}:J \times X \times X \longrightarrow X,$ $k= {1,\ldots,m}$ is such that:\\ \hspace*{20pt} {$($a$)$}The partial function $t \mapsto h_i^k(t,x,y)$ is Riemann integrable,\\ \hspace*{20pt} {$($b$)$} The partial function $(x,y) \mapsto h_i^k(t,x,y)$ is weakly sequentially continuous,\\ \hspace*{20pt} {$($c$)$} There is nonnegative real numbers ${b}_{i1}^{k}$ and ${b}_{i2}^{k}$ such that \\
$$\|h_i^k(t,x,y)-h_i^k(t,\tilde {x},\tilde{y})\|\leq {b}_{i1}^{k}\|x-\tilde{x}\|+{b}_{i2}^{k}\|y-\tilde{y}\|.$$
\begin{Theorem} \label{mple} Suppose that the assumptions $(\mathcal{H}_{0})-(\mathcal{H}_3)$ are satisfied. Moreover, assume that there exists a real number $r_0> 0$ and $P \in \mathbb{R}^{\ast}_+$ such that \begin{eqnarray}\label{app}
\displaystyle \left\{
\begin{array}{lll}
\|g_i(t, s, x, y)\|\leq P \ \ \text{ with } \|x\|\leq r_0 \text{ and }\|y\|\leq r_0 \\\\
\rho\big[ P\big(\frac{T^{\alpha_1}}{\Gamma(\alpha_1+1)}+\frac{T^{\alpha_2}}{\Gamma(\alpha_2+1)} \big)+\sum_{k=1}^{m} \big(\frac{T^{\beta_1^k}}{\Gamma(\beta_1^k+1)}+\frac{T^{\beta_2^k}}{\Gamma(\beta_2^k+1)}\big)\big]< 1, \textrm{ and } \\\\
M_A, M_{C} \textrm{ and } \|B(S)\|M_A+M_C \textrm{ are three matrices converging to zero, where }
\end{array}
\right. \end{eqnarray}
$$\rho=\max\{ a_{11},a_{12},a_{21},{a_{22}},{b}_{11}^{k},{b}_{12}^k,{b}_{21}^K,{b}_{21}^k,{b}_{22}^k ;\ \ k=1,\ldots,m \},$$ $$M_A=\left( \begin{array}{cc}
a_{11} & a_{12} \\
a_{21} & a_{22} \\ \end{array} \right) , \ \ M_{C}=\left( \begin{array}{cc}
\sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}b{_{11}^{k} } & \sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}b{_{12}^{k} } \\
\sum_{k=1}^{m}\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}b{_{21}^{k} } & \sum_{k=1}^{m}\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}b{_{22}^{k} } \\ \end{array}
\right) \texttt{ and }$$ $$\ \ \|B(S)\|M_A+M_C= \left(
\begin{array}{cc}
\dfrac{PT^{\alpha_1}}{\Gamma(\alpha_1+1)}a_{11}+\sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}b{_{11}^{k} } & \dfrac{PT^{\alpha_2}}{\Gamma(\alpha_2+1)}a_{12}+\sum_{k=1}^{m}\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}b{_{12}^{k} } \\
\dfrac{PT^{\alpha_1}}{\Gamma(\alpha_1+1)}a_{21}+\sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}b{_{21}^{k} } & \dfrac{PT^{\alpha_2}}{\Gamma(\alpha_2+1)}a_{22}+\sum_{k=1}^{m}\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}b{_{22}^{k}} \\
\end{array}
\right).$$
Then the problem $(\ref{Sy3})$ has a solution in $C(J,X)\times C(J,X).$
$\diamondsuit$
\end{Theorem}
\begin{pf} We recall that the problem $(\ref{Sy3} $) is equivalent to the operator equation $$(x,y)=(A_1(x,y),A_2(x,y))(B_1(x,y),B_2(x,y))+(C_1(x,y),C_2(x,y)),$$ where the operators $A_i,B_i$ and $C_i, \ \ i=1,2$ are defined by $$\left\{
\begin{array}{ll}
A_i(x,y)(t)=f_i(t,x(t),y(t))\\\\
B_i(x,y)(t)= \int_{0}^{t}\frac{(t-s)^{\alpha_1-1}}{\Gamma(\alpha_1)}g_i(s,x(s),y(s))ds\\\\
C_i(x,y)(t)=\sum _{k=1}^{m}I^{\beta_{1}^{i}}h_{i}^{k}(t,x(t),y(t)).
\end{array} \right.$$
\noindent Let us define the subset $S$ of $C(J,X)\times C(J,X)$ by
$$S=\left\{(x,y)\in C(J,X)\times C(J,X), \ \ \|(x,y)\| = \left( \begin{array}{cc}
\|x\|_\infty \\
\|y\|_\infty \\ \end{array} \right) \leq \left( \begin{array}{cc}
r_0 \\
r_0\\ \end{array} \right) \right\},$$ where $$ r_0\geq\frac{F_0P\big(\frac{T^{\alpha_1}}{\Gamma(\alpha_1+1)}+\frac{T^{\alpha_2}}{\Gamma(\alpha_2+1)}\big)+H_0\sum_{k=1}^{m}\big(\frac{T^{\beta_1^k}}{\Gamma(\beta_1^k+1)}+\frac{T^{\beta_2^k}}{\Gamma(\beta_2^k+1)}\big)
}{1-\rho\big[P \big(\frac{T^{\alpha_1}}{\Gamma(\alpha_1+1)}+\frac{T^{\alpha_2}}{\Gamma(\alpha_2+1)} \big)+\sum_{k=1}^{m} \big(\frac{T^{\beta_1^k}}{\Gamma(\beta_1^k+1)}+\frac{T^{\beta_2^k}}{\Gamma(\beta_2^k+1)}\big)\big]} , $$
with $$\left\{
\begin{array}{ll}
F_0=\max\ \big\{ \|f_1(\cdot,0,0)\|,\|f_2(\cdot,0,0)\| \big\} \\
H_0= \max\ \big\{ \|h_1^k(\cdot,0,0)\|,\|h_2^k(\cdot,0,0)\|;\ \ k=1,\ldots,m \big\}\\
\end{array}
\right. $$ Our strategy is to apply Theorem \ref{3} to prove the existence of a fixed point for the nonlinear equation $(\ref{Sy3})$ in $S.$ Then, we need to verify the following steps:\\ \textbf{Claim 1:} We start by showing that the operators $A,C: C(J,X) \times C(J,X)\longrightarrow C(J,X) \times C(J,X)$ and $B: S\longrightarrow C(J,X) \times C(J,X)$ are weakly sequentially continuous. Firstly, we verify that the operator $A_i(x,y), \text{ for } i=1,2$ is continuous on $J$ for all $(x,y) \in C(J,X) \times C(J,X).$ To see this, let $\{t_n, \ \ n \in \mathbb{N}\}$ be any sequence in J converging to a point $t$ in $J.$ Then, $$\begin{array}{rcl}
\|A_i(x,y)(t_n)-A_i(x,y)(t)\| &=& \|f_i(t_n,x(t_n),y(t_n))-f_i(t,x(t),y(t))\|
\\\\&\leq & a_{i1}\|x(t_n)-x(t)\|+a_{i2}\|y(t_n)-y(t)\| \\\\&+& \|f_i(t_n,x(t),y(t)-f_i(t,x(t),y(t)\|. \end{array}$$ The continuity of $x,y$ and $t\mapsto f_i(t,x,y)$ on $[0,1]$
implies that the function $A_i(x,y)$ is continuous.
Let $\{(x_n,y_n), n\in \mathbb{N}\}$ be a weakly convergent sequence of $C(J,X) \times C(J,X)$ to a point $(x,y).$ In this case, the set $\{(x_n,y_n), n\in \mathbb{N}\}$
is bounded and so, we can apply the Dobrakov's theorem \cite {Dob} in order to get $$(x_n(t),y_n(t))\rightharpoonup (x(t),y(t)) \text{ in } X\times X.$$ Based on assumption $(\mathcal{H}_0)(d),$ it is shown that $A_i(x_n,y_n)(t)\rightharpoonup A_i(x,y)(t)$ and then, we can again apply the Dobrakov's theorem to obtain the weak sequential continuity of the operator $A.$ Besides, the use of assumption $(\mathcal{H}_{1})(b) $ and assumption $(\mathcal{H}_2)(b)$ as well as the Dobrakov's theorem \cite[page 36]{Dob} leads to the two maps $B$ and $C$ are weakly sequentially continuous.\\
\textbf{Claim 2:} The operators $A$ and $C$ are contractive. The claim regarding the operator $A$ is immediate, from assumption $(\mathcal{H}_0)(e).$ Let us fix arbitrary $(x,y),(\tilde{x},\tilde{y}) \in C(J,X) \times C(J,X).$ If we take an arbitrary $t\in[0, 1],$ then we get
$$\begin{array}{rcl} \|C_i(x,y)(t)-C_i(\tilde{x},\tilde{y})(t)\|&=& \left\|\displaystyle\sum_{k=1}^{m}I^{\beta_{i}^{k}}h_i^k(t,x(t),y(t)-\displaystyle\sum_{k=1}^{m}I^{\beta_{i}^{k}}h_i^k(t,\tilde{x}(t),\tilde{y}(t))\right\| \\\\&\leq& \displaystyle\sum_{k=1}^{m}I^{\beta_{i}^{k}}\|h_i^k(t,x(t),y(t)-h_i^k(t,\tilde{x}(t),\tilde{y}(t))\| \\\\&\leq& \displaystyle\sum_{k=1}^{m}I^{\beta_{i}^{k}}\left(b_{i1}^{k}\|x(t)-\tilde{x}(t)\| + b_{i1}^{k}\|y(t)-\tilde{y}(t)\| \right) \\\\&\leq& \displaystyle\sum_{k=1}^{m}\frac{T^{\beta_{i}^{k}}}{\Gamma(\beta_{i}^{k}+1)}\left(b_{i1}^{k}\|x(t)-\tilde{x}(t)\| + b_{i1}^{k}\|y(t)-\tilde{y}(t)\| \right). \end{array}$$
This implies that
$\left\|C(x,y)-C(\tilde{x},\tilde{y})\right\| \leq M_C\left\|(x,y)-(\tilde{x},\tilde{y})\right\|,$ where $$M_C= \left( \begin{array}{cc}
\displaystyle\sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}b{_{11}^{k} } & \displaystyle\sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}b{_{12}^{k} } \\ \\
\displaystyle\sum_{k=1}^{m}\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}b{_{21}^{k} } & \displaystyle\sum_{k=1}^{m}\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}b{_{22}^{k} } \end{array} \right).$$
\noindent \textbf{Claim 3:} Let $\varepsilon > 0,\ \ (x,y) \in S$ and $t,t' \in J$ such that
$|t'-t|<\varepsilon.$ From the Hahn-Banach theorem there exists a linear function $\phi \in X^{\ast}$ with $\|\phi\|=1$ such that $$\left\|B_i(x,y)(t')-B_i(x,y)(t)\right\|=\phi\left(B_i(x,y)(t)-B_i(x,y)(t)\right), \ \ i=1,2.$$ Based on the first inequality in \eqref{app} it is shown that $$\begin{array}{rcl}
\left\|B_i(x,y)(t')-B_i(x,y)(t)\right\| &=& \phi \left(\displaystyle\int_{0}^{t'}\frac{(t-s)^{\alpha_i-1}}{\Gamma(\alpha_j)}g_i(s,x(s),y(s))ds-\displaystyle\int_{0}^{t}\frac{(t-s)^{\alpha_i-1}}{\Gamma(\alpha_i)}g_i(s,x(s),y(s))\right) \\\\&\leq& \displaystyle\int_{0}^{t}\frac{|(t'-s)^{\alpha_i-1}-(t-s)^{\alpha_i-1}|}{\Gamma(\alpha_i)}\phi(g_i(s,x(s),y(s)))ds \\\\& +& \displaystyle\int_{t}^{t'}\frac{(t'-s)^{\alpha_i-1}}{\Gamma(\alpha_i)}\phi(g_i(s,x(s),y(s)))ds \\\\&\leq& P\displaystyle\int_{0}^{t}\frac{|(t'-s)^{\alpha_i-1}-(t-s)^{\alpha_i-1}|}{\Gamma(\alpha_i)} ds + P\displaystyle\int_{t}^{t'}\frac{(t'-s)^{\alpha_i-1}}{\Gamma(\alpha_i)}ds. \end{array}$$ This implies that
$\left\|B_i(x,y)(t')-B_i(x,y)(t)\right\|\to 0 \text{ as } \varepsilon \to 0,$ and consequently $B(S)$ is a weakly equi-continuous subset.
Let now $\{(x_n,y_n), n\in \mathbb{N}\}$ be any sequence in $S.$ From the first inequality in \eqref{app}, it follows that $$\begin{array}{rcl}
\left\|B_i(t,x_n(t),y_n(t))\right\| &\leq& \displaystyle\int_{0}^{t}\frac{(t-s)^{\alpha_i-1}}{\Gamma(\alpha_i)}\left\|g_i(s,x_n(s),y_n(s))\right\|ds \\\\&\leq& \displaystyle\frac{PT^{\alpha_i}}{\Gamma(\alpha_i+1)}, \end{array}$$ for all $t\in [0,1].$ This demonstrate that $\{B_i(x_n,y_n), \ \ n \in \mathbb{N}\}$ is a uniformly bounded sequence in $B(S)$ and so, $B(S)(t)$ is sequentially relatively weakly compact. Hence, $B(S)$ is sequentially relatively weakly compact in light of the Arzel\`{a}-Ascoli's theorem \cite{Ascoli}. An application of Eberlein-\u{S}mulian's theorem \cite{kes} yields that $B(S)$ is relatively weakly compact.
\noindent \textbf{Claim 4:} The operators $A,B$ and $ C $ satisfy assumption $(iv)$ of Theorem $\ref{3}.$ To see this, let $(x,y)\in C(J,X) \times C(J,X)$ and $(u,v) \in S$ with $(x,y)=A(x,y)\cdot B(u,v)+C(x,y).$ We shall show that $(x,y)\in S.$ For all $t\in[0,1],$ we have
$$ \begin{array}{rcl}
\|x(t)\| &\leq& \left\|f_1(t,x(t),y(t))\right\|\displaystyle\int_0^t \displaystyle\frac{(t-s)^{\alpha_1-1}}{\Gamma(\alpha_1)}\left\|g_1(s,u(s),v(s))\right\|ds+ \displaystyle\sum_{i=1}^m I^{\beta_1^i}\|h_1^i(t,x(t),y(t))\| \\\\&\leq& \left[\rho( \|x\|_\infty+\|y\|_\infty)+F_0\right]\displaystyle\frac{PT^{\alpha_1}}{\Gamma(\alpha_1+1)}+\displaystyle\sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}\left[\rho(\|x\|_{\infty}+\|y\|_{\infty})+H_0 \right] \end{array}$$ This implies that $$\begin{array}{rcl}
\|x(t)\|+\|y(t)\| &\leq& \big[\rho( \|x\|_\infty+\|y\|_\infty)+F_0\big]P\left(\frac{T^{\alpha_1}}{\Gamma(\alpha_1+1)}+ \frac{T^{\alpha_2}}{\Gamma(\alpha_2+1)}\right)\ \ \\ &+& \sum_{k=1}^{m}\left(\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}+\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}\right) \big[\rho(\|x\|_\infty+\|y\|_\infty)+H_0 \big] \end{array}$$
Consequently, $$\|x\|_\infty\leq r_0 \text{ and } \|y\|_\infty \leq r_0.$$ To end the proof, we apply Theorem $(\ref{mple})$, we deduce that the problem $(\ref{Sy3})$ has, at least, one solution in $ C(J,X) \times C(J,X).$\\
\textbf{Example :} Let $C(J,\mathbb{R})$ be the Banach algebra of all continuous functions from $J$ to $\mathbb{R} $ endowed with the sup-norm $\|\cdot\|$ defined by $\|x\|_\infty = \sup \limits_{0 \leq t \leq T}|x(t)|, $ for each $x \in C(J, \mathbb{R}).$
\begin{equation}\label{exe} \left\{
\begin{array}{lll}
D^{\frac{1}{2}}\big[\frac{x(t)- \sum _{k=1}^{2}I^{\beta_{1}^{k}}h_{1}^{k}(t,x(t),y(t))}{f_1(t,x(t),y(t))}\big]=\frac{3}{35(13-t^2)}(7|x(t)|+15|y(t)|), & \hbox{} \\ \\
D^{\frac{1}{2}}\big [\frac{y(t)- \sum _{k=1}^{2}I^{\beta_{2}^{k}}h_{2}^{k}(t,x(t),y(t))}{f_2(t,x(t),y(t))}\big]=\frac{3}{35(13-t^2)}(7|x(t)|+15|y(t)|), & \hbox{} \\ \\
x(0)=y(0)=0, \ \
{ t \in[0,1]}.
\end{array} \right. \end{equation} Note that this problem may be transformed into the fixed point problem $(\ref{Sy3})$ in view of lemma $2.5$ in \cite{D},
where $$\sum_{k=1}^{2}I^{\beta_{1}^{k}}h_{1}^{k}(t,x(t),y(t))=I^{ 1 / 3}\frac{2te^{-3t}}{15(3+t)}\left(\frac{x^2(t)+9|x(t)|}{|x(t)|+5}+\frac{12e^{3t}}{5} \right) + I^{ 10/ 3}\frac{2t \sin \pi t}{14+t^2}\left(\frac{x^2(t)+5|x(t)|}{|x(t)|+8}+\frac{1}{3}\right), $$
$$\sum_{k=1}^{2}I^{\beta_{2}^{k}}h_{2}^{k}(t,x(t),y(t))=I^{ 7 / 4}\frac{t \sin t}{7(4+e^t)}\left(\frac{y^2(t)+4|y(t)|}{|y(t)|+3}+\cos t \right) + I^{ 29/ 6}\frac{3t \cos t}{10(4-t^2)}\left(\frac{y^2(t)+5|y(t)|}{|y(t)|+4}+\frac{t}{t+2}\right), $$
and $$f_1(t,x(t),y(t))=\dfrac{3cos \pi t +2t}{5(2+10t^2)(|x(t)|+3)},f_2(t,x(t),y(t))=\dfrac{4cos \pi t +3t}{7(3+8t^2)(|y(t)|+6)},$$ and
$$g_1(t,x(t),y(t))=g_2(t,x(t),y(t))=\frac{3}{35(13-t^2)}(7|x(t)|+15|y(t)|,$$ here $\alpha_1=\alpha_2 = \dfrac{1}{2}, T = 1, m = 2, \beta_1^1=\dfrac{1}{3} ,\beta_1^2=\dfrac{10}{3}, \beta_2^1=\dfrac{7}{4}$ and $ \beta_2^2=\dfrac{29}{6}.$ \\We can show that \begin{eqnarray*}
|f_1(t,x,y)-f_1(t,\tilde{x},\tilde{y})|\leq \frac{3+2t}{5(2+10t^2)}|x-\tilde{x}|, \end{eqnarray*} \begin{eqnarray*}
|f_2(t,x,y)-f_2(t,\tilde{x},\tilde{y})|\leq \frac{4+3t}{7(1+5t^2)}|y-\tilde{y}|, \end{eqnarray*} and \begin{eqnarray*}
|h_1^1(t,x,y)-h_1^1(t,\tilde{x},\tilde{y})|\leq \frac{18 t}{75(3+t)}|x-\tilde{x}|, \end{eqnarray*} \begin{eqnarray*}
|h_1^2(t,x,y)-h_1^2(t,\tilde{x},\tilde{y})|\leq \frac{10 t}{8(14+t^2)}|x-\tilde{x}|,
\end{eqnarray*}
\begin{eqnarray*}
|h_2^1(t,x,y)-h_2^1(t,\tilde{x},\tilde{y})|\leq \frac{4 t}{21(4+e^t)}|y-\tilde{y}|, \end{eqnarray*} \begin{eqnarray*}
|h_2^2(t,x,y)-h_2^2(t,x,\tilde{y})|\leq \frac{3 t}{8(4-t^2)}|y-\tilde{y}|. \end{eqnarray*} It follows that $a_{11}=\dfrac{1}{12},\ \ a_{21}=0,\ \ a_{12}=0, \ \ a_{22}= \dfrac{ 1}{6},\ \ b^{1}_{11}=\dfrac{3}{50},\ \ b^{1}_{12}=0,\ \ b_{21}^{1}=0,\ \ b_{22}^{1}=\dfrac{4}{21(4+e)},\ \ b_{11}^{2}=\dfrac{1}{12},\ \ b_{12}^{2}=0,\ \ b_{21}^{2}=0,\ \ b_{22}^{2}= \dfrac{1}{8}. $
Now $$|g_1(t,x,y)|=|g_2(t,x,y)|\leq \dfrac{1}{4} $$
It is easy to verify that $ \rho= \dfrac{1}{6},\ \ F_0=\dfrac{1}{36},\ \ H_0=\dfrac{2}{25}.$ We see that condition $(\mathcal{H}_3)$ is followed with a number $r_0=2.$ Moreover, $ \dfrac{PT^{\alpha_1}}{\Gamma(\alpha_1+1)}a_{11}+\sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}b{_{11}^{k} }\simeq 0.0990,\ \
\dfrac{PT^{\alpha_2}}{\Gamma(\alpha_2+1)}a_{12}+\sum_{k=1}^{m}\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}b{_{12}^{k} }= \dfrac{PT^{\alpha_1}}{\Gamma(\alpha_1+1)}a_{21}+\sum_{k=1}^{m}\frac{T^{\beta_1^{k}}}{\Gamma(\beta_1^{k}+1)}b{_{21}^{k} }=0, \ \
\dfrac{PT^{\alpha_2}}{\Gamma(\alpha_2+1)}a_{22}+\sum_{k=1}^{m}\frac{T^{\beta_2^{k}}}{\Gamma(\beta_2^{k}+1)}b{_{22}^{k}} \simeq 0.0653.$ Theorem $(\ref{mple})$ proves the existence of a solution to system $(\ref{exe})$ . \end{pf}
\end{document} | arXiv |
\begin{document}
\title{Synchronisation in Invertible Random Dynamical Systems on the Circle} \author{Julian Newman} \maketitle
\begin{abstract} \noindent In this paper, we study geometric features of orientation-preserving random dynamical systems on the circle driven by memoryless noise that exhibit stable synchronisation: we consider crack points, invariant measures, and the link between synchronisation and compressibility of arcs; we also characterise stable synchronisation in additive-noise stochastic differential equations on the circle, in terms of ``subperiodicity'' of the vector field. \end{abstract}
\section{Introduction}
It is well-known that any ``sufficiently noisy'' invertible random dynamical system on the circle driven by memoryless noise exhibits ``contraction of orbits'' or ``synchronisation'', in the sense that the distance between the trajectories of two given initial conditions almost surely converges to $0$ as time tends to $\infty$. In discrete time, we have the following:
\begin{propn}[{[Ant84], [Mal14]}] Given a set $F \subset \mathrm{Homeo}^+(\mathbb{S}^1)$, equipped with the uniform topology, and a probability measure $\nu$ on $F$ with full support, if either: \begin{enumerate}[\indent (a)] \item there is no finite-order orientation-preserving homeomorphism other than $\mathrm{id}_{\mathbb{S}^1}$ that commutes with every $f \in F$, and for every $x \in \mathbb{S}^1$ and open $U \subset \mathbb{S}^1$, there exist $f_1,\ldots,f_n,\tilde{f}_1,\ldots,\tilde{f}_m \in F$ such that \[ f_n \circ \ldots \circ f_1(x) \in U \hspace{4mm} \textrm{and} \hspace{4mm} x \in \tilde{f}_m \circ \ldots \circ \tilde{f}_1(U) \, ; \textrm{ or} \] \item for every distinct $x,y \in \mathbb{S}^1$ there exist $f_1,\ldots,f_n \in F$ such that \[ d( \, f_n \circ \ldots \circ f_1(x) \, , \, f_n \circ \ldots \circ f_1(y) \, ) \ < \ d(x,y) \] and there does not exist $p \in \mathbb{S}^1$ such that for every $f \in F$, $f(p)=p$; \end{enumerate} then given any $x,y \in \mathbb{S}^1$, we have that for $\nu^{\otimes \mathbb{N}}$-almost all $(f_n)_{n \geq 1}$, \[ d( \, f_n \circ \ldots \circ f_1(x) \, , \, f_n \circ \ldots \circ f_1(y) \, ) \ \to \ 0 \ \textrm{ as } n \to \infty. \] \end{propn}
\noindent In case~(a), the result is due to [Ant84]; in case~(b), the result is due to [Mal14]. Moreover, it is shown in [Mal14] that the convergence occurs at an exponential rate. \\ \\ Now by [Mal14, Theorem~A], we can add the following to the conclusion in the above proposition: \emph{given any $x \in \mathbb{S}^1$, we have that for $\nu^{\otimes \mathbb{N}}$-almost every $(f_n)_{n \geq 1}$, there exists a neighbourhood $U$ of $x$ such that} \[ \mathrm{diam}( \, f_n \circ \ldots \circ f_1(U) \, ) \ \to \ 0 \ \textit{ as } n \to \infty. \] (Again, the convergence is at an exponential rate.) This additional property implies physically that small unexpected perturbations to the evolution of the trajectories are unlikely to destroy the synchronisation described in the above proposition. Hence, we refer to synchronisation combined with this additional property as ``stable synchronisation''. \\ \\ Synchronisation in continuous-time systems on the circle has been studied in [Crau02], and some specific examples in [Bax86]. Necessary and sufficient conditions for stable synchronisation in a more general context have been given in [New17]. \\ \\ The goal of this paper is to describe certain geometrical features of orientation-preserving random dynamical systems on the circle exhibiting stable synchronisation. Our results apply in both discrete and continuous time. In Section~2, we will introduce our setting. In Section~3, we will present a characterisation of stable synchronisation in terms of ``crack points''. On the basis of this, in Section~4 we will describe the ``invariant measures'' of systems exhibiting stable synchronisation. In Section~5, we will present a result linking contractibility for pairs of trajectories, compressibility for arcs, and stable synchronisation; we will then characterise stable synchronisation in additive-noise stochastic differential equations on the circle, in terms of ``subperiodicity'' of the vector field.
\section{Our setting}
Let $\mathbb{T}^+$ denote either $\mathbb{N} \cup \{0\}$ or $[0,\infty)$. Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \in \mathbb{T}^+},\mathbb{P})$ be a filtered probability space (the ``noise space''), and write $\mathcal{F}_+:=\sigma(\mathcal{F}_t:t \in \mathbb{T}^+)$. Let $(\theta^t)_{t \in \mathbb{T}^+}$ be a family of $\mathbb{P}$-preserving $(\mathcal{F},\mathcal{F})$-measurable functions $\theta^t:\Omega \to \Omega$ such that $\theta^0=\mathrm{id}_\Omega$ and $\theta^{s+t}=\theta^t \circ \theta^s$ for all $s,t \in \mathbb{T}^+$. Suppose moreover that: \begin{enumerate}[\indent (i)] \item $\theta^t$ is $(\mathcal{F}_{s+t},\mathcal{F}_s)$-measurable (i.e.~$\theta^{-t}\mathcal{F}_s \subset \mathcal{F}_{s+t}$) for all $s,t \in \mathbb{T}^+$; \item for each $t \in \mathbb{T}^+$, $\mathcal{F}_t$ and $\theta^{-t}\mathcal{F}_+$ are independent $\sigma$-algebras according to $\mathbb{P}$ (i.e. $\mathbb{P}(E \cap \theta^{-t}(F)) \, = \, \mathbb{P}(E)\mathbb{P}(F)$ for all $E \in \mathcal{F}_t$ and $F \in \mathcal{F}_+$). \end{enumerate}
\noindent (Here, we use the convention $\theta^{-t}(E):=(\theta^t)^{-1}(E)$.) \\ \\ Heuristically, as suggested by (i), $\theta^t\omega$ represents a \emph{time-shift} of the noise realisation $\omega$ forward by time $t$. The fact that $\mathbb{P}$ is invariant under $(\theta^t)$ represents the assumption that the noise is strictly stationary, and condition~(ii) represents the assumption that the noise is memoryless. We emphasise that, whether we are considering a one-sided-time noise process or a two-sided-time noise process, $\mathcal{F}_t$ always represents the information available between time 0 and time $t$.
\begin{ex}[Gaussian white noise] Following sections~A.2 and A.3 of [Arn98], an ``eternal'' one-dimensional Gaussian white noise process may be described according to the framework above as follows: Let $\Omega:=\{\omega \in C(\mathbb{R},\mathbb{R}) : \omega(0)=0\}$. For each $t \in [0,\infty)$, let $\mathcal{F}_t$ be the smallest $\sigma$-algebra on $\Omega$ with respect to which the projection $W_s:\omega \mapsto \omega(s)$ is measurable for every $s \in [0,t]$. Let $\mathcal{F}$ be the smallest $\sigma$-algebra on $\Omega$ with respect to which the projection $W_s:\omega \mapsto \omega(s)$ is measurable for every $s \in \mathbb{R}$. Let $\mathbb{P}$ be the \emph{Wiener measure} on $(\Omega,\mathcal{F})$---that is, $\mathbb{P}$ is the unique probability measure under which the stochastic processes $(W_t)_{t \geq 0}$ and $(W_{\!-t})_{t \geq 0}$ are independent Wiener processes. Finally, for each $\tau \geq 0$ and $s \in \mathbb{R}$, set $\theta^\tau\omega(s):=\omega(\tau+s)-\omega(\tau)$. \end{ex}
\noindent Now let $\mathbb{S}^1$ be the unit circle, which we identify with $^{\mathbb{R}\!}/_{\!\mathbb{Z}\,}$ in the obvious manner, and let $l$ denote the Lebesgue measure on $\mathbb{S}^1$ (with $l(\mathbb{S}^1)=1$). Let $\pi:\mathbb{R} \to \mathbb{S}^1$ denote the natural projection, i.e.~$\pi(x)\,=x+\mathbb{Z} \, \in \, \mathbb{S}^1$; a \emph{lift} of a point $x \in \mathbb{S}^1$ is a point $x' \in \mathbb{R}$ such that $\pi(x')=x$, and a lift of a set $A \subset \mathbb{S}^1$ is a set $B \subset \mathbb{R}$ such that $\pi(B)=A$. Define the metric $d$ on $\mathbb{S}^1$ by
\[ d(x,y) \ = \ \min\{|x'-y'| : \, x' \textrm{ is a lift of } x, \, y' \textrm{ is a lift of } y \}. \] \noindent Note that under this metric, for any connected $J \subset \mathbb{S}^1$, \[ \mathrm{diam}\,J \ = \ \min\!\left(l(J),\tfrac{1}{2}\right)\!. \] \noindent Let $\,\varphi \!\! = \!\! \left(\varphi(t,\omega)\right)_{t \in \mathbb{T}^+ \! , \, \omega \in \Omega}\,$ be a $(\mathbb{T}^+ \! \times \Omega)$-indexed family of orientation-preserving homeomorphisms $\varphi(t,\omega):\mathbb{S}^1 \to \mathbb{S}^1$ such that: \begin{enumerate}[\indent (a)] \item the map $(\omega,x) \mapsto \varphi(t,\omega)x$ is $(\mathcal{F}_t \otimes \mathcal{B}(\mathbb{S}^1),\mathcal{B}(\mathbb{S}^1))$-measurable for each $t \in \mathbb{T}^+$; \item $\varphi(0,\omega) \, = \, \mathrm{id}_{\mathbb{S}^1}$ for all $\omega \in \Omega$; \item $\varphi(s+t,\omega) \, = \, \varphi(t,\theta^s\omega) \circ \varphi(s,\omega)\,$ for all $s,t \in \mathbb{T}^+$ and $\omega \in \Omega$; \item for any decreasing sequence $(t_n)$ in $\mathbb{T}^+$ converging to a time $t$, and any sequence $(x_n)$ in $\mathbb{S}^1$ converging to a point $x$, $\,\varphi(t_n,\omega)x_n \to \varphi(t,\omega)x\,$ as $n \to \infty$ for all $\omega \in \Omega$; \item there exists a function $\varphi_-:\mathbb{T}^+ \times \Omega \times \mathbb{S}^1 \to \mathbb{S}^1$ such that for any strictly increasing sequence $(t_n)$ in $\mathbb{T}^+$ converging to a time $t$, and any sequence $(x_n)$ in $\mathbb{S}^1$ converging to a point $x$, $\,\varphi(t_n,\omega)x_n \to \varphi_-(t,\omega,x)\,$ as $n \to \infty$ for all $\omega \in \Omega$. \end{enumerate}
\noindent We refer to $\varphi$ as a \emph{random dynamical system} (RDS) on $\mathbb{S}^1$; more specifically, since $\varphi(t,\omega)$ is a homeomorphism for all $t$ and $\omega$, we refer to $\varphi$ as an \emph{invertible RDS}. Conditions~(d) and (e) constitute the ``c\`{a}dl\`{a}g'' property, with (d) being right-continuity and (e) being left limits.\footnote{The left-limits property is included simply to ensure that ``asymptotic stability'' (defined as the existence of a neighbourhood of the initial condition that contracts in diameter to $0$ under the flow) implies stability in the sense of Lyapunov.} It is not hard to show that property~(d) implies the following: \begin{enumerate}[\indent (d')] \item for any decreasing sequence $(t_n)$ in $\mathbb{T}^+$ converging to a time $t$, and any sequence $(x_n)$ in $\mathbb{S}^1$ converging to a point $x$, $\,\varphi(t_n,\omega)^{-1}(x_n) \to \varphi^{-1}(t,\omega)(x)\,$ as $n \to \infty$ for all $\omega \in \Omega$. \end{enumerate}
\noindent We will say that $\varphi$ is a \emph{continuous RDS} if for all $\omega \in \Omega$ the map $(t,x) \mapsto \varphi(t,\omega)x$ is jointly continuous. (In this case, $(t,x) \mapsto \varphi(t,\omega)^{-1}(x)$ is also jointly continuous for all $\omega$.)
\begin{defi} We say that $\varphi$ is \emph{synchronising} if for all $x,y \in \mathbb{S}^1$, \[ \mathbb{P}( \, \omega \, : \, d(\varphi(t,\omega)x,\varphi(t,\omega)y) \to 0 \textrm{ as } t \to \infty \, ) \ = \ 1. \] \end{defi}
\begin{defi} We say that $\varphi$ is \emph{everywhere locally stable} if for all $x \in \mathbb{S}^1$, \[ \mathbb{P}( \, \omega \, : \, \exists \, \textrm{open } U \! \ni x \, \textrm{ s.t.~} l(\varphi(t,\omega)U) \to 0 \textrm{ as } t \to \infty \, ) \ = \ 1 \, ; \] \noindent and we say that $\varphi$ is \emph{stably synchronising} if $\varphi$ is both synchronising and everywhere locally stable. \end{defi}
\noindent An example of a system that is synchronising but not stably synchronising is the following: Within a \emph{deterministic} setting (i.e.~taking $\Omega$ to be just a singleton $\{\omega\}$), let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism with a unique fixed point $p$, and (working in discrete time) take $\varphi(n,\omega):=f^n$. For every $x \in \mathbb{S}^1$, $f^n(x) \to p$ as $n \to \infty$. Hence in particular, for all $x,y \in \mathbb{S}^1$, $d(f^n(x),f^n(y)) \to 0$ as $n \to \infty$; and yet for every neighbourhood $U$ of $p$, $l(f^n(U))$ tends to 1 rather than to 0 as $n \to \infty$.
\section{Crack points}
Observe that for any $\omega \in \Omega$, the binary relation $\sim_\omega$ on $\mathbb{S}^1$ defined by \[ x \sim_\omega y \hspace{3mm} \Longleftrightarrow \hspace{3mm} d(\varphi(t,\omega)x,\varphi(t,\omega)y) \to 0 \textrm{ as } t \to \infty \] \noindent is an equivalence relation. It is easy to show (by considering only rational times) that the set $\,\{(x,y,\omega) \in \mathbb{S}^1 \times \mathbb{S}^1 \times \Omega \, : \, x \sim_\omega y \}\,$ is a $(\mathcal{B}(\mathbb{S}^1 \times \mathbb{S}^1) \otimes \mathcal{F}_+)$-measurable set.
\begin{defi}[c.f.~\textrm{[Kai93]}] Given a point $r \in \mathbb{S}^1$ and a sample point $\omega \in \Omega$, we will say that $r$ is a \emph{crack point} of $\omega$ if the following equivalent statements hold: \begin{itemize} \item for every $A \subset \mathbb{S}^1$ with $r \nin \bar{A}$, $\mathrm{diam}(\varphi(t,\omega)A) \to 0$ as $t \to \infty$; \item for every closed $G \subset \mathbb{S}^1$ with $r \nin G$, $l(\varphi(t,\omega)G) \to 0$ as $t \to \infty$; \item for every open $U \subset \mathbb{S}^1$ with $r \in U$, $l(\varphi(t,\omega)U) \to 1$ as $t \to \infty$. \end{itemize} \end{defi}
\noindent Obviously, any sample point admits at most one crack point. If a sample point $\omega$ admits a crack point, then we will say that $\omega$ is \emph{contractive}. \\ \\ Now if a sample point $\omega$ admits a crack point $r$, then either (a)~the equivalence relation $\sim_\omega$ has two equivalence classes, namely $\{r\}$ and $\mathbb{S}^1 \setminus \{r\}$, or (b)~the equivalence relation $\sim_\omega$ has one equivalence class (the whole of $\mathbb{S}^1$). In case~(a), we say that $r$ is a \emph{repulsive} crack point of $\omega$.
\begin{defi} Let $\Omega_c \subset \Omega$ be the set of contractive sample points, and let $\tilde{r}:\Omega_c \to \mathbb{S}^1$ denote the function sending a contractive sample point $\omega$ onto its crack point $\tilde{r}(\omega)$. \end{defi}
\begin{lemma} \label{crrfp} $\Omega_c$ is $\mathcal{F}_+$-measurable, and $\tilde{r}:\Omega_c \to \mathbb{S}^1$ is measurable with respect to the $\sigma$-algebra $\mathcal{F}_c$ of $\mathcal{F}_+$-measurable subsets of $\Omega_c$. For each $t \in \mathbb{T}^+$, $\theta^{-t}(\Omega_c) = \Omega_c$ and $\tilde{r}(\theta^t\omega)=\varphi(t,\omega)\tilde{r}(\omega)$ for all $\omega \in \Omega_c$. \end{lemma}
\begin{proof} Let $R$ be a countable dense subset of $\mathbb{S}^1$. For any connected $J \subset \mathbb{S}^1$, it is clear (by considering rational times) that \begin{equation} \label{meas} \{ \omega \in \Omega \, : \, l(\varphi(t,\omega)J) \to 0 \textrm{ as } t \to \infty \} \; \in \, \mathcal{F}_+. \end{equation} \noindent So then, in order to show that $\Omega_c \in \mathcal{F}_+$, it suffices to prove the following statement: a sample point $\omega \in \Omega$ is contractive if and only if for every $n \in \mathbb{N}$ there is a connected open set $U_n \subset \mathbb{S}^1$ with endpoints in $R$ such that $1-\frac{1}{n} < l(U_n) < 1$ and $l(\varphi(t,\omega)U_n) \to 0$ as $t \to \infty$. Now the ``only if'' direction is obvious. For the ``if'' direction: suppose that for every $n \in \mathbb{N}$ there exists a connected open set $U_n \subset \mathbb{S}^1$ with endpoints in $R$ such that $1-\frac{1}{n} < l(U_n) < 1$ and $l(\varphi(t,\omega)U_n) \to 0$ as $t \to \infty$; and let $U:=\bigcup_{n=1}^\infty U_n$. Since $U_n$ is connected for all $n$ and $l(U_n) \to 1$ as $n \to \infty$, we clearly have that either $U=\mathbb{S}^1$ or $\mathbb{S}^1 \setminus \{U\}$ is a singleton. Now suppose, for a contradiction, that $U=\mathbb{S}^1$. Then, since $\mathbb{S}^1$ is compact, there is a finite subset $\{n_1,\ldots,n_k\}$ of $\mathbb{N}$ such that $\mathbb{S}^1=\bigcup_{i=1}^k U_{n_i}$; but since $l(\varphi(t,\omega)U_{n_i}) \to 0$ as $t \to \infty$ for each $i$, we then have that $l(\varphi(t,\omega)\mathbb{S}^1) \to 0$ as $t \to \infty$, which is absurd. So then, we must have that $\mathbb{S}^1 \setminus U$ is equal to a singleton $\{r\}$. We now show that $r$ is a crack point. Fix any closed $G \subset \mathbb{S}^1$ with $r \nin G$. Take $n$ such that $l(U_n)>1-d(r,G)$; then $G \subset U_n$ and so $l(\varphi(t,\omega)G) \to 0$ as $t \to \infty$. Hence $r$ is a crack point of $\omega$. \\ \\ Thus we have shown that $\Omega_c$ is $\mathcal{F}_+$-measurable. Now for any $\omega \in \Omega_c$ and any non-empty closed connected $K \subset \mathbb{S}^1$, observe that $\tilde{r}(\omega) \in K$ if and only if for every closed connected $G \subset \mathbb{S}^1 \setminus K$ with $\partial G \subset R$, $l(\varphi(t,\omega)G) \to 0$ as $t \to \infty$. So by (\ref{meas}) and the countability of $R$, $\tilde{r}^{-1}(K) \in \mathcal{F}_c$ for every closed connected $K \subset \mathbb{S}^1$. Hence $\tilde{r}$ is $\mathcal{F}_c$-measurable. \\ \\ Now fix any $t \in \mathbb{T}^+$ and $\omega \in \Omega$. First suppose that $\omega$ admits a crack point $r$:~then for any closed $G \not\ni \varphi(t,\omega)r$, \[ \mathrm{diam}(\varphi(s,\theta^t\omega)G) \ = \ \mathrm{diam}(\varphi(s+t,\omega) \, (\varphi(t,\omega)^{-1}(G)) \, ) \to 0 \textrm{ as } s \to \infty \] \noindent since $\varphi(t,\omega)^{-1}(G)$ is a closed set not containing $r$; so $\varphi(t,\omega)r$ is a crack point of $\theta^t\omega$. Now suppose that $\theta^t\omega$ admits a crack point $q$:~then for any closed $G \not\ni \varphi(t,\omega)^{-1}(q)$, \[ \mathrm{diam}(\varphi(s+t,\omega)G) \ = \ \mathrm{diam}(\varphi(s,\theta^t\omega) \, (\varphi(t,\omega)G) \, ) \to 0 \textrm{ as } s \to \infty \] \noindent since $\varphi(t,\omega)G$ is a closed set not containing $q$; so $\varphi(t,\omega)^{-1}(q)$ is a crack point of $\omega$. Thus we have proved that $\omega$ admits a crack point if and only if $\theta^t\omega$ admits a crack point, and that in this case, $\tilde{r}(\theta^t\omega)=\varphi(t,\omega)\tilde{r}(\omega)$. \end{proof}
\begin{thm} \label{cr char} $\mathbb{P}(\Omega_c)$ is equal to either $0$ or $1$. In the case that $\mathbb{P}(\Omega_c)=1$, either: \begin{enumerate}[\indent (a)] \item for every $x \in \mathbb{S}^1$, $\,\mathbb{P}(\omega \in \Omega_c \, : \, \tilde{r}(\omega) = x) \ = \ 0$; or \item there exists a deterministic fixed point $p \in \mathbb{S}^1$ such that $\,\mathbb{P}(\omega \in \Omega_c \, : \, \tilde{r}(\omega) = p) \ = \ 1$. \end{enumerate} \noindent $\varphi$ is stably synchronising if and only if $\mathbb{P}(\Omega_c)=1$ and case~(a) holds. In this case, we also have that for $\mathbb{P}$-almost every $\omega \in \Omega_c$, $\tilde{r}(\omega)$ is a repulsive crack point of $\omega$. \end{thm}
\begin{rmk} In the case that there is no deterministic fixed point, the fact that stable synchronisation implies $\mathbb{P}(\Omega_c)=1$ can also be derived using results from [Mal14]. \end{rmk}
\subsection*{Proof of Theorem~\ref{cr char}}
\begin{lemma} \label{empf}
The measure-preserving flow $(\Omega,\mathcal{F}_+,\mathbb{P}|_{\mathcal{F}_+},(\theta^t)_{t \in \mathbb{T}^+})$ is ergodic. \end{lemma}
\noindent For a proof, see e.g.~[New15, Corollary~133].
\begin{cor} $\mathbb{P}(\Omega_c)$ is equal to either $0$ or $1$. \end{cor}
\begin{proof} Follows immediately from Lemmas~\ref{crrfp} and \ref{empf}. \end{proof}
\noindent Now for each $x \in \mathbb{S}^1$ and $t \in \mathbb{T}^+$, define the probability measure $\bar{\varphi}_x^t$ on $\mathbb{S}^1$ by \[ \bar{\varphi}_x^t(A) \ \, := \ \, \mathbb{P}( \, \omega \, : \, x \in \varphi(t,\omega)A \, ) \ = \ \mathbb{P}( \, \omega \, : \, x \in \varphi(t,\theta^s\omega)A \, ) \] \noindent for all $A \in \mathcal{B}(\mathbb{S}^1)$ and any $s \in \mathbb{T}^+$. Given any $s,t \in \mathbb{T}^+$, observe that \begin{itemize} \item under the random map $\varphi(s,\omega)^{-1}\!:\mathbb{S}^1 \!\to \mathbb{S}^1$, the transition probability from a point $y$ to a set $A$ is precisely $\bar{\varphi}_y^s(A)$; \item under the random map $\varphi(t,\theta^s\omega)^{-1}\!:\mathbb{S}^1 \!\to \mathbb{S}^1$, the transition probability from a point $x$ to a set $Y$ is precisely $\bar{\varphi}_x^t(Y)$; \item under the random map $\varphi(s+t,\omega)^{-1}\!:\mathbb{S}^1 \!\to \mathbb{S}^1$, the transition probability from a point $x$ to a set $A$ is precisely $\bar{\varphi}_x^{s+t}(A)$. \end{itemize} \noindent Therefore, since the $\sigma$-algebras $\mathcal{F}_s$ and $\theta^{-s}\mathcal{F}_t$ are independent, the Chapman-Kolmogorov equation \[ \bar{\varphi}_x^{s+t}(A) \ = \ \int_{\mathbb{S}^1} \bar{\varphi}_y^s(A) \; \bar{\varphi}_x^t(dy) \] \noindent is satisfied for any $x \in \mathbb{S}^1$, $A \in \mathcal{B}(\mathbb{S}^1)$ and $s,t \in \mathbb{T}^+$. Moreover, since the map $(t,x) \mapsto \varphi(t,\omega)^{-1}(x)$ is jointly continuous in $x$ and right-continuous in $t$ for every $\omega \in \Omega$, the dominated convergence theorem gives that the map $(t,x) \mapsto \bar{\varphi}_x^t$ is (with respect to the topology of weak convergence) jointly continuous in $x$ and right-continuous in $t$. \\ \\ We will say that a probability measure $\rho$ on $\mathbb{S}^1$ is \emph{reverse-stationary (with respect to $\varphi$)} if for all $t \in \mathbb{T}^+$ and $A \in \mathcal{B}(\mathbb{S}^1)$, \[ \rho(A) \ = \ \int_\Omega \rho(\varphi(t,\omega)A) \, \mathbb{P}(d\omega). \hspace{0.2mm}\footnotemark \]
\footnotetext{If $\theta^\tau$ is a measurable automorphism of $(\Omega,\mathcal{F})$ for all $\tau \in \mathbb{T}^+$, then one can naturally define $\varphi(t,\omega)$ for negative $t$ by $\varphi(t,\omega):=\varphi(|t|,\theta^t\omega)^{-1}$. However, we emphasise that even in this case, in the definition of reverse-stationarity we must restrict to nonnegative $t$.}Note that for any $s \in \mathbb{T}^+$, since $\mathbb{P}$ is $\theta^s$-invariant this is equivalent to saying that for all $t \in \mathbb{T}^+$ and $A \in \mathcal{B}(\mathbb{S}^1)$, \[ \rho(A) \ = \ \int_\Omega \rho(\varphi(t,\theta^s\omega)A) \, \mathbb{P}(d\omega). \] Note also that $\rho$ is reverse-stationary if and only if $\rho$ is a stationary measure of the family of transition probabilities $(\bar{\varphi}_x^t)_{x \in \mathbb{S}^1 \! , \, t \in \mathbb{T}^+}$, i.e. \[ \rho(A) \ = \ \int_{\mathbb{S}^1} \bar{\varphi}_x^t(A) \, \rho(dx) \] for all $t \in \mathbb{T}^+$ and $A \in \mathcal{B}(\mathbb{S}^1)$. Therefore, by the Krylov-Bogolyubov theorem (e.g.~[New15, Theorem~114] or [Kif86, Lemma~5.2.1]), there must exist at least one reverse-stationary probability measure.
\begin{defi} We say that a point $p \in \mathbb{S}^1$ is a \emph{deterministic fixed point} (\emph{of $\varphi$}) if $\mathbb{P}$-almost every $\omega \in \Omega$ has the property that for all $t \in \mathbb{T}^+$, $\varphi(t,\omega)p=p$. \end{defi}
\begin{defi} We say that a set $A \subset \mathbb{S}^1$ is \emph{forward-invariant} (\emph{under $\varphi$}) if $\mathbb{P}$-almost every $\omega \in \Omega$ has the property that for all $t \in \mathbb{T}^+$, $\varphi(t,\omega)A \subset A$. \end{defi}
\noindent Note that a \emph{finite} set $P \subset \mathbb{S}^1$, the following statements are equivalent: \begin{itemize} \item $P$ is forward-invariant; \item $\mathbb{S}^1 \setminus P$ is forward-invariant; \item for each $t \in \mathbb{T}^+$, \[ \mathbb{P}( \, \omega \, : \, \varphi(t,\omega)P = P ) \ = \ 1. \] \end{itemize} Now we say that a probability measure $\rho$ on $\mathbb{S}^1$ is \emph{atomless} if for all $x \in \mathbb{S}^1$, $\rho(\{x\})=0$.
\begin{lemma} \label{atomic}
Let $\rho$ be a probability measure that is ergodic with respect to the family of transition probabilities $(\bar{\varphi}_x^t)_{x \in \mathbb{S}^1 \! , \, t \in \mathbb{T}^+}$. Then either $\rho$ is atomless, or $\rho \, = \, \frac{1}{|P|}\sum_{x \in P} \delta_x$ for some finite forward-invariant set $P \subset \mathbb{S}^1$. \end{lemma}
\begin{proof} Suppose that $\rho$ is not atomless. Let $m:=\max\{ \rho(\{x\}) : x \in \mathbb{S}^1 \}$ and let $P:=\{x \in \mathbb{S}^1 : \rho(\{x\})=m\}$. For any $t \in \mathbb{T}^+$ and $\omega \in \Omega$, if $P \neq \varphi(t,\omega)P$ then $\rho(\varphi(t,\omega)P)<\rho(P)$; so since $\rho$ is reverse-stationary, we have that for each $t \in \mathbb{T}^+$, \[ \mathbb{P}( \, \omega \, : \, P = \varphi(t,\omega)P ) \ = \ 1, \] i.e.\ $P$ is forward-invariant. Note that $\bar{\varphi}_x^t(P)=1$ for each $x \in P$ and $t \in \mathbb{T}^+$. So since $\rho$ is ergodic with respect to $(\bar{\varphi}_x^t)_{x \in \mathbb{S}^1 \! , \, t \in \mathbb{T}^+}$ and $\rho(P)>0$, it follows that $\rho(P)=1$. \end{proof}
\begin{cor} \label{Dirac} Let $\rho$ be as in Lemma~\ref{atomic}, and suppose moreover that $\varphi$ is synchronising. Then $\rho$ is either atomless or a Dirac mass at a deterministic fixed point. \end{cor}
\begin{proof} Since $\varphi$ is synchronising, any finite forward-invariant set $P$ must be a singleton. So the result is immediate. \end{proof}
\begin{lemma} \label{Dirrfp} Suppose we have an $\mathcal{F}_+$-measurable function $q:\Omega \to \mathbb{S}^1$ with the property that for each $t \in \mathbb{T}^+$, for $\mathbb{P}$-almost all $\omega \in \Omega$, $\varphi(t,\omega)q(\omega)=q(\theta^t\omega)$. Then $q_\ast\mathbb{P}$ is an ergodic measure of the family of transition probabilities $(\bar{\varphi}_x^t)_{x \in \mathbb{S}^1 \! , \, t \in \mathbb{T}^+}$, and is either atomless or a Dirac mass at a deterministic fixed point. \end{lemma}
\begin{proof} First we show that $q_\ast\mathbb{P}$ is stationary with respect to $(\bar{\varphi}_x^t)$ (i.e.~is reverse-stationary with respect to $\varphi$). Note that for each $t$, the map $\omega \mapsto q(\theta^t\omega)$ is $\theta^{-t}\mathcal{F}_+$-measurable. For any $t \in \mathbb{T}^+$ and $A \in \mathcal{B}(\mathbb{S}^1)$, \begin{align*} \int_{\mathbb{S}^1} \bar{\varphi}_x^t(A) \; q_\ast\mathbb{P}(dx) \ &= \ \int_{\mathbb{S}^1} \bar{\varphi}_x^t(A) \ (q \circ \theta^t)_\ast\mathbb{P}(dx) \hspace{3mm} \textrm{(since $\mathbb{P}$ is $\theta^t$-invariant)} \\ &= \ \int_\Omega \bar{\varphi}_{q(\theta^t\omega)}^t(A) \, \mathbb{P}(d\omega) \\ &= \ \int_\Omega \mathbb{P}( \hspace{0.2mm} \tilde{\omega} \, : \, \varphi(t,\tilde{\omega})^{-1}(q(\theta^t\omega)) \in A \hspace{0.2mm} ) \; \mathbb{P}(d\omega) \\ &= \ \mathbb{P}( \hspace{0.2mm} \omega \, : \, \varphi(t,\omega)^{-1}(q(\theta^t\omega)) \in A \hspace{0.2mm} ) \\ &\hspace{10mm} \textrm{since $\mathcal{F}_t$ and $\theta^{-t}\mathcal{F}_+$ are independent $\sigma$-algebras} \\ &= \ \mathbb{P}( \hspace{0.2mm} \omega \, : \, q(\omega) \in A \hspace{0.2mm} ) \\ &= \ q_\ast\mathbb{P}(A). \end{align*} \noindent Hence $q_\ast\mathbb{P}$ is stationary with respect to $(\bar{\varphi}_x^t)$. Now let $A \in \mathcal{B}(\mathbb{S}^1)$ be a set such that for each $t \in \mathbb{T}^+$, for $(q_\ast\mathbb{P})$-almost every $x \in A$, $\bar{\varphi}_x^t(A)=1$; to prove that $q_\ast\mathbb{P}$ is ergodic with respect to $(\bar{\varphi}_x^t)$, we need to show that $q_\ast\mathbb{P}(A) \in \{0,1\}$. Let $E:=q^{-1}(A) \in \mathcal{F}_+$, and for each $t \in \mathbb{T}^+$ let \[ \tilde{E}_t \ := \ \{ \omega \, : \, \varphi(t,\omega)^{-1}(q(\theta^t\omega)) \in A \}. \] \noindent Obviously $\mathbb{P}(E \triangle \tilde{E}_t)=0$ for each $t$. So then \begin{align*}
\mathbb{P}( \, E \, \cap \, \theta^{-t}(E) \, ) \ &= \ \int_{\theta^{-t}(E)} \mathbb{P}(E|\theta^{-t}\mathcal{F}_+)(\omega) \; \mathbb{P}(d\omega) \\
&= \ \int_{\theta^{-t}(E)} \mathbb{P}(\tilde{E}_t|\theta^{-t}\mathcal{F}_+)(\omega) \; \mathbb{P}(d\omega) \\ &= \ \int_{\theta^{-t}(E)} \mathbb{P}( \hspace{0.2mm} \tilde{\omega} \, : \, \varphi(t,\tilde{\omega})^{-1}(q(\theta^t\omega)) \in A \hspace{0.2mm} ) \; \mathbb{P}(d\omega) \\ &\hspace{10mm} \textrm{since $\mathcal{F}_t$ and $\theta^{-t}\mathcal{F}_+$ are independent $\sigma$-algebras} \\ &= \ \int_{\theta^{-t}(E)} \bar{\varphi}_{q(\theta^t\omega)}^t(A) \, \mathbb{P}(d\omega) \\ &= \ \int_A \bar{\varphi}_x^t(A) \; q_\ast\mathbb{P}(dx) \\ &= \ q_\ast\mathbb{P}(A) \\ &= \ \mathbb{P}(E). \end{align*} \noindent Hence $\mathbb{P}(E \setminus \theta^{-t}(E))=0$ for each $t$. Therefore, since $E \in \mathcal{F}_+$, Lemma~\ref{empf} gives that $\mathbb{P}(E) \in \{0,1\}$. So $q_\ast\mathbb{P}(A) \in \{0,1\}$, as required. \\ \\
Thus we have shown that $q_\ast\mathbb{P}$ is ergodic with respect to $(\bar{\varphi}_x^t)$. Now suppose that $q_\ast\mathbb{P}$ is not atomless. Then by Lemma~\ref{atomic}, there is a finite forward-invariant set $P \subset \mathbb{S}^1$ such that $q_\ast\mathbb{P} \, = \, \frac{1}{|P|}\sum_{x \in P} \delta_x$. Fix an arbitrary $x \in P$. Let $E:=q^{-1}(\{x\}) \in \mathcal{F}_+$, and for each $t \in \mathbb{T}^+$ let \[ \tilde{E}_t \ := \ \{ \omega \, : \, q(\theta^t\omega) = \varphi(t,\omega)x \}. \] \noindent Once again, $\mathbb{P}(E \triangle \tilde{E}_t)=0$ for each $t$. For each $n \in \mathbb{N}$, for any $F \in \mathcal{F}_n$, we have that \begin{align*}
\mathbb{P}(E \cap F) \ &= \ \int_F \mathbb{P}(E|\mathcal{F}_n)(\omega) \; \mathbb{P}(d\omega) \\
&= \ \int_F \mathbb{P}(\tilde{E}_n|\mathcal{F}_n)(\omega) \; \mathbb{P}(d\omega) \\ &= \ \int_F \mathbb{P}( \hspace{0.2mm} \tilde{\omega} \, : \, q(\theta^n\tilde{\omega}) = \varphi(n,\omega)x \hspace{0.2mm} ) \; \mathbb{P}(d\omega) \\ &\hspace{10mm} \textrm{since $\mathcal{F}_n$ and $\theta^{-n}\mathcal{F}_+$ are independent $\sigma$-algebras} \\ &= \ \int_F q_\ast\mathbb{P} \hspace{0.2mm} ( \hspace{0.2mm} \{\varphi(n,\omega)x\} \hspace{0.2mm} ) \; \mathbb{P}(d\omega) \\
&= \ \int_F \, \tfrac{1}{|P|} \; \mathbb{P}(d\omega) \\
&= \ \tfrac{1}{|P|} \mathbb{P}(F) \\ &= \ \mathbb{P}(E)\mathbb{P}(F). \end{align*}
\noindent So $E$ is independent of $\mathcal{F}_n$ for each $n \in \mathbb{N}$, and therefore $E$ is independent of $\mathcal{F}_+$. In particular, $E$ is independent of itself, and so $\mathbb{P}(E)=1$. Hence $|P|=1$, i.e.~$q_\ast\mathbb{P}$ is a Dirac mass at a deterministic fixed point. \end{proof}
\begin{cor} \label{a or b} If $\mathbb{P}(\Omega_c)=1$ then either case~(a) or case~(b) in the statement of Theorem~\ref{cr char} holds. \end{cor}
\begin{proof} Fix an arbitrary $k \in \mathbb{S}^1$, and define the function $q:\Omega \to \mathbb{S}^1$ by \[ q(\omega) \ = \ \left\{ \begin{array}{c l} \tilde{r}(\omega) & \omega \in \Omega_c \\ k & \omega \in \Omega \setminus \Omega_c. \end{array} \right. \] \noindent By Lemma~\ref{crrfp}, $q$ is an $\mathcal{F}_+$-measurable function. If $\mathbb{P}(\Omega_c)=1$ then by Lemma~\ref{crrfp}, for each $t \in \mathbb{T}^+$, for $\mathbb{P}$-almost all $\omega \in \Omega$, $q(\theta^t\omega)=\varphi(t,\omega)q(\omega)$. Hence Lemma~\ref{Dirrfp} gives the desired result. \end{proof}
\begin{lemma} \label{atomless} If $\varphi$ is stably synchronising then $\varphi$ admits at least one atomless reverse-stationary probability measure. \end{lemma}
\begin{proof} Suppose $\varphi$ is stably synchronising. First suppose that $\varphi$ does \emph{not} have a deterministic fixed point. We know that there exists at least one probability measure $\rho$ that is ergodic with respect to $(\bar{\varphi}_x^t)$; by Corollary~\ref{Dirac}, such a probability measure must be atomless. So now suppose that $\varphi$ \emph{does} have a deterministic fixed point $p$. Let $p' \in \mathbb{R}$ be a lift of $p$, and for each $v \in [0,1]$, let $J_v:=\pi([p',p'+v])$. Define the function $h:\Omega \to [0,1]$ by \[ h(\omega) \ = \ \sup\{ v \in [0,1) \, : \, l(\varphi(t,\omega)J_v) \to 0 \textrm{ as } t \to \infty \}. \] \noindent For any $c \in [0,1)$ and $\omega \in \Omega$, $h(\omega) > c$ if and only if there exists $v \in (c,1) \cap \mathbb{Q}$ such that $l(\varphi(t,\omega)J_v) \to 0$ as $t \to \infty$. Hence $h$ is $\mathcal{F}_+$-measurable. Now we know that for $\mathbb{P}$-almost every $\omega \in \Omega$ there exists an open neighbourhood $U$ of $p$ such that $l(\varphi(t,\omega)U) \to 0$ as $t \to \infty$. Hence $h(\omega) \in (0,1)$ for $\mathbb{P}$-almost all $\omega \in \Omega$. \\ \\ Now define the function $q:\Omega \to \mathbb{S}^1$ by \[ q(\omega) \ = \ \pi(p' + h(\omega)). \] \noindent Since $h$ is $\mathcal{F}_+$-measurable, $q$ is $\mathcal{F}_+$-measurable. Given any $t \in \mathbb{T}^+$ and $\omega \in \Omega$, we have that for all $v \in [0,1)$, \[ l(\varphi(s,\omega)J_v) \to 0 \textrm{ as } s \to \infty \hspace{4mm} \Longleftrightarrow \hspace{4mm} l(\varphi(s,\theta^t\omega) \, (\varphi(t,\omega)J_v) \, ) \to 0 \textrm{ as } s \to \infty \] \noindent and therefore $q(\theta^t\omega)=\varphi(t,\omega)q(\omega)$. Hence, by Lemma~\ref{Dirrfp}, $q_\ast\mathbb{P}$ is ergodic with respect to $(\bar{\varphi}_x^t)$. Moreover, since $h(\omega) \in (0,1)$ for $\mathbb{P}$-almost all $\omega \in \Omega$, $q_\ast\mathbb{P}$ is not equal to $\delta_p$. Since $\varphi$ is synchronising, $\varphi$ cannot have more than one deterministic fixed point, and therefore by Corollary~\ref{Dirac} (or the second statement in Lemma~\ref{Dirrfp}) $q_\ast\mathbb{P}$ must be atomless. \end{proof}
\begin{lemma} \label{mart} Suppose we have an atomless\footnote{The condition that $\rho$ is atomless can in fact be dropped, although the proof then becomes significantly longer, as it is harder to justify that the martingale $(h_t)_{t \in \mathbb{T}^+}$ almost surely has right-continuous sample paths. In any case, we will not need this for our purposes.} reverse-stationary probability measure $\rho$. Then for any connected $J \subset \mathbb{S}^1$, for $\mathbb{P}$-almost all $\omega \in \Omega$, $\rho(\varphi(t,\omega)J)$ is convergent as $t \to \infty$. \end{lemma}
\noindent The main idea of the proof is the same as in [LeJ87, Lemme~1].
\begin{proof} Fix a connected $J \subset \mathbb{S}^1$, and for each $t$ and $\omega$ let $h_t(\omega)=\rho(\varphi(t,\omega)J)$. Note that for each boundary point $x$ of $J$, the map $t \mapsto \varphi(t,\omega)x$ is right-continuous for all $\omega$. Hence, since $\rho$ is atomless, the map $t \mapsto h_t(\omega)$ is right-continuous for all $\omega$. So if we can show that $(h_t)_{t \in \mathbb{T}^+}$ is an $(\mathcal{F}_t)_{t \in \mathbb{T}^+}$-adapted martingale, then the martingale convergence theorem will give the desired result. Fix any $s,t \in \mathbb{T}^+$. We have that \begin{align*}
\mathbb{E}[h_{s+t}|\mathcal{F}_s](\omega) \ &= \ \mathbb{E}[\,\tilde{\omega} \mapsto \rho(\varphi(s+t,\tilde{\omega})J)\,|\,\mathcal{F}_s\,](\omega) \\
&= \ \mathbb{E}[\,\tilde{\omega} \mapsto \rho( \hspace{0.2mm} \varphi(t,\theta^s\tilde{\omega})(\varphi(s,\tilde{\omega})J) \hspace{0.2mm} )\,|\,\mathcal{F}_s\,](\omega) \\ &= \ \mathbb{E}[\,\tilde{\omega} \mapsto \rho( \hspace{0.2mm} \varphi(t,\theta^s\tilde{\omega})(\varphi(s,\omega)J) \hspace{0.2mm} )\,] \\ & \hspace{10mm} \textrm{since $\mathcal{F}_s$ and $\theta^{-s}\mathcal{F}_t$ are independent $\sigma$-algebras} \\ &= \ \rho(\varphi(s,\omega)J) \\ & \hspace{10mm} \textrm{since $\rho$ is reverse-stationary} \\ &= \ h_s(\omega). \end{align*} \noindent So we are done. \end{proof}
\begin{lemma} \label{1atomless} If $\varphi$ is stably synchronising then $\varphi$ admits at least one atomless reverse-stationary probability measure. \end{lemma}
\begin{proof} Suppose $\varphi$ is stably synchronising. First suppose that $\varphi$ does \emph{not} have a deterministic fixed point. We know that there exists at least one probability measure $\rho$ that is ergodic with respect to $(\bar{\varphi}_x^t)$; by Corollary~\ref{Dirac}, such a probability measure must be atomless. So now suppose that $\varphi$ \emph{does} have a deterministic fixed point $p$. Let $p' \in \mathbb{R}$ be a lift of $p$, and for each $v \in [0,1]$, let $J_v:=\pi([p',p'+v])$. Define the function $h:\Omega \to [0,1]$ by \[ h(\omega) \ = \ \sup\{ v \in [0,1) \, : \, l(\varphi(t,\omega)J_v) \to 0 \textrm{ as } t \to \infty \}. \] \noindent For any $c \in [0,1)$ and $\omega \in \Omega$, $h(\omega) > c$ if and only if there exists $v \in (c,1) \cap \mathbb{Q}$ such that $l(\varphi(t,\omega)J_v) \to 0$ as $t \to \infty$. Hence $h$ is $\mathcal{F}_+$-measurable. Now we know that for $\mathbb{P}$-almost every $\omega \in \Omega$ there exists an open neighbourhood $U$ of $p$ such that $l(\varphi(t,\omega)U) \to 0$ as $t \to \infty$. Hence $h(\omega) \in (0,1)$ for $\mathbb{P}$-almost all $\omega \in \Omega$. \\ \\ Now define the function $q:\Omega \to \mathbb{S}^1$ by \[ q(\omega) \ = \ \pi(p' + h(\omega)). \] \noindent Since $h$ is $\mathcal{F}_+$-measurable, $q$ is $\mathcal{F}_+$-measurable. Given any $t \in \mathbb{T}^+$ and $\omega \in \Omega$, we have that for all $v \in [0,1)$, \[ l(\varphi(s,\omega)J_v) \to 0 \textrm{ as } s \to \infty \hspace{4mm} \Longleftrightarrow \hspace{4mm} l(\varphi(s,\theta^t\omega) \, (\varphi(t,\omega)J_v) \, ) \to 0 \textrm{ as } s \to \infty \] \noindent and therefore $q(\theta^t\omega)=\varphi(t,\omega)q(\omega)$. Hence, by Lemma~\ref{Dirrfp}, $q_\ast\mathbb{P}$ is ergodic with respect to $(\bar{\varphi}_x^t)$. Moreover, since $h(\omega) \in (0,1)$ for $\mathbb{P}$-almost all $\omega \in \Omega$, $q_\ast\mathbb{P}$ is not equal to $\delta_p$. Since $\varphi$ is synchronising, $\varphi$ cannot have more than one deterministic fixed point, and therefore by Corollary~\ref{Dirac} (or the second statement in Lemma~\ref{Dirrfp}) $q_\ast\mathbb{P}$ must be atomless. \end{proof}
\begin{lemma} \label{cr1} Suppose $\varphi$ is synchronising and admits an atomless reverse-stationary probability measure $\rho$. Then $\mathbb{P}(\Omega_c)=1$. \end{lemma}
\begin{proof} Given any connected $J \subset \mathbb{S}^1$ with $0<l(J)<1$, writing $\partial J \! =: \! \{x,y\}$, for $\mathbb{P}$-almost all $\omega \in \Omega$ we have the following: $d(\varphi(t,\omega)x,\varphi(t,\omega)y) \to 0$ as $t \to \infty$ (since $\varphi$ is synchronising), and $\rho(\varphi(t,\omega)J)$ is convergent as $t \to \infty$ (by Lemma~\ref{mart}); and therefore (since $\rho$ is atomless) $l(\varphi(t,\omega)J)$ converges to either 0 or 1 as $t \to \infty$. Now fix an arbitrary $k \in \mathbb{R}$, and for each $v \in [0,1]$, let $J_v:=\pi([k,k+v])$. Let $\Omega' \subset \Omega$ be a $\mathbb{P}$-full set such that for each $\omega \in \Omega'$ and $v \in [0,1] \cap \mathbb{Q}$, $l(\varphi(t,\omega)J_v)$ converges to either 0 or 1 as $t \to \infty$. For each $\omega \in \Omega'$, let \begin{align*} c(\omega) \ :=& \ \sup \{ v \in [0,1] : l(\varphi(t,\omega)J_v) \to 0 \textrm{ as } t \to \infty \} \\ =& \ \hspace{1.2mm} \inf \{ v \in [0,1] : l(\varphi(t,\omega)J_v) \to 1 \textrm{ as } t \to \infty \}. \end{align*} \noindent It is clear that for each $\omega \in \Omega'$, $\pi(k+c(\omega))$ is a crack point of $\omega$. So we are done. \end{proof}
\noindent Combining Lemmas~\ref{1atomless} and \ref{cr1} gives that if $\varphi$ is stably synchronising then $\mathbb{P}(\Omega_c)=1$.
\begin{lemma} Suppose $\mathbb{P}(\Omega_c)=1$. Then $\varphi$ is stably synchronising if and only if case~(a) in the statement of Theorem~\ref{cr char} holds. \end{lemma}
\begin{proof} For any $x,y \in \mathbb{S}^1$ and $\omega \in \Omega_c$, if $\tilde{r}(\omega) \neq x$ and $\tilde{r}(\omega) \neq y$ then $x \sim_\omega y$. Hence it is clear that in case~(a) in the statement of Theorem~\ref{cr char}, $\varphi$ is synchronising. For any $x \in \mathbb{S}^1$ and $\omega \in \Omega_c$, if $\tilde{r}(\omega) \neq x$ then there obviously exists a neighbourhood $U$ of $x$ such that $\mathrm{diam}(\varphi(t,\omega)U) \to 0$ as $t \to \infty$. Hence, in case~(a) in the statement of Theorem~\ref{cr char}, $\varphi$ is everywhere locally stable. Thus, in case~(a) in the statement of Theorem~\ref{cr char}, $\varphi$ is stably synchronising. Now if there exists $p \in \mathbb{S}^1$ such that $\mathbb{P}(\omega \in \Omega_c \, : \, \tilde{r}(\omega) = p) \, > \, 0$, then $\varphi$ is clearly not everywhere locally stable (since the point $p$ serves as a counterexample), and so $\varphi$ is not stably synchronising. \end{proof}
\noindent Now we will say that a probability measure $\rho$ on $\mathbb{S}^1$ is \emph{forward-stationary} if for all $t \in \mathbb{T}^+$ and $A \in \mathcal{B}(\mathbb{S}^1)$, \[ \rho(A) \ = \ \int_\Omega \rho(\varphi(t,\omega)^{-1}(A)) \, \mathbb{P}(d\omega). \] Again, by the Krylov-Bogolyubov theorem, there is at least one forward-stationary probability measure.
\begin{lemma} \label{rep} Let $q:\Omega \to \mathbb{S}^1$ be as in Lemma~\ref{Dirrfp}, and suppose that $q_\ast\mathbb{P}$ is atomless. Let $\rho$ be any forward-stationary probability measure. Then for $\mathbb{P}$-almost every $\omega \in \Omega$, $\rho(x \in \mathbb{S}^1 : x \sim_\omega q(\omega))=0$. \end{lemma}
\begin{proof} Define the functions \begin{align*} \Theta \, &: \, \Omega \times \mathbb{S}^1 \, \to \, \Omega \times \mathbb{S}^1 \\ \Theta_2 \, &: \, \Omega \times \mathbb{S}^1 \times \mathbb{S}^1 \, \to \, \Omega \times \mathbb{S}^1 \times \mathbb{S}^1 \end{align*} \noindent
by \begin{align*} \Theta(\omega,x) \ &= \ (\theta^1\omega,\varphi(1,\omega)x) \\ \Theta_2(\omega,x,y) \ &= \ (\theta^1\omega,\varphi(1,\omega)x,\varphi(1,\omega)y). \end{align*}
\noindent It is not hard to show (e.g.~as in [New15, Theorem~143(i)] or [Kif86, Lemma~1.2.3]) that $\Theta$ is a measure-preserving transformation of $\left(\Omega \times \mathbb{S}^1 \, , \, \mathcal{F}_+ \otimes \mathcal{B}(\mathbb{S}^1) \, , \, \mathbb{P}|_{\mathcal{F}_+} \otimes \rho\right)$. Now define the probability measure $\mathfrak{p}$ on the measurable space $\left(\Omega \times \mathbb{S}^1 \times \mathbb{S}^1 \, , \, \mathcal{F}_+ \otimes \mathcal{B}(\mathbb{S}^1 \times \mathbb{S}^1)\right)$ by \[ \mathfrak{p}(A) \ := \ \mathbb{P} \otimes \rho( \, (\omega,x) \in \Omega \times \mathbb{S}^1 \, : \, (\omega,x,q(\omega)) \in A \, ). \] \noindent For any $A \in \mathcal{F}_+ \otimes \mathcal{B}(\mathbb{S}^1 \times \mathbb{S}^1)$, since $q$ is $\mathcal{F}_+$-measurable, the set $\{(\omega,x) : (\omega,x,q(\omega)) \in A\}$ is $(\mathcal{F}_+ \otimes \mathcal{B}(\mathbb{S}^1))$-measurable. With this, we have \begin{align*} \mathfrak{p}(\Theta_2^{-1}(A)) \ &= \ \mathbb{P} \otimes \rho( \, (\omega,x) \in \Omega \times \mathbb{S}^1 \, : \, (\theta^1\omega,\varphi(1,\omega)x,\varphi(1,\omega)q(\omega)) \in A \, ) \\ &= \ \mathbb{P} \otimes \rho( \, (\omega,x) \in \Omega \times \mathbb{S}^1 \, : \, (\theta^1\omega,\varphi(1,\omega)x,q(\theta^1\omega)) \in A \, ) \\ &= \ \mathbb{P} \otimes \rho( \, \Theta^{-1\!}\{(\omega,x) \in \Omega \times \mathbb{S}^1 \, : \, (\omega,x,q(\omega)) \in A\} \, ) \\ &= \ \mathbb{P} \otimes \rho( \, (\omega,x) \in \Omega \times \mathbb{S}^1 \, : \, (\omega,x,q(\omega)) \in A \, ) \\ &= \ \mathfrak{p}(A). \end{align*} \noindent So $\mathfrak{p}$ is $\Theta_2$-invariant. Now let $\Delta:=\{(x,x):x \in \mathbb{S}^1\}$. Since $q_\ast\mathbb{P}$ is atomless, we have that \begin{align*} \mathfrak{p}(\Omega \times \Delta) \ &= \ \mathbb{P} \otimes \rho( \, (\omega,x) \in \Omega \times \mathbb{S}^1 \, : \, q(\omega)=x \, ) \\ &= \ \int_{\mathbb{S}^1} \mathbb{P}(\omega \in \Omega : q(\omega)=x) \, \rho(dx) \\ &= \ 0. \end{align*} \noindent So letting $U_\varepsilon := \{(x,y) \in \mathbb{S}^1 \times \mathbb{S}^1 : d(x,y) < \varepsilon\}$ for each $\varepsilon>0$, we have that $\mathfrak{p}(\Omega \times U_\varepsilon) \to 0$ as $\varepsilon \to 0$. Hence the set \begin{align*} K \ :=& \ \{ \, (\omega,x,y) \in \Omega \times \mathbb{S}^1 \times \mathbb{S}^1 \, : \, d(\varphi(n,\omega)x,\varphi(n,\omega)y) \to 0 \textrm{ as } n \to \infty \, \} \\ =& \ \bigcap_{n=1}^\infty \, \bigcup_{i=1}^\infty \, \bigcap_{j=i}^\infty \; \Theta_2^{-j}(U_{\frac{1}{n}}) \end{align*} \noindent is a $\mathfrak{p}$-null set. Hence the set \[ L \ := \ \{ \, (\omega,x) \in \Omega \times \mathbb{S}^1 \, : \, d(\varphi(n,\omega)x,\varphi(n,\omega)q(\omega)) \to 0 \textrm{ as } n \to \infty \, \} \] \noindent is a $(\mathbb{P} \otimes \rho)$-null set. So (with Fubini's theorem) we are done. \end{proof}
\begin{cor} If $\varphi$ is stably synchronising then for $\mathbb{P}$-almost all $\omega \in \Omega_c$, $\tilde{r}(\omega)$ is a repulsive crack point of $\omega$. \end{cor}
\begin{proof} Taking $q$ to be as in the proof of Corollary~\ref{a or b}, the result follows immediately from Lemma~\ref{rep} and the existence of a forward-stationary probability measure. \end{proof}
\section{Invariant measures}
We will describe the set of invariant random probability measures for $\varphi$ in the case that $\mathbb{P}(\Omega_c)=1$ (and in particular, in the case that $\varphi$ is stably synchronising). We begin with motivation from the deterministic setting: \\ \\ An \emph{autonomous flow of the circle} is a $\mathbb{T}^+$-indexed family $(f^t)_{t \in \mathbb{T}^+}$ of orientation-preserving homeomorphisms $f^t:\mathbb{S}^1 \to \mathbb{S}^1$ such that $f^{s+t}=f^t \circ f^s$ for all $s,t \in \mathbb{T}^+$ and $f^0$ is the identity function. Given an autonomous flow of the circle $(f^t)$, we will say that a point $p \in \mathbb{S}^1$ is a \emph{fixed point} if $f^t(p)=p$ for all $t \in \mathbb{T}^+$. Note that if $\Omega$ is a singleton $\{\omega\}$ then $(\varphi(t,\omega))_{t \in \mathbb{T}^+}$ is an autonomous flow of the circle. \\ \\ Let $\mathcal{M}_1$ be the set of probability measures on $\mathbb{S}^1$. Now, heuristically, the map $x \mapsto \delta_x$ serves as a natural way of identifying points in $\mathbb{S}^1$ with measures on $\mathbb{S}^1$. On the basis of this identification, given an autonomous flow of the circle $(f^t)$, for each $t \in \mathbb{T}^+$ the map $\rho \mapsto f^t_\ast\rho$ on $\mathcal{M}_1$ serves as a natural way of ``lifting'' $f^t$ from $\mathbb{S}^1$ to $\mathcal{M}_1$, since $f^t_\ast\delta_x=\delta_{f^t(x)}$ for all $x \in \mathbb{S}^1$. In particular, a point $p \in \mathbb{S}^1$ is a fixed point of $(f^t)$ if and only if $\delta_p$ is an invariant measure (and in fact, an ergodic invariant measure) of $(f^t)$. \\ \\ We will say that an autonomous flow of the circle $(f^t)$ is \emph{simple} if there exist distinct points $r,a \in \mathbb{S}^1$ (respectively called the \emph{repeller} and the \emph{attractor} of $(f^t)$) such that $r$ is a fixed point and for all $x \in \mathbb{S}^1 \setminus \{r\}$, $f^t(x) \to a$ as $t \to \infty$. (It follows that $a$ is also a fixed point.) In this case, it is easy to show that the set of invariant probability measures for $(f^t)$ is given by $\{\lambda \delta_r + (1-\lambda) \delta_a : \lambda \in [0,1]\}$. We will refer to the pair $(a,r)$ as a \emph{global attractor-repeller pair} for $(f^t)$; the basis for this terminology that ``$r$ repels all points in $\mathbb{S}^1$ (other than itself) towards $a$''. \\ \\ If $\Omega$ is a singleton $\{\omega\}$, the flow $(\varphi(t,\omega))_{t \in \mathbb{T}^+}$ is simple if and only if $\omega$ admits a repulsive crack point $r$, in which case $r$ is the repeller of $(\varphi(t,\omega))_{t \in \mathbb{T}^+}$. Obviously, in this case, $\varphi$ is not synchronising, since the repeller and attractor of $(\varphi(t,\omega))_{t \in \mathbb{T}^+}$ are distinct deterministic fixed points of $\varphi$. (Note that it is impossible for $\varphi$ to be stably synchronising when $\Omega$ is a singleton.) \\ \\ We now go on to extend the notion of ``simplicity'' to the random setting, and show that if $\varphi$ is stably synchronising then $\varphi$ is simple. We will need a couple of assumptions. \\ \\ \textbf{Assumption A:} For all $t \in \mathbb{T}^+$, $\theta^t$ is a measurable automorphism of the measurable space $(\Omega,\mathcal{F})$. \\ \\ The heuristic interpretation of this assumption is that the underlying noise process has been going on since eternity past. We refer to $\mathcal{F}_+$ as the \emph{future $\sigma$-algebra}, and we define the \emph{past $\sigma$-algebra} by $\mathcal{F}_- \, := \, \sigma(\theta^t\mathcal{F}_t:t \in \mathbb{T}^+) \subset \mathcal{F}$. It is not hard to show that $\mathcal{F}_+$ and $\mathcal{F}_-$ are independent (according to $\mathbb{P}$). \\ \\ Let $\tilde{\mathcal{F}}:=\sigma(\theta^t\mathcal{F}_s:s,t \in \mathbb{T}^+)$. Note that $\mathcal{F}_+$ and $\mathcal{F}_-$ are both sub-$\sigma$-algebras of $\tilde{\mathcal{F}}$, and that for all $t \in \mathbb{T}^+$, $\theta^t$ is a measurable automorphism of $(\Omega,\tilde{\mathcal{F}})$. Moreover, we have the following:\footnote{Lemma~5.1 is a generalisation of some of the content of the section ``Invariant and Tail $\sigma$-Algebra'' on p547 of [Arn98].}
\begin{lemma}
For all $t \in \mathbb{T}^+ \setminus \{0\}$, $\theta^t$ is an ergodic measure-preserving transformation of the probability space $(\Omega,\tilde{\mathcal{F}},\mathbb{P}|_{\tilde{\mathcal{F}}})$. \end{lemma}
\begin{proof}
Fix $t \in \mathbb{T}^+ \setminus \{0\}$. For each $n \in \mathbb{Z}$, let $\mathcal{G}_n=\theta^{nt}\mathcal{F}_+$. (Note that $\mathcal{G}_n$ is increasing in $n$.) Let $\mathcal{G}_{-\infty}:=\bigcap_{n \in \mathbb{Z}} \mathcal{G}_n$, and observe that $\tilde{\mathcal{F}}=\sigma(\bigcup_{n \in \mathbb{Z}} \mathcal{G}_n)$. Let $E \in \tilde{\mathcal{F}}$ be a set with $\theta^{-t}(E)=E$, and let $g:\Omega \to [0,1]$ be a version of $\mathbb{P}(E|\mathcal{F}_+)$; so for every $n \in \mathbb{Z}$, $g \circ \theta^{nt}$ is a version of $\mathbb{P}(E|\mathcal{G}_{-n})$. By a version of the Kolmogorov 0-1 law (e.g.~[New15, Proposition~132]), $\mathcal{G}_{-\infty}$ is a $\mathbb{P}$-trivial $\sigma$-algebra, and so the constant map $\,\omega \mapsto \mathbb{P}(E)$ is a version of $\mathbb{P}(E|\mathcal{G}_{-\infty})$. Therefore, by L\'{e}vy's downward theorem ([Will91, Theorem~14.4]), $g \circ \theta^{nt}(\omega) \to \mathbb{P}(E)$ as $n \to \infty$ for $\mathbb{P}$-almost all $\omega \in \Omega$. But since $\theta^t$ is itself $\mathbb{P}$-preserving, it follows that for each $n \in \mathbb{Z}$, $g \circ \theta^{nt}(\omega) = \mathbb{P}(E)$ for $\mathbb{P}$-almost all $\omega \in \Omega$. So the constant map $\,\omega \mapsto \mathbb{P}(E)$ is a version of $\mathbb{P}(E|\mathcal{G}_n)$ for each $n$, i.e.~$E$ is independent of $\mathcal{G}_n$ for each $n$. It follows that $E$ is independent of $\tilde{\mathcal{F}}$. In particular, $E$ is independent of itself, and so $\mathbb{P}(E) \in \{0,1\}$. \end{proof}
\noindent Hence (by reducing our underlying probability space to $(\Omega,\tilde{\mathcal{F}},\mathbb{P}|_{\tilde{\mathcal{F}}})$ if necessary) we may add the following assumption without loss of generality: \\ \\ \textbf{Assumption B:} $\mathbb{P}$ is an ergodic measure of the dynamical system $(\theta^t)_{t \in \mathbb{T}^+}$ on $(\Omega,\mathcal{F})$. \\ \\ \textbf{Throughout Section~4, we will work with Assumptions~A and B.} \\ \\ We equip $\mathcal{M}_1$ with the ``evaluation $\sigma$-algebra'' $\mathfrak{K}:=\sigma(\rho \mapsto \rho(A) : A \in \mathcal{B}(\mathbb{S}^1))$. So for any measurable space $(E,\mathcal{E})$, a function $f:E \to \mathcal{M}_1$ is measurable if and only if the map $\,\xi \mapsto f(\xi)(A)\,$ from $E$ to $[0,1]$ is measurable for all $A \in \mathcal{B}(\mathbb{S}^1)$. It is not hard to show that the map $x \mapsto \delta_x$ is a measurable embedding of $\mathbb{S}^1$ into $\mathcal{M}_1$---that is to say, a set $A \subset \mathbb{S}^1$ is $\mathcal{B}(\mathbb{S}^1)$-measurable if and only if the set $\{\delta_x : x \in A\}$ is $\mathfrak{K}$-measurable.\footnote{On the one hand, the map $x \mapsto \delta_x$ is obviously measurable, and so if $\{\delta_x : x \in A\}$ is measurable then $A$ is measurable. On the other hand, for any $A \in \mathcal{B}(\mathbb{S}^1)$, $\{\delta_x:x \in A\}$ is precisely the set of probability measures $\rho$ for which $\rho \otimes \rho(\{(x,x): x \in A\})=1$. Now the set $\{(x,x): x \in A\}=\Delta \cap (A \times A)$ is obviously measurable, and it is not hard to show that the map $\rho \mapsto \rho \otimes \rho$ is measurable (with respect to the respective evaluation $\sigma$-algebras); therefore $\{\delta_x:x \in A\}$ is measurable.}
\begin{defi} A \emph{random probability measure} (\emph{on $\mathbb{S}^1$}) is an $\Omega$-indexed family $(\mu_\omega)_{\omega \in \Omega}$ of probability measures $\mu_\omega$ on $\mathbb{S}^1$ such that the map $\omega \mapsto \mu_\omega$ from $\Omega$ to $\mathcal{M}_1$ is measurable (i.e.~such that for each $A \in \mathcal{B}(\mathbb{S}^1)$, the map $\omega \mapsto \mu_\omega(A)$ from $\Omega$ to $[0,1]$ is measurable). We say that two random probability measures $(\mu_\omega)_{\omega \in \Omega}$ and $(\mu'_\omega)_{\omega \in \Omega}$ are \emph{equivalent} if $\mu_\omega=\mu'_\omega$ for $\mathbb{P}$-almost all $\omega \in \Omega$. \end{defi}
\begin{defi} We will say that a probability measure $\mu$ on the measurable space $(\Omega \times \mathbb{S}^1,\mathcal{F} \otimes \mathcal{B}(\mathbb{S}^1))$ is \emph{compatible} if $\mu(E \times \mathbb{S}^1)=\mathbb{P}(E)$ for all $E \in \mathcal{F}$. We write $\mathcal{M}_1^\mathbb{P}$ for the set of compatible probability measures. \end{defi}
\noindent The \emph{disintegration theorem} (e.g.~[Crau02a, Proposition~3.6]) states that for every compatible probability measure $\mu$ there exists a random probability measure $(\mu_\omega)_{\omega \in \Omega}$ (unique up to equivalence) such that \[ \mu(A) \ = \ \int_\Omega \mu_\omega(A_\omega) \, \mathbb{P}(d\omega) \] \noindent for all $A \in \mathcal{F} \otimes \mathcal{B}(\mathbb{S}^1)$, where $A_\omega$ denotes the $\omega$-section of $A$. The random probability measure $(\mu_\omega)$ is called \emph{a (version of the) disintegration of $\mu$}, and we will refer to $\mu$ as \emph{the integrated form of $(\mu_\omega)$}.
\begin{lemma} \label{ms} Let $\mu^1$ and $\mu^2$ be compatible probability measures, with $(\mu^1_\omega)$ and $(\mu^2_\omega)$ being disintegrations of $\mu^1$ and $\mu^2$ respectively. If $\mu^1$ and $\mu^2$ are mutually singular then for $\mathbb{P}$-almost every $\omega \in \Omega$, $\mu^1_\omega$ and $\mu^2_\omega$ are mutually singular. \end{lemma}
\begin{proof} Suppose we have a set $A \in \mathcal{F} \otimes \mathcal{B}(\mathbb{S}^1)$ such that $\mu^1(A)=1$ and $\mu^2(A)=0$. Then it is clear that for $\mathbb{P}$-almost all $\omega \in \Omega$, $\mu^1_\omega(A_\omega)=1$ and $\mu^2_\omega(A_\omega)=0$. So we are done. \end{proof}
\begin{defi} For any measurable function $q:\Omega \to \mathbb{S}^1$, we define the probability measure $\boldsymbol{\delta}_q$ on $(\Omega \times \mathbb{S}^1,\mathcal{F} \otimes \mathcal{B}(\mathbb{S}^1))$ by $\boldsymbol{\delta}_q(A)=\mathbb{P}(\omega \in \Omega : (\omega,q(\omega)) \in A)$. \end{defi}
\noindent Note that $\boldsymbol{\delta}_q$ is a compatible probability measure, and admits the disintegration $(\delta_{q(\omega)})_{\omega \in \Omega}$. \\ \\ We will say that two measurable functions $q,q':\Omega \to \mathbb{S}^1$ are \emph{equivalent} if $q(\omega)=q'(\omega)$ for $\mathbb{P}$-almost all $\omega \in \Omega$. Let $L^0(\mathbb{P},\mathbb{S}^1)$ denote the set of equivalence classes of measurable functions from $\Omega$ to $\mathbb{S}^1$; and for any measurable $q:\Omega \to \mathbb{S}^1$, let $\hat{q} \in L^0(\mathbb{P},\mathbb{S}^1)$ denote the equivalence class represented by $q$. For any sub-$\sigma$-algebra $\mathcal{G}$ of $\mathcal{F}$, we say that an element $\hat{q}$ of $ L^0(\mathbb{P},\mathbb{S}^1)$ is $\mathcal{G}$-measurable if it admits a representative $q:\Omega \to \mathbb{S}^1$ that is $\mathcal{G}$-measurable. \\ \\ Heuristically, $L^0(\mathbb{P},\mathbb{S}^1)$ can be viewed as the set of ``random points in $\mathbb{S}^1$'', identified up to equivalence; and by the disintegration theorem, $\mathcal{M}_1^\mathbb{P}$ can be regarded as the set of random probability measures on $\mathbb{S}^1$, identified up to equivalence. Note that the map $\hat{q} \mapsto \boldsymbol{\delta}_q$ from $L^0(\mathbb{P},\mathbb{S}^1)$ to $\mathcal{M}_1^\mathbb{P}$ is well-defined and injective. Thus, heuristically, this map serves as a natural way of identifying random points in $\mathbb{S}^1$ with random measures on $\mathbb{S}^1$. \\ \\ Now for each $t \in \mathbb{T}^+$, define the map $\Phi^t:L^0(\mathbb{P},\mathbb{S}^1) \to L^0(\mathbb{P},\mathbb{S}^1)$ by \[ r \in \Phi^t(\hat{q}) \hspace{3mm} \Longleftrightarrow \hspace{3mm} r(\omega) \, = \, \varphi(t,\theta^{-t}\omega)q(\theta^{-t}\omega) \textrm{ for $\mathbb{P}$-almost all $\omega \in \Omega$}. \] \noindent (It is easy to show that since $\mathbb{P}$ is $\theta^t$-invariant, this is indeed a well-defined map.) One can easily check that $\Phi^0$ is the identity function and $\Phi^{s+t}=\Phi^t \circ \Phi^s$ for all $s,t \in \mathbb{T}^+$. If $\Omega$ is a singleton $\{\omega\}$ then we may identify $L^0(\mathbb{P},\mathbb{S}^1)$ with $\mathbb{S}^1$ in the obvious manner, in which case $\Phi^t$ is simply equal to $\varphi(t,\omega)$ for all $t$.
\begin{defi} An element $p$ of $L^0(\mathbb{P},\mathbb{S}^1)$ is called a \emph{random fixed point} (\emph{of $\varphi$}) if $\Phi^t(p)=p$ for all $t \in \mathbb{T}^+$. In this case, for convenience, we will also refer to any representative of $p$ as a random fixed point. \end{defi}
\noindent Note that a measurable function $q:\Omega \to \mathbb{S}^1$ is a random fixed point if and only if for each $t \in \mathbb{T}^+$, for $\mathbb{P}$-almost all $\omega \in \Omega$, $\varphi(t,\omega)q(\omega)=q(\theta^t\omega)$. (This is precisely the property described in the statement of Lemma~\ref{Dirrfp}.) \\ \\ We now describe how to ``lift'' the dynamics of $(\Phi^t)_{t \in \mathbb{T}^+}$ from $L^0(\mathbb{P},\mathbb{S}^1)$ onto $\mathcal{M}_1^\mathbb{P}$. For each $t \in \mathbb{T}^+$, define the map $\Theta^t:\Omega \times \mathbb{S}^1 \to \Omega \times \mathbb{S}^1$ by \[ \Theta^t(\omega,x) \ = \ (\theta^t\omega,\varphi(t,\omega)x). \]
\noindent Note that $(\Theta^t)_{t \in \mathbb{T}^+}$ is a dynamical system on the measurable space $(\Omega \times \mathbb{S}^1,\mathcal{F} \otimes \mathcal{B}(\mathbb{S}^1))$---that is to say: $\Theta^0$ is the identity function; $\Theta^{s+t}=\Theta^t \circ \Theta^s$ for all $s,t \in \mathbb{T}^+$; and $\Theta^t$ is a measurable self-map of $\Omega \times \mathbb{S}^1$ for all $t \in \mathbb{T}^+$.
\begin{lemma} \label{lift} For any compatible probability measure $\mu$ with disintegration $(\mu_\omega)$ and any $t \in \mathbb{T}^+$, $\Theta^t_\ast\mu$ is a compatible probability measure with disintegration $(\varphi(t,\theta^{-t}\omega)_\ast\mu_{\theta^{-t}\omega})$. \end{lemma}
\noindent In particular, for any measurable $q:\Omega \to \mathbb{S}^1$ and any $t \in \mathbb{T}^+$, we have that $\Theta^t_\ast\boldsymbol{\delta}_q=\boldsymbol{\delta}_r$ where $r$ is a representative of $\Phi_t(\hat{q})$. Thus the semigroup $(\Theta^t_\ast)_{t \in \mathbb{T}^+}$ on $\mathcal{M}_1^\mathbb{P}$ serves as a natural ``lift'' of the semigroup $(\Phi^t)_{t \in \mathbb{T}^+}$ on $L^0(\mathbb{P},\mathbb{S}^1)$. For a proof of Lemma~\ref{lift}, see [Arn98, Lemma~1.4.4].
\begin{defi} A probability measure $\mu$ on $(\Omega \times \mathbb{S}^1,\mathcal{F} \otimes \mathcal{B}(\mathbb{S}^1))$ is called an \emph{invariant (probability) measure of $\varphi$} if $\mu$ is compatible and is an invariant measure of the dynamical system $(\Theta^t)_{t \in \mathbb{T}^+}$. In this case, we will refer to any disintegration $(\mu_\omega)$ of $\mu$ as an \emph{invariant random probability measure of $\varphi$}. \end{defi}
\noindent Note that a random probability measure $(\mu_\omega)$ is invariant if and only if for each $t \in \mathbb{T}^+$, for $\mathbb{P}$-almost all $\omega \in \Omega$, $\varphi(t,\omega)_\ast\mu_\omega=\mu_{\theta^t\omega}$. \\ \\ Obviously, a measurable function $q:\Omega \to \mathbb{S}^1$ is a random fixed point if and only if $\boldsymbol{\delta}_q$ is an invariant measure. Moreover (as a consequence of Assumption~B) we have the following:
\begin{lemma} \label{rfperg} For any random fixed point $q:\Omega \to \mathbb{S}^1$, $\boldsymbol{\delta}_q$ is ergodic with respect to $(\Theta^t)_{t \in \mathbb{T}^+}$. \end{lemma}
\begin{proof} Fix any $A \in \mathcal{F} \otimes \mathcal{B}(\mathbb{S}^1)$, and for each $t \in \mathbb{T}^+$, let \[ E_t \ := \ \{ \omega \in \Omega : \Theta^t(\omega,q(\omega)) \in A \}. \] \noindent Note that for every $t \in \mathbb{T}^+$, $\boldsymbol{\delta}_q(\Theta^{-t}(A) \triangle A)=\mathbb{P}(E_t \triangle E_0)$. Now since $q$ is a random fixed point, we have that for each $t \in \mathbb{T}^+$, $\mathbb{P}(E_t \triangle \theta^{-t}(E_0))=0$. \\ \\ So, if $\boldsymbol{\delta}_q(\Theta^{-t}(A) \triangle A)=0$ for all $t \in \mathbb{T}^+$, then $\mathbb{P}(\theta^{-t}(E_0) \triangle E_0)=0$ for all $t \in \mathbb{T}^+$, so $\mathbb{P}(E_0) \in \{0,1\}$ (since $\mathbb{P}$ is $(\theta^t)$-ergodic), so $\boldsymbol{\delta}_q(A) \in \{0,1\}$. Thus $\boldsymbol{\delta}_q$ is $(\Theta^t)$-ergodic. \end{proof}
\noindent We will now describe the set of invariant measures of $\varphi$ when $\mathbb{P}(\Omega_c)=1$.
\begin{thm} \label{simple} Suppose $\mathbb{P}(\Omega_c)=1$, and let $r:\Omega \to \mathbb{S}^1$ be a measurable function with $r(\omega)=\tilde{r}(\omega)$ for all $\omega \in \Omega_c$. (By Lemma~\ref{crrfp}, $\hat{r}$ is an $\mathcal{F}_+$-measurable random fixed point.) Either: \begin{enumerate}[\indent (A)] \item $\boldsymbol{\delta}_r$ is the only $\varphi$-invariant probability measure; or \item there exists an $\mathcal{F}_-$-measurable random fixed point $a:\Omega \to \mathbb{S}^1$, with $a(\omega) \neq r(\omega)$ for $\mathbb{P}$-almost every $\omega \in \Omega$, such that the set of invariant measures of $\varphi$ is given by $\{\lambda \boldsymbol{\delta}_r + (1-\lambda) \boldsymbol{\delta}_a : \lambda \in [0,1]\}$. \end{enumerate} \end{thm}
\noindent Obviously, in case~(B), we have that for $\mathbb{P}$-almost every $\omega \in \Omega_c$, for all $x \in \mathbb{S}^1 \setminus \{r(\omega)\}$, $x \sim_\omega a(\omega)$.
\begin{defi} If $\mathbb{P}(\Omega_c)=1$ and case~(B) of Theorem~\ref{simple} holds, then we will say that $\varphi$ is \emph{simple}. In this case, letting $r$ and $a$ be as in Theorem~\ref{simple}, we refer to the pair $(\hat{a},\hat{r})$ as the \emph{global random attractor-repeller pair} of $\varphi$. \end{defi}
\noindent Before proving Theorem~\ref{simple}, it will be useful to introduce the following definition (taken from [KN04]):
\begin{defi} The \emph{spread} $D(\rho)$ of a probability measure $\rho$ on $\mathbb{S}^1$ is defined as \[ D(\rho) \ := \ \inf \{ v > 0 \, : \, \exists \, \textrm{closed connected } J \subset \mathbb{S}^1 \textrm{ with } l(J) < v \textrm{ and } \rho(J)>1-v \}. \] \end{defi}
\noindent It is not hard to show (by considering connected sets with rational endpoints) that $D(\cdot)$ serves as a measurable map from $\mathcal{M}_1$ to $[0,\frac{1}{2}]$, and that $D(\rho)=0$ if and only if $\rho$ is a Dirac mass.
\begin{proof}[Proof of Theorem~\ref{simple}] Suppose $\boldsymbol{\delta}_r$ is not the only invariant measure, and let $\mu$ be an invariant measure distinct from $\boldsymbol{\delta}_r$. Let $\mu^a$ and $\mu^s$ denote respectively the absolutely continuous and singular parts of the Radon-Nikodym decomposition of $\mu$ with respect to $\boldsymbol{\delta}_r$; note that $\mu^a$ and $\mu^s$ are themselves invariant under the dynamical system $(\Theta^t)_{t \in \mathbb{T}^+}$. By Lemma~\ref{rfperg}, $\boldsymbol{\delta}_r$ is ergodic with respect to $(\Theta^t)$ and therefore $\mu^a$ must be a scalar multiple of $\boldsymbol{\delta}_r$. Hence the probability measure $\nu$ on $\Omega \times \mathbb{S}^1$ given by $\nu(A)=\frac{1}{\mu^s(\Omega \times \mathbb{S}^1)}\mu^s(A)$ is compatible, and is therefore an invariant measure of $\varphi$. Let $(\nu_\omega)_{\omega \in \Omega}$ be a disintegration of $\nu$. By Lemma~\ref{ms}, $\nu_\omega(\{r(\omega)\})=0$ for $\mathbb{P}$-almost all $\omega \in \Omega$. Hence $D(\varphi(t,\omega)_\ast\nu_\omega) \to 0$ as $t \to \infty$ for $\mathbb{P}$-almost every $\omega \in \Omega$. Since $\nu$ is invariant, this implies that for $\mathbb{P}$-almost all $\omega \in \Omega$, $D(\nu_{\theta^n\omega}) \to 0$ as $n \to \infty$ (in the integers). But since $\mathbb{P}$ is $\theta^1$-invariant, it follows that $D(\nu_\omega)=0$ for $\mathbb{P}$-almost all $\omega \in \Omega$, i.e.~$\nu_\omega$ is a Dirac mass for $\mathbb{P}$-almost all $\omega \in \Omega$. So there exists a measurable function $\tilde{a}:\Omega \to \mathbb{S}^1$ such that $\nu_\omega=\delta_{\tilde{a}(\omega)}$ for $\mathbb{P}$-almost all $\omega \in \Omega$ (and so $\nu=\boldsymbol{\delta}_{\tilde{a}}$). Since $\nu$ is $\varphi$-invariant, it follows that $\tilde{a}$ is a random fixed point. \\ \\ So far, we have seen that any invariant measure $\mu$ is a convex combination of $\boldsymbol{\delta}_r$ and $\boldsymbol{\delta}_{\tilde{a}}$ for some random fixed point $\tilde{a}:\Omega \to \mathbb{S}^1$ such that $\tilde{a}(\omega) \neq r(\omega)$ for $\mathbb{P}$-almost all $\omega \in \Omega$. We next show that up to equivalence, there is \emph{only one} random fixed point that is $\mathbb{P}$-almost everywhere distinct from $r$. Let $a,b:\Omega \to \mathbb{S}^1$ be two random fixed points that are $\mathbb{P}$-almost everywhere distinct from $r$. It is clear that for $\mathbb{P}$-almost every $\omega \in \Omega$, $d(b(\theta^n\omega),a(\theta^n\omega)) \to 0$ as $n \to \infty$ (in the integers). But since $\mathbb{P}$ is $\theta^1$-invariant, it follows that $d(b(\omega),a(\omega))=0$, i.e.~$b(\omega)=a(\omega)$, for $\mathbb{P}$-almost all $\omega \in \Omega$. \\ \\ So then, the set of all invariant measures takes the form $\{\lambda \boldsymbol{\delta}_r + (1-\lambda) \boldsymbol{\delta}_{\tilde{a}} : \lambda \in [0,1]\}$ for some random fixed point $\tilde{a}:\Omega \to \mathbb{S}^1$ that is $\mathbb{P}$-almost everywhere distinct from $r$. It remains to show that one can modify $\tilde{a}$ on a $\mathbb{P}$-null set to obtain an $\mathcal{F}_-$-measurable function. Fix any point $y \in \mathbb{S}^1$ such that $\mathbb{P}_\ast r(\{y\})=0$. For $\mathbb{P}$-almost every $\omega \in \Omega$, $d(\varphi(n,\omega)y,\tilde{a}(\theta^n\omega)) \to 0$ as $n \to \infty$ (in the integers). Since almost sure convergence implies convergence in probability, it follows that the random variable $\,\omega \mapsto d(\varphi(n,\omega)y,\tilde{a}(\theta^n\omega))\,$ converges in probability to 0 as $n \to \infty$. But since $\mathbb{P}$ is $\theta^n$-invariant for all $n$, this is the same as saying that the random variable $\omega \mapsto \varphi(n,\theta^{-n}\omega)y$ converges in probability to $\tilde{a}$ as $n \to \infty$. Hence in particular, there exists an unbounded increasing sequence $(m_n)_{n \in \mathbb{N}}$ in $\mathbb{N}$ such that for $\mathbb{P}$-almost all $\omega \in \Omega$, $\varphi(m_n,\theta^{-m_n}\omega)y \to \tilde{a}(\omega)$ as $n \to \infty$. So, fixing an arbitrary $k \in \mathbb{S}^1$, the function $a:\Omega \to \mathbb{S}^1$ given by \[ a(\omega) \ = \ \left\{ \begin{array}{c l} \underset{n \to \infty}{\lim} \; \varphi(m_n,\theta^{-m_n}\omega)y & \textrm{if this limit exists} \\ k & \textrm{otherwise} \end{array} \right. \] \noindent is $\mathcal{F}_-$-measurable and agrees with $\tilde{a}$ $\mathbb{P}$-almost everywhere. So we are done. \end{proof}
\noindent Now by Theorem~\ref{cr char}, $\varphi$ is stably synchronising if and only if $\mathbb{P}(\Omega_c)=1$ and $\mathbb{P}_\ast r$ is atomless.
\begin{thm} \label{ssis} If $\varphi$ is stably synchronising then $\varphi$ is simple. \end{thm}
\noindent To prove Theorem~\ref{ssis}, we introduce the following:
\begin{defi} We will say that a compatible probability measure $\mu$ is \emph{past-measurable} if there is a version $(\mu_\omega)$ of the disintegration of $\mu$ such that the map $\omega \mapsto \mu_\omega$ is $(\mathcal{F}_-,\mathfrak{K})$-measurable. \end{defi}
\begin{lemma} \label{ks} For every forward-stationary probability measure $\rho$, there exists a past-measurable invariant measure $\mu^\rho$ such that $\rho(A)=\mu^\rho(\Omega \times A)$ for all $A \in \mathcal{B}(\mathbb{S}^1)$. \end{lemma}
\noindent For a proof (detailing the explicit construction of $\mu^\rho$), see [KS12, Theorem~4.2.9(ii)].
\begin{proof}[Proof of Theorem \ref{ssis}] Suppose $\mathbb{P}(\Omega_c)=1$ and $\varphi$ is not simple. We know that there exists a forward-stationary probability measure $\rho$. Since $\boldsymbol{\delta}_r$ is the \emph{only} invariant measure of $\varphi$, Lemma~\ref{ks} then implies that $\boldsymbol{\delta}_r$ is past-measurable. So $r$ has an $\mathcal{F}_-$-measurable modification. But $r$ also obviously has an $\mathcal{F}_+$-measurable modification. Since $\mathcal{F}_-$ and $\mathcal{F}_+$ are independent, it follows that $\mathbb{P}_\ast r$ is a Dirac mass, so $\varphi$ is not stably synchronising. \end{proof}
\section{Contractibility and compressibility}
\begin{defi} We say that $\varphi$ is \emph{contractible} if for any distinct $x,y \in \mathbb{S}^1$, \[ \mathbb{P}( \, \omega \, : \, \exists \, t \in \mathbb{T}^+ \textrm{ s.t.~} d(\varphi(t,\omega)x,\varphi(t,\omega)y) < d(x,y) \, ) \ > \ 0. \] \end{defi}
\begin{lemma} \label{contr} Suppose $\varphi$ is contractible, and fix any $x,y \in \mathbb{S}^1$. For $\mathbb{P}$-almost all $\omega \in \Omega$ there is an unbounded increasing sequence $(t_n)$ in $\mathbb{T}^+$ such that $d(\varphi(t_n,\omega)x,\varphi(t_n,\omega)y) \to 0$ as $n \to \infty$. \end{lemma}
\noindent For a proof, see see Section~4.1 of [New17]. (A similar statement can also be found in [BS88, Proposition~4.1].) \\ \\ Now define the \emph{anticlockwise distance function} $d_+:\mathbb{S}^1 \times \mathbb{S}^1 \to [0,1)$ by \[ d_+(x,y) \ = \ \min\{ r \geq 0 : \, \pi(x' + r) = y \} \] \noindent where $x'$ may be any lift of $x$. Obviously $d_+$ is not symmetric, but rather satisfies the relation \[ d_+(y,x) \ = \ 1 - d_+(x,y). \] \noindent It is clear that for all $x,y \in \mathbb{S}^1$, \[ d(x,y) \ = \ \left\{ \! \begin{array}{c l} d_+(x,y) & \textrm{if } d_+(x,y) \leq \frac{1}{2} \\ d_+(y,x) & \textrm{if } d_+(x,y) \geq \frac{1}{2}. \end{array} \right. \] \noindent Note that $d_+$ is continuous on the set $\{(x,y) \in \mathbb{S}^1 \times \mathbb{S}^1 : x \neq y\}$. For any interval $I \subset \mathbb{R}$ of positive length less than 1, letting $x_1:=\pi(\inf I)$, $x_2:=\pi(\sup I)$ and $J:=\pi(I)$, we have that \[ l(\varphi(t,\omega)J) \ = \ d_+(\varphi(t,\omega)x_1,\varphi(t,\omega)x_2) \] \noindent for all $t$ and $\omega$.
\begin{defi} We say that $\varphi$ is \emph{compressible} if for any distinct $x,y \in \mathbb{S}^1$, \[ \mathbb{P}( \, \omega \, : \, \exists \, t \in \mathbb{T}^+ \textrm{ s.t.~} d_+(\varphi(t,\omega)x,\varphi(t,\omega)y) < d_+(x,y) \, ) \ > \ 0. \] \end{defi}
\noindent By reversing the order of inputs, this is equivalent to saying that for any distinct $x,y \in \mathbb{S}^1$, \[ \mathbb{P}( \, \omega \, : \, \exists \, t \in \mathbb{T}^+ \textrm{ s.t.~} d_+(\varphi(t,\omega)x,\varphi(t,\omega)y) > d_+(x,y) \, ) \ > \ 0. \]
\noindent Perhaps more intuitively, we can also define compressibility in terms of connected subsets of $\mathbb{S}^1$: $\varphi$ is compressible if and only if for every connected set $J \subset \mathbb{S}^1$ with $0<l(J)<1$, \[ \mathbb{P}( \, \omega \, : \, \exists \, t \in \mathbb{T}^+ \textrm{ s.t.~} l(\varphi(t,\omega)J) < l(J) \, ) \ > \ 0. \] \noindent Again, this is equivalent to saying that for every connected set $J \subset \mathbb{S}^1$ with $0<l(J)<1$, \[ \mathbb{P}( \, \omega \, : \, \exists \, t \in \mathbb{T}^+ \textrm{ s.t.~} l(\varphi(t,\omega)J) > l(J) \, ) \ > \ 0. \]
\noindent Obviously, if $\varphi$ is compressible then $\varphi$ is contractible.
\begin{prop} If $\varphi$ is compressible then for any $x,y \in \mathbb{S}^1$ and $\varepsilon>0$ there exists $t \in \mathbb{T}^+$ such that \[ \mathbb{P}( \, \omega \, : \, d_+(\varphi(t,\omega)x,\varphi(t,\omega)y) < \varepsilon \, ) \ > \ 0. \] \end{prop}
\begin{proof} Suppose we have $x,y \in \mathbb{S}^1$ and $\varepsilon>0$ such that for all $t \in \mathbb{T}^+$, \[ \mathbb{P}( \, \omega \, : \, d_+(\varphi(t,\omega)x,\varphi(t,\omega)y) < \varepsilon \, ) \ = \ 0. \] \noindent Let $\Delta$ be the diagonal in $\mathbb{S}^1 \times \mathbb{S}^1$, i.e.~$\Delta=\{(\xi,\xi):\xi \in \mathbb{S}^1\}$. (From now on, we follow the terminology of Section~2.2 of [New17].) Let $G_{(x,y)} \subset \mathbb{S}^1 \times \mathbb{S}^1$ be the smallest closed set containing $(x,y)$ that is forward-invariant under the two-point motion of $\varphi$. The open set $\{(u,v) \in \mathbb{S}^1 \times \mathbb{S}^1 : 0 < d_+(u,v) < \varepsilon\}$ is not accessible from $(x,y)$, and therefore (e.g.~by [New17, Lemma~2.2.3]) $G_{(x,y)}$ does not intersect this set. So if we let $(\bar{u},\bar{v})$ be a point from the compact set $K:=\{ (u,v) \in G_{(x,y)} : \varepsilon \leq d_+(u,v) \leq d_+(x,y) \}$ which minimises $d_+$ on $K$, then $(\bar{u},\bar{v})$ will minimise $d_+$ on the whole of $G_{(x,y)} \setminus \Delta$. Since $G_{(x,y)}$ is forward-invariant, we then have that \[ \mathbb{P}( \, \omega \, : \, \exists \, t \in \mathbb{T}^+ \textrm{ s.t.~} 0 < d_+(\varphi(t,\omega)\bar{u},\varphi(t,\omega)\bar{v}) < d_+(\bar{u},\bar{v}) \, ) \ = \ 0. \] \noindent Obviously, since $\varphi(t,\omega)$ is bijective for all $t$ and $\omega$, $d_+(\varphi(t,\omega)\bar{u},\varphi(t,\omega)\bar{v})$ cannot be 0; so we can simply write that \[ \mathbb{P}( \, \omega \, : \, \exists \, t \in \mathbb{T}^+ \textrm{ s.t.~} d_+(\varphi(t,\omega)\bar{u},\varphi(t,\omega)\bar{v}) < d_+(\bar{u},\bar{v}) \, ) \ = \ 0. \] \noindent Thus $\varphi$ is not compressible. \end{proof}
\begin{defi} We say that $\varphi$ \emph{has reverse-minimal dynamics} if the only open forward-invariant sets are $\emptyset$ and $\mathbb{S}^1$. \end{defi}
\noindent Obviously if $\varphi$ has a deterministic fixed point $p$ then $\mathbb{S}^1 \setminus \{p\}$ is forward-invariant, and so $\varphi$ does not have reverse-minimal dynamics.
\begin{prop} \label{cts} If $\mathbb{T}^+=[0,\infty)$ and $\varphi$ is a continuous RDS, then the following are equivalent: \begin{enumerate}[\indent (i)] \item $\varphi$ has reverse-minimal dynamics; \item the only \emph{closed} forward-invariant sets are $\emptyset$ and $\mathbb{S}^1$. \end{enumerate} \end{prop}
\begin{rmk} In general, when $\emptyset$ and $\mathbb{S}^1$ are the only closed forward-invariant sets, we say that $\varphi$ \emph{has minimal dynamics}. So Proposition~\ref{cts} says that for continuous RDS in continuous time, minimal dynamics and reverse-minimal dynamics are equivalent. \end{rmk}
\begin{proof}[Proof of Proposition \ref{cts}] We first show that (i)$\Rightarrow$(ii). Suppose we have a closed forward-invariant non-empty proper subset $G$ of $\mathbb{S}^1$; we need to show that there exists an open forward-invariant non-empty proper subset $U$ of $\mathbb{S}^1$. Firstly, if $G$ is a singleton $\{p\}$ then $U:=\mathbb{S}^1 \setminus \{p\}$ is clearly forward-invariant. Now consider the case that $G$ is not a singleton, and let $V$ be a connected component of $\mathbb{S}^1 \setminus G$; we will show that $U:=\mathbb{S}^1 \setminus \bar{V}$ is forward-invariant. (Note that $U$ is non-empty, since $G$ is not a singleton.) Fix any $\omega$ with the property that $\varphi(t,\omega)G \subset G$ for all $t \in \mathbb{T}^+$. Since $\partial V \subset G$, we have that for all $t$, $\varphi(t,\omega)\partial V \subset G$ and therefore in particular $\varphi(t,\omega)\partial V \, \cap \; V = \emptyset$. Now since $\varphi$ is a continuous RDS, it is clear that we can define continuous functions $a,b:[0,\infty) \to \mathbb{R}$ with $a<b$ such that $[a(t),b(t)]$ is a lift of $\varphi(t,\omega)\bar{V}$ for all $t$. (So $\{a(t),b(t)\}$ projects onto $\varphi(t,\omega)\partial V$ for all $t$.) For all $t$, since $\varphi(t,\omega)\partial V \, \cap \; V = \emptyset$, we have that $a(t),b(t) \nin (a(0),b(0))$. Therefore (due to the intermediate value theorem), $a(t) \leq a(0)$ for all $t$ and $b(t) \geq b(0)$ for all $t$. Hence $\bar{V} \subset \varphi(t,\omega)\bar{V}$ for all $t$. Since $\varphi(t,\omega)$ is bijective for all $t$, it follows that $\varphi(t,\omega)U \subset U$ for all $t$. So $U$ is forward-invariant. \\ \\ Now, in order to show that (ii)$\Rightarrow$(i), first observe that a set $A \subset \mathbb{S}^1$ is forward-invariant if and only if $\mathbb{P}$-almost every $\omega \in \Omega$ has the property that for all $t \in \mathbb{T}^+$, \[ \varphi(t,\omega)^{-1}(X \setminus A) \ \subset \ X \setminus A. \] \noindent Hence the fact that (ii)$\Rightarrow$(i) follows from the fact that (i)$\Rightarrow$(ii), except with $\varphi(t,\omega)$ replaced by $\varphi(t,\omega)^{-1}$. \end{proof}
\begin{prop} \label{comthm} Suppose $\varphi$ is contractible and has no deterministic fixed point. Then $\varphi$ is compressible if and only if $\varphi$ has reverse-minimal dynamics; and in this case, $\varphi$ is stably synchronising. \end{prop}
\begin{rmk} At least in discrete time, if $\varphi$ is contractible and has no deterministic fixed point, then it actually follows automatically that $\varphi$ is stably synchronising, and moreover at an exponential rate; see [Mal14]. (Nonetheless, our proof of stable synchronisation under the additional assumption of compressibility/reverse-minimality is simpler and more elementary than the proof for the results in [Mal14].) \end{rmk}
\subsection*{Proof of Proposition~\ref{comthm}}
\begin{lemma} Suppose $\varphi$ is compressible and has no deterministic fixed point. Then $\varphi$ has reverse-minimal dynamics. \end{lemma}
\begin{proof} Suppose for a contradiction that $\varphi$ does not have reverse-minimal dynamics, and let $U$ be an open forward-invariant non-empty proper subset of $\mathbb{S}^1$. Let $V$ be a maximal-length connected component of $U$. Since there are no deterministic fixed points, $\mathbb{S}^1 \setminus U$ is not a singleton and so $l(V)<1$. Hence, since $\varphi$ is compressible, there is a positive-measure set of sample points $\omega \in \Omega$ for each of which, for some $t \in \mathbb{T}^+$, $l(\varphi(t,\omega)V)>l(V)$. However, $\varphi(t,\omega)V$ is connected for all $t$ and $\omega$, and so if $l(\varphi(t,\omega)V)>l(V)$ then $\varphi(t,\omega)V$ cannot be a subset of $U$. This contradicts the fact that $U$ is forward-invariant. \end{proof}
\begin{lemma} \label{diffuse} Suppose $\varphi$ is contractible and admits a reverse-stationary probability measure $\rho$ that is atomless and has full support. Then $\varphi$ is synchronising. \end{lemma}
\noindent (We will soon prove that under these same conditions, $\varphi$ is in fact \emph{stably} synchronising.)
\begin{proof} Fix any distinct $x,y \in \mathbb{S}^1$. Let $J \subset \mathbb{S}^1$ be a connected set with $\partial J = \{x,y\}$. By Lemmas~\ref{mart} and \ref{contr}, there is a $\mathbb{P}$-full set of sample points $\omega$ with the properties that \begin{enumerate}[\indent (a)] \item there exists an unbounded increasing sequence $(t_n)$ in $\mathbb{T}^+$ such that \[ d(\varphi(t_n,\omega)x,\varphi(t_n,\omega)y) \to 0 \hspace{3mm} \textrm{as } n \to \infty \, ; \] \item $\rho(\varphi(t,\omega)J)$ is convergent as $t \to \infty$. \end{enumerate} \noindent Fix any $\omega$ with both these properties, and let $(t_n)$ be as in (a). For any $n$, $d(\varphi(t_n,\omega)x,\varphi(t_n,\omega)y)$ is precisely the smaller of $l(\varphi(t_n,\omega)J)$ and $1 - l(\varphi(t_n,\omega)J)$. Hence there must exist a subsequence $(t_{m_n})$ of $(t_n)$ such that either $l(\varphi(t_{m_n},\omega)J) \to 0$ as $n \to \infty$ or $l(\varphi(t_{m_n},\omega)J) \to 1$ as $n \to \infty$. Since $\rho$ is atomless, it follows that either $\rho(\varphi(t_{m_n},\omega)J) \to 0$ as $n \to \infty$ or $\rho(\varphi(t_{m_n},\omega)J) \to 1$ as $n \to \infty$. Since $\rho(\varphi(t,\omega)J)$ is convergent as $t \to \infty$, it follows that either $\rho(\varphi(t,\omega)J) \to 0$ as $t \to \infty$ or $\rho(\varphi(t,\omega)J) \to 1$ as $t \to \infty$. Since $\rho$ has full support, it follows that either $l(\varphi(t,\omega)J) \to 0$ as $t \to \infty$ or $l(\varphi(t,\omega)J) \to 1$ as $t \to \infty$. Hence $d(\varphi(t,\omega)x,\varphi(t,\omega)y) \to 0$ as $t \to \infty$. \end{proof}
\begin{lemma} \label{diffusecom} Under the hypotheses of Lemma~\ref{diffuse}, for any connected $J \subset \mathbb{S}^1$, \[ \mathbb{P}( \, \omega \, : \, l(\varphi(t,\omega)J) \to 0 \ \mathrm{as} \ t \to \infty \, ) \ = \ 1 - \rho(J). \] \end{lemma}
\begin{proof} Fix any connected $J \subset \mathbb{S}^1$. As in the proof of Lemma~3.3, we have that for $\mathbb{P}$-almost every $\omega \in \Omega$, either \[ \rho(\varphi(t,\omega)J) \to 0 \hspace{3mm} \textrm{and} \hspace{3mm} l(\varphi(t,\omega)J) \to 0 \hspace{3mm} \textrm{as } t \to \infty. \] \noindent or \[ \rho(\varphi(t,\omega)J) \to 1 \hspace{3mm} \textrm{and} \hspace{3mm} l(\varphi(t,\omega)J) \to 1 \hspace{3mm} \textrm{as } t \to \infty. \] \noindent So then, letting $E$ denote the set of sample points $\omega$ for which the latter scenario holds, the dominated convergence theorem gives that as $t \to \infty$, \[ \int_\Omega \rho(\varphi(t,\omega)J) \, \mathbb{P}(d\omega) \ \to \ \int_\Omega \mathbbm{1}_E(\omega) \, \mathbb{P}(d\omega) \ = \ \mathbb{P}(E). \] \noindent But we also know that for any $t$, \[ \int_\Omega \rho(\varphi(t,\omega)J) \, \mathbb{P}(d\omega) \ = \ \rho(J). \] \noindent Hence $\mathbb{P}(E)=\rho(J)$, i.e.~the probability of the latter scenario is $\rho(J)$ and the probability of the former scenario is $1-\rho(J)$. \end{proof}
\noindent Combining Lemmas~\ref{diffuse} and \ref{diffusecom}, we have:
\begin{cor} \label{diffusess} Under the hypotheses of Lemma~\ref{diffuse}, $\varphi$ is stably synchronising. \end{cor}
\begin{proof} We already know (from Lemma~\ref{diffuse}) that $\varphi$ is synchronising. Now fix any $x \in X$. Let $(U_n)_{n \in \mathbb{N}}$ be a nested sequence of connected neighbourhoods of $x$ such that $\bigcap_n \hspace{-0.2mm} U_n = \{x\}$. For each $n$, \begin{align*} \mathbb{P}( \, \omega \, : \, \exists \, \textrm{open } & U \! \ni x \, \textrm{ s.t.~} l(\varphi(t,\omega)U) \to 0 \textrm{ as } t \to \infty \, ) \\ &\geq \ \mathbb{P}( \, \omega \, : \, l(\varphi(t,\omega)U_n) \to 0 \ \mathrm{as} \ t \to \infty \, ) \\ &= \ 1 - \rho(U_n). \end{align*} \noindent But since $\rho$ is atomless, $\rho(U_n) \to 0$ as $n \to \infty$. Hence \[ \mathbb{P}( \, \omega \, : \, \exists \, \textrm{open } U \! \ni x \, \textrm{ s.t.~} l(\varphi(t,\omega)U) \to 0 \textrm{ as } t \to \infty \, ) \ = \ 1. \] \noindent So we are done. \end{proof}
\begin{lemma} \label{open} Given a dense subset $D$ of $\mathbb{T}^+$, an open set $A \subset \mathbb{S}^1$ is forward-invariant if and only if for each $t \in D$, for $\mathbb{P}$-almost all $\omega \in \Omega$, $\varphi(t,\omega)A \subset A$. \end{lemma}
\begin{proof} The ``only if'' direction is clear. Now fix an open set $A \subset \mathbb{S}^1$, and let $D$ be a dense subset of $\mathbb{T}^+$ such that for each $t \in D$, for $\mathbb{P}$-almost all $\omega \in \Omega$, $\varphi(t,\omega)A \subset A$. Let $\tilde{D}$ be a countable dense subset of $D$. $\mathbb{P}$-almost every $\omega \in \Omega$ has the property that for every $t \in \tilde{D}$, $\varphi(t,\omega)A \subset A$ and so $\varphi(t,\omega)^{-1}(\mathbb{S}^1 \setminus A) \subset \mathbb{S}^1 \setminus A$. But since $\mathbb{S}^1 \setminus A$ is closed and the map $t \mapsto \varphi(t,\omega)^{-1}(x)$ is right-continuous for each $x$ and $\omega$, it follows that $\mathbb{P}$-almost every $\omega$ has the property that for every $t \in \mathbb{T}^+$, $\varphi(t,\omega)^{-1}(\mathbb{S}^1 \setminus A) \subset \mathbb{S}^1 \setminus A$ and so $\varphi(t,\omega)A \subset A$. So we are done. \end{proof}
\begin{lemma} \label{statinv} For any reverse-stationary probability measure $\rho$, $\mathbb{S}^1 \setminus \mathrm{supp}\,\rho$ is forward-invariant. \end{lemma}
\begin{proof} Let $\rho$ be a reverse-stationary probability measure, and let $U:=\mathbb{S}^1 \setminus \mathrm{supp}\,\rho$. For each $t \in \mathbb{T}^+$, \[ 0 \ = \ \rho(U) \ = \ \int_\Omega \rho(\varphi(t,\omega)U) \, \mathbb{P}(d\omega). \] \noindent Therefore, for each $t \in \mathbb{T}^+$, for $\mathbb{P}$-almost all $\omega$, $\rho(\varphi(t,\omega)U)=0$ and so $\varphi(t,\omega)U \subset U$. Hence, by Lemma~\ref{open}, $U$ is forward-invariant. \end{proof}
\begin{lemma} \label{rmdiff} If $\varphi$ has reverse-minimal dynamics then every reverse-stationary probability measure is atomless and has full support. \end{lemma}
\begin{proof}
By Lemma~\ref{statinv}, if $\varphi$ has reverse-minimal dynamics then every reverse-stationary probability measure has full support. Now suppose that $\varphi$ has reverse-minimal dynamics and let $\rho$ be a probability measure on $\mathbb{S}^1$ that is not atomless; we will show that $\rho$ is not reverse-stationary. Let $m:=\max\{ \rho(\{x\}) : x \in \mathbb{S}^1 \}$ and let $P:=\{x \in \mathbb{S}^1 : \rho(\{x\})=m \}$. (So $\rho(P)=m|P|$.) Since $\varphi$ has reverse-minimal dynamics, $\mathbb{S}^1 \setminus P$ is not forward-invariant and so (by Lemma~\ref{open}) there must exist $t_0 \in \mathbb{T}^+$ such that \[ \mathbb{P}( \, \omega \, : \, P \neq \varphi(t_0,\omega)P ) \ > \ 0. \] \noindent Obviously, for any $\omega$, if $P \neq \varphi(t_0,\omega)P$ then $\rho(\varphi(t_0,\omega)P)<\rho(P)$. Hence we have that \[ \int_\Omega \rho(\varphi(t_0,\omega)P) \, \mathbb{P}(d\omega) \ < \ \rho(P). \] \noindent Thus $\rho$ is not reverse-stationary. \end{proof}
\noindent Combining Lemma~\ref{rmdiff} with Corollary~\ref{diffusess}, we have that if $\varphi$ has reverse-minimal dynamics then $\varphi$ is stably synchronising.
\begin{lemma} Under the hypotheses of Lemma~\ref{diffuse}, $\varphi$ is compressible. \end{lemma}
\begin{proof} For any connected $J \subset \mathbb{S}^1$ with $0 < l(J) < 1$, since $\rho$ has full support, $\rho(J)<1$. Hence, by Lemma~\ref{diffusecom}, there is a positive-measure set of sample points $\omega$ such that $l(\varphi(t,\omega)J) \to 0$ as $t \to \infty$. So in particular, $\varphi$ is compressible. \end{proof}
\noindent So we are done.
\subsection*{Additive-noise SDE}
Large classes of ordinary differential equations in Euclidean space have been proven to exhibit synchronous behaviour after the addition of Gaussian white noise to the right-hand side (see [CF98] for the one-dimensional case, and [FGS17] for higher-dimensional cases). We shall now do the same for ODEs on $\mathbb{S}^1$. Let $(\Omega,\mathcal{F},\mathbb{P},(\theta^t),(W_t))$ be as in Example~2.1. Let $b:\mathbb{R} \to \mathbb{R}$ be a 1-periodic Lipschitz function, and let $\varphi$ be the RDS on $\mathbb{S}^1$ with trajectories $(\varphi(t,\omega)x)_{t \geq 0}$ whose lifts to $\mathbb{R}$ satisfy the integral equation \begin{equation} \label{sde} X_t(\omega) \ = \ X_0(\omega) \, + \, \int_0^t b(X_s(\omega)) \hspace{0.1mm} ds \, + \, \sigma \hspace{0.1mm} W_t(\omega) \end{equation} \noindent (where $\sigma > 0$). It is clear that if $b$ is $\frac{1}{n}$-periodic for some $n > 1$ then $\varphi$ is not synchronising, since any two trajectories starting at a distance $\frac{1}{n}$ apart will remain of distance $\frac{1}{n}$ apart. In the converse direction, Proposition~\ref{comthm} yields the following result:
\begin{prop} If the least period of $b$ is $1$ then $\varphi$ is stably synchronising. \end{prop}
\begin{proof} $\varphi$ clearly has no deterministic fixed points, so we just need to show that $\varphi$ is compressible. Fix a connected $J \subset \mathbb{S}^1$ with $0 < l(J) < 1$. Now, regarding $\Omega$ as being equipped with the topology of uniform convergence on bounded intervals, it is known ([Bur83, Theorem~3.4.1]) that for any $x \in \mathbb{S}^1$ and $t \geq 0$ the map $\omega \mapsto \varphi(t,\omega)x$ is continuous, and it is also known ([Fre13, Proposition~477F]) that the Wiener measure $\mathbb{P}$ assigns strictly positive measure to every non-empty open subset of $\Omega$. In view of these facts, in order to prove compressibility, we just need to find one sample point $\omega_0 \in \Omega$ with the property that at some time $t>0$, $l(\varphi(t,\omega_0)J)<l(J)$. Let $[c_1,c_2] \subset \mathbb{R}$ be a lift of $\bar{J}$ (so $c_2-c_1=l(J)$). Since $b$ is continuous and periodic but \emph{not} $l(J)$-periodic, there must exist $a > c_1$ such that $b(a+l(J))<b(a)$. Now for large $\eta > 0$, consider a sample point $\omega_0^{(\eta)}$ whose path on the time-interval $[0,1]$ is given by \[ \omega_0^{(\eta)}(t) \ = \ \left\{ \!\! \begin{array}{c l} \eta t & t \in [0,\frac{a - c_1}{\sigma\eta}] \\ \frac{a - c_1}{\sigma} & t \in [\frac{a - c_1}{\sigma\eta},1]. \end{array} \right. \] \noindent Let $a_1(t)$ be the solution of (\ref{sde}) with $\omega:=\omega_0^{(\eta)}$ and $X_0:=c_1$. Let $a_2(t)$ be the solution of (\ref{sde}) with $\omega:=\omega_0^{(\eta)}$ and $X_0:=c_2$. (So $[a_1(t),a_2(t)]$ is a lift of $\varphi(t,\omega_0^{(\eta)})\bar{J}$ for all $t$.) Provided $\eta$ is sufficiently large: $a_1(\frac{a - c_1}{\sigma\eta})$ will be very close to $a$, and $a_2(\frac{a - c_1}{\sigma\eta})$ will be very close to $a+l(J)$; hence in particular, $b(a_2(\frac{a - c_1}{\sigma\eta}))<b(a_1(\frac{a - c_1}{\sigma\eta}))$, and so there will exist $\delta>0$ such that at time $t:=\frac{a - c_1}{\sigma\eta}+\delta$, we have $l(\varphi(t,\omega_0^{(\eta)})J)=a_2(t)-a_1(t)<l(J)$. Thus $\varphi$ is compressible. Hence Proposition~\ref{comthm} gives that $\varphi$ is stably synchronising. \end{proof}
\subsubsection*{References:}
{[Ant84]} Antonov, V.~A., Modeling of processes of cyclic evolution type. Synchronization by a random signal, \emph{Vestnik Leningradskogo Universiteta Matematika, Mekhanika, Astronomiya} vyp.~2, pp67--76. \\ \\ {[Arn98]} Arnold, L., \emph{Random Dynamical Systems}, Springer, Berlin Heidelberg New York. \\ \\ {[Bax86]} Baxendale, P.~H., Asymptotic behaviour of stochastic flows of diffeomorphisms, \emph{Stochastic processes and their applications}, pp1--19. \\ \\ {[BS88]} Baxendale, P.~H., Stroock, D.~W., Large Deviations and Stochastic Flows of Diffeomorphisms, \emph{Probability Theory and Related Fields} \textbf{80}(2), pp169--215. \\ \\ {[Bur83]} Burton, T.~A., \emph{Volterra integral and differential equations}, Academic Press, New~York London. \\ \\ {[CF98]} Crauel, H., Flandoli, F., Additive Noise Destroys a Pitchfork Bifurcation, \emph{Journal of Dynamics and Differential Equations} \textbf{10}(2), pp259--274. \\ \\ {[Crau02]} Crauel, H., Invariant measures for random dynamical systems on the circle, \emph{Archiv der Mathematik} \textbf{78}(2), pp145--154. \\ \\ {[Crau02a]} Crauel, H., \emph{Random Probability Measures on Polish Spaces}, Stochastics Monographs 11, Taylor \& Francis, London. \\ \\ {[FGS17]} Flandoli, F., Gess, B., Scheutzow, M., Synchronization by noise, \emph{Probability Theory and Related Fields} \textbf{168}(3--4), pp511--556. \\ \\ {[Fre13]} Fremlin, D.~H., \emph{Measure Theory, Volume 4: Topological Measure Spaces} (Chapter~47, version of 8th~April 2013), \url{https://www.essex.ac.uk/maths/people/fremlin/chap47.pdf}. \\ \\ {[Kai93]} Kaijser, T., On stochastic perturbations of iterations of circle maps, \emph{Physica~D} \textbf{68}(2), pp201--231. \\ \\ {[Kif86]} Kifer, Y., \emph{Ergodic Theory of Random Transformations}, Birkh\"{a}user, Boston. \\ \\ {[KN04]} Kleptsyn, V.~A., Nalskii, N.~B., \emph{Contraction of orbits in random dynamical systems on the circle}, Functional Analysis and Its Applications \textbf{38}(4), pp267--282. \\ \\ {[KS12]} Kuksin, S., Shirikyan, A., \emph{Mathematics of Two-Dimensional Turbulence}, Cambridge Tracts in Mathematics 194, Cambridge University Press, Cambridge~New~York. \\ \\ {[LeJ87]} Le Jan, Y., \'{E}quilibre statistique pour les produits de diff\'{e}omorphismes al\'{e}atoires ind\'{e}pendants, \emph{Annales de l'Institut Henri Poincar\'{e} Probabilit\'{e}s et Statistiques} \textbf{23}(1), pp111--120. \\ \\ {[Mal14]} Malicet, D., Random walks on $\mathrm{Homeo}(S^1)$, \url{http://perso.crans.org/mdominique/randomwalks.pdf}. \\ \\ {[New15]} Newman, J., Ergodic Theory for Semigroups of Markov Kernels (version of 5th~July 2015), \url{http://wwwf.imperial.ac.uk/~jmn07/Ergodic_Theory_for_Semigroups_of_Markov_Kernels.pdf}. \\ \\ {[New17]} Newman, J., Necessary and sufficient conditions for stable synchronization in random dynamical systems, \emph{Ergodic Theory and Dynamical Systems}, 1-19. doi:10.1017/etds.2016.109. \\ \\ {[Will91]} Williams, D., \emph{Probability with Martingales}, Cambridge University Press, Cambridge.
\end{document} | arXiv |
\begin{document}
\title{Time Series Classification using Convolutional Neural Network on Imbalanced Datasets}
\section*{Abstract} \underline{T}ime \underline{S}eries \underline{C}lassification (TSC) has drawn a lot of attention in literature because of its broad range of applications for different domains, such as medical data mining, weather forecasting. Although TSC algorithms are designed for balanced datasets, most real-life time series datasets are imbalanced. The Skewed distribution is a problem for time series classification both in distance-based and feature-based algorithms under the condition of poor class separability. To address the imbalance problem, both sampling-based and algorithmic approaches are used in this paper. Different methods significantly improve time series classification's performance on imbalanced datasets. Despite having a high imbalance ratio, the result showed that F score could be as high as 97.6\% for the simulated TwoPatterns Dataset.
\section{Introduction}\label{chap:introduction}
\underline{T}ime \underline{s}eries \underline{c}lassification (TSC) is one of the main tasks among all time series data operations, which has a vast number of applications in our daily life. For instance, A real-time warning system based on TSC has achieved significant performance compared with traditional clinical approaches and is applied in smart hospitals.
Most of the datasets related to time series are imbalanced. However, classification algorithms are designed for balanced datasets. Classification on a skewed distribution is always challenging because it is biassed towards the majority classes while ignoring minority classes. Although miss classification of a majority class is acceptable, miss classification of a minority class is dangerous.
1NN DTW is state-of-the-art in time series classification (TSC). A distance-based method might suffer despite having high-class separability in an imbalanced dataset. There is a high probability that a lot of majority class samples surround a minority class sample, and a minority class sample will be miss-classified. Nowadays, CNN is becoming popular in time series classification because of its ability to find essential features without human supervision. We can mitigate the problem of imbalanced dataset by modifying the loss function to obtain equal contribution from majority and minority classes in loss function.
We did some literature review of the algorithms and techniques that address an imbalanced dataset problem. The algorithms are re-evaluated on the datasets from UCR archives by making them imbalanced. We have reported which algorithm is suitable for what amount of imbalance.
\subsection{Overview} The rest of the paper is organized as follows. A short review of related work is described in \autoref{chap:relatedwork}. \autoref{chap:background} depicts the forms, parameters and background information of imbalanced distribution. In \autoref{chap:dataset} seven different time series datasets from the UCR archive are described. We have illustrated methodologies to address the imbalance problem in \autoref{chap:methods}. Our experimental report on seven real-world data sets is described in \autoref{chap:results}. Finally, the paper is concluded in \autoref{chap:conclusion}. \section{Related Work}\label{chap:relatedwork}
This section discusses the papers related to addressing imbalanced dataset.
\cite{class_cost_sensitive_cnn_itsc} described the difficulties of time series classification on an imbalanced dataset. The author proposed to set the learning rate based on the ratio of minority samples in each mini-batch. Another approach, namely modified loss function, was proposed, ensuring that each class's contribution is equal in mini-batch. We have adopted this paper to mitigate the imbalance problem.
\cite{class_imb_cnn} described the taxonomy and parameters of an imbalanced dataset. To address the imbalance problem, sampling-based methods were proposed. In \autoref{chap:results}, we argue our results with the imbalance parameters.
\cite{class_mfe_msfe} proposed two novel loss functions, Mean False Error (MFE) and Mean Square False Error (MSFE), for the imbalanced datasets, which ensured equal contribution in loss from each class in a mini-batch. Experimental results exposed that better gradient descent is possible with the proposed loss function.
\cite{class_bootstrapping} proposed an algorithmic approach, namely bootstrapping, to address imbalanced distribution by creating a balanced mini-batch. The author also prioritized recall over precision in imbalanced dataset as false negatives are more important than false positives. Bootstrapping method is adopted in this paper to mitigate imbalanced distribution.
\cite{class_adaptive} proposed a developed version of Mean Square Error (MSE), namely Global Mean Square Error (GMSE), which ensures the contribution of loss for each class is equal, and the loss function is proportional to the class separability score. The proposed loss function is used to address the imbalanced distribution. Class separability score is used in \autoref{chap:results}, to argue the reason for poor classification.
\cite{class_cost_aware} proposed a novel loss function, namely, cost-aware loss, that embeds the training stage's cost information. The author showed that the loss function could also be integrated into the pre-training stage to conduct cost-aware feature extraction more effectively. Experimental results justified the validity of the novel loss function by making existing deep learning models cost-sensitive and demonstrated that the proposed model outperforms other deep models.
\section{Background}\label{chap:background}
We have discussed forms and parameters of an imbalanced dataset firstly on this section. Finally, the section has discussed class separability score and its implementation.
There are two main \textit{forms of an imbalanced distribution}. Those are \cite{class_imb_cnn}:
\begin{enumerate} \item \textit{Step imbalance} refers to a class distribution with majority class instances and minority class instances, where all the majority classes has an equal number of instances, and all the minority classes has an equal number of instances. However, the number of instances in majority classes and minority classes is not equal . \autoref{fig:step_imbalance_1} shows a histogram representation of a step imbalance dataset where all the majority and minority classes have 500 and 5000 instances respectively.
Step imbalance can be defined by two parameters as follows:
\begin{enumerate}
\item \textit{Fraction of minority class} $\mu$ is the ratio between the number of minority classes and the total number of classes. It can be expressed by \autoref{eq:fraction_minor_class} .
\begin{equation}
\label{eq:fraction_minor_class}
\mu = \frac{number\; of\; minority\; class}{total\; number\; of\; class}
\end{equation}
According to \autoref{fig:step_imbalance_1}, in a 10 class dataset, we have 5 minority classes. So, $\mu = \frac{5}{10} = .5$. However, According to \autoref{fig:step_imbalance_2}, we have 9 minority classes out of 10 classes. So, $\mu = \frac{9}{10} = .9$
\item \textit{Imbalance ratio} $\rho$ is the ratio between the number of instances in majority and minority classes. It can be expressed by \autoref{ir} .
\begin{equation}
\label{ir}
\rho = \frac{number\; of\; instances\; in\; majority\; class}{number\; of\; instances\; in\; minority\; class}
\end{equation}
According to \autoref{fig:step_imbalance_1}, majority and minority class has 5000 and 500 instances respectively. So, $\rho = \frac{5000}{500} = 10$. On the other hand, In \autoref{fig:step_imbalance_2}, majority and minority class has 5000 and 2500 instances respectively. So, $\rho = \frac{5000}{2500} = 2$ \end{enumerate}
\item \textit{Linear imbalance} refers to a class distribution where the number of instances of any two classes is not equal and increases gradually over classes. \autoref{fig:linear_imbalance} is a histogram representation of a linear imbalance where the number of instances increased gradually from class 1 to class 10 . Linear imbalance can be defined with parameter imbalance ratio $\rho$, which is the ratio of maximum and minimum number of instances among all classes. The number of instances in intermediate classes can be interpolated. In \autoref{fig:linear_imbalance}, we have 5000 and 500 instances in majority and minority classes respectively. So, $\rho = \frac{5000}{500} = 10$ . \end{enumerate}
\begin{figure}
\caption{Step imbalance; $\rho=10, \mu=0.5$}
\label{fig:step_imbalance_1}
\caption{Step imbalance; $\rho=2, \mu=0.9$}
\label{fig:step_imbalance_2}
\caption{Linear imbalance; $\rho=10$}
\label{fig:linear_imbalance}
\caption{Forms and parameters of imbalance; source \cite{class_imb_cnn}}
\label{fig:forms_of_imbalance}
\end{figure}
\textit{Class separability} is a quantitative measure that defines how well each data point falls into its own class \cite{class_adaptive}. For instance, in a binary classification problem, the class separability score for a positive instance $i$ can be defined by \autoref{class_sep} \cite{class_adaptive}. \begin{equation}
\label{class_sep}
s(i)=\frac{n(i)-p(i)}{\max \{n(i), p(i)\}} \end{equation} Here, $p(i)$ represents the average distance of i from all positive class instances, and $n(i)$ represents the average distance of i from all negative class instances.
The class separability score was proposed for binary classification, and adopted in multi-class problem using binarization. The overall class separability score of a dataset is the average separability score of all instances and ranged from -1 to 1. In \autoref{fig:class_sep_1}, class separability score is nearly $1$ as positive instances and negative instances lie with their class instances. However, the class separability score is nearly $-1$ in \autoref{fig:class_sep_minus_1} because the positive and negative instances are positioned completely with different class instances. In an imbalanced dataset, majority classes dominate over the score. To address this problem, majority and minority classe's contribution is considered equally by taking the average.
\begin{figure}
\caption{S $\approx$ 1}
\label{fig:class_sep_1}
\caption{S $\approx$ -1}
\label{fig:class_sep_minus_1}
\caption{Class separability score (S)}
\label{fig:class_separability_score}
\end{figure}
\section{Dataset}\label{chap:dataset}
To compare the time series classification performance, seven different datasets from the UCR archive are used in this paper \cite{UCRArchive}. Out of the seven datasets, four are EEG dataset, one is ECG dataset, one is a simulated dataset, and one is human activity recognition dataset. A summary of the datasets is given in \autoref{tab:dataset}. \begin{table}[H]
\caption{Dataset summary; source: \cite{ds_ucr} [modified]}
\label{tab:dataset}
\resizebox{\textwidth}{!}{
\begin{tabular}{|llllll|}
\hline
\hline
Dataset &Type & Train Cases & Dimensions & Length & \shortstack{Class separability\\ score} \\
ECG5000 &ECG & 500 & 1 & 140 & 0.38 \\
TwoPatterns &Simulated & 1000 & 1 & 128 & 0.49 \\
HandMovementDirection &EEG & 320 & 10 & 400 & 0.0034 \\
BasicMotions &Human Activity Recognition & 40 & 6 & 100 & 0.54 \\
SelfRegulationSCP1 &EEG & 268 & 6 & 896 & 0.098 \\
SelfRegulationSCP2 &EEG & 200 & 7 & 1152 & -0.0012 \\
MotorImagery &EEG & 278 & 64 & 3000 & 0.3 \\
\hline
\end{tabular}
}
\end{table}
\section{Methods}\label{chap:methods}
This section has discussed different methods to address imbalanced time series dataset.
An imbalanced dataset can be addressed in two different ways as follows \cite{class_imb_cnn}:
\begin{enumerate}
\item \textit{Data level approach} addresses an imbalanced distribution by modifying the dataset to create a balanced distribution of the classes, and can be performed in two ways:
\begin{enumerate}
\item \textit{Under-sampling} removes samples from the majority classes and creates a balanced distribution for the class label. Although we are losing information, less training time is required for fewer data. Under-sampling is not suitable for a dataset with a high imbalance ratio as we are losing too much information and have very few samples to train the network.
\item \textit{Oversampling} creates a balanced distribution of the class level by developing artificial instances of the minority classes. Different oversampling approaches are given below :
\begin{enumerate}
\item \textit{\underline{S}ynthetic \underline{M}inority \underline{O}ver-sampling \underline{T}echnique (SMOTE)} augments new samples by interpolating neighbors \cite{pre_data_smote}.
\item \textit{Cluster-based oversampling} cluster the dataset first and then oversample each cluster separately \cite{sampling_cluster_based}.
\end{enumerate}
For a dataset with a high imbalance ratio, oversampling will overfit because we replicate the same sample multiple times to develop artificial samples, hence losing the model’s generalization capacity. Moreover, developing an artificial sample is a time-consuming process \cite{class_cost_sensitive_cnn_itsc}.
\end{enumerate}
\item \textit{Algorithmic approach} addresses the imbalance problem by modifying the loss function such that contribution to the loss function from majority classes and minority classes are equal. Different algorithmic approaches are given below:
\begin{enumerate}
\item \textit{Weighted loss in mini-batch} is an algorithmic procedure, where the contribution of loss in mini-batch from each class is considered equally \cite{class_cost_sensitive_cnn_itsc}. \autoref{weighted_loss} shows the loss function of this approach.
\begin{eqnarray}
\label{weighted_loss}
E_{mini-batch}(\theta) = \dfrac{1}{|C|} \sum_{c \in All\, class} E_c(\theta)
\end{eqnarray}
Here, $E_c$ represents the average loss for class $c$, and $\theta$ represents the network's weights.
\autoref{E_c} expresses the loss for a particular class $c$ in mini-batch.
\begin{equation}
\label{E_c}
E_{c}(\boldsymbol{\theta})=\frac{1}{N_{c}} \sum_{i=1}^{N_{c}} L O S S_{i}
\end{equation}
Here, $N_c$ implies the number of instances in class $c$ and $LOSS_{i}$ represents categorical cross entropy loss for instance $i$.
\item \textit{Bootstrapping approach} is an algorithmic approach that addresses imbalanced dataset by creating a balanced mini-batch from the majority and minority classes \cite{class_bootstrapping}. Let us assume, in a dataset, there are $n$, $m$ instances from majority and minority class respectively, where $n>>m$. the batch size $s$ is selected in a way that it can be representative of all classes.
The number of samples in majority and minority classes per batch is defined as $s_n$ and $s_p$ respectively, where $s_{p} \approx s_{n}$ . Total number of batch per epoch $N$ is $\dfrac{n}{s_n}$ . Some negative samples might be removed from our training set in each epoch because $n$ might not be divisible by $s_n$. Those ignored samples will not create a detrimental impact on training's quality because there is a lot of samples from the majority classes.
In each mini-batch, $s_n$ distinct majority class samples are selected. One sample from the majority class can be selected maximum of one time per epoch. However, $s_p$ samples from the minority classes are selected randomly, and a sample from the minority class can be selected multiple times in an epoch. All the samples in the minority class have an equal probability of being chosen in each mini-batch. Such a random process guarantees that each positive instance has an equal probability of being trained with different negative instances and avoid overfitting \cite{class_bootstrapping}.
\autoref{alg:bootstrapping} describes training a CNN model with bootstrapping approach.
\begin{algorithm}[H]
\caption{Train CNN with Bootstrapping \cite{class_bootstrapping}}
\KwIn{A dataset $D$; Number of samples from positive instances in mini-batch $s_p$;
\\ Number of samples from negative instances in mini-batch $s_n$;}
\KwOut{A trained CNN model}
Divide majority class samples into $N$ batches, each with $s_n$ instances\\
$model$ $\gets$ an untrained CNN model\\
\ForEach{epoch}
{
\For{$1:N$}
{
$batch_{majority} \gets s_{n}\: distinct\: instances\: from\: majority\: class$\\
$batch_{minority} \gets s_{p}\: random\: instances\: from\: minority\: class$\\
$batch_{balance} \gets batch_{majority} + batch_{minority}$\\
Forward pass\\
compute categorical cross entropy loss\\
compute gradients\\
Update weights $\theta$ of $model$ using backpropagation
}
}
\KwRet{$model$}\;
\label{alg:bootstrapping}
\end{algorithm}
\item \textit{\underline{M}ean \underline{F}alse \underline{E}rror (MFE) and \underline{M}ean \underline{S}quare \underline{F}alse \underline{E}rror (MSFE)} are two improve loss functions of the existing \underline{M}ean \underline{S}quare \underline{E}rror (MSE), which address the imbalanced dataset \cite{class_mfe_msfe}. Let us assume, in a dataset, we have total 100 samples where 10 samples are from minority class and 90 samples are from majority class. In an imbalanced dataset, minority class instances are more important than majority class instances. Therefore, minority class instances are regarded as positive instances and majority class instances are regarded as negative instances. \autoref{tab:example-of-mfe-and-msfe} shows a confusion matrix for classification. All the loss functions are discussed based on this confusion matrix.
\begin{table}[]
\centering
\caption{Confusion matrix;source:\cite{class_mfe_msfe}[modified]}
\label{tab:example-of-mfe-and-msfe}
\begin{tabular}{|llrrr|}
\hline
\hline
\multicolumn{2}{|l}{\multirow{2}{*}{}} & \multicolumn{2}{l}{Prediction} & \\
\multicolumn{2}{|l}{} & P & N & \textbf{Total} \\
\hline
\multirow{2}{*}{\begin{turn}{90}Truth\end{turn}} & P & 85 & 5 & \textbf{90} \\
& N & 5 & 5 & \textbf{10} \\
\multicolumn{2}{|l}{\textbf{Total}} & \textbf{90} & \textbf{10} & \textbf{100}\\
\hline
\end{tabular}
\end{table}
\cite{class_mfe_msfe} discussed 3 different loss function as follows:
\begin{enumerate}
\item \textit{MSE} reduces the square error between prediction and ground truth and can be expressed using \autoref{mse} \cite{class_mfe_msfe}.
\begin{equation}
\label{mse}
l=\frac{1}{M} \sum_{i} \sum_{n} \frac{1}{2}\left(d_{n}^{(i)}-y_{n}^{(i)}\right)^{2}
\end{equation}
Here, $M$ is the total number of samples. $d_{n}^{(i)}$ represents the
ground truth value of $i^{th}$ sample on $n^{th}$ neuron while $y_{n}^{(i)}$ is the
corresponding prediction. For instance, in the scenario of binary classification, if the $4^{th}$ sample belongs to the second class. But it is miss-classified as first class, then the ground-truth vector and prediction vector for this sample is
$d^{(4)}=[0,1]^{T}$ and $y^{(4)}=[1,0]^{T}$ respectively.
Here, $d_{1}^{(4)}=0$ and $d_{2}^{(4)}=1$ while $y_{1}^{(4)}=1$ and $y_{2}^{(4)}=0$. So the error of this sample is $1 / 2^{*}\left((0-1)^{\wedge} 2+(1-O)^{\wedge} 2\right)=1$\cite{class_mfe_msfe}.
According to the confusion matrix located in \autoref{tab:example-of-mfe-and-msfe} , $ l_{MSE} = \dfrac{1}{100}(5+5) = .1$
\item \textit{Mean False Error (MFE)} is the summation of mean false positive error (FPE) and mean false negative error (FNE) \cite{class_mfe_msfe}. FPE and FNE capture errors from negative and positive class respectively. \autoref{mfe} defines MFE \cite{class_mfe_msfe}.
\begin{equation}
\label{mfe}
l^{\prime}=F P E+F N E
\end{equation}
and
\begin{equation}
\begin{array}{c}
F P E=\frac{1}{N} \sum_{i=1}^{N} \sum_{n} \frac{1}{2}\left(d_{n}^{(i)}-y_{n}^{(i)}\right)^{2} \\
F N E=\frac{1}{P} \sum_{i=1}^{P} \sum_{n} \frac{1}{2}\left(d_{n}^{(i)}-y_{n}^{(i)}\right)^{2}
\end{array}
\end{equation}
$N$ and $P$ represent the numbers of instances in negative class and positive class respectively. According to the confusion matrix located in \autoref{tab:example-of-mfe-and-msfe}, $l^{\prime}=\frac{5}{10}+\frac{5}{90}=0.55$ .
\item \textit{MSFE} is an improved version of MFE and can be expressed by \autoref{msfe} \cite{class_mfe_msfe}.
\begin{equation}
\label{msfe}
l^{\prime \prime}=F P E^{2}+F N E^{2}
\end{equation}
According to the confusion matrix located in \autoref{tab:example-of-mfe-and-msfe}, $l^{\prime \prime}=\left(\frac{5}{10}\right)^{2}+\left(\frac{5}{90}\right)^{2}=0.25$.
\end{enumerate}
Minimization of MFE implies minimization of the sum of False Positive Error (FPE) and False Negative Error(FNE) \cite{class_mfe_msfe}. FPE contributes more in an imbalanced distribution because the number of negative samples (majority samples) is much higher than the number of positive samples(minority samples). For instance, in the confusion matrix, we have only 10 positive samples whereas 90 negative samples. To achieve higher performance in a positive class, FNE should be pretty low. We want a high performance on difficult calf birth in our scenario. MFE is not sensitive to the positive class (minority class). MSFE solves the problem effectively \cite{class_mfe_msfe}.
\autoref{MSFE_BETTER} expresses MSFE \cite{class_mfe_msfe}.
\begin{equation}
\label{MSFE_BETTER}
\begin{array}{l}
MSFE=F P E^{2}+F N E^{2} \\
\quad=\frac{1}{2}\left((F P E+F N E)^{2}+(F P E-F N E)^{2}\right)
\end{array}\end{equation}
The minimization operation of MSFE can find
a minimal sum of FPE and FNE and a minimal difference. In other words, it reduces error from positive and negative classes simultaneously \cite{class_mfe_msfe}.
\item \textit{\underline{G}lobal \underline{M}ean \underline{S}quare \underline{E}rror (GMSE)} is an improved version of Mean Square Error (MSE), which considers loss equally from the majority and minority classes and the loss can be defined with \autoref{adaptive_loss} \cite{class_adaptive}.
\begin{equation}
\label{adaptive_loss}
E=\frac{1}{n} \sum_{p=1}^{n} \kappa * \left(d_{p}- y_{p}\right)^{2} \text { where } \kappa=\left\{\begin{array}{ll}
1, & \text { if } p \in \text { majority } \\
k^{*}, & \text {otherwise }
\end{array}\right.\end{equation}
Here $d_{p}$ and $y_{p}$ represent the categorical value of ground truth and predicted output of $p^{th}$ instance respectively, and $n$ is the total number of training samples. $k^{*}$ depends on class separability score, evaluation metrics and imbalance ratio. Instead of punishing the network equally, GMSE imposes punishment based on class separability. The goal of this algorithm is to learn network weights and $k^*$ jointly. Network weights are updated after each mini-batch, whereas $k^*$ are updated after each epochs. $k^*$ is updated as follows \cite{class_adaptive}:
\begin{equation}\kappa^{*}=\operatorname{argmin} F(\kappa) ; \quad F(\kappa)=\|T-\kappa\|^{2}\end{equation}
and the gradient descent can be expressed as follows \cite{class_adaptive}:
\begin{equation}
\label{eqgradientk}
\nabla F(\kappa)=\nabla\|T-\kappa\|^{2}=-(T-\kappa)
\end{equation}
Three variants of T is given below \cite{class_adaptive}:
\autoref{eq_T1} \cite{class_adaptive} expresses T, to optimize G-Mean and Accuracy jointly.
\begin{equation}
\label{eq_T1}
T_{1}=H * \exp \left(-\frac{G M e a n}{2}\right) * \exp \left(-\frac{\text {Accuracy}}{2}\right)
\end{equation}
\autoref{eq_T2} \cite{class_adaptive} expresses T, to Optimize G-Mean only.
\begin{equation}
\label{eq_T2}
T_{2}=H * \exp \left(-\frac{G M e a n}{2}\right)
\end{equation}
\autoref{eq_T3} \cite{class_adaptive} expresses T, to Optimize G-Mean and validation errors $(1-Accuracy)$ jointly.
\begin{equation}
\label{eq_T3}
T_{3}=H * \exp \left(-\frac{G M e a n}{2}\right) * \exp \left(-\frac{(1-A c c u r a c y)}{2}\right)
\end{equation}
The idea of \autoref{eq_T3} is to see whether bringing down the accuracy would help improve G-mean.
Here $H$ represents the maximum cost for a minority class instance. $H$ depends on class separability score, imbalance ratio and can be defined as $H=I R(1+S)$.
Here, $IR$ represents the imbalance ratio, and $S$ represents the class separability score.
The value of $H$ can be ranged from $0$ to $2 \times IR$ based on class separability score. The value of $H = 2\times IR$, when the class separability score is highest $(S = 1)$ and $H = 0$, when the class separability score is lowest $(S = -1)$. For example, GMSE punishes the network highly for miss-classifying the two patterns dataset because class separability score for this dataset is high (.54). On the other hand, it would not punish the network highly for the SelfRegulationSCP2 dataset for miss-classification for having poor class separability (-.0012). However, if the Imbalance ratio is 6 then a minority class sample contributes six times more than majority class sample to loss.
\cite{class_adaptive} discovered, $T_2$ is the most suitable measure by grid search.
\autoref{alg:AdaptiveGMSE} describes training a CNN model with GMSE loss.
\begin{algorithm}[H]
\caption{GMSE Algorithm (learnable weight); Source: \cite{class_adaptive}}
\KwIn{A dataset $D$;}
\KwOut{A trained CNN model}
$model$ $\gets$ an untrained CNN model\\
$k \gets 1$\\
\ForEach{epoch}
{
\ForEach{mini-batch}
{
Forward pass\\
Compute loss with \autoref{adaptive_loss}\\
Calculate gradients for error\\
Update weights $\theta$ of $model$ using backpropagation
}
Compute gradients for $k$ using \autoref{eqgradientk}, with $T$ either $T_1,T_2 or T_3$ \\
Update $k$
}
\KwResult{model}
\label{alg:AdaptiveGMSE}
\end{algorithm}
\item \textit{Adaptive learning rate} addresses the imbalanced dataset problem by changing the learning rate based on the number of minority samples in the mini-batch. It increases the learning rate if there are many minority class instances in a mini-batch \cite{class_cost_sensitive_cnn_itsc}.
\end{enumerate} \end{enumerate}
\section{Results} \label{chap:results}
In this section we have discussed the effect of different methods to address imbalanced datasets with four fold cross-validation. Imbalance ratio of all the dataset is set to 4.
According to \autoref{tab:method_to_address_imb_f}, \autoref{tab:method_to_address_imb_auc}, GMSE method is outperforming on dataset HandMovementDirection, SelfRegulationSCP2 and SelfRegulationSCP2, which has poor class separability. For instance, the class separability score for SelfRegulationSCP2 dataset is only -.00123, and our F3 and AUC are 55.65$\%$ and 60$\%$ respectively, which is higher than other methods. This is because only GMSE method is considering class separability score while punishing the network. However,according to \autoref{tab:method_to_address_imb_training_time} training time for the GMSE method is higher than other methods in all datasets. For instance, in SelfRegulationSCP2 dataset, the training time for GMSE method is 127.13 seconds, whereas training time for unweighted loss, weighted loss and bootstrapping is 109,110 and 119 seconds respectively. GMSE is taking a high training time because computing class separability score which is a time-consuming operation.
\begin{table}[]
\caption{Performance (F3) comparison between different method to address imbalance dataset in percentage($\%$)}
\label{tab:method_to_address_imb_f}
\resizebox{\textwidth}{!}{
\begin{tabular}{|ll|llll|}
\hline\hline
Dataset&Class Seperability & Unweighted loss & Weighted loss & GMSE & Bootstrapping \\
\hline
ECG5000 & 0.386 & 39.09 $\pm$ 0.26 & 46.15 $\pm$ 0.58 & 21.42 $\pm$ 0.58 & \textbf{51.17 $\pm$ 2.19} \\
TwoPatterns & 0.49 & 92.06 $\pm$ 3.46 & \textbf{99.29 $\pm$ 0.03} & 72.49 $\pm$ 0.05 & 98.33 $\pm$ 0.64 \\
HandMovementDirection & 0.0034 & 2.85 $\pm$ 2.85 & 3.8 $\pm$ 0.07 & \textbf{25.0 $\pm$ 0.0} & 16.51 $\pm$ 3.37 \\
BasicMotions & 0.54 & 20.54 $\pm$ 3.04 & \textbf{66.97 $\pm$ 19.25} & 11.36 $\pm$ 11.36 & 50.77 $\pm$ 27.62 \\
SelfRegulationSCP1 & 0.098 & 49.21 $\pm$ 3.33 & 62.09 $\pm$ 1.34 & \textbf{65.88 $\pm$ 0.0} & 56.47 $\pm$ 3.19 \\
SelfRegulationSCP2 & -0.00123 & 45.65 $\pm$ 0.0 & 47.26 $\pm$ 0.01 &\textbf{55.65 $\pm$ 0.0} & 47.72 $\pm$ 2.07 \\
MotorImagery & 0.3 & 45.95 $\pm$ 0.99 & 52.09 $\pm$ 1.68 & 44.3 $\pm$ 0.66 & \textbf{57.8 $\pm$ 10.51} \\ \hline \end{tabular}
} \end{table}
\begin{table}[]
\caption{Performance (AUC) comparison between different method to address imbalance dataset in percentage($\%$)}
\label{tab:method_to_address_imb_auc}
\resizebox{\textwidth}{!}{
\begin{tabular}{|ll|llll|}
\hline\hline
Dataset& Class Seperability & Unweighted loss & Weighted loss & GMSE & Bootstrapping \\
\hline
ECG5000 & 0.386 & 68.39 $\pm$ 0.07 & 72.05 $\pm$ 0.66 & 52.12 $\pm$ 1.05 & \textbf{75.46 $\pm$ 0.43} \\
TwoPatterns & 0.49 & 94.74 $\pm$ 2.28 & \textbf{99.53 $\pm$ 0.02} & 83.21 $\pm$ 0.01 & 98.88 $\pm$ 0.42 \\
HandMovementDirection & 0.0034 & 49.79 $\pm$ 0.21 & 49.63 $\pm$ 1.51 & \textbf{50.0 $\pm$ 0.0} & 49.04 $\pm$ 1.41 \\
BasicMotions & 0.54 & 58.61 $\pm$ 2.5 & \textbf{83.15 $\pm$ 6.48} & 57.5 $\pm$ 7.5 & 69.14 $\pm$ 15.81 \\
SelfRegulationSCP1 & 0.098 & 52.84 $\pm$ 2.84 & 64.12 $\pm$ 1.57 & \textbf{70.0 $\pm$ 0.0} & 58.92 $\pm$ 3.58 \\
SelfRegulationSCP2 & -0.00123 & 50.0 $\pm$ 0.0 & 49.88 $\pm$ 1.02 & \textbf{60.0 $\pm$ 0.0} & 49.27 $\pm$ 2.01 \\
MotorImagery & 0.3 & 50.59 $\pm$ 0.59 & 55.0 $\pm$ 1.27 & 48.16 $\pm$ 1.84 & \textbf{57.84 $\pm$ 10.46} \\
\hline
\end{tabular}
} \end{table}
\begin{table}[]
\caption{Training time comparison between different method to address imbalance dataset in second}
\label{tab:method_to_address_imb_training_time}
\resizebox{\textwidth}{!}{
\begin{tabular}{|ll|llll|}
\hline\hline
Dataset & Class Seperability & Unweighted loss & Weighted loss & GMSE & Bootstrapping \\
\hline
ECG5000 & 0.386 & 18.56 $\pm$ 4.52 & 25.65 $\pm$ 0.61 & 34.29 $\pm$ 1.35 & 26.41 $\pm$ 0.27 \\
TwoPatterns & 0.49 & 32.86 $\pm$ 0.3 & 43.82 $\pm$ 3.83 & 45.38 $\pm$ 0.74 & 40.71 $\pm$ 2.13 \\
HandMovementDirection & 0.0034 & 48.35 $\pm$ 0.49 & 48.94 $\pm$ 1.83 & 56.69 $\pm$ 0.2 & 50.25 $\pm$ 3.11 \\
BasicMotions & 0.54 & 54.33 $\pm$ 4.52 & 59.72 $\pm$ 1.51 & 71.55 $\pm$ 3.62 & 74.61 $\pm$ 2.74 \\
SelfRegulationSCP1 & 0.098 & 97.37 $\pm$ 0.53 & 107.53 $\pm$ 10.1 & 125.5 $\pm$ 5.94 & 116.61 $\pm$ 22.52 \\
SelfRegulationSCP2 & -0.00123 & 109.59 $\pm$ 5.33 & 110.22 $\pm$ 5.36 & 127.13 $\pm$ 0.51 & 119.47 $\pm$ 0.47 \\
MotorImagery & 0.3 & 163.63 $\pm$ 48.53 & 274.08 $\pm$ 18.52 & 367.86 $\pm$ 58.68 & 238.47 $\pm$ 23.97\\
\hline
\end{tabular}
} \end{table}
Although we have proposed some effective measures to classify time series dataset, our methods have some bottlenecks also. The Strength and weaknesses of the paper are discussed in this chapter.
According to chapter \nameref{chap:results}, performance is improved, and we are getting a 4.6$\%$ boost after applying weighted loss on the cow dataset. Imbalanced dataset with poor class separability like SelfRegulationSCP2, 9.3 $\%$ boost is achieved after applying GMSE loss. Moreover, GMSE is outperforming any other method in a dataset with poor class separability, e.g., HandMovementDirection, Cow dataset. Finally, the test time complexity of our model is low comparing 1NN DTW.
Training time complexity is high in CNN; comparing 1NN DTW and our model can not classify time series with variable lengths. A lot of training samples are required to train the network perfectly. Cow dataset is not big enough to make some of the class representatives of the training, test and validation set, which creates a detrimental impact on classification's quality. If a class instance is not available in the validation set, then early stopping will not work perfectly. Hence, it will lead to overfitting. On the other hand, if a class instance is only available in the test set, we can not train the network at all.
\section{Conclusion}\label{chap:conclusion}
We have addressed imbalanced distribution by different algorithmic approaches where we have modified the loss function.
In \textit{summary}, we can say GMSE is suitable for a dataset with poor class separability as it considers class separability score while calculating loss. However, GMSE needs high training time because we need to compute class separability score. Bootstrapping approach is less prone to overfitting and better gradient descents is possible.
In the GMSE method, we find class separability scores only at the beginning of a training. Computing class separability score at each mini-batch can improve the performance, but it is computationally demanding. Therefore, we are leaving this for future work. In this paper, all the classes from a dataset have an equal contribution to the evaluation metrics. Developing a cost-sensitive evaluation metrics based on the class label's importance can be a possible future work.
\printbibliography
\end{document} | arXiv |
\begin{document}
\title{Symmetry analysis of an elastic beam with axial load}
\author{Bidisha Kundu$^*$, Ranjan Ganguli\\ Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India\\ [email protected], [email protected]\\
$^*$Corresponding author }
\begin{abstract} We construct the closed form solution of an elastic beam with axial load using Lie symmetry method. A beam with spatially varying physical properties such as mass and second moment of inertia is considered. The governing fourth order partial differential equation with variable coefficients which is not amenable to simple methods of solution, is solved using Lie symmetry. We incorporate boundary conditions and then compare with the numerical solution. \end{abstract}
\maketitle
\textit{Keywords}: Lie symmetry, elastic beam, closed-form solution
\section{Introduction} An elastic beam is a three dimensional structure whose axial extension is more than any other dimension orthogonal to it. It is a fundamental model which pervades every corner of physics and engineering.
However, the integrability in finite numbers of terms or exact solution of the governing equation of this model is an open question.
The deflection is dependent not only on the mass of the beam or external applied force but also on the elastic properties of the material and geometry of the beam.
The equation of motion of the beam is a fourth-order linear partial differential equation with variable coefficients. The equation can be derived from generalised Hooke's law and force balance or by minimizing the energy of the system. Though the equation is linear in nature, due to the presence of variable coefficients, it is very arduous to get the solution analytically. In \cite{guede2001apparentlynew}, the closed form solution of vibrating axially loaded beam was studied with different boundary conditions. However, to the best of our knowledge, in this work, for the first time, Euler-Bernoulli beam with axial force is studied using the Lie-method. Here we follow the Lie-group method to get the similarity solution.
\par The main motivation to use the Lie method is to get the solution of the mathematical model in different forms which are very easy to use. First, Sophus Lie applied this method to partial differential equations and later this method was further developed by Ovsjannikov \cite{ovsjannikov1962group} and Matschat and M\"{u}ller\cite{muller1962and}.
Lie method has been applied to various types of ordinary differential equations (ODE) and partial differential equations(PDE). A systematic approach for applying Lie method to ODE and PDE is found in Refs. [\cite{olver2012applications},\cite{bluman2013symmetries},\cite{ibragimov1995crc}, \cite{hydon2000symmetrynew}]. Bluman et al. used this method for different types of mathematical physics problems such as wave equation, diffusion equation etc. \cite{bluman1980remarkable,bluman1987invariance}. Torrisi et al. studied diffusion equation by equivalence transformation \cite {torrisi1996group}. Ibragimov applied this method to some real life problems \cite{ibragimov2004equivalence,ibragimov2011lie} related to tumour growth model and also to metallurgical industry mathematical models. In \cite{gray2015calculate}, the procedure to calculate all point symmetries of linear and linearizable differential equations was studied. An algorithm for integrating systems of two second-order ordinary differential equations with four symmetries was given in \cite{gainetdinova2017integrability}. The Lie symmetry method is also employed in ordinary difference equation \cite{hydon2000symmetries}.
\par The Lie-method is also available in recent research addressing equations in mathematical physics \cite{kang2012symmetry,singla2017invariant,hau2017optimal}. In Ref. \cite{kang2012symmetry}, Kang and Qu have studied the relationship between Lie point symmetry and fundamental solution for systems of parabolic equations. The invariant analysis of space-time fractional nonlinear systems of partial differential equations is performed using Lie symmetry method in Ref. \cite{singla2017invariant}. Hau, et al. presented a unifying solution framework for the linearized compressible equations for two-dimensional linearly sheared unbounded flows \cite{hau2017optimal}.
The symmetry analysis for a physical problem is very important. In Lorentz transformation used in special relativity, Yang-Mills theory \cite{strobl2004algebroid,beisert2017yangian}, and Schr\"{o}dinger equation, the Lie symmetry analysis has been used extensively. In Ref. \cite{belmonte2007lie}, Lie symmetries and canonical transformations are applied to construct the explicit solutions of Schr\"{o}dinger equation with a spatially inhomogeneous nonlinearity from those of the homogeneous nonlinear Schr\"{o}dinger equation. In Ref. \cite{budanur2015reduction}, symmetry analysis is used to understand the fluid flow in a pipe or channel structure. \par The Lie-method has also been employed in the field of mechanics \cite{bocko2012symmetries}. In Ref. \cite{ozkaya2002group}, a beam moving with time-dependent axial velocity is studied using equivalence transformation. Wafo discussed the Euler-Bernoulli (EB) beam from a symmetry stand-point \cite{soh2008euler}. The general beam equation is studied by Bokhari et al. for symmetries and integrability with Lie method \cite{bokhari2010symmetries}. Bokhari et al also found the complete Lie symmetry classification of the fourth-order dynamic EB beam equation with load dependent on normal displacement. Johnpillai et al studied the EB beam equation from the Noether symmetry viewpoint \cite{johnpillai2016noether}.
\par
We search for one-parameter group of transformation which leaves the governing PDE invariant and we get the corresponding Lie-algebras. Using the infinitesimal generator of this transformation, we solve the newly reduced differential equation. In Section $2$, we describe our problem. The mathematical theories for the Lie-symmetry approach, the procedure to get the invariant solution and application of the Lie method to our problem are described in Section $3$. The closed form solutions for different cases and boundary condition imposition are given in Section $4$ and Section $5$, respectively.
\section{Formulation}
We fix the coordinate axes $X$, $Y, Z$ along the length, breadth and height of the beam, respectively. We consider that the beam is slender and of length $l$. Here, $u(x,t)$ and $M(x,t)$ are the out-plane bending displacement along Z and bending moment, respectively, at the point $x$ and the instant $t$ where $x\in [0,l]$ and $t\in \mathbb{R}^{+}$. We assume the stiffness functions $EI:[0,l]\rightarrow \mathbb{R}^{+}$ and $m:[0,l]\rightarrow \mathbb{R}^{+}$ are continuously differentiable functions. Here $\mathbb{R}^{+}$ is the set of all positive real numbers. \par
The equation of motion of the Euler-Bernoulli beam with axial force is given by \begin{eqnarray} \label{first equation} \frac{\partial{^2}}{\partial x{^2}}\left(EI(x)\frac{\partial{^2}u}{\partial x{^2}}\right)+ m(x)\frac{\partial{^2}u}{\partial t{^2}}- \frac{\partial}{\partial x}\left(T(x)\frac{\partial u}{\partial x}\right)=0.
\end{eqnarray}
For rotating beam $T(x)=\int_{x}^{l}m(x)\Omega^{2}x dx$ where $\Omega$ is the rotating speed. Another important practical case is the gravity loaded beam. For this problem $T(x)=\int_{x}^{l}m(x)g dx$ where $g$ is the gravitational force acting on the system. In the case of a stiff-string or piano string $T$ is constant \cite{gunda2008stiff}.
This formulation is valid for all tensile loads and for compressive loads less than the critical buckling load.
In general, for this type of beam, cantilever boundary condition at the left end and free at the right end is considered. Here, a schematic diagram of an axially loaded beam (under tension), is shown in Figure (\ref{beam}). \vspace*{-7pt} \begin{figure}
\caption{Schematic diagram of an axially loaded beam (under tension).}
\label{beam}
\end{figure} \vspace*{-5pt} \section{The Lie symmetry method} Eq. (\ref{first equation}) is a linear PDE of order four in $x$ and order two in $t$.
The Lie symmetry method will help us to extract the closed form solution via the symmetry analysis. We define the original Eq. (\ref{first equation}) in geometric form. \par \par
Consider $X=\{(x,t)|x,t\in \mathbb{R}\}$ as the two-dimensional manifold of the independent variables and $U\subset \mathbb{R}$ as the one dimensional manifold of dependent variable. Equation (\ref{first equation}) is defined on the manifold $\mathbb{M}\subset X\times U$ of dimension three.
In order to establish a concrete geometric structure of the differential equation, we have to construct a manifold $\mathfrak{J}$ of dimension $p$ which includes the independent variables, dependent variables, and all possible partial derivatives of the dependent variable with respect to independent variables up to order of the given differential equation. Here the dimension $p$ depends on the dimension of $X, U$ as well as on the order of the differential equation given. We ``prolong'' the original manifold $\mathbb{M}\subset X\times U$ to $\mathfrak{J}$ which also has manifold structure. \par The main motivation to study a differential equation with Lie symmetry method is to find a favourable coordinate transformation which can produce another equivalent form of the given equation. In the new coordinate systems, the new equation may have closed form solution. To get the favourable coordinate transformations, we should take care of the derivatives of the dependent variables which is related to the vector fields of $\mathbb{M}$. We also want to measure the contribution of the vector fields related to the given differential equation in the prolonged space $\mathfrak{J}$ which is called the prolongation of the vector field. \par As this is a PDE of order four, we need the prolongation of order four, i.e. $pr^{(4)}{\bf v}$. For a PDE with one dependent variable and two independent variables, the one-parameter Lie group of transformations is \begin{eqnarray} \label{one-parameter}
x^{*}=X(x,t,u;\epsilon )= x+\epsilon {\xi}(x,t,u) + O (\epsilon^{2}) \nonumber \\
t^{*}=T(x,t,u;\epsilon )= t+\epsilon \tau(x,t,u) + O (\epsilon^{2})\nonumber \\
u^{*}=U(x,t,u;\epsilon )= u+\epsilon {\eta}(x,t,u) + O (\epsilon^{2})
\end{eqnarray}
The prolongation of order four $pr^{(4)}{\bf v}$ is given by
\begin{eqnarray} pr^{(4)}{\bf v}={\xi}\frac{\partial}{\partial x}+\tau \frac{\partial}{\partial t}+{\eta} \frac{\partial}{\partial u} +{\eta^x} \frac{\partial}{\partial u_{x}}+{\eta^t} \frac{\partial}{\partial u_{t}}+{\eta^{xx}} \frac{\partial}{\partial u_{xx}}+...
\nonumber \\
{\eta^{xxt}} \frac{\partial}{\partial u_{xxt}}+...+{\eta^{ttt}} \frac{\partial}{\partial u_{ttt}} +{\eta^{xxxx}} \frac{\partial}{\partial u_{xxxx}}+{\eta^{xxxt}} \frac{\partial}{\partial u_{xxxt}}+...+{\eta^{tttt}}\frac{\partial}{\partial u_{tttt}} \end{eqnarray}
\par
From the Fundamental theorem [Theorem $2.31$, \cite{olver2012applications}] of transformations (\ref{one-parameter}) admitted by the PDE (\ref{first equation}), applying $pr^{(4)}{\bf v}$ on Eq.(\ref{first equation}) yields
\begin{gather}
\xi (x,t,u(x,t)) \left( EI^{(3)}(x) u_{xx}(x,t)+2 EI''(x)
u_{xxx}(x,t)\right.\nonumber\\
\left.
+EI'(x) u_{xxxx}(x,t)+
m'(x) u_{tt}(x,t)
- T''(x)u_{x}(x,t)\right.\nonumber\\
\left.
- T'(x) u_{xx}(x,t) \right) +
\eta^{ xx} EI''(x)+2
\eta ^{xxx} EI'(x)\nonumber\\+\eta^{xxxx} EI(x)+
\eta ^{tt}m(x)
-\eta^{x} T'(x)- \eta ^{xx} T(x)=0
\label{equaprolong} \end{gather}
Eq. (\ref{equaprolong}) should be satisfied to yield the admissible transformations. The value of $u_{xxxx}(x,t)$ from Eq. (\ref{first equation}) is substituted into the above Eq. (\ref{equaprolong}).
Now substituting the values of $\eta^{x}, \eta^{xx}, \eta^{tt}, \eta^{xxx}, \eta^{xxxx}$ from
Theorem 32.3.5 \cite{hassani2013mathematical}, Eq. (\ref{equaprolong}) is simplified
and a polynomial equation is formed in all possible derivatives $u^{i,j}$ of $u$ upto order four with respect to $x$, $t$ where \begin{eqnarray}
u^{i,j}=\frac{\partial^{i+j} u}{\partial x^{i}\partial t^{j}} \end{eqnarray} This polynomial equation should be satisfied for arbitrary $x$, $t$ and $u^{i,j}$ which requires the coefficients of all $u^{i,j}$ and all its product terms to be equal to zero.
The determining equations for $\xi, \tau, \eta$ are \begin{footnotesize} \begin{gather}
\frac{\partial \tau }{\partial x}=0
\label{eq1}\\
\frac{\partial \tau }{\partial u}=0
\label{eq2}\\
\frac{\partial \xi }{\partial t}=0
\label{eq3}\\
\frac{\partial \xi }{\partial u}=0
\label{eq4}\\
m(x) \frac{\partial^{2}\eta}{\partial t^{2}}-T'(x)\frac{\partial \eta}{\partial x}-T(x)\frac{\partial^{2}\eta}{\partial x^{2}}+EI''(x)\frac{\partial^{2}\eta}{\partial x^{2}}+2EI'(x)\frac{\partial^{3}\eta}{\partial x^{3}}+EI(x)\frac{\partial^{4}\eta}{\partial x^{4}}=0
\label{eq5}\\
-\frac{2\xi (x,t,u)(EI'(x))^{2}}{EI(x)}+2 \xi (x,t,u)EI''(x)+2EI'(x)\frac{\partial \xi}{\partial x}+4 EI(x)\frac{\partial^{2}\eta}{\partial x \partial u}-6 EI(x)\frac{\partial^{2}\xi}{\partial x^{2}}=0
\label{eq6}\\
\frac{T(x)\xi (x,t,u)EI'(x)}{EI(x)}-\xi (x,t,u)T'(x)-\frac{\xi (x,t,u)EI'(x)EI''(x)}{EI(x)}+\xi (x,t,u)EI'''(x)\nonumber \\-2T(x) \frac{\partial \xi}{\partial x} +2 EI''(x)\frac{\partial \xi}{\partial x}
+6EI'(x)\frac{\partial^{2}\eta}{\partial x \partial u}+6EI'(x)\frac{\partial^{2}\xi}{\partial x^{2}}+6EI(x)\frac{\partial^{3}\eta}{\partial x^{2}\partial u}
+4EI(x)\frac{\partial^{3}\xi}{\partial x^{3}}=0
\label{eq7} \end{gather}
\begin{gather}
\frac{T'(x)\xi (x,t,u)EI'(x)}{EI(x)}-T''(x)\xi (x,t,u)-m(x)\frac{\partial^{2}\xi}{\partial t^{2}}
-3T'(x)\frac{\partial \xi}{\partial x}
-2T(x)\frac{\partial^{2}\eta}{\partial x \partial u}
+2EI''(x)\frac{\partial^{2}\eta}{\partial x \partial u}\nonumber \\+T(x)\frac{\partial^{2}\xi}{\partial x^{2}}-EI(x)\frac{\partial^{2}\xi}{\partial x^{2}}+6EI'(x)\frac{\partial^{3}\eta}{\partial x^{2}\partial u}-2EI'(x)\frac{\partial^{3}\xi}{\partial x^{3}}+4EI(x)\frac{\partial^{4}\eta}{\partial x^{3}\partial u}+EI(x)\frac{\partial^{4}\xi}{\partial x^{4}}=0\label{eq8}\\
-\frac{m(x)\xi (x,t,u)EI'(x)}{EI(x)}+\xi (x,t,u)m'(x)-2 m(x)\frac{\partial \tau}{\partial t}+4\frac{\partial \xi}{\partial x}=0\label{eq9}\\
2 m(x)\frac{\partial^{2}\eta}{\partial t\partial u}-m(x)\frac{\partial^{2} \tau}{\partial t^{2}}+T'(x)\frac{\partial \tau}{\partial x}+T(x)\frac{\partial^{2} \tau}{\partial x^{2}}-EI''(x)\frac{\partial^{2} \tau}{\partial x^{2}}-2EI'(x)\frac{\partial^{3} \tau}{\partial x^{3}}-EI(x)\frac{\partial^{4} \tau}{\partial x^{4}}=0\label{eq10}\\
\frac{\partial^{2}\eta}{\partial u^{2}}=0\label{etauu}
\end{gather} \end{footnotesize} From Eqs.(\ref{eq1}, \ref{eq2}) $\tau(x,t,u)=\tau(t)$, from Eqs. (\ref{eq3}, \ref{eq4}) $\xi(x,t,u)=\xi(x)$ and from Eq. (\ref{etauu}) $\eta(x,t,u)=A(x,t)u+B(x,t)$ for some arbitrary functions $A(x,t)$, $B(x,t)$. We assume $\frac{\partial B}{\partial x}=0$ and $\frac{\partial^{2}B}{\partial t^{2}}=0$, i.e., $B(x,t)=d_{1}+d_{2}t$ where $d_{1},d_{2}$ are constants. From the Eq.(\ref{eq6}), we observe that $\frac{\partial^{2}\eta}{\partial x \partial u}=\frac{\partial A}{\partial x}$ should be free from time variable $t$. Hence, $A(x,t)=f_{1}(x)+f_{2}(t)$. \par Now from Eq. (\ref{eq10}), we see that \begin{equation} 2 m(x)\frac{\partial^{2}\eta}{\partial t\partial u}-m(x)\frac{\partial^{2} \tau}{\partial t^{2}}=0 \label{reduced10} \end{equation} Again from Eq. (\ref{eq9}), $\frac{\partial \tau}{\partial t}$ should be a constant. Assume, $2\frac{\partial \tau}{\partial t}=\omega$ which implies $\tau(t)=\frac{\omega}{2}t+t_{0}$. As from Eq. (\ref{reduced10}), \begin{equation} 2 f_{2}(t)=\frac{\partial \tau}{\partial t}=\frac{\omega}{2} \end{equation} So, $f_{2}(t)=\frac{\omega}{4}$ and $\eta(x,t,u)=(f_{1}(x)+\frac{\omega}{4})u+d_{1}+d_{2}t$. \par Now we consider two cases; $(a)$ when $f_{1}(x)$ is constant, $(b)$ when $f_{1}(x)$ is not constant. For the first case, there are two subcases; $(a.1)$ when $\frac{d^{3}\xi}{dx^{3}}=k$ for some constant $k$ i.e., $\frac{d^{3}\xi}{dx^{3}}=0,\frac{d^{4}\xi}{dx^{4}}=0$,$(a.2)$ when$\frac{d^{3}\xi}{dx^{3}}$ is not a constant. All the combinations of the stiffness, mass, axial force, and transformations are given in Table \ref{symmetryTab1}. \paragraph*{Case $(a)$ when $f_{1}(x)$ is constant.$-$} For the case $(a.1)$, from Eq. (\ref{eq6}), it can be shown that \begin{equation} \xi(x)=f_{0}(EI(x))^{\frac{1}{3}} \end{equation} and for the assumption $\frac{d^{2}\xi}{dx^{2}}=k$, $EI(x)=\frac{k^3 x^6}{8 f_{0}^3}$. For the case $(a.2)$, $\frac{d^{2}\xi}{dx^{2}}$ is not a constant. Here, we assume the coefficients of $\xi(x)$ are zero in the Eqs. (\ref{eq6}), (\ref{eq7}), (\ref{eq8}) and the form of $EI(x), m(x)$, and $T(x)$ are evaluated. \paragraph*{Case $(b)$ when $f_{1}(x)$ is not a constant.$-$} In this case also, if we assume the coefficients of $\xi(x)$ are zero in the Eqs. (\ref{eq6}), (\ref{eq7}), (\ref{eq8}) and the form of $EI(x), m(x)$ and $T(x)$ are evaluated.
\par Based on the assumptions, there may be more than these combinations of physical properties and coordinate transformations which can lead us to find a closed form solution. In Table \ref{symmetryTab1}, the possible list of these combinations are listed. It is observed that the transformation rule for the spatial coordinate is strongly dependent on the stiffness of the beam, i.e., the geometry of beam of certain material. It can be shown that for a beam of polynomial varying stiffness $EI(x)=(a_{0}+a_{1}x)^{n}$ where $a_{1}\neq 0$ any real number and $n> 3$ a nonzero positive integer, the proper transformation exist which leads to exact solution given in Table \ref{symmetryTab1} where $G(x)$ is
\begin{gather} G(x)=(a_{0}+a_{1} x)^{-n} \left(-\frac{2 \sqrt{a_{1}} A_{1} \sqrt{n-2}
(a_{0}+a_{1} x)^{\frac{1}{2} \left(-\frac{\sqrt{a_{1}^3 (n-2)
(n-1)^2+4 T_{1}}}{a_{1}^{3/2} \sqrt{n-2}}+n+3\right)}}{a_{1}^{3/2}
(n-3) \sqrt{n-2}+\sqrt{a_{1}^3 (n-2) (n-1)^2+4 T_{1}}}
-\right.\nonumber\\
\left.
\frac{2\sqrt{a_{1}} A_{2} \sqrt{n-2} (a_{0}+a_{1} x)^{\frac{1}{2}
\left(\frac{\sqrt{a_{1}^3 (n-2) (n-1)^2+4 T_{1}}}{a_{1}^{3/2}
\sqrt{n-2}}+n+3\right)}}{a_{1}^{3/2} (n-3) \sqrt{n-2}-\sqrt{a_{1}^3
(n-2) (n-1)^2+4 T_{1}}}-
\frac{A_{3} (a_{0}+a_{1} x)^3}{a_{1}
(n-3)}\right)
\label{G(x)} \end{gather}
\begin{table}[h!]
\centering
\begin{tabular}{|c|l|p{3cm}|p{4cm}|}
\hline Case & Physical Properties & Transformations & $u(x,t)$ \\ \hline \multirow{3}{*}{$(a.1)$} & $EI(x)=\frac{k^3 x^6}{8 f_{0}^3}$ & $\xi(x)=\frac{k x^2}{2}$ & $ e^{-2/x} \left(A_{1} e^{\lambda t}+A_{2} e^{-\lambda t}\right)
$\\
& $m(x)=m_{0}\frac{ e^{\frac{-4 c_{2}}{k x}}}{x^2}$ & $\tau(t)=\frac{\omega}{2}t+t_{0}$& \\
& $T(x)=T_{0} x^2-\frac{3 k^3 x^4}{4 f_{0}^3}$ & $\eta=(\alpha+\frac{\omega}{4})u+d_{1}+d_{2}t$ &\\ \hline \multirow{3}{*}{$(a.2)$} & $EI(x)=a_{1} e^{a_{0} x}$ & $\xi(x)=\frac{3 f_{0} e^{\frac{a_{0} x}{3}}}{a_{0}}$& $e^{\frac{x}{r_{0}}} (A_{1}+A_{2} t) $\\
& $m(x)=m_{0} e^{\frac{1}{3} a_{0} \left(-\frac{6 c_{2} e^{-\frac{a_{0}
x}{3}}}{a_{0} f_{0}}-x\right)}$& $\tau(t)=\frac{\omega}{2}t+t_{0}$ &\\
& $T(x)=-\frac{2}{9} a_{0}^2 a_{1} e^{a_{0} x}$ & $\eta=(\alpha+\frac{\omega}{4})u+d_{1}+d_{2}t$ &\\ \hline \multirow{3}{*}{$(b)$} & $EI(x)=a_{1} e^{-v x}$ & $\xi(x)=\frac{e^{v x}}{2 v^2}$ &$e^{2 v x} (A_{1}+A_{2} t) $\\
& $m(x)= m_{0}e^{v \left(-4 c_{2} e^{-v x}-5 x\right)}$ & $\tau(t)=\frac{\omega}{2}t+t_{0}$&\\
& $T(x)=2 a_{1} v^2 e^{-v x}$ & $\eta=\left(\frac{e^{v x}}{v}+\frac{\omega}{4}\right)u+d_{1}+d_{2}t$ &\\ \hline \multirow{3}{*}{$(c)$} & $EI(x)=(a_{0}+a_{1}x)^{n}$ & $\xi(x)=\frac{(a_{0}+a_{1} x)}{a_{1} n}$ &$2t+G(x)$\\
& $m(x)=m_{0} (a_{0}+a_{1} x)^{\frac{f_{0} (n-4)+n \omega }{f_{0}}}$ & $\tau(t)=\frac{\omega}{2}t+t_{0}$ &\\
& $T(x)=\frac{T_{1} (a_{0}+a_{1} x)^{n-2}}{a_{1} (n-2)}$ & $\eta=\frac{\omega}{4}u+d_{1}+d_{2}t$& \\ \hline \end{tabular}
\caption{The transformations and solutions for the corresponding physical properties. Here $\lambda=-\frac{\sqrt{2} \sqrt{2 f_{0}^3
T_{0}-k^3}}{f_{0}^{3/2} \sqrt{m_{0}}}$ and $G(x)$ is given in (\ref{G(x)}).
}
\label{symmetryTab1} \end{table} \section{Closed form solution} For the case $(a.1)$, the acting vector field is \begin{eqnarray} v=\frac{k x^2}{2}\frac{\partial}{\partial x}+(\frac{\omega}{2}t+t_{0})\frac{\partial}{\partial t}+((\alpha+\frac{\omega}{4})u+d_{1}+d_{2}t)\frac{\partial}{\partial u} \end{eqnarray} If we choose, $(\alpha+\frac{\omega}{4})=k$, then for vector field $X=\frac{x^2}{2}\frac{\partial}{\partial x}+u\frac{\partial}{\partial u}$ the characteristic equation is \begin{eqnarray} \frac{dx}{\frac{x^2}{2}}=\frac{dt}{0}=\frac{du}{u} \end{eqnarray} which implies \begin{eqnarray} u(x,t)=exp\left(-\frac{2}{x}\right)F(t) \end{eqnarray} for some arbitrary function $F(t)$. Now substituting this $u(x,t)$ in the original equation Eq. (\ref{first equation}) for $EI(x)=\frac{k^3 x^6}{8 f_{0}^3}$, $T(x)=T_{0} x^2-\frac{3 k^3 x^4}{4 f_{0}^3}$ and $m(x)=m_{0}\frac{ e^{\frac{-4 c_{2}}{k x}}}{x^2}$, a second order ordinary differential equation in $F(t)$ is found. Solving this equation for $F(t)$, the given $u(x,t)$ in Table \ref{symmetryTab1} is found. Following the similar way, we get the invariant solutions for other cases given in the Table \ref{symmetryTab1}.
\section{Boundary condition imposition} The symmetry analysis of a boundary value problem (BVP) may not always be successful \cite{hydon2005symmetry}. For a BVP, the domain is fixed and the symmetry of a BVP requires not only the invariance of the given differential equation, but also the invariance of the boundary data. There is no available systematic procedure for the symmetry analysis of a BVP \cite{clarkson2001open}. However, in this paper, the invariant solution is found by weakening the conditions given by Bluman and Kumei in section $4.4.1$ \cite{bluman2013symmetries} for the invariance of the boundary. We consider a beam with unit length. The boundary conditions are; $u(0,t)=0, \frac{\partial u}{\partial x}(0,t)=0, \frac{\partial^{2} u}{\partial x ^{2}}(1,t)=0, \frac{\partial^{3} u}{\partial x ^{3}}(1,t)=0,$ and the initial conditions are $u(x,0)=h(x)$,
$\frac{\partial u}{\partial t}(x,0)=0$. Satisfying all the boundary condition for the deflection, i.e. for $u$, of an elastic beam which is fixed at the left end and free at the right end, the feasible stiffness, mass, and the axial load are given by;
\begin{figure}
\caption{Comparison of numerical and analytical solutions }
\label{comparison}
\end{figure} \begin{eqnarray} EI(x)=\frac{g_{1} x^4 (6 m_{0}+x (3 m_{1}+2 m_{2} x))^4}{(m_{0}+x
(m_{1}+m_{2} x))^3}
\end{eqnarray}
\begin{eqnarray}
m(x)= m_{0}+m_{1} x+m_{2} x^2
\end{eqnarray}
\begin{gather}
T(x)=\frac{g_{1} x^2 (6 m_{0}+x (3 m_{1}+2 m_{2} x))^2 }{(m_{0}+x (m_{1}+m_{2} x))^5}
\left(-56
m_{0}^4+8 m_{0}^3 x (17 m_{2} x-10 m_{1})
\right.\nonumber\\\left. +12 m_{0}^2 x^2
\left(-7 m_{1}^2+8 m_{1} m_{2} x+2 m_{2}^2 x^2\right)
+2m_{0} x^3 \left(-22 m_{1}^3+9 m_{1}^2 m_{2} x+24 m_{1}
m_{2}^2 x^2+12 m_{2}^3 x^3\right)
\right.\nonumber\\\left.
-m_{1}^2 x^4 \left(11
m_{1}^2+14 m_{1} m_{2} x+6 m_{2}^2
x^2\right)\right) \end{gather} and the transformations are; \begin{eqnarray} \xi(x)=\frac{m_{0} x+\frac{m_{1} x^2}{2}+\frac{m_{2} x^3}{3}}{m_{0}+x(m_{1}+m_{2} x)}\\
\tau(t)=c_{1}\\
\eta=\omega u+d_{1}+d_{2}t
\end{eqnarray} where $m_{1}, m_{2}$ are dependent on $m_{0}$ and the general expressions are \begin{equation*} m_{1}=\frac{\sqrt[3]{\sqrt{6} m_{0}^3+9m_{0}^3}}{3^{2/3}
\sqrt[3]{5}}+\frac{\sqrt[3]{\frac{5}{3}} m_{0}^2}{\sqrt[3]{\sqrt{6}
m_{0}^3+9 m_{0}^3}}-2 m_{0} \end{equation*} $ m_{2}= \frac{1}{240 m_{0}^2}( 5 \sqrt{6} m_{0}^3+195 m_{0}^3+8 \sqrt[3]{5} \left(3
\left(9+\sqrt{6}\right)\right)^{2/3} \left(m_{0}^3\right)^{2/3}
m_{0}+\frac{375 m_{0}^6}{\sqrt{6} m_{0}^3+9 m_{0}^3}+\frac{120
\sqrt[3]{3} 5^{2/3} m_{0}^5}{\left(9+\sqrt{6}\right)^{2/3}
\left(m_{0}^3\right)^{2/3}}-\frac{120\ 3^{2/3}
\sqrt[3]{\frac{5}{9+\sqrt{6}}} m_{0}^4}{\sqrt[3]{m_{0}^3}}-24\ 5^{2/3}
\sqrt[3]{3 \left(9+\sqrt{6}\right)} \sqrt[3]{m_{0}^3} m_{0}^2) $
\par We choose $\omega = 2$, $g_{0} = 1$, $g_{1} = \frac{1}{3000}$, $m_{0}=1$. Then $m_{1}=-0.840295, m_{2}=0.277816$. Here $\xi(0)=0$ but it is not possible to impose $\xi(1)=0$ which provides non-feasible physical properties. The corresponding analytical solution is ;
\begin{eqnarray} u(x,t)=\left(m_{0} x+\frac{m_{1} x^2}{2}+\frac{m_{2} x^3}{3}\right)^2
\left(A_{1} \cos \left(120 \sqrt{3} \sqrt{g_{1}} t\right)+A_{2} \sin
\left(120 \sqrt{3} \sqrt{g_{1}} t\right)\right)
\end{eqnarray} For zero initial velocity, $A_{2} =0$. Considering $A_{1}=1$,
\begin{eqnarray} u(x,t)=\left(m_{0} x+\frac{m_{1} x^2}{2}+\frac{m_{2} x^3}{3}\right)^2
\left( \cos \left(120 \sqrt{3} t\right)+ \sin
\left(120 \sqrt{3} t\right)\right)
\end{eqnarray}
The comparison between the analytical form of the solution is matching well with the numerical solution given in Fig. \ref{comparison}. For the numerical solution, NDSolve command in $Mathematica$ $9.0$ is used with $h(x)=\left(m_{0} x+\frac{m_{1} x^2}{2}+\frac{m_{2} x^3}{3}\right)^2$.
\section{Conclusion} In conclusion, Lie symmetry method is applied to analyze the symmetry and the exact or closed form solution of the Euler-Bernoulli beam with axial load. Some combinations of the coordinate transformations dependent on the system properties are found which provide the closed form solution.
Different combinations of stiffness, mass and axial force are also found which yield
a closed form solution. It is observed that the crucial spatial transformations are dependent on the stiffness of the beam.
There are some nanobeam type structures such as nanoropes, nanorod etc. \cite{fan2009superelastic,lu1997elastic,manghi2006propulsion} where this analytical result can be used directly to measure the strain-displacement relation.
The imposition of boundary conditions for the symmetry analysis of an elastic beam is carried out successfully.
\end{document} | arXiv |
How many prime numbers are between 30 and 50?
We check odd numbers between $30$ and $50$ and find that the prime numbers are $31,37,41,43,47$. There are $\boxed{5}$ prime numbers between $30$ and $50$. | Math Dataset |
\begin{document}
\title{{F}robenius-Unstable Bundles and $p$-Curvature} \author{Brian Osserman} \begin{abstract} We use the theory of $p$-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for $p$-curvature to analyze low-characteristic cases, and then using degeneration techniques to obtain an answer for a general curve by degenerating to an irreducible rational nodal curve, and applying the results of \cite{os10} and \cite{os7}. We also apply our explicit formulas to give a new description of the strata of curves of genus 2 of different $p$-rank. \end{abstract} \thanks{This paper was partially supported by fellowships from the National Science Foundation and Japan Society for the Promotion of Science.} \maketitle
\section{Introduction}
The primary theme of this paper is to use the following question as an invitation to a detailed study of the theory of $p$-curvature of connections:
\begin{ques} Given a smooth curve $C$ of genus 2 over an algebraically closed field $k$ of positive characteristic, what is the number of {\bf Frobenius-unstable} vector bundles of rank 2 and trivial determinant on $C^{(p)}$? That is, if $F:C \rightarrow C^{(p)}$ denotes the relative Frobenius morphism from $C$ to its $p$-twist, how many vector bundles ${\mathscr F}$ are there on $C^{(p)}$ (of the stated rank and determinant) which are themselves semistable, but for which $F^* {\mathscr F}$ is unstable? \end{ques}
Because semistability is preserved by pullback under separable morphisms (see \cite[Lem. 3.2.2]{h-l}), the Frobenius-unstable case is in some sense a universal case for destabilization. Furthermore, Frobenius-unstable bundles are closely related to the study of the generalized Verschiebung, and its relationship to $p$-adic representations of the fundamental group of $C$, in the case that $C$ is defined over a finite field; see \cite{os9} for details.
The analysis of our question is in two parts: first, we use explicit formulas for $p$-curvature to calculate the answer directly for odd characteristics $\leq 7$; and second, we use the abstract theory of $p$-curvature to give a new proof of the answer for a general curve of genus 2 in any odd characteristic, via degeneration to an irreducible rational nodal curve and application of the results of \cite{os10} and \cite{os7}. The latter result is originally due to Mochizuki; see \cite{mo3} and \cite{os6}. The main advantage of the explicit approach, as compared to the more general degeneration argument, is that the $p$-curvature formulas may be used to study arbitrary smooth curves, and do not give results only for general curves. This distinction is underscored by an algorithm derived via the same techniques to explicitly describe the loci of curves of genus $2$ and $p$-ranks $0$ or $1$ in any specified characteristic. Additionally, the explicit approach is useful for computing examples in order to formulate conjectures; one aim of this paper is therefore to serve as an illustration of how $p$-curvature may be used very concretely for experimental purposes, and more theoretically for more general results.
Our main theorem is:
\begin{thm}\label{exp-main} Let $C$ be a smooth, proper curve of genus $2$ over an algebraically closed field $k$ of characteristic $p$; it may be described on an affine part by $y^2=g(x)$ for some quintic $g$. Then the number of semistable vector bundles on $C$ with trivial determinant which pull back to unstable vector bundles under the relative Frobenius morphism is: \begin{itemize} \item[$p=3$:] $16 \cdot 1$; \item[$p=5$:] $16 \cdot e_5$, where $e_5=5$ for $C$ general, and is given for an arbitrary $C$ as the number of distinct roots of a quintic polynomial with coefficients in terms of the coefficients of $g$; \item[$p=7$:] $16 \cdot e_7$, where $e_7=14$ for $C$ general, and is given for an arbitrary $C$ as the number of points in the intersection of four curves in ${\mathbb A}^2$ whose coefficients are in terms of the coefficients of $g$. \item[$p>2$:] (Mochizuki \cite{mo3}, \cite{os6}) $16 \cdot \frac{p^3-p}{24}$ for $C$ general. \end{itemize}
Furthermore, when $C$ is general, any Frobenius-unstable bundle ${\mathscr F}$ has no non-trivial deformations which yield the trivial deformation of $F^* {\mathscr F}$. \end{thm}
There is a considerable amount of literature on Frobenius-unstable vector bundles. Gieseker and Raynaud produced certain examples of Frobenius-unstable bundles in \cite{gi2} and \cite[p. 119]{ra2}, but, aside from the results of Mochizuki discussed below, the first classification-type result is due to Laszlo and Pauly, who answered our main question in characteristic 2: there is always a single Frobenius-unstable bundle (see \cite{l-p}, argument for Prop. 6.1 2.; the equations for an ordinary curve are not used). Joshi, Ramanan, Xia and Yu obtain results on the Frobenius-unstable locus in characteristic $2$ for higher-genus curves in \cite{j-r-x-y}. Most recently, and concurrently with the initial preparation of the present paper, Lange and Pauly \cite{l-p3} have recovered the formula of Theorem \ref{exp-main} for general $C$ in the case of ordinary curves via a completely different approach, although they obtain only an inequality, rather than an equality.
However, the most comprehensive results to date follow from Mochizuki's work (see \cite{mo3} and \cite{os6}), which was carried out in the context of ${\mathbb P}^1$-bundles on curves in any odd characteristic, via degeneration techniques quite similar to those which we pursue in Sections 8 and 9. Indeed, key results and their arguments in Sections \ref{s-exp-det}, \ref{s-def-background}, and \ref{s-def-deform} are essentially the same as Mochizuki's; in the first case, the argument presented here was discovered independently, while in the other cases, the author's original arguments were more complicated and less general than Mochizuki's, and have thus been replaced. There are several justifications for the logical redundancy: the arguments in question are all quite short, and it seems desirable to have a self-contained proof of the main theorem, without translating to projective bundles and back; the argument of Section \ref{s-exp-det} is actually substantially simpler in our case of curves of genus $2$; and finally, the gluing statements of Section \ref{s-def-background} require some ridigifying hypotheses in the context of vector bundles that do not arise in Mochizuki's work.
Lastly, we remark that as discussed in \cite{os6}, Mochizuki's strategy is to degenerate to totally degenerate curves, while our strategy is to degenerate to irreducible nodal curves. Aside from allowing one to make more naive arguments in terms of explicit degenerations, ours is a substantially more difficult approach, since after reducing the problem to self-maps of ${\mathbb P}^1$ with prescribed ramification, in Mochizuki's case it suffices to handle the case of three ramification points, while our argument requires four, and is therefore far more complicated; see \cite{os7} for details. However, degenerating to irreducible curves is helpful for studying Frobenius-unstable bundles in higher genus; see \cite{os6}.
We begin in Section \ref{s-exp-background} by relating our main question to $p$-curvature, and Section \ref{s-exp-gen-pcurve} is then devoted to developing explicit and completely general combinatorial formulas for $p$-curvature. We make certain necessary computations for genus $2$ curves in Section \ref{s-exp-ftheta}, which we also apply to obtain an explicit algorithm for generating $p$-rank formulas in any given odd characteristic. Section \ref{s-exp-conns} is devoting to computing the space of connections on a certain unstable bundle, and in Section \ref{s-exp-pcurve} we conclude the computation with explicit descriptions of the locus of vanishing $p$-curvature in characteristics $3, 5 \text{ and } 7$. The space of connections on the same bundle having nilpotent $p$-curvature is shown to be finite and flat in Section \ref{s-exp-det}, again by explicit computation; this completes the proof of Theorem \ref{exp-main} for $p\leq 7$, and also provides a key step of the general case. In Section \ref{s-def-background} we discuss the relationship between connections on nodal curves and their normalizations, and finally in Section \ref{s-def-deform} we show that connections on nodal curves deform, and apply the results of \cite{os10} and \cite{os7} to conclude our main theorem.
Computations were carried out in Maple and Mathematica, and in the case of the $p$-curvature formulas of Section \ref{s-exp-gen-pcurve}, using simple C code.
The contents of this paper form a portion of the author's 2004 PhD thesis at MIT, under the direction of Johan de Jong.
\section{From {F}robenius-instability to $p$-curvature}\label{s-exp-background}
We begin by explaining how classification of Frobenius-unstable vector bundles is related to $p$-curvature of connections. For the basic theory of connections and $p$-curvature, we refer the reader to \cite[\S 1, \S 5]{ka1}. Note that the induced connection on tensor products descends to wedge products, so that for a vector bundle ${\mathscr E}$ with connection, we obtain an induced {\bf determinant connection} on $\det {\mathscr E}$. Additionally, given $\varphi \in \operatorname{Aut}({\mathscr E})$ and a $\nabla$ on ${\mathscr E}$, we refer to the operation of conjugation by $\varphi$ on $\nabla$ as {\bf transport}. We summarize the basic results relating Frobenius with $p$-curvature, due to to Katz \cite{ka1}.
\begin{thm}\label{exp-cartier-vect} Let $X$ be a smooth $S$-scheme, with $S$ having characteristic $p$, and let $F: X \rightarrow X^{(p)}$ be the relative Frobenius morphism. Then for any vector bundle ${\mathscr F}$ on $X^{(p)}$, $F^* {\mathscr F}$ is equipped with a canonical connection $\nabla ^{\text{can}}$. For any vector bundle ${\mathscr E}$ with connection $\nabla$ on $X$, the kernel of $\nabla$, denoted ${\mathscr E}^\nabla$, is naturally an ${\mathscr O}_{X^{(p)}}$-module.
The operations ${\mathscr F} \mapsto (F^* {\mathscr F}, \nabla ^{\text{can}})$ and $({\mathscr E}, \nabla) \mapsto {\mathscr E}^{\nabla}$, are mutually inverse functors, giving an equivalence of categories between the category of vector bundles of rank $n$ on $X^{(p)}$ and the full subcategory of the category of vector bundles of rank $n$ with integrable connection on $X$ consisting of objects whose connection has $p$-curvature zero.
Furthermore, the same statement holds when restricted to the full subcategories of vector bundles with trivial determinant on $X^{(p)}$, and vector bundles with connection both having trivial determinant on $X$. \end{thm}
\begin{proof} See \cite[\S 5]{ka1}, and in particular \cite[Thm. 5.1]{ka1}. It only remains to check that the categorical equivalence on coherent sheaves gives an equivalence on vector bundles, and again in the case of trivial determinant. The first assertion follows from the fact that $F$ is faithfully flat when $X/S$ is smooth. The second is easily checked by verifying that the operation ${\mathscr F} \mapsto (F^*{\mathscr F}, \nabla ^{\text{can}})$ commutes with taking determinants. \end{proof}
Thus, $p$-curvature is naturally related to the study of Frobenius-pullbacks. The categorical equivalence implies that isomorphism classes of ${\mathscr F}$ will correspond to transport equivalence classes of connections with vanishing $p$-curvature. In the case of our particular question, the relationship is particularly helpful. We assume we are in the following situation.
\begin{sit}\label{exp-genus} $C$ is a smooth, proper curve of genus 2, over an algebraically closed field $k$ of characteristic $p$. \end{sit}
In this situation, Joshi and Xia showed that there are at most finitely many Frobenius-unstable vector bundles of rank $2$ and trivial determinant on $C$ (see \cite[Thm. 3.2]{j-x}, although we will also obtain a more direct proof from Corollary \ref{exp-finite}), and also gave the following description of them (see \cite[Prop. 3.3]{j-x}):
\begin{prop}\label{exp-unstable} (Joshi-Xia) Let ${\mathscr F}$ be a semistable rank $2$ vector bundle on $C$ with trivial determinant, and suppose ${\mathscr E} = F^* {\mathscr F}$ is unstable. Then there is a non-split exact sequence $$0 \rightarrow {\mathscr L} \rightarrow {\mathscr E} \rightarrow {\mathscr L}^{-1} \rightarrow 0$$ where ${\mathscr L}$ is a {\bf theta characteristic}, that is, ${\mathscr L} ^{\otimes 2} \cong \Omega^1 _C$. \end{prop}
We thus have a natural set of unstable vector bundles upon which to look for connections with vanishing $p$-curvature. Indeed, it is easy to see that the proposition is sharp.
\begin{cor} Frobenius-unstable vector bundles of rank $2$ and trivial determinant on $C$ are necessarily stable, and in one-to-one correspondence with transport-equivalence classes of connections on vector bundles ${\mathscr E}$ as in the above proposition, having trivial determinant and vanishing $p$-curvature. This correspondence is functorial in the sense that after arbitrary base change $C' \rightarrow C$, vector bundles ${\mathscr F}$ with trivial determinant and $F^* {\mathscr F} \cong {\mathscr E}'$ are in one-to-one correspondence with transport-equivalence classes of connections on ${\mathscr E}'$ having trivial determinant and vanishing $p$-curvature. \end{cor}
\begin{proof} The functoriality is the more obvious statement, in light Theorem \ref{exp-cartier-vect}. For the rest, all we need check is that if $F^* {\mathscr F} \cong {\mathscr E}$ for some ${\mathscr F}$, we necessarily have that ${\mathscr F}$ is stable. But if ${\mathscr M} \subset {\mathscr F}$ is a non-negative line sub-bundle, $F^* {\mathscr M} \subset {\mathscr F}$ is non-negative with degree a multiple of $p$, which cannot occur when $F^* {\mathscr F} \cong {\mathscr E}$ by the following standard lemma. \end{proof}
We state the lemma in more generality than immediately necessary, for later use. The argument for the $\mathcal{E}nd^0({\mathscr E})$ case is taken from \cite[Lem. I.3.5, p. 105]{mo3}.
\begin{lem}\label{exp-destab-unique} Let ${\mathscr E}$ be a rank $2$ vector bundle of degree $0$ on a possibly nodal curve $C$, and suppose ${\mathscr L}$ is a positive line bundle giving an exact sequence $$0 \rightarrow {\mathscr L} \rightarrow {\mathscr E} \rightarrow {\mathscr E}/{\mathscr L} \rightarrow 0$$ Then ${\mathscr L}$ is unique, and is the maximal degree line bundle inside ${\mathscr E}$, and ${\mathscr E}$ has no quotient line bundle of degree $0$. Furthermore, the same statement holds for positive sub-bundles of given rank of the traceless endomorphisms $\mathcal{E}nd^0({\mathscr E})$. \end{lem}
\begin{proof}One checks this simply by considering maps of the form ${\mathscr L}\rightarrow {\mathscr E} \rightarrow {\mathscr E}/{\mathscr L}'$, and considering the degrees of the line bundles in question. For the $\mathcal{E}nd^0({\mathscr E})$ case, because $\mathcal{E}nd^0({\mathscr E})$ is self-dual it suffices to consider the case of line sub-bundles, and to show that the existence of a positive sub-bundle precludes the existence of a line sub-bundle of degree $0$. But if we have ${\mathscr L} \subset \mathcal{E}nd^0({\mathscr E})$ positive, and ${\mathscr L}' \subset \mathcal{E}nd^0({\mathscr E})$ non-negative, first by considering ${\mathscr L}' \rightarrow \mathcal{E}nd^0({\mathscr E}) \rightarrow {\mathscr L}^{-1}$ we find that the composition must be zero, so that we have a map $\mathcal{E}nd^0({\mathscr E})/{\mathscr L}' \rightarrow {\mathscr L}^{-1}$. But then considering the natural ${\mathscr O}_C \subset \mathcal{E}nd^0({\mathscr E})/{\mathscr L}'$, composing with the map to ${\mathscr L}^{-1}$ must again give zero, so that in fact the map $\mathcal{E}nd^0({\mathscr E}) \rightarrow {\mathscr L}^{-1}$ factors through $(\mathcal{E}nd^0({\mathscr E})/{\mathscr L}')/{\mathscr O}_C \cong {\mathscr L}'^{-1}$, from which one can conclude the desired statement. \end{proof}
Next, we note that the ${\mathscr E}$ of Proposition \ref{exp-unstable} are nearly unique.
\begin{prop}\label{exp-unstable-unique} There are only $16$ choices for ${\mathscr E}$ as described in Proposition \ref{exp-unstable}, one for each choice of ${\mathscr L}$. \end{prop}
\begin{proof} Any two choices of ${\mathscr L}$ differ by one of the $2^{2g}=16$ line bundle of order $2$ on $C$. With ${\mathscr L}$ chosen, we calculate that $\operatorname{Ext}^1({\mathscr L}^{-1}, {\mathscr L}) \cong H^0(C, {\mathscr O}_C) \cong k$, so the isomorphism class of ${\mathscr E}$ is uniquely determined. \end{proof}
Lastly, we observe that it suffices to handle a single choice of ${\mathscr E}$.
\begin{cor}\label{exp-unstable-same} For any ${\mathscr E}, {\mathscr E}'$ as in Proposition \ref{exp-unstable-unique}, there is a canonical funtorial equivalence between the vector bundles ${\mathscr F}$ of trivial determinant with $F^* {\mathscr F} \cong {\mathscr E}$, and those with $F^* {\mathscr F} \cong {\mathscr E}'$. \end{cor}
\begin{proof} From Proposition \ref{exp-unstable-unique} we see that ${\mathscr E}$ and ${\mathscr E}'$ are related by tensoring by a $2$-torsion line bundle. The corollary is then easily verified by the bijectivity of $F^*$ on $2$-torsion line bundles. \end{proof}
Having reduced our main question to a matter of classifying connections with vanishing $p$-curvature on a certain vector bundle, we briefly develop the formal properties of $p$-curvature, which we will not need to use until Section \ref{s-exp-det} and the following sections. The statement is:
\begin{prop}\label{exp-pcurve-formal} Given a connection $\nabla$ on a vector bundle ${\mathscr E}$ on a smooth $X$ over $S$, we have the following description of the $p$-curvature $\psi_{\nabla}$ of $\nabla$. \begin{ilist} \item We may describe $\psi_{\nabla}$ as an element of $$\Gamma(X, \mathcal{E}nd({\mathscr E}) \otimes F^* \Omega^1_{X^{(p)}/S})^{\nabla^{\operatorname{ind}}},$$ where the superscript denotes the subspace of sections horizontal for $\nabla^{\operatorname{ind}}$; \item If ${\mathscr E}$ and $\nabla$ have trivial determinant, we find that $\psi_{\nabla}$ lies in $$\Gamma(X, \mathcal{E}nd^0({\mathscr E}) \otimes F^* \Omega^1_{X^{(p)}/S})^{\nabla^{\operatorname{ind}}},$$ where $\mathcal{E}nd^0({\mathscr E})$ denotes the sheaf of traceless endomorphisms of ${\mathscr E}$. \item Assuming ${\mathscr E}$ has a connection, we may also consider $p$-curvature as giving maps between affine spaces $$\psi: \Gamma(X, \operatorname{Conn}({\mathscr E})) \rightarrow \Gamma(X, \mathcal{E}nd({\mathscr E}) \otimes F^* \Omega^1_{X^{(p)}}),$$ $$\psi^0: \Gamma(X, \operatorname{Conn}^0({\mathscr E})) \rightarrow \Gamma(X, \mathcal{E}nd^0({\mathscr E}) \otimes F^* \Omega^1_{X^{(p)}}),$$ where $\operatorname{Conn}({\mathscr E})$ and $\operatorname{Conn}^0({\mathscr E})$ denote the sheaves of connections on ${\mathscr E}$, and of connections with trivial determinant (when ${\mathscr E}$ has trivialized determinant) respectively. \item We may take the determinant of the previous maps, and in the case that ${\mathscr E}$ has trivial determinant, we obtain a map $$\det \psi^0: \Gamma(X, \operatorname{Conn}^0({\mathscr E})) \rightarrow \Gamma(X^{(p)}, (\Omega^1_{X^{(p)}})^{\otimes n}).$$ \end{ilist} \end{prop}
\begin{proof} Assertion (i) follows directly from the linearity and $p$-linearity results of Katz \cite[5.0.5, 5.2.0]{ka1}, together with the fact that $\psi_\nabla (\theta)$ commutes with $\nabla_{\theta'}$ for any $\theta'$, by\cite[5.2.3]{ka1}. Assertion (ii) follows from explicit computation, in Corollary \ref{exp-pcurve-cors} (ii). We then obtain assertion (iii) formally: since we are working over an arbitrary scheme, we obtain the map on arbitrary $T$-valued points, and if ${\mathscr E}$ has a connection, the space of connections is a torsor over $\Gamma(X, {\mathscr E}\otimes \Omega^1_X)$, and likewise after arbitrary pull-back, and hence representable by an affine space. Finally, for assertion (iv), we just put together assertions (ii) and (iii), checking that in the trivial determinant case, the induced connection on the determinant of $\operatorname{End}^0({\mathscr E})$ is likewise trivial. \end{proof}
\section{Explicit $p$-curvature Formulas}\label{s-exp-gen-pcurve}
In this section, we develop general combinatorial formulas which may be used to explicitly compute the $p$-curvature of a connection for any given $p$, and in any dimension, although it will be easiest to compute in the case of curves, where it suffices to consider a single derivation. We specify our notation for the section.
\begin{sit}\label{exp-single-open} $U$ denotes an affine open on a smooth $X/S$. We are given a vector bundle ${\mathscr E}$ trivialized on $U$, and a derivation $\theta$ on $U$. We thus obtain a connection matrix $\bar{T}$ on $U$ associated to any connection $\nabla$ on ${\mathscr E}$, such that $\nabla _{\theta} (s) = \bar{T} s + \theta s$. Denote also by $\bar{T}_{(p)}$ the connection matrix associated to $\nabla$ and $\theta^p$. \end{sit}
One can then easily check the following explicit formula for the $p$-curvature associated to $\nabla$ and $\theta$.
\begin{lem} We have $\psi_{\nabla} (\theta) = (\bar{T} + \theta)^p - \bar{T}_{(p)} - \theta ^p$ \end{lem}
We now describe the expansion of $(\bar{T} + \theta)^n$ using the commutation relation $\theta \bar{T} = (\theta \bar{T}) + \bar{T} \theta$, where, in order to make formulas easier to parse, $(\theta \bar{T})$ denotes the application of $\theta$ to the coordinates of $\bar{T}$.
\begin{prop}Given ${\mathfrak i} = (i_1, \dots, i_\ell) \in {\mathbb N}^{\ell-1}\times ({\mathbb N} \cup \{0\})$ with $\sum _{j=1} ^\ell i_j = n$, denote by $\hat{n}_{\mathfrak i}$ the coefficient of $\bar{T}_{{\mathfrak i}}:=(\theta ^{i_1-1} \bar{T})\dots (\theta ^{i_{\ell-1}-1} \bar{T}) \theta ^{i_\ell}$ in the full expansion of $(\bar{T} + \theta)^n$. Also denote by ${\mathfrak i}_0$ the vector $(i_1, \dots, i_{\ell-1}, 0)$. Then we have: $$\hat{n}_{{\mathfrak i}} = \binom{n}{i_\ell} \hat{n}_{{\mathfrak i}_0}$$ \end{prop}
\begin{proof} Although this formula may be seen directly, the proof is expressed most clearly by induction on $n$, which we sketch. We may assume that $i_{\ell}>0$, or the statement is trivial. By definition, we have
$$(\bar{T}+\theta)^n= (\bar{T}+\theta)(\sum _{\ell'} \sum _{|{\mathfrak i}'|=n-1} \hat{n}_{{\mathfrak i}'} \bar{T}_{{\mathfrak i}'}),$$
where ${\mathfrak i}' = (i'_1, \dots, i'_{\ell'})$, and $|{\mathfrak i}'|:= \sum _j i'_j$. Multiplying out and commuting the $\theta$ from left to right until we obtain another such expression, we find two cases: $i_1 =1$ and $i_1>1$; we handle the case $i_1>1$, the other being essentially the same. In this case, we obtain the inductive formula $\hat{n}_{{\mathfrak i}} = \sum_j \hat{n}_{{\mathfrak i} - 1_j}$, where $1_j$ denotes the vector which is $1$ in the $j$th position and $0$ elsewhere, and where $j$ is allowed to range only over values where $i_j > 1$. We then have also that $\hat{n}_{{\mathfrak i}_0} = \sum_{j<\ell} \hat{n}_{{\mathfrak i}_0-1_j}$, so that if we induct on $n$, we have $\hat{n}_i = \sum _j \hat{n}_{i-1_j} = \sum _{j <\ell} \binom{n-1}{i_\ell} \hat{n}_{({\mathfrak i}-1_j)_0} + \binom{n-1}{i_{\ell-1}} \hat{n}_{({\mathfrak i} - 1_\ell)_0} = (\binom{n-1}{i_\ell}+\binom{n-1}{i_\ell-1}) \hat{n}_{{\mathfrak i}_0},$ where the last equality makes use of the observation that ${\mathfrak i}_0 = ({\mathfrak i}-1_\ell)_0$. Then the identity $\binom{n-1}{r}+ \binom{n-1}{r-1} = \binom{n}{r}$ completes the proof. \end{proof}
It follows that if $n=p$, $\hat{n}_{{\mathfrak i}}$ is nonzero mod $p$ only if $i_\ell=0$ or $i_\ell =p$, and in the latter case, we have $\ell=1$, $i_1 =p$, and $\hat{n}_{{\mathfrak i}} =1$, which precisely cancels the $\theta ^p$ subtracted off in the formula for $\psi_{\nabla}(\theta)$. We immediately see that $\psi_{\nabla}(\theta)$ is in fact given entirely by linear terms. In particular, this explicitly recovers the statement we already knew to be true that $p$-curvature takes values in the space of ${\mathscr O}_C$-linear endomorphisms of ${\mathscr E}$. We may now restrict our attention to the linear terms in the expansion, and will shift our notation accordingly:
\begin{prop}\label{exp-gen-pcurve} Given ${\mathfrak i} = (i_1, \dots, i_\ell) \in {\mathbb N}^\ell$ with $\sum _{j=1} ^\ell i_j = n$, denote by $n_{\mathfrak i}$ the coefficient of $\bar{T}_{{\mathfrak i}}=(\theta ^{i_1-1} \bar{T})\dots (\theta ^{i_\ell-1} \bar{T})$ in the full expansion of $(\bar{T} + \theta)^n$. Also denote by $\hat{{\mathfrak i}}$ the truncated vector $(i_1, \dots, i_{\ell-1})$. Then we have: $$n_{{\mathfrak i}} = \binom{n-1}{i_\ell-1} n_{\hat{{\mathfrak i}}}$$ We thus get $$n_{{\mathfrak i}} = \prod _{j=1} ^\ell \binom{n-1 - \sum _{m=j+1} ^\ell i_m}{i_j -1} = \frac{(n-1)!}{(\prod_{j=1}^\ell (i_j-1)!)(\prod _{j=1} ^{\ell-1} (\sum _{m=1} ^j i_m))}$$ \end{prop}
\begin{proof} This follows from the same induction argument as the previous proposition. \end{proof}
We note that this implies that every such term in the expansion of $(\bar{T}+\theta)^n$ is nonzero mod $n$ when $n=p$, since the numerator in the resulting formula is simply $(n-1)!$. Thus, the $p$-curvature formula is always maximally complex, having an exponential number of terms. However, when some of the terms commute, the formulas tend to simplify considerably.
\begin{prop}\label{exp-pcurve-commute} Given $\ell>0$ and a subset $\Lambda \subset \{1, \dots, \ell\}$, denote by $S^{\Lambda}_{\ell}$ the subset of the permutation group $S_{\ell}$ which preserves the order of the elements of $\Lambda$; that is, $S^{\Lambda}_{\ell}:=\{\sigma \in S_{\ell}:\forall j<j' \in \Lambda, \sigma(j)<\sigma(j')\}$. Given also ${\mathfrak i} = (i_1, \dots, i_\ell) \in {\mathbb N}^\ell$ with $\sum _{j=1} ^\ell i_j = n$, we denote by $n^{\Lambda}_{\mathfrak i}$ the sum over all $\sigma \in S^\Lambda_{\ell}$ of $n_{\sigma({\mathfrak i})}$, where $\sigma({\mathfrak i})$ denotes the vector $(i_{\sigma^{-1}(1)}, \dots, i_{\sigma^{-1}(\ell)})$ obtained from ${\mathfrak i}$ by permuting the coordinates under $\sigma$. Then we have: $$n^{\Sigma}_{{\mathfrak i}} = \frac{n!}{ \prod _{j=1} ^\ell (i_j-1)! \prod _{j=1}^{\ell}(i_j+\sum_{m<j}^{m,j \in \Lambda} i_m)}.$$ Note that the last sum in the denominator is non-empty only for $j \in \Lambda$. \end{prop}
\begin{proof} First note that if we want the entries of ${\mathfrak i}$ with indices in $\Lambda$ to have the same order in $\sigma({\mathfrak i})$, we must apply $\sigma^{-1}$ rather than $\sigma$ to the indices, as in our definition. Applying our previous formula, we really just want to show that $$\sum _{\sigma \in S^{\Lambda}_\ell} \prod _{j=1} ^{\ell-1} \frac{1}{\sum _{m=1} ^j i_ {\sigma^{-1}(m)}} = \frac{n}{\prod _{j=1} ^\ell (i_j + \sum_{m<j}^{m,j \in \Lambda} i_m)} = \frac{\sum_{j=1} ^\ell i_j}{\prod _{j=1} ^\ell (i_j + \sum_{m<j}^{m,j \in \Lambda} i_m)}.$$ Dividing through by $\sum_{j=1}^\ell i_j$ reduces the identity to \begin{equation}\label{exp-perm-comb} \sum _{\sigma \in S^{\Lambda}_\ell} \prod _{j=1} ^{\ell} \frac{1}{\sum _{m=1} ^j i_ {\sigma^{-1}(m)}} = \frac{1}{\prod _{j=1} ^\ell (i_j + \sum_{m<j}^{m,j \in \Lambda} i_m)}. \end{equation} We show this by induction on $\ell$ (noting that it is rather trivial in the case $\ell=1$, whether or not $\Lambda$ is empty), breaking up the first sum
over $S^{\Lambda}_\ell$ into $\ell-|\Lambda|+1$ pieces, depending on which $i_r$ ends up in the final place. There are two cases to consider: $r \not\in \Lambda$, or $r = \Lambda_{\max}$. In either case, the relevant part of the sum on the left hand side becomes $\sum _{\sigma \in S^{\Lambda,r}_{\ell}} \prod _{j =1} ^{\ell} \frac{1}{\sum _{m=1}^j i_{\sigma^{-1}(m)}}$, where $S^{\Lambda,r}_{\ell}$ denotes the subset of $S^{\Lambda}_{\ell}$ sending $r$ to $\ell$. Now, the point is that for our sums, this will be equivalent to an order-preserving subset of the symmetric group acting on a set of $\ell-1$ elements, allowing us to apply induction. In the case that $r \not \in \Lambda$, $\Lambda$ is in essence unaffected, and we find that $$\sum _{\sigma \in S^{\Lambda,r}_{\ell}} \prod _{j =1} ^{\ell} \frac{1}{\sum _{m=1}^j i_{\sigma^{-1}(m)}}= \frac{1}{n} \sum _{\sigma \in S^{\Lambda,r}_{\ell}} \prod _{j = 1}^{\ell-1} \frac{1}{\sum _{m=1}^{j} i_{\sigma^{-1}(m)}},$$ and one checks that this sum is of the same form as Equation \ref{exp-perm-comb}, with $i_r$ ommitted, so by induction we find that this sum is equal to $$ \frac{1}{n}\frac{1}{\prod_{j \neq r} (i_j + \sum_{m<j}^{m,j \in \Lambda} i_m)} = \frac{i_r+\sum_{m<r}^{m,r \in \Lambda}i_m}{n\prod_{j=1}^\ell (i_j + \sum_{m<j}^{m,j \in \Lambda} i_m)} = \frac{i_r}{n \prod_{j=1}^\ell (i_j + \sum_{m<j}^{m,j \in \Lambda} i_m)},$$ since $r \not \in \Lambda$. In the case that $r=\Lambda_{\max}$, we effectively reduce the size of $\Lambda$ by one, but because $r$ is maximal in $\Lambda$, for $j \neq r$ the term $\sum _{m <j}^{m,j \in \Lambda}i_m $ is unaffected by omitting $r$ from $\Lambda$. We thus find, arguing as before, $$\sum _{\sigma \in S^{\Lambda,r}_{\ell}} \prod _{j =1} ^{\ell} \frac{1}{\sum _{m=1}^j i_{\sigma^{-1}(m)}} = \frac{i_r+\sum_{m<r}^{m,r \in \Lambda}i_m}{n \prod_{j=1}^\ell (i_j + \sum_{m<j}^{m,j \in \Lambda} i_m)} = \frac{\sum_{j \in \Lambda} i_j}{n \prod_{j=1}^\ell (i_j + \sum_{m<j}^{m,j \in \Lambda} i_m)}.$$ Adding these up as $r$ ranges over $\Lambda_{\max}$ and all values not in $\Lambda$, and using $n = \sum _j i_j$, we get the desired identity. \end{proof}
We give some specific applications of this formula.
\begin{cor}\label{exp-pcurve-cors} Let ${\mathscr E}$ be a vector bundle of rank $r$ on a smooth variety $X$ over a field $k$, with $\nabla$ an integrable connection on ${\mathscr E}$ and $\theta$ a derivation on an open set $U$ which also trivializes ${\mathscr E}$. We have: \begin{ilist} \item If $r=1$, $p$-curvature is given by: $$\psi_{\nabla}(\theta) = \bar{T}^p + (\theta ^{p-1} \bar{T}) - \bar{T}_{(p)}.$$ \item Suppose ${\mathscr E}$ has trivialized determinant, and $\nabla$ has trivial determinant. Then the $p$-curvature of $\nabla$ has image in the traceless endomorphisms of ${\mathscr E}$. \item Suppose $\nabla'$ is a connection on $U$ with $\nabla'-\nabla = \omega \operatorname{I}$ a scalar endomorphism. Then we have $$\psi_{\nabla'}(\theta)-\psi_{\nabla}(\theta) = ((\hat{\theta}(\omega))^p + \theta^{p-1}(\hat{\theta}(\omega))-\hat{\theta^p}(\omega))\operatorname{I},$$ where $\hat{\theta}$ denotes the unique linear map $\Omega^1_C\rightarrow {\mathscr O}_C$ such that $\theta = \hat{\theta} \circ d$. \end{ilist} \end{cor}
\begin{proof} From the previous proposition, we see that when $n=p$ and $\Lambda$ is empty, so that all the involved matrices commute, we have $$n^{\varnothing}_{{\mathfrak i}} = \frac{p!}{\prod_{j=1}^{\ell}i_j!},$$ but the actual coefficient will be $n^{\varnothing}_{{\mathfrak i}}/P_{{\mathfrak i}}$, where $P_{{\mathfrak i}}$ is the number of permutations fixing the vector ${\mathfrak i}$, since summing up over all permutations will count each term $P_{{\mathfrak i}}$ times. We see that this expression can be non-zero mod $p$ only if either $P_{{\mathfrak i}}$ is a multiple of $p$, or some $i_j$ is. Since $P_{{\mathfrak i}}$ is the order of a subgroup of $S_\ell$, it can be a multiple of $p$ if and only if $\ell=p$ and each $i_j=1$. On the other hand, an $i_j$ can be a multiple of $p$ if and only if $\ell=1$ and $i_1=p$; these two terms simply reiterate that the coefficients of $\bar{T}^p$ and $(\theta ^{p-1} \bar{T})$ are both $1$, and we see that every other coefficient vanishes mod $p$. We immediately conclude (i), and for (ii) we see similarly that we have $$\operatorname{Tr} \psi_{\nabla}(\theta) = \operatorname{Tr} \bar{T}^p + \operatorname{Tr} (\theta ^{p-1} \bar{T}) - \operatorname{Tr} f_{\theta ^p} \bar{T}.$$ The second and third terms visibly have vanishing trace because $\bar{T}$ does, while it is easy to see (for instance, by passing to the algebraic closure of $k$ and taking the Jordan normal form) that $\operatorname{Tr} (\bar{T}^p) = (\operatorname{Tr} \bar{T})^p =0$.
For (iii), we can compare the $p$-curvatures of $\nabla$ and $\nabla'$ term by term; we have $\bar{T}+\hat{\theta}(\omega) \operatorname{I}$ as the matrix for $\nabla'$, and we see that if we expand each term of $\psi_{\nabla'}(\theta)$, we get $\psi_{\nabla}(\theta)$ from expanding out only terms involving $\bar{T}$ and $\theta$, and $((\hat{\theta}(\omega))^p + \theta^{p-1}(\hat{\theta}(\omega))-\hat{\theta^p}(\omega))\operatorname{I}$ from expanding out terms involving only $\hat{\theta}(\omega)$ and $\theta$, since these last all commute with one another. We thus want to show that all of the coefficients of the cross terms are always zero mod $p$. If we consider a particular term $(\theta_0 ^{i_1-1} (\bar{T}+\hat{\theta}(\omega)\operatorname{I}))\dots (\theta_0 ^{i_\ell-1} (\bar{T}+\hat{\theta}(\omega)\operatorname{I}))$ corresponding to a vector ${\mathfrak i}$, a cross term will arise by choosing a subset $\Lambda \subset \{1, \dots, \ell\}$ from which the $\bar{T}$ terms will be chosen, with the $\hat{\theta}(\omega)\operatorname{I}$ term being chosen for all indices outside $\Lambda$. To compute the relevant coefficient we can essentially sum over all permutations in the $S^{\Lambda}_{\ell}$ of Proposition \ref{exp-pcurve-commute}. The only caveat is that if $\sigma \in S^{\Lambda}_{\ell}$ fixes $\Lambda$ and leaves the vector ${\mathfrak i}$ unchanged, then it will give the same term in the expansion as the identity. Such $\sigma$ form a subgroup of $S_{\ell}$, and if we denote the order of this subgroup by $P^{\Lambda}_{{\mathfrak i}}$, we find that the coefficient we want to compute is given by, still in the notation of Proposition \ref{exp-pcurve-commute}, the expression $n^{\Lambda}_{{\mathfrak i}}/P^{\Lambda}_{{\mathfrak i}}$. Now, the only way to cancel the $p$ in the numerator of $n^{\Lambda}_{{\mathfrak i}}$ would be for either $P^{\Lambda}_{{\mathfrak i}}$ or the denominator of $n^{\Lambda}_{{\mathfrak i}}$ to also be divisible by $p$. The denominator of $n^{\Lambda}_{{\mathfrak i}}$ cannot be divisible by $p$, since the $i_j$ add up to $p$, and the only way that $p$ could appear in the denominator would therefore be when $\Lambda$ is all of $\{1, \dots, \ell\}$, which corresponds to the terms which only involve $\bar{T}$, or when $\ell=1$, which gives the $\theta^{p-1}(\hat{\theta}(\omega))$ term. Similarly, $P^{\Lambda}_{{\mathfrak i}}$ is the order of a subgroup of $S_{\ell}$ which fixes $\Lambda$, so can be a
multiple of $p$ only if $\ell=p$ and $|\Lambda|=0$, which corresponds to the term $((\hat{\theta}(\omega))^p$. This yields the desired result. \end{proof}
We do not use the last statement of the corollary in this paper, but it could be used to show, for instance, that when $r$ is prime to $p$, and ${\mathscr E}$ has rank $r$, then if any representative on ${\mathscr E}$ of a projective connection $\nabla$ on ${\mathbb P}({\mathscr E})$ has vanishing $p$-curvature, then the unique representative on ${\mathscr E}$ of $\nabla$ with vanishing trace must likewise have vanishing $p$-curvature. We also remark that results such as statement (ii) above may generally be obtained more abstractly via general functoriality statements on $p$-curvature, but such a point of view requires familiarity with Grothendieck's abstract theory of connections; see \cite{os6}.
We conclude with some observations in the case of curves.
\begin{lem} In the case that $X$ is a curve, the $p$-curvature of a connection $\nabla$ is identically $0$ if and only if $\psi_{\nabla} (\theta)=0$ for any non-zero derivation $\theta$. In addition, $\bar{T}_{(p)}=f_{\theta^p} \bar{T}$ for some function $f_{\theta^p}$, satisfying $f_{\theta^p}\theta = \theta^p$. \end{lem}
\begin{proof} These statements follow trivially from the fact that the sheaf of derivations is invertible, and the $p$-linearity of the $p$-curvature map with respect to derivations. \end{proof}
Finally, we record in this situation the general $p$-curvature formulas in characteristics $3$, $5$, and $7$, for later use.
Characteristic 3: \begin{equation}\label{exp-pcurve-3} \psi_\nabla (\theta)= \bar{T}^3+(\theta \bar{T})\bar{T} + 2 \bar{T} (\theta \bar{T}) +(\theta^2 \bar{T})-f_{\theta ^3} \bar{T} \end{equation}
Characteristic 5: \begin{multline}\label{exp-pcurve-5} \psi_\nabla (\theta)= \bar{T} ^5 + 4 \bar{T} ^3(\theta^1 \bar{T}) + 3 \bar{T} ^2(\theta^1 \bar{T})\bar{T} + \bar{T} ^2(\theta^2 \bar{T}) + 2 \bar{T}(\theta^1 \bar{T})\bar{T} ^2 \\ + 3 \bar{T}(\theta^1 \bar{T}) ^2 + 3 \bar{T}(\theta^2 \bar{T})\bar{T} + 4 \bar{T}(\theta^3 \bar{T}) + (\theta^1 \bar{T})\bar{T} ^3 \\ + 4 (\theta^1 \bar{T})\bar{T}(\theta^1 \bar{T}) + 3 (\theta^1 \bar{T}) ^2\bar{T} + (\theta^1 \bar{T})(\theta^2 \bar{T}) + (\theta^2 \bar{T})\bar{T} ^2 \\ + 4 (\theta^2 \bar{T})(\theta^1 \bar{T}) + (\theta^3 \bar{T})\bar{T} + (\theta^4 \bar{T}) - f_{\theta^5} \bar{T} \end{multline}
Characteristic 7: \begin{multline}\label{exp-pcurve-7} \psi_\nabla (\theta)= \bar{T} ^7 + 6 \bar{T} ^5(\theta^1 \bar{T}) + 5 \bar{T} ^4(\theta^1 \bar{T})\bar{T} + \bar{T} ^4(\theta^2 \bar{T}) + 4 \bar{T} ^3(\theta^1 \bar{T})\bar{T} ^2 \\ + 3 \bar{T} ^3(\theta^1 \bar{T}) ^2 + 3 \bar{T} ^3(\theta^2 \bar{T})\bar{T} + 6 \bar{T} ^3(\theta^3 \bar{T}) + 3 \bar{T} ^2(\theta^1 \bar{T})\bar{T} ^3 \\ + 4 \bar{T} ^2(\theta^1 \bar{T})\bar{T}(\theta^1 \bar{T}) + \bar{T} ^2(\theta^1 \bar{T}) ^2\bar{T} + 3 \bar{T} ^2(\theta^1 \bar{T})(\theta^2 \bar{T}) + 6 \bar{T} ^2(\theta^2 \bar{T})\bar{T} ^2 \\ + \bar{T} ^2(\theta^2 \bar{T})(\theta^1 \bar{T}) + 3 \bar{T} ^2(\theta^3 \bar{T})\bar{T} + \bar{T} ^2(\theta^4 \bar{T}) + 2 \bar{T}(\theta^1 \bar{T})\bar{T} ^4 \\ + 5 \bar{T}(\theta^1 \bar{T})\bar{T} ^2(\theta^1 \bar{T}) + 3 \bar{T}(\theta^1 \bar{T})\bar{T}(\theta^1 \bar{T})\bar{T} + 2 \bar{T}(\theta^1 \bar{T})\bar{T}(\theta^2 \bar{T}) \\ + \bar{T}(\theta^1 \bar{T}) ^2\bar{T} ^2 + 6 \bar{T}(\theta^1 \bar{T}) ^3 + 6 \bar{T}(\theta^1 \bar{T})(\theta^2 \bar{T})\bar{T} + 5 \bar{T}(\theta^1 \bar{T})(\theta^3 \bar{T}) \\ + 3 \bar{T}(\theta^2 \bar{T})\bar{T} ^3 + 4 \bar{T}(\theta^2 \bar{T})\bar{T}(\theta^1 \bar{T}) + \bar{T}(\theta^2 \bar{T})(\theta^1 \bar{T})\bar{T} + 3 \bar{T}(\theta^2 \bar{T}) ^2 \\ + 4 \bar{T}(\theta^3 \bar{T})\bar{T} ^2 + 3 \bar{T}(\theta^3 \bar{T})(\theta^1 \bar{T}) + 5 \bar{T}(\theta^4 \bar{T})\bar{T} + 6 \bar{T}(\theta^5 \bar{T}) + (\theta^1 \bar{T})\bar{T} ^5 \\ + 6 (\theta^1 \bar{T})\bar{T} ^3(\theta^1 \bar{T}) + 5 (\theta^1 \bar{T})\bar{T} ^2(\theta^1 \bar{T})\bar{T} + (\theta^1 \bar{T})\bar{T} ^2(\theta^2 \bar{T}) \\ + 4 (\theta^1 \bar{T})\bar{T}(\theta^1 \bar{T})\bar{T} ^2 + 3 (\theta^1 \bar{T})\bar{T}(\theta^1 \bar{T}) ^2 + 3 (\theta^1 \bar{T})\bar{T}(\theta^2 \bar{T})\bar{T} \\ + 6 (\theta^1 \bar{T})\bar{T}(\theta^3 \bar{T}) + 3 (\theta^1 \bar{T}) ^2\bar{T} ^3 + 4 (\theta^1 \bar{T}) ^2\bar{T}(\theta^1 \bar{T}) + (\theta^1 \bar{T}) ^3\bar{T} \\ + 3 (\theta^1 \bar{T}) ^2(\theta^2 \bar{T}) + 6 (\theta^1 \bar{T})(\theta^2 \bar{T})\bar{T} ^2 + (\theta^1 \bar{T})(\theta^2 \bar{T})(\theta^1 \bar{T}) \\ + 3 (\theta^1 \bar{T})(\theta^3 \bar{T})\bar{T} + (\theta^1 \bar{T})(\theta^4 \bar{T}) + (\theta^2 \bar{T})\bar{T} ^4 + 6 (\theta^2 \bar{T})\bar{T} ^2(\theta^1 \bar{T}) \\ + 5 (\theta^2 \bar{T})\bar{T}(\theta^1 \bar{T})\bar{T} + (\theta^2 \bar{T})\bar{T}(\theta^2 \bar{T}) + 4 (\theta^2 \bar{T})(\theta^1 \bar{T})\bar{T} ^2 \\ + 3 (\theta^2 \bar{T})(\theta^1 \bar{T}) ^2 + 3 (\theta^2 \bar{T}) ^2\bar{T} + 6 (\theta^2 \bar{T})(\theta^3 \bar{T}) + (\theta^3 \bar{T})\bar{T} ^3 \\ + 6 (\theta^3 \bar{T})\bar{T}(\theta^1 \bar{T}) + 5 (\theta^3 \bar{T})(\theta^1 \bar{T})\bar{T} + (\theta^3 \bar{T})(\theta^2 \bar{T}) + (\theta^4 \bar{T})\bar{T} ^2 \\ + 6 (\theta^4 \bar{T})(\theta^1 \bar{T}) + (\theta^5 \bar{T})\bar{T} + (\theta^6 \bar{T}) - f_{\theta^7} \bar{T} \end{multline}
\section{On $f_{\theta^p}$ and $p$-rank in Genus $2$}\label{s-exp-ftheta}
In this section, we give an explicit formula for $f_{\theta^p}$ on a genus $2$ curve $C$, and note that we can use these ideas to derive explicit formulas for the $p$-rank of the Jacobian of $C$. Throughout, we work under the hypotheses and notation of Situations \ref{exp-genus} and \ref{exp-single-open}, with $X=C$.
We first note that (irrespective of the genus of $C$), although $f_{\theta ^p}$ will be $0$ only if $\theta(f)=1$ for some $f$ on $U$, we will always have:
\begin{lem}\label{exp-thetaftheta} $\theta f_{\theta ^p}=0$. \end{lem}
\begin{proof} Given any $f$, $\theta ^p f = f_{\theta^p}\theta(f)$, so $\theta ^{p+1} f = \theta (f_{\theta ^p} \theta(f)) = \theta (f_{\theta ^p}) \theta(f) + f_{\theta ^p} \theta^2(f) = \theta (f_{\theta ^p}) \theta(f) + \theta ^{p+1}(f)$. Since this is true for all $f$, we must have $\theta(f_{\theta^p})=0$, as desired. \end{proof}
We now specify some normalizations and notational conventions special to genus $2$ which we will follow through the end of our explicit calculations in Section \ref{s-exp-det}.
\begin{sit}\label{exp-presented} $C$ is a smooth, proper genus $2$ curve over an algebraically closed field $k$. It is presented explicitly on an affine open set $U_2$ by $$y^2 = g(x) = x^5 +a_1 x^4 + a_2 x^3 + a_3 x^2 + a_4 x + a_5,$$ with the complement of $U_2$ being a single, smooth, Weierstrass point $w$ at infinity. We also have the form $\omega_2 = y^{-1} dx$ trivializing $\Omega^1_C$ on $U_2$, and the derivation $\theta$ on $U_2$ given by $\theta f = y \frac{df}{dx}$. Equivalently, $\hat{\theta}(\omega_2)=1$, where $\hat{\theta}$ denotes the map $\Omega^1_C \rightarrow {\mathscr O}_C$ such that $\theta= \hat{\theta} \circ d$. \end{sit}
For this section only, we set $U=U_2$ and $\omega=\omega_2$. We set $g_k(x) = \theta ^{k-1} x$; we see by induction that this is a polynomial in $x$ for $k$ odd. Noting that $\theta(p(x))=yp'(x)$ for $p(x)$ any polynomial in $x$, and $\theta(y)=\frac{1}{2}g'(x)$, we have that for $k$ odd, $g_k(x) = \theta^2 (g_{k-2}(x))= \theta (y g'_{k-2}(x))$, and we get the recursive formula: \begin{equation}\label{exp-gk} g_k(x)=g''_{k-2}(x)g(x) + \frac{1}{2}g'_{k-2}(x)g'(x) \end{equation} for $k$ odd.
But $f_{\theta^p} = \hat{\theta}^p(y^{-1} dx)$ by definition, which is just $y^{-1} \theta ^p (x)$, so we also find \begin{equation}\label{exp-ftheta} f_{\theta^p} = y^{-1} \theta g_p (x) = g'_p(x) \end{equation} In particular, $f_{\theta^p}$ is a polynomial in $x$, and can therefore only have nonzero terms mod $p$ in degrees which are multiples of $p$. However, we see by induction that the degree of $g_p(x)$ is always less than $2p$, so the only nonzero terms of $f_{\theta^p}$ are the constant term and the $p$th power term (from which it follows that the only nonzero terms of $g_p(x)$ are the constant, linear, $p$th power, and $(p+1)$st power terms).
For later use, we note the formulas for characteristics $3$, $5$, and $7$ obtained by combining equations \ref{exp-gk} and \ref{exp-ftheta}:
Characteristic 3: \begin{equation}\label{exp-ftheta-3}f_{\theta^3} = x^3+a_3 \end{equation}
Characteristic 5: \begin{equation}\label{exp-ftheta-5}f_{\theta^5} = 2 a_1 x^5 + a_3^2 + 2 a_2 a_4 + 2 a_1 a_5 \end{equation}
Characteristic 7: \begin{equation}\label{exp-ftheta-7}f_{\theta^7} = (3 a_1 ^2 + 3 a_2) x ^7 + a_3 ^3 + 6 a_2 a_3 a_4 + 3 a_1 a_4 ^2 + 3 a2 ^2 a_5 + 6 a_1 a_3 a_5 + 6 a_4 a_5 \end{equation}
As a final note, we can use this to derive explicit formulas for the $p$-rank of the Jacobian of $C$ in terms of the coefficients of $g(x)$.
\begin{prop}\label{exp-prank} If we denote by $h_1, h_2, h_3, h_4$ the polynomials in the coefficients of $g(x)$ giving the constant, linear, $p$th power, and $(p+1)$st power terms of $g_p(x)$, then the $p$-rank of the Jacobian of $C$ is: \begin{itemize} \item[$2$ if:] $h_1 h_4- h_2 h_3 \neq 0$; \item[$1$ if:] $h_1 h_4- h_2 h_3 =0$ but either $h_3^p - h_2 h_4 ^{p-1} \neq 0$ or $h_1^p h_4 - h_2 ^{p+1} \neq 0$; \item[$0$ if:] $h_1 h_4- h_2 h_3 = h_3^p - h_2 h_4 ^{p-1} = h_1^p h_4 - h_2^{p+1} = 0$. \end{itemize} \end{prop}
\begin{proof} The $p$-torsion of the Jacobian is simply the number of (transport equivalence classes of, but endomorphisms of a line bundle are only scalars, and hold connections fixed) connections with $p$-curvature $0$ on the trivial bundle. We note that the space of connections on ${\mathscr O}_C$ can be written explicitly as $f \mapsto df + f (c_1 +c_2 x) \omega$, meaning the connection matrix on $U$ with respect to $\theta$ is given simply by the function $\bar{T}= c_1+c_2 x$. Using the $p$-curvature formula for rank $1$ given by Corollary \ref{exp-pcurve-cors} (i), we find \begin{gather} \psi_{\nabla}(\theta_0) = (c_1 + c_2 x)^p + \theta_0^{p-1}(c_1+c_2 x) - f_{\theta_0 ^p} (c_1 + c_2 x) \\ = c_1 ^p + c_2 ^p x^p + c_2 g_p(x) - g'_p(x) (c_1 +c_2 x) \\ = (c_1^p + c_2 h_1 - c_1 h_2) + (c_2 ^p + c_2 h_3 - c_1 h_4) x^p. \end{gather} Setting the $p$-curvature to zero, we obtain: $$0 = (c_1^p + c_2 h_1 - c_1 h_2) + (c_2 ^p + c_2 h_3 - c_1 h_4) x^p.$$
We first consider this equation in the case that $h_4 \neq 0$. In this case, we find that we can write $c_1= \frac{c_2^p+c_2 h_3}{h_4}$, and substituting in, we find we get $p^2$ solutions if $h_1 h_4^p - h_2 h_3 h_4^{p-1} \neq 0$, and otherwise, $p$ solutions if $h_3 ^p - h_2 h_4^{p-1} \neq 0$, and finally $1$ solution if both vanish. On the other hand, in the case that $h_4=0$, we see that $c_2$ becomes independent of $c_1$, we get $p^2$ solutions if and only if both $h_2$ and $h_3$ are nonzero; $p$ solutions if either but not both are nonzero, and $1$ solution if they are both $0$. One can then check that both these casese are expressed by the asserted polynomial conditions in the $h_i$. \end{proof}
For $p=3$, we have $$g_p(x) = 1 x^4 - a_1 x^3 + a_3 x - a_4,$$ so $h_4$ is always nonzero, and we find that the $p$-rank of $C$ is $2$ when $a_4 - a_1 a_3 \neq 0$, is $1$ when $a_4 - a_1 a_3 = 0$ but $a_1 ^3 - a_3 \neq 0$, and is $0$ when $a_4 - a_1 a_3 = a_1 ^3 - a_3 = 0$.
For $p=5$, we have $$g_p(x) = 2 a_1 x_6 + (4 a_1 ^2 + 3 a_2) x^5 + (a_3 ^2 + 2 a_2 a_4 + 2 a_1 a_5) x +(3 a_3 a_4 + 3 a_2 a_5),$$ so the $p$-rank of $C$ is $2$ when $$a_1 (a_3 a_4 + a_2 a_5) - (4 a_1 ^2 + 3 a_2)(a_3^2 + 2 a_2 a_4 + 2 a_1 a_5) \neq 0.$$ The $p$-rank is $1$ when $$a_1 (a_3 a_4 + a_2 a_5) - (4 a_1 ^2 + 3 a_2)(a_3^2 + 2 a_2 a_4 + 2 a_1 a_5) = 0$$ but either $$4 a_1 ^{10} + 3 a_2 ^5 - (a_3 ^2 + 2 a_2 a_4 + 2 a_1 a_5) a_1^4 \neq 0$$ or $$(3 a_3 ^5 a_4 ^5 + 3 a_2^5 a_5^5) 2 a_1 - (a_3^2 + 2 a_2 a_4 + 2 a_1 a_5)^6 \neq 0.$$ Lastly, the $p$-rank is $0$ when \begin{align*}0= a_1 (a_3 a_4 + a_2 a_5) - (4 a_1 ^2 + 3 a_2)(a_3^2 + 2 a_2 a_4 + 2 a_1 a_5) \notag \\ = 4 a_1 ^{10} + 3 a_2 ^5 - (a_3 ^2 + 2 a_2 a_4 + 2 a_1 a_5) a_1^4 \\ = (3 a_3 ^5 a_4 ^5 + 3 a_2^5 a_5^5) 2 a_1 - (a_3^2 + 2 a_2 a_4 + 2 a_1 a_5)^6.\notag \end{align*}
While explicit computations of the $p$-rank of the Jacobian of a curve are not hard in general, it is perhaps worth mentioning that this method, aside from providing a complete and explicit solution for genus $2$ curves, does so in a sufficiently elementary way that it can be presented as a calculation of the $p$-torsion of $\operatorname{Pic}(C)$ without knowing any properties of the Jacobian, or even that it exists.
\section{The Space of Connections}\label{s-exp-conns}
In this section we carry out the first portion of the necessary computations for the explicit portion of Theorem \ref{exp-main}, by calculating the space of transport-equivalence classes of connections on a particular vector bundle ${\mathscr E}$. We suppose:
\begin{sit}\label{exp-specific-e} With the notation and hypotheses of Situation \ref{exp-presented}, we further declare that ${\mathscr E}$ is the bundle determined by Propositions \ref{exp-unstable} and \ref{exp-unstable-unique} for the choice ${\mathscr L} = {\mathscr O}_C([w])$. \end{sit}
In this situation, if $U_1, U_2$ are a trivializing cover for ${\mathscr L}$, with transition function $\varphi _{12}$, then ${\mathscr L}^{-1}, {\mathscr L}^{\otimes 2} = \Omega^1_C,$ and ${\mathscr E}$ are all trivialized by this cover as well, and ${\mathscr E}$ can be represented with a transition matrix of the form $$E=\begin{bmatrix} \varphi _{12} & \varphi _{\mathscr E} \\ 0 & \varphi _{12} ^{-1} \end{bmatrix}$$ for some $\varphi _{\mathscr E}$ regular on $U_1 \cap U_2$.
We see immediately that we can choose $\varphi_{12}$ and $U_1$ so that $\varphi_{12}$ is regular on $U_1$ with a simple zero at $w$, and non-vanishing elsewhere: we simply set $\varphi_{12}$ to be any function with a simple zero at $w$, and take $U_1$ to be the complement of any other zeroes and poles. For compatibility of trivializations of ${\mathscr L}$ and $\Omega_C^1$, we must then set $\omega_1 = \varphi_{12}^{-2} \omega _2$. Beyond these properties, our specific choice of $\varphi_{12}$ will be completely irrelevant, but we note that it is possible to choose $\varphi_{12}$ to vary algebraically (in fact, to be in some sense invariant) as our $a_i$ and the corresponding curves vary: we can simply set $\varphi_{12} = \frac{x^2}{y}$.
\begin{prop}\label{exp-extension} The unique non-trivial isomorphism class for ${\mathscr E}$ may be realized by setting $\varphi _{\mathscr E} = \varphi_{12} ^{-2}$. \end{prop}
\begin{proof}We claim that there cannot be a splitting map from ${\mathscr E}$ back to ${\mathscr L}$. Indeed, one checks explicitly that such a splitting would require the existence of a rational function on $C$ having a pole of order exactly $3$ at $w$, and regular elsewhere, which is not possible. \end{proof}
We now note that since $\varphi_{12}$ has a simple zero at $w$, and $\omega_1$ is invertible at $w$, if we further restrict $U_1$ we can guarantee that $\frac{d\varphi_{12}}{\omega_1}$ is likewise everywhere invertible on $U_1$. Having done so, $\varphi_{\mathscr E} = \varphi_{12}^{-2}$, so $d\varphi_{\mathscr E}= -2\varphi_{12}^{-3} d\varphi_{12}$, and $\frac{d\varphi_{\mathscr E}}{\omega_1}$ is regular and nonvanishing on $U_1$ except for a pole of order $3$ at $w$.
Now, we can trivialize ${\mathscr E} \otimes \Omega^1_C$ on the $U_i$ with transition matrix $\varphi _{12} ^2 E$. We can then represent a connection $\nabla: {\mathscr E} \rightarrow {\mathscr E} \otimes \Omega^1_C$ by $2 \times 2$ connection matrices $\bar{T}_1$ and $\bar{T}_2$ of functions regular on $U_1$ and $U_2$ respectively. These act by sending $s_i \mapsto \bar{T}_i s_i + \frac{ds_i}{\omega_i}$ on $U_i$, where the $s_i$ are given as vectors under the trivialization, so one checks that $\bar{T}_1$ and $\bar{T}_2$ must be related by: $$\bar{T}_1 = \varphi _{12}^2 E \bar{T}_2 E^{-1}+E \frac{dE^{-1}}{\omega _1}$$
We now explicitly compute $\bar{T}_2$ in terms of $\bar{T}_1$ in preparation for computing the space of connections. If $\bar{T}_2= \begin{bmatrix} f_{11} & f_{12} \\ f_{21} & f_{22} \end{bmatrix}$, then:
\begin{equation}\label{exp-conneq} \bar{T}_1 = \begin{bmatrix} \varphi _{12}^2f_{11}+f_T & \varphi_{12}^4 f_{12} + \varphi_{12}^3 \varphi_{\mathscr E} (f_{22} - f_{11}) - \varphi _{12}\varphi_{\mathscr E} f_T - \varphi_{12}\frac{d\varphi_{\mathscr E}}{\omega_1} \\ f_{21} & \varphi _{12}^2f_{22}-f_T \end{bmatrix} \end{equation} where $f_T=\varphi_{12} \varphi_{\mathscr E} f_{21} - \varphi_{12}^{-1} \frac{d \varphi_{12}}{\omega_1}$
Note that this implies $f_{21}$ is everywhere regular and hence constant.
We now show:
\begin{prop}\label{exp-conns} The space of connections on ${\mathscr E}$ is given by $f_{21}=C_1$, $f_{11}=c_1+c_2 x$, $f_{22}=c_3+c_4 x$, and $f_{12}=c_5+c_6 x + c_7 x^2 +c_8 y +C_2 x^3$, where the $c_i$ are arbitrary constants subject to the single linear relation $c_8 = C_2 (c_2-c_4)$, and $C_1 \text{ and } C_2$ are predetermined nonzero constants satisfying $C_1 C_2 =\frac{-1}{2}$. \end{prop}
\begin{proof}We begin by looking at the lower right entry of the matrix for $\bar{T}_1$ in Equation \ref{exp-conneq}, and note the $\varphi_{12}^{-1} \frac{d\varphi_{12}}{\omega_1}$ has a simple pole at $w$ which must be cancelled by one of the other terms. We also note that since $\varphi_{\mathscr E}=\varphi_{12}^{-2}$, and $f_{21}$ must be constant, the term $\varphi_{12} \varphi_{\mathscr E} f_{21} = \varphi _{12}^{-1}f_{21}$ is regular on $U_1$ away from $w$, where it can have at most a simple pole. Thus the $\varphi_{12}^2 f_{22}$ term must likewise be regular on $U_1$ away from $w$, with at most a simple pole at $w$. Since $f_{22}$ must be regular on $U_2$ by hypothesis, we conclude it is regular on $C$ except possibly for a pole of order at most $3$ at $w$. But such a pole of order $3$ isn't possible, so $f_{22} \in \Gamma({\mathscr O}_C(2[w]))$. This means that the simple poles of the other two terms must cancel, and $f_{21}$ is determined as a (nonzero) constant $C_1$: explicitly, $C_1=\frac{d\varphi _{12}}{\omega_1}(w)$. Precisely the same argument applies to the upper right entry, placing $f_{11} \in \Gamma({\mathscr O}_C(2[w]))$, so it only remains to analyze the upper right entry of the matrix.
We immediately observe that on $U_1$, each term (excluding the $\varphi_{12}^4 f_{12}$ term) is regular except possibly for a pole of order at most $2$ at $w$, which of course implies that $\varphi_{12}^4 f_{12}$ is also, and we can conclude that $f_{12}$ is regular on $C$ except for a pole of order at most $6$ at $w$. Then we have $f_{21}=C_1 \in k^*$, $f_{11}=c_1+c_2 x$, $f_{22}=c_3+c_4 x$, and $f_{12}=c_5+c_6 x + c_7 x^2 +c_8 y +C_2 x^3$, and we claim that $C_2$ is also determined: the only other terms which can have double poles are $-\varphi_{12}^2\varphi_{\mathscr E} ^2 f_{21}+\varphi_{\mathscr E} \frac{d\varphi_{12}}{\omega_1} - \varphi_{12}\frac{d \varphi_{\mathscr E}}{\omega_1} =-\varphi_{12}^{-2} f_{21}+3\varphi_{12}^{-2}\frac{d\varphi_{12}}{\omega_1}$ which are now predetermined, so $C_2$ is also determined, explicitly as $-2 (\varphi_{12}^{-6} x ^{-3})(w)C_1$. Lastly, we note that there is a linear relation on $c_2, c_4, \text{ and } c_8$ to insure that the simple poles cancel.
To conclude the proof, we use formal local analysis at $w$ to obtain the desired statements on this linear relation and $C_1$ and $C_2$. Explicitly, our linear relation is given as $c_8=((\varphi_{12}^{-3}y^{-1}x)(w))(c_2-c_4)+ ((y^{-1}\varphi_{12}^{-5})(w))((\varphi_{12}^{-1}(C_1-3\frac{d\varphi_{12}}{\omega_1}) -C_2 x^3 \varphi_{12}^5)(w))$. Now, choose a local coordinate $z$ at $w$; we will denote by $\ell_z(f)$ and $\ell_z'(f)$ the leading and second terms of the Laurent series expansion for $f$ in terms of $z$. From our relation between $x$ and $y$, we have $\ell_z(x)^5=\ell_z(y)^2$ and $2 \ell_z(y) \ell_z'(y) = 5 \ell_z(x)^4 \ell_z'(x)$. Simply considering leading terms, we find that since $\omega_1 = \varphi_{12}^{-2}y^{-1}dx$, we have $C_1 = \frac{-\ell_z(\varphi_{12})^{3}\ell_z(y)}{2\ell_z(x)}$, and $C_2 = \frac{\ell_z(x)}{\ell_z(\varphi_{12})^3 \ell_z(y)}$. Thus, we have that $C_1 C_2 = \frac{-1}{2}$, and also that $C_2$ is the coefficient of $(c_2-c_4)$ in our linear relation. It only remains to show that the constant term in that relation is in fact $0$. We may write it as $((y^{-1}\varphi_{12}^{-5})(w))((\varphi_{12}^{-1}(C_1-3\frac{d\varphi_{12}}{\omega_1} -C_2 x^3 \varphi_{12}^6))(w)$, so it suffices to show that $C_1- 3\frac{d\varphi_{12}}{\omega_1} -C_2 x^3 \varphi _{12}^6$, which we know must vanish at $w$, in fact vanishes to order at least $2$ at $w$. For this, it is convenient to specialize to $z = \varphi_{12}$, whereupon our earlier relation simplifies to $\ell_z(y)=C_2^{-1} \ell_z(x)$. We also compute $\ell_z(x)^3 = C_2^{-2}$, from which it follows that we can write $\ell_z'(y) = \frac{5}{2} C_2^{-1} \ell_z'(x)$. We can now write everything in terms of $\ell_z(x), \ell_z'(x)$ and $C_2$, and check directly that we get the desired cancellation to order $2$. \end{proof}
We also consider the endomorphisms of ${\mathscr E}$, so that we can normalize our connections via transport to simplify calculations. An endomorphism is given by matrices $S_i$ regular on $U_i$, satisfying the relationship $S_1 = E S_2 E^{-1}$. If we write $S_2 = \begin{bmatrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{bmatrix}$ we find that \begin{equation}\label{exp-end-eq} S_1 = \begin{bmatrix} g_{11} + \varphi _{12} ^{-1} \varphi_{{\mathscr E}} g_{21} & \varphi_{12} ^2 g_{12} + \varphi _{12} \varphi _{{\mathscr E}} g_{22} - \varphi _{12} \varphi _{{\mathscr E}} g_{11} - \varphi_{{\mathscr E}}^2 g_{21} \\ \varphi _{12} ^{-2} g_{21} & g_{22} - \varphi_{12} ^{-1} \varphi _{{\mathscr E}} g_{21} \end{bmatrix} \end{equation}
We can now compute directly:
\begin{prop}\label{exp-ends} The space of endomorphisms of ${\mathscr E}$ is given by $g_{21}=0$, $g_{11}=g_{22} \in k$, and $g_{12} = g_{12}^0 + g_{12}^1 x \in \Gamma({\mathscr O}_C(2[w]))$. Every connection on ${\mathscr E}$ has a unique transport-equivalent connection with $f_{11}=0$. \end{prop}
\begin{proof}Noting that the lower left entry for $S_1$ in equation \ref{exp-end-eq} is $\varphi _{12} ^{-2} g_{21}$, we see that $g_{21}$ has to be regular everywhere on $C$, and vanishes to order at least $2$ at $w$; hence, it is $0$. We then see that the upper left and lower right entries are just $g_{11}$ and $g_{22}$ respectively, meaning that these are both everywhere regular and hence constant. Finally, the upper right term is then $\varphi _{12} ^2 g_{12} + \varphi _{12} ^{-1} (g_{22}-g_{11})$; the second term will have a simple pole at $w$ if and only if $g_{22} \neq g_{11}$, and since $g_{12}$ cannot have a triple pole at $w$, we conclude that $g_{22} = g_{11}$, and finally that $g_{12} \in \Gamma({\mathscr O}_C(2[w]))$, giving the description of the endomorphisms of ${\mathscr E}$.
Such an endomorphism is invertible if and only if $g_{11} \neq 0$. Since transport along an automorphism is invariant under scaling the automorphism, we can then set $g_{11}=g_{22}=1$ without loss of generality. Now, since $S_2$ is upper triangular, with constant diagonal coefficients, $S_2^{-1} \frac{dS_2}{\omega_2}$ has only its upper right coefficient non-zero. Moreover, conjugating $\bar{T}_2$ by $S_2$ will simply substract $f_{21} g_{12}$ from the upper left coefficient of $\bar{T}_2$. Since we know $f_{21}$ is a determined nonzero constant, and $g_{12}$ and $f_{11}$ can both be arbitrary in $\Gamma({\mathscr O}_C(2[w]))$, this means that each connection has a unique transport class with $f_{11}=0$, as desired. \end{proof}
Thus, from now on we will normalize our calculations as follows: set $f_{11} = 0$ by transport; set $f_{22} =0$ since we want the determinant connection (obtained by taking the trace) to be $0$; and set $f_{21} =1$. We accomplish the last by scaling $\varphi_{12}$ appropriately: we saw that $f_{21}= \frac{d\varphi_{12}}{\omega_1}(w)$, and recalling that $\omega_1 = \varphi_{12}^{-2} y^{-1} dx$, it suffices to scale $\varphi_{12}$ by a cube root of $f_{21}$. We also note that this does not pose any problems for our prior choice of $\varphi_{12} = \frac{x^2}{y}$; one can check that for this choice, we have $f_{21}$ invariant as $\frac{-1}{2}$, and the scaling factor for $\varphi_{12}$ is independent of the $a_i$. Lastly, since $c_8=0$ now that $c_2=c_4=0$, we conclude that we are reduced to considering the case:
\begin{sit}\label{exp-normalized} Our connection matrix $T_2$ on $U_2$ is of the form $T_2 = \begin{bmatrix}0 & f_{12} \\ 1 & 0 \end{bmatrix}$, with $f_{12} = c_5+c_6 x + c_7 x^2 - \frac{1}{2}x^3$. \end{sit}
Finally, for later use we formally generalize our results.
\begin{prop}\label{exp-gen} Propositions \ref{exp-conns} and \ref{exp-ends} hold in the following more more general settings: \begin{ilist} \item After base change to an arbitrary $k$-algebra $A$, if we replace the $k$-valued constants by $A$-valued constants; \item When we allow our defining polynomial $g(x)$ to degenerate to produce nodes away from $w$, if we replace $\Omega^1_C$ by the dualizing sheaf $\omega_C$ in the definition of connections; \item When we consider families of curves obtained from maps $k[a_1, \dots a_5] \rightarrow A$ taking values in the open subset $U_{\operatorname{nod}} \subset {\mathbb A}^5$ corresponding to at worst nodal curves. \end{ilist} \end{prop}
\begin{proof} For (i), if we denote by $f$ the map $\operatorname{Spec} A \rightarrow \operatorname{Spec} k$, and $\pi$ the structure map $C \rightarrow \operatorname{Spec} k$, this coefficient replacement corresponds to the natural map $f^* \pi_* {\mathscr F} \rightarrow \pi_{f*} f_{\pi}^* {\mathscr F}$ for the sheaves $\mathcal{E}nd({\mathscr E}) \otimes \Omega^1_C$ and $\mathcal{E}nd({\mathscr E})$. But since the base is a point, every base change is flat, and it immediately follows \cite[Prop. III.9.3]{ha1} that this natural map is always an isomorphism, giving the desired statement.
For (ii), we need only note that our arguments go through unmodified, since $\omega_C$ is still isomorphic to ${\mathscr O}(2[w])$, and the same standard Riemann-Roch argument as in the smooth case still shows it that there can be no function in $\Gamma({\mathscr O}(3[w]))\smallsetminus \Gamma({\mathscr O}(2[w]))$.
Finally, for (iii) we make use of the fact that, as remarked immediately above, we can choose $\varphi_{12}$ to be a specific function varying algebraically in the whole family. Once again, if we denote by ${\mathscr F}$ the sheaf $\mathcal{E}nd({\mathscr E}) \otimes \omega_C$ or $\mathcal{E}nd({\mathscr E})$ as appropriate, but this time in the universal setting over $U_{\operatorname{nod}}$, the theory of cohomology and base change gives that since $h^0(C, {\mathscr F})$ is constant on fibers, $\pi_* {\mathscr F}$ is locally free of the same rank, and pushforward commutes with base change. Now, if we let our constants describing sections of ${\mathscr F}$ lie in $k[a_1, \dots, a_5]$, we clearly obtain a subsheaf of $\pi_* {\mathscr F}$ of the correct rank; further, the inclusion map is an isomorphism when restricted to every fiber, so it must in fact be an isomorphism, which yields the desired result for arbitrary $A$ via base change. \end{proof}
It follows formally that the closed subschemes we describe explicitly corresponding to vanishing $p$-curvature in Section \ref{s-exp-pcurve} and nilpotent $p$-curvature in Section \ref{s-exp-det} are also functorial descriptions which hold for nodal curves.
\section{Calculations of $p$-curvature}\label{s-exp-pcurve}
Continuing with the situation and notations of the previous section, and in particular that of Situation \ref{exp-presented}, we conclude with the $p$-curvature calculations to complete the proof of Theorem \ref{exp-main} for $p \leq 7$, except for the statement on the general curve in characteristic $7$, which depends on the results of the subsequent section.
We write: $$\psi_\nabla (\theta) = \begin{bmatrix} h_{11} & h_{12} \\ h_{21} & h_{22} \end{bmatrix}$$
The first case we handle is $p=3$. Equation \ref{exp-ftheta-3} gave us $f_{\theta ^3} = x^3 +a_3$. We show:
\begin{prop}In characteristic $3$, ${\mathscr E}$ has a unique transport equivalence class of connections with $p$-curvature zero and trivial determinant. \end{prop}
\begin{proof}With all of our normalizations from Situation \ref{exp-normalized}, the $p$-curvature matrix given by Equation \ref{exp-pcurve-3} becomes rather tame:
$$\psi_\nabla (\theta) = \begin{bmatrix} \theta f_{12} & f_{12}^2 + \theta ^2 f_{12} - f_{\theta^3} f_{12} \\ f_{12}- f_{\theta^3} & -\theta f_{12} \end{bmatrix}$$
Even better, we note that we have $$h_{12} = \theta (h_{11}) + f_{12} h_{21},$$ so $h_{12}$ vanishes if $h_{11}$ and $h_{21}$ do. Similarly, recalling that by Lemma \ref{exp-thetaftheta}, $\theta f_{\theta ^3} = 0$, we see that $h_{11} = \theta (h_{21})$, and $h_{22} = - h_{11}$. Hence, to check if the $p$-curvature vanishes, it suffices to check that $h_{21}$ vanishes.
But this is a triviality, as we simply get that $h_{21}=0$ if and only if $f_{12} = a_3 + x^3$. Recalling that after normalization $f_{12}$ was given by $c_5 + c_6 x + c_7 x^2 - \frac{1}{2} x^3$, we get the unique solution $c_5 = a_3, c_6=c_7=0$. \end{proof}
We now handle the case $p=5$. We had from Equation \ref{exp-ftheta-5} that $f_{\theta^5} = 2 a_1 x^5 + a_3^2 + 2 a_2 a_4 + 2 a_1 a_5$.
\begin{prop}In characteristic $5$, the number of transport equivalence classes of connections with $p$-curvature zero and trivial determinant is given as the number of roots of the quintic polynomial: \begin{multline*} (3 a_1 a_2 ^2 + 3 a_2 a_3 + a_5) + (a_1 ^2 a_2 + a_2 ^2 + 3 a_1 a_3 + 4 a_4) c_5 \\ + (3 a_1 ^3 + 4 a_1 a_2 + a_3) c_5 ^2 + (3 a_1 ^2 + 4 a_2) c_5 ^3 + a_1 c_5 ^4 + 4 c_5 ^5 \end{multline*} \end{prop}
\begin{proof} With our normalizations as above, in terms of $f_{12}$ and $f_{\theta^5}$, the $p$-curvature matrix obtained from Equation \ref{exp-pcurve-5} is
$$\psi_\nabla (\theta) = \begin{bmatrix} 4 f_{12} \theta (f_{12}) + \theta^3(f_{12})& f_{12}^3 + 4 (\theta(f_{12}))^2 + 2 f_{12} \theta^2(f_{12}) + \theta^4 (f_{12}) + 4 f_{12} f_{\theta^5} \\ f_{12}^2 + 3 \theta^2(f_{12}) + 4 f_{\theta ^5} & f_{12} \theta(f_{12}) + 4 \theta^3 (f_{12}) \end{bmatrix}$$
Conveniently, we note that as before it actually suffices to check that $h_{21}$ is $0$, since we see that $h_{22}= 3 \theta _0 (h_{21})$, that $h_{11} = -h_{22}$, and that $h_{12} = f_{12} h_{21} + 2 \theta ^2 (h_{21})$.
Substituting in for $f_{12}$ and $f_{\theta^5}$, we get that the remaining (lower left) term is given by
\begin{multline*} (4 a_3 ^2 + 3 a_2 a_4 + 3 a_1 a_5 + c_3 ^2 + a_5 c_5) + (a_5 + 3 a_3 c_4 + 2 c_3 c_4 + 4 a_4 c_5) x \\ + (2 a_2 c_4 + c_4 ^2 + 2 a_3 c_5 + 2 c_3 c_5) x ^2 \\ + (4 a_3 + 4 c_3 + a_1 c_4 + 2 c_4 c_5) x ^3 + (3 a_2 + 4 c_4 + 3 a_1 c_5 + c_5^2) x ^4 \end{multline*}
Setting the $x^3$ and $x^4$ terms to $0$ allows us to solve for $c_4$ and $c_3$. Substituting in, we find that the $x^2$ term drops out, while the coefficient of $x$ is:
\begin{multline*} (3 a_1 a_2 ^2 + 3 a_2 a_3 + a_5) + (a_1 ^2 a_2 + a_2 ^2 + 3 a_1 a_3 + 4 a_4) c_5 \\ + (3 a_1 ^3 + 4 a_1 a_2 + a_3) c_5 ^2 + (3 a_1 ^2 + 4 a_2) c_5 ^3 + a_1 c_5 ^4 + 4 c_5 ^5 \end{multline*}
The constant coefficient is $c_5+ 3 a_1$ times the $x$ coefficient, so we get that the connections with $p$-curvature $0$ correspond precisely to the roots of the above polynomial, as asserted. \end{proof}
Lastly, we take a look at the case $p=7$. Equation \ref{exp-ftheta-7} gave us: $$f_{\theta^7} = a_3 ^3 + 6 a_2 a_3 a_4 + 3 a_1 a_4 ^2 + 3 a_2 ^2 a_5 + 6 a_1 a_3 a_5 + 6 a_4 a_5 + (3 a_1 ^2 + 3 a_2) x ^7.$$
We will show:
\begin{prop}In characteristic 7, the number of transport equivalence classes of connections on ${\mathscr E}$ with $p$-curvature $0$ and trivial determinant is given as the intersection of four plane curves in ${\mathbb A}^2$. For a general curve, it is positive. The locus $F_{2,7}$ of transport equivalence classes of connections on ${\mathscr E}$ with $p$-curvature $0$ and trivial determinant considered over the ${\mathbb A}^5$ with which we parametrize genus $2$ curves is cut out by $4$ hypersurfaces in ${\mathbb A}^5 \times {\mathbb A}^2$. \end{prop}
\begin{proof} Here, even with our normalizations the $p$-curvature matrix obtained from Equation \ref{exp-pcurve-7} is rather messy, but we find its coefficients are given by:
$$h_{11} = 2 f_{12} ^2 \theta (f_{12}) + \theta (f_{12}) \theta ^2 (f_{12}) - 3 f_{12} \theta ^3 (f_{12}) + \theta ^5 (f_{12})$$
$$h_{21} = -f_{\theta^7} + f_{12} ^3 + 3 (\theta (f_{12})) ^2 - f_{12} \theta ^2 (f_{12}) - 2 \theta ^4 (f_{12})$$
$$h_{12} = -f_{\theta^7} f_{12} + f_{12} ^4 + f_{12} ^2 \theta ^2 (f_{12}) + (\theta ^2 (f_{12})) ^2 - 2 \theta (f_{12}) \theta ^3 (f_{12}) + 2 f_{12} \theta ^4 (f_{12}) + \theta ^6 (f_{12})$$
$$h_{22} = -2 f_{12} ^2 \theta (f_{12}) - \theta (f_{12}) \theta ^2 (f_{12}) + 3 f_{12} \theta ^3 (f_{12}) - \theta ^5 f_{12}$$
Once again, it is enough to consider a single one of these coefficients, as we see that $h_{11} = 3 \theta(h_{21})$, that $h_{12} = f_{12} h_{21} + 3 \theta^2(h_{21})$, and that $h_{22} = -h_{11}$.
Looking then at the formula for $h_{21}$, substituting in for $f_{12}$ and $f_{\theta^7}$ gives a polynomial of degree $6$ in $x$. The $x^6$ term lets us solve for $c_3$:
$$c_3 = 5 a_1 a_2 + a_3 + 4 a_1 c_4 + 4 a_1^2 c_5 + c_4 c_5 + 2 a_1 c_5^2 + 5 c_5^3$$
The $x^5$ term is then
\begin{multline}h_{7,1}=2 a_1 ^2 a_2 + a_1 a_3 + 5 a_4 + 4 a_1 ^2 c_4 + 5 a_2 c_4 + 6 c_4 ^2 \\ + 3 a_1 ^3 c_5 + 6 a_1 a_2 c_5 + 3 a_3 c_5 + 5 a_1 c_4 c_5 + 3 a_1 c_5 ^3 + 6 c_5 ^4. \end{multline} while the $x^4$ term is $-c_5$ times the $x^5$ term, and the $x^3$ term is $-(c_5^2+a_1 c_5 + 3a_2 + c_4)$ times the $x^5$ term. Taking the $x^2$ term minus $-(5 c_5^3+ 5 a_1 c_5^2 + 2 c_4 c_5 + 5 a_1 a_2 + 4 a_3 + 2 a_1 c_4)$ times the $x^5$ term leaves:
\begin{multline}h_{7,2}= 3 a_1 ^3 a_2 ^2 + 6 a_1 ^2 a_2 a_3 + 4 a_1 a_3 ^2 + 4 a_3 a_4 + 2 a_2 a_5 + 3 a_1 ^3 a_2 c_4 + 4 a_1 ^2 a_3 c_4 \\ + 2 a_1 a_4 c_4 + 4 a_5 c_4 + a_1 ^3 c_4 ^2 + a_3 c_4^2 + 3 a_1 c_4^3 + a_1 ^4 a_2 c_5 \\ + 5 a_1 ^3 a_3 c_5 + a_1 ^2 a_4 c_5 + 3 a_1 a_5 c_5 + 6 a_1 ^4 c_4 c_5 + a_1 ^2 a_2 c_4 c_5 \\ + a_1 a_3 c_4 c_5 + 3 a_4 c_4 c_5 + a_1 ^2 c_4 ^2 c_5 + 5 a_2 c_4 ^2 c_5 + c_4 ^3 c_5 \\ + 4 a_1 ^3 a_2 c_5 ^2 + 6 a_1 ^2 a_3 c_5 ^2 + a_1 a_4 c_5 ^2 + 3 a_5 c_5 ^2 + 3 a_1 ^3 c_4 c_5 ^2 \\ + a_1 a_2 c_4 c_5 ^2 + a_3 c_4 c_5 ^2 + 3 a_1 ^2 a_2 c_5 ^3 + a_1 a_3 c_5 ^3 + 4 a_1 ^2 c_4 c_5 ^3 + 6 c_4 ^2 c_5 ^3. \end{multline}
Similarly, taking the $x$ term minus $-(5 c_4 c_5^2+5 a_1 c_4 c_5 +6 a_4 + 2 c_4^2)$ times the $x^5$ term leaves:
\begin{multline}h_{7,3} = 5 a_1 ^2 a_2 a_4 + 6 a_1 a_3 a_4 + a_1 a_2 a_5 + 5 a_1 ^2 a_2 ^2 c_4 + 4 a_1 a_2 a_3 c_4 + 3 a_1 ^2 a_4 c_4 \\ + 2 a_1 a_5 c_4 + 5 a_1 ^2 a_2 c_4 ^2 + a_1 a_3 c_4 ^2 + 3 a_4 c_4 ^2 + 3 a_2 c_4 ^3 \\ + 5 c_4 ^4 + 4 a_1 ^3 a_4 c_5 + 6 a_3 a_4 c_5 + 5 a_1 ^2 a_5 c_5 + 3 a_2 a_5 c_5 \\ + 4 a_1 ^3 a_2 c_4 c_5 + 4 a_1 ^2 a_3 c_4 c_5 + a_1 a_4 c_4 c_5 + 6 a_5 c_4 c_5 \\ + 3 a_1 ^3 c_4 ^2 c_5 + 4 a_1 a_2 c_4 ^2 c_5 + 4 a_3 c_4 ^2 c_5 + a_1 c_4 ^3 c_5 \\ + 2 a_1 ^2 a_4 c_5 ^2 + 6 a_1 a_5 c_5 ^2 + 2 a_1 ^2 a_2 c_4 c_5 ^2 + 2 a_1 a_3 c_4 c_5 ^2 \\ + a_4 c_4 c_5 ^2 + 5 a_1 ^2 c_4 ^2 c_5 ^2 + 4 a_2 c_4 ^2 c_5 ^2 + 5 c_4 ^3 c_5 ^2 \\ + 5 a_1 a_4 c_5 ^3 + a_5 c_5 ^3 + 5 a_1 a_2 c_4 c_5 ^3 + 5 a_3 c_4 c_5 ^3 + 2 a_1 c_4 ^2 c_5 ^3. \end{multline}
Lastly, taking the constant term minus \begin{multline*}-(6 c_5^5+5 c_4 c_5^3 + 3 a_1 ^2 c_5^3 +2 a_1^3 c_5^2 + 5 a_1 a_2 c_5^2 + 2 a_3 c_5^2 + 6 a_1 c_4 c_5^2 \\ + 5 a_1^2 a_2 c_5 + 2 a_1 a_3 c_5 + 2 a_4 c_5 + a_1^2 c_4 c_5 \\ + 2 a_2 c_4 c_5 + 2 c_4^2 c_5 + 6 a_1 a_4 + 4 a_5 + 4 a_1 a_2 c_4 + 3 a_3 c_4 + 3 a_1 c_4^2) \end{multline*} times the $x^5$ term leaves:
\begin{multline}h_{7,4}=6 a_1 ^3 a_2 ^3 + 5 a_1 ^2 a_2 ^2 a_3 + a_1 a_2 a_3 ^2 + 5 a_1 ^3 a_2 a_4 + 6 a_1 ^2 a_3 a_4 + a_2 a_3 a_4 \\ + 6 a_1 a_4 ^2 + a_1 ^2 a_2 a_5 + 4 a_2 ^2 a_5 + 5 a_1 a_3 a_5 + 4 a_4 a_5\\ + 4 a_1 ^2 a_2 a_3 c_4 + a_1 a_3 ^2 c_4 + 3 a_1 ^3 a_4 c_4 + 2 a_1 a_2 a_4 c_4 \\ + 3 a_3 a_4 c_4 + 2 a_1 ^2 a_5 c_4 + 3 a_1 ^3 a_2 c_4 ^2 + 6 a_1 a_2 ^2 c_4 ^2 \\ + a_2 a_3 c_4 ^2 + 6 a_5 c_4 ^2 + 6 a_1 ^3 c_4 ^3 + 4 a_1 a_2 c_4 ^3 + 4 a_3 c_4 ^3 \\ + 4 a_1 c_4 ^4 + 2 a_1 ^4 a_2 ^2 c_5 + 3 a_1 ^3 a_2 a_3 c_5 + 4 a_1 ^4 a_4 c_5 \\ + 2 a_1 ^2 a_2 a_4 c_5 + 2 a_1 a_3 a_4 c_5 + 5 a_1 ^3 a_5 c_5 + a_3 a_5 c_5 + 3 a_1 ^4 a_2 c_4 c_5 \\ + 2 a_1 ^2 a_2 ^2 c_4 c_5 + 2 a_1 ^3 a_3 c_4 c_5 + 2 a_1 a_2 a_3 c_4 c_5 + 5 a_3 ^2 c_4 c_5 \\ + 6 a_1 ^2 a_4 c_4 c_5 + 6 a_2 a_4 c_4 c_5 + 5 a_1 a_5 c_4 c_5 + 2 a_1 ^4 c_4 ^2 c_5 + 2 a_1 ^2 a_2 c_4 ^2 c_5 \\ + 3 a_2 ^2 c_4 ^2 c_5 + 6 a_1 a_3 c_4 ^2 c_5 + 4 a_4 c_4 ^2 c_5 + a_2 c_4 ^3 c_5 + 5 c_4 ^4 c_5 \\ + a_1 ^3 a_2 ^2 c_5 ^2 + 5 a_1 ^2 a_2 a_3 c_5 ^2 + 2 a_1 ^3 a_4 c_5 ^2 + 2 a_1 a_2 a_4 c_5 ^2 \\ + 2 a_3 a_4 c_5 ^2 + 6 a_1 ^2 a_5 c_5 ^2 + 5 a_1 ^3 a_2 c_4 c_5 ^2 + 2 a_1 a_2 ^2 c_4 c_5 ^2 \\ + a_1 ^2 a_3 c_4 c_5 ^2 + 2 a_2 a_3 c_4 c_5 ^2 + 4 a_1 a_4 c_4 c_5 ^2 + 5 a_5 c_4 c_5 ^2 \\ + a_1 ^3 c_4 ^2 c_5 ^2 + 6 a_1 a_2 c_4 ^2 c_5 ^2 + 2 a_1 c_4 ^3 c_5 ^2 + 6 a_1 ^2 a_2 ^2 c_5 ^3 \\ + 2 a_1 a_2 a_3 c_5 ^3 + 5 a_1 ^2 a_4 c_5 ^3 + a_1 a_5 c_5 ^3 + 2 a_1 ^2 a_2 c_4 c_5 ^3 \\ + 6 a_1 a_3 c_4 c_5 ^3 + 5 a_4 c_4 c_5 ^3 + 6 a_1 ^2 c_4 ^2 c_5 ^3 + 4 a_2 c_4 ^2 c_5 ^3 + 3 c_4 ^3 c_5 ^3. \end{multline}
These four polynomials are then the defining equations in characteristic $7$, describing the locus as an intersection of $4$ affine plane curves, as desired. By direct computation in Macaulay 2, the coordinate ring of the affine algebraic set cut out by these equations has dimension 5. Since we know that it can only have dimension $0$ over any given choice for the $a_i$, this implies that it has a non-empty fiber for a general choice of $a_i$, yielding the positivity assertion. \end{proof}
Finally, we compute an example which will allow us to deduce the characteristic $7$ case of Theorem \ref{exp-main} in the next section.
\begin{lem}\label{exp-ch7-ex} For the curve given by $a_1=a_2=a_3=0, a_4=1$, and $a_5=3$, there are $14$ solutions to our equations, all reduced. Further, the local rings of $F_{2,7}$ at each of these points are all isomorphic. \end{lem}
\begin{proof} First, we set $a_1=a_2=a_3=0$, $a_4=1$ and $a_5=3$, and our defining equations become considerably simpler:
$$h_{7,1}= 5 + 6 c_4^2 + 6 c_5^4$$ $$h_{7,2}= 5 c_4 + 3 c_4 c_5 + c_4^3 c_5 + 2 c_5^2 + 6 c_4 ^2 c_5^3$$ $$h_{7,3}= 3 c_4^2 + 5 c_4^4 + 4 c_4 c_5 + c_4 c_5^2 + 5 c_4^3 c_5^2 + 3 c_5^3$$ $$h_{7,4}= 5+ 4 c_4^2 + 4 c_4 ^2 c_5 + 5 c_4 ^4 c_5 + c_4 c_5^2 + 5 c_4 c_5^3 + 3 c_4^3 c_5^3$$
If we use $h_{7,1}$ to substitute for $c_4^2$ in $h_{7,2}$, we get:
$$c_4(5+c_5+ 6 c_5^5) + c_5^2(2+ 2 c_5 + c_5^5)$$
We check that we cannot have $5 + c_5 + 6 c_5^5=0$, so we can localize away from $5+c_5 + 6 c_5^5$, setting $c_4 = \frac{c_5^2(2+ 2 c_5 + c_5^5)}{5 + c_5 + 6 c_5^5}$. Making this substitution and taking numerators, the $h_{7,i}$ give four polynomials in $c_5$. However, they are multiples of the polynomial given by $h_{7,1}$, which is:
$$6 + c_5 + 5 c_5 ^2 + 6 c_5 ^4 + 2 c_5 ^5 + 6 c_5 ^6 + 6 c_5 ^9 + 3 c_5 ^{10} + 5 c_5 ^{14}$$
This then gives the $14$ reduced solutions, and the fact that the local rings of $F_{2,7}$ at each of these points are isomorphic follows from the fact that this degree $14$ polynomial is irreducible over ${\mathbb F}_7$, since the $14$ points are then Galois conjugate in $F_{2,7}$, which is defined over ${\mathbb F}_7$. \end{proof}
\section{On The Determinant of the $p$-Curvature Map}\label{s-exp-det}
In this section we explicitly calculate the highest degree terms of $\det \psi$, the determinant of the $p$-curvature map in the case of a genus 2 curve and the specific unstable vector bundle of Situation \ref{exp-specific-e}. We use the calculation to prove that $\det \psi$ is finite flat, of degree $p^3$, and therefore conclude that in families of curves, the kernel of $\det \psi$ is finite flat. This has immediate implications for the connections on ${\mathscr E}$ of vanishing $p$-curvature as well, in particular allowing us to finish the proof of the characteristic-specific portion of Theorem \ref{exp-main}. The results here are a special case of Mochizuki's work (see \cite[Thm. II.2.3, p. 129]{mo1}), obtained by an argument which is essentially the same, but discovered independently, and significantly simpler in the special case handled here.
We wish to compute in our specific situation the morphism $\det \psi^0$ (Proposition \ref{exp-pcurve-formal} (iv)), which is to say, the morphism obtained from $\psi^0$ (Proposition \ref{exp-pcurve-formal} (iii)) by taking the determinant. In fact, we take $\psi^0$ to be the induced map on transport-equivalence classes of connections. We remark that in the situation of rank $2$ vector bundles with trivial determinant, and after restricting to connections with trivial determinant, because the image of $\psi^0$ is contained among the traceless endomorphisms, the vanishing of the determinant of the $p$-curvature is then equivalent to nilpotence of the endomorphisms given by the $p$-curvature map. Such connections are frequently called {\bf nilpotent} in the literature (see, for instance, \cite{ka1} or \cite{mo3}).
We now take our curve $C$ of genus $2$ from before, with ${\mathscr E}$ the particular unstable bundle of rank $2$ we had been studying, as in Situations \ref{exp-presented}, \ref{exp-specific-e}, and \ref{exp-normalized}. We also take the particular $\theta$ from before, with $\hat{\theta}(\omega_2)=1$. Since $\omega_2$ has a double zero at $w$, we see that $\theta$ has a double pole there, so that our explicit identification of $\Omega^1_C$ is as ${\mathscr O}(2 [w])$. We know that our space of connections with trivial determinant on ${\mathscr E}$ is (modulo transport) $3$-dimensional, and of course $h^0(C^{(p)}, (\Omega^1_{C^{(p)}})^{\otimes 2})= \deg (\Omega^1_{C^{(p)}})^{\otimes 2}+1-g= 4g-4+1-g = 3g-3 = 3$, so we have a map from ${\mathbb A}^3$ to ${\mathbb A}^3$. We choose coordinates on the first space to be given by the $(c_5, c_6, c_7)$ determining $f_{12}$, while the function we will get will be of the form $f_1(c_5, c_6, c_7)+f_2(c_5, c_6, c_7)x^p+f_3(c_5, c_6, c_7)x^{2p}$, and we obtain coordinates on the image space as the monomials $(1,x^p, x^{2p})$.
We will use our earlier calculations to recover, in a completely explicit and elementary fashion, the genus $2$ case of Mochizuki's result:
\begin{thm}\label{exp-nilp} On the unstable vector bundle ${\mathscr E}$ described by Situation \ref{exp-specific-e} for a smooth proper genus $2$ curve $C$ as in Situation \ref{exp-presented}, the map $\det \psi^0$ is a finite flat morphism from ${\mathbb A}^3$ to ${\mathbb A}^3$, of degree $p^3$. Further, $\det \psi^0$ remains finite flat when considered as a family of maps over the open subset $U_{\operatorname{ns}} \subset {\mathbb A}^5$ corresponding to nonsingular curves. Lastly, the induced map from $\ker \det \psi^0$ to $U_{\operatorname{ns}}$ is finite flat. \end{thm}
\begin{proof} It suffices to prove the asserted finite flatness for the family of maps ${\mathbb A}^3 \times U_{\operatorname{ns}} \rightarrow {\mathbb A}^3 \times U_{\operatorname{ns}}$ over $U_{\operatorname{ns}}$, since the statements on individual curves and on the kernel of $\det \psi$ both follow from restriction to fibers. This is turn will follow from the claim that the leading term of $f_i$ is $-c_{i+4}^p$, with all other terms of strictly lesser total degree in the $c_i$. We prove this by direct calculation.
If $T = \begin{bmatrix} 0 & f_{12} \\ 1 & 0 \end{bmatrix}$ is the connection matrix for $\nabla$, we claim that the leading term will come from the $T^p$ term in the $p$-curvature formula. Now, $T^2 = \begin{bmatrix} f_{12} & 0\\ 0 & f_{12} \end{bmatrix}$, so we find $$T^p = \begin{bmatrix} 0 & (f_{12})^{\frac{p+1}{2}} \\ (f_{12})^{\frac{p-1}{2}} & 0 \end{bmatrix}$$
Next, $f_{12}$ is linear in the $c_i$, as are $\theta ^i f_{12}$ for all $i$. Considering the $p$-curvature formula coefficients as polynomials in $\theta ^i f_{12}$, we will show that the degree of the remaining terms are all less than or equal to $\frac{p-1}{2}$, with the degree of the terms in the lower left strictly less. This will imply that the leading term of the determinant is given by $$-(f_{12})^p = -(c_5+c_6 x + c_7 x^2 - \frac{1}{2}x^3)^p = -c_5^p - c_6^p x^p - c_7^p x^{2p}+ \frac{1}{2^p}x^{3p}$$ giving the desired formula for the leading terms of the constant, $x^p$, and $x^{2p}$ terms.
We observe that since $\theta^i T = \begin{bmatrix} 0 & \theta ^i f_{12} \\ 0 & 0 \end{bmatrix}$ for all $i>0$, $(\theta^i T)( \theta ^j T)=0$ for any $i,j>0$. We use this and the fact that $T^2$ is diagonal to write any term in the $p$-curvature as one of the following: \begin{nlist} \item $T^{2 i_0} (\theta^{i_1}T) T \dots (\theta^{i_k}T)$ \item $T^{2 i_0} T (\theta^{i_1}T) T \dots (\theta^{i_k}T)$ \item $T^{2 i_0} (\theta^{i_1}T) T \dots (\theta^{i_k}T) T$ \item $T^{2 i_0} T (\theta^{i_1}T) T \dots (\theta^{i_k}T) T$ \end{nlist} where $2 i_0 + \sum _{j>0} (i_j+2) = p+1, p, p, p-1$ respectively.
We observe that these correspond to non-zero upper right, lower right, upper left, and lower left coefficients, respectively (in particular, at most one is non-zero). We know that the first term is a scalar matrix of degree $i_0$ in $f_{12}$. We see that $T (\theta^{i_j} T) = \begin{bmatrix} 0 & 0 \\ 0 & \theta ^i f_{12} \end{bmatrix}$, so a product of $k-1$ such terms has total degree $k-1$ in the $\theta ^i f_{12}$. Lastly, multiplying on the left by $(\theta ^{i_1}T)$ raises the degree by one and moves the nonzero coefficient back to the upper right. Thus, in the first case, we get total degree $i_0+k$. But we see that this is actually the same in the other cases, as multiplying on the left or right by $T$ just moves the nonzero coefficient, without changing it. Finally, with $k>0$, we have $i_0 +k < \frac{1}{2}( 2 i_0 + \sum _{j>0} (i_j+2)) $, which is $\frac{1}{2}$ times $p+1, p, p$ or $p-1$ depending on the case. But this is precisely what we wanted to show, since it forces the degree to be less than or equal to $\frac{p-1}{2}$ in the first three cases, and strictly less in the fourth.
Lastly, $-f_{\theta^p} T$ is linear in the $c_i$ in the upper right term, and constant in the rest, so doesn't cause any problems for $p\geq 3$. \end{proof}
We can immediately conclude:
\begin{cor}\label{exp-finite} The subscheme of $U_{\operatorname{ns}} \times {\mathbb A}^3$ giving connections with $p$-curvature $0$ is finite over $U_{\operatorname{ns}}$. \end{cor}
We are now ready to put together previous results to finish the proof of our main theorem in the case of characteristic $7$:
\begin{proof}[Proof of Theorem \ref{exp-main}, $p=7$ case] We simply apply our finiteness result to our explicit example from Lemma \ref{exp-ch7-ex}. We calculated that $F_{2,7}$ has dimension $5$, so by properness the local ring of at least one point in our example has dimension $5$, hence they all do. By the reducedness of our example, all its points are unramified over the base, and by finiteness, we conclude that on an open subset of the base containing our chosen point, $F_{2,7}$ is finite and unramified, and everywhere $5$-dimensional. Then, by the regularity of the base, we find that over this open set, $F_{2,7}$ must be regular, hence flat, hence \'etale, so we conclude the desired statement for a general curve in characteristic $7$. \end{proof}
\section{Connections and Nodes}\label{s-def-background}
This section and the next draw heavily on the results and ideas of Sections 2 and 3 of \cite{os10}.
In this section, we discuss connections on nodal curves, and classify them in terms of gluings of connections on the normalization. For the sake of simplicity and generality, we follow Mochizuki's argument for the gluing, with the only difference being that because we are not working with projective bundles, we must rigidify our situation by specifying glued line sub-bundles ${\mathscr L}$, as in Proposition \ref{def-glue-prop}.
Let $C$ be a proper nodal curve, and ${\mathscr E}$ a vector bundle on $C$. We begin by fixing some terminology:
\begin{defn}A {\bf logarithmic connection} on ${\mathscr E}$ is a $k$-linear map $\nabla: {\mathscr E} \rightarrow {\mathscr E} \otimes \omega _C$, where $\omega_C$ is the dualizing sheaf on $C$, and $\nabla$ satisfies the Liebnitz rule induced by the canonical map $\Omega^1_C \rightarrow \omega_C$. Given a reduced divisor $D$ supported on the smooth locus of $C$, a {\bf $D$-logarithmic connection} on ${\mathscr E}$ is a $k$-linear map $\nabla: {\mathscr E} \rightarrow {\mathscr E} \otimes \omega _C(D)$ satisfying the Liebnitz rule. \end{defn}
We note that with the exception of the Cartier isomorphism, all the properties of connections and $p$-curvature which we have used still hold if one replaces $\Omega^1_C$ by $\omega_C$ (and in particular, the sheaf of derivations by $\omega_C^\vee$) throughout. We summarize as follows.
\begin{prop}\label{def-background} All statements on induced connections for operations of vector bundles, and the basic properties of the $p$-curvature map of Proposition \ref{exp-pcurve-formal}, hold in the case of logarithmic connections on nodal curves, with $\omega_C$ in place of $\Omega_C^1$. One still has a canonical connection on a Frobenius pullback with vanishing $p$-curvature whose kernel recovers the original sheaf. \end{prop}
Although it is true that taking the kernel of the canonical connection of a Frobenius pullback still recovers the original sheaf on $C^{(p)}$ when $C$ is singular, the Cartier isomorphism fails because given a logarithmic connection with vanishing $p$-curvature on $C$, the Frobenius pullback of the kernel will not in general map surjectively onto the original sheaf at the singularities of $C$.
\begin{notn}Let $\tilde{C}$ be the normalization of $C$, and $\tilde{{\mathscr E}}$ the pullback of ${\mathscr E}$ to $\tilde{C}$. Given a logarithmic connection $\nabla$ on ${\mathscr E}$, we get a $D_C$-logarithmic connection $\tilde{\nabla}$ on $\tilde{{\mathscr E}}$, where $D_C$ is the divisor of points lying above the nodes of $C$. \end{notn}
We want a complete description of connections on $\tilde{{\mathscr E}}$ arising this way, and a correspondence between these and connections on ${\mathscr E}$. We claim:
\begin{prop}\label{def-glue-basic} Logarithmic connections $\nabla$ on ${\mathscr E}$ are equivalent to connections on $\tilde{{\mathscr E}}$ having simple poles at the points $P_1, Q_1, \dots, P_\delta, Q_\delta$ lying above the nodes of $C$, and such that under
the gluing maps $G_i : \tilde{{\mathscr E}}|_{P_i} \rightarrow \tilde{{\mathscr E}}|_{Q_i}$ giving ${\mathscr E}$, for each $i$ we have $\operatorname{Res}_{P_i}(\nabla) = - G_i^{-1} \circ \operatorname{Res} _{Q_i}(\nabla) \circ G_i$. The properties of having trivial determinant and vanishing $p$-curvature are preserved under this correspondence. \end{prop}
\begin{proof} The main assertion follows easily from \cite[Thm. 5.2.3]{co2} together with the remark \cite[p. 226]{co2} for nodal curves, which together state that sections of $\omega_C$ correspond to sections of $\Omega^1_{\tilde{C}}(D_C)$ with residues at the pair of points above any given node adding to zero.
Since vanishing $p$-curvature can be verified on open sets, and the normalization map is an isomorphism away from the nodes, it is clear that logarithmic connections with vanishing $p$-curvature on $C$ will correspond to logarithmic connections with vanishing $p$-curvature on $\tilde{C}$. The same argument also works for trivial determinant. \end{proof}
We can in particular conclude:
\begin{cor}\label{def-conn-line} Let ${\mathscr L}$ be a line bundle on $C$. Then ${\mathscr L}$
can have a logarithmic connection $\nabla$ with vanishing $p$-curvature only if $p | \deg \tilde{{\mathscr L}}$. \end{cor}
\begin{proof}Applying the previous proposition, if we pull back to $\tilde{\nabla}$ on $\tilde{{\mathscr L}}$ we find that the residues of $\tilde{\nabla}$ come in additive inverse pairs mod $p$. We obviously have
$p | {\mathscr F}^* (\tilde{{\mathscr L}}^{\tilde{\nabla}})$, and then by \cite[Cor. 2.11]{os10} we have that the determinant of the inclusion map ${\mathscr F}^* (\tilde{{\mathscr L}}^{\tilde{\nabla}}) \hookrightarrow \tilde{{\mathscr L}}$ has total order equal to the sum of the residues mod $p$, which is zero, so we conclude that $\deg \tilde{{\mathscr L}}$ must also vanish mod $p$, as asserted. \end{proof}
We now restrict to the situation:
\begin{sit} Suppose that ${\mathscr E}$ has rank 2 and trivial determinant, and we have fixed an exact sequence $$0 \rightarrow {\mathscr L} \rightarrow {\mathscr E} \rightarrow {\mathscr L}^{-1} \rightarrow 0.$$ The same statements then hold for $\tilde{{\mathscr E}}$. \end{sit}
We introduce some terminology in this situation:
\begin{defn} Given a logarithmic connection $\nabla$ on ${\mathscr E}$ (resp., a $D$-logarithmic connection $\nabla$ on $\tilde{{\mathscr E}}$), the {\bf Kodaira-Spencer map} associated to $\nabla$ and a sub-line-bundle ${\mathscr L}$ (resp., $\tilde{{\mathscr L}}$) is the natural map ${\mathscr L} \rightarrow {\mathscr L}^{-1} \otimes \omega_C$ (respectively, $\tilde{{\mathscr L}} \rightarrow \tilde{{\mathscr L}}^{-1} \otimes \Omega^1_{\tilde{C}}(D)$) obtained by composing the inclusion map, $\nabla$, and the quotient map. One verifies directly that this is a linear map.
In the case that ${\mathscr E}$ (resp., $\tilde{{\mathscr E}}$) is unstable, we will refer to the Kodaira-Spencer map of $\nabla$ to mean the map associated to $\nabla$ and its destabilizing line bundle. \end{defn}
Recall that by Lemma \ref{exp-destab-unique}, the destabilizing line bundle is unique, so the last part of the definition is justified. Note that with this terminology, Joshi and Xia's proof of \ref{exp-unstable} boils down to the statement that the Frobenius-pullback of a Frobenius-unstable bundle necessarily has a connection such that the Kodaira-Spencer map of the destabilizing line bundle is an isomorphism. It should perhaps therefore not be surprising that we will consider connections for which the Kodaira-Spencer is an isomorphism. We note:
\begin{lem}\label{def-arith-genus} Suppose that the arithetmic genus of $C$ (resp., the genus of $\tilde{C}$ plus $\frac{\deg D}{2}$) is greater than or equal to $3/2$; that is to say, we are in the ``hyperbolic'' case. Then if the Kodaira-Spencer map associated to $(\nabla, {\mathscr L})$ is an isomorphism for any $\nabla$, then ${\mathscr L}$ is a destabilizing line bundle for ${\mathscr E}$ (resp., $\tilde{{\mathscr E}}$), and is thus uniquely determined even independent of $\nabla$. \end{lem}
\begin{proof} The Kodaira-Spencer isomorphism gives ${\mathscr L}^{\otimes 2} \cong \omega_C$ (resp., ${\mathscr L}^{\otimes 2} \cong \Omega^1_{\tilde{C}}(D)$), which from the hypotheses has positive degree. \end{proof}
One can approach the issue of gluing connections from two perspectives: either fixing the glued bundle ${\mathscr E}$ on $C$, and exploring which connections on $\tilde{{\mathscr E}}$ will glue to yield connections on ${\mathscr E}$, or allowing the gluing of ${\mathscr E}$ itself to vary as well. The author had originally intended to use the first approach, since we ultimately wish to classify the connections on a particular unstable bundle on a nodal curve. However, the second approach, pursued by Mochizuki \cite[p. 118]{mo3}, offers a more transparent view of the more general setting, and ultimately yields a cleaner argument even for our specific application. As such, we now fix $\tilde{{\mathscr E}}$ on $\tilde{C}$, but do not assume a fixed gluing ${\mathscr E}$ on $C$. That is to say:
\begin{sit}\label{def-glue} Fix $\tilde{{\mathscr E}}$ of rank 2 and trivial determinant, together with an exact sequence $$0 \rightarrow \tilde{{\mathscr L}} \rightarrow \tilde{{\mathscr E}} \rightarrow \tilde{{\mathscr L}}^{-1} \rightarrow 0.$$ \end{sit}
The main statement on gluing is:
\begin{prop}\label{def-glue-prop} In Situation \ref{def-glue}, let $\tilde{\nabla}$ be a $D_C$-logarithmic connection on $\tilde{C}$ with trivial determinant and vanishing $p$-curvature, such that the Kodaira-Spencer map associated to $\tilde{{\mathscr L}}$ is an isomorphism. Further suppose that the $e_1,e_2$ of \cite[Cor. 2.10]{os10} match one another (up to permutation) for pairs of points lying above given nodes of $C$. Then if one fixes a gluing ${\mathscr L}$ of $\tilde{{\mathscr L}}$ with ${\mathscr L}^{\otimes 2} \cong \omega_C$, there is a unique gluing of $(\tilde{{\mathscr E}}, \tilde{\nabla})$ to a pair $({\mathscr E}, \nabla)$ on $C$, such that one obtains a sequence $$0 \rightarrow {\mathscr L} \rightarrow {\mathscr E} \rightarrow {\mathscr L}^{-1} \rightarrow 0,$$ and the resulting $({\mathscr E}, \nabla)$ will also have Kodaira-Spencer map an isomorphism. If $C$ has arithmetic genus at least $2$, transport equivalence is preserved under this correspondence. \end{prop}
\begin{proof} We first claim that the condition that the Kodaira-Spencer map for $\tilde{{\mathscr L}}$ be an isomorphism implies that for any $P \in \{P_i, Q_i\}$, $\tilde{{\mathscr L}}|_P$ is not contained in an eigenspace of $\operatorname{Res}_P \tilde{\nabla}$, and that the eigenvalues are both non-zero. But due to the triviality of the determinant, the sum of the eigenvalues is zero, so because the residue matrices are diagonalizable (see \cite[Cor. 2.11]{os10}), the latter assertion is actually a consequence of the former. Now, considering the definition of the Kodaira-Spencer map $\tilde{{\mathscr L}} \rightarrow \tilde{{\mathscr L}}^{-1}\otimes \Omega^1_{\tilde{C}}(D_C)$, if we restrict to $P$ we get a map which is clearly equal to zero if and only if
$\nabla(\tilde{{\mathscr L}}) |_P \subset \tilde{{\mathscr L}}\otimes \Omega^1_{\tilde{C}}|_P$,
which is the case precisely when $\tilde{{\mathscr L}}|_P$ is contained in an eigenspace of $\operatorname{Res}_P \tilde{\nabla}$, as desired.
Given this, for each pair $P_i, Q_i$, Proposition \ref{def-glue-basic} and our hypothesis on the matching eigenvalues of the residue matrices at $P_i, Q_i$ imply that in order to glue the connection, it is necessary and sufficient to map eigenspaces of opposing sign to each other. To glue $\tilde{{\mathscr L}}$, we also map its image at $P_i$ to its image at $Q_i$. We thus see that the two eigenspaces of $\operatorname{Res}_{P_i} \tilde{\nabla}$ and $\operatorname{Res}_{Q_i} \tilde{\nabla}$ and the images of $\tilde{{\mathscr L}}$ form a set of three one-dimensional subspaces which must be matched under $G_i$, and it is easy to see that this determines $G_i$ up to scaling. But finally, scaling of $G_i$ is equivalent to scaling the induced gluing map on $\tilde{{\mathscr L}}$, which is precisely what determines the isomorphism class of the glued ${\mathscr L}$; thus, ${\mathscr L}$ may be specified arbitrarily, and given a choice of ${\mathscr L}$, the $G_i$ and hence the pair $({\mathscr E}, \nabla)$ are uniquely determined, as desired. Lastly, we observe that since the Kodaira-Spencer map gives an isomorphism ${\mathscr L} \otimes ({\mathscr E}/{\mathscr L})^{-1} \cong \omega_C$, the hypothesis that ${\mathscr L}^{\otimes 2} \cong \omega_C$ is equivalent to the condition that the glued ${\mathscr E}$ have trivial determinant.
Considering transport, it is trivial that if two connections on ${\mathscr E}$ are transport-equivalent, then their pullbacks to $\tilde{{\mathscr E}}$ are, and for the converse, the uniqueness of the gluing makes it clear that if two connections $\tilde{\nabla}$ and $\tilde{\nabla'}$ on $\tilde{{\mathscr E}}$ are transport-equivalent under an automorphism $\varphi$ of $\tilde{{\mathscr E}}$, then $\varphi$ naturally gives an isomorphism of the two gluings ${\mathscr E}$ and ${\mathscr E}'$ which takes $\nabla$ to $\nabla'$. Finally, the hypothesis that the arithmetic genus of $C$ is at least $2$ implies that ${\mathscr L}$ and $\tilde{{\mathscr L}}$ are uniquely determined as the destabilizing sub-bundles of ${\mathscr E}$ and $\tilde{{\mathscr E}}$, so there is no concern that they might change under transport. \end{proof}
Putting together the previous propositions, we finally conclude:
\begin{cor}\label{def-glue-main} Let $\tilde{{\mathscr E}}$ be a vector bundle on $\tilde{C}$ of rank 2, with the arithmetic genus of $C$ being at least 2, and suppose there exists an exact sequence $$0 \rightarrow \tilde{{\mathscr L}} \rightarrow \tilde{{\mathscr E}} \rightarrow \tilde{{\mathscr L}}^{-1} \rightarrow 0.$$ Fix a gluing of $\tilde{{\mathscr L}}$ to a line bundle ${\mathscr L}$ on $C$ satisfying ${\mathscr L}^{2} \cong \omega_C$. Then there exists a bijective equivalence between transport-equivalence classes of $D_C$-logarithmic connections $\tilde{\nabla}$ on $\tilde{{\mathscr E}}$ with trivial determinant and vanishing $p$-curvature, the eigenvalues of the residues of $\tilde{\nabla}$ matching at the pairs of points above each node, and having the Kodaira-Spencer map an isomorphism on one side, and on the other side, pairs $({\mathscr E}, \nabla)$ of gluings of ${\mathscr E}$ preserving an exact sequence $$0 \rightarrow {\mathscr L} \rightarrow {\mathscr E} \rightarrow {\mathscr L}^{-1} \rightarrow 0,$$ together with logarithmic connections $\nabla$ on ${\mathscr E}$ with vanishing $p$-curvature and trivial determinant and having the Kodaira-Spencer map an isomorphism, up to isomorphism and transport equivalence.
Further, this correspondence holds for first-order infinitesmal deformations. \end{cor}
\begin{proof} We can immediately conclude the statement over a field from our previous propositions. For first-order deformations, the same arguments will go through, with the aid of the following facts: first and most substantively, it follows from \cite[Cor. 3.6]{os10} that the residue matrices on $\tilde{C}$ will still be diagonalizable over $k[\epsilon]/\epsilon^2$, with the eigenvalues $e_i$ the same as for the connection being deformed. Next, since we are simply taking a base change of our original situation over $k$, the general gluing description given by Proposition \ref{def-glue-basic} still holds for formal reasons. Finally, one can easily verify that even over an arbitrary ring, it is still the case that an automorphism of a rank two free module is determined uniquely by sending any three pairwise independent lines to any other three. We therefore conclude the desired statement for first-order deformations as well. \end{proof}
\section{Deforming to a Smooth Curve}\label{s-def-deform}
The ultimate goal of this section is to prove that the connections we are interested in can always be smoothed from a general irreducible rational nodal curve, which together with the finiteness result of Section \ref{s-exp-det} and the main results of \cite{os7}, \cite{os10}, will allow us to finish the proof of the characteristic-independent portion of Theorem \ref{exp-main}. We begin with some general observations on when the space of connections with vanishing $p$-curvature is smooth over a given deformation of the curve and vector bundle. We then make a key dimension computation using the techniques of \cite{os10} and of the previous section, once again following arguments of Mochizuki \cite[Cor. II.2.5, p. 150]{mo3} rather than the original approach of the author, for the sake simplicity and generality.
\begin{sit}\label{def-sit} We suppose that $C_0$ is an irreducible, rational proper curve with two nodes, $\tilde{C_0} \cong {\mathbb P}^1$ its normalization, with $P_1, Q_1, P_2, Q_2$ being the points lying above the two nodes. We let ${\mathscr E}_0$ be the vector bundle described by Situation \ref{exp-specific-e}, and $\nabla_0$ a logarithmic connection on ${\mathscr E}_0$ with trivial determinant and vanishing $p$-curvature. \end{sit}
By Proposition \ref{def-background}, $p$-curvature gives an algebraic morphism $\psi_p : H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes \omega_{C_0}) \rightarrow H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes F^* \omega_{C_0^{(p)}})$ such that for $\varphi \in H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes \omega_{C_0})$, $\psi_p (\nabla_0 + \varphi)$ in fact lies in $H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes F^* \omega_{C_0^{(p)}})^{(\nabla_0+\varphi)^{\operatorname{ind}}}$. Now, we first claim:
\begin{lem}\label{def-dpsi} If $\nabla_0$ has vanishing $p$-curvature, the differential of $\psi_p$ at $0$ gives a linear map $$d\psi_p: H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes \omega_{C_0}) \rightarrow H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes F^* \omega_{C_0^{(p)}})^{\nabla_0^{\operatorname{ind}}}.$$ \end{lem}
\begin{proof} We simply consider the induced map on first-order deformations of $\nabla_0$. Denoting for the moment by $C_1$, ${\mathscr E}_1$ the base change of $C_0$, ${\mathscr E}_0$ to $k[\epsilon]/(\epsilon^2)$, suppose that $\varphi \in \epsilon H^0(\mathcal{E}nd^0({\mathscr E}_1) \otimes \omega_{C_1}) \cong H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes \omega_{C_0})$, and consider $\nabla_0 + \varphi$. Since $\nabla_0$ has vanishing $p$-curvature, the image under $\psi_p$ is in $\epsilon H^0(\mathcal{E}nd^0({\mathscr E}_1) \otimes F^* \omega_{C_1^{(p)}}) ^{(\nabla_0+ \epsilon \varphi)^{\operatorname{ind}}}$, which is naturally isomorphic to $H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes F^* \omega_{C_0^{(p)}})^{\nabla_0^{\operatorname{ind}}}$, giving the desired result. \end{proof}
Our main assertion is:
\begin{prop} If the map $d\psi_p$ of the previous lemma is surjective, then given a deformation $C$ of $C_0$ and ${\mathscr E}$ of ${\mathscr E}_0$ on $C$, such that the functor of connections on ${\mathscr E}$ with trivial determinant is formally smooth at $\nabla_0$, then the functor of connections on ${\mathscr E}$ with trivial determinant and vanishing $p$-curvature is formally smooth at $\nabla_0$. \end{prop}
\begin{proof} By hypothesis, there is no obstruction to deforming $\nabla_0$ as a connection with trivial determinant. Following \cite[Def. 1.2, Rem. 2.3]{sc2}, we say that a map $B \twoheadrightarrow A$ of local Artin rings over the base ring of our deformation and having residue field $k$ is a {\bf small extension} if the kernel is a principal ideal $(\epsilon)$ with $(\epsilon) {\mathfrak m}_B = 0$; it follows then that $\epsilon B \subset B$ is isomorphic to $k$. To verify (formal) smoothness, by virtue of \cite[Prop. 17.14.2]{ega44} it is easily checked inductively that it is enough to check on small extensions. We show therefore that for such a small extension, when $d\psi_p$ is surjective there is no obstruction to lifting a deformation of $\nabla_0$ over $A$ to a deformation over $B$, even with the addition of the vanishing $p$-curvature hypothesis. Let $C_B, {\mathscr E}_B$ be the given deformations over $B$ of $C_0, {\mathscr E}_0$ respectively, with $C_A, {\mathscr E}_A$ the induced deformations over $A$, and suppose that $\nabla_B$ is a connection on ${\mathscr E}_B$ such that $\nabla_A$ has vanishing $p$-curvature. The main point is that it is straightforward to check that the hypothesis that $\epsilon B \cong k$ implies that $\epsilon H^0(\mathcal{E}nd^0({\mathscr E}_B) \otimes \omega_{C_B}) \cong H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes \omega_{C_0})$, and for any $\varphi \in \epsilon H^0(\mathcal{E}nd^0({\mathscr E}_B) \otimes \omega_{C_B})$, we have $\epsilon H^0(\mathcal{E}nd^0({\mathscr E}_B) \otimes F^* \omega_{C_B^{(p)}}) ^{(\nabla_B+ \varphi)^{\operatorname{ind}}} \cong H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes F^* \omega_{C_0^{(p)}})^{\nabla_0^{\operatorname{ind}}}$. We want to show that for some choice of $\varphi \in \epsilon H^0(\mathcal{E}nd^0({\mathscr E}_B) \otimes \omega_{C_B})$, $\nabla_B + \varphi$ has vanishing $p$-curvature. But as before, since $\nabla_A$ has vanishing $p$-curvature, the image under $\psi_p$ of $\nabla_B + \varphi$ is in $\epsilon H^0(\mathcal{E}nd^0({\mathscr E}_B) \otimes F^* \omega_{C_B^{(p)}}) ^{(\nabla_B+ \varphi)^{\operatorname{ind}}}$, and under the above isomorphisms, the induced map is equal to $d\psi_p + \frac{1}{\epsilon} \psi_p(\nabla_B)$, where $\frac{1}{\epsilon}$ is simply shorthand for the isomorphism $\epsilon H^0(\mathcal{E}nd^0({\mathscr E}_B) \otimes F^* \omega_{C_B^{(p)}}) ^{(\nabla_B+ \varphi)^{\operatorname{ind}}}\overset{\sim}{\rightarrow} H^0(\mathcal{E}nd^0({\mathscr E}_0) \otimes F^* \omega_{C_0^{(p)}})^{\nabla_0^{\operatorname{ind}}}$. Hence if $d \psi_p$ is surjective, we can choose $\varphi$ so that $\nabla_B + \varphi$ has vanishing $p$-curvature, as desired. \end{proof}
We observe that in our situation, the normalization $\tilde{{\mathscr E}_0}$ of ${\mathscr E}_0$ is isomorphic to ${\mathscr O}(1) \oplus {\mathscr O}(-1)$: we certainly have $\tilde{{\mathscr L}}\cong {\mathscr O}(1)$, so by Lemma \ref{exp-destab-unique}, ${\mathscr O}(1)$ is the maximal line bundle in $\tilde{{\mathscr E}_0}$, and then the desired splitting follows from \cite[Proof of Thm. 1.3.1]{h-l}. Also, by Proposition \ref{def-glue-basic} $\tilde{\nabla_0}$ is a $D_{C_0}$-logarithmic connection on $\tilde{{\mathscr E}_0}$ with trivial determinant and vanishing $p$-curvature. For the sake of cleanness and generality, we use Mochizuki's arguments \cite[Cor. II.2.5, p. 150]{mo3} to prove the following.
\begin{prop}\label{def-aux} If $\nabla_0$ has a non-zero Kodaira-Spencer map, then the space of sections of $\mathcal{E}nd^0({\mathscr E}_0) \otimes F^* \omega_{C_0^{(p)}}$ horizontal with respect to the connection $\nabla^{\operatorname{ind}}_0$ induced by $\nabla_0$ on ${\mathscr E}_0$ and $\nabla ^{\text{can}}$ on $F^* \omega_{C_0^{(p)}}$ has dimension $3$. \end{prop}
\begin{proof} The proof proceeds in two parts: we show that $H^1(C_0, (\mathcal{E}nd^0({\mathscr E}_0) \otimes F^* \omega_{C_0^{(p)}})^{\nabla_0^{\operatorname{ind}}})=0$, and then compute the Euler characteristic. Both computations require formal local computations, so we begin by setting out the situation formally locally at a node of $C_0$. First, note that although taking kernels and tensor products of connections do not commute in general, there is no problem when one connection is obtained as the canonical connection of a Frobenius pullback, so we have $(\mathcal{E}nd^0({\mathscr E}_0) \otimes {\mathscr F}^* \omega_{C_0^{(p)}})^{\nabla_0^{\operatorname{ind}}} = \mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}} \otimes \omega_{C_0^{(p)}}$. Formally locally at the node, $C_0$ is isomorphic to $k[[x,y]]/(x,y)$; moreover, we claim that if we choose $x,y$ correctly, we can trivialize ${\mathscr E}_0$ so that $\nabla_0^{\mathcal{E}nd}$ has connection matrix $\begin{bmatrix}e (\frac{dx}{x} - \frac{dy}{y}) & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & e (\frac{dy}{y} - \frac{dx}{x}) \end{bmatrix}$ for some $e$ with $0<e<p$. Indeed, this follows from Proposition \ref{def-glue-basic} together with the formal local diagonalizability result of \cite[Cor. 2.10]{os10} applied to $\tilde{C}_0$, noting that if the residue of $\nabla_0$ has eigenvalues $e', -e'$, then the residue of $\nabla_0^{\mathcal{E}nd}$ has eigenvalues $2e', 0, -2e'$. By the same token, the pullback to the normalization gives connection matrices $\begin{bmatrix}e \frac{dx}{x} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -e \frac{dx}{x}\end{bmatrix}$ and $\begin{bmatrix}-e \frac{dy}{y} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & e \frac{dy}{y}\end{bmatrix}$. Finally, we note that the kernel of the connection on $C_0$ is given over ${\mathscr O}_{C_0^{(p)}}$ by $(x^{p-e}, y^e) \oplus (1) \oplus (x^e, y^{p-e})$, and by $(x^{p-e}) \oplus (1) \oplus (x^{e})$ and $(y^{e}) \oplus (1) \oplus (y^{p-e})$ on the normalization. The formal local calculations of the following paragraphs are justified by the following facts: given a sheaf map, surjectivity, and more generally factoring through a given subsheaf, may be checked after completion; completion commutes with pullback, with taking kernels of connections in characteristic $p$, and with modding out by torsion over a DVR; finally, completion is well-behaved with respect to pushforward under the normalization map by the theorem on formal functions.
Now, to check that $H^1$ vanishes, by Grothendieck duality on $C_0^{(p)}$ it suffices to check that $\operatorname{Hom}(\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}} \otimes \omega_{C_0^{(p)}}, \omega_{C_0^{(p)}}) = \operatorname{Hom}(\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}}, {\mathscr O}_{C_0^{(p)}})=0$. Although a section of the latter need not come from a map $\mathcal{E}nd^0({\mathscr E}_0)\rightarrow {\mathscr O}_{C_0}$ which is horizontal with respect to $\nabla_0^{\mathcal{E}nd}$, we claim that it does after normalization. We have a natural map $\mathcal{H}om(\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}},
{\mathscr O}_{C_0^{(p)}})|_{\tilde{C}_0} \rightarrow
\mathcal{H}om(\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0}^{\nabla_0^{\mathcal{E}nd}}, {\mathscr O}_{\tilde{C}_0^{(p)}})$, and a natural inclusion $\mathcal{H}om(\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0}, {\mathscr O}_{\tilde{C}_0})^{\nabla_0^{\mathcal{E}nd}} \hookrightarrow
\mathcal{H}om(\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0}^{\nabla_0^{\mathcal{E}nd}}, {\mathscr O}_{\tilde{C}_0^{(p)}})$. These are both isomorphisms away from the points above the nodes, for trivial reasons in the first case, and because of Theorem \ref{exp-cartier-vect} for the second. We want to show that the first map factors through the second. Examining the formal local situation at a node, we first note that if $e_1, e_2>0$, any map from $(x^{e_1}, y^{e_2})$ to ${\mathscr O}_{C_0^{(p)}}$ necessarily vanishes, and more specifically, sends $x^{e_1}$ and $y^{e_2}$ to positive ($p$th) powers of $x$ and $y$ respectively. It is thus clear that give a map $\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}}
\rightarrow {\mathscr O}_{C_0^{(p)}}$, after normalization we can divide through to get a map formally locally $\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0} \rightarrow {\mathscr O}_{\tilde{C}_0}$ which commutes with the induced connection, completing the proof of the claim. Next, we claim that such a map must be $0$. Indeed, if we consider the line sub-bundle ${\mathscr L}^0 \subset \mathcal{E}nd^0({\mathscr E}_0)$ which sends ${\mathscr L} \subset {\mathscr E}_0$ to $0$, we see that it is isomorphic to ${\mathscr L}^{\otimes 2}$, and is not horizontal for $\nabla_0^{\mathcal{E}nd}$, since ${\mathscr L}$ is not horizontal for $\nabla_0$, and we have the same situation after normalization. But having such a destabilizing line sub-bundle precludes the existence of a horizontal morphism
$\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0} \rightarrow {\mathscr O}_{\tilde{C}_0}$ by Proposition \ref{exp-destab-unique}, so we conclude the desired vanishing statement.
Thus, it remains to compute the Euler characteristic of $\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}}$. Since we only have two non-zero eigenvalues at each $P_i$ or $Q_i$, it follows from \cite[Cor. 2.11]{os10} that the cokernel of
$F^*((\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0})^{\nabla_0^{\mathcal{E}nd}}) \rightarrow
\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0}$ is supported at the $P_i,Q_i$, with length $p$ at each point. Since $\deg(\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0})=0$, we find that
$\deg(F^*((\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0})^{\nabla_0^{\mathcal{E}nd}}))=-4p$. Next, we claim that $(\mathcal{E}nd^0({\mathscr E}_0)|_{\tilde{C}_0})^{\nabla_0^{\mathcal{E}nd}}$ is isomorphic to the quotient of $(\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}})|_{\tilde{C}_0}$ by its torsion, which we denote by ${\mathscr F}$; indeed, we clearly have a morphism from the latter to the former, which is an isomorphism away from the points above the nodes, hence gives an injection since we modded out by torsion. Surjectivity above the nodes is then checked formally locally from our above description, so we have $\deg({\mathscr F}) = -4$, and $\chi({\mathscr F})=-1$. Finally, if $\nu$ denotes the normalization map, we claim that the natural injection $\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}} \hookrightarrow \nu_* {\mathscr F}$ has cokernel of length $1$ at each node; again, this is checked formally locally, noting that the cokernel will arise only from the summand at each node on which the connection vanishes. We conclude therefore that $\chi (\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}}) = -3$, so $H^0 (\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}}\otimes \omega_{C_0^{(p)}}) = \chi (\mathcal{E}nd^0({\mathscr E}_0)^{\nabla_0^{\mathcal{E}nd}}\otimes \omega_{C_0^{(p)}}) = 3$, completing the proof of the proposition. \end{proof}
Finally, we put these results together in our specific situation:
\begin{thm}\label{def-deform} Let $C_0$ be a nodal rational curve of genus $2$, and ${\mathscr E}_0$ as in Situation \ref{exp-specific-e}. Let $\nabla_0$ have vanishing $p$-curvature and trivial determinant, and suppose that $\nabla_0$ has no deformations preserving the $p$-curvature and not arising from transport. Then the map $d\psi_p$ of Lemma \ref{def-dpsi} is surjective; in particular, given any deformation $C$ of $C_0$, if ${\mathscr E}$ is the corresponding deformation of ${\mathscr E}_0$, then the space of connections with trivial determinant and vanishing $p$-curvature on ${\mathscr E}$ is formally smooth at $\nabla_0$. \end{thm}
\begin{proof} The main point is that by Remark \ref{exp-gen}, the space of transport-equivalence classes of connections with trivial determinant on ${\mathscr E}_0$ or ${\mathscr E}$ is explicitly parametrized by ${\mathbb A}^3$ over the appropriate base. In particular, deformations of $\nabla_0$ as a connection with trivial determinant are unobstructed, and it also follows that the space of first-order deformations of $\nabla_0$ with trivial determinant, modulo those arising from transport, is three-dimensional. By Proposition \ref{def-aux}, the image space of $d\psi_p$ is three-dimensional. We therefore get surjectivity precisely when transport accounts for the entire kernel, which is to say, when there are no deformations of $\nabla_0$ having vanishing $p$-curvature and trivial determinant other than those obtained by transport. We can thus apply the previous proposition to conclude smoothness. \end{proof}
It is now a matter of some simple combinatorics to complete the proof of the characteristic-independent portion of Theorem \ref{exp-main}.
\begin{proof}[Proof of Theorem \ref{exp-main}, $p>2$ case] By the results of Section \ref{s-exp-background} it suffices to show that, for the particular ${\mathscr E}$ of Situation \ref{exp-specific-e}, there are precisely $\frac{1}{24}p(p^2-1)$ transport-equivalence classes of connections with trivial determinant and vanishing $p$-curvature on ${\mathscr E}$, and that none of these have any non-trivial deformations. We will show that this statement holds in the situation that $C$ is a general rational nodal curve, and then conclude the same result must hold for a general smooth curve.
We observe that even in the situation of a nodal curve, there is a unique extension ${\mathscr E}$ of ${\mathscr L}^{-1}$ by ${\mathscr L}$; indeed, the proof of Proposition \ref{exp-unstable-unique} goes through with $\omega_C$ in place of $\Omega^1_C$. We also note that by Corollary \ref{def-conn-line}, the argument of Proposition \ref{exp-unstable} still shows that any connection must have its Kodaira-Spencer map be an isomorphism. It then follows from Corollary \ref{def-glue-main} that it suffices to prove the same result for $D$-logarithmic connections on ${\mathscr O}(1) \oplus {\mathscr O}(-1)$ on ${\mathbb P}^1$ satisfying the hypotheses of \cite[Sit. 2.12]{os10} and having the Kodaira-Spencer map an isomorphism, where $D$ is made up of four general points on ${\mathbb P}^1$, and the eigenvalues of the residues at the points match in the appropriate pairs. We note that by degree considerations, the Kodaira-Spencer map in this case is always either zero or an isomorphism, so if we fix eigenvalues $\alpha_i$ for each pair $(P_i, Q_i)$, by \cite[Thm. 1.1]{os10} we find that we are looking for separable rational functions on ${\mathbb P}^1$ of degree $2p-1 - 2 \sum \alpha_i$, and ramified to order at least $p- 2\alpha_i$ at $P_i$ and $Q_i$ (note that the coefficient doubling for the degree is due to our use of a single, matching $\alpha_i$ for both $P_i$ and $Q_i$). We could use the second formula of \cite[Cor. 8.1]{os7} to compute the answer directly, but the first formula yields a more elegant solution. In either case, we are already given the lack of non-trivial deformations, so it suffices to show that the number of maps is correct. The formula gives that for each $(\alpha_1, \alpha_2)$ there are $$\min\{\{p-2 \alpha_i\}_i, \{p-2 \alpha_{3-i}\}_i, \{2\alpha_i\}_i, \{2\alpha_{3-i}\}_i\}$$ such maps, which reduces to $$\min\{\{p-2 \alpha_i\}_i, \{2 \alpha_i\}_i\}.$$ Rather than summing up over all $\alpha_i$, as we would with the second formula, we note that the number of maps will also be given by: $$\sum_{1 \leq j \leq (p-1)/2} \#\{(\alpha_1, \alpha_2): j \leq 2 \alpha_i, j \leq p -2 \alpha_i\}$$ which then reduces to $$ \sum_{1 \leq j \leq (p-1)/2} \!\!\!\!\!\!\!\! (\frac{p+1}{2}-j)^2 = \!\!\!\!\!\!\!\! \sum_{1 \leq j \leq (p-1)/2} \!\!\!\!\!\!\!\! j^2 = \sum_{1 \leq j \leq (p-1)/2} \!\!\!\!\!\!\!\! (2 \binom{j}{2} + j)$$ $$= 2 \binom{(p+1)/2}{3} + \frac{p+1}{2}\frac{p-1}{4} = \frac{1}{24}(p+1)((p-1)(p-3)+ 3(p-1)) = \frac{(p+1)(p-1)p}{24},$$ giving the desired result for a general nodal curve.
We can now apply Theorem \ref{def-deform} to conclude that since none of our connections on the general nodal curve have non-trivial deformations, the space of connections with trivial determinant and vanishing $p$-curvature on our chosen bundle over our parameter space of genus $2$ curves is formally smooth at each connection on the general nodal curve. Furthermore, by Corollary \ref{exp-finite} (in light of Remark \ref{exp-gen}), this space of connections is finite, so we conclude that it is finite \'etale at the general nodal curve, and finite everywhere, which then implies (i) for a general smooth curve, as desired. \end{proof}
\end{document} | arXiv |
Class 9 Science CBSE Objective type for Atoms and Molecules
NCERT solutions for Atoms and Molecules
Short Questions
Long Questions
Shorts questions -2
Objective Questions
Given below are the Class 9 Science CBSE Objective type or MCQ for Atoms and Molecules
Hope you like them and do not forget to like , social share and comment at the end of the page.
Multiple Choice Questions
Question 1.
Which of the following has largest number of particles?
a. 8g of $CH_4$
b. 4.4g of $CO_2$
c. 34.2g of $C_{12}H_{22}O_{11}$
d.2g of $H_2$
The number of molecules in 16.0g of oxygen is:
a.$6.02 \times 10^{23}$
b.$6.02 \times 10^{-23}$
c.$3.01 \times 10^{-23}$
d.$3.01 \times 10^{23}$
The percentage of hydrogen in $H_{2}O$ is:
a.8.88
b.11.12
c.20.60
d.80.0
. Find the mass of oxygen contained in 1 kg of potassium nitrate $KNO_{3}$.
a.475.5g
b.485.5g
c.475.2g
d.485.2g
How many moles of electron weight one kilogram?
a. $6.023 \times 10^{23}$
b. $6.023 \times 10^{8}$
c. $9.108 \times 10^{54}$
d. $1.82 \times 10^{8}$
25.4g of iodine and 14.2g of chlorine are made to react completely to yield a mixture of $ICl$ and $ICl_{3}$. Calculate the ratio of moles of $ICl$ and $ICl_{3}$:
a.1: 1
b.2: 1
c.3: 1
d.1: 2
The mass of sodium in 11.7 g of sodium chloride is:
a.2.3g
b.4.6g
c.6.9g
d.7.1g
The formula of a chloride of a metal M is $MCl_{3}$, the formula of the phosphate of metal M will be:
a.$MPO_{4}$
b.$M_2PO_{4}$
c.$M_{3}PO_{4}$
d.$M_2(PO_{4})_{3}$
Which of the following contains the largest number of molecules?
a.0.2 mol $H_{2}$
b.8.0g $H_{2}$
c.17g of $H_{2 }O$
d.6.0 g of $CO_{2}$
Question 10.
One gram of which of the following contains largest number of oxygen atoms?
a.O
b.$O_{2}$
c.$O_{3}$
d.All contains same
The percentage by weight of $O_{2}$ in $CaO_{4}$ ( O = 16, S = 32, Ca = 40) is:
b.28.2
c.47.2
The percentage by weight of Zn in white vitriol, $ZnSO_{4}.7H_{2}O$ (Zn = 65, S = 32, O = 16, H = 1) is approximately:
The formation of $SO_{2}$ and $SO_{3}$ explain:
a.The law of conservation of mass
b.The law of multiple proportions
c.The law of definite properties
d.Boyle's law
The law of definite proportions was given by:
a.John Dalton
b.Humphry davy
c.Proust
d.Michael Faraday
Molecular mass is defined as the:
a.Mass of one atom compared with the mass of one molecule
b.Mass f one atom compared with the mass of one atom of hydrogen
c.Mass of one molecule of any substance compared with the mass of one atom of C – 12
d.None of the above
0.001 g of C is required t write a letter with a graphite pencil. The total number of C atoms used in writing the letter is:
b.$5 \times 10^{19}$
c.$5.0 \times 10^{24}$
d.$6.023 \times 10^{23}$
One mole of a gas occupies a volume of 22.4 L. This is derived from:
a.Berzelius's hypothesis
b.Gay- Lussac's law
c.Avogadro's law
d.Dalton's law
The mass of one C atom is:
a.$6.023 \times 10^{23}$ g
b.$1.99 \times 10^{-23}$g
c.2.00 g
d.12 g
The chemical symbol for barium is:
b.Ba
c.Be
d.Bi
The chemical symbol P stands for:
a.Phosphorus
b.Potassium
c.Polonium
d.Promethium
A group of atoms chemically bonded together is a (an):
a.Molecule
b.On
c.Salt
d.Element
Adding electrons to an atom will result in a (an):
b.Anion
c.Cation
d.Salt
When an atom loses electrons, it is called a (an) ______ and has a _____ charge.
a.Anion, positive
b.Cation, positive
c.Anion, negative
d.Cation, negative
The molecule formula $P_{2}O_{5}$ means that:
a.A molecule contains 2 atoms of P and 5 atoms of O
b.The ratio of the mass of P to the mass of O in the molecule is 2:5
c.There are twice as many P atoms in the molecule as there are O atoms
d.The ratio of the mass of P to the ass of O in the molecule is 5:2
The correct symbol for silver is:
a.Ag
c.Ar
d.AI
A spartame, an artificial sweetener, has the molecular formula $C_{14}H_{18}N_{2}O_{5}$. What is the mass in grams of one molecule? (Atomic weights: C = 12.01, H = 1.008, N = 14.01, O = 16.00)
a.$4.89 \times 10^{-21}$
d.$4.89 \times 10^{-22}$
Morphine, an addictive drug, has the molecular formula $C_{17}H_{19}NO_{3}$. What is the mass in grams of one molecule?(Atomic weights: C = 12.01, H = 1.008, N = 14..1, O = 16.00)
a. $2.24 \times 10^{-22}$
b. $3.85 \times 10^{-22}$
c. $3.85 \times 10^{-21}$
d. $4.74 \times 10^{-22}$
The controversial artificial sweetener saccharin has the molecular formula $C_{3}H_{5}O_{5}NS$. What is the mass in grams of one molecule? Atomic weights: C = 12.01, H = 1.008, O = 16.00, N = 14.01, S = 32.06.
The statue of Liberty is made of 2.0 x 105 Ibs of copper sheets bolted to a framework. (1Ib = 454 g) . How many atoms of copper are on the statue? (Atomic weight: Cu = 63.5.)
a.$2.1 \times 10^{27}$
b.$8.6 \times 10^{29}$
d.$8.6 \times 10^{26}$
Selenium ingested in the amount of 90 microgarms per day causes loss of hair. How many selenium atoms are n this size sample? (Atomic weight: Se = 78.96.)
Novocain is $C_{13}H_{16}N_{2}O_{2}$ What is the total number of moles of atoms in 0.020 moles of Novocain?
(Atomic weights: C = 12.01, O = 16.00, N = 14.01, H = 1.008.)
a.0.033
b.3.3
c.0.66
d.0.33
Methoxychlor, a garden insecticide, has the molecular formula $C_{16}H_{15}C_{13}O$. What is the total number of moles of atoms in a 3.0 mg sample? (Atomic weights: C = 12.01, H = 1.008, Cl = 35.45, O = 16.00)
a.$8.7 \times 10^{-6}$
b.$3.0 \times 10^{-5}$
c.$3.1 \times 10^{-1}$
d.$3.1 \times 10^{-4}$
What are the total number of moles of atoms in 4.32 g of $Sc(NO_{3})_{3}$?
(Atomic weights: Sc = 45.0, O = 16.00, N = 14.01.)
a.0.0132
b.0.324
c.0.0187
d.0.243
Two samples of lead oxide were separately reduced to metallic lead by heating in a current of hydrogen. The weight of lead from one oxide was half the weight of lead obtained from the other oxide. The data illustrates:
a.Law of reciprocal proportions
b.Law of constant proportions
c.Law of multiple proportions
d.Law of equivalent proportions
The percentage of copper and oxygen in samples of CuO obtained by different methods were found to be the same. The illustrate the law of:
a.Constant proportion
b.Conservation of mass
c.Multiple proportions
d.Reciprocal proportions
The total number of atoms represented by the compound $CuSO_4.5H_2O$ is:
In compound A, 1.00g of nitrogen unites with 0.57 g of oxygen. In compound B, 2.00g of nitrogen combines with 2.24g of oxygen. In compound C, 3.00g of nitrogen combines with 5.11g of oxygen. These results obey the following law:
a.Law of constant proportion
b.Law of multiple proportion
c.Law of reciprocal proportion
d. Dalton's law of partial pressure
The weight of a molecule of the compound $C_{60}H_{122}$ is:
a.$1.4 \times 10^{-21}$ g
b.$1.09 \times 10^{-21}$ g
c.$5.025 \times 10^{23}$ g
d.$16.023 \times 10^{23}$ g
The mass of a molecule of water is:
a.$3 \times 10^{-26}$ kg
b.$3 \times 10^{-25 }$kg
c.$1.5 \times 10^{-26}$ kg
d.$2.5 \times 10^{-26}$ kg
The number of atom in 4.25g of $NH_{3}$ is approximately:
a.$1 \times 10^{23}$
c.$4 \times 10^{23}$
d.$6 \times 10^{23}$
Volume of a gas at STP is $1.12 \times 10^{-7}$ cc. Calculate the number of molecules in it:
b.$3.01 \times 10^{12}$
c.$3.01 \times 10^{23}$
The number of molecules of $CO_{2}$ present in 44g of $CO_{2}$ is:
c.$12 \times 10^{23}$
The volume occupied by 4.4g of $CO_2$ at STP is:
a.22.4 L
b.2.24 L
c.0.224 L
d.0.1 L
How many molecules are present in one gram of hydrogen?
1.(d)
3.(b)
6.(a)
10.(c)
12.(a)
13.(b)
26.(d) | CommonCrawl |
\begin{document}
\title[Incompressible Navier-Stokes equation with variable density] {Global unique solutions for the inhomogeneous Navier-Stokes equation with only bounded density, in critical regularity spaces} \author{ Rapha\"el Danchin} \author{Shan Wang} \begin{abstract} We here aim at proving the global existence and uniqueness of solutions to the inhomogeneous incompressible Navier-Stokes system in the case where \emph{the initial density $\rho_0$ is discontinuous and the initial velocity $u_0$ has critical regularity.}
Assuming that $\rho_0$ is close to a positive constant, we obtain global existence and uniqueness in the two-dimensional case whenever the initial velocity $u_0$ belongs to the critical homogeneous Besov space $\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2})$ $(1<p<2)$ and, in the three-dimensional case, if $u_0$ is small in $\dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})$ $(1<p<3).$
Next, still in a critical functional framework, we establish a uniqueness statement that is valid in the case of large variations of density with, possibly, vacuum. Interestingly, our result implies that the Fujita-Kato type solutions constructed by P. Zhang in \cite{Zhang19} are unique.
Our work relies on interpolation results, time weighted estimates and maximal regularity estimates in Lorentz spaces (with respect to the time variable) for the evolutionary Stokes system. \end{abstract}
\date{} \keywords{Lorentz spaces, critical regularity, uniqueness, global solutions, inhomogeneous Navier-Stokes equations, bounded density, vacuum}
\maketitle \section*{Introduction} We are concerned with the initial value problem for the following inhomogeneous incompressible Navier-Stokes system:
\begin{equation*} \left\{\begin{aligned} &\rho_{t}+u\cdot \nabla \rho=0, \\
&\rho u_{t}+\rho u\cdot \nabla u-\mu\Delta u+\nabla P=0, \\ &\div u=0,\\
&(\rho,u)|_{t=0}=(\rho_{0},u_{0}), \end{aligned}\right.\eqno(INS) \end{equation*} where $\rho=\rho(t,x)\geq0,$ $P=P(t,x)\in\mathbb R$ and $u=u(t,x)\in\mathbb R^d$ stand for the density, pressure and velocity field of the fluid, respectively. We consider the evolution for positive times $t$ in the case where the space variable $x$ belongs to the whole space $\mathbb R^d$
with $d=2,3.$
\medbreak It has long been observed that smooth enough solutions obey the following energy balance: \begin{equation}\label{eq:energy}\frac 12\norm{\sqrt{\rho(t)}u(t)}^2_{L_2}+\int_0^t \norm{\nabla u(t)}^2_{L_2} \,d\tau = \frac 12 \norm{\sqrt{\rho_0}u_0}^2_{L_2},\end{equation} and that, as a consequence of the divergence free property of the velocity field,
the Lebesgue measure of \begin{equation}\label{eq:density}\bigl\{ x\in \mathbb R^d : \alpha \leq \rho(t,x)\leq \beta\bigr\}\end{equation} is independent of $t,$ for any $0\leq\alpha\leq\beta.$ \medbreak In 1974, by combining these relations with Galerkin approximation and compactness arguments, Kazhikhov \cite{AVK} established that for any data $(\rho_0,u_0)$ such that $\rho_0 \in L_\infty,$ $\div u_0=0$ and $\sqrt{\rho_0}u_0 \in L_2,$ and \emph{provided $\rho_{0}$ is bounded away from vacuum} (that is $\inf\rho_0(x)>0$), (INS) has at least one global distributional solution satisfying \eqref{eq:energy} with an inequality. The no vacuum assumption was removed later by J. Simon in \cite{JS}, then, by taking advantage of the theory developed in \cite{DL1989}, P.-L. Lions \cite{PL} extended the previous results to the case of a density dependent viscosity, proved that the mass equation of (INS) is satisfied in the renormalized meaning, that the velocity field admits a unique generalized flow and, finally, that \eqref{eq:density} is true. However, from that time whether these weak solutions are unique is an open question, even in dimension two. \smallbreak By using totally different approaches, a number of authors proved that in the case of smooth enough data, (INS) admits a unique solution at least locally in time. In fact, as for the classical incompressible Navier-Stokes equations (that is (INS) with constant positive density), the general picture is that provided the initial density is sufficiently smooth, bounded and bounded away from zero, there exists a global unique solution if the initial velocity is small in the sense of some `critical norm', and that it can be arbitrarily large in dimension $d=2.$ This general fact has been first observed by O.~Ladyzhenskaya and V.~Solonnikov \cite{OV2} in the case where the fluid domain $\Omega$ is a smooth bounded subset of $\mathbb R^d$ ($d=2,3$) and the velocity vanishes at the boundary. More precisely, assuming that $u_0$ is in the Sobolev–Slobodeckij space $W^{2-\frac 2p,p}(\Omega)$ with $p>d,$ is divergence free and has null trace on $\partial \Omega,$ and that $\rho_0$ is $C^1$ and is bounded away from zero, they proved: \begin{itemize}
\item[--] the global well-posedness in dimension $d=2$,\smallbreak
\item[--] the local well-posedness in dimension $d=3$
(and global well-posedness if $u_0$ is small in $W^{2-\frac 2p,p}(\Omega)$).
\end{itemize}
Results in the same spirit in other functional frameworks have been proved by a number of authors (see e.g. the survey paper \cite{RD}).
Still for smooth enough data,
the non vacuum assumption has been weakened by Choe and Kim in \cite{CK}
\medbreak
A natural question is the minimal regularity requirement for the data ensuring
(at least local) existence and uniqueness. It has been observed by Fujita and Kato \cite{FK}
in the constant density case (and later for a number of evolutionary equations) that this
issue is closely linked to the scaling invariance of the system under consideration. Here it is obvious that
if $(\rho, u, P)$ is a solution of (INS) on $\mathbb R_{+}\times \mathbb R^{d}$ for data $(\rho_0,u_0)$
then, for all $\lambda>0,$ the rescaled triplet $(\rho, u, P)\to (\rho_{\lambda}, u_{\lambda}, P_{\lambda})$ defined by \begin{equation}\label{isi} (\rho_{\lambda}, u_{\lambda}, P_{\lambda})\overset{\text{def}}{=}(\rho(\lambda^{2}t, \lambda x), \lambda u(\lambda^{2}t, \lambda x), \lambda^{2}P(\lambda^{2}t, \lambda x)) \end{equation} is a solution of (INS) on $\mathbb R_{+}\times \mathbb R^{d},$ with data $(\rho_0(\lambda\cdot),\lambda u_0(\lambda\cdot)).$
\medbreak
A number of works have been dedicated to the well-posedness of (INS) in $\mathbb R^d,$
in so-called critical framework, that is to say in functional spaces with the above scaling invariance. Restricting our attention to the case where the density tends to some positive constant at infinity (say $1$ for notational simplicity) and setting $a\overset{\text{def}}{=}{1}/{\rho}-1,$ System (INS) rewrites in terms of $(a,u,P)$ as follows: \begin{equation}\label{rins2} \left\{\begin{aligned} &a_t+u\cdot\nabla a=0, \\ &u_t+u\cdot\nabla u-(1+a)(\mu\Delta u-\nabla P)=0,\\ &\div u=0,\\
&(a,u)|_{t=0}=(a_{0},u_{0}). \end{aligned}\right. \end{equation} In \cite{DR2003}, the first author established the existence and uniqueness of a solution to \eqref{rins2} in critical Besov spaces. More precisely, in the case where $a_{0}\in \dot{B}^{d/2}_{2,1}(\mathbb R^d)$ and $u_{0}\in \dot{B}^{d/2-1}_{2,1}(\mathbb R^d)$ with $\div u_{0}=0,$ he proved that there exists a constant $c$ depending only on $d$ such that, if \begin{equation*}
\norm{a_{0}}_{\dot{B}^{1}_{2,1}} \leq c,\end{equation*} then \eqref{rins2} admits a unique local solution $(a,u,\nabla P)$ with $$a\!\in\! C_{b}([0,T);\dot{B}^{d/2}_{2,1}),\quad\!\! u\!\in\! C_{b}([0,T);\dot{B}^{d/2}_{2,1})\cap L_{1}(0,T;\dot{B}^{d/2+1}_{2,1}) \!\quad\hbox{and}\quad\!
\nabla P\!\in\! L_{1}(0,T;\dot{B}^{d/2-1}_{2,1})$$ and that there exists $c'>0$ such that this solution is global (i.e. one can take $T=\infty$) if $\norm{u_{0}}_{\dot{B}^{d/2-1}_{2,1}}\leq c'\mu.$ \smallbreak Shortly after, these results have been extended by H. Abidi in \cite{AH}, then H. Abidi and M. Paicu \cite{AP} to critical Besov spaces of type $\dot B^s_{p,1}$ with $p>1.$ \medbreak Again in the critical functional framework,
J.~Huang, M.~Paicu and P.~Zhang noticed
that, somehow, only $d-1$ components of $u_0$ need
to be small for global existence of weak solutions:
in \cite{HPZ2013}, they just required that $$(\mu \norm{a_{0}}_{L_{\infty}}+\norm{u_{0}^{h}}_{\dot{B}^{-1+\frac{d}{p}}_{p,r}})\exp(C_{r}\mu^{-2r}\norm{u^{d}_{0}}^{2r}_{\dot{B}^{-1+\frac{d}{p}}_{p,r}})\leq c_{0}\mu$$ for some positive constants $c_{0}$ and $C_{r}$. \medbreak Achieving results in the critical functional framework when the density has large variations requires techniques that are not just based on perturbation arguments. In \cite{DR2004}, the first author investigated the problem in Sobolev spaces but failed to reach the critical exponent. Recently, H.~Abidi and G.~Gui \cite{AG} proved the global unique solvability of the 2-D incompressible inhomogeneous Navier-Stokes equations
whenever $\rho^{-1}_0-1$ is in $\dot{B}^{2/p}_{p,1}(\mathbb R^2)$ for some $2<p<\infty,$ and $u_0$ is in $\dot{B}^0_{2,1}(\mathbb R^2)$.
This is, to our knowledge, the first global
well-posedness result at the critical level of regularity, that
does not require any smallness condition
(see also the work by H. Xu in \cite{Xu}, based on different techniques). \medbreak A number of recent works aimed at proving existence and uniqueness results in the case where the density is only bounded (and not continuous). In this respect, significant progress has been done by M. Paicu, P. Zhang and Z.~Zhang in \cite{PZZ} where the global existence and uniqueness of solution to (INS) is shown in $\mathbb R^d,$ $d=2,3$ assuming only that $\rho_0^{\pm1}$ is bounded and that $u_{0}\in H^{s}(\mathbb R^{2})$ for some $s>0$ (2D case) or $u_{0}\in H^{1}(\mathbb R^{3})$ with $\norm{u_{0}}_{L_{2}}\norm{\nabla u_{0}}_{L_{2}}$ sufficiently small (3D case). This result was extended to velocities in $H^s(\mathbb R^3)$ with $s>1/2$ by D. Chen, Z. Zhang and W. Zhao in \cite{DZW}.
Finally, the lower bound assumption was totally removed by the first author and P.B. Mucha in \cite{DM1} in the case where the fluid domain $\Omega$ is either bounded or the torus. There, it is only needed that $u_0$ is
$H^1_0(\Omega)$ and that $\rho_0$ is bounded.
\smallbreak Very recently, in the 3D case, P.~Zhang \cite{Zhang19} established the global existence of weak solutions to the 3D inhomogeneous incompressible Navier-Stokes system with initial density in $L_\infty(\mathbb R^3),$ bounded away from zero, and initial velocity sufficiently small in the critical Besov space $\dot{B}^{1/2}_{2,1}(\mathbb R^{3}).$ This is the first example of a global existence result within a Besov critical framework for the velocity and no regularity for the density, in the large variations case. Note however that the uniqueness of these solutions has not been proved so far. \medbreak The primary goal of our paper is to establish the global existence of solutions of (INS) \emph{that are unique} in a critical regularity framework, in the case where the initial density is close to a positive constant in $L_\infty$ but has no regularity whatsoever. To our knowledge, no result of this type has been proved before. In accordance with the state-of-the art for the homogeneous Navier-Stokes equations (that is, with constant density), smallness of the velocity will be required if $d=3,$ but not if $d=2.$
The uniqueness part of our statements will come up has an easy consequence of a much more general
result within a critical regularity framework, that allows for density with large variations
(that is even allowed to vanish on arbitrary sets if the dimension is $3$). As a by-product, we shall obtain that the global solutions constructed by P. Zhang in \cite{Zhang19} (that are allowed to have large density variations), are actually unique. \medbreak Our existence results are strongly based on a novel maximal regularity estimate for the Stokes system equation originating from the recent paper \cite{DM2} by P.B.~Mucha, P.~Tolksdorf and the first author, where the time regularity
is measured in \emph{Lorentz spaces}.
Time weighted estimates will also play an important role (see the end
of the next section for more explanation).
\medbreak
\section{Tools, results and approach}\label{section1}
Before stating our main existence results for (INS), introducing a few notations and recalling some results is in order. \medbreak First, throughout the text, $A\lesssim B$ means that $A\leq CB$, where $C$ designates
various positive real numbers the value of which does not matter.
\smallbreak For any Banach space $X,$ index $q$ in $[1,\infty]$ and time $T\in[0,\infty],$ we use the notation
$\|z\|_{L_q(0,T;X)}\overset{\text{def}}{=}
\bigl\| \|z\|_{X}\bigr\|_{L_q(0,T)}.$
If $T=\infty$, then we just write $\|z\|_{L_q(X)}.$ In the case where $z$ has $n$ components $z_k$ in $X,$ we
keep the notation $\norm{z}_X$ to mean $\sum_{k\in\{1,\cdots,n\}} \norm{z_k}_X$. \smallbreak We shall use the following notation for the \emph{convective derivative}: \begin{equation}\label{def:convective} \frac{D}{Dt}\overset{\text{def}}{=}\partial_{t}+u\cdot \nabla \quad\hbox{and}\quad \dot{u}\overset{\text{def}}{=}u_{t}+u\cdot \nabla u.\end{equation}
Next, let us recall the definition of Besov spaces on $\mathbb R^d.$ Following \cite[Chap. 2]{BCD}, we fix two smooth functions $\chi$ and $\varphi$
such that $$\displaylines{\text{Supp}\ \varphi \subset \{\xi\in\mathbb R^{d},\: 3/4\leq \abs{\xi}\leq 8/3\}\ \ \text{and } \ \ \forall\xi\in\mathbb R^{d}\setminus\{0\}, \;\underset{j\in\mathbb Z}{\sum}\varphi(2^{-j}\xi)=1,\cr \text{Supp}\ \chi\subset\{\xi\in \mathbb R^{d},\: \abs{\xi}\leq 4/3\} \ \text{and}\ \ \forall \xi\in \mathbb R^{d},\; \chi(\xi)+\sum_{j\geq 0}\varphi(2^{-j}\xi)=1,}$$ and set for all $j\in\mathbb Z$ and tempered distribution $u,$
$$\dot{\Delta}_{j}u\overset{\text{def}}{=}\mathcal{F}^{-1}(\varphi(2^{-j}\cdot)\widehat{u})\overset{\text{def}}{=}2^{jd} \widetilde{h}(2^{j}\cdot)\star u\quad\hbox{with}\quad\widetilde{h}\overset{\text{def}}{=}\mathcal{F}^{-1}\phi,$$ \begin{equation}\label{eq:Sj}\dot{S}_{j}u\overset{\text{def}}{=}\mathcal{F}^{-1}(\chi(2^{-j}\cdot)\widehat{u})\overset{\text{def}}{=}2^{jd}h(2^{j}\cdot)\star u\quad\hbox{with}\quad h \overset{\text{def}}{=}\mathcal{F}^{-1}\chi,\end{equation}
where $\mathcal{F}u$ and $\widehat{u}$ denote the Fourier transform of $u.$
\begin{definition}[Homogeneous Besov spaces] Let $(p,r)\in [1,\infty]^{2}$ and $s\in\mathbb R.$ We set $$\norm{u}_{\dot{B}^{s}_{p,r}(\mathbb R^{d})}\overset{\text{def}}{=}\norm{(2^{js}\norm{\dot{\Delta}_{j} u}_{L_{p}(\mathbb R^{d})})_{j\in \mathbb Z}}_{\mathit{\ell}_{r}(\mathbb Z)}.$$ We denote by $\dot{B}^{s}_{p,r}(\mathbb R^{d})$ the set of tempered distributions $u$
such that $\norm{u}_{\dot{B}^{s}_{p,r}(\mathbb R^{d})}<\infty$ and
\begin{equation}\label{eq:lf}
\underset{j\to -\infty}{\lim}\norm{\dot S_{j}u}_{L_{\infty}(\mathbb R^{d})}=0.
\end{equation} \end{definition}
It is classical that the scaling invariance condition for $u_0$ pointed out in \eqref{isi} is satisfied for all elements of $\dot B^{-1+d/p}_{p,r}(\mathbb R^d)$ with $1\leq p,r\leq\infty.$
\medbreak
Next, we define Lorentz spaces, and recall a useful characterization. \begin{definition} Given $f$ a measurable function on a measure space $(X,\mu)$ and $1\leq p,r\leq \infty$, we define
$$\widetilde\|{f}\|_{L_{p,r}(X,\mu)}:= \begin{cases} (\int_{0}^{\infty}(t^{\frac{1}{p}}f^{*}(t))^{r}\,\frac{dt}{t})^{\frac1r} & \text{if $r<\infty$},\\ \underset{t>0}{\sup} t^{\frac{1}{p}}f^{*}(t)& \text{if $r=\infty$}, \end{cases}$$
where $$f^{*}(t):=\inf\bigl\{s\geq 0:|\{\abs{f}>s\}|\leq t\bigr\}\cdotp$$
The set of all $f$ with $\widetilde\|{f}\|_{L_{p,r}(X,\mu)}<\infty$ is called the Lorentz space with indices $p$ and $r$. \end{definition} \begin{remark}\label{lorentzdef2} It is well known that $L_{p,p}(X,\mu)$ coincides with the Lebesgue space $L_p(X,\mu).$ Furthermore, according to \cite[Prop.1.4.9] {LG}, the Lorentz spaces may be endowed with the following (equivalent) quasi-norm: \begin{equation*} \norm{f}_{L_{p,r}(X,\mu)}:=
\begin{cases}
p^{\frac{1}{r}}\biggl(\displaystyle \int_{0}^{\infty}\bigl(s|\{\abs{f}>s\}|^{\frac{1}{p}}\bigr)^{r}\,\frac{ds}{s}\biggr)^{\frac{1}{r}} & \text{if $r<\infty$}\\
\underset{s>0}{\sup} s|\{\abs{f}>s\}|^{\frac{1}{p}}& \text{if $r=\infty$}.
\end{cases} \end{equation*} \end{remark}
Our results will strongly rely on a maximal regularity property for the following evolutionary Stokes system:
\begin{equation}\label{eq:stokes} \left\{\begin{aligned}
&u_{t}-\mu \Delta u+\nabla P=f &\ \ \text{in }\ \mathbb R_+\times \mathbb R^{d}, \\
&\div u=0 &\ \ \text{in }\ \mathbb R_+\times \mathbb R^{d}, \\
&u|_{t=0}=u_{0} &\ \ \text{in }\ \mathbb R^{d}. \end{aligned}\right. \end{equation}
It has been pointed out in \cite[Prop. 2.1]{DM2}
that for the free heat equation
supplemented with initial data $u_0$ in
$\dot B^{2-2/q}_{q,r}(\mathbb R^d),$
the solution $u$ is such that $u_t$ and $\nabla^2u$ are in $L_{q,r}(\mathbb R_+;L_p(\mathbb R^d))$ and
that, conversely, the Besov regularity
$\dot B^{2-2/q}_{q,r}(\mathbb R^d)$
corresponds to the regularity of the trace at $t=0$
of functions $u:\mathbb R_+\times\mathbb R^d\to\mathbb R$
such that $u_t,\nabla^2u\in L_{q,r}(\mathbb R_+;L_p(\mathbb R^d)).$
This motivates us to introduce the following function space:
\begin{equation}\label{eq:W}
\dot{W}^{2,1}_{p,(q,r)}(\mathbb R_+\times \mathbb R^{d}):=\bigl\{u\in \mathcal{C}(\mathbb R_+;\dot{B}^{2-2/q}_{p,r}( \mathbb R^{d})):u_{t}, \nabla^{2}u\in L_{q,r}(\mathbb R_+;L_{p}( \mathbb R^{d})) \bigr\}\cdotp\end{equation}
Back to (INS), in accordance with \eqref{isi}, we need $2-2/q=-1+d/p.$ Furthermore, for reasons that will be explained later on
(in particular the fact $\dot B^{d/p}_{p,r}(\mathbb R^d)$
embeds in $L_\infty(\mathbb R^d)$ if and only if $r=1$),
we shall only consider Besov spaces of type
$\dot B^{d/p-1}_{p,1}(\mathbb R^d).$ \medbreak It is now time to state the main results of the paper. In the two-dimensional case, our global existence result reads: \begin{theorem}\label{themd2} Let $p\in(1,2)$ and $q$ be defined by $1/q+1/p=3/2.$ Denote by $s$ and $m$ the conjugate Lebesgue exponents of $p$ and $q,$ respectively. Assume that the initial divergence-free velocity $u_{0}$ is in $\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2}),$
and that $\rho_{0}$ belongs to $L_{\infty}(\mathbb R^{2})$. There exists a constant $c>0$ such that if \begin{equation}\label{inidr}
\norm{\rho_{0}-1}_{L_{\infty}(\mathbb R^{2})}< c, \end{equation}
then (INS) has a unique global-in-time solution
$(\rho,u,\nabla P)$ satisfying the energy balance~\eqref{eq:energy},
$u\in \dot W^{2,1}_{p,(q,1)}(\mathbb R_+\times\mathbb R^2),$
$\nabla P\in L_{q,1}(\mathbb R_+;L_p(\mathbb R^2)),$
\begin{equation}\label{eq:smallrho}\norm{\rho-1}_{L_{\infty}(\mathbb R_+\times\mathbb R^{2})}
= \norm{\rho_{0}-1}_{L_{\infty}(\mathbb R^{2})}< c,
\end{equation}
and the following properties:
\begin{itemize}
\item $\nabla u\in L_1(\mathbb R_+;L_\infty(\mathbb R^2))$ and $u\in L_2(\mathbb R_+; L_\infty(\mathbb R^2))$;
\item $tu\in L_\infty(\mathbb R_+;\dot B^{1+2/m}_{m,1}(\mathbb R^2))$ and $\bigl(u,(tu)_t, \nabla^2(tu),\nabla(tP)\bigr)
\in L_{s,1}(\mathbb R_+;L_m(\mathbb R^2))$;
\item $t\dot u\in \dot W^{2,1}_{p,(q,1)}(\mathbb R_+\times\mathbb R^2)\,$ and $\,t\dot u\in L_2(\mathbb R_+;L_\infty(\mathbb R^2))$;
\item $t^{\frac k2}\nabla^k u\in L_\infty(\mathbb R_+;L_2(\mathbb R^2))$
and $t^{\frac k2}\nabla^{k+1}u\in L_2(\mathbb R_+\times\mathbb R^2)$
for $k=0,1,2,$
\item $t^{\frac{k+2}2}\nabla^{k}\dot u\in L_\infty(\mathbb R_+;L_2(\mathbb R^2))$ for $k=0,1$ and
$t^{\frac{k+1}2}\nabla^{k}\dot u\in L_2(\mathbb R_+\times\mathbb R^2)$
if $k=0,1,2,$
\item $t^{\frac12}\nabla P\in L_2(\mathbb R_+\times\mathbb R^2)$
and $t\nabla P\in L_\infty(\mathbb R_+; L_2(\mathbb R^2)).$
\end{itemize}
\end{theorem}
In dimension three, our global existence result reads~: \begin{theorem}\label{them1d3} Let $p\in(1,3)$ and $q\in(1,\infty)$ such that $3/p+2/q=3.$ There exist a
positive constant $c$ such that if the initial density is such that \begin{equation}\label{inidr3}
\norm{\rho_{0}-1}_{L_{\infty}(\mathbb R^{3})}< c, \end{equation} and if the initial divergence-free velocity satisfies $$u_{0}\in \dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})\quad (1<p\leq 2)\ \hbox{ or }\ u_{0}\in \dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})\cap L_2(\mathbb R^3)\quad (2<p<3)$$ with \begin{equation}\label{ini1d3} \norm{u_{0}}_{\dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})}<c\mu, \end{equation}
then (INS) has a unique global-in-time solution $(\rho,u,\nabla P)$
with $\nabla P\in L_{q,1}(\mathbb R_+;L_p(\mathbb R^3))$ and
$u\in\dot W_{p,(q,1)}^{2,1}(\mathbb R_+\times\mathbb R^3),$
satisfying the energy balance \eqref{eq:energy} if $p>2$,
\begin{equation}\label{eq:smallrhod3}\norm{\rho-1}_{L_{\infty}(\mathbb R_+\times\mathbb R^{3})}
= \norm{\rho_{0}-1}_{L_{\infty}(\mathbb R^{3})}< c,
\end{equation}
and, furthermore, the following properties:
\begin{itemize}
\item $\nabla u\in L_1(\mathbb R_+;L_\infty(\mathbb R^3))$ and $u\in L_2(\mathbb R_+; L_\infty(\mathbb R^3))$;
\item $(tu)\in W_{m,(s,1)}^{2,1}(\mathbb R_+\times\mathbb R^3)$ and $t\nabla P\in L_{s,1}(\mathbb R_+;L_m(\mathbb R^3))$
for all $3<m<\infty$ and $q<s<\infty$ such that $3/m+2/s=1$;
\item $t\dot u\in\dot W_{p,(q,1)}^{2,1}(\mathbb R_+\times\mathbb R^3)$;
\item $(u, t\dot u)\in L_{s,1}(\mathbb R_+;L_m(\mathbb R^3)).$
\end{itemize}
\end{theorem}
\begin{remark} If $p>2,$ then the (subcritical) assumption $u_0\in L_2(\mathbb R^3)$ ensures the constructed solution to have finite energy. It is only required for proving uniqueness, and it is not needed if $p\leq2.$
At the same time, the priori estimates leading to global existence are performed in critical spaces, and do not require the energy to be finite.
Like in the two-dimensional case, higher order time weighted energy estimates may be proved. However, since they are not needed for getting uniqueness, we refrain from stating them. \end{remark}
The uniqueness part of the above two theorems is a consequence of the following much more general result. \begin{theorem}\label{thm:uniqueness}
Let $(\rho_1,u_1,P_1)$ and $(\rho_2,u_2,P_2)$ be two solutions of $(INS)$ on $[0,T]\times\mathbb R^d$ corresponding to the same initial data. Assume in addition that: \begin{itemize} \item $\sqrt{\rho_1}(u_2-u_1)\in L_\infty(0,T;L_2(\mathbb R^d))$; \item $(\nabla u_2-\nabla u_1)\in L_2(0,T\times\mathbb R^d)$; \item $\nabla u_2\in L_1(0,T;L_\infty(\mathbb R^d))$; \item $t\dot u_2\in L_2(0,T;L_\infty(\mathbb R^d))$; \item Case $d=2$: $\rho_0$ is bounded away from zero and $t\nabla^2\dot u_2\in L_q(0,T;L_p(\mathbb R^2))$ for some $1<p,q<2$ such that $1/p+1/q=3/2,$
\item Case $d=3$: $t\nabla\dot u_2\in L_2(0,T;L_3(\mathbb R^3)).$ \end{itemize} Then, $(\rho_1,u_1,P_1)\equiv (\rho_2,u_2,P_2)$ on $[0,T]\times\mathbb R^d.$ \end{theorem} \begin{remark} Although the density may have large variations (and even vanish in the three-dimensional case), the regularity requirements in the above
uniqueness result are all at the critical level
in the sense of \eqref{isi}.
In the last section of the paper, we shall present
another uniqueness statement in dimension two,
that allows for vacuum, but require
a slightly supercritical regularity assumption. \end{remark}
We shall also see that the Fujita-Kato type global solutions constructed by P. Zhang in \cite{Zhang19} satisfy $\nabla u\in L_1(\mathbb R_+;L_\infty(\mathbb R^3)).$ As a consequence, Theorem \ref{thm:uniqueness} will ensure uniqueness. This leads to the following global well-posedness statement. \begin{theorem}\label{them:PZuniqueness} Let $(\rho_0,u_0)$ satisfy $$0<c_0\leq \rho_0\leq C_0<\infty \quad\hbox{and}\quad u_{0}\in \dot{B}^{{1}/{2}}_{2,1}(\mathbb R^3).$$ Then, there exists a constant $\varepsilon_0>0$ depending only on $c_0,C_0$ such that if \begin{equation}\label{eq:Z1} \norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}(\mathbb R^3)}\leq \varepsilon_0\mu, \end{equation} then System (INS) has a unique global solution $(\rho,u, \nabla P)$ with $u\in \mathcal C(\mathbb R_+;\dot{B}^{1/2}_{2,1}(\mathbb R^3))\cap L_2(\mathbb R_+;\dot{B}^{3/2}_{2,1}(\mathbb R^3))$ which satisfies \begin{equation}\label{eq:Z2}c_0\leq \rho \leq C_0 \ \hbox{ on }\ \mathbb R_+\times \mathbb R^3,\end{equation} and, for some absolute constant $C,$ \begin{multline}\label{eq:PZsolutions}
\norm{u}_{L_{\infty}(\mathbb R_+;\dot{B}^{1/2}_{2,1})}+
\sqrt\mu\,\norm{(u,t\dot{u})}_{ L_{2}(\mathbb R_+;\dot{B}^{3/2}_{2,1})}+\norm{\sqrt{\mu t}\,u}_{ L_{\infty}(\mathbb R_+;\dot{B}^{3/2}_{2,1})}\\+\norm{\sqrt{t}(\mu\nabla u,P )}_{ L_{2}(\mathbb R_+;\dot{B}^{1/2}_{6,1})}
+\mu\norm{\sqrt{t}\,u_t}_{ L_{2}(\mathbb R_+;\dot{B}^{1/2}_{2,1})}\\
+\norm{\sqrt{t}(\mu\nabla^2 u,\nabla P )}_{ L_{2}(\mathbb R_+;L_3)}+\norm{tu_t}_{L_{\infty}(\mathbb R_+;\dot{B}^{1/2}_{2,1})}
\leq C\norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}}. \end{multline} Furthermore, we have $\nabla u$ is in $L_1(\mathbb R_+;L_\infty(\mathbb R^3))$ with $$\mu\norm{\nabla u}_{L_1(\mathbb R_+;L_\infty(\mathbb R^3))}\leq C\norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}(\mathbb R^3)}\cdotp $$ \end{theorem}
Let us shortly present the main ingredients leading to the above statements.
The common starting point for proving the existence part in Theorems \ref{themd2} and \ref{them1d3} is the maximal regularity result in Lorentz spaces stated in Proposition \ref{propregularity}. In fact, in parabolic spaces, the Besov regularity that is required for the initial velocity exactly corresponds to the trace at $t=0$ of functions $u:\mathbb R_+\times\mathbb R^d\to\mathbb R$ such that $u_t$ and $\nabla^2u$ are in $L_{q,1}(\mathbb R_+;L_p(\mathbb R^d)).$ Then, proving estimates for (INS) is based on a perturbation argument from the Stokes system (this is the only place where we need the density to be close to some positive constant). In dimension $d=2,$ the space $L_2(\mathbb R^2)$ turns out to be critical, and one can combine these estimates with the energy balance \eqref{eq:energy} so as to discard any smallness assumption for the velocity.
Since the first part of (INS) is a transport equation, in order to prove the uniqueness, it is essentially mandatory to have at least $\nabla u\in L_{1,loc}(\mathbb R_+;L_\infty(\mathbb R^d)).$ This property will be achieved by combining critical estimates for $u$ and $tu,$ with an interpolation argument involving, again, Lorentz norms for the time variable.
In our setting, it is not clear whether knowing only $\nabla u\in L_{1,loc}(\mathbb R_+;L_\infty(\mathbb R^d))$ is enough to get uniqueness. Here, to conclude, we establish a number of time weighted estimates of energy type (still involving only critical norms). We will in particular get accurate enough information on $\dot u,$ which will spare us going to Lagrangian coordinates. In fact, in contrast with recent works on similar issues (see e.g.\cite{DM1,DM2}) our proof of uniqueness is performed directly on the original system (INS): we estimate the difference of velocities in the energy space and, by means of a duality argument, the difference of densities in $\dot H^{-1}(\mathbb R^d).$ In dimension $d=3,$ we do not need the density to be positive. In the two-dimensional case, the space $\dot H^{1}(\mathbb R^2)$ fails to be embedded in any Lebesgue space, which complicates the proof, unless the density has a positive lower bound. If it is not the case, then one can combine a suitable logarithmic interpolation inequality with Osgood lemma so as to get a uniqueness result in some cases where the density vanishes. However, we have to strengthen slightly our regularity requirement on the velocity (see the end of Section 5). \medbreak The rest of the paper unfolds as follows. The a priori estimates leading to global existence for Theorems \ref{themd2} and \ref{them1d3} are performed in the next two sections.
Section \ref{section4} is devoted to the proof of the global existence. Section \ref{section5} is dedicated to the proof of
various stability estimates and uniqueness statements
that, in particular, imply Theorem \ref{thm:uniqueness} and
the uniqueness part of Theorems \ref{themd2}, \ref{them1d3} and \ref{them:PZuniqueness}.
For reader's convenience, we present in Appendix
the maximal regularity result in Lorentz spaces
of \cite{DM2} adapted to the Stokes system,
recall a few properties of Besov and Lorentz spaces
and prove a critical bilinear estimate with a logarithmic loss
that is needed for uniqueness in dimension $d=2.$
\section{A priori estimates in the 2D case} \label{section2}
This part is devoted to the proof of a priori estimates for (INS) in the 2D case. We shall first establish estimates for $u$ in the critical regularity space $\dot W^{2,1}_{p,(q,1)}(\mathbb R_+\times\mathbb R^2)$ with $1/q+1/p=3/2$ defined in \eqref{eq:W}, which actually suffices to get the global existence of a solution. Then, we will prove time weighted estimates both of energy type and in critical Besov spaces that are needed for uniqueness. The last statement of the section points out higher order time weighted estimates, of independent interest.
\smallbreak
\begin{proposition}\label{prop0d2} Let $(\rho,u)$ be a smooth solution of (INS) on $[0,T]\times\mathbb R^2$ with sufficiently decaying velocity, and density satisfying \begin{equation}\label{eq:smallrho1}\sup_{t\in[0,T]}\norm{\rho(t)-1}_{L_{\infty}( \mathbb R^{2})}\leq c\ll1.\end{equation}
Then, it holds that \begin{equation}\label{eq:L2} \norm{u}^{2}_{L_{\infty}(0,T;L_{2}(\mathbb R^{2}))}+ 2\mu\norm{\nabla u}^{2}_{L_{2}(0,T\times\mathbb R^{2})}\leq \norm{u_{0}}^{2}_{L_{2}(\mathbb R^{2})} \end{equation} and, for all $1<p,q<2$ with $1/p+1/q=3/2,$ \begin{multline}\label{eq:u} \mu^{\frac1p-\frac12}\norm{u}_{L_{\infty}(0,T; \dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2}))}+\norm{u_{t}, \mu\nabla^{2} u,\nabla P}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}+\mu^{\frac12}\norm{u}_{L_{s,1}(0,T;L_{m}(\mathbb R^{2}))} \\ \leq C \mu^{\frac1p-\frac12}\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2})} e^{C\mu^{-2}\norm{u_{0}}^{2}_{L_2(\mathbb R^{2})}},\end{multline} for a constant $C$ independent of $T$ and $\mu,$ with $m$ and $s$ being the conjugate exponents of $q$ and $p,$ respectively. Furthermore, we have
\begin{equation}\label{eq:dotu2}
\norm{\dot{u}}_{L_{q,1}(0,T;L_{p}(\mathbb R^{2}))} \leq C \mu^{\frac1p-\frac12}\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2})}
e^{C\mu^{-2}\norm{u_{0}}_{L_2(\mathbb R^{2})}^2} \end{equation} and
\begin{equation}\label{eq:uLinfty}\mu^{\frac12}\norm{u}_{L_2(0,T;L_{\infty}(\mathbb R^{2}))} \leq C\mu^{\frac1p-\frac12}\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2})}
e^{C\mu^{-2}\norm{u_{0}}^{2}_{L_{2}(\mathbb R^{2})}}\cdotp\end{equation} \end{proposition} \begin{proof} Putting together the energy balance \eqref{eq:energy} and \eqref{eq:smallrho1} clearly ensures \eqref{eq:L2} provided~$c$ has been chosen small enough. \medbreak For proving the other inequalities, note that, thanks to the following rescaling:
\begin{equation}\label{eq:rescaling}
(\widetilde\rho,\widetilde u,\widetilde P)(t,x):= (\rho,\mu^{-1}u, \mu^{-2} P)(\mu^{-1}t,x),\qquad
(\widetilde\rho_0,\widetilde u_0)(x):= (\rho_0,\mu^{-1}u_0)(x),
\end{equation}
one may assume with no loss of generality that $\mu=1.$ \medbreak In order to prove \eqref{eq:u}, let us observe that \begin{equation}\label{s4e1} u_{t}-\Delta u+\nabla P=-(\rho-1)u_{t}-\rho u\cdot \nabla u,\qquad\div u=0. \end{equation}
Looking at \eqref{s4e1} as a Stokes equation with source term, Proposition \ref{propregularity} gives us
\begin{multline}\label{esb2d}
\norm{u}_{L_{\infty}(0,T; \dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2}))}+\norm{u_{t}, \nabla^{2} u,\nabla P}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}+\norm{u}_{L_{s,1}(0,T;L_{m}(\mathbb R^{2}))}\\ \leq C\bigl(\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2})}+\norm{(\rho-1)u_{t}+\rho u\cdot \nabla u}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}\bigr)\cdotp
\end{multline} By H\"older inequality, we have
$$\displaylines{\quad \norm{(\rho-1)u_{t}+\rho u\cdot \nabla u}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}\leq
\norm{\rho-1}_{L_{\infty}(0,T\times\mathbb R^{2})}\norm{u_{t}}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}
\cr
+\norm{\rho}_{L_{\infty}(0,T\times\mathbb R^{2})}\norm{u\cdot \nabla u}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}.\quad} $$ If $c$ is small enough in \eqref{eq:smallrho1}, then the first part in the right-hand side can be absorbed by the left-hand side of \eqref{esb2d}.
For the last term, we have by H\"older inequality, $$\norm{u\cdot \nabla u}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}\leq \norm{u}_{L_{s,1}(0,T;L_{m}(\mathbb R^{2}))}\norm{\nabla u}_{L_{2}(0,T;L_{2}(\mathbb R^{2}))}.$$ Hence, there exists a (small) constant $\alpha>0$ such that if \begin{equation}\label{gub}
\norm{\nabla u}_{L_{2}(0,T;L_{2}(\mathbb R^{2}))}\leq \alpha, \end{equation} then \eqref{esb2d} implies that $$\norm{u}_{L_{\infty}(0,T; \dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2}))}\!+\!\norm{u_{t}, \nabla^{2}\!u,\nabla\! P}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}+\norm{u}_{L_{s,1}(0,T;L_{m}(\mathbb R^{2}))} \!\lesssim\! \norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2})}.$$ If \eqref{gub} is not satisfied then we follow the method used for proving \cite[Theorem 3.1]{DM2} and split $[0,T]$ into a finite number $K$ of intervals $[T_{k-1},T_{k})$ with $T_{0}=0,$ $T_{K}=T$, and $T_1,\cdots,T_{K-1}$ defined by: \begin{equation*}
\begin{array}{cc}
\norm{\nabla u}_{L_{2}((T_{k-1},T_{k})\times \mathbb R^{2})}=\alpha &\hbox{if } 1\leq k\leq K-1; \\[1ex]
\norm{\nabla u}_{L_{2}((T_{k-1},T_{k})\times \mathbb R^{2})}\leq \alpha &\hbox{for } k=K.
\end{array} \end{equation*} For fixed $\alpha,$ we calculate the value of $K$ by $$\begin{aligned} K\alpha^{2}\geq \sum^{K}_{k=1}\norm{\nabla u}^{2}_{L_{2}((T_{k-1},T_{k})\times \mathbb R^{2})}&=\norm{\nabla u}^{2}_{L_{2}(0,T\times \mathbb R^{2})}\\ &>\sum^{K-1}_{k=1}\norm{\nabla u}^{2}_{L_{2}((T_{k-1},T_{k})\times \mathbb R^{2})}=(K-1)\alpha^{2},\end{aligned}$$ which gives \begin{equation}\label{valueK}
K=\lceil \alpha^{-2}\norm{\nabla u}^{2}_{L_{2}(0,T\times \mathbb R^{2})}\rceil. \end{equation} Then, we adapt \eqref{esb2d} to each interval $[T_k,T_{k+1})$ getting $$\displaylines{ \quad \norm{u}_{L_{\infty}(T_{k},T_{k+1}; \dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2}))}+\norm{u_{t}, \nabla^{2} u,\nabla P}_{L_{q,1}(T_{k},T_{k+1};L_{p}( \mathbb R^{2}))}+\norm{u}_{L_{s,1}(T_{k},T_{k+1};L_{m}(\mathbb R^{2}))}
\cr
\leq C \norm{u(T_{k})}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2})}.}$$ Arguing by induction, taking $K$ according to \eqref{valueK} and using \eqref{eq:L2}
so as to bound $\|\nabla u\|_{L_2(0,T\times\mathbb R^2)}$, we conclude that \begin{multline}\label{eq:u2} \norm{u}_{L_{\infty}(0,T; \dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2}))}+\norm{u_{t},\nabla^{2} u,\nabla P}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}+\norm{u}_{L_{s,1}(0,T;L_{m}(\mathbb R^{2}))} \\ \leq C \norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^{2})}\exp(C\norm{u_{0}}^{2}_{L_2(\mathbb R^{2})}).\end{multline}
In order to prove \eqref{eq:dotu2}, it suffices to use the fact that $$\begin{aligned}
\norm{\dot{u}}_{L_{q,1}(0,T;L_{p}(\mathbb R^{2}))} &\leq \norm{u_{t}}_{L_{q,1}(0,T;L_{p}(\mathbb R^{2}))}+\norm{u\cdot \nabla u}_{L_{q,1}(0,T;L_{p}(\mathbb R^{2}))}\\
&\leq \norm{u_{t}}_{L_{q,1}(0,T;L_{p}(\mathbb R^{2}))}+\norm{u}_{L_{s,1}(0,T;L_{m}(\mathbb R^{2}))}\norm{\nabla u}_{L_{2}(0,T;L_{2}(\mathbb R^{2}))}.
\end{aligned}$$
Then, bounding the right-hand side according to \eqref{eq:L2} and \eqref{eq:u2}
yields \eqref{eq:dotu2}.
\smallbreak
Finally, as a consequence of Gagliardo-Nirenberg inequality and embedding, we have:
\begin{equation}\label{eq:lil2esd2}
\norm{z}_{L_{\infty}(\mathbb R^2)}\lesssim \norm{z}^{1-q/2}_{L_{2}(\mathbb R^2)}\norm{\nabla ^{2}z}^{q/2}_{L_{p}(\mathbb R^2)}\lesssim \norm{z}^{1-q/2}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^2)}\norm{\nabla ^{2}z}^{q/2}_{L_{p}(\mathbb R^2)}.
\end{equation}
Hence, using Inequality \eqref{eq:u2}, we find that \begin{equation}\label{esulinfty}
\begin{aligned}
\int_{0}^{T}\norm{u}^{2}_{L_{\infty}(\mathbb R^2)}\,dt &\leq C\int_{0}^{T}\norm{u}^{2-q}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^2)}\norm{\nabla^{2}u}^{q}_{L_{p}(\mathbb R^2)}\,dt\\
&\leq C\norm{u}^{2-q}_{L_{\infty}(0,T;\dot{B}^{-1+2/p}_{p,1}(\mathbb R^2))}\norm{\nabla^{2}u}^{q}_{L_{q}(0,T;L_{p}(\mathbb R^2))} \\
&\leq C\norm{u_{0}}^{2}_{\dot{B}^{-1+2/p}_{p,1}(\mathbb R^2)}\exp{(C\norm{u_{0}}^{2}_{L_{2}(\mathbb R^2)})}. \end{aligned}\end{equation}
This completes the proof of the proposition.\end{proof} \medbreak {}For better readability, we drop from now on $\mathbb R^2$ in the norms. \begin{proposition}\label{prop1} Under the assumptions of Proposition \ref{prop0d2}, we have $$\displaylines{
\mu\norm{tu}_{L_{\infty}(0,T;\dot{B}^{2-2/s}_{m,1})}+\mu^{\frac1s}\norm{(tu)_{t}, \mu\nabla^{2}(tu),\nabla (tP)}_{L_{s,1}(0,T;L_{m})}
+\mu^{\frac1s}\norm{t\dot u}_{L_{s,1}(0,T;L_{m})}
\cr
\leq C\mathcal E_0\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}}\quad\hbox{with}\quad \mathcal E_0:=\exp\Bigl(C\mu^{-s}\norm{u_{0}}^{s}_{\dot{B}^{-1+2/p}_{p,1}} e^{C\mu^{-2}\norm{u_{0}}^{2}_{L_2}}\Bigr)
\cdotp}$$ \end{proposition} \begin{proof} Again, we use the rescaling \eqref{eq:rescaling} to reduce the proof to the case $\mu=1.$ Now, multiplying both sides of \eqref{s4e1} by time $t$ yields $$(tu)_{t}-\Delta (tu)+\nabla(tP)=-(\rho-1)(tu)_{t}+\rho u-\rho u\cdot \nabla tu,\qquad \div(tu)=0.$$ Then, taking advantage of of Proposition \ref{propregularity} with Lebesgue indices $m$ and $s$ gives $$\displaylines{\quad \norm{tu}_{L_{\infty}(0,T;\dot{B}^{2-2/s}_{m,1})}+\norm{(tu)_{t}, \nabla^{2}(tu),\nabla (tP)}_{L_{s,1}(0,T;L_{m})}
\cr
\leq\norm{\rho-1}_{L_{\infty}(0,T\times\mathbb R^{2})}\norm{(tu)_{t}}_{L_{s,1}(0,T;L_{m})}
\cr
+\norm{\rho}_{L_{\infty}(0,T\times\mathbb R^{2})} \left (\norm{u}_{L_{s,1}(0,T;L_{m})}+\norm{tu\cdot \nabla u}_{L_{s,1}(0,T;L_{m})}\right)\cdotp}$$ Owing to \eqref{eq:smallrho1}, the second line may be absorbed by the first one. Next, as $2/m=1-2/s,$ combining H\"older inequality and the following embedding: \begin{equation}\label{eq:embed} \dot B^{2/m}_{m,1}(\mathbb R^2)\hookrightarrow L_\infty(\mathbb R^2) \end{equation} yields \begin{equation*}
\begin{aligned}
\norm{tu\cdot \nabla u}_{L_{s,1}(0,T;L_{m})}&\leq \norm{t \nabla u}_{L_{\infty}(0,T\times \mathbb R^{2})} \norm{u}_{L_{s,1}(0,T;L_{m})}\\
&\lesssim \norm{t u}_{L_{\infty}(0,T;\dot{B}^{2-2/s}_{m,1})} \norm{u}_{L_{s,1}(0,T;L_{m})}.
\end{aligned} \end{equation*} Hence, there exists a (small) positive constant $\beta$ such that, if $$\norm{u}_{L_{s,1}(0,T;L_{m})}\leq \beta,$$ then we have \begin{equation}\label{estud2}
\norm{tu}_{L_{\infty}(0,T;\dot{B}^{2-2/s}_{m,1})}+\norm{(tu)_{t}, \nabla^{2}(tu),\nabla (tP)}_{L_{s,1}(0,T;L_{m})} \leq C\norm{u}_{L_{s,1}(0,T;L_{m})}. \end{equation} If $\norm{u}_{L_{s,1}(0,T;L_{m})}>\beta,$ then one can argue as in the proof of the previous proposition: there exists a finite sequence $0=T_{0}<T_{1}<\cdots<T_{K-1}<T_{K}=T$ such that \begin{equation}\label{ulsmtk}
\begin{array}{cc}
\norm{\nabla u}_{L_{s,1}((T_{k-1},T_{k});L_m)}=\beta &\hbox{if } 1\leq k\leq K-1; \\[1ex]
\norm{\nabla u}_{L_{s,1}((T_{k-1},T_{k};L_m))}\leq \beta &\hbox{for } k=K.
\end{array} \end{equation}
Indeed, from Remark \ref{lorentzdef2}, we have $$\norm{U(t)}_{L_{s,1}(0,T)}=s\int_{0}^{\infty} \abs{\{t\in(0,T): \abs{U(t)}>\lambda\}}^{1/s}\,d\lambda\quad\hbox{with}\quad U(t):=\norm{u(t,\cdot)}_{L_{m}}$$ which, together with Lebesgue dominated theorem gives $$ \int_{0}^{\infty} \abs{\{t\in (T_1,T_2): \abs{U(t)}>\lambda\}}^{1/s}\,d\lambda\to 0\quad \text{as}\quad T_2-T_1\to 0,$$ which allows to construct a family $(T_k)_{0\leq k\leq K}$ satisfying \eqref{ulsmtk}. Now, by H\"older inequality (with exponents $s$ and $p$) we have for all $\lambda>0,$ $$\begin{aligned}\sum_{k=1}^K \abs{\{t\in (T_{k-1},T_k): \abs{U(t)}>\lambda\}}^{1/s} &\leq K^{1/p}\biggl(\sum_{k=1}^K \abs{\{t\in (T_{k-1},T_k): \abs{U(t)}>\lambda\}}\biggr)^{1/s}\\ &= K^{1/p}\abs{\{t\in (0,T): \abs{U(t)}>\lambda\}}^{1/s}. \end{aligned}$$ Hence, integrating with respect to $\lambda$ and using \eqref{ulsmtk} yields $K\lesssim \beta^{-s}\norm{u}_{L_{s,1}(0,T;L_{m})}^s.$
Arguing by induction, we thus obtain
$$
\norm{tu}_{L_{\infty}(0,T;\dot{B}^{2-2/s}_{m,1})}+\norm{(tu)_{t}, \nabla^{2}(tu),\nabla (tP)}_{L_{s,1}(0,T;L_{m})}
\leq C\norm{u}_{L_{s,1}(0,T;L_{m})}e^{C\norm{u}_{L_{s,1}(0,T;L_{m})}^s}\cdotp$$
In the end, using the first estimate of Proposition \ref{prop0d2}, one may conclude that \begin{multline}\label{eq:uu} \norm{tu}_{L_{\infty}(0,T;\dot{B}^{2-2/s}_{m,1})}+\norm{(tu)_{t}, \nabla^{2}(tu),\nabla (tP)}_{L_{s,1}(0,T;L_{m})}\\ \leq C \norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}}\: \exp\Bigl(C\norm{u_{0}}^{s}_{\dot{B}^{-1+2/p}_{p,1}} \exp\bigl(C\norm{u_{0}}^{2}_{L_2}\bigr)\Bigr)\cdotp\end{multline} To bound $t\dot u,$ we just have to observe that $t\dot u= (tu)_t - u + tu\cdot\nabla u.$ Hence, by H\"older inequality and \eqref{eq:embed}, we get $$\norm{t\dot{u}}_{L_{s,1}(0,T;L_{m})}\leq \norm{(tu)_{t}}_{L_{s,1}(0,T;L_{m})}+\norm{u}_{L_{s,1}(0,T;L_{m})}+ \norm{tu}_{L_\infty(0,T; \dot B^{2-2/s}_{m,1})} \norm{u}_{L_{s,1}(0,T;L_{m})}.$$ At this stage, using Inequalities \eqref{eq:u} and \eqref{eq:uu} gives the desired result. \end{proof} \begin{corollary}\label{coro1d2} With the notation of Proposition \ref{prop1}, we have: \begin{align}\label{eq:tnablau1} \mu\int_{0}^{T} \norm{\nabla u}_{L_{\infty}}\,dt&\leq C \norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}}\,\mathcal E_0\, e^{C\mu^{-2}\norm{u_{0}}^{2}_{L_2}},\\ \label{eq:tnablau2}
\mu\biggl(\int_{0}^T t\norm{\nabla u}^{2}_{L_{\infty}}\,dt\biggr)^{1/2}&\leq C\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}}\,\mathcal E_0\, e^{C\mu^{-2}\norm{u_{0}}^{2}_{L_2}}, \\ \label{estulilid2}\underset{t\in[0,T]}
\sup (\mu t)^{1/2} \|u(t)\|_{L_\infty}&\leq C\|u_0\|_{L_2}^{1/2} \norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}}^{1/2} \,\mathcal E_0.\end{align} \end{corollary} \begin{proof} Just consider the case $\mu=1.$
From the following Gagliardo-Nirenberg inequality $$\norm{z}_{L_{\infty}}\lesssim \norm{ \nabla z}^{1-2/m}_{L_{p}}\norm{\nabla z}^{2/m}_{L_{m}},$$ and H\"older estimates in Lorentz spaces (see Proposition \ref{p:lorentz}), we gather that \begin{equation*} \begin{aligned} \int^{T}_{0}\norm{\nabla u}_{L_{\infty}}\,dt&\lesssim \int^{T}_{0} t^{-2/m} \norm{ \nabla^{2} u}^{1-2/m}_{L_{p}}\norm{t\nabla^{2} u}^{2/m}_{L_{m}}\,dt\\ &\lesssim\norm{t^{-2/m}}_{L_{m/2,\infty}(0,T)}\norm{\nabla^{2} u}^{1-2/m}_{L_{q,1}(0,T;L_{p})}\norm{t \nabla^{2} u}^{2/m}_{L_{s,1}(0,T;L_{m})}.\end{aligned} \end{equation*} As $t\mapsto t^{-2/m}\in L_{m/2,\infty}(\mathbb R_+)$ and the other terms of the right-hand side may be bounded by means of Propositions \ref{prop0d2} and \ref{prop1}, we get \eqref{eq:tnablau1}. Next, by virtue of \eqref{eq:embed}, we have \begin{equation*} \begin{aligned} \int_{0}^{T}t\norm{\nabla u}^{2}_{L_{\infty}}\,dt& \leq \int_{0}^{T} t\norm{\nabla u}_{\dot{B}^{2/m}_{m,1}}\norm{\nabla u}_{L_{\infty}}\,dt\\ &\lesssim \int_{0}^{T} \norm{tu}_{\dot{B}^{2-2/s}_{m,1}}\norm{\nabla u}_{L_{\infty}}\,dt\\ &\lesssim\norm{tu}_{L_{\infty}(0,T;\dot{B}^{2-2/s}_{m,1})} \norm{\nabla u}_{L_{1}(0,T;L_{\infty})}, \end{aligned} \end{equation*} whence the second inequality. \medbreak
Finally, by interpolation, we have for all $t\in[0,T],$
$$t^{1/2} \|u(t)\|_{L_\infty}\lesssim \|u(t)\|_{L_2}^{1/2}\|t u(t)\|_{\dot B^{1+2/m}_{m,1}}^{1/2}$$ which, in light of \eqref{eq:L2} and of Proposition \ref{prop1} completes the proof. \end{proof} \bigbreak The rest of this section is devoted to establishing supplementary time weighted estimates of energy type that will be needed to prove the uniqueness of solutions of (INS). For expository purpose, we shall always assume that $\mu=1.$ \begin{proposition}\label{prop2} Under the assumptions of Proposition \ref{prop0d2}, we have for all $t\in [0, T],$ $$\displaylines{\quad t\int_{\mathbb R^{2}}\abs{\nabla u(t)}^{2}\,dx+\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau \bigl(\rho \abs{\dot{u}}^{2} +\abs{\nabla^{2}u}^{2}+\abs{\nabla P}^{2}\bigr)\,dx \,d\tau
\cr
\leq C\norm{u_{0}}^{2}_{L_{2}}\, \exp\left(C\norm{u_{0}}_{L_{2}}\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}} \mathcal E_0^2\right)\cdotp}$$ \end{proposition}
\begin{proof} Let us rewrite the velocity equation as:
\begin{equation}\label{edu1}
\rho \dot{u}=\Delta u -\nabla P \quad\hbox{with}\quad \dot{u}:= u_t+u\cdot\nabla u. \end{equation} As $\div u=0,$ testing \eqref{edu1} by $t\dot{u}$ yields $$
\int_{\mathbb R^2}\rho t|\dot{u}|^2\,dx=t\int_{\mathbb R^2}\Delta u\cdot u_t\,dx -t\int_{\mathbb R^2}\nabla P\cdot u_t\,dx + t\int_{\mathbb R^2}\bigl(\Delta u-\nabla P)\cdot(u\cdot\nabla u)\,dx$$ whence, integrating by parts and using again \eqref{edu1},
$$\frac{1}{2}\frac{d}{dt}\int_{\mathbb R^{2}}t\abs{\nabla u}^{2}\,dx+\int_{\mathbb R^{2}} \rho t\abs{\dot{u}}^{2}\,dx=\int_{\mathbb R^{2}}\rho t\dot{u}\cdot(u\cdot \nabla u)\,dx + \frac12\int_{\mathbb R^2}|\nabla u|^2\,dx.$$ Performing a time integration, we get for all $0\leq t\leq T,$ $$\frac{t}{2}\int_{\mathbb R^{2}}\!\abs{\nabla u(t)}^{2}dx+\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau \rho \abs{\dot{u}}^{2}dx\,d\tau=\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\!\! \tau \rho \dot{u}\cdot(u\cdot \nabla u)\,dx\, d\tau+\frac{1}{2}\int_{0}^{t} \!\!\int_{\mathbb R^{2}} \abs{\nabla u(\tau)}^{2}dx d\tau.$$ To bound the right-hand side, we use the fact that
$$ \begin{aligned}
\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\! \tau \rho \dot{u}\cdot
(u\cdot \nabla u)\,dx\,d\tau&\leq \int_{0}^{t} \norm{\sqrt{\rho\tau}\dot{u}}_{L_{2}}\norm{\sqrt{\rho\tau}\,u\cdot \nabla u}_{L_{2}}\,d\tau\\
&\leq \frac12 \int_{0}^{t} \norm{\sqrt{\rho \tau}\dot{u}}^{2}_{L_{2}}\,d\tau
+\frac{\|\rho_0\|_{L_\infty}}{2}\int_{0}^{t}\! \norm{u}^{2}_{L_{\infty}}\norm{\tau^{1/2}\nabla u}^{2}_{L_{2}}\,d\tau.
\end{aligned}$$
Observe that, thanks to \eqref{edu1}, we have for some constant $C$ depending only on $\|\rho_0\|_{L_\infty},$ \begin{equation}\label{ed2}\norm{\nabla^{2}u}_{L_{2}}^2+\norm{\nabla P}_{L_{2}}^2\leq C\norm{\sqrt{\rho}\dot{u}}_{L_{2}}^2.\end{equation} Hence, applying Gronwall lemma yields some constant $C$
depending only on $\|\rho_0\|_{L_\infty},$ and such that $$\displaylines{\quad t\int_{\mathbb R^{2}}\abs{\nabla u(t)}^{2}\,dx+\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau \rho \abs{\dot{u}}^{2}\,dx\, d\tau +\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau\bigl(\abs{\nabla^{2}u}^{2}+\abs{\nabla P}^{2}\bigr)dx\, d\tau
\cr
\leq C \int_0^t\norm{\nabla u}^{2}_{L_{2}}\exp \biggl(C\int_{\tau}^{t} \norm{u}^{2}_{L_{\infty}}\,d\tau'\biggr)d\tau.}$$ Putting together with \eqref{eq:L2} and \eqref{esulinfty} completes the proof of the proposition.
\end{proof}
\begin{proposition}\label{prop3} Under the assumptions of Proposition \ref{prop0d2}, there exists a constant $C_0$ depending only on $p$ and on $\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}},$ such that for all $t\in [0,T],$ $$\int_{\mathbb R^{2}} t^{2}\Bigl(\rho\bigl(\abs{u_t}^{2}+\abs{\dot{u}}^{2}\bigr)+ \abs{\nabla^{2} u}^{2}+\abs{\nabla P}^{2}\Bigr)dx +\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau^{2} \bigl(\abs{\nabla u}^{2}+\abs{\nabla \dot{u}}^{2}\bigr)dx\,d\tau\leq C_0.$$ \end{proposition} \begin{proof} From \eqref{eq:embed}, the definition of $\dot{u}$ and H\"older inequality, one can write \begin{equation*}
\begin{aligned}
\norm{t\dot{u}- t{u}_t}_{L_{\infty}(0,T;L_{2})}
&\leq \norm{t \nabla u}_{L_{\infty}(0,T\times\mathbb R^2)}\norm{u}_{L_{\infty}(0,T;L_{2})}\\
&\leq C\norm{t u}_{L_{\infty}(0,T;\dot{B}^{1+2/m}_{m,1})}\norm{u}_{L_{\infty}(0,T;L_{2})}
\end{aligned} \end{equation*} and \begin{equation*}
\begin{aligned}
\norm{t\nabla \dot{u}-t\nabla u_t&}_{L_{2}(0,T\times\mathbb R^2)}
\leq \norm{t\nabla u\otimes \nabla u}_{L_{2}(0,T\times\mathbb R^2)}+\norm{t u\otimes \nabla^{2} u}_{L_{2}(0,T\times\mathbb R^2)}\\
&\leq \norm{t\nabla u}_{L_{\infty}(0,T\times\mathbb R^2)}\norm{\nabla u}_{L_{2}(0,T\times\mathbb R^2)}+\norm{t\nabla^{2} u}_{L_{\infty}(0,T;L_{2})}\norm{u}_{L_{2}(0,T;L_{\infty})}\\
&\leq C\norm{t u}_{L_{\infty}(0,T;\dot{B}^{1+2/m}_{m,1})}\norm{\nabla u}_{L_{2}(0,T\times\mathbb R^2)}+\norm{t\nabla^{2} u}_{L_{\infty}(0,T;L_{2})}\norm{u}_{L_{2}(0,T;L_{\infty})}.
\end{aligned} \end{equation*} Furthermore, \eqref{ed2} implies that \begin{equation}\label{ed2b}\norm{t\nabla^{2}u}_{L_{2}}^2+\norm{t\nabla P}_{L_{2}}^2\leq C\norm{t\sqrt{\rho}\dot{u}}_{L_{2}}^2. \end{equation} Hence, to complete the proof, it is only a matter of showing that $$\norm{t{u}_{t}}_{L_\infty(0,T;L_2)} +\norm{t\nabla {u}_{t}}_{L_{2}(0,T\times \mathbb R^2)}\leq C_0.$$ To do so, apply $\partial_{t}$ to the momentum equation of (INS). We get \begin{equation}\label{eq:utt}
\rho u_{tt}+\rho u\cdot \nabla u_{t}-\Delta u_{t}+\nabla P_{t}=-\rho_{t}\dot{u}-\rho u_{t}\cdot \nabla u. \end{equation} As $\div u_t=0,$ by taking the $L_{2}(\mathbb R^2;\mathbb R^2)$ scalar product of \eqref{eq:utt} with $t^{2}u_t,$ we obtain $$\displaylines{ \quad \frac{1}{2}\frac{d}{dt}\int_{\mathbb R^{2}}\rho t^{2}\abs{u_{t}}^{2}\,dx+\int_{\mathbb R^{2}}t^{2}\abs{\nabla u_{t}}^{2}\,dx
\cr
\leq \int_{\mathbb R^{2}} t\rho\abs{u_{t}}^{2}\,dx-\int_{\mathbb R^{2}}t^{2} \rho_{t}\dot{u}\cdot u_{t}\,dx-\int_{\mathbb R^{2}}t^{2}\rho (u_{t}\cdot \nabla u)\cdot u_{t}\,dx.}$$ Then, integrating with respect to time yields for all $t\in[0,T],$ \begin{multline}\label{eq:II2} \frac12\underset{\tau\leq t}{\sup}\norm{t\sqrt{\rho} u_{t}}^{2}_{L_{2}}+\norm{t\nabla u_{t}}^{2}_{L_{2}(0,t\times\mathbb R^2} \leq \int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau\rho\abs{u_{\tau}}^{2}\,dx\,d\tau\\-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} \rho_{\tau}\dot{u}\cdot u_{\tau}\,dx\,d\tau-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2}\rho (u_{\tau}\cdot \nabla u)\cdot u_{\tau}\,dx\,d\tau=:I_{1}+I_{2}+I_{3}.\end{multline} For term $I_{2},$ the mass equation of (INS) and integration by parts yield \begin{equation*}
\begin{aligned}
I_{2}
&=\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} \div(\rho u)\,\dot{u}\cdot u_{\tau}\,dx\,d\tau\\
&=-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} (\rho u\cdot \nabla{\dot{u}}) \cdot u_{\tau}\,dx\,d\tau
-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} (\rho u\cdot \nabla{u_{\tau})\cdot \dot{u} }\,dx\,d\tau\\
&=:I_{21}+I_{22}.
\end{aligned} \end{equation*}
Since $\dot{u}=u_{t}+u\cdot \nabla u$, we may write \begin{equation*}
\begin{aligned}
I_{21}=&-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} (\rho u\cdot \nabla{u}_{\tau}) \cdot u_{\tau}\,dx\,d\tau
-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} (\rho u\cdot \nabla(u\cdot \nabla u)) \cdot u_{\tau}\,dx\,d\tau\\
=&-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} (\rho u\cdot \nabla u_{\tau}) \cdot u_{\tau}\,dx\,d\tau
-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} \bigl(\rho u\cdot (\nabla^{2}u\cdot u) \cdot u_{\tau}\bigr)dx\,d\tau\\
&\hspace{6cm}-\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2} \bigl(\rho u\cdot
(\nabla u\cdot\nabla u)\bigr) \cdot u_{\tau}\,dx\,d\tau
\end{aligned} \end{equation*}
and
$$ I_{22}=-\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau^{2}(\rho u\cdot \nabla u_{\tau})\cdot u_{\tau}\,dx\,d\tau
-\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau^{2}(\rho u\cdot \nabla u_\tau)\cdot ( u\cdot \nabla u)\,dx\,d\tau.$$ Applying Young's inequality and remembering that $\rho$ is bounded gives for all $\varepsilon>0,$
\begin{equation*}
\begin{aligned}
I_{21} &\lesssim \int_{0}^{t} \norm{u}_{L_{\infty}}\norm{\tau \nabla u_{\tau}}_{L_{2}}
\norm{\tau\sqrt{\rho} u_{\tau}}_{L_{2}}\,d\tau\\
&\qquad\qquad+\int_{0}^{t}\tau\norm{u}^{2}_{L_{\infty}}\norm{\nabla ^{2} u}_{L_{2}}\norm{\tau \sqrt{\rho} u_{\tau}}_{L_{2}}\,d\tau\\
&\qquad\qquad+\int_{0}^{t} \norm{\tau^{1/2}\nabla u}_{L_{2}}\norm{\tau^{1/2}\nabla u}_{L_{\infty}} \norm{\tau\sqrt{\rho}u_{\tau}}_{L_{2}} \norm{ u}_{L_{\infty}}\,d\tau\\
&\leq C\varepsilon^{-1} \int_{0}^{t} \norm{u}^{2}_{L_{\infty}}
\norm{\tau \sqrt{\rho} u_{\tau}}^{2}_{L_{2}}\,d\tau+C\int_{0}^{t}\norm{\tau^{1/2}u}^{2}_{L_{\infty}}\norm{\tau^{1/2}\nabla ^{2} u}^{2}_{L_{2}}\,d\tau\\
&\qquad\qquad+C\int_{0}^{t} \norm{\tau^{1/2}\nabla u}^{2}_{L_{2}}
\norm{\tau^{1/2}\nabla u}^{2}_{L_{\infty}}\,d\tau+\varepsilon \int_{0}^{t}\norm{\tau \nabla u_{\tau}}^{2}_{L_{2}}\,d\tau,
\end{aligned} \end{equation*} and \begin{equation*}
\begin{aligned}
I_{22} &\lesssim \int_{0}^{t} \norm{\tau \sqrt{\rho} u_{\tau}}_{L_{2}}\norm{\tau \nabla u_{\tau}}_{L_{2}}\norm{u}_{L_{\infty}} \,d\tau
+\int_{0}^{t}\tau\norm{\tau \nabla u_{\tau}}_{L_{2}}\norm{u}^{2}_{L_{\infty}}\norm{\nabla u}_{L_{2}}\,d\tau\\
&\leq C\varepsilon^{-1}\biggl(\int_{0}^{t} \norm{\tau \sqrt{\rho} u_{\tau}}^{2}_{L_{2}}\norm{u}^{2}_{L_{\infty}}\,d\tau\\
&\qquad\qquad\qquad+\int_{0}^{t}\norm{u}^{2}_{L_{\infty}}\norm{\tau^{1/2}u}^{2}_{L_{\infty}}\norm{\tau^{1/2}\nabla u}^{2}_{L_{2}} \,d\tau\biggr) +\varepsilon \int_{0}^{t}\norm{\tau \nabla u_{\tau}}^{2}_{L_{2}}\,d\tau.
\end{aligned} \end{equation*} For $I_{3}$, one has $$ I_{3}=-\int_{0}^{t}\int_{\mathbb R^{2}}\tau ^{2}(\rho u_{\tau}\cdot \nabla u)\cdot u_{\tau}\,dx\,d\tau
\leq \int_{0}^{t} \norm{\tau\sqrt{\rho} u_{\tau}}^{2}_{L_{2}}\norm{\nabla u}_{L_{\infty}}\,d\tau.$$
Taking $\varepsilon$ small enough, then
reverting to \eqref{eq:II2} and applying Gronwall inequality gives $$\displaylines{ \quad \underset{\tau\leq t}{\sup}\norm{t\sqrt{\rho}u_{t}}^{2}_{L_{2}}+\int_0^t\norm{t\nabla u_{t}}^{2}_{L_{2}}\,d\tau \leq C\exp\biggl(\int_{0}^{t} \norm{u}^{2}_{L_{\infty}}+\norm{\nabla u}_{L_{\infty}}\,d\tau\biggr)
\cr
\biggl(\int_{0}^{t} \norm{\tau^{1/2}\nabla u}^{2}_{L_{2}}
\norm{\tau^{1/2}\nabla u}^{2}_{L_{\infty}}\,d\tau+\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau\rho\abs{u_{\tau}}^{2}\,dx\,d\tau
\cr
+\int_{0}^{t}\!\norm{u}^{2}_{L_{\infty}}\norm{\tau^{1/2}u}^{2}_{L_{\infty}}\norm{\tau^{1/2}\nabla u}^{2}_{L_{2}} \,d\tau +\int_{0}^{t}\!\norm{\tau^{1/2}u}^{2}_{L_{\infty}}\norm{\tau^{1/2}\nabla ^{2} u}^{2}_{L_{2}}\,d\tau\biggr)\cdotp}$$ Combining with Propositions \ref{prop1} and \ref{prop2}, Inequality \eqref{esulinfty} and Corollary \ref{coro1d2} allows to bound the right-hand side by $C_0$
for all $t\in[0,T],$ and using also \eqref{ed2b} completes the proof. \end{proof}
In order to get a higher order time weighted estimate, one has
to consider the evolutionary equation for $\dot u.$ So we take the convective derivative of \eqref{edu1}, getting
$$\frac{D}{Dt} (\rho \dot{u})-\frac{D}{Dt}\Delta u+\frac{D}{Dt}\nabla P=0.$$ Observe that
$$\begin{aligned}-\frac{D}{Dt} \Delta u&=-\Delta \dot{u}+\Delta u \cdot \nabla u+2\nabla u \cdot \nabla^{2} u
\quad\hbox{with}\quad(\nabla u \cdot \nabla^{2} u)^{i}:=\underset{1\leq j,k\leq d}{\displaystyle \sum} \partial_{k}u^{j} \,\partial_{j}\partial_{k} u^{i},\\
\frac{D}{Dt}\nabla P&=\nabla \dot{P}-\nabla u\cdot \nabla P,\\
\frac{D}{Dt}(\rho \dot{u})&=\rho\ddot u\quad\hbox{with}\quad
\ddot u:=\frac D{Dt}\dot u.\end{aligned}$$
Hence, we have
\begin{equation}\label{doubleD}
\rho \ddot u-\Delta \dot{u}+\nabla \dot{P}=
f\quad\hbox{with}\quad f:=-\Delta u\cdot \nabla u-2\nabla u \cdot \nabla^{2} u+\nabla u\cdot \nabla P.\end{equation}
\begin{proposition}\label{prop5}
Under the assumptions of Proposition \ref{prop1}, it holds that
$$\norm{t\dot{u}}_{L_{\infty}(0,T; \dot{B}^{-1+2/p}_{p,1}(\mathbb R^2))}+\norm{(t\dot{u})_{t}, t\nabla^{2} \dot{u}}_{L_{q,1}(0,T;L_{p}( \mathbb R^{2}))}+\norm{t\dot u}_{L_2(0,T;L_\infty(\mathbb R^2))}\leq C_{0}.$$
\end{proposition}
\begin{proof} From \eqref{doubleD}, we get
the following equation for $t\dot{u}$: \begin{equation}\label{eq:ttu}\rho(t\dot{u})_t-\Delta(t\dot{u})+\nabla(t\dot{P}) = -t\rho u\cdot\nabla\dot{u}+\rho\dot{u} + tf.\end{equation} Since $\div \dot u \not=0,$ one cannot apply directly Proposition \ref{propregularity}. Now, let us introduce the Helmholtz projectors on divergence free and gradient like vector-fields, namely, \begin{equation}\label{eq:PQ}\mathbb P:={\rm Id}+\nabla (-\Delta)^{-1}\div\quad\hbox{and}\quad \mathbb Q:=-\nabla (-\Delta)^{-1}\div \end{equation} We observe that $$\nabla(t\dot{P})=\mathbb Q\Bigl( -t\rho u\cdot\nabla\dot{u}+\rho\dot{u} + tf - \rho(t\dot{u})_t+\Delta(t\dot{u})\Bigr)\cdotp$$ Hence, reverting to \eqref{eq:ttu} implies that \begin{equation}\label{eq:nnnn} \rho(t\dot{u})_t-\Delta(t\dot{u}) =\mathbb P\bigl(\rho\dot u+tf -t\rho u\cdot\nabla\dot u\bigr) +\mathbb Q\bigl(\rho(t\dot{u})_t-\Delta(t\dot{u}) \bigr). \end{equation} Using the fact that $\div u=0,$ we easily get
\begin{equation}\label{divtu}\div \dot{u}=\underset{1\leq i,j\leq d}{\sum}\partial_{i} u^{j}\partial_{j} u^{i}=
{\rm Tr}(\nabla u \cdot \nabla u),\end{equation} whence $$\mathbb Q(t\Delta\dot u) = t\nabla{\rm Tr} (\nabla u\cdot\nabla u)$$ and since $$\mathbb Q((\rho(t\dot u)_t) =\mathbb Q\bigl( (\rho-1)(t\dot u)_t +\dot u + tu\cdot\nabla u_t +tu_t\cdot\nabla u \bigr),$$ we get in the end, \begin{multline}\label{eqttu}(t\dot{u})_{t}-\Delta t \dot{u}
=\mathbb P[ (1-\rho)(t\dot{u})_{t}-t\rho u\cdot \nabla \dot{u}+\rho\dot u+tf]\\
+\mathbb Q(\dot u+ t u_{t}\cdot\nabla u+t u\cdot \nabla u_{t})- \nabla{\rm Tr} (t\nabla u\cdot\nabla u).\end{multline} At this point, we use the maximal regularity estimate \emph{for the heat equation} stated in \cite[Prop. 2.1]{DM2} as well as the continuity of $\mathbb P$ and $\mathbb Q$ on $L_p$ to conclude that $$\displaylines{
\norm{t\dot{u}}_{L_{\infty}(0,T;\dot{B}^{-1+2/p}_{p,1})}+\norm{(t\dot{u})_{t}, \nabla^{2} t\dot{u}}_{L_{q,1}(0,T;L_{p})}
\cr
\lesssim \norm{(1-\rho)(t\dot{u})_{t}-t\rho u\cdot \nabla \dot{u}+\rho\dot{u}+tf}_{L_{q,1}(0,T;L_{p})}
\cr
+\norm{\dot u + t u_{t}\cdot\nabla u+t u\cdot \nabla u_{t}}_{L_{q,1}(0,T;L_{p})} +\norm{t\nabla u\otimes \nabla^2u}_{L_{q,1}(0,T;L_{p})}.}$$ As usual, owing to \eqref{eq:smallrho}, the first term in the right-hand side may be absorbed by the left-hand side. Now, using \eqref{eq:embed} and the definition of $f$ in \eqref{doubleD}, we get
\begin{equation*}
\begin{aligned}
\norm{t f}_{L_{q,1}(0,T;L_{p})} &\leq C\norm{t\nabla u}_{L_{\infty}(0,T\times\mathbb R^2)}
(\norm{ \nabla^{2}u}_{L_{q,1}(0,T;L_{p}))}+\norm{ \nabla P}_{L_{q,1}(0,T;L_{p})})\\
&\leq C \norm{tu}_{L_{\infty}(0,T;\dot{B}^{1+2/m}_{m,1})}
(\norm{ \nabla^{2}u}_{L_{q,1}(0,T;L_{p})}+\norm{ \nabla P}_{L_{q,1}(0,T;L_{p})}).
\end{aligned} \end{equation*} Next, $\norm{\dot{u}}_{L_{q,1}(0,T;L_{p})}$ may be bounded according to
Inequality \eqref{eq:dotu2}.
Finally, we have $$\begin{aligned}\norm{ t \rho u\cdot \nabla \dot{u}}_{L_{q,1}(0,T;L_{p})}&\leq C \norm{ t \nabla \dot{u}}_{L_{2}(0,T\times \mathbb R^{2})}\norm{u}_{L_{s,1}(0,T;L_{m})},\\ \norm{ t u\cdot \nabla u_{t}}_{L_{q,1}(0,T;L_{p})}&\leq C \norm{t\nabla u_{t}}_{L_{2}(0,T\times\mathbb R^{2})}\norm{u}_{L_{s,1}(0,T;L_{m})},\\ \norm{t u_{t}\cdot \nabla u}_{L_{q,1}(0,T;L_{p})}&\leq C \norm{tu_{t}}_{L_{s,1}(0,T;L_{m})}\norm{\nabla u}_{L_{2}(0,T\times\mathbb R^{2})},\\
\norm{t\nabla u\otimes\nabla^2 u}_{L_{q,1}(0,T;L_{p})}&\leq C\norm{t \nabla u}_{L_{\infty}(0,T\times \mathbb R^{2})}\norm{\nabla^{2} u}_{L_{q,1}(0,T;L_{p})}\\
&\leq C\norm{t u}_{L_{\infty}(0,T;\dot{B}^{1+2/m}_{m,1})}\norm{\nabla^{2} u}_{L_{q,1}(0,T;L_{p})}.
\end{aligned}$$ Then, putting all together with Proposition \ref{prop0d2}, Inequality \eqref{eq:dotu2}, Proposition \ref{prop1} and Proposition \ref{prop3}, we discover that $$\norm{t\dot{u}}_{L_{\infty}(0,T; \dot{B}^{-1+2/p}_{p,1})}+\norm{(t\dot{u})_{t}, t\nabla^{2} \dot{u}}_{L_{q,1}(0,T;L_{p})}\leq C_0\cdotp$$
Finally, Inequality \eqref{eq:lil2esd2} enables us to conclude that
$$\|t\dot u\|_{L_2(0,T;L_\infty)}\leq \norm{t\dot u}^{\frac{2-q}{2}}_{L_{\infty}(0,T;\dot{B}^{-1+2/p}_{p,1})}\norm{t\nabla^{2}\dot u}^{\frac 2q}_{L_{q,1}(0,T;L_{p})}\leq C_0,$$ which completes the proof.
\end{proof}
\medbreak
We end this section by stating higher order
energy type time weighted estimates (that are not required
for proving the uniqueness).
\begin{proposition}\label{prop4}
Under the assumptions of Proposition \ref{prop1}, we have for all $t\in[0,T],$
$$\underset{\tau\in[0,t]}\sup\norm{\tau^{3/2}\nabla \dot{u}}^{2}_{L_{2}}+ \int_0^t\norm{\tau^{3/2}\nabla^{2} \dot{u},t^{3/2}\nabla \dot{P},t^{3/2}\sqrt{\rho}\ddot u}^{2}_{L_{2}}\,d\tau\leq C_0
$$ where $C_0$ depends only on $p$ and on $\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}}.$
\end{proposition}
\begin{proof}
Taking the $L_{2}(\mathbb R^2;\mathbb R^2)$ inner product of \eqref{doubleD} with $t^{3}\ddot u$ then integrating on $[0,t]$ yields
\begin{multline}\label{eq:nabladotu}
\frac{t^{3}}{2}\int_{\mathbb R^{2}}\abs{\nabla \dot{u}}^{2}\,dx+\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau^{3} \rho \abs{\ddot u}^{2}\,dx\,d\tau
= \int_{0}^{t}\!\!\int_{\mathbb R^{2}}\frac{3\tau^{2}}{2}\abs{\nabla \dot{u}}^{2}\,dx\,d\tau\\+\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\Delta \dot{u}\cdot \tau^{3} u\cdot \nabla \dot{u}\,dx\,d\tau-\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \nabla \dot{P} \cdot \bigl(\tau^{3}u\cdot \nabla \dot{u}\bigr)\,dx\,d\tau\\
\qquad+\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \nabla \dot{P} \cdot \bigl(\tau^{3}u_{t}\cdot \nabla u\bigr)\,dx\,d\tau
+\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \nabla \dot{P} \cdot \bigl(\tau^{3}u\cdot \nabla u_t\bigr)\,dx\,d\tau\\
+\int_{0}^{t}\int_{\mathbb R^{2}}f\cdot \tau^{3} \ddot u\,dx\,d\tau=:\underset{1\leq k\leq 6}{J_{k}}.
\end{multline}
In order to bound $J_{2},J_{3},J_{4},J_{5},$ we proceed as follows:
\begin{equation*}
\begin{aligned}
J_{2}&=\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\Delta \dot{u}\cdot \tau^{3} u\cdot \nabla \dot{u}\,dx\,d\tau\\
&\leq \norm{\tau^{3/2} \nabla^{2} \dot{u}}_{L_{2}(0,t\times\mathbb R^2)}\norm{\tau \nabla \dot{u}}_{L_{2}(0,t\times\mathbb R^2)}\norm{\tau^{1/2}u}_{L_{\infty}(0,t\times\mathbb R^{2})},
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
J_{3}&=-\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \nabla \dot{P} \cdot
(\tau^{3}u\cdot \nabla \dot{u})\,dx\,d\tau\\
&\leq \norm{\tau^{3/2} \nabla \dot{P}}_{L_{2}(0,t\times\mathbb R^2)}\norm{\tau \nabla \dot{u}}_{L_{2}(0,t\times\mathbb R^2)}
\norm{\tau^{1/2}u}_{L_{\infty}(0,T\times\mathbb R^{2})},
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
J_{4}&=\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \nabla \dot{P} \cdot (\tau^{3}u_{\tau}\cdot \nabla u)\,dx\,d\tau\\
&\leq \norm{\tau^{3/2} \nabla \dot{P}}_{L_2(0,t\times\mathbb R^2)}\norm{\tau u_{\tau}}_{L_{\infty}(0,t;L_{2})}\norm{\tau^{1/2}\nabla u}_{L_{2}(0,t;L_{\infty})},
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
J_{5}&=\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \nabla \dot{P} \cdot \bigl(\tau^{3}u\cdot \nabla u_{t}\bigr)dx\,d\tau\\
&\leq \norm{\tau^{3/2} \nabla \dot{P}}_{L_2(0,t\times\mathbb R^2)} \norm{\tau \nabla u_{t}}_{L_{2}(0,t\times\mathbb R^2)}
\norm{\tau^{1/2}u}_{L_{\infty}(0,t\times\mathbb R^{2})}.
\end{aligned}
\end{equation*}
At this point, we have to explain how to bound $t^{3/2}\nabla^{2} \dot{u}$ and
$t^{3/2}\nabla \dot{P}$ in $L_2(0,T\times\mathbb R^2).$
Observe that \eqref{doubleD} and \eqref{divtu}
ensure that
\begin{equation}\label{eq:dotP} \nabla \dot{P}=\mathbb Q f+\mathbb Q\bigl(\rho \ddot u\bigr)+\nabla {\rm Tr}(\nabla u \cdot\nabla u). \end{equation} Hence, owing to the continuity of $\mathbb Q$ on $L_2,$
we have for all $t\in[0,T],$
$$\|t^{3/2}\nabla \dot{P}(t)\|_{L_2} \lesssim \|\sqrt\rho\, t^{3/2}\ddot u(t)\|_{L_2}+\|t^{3/2}(\nabla u\otimes\nabla^2u)(t)\|_{L_2}
+\|t^{3/2}f(t)\|_{L_2}.$$
Hence, since
$$t^{3/2}\Delta\dot{u} = t^{3/2}\nabla\dot{P} + \rho t^{3/2}\ddot u + t^{3/2}\Delta u\cdot \nabla u + 2t^{3/2}\nabla u\cdot\nabla^2u
-t^{3/2}\nabla u\cdot\nabla P,$$ we easily get
$$\begin{aligned}
\norm{t^{3/2}\nabla^{2} \dot{u},t^{3/2}\nabla \dot{P}}_{L_{2}(0,t\times\mathbb R^2)}
&\lesssim\norm{t^{3/2}\sqrt{\rho}\ddot u}_{L_{2}(0,t\times\mathbb R^2)}
+\norm{t^{3/2}\nabla u\otimes \nabla^{2}u}_{L_2(0,t\times\mathbb R^2)}\\
&\hspace{4cm}+\norm{t^{3/2}\nabla u\cdot \nabla P}_{L_{2}(0,t\times\mathbb R^2)}\\
&\lesssim\norm{t^{3/2}\sqrt{\rho}\ddot u}_{L_{2}(0,t\times\mathbb R^2)}\\
&\quad\qquad+\norm{t\nabla^{2} u, t\nabla P}_{L_{\infty}(0,t;L_{2})}\norm{t^{1/2}\nabla u}_{L_2(0,t;L_{\infty})}.
\end{aligned}$$
Thanks to Corollary \ref{coro1d2} and Proposition \ref{prop3}, we thus end up with
\begin{equation}\label{tn2tues}
\norm{t^{3/2}\nabla^{2} \dot{u},t^{3/2}\nabla \dot{P}}_{L_{2}(0,t\times\mathbb R^2)}\lesssim\norm{t^{3/2}\sqrt{\rho}\ddot u}_{L_{2}(0,t\times\mathbb R^2)}+C_{0}. \end{equation} Reverting to the above inequalities for $J_2$ to $J_5$ and taking advantage of Corollary \ref{coro1d2}, Proposition \ref{prop2} and Proposition \ref{prop3}, we conclude that there exists some constant $C_0$ depending only on $p$ and on $\norm{u_{0}}_{\dot{B}^{-1+2/p}_{p,1}},$ and such that
\begin{eqnarray}\label{esj2345}
\sum_{k=2}^5J_{k} &\!\!\!\leq\!\!\!& C_0 \left(\norm{t^{3/2}\sqrt{\rho}\,\ddot u}_{L_{2}(0,t\times\mathbb R^2)}+C_{0}\right)\nonumber\\
&\!\!\!\leq\!\!\!& \frac{1}{4}\norm{t^{3/2}\sqrt{\rho}\,\ddot u}^{2}_{L_{2}(0,t\times\mathbb R^2)}+2C_{0}^2.
\end{eqnarray} For $J_{6},$ we write that \begin{equation*}
\begin{aligned}
J_{6}&=\int_{0}^{t}\!\!\int_{\mathbb R^{2}}f \cdot \tau^{3} \ddot u\,dx\,d\tau\\
&=\int_{0}^{t}\!\!\int_{\mathbb R^{2}}
(-\Delta u\cdot \nabla u-2\nabla u \cdot \nabla^{2} u+\nabla u\cdot \nabla P)\cdot \tau^{3} \ddot u\,dx\,d\tau\\
&\lesssim\norm{\tau^{3/2} \ddot u}_{L_{2}(0,t\times\mathbb R^2)}
\norm{\tau^{1/2} \nabla u}_{L_2(0,t;L_{\infty})}\bigl(\norm{\tau \nabla^{2}u}_{L_{\infty}(0,t;L_{2})}
+\norm{\tau \nabla P}_{L_{\infty}(0,t;L_{2})}\bigr),
\end{aligned} \end{equation*} which along with Proposition \ref{prop3}, Corollary \ref{coro1d2} and \eqref{eq:smallrho} gives $$J_{6}\leq C_0\norm{\tau^{3/2} \sqrt\rho\ddot u}_{L_{2}(0,t\times\mathbb R^2)} \leq \frac14 \norm{\tau^{3/2} \sqrt\rho \ddot u}^{2}_{L_{2}(0,t\times\mathbb R^2)} +2C_{0}^2.$$ Inserting the above inequality and \eqref{esj2345} in \eqref{eq:nabladotu}, we get
$$t^{3}\int_{\mathbb R^{2}}\abs{\nabla \dot{u}}^{2}\,dx+\int_{0}^{t}\!\!\int_{\mathbb R^{2}} \tau^{3} \rho \abs{\ddot u}^{2}\,dx\,d\tau
\leq 3\int_{0}^{t}\!\!\int_{\mathbb R^{2}}\tau^{2}\abs{\nabla \dot{u}}^{2}\,dxd\tau+C_{0}$$
which, by virtue of Proposition \ref{prop3}, completes the proof.
\end{proof}
\section{Estimates in the three-dimensional case \label{section3} }
Here we establish the inequalities
that are needed to prove Theorem \ref{them1d3}.
The first two propositions are required for proving the existence
of a global solution, while the last one is needed for uniqueness.
\begin{proposition}\label{propcsd3}
Let $(\rho,u)$ be a smooth solution of $(INS)$ on $[0,T]\times\mathbb R^3,$
with $u$ sufficiently decaying at infinity and $\rho$ such that
\begin{equation}\label{eq:smallrho3}
\sup_{t\in[0,T]}\|\rho(t)-1\|_{L_\infty(\mathbb R^3)}\leq c\ll1.
\end{equation}
Then, for all indices $1<m,p,q,s<\infty$ satisfying
\begin{equation}\label{eq:relation}
\frac 3p+\frac2q=3\!\quad\hbox{and}\quad\!\frac3m+\frac2s=1,\quad\hbox{with}\quad
p<m<\infty\quad\hbox{and}\quad q<s<\infty,\end{equation}
the following
inequalities hold true:
\begin{multline}\label{eq:u3d1} \mu^{\frac3{2p}-\frac12}\norm{u}_{L_{\infty}(0,T; \dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3}))} +\mu^{\frac3{2p}-\frac12+\frac1s}\norm{ u}_{L_{s,1}(0,T;L_{m}(\mathbb R^{3}))}\\+\norm{\dot{u}, u_{t}, \mu\nabla^{2} u,\nabla P}_{L_{q,1}(0,T;L_{p}( \mathbb R^{3}))}\leq C\mu^{\frac3{2p}-\frac12} \norm{u_{0}}_{\dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})},\end{multline} \begin{equation}\label{eq:u3d2} \quad\hbox{and}\quad\mu^{\frac12}\norm{u}_{L_2(0,T;L_{\infty}(\mathbb R^{3}))}\leq C \norm{u_{0}}_{\dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})}\cdotp\qquad\qquad\end{equation}
\end{proposition}
\begin{proof}
For notational simplicity, we omit to specify the dependence of the norms with respect to $\mathbb R^3$ in the proof. As usual, we only consider
the case $\mu=1.$
Now, applying Proposition \ref{propregularity} to System \eqref{s4e1} yields \begin{multline}\label{esb3d}
\norm{u}_{L_{\infty}(0,T;\dot{B}^{-1+3/p}_{p,1})}+\norm{u_{t}, \nabla^{2} u,\nabla P}_{L_{q,1}(0,T;L_{p})}+\|u\|_{L_{s,1}(0,T;L_{m})}
\\ \leq C\bigl(\norm{u_{0}}_{\dot{B}^{-1+3/p}_{p,1}}+\norm{(\rho-1)u_{t} +\rho u\cdot \nabla u}_{L_{q,1}(0,T;L_{p})}\bigr)\cdotp \end{multline} By H\"older inequality, we have $$\displaylines{ \norm{(\rho-1)u_{t}+\rho u\cdot \nabla u}_{L_{q,1}(0,T;L_{p})}
\cr
\leq
\norm{\rho-1}_{L_{\infty}(0,T\times\mathbb R^{3})}\norm{u_{t}}_{L_{q,1}(0,T;L_{p})} +\norm{\rho}_{L_{\infty}(0,T\times\mathbb R^{3})}\norm{u\cdot \nabla u}_{L_{q,1}(0,T;L_{p})}.}$$ Owing to \eqref{eq:smallrho3}, the first term can be absorbed by the left-hand side of \eqref{esb3d}.
For term $\norm{u\cdot \nabla u}_{L_{q,1}(0,T;L_{p}( \mathbb R^{3}))}$, by embedding
\begin{equation}\label{eq:embedL3}
\dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})\hookrightarrow L_{3}(\mathbb R^{3})\end{equation} and
\begin{equation}\label{eq:W1p}
\dot{W}^{1}_{p}(\mathbb R^{3})\hookrightarrow L_{p^{*}}(\mathbb R^{3}) \quad\hbox{with}\quad \frac{1}{p^{*}}=\frac{1}{p}-\frac{1}{3},\end{equation} we obtain \begin{equation*} \begin{aligned} \norm{u\cdot \nabla u}_{L_{q,1}(0,T;L_{p})}&\leq \norm{u}_{L_{\infty}(0,T;L_{3})}\norm{\nabla u}_{L_{q,1}(0,T;L_{p^{*}})}\\ &\lesssim \norm{u}_{L_{\infty}(0,T;\dot{B}^{-1+3/p}_{p,1})}\norm{\nabla^{2} u}_{L_{q,1}(0,T;L_{p})}. \end{aligned} \end{equation*}
Denoting $\Phi_0:= \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}$ and $$\Phi:=\norm{u}_{L_{\infty}(0,T; \dot{B}^{-1+3/p}_{p,1})}+\norm{u_{t}, \nabla^{2} u,\nabla P}_{L_{q,1}(0,T;L_{p})}+
\|u\|_{L_{s,1}(0,T;L_{m})},$$
we can conclude that $$\Phi\leq C(\Phi_{0}+\Phi^{2}).$$ Hence, if \begin{equation}\label{ini2d3}
4C\Phi_{0} < 1, \end{equation} then one can assert that \begin{equation}\label{eq:Phi} \Phi\leq 2 \Phi_{0}.\end{equation} Clearly, $\dot{u}$ satisfies the same inequality since $\Phi$ is small and, by H\"older inequality, \begin{equation*} \norm{\dot{u}}_{L_{q,1}(0,T;L_{p})} \leq \norm{u_{t}}_{L_{q,1}(0,T;L_{p})}+
\norm{u}_{L_{\infty}(0,T;\dot{B}^{-1+3/p}_{p,1})}\norm{\nabla^{2} u}_{L_{q,1}(0,T;L_{p})}\leq C\Phi(1+\Phi). \end{equation*} Finally, as a consequence of Gagliardo-Nirenberg inequality and embedding, we have:
\begin{equation}\label{eq:lil2esd3}
\norm{z}_{L_{\infty}}\lesssim \norm{z}^{1-q/2}_{L_{3}}\norm{\nabla ^{2}z}^{q/2}_{L_{p}}\lesssim \norm{z}^{1-q/2}_{\dot{B}^{-1+3/p}_{p,1}}\norm{\nabla ^{2}z}^{q/2}_{L_{p}},
\end{equation} whence \begin{equation}\label{esulinfty3d}
\begin{aligned}
\int_{0}^{T}\norm{u}^{2}_{L_{\infty}}\,dt &\leq C\int_{0}^{T}\norm{u}^{2-q}_{\dot{B}^{-1+3/p}_{p,1}}\norm{\nabla^{2}u}^{q}_{L_{p}}\,d\tau\\
&\leq C\norm{u}^{2-q}_{L_{\infty}(0,T;\dot{B}^{-1+3/p}_{p,1})}\norm{\nabla^{2}u}^{q}_{L_{q,1}(0,T;L_{p})}\\
&\leq C \Phi^{2}\cdotp
\end{aligned} \end{equation} Owing to \eqref{eq:Phi}, this yields \eqref{eq:u3d2}. \end{proof}
\begin{proposition}\label{prop1d3} Under the assumptions Proposition \ref{propcsd3}, we have $$\mu\norm{tu}_{L_{\infty}(0,T;\dot{B}^{1+3/m}_{m,1}(\mathbb R^{3}))}
+\mu^{\frac1s}\norm{(tu)_{t}, \mu\nabla^{2}(tu),\nabla(tP)}_{L_{s,1}(0,T;L_{m}(\mathbb R^{3}))}\leq C \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}(\mathbb R^3)}.$$ Moreover, the following inequalities hold true:
$$\mu\!\int_{0}^{T}\!\! \norm{\nabla u}_{L_{\infty}(\mathbb R^{3})}\,dt\leq C \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})}\!\quad\hbox{and}\quad\! \mu\!\int_{0}^{T} \!\!t\norm{\nabla u}^{2}_{L_{\infty}(\mathbb R^{3})}\,dt\leq C \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})}^2.$$ \end{proposition}
\begin{proof} Assume that $\mu=1.$ Multiplying both sides of \eqref{s4e1} by time $t$ yields $$(tu)_{t}-\Delta (tu)+\nabla(tP)=(1-\rho)(tu)_{t}+\rho u- \rho u\cdot \nabla tu.$$ Then, taking advantage of Proposition \ref{propregularity}, we get: $$\displaylines{\quad \norm{tu}_{L_{\infty}(0,T;\dot{B}^{1+3/m}_{m,1})}+\norm{(tu)_{t}, \nabla^{2}(tu),\nabla(tP)}_{L_{s,1}(0,T;L_{m})}
\cr
\lesssim\norm{\rho-1}_{L_{\infty}(0,T\times\mathbb R^{3})}\norm{(tu)_{t}}_{L_{s,1}(0,T;L_{m})}
\cr
+\norm{\rho}_{L_{\infty}(0,T\times\mathbb R^{3})} \left (\norm{u}_{L_{s,1}(0,T;L_{m})}+\norm{tu\cdot \nabla u}_{L_{s,1}(0,T;L_{m})}\right)\cdotp}$$ Owing to \eqref{eq:smallrho3}, the first term of the right-hand side may be bounded by the left-hand side, and we deduce from H\"older inequality and the embedding \begin{equation}\label{eq:embed3}\dot{B}^{3/m}_{m,1}(\mathbb R^{3})\hookrightarrow L_{\infty}(\mathbb R^{3})\end{equation} that \begin{equation*} \begin{aligned}
\norm{tu\cdot \nabla u}_{L_{s,1}(0,T;L_{m})}
&\leq \norm{ u}_{L_{s,1}(0,T;L_{m})} \norm{t\nabla u}_{L_{\infty}(0,T\times\mathbb R^{3})}\\
& \leq C\norm{ u}_{L_{s,1}(0,T;L_{m})}\norm{tu}_{L_{\infty}(0,T;\dot{B}^{1+3/m}_{m,1})}. \end{aligned} \end{equation*} Remember that Proposition \ref{propcsd3} allows to bound $u$ in $L_{s,1}(0,T;L_{m}(\mathbb R^{3}))$ by $\Phi_0.$ Hence, setting $$\Pi:=\norm{tu}_{L_{\infty}(0,T;\dot{B}^{1+3/m}_{m,1})} +\norm{(tu)_{t},\mu \nabla^{2}(tu),\nabla(tP)}_{L_{s,1}(0,T;L_{m})},$$ the above calculations imply that
$\Pi\leq C(1+\Pi)\Phi_0$ and, as $ \Phi_0$ is small, this completes the proof of the first part of the proposition.
\medbreak
Bounding $\nabla u$ relies on the following interpolation inequality
(as \eqref{eq:relation} implies that $p<3<m$): $$\norm{u}_{L_{\infty}(\mathbb R^{3})}\leq \norm{\nabla u}^{\frac{p(m-3)}{3(m-p)}}_{L_{p}(\mathbb R^{3})}\norm{\nabla u}^{\frac{m(3-p)}{3(m-p)}}_{L_{m}(\mathbb R^{3})}.$$ Hence, applying H\"older inequality in Lorentz spaces with exponents: $$(p_1,r_1)=\biggl(\frac{3(m-p)}{m(3-p)},\infty\biggr),\!\quad (p_2,r_2)=\biggl(\frac{3q(m-p)}{p(m-3)},\frac{p_2}q\biggr),\!\quad (p_3,r_3)=\biggl(\frac{3s(m-p)}{m(3-p)},\frac{p_3}s\biggr),$$
using the fact that $t^{-\alpha}$ with $\alpha=m(3-p)/(3(m-p))$
is in $L_{1/\alpha,\infty}(\mathbb R_+),$
\eqref{eq:u3d1} and the first inequality of Proposition \ref{prop1d3},
we end up with \begin{equation*}
\begin{aligned}
\int_{0}^{T}\norm{\nabla u}_{L_{\infty}}\,dt
&\leq \int_{0}^{T} t^{-\frac{m(3-p)}{3(m-p)}} \norm{\nabla^{2} u}^{\frac{p(m-3)}{3(m-p)}}_{L_{p}}\norm{t\nabla^{2} u}^{\frac{m(3-p)}{3(m-p)}}_{L_{m}}\,dt\\
&\leq C \norm{\nabla^{2} u}^{\frac{p(m-3)}{3(m-p)}}_{L_{q,1}(0,T;L_{p})}\norm{t\nabla^{2} u}^{\frac{m(3-p)}{3(m-p)}}_{L_{s,1}(0,T;L_{m})}\\
&\leq C \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}.
\end{aligned} \end{equation*} Furthermore, we deduce from \eqref{eq:embed3} that
$$ \begin{aligned}
\int_{0}^{T} t\norm{\nabla u}^{2}_{L_{\infty}}\,dt
&\leq \int_{0}^{T}t\norm{\nabla u}_{\dot{B}^{3/m}_{m,1}}\norm{\nabla u}_{L_{\infty}}\,dt\\
&\leq \norm{tu}_{L_{\infty}(0,T;\dot{B}^{1-3/m}_{m,1})}\int_{0}^{T}\norm{\nabla u}_{L_{\infty}}\,dt\\&\leq
C \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}^{2},
\end{aligned}$$
by virtue of the inequality we proved just before. \end{proof}
To prove the uniqueness, the following time weighted estimate is required.
\begin{proposition}\label{prop2d3}
Under the assumptions of Proposition \ref{propcsd3}, it holds that
\begin{multline}\label{eq:prop33}\mu^{\frac3{2p}-\frac12}\norm{t\dot{u}}_{L_{\infty}(0,T; \dot{B}^{-1+3/p}_{p,1})}+\norm{(t\dot{u})_{t}, \mu t\nabla^{2} \dot{u}}_{L_{q,1}(0,T;L_{p})}\\
+\mu^{\frac3{2p}-\frac12+\frac1s}\norm{ t\dot{u}}_{L_{s,1}(0,T;L_{m})}\leq C\mu^{\frac3{2p}-\frac12}\|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}.\end{multline}
\medbreak Furthermore, we have \begin{equation}\label{eq:coro2d3}
\mu^{\frac12}\norm{t\nabla \dot{u}}_{L_{2}(0,T;L_{3}(\mathbb R^{3}))}+
\mu^{\frac12}\norm{t\dot{u}}_{L_{2}(0,T;L_{\infty}(\mathbb R^{3}))}\leq C \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}(\mathbb R^{3})}. \end{equation}
\end{proposition}
\begin{proof}
We know that $t\dot{u}$ satisfies \eqref{eqttu} and we also observe, owing to $\div u=\div u_t=0,$ that
$$\mathbb Q(u\cdot\nabla u_t)=\mathbb Q(u_t\cdot\nabla u).$$
Hence, using the maximal regularity estimates
in Lorentz spaces for the heat equation
(cf \cite[Prop. 2.1]{DM2}) and the continuity of the Helmholtz projectors on ${L_{q,1}(0,T;L_{p})},$ we
get
$$\displaylines{\norm{t\dot{u}}_{L_{\infty}(0,T; \dot{B}^{-1+2/p}_{p,1})}+\norm{(t\dot{u})_{t}, t\nabla^{2}\dot{u}}_{L_{q,1}(0,T;L_{p})}+\norm{ t\dot{u}}_{L_{s,1}(0,T;L_{m})}\lesssim
\|(1-\rho)(t\dot{u})_{t}\|_{L_{q,1}(0,T;L_{p})}
\cr
+\|t\rho u\cdot \nabla \dot{u}\|_{L_{q,1}(0,T;L_{p})}
+\|\rho\dot u\|_{L_{q,1}(0,T;L_{p})}+\|tf\|_{L_{q,1}(0,T;L_{p})}
\cr
+\|\dot u+ t u_{t}\cdot\nabla u\|_{L_{q,1}(0,T;L_{p})}
+ \|t\nabla{\rm Tr} (\nabla u\cdot\nabla u)\|_{L_{q,1}(0,T;L_{p})}.}$$
Owing to \eqref{eq:smallrho3}, the first term in the right-hand side may be absorbed by the left-hand side,
and Proposition \ref{propcsd3} allows to bound $\norm{\dot{u}}_{L_{q,1}(0,T;L_{p})}.$
Also recall that $$f=-\Delta u\cdot \nabla u-2\nabla u \cdot \nabla^{2} u+\nabla u\cdot \nabla P.$$
Hence, thanks to \eqref{eq:embed3} and to Propositions \ref{propcsd3},
\ref{prop1d3},
$$ \begin{aligned}
\norm{t f}_{L_{q,1}(0,T;L_{p})}\!+\!\|t\nabla{\rm Tr} (\nabla u\cdot\nabla u)\|_{L_{q,1}(0,T;L_{p})} &\lesssim \norm{t\nabla u}_{L_{\infty}(0,T\times \mathbb R^{3})} \norm{ \nabla^{2}u,\nabla P}_{L_{q,1}(0,T;L_{p})}\\
&\lesssim \norm{tu}_{L_{\infty}(0,T;\dot{B}^{1+3/m}_{m,1})}\norm{ \nabla^{2}u,\nabla P}_{L_{q,1}(0,T;L_{p})}\\
&\lesssim \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}^2.
\end{aligned}$$
Using the H\"older inequality in Lorentz spaces, the embeddings \eqref{eq:W1p} and
\eqref{eq:embedL3}, and Propositions \ref{propcsd3}, \ref{prop1d3}, we obtain \begin{equation*}
\begin{aligned}
\norm{ t \rho u\cdot \nabla \dot{u}}_{L_{q,1}(0,T;L_{p})}&\lesssim\norm{ t \nabla \dot{u}}_{L_{q,1}(0,T;L_{p^{*}})}\norm{u}_{L_{\infty}(0,T;L_{3})}\\
&\lesssim\norm{ t \nabla^{2} \dot{u}}_{L_{q,1}(0,T;L_{p})} \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}},
\end{aligned} \end{equation*} \begin{equation*}
\begin{aligned}
\norm{t u_{t}\cdot \nabla u}_{L_{q,1}(0,T;L_{p})}&\leq \norm{u_{t}}_{L_{q,1}(0,T;L_{p})}\norm{t\nabla u}_{L_{\infty}(0,T\times\mathbb R^{3})}\\
&\lesssim\norm{u_{t}}_{L_{q,1}(0,T;L_{p})}\norm{t u}_{L_{\infty}(0,T;\dot{B}^{1+3/m}_{m,1})}\\
&\lesssim \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}^2.
\end{aligned} \end{equation*} Putting the above inequalities together, we conclude that
$$\displaylines{\norm{t\dot{u}}_{L_{\infty}(0,T; \dot{B}^{-1+3/p}_{p,1})}+\norm{(t\dot{u})_{t}, t\nabla^{2}\dot{u}}_{L_{q,1}(0,T;L_{p})}+\norm{ t\dot{u}}_{L_{s,1}(0,T;L_{m})}
\cr
\lesssim \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}^2
+\bigl(1+ \norm{ t \nabla^{2} \dot{u}}_{L_{q,1}(0,T;L_{p})}\bigr) \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}.}$$
Since $ \|u_0\|_{\dot{B}^{-1+3/p}_{p,1}}$ is small, we have
\eqref{eq:prop33}.
\medbreak
In order to prove Inequality \eqref{eq:coro2d3},
let us first consider the case $3/2<p<3$ (which implies that $1<q<2$). Then, Inequality \eqref{eq:lil2esd3} ensures that $$\norm{\dot u}_{L_{\infty}}\leq C\norm{\dot u}^{1-\frac{q}{2}}_{L_{3}}\norm{\nabla^{2}\dot u}^{\frac{q}{2}}_{L_p}\leq C\norm{\dot u}^{1-\frac{q}{2}}_{\dot{B}^{-1+3/p}_{p,1}}\norm{\nabla^{2}\dot u}^{\frac{q}{2}}_{L_{p}}.$$ Consequently, $$\norm{t\dot{u}}_{L_{2}(0,T;L_{\infty})}\leq C\norm{t\dot{u}}^{1-\frac{q}{2}}_{L_{\infty}(0,T;\dot{B}^{-1+3/p}_{p,1})}\norm{\nabla^{2}(t\dot{u})}^{\frac{q}{2}}_{L_{q}(0,T;L_p)},$$ Then, applying \eqref{eq:prop33} gives the second part of \eqref{eq:coro2d3}. \medbreak In order to complete the proof of \eqref{eq:coro2d3}, it suffices to apply Proposition \ref{prop:for existence} with $r=3$ to $t\dot{u}$ (keeping in mind that $-1+3/p=2-2/q$) then H\"older inequality with respect to the time variable. In the end, as $p\in(3/2,3),$ we get $$\norm{t\nabla \dot{u}}_{L_{2}(0,T;L_{3})}\lesssim \norm{t\dot{u}}_{L_{\infty}(0,T;\dot{B}^{-1+3/p}_{p,1})}^\theta \norm{t\nabla^{2}\dot{u}}_{L_{q}(0,T;L_p)}^{1-\theta} \quad\hbox{with}\quad \theta=\frac{2p-3}{3p-3}\cdotp$$ Then, applying the first part of the proposition gives the desired result. \smallbreak The case $1<p\leq 3/2$ reduces to the case we treated before since $\dot{B}^{-1+\frac{3}{p}}_{p,1}\hookrightarrow \dot{B}^{-1+\frac{3}{p_{1}}}_{p_{1},1}$ for some $p_1\in(3/2,3).$ \end{proof}
\section{Existence} \label{section4} This section is devoted to the proof of existence of a global solution under our assumptions (both in dimensions $2$ and $3$).
As a first step, we shall smooth out the data so as to apply prior results ensuring the existence of a sequence $(a^n,u^n,\nabla P^n)_{n\in\mathbb N}$ of strong (relatively) smooth solutions to \eqref{rins2}. The estimates of Sections \ref{section2} and \ref{section3} will guarantee that the solution $(a^n,u^n,\nabla P^n)_{n\in\mathbb N}$ is global and uniformly bounded in the expected spaces. In order to pass to the limit, we shall take advantage of compactness arguments. A technical point is that Lorentz spaces $L_{q,1}$ are \emph{nonreflexive}, so that one cannot directly use the classical results, like Aubin-Lions' lemma. To overcome the difficulty, we shall look at the approximate solutions in the slightly larger (but reflexive)
space
$$\dot{W}^{2,1}_{p,r}(\mathbb R_+ \times \mathbb R^d):=
\bigl\{u\in\mathcal C_b(\mathbb R_+;\dot B^{2-2/r}_{p,r}(\mathbb R^d)\,:\,
u_t,\nabla^2u\in L_r(\mathbb R_+;L_p(\mathbb R^d))\bigr\}
$$ for some $ 1<r<\infty,$
then check afterward that the constructed solution has the desired
regularity.
\medbreak
As a first, let us smooth out the initial
data $a_0$ and $u_0$ by means of non-negative mollifiers, to get a
sequence $(a_0^n, u_0^n)_{n\in{\mathbb N}}$ of smooth data such that
\begin{equation}\label{eq:unifbound} \norm{a^n_{0}}_{L_\infty}\leq \norm{a_{0}}_{L_\infty},\quad \norm{u^n_{0}}_{\dot{B}^{-1+d/p}_{p,1}}\leq C \norm{u_{0}}_{\dot{B}^{-1+d/p}_{p,1}} \end{equation} with, in addition, $$a^{n}_{0}\rightharpoonup a_0 \quad \text{weak * in}\quad L_\infty \quad\hbox{and}\quad u_{0}^n \to u_{0} \quad \text{strongly in}\quad \dot{B}^{-1+d/p}_{p,1}. $$
According to e.g. \cite{DR2004}, there exists $T>0$ such that System \eqref{rins2} supplemented with initial data $(a_0^n, u_0^n)$ admits a unique smooth local solution $(a^n,u^n,\nabla P^n)$ on $[0,T]\times\mathbb R^d.$ In particular, the energy balance is satisfied (in the cases where $u_0$ is in $L_2(\mathbb R^d)$), $a^n\in\mathcal C_b([0,T]\times\mathbb R^d)$ and $(u^n,\nabla P^n)$ is in the space
$$E^{p,r}_{T}=\bigl\{(u,\nabla P)\quad\hbox{with}\quad u\in \dot{W}^{2,1}_{p,r}(0,T\times \mathbb R^d) \quad\hbox{and}\quad \nabla P \in L_{r}(0,T;L_p)\bigr\}\quad\hbox{for all }\ r\geq1.$$ Let us denote by $T^n$ the maximal time of existence of $(a^n,u^n,\nabla P^n).$
Since the calculations of the previous sections just follow from the properties of the heat flow and transport equation, basic functional analysis and integration
by parts, each $(a^{n},u^{n},\nabla P^n)$ satisfies the estimates
therein \emph{up to time $T^n$}, and thus
\begin{equation}\label{eq:anbound}
\norm{a^{n}(t)}_{L_\infty}= \norm{a^{n}_0}_{L_\infty} \leq \norm{a_0}_{L_\infty} \quad \text{for all} \quad t\in [0,T^n)
\end{equation} and
\begin{equation}\label{eq:unlorentz}
\norm{u^n}_{\dot{W}^{2,1}_{p,(q,1)}(0,T^n\times\mathbb R^d)}+\norm{\nabla P^n}_{L_{q,1}(0,T^n;L_p)}\leq C\norm{u^{n}_0}_{\dot{B}^{-1+d/p}_{p,1}}\leq C\norm{u_0}_{\dot{B}^{-1+d/p}_{p,1}}\cdotp
\end{equation} Furthermore, taking any $r\in(1,\infty)$ and applying Proposition \ref{propregularity} with $q=r$ to $$\partial_{t} u^{n}-\Delta u^n+\nabla P^n=-a^n \partial_t u^n-(1+a^n) u^n\cdot \nabla u^n,\qquad \div u^n=0,$$ yields for all $T<T^n,$ \begin{equation*} \begin{aligned} \norm{u^n, \nabla P^n}_{E^{p,r}_T}&\leq C(\norm{u^n_0}_{\dot{B}^{2-2/r}_{p,r}}+\norm{a^n \partial_t u^n+(1+a^n) u^n\cdot \nabla u^n}_{L_{r}(0,T;L_p)})\\
&\leq C (\norm{u^{n}_0}_{\dot{B}^{2-2/r}_{p,r}}+\norm{a^n}_{L_{\infty}(0,T\times \mathbb R^d)}\norm{\partial_t u^n}_{L_{r}(0,T;L_p)}\\ &\qquad\qquad\qquad+(1+\norm{a^n}_{L_{\infty}(0,T\times \mathbb R^d)})\norm{u^n\cdot \nabla u^n}_{L_{r}(0,T;L_p)}). \end{aligned} \end{equation*} In light of \eqref{inidr} or \eqref{ini1d3}, and \eqref{eq:anbound}, the above inequality becomes
$$\norm{u^n, \nabla P^n}_{E^{p,r}_T}\lesssim \norm{u^n_0}_{\dot{B}^{2-2/r}_{p,r}}+ \norm{u^n\cdot \nabla u^n}_{L_{r}(0,T;L_p)},$$
whence
\begin{equation*}
\begin{aligned}
\norm{u^n, \nabla P^n}^r_{E^{p,r}_T}&\lesssim \norm{u^n_0}^{r}_{\dot{B}^{2-2/r}_{p,r}}+\int_{0}^{T} \norm{u^n\cdot \nabla u^n}^{r}_{L_p}\,dt\\
&\lesssim \norm{u^n_0}^{r}_{\dot{B}^{2-2/r}_{p,q}}+\int_{0}^{T} \norm{u^n}^{r}_{L_{\frac{dr}{r-1}}}\norm{\nabla u^n}^{r}_{L_{\beta}}\,dt
\quad\hbox{with}\quad \frac{1}{\beta}+\frac{1}{d}-\frac{1}{dr}=\frac{1}{p}\cdotp
\end{aligned}
\end{equation*} Combining with Proposition \ref{prop:for existence} and Young's inequality, one gets $$\norm{u^n, \nabla P^n}^r_{E^{p,r}_T}\lesssim \norm{u^n_0}^r_{\dot{B}^{2-2/r}_{p,r}} +\varepsilon \int_0^T
\norm{\nabla^{2} u^n}^{r}_{L_p}\,dt+C_\varepsilon \int_0^T \norm{u^n}^{2r}_{L_{\frac{dr}{r-1}}}
\norm{u^n}^{r}_{\dot{B}^{2-2/r}_{p,r}}\,dt\cdotp$$ Then, taking $\varepsilon$ small enough and using Gronwall's inequality yields \begin{equation}\label{eq:unest} \norm{u^n, \nabla P^n}^r_{E^{p,r}_T}\leq C \norm{u^n_0}^r_{\dot{B}^{2-2/r}_{p,r}} \exp\biggl(C\int_0^T \norm{u^n}^{2r}_{L_{\frac{dr}{r-1}}} \,dt \biggr) \cdotp \end{equation} In the end, Gagliardo-Nirenberg inequality and embedding give $$\norm{u^n}_{L_{\frac{dr}{r-1}}}\leq \norm{u^n}^{\frac 1r}_{L_{\infty}}\norm{u^n}^{1-\frac 1r}_{\dot{B}^{-1+d/p}_{p,1}},$$ which implies that \begin{equation}\label{eq:new}\int_0^T \norm{u^n}^{2r}_{L_{\frac{dr}{r-1}}} \,dt \leq \norm{u^n}^2_{L_2(0,T;L_\infty)}\norm{u^n}^{2(r-1)}_{L_\infty(0,T;\dot{B}^{-1+d/p}_{p,1})}.\end{equation} Now, we deduce from
Proposition \ref{prop0d2} (case $d=2$) or
Proposition \ref{propcsd3} (case $d=3$)
that the two factors of the right-hand side of \eqref{eq:new}
are bounded by $\|u_0\|_{\dot B^{-1+d/p}_{p,1}}.$
Hence, reverting to \eqref{eq:unest} and
using a classical continuation argument allows
to conclude that the solution is global and
belongs to all spaces
$W^{2,1}_{p,r}(\mathbb R_+\times\mathbb R^d)$ with $1<r<\infty.$
Furthermore, since the solution is smooth and
\eqref{eq:unifbound} is satisfied, all the a priori estimates
of Sections \ref{section2} and \ref{section3}
are satisfied uniformly with respect to $n.$
In particular, $(u^{n},\nabla P^n)_{n\in\mathbb N}$ is bounded in $E^{p,q}_T$ for all $T\geq0.$
This, together with \eqref{eq:anbound} ensures that
there exists a subsequence, still denoted by $(a^{n},u^{n},\nabla P^{n})_{n\in{\mathbb N}},$ and $(a,u,\nabla P)$ with
$$a\in L_{\infty}(\mathbb R_{+}\times\mathbb R^{d}),\quad \nabla P\in L_{q}(\mathbb R_{+};L_{p}(\mathbb R^{d}))\quad\hbox{and}\quad u\in \dot{W}^{2,1}_{p,q}(\mathbb R_{+}\times \mathbb R^d)$$
such that
\begin{equation}\label{eq:weak limit}
\begin{aligned}
&a^{n}\rightharpoonup a\quad \text{weak * in } \quad L_{\infty}(\mathbb R_{+}\times\mathbb R^{d}),\\
&u^{n}\rightharpoonup u\quad \text{weak * in } \quad L_{\infty}(\mathbb R_{+};\dot B^{2-2/q}_{p,q}),\\
&(\partial_{t}u^{n}, \nabla^{2}u^{n})\rightharpoonup (\partial_{t}u, \nabla^{2}u)\quad \text{weakly in} \quad
L_{q}(\mathbb R_{+};L_{p}),\\
&\nabla P^{n} \rightharpoonup \nabla P\quad \text{weakly in} \quad
L_{q}(\mathbb R_{+};L_{p}).
\end{aligned}
\end{equation}
Furthermore, as all the spaces under consideration in the previous sections have the Fatou property, the estimates proved therein as still valid. For example, one gets
$$u\in \dot{W}^{2,1}_{p,(q,1)}(\mathbb R_{+}\times \mathbb R^d)\quad \text{and} \quad \nabla P\in L_{q,1}(\mathbb R_+;L_p (\mathbb R^d)).$$ Note that the fact that $(\partial_t u^n)_{n\in {\mathbb N}}$ is bounded in $L_q(\mathbb R_{+};L_p)$ enables us to take advantage of Arzel\`a-Ascoli Theorem in order to get strong convergence results for $u$ like, for instance, for all small enough $\varepsilon>0,$ \begin{equation}\label{eq:strong limit} \begin{aligned} &u^{n}\rightarrow u \ \text{strongly in}\ L_{\infty,loc}(\mathbb R_{+};L_{d-\varepsilon,loc}(\mathbb R^d)),\\ &\nabla u^{n}\rightarrow \nabla u \ \text{strongly in }\ L_{q,loc}(\mathbb R_{+};L_{p^*-\varepsilon,loc})\quad\hbox{with}\quad\frac1{p^*} =\frac1p-\frac1d\cdotp\end{aligned} \end{equation} This allows to pass to the limit in the convection term of the velocity equation of \eqref{rins2}. In order to pass to the limit in the terms containing $a^n$ and conclude that $(a,u,\nabla P)$ is a global weak solution, one can argue as in \cite{HPZ2013, DFM}. Since $a\in L_\infty(\mathbb R_+\times\mathbb R^d)$ and $\nabla u\in L_{2q}(\mathbb R_{+};L_{\frac{dq}{2q-1}}),$ the Di Perna - Lions theory in \cite{DL1989} ensures that $a$ is the only solution to the mass equation of (INS) and that $$a^{n}\rightarrow a \ \text{strongly in }\ L_{\alpha,loc}(\mathbb R_{+}\times \mathbb R^d)\quad\hbox{for all }\ 1<\alpha<\infty.$$ Then, using the uniform bounds for $(a^n,u^n,P^n)_{n\in\mathbb N}$ one can pass to the limit in all the terms of the following equations, that are satisfied by construction of $(a^{n},u^{n},\nabla P^{n})$:
$$\int_{0}^{\infty}\!\!\!\int_{\mathbb R^{d}}a^{n}(\partial_{t}\phi+u^{n}\cdot \nabla \phi)\,dx\,dt+\int_{\mathbb R^{d}}\phi(0,x)a^n_{0}(x)\,dx=0,$$
$$\int_{0}^{\infty}\!\!\!\int_{\mathbb R^{d}}\phi \,\div{u^{n}}\,dx\,dt=0$$
for all functions $\phi\in \mathcal{C}^{\infty}_{c}(\mathbb R_+\times \mathbb R^{d}),$ and
$$\displaylines{ \quad \int_{0}^{\infty}\!\!\!\int_{\mathbb R^{d}}\bigl((1+a^{n})(u^{n}\cdot \partial_{t}\Phi+(u^{n}\otimes u^{n}:\nabla \Phi))+(\Delta u^{n}-\nabla P^{n})\cdot \Phi\bigr)\,dx\,dt
\cr
+\int_{\mathbb R^{d}}u^{n}_{0}\cdot \Phi(0,x)\,dx=0,}$$
for all $\Phi\in \mathcal{C}^{\infty}_{c}(\mathbb R_+\times \mathbb R^{d};\mathbb R^d)$ with $\div \Phi\equiv 0.$ \smallbreak This ensures that $(a,u,\nabla P)$ is a distributional solution of \eqref{rins2}, which completes the proof of the existence parts of Theorems \ref{themd2} and \ref{them1d3}.
\section{Uniqueness results}\label{section5}
The goal of this section is to prove Theorem \ref{thm:uniqueness}, and to show that it implies the uniqueness part of
Theorems \ref{themd2}, \ref{them1d3} and \ref{them:PZuniqueness}.
Theorem \ref{thm:uniqueness} will come up as a consequence
of the stability estimates of Proposition \ref{propunid3} (in the 3D case)
and of Propositions \ref{propunid2}, and \ref{propunid2bis} (2D case).
\smallbreak
Throughout this part, we denote \begin{equation}\label{eq:notdensity} r_0:=\inf_{x\in\mathbb R^d} \rho_0(x)\quad\hbox{and}\quad R_0:= \sup_{x\in\mathbb R^d} \rho_0(x). \end{equation}
\subsection{Uniqueness in the three-dimensional case}
Let us first state general stability estimates in the 3D case. \begin{proposition} \label{propunid3}Let $(\rho_1,u_1,P_1)$ and $(\rho_2,u_2,P_2)$ be two finite energy solutions of $(INS)$ on $[0,T]\times\mathbb R^3$ corresponding to the same initial density $\rho_0$ but, possibly, two different initial velocities $u_{1,0}$ and $u_{2,0}.$ Let $\delta\!\rho:=\rho_1-\rho_2,$ $\delta\!u:=u_1-u_2,$
$$ f(t):=\|t\dot u_2\|_{L_\infty(\mathbb R^3)}+\|t\nabla \dot u_2\|_{L_3(\mathbb R^3)}
\quad\hbox{and}\quad g(t):=\|\nabla u_2\|_{L_\infty(\mathbb R^3)}.$$ There exists an absolute constant $C$ such that the following inequality holds true for all $t\in[0,T]$: $$\displaylines{ \underset{\tau\in [0,t]}{\sup} \tau^{-1}\norm{\delta\!\rho(\tau)}_{\dot{H}^{-1}(\mathbb R^{3})}
\leq R_0^{1/2} \|\sqrt{\rho_0}\,\delta\!u_{0}\|_{L_2(\mathbb R^3)} e^{\frac C2R_0\int_0^tf^2 \,d\tau\: \exp(2\int_0^tg\,d\tau)} e^{2\int_0^t g\,d\tau},\cr
\|\sqrt{\rho_1(t)}\delta\!u(t)\|^2_{L_2(\mathbb R^3)} + \!\int_0^t\!\|\nabla\delta\!u\|^2_{L_2(\mathbb R^3)}\,d\tau
\leq \|\sqrt{\rho_0}\,\delta\!u_{0}\|^2_{L_2(\mathbb R^3)}e^{CR_0\int_0^t\!f^2 \,d\tau\: \exp(2\!\int_0^tg\,d\tau)} e^{2\!\int_0^t g\,d\tau} \cdotp}$$ \end{proposition} \begin{proof} The beginning of the proof is independent of the dimension $d.$ In sharp contrast with \cite{DM1,RD,DM2}, our stability estimates
are performed directly in Eulerian coordinates: we consider
the following system that is satisfied by $\delta\!\rho$, $\delta\!u$ and $\delta\!P:=P_1-P_2,$ denoting $\dot u_2:=(u_2)_t+u_2\cdot\nabla u_2,$ \begin{equation}\label{insu} \left\{\begin{aligned} &(\delta\!\rho)_{t}+\delta\!u\cdot \nabla \rho_{1}+ u_{2}\cdot \nabla \delta\!\rho=0, \\
& \rho_1(\delta\!u)_{t}+\rho_1u_{1}\cdot \nabla \delta\!u- \Delta \delta\!u+\nabla \delta\!P=-\delta\!\rho \,\dot{u_{2}}-\rho_1\delta\!u\cdot \nabla u_{2},\\
&\delta\!\rho|_{t=0}=0, \qquad\delta\!u|_{t=0}=\delta\!u_0. \end{aligned}\right. \end{equation}
Let us set $\phi:=-(-\Delta)^{-1}\delta\!\rho$ (so that $\norm{\delta\!\rho}_{\dot{H}^{-1}(\mathbb R^{d})}=\norm{\nabla \phi}_{L_{2}(\mathbb R^{d})}$). Testing the first equation of \eqref{insu} by $\phi$ yields after integrating by parts and using that $\div u_1=\div u_2=0,$ \begin{equation*} \begin{aligned} \frac{1}{2}\frac{d}{dt}\norm{\nabla \phi}^{2}_{L_{2}(\mathbb R^{d})}&\!\leq \int_{\mathbb R^{d}} \nabla u_{2}:(\nabla \phi \otimes\nabla \phi)\,dx-\int_{\mathbb R^{d}} \rho_{1}\delta\!u\cdot \nabla \phi\,dx\\
&\!\leq \norm{\nabla u_{2}}_{L_{\infty}(\mathbb R^{d})}\norm{\nabla \phi\! \otimes \!\nabla \phi}_{L_{1}(\mathbb R^{d})}+\|\rho_1\|_{L_\infty(\mathbb R^d)}^{1/2}\norm{\sqrt{\rho_1}\,\delta\!u}_{L_{2}(\mathbb R^{d})}\norm{\nabla \phi}_{L_{2}(\mathbb R^{d})}. \end{aligned} \end{equation*} After time integration, we find that for all $t\in [0,T]$, \begin{equation}\label{eq:dphi} \norm{\nabla \phi(t)}_{L_{2}(\mathbb R^{d})}\leq \int_{0}^{t} \norm{\nabla u_{2}}_{L_{\infty}(\mathbb R^{d})}\norm{\nabla \phi}_{L_{2}(\mathbb R^{d})}\,d\tau+\int_{0}^{t} \norm{\rho_{1}}^{1/2}_{L_{\infty}(\mathbb R^{d})}\norm{\sqrt{\rho_1}\,\delta\!u}_{L_{2}(\mathbb R^{d})}\,d\tau.\end{equation} For all $t\in[0,T],$ set \begin{align}\label{def:XY} X(t)&:=\underset{\tau\in [0,t]}{\sup} \tau^{-1}\norm{\delta\!\rho(\tau)}_{\dot{H}^{-1}(\mathbb R^{d})}\nonumber\\ \quad\hbox{and}\quad Y(t)&:=\Bigl(\underset{\tau\in[0,t]}\sup\norm{(\sqrt{\rho_1}\delta\!u)(\tau)}_{L_{2}(\mathbb R^{d})}^2 +\norm{\nabla \delta\!u}_{L_{2}(0,t\times \mathbb R^{d})}^2\Bigr)^{1/2}.\end{align} From \eqref{eq:dphi} and the mass conservation, we end up with \begin{equation}\label{esxt}X(t) \leq \int_0^t g X\,d\tau + R_0^{1/2} Y(t) \end{equation} and that Gronwall lemma thus gives (since $Y$ is nondecreasing) \begin{equation}\label{eq:XY} X(t)\leq R_0^{1/2}Y(t) e^{\int_0^t g\,d\tau}\cdotp \end{equation}
In order to control $Y,$ we test \eqref{insu} by $\delta\!u$ and find that \begin{equation}\label{eq:Y} \frac{1}{2}\frac{d}{dt} \int_{\mathbb R^{d}}\rho_1\abs{\delta\!u}^{2}\,dx+\int_{\mathbb R^{d}}\abs{\nabla \delta\!u}^{2}\,dx
= -\int_{\mathbb R^{d}}\delta\!\rho \, \dot{u_{2}}\cdot \delta\!u\,dx-\int_{\mathbb R^{d}}\rho_1(\delta\!u\cdot \nabla u_{2})\cdot \delta\!u\,dx.\end{equation}
Bounding the last term is straightforward: we just write that
\begin{equation}\label{eq:B2}
-\int_{\mathbb R^{d}}\rho_1(\delta\!u\cdot \nabla u_{2})\cdot \delta\!u\,dx \leq \norm{\nabla u_{2}}_{L_{\infty}(\mathbb R^{d})}\norm{\sqrt{\rho_1}\,\delta\!u}^{2}_{L_{2}(\mathbb R^{d})}.
\end{equation} In order to estimate the term with $\delta\!\rho \,\dot u_2\cdot\delta\!u,$ we argue by duality, writing that \begin{eqnarray}\label{eq:duality}
-\int_{\mathbb R^{d}}\delta\!\rho \: \dot{u_{2}}\cdot \delta\!u\,dx&\!\!\!\leq\!\!\!& \|\delta\!\rho\|_{\dot H^{-1}(\mathbb R^d)} \|{\dot u_{2}}\cdot \delta\!u\|_{\dot H^1(\mathbb R^d)}\nonumber\\
&\!\!\!\leq\!\!\!& X\bigl(\|\tau\nabla \dot{u}_2 \cdot \delta\!u\|_{L_2(\mathbb R^d)} + \|\tau \dot{u}_2\cdot\nabla\delta\!u\|_{L_2(\mathbb R^d)}\bigr)\cdotp\end{eqnarray} Assuming in the rest of the proof that $d=3,$ and using H\"older inequality and the embedding $\dot{H}^{1}(\mathbb R^{3})\hookrightarrow L_{6}(\mathbb R^{3}),$ we get $$\begin{aligned}
-\int_{\mathbb R^{3}}\delta\!\rho \: \dot{u_{2}}\cdot \delta\!u\,dx&\leq X \bigl(\norm{\tau\nabla \dot{u}_{2}}_{L_{3}(\mathbb R^{3})}\norm{\delta\!u}_{L_{6}(\mathbb R^{3})}
+ \norm{\tau\dot{u_{2}}}_{L_{\infty}(\mathbb R^{3})} \norm{\nabla \delta\!u}_{L_{2}(\mathbb R^{3})}\bigr)\\
&\leq C X \norm{\nabla \delta\!u}_{L_{2}(\mathbb R^{3})} \bigl(\tau\norm{\nabla \dot{u}_{2}}_{L_{3}(\mathbb R^{3})}
+ \norm{\tau\dot{u_{2}}}_{L_{\infty}(\mathbb R^{3})}\bigr)\\
&\leq \frac12\|\nabla\delta\!u\|_{L_2(\mathbb R^3)}^2 + \frac{C^2}2 X^2 \bigl(\norm{\tau\nabla \dot{u}_{2}}_{L_{3}(\mathbb R^{3})}
+ \norm{\tau\dot{u_{2}}}_{L_{\infty}(\mathbb R^{3})}\bigr)^2
\cdotp
\end{aligned}$$
Hence, plugging \eqref{eq:B2} and the above inequality in \eqref{eq:Y} and using the notation of the statement, we get
(for another constant $C$):
$$\frac{d}{dt} \int_{\mathbb R^{3}}\rho_1\abs{\delta\!u}^{2}\,dx+\int_{\mathbb R^{3}}\abs{\nabla \delta\!u}^{2}\,dx\leq
2 \norm{\nabla u_{2}}_{L_{\infty}(\mathbb R^{d})}\norm{\sqrt{\rho_1}\,\delta\!u}^{2}_{L_{2}(\mathbb R^{3})} + C f^2 X^2.$$
Integrating on $[0,t],$ the above inequality becomes $$Y^2(t)\leq Y^2(0) + 2\int_0^t gY^2\,d\tau + C\int_0^t f^2 X^2\,d\tau\cdotp$$ Hence, Gronwall lemma gives $$Y^2(t)\leq e^{2\int_0^tg}\biggl( Y^2(0) + C\int_0^t e^{-2\int_0^\tau g\,d\tau'} f^2 X^2\,d\tau\biggr)\cdotp$$ Plugging \eqref{eq:XY} in the above inequality, we discover that $$ Y^2(t)\leq e^{2\int_0^tg\,d\tau}\biggl( Y^2(0) + CR_0\int_0^t f^2 Y^2\,d\tau\biggr)\cdotp$$ Hence, applying again Gronwall inequality, we end up with $$Y^2(t) \leq e^{CR_0(\int_0^tf^2 \,d\tau)\: \exp(2\int_0^tg\,d\tau)} e^{2\int_0^t g\,d\tau} Y^2(0).$$ Inserting this latter inequality in \eqref{eq:XY} completes the proof.
\end{proof}
We claim that the above proposition implies the uniqueness part of Theorem \ref{them1d3}.
As a first, we have to explain why the map $t\mapsto t^{-1}{\delta\!\rho}$ belongs to $L_{\infty}(0,T;\dot{H}^{-1}(\mathbb R^{3})).$
In fact, for all $t\in[0,T],$ integrating the mass equation of (INS) on $[0,t]$ yields $$\rho_i(t)-\rho_{0}=-\int_{0}^{t}\div(\rho_i u_i)\,d\tau,\qquad i=1,2.$$ Hence, $$\begin{aligned}\norm{\rho_i(t)-\rho_{0}}_{\dot{H}^{-1}(\mathbb R^{3})}&\leq \int_{0}^{t}\norm{\div(\rho_i u_i)}_{\dot{H}^{-1}(\mathbb R^{3})}\,d\tau\\ &\leq t\norm{\sqrt{\rho_i}u_i}_{L_{\infty}(\mathbb R_{+};L_{2}(\mathbb R^{3}))}\norm{\sqrt{\rho_i}}_{L_{\infty}(\mathbb R_{+}\times\mathbb R^{3})},\end{aligned}$$ and thus, thanks to the energy balance and the mass equation,
$$t^{-1}\|\delta\!\rho(t)\|_{\dot H^{-1}(\mathbb R^3)}\leq \sqrt{R_0}\,\|\sqrt{\rho_0}u_0\|_{L_2(\mathbb R^3)}.$$
Next, we have to show that $\delta\!u$ is in the energy space.
If $2<p<3,$ then this is guaranteed by the assumption of Theorem \ref{them1d3}. If $1<p\leq2,$ then we argue as follows. By construction, $\partial_t{u}$ is in $L_{q,1}([0,T];L_p(\mathbb R^3))$ and $u$
is in $C_b([0,T];\dot{B}^{-1+3/p}_{p,1}(\mathbb R^3))$, whence
$$u(t)-u_0 \in C([0,T];L_p(\mathbb R^3))\cap C([0,T];\dot{B}^{-1+3/p}_{p,1}(\mathbb R^3)).$$
Hence $u(t)-u_0 \in C([0,T]; B^{-1+3/p}_{p,1}(\mathbb R^3))$
(nonhomogeneous Besov space). Owing to the classical
embedding
$$B^{-1+3/p}_{p,1}(\mathbb R^3)\hookrightarrow H^{1/2}(\mathbb R^3)\quad\hbox{for }\ 1<p\leq2,$$ we obtain $$u(t)-u_0 \in C([0,T];L_2(\mathbb R^3)).$$
Now, taking $(s,m)=(4,2)$ in
\eqref{eq:maxreg2}, we
see that $\nabla u_i$ belongs to $L_{4,1}([0,T];L_2),$ hence to
$L_2(0,T\times\mathbb R^d).$ From this, we eventually conclude
that
$\delta\!u$ is in $L_{\infty}(0,T; L_2)\cap L_{2}(0,T;\dot H^1).$
\smallbreak
Finally, the solution constructed in Theorem \ref{them1d3} satisfies $t\nabla\dot u\in L_2(\mathbb R_+;L_3(\mathbb R^3))$ and $\nabla u\in L_1(\mathbb R_+;L_\infty(\mathbb R^d)).$ Hence, all the assumptions of Theorem \ref{thm:uniqueness} are satisfied by the solutions constructed in Theorem \ref{them1d3}, which are thus unique.
\bigbreak
Another corollary Proposition \ref{propunid3} is the uniqueness of P. Zhang's
solutions constructed in \cite{Zhang19}.
Indeed, if $(\rho,u)$ stands for a solution of Theorem \ref{them:PZuniqueness} then it satisfies \eqref{eq:PZsolutions}.
Therefore, thanks to the embeddings $$\dot{B}^{1/2}_{2,1}(\mathbb R^{3})\hookrightarrow L_3(\mathbb R^3), \quad \dot{B}^{3/2}_{2,1}(\mathbb R^{3})\hookrightarrow L_\infty(\mathbb R^3),$$
on can write that for all $T>0,$ we have
$$t\nabla\dot u \in L_2(\mathbb R_+;L_3(\mathbb R^3))\quad\hbox{and}\quad t\dot u \in L_2(\mathbb R_+;L_\infty(\mathbb R^3)).$$
Hence, if we prove in addition that
$\nabla u\in L_1(0,T;L_\infty(\mathbb R^3))$ for all $T>0,$
then Proposition \ref{propunid3} will ensure uniqueness. \smallbreak In order to prove this latter property, let us observe as in \cite{Zhang19} that if $(\rho,u,\nabla P)$ is a solution to (INS) on $[0,T]\times\mathbb R^3,$
and if we look at the following \emph{linear} Stokes system with convection:
\begin{equation*} \left\{\begin{aligned}
&\rho \partial_t u_j+\rho u\cdot \nabla u_j-\Delta u_j+\nabla P_j=0, \\ &\div u_j=0,\\
&u_j|_{t=0}=\dot \Delta_j u_0, \end{aligned}\right. \end{equation*} then, by uniqueness, we have
\begin{equation}\label{eq:sumup} u=\sum_{j\in\mathbb Z}u_j \quad\hbox{and}\quad \nabla P=\sum_{j\in \mathbb Z}\nabla P_j \cdotp \end{equation}
In \cite{Zhang19}, under assumptions \eqref{eq:Z1} and \eqref{eq:Z2},
the following time weighted estimates
have been proved
(see Corollaries 3.1, 3.2 and 4.2 and Inequalities
(2.10), (3.8) and (3.23)): \begin{align}\label{eq:timeweighted1}
\|\sqrt{t}\nabla^2 \!u_j\|_{L_2(0,T\times\mathbb R^3)}\!+\!\|t\nabla \partial_t u_j\|_{L_2(0,T\times\mathbb R^3)}\!+\!\|t \nabla^2\! u_j\|_{L_\infty(0,T;L_2)}&\lesssim d_j^1 2^{-\frac j2}\norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}} \\\label{eq:timeweighted2}
\|\nabla^2 u_j\|_{L_2(0,T\times\mathbb R^3)}+\|\sqrt{t}\nabla^2 u_j\|_{L_\infty(0,T;L_2)}+\|t\nabla \partial_t u_j\|_{L_\infty(0,T;L_2)}&\lesssim d_j^2 2^{\frac j2}\norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}}, \end{align} with $\{d_j^1\}_{j\in \mathbb Z}$ and
$\{d_j^2\}_{j\in \mathbb Z}$ in the unit ball of $\ell_1(\mathbb Z).$ \medbreak This, together with the following interpolation result (see \cite[Th2:1.18.6]{HT}): $$L_{4,1}(0,T;L_2) =\bigl(L_{2}(0,T;L_2),L_{\infty}(0,T;L_2)\bigr)_{1/2,1}$$ yields \begin{equation}\label{eq:esl41l2}
\|\sqrt{t}\nabla^2 u_j\|_{L_{4,1}(0,T;L_2)}\leq C d_j \norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}}\quad\hbox{with}\quad \sum_{j\in\mathbb Z} d_j=1.\end{equation} Next, since $$-\Delta u_j+\nabla P_j=-\rho\partial_tu_j-\rho v\cdot\nabla u_j, $$ we have in light of the standard elliptic regularity result for the Stokes system:
$$\|t\nabla^2 u_j, t\nabla P_j\|_{L_{2}(0,T;L_6)} \leq C\bigl(\|\partial_tu_j\|_{L_{2}(0,T;L_6)}
+ \|v\cdot\nabla u_j\|_{L_{2}(0,T;L_6)}\bigr)\cdotp$$ Hence, as $\norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}}$ is small, using H\"older inequality,
$\dot H^1(\mathbb R^3)\hookrightarrow L_6(\mathbb R^3)$ and $\dot B^{3/2}_{2,1}(\mathbb R^3)\hookrightarrow L_\infty(\mathbb R^3)$ and, eventually, \eqref{eq:PZsolutions} and \eqref{eq:timeweighted1}, we get $$\begin{aligned}
\|t\nabla^2 u_j, t\nabla P_j\|_{L_{2}(0,T;L_6)}
&\leq C \bigl(\|t\partial_t u_j\|_{L_{2}(0,T;L_6)}+\|tu\cdot \nabla u_j\|_{L_{2}(0,T;L_6)} \bigr)\\
&\leq C \bigl(\|t\nabla \partial_t u_j\|_{L_{2}(0,T\times \mathbb R^3)}+\|u\|_{L_{2}(0,T;L_\infty)}\|t\nabla^2 u_j\|_{L_{\infty}(0,T;L_2)} \bigr)\\
&\leq C d_j^12^{-j/2}\norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}}. \end{aligned}$$ Similarly, we deduce from elliptic regularity, embedding, \eqref{eq:PZsolutions}
and \eqref{eq:timeweighted2}
that $$\begin{aligned}
\|t\nabla^2 u_j, t\nabla P_j\|_{L_{\infty}(0,T;L_6)}
&\!\leq\! C \bigl(\|t\partial_t u_j\|_{L_{\infty}(0,T;L_6)}+\|tu\cdot \nabla u_j\|_{L_{\infty}(0,T;L_6)} \bigr)\\
&\!\leq\! C \bigl(\|t\nabla\partial_t u_j\|_{L_{\infty}(0,T;L_2)}\!+\!\|\sqrt t u\|_{L_{\infty}(0,T\times\mathbb R^{3})}\|\sqrt t \nabla^2 u_j\|_{L_{\infty}(0,T;L_2)} \bigr)\\
&\leq C
d_j^22^{j/2}\norm{u_0}_{\dot{B}^{{1}/{2}}_{2,1}}. \end{aligned}$$ Together with the interpolation property $$L_{4,1}(0,T;L_6)=\bigl(L_2(0,T;L_6),L_{\infty}(0,T;L_6)\bigr)_{1/2,1},$$ this yields \begin{equation}\label{eq:l41l6es}
\|t\nabla^2 u_j\|_{L_{4,1}(0,T;L_6)}\leq C d_j \|u_0\|_{\dot{B}^{1/2}_{2,1}}\quad\hbox{with}\quad\sum_{j\in\mathbb Z} d_j=1. \end{equation} Summing up on all $j\in \mathbb Z$ we deduce from \eqref{eq:sumup}, \eqref{eq:esl41l2} and \eqref{eq:l41l6es} that
\begin{equation}\label{eq:last}\|t \nabla^2 u\|_{L_{4,1}(0,T;L_6)} + \| \sqrt{t} \nabla^2 u\|_{L_{4,1}(0,T;L_2)}\lesssim \norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}}.\end{equation} It is now easy to bound $\nabla u$ in $L_1(0,T;L_\infty)$ for all $T>0.$ Recall the following Gagliardo-Nirenberg inequality:
$$\|z\|_{L_\infty}\leq \norm{\nabla z}^{1/2}_{L_2}\norm{\nabla z}^{1/2}_{L_6},$$ Then, combining with Proposition \ref{p:lorentz} (items (iii), (iv) and (v)) and \eqref{eq:last} yields $$\begin{aligned}
\int^{T}_{0} \|\nabla u\|_{\infty}\,dt
&\leq C \int_0^T \norm{\nabla^2 u}^{1/2}_{L_2}\norm{\nabla^2 u}^{1/2}_{L_6}\,dt\\
&\leq C \int_0^T t^{-3/4}\norm{\sqrt{t}\nabla^2 u}^{1/2}_{L_2}\norm{t\nabla^2 u}^{1/2}_{L_6}\,dt\\
&\leq C\|t^{-3/4}\|_{L_{4/3,\infty}(\mathbb R_+)}\norm{\sqrt{t}\nabla^2 u}^{1/2}_{L_{4,1}(0,T;L_2)}\norm{t\nabla^2 u}^{1/2}_{L_{4,1}(0,T;L_6)}\\
&\leq C\norm{u_0}_{ \dot{B}^{{1}/{2}}_{2,1}}\cdotp \end{aligned}$$ This completes the proof of the Lipschitz regularity for the velocity. Now, applying Proposition \ref{propunid3} yields uniqueness in Theorem
\ref{them:PZuniqueness}.
\subsection{Stability and uniqueness in the two-dimensional case}
Let us present a first result that requires the density to be bounded away from zero (that is $r_0>0$ in \eqref{eq:notdensity}). \begin{proposition} \label{propunid2} Let $(\rho_1,u_1,P_1)$ and $(\rho_2,u_2,P_2)$ be two solutions of $(INS)$ on $[0,T]\times\mathbb R^2$ corresponding to the same initial density $\rho_0$ \emph{bounded away from $0$} and, possibly, two different initial velocities $u_{1,0}$ and $u_{2,0}.$
Denote $g(t):=\|\nabla u_2(t)\|_{L_\infty(\mathbb R^2)}$ and
$$ f_1(t):=\| t\dot u_2\|^2_{L_\infty(\mathbb R^2)}+\|t\nabla^2\dot u_2\|^q_{L_p(\mathbb R^2)}\ \hbox{ for some }\ 1<p,q<2\ \hbox{ such that }\ \frac 1p+\frac 1q=\frac 32\cdotp$$
There exists a constant $C$ depending only $p$ such that the functions $\delta\!\rho:=\rho_1-\rho_2$ and
$\delta\!u:=u_1-u_2$ satisfy for all $t\in[0,T]$: $$\displaylines{ \underset{\tau\in [0,t]}{\sup} \tau^{-1}\norm{\delta\!\rho(\tau)}_{\dot{H}^{-1}(\mathbb R^{2})}
\cr
\leq R_0^{1/2} \|\sqrt{\rho_0}\,\delta\!u_{0}\|_{L_2(\mathbb R^2)} \exp\biggl(\int_0^t(2g+{\textstyle\frac{Cf_1}{2r_0}})\,d\tau\biggr) \exp\biggl({\textstyle\frac{CR_0}2}e^{2\int_0^tg\,d\tau}\int_0^tf_1\,d\tau\biggr),\cr
\|\sqrt{\rho_1(t)}\delta\!u(t)\|^2_{L_2(\mathbb R^2)} + \!\int_0^t\!\|\nabla\delta\!u\|^2_{L_2(\mathbb R^2)}\,d\tau
\cr
\leq \|\sqrt{\rho_0}\,\delta\!u_{0}\|^2_{L_2(\mathbb R^2)} \exp\biggl(\int_0^t(2g+Cr_0^{-1}f_1)\,d\tau\biggr) \exp\biggl(CR_0e^{2\int_0^tg\,d\tau}\int_0^tf_1\,d\tau\biggr)\cdotp
}$$ \end{proposition} \begin{proof}
Let us define the functions $X$ and $Y$ according to \eqref{def:XY}. Compared to the three-dimensional case, the only change is in the treatment of the first term of the right-hand side of \eqref{eq:duality}. Now, using
H\"older inequality, the embedding
$$\dot{W}^{1}_{p}(\mathbb R^2)\hookrightarrow L_{m}(\mathbb R^2)\quad\hbox{with}\quad\frac 1m=\frac 1p-\frac 12,$$
and Gagliardo-Nirenberg inequality, we get (with $1/s:=1-1/p$):
$$\begin{aligned}
\|\tau\nabla\dot{u_{2}}\cdot \delta\!u\|_{L_2(\mathbb R^2)} \leq & \norm{\tau\nabla \dot{u}_{2}}_{L_{m}(\mathbb R^{2})}\norm{\delta\!u}_{L_{s}(\mathbb R^{2})}\\
\leq & C \norm{t\nabla^{2}\dot{u_2}}_{L_p(\mathbb R^2)}\norm{\delta\!u}^{\frac 2s}_{L_2(\mathbb R^2)}\norm{\nabla\delta\!u}^{\frac 2m}_{L_2(\mathbb R^2)}.
\end{aligned}$$
Hence, reverting to \eqref{eq:duality} and using Young inequality, we obtain that
$$\begin{aligned}
-\int_{\mathbb R^{2}}\delta\!\rho \: \dot{u_{2}}\cdot \delta\!u\,dx \leq &\frac 12 \norm{\nabla\delta\!u}_{L_2(\mathbb R^2)}^2+C X^2\norm{t \dot{u_2}}^{2}_{L_\infty(\mathbb R^2)}
+C X^q \norm{t\nabla^{2}\dot{u_2}}^q_{L_p(\mathbb R^2)}\norm{\delta\!u}^{2-q}_{L_2(\mathbb R^2)}\\
\leq &\frac 12 \norm{\nabla\delta\!u}_{L_2(\mathbb R^2)}^2+C X^2 \bigl(\norm{t \dot{u_2}}^{2}_{L_\infty(\mathbb R^2)}+\norm{t\nabla^{2}\dot{u_2}}^q_{L_p(\mathbb R^2)}\bigr)\\
&\hspace{5cm}+C\norm{t\nabla^{2}\dot{u_2}}^q_{L_p(\mathbb R^2)}\norm{\delta\!u}^{2}_{L_2(\mathbb R^2)}\cdotp
\end{aligned}$$ Then, substituting \eqref{eq:B2} and the above inequality into \eqref{eq:Y} yields
$$\frac{d}{dt} \int_{\mathbb R^{2}}\rho_1\abs{\delta\!u}^{2}\,dx+\int_{\mathbb R^{2}}\abs{\nabla \delta\!u}^{2}\,dx\leq
2 g \norm{\sqrt{\rho_1}\,\delta\!u}^{2}_{L_{2}(\mathbb R^{2})} + C f_1 X^2+C f_1 \norm{\delta\!u}^{2}_{L_{2}(\mathbb R^{2})}\cdotp$$ As the density is bounded from below by $r_0>0,$ after integrating on $[0,t],$ we get $$Y^2(t)\leq \int_0^t (2 g(\tau)+Cr_0^{-1} f_1(\tau))Y^{2}(\tau) \,d\tau+C\int_0^t X^2(\tau) f_{1}(\tau)\,d\tau+Y^2(0).$$ Hence, applying Gronwall's inequality yields $$Y^{2}(t)\leq e^{\int_0^t (2 g+Cr_0^{-1} f_1) \,d\tau}\bigl(Y^2(0)+\int_0^t C X^2 f_1 e^{-\int_0^\tau (2 g+C r_0^{-1}f_1) \,d\tau'}\,d\tau\bigr),$$ which together with \eqref{eq:XY} implies $$Y^{2}(t)\leq e^{\int_0^t (2 g+Cr_0^{-1} f_1) \,d\tau}\bigl(Y^2(0)+C R_{0}\int_0^t Y^2 f_1 e^{-\int_0^\tau Cr_0^{-1}f_1\,d\tau'}\,d\tau\bigr)\cdotp$$ Hence, we deduce from Gronwall's inequality that $$Y^{2}(t)\leq Y^2(0)\, \exp\biggl(\int_0^t(2g+Cr_0^{-1}f_1)\,d\tau\biggr) \exp\biggl(CR_0e^{2\int_0^tg\,d\tau}\int_0^tf_1\,d\tau\biggr)\cdotp $$ Inserting this latter inequality in the inequality for $X$ completes the proof. \end{proof} \smallbreak Proposition \ref{propunid2} implies the uniqueness part of Theorem \ref{themd2}. Indeed, the density of the solutions constructed therein is bounded away from zero, the gradient of their velocity is in $L_1(\mathbb R_+;L_\infty(\mathbb R^d)),$ we have $t\nabla^2\dot u\in L_q(\mathbb R_+;L_p(\mathbb R^2))$ with $1<p,q<2$ such that $1/p+1/q=3/2$ and $t\dot u\in L_2(\mathbb R_+;L_\infty(\mathbb R^2)),$ the solutions have finite energy, the map $t\mapsto t^{-1}\delta\!\rho$ is in $L_\infty(0,T;\dot H^{-1}(\mathbb R^3))$ for all $T>0$ (the proof is exactly the same as in the 3D case).
\medbreak Having in mind the results in the three-dimensional case, it is natural to address the uniqueness issue \emph{without assuming that the density has a positive lower bound}. The following result ensures uniqueness in the case of periodic boundary conditions, without making any particular assumption on the density. \begin{proposition} \label{propunid2bis} Let $(\rho_1,u_1,P_1)$ and $(\rho_2,u_2,P_2)$ be two solutions of $(INS)$ on $[0,T]\times\mathbb T^2$ corresponding to the same data $(\rho_0,u_0)$ such that $M:=\int_{\mathbb T^2}\rho_0\,dx$ is positive. \smallbreak
If, in addition,
$$\nabla u_2\in L_1(0,T;L_\infty(\mathbb T^2))\quad\hbox{and}\quad \bigl[t\mapsto t\,\log(e+t^{-1}) \dot{u}_2\bigr]\in L_{2}(0,T;L_\infty(\mathbb T^2)\cap H^1(\mathbb T^2)),$$
then $(\rho_1,u_1,P_1)=(\rho_2,u_2,P_2)$ on $[0,T]\times\mathbb T^2.$ \end{proposition} \begin{proof} Compared to the previous proposition, the only change is in the treatment of the first term of the right-hand side of \eqref{eq:duality}. Thanks to Inequality \eqref{eq:product1} adapted to the periodic setting\footnote{The Littlewood-Paley decomposition that is required for proving \eqref{eq:product1} may be adapted to the periodic setting, see e.g. \cite{D-cours}.}, we have \begin{align}\label{eq:uniq23}
-\int_{\mathbb T^{2}}\delta\!\rho \: \dot{u_{2}}\cdot \delta\!u\,dx &\leq \|\delta\!u\|_{H^1(\mathbb T^2)}\norm{\delta\!\rho \cdot \dot u_2}_{H^{-1}(\mathbb T^2)}\nonumber\\
&\lesssim \|\delta\!u\|_{H^1(\mathbb T^2)}\|\delta\!\rho\|_{H^{-1}(\mathbb T^2)}
\log^{\frac12}\biggl(1+\frac{\|\delta\!\rho\|_{L_2(\mathbb T^2)}}{\|\delta\!\rho\|_{H^{-1}(\mathbb T^2)}}\biggr)
\|\dot u_2\|_{H^1(\mathbb T^2)\cap L_\infty(\mathbb T^2)}.\end{align}
Note that one cannot bound directly $\|\delta\!u\|_{H^1(\mathbb T^2)}$ from $\|\nabla\delta\!u\|_{L_2(\mathbb T^2)}$ since $\int_{\mathbb T^2} \delta\!u\,dx$ need not be zero and, in the periodic setting,
$$\|\delta\!u\|_{H^1(\mathbb T^2)}\simeq \biggl|\int_{\mathbb T^2} \delta\!u\,dx\biggr| +\|\nabla\delta\!u\|_{L_2(\mathbb T^2)}.$$ To bound the first term we write that by virtue of Cauchy-Schwarz and Poincar\'e inequalities, $$\begin{aligned}
\biggl|M\int_{\mathbb T^2} \delta\!u\,dx\biggr|
&=\biggl| \int_{\mathbb T^2}\rho_1\delta\!u\,dx +\int_{\mathbb T^2}(M-\rho_1)\biggl(\delta\!u-\int_{\mathbb T^2}\delta\!u\,dx\biggr)dx\biggr|\\
&\leq \sqrt M \,\|\sqrt{\rho_1} \delta\!u\|_{L_2(\mathbb T^2)}
+ C\|M-\rho_1\|_{L_2(\mathbb T^2)}\|\nabla\delta\!u\|_{L_2(\mathbb T^2)}. \end{aligned} $$ Therefore, there exists a constant $C$ depending only on $M$ and on $R_0,$ and such that
$$\|\delta\!u\|_{H^1(\mathbb T^2)}\leq C\bigl(\|\sqrt{\rho_1} \delta\!u\|_{L_2(\mathbb T^2)}+\|\nabla\delta\!u\|_{L_2(\mathbb T^2)}\bigr)\cdotp$$
Let us denote
\begin{equation}\label{eq:f2}
g(t):=\|\nabla u_2(t)\|_{L_\infty(\mathbb T^2)}\quad\hbox{and}\quad f_2(t):=\|t \dot u_2\|_{L_\infty(\mathbb T^2)\cap H^1(\mathbb T^2)}.\end{equation} Plugging the above inequality in \eqref{eq:uniq23} and using \eqref{eq:f2}, \eqref{def:XY} and \eqref{eq:notdensity}, and, finally, Young's inequality for the second line yields
$$\begin{aligned}
-\int_{\mathbb T^{2}}\delta\!\rho \: \dot{u_{2}}\cdot \delta\!u\,dx &\leq
C\bigl(\|\sqrt{\rho_1} \delta\!u\|_{L_2(\mathbb T^2)}+\|\nabla\delta\!u\|_{L_2(\mathbb T^2)}\bigr)
X\,\log^{\frac12}\biggl(1+\frac{R_0}{\|\delta\!\rho\|_{H^{-1}(\mathbb T^2)}}\biggr) f_2\\
&\leq \frac12\|\nabla\delta\!u\|_{L_2(\mathbb T^2)}^2+Y^2+CX^2f_2^2\,\log\biggl(1+\frac{R_0}{tX}\biggr)\cdotp
\end{aligned}$$ Still thanks to \eqref{eq:notdensity}, we see that there exists a constant $C$ such that
$$\sup_{t\in[0,T]} \|\delta\!\rho(t)\|_{L_2}\leq CR_0.$$
Hence, reverting to \eqref{eq:Y} and using the notation of \eqref{def:XY} yields
$$ \frac{d}{dt}Y^2\lesssim (1+g)Y^2 + X^2f_2^2\, \log\biggl(1+\frac{R_0}{tX}\biggr)\cdotp$$
Remembering \eqref{eq:XY} and integrating the above inequality yields
\begin{equation*} Y^2(t)\lesssim \int_0^t (1+g) Y^2\,d\tau +R_0\int_0^t e^{2\int_0^\tau g\,d\tau'} f^2_2 \, Y^2 \log\biggl(1+\frac{R_0^{1/2}}{Y \tau e^{\int_0^\tau g\,d\tau'}}\biggr)d\tau. \end{equation*}
Hence, taking advantage of the following basic inequality: $$\log(1+aY^{-1})\leq \log(1+a)(1-\log Y), \quad a\geq 0,\quad Y\in (0,1),$$ we get
\begin{equation*} Y^2(t)\lesssim \int_0^t (1+g) Y^2\,d\tau +CR_0\int_0^t \log(1+R_0^{1/2}\tau^{-1})\, e^{2\int_0^\tau g\,d\tau'} f^2_2 \, Y^2 \,(1-\frac12\log Y^2)\,d\tau. \end{equation*} Our assumptions ensure that both $g$ and $\tau\mapsto \log(1+R_0^{1/2}\tau^{-1})\, e^{2\int_0^\tau g\,d\tau'} f^2_2(\tau)$ are integrable on $[0,T].$ Furthermore, the function $r\to r(1-\frac12\log r)$ is increasing near $0^+$ and satisfies $$\int_0^1\frac{dr}{ r(1-\frac12\log r)}=\infty.$$ Hence
one can apply Osgood lemma (see e.g. \cite[Lemma 3.4]{BCD}) so as to conclude that $Y\equiv 0$ on $[0,T],$ and thus, owing to \eqref{eq:XY}, we have $X\equiv 0,$ too. \end{proof}
\subsection* {Acknowledgments:}
The first author is partially supported by the ANR project INFAMIE (ANR-15-CE40-0011).
The second author has been partly funded by the B\'ezout Labex, funded by ANR, reference ANR-10-LABX-58.
\appendix \section{} For the reader's convenience, we here list some results involving Besov spaces and Lorentz spaces, prove maximal regularity estimates in Lorentz spaces for \eqref{eq:stokes}, and product estimates that were needed at the end of the last section. \medbreak The following properties of Lorentz spaces may be found in e.g. \cite{LG}: \begin{proposition}[Properties of Lorentz spaces] \label{p:lorentz} There holds: \begin{enumerate} \item {\rm Interpolation}: For all $1\leq r,q\leq \infty$ and $\theta\in (0,1)$, we have $$\left(L_{p_{1}}(\mathbb R_{+};L_{q}(\mathbb R^{d}));L_{p_{2}}(\mathbb R_{+};L_{q}(\mathbb R^{d}))\right)_{\theta,r}=L_{p,r}(\mathbb R_{+};L_{q}(\mathbb R^{d}))),$$ where $1<p_{1}<p<p_{2}<\infty$ are such that $\frac{1}{p}=\frac{(1-\theta)}{p_{1}}+\frac{\theta}{p_{2}}\cdotp$ \item {\rm Embedding}: $L_{p,r_{1}}\hookrightarrow L_{p,r_{2}} \ \text{if}\ r_{1}\leq r_{2},$ and $L_{p,p}=L_{p}.$ \item {\rm H\"older inequality}: for $1<p,p_{1},p_{2}<\infty$ and $1 \leq r,r_{1},r_{2}\leq \infty,$
we have $$\norm{fg}_{L_{p,r}}\lesssim \norm{f}_{L_{p_{1},r_{1}}}\norm{g}_{L_{p_{2},r_{2}}}\quad\hbox{if}\quad \frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}\quad\hbox{and}\quad\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}}\cdotp$$
This still holds for couples $(1,1)$ and $(\infty,\infty)$ with the convention $L_{1,1}=L_{1}$ and $L_{\infty,\infty}=L_{\infty}.$ \item For any $\alpha>0$ and nonnegative measurable function $f,$ we have $\norm{f^{\alpha}}_{L_{p,r}}=\norm{f}^{\alpha}_{L_{p\alpha,r\alpha}}$. \item For any $k>0$, we have $\norm{x^{-k}1_{\mathbb R_+}}_{L_{1/k,\infty}}=1.$ \end{enumerate} \end{proposition} Next, let us state a few classical properties of Besov spaces. \begin{proposition}[Besov embedding]\label{p:A1} There holds: \begin{enumerate} \item For any $(p,q)$ in $[1,\infty]^{2}$ such that $p\leq q,$ we have
$$\dot{B}^{d/p-d/q}_{p,1}(\mathbb R^{d})\hookrightarrow L_{q}(\mathbb R^{d}).$$ \item Let $1\leq p_{1}\leq p_{2}\leq \infty$ and $1\leq r_{1}\leq r_{2}\leq \infty.$ Then, for any real number $s$, $$\dot{B}^{s}_{p_{1},r_{1}}(\mathbb R^{d})\hookrightarrow \dot{B}^{s-d(\frac{1}{p_{1}}-\frac{1}{p_{2}})}_{p_{2},r_{2}}(\mathbb R^{d}).
$$ \end{enumerate} \end{proposition} The interpolation theory in Besov spaces played an important role in our paper. Below are listed some results that we used (see details in \cite[Prop. 2.22]{BCD} or in
\cite[chapter 2.4.2]{HT}). \begin{proposition}[Interpolation]\label{interpolation} A constant $C$ exists that satisfies the following properties. If $s_{1}$ and $s_{2}$ are real numbers such that $s_{1}<s_{2}$ and $\theta\in ]0,1[,$ then we have, for any $(p,r)\in [1,\infty]^{2}$ and any tempered distribution $u$ satisfying \eqref{eq:lf}, $$\norm{u}_{\dot{B}^{\theta s_{1}+(1-\theta)s_{2}}_{p,r}(\mathbb R^{d})}\leq \norm{u}^{\theta}_{\dot{B}^{s_{1}}_{p,r}}\norm{u}^{1-\theta}_{\dot{B}^{s_{2}}_{p,r}(\mathbb R^{d})} $$ and, for some constant $C$ depending only on $\theta$ and $s_2-s_1,$ $$\norm{u}_{\dot{B}^{\theta s_{1}+(1-\theta)s_{2}}_{p,1}(\mathbb R^{d})}\leq C\norm{u}^{\theta}_{\dot{B}^{s_{1}}_{p,\infty}(\mathbb R^{d})}\norm{u}^{1-\theta}_{\dot{B}^{s_{2}}_{p,\infty}(\mathbb R^{d})}. $$ Furthermore, we have for all $s\in(0,1)$ and $(p,q)\in[1,\infty]^2:$ $$\dot{B}^{s}_{p,q}(\mathbb R^d)=\bigl(L_{p}(\mathbb R^d);\dot{W}^{1}_{p}(\mathbb R^d)\bigr)_{s,q}.$$ \end{proposition}
The following proposition has been used several times. \begin{proposition}\label{prop:for existence} Let $1\leq q<\infty,$ $1\leq p<r\leq\infty$ and $\theta\in(0,1)$ such that \begin{equation}\label{eq:q}
\frac{1}{r}+\frac{1}{d}-\frac{2\theta}{dq}=\frac{1}{p}\cdotp \end{equation}
Then, there exists $C$ so that the following inequality holds true
$$\norm{\nabla u}_{L_r(\mathbb R^d)}\leq C\norm{\nabla^2 u}^{\theta}_{L_p(\mathbb R^d)}\norm{u}^{1-\theta}_{\dot{B}^{2-2/q}_{p,\infty}(\mathbb R^d)}\cdotp$$
\end{proposition} \begin{proof} Proposition \ref{interpolation} tells us in particular that
$$\|u\|_{\dot B^{2-\frac{2\theta}q}_{p,1}}\lesssim
\|u\|_{\dot B^2_{p,\infty}}^{1-\theta}
\|u\|_{\dot B^{2-\frac2q}_{p,\infty}}^{\theta}.$$ It is obvious that
$$\|u\|_{\dot B^2_{p,\infty}}\lesssim \|\nabla^2 u\|_{L_p}$$ and, according to Proposition \ref{p:A1} and to the definition of $\theta,$ we have $$\dot B^{2-\frac{2\theta}q}_{p,1}(\mathbb R^d)\hookrightarrow \dot B^{2+\frac dr-\frac dp-\frac{2\theta}q}_{r,1}(\mathbb R^d) =\dot B^1_{r,1}(\mathbb R^d).$$ As $\dot B^1_{r,1}(\mathbb R^d)\hookrightarrow \dot W^1_r(\mathbb R^d),$ we get the desired inequality.
\end{proof}
The following result that is an easy
adaptation of \cite[Prop. 2.1]{DM2} played a key role in Sections \ref{section2} and \ref{section3}. \begin{proposition}\label{propregularity} Let $1<p,q< \infty$ and $1\leq r\leq \infty.$ Then, for any $u_{0}\in \dot{B}^{2-2/q}_{p,r}(\mathbb R^{d})$ with $\div u_0=0,$ and any $f\in L_{q,r}(0,T;L_{p}(\mathbb R^{d})),$ the Stokes system \eqref{eq:stokes} has a unique solution $(u,\nabla P)$ with $\nabla P\in L_{q,r}(0,T;L_p(\mathbb R^d))$ and\footnote{Only weak continuity holds if $r=\infty.$} $u$ in the space $$\dot{W}^{2,1}_{p,(q,r)}((0,T)\times \mathbb R^{d}):=\bigl\{u\in \mathcal{C}([0,T];\dot{B}^{2-2/q}_{p,r}( \mathbb R^{d})):u_{t}, \nabla^{2}u\in L_{q,r}(0,T;L_{p}( \mathbb R^{d})) \bigr\}\cdotp$$ Furthermore, there exists a constant $C$ \emph{independent of $T$} such that \begin{multline}\label{eq:maxreg1} \mu^{1-1/q}\norm{u}_{L_{\infty}(0,T;\dot{B}^{2-2/q}_{p,r}(\mathbb R^{d}))}+\norm{u_{t}, \mu\nabla^{2}u,\nabla P}_{L_{q,r}(0,T;L_{p}(\mathbb R^{d}))} \\ \leq C\bigl(\mu^{1-1/q}\norm{u_{0}}_{\dot{B}^{2-2/q}_{p,r}(\mathbb R^{d})}+\norm{f}_{L_{q,r}(0,T;L_{p}(\mathbb R^{d}))}\bigr)\cdotp\end{multline} Let $\widetilde s>q$ be such that $$\frac1q-\frac1{\widetilde s}\leq \frac12\quad\hbox{and}\quad \frac d{2p}+\frac1q-\frac1{\widetilde s}>\frac12,$$
and define $\widetilde m\geq p$ by the relation $$\frac{d}{2\widetilde m}+\frac{1}{\widetilde s}=\frac{d}{2p}+\frac{1}{q}-\frac{1}{2}\cdotp$$ Then, the following inequality holds true: \begin{multline}\label{eq:maxreg2}\mu^{1+\frac{1}{\widetilde s}-\frac{1}{q}}\norm{\nabla u}_{L_{\widetilde s,r}(0,T;L_{\widetilde m}(\mathbb R^{d}))}\\\leq C(\mu^{1-1/q}\norm{u}_{L_{\infty}(0,T;\dot{B}^{2-2/q}_{p,r}(\mathbb R^{d}))}+\norm{u_{t}, \mu\nabla^{2}u}_{L_{q,r}(0,T;L_{p}(\mathbb R^{d}))}). \end{multline} Finally, if $2/q+d/p>2,$ then for all $s\in(q,\infty)$ and $m\in(p,\infty)$ such that $$\frac{d}{2m}+\frac{1}{s}=\frac{d}{2p}+\frac{1}{q}-1,$$ it holds that \begin{multline}\label{eq:maxreg3}\mu^{1+\frac{1}{s}-\frac{1}{q}}\norm{u}_{L_{s,r}(0,T;L_{m}(\mathbb R^{d}))}\\\leq C\bigl(\mu^{1-1/q}\norm{u}_{L_{\infty}(0,T;\dot{B}^{2-2/q}_{p,r}(\mathbb R^{d}))}+\norm{u_{t}, \mu\nabla^{2}u}_{L_{q,r}(0,T;L_{p}(\mathbb R^{d}))}\bigr)\cdotp \end{multline} \end{proposition} \begin{proof}
Let $\mathbb P$ and $\mathbb Q$ be the Helmholtz projectors defined in \eqref{eq:PQ}. As $u=\mathbb P u,$ we have
$$u_t-\mu\Delta u=\mathbb P f,\qquad u|_{t=0}= u_0.$$ Hence applying \cite[Prop 2.1]{DM2} and using that $\mathbb P$ is continuous on $L_{q,r}(0,T;L_{p}( \mathbb R^{d}))$ gives \eqref{eq:maxreg1} and \eqref{eq:maxreg3} for $u.$ Since $\nabla P=\mathbb Q f,$ and $\mathbb Q$ is also continuous on $L_{q,r}(0,T;L_{p}( \mathbb R^{d})),$
$\nabla P$ satisfies \eqref{eq:maxreg1} too.
\smallbreak
In order to prove \eqref{eq:maxreg2}, take $q_0$ and $q_1$ such that $1<q_{0}<q<q_{1}<\infty$ and $2/q=1/q_{0}+1/q_{1}.$ From the mixed derivative theorem we have for all $\gamma \in (0,1)$ and $i=0,1,$ $$ \dot{W}^{2,1}_{p,q_{i}}((0,T)\times \mathbb R^{d}):= \dot{W}^{2,1}_{p,(q_{i},q_{i})}((0,T)\times \mathbb R^{d})\hookrightarrow \dot{W}^{\gamma}_{q_{i}}(0,T;\dot{W}^{2-2\gamma}_{p}(\mathbb R^{d})).$$ Let $\gamma:=1/q-1/\widetilde s$ (so that $d/\widetilde m=d/p+2\gamma-1$).
As $\gamma\in (0,\frac{1}{2}]$ and $1-2\gamma<d/p,$ one can use the Sobolev embedding
\begin{equation}\label{eq:embed1}\dot{W}^{\gamma}_{q_{i}}(0,T;\dot{W}^{2-2\gamma}_{p}(\mathbb R^{d}))\hookrightarrow L_{\widetilde s_{i}}(0,T;\dot{W}^{1}_{\widetilde m}(\mathbb R^{d}))\quad\hbox{with}\quad \frac{1}{\widetilde s_{i}}=\frac{1}{q_{i}}-\gamma.\end{equation}
In the proof of \cite[Prop. 2.1]{DM2}, it is pointed out that
$$\dot{W}^{2,1}_{p,(q,r)}((0,T)\times \mathbb R^{d})=\bigl(\dot{W}^{2,1}_{p,q_0}((0,T)\times \mathbb R^{d});\dot{W}^{2,1}_{p,q_1}((0,T)\times \mathbb R^{d})\bigr)_{\frac12,r}.$$
Consequently, the embeddings \eqref{eq:embed1} with $i=0$ and $i=1$ imply that
\begin{equation}\label{eq:embed2}
\dot{W}^{2,1}_{p,(q,r)}((0,T)\times \mathbb R^{d})\hookrightarrow \bigl( L_{\widetilde s_{0}}(0,T;\dot{W}^{1}_{\widetilde m}(\mathbb R^{d})); L_{\widetilde s_{1}}(0,T;\dot{W}^{1}_{\widetilde m}(\mathbb R^{d}))\bigr)_{\frac 12,r}.\end{equation}
Note that our definition of $\gamma,$ $\widetilde s_0,$ $\widetilde s_1,$ $q_0$ and $q_1$ ensures that
$$\frac{1}{2}\left(\frac{1}{\widetilde s_{0}}+\frac{1}{\widetilde s_{1}}\right)=\frac{1}{2}\left(\frac{1}{q_{0}}+\frac{1}{q_{1}}\right)-\gamma=\frac{1}{\widetilde s}\cdotp$$ Hence the real interpolation space in the right of \eqref{eq:embed2} is nothing but $L_{q,r}(0,T;\dot{W}^1_{\widetilde m}(\mathbb R^{d})),$ which completes the proof.
\end{proof}
The usual product is continuous in many Besov spaces (see e.g. \cite{AH, DR2003, RS1996}). We here present a result that played a key role in the proof of uniqueness in dimension two. In order to prove it, we need to introduce following so-called Bony's decomposition (see \cite{Bony}):
$$uv= T_u v+ T_v u+ R(u,v)$$ with $$ T_u v\triangleq\sum_{j\geq1} S_{j-1}u \Delta_j v \quad\hbox{and}\quad R(u,v)\triangleq\sum_{j\geq-1}\sum_{\abs{k-j}\leq 1} \Delta_j u \Delta_k v.$$
Above, we used the notation $\Delta_j:=\dot\Delta_j$ for $j\geq0,$ $\Delta_{-1}:=\dot S_0,$ $\Delta_j=0$ if $j\leq-2$ and $S_j:=\sum_{j'\leq j-1} \Delta_j.$ \smallbreak The above operators $T$ and $R$ are called paraproduct and remainder, respectively. Their general properties of continuity may be found in \cite{BCD,BL, HT}. The last inequality is new to the best of our knowledge. \begin{proposition}\label{prop:h-1} Let $2\leq p\leq \infty$ and $1\leq r_1,r_2\leq \infty$ satisfy $\frac{1}{r_1}+\frac{1}{r_2}=1.$ Then, the following inequality holds true:
\begin{equation}\label{reminder1}
\| R(u,v)\|_{B^{-\frac{d}{p}}_{p,\infty}(\mathbb R^d)}\lesssim \|u\|_{B^{\frac dp}_{p,r_1}(\mathbb R^d)}\|v\|_{B^{-\frac dp}_{p,r_2}(\mathbb R^d)}. \end{equation}
In $\mathbb R^2$, it holds that \begin{equation}\label{eq:product1}
\|uv\|_{H^{-1}(\mathbb R^2)}\lesssim \log^{\frac 12}\Bigl(1+\frac{\|v\|_{L_2(\mathbb R^2)}}{\|v\|_{H^{-1}(\mathbb R^2)}}\Bigr)\|u\|_{H^1(\mathbb R^2)\cap L_\infty(\mathbb R^2)}\|v\|_{H^{-1}(\mathbb R^2)}. \end{equation}
\end{proposition} \begin{proof} To prove the first statement, we use that, by definition of the homogeneous remainder operator $$R(u,v)=\sum_{j\geq-1}\widetilde \Delta_{j} u\Delta_j v\quad\hbox{with}\quad \widetilde \Delta_{j}\triangleq \Delta_{j-1}+\Delta_{j} +\Delta_{j+1}.$$ Hence, owing to the support properties of the dyadic partition of unity, there exists an integer $N_0$ such that \begin{equation}\label{eq:decompose}
\Delta_k R(u,v)=\sum_{j\geq k-N_0} \Delta_k (\Delta_j u\widetilde\Delta_j v) =\sum_{\nu\leq N_0} \dot \Delta_k(\widetilde\Delta_{k-\nu} u\Delta_{k-\nu} v) \cdotp
\end{equation} As $2\leq p\leq \infty,$ thanks to Bernstein's inequality, we have
$$\|\Delta_k R(u,v)\|_{L_p(\mathbb R^d)}\leq 2^{k\frac{d}{p}}\|\Delta_k R(u,v)\|_{L_{p/2}(\mathbb R^d)}\cdotp$$ Therefore, using convolution inequalities and \eqref{eq:decompose}, we discover that $$\begin{aligned}
2^{-k\frac dp}\|\Delta_k R(u,v)\|_{L_p(\mathbb R^d)}
&\leq C\sum_{\nu \leq N_0}\|\widetilde\Delta_{k-\nu} u\Delta_{k-\nu} v\|_{L_{p/2}(\mathbb R^d)}\\
&\leq C\sum_{\nu\leq N_0} 2^{\frac{(k-\nu)d}{p}}\|\widetilde\Delta_{k-\nu} u\|_{L_p(\mathbb R^d)} 2^{-\frac{(k-\nu)d}{p}}\|\Delta_{k-\nu} v \|_{L_p(\mathbb R^d)}, \end{aligned}$$ which gives \eqref{reminder1}.
\smallbreak In order to prove \eqref{eq:product1}, we start from the following properties of continuity of the paraproduct operator (see the details in \cite[Chapter 2]{BCD}):
\begin{align}\| T_{u}v\|_{H^{-1}(\mathbb R^2)}&\lesssim \|u\|_{L_\infty(\mathbb R^2)}\|v\|_{H^{-1}(\mathbb R^2)},\label{eq:tuv1}\\
\label{eq:tuv2}\| T_{v}u\|_{H^{-1}(\mathbb R^2)}&\lesssim
\| v\|_{H^{-1}(\mathbb R^2)}\|u\|_{H^1(\mathbb R^2)}\cdotp \end{align} Next, we decompose $R(u,v)$ into low and high frequencies, using \eqref{reminder1}, to get $$\begin{aligned}
\|R(u,v)\|^2_{H^{-1}(\mathbb R^2)}&=\sum_{j\geq -1} 2^{-2j}\|\Delta_j R(u,v)\|^2_{L_2(\mathbb R^2)}\\
&=\sum_{-1\leq j\leq N} 2^{-2j}\|\Delta_j R(u,v)\|^2_{L_2(\mathbb R^2)}+\sum_{j>N} 2^{-2j}\| \Delta_j R(u,v)\|^2_{L_2(\mathbb R^2)}\\
&\lesssim N \|R(u,v)\|^2_{B^{-1}_{2,\infty}(\mathbb R^2)}+2^{-2N}\|R(u,v)\|^2_{B^0_{2,\infty}(\mathbb R^2)}\\
&\lesssim N \|u\|^2_{H^1(\mathbb R^2)}\|v\|^2_{H^{-1}(\mathbb R^2)}+2^{-2N}\|u\|^2_{H^1(\mathbb R^2)}\|v\|^2_{B^{-1}_{\infty,2}(\mathbb R^2)}\cdotp \end{aligned}$$
Then, choose $N$ be the closest integer larger than $\log_2 \Bigl(1+\frac{\|v\|_{L_2}}{\|v\|_{H^{-1}}}\Bigr)$, by virtue of embedding $B^0_{2,2}(\mathbb R^2)\hookrightarrow B^{-1}_{\infty,2}(\mathbb R^2)$ we infer that
$$\|R(u,v)\|_{H^{-1}(\mathbb R^2)}\lesssim \log^{\frac 12}\Bigl(1+\frac{\|v\|_{L_2(\mathbb R^2)}}{\|v\|_{H^{-1}(\mathbb R^2)}}\Bigr)\|u\|_{H^1(\mathbb R^2)}\|v\|_{H^{-1}(\mathbb R^2)}\cdotp $$ Together with \eqref{eq:tuv1} and \eqref{eq:tuv2}, this completes the proof of \eqref{eq:product1}.
\end{proof}
\begin{small}
\end{small}
\bigbreak\bigbreak \noindent\textsc{Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, LAMA UMR8050, F-94010 Creteil, France} \par\nopagebreak E-mail addresses: [email protected], [email protected]
\end{document} | arXiv |
\begin{definition}[Definition:Multiplication/Multiplier]
Let $a \times b$ denote the operation of multiplication on two objects.
The object $a$ is known as the '''multiplier of $b$'''.
That is, it is the object which is to multiply the multiplicand.
Note that the nature of $a$ and $b$ has deliberately been left unspecified.
They could be, for example, numbers, matrices or more complex expressions constructed from such elements.
\end{definition} | ProofWiki |
\begin{document}
\begin{abstract} The rate of a standard graded $K$-algebra $A$ is a measure of the growth of the shifts in a minimal free resolution of $K$ as an $A$-module. In particular $A$ has rate one if and only if it is Koszul. It is known that a generic Artinian Gorenstein algebra of embedding dimension $n \geq 3$ and socle degree $s=3$ is Koszul. We prove that a generic Artinian Gorenstein algebra with $n\geq 4$ and $s \ge 3 $ has rate $ \lfloor \frac{s}{2} \rfloor. $ In the process we show that such an algebra is generated in degree $\lfloor \frac{s}{2} \rfloor +1. $ This gives a partial positive answer to a longstanding conjecture stated by the first author on the minimal free resolution of a generic Artinian Gorenstein ring of odd socle degree.
\end{abstract}
\title{On the rate of generic Gorenstein $K$-algebras}
\section{Introduction and preliminary results}
Let $K$ be a field and $A=\bigoplus_{i\in \mathbb{N}}A_i$ be a finitely generated $\mathbb{N}$-graded $K$-algebra. We recall that $A$ is said to be standard graded if $A_0= K$ and $A$ is generated (as a $K$-algebra) by elements of degree $1, $ that is, $A=K[A_1]$. We denote by ${ \mathfrak{m}} $ the homogeneous maximal ideal $\bigoplus_{i\ge 1}A_i$ of $A$. We let $\edim(A)=\dim_K(A_1)$ be the embedding dimension of $A$, $\operatorname{soc}(A) = 0:_A\mathfrak{m}$ be its socle, which is a homogeneous ideal of $A$, and $\socdeg(A) = \sup\{i \in \mathbb{N} \mid \operatorname{soc}(A)_i \ne 0\}$. Observe that, if $A$ is Artinian, then $\socdeg(A) = \max\{i \in \mathbb{N} \mid A_i\ne 0\}$. Consider a minimal graded free resolution $$ \mathbb{F}: \dots \to F_i \to F_{i-1}\to \dots \to F_0\to K\to 0 $$ of $K=A/{\mathfrak{m}}$ as an $A$-module. It is well-known that $\mathbb{F}$ plays an important role in the study of the homological properties of $A$. For instance, by the Auslander-Buchsbaum-Serre's Theorem, $F_i = 0$ for $i\gg 0$ if and only if $A$ is regular, i.e., $A$ is isomorphic to a polynomial ring over $K$.
There are several invariants attached to $\mathbb{F}$. An important one is the Backelin rate, which measures the growth of the shifts in $\mathbb{F}$; we now recall its definition. For any integer $i \geq 0$ we let $t_i^A(K) = \sup \{j \mid\mathop{\kern0pt\fam0Tor}\nolimits_i^A(K,K)_j\neq 0 \}$ where, by convention, $t_i^A(K) = -\infty$ whenever $\mathop{\kern0pt\fam0Tor}\nolimits_i^A(K,K)=0$. We observe that $\mathop{\kern0pt\fam0Tor}\nolimits_i^A(K,K)$ is a finitely generated graded $K$-vector space, therefore the integer $t_i^A(K)$ is indeed well-defined. It readily follows from the given definitions and the Auslande-Buchsbaum-Serre's Theorem that, if $A$ is not regular, then $t_i^A(K) \geq i$ for all $i \geq 0$. In the following we assume $A$ not regular, hence $\mathbb{F}$ is an infinite resolution. The Backelin rate of $A$ (see \cite{Backelin}) is defined as \[
\mathop{\kern0pt\fam0rate}\nolimits(A)= \sup_{i \geq 2} \left\{\frac{t_i^A(K) - 1}{i-1}\right\}. \]
It turns out that the rate of any standard graded $K$-algebra is finite, see \cite{Anick,ABH}. Moreover, it follows from our previous considerations that $\mathop{\kern0pt\fam0rate}\nolimits(A) \geq 1$ always holds, and one has that $\mathop{\kern0pt\fam0rate}\nolimits(A)=1$ if and only if $A$ is Koszul. In this sense, the rate of $A$ can be regarded as a measure of how far is $A$ from being Koszul. Consider a minimal presentation of $A$ as a quotient of a polynomial ring, i.e.
$ A\cong R/I
$ where $R=K[x_1,\dots,x_n]$ is a polynomial ring and $I$ is an ideal generated by homogeneous elements of degree $>1$ (or, equivalently, $n=\edim(A)$). Let $m(I)$ be the maximum degree of a minimal homogeneous generator of $I$. It follows from (the graded version of) \cite[Thm. 2.3.2]{BH} that $t_2(K)=m(I)$, and therefore one has $$ \mathop{\kern0pt\fam0rate}\nolimits(R/I) \ge m(I)-1. $$ In \cite{Backelin} Backelin proved that $\mathop{\kern0pt\fam0rate}\nolimits(R/I)=m(I)-1$ for any monomial ideal $I$ (see also \cite{ERT}). Another interesting class of standard graded $K$-algebras with minimal rate is given by generic toric rings, as proved by Gasharov, Peeva and Welker in \cite{GPW}. Extending a result of Kempf \cite{Kempf}, Conca, De Negri and Rossi in \cite{CDR} proved that, under suitable conditions, the coordinate ring of generic points in $\mathbb{P}^n$ has minimal rate. It is an interesting problem to provide further classes of algebras of minimal rate.
\vskip 2mm Artinian Gorenstein $K$-algebras of fixed embedding dimension $n$ and socle degree $s$ provide a good setting for studying this question, as they can be parametrized by points in a suitable projective space via the correspondence given by Macaulay's inverse system. In fact they are in bijective correspondence with polynomials in $n$ variables which are homogeneous of degree $s$. Artinian Gorenstein $K$-algebras which have maximal length in this family are called compressed; it is known that, when $A$ is a generic element of such a family, $A$ is compressed \cite{Iarrobino}.
Conca, Rossi and Valla proved that a generic Artinian Gorenstein algebra $A$ of socle degree three and $n \ge 3 $ is Koszul, see \cite{CRV}. The result stimulated several interesting questions and recent results on being $A=R/I$ Koszul, provided $I$ is generated by quadrics (see for instance \cite{MSS1,MSS2}).
Observe that if the socle degree of $A$ is two, then quadratic always implies Koszul. The main goal of this paper is to prove the analogue of Conca, Rossi and Valla's result for higher socle degrees in terms of the rate. We prove the following result.
\begin{theoremx} \label{thmA} Let $A$ be a generic Artinian Gorenstein graded $K$-algebra of socle degree $s \geq 3$, and assume that $ \edim(A) \geq 4$. Then $\mathop{\kern0pt\fam0rate}\nolimits(A) = \lfloor \frac{s}{2} \rfloor$. \end{theoremx}
A first important step for proving Theorem \ref{thmA} is to provide an upper bound on $t_i^A(K)$ for all $i \geq 0$ in terms of $m(I), $ see Proposition \ref{prop compressed}. Several interesting bounds on $t_i^A(K)$ have been proved in the literature, see for instance \cite{ACI1,ACI2,DHS}. However, the one which is needed for our purposes is obtained from an explicit computation of the bigraded Poincar\'e series of $K$, see Proposition \ref{Proposition PS}. The proof is based on the graded version of an argument due to Rossi and \c{S}ega, used in \cite{RS} in the local case. As a consequence of the above results, we prove that if $A=R/I$ is Gorenstein compressed with socle degree $s \geq 4$ and if $I$ is generated by homogeneous polynomials of the same degree, then the rate is minimal, that is $\lfloor \frac{s}{2} \rfloor.$ Then the key step in the proof of our main result is to show that a $K$-algebra satisfying the assumptions of Theorem \ref{thmA} has generators in the same degree.
To this end, recall that a generic Artinian Gorenstein $K$-algebra $A$ of socle degree $s=3$ and embedding dimension $n \geq 3$ is generated by quadrics. An Artinian compressed Gorenstein algebra of even socle degree $s=2(t-1)\ge 2$ has almost linear resolution; as a consequence, it is minimally generated only in degree $t$, see \cite[Example 4.7]{Iarrobino}. On the other hand, an Artinian compressed Gorenstein $K$-algebra $A=R/I$ of odd socle degree $s=2t-1 \geq 3$ always has generators in degree $t$, and it has no generators in degrees different than $t$ or $t+1$ \cite[Proposition 3.2]{B99}. If $A=R/I$ is a generic Artinian Gorenstein $K$-algebras of odd socle degree $s=2t-1 \geq 3$ and $n \geq 4$, the graded Betti numbers in a minimal free resolution of $A$ as $R$-module were conjectured by Boij \cite[Conjecture 3.13]{B99} in terms of $n$ and $t. $ For every homological position there exists a Zariski open set (depending on the coefficients of the form $F$ of degree $s$) where the Artinian compressed Gorenstein algebras have the same graded Betti numbers. The conjecture in particular predicts that $I$ should be minimally generated in degree $t$.
In this article we give a positive answer to this part of the conjecture. In order to do this, we need to show that the Zariski open set parametrizing the Artinian compressed Gorenstein algebras generated in the same degree $t$ is not empty.
\begin{theoremx} \label{thmB} For all integers $t \ge 2$ and $n \ge 4$ there exists an Artinian compressed
Gorenstein $K$-algebra $A= R/I$ with socle degree $s=2t-1$ and embedding dimension $n$ such that $I$ is generated
in degree $t$. \end{theoremx}
The first step in our proof is to produce an Artinian level algebra with socle degree $s$ defined by monomials of degree $s$, see Lemma \ref{lemma:1}. The required Artinian compressed Gorenstein $K$-algebra is then obtained from it using a tricky construction, see Theorem \ref{generation}.
We point out that Boij's conjecture fails in general for generic Artinian Gorenstein algebras but in the middle of the resolution, not at the level of generators, consistently with Theorem \ref{thmB}. In fact, counterexamples were exhibited by Kunte \cite{Ku}, see Corollary 5.6 and Remark 5.7.
\section{LG$_t$-algebras and rate}
In this section we present some basic results on the rate of standard graded $K$-algebras. Consider a minimal presentation of $A$ as a quotient of a polynomial ring, i.e.
$ A \cong R/I
$ where $R=K[x_1,\dots,x_n]$ is a polynomial ring and $I \ne 0$ is a homogeneous ideal generated in degree at least two.
Let $\tau$ be a monomial order on $R$. Denote by $\mathop{\kern0pt\fam0in}\nolimits_\tau(I)$ the initial ideal of $I$ with respect to $\tau$. In \cite[Cor. 2.4, Thm.2.2]{BHV} it is proved that: $$ \mathop{\kern0pt\fam0rate}\nolimits(A)\le\mathop{\kern0pt\fam0rate}\nolimits(R/\mathop{\kern0pt\fam0in}\nolimits_\tau(I)). $$ It is known that if $0 \ne J \subseteq R$ is any monomial ideal, then $\mathop{\kern0pt\fam0rate}\nolimits(R/J)=m(J)-1$ (see \cite{Backelin, ERT}). Putting all these facts together one has that $$ m(I)-1\le \mathop{\kern0pt\fam0rate}\nolimits(A)\le m(\mathop{\kern0pt\fam0in}\nolimits_\tau(I)) -1. $$ It follows that if $I$ is generated by a Gr\"obner basis of forms of degree at most $m(I)$ with respect to some coordinate system and some monomial order, then the rate is minimal, that is, $\mathop{\kern0pt\fam0rate}\nolimits(A)= m(I)-1$. Another interesting class of standard graded $K$-algebras with minimal rate is given by generic toric rings, see \cite{GPW}.
Motivated by the previous observations, we make the following definitions.
\begin{definition} Let $A$ be a standard graded $K$-algebra. For an integer $t >1$ we say that $A$ is a G$_t$-algebra if it has a defining ideal $I$ with respect to some minimal presentation $A\cong R/I$ possessing a Gr{\"o}bner basis of forms of degree at most $t$. We say that $A$ is an LG$_t$-algebra if there exists a graded G$_t$-algebra $S$ and a regular sequence $y_1,\ldots,y_r$ of linear forms of $S$ such that $A \cong S/(y_1,\ldots,y_r)$. \end{definition}
Observe that being a G$_2$-algebra (resp. an LG$_2$-algebra) is equivalent to being G-quadratic (resp. LG-quadratic). LG-quadratic algebras are Koszul, see \cite{Conca, CDR2}. The following proposition extends this result to the case $t \geq 2$.
\begin{proposition} \label{prop LGt} Let $A$ be an LG$_t$-algebra. Then $\mathop{\kern0pt\fam0rate}\nolimits(A) \leq t-1$. \end{proposition} \begin{proof} First, we claim that if $A=S/(y)$ where $y \in S$ is a regular element of degree one, then $\mathop{\kern0pt\fam0rate}\nolimits(A) = \mathop{\kern0pt\fam0rate}\nolimits(S)$. By the graded version of \cite[Satz 1]{Scheja} we have that $P^S_K(u,v) = (1+uv)P^A_K(u,v)$. In particular, for all $i \geq 2$ we have that $\beta_{ij}^S(K) = \beta_{i-1j}^A(K) + \beta_{ij}^A(K)$. Thus, for all $i \geq 2$ one has $t_i^S(K) = \max\{t_i^A(K),t_{i-1}^A(K)+1\}$. Note that for every $i \geq 3$ one has $t_{i-1}^A(K)/(i-1) \leq (t_{i-1}^A(K)-1)/(i-2)$. Indeed, this is equivalent to the fact that $t_{i-1}^A(K) \geq i-1$, which holds true since $A$ is assumed to be not regular. This together with the fact that $t_1^A(K)=1$ gives \[ \mathop{\kern0pt\fam0rate}\nolimits(S) = \sup_{i \geq 2} \left\{\frac{t_1^A(K)-1}{i-1},\frac{t_{i-1}^A(K)}{i-1}\right\} = \sup_{i \geq 2} \left\{\frac{t_i^A(K)-1}{i-1}\right\} = \mathop{\kern0pt\fam0rate}\nolimits(A). \] Now assume that $A$ is an LG$_t$-algebra, i.e., there exists a G$_t$-algebra $S$ such that $A \cong S/(y_1,\ldots,y_r)$, with $y_1,\ldots,y_r$ a regular sequence of linear forms. A repeated application of the above claim together with the fact that G$_t$-algebras have rate at most $t-1$ gives that $\mathop{\kern0pt\fam0rate}\nolimits(A) \leq t-1$. \end{proof}
A standard application of Proposition \ref{prop LGt} gives a proof of the following generalization of a result of Tate \cite{Tate}, which states that every quadratic complete intersection is Koszul.
\begin{corollary} Let $A$ be a standard graded $K$-algebra which is a complete intersection. Assume that the defining ideal of $A$ is generated by a regular sequence of degrees at most $t$. Then $\mathop{\kern0pt\fam0rate}\nolimits(A) = t-1$. \end{corollary} \begin{proof} Consider a minimal presentation $A\cong R/I$ with $R=K[x_1,\ldots,x_n]$ a standard graded polynomial ring, and $I=(f_1,\ldots,f_c)$ an ideal generated by a regular sequence with $m(I) = t$. Let $d_i=\deg(f_i)$. Let $S=R[y_1,\ldots,y_c]$, $g_i = y_i^{d_i}-f_i$ for $i=1,\ldots,c$, and set $J=(g_1,\ldots,g_c)$. Fix a monomial order $\tau$ on $S$ such that $y_1>y_2>\ldots>y_c>x_1>\ldots>x_n$. Then the leading terms $\mathop{\kern0pt\fam0in}\nolimits_\tau(g_1) = y_1^{d_1},\ldots,\mathop{\kern0pt\fam0in}\nolimits_\tau(g_c) = y_c^{d_c}$ form a regular sequence, and thus $g_1,\ldots,g_c$ are a Gr{\"o}bner basis of forms of degree at most $t$. As $y_1,\ldots,y_c$ are a regular sequence of linear forms in $S$, it follows that $A \cong S/(y_1,\ldots,y_c)$ is an LG$_t$-algebra, and the result follows from Proposition \ref{prop LGt} and the general fact that $\mathop{\kern0pt\fam0rate}\nolimits(A) \ge m(I)-1 = t-1$. \end{proof} \section{The graded Poincar\'e series of a compressed Gorenstein $K$-algebra} \begin{comment} Let $\mathbb{F}_\bullet $ the minimal graded free resolution $$ \mathbb{F}_\bullet: \dots \to F_i \to F_{i-1}\to \dots \to F_0\to K\to 0 $$ of $K=A/{ \mathfrak{m}}$ as an $A$-module. Assume throughout this article that $A$ is not regular.
The Poincar\'e series of $K$ over $A$, denoted $P^A_K(u) \in \mathbb{Z}\ps{u}$, is defined as the formal power series whose $i$-th coefficient is the rank of the free module $F_i$, that is, the $K$-vector space dimension of $\mathop{\kern0pt\fam0Tor}\nolimits_i^A(K,K)$. More generally, given \end{comment}
Let $A$ be a standard graded $K$-algebra and $M$ be a finitely generated $\mathbb{Z}$-graded $A$-module.
The graded Poincar\'e series of $M$ is defined as \[ P^A_M(u,v) = \sum_{i \geq 0}\left(\sum_{j \in \mathbb{Z}}\beta_{ij}^A(M)v^j\right)u^i \in \mathbb{Z}[v,v^{-1}]\ps{u}, \] where $\beta_{ij}^A(M) = \dim_K(\mathop{\kern0pt\fam0Tor}\nolimits_i^A(M,K)_j)$. Note that, fixed an integer $i \geq 0$, one has that $\mathop{\kern0pt\fam0Tor}\nolimits_i^A(M,K)_j=0$ for all but finitely many $j \in \mathbb{Z}$. Therefore each sum $\sum_{j \in \mathbb{Z}} \beta_{ij}^A(M)v^j$ is finite. \vskip 2mm
Given a minimal presentation $A=R/I$ with $R=K[x_1,\ldots,x_n]$, there is a point-wise inequality \begin{equation} \label{eqGolod} P^A_K(u,v) \leq \frac{P^R_K(u,v)}{1-u(P^R_A(u,v)-1)}. \end{equation} \begin{definition} A standard graded $K$-algebra $A$ is said to be Golod in the graded sense if equality holds in (\ref{eqGolod}). \end{definition} \begin{remark} \label{remark Golod} Since $\beta_i^A(K) = \beta_i^{A_\mathfrak{m}}(A_\mathfrak{m}/\mathfrak{m} A_\mathfrak{m})$ for all $i$, and because one always has a point-wise inequality between graded Betti numbers in (\ref{eqGolod}), it turns out that $A$ is Golod in the graded sense if and only if $A_\mathfrak{m}$ is a Golod local ring. \end{remark}
With the aim of bounding $t^i_A(K)$ for every $i>0$, the goal of this section is to compute the graded Poincar\'e series of $K = A/\mathfrak{m}$ for any Artinian compressed Gorenstein algebra $A$.
Assume that $A$ is an Artinian Gorenstein standard graded $K$-algebra. Then we can write $A=\bigoplus_{i=0}^s A_i$ with $A_s \cong K$. The integer $s$ is called the socle degree of $A$. The Hilbert function of $A$ is by definition the function $\HF_A:\mathbb{N} \to \mathbb{N}$, defined as $\HF_A(i) = \dim_K(A_i)$.
Artinian Gorenstein algebras of fixed embedding dimension $n$ and socle degree $s$ can be parametrized by points in projective space via the Macaulay inverse system and from this one can deduce that for every integer $i$
\[ \HF_A(i) \le \min\left\{ \binom{n+i-1}{i}, \binom{n+s-i-1}{s-i} \right\}. \]
\begin{definition} An Artinian Gorenstein $K$-algebra $A$ of socle degree $s$ is said to be {\it{ compressed}} if for all $i \in \{0,\ldots,s\}$ one has \[ \HF_A(i) = \min\left\{ \binom{n+i-1}{i}, \binom{n+s-i-1}{s-i} \right\}. \] \end{definition} There exists a non-empty Zariski open set of Artinian Gorenstein algebras with maximal Hilbert function and several authors investigated Artinian Gorenstein compressed algebras because they provide a good setting for studying interesting questions (see \cite{EI78,Iarrobino,IK,B99,RS}). \vskip 2mm
The following result is a graded version of \cite[Proposition 6.2]{RS}.
\begin{proposition} \label{Proposition PS} Let $A$ be an Artinian compressed Gorenstein $K$-algebra of embedding dimension $n \ge 2 $ and socle degree $s$, with $2 \leq s \ne 3$. Then \[ P^A_K(u,v) = \frac{(1+uv)^n}{1-u(P^R_A(u,v) - 1) + u^{n+1}v^{n+s}(1+u)}. \] \end{proposition} \begin{proof} By \cite{LevinAvramov}, the natural map $A \to T = A/\operatorname{soc}(A)$ is Golod, therefore \[ P^T_K(u,v) = \frac{P^A_K(u,v)}{1-u(P^A_T(u,v)-1)}. \] From the graded short exact sequence $0 \to \operatorname{soc}(A) \cong K(-s) \to A \to T \to 0$, we see that $P^A_T(u,v) = 1 + uv^sP^A_K(u,v)$, and therefore \[ P^T_K(u,v) = \frac{P^A_K(u,v)}{1-u^2v^sP^A_K(u,v)}, \] from which we conclude \[ P^A_K(u,v) = \frac{P^T_K(u,v)}{1+u^2v^sP^T_K(u,v)}. \] Since $A_\mathfrak{m}$ satisfies the assumptions of \cite[Theorem 5.1]{RS}, we have that $T_\mathfrak{m}$ is a Golod ring (see the paragraph before \cite[Proposition 6.3]{RS}). Thus, we have that $T$ is Golod in the graded sense by Remark \ref{remark Golod}, and therefore \[ P^T_K(u,v) = \frac{P^R_K(u,v)}{1-u(P^R_T(u,v)-1)} = \frac{(1+uv)^n}{1-u(P^R_T(u,v)-1)}. \] Consider again the graded short exact sequence $0 \to \operatorname{soc}(A) \cong K(-s) \to A \to T \to 0$. As the map $\mathop{\kern0pt\fam0Tor}\nolimits_i^R(\operatorname{soc}(A),K) \to \mathop{\kern0pt\fam0Tor}\nolimits_i^R(A,K)$ is zero for $i \ne n$, we obtain the following exact sequences \[ \xymatrix{ 0 \ar[r] & \mathop{\kern0pt\fam0Tor}\nolimits_i^R(A,K) \ar[r] & \mathop{\kern0pt\fam0Tor}\nolimits_i^R(T,K) \ar[r] & \mathop{\kern0pt\fam0Tor}\nolimits_{i-1}^R(K,K)(-s) \ar[r] & 0 } \] for all $0 < i < n$, and \[ \xymatrix{ 0 \ar[r] & \mathop{\kern0pt\fam0Tor}\nolimits_n^R(K,K)(-s) \ar[r] & \mathop{\kern0pt\fam0Tor}\nolimits_n^R(A,K) \ar[r] & \mathop{\kern0pt\fam0Tor}\nolimits_n^R(T,K) \ar[r] & \mathop{\kern0pt\fam0Tor}\nolimits_{n-1}^R(K,K)(-s) \ar[r] & 0. } \] Therefore for $0<i<n$ we have \[ \beta_{i,j}^R(T) = \beta_{i,j}^R(A) + \beta_{i-1,j-s}^R(K), \] while for $i=n$ we have \[ \beta_{n,j}^R(T) = \beta_{n,j}^R(A) + \beta_{n-1,j-s}^R(K) - \beta_{n,j-s}^R(K). \] Putting all together we have \begin{align*} P^R_T(u,v) &= \sum_{i,j} \beta_{i,j}^R(T)u^iv^j \\ & = \sum_{\substack{0 \leq i \leq n \\ j \in \mathbb{Z}}} \beta_{i,j}^R(A) u^iv_j + uv^s \sum_{\substack{1 \leq i \leq n \\ j \in \mathbb{Z}}} \beta_{i-1,j-s}^R(K)u^{i-1}v^{j-s} - \sum_{j \in \mathbb{Z}} \beta_{n,j-s}^R(K) \\ & = P^R_A(u,v) + uv^sP^R_K(u,v) - u^n \sum_{j \in \mathbb{Z}} \left(\beta_{n,j-s}^R(K)u v^j + \beta_{n,j-s}^R(K)v^j\right) \\ & = P^R_A(u,v) + uv^sP^R_K(u,v) - u^n(u+1)v^{n+s} = P^R_A(u,v) + uv^s(1+uv)^n - u^n(u+1)v^{n+s} \end{align*} where we used that $\beta_{n,j-s}^R(K) = \begin{cases} 1 & \text{ if } j-s = n \\ 0 & \text{ otherwise.} \end{cases}$
Substituting the identities obtained so far in the formula for $P^A_K(u,v)$, we finally get that \begin{align*} P^A_K(u,v) & = \frac{P^T_K(u,v)}{1+u^2v^sP^T_K(u,v)} \\ & = \frac{(1+uv)^n}{1-u(P^R_T(u,v)-1)+u^2v^s(1+uv)^n} \\ & = \frac{(1+uv)^n}{1-u(P^R_A(u,v)-1) + u^{n+1}v^{n+s}(1+u)}. \qedhere \end{align*} \end{proof}
\begin{proposition} \label{prop compressed} Let $A$ be an Artinian compressed Gorenstein graded $K$-algebra of embedding dimension $n \geq 2$ and socle degree $s \geq 4$, and let $A=R/I$ be a minimal presentation. Then $t_i^A(K) \leq (m(I)-1)(i-1)+1$. \end{proposition} \begin{proof} Let $t= \lfloor \frac{s}{2} \rfloor+1$. Since $A$ is a compressed Gorenstein $K$-algebra, by \cite{FrobergLaksov} (see also \cite{B99}) a minimal free resolution of $A$ over $R$ has the following shape: \[ \xymatrixrowsep{1mm} \xymatrix{ &&R(-(n+t-2))^{a_{n-1}} && R(-t)^{a_1} \\ 0 \ar[r] & R(-(n+s)) \ar[r] & \qquad \bigoplus \qquad \ar[r] & \ldots \ar[r] & \qquad \bigoplus \qquad \ar[r]& R \ar[r] & A \ar[r]&0. \\ &&R(-(n+t-1))^{b_{n-1}} &&R(-(t+1))^{b_1} } \] By Proposition \ref{Proposition PS} we therefore obtain \begin{align*} P^A_K(u,v) & = \displaystyle \frac{(1+uv)^n}{1-u(\sum_{j=1}^{n-1} (a_j+b_jv)u^jv^{t+j-1}) + u^nv^{n+s}) + u^{n+1}v^{n+s}(1+u)} \\ & = \frac{(1+uv)^n}{1-\sum_{j=1}^{n-1} (a_j+b_jv) u^{j+1}v^{t+j-1} + u^{n+2}v^{n+s}}. \end{align*} Write $P^A_K(u,v) = \sum_{i \geq 0} \beta_i(v)u^i$, where $\beta_i(v) \in \mathbb{Z}[v]$. For simplicity, for all $i \in \mathbb{N}$ we let $\delta_i=\deg(\beta_i(v))=t_i^A(K)$, where $\deg(-)$ denotes the degree of a polynomial in the variable $v$. By clearing denominators and equating the coefficients of $u^i$ in both sides of the previous identity we deduce that \begin{equation} \label{casesBetti} \beta_i = \begin{cases} 1 & \text{ if } i=0 \\ nv & \text{ if } i=1 \\ {n \choose i}v^i + \sum_{j=1}^{i-1} \beta_{i-j-1}(a_j+b_jv)v^{t+j-1} & \text{ if } 2 \leq i \leq n \\ \sum_{j=1}^{n-1} \beta_{n-j+1}(a_j+b_jv)v^{t+j-1} + v^{n+s} & \text{ if } i=n+2 \\ \sum_{j=1}^{n-1} \beta_{i-j-1}(a_j+b_jv)v^{t+j-1} & \text{ otherwise} \end{cases}. \end{equation}
We distinguish two cases. If $b_1=0$, then by induction on $i \geq 1$ we prove that $\delta_i \leq (t-1)(i-1)+1$. Note that for $i=1,2$ the inequality is satisfied since $\delta_1=1$ and $\delta_2=t$; thus we may assume $i \geq 3$. For $2 \leq j=i-1 \leq n-1$ we have that $\deg(\beta_0(a_{i-1}+b_{i-1}v)v^{t+i-2}) \leq t+i-1 \leq (t-1)(i-1)+1$. For $1 \leq j \leq n-1$ with $j\ne i-1$, if we assume that $\delta_{i-j-1} \leq (t-1)(i-j-2)+1$ then \begin{equation} \label{Deg1} \deg(\beta_{i-j-1}(a_j+b_jv)v^{t+j-1}) \leq (t-1)(i-j-2)+1+t+j \leq (t-1)(i-1)+1, \end{equation} where the latter holds since $t \geq 2$ in our assumptions. One can check directly that $i \leq (t-1)(i-1)+1$ holds. Finally, since $s \leq 2t-1$, $n \geq 2$ and $t \geq 3$, we also have that $n+s \leq (t-1)(n+1)-1$. All these inequalities, together with (\ref{casesBetti}) and the induction hypothesis which allows to use (\ref{Deg1}) for appropriate values of $j$, give that $\delta_i \leq (t-1)(i-1)+1$. Note that the assumption $b_1=0$ guarantees that $m(I)=t$, and since $\delta_i=t_i^A(K)$ we proved that $t_i^A(K) \leq (m(I)-1)(i-1)+1$, as desired.
Now assume that $b_1\ne 0$, so that $m(I)=t+1$. In this case, we prove by induction on $i \geq 1$ that $\delta_i \leq t(i-1)+1$. The strategy is similar to the one used in the previous case. Since $\delta_1=1$ the claimed inequality is true for the base case, and we may assume $i \geq 2$. For $1 \leq j=i-1 \leq n-1$ we have that $\deg(\beta_0(a_{i-1}+b_{i-1}v)v^{t+i-2}) \leq t+i-1 \leq t(i-1)+1$. Observe that, for $1 \leq j \leq n-1$ with $j \ne i-1$, if we assume that $\delta_{i-j-1} \leq t(i-j-2)+1$ then \begin{equation} \label{Deg2} \deg(\beta_{i-j-1}(a_j+b_jv)v^{t+j-1}) \leq \delta_{i-j-1}+t+j \leq t(i-j-2)+1+t+j \leq t(i-1)+1. \end{equation} Finally, we have that $i \leq t(i-1)+1$ for all $i \geq 2$ and $n+s \leq t(n+1)-1$ since $s \leq 2t-1$. These inequalities, together with (\ref{casesBetti}) and the induction hypothesis which allows to use (\ref{Deg2}) for all appropriate values of $j$, give that $t_i^A(K) = \delta_i \leq t(i-1)+1 = (m(I)-1)(i-1)+1$. \end{proof}
\begin{comment}
If $i \leq n$, by (\ref{eqPoincare}) we have \[ \beta_i = {n \choose i}v^i + \sum_{j=1}^{i-1} \beta_{i-j-1}(a_j+b_jv)v^{t+j-1}. \] By induction, for $j \ne i-1$ we have that $\deg(\beta_{i-j-1}(a_j+b_jv)v^{t+j-1}) \leq \delta_{i-j-1}+t+j \leq t(i-j-2)+1+t+j$, and a direct computation shows that the latter is at most $t(i-1)+1$. Also, for $j=i-1$ we have that $\beta_0=1$, and thus $\deg(\beta_0(a_{i-1}+b_{i-1}v)v^{t+i-2}) \leq t+i-1$, and the latter can be shown to be at most $t(i-1)+1$. Since $i \leq t(i-1)+1$ as well, we conclude that $\delta_i \leq t(i-1)+1$ in this case. For $i=n+2$ the equality (\ref{eqPoincare}) gives \[ \beta_{n+2} =\sum_{j=1}^{n-1} \beta_{n-j+1}(a_j+b_jv)v^{t+j-1} + v^{n+s}. \] Using that $\delta_i \leq t(i-1)+1$ for all $i \leq n$ by induction, a computation analogous to the one carried out before shows that for all $1 \leq j \leq n-1$ one has $\deg(\beta_{n-j+1}(a_j+b_jv)v^{t+j-1}) \leq \delta_{n-j+1}+t+j \leq t(n-j)+1+t+j \leq t(n+1)+1$. Since $s \leq 2t$, we have that $n+s \leq n+2t \leq t(n+1)+1$. In conclusion, $\delta_{n+2} \leq t(n+1)+1$, as desired. Finally, for $i=n+1$ or $i>n+2$, we have \[ \beta_i =\sum_{j=1}^{n-1} \beta_{i-j-1}(a_j+b_jv)v^{t+j-1}. \] As before, $\deg(\beta_{i-j-1}(a_j+b_jv)v^{t+j-1}) \leq \delta_{i-j-1}+t+j \leq t(i-j-2)+1+t+j \leq t(i-1)+1$, and thus $\delta_i \leq t(i-1)+1$ as desired. \end{comment}
In \cite[Question 6.10]{CRV} the authors ask if any quadratic Gorenstein algebra of socle degree three is Koszul. However, for any embedding dimension $n \geq 8$ there are examples of Gorenstein quadratic algebras of regularity three which are not Koszul, see \cite{MSS1,MCS}.
A surprising consequence of Proposition \ref{prop compressed} is that compressed Gorenstein algebras with $s \geq 4$ and generated in one degree have minimal rate.
\begin{corollary} \label{corollary t-generated} Let $A$ be an Artinian compressed Gorenstein graded $K$-algebra of socle degree $s \geq 4$, and let $A=R/I$ be a minimal presentation. Assume $I$ is generated in degree $\lfloor \frac{s}{2} \rfloor +1 $. Then $\mathop{\kern0pt\fam0rate}\nolimits(A) = \lfloor \frac{s}{2} \rfloor$. \end{corollary}
\section{The rate of a generic Gorenstein $K$-algebra}
Every Artinian Gorenstein graded algebra $A=R/I$ of socle degree $s,$ corresponds, up to scalars, to a form $F$ of degree $s$ in another set of variables as follows. Let $R=K[x_1, \dots, x_n] $ and $B=K[y_1, \dots, y_n]. $ Regard $B$ as a $R$-module via the action $x_i \circ F= \partial F/\partial y_i; $ this action can be extended to every element $G \in R $ acting as a differential operator on the elements of $B.$ Given a form $F$ of degree $s, $ we denote by $I_F$ the ideal of the elements of $R$ which annihilate $F$:
$$ I_F= \{G\in R \ | \ G \circ F=0\}.$$ Set $A_F= R/I_F.$ Then $A_F$ is a standard Artinian Gorenstein algebra of socle degree $s.$ Moreover, every ideal $I \subset R$ which defines a standard graded Artinian Gorenstein algebra of socle degree $s$ arises in this way. This correspondence is known as Macaulay's Inverse System. For more information we refer to the book of Iarrobino and Kanev \cite{IK} .
\begin{definition} Assume that $K$ is infinite. We say that an Artinian Gorenstein $K$-algebra $A$ of socle degree $s$ is generic if $A=A_F$ with $F \in B$ a generic form of degree $s$. \end{definition}
In particular generic Artinian Gorenstein $K$-algebras are compressed, see \cite[Proposition 3.12]{IK}.
\subsection{Generators of generic Artinian Gorenstein algebras}
It is known that Artinian compressed Gorenstein $K$-algebras of even socle degree $s=2(t-1)>1 $ have almost linear resolution (see \cite[4.7]{Iarrobino}). That is, if $n=\edim(A)$ and $A=R/I$ is a minimal presentation, then a minimal free resolution of $A$ over $R$ is almost linear: \[ \xymatrixcolsep{4.5mm} \xymatrix{ 0 \ar[r] & R(-(n+s)) \ar[r] & R^{a_{n-1}}(-(n+t-2)) \ar[r] & \ldots \ar[r] & R(-(t+1)) \ar[r] & R^{a_1}(-t) \ar[r] & R \ar[r] & A \ar[r] & 0. } \] In particular \[ m(I)=t=s/2 + 1. \] \vskip 2mm
If $s=2t-1$ and $n=3$ the Betti numbers of generic Gorenstein algebras are known, see for instance \cite[Example 3.16]{B99}, and the following holds: \[ m(I) = \begin{cases} t & \text{ if } t \text{ is even}\\ t +1 & \text{ if } t \text{ is odd} \end{cases}. \]
\vskip 2mm
The values of the graded Betti numbers of generic Artinian Gorenstein algebras with $n>3$ and odd socle degree $s=2t-1$ were conjectured by Boij \cite[Conjecture 3.13]{B99}. In fact, Boij proved that there exists a Zariski open set of Artinian compressed Gorenstein algebras all sharing the same numerical resolution, and he conjectured that such a set is non-empty. Computer generated evidence was given in \cite[Section 4]{B99}, but the conjecture was later disproved for special values of $n$ and $s$ by Kunte \cite[Corollary 5.6,\ Remark 5.7]{Ku}. However, in all known counterexamples the failure is in the middle of the resolution. The aim of this section is to prove that Boij's conjecture holds true at the beginning of the resolution. Namely, we prove that generic Artinian Gorenstein algebras of socle degree $s=2t-1$ are minimally generated in degree $t$.
We recall that an Artinian $K$-algebra $A$ of socle degree $s$ is called level if $\operatorname{soc}(A) \subseteq A_s$. The construction of the following lemma will be crucial in what follows.
\begin{lemma}\label{lemma:1}
For all integers $t \geq 2$ and $m\geq 3$ there exists a monomial ideal $J \subseteq S=K[x_1,\ldots,x_m]$ such that $m(J)=t$ and $A=S/J$ is an Artinian level algebra with socle degree $t$. \end{lemma} \begin{proof}
We start by considering the even case $t=2u$ with $u \geq 1$. Given $m \geq 3$, we let $j$ be an integer such that $1 \leq j < m$. We set $X=(x_1,\ldots,x_j)$, $Y=(x_{j+1},\ldots,x_m)$ and $\mathfrak{m}=X+Y$. We claim that the ideal $J=(X^2+Y^2)^u$ satisfies the desired conditions. It is clear that $J$ is a monomial ideal generated in degree $2u=s$. In order to show that $A=S/J$ is level of socle degree $s$, we first set up some notation: we let ${\bf x}=x_1\cdots x_j$, and given ${\bf a}=(a_1,\ldots,a_j) \in \mathbb{N}^j$ and we set ${\bf x}^{{\bf a}} = x_1^{a_1} \cdots x_j^{a_j}$. Similarly we let ${\bf y}=x_{j+1}\cdots x_m$, and given ${\bf b}=(b_{j+1},\ldots,b_m) \in \mathbb{N}^{m-j}$ we set ${\bf y}^{{\bf b}} = x_{j+1}^{b_{j+1}} \cdots x_m^{b_{m}}$. Note that $S$ is bigraded by setting $\deg(x_i)=(1,0)$ for all $x_i \in X$, and $\deg(x_i) = (0,1)$ for all $x_i \in Y$. In this way, if we let $a=|{\bf a}| = \sum_{i=1}^j a_i$ and $b=|{\bf b}| = \sum_{i=j+1}^m b_i$, then the monomial ${\bf x}^{\bf a}{\bf y}^{\bf b}$ is bigraded of degree $(a,b)$. Observe the generators of $J$ are precisely the monomials ${\bf x}^{\bf a}{\bf y}^{\bf b}$ of degree $a+b=s$ and such that both $a$ and $b$ are even.
To show that $A$ is level with $\socdeg(A)=t$ we first prove that $\mathfrak{m}^{t+1} \subseteq J$. Let ${\bf x}^{\bf a}{\bf y}^{\bf b}$ be a monomial of degree $a+b \geq t+1 = 2u+1$. It is easy to see that there exist vectors ${\bf a}' \leq {\bf a}$ and ${\bf b}' \leq {\bf b}$ (where the inequality is intended point-wise) such that $|{\bf a}'| =2h$ and $|{\bf b}'|=2(u-h)$ for some $h \in \mathbb{N}$. It follows that ${\bf x}^{\bf a} \in ({\bf x}^{{\bf a}'}) \subseteq X^{2h} = (X^2)^h$ and ${\bf y}^{\bf b} \in ({\bf y}^{{\bf b}'}) \subseteq Y^{2(u-h)} = (Y^2)^{u-h}$. Therefore, we conclude that ${\bf x}^{\bf a}{\bf y}^{\bf b} \in (X^2)^h(Y^2)^{u-h} \subseteq J$. This shows that $\socdeg(A) \leq t$. Now we show that $\socdeg(A) \geq t$. Since $J$ is a monomial ideal, so is $J:\mathfrak{m}$. Therefore, by degree considerations we only have to show that no monomial of degree $t-1=2u-1$ belongs to $J:\mathfrak{m}$. But if ${\bf x}^{\bf a} {\bf y}^{\bf b}$ has degree $a+b=t-1$, then either $a$ is odd and $b$ is even or viceversa. Assume without loss of generality that we are in the first case; if by contradiction we assume that ${\bf x}^{\bf a} {\bf y}^{\bf b} \in J:\mathfrak{m}$, then ${\bf x}^{\bf a} {\bf y}^{\bf b} \cdot x_{j+1} = {\bf x}^{\bf a}{\bf y}^{{\bf b}'} \in J$, where ${\bf b}'= {\bf b} + (1,0\ldots,0)$. By degree considerations, ${\bf x}^{\bf a}{\bf y}^{{\bf b}'}$ must then be one of the minimal generators of $J$; however, this contradicts our previous characterization of such generators, since $a$ is odd. This concludes the proof in the case $s$ is even.
We now focus on the case in which $t=2u+1$ is odd, with $u \geq 1$. Given $m \geq 3$, we let $j,k$ be integers such that $1 \leq j < k \leq m$, and we let $X=(x_1,\ldots,x_j), Y=(x_{j+1},\ldots,x_k)$, and $Z=(x_{k+1},\ldots,x_m)$. We also let $\mathfrak{m}=X+Y+Z$. We claim that the ideal $J=X(X^2+Y^2)^u+Y(Y^2+Z^2)^u+Z(Z^2+X^2)^u+XYZ((X+Y)^2+Z^2)^{u-1}$ satisfies the desired conditions. It is again clear that $J$ is a monomial ideal generated in degree $t$.
To show that $A$ is level of socle degree $t$, we consider a trigrading on $S$ by setting $\deg(x_i)=(1,0,0)$ for $x_i \in X$, $\deg(x_i) = (0,1,0)$ for $x_i \in Y$, and $\deg(x_i)=(0,0,1)$ for $x_i \in Z$. We let ${\bf x}=x_1\cdots x_j$, ${\bf y}=x_{j+1} \cdots x_k$ and ${\bf z}=x_{k+1}\cdots x_m$. Given vectors ${\bf a},{\bf b}$ and ${\bf c}$ of appropriate length and with non-negative entries, we set ${\bf x}^{\bf a}, {\bf y}^{\bf b}$ and ${\bf z}^{\bf c}$ in analogy with the previous case.
Observe that the minimal monomial generators of $J$ are all the monomials ${\bf x}^{\bf a}{\bf y}^{\bf b}{\bf z}^{\bf c}$ of degree $a+b+c=2u+1$ which satisfy one of the following four conditions: \begin{enumerate} \item $c=0$, $a$ is odd and $b$ is even. \item $a=0$, $b$ is odd and $c$ is even. \item $b=0$, $c$ is odd and $a$ is even. \item $abc \ne 0$, $a+b$ is even and $c$ is odd. \end{enumerate}
We start by showing that $\mathfrak{m}^{t+1} \subseteq J$. Assume that ${\bf x}^{\bf a}{\bf y}^{\bf b}{\bf z}^{\bf c}$ has degree $a+b+c \geq t+1 = 2(u+1)$; we distinguish some cases: if $abc \ne 0$, then we can find vectors ${\bf a}' \leq {\bf a}$, ${\bf b}' \leq {\bf b}$ and ${\bf c}' \leq {\bf c}$ with $|{\bf a}'|+|{\bf b}'|+|{\bf c}'| = 2u-1$ and such that ${\bf x}^{\bf a}{\bf y}^{\bf b}{\bf z}^{\bf c} \in XYZ \cdot ({\bf x}^{{\bf a}'}{\bf y}^{{\bf b}'}{\bf z}^{{\bf c}'})$. We have already shown in the even case that $\mathfrak{m}^{2u-1} \subseteq ((X+Y)^2+Z^2)^{u-1}$, and in particular we have that ${\bf x}^{{\bf a}'}{\bf y}^{{\bf b}'}{\bf z}^{{\bf c}'} \in ((X+Y)^2+Z^2)^{u-1}$. It follows that ${\bf x}^{\bf a}{\bf y}^{\bf b}{\bf z}^{\bf c} \in J$ in this case. Now assume that $abc=0$. If $a=0$, then since $b+c \geq 2(u+1)$ we can find ${\bf b}' \leq {\bf b}$ and ${\bf c}' \leq {\bf c}$ such that $|{\bf b}'|$ is odd, $|{\bf c}'|$ is even, and $|{\bf b}'|+|{\bf c}'| = 2u+1$. It follows that ${\bf x}^{\bf a}{\bf y}^{\bf b}{\bf z}^{\bf c} \in ({\bf y}^{{\bf b}'}{\bf z}^{{\bf c}'}) \subseteq Y(Y^2+Z^2)^u \subseteq J$. The remaining cases in which either $b=0$ or $c=0$ are handled similarly. This concludes the proof that $\socdeg(A) \leq t$.
To see that equality holds, as above it suffices to show that no monomial $w={\bf x}^{\bf a}{\bf y}^{\bf b}{\bf z}^{\bf c}$ of degree $a+b+c=t-1=2u$ belongs to $J:\mathfrak{m}$. Assume by way of contradiction that such a monomial $w$ belongs to $J:\mathfrak{m}$. If two of the exponents $a,b,c$ are equal to zero, for instance if $a=2u$ and $b=c=0$, then observe that $w \cdot x_{j+1} \in J$ must be a minimal monomial generator of $J$, by degree considerations. But $w \cdot x_{j+1} = {\bf x}^{\bf a} \cdot x_{j+1} = {\bf x}^{\bf a}{\bf y}^{{\bf b}'}{\bf z}^{\bf 0}$ with ${\bf b}'=(1,0,\ldots,0)$ does not belong to $J$, thanks to the above characterization of the minimal generators of $J$. This contradicts our choice of $w \in J:\mathfrak{m}$. The cases in which $a=b=0$ or $a=c=0$ are handled similarly. Now assume that only one among $a,b,c$ is equal to zero. For instance, say $a=0$ and $b+c=2u$. If $b$ is even, then we consider $w\cdot x_{k+1} = {\bf x}^{\bf 0}{\bf y}^{{\bf b}}{\bf z}^{{\bf c}'}$ with ${\bf c}'={\bf c}+(1,0,\ldots,0)$, which must be a minimal generator of $J$, necessarily of type (2). However $b$ is even, again a contradiction. If $b$ is odd, then we consider $w \cdot x_{j+1} = {\bf x}^{\bf 0}{\bf y}^{{\bf b}'}{\bf z}^{\bf c}$ with $|{\bf b}'| = b+1$ odd, and we reach again a similar contradiction. The other cases in which only one of the three exponents is equal to zero are tackled in a similar fashion.
Finally, assume that $abc\ne 0$. If $c$ is odd, then $w \cdot x_{k+1} = {\bf x}^{{\bf a}}{\bf y}^{\bf b}{\bf z}^{{\bf c}'}$ with $|{\bf c}'| = c+1$ must be a minimal generator of $J$, necessarily of type (4); but $c+1$ is even, a contradiction. If $c$ is even, then so is $a+b = 2u-c$. Then $w \cdot x_1 = {\bf x}^{{\bf a}'}{\bf y}^{\bf b}{\bf z}^{\bf c}$ with $|{\bf a}'| = a+1$ must be a minimal generator of $J$ of type (4); however $|{\bf a}'| + |{\bf b}| = a+1+b$ is odd, a contradiction. \end{proof}
\begin{theorem} \label{generation}
For all odd integers $s=2t-1 \geq 3$ and all $n\ge 4$ there exists an Artinian compressed
Gorenstein algebra $A= K[x_1,\ldots,x_n]/I$ with socle degree $s$ and such that $m(I)=t$. \end{theorem}
\begin{proof}
From Lemma~\ref{lemma:1} we can construct a monomial ideal $J \subseteq S= K[x_1,\dots,x_{n-1}]$ with $m(J)=t$ and $\operatorname{soc}(S/J)$ concentrated only in degree $t$. From the contruction by Hartshorne, there is a reduced set of
points $X$ in $\mathbb P^{n-1}$ such that the Artinan reduction
$K[x_1,\dots,x_n]/(I_X + (x_n))$ is isomorphic to $S/J$. Now, the canonical module $\omega_X$ can be embedded as an ideal in $R_X =K[x_1,\dots,x_n]/I_X$. This can be done in all degrees $i$ where $\dim_k((R_X)_i) = \deg(X)$ holds \cite{Kr00}. Since $t$ is the socle degree of the Artinian reduction $S/J$ of $R_X$ we have the equality $\dim_k((R_X)_t) =\deg(X)$, and therefore such an embedding can be obtained in degree $t$. Let $\mathfrak{a}$ be such a canonical ideal. If we let $I \subseteq R=K[x_1,\ldots,x_n]$ be the ideal such that $I/I_X=\mathfrak{a}$, then since both $\mathfrak{a} \subseteq R_X$ and $I_X \subseteq R$ are generated in degree $t$ we conclude that $I$ is generated in degree $t$ as well. Thus $A=R_X/\mathfrak{a} \cong R/I$ is an Artinian Gorenstein $K$-algebra with defining ideal $I$ generated only in degree $t$. Note that $\socdeg(A) = 2t-1$. In fact, $\omega_X$ has regularity $t$, and since $\mathfrak{a}$ is a canonical ideal generated in degree $t$ we have that $\dim_k(\mathfrak{a}_i) = \deg(X)$ if and only if $i \geq 2t$. It follows that $A_i=0$ for all $i \geq 2t$, while $A_{2t-1} \ne 0$. Finally, since $I$ has no generators in
degree less than $t$, we conclude that $A$ is compressed. \end{proof}
\begin{remark}
The construction of Theorem \ref{generation} also works for $n=3$ and $s$ an even integer. However, we point out that for $s =2t-1$ with $t \geq 3$ odd there is no Artinian Gorenstein compressed algebra $K[x_1,x_2,x_3]/I$ generated only in degree $t$. In fact, if such an algebra existed, the ideal $I $ would have an even number of generators, contradicting the structure theorem of Buchsbaum and Eisenbud for ideals of codimension three defining Gorenstein rings \cite{BE}, Corollary 2.2. For this reason we assume $n\ge 4.$ \end{remark}
As a consequence of the above results, we may conclude that Boij's conjecture holds true for the first Betti number. Since generic Artinian Gorenstein graded $K$-algebras are compressed, it is enough to observe that, by Theorem \ref{generation}, the Zariski open set parameterizing the Artinian compressed Gorenstein algebras generated in one degree is non-empty.
\begin{corollary} \label{generic} A generic Artinian Gorenstein $K$-algebra $A$ with $\edim(A) \geq 4$ and socle degree $s \geq 3 $ is generated in degree $\lfloor \frac{s}{2} \rfloor +1$. \end{corollary}
\subsection{Rate of of generic Artinian Gorenstein algebras} { Thanks to Corollary \ref{generic} we have the following result.}
\begin{theorem} \label{main} Let $A$ be a generic Artinian Gorenstein graded $K$-algebra of socle degree $s \geq 3$, and assume that $\edim(A) \geq 4$. Then $\mathop{\kern0pt\fam0rate}\nolimits(A) = \lfloor \frac{s}{2} \rfloor $. \end{theorem} \begin{proof} Since generic Artinian Gorenstein graded $K$-algebras are compressed, by Proposition \ref{prop compressed} we have that the generic Artinian Gorenstein $K$-algebra $A$ satisfies $\mathop{\kern0pt\fam0rate}\nolimits(A) \leq m(I)-1$, with $A=R/I$ a minimal presentation. On the other hand, the reverse inequality always holds, and therefore $\mathop{\kern0pt\fam0rate}\nolimits(A)=m(I)-1$. Finally, by Theorem \ref{generation}, $m(I) = \lfloor \frac{s}{2} \rfloor+1$, and the result follows. \end{proof}
\begin{remark} Artinian compressed Gorenstein $K$-algebras of even socle degree $s=2(t-1)$ have almost linear resolution and, in particular, they are generated in degree $t$. Hence the conclusion of Theorem \ref{main} holds for any Artinian Gorenstein compressed $K$-algebra of even socle degree, not necessarily generic. \end{remark}
\begin{remark} When $A=R/I$ is a generic Artinian Gorenstein $K$-algebra of odd socle degree $s=2t-1$ with $\edim(A)=3$ we have that \[ \mathop{\kern0pt\fam0rate}\nolimits(A) = \begin{cases} t-1 & \text{ if } t \text{ is even}\\ t & \text{ if } t \text{ is odd} \end{cases}. \] In fact, if $s=3$ then $A$ is Koszul by \cite{CRV}. If $s \geq 5$ then in the first case $I$ is generated in degree $t$, while in the second there is one generator in degree $t+1$. In both cases we conclude by Proposition \ref{prop compressed} since $A$ is compressed. \end{remark}
\end{document} | arXiv |
Proof theory
Proof theory is a major branch[1] of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
Some of the major areas of proof theory include structural proof theory, ordinal analysis, provability logic, reverse mathematics, proof mining, automated theorem proving, and proof complexity. Much research also focuses on applications in computer science, linguistics, and philosophy.
History
Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of Mathematics. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable $\Pi _{1}^{0}$ sentences) are finitarily true; once so grounded we do not care about the non-finitary meaning of their existential theorems, regarding these as pseudo-meaningful stipulations of the existence of ideal entities.
The failure of the program was induced by Kurt Gödel's incompleteness theorems, which showed that any ω-consistent theory that is sufficiently strong to express certain simple arithmetic truths, cannot prove its own consistency, which on Gödel's formulation is a $\Pi _{1}^{0}$ sentence. However, modified versions of Hilbert's program emerged and research has been carried out on related topics. This has led, in particular, to:
• Refinement of Gödel's result, particularly J. Barkley Rosser's refinement, weakening the above requirement of ω-consistency to simple consistency;
• Axiomatisation of the core of Gödel's result in terms of a modal language, provability logic;
• Transfinite iteration of theories, due to Alan Turing and Solomon Feferman;
• The discovery of self-verifying theories, systems strong enough to talk about themselves, but too weak to carry out the diagonal argument that is the key to Gödel's unprovability argument.
In parallel to the rise and fall of Hilbert's program, the foundations of structural proof theory were being founded. Jan Łukasiewicz suggested in 1926 that one could improve on Hilbert systems as a basis for the axiomatic presentation of logic if one allowed the drawing of conclusions from assumptions in the inference rules of the logic. In response to this, Stanisław Jaśkowski (1929) and Gerhard Gentzen (1934) independently provided such systems, called calculi of natural deduction, with Gentzen's approach introducing the idea of symmetry between the grounds for asserting propositions, expressed in introduction rules, and the consequences of accepting propositions in the elimination rules, an idea that has proved very important in proof theory.[2] Gentzen (1934) further introduced the idea of the sequent calculus, a calculus advanced in a similar spirit that better expressed the duality of the logical connectives,[3] and went on to make fundamental advances in the formalisation of intuitionistic logic, and provide the first combinatorial proof of the consistency of Peano arithmetic. Together, the presentation of natural deduction and the sequent calculus introduced the fundamental idea of analytic proof to proof theory.
Structural proof theory
Main article: Structural proof theory
Structural proof theory is the subdiscipline of proof theory that studies the specifics of proof calculi. The three most well-known styles of proof calculi are:
• The Hilbert calculi
• The natural deduction calculi
• The sequent calculi
Each of these can give a complete and axiomatic formalization of propositional or predicate logic of either the classical or intuitionistic flavour, almost any modal logic, and many substructural logics, such as relevance logic or linear logic. Indeed, it is unusual to find a logic that resists being represented in one of these calculi.
Proof theorists are typically interested in proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. Much of the interest in cut-free proofs comes from the subformula property: every formula in the end sequent of a cut-free proof is a subformula of one of the premises. This allows one to show consistency of the sequent calculus easily; if the empty sequent were derivable it would have to be a subformula of some premise, which it is not. Gentzen's midsequent theorem, the Craig interpolation theorem, and Herbrand's theorem also follow as corollaries of the cut-elimination theorem.
Gentzen's natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz. The definition is slightly more complex: we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Girard's proof nets also support a notion of analytic proof.
A particular family of analytic proofs arising in reductive logic are focused proofs which characterise a large family of goal-directed proof-search procedures. The ability to transform a proof system into a focused form is a good indication of its syntactic quality, in a manner similar to how admissibility of cut shows that a proof system is syntactically consistent.[4]
Structural proof theory is connected to type theory by means of the Curry–Howard correspondence, which observes a structural analogy between the process of normalisation in the natural deduction calculus and beta reduction in the typed lambda calculus. This provides the foundation for the intuitionistic type theory developed by Per Martin-Löf, and is often extended to a three way correspondence, the third leg of which are the cartesian closed categories.
Other research topics in structural theory include analytic tableau, which apply the central idea of analytic proof from structural proof theory to provide decision procedures and semi-decision procedures for a wide range of logics, and the proof theory of substructural logics.
Ordinal analysis
Main article: Ordinal analysis
Ordinal analysis is a powerful technique for providing combinatorial consistency proofs for subsystems of arithmetic, analysis, and set theory. Gödel's second incompleteness theorem is often interpreted as demonstrating that finitistic consistency proofs are impossible for theories of sufficient strength. Ordinal analysis allows one to measure precisely the infinitary content of the consistency of theories. For a consistent recursively axiomatized theory T, one can prove in finitistic arithmetic that the well-foundedness of a certain transfinite ordinal implies the consistency of T. Gödel's second incompleteness theorem implies that the well-foundedness of such an ordinal cannot be proved in the theory T.
Consequences of ordinal analysis include (1) consistency of subsystems of classical second order arithmetic and set theory relative to constructive theories, (2) combinatorial independence results, and (3) classifications of provably total recursive functions and provably well-founded ordinals.
Ordinal analysis was originated by Gentzen, who proved the consistency of Peano Arithmetic using transfinite induction up to ordinal ε0. Ordinal analysis has been extended to many fragments of first and second order arithmetic and set theory. One major challenge has been the ordinal analysis of impredicative theories. The first breakthrough in this direction was Takeuti's proof of the consistency of Π1
1
-CA0 using the method of ordinal diagrams.
Provability logic
Main article: Provability logic
Provability logic is a modal logic, in which the box operator is interpreted as 'it is provable that'. The point is to capture the notion of a proof predicate of a reasonably rich formal theory. As basic axioms of the provability logic GL (Gödel-Löb), which captures provable in Peano Arithmetic, one takes modal analogues of the Hilbert-Bernays derivability conditions and Löb's theorem (if it is provable that the provability of A implies A, then A is provable).
Some of the basic results concerning the incompleteness of Peano Arithmetic and related theories have analogues in provability logic. For example, it is a theorem in GL that if a contradiction is not provable then it is not provable that a contradiction is not provable (Gödel's second incompleteness theorem). There are also modal analogues of the fixed-point theorem. Robert Solovay proved that the modal logic GL is complete with respect to Peano Arithmetic. That is, the propositional theory of provability in Peano Arithmetic is completely represented by the modal logic GL. This straightforwardly implies that propositional reasoning about provability in Peano Arithmetic is complete and decidable.
Other research in provability logic has focused on first-order provability logic, polymodal provability logic (with one modality representing provability in the object theory and another representing provability in the meta-theory), and interpretability logics intended to capture the interaction between provability and interpretability. Some very recent research has involved applications of graded provability algebras to the ordinal analysis of arithmetical theories.
Reverse mathematics
Main article: Reverse mathematics
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics.[5] The field was founded by Harvey Friedman. Its defining method can be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.
In reverse mathematics, one starts with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems. For example, to study the theorem "Every bounded sequence of real numbers has a supremum" it is necessary to use a base system that can speak of real numbers and sequences of real numbers.
For each theorem that can be stated in the base system but is not provable in the base system, the goal is to determine the particular axiom system (stronger than the base system) that is necessary to prove that theorem. To show that a system S is required to prove a theorem T, two proofs are required. The first proof shows T is provable from S; this is an ordinary mathematical proof along with a justification that it can be carried out in the system S. The second proof, known as a reversal, shows that T itself implies S; this proof is carried out in the base system. The reversal establishes that no axiom system S′ that extends the base system can be weaker than S while still proving T.
One striking phenomenon in reverse mathematics is the robustness of the Big Five axiom systems. In order of increasing strength, these systems are named by the initialisms RCA0, WKL0, ACA0, ATR0, and Π1
1
-CA0. Nearly every theorem of ordinary mathematics that has been reverse mathematically analyzed has been proven equivalent to one of these five systems. Much recent research has focused on combinatorial principles that do not fit neatly into this framework, like RT2
2
(Ramsey's theorem for pairs).
Research in reverse mathematics often incorporates methods and techniques from recursion theory as well as proof theory.
Functional interpretations
Functional interpretations are interpretations of non-constructive theories in functional ones. Functional interpretations usually proceed in two stages. First, one "reduces" a classical theory C to an intuitionistic one I. That is, one provides a constructive mapping that translates the theorems of C to the theorems of I. Second, one reduces the intuitionistic theory I to a quantifier free theory of functionals F. These interpretations contribute to a form of Hilbert's program, since they prove the consistency of classical theories relative to constructive ones. Successful functional interpretations have yielded reductions of infinitary theories to finitary theories and impredicative theories to predicative ones.
Functional interpretations also provide a way to extract constructive information from proofs in the reduced theory. As a direct consequence of the interpretation one usually obtains the result that any recursive function whose totality can be proven either in I or in C is represented by a term of F. If one can provide an additional interpretation of F in I, which is sometimes possible, this characterization is in fact usually shown to be exact. It often turns out that the terms of F coincide with a natural class of functions, such as the primitive recursive or polynomial-time computable functions. Functional interpretations have also been used to provide ordinal analyses of theories and classify their provably recursive functions.
The study of functional interpretations began with Kurt Gödel's interpretation of intuitionistic arithmetic in a quantifier-free theory of functionals of finite type. This interpretation is commonly known as the Dialectica interpretation. Together with the double-negation interpretation of classical logic in intuitionistic logic, it provides a reduction of classical arithmetic to intuitionistic arithmetic.
Formal and informal proof
Main article: Formal proof
The informal proofs of everyday mathematical practice are unlike the formal proofs of proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, given enough time and patience. For most mathematicians, writing a fully formal proof is too pedantic and long-winded to be in common use.
Formal proofs are constructed with the help of computers in interactive theorem proving. Significantly, these proofs can be checked automatically, also by computer. Checking formal proofs is usually simple, whereas finding proofs (automated theorem proving) is generally hard. An informal proof in the mathematics literature, by contrast, requires weeks of peer review to be checked, and may still contain errors.
Proof-theoretic semantics
In linguistics, type-logical grammar, categorial grammar and Montague grammar apply formalisms based on structural proof theory to give a formal natural language semantics.
See also
• Intermediate logic
• Model theory
• Proof (truth)
• Proof techniques
• Sequent calculus
Notes
1. According to Wang (1981), pp. 3–4, proof theory is one of four domains mathematical logic, together with model theory, axiomatic set theory, and recursion theory. Barwise (1978) consists of four corresponding parts, with part D being about "Proof Theory and Constructive Mathematics".
2. Prawitz (2006, p. 98) harvtxt error: no target: CITEREFPrawitz2006 (help).
3. Girard, Lafont, and Taylor (1988).
4. Chaudhuri, Kaustuv; Marin, Sonia; Straßburger, Lutz (2016), "Focused and Synthetic Nested Sequents", Lecture Notes in Computer Science, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 390–407, doi:10.1007/978-3-662-49630-5_23, ISBN 978-3-662-49629-9
5. Simpson 2010
References
• J. Avigad and E.H. Reck (2001). "'Clarifying the nature of the infinite': the development of metamathematics and proof theory". Carnegie-Mellon Technical Report CMU-PHIL-120.
• J. Barwise, ed. (1978). Handbook of Mathematical Logic. North-Holland.
• S. Buss, ed. (1998) Handbook of Proof Theory. Elsevier.
• G. Gentzen (1935/1969). "Investigations into logical deduction". In M. E. Szabo, ed. Collected Papers of Gerhard Gentzen. North-Holland. Translated by Szabo from "Untersuchungen über das logische Schliessen", Mathematisches Zeitschrift v. 39, pp. 176–210, 405 431.
• J.-Y. Girard, P. Taylor, Y. Lafont (1988). "Proofs and types". Cambridge University Press. ISBN 0-521-37181-3
• D. Prawitz (1965). Natural deduction: A proof-theoretical study, Dover Publications, ISBN 978-0-486-44655-4
• S.G. Simpson (2010). Subsystems of Second-order Arithmetic, second edition. Cambridge University Press, ISBN 978-0521150149.
• A. S. Troelstra and H. Schwichtenberg (1996). Basic Proof Theory, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, ISBN 0-521-77911-1.
• H. Wang (1981). Popular Lectures on Mathematical Logic, Van Nostrand Reinhold Company, ISBN 0-442-23109-1.
External links
Wikimedia Commons has media related to Proof theory.
• "Proof theory", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• J. von Plato (2008). The Development of Proof Theory. Stanford Encyclopedia of Philosophy.
Mathematical logic
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Chapter 6 - Ibn Ṭūlūn's Pacification Campaign: Sedition, Authority, and Empire in Abbasid Egypt
from Part I - Political and Administrative Connections
By Matthew S. Gordon
Edited by Jelle Bruning, Janneke H. M. de Jong, Petra M. Sijpesteijn
Book: Egypt and the Eastern Mediterranean World
Published online: 01 December 2022
Print publication: 15 December 2022, pp 169-202
Developments in third/ninth-century Egypt relate to the wider history of the Abbasid imperial realm in a number of ways. These had to do, in one sense or another, with the fraying of the Arab Islamic empire, governed, at this point, by the Abbasid house.1 This chapter considers one such development: the turn to control over Egypt by the Turkic–Central Asian military command in Samarra. My argument is that, at a moment in which the Abbasid state was struggling to sustain its hold over a once far-flung but now shrinking domain, it ceded authority over Egypt to those same military/political circles. Egypt, in this scenario, was a key interest of the Samarran commanders and in defense of which they devoted considerable energy and resources. It was a matter of consolidating authority over the province's considerable public wealth, to be sure, but the sources point to apparent private interests – specifically, landholdings – on the part of the commanders as well.
The Evolutionary Map of the Universe Pilot Survey – ADDENDUM
Ray P. Norris, Joshua Marvil, J. D. Collier, Anna D. Kapińska, Andrew N. O'Brien, L. Rudnick, Heinz Andernach, Jacobo Asorey, Michael J. I. Brown, Marcus Brüggen, Evan Crawford, Jayanne English, Syed Faisal ur Rahman, Miroslav D. Filipović, Yjan Gordon, Gülay Gürkan, Catherine Hale, Andrew M. Hopkins, Minh T. Huynh, Kim HyeongHan, M. James Jee, Bärbel S. Koribalski, Emil Lenc, Kieran Luken, David Parkinson, Isabella Prandoni, Wasim Raja, Thomas H. Reiprich, Christopher J. Riseley, Stanislav S. Shabala, Jaimie R. Sheil, Tessa Vernstrom, Matthew T. Whiting, James R. Allison, C. S. Anderson, Lewis Ball, Martin Bell, John Bunton, T. J. Galvin, Neeraj Gupta, Aidan Hotan, Colin Jacka, Peter J. Macgregor, Elizabeth K. Mahony, Umberto Maio, Vanessa Moss, M. Pandey-Pommier, Maxim A. Voronkov
Journal: Publications of the Astronomical Society of Australia / Volume 39 / 2022
Published online by Cambridge University Press: 02 November 2022, e055
ʿUthman ibn ʿAffan: Legend or Liability? Heather N. Keaney (London: Oneworld Academic, 2021). Pp. 159. $30.00 cloth. ISBN: 9781786076977
Matthew S. Gordon
Journal: International Journal of Middle East Studies / Volume 54 / Issue 2 / May 2022
Securely sharing DSMB reports to speed decision making from multiple, concurrent, independent studies of similar treatments in COVID-19
Natalie A. Dilts, Frank E. Harrell, Christopher J. Lindsell, Samuel Nwosu, Thomas G. Stewart, Matthew S. Shotwell, Jill M. Pulley, Terri L. Edwards, Emily Sheffer Serdoz, Katelyn Benhoff, Gordon R. Bernard
Journal: Journal of Clinical and Translational Science / Volume 6 / Issue 1 / 2022
Published online by Cambridge University Press: 11 April 2022, e49
As clinical trials were rapidly initiated in response to the COVID-19 pandemic, Data and Safety Monitoring Boards (DSMBs) faced unique challenges overseeing trials of therapies never tested in a disease not yet characterized. Traditionally, individual DSMBs do not interact or have the benefit of seeing data from other accruing trials for an aggregated analysis to meaningfully interpret safety signals of similar therapeutics. In response, we developed a compliant DSMB Coordination (DSMBc) framework to allow the DSMB from one study investigating the use of SARS-CoV-2 convalescent plasma to treat COVID-19 to review data from similar ongoing studies for the purpose of safety monitoring.
The DSMBc process included engagement of DSMB chairs and board members, execution of contractual agreements, secure data acquisition, generation of harmonized reports utilizing statistical graphics, and secure report sharing with DSMB members. Detailed process maps, a secure portal for managing DSMB reports, and templates for data sharing and confidentiality agreements were developed.
Four trials participated. Data from one trial were successfully harmonized with that of an ongoing trial. Harmonized reports allowing for visualization and drill down into the data were presented to the ongoing trial's DSMB. While DSMB deliberations are confidential, the Chair confirmed successful review of the harmonized report.
It is feasible to coordinate DSMB reviews of multiple independent studies of a similar therapeutic in similar patient cohorts. The materials presented mitigate challenges to DSMBc and will help expand these initiatives so DSMBs may make more informed decisions with all available information.
The Evolutionary Map of the Universe pilot survey
Australian SKA Pathfinder
Published online by Cambridge University Press: 07 September 2021, e046
We present the data and initial results from the first pilot survey of the Evolutionary Map of the Universe (EMU), observed at 944 MHz with the Australian Square Kilometre Array Pathfinder (ASKAP) telescope. The survey covers $270 \,\mathrm{deg}^2$ of an area covered by the Dark Energy Survey, reaching a depth of 25–30 $\mu\mathrm{Jy\ beam}^{-1}$ rms at a spatial resolution of $\sim$ 11–18 arcsec, resulting in a catalogue of $\sim$ 220 000 sources, of which $\sim$ 180 000 are single-component sources. Here we present the catalogue of single-component sources, together with (where available) optical and infrared cross-identifications, classifications, and redshifts. This survey explores a new region of parameter space compared to previous surveys. Specifically, the EMU Pilot Survey has a high density of sources, and also a high sensitivity to low surface brightness emission. These properties result in the detection of types of sources that were rarely seen in or absent from previous surveys. We present some of these new results here.
Chapter 14 - Slavery in the Islamic Middle East (c. 600–1000 CE)
from Part IV - The Islamic World
Edited by Craig Perry, Emory University, Atlanta, David Eltis, Emory University, Atlanta, Stanley L. Engerman, University of Rochester, New York, David Richardson, University of Hull
Book: The Cambridge World History of Slavery
Print publication: 12 August 2021, pp 337-361
Much evidence – textual, material and documentary – points to slavery in the early and medieval Islamic Middle East (c. 600-1000 CE) as a social fact, persistent and multivalent. This is especially true for the urban landscape: the presence of enslaved and freed persons would have been impossible to miss. More difficult is the reconstruction of Middle Eastern agrarian slavery. This is a survey essay with particular reference to the early Abbasid Caliphate (c. 750-950) and select questions around which debate in modern scholarship has grown. One must comb medieval Arabic texts (literary and documentary) to reconstruct patterns of early Islamic-era enslavement; the organization and dynamics of slave commerce; the demands on slave and freed labor; and the (relative) social integration of the enslaved. The Arabic/Islamic library illuminates all manner of topics, religious and secular alike. Literary references to slavery and/or enslaved persons therein are plentiful and of a great variety. One has references in works of poetry and adab, an elastic term used for a variety of Arabic prose writings. Equally numerous are references in chronicles, biographical dictionaries, and works of geography and political thought. Medieval Arabic legal and religious writings provide a considerable number of references as well.
1 - The early Islamic empire and the introduction of military slavery
from Part I - Foundations, c.600–1000 ce
Edited by Anne Curry, University of Southampton
David A. Graff, Kansas State University
Book: The Cambridge History of War
Print publication: 01 October 2020, pp 17-49
The inhabitants of seventh-century Arabia mobilized for warfare in a manner new to that region of the Near East: the effort fell, albeit gradually, under central authority.1 Arabia had long been a highly variegated cultural zone, encompassing the Syrian Desert, southern Mesopotamia, and the Arabian Peninsula. Acting in tandem, largely nomadic tribal forces accepted the leadership of sedentary townsmen, the great number of whom belonged to the Quraysh, an influential tribe of two towns of the Hejaz region, Mecca and Medina. If Yemen and south Arabia had long known the rule of kings, and, hence, more formal military organization, only now did the central and northern stretches of the Peninsula and southern Syria experience what can thereby be considered as early state formation. The effort was driven by an equally untested set of ethical and spiritual teachings. A charismatic figure, Muḥammad ibn ʿAbdallāh (570–632), preached a strict monotheism; these teachings served as the seedbed of what would soon be known as Islam.
Part I - Beyond Warfare: Armies, Tribes and Lords
Edited by Matthew S. Gordon, University of Miami, Richard W. Kaeuper, University of Rochester, New York, Harriet Zurndorfer, Universiteit Leiden
Book: The Cambridge World History of Violence
Print publication: 26 March 2020, pp 17-120
Part II - The Violence of Governments and Rulers
Print publication: 26 March 2020, pp 121-204
Print publication: 26 March 2020, pp iv-iv
Contributors to Volume II
Print publication: 26 March 2020, pp xii-xiv
Part III - Social, Interpersonal and Collective Violence
Part IV - Religious, Sacred and Ritualised Violence
Figures and Table
Print publication: 26 March 2020, pp ix-xi
Print publication: 26 March 2020, pp v-viii
Introduction to Volume II
By Matthew S. Gordon, Richard W. Kaeuper, Harriet Zurndorfer
Print publication: 26 March 2020, pp 1-16
This introductory chapter of volume II of the Cambridge World History of Violence, which focuses on the thousand years between 500 and 1500, or what is also known as the Middle Millennium, examines .institutions and forms of violence in the geographical area including Japan and China, Central Asia, North Africa, and Europe, with two additional chapters extending coverage into Aztec and Mayan culture. The topics of this introduction are set in four contexts in which violence occurred across this broad chronology and vast territory. They are: the formation of centralized polities through war and conquest; institution building and ideological expression by these same polities; control of extensive trade networks; and the emergence and dominance of religious ecumenes. Attention is also given to the idea of how theories of violence are relevant to the specific historical circumstances discussed in the volume's chapters. A final section on the depiction of violence, both visual and literary, demonstrates the ubiquity of societal efforts to confront meanings of violence during this longue durée.
Part V - Depictions of Violence
The Cambridge World History of Violence
Volume 2, AD 500–AD 1500
Edited by Matthew S. Gordon, Richard W. Kaeuper, Harriet Zurndorfer
Print publication: 26 March 2020
Buy the print book
Violence permeated much of social life across the vast geographical space of the European, American, Asian and Islamic lands and through the broad sweep of what is often termed the Middle Millennium (roughly 500 to 1500). Focusing on four contexts in which violence occurred across this huge area, the contributors to this volume explore the formation of centralised polities through war and conquest; institution building and ideological expression by these same polities; control of extensive trade networks; and the emergence and dominance of religious ecumenes. Attention is also given to the idea of how theories of violence are relevant to the specific historical circumstances discussed in the volume's chapters. A final section on the depiction of violence, both visual and literary, demonstrates the ubiquity of societal efforts to confront meanings of violence during this longue durée.
The Taipan Galaxy Survey: Scientific Goals and Observing Strategy
Elisabete da Cunha, Andrew M. Hopkins, Matthew Colless, Edward N. Taylor, Chris Blake, Cullan Howlett, Christina Magoulas, John R. Lucey, Claudia Lagos, Kyler Kuehn, Yjan Gordon, Dilyar Barat, Fuyan Bian, Christian Wolf, Michael J. Cowley, Marc White, Ixandra Achitouv, Maciej Bilicki, Joss Bland-Hawthorn, Krzysztof Bolejko, Michael J. I. Brown, Rebecca Brown, Julia Bryant, Scott Croom, Tamara M. Davis, Simon P. Driver, Miroslav D. Filipovic, Samuel R. Hinton, Melanie Johnston-Hollitt, D. Heath Jones, Bärbel Koribalski, Dane Kleiner, Jon Lawrence, Nuria Lorente, Jeremy Mould, Matt S. Owers, Kevin Pimbblet, C. G. Tinney, Nicholas F. H. Tothill, Fred Watson
Published online by Cambridge University Press: 24 October 2017, e047
The Taipan galaxy survey (hereafter simply 'Taipan') is a multi-object spectroscopic survey starting in 2017 that will cover 2π steradians over the southern sky (δ ≲ 10°, |b| ≳ 10°), and obtain optical spectra for about two million galaxies out to z < 0.4. Taipan will use the newly refurbished 1.2-m UK Schmidt Telescope at Siding Spring Observatory with the new TAIPAN instrument, which includes an innovative 'Starbugs' positioning system capable of rapidly and simultaneously deploying up to 150 spectroscopic fibres (and up to 300 with a proposed upgrade) over the 6° diameter focal plane, and a purpose-built spectrograph operating in the range from 370 to 870 nm with resolving power R ≳ 2000. The main scientific goals of Taipan are (i) to measure the distance scale of the Universe (primarily governed by the local expansion rate, H0) to 1% precision, and the growth rate of structure to 5%; (ii) to make the most extensive map yet constructed of the total mass distribution and motions in the local Universe, using peculiar velocities based on improved Fundamental Plane distances, which will enable sensitive tests of gravitational physics; and (iii) to deliver a legacy sample of low-redshift galaxies as a unique laboratory for studying galaxy evolution as a function of dark matter halo and stellar mass and environment. The final survey, which will be completed within 5 yrs, will consist of a complete magnitude-limited sample (i ⩽ 17) of about 1.2 × 106 galaxies supplemented by an extension to higher redshifts and fainter magnitudes (i ⩽ 18.1) of a luminous red galaxy sample of about 0.8 × 106 galaxies. Observations and data processing will be carried out remotely and in a fully automated way, using a purpose-built automated 'virtual observer' software and an automated data reduction pipeline. The Taipan survey is deliberately designed to maximise its legacy value by complementing and enhancing current and planned surveys of the southern sky at wavelengths from the optical to the radio; it will become the primary redshift and optical spectroscopic reference catalogue for the local extragalactic Universe in the southern sky for the coming decade. | CommonCrawl |
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Origin of Devonian mafic magmatism in the East Kunlun orogenic belt, northern Tibetan Plateau: implications for continental exhumation
You-Jun Tang, Bin Liu, Mei-Jun Li, Yue Wu, Jian Huang, Shao-Qing Zhao, Yang Sun
Journal: Geological Magazine , First View
Published online by Cambridge University Press: 09 January 2020, pp. 1-16
This paper presents a comprehensive study of the zircon geochronology, geochemistry and Sr–Nd isotope geology of Devonian mafic rocks developed in the East Kunlun orogenic belt, northern Tibetan Plateau, and reveals their mantle sources, petrogenesis and geodynamic implications for continental exhumation. The zircon geochronology of typical samples indicates that these mafic rocks crystallized at 406∼408 Ma. They can be classified into two different groups based on petrographic observations and geochemical compositions. Group 1 rocks exhibit low TiO2 and FeOt contents and Nb/Y ratios and have enriched mid-ocean ridge basalt (E-MORB)-like compositions with slight negative Nb and Ta anomalies. However, Group 2 rocks have distinctly high TiO2 and FeOt contents and Nb/Y ratios, comparable to typical Fe–Ti-rich mafic rocks worldwide. All the samples exhibit weak enrichments in light rare earth elements, Nb and Ta relative to the primitive mantle. Based on geochemical and isotopic studies, Group 1 rocks are suggested to be derived from depleted asthenospheric mantle that was metasomatized by c. 3–5 % continental crustal components, while Group 2 rocks originated from partial melting of enriched lithospheric mantle. The high contents of Fe, Ti and Nb for Group 2 rocks could be attributed to a high degree of olivine crystallization under low fO2 conditions with delayed nucleation of Fe–Ti oxides. Combining those results with other geological data, we conclude that slab break-off was the key factor causing exhumation of eclogites and triggering flare-up of the Devonian magmatism, and that continental collision or continental subduction may have initiated at 431∼436 Ma.
Seasonality of the transmissibility of hand, foot and mouth disease: a modelling study in Xiamen City, China
Zehong Huang, Mingzhai Wang, Luxia Qiu, Ning Wang, Zeyu Zhao, Jia Rui, Yao Wang, Xingchun Liu, Mikah Ngwanguong Hannah, Benhua Zhao, Yanhua Su, Bin Zhao, Tianmu Chen
Journal: Epidemiology & Infection / Volume 147 / 2019
Published online by Cambridge University Press: 30 December 2019, e327
This study attempts to figure out the seasonality of the transmissibility of hand, foot and mouth disease (HFMD). A mathematical model was established to calculate the transmissibility based on the reported data for HFMD in Xiamen City, China from 2014 to 2018. The transmissibility was measured by effective reproduction number (Reff) in order to evaluate the seasonal characteristics of HFMD. A total of 43 659 HFMD cases were reported in Xiamen, for the period 2014 to 2018. The median of annual incidence was 221.87 per 100 000 persons (range: 167.98/100,000–283.34/100 000). The reported data had a great fitting effect with the model (R2 = 0.9212, P < 0.0001), it has been shown that there are two epidemic peaks of HFMD in Xiamen every year. Both incidence and effective reproduction number had seasonal characteristics. The peak of incidence, 1–2 months later than the effective reproduction number, occurred in Summer and Autumn, that is, June and October each year. Both the incidence and transmissibility of HFMD have obvious seasonal characteristics, and two annual epidemic peaks as well. The peak of incidence is 1–2 months later than Reff.
Malnutrition Screening and Acute Kidney Injury in Hospitalized Patients: a Retrospective Study Over 5-year Period from China
Chenyu Li, Lingyu Xu, Chen Guan, Long Zhao, Congjuan Luo, Bin Zhou, Xiaosu Zhang, Jing Wang, Jun Zhao, Junyan Huang, Dan Li, Hong Luan, Xiaofei Man, Lin Che, Yanfei Wang, Hui Zhang, Yan Xu
Journal: British Journal of Nutrition /
Published online by Cambridge University Press: 28 October 2019, pp. 1-26
Malnutrition and acute kidney injury (AKI) are common complications in hospitalized patients, and both increase mortality; however, the relationship between them is unknown. This is a retrospective propensity score matching study enrolling 46,549 inpatients, aimed to investigate the association between Nutritional Risk Screening 2002 (NRS-2002) and AKI, and to assess the ability of NRS-2002 and AKI in predicting prognosis. In total, 37,190 (80%) and 9,359 (20%) patients had NRS-2002 scores < 3 and ≥ 3, respectively. Patients with NRS-2002 scores ≥ 3 had longer lengths of stay (12.6±7.8 days vs. 10.4±6.2 days, P < 0.05), higher mortality rates (9.6% vs. 2.5%, P<0.05), and higher incidence of AKI (28% vs. 16%, P < 0.05) than normal nutritional patients. The NRS-2002 showed a strong association with AKI, that is, the risk of AKI changed in parallel with the score of the NRS-2002. In short- and long-term survival, patients with a lower NRS-2002 score or who did not have AKI achieved a significantly lower risk of mortality than those with a high NRS-2002 score or AKI. Univariate Cox regression analyses indicated that both the NRS-2002 and AKI were strongly related to long-term survival (area under the curve (AUC) 0.79 and 0.71) and that the combination of the two showed better accuracy (AUC 0.80) than the individual variables. In conclusion, malnutrition can increase the risk of AKI, and both AKI and malnutrition can worsen the prognosis, that the undernourished patients who develop AKI yield far worse prognosis than normal nutritional patients.
The Curious Case of Ghana/Côte d'Ivoire: A Consistent Approach to Hydrocarbon Activities in the Disputed Area?
Bin ZHAO
Journal: Asian Journal of International Law , First View
Published online by Cambridge University Press: 01 August 2019, pp. 1-31
The Ghana/Côte d'Ivoire case is the first time that an ITLOS ad hoc chamber has addressed the lawfulness of unilateral hydrocarbon activities in a disputed maritime area. This paper analyzes the Chamber's ruling on Côte d'Ivoire's submission no. 2, which covers several important issues: the jurisdiction of the Chamber to decide on submission no. 2, the alleged violation of sovereign rights, and the alleged violation of Article 83(3) of UNCLOS. The paper argues that the Chamber's jurisdictional basis of forum prorogatum is questionable, and that there are inconsistent approaches between the Judgment and the Order of the provisional measures. Additionally, there are not only inconsistent understandings of the "Ivorian maritime area" within the same submission no. 2 by the Chamber, but also a logical gap in the Chamber's reasoning with regard to submission no. 2(iii). Consequently, the Chamber's inconsistent approaches may jeopardize the persuasiveness of its judgment.
In vitro efficacies of solubility-improved mebendazole derivatives against Echinococcus multilocularis
Shuo Xu, Liping Duan, Haobing Zhang, Bin Xu, Junhu Chen, Wei Hu, Weifeng Gui, Fuqiang Huang, Xu Wang, Zhisheng Dang, Yumin Zhao
Journal: Parasitology / Volume 146 / Issue 10 / September 2019
Recently, we introduced an epoxy group to mebendazole by a reaction with epichlorohydrin and obtained two isoforms, mebendazole C1 (M-C1) and mebendazole C2 (M-C2). The in vitro effects of mebendazole derivatives at different concentrations on Echinococcus multilocularis protoscoleces and metacestodes as well as cytotoxicity in rat hepatoma (RH) cells were examined. The results demonstrated that the solubility of the two derivatives was greatly improved compared to mebendazole. The mortality of protoscoleces in vitro reached to 70–80% after 7 days of exposure to mebendazole or M-C2, and M-C2 showed higher parasiticidal effects than mebendazole (P > 0.05). The parasiticidal effect of M-C1 was low, even at a concentration of 30 µm. The percentage of damaged metacestodes that were treated with mebendazole and M-C2 in vitro at different concentrations were similar, and M-C1 exhibited insignificant effects on metacestodes. Significant morphological changes on protoscoleces and metacestodes were observed after treatment with mebendazole and M-C2. In addition, the introduction of an epoxy group to mebendazole also reduced its cytotoxicity in RH cells. Our results demonstrate that the introduction of an epoxy group not only improved the solubility of mebendazole, but also increased its parasiticidal effects on E. multilocularis and reduced its cytotoxicity in RH cells.
Investigation of hydrogen embrittlement in 12Cr2Mo1R(H) steel
Xiaowei Luo, Bin Bian, Kun Zhang, Danlei Tian, Min Pan, Xiaolang Chen, Heming Zhao
Journal: Journal of Materials Research / Volume 33 / Issue 20 / 29 October 2018
Published online by Cambridge University Press: 24 September 2018, pp. 3501-3511
Print publication: 29 October 2018
The hydrogen embrittlement of 12Cr2Mo1R(H) steel at different strain rates were investigated after hydrogen precharging for 4 h in a 0.5 M H2SO4 solution with 2 g/L ammonium thiocyanate. Results showed that the embrittlement index increased and gradually reached a relative stable value of about 20% at the strain rate of 5 × 10−5 s−1 with the decrease of strain rates. SEM images depicted small and deep flakes at high strain rates, while flakes grew larger at slow strain rates. Most hydrogen-induced cracks (HICs) were transgranular fracture through lath grain of bainitic ferrite. High strain field surrounds the crack tips, which makes the crack tips of two close and parallel cracks deflect toward each another and even form crack coalescence. The electron backscatter diffraction technique was used to investigate the effects of grain boundaries, recrystallization fraction, kernel average misorientation map, texture component, and coincidence site lattice boundaries on the HIC propagation. High densities of dislocations and strain concentrations were found around the cracks, where grains are highly sensitive to HIC.
Prevalence and risk factors of impaired fasting glucose and diabetes among Chinese children and adolescents: a national observational study
Zhenghe Wang, Zhiyong Zou, Haijun Wang, Jin Jing, Jiayou Luo, Xin Zhang, Chunyan Luo, Haiping Zhao, Dehong Pan, Jun Ma, Bin Dong, Yinghua Ma
Journal: British Journal of Nutrition / Volume 120 / Issue 7 / 14 October 2018
Published online by Cambridge University Press: 29 August 2018, pp. 813-819
The prevalence of impaired fasting glucose (IFG) and diabetes mellitus (DM) has reportedly increased significantly among Chinese children and adolescents. We aimed to examine the prevalence of IFG and DM, the disparities in sex and region and related risk factors among Chinese children and adolescents. A total of 16 434 Chinese children aged 6–17 years were selected from a national cross-sectional survey, and fasting glucose was measured for all participants. Overall, mean fasting plasma glucose (FPG) concentration was (4·64 (sd 0·51)) mmol/l, and the prevalence of DM and IFG was 0·10 and 1·89 %, respectively. Compared with girls, boys had higher FPG concentration (4·69 v. 4·58 mmol/l, r 0·107, P<0·001) and IFG prevalence (2·67 v. 1·07 %, rφ 0·059, P<0·001). Compared with rural children and adolescents, urban children and adolescent had higher FPG concentration (4·65 v. 4·62 mmol/l, r 0·029, P<0·001) and DM prevalence (0·15 v. 0·05 %, rφ 0·016, P<0·01). In addition, self-reported fried foods intake and overweight/obesity were positively associated with IFG, and the proportion of consuming fried foods more than or equal to once per week and overweight/obesity prevalence in boys and urban children and adolescents were significantly higher than girls and rural children and adolescents, respectively (P<0·05). Although the prevalence of IFG and DM was relatively low in Chinese children and adolescents, sex and region disparities were observed, which may be associated with differences in overweight/obesity prevalence and dietary factors.
Invasive Smooth Cordgrass (Spartina alterniflora) Eradication and Native Crab Recovery
Long Tang, Bo Li, Bin Zhao, Peng Li, Zhanbin Li, Yang Gao
Journal: Invasive Plant Science and Management / Volume 11 / Issue 2 / June 2018
Published online by Cambridge University Press: 06 July 2018, pp. 89-95
Print publication: June 2018
Invasive smooth cordgrass (Spartina alterniflora Loisel) eradication is important for the health of many coastal ecosystems. An integrated regime of continuous submergence after clear mowing, with three interval levels between mowing and submergence (5, 10, and 15 d) and three submergence depths (20, 30, and 50 cm), was implemented in cofferdams enclosing invader populations along a Chinese coast. In July of the following year, after the roots of mowed S. alterniflora had been submerged for 12 mo, some ramets grew under the regime with an interval of 15 d and the regime with a submergence depth of 20 cm, but no ramets occurred under the regimes with submergence depths of 30 or 50 cm and intervals of 5 or 10 d. Four crab species were documented: Helice tridens tientsinensis Rathbun, Sesarma dehaani H. Milne-Edwards, Ocypode stimpsoni Ortmann, and Chiromantes haematocheir de Haan. Biomass and abundance values of crab species in the cofferdams were similar to those in the mudflats but different from those in smooth cordgrass populations. Thus, the treatment of submergence after mowing, which was implemented in the cofferdams, can control S. alterniflora and provide a mudflat-like habitat that promotes crab recovery if this treatment uses the proper combination of submergence depth and interval between mowing and submergence.
A coaxial-output rolled strip pulse forming line based on multi-layer films
Jian-Cang Su, Rui Li, Jie Cheng, Bin-Xiong Yu, Xi-Bo Zhang, Liang Zhao, Wen-Hua Huang
Journal: Laser and Particle Beams / Volume 36 / Issue 1 / March 2018
Published online by Cambridge University Press: 24 January 2018, pp. 69-75
Print publication: March 2018
A coaxial-output rolled strip pulse-forming line (RSPFL) with a dry structure is researched for the purpose of miniaturization and all-solid state of pulse-forming lines (PFL). The coaxial-output RSPFL consists of a coaxial-output electrode (COE) and a rolled strip line (RSL). The COE is characterized by quasi-coaxial structure, making the output pulse propagate along the axial direction with a small output inductance. The RSL is rolled on the COE, whose transmission characteristics are analyzed theoretically. It shows that the RSL can be regarded as a planar strip line when the rolling radius of the strip line is larger than 60 times of the thickness of the insulation dielectric layer of RSL. CST modeling was carried out to simulate the discharging characteristic of the coaxial-output RSPFL. It shows that the coaxial-output RSPFL can deliver a discharging pulse with a rise time <6 ns when the impedance of the RSL matches that of the COE, which confirms the theoretical analysis. A prototype of the coaxial-output RSPFL was developed. A 49-kV discharging pulse on a matched load was achieved when it was charged to 100 kV. The discharging waveform has a pulse width of 32 ns, with a rise time of 6 ns, which is consistent with the simulation waveform. An energy-storage density of 1.9 J/L was realized in the coaxial-output RSPFL. By the method of multi-stage connection in series, a much higher output voltage is convenient to be obtained.
Quantitative Studies of Endothelial Cell Fibronectin and Filamentous Actin (F-Actin) Coalignment in Response to Shear Stress
Xianghui Gong, Xixi Zhao, Bin Li, Yan Sun, Meili Liu, Yan Huang, Xiaoling Jia, Jing Ji, Yubo Fan
Journal: Microscopy and Microanalysis / Volume 23 / Issue 5 / October 2017
Print publication: October 2017
Both fibronectin (FN) and filamentous actin (F-actin) fibers play a critical role for endothelial cells (ECs) in responding to shear stress and modulating cell alignment and functions. FN is dynamically coupled to the F-actin cytoskeleton via focal adhesions. However, it is unclear how ECs cooperatively remodel their subcellular FN matrix and intracellular F-actin cytoskeleton in response to shear stress. Current studies are hampered by the lack of a reliable and sensitive quantification method of FN orientation. In this study, we developed a MATLAB-based feature enhancement method to quantify FN and F-actin orientation. The role of F-actin in FN remodeling was also studied by treating ECs with cytochalasin D. We have demonstrated that FN and F-actin codistributed and coaligned parallel to the flow direction, and that F-actin alignment played an essential role in regulating FN alignment in response to shear stress. Our findings offer insight into how ECs cooperatively remodel their subcellular ECM and intracellular F-actin cytoskeleton in response to mechanical stimuli, and are valuable for vascular tissue engineering.
Observations and modelling of first-year ice growth and simultaneous second-year ice ablation in the Prydz Bay, East Antarctica
Jiechen Zhao, Bin Cheng, Qinghua Yang, Timo Vihma, Lin Zhang
Journal: Annals of Glaciology / Volume 58 / Issue 75pt1 / July 2017
Published online by Cambridge University Press: 27 November 2017, pp. 59-67
Print publication: July 2017
The seasonal cycle of fast ice thickness in Prydz Bay, East Antarctica, was observed between March and December 2012. In March, we observed a 0.16 m thickness gain of 0.22 m-thick first-year ice (FYI), while 1.16 m-thick second-year ice (SYI) nearby simultaneously ablated by 0.59 m. A 1-D thermodynamic sea-ice model was applied to identify the factors that led to the simultaneous growth of FYI and melt of SYI. The different evolutions were explained by the difference in the conductive heat flux between the FYI and SYI. As the FYI was thin, there was a large temperature gradient between the ice base and the colder ice surface. This generated an upward conductive heat flux, which was larger than the heat flux from the ocean to the ice base, yielding basal growth of ice. In the case of the thicker SYI the temperature gradient and, hence, the conductive heat flux were smaller, and not sufficient to balance the oceanic heat flux at the ice base, yielding basal ablation. Penetration of solar radiation affected the conductive heat flux in both cases, and the model results were sensitive to the initial ice temperature profile and the uncertainty of the oceanic heat flux.
The Wuhan Twin Birth Cohort (WTBC)
Jinzhu Zhao, Shaoping Yang, Anna Peng, Zhengmin Qian, Hong Xian, Tianjiao Chen, Guanghui Dong, Yiming Zhang, Xijiang Hu, Zhong Chen, Jiangxia Cao, Xiaojie Song, Shunqing Xu, Tongzhang Zheng, Bin Zhang
Journal: Twin Research and Human Genetics / Volume 20 / Issue 4 / August 2017
Published online by Cambridge University Press: 27 June 2017, pp. 355-362
Print publication: August 2017
The Wuhan Pre/Post-Natal Twin Birth Registry (WPTBR) is one of the largest twin birth registries with comprehensive medical information in China. It recruits women from the first trimester of pregnancy and their twins from birth. From January 2006 to May 2016, the total number of twins enrolled in WPTBR is 13,869 twin pairs (27,553 individuals). The WPTBR initiated the Wuhan Twin Birth Cohort (WTBC). The WTBC is a prospective cohort study carried out through incorporation of three samples. The first one comprises 6,920 twin pairs, and the second one, 6,949 twin pairs. Both are population-based samples linked to the WPTBR and include pre- and post-natal information from WPTBR. The second sample includes neonatal blood spots as well. Using a hospital-based approach, we recently developed a third sample with a target enrolment of 1,000 twin pairs and their mothers. These twins are invited, via their parents, to participate in a periodic health examination from the first trimester of pregnancy to 18 years. Biological samples are collected initially from the mother, including blood, urine, cord blood, cord, amniotic fluid, placenta, breast milk and meconium, and vaginal secretions, and later from the twins, including meconium, stool, urine, and blood. This article describes the design, recruitment, follow-up, data collection, and measures, as well as ongoing and planned analyses at the WTBC. The WTBC offers a unique opportunity to follow women from prenatal to postnatal, as well as follow-up of their twins. This cohort study will expand the understanding of genetic and environmental influences on pregnancy and twins' development in China.
On a Yamabe Type Problem in Finsler Geometry
Bin Chen, Lili Zhao
Journal: Canadian Mathematical Bulletin / Volume 60 / Issue 2 / 01 June 2017
Print publication: 01 June 2017
In this paper, a newnotion of scalar curvature for a Finsler metric $F$ is introduced, and two conformal invariants $Y(M,F)$ and $C(M,F)$ are defined. We prove that there exists a Finsler metric with constant scalar curvature in the conformal class of $F$ if the Cartan torsion of $F$ is sufficiently small and $Y(M,F)C(M,F)<Y({{\mathbb{S}}^{n}})$ where $Y({{\mathbb{S}}^{n}})$ is the Yamabe constant of the standard sphere.
Insights into the velocity-dependent geometry and internal strain in accretionary wedges from analogue models
BIN DENG, LEI JIANG, GAOPING ZHAO, RUI HUANG, YUANBO WANG, SHUGEN LIU
Journal: Geological Magazine / Volume 155 / Issue 5 / July 2018
Published online by Cambridge University Press: 25 January 2017, pp. 1089-1104
Although the brittle material in analogue models is characterized by a linear Navier-Coulomb behaviour and rate-independent deformation, the geometry and style of deformation in accretionary wedges is sensitive to shortening velocity. In this study we have constructed a series of analogue models with various shortening velocities in order to study the influence of shortening velocity on the geometry and kinematics of accretionary wedges. Model results illustrate how shortening velocity has an important influence on the geometry and kinematics of the resulting wedge. In general, for models having similar bulk shortening, the accretionary wedges with higher velocities of shortening are roughly steeper, higher and longer, as well as having larger critical wedge angles and height. It accommodates a number of foreland-vergent thrusts, larger fault spacing and displacement rates than those of low- to medium-velocity shortening, which indicates a weak velocity-dependence in geometry of the wedge. Moreover, models with a high velocity of shortening undergo larger amounts of volumetric strain and total layer-parallel shortening than models with low- to medium-velocity shortening. The former accommodate a greater development of back thrusts and asymmetric structures; a backwards-to-forwards style of wedge growth therefore occurs in the frontal zone under high-velocity shortening.
Size-controlled Intercalation to Conversion Transition in Lithiation of Transition Metal Chalcogenides–NbSe3
Langli Luo, Benliang Zhao, Bin Xiang, Chong-Min Wang
Journal: Microscopy and Microanalysis / Volume 22 / Issue S3 / July 2016
The diagnostic value of plasma N-terminal connective tissue growth factor levels in children with heart failure
Gang Li, Xueqing Song, Jiyi Xia, Jing Li, Peng Jia, Pengyuan Chen, Jian Zhao, Bin Liu
Journal: Cardiology in the Young / Volume 27 / Issue 1 / January 2017
Published online by Cambridge University Press: 16 March 2016, pp. 101-108
Print publication: January 2017
The aim of this study was to assess the diagnostic value of plasma N-terminal connective tissue growth factor in children with heart failure.
Methods and results
Plasma N-terminal connective tissue growth factor was determined in 61 children, including 41 children with heart failure, 20 children without heart failure, and 30 healthy volunteers. The correlations between plasma N-terminal connective tissue growth factor levels and clinical parameters were investigated. Moreover, the diagnostic value of N-terminal connective tissue growth factor levels was evaluated. Compared with healthy volunteers and children without heart failure, plasma N-terminal connective tissue growth factor levels were significantly elevated in those with heart failure (p<0.01). N-terminal pro-brain natriuretic peptide and left ventricular end-diastolic dimension were positively correlated with plasma N-terminal connective tissue growth factor levels (r=0.364, p=0.006; r=0.308, p=0.016), whereas there was a negative correlation between left ventricular ejection fraction and plasma N-terminal connective tissue growth factor (r=−0.353, p=0.005). Connective tissue growth factor was significantly correlated with the severity of heart failure (p<0.001). Moreover, addition of connective tissue growth factor to N-terminal pro-brain natriuretic peptide did not significantly increase area under curve for diagnosing heart failure (area under curve difference 0.031, p>0.05), but it obviously improved the ability of diagnosing heart failure in children, as demonstrated by the integrated discrimination improvement (6.2%, p=0.013) and net re-classification improvement (13.2%, p=0.017) indices.
Plasma N-terminal connective tissue growth factor is a promising diagnostic biomarker for heart failure in children.
Laser performance of the SG-III laser facility
Wanguo Zheng, Xiaofeng Wei, Qihua Zhu, Feng Jing, Dongxia Hu, Jingqin Su, Kuixing Zheng, Xiaodong Yuan, Hai Zhou, Wanjun Dai, Wei Zhou, Fang Wang, Dangpeng Xu, Xudong Xie, Bin Feng, Zhitao Peng, Liangfu Guo, Yuanbin Chen, Xiongjun Zhang, Lanqin Liu, Donghui Lin, Zhao Dang, Yong Xiang, Xuewei Deng
Journal: High Power Laser Science and Engineering / Volume 4 / 2016
Published online by Cambridge University Press: 13 July 2016, e21
SG-III laser facility is now the largest laser driver for inertial confinement fusion research in China. The whole laser facility can deliver 180 kJ energy and 60 TW power ultraviolet laser onto target, with power balance better than 10%. We review the laser system and introduce the SG-III laser performance here.
Late Quaternary carbon cycling responses to environmental change revealed by multi-proxy analyses of a sediment core from an upland lake in southwest China
Enlou Zhang, Weiwei Sun, Ming Ji, Cheng Zhao, Bin Xue, Ji Shen
Journal: Quaternary Research / Volume 84 / Issue 3 / November 2015
Published online by Cambridge University Press: 20 January 2017, pp. 415-422
Print publication: November 2015
Stable carbon isotope (δ13C) values of organic matter in lacustrine sediments are commonly used to trace past changes in terrestrial and aquatic carbon cycles. Here we use a high-resolution, well-dated δ13C record from Lake Tengchongqinghai (TCQH) in southwestern China, together with other proxy indices, to reconstruct the paleolimnological history over the past 18.5 ka. Organic matter in the sediments of Lake TCQH is derived predominately from aquatic macrophytes. The lacustrine primary productivity is closely linked with lake-level changes affected by variations in the strength of the Asian summer monsoon and modified by evapotranspiration. Similar to lake sediments world-wide, a ca. − 3‰ shift occurred in the δ13C values of Lake TCQH in response to the significant increase in atmospheric CO2 concentration during the last deglaciation. In the Holocene, the availability of dissolved CO2 in the lake water of Lake TCQH was determined by variations in hydraulic energy: low water turbulence creates a thick, stagnant boundary layer around aquatic plants, which will restrict the rate of CO2 diffusion and result in more positive δ13C values of aquatic plants. In contrast, significant water turbulence dramatically reduces the boundary layer thickness leading to more negative δ13C values of aquatic plants.
Design, analysis, and control of a cable-driven parallel platform with a pneumatic muscle active support
Xingwei Zhao, Bin Zi, Lu Qian
Journal: Robotica / Volume 35 / Issue 4 / April 2017
Published online by Cambridge University Press: 19 October 2015, pp. 744-765
Print publication: April 2017
The neck is an important part of the body that connects the head to the torso, supporting the weight and generating the movement of the head. In this paper, a cable-driven parallel platform with a pneumatic muscle active support (CPPPMS) is presented for imitating human necks, where cable actuators imitate neck muscles and a pneumatic muscle actuator imitates spinal muscles, respectively. Analyzing the stiffness of the mechanism is carried out based on screw theory, and this mechanism is optimized according to the stiffness characteristics. While taking the dynamics of the pneumatic muscle active support into consideration as well as the cable dynamics and the dynamics of the Up-platform, a dynamic modeling approach to the CPPPMS is established. In order to overcome the flexibility and uncertainties amid the dynamic model, a sliding mode controller is investigated for trajectory tracking, and the stability of the control system is verified by a Lyapunov function. Moreover, a PD controller is proposed for a comparative study. The results of the simulation indicate that the sliding mode controller is more effective than the PD controller for the CPPPMS, and the CPPPMS provides feasible performances for operations under the sliding mode control.
DEFOLIATION ENHANCES GREEN FORAGE PERFORMANCE BUT INHIBITS GRAIN YIELD IN BARLEY (HORDEUM VULGARE L.)
XIAODONG CHEN, BIN ZHAO, LIANG CHEN, RUI WANG, CHANGHAO JI
Journal: Experimental Agriculture / Volume 52 / Issue 3 / July 2016
Published online by Cambridge University Press: 16 September 2015, pp. 391-404
To evaluate the effects of defoliation on green forage performance and grain yield (GY) variation in barley, five barley genotypes were imposed on three levels of defoliation treatments over two consecutive growing seasons in this study. The results indicated that green forage yields were significantly improved by repeated defoliation. The traits of green forage quality, including the ratio of dry weight to fresh weight, crude ash and calcium content were improved, while crude protein and crude fat were reduced, and crude fiber and phosphorus contents appeared not to be influenced by repeated defoliation. Plant height (PH), GY and other yield components, grain number per spike and thousand kernel weight, were significantly reduced by defoliation over the two growing seasons, while internode length below spike was less affected. Reduction in spike length and the number of spikes per plant were identified in only one year. Correlation analysis revealed that only PH exhibited a positive correlation with GY. Effects of genotype, interaction between genotype and defoliation, and environments on changes of forage yield and quality and GY were also discussed. Our current work provides a feasible approach to select elite barley cultivars with optimal defoliation treatments for both forage and grain uses in barley breeding programme. | CommonCrawl |
Uncertain Sequences
March 14, 2019 August 12, 2021 / Puzzles / Ambiguity, Assumptions / By Dave Peterson
We've often pointed out that pattern or sequence problems, when nothing is given but a list of numbers, are not really math, in the sense that there is no one correct answer. They are psychology questions: What would a math teacher think is an interesting sequence to ask about? Mathematically, any number could come next, and infinitely many formulas or patterns could be found that produce the given terms. Let's look at some questions we've had that took this to an extreme.
Three terms, and an unsatisfying answer
First, from 1997:
Alternating Sequence
Dear Dr. Math,
I need help. I have a problem that neither my friends nor I can get. I thought you could help. We have to find a pattern and find the next three numbers for this: 0, 8, 27, _, _, _. I hope you can find the answer.
Shoushou
Three terms are not enough to determine a sequence, unless something very obvious is going on. If this had been 1, 8, 27, it would have been obvious: Even though there are many options, ranging from the repeating sequence
1, 8, 27, 1, 8, 27, … ,
to the quadratic sequence \(a_n = 6n^2 – 11x + 6\), which produces
1, 8, 27, 58, 101, 156, … ,
any reasonable person would guess that the intended sequence is \(a_n = n^3\).
But what of the sequence that was given? It isn't at all obvious, and therefore is really impossible to know. Doctor Wallace answered, suggesting one idea to start with:
For this series, the first thing to notice is that all of the numbers are perfect cubes: 0 = 0^3, 8 = 2^3, and 27 = 3^3.
If we list the numbers that are the cube roots of these numbers, we get this series:
Now, all you have to do is find a pattern among these three. The one I see is an alternating series. There may be others. Let me know how your work goes on this problem.
Since the cubes are so obvious, his idea is that the sequence might be the cubes of this sequence. But what is this sequence? He suggests what may be the simplest idea, an alternating sequence (that is, alternating terms follow different rules). We could instead use \(\displaystyle -\frac{x^2-7x+6}{2}\), which generates
0, 2, 3, 3, 2, 0, …,
so that the cubes would be
0, 8, 27, 27, 8, 0, …;
but that certainly isn't the first thing that comes to mind.
Shoushou wrote back:
Thanks for giving me that great hint. I'm not sure if I have it right, but I think it might be +2, +1, +2, +1. This doesn't quite satisfy me, but it's the best I can come up with for now.
His suggestion is that the sequence is formed by starting with 0 and alternately adding 2, then 1, then repeating, so that the sequence would continue
0, 2, 3, 5, 6, 8, … ,
and the sequence of cubes would be
0, 8, 27, 125, 216, 512, … .
Doctor Wallace agreed:
I think you're right. That's what I came up with, too. And I feel the same way - I wasn't quite satisfied. Somehow, it didn't feel just right. I think it was because there were only 3 numbers given, and it's hard to come to just one pattern with only three numbers. But it seems that, since the number 1 was missing, +2 +1 +2 +1 ... is the only thing that makes sense.
Then the next two numbers in the pattern would be 5^3 and 6^3. What did your teacher have in mind for the answer? If you can come up with another pattern that fits the series, please write and let me know.
We never heard what the teacher intended; I certainly hope that any answer that made some sort of sense was accepted.
I've mentioned a couple other possibilities. If we find a quadratic function that gives the required values, it's \(\displaystyle a_n = \frac{11x^2-17x+6}{2}\), and continues
0, 8, 27, 57, 98, 150, … .
Or, we could continue decreasing the amount added, so that we next add 0, then -1. The base sequence is then 0, 2, 3, 3, 2, 0 — which is my quadratic sequence above!
To find other possibilities, we could go to OEIS, the Online Encyclopedia of Integer Sequences, which catalogs sequences that people have found interesting. The site recommends entering about 6 terms, which will filter out many alternatives. Entering "0, 8, 27" gives me sequences like "Sum of cubes of primes dividing n," or "Numbers that are the sum of cubes of distinct primes," to list only the most comprehensible of those that start with the 0. And it doesn't include any of those I've mentioned.
But my guess is that the zero was a typo.
Four terms, still insufficient
Here's another from 2001:
Finding a Pattern
My daughter received this in a homework assignment, and I don't believe there is enough specific information to logically give the next four numbers in the sequence: 2, 8, 7, 28.
Four terms might be enough to support a very simple sequence; for example, if the difference from one term to the next was the same, we'd have three identical numbers and a reasonably strong case for an arithmetic progression. But the differences here are 6, -1, 21, which reveals nothing.
I agreed with Ray, but I had the same idea Doctor Wallace had above, and with a little more justification:
I agree, there really is not enough information here. I can guess what they probably want, however; most likely they have had other examples where they alternated two simple operations to get successive terms, and you are expected to assume that this pattern is similar.
If so, then we are first multiplying by 4 (2*4 = 8), then subtracting 1 (8-1 = 7), then multiplying by 4 again (7*4 = 28), so you would continue in the same way: 27, 108, 107, 428.
The evidence we have is that multiplication by 4 appears twice, and subtraction of 1 in between is a simple thing to do. So I felt justified in using the word "probably". My sequence, then, is
2, 8, 7, 28, 27, 108, 107, 428, … .
But there isn't a lot of evidence for the subtraction step, as we see only one instance of it; and some other alternation is quite reasonable:
But another perfectly valid pattern would be "for odd terms, add 5 each time; for even terms, add 20." That would give 12, 48, 17, 68.
That is, the sequence could arise from merging the arithmetic progressions 2, 7, 12, 17, … (common difference 5) and 8, 28, 48, 68, … (common difference 20). This is a different kind of alternating sequence:
2, 8, 7, 28, 12, 48, 17, 68, …
On the other hand, if we're merging sequences, we have only two terms of each, so they could be just about anything. What if the even terms were a geometric progression where we multiply each term by 28/8 = 3.5 to get the next: 8, 28, 98, 343, … ? And, interestingly, the odd terms would have the same common ratio, 7/2 = 3.5, and we'd get 2, 7, 24.5, 85.75, … . So our answer would be
2, 8, 7, 28, 24.5, 98, 85.75, 68, … .
Not very pretty, but just as logical!
Who is to say whether any of these is "correct"? All yield the same first four terms, so a teacher would have to give credit to all of them — unless there is some information we haven't been given, such as that the class has only learned about merging arithmetic progressions, or all the examples given had alternated addition and multiplication.
I continued:
If a problem merely says "give the next four numbers" or "find the pattern in this sequence," there are infinitely many possible answers, since the word "pattern" has no precise definition; it's really a matter of guessing what pattern they had in mind, which is not math but psychology or ESP. To make this a valid problem, they should say something at least as clear as "This sequence was formed by a pattern similar to those you saw in this chapter. Make a reasonable guess as to what the pattern is, and show how it continues." Or, I suppose, they could say "Find a pattern in this sequence, explain how it works, and use that pattern to predict the next four numbers. There may be more than one correct answer."
But to imply that students can determine _the_ correct answer by looking at four numbers is a misleading lesson in what math is all about. It's not a guessing game.
Four terms, and a thoughtful student
Here is another with four terms, from 2002:
The Perils of Predicting Patterns
What is the next number in the pattern 1,3,6,10 ___?
If the pattern is to add 2, 3, 4, and then 2, 3, 4 again and again, it should be 12. But if the pattern is to add 2, 3, 4, 5, and so on, then it should be 15. Which is correct?
To my mind, it's "obvious" what they intended, because this is a well-known arithmetic progression called the triangular numbers (1, 1+2, 1+2+3, 1+2+3+4, …). If so, then Su's second guess is "correct". But Su has the makings of a mathematician, and sees that there are other possibilities, one being a mere repetition of the pattern of differences. Doctor Greenie had an excellent answer:
There can't be a single "correct answer" for any question like this, since no matter what list of numbers you give me, I can find a formula that will fit them to any following number.
Usually what is desired is the "simplest" answer, and unfortunately, different people's definition of what it means to be "simple" varies.
Obviously, adding 2, then 3, then 4, then 2, then 3, then 4, then 2, and so on, is one way to extend the sequence; and adding 2, then 3, then 4, then 5, then 6 is another.
To my mind, the second is slightly simpler but only because the pattern presented so far does not give us any reason to think that we should go back to adding 2 to obtain the next number.
In other words, in line with my thoughts above, in the absence of evidence that a change is to be made (such as going back and repeating), it is more likely that the author intended just to continue the existing pattern. But that doesn't make either answer more correct than the other.
These sorts of patterns are used in intelligence tests, and the "correct" answer is "whatever very intelligent people think the correct answer is". That's not much help, is it?
I remember a wonderful example shown to me once that illustrated how silly this sort of question is. Here it is:
What comes next in this sequence?
33, 23, 14, 9, ___
The answer is "Christopher Street". The reason is that the numbers are the exits of the 6th Avenue subway in New York City.
(Looking at a current subway map, it looks like things have changed a little, but the idea is clear!)
Four terms, a proof, and a guess
A 1996 question provided a chance for a different approach to the issue:
Predicting the Next Number
When given a series of numbers and asked to predict the next number, what is the formula for doing so? Example: 2,5,12,23, ?
This question appears on psychological exams, federal employment exams and many others. Is there a mathematical way of determining the next number in this series?
As we've been saying, there is no truly mathematical way to do it; but Doctor Jerry provided a very mathematical way to show that it can't really be done. He supposed that we knew a polynomial formula for the sequence, and showed that we can find another polynomial that will give the very same four terms, AND whatever fifth term we choose! This is the idea Doctor Greenie mentioned in his first sentence above.
First, if the first several terms of a sequence are given, there is no method for determining the general term. Suppose I'm given the numbers a, b, c, d and asked to determine the fifth and sixth terms of the sequence. I'll show that any number of different solutions is possible.
I start by determining a polynomial
p(x) = Ax^3 + Bx^2 + Cx + D
such that p(1) = a, p(2) = b, p(3) = c, and p(4) = d. Then consider
f(n) = p(n) + (n-1)(n-2)(n-3)(n-4) or
g(n) = p(n) + (n-1)(n-2)(n-3)(n-4)(n-5).
Notice that both f and g determine sequences whose first four terms are a, b, c, and d. Remaining terms are wildly different. This idea could be elaborated.
His function f is identical to p for n = 1, 2, 3, 4, because the added product is zero in all four cases. But for n = 5, he is adding 24 to the value of p(5), giving a new polynomial with a different fifth term; and we could adjust that to make it any number we want. And his function g will match p for the first 5 terms, but differ in the sixth.
So even if we required a really "mathematical" answer in the form of a polynomial, there are infinitely many "correct" answers. You'd have to specify that the function must have the lowest possible degree, to make only one answer correct.
But, of course, that is not what problems like this are looking for:
You can, however, try to guess what was most likely in the mind of the person who made up the question. For the sequences
2,4,6,8,...
1,4,9,16,...
I suppose most persons would say 10,12 or 25,36.
The two examples he gives are "skip counting" (an arithmetic progression) and a list of perfect squares, both of which most of us would recognize and suppose that anyone writing a test would be more likely to think of than of other possibilities.
For the sequence you gave, 2, 5, 12, 23, I noticed that 5 - 2 = 3, 12 - 5 = 7, and 23 - 12 = 11. Since 3, 7, and 11 can be viewed as the odd numbers, leaving every other one out, one could argue that the next two terms are 23 + 15 and 38 + 19. Other, more or less natural answers are possible. However, one has no choice but to accept whatever the text makers decree is the correct answer!
Here, he observes that the differences between successive terms are 3, 7, 11, which increase by 4 each time; so the natural thing to do is to continue with 15, 19.
The problem (like all the others in this post) didn't ask for a formula, just for the next term, so that is all that is needed. The differences imply (as we'll see when we get to the Method of Finite Differences) that the sequence has a second-degree polynomial (quadratic) formula, which turns out to be \(2x^2 – 3x + 3\), which generates the terms
2, 5, 12, 23, 38, 57, … ,
just as Doctor Jerry found by adding 15 and 19.
If we take that as \(p(x)\), then the \(f(x)\) above is $$f(n) = p(n) + (n-1)(n-2)(n-3)(n-4) =\\ 2x^2 – 3x + 3 + n^4-10n^3+35n^2-50n+24 = n^4-10n^3+37n^2-53n+27.$$ The first six terms of this are
2, 5, 12, 23, 62, 177.
As intended, the first four terms agree, but the next terms are different. | CommonCrawl |
\begin{definition}[Definition:Inverse Cosecant/Complex/Definition 2]
Let $z \in \C_{\ne 0}$ be a non-zero complex number.
The '''inverse cosecant of $z$''' is the multifunction defined as:
:$\csc^{-1} \left({z}\right) := \left\{{\dfrac 1 i \ln \left({\dfrac {i + \sqrt{\left|{z^2 - 1}\right|} e^{\left({i / 2}\right) \arg \left({z^2 - 1}\right)}} z}\right) + 2 k \pi: k \in \Z}\right\}$
where:
: $\sqrt{\left|{z^2 - 1}\right|}$ denotes the positive square root of the complex modulus of $z^2 - 1$
: $\arg \left({z^2 - 1}\right)$ denotes the argument of $z^2 - 1$
: $\ln$ denotes the complex natural logarithm considered as a multifunction.
\end{definition} | ProofWiki |
Modified Schmidt games and a conjecture of Margulis
JMD Home
July 2013, 7(3): 461-488. doi: 10.3934/jmd.2013.7.461
Ergodic properties of $k$-free integers in number fields
Francesco Cellarosi 1, and Ilya Vinogradov 2,
Department of Mathematics, Altgeld Hall, 1409 W Green Street, Urbana, IL 61801, United States
School of Mathematics, University Walk, Bristol, BS8 1TW, United Kingdom
Received March 2013 Revised September 2013 Published December 2013
Let $K/\mathbf{Q}$ be a degree-$d$ extension. Inside the ring of integers $\mathscr O_K$ we define the set of $k$-free integers $\mathscr F_k$ and a natural $\mathscr O_K$-action on the space of binary $\mathscr O_K$-indexed sequences, equipped with an $\mathscr O_K$-invariant probability measure associated to $\mathscr F_k$. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the work of Cellarosi and Sinai [J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343--1374] that considered the case $K=\mathbf{Q}$ and $k=2$.
Keywords: ergodicity, group actions with pure point spectrum, isomorphism of group actions., Square-free and $k$-free integers in number fields, correlation functions.
Mathematics Subject Classification: Primary: 37A35, 37A45; Secondary: 11R04, 11N25, 37C85, 28D1.
Citation: Francesco Cellarosi, Ilya Vinogradov. Ergodic properties of $k$-free integers in number fields. Journal of Modern Dynamics, 2013, 7 (3) : 461-488. doi: 10.3934/jmd.2013.7.461
M. Baake, R. V. Moody and P. A. B. Pleasants, Diffraction from visible lattice points and $k$th power free integers,, Discrete Math., 221 (2000), 3. doi: 10.1016/S0012-365X(99)00384-2. Google Scholar
V. Bergelson and A. Gorodnik, Weakly mixing group actions: A brief survey and an example,, in Modern Dynamical Systems and Applications, (2004), 3. Google Scholar
F. Cellarosi and Ya. G. Sinai, Ergodic properties of square-free numbers,, J. Eur. Math. Soc. (JEMS), 15 (2013), 1343. doi: 10.4171/JEMS/394. Google Scholar
R. R. Hall, The distribution of squarefree numbers,, J. Reine Angew. Math., 394 (1989), 107. doi: 10.1515/crll.1989.394.107. Google Scholar
P. R. Halmos and J. von Neumann, Operator methods in classical mechanics. II,, Ann. of Math. (2), 43 (1942), 332. doi: 10.2307/1968872. Google Scholar
D. R. Heath-Brown, The square sieve and consecutive square-free numbers,, Math. Ann., 266 (1984), 251. doi: 10.1007/BF01475576. Google Scholar
E. Hewitt and K. A. Ross, Abstract Harmonic Analysis. Vol. I. Structure of Topological Groups, Integration Theory, Group Representations,, Second edition, (1979). Google Scholar
I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus,, Second edition, (1991). Google Scholar
L. B. Koralov and Y. G. Sinai, Theory of probability and random processes,, Second edition, (2007). Google Scholar
J. Liu and P. Sarnak, The Möbius function and distal flows,, preprint., (). Google Scholar
G. W. Mackey, Ergodic transformation groups with a pure point spectrum,, Illinois J. Math., 8 (1964), 593. Google Scholar
L. Mirsky, Arithmetical pattern problems relating to divisibility by $r$th powers,, Proc. London Math. Soc. (2), 50 (1949), 497. Google Scholar
W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers,, Third edition, (2004). Google Scholar
J. Neukirch, Algebraic Number Theory,, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], (1999). Google Scholar
R. Peckner, Uniqueness of the measure of maximal entropy for the squarefree flow,, preprint, (2012). Google Scholar
P. A. B. Pleasants and C. Huck, Entropy and diffraction of the $k$-free points in $n$-dimensional lattices,, Discrete Comput. Geom., 50 (2013), 39. doi: 10.1007/s00454-013-9516-y. Google Scholar
V. A. Rokhlin, On the problem of the classification of automorphisms of Lebesgue spaces,, Doklady Akad. Nauk SSSR (N. S.), 58 (1947), 189. Google Scholar
V. A. Rokhlin, Unitary rings,, Doklady Akad. Nauk SSSR (N. S.), 59 (1948), 643. Google Scholar
P. Sarnak, Three lectures on the Möbius function randomness and dynamics (Lecture 1)., Available at: , (). Google Scholar
K. Schmidt, Dynamical Systems of Algebraic Origin,, [2011 reprint of the 1995 original] [MR1345152], (1995). Google Scholar
K. M. Tsang, The distribution of $r$-tuples of squarefree numbers,, Mathematika, 32 (1985), 265. doi: 10.1112/S0025579300011049. Google Scholar
J. von Neumann, Zur Operatorenmethode in der klassischen Mechanik,, Ann. of Math. (2), 33 (1932), 587. doi: 10.2307/1968537. Google Scholar
R. J. Zimmer, Ergodic actions with generalized discrete spectrum,, Illinois J. Math., 20 (1976), 555. Google Scholar
Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
Dariusz Skrenty. Absolutely continuous spectrum of some group extensions of Gaussian actions. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 365-378. doi: 10.3934/dcds.2010.26.365
Franz W. Kamber and Peter W. Michor. Completing Lie algebra actions to Lie group actions. Electronic Research Announcements, 2004, 10: 1-10.
Dongmei Zheng, Ercai Chen, Jiahong Yang. On large deviations for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7191-7206. doi: 10.3934/dcds.2016113
Kengo Matsumoto. K-groups of the full group actions on one-sided topological Markov shifts. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3753-3765. doi: 10.3934/dcds.2013.33.3753
A. Katok and R. J. Spatzier. Nonstationary normal forms and rigidity of group actions. Electronic Research Announcements, 1996, 2: 124-133.
Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
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Jean-Paul Thouvenot. The work of Lewis Bowen on the entropy theory of non-amenable group actions. Journal of Modern Dynamics, 2019, 15: 133-141. doi: 10.3934/jmd.2019016
Jean-François Biasse. Subexponential time relations in the class group of large degree number fields. Advances in Mathematics of Communications, 2014, 8 (4) : 407-425. doi: 10.3934/amc.2014.8.407
Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068
Yoshikazu Katayama, Colin E. Sutherland and Masamichi Takesaki. The intrinsic invariant of an approximately finite dimensional factor and the cocycle conjugacy of discrete amenable group actions. Electronic Research Announcements, 1995, 1: 43-47.
Danijela Damjanović, Anatole Katok. Periodic cycle functions and cocycle rigidity for certain partially hyperbolic $\mathbb R^k$ actions. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 985-1005. doi: 10.3934/dcds.2005.13.985
Yves Guivarc'h. On the spectrum of a large subgroup of a semisimple group. Journal of Modern Dynamics, 2008, 2 (1) : 15-42. doi: 10.3934/jmd.2008.2.15
Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775
Thierry Barbot, Carlos Maquera. On integrable codimension one Anosov actions of $\RR^k$. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 803-822. doi: 10.3934/dcds.2011.29.803
Federico Rodriguez Hertz, Jana Rodriguez Hertz. Cohomology free systems and the first Betti number. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 193-196. doi: 10.3934/dcds.2006.15.193
Livio Flaminio, Miguel Paternain. Linearization of cohomology-free vector fields. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1031-1039. doi: 10.3934/dcds.2011.29.1031
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Anatole Katok, Federico Rodriguez Hertz. Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data. Journal of Modern Dynamics, 2007, 1 (2) : 287-300. doi: 10.3934/jmd.2007.1.287
Francesco Cellarosi Ilya Vinogradov | CommonCrawl |
Random reals and strongly meager sets
Adding a single Cohen real makes the set of reals from the ground model strong measure zero (see this question).
The notion of strong measure zero sets has its dual concept in the category branch -- strongly meager sets. A set $X\subseteq \mathbb{R}$ is strongly meager if for any null set $Y$ there exists $t\in \mathbb{R}$ such that $(t+X)\cap Y=\varnothing$. One can see duality of these notions due to Galvin-Mycielski-Solovay Theorem which states that a set $X\subseteq \mathbb{R}$ is strong measure zero if and only if for any meager set $Y$ there exists $t\in \mathbb{R}$ such that $(t+X)\cap Y=\varnothing$.
Random real forcing is dual to Cohen forcing in the sense of measure and category. Therefore it makes sense to ask, whether:
The set of reals from generic model $ \mathbb{R}\cap V$ is strongly meager after adding a single random real?
I have heard that the answer is affirmative, but I have not been able to find any published proof. Note that $\mathbb{R}\cap V$ is meager after adding a random real (see this question).
set-theory forcing lo.logic
m_korchm_korch
As I have written above the affimative answer itself was known to many people including T. Bartoszyński. The following proof is due to T. Weiss (my advisor).
Proof. We follow closely the proof and notation of Lemma 3.2.42 from [1]. Let $A$ be a Borel measure zero set in $M[r]$, where $r$ is a random real over $M$. There exists $\dot{A}\subseteq 2^{\omega}\times 2^{\omega}$ measure zero set coded in $M$, such that $\dot{A}_{r}=A$ (notation: $\dot{A}_{r}=\{y\colon \left<r,y\right>\in \dot{A}\}$).
Then $$\dot{A}\subseteq\bigcap_{m\in\omega}\bigcup_{n\geq m} [s_{n}]\times[t_{n}]$$ where $s_n, t_n\in 2^{<\omega}$, $\sum_{n=0}^{\infty}\frac{1}{2^{2|s_{n}|}}<\infty$ and we can assume that $|t_{n}|=|s_{n}|$ for any $n\in\omega$.
Let $z\in 2^{\omega}\cap M$ and $f\in\omega^\omega$ be increasing. Then $$\mu(\{x\colon\left<x,x_f+z\right>\in [s]\times[t]\})\leq \frac{2^{f^{-1}(|s|)}}{2^{|s|+|t|}}$$ (where $x_f\in 2^{\omega}$ such that $x_{f}(n)=x(f(n))$). By induction on length $|s_{n}|$ we define an increasing function $f_{A}\in\omega^{\omega}$ such that $$\sum_{n=0}^{\infty}\frac{2^{f_{A}^{-1}(|s_{n}|)}}{2^{2|s_{n}|}}<\infty.$$ It is easy to see that such function exists as for any $\varepsilon>0$ we can find $N_{\varepsilon}\in\omega$ such that $\sum_{n\geq N_{\varepsilon}}\frac{1}{2^{2|s_{n}|}}<\varepsilon$.
Notice also that $\left< x,x_{f}+z\right>\in [s]\times[t] $ if and only if $\left<x,x_{f}\right>\in[s]\times [t+z]$.
The set $$H_{z}=\{x\in\left<x,x_{f_A}+z\right>\in\bigcap_{m\in\omega}\bigcup_{n\geq m} [s_{n}]\times[t_{n}]\}$$ has measure zero and is coded in $M$ for every $z\in 2^{\omega}\cap M$. Thus $r\notin H_{z}$ and $\left<r,r_{f_{A}}+z\right>\notin \dot{A}$. This implies that $r_{f_{A}}\notin A+z$ for every $z\in 2^{\omega}\cap M$, so $(2^{\omega}\cap M)+A\neq 2^{\omega}$ and so $2^{\omega}\cap M$ is strongly meager.
$\square$
[1] T. Bartoszyński, H. Judah, Set thoery: on the structure of the real line, A K Peters, 1995
Not the answer you're looking for? Browse other questions tagged set-theory forcing lo.logic or ask your own question.
Adding a random real makes the set of ground model reals meager
Cohen reals and strong measure zero sets
Dual Borel conjecture in Laver's model
Restrictions of null/meager ideal
Antirandom reals
Representation of meager sets in Cohen extensions
On a strengthening of strong measure zero
Solovay-random pairs of reals
Characterization of Cohen reals
Laver property, non-meager reals and cardinal characteristics | CommonCrawl |
Raymond Louis Wilder
Raymond Louis Wilder (3 November 1896 in Palmer, Massachusetts – 7 July 1982 in Santa Barbara, California) was an American mathematician, who specialized in topology and gradually acquired philosophical and anthropological interests.
Life
Wilder's father was a printer. Raymond was musically inclined. He played cornet in the family orchestra, which performed at dances and fairs, and accompanied silent films on the piano.
He entered Brown University in 1914, intending to become an actuary. During World War I, he served in the U.S. Navy as an ensign. Brown awarded him his first degree in 1920, and a master's degree in actuarial mathematics in 1921. That year, he married Una Maude Greene; they had four children, thanks to whom they have ample descent.
Wilder chose to do his Ph.D. at the University of Texas at Austin, the most fateful decision of his life. At Texas, Wilder discovered pure mathematics and topology, thanks to the remarkable influence of Robert Lee Moore, the founder of topology in the US and the inventor of the Moore method for teaching mathematical proof. Moore was initially unimpressed by the young actuary, but Wilder went on to solve a difficult open problem that Moore had posed to his class. Moore suggested Wilder write up the solution for his Ph.D. thesis, which he did in 1923, titling it Concerning Continuous Curves. Wilder thus became the first of Moore's many doctoral students at the University of Texas.
After a year as an instructor at Texas, Wilder was appointed assistant professor at the Ohio State University in 1924. That university required that its academic employees sign a loyalty oath, which Wilder was very reluctant to sign because doing so was inconsistent with his lifelong progressive political and moral views.
In 1926, Wilder joined the faculty of the University of Michigan at Ann Arbor, where he supervised 26 Ph.Ds and became a research professor in 1947. During the 1930s, he helped settle European refugee mathematicians in the United States. Mathematicians who rubbed shoulders with Wilder at Michigan and who later proved prominent included Samuel Eilenberg, the cofounder of category theory, and the topologist Norman Steenrod. After his 1967 retirement from Michigan at the rather advanced age of 71, Wilder became a research associate and occasional lecturer at the University of California at Santa Barbara.
Wilder was vice president of the American Mathematical Society, 1950–1951, president 1955–1956, and the Society's Josiah Willard Gibbs Lecturer in 1969. He was president of the Mathematical Association of America, 1965–1966, which awarded him its Distinguished Service Medal in 1973.[1] He was elected to the American National Academy of Sciences in 1963. Brown University (1958) and the University of Michigan (1980) awarded him honorary doctorates. The mathematics department at the University of California annually bestows one or more graduating seniors with an award in Wilder's name.
The historical, philosophical, and anthropological writings of Wilder's later years suggest a warm, colorful personality. Raymond (2003) attests to this having been the case. For instance:
"[Wilder] was a devoted student of southwestern Native American culture. One day he told me that after retiring he would like to be a bartender in a rural area of Arizona or New Mexico, because he found the stories of the folk he met in bars there so fascinating."
The topologist
Wilder's thesis set out a new approach to the Schönflies programme, which aimed to study positional invariants of sets in the plane or 2-sphere. A positional invariant of a set A with respect to a set B is a property shared by all homeomorphic images of A contained in B. The best known example of such a positional invariant is embodied in the Jordan curve theorem: A simple closed curve in the 2-sphere has precisely two complementary domains and is the boundary of each of them. A converse to the Jordan curve theorem, proved by Schönflies, states that a subset of the 2-sphere is a simple closed curve if it:
• Has two complementary domains;
• Is the boundary of each of these domains;
• Is accessible from each of these domains.
In his "A converse of the Jordan-Brouwer separation theorem in three dimensions" (1930), Wilder showed that a subset of Euclidean 3-space whose complementary domains satisfied certain homology conditions was a 2-sphere.
Around 1930, Wilder moved from set-theoretic topology to algebraic topology, calling in 1932 for the unification of the two areas. He then began an extensive investigation of the theory of manifolds, e.g., his "Generalized closed manifolds in n-space" (1934), in effect extending the Schönflies programme to higher dimensions. This work culminated in his Topology of Manifolds (1949), twice reprinted, whose last three chapters discuss his contributions to the theory of positional topological invariants.
The philosopher
During the 1940s, Wilder met and befriended the University of Michigan anthropologist Leslie White, whose professional curiosity included mathematics as a human activity (White 1947). This encounter proved fateful, and Wilder's research interests underwent a major change, towards the foundations of mathematics. This change was foreshadowed by his 1944 article "The nature of mathematical proof," and heralded by his address to the 1950 International Congress of Mathematicians, titled "The cultural basis of mathematics," which posed the questions:
• "How does culture (in its broadest sense) determine a mathematical structure, such as a logic?"
• "How does culture influence the successive stages of the discovery of a mathematical structure?"
In 1952, he wrote up his course on foundations and the philosophy of mathematics into a widely cited text, Introduction to the foundations of mathematics.
Wilder's Evolution of mathematical concepts. An elementary study (1969) proposed that "we study mathematics as a human artifact, as a natural phenomenon subject to empirical observation and scientific analysis, and, in particular, as a cultural phenomenon understandable in anthropological terms." In this book, Wilder wrote:
"The major difference between mathematics and the other sciences, natural and social, is that whereas the latter are directly restricted in their purview by environmental phenomena of a physical or social nature, mathematics is subject only indirectly to such limitations. ... Plato conceived of an ideal universe in which resided perfect models ... the only reality mathematical concepts have is as cultural elements or artifacts."
Wilder's last book, Mathematics as a cultural system (1981), contained yet more thinking in this anthropological and evolutionary vein.
Wilder's eclectic and humanist perspective on mathematics appears to have had little influence on subsequent mathematical research. It has, however, had some influence on the teaching of mathematics and on the history and philosophy of mathematics. In particular, Wilder can be seen as a precursor to the work of Howard Eves, Evert Willem Beth, and Davis and Hersh (1981). Wilder's call for mathematics to be scrutinized by the methods of social science anticipates some aspects of Where Mathematics Comes From, by George Lakoff and Rafael Nunez. For an introduction to the limited anthropological research on mathematics, see the last chapter of Hersh (1997).
Bibliography
Books by Wilder:
• 1949. Topology of Manifolds.[2]
• 1965 (1952). Introduction to the foundations of mathematics.[3]
• 1969. Evolution of mathematical concepts. An elementary study.
• 1981. Mathematics as a cultural system. (ISBN 0-08-025796-8)
Biographical:
• Raymond, F., 2003, " Raymond Louis Wilder" in Biographical Memoirs National Academy of Sciences 82: 336–51.
Related work cited in this entry:
• Philip J. Davis and Reuben Hersh, 1981. The Mathematical Experience.
• Reuben Hersh, 1997. What Is Mathematics, Really? Oxford Univ. Press.
• Leslie White, 1947, "The Locus of Mathematical Reality: An Anthropological Footnote," Philosophy of Science 14: 289–303. Reprinted in Reuben Hersh, ed., 2006. 18 Unconventional Essays on the Nature of Mathematics. Springer: 304–19.
References
1. MAA presidents: Raymond Louis Wilder
2. Eilenberg, Samuel (1950). "Review: Topology of manifolds, by R. L. Wilder". Bull. Amer. Math. Soc. 56 (1, Part 1): 75–77. doi:10.1090/s0002-9904-1950-09349-5.
3. Frink, Orrin (1953). "Review: Introduction to the foundations of mathematics, by R. L. Wilder". Bull. Amer. Math. Soc. 59 (6): 580–582. doi:10.1090/s0002-9904-1953-09770-1.
External links
• J J O'Connor and E F Robertson, MacTutor: Raymond Louis Wilder. The source for this entry.
• Wilder papers at the University of Texas.
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\begin{document}
\title{Quantum Liang Information Flow as Causation Quantifier}
\newcommand{Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, United Kingdom.}{Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, United Kingdom.}
\author{Bin Yi} \affiliation{Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, United Kingdom.}
\author{Sougato Bose} \affiliation{Department of Physics and Astronomy, University College London, Gower Street, WC1E 6BT London, United Kingdom.}
\date{\today}
\begin{abstract}
Liang information flow is a quantity widely used in classical network theory to quantify causation, and has been applied widely, for example, to finance and climate. The most striking aspect here is to freeze/subtract a certain node of the network to ascertain its causal influence to other nodes of the network. Such an approach is yet to be applied to quantum network dynamics. Here we generalize Liang information flow to the quantum domain using the von-Neumann entropy. Using that we propose to assess the relative importance of various nodes of a network to causally influence a target node. We exemplify the application by using small quantum networks.
\end{abstract}
\maketitle
\paragraph*{Introduction} The significance of information flow lies not only in communication, but also in its logical implication of causation\cite{hlavavckova2007causality,pearl2000models,spirtes2000causation,bezruchko2010extracting,schreiber2000measuring}. Established in the context of classical physics, the mathematical theory of causality has been widely applied to a variety of disciplines, e.g., climate science\cite{wang2004relation,runge2012quantifying}, network dynamics\cite{sun1994neural,ay2008information,sommerlade2009estimating,timme2014revealing}, neuroscience\cite{pereda2005nonlinear,friston2003dynamic,schelter2006testing,staniek2008symbolic,andrzejak2011characterizing,wu2008detecting}, fiance\cite{marschinski2002analysing,lee2012jumps}, turbulence\cite{tissot2014granger,materassi2014information}. Historically, various measures of classical information flow were proposed\cite{runge2012quantifying,vastano1988information,sun2014causation,schreiber2000measuring,marschinski2002analysing,staniek2008symbolic,duan2013direct}, nonetheless, limitations were pointed out\cite{hahs2011distinguishing,smirnov2013spurious,sun2014causation}. In light of the limitations, Liang and Kleeman established a universally applicable formalism within the framework of classical dynamical systems\cite{san2005information,san2008information,san2014unraveling,san2016information,liang2021measuring,liang2021normalized}. The series of work puts the notion of information flow and causation on a rigorous footing, as Liang(2016)\cite{san2016information} argued: "Information flow and causality can be derived \textit{ab initio}" The formalism has been validated with various benchmark cases\cite{san2016information}, and successfully applied to many realistic problems: glaciology\cite{vannitsem2019testing}, neuro-science\cite{hristopulos2019disrupted}, El Ni\~{n}o-Indian Ocean Dipole relation\cite{san2014unraveling}, precipitation-soil moisture interaction\cite{hagan2019time}, global climate change\cite{stips2016causal}, etc.
The discussion of causality in quantum physics goes back to the paradigmatic Bell experiment\cite{bell1964einstein}. Causal structure places constraints on the correlations that can be generated in any classical hidden variable theories, which quantum physics violates\cite{freedman1972experimental,aspect1982experimental,christensen2013detection,rowe2001experimental,giustina2013bell}. Motivated by the connection between causality and correlations, various attempts have been made to estimate causal influences in certain quantum environments\cite{gachechiladze2020quantifying,henson2014theory,chaves2015information,costa2016quantum,fritz2016beyond,barrett2019quantum,wolfe2021quantum,aaberg2020semidefinite}. The quantification of causal effects in quantum regime shed new light on quantum communication\cite{fitzsimons2015quantum,pisarczyk2019causal}. Furthermore, a information-flux approach was introduced for many-body systems to quantify the influence from a specific element to another, aiming to facilitate the design of quantum processors equipped with large registers\cite{di2007information,di2008deeper}. In quantum mechanics, correlation functions of Heisenberg picture evolving operators are often used to ascertain casual influences, but one has to be careful that correlations do not imply causation.
Somewhat counterintuitively, the most straightforward approach to ascertaining causality, for example, one which an experimentalist will naturally employ, namely, to subtract a given component from a network to quantify its influence on other subsystems, remain unexplored. Motivated by that, in this work, we adopt Liang's methodology to establish a formalism of quantum information flow. As opposed to all the approaches mentioned above in the quantum context, here one detaches or freezes a certain subsystem of a network (sender) in order to ascertain its causal influence on other subsystems (target). The change of a target element's von-Neumann entropy, which possess various interpretations\cite{nielsen2002quantum}, then defines the information flow from the sender. When the sending and receiving elements evolve independently, then the information flow measure vanishes.
\paragraph*{Definition:} Consider arbitrary multi-partite system with density state $\rho$, evolving under unitary operator $U(t)=e^{-iHt/\hbar}$, with Hamiltonian $H$. Following Liang's methodology (briefly reviewed in the Supplementary Material[SM]), we decompose the time rate change of von-Neumann entropy of subsystem A, $dS_A/dt$, into two parts: $T_{B\rightarrow A}$, the rate of information flow from subsystem B to A, and $\frac{dS_{A\not{B}}}{dt}$, the entropic evolution rate of subsystem A with influence from B excluded: \begin{equation}\label{flow} T_{B\rightarrow A}=\frac{dS_A}{dt}-\frac{dS_{A\not{B}}}{dt} \end{equation} $S$ is von-Neumann entropy given by $S=-\mathrm{Tr}(\sigma \mathrm{log}\sigma)$ for arbitrary density state $\sigma$. $S_{A\not{B}}=S(\rho_{A\not{B}})=S[\varepsilon(t)_{A\not{B}} \rho_A(0)]$, where $\varepsilon(t)_{A\not{B}}$ is a map denoting the evolution of A with B frozen. We will discuss the definition and properties of $\varepsilon(t)_{A\not{B}}$ in the following section. If we consider time evolution as a discrete mapping during interval $\Delta t$, the cumulative information flow is then: \begin{equation} \mathbb{T}_{B\rightarrow A}=\int T_{B\rightarrow A} dt= \Delta (S_A-S_{A\not{B}}) \label{discreteT} \end{equation} Note that von-Neumann entropy, therefore the information flow formalism, possess various interpretations\cite{nielsen2002quantum}. Particularly distinct from classical Shannon entropy, von-Neumann entropy quantifies the entanglement within a pure bipartite quantum system. $S_{A\not{B}}$ (or $S_A$) can then be interpreted as the entanglement between A and the rest of the universe with (or without) B frozen. The term $(S_A-S_{A\not{B}})$ that appears in eq\ref{flow},\ref{discreteT} is then the difference of these two entanglement measures, in units of ebits. $\mathbb{T}_{B\rightarrow A}$ then quantifies the causal influence of B on A in the sense of how much it causes the entanglement of A with the rest of the universe to change. Similarly, other interpretations of von-Neumann entropy, such as uncertainty of a given state, also applies here.
\paragraph*{Evolution of subsystem A with B frozen} Since $\varepsilon(t)_{A\not{B}}$ is a mapping of density states, it can be interpreted as a quantum channel acting on subsystem A\cite{nielsen2002quantum}: $\rho_A(0)\overset{\varepsilon(t)_{A\not{B} }}{\rightarrow}\rho_{A\not{B}}(t)$. We further require that $\varepsilon(t)_{A\not{B}}$ corresponds to a physical process, therefore it can be obtained from taking the partial trace of the full system, which evolves unitarily. For tripartite system $\rho_{ABC}$: \begin{equation} \rho_{A\not{B}}(t)=\mathrm{Tr}_{BC}\{U_{A\not{B}C}(t)\rho_{ABC}(0 )U^\dagger_{A\not{B}C}(t)\} \end{equation} for some unitary operator $U_{A\not{B}C}$.
Moreover, we require that the evolution mechanism with some subsystems frozen takes product form between the frozen qubits and the rest of the system: \begin{equation} U_{A\not{B}C}(t)=\mathcal{V}_{AC}\otimes \mathcal{W}_B \label{UnotB} \end{equation} where $\mathcal{V}_{AC}$ and $\mathcal{W}_B$ are unitary operators acting on subsystems AC and B respectively. Frozen mechanism of the form Eq\ref{UnotB} guarantees what Liang referred to as \textit{the principle of nil causality}\cite{san2016information} (See SM for proof):
\emph{$T_{B\rightarrow A}=0$ if the evolution of A is independent of B, that is, the unitary evolution operator $U_{ABC}(t)$ takes separable form $\mathcal{M}_A \otimes \mathcal{N}_{BC}$ or $\mathcal{O}_{AC} \otimes \mathcal{Q}_{B}$. }
Therefore, the causal structure of space-time in physics is embedded in the formalism. If quantum operations, conducted at 4-dimensional coordinate $x$ and $y$, are space-like separated, hence non-causal, then the operations acting at $x$ does not affect the state located at $y$ and vice versa. The quantum operations at $x$ and $y$ commute and the joint evolution is in product form. Thus the quantum Liang information flow from one coordinate to another vanishes.
\subparagraph*{Bipartite system}
Consider bipartite state $\rho_{AB}$ under unitary evolution $U_{AB}(t)$. Consulting with eq\ref{UnotB}, $U_{A\not{B}}$ takes the form $\mathcal{V}_A\otimes \mathcal{W}_B$ in 2 dimensions. Since von-Neumann entropy is invariant under unitary change of basis, $\rho_{A\not{B}}=\mathcal{V}_A \rho_A(0) \mathcal{V}_A^\dagger$ and $\frac{dS_{A\not{B}}}{dt}=0$. Therefore, the rate of information flow from B to A: $T_{B\rightarrow A}=\frac{dS_A}{dt}$. Similarly,$T_{A\rightarrow B}=\frac{dS_B}{dt}$. If the initial state $\rho_{AB}(0)$ is pure, that is, the system is closed, by Schmidt decomposition, $\rho_A$ and $\rho_B$ share the same set of eigenvalues. Since closed bipartite system is symmetric, $S_A(t)=S_B(t)$ and $T_{B\rightarrow A}=T_{A\rightarrow B}$. In general, if the initial state $\rho_{AB}(0)$ is mixed, which can arise from entanglement with some external system, then we no longer have the symmetry $T_{A\rightarrow B}\neq T_{B\rightarrow A}$. Consider CNOT gate with controlled qubit A and target qubit B acts on the initial state $\rho_{AB}(0)=(1/2|0\rangle \langle 0|_A+1/2|1\rangle \langle 1|_A)\otimes|0\rangle \langle 0|_B$, the system evolves to $1/2|0\rangle \langle 0|_A\otimes|0\rangle \langle 0|_B+1/2|1\rangle \langle 1|_A\otimes|1\rangle \langle 1|_B$. The cumulative information flow for this discrete mapping $\mathbb{T}_{B\rightarrow A}=\Delta S_A=0$ and $\mathbb{T}_{A\rightarrow B}=\Delta S_B=1bit$. The asymmetric quantum information flow obtained for initially mixed bipartite system parallels its classical counterpart (see SM for details). For multi-partite system $\rho_{ABCD\cdots}$, the information flow from the rest of a closed system towards a particular unit, say A, is equivalent to the bipartite scenario: $T_{BCD\cdots \rightarrow A}=\frac{dS_A}{dt}$, $\mathbb{T}_{BCD\cdots \rightarrow A}=\Delta S_A$. \subparagraph{Multipartite system} In general, evaluation of the information flow in multipartite system requires a method to fix $\mathcal{V}_{AC}$ in eq\Ref{UnotB}. In this section, we illustrate such an approach with tripartite system $\rho_{ABC}$. Consider time evolution operator $U(t)=e^{-iHt/\hbar}$. We define the evolution of A with B frozen by replacing the interaction terms relevant to B in the Hamiltonian $H$, with identity operator. For instance, take Hamiltonian: \begin{equation} H_{ABC}=H_{0A}+H_{0B}+H_{0C}+\mathcal{A}\otimes \mathcal{C}+\mathcal{B} \otimes \mathcal{C} \label{Hamiltonian} \end{equation} where $H_{0i}$,with $i=A,B,C$, is the free Hamiltonian. And $\mathcal{A},\mathcal{B},\mathcal{C}$, which describe their interactions, are hermitian operators acting on subsystem A, B, C respectively. The evolution mechanism with B frozen is then: $U_{A\not{B}C}=e^{-iH_{A\not{B}C} t/\hbar}$, where \begin{equation} H_{A\not{B}C}\equiv H_{0A}+H_{0C}+\mathcal{A}\otimes \mathcal{C}+I_B \end{equation} $U_{A\not{B}C}$ is clearly of the product form given in eq\ref{UnotB}, with $\mathcal{W}_B=I$ and $\mathcal{V}_{AC}$ generated by hermitian operator $H_{0A}+H_{0C}+\mathcal{A}\otimes \mathcal{C}$. The operational meaning of $U_{A\not{B}C}$ is then:
\emph{evolution of the system if subsystem B is removed from the original evolution mechanism.}
The operational meaning of the frozen mechanism guarantees that this definition is basis(observable) independent. Now, we are equipped with the tools needed to evaluate quantum Liang information flow. In the next section, we will elucidate this formalism with applications. \paragraph*{Application: multi-qubit spin system} Consider a multi-qubit spin chain, the interaction Hamiltonian between any two interacting qubits i,j is given by\cite{yung2003exact}: \begin{equation} H_{spin,ij}=\eta_{ij} (\sigma_{+i}\sigma_{-j}+\sigma_{-i}\sigma_{+j}) \label{spinH} \end{equation} where $\sigma_{\pm}$ can be expressed in terms of Pauli matrices $\{\sigma_{x,y,z}\}$, $\sigma_{\pm}=\frac{1}{2}(\sigma_x\pm i \sigma_y)$. $\eta$ is relative coupling strength. The Hamiltonian for 3 interacting qubits, labeled A, B, C, of the form eq\ref{Hamiltonian} is given by: \begin{equation} \eta_{AC} (\sigma_{+A}\sigma_{-C}+\sigma_{-A}\sigma_{+C })+\eta_{BC}(\sigma_{+B}\sigma_{-C}+\sigma_{-B}\sigma_{+C}) \label{H3spin} \end{equation} \subparagraph*{Relative coupling strength variation}
In this section, we investigate cumulative Information flow $\mathbb{T}$ from A, B to C with different coupling strength. We set the initial state of the sending qubits A, B being maximally mixed while the receiving qubits C pure: $\rho(0)=I_A\otimes I_{B}\otimes|0\rangle\langle 0|_C$. So the sending qubits are competing to propagate uncertainty towards the target qubit. The Hamiltonian with one qubit frozen, say A, is obtained by erasing the terms involving qubit A in Hamiltonian eq\ref{H3spin}: \begin{equation} H_{\not{A}BC}=\lambda\sigma_{+C}\sigma_{-B}+\sigma_{-C}\sigma_{+B}+I_A \label{spinAnotB} \end{equation} The evolution of $\rho_{\not{A}\not{B}C}$ is defined similarly by removing hermitian terms relevant to qubits A,B altogether. Therefore, $\Delta S_{\not{A}\not{B}C}$ vanishes and the joint cumulative information flow from AB to C is: $\mathbb{T}_{AB\rightarrow C}=\Delta S_C$. Set $\eta_{AC}=1$, $\eta_{BC}=3$, at time $t\sim 0.49$, the entropy of C reaches its maxima of 1 bit for the first time. This is the maximum uncertainty qubit C can receive, determined by its dimension. For the purpose of illustration, we compare the cumulative information flow from different sending qubits before this capacity is reached. The early time behavior of cumulative information flow $\mathbb{T}_{AB\rightarrow C}(t)$, $\mathbb{T}_{A\rightarrow C}(t)$, $\mathbb{T}_{B\rightarrow C}(t)$ is plotted in figure \ref{3spin1}.
\begin{figure}
\caption{\footnotesize \textbf{3-qubit spin chain} (a) From top to bottom (measured in Bits): $\mathbb{T}_{AB\rightarrow C}$, $\mathbb{T}_{B\rightarrow C}+\mathbb{T}_{A\rightarrow C}$, $\mathbb{T}_{B\rightarrow C}$, $\mathbb{T}_{A\rightarrow C}$. Coupling strength: $\eta_{AC}=1$, $\eta_{BC}=3$. Initial state: $\rho(0)=I_A\otimes I_B\otimes |0\rangle \langle 0|_C$. (b)Blue curves: $\mathbb{T}_{A\rightarrow C}$, Orange curves: $\mathbb{T}_{B\rightarrow C}$. Solid curves: Initial state $\rho_{0(1)}=I_A\otimes (0.9|0\rangle \langle 0|+0.1|1\rangle \langle 1|)_{B}\otimes|0\rangle \langle 0|_C$, Dashed curves: Initial state $\rho_{0(2)}=I_A\otimes (0.1|0\rangle \langle 0|+0.9|1\rangle \langle 1|)_{B}\otimes|0\rangle \langle 0|_C$. Coupling strength: $\eta_{AC}=\eta_{BC}=1$.}
\label{3spin1}
\label{3spininitial}
\end{figure} From figure\ref{3spin1}, we notice that: The cumulative information flow from B to C is greater than that from A to C: $\mathbb{T}_{B\rightarrow C}>\mathbb{T}_{A\rightarrow C}$. This formalism is consistent with the intuition that the strongly coupled qubit has greater impact towards the target. The direct addition of cumulative information flow from individual qubit A, B is smaller than the joint value: $\mathbb{T}_{B\rightarrow C}+\mathbb{T}_{A\rightarrow C}<\mathbb{T}_{AB\rightarrow C}$ in this example. It means that turning off qubit A and B altogether has more impact on qubit C than the direct addition of turning A, B off one at a time. Similar result is obtained for the early time behavior of 5 qubit spin chain (See SM). \subparagraph*{Initial configuration dependence}
Note that the information flow formalism also depends on the initial configuration. To see how different initial states affect the information flow, set the coupling constant equal: $\eta_{AC}=\eta_{BC}=1$, with initial state $\rho_{0(1)}=I_A\otimes (0.9|0\rangle \langle 0|+0.1|1\rangle \langle 1|)_{B}\otimes|0\rangle \langle 0|_C$ and $\rho_{0(2)}=I_A\otimes (0.1|0\rangle \langle 0|+0.9|1\rangle \langle 1|)_{B}\otimes|0\rangle \langle 0|_C$. In both cases, the initial entropy of qubit B is $\sim0.47$bit while A is 1 bit. At a first glance, one may be expecting that A is transmitting more uncertainty to C than qubit B. From figure\ref{3spininitial}, we see this is indeed the case for initial state $\rho_{0(1)}$. But when the initial state is switched to $\rho_{0(2)}$, we have $\mathbb{T}_{B\rightarrow C}>\mathbb{T}_{A\rightarrow C}$. This is because increasing in von-Neumann entropy could result from not only classical uncertainty propagation but also entanglement generation. The qubit interaction given in eq\ref{spinH} entangles state $|10\rangle$ ($|01\rangle$), while it does not act on state $|00\rangle$($|11\rangle$): \begin{eqnarray}
&(\sigma_{+}\sigma_{-}+\sigma_{-}\sigma_{+})|00\rangle=0\nonumber ,\,\,\ (\sigma_{+}\sigma_{-}+\sigma_{-}\sigma_{+})|10\rangle=|01\rangle \nonumber \end{eqnarray}
For initial state $\rho_{0(2)}$, qubit B and C has 90\% probability in $|10\rangle_{BC}$ state, the entangling mechanism greatly increases $\mathbb{T}_{B\rightarrow C}$ compare to $\rho_{0(1)}$, for which the probability is only 10\%. Changing the initial state to $\rho_{0(2)}$ also suppresses $\mathbb{T}_{A\rightarrow C}$ due to growing competition from B.
\subparagraph*{Quantum super-exchange}
Add constant magnetic field along the z-axis with strength $\mathbf{B}$ on the intermediate qubit C so that its energy is lifted by an amount $\mathbf{B}\sigma_z$, while qubit A and B remains unaffected. The total Hamiltonian acting on the system then adds up an additional term:
\begin{equation} H_{additional}=I_A\otimes I_B \otimes \mathbf{B}\sigma_{z(C)}
\end{equation}
Set coupling strength $\eta_{AC}=\eta_{BC}=1$ and initial state $\rho(0)=I_A\otimes|0\rangle \langle 0|_B\otimes I_{C}$. We wish to compare information flow from A,C to B with various magnetic field strength. Note that when $\mathbf{B}=0$, the dynamics of information flow in the XY model (eq\ref{spinH}), which is not apriori obvious, can be pictured from fig\ref{superexchange1} The cumulative information flow is initially from C to B and it reaches a high value of 1 bit before it declines and is overtaken by the cumulative information flow from A to B. As the magnetic field strength increases, super-exchange process \cite{benjamin2004quantum} between A and B becomes progressively dominant. Thus, we see that information flow from C to B goes down while that from A to B becomes that dictated by an effective weaker super-exchange coupling $\eta_{AC}^2/\mathbf{B}$ between A and B $(\sigma_{+A}\sigma_{-B}+h.c.)$\cite{benjamin2004quantum}. \begin{figure}
\caption{\footnotesize \textbf{Quantum super-exchange}: (In Bits) Blue curves: $\mathbb{T}_{A\rightarrow B}$, Orange curves: $\mathbb{T}_{C\rightarrow B}$. In \ref{superexchange1},\ref{superexchange2},\ref{superexchange3},\ref{superexchange4}, Magnetic field strength set to $\mathbf{B}=0, 3, 5, 15$ respectively. Coupling strength: $\eta_{AC}=\eta_{BC}=1$. Initial state: $I_A\otimes|0\rangle \langle 0|_B\otimes I_{C}$. }
\label{superexchange1}
\label{superexchange2}
\label{superexchange3}
\label{superexchange4}
\label{superexchange}
\end{figure}
\subparagraph*{5-qubit network} Consider 5-qubit spin system, labeled A,B,C,D,E, with E in the center, we wish to investigate information flow towards E. The total Hamiltonian for the 5-qubit spin chain is \begin{equation} H_{spin,tot}=\sum_{i} H_{spin,iE} \end{equation} with $i=A,B,C,D$. Set all the coupling strength with E identical: $\eta_{DE}=\eta_{CE}=\eta_{BE}=\eta_{AE}=1$, and initial state of sending qubits A,B,C,D maximally mixed, receiving qubit E pure. At time $t\sim0.69$, the entropy of E reaches its maximum of 1 bit for the first time. The cumulative information flow from each sending qubit, which is identical $\mathbb{T}_{A\rightarrow E}=\mathbb{T}_{B\rightarrow E}=\mathbb{T}_{C\rightarrow E}=\mathbb{T}_{D \rightarrow E}$, is plotted for the time interval $t\in [0,0.69]$ in figure\ref{5qubit1}.
\begin{figure}
\caption{\footnotesize \textbf{5-qubit network} Cumulative information flow (in Bits) (a) from any sending qubit towards E with identical coupling strength: $\eta_{DE}=\eta_{CE}=\eta_{BE}=\eta_{AE}=1$. (b) with additional coupling $\eta_{CD}=5$. Orange curve: A(or B) to E, Blue curve: C(or D) to E.}
\label{5qubit1}
\label{5qubit2}
\end{figure}
Now let us add mutual interaction between C, D with relative coupling strength $\eta_{CD}=5$ and observe how does the information flow towards the center qubit E changes (schematic diagram of the interaction pattern is shown in SM). The total Hamiltonian is now given by: \begin{equation} \sum_{i} H_{spin,iE}+H_{spin,CD} \end{equation}
With presence of this additional interaction term, the cumulative information flow from each sending qubit to E is plotted in figure\ref{5qubit2}. Compare figure\ref{5qubit2} with figure\ref{5qubit1}, the presence of the additional interaction term between C,D greatly reduces the transmitted uncertainty from qubit C (D) to qubit E, while increases that from qubit A (B) to qubit E. After time $t\sim0.49$, $\mathbb{T}_{C\rightarrow E}$ reaches negative value, that is, the presence of qubit C (D) reduces the uncertainty of qubit E. The uncertainty from qubit C (D) now has two routes to propagate, either towards E or D (C). Also, the relative coupling strength $\eta_{CD}$ is five times stronger than $\eta_{CE},\eta_{DE}$. The strongly coupled route connecting C and D then diverts the uncertainty propagation away from the original path between C (D) and E, so that $\mathbb{T}_{C\rightarrow E}$($\mathbb{T}_{D\rightarrow E}$) decreases. Qubit A and B now has less competition from qubit C and D to propagate uncertainty towards qubit E. Then, $\mathbb{T}_{A\rightarrow E}$($\mathbb{T}_{B\rightarrow E}$) increases. The presence of certain coupling could preserve information, which may be exploited to design robust quantum circuits.
\paragraph*{Application: Two-qubit system in bosonic bath}
Let subsystem A and B indicate two non-interacting qubits with ground and excited states $|0\rangle$, $|1\rangle$, embedded in a common zero-temperature bosonic reservoir labeled C. We wish to compare information flow between the two qubits. The Hamiltonian governing the mechanism is given by $H_{SB}=H_0+H_{int}$, with: \begin{eqnarray} H_0&=&\omega_0 \sigma_+^A \sigma_-^A+ \omega_0 \sigma_+^B \sigma_-^B+\sum_{k}\omega_k b_k^\dagger b_k \nonumber\\ H_{int}&=&\alpha_A \sigma_+^A \sum_{k}g_k b_k+\alpha_B \sigma_+^B \sum_{k}g_k b_k+h.c. \label{Hsb} \end{eqnarray} where $\sigma_{\pm}^{A(B)}$ and $\omega_0$ are the inversion operator and transition frequency of qubit A(B). $b_k,b_k^\dagger$ are annihilation and creation operator of the environment C. $\alpha_{A(B)}$ measures the coupling between each qubit and the reservoir. In the limit $\alpha_B$ or $\alpha_A$ goes to 0, that is, when one of the qubit decouples from the setup, then $\rho_A$ and $\rho_{A\not{B}}$ obeys the same equation of motion and $\rho_A(t)=\rho_{A\not{B}}(t)$. Therefore, $T_{B\rightarrow A}=T_{A\rightarrow B}=0$.
We adopt the lossy cavity model given in Ref\cite{franco2013dynamics}. The two-qubit dynamics is solved exactly at zero temperature. Take initial state $\rho_{AB}(0)=|\psi_0\rangle \langle \psi_0|$, where $|\psi_0\rangle=\frac{1}{\sqrt{3}}(|01\rangle +\sqrt{2}|10\rangle)$. Let $\lambda=1$, $\hbar=1$, $\alpha_A/\alpha_B=10/1$ and take strong coupling limit $R=10$, where $\lambda$ defines the spectral width of the coupling and $R$ determines the collective coupling strength. The rate of information flow from B to A versus that from A to B is plotted in figure \ref{spinboson1}. The cumulative information flow is shown in figure \ref{spinboson2}. \begin{figure}
\caption{\footnotesize \textbf{Two-qubit system in a lossy cavity} Blue curves: From B to A. Orange curves: From A to B. Coupling strength ratio: $\alpha_A/\alpha_B=10/1$. (a) Rate of information flow (Bits per unit time) (b) Cumulative information flow (Bits).}
\label{spinboson1}
\label{spinboson2}
\end{figure} From Fig\ref{spinboson1}, we see that the rate of information flow from the weakly coupled qubit (B) towards the strongly coupled qubit (A) possess higher peak than that from A to B. On the other hand, as shown in fig\ref{spinboson2}, the cumulative information flow from A to B grows steadily and surpass that from B to A as the system approaches equilibrium. Note that the information flow formalism is generically asymmetric $T_{B\rightarrow A}\neq T_{A\rightarrow B}$ as opposed to most quantum correlation measures.
\paragraph*{Conclusions:}
In this paper, we have generalized Liang's methodology to quantify the causal influences in a quantum network. A unique feature of quantum networks is the possiblity of entanglement between its components. Thus, there are two ways to increase the entropy of a node: classical uncertainty propagation, as well as the growth of entanglement. We have verified the formalism through simple networks. The information flow between two qubits through a common bath could be nontrivial in the sense that the weakly coupled qubit has higher rate of information flow, while in the long run, the strongly coupled qubit has more impact on the weakly coupled one. Another non-trivial result obtained for a 5-qubit network reveals that an additional strong coupling diverts the directions of uncertainty propagation. Causal influences in generic complex quantum networks may be intricate and a picturization in terms of information flow will certainly aid their understanding. Note that definition of the information flow formalism is based on 1.full knowledge of the dynamics, 2.an intervention (frozen mechanism) act upon the system. For its classical counterpart, Liang has showed that the information flow can be estimated with local statistics for a broad range of subjects\cite{san2014unraveling,san2016information,liang2021measuring,liang2021normalized,vannitsem2019testing,hristopulos2019disrupted,hagan2019time,stips2016causal}. To what extent can the quantum version be estimated without knowing the dynamics apriori or doing the intervention on the system remains a subject for further investigation.
\acknowledgements B.Yi would like to thank S.X.Huang for presenting the problem as well as inspiring discussions with X.S.Liang and A.J.Leggett. S.Bose acknowledges the EPSRC grant Nonergodic quantum manipulation EP/R029075/1.
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{Pearl}} \emph {et~al.},\ }\href@noop {} {\bibfield {journal} {\bibinfo
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{volume} {19}} (\bibinfo {year} {2000})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Spirtes}\ \emph {et~al.}(2000)\citenamefont
{Spirtes}, \citenamefont {Glymour}, \citenamefont {Scheines},\ and\
\citenamefont {Heckerman}}]{spirtes2000causation}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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}\href@noop {} {\emph {\bibinfo {title} {Causation, prediction, and
search}}}\ (\bibinfo {publisher} {MIT press},\ \bibinfo {year}
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\bibitem [{\citenamefont {Bezruchko}\ and\ \citenamefont
{Smirnov}(2010)}]{bezruchko2010extracting}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.~P.}\ \bibnamefont
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(\bibinfo {publisher} {Springer Science \& Business Media},\ \bibinfo {year}
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\bibitem [{\citenamefont {Schreiber}(2000)}]{schreiber2000measuring}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont
{Schreiber}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
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\bibitem [{\citenamefont {Wang}\ \emph {et~al.}(2004)\citenamefont {Wang},
\citenamefont {Anderson}, \citenamefont {Kaufmann},\ and\ \citenamefont
{Myneni}}]{wang2004relation}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {W.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {R.~B.}\ \bibnamefont {Myneni}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Journal of
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(\bibinfo {year} {2004})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Runge}\ \emph {et~al.}(2012)\citenamefont {Runge},
\citenamefont {Heitzig}, \citenamefont {Marwan},\ and\ \citenamefont
{Kurths}}]{runge2012quantifying}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont {Marwan}}, \ and\ \bibinfo
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{\bibfield {journal} {\bibinfo {journal} {Physical Review E}\ }\textbf
{\bibinfo {volume} {86}},\ \bibinfo {pages} {061121} (\bibinfo {year}
{2012})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Sun}(1994)}]{sun1994neural}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
{Sun}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {IEEE
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{pages} {604} (\bibinfo {year} {1994})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Ay}\ and\ \citenamefont
{Polani}(2008)}]{ay2008information}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {N.}~\bibnamefont
{Ay}}\ and\ \bibinfo {author} {\bibfnamefont {D.}~\bibnamefont {Polani}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Advances in
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(\bibinfo {year} {2008})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Sommerlade}\ \emph {et~al.}(2009)\citenamefont
{Sommerlade}, \citenamefont {Eichler}, \citenamefont {Jachan}, \citenamefont
{Henschel}, \citenamefont {Timmer},\ and\ \citenamefont
{Schelter}}]{sommerlade2009estimating}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {L.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Jachan}}, \bibinfo
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{\bibfnamefont {B.}~\bibnamefont {Schelter}},\ }\href@noop {} {\bibfield
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{volume} {80}},\ \bibinfo {pages} {051128} (\bibinfo {year}
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\bibitem [{\citenamefont {Timme}\ and\ \citenamefont
{Casadiego}(2014)}]{timme2014revealing}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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{2014})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Pereda}\ \emph {et~al.}(2005)\citenamefont {Pereda},
\citenamefont {Quiroga},\ and\ \citenamefont
{Bhattacharya}}]{pereda2005nonlinear}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {E.}~\bibnamefont
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Progress in
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(\bibinfo {year} {2005})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Friston}\ \emph {et~al.}(2003)\citenamefont
{Friston}, \citenamefont {Harrison},\ and\ \citenamefont
{Penny}}]{friston2003dynamic}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {K.~J.}\ \bibnamefont
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Neuroimage}\
}\textbf {\bibinfo {volume} {19}},\ \bibinfo {pages} {1273} (\bibinfo {year}
{2003})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Schelter}\ \emph {et~al.}(2006)\citenamefont
{Schelter}, \citenamefont {Winterhalder}, \citenamefont {Eichler},
\citenamefont {Peifer}, \citenamefont {Hellwig}, \citenamefont {Guschlbauer},
\citenamefont {L{\"u}cking}, \citenamefont {Dahlhaus},\ and\ \citenamefont
{Timmer}}]{schelter2006testing}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {B.}~\bibnamefont {Hellwig}}, \bibinfo
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{\bibinfo {volume} {152}},\ \bibinfo {pages} {210} (\bibinfo {year}
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\bibitem [{\citenamefont {Staniek}\ and\ \citenamefont
{Lehnertz}(2008)}]{staniek2008symbolic}
\BibitemOpen
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{Physical review letters}\ }\textbf {\bibinfo {volume} {100}},\ \bibinfo
{pages} {158101} (\bibinfo {year} {2008})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Andrzejak}\ and\ \citenamefont
{Kreuz}(2011)}]{andrzejak2011characterizing}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.~G.}\ \bibnamefont
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{Kreuz}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {EPL
(Europhysics Letters)}\ }\textbf {\bibinfo {volume} {96}},\ \bibinfo {pages}
{50012} (\bibinfo {year} {2011})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Wu}\ \emph {et~al.}(2008)\citenamefont {Wu},
\citenamefont {Liu},\ and\ \citenamefont {Feng}}]{wu2008detecting}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
{Wu}}, \bibinfo {author} {\bibfnamefont {X.}~\bibnamefont {Liu}}, \ and\
\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont {Feng}},\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {Journal of Neuroscience Methods}\
}\textbf {\bibinfo {volume} {167}},\ \bibinfo {pages} {367} (\bibinfo {year}
{2008})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Marschinski}\ and\ \citenamefont
{Kantz}(2002)}]{marschinski2002analysing}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {R.}~\bibnamefont
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{\bibinfo {volume} {30}},\ \bibinfo {pages} {275} (\bibinfo {year}
{2002})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Lee}(2012)}]{lee2012jumps}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~S.}\ \bibnamefont
{Lee}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {The
Review of Financial Studies}\ }\textbf {\bibinfo {volume} {25}},\ \bibinfo
{pages} {439} (\bibinfo {year} {2012})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Tissot}\ \emph {et~al.}(2014)\citenamefont {Tissot},
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\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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{pages} {012006}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Materassi}\ \emph {et~al.}(2014)\citenamefont
{Materassi}, \citenamefont {Consolini}, \citenamefont {Smith},\ and\
\citenamefont {De~Marco}}]{materassi2014information}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.}~\bibnamefont
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and\ \bibinfo {author} {\bibfnamefont {R.}~\bibnamefont {De~Marco}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Entropy}\ }\textbf
{\bibinfo {volume} {16}},\ \bibinfo {pages} {1272} (\bibinfo {year}
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\bibitem [{\citenamefont {Vastano}\ and\ \citenamefont
{Swinney}(1988)}]{vastano1988information}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~A.}\ \bibnamefont
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{Physical Review Letters}\ }\textbf {\bibinfo {volume} {60}},\ \bibinfo
{pages} {1773} (\bibinfo {year} {1988})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Sun}\ and\ \citenamefont
{Bollt}(2014)}]{sun2014causation}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.}~\bibnamefont
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D: Nonlinear Phenomena}\ }\textbf {\bibinfo {volume} {267}},\ \bibinfo
{pages} {49} (\bibinfo {year} {2014})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Duan}\ \emph {et~al.}(2013)\citenamefont {Duan},
\citenamefont {Yang}, \citenamefont {Chen},\ and\ \citenamefont
{Shah}}]{duan2013direct}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {P.}~\bibnamefont
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\bibitem [{\citenamefont {Hahs}\ and\ \citenamefont
{Pethel}(2011)}]{hahs2011distinguishing}
\BibitemOpen
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\bibitem [{\citenamefont {Smirnov}(2013)}]{smirnov2013spurious}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~A.}\ \bibnamefont
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\bibitem [{\citenamefont {San~Liang}\ and\ \citenamefont
{Kleeman}(2005)}]{san2005information}
\BibitemOpen
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{Physical review letters}\ }\textbf {\bibinfo {volume} {95}},\ \bibinfo
{pages} {244101} (\bibinfo {year} {2005})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {San~Liang}(2008)}]{san2008information}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
{San~Liang}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
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{031113} (\bibinfo {year} {2008})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {San~Liang}(2014)}]{san2014unraveling}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
{San~Liang}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Physical Review E}\ }\textbf {\bibinfo {volume} {90}},\ \bibinfo {pages}
{052150} (\bibinfo {year} {2014})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {San~Liang}(2016)}]{san2016information}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {X.}~\bibnamefont
{San~Liang}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
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{052201} (\bibinfo {year} {2016})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Liang}(2021{\natexlab{a}})}]{liang2021measuring}
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{2021}{\natexlab{a}})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Liang}(2021{\natexlab{b}})}]{liang2021normalized}
\BibitemOpen
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(\bibinfo {year} {2021}{\natexlab{b}})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Vannitsem}\ \emph {et~al.}(2019)\citenamefont
{Vannitsem}, \citenamefont {Dalaiden},\ and\ \citenamefont
{Goosse}}]{vannitsem2019testing}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Geophysical
Research Letters}\ }\textbf {\bibinfo {volume} {46}},\ \bibinfo {pages}
{12125} (\bibinfo {year} {2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Hristopulos}\ \emph {et~al.}(2019)\citenamefont
{Hristopulos}, \citenamefont {Babul}, \citenamefont {Babul}, \citenamefont
{Brucar},\ and\ \citenamefont {Virji-Babul}}]{hristopulos2019disrupted}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~T.}\ \bibnamefont
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\bibinfo {author} {\bibfnamefont {S.}~\bibnamefont {Babul}}, \bibinfo
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{\bibfield {journal} {\bibinfo {journal} {Frontiers in human neuroscience}\
}\textbf {\bibinfo {volume} {13}},\ \bibinfo {pages} {419} (\bibinfo {year}
{2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Hagan}\ \emph {et~al.}(2019)\citenamefont {Hagan},
\citenamefont {Wang}, \citenamefont {San~Liang},\ and\ \citenamefont
{Dolman}}]{hagan2019time}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {D.~F.~T.}\
\bibnamefont {Hagan}}, \bibinfo {author} {\bibfnamefont {G.}~\bibnamefont
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Journal of
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(\bibinfo {year} {2019})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Stips}\ \emph {et~al.}(2016)\citenamefont {Stips},
\citenamefont {Macias}, \citenamefont {Coughlan}, \citenamefont
{Garcia-Gorriz},\ and\ \citenamefont {San~Liang}}]{stips2016causal}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
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\bibinfo {author} {\bibfnamefont {C.}~\bibnamefont {Coughlan}}, \bibinfo
{author} {\bibfnamefont {E.}~\bibnamefont {Garcia-Gorriz}}, \ and\ \bibinfo
{author} {\bibfnamefont {X.}~\bibnamefont {San~Liang}},\ }\href@noop {}
{\bibfield {journal} {\bibinfo {journal} {Scientific reports}\ }\textbf
{\bibinfo {volume} {6}},\ \bibinfo {pages} {1} (\bibinfo {year}
{2016})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Bell}(1964)}]{bell1964einstein}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {J.~S.}\ \bibnamefont
{Bell}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physics
Physique Fizika}\ }\textbf {\bibinfo {volume} {1}},\ \bibinfo {pages} {195}
(\bibinfo {year} {1964})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Freedman}\ and\ \citenamefont
{Clauser}(1972)}]{freedman1972experimental}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~J.}\ \bibnamefont
{Freedman}}\ and\ \bibinfo {author} {\bibfnamefont {J.~F.}\ \bibnamefont
{Clauser}},\ }\href@noop {} {\bibfield {journal} {\bibinfo {journal}
{Physical Review Letters}\ }\textbf {\bibinfo {volume} {28}},\ \bibinfo
{pages} {938} (\bibinfo {year} {1972})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Aspect}\ \emph {et~al.}(1982)\citenamefont {Aspect},
\citenamefont {Grangier},\ and\ \citenamefont
{Roger}}]{aspect1982experimental}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {A.}~\bibnamefont
{Aspect}}, \bibinfo {author} {\bibfnamefont {P.}~\bibnamefont {Grangier}}, \
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}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physical review
letters}\ }\textbf {\bibinfo {volume} {49}},\ \bibinfo {pages} {91} (\bibinfo
{year} {1982})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Christensen}\ \emph {et~al.}(2013)\citenamefont
{Christensen}, \citenamefont {McCusker}, \citenamefont {Altepeter},
\citenamefont {Calkins}, \citenamefont {Gerrits}, \citenamefont {Lita},
\citenamefont {Miller}, \citenamefont {Shalm}, \citenamefont {Zhang},
\citenamefont {Nam} \emph {et~al.}}]{christensen2013detection}
\BibitemOpen
\bibfield {author} {\bibinfo {author} {\bibfnamefont {B.~G.}\ \bibnamefont
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\bibinfo {author} {\bibfnamefont {T.}~\bibnamefont {Gerrits}}, \bibinfo
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{\bibfnamefont {A.}~\bibnamefont {Miller}}, \bibinfo {author} {\bibfnamefont
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\bibnamefont {Nam}}, \emph {et~al.},\ }\href@noop {} {\bibfield {journal}
{\bibinfo {journal} {Physical review letters}\ }\textbf {\bibinfo {volume}
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{NoStop}
\bibitem [{\citenamefont {Rowe}\ \emph {et~al.}(2001)\citenamefont {Rowe},
\citenamefont {Kielpinski}, \citenamefont {Meyer}, \citenamefont {Sackett},
\citenamefont {Itano}, \citenamefont {Monroe},\ and\ \citenamefont
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\bibitem [{\citenamefont {Chaves}\ \emph {et~al.}(2015)\citenamefont {Chaves},
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\bibitem [{\citenamefont {Costa}\ and\ \citenamefont
{Shrapnel}(2016)}]{costa2016quantum}
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\bibitem [{\citenamefont {Fritz}(2016)}]{fritz2016beyond}
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{NoStop}
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\bibitem [{\citenamefont {{\AA}berg}\ \emph {et~al.}(2020)\citenamefont
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}\textbf {\bibinfo {volume} {125}},\ \bibinfo {pages} {110505} (\bibinfo
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\bibitem [{\citenamefont {Fitzsimons}\ \emph {et~al.}(2015)\citenamefont
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\BibitemOpen
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{Scientific reports}\ }\textbf {\bibinfo {volume} {5}},\ \bibinfo {pages} {1}
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\bibitem [{\citenamefont {Pisarczyk}\ \emph {et~al.}(2019)\citenamefont
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}\textbf {\bibinfo {volume} {123}},\ \bibinfo {pages} {150502} (\bibinfo
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\bibitem [{\citenamefont {Di~Franco}\ \emph {et~al.}(2007)\citenamefont
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\citenamefont {Kim}}]{di2007information}
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\ and\ \bibinfo {author} {\bibfnamefont {M.}~\bibnamefont {Kim}},\
}\href@noop {} {\bibfield {journal} {\bibinfo {journal} {Physical Review
A}\ }\textbf {\bibinfo {volume} {76}},\ \bibinfo {pages} {042316} (\bibinfo
{year} {2007})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Di~Franco}\ \emph {et~al.}(2008)\citenamefont
{Di~Franco}, \citenamefont {Paternostro},\ and\ \citenamefont
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\bibitem [{\citenamefont {Nielsen}\ and\ \citenamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.~A.}\ \bibnamefont
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\bibitem [{\citenamefont {Yung}\ \emph {et~al.}(2003)\citenamefont {Yung},
\citenamefont {Leung},\ and\ \citenamefont {Bose}}]{yung2003exact}
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {M.-H.}\ \bibnamefont
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quant-ph/0312105}\ } (\bibinfo {year} {2003})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Benjamin}\ and\ \citenamefont
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\bibfield {author} {\bibinfo {author} {\bibfnamefont {S.~C.}\ \bibnamefont
{Benjamin}}\ and\ \bibinfo {author} {\bibfnamefont {S.}~\bibnamefont
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Review A}\ }\textbf {\bibinfo {volume} {70}},\ \bibinfo {pages} {032314}
(\bibinfo {year} {2004})}\BibitemShut {NoStop}
\bibitem [{\citenamefont {Franco}\ \emph {et~al.}(2013)\citenamefont {Franco},
\citenamefont {Bellomo}, \citenamefont {Maniscalco},\ and\ \citenamefont
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\appendix \section{Brief review of classical information flow-based causality analysis}\label{classicalLianginfo} Liang information flow quantitatively defines causality. The series of work starts with the investigation of bi-variate deterministic systems and is originally based on a heuristic argument\cite{san2005information}. Later on, the formalism is put on a rigorous footing and generalized to stochastic and multi-variate systems\cite{san2008information,san2016information,liang2021normalized}. To present this fundamental idea in its simplest form, we will focus on bi-variate autonomous system with equation of motion given by: \begin{equation} \frac{d \mathbf{x}}{dt}=\mathbf{F}(\mathbf{x})\label{eom} \end{equation} where $\mathbf{x}=(x_1,x_2)\in \Omega$ and the sample space $\Omega$ is a direct product of subspace $\Omega_1 \otimes \Omega_2$. $\mathbf{X}=(X_1,X_2)$ is the random variable of subsystem 1 and 2. $\{\mathbf{X},t\}$ is assumed a stochastic process and the joint probability density distribution at time t is denoted $\rho(x_1,x_2,t)$. $\mathbf{F}=(F_1,F_2)$ may be interpreted as the force acting on the system. Shannon entropy of this system is given by: \begin{equation} S_{(classical)}(t)=-\int_{\Omega} \rho log(\rho) dx_1 dx_2 \label{classical entropy} \end{equation} Substitute eq\ref{eom} into eq\ref{classical entropy}, one obtains the time rate change of entropy, provided that $\rho$ vanishes at boundaries\cite{san2005information}: \begin{equation} \frac{dS_{(classical)}}{dt}=E(\mathbf{\nabla} \cdot \mathbf{F})\label{dS} \end{equation} The right hand side is the expectation value of the divergence of force $\mathbf{F}$. The physics revealed by eq\ref{dS} is that the expansion and contraction of the phase space governs the change of entropy.
The probability distribution of a subsystem, say subsystem 1, can be obtained by taking the marginal density $\rho_1(x_1,t)=\int_{\Omega_2}\rho(x_1,x_2,t)dx_2$. Its entropy can be calculated: \begin{equation} \frac{dS_{1(classical)}}{dt}=-\int_{\Omega}\rho[\frac{F_1}{\rho_1}\frac{\partial \rho_1}{\partial x_1}]dx_1dx_2 \label{dS1} \end{equation} Liang and Kleeman identified that the entropy change of subsystem 1 given by eq\ref{dS1} can be decomposed into two parts: the evolution due to $X_1$ alone, with effect from subsystem 2 excluded, denoted as $\frac{dS_{1\not{2}(classical)}}{dt}$. Another part is the influence from $X_2$ through the coupling with external force. Through heuristic reasoning based on the interpretation of eq\ref{dS}, Liang and Kleeman argue that if subsystem 1 evolves on its own, the entropy change of subsystem 1 would depend only on $\partial F_1/\partial x_1$: \begin{equation} \frac{dS_{1\not{2}(classical)}}{dt}=E(\frac{\partial F_1}{\partial x_1})=\int_{\Omega}\rho \frac{\partial F_1}{\partial x_1} dx_1 dx_2 \label{1not2} \end{equation} Later on, Liang(2016\cite{san2016information}) proved that the above result eq\ref{1not2} can be derived by treating $x_2$ as a fixed parameter at time $t$, rather than a variate.
The rate of information flow from $X_2$ to $X_1$ is then: \begin{eqnarray} T_{2\rightarrow 1}&=&\frac{dS_{1(classical)}}{dt}-\frac{dS_{1\not{2}(classical)}}{dt}\nonumber \\ &=&-\int_{\Omega}\rho[\frac{F_1}{\rho_1}\frac{\partial \rho_1}{\partial x_1}+\frac{\partial F_1}{\partial x_1}]dx_1dx_2 \label{Tc} \end{eqnarray} This formula verifies what Liang refers to as \textit{the principle of nil causality}:
\emph{If $F_1$ is independent of $x_2$, then the information flow from 2 to 1 vanishes: $T_{2\rightarrow 1}=0$.}
If $T_{2\rightarrow 1}$ is negative (positive), the interpretation is that system 2 is making system 1 more (less) certain. Note that the information flow formalism eq\ref{Tc} is asymmetric, that is $T_{2\rightarrow 1}\neq T_{1\rightarrow 2}$. When the information flow from 2 to 1 vanishes, that from 1 to 2 maybe non-zero. The asymmetry feature distinguishes the information flow formalism with classical correlation measures.
It should be pointed out that the evaluation of eq\ref{Tc} requires full knowledge of the dynamics. In 2014\cite{san2014unraveling}, Liang showed that $T_{2\rightarrow 1}$ can be estimated with local statistics. The maximum-likelihood estimator of eq\ref{Tc} is shown to be a combination of some sample covariances, which greatly facilitates the implementation of the causality analysis.
This formalism has been widely applied to realistic schemes\cite{san2014unraveling,stips2016causal,hagan2019time,vannitsem2019testing,hristopulos2019disrupted}. Among them, we will briefly mention its application to a network consisting of Stuart-Landau oscillators\cite{liang2021measuring}, a typical model for many biological phenomena\cite{strogatz2001exploring}. The magnitude of Liang information flow quantifies the influence of individual components to produce the collective behavior of the whole system. The direct addition of individual contributions does not equal the cumulative information flow, demonstrating its collective property. Moreover, the node with greatest information flow is verified to be the most crucial as its suppression leads to shut down of the entire network. Surprisingly, such a node may be sparsely connected, rather than a center of network. The information-flow based causality analysis successfully explains why small defects at local node could severely damage structural integrity.
\section{Classical closed bivariate system}\label{classicalbipartite} The classical model considered in eq\ref{eom} is dissipative. System 1 and 2 exchanges energy with the environment through external force $\mathbf{F}$. If system 1 and 2 is closed, the divergence of force $\mathbf{F}$ vanishes: $\mathbf{\nabla} \cdot \mathbf{F}=0$. As a result, eq\ref{dS}, eq\ref{1not2} becomes: $dS_{(classical)}/dt=E(\mathbf{\nabla} \cdot \mathbf{F})=0$, $dS_{1\not{2}(classical)}/dt=E(\frac{\partial F_1}{\partial x_1})=0$, therefore, \begin{equation} T_{2\rightarrow 1}=\frac{dS_{1(classical)}}{dt}\label{eq10} \end{equation} Eq\ref{eq10} is completely in agreement with the quantum formalism obtained for initially mixed bipartite system.
\section{\textit{the principle of nil causality in Quantum regime}}\label{causality} For tripartite system, if $U_{A\not{B}C}(t)$ takes the form of eq(4) in the main text ($\mathcal{V}_{AC}\otimes \mathcal{W}_B$), then the statement of causality is satisfied, that is, $T_{B\rightarrow A}=0$ when A evolves independent of B. \begin{proof}
If $U_{ABC}=\mathcal{M}_A \otimes \mathcal{N}_{BC}$, the evolution of A is solely determined by unitary operator $\mathcal{M}_A$. Excluding B from the joint evolution of subsystem BC, denoted $\mathcal{N}_{\not{B}C}$, has no effect on A. Therefore, $\rho_A(t)=\rho_{A\not{B}}(t)=\mathcal{M}_A \rho_A(0) \mathcal{M}_A^\dagger$. By the unitary invariance of von-Neumann entropy, $\frac{dS_A}{dt}=\frac{dS_{A\not{B}}}{dt}=0$, thus $T_{B\rightarrow A}=0$.
If $U_{ABC}(t)=\mathcal{O}_{AC} \otimes \mathcal{Q}_{B}$, it is already of the form given in eq\ref{UnotB}. Therefore, excluding B or not has no impact on the joint evolution of system AC. That is,
\begin{eqnarray}
\rho_A(t)&=&\mathrm{Tr}_{BC}\{U_{ABC}(t)\rho_{ABC}(0 )U^\dagger_{ABC}(t)\}\nonumber\\
&=&\mathrm{Tr}_C[\mathcal{O}_{AC} \rho_{AC}(0) \mathcal{O}_{AC}^\dagger]\nonumber\\
&=&\mathrm{Tr}_{BC}\{U_{A\not{B}C}(t)\rho_{ABC}(0 )U^\dagger_{A\not{B}C}(t)\}=\rho_{A\not{B}}(t)\nonumber
\end{eqnarray}
Therefore, $T_{B\rightarrow A}=\frac{dS_A}{dt}-\frac{dS_{A\not{B}}}{dt}=0$. \end{proof} This results obtained above can be easily extended to multi-dimensions. Whether the converse proof also holds remains an open question.
\section{Relative coupling strength variation for 5 qubits}\label{5qubit}
For 5 qubits, labeled A,B,C,D,E, with E in the center and interacting with other qubits independently. To check if stronger coupled sending qubit delivers more information towards the receiving qubit, we set $\eta_{DE}=1$, $\eta_{CE}=2$, $\eta_{BE}=3$, $\eta_{AE}=4$ and let the initial state of the sending qubits A,B,C,D being maximally mixed and the receiving qubit E pure, so that $\rho_0=I_A/2\otimes I_B/2\otimes I_C/2\otimes I_D/2\otimes |0\rangle \langle 0|_E$.
Calculation of information flow from the $k^{th}$ qubit to E, where k runs through the sending qubits, requires the evolution mechanism with the $k^{th}$ qubit frozen: \begin{equation} H_{spin,\not{k}}=\sum_{i,i\neq k}H_{spin,iE} \end{equation} The joint information flow from A,B,C,D to E is simply the change of $S_E$: \begin{equation} \mathbb{T}_{ABCD\rightarrow E}=\Delta S_E \label{ABCDtoE} \end{equation} At time $t\sim0.26$, the entropy of E reaches its maxima $S_E=1bit$ for the first time. The Information flow from each sending qubit to E is plotted in figure \ref{fig4}, before the capacity is reached. The stronger coupled qubit delivers more information to E at all time during $t\in [0,0.26]$: \begin{equation} \mathbb{T}_{A\rightarrow E}>\mathbb{T}_{B\rightarrow E}>\mathbb{T}_{C\rightarrow E}>\mathbb{T}_{D \rightarrow E} \end{equation} \begin{figure}
\caption{\footnotesize \textbf{Cumulative Information flow towards qubit E (in Bits)} from top to bottom: $\mathbb{T}_{ABCD\rightarrow E}$, $\mathbb{T}_{A\rightarrow E}+\mathbb{T}_{B\rightarrow E}+\mathbb{T}_{C\rightarrow E}+\mathbb{T}_{D \rightarrow E}$,$\mathbb{T}_{A\rightarrow E}$, $\mathbb{T}_{B\rightarrow E}$, $\mathbb{T}_{C\rightarrow E}$, $\mathbb{T}_{D\rightarrow E}$}
\label{fig4}
\end{figure} At $t=0.26$, $\mathbb{T}_{A\rightarrow E}\sim 0.0731$bits, $\mathbb{T}_{B\rightarrow E}\sim 0.0132$bits, $\mathbb{T}_{C\rightarrow E}\sim 0.0022$bits, $\mathbb{T}_{D\rightarrow E}\sim 0.0001$bits. Similar to the results obtained for 3 qubit system in the main text, here we also observe superadditivity of quantum Liang information flow: \begin{equation} \mathbb{T}_{ABCD\rightarrow E}>\mathbb{T}_{A\rightarrow E}+\mathbb{T}_{B\rightarrow E}+\mathbb{T}_{C\rightarrow E}+\mathbb{T}_{D \rightarrow E} \end{equation} \section{Schematic diagram of a 5-qubit network} With the Hamiltonian given by eq(12) in the main text, the corresponding schematic diagram is shown below: \begin{figure}
\caption{\footnotesize \textbf{schematic diagram:} A,B couples solely with E, while C,D also interacts with each other. $\eta_{DE}=\eta_{CE}=\eta_{BE}=\eta_{AE}=1$, $\eta_{CD}=5$}
\label{fig7}
\end{figure}
\end{document} | arXiv |
\begin{definition}[Definition:Spheroid/Prolate]
A '''prolate spheroid''' is an ellipsoid which is the solid of revolution formed by rotating an ellipse about its major axis.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Lower Bound of Sequence]
A special case of a lower bound of a mapping is a '''lower bound of a sequence''', where the domain of the mapping is $\N$.
Let $\struct {T, \preceq}$ be an ordered set.
Let $\sequence {x_n}$ be a sequence in $T$.
Let $\sequence {x_n}$ be bounded below in $T$ by $L \in T$.
Then $L$ is a '''lower bound of $\sequence {x_n}$'''.
\end{definition} | ProofWiki |
Asian Pacific Journal of Cancer Prevention
Pages.2433-2438
Asian Pacific Organization for Cancer Prevention (아시아태평양암예방학회)
Implementation of Proteomics for Cancer Research: Past, Present, and Future
Karimi, Parisa ;
Shahrokni, Armin ;
Nezami Ranjbar, Mohammad R.
https://doi.org/10.7314/APJCP.2014.15.6.2433
Cancer is the leading cause of the death, accounts for about 13% of all annual deaths worldwide. Many different fields of science are collaborating together studying cancer to improve our knowledge of this lethal disease, and find better solutions for diagnosis and treatment. Proteomics is one of the most recent and rapidly growing areas in molecular biology that helps understanding cancer from an omics data analysis point of view. The human proteome project was officially initiated in 2008. Proteomics enables the scientists to interrogate a variety of biospecimens for their protein contents and measure the concentrations of these proteins. Current necessary equipment and technologies for cancer proteomics are mass spectrometry, protein microarrays, nanotechnology and bioinformatics. In this paper, we provide a brief review on proteomics and its application in cancer research. After a brief introduction including its definition, we summarize the history of major previous work conducted by researchers, followed by an overview on the role of proteomics in cancer studies. We also provide a list of different utilities in cancer proteomics and investigate their advantages and shortcomings from theoretical and practical angles. Finally, we explore some of the main challenges and conclude the paper with future directions in this field.
Proteomics;cancer;biomarkers;mass spectrometry;bioinformatics
Hanash SM, Baik CS, Kallioniemi O (2011). Emerging molecular biomarkers-blood-based strategies to detect and monitor cancer. Nat Rev Clin Oncol, 8, 142-50. https://doi.org/10.1038/nrclinonc.2010.220
Gray J, Chattopadhyay D, Beale GS, et al (2009). A proteomic strategy to identify novel serum biomarkers for liver cirrhosis and hepatocellular cancer in individuals with fatty liver disease. BMC Cancer, 9, 271. https://doi.org/10.1186/1471-2407-9-271
Gstaiger M, Aebersold R (2009). Applying mass spectrometrybased proteomics to genetics, genomics and network biology. Nat Rev Genet, 10, 617-27. https://doi.org/10.1038/nrg2633
Hainaut P, Plymoth A (2011). Biomarkers in cancer research and treatment: promises and challenges. Curr Opin Oncol, 23, 61. https://doi.org/10.1097/CCO.0b013e328341623e
Hanash SM, Pitteri SJ, Faca VM (2008). Mining the plasma proteome for cancer biomarkers. Nature, 452, 571-9. https://doi.org/10.1038/nature06916
Hatakeyama H, Kondo T, et al (2006). Protein clusters associated with carcinogenesis, histological differentiation and nodal metastasis in esophageal cancer. Proteomics, 6, 6300-16. https://doi.org/10.1002/pmic.200600488
Hooshmand S, Ghaderi A, Yusoff K, et al (2013). Downregulation of RhoGDI$\alpha$ increased migration and invasion of ER+MCF7 and ER? MDA-MB-231 breast cancer cells. Cell Adh Migr, 7, 297-303. https://doi.org/10.4161/cam.24204
International Cancer Genome Consortium, Hudson TJ, Anderson W, et al (2010). International network of cancer genome projects. Nature, 464, 993-8. https://doi.org/10.1038/nature08987
IBM Corp. Released 2011. IBM SPSS Statistics for Windows, V. A., NY: IBM Corp.
Jacquemier J, Ginestier C, Rougemount J, et al (2005). Protein expression profiling identifies subclasses of breast cancer and predicts prognosis. Cancer Res, 65, 767-79.
Ferlay J, Shin HR, Bray F, et al (2010). Estimates of worldwide burden of cancer in 2008: GLOBOCAN 2008. Int J Cancer, 127, 2893-917. https://doi.org/10.1002/ijc.25516
Edwards BK, Ward E, Kohler BA, et al (2010). Annual report to the nation on the status of cancer, 1975-2006, featuring colorectal cancer trends and impact of interventions (risk factors, screening, and treatment) to reduce future rates. Cancer, 116, 544-73. https://doi.org/10.1002/cncr.24760
Etzioni R, Urban N, Ramsey S, et al (2003). The case for early detection. Nat Rev Cancer, 3, 243-52. https://doi.org/10.1038/nrc1041
Fan Y, Wang J, Yang Y, et al (2010). Detection and identification of potential biomarkers of breast cancer. J Cancer Res Clin Oncol, 136, 1243-54. https://doi.org/10.1007/s00432-010-0775-1
Ferrari M (2005). Cancer nanotechnology: opportunities and challenges. Nat Rev Cancer, 5, 161-71. https://doi.org/10.1038/nrc1566
Fujita Y, Nakanishi T, Hiramatsu M, et al (2006). Proteomicsbased approach identifying autoantibody against peroxiredoxin VI as a novel serum marker in esophageal squamous cell carcinoma. Clin Cancer Res, 12, 6415-20. https://doi.org/10.1158/1078-0432.CCR-06-1315
Ghabaee M, Bayati A, Amir Saroukolaei, et al (2009). Analysis of HLA DR2&DQ6 (DRB1* 1501, DQA1* 0102, DQB1* 0602) haplotypes in Iranian patients with multiple sclerosis. Cell Mol Neurobiol, 29, 109-14. https://doi.org/10.1007/s10571-008-9302-1
Gold L, Ayers D, Bertino J, et al (2010). Aptamer-based multiplexed proteomic technology for biomarker discovery. PloS One, 5, 15004. https://doi.org/10.1371/journal.pone.0015004
Goncalves A, Charafe-Jauffret E, Bertucci F, et al (2008). Protein profiling of human breast tumor cells identifies novel biomarkers associated with molecular subtypes. Mol Cell Proteomics, 7, 1420-33. https://doi.org/10.1074/mcp.M700487-MCP200
Gortzak-Uzan L, Ignatchenko A, Evangelou Al, et al (2007). A proteome resource of ovarian cancer ascites: integrated proteomic and bioinformatic analyses to identify putative biomarkers. J Proteome Res, 7, 339-51.
Boja ES, Rodriguez H (2012). Mass spectrometry based targeted quantitative proteomics: Achieving sensitive and reproducible detection of proteins. Proteomics, 12, 1093-110. https://doi.org/10.1002/pmic.201100387
Bai Z, Ye Y, Liang B, et al (2011). Proteomics-based identification of a group of apoptosis-related proteins and biomarkers in gastric cancer. Int J Oncol, 38: 375-83.
Baker ES, Liu T, Petyuk VA, et al (2012). Mass spectrometry for translational proteomics: progress and clinical implications. Genome Med, 4, 1-11. https://doi.org/10.1186/gm300
Bhati A, Garg H, Gupta A, et al (2012). Omics of cancer. Asian Pac J Cancer Prev, 13, 4229-33. https://doi.org/10.7314/APJCP.2012.13.9.4229
Brower V (2011). Epigenetics: Unravelling the cancer code. Nature, 471, 12-3. https://doi.org/10.1038/471S12a
Cao Y, DePinho RA, Ernst M, Vousden (2011). Cancer research: past, present and future. Nat Rev Cancer, 11, 749-54. https://doi.org/10.1038/nrc3138
Castronovo V, Kischel P, Guillonneau F, et al (2007). Identification of specific reachable molecular targets in human breast cancer using a versatile ex vivo proteomic method. Proteomics, 7, 1188-96. https://doi.org/10.1002/pmic.200600888
Chandra H, Reddy PJ, Srivastava S (2011). Protein microarrays and novel detection platforms. Expert Rev Proteomics, 8, 61-79. https://doi.org/10.1586/epr.10.99
Chang R, Shoemaker R, Wang W (2011). Systematic search for recipes to generate induced pluripotent stem cells. PLoS Comput Biol, 7, 1002300. https://doi.org/10.1371/journal.pcbi.1002300
Coghlin C, Murray GI (2013). Progress in the identification of plasma biomarkers of colorectal cancer. Proteomics, 13 2227-8. https://doi.org/10.1002/pmic.201300245
Collins BC, Lau TY, O'Connor DP, Hondermarck H (2009). Cancer proteomics-an evolving battlefield. Conference on Cancer Proteomics 2009: Mechanistic Insights, Technological Advances & Molecular Medicine. EMBO reports, 10, 1202-5. https://doi.org/10.1038/embor.2009.222
Ardekani AM, Akhondi MM, Sadeghi MR (2008). Application of genomic and proteomic technologies to early detection of cancer. Arch Iran Med, 11, 427-34.
Moghanibashi M, Rastgar Jazii F, Soheili ZS, et al (2013). Esophageal cancer alters the expression of nuclear pore complex binding protein Hsc70 and eIF5A-1. Funct Integr Genomics, 13, 253-60. https://doi.org/10.1007/s10142-013-0320-9
Maurer HH (2010). Perspectives of liquid chromatography coupled to low-and high-resolution mass spectrometry for screening, identification, and quantification of drugs in clinical and forensic toxicology. Ther Drug Monit, 32, 324-7. https://doi.org/10.1097/FTD.0b013e3181dca295
Mishra A, Verma M (2010). Cancer biomarkers: are we ready for the prime time? Cancers, 2, 190-208. https://doi.org/10.3390/cancers2010190
Moghanibashi M, Jazii FR, Soheili ZS, et al (2012). Proteomics of a new esophageal cancer cell line established from Persian patient. Gene, 500, 124-33. https://doi.org/10.1016/j.gene.2012.03.038
Martin KJ, Fournier MV, Reddy GP, Pardee AB (2010). A need for basic research on fluid-based early detection biomarkers. Cancer Res, 70, 5203-6. https://doi.org/10.1158/0008-5472.CAN-10-0987
Montazery-Kordy H, Miran-Baygi MH, Moradi MH (2008). A data-mining approach to biomarker identification from protein profiles using discrete stationary wavelet transform. J Zhejiang Univ Sci B, 9, 863-70. https://doi.org/10.1631/jzus.B0820163
Nezami Ranjbar MR, Zhao Y, Tadesse MG, Wang Y, Ressom HW (2013) Gaussian process regression model for mormalization of LC-MS data using scan-level information. Proteome Sci, 11, 13. https://doi.org/10.1186/1477-5956-11-13
Nicolini C, Pechkova E (2010). An overview of nanotechnologybased functional proteomics for cancer and cell cycle progression. Anticancer Res, 30, 2073-80.
Orvisky ES, Drake K, Martin BM, et al (2006). Enrichment of low molecular weight fraction of serum for MS analysis of peptides associated with hepatocellular carcinoma. Proteomics, 6, 2895-902. https://doi.org/10.1002/pmic.200500443
Karimi P, Modarresi SZ, Sahraian MA, et al (2013). The relation of multiple sclerosis with allergy and atopy: a case control study. Iran J Allergy Asthma Immunol, 12, 182-9.
Jazii FR, Najafi Z, Malekzadeh R, et al (2006). Identification of squamous cell carcinoma associated proteins by proteomics and loss of beta tropomyosin expression in esophageal cancer. World J Gastroenterol, 12, 7104-12.
Kamangar F, Karimi P (2013). The state of nutritional epidemiology: why we are still unsure of what we should eat?. Arch Iran Med, 16, 483-6.
Karimi P, Islami F, Anandasabapathy S, Freedman ND, Kamangar F (2014). Gastric cancer: descriptive epidemiology, risk factors, screening, and prevention. Cancer Epidemiol Biomarkers Prev, In Press.
Karimi P, Shahrokni A, Moradi S (2013). Evidence for U.S. Preventive Services Task Force (USPSTF) Recommendations against routine mammography for females between 40-49 years of age. Asian Pac J Cancer Prev, 14, 2137-9. https://doi.org/10.7314/APJCP.2013.14.3.2137
Kent W J, Sugnet CW, Furey TS, et al (2002). The human genome browser at UCSC. Genome Res, 12, 996-1006. https://doi.org/10.1101/gr.229102.ArticlepublishedonlinebeforeprintinMay2002
Kocevar N, Odreman F, Vindigni A, Grazio SF, Komel R (2012). Proteomic analysis of gastric cancer and immunoblot validation of potential biomarkers. World J Gastroenterol, 18, 1216-28. https://doi.org/10.3748/wjg.v18.i11.1216
Li G, Xiao Z, Liu J, et al (2011). Cancer: a proteomic disease. Sci China Life Sci, 54, 403-8.
Li J, Zhang Z, Rosenzweig J, Wang YY, Chan DW (2002). Proteomics and bioinformatics approaches for identification of serum biomarkers to detect breast cancer. Clin Chem, 48, 1296-304.
Lopez E, Wesselink JJ, Lopez I, et al (2011). Technical phosphoproteomic and bioinformatic tools useful in cancer research. J Clin Bioinforma, 1, 26. https://doi.org/10.1186/2043-9113-1-26
Uemura N, Nakanishi Y, Kato H, et al (2009). Transglutaminase 3 as a prognostic biomarker in esophageal cancer revealed by proteomics. Int J Cancer, 124, 2106-15. https://doi.org/10.1002/ijc.24194
Yanagisawa K, Tomida S Shimada Y, et al (2007). A 25-Signal proteomic signature and outcome for patients with resected non-small-cell lung cancer. J Natl Cancer Inst, 99, 858-67. https://doi.org/10.1093/jnci/djk197
Yang F, Xiao ZQ, Zhang XZ, et al (2007). Identification of tumor antigens in human lung squamous carcinoma by serological proteome analysis. J Proteome Res, 6, 751-8. https://doi.org/10.1021/pr0602287
Yousefi Z, Sarvari J, Nakamura K, et al (2012). Secretomic analysis of large cell lung cancer cell lines using two-dimensional gel electrophoresis coupled to mass spectrometry. Folia Histochem Cytobiol, 50, 368-74. https://doi.org/10.5603/FHC.2012.0050
Tan HT, Lee YH, Chung MC (2012). Cancer proteomics. Mass Spectrom Rev, 31, 583-605. https://doi.org/10.1002/mas.20356
Thiviyanathan V, Gorenstein DG (2012). Aptamers and the next generation of diagnostic reagents. Proteomics Clin Appl, 6, 563-73. https://doi.org/10.1002/prca.201200042
Varghese RS, Zhou B, Nezami Ranjbar MR, Zhao Y, Resson HW (2012). Ion annotation-assisted analysis of LC-MS based metabolomic experiment. Proteome Sci, 21, 1477-5956.
Walther TC, Mann M (2010). Mass spectrometry-based proteomics in cell biology. J Cell Biol, 190, 491-500. https://doi.org/10.1083/jcb.201004052
Wang J-J, Liu Y, Zheng Y (2012). Comparative proteomics analysis of colorectal cancer. Asian Pac J Cancer Prev, 13, 1663-6. https://doi.org/10.7314/APJCP.2012.13.4.1663
Wang P, Whiteaker JR, Paulovich AG (2009). The evolving role of mass spectrometry in cancer biomarker discovery. Cancer Biol Ther, 8, 1083-94.
Wu CC, Chen HC, Chen SJ, et al (2008). Identification of collapsin response mediator protein-2 as a potential marker of colorectal carcinoma by comparative analysis of cancer cell secretomes. Proteomics, 8, 316-32. https://doi.org/10.1002/pmic.200700819
Wulfkuhle JD, Liotta LA, Petriccoin EF (2003). Proteomic applications for the early detection of cancer. Nat Rev Cancer, 3, 267-75. https://doi.org/10.1038/nrc1043
Wulfkuhle JD, Paweletz CP, Steeg PS, Petricoin EF 3rd, Liotta L (2003). Proteomic approaches to the diagnosis, treatment, and monitoring of cancer. Adv Exp Med Biol, 532, 59-68. https://doi.org/10.1007/978-1-4615-0081-0_7
Xiao JF, Varghese RS, Zhou B, et al (2012). LC-MS based serum metabolomics for identification of hepatocellular carcinoma biomarkers in Egyptian Cohort. J Proteome Res, 11, 5914-23.
Smith RA, Cokkinides V, von Wschenbach AC, et al (2002). American Cancer Society guidelines for the early detection of cancer. CA Cancer J Clin, 52, 8-22. https://doi.org/10.3322/canjclin.52.1.8
Sadat Tabatabaei Mirakabad F, Nejati-Koshki K, Akbarzadeh A, et al (2014) PLGA-Based nanoparticles as cancer drug delivery systems. Asian Pac J Cancer Prev, 15, 517-35. https://doi.org/10.7314/APJCP.2014.15.2.517
Shahrokni A, Karimi P (2012). Geriatric assessment in geriatric oncology; the evolution of authors' network from 2000 to 2012. J Geriatr Oncol, 3, 78-9.
Siegel R, Naishadham D, Jemal A (2013). Cancer statistics, 2013. CA Cancer J Clin, 63, 11-30. https://doi.org/10.3322/caac.21166
Sousa JF, Ham AJ, Whitwell C, et al (2012). Proteomic profiling of paraffin-embedded samples identifies metaplasia-specific and early-stage gastric cancer biomarkers. Am J Pathol, 181, 1560-72. https://doi.org/10.1016/j.ajpath.2012.07.027
Paul D, Kumar A, Gajbhiye A, Santra MK, Srikanth R (2013). Mass spectrometry-based proteomics in molecular diagnostics: discovery of cancer biomarkers using tissue culture. Biomed Res Int, 2013, 783131.
Srinivas PR, Verma M, Zhao Y, Srivastava S (2002). Proteomics for cancer biomarker discovery. Clin Chem, 48, 1160-9.
Stoevesandt O, Taussig MJ, He M (2009). Protein microarrays: high-throughput tools for proteomics. Expert Rev Proteomics, 6, 145-57. https://doi.org/10.1586/epr.09.2
Strassberger V, Fugmann T, Neri D, Roesli C (2010). Chemical proteomic and bioinformatic strategies for the identification and quantification of vascular antigens in cancer. J Proteomics, 73, 1954-73. https://doi.org/10.1016/j.jprot.2010.05.018
Sun W, Xing B, Sun Y, et al (2007). Proteome analysis of hepatocellular carcinoma by two-dimensional difference gel electrophoresis novel protein markers in hepatocellular carcinoma tissues. Mol Cell Proteomics, 6, 1798-808. https://doi.org/10.1074/mcp.M600449-MCP200
Rahman SM, Gonzalez AL Li M, et al (2011). Lung cancer diagnosis from proteomic analysis of preinvasive lesions. Cancer Res, 71, 3009-17. https://doi.org/10.1158/0008-5472.CAN-10-2510
Pei H, Zhu H, Zeng S, et al (2007). Proteome analysis and tissue microarray for profiling protein markers associated with lymph node metastasis in colorectal cancer. J Proteome Res, 6, 2495-501. https://doi.org/10.1021/pr060644r
Pepe MS, Etzioni R, Feng Z, et al (2001). Phases of biomarker development for early detection of cancer. J Natl Cancer Inst, 93, 1054-61. https://doi.org/10.1093/jnci/93.14.1054
Poste G (2012). Biospecimens, biomarkers, and burgeoning data: the imperative for more rigorous research standards. Trends Mol Med, 18, 717-22. https://doi.org/10.1016/j.molmed.2012.09.003
Ransohoff DF, Gourlay ML (2010). Sources of bias in specimens for research about molecular markers for cancer. J Clin Oncol, 28, 698-704. https://doi.org/10.1200/JCO.2009.25.6065
Ray S, Reddy PJ, Jain R, et al (2011). Proteomic technologies for the identification of disease biomarkers in serum: advances and challenges ahead. Proteomics, 11, 2139-61. https://doi.org/10.1002/pmic.201000460
Ressom HW, Xiao JF Tuli L, et al (2012). Utilization of metabolomics to identify serum biomarkers for hepatocellular carcinoma in patients with liver cirrhosis. Anal Chim Acta, 19, 90-100.
Rezaee AR, Azadi A, Houshmand M, et al (2013). Mitochondrial and nuclear genes as the cause of complex I deficiency. Genet Mol Res, 12, 3551-4. https://doi.org/10.4238/2013.September.12.1
Rual JF, Venkatesan K, Hao T, et al (2005). Towards a proteomescale map of the human protein-protein interaction network. Nature, 437, 1173-8. https://doi.org/10.1038/nature04209
Rusling JF, Kumar CV, Gutkind JS, et al (2010). Measurement of biomarker proteins for point-of-care early detection and monitoring of cancer. Analyst, 135, 2496-511. https://doi.org/10.1039/c0an00204f
Proteome–Metabolome Profiling of Ovarian Cancer Ascites Reveals Novel Components Involved in Intercellular Communication vol.13, pp.12, 2014, https://doi.org/10.1074/mcp.M114.041194
Effects of Hypobaric Conditions on Apoptosis Signalling Pathways in HeLa Cells vol.15, pp.12, 2014, https://doi.org/10.7314/APJCP.2014.15.12.5043
Pemetrexed Induces G1 Phase Arrest and Apoptosis through Inhibiting Akt Activation in Human Non Small Lung Cancer Cell Line A549 vol.16, pp.4, 2015, https://doi.org/10.7314/APJCP.2015.16.4.1507
Antioxidant, Anticancer and Anticholinesterase Activities of Flower, Fruit and Seed Extracts of Hypericum amblysepalum HOCHST vol.16, pp.7, 2015, https://doi.org/10.7314/APJCP.2015.16.7.2763
Polycyclic aromatic hydrocarbons and childhood asthma vol.30, pp.2, 2015, https://doi.org/10.1007/s10654-015-9988-6 | CommonCrawl |
Using the artificial neural networks for prediction and validating solar radiation
Zahraa E. Mohamed1
The main objective of this paper is to employ the artificial neural network (ANN) models for validating and predicting global solar radiation (GSR) on a horizontal surface of three Egyptian cities. The feedforward backpropagation ANNs are utilized based on two algorithms which are the basic backpropagation (Bp) and the Bp with momentum and learning rate coefficients respectively. The statistical indicators are used to investigate the performance of ANN models. According to these indicators, the results of the second algorithm are better than the other. Also, model (6) in this method has the lowest RMSE values for all cities in this study. The study indicated that the second method is the most suitable for predicting GSR on a horizontal surface of all cities in this work. Moreover, ANN-based model is an efficient method which has higher precision.
The solar energy is considered as one of renewable energy sources that are from the most promising sources to supply the world's energy demand. Accurate knowledge of the solar radiation (SR) data is considered the first stage in solar energy availability assessment. It is used as the basic input for many solar energy applications. But there is unavailability of the solar radiation measurements for different sites, due to the high cost of measuring equipments and their maintenance [1,2,3,4].
Many studies are implemented to develop for predicting the GSR using different techniques such as ANN, fuzzy control, and empirical models. These techniques depended on different types of datasets (such as meteorological and geographical). For example, Fadare [3] used several models which depended on feedforward and multilayered ANN for estimating GSR in 195 cities in Nigeria. He used some meteorological data as inputs in ANN models. The study demonstrated the ability of ANN to predict GSR in most of these cities in Nigeria. Elminir et al. [5] implemented ANN to estimate the GSR in some cities in Egypt. The authors used the different combinations of inputs of meteorological data as input of ANN models. The outcomes showed that the ANN models donate excellent predictions. Koca et al. [6] utilized an ANN-based model for assessment of GSR for seven cities in Turkey. They applied linear and nonlinear activation functions in the hidden layer in ANN models. The study showed that the ANN models are suitable for evaluating solar radiation in Turkey. Mohandes et al. [7] designed the ANN-based models for forecasting the GSR in Saudi Arabia. They used Bp algorithms for training the different pattern of multilayer feedforward NN. The outcomes indicated that MAPE is the best in all models. Rehman and Mohandes [8] employed the temperature, day of the year, and relative humidity values as input in ANN models for estimating GSR in Saudi Arabia. The outcomes showed that the ANN models are highly accurate for prediction of solar radiation. Jiang [9] employed the ANN model to forecast GSR in China and found the ANN model is better than regression models.
Khatib et al. [10] used many models to estimate GSR. These models are linear, nonlinear, adaptive neuron fuzzy inference system (ANFIS) and ANN models. The study showed that the most accurate methods for prediction of SR were ANN models. Mellit et al. [11] presented new models for the estimation of GSR, and these models combined the neural network and the fuzzy logic. The correlation coefficient obtained for the validation dataset is 98%. These models can be used for the estimation of the GSR for any locations in Algeria. Hassan et al. [12] introduced a new temperature-based model for predicting GSR. The results showed that the new proposed models have accurate estimations for GSR at different sites in Egypt.
The main aim of this paper is to predict and assess the daily GSR based on set of inputs used in all ANN models. The datasets of three cities Borg El-Arab, Cairo, and Aswan are used for training and testing. All ANN models used two different Bp algorithms which are basic Bp and Bp with momentum and learning rate coefficients. We calculated these statistical indicators to evaluate the performance of the proposed models. These indicators are root mean square error (RMSE), the mean absolute percentage error (MAPE), mean absolute bias error (MABE), correlation coefficient (r) and coefficient of determination (R2). Accurate ANN models were based on the minimum values to RMSE, MABE, and MAPE and maximum values of r and R2. In the present work, ANN models are compared with other similar works to establish the performance of our models with others.
Most regions of Egypt obtain enormous amount of solar energy due to their valuable geographical place. The data used in this study are the global solar radiation (GSR), maximum temperature (T max), minimum temperature (T min), averages temperature (T avg), relative humidity (RH), and atmospheric pressure (Atm.p) of three different locations which are Borg El-Arab, Cairo, and Aswan. These locations varied in climatic condition across Egypt and data were collected for a period of 14 years from 1 January 2002 to 31 December 2015 which we obtained from the NASA Surface meteorology and Solar Energy web siteFootnote 1.
The city of Borg El-Arab is a coastal region located in the Mediterranean, Cairo city is a moderate region and has dry climate in summer season, and Aswan city is located in the Upper Egypt and has the dry desert climate. The selected locations in the work are presented in Table 1. In addition, Egypt is one of the countries located in the most favorable solar belt, which enjoys abundant solar radiation almost 4500 h of sunshine per year, 13–35 MJ/m2/day of solar energy magnitude [12,13,14].
Table 1 Geographical locations for selected cities
The description of artificial neural network technique
The ANN model is an intelligent system and is used to solve complicated problems in many applications such as optimization, prediction, modeling, clustering, pattern recognitions, simulation, and others. The ANN structure consists of three layers: the input layer which has collected data, an output layer which produces computed information, and one or more hidden layers suitable to connect the input and output layer. A neuron is a basic processing unit of a NN and performs two functions: the collecting of the inputs and producing of the output. Each input is multiplied by connection weights, and its products and biases are added and then passed through an activation function to produce an output as shown in Fig. 1.
Simple structure of ANN
A multilayer feedforward network has neurons structured into layers and can pass in one direction without feedback connection. The Bp algorithm is the most method used for training feedforward ANNs which is dependent on the gradient descent optimization technique. Bp is a technique based on supervised learning [15] that is used for training NN, and it is processed to learn samples iteratively. Therefore, it compared the network predicted for each input with the actual value. To minimize the mean squared error between the network estimated and the measured data, the weights are adjusted for each training model [5].
In Fig. 2, each input xi in the input layer is multiplied by a connection weight wij between the neuron i in the input layer to the neuron j of the hidden layer and get its products, and bias bj are summed formally net input Ij in Eq. (1) and then passed to the hidden layer by a nonlinear sigmoid activation function in Eq. (2) to produce an output yj as in Eq. (3):
$$ {I}_j=\sum \limits_{i=1}^n{x}_i{w}_{ij}+{b}_j $$
$$ f(I)=\raisebox{1ex}{$\left(1-{e}^{-2I}\right)$}\!\left/ \!\raisebox{-1ex}{$\left(1+{e}^{-2I}\right)$}\right. $$
$$ {y}_j=f\left({\mathrm{I}}_j\right) $$
The ANN model of the present study
Then, it sends the out signal yj from the hidden layer to all k neuron in the output layer ok and computes the input to the k neuron of the output layer is o′k in Eq. (4).
$$ o{\hbox{'}}_k=\sum \limits_{j=1}^h{y}_jw{\hbox{'}}_{jk}+b{\hbox{'}}_k $$
where w′jk is the weight of the connected between the j neuron in the hidden layer to the k neuron of the output layer; b′k is the bias.
Finally, it computes the output layer signals by the sigmoid activation function in Eq. (5).
$$ {o}_k=f\left(o{\hbox{'}}_k\right) $$
The error training between the target output and the measured data is in Eq. (6).
$$ {e}_k=\frac{1}{2}\sum \limits_{k=1}^n\left({t}_k-{o}_k\right) $$
This procedure is applied to all pairs in the training set and the training cycle which is known an epoch and is repeated until the error reduces to the limit value [16, 17].
The error gradient for output layer is completed in Eq. (7).
$$ {\updelta}_k=\left({t}_k-{o}_k\right)f\hbox{'}\left(o{\hbox{'}}_k\right) $$
The difference between two algorithms which are the basic Bp and Bp with momentum and learning rate is in the updates of weights.
In Bp algorithm with momentum and learning rate, the weights between the different layers may be updated. And the error was then propagated backward from the output layer to the input layer in Eq. (8).
$$ {w}_{jk}^{\prime}\left(t+1\right)={w}_{jk}^{\prime }(t)+\eta {\updelta}_k{y}_j+\alpha \left({w}_{jk}^{\prime }(t)-{w}_{jk}^{\prime}\left(t-1\right)\right) $$
where η is the learning rate and α is the momentum coefficient.
But in the basic Bp algorithm, the weight update between neuron k from the output layer and neuron j from the hidden layer is as follows (Eq. (8′)):
w′jk(t + 1) = w′jk(t) + ηδkyj (8′)
Both of the coefficients η and α are used at the start of learning process and demonstrate the speed and stability of the network [15, 18]. Selecting small learning rate η may lead to slow rate of convergence, but the large η may show oscillation. The solution to make the convergence of the network fast without oscillation is used in the momentum coefficient. It is used for accelerating the convergence of the algorithm, and it contains the effect of previous weight changes on the current direction of movement.
The \( \Delta {w}_{jk}^{\prime } \) and ∆b′k are computed to update \( {w}_{jk}^{\prime } \) and b′k as shown in Eqs. (9) and (10) respectively.
$$ \varDelta {w}_{jk}^{\prime }=\eta {\updelta}_k{y}_j $$
$$ \varDelta b{\hbox{'}}_k=\eta {\updelta}_k $$
And the error gradient for the hidden layer is computed in Eq. (11).
$$ {\updelta}_{\mathrm{j}}=\sum \limits_{k=1}^n{\updelta}_kw{\hbox{'}}_{jk}f\hbox{'}\left(o{\hbox{'}}_k\right) $$
∆wij and ∆bj are calculated to update wij and bj as shown in Eqs. (12) and (13) respectively.
$$ \varDelta {w}_{ij}=\eta {\updelta}_j{x}_i $$
$$ \varDelta {b}_j=\eta {\updelta}_j $$
The weights and the biases are updated in Eqs. (14)–(17) respectively:
$$ {w}_{jk}^{\prime}\left(t+1\right)={w}_{jk}^{\prime }(t)+\varDelta {w}_{jk}^{\prime }+\alpha \left({w}_{jk}^{\prime }(t)-{w}_{jk}^{\prime}\left(t-1\right)\right) $$
$$ b{\hbox{'}}_k\left(t+1\right)=b{\hbox{'}}_k(t)+\varDelta b{\hbox{'}}_k+\alpha \left({b}_k^{\prime }(t)-{b}_k^{\prime}\left(t-1\right)\right) $$
$$ {w}_{ij}\left(t+1\right)={w}_{ij}(t)+\varDelta {w}_{ij}+\alpha \left({w}_{ij}(t)-{w}_{\mathrm{ij}}\left(t-1\right)\right) $$
$$ {b}_j\left(t+1\right)={b}_j(t)+\varDelta {b}_j+\alpha \left({b}_j(t)-{b}_j\left(t-1\right)\right) $$
In this study, the Bp algorithm with momentum and learning rate is applied as the following:
Initialize all weights and biases and normalize the training data.
Set the values of learning rate and momentum coefficients.
Compute the output of neurons in the hidden layer and in the output layer.
Compute the error by comparing the actual and predicted values.
Update all weights and biases by Eqs. (14)–(17).
Repeat steps 3 to 5 for all training data until the error converges to limit level.
To measure the performance of the network and show the error rate of proposed methods, there are six indicators used as error assessment techniques: MSE, RMSE, MAPE, MABE, r, and R2 [5, 12,13,14, 16, 19]. These indicators are calculated by the following equations (Eqs. (18)–(23)).
$$ \mathrm{MSE}=\frac{1}{n}\ \sum \limits_{i=1}^n{\left({X}_t-X{\hbox{'}}_t\right)}^2 $$
$$ \mathrm{RMSE}=\sqrt{\frac{1}{n}\ \sum \limits_{i=1}^n{\left({X}_t-X{\hbox{'}}_t\right)}^2} $$
$$ \mathrm{MAPE}=\frac{1}{n}\sum \limits_{i=1}^n\;\frac{\mid {X}_t-X{\hbox{'}}_t\mid }{X_t}\times 100 $$
$$ \mathrm{MABE}=\sum \limits_{i=1}^n\kern0.1em \frac{\mid {X}_t-X{\hbox{'}}_t\mid }{n} $$
$$ r=\frac{\sum_{i=1}^n\left({X}_t-\overline{X_t}\right)\left(X{\hbox{'}}_t-\overline{X{\hbox{'}}_t}\right)}{\sqrt{\sum_{i=1}^n{\left({X}_t-\overline{X_t}\right)}^2{\sum}_{i=1}^n{\left(X{\hbox{'}}_t-\overline{X{\hbox{'}}_t}\right)}^2}} $$
$$ {R}^2=1-\frac{\sum_{i=1}^n{\left({X}_t-X{\hbox{'}}_t\right)}^2}{\sum_{i=1}^n{\left(X{\hbox{'}}_t\right)}^2} $$
where Xt,and X′t are the values of tth actual and predicted global solar radiation respectively. \( \overline{X_t} \) is the average value of actual global solar radiation, \( \overline{X{\prime}_t} \) is the average value of the predicted global solar radiation, and n is the total number of observations.
The statistical error RMSE is used as a metric to measure model performance; its value is always positive and zero is the ideal case. The accuracy of the model is evaluated by the indicator MAPE, and the minimum value means that the model is with high accuracy. MABE is another metric to measure how close the predicted values are to the measured values, and the best model performance is with minimum value. The correlation coefficient (r) is used to estimate the correlation between model and observations. If r = 1, it means that there is an exact linear relationship between measured and predicted values; the largest value is better. Determination (R2) shows information about the goodness of fit; its values are between zero and one (0 ≤ R2 ≤ 1), and the largest value is the best value [2, 20].
Artificial neural network implementation
In this paper, we applied two Bp algorithms: the basic Bp and Bp with momentum and learning rate, respectively. ANN is employed to estimate the daily mean and yearly mean global solar radiation using the following parameters: maximum temperature, minimum temperature, average temperature, relative humidity, and atmospheric pressure of three cities in Egypt in the period from 2002 to 2015.
The performance and accuracy of the model depends on input dataset, the number of neurons in the hidden layer, the number of hidden layer, and learning algorithm. RMSE, MAPE, MABE, r, and R2 are used to evaluate and measure the performance and accuracy of the ANN models and the correlation between the model and the observations. MATLAB software is working to apply the proposed models. To train and test the neural network, we got the measured meteorological data from NASA Surface meteorology and Solar Energy web site containing the time period of 14 years (from 2002 to 2015). That dataset from 2002 to 2012 is utilized to train the network, and the period from 2013 to 2015 is employed to test the network.
In the training phase, we determined the input dataset, and it should be normalized in the range (− 1, 1) [21]. For assurance, use all parameters with small magnitudes due to consistency in the learning algorithm and then return to original values after the simulation using Eq. (24).
$$ {x}_{\mathrm{nor}}=\frac{3\left({X}_i-{X}_{\mathrm{min}}\right)}{2\left({X}_{\mathrm{max}}-{X}_{\mathrm{min}}\right)}-1 $$
where Xi is original value, Xmin and Xmax are the minimum and the maximum value of original values, and Xnor is the normalized value.
After creating the normalization values from the previous equation, the datasets in the training and testing phases are divided randomly into three subsets: training, validation, and testing [3]. The numbers of hidden neurons are 20 for estimating the performance of the model with different values of learning rate which are 0.01, 0.2, and 0.3 and of momentum coefficient which are 0.7 and 0.9. The minimum performance gradient is 10−6, maximum number of epochs to train is 2000, maximum validation failure is 60. The values of the weights and the biases are set randomly in the first; the results of the fitting tool differ every time during it runs. Hence, the network with the same number of neurons is retrained through 80 runs to detect the run number for best validation performance.
The results and discussions
To indicate the performance of the ANN models, we implemented the first algorithm (basic Bp) in three models with different values of learning rate which are 0.01, 0.2, and 0.3 respectively for both training and testing samples. The second algorithm (Bp with learning rate and momentum coefficients) is applied in six models with the same inputs in the experimental and with different values of two parameters, learning rate and momentum, which are 0.01, 0.2, 0.3, and 0.7, 0.9 respectively for both training and testing samples.
The statistical errors RMSE, MAPE, MABE, r, and R2 are calculated using Eqs. (18)–(23). The acceptable models are illustrated and the best model is identified by comparing the statistical errors associated with all models, and the best model is shown in italics as presented in Table 2. Based on the obtained results, all ANN models are ranked according to their RMSE values. The best model has the lowest value of RMSE for every city of two methods in the training and the testing.
Table 2 The statistical data for training and testing of the first method
Based on these results, the ANN models are ranked according to their RMSE values and the best model has the lowest value [5, 12, 13]. Also, in each city, the best model is recognized by comparing the statistical errors with all models in the two algorithms, and it is specified in italics as displayed in Table 2. Moreover, all models are arranged according to their performance.
According to the values of statistical indictors of the first algorithm, the proposed model (3) is the best model in this method in all testing cities. Its different error values are RMSE of 2.662 MJ/m2/day, 2.824 MJ/m2/day, and 1.455 MJ/m2/day; MAPE values are in the range of 10.852–14.106%, 11.875–15.925%, and 5.642–6.362%; the values of MABE are 2.026 MJ/m2/day, 2.144 MJ/m2/day, and 1.073 MJ/m2/day. The values of correlation coefficient (r) are 89.90%, 92.83%, and 96.09% respectively in testing cities. Coefficient of determination R2 showed the goodness fitting of data based on testing dataset; all values of R2 are greater than 0.99 in the testing cities as presented in Table 2. The performances of the other models in training cities demonstrated a good estimation for GSR with R2 values larger than 0.99.
Table 3 summarizes the results of the second algorithm Bp with learning rate and monument coefficients, and the proposed model (6) is the best model in this method in all testing cities. The various errors values of proposed model (6) are RMSE of 2.245 MJ/m2/day, 2.470 MJ/m2/day, and 1.439 MJ/m2/day; the values of MAPE are 10.684%, 11.466%, and 5.282%; the values of MABE are 1.715 MJ/m2/day, 1.878 MJ/m2/day, and 1.180 MJ/m2/day. The r values are 92.23%, 94.76%, and 96.55% respectively in the testing cities. Coefficient of determination R2 is larger than 0.99 in all testing cities as shown in Table 3.
Table 3 The statistical data for training and testing of the second method
In general, there is a good agreement between the measurements and predictions. Also, Bp algorithm with momentum and learning rate is better and more accurate than basic Bp algorithm, and it has needed less computation time than other methods.
The comparison between the predicted global solar radiations by ANN-based models and the measured global solar radiation of the three cities Cairo, Borg Al Arab, and Aswan is presented in Fig. 3 and Fig. 4 respectively. In the first algorithm, the proposed model (3) has the minimum value of RMSE in testing stage in all testing cities, and the values of least RMSE of these cities are 2.662 MJ/m2/day, 2.824 MJ/m2/day, and 1.555 MJ/m2/day as shown in Fig. 3.
Predicted and measured GSR on testing data of the first method
Predicted and measured GSR on testing data of the second method
Figure 4 displays results of model (6) which has the minimum overall RMSE of 2.245 MJ/m2/day, 2.470 MJ/m2/day, and 1.439 MJ/m2/day of the Bp with learning rate and momentum algorithm. The coefficient of determination R2 obtained for the datasets is almost 0.9999. This showed that there is a good agreement between measured and predicted datasets. We observed from the chart that ANN-predicted results of GSR of the second algorithm are better than the ANN-predicted results of GSR of the first algorithm and are considered more consistent with measured data for almost all the datasets.
Table 4 displays the values of RMSEave.( ave. is referred to the average test values for selected test cities), MAPEave,. and R2 in the present study and other similar studies in literature [3, 6,7,8].
Table 4 The comparison of similar studies in the literature
In this paper, ANN-based models were employed for evaluating and predicting of global solar radiation for three cities in Egypt. According to the statistical indicators, the second algorithm is better than the other ANN models in the testing data. Moreover, in all cases, R2 is greater than 99% and RMSE values are small. This indicated that the Bp with momentum and learning rate algorithm is better than the basic Bp algorithm, and the performance of the second algorithm is the best in all cities. These results showed that the developed ANN model can be the best alternative to the traditional estimation models with acceptable accuracy.
The datasets used and analyzed during the current study are available from the corresponding author on reasonable request.
www.nasa.gov
ANFIS:
Adaptive neuron fuzzy inference system
ANN:
Artificial neural network
Bp:
Backpropagation
GSR:
Global solar radiation
MABE:
Mean absolute bias error
MAPE:
Mean absolute percentage error
r :
Correlation coefficient
R 2 :
Coefficient of determination
RMSE:
Root mean square error
Amrouche, B., Le Pivert, X.: Artificial neural network based daily local forecasting for global solar radiation. Applied Energy. 130, 333–341 (2014). https://doi.org/10.1016/j.apenergy.2014.05.055
Hassan, G.E., Youssef, M.E., Ali, M.A., Mohamed, Z.E., Hanafy, A.A.: Evaluation of different sunshine-based models for predicting global solar radiation – case study: New Borg El-Arab city. Thermal Science. 22(2), 979–992 (2018). https://doi.org/10.2298/TSCI160803085H
Fadare, D.A.: Modelling of solar energy potential in Nigeria using an artificial neural network model. Applied Energy. 86, 1410–1422 (2009)
Voyant, C., Darras, C., Muselli, M., Paoli, C., Nivet, M.L., Poggi, P.: Bayesian rules and stochastic models for high accuracy prediction of solar radiation. Applied Energy. 114, 218–226 (2014). https://doi.org/10.1016/j.apenergy.2013.09.051
Elminir, H.K., Azzam, Y.A., Younes, F.I.: Prediction of hourly and daily diffuse fraction using neural network, as compared to linear regression models. Energy. 32, 1513–1523 (2007). https://doi.org/10.1016/j.energy.2006.10.010
Koca, A., Oztop, H.F., Varol, Y., Koca, G.O.: Estimation of solar radiation using artificial neural networks with different input parameters for Mediterranean region of Anatolia in Turkey. Expert Systems with Applications. 38(7), 8756–8762 (2011). https://doi.org/10.1016/j.eswa.2011.01.085
Mohandes, M., Balghonaim, A., Kassas, M., Rehman, S., Halawani, T.O.: Use of radial basis functions for estimating monthly mean daily solar radiation. Solar Energy. 68(2), 161–168 (2000)
Rehman, S., Mohandes, M.: Artificial neural network estimation of global solar radiation using air temperature and relative humidity. Energy Policy. 36(2), 571–576 (2008). https://doi.org/10.1016/j.enpol.2007.09.033
Jiang, Y.: Computation of monthly mean daily global solar radiation in China using artificial neural networks and comparison with other empirical models. Energy. 34, 1276–1283 (2009). https://doi.org/10.1016/j.energy.2009.05.009
Khatib, T., Mohamed, A., Sopian, K.: A review of solar energy modeling techniques. Renewable and Sustainable Energy Reviews. 16, 2864–2869 (2012)
Mellit, A., Kalogirou, S.A.: ANFIS-based modelling for photovoltaic power supply system: a case study. Renewable energy. 36(1), 250–258 (2011)
Hassan, G.E., Youssef, M.E., Mohamed, Z.E., Ali, M.A., Hanafy, A.A.: New temperature-based models for predicting global solar radiation. Applied Energy. 179, 437–450 (2016). https://doi.org/10.1016/j.apenergy.2016.07.006
Hassan, G.E., Youssef, M.E., Mohamed, Z.E., Ali, M.A., Shehata, A.I.: Performance assessment of different day-of-the-year based models for estimating global solar radiation - Case study: Egypt. Journal of Atmospheric and Solar-Terrestrial Physics. 149, 69–80 (2016). https://doi.org/10.1016/j.jastp.2016.09.011
El-Metwally, M.: Simple new methods to estimate global solar radiation based on meteorological data in Egypt. Atmospheric Research. 69, 217–239 (2004). https://doi.org/10.1016/j.atmosres.2003.09.002
R. Rojas Neural networks: a systematic introduction, Springer, ISBN 3-540-60505-3, Germany, (1996).
Çelik, O., Teke, A., Yıldırıma, H.B.K.: The optimized artificial neural network model with Levenberge Marquardt algorithm for global solar radiation estimation in Eastern Mediterranean Region of Turkey. Journal of Cleaner Production. 116, 1–12 (2016). https://doi.org/10.1016/j.jclepro.2015.12.082
Hasni, A., Sehli, A., Draoui, B., Bassou, A., Amieur, B.: Estimating global solar radiation using artificial neural network and climate data in the south-western region of Algeria. Energy Procedia. 18, 531–537 (2012). https://doi.org/10.1016/j.egypro.2012.05.064
Qazi, A., Fayaz, H., Wadi, A., Raj, R.G., Rahim, N.A., Khan, W.A.: The artificial neural network for solar radiation prediction and designing solar systems: a systematic literature review. Journal of Cleaner Production. 104, 1–12 (2015). https://doi.org/10.1016/j.jclepro.2015.04.041
Azadeh, A., Babazadeh, R., Asadzadeh, S.M.: Optimum estimation and forecasting of renewable energy consumption by artificial neural networks. Renewable and Sustainable Energy Reviews. 27, 605–612 (2013). https://doi.org/10.1016/j.rser.2013.07.007
Khorasanizadeh, H., Mohammadi, K.: Prediction of daily global solar radiation by day of the year in four cities located in the sunny regions of Iran. Energy Conversion and Management. 76, 385–392 (2013). https://doi.org/10.1016/j.enconman.2013.07.073
Rahimikhoob, A.: Estimating global solar radiation using artificial neural network and air temperature data in a semi-arid environment. Renewable energy. 35, 2131–2135 (2010). https://doi.org/10.1016/j.renene.2010.01.029
The author would like to thank the Informatics Research Institute, City for Scientific Research and Technological Applications, New Borg El-Arab City, 21934 Alexandria, Egypt, for providing the weather data.
Department of Mathematics, Faculty of Science, Zagazig University, P.O. Box 44519, Zagazig, Egypt
Zahraa E. Mohamed
Zahraa Elsayed Mohamed received M.Sc. and PhD degrees in computer science from Faculty of Science, Zagazig University, Egypt. Her research interests are in the areas of computer science and their applications, distributed data base, and wireless sensor.
Correspondence to Zahraa E. Mohamed.
Mohamed, Z.E. Using the artificial neural networks for prediction and validating solar radiation. J Egypt Math Soc 27, 47 (2019). https://doi.org/10.1186/s42787-019-0043-8
Backpropagation algorithm
Mathematics subject classification
97P10, 97R20, 97R30, 97R40 | CommonCrawl |
Can the inverse of this function be expressed in closed form?
So there is such a visual way to show that set of integers has the same cardinality as the set of natural numbers. $$0,1,-1,2,-2,3,-3,4,-4,5,-5...$$ But I think it is not really rigorous proof (and I think it does not pretend to be so) of the fact that integers and natural numbers are equipotent, because it doesn't really show the bijective function between $\mathbb{Z}$ and $\mathbb{N}$ . So I tried to find such a function which will represent the sequence above. So what I found was this: $$f(n)=\sum_{i=0}^{n-1} i\cdot(-1)^{i+1}$$ The domain of course is $\mathbb{N}$. And although this function associates some integer to every natural number, I do not know how to show that for any integer $b$ I can find some $m$ such that $f(m)=b$. So to do this I need to find the inverse function of $f$ or at least show that this inverse exists, but can this inverse function be expressed in closed form? If not, is it possible to prove the inverse exists?
inverse-function
Kenta S
Юрій ЯрошЮрій Ярош
$\begingroup$ By induction on $n\in \Bbb Z^+,$ if $f(2n-1)=n$ then $f(2n)=-n$ and $f(2(n+1)-1)=n+1.$ $\endgroup$ – DanielWainfleet Jun 27 '18 at 11:04
$\begingroup$ In what sense is this not adequately rigorous? It is a somewhat trivial lemma that if a repetition-free sequence $x_1, x_2, x_3, ...$ enumerates a set $A$ then that set is countably infinite with the map $f: \mathbb{N} \rightarrow A$ given by $f(i) = x_i$ a bijection. Asking for an explicit formula for the inverse makes it seem more complicated than it is. $\endgroup$ – John Coleman Jun 27 '18 at 13:49
$\begingroup$ I second @JohnColeman 's comment. As an instructor I would much rather see the equicardinality proved by your sequence than by any cooked up formula.and its inverse. $\endgroup$ – Ethan Bolker Jun 27 '18 at 14:06
$\begingroup$ @JohnColeman Because you need to define the sequence somehow and prove it is repetition-free. Because of that I was trying to find this function to define this sequence. $\endgroup$ – Юрій Ярош Jun 27 '18 at 14:49
$\begingroup$ The point is that a description like "zero, and then each natural number followed by its negative in increasing order of size" is perfectly precise, and you can easily argue that it contains every integer exactly once. You don't need to be able to express it as a formula or give an explicit inverse to make those arguments. $\endgroup$ – Ben Millwood Jun 27 '18 at 15:43
$f^{-1}(n) = |2n - \frac{1}{2}| + \frac{1}{2}$
gandalf61gandalf61
I am not sure what you consider to be a closed form, but I would consider a simpler version of the function $f$ to be: $$ f(n)=-\lfloor n/2\rfloor\cdot(-1)^n $$ and the inverse could be expressed as: $$ f^{-1}(n)= \begin{cases} 2n & \text{if }n>0\\ -2n+1 & \text{otherwise} \end{cases} $$
StringString
$\begingroup$ By closed form I mean expression in terms of polynomials, exponents, logarithms, trigonometric functions. $\endgroup$ – Юрій Ярош Jun 27 '18 at 10:00
$\begingroup$ @ЮрійЯрош: But in that sense your suggested form of $f$ is not a closed form, since it has a variable number of terms. What I mean by that is that summation still involves something similar to dividing into cases, since checking the condition $i\leq n-1$ is some sort of if-statement. $\endgroup$ – String Jun 27 '18 at 10:01
$\begingroup$ Yes, but I was asking about the inverse. Of course it's hard to imagine closed form of the inverse of the function which itself has no closed form. $\endgroup$ – Юрій Ярош Jun 27 '18 at 10:06
$\begingroup$ If you plug in $2n$ for positive $n$ in $f$ you get $-n$, but you should get $n$. $\endgroup$ – Юрій Ярош Jun 27 '18 at 10:24
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Inverse function of $x-\lfloor x \rfloor $ and $(x-\lfloor x \rfloor)^2$ | CommonCrawl |
Monday–Friday, March 14–18, 2016; Baltimore, Maryland
Session S34: Small Molecule Glasses
Hide Abstracts
Sponsoring Units: GSOFT
Chair: Kate Jensen, Yale University
S34.00001: Point-to-set correlations and rugged landscapes
Sho Yaida, Ludovic Berthier, Patrick Charbonneau, Gilles Tarjus
Upon approaching the glass transition a liquid gets sluggish without obvious structural changes. The glassy slowdown is instead attributed to an increasing roughness in the underlying free-energy landscape. Cavity point-to-set (PTS) correlations are real-space tools for characterizing the evolution of this rugged landscape, but their measurement is a serious computational challenge. Here, we first describe how advanced Monte Carlo techniques can be used to dramatically enhance sampling in cavities, extending the range over which PTS correlations can be obtained. By suitably generalizing the notion of PTS correlations to capture any type of growing order in liquids, be it local or amorphous, we then establish a criterion for distinguishing a dynamical slowdown due to critical ordering from one due to glassiness. These methodological advances shed a new light on the interplay between structure and dynamics in model glass formers, and tie in with recent field-theoretic results about the nature of jumps between metastable minima in rough landscapes. [Preview Abstract]
S34.00002: Evidence for a second-order phase transition to a low-entropy glass
C. Patrick Royall, Francesco Turci, Thomas Speck
The physics underlying the glass transition is a major outstanding. Central its solution is whether there is some kind of thermodynamic transition to a "ideal glass", a disordered state with extremely low entropy, or whether in principle a liquid may be supercooled to arbitrary low temperature. Among the challenges that lie in tackling the glass transition are the immense timescales involved. Computer simulation, which might otherwise be able to pick up hints of a thermodynamic transition is limited by the small time-window over which a liquid can be equilibrated. Here we address this challenge using trajectory sampling in a system undergoing a first order nonequilibrium phase transition to a glassy state rich in low-energy geometric motifs. Extrapolation to equilibrium indicates that the transition would occur at a similar temperature at which the ideal glass transition is expected from extrapolation of dynamic and thermodynamic measurements. We further reweight nonequilibrium data to equilibrium leading to configurations representative of extremely low temperature, which indicate a transition to a low energy state at the ideal glass transition temperature. We thus interpret the ideal glass transition as the lower critical endpoint of this nonequilibrium transition. [Preview Abstract]
S34.00003: Softness Correlations Across Length Scales
Robert Ivancic, Amit Shavit, Jennifer Rieser, Samuel Schoenholz, Ekin Cubuk, Douglas Durian, Andrea Liu, Robert Riggleman
In disordered systems, it is believed that mechanical failure begins with localized particle rearrangements. Recently, a machine learning method has been introduced to identify how likely a particle is to rearrange given its local structural environment, quantified by \textit{softness}. We calculate the softness of particles in simulations of atomic Lennard-Jones mixtures, molecular Lennard-Jones oligomers, colloidal systems and granular systems. In each case, we find that the length scale characterizing spatial correlations of softness is approximately a particle diameter. These results provide a rationale for why localized rearrangements---whose size is presumably set by the scale of softness correlations---might occur in disordered systems across many length scales. [Preview Abstract]
S34.00004: Onset of cooperative dynamics in equilibrium glass-forming metallic liquids
Abhishek Jaiswal, Yang Zhang
Onset of cooperative dynamics has been observed in the metastable regime of many molecular liquids, colloids, and granular materials approaching their respective glass or jamming transition points. It is also considered to play a significant role in the emergence of slow dynamics. However, the nature of such dynamical cooperativity remains elusive in multicomponent metallic liquids characterized by complex many-body interactions and high mixing entropy. Herein, we report indications of the onset of cooperative dynamics in an equilibrium glass-forming metallic liquid (ZrCuNiAl). This is revealed by deviation of the experimentally measured mean diffusion coefficient from its high temperature Arrhenius behavior below $T_{o}$~$\approx $ 1300~K, i.e., a crossover from uncorrelated dynamics above $T_{o}$~to landscape-influenced correlated dynamics below $T_{o}$. The onset/crossover in this system is observed at approximately twice of its calorimetric glass transition temperature ($T_{g}$~$\approx $ 697~K) and in the stable liquid phase, unlike many molecular liquids. Furthermore, we show the presence of such a dynamical onset phenomenon in ten other glass-forming metallic liquids, universally occurring at approximately twice of their $T_{g}$ and in their liquid phases. [Preview Abstract]
S34.00005: Differences in dynamic heterogeneity in strong and fragile glass formers
Hannah Staley, Elijah Flenner, Grzegorz Szamel
We study dynamic heterogeneity in a model strong glass former. We examine the spatial extent $\xi_4^a(t)$ and the strength $\chi_4^a(t)$ of the heterogeneity of the dynamics at two length scales $a$. One length scale corresponds to the nearest neighbor separation and the other length scale corresponds to the length scale of the tetrahedral network. We find that the dynamic correlation length $\xi_4^a$ grows much slower with increasing relaxation time at both length scales than for model fragile glass formers. We also find that the dynamically correlated regions are more ramified for the strong glass former than for model fragile glass formers. However, we do find that Stokes-Einstein violation indicates a change in the character of the dynamic heterogeneities for the strong glass former and the fragile glass formers. [Preview Abstract]
S34.00006: Correlating structural and dynamic fragility in glass-forming liquids
Dmitry Voylov, Philip Griffin, Brandon Mercado, Jong Keum, Vladimir Novikov, Alexei Sokolov
The glass transition was attracting wide interest over the last several decades, but still remains the topic of intensive research and discussions. One of the most intriguing and well-known observations is a drastic change of dynamic properties with only slight variations of structure upon cooling down to the glass transition temperature Tg. This has led many to believe that the changes of dynamics during approach to Tg have no structural signatures which would be significant and common to different types of glass-forming liquids. Here we demonstrate analysis of temperature dependence of the main diffraction peak in a static structure factor of various glass-formers. We show that the relative changes of its width with temperature correlates with fragility of these materials. This observation was analyzed using Adam-Gibbs approach establishing a connection between the structural and dynamical properties of glass-forming materials. [Preview Abstract]
S34.00007: Aging and random-field magnetism in ferromagnet/antiferromagnet bilayers.
Tianyu Ma, Ryan Freeman, Xiang Cheng, Stefan Boettcher, Sergei Urazhdin
Exchange interaction at the interface between a ferromagnet (F) and an antiferromagnet (AF) results in a random effective exchange field acting on both F and AF [1], which can produce complex equilibrium and dynamical states. We utilized anisotropic magnetoresistance to look for signatures of such states in epitaxial Py$=$Permalloy/Fe50Mn50 and polycrystalline CoO/Py bilayers. For thin AF layers, both systems exhibit slow cooperative aging indicative of a complex glassy state [2]. Aging follows the same small power-law or logarithmic dependence and is observed over a wide range of temperatures and fields, suggesting a universal aging mechanism. Glassy relaxation is not observed at any temperature for AF thickness above 3.5nm. We argue that these observations are inconsistent with the usual ``granular'' and ``domain-state'' models of F/AF systems. We discuss the implications of our results for the random field magnetism, and the relationship between the dimensionality and the topological properties of magnetic systems. \begin{enumerate} \item A.P. Malozemoff, Phys. Rev. B 35, 3679(R) (1987). \item T.C. Proctor, D.A. Garanin, and E.M. Chudnovsky, Phys. Rev. Lett. 112, 097201 (2014). \end{enumerate} [Preview Abstract]
S34.00008: Aging in the two-dimensional random-field systems
Xiang Cheng, Tianyu Ma, Sergei Urazhdin, Stefan Boettcher
Random fields introduced into the classical Ising and Heisenberg spin models can roughen the energy landscape, leading to complex nonequilibrium dynamics. The effects of random fields on magnetism have been previously studied in the context of dilute antiferromagnets (AF), impure substrates, and magnetic alloys $[1]$. We utilized random-field spin models to simulate the observed magnetic aging in thin-film ferromagnet/antiferromagnet (F/AF) bilayers. Our experiments show extremely slow cooperative relaxation over a wide range of temperatures and magnetic fields $[2]$. In our simulations, the experimental system is coarse-grained into a random field Ising model on a 2D square lattice. Monte Carlo simulations indicate that aging processes may be associated with the glassy evolution of the magnetic domain walls, due to the pinning by the random fields. The scaling of the simulated aging agrees well with experiments. Both are consistent with either a small power-law or logarithmic dependence on time. We further discuss the topological effects on aging due to the dimensional crossover from the Ising to the Heisenberg regime.\\ $[1]$T. Nattermann, Spin glasses and random fields, 12 (1997):277\\ $[2]$ S. Urazhdin, arXiv:1503.08380 (2015)(arxiv.org/pdf/1503.08380.pdf) [Preview Abstract]
S34.00009: Qualitative change in structural dynamics of some glass-forming systems
Vladimir Novikov, Alexei Sokolov
Analysis of temperature dependence of structural relaxation time $\tau (T)$ in supercooled liquids revealed a qualitatively distinct feature - a sharp, cusp-like maximum in the second derivative of log $\tau _{\mathrm{\alpha }}(T)$ at some $T_{max}$. It suggests that the super-Arrhenius temperature dependence of $\tau_{\mathrm{\alpha }}(T)$ in glass-forming liquids eventually crosses over to an Arrhenius behavior at \textit{T\textless T}$_{max}$, and there is no divergence of $\tau_{\mathrm{\alpha }}(T)$ at non-zero $T$. $T_{max}$ can be above or below $T_{g}$, depending on sensitivity of $\tau (T)$ to change in liquid's density quantified by the exponent $\gamma $ in the scaling $\tau_{\mathrm{\alpha }}(T)$ \textasciitilde exp($A/T\rho^{\mathrm{-\gamma }})$. These results might turn the discussion of the glass transition to the new avenue -- the origin of the limiting activation energy for structural relaxation at low $T$. [Preview Abstract]
S34.00010: Percolation Thresholds in Angular Grain media: Drude Directed Infiltration
Donald Priour
Pores in many realistic systems are not well delineated channels, but are void spaces among grains impermeable to charge or fluid flow which comprise the medium. Sparse grain concentrations lead to permeable systems, while concentrations in excess of a critical density block bulk fluid flow. We calculate percolation thresholds in porous materials made up of randomly placed (and oriented) disks, tetrahedrons, and cubes. To determine if randomly generated finite system samples are permeable, we deploy virtual tracer particles which are scattered (e.g. specularly) by collisions with impenetrable angular grains. We hasten the rate of exploration (which would otherwise scale as $n_{\mathrm{coll}}^{1/2}$ where $n_{\mathrm{coll}}$ is the number of collisions with grains if the tracers followed linear trajectories) by considering the tracer particles to be charged in conjunction with a randomly directed uniform electric field. As in the Drude treatment, where a succession of many scattering events leads to a constant drift velocity, tracer displacements on average grow linearly in $n_{\mathrm{coll}}$. By averaging over many disorder realizations for a variety of systems sizes, we calculate the percolation threshold and critical exponent which characterize the phase transition. [Preview Abstract]
S34.00011: Influence of Hydrogen Bonding on the Kinetic Stability of Vapor Deposited Triazine Glasses
Audrey Laventure, Ankit Gujral, Olivier Lebel, Christian Pellerin, Mark D. Ediger
Physical vapor deposition (PVD) can produce glasses with enhanced kinetic stability, high density and anisotropy. However, the influence of hydrogen bonding on these properties has not been fully explored. We vapor deposit a series of triazine derivatives containing functional groups with different H-bonding capability, i.e. NHMe (H-bond donor), OMe (H-bond acceptor) and Et (none) using a wide range of substrate temperatures, from 0.60 to 1.05Tg. PVD glasses of the NHMe derivative have inferior kinetic stability compared to its OMe and Et analogues. This behavior can be rationalized by the higher average number of bonded NH per molecule found in PVD glasses of the NHMe derivative, as quantified by infrared spectroscopy (IR). Despite this difference in H-bonding, IR and wide angle X-ray scattering reveal that all three compounds show a tendency to orient parallel to the substrate at low substrate temperatures. Our results support the hypothesis that strong intermolecular interactions, such as H-bonds, can hinder mobility of the molecules at the interface and thus limit their possibility to sample the potential energy landscape to produce stable glasses. [Preview Abstract]
S34.00012: Thermal properties of composite materials: a complex systems approximation
J. L. Carrillo, Beatriz Bonilla, J. J. Reyes, Victor Dossetti
We propose an effective media approximation to describe the thermal diffusivity of composite samples made of polyester resin and magnetite inclusions. By means of photoacoustic spectroscopy, the thermal diffusivity of the samples were experimentally measured. The volume fraction of the inclusions was systematically varied in order to study the changes in the effective thermal diffusivity of the composites. For some samples, a static magnetic field was applied during the polymerization process, resulting in anisotropic inclusion distributions. Our results show a significant difference in the thermal properties of the anisotropic samples, compared to the isotropic randomly distributed. We correlate some measures of the complexity of the inclusion structure with the observed thermal response through a multifractal analysis. In this way, we are able to describe, and at some extent predict, the behavior of the thermal diffusivity in terms of the lacunarity and other measures of the complexity of these samples [1]. [1] F. Cervantes-Alvarez, J J Reyes-Salgado, V Dossetti, and J L Carrillo, J. Phys. D: Appl. Phys. 47 (2014) 235303; J. J. Reyes-Salgado, B. Bonilla, V. Dossetti, and J L Carrillo, J. Phys. D: Appl. Phys. 48, (2015) [Preview Abstract]
S34.00013: Preparing anisotropic glasses from structural analogs of liquid crystal formers by physical vapor deposition
Jaritza Gomez, Mark Ediger
Physical vapor deposition (PVD) can be used to tune molecular orientation in glasses by depositing at substrate temperatures (T$_{\mathrm{substrates}})$ just below the glass transition temperature (T$_{\mathrm{g}})$. Glasses of a smectic A liquid crystal (LC) former, itraconazole, deposited at a T$_{\mathrm{substrate\thinspace }}=$ T$_{\mathrm{g}}$ have been shown to inherit the structure of the equilibrium smectic liquid and orient nearly perpendicular to the substrate. Here we report the deposition of glasses prepared from molecules that are structural analogs to known LC formers: posaconazole and a functionalized perylenemonoimide (PMI), analogs to itraconazole and a previously reported columnar LC, respectively. Spectroscopic ellipsometry and infrared spectroscopy are used to characterize average molecular orientation in the as-deposited glasses. Surprisingly, we find that molecular orientation in glasses of posaconazole deposited at different T$_{\mathrm{substrates}}$ does not follow the previously observed trends for linear molecules without LC states, but more closely follows itraconazole. In addition, we find that glasses deposited at T$_{\mathrm{g}}$ are not isotropic, even though liquid-cooled glasses do not show preferential molecular orientation. Similarly, glasses from a functionalized PMI, structural analog to a known columnar LC, show molecular orientation at T$_{\mathrm{substrate\thinspace }}=$ T$_{\mathrm{g}}$. These results may provide insights into the mechanism by which physical vapor deposition can produce glasses with tunable molecular orientation. [Preview Abstract]
S34.00014: Fragility of Ionic Liquids Measured by Flash Differential Scanning Calorimetry
Ran Tao, Eshan Gurung, Edward L. Quitevis, Sindee L. Simon
Ionic liquids are a class of materials that possess attractive properties. They generally have low rates of crystallization due to their bulky and asymmetrical ion structure, and are often considered as good glass-forming materials. In this work, a series of imidazolium-based ionic liquids with varying functionalities from aliphatic to aromatic groups and a fixed anion are characterized using fast scanning differential scanning calorimetry. The limiting fictive temperature Tf', which is equivalent to the glass transition temperature Tg, is measured on heating as a function of cooling rate using Flash differential scanning calorimetry. Different calculation methods are employed and compared for the determination of Tf'. The dynamic fragility is obtained for the series of ionic liquids, and using this data along with a compilation of data from the literature reveals the relationship between molecular structure and fragility for ionic liquids. [Preview Abstract]
S34.00015: Generating tunable structures in glassy materials: Smectic-like layering in glasses of a liquid crystal system prepared by vapor deposition
Ankit Gujral, Jaritza Gomez, Jing Jiang, Chengbin Huang, Kathryn O'Hara, Michael Toney, Michael Chabinyc, Lian Yu, Mark Ediger
Anisotropic packing, particularly in highly ordered liquid crystalline configurations, has been shown to be useful in organic electronic and optoelectronic applications. In this work, vapor deposited glasses of a model smectic liquid crystal-forming molecule, itraconazole, are investigated. The films are characterized using x-ray scattering, FTIR and spectroscopic ellipsometry, and are found to exhibit unprecedented structural and optical anisotropy for a macroscopically homogeneous solid. A smectic-like layered structure is observed in the glasses that are prepared by depositing the glass at a substrate temperature during deposition (T$_{sub}$) maintained below the glass transition temperature, T$_{g}$, of the molecule. The layer spacing, and the associated average tilt angle of the molecules, is found to be tunable as a function of T$_{sub}$. The layer spacing reduces by 16\% as T$_{sub}$ is lowered. These features are retained in the films when heated to at least T$_{g}$ of the molecule. [Preview Abstract] | CommonCrawl |
If $g(x) = x^2$ and $f(x) = 2x - 1$, what is the value of $f(g(2))$?
\[
f(g(2))=f\left(2^2\right)=f(4)=2\cdot4-1=\boxed{7}
\] | Math Dataset |
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17th/18th Century Philosophy > 17th/18th Century French Philosophy > Condorcet
Condorcet
Antoine Arnauld (95)
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Condorcet's Jury Theorem and the Optimum Number of Voters.Jason Brennan - forthcoming - POLITICS.details
Many political theorists and philosophers use Condorcet's Jury Theorem to defend democracy. This paper illustrates an uncomfortable implication of Condorcet's Jury Theorem. Realistically, when the conditions of Condorcet's Jury Theorem hold, even in very high stakes elections, having more than 100,000 citizens vote does no significant good in securing good political outcomes. On the Condorcet model, unless voters enjoy voting, or unless they produce some other value by voting, then the cost to most voters of voting exceeds the expected epistemic (...) benefits to the common good of their casting a vote. Anyone who is committed to democracy on the basis of the Jury Theorem ought also to hold that widespread voting is wasteful, at least unless she can provide some further justification of mass democratic participation. (shrink)
Condorcet in 17th/18th Century Philosophy
Participatory Democracy in Social and Political Philosophy
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Jury Theorems.Franz Dietrich & Kai Spiekermann - forthcoming - The Stanford Encyclopedia of Philosophy.details
Jury theorems are mathematical theorems about the ability of collectives to make correct decisions. Several jury theorems carry the optimistic message that, in suitable circumstances, 'crowds are wise': many individuals together (using, for instance, majority voting) tend to make good decisions, outperforming fewer or just one individual. Jury theorems form the technical core of epistemic arguments for democracy, and provide probabilistic tools for reasoning about the epistemic quality of collective decisions. The popularity of jury theorems spans across various disciplines such (...) as economics, political science, philosophy, and computer science. This entry reviews and critically assesses a variety of jury theorems. It first discusses Condorcet's initial jury theorem, and then progressively introduces jury theorems with more appropriate premises and conclusions. It explains the philosophical foundations, and relates jury theorems to diversity, deliberation, shared evidence, shared perspectives, and other phenomena. It finally connects jury theorems to their historical background and to democratic theory, social epistemology, and social choice theory. (shrink)
Collective Epistemology in Epistemology
Epistemology of Disagreement in Epistemology
Formal Social Epistemology in Epistemology
Judgment Aggregation in Social and Political Philosophy
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One Standard to Rule Them All?Marc-Kevin Daoust - 2019 - Ratio 32 (1):12-21.details
It has been argued that an epistemically rational agent's evidence is subjectively mediated through some rational epistemic standards, and that there are incompatible but equally rational epistemic standards available to agents. This supports Permissiveness, the view according to which one or multiple fully rational agents are permitted to take distinct incompatible doxastic attitudes towards P (relative to a body of evidence). In this paper, I argue that the above claims entail the existence of a unique and more reliable epistemic standard. (...) My strategy relies on Condorcet's Jury Theorem. This gives rise to an important problem for those who argue that epistemic standards are permissive, since the reliability criterion is incompatible with such a type of Permissiveness. (shrink)
Epistemic Permissivism in Epistemology
Evidentialism in Epistemology
Formal Social Epistemology, Misc in Epistemology
Rational Requirements in Epistemology
Reliabilism about Justification in Epistemology
A Condorcet Jury Theorem for Couples.Ingo Althöfer & Raphael Thiele - 2016 - Theory and Decision 81 (1):1-15.details
The agents of a jury have to decide between a good and a bad option through simple majority voting. In this paper the jury consists of N independent couples. Each couple consists of two correlated agents of the same competence level. Different couples may have different competence levels. In addition, each agent is assumed to be better than completely random guessing. We prove tight lower and upper bounds for the quality of the majority decision. The lower bound is the same (...) as the competence of majority voting of N independent agents. The upper bound cases for negatively correlated couples can be much better than the value for 2N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \, N$$\end{document} independent agents. (shrink)
Social Choice Theory in Social and Political Philosophy
Is the Equal-Weight View Really Supported by Positive Crowd Effects?Christian J. Feldbacher-Escamilla - 2015 - In Uskali Mäki, Ioannis Votsis, Stephanie Ruphy & Gerhard Schurz (eds.), Recent Developments in the Philosophy of Science: EPSA13 Helsinki. Heidelberg: Springer. pp. 87-98.details
In the debate of epistemic peer disagreement the equal-weight view suggests to split the difference between one's own and one's peer's opinions. An argument in favour of this view---which is prominently held by Adam Elga---is that by such a difference-splitting one may make some use of a so-called wise-crowd effect. In this paper it is argued that such a view faces two main problems: First, the problem that the standards for making use of a wise-crowd effect are quite low. And (...) second, the problem that following the equal-weight view decreases such effects and by this the argument's own basis is defeated. We therefore come to the conclusion that an argument for the equal-weight view with the help of wise-crowd effects as provided more or less explicitly by Elga does not succeed. (shrink)
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Condorcet and Progress.Ira O. Wade - 2015 - In The Structure and Form of the French Enlightenment, Volume 2: Esprit Revolutionnaire. Princeton University Press. pp. 363-387.details
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On Bernard Bosanquet's "The Reality of the General Will".Robert Stern - 2014 - Ethics 125 (1):192-195,.details
This article is a discussion of Bernard Bosanquet's paper 'The Reality of the General Will', in which its main arguments and motivations are explained. His position is compared to Rousseau's on the general will.
19th Century British Philosophy, Misc in 19th Century Philosophy
Democracy, Misc in Social and Political Philosophy
Jean-Jacques Rousseau in 17th/18th Century Philosophy
Epistemic Democracy with Defensible Premises.Franz Dietrich & Kai Spiekermann - 2013 - Economics and Philosophy 29 (1):87--120.details
The contemporary theory of epistemic democracy often draws on the Condorcet Jury Theorem to formally justify the 'wisdom of crowds'. But this theorem is inapplicable in its current form, since one of its premises – voter independence – is notoriously violated. This premise carries responsibility for the theorem's misleading conclusion that 'large crowds are infallible'. We prove a more useful jury theorem: under defensible premises, 'large crowds are fallible but better than small groups'. This theorem rehabilitates the importance of deliberation (...) and education, which appear inessential in the classical jury framework. Our theorem is related to Ladha's (1993) seminal jury theorem for interchangeable ('indistinguishable') voters based on de Finetti's Theorem. We also prove a more general and simpler such jury theorem. (shrink)
Justification of Democracy in Social and Political Philosophy
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Condorcet: Political Writings.Jean-Antoine-Nicolas de Caritat Condorcet (ed.) - 2012 - Cambridge University Press.details
Machine generated contents note: Editors' introduction; Published works by Condorcet; Suggestions for further reading; Principal events in Condorcet's life; Notes on the texts; 1. The Sketch; 2. On slavery; 3. On the emancipation of women; 4. On despotism; 5. On freedom; 6. On revolution; 7. Advice to his daughter and testament; Index.
History of Political Philosophy in Social and Political Philosophy
A Statistical Approach to Epistemic Democracy.Marcus Pivato - 2012 - Episteme 9 (2):115-137.details
We briefly review Condorcet's and Young's epistemic interpretations of preference aggregation rules as maximum likelihood estimators. We then develop a general framework for interpreting epistemic social choice rules as maximum likelihood estimators, maximum a posteriori estimators, or expected utility maximizers. We illustrate this framework with several examples. Finally, we critique this program.Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage (...) Your Content and Devices page of your Amazon account. Then enter the 'name' part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. '@free.kindle.com' emails are free but can only be sent to your device when it is connected to wi-fi. '@kindle.com' emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.A STATISTICAL APPROACH TO EPISTEMIC DEMOCRACYVolume 9, Issue 2Marcus PivatoDOI: https://doi.org/10.1017/epi.2012.4Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. A STATISTICAL APPROACH TO EPISTEMIC DEMOCRACYVolume 9, Issue 2Marcus PivatoDOI: https://doi.org/10.1017/epi.2012.4Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. A STATISTICAL APPROACH TO EPISTEMIC DEMOCRACYVolume 9, Issue 2Marcus PivatoDOI: https://doi.org/10.1017/epi.2012.4Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)
Social Epistemology in Epistemology
Condorcet Vs. Borda in Light of a Dual Majoritarian Approach.Eyal Baharad & Shmuel Nitzan - 2011 - Theory and Decision 71 (2):151-162.details
Many voting rules and, in particular, the plurality rule and Condorcet-consistent voting rules satisfy the simple-majority decisiveness property. The problem implied by such decisiveness, namely, the universal disregard of the preferences of the minority, can be ameliorated by applying unbiased scoring rules such as the classical Borda rule, but such amelioration has a price; it implies erosion in the implementation of the widely accepted majority principle . Furthermore, the problems of majority decisiveness and of the erosion in the majority principle (...) are not necessarily severe when one takes into account the likelihood of their occurrence. This paper focuses on the evaluation of the severity of the two problems, comparing simple-majoritarian voting rules that allow the decisiveness of the smallest majority larger than 1/2 and the classical Borda method of counts. Our analysis culminates in the derivation of the conditions that determine, in terms of the number of alternatives k, the number of voters n, and the relative (subjective) weight assigned to the severity of the two problems, which of these rules is superior in light of the dual majoritarian approach. (shrink)
Enfranchising Incompetents: Suretyship and the Joint Authorship of Laws.Robert E. Goodin & Joanne C. Lau - 2011 - Ratio 24 (2):154-166.details
Proposals to lower the age of voting, to 15 for example, are regularly met with worries that people that age are not sufficiently 'competent'. Notice however that we allow people that age to sign binding legal contracts, provided that those contracts are co-signed by their parents. Notice, further, that in a democracy voters are collectively 'joint authors' of the laws that they enact. Enfranchising some less competent voters is no worry, the Condorcet Jury Theorem assures us, so long as the (...) electorate's competence averaging across all voters remains better-than-random. (shrink)
Democracy in Social and Political Philosophy
Government and Democracy in Social and Political Philosophy
The Logical Space of Democracy.Christian List - 2011 - Philosophy and Public Affairs 39 (3):262-297.details
Can we design a perfect democratic decision procedure? Condorcet famously observed that majority rule, our paradigmatic democratic procedure, has some desirable properties, but sometimes produces inconsistent outcomes. Revisiting Condorcet's insights in light of recent work on the aggregation of judgments, I show that there is a conflict between three initially plausible requirements of democracy: "robustness to pluralism", "basic majoritarianism", and "collective rationality". For all but the simplest collective decision problems, no decision procedure meets these three requirements at once; at most (...) two can be met together. This "democratic trilemma" raises the question of which requirement to give up. Since different answers correspond to different views about what matters most in a democracy, the trilemma suggests a map of the "logical space" in which different conceptions of democracy are located. It also sharpens our thinking about other impossibility problems of social choice and how to avoid them, by capturing a core structure many of these problems have in common. More broadly, it raises the idea of "cartography of logical space" in relation to contested political concepts. (shrink)
Arrow's Theorem in Social and Political Philosophy
Condorcet's Paradox in Social and Political Philosophy
Deliberative Democracy in Social and Political Philosophy
Social Choice Theory, Misc in Social and Political Philosophy
Learning Juror Competence: A Generalized Condorcet Jury Theorem.Jan-Willem Romeijn & David Atkinson - 2011 - Politics, Philosophy and Economics 10 (3):237-262.details
This article presents a generalization of the Condorcet Jury Theorem. All results to date assume a fixed value for the competence of jurors, or alternatively, a fixed probability distribution over the possible competences of jurors. In this article, we develop the idea that we can learn the competence of the jurors by the jury vote. We assume a uniform prior probability assignment over the competence parameter, and we adapt this assignment in the light of the jury vote. We then compute (...) the posterior probability, conditional on the jury vote, of the hypothesis voted over. We thereby retain the central results of Condorcet, but we also show that the posterior probability depends on the size of the jury as well as on the absolute margin of the majority. (shrink)
Philosophy of Education in Philosophy of Social Science
Meta-Analysis as Judgment Aggregation.Berna Kilinc - 2010 - In Henk W. de Regt (ed.), Epsa Philosophy of Science: Amsterdam 2009. Springer. pp. 123--135.details
My goal in this paper is to see the extent to which judgment aggregation methods subsume meta-analytic ones. To this end, I derive a generalized version of the classical Condorcet Jury Theorem, the aggregative implications of which have been widely exploited in the area of rational choice theory, but not yet in philosophy of science. I contend that the generalized CJT that I prove below is useful for modelling at least some meta-analytic procedures.
The History of Feminism: Marie-Jean-Antoine-Nicolas de Caritat, Marquis de Condorcet.Joan Landes - 2010 - Stanford Encyclopedia of Philosophy.details
Feminist History of Philosophy in Philosophy of Gender, Race, and Sexuality
The Silence of Progress: The Historie Triple-Space Reduction of Condorcet.Hernan Neira - 2010 - Pensamiento 66 (249):771-790.details
Independence and Interdependence in Collective Decision Making: An Agent-Based Model of Nest-Site Choice by Honey Bee Swarms.Christian List, Christian Elsholtz & Thomas Seeley - 2009 - Philosophical Transactions of the Royal Society B 364:755-762.details
Condorcet's classic jury theorem shows that when the members of a group have noisy but independent information about what is best for the group as a whole, majority decisions tend to outperform dictatorial ones. When voting is supplemented by communication, however, the resulting interdependencies between decision-makers can strengthen or undermine this effect: they can facilitate information pooling, but also amplify errors. We consider an intriguing non-human case of independent information pooling combined with communication: the case of nest-site choice by honey (...) bee swarms. It is empirically well-documented that when there are different nest sites that vary in quality, the bees usually choose the best one. We develop a new agent-based model of the bees' decision process and show that its remarkable reliability stems from a particular interplay of independence and interdependence between the bees. (shrink)
Biological Modeling in Philosophy of Biology
Philosophy of Biology, Misc in Philosophy of Biology
Political Science in Social Sciences
Condorcet and Communitarianism: Boghossian's Fallacious Inference.Armin Schulz - 2009 - Synthese 166 (1):55 - 68.details
This paper defends the communitarian account of meaning against Boghossian's (Wittgensteinian) arguments. Boghossian argues that whilst such an account might be able to accommodate the infinitary characteristic of meaning, it cannot account for its normativity: he claims that, since the dispositions of a group must mirror those of its members, the former cannot be used to evaluate the latter. However, as this paper aims to make clear, this reasoning is fallacious. Modelling the issue with four (justifiable) assumptions, it shows that (...) Condorcet's 'Jury Theorem' can be used to prove that the dispositions of the majority of the members of a group can differ from those of any individual member in a way that makes it possible to use communal dispositions as a standard with which individual dispositions can be assessed. Moreover, the argument of the paper is also shown to have general implications for the use of formal methods in the explanation of the nature of certain fallacious inferences. (shrink)
Communitarianism in Social and Political Philosophy
The Premises of Condorcet's Jury Theorem Are Not Simultaneously Justified.Franz Dietrich - 2008 - Episteme 5 (1):56-73.details
Condorcet's famous jury theorem reaches an optimistic conclusion on the correctness of majority decisions, based on two controversial premises about voters: they are competent and vote independently, in a technical sense. I carefully analyse these premises and show that: whether a premise is justi…ed depends on the notion of probability considered; none of the notions renders both premises simultaneously justi…ed. Under the perhaps most interesting notions, the independence assumption should be weakened.
Interpretation of Probability in Philosophy of Probability
Condorcet: Communication/Science/Democracy.György Márkus - 2007 - Critical Horizons 8 (1):18-32.details
Condorcet's arguments concerning the dependence of unhindered scientific development on the presence of democratic conditions still sounds relevant today, because they are based on specific and complex considerations concerning the character of the social enterprise of science that articulates problems that still continue. The implicit dispute between Condorcet and Rousseau is also the first great historical example of the conflict between the Enlightenment and Romanticism, which accompanies the history of modernity, as an unresolved and indeed irresolvable opposition that belongs to (...) the prehistory of our own confusions and quandaries concerning the relations between culture, science, politics and society. (shrink)
The Epistemology of Democracy.Elizabeth Anderson - 2006 - Episteme 3 (1-2):8-22.details
Th is paper investigates the epistemic powers of democratic institutions through an assessment of three epistemic models of democracy : the Condorcet Jury Th eorem, the Diversity Trumps Ability Th eorem, and Dewey's experimentalist model. Dewey's model is superior to the others in its ability to model the epistemic functions of three constitutive features of democracy : the epistemic diversity of participants, the interaction of voting with discussion, and feedback mechanisms such as periodic elections and protests. It views democracy as (...) an institution for pooling widely distributed information about problems and policies of public interest by engaging the participation of epistemically diverse knowers. Democratic norms of free discourse, dissent, feedback, and accountability function to ensure collective, experimentallybased learning from the diverse experiences of diff erent knowers. I illustrate these points with a case study of community forestry groups in South Asia, whose epistemic powers have been hobbled by their suppression of women's participation. (shrink)
Epistemology, Misc in Epistemology
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Democratic Answers to Complex Questions – An Epistemic Perspective.Luc Bovens & Wlodek Rabinowicz - 2006 - Synthese 150 (1):131-153.details
This paper addresses a problem for theories of epistemic democracy. In a decision on a complex issue which can be decomposed into several parts, a collective can use different voting procedures: Either its members vote on each sub-question and the answers that gain majority support are used as premises for the conclusion on the main issue, or the vote is conducted on the main issue itself. The two procedures can lead to different results. We investigate which of these procedures is (...) better as a truth-tracker, assuming that there exists a true answer to be reached. On the basis of the Condorcet jury theorem, we show that the pbp is universally superior if the objective is to reach truth for the right reasons. If one instead is after truth for whatever reasons, right or wrong, there will be cases in which the cbp is more reliable, even though, for the most part, the pbp still is to be preferred. (shrink)
No Testimonial Route to Consensus.Philip Pettit - 2006 - Episteme 3 (3):156-165.details
The standard image of how consensus can be achieved is by pooling evidence and reducing if not eliminating disagreements. But rather than just pooling substantive evidence on a certain question, why not also take into account the formal, testimonial evidence provided by the fact that a majority of the group adopt a particular answer? Shouldn't we be reinforced by the discovery that we are on that majority side, and undermined by the discovery that we are not? Shouldn't this be so, (...) in particular, when Condorcet's jury theorem applies? It turns out not. There are serious problems attending any strategy of majoritarian deference. (shrink)
Epistemology of Testimony in Epistemology
Death in Condorcet's Eloges des Académiciens de l'Académie Royale des Sciences.Timothy Reeve - 2006 - In G. J. Mallinson (ed.), Interdisciplinarity: Qu'est-Ce Que les Lumières: La Reconnaissance au Dix-Huitième Siècle. Voltaire Foundation.details
De doctrinale paradox.Luc Bovens & Wlodek Rabinowicz - 2005 - Algemeen Nederlands Tijdschrift voor Wijsbegeerte 97 (1):XX.details
Suppose a committee or a jury confronts a complex question, the answer to which requires attending to several sub-questions. Two different voting procedures can be used. On one, the committee members vote on each sub-question and the voting results are used as premises for the committee's conclusion on the main issue. This premise-based procedure can be contrasted with the conclusion-based approach, which requires the members to directly vote on the conclusion, with the vote of each member being guided by her (...) views on the relevant sub-questions. The two procedures are not equivalent: There may be a majority of voters supporting each of the premises, but if these majorities do not significantly overlap, there will be a majority against the conclusion. Pettit (2001) connects the choice between the two procedures with the discussion of deliberative democracy. The problem we want to examine instead concerns the relative advantages and disadvantages of the two procedures from the epistemic point of view. Which of them is better when it comes to tracking truth? As it turns out, the answer is not univocal. On the basis of Condorcet's jury theorem, the premise-based procedure can be shown to be superior if the objective is reach truth for the right reasons, without making any mistakes on the way. However, if the goal instead is to reach truth for whatever reasons, right or wrong, there will be cases in which using the conclusion-based procedure turns out to be more reliable, even though, for the most part, the premise-based procedure will retain its superiority. (shrink)
Paradoxes in Logic and Philosophy of Logic
Which Scoring Rule Maximizes Condorcet Efficiency Under Iac?Davide P. Cervone, William V. Gehrlein & William S. Zwicker - 2005 - Theory and Decision 58 (2):145-185.details
Consider an election in which each of the n voters casts a vote consisting of a strict preference ranking of the three candidates A, B, and C. In the limit as n→∞, which scoring rule maximizes, under the assumption of Impartial Anonymous Culture (uniform probability distribution over profiles), the probability that the Condorcet candidate wins the election, given that a Condorcet candidate exists? We produce an analytic solution, which is not the Borda Count. Our result agrees with recent numerical results (...) from two independent studies, and contradicts a published result of Van Newenhizen (Economic Theory 2, 69–83. (1992)). (shrink)
A Simple Proof of Sen's Possibility Theorem on Majority Decisions.Christian Elsholtz & Christian List - 2005 - Elemente der Mathematik 60:45-56.details
Condorcet's voting paradox shows that pairwise majority voting may lead to cyclical majority preferences. In a famous paper, Sen identified a general condition on a profile of individual preference orderings, called triplewise value-restriction, which is sufficient for the avoidance of such cycles. This note aims to make Sen's result easily accessible. We provide an elementary proof of Sen's possibility theorem and a simple reformulation of Sen's condition. We discuss how Sen's condition is logically related to a number of precursors. Finally, (...) we state a necessary and sufficient condition for the avoidance of cycles, and suggest that, although there is still some logical space between that condition and Sen's sufficient condition, Sen's condition cannot be further generalized in an appealing way. (shrink)
The Probability of Inconsistencies in Complex Collective Decisions.Christian List - 2005 - Social Choice and Welfare 24 (1):3-32.details
Many groups make decisions over multiple interconnected propositions. The "doctrinal paradox" or "discursive dilemma" shows that propositionwise majority voting can generate inconsistent collective sets of judgments, even when individual sets of judgments are all consistent. I develop a simple model for determining the probability of the paradox, given various assumptions about the probability distribution of individual sets of judgments, including impartial culture and impartial anonymous culture assumptions. I prove several convergence results, identifying when the probability of the paradox converges to (...) 1, and when it converges to 0, as the number of individuals increases. Drawing on the Condorcet jury theorem and work by Bovens and Rabinowicz , I use the model to assess the "truth-tracking" performance of two decision procedures, the premise- and conclusion-based procedures. I compare the present results with existing results on the probability of Condorcet's paradox. I suggest that the doctrinal paradox is likely to occur under plausible conditions. (shrink)
Collective Intentionality in Philosophy of Mind
Formal Epistemology, Misc in Epistemology
Voting Procedures for Complex Collective Decisions. An Epistemic Perspective.Luc Bovens & Wlodek Rabinowicz - 2004 - Ratio Juris 17 (2):241-258.details
Legal Process in Philosophy of Law
Complex Collective Decisions: An Epistemic Perspective.Luc Bovens & Wlodek Rabinowicz - 2004 - Associations: Journal for Social and Legal Theory 7 (X).details
A Model of Jury Decisions Where All Jurors Have the Same Evidence.Franz Dietrich & Christian List - 2004 - Synthese 142 (2):175 - 202.details
Under the independence and competence assumptions of Condorcet's classical jury model, the probability of a correct majority decision converges to certainty as the jury size increases, a seemingly unrealistic result. Using Bayesian networks, we argue that the model's independence assumption requires that the state of the world (guilty or not guilty) is the latest common cause of all jurors' votes. But often – arguably in all courtroom cases and in many expert panels – the latest such common cause is a (...) shared 'body of evidence' observed by the jurors. In the corresponding Bayesian network, the votes are direct descendants not of the state of the world, but of the body of evidence, which in turn is a direct descendant of the state of the world. We develop a model of jury decisions based on this Bayesian network. Our model permits the possibility of misleading evidence, even for a maximally competent observer, which cannot easily be accommodated in the classical model. We prove that (i) the probability of a correct majority verdict converges to the probability that the body of evidence is not misleading, a value typically below 1; (ii) depending on the required threshold of 'no reasonable doubt', it may be impossible, even in an arbitrarily large jury, to establish guilt of a defendant 'beyond any reasonable doubt'. (shrink)
Bayesian Reasoning, Misc in Philosophy of Probability
The Persuasiveness of Democratic Majorities.Robert E. Goodin & David Estlund - 2004 - Politics, Philosophy and Economics 3 (2):131-142.details
Under the assumptions of the standard Condorcet Jury Theorem, majority verdicts are virtually certain to be correct if the competence of voters is greater than one-half, and virtually certain to be incorrect if voter competence is less than one-half. But which is the case? Here we turn the Jury Theorem on its head, to provide one way of addressing that question. The same logic implies that, if the outcome saw 60 percent of voters supporting one proposition and 40 percent the (...) other, then average voter competence must either be 0.60 or 0.40. We still have to decide which, but limiting the choice to those two values is a considerable aid in that. Key Words: Condorcet Jury Theorem • epistemic democracy • voter competence. (shrink)
On the Significance of the Absolute Margin.Christian List - 2004 - British Journal for the Philosophy of Science 55 (3):521-544.details
Consider the hypothesis H that a defendant is guilty, and the evidence E that a majority of h out of n independent jurors have voted for H and a minority of k:=n-h against H. How likely is the majority verdict to be correct? By a formula of Condorcet, the probability that H is true given E depends only on each juror's competence and on the absolute margin between the majority and the minority h-k, but neither on the number n, nor (...) on the proportion h/n. This paper reassesses that result and explores its implications. First, using the classical Condorcet jury model, I derive a more general version of Condorcet's formula, confirming the significance of the absolute margin, but showing that the probability that H is true given E depends also on an additional parameter: the prior probability that H is true. Second, I show that a related result holds when we consider not the degree of belief we attach to H given E, but the degree of support E gives to H. Third, I address the implications for the definition of special majority voting, a procedure used to capture the asymmetry between false positive and false negative decisions. I argue that the standard definition of special majority voting in terms of a required proportion of the jury is epistemically questionable, and that the classical Condorcet jury model leads to an alternative definition in terms of a required absolute margin between the majority and the minority. Finally, I show that the results on the significance of the absolute margin can be resisted if the so-called assumption of symmetrical juror competence is relaxed. (shrink)
An Epistemic Free-Riding Problem?Christian List & Philip Pettit - 2004 - In Philip Catton & Graham Macdonald (eds.), Karl Popper: Critical Appraisals. Routledge. pp. 128-158.details
One of the hallmark themes of Karl Popper's approach to the social sciences was the insistence that when social scientists are members of the society they study, then they are liable to affect that society. In particular, they are liable to affect it in such a way that the claims they make lose their validity. "The interaction between the scientist's pronouncements and social life almost invariably creates situations in which we have not only to consider the truth of such pronouncements, (...) but also their actual influence on future developments. The social scientist may be striving to find the truth; but, at the same time, he must always be exerting a definite influence upon society. The very fact that his pronouncements do exert an influence destroys their objectivity." (Popper 1963. (shrink)
Popper: Philosophy of Social Science in 20th Century Philosophy
Aggregating Sets of Judgments: Two Impossibility Results Compared.Christian List & Philip Pettit - 2004 - Synthese 140 (1-2):207 - 235.details
The ``doctrinal paradox'' or ``discursive dilemma'' shows that propositionwise majority voting over the judgments held by multiple individuals on some interconnected propositions can lead to inconsistent collective judgments on these propositions. List and Pettit (2002) have proved that this paradox illustrates a more general impossibility theorem showing that there exists no aggregation procedure that generally produces consistent collective judgments and satisfies certain minimal conditions. Although the paradox and the theorem concern the aggregation of judgments rather than preferences, they invite comparison (...) with two established results on the aggregation of preferences: the Condorcet paradox and Arrow's impossibility theorem. We may ask whether the new impossibility theorem is a special case of Arrow's theorem, or whether there are interesting disanalogies between the two results. In this paper, we compare the two theorems, and show that they are not straightforward corollaries of each other. We further suggest that, while the framework of preference aggregation can be mapped into the framework of judgment aggregation, there exists no obvious reverse mapping. Finally, we address one particular minimal condition that is used in both theorems – an independence condition – and suggest that this condition points towards a unifying property underlying both impossibility results. (shrink)
Condorcet and Modernity.David Williams - 2004 - Cambridge University Press.details
David Williams explores the complex links between Condorcet as visionary ideologist and pragmatic legislator, and between his concept of modernity and the management of change. The Marquis de Condorcet was one of the few Enlightenment thinkers to witness and participate in the French Revolution. Based on an extensive array of printed and original manuscript sources, Williams' analysis of Condorcet's politics will be a major contribution to Enlightenment studies.
$35.33 new $46.47 used Amazon page
Democracy and Argument: Tracking Truth in Complex Social Decisions.Luc Bovens & Wlodek Rabinowicz - 2003 - In Anne van Aaken, Christian List & Christoph Luetge (eds.), Deliberation and Decision: Economics, Constitutional Theory, and Deliberative Democracy. Law, ethics and economics. Aldershot, UK: Ashgate. pp. 143-157.details
Suppose a committee has to take a stand on a complex issue, where the decision presupposes answering a number of sub-questions. There is an agreement within the committee which sub-questions should be posed. All questions are of the "yes or no?"-type and the main question is to be given the yes-answer if and only if each sub-question is answered with "yes". Two different voting procedures can be used. On one procedure, the committee members vote on each sub-question and the voting (...) results then determine the committee's conclusion on the main issue. This premise-based procedure (or pbp, for short) can be contrasted with the conclusion-based procedure (cbp), on which the members directly vote on the conclusion, with the vote of each member being guided by her views on the relevant sub-questions. The problem we want to examine concerns the relative advantages and disadvantages of the two procedures from the epistemic point of view. In some cases one can assume that the question before the committee has a right answer, which the committee is trying to reach. Is one of the two procedures better when it comes to tracking the truth? As it turns out, the answer to this query is not univocal: On the basis of Condorcet's jury theorem we shall show that the premise-based procedure is clearly superior if we want to reach truth for the right reasons, i.e. without making any mistakes on the road to the conclusion. However, if the goal instead is to reach truth for whatever reasons, right or wrong, there will be special cases in which using the conclusion-based procedure turns out to be more reliable. But for the most part, the premise-based procedure will still retain its superiority. In this respect, our results disconfirm the tentative conjectures that have been put forward in Pettit and Rabinowicz (2001). (shrink)
Rational Aggregation.Bruce Chapman - 2002 - Politics, Philosophy and Economics 1 (3):337-354.details
In two recent papers, Christian List and Philip Pettit have argued that there is a problem in the aggregation of reasoned judgements that is akin to the aggregation of the preference problem in social choice theory. 1 Indeed, List and Pettit prove a new general impossibility theorem for the aggregation of judgements, and provide a propositional interpretation of the social choice problem that suggests it is a special case of their impossibility result. 2 Specifically, they show that no judgement aggregation (...) function for a group is possible if the group seeks to satisfy certain `minimal conditions' designed to ensure that the function is both responsive to the individually rational views of its members and collectively rational in the set of judgements it holds. In this article, I resist the List and Pettit claim that there is the same propensity for collective irrationality or incoherence in the aggregation of reasoned judgements as there is in the aggregation of preference. I argue that reason, because it has a logical structure that is lacking in mere preference, has the effect of giving priority to some aggregations over others, a priority that is not permitted by one of the conditions imposed by List and Pettit. This avoids the incoherence that would otherwise exist if these different aggregations, not consistent with one another, were to compete at the same level of priority. The priority of some aggregations is particularly apparent, I shall argue, if one views the aggregation of judgements through the lens of common law decision-making. Key Words: social choice • judgement • Condorcet jury theorem • collective rationality • public reason • doctrinal paradox • discursive dilemma. (shrink)
Social and Political Philosophy, Miscellaneous in Social and Political Philosophy
Condorcet's Paradox and the Likelihood of its Occurrence: Different Perspectives on Balanced Preferences.William V. Gehrlein - 2002 - Theory and Decision 52 (2):171-199.details
Many studies have considered the probability that a pairwise majority rule (PMR) winner exists for three candidate elections. The absence of a PMR winner indicates an occurrence of Condorcet's Paradox for three candidate elections. This paper summarizes work that has been done in this area with the assumptions of: Impartial Culture, Impartial Anonymous Culture, Maximal Culture, Dual Culture and Uniform Culture. Results are included for the likelihood that there is a strong winner by PMR, a weak winner by PMR, and (...) the probability that a specific candidate is among the winners by PMR. Closed form representations are developed for some of these probabilities for Impartial Anonymous Culture and for Maximal Culture. Consistent results are obtained for all cultures. In particular, very different behaviors are observed for odd and even numbers of voters. The limiting probabilities as the number of voters increases are reached very quickly for odd numbers of voters, and quite slowly for even numbers of voters. The greatest likelihood of observing Condorcet's Paradox typically occurs for small numbers of voters. Results suggest that while examples of Condorcet's Paradox are observed, one should not expect to observe them with great frequency in three candidate elections. (shrink)
On the (Sample) Condorcet Efficiency of Majority Rule: An Alternative View of Majority Cycles and Social Homogeneity.Michel Regenwetter, James Adams & Bernard Grofman - 2002 - Theory and Decision 53 (2):153-186.details
The Condorcet efficiency of a social choice procedure is usually defined as the probability that this procedure coincides with the majority winner (or majority ordering) in random samples, given a majority winner exists (or given the majority ordering is transitive). Consequently, it is in effect a conditional probability that two sample statistics coincide, given certain side conditions. We raise a different issue of Condorcet efficiencies: What is the probability that a social choice procedure applied to a sample matches with the (...) majority preferences of the population from which the sample was drawn? We investigate the canonical case where the sample statistic is itself also majority rule and the samples are drawn from real world distributions gathered from national election surveys in Germany, France, and the United States. We relate the results to the existing literature on majority cycles and social homogeneity. We find that these samples rarely display majority cycles, whereas the probability that a sample misrepresents the majority preferences of the underlying population varies dramatically and always exceeds the probability that the sample displays cyclic majority preferences. Social homogeneity plays a fundamental role in the type of Condorcet efficiency investigated here. (shrink)
Public Discourse: Creating the Conditions for Dialogue Concerning the Common Good in a Postmodern Heterogeneous Democracy.Peggy Ruth Geren - 2001 - Studies in Philosophy and Education 20 (3):191-199.details
This paper offers a philosophical `history' of the nature of`public discourse' – a basic element of human rights. It beginswith Enlightenment views from Condorcet and Jefferson, turns to Dewey,and then to Habermas. Over a couple of centuries not only does thecentral character of discourse change but so too does the definition ofa public person.
The Set Theoretic Ambit Of Arrow's Theorem.Louis M. Guenin - 2001 - Synthese 126 (3):443-472.details
Set theoretic formulation of Arrow's theorem, viewed in light of a taxonomy of transitive relations, serves to unmask the theorem's understated generality. Under the impress of the independence of irrelevant alternatives, the antipode of ceteris paribus reasoning, a purported compiler function either breaches some other rationality premise or produces the "effet Condorcet". Types of cycles, each the seeming handiwork of a virtual voter disdaining transitivity, are rigorously defined. Arrow's theorem erects a dilemma between cyclic indecision and dictatorship. Maneuvers responsive thereto (...) are explicable in set theoretic terms. None of these gambits rival in simplicity the unassisted escape of strict linear orderings, which, by virtue of the Arrow-Sen reflexivity premise, are not captured by the theorem. Yet these are the relations among whose n-tuples the "effet Condorcet" is most frequent. A generalization and stronger theorem encompasses these and all other linear orderings and total tierings. Revisions to the Arrow-Sen definitions of 'choice set' and 'rationalization' similarly enable one to generalize Sen's demonstration that some rational choice function always exists. Similarly may one generalize Debreu's theorems establishing conditions under which a binary relation may be represented by a continuous real-valued order homomorphism. (shrink)
Some Remarks on the Probability of Cycles - Appendix 3 to 'Epistemic Democracy: Generalizing the Condorcet Jury Theorem'.Christian List - 2001 - Journal of Political Philosophy 9 (3):277-306.details
This item was published as 'Appendix 3: An Implication of the k-option Condorcet jury mechanism for the probability of cycles' in List and Goodin (2001) http://eprints.lse.ac.uk/705/. Standard results suggest that the probability of cycles should increase as the number of options increases and also as the number of individuals increases. These results are, however, premised on a so-called "impartial culture" assumption: any logically possible preference ordering is assumed to be as likely to be held by an individual as any other. (...) The present chapter shows, in the three-option case, that given suitably systematic, however slight, deviations from an impartial culture situation, the probability of a cycle converges either to zero (more typically) or to one (less typically) as the number of individuals increases. (shrink)
Epistemic Democracy: Generalizing the Condorcet Jury Theorem.Christian List & Robert E. Goodin - 2001 - Journal of Political Philosophy 9 (3):277–306.details
This paper generalises the classical Condorcet jury theorem from majority voting over two options to plurality voting over multiple options. The paper further discusses the debate between epistemic and procedural democracy and situates its formal results in that debate. The paper finally compares a number of different social choice procedures for many-option choices in terms of their epistemic merits. An appendix explores the implications of some of the present mathematical results for the question of how probable majority cycles (as in (...) Condorcet's paradox) are in large electorates. (shrink)
Political Ethics in Applied Ethics
Scoring Rules, Condorcet Efficiency and Social Homogeneity.Dominique Lepelley, Patrick Pierron & Fabrice Valognes - 2000 - Theory and Decision 49 (2):175-196.details
In a three-candidate election, a scoring rule s (s in [0,1]) assigns 1, s, and 0 points (respectively) to each first, second and third place in the individual preference rankings. The Condorcet efficiency of a scoring rule is defined as the conditional probability that this rule selects the winner in accordance with Condorcet criteria (three Condorcet criteria are considered in the paper). We are interested in the following question: What rule s has the greatest Condorcet efficiency? After recalling the known (...) answer to this question, we investigate the impact of social homogeneity on the optimal value of ?. One of the most salient results we obtain is that the optimality of the Borda rule (s=1/2) holds only if the voters act in an independent way. (shrink)
Scoring Rules in Philosophy of Probability
"Instruction Publique" in the French Eighteenth-Century Discourse of Modernisation.Ursula Hofer - 1999 - Studies in Philosophy and Education 18 (1):15-24.details
Towards the end of eighteenth century in France, the newly acquired rights of people as citizens needed assuring. This article traces the principles through which Condorcet tried to realise this on an institutional level. Condorcet did not view the Enlightenment ideas of progress as primarily referring to the state. Rather, he focused on the rights of individuals, particularly on their right to develop their own potential. He bound this perception with the unconditional demand for recognition of the rights of all (...) people, in particular also for a clear renunciation of any gender-specific interpretation of this fundamental idea. Evidence for this is found in Condorcet's writings on Instruction Publique. In contrast to other educational programmes of the revolutionary period, these evince a discriminating interpretation of freedom and equality, and recognise the numerous threats to a self-determined human existence. (shrink)
Condorcet Et la Question de L'Égalité.Charles Coutel - 1998 - Dialogue 37 (4):681-.details
This paper intends to focus on Condorcet's approach to the Principle of Equality. Condorcet, in effect, strenuously strives to counter the risks of equalitarianism, such as élitism. According to him, it is in the interest of the Republic and of public instruction to favour the diversity of talents and the spreading of enlightenment, since, in the end, it will benefit all citizens.
Egalitarianism in Social and Political Philosophy
Opinion Leaders, Independence, and Condorcet's Jury Theorem.David M. Estlund - 1994 - Theory and Decision 36 (2):131-162.details
Intellectual Subversion in Eighteenth Century Political Thought: Condorcet's Philosophy of History.Jennifer Paige Montana - 1994 - Dissertation, Harvard Universitydetails
Through a careful reconstruction of Condorcet's highly fragmented historical thought I analyze the political implications of his philosophy of history. I argue that the specific historical tendency developed in Condorcet's philosophy of history links progress with collective self-interpretation rather than material productivity. I contend that Condorcet's philosophy of history does not represent a symbolic glorification of the domination of nature by reason. His philosophy of history does not purport to offer a science of futurology that can quantify the conquest of (...) the external world by transforming the prediction of the future into an empirical matter. It is not a disguised sociology. Finally, the purpose of history does not consist in the glorification of the discrete individual's domination of the external world--posing as an accurate collective mirror of the heroic mastery and acquisition of nature by society. Rather the purpose of history lies in a form of self-understanding as a culture that occurs through the active collective participation in its creation. The latter can only be facilitated through the emancipation of the intellect. ;Condorcet's attempt to free the intellect entails a process of subversion. He tries to "redirect" political and cultural identity by subverting or "refashioning... intelligence." This fundamental alteration in thinking represents "a type of revolution in the mind," as he argues. Through a process of public instruction that reflects a re-conceptualization of the meanings, uses, and purpose of history, the individual's relationship to the "nation" is redefined. His/her political identity is displaced by a broader cultural consciousness. Condorcet liberates knowledge and the intellect through this process. ;His re-conceptualization of history functions to subvert traditional political and social institutions and the modes of thinking that underlie them in order to focus upon the open building of knowledge. Condorcet's philosophy of history is underlaid by an epistemology. In essence, the former is, in part, constructed to illuminate and support this theory of knowledge and its transformative political implications. The underlying purpose of knowledge itself is redefined. It is circumscribed by what I argue signifies the humanization of the subject--the self's recognition of its "sentient-rational" nature. It represents what I call a knowledge that humanizes. ;To view history as an expression of self-interpretation through which greater self-understanding is possible reflects a substantive philosophy of history. This view has not been acknowledged in Condorcet's work, in particular, or as a theme in Enlightenment historical thinking, in general. The re-conceptualization of the meaning, uses, and purpose of history pushes into prominence the introspective dimension of knowledge in which knowledge transforms the subject rather than knowledge being used to master and/or acquire an object external to the subject. The inward looking or introspective nature of this form of consciousness provides the foundation for a free society in which knowledge enhances rather than oppresses the human spirit. (shrink) | CommonCrawl |
Evolution Equations and Control Theory
2022, Volume 11, Issue 5: 1701-1744. Doi: 10.3934/eect.2021061
This issue Previous Article Stability estimate for a partial data inverse problem for the convection-diffusion equation Next Article Two simple criterion to obtain exact controllability and stabilization of a linear family of dispersive PDE's on a periodic domain
Convergence of random attractors towards deterministic singleton attractor for 2D and 3D convective Brinkman-Forchheimer equations
Kush Kinra and
Manil T. Mohan,
Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand 247667, INDIA
*Corresponding author: Manil T. Mohan
Early access: December 2021
The second author is supported by DST INSPIRE Faculty Award (IFA17-MA110)
This work deals with the asymptotic behavior of the two as well as three dimensional convective Brinkman-Forchheimer (CBF) equations in an $ n $-dimensional torus ($ n = 2, 3 $):
$ \frac{\partial\boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p = \boldsymbol{f}, \ \nabla\cdot\boldsymbol{u} = 0, $
where $ r\geq1 $. We prove that the global attractor of the above system is singleton under small forcing intensity ($ r\geq 1 $ for $ n = 2 $ and $ r\geq 3 $ for $ n = 3 $ with $ 2\beta\mu\geq 1 $ for $ r = n = 3 $). But if one perturbs the above system with an additive or multiplicative white noise, there is no sufficient evidence that the random attractor keeps the singleton structure. We obtain that the random attractor for 2D stochastic CBF equations forced by additive and multiplicative white noise converges towards the deterministic singleton attractor for all $ 1\leq r<\infty $, when the coefficient of random perturbation converges to zero (upper and lower semicontinuity). For the case of 3D stochastic CBF equations perturbed by additive and multiplicative white noise, we are able to establish that the random attractor converges towards the deterministic singleton attractor for $ 3\leq r<\infty $ ($ 2\beta\mu\geq 1 $ for $ r = 3 $), when the coefficient of random perturbation converges to zero.
Deterministic and stochastic convective Brinkman-Forchheimer equations,
small forcing intensity,
singleton attractor,
upper semicontinuity,
lower semicontinuity.
Mathematics Subject Classification: Primary 35B41, 35Q35; Secondary 37L55, 37N10, 35R60.
[1] S. N. Antontsev and H. B. de Oliveira, The Navier-Stokes problem modified by an absorption term, Appl. Anal., 89 (2010), 1805-1825. doi: 10.1080/00036811.2010.495341.
[2] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.
[3] A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North-Holland, 1992.
[4] J. Babutzka and P. C. Kunstmann, $L^q$-Helmholtz decomposition on periodic domains and applications to Navier-Stokes equations, J. Math. Fluid Mech., 20 (2018), 1093-1121. doi: 10.1007/s00021-017-0356-z.
[5] P. Bates, H. Lisei and K. Lu, Attractors for stochastic lattice dynamical systems, Stoch. Dyn., 6 (2006), 1-21. doi: 10.1142/S0219493706001621.
[6] P. Bates, K. Lu and B. Wang, Random attractors for stochastic reaction-diffusion equations on unbounded domains, J. Differential Equations, 246 (2009), 845-869. doi: 10.1016/j.jde.2008.05.017.
[7] Z. Brzeźniak, B. Goldys and Q. T. Le Gia, Random attractors for the stochastic Navier-Stokes equations on the 2D unit sphere, J. Math. Fluid Mech., 20 (2018), 227-253. doi: 10.1007/s00021-017-0351-4.
[8] T. Caraballo, H. Crauel, J. A. Langa and J. C. Robinson, The effect of noise on the Chafee-Infante equation: A nonlinear case study, Proc. Amer. Math. Soc., 135 (2007), 373-382. doi: 10.1090/S0002-9939-06-08593-5.
[9] T. Caraballo, J. A. Langa and J. C. Robinson, Upper semicontinuity of attractors for small random perturbations of dynamical systems, Comm. Partial Differential Equations, 23 (1998), 1557-1581. doi: 10.1080/03605309808821394.
[10] A. N. Carvalho, J. A. Langa and J. C. Robinson, On the continuity of pullback attractors for evolution processes, Nonlinear Anal., 71 (2009), 1812-1824. doi: 10.1016/j.na.2009.01.016.
[11] I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, Universitext. Springer, Cham, 2015. doi: 10.1007/978-3-319-22903-4.
[12] H. Crauel, Global random attractors are uniquely determined by attracting deterministic compact sets, Ann. Mat. Pura Appl., 176 (1999), 57-72. doi: 10.1007/BF02505989.
[13] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynamics Differential Equations, 9 (1997), 307-341. doi: 10.1007/BF02219225.
[14] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Prob. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705.
[15] H. Cui and J. A. Langa, Uniform attractors for non-autonomous random dynamical systems, J. Differential Equations, 263 (2017), 1225-1268. doi: 10.1016/j.jde.2017.03.018.
[16] H. Cui and P. E. Kloeden, Convergence rate of random attractors for 2D Navier-Stokes equation towards the deterministic singleton attractor, Contemporary Approaches and Methods in Fundamental Mathematics and Mechanics, (2021), 233–255. doi: 10.1007/978-3-030-50302-4_10.
[17] X. Fan, Attractors for a damped stochastic wave equation of the sine-Gordon type with sublinear multiplicative noise, Stoch. Anal. Appl., 24 (2006), 767-793. doi: 10.1080/07362990600751860.
[18] C. L. Fefferman, K. W. Hajduk and J. C. Robinson, Simultaneous approximation in Lebesgue and Sobolev norms via eigenspaces, https://arXiv.org/pdf/1904.03337.pdf.
[19] X. Feng and B. You, Random attractors for the two-dimensional stochastic g-Navier-Stokes equations, Stochastics, 92 (2020), 613-626. doi: 10.1080/17442508.2019.1642340.
[20] F. Flandoli and B. Schmalfuss, Random attractors for the 3D stochastic Navier- Stokes equation with multiplicative white noise, Stochastics Stochastic Rep., 59 (1996), 21-45. doi: 10.1080/17442509608834083.
[21] C. Foias, O. P. Manley, R. Temam and Y. M. Tréve, Asymptotic analysis of the Navier-Stokes equations, Phys. D, 9 (1983), 157-188. doi: 10.1016/0167-2789(83)90297-X.
[22] D. Fujiwara and H. Morimoto, An $L^r$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700.
[23] B. Gess, W. Liu and A. Schenke, Random attractors for locally monotone stochastic partial differential equations, J. Differential Equations, 269 (2020), 3414-3455. doi: 10.1016/j.jde.2020.03.002.
[24] K. W. Hajduk and J. C. Robinson, Energy equality for the 3D critical convective Brinkman-Forchheimer equations, J. Differential Equations, 263 (2017), 7141-7161. doi: 10.1016/j.jde.2017.08.001.
[25] J. K. Hale and G. Raugel, Lower semicontinuity of attractors of gradient systems and applications, Ann. Mat. Pura Appl., 154 (1989), 281-326. doi: 10.1007/BF01790353.
[26] X. Jia and X. Ding, Random attractors for stochastic retarded 2D-Navier-Stokes equations with additive noise, J. Funct. Spaces, 2018 (2018), Art. ID 3105239, 14pp. doi: 10.1155/2018/3105239.
[27] V. K. Kalantarov and S. Zelik, Smooth attractors for the Brinkman-Forchheimer equations with fast growing nonlinearities, Commun. Pure Appl. Anal., 11 (2012), 2037-2054. doi: 10.3934/cpaa.2012.11.2037.
[28] K. Kinra and M. T. Mohan, Random attractors for 2D and 3D stochastic convective Brinkman-Forchheimer equations in some unbounded domains, Submitted, https://arXiv.org/pdf/2010.08753.pdf.
[29] K. Kinra and M. T. Mohan, Existence and upper semicontinuity of random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in bounded domains, Submitted, https://arXiv.org/pdf/2011.06206.pdf.
[30] K. Kinra and M. T. Mohan, ${\mathbb{H}}^1$-random attractors for the 2D stochastic convective Brinkman-Forchheimer equations in some unbounded domains, Submitted, https://arXiv.org/pdf/2111.07841.pdf.
[31] K. Kinra and M. T. Mohan, Large time behavior of the deterministic and stochastic 3D convective Brinkman-Forchheimer equations in periodic domains, J. Dynamics and Differential Equations, 2021 doi: 10.1007/s10884-021-10073-7.
[32] K. Kinra and M. T. Mohan, Weak pullback mean random attractors for the stochastic convective Brinkman-Forchheimer equations and locally monotone stochastic partial differential equations, Submitted, https://arXiv.org/pdf/2012.12605.pdf.
[33] Y. Li, H. Cui and J. Li, Upper semi-continuity and regularity of random attractors on p-times integrable spaces and applications, Nonlinear Anal., 109 (2014), 33-44. doi: 10.1016/j.na.2014.06.013.
[34] D. Li and P. Kloeden, Equi-attraction and the continuous dependence of attractors on parameters, Glasg. Math. J., 46 (2004), 131-141. doi: 10.1017/S0017089503001605.
[35] L. Liu and X. Fu, Existence and upper semicontinuity of $(L^2, L^q)$ pullback attractors for a stochastic p-laplacian equation, Commun. Pure Appl. Anal., 16 (2017), 443-474. doi: 10.3934/cpaa.2017023.
[36] H. Liu and H. Gao, Stochastic 3D Navier-Stokes equations with nonlinear damping: Martingale solution, strong solution and small time LDP, Interdiscip. Math. Sci., World Sci. Publ., 20 (2019), 9-36.
[37] P. A. Markowich, E. S. Titi and S. Trabelsi, Continuous data assimilation for the three-dimensional Brinkman-Forchheimer-extended Darcy model, Nonlinearity, 29 (2016), 1292-1328. doi: 10.1088/0951-7715/29/4/1292.
[38] M. T. Mohan, On the convective Brinkman-Forchheimer equations, Submitted.
[39] M. T. Mohan, Stochastic convective Brinkman-Forchheimer equations, Submitted, 2020, https://arXiv.org/abs/2007.09376.
[40] M. T. Mohan, Asymptotic analysis of the 2D convective Brinkman-Forchheimer equations in unbounded domains: Global attractors and upper semicontinuity, Submitted, https://arXiv.org/abs/2010.12814.
[41] M. T. Mohan, The ${\mathbb{H}}^1$-compact global attractor for the two dimentional convective Brinkman-Forchheimer equations in unbounded domains, J. Dynamical and Control Systems, 2021. doi: 10.1007/s10883-021-09545-2.
[42] M. T. Mohan, Well-posedness and asymptotic behavior of stochastic convective Brinkman-Forchheimer equations perturbed by pure jump noise, Stoch PDE: Anal Comp, (2021). doi: 10.1007/s40072-021-00207-9.
[43] M. T. Mohan, ${\mathbb{L}}^p$-solutions of deterministic and stochastic convective Brinkman-Forchheimer equations, Anal. Math. Phys., 11 (2021), Ar. No.: 164, 33pp. doi: 10.1007/s13324-021-00595-0.
[44] M. T. Mohan, Martingale solutions of two and three dimensional stochastic convective Brinkman-Forchheimer equations forced by Lévy noise, Submitted, https://arXiv.org/pdf/2109.05510.pdf.
[45] J. C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, 2001.
[46] J. C. Robinson, Attractors and finite-dimensional behavior in the 2D Navier-Stokes equations, ISRN Math. Anal., 2013 (2013), Article ID 291823, 29pp. doi: 10.1155/2013/291823.
[47] J. C. Robinson, J. L. Rodrigo and W. Sadowski, The Three-Dimensional Navier-Stokes Equations, Classical Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 2016. doi: 10.1017/CBO9781139095143.
[48] J. C. Robinson and W. Sadowski, A local smoothness criterion for solutions of the 3D Navier-Stokes equations, Rend. Semin. Mat. Univ. Padova, 131 (2014), 159-178. doi: 10.4171/RSMUP/131-9.
[49] M. Röckner and X. Zhang, Tamed 3D Navier-Stokes equation: Existence, uniqueness and regularity, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009), 525-549. doi: 10.1142/S0219025709003859.
[50] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, 68. Springer-Verlag, New York, 1988. doi: 10.1007/978-1-4684-0313-8.
[51] R. Temam, Navier-Stokes Equations and Nonlinear Functional Analysis, 2$^{nd}$ edition, CBMS-NSF Regional Conference Series in Applied Mathematics, 1995. doi: 10.1137/1.9781611970050.
[52] B. Wang, Upper semicontinuity of random attractors for non-compact random dynamical systems, Electron. J. Differential Equations, (2009), 1–18.
[53] J. Wang, C. Li, L. Yang and M. Jia, Upper semi-continuity of random attractors and existence of invariant measures for nonlocal stochastic Swift-Hohenberg equation with multiplicative noise, J. Math. Phys., 62 (2021), 111507, 31pp. doi: 10.1063/5.0039187.
[54] B. You, The existence of a random attractor for the three dimensional damped Navier-Stokes equations with additive noise, Stoch. Anal. Appl., 35 (2017), 691-700. doi: 10.1080/07362994.2017.1311794.
[55] S. Zhou and Z. Wang, Upper-semicontinuity of attractors for reaction-diffusion equation and damped wave equation in $\mathbb{R}^n$ perturbed by small multiplicative noises, Dynam. Systems Appl., 22 (2013), 15-31.
Kush Kinra
Manil T. Mohan | CommonCrawl |
\begin{document}
\title{The Generalization of the Decomposition of Functions by Energy Operators}
\maketitle
\author{J.-P.~Montillet \footnote{Dr. J.P. Montillet is a Research Fellow at the Research School of Earth Sciences at the Australian National University}, [email protected]$}
\begin{abstract}
\boldmath{This work starts with the introduction of a family of differential energy operators. Energy operators (${\Psi}_{R}^{+}$, ${\Psi}_{R}^{-}$) were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives ($\Psi^{-}_k$, $\Psi^{+}_k$, $k=\{0,\pm 1,\pm 2,...\}$). The main part of the work demonstrates for any smooth real-valued function $f$ in the Schwartz space ($\mathbf{S}^{-}(\mathbb{R})$), the successive derivatives of the $n$-th power of $f$ ($n \in \mathbb{Z}$ and $n\neq 0$) can be decomposed using only $\Psi^{+}_k$ (Lemma); or if $f$ in a subset of $\mathbf{S}^{-}(\mathbb{R})$, called $\mathbf{s}^{-}(\mathbb{R})$, $\Psi^{+}_k$ and $\Psi^{-}_k$ ($k\in \mathbb{Z}$) decompose in a unique way the successive derivatives of the $n$-th power of $f$ (Theorem). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.
}
\end{abstract}
\section{Introduction}\label{Introduction part}
Two decades ago, an energy operator (${\Psi}_{R}^{-}$) was first defined in \cite{Kaiser90}. Since then, this work has been extensively used in telecommunications (see for example \cite{Bovik93} or \cite{Hamila1999}). The bilinearity properties of this operator were studied in \cite{Boudraa et al.2009}. More recently, the author in \cite{JPMontillet2010} introduced the energy operators (${\Psi}_{R}^{+}$, ${\Psi}_{R}^{-}$) in time and space. This was part of a general method for separating the energy of finite energy functions in time and space with application to the wave equation. Note that calling ${\Psi}_{R}^{+}$ an energy operator is an abuse of language. The term was already used in \cite{JPMontillet2010} as the definition of ${\Psi}_{R}^{+}$ is very close to the Teager-Kaiser energy operator ${\Psi}_{R}^{-}$.
\newline This work focuses on the decomposition of a smooth real-valued function $f$ using family of differentiable energy operators based on the energy operators ${\Psi}_{R}^{\pm}$. Throughout this work, $f$ is supposed to be in the Schwartz space $\mathbf{S}^{-}(\mathbb{R})$ defined as:
\begin{equation}
\mathbf{S}^{-}(\mathbb{R}) =\{f \in \mathbf{C}^{\infty}(\mathbb{R}), \qquad {sup}_{t<0} |t^k||\partial_t^j f(t)|<\infty,\qquad \forall k \in \mathbb{Z}^+, \qquad \forall j \in \mathbb{Z}^+ \} \end{equation}
Note that $\partial_t$ is the derivative related to the variable $t$. $\mathbb{Z}^+$ is the set of positive integers including $0$. In the following, let us call the set $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$ all functions defined such as $F:$ $\mathbf{S}^{-}(\mathbb{R})$ $\rightarrow$ $\mathbf{S}^{-}(\mathbb{R})$.
\newline First, two families of differential energy operators $\Psi^{+}_k$ and $\Psi^{-}_k$ ($k=\{0,\pm 1,\pm 2,...\}$) are introduced with the notations following \cite{Maragos1995} and \cite{JPMontillet2010} where $k$ is the degree of their derivatives. Note that for $f$ in $\mathbf{S}^{-}(\mathbb{R})$, $\Psi^{+}_k(f)$ and $\Psi^{-}_k(f)$ ($k=\{0,\pm 1,\pm 2,...\}$) are in $\mathbf{S}^{-}(\mathbb{R})$ ($(\Psi^{\pm}_k)_{k \in \mathbb{Z} }$ $\subseteq$ $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$. If not explicitly written, any families of operator in $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$ in this work follow the derivative chain rule property:
\begin{equation}\label{derivative chain rules0111} \forall f \in \mathbf{S}^{-}(\mathbb{R}), \qquad \partial_t \Psi_k(f) = \Psi_{k+1}(f) + \Psi_{k-1}(\partial_t f) \end{equation}
In addition, the term \emph{decompose} is defined as:
\newline $\bold{Definition}$ $1$: for all $f$ in $\mathbf{S}^{-}(\mathbb{R})$, for all $v\in\mathbb{Z}^+-\{0\}$, for all $n\in\mathbb{Z}^+$ and $n>1$, the family of operators $(\Psi_k)_{k \in \mathbb{Z}}$ (with $(\Psi_k)_{k \in \mathbb{Z}}$ $\subseteq$ $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$) decomposes $\partial_t^v$$f^n$ in $\mathbb{R}$, if it exists $(N_j)_{j\in \mathbb{Z}^+ \cup \{0\}}$ $\subseteq$ $\mathbb{Z^+}$, $(C_i)_{i=-N_j}^{N_j}$ $\subseteq$ $\mathbb{R}$, and it exists $(\alpha_j)$ and $l$ in $\mathbb{Z^+}\cup\{0\}$ (with $l<v$) \\ such as $\partial_t^v$$f^n = \sum_{j=0}^{v-1} \big(_{j}^{v-1} \big) \partial_t^{v-1-j} f^{n-l} \sum_{k=-N_j}^{N_j} C_k \Psi_k(\partial_t^{\alpha_k}f)$.
\\ Definition $1$ is based on the general Leibniz rule for the $n-th$ derivative of a product of functions \cite{BruceWest}. One can define the image $Im(\Psi^{+}_k)$ and kernel $Ker(\Psi^{+}_k)$ (for $k$ in $\mathbb{Z}$) of an energy operator such as:
\begin{equation}\label{Impsik+}
Im(\Psi^{+}_k) = \{\Psi^{+}_k(f) \in \mathbb{R} | \qquad f \in \mathbf{S}^{-}(\mathbb{R}) \} \end{equation}
and
\begin{equation}
Ker(\Psi^{+}_k) = \{f\in \mathbf{S}^{-}(\mathbb{R})| \qquad \Psi^{+}_k(f)=0 \} \end{equation}
Obviously, the null function ($f: \mathbb{R}\rightarrow 0$) belongs to $Ker(\Psi^{+}_k)$.
One can define also $Im(\Psi^{-}_k)$ and $Ker(\Psi^{-}_k)$ associated with the family of DEOs $\Psi^{-}_k$ ($k$ in $\mathbb{Z}$).
Now, let us define a subset $\mathbf{s}^{-}(\mathbb{R})$ $\subseteq$ $\mathbf{S}^{-}(\mathbb{R})$ such as:
\begin{eqnarray}
\mathbf{s}^{-}(\mathbb{R}) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \hspace{0.2em}\forall\hspace{0.2em} k \in \mathbb{Z}, \hspace{0.2em} \Psi^{+}_k(f) \neq \{0\} \hspace{0.2em} \nonumber \\
& & \hspace{0.2em} \forall\hspace{0.2em} k \in \mathbb{Z}-\{1\}, \hspace{0.2em}\Psi^{-}_k(f) \neq \{0\} \} \nonumber
\end{eqnarray}
Note that a possible way to define $\mathbf{s}^{-}(\mathbb{R})$ is:
\begin{equation}
\mathbf{s}^{-}(\mathbb{R}) = \{ f \in \mathbf{S}^{-}(\mathbb{R}) | f \notin (\cup_{k \in \mathbb{Z}} Ker(\Psi^{+}_k))\cup(\cup_{k \in \mathbb{Z}-\{1\}} Ker(\Psi^{-}_k))\} \end{equation}
The definition of the subset $\mathbf{s}^{-}(\mathbb{R})$ excludes $\Psi^{-}_1$ as by definition of this operator $Im(\Psi^{-}_1)$ equal $\{0\}$ for all $f$ in $\mathbf{S}^{-}(\mathbb{R})$.
Following Definition $1$, the \emph{uniqueness} of the decomposition in $\mathbf{s}^{-}(\mathbb{R})$ with the families of differential operators can be stated as:
\newline $\bold{Definition}$ $2$: for all $f$ in $\mathbf{s}^{-}(\mathbb{R})$, for all $v\in\mathbb{Z}^+-\{0\}$, for all $n\in\mathbb{Z}^+$ and $n>1$, the families of operators $(\Psi^{+}_k)_{k \in \mathbb{Z}}$ and $(\Psi^{-}_k)_{k \in \mathbb{Z}}$ ($(\Psi^{+}_k)_{k \in \mathbb{Z}}$ and $(\Psi^{-}_k)_{k \in \mathbb{Z}}$$\subseteq$ $\mathcal{F}(\mathbf{s}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R}))$) decompose uniquely $\partial_t^v$ $f^n$ in $\mathbb{R}$, if for any family of operators $(S_k)_{k \in \mathbb{Z}}$ $\subseteq$ $\mathcal{F}(\mathbf{S}^{-}(\mathbb{R}),\mathbf{S}^{-}(\mathbb{R})$) decomposing $\partial_t^v$$f^n$ in $\mathbb{R}$, there exists a unique couple $(\beta_1,\beta_2)$ in $\mathbb{R}^2$ such as:
\begin{equation} S_k(f) = \beta_1 \Psi^{+}_k(f) + \beta_2 \Psi^{-}_k(f), \qquad \forall k\in\mathbb{Z} \end{equation}
\\ The main goal of this work is to give a proof of the following lemma and theorem:
\\$\bold{Lemma}$: for $f$ in $\mathbf{S}^{-}(\mathbb{R})$, the family of DEO ${\Psi}_{k}^{+}$ ($k=\{0,\pm 1,\pm 2,...\}$) decomposes the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$.
\\$\bold{Theorem}$: for $f$ in $\mathbf{s}^{-}(\mathbb{R})$, the families of DEO ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$ ($k=\{0,\pm 1,\pm 2,...\}$) decompose uniquely the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$.
\\ The proofs of the lemma and theorem are given for the $n$-th power of $f$ with $n\in\mathbb{Z}^+$ and $n>1$. A discussion takes place for the special case $n=1$ and $n<0$. In addition, we assume a function $f$ of a variable $t$ with values in $\mathbb{R}$, which can be time or one of the dimension in space ($x,y,z$).
Note that in Section \ref{SectionProperties}, the study of the properties of the images helps to simplify the formulas shown in Lemma and Theorem.
\\ Finally, the last part is dedicated to applying the development in the previous sections to the energy function $\mathcal{E}$ defined as:
\begin{equation}\label{EnergyfunDefine02} \mathcal{E}(f_1(\tau)) = \int_a^{\tau} f_1(t)^2dt < \infty \end{equation}
with $a$ and $\tau$ in $\mathbb{R}$. $f_1$ is assumed to be in $\mathbf{S}^{-}(\mathbb{R})$ and analytic.
\section{Family of Energy Operators}
Following the general description in \cite{Kaiser90}, the general formula of the operator can be written as $P_{-}$ a bilinear form of $\mathbb{R}$ $ \times $ $\mathbb{R}$ $\rightarrow$ $\mathbb{R}$ defined in the real domain for all functions $f$ and $g$ in $C^{\infty}(\mathbb{R})$ as:
\begin{equation} {P}_{-}[f(t),g(t)]=\frac{1}{2}[ f(t)\partial_t g(t)+g(t)\partial_t f(t)]-\frac{1}{2}[f(t)\partial_t^2g(t)+\partial_t^2f(t)g(t)] \label{Equation Energy Operator in R} \end{equation}
Some years after, the authors in \cite{Maragos1995} introduced the $k$-th differential energy operator (DEO):
\begin{equation}\label{familyDEOdefinition} {\Psi}_{k}^{-}(f(t)) = \dot{f}(t){f}^{(k-1)}(t)- f(t) {f}^{(k)}(t), \qquad k \in \mathbb{Z} \end{equation}
Note that $k$ is the order of the operators. Here the derivation is following the variable $t$ and ${f}^{(k)}$ means $\partial_t^k f$. One can see that ${\Psi}_{k}^{-}$ is the quadratic form of the bilinear form $P_{-}$.
To explicitly define ${f}^{(k)}$ for all k in $\mathbb{Z}-\{0\}$:
\begin{eqnarray} f^{(k)}(t) &=& \partial_t^k f(t), \qquad \forall k \in \mathbb{Z}^+ -\{0\} \nonumber \\ f^{(k)}(t) &=& \int_{-\infty}^{t}(\hdots(\int_{-\infty}^{\tau_1}f(\tau_1)d\tau_1)...)d\tau_k, \qquad \forall k \in \mathbb{Z}^- -\{0\}\nonumber \\ f^{(k)}(t) &=& f(t), \qquad k = 0 \end{eqnarray}
With this definition it is important to underline that we are interested in the function such that:
\begin{eqnarray} \partial_t(\int_{-\infty}^{t}f(s)ds) &=& f(t) \nonumber \\ \int_{-\infty}^{t}\partial_{t}f(s)ds &=& f(t) \end{eqnarray}
This explains why we choose $f$ in the Schwartz space $\mathbf{S}^{-}(\mathbb{R})$.
Based on the definition given in \cite{JPMontillet2010}, one can define the DEO family in the same way:
\begin{equation}\label{Psik+defdefef} {\Psi}_{k}^{+}(f) = \dot{f}{f}^{(k-1)} + f {f}^{(k)}, \qquad \forall k \in \mathbb{Z} \end{equation}
${\Psi}_{k}^{+}$ is also a quadratic form with the same properties as ${\Psi}_{k}^{-}$. It is easy to show that the derivative chain rule in Equation \eqref{derivative chain rules0111} remains the same for the DEO family ${\Psi}_{k}^{+}$.
\section{Proof of the Lemma and Theorem}\label{LemmasProofs}
$\bold{Lemma}$: for $f$ in $\mathbf{S}^{-}(\mathbb{R})$, the family of DEO ${\Psi}_{k}^{+}$ ($k=\{0,\pm 1,\pm 2,...\}$) decomposes the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$.
\begin{proof}
The general proof is structured via an induction for the existence of the decomposition. First, the study of the four first derivatives of some selected powers of $f$ ($n=\{2, 3\}$) shows how the differential energy operators (or DEOs) are defined for each selected power of $f$. Furthermore, it gives a the method to find the energy operator family to decompose the successive derivatives. The case $n=p$ ($p>1$) ends the induction proof.
The non-uniqueness of the decomposition is justified in a separate paragraph with a counter example.
\newline A- $\bold{Existence}$ $\bold{of}$ $\bold{the}$ $\bold{Decomposition}$ $\bold{of}$ $\bold{the}$ $\bold{Power}$ $\bold{of}$ $f$ $\bold{via}$ $\bold{DEOs}$
\\ $\bold{Case}$ $n=2$:
\\Let $f$ be a function in $\mathbf{S}^{-}(\mathbb{R})$.
One can write the coefficient $\partial_t^v f^2$ for $v=\{1,2,3,4\}$ as:
\begin{eqnarray}\label{Coeff2n}
\partial_t f^2 &=& 2 f \partial_t f \nonumber \\ \partial_t f^2 &=& {\Psi}_{1}^{+}(f) \nonumber \\ & & \nonumber \\ \partial_t^2 f^2 &=& 2 (\partial_t f)^2 + 2 f \partial_t^2 f \nonumber \\ \partial_t^2 f^2 &=& \partial_t {\Psi}_{1}^{+}(f) \nonumber \\ \partial_t^2 f^2 &=& {\Psi}_{2}^{+}(f) + {\Psi}_{0}^{+}(\partial_t f) \nonumber \\ & & \nonumber \\ \partial_t^3 f^2 &=& {\Psi}_{3}^{+}(f) +2 {\Psi}_{1}^{+}(\partial_t f) + {\Psi}_{-1}^{+}(\partial_t^2 f) \nonumber \\ & & \nonumber \\ \partial_t^4 f^2 &=& {\Psi}_{4}^{+}(f) +3 {\Psi}_{2}^{+}(\partial_t f) +3 {\Psi}_{0}^{+}(\partial_t^2 f) + {\Psi}_{-2}^{+}(\partial_t^3 f) \nonumber \\
\end{eqnarray}
In this example, the successive coefficients are calculated using the derivative chain rule (e.g. Equation \eqref{derivative chain rules0111}). The scalars for each coefficient follow the Pascal's triangle rule and the order of the derivatives. Moreover, it is possible to develop the same type of Pascal's triangle rule to predict not only the scalar coefficients but also the order of the energy operators ($k$) involved for each derivative.
\begin{figure}
\caption{\footnotesize{Pascal's triangle rule for the energy operators}}
\label{Pascaletrainglerules}
\end{figure}
Figure \ref{Pascaletrainglerules} summarizes this general rule. It is straightforward to see that for the $v$-th derivative order, the highest DEO order is equal to $v$. Then, the remaining energy operator orders are calculated by decreasing the order by one for each previous energy operator involved in the ($v-1$)-th derivative. Secondly, it is important to see that all the DEOs involved to approximate the $v$-th derivative of $f^2$ are applied to the $s$-th derivatives of $f$ with $s=\{1,...,v-1\}$, except the DEO with the highest order (which is applied directly to $f$).
For simplicity using Equation \eqref{Coeff2n}, one can write ($\forall s \in \{1,2,...,m\}$):
\begin{eqnarray}\label{Aocorf1N2}
a_s^+(f) &=& \partial_t^s f^2 \nonumber \\ a_s^+(f) &=& \partial_t^{s-1} \Psi_{1}^{+} (f) \end{eqnarray}
$a_s^+$ is a sum of DEOs. The generalization of $a_s^+(f)$ is given via the formula:
\begin{eqnarray}\label{coefficientsAPgeneral} a_s^+(f) &=& \sum_{k=0}^{s-1} \big(_{k}^{s-1} \big) \Psi_{2(k+1)-s}^{+}(\partial_t^{s-k-1}f), \qquad \forall s\in \mathbb{Z}^{+}-\{0\} \end{eqnarray}
This formula can be checked for $s=\{1,2,3,4\}$ using Equation \eqref{Coeff2n}.
Let us assume the formula is true for $s=m$. One can write:
\begin{eqnarray}\label{coefficientsAPgeneralbis} a_{m}^+(f) &=& \partial_t a_{m-1}^+(f)\nonumber \\ a_{m}^+(f) &=& \partial_t \sum_{k=0}^{m-2} \big(_{k}^{m-2} \big) \Psi_{2(k+1)-m+1}^{+}(\partial_t^{m-k-2}f)\nonumber \\ a_{m}^+(f) &=& \sum_{k=0}^{m-2} \big(_{k}^{m-2} \big) [ \Psi_{2(k+1)-m+2}^{+}(\partial_t^{m-k-2}f) + \Psi_{2(k+1)-m}^{+}(\partial_t^{m-k-1}f)] \nonumber \\
\end{eqnarray}
Now using Equations \eqref{coefficientsAPgeneralbis} and the derivative chain rule property, the case $m+1$ is:
\begin{eqnarray}\label{coefficientsAPgeneralbis2} a_{m+1}^+(f) &=& \partial_t a_{m}^+(f)\nonumber \\ a_{m+1}^+(f) &=& \sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) [ \Psi_{2(k+1)-m+1}^{+}(\partial_t^{m-k-1}f) + \Psi_{2(k+1)-m-1}^{+}(\partial_t^{m-k}f)] \nonumber \\
a_{m+1}^+(f) &=& \sum_{k=0}^{m} \big(_{k}^{m} \big) \Psi_{2(k+1)-m-1}^{+}(\partial_t^{m-k}f)\nonumber \\ \end{eqnarray}
The above equations finish the induction proof for this particular case.
\newline $\bold{Case}$ $n=3$:
\\ In the same way as in the previous case, we write the development of the first coefficient:
\begin{eqnarray}\label{f3coef1}
\partial_t f^3 &=& 3 f^2 \partial_t f \nonumber \\
\end{eqnarray}
Let us introduce a DEO which is just a product of ${\Psi}_{k}^{+}$ by a constant as:
\begin{equation}\label{GammaOperatorfamily}
\Gamma_{k}^{+} (f) = \frac{3}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}), \qquad \forall k\in \mathbb{Z} \end{equation}
Clearly, by definition $\Gamma_{k}^{+}$ is a quadratic form and a DEO as it is proportional to ${\Psi}_{k}^{+}$. Note that the derivative properties shown in Equation \eqref{derivative chain rules0111} hold for this DEO. Using Equation \eqref{Aocorf1N2}, it is possible to write a similar equality with $a_k^{+}:$ $A_i^+(f) = 3/2 \partial_t^{i-1} a_i^+(f)$ (with $i$ in $\mathbb{Z}^{+}-\{0\}$). It is then possible to write the Equation \eqref{f3coef1} and the successive derivatives of $f^3$ as :
\begin{eqnarray}
\partial_t f^3 &=& f \Gamma_{1}^{+}(f) \\ \partial_t f^3 &=& f A_1^{+}(f) \nonumber \\
& & \nonumber \\
\partial^2_t f^3 &=& f (\Gamma_{2}^{+}(f) + \Gamma_{0}^{+}(\partial_t f)) + \partial_t f \Gamma_{1}^{+} (f) \nonumber \\
\partial^2_t f^3 &=& f A_2^{+}(f) + \partial_t f A_1^{+}(f) \\
& & \nonumber \\
\partial^3_t f^3 &=& f (\Gamma_{3}^{+}(f) + 2 \Gamma_{1}^{+}(\partial_t f) + \Gamma_{-1}^{+}(\partial^2_t f)) + 2 \partial_t f (\Gamma_{2}^{+}(f) + \Gamma_{0}^{+}(\partial_t f)) + \partial^2_t f \Gamma_{1}^{+} (f) \nonumber \\
\partial^3_t f^3 &=& f A_3^{+}(f) + 2 \partial_t f A_2^{+}(f) + \partial^2_t f A_1^{+}(f) \\
& & \nonumber \\
\partial^4_t f^3 &=& f (\Gamma_{4}^{+}(f) + 3 \Gamma_{2}^{+}(\partial_t f) + 3 \Gamma_{0}^{+}(\partial^2_t f) + \Gamma_{-2}^{+}(\partial^2_t f)) \nonumber \\ & & + 3 \partial_t f (\Gamma_{3}^{+}(f) + 2 \Gamma_{1}^{+}(\partial_t f) + \nonumber \\
&& \Gamma_{-1}^{+}(\partial^2_t f)) + 3 \partial^2_t f (\Gamma_{2}^{+}(f) + \nonumber \\
& & \Gamma_{0}^{+}(\partial_t f)) + \partial^3_t f \Gamma_{1}^{+} (f) \nonumber \\
\partial^4_t f^3 &=& f A_4^{+}(f) + 3 \partial_t f A_3^{+}(f) + 3 \partial^2_t f A_2^{+}(f) + \partial^3_t f A_1^{+}(f) \nonumber \\
\end{eqnarray}
\\ There is a certain symmetry between the above equations and Equations \eqref{Coeff2n}. The decomposition of $\partial_t^s f^3$ is performed using the DEO $\Gamma_{k}^{+}$. Using the equations above and the general definition of $A_s^{+}$ ($\forall s \in \mathbb{Z}^+ -\{0\}$) in Equation \eqref{coefficientsAPgeneral}, it is then possible to generalize the formula:
\begin{equation}\label{dtp3fwithAp} \partial_t^{m+1} f^3 =\sum_{k=0}^m \big(_{k}^m \big) A_{k+1}^+(f) \partial_t^{m-k} f, \qquad \forall m\in \mathbb{Z}^+ \end{equation}
This formula is checked with $m=\{0,1,2,3\}$. In order to finish the proof by induction, let us then assume the formula true for $m+1$. Following Equation \eqref{dtp3fwithAp} and the previous development, one can write:
\begin{eqnarray}\label{dtp3fwithAp1b} \partial_t^{m+1} f^3 &=& \partial_t (\partial_t^{m} f^3)\nonumber \\ \partial_t^{m+1} f^3 &=&\sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) [A_{k+2}^+(f) \partial_t^{m-k-1} f +A_{k+1}^+(f) \partial_t^{m-k} f] \end{eqnarray}
Finally using Equation \eqref{dtp3fwithAp1b}, the development for $m+2$ is:
\begin{eqnarray}\label{dtp3fwithAp02} \partial_t^{m+2} f^3 &=& \partial_t(\partial_t^{m+1} f^3)\nonumber \\ \partial_t^{m+2} f^3 &=& \partial_t (\sum_{k=0}^m \big(_{k}^m \big) A_{k+1}^+(f) \partial_t^{m-k} f) \nonumber \\ \partial_t^{m+2} f^3 &=& \sum_{k=0}^m \big(_{k}^m \big) [A_{k+2}^+(f) \partial_t^{m-k} f +A_{k+1}^+(f) \partial_t^{m-k+1} f] \nonumber \\ \partial_t^{m+2} f^3 &=& \sum_{k=0}^{m+1} \big(_{k}^{m+1} \big) A_{k+1}^+(f) \partial_t^{m-k+1} f \nonumber \\ \end{eqnarray}
This ends the induction proof to show Equation \eqref{dtp3fwithAp}, and also this particular case.
\\ $\bold{Case}$ $n=p$ with $p>1$:
\\ First, let us assume that the family of DEO ${\theta}_{k}^{+}$ ($k\in\mathbb{Z}$) proportional to the family of DEO ${\Psi}_{k}^{+}$ ($k\in\mathbb{Z}$) which decomposes the successive derivatives of ${f}^{p-1}$ with the definition:
\begin{eqnarray} {\theta}_{k}^{+}(f) &=& \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}) \end{eqnarray}
In addition, one can define $B_i^+(f) = \partial_t^{i-1} {\theta}_1^+(f)$ (with $i$ in $\mathbb{Z}^{+}-\{0\}$). Let us then write the first derivatives of ${f}^{p}$:
\begin{eqnarray}\label{NONOVO}
\partial_t f^p &=& p f^{p-1} \partial_t f \nonumber \\
\partial_t f^p &=& \frac{p}{2} f^{p-2} \Psi_1^+(f) \nonumber \\ \partial_t f^p &=& \frac{p}{p-1} {\theta}_{1}^{+}(f) f^{p-2} \nonumber \\ \partial_t f^p &=& \frac{p}{p-1} B_1^+(f) f^{p-2}\nonumber \\
& & \nonumber \\
\partial_t^2 f^p &=& \frac{p}{p-1} B_1^+(f) \partial_t f^{p-2}+ \frac{p}{p-1} B_2^+(f) f^{p-2} \nonumber \\
\partial_t^3 f^p &=& \frac{p}{p-1} B_1^+(f) \partial_t^2 f^{p-2}+ 2\frac{p}{p-1}B_2^+(f) \partial_t f^{p-2} \nonumber \\ & & + \frac{p}{p-1}B_3^+(f) f^{p-2} \end{eqnarray}
As shown in the previous case, we can generalize the formula for the $s$-th derivative as:
\begin{equation}\label{dtppfwithAps} \partial_t^{s+1} f^p =\sum_{k=0}^s \big(_{k}^s \big) \frac{p}{p-1}B_{k+1}^+(f) \partial_t^{s-k} f^{p-2}, \qquad \forall s \in \mathbb{Z}^+ \end{equation}
The formula in Equation \eqref{dtppfwithAps} has just been verified for $s=\{0,1,2\}$. Furthermore, one can write for the case $s=m$:
\begin{eqnarray}\label{dtp3fwithAp1b} \partial_t^{m+1} f^p &=& \partial_t (\partial_t^{m} f^p) \nonumber \\ \partial_t^{m+1} f^p &=&\frac{p}{p-1} \sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) [B_{k+2}^+(f) \partial_t^{m-k-1} f^{p-2} + B_{k+1}^+(f) \partial_t^{m-k} f^{p-2}] \nonumber \\ \end{eqnarray}
Let us assume that the Equation \eqref{dtppfwithAps} is true for $s=m+1$. Using the previous equation, the case $s=m+2$ is developed as:
\begin{eqnarray}\label{dtp3fwithAp02} \partial_t^{m+2} f^p &=& \frac{p}{p-1} \partial_t (\sum_{k=0}^m \big(_{k}^m \big) B_{k+1}^+(f) \partial_t^{m-k} f^{p-2}) \nonumber \\ \partial_t^{m+2} f^p &=& \frac{p}{p-1} \sum_{k=0}^v \big(_{k}^m \big) [B_{k+2}^+(f) \partial_t^{m-k} f^{p-2} +B_{k+1}^+(f) \partial_t^{m-k+1} f^{p-2}] \nonumber \\ \partial_t^{m+2} f^p &=& \frac{p}{p-1} \sum_{k=0}^{m+1} \big(_{k}^{m+1} \big) B_{k+1}^+(f) \partial_t^{m-k+1} f^{p-2} \nonumber \\ \end{eqnarray}
$(B_k^+)_{k\in \mathbb{Z}}$ and by definition, $(\theta_k^+)_{k\in \mathbb{Z}}$ decompose $\partial_t^{s} f^p$ ($s \in \mathbb{Z}^+$, $p\in \mathbb{Z}^+ -\{0,1\}$). This ends the induction proof to confirm Equation \eqref{dtppfwithAps}.
Finally as ${\theta}_{k}^{+}$ is proportional to ${\Psi}_{k}^{+}$, one can conclude that the family of DEO ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$) decomposes the successive derivatives of ${f}^{p}$ ($p \in \mathbb{Z}^+$, and $p> 1$). This ends the induction proof on the $n$-th power of $f$.
\newline B -$\bold{Uniqueness}$ $\bold{of}$ $\bold{the}$ $\bold{Decomposition}$
\\ In the previous proof, it has been shown that given a family of DEOs ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$), it is possible to decompose the successive derivatives $\partial_t^i f^n$ ($n\in \mathbb{Z}^+$, $n>1$, $i\in\mathbb{Z}^+$).
However, there is no uniqueness of the decomposition of $f^n$ with the DEO family ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$). A simple counter example can be found using the DEO family:
\begin{equation}\label{definitionOfeta}
\eta_k(f) = 3(\partial_t f \partial_t^{k-1} f)-f\partial_t^k f , \qquad \forall k \in \mathbb{Z} \end{equation}
Note that the derivative chain rule property is applied to this operator. One can verify:
\begin{eqnarray}\label{Coeff2nnn}
\partial_t f^2 &=& 2 f \partial_t f \nonumber \\ \partial_t f^2 &=& \eta_1(f) \nonumber \\ \eta_1(f)&=&{\Psi}_{1}^{+}(f) \nonumber \\ & & \nonumber \\ \partial_t^2 f^2 &=& 2 (\partial_t f)^2 + 2 f \partial_t^2 f \nonumber \\ \partial_t^2 f^2 &=& \partial_t {\eta}_{1}^{+}(f) \nonumber \\ \partial_t {\Psi}_{1}^{+}(f) &=& \partial_t {\eta}_{1}^{+}(f) \nonumber \\ \partial_t^2 f^2 &=& {\eta}_{2}^{+}(f) + {\eta}_{0}^{+}(\partial_t f) \nonumber \\ & & \nonumber \\ \partial_t^3 f^2 &=& \partial_t^2 {\eta}_{1}^{+}(f) \nonumber \\ \partial_t^2 {\Psi}_{1}^{+}(f) &=& \partial_t^2 {\eta}_{1}^{+}(f) \nonumber \\ \partial_t^3 f^2 &=& {\eta}_{3}^{+}(f) +2 {\eta}_{1}^{+}(\partial_t f) + {\eta}_{-1}^{+}(\partial_t^2 f) \nonumber \\ & & \nonumber \\ \partial_t^4 f^2 &=& \partial_t^3 {\eta}_{1}^{+}(f) \nonumber \\ \partial_t^3 {\Psi}_{1}^{+}(f) &=& \partial_t^3 {\eta}_{1}^{+}(f) \nonumber \\ \partial_t^4 f^2 &=& {\eta}_{4}^{+}(f) +3 {\eta}_{2}^{+}(\partial_t f) +3 {\eta}_{0}^{+}(\partial_t^2 f) + {\eta}_{-2}^{+}(\partial_t^3 f)
\end{eqnarray}
\end{proof}
It is important to underline the fact that $(a_k^+)_{k\in\mathbb{Z}^+-\{0\}}$, $(A_k^+)_{k\in\mathbb{Z}^+-\{0\}}$ and $(B_k^+)_{k\in\mathbb{Z}^+-\{0\}}$ are not included in the definitions $1$ and $2$ as they do not follow the derivative chain rule. In addition, $Definition$ $1$ can also be simplified fixing $l=2$ based on the definition of the energy operator in Equation \eqref{Psik+defdefef}.
\newline $\bold{Theorem}$: for $f$ in $\mathbf{s}^{-}(\mathbb{R})$, the family of DEOs ${\Psi}_{k}^{-}$ and ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$) decomposes uniquely the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$.
\begin{proof}
The proof is separated into three parts. The preliminary part justifies by induction the decomposition of the successive derivatives of the $n$-th power of $f$ for $n\in\mathbb{Z}^+$ and $n>1$ with ${\Psi}_{k}^{-}$ and ${\Psi}_{k}^{+}$ ($k\in \mathbb{Z}$). The proof is similar to the one in the Lemma, hence some parts are shorten to avoid long repetitions. \\The second section focuses on the existence of the decomposition as defined in Definition $1$ and Definition $2$. This is also shown by induction on the successive derivatives of $f^n$ ($n\in\mathbb{Z}^+$ and $n>1$) reusing the examples of different families of energy operator previously seen. Finally, the uniqueness is detailed in the last part.
\newline A- $\bold{Preliminary}$
\newline Recall the formula of ${\Psi}_{1}^{-}(f)$:
\begin{equation} {\Psi}_{1}^{-}(f) = \dot{f}f - f \dot{f} =0 \end{equation}
One can see from the definition of ${\Psi}_{1}^{-}$, that this DEO projects any smooth real-valued function $f$ (in $\mathbf{s}^{-}(\mathbb{R})$) onto the null space (${\Psi}_{1}^{-}(f)$: $\mathbb{R}\times\mathbb{R}$ $\rightarrow$ $0$).
For any $f$ in $\mathbf{s}^{-}(\mathbb{R})$,
\newline $\bold{Case}$ $n=2$:
\begin{eqnarray}\label{equationPsikm01} \partial_t f^2 &=& f \partial_t f + f\partial_t f + f\partial_t f - f\partial_t f \nonumber \\ \partial_t f^2 &=& {\Psi}_{1}^{+}(f) + {\Psi}_{1}^{-}(f)\nonumber \\ & & \nonumber \\ \partial_t^2 f^2 &=& 2 (\partial_t f)^2 + 2 f \partial_t^2 f \nonumber \\ \partial_t^2 f^2 &=& \partial_t ({\Psi}_{1}^{+}(f) +{\Psi}_{1}^{-}(f)) \nonumber \\ \partial_t^2 f^2 &=& {\Psi}_{2}^{+}(f) + {\Psi}_{0}^{+}(\partial_t f) + \nonumber \\
&& {\Psi}_{2}^{-}(f) + {\Psi}_{0}^{-}(\partial_t f) \nonumber \\ & & \nonumber \\ \partial_t^3 f^2 &=& ({\Psi}_{3}^{+}(f) + {\Psi}_{3}^{-}(f)) +2 ({\Psi}_{1}^{+}(\partial_t f)+{\Psi}_{1}^{-}(\partial_t f)) + \nonumber \\ & & ({\Psi}_{-1}^{+}(\partial_t^2 f) +{\Psi}_{-1}^{-}(\partial_t^2 f)) \nonumber \\ \end{eqnarray}
In this development, there is a symmetry with the proof of the previous lemma (e.g., Equation \eqref{Coeff2n}). Thus, this allows to shorten the proof by induction of the decomposition of the successive derivatives of the $n$-th power of $f$. Therefore, a necessary step as shown in the demonstration of the existence in the Lemma, is the demonstration of a similar formula for $\partial_t^k ({\Psi}_{1}^{-}(f))$ (e.g., Equation \eqref{coefficientsAPgeneralbis2}). It ends with the generalization for $n=p$.
\\ From Equations \eqref{equationPsikm01}, one can define $a_s^-(f)$ in the same way that $a_s^+(f)$ was defined in Equations \eqref{Aocorf1N2} as:
\begin{equation}\label{Aocorf1N3} a_s^-(f) = \partial_t^{s-1} \Psi_{1}^{-} (f), \qquad \forall s\in \mathbb{Z}^+-\{0\} \end{equation}
With the property of the derivative chain rule in Equation \eqref{derivative chain rules0111} and Equation \eqref{Aocorf1N2}, it is easy to calculate the first terms of the DEO family $a_s^-(f)$ such as :
\begin{eqnarray}\label{Coeff2nmoins}
a_1^-(f)={\Psi}_{1}^{-}(f) &=& 0 \nonumber \\ & & \nonumber \\ a_2^-(f)=\partial_t {\Psi}_{1}^{-}(f) &=& {\Psi}_{2}^{-}(f) + {\Psi}_{0}^{-}(\partial_t f) \nonumber \\ {\Psi}_{2}^{-}(f) &=& -{\Psi}_{0}^{-}(\partial_t f) \nonumber \\ & & \nonumber \\ a_3^-(f)=\partial_t^2 {\Psi}_{1}^{-}(f) &=& {\Psi}_{3}^{-}(f) +2 {\Psi}_{1}^{-}(\partial_t f) + {\Psi}_{-1}^{-}(\partial_t^2 f) \nonumber \\ {\Psi}_{3}^{-}(f) &=& -{\Psi}_{-1}^{-}(\partial_t^2 f) \nonumber \\ & & \nonumber \\ a_4^-(f)=\partial_t^3 {\Psi}_{1}^{-}(f) &=& {\Psi}_{4}^{-}(f) +3 {\Psi}_{2}^{-}(\partial_t f) +3 {\Psi}_{0}^{-}(\partial_t^2 f) + {\Psi}_{-2}^{-}(\partial_t^3 f) \nonumber \\ {\Psi}_{4}^{-}(f) &=& -{\Psi}_{-2}^{-}(\partial_t^3 f)
\end{eqnarray}
The above equations are similar to the development in Equation \eqref{Coeff2n}.
In addition, the family of DEO ${\Psi}_{k}^{-}$ ($k\in\mathbb{Z}$) has the same derivative properties as ${\Psi}_{k}^{+}$. A similar equation can then be established for $a_s^-(f)$ following the development written in Equation \eqref{Coeff2n} as:
\begin{eqnarray}\label{coefficientsAPgeneral2bis} a_s^-(f) &=& \sum_{k=0}^{s-1} \big(_{k}^{s-1} \big) \Psi_{2(k+1)-s}^{-}(\partial_t^{s-k-1}f), \forall s\in \mathbb{Z}^{+} -\{0\} \end{eqnarray}
and,
\begin{equation}
a_s^-(f) = \partial_t^s \Psi_1^{-}(f) =0, \qquad \forall s \in \mathbb{Z}^+-\{0\}, \qquad \forall f\in\mathbf{s}^{-}(\mathbb{R}) \end{equation}
This formula has just been checked for $s=\{1,2,3,4\}$ with the Equation \eqref{Coeff2nmoins}.
The generalization of the formula for $s=m$ is very similar to that described in the Equations \eqref{coefficientsAPgeneralbis} literally by changing $+$ and $-$ in the definition of the energy operator. It follows that the decomposition of the successive derivatives of $f^2$ is generalized for any $p$ in $\mathbb{Z}^+-\{0\}$ as:
\begin{eqnarray}\label{Coeff2nmoinspp} \partial_t^m f^2 &=& \partial_t^{m-1} ({\Psi}_{1}^{+}(f) +{\Psi}_{1}^{-}(f)) \nonumber \\
&=& \sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) \Psi_{2(k+1)-m}^{+}(\partial_t^{m-k-1}f) + \nonumber \\
& & \sum_{k=0}^{m-1} \big(_{k}^{m-1} \big) \Psi_{2(k+1)-m}^{-}(\partial_t^{m-k-1}f) \nonumber \\ \end{eqnarray}
In this case, $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ directly decompose $\partial_t^m f^2$. Note that the family $(a_k^-)_{k\in\mathbb{Z}^+-\{0\}}$ does not follow the derivative chain rule, and thus cannot be defined using definitions $1$ and $2$. In addition, the Equation \eqref{coefficientsAPgeneral2bis} can be easily extended for all $f$ in $\mathbf{S}^{-}(\mathbb{R})$ if we follow the induction proof in the Lemma (e.g., Equations \eqref{coefficientsAPgeneralbis}) as there is no restriction on the Images of the DEOs $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$.
\\$\bold{Case}$ $n=p$ with $p>1$:
\\ Following the same step as in the proof of the previous lemma, let us define the families of DEO ${\theta}_{k}^{-}$ proportional to the family of DEO ${\Psi}_{k}^{-}$ and ${\theta}_{k}^{+}$ proportional to the family of DEO ${\Psi}_{k}^{+}$ ($k\in\mathbb{Z}$) with the assumption that they decompose the successive derivatives of ${f}^{p-1}$ as:
\begin{eqnarray}\label{psi+0008} {\theta}_{k}^{+}(f) &=& \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}) \nonumber \\ &= & \frac{(p-1)}{2} \Psi_k^+(f) \nonumber \\ {\theta}_{k}^{-}(f) &= & \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} - f {f}^{(k)}) \nonumber \\ &= &\frac{(p-1)}{2} \Psi_k^-(f) \end{eqnarray}
Following the same development as in Equation \eqref{NONOVO}, one can define $B_i^+(f) = \partial_t^{i-1} {\theta}_1^+(f)$ and $B_i^-(f) = \partial_t^{i-1} {\theta}_1^-(f)$ (with $i$ in $\mathbb{Z}^{+}-\{0\}$). Let us then write the first derivatives of ${f}^{p}$ such as:
\begin{eqnarray}\label{NONOVO2}
\partial_t f^p &=& p f^{p-1} \partial_t f \nonumber \\
\partial_t f^p &=& \frac{p}{2} f^{p-2} (\Psi_k^+(f) +\Psi_k^-(f)) \nonumber \\
\partial_t f^p &=& \frac{p}{p-1} ({\theta}_{1}^{+}(f)+{\theta}_{1}^{-}(f)) f^{p-2} \nonumber \\ \partial_t f^p &=& \frac{p}{p-1} (B_1^+(f) + B_1^-(f)) f^{p-2} \nonumber \\
& & \nonumber \\
\partial_t^2 f^p &=& \frac{p}{p-1} (B_1^+(f)+B_1^-(f)) \partial_t f^{p-2}+ \frac{p}{p-1} (B_2^+(f)+B_2^-(f)) f^{p-2} )\nonumber \\
\end{eqnarray}
There is again a symmetry with the proof in the lemma. Following Equation \eqref{dtppfwithAps}, we can define the $s+1$-th derivative of $f^p$ using $B_{k+1}^-(f)$ and $B_{k+1}^+(f)$:
\begin{equation}\label{dtppfwithApsd2} \partial_t^{s+1} f^p = \sum_{k=0}^s \big(_{k}^s \big) \frac{p}{p-1}(B_{k+1}^-(f) +B_{k+1}^+(f)) \partial_t^{s-k} f^{p-2}, \qquad \forall s \in \mathbb{Z}^+ \end{equation}
This equation has just been checked for $s=\{0,1\}$. As the induction proof follows exactly the same development as in Equation \eqref{dtp3fwithAp02} by only adding $B_{k+1}^-(f)$ (with the same properties as $B_{k+1}^+(f)$ in Equation \eqref{NONOVO2}), it allows then to assume the generalization to the case $s+2$.
\newline Thus, $(B_{k}^+)_{k\in\mathbb{Z}}$ and $(B_{k}^-)_{k\in\mathbb{Z}}$ decompose the $s$-th derivative of $f^p$. From their definition, one can conclude that $(\theta_{k}^+)_{k\in\mathbb{Z}}$ and $(\theta_{k}^-)_{k\in\mathbb{Z}}$ decompose $\partial_t^sf^p$.
$({\theta}_{k}^{+})_{k\in\mathbb{Z}}$ and $({\theta}_{k}^{-})_{k\in\mathbb{Z}}$ are proportional to $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ respectively. Finally, $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ decompose the successive derivatives of ${f}^{p}$ ($p \in \mathbb{Z}^+$, and $p> 1$).
\newline B- $\bold{Existence}$ $\bold{of}$ $\bold{the}$ $\bold{Decomposition}$
\newline The proof is also structured as an induction on the $n$-th power of $f$. The different cases revisit some families of operator defined in the previous sections of this work (e.g., proof of the Lemma).
\\$\bold{Case}$ $n=2$:
It was shown that the family of operator $(\eta_k)_{k\in\mathbb{Z}}$ (see definition in the proof of the Lemma), decomposes $\partial^s_t f^{2}$ ($s\in\mathbb{Z}^+-\{0\}$). As defined in Equation \eqref{definitionOfeta}, one can rewrite it as a sum of the DEO family $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$ as:
\begin{equation} \eta_k(f) = \Psi_k^+(f) + 2 \Psi_k^-(f), \qquad k \in \mathbb{Z} \end {equation}
\\$\bold{Case}$ $n=3$:
In the lemma, the family of operator $(\Gamma_{k}^{+})_{k\in\mathbb{Z}}$ was defined in Equation \eqref{GammaOperatorfamily} and decomposes $\partial^s_t f^{3}$ ($s\in\mathbb{Z}^+-\{0\}$). One can rewrite the definition as:
\begin{eqnarray}\label{GammaOperatorfamilyPi}
\Gamma_{k}^{+} (f) &=& \frac{3}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}) \nonumber \\ \Gamma_{k}^{+} (f) &=& \frac{3}{2} \Psi_k^+(f) + 0 \Psi_k^-(f)
\end{eqnarray}
\\$\bold{Case}$ $n=p$ with $p>1$:
Previously, Equations \eqref{psi+0008} defined the operators ${\theta}_{k}^{+}$ and ${\theta}_{k}^{-}$ decomposing $\partial^{s}_t f^{p}$ ($s\in\mathbb{Z}^+-\{0\}$) and proportional to $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$ as:
\begin{eqnarray}\label{psi+000100} {\theta}_{k}^{+}(f) &=& \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} + f {f}^{(k)}) \nonumber \\ &= & \frac{(p-1)}{2} \Psi_k^+(f) \nonumber \\ {\theta}_{k}^{-}(f) &= & \frac{(p-1)}{2} (\dot{f}{f}^{(k-1)} - f {f}^{(k)}) \nonumber \\ &= &\frac{(p-1)}{2} \Psi_k^-(f) \end{eqnarray}
By induction it was also shown in the same section that ${\theta}_{k}^{-}$ and ${\theta}_{k}^{+}$ decompose $\partial^{s}_t f^{p+1}$ ($s\in\mathbb{Z}^+-\{0\}$). It is then possible to conclude the existence of the decomposition of any operator by using $({\Psi}_{k}^{-})_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^{+})_{k\in\mathbb{Z}}$.
\newline C- $\bold{About}$ $\bold{Uniqueness}$ $\bold{of}$ $\bold{the}$ $\bold{Decomposition}$
\\ With the previous section, let us show by induction the uniqueness of the decomposition of any family of operators decomposing $\partial_t^s f^n$ ($s$ in $\mathbb{Z}^+$, $n$ in $\mathbb{Z}^+$ and $n > 1$). The induction is focused on the proof of the uniqueness of the decomposition of a family of operator $({S}_{k})_{k\in\mathbb{Z}}$ (following the derivative chain rule property) by $({\Psi}_{k}^+)_{k\in\mathbb{Z}}$ and $({\Psi}_{k}^-)_{k\in\mathbb{Z}}$. In other words, the induction is on the $k$-th order of the operator.
\\$\bold{Case}$ $k=2$:
For $f$ in $\mathbf{s}^{-}(\mathbb{R})$ and $n$ in $\mathbb{Z}^+$ and $n > 1$, one can assume that $(\alpha_1,\alpha_2,\beta_1,\beta_2)$ exist in $\mathbb{R}^4$ such as:
\begin{eqnarray}\label{eqrefpartial0102}
\partial_t^s f^n &=& \partial_t^{s-1} (f^{n-2} S_1(f)) \nonumber \\ \partial_t^s f^n &=& \partial_t^{s-1} ( f^{n-2} (\alpha_1 \Psi^{+}_1(f) +\alpha_2 \Psi^{-}_1(f))) \nonumber \\ \partial_t^s f^n &=& \partial_t^{s-1} ( f^{n-2} (\beta_1 \Psi^{+}_1(f) +\beta_2 \Psi^{-}_1(f))) \nonumber \\ \end{eqnarray}
\\ As with the operator family $({S}_{k})_{k\in\mathbb{Z}}$ follows the derivative chain rule property:
\begin{eqnarray}\label{equationalphabeta1}
\partial_t S_1(f) &=& S_2(f) +S_0(\partial_t f)\nonumber \\ \partial_t S_1(f) &=&\alpha_1 \partial_t \Psi^{+}_1(f) +\alpha_2 \partial_t \Psi^{-}_1(f) \nonumber \\ \partial_t S_1(f) &=&\alpha_1 (\Psi^{+}_2(f) +\Psi^{+}_0(\partial_t f)) +\alpha_2 ( \Psi^{-}_2(f) +\Psi^{-}_0(\partial_t f)) \nonumber \\ \end{eqnarray}
And then,
\begin{eqnarray}\label{equationalphabeta1b} S_2(f) &=& \alpha_1 \Psi^{+}_2(f) +\alpha_2 \Psi^{-}_2(f) \nonumber \\ S_2(f) &=& \beta_1 \Psi^{+}_2(f) +\beta_2 \Psi^{-}_2(f) \nonumber \\ (\alpha_1-\beta_1) \Psi^{+}_2(f) +(\alpha_2-\beta_2) \Psi^{-}_2(f) &=& 0
\end{eqnarray}
As $Im(\Psi^{+}_2)$ and $Im(\Psi^{-}_2)$ are not reduced to $\{0\}$ by definition, it follows that $\alpha_1=\beta_1$ and $\alpha_2=\beta_2$. Note that it is not possible to do this simple check for $k=1$ as $Im(\Psi^{-}_1)=\{0\}$.
\\$\bold{Case}$ $k=p$:
Let us assume the uniqueness of the decomposition for $k=p-1$ (with $k\neq1$). For $k=p$, following Equation \eqref{equationalphabeta1}:
\begin{eqnarray}\label{equationalphabeta1kn}
\partial_t S_{p-1}(f) &=& S_p(f) +S_{p-2}(\partial_t f)\nonumber \\ \partial_t S_{p-1}(f) &=&\alpha_1 \partial_t \Psi^{+}_{p-1}(f) +\alpha_2 \partial_t \Psi^{-}_{p-1}(f) \nonumber \\ \partial_t S_{p-1}(f) &=&\alpha_1 (\Psi^{+}_p(f) +\Psi^{+}_{p-2}(\partial_t f)) +\alpha_2 ( \Psi^{-}_p(f) +\Psi^{+}_{p-2}(\partial_t f)) \nonumber \\ \end{eqnarray}
And then,
\begin{eqnarray}\label{equationalphabeta1knb}
S_p(f) &=& \alpha_1 \Psi^{+}_p(f) +\alpha_2 \Psi^{-}_p(f) \nonumber \\ S_p(f) &=& \beta_1 \Psi^{+}_p(f) +\beta_2 \Psi^{-}_p(f) \nonumber \\ (\alpha_1-\beta_1) \Psi^{+}_p(f) +(\alpha_2-\beta_2) \Psi^{-}_p(f) &=& 0
\end{eqnarray}
By definition for $p\neq1$, $Im(\Psi^{+}_p)$ and $Im(\Psi^{-}_p)$ are not reduced to $\{0\}$, and it follows that $\alpha_1=\beta_1$ and $\alpha_2=\beta_2$.
\\$\bold{Special}$ $\bold{Case}$ $k=1$:
To complete the proof with the assumption that $\alpha_1=\beta_1$ and $\alpha_2=\beta_2$ for $k \in\mathbb{Z}$ and $k \neq 1$ , the special case $k=1$ can be solved as:
\begin{eqnarray}\label{equationalphabeta1c} \partial_t(\alpha_1 \Psi^{+}_1(f)) &=&\alpha_1(\Psi^{+}_2(f) + \Psi^{+}_0(\partial_t f)) \nonumber \\ &=&\beta_1(\Psi^{+}_2(f) + \Psi^{+}_0(\partial_t f)) \nonumber \\ &=&\partial_t(\beta_1\Psi^{+}_1(f))\nonumber \\ & & \nonumber \\ \partial_t(\alpha_2 \Psi^{-}_1(f)) &=&\alpha_2(\Psi^{-}_2(f) + \Psi^{-}_0(\partial_t f)) \nonumber \\ &=&\beta_2(\Psi^{-}_2(f) + \Psi^{-}_0(\partial_t f)) \nonumber \\ &=&\partial_t(\beta_2\Psi^{-}_1(f))\nonumber \\ \nonumber\\ \end{eqnarray}
Thus, $\alpha_1=\beta_1$ and $\alpha_2=\beta_2$. Let us finish the proof with a remark for the case $\partial_t f^2$. Rewriting Equation \eqref{eqrefpartial0102} as:
\begin{eqnarray}
\partial_t f^2 &=& S_1(f) \nonumber \\ \partial_t f^2 &=& \alpha_1 \Psi^{+}_1(f) +\alpha_2 \Psi^{-}_1(f) \nonumber \\ \partial_t f^2 &=& \beta_1 \Psi^{+}_1(f) +\beta_2 \Psi^{-}_1(f) \nonumber \\ &=& \Psi^{+}_1(f) \end{eqnarray}
Thus, we conclude that in this case, $\alpha_1 = \beta_1 =1$.
\end{proof}
Note that in \cite{Boudraa et al.2009}, the authors based their work on the energy operator defined as $LP_2(f,g)=\partial_t f \partial_t g - f \partial_t^2 g$ ($LP_2$ : $\mathbb{R}^2$ $\rightarrow$ $\mathbb{R}$) which was then generalized to the complex set ($LP_2^{\mathbb{C}}(f,g)$). In the development of their work, they found a similar type of formula as found in Equation \eqref{coefficientsAPgeneral}, but restricted to the definition of their energy operator. In addition, one can underline that the generalization of the decomposition of the successive derivatives of the $n$-th power of $f$ with the DEOs ${(\Psi^{+}_k)}_{k\in \mathbb{Z}}$ and ${(\Psi^{-}_k)}_{k\in \mathbb{Z}}$ (e.g., Equations \eqref{coefficientsAPgeneral} and \eqref{Coeff2nmoinspp}) follows the general Leibniz derivative rules \cite{BruceWest}.
\newline $\bold{Discussion}$ $n<-1$: This case focuses on the decomposition using the DEO family of the quotient of the function $f$:($\mathbb{R}\rightarrow\mathbb{R}$) defined as:
\begin{equation} \forall f \in \mathbf{S}^{-}(\mathbb{R}), \qquad \forall t \in \mathbb{R}, \qquad f(t)\neq 0, \qquad \forall n \in \mathbb{Z}^+, n>1, \frac{1}{f^n}
\end{equation}
They are just a particular case of $f^n$ ($n>1$, $n\in \mathbb{Z}^+$). Using an intermediary function, $h$ such as $h = \frac{1}{f}$, the problem of decomposing $\partial_t^s f^{-n}$ ($s \in \mathbb{Z}^+$) is equivalent to resolving $\partial_t^s h^{n}$, which has been demonstrated in the Lemma and Theorem.
\newline $\bold{Discussion}$ $n=1$: This case does not make sense to decompose $f$ and its $k$-th derivative ($k\in\mathbb{Z}^+$) as a sum of energy operators based on the general definition of the DEO from \cite{Maragos1995}: the $k$-th order DEO is the cross energy between a function and its $k-1$ derivatives. However, one can use a general formula in the special case:
\begin{equation} \forall t\in \mathbb{R} , \qquad f\in \mathbf{S}^{-}(\mathbb{R}),\qquad f(t) \neq 0 \nonumber \end{equation}
\begin{eqnarray}\label{discussion2a} \partial_t^k f &=& \partial_t^k \big(\frac{f^3}{f^2}\big) \nonumber \\ k=1, \qquad \partial_t f &=& f^{-2} \partial_t f^3 + f^3 \partial_t f^{-2} \nonumber \\ k=2, \qquad \partial_t^2 f &=& 2 \partial_t f^{-2} \partial_t f^3 + f^3 \partial_t^2 f^{-2} + f^{-2} \partial_t^2 f^3 \end{eqnarray}
The example for $k =\{1,2\}$ in Equation \eqref{discussion2a} shows that $\partial_t f$ can be decomposed into a product of successive derivatives of $f^{3}$ and $f^{-2}$. Those derivatives can be decomposed into a sum of DEOs based on the development of the proof of the Lemma and the previous discussion (for $n<-1$ ).
\newline Note that if we restrict $f$ in $\mathbf{S}^{-}(\mathbb{R})$ and to be expandable in Taylor-Series in an interval $[a,b]$ in $\mathbb{R}$ such as:
\begin{equation} f(t) = \Sigma_{k=0}^{\infty} (\partial_t^k f(t_0)) \frac{(t-t_0)^k}{k!}, \qquad \forall t \in [a,b], \qquad t_0 \in [a,b] \end{equation}
One can then decompose the coefficients of the Taylor-Series of $f$ as a sum of DEO via the method shown in the Lemma and Theorem.
\section{Some Properties of the Images and Kernels of the DEO Families}\label{SectionProperties}
In this part, we study the relationships between $Im(\Psi^{+}_k)$ and $Im(\Psi^{-}_k)$, and $Ker(\Psi^{+}_k)$ and $Ker(\Psi^{-}_k)$ via the demonstration of the following properties.
\\$\mathbf{Properties}$ $1$: for $k$ in $\mathbb{Z}$ and $f$ in $\mathbf{S}^{-}(\mathbb{R})$, \begin{eqnarray}\label{Property01} a) \Psi^{+}_k(f) &=& \Psi^{+}_{-k+2}(\partial_t^{k-1}f) \nonumber \\ b) \Psi^{-}_k(f) &=& -\Psi^{-}_{-k+2}(\partial_t^{k-1}f) \nonumber \\ c) \Psi^{+}_k(f)+\Psi^{-}_k(f) &=& \Psi^{+}_{-k+2}(\partial_t^{k-1}f) -\Psi^{-}_{-k+2}(\partial_t^{k-1}f) \nonumber \\ d) Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k) & \subseteq & Im(\Psi^{+}_k - \Psi^{-}_k) \nonumber \\ e) Im(\Psi^{+}_k) \bigcap Im(-\Psi^{-}_k) & \subseteq & Im(\Psi^{+}_k + \Psi^{-}_k) \nonumber \\
\end{eqnarray}
\begin{proof} To show Equation (\ref{Property01}-a), the definition of $\Psi^{+}_{-k+2}(\partial_t^{k-1}f)$ is:
\begin{equation} \Psi^{+}_{-k+2}(f) = \partial_t f \partial_t^{-k+1}f + f \partial_t^{-k+2}f \end{equation}
and then,
\begin{eqnarray}\label{equationproperty01b} \Psi^{+}_{-k+2}(\partial_t^{k-1}f) &=& \partial_t^k f f + \partial_t^{k-1}f \partial_t f \nonumber \\ \Psi^{+}_{-k+2}(\partial_t^{k-1}f) &=& \Psi^{+}_k(f) \end{eqnarray}
This last equation then shows the first assertion $\Psi^{+}_k(f) = \Psi^{+}_{-k+2}(\partial_t^{k-1}f)$. Following the same development as in Equation \eqref{equationproperty01b}, one can derive directly Equation (\ref{Property01}-b) as:
\begin{eqnarray}\label{equationproperty01c} \Psi^{-}_{-k+2}(\partial_t^{k-1}f) &=& \partial_t^k f f - \partial_t^{k-1}f \partial_t f \nonumber \\ \Psi^{-}_{-k+2}(\partial_t^{k-1}f) &=& -\Psi^{-}_k(f) \end{eqnarray}
Equation (\ref{Property01}-c) is a consequence of Equation (\ref{Property01}-a) and Equation (\ref{Property01}-b) by simply replacing $\Psi^{+}_k(f)$ and $\Psi^{-}_k(f)$ by respectively $\Psi^{+}_{-k+2}(\partial_t^{k-1}f)$ and $-\Psi^{-}_{-k+2}(\partial_t^{k-1}f)$.
\newline Let us write the definition of $Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k) $ and $Im(\Psi^{+}_k - \Psi^{-}_k)$ such as:
\begin{eqnarray}
Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k) &=& \{\Psi^{+}_k(f) = \Psi^{-}_k(f) |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\
&=& \{ 2 f\partial_t^k f =0 |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\
Im(\Psi^{+}_k - \Psi^{-}_k) &=& \{\Psi^{+}_k(f) - \Psi^{-}_k(f) |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\
&=& \{ 2 f\partial_t^k f |f \in \mathbf{S}^{-}(\mathbb{R})\} \end{eqnarray}
$Im(\Psi^{+}_k - \Psi^{-}_k)$ is non-empty as it contains $0$ when $f$ is the null function. Thus, the above definitions show the inclusion of the subset $Im(\Psi^{+}_k) \bigcap Im(\Psi^{-}_k)$ into $Im(\Psi^{+}_k - \Psi^{-}_k)$. In the same way, one can show Equation (\ref{Property01}-e) through the definition of each subset.
\begin{eqnarray}
Im(\Psi^{+}_k) \bigcap Im(-\Psi^{-}_k) &=& \{\Psi^{+}_k(f) +\Psi^{-}_k(f) =0 |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\
Im(\Psi^{+}_k + \Psi^{-}_k) &=& \{\Psi^{+}_k(f) + \Psi^{-}_k(f) |f \in \mathbf{S}^{-}(\mathbb{R})\} \nonumber \\ \end{eqnarray}
As before, $0$ is included in $Im(\Psi^{+}_k + \Psi^{-}_k)$ (e.g. $\Psi^{+}_k (0) = \Psi^{-}_k (0) =0$, for $k$ in $\mathbb{Z}$). Thus, $Im(\Psi^{+}_k) \bigcap Im(-\Psi^{-}_k) \subset Im(\Psi^{+}_k + \Psi^{-}_k)$.
Note that Equations (\ref{Property01}-a), (\ref{Property01}-b) and (\ref{Property01}-c) are important as they are directly linked to $Im(\partial_t^p\Psi^{+}_1)$ and $Im(\partial_t^p\Psi^{-}_1)$ ($p$ in $\mathbb{Z}-\{0\}$).
\end{proof}
Similar properties can be shown from the definitions of $Ker(\Psi^{+}_k)$ and $Ker(\Psi^{-}_k)$:
\\$\mathbf{Properties}$ $2$: for $k$ in $\mathbb{Z}$ and $f$ in $\mathbf{S}^{-}(\mathbb{R})$, \begin{eqnarray}\label{Property02Kernel}
a) Ker(\Psi^{+}_k) \bigcap Ker(\Psi^{-}_k) & \subseteq & Ker(\Psi^{+}_k - \Psi^{-}_k) \nonumber \\ b) Ker(\Psi^{+}_k) \bigcap Ker(-\Psi^{-}_k) & \subseteq & Ker(\Psi^{+}_k + \Psi^{-}_k) \nonumber \\ c) Ker(-\Psi^{+}_k) &=& Ker(\Psi^{+}_k) \nonumber \\ d) Ker(-\Psi^{-}_k) &=& Ker(\Psi^{-}_k) \nonumber \\ \end{eqnarray}
\begin{proof}
The demonstration follows the definition of the kernels in the same way as in the previous properties:
\begin{eqnarray} \label{defKer01}
Ker(\Psi^{+}_k) \bigcap Ker(\Psi^{-}_k) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \qquad \Psi^{-}_k(f)=\Psi^{+}_k(f) =0\} \nonumber \\
\end{eqnarray}
\begin{eqnarray}\label{defKer02}
Ker(\Psi^{+}_k - \Psi^{-}_k) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{+}_k(f) -\Psi^{-}_k(f) =0\} \nonumber \\ \end{eqnarray}
And, \begin{eqnarray} \label{defKer01b}
Ker(\Psi^{+}_k) \bigcap Ker(-\Psi^{-}_k) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \qquad \Psi^{+}_k(f)=-\Psi^{-}_k(f) =0\} \nonumber\\ \end{eqnarray}
\begin{eqnarray}\label{defKer02b}
Ker(\Psi^{+}_k + \Psi^{-}_k) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{+}_k(f) +\Psi^{-}_k(f) =0\}\nonumber \\ \end{eqnarray}
It is necessary to underline that $Ker(\Psi^{+}_k)$, $Ker(\Psi^{-}_k)$ , $Ker(\Psi^{+}_k + \Psi^{-}_k)$ and $Ker(\Psi^{+}_k - \Psi^{-}_k)$ are all non empty sets as the null function is included in all of them. This then demonstrates Equations (\ref{Property02Kernel}-a) and (\ref{Property02Kernel}-b).
\newline Similarly we have,
\begin{eqnarray}\label{defKer03}
Ker(\Psi^{+}_k ) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{+}_k(f) = -\Psi^{+}_k(f) =0\} \nonumber \\
Ker(\Psi^{-}_k ) &=& \{f\in \mathbf{S}^{-}(\mathbb{R})| \Psi^{-}_k(f) = -\Psi^{-}_k(f) =0\} \nonumber \\ \end{eqnarray}
This directly shows Equations (\ref{Property02Kernel}-c) and (\ref{Property02Kernel}-d).
\end{proof}
Following the remarks at the end of Properties $1$, we can rewrite the formulas in Equations \eqref{coefficientsAPgeneral} based on the relationship of the images. With the Equation (\ref{Property01}-a), the formula \eqref{coefficientsAPgeneral} can be rewritten as:
\begin{eqnarray}\label{Coeff2nRewritten}
\partial_t {\Psi}_{1}^{+}(f) &=& {\Psi}_{2}^{+}(f) + {\Psi}_{0}^{+}(\partial_t f) \nonumber \\ \partial_t {\Psi}_{1}^{+}(f) &=& 2{\Psi}_{2}^{+}(f) \nonumber \\
\partial_t^2 {\Psi}_{1}^{+}(f) &=& {\Psi}_{3}^{+}(f) + 2{\Psi}_{1}^{+}(\partial_t f) + {\Psi}_{-1}^{+}(\partial_t^2 f) \nonumber \\ &=& 2{\Psi}_{3}^{+}(f) + 2{\Psi}_{1}^{+}(\partial_t f) \nonumber \\
\end{eqnarray}
Thus for $p$ in $\mathbb{Z}^{+}-\{0\}$, $k$ in $\mathbb{Z}$ and $s$ in in $\mathbb{Z}^+-\{0\}$,
\begin{equation*}\label{coefficientsAPgeneralRewrittenA}
a_p^+(f) =\left\{
\begin{array}{rl}
\Psi_{1}^{+}(f),& p=1 \nonumber \\ 2 \sum_{k > \frac{2s-1}{2}}^{2s-1} \big(_{k}^{2s-1} \big)\Psi_{2(k+1)-2s}^{+}(\partial_t^{2s-k-1}f), & p=2s \\ 2 \sum_{k > s}^{2s} \big(_{k}^{2s} \big) \Psi_{2(k+1)-2s-1}^{+}(\partial_t^{2s-k}f) + \big(_{s}^{2s} \big) \Psi_{1}^{+}(\partial_t^{s}f),& p=2s+1 \\ \end{array} \right . \end{equation*}
Consequently, the successive derivatives of $f^n$ ($n$ in $\mathbb{Z}^{+}-\{0\}$) can be decomposed with the DEO family $(\Psi_{k}^{+})$ with $k$ in $\mathbb{Z}^+-\{0\}$.
\newline In the same way following Equation (\ref{Property01}-b), Equation \eqref{coefficientsAPgeneral2bis} is simplified to:
\begin{equation}\label{coefficientsAPgeneralRewrittenA}
a_p^-(f) = 0 \qquad \forall p \in \mathbb{Z}^{+}-\{0\} \end{equation}
\section{Application to the Energy Function}
The energy function $\mathcal{E}$ considered in this section is the one previously defined in Equation \eqref{EnergyfunDefine02}.
$f_1$ and $f_1^2$ are considered to be in $\mathbf{S}^{-}(\mathbb{R})$, analytic and with a finite energy.
Note that the choice of this example is also based on the discussion (case $n=1$) after the Lemma.
\\ One can develop the energy function in Taylor-Series on the interval of definition $[0,\tau]$ ($\tau$ in $\mathbb{R}$) for a nominated $\tau_0$ in the defined interval,
\begin{eqnarray}\label{EquationEfunction} \mathcal{E}(f_1(\tau) ) &=& \mathcal{E}(f_1(\tau_0)) + \sum_{k=1}^\infty \partial_t^k \mathcal{E}(f_1(\tau_0)) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\
&=& \mathcal{E}(f_1(\tau_0)) + \sum_{k=1}^\infty \partial_t^{k-1} f_1^2(\tau_0) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\
&=& \mathcal{E}(f_1(\tau_0)) + \sum_{k=1}^\infty \partial_t^{k-1} f_1^2(\tau_0) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\
&=& \mathcal{E}(f_1(\tau_0)) +f_1^2(\tau_0) (\tau-\tau_0) + \sum_{k=2}^\infty \partial_t^{k-2} (\Psi_1^+(f_1)(\tau_0) \nonumber \\ & &+ \Psi_1^-(f_1)(\tau_0))\frac{(\tau-\tau_0)^k}{k!} \nonumber \\ &=& \mathcal{E}(f_1(\tau_0)) +f_1^2(\tau_0) (\tau-\tau_0) + \sum_{k=2}^\infty \partial_t^{k-2} \Psi_1^+(f_1)(\tau_0) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\ \end{eqnarray}
In the special case that the series is absolutely convergent, one can write for $p$ in $\mathbb{Z}^+-\{0\}$:
\begin{eqnarray}
\big| \frac{\partial_t^{p+1} \mathcal{E}(f_1(\tau_0)) }{\partial_t^{p} \mathcal{E}(f_1(\tau_0))} \big| \big| \frac{(\tau-\tau_0)}{p+1} \big|&<& 1 \nonumber \\ \end{eqnarray}
or for $p$ in $\mathbb{Z}^+-\{0,1\}$
\begin{eqnarray}\label{equationenergyseries1}
\big| \frac{\partial_t^{p+1} (\Psi_1^+(f_1)(\tau_0) }{\partial_t^{p} \Psi_1^+(f_1)(\tau_0) } \big| &<& \big| \frac{p+1}{(\tau-\tau_0)}\big| \end{eqnarray}
\newline Let us take an example with the periodic function :
\begin{equation}\label{funGt} g(t) = A \cos(t), \qquad t \in [-\pi,\pi], \qquad A \in \mathbb{R} \end{equation}
We are interested in the development in Taylor-Series of the energy of $g$ following Equation \eqref{EquationEfunction} with $\tau$ and $\tau_0$ in $[-\pi,\pi]$ such that:
\begin{eqnarray} \mathcal{E}(g(\tau))&=& \mathcal{E}(g(\tau_0)) +g^2(\tau_0) (\tau-\tau_0) + \sum_{k=2}^\infty \partial_t^{k-2} \Psi_1^+(g)(\tau_0) \frac{(\tau-\tau_0)^k}{k!} \nonumber \\ \end{eqnarray}
In addition, with the definition of $\Psi_1^+$ (e.g. Equation \eqref{Psik+defdefef}) one can write:
\begin{eqnarray}\label{cossinexample} \Psi_1^+(g(t)) &=& -2A \cos(t) \sin(t)\nonumber \\ \partial_t \Psi_1^+(g(t)) &=& -2A (\cos^2(t) - \sin^2(t)) \nonumber \\
&=& 2A (2\sin^2(t)-1) \nonumber \\ \partial_t^2 \Psi_1^+(g(t)) &=& 8A \sin(t) \cos(t) \nonumber \\ \partial_t^3 \Psi_1^+(g(t)) &=& -8A (2\sin^2(t)-1) \nonumber \\ \end{eqnarray}
One can deduce a general formula for the derivatives of $g$ from the above equations as:
\begin{equation} \forall k \in \mathbb{Z}^+,\left\{ \begin{array}{rcl}
\partial_t^{2k+1} \Psi_1^+(g(t)) &=& (-1)^{k+1} 2^{2k+1} A (\cos^2(t) - \sin^2(t))\\
\partial_t^{2k} \Psi_1^+(g(t)) &=& (-1)^{k+1} 2^{2k+1} A (\cos(t) \sin(t))\\ \end{array} \right. \end{equation}
From those equations and the general trigonometric properties of the functions $cosines$ and $sines$, the upper bound of $\partial_t^{p} \Psi_1^+(g(t))$ is:
\begin{equation} \forall k \in \mathbb{Z}^+,\left\{ \begin{array}{rcl}
\partial_t^{2k+1} \Psi_1^+(g(t)) &\leq& | 2^{2k+1} A | \\
\partial_t^{2k} \Psi_1^+(g(t)) &\leq& | 2^{2k+1} A | \\ \end{array} \right. \label{example02eq} \end{equation}
Let us now examine the convergence of the Taylor-Series of $\mathcal{E}(g)$ using the ratio test (see \cite{Kreyszig}) as described in Equation \eqref{equationenergyseries1}. One can write for $p$ in $\mathbb{Z}^+-\{0,1\}$:
\begin{eqnarray}
lim_{p\rightarrow + \infty}\big| \frac{\partial_t^{p+1} (\Psi_1^+(g)(\tau_0) }{\partial_t^{p} \Psi_1^+(g)(\tau_0) } \big| &=& \nonumber \\
lim_{p\rightarrow + \infty}\big| 2^2 \frac{(\tau-\tau_0)}{p+1}\big| &=& 0 \end{eqnarray}
Thus, the Taylor-Series of $\mathcal{E}(g)$ is absolutely convergent. Moreover, we can find the distances between the successive derivatives of $\partial_t^{p} \Psi_1^+(g)(\tau_0)$ with $\tau$ and $\tau_0$ in $[-\pi,\pi]$.
\begin{eqnarray}
|\partial_t^{p+1} (\Psi_1^+(g)(\tau_0)| - |\partial_t^{p} (\Psi_1^+(g)(\tau_0)| &=& \big( 2^2 \frac{|\tau-\tau_0|}{p+1} -1 \big) |\partial_t^{p} (\Psi_1^+(g)(\tau_0)| \nonumber \\
|\partial_t^{p+1} (\Psi_1^+(g)(\tau_0)| - |\partial_t^{2} (\Psi_1^+(g)(\tau_0)| & =& \sum_{i=2}^{p+1} \big( 2^2 \frac{|\tau-\tau_0|}{i+1} -1 \big) |\partial_t^{i} (\Psi_1^+(g)(\tau_0)|\nonumber \\
\end{eqnarray}
With Equations \eqref{example02eq}, the distance can be upper bounded as:
\begin{equation}
|\partial_t^{p+1} (\Psi_1^+(g)(\tau_0)| - |\partial_t^{2} (\Psi_1^+(g)(\tau_0)| \left\{ \begin{array}{rl}
\leq & \sum_{k=1}^{(p+1)/2} \big( 2^2 \frac{|\tau-\tau_0|}{2 k+1} -1 \big) | 2^{2k+1} A | , \qquad p =2k\\
\leq & \sum_{k=1}^{p/2} \big( 2^2 \frac{|\tau-\tau_0|}{2 k+2} -1 \big) | 2^{2k+1} A | , \qquad p= 2k+1 \end{array} \right. \end{equation}
Note that the distance for the case $p =\{0,1\}$ should be calculated with the equations in \eqref{cossinexample}.
\section{Conclusions}
This work defined two families of DEOs ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$($k \in \mathbb{Z}$).
A lemma and theorem were developed to decompose any function $f$ in $\mathbf{S}^{-}(\mathbb{R})$ and its $n$-th power ($n$ in $\mathbb{Z}$ and $n \neq 0$) using the DEO families ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$. In the demonstrations, some close-form formulas are demonstrated such as the decomposition of $f^2$ and $f^3$ with the DEO families ${\Psi}_{k}^{+}$ alone (Lemma) or with ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$ (Theorem) if $f$ is chosen in the subset $\mathbf{s}^{-}(\mathbb{R})$. In addition, the theorem shows the existence and uniqueness of the decomposition with the energy operator families, whereas the lemma only shows the existence of the decomposition when using only $({\Psi}_{k}^{+})_{k \in \mathbb{Z}}$. Note that the lemma and theorem justify the decomposition with the family of energy operators in the case of $f^n$ with $n \in \mathbb{Z}$ and $n> 1$. However, the discussions following the proof of the theorem deal with the cases $n <0$ and $n=1$. In the following section, the study of the images and kernels helped to simplify some of the formulas decomposing $f^2$ with ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$. The last part applied this to the energy function and how ${\Psi}_{k}^{+}$ and ${\Psi}_{k}^{-}$ can appear in the development in Taylor-Series of the energy function.
\newline This work is an extension of the recently published method to decompose the wave equation with energy operators and a necessary step to extend the method to other linear partial differential equations.
\section*{Acknowledgment}
A special thanks is addressed to Professor Alan McIntosh and Dr. Pierre Portal at the Centre for Mathematics and its Applications at the Australian National University for their inputs and discussions when writing this manuscript. The author also acknowledges the comments from Dr. Ryan Loxton from the Department of Mathematics and Statistics at Curtin University, and from Dr. Herb McQueen from the Research School of Earth Sciences at the Australian National University.
\end{document} | arXiv |
arXiv.org > math > arXiv:2105.00135
Mathematics > History and Overview
arXiv:2105.00135 (math)
[Submitted on 1 May 2021 (v1), last revised 15 Sep 2021 (this version, v4)]
Title:Exact and approximate solutions to the minimum of $1+x+\cdots+x^{2n}$
Authors:Aaron Hendrickson, Claude F. Leibovici
Abstract: The polynomial $f_{2n}(x)=1+x+\cdots+x^{2n}$ and its minimizer on the real line $x_{2n}=\operatorname{arg\,inf} f_{2n}(x)$ for $n\in\Bbb N$ are studied. Results show that $x_{2n}$ exists, is unique, corresponds to $\partial_x f_{2n}(x)=0$, and resides on the interval $[-1,-1/2]$ for all $n$. It is further shown that $\inf f_{2n}(x)=(1+2n)/(1+2n(1-x_{2n}))$ and $\inf f_{2n}(x)\in[1/2,3/4]$ for all $n$ with an exact solution for $x_{2n}$ given in the form of a finite sum of hypergeometric functions of unity argument. Perturbation theory is applied to generate rapidly converging and asymptotically exact approximations to $x_{2n}$. Numerical studies are carried out to show how many terms of the perturbation expansion for $x_{2n}$ are needed to obtain suitably accurate approximations to the exact value.
Comments: 12 pages, 2 figures
Subjects: History and Overview (math.HO); Classical Analysis and ODEs (math.CA)
MSC classes: 65H04 (Primary) 65Q30 (Secondary)
Cite as: arXiv:2105.00135 [math.HO]
(or arXiv:2105.00135v4 [math.HO] for this version)
From: Aaron Hendrickson [view email]
[v1] Sat, 1 May 2021 00:47:41 UTC (70 KB)
[v3] Fri, 14 May 2021 14:16:10 UTC (73 KB)
[v4] Wed, 15 Sep 2021 14:56:15 UTC (132 KB)
math.HO
math.CA | CommonCrawl |
Joseph L. Doob
Joseph Leo Doob (February 27, 1910 – June 7, 2004) was an American mathematician, specializing in analysis and probability theory.
Joseph L. "Joe" Doob
Tokyo, 1969 (courtesy MFO)
Born(1910-02-27)February 27, 1910
Cincinnati, Ohio, U.S.
DiedJune 7, 2004(2004-06-07) (aged 94)
Urbana, Illinois, U.S.
NationalityAmerican
Alma materHarvard University
Known forDoob's martingale inequality
Doob decomposition theorem
Scientific career
FieldsMathematician
InstitutionsUniversity of Illinois at Urbana-Champaign
Doctoral advisorJoseph L. Walsh
Doctoral students
• Warren Ambrose
• David Blackwell
• Yuan-Shih Chow
• Paul Halmos
• J. Laurie Snell
• Mary E. Thompson
InfluencesAndrey Kolmogorov
The theory of martingales was developed by Doob.
Early life and education
Doob was born in Cincinnati, Ohio, February 27, 1910, the son of a Jewish couple, Leo Doob and Mollie Doerfler Doob. The family moved to New York City before he was three years old. The parents felt that he was underachieving in grade school and placed him in the Ethical Culture School, from which he graduated in 1926. He then went on to Harvard where he received a BA in 1930, an MA in 1931, and a PhD (Boundary Values of Analytic Functions, advisor Joseph L. Walsh) in 1932. After postdoctoral research at Columbia and Princeton, he joined the department of mathematics of the University of Illinois in 1935 and served until his retirement in 1978. He was a member of the Urbana campus's Center for Advanced Study from its beginning in 1959. During the Second World War, he worked in Washington, D.C., and Guam as a civilian consultant to the Navy from 1942 to 1945; he was at the Institute for Advanced Study for the academic year 1941–1942[1] when Oswald Veblen approached him to work on mine warfare for the Navy.
Work
Doob's thesis was on boundary values of analytic functions. He published two papers based on this thesis, which appeared in 1932 and 1933 in the Transactions of the American Mathematical Society. Doob returned to this subject many years later when he proved a probabilistic version of Fatou's boundary limit theorem for harmonic functions.
The Great Depression of 1929 was still going strong in the thirties and Doob could not find a job. B.O. Koopman at Columbia University suggested that statistician Harold Hotelling might have a grant that would permit Doob to work with him. Hotelling did, so the Depression led Doob to probability.
In 1933 Kolmogorov provided the first axiomatic foundation for the theory of probability. Thus a subject that had originated from intuitive ideas suggested by real life experiences and studied informally, suddenly became mathematics. Probability theory became measure theory with its own problems and terminology. Doob recognized that this would make it possible to give rigorous proofs for existing probability results, and he felt that the tools of measure theory would lead to new probability results.
Doob's approach to probability was evident in his first probability paper,[2] in which he proved theorems related to the law of large numbers, using a probabilistic interpretation of Birkhoff's ergodic theorem. Then he used these theorems to give rigorous proofs of theorems proven by Fisher and Hotelling related to Fisher's maximum likelihood estimator for estimating a parameter of a distribution.
After writing a series of papers on the foundations of probability and stochastic processes including martingales, Markov processes, and stationary processes, Doob realized that there was a real need for a book showing what is known about the various types of stochastic processes, so he wrote the book Stochastic Processes.[3] It was published in 1953 and soon became one of the most influential books in the development of modern probability theory.
Beyond this book, Doob is best known for his work on martingales and probabilistic potential theory. After he retired, Doob wrote a book of over 800 pages: Classical Potential Theory and Its Probabilistic Counterpart.[4] The first half of this book deals with classical potential theory and the second half with probability theory, especially martingale theory. In writing this book, Doob shows that his two favorite subjects, martingales and potential theory, can be studied by the same mathematical tools.
The American Mathematical Society's Joseph L. Doob Prize, endowed in 2005 and awarded every three years for an outstanding mathematical book, is named in Doob's honor.[5] The postdoctoral members of the department of mathematics of the University of Illinois are named J L Doob Research Assistant Professors.
Honors
• President of the Institute of Mathematical Statistics in 1950.
• Elected to National Academy of Sciences 1957.
• President of the American Mathematical Society 1963–1964.
• Elected to American Academy of Arts and Sciences 1965.
• Associate of the French Academy of Sciences 1975.
• Awarded the National Medal of Science by the President of the United States Jimmy Carter 1979.[6]
• Awarded the Steele Prize by the American Mathematical Society. 1984.
Publications
Books
• — (1953). Stochastic Processes. John Wiley & Sons. ISBN 0-471-52369-0.[7]
• — (1984). Classical Potential Theory and Its Probabilistic Counterpart. Berlin Heidelberg New York: Springer-Verlag. ISBN 3-540-41206-9.[8]
• — (1993). Measure Theory. Berlin Heidelberg New York: Springer-Verlag.[9]
Articles
• Joseph Leo Doob (1 June 1934). "Stochastic Processes and Statistics". Proceedings of the National Academy of Sciences of the United States of America. 20 (6): 376–379. Bibcode:1934PNAS...20..376D. doi:10.1073/PNAS.20.6.376. ISSN 0027-8424. PMC 1076423. PMID 16587907. Wikidata Q33740310.
• — (1934). "Probability and statistics". Transactions of the American Mathematical Society. American Mathematical Society. 36 (4): 759–775. doi:10.2307/1989822. JSTOR 1989822.
• — (1957). "Conditional brownian motion and the boundary limits of harmonic functions" (PDF). Bulletin de la Société Mathématique de France. 85: 431–458. doi:10.24033/bsmf.1494.
• — (1959). "A non probabilistic proof of the relative Fatou theorem" (PDF). Annales de l'Institut Fourier. 9: 293–300. doi:10.5802/aif.93.
• — (1962). "Boundary properties of functions with finite Dirichlet integrals" (PDF). Annales de l'Institut Fourier. 12: 573–621. doi:10.5802/aif.126.
• — (1963). "Limites angulaires et limites fines" (PDF). Annales de l'Institut Fourier. 13 (2): 395–415. doi:10.5802/aif.152.
• — (1965). "Some classical function theory theorems and their modern versions" (PDF). Annales de l'Institut Fourier. 15 (1): 113–135. doi:10.5802/aif.200.
• — (1967). "Erratum: Some classical function theory theorems and their modern versions" (PDF). Annales de l'Institut Fourier. 17 (1): 469. doi:10.5802/aif.264.
• — (1973). "Boundary approach filters for analytic functions" (PDF). Annales de l'Institut Fourier. 23 (3): 187–213. doi:10.5802/aif.476.
• — (1975). "Stochastic process measurability conditions" (PDF). Annales de l'Institut Fourier. 25 (3–4): 163–176. doi:10.5802/aif.577.
See also
• Martingale (probability theory)
• Doob–Dynkin lemma
• Doob martingale
• Doob's martingale convergence theorems
• Doob's martingale inequality
• Doob–Meyer decomposition theorem
• Optional stopping theorem
Notes
1. Doob, Joseph Leo, Community of Scholars Profile, IAS Archived 2013-10-10 at the Wayback Machine
2. J.L. Doob Probability and statistics
3. Doob J.L., Stochastic Processes
4. Doob J.L., Classical Potential Theory and Its Probabilistic Counterpart
5. Joseph L. Doob Prize. American Mathematical Society. Accessed September 1, 2008
6. National Science Foundation – The President's National Medal of Science
7. Chung, K. L. (1954). "Review of Stochastic processes by J. L. Doob". Bull. Amer. Math. Soc. 60: 190–201. doi:10.1090/S0002-9904-1954-09801-4.
8. Meyer, P. A. (1985). "Review of Classical potential theory and its probabilistic counterpart by J. L. Doob". Bull. Amer. Math. Soc. (N.S.). 12: 177–181. doi:10.1090/S0273-0979-1985-15340-6.
9. Meyer, P. A. (1994). "Review of Measure theory by J. L. Doob". Bull. Amer. Math. Soc. (N.S.). 31: 233–235. doi:10.1090/S0273-0979-1994-00541-5.
External links
• Joseph L. Doob at the Mathematics Genealogy Project
• A Conversation with Joe Doob Archived 2018-06-22 at the Wayback Machine
• Doob biography
• Record of the Celebration of the Life of Joseph Leo Doob
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| Wikipedia |
Signal quality of simultaneously recorded endovascular, subdural and epidural signals are comparable
Sam E. John ORCID: orcid.org/0000-0003-3780-22101,2,3,7,
Nicholas L. Opie2,3,7,
Yan T. Wong1,6,
Gil S. Rind2,3,7,
Stephen M. Ronayne2,3,7,
Giulia Gerboni1,2,3,
Sebastien H. Bauquier5,
Terence J. O'Brien2,3,
Clive N. May3,
David B. Grayden ORCID: orcid.org/0000-0002-5497-72341,4 &
Thomas J. Oxley2,3,7
Scientific Reports volume 8, Article number: 8427 (2018) Cite this article
Neural decoding
Sensors and biosensors
An Author Correction to this article was published on 27 November 2018
This article has been updated
Recent work has demonstrated the feasibility of minimally-invasive implantation of electrodes into a cortical blood vessel. However, the effect of the dura and blood vessel on recording signal quality is not understood and may be a critical factor impacting implementation of a closed-loop endovascular neuromodulation system. The present work compares the performance and recording signal quality of a minimally-invasive endovascular neural interface with conventional subdural and epidural interfaces. We compared bandwidth, signal-to-noise ratio, and spatial resolution of recorded cortical signals using subdural, epidural and endovascular arrays four weeks after implantation in sheep. We show that the quality of the signals (bandwidth and signal-to-noise ratio) of the endovascular neural interface is not significantly different from conventional neural sensors. However, the spatial resolution depends on the array location and the frequency of recording. We also show that there is a direct correlation between the signal-noise-ratio and classification accuracy, and that decoding accuracy is comparable between electrode arrays. These results support the consideration for use of an endovascular neural interface in a clinical trial of a novel closed-loop neuromodulation technology.
The endovascular neural interface, known as the StentrodeTM, provides a minimally-invasive method for recording brain signals and, potentially, stimulating cortical tissue without the need for risky, open-brain surgery1. Methods to achieve endovascular brain recordings have progressed significantly from the earliest use of a wire in a cerebral blood vessel2, to a catheter mounted device3,4 and recently to the development of the Stentrode device1,5. Chronically-implantable endovascular devices are a promising method to achieve brain recordings without the need for craniotomy.
The minimally-invasive nature of implantation makes the endovascular (EV) approach desirable for use as a brain-machine interface (BMI). Previous applications of BMI based on cortical surface brain recording have used subdural (SD) arrays, which are placed under the dura, or epidural (ED) arrays, which are placed above the dura6,7,8,9,10,11,12,13. Despite many successful studies9,14,15,16, SD and ED devices require a craniotomy for implantation and are associated with a risk of infection, surgical complications, and mortality17. The EV array avoids the use of a craniotomy while still recording surface potentials from the brain. However, the clinical significance, quality, and efficacy of signals recorded is not clearly understood.
EV neural interfaces are delivered to the target area via cortical blood vessels using concentric catheters1,3,4,5,18,19,20,21,22,23,24,25,26. Studies have shown that neural signals can be recorded from microwires, catheter mounted electrodes, wire mounted electrodes, or stent mounted electrodes. A thorough review of endovascular technology was provided by Sefcik et al.5. The first EV neural interface consisted of a 0.6 mm electrode mounted on a guidewire tip2. This was followed by several short reports of similar recordings with guide wires3,5,27,28,29,30,31,32,33. The next major advance was two decades later, also recording with a microwire31, closely followed by a landmark study showing the feasibility of a multi-channel EV array with 16 electrodes3. Another recent study in the field showed recordings from a nanowire electrode array (0.6 µm diameter) in a capillary32. Most studies prior to 2016 were performed in humans undergoing surgery or in animal models acutely and therefore only lasted a few hours. The next major study was published in 2016 with electrodes mounted on a self-expanding stent1. This study was the first to show the ability to chronically implant a stent mounted electrode array into a blood vessel and record neural information over periods up to 6 months. Endovascular technology has progressed significantly from the earliest use of a wire in the brain to record brain signal to catheter mounted devices and now stent mounted devices5. In the last 10 years, there have been few reports of brain signal recording from acute implantation of catheters or wires in a cortical blood vessel4,18,20,21,22,33. Two studies, He et al.18 and Bower et al.4 also evaluated the signal quality of the recordings using electrodes acutely placed in blood vessels in acute implantation lasting a few hours and added significantly to the field.
Bower et al.4 showed for the first time that microelectrodes (40 µm diameter) could record cortical signals from within a blood vessel. Using a porcine model, a catheter-based electrode array was placed into the superior sagittal sinus (SSS) via a small incision in the vein. High amplitude spikes (>0.5 mV) were generated using penicillin injections into the cortex and acute recordings from an endovascular catheter array (macro ring electrode and micro disc electrodes) were compared to subdural arrays. The authors noted that epileptiform spikes from 40 µm disc electrodes and a 1 mm wide ring (other dimensions not reported) placed endovascularly had the same amplitudes as 40 µm and 2 mm disc electrodes placed subdurally. They also noted spatially localized 'microspikes' recorded by the SD and EV microelectrode arrays, but not on the macro arrays. However, the paper did not quantify the SNR of the recording or the spatial resolutions obtainable by the electrodes. Typical oscillations in the brain recorded by subdural and epidural arrays range in the order of 10–500 uV34 while, during an epileptic event, the synchronized high amplitude signals may not be easily differentiated. The high amplitude spikes would also make it difficult to quantitatively differentiate recording properties between electrode sizes or the effect of the tissue surrounding the electrodes. It is noteworthy that similar qualitative patterns were noted on both microarrays that were different from macroarrays. The study showed the feasibility of electrodes within a blood vessel in recording epileptiform spiking, leading to the conclusion that endovascular arrays would be useful in recording neural signals toward localization of epileptogenic foci.16
He et al.18 used a guidewire electrode similar to previous studies5 in a porcine model and showed that guidewire electrode recording quality defined by the SNR of auditory and visual evoked potentials was superior to (scalp) electroencephalography (EEG). The comparatively superior SNR of guidewire electrode recording was not surprising as the skull is a strong attenuator of cortical signals and the guidewire electrodes were under the skull. It is more important to evaluate whether endovascularly placed electrodes are comparable to SD and ED electrodes1,4 which are all implanted under the skull and closer to the brain than EEG. He et al. found a dependence of SNR on location of the wire; however, the spatial resolution was not quantified. Interpolation of the figures appears to indicate spatial resolution in the order of 10's of millimeters, though it would be expected that the spatial resolution of endovascular electrodes would be close to that of subdural and epidural arrays in the order of 2–6 mm.
While both Bower et al.4 and He et al.18 showed some quantification of signal quality, both studies used catheters or guidewire mounted electrodes to perform recordings5. Furthermore, both studies show recordings over an acute implantation period of a few hours. Until now, only Oxley et al.1 has demonstrated a stent-based device that can be chronically implanted into the blood vessel and record neural activity. Furthermore, a chronic six month study1,23,25 used electrodes opposing the blood vessel wall, which showed that the process of incorporation of the electrode takes approximately 14 days and recordings before this time were highly variable and, in some cases, not differentiable from noise. To date, the stent mounted technique is the most feasible technique for chronic implantation and recording, making it possible to envision multiple applications of the minimally invasive EV neural interface1,5,35.
In our previous work1, we compared the signal bandwidths and artefacts of the first generation chronically implantable stent-based EV device with macro SD and ED electrodes. The SD and ED electrodes used in the study were much larger than the EV electrodes, leading to a skewed view toward the larger electrode sizes. In this previous work, we alluded to the potential of high spatial resolution with the EV arrays, but this was not evaluated. Furthermore, the SNR and the effect of noise on the signals were not investigated. Further work is required to understand the clinical utility and to establish comparability to conventional electrodes to be considered as a feasible method neural interfacing.
In the present manuscript, we evaluate the SNR and the ability to detect a signal with the EV array (Fig. 1a) in comparison to SD and ED arrays (Fig. 1b). Measurements were made three weeks after implantation, providing sufficient time for incorporation of the devices into the blood vessel wall. In all previous studies except Oxley et al. measurements were made within minutes/hours after implantation and, therefore, prior to incorporation of the electrodes into the tissue. Efficacy of brain recordings, such as those obtained using EV, SD, or ED arrays, can be characterized by the recording bandwidth, signal-to-noise ratio (SNR), and spatial resolution. The bandwidth provides an estimate of the maximum frequency range over which neural information can be used for useful interpretation of the brain signals. The SNR is a critical feature of any clinical neural interface as it has a strong correlation with decoding performance for a BMI36,37,38. The spatial resolution achievable with any given device demonstrates its ability to record spatially localized activity. Arrays with better spatial resolution can record spatially specific information that is vital in accurately decoding movement intent in a BMI36,39,40.
Implanted devices, their placement, and example recordings. (a) EV array. (b) ED/SD array (Cortec gmbh, Freiburg, Germany). (c) X-ray image of the ED array (left), EV array (middle, implanted in the superior sagittal sinus), and SD array (right) implanted in a sheep brain. R, rostral. (d) MRI reconstruction of the sheep brain with the major veins shown. The box region shows the implantation site of the arrays, with the superior sagittal sinus marked in green. All scale bars are 5 mm. Averaged electrically evoked potentials were obtained by stimulating the median nerve at different current levels using cathodal monophasic constant current pulse, while simultaneously recording from the (e) SD, (f) EV, and (g) ED arrays. Red * indicates threshold level of stimulation.
In the present study, we systematically investigated the effects of electrode size and location on the signal bandwidth, sensitivity, SNR, spatial profile, and ability to decode the recordings. We compared the signal quality of recordings obtained with EV arrays to those from conventional SD and ED arrays implanted in sheep. The results demonstrate that that the EV array is a suitable candidate to decode neural information that may be used in a BMI.
Figure 1 shows the electrode arrays used in this study (Fig. 1a,b), placed in the superior sagittal sinus of sheep (Fig. 1c,d). Example recordings during median nerve stimulation are shown in for SD, EV and ED arrays in Fig. 1e–g respectively. The red asterixis shows where a discernible response to the stimulation of the median nerve was detected. The evoked potential waveforms of Fig. 1(e–g) showed differences in the waveform shapes when visually assessed. The differences in the shapes were not consistent across all animals and, since electrode positions across experiments varied in each animal, the waveform shapes were thus not assessed. However, waveform shapes may hold additional information regarding the underlying neural of the response and would be better suited to be addressed in an animal model that is better understood, with comprehensive literature and understanding of the structure and function of cerebral cortex.
Comparison of bandwidths of SD, EV and ED arrays
The bandwidth of surface local field potentials (LFP), such as those that are recorded by the SD, EV, and ED arrays, have been reported to be less than 500 Hz1,34,41. The limit of the amount of information that can be recorded is thought to be related to the distance between the recording electrodes and the target neurons and to the sizes of the electrodes36,37. The bandwidth of recorded signals provides an estimate of the quantity of information that can be obtained using the SD, EV, and ED arrays. Figure 2a shows frequency spectra and bandwidth estimations from representative electrodes of the three arrays from baseline recordings in awake, resting animals. The power spectra in Fig. 2a show that the differences in the powers were evident across all frequency bands and not limited to the frequencies representing the noise. The raw power spectra were not normalized for comparison of the maximum bandwidth.
Electrode bandwidths. (a) Frequency spectra from representative 500 µm SD, EV, and ED electrodes, displaying characteristic (1/f) frequency responses. Band powers were calculated in individual 2 s windows using the Thompson multitaper method with a centre frequency of 1 Hz (2 Hz resolution). Dashed vertical lines and numbers indicate calculated maximum bandwidths. Grey bars indicate respective noise floors. (b) Maximum bandwidths for ED, EV, and SD arrays for electrode sizes 500, 750, and 1000 µm. Circles show individual values, centre lines show mean values, error bars show standard error of the mean. Two-way ANOVA showed no significant effect of either the array location (p = 0.75) or electrode size (p = 0.15) on the bandwidth.
Figure 2b shows bandwidths measured from each array and with each electrode size. Bandwidth from all arrays were normally distributed and showed large variability in bandwidth across electrodes. Two-way ANOVA showed no significant effect of either the array location (F(1,147) = 0.1, p = 0.75) or electrode size (F(1,147) = 2.08, p = 0.15) on the bandwidth. There was no significant interaction between recording location and size of electrode (F(3,147) = 0.38, p = 0.76). In our previous study1, the SD and ED electrodes were larger in size (4 mm diameter) than the EV (0.75 mm diameter), which possibly influenced the bandwidths recorded42.
Single trial signal-to-noise ratios of SD, EV and ED arrays
The efficacy of neural recording in detecting neural events and decoding activity is improved with greater signal amplitude relative to background noise. Therefore, signal-to-noise ratio (SNR) is a useful measure of signal quality, where SNR = 1 indicates equal signal and noise levels. For a BMI, it is important that the SNR is as high as possible to ensure recordings yield high decoding accuracy43. Figure 3a shows change in SNR of 750 µm electrodes with different numbers of trials included in the average response. Slope here shows the rate of change of the SNR with number of trials. Figure 3b shows the single trial SNR, Fig. 3c shows the SNR calculated on the signal after 10 consecutive trials were averaged together, and Fig. 3d shows the slope of the fits in Fig. 3a for all arrays and animals.
Signal-to-noise ratios (SNRs). (a) Example of SNRs vs. number of trials (repetitions of stimuli) for the ED (green), EV (purple), and SD (orange) 750 µm diameter electrodes. Example traces were taken from electrodes closest to each other as determined on an x-ray image. Shaded areas indicate standard error of the mean. The lines are straight line fits (y = P*x + Q) for each array, where x is the number of trials, P is the slope of fit line, and Q is the intercept. (b) Single trial SNR. (c) SNR averaged over 10 trials. (d) The slopes 'P' of the fit lines shown in (a). Symbols show individual values, centre lines show mean values, error bars show standard error of the mean. Mood median test showed no significant effect of electrode size or location on either; (b) single trial SNR, (c) Trial averaged SNR or (d) the slope of fit line (p > 0.05).
The data were not normally distributed with differences in standard deviations and varying distribution shapes; therefore, a Mood median test was performed to quantify the effect of electrode size and array location on the SNR. The single trial SNR (Fig. 3b) did not show a statistically significant effect of electrode size (χ2 = 2.17, DOF = 2, p = 0.33) or array location (χ2 = 1.79, DOF = 2, p = 0.40). Similarly, the trial averaged SNR (10 trials) in Fig. 3c showed the effects of electrode size (χ2 = 5.07, DOF = 2, p = 0.07) or array location (χ2 = 3.24, DOF = 2, p = 0.19) were not statistically significant. However, it should be noted that the SNR of the SD showed a wide range of values (interquartile range (IQR) = 15.89) with a bimodal distribution compared to the those of the EV (IQR = 1.71) and ED arrays (IQR = 2.45), which showed unimodal distributions. The wide range and bimodal distributions of SD-SNR values in Fig. 3b–d, indicate that some SD electrodes outperformed the EV and ED arrays. The rate of change of SNR (Fig. 3d) from 1 to 10 trial averages given by the slope also showed no significant effect of electrode size (χ2 = 2.63, DOF = 2, p = 0.26) or array location (χ2 = 0.14, DOF = 2, p = 0.93). Since the SD array is closer to the neural tissue than the ED and EV arrays, a higher SNR would be anticipated. The thickness of the dura in sheep measured 80–100 µm, with cerebrospinal fluid (CSF) separating the dura and the brain; in previous work, we showed the SSS vessel wall thickness varied from 200–600 µm25. The binomial distribution of the SNR of SD arrays may have also resulted from electrode locations where some electrodes on the SD array were closer to the source of the evoked potentials. The binomial distribution of the SNR of SD arrays may have also resulted electrode locations where some electrodes on the SD array were closer to the source of the evoked potentials. Previous work has also shown SD electrodes tend to have higher amplitudes than ED electrodes42. Our results indicate that, four weeks after implantation, the SNR of SD, EV, and ED electrodes were not significantly affected by electrode size. However, some electrodes on the SD array clearly outperformed the SNR of the EV and ED arrays, whereas other SD electrodes showed an SNR comparable to those of the EV and ED arrays.
Spatial resolutions of SD, EV and ED arrays
The spatial resolution of an array refers to the ability of the array to localize discriminable neural signals. Spatial resolution is largely a function of distances between recording electrodes on an array, distances between electrodes and neural signals of interest, and electrode sizes. Higher spatial resolution provides greater specificity for a BMI44.
Our results showed that spatial resolution was frequency dependent and was dominated by array location at lower frequencies but not by electrode size for electrodes between 500-100 µm in diameter. Figure 4 (a–c) show a reduction in the magnitude squared coherence with increasing inter-electrode distances for ED, EV, and SD arrays. The data have been fitted with exponential curves39. The dashed horizontal line at coherence = 0.3 shows the threshold level above which the data from the two electrodes were said to be arising from a common source.
Spatial resolution. Representative data showing the estimation of spatial resolution using magnitude squared coherence versus the inter-electrode distance for (a) ED, (b) EV, and (c) SD electrodes. Fits were estimated as an exponential function of the magnitude squared coherence and were weighted to the inverse of the inter-electrode distance. The dashed horizontal line at 0.3 shows the level at which the signals between signals were considered independent. (d) Spatial resolutions at 8–24 Hz. (e) Spatial resolutions at 25–80 Hz. (f) Spatial resolutions at 81–120 Hz. (g) Spatial resolutions at 121–180 Hz. Symbols show individual values, centre lines show mean, error bars show standard error of the mean. Kruskal-Wallis test showed a significant effect of electrode location in the low frequency (d) (p = 0.003). However, there was effect of electrode size at any frequency band (p > 0.05) or electrode location at frequencies greater than 24 Hz (e–g). (h) Frequency dependence of spatial resolution, symbols indicate electrode size and lines are global fits at each electrode size and array. Pearson's correlation analysis showed strong negative correlation between spatial resolution and frequency for all electrode sizes with SD arrays ρ > 0.6 (p < 0.05); moderate negative correlation 750 µm diameter EV electrodes ρ = 0.45 (p < 0.05); and weak correlations not significantly different to zero for ED electrodes and 500 µm diameter EV electrodes (p > 0.1).
Figure 4 (d–g) show the mean spatial resolutions for all the arrays at each electrode size in four frequency bands: 8–24 Hz, 25–80 Hz, 81–120 Hz, and 121–180 Hz. The data were normally distributed but unbalanced with a sample size limited to the number of animals – ED (N = 5 animals), SD (N = 5 animals), and EV (N = 6 animals), so we used a Kruskal-Wallis test to compare medians of spatial resolutions between groups. Spatial resolution measurements were repeated in the 8–24 Hz frequency band (Fig. 4d), the median spatial resolutions at different array location (SD, EV, and ED) was statistically significant (χ2(2) = 11.42, p = 0.003) but the median spatial resolutions did not vary significantly with effect of electrode size (χ2(2) = 1.28, p = 0.52). The median spatial resolution from EV electrodes varied least from the median of all groups (z = 0.45) while the median spatial resolution of the ED arrays was lower (z = −2.29) and the median SNR of the SD arrays was higher (z = 2.78) than the median of all groups.
In the 28–80 Hz frequency band (Fig. 4e), the median spatial resolution did not vary significantly with either the array location (χ2(2) = 5.83, p = 0.054) or electrode size (χ2(2) = 4.97, p = 0.083). Similarly, in the 81–120 Hz frequency band (Fig. 4f), median spatial resolution did not vary significantly with either the array location (χ2(2) = 5.63, p = 0.06), or electrode size (χ2(2) = 1.81, p = 0.40). Likewise, in the 121–180 Hz frequency band (Fig. 4g), there was no statistically significant effects of array location (χ2(2) = 2.22, p = 0.33) or electrode size (χ2(2) = 0.88, p = 0.64) on the spatial resolution.
Figure 4h shows the changes in spatial resolution with frequency for SD, EV, and ED arrays at all electrode sizes. Fit lines shown are global linear regressions for each electrode size in all arrays. Pearson's correlation showed there was a moderate to strong negative correlation between spatial resolution and frequency, at all electrode sizes for SD arrays (500 µm, ρ (2) = −0.60, p = 0.005; 750 µm, ρ (2) = −0.78, p = 0 0.0001; 1000 µm, ρ (2) = −0.62, p = 0.003). There was also a moderate negative correlation between spatial resolution and frequency for 750 µm EV arrays (ρ (2) = −0.46, p = 0.02). There were weak negative correlations observed between spatial resolution and frequency at all electrode sizes for ED (500 µm, ρ (2) = −0.13, p = 0.56; 750 µm, ρ (2) = −0.30, p = 0.20; 1000 µm, ρ (2) = −0.33, p = 0.15) and 500 µm EV arrays (ρ (2) = −0.25, p = 0.25); however, these weak negative correlations were not significantly different from zero.
Results showed that the spatial resolution was frequency dependent and varied with the array location and to a lesser extent on the electrode size. At the onset of the study, it was expected that SD arrays would have the highest spatial resolution since the SD electrodes are closest to the brain and in contact with the cortical surface. However, there were minimal effects of array location on the spatial resolution at frequencies greater than 24 Hz in the resting state. In all three arrays, there was no effect of electrode size in electrodes between 500–1000 µm in diameters.
Single trial decoding performance of evoked potential in SD, EV and ED arrays
The goal of BMI technology is to decode neural signals accurately to control external interfaces. Accuracy of control provides a measure of the reliability of the decoding. The accuracy of decoding is a key indicator of the ability to decode discrete user activity and is dependent on several factors. One key factor that could enhance decoding accuracy is the effect of the SNR on the signal. We therefore measured the accuracy of decoding discrete evoked potentials in sheep.
Figure 5a, shows the dependence of the decoding accuracy in detecting an evoked potential on the number of trial averages for the SD (N = 5 animals), EV (N = 6 animals) and ED (N = 5 animals), arrays. A global linear regression applied to the decoding accuracy in Fig. 5a, from all animals for each array, showed the performances of the three arrays were comparable with overlapping confidence intervals. Slopes of the global fits were 2.642 (r2 = 0.77, p = 0.0001) for ED arrays, 2.97 (r2 = 0.74, p = 0.0001) for EV arrays, and 2.60 (r2 = 0.72, p = 0.0001) for SD arrays. Tukey corrected multiple paired t-tests between the accuracies of SD, EV, and ED arrays also showed no statistically significant differences in the decoding accuracies (p > 0.05 for all comparisons).
Decoding accuracies. (a) Decoding accuracy of evoked potentials versus number of trials used in calculating the average. Symbols indicate different animals and colours indicate the ED (green), EV (purple), and SD (orange) arrays. The fit lines are global fits on decoding accuracy across all animal; ED (N = 5 animals), SD (N = 5 animals), and EV (N = 6 animals) for each array. The symbols for each array have been adjusted in the x-axis direction to improve visualization. Dashed lines show 95% confidence intervals. Global Fit lines showed a good fit to all arrays r2 > 0.7. (b) Dependence of the decoding accuracy on SNR. Both decoding accuracy and SNR are normalized to the max of decoding accuracy and SNR, respectively, in each animal Dotted lines show global fits across all animals. Pearson's correlation showed strong (ρ > 0.7, p < 0.05). correlation between SNR and decoding accuracy for all arrays.
Figure 5b shows the dependence of the decoding accuracy on the SNR. Both SNR and decoding accuracy were normalized by their maximum value within each animal due to large inter-animal variability. There was a linearly increasing improvement in decoding accuracy with increasing SNR. Pearson's correlation calculated for each animal showed a strong correlation [ρ (degrees of freedom)] in all arrays. Mean correlations ρ (2) (N, standard deviations) were: ED, ρ (2) = 0.77 (N = 5, σ = 0.393), p = 0.048; EV, ρ (2) = 0.80 (N = 6, σ = 0.11), p = 0.041 and SD ρ (2) = 0.92 (N = 5, σ = 0.03), p = 0.001.
Results showed that there were no differences in decoding accuracies from the ED, EV, or SD arrays. However, we found that there was a strong correlation between SNR and the decoding accuracy in all three arrays.
To date literature in endovascular electrocorticography has been sparse5. However, since 2016 there has been a steady increase in the literature in endovascular electrocorticography owing to the steady improvements in the technology5. Thus far previous work has shown that: endovascular electrodes may be placed in a cortical blood vessel to record and stimulate the brain1,3,5,18,19,20,21,22,45,46; signals quality is likely comparable to that of other intracortical arrays1,4; electrode size (micro versus macro electrode) affects the type of signal recorded (i.e. smaller electrodes <100 µm diameter can recording local field potentials)4. However there are notable discrepancy between studies1,4,42 relating the effects of dura on the signal recorded.
Here, we show that: (1) The decoding accuracy is comparable between the EV, SD and ED arrays; (2) There is a correlation between the signal-noise-ratio and classification accuracy in all three recording modalities; (3) Bandwidth spatial resolution and signal to noise ratio of recordings from the EV arrays are comparable to SD and ED recordings; (4) The dura does not significantly reduce the signal to noise ratio; and (5) SD arrays had the best spatial resolution of the three arrays only at frequencies below 25 Hz. These results extend prior work by comprehensively addressing equivalence of EV recordings to standard ED and SD arrays with regard the bandwidth, signal-to-noise ratio, the spatial resolution and decoding ability of these devices for potential BCI applications.
Neither the dura nor the blood vessel significantly affect signal quality or performance. While some differences were observed in the signal waveform and the signal powers between electrodes on the SD, EV, and ED arrays, the SNR was similar across the three arrays; i.e., the absolute power was smaller but not the SNR. This indicates that the noise and the signal changed proportionally in the three arrays. For example, the EV array showed the lowest raw power, resulting from the low signal amplitudes seen on the EV array; however, the SNR was comparable with the SD and ED arrays. Contrary to some literature42,47,48, our results indicate that there was negligible effect of dura or blood vessel on the quality of the signals. This was also seen in the decoding performance, where there were no significant differences between ED and SD recordings49. Bower et al.4 also showed a similar result where there were no discernible differences between SD and ED electrodes. The finding that there was no discernible difference between SD and ED was surprising as the SD arrays which are closer to the brain than the ED arrays. The ED similar to the EV array is separated from the brain by the dura and CSF. In addition, the EV array has an additional layer of the blood vessel wall in the case of EV. It would be expected that some differences in signals would have been evident that relate to the distance from the brain or the tissue. The result of Bower et al.4 may have been due to the large epileptiform spiking amplitudes (0.5–1 mV) used to assess the signal which are much greater than typical intracortical brain signals of 10–50 µV amplitudes. At these high amplitudes typical intracortical brain signals may not be discernible as by Bower et al.4. In the present study we showed that the electrode location affected the spatial resolution at low frequencies but there were no significant differences in the signal quality measured by the bandwidth, SNR and decoding ability. The lack of difference in signal quality in particular SNR of the three array types in the present study may be due to the time points at which signal quality was assessed. The time points used in this study are 25 days post-implantation for the EV array and 21 days post-implantation for the ED and SD arrays. Chronic histological studies have also shown that the immune response to SD and ED arrays results in an immediate fibrous tissue encapsulation occurring over 14 days, followed by long-term tissue responses50,51. The EV arrays are not encapsulated with a fibrous layer, instead, they are covered by a thin layer of endothelium1,25. It is reasonable to consider that fibrous tissue encapsulation in SD arrays would result in a migration of implanted electrodes further away from the surface of the brain. It is possible that the advantage of the SD array being close to the brain may be circumvented by the fibrous encapsulation post-implantation, making the SNR and bandwidth of the ED, SD, and EV recordings more comparable.
Previous work47 also noted that the dura itself did not change the spatial resolution but rather it was the thickness of the CSF that made the greatest difference. Furthermore, conductivity measures of CSF, endothelium and dura are higher than that of fibrous tissue52,53 such as would be expected to be encapsulated around SD and ED electrode. It can therefore be concluded that neither the dura nor the blood vessel significantly affect the recorded signal quality or performance.
Llinás, 2005 and Watanabe, 200932,33 suggested the possibility of using a 'nanoprobe' that can endovascularly record the electrical activity of a single neuron, or small group of neurons. Thus far, work in endovascular neural interfaces suggests that high frequency brain signals1,4,31 such as signals in the high gamma range, fast ripples and high frequency oscillations are possible from electrodes placed on the surface of the brain, but the ability to record multiunit activity using the endovascular approach has not been shown. Two key factors that determine electrodes ability to record multiunit activity are electrode size and distance from neurons36,54,55,56. Modelling studies have predicted that electrodes <50 μm in diameter will be able to record brain signals in the range of local field potentials and multiunit or single unit spiking while larger electrodes can record fast ripples and cortical oscillations including cortical oscillations36,43,47,57. With small electrode, the "listening sphere" is small. Since recording from a neuron is a function of distance; the electric field falls as a square of the distance from the source (1/r2). This would imply that to record multiunit activity recording electrodes would need to be closer to 200 μm from target neurons55,58,59,60. The distance between the electrode within a blood vessel in the superior sagittal sinus was calculated between 260–680 μm. It could be speculated that high resolution recordings such as LFP's may be theoretically possible if electrodes were small <50 μm in diameter and less than 200 μm from the region of interest. While theoretically possible to achieve multiunit recordings endovascularly, present generation devices will require significant design changes to achieve this. Furthermore, electrodes would need to be located within the cortical blood vessels, such as the central sulcal vein, where the distance between the electrode and the brain were was minimized to under 200 μm. Further work, should investigate the possibility of high resolution recording from the brain and the possibility of recording multiunit activity from within a blood vessel.
The SNR of the EV array is a strong indicator of decoder performance. The SNR of the signal was strongly correlated with offline decoder performance, with higher SNR resulting in greater decoding accuracy. As expected when SNR was close to 1, the decoding performance was close to chance. This is not surprising considering the many noise sources in neural recordings including thermal noise, movement artefacts, ambient electrical noise, etc. Results showed that the signal and the noise scale together, irrespective of the separate locations of implantation (SD, EV, and ED). Averaging trials has been used as one method of improving the SNR, however, in an ideal BMI it may not be viable to average multiple trials, as this will limit the speed at which assistive technologies can be commanded and controlled. Therefore, methods of improving SNR, such as active grounding, referencing, etc., should be considered more thoroughly.
The EV array is a potential minimally-invasive alternative to SD and ED as a brain-activated switch. The aim of a BMI is to create a bridge between the brain and the external world via prostheses. Recent work has shown the transforming nature of a brain-activated switch in one person who was completely locked-in by providing thought-controlled spellers and cursor control9,61. Brain-activated switch-type BMIs utilizing brain surface potentials, such as those obtained from SD and ED arrays, have emerged as a viable signal for long-term neural interfacing in BMI. Previous work with SD and ED arrays showed no difference in their abilities to decode neural signals49. Our results show that the signal quality, defined by the bandwidth of recording and the signal-to-noise ratio, is not significantly different between the ED, EV, and SD arrays. Our results also show that the decoding accuracy in detecting an evoked potential is also comparable between ED, EV, and SD arrays. The promising prior results from ED and SD arrays suggest that the EV array may be a viable location of BMI control that bypasses the need for a craniotomy. These results shed further light on the findings of Bower et al.4 who found SD and EV electrodes performed reliably and concluded similarities in the recordings. These results further motivate a clinical trial where movement intent may be decoded to control external devices such as a speller or a wheelchair.
Critical factors that affect decoding neural activity are the bandwidth of information, the distance from target neurons, density of electrodes, and noise in the recording. Regarding the quality of data recorded (Bandwidth and SNR), the EV array is comparable to ED and SD arrays. However, the coverage that can be achieved by present EV arrays is limited to a few centimetres and can only be near brain areas adjacent to sizeable blood vessels; the impact of these limitations remains to be evaluated. The spatial resolution in this study was limited by the inter-electrode distances and electrode sizes. Larger reductions in the electrode size could increase spatial resolutions and allow for the implantation of electrode arrays with greater electrode numbers.
Previous work in BMI showed that the medial wall along the superior sagittal sinus contains a wealth of information about movement and movement intent62. It would be ideal to resolve this information within the medial wall in the posterior parietal cortex, primary motor cortex, and supplementary motor area to control movement of external objects. The medial wall would be an ideal location for the location of the EV array in humans. With the spatial resolutions, presently achievable with the EV along with the signal quality, it would be feasible to achieve discrete control with a small number of electrodes within a blood vessel in the brain.
Recent technological advances have led to the emergence of endovascular arrays for chronic recording of brain signals. We have demonstrated that the quality of recordings from endovascular arrays is comparable to recordings from epidural and subdural arrays, with reduction in electrode size resulting in enhanced spatial resolution across all arrays. These findings indicate that the endovascular array provides hope for a minimally-invasive technique for recording neural activity from the brain without the need for craniotomy. Importantly, the finding that the performance of the endovascular array is comparable to subdural and epidural arrays provides support for a minimally-invasive brain sensor with potential for use in a closed-loop neuromodulation system, such as a brain-machine interface.
Six adult Corriedale ewes weighing 60–70 kg were used in this study. Experiments were conducted at The Florey Institute of Neuroscience and Mental Health and were approved by the Florey Institute Animal Ethics Committee. Studies were in accordance with the NHMRC Principles of Laboratory Animal Care, Prevention of Cruelty to Animals Act, Australia, 2004, and the NHMRC Australian Code of Practice for the Care & Use of Animals for Scientific Purpose (seventh edition, 2004).
Six animals were implanted with ED, EV, and SD arrays, however two of the SD and ED arrays and one of the EV arrays developed faults at the connector. Therefore, we evaluated the quality of recordings obtained from the ED (4 arrays), EV (5 arrays), and SD (4 arrays) arrays from six animals. As illustrated in Fig. 1, the ED and SD arrays were placed on different hemispheres and were adjacent to the EV array. The SD and ED arrays were manufactured by Cortec GMBH, Germany and the EV arrays were made in-house1,25. The SD and ED arrays comprised 24 electrodes, with eight electrodes each of 500 µm, 750 µm, and 1000 µm diameter. Inter-electrode distance between similar electrode sizes was 4 mm and adjacent dissimilar sizes was 1.5 mm. The EV array had two sizes of electrodes, 500 µm and 750 µm; inter-electrode distance varied between electrodes (~2–6 mm). All electrodes were made of platinum. In this study, we used three electrode sizes for the ED and SD arrays and two sizes for the EV array. The largest size on the ED and SD arrays of 1 mm diameter would not fit within the dimensions of the blood vessel to be implanted and so was not used on an EV array. The 1 mm diameter was used in the ED and SD arrays based on previous studies showing that this is the optimum size to improve noise susceptibility while maintaining acceptable spatial resolution for a brain-computer interface36. Device positions were assessed immediately after deployment, and prior to termination of the experiment.
Animals were administered antiplatelet medication (Aspirin, 100 mg) daily from two days prior to implantation to minimize thrombosis and this was continued until the termination of the experiment. To induce anaesthesia, animals were premedicated with sodium thiopentone followed by intubation and ventilation with Isoflurane in air/O2. A cut-down and direct cannulation of the jugular vein was followed by advancement of a coaxial catheter system into the superior sagittal sinus, adjacent to the motor cortex1,25. Implantation of the EV array was performed under visual guidance using digital subtraction angiography (Arcadis Avantic, Siemens, Munich, Germany) as reported previously1,25. Percutaneous leads of the arrays exited the skin at the back of the neck.
After a 3–4 days recovery period, the animals underwent a second surgery to implant the cortical surface arrays. Under anesthesia (Isoflurane), the SD and ED arrays were implanted via craniotomy (1.4 × 0.8 cm) over the motor and somatosensory areas (Fig. 1). The exposed dura was covered with silicone sheet and then with dental cement. Percutaneous leads of the arrays exited the skin at the back of the neck similar to the EV device. The animals were kept in individual pens for periods of 3–4 weeks, which we have shown is sufficient for all the electrodes to be incorporated into the tissue1,25,63.
Prior to termination of the animal, in an acute experiment, animals were pre-medicated with a bolus of Sodium Thiopental followed by maintenance with Propofol and ketamine. Following intubation, the animals were ventilated with air. The median nerve in the sheep was exposed and stimulating needle electrodes were implanted 5 mm apart, and cortical evoked potentials in response to median nerve stimulation were recorded. Both left and right legs were stimulated. Animals were euthanized with an overdose of pentobarbital.
Cortical Recordings
Cortical electrophysiological signals (ECoG) were recorded using a g.tec USB amplifier (g.Tec, GMBH, Germany) at a sampling rate of 4800 Hz. ECoG recordings were analyzed using MATLAB (MathWorks Inc., Natwick, USA). Signals were band-pass filtered between 4–2400 Hz using a 4th order butterworth filter and 50 Hz noise and associated harmonics were removed using a IIR comb filter. Broken electrodes were identified when the impedances were greater than 1 MΩ at 1 kHz and these were removed from the analysis. Artefacts, such as spikes due to the electronics or cable movement, were seen in awake recordings. Electrical artefacts were identified where the RMS amplitudes in 0.1 s segments with 0.05 s overlap between segments were greater than 20 times the average RMS amplitude of all electrodes of the array. Artefacts caused by chewing were identified using a Hilbert transform to obtain an envelope of the artefact and finding prominent peaks in the envelope. Segments with artefacts were visually verified and removed from further analysis. Recordings were common average re-referenced to the average of all electrodes of the same size on each array. For example, each 500 µm electrode within the EV array was common average referenced to all 500 µm electrodes on the EV array.
Signal Bandwidth
The bandwidth of the recorded signal is a meaningful indicator of electrode performance since ECoG follows a typical 1/f decrease in the signal amplitude ending in a flat response equal to the noise floor, where the signal is indistinguishable from the noise. To evaluate the bandwidth of recordings, the signals were recorded in awake animals 3 weeks after implantation. The animals were either standing or sitting in their pen with minimal interaction with the surroundings. Recordings were separated into 2 s windows and the power spectra were calculated for each electrode using the Thompson multitaper method with a centre frequency of 1 Hz (2 Hz resolution). The noise of each electrode was estimated using the spectral power between 800 Hz and 1200 Hz. This frequency band was chosen as it was the highest frequency band located below the Nyquist frequency and was where the asymptote of the 1/f spectral profile was clearly seen1,42. Regions of 10 Hz around each harmonic of 50 Hz were ignored in the analysis. Median spectral content in each 10 Hz bin (between 4 Hz to 1200 Hz) were compared to the noise estimate. A 10 Hz bin was considered dissimilar from noise if the median power was greater than the upper boundary of the noise (3rd quartile + 1.5 IQR).
Signal-to-Noise Ratio (SNR)
The signal-to-noise ratio (SNR) offers a relative measure of quality of the recorded signal relative to the noise. We computed the SNR from electrically evoked cortical potentials by stimulating the median nerve in anaesthetized animals. Stimulation pulses were generated using a NI-myDAQ and LabView (National Instruments Corp., Austin USA) and passed through an AM-2200 stimulus isolator (AM systems, Sequim, USA).
Stimulation of the median nerve comprised constant current, monophasic, cathodal pulses between two electrodes placed 2 cm apart. Current amplitudes were varied randomly between 0 mA and 6 mA (0.5 mA steps). The maximum amplitudes used for stimulation varied between animals and were between 3 mA and 6 mA to reach maximum visible movement of the leg on stimulation. Stimuli were presented at 0.73 Hz and 10 repetitions were performed at each current level.
We measured the dependence of the SNR on the number of averaged trials for the three recording modalities. Averaging trials in electrophysiology is a strong tool, reducing the effect of noise with each additional trial. The slope (rate of change in the SNR with increasing number of averages) of the SNR, single trial SNR and 10 trial averaged SNR were expected to be different for varying array locations and electrode sizes. Both left and right limbs were stimulated, however only the stimulation with the lowest threshold of the two for each array was used in the SNR analysis. This enabled reducing any bias due where the stimulation was always contralateral to the one array while ipsilateral to another.
The evoked response to median nerve stimulation was analyzed by first segmenting the data into individual current levels and then averaging within each current level. The threshold current level of the evoked potential (Th) was detected by finding peaks in the data where the amplitude after the maximum stimulation \({\rm{\max }}\,{A}_{t0}^{t2}\,\,\)was greater than twice the maximum amplitude of the background,
$$\begin{array}{c}Th=\,{\rm{\max }}\,{A}_{{t}_{0}}^{{t}_{200}} > 2\times \,{\rm{\max }}\,{A}_{{t}_{-200}}^{{t}_{0}},\end{array}$$
where, t0 was the stimulation time, t-200 was 200 ms prior to t0 and t2 was 200 ms after t0.
SNR was defined as the ratio of the root mean square (RMS) at threshold current level (Th) at time 100 ms (t100) after stimulation t0 (\({{\rm{RMS}}}_{T{h}_{t0}}^{{t}_{100}}\)) to the RMS at threshold current level (Th) at time 100 ms (t−100) before stimulation (t0) (\({{\rm{RMS}}}_{T{h}_{{t}_{-100}}}^{t0}\)),
$$SN{R}_{i}={{\rm{RMS}}}_{T{h}_{{t}_{0}}}^{{t}_{100}}/{{\rm{RMS}}}_{T{h}_{{t}_{-100}}}^{{t}_{0}}.$$
The SNR was calculated on the trial averaged signal in varying combinations of trials (nCr) without replication, where n is the total number of trials (10) and r is the number of trials (1, 2, 3, …, 10) used for each average. For example, C(1, 10) = 10 combinations in total i.e. SNR is calculated on single trials; C(2, 10) = 45 combination with 2 trials in each combination and SNR is calculated on each combination; C(5, 10) = 252 combinations with 5 trials in each combination and SNR is calculated on each combination and C(10,10) = 1 combination of 10 trials and SNR is calculated on the average of 10 trials.
Spatial resolution is a measure of the shared neural activity between adjacent electrodes. The spatial resolution achievable by neural recordings depends largely on the array location and size. To evaluate the effect of array location on spatial resolutions, we calculated the magnitude squared coherence64 between electrodes. Recordings were taken in awake animals while the animals were in their individual pens. Coherence is known to vary with distance36,44 that can be approximated by an exponential function (\(Y=a{e}^{-bx}+c\))39. The function 'Y' approximates the rate of change 'b' of the coherence between electrode from a base constant of 'a' at distance 'x' and an offset of 'c'. If the coefficient associated with b and/or d is negative, y represents exponential decay. The spatial resolution is approximated as the inter-electrode distance where magnitude squared coherence equals 0.336,39,44.
In this study, we used magnitude squared coherence as a measure of spatial resolution during rest while the animal was awake and in its cage. In this period, the coherence would be dominated by the magnitude of the electric fields, which decrease as the inverse of the square of distance, and to a lesser extent functional connection in the brain. Since the correlation was not driven by an evoked stimulus, the correlation would be assumed to be mediated by the electrode's ability to capture the change in the spatial localized brain rhythms. We measured the magnitude squared coherence in frequency bands relevant to motor induced oscillations in the brain (8–24, 25–80, 81–120, 121–180 Hz) as these are most commonly used in ECoG-based BCIs65,66,67. Magnitude squared coherence was calculated in frequency bands as previous work with ECoG showed that coherence is frequency dependent39 and provides a robust measure of signal quality for the SD, EV, and ED arrays.
Evoked Potential Decoding
The decoding performance on the different arrays was assessed by classifying the presence or absence of an evoked potential using the recording features from cortical signals. All analysis was performed at the threshold current level identified earlier. Trials were bootstrap averaged in every combination of trials without replication (nPr), where n is the total number of trials (10) and r is the number of trials (1, 2, 3, …, 10) used for each average. We estimated the average power in the time in 4–24, 30–45, 55–95 and 105–145 Hz bands using Welch power estimate, on each trial combination. Evoked potential related features (i.e. power in the defined frequency band mentioned above) were extracted from all electrodes on each array and used to detect the presence of an evoked potential using a linear discriminant analysis and a leave half out cross validation where the classifier is trained on 50 percent of trials and then tested on the remaining 50 percent of trials. The process was repeated for every non-repeating combination and the average accuracy is reported.
Statistical tests were performed using MATLAB (MathWorks Inc., MA, USA) or Minitab (Minitab Inc.). Between groups comparison was made using ANOVA's where assumptions of normality and variance were satisfied. Where assumptions of normality and variance were violated Kruskal-Wallis test was used when data distributions between groups were similar and Mood median test was if distribution shapes were different. Pearsons correlation analysis was used to analyse trends in the data and were defined weak (0–0.4), moderate (0.41–0.8) and strong (0.81–1.0).
The datasets generated during the current study are available from the corresponding author on reasonable request.
A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has not been fixed in the paper.
Oxley, T. J. et al. Minimally invasive endovascular stent-electrode array for high-fidelity, chronic recordings of cortical neural activity. Nat. Biotechnol. 34, 320–327 (2016).
Penn, R. D., Hilal, S. K., Michelsen, W. J., Goldensohn, E. S. & Driller, J. Intravascular intracranial EEG recording. Technical note. J. Neurosurg. 38, 239–43 (1973).
Thomke, F., Stoeter, P. & Stader, D. Endovascular electroencephalography during an intracarotid amobarbital test with simultaneous recordings from 16 electrodes. J. Neurol. Neurosurg. Psychiatry 64, 565–565 (1998).
Bower, M. R. et al. Intravenous recording of intracranial, broadband EEG. J. Neurosci. Methods 214, 21–6 (2013).
ADS Article Google Scholar
Sefcik, R. K. et al. The evolution of endovascular electroencephalography: historical perspective and future applications. Neurosurg. Focus 40, 1–8 (2016).
Yanagisawa, T. et al. Neural decoding using gyral and intrasulcal electrocorticograms. Neuroimage 45, 1099–1106 (2009).
Schalk, G. et al. Decoding two-dimensional movement trajectories using electrocorticographic signals in humans. J. Neural Eng. 4, 264–75 (2007).
ADS CAS Article Google Scholar
Miller, K. J. et al. Three cases of feature correlation in an electrocorticographic BCI. Conf. Proc. IEEE Eng. Med. Biol. Soc. 2008, 5318–21 (2008).
Vansteensel, M. J. et al. Fully Implanted Brain–Computer Interface in a Locked-In Patient with ALS. N. Engl. J. Med. 375, 2060–2066 (2016).
Williams, J. J., Rouse, A. G., Thongpang, S., Williams, J. C. & Moran, D. W. Differentiating closed-loop cortical intention from rest: building an asynchronous electrocorticographic BCI. J. Neural Eng. 10, 46001 (2013).
Martens, S. et al. Epidural electrocorticography for monitoring of arousal in locked-in state. Front. Hum. Neurosci. 8, 861 (2014).
Rouse, A. G., Williams, J. J., Wheeler, J. J. & Moran, D. W. Cortical adaptation to a chronic micro-electrocorticographic brain computer interface. J. Neurosci. 33, 1326–30 (2013).
Eliseyev, A. & Aksenova, T. Stable and artifact-resistant decoding of 3D hand trajectories from ECoG signals using the generalized additive model. J. Neural Eng. 11, 66005 (2014).
Shimoda, K., Nagasaka, Y., Chao, Z. C. & Fujii, N. Decoding continuous three-dimensional hand trajectories from epidural electrocorticographic signals in Japanese macaques. J. Neural Eng. 9, 36015 (2012).
Yanagisawa, T. et al. Real-time control of a prosthetic hand using human electrocorticography signals. J. Neurosurg. 114, 1715–22 (2011).
Milekovic, T. et al. An online brain-machine interface using decoding of movement direction from the human electrocorticogram. J. Neural Eng. 9, 46003 (2012).
Tebo, C. C., Evins, A. I., Christos, P. J., Kwon, J. & Schwartz, T. H. Evolution of cranial epilepsy surgery complication rates: a 32-year systematic review and meta-analysis. J. Neurosurg. 120, 1415–1427 (2014).
He, B. D., Ebrahimi, M., Palafox, L. & Srinivasan, L. Signal quality of endovascular electroencephalography. J. Neural Eng. 13, 16016 (2016).
Boniface, S. J. & Antoun, N. Endovascular electroencephalography: the technique and its application during carotid amytal assessment. J. Neurol. Neurosurg. Psychiatry 62, 193–5 (1997).
Kara, T. et al. Endovascular brain intervention and mapping in a dog experimental model using magnetically-guided micro-catheter technology. Biomed. Pap. Med. Fac. Univ. Palacky. Olomouc. Czech. Repub. 158, 221–6 (2014).
Henz, B. D. et al. Advances in radiofrequency ablation of the cerebral cortex in primates using the venous system: Improvements for treating epilepsy with catheter ablation technology. Epilepsy Res. 108, 1026–1031 (2014).
Henz, B. D. et al. Successful radiofrequency ablation of the cerebral cortex in pigs using the venous system: Possible implications for treating CNS disorders. Epilepsy Res. 80, 213–218 (2008).
Opie, N. L. et al. Chronic impedance spectroscopy of an endovascular stent-electrode array. J. Neural Eng. 13, 46020 (2016).
Oxley, T. J. et al. An ovine model of cerebral catheter venography for implantation of an endovascular neural interface. J. Neurosurg. 128, 1020-1027 (2018).
Opie, N. L. et al. Micro-CT and Histological Evaluation of a Neural Interface Implanted within a Blood Vessel. IEEE Trans. Biomed. Eng. 64, 928–934 (2017).
Wong, Y. T. et al. Suitability of nitinol electrodes in neural prostheses such as endovascular neural interfaces. In 2016 38th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC) 4463–4466 (IEEE, 2016).
Nakase, H. et al. An intra-arterial electrode for intracranial electro-encephalogram recordings. Acta Neurochir. (Wien). 136, 103–105 (1995).
Stoeter, P., Dieterle, L., Meyer, a & Prey, N. Intracranial electroencephalographic and evoked-potential recording from intravascular guide wires. AJNR. Am. J. Neuroradiol. 16, 1214–7 (1995).
Mikuni, N. et al. 'Cavernous sinus EEG': a new method for the preoperative evaluation of temporal lobe epilepsy. Epilepsia 38, 472–482 (1997).
Kunieda, T. et al. Use of Cavernous Sinus EEG in the Detection of Seizure Onset and Spread in Mesial Temporal Lobe Epilepsy. Epilepsia 41, 1411–1419 (2000).
Ishida, S. et al. Intracranial EEG Recording from Intravascular Electrodes in Patients with Temporal Lobe Epilepsy. Epilepsia 39, 77–77 (1998).
Llinás, R. R., Walton, K. D., Nakao, M., Hunter, I. & Anquetil, P. A. Neuro-vascular central nervous recording/stimulating system: Using nanotechnology probes. J. Nanoparticle Res. 7, 111–127 (2005).
Watanabe, H., Takahashi, H., Nakao, M., Walton, K. & Llinás, R. R. Intravascular Neural Interface with Nanowire Electrode. Electron. Commun. Japan 92, 29–37 (2009).
Buzsáki, G., Anastassiou, C. A. & Koch, C. The origin of extracellular fields and currents–EEG, ECoG, LFP and spikes. Nat. Rev. Neurosci. 13, 407–20 (2012).
Weiner, G. M., Ozpinar, A. & Ducruet, A. Endovascular Access for Cortical Mapping. Neurosurgery 79, N22–3 (2016).
Wodlinger, B., Degenhart, A. D., Collinger, J. L., Tyler-Kabara, E. C. & Wang, W. The impact of electrode characteristics on electrocorticography (ECoG). Conf. Proc. IEEE Eng. Med. Biol. Soc. 2011, 3083–6 (2011).
Rohde, M. M. et al. Quality estimation of subdurally recorded, event-related potentials based on signal-to-noise ratio. IEEE Trans. Biomed. Eng. 49, 31–40 (2002).
Lahr, J. et al. Invasive brain–machine interfaces: a survey of paralyzed patients' attitudes, knowledge and methods of information retrieval. J. Neural Eng. 12, 43001 (2015).
Muller, L., Hamilton, L. S., Edwards, E., Bouchard, K. E. & Chang, E. F. Spatial resolution dependence on spectral frequency in human speech cortex electrocorticography. J. Neural Eng. 13, 56013 (2016).
Gunduz, A., Sanchez, J. C., Carney, P. R. & Principe, J. C. Mapping broadband electrocorticographic recordings to two-dimensional hand trajectories in humans Motor control features. Neural Netw. 22, 1257–70 (2009).
Warren, D. J. et al. Recording and Decoding for Neural Prostheses. Proc. IEEE 104, 374–391 (2016).
Bundy, D. T. et al. Characterization of the effects of the human dura on macro- and micro-electrocorticographic recordings. J. Neural Eng. 11, 16006 (2014).
Wilson, J. A., Felton, E. A., Garell, P. C., Schalk, G. & Williams, J. C. ECoG factors underlying multimodal control of a brain-computer interface. IEEE Trans. Neural Syst. Rehabil. Eng. 14, 246–250 (2006).
Wang, W. et al. Human motor cortical activity recorded with Micro-ECoG electrodes, during individual finger movements. In Annual International Conference of the IEEE Engineering in Medicine and Biology Society 2009, 586–9 (2009).
Wong, Y. T. et al. Suitability of nitinol electrodes in neural prostheses such as endovascular neural interfaces. In Proceedings of the Annual International Conference of the IEEE Engineering in Medicine and Biology Society, EMBS 2016–Octob (2016).
Rousseau, H. et al. Self-expanding endovascular prosthesis: an experimental study. Radiology 164, 709–714 (1987).
Slutzky, M. W., Jordan, L. R. & Miller, L. E. Optimal spatial resolution of epidural and subdural electrode arrays for brain-machine interface applications. Conf. Proc. IEEE Eng. Med. Biol. Soc. 2008, 3771–4 (2008).
Baek, D.-H. et al. A thin film polyimide mesh microelectrode for chronic epidural electrocorticography recording with enhanced contactability. J. Neural Eng. 11, 46023 (2014).
Flint, R. D., Rosenow, J. M., Tate, M. C. & Slutzky, M. W. Continuous decoding of human grasp kinematics using epidural and subdural signals. J. Neural Eng. 14, 16005 (2017).
Degenhart, A. D. et al. Histological evaluation of a chronicallyimplanted electrocorticographic electrode grid in a non-human primate. J. Neural Eng. 13, 46019 (2016).
Vistnes, L. M., Ksander, G. A. & Kosek, J. Study of encapsulation of Silicone Rubber Implants in Animals. Plast. Reconstr. Surg. 64, 580–588 (1978).
Gabriel, S., Lau, R. W. & Gabriel, C. The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues. Phys. Med. Biol. 41, 2271–93 (1996).
Gabriel, C., Peyman, A. & Grant, E. H. Electrical conductivity of tissue at frequencies below 1 MHz. Phys. Med. Biol. 54, 4863–4878 (2009).
Rubehn, B., Bosman, C., Oostenveld, R., Fries, P. & Stieglitz, T. A MEMS-based flexible multichannel ECoG-electrode array. J. Neural Eng. 6, 36003 (2009).
Khodagholy, D. et al. NeuroGrid: recording action potentials from the surface of the brain. Nat Neurosci 18, 310–315 (2015).
Kim, J., Wilson, J. A. & Williams, J. C. A Cortical Recording Platform Utilizing µECoG Electrode Arrays. In 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 2721, 5353–5357 (IEEE, 2007).
Wang, X. et al. Mapping the fine structure of cortical activity with different micro-ECoG electrode array geometries. J. Neural Eng. 14, 56004 (2017).
Einevoll, G. T., Kayser, C., Logothetis, N. K. & Panzeri, S. Modelling and analysis of local field potentials for studying the function of cortical circuits. Nat. Rev. Neurosci. 14, 770–85 (2013).
Lindén, H. et al. Modeling the spatial reach of the LFP. Neuron 72, 859–872 (2011).
Kajikawa, Y. & Schroeder, C. E. How local is the local field potential? Neuron 72, 847–858 (2011).
Moritz, C. T., Perlmutter, S. I. & Fetz, E. E. Direct control of paralysed muscles by cortical neurons. Nature 456, 639–642 (2008).
Yoo, P. E. et al. 7T-fMRI: Faster temporal resolution yields optimal BOLD sensitivity for functional network imaging specifically at high spatial resolution. Neuroimage 164, 214–229 (2018).
Sillay, K. A. et al. Long-term measurement of impedance in chronically implanted depth and subdural electrodes during responsive neurostimulation in humans. Brain Stimul. 6, 718–26 (2013).
Kay, S. M. Modern Spectral Estimation. Book (Prentice-Hall, 1988).
Collinger, J. L. et al. Motor-related brain activity during action observation: a neural substrate for electrocorticographic brain-computer interfaces after spinal cord injury. Front. Integr. Neurosci. 8, 17 (2014).
Morris, S. et al. Patient Specific Cortical Electrodes for Sulcal and Gyral Implantation. IEEE Trans. Biomed. Eng. 62, 1034–1041 (2014).
Ball, T., Schulze-Bonhage, A., Aertsen, A. & Mehring, C. Differential representation of arm movement direction in relation to cortical anatomy and function. J. Neural Eng. 6, 16006 (2009).
We thank Covidien (Medtronic) for provision of 30 Solitaire stentriever devices as a product research grant. T. Dornum, A. McDonald and T. Vale for surgical and animal handling assistance. We also acknowledge Prof. A. Burkitt, Mr. A. Meltzer, Mr. Stefan Wilson, and Mr. T. Scordas for their support and input. This work was funded by the National Health and Medical Research Council of Australia (NHMRC) Project Grant APP1062532, US Defense Advanced Research Projects Agency (DARPA) Microsystems Technology Office contract N66001-12-1-4045; Office of Naval Research (ONR) Global N62909-14-1-N020; Defence Health Foundation, Australia (Booster Grant); and the Victorian Government's Operational Infrastructure Support Program.
Department of Biomedical Engineering, The University of Melbourne, Parkville, Australia
Sam E. John, Yan T. Wong, Giulia Gerboni & David B. Grayden
Vascular Bionics Laboratory, Department of Medicine, Royal Melbourne Hospital, (RMH), The University of Melbourne, Parkville, Australia
Sam E. John, Nicholas L. Opie, Gil S. Rind, Stephen M. Ronayne, Giulia Gerboni, Terence J. O'Brien & Thomas J. Oxley
Florey Institute of Neuroscience and Mental Health, Parkville, Australia
Sam E. John, Nicholas L. Opie, Gil S. Rind, Stephen M. Ronayne, Giulia Gerboni, Terence J. O'Brien, Clive N. May & Thomas J. Oxley
Centre for Neural Engineering, The University of Melbourne, Carlton, Australia
David B. Grayden
Department of Veterinary Science, The University of Melbourne, Werribee, Australia
Sebastien H. Bauquier
Department of Physiology and Department of Electrical and Computer Systems Engineering, Monash University, Clayton, Australia
Yan T. Wong
SmartStent Pty Ltd, Parkville, Australia
Sam E. John, Nicholas L. Opie, Gil S. Rind, Stephen M. Ronayne & Thomas J. Oxley
Sam E. John
Nicholas L. Opie
Gil S. Rind
Stephen M. Ronayne
Giulia Gerboni
Terence J. O'Brien
Clive N. May
Thomas J. Oxley
S.E.J., N.L.O., T.J.O., G.S.R. and S.M.R., conceived the experiments and hypotheses. S.E.J., N.L.O., G.S.R., T.J.O. and S.H.B. designed experiments. S.E.J., N.L.O., G.S.R., G.G., S.M.R., Y.T.W. performed experiments. S.E.J. developed the analytical tools for the analysis, analyzed data. T.J.B., C.N.M., D.B.G., T.J.O. supervised the analysis, writing and edited the manuscript. All authors contributed extensively to the writing of the paper including analysis and interpretation of the data.
Correspondence to Sam E. John.
Dr's. Opie and Oxley are directors of Smarstent Pty Limited and Synchron Inc. Dr. Oxley and a company associated with Dr. Opie are shareholders of Synchron Inc. Dr. John, Mr. Rind and Mr. Ronayne hold share options in Synchron Inc. Dr. John, Mr. Rind and Mr. Ronayne are employed by SmartStent Pty. Ltd as well as the University of Melbourne. This work was performed under employment at the University of Melbourne. Dr John, Dr Opie, Dr Wong, Mr Rind, Mr Ronayne, Dr. O'Brien, Dr. May, Dr. Grayden, Dr Oxley are listed as inventors on published or submitted patents on stentrode design and/or application. A company associated with Dr. Opie receives a consulting fee from SmartStent. SmartStent did not contribute financially or in kind to the experiments and analyses described in this work. Medtronic PLC (Covidien), Minneapolis, MN provided Solitaire stentriever devices used in this study as a gift. Medtronic was not involved in developing the devices, performing the study or analysis and it did not provide financial contributions to this work. All work was performed under the funding sources mentioned below. No other authors have any financial ties to SmartStent Pty. Ltd.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
John, S.E., Opie, N.L., Wong, Y.T. et al. Signal quality of simultaneously recorded endovascular, subdural and epidural signals are comparable. Sci Rep 8, 8427 (2018). https://doi.org/10.1038/s41598-018-26457-7
Decoding Accuracy
Local Array
Neural Interface
Electrode Size
Medium Spatial Resolution
Focal stimulation of the sheep motor cortex with a chronically implanted minimally invasive electrode array mounted on an endovascular stent
Nature Biomedical Engineering (2018) | CommonCrawl |
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Journal of Agricultural, Biological, and Environmental Statistics (2002-03-01) 7: 74-94 , March 01, 2002
By Smith, Robert W.
When appropriate data from regional reference locations are available, tolerance-interval bounds can be computed to provide criteria or limits distinguishing reference from nonreference conditions. If the limits are to be to applied to locations and times beyond the original data, the data should include temporal and spatial variation and the tolerance interval calculations should utilize a random crossed or nested ANOVA statistical design. Two computational methods for such designs are discussed and evaluated with simulations. Both methods are shown to perform well, and the adverse effect of using an improper design model is demonstrated. Three real-world applications are shown, where tolerance intervals are used to (1) establish a reference threshold for a benthic community pollution index, (2) set criteria for chemicals in sediments, and (3) establish background thresholds for survival rates in sediment bioassay tests. Some practical considerations in the use of the tolerance intervals are discussed.
An 'apples to apples' comparison of various tests for exponentiality
Computational Statistics (2017-12-01) 32: 1241-1283 , December 01, 2017
By Allison, J. S.; Santana, L.; Smit, N.; Visagie, I. J. H. Show all (4)
The exponential distribution is a popular model both in practice and in theoretical work. As a result, a multitude of tests based on varied characterisations have been developed for testing the hypothesis that observed data are realised from this distribution. Many of the recently developed tests contain a tuning parameter, usually appearing in a weight function. In this paper we compare the powers of 20 tests for exponentiality—some containing a tuning parameter and some that do not. To ensure a fair 'apples to apples' comparison between each of the tests, we employ a data-dependent choice of the tuning parameter for those tests that contain these parameters. The comparisons are conducted for various samples sizes and for a large number of alternative distributions. The results of the simulation study show that the test with the best overall power performance is the Baringhaus and Henze test, followed closely by the test by Henze and Meintanis; both tests contain a tuning parameter. The score test by Cox and Oakes performs the best among those tests that do not include a tuning parameter.
Partially linear varying coefficient models with missing at random responses
Annals of the Institute of Statistical Mathematics (2013-08-01) 65: 721-762 , August 01, 2013
By Bravo, Francesco
This paper considers partially linear varying coefficient models when the response variable is missing at random. The paper uses imputation techniques to develop an omnibus specification test. The test is based on a simple modification of a Cramer von Mises functional that overcomes the curse of dimensionality often associated with the standard Cramer von Mises functional. The paper also considers estimation of the mean functional under the missing at random assumption. The proposed estimator lies in between a fully nonparametric and a parametric one and can be used, for example, to obtain a novel estimator for the average treatment effect parameter. Monte Carlo simulations show that the proposed estimator and test statistic have good finite sample properties. An empirical application illustrates the applicability of the results of the paper.
Notes on the dimension dependence in high-dimensional central limit theorems for hyperrectangles
Japanese Journal of Statistics and Data Science (2020-10-26): 1-41 , October 26, 2020
By Koike, Yuta
Let $$X_1,\ldots ,X_n$$ be independent centered random vectors in $${\mathbb {R}}^d$$ . This paper shows that, even when d may grow with n, the probability $$P(n^{-1/2}\sum _{i=1}^nX_i\in A)$$ can be approximated by its Gaussian analog uniformly in hyperrectangles A in $${\mathbb {R}}^d$$ as $$n\rightarrow \infty$$ under appropriate moment assumptions, as long as $$(\log d)^5/n\rightarrow 0$$ . This improves a result of Chernozhukov et al. (Ann Probab 45:2309–2353, 2017) in terms of the dimension growth condition. When $$n^{-1/2}\sum _{i=1}^nX_i$$ has a common factor across the components, this condition can be further improved to $$(\log d)^3/n\rightarrow 0$$ . The corresponding bootstrap approximation results are also developed. These results serve as a theoretical foundation of simultaneous inference for high-dimensional models.
Reducing bias in parameter estimates from stepwise regression in proportional hazards regression with right-censored data
Lifetime Data Analysis (2008-03-01) 14: 65-85 , March 01, 2008
By Soh, Chang-Heok; Harrington, David P.; Zaslavsky, Alan M.
When variable selection with stepwise regression and model fitting are conducted on the same data set, competition for inclusion in the model induces a selection bias in coefficient estimators away from zero. In proportional hazards regression with right-censored data, selection bias inflates the absolute value of parameter estimate of selected parameters, while the omission of other variables may shrink coefficients toward zero. This paper explores the extent of the bias in parameter estimates from stepwise proportional hazards regression and proposes a bootstrap method, similar to those proposed by Miller (Subset Selection in Regression, 2nd edn. Chapman & Hall/CRC, 2002) for linear regression, to correct for selection bias. We also use bootstrap methods to estimate the standard error of the adjusted estimators. Simulation results show that substantial biases could be present in uncorrected stepwise estimators and, for binary covariates, could exceed 250% of the true parameter value. The simulations also show that the conditional mean of the proposed bootstrap bias-corrected parameter estimator, given that a variable is selected, is moved closer to the unconditional mean of the standard partial likelihood estimator in the chosen model, and to the population value of the parameter. We also explore the effect of the adjustment on estimates of log relative risk, given the values of the covariates in a selected model. The proposed method is illustrated with data sets in primary biliary cirrhosis and in multiple myeloma from the Eastern Cooperative Oncology Group.
Testing for one-sided alternatives in nonparametric censored regression
TEST (2012-09-01) 21: 498-518 , September 01, 2012
By Heuchenne, Cédric; Pardo-Fernández, Juan Carlos
Assume that we have two populations (X1,Y1) and (X2,Y2) satisfying two general nonparametric regression models Yj=mj(Xj)+εj, j=1,2, where m(⋅) is a smooth location function, εj has zero location and the response Yj is possibly right-censored. In this paper, we propose to test the null hypothesis H0:m1=m2 versus the one-sided alternative H1:m1<m2. We introduce two test statistics for which we obtain the asymptotic normality under the null and the alternative hypotheses. Although the tests are based on nonparametric techniques, they can detect any local alternative converging to the null hypothesis at the parametric rate n−1/2. The practical performance of a bootstrap version of the tests is investigated in a simulation study. An application to a data set about unemployment duration times is also included.
Empirical process approach to some two-sample problems based on ranked set samples
Annals of the Institute of Statistical Mathematics (2007-12-01) 59: 757-787 , December 01, 2007
By Ghosh, Kaushik; Tiwari, Ram C.
We study the asymptotic properties of both the horizontal and vertical shift functions based on independent ranked set samples drawn from continuous distributions. Several tests derived from these shift processes are developed. We show that by using balanced ranked set samples with bigger set sizes, one can decrease the width of the confidence band and hence increase the power of these tests. These theoretical findings are validated through small-scale simulation studies. An application of the proposed techniques to a cancer mortality data set is also provided.
Model-based INAR bootstrap for forecasting INAR(p) models
By Bisaglia, Luisa ; Gerolimetto, Margherita
In this paper we analyse some bootstrap techniques to make inference in INAR(p) models. First of all, via Monte Carlo experiments we compare the performances of these methods when estimating the thinning parameters in INAR(p) models; we state the superiority of model-based INAR bootstrap approaches on block bootstrap in terms of low bias and Mean Square Error. Then we adopt the model-based bootstrap methods to obtain coherent predictions and confidence intervals in order to avoid difficulty in deriving the distributional properties. Finally, we present an empirical application.
Coverage plots for assessing the variability of estimated contours of a density
Statistics and Computing (1996-12-01) 6: 325-336 , December 01, 1996
By Lin, Xun-Guo; Pope, Alun
Methods for assessing the variability of an estimated contour of a density are discussed. A new method called the coverage plot is proposed. Techniques including sectioning and bootstrap techniques are compared for a particular problem which arises in Monte Carlo simulation approaches to estimating the spatial distribution of risk in the operation of weapons firing ranges. It is found that, for computational reasons, the sectioning procedure outperforms the bootstrap for this problem. The roles of bias and sample size are also seen in the examples shown.
Changepoint in dependent and non-stationary panels
Statistical Papers (2020-08-01) 61: 1385-1407 , August 01, 2020
By Maciak, Matúš ; Pešta, Michal ; Peštová, Barbora
Detection procedures for a change in means of panel data are proposed. Unlike classical inference tools used for the changepoint analysis in the panel data framework, we allow for mutually dependent and generally non-stationary panels with an extremely short follow-up period. Two competitive self-normalized test statistics are employed and their asymptotic properties are derived for a large number of available panels. The bootstrap extensions are introduced in order to handle such a universal setup. The novel changepoint methods are able to detect a common break point even when the change occurs immediately after the first time point or just before the last observation period. The developed tests are proved to be consistent. Their empirical properties are investigated through a simulation study. The invented techniques are applied to option pricing and non-life insurance. | CommonCrawl |
Modelling and analysis of the dynamics of adaptive temporal–causal network models for evolving social interactions
Jan Treur1
Network-Oriented Modelling based on adaptive temporal–causal networks provides a unified approach to model and analyse dynamics and adaptivity of various processes, including mental and social interaction processes.
Adaptive temporal–causal network models are based on causal relations by which the states in the network change over time, and these causal relations are adaptive in the sense that they themselves also change over time.
It is discussed how modelling and analysis of the dynamics of the behaviour of these adaptive network models can be performed. The approach is illustrated for adaptive network models describing social interaction.
In particular, the homophily principle and the 'more becomes more' principles for social interactions are addressed. It is shown how the chosen Network-Oriented Modelling method provides a basis to model and analyse these social phenomena.
Network-Oriented Modelling has been proposed as a modelling perspective suitable for processes that are highly dynamic, circular and interactive; e.g. [1, 2]. In different application areas, this modelling perspective has been proposed in different forms: in the contexts of modelling organisations and social systems (e.g. [3,4,5]), of modelling metabolic processes (e.g. [6]), and of modelling electromagnetic systems (e.g. [7,8,9]. To address dynamics well, Network-Oriented Modelling based on adaptive temporal–causal networks has been developed [1, 2, 10]. This approach incorporates a continuous (real) time dimension. Adaptive temporal–causal network models are dynamic in two ways: their states change over time based on the causal relations in the network, but these causal relations may also change over time. As, in such networks many interrelating cycles often occur, their emerging behaviour patterns are not always easy to predict or analyse. This may make it hard to evaluate whether observed outcomes of simulations are plausible or might be due to implementation errors.
However, some specific types of properties can also be analysed by calculations in a mathematical manner, without performing simulations; see, for example [11,12,13,14,15,16]. Such properties that are found in an analytical mathematical manner can be used for verification of the model by checking them for the values observed in simulation experiments. If one of these properties is not fulfilled (and the mathematical analysis was done in a correct manner), then there will be some error in the implementation of the model. In this paper, methods to analyse such properties of temporal–causal network models will be described. They will be illustrated for two types for dynamic connection weights in adaptive temporal–causal network models modelling evolving social interaction: one based on the homophily principle ("Modelling evolving social interactions by adaptive networks based on the homophily principle" section), and one based on the more becomes more principle ("Modelling evolving social interactions by adaptive networks based on the 'more becomes more' principle" section). A preliminary, shorter presentation of part of the work described here can be found in [17].
Network-Oriented Modelling by temporal–causal networks
The Network-Oriented Modelling approach based on temporal–causal networks, described in more detail in [1, 10] is a generic and declarative dynamic modelling approach based on networks of causal relations. Dynamics is addressed by incorporating a continuous time dimension. This temporal dimension enables modelling by networks that inherently contain cycles, such as networks modelling mental or brain processes, or social interaction processes, and also enables to address the timing of the processes in a differentiated manner. The modelling perspective can incorporate ingredients from different modelling approaches: for example, ingredients that are sometimes used in neural network models, and ingredients that are sometimes used in probabilistic or possibilistic modelling. It is more generic than such methods in the sense that a much wider variety of modelling elements are provided, enabling the modelling of many types of dynamical systems, as described in [1, 10]. The Network-Oriented Modelling approach is supported by a few modelling environments (in Matlab, or in Python, for example) that can be used to model conceptually in a declarative manner, without the need of programming. This code is in principle structure-preserving and follows the concepts described in the conceptual description presented in "Conceptual representations of temporal–causal network models" section below. It calculates simulation traces numerically based on the formulae discussed in "From a conceptual representation to a numerical representation" section, and in particular by means of the difference equations. A number of options for often-used combination functions are available within this software and can just be selected. However, for large-scale networks also, dedicated implementations can be developed directly using more efficient programming languages, or dedicated, optimised differential equation solvers developed to handle large systems of differential equations.
Conceptual representations of temporal–causal network models
Temporal–causal network models can be represented at two levels: by a conceptual representation and by a numerical representation. A conceptual representation of a temporal–causal network model can have a (labelled) graphical form (or an equivalent matrix form), as shown in the examples presented below. The following three elements define temporal–causal networks, and are part of a conceptual representation of a temporal–causal network model:
connection weight ω X,Y Each connection from a state X to a state Y has a connection weight ω X,Y representing the strength of the connection, often between 0 and 1, but sometimes also below 0 (negative effect).
combination function c Y (..) For each state Y (a reference to) a combination function c Y (..) is chosen to aggregate the causal impacts of other states on state Y. This can be a standard function from a library (e.g. a scaled sum function) or an own-defined function.
speed factor η Y For each state Y, a speed factor η Y is used to represent how fast a state is changing upon causal impact, usually in the [0, 1] interval.
In the first place, a conceptual representation of a temporal–causal network model involves representing in a declarative manner states and connections between them. The connections represent (causal) impacts of states on each other, as assumed to hold for the application domain addressed. Each state X is assumed to have an (activation) level that varies over time, indicated in the numerical representation by a real number X(t). In reality, not all causal relations are equally strong, so some notion of strength of a connection from a state X to a state Y is used: a connection weight ω X,Y . Combination functions can have different forms. The applicability of a specific combination rule may depend much on the type of application addressed, and even on the type of states within an application. Therefore, for the Network-Oriented Modelling approach based on temporal–causal networks a number of standard combination functions are available as options and a number of relevant properties of such combination functions have been identified; e.g. see [10], Table 10, or [1], Chapter 2, Table 2.10. Some of these standard combination functions are scaled sum, product, complementary product, max, min, and simple and advanced logistic sum functions (for some of these examples of combination functions the numerical representations are discussed in "From a conceptual representation to a numerical representation" section). These options cover elements from different existing approaches, varying from approaches considered for reasoning with uncertainty, probability, possibility or vagueness, to approaches based on neural networks; e.g. [18,19,20,21,22,23,24,25,26]. In addition, there is still the option to specify any other (non-standard) combination function.
Conceptual representations for an adaptive network
The above three concepts (connection weight, combination function, speed factor) can be considered as parameters representing characteristics in a network model. In a non-adaptive network model, these parameters are fixed over time. But to model processes by adaptive networks, not only the state levels, but also these parameters can change over time. For example, the connection weights can change over time to model evolving connections in network models. For modelling processes as adaptive networks, some of the parameters (such as connection weights) are handled in a similar manner as states. For example, see Fig. 1, where the states affect the connection between them, as happens, for example, in adaptive social networks based on the homophily principle (see "Modelling evolving social interactions by adaptive networks based on the homophily principle" section).
Conceptual representation of an example with an adaptive connection weight
This can be represented differently by considering the connection weight \( \upomega_{X,Y} \) as a state \( \Omega_{X,Y} \) that changes over time, represented by an extra node in the network. As a first step, this node for the state \( \Omega_{X,Y} \) representing \( \upomega_{X,Y} \) is added and connected; see Fig. 2 for a conceptual representation. In the new situation depicted in Fig. 2, the weight \( \upomega_{X,Y} \) is represented by a state \( \Omega_{X,Y} \) with activation values \( \Omega_{X,Y} \left( t \right) \) the same as the connection weight values ω X,Y (t) in the old situation for each t: \( \Omega_{X,Y} \left( t \right) = \upomega_{X,Y} \left( t \right) \). This state \( \Omega_{X,Y} \) is affected by both X and Y, so connections from these states to \( \Omega_{X,Y} \) are incorporated. Moreover, a connection from \( \Omega_{X,Y} \) to Y is used to represent the effect of the connection strength on Y, and a connection from \( \Omega_{X,Y} \) to itself for persistence. The weights of all these connections are assumed 1; see Fig. 2. As a next step, it is explored what combination functions are needed for \( \Omega_{X,Y} \) and Y in this new situation depicted in Fig. 2.
Graphical conceptual representation with state Ω X,Y representing an adaptive connection weight ω X,Y
First, a combination function \( {\mathbf{c}}_{{\Omega}_{X,Y}} \varvec{(..)} \) for the state \( \Omega_{X,Y} \) has to be assumed, to aggregate the impacts of X and Y, and \( \Omega_{X,Y} \) on \( \Omega_{X,Y} \). This will depend on the adaptation principle that is chosen. Next, the new combination function for Y has to be determined. Below the corresponding combination functions will be discussed in more numerical detail.
From a conceptual representation to a numerical representation
Based on a conceptual representation of a temporal–causal network model, in order to obtain a numerical representation of the network model the following concepts can be defined:
\( \text{The}\;impact\; \text{of \; state}\;X\;\text{on\;state}\;Y\;\text{at\;time}\;t\quad\quad\quad\quad\quad{\mathbf{impact}}_{X,Y} (t) \)
The impact of state X on state Y at time t is defined by
$$ {\mathbf{impact}}_{X,Y} (t) = {\boldsymbol{\upomega}}_{\varvec{X,Y}} X(t). $$
Here X(t) is the activation level of state X at t. Note that also a connection from a state Y to itself is allowed. The weight \( \upomega_{Y,Y} \) of such a connection can, for example, be used to model persistence of state Y.
\( \text{The}\;aggregated\; impact\; \text{on\;state}\;Y\;\text{at\;time}\;t\quad\quad\quad\quad\quad{\mathbf{aggimpact}}_{Y} (t) \)
When more than one causal relation affects a given state Y, these causal effects have to be combined. To this end, some way to aggregate multiple causal impacts on a state is used; this is done using a combination function c Y (..) that uses the impacts \( {\mathbf{impact}}_{{X_{i} ,Y}} (t) \) from states X 1 , …, X k on Y as input and provides one aggregated impact value out of them:
$$ {\mathbf{aggimpact}}_{Y} (t) = {\mathbf{c}}_{Y} ({\mathbf{impact}}_{{X_{1} ,Y}} (t), \ldots ,{\mathbf{impact}}_{{X_{k} ,Y}} (t)). $$
Moreover, not every state has the same extent of flexibility in responding to impact; some states respond fast, and other states may be more rigid and may respond more slowly. Therefore, a speed factor \( \boldsymbol{\upeta}_{\varvec{Y}} \) of a state Y is used for timing of effectuation of causal impacts, as shown in the following difference and differential equations:
$$ \begin{aligned} Y(t +\Delta t) & = Y(t) + {\boldsymbol{\upeta}}_{Y} \left[ {{\mathbf{aggimpact}}_{Y} (t) - Y(t)} \right]\Delta t \\ {\text{d}}Y(t)/{\text{d}}t & = {\boldsymbol{\upeta}}_{Y} \left[ {{\mathbf{aggimpact}}_{Y} (t) - Y(t)} \right]. \\ \end{aligned} $$
Given the above concepts, a conceptual representation of a temporal–causal network model can be transformed in a systematic and automated manner into a numerical representation of the model, thus obtaining the following difference and differential equation for each state Y, expressed using the basic elements \( {\boldsymbol{\upomega}}_{\varvec{X,Y}} ,{\mathbf{c}}_{\varvec{Y}} ( \ldots ) \), and η Y of a conceptual representation of the model:
$$ \begin{aligned} Y(t +\Delta t) & = Y(t) + {\boldsymbol{\upeta}}_{{\varvec{Y}}} \left[ {{\mathbf{c}}_{Y} \left( {{\boldsymbol{\upomega}}_{{{\varvec{X}}_{{\varvec{1}}} ,{\varvec{Y}}}} X_{ 1} \left( t \right), \ldots ,{\boldsymbol{\upomega}}_{{{\varvec{X}}_{{\varvec{k}}} ,{\varvec{Y}}}} X_{k} \left( t \right)} \right) - Y\left( t \right)} \right]\Delta t \\ {\text{d}}Y(t)/{\text{d}}t & = {\boldsymbol{\upeta}}_{{\varvec{Y}}} \left[ {{\mathbf{c}}_{{\varvec{Y}}} \left( {{\boldsymbol{\upomega}}_{{{\varvec{X}}_{{\varvec{1}}} ,{\varvec{Y}}}} X_{ 1} \left( t \right), \ldots ,{\boldsymbol{\upomega}}_{{{\varvec{X}}_{{\varvec{k}}} ,{\varvec{Y}}}} X_{k} \left( t \right)} \right) - Y\left( t \right)} \right]. \\ \end{aligned} $$
The numerical representations of some example combination functions are as follows:.
Numerical representation of a scaled sum combination function
In some cases, it is useful to apply a scaling factor to the sum combination function by dividing it by some scaling factor \( \lambda \):
$$ {\mathbf{c}}(V_{ 1} ,\ldots ,V_{k} ) = {\mathbf{ssum}}_{\uplambda} (V_{ 1} , \ldots ,V_{k} ) = (V_{ 1} + \cdots + V_{k} )/\uplambda\text{.} $$
In cases where this combination function is used for a state Y with \( X_{ 1} , \ldots , X_{k} \) connected to Y, then this function works as follows on the \( X_{i} \):
$$ {\mathbf{ssum}}_{\uplambda} ({\boldsymbol{\upomega}}_{{\varvec{X}}_{\varvec{1}} ,{\varvec{Y}}} X_{ 1} , \ldots , {\boldsymbol{\upomega}}_{{\varvec{X}}_{\varvec{k}} ,{\varvec{Y}}} X_{k} ) = ({\boldsymbol{\upomega}}_{{\varvec{X}}_{\varvec{1}} ,{\varvec{Y}}} X_{ 1} + \ldots + {\boldsymbol{\upomega}}_{{\varvec{X}}_{\varvec{k}} ,{\varvec{Y}}} X_{k} )/\uplambda\text{.} $$
Numerical representation of a simple logistic sum combination function
The logistic sum combination function has two closely related variants, the simple variant and the more advanced variant (see below). In these functions, τ is a threshold parameter and σ a steepness parameter. The simple logistic function is defined as:
$$ {\mathbf{c}}(V_{ 1} , \ldots , V_{k} ) = {\mathbf{slogistic}}\;(V_{ 1} , \ldots, V_{k} ) = \frac{1}{{1 + {\mathbf{e}}\varvec{ }^{{ -\upsigma(V_{1} + \cdots + V_{\text{k}} -\uptau) }} }}. $$
To indicate the dependence of \( \sigma \) and \( \tau, \) sometimes these are used as subscripts: \( {\mathbf{slogistic}}_{\sigma ,\tau } (V_{ 1} , \ldots ,V_{k} ). \)
In cases where this combination function is used for a state Y with \( X_{ 1} , \ldots , X_{k} \) connected to Y, then this function works as follows on the X i :
$$ {\mathbf{slogistic}}({\boldsymbol{\upomega}}_{{{\mathbf{X}}_{{\mathbf{1}}} {\mathbf{,Y}}}} X_{ 1} , \ldots , {\boldsymbol{\upomega}}_{{{\mathbf{X}}_{{\mathbf{k}}} {\mathbf{,Y}}}} X_{k} ) = 1/\left( {1 + {\text{e}}^{{ -\upsigma\left( {{\boldsymbol{\upomega}}_{{{\mathbf{X}}_{{\mathbf{1}}} {\mathbf{,Y}}}} X_{ 1} + \cdots + {\boldsymbol{\upomega}}_{{{\mathbf{X}}_{{\mathbf{k}}} {\mathbf{,Y}}}} X_{k} -\uptau} \right)}} } \right). $$
Numerical representation of an advanced logistic sum combination function
In the simple logistic variant, it holds slogistic \( (0, \ldots ,0) = 1/( 1+ {\mathbf{e}}^{\sigma \tau } ) \), and this is nonzero, which is undesirable property as it creates in an unintended manner activation out of no activation. This issue is compensated for in the advanced variant. The advanced logistic sum combination function is defined as
$$ {\mathbf{c}}(V_{ 1} , \ldots , V_{k} ) = {\mathbf{alogistic}}(V_{ 1} , \ldots , V_{k} ) = \left[ {\frac{1}{{1 + {\text{e}}^{{ - {\sigma (}V_{ 1} + \ldots + V_{k} { - \tau )}}} }} - \frac{1}{{1 + {\text{e}}^{{ \upsigma \uptau }} }}} \right]\left( {1 + {\text{e}}^{{ -\upsigma \uptau }} } \right) $$
To indicate the dependence of \( \sigma \) and \( \tau, \) sometimes these are used as subscripts:
$$ {\mathbf{alogistic}}_{{\upsigma, \uptau }} (V_{ 1} , \ldots , V_{k} ) $$
For an overview of a number of standard combination functions, see Table 1.
Table 1 Overview of a number of standard combination functions
Numerical representations for an adaptive network
In the simple example depicted in Fig. 1, Y has another impact from Z, besides the impact from X. Then in the new situation depicted in Fig. 2, there are not just two but three states with impact on Y, namely X, Z and \( \Omega_{X,Y} \). This requires a new combination function \( {\mathbf{c}}^*_{\varvec{Y}} (V_{1} , V_{2} , W) \) for Y with three arguments, which is applied to the impacts \( X(t),\varvec{\upomega}_{Z,X} Z(t) \) and \( \Omega_{X,Y} \left( t \right) \) on Y, obtaining aggregated impact \( {\mathbf{c}}^*_{Y} (X\left( t \right),\upomega_{Z,X} Z(t),\Omega_{X,Y} \left( t \right)) \). This aggregated impact is equal to \( {\mathbf{c}}_{Y} (\upomega_{X,Y} (t)X(t), {\boldsymbol{\upomega}}_{Z,Y} Z(t)) \) in the previous model representation depicted in Fig. 1. Therefore,
$$ {\mathbf{c}}^*_{\varvec{Y}} (V_{ 1} ,\;V_{ 2} ,\;W) \, = \, {\mathbf{c}}_{\varvec{Y}} (WV_{ 1} ,\;V_{ 2} ) $$
For example, if in the situation of Fig. 1 \( {\mathbf{c}}_{\varvec{Y}} (V_{ 1} , \, V_{ 2} ) \) is the sum function \( V_{ 1} + \, V_{ 2} \), then \( {\mathbf{c}}^*_{\varvec{Y}} (V_{ 1} , \, V_{ 2} , \, W) = \, WV_{ 1} + \, V_{ 2} \), which is a combination of a product and a sum function. More in general, suppose in total there are k states X i with impact on Y, according to combination function \( {\mathbf{c}}_{\varvec{Y}} (V_{ 1} , \ldots , V_{k} ) \). If all these connections are adaptive, then the new combination function becomes
$$ {\mathbf{c}}^*_{\varvec{Y}} (V_{ 1} ,\ldots , V_{k} , W_{ 1} ,\ldots , W_{k} ) = {\mathbf{c}}_{\varvec{Y}} (W_{ 1} V_{ 1} ,\ldots , W_{k} V_{k} ) $$
Modelling evolving social interactions by adaptive networks based on the homophily principle
Next an adaptive temporal–causal network model is discussed to model evolving social interactions based on the homophily principle. According to this principle, also indicated as 'birds of a feather flock together', connections are strengthened if the connected states are similar. For example, when two persons both like the same type of music, movies, drinks, and parties, they may strengthen their connection. For the current model, the dynamic connection weights \( \upomega_{{X_{A} ,X_{B} }} \) from state X A of person A to state X B of person B are assumed to change over time based on the principle that the closer the activation levels of the states of the interacting persons, the stronger the mutual connections between the persons will become, and the higher the difference between the activation levels, the weaker they will become. For a conceptual representation, see Fig. 3.
Graphical conceptual representation of an adaptive temporal–causal network model for the homophily principle
As discussed in "Network-Oriented Modelling by temporal–causal networks" section, \( \upomega_{{X_{A} ,X_{B} }} \) can be represented by state \( \Omega_{{X_{A} ,X_{B} }} \) and the weights of the connections involving \( \Omega_{{X_{A} ,X_{B} }} \) are assumed 1: the weights of the connections from \( X_{A} \) and \( X_{B} \) to \( \Omega_{{X_{A} ,X_{B} }} \), and from \( \Omega_{{X_{A} ,X_{B} }} \) to \( X_{B} \) and to itself. Based on this according to the temporal–causal network approach, the homophily principle may be formalised using the following general format and a combination function \( {\mathbf{c}}_{A,B} (V_{ 1} , \, V_{ 2} , \, W) \) that still has to be determined:
$$ \begin{aligned} \Omega_{{X_{A} ,X_{B} }} (t + \Delta t) & = \Omega_{{X_{A} ,X_{B} }} \left( t \right) + {\boldsymbol{\upeta}}_{{\Omega_{X_{A} ,X_{B}} }} \left[ {{\mathbf{c}}_{\Omega_{{X_{A} ,X_{B} }}} (X_{A} \left( t \right),\;X_{B} \left( t \right),\;\Omega_{{X_{A} ,X_{B} }} ) - \Omega_{{X_{A} ,X_{B} }} } \right]\Delta t \\ {{{\mathbf{d}}\Omega_{{X_{A} ,X_{B} }} } \mathord{\left/ {\vphantom {{{\mathbf{d}}\Omega_{{X_{A} ,X_{B} }} } {{\mathbf{d}}t}}} \right. \kern-0pt} {{\mathbf{d}}t}} & = {\boldsymbol{\upeta}}_{\Omega_{{X_{A} ,X_{B} }}} \left[ {{\mathbf{c}}_{{\Omega_{{X_{A} ,X_{B} }} }} (X_{A} ,\;X_{B} ,\;\Omega_{{X_{A} ,X_{B} }} ) - \Omega_{{X_{A} ,X_{B} }} } \right] \\ \end{aligned} $$
Note that the connection weight \( \Omega_{{X_{A} ,X_{B} }} \) increases when \( {\mathbf{c}}_{{\Omega_{{X_{A}, X_{B} }} }} \left( {X_{A} \left( t \right),\;X_{B} \left( t \right),\Omega_{{X_{A} ,X_{B} }} \left( t \right)} \right) > \Omega_{{X_{A} ,X_{B} }} \left( t \right), \) decreases when \( {\mathbf{c}}_{{\Omega_{{X_{A}, X_{B} }} }} \left( {X_{A} \left( t \right),\;X_{B} \left( t \right),\Omega_{{X_{A} ,X_{B} }} \left( t \right)} \right) < \Omega_{{X_{A} ,X_{B} }} \left( t \right) \) and stays the same when \( {\mathbf{c}}_{{\Omega_{{X_{A}, X_{B} }} }} \left( {X_{A} \left( t \right),\;X_{B} \left( t \right),\Omega_{{X_{A} ,X_{B} }} \left( t \right)} \right) = \Omega_{{X_{A} ,X_{B} }} \left( t \right). \)
Examples of such combination functions can be obtained when a threshold value \( \tau_{{\Omega_{{X_{A} ,X_{B} }} }} \) is assumed such that the connection weight \( \Omega_{{X_{A} ,X_{B} }} \) becomes stronger when \( \left| {X_{A} \left( t \right) - X_{B} \left( t \right)} \right| < \tau_{{\Omega_{{X_{A} ,X_{B} }} }} \) (levels of \( X_{A} \) and \( X_{B} \) close to each other) and weaker when \( \left| {X_{A} \left( t \right) - X_{B} \left( t \right)} \right| > \tau_{{\Omega_{{X_{A} ,X_{B} }} }} \) (levels of \( X_{A} \) and \( X_{B} \) not so close to each other). The following is an example which is linear in \( X_{A} \left( t \right)\,{\text{and}}\,X_{B} (t) \):
$$ {\mathbf{c}}_{{\Omega_{{X_{A} ,X_{B} }} }} (X_{A} \left( t \right), \, X_{B} \left( t \right),\Omega_{{X_{A} ,X_{B} }} \left( t \right)) = \Omega_{{X_{A} ,X_{B} }} \left( t \right) \, + \beta (\tau_{{\Omega_{{X_{A} ,X_{B} }} }} - |X_{A} \left( t \right) - X_{B} \left( t \right)|) $$
The factor \( \beta \) can be made dependent on \( \Omega_{{X_{A} ,X_{B} }} \left( t \right) \), to keep values of \( \Omega_{{X_{A} ,X_{B} }} \left( t \right) \) within the [0, 1] interval: \( \beta = \alpha \Omega_{{X_{A} ,X_{B} }} \left( t \right)\left( {1 - \Omega_{{X_{A} ,X_{B} }} \left( t \right)} \right) \), with α an amplification parameter. This makes the combination function
$$ {\mathbf{c}}_{{\Omega_{{X_{A} ,X_{B} }} }} (V_{ 1} ,V_{ 2} , \, W) = W + \alpha W( 1- W) \, (\tau_{{\Omega_{{X_{A} ,X_{B} }} }} - \, |V_{ 1} - V_{ 2} |) $$
where \( V_{ 1} ,V_{ 2} \) refer to \( X_{A} ,X_{B} \) and W to \( \Omega_{{X_{A} ,X_{B} }} \). Thus, we obtain the following:
$$ \begin{aligned} \Omega_{{X_{A} ,X_{B} }} (t + \Delta t) & = \Omega_{{X_{A} ,X_{B} }} (t) + {\boldsymbol{\upeta}}_{{\Omega_{{X_{A} ,X_{B} }} }} [\alpha \Omega_{{X_{A} ,X_{B} }} \left( t \right)( 1- \Omega_{{X_{A} ,X_{B} }} \left( t \right)) \, (\tau_{{\Omega_{{X_{A} ,X_{B} }} }} - |X_{A} (t) - X_{B} \left( t \right)|)]\Delta t \\ {\text{d}}\Omega_{{X_{A} ,X_{B} }} / {\text{d}}t \, & = {\boldsymbol{\upeta}}_{{\Omega_{{X_{A} ,X_{B} }} }} [\alpha \Omega_{{X_{A} ,X_{B} }} \left( t \right)( 1- \Omega_{{X_{A} ,X_{B} }} \left( t \right)) \, (\tau_{{\Omega_{{X_{A} ,X_{B} }} }} - |X_{A} (t) - X_{B} \left( t \right)|)]. \\ \end{aligned} $$
The combination function for \( X_{B} \) can be found in the same way as in the "Network-Oriented Modelling by temporal–causal networks" section for Y.
In Figs. 4 and 5, as an illustration, an example simulation for this homophily model is shown, based on a (fully connected) example network of 10 states X 1 to X 10, with the initial values of the connection weights shown in Table 2. For the contagion between states, a dynamic scaled sum function has been used in which, at each point in time, the scaling factor is equal to the sum of the connection weights involved. The homophily threshold \( \tau \) was set at 0.025, and the amplification factor α at 20. Speed factors for states were 0.5 and for connections 0.3.
State values for the homophily example simulation showing emerging clusters
Some of the connection weights for the homophily example simulation
Table 2 Initial connection weights for the homophily example simulation
All connection weights approximate either 0 or 1, as can be seen for a few examples (of the 90 connections) in Fig. 5. In Fig. 4, it is shown that clustering emerges, in this case in 3 clusters that in the end each are fully connected by connection weights 1, and the connection weights between states from different clusters have become practically 0. That such patterns always occur will be analysed further in the "Mathematical analysis for the homophily principle" section.
Modelling evolving social interactions by adaptive networks based on the 'more becomes more' principle
Another type of model for a dynamic connection from a person B to A takes into account to which extent other persons C connect to person A. The idea behind this is that somebody who is very popular seems worth connecting to. Sometimes this is called the 'more becomes more' principle, and in a wider context it relates to what sometimes is called 'the rich get richer' (Simon [27]), 'cumulative advantage' (Price [28]), 'the Matthew effect' (Merton [29]) or 'preferential attachment' (Barbasi [30]). For example, if B is followed by many others C on Twitter, then B seems to be interesting to follow for A as well. As the connections of others to B may change over time, this will imply that A also will have a dynamic connection to B, and in turn this connection will affect the connection of others to B over time as well. This can be modelled taking into account the weights \( \upomega_{{C_{i} ,B}} \) for i = 1,…, k of all connections from others C i to B as shown in Fig. 6 in conceptual representation and in numerical representation as follows:
$$ \begin{aligned} {\mathbf{d}}\upomega_{A,B} /{\mathbf{d}}t & = {\boldsymbol{\upeta}}_{\varvec{A,B}} [{\mathbf{c}}_{\varvec{A,B}} (\upomega_{{C_{1} ,B}} \ldots ,\upomega_{{C_{k} ,B}} ) - \upomega_{A,B} ] \\ \upomega_{A,B} (t + \Delta t) & = \upomega_{A,B} \left( t \right) + {\boldsymbol{\upeta}}_{\varvec{A,B}} [{\mathbf{c}}_{\varvec{A,B}} (\upomega_{{C_{1} ,B}} \left( t \right), \ldots ,\upomega_{{C_{k} ,B}} \left( t \right)) - \upomega_{A,B} \left( t \right)]. \\ \end{aligned} $$
Conceptual representation of an adaptive temporal–causal network model for the 'more becomes more' principle
Here c A,B (…) is a combination function for the values \( \upomega_{{C_{1} ,B}} , \ldots ,\upomega_{{C_{k} ,B}} \); for example, a logistic sum function or a scaled sum function with scale factor the number k of other persons connected to B. Note that the latter combination function only takes into account the average strengths of the connections, not the total number of them.
$$ \begin{aligned} {\mathbf{ssum}}_{\uplambda} (V_{ 1} , \ldots ,V_{k} ) & = (V_{ 1} + \cdots + V_{k} )/{\uplambda} \\ {\mathbf{alogistic}}(V_{ 1} , \ldots ,V_{k} ) & = \left[ {\frac{1}{{1 + {\mathbf{e}}\varvec{ }^{{ - \sigma (V_{1} + \cdots + V_{\text{k}} - \tau )}} }} - \frac{1}{{1 + {\mathbf{e}}^{\sigma \tau } \varvec{ } }}} \right]( 1+ {\mathbf{e}}^{ - \sigma \tau } ). \\ \end{aligned} $$
Note that a network modelling the initiation of connections is not automatically a network indicating social contagion; this will depend on the application considered. For example, a network modelling a connection from A to B when A is following B on Twitter will not play a role in social contagion from A to B. For social contagion, the opposite network plays a role where a connection from A to B occurs when A is followed by B, which is not initiated by A but by B: on Twitter and most other social media you cannot appoint your own followers. As another example, when A often contacts B for advice, and this advice is often taken over by A, then the initiation connection is from A to B but the contagion connection is from B to A. In other cases, it may be different. For example, if A wants to announce an event or new product, he or she can choose an occasion where many others will see the message; for example, posting it on a suitable forum; in such a case both the initiation and the social contagion are directed from A to the others.
Mathematical analysis of temporal–causal network models
In this section, we discuss how some types of dynamic properties of adaptive temporal–causal network models can be analysed mathematically, in particular, stationary points and monotonicity. These are basic concepts that also can be found in [1], chapter 12 or [31] . A stationary point of a state occurs at some point in time if for this time point no change occurs: the graph is horizontal at that point. Stationary points are usually maxima or minima (peaks or dips) but sometimes also other stationary points may occur. An equilibrium occurs when for all states no change occurs. From the difference or differential equations describing the dynamics for a model, it can be analysed when stationary points or equilibria occur. Moreover, it can be found when a certain state is increasing or decreasing, when a state is not in a stationary point or equilibrium. First a definition for these notions.
Definition (stationary point, increase, decrease, and equilibrium)
A state Y has a stationary point at t if \( {\mathbf{d}}Y\left( t \right)/{\mathbf{d}}t = 0. \)
A state Y is increasing at t if \( {\mathbf{d}}Y\left( t \right)/{\mathbf{d}}t > 0. \)
A state Y is decreasing at t if \( {\mathbf{d}}Y\left( t \right)/{\mathbf{d}}t < 0. \)
The model is in equilibrium at t if every state Y of the model has a stationary point at t. This equilibrium is attracting when for any state Y, all values of Y in some neighbourhood of the equilibrium value increase when the value is below the equilibrium value and decrease when the value is above the equilibrium value.
A question that can be addressed is whether observations based on one or more simulation experiments are in agreement with a mathematical analysis. If it is found out that the observations are in agreement with the mathematical analysis, then this provides some extent of corroboration that the implemented model is correct. If they turn out not to be in agreement with the mathematical analysis, then this indicates that probably there is something wrong, and further inspection and correction has to be initiated. Considering the differential equation for a temporal–causal network model, more specific criteria can be found:
$$ {\text{d}}Y\left( t \right)/{\text{d}}t \, = {\boldsymbol{\upeta}}_{Y} [{\mathbf{c}}_{Y} ({\boldsymbol{\upomega}}_{{X_{1} ,Y}} X_{ 1} \left( t \right), \, \ldots ,{\boldsymbol{\upomega}}_{{X_{k} ,Y}} X_{k} \left( t \right)) \, - Y\left( t \right)] $$
where X 1,…, X k are the states with connections to Y. For example, it can be concluded that
$$ {\text{d}}Y\left( t \right)/{\text{d}}t > 0 \Leftrightarrow {\mathbf{c}}_{Y} ({\boldsymbol{\upomega}}_{{X_{1} ,Y}} X_{ 1} \left( t \right), \, \ldots ,{\boldsymbol{\upomega}}_{{X_{k} ,Y}} X_{k} \left( t \right)) > Y\left( t \right). $$
In this manner, the following criteria can be found:
Criteria for increase, decrease, stationary point and equilibrium
Let Y be a state and \( X_{ 1} , \ldots ,X_{k} \) the states connected toward Y. Then, the following hold:
$$\begin{aligned}Y\;\text{has\;a\;stationary\;point\;at}\;t & \Leftrightarrow{\mathbf{c}}_{Y} ({\boldsymbol{\upomega}}_{{X_{1} ,Y}} X_{ 1} \left( t \right), \, \ldots ,{\boldsymbol{\upomega}}_{{X_{k} ,Y}} X_{k} \left( t \right)) = Y(t) \\Y\;\text{is\;increasing\; at}\;t& \Leftrightarrow{\mathbf{c}}_{Y} ({\boldsymbol{\upomega}}_{{X_{1} ,Y}} X_{ 1} \left( t \right), \, \ldots ,{\boldsymbol{\upomega}}_{{X_{k} ,Y}} X_{k} \left( t \right)) > Y(t) \\Y\;\text{is\;decreasing\; at}\;t& \Leftrightarrow{\mathbf{c}}_{Y} ({\boldsymbol{\upomega}}_{{X_{1} ,Y}} X_{ 1} \left( t \right), \, \ldots ,{\boldsymbol{\upomega}}_{{X_{k} ,Y}} X_{k} \left( t \right)) < Y(t)\\\text{The\;model\;is\;in\;equilibrium\;a}\;t& \Leftrightarrow {\mathbf{c}}_{Y} ({\boldsymbol{\upomega}}_{{X_{1} ,Y}} X_{ 1} \left( t \right), \, \ldots ,{\boldsymbol{\upomega}}_{{X_{k} ,Y}} X_{k} \left( t \right)) = Y(t)\; \text{for}\;\boldsymbol{\it{every}}\;\text{state}\;Y(\text{i.e.\;a\;joint\;stationary\;state})\\\end{aligned}$$
Note that these criteria can immediately be found from a conceptual representation of a temporal–causal network model, as long as the referred combination function is known. Using the above criteria, no further numerical representation is needed of the difference or differential equations, for example. From these criteria, more insight can be obtained about the behaviour of the network model, in particular which stationary points are possible for a state in the model, and which equilibria are possible for the whole model. Sometimes, the stationary point equation can be rewritten into an equation of the form Y(t) = .. such that Y(t) does not occur on the right-hand side. In the "Mathematical analysis for the homophily principle" and "Mathematical analysis for the 'more becomes more' principle" sections, examples of this are shown.
The criteria can also be used to verify (the implementation of) the model based on inspection of stationary points or equilibria, in two different manners A and B. Note that in a given simulation the stationary points that are identified are usually approximately stationary; how closely they are approximated depends on different aspects, for example, on the step size, or on how long the simulation is done.
A. Verification by checking stationary points through substitution of the values from a simulation in the criterion
Generate a simulation.
Consider any state Y with a stationary point at any time point t and states X 1 , …, X k affecting it.
Substitute the values \( Y\left( t \right) \) and \( X_{ 1} \left( t \right) \) , …, \( X_{k} \left( t \right) \) in the criterion \( {\mathbf{c}}_{Y} (\upomega_{{X_{1} ,Y}} X_{ 1} (t), \ldots ,\upomega_{{X_{k} ,Y}} X_{k} (t)) = Y(t). \)
If the equation holds (for example, with an accuracy <0.05), then this test succeeds, otherwise it fails
If this test fails, then it has to be explored were the error can be found
Note that this method A. works without having to solve the equations, only substitution takes place; therefore it works for any choice of combination function. Moreover, note that the method also works when the values of the states fluctuate, for example according to a recurring pattern (a limit cycle). In such cases for each state, there are maxima (peaks) and minima (dips), which also are stationary points to which the method can be applied; here it is important to choose a small step size as each stationary point occurs at one time point only. There is still another method B. possible that can be applied sometimes; it is based on solving the equations for the stationary point values by symbolic rewriting. This can provide explicit expressions for stationary point values in terms of the parameters of the model. Such expressions can be used to predict equilibrium values for specific simulations, based on the choice of parameter values.
B. Verification by solving the equilibrium equations and comparing predicted equilibrium values to equilibrium values in a simulation
Consider the equilibrium equations for all states Y:
$$ {\mathbf{c}}_{Y} (\upomega_{{X_{1} ,Y}} X_{ 1} (t), \ldots ,\upomega_{{X_{k} ,Y}} X_{k} (t)) = Y(t). $$
Leave the t out and denote the values as constants
$$ {\mathbf{c}}_{Y} (\upomega_{{X_{1} ,Y}} \underline{\mathbf{X}}_{ 1} , \ldots ,\upomega_{{{X}_{k} ,Y}} \underline{\mathbf{X}}_{ k} ) = \underline{Y} . $$
An equilibrium is a solution \( \underline{\mathbf{X}}_{ 1} , \ldots ,\underline{\mathbf{X}}_{ k} \) of the following set of n equilibrium equations in the n states \( X_{ 1} , \, \ldots ,X_{n} \) of the model:
$$ \begin{aligned} & {\mathbf{c}}_{{X_{1} }} (\upomega_{{X_{1}, X_{1} }} \underline{{{\mathbf{X}}_{1} }} , \ldots ,\upomega_{{X_{n} ,X_{1} }} \underline{{{\mathbf{X}}_{n} }} ) = \underline{{{\mathbf{X}}_{1} }} \hfill \\ & \ldots \hfill \\ & {\mathbf{c}}_{{X_{n} }} (\upomega_{{X_{ 1} ,X_{n} }} \underline{{{\mathbf{X}}_{1} }} , \ldots ,\upomega_{{X_{n} ,X_{n} }} \underline{{{\mathbf{X}}_{n} }} ) = \underline{{{\mathbf{X}}_{n} }} \hfill \\ \end{aligned} $$
Solve these equations mathematically in an explicit analytical form: for each state X i a mathematical formula X i = … in terms of the parameters of the model (connection weights and parameters in the combination function \( {\mathbf{c}}_{{X}_{i}}\) (..), such as the steepness σ and threshold τ in a logistic sum combination function); more than one solution is possible.
Identify equilibrium values in this simulation.
If for all states Y, the predicted value Y from a solution of the equilibrium equations equals the value for Y obtained from the simulation (for example, with an accuracy <0.05), then this test succeeds, otherwise it fails.
If this test fails, then it has to be explored where the error can be found.
For more details, see [1], chapter 12, or [31]. This method B. provides more, but a major drawback is that it cannot be applied in all situations; this depends on the chosen combination functions: e.g. for logistic functions, it does not work.
Mathematical analysis for the homophily principle
In the "Modelling evolving social interactions by adaptive networks based on the 'more becomes more' principle" section, it was shown how the homophily principle for evolving social interaction may be modelled using a combination function:
$$ {\mathbf{c}}_{{\Omega_{{X_{A} ,X_{B} }} }} (V_{ 1} ,V_{ 2} ,W) = W + W\left( { 1- W} \right) \, \left( {\tau_{{\Omega_{{X_{A} ,X_{B} }} }} - |V_{ 1} - V_{ 2} |} \right) $$
In this section, we analyse which stationary points can occur for \( \Omega_{{X_{A} ,X_{B} }} \), according to the approach described in "Mathematical analysis of temporal–causal network models". For this case, the criterion from the "Mathematical analysis of temporal–causal network models" section for a stationary point is
$$ \begin{aligned} & {\mathbf{c}}_{{\Omega_{{X_{A} ,X_{B} }} }} \left( {X_{A} \left( t \right), \, X_{B} \left( t \right),\Omega_{{X_{A} ,X_{B} }} \left( t \right)} \right) = \Omega_{{X_{A} ,X_{B} }} \left( t \right) \Leftrightarrow \hfill \\ & \Omega_{{X_{A} ,X_{B} }} \left( t \right)\left( { 1- \Omega_{{X_{A} ,X_{B} }} \left( t \right)} \right)\left( {\tau_{{\Omega_{{X_{A} ,X_{B} }} }} - |X_{A} \left( t \right) - X_{B} \left( t \right)|} \right) = 0 \hfill \\ \end{aligned} $$
Clearly, for \( \Omega_{{X_{A} ,X_{B} }} (t) = 0 \) or \( \Omega_{{X_{A} ,X_{B} }} (t) = 1, \) one of the left-hand side factors in this condition is 0. In contrast, when \( 0 < \Omega_{{X_{A} ,X_{B} }} (t) < 1, \) the right-hand factor should equal 0:
$$ \tau_{{\Omega_{{X_{A} ,X_{B} }} }} - |X_{A} \left( t \right) - X_{B} \left( t \right)| = 0 \Leftrightarrow \left| {X_{A} \left( t \right) - X_{B} \left( t \right) \, } \right| = \tau_{{\Omega_{{X_{A} ,X_{B} }} }} . $$
Therefore, in principle, there are three types of stationary points for \( \Omega_{{X_{A} ,X_{B} }} (t) \).
Stationary points for \( \Omega_{{X_{A} ,X_{B} }} (t) \):
$$ \Omega_{{X_{A} ,X_{B} }} (t) = 0\,{\text{or}}\,\Omega_{{X_{A} ,X_{B} }} (t) = 1\,{\text{or}}\,|X_{A} \left( t \right) - X_{B} \left( t \right)| = \tau_{{\Omega_{{X_{A} ,X_{B} }} }} \,{\text{and}}\,\Omega_{{X_{A} ,X_{B} }} (t) \text{\,have\,any\,value}.$$
Similarly, the following can be found.
Increasing \( \Omega_{{X_{A} ,X_{B} }} (t) \)
$$ {\text{d}}\Omega_{{X_{A} ,X_{B} }} (t)/{\text{d}}t > 0 \Leftrightarrow (\tau_{{\Omega_{{X_{A} ,X_{B} }} }} - |X_{A} \left( t \right) - X_{B} \left( t \right)|) > 0 \Leftrightarrow \left| {X_{A} \left( t \right) - X_{B} \left( t \right) \, } \right| < \tau_{{\Omega_{{X_{A} ,X_{B} }} }} $$
Decreasing \( \Omega_{{X_{A} ,X_{B} }} (t) \)
$$ {\text{d}}\Omega_{{X_{A} ,X_{B} }} \left( t \right)/{\text{d}}t < 0\, \Leftrightarrow \left( {\tau_{{\Omega_{{X_{A} ,X_{B} }} }} - \left| {X_{A} \left( t \right) - X_{B} \left( t \right)} \right|} \right) < 0 \Leftrightarrow \left| {X_{A} \left( t \right) - X_{B} \left( t \right)} \right| > \tau_{{\Omega_{{X_{A} ,X_{B} }} }} $$
This shows that for cases that \( \left| {X_{A} \left( t \right) - X_{B} \left( t \right) \, } \right| < \tau_{{\Omega_{{X_{A} ,X_{B} }} }} \) the connection keeps on becoming stronger until \( \Omega_{{X_{A} ,X_{B} }} (t) \) approaches 1. Similarly for cases that \( |X_{A} \left( t \right) - X_{B} \left( t \right)| > \tau_{{\Omega_{{X_{A} ,X_{B} }} }} \) the connection keeps on becoming weaker until \( \Omega_{{X_{A} ,X_{B} }} (t) \) approaches 0. This implies that \( \Omega_{{X_{A} ,X_{B} }} (t) = \, 0 \) and \( \Omega_{{X_{A} ,X_{B} }} (t) = { 1} \) can both become attracting, but under different circumstances concerning the values of \( X_{A} \left( t \right) \) and \( X_{B} \left( t \right) \). In [1], chapter 11, section 11.7 for such an adaptive network model, an example simulation is shown where indeed the connection weights all converge to 0 or 1, and during this process clusters are formed of persons with equal levels of their state; see also [32].
Mathematical analysis for the 'more becomes more' principle
The criterion for stationary points applied to the adaptive network model for the 'more becomes more' principle is the following:
$$ {\mathbf{c}}_{A,B} \left( {\upomega_{{C_{ 1} ,B}} \left( t \right), \ldots ,\upomega_{{C_{k} , \, B}} \left( t \right)} \right) = \upomega_{A,B} \left( t \right) $$
where \( C_{ 1} , \ldots ,C_{k} \), and A are the states connected to B. For a joint stationary point, this criterion applies to any state connected to B. Renaming A by \( C_{k + 1} \) this can also be formulated by the following set of \( k + 1 \) equations for \( i = { 1}, \ldots ,k + 1 \):
$$ {\mathbf{c}}_{{C_{i} ,B}} \left( {\upomega_{{C_{ 1} ,B}} \left( t \right), \ldots ,\upomega_{{C_{i - 1} ,B}} \left( t \right),\upomega_{{C_{i + 1} ,B}} \left( t \right), \ldots ,\upomega_{{C_{k + 1} ,B}} \left( t \right)} \right) = \upomega_{{C_{i} ,B}} \left( t \right) $$
or written out:
$$ \begin{aligned} & {\mathbf{c}}_{{C_{{\mathbf{1}}} ,B}} (\upomega_{{C_{ 2} ,B}} (t), \ldots ,\upomega_{{C_{{k + {\mathbf{1}}}} ,B}} (t)) = \upomega_{{C_{{\mathbf{1}}} ,B}} (t) \hfill \\ & {\mathbf{c}}_{{C_{{\mathbf{2}}} ,B}} (\upomega_{{C_{{\mathbf{1}}} ,B}} (t),\upomega_{{C_{ 3} ,B}} (t), \ldots ,\upomega_{{C_{{k + {\mathbf{1}}}} ,B}} (t)) = \upomega_{{C_{2} ,B}} (t) \hfill \\ & \ldots \hfill \\ & {\mathbf{c}}_{{C_{{k + {\mathbf{1}}}} ,B}} (\upomega_{{C_{{\mathbf{1}}} ,B}} (t), \ldots ,\upomega_{Ck,B} (t)) = \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} (t) \hfill \\ \end{aligned} $$
If for the combination function \( {\mathbf{c}}_{\boldsymbol{{C_{i} ,B}}} \) (..) the scaled sum function is chosen with scaling factor the number k, this provides the following set of k + 1 linear equations for a joint stationary state for the connections to B:
$$ \begin{aligned} & (\upomega_{{C_{{\mathbf{2}}} ,B}} (t) + \cdots + \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} (t))/k = \upomega_{{C_{{\mathbf{1}}} ,B}} (t) \hfill \\ & (\upomega_{{C_{{\mathbf{1}}} ,B}} (t) + \upomega_{{C_{ 3} ,B}} (t) + \cdots + \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} (t))/k = \upomega_{{C_{2} ,B}} (t) \hfill \\ & \ldots \hfill \\ & (\upomega_{{C_{ 1} ,B}} (t) + \cdots + \upomega_{{C_{k} ,B}} (t))/k = \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} (t) \hfill \\ \end{aligned} $$
By multiplying both sides by k this provides
$$ \begin{aligned} & (\upomega_{{C_{{\mathbf{2}}} ,B}} (t) + \cdots + \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} (t)) = k\upomega_{{C_{ 1} ,B}} (t) \hfill \\ & (\upomega_{{C_{{\mathbf{1}}} ,B}} (t) + \upomega_{{C_{ 3} ,B}} (t) + \cdots + \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} (t)) = k\upomega_{{C_{ 2} ,B}} (t) \hfill \\ & \ldots \hfill \\ & (\upomega_{{C_{ 1} ,B}} (t) + \cdots + \upomega_{{C_{k} ,B}} (t)) = k\upomega_{{C_{{k + {\mathbf{1}}}} ,B}} t \hfill \\ \end{aligned} $$
This set of equations can be solved easily. For each i, adding \( \upomega_{{C_{i} ,B}} \left( t \right) \) to both sides of the ith equation yields
$$ \begin{aligned} & \upomega_{{C_{ 1} ,B}} \left( t \right) + \upomega_{{C_{2} ,B}} \left( t \right) + \cdots + \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} \left( t \right) = k\upomega_{{C_{ 1} ,B}} \left( t \right) + \upomega_{{C_{ 1} ,B}} \left( t \right)) = (k + 1)\upomega_{{C_{ 1} ,B}} \left( t \right) \\ & \upomega_{{C_{ 1} ,B}} \left( t \right) + \upomega_{{C_{2} ,B}} \left( t \right) + \cdots + \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} \left( t \right) = k\upomega_{{C_{2} ,B}} \left( t \right) + \upomega_{{C_{2} ,B}} \left( t \right) = (k + 1)\upomega_{{C_{2} ,B}} \left( t \right) \\ & \ldots \\ & \upomega_{{C_{ 1} ,B}} \left( t \right) + \upomega_{{C_{2} ,B}} \left( t \right) + \cdots + \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} \left( t \right) = k\upomega_{{C_{{k + {\mathbf{1}}}} ,B}} \left( t \right) + \upomega_{{C_{{k + {\mathbf{1}}}} ,B}} \left( t \right) = \, (k + 1)\upomega_{{C_{{k + {\mathbf{1}}}} ,B}} \left( t \right). \\ \end{aligned} $$
As all left-hand sides are equal now, it follows that the right-hand sides are equal as well, so for a joint stationary point
$$ \upomega_{{C_{i} ,B}} \left( t \right) = \upomega_{{C_{j} ,B}} \left( t \right) $$
for all i and j. Therefore in a joint stationary state for all connections \( \upomega_{{C_{i} ,B}} \) to B they have the same weight value.
By a slightly different argument a similar conclusion can be drawn when not a scaled sum combination function but a logistic combination function is chosen.
The aggregated impact on the connection weight \(\upomega_{{C_{i}}, B} \) is given by
$$ \begin{aligned}& {\mathbf{alogistic}}\left( {\upomega_{{C_{ 1} ,B}} (t), \ldots ,\upomega_{{C_{{i - {\mathbf{1}}}} ,B}} (t),\upomega_{{C_{{i + {\mathbf{1}}}} ,B}} (t), \ldots ,\upomega_{{C_{k} ,B}} \left( t \right)} \right) \\ &\quad = \left[ {\left( {{1 \mathord{\left/ {\vphantom {1 {\left( {1 + {\mathbf{e}}^{{ - \sigma \left( {\upomega_{{C_{ 1} ,B}} + \cdots + \upomega_{{C_{i - 1} ,B}} + \upomega_{{C_{i + 1} ,B}} + \cdots + \upomega_{{C_{k} ,B}} - \tau } \right)}} } \right)}}} \right. \kern-0pt} {\left( {1 + {\mathbf{e}}^{{ - \sigma \left( {\upomega_{{C_{ 1} ,B}} + \cdots + \upomega_{{C_{i - 1} ,B}} + \upomega_{{C_{i + 1} ,B}} + \cdots + \upomega_{{C_{k} ,B}} - \tau } \right)}} } \right)}}} \right) - \left( {{1 \mathord{\left/ {\vphantom {1 {( 1+ {\mathbf{e}}^{\sigma \tau } )}}} \right. \kern-0pt} {( 1+ {\mathbf{e}}^{\sigma \tau } )}}} \right)} \right]\left( { 1+ {\mathbf{e}}^{ - \sigma \tau } } \right) \\ &\quad = \left[ {\left( {{1 \mathord{\left/ {\vphantom {1 {\left( {1 + {\mathbf{e}}^{{ - \sigma \left( {\upomega_{{C_{ 1} ,B}} + \cdots + \upomega_{{C_{i - 1} ,B}} + \upomega_{{C_{i} ,B}} + \upomega_{{C_{i + 1} ,B}} + \cdots + \upomega_{{C_{k} ,B}} - \tau - \upomega_{{C_{i} ,B}} } \right)}} } \right)}}} \right. \kern-0pt} {\left( {1 + {\mathbf{e}}^{{ - \sigma \left( {\upomega_{{C_{ 1} ,B}} + \cdots + \upomega_{{C_{i - 1} ,B}} + \upomega_{{C_{i} ,B}} + \upomega_{{C_{i + 1} ,B}} + \cdots + \upomega_{{C_{k} ,B}} - \tau - \upomega_{{C_{i} ,B}} } \right)}} } \right)}}} \right) - \left( {{1 \mathord{\left/ {\vphantom {1 {( 1+ {\mathbf{e}}^{\sigma \tau } )}}} \right. \kern-0pt} {( 1+ {\mathbf{e}}^{\sigma \tau } )}}} \right)} \right] \\ &\quad\quad \times \left( { 1+ {\mathbf{e}}^{ - \sigma \tau } } \right) \\ &\quad = \left[ {\left( {{1 \mathord{\left/ {\vphantom {1 {\left( { 1+ {\mathbf{e}}^{{ - \sigma \left( {\Omega - \upomega_{{C_{i} ,B}} } \right)}} } \right)}}} \right. \kern-0pt} {\left( { 1+ {\mathbf{e}}^{{ - \sigma \left( {\Omega - \upomega_{{C_{i} ,B}} } \right)}} } \right)}}} \right) -\upmu} \right]v \\ &\quad = \left[ {\left( {{1 \mathord{\left/ {\vphantom {1 {\left( { 1+ {\mathbf{e}}^{ - \sigma \Omega } {\mathbf{e}}^{{\sigma \upomega_{{C_{i} ,B}} }} } \right)}}} \right. \kern-0pt} {\left( { 1+ {\mathbf{e}}^{ - \sigma \Omega } {\mathbf{e}}^{{\sigma \upomega_{{C_{i} ,B}} }} } \right)}}} \right) -\upmu} \right]v \\ &\quad = \left[ {\left( {{1 \mathord{\left/ {\vphantom {1 {\left( { 1+ \kappa {\mathbf{e}}^{{\sigma \Omega_{{_{{C_{i} ,B}} }} }} } \right)}}} \right. \kern-0pt} {\left( { 1+ \kappa {\mathbf{e}}^{{\sigma \upomega_{{_{{C_{i} ,B}} }} }} } \right)}}} \right) -\upmu} \right]v \\ &\quad = {\text{f}}(\upomega_{{C_{i} ,B}} ) \\ \end{aligned} $$
with \( {\text{f}}\left( V \right) \) a function defined by
$$ {\text{f}}\left( V \right) = \left[ {\frac{1}{{1 + \kappa {\mathbf{e}}^{\sigma V} }} -\upmu} \right]\nu $$
Here κ, μ, ν are positive constants:
$$ \begin{aligned} \kappa & = {\mathbf{e}}^{ - \sigma \Omega } {\text{ with }}\Omega = \upomega_{{C_{ 1} ,B}} + \cdots + \upomega_{{C_{k} ,B}} - \tau \\\upmu & = 1 /( 1+ {\mathbf{e}}^{\sigma \tau } ) \\ \nu & = ( 1+ {\mathbf{e}}^{ - \sigma \tau } ). \\ \end{aligned} $$
Using this function, for this case, the stationary point equations get the following uniform form:
$$ \begin{aligned} & {\text{f}}(\upomega_{{C_{ 1} ,B}} (t)) = \upomega_{{C_{ 1} ,B}} (t) \\ & \qquad \qquad \ldots \\ & \qquad \qquad \ldots \\ & \qquad \qquad \ldots \\ & {\text{f}}(\upomega_{{C_{k} ,B}} (t)) = \upomega_{{C_{k} ,B}} (t). \\ \end{aligned} $$
Therefore, the question becomes how many solutions the equation \( {\text{f}}\left( V \right) \, = V \) has. Now \( {\mathbf{e}}^{\sigma V} \) is monotonically increasing in V, and therefore \( 1/( 1+ \kappa {\mathbf{e}}^{\sigma V} )) \) and also \( {\text{f}}\left( V \right) \) are monotonically decreasing: \( V_{ 1} \le V_{ 2} \Rightarrow {\text{f}}\left( {V_{ 1} } \right) \ge {\text{f}}\left( {V_{ 2} } \right) \). Suppose \( V_{ 1} \) and \( V_{ 2} \) are two solutions of the equation \( {\text{f}}\left( V \right) = V \), and assuming \( V_{ 1} \le V_{ 2} \), it follows \( V_{ 1} = {\text{f}}\left( {V_{ 1} } \right) \ge {\text{f}}\left( {V_{ 2} } \right) = V_{ 2} ,\;{\text{so}}\,V_{ 1} = V_{ 2} \). This implies that the equation \( {\text{f}}\left( V \right) = V \) has at most one solution. From this, it follows that also for the alogistic function as combination function in a joint stationary point all \( \upomega_{{C_{i} ,B}} \) values will be equal.
There is also an abstract general argument possible for a whole class of combination functions, namely, the combination functions that are (1) symmetric in their arguments and that are (2) monotonic:
If \( U_{ 1} , \ldots ,U_{k} \) is a permutation of \( V_{ 1} , \ldots ,V_{k} \), then \( {\mathbf{c}}(U_{ 1} , \ldots ,U_{k} ) = {\mathbf{c}}(V_{ 1} , \ldots ,V_{k} ) \)
If it holds \( U_{i} \le V_{i} \) for all i, then \( {\mathbf{c}}(U_{ 1} , \ldots ,U_{k} ) \le {\mathbf{c}}(V_{ 1} , \ldots ,V_{k} ) \)
If in a fully connected network a combination function c(..) is used that is symmetric and monotonic and all connection weights between different states are the same (for example, assume all of them 1), and no connections occur from states to themselves, then the argument is as follows. Suppose all states have the same combination function and joint stationary points are given, so that for all i and j (assume i < j):
$$ \begin{aligned} X_{i} & = {\mathbf{c}}(X_{ 1} , \ldots ,X_{i - 1} ,X_{i + 1} , \ldots \ldots ,X_{k} ) \\ X_{j} & = {\mathbf{c}}(X_{ 1} , \ldots \ldots ,X_{j - 1} ,X_{j + 1} , \ldots ,X_{k} ) \\ \end{aligned} $$
then by symmetry
$$ \begin{aligned} X_{i} & = {\mathbf{c}}(X_{ 1} , \ldots ,X_{i - 1} ,X_{i + 1} , \ldots ,X_{j - 1} ,X_{j + 1} , \ldots ,X_{k} ,X_{j} ) \\ X_{j} & = {\mathbf{c}}(X_{ 1} , \ldots ,X_{i - 1} ,X_{i + 1} , \ldots ,X_{j - 1} ,X_{j + 1} , \ldots ,X_{k} ,X_{i} ) \\ \end{aligned} $$
Now suppose \( X_{i} \le X_{j} \) then by monotonicity
$$ \begin{aligned} X_{i} &= {\mathbf{c}}(X_{ 1} , \ldots ,X_{i - 1} ,X_{i + 1} , \ldots ,X_{j - 1} ,X_{j + 1} , \ldots ,X_{k} ,X_{j} ) \\ & \ge {\mathbf{c}}(X_{ 1} , \ldots ,X_{i - 1} ,X_{i + 1} , \ldots ,X_{j - 1} ,X_{j + 1} , \ldots ,X_{k} ,X_{i} ) \\ & = X_{j} \\ \end{aligned} $$
From the above, it follows that \( X_{i} = X_{j} \). The same argument applies when it is assumed \( X_{i} \ge X_{j} \). Therefore in this case, in a joint stationary point all state values are equal, which was also found above by more specific methods for the special cases of a scaled sum and an advanced logistic sum combination function, which indeed both are symmetric and monotonic combination functions. Thus, we obtain the following theorem:
When in a fully connected network with equal connection weights a combination function is used that is symmetric and monotonic, then in a joint stationary point all state values are equal.
'More becomes more' and scale-free networks
The 'more becomes more' principle has also been used to provide an explanation for the empirical evidence that most real-world networks are scale-free. The idea is that the typical distribution of degrees according to a power law emerges from an evolving network when it is assumed that the network dynamics is based on some form of a 'more become more' principle (also called preferential attachment); see, for example, [30, 33,34,35]; see also [36, 37]. An indication of the type of argument followed is illustrated in Fig. 7. Here the distribution of nodes (vertical axis) over degrees (horizontal axis) is depicted; this distribution is assumed stable over time. A time point t is considered and the focus is at the nodes with some degree d t at t (see at the horizontal axis). There is a (relative) number or density n t of them (vertical axis). Moreover, the nodes with degree between d t and a bit higher \( d_{t} + \Delta d_{t} \) are considered, an interval of length \( \Delta d_{t} \) at the horizontal axis. The (relative) number of nodes with degree within this interval is represented in Fig. 7 by the area of the (left) rectangle above that interval. This area is approximated by \( n_{t} \Delta d_{t} \).
Emerging scale-free network from an adaptive network
Now consider a time step from t to \( t + \Delta t \). Due to growth of the number of connections, the nodes with degree d t at time t will have a higher degree \( d_{t + \Delta t} \) at \( t + \Delta t \), and the nodes with degree \( d_{t} + \Delta d_{t} \) at time t will have a higher degree \( d_{t + \Delta t} + \Delta d_{t + \Delta t} \) at \( t + \Delta t \). Due to the 'more becomes more' principle, from \( d_{t} < d_{t} + \Delta d_{t} \) it follows that from t to t + ∆t the nodes with degree \( d_{t} + \Delta d_{t} \) at time t will get more new connections than the nodes with degree d t at time t. Therefore the increase in degree of these nodes with degree \( d_{t} + \Delta d_{t} \) at time t will be higher:
$$ \Delta d_{t + \Delta t} > \Delta d_{t} $$
The numbers of nodes previously represented at t by the left rectangle are represented at \( t + \Delta t \) by the right rectangle. Moreover, because they describe the same nodes, the areas indicated as shaded are the same:
$$ n_{t} \Delta d_{t} = n_{t + \Delta t} \Delta d_{t + \Delta t} $$
Given this equality, from \( \Delta d_{t + \Delta t} > \Delta d_{t} \) ('more becomes more' principle) it follows that \( n_{t + \Delta t} < n_{t} \). Therefore the distribution is monotonically decreasing. By a more complex argument it has been derived that based on some more precise assumptions on the formalisation of the 'more becomes more' principle, a distribution is obtained that is monotonically decreasing according to a power law; for example, see [30, 33,34,35] for more details.
The Network-Oriented Modelling approach based on adaptive temporal–causal networks, as described here (see also [1, 10]), provides a dynamic modelling approach that enables a modeller to design high-level conceptual model representations in the form of cyclic graphs (or connection matrices). These conceptual representations can be systematically transformed in an automated manner into executable numerical representations that can be used to perform simulation experiments. The modelling approach makes it easy to take into account. on the one hand. theories and findings from any domain from, for example, biological, psychological, neurological or social sciences, as such theories and findings are often formulated in terms of causal relations. This applies, among others, to mental processes based on complex brain networks, which, for example, often involve dynamics based on interrelating and adaptive cycles, but equally well it applies to the adaptive dynamics of social interactions. For a more detailed theoretical analysis on the wide applicability of the approach, see [38, 39]; for example, there it is shown that any smooth (state-determined) dynamical system can be modelled by a temporal–causal network model.
This enables to address complex adaptive phenomena within all kinds of integrated cognitive, affective and social processes. By using temporal–causal relations from those domains as a main vehicle and structure for network models, the obtained network models get a strong relation to the large body of empirically founded knowledge from the Neurosciences and Social Sciences. This makes them scientifically justifiable to an extent that is not attainable for black box models which lack such a relation.
In this paper, we have discussed in some detail how mathematical analysis can be used to find out some properties of the dynamics of a network model designed according to a Network-Oriented Modelling approach based on temporal–causal networks; see also [1], chapter 12, or [31]. An advantage is that such an analysis is done without performing simulations. This advantage makes that it can be used as an additional source of knowledge, independent of a specific implementation of the model. By comparing properties found by mathematical analysis and properties observed in simulation experiments a form of verification can be done. If a discrepancy is found, for example, in the sense that the mathematical analysis predicts a certain property but some simulation does not satisfy this property, this can be a reason to inspect the implementation of the model carefully (and/or check whether the mathematical analysis is correct). Having such an option can be fruitful during a development process of a model, as to acquire empirical data for validation of a model may be more difficult or may take a longer time.
Adaptive network models combining the homophily and the 'more becomes more' principle also have been studied recently, in particular in [40, 41]. The methods described in the current paper can and actually have also be applied to such integrated cases. Moreover, it has been shown in [40, 41] how the modelling approach can be related to empirical real-world data on evolving friendship networks.
Mental processes can also be modelled by temporal–causal networks in an adaptive manner. The parameters that can change can be modelled in the same way as states, following the approach in "Network-Oriented Modelling by temporal–causal networks" section. This can be applied, for example to the way in which connection strengths can change based on Hebbian learning. Hebbian learning [42], is based on the principle that strengthening of a connection between neurons over time may take place when both states are often active simultaneously ('neurons that fire together, wire together'). The principle itself goes back to Hebb [42], but see also, e.g. [43]. For some more details on this, see [31].
Treur J. Network-oriented modelling: addressing complexity of cognitive, affective and social interactions. Understanding complex systems series. Berlin: Springer; 2016.
Treur J. Network-oriented modelling and its conceptual foundations. In: Proceedings of the 8th international conference on social informatics, SocInfo'16. Lecture Notes in AI. Springer Publishers; 2016.
Chung B, Choi H, Kim S. Workflow-enabled internet service delivery for a variety of access networks. In: Proceeding APNOMS'03; 2003.
Elzas MS. Organizational structures for facilitating process innovation. Real time control of large scale systems. Heidelberg: Springer; 1985. p. 151–63.
Naudé A, Le Maitre D, de Jong T, Mans GFG, Hugo W. Modelling of spatially complex human–ecosystem, rural–urban and rich–poor interactions. 2008. https://www.researchgate.net/profile/Tom_De_jong/publication/30511313_Modelling_of_spatially_complex_human-ecosystem_rural-urban_and_rich-poor_interactions/links/02e7e534d3e9a47836000000.pdf. Accessed 16 June 2017.
Cottret L, Jourdan F. Graph methods for the investigation of metabolic networks in parasitology. Parasitology. 2010;137(09):1393–407.
Felsen LB, Mongiardo M, Russer P. Electromagnetic field representations and computations in complex structures I: complexity architecture and generalized network formulation. Int J Numer Model Electron Netw Devices Fields. 2002;15(1):93–107.
Article MATH Google Scholar
Felsen LB, Mongiardo M, Russer P. Electromagnetic field computation by network methods. Berlin: Springer Science and Business Media; 2009.
Russer P, Cangellaris AC. Network oriented modeling, complexity reduction and system identification techniques for electromagnetic systems. In: Proceedings 4th international workshop on computational electromagnetics in the time-domain; 2001. p. 105–22.
Treur J. Dynamic modelling based on a temporal–causal network modelling approach. Biol Inspi Cognit Archit. 2016;16:131–68.
Brauer F, Nohel JA. Qualitative theory of ordinary differential equations. Berlin: Benjamin; 1969.
Lotka AJ. Elements of physical biology. Dover: Williams and Wilkins (1924); 1956.
Picard E. Traité d'Analyse. 1891;1.
Poincaré H. Mémoire sur les courbes défine par une équation différentielle. 1881–1882.
Poincaré H. New Methods of Celestial Mechanics, 3 vols (English translation). 1967.
Treur J. Network-oriented modeling and analysis of dynamics based on adaptive temporal–causal networks. In: Complex Networks & their Applications V. Proceedings of the 5th international workshop on complex networks and their applications. Studies in Computational Intelligence, vol. 693. Berlin: Springer Publishers; 2016. p. 69–82.
Beer RD. On the dynamics of small continuous-time recurrent neural networks. Adapt Behav. 1995;3:469–509.
Dubois D, Lang J, Prade H. Fuzzy sets in approximate reasoning, part 2: logical approaches. Fuzzy Sets Syst. 1991;40:203–44.
Dubois D, Prade H. Possibility theory, probability theory and multiple-valued logics: a clarification. Ann Math Artif Intell. 2002;32:35–66.
MathSciNet Article MATH Google Scholar
Giangiacomo G. Fuzzy logic: mathematical tools for approximate reasoning. Dordrecht: Kluwer Academic Publishers; 2001.
Grossberg S. On learning and energy-entropy dependence in recurrent and nonrecurrent signed networks. J Stat Phys. 1969;1:319–50.
Hirsch M. Convergent activation dynamics in continuous-time networks. Neural Netw. 1989;2:331–49.
Hopfield JJ. Neural networks and physical systems with emergent collective computational properties. Proc Nat Acad Sci (USA). 1982;79:2554–8.
Hopfield JJ. Neurons with graded response have collective computational properties like those of two-state neurons. Proc Nat Acad Sci (USA). 1984;81:3088–92.
Zadeh L. Fuzzy sets as the basis for a theory of possibility. Fuzzy Sets Syst. 1978;1:3–28 (Reprinted in Fuzzy Sets Syst 1999;100(Supplement): 9–34).
Simon HA. On a class of skew distribution functions. Biometrika. 1955;42:425–40.
de Price DJ. A general theory of bibliometric and other cumulative advantage processes. J Am Soc Inform Sci. 1976;27:292–306.
Merton RK. The Matthew effect in science. Science. 1968;159:56–63.
Barabási AL, Albert R. Emergence of scaling in random networks. Science. 1999;286:509–12.
Treur J. Verification of temporal–causal network models by mathematical analysis. Vietnam J Comput Sci. 2016;3:207–21. doi:10.1007/s40595-016-0067-z.
Sharpanskykh A, Treur J. Modelling and analysis of social contagion in dynamic networks. Neurocomput J. 2014;146:140–50.
Krapivsky PL, Redner S, Leyvraz F. Connectivity of growing random networks. Phys Rev Lett. 2000;85(21):4629–32.
Krapivsky PL, Redner S. Organization of growing random networks. Phys Rev E. 2001;63(6):066123.
Krapivsky PL, Redner S. Rate equation approach for growing networks. In: Pastor-Satorras R, Rubi M, Diaz-Guilera A, editors. Statistical mechanics of complex networks. Lecture Notes in Physics, vol. 625; 2003. p. 3–22.
Bornholdt S, Ebel H. World Wide Web scaling exponent from Simon's 1955 model. Phys Rev E. 2001;64(3):035104.
Newman MEJ. The structure and function of complex networks. Siam Rev. 2003;45(2):167–256.
Treur J. Do network models just model networks? On the applicability of network-oriented modeling. In: Proceedings of the international conference on network science, NetSci-X-2017. Lecture Notes in Computer Science. Springer Publishers; 2017.
Treur J. On the applicability of network-oriented modeling based on temporal–causal networks: why network models do not just model networks. J Inform Telecommun. 2017;1:23–40.
Blankendaal R, Parinussa S, Treur J. A temporal–causal modelling approach to integrated contagion and network change in social networks. In: Proceedings of the 22nd European conference on artificial intelligence, ECAI 2016. Frontiers in artificial intelligence and applications, vol. 285. IOS Press; 2016. p. 1388–96.
van den Beukel S, Goos S, Treur J. Understanding homophily and more-becomes-more through adaptive temporal–causal network models. In: Proceedings of the 15th international conference on practical applications of agents and multi-agent systems, PAAMS'17. Lecture Notes in Computer Science. Springer Publishers; 2017.
Hebb D. The Organisation of Behavior. Hoboken: Wiley; 1949.
Gerstner W, Kistler WM. Mathematical formulations of Hebbian learning. Biol Cybern. 2002;87:404–15.
The author declares no competing interests.
Behavioural Informatics Group, Vrije Universiteit Amsterdam, Amsterdam, The Netherlands
Jan Treur
Correspondence to Jan Treur.
Treur, J. Modelling and analysis of the dynamics of adaptive temporal–causal network models for evolving social interactions. Comput Soc Netw 4, 4 (2017). https://doi.org/10.1186/s40649-017-0039-1
Accepted: 31 May 2017
Temporal-causal Network Model
Adaptive Network
Homophily Principle
Combination Function
Stationary Point Value
Special Issue of the 5th International Workshop on Complex Networks and Their Applications | CommonCrawl |
Szegő kernel
In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions. It is named for its discoverer, the Hungarian mathematician Gábor Szegő.
Let Ω be a bounded domain in Cn with C2 boundary, and let A(Ω) denote the space of all holomorphic functions in Ω that are continuous on ${\overline {\Omega }}$. Define the Hardy space H2(∂Ω) to be the closure in L2(∂Ω) of the restrictions of elements of A(Ω) to the boundary. The Poisson integral implies that each element ƒ of H2(∂Ω) extends to a holomorphic function Pƒ in Ω. Furthermore, for each z ∈ Ω, the map
$f\mapsto Pf(z)$
defines a continuous linear functional on H2(∂Ω). By the Riesz representation theorem, this linear functional is represented by a kernel kz, which is to say
$Pf(z)=\int _{\partial \Omega }f(\zeta ){\overline {k_{z}(\zeta )}}\,d\sigma (\zeta ).$
The Szegő kernel is defined by
$S(z,\zeta )={\overline {k_{z}(\zeta )}},\quad z\in \Omega ,\zeta \in \partial \Omega .$
Like its close cousin, the Bergman kernel, the Szegő kernel is holomorphic in z. In fact, if φi is an orthonormal basis of H2(∂Ω) consisting entirely of the restrictions of functions in A(Ω), then a Riesz–Fischer theorem argument shows that
$S(z,\zeta )=\sum _{i=1}^{\infty }\phi _{i}(z){\overline {\phi _{i}(\zeta )}}.$
References
• Krantz, Steven G. (2002), Function Theory of Several Complex Variables, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2724-6
| Wikipedia |
\begin{document}
\title{Electromagnetic field quantization in an anisotropic and inhomogeneous
magnetodielectric} \begin{abstract}
The electromagnetic field in an anisotropic and inhomogeneous magnetodielectric is quantized by modelling the medium with two independent quantum fields. Some coupling tensors coupling the electromagnetic field with the medium are introduced. Electric and magnetic polarizations are obtained in terms of the ladder operators of the medium and the coupling tensors explicitly. Using a minimal coupling scheme for electric and magnetic interactions, the Maxwell equations and the constitutive equations of the medium are obtained. The electric and magnetic susceptibility tensors of the medium are calculated in terms of the coupling tensors. Finally the efficiency of the approach is elucidated by some examples.
{\bf Keywords: Field quantization, Magnetodielectric, Anisotropic, Inhomogeneous, Coupling tensor, E-M Quantum fields}
{\bf PACS number: 12.20.Ds}
\end{abstract} \section{Introduction} The quantization of electromagnetic field in an absorptive dielectric, represents one of the most and interesting problems in quantum optics, because it gives a rigorous test of our understanding of the interaction of light with matter. One of the important methods to quantize the electromagnetic field in the presence of an absorptive medium is known as Green function method \cite{.1}-\cite{.7}. In this method by adding the noise electric and magnetic polarization densities to classical constitutive equations of the medium, these equations are considered as definitions of electric and magnetic polarization operators. The noise polarizations are related to two independent sets of bosonic operators. Combination of the Maxwell equations and the constitutive equations in frequency domain, give the electromagnetic field operators in terms of the noise polarizations and classical Green tensor. Suitable commutation relations are imposed on the bosonic operators such that the commutation relations between electromagnetic field operators become identical with those in free space.\\ An interesting quantization scheme of electromagnetic field in the presence of an absorptive dielectric medium is based on the Hopfield model of a dielectric \cite{1}, where the polarization of the dielectric is represented by a damped quantum field \cite{2}. Huttner and Barnett \cite{3} for a homogeneous medium and after Suttorp and Wubs \cite{4} for an inhomogeneous medium in the framework of the damped polarization model have presented a canonical quantization for the electromagnetic field inside an absorptive dielectric. This scheme is based on a microscopic model in which the medium is represented by a collection of interacting matter fields. The absorptive character of the medium is modelled through the interaction of the matter fields with a reservoir consisting of a continuum of the Klein-Gordon fields. In this model, eigen-operators for the coupled systems are calculated and the electromagnetic field operators have been expressed in terms of these eigen-operators. Also, the dielectric function is derived and it is shown that it satisfies
the Kramers-Kronig relations \cite{5}.\\ Another approach to quantizing a dissipative system is by considering the dissipation as a result of interaction between the system and a heat bath consisting of a set of harmonic oscillators \cite{5.9}-\cite{17}. In this method the whole system is composed of two parts, the main system and a heat bath which interacts with the main system and causes the dissipation of energy on it.
In a recent approach to electromagnetic field quantization the present authors have quantized the electromagnetic
field in an isotropic magnetodielectric \cite{18}. In this approach: (i) the electromagnetic field is taken as the main
quantum system and the medium as a heat bath. (ii) The polarizability of the medium is defined in terms of
dynamical variables of the medium. (iii) The polarizability and absorptivity of the medium are not independent of each other, as expected, this is contrary to the damped polarization model where polarizability and absorptivity are treated independently \cite{3,4}. (iv) If the medium is both magnetizable and polarizable, one must models the medium with two independent collections of harmonic oscillators, where one collection describes electric properties and the other one describes magnetic properties of the medium. This scheme leads to a consistent quantization of the electromagnetic field in the presence of an absorptive magnetodielectric \cite{18}.\\ In the present article, the idea introduced in the previous work \cite{18} is generalized to the case of an anisotropic and inhomogeneous magnetodielectric.
\section{ Quantum dynamics}
Electromagnetic field quantization can be achieved in an anisotropic magnetodielectric by modelling the medium with two independent quantum fields. Let us call these fields E and M quantum fields, describing the polarizability and magnetizability of the medium respectively. These quantum fields couple the medium with electromagnetic field through some coupling tensors. The electric and magnetic polarization densities of the medium are defined as linear expansions in terms of the ladder operators of the E and M fields. The coefficients of these expansions are real valued coupling tensors. We will see that the electric and magnetic susceptibility tensors can be obtained in terms of the coupling tensors. In the following we use the Coloumb gauge and assume the periodic boundary conditions with no loss of generality of the approach.
The electromagnetic vector potential $\vec{A}$ inside a box with volume $V=L_1L_2L_3$ can be expanded in terms of plane waves as \begin{equation} \label{d1} \vec{A}(\vec{r},t)=\sum_{\vec{n}} \sum_{\lambda=1}^2 \sqrt{\frac{\hbar} {2\varepsilon_0V\omega_{\vec{n}}}}\left[a_{\vec{n}\lambda}(t)e^{i\vec{k}_{\vec{n}}\cdot\vec{r}}+a_{\vec{n} \lambda}^\dag(t)e^{-i\vec{k}_{\vec{n}}\cdot\vec{r}}\right]\vec{e}_{\vec{n}\lambda}, \end{equation}
where $\omega_{\vec{n}}=c|\vec{k}_{\vec{n}}|$ is the frequency corresponding to the mode $\vec{n}$, the vector $\vec{n}$ is a triplet of integer numbers $ (n_1,n_2,n_3) $, $ \sum_{\vec{n}}$ means $\displaystyle\sum_{n_1,n_2,n_3=-\infty}^{+\infty} $, $\varepsilon_0$ is the permittivity of the vacuum, $\vec{k}_{\vec{n}}=\frac{2n_1\pi}{L_1}\vec{i}+\frac{2n_2\pi}{L_2}\vec{j}+\frac{2n_3\pi}{L_3}\vec{k}$
is the wave vector and $\vec{e}_{\vec{n}\lambda}$ are polarization unit vectors for each $\vec{n}$, satisfying
\begin{eqnarray}\label{d2} \vec{e}_{\vec{n}\lambda}.\vec{e}_{\vec{n}\lambda'}&=&\delta_{\lambda\lambda'},\nonumber\\
\vec{e}_{\vec{n}\lambda}.\vec{k}_{\vec{n}}&=&0. \end{eqnarray}
The operators $a_{\vec{n}\lambda}(t)$ and $a_{\vec{n}\lambda}^\dag(t) $ are annihilation and creation operators of the electromagnetic field and satisfy the following equal time commutation rules \begin{equation}\label{d3} [a_{\vec{n}\lambda}(t),a_{\vec{m}\lambda'}^\dag(t)]= \delta_{\vec{n},\vec{m}}\delta_{\lambda\lambda'}. \end{equation} Quantization in Coloumb gauge usually needs resolution of a vector field in its transverse and longitudinal parts. Any vector field $\vec{F}(\vec{r})$ can be resolved in two components, transverse and longitudinal component which are denoted by $ \vec{F}^\bot$
and $\vec{F}^\|$ respectively. The transverse part satisfy the coloumb condition $ \nabla\cdot\vec{F}^\bot=0$ and the longitudinal component is a conservative field $
\nabla\times\vec{F}^\|=0$. For a periodic boundary condition these two parts are defined as \begin{eqnarray}\label{d4} && \vec{F}^\bot(\vec{r},t)=\vec{F}(\vec{r},t)+\int_V d^3r' \nabla'\cdot\vec{F}(\vec{r'},t)\vec{\nabla} G(\vec{r},\vec{r'}), \\
&&\vec{F}^\|(\vec{r},t)=-\int_V d^3r' \nabla'\cdot\vec{F}(\vec{r'},t)\vec{\nabla} G(\vec{r},\vec{r'}), \end{eqnarray} where \begin{equation}\label{d5}
G(\vec{r},\vec{r'})=\sum_{\vec{n}}\frac{1}{|\vec{k}_{\vec{n}}|^2}e^{
\imath\vec{k}_{\vec{n}}\cdot(\vec{r}-\vec{r'})}, \end{equation}
is the Green function and satisfies the Poison equation \begin{equation}\label{d6} \nabla^2G(\vec{r},\vec{r'})=-\delta(\vec{r}-\vec{r'}). \end{equation}
In absence of external charges the displacement field is purely transverse, and we can expand it in terms of the plane waves \begin{equation}\label{d7} \vec{D}(\vec{r},t)=-i\varepsilon_0\sum_{\vec{n}}\sum_{\lambda=1}^2 \sqrt{\frac{\hbar\omega_{\vec{n}}}{2\varepsilon_0V}} \left[a_{\vec{n}\lambda}^\dag(t)e^{-\imath\vec{k}_{\vec{n}}\cdot\vec{r}}-a_{\vec{n}\lambda}(t)e^{\imath\vec{k}_{\vec{n}}\cdot\vec{r}}\right]\vec{e}_{\vec{n}\lambda}. \end{equation}
The commutation relations (\ref{d3}) lead to the following commutation relations between the components of the vector potential $\vec{A}$ and the displacement operator $\vec{D}$ \begin{equation}\label{d8} [A_l(\vec{r},t),-D_j(\vec{r'},t)]= \imath\hbar\delta_{lj}^\bot(\vec{r}-\vec{r'}), \end{equation} where \begin{equation}\label{d9}
\delta_{lj}^\bot(\vec{r}-\vec{r'})=\frac{1}{V} \sum_{\vec{n}}(\delta_{lj}-\frac{k_{\vec{n}l} k_{\vec{n}j}
}{|\vec{k}_{\vec{n}}|^2})e^{\imath\vec{k}_{\vec{n}}\cdot(\vec{r}-\vec{r'})}, \end{equation} is the transverse delta function. From (\ref{d9}), we see that $-\vec{D}$ plays the role of the momentum density of electromagnetic field. The Hamiltonian of the electromagnetic field inside the box is given by
\begin{eqnarray}\label{d10} &&H_F(t)=\int_V d^3r \left[\frac{ \vec{D}^2}{2\varepsilon_0}+ \frac{(\nabla\times\vec{A})^2}{2\mu_0}\right]= \sum_{\vec{n}}\sum_{\lambda=1}^2\hbar \omega_{\vec{n}}a_{\vec{n}\lambda}^\dag(t) a_{\vec{n}\lambda}(t).\nonumber\\ && \end{eqnarray}
where $\mu_0$ is the magnetic permittivity of the vacuum and we have used normal ordering for $ a_{\vec{n}\lambda}^\dag(t) $ and $ a_{\vec{n}\lambda}(t) $.
Now we include the medium in the process of quantization. For this purpose let the Hamiltonian corresponding to E and M quantum fields be denoted by $H_e$ and $H_m$ respectively. Then the medium Hamiltonian can be written as
\begin{eqnarray}\label{d11} &&H_d=H_e+H_m,\nonumber\\ && H_e(t)=\sum_{\vec{n}}\sum_{\nu=1}^3\int_{-\infty}^{+\infty}d^3q \hbar\omega_{\vec{q}} d_{\vec{n}\nu}^\dag(\vec{q},t)d_{\vec{n}\nu}(\vec{q},t), \nonumber\\ &&H_m(t)=\sum_{\vec{n}}\sum_{\nu=1}^3\int_{-\infty}^{+\infty}d^3q \hbar\omega_{\vec{q}} b_{\vec{n}\nu}^\dag(\vec{q},t)b_{\vec{n}\nu}(\vec{q},t), \end{eqnarray}
where the annihilation and creation operators $ d_{\vec{n}\nu}(\vec{q},t)$, $d_{\vec{n}\nu}^\dag(\vec{q},t)$, $ b_{\vec{n}\nu}(\vec{q},t)$ and $b_{\vec{n}\nu}^\dag(\vec{q},t)$ satisfy the following equal-time commutation relations \begin{eqnarray}\label{d12} &&[d_{\vec{n}\nu}(\vec{q},t),d_{\vec{m}\nu'}^\dag(\vec{q}',t)]= \delta_{\vec{n},\vec{m}}\delta_{\nu\nu'}\delta(\vec{q}-\vec{q'}),\nonumber\\ &&[b_{\vec{n}\nu}(\vec{q},t),b_{\vec{m}\nu'}^\dag(\vec{q'},t)]= \delta_{\vec{n},\vec{m}}\delta_{\nu\nu'}\delta(\vec{q}-\vec{q'}). \end{eqnarray}
In relations (\ref{d11}) $\omega_{\vec{q}}$ is the dispersion relation of the magnetodielectric. It is remarkable to note that, although the medium is anisotropic in its electric and magnetic properties, we do not need to take the dispersion relation as a tensor. As discussed in \cite{18}, we can assume a linear dispersion relation $\omega_{\vec{q}}=c|\vec{q}|$ with no loss of generality, but taking a linear dispersion relation simplifies the formulas considerably. Therefore from now on we choose the dispersion relation as $\omega_{\vec{q}}=c|\vec{q}|$ where $c=\frac{1}{\sqrt{\varepsilon_0\mu_0}}$ is the proportionality constant.
The basic idea in this quantization method is that the electric and magnetic properties of an anisotropic magnetodielectric can be described by E and M quantum fields. This means that we can define the electric and magnetic polarization densities of a linear but anisotropic medium as linear combinations of the ladder operators of the E and M quantum fields, respectively. Therefore
\begin{eqnarray}\label{d13} &&P_i(\vec{r},t)=\frac{1}{\sqrt{V}}\sum_{\vec{n}}\sum_{\nu=1}^3\int d^3\vec{q}f_{ij}(\omega_{\vec{q}},\vec{r})\left[d_{\vec{n}\nu}(\vec{q},t)e^{\imath \vec{k}_{\vec{n}}\cdot\vec{r}} +h.c.\right] v^j_{\vec{n}\nu},\nonumber\\ && \end{eqnarray} \begin{eqnarray}\label{d14} &&M_i(\vec{r},t)=\frac{\imath}{\sqrt{V}}\sum_{\vec{n}}\sum_{\nu=1}^3\int d^3\vec{q}g_{ij}(\omega_{\vec{q}},\vec{r})\left[b_{\vec{n}\nu}(\vec{q},t)e^{\imath \vec{k}_{\vec{n}}\cdot\vec{r}} -h.c.\right] s^j_{\vec{n}\nu},\nonumber\\ && \end{eqnarray}
where $\vec{P}$ and $\vec{M}$ are electric and magnetic polarization densities of the medium and
\begin{eqnarray}\label{d15} &&\vec{v}_{\vec{n}\nu}=\vec{e}_{\vec{n}\nu},\hspace{1.50cm}\nu=1,2\nonumber\\ &&\vec{s}_{\vec{n}\nu}=\hat{k}_{\vec{n}}\times\vec{e}_{\vec{n}\nu},\hspace{1cm}\nu=1,2\nonumber\\ &&\vec{v}_{\vec{n}3}=\vec{s}_{\vec{n}3}=\hat{k}_{\vec{n}}\hspace{1.50cm}
\hat{k}_{\vec{n}}=\frac{\vec{k}_{\vec{n}}}{|\vec{k}_{\vec{n}}|}.\nonumber\\ \end{eqnarray}
In definitions of polarization densities (\ref{d13}) and (\ref{d14}), the real valued tensors
$ f_{ij}(\omega_{\vec{q}},\vec{r})$ and $g_{ij}(\omega_{\vec{q}},\vec{r}) $, are called the coupling
tensors of the electromagnetic field and the medium which are dependent (independent) on position $ \vec{r} $ for
inhomogeneous (homogeneous) magnetodielectrics. The coupling tensors play the key role in this method
and are a measure for the strength of the polarizability and magnetizability of the
medium macroscopically. We will see that the imaginary parts of the electric and magnetic
susceptibilty in frequency domain can be obtained in terms of
these coupling tensors. Also, explicit forms for the noise polarization densities
can be obtained in terms of the coupling tensors and the ladder operators of the medium. The coupling tensors are common factors in the noise densities and the electric and magnetic susceptibilities, and so the strength of the noise densities are dependent on the strength of the electric and magnetic susceptibility. It can be shown that for a non absorptive medium, the noise densities tend to zero as expected and this quantization scheme reduces to the usual quantization in such media.
A consistent quantization scheme must lead to the correct equations of motion
of the system and the medium. These equations are macroscopic Maxwell and constitutive equations of the
medium and we will see that these equations can be obtained from the Heisenberg equations using the total Hamiltonian defined by
\begin{eqnarray}\label{d16} &&\tilde{H}(t)=\int d^3r \left\{\frac{[ \vec{D}(\vec{r},t)-\vec{P}(\vec{r},t)]^2}{2\varepsilon_0}+ \frac{(\nabla\times\vec{A})^2(\vec{r},t)}{2\mu_0} -\nabla\times\vec{A}(\vec{r},t).\vec{M}(\vec{r},t)\right\}\nonumber\\ &&+H_e+H_m. \end{eqnarray}
\subsection{Maxwell equations} Using the commutation relations (\ref{d8}) the Heisenberg equations for the vector potential $\vec{A}$ and the displacement field $\vec{D}$ are \begin{equation}\label{d17} \frac{\partial\vec{A}(\vec{r},t)}{\partial t}=\frac{\imath}{\hbar}[\tilde{H},\vec{A}(\vec{r},t)]= -\frac{\vec{D}(\vec{r},t)-\vec{P}^\bot(\vec{r},t)}{\varepsilon_0}, \end{equation} \begin{equation}\label{d18} \frac{\partial\vec{D}(\vec{r},t)}{\partial t}=\frac{\imath}{\hbar}[\tilde{H},\vec{D}(\vec{r},t)]= \frac{\nabla\times\nabla\times\vec{A}(\vec{r},t)}{\mu_0}-\nabla\times\vec{M}(\vec{r},t), \end{equation} where $\vec{P}^\bot$ is the transverse component of $\vec{P}$. The transverse electrical field $ \vec{E}^\bot $, magnetic induction $\vec{B}$ and magnetic field $\vec{H} $ are defined by
\begin{equation}\label{d19}
\vec{E}^\bot=-\frac{\partial\vec{A}}{\partial t},\hspace{1.00
cm}\vec{B}=\nabla\times\vec{A},\hspace{1.00
cm}\vec{H}=\frac{\vec{B}}{\mu_0}-\vec{M}.
\end{equation}
Using these recent relations, (\ref{d17}) and (\ref{d18}) can be rewritten as \begin{equation} \label{d20} \vec{D}=\varepsilon_0 \vec{E}^\bot+\vec{P}^\bot, \end{equation} \begin{equation}\label{d21} \frac{\partial \vec{D}}{\partial t}=\nabla\times\vec{H}, \end{equation} which are the definitions of the displacement field and the macroscopic Maxwell equation, in the presence of a magnetodielectric, respectively.
\subsection{Constitutive equations of the medium} A magnetodielectric subjected to electromagnetic field can be polarized and magnetized in consequence of interaction of the medium with the field. The macroscopic electric and magnetic polarizations is related to electric and magnetic fields, respectively by the constitutive equations of the medium. Therefore a quantization scheme must be able to give the constitutive equations in the Heisenberg picture. In this section by applying the Heisenberg equations to the ladder operators of the medium we find the correct constitutive equations of the medium.
the The time evolution of the operators $ d_{\vec{n}\nu}(\vec{q},t) $ and $ b_{\vec{n}\nu}(\vec{q},t)$ can be obtained from the commutation relations (\ref{d12}) and the Hamiltonian (\ref{d16}) as follows \begin{eqnarray}\label{d22} &&\dot{d}_{\vec{n}\nu}(\vec{q},t)= \frac{\imath}{\hbar}[\tilde{H},d_{\vec{n}\nu}(\vec{q},t)]=\nonumber\\ &&-\imath\omega_{\vec{q}}d_{\vec{n}\nu}(\vec{q},t)+\frac{\imath}{\hbar \sqrt{V}}\int_V d^3\vec{r'}e^{-\imath\vec{k}_{\vec{n}}\cdot\vec{r'}}f_{ij}(\omega_{\vec{q}},\vec{r'}) E^i(\vec{r'},t)v^j_{\vec{n}\nu}, \end{eqnarray}
\begin{eqnarray}\label{d23} &&\dot{b}_{\vec{n}\nu}(\vec{q},t)= \frac{\imath}{\hbar}[\tilde{H},b_{\vec{n}\nu}(\vec{q},t)]=\nonumber\\ &&-\imath\omega_{\vec{q}}b_{\vec{n}\nu}(\vec{q},t)+\frac{1}{\hbar \sqrt{V}}\int_V d^3\vec{r'}e^{-\imath\vec{k}_{\vec{n}}\cdot\vec{r'}}g_{ij}(\omega_{\vec{q}},\vec{r'}) B^i(\vec{r'},t)s^j_{\vec{n}\nu}. \end{eqnarray}
It is easy to show that these equations have the following formal solutions
\begin{eqnarray}\label{d24} &&{d}_{\vec{n}\nu}(\vec{q},t)= d_{\vec{n}\nu}(\vec{q},0)e^{-\imath\omega_{\vec{q}}t}+\nonumber\\ &&\frac{\imath}{\hbar\sqrt{V}}\int_0^t dt'e^{-\imath\omega_{\vec{q}}(t-t')} \int_V d^3r'e^{-\imath\vec{k}_{\vec{n}}\cdot\vec{r'}}f_{ij}(\omega_{\vec{q}},\vec{r'}) E^i(\vec{r'},t')v^j_{\vec{n}\nu}, \end{eqnarray}
\begin{eqnarray}\label{d25} &&{b}_{\vec{n}\nu}(\vec{q},t)= b_{\vec{n}\nu}(\vec{q},0)e^{-\imath\omega_{\vec{q}}t}+\nonumber\\ &&\frac{1}{\hbar\sqrt{V}}\int_0^t dt'e^{-\imath\omega_{\vec{q}}(t-t')} \int_V d^3r'e^{-\imath\vec{k}_{\vec{n}}\cdot\vec{r'}}g_{ij}(\omega_{\vec{q}},\vec{r'}) B^i(\vec{r'},t')s^j_{\vec{n}\nu}. \end{eqnarray}
Substituting (\ref{d24}) in (\ref{d13}) and (\ref{d25}) in (\ref{d14}) we obtain
the macroscopic constitutive equations of the anisotropic polarizable and magnetizable medium, \begin{equation}\label{d26}
\vec{P}(\vec{r},t)=\vec{P}_N(\vec{r},t)+\varepsilon_0\int_0^{|t|}
d t' \chi^e(\vec{r},|t|-t')\vec{E}(\vec{r},\pm t'), \end{equation} \begin{equation}\label{d27}
\vec{M}(\vec{r},t)=\vec{M}_N(\vec{r},t)+\frac{1}{\mu_0}\int_0^{|t|}
d t' \chi^m(\vec{r},|t|-t')\vec{B}(\vec{r},\pm t'), \end{equation} where the upper (lower) sign corresponds to $t>0$ ($ t<0 $) and
$\vec{E}=-\frac{\partial\vec{A}}{\partial t}-\frac{\vec{P}^\|}{\varepsilon_0} $ is the total electrical field.
The memory tensors \begin{equation}\label{d28}
\chi^e(\vec{r},t) =\left\{\begin{array}{cc}
\frac{8\pi}{\hbar c^3 \varepsilon_0}\int_0^\infty d\omega\omega^2(ff^t)(\omega,\vec{r})\sin\omega t & \hspace{1.00cm}t>0, \\ \\ 0& \hspace{1.00cm}t\leq 0, \
\end{array}\right. \end{equation} \begin{equation}\label{d29}
\chi^m(\vec{r},t) =\left\{\begin{array}{cc}
\frac{8\pi\mu_0}{\hbar c^3 }\int_0^\infty d\omega\omega^2(gg^t)(\omega,\vec{r})\sin\omega t & \hspace{1.00cm}t>0, \\ \\ 0& \hspace{1.00cm}t\leq 0, \
\end{array}\right. \end{equation}
are called the electric and magnetic susceptibility tensors
of the magnetodielectric, respectively and $f^t$, $g^t$ denote the transpose of the coupling tensors $f$, $g$. If we are given a definite pair of tensors $\chi^e(\vec{r},t)$, $\chi^m(\vec{r},t)$ which are zero for $t\leq 0 $, then we can inverse (\ref{d28}) and (\ref{d29}) and obtain the corresponding tensors $(ff^t)$ and $(gg^t)$ as,
\begin{eqnarray}\label{d30} &&(ff^t)(\omega,\vec{r})=\nonumber\\ &&\nonumber\\ &&\left\{\begin{array}{cc}
\frac{\hbar c^3 \varepsilon_0 }{4\pi^2\omega^2}\int_0^\infty dt\chi^e(\vec{r},t) \sin\omega t=\frac{\hbar c^3 \varepsilon_0 }{4\pi^2\omega^2}Im\left[\underline{\chi}^e(\vec{r},\omega)\right] & \hspace{1.00cm} \omega>0,\\ \\
0 & \hspace{1.00cm}\omega=0, \end{array}\right.\nonumber\\ &&\nonumber\\ && \end{eqnarray} \begin{eqnarray}\label{d31} &&(gg^t)(\omega,\vec{r})=\nonumber\\ &&\nonumber\\ &&\left\{\begin{array}{cc}
\frac{\hbar c^3 }{4\pi^2\mu_0\omega^2}\int_0^\infty dt\chi^m(\vec{r},t) \sin\omega t=\frac{\hbar c^3 }{4\pi^2\mu_0\omega^2}Im\left[\underline{\chi}^m(\vec{r},\omega)\right] & \hspace{1.00cm} \omega>0,\\ \\
0 & \hspace{1.00cm}\omega=0, \end{array}\right.\nonumber\\ &&\nonumber\\ && \end{eqnarray} where $\underline{\chi}^e(\vec{r},\omega)$ and $\underline{\chi}^m(\vec{r},\omega) $ are the susceptibility tensors in the frequency domain. The operators $\vec{P}_N $ and $\vec{M}_N$ in (\ref{d26}) and (\ref{d27}) are the noise electric and magnetic polarization densities
\begin{eqnarray}\label{d32} &&P_{Ni}(\vec{r},t)= \frac{1}{\sqrt{V}}\sum_{\vec{n}}\sum_{\nu=1}^3\int d^3qf_{ij}(\omega_{\vec{q}},\vec{r})\left[d_{\vec{n}\nu}(\vec{q},0) e^{-\imath\omega_{\vec{q}}t+\imath\vec{k}_{\vec{n}}\cdot\vec{r}}+h.c.\right] v^j_{\vec{n} \nu},\nonumber\\ && \end{eqnarray}
\begin{eqnarray}\label{d33} &&M_{Ni}(\vec{r},t)= \frac{\imath}{\sqrt{V}}\sum_{\vec{n}}\sum_{\nu=1}^3\int d^3qg_{ij}(\omega_{\vec{q}},\vec{r})\left[b_{\vec{n}\nu}(\vec{q},0) e^{-\imath\omega_{\vec{q}}t+\imath\vec{k}_{\vec{n}}\cdot\vec{r}}-h.c.\right] s^j_{\vec{n} \nu}.,\nonumber\\ && \end{eqnarray}
These noises are necessary for a consistent quantization of the electromagnetic field in the presence of an absorptive medium.
From (\ref{d30}) and (\ref{d31}) it is clear that for a given pair
of the susceptibility tensors $ \chi^e$ and $\chi^m$ there are infinite number of coupling tensors $ f$ and $g$ satisfying the equations (\ref{d30}) and (\ref{d31}). In fact for a given pair of $ \chi^e$ and $\chi^m$ if the tensors $f$ and $g$ satisfy equations (\ref{d30}) and (\ref{d31}), then the coupling tensors $fA$ and $gA$, for any orthogonal matrix $(AA^t=1)$, are also a solution. Certainly this affect the space-time dependence of the noise polarizations and therefore the space-time dependence of the electromagnetic field operators, but all of these choices are equivalent. This means that the various choices of the coupling tensors $ f$ and $g$ satisfying (\ref{d30}) and (\ref{d31}), for a given pair of the susceptibilities $\chi^e$ and $\chi^m$, do not affect the commutation relations between the field operators and hence the physical observables. This becomes more clear if we compute the commutation relations between the components of the Fourier transform of the noise polarizations
\begin{eqnarray}\label{d34} &&[\underline{P}_{Ni}(\vec{r},\omega) , \underline{P}_{Nj}^\dag(\vec{r'},\omega')]=\frac{\hbar\varepsilon_0}{\pi}Im\left[ \underline{\chi}^e_{ij}(\vec{r},\omega)\right]\delta(\vec{r}-\vec{r'}) \delta(\omega-\omega'),\nonumber\\ &&[\underline{M}_{Ni}(\vec{r},\omega) , \underline{M}_{Nj}^\dag(\vec{r'},\omega')]=\frac{\hbar}{\mu_0\pi}Im\left[ \underline{\chi}^m_{ij}(\vec{r},\omega)\right]\delta(\vec{r}-\vec{r'}) \delta(\omega-\omega').\nonumber\\ && \end{eqnarray}
These relations are generalization of those in reference \cite{19}.
For a given pair of $\chi^e $ and $\chi^m $, various choices of the coupling tensors $f$ and $g$ satisfying the relations
(\ref{d30}) and (\ref{d31}), do not affect these commutation
relations and accordingly the commutation relations between the
electromagnetic field operators. Hence, all of the field
operators which are obtained by using a definite pair of the susceptibilities
$\chi^e $ and $\chi^m$, with different coupling tensors, satisfying (\ref{d30})
and (\ref{d31}), are equivalent.
\section{ Solution of Heisenberg equations} The Maxwell and constitutive equations of the medium constitute a set of coupled equations. In this section we solve them in terms of their initial conditions using the Laplace transformation technique. For any time-dependent operator $ g(t)$ the forward and backward Laplace transformation of $ g(t)$ are defined by \begin{eqnarray}\label{d35} &&g^f(s)=\int_0^\infty dt g(t)e^{-st},\nonumber\\ &&g^b(s)=\int_0^\infty dt g(-t)e^{-st}, \end{eqnarray}
respectively. Carrying out the forward and backward Laplace transformation of the Maxwell equation (\ref{d21}) and the constitutive equations (\ref{d20}), (\ref{d26}) and (\ref{d27}) we find
\begin{eqnarray}\label{d36} &&\nabla\times\nabla\times\vec{E}^{f,b}(\vec{r},s)+\mu_0\varepsilon_0 s^2\tilde{\varepsilon}(\vec{r},s)\vec{E}^{f,b}(\vec{r},s)-\nonumber\\ &&\nabla\times\tilde{\chi}^m(\vec{r},s) \nabla\times\vec{E}^{f,b}(\vec{r},s)= \vec{J}^{f,b}(\vec{r},s), \end{eqnarray}
where $\tilde{\varepsilon}(\vec{r},s)=1+\tilde{\chi}^e(\vec{r},s)$ and $\tilde{\chi}^m(\vec{r},s)$ are the Laplace transformations of the electric permeability tensor and magnetic susceptibility tensor of the medium, respectively and
\begin{eqnarray}\label{d37} &&\vec{J}^{f,b}(\vec{r},s)=\pm\nabla\times\vec{B}(\vec{r},0)-\mu_0s^2 \vec{P}^{f,b}_N(\vec{r},s)\mp\nonumber\\ &&\mu_0s \nabla\times\vec{M}^{f,b}_N(\vec{r},s)\mp\nabla\times\tilde{\chi}^m(\vec{r},s) \vec{B}(\vec{r},0)+ \mu_0 s \vec{D}(\vec{r},0), \end{eqnarray}
is the forward and backward Laplace transformation of the noise current where upper(lower) sign corresponds to $ \vec{J}^f(\vec{r},s)$ ($\vec{J}^f(\vec{r},s)$). The wave equation (\ref{d36}) can be solved using the Green tensor method \cite{19}. To see the space-time dependence of electric field more explicitly, let us consider a homogeneous but anisotropic bulk medium. In this case by expanding $\vec{E}^{f,b}(\vec{r},s)$ and $\vec{J}^{f,b}(\vec{r},s)$ in terms of plane waves as
\begin{eqnarray}\label{d38} &&\vec{E}^{f,b}(\vec{r},s)=\frac{1}{\sqrt{V}}\sum_{\vec{n}}\underline{\vec{E}}^{f,b}(\vec{k}_{\vec{n}},s) e^{\imath\vec{k}_{\vec{n}}\cdot\vec{r}},\nonumber\\ &&\vec{J}^{f,b}(\vec{r},s)=\frac{1}{\sqrt{V}}\sum_{\vec{n}}\underline{\vec{J}}^{f,b}(\vec{k}_{\vec{n}},s) e^{\imath\vec{k}_{\vec{n}}\cdot\vec{r}},\nonumber\\ \end{eqnarray} and inserting this expansions in the wave equation (\ref{d36}) we obtain \begin{eqnarray}\label{d39} &&\Lambda(\vec{k}_{\vec{n}},s)\underline{\vec{E}}^{f,b}(\vec{k}_{\vec{n}},s)= \underline{\vec{J}}^{f,b}(\vec{k}_{\vec{n}},s),\nonumber\\ &&\Lambda_{ij}(\vec{k}_{\vec{n}},s)=-\varepsilon_{i\mu\nu}\varepsilon_{\alpha\beta j}\left[\delta_{\nu\alpha}-\tilde{\chi}^m_{ \nu\alpha}(s)\right]k^\mu_{\vec{n}}k^\beta_{\vec{n}}+\mu_0\varepsilon_0 s^2\tilde{\varepsilon}_{ij}(s), \end{eqnarray} where $\varepsilon_{i\mu\nu}$ is the Levi-Civita symbol. we can use the expansions (\ref{d1}), (\ref{d7}), (\ref{d32}) and (\ref{d33}) to obtain the operator $\underline{\vec{J}}^{f,b}(\vec{k}_{\vec{n}},s)$ in terms of the ladder operators of the electromagnetic field and the magnetodielectric medium. Finally using (\ref{d38}) and (\ref{d39}) after some elaborated calculations we obtain the space-time dependence of the electric field in terms of the ladder operators of the field and medium as \begin{eqnarray}\label{d40} &&E_i(\vec{r},t)=\sum_{\vec{n}}\sum_{\lambda=1}^2\sqrt{\frac{\hbar\omega_{\vec{n}} \mu_0}{2cV}} \left[ \eta^\pm_{ij}(\vec{k}_{\vec{n}},t)a_{\vec{n}\lambda}(0)e^{\imath\vec{k}_{\vec{n}} \cdot\vec{r}}+h.c.\right]e^j_{\vec{n}\lambda}\nonumber\\ &&-\frac{\mu_0}{\sqrt{V}}\sum_{\vec{n}}\sum_{\lambda=1}^3\int d^3q\left[ \xi^{\pm}_{ij}(\omega_{\vec{q}},\vec{k}_{\vec{n}},t)d_{\vec{n}\nu}(\vec{q},0) e^{\imath\vec{k}_{\vec{n}}\cdot\vec{r}}+h.c.\right]v^j_{\vec{n}\nu}\nonumber\\ &&\pm\frac{\mu_0}{\sqrt{V}}\sum_{\vec{n}}\sum_{\lambda=1}^3\int d^3q\left[ \zeta^{\pm}_{ij}(\omega_{\vec{q}},\vec{k}_{\vec{n}},t)b_{\vec{n}\nu}(\vec{q},0) e^{\imath\vec{k}_{\vec{n}}\cdot\vec{r}}+h.c.\right]s^j_{\vec{n}\nu},\nonumber\\ && \end{eqnarray}
where the upper (lower) sign corresponds to $t>0$ $(t<0)$ and $\eta^{+}_{ij}(\vec{k}_{\vec{n}},+t)$, $\eta^{-}_{ij}(\vec{k}_{\vec{n}},-t)$, $\xi^{+}_{ij}(\omega_{\vec{q}},\vec{k}_{\vec{n}},+t)$, $ \xi^{-}_{ij}(\omega_{\vec{q}},\vec{k}_{\vec{n}},-t)$, $ \zeta^{+}_{ij}(\omega_{\vec{q}},\vec{k}_{\vec{n}},+t)$ and $ \zeta^{-}_{ij}(\omega_{\vec{q}},\vec{k}_{\vec{n}},-t)$ for $ t>0$ are given by
\begin{eqnarray}\label{d41} &&\eta^{\pm}_{ij}(\vec{k}_{\vec{n}},\pm t)=L^{-1}\left\{ \Lambda_{il}^{-1}(\vec{k}_{\vec{n}},s)\left[\left( \imath s \pm \omega_{\vec{n}}\right)\delta_{lj}\pm \omega_{\vec{n}}\varepsilon_{l\mu\nu}\varepsilon_{\alpha\beta j}\hat{k}^\mu_{\vec{n}}\hat{k}^\beta_{\vec{n}}\tilde{\chi}^m_{ \nu\alpha}(s)\right]\right\},\nonumber\\
&&\xi^{\pm}_{ij}(\vec{k}_{\vec{n}},\pm t)=L^{-1}\left\{ \Lambda_{il}^{-1}(\vec{k}_{\vec{n}},s)\frac{s^2}{s\pm\imath\omega_{\vec{q}}} \right\}f_{lj}(\omega_{\vec{q}}),\nonumber\\ &&\zeta^{\pm}_{ij}(\vec{k}_{\vec{n}},\pm t)=L^{-1}\left\{ \Lambda_{il}^{-1}(\vec{k}_{\vec{n}},s)\frac{s}{s\pm\imath\omega_{\vec{q}}}\right\} \varepsilon_{l\alpha\beta}k_{\vec{n}}^\alpha g_{\beta j}(\omega_{\vec{q}}),\nonumber\\ && \end{eqnarray} and $L^{-1}\{h(s)\}$ denotes the inverse Laplace transformation of $ h(s)$ and $ \Lambda^{-1}$ is the inverse of the matrix $\Lambda$.\\ \textbf{Example 1:}\\ In the first example we show that in the absence of any medium this quantization scheme reduces to the usual quantization in the vacuum. In free space the electric and magnetic susceptibility tensors are zero and from (\ref{d30}) and (\ref{d31}) we deduce that the coupling tensors $ f $ and $g$ are also zero. Therefore from (\ref{d39}), (\ref{d40}) and (\ref{d41}) one finds
\begin{equation}\label{d42} \vec{E}(\vec{r},t)=\imath\sum_{\vec{n}}\sum_{\lambda=1}^2 \sqrt{\frac{\hbar\omega_{\vec{n}}}{2\varepsilon_0 V}}\left[ a_{\vec{n}\lambda}(0)e^{-\imath\omega_{\vec{n}} t+\imath\vec{k}_{\vec{n}}\cdot\vec{r}}-h.c.\right]\vec{e}_{\vec{n}\lambda}, \end{equation}
which is the electric field in the free space. So in this case, quantization of electromagnetic field is reduced to the usual quantization in the vacuum as expected.\\ \textbf{Example 2 :}\\ Take the susceptibility tensors $\chi^e$ and $\chi^m$ as follows \begin{displaymath}\label{d43} \chi^e(\vec{r},t)=\chi^e_0(\vec{r})\times\left\{\begin{array}{ll} \frac{1}{\triangle} & 0<t<\triangle,\\ \\ 0 & \textrm{otherwise}, \end{array}\right. \end{displaymath} \begin{displaymath}\label{d44} \chi^m(\vec{r},t)=\chi^m_0(\vec{r})\times\left\{\begin{array}{ll} \frac{1}{\triangle} & 0<t<\triangle,\\ \\ 0 & \textrm{otherwise}, \end{array} \right. \end{displaymath}
where $\chi^e_0(\vec{r})$ and $\chi^m_0(\vec{r})$ are some time independent but position dependent tensors and $\triangle $ is a real positive constant, using (\ref{d30}) and (\ref{d31}) we find
\begin{eqnarray}\label{d45} (ff^t)(\omega,\vec{r})=\frac{\hbar c^3\varepsilon_0}{4\pi^2 \omega^2}\frac{\sin^2\frac{\omega\triangle}{2}} {\frac{\omega\triangle}{2}}\chi^e_0(\vec{r}),\nonumber\\ (gg^t)(\omega,\vec{r})=\frac{\hbar c^3}{4\pi^2 \omega^2\mu_0}\frac{\sin^2\frac{\omega\triangle}{2}} {\frac{\omega\triangle}{2}}\chi^m_0(\vec{r}),\nonumber\\ \end{eqnarray}
and from (\ref{d26}) and (\ref{d27})
\begin{eqnarray}\label{d46} &&\vec{P}(\vec{r},t)=\vec{P}_N(\vec{r},t)+\chi^e_0(\vec{r})\frac{\varepsilon_0} {\triangle}
\int_{|t|-\triangle}^{|t|} d t'\vec{E}(\vec{r},\pm t'),\nonumber\\ &&\vec{M}(\vec{r},t)=\vec{M}_N(\vec{r},t)+\chi^m_0(\vec{r})\frac{1}{\mu_0
\triangle}\int_{|t|-\triangle}^{|t|} d t'\vec{B}(\vec{r},\pm t'), \end{eqnarray}
where $\vec{P}_N(\vec{r},t),\vec{M}_N(\vec{r},t)$ are the noise polarization densities correspond to a pair coupling tensors $f$ and $g$ satisfying (\ref{d45}). In the limit $ \triangle \rightarrow 0 $, from (\ref{d45}) we deduce that the coupling tensors and therefore the noise polarization densities defined by (\ref{d32}) and (\ref{d33}) tend to zero. In this case the constitutive equations (\ref{d46}) are
\begin{eqnarray}\label{d47} \vec{P}(\vec{r},t)&=&\varepsilon_0\chi^e_0(\vec{r})\vec{E}(\vec{r},t),\nonumber\\ \vec{M}(\vec{r},t)&=&\frac{1}{\mu_0}\chi^m_0(\vec{r})\vec{B}(\vec{r},t),\nonumber\\ \end{eqnarray}
and the wave equation (\ref{d36}) becomes
\begin{eqnarray}\label{d48} &&\nabla\times\nabla\times\vec{E}^{f,b}(\vec{r},s)+\mu_0\varepsilon_0 s^2\left[1+\chi^e_0(\vec{r})\right]\vec{E}^{f,b}(\vec{r},s)-\nonumber\\ &&\nabla\times\chi^m_0(\vec{r}) \nabla\times\vec{E}^{f,b}(\vec{r},s)= \vec{J}^{f,b}(\vec{r},s), \end{eqnarray}
where the noise current density (\ref{d37}) is
\begin{eqnarray}\label{d49} &&\vec{J}^{f,b}(\vec{r},s)=\pm\nabla\times\vec{B}(\vec{r},0)\mp \nabla\times\chi^m_0(\vec{r})\vec{B}(\vec{r},0)+ \mu_0 s\vec{D}(\vec{r},0).\nonumber\\ && \end{eqnarray}
We see that the noise operators have vanished. This is because in the limit $\triangle\rightarrow 0$, the absorption coefficients tend to zero. The solution of the wave equation (\ref{d48}) can be expressed in terms of the Green tensor as
\begin{equation}\label{d50} \vec{E}^{f,b}(\vec{r},s)=\int d^3r'G(\vec{r},\vec{r'},s)\vec{J}^{f,b}(\vec{r'},s), \end{equation}
where the Green tensor satisfies the equation \begin{eqnarray}\label{d51} &&\nabla\times\nabla\times G(\vec{r},\vec{r'},s)+\mu_0\varepsilon_0 s^2\left[1+\chi^e_0(\vec{r})\right]G(\vec{r},\vec{r'},s)-\nonumber\\ &&\nabla\times\chi^m_0(\vec{r}) \nabla\times G(\vec{r},\vec{r'},s)=\delta(\vec{r}-\vec{r'}), \end{eqnarray}
together with some boundary conditions. These boundary conditions
guarantee the continuity of the tangential component of electric field and the normal component of magnetic field at some surfaces where the susceptibilities of the medium become discontinuous. For an anisotropic homogeneous medium using (\ref{d39}), (\ref{d40}) and (\ref{d41}) we can write the electric field as follows
\begin{eqnarray}\label{d51.1} &&E_i(\vec{r},t)=\sum_{\vec{n}}\sum_{\lambda=1}^2\sqrt{\frac{\hbar\omega_{\vec{n}} \mu_0}{2cV}} \left[ \eta^\pm_{ij}(\vec{k}_{\vec{n}},t)a_{\vec{n}\lambda}(0)e^{\imath\vec{k}_{\vec{n}} \cdot\vec{r}}+h.c.\right]e^j_{\vec{n}\lambda},\nonumber\\ \end{eqnarray}
where
\begin{eqnarray}\label{d51.2} &&\eta^{\pm}_{ij}(\vec{k}_{\vec{n}},\pm t)=L^{-1}\left\{ \Lambda_{il}^{-1}(\vec{k}_{\vec{n}},s)\left[\left( \imath s \pm \omega_{\vec{n}}\right)\delta_{lj}\pm \omega_{\vec{n}}\varepsilon_{l\mu\nu}\varepsilon_{\alpha\beta j}\hat{k}^\mu_{\vec{n}}\hat{k}^\beta_{\vec{n}}(\chi^m_0)_{ \nu\alpha}\right]\right\},\nonumber\\ &&\Lambda_{il}(\vec{k}_{\vec{n}},s)=-\varepsilon_{i\mu\nu}\varepsilon_{\alpha\beta l}\left[\delta_{\nu\alpha}-(\chi^m_0)_{ \nu\alpha}\right]k^\mu_{\vec{n}}k^\beta_{\vec{n}}+\mu_0\varepsilon_0 s^2\left(1+(\chi^e_0)_{il}\right). \end{eqnarray}
This example shows that the present quantization scheme is reduced to the usual quantization in a nonabsorptive medium.\\ \textbf{Example 4: A simple model for electric susceptibility tensor}\\ If we neglect the difference between local and macroscopic electric field for substances with a low density, then the classical equation of a bound atomic electron in an external electric field for small oscillation can be written as
\begin{equation}\label{d52} \ddot{x}_i+2\gamma\dot{x}_i+K_{ij}x_j=-\frac{e}{m}\vec{E}_i(t),\hspace{2.00 cm}i=1,2,3, \end{equation}
where $\vec{E}(t)$ is the electric field in the place of the atom and the magnetic force has been neglected in comparison with the electric one and $\gamma$ is a damping coefficient. We have assumed that for sufficiently small oscillations the ith component of the force exerted on the bound electron by nucleus, can be expressed as a linear combination of the coordinates of the electron with constant coefficients $K_{ij}$. Therefore in this simple model the motion of the bound electron is as a forced anisotropic harmonic oscillator. Let $\underline{\vec{E}}(\omega) $ and $\underline{\vec{r}}(\omega)$ be Fourier transforms of the electrical field $\vec{E}(t)$ and position $\vec{r}(t) $ respectively. From (\ref{d52}) we find
\begin{equation}\label{d53} \underline{\vec{r}}(\omega)= =-\frac{e}{m}\left[(-\omega^2+2\imath\gamma \omega)1+K\right]^{-1}\underline{\vec{E}}(\omega). \end{equation}
Now let there be $ N $ molecules per unit volume of the medium with $ z $ electrons per molecule. We assume that the damping coefficient ($\gamma$) and the tensor $K$ are identical for each electron. Then for the Fourier transform of the electric polarization density we find
\begin{equation}\label{d54} \underline{\vec{P}}(\vec{r},\omega)= \frac{Ne^2}{m}\left[(-\omega^2+2\imath\gamma \omega)1+K\right]^{-1}\underline{\vec{E}}(\vec{r},\omega). \end{equation}
From (\ref{d54}) we find the electric susceptibility tensor of the medium in frequency domain
\begin{equation}\label{d55} \underline{\chi}^e(\omega)=\frac{Ne^2}{m\varepsilon_0}\left[(-\omega^2+2\imath\gamma \omega)1+K\right]^{-1}. \end{equation}
From (\ref{d30}) we have
\begin{eqnarray}\label{d56}
(ff^t)(\omega)=\left\{\begin{array}{cc}
\frac{\hbar c^3\varepsilon_0}{4\pi^2\omega^2}2\gamma\omega\left[
\left(-\omega^21+K\right)^2+4\gamma^2\omega^21\right]^{-1} & \omega\neq0, \\
\\
0 & \omega=0 \
\end{array}\right.
\end{eqnarray}
In the special case $\gamma=0 $, the anisotropic dielectric substance is a nonabsorptive one and this relation for $\omega\neq0$ becomes \begin{equation}\label{d57} (ff^t)(\omega)=\frac{\hbar c^3\varepsilon_0}{4\pi\omega^2}\sum_{i=1}^3\delta(\omega-\omega_i) \left(\begin{array}{ccc}
x_i^2 & x_iy_i & x_iz_i \\
y_ix_i & y_i^2 & y_iz_i \\
z_ix_i & z_iy_i & z_i^2 \end{array}\right). \end{equation}
where $\omega_i$, are eigenvalues of the tensor $K$ corresponding to eigenvectors $ R_i= \left(\begin{array}{c}
x_i \\
y_i \\
z_i \end{array}\right)$,$\hspace{0.3cm}(i=1,2,3)$. In this case the coupling tensor $f$ is nonzero only for frequencies $\omega_1,\omega_2,\omega_3 $,. These frequencies are
the resonance frequencies of the equation (\ref{d52}). Therefore
when $\gamma=0$, that is for a non absorptive medium, the coupling tensor $f$ and therefore the noise electrical polarization density is equal to zero except in resonance frequencies, where the energy of electromagnetic field is absorbed by the oscillator. This example explicitly shows that this quantization scheme is also applicable to anisotropic dispersive but non absorptive media. \section{Summary} The electromagnetic field quantization in the presence of an anisotropic magnetodielectric is investigated consistently by modelling the magnetodielectric with two independent quantum fields namely E and M quantum fields. For a given pair of the electric and magnetic susceptibility tensors $\chi^e$ and $\chi^m$, we have found the corresponding coupling tensors $f$ and $g$, which couple electromagnetic field to E and M quantum fields respectively. The explicit space-time dependence of the noise polarizations are obtained in terms of the ladder operators of the medium and the coupling tensors as a consequence of Heisenberg equations. Maxwell and constitutive equations are obtained directly from Heisenberg equations. In the limiting case, i.e., when there is no medium, the approach tends to the usual method of quantization of the electromagnetic field in vacuum. Also when the medium is a non absorptive one, the noise polarizations tend to zero and in this case the approach is equivalent to the previous methods, as expected.
\end{document} | arXiv |
How do we know the LHC results are robust?
Nature article on reproducibility in science.
According to that article, a (surprisingly) large number of experiments aren't reproducible, or at least there have been failed attempted reproductions. In one of the figures, it's said that 70% of scientists in physics & engineering have failed to reproduce someone else's results, and 50% have failed to reproduce their own.
Clearly, if something cannot be reproduced, its veracity is called into question. Also clearly, because there's only one particle accelerator with the power of the LHC in the world, we aren't able to independently reproduce LHC results. In fact, because 50% of physics & engineering experiments aren't reproducible by the original scientists, one might expect there's a 50% chance that if the people who originally built the LHC built another LHC, they will not reach the same results. How, then, do we know that the LHC results (such as the discovery of the Higgs boson) are robust? Or do we not know the LHC results are robust, and are effectively proceeding on faith that they are?
EDIT: As pointed out by Chris Hayes in the comments, I misinterpreted the Nature article. It says that 50% of physical scientists have failed to reproduce their own results, which is not the same statement as 50% of physics experiments aren't reproducible. This significantly eases the concern I had when I wrote the question. I'm leaving the question here however, because the core idea - how can we know the LHC's results are robust when we only have one LHC? - remains the same, and because innisfree wrote an excellent answer.
particle-physics large-hadron-collider data-analysis
$\begingroup$ I think it is worth pointing out that the LHC doesn't just do one particle collision and then say the experiment is completed. How much do you know about what goes into such experiments, how many times they are actually repeated, and then how the data is analyzed from there? $\endgroup$ – Aaron Stevens Mar 27 at 5:11
$\begingroup$ I was asking if you had looked into the efforts taken to make sure the results from the LHC are good results and not just mistakes. Also the LHC isn't the only particle collider in existence. $\endgroup$ – Aaron Stevens Mar 27 at 5:19
$\begingroup$ For more about this important question than you may have bargained for, look for discussions about the "look-elsewhere effect" in the statistical analysis of the data from the mostly-independent ATLAS and CMS experiments at the LHC, especially in the context of their joint discovery of the Higgs particle. $\endgroup$ – rob♦ Mar 27 at 5:52
$\begingroup$ @Allure "Half [of scientists] have failed to reproduce their own experiments" is an enormously different statement from "half of all experiments are non-reproducible". The former statement (from the Nature article) includes scientists who have failed to reproduce a single one of their experiments, even if they successfully reproduced 99 out of 100. See page 10 of the questionnaire for the exact wording. $\endgroup$ – Chris Hayes Mar 27 at 9:46
$\begingroup$ @ChrisHayes My thought exactly! Furthermore, a failed attempt to replicate an experiment doesn't necessarily mean that the original experiment is "non-reproducible". $\endgroup$ – jkej Mar 27 at 10:13
That's a really great question. The 'replication crisis' is that many effects in social sciences (and, although to a lesser extent, other scientific fields) couldn't be reproduced. There are many factors leading to this phenomenon, including
Weak standards of evidence, e.g., $2\sigma$ evidence required to demonstrate an effect
Researchers (subconsciously or otherwise) conducting bad scientific practice by selectively reporting and publishing significant results. E.g. considering many different effects until they find a significant effect or collecting data until they find a significant effect.
Poor training in statistical methods.
I'm not entirely sure about the exact efforts that the LHC experiments are making to ensure that they don't suffer the same problems. But let me say some things that should at least put your mind at ease:
Particle physics typically requires a high-standard of evidence for discoveries ($5\sigma$). To put that into perspective, the corresponding type-1 error rates are $0.05$ for $2\sigma$ and about $3\times10^{-7}$ for $5\sigma$
The results from the LHC are already replicated!
There are several detectors placed around the LHC ring. Two of them, called ATLAS and CMS, are general purpose detectors for Standard Model and Beyond the Standard Model physics. Both of them found compelling evidence for the Higgs boson. They are in principle completely independent (though in practice staff switch experiments, experimentalists from each experiment presumably talk and socialize with each other etc, so possibly a very small dependence in analysis choices etc).
The Tevatron, a similar collider experiment in the USA operating at lower-energies, found direct evidence for the Higgs boson.
The Higgs boson was observed in several datasets collected at the LHC
The LHC (typically) publishes findings regardless of their statistical significance, i.e., significant results are not selectively reported.
The LHC teams are guided by statistical committees, hopefully ensuring good practice
The LHC is in principle committed to open data, which means a lot of the data should at some point become public. This is one recommendation for helping the crisis in social sciences.
Typical training for experimentalists at the LHC includes basic statistics (although in my experience LHC experimentalits are still subject to the same traps and misinterpretations as everyone else).
All members (thousands) of the experimental teams are authors on the papers. The incentive for bad practices such as $p$-hacking is presumably slightly lowered, as you cannot 'discover' a new effect and publish it only under your own name, and have improved job/grant prospects. This incentive might be a factor in the replication crisis in social sciences.
All papers are subject to internal review (which I understand to be quite rigorous) as well as external review by a journal
LHC analyses are often (I'm not sure who plans or decides this) blinded. This means that the experimentalists cannot tweak the analyses depending on the result. They are 'blind' to the result, make their choices, then unblind it only at the end. This should help prevent $p$-hacking
LHC analysis typically (though not always) report a global $p$-value, which has beeen corrected for multiple comparisons (the look-elsewhere effect).
The Higgs boson (or similar new physics) was theoretically required due to a 'no-lose' theorem about the breakdown of models without a Higgs at LHC energies, so we can be even more confident that it is a genuine effect. The other new effects that are being searched for at the LHC, however, arguably aren't as well motivated, so this doesn't apply to them. E.g., there was no a priori motivation for a 750 GeV resonanace that was hinted at in data but ultimately disappeared.
If anything, there is a suspicion that the practices at the LHC might even result in the opposite of the 'replication crisis;' analyses that find effects that are somewhat significant might be examined and tweaked until they decrease. In this paper it was argued this was the case for SUSY searches in run-1.
innisfreeinnisfree
$\begingroup$ This is an excellent answer! I think it should be further emphasized how different $2\sigma$ is from $5 \sigma$. Using the standard $2 \sigma$ conventions of social science, you have a 5% chance of getting a significant result every time you test a completely false hypothesis. (And this can easily be boosted by a factor of 10 by $p$-hacking techniques, plus you can report something like $p = 0.1$ as "trending towards significance".) Asking for $5 \sigma$ is not merely being $5/2$ as rigorous, the corresponding $p$-value cutoff is roughly $0.0000003$. $\endgroup$ – knzhou Mar 27 at 10:32
$\begingroup$ I think saying the "social sciences" is possibly a bit overly specific. There's been much talk and news of reproducibility problems in biology and chemistry as well, at least in semi-recent years, though perhaps not as bad as the social sciences are experiencing. $\endgroup$ – mbrig Mar 27 at 17:05
$\begingroup$ Although it is a slightly different issue than the statistical considerations that are the focus of this answer, laymen also often don't appreciate that the LHC has necessarily reproduced many previous discoveries: atlas.cern/updates/atlas-blog/art-rediscovery . From these and similar studies, we can directly evaluate whether the reproducibility crisis seems to be present in particle physics... and, not surprisingly given the extensive measures described in your answer, it appears that it does better than many (all?) other fields, so far. $\endgroup$ – Rococo Mar 27 at 17:22
$\begingroup$ A minor addition is that the prior probability for the Higgs boson existing and to a much lesser extent being in the range it was found is likely higher than "surprising" results in social science. Which is just to say it wasn't a surprise that the Higgs boson existed; that's what the theory predicted. Some new non-Higgs particle would warrant much more skepticism. $\endgroup$ – Derek Elkins Mar 28 at 6:49
In addition to innisfree's excellent list, there's another fundamental difference between modern physics experiments and human-based experiments: While the latter tend to be exploratory, physics experiments these days are primarily confirmatory.
In particular, we have theories (sometimes competing theories) that model our idea of how physics works. These theories make specific predictions about the kinds of results we ought to see, and physics experiments are generally then built to discriminate between the various predictions, which are typically either of the form "this effect happens or doesn't" (jet quenching, dispersion in the speed of light due to quantized space), or "this variable has some value" (the mass of the Higgs boson). We use computer simulations to produce pictures of what the results would look like in the different cases and then match the experimental data with those models; nearly always, what we get matches one or the other of the suspected cases. In this way, experimental results in physics are rarely shocking.
Occasionally, however, what we see is something really unexpected, such as the time OPERA seemed to have observed faster-than-light motion—or, for that matter, Rutherford's gold-foil experiment. In these cases, priority tends to go toward reproducing the effect if possible and explaining what's going on (which usually tends to be an error of some sort, such as the miswired cable in OPERA, but does sometimes reveal something totally new, which then tends to become the subject of intense research itself until the new effect is understood well enough to start making models of it again).
chrylischrylis
$\begingroup$ I understand what you mean, but "match experimental data with models" sounds like there is ample reason to expect confirmation bias if not done properly. $\endgroup$ – Scrontch Mar 27 at 8:45
$\begingroup$ @Scrontch If not done properly, of course, but the useful property of these two questions (yes/no and value-in-range) is that we can run simulations ahead of time and define with clarity what the results should look like in the different possible universes, including such information as how wide the margins need to be to give us confidence. There are (fairly) standard ways of doing this. $\endgroup$ – chrylis Mar 27 at 20:31
The paper seems to be a statistical analysis of opinions, and in no way is rigorous enough to raise a question about the LHC. It is statistics about undisclosed statistics.
Here is a simpler example for statistics of failures: Take an Olympics athlete. How many failures before breaking the record? Is the record not broken because there may have been a thousand failures before breaking it?
What about the hundreds of athletes who try to reproduce and get a better record? Should they not try?
The statistics of failed experiments is similar: There is a goal (actually thousands of goals depending on the physics discipline), and a number of trials to reach the goal, though the olympics record analogy should not be taken too far, only to point out the difficulty of combining statistics from a large number of sets. In physics there may be wrong assumptions, blind alleys, logical errors... that may contribute to the failure of reproducibility. The confidence level from statistical and systematic errors are used to define the robustness of a measurement.
from the question:
"because 50% of physics & engineering experiments aren't reproducible by the original scientists",
This is a fake statement from a dubious poll. The statistical significance of the "not reproducible " has not been checked in the poll. Only if it were a one standard deviation result , there exists almost a 50% a probability of the next trial not to reproduce.
one might expect there's a 50% chance that if the people who originally built the LHC built another LHC, they will not reach the same results
No way, because engineering and physics analysis at the LHC are over the 4 sigma level, and the probability of negation is small. Even a 3sigma level has confidence 99% , so the chance is in no way 50%.
We know the LHC results are robust because there are two major and many smaller experiments trying for the same goals. The reason there are two experiments is so that systematic errors in one will not give spurious results. We trust that the measurement statistics that give the end results are correct, as we trust for the record breaking run that the measured times and distances are correct.
(And LHC is not an experiment. It is where experiments can be carried out depending on the efforts and ingenuity of researchers, it is the field where the Olympics takes place.)
The robustness of scientific results depends on the specific experimental measurements, not on integrating over all disparate experiments ever made. Bad use of statistics. For statistics of statistics, i.e. the confidence level of the "failed experiments" have to be done rigorously and the paper is not doing that.
Another way to look at it: If there were no failures , would the experiments mean anything? They would be predictable by pen and paper.
$\begingroup$ I'm not sure I buy the Olympics analogy. Failed attempts at breaking a record isn't the same thing as a failed attempt to reproduce an experiment. It also sounds like you are saying we should just cherry pick what does work and ignore when it fails. $\endgroup$ – Aaron Stevens Mar 27 at 5:14
$\begingroup$ @AaronStevens " cherry pick what does work" but is not that evolution in general? and "ignore when it fails" one learns from failure to design better experiments. $\endgroup$ – anna v Mar 27 at 5:35
$\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – ACuriousMind♦ Mar 29 at 22:24
Any one experiment is repeated many times on the same equipment. They look for rare events, and it takes a lot of rare events to be sure that they aren't just coincidence.
The question about how many LHCs it takes to be sure, is different.
Each LHC component had to be carefully tested to make sure it was in spec. Remember the example of the experiment that seemed to get a result slightly faster than light. Because it was so important, they went to great expense to test everything, components around the world, until they found two components that were out of spec, that created the small error. If the error had been in the other direction would they have done that testing? No. They wouldn't even notice the error. It wouldn't be important. What made this one important was faster than light. Did they carefully record every out-of-spec component they found that would tend to slow the signal, that might cancel the positive errors they found? Maybe. That wasn't what they were looking for, though. That was a complication and not a solution to the problem.
After the tested LHC components are installed they must be tested again in case they were changed while being handled.
Then they must be calibrated. Every analog output could have a baseline that's a little bit off, because of random things. A solder joint that's slightly different. An AC circuit nearby that changes things a little bit every 120th of a second. The baseline must be calibrated for every one of them. Once the signal has been converted to digital then it's OK. Errors smaller than the cutoff are ignored, and larger errors make one bit difference. For the calibration, you know what the outcome is supposed to be, so you set it to that.
Could all this have somehow changed the outcomes so that some extremely unlikely results are falsely reported more often than they should be?
There's no theoretical reason to expect it. And the engineers who assembled the LHC were very very careful. But how could we test it? The obvious way is to build at least 2 more LHCs and notice how consistent their results are. That would be very expensive. It will not be done.
We can get some confidence by looking at results from other machinery. It's like -- the LHC was used to scan for a wide range of possible results that could be called the Higgs boson. They could do in years what a lesser machine might take centuries to do. But once we have a specific Higgs boson to look for, some of the others can look for that specifically and see whether they find it. If they do, then there's probably something there beyond equipment error.
Something else they can do (which I think they are doing part of the time) is look for things that are supposed to not happen that nobody predicts will happen. When they find one for sure then everybody will get excited. People will say there's something wrong, and insist that they check for every possible error that could give them that result. Like with the faster-than-light thing.
J ThomasJ Thomas
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Football free-kicks… taken by Newton
Discover the mysteries behind the most spectacular free-kick ever scored and how Newton can help us to simulate it.
Image: Flickr user Akshay Davis, CC BY 2.0
by Hugo Castillo Sánchez. Published on 1 June 2017.
This Saturday might be a normal day for many people, but football fans around the world are looking forward to it, as two of the best teams in the the world are playing in the final of the best club football competition, the Champions League, at the National Stadium of Wales in Cardiff.
Here at Chalkdust, we're quite excited too, and so we decided to analyse the mathematics and physics behind one of the best goals ever scored… and go on to reproduce it mathematically! You might find it provides an excellent topic of conversation in your preferred pub before the match or during half-time.
The impossible football free-kick
Roberto Carlos, considered one of the best left-backs of all time. Flickr user Nazionale Calcio, CC BY 2.0.
In 1997, the Tournoi de France (French for "Tournament of France") was an international football tournament organised by France, held the year before the World Cup was hosted by the same country. Four national teams participated: France, Italy, England and Brazil. The teams played against each other and England, having won the most points, finished as the champions. This might sound like any other normal tournament, but something happened that no one had seen before.
On 3rd June, the inaugural match took place in Lyon: France played against the reigning World Cup champions Brazil—a match that every football fan would have loved to watch, as many of the most gifted and decorated players of their generation (even of all time) were playing. These included Zinedine Zidane (one of the best French players in history), Fabian Barthez, Ronaldo Nazario (the best and most skilful striker at the time), Romario, Cafu and Roberto Carlos.
One of the French players committed a foul towards the middle of the field, 35 metres away from the goal France was defending. The Brazilian left-back, Roberto Carlos, who was known to be an excellent free-kick taker, took the ball. Although it was a long range free-kick, the goalkeeper Fabian Barthez knew all about Roberto Carlos' ability (his powerful free-kicks had been measured at 169km/h), and so he placed four players in the wall to block the shot (normally, from long distance, a goalkeeper would place only two players). Roberto Carlos took a long run-up in order to hit his shot with lots of power and then… he made it. It was astonishing! Take a look for yourself:
Impressive, isn't it? What a goal. So powerful. But now, let's change the angle of the camera:
I would use many adjectives to describe this goal, but the space I have is simply too short to write them all down. Simply incredible. Many (older) football fans will recognise this goal, but if this is the first time you watch it, I hope it gives you the same emotions it gave me when I watched this beauty at 11 years of age. This is by far the most spectacular goal I have seen so far in my life: from 35m out the ball was heading towards the corner flag, but it then curved like a "banana" and landed in the back of the net. What made it even more impressive is that the bend was so pronounced the French goalkeeper never even moved. You can actually see that in the GIF above. Even the camera man was tricked by Roberto Carlos' shot.
This free-kick caught the attention of many scientists, whose main concern was to explain the banana effect of the ball's trajectory and under what conditions the famous free-kick could be reproduced. Match commentators at the time named it the impossible free kick, or the goal that defied the laws of physics. We will see that in fact there was nothing impossible about it at all and that it actually perfectly obeyed the laws of physics.
The Magnus effect
The phenomenon responsible for the banana effect is actually very common in ball sports. In fact, here at Chalkdust we have already written about its appearance in the world of tennis. The phenomenon is known as the Magnus effect, which can be explained using a simple illustration.
Diagram that shows a rotating ball moving through air. By Rdurkacz – using inkscape, inspired by original version of the Wikipedia diagram, CC BY-SA 3.0,
In the diagram, the football is moving to the left and is also rotating clockwise at the same time. When this happens, there is a thin layer of fluid (in our case, air) that moves with the surface of the ball. This causes a difference in the velocities of the air at the upper (1) and lower (2) sides of the ball.
At position 1, the football is rotating away from the direction of the ball's flight. If we measured the speed of the air flow at that point, we would observe that the velocity would always be smaller than the velocity on the bottom side. This is because at position 2, the ball is moving faster due to the combined effect of the flight direction and rotation. The difference in velocities on the two sides of the ball will cause a pressure difference resulting in a net force acting perpendicularly to the direction of flight. This is known as the Magnus force.
Roberto Carlos' physique
We can now give a simple explanation of Roberto Carlos' free-kick: he struck the ball with the outer side of his boot's laces, which caused the ball to move through the air with a large amount of rotation. This generated the Magnus force that pushed the ball to the left and made it enter the goal.
Roberto Carlos' big thighs allowed him to hit the ball violently. Flickr, user: I30_, Public Domain
However, a spinning ball is not enough: the ball has to be hit strongly enough to induce enough spin, which is something that Roberto Carlos knew how to do perfectly. If a free-kick like Roberto Carlos' is to be replicated, it needs to be from long range to ensure that the Magnus force has enough of an effect on the ball before it reaches the goalpost. If the distance from the free-kick taker and the goal are not large enough, the football would just end up going well wide of the goal.
Newton's second law
Forces on a ball
Now that we know the science behind Roberto Carlos' free kick, it is time to start using mathematics to model the ball's trajectory. First of all, we need to determine the forces acting on the ball. These are illustrated in the diagram:
Forces on a spinning ball
The ball (with mass $m$) is moving through the air (of density $\rho$) with vector velocity $\vec{v}=(v_x, v_y, v_z)$. The simplest force acting on the ball is the gravitational force ($m\vec{g}$), which is pointing downwards and is assumed to be constant.
There are two additional forces that only exist when the ball is moving through a fluid: one of them is called the drag force ($\vec{F}_D$) and it is related to the air resistance. The main characteristic of this force is that it acts in the opposite direction to the motion of the ball. Mathematically speaking, the drag force adopts the following form:
$\vec{F}_D = -\frac{1}{2}\rho A C_D|\vec{v}|\vec{v},$
where $A$ is the cross-sectional area of the ball; $|\vec{v}|$ is the magnitude of the vector velocity; and $C_D$, a dimensionless number, is the drag coefficient, which has to be determined experimentally and depends on many factors, such as the characteristics of the ball and the temperature, among others. Greater values of $C_D$ mean higher air resistance, and in the absence of air, $\vec{F}_D=0$.
The other force is the Magnus force, $\vec{F}_L$ (also called the lift force). As we stated, this force is the result of the ball interacting with air: the main difference with the drag force is that the spin of the football plays an important role here. The lift force is perpendicular to the vector velocity, and can be calculated using the following relation:
$\vec{F}_L = \frac{1}{2}\rho A C_L|\vec{v}| f(\theta)\vec{v}.$
As we can see, the Magnus force depends on almost the same variables as the drag force, except that instead of a drag coefficient, now we have a lift coefficient $C_L$ (which also has to be calculated experimentally) and there is a function $f(\theta)$ that relates the Magnus force with the angle of spin of the ball.
Newton's second law has a ball!
As we all know, Newton's second law simply states that the rate of change of momentum ($m\vec{a}$, when $m$ is constant) is equal to the forces acting on a body (in our case, the ball). Summing all the forces, we end up with the general equation we are looking for:
$m\vec{a} = -m\vec{g} + \vec{F}_D + \vec{F}_L.$
All the terms of the equation are in vector form. We wish to obtain and describe the trajectory of a spinning ball in three dimensions, $\vec{x}=(x,y,z)$. Assuming that the angle of spin is $-90$º, the simplest system of equations to trace the football's trajectory is:
$$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} = -|\vec{v}|H \Big(C_D\frac{\mathrm{d}x}{\mathrm{d}t} + C_L \frac{\mathrm{d}y}{\mathrm{d}t}\Big)$$
$$\frac{\mathrm{d}^2 y}{\mathrm{d}t^2} = -|\vec{v}|H \Big(C_D\frac{\mathrm{d}y}{\mathrm{d}t} – C_L \frac{\mathrm{d}x}{\mathrm{d}t}\Big)$$
$$\frac{\mathrm{d}^2 z}{\mathrm{d}t^2} = -|\vec{v}|H C_D\frac{\mathrm{d}z}{\mathrm{d}t}- g,$$
where $H$ and $|\vec{v}|$ are:
$$H= \frac{\rho A}{2m}$$
$|\vec{v}|= \sqrt{ (\frac{\mathrm{d}x}{\mathrm{d}t})^2 + (\frac{\mathrm{d}y}{\mathrm{d}t})^2 + (\frac{\mathrm{d}z}{\mathrm{d}t})^2}$.
We have a system of three ordinary differential equations (ODEs) in three dimensions whose dependent and independent variables are position and time, respectively. Notice that the drag forces ($C_D$) will affect the motion in all three dimensions. On the other hand, the Magnus force ($C_L$) will only have an effect on the $x$-$y$ plane because these equations consider that the ball is spinning about its vertical axis ($z$). The equations also assume that the gravitational force will just affect the motion in $z$. Now that we have the necessary equations, it is time to solve them.
Simulating Roberto Carlos' free-kick
In order to completely solve the system of ODEs, we need the initial conditions ($\vec{x}_0$, $\vec{v}_0$), empirical equations to calculate the coefficients $C_D$ and $C_L$, the characteristics of the ball ($m$ and $A$) and the density of air ($\rho$).
Drag and lift coefficients
The drag and lift coefficients need to be determined experimentally: in a laboratory, the trajectory of a football is studied and its velocity and acceleration are then calculated. Once this information is known, equations for $C_D$ and $C_L$ as a function of the speed and other parameters are derived. Many equations for these coefficients can be found in the literature. Here we will use the simplest ones: for instance, many authors have proposed an equation for the lift coefficient that adopts the following form:
$$C_L = \frac{\omega R}{|\vec{v}|},$$
where $R$ and $|\vec{v}|$ are the ball's radius and velocity, respectively, and $\omega$ is the angular velocity of the ball, which is dependent on time:
$$\omega = \omega_0 \exp{(-t/7)}.$$
Here $w_0$ is the angular velocity that the ball has when it is first hit by the player. We can notice that the angular velocity decreases as time increases; and if the lift coefficient is proportional to the angular velocity, then $C_L$ will also decrease, and therefore we can say that the Magnus force will affect the trajectory of the ball at short to medium times, and at large times its effect will vanish. It has been reported that the initial angular velocity of Roberto Carlos' free kick was 88 rad/s, so we will use this value later.
Angular velocity as a function of time. $w_0=88$ rad/s
The equations for estimating the drag coefficient are more complex and more complicated to obtain experimentally, as $C_D$ depends on many factors. For instance, in this paper they determined the following equation for a Teamgeist ball (used in the World Cup of Germany, 2006):
$$C_D= \begin{cases} 0.4127(C_L)^{0.3056} &\mbox{if } |\vec{v}|> v_c &\mbox{and} \qquad C_L >0.05 , \\ 0.155 + \frac{0.346}{1+ \exp{(|\vec{v}|- v_c)/v_s}} &\mbox{otherwise,} \end{cases} $$
where $v_c=12.19$m/s and $v_s=1.309$m/s. The sudden change in the values of the drag coefficient is due to a phenomenon called drag crisis, which is commonly observed in fluid mechanics problems.
Characteristics of the ball
We will be using the following values for the radius $R$ and mass $m$ of a standard football, and the density of the air (at standard temperature and assuming that it is a constant):
$m=0.442 \mbox{kg} \qquad R=0.11 \mbox{m} \qquad \rho = 1.225 \mbox{kg}/\mbox{m}^3$
Initial conditions
We have a system of three ordinary differential equations of second order, which means we need six initial conditions in order to find the solution for the system: three of them are for the initial vector position $\vec{x}_0$ and the other three for the initial vector velocity $\vec{v}_0$.
For practical purposes, using a Cartesian coordinate system, we choose the zero vector $\vec{x}_0=(0,0,0)$ to be the point from which the free kick is going to be taken. The initial vector position, on the other hand, is not as simple to obtain. We will make use of the following figure, which illustrates our coordinate system:
Initial position and velocity vectors.
The ball will move with an initial vector velocity $\vec{v}_0=(v_{x0}, v_{y0},v_{z0})$, whose magnitude is defined as:
$|\vec{v}_0|= \sqrt{ (v_{x0})^2 + (v_{y0})^2 + (v_{z0})^2}$.
It has been reported that the initial velocity of the ball in Roberto Carlos' free-kick was approximately $38$m/s. The values of the angle with the horizontal, $\alpha$, and the angle to the $y$-axis, $\gamma$, were also determined by scientists and their values are
$\alpha = 16º \qquad \gamma=5º.$
These values are highly important as they will allow us to calculate the elements of the initial vector velocity. Using a bit of geometry and trigonometry, and taking the values of the angles $\alpha$, $\gamma$ and $|\vec{v}_0|=$38 m/s, we can calculate the values of the components of the initial vector velocity, which are:
$$\vec{v}_0=(10.47, 35.11, 10.07)\mbox{m/s}$$
Now we have all the elements necessary to solve our system of ODEs. However, an analytic solution is not possible to obtain, as there are non-linear terms and the system of equations is strongly coupled: thus, numerical methods are required. We will use one of the most popular software packages, MATLAB, to find a numerical solution to our system. More specifically, we will use the solver called ode45, which, for a given set of initial conditions, will help us to integrate our equations.
Simulation results
Once we have introduced our system of equations, the parameters (lift and drag coefficients, characteristics of the ball) and the initial conditions, we make use of the solver ode45 and we then plot the solutions in our coordinate system. In our first calculation we consider the initial angular velocity as zero ($\omega_0=0$). Our code gives us the following results:
$v_0=38$m/s and $w_0=0$
The red lines represent the dimensions of the goal according to official rules of football (just half of it is shown as the free kick is taken from a position roughly at the centre of the goal). For this case, notice that the trajectory of the ball, which adopts a nice parabolic form, is not even close to the goal, which is 35m away along the $y$-axis from the initial point $\vec{x}_0=(0,0,0)$. Now we show the results considering the exact conditions of Roberto Carlos' free-kick, setting the angular velocity to be 88 rad/s.
$v_0=38$m/s and $w_0=88$rad/s
Goal! (Kind of!) The banana effect described at the beginning of this blog post can be replicated using a simple model that has been derived from Newton's second law. According to our calculations, it took approximately 1.34s for the ball to reach the goal, which is consistent with the actual free kick. Below, we can appreciate the ball deflection in the $x$-$y$ plane:
Assuming that the the wall formed by four players is located 9.1m from the point where the free kick is taken (according to the official rules of football) and has a width of 1m width, our predictions show the ball sidestepping the wall and beginning to bend inwards approximately 20m away from our point of origin. It has been reported that the deflection in Roberto Carlos' free kick was approximately 3m, which is also consistent with these results.
However, for the ball height, things are slightly different. In the diagram below, we show the ball height in the $y$-$z$ plane:
The figure shows that the ball reaches a maximum height of approximately 4m, but that its final height (when the ball has travelled 35m in the $y$-direction) is around 2.5m, which is 1 metre greater than the final height seen in the real free-kick and higher than the actual crossbar! The difference is mostly due to the equations for the drag coefficient $C_D$, which have been calculated for a Teamgeist ball, a football with different characteristics than the one used in the match played in 1997. In addition, we are using a simple model derived from Newton's second law, and we are ignoring a possible contribution of the Magnus force in the equation for the acceleration in $z$ direction.
Although there are many more sophisticated and complex models (some of them using the approach of fluid mechanics), the model derived from classical mechanics can replicate nicely the trajectory of the ball and the banana effect observed in the famous Roberto Carlos free-kick.
As mathematicians and scientists, it would be great to think that Roberto Carlos spent weeks solving differential equations and studying the physics behind the ball's motion. But probably not. In reality, professional players spend months and years training, and they manage to learn and understand through pratice phenomena such as the Magnus force and how air resistance can affect their shots: the result is that we have witnessed some spectacular goals.
For further information on this topic, I highly recommend the article Football curves, written by Gillaume Dupeux and colleagues, in which they analysed all the curves observed in football. In 2010, they carried out a more in-depth analysis into the problem and simulated the free kick in question perfectly. In addition, they demonstrated that if gravity had not taken over, the path the ball would have taken would have adopted the form of an ideal spiral.
Now that you know a bit about the physics and mathematics behind free-kicks, it's time to go to the pub and support your favourite team and, maybe, see the Magnus effect in action!
Sandhu, J., Edgington, A., Grant, M., Rowe-Gurney, N., How to score a goal, Journal of Physics Special Topics, 2011.
Goff, J.E., Carré, M., Soccer ball lift coefficients via trajectory analysis, European Journal of Physics, May 2010, Vol 31, No 4.
Goff, J.E., Carré, Trajectory analysis of a soccer ball, American journal of Physics, 77, 1020 (2009).
Bray, K., Kerwin, D., Modelling the flight of a soccer ball in a direct free kick, Journal of Sports Science, Vol. 21, Issue 2, 2003.
Dupeux, G., Cohen C., Le Goff., A., Quére, D., Clanet, C., Football curves, Journal of Fluids and Structures, Vol. 27, Issues 5-6, 2011.
Hazar, Alexandra, Arsic, D., Diaz, J.J., Hakansson, J., Carvalho, P.F., Paas, R., Reis, T., Phantom footballs and impossible free kicks: modelling the flight of modern soccer balls, Modelling week, final report, TUD, ECMI, 2012.
[France vs Brazil, 1997, Public Domain. Pictures: 1- Banner picture: adapted from Flickr.com – BrazilvWales 136 by Akshay Davis CC-BY 2.0; 2- adapted from Flickr.com – Leicester City (and Denmark) goalkeeper Kasper Schmeichel by Ben Sutherland, CC-BY 2.0; 3- adapted from pixabay – by stux, Public domain; other pictures by Chalkdust]
Hugo Castillo Sánchez
Hugo is a chemical engineer doing a PhD in Mathematics at University College London. He is currently working on non-Newtonian fluid dynamics. He is also interested in transport phenomena and rheology (the science of deformation).
hugocastillocom.wordpress.com + More articles by Hugo
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Applied Water Science
August 2019 , 9:144 | Cite as
Evaluation of energy dissipation on stepped spillway using evolutionary computing
Abbas Parsaie
Amir Hamzeh Haghiabi
Part of the following topical collections:
Water and Energy
In this study, using the M5 algorithm and multilayer perceptron neural network (MLPNN), the capability of stepped spillways regarding energy dissipation (ED) was approximated. For this purpose, relevant data was collected from valid sources. The study of the developed model based on the M5 algorithm showed that the Drop and Froude numbers play important roles in modeling and approximating the ED. The error indices of M5 algorithm in training were R2 = 0.99 and RMSE = 2.48 and in testing were R2 = 0.99 and RMSE = 2.23. The study of developed MLPNN revealed that this model has one hidden layer which includes five neurons. Among the tested transfer functions, the great efficiency was related to the Tansing function. The error indices of MLPNN in training were R2 = 0.97 and RMSE = 3.73 and in testing stages were R2 = 0.97 and RMSE = 3.98. Evaluation of the results of both applied methods shows that the accuracy of the MLPNN is a bit less than the M5 algorithm.
Energy dissipation Soft computing Drop number Spillways M5 algorithm
ANFIS
Adaptive neuro-fuzzy inference system
Computational fluid dynamic
Drop number
Energy dissipation
Froude number
Gravity acceleration
Genetic expression programming
GMDH
Group method of data handling
Total head of flow
Height of steps
Length of crest
Length of steps
Multivariate adaptive regression splines
MLPNN
Multilayer perceptron neural network
Number of steps
Discharge per width
Flow depth over the crest
y1 and y2
Conjugated depths of hydraulic jump
This article is part of the Topical Collection "Water and Energy" guest edited by Enrico Drioli.
Spillways are structures that are extensively used to evacuate surplus flow over reservoir capacity of dams. One of the most important hydraulic problems of spillways is the high flow velocity, which causes cavitation and scouring at its downstream. Hence, dissipation of the energy of flow is the primary issue of spillways. In this way, using the baffles along the chute of the spillway, stepped spillway, ski jump buckets and stilling basin at the toe of spillways have been suggested (Heller et al. 2005; Movahedi et al. 2019; Xiao et al. 2015). Baffles usually are used in the small dam projects. The use of other mentioned structures is common in large dams projects (Erfanain-Azmoudeh and Kamanbedast 2013). An economic examination of three options including stilling basin, flip bucket and stepped spillway for designing the energy dissipater in large dams indicated that the use of stepped spillway is a logical decision (Christodoulou 1993). The advantage of stepped spillway in comparison with other energy dissipater structures is related to reduce or remove the probability of cavitation occurrence on the spillway (Frizell et al. 2013; Pfister and Hager 2011). The flow pattern over the stepped spillways was classified into three as napped, transition and skimming flows. The napped flow appears in the low discharge, and in the skimming flow, there is a virtual boundary between the jet stream and the steps. The transition regime is a status between the napped and skimming flow (Shahheydari et al. 2014). Although the energy lost in the nape regime is more than the skimming flow, but due economic reasons, the stepped spillways are designed under skimming flow condition. By advancing the computer technology, the investigators have tried to study the properties of flow over the spillway using the numerical methods (Parsaie et al. 2016a, 2018b). Numerical modeling is divided into two main groups as computational fluid dynamic (CFD) and soft computing. In the field of CFD, the governing equations which are usually Navier–Stokes equations are solved along the turbulence models such as K-epsilon and RNG . (Kim and Park 2005). Fortunately, nowadays, a number of user-friendly CFD packages have proposed to easily apply the CFD techniques along the physical modeling to reduce the cost of experiments (Salmasi and Samadi 2018). Along the physical and CFD modeling, another field of numerical modeling, i.e., soft computing techniques have been implemented for accurately present the results of experimental studies which are based on the defining the depended desired parameter with correspond to measuring the independent variables (Azamathulla et al. 2016; Maghsoodi et al. 2012; Najafzadeh et al. 2017; Najafzadeh and Zeinolabedini 2019; Sihag et al. 2019; Wu 2011). Among the soft computing models using the ANFIS by Salmasi and Özger (2014) and the GEP by Roushangar et al. (2014), MARS and GMDH methods by Parsaie et al. (2018a, c) have been reported to predict the energy dissipation over the stepped spillway. Reviewing the literature shows that the M5 algorithm for modeling the capabilities of stepped spillway has not been test; hence, in this a formula based on the M5 algorithm for modeling and predicting the performance of stepped spillways regarding energy dissipation is proposed.
Energy dissipation involved parameters
The scheme of stepped spillway is shown in Fig. 1. In this figure, the size of steps (height and length) is cleared via hs and Ls, respectively. Hw is the height of dam, y0 is the depth of flow over the crest, Lc is the length of crest, and y1 and y2 are the conjugated depths of hydraulic jump, respectively. The energy dissipation over the stepped spillway is estimated using the Bernoulli equation in the upstream and downstream of the spillway. As given in Eq. (1), the total upstream energy is cleared with \(H_{0}\) and downstream total energy as presented in Eq. (2) is calculated with \(H_{1}\).
The sketch of stepped spillway
$$H_{0} = H_{\text{w}} + y_{0} + \frac{{V_{0}^{2} }}{2g} = H_{\text{w}} + y_{0} + \frac{{q^{2} }}{{2g\left( {H_{\text{w}} + y_{0} } \right)}}$$
$$H_{0} = y_{1} + \frac{{V_{1}^{2} }}{2g} = y_{1} + \frac{{q^{2} }}{{2gy_{1}^{2} }}$$
The total head loss is evaluated using Eq. (3).
$$\frac{\Delta H}{{H_{0} }} = \frac{{H_{0} - H_{1} }}{{H_{0} }} = 1 - \frac{{H_{1} }}{{H_{0} }}$$
The involved geometrical and hydraulically parameters on the energy dissipation are collected in Eq. (4).
$$\frac{\Delta H}{{H_{0} }} = f\left( {q,L_{\text{s}} ,h_{\text{s}} ,H_{\text{w}} ,g,N} \right)$$
Using the Buckingham \(\Pi\) as dimensional analysis technique, the most important parameters on the energy dissipation are derived and given in Eq. (5).
$$\frac{\Delta H}{{H_{0} }} = f\left( {\frac{{q^{2} }}{{gH_{\text{w}}^{3} }},\frac{{h_{\text{s}} }}{{L_{\text{s}} }},N,\frac{{y_{\text{c}} }}{{h_{\text{s}} }},Fr_{1} } \right)$$
With assuming the \({\text{DN}} = {{q^{2} } \mathord{\left/ {\vphantom {{q^{2} } {gH_{\text{w}}^{3} }}} \right. \kern-0pt} {gH_{\text{w}}^{3} }}\) and \(S = {{h_{\text{s}} } \mathord{\left/ {\vphantom {{h_{\text{s}} } {L_{\text{s}} }}} \right. \kern-0pt} {L_{\text{s}} }}\), Eq. (5) can be rewritten as Eq. (6).
$${\text{EDR}} = \frac{\Delta H}{{H_{0} }} = f\left( {{\text{DN}},S,N,y_{c}/h_{s},Fr_{1} } \right)$$
As mentioned in the past, developing the soft computing techniques is based on the dataset. Therefore to prepare the M5 algorithm and MLPNN, the dataset published by Salmasi and Özger (2014) was used and their range are presented in Table 1.
Range of dataset assigned to stages of prepared soft computing models (Salmasi and Özger 2014)
yc/h
AVEG
STDV
STDEV
M5 tree model
The M5 model at the first time which was proposed by the Quinlan (1992) is based on the classification tree method. The M5 model uses for the mapping the relation between the independent variables to the dependent variable and unlike the decision tree model in addition to qualities' data uses for the quantitative data. The M5 model is similar to the piecewise linear functions method which is a combination of the linear regression and tree regression method. The M5 model is widely used in most area of the science. A linear or nonlinear regression proposes an equation for all the data which researchers attempt to mathematical modeling, whereas the M5 tree model tries to divide data into several categories which named leaf. Modeling the relationship between input and output data which categorized in each leaf by the linear regression is the main process which is conducted in the M5 tree method. This approach can be used for continuous data. The structure of the M5 tree model is similar to the natural tree which include the root, stem, leafs and nodes. Decision tree models are drawn from up to down. The root is considered as first node at the top of the graph, and during the model development the tree branches and leafs are drowned. Each of the nodes has been considered as independent variables. Constructing the M5 tree model included two stages, one developing the decision tree by data categorizing (the main criteria for data categorizing are increasing the covariance or reducing the standard deviation). Equation (4) is the criteria for the standard deviation in the each of the leafs (Kumar and Sihag 2019; Sihag et al. 2018, 2019).
$${\text{SDR}} = {\text{SD}}\left( T \right) - \sum {\frac{{\left| {{\text{T}}_{\text{i}} } \right|}}{{\left| {\text{T}} \right|}}} {\text{SD}}\left( {T_{i} } \right)$$
where SDR is the standard deviation reduction, T the dataset inputs into the tree branches, and Ti the dataset in leafs. SD is the standard deviation. With the growth and development of M5 tree model, it is feared that the performance of the model leads to have so local behavior so usually in the second stage of the model development the pruning the tree is considered. To this purpose, the Quinlan algorithm is used. In this algorithm allowed to the tree to have enough growing then the branched which has not influence effect on the precision improvement is pruned. Figure 2 shows a schematic shape of the M5 tree model development. In Fig. 2a, the X1 and X2 are the input variables (independent parameters) and Y is the output data (dependent parameter), and Fig. 2b shows the tree model development for mapping the input and outputs data.
The sketch of the M5 tree model development (Etemad-Shahidi and Mahjoobi 2009)
Review on ANNs
The idea of ANN was given form human brain. Therefore, for the modeling of the knowledge behind the data recorded from the desired phenomenon to be transmitted by the neurons. The sketch network and a neuron are shown in Fig. 3. As shown in this figure, the collaboration of these neurons in parallel leads to the formation of a network, which may include one or more hidden layers. These types of networks are now introduced into the multilayered perceptron neural network (MLPNN). Investigating the structure of the neuron shows that the information is firstly multiplied in a coefficient and then are summarized, and finally, its result is introduced into a function that governs the behavior of the neuron. This function is named transfer or active function. So far, several types of transfer functions have been proposed for multilayer neural networks. A number of well-known transfer functions which are used for developing the MLP are present as below. The purpose of the calibration of a MLPNN is that both the coefficients multiplied in the input information and the coefficients used in the governing functions of the neurons (transfer function) are justified. This is performed via powerful methods such as Levenberg–Marquardt method (Sihag et al. 2019; Tiwari et al. 2019).
The structure of developed the MLPNN for predicting the EDR
Gaussian: \(F(x) = a\exp \left( { - \frac{{\left( {x - b} \right)^{2} }}{{c^{2} }}} \right)\)
Sigmoidal: \(F(x) = \frac{1}{1 + \exp ( - x)}\)
Tansing: \(F(x) = \frac{2}{{\left( {1 + \exp ( - 2x)} \right)}} - 1\)
Here, results of modeling and predicting the energy dissipation of flow over the stepped spillways using the M5 algorithm and MLPNN are presented. The first stage of modeling each phenomenon using soft computing techniques is data preparation. The purpose of data preparation is to divide them into two categories of training and testing. The training and testing dataset are utilized for development and validation of model, respectively. In this study, 80% of the data was used to train and the rest was assigned to test the models. The data shuffling method has been used to assign data to training and testing categories. In the phase of preparation and development models, the training data are used for calibration of the final model. For example, in developing phase the M5 algorithm for modeling and prediction of energy dissipation, firstly the space of inputs features based on the training data, are classified. Then, a linear function is fitted on the each class. The results of feature classification of training dataset of energy dissipation using M5 algorithm are given in Eq. (8). As presented in this equation, the first and most important parameter on which the first classification of training dataset is performed is the Drop number. The second important parameter is the Froude number. The examination of the above points shows that these two parameters (Drop and Froude numbers) are the most important parameters in estimating the amount of energy dissipation of flow passing over the stepped spillways. Of course, this also confirms the results obtained by previous studies. It is worth noting that the threshold criterion is considered for branching and classification of 0.05. As outlined in the materials and methods, the development of the M5 algorithm consists of two steps: the first stage involves the growth and development of the model, and the second stage involves pruning the additional branches produced in the first stage. The results of developed M5 algorithm in training and testing stages are shown in Fig. 4. As shown in this figure, the error indices of the M5 algorithm in training were RMSE = 2.48 and R2 = 0.99 and in testing stages were RMSE = 2.23 and R2 = 0.99.
The performance of applied models in training (a) and testing (b) stages
To examine the efficiency of the M5 algorithm, its performance was compared with MLPNN a common model of soft computing methods. The same dataset was used for training and testing of MLPNN. The MLPNN which was proposed by Haghiabi et al. (2018) was considered. They recommend that, in order to reduce the trial and error process in designing the structure of the MLPNN, first, a single-layer network, which contains a number of neurons equal to the number of input features, is considered. Then in next stage, the different type transfer functions can be tested to define the best of them. In this study, the structure of developed MLPNN model is shown in Fig. 3. As shown in this figure, the developed MLPNN has one hidden layer which includes five neurons. The best performance of transfer function is related to Tansing function. The performance of developed MLPNN in training and testing stages is shown in Fig. 4. The error indices of MLPNN in training were R2 = 0.97 and RMSE = 3.73 and in testing stages were R2 = 0.97 and RMSE = 3.98. Comparing the performance of M5 algorithm with MLPNN shows that the accuracy of M5 algorithm is a bit more than MLPNN. The performance of stepped spillways regarding energy dissipation has been predicted using group method of data handling (GMDH), genetic programming (GP), support vector machine (SVM) and multivariate adaptive regression splines (MARS) (Parsaie et al. 2016b, 2018a, c). According to the reports, the error indices of MARS technique in preparation stages were R2 = 0.99 and RMSE = 0.65. The error indices of GMDH in development stages (training and testing stages) were R2 = 0.95 and RMSE = 5.4. The performance of SVM and GP in training and testing was R2 = 0.98, RMSE = 2.61 and R2 = 0.96, RMSE = 4.94, respectively. Comparing the performance of developed M5 algorithm with previous applied models shows that the accuracy of M5 algorithm is a bit more than the GP and GMDH, and is a bit less than the MARS and SVM.
Stepped spillways are hydraulic structures that are commonly used in water engineering and watershed projects. These structures have been very much considered due to the economics, easy in construction and proper operation of energy dissipation and elimination of probability of cavitation. In watershed projects, this type of spillways can also be constructed from local materials such as loose rock dams. In this paper, new formula based on the M5 algorithm was proposed for estimating the performance of stepped spillways regarding energy dissipation. To compare the performance of M5 algorithm with other type of soft computing methods, the MLPNN was chosen. Results of M5 algorithm showed that there is good agreement between observed data and M5 algorithm outputs. Reviewing the structure of formula derived from the M5 algorithm declared that the Drop and Froude numbers are the main parameters used for feature classification in M5 algorithm. Results of MLPNN model showed that this model also has good performance in predicting the performance of stepped spillways regarding energy dissipation of flow. However, the accuracy of M5 algorithm is a bit more than the MLPNN. The best performance of tested transfer function during the development of the MLPNN model is related to Tansing function.
The authors declare that they have no conflict of interest.
This article does not contain any studies with human participants or animals performed by any of the authors.
Azamathulla HM, Haghiabi AH, Parsaie A (2016) Prediction of side weir discharge coefficient by support vector machine technique. Water Sci Technol Water Supply 16(4):1002–1016CrossRefGoogle Scholar
Christodoulou GC (1993) Energy dissipation on stepped spillways. J Hydraul Eng 119(5):644–650CrossRefGoogle Scholar
Erfanain-Azmoudeh M-H, Kamanbedast AA (2013) Determine the appropriate location of aerator system on Gotvandoliadam's spillway using flow 3D. Am Eur J Agric Environ Sci 13(3):378–383Google Scholar
Etemad-Shahidi A, Mahjoobi J (2009) Comparison between M5′ model tree and neural networks for prediction of significant wave height in Lake Superior. Ocean Eng 36(15):1175–1181. https://doi.org/10.1016/j.oceaneng.2009.08.008 CrossRefGoogle Scholar
Frizell KW, Renna FM, Matos J (2013) Cavitation potential of flow on stepped spillways. J Hydraul Eng 139(6):630–636CrossRefGoogle Scholar
Haghiabi AH, Parsaie A, Ememgholizadeh S (2018) Prediction of discharge coefficient of triangular labyrinth weirs using adaptive neuro fuzzy inference system. Alex Eng J 57(3):1773–1782. https://doi.org/10.1016/j.aej.2017.05.005 CrossRefGoogle Scholar
Heller V, Hager WH, Minor H-E (2005) Ski jump hydraulics. J Hydraul Eng 131(5):347–355CrossRefGoogle Scholar
Kim D, Park J (2005) Analysis of flow structure over ogee-spillway in consideration of scale and roughness effects by using CFD model. KSCE J Civ Eng 9(2):161–169CrossRefGoogle Scholar
Kumar M, Sihag P (2019) Assessment of infiltration rate of soil using empirical and machine learning-based models. Irrig Drain. https://doi.org/10.1002/ird.2332 CrossRefGoogle Scholar
Maghsoodi R, Roozgar MS, Sarkardeh H, Azamathulla HM (2012) 3D-simulation of flow over submerged weirs. Int J Model Simul 32(4):237Google Scholar
Movahedi A, Kavianpour M, Aminoroayaie Yamini O (2019) Experimental and numerical analysis of the scour profile downstream of flip bucket with change in bed material size. ISH J Hydraul Eng 25(2):188–202CrossRefGoogle Scholar
Najafzadeh M, Zeinolabedini M (2019) Prognostication of waste water treatment plant performance using efficient soft computing models: an environmental evaluation. Measurement 138:690–701CrossRefGoogle Scholar
Najafzadeh M, Tafarojnoruz A, Lim SY (2017) Prediction of local scour depth downstream of sluice gates using data-driven models. ISH J Hydraul Eng 23(2):195–202CrossRefGoogle Scholar
Parsaie A, Dehdar-Behbahani S, Haghiabi AH (2016a) Numerical modeling of cavitation on spillway's flip bucket. Front Struct Civ Eng 10(4):438–444CrossRefGoogle Scholar
Parsaie A, Haghiabi AH, Saneie M, Torabi H (2016b) Prediction of energy dissipation on the stepped spillway using the multivariate adaptive regression splines. ISH J Hydraul Eng 22(3):281–292CrossRefGoogle Scholar
Parsaie A, Haghiabi AH, Saneie M, Torabi H (2018a) Prediction of energy dissipation of flow over stepped spillways using data-driven models. Iran J Sci Technol Trans Civ Eng 42(1):39–53CrossRefGoogle Scholar
Parsaie A, Moradinejad A, Haghiabi AH (2018b) Numerical modeling of flow pattern in spillway approach channel. Jordan J Civ Eng 12(1):1–9Google Scholar
Parsaie A, Haghiabi AH, Saneie M, Torabi H (2018c) Applications of soft computing techniques for prediction of energy dissipation on stepped spillways. Neural Comput Appl 29(12):1393–1409CrossRefGoogle Scholar
Pfister M, Hager WH (2011) Self-entrainment of air on stepped spillways. Int J Multiphase Flow 37(2):99–107CrossRefGoogle Scholar
Quinlan JR (1992) Learning with continuous classes. In: Proceedings of the 5th Australian joint conference on artificial intelligence. World Scientific, pp 343–348Google Scholar
Roushangar K, Akhgar S, Salmasi F, Shiri J (2014) Modeling energy dissipation over stepped spillways using machine learning approaches. J Hydrol 508:254–265CrossRefGoogle Scholar
Salmasi F, Özger M (2014) Neuro-fuzzy approach for estimating energy dissipation in skimming flow over stepped spillways. Arab J Sci Eng 39(8):6099–6108CrossRefGoogle Scholar
Salmasi F, Samadi A (2018) Experimental and numerical simulation of flow over stepped spillways. Appl Water Sci 8(8):229CrossRefGoogle Scholar
Shahheydari H, Nodoshan EJ, Barati R, Moghadam MA (2014) Discharge coefficient and energy dissipation over stepped spillway under skimming flow regime. KSCE J Civ Eng 19(4):1174–1182CrossRefGoogle Scholar
Sihag P, Tiwari NK, Ranjan S (2018) Prediction of cumulative infiltration of sandy soil using random forest approach. J Appl Water Eng Res 7:1–25Google Scholar
Sihag P, Mohsenzadeh Karimi S, Angelaki A (2019) Random forest, M5P and regression analysis to estimate the field unsaturated hydraulic conductivity. Appl Water Sci 9(5):129. https://doi.org/10.1007/s13201-019-1007-8 CrossRefGoogle Scholar
Tiwari NK, Sihag P, Singh BK, Ranjan S, Singh KK (2019) Estimation of tunnel desilter sediment removal efficiency by ANFIS. Iran J Sci Technol Trans Civ Eng. https://doi.org/10.1007/s40996-019-00261-3 CrossRefGoogle Scholar
Wu F (2011) Support vector machine approach to for longitudinal dispersion coefficients in streams. Appl Soft Comput 11(2):2902–2905CrossRefGoogle Scholar
Xiao Y, Wang Z, Zeng J, Zheng J, Lin J, Zhang L (2015) Prototype and numerical studies of interference characteristics of two ski-jump jets from opening spillway gates. Eng Comput 32(2):289–307CrossRefGoogle Scholar
1.Hydro-Structure EngineeringShahid Chamran University of AhvazAhvazIran
2.Water Engineering DepartmentLorestan UniversityKhorramabadIran
Parsaie, A. & Haghiabi, A.H. Appl Water Sci (2019) 9: 144. https://doi.org/10.1007/s13201-019-1019-4
Publisher Name Springer International Publishing
King Abdulaziz City for Science and Technology | CommonCrawl |
\begin{document}
\title{\textbf{Two-bubble nodal solutions for slightly subcritical Fractional Laplacian} \thanks{The second author was supported by the National Natural Science Foundation of China (Grant Nos.11271299, 11001221) and the Fundamental Research Funds for the Central Universities (Grant No. 3102015ZY069).}} \author{Qianqiao Guo, Yunyun Hu} \date{} \maketitle
\noindent \begin{abs}
In this paper, we consider the existence of nodal solutions with two bubbles to the slightly subcritical problem with the fractional Laplacian \begin{equation*} \left\{\aligned
&(-\Delta)^su=|u|^{p-1-\varepsilon}u\ \ \mbox{in}\ \Omega,\\ &u=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial\Omega, \endaligned \right. \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N>2s$, $0<s<1$, $ p=\frac{N+2s}{N-2s}$ and $\varepsilon>0$ is a small parameter, which can be seen as a nonlocal analog of the results of Bartsch, Micheletti and Pistoia (2006) \cite{Bartsch1}. \end{abs}
\noindent \begin{key words} Fractional Laplacian, Nodal solutions, Slightly subcritical problem, Lyapunov-Schmidt reduction \end{key words} \indent
\section{\textbf{Introduction}\label{Section 1}} This paper is devoted to the problem involving Fractional Laplacian \begin{equation}\label{pro} \left\{\aligned
&(-\Delta)^su=|u|^{p-1-\varepsilon}u\ \ \ \mbox{in}\ \Omega,\\ &u=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial\Omega, \endaligned \right. \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N$, $N>2s$, $0<s<1$, $ p=\frac{N+2s}{N-2s}$ and $\varepsilon>0$ is a small parameter, $(-\Delta)^s$ stands for the fractional Laplacian operator.
The fractional Laplacian appears in physics, biological modeling, probability and mathematical finance, which is a nonlocal operator. Therefore it is difficult to handle and has attracted much attention in recent years. Importantly, Caffarelli, Silvestre \cite{Caffarelli2} developed an extension method to transform the nonlocal problem into a local one, which helps to study the fractional Laplacian by purely local arguments. By using their extension, many authors studied the existence of solutions to problem $(-\Delta)^su=f(u)$ with $f: \mathbb R^N\rightarrow \mathbb R$. For example, when $s=\frac{1}{2}$, Cabr\'{e} and Tan \cite{Tan17} and Tan \cite{Jing13} established the existence of positive solutions for nonlinear equations having subcritical growth.
Then it is interesting to study the blow-up phenomenon of solutions to (\ref{pro}) as $\epsilon \to 0^+$. For positive solutions, Chio, Kim and Lee \cite{Chio12} established the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions. Rois and Luis \cite{Rois15} generalized the work of Chio, Kim and Lee \cite{Chio12}, and took into account both subcritical and supercritical case. These papers are however not concerned with the nodal solutions involving the Fractional Laplacian.
If $s=1$, problem (\ref{pro}) was extensively studied about the blow-up phenomenon of positive and nodal solutions. It was proved in \cite{Rey3,Han4,Brezis5} that as $\varepsilon$ goes to zero, positive solution $u$ to problem (\ref{pro}) blows up and concentrates at a critical point of the Robin's function. Rey \cite{Rey6} considered the positive solutions with double blow-up and showed that the two concentration points $\sigma_1^*$ and $\sigma_2^*$ must be such that $(\sigma_1^*, \sigma_2^*)$ is a critical point of the function \begin{equation} \Phi(\sigma_1, \sigma_2)=H^\frac{1}{2}(\sigma_1, \sigma_1)H^\frac{1}{2}(\sigma_2, \sigma_2)-G(\sigma_1, \sigma_2),\ \ \ \ (\sigma_1, \sigma_2)\in\Omega\times\Omega \end{equation} and satisfies $\Phi(\sigma_1^*, \sigma_2^*)\geq0$. Here $G$ is the Green's function of the Dirichlet Laplacian and $H$ is its regular part. See also \cite{Bahri7} for the existence of positive solutions with multiple bubbles. In a convex domain, it proved in \cite{Grossi8} that no positive solutions have multiple bubbles for problem (\ref{pro}). On the other hand, nodal solutions with multiple-bubbles also exist for the problem (\ref{pro}) with $s=1$ in a general smooth bounded domain $\Omega$. As the parameter $\varepsilon$ goes to zero, Bartsch, Micheletti and Pistoia \cite{Bartsch1} proved the existence of nodal solutions which blow up positively at a point $\sigma_1^*\in\Omega$ and blow up negatively at a point $\sigma_2^*\in\Omega$, with $\sigma_1^*\neq\sigma_2^*$. In \cite{Ben9} Ben Ayed, Mehdi and Pacella classified the nodal solutions according to the concentration speeds of the positive and negative part. Bartsch, D'Aprile and Pistoia in \cite{Bartsch10} studied the existence of nodal solutions with four bubbles in a smooth bounded domain $\Omega$. When $\Omega$ is a ball, they also proved the nodal with three bubbles in \cite{Bartsch11}.
In this paper, we are interested in the existence of nodal solutions which blow-up and concentrate at two different points of the domain $\Omega$.
In order to state our result, we introduce some well known notations. Let $G$ be the Green's function of $(-\Delta)^s$ in $\Omega$ with Dirichlet boundary conditions, that is, $G$ satisfies \begin{equation}\label{Green} \left\{\aligned &(-\Delta)^sG(\cdot,y)=\delta_y\ \ \ \mbox{in}\ \Omega,\\ &G(\cdot,y)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial\Omega, \endaligned \right. \end{equation} where $\delta_y$ denotes the Dirac mass at the point $y$. The regular part of $G$ is given by
\begin{equation}\label{Green-regular}
H(x,y)=\frac{c_{N,s}}{|x-y|^{N-2s}}-G(x,y)\ \ \ \mbox{where}\ \ c_{N,s}=\frac{2^{1-2s}\Gamma(\frac{N-2s}{2})}{2\pi^{\frac{N}{2}}\Gamma(s)}. \end{equation} The diagonal $H(x,x)$ is usually called the Robin's function of the domain $\Omega$.
Now we can state the main result. Let us consider the function $\varphi:\Omega\times\Omega\rightarrow\mathbb R$ defined by \begin{equation}\label{phi-function} \varphi(\sigma_1,\sigma_2)=H^\frac{1}{2}(\sigma_1, \sigma_1)H^\frac{1}{2}(\sigma_2, \sigma_2)+G(\sigma_1, \sigma_2), \end{equation} which will play a crucial role in our analysis.
\begin{theorem}\label{theorem-1 bubbles} Suppose that $0<s<1$ and $N>2s$, then there exists a small number $\varepsilon_0>0$ such that for $0<\varepsilon<\varepsilon_0$, problem (\ref{pro}) has a pair of solutions $u_\varepsilon$ and $-u_\varepsilon$. As $\varepsilon$ goes to zero, $u_\varepsilon$ blows up positively at a point $\sigma_1^*\in\Omega$ and negatively at a point $\sigma_2^*\in\Omega$, where $\varphi(\sigma_1^*,\sigma_2^*)=\min\limits_{\Omega\times\Omega}\varphi$. \end{theorem}
The proof of Theorem \ref{theorem-1 bubbles} is motivated the result of Bartsch, Micheletti and Pistoia \cite{Bartsch1} on the local problem, based on a Lyapunov-Schmidt reduction scheme. The main point is to find critical points of the finite dimensional reduced functional corresponding to critical points of the energy function of problem (\ref{pro}). The reduced functional is given in terms of the Green's and Robin's functions. In subcritical case, the role of Green's and Robin's functions in the concentration phenomena associated to the critical exponent has already been considered in several works, see \cite{Rey3,Han4,Brezis5,Chio12} and \cite{Rey6,Bahri7}. The proofs here borrow ideas of the above mentioned works.
This paper is organized as follows. In section 2, we present some definitions and the basic properties of the fractional Laplacian in bounded domains and in the whole $\mathbb R^N$. Section 3 is devoted to developing the analytical tools toward the main results. Moreover, nodal solutions are constructed by the Lyapunov-Schmidt reduction method. Finally, in the Appendix, some necessary estimates for the construction of the nodal solutions are exhibited.
\section{\textbf{Preliminary }\label{Section 2}}
In this section we review some basic definitions and properties of the fractional Laplacian. We refer to \cite{Colorado16,Tan17,Capella18,Kim19,Tan20,Stinga21,Barrios22} for the details.
Let $\Omega$ be a smooth bounded domain in $\mathbb R^N$. We define $(-\Delta)^s$ through the spectral decomposition using the powers of the eigenvalues of the Laplacian operator $-\Delta$ in $\Omega$. Let $\{\lambda_i,\phi_i\}_{i=1}^\infty$ denote the eigenvalues and eigenfunctions of $-\Delta$ in $\Omega$ with zero Dirichlet boundary condition,
\begin{equation}\label{Dirichlet} \left\{\aligned &-\Delta\phi_i=\lambda_i\phi_i\ \ \ \ \mbox{in}\ \Omega,\\ &\phi_i=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial\Omega. \endaligned \right. \end{equation}
The fractional Laplacian is well defined in the fractional Sobolev space $H_{0}^s(\Omega)$, \[H_{0}^s(\Omega):=\left\{u=\sum_{i=1}^\infty a_i\phi_i \in L^2(\Omega):\sum_{i=1}^\infty a_{i}^2\lambda_{i}^s<\infty\right\}, \] which is a Hilbert space whose inner product is defined by \[\left\langle \sum_{i=1}^\infty a_i\phi_i, \sum_{i=1}^\infty b_i\phi_i\right\rangle_{H_{0}^s(\Omega)}=\sum_{i=1}^\infty a_i b_i\lambda_{i}^s. \] Moreover, we define fractional Laplacian $(-\Delta)^s:H_{0}^s(\Omega)\rightarrow H_{0}^s(\Omega)$ as: \[(-\Delta)^s\left(\sum_{i=1}^\infty a_i\phi_i\right)=\sum_{i=1}^\infty a_i \lambda_{i}^s\phi_i. \] Note that by the above definitions, we have the following expression for the inner product: \begin{equation}\label{inner product} \langle u,v\rangle_{H_{0}^s(\Omega)}=\int_\Omega(-\Delta)^{\frac{s}{2}}u(-\Delta)^{\frac{s}{2}}v=\int_\Omega(-\Delta)^suv,\ \ \ \ u,v \in H_{0}^s(\Omega). \end{equation}
We will recall an equivalent definition based on an extension problem introduced by Caffarelli and Silvestre \cite{Caffarelli2}. For the sake of simplicity, we denote $\Omega\times(0,\infty)$ by $\mathcal{C}$ and its lateral boundary $\partial\Omega\times(0,\infty)$ by $\partial_\mathcal{C}$, where $\Omega$ is either a smooth bounded domain or $\mathbb R^N$. If $\Omega$ is a smooth bounded domain, the function space $H_{0,L}^s(\mathcal{C})$ is defined as the completion of
\[C_{\mathcal{C},L}^\infty(\mathcal{C}):=\left\{U\in C^\infty(\bar{\mathcal{C}}):U=0\ \mbox{on}\ \partial_L \mathcal{C}\right\} \] with respect to the norm \begin{equation}\label{norm}
\parallel U\parallel_\mathcal{C}=\left(\frac{1}{k_s}\int_\mathcal{C} t^{1-2s}|\nabla U|^2\right)^\frac{1}{2}, \end{equation} where $t>0$ represents the last variable in $\mathbb R^{N+1}$ and $k_s$ is a normalization constant (see \cite{Barrios22,Chio12}). This is a Hilbert space endowed with the following inner product \[(U,V)_\mathcal{C}=\frac{1}{k_s}\int_\mathcal{C}t^{1-2s}\nabla U\cdot\nabla V \quad\mbox{for\ all}\ \ U,V\in H_{0,L}^s(\mathcal{C}). \] Moreover, in the entire space, we define $\mathcal{D}^s(\mathbb R_{+}^{N+1})$ as the completion of $C_{\mathcal{C}}^\infty(\overline{\mathbb R_{+}^{N+1}})$ with respect to the norm $\parallel U\parallel_{\mathbb R_{+}^{N+1}}$ (defined as in (\ref{norm}) by putting $\mathcal{C}=\mathbb R_{+}^{N+1}$). We know that if $\Omega$ is a smooth bounded domain, then \begin{equation}\label{trace}
H_{0}^s(\Omega)=\left\{u=\mbox{tr}|_{\Omega\times\{0\}}U:U\in H_{0,L}^s(\mathcal{C})\right\}. \end{equation} It also holds that \[\parallel U(\cdot,0)\parallel_{H^s(\mathbb R^N)}\leq C\parallel U\parallel_{\mathbb R_{+}^{N+1}} \] for some $C>0$ independent of $U\in \mathcal{D}^s(\mathbb R_{+}^{N+1})$.
Now we consider the harmonic extension problem. For some given functions $u\in H_{0}^s(\Omega)$, $U\in H_{0,L}^s(\mathcal{C})$ solves the equation \begin{equation}\label{extension} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla U)=0\ \ \ \ \ \mbox{in}\ \mathcal{C},\\ &U=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial_L \mathcal{C},\\ &U(x,0)=u(x)\ \ \ \ \ \ \ \ \ \mbox{on}\ \Omega\times\{0\} \endaligned \right. \end{equation} as a unique solution. The relevance of the extension function $U$ is that it is related to the fractional Laplacian of the original function $u$ through the formula \begin{equation}\label{formula} -\frac{1}{k_s}\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial U}{\partial t}(x,t)=(-\Delta)^su(x), \end{equation} where $k_s>0$ depends on $N$ and $s$ (see \cite{Caffarelli2} and \cite{Capella18} for the entire and bounded domain case, respectively). By the above extension, the problem (\ref{pro}) is transformed into its equivalence problem \begin{equation}\label{equi} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla U)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \mathcal{C},\\ &U=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial_L \mathcal{C},\\
&-\frac{1}{k_s}\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial U}{\partial t}(x,t)=|U|^{p-1-\varepsilon}U\ \ \ \ \mbox{on}\ \Omega\times\{0\}. \endaligned \right. \end{equation}
In a completely analogous extension procedure, the Green's function $G$ of the fractional Laplacian $(-\Delta)^s$ defined in (\ref{Green}) can be regarded as the trace of the solution $G_\mathcal{C}(z,y)$ ($z=(x,t)\in \mathcal{C}$, $y\in\Omega$) for the following extended Dirichlet-Neumann problem \begin{equation} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla G_\mathcal{C}(\cdot,y))=0\ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \mathcal{C},\\ &G_\mathcal{C}(\cdot,y)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial_L \mathcal{C},\\ &-\frac{1}{k_s}\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial G_\mathcal{C}(\cdot,y)}{\partial t}=\delta_y\ \ \ \ \mbox{on}\ \Omega\times\{0\}. \endaligned \right. \end{equation} Moreover, if a function $U$ satisfies \begin{equation} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla U)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \mathcal{C},\\ &U=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial_L \mathcal{C},\\ &-\frac{1}{k_s}\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial U_\mathcal{C}(\cdot,y)}{\partial t}=g(x)\ \ \ \ \mbox{on}\ \Omega\times\{0\}, \endaligned \right. \end{equation} then we have the following expression \[U(z)=\int_\Omega G_\mathcal{C}(z,y)g(y)dy=\int_\Omega G_\mathcal{C}(z,y)(-\Delta)^su(y)dy\ \ \ \ \ \ \mbox{for all}\ \ z\in \mathcal{C}, \]
where $u=\mbox{tr}|_{\Omega\times\{0\}}U$.
Green's function $G_\mathcal{C}$ can be partitioned to the singular part and the regular part on $\mathcal{C}$. For the singular part of the Green's function $G_\mathcal{C}$, it can be given by \begin{equation}
G_{\mathbb R_{+}^{N+1}}(z,y):=\frac{c_{N,s}}{|y-z|^{N-2s}}, \end{equation} where $c_{N,s}$ is defined in (\ref{Green-regular}), and $G_{\mathbb R_{+}^{N+1}}$ solves the problem $$\aligned\quad \left\{\aligned &\mbox{div}(t^{1-2s}\nabla G_{\mathbb R_{+}^{N+1}}(z,y))=0\ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \mathbb R_{+}^{N+1},\\ &-\frac{1}{k_s}\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial G_{\mathbb R_{+}^{N+1}}(z,y)}{\partial t}=\delta_y\ \ \ \ \mbox{on}\ \Omega\times\{0\} \endaligned \right. \endaligned$$ for $y\in \mathbb R^N$. The regular part can be seen as the unique solution to $$\aligned\quad \left\{\aligned &\mbox{div}(t^{1-2s}\nabla H_\mathcal{C}(z,y))=0\ \ \ \ \ \ \ \ \ \ \mbox{in}\ \mathcal{C},\\
&H_\mathcal{C}(z,y)=\frac{c_{N,s}}{|y-z|^{N-2s}}\ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial_L \mathcal{C},\\ &-\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial H_\mathcal{C}(z,y)}{\partial t}=0\ \ \ \ \ \ \mbox{on}\ \Omega\times\{0\}. \endaligned \right. \endaligned$$ Then we have \begin{equation}\label{regular} G_\mathcal{C}(z,y)=G_{\mathbb R_{+}^{N+1}}(z,y)-H_\mathcal{C}(z,y). \end{equation}
Next we present the Sharp Sobolev and trace inequalities (see \cite{Chio12,Cotsiolis23}). Given any $\lambda>0$ and $\xi\in\mathbb R^N$, here \begin{equation}\label{solution}
w_{\lambda,\xi}(x)=a_{N,s}\left(\frac{\lambda}{\lambda^2+|x-\xi|^2}\right)^\frac{N-2s}{2} \end{equation} is an explicit family of solutions to \begin{equation}\label{critical} (-\Delta)^su=u^p\ \ \ \ \ \mbox{in}\ \mathbb R^N,\\ \end{equation} where $a_{N,s}>0$ (see \cite{Chen24,Li25,Li26} for details). Then the sharp Sobolev inequality from \cite{Cotsiolis23} is the following: \begin{equation}\label{Sobolev}
\left(\int_{\mathbb R^N}|u|^{p+1}dx\right)^\frac{1}{p+1}\leq S_{N,s}\left(\int_{\mathbb R^N}|(-\Delta)^\frac{s}{2}u|^2dx\right)^\frac{1}{2}. \end{equation} The equality is attained if and only if $u(x)=cw_{\lambda,\xi}(x)$ for any $c>0$, $\lambda>0$ and $\xi\in\mathbb R^N$, where \[S_{N,s}=2^{-s}\pi^{-s/2}\left[\frac{\Gamma(\frac{N-2s}{2})}{\Gamma(\frac{N+2s}{2})}\right]^\frac{1}{2}\left[\frac{\Gamma(N)}{\Gamma(N/2)}\right]^\frac{s}{N}, \] (refer to \cite{Carlen27,Frank28,Lieb29}). Now let $W_{\lambda,\xi}\in \mathcal{D}^s(\mathbb R_{+}^{N+1})$ be the s-harmonic extension of $w_{\lambda,\xi}$ satisfying \begin{equation}\label{s-harmonic} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla W_{\lambda,\xi}(x,t))=0\ \ \ \ \ \ \ \ \ \mbox{in}\ \ \mathbb R_{+}^{N+1},\\ &W_{\lambda,\xi}(x,0)=w_{\lambda,\xi}(x)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{for}\ x\in\mathbb R^N.\\ \endaligned \right. \end{equation} It implies that the Sobolev trace inequality \begin{equation}\label{inequality}
\left(\int_{\mathbb R^N}|U(x,0)|^{p+1}dx\right)^\frac{1}{p+1}\leq \frac{S_{N,s}}{\sqrt{k_s}}\left(\int_0^\infty\int_{\mathbb R^N}t^{1-2s}|\nabla U(x,t)|^2dxdt\right)^\frac{1}{2} \end{equation} gets the equality if and only if $U(x,t)=cW_{\lambda,\xi}(x,t)$ for any $c>0$, $\lambda>0$ and $\xi\in\mathbb R^N$, where $k_s>0$ is given in (\ref{formula}) (see \cite{Xiao30}).\\
\section{\textbf{The finite dimensional reduction}\label{Section 3}}
In this section we are devoted to proving Theorem \ref{theorem-1 bubbles} by applying the Lyapunov-Schmidt reduction method. Similar methods are used in \cite{Bartsch1,Rey6,Bahri7,Musso31}.
Let $\Omega$ be a smooth bounded domain in $\mathbb R^N$. Set \begin{equation}\label{set} \Omega_\varepsilon=\varepsilon^{-\frac{1}{N-2s}}\Omega=\{\varepsilon^{-\frac{1}{N-2s}}x:\ \ x\in\Omega\}, \end{equation} then the changing variables \begin{equation}\label{variables} v(x)=\varepsilon^\frac{1}{2-\frac{\varepsilon(N-2s)}{2s}}u(\varepsilon^\frac{1}{N-2s}x)\ \ \ \ \mbox{for}\ \ x\in\Omega_\varepsilon \end{equation} transforms equation (\ref{pro}) into \begin{equation}\label{transform} \left\{\aligned
&(-\Delta)^sv=|v|^{p-1-\varepsilon}v\ \ \ \mbox{in}\ \Omega_\varepsilon,\\ &v=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial\Omega_\varepsilon. \endaligned \right. \end{equation} It follows that $u(x)$ is a solution to (\ref{pro}) if and only if $v(x)=\varepsilon^\frac{1}{2-\frac{\varepsilon(N-2s)}{2s}}u(\varepsilon^\frac{1}{N-2s}x)$ is a solution of equation (\ref{transform}). In the proof of Theorem 1.1, solutions to (\ref{transform}) are close related to the following dilated equation \begin{equation}\label{dilated equation} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla V)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \mathcal{C}_\varepsilon,\\ &V=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial_L \mathcal{C}_\varepsilon,\\
&-\frac{1}{k_s}\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial V}{\partial t}=|V|^{p-1-\varepsilon}V\ \ \ \ \ \ \ \ \ \mbox{on}\ \Omega_\varepsilon\times\{0\}. \endaligned \right. \end{equation} It is easy to know that if $V$ is the solution of (\ref{dilated equation}), then $U(x)=\varepsilon^{-\frac{1}{2-\frac{\varepsilon(N-2s)}{2s}}}V(\varepsilon^{-\frac{1}{N-2s}}x)$ solve the problem (\ref{equi}). To look for the solutions that satisfy the equation (\ref{equi}), it suffices to apply the Lyapunov-Schmidt reduction method to the extended problem (\ref{dilated equation}). Moreover, it is easy to know that the harmonic extension $V$ of function $v$ satisfies the problem \begin{equation}\label{harmonic extension} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla V)=0\ \ \ \ \mbox{in}\ \mathcal{C}_\varepsilon,\\ &V=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial_L \mathcal{C}_\varepsilon,\\ &V(x,0)=v(x)\ \ \ \ \ \ \ \ \mbox{on}\ \Omega_\varepsilon\times\{0\}, \endaligned \right. \end{equation} where $\mathcal{C}_\varepsilon=\varepsilon^{-\frac{1}{N-2s}}\mathcal{C}=\{\varepsilon^{-\frac{1}{N-2s}}(x,t):\ \ (x,t)\in\mathcal{C}\}$.
Let us recall the the functions $w_{\lambda,\xi}$ and $W_{\lambda,\xi}$ defined in (\ref{solution}) and (\ref{s-harmonic}). By the result of \cite{del32}, it is known that the kernel of the operator $(-\Delta)^s-pw_{\lambda,\xi}^{p-1}$ is spanned by the functions \begin{equation}\label{span} \frac{\partial w_{\lambda,\xi}}{\partial\xi_1},\ \cdot\cdot\cdot,\frac{\partial w_{\lambda,\xi}}{\partial\xi_N}\ \ \mbox{and}\ \ \frac{\partial w_{\lambda,\xi}}{\partial\lambda}, \end{equation} namely they satisfy the equation \begin{equation}\label{namely} (-\Delta)^s\phi=pw_{\lambda,\xi}^{p-1}\phi \ \ \ \ \mbox{in}\ \mathbb R^N, \end{equation} where $\xi=(\xi_1,\cdot\cdot\cdot\cdot\cdot,\xi_N)$ in $\mathbb R^N$. We also have that all bounded solutions of the extended problem of (\ref{namely}) \begin{equation}\label{bounded solutions} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla \Phi)=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \mathbb R_{+}^{N+1},\\ &-\frac{1}{k_s}\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial \Phi}{\partial t}=pw_{\lambda,\xi}^{p-1}\Phi\ \ \ \ \ \ \mbox{on}\ \mathbb R^N\times\{0\} \endaligned \right. \end{equation} consist of the linear combinations of the functions \begin{equation}\label{linear combinations} \frac{\partial W_{\lambda,\xi}}{\partial\xi_1},\ \ \cdot\cdot\cdot,\frac{\partial W_{\lambda,\xi}}{\partial\xi_N}\ \ \mbox{and}\ \ \frac{\partial W_{\lambda,\xi}}{\partial\lambda}. \end{equation} In order to construct the multi-bubble nodal solutions of (\ref{equi}), for $\eta\in(0,1)$, we define the admissible set \begin{align}\label{admissible set} &\mathcal{O}_\eta=\{(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}):=((\lambda_1,...,\lambda_k), (\sigma_1,...,\sigma_k))\in\mathbb R_{+}^{k}\times\Omega^k, \ \ \sigma_i=(\sigma_{i}^1,...,\sigma_{i}^N),\nonumber\\
&\mbox{dist}(\sigma_i,\partial\Omega)>\eta, \ \ \eta<\lambda_i<\frac{1}{\eta}, \ \ |\sigma_i-\sigma_j|>\eta, \ \ i\neq j, \ \ i,j=1,...,k \}. \end{align}
It is useful to rewrite problem (\ref{pro}) in a different setting. To this end, let us introduce the following operator.
\begin{definition}\label{definintion-operator L-1} Let the map \begin{equation}\label{map} i_{\varepsilon}^*:L^\frac{2N}{N+2s}(\Omega_\varepsilon)\rightarrow H_{0,L}^s(\mathcal{C}_\varepsilon) \end{equation} be the adjoint operator of the Sobolev trace embedding \begin{equation} i_{\varepsilon}:H_{0,L}^s(\mathcal{C}_\varepsilon)\rightarrow L^\frac{2N}{N-2s}(\Omega_\varepsilon)\nonumber \end{equation} defined by the \begin{equation}
i_{\varepsilon}(V)=\mbox{tr}|_{\Omega_\varepsilon\times\{0\}}(V)\nonumber\ \ \ \ \ \mbox{for}\ \ V\in H_{0,L}^s(\mathcal{C}_\varepsilon), \end{equation} which comes from the inequality (\ref{inequality}), that is, for some $v\in L^\frac{2N}{N+2s}(\Omega_\varepsilon)$ and $Z\in H_{0,L}^s(\mathcal{C}_\varepsilon)$, \begin{equation} i_{\varepsilon}^*(v)=Z\nonumber \end{equation} if and only if \begin{equation} \left\{\aligned &\mbox{div}(t^{1-2s}\nabla Z)=0\ \ \ \ \ \ \ \ \ \ \ \ \mbox{in}\ \mathcal{C}_\varepsilon,\\ &Z=0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{on}\ \partial_L \mathcal{C}_\varepsilon,\\ &-\frac{1}{k_s}\lim\limits_{t\rightarrow0^{+}}t^{1-2s}\frac{\partial Z}{\partial t}=v\ \ \ \ \mbox{on}\ \Omega_\varepsilon\times\{0\}.\nonumber \endaligned \right. \end{equation} \end{definition} By the definition of the operator $i_{\varepsilon}^*$, solving problem (\ref{dilated equation}) is equivalent to find a solution of the fixed point problem \begin{equation} V=k_s i_{\varepsilon}^*\left(f_\varepsilon\left(i_{\varepsilon}(V)\right)\right),\ \ \ V\in H_{0,L}^s(\mathcal{C}_\varepsilon), \end{equation}
where $f_\varepsilon(s)=|s|^{p-1-\varepsilon}s$. Notice that from (\ref{trace}) we have $i_{\varepsilon}:H_{0,L}^s(\mathcal{C}_\varepsilon)\rightarrow H_{0}^s(\Omega_\varepsilon)\subset L^\frac{2N}{N-2s}(\Omega_\varepsilon)$ and so $(-\Delta)^s(i_{\varepsilon}(U))$ makes sense.
We look for solutions of (\ref{transform}) of the form \begin{equation} v=\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i,\delta_i}+\phi_\varepsilon,\nonumber \end{equation} for $k\geq1$ a fixed integer and $a_1,...,a_k\in\{\pm1\}$ fixed, where $\phi_\varepsilon$ is a lower order term and $\mathcal{P}_\varepsilon: H^s(\mathbb R^N)\rightarrow H_{0}^s(\Omega_\varepsilon)$ is the projection defined by the equation \begin{equation} \left\{\aligned &(-\Delta)^s\mathcal{P}_\varepsilon w_i=(-\Delta)^sw_i \ \ \ \mbox{in}\ \Omega_\varepsilon,\\ &\mathcal{P}_\varepsilon w_i=0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox {on} \ \partial\Omega_\varepsilon,\nonumber \endaligned \right. \end{equation} where $w_i=w_{\lambda_i,\delta_i}$, $\delta_i=\varepsilon^{-\frac{1}{N-2s}}\sigma_i\in\Omega_\varepsilon$.
Let us introduce some notations. For $\xi=(\xi^1,...,\xi^N)\in\mathbb R^N$ and $j=1,2,...,N$, we define the functions \begin{equation}\label{functions} \Psi_{\lambda,\xi}^0=\frac{\partial W_{\lambda,\xi}}{\partial\lambda},\ \ \Psi_{\lambda,\xi}^j=\frac{\partial W_{\lambda,\xi}}{\partial\xi^j},\ \ \psi_{\lambda,\xi}^0=\frac{\partial w_{\lambda,\xi}}{\partial\lambda},\ \ \psi_{\lambda,\xi}^j=\frac{\partial w_{\lambda,\xi}}{\partial\xi^j} \end{equation} and \begin{equation}\label{notations} \mathcal{P}_\varepsilon W_{\lambda,\xi}=i_{\varepsilon}^*(w_{\lambda,\xi}^p),\ \ \mathcal{P}_\varepsilon \Psi_{\lambda,\xi}^j=i_{\varepsilon}^*(pw_{\lambda,\xi}^{p-1}\psi_{\lambda,\xi}^j),\ \ \ \ j=0,1,...,N. \end{equation} Moreover, we let the functions $\mathcal{P}_\varepsilon w_{\lambda,\xi}$ and $\mathcal{P}_\varepsilon \psi_{\lambda,\xi}^j$ be \begin{equation}\label{define} \mathcal{P}_\varepsilon w_{\lambda,\xi}=i_{\varepsilon}(\mathcal{P}_\varepsilon W_{\lambda,\xi}),\ \ \mathcal{P}_\varepsilon \psi_{\lambda,\xi}^j=i_{\varepsilon}(\mathcal{P}_\varepsilon\Psi_{\lambda,\xi}^j),\ \ \ \ \ j=0,1,...,N \end{equation} which satisfy the equations $(-\Delta)^su=w_{\lambda,\xi}^p$ and $(-\Delta)^su=pw_{\lambda,\xi}^{p-1}\psi_{\lambda,\xi}^j$ in $\Omega_\varepsilon$, respectively. For the sake of simplicity, we denote \begin{equation}\label{simplicity} W_i=W_{\lambda_i,\delta_i},\ \mathcal{P}_\varepsilon W_i=\mathcal{P}_\varepsilon W_{\lambda_i,\delta_i},\ \Psi_i^j=\Psi_{\lambda_i,\delta_i}^j,\ \mathcal{P}_\varepsilon\Psi_i^j=\mathcal{P}_\varepsilon\Psi_{\lambda_i,\delta_i}^j,\ i=1,2,...,k,\ \ j=0,1,...,N. \end{equation} for $\delta_i=\varepsilon^{-\frac{1}{N-2s}}\sigma_i\in\Omega_\varepsilon$ and $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$. Similary, we denote \begin{equation}\label{Similary} \mathcal{P}_\varepsilon w_i=\mathcal{P}_\varepsilon w_{\lambda_i,\delta_i},\ \ \mathcal{P}_\varepsilon\psi_i^j=\mathcal{P}_\varepsilon\psi_{\lambda_i,\delta_i}^j,\ \ \ \ \ \ i=1,2,...,k,\ \ j=0,1,...,N. \end{equation} Set the space \begin{equation}\label{space} \mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon=\{u\in H_{0,L}^1(\mathcal{C}_\varepsilon):\ \ (u,\mathcal{P}_\varepsilon\psi_i^j)_{\mathcal{C}_\varepsilon}=0 ,\ \ i=1,2,...,k,\ \ j=0,1,...,N\}, \end{equation} where $\varepsilon>0$ and $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$. We also need the following orthogonal projections \begin{equation}\label{projection} \Pi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon:\ \ H_{0,L}^s(\mathcal{C}_\varepsilon)\rightarrow\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon. \end{equation}
Now if we let the linear operator $L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon:\ \mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\rightarrow\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ be defined by \begin{equation}\label{operator} L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon(\Phi)=\Phi-\Pi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon i_{\varepsilon}^*\left[f'_0(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i,\delta_i})\cdot i_{\varepsilon}(\Phi)\right], \end{equation} then we can give an a-prior estimate for $\Phi\in\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$.
\begin{lemma}\label{lemma-estimate of error-1} For any $\eta>0$ there exists sufficiently small $\varepsilon>0$ and a constant $C=C(N,\eta)$ such that, for every $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$, the operator $L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ satisfies \begin{equation}\label{lemma 3.2}
\| L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon(\Phi)\|_{\mathcal{C}_\varepsilon}\geq C\|\Phi\|_{\mathcal{C}_\varepsilon}\ \ \ \ \forall\ \Phi\in\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon. \end{equation} \end{lemma} \begin{proof} We omit it since it is similarly to Lemma 5.1 in \cite{Chio12}. \end{proof}
\begin{proposition} \label{proposition-invertibility} The inverse $(L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon)^{-1}$ of $L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon:\ \mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\rightarrow\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ exists for any $\varepsilon>0$ small and $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$. Besides, if $\varepsilon$ is small enough, its operator norm is uniformly bounded in $\varepsilon$ and $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$. \end{proposition} \begin{proof} The proof is similarly to Proposition 5.2 in \cite{Chio12} and thus is omitted here. \end{proof}
\begin{proposition}\label{proposition-reducement-1} For any sufficiently small $\eta>0$, there exist $\varepsilon_0>0$ and a constant $C>0$ such that for any $\varepsilon\in(0,\varepsilon_0)$ and any $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$, there exists a unique solution $\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\in\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ satisfying \begin{equation}\label{proposition 3.4.1} \Pi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\left\{\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i+\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon-i_\varepsilon^*\left[f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon)\right)\right]\right\}=0, \end{equation} and \begin{equation}\label{proposition 3.4.2}
\|\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\|_{\mathcal{C}_\varepsilon}\leq \left\{\aligned &C\varepsilon^{\frac{N+2s}{2}\alpha_0}\ \ \ \ \ \ \ \ \ \ \mbox{if}\ \ N>6s,\\
&C({\varepsilon+\varepsilon|\ln\varepsilon|})\ \ \ \ \mbox{if}\ \ N=6s,\\ &C\varepsilon\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \ 2s<N<6s, \endaligned \right. \end{equation} where $\alpha_0=\frac{1}{N-2s}$. Furthermore, the map $\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon:\ \ \mathcal{O}_\eta\rightarrow\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ is $C^1(\mathcal{O}_\eta)$. \end{proposition} \begin{proof} First of all we point out that $\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ is a solution of equation (\ref{proposition 3.4.1}) if and only if $\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ is a fixed point of operator $T_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon:\ \mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\rightarrow\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ defined by \begin{align*} &T_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon(\Phi)= (L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon)^{-1}N_\varepsilon(\Phi)\ \ \ \ \mbox{for}\ \ \Phi\in\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon, \end{align*} where \begin{align*} &N_\varepsilon(\Phi)= \Pi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon i_\varepsilon^*\left[f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi)\right)-\sum_{i=1}^ka_if_0(w_i)-f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)i_\varepsilon(\Phi)\right]. \end{align*} The claim will follow by showing that $T_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ is a contraction mapping on $\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon:=\{\Phi\in\mathcal{K}_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon:\ \Phi\ \mbox{satisfies}\ (\ref{proposition 3.4.2})\}$. By Lemma \ref{lemma-estimate of error-1}, Lemma \ref{(M17)-Lemma A} and (\ref{lemma A.7-1}) in Lemma \ref{lemma A.7}, we get \begin{align}\label{estimation-1}
&\|T_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon(\Phi)\|_{\mathcal{C}_\varepsilon}\leq C\left\|f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi)\right)-\sum_{i=1}^ka_if_0(w_i)-f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)i_\varepsilon(\Phi)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \leq C\Bigg\|f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi)\right)-f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)i_\varepsilon(\Phi)\Bigg\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ \quad\quad\quad +C\left\|\left[f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right]i_\varepsilon(\Phi)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ \quad\quad\quad +C\left\| f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ \quad\quad\quad +C\left\| f_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-\sum_{i=1}^ka_if_0 (w_i)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}. \end{align} It is easy to see that \begin{align}\label{estimation-2}
&\left\|f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi)\right)-f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)i_\varepsilon(\Phi)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\leq C\| i_\varepsilon(\Phi)\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}^{min\{2,p\}}\nonumber\\
&\leq C\|\Phi\|_{\mathcal{C}_\varepsilon}^{min\{2,p\}} \end{align} and by (\ref{lemma A.7-2}) of Lemma {\ref{lemma A.7}} that \begin{align}\label{estimation-3}
&\left\|\left[f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right]i_\varepsilon(\Phi)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\leq\left\| f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right\|_{L^\frac{N}{2s}(\Omega_\varepsilon)}\| i_\varepsilon(\Phi)\|_{L^\frac{2N}{N-2s}(\Omega_\varepsilon)}\nonumber\\
&\leq C\varepsilon|\ln\varepsilon|\|\Phi\|_{\mathcal{C}_\varepsilon}. \end{align}
By using Lemma \ref{(M17)-Lemma A}, (\ref{estimation-1}), (\ref{estimation-2}) and (\ref{estimation-3}), we deduce that if $\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ satisfies (\ref{proposition 3.4.2}), that is , $\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\leq C_1(\varepsilon+\gamma(\varepsilon))$, then there exists $C_1>0$ such that $\| T_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon(\Phi)\|_{\mathcal{C}_\varepsilon}\leq C_1(\varepsilon+\gamma(\varepsilon))$. Arguing as in the previous step, we can prove that if $\Phi_1$ and $\Phi_2$ satisfy (\ref{proposition 3.4.2}) then \begin{align*}
&\|T_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon(\Phi_1)-T_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon(\Phi_2)\|_{\mathcal{C}_\varepsilon}\nonumber\\
&=\|(L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon)^{-1}(N_\varepsilon(\Phi_1))-(L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon)^{-1}(N_\varepsilon(\Phi_2))\|_{\mathcal{C}_\varepsilon}\nonumber\\
&=\Bigg\|(L_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon)^{-1}\Pi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon i_\varepsilon^*\Bigg[f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi_1)\right) -f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi_2)\right)\nonumber\\
&\ \ \ -f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)(i_\varepsilon(\Phi_1)-i_\varepsilon(\Phi_2))\Bigg]\Bigg\|_{\mathcal{C}_\varepsilon}\nonumber\\
&\leq C_2\Bigg\|f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi_1)\right) -f_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi_2)\right)\nonumber\\
&\ \quad-f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi_2)\right)(i_\varepsilon(\Phi_1)-i_\varepsilon(\Phi_2))\Bigg\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\ \quad+C_2\left\| \left[f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i+i_\varepsilon(\Phi_2)\right)-f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right](i_\varepsilon(\Phi_1)-i_\varepsilon(\Phi_2))\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\ \quad+C_2\left\| \left[f'_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right](i_\varepsilon(\Phi_1)-i_\varepsilon(\Phi_2))\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\nonumber\\
&\leq L\|\Phi_1-\Phi_2\|_{\mathcal{C}_\varepsilon}. \end{align*} for some $L\in(0,1)$. The remaining parts are obtained by standard arguments, see \cite{Musso31}. \end{proof}
It is easy to know that for any fixed $\varepsilon>0$, $V\in H_{0,L}^s(\Omega_\varepsilon)$ is a weak solution to (\ref{dilated equation}) if and only if it is a critical point of the energy functional $E_\varepsilon: H_{0,L}^s(\mathcal{C}_\varepsilon)\rightarrow\mathbb R$ defined by \begin{equation}\label{energy functional}
E_\varepsilon(V)=\frac{1}{2k_s}\int_{\mathcal{C}_\varepsilon} t^{1-2s}|\nabla V|^2dxdt-\int_{\Omega_\varepsilon\times\{0\}}F_\varepsilon(i_\varepsilon(V))dx, \end{equation} where $F_\varepsilon(t)=\int_0^tf_\varepsilon(t)dt$. Notice that $E_\varepsilon(V)$ is a $C^1$-functional and \begin{equation}\label{C1-functional} E'_\varepsilon(V)\Phi=\frac{1}{k_s}\int_{\mathcal{C}_\varepsilon} t^{1-2s}\nabla V\cdot\nabla \Phi dxdt-\int_{\Omega_\varepsilon\times\{0\}}f_\varepsilon(i_\varepsilon(V))i_\varepsilon(\Phi)dx\ \ \ \mbox{for any}\ \ \Phi\in H_{0,L}^s(\mathcal{C}_\varepsilon). \end{equation} Now we introduce the function \begin{equation}\label{func} I_\varepsilon({\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}})=E_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i+\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\right) \end{equation} for $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})=((\lambda_1,...,\lambda_k), (\sigma_1,...,\sigma_k))\in\mathcal{O}_\eta$.
Let $\alpha_0=\frac{1}{N-2s}$ here and in the sequel. Arguing as Proposition 5.4 in \cite{Chio12} and Lemma 2.6 in \cite{Rois15}, we can obtain the following result. \begin{proposition}\label{proposition-expansion of L-1} (1) Suppose $\varepsilon>0$ is sufficiently small. If $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})$ is a critical point of the function $I_\varepsilon({\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}})$, then the function $V=\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon W_i+\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon$ is a solution to (\ref{dilated equation}). Hence the changing variables $U(z)=\varepsilon^{-\frac{1}{2-\frac{\varepsilon(N-2s)}{2s}}}V_\varepsilon(\varepsilon^{-\frac{1}{N-2s}}z)$ is the solution of (\ref{equi}) for $z\in\mathcal{C}$.
(2) For $\varepsilon\rightarrow0$, there holds \begin{equation}\label{equa-1} I_\varepsilon({\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}})=\frac{ksc_0}{N}-\frac{\varepsilon kc_0}{(p+1)^2}+\frac{1}{2}\varepsilon\Upsilon_k(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})+\frac{\varepsilon k}{p+1}\int_{\mathbb R^N}w^{p+1}\log wdx+o(\varepsilon) \end{equation} in $C^1$-uniformly with respect to $({\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}})\in\mathcal{O}_\eta$. Here \begin{align}\label{equa-2} &\Upsilon_k(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})=c_1^2\left(\sum_{i=1}^k\lambda_i^{N-2s}H(\sigma_i,\sigma_i)-\sum\limits_{i,h=1,i\neq h}^ka_ia_hG(\sigma_i,\sigma_h)(\lambda_i\lambda_h)^\frac{N-2s}{2}\right)\nonumber\\ &\ \ \ \ \ \ \ \ \ \quad\quad-\frac{c_0(N-2s)}{p+1}\log(\lambda_1\cdot\cdot\cdot\cdot\lambda_k), \end{align} \begin{equation}\label{equa-3} c_0=\int_{\mathbb R^N}w^{p+1}dx \end{equation} and \begin{equation}\label{equa-4} c_1=\int_{\mathbb R^N}w^pdx, \end{equation} where $w:=w_{1,0}$. \end{proposition}
\begin{proof}
We first prove (1). Setting $\overline{V}=\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i$ for the sake of simplicity. Applying $I'_\varepsilon({\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}})=0$, (\ref{proposition 3.4.1}) and (\ref{proposition 3.4.2}), we get \begin{align*} &\frac{\partial I_\varepsilon }{\partial\varrho}=E'_\varepsilon(\overline{V}+\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon)\cdot\left(\frac{\partial\overline{V}}{\partial\varrho}+\frac{\partial\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon}{\partial\varrho}\right)\nonumber\\ &\ \ \quad=\left(\overline{V}+\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon-i_{\varepsilon}^*f_\varepsilon(\overline{V}+i_{\varepsilon}(\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}})), \frac{\partial\overline{V}}{\partial\varrho}+\frac{\partial\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon}{\partial\varrho}\right)_\mathcal{C_\varepsilon}\nonumber\\ &\ \ \quad=\sum_{h=1}^k\sum_{l=0}^Nc_{hl}\left( \mathcal{P}_\varepsilon\Psi_h^l, \frac{\partial\overline{V}}{\partial\varrho}+\frac{\partial\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon}{\partial\varrho}\right)_\mathcal{C_\varepsilon}\nonumber\\ &\ \ \quad=\sum_{h=1}^k\sum_{l=0}^Nc_{hl}\left[\left(\mathcal{P}_\varepsilon\Psi_h^l, \sum_{i=1}^ka_i\mathcal{P}_\varepsilon \frac{\partial W_i}{\partial\varrho}\right)_\mathcal{C_\varepsilon}-\left(\mathcal{P}_\varepsilon \frac{\partial \Psi_h^l}{\partial\varrho},\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\right)_\mathcal{C_\varepsilon}\right]\nonumber\\ &\ \ \quad=0, \end{align*} where $\varrho$ is one of $\lambda_i$ and $\sigma_i^j$ with $ i=1,2,...,k$ and $j=1,...,N$ , $c_{hl}\in\mathbb R$. We also can conclude that $c_{hl}=0$ for all $h$ and $l$, which implies that the function $V$ is a solution of the equation (\ref{dilated equation}), and hence $U(x)$ is a solution to (\ref{equi}) for $\varepsilon>0$ sufficiently small.
Now we give the proof of (2). Using (\ref{proposition 3.4.2}), we can obtain that \begin{align*} &I_\varepsilon({\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}})=E_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i+\Phi_{\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$}}^\varepsilon\right)=E_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)+o(\varepsilon)\nonumber\\
&\ \ \ \ \ \ \ \ \quad=\frac{1}{2k_s}\int_{\mathcal{C}_\varepsilon} t^{1-2s}\left|\nabla \left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)\right|^2dxdt-\frac{1}{p+1-\varepsilon}\int_{\Omega_\varepsilon\times\{0\}}\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right|^{p+1-\varepsilon}dx\nonumber\\ &\ \ \ \ \ \ \ \ \ \ \ \quad+o(\varepsilon). \end{align*} We decompose \begin{equation}\label{decomp} E_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)=E_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)+\left[E_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)-E_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)\right], \end{equation} so it suffice to estimate the above two terms. It is easy to see that \begin{align}\label{terms} &E_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)
=\frac{1}{2k_s}\int_{\mathcal{C}_\varepsilon} t^{1-2s}\left|\nabla\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)\right|^2dxdt-\frac{1}{p+1}\int_{\Omega_\varepsilon\times\{0\}}\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right|^{p+1}dx. \end{align}
Setting $B_i=B_N(\sigma_i,\eta/2)$, where $\eta$ is defined in (\ref{admissible set}), applying Lemma \ref{lemma A.1} and Lemma \ref{lemma A.2}, we can deduce that \begin{align*} &\int_{\Omega_\varepsilon}w_i^p\mathcal{P}_\varepsilon w_idx=\int_{\Omega_\varepsilon}w_i^{p+1}dx+\int_{\Omega_\varepsilon}w_i^p(\mathcal{P}_\varepsilon w_i-w_i)dx\nonumber\\ &\ \ \ \ \ \ \ \quad\quad\quad\quad=c_0-\varepsilon c_1\lambda_i^\frac{N-2s}{2}\int_{\Omega_\varepsilon}w_i^pH(\varepsilon^{\alpha_0}x,\sigma_i)dx+o(\varepsilon)\nonumber\\ &\ \ \ \ \ \ \ \quad\quad\quad\quad=c_0-\varepsilon c_1^2\lambda_i^{N-2s}H(\sigma_i,\sigma_i)+o(\varepsilon),\nonumber\\ &\int_{\Omega_\varepsilon}w_h^p\mathcal{P}_\varepsilon w_idx=\int_\frac{B_i}{\varepsilon^{\alpha_0}}w_h^p\mathcal{P}_\varepsilon w_idx+o(\varepsilon)\nonumber\\ &\ \ \ \ \ \ \ \quad\quad\quad\quad=\int_\frac{B_i}{\varepsilon^{\alpha_0}}\varepsilon c_1\lambda_i^\frac{N-2s}{2}w_h^pG(\varepsilon^{\alpha_0}x,\sigma_i)dx+o(\varepsilon)\nonumber\\ &\ \ \ \ \ \ \ \quad\quad\quad\quad=\varepsilon c_1^2(\lambda_i\lambda_h)^\frac{N-2s}{2}G(\sigma_i,\sigma_h)+o(\varepsilon),\nonumber\\ \end{align*} for $i,h=1,2,...,k$ and $i\neq h$, where $G$ and $H$ are the functions defined in (\ref{Green}) and (\ref{Green-regular}), $c_0$ and $c_1$ are defined in (\ref{equa-3}) and (\ref{equa-4}), respectively. Integrating by parts and then the estimates obtained above yield that \begin{align}\label{estim-1}
&\frac{1}{2k_s}\int_{\mathcal{C}_\varepsilon} t^{1-2s}\left|\nabla \left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)\right|^2dxdt\nonumber\\ &=\frac{1}{2}\sum_{i=1}^k\int_{\Omega_\varepsilon}w_i^p\mathcal{P}_\varepsilon w_idx+\frac{1}{2} \sum\limits_{i,h=1,i\neq h}^k a_ia_h\int_{\Omega_\varepsilon}w_h^p\mathcal{P}_\varepsilon w_idx\nonumber\\ &=\frac{1}{2}\sum_{i=1}^k[c_0-\varepsilon c_1^2\lambda_i^{N-2s}H(\sigma_i,\sigma_i)+o(\varepsilon)]+\frac{1}{2}\sum\limits_{i,h=1,i\neq h}^k\varepsilon a_ia_h c_1^2(\lambda_i\lambda_h)^\frac{N-2s}{2}G(\sigma_i,\sigma_h)+o(\varepsilon)\nonumber\\ &=\frac{kc_0}{2}-\frac{c_1^2\varepsilon}{2}\left[\sum_{i=1}^k\lambda_i^{N-2s}H(\sigma_i,\sigma_i)-\sum\limits_{i,h=1,i\neq h}^ka_ia_h (\lambda_i\lambda_h)^\frac{N-2s}{2}G(\sigma_i,\sigma_h)\right]+o(\varepsilon). \end{align} On the other hand, we see that \begin{align}\label{estim-2}
&\int_{\Omega_\varepsilon}\left|a_i\mathcal{P}_\varepsilon w_i\right|^{p+1}dx=\int_{\Omega_\varepsilon}\left|\mathcal{P}_\varepsilon w_i\right|^{p+1}dx =c_0-\varepsilon (p+1)c_1^2\lambda_i^{N-2s}H(\sigma_i,\sigma_i)+o(\varepsilon), \end{align} \begin{align}\label{estim-3}
&\int_{\Omega_\varepsilon}\left(\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right|^{p+1}-\sum_{i=1}^k\left|a_i\mathcal{P}_\varepsilon w_i\right|^{p+1}\right)dx\nonumber\\ &=\varepsilon c_1^2(p+1)\sum\limits_{i,h=1,i\neq h}^ka_ia_h (\lambda_i\lambda_h)^\frac{N-2s}{2}G(\sigma_i,\sigma_h)+o(\varepsilon). \end{align} From the estimates obtained in the previous paragraph, we can conclude that \begin{align}\label{estim-4} &E_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)=\frac{ksc_0}{N}+\frac{1}{2}\varepsilon c_1^2\sum_{i=1}^k\lambda_i^{N-2s}H(\sigma_i,\sigma_i)\nonumber\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\frac{1}{2}\varepsilon c_1^2\sum\limits_{i,h=1,i\neq h}^ka_ia_hG(\sigma_i,\sigma_h)(\lambda_i\lambda_h)^\frac{N-2s}{2}+o(\varepsilon). \end{align}
As we have seen, it has \begin{align}\label{estim-5}
&\int_{\Omega_\varepsilon}\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right|^{p+1}dx=kc_0+o(1), \end{align} \begin{align}\label{estim-6}
&\int_{\Omega_\varepsilon}\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right|^{p+1}\log\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right|dx\nonumber\\ &=-\frac{c_0(N-2s)}{2}\log(\lambda_1\cdot\cdot\cdot\cdot\lambda_k)+k\int_{\mathbb R^N}w^{p+1}\log wdx+o(1). \end{align} The second equality (\ref{estim-6}) can be computed as Lemma 2.6 in \cite{Rois15}, Lemma 6.2 \cite{Musso34} and \cite{Del33}. Moreover, by using Taylor's expansion, (\ref{estim-5}) and (\ref{estim-6}), we can conclude that \begin{align}\label{estim-7} &E_\varepsilon\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)-E_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right)\nonumber\\
&=\frac{1}{p+1}\int_{\Omega_\varepsilon}\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right|^{p+1}dx-\frac{1}{p+1-\varepsilon}\int_{\Omega_\varepsilon}\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right|^{p+1-\varepsilon}dx\nonumber\\
&=-{\frac{\varepsilon}{(p+1)^2}}\int_{\Omega_\varepsilon}\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right|^{p+1}dx+\frac{\varepsilon}{p+1}\int_{\Omega_\varepsilon}\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right|^{p+1}\log\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon W_i\right|dx+o(\varepsilon)\nonumber\\ &=-\frac{\varepsilon kc_0}{(p+1)^2}+\frac{\varepsilon k}{p+1}\int_{\mathbb R^N}w^{p+1}\log wdx -\frac{c_0\varepsilon(N-2s)}{2(p+1)}\log(\lambda_1\cdot\cdot\cdot\cdot\lambda_k)+o(\varepsilon). \end{align} Then by (\ref{terms}) and (\ref{estim-7}), the proof is complete. \end{proof}
Now we consider the case $k=2$ and suppose $a_1=1$ and $a_2=-1$. We introduce the set \begin{equation}\label{set} \Lambda:=\{(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})=(\lambda_1,\ \lambda_2,\ \sigma_1,\ \sigma_2): \lambda_1>0,\ \lambda_2>0,\ \sigma_1\in\Omega,\ \sigma_2\in\Omega\ \mbox{and}\ \sigma_1\neq\sigma_2\} \end{equation} and the function $\Upsilon_2:\Lambda\rightarrow \mathbb R$ defined by \begin{align}\label{define} &\Upsilon_2(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})=c_1^2[H(\sigma_1,\sigma_1)\lambda_1^{N-2s}+H(\sigma_2,\sigma_2)\lambda_2^{N-2s}+2G(\sigma_1,\sigma_2)\lambda_1^\frac{N-2s}{2}\lambda_2^\frac{N-2s}{2}]\nonumber\\ &\ \ \ \ \ \ \ \ \ \quad\quad-\frac{c_0(N-2s)}{p+1}\log(\lambda_1\lambda_2). \end{align}
\begin{lemma}\label{lemma-expansion of L-1} If $(\lambda^*,\sigma^*)$ is a critical point of $\Upsilon_2$, then $\sigma^*$ is a critical point of $\varphi$. If $(\lambda^*,\sigma^*)$ is a minimal point of $\Upsilon_2$, then $\sigma^*$ is a minimal point of $\varphi$. \end{lemma} \begin{proof} The proof is similarly to Lemma 3.2 in \cite{Bartsch1} and thus is omitted here. \end{proof}
\noindent\textbf{Proof of Theorem 1.1.} Similarly as Theorem 1.1 of \cite{Bartsch1}, the above lemmas and propositions give the result.
$\Box$
\appendix \section{\textbf{Appendix}\label{Appendix A}}
In this section, we collect some technical lemmas from \cite{Chio12} and give some basic estimations needed.
By using the definition of $w_{\lambda,\xi}$, $\psi_{\lambda,\xi}^j$, $\mathcal{P}_\varepsilon w_{\lambda,\xi}$ and $\mathcal{P}_\varepsilon \psi_{\lambda,\delta}^j$ $(\mbox{for}\ \ i=1,...,k\ \mbox{and}\ \ j=1,...,N)$, we get \begin{align}\label{Appendix A.1} &\psi_{\lambda,\sigma}^0(x)=\frac{\partial w_{\lambda,\sigma}}{\partial \lambda}(x)\nonumber\\
&\ \ \ \ \ \ \ \ \ \ =a_{N,s}\frac{(N-2s)}{2}\lambda^\frac{N-2s-2}{2}\frac{|x-\sigma|^2-\lambda^2}{(\lambda^2+|x-\sigma|^2)^\frac{N-2s+2}{2}}\ \ \ \ x\in\mathbb R^N, \end{align} \begin{align}\label{Appendix A.2} &\psi_{\lambda,\sigma}^j(x)=\frac{\partial w_{\lambda,\sigma}}{\partial \sigma_j}(x)\nonumber\\
&\ \ \ \ \ \ \ \ \ \ =-a_{N,s}(N-2s)\lambda^\frac{N-2s}{2}\frac{x_j-\sigma_j}{(\lambda^2+|x-\sigma|^2)^\frac{N-2s+2}{2}}\ \ \ \ \ x\in\mathbb R^N. \end{align} In particular it holds \begin{align}\label{Appendix A.3} &\mathcal{P}_\varepsilon w_{\varepsilon^{\alpha_0}\lambda,\sigma}(x)=\varepsilon^{-\frac{(N-2s)\alpha_0}{2}}\mathcal{P}_\varepsilon w_{\lambda,\sigma\varepsilon^{-\alpha_0}}\left(\frac{x}{\varepsilon^{\alpha_0}}\right)\ \ \ \ \ x\in \Omega, \end{align} \begin{align}\label{Appendix A.4} &\mathcal{P}_\varepsilon \psi_{\varepsilon^{\alpha_0}\lambda,\sigma}^j(x)=\varepsilon^{-\frac{(N-2s+2)\alpha_0}{2}}\mathcal{P}_\varepsilon \psi_{\lambda,\sigma\varepsilon^{-\alpha_0}}^j\left(\frac{x}{\varepsilon^{\alpha_0}}\right)\ \ \ \ \ x\in \Omega. \end{align}
The first four lemmas are from Lemma C.1-C.4 in \cite{Chio12}. \begin{lemma}\label{lemma A.1}
Let $\lambda>0$ and $\sigma=(\sigma^1,...,\sigma^N)\in\Omega$. For any $x\in\Omega_\varepsilon$, there hold \begin{eqnarray*} &\mathcal{P}_\varepsilon w_{\lambda,\sigma\varepsilon^{-\alpha_0} }(x)=w_{\lambda,\sigma\varepsilon^{-\alpha_0}}(x)-c_1\lambda^\frac{N-2s}{2}H(\varepsilon^{\alpha_0}x,\sigma)\varepsilon^{(N-2s)\alpha_0}+o(\varepsilon^{(N-2s)\alpha_0}), \end{eqnarray*} \begin{eqnarray*} &\ \ \ \ \ \ \ \mathcal{P}_\varepsilon \psi_{\lambda,\sigma\varepsilon^{-\alpha_0}}^j(x)=\psi_{\lambda,\sigma\varepsilon^{-\alpha_0}}^j(x)-c_1\lambda^\frac{N-2s}{2}\frac{\partial H}{\partial\sigma^j}(\varepsilon^{\alpha_0}x,\sigma)\varepsilon^{(N-2s+1)\alpha_0}+o(\varepsilon^{(N-2s+1)\alpha_0}), \end{eqnarray*} \begin{eqnarray*} &\ \ \ \ \ \ \ \ \ \ \mathcal{P}_\varepsilon \psi_{\lambda,\sigma\varepsilon^{-\alpha_0}}^0(x)=\psi_{\lambda,\sigma\varepsilon^{-\alpha_0}}^0(x)-\frac{c_1(N-2s)}{2}\lambda^\frac{N-2s-2}{2} H(\varepsilon^{\alpha_0}x,\sigma)\varepsilon^{(N-2s)\alpha_0}+o(\varepsilon^{(N-2s)\alpha_0}), \end{eqnarray*} where $c_1$ is defined in (\ref{equa-4}). As $\varepsilon\rightarrow0$, $o\to 0$ uniformly in $x\in\Omega_\varepsilon$ and $\sigma\in\Omega$ provided $\mbox{dist}(\sigma,\partial\Omega)>\bar{C}$ for some constant $\bar{C}>0$. \end{lemma}
\begin{lemma}\label{lemma A.2} Let $\lambda>0$ and $\sigma=(\sigma^1,...,\sigma^N)\in\Omega$, there hold \begin{eqnarray*}\label{Appendix A.8} &\mathcal{P}_\varepsilon w_{\lambda,\sigma\varepsilon^{-\alpha_0}}(x)=c_1\lambda^\frac{N-2s}{2}G(\varepsilon^{\alpha_0}x,\sigma)\varepsilon^{(N-2s)\alpha_0}+o(\varepsilon^{(N-2s)\alpha_0}), \end{eqnarray*} \begin{eqnarray*}\label{Appendix A.9} &\ \ \ \ \ \ \ \ \mathcal{P}_\varepsilon \psi_{\lambda,\sigma\varepsilon^{-\alpha_0}}^j(x)=c_1\lambda^\frac{N-2s}{2}\frac{\partial G}{\partial\sigma^j}(\varepsilon^{\alpha_0}x,\sigma)\varepsilon^{(N-2s+1)\alpha_0}+o(\varepsilon^{(N-2s+1)\alpha_0}), \end{eqnarray*} \begin{eqnarray*}\label{Appendix A.10} &\ \ \ \ \ \ \ \ \ \ \ \mathcal{P}_\varepsilon \psi_{\lambda,\sigma\varepsilon^{-\alpha_0}}^0(x)=\frac{c_1(N-2s)}{2}\lambda^\frac{N-2s-2}{2} G(\varepsilon^{\alpha_0}x,\sigma)\varepsilon^{(N-2s)\alpha_0}+o(\varepsilon^{(N-2s)\alpha_0}), \end{eqnarray*}
where $c_1>0$ is the constant defined in (\ref{equa-4}). As $\varepsilon\rightarrow0$, $o\to 0$ uniformly in $x\in\Omega_\varepsilon$ and $\sigma\in\Omega$ provided $|\sigma-\varepsilon^{\alpha_0}x|>C$ and $\mbox{dist}(\partial\Omega,\varepsilon^{\alpha_0}x)>C$ for fixed $C>0$. \end{lemma}
\begin{lemma}\label{lemma A.3} For any $\varepsilon>0$, $i=1,...,k$ and $j=1,...,N$, there exists $C>0$ such that \begin{eqnarray}\label{Appendix A.11}
&\|\mathcal{P}_\varepsilon w_i\|_{L^\frac{2N}{N-2s}(\Omega_\varepsilon)}\leq\|w_i\|_{L^\frac{2N}{N-2s}(\Omega_\varepsilon)}\leq C, \end{eqnarray} \begin{eqnarray}\label{Appendix A.12}
&\|\mathcal{P}_\varepsilon \psi_i^j\|_{L^\frac{2N}{N-2s}(\Omega_\varepsilon)}\leq C. \end{eqnarray} Moreover, we have \begin{eqnarray}\label{Appendix A.13}
&\|\mathcal{P}_\varepsilon \psi_i^j\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\leq C, \end{eqnarray} \begin{eqnarray}\label{Appendix A.14}
&\|\mathcal{P}_\varepsilon w_i\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\leq \left\{\aligned &C\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \ N>6s,\\
&C\varepsilon^{-\frac{(6s-N)\alpha_0}{2}}|\log\varepsilon|\ \ \ \ \ \mbox{if}\ \ N\leq 6s \endaligned \right. \end{eqnarray} and \begin{eqnarray}\label{Appendix A.15}
&\|\mathcal{P}_\varepsilon \psi_i^0\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\leq \left\{\aligned &C\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \ N>6s,\\
&C\varepsilon^{-\frac{(6s-N)\alpha_0}{2}}|\log\varepsilon|\ \ \ \ \ \mbox{if}\ \ N\leq 6s. \endaligned \right. \end{eqnarray} \end{lemma}
\begin{lemma}\label{lemma A.4} For $i=1,...,k$ and $j=1,...,N$, we have \begin{eqnarray}\label{Appendix A.16}
&\|\mathcal{P}_\varepsilon \psi_i^j-\psi_i^j\|_{L^\frac{2N}{N-2s}(\Omega_\varepsilon)}\leq C\varepsilon^{\alpha_0\frac{N-2s+2}{2}} \end{eqnarray} and \begin{eqnarray}\label{Appendix A.17}
&\|\mathcal{P}_\varepsilon \psi_i^0-\psi_i^0\|_{L^\frac{2N}{N-2s}(\Omega_\varepsilon)}\leq C\varepsilon^{\alpha_0\frac{N-2s}{2}}. \end{eqnarray} \end{lemma}
Similarly to Lemma A.2 in \cite{Bartsch1}, Lemma A.3 in \cite{Musso31} and Lemma C.5 in \cite{Chio12}, we obtain the following Lemma.
\begin{lemma}\label{(M17)-Lemma A} For any $\eta>0$ and for any $\varepsilon_0>0$ there exists $C>0$ such that for any $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$ and $\varepsilon\in(0,\varepsilon_0)$ we have \begin{eqnarray}\label{Appendix A.18}
&\left\|f_0\left(\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum\limits_{i=1}^ka_if_0(w_i)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\leq \left\{\aligned &C\varepsilon^{\frac{N+2s}{2}\alpha_0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \ N>6s,\\
&C(\varepsilon+\varepsilon^{(N-2s)\alpha_0}|\ln\varepsilon|)\ \ \ \ \ \mbox{if}\ \ N=6s,\\ &C\varepsilon^{(N-2s)\alpha_0}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mbox{if}\ \ N<6s, \endaligned \right. \end{eqnarray} \begin{eqnarray}\label{Appendix A.19}
&\left\|f'_0\left(\sum\limits_{i=1}^k a_i\mathcal{P}_\varepsilon w_i\right)- \sum\limits_{i=1}^ka_if'_0(w_i)\right\|_{L^\frac{N}{2s}(\Omega_\varepsilon)}\leq C\varepsilon^{2s\alpha_0}. \end{eqnarray} \end{lemma}
\begin{proof} We just prove the first inequality by using Lemma A.1 in \cite{Musso34}. The proof of (\ref{Appendix A.19}) is similar. Since $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$ it holds $|\sigma_i-\sigma_j|>\eta$ for any $i\neq j (i, j=1,...,k)$. We have \begin{align*}
&\left\|f_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum_{i=1}^ka_if_0(w_i)\right\|^{\frac{2N}{N+2s}}_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\\
&=\int_{\Omega_\varepsilon}\left|\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i(y)\right|^{p-1}\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i(y)\right)-\sum_{i=1}^ka_iw_i^p(y)\right|^\frac{2N}{N+2s}dy\ \ \ ( \mbox{set}\ \ x=\varepsilon^{\alpha_0}y)\\
&=\int_\Omega\left|\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right|^{p-1}\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right)-\sum_{i=1}^ka_iw_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^p(x)\right|^\frac{2N}{N+2s}dx\\
&=\sum_{j=1}^k\int_{B(\sigma_j,\eta/2)}\left|\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right|^{p-1}\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right)-\sum_{i=1}^ka_iw_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^p(x)\right|^\frac{2N}{N+2s}dx\\
&+\int_{\Omega\setminus{\bigcup\limits_{j=1}^kB(\sigma_j,\eta/2)}}\left|\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right|^{p-1}\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right)-\sum_{i=1}^ka_iw_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^p(x)\right|^\frac{2N}{N+2s}dx.\\ \end{align*} Firstly, \begin{align*}
&\int_{\Omega\setminus{\bigcup\limits_{j=1}^kB(\sigma_j,\eta/2)}}\left|\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right|^{p-1}\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right)-\sum_{i=1}^ka_iw_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^p(x)\right|^\frac{2N}{N+2s}dx\\ &\leq C\sum_{i=1}^k\int_{\Omega\setminus{\bigcup\limits_{j=1}^kB(\sigma_j,\eta/2)}}w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^{p\frac{2N}{N+2s}}dx\\ &\leq C\sum_{i=1}^k(\lambda_i\varepsilon^{\alpha_0})^N\\ &\leq C\varepsilon^{N{\alpha_0}}. \end{align*} Secondly, \begin{align*}
&\int_{B(\sigma_j,\eta/2)}\left|\left|\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right|^{p-1}\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right)-\sum_{i=1}^ka_iw_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^p(x)\right|^\frac{2N}{N+2s}dx\\
&\leq C\int_{B(\sigma_j,\eta/2)}\left|\left(\sum_{i=1}^k\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}(x)\right)^p-w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}^p(x)\right|^\frac{2N}{N+2s}dx\\
&\ \ \ +C\sum\limits_{i=1, i\neq j}^k\int_{B(\sigma_j,\eta/2)}|w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^p(x)|^\frac{2N}{N+2s}dx\\
&\leq C\int_{B(\sigma_j,\eta/2)}\left|\mathcal{P}_\varepsilon w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}^p(x) -w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}^p(x)\right|^\frac{2N}{N+2s}dx\\
&\ \ \ +C\sum\limits_{i=1, i\neq j}^k\int_{B(\sigma_j,\eta/2)}|w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^p(x)|^\frac{2N}{N+2s}dx+C\varepsilon^{N{\alpha_0}}. \end{align*} It is easy to see that \begin{align*}
&\sum\limits_{i=1, i\neq j}^k\int_{B(\sigma_j,\eta/2)}|w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}^p(x)|^\frac{2N}{N+2s}dx\\
&\leq \sum\limits_{i=1, i\neq j}^k\int_{B(\sigma_j,\eta/2)}\left|\frac{\lambda_i\varepsilon^{\alpha_0}}{(\lambda_i\varepsilon^{\alpha_0})^2+|x-\sigma_i|^2}\right|^Ndx\\ &\leq C\varepsilon^{N{\alpha_0}}. \end{align*} For $N>6s$, by using Lemma \ref{lemma A.1} and the mean value theorem, we get that \begin{align*}
&\int_{B(\sigma_j,\eta/2)}\left|\mathcal{P}_\varepsilon w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}^p(x) -w_{\lambda_j\varepsilon^{{\alpha_0},\sigma_j}}^p(x)\right|^\frac{2N}{N+2s}dx\\
&=p\int_{B(\sigma_j,\eta/2)}|(w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}+\theta(x)(\mathcal{P}_\varepsilon w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}(x) -w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}(x)))^{p-1}\\
&\ \ \ \ \times(\mathcal{P}_\varepsilon w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}(x)-w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}(x))|^\frac{2N}{N+2s}dx\\ &\leq C\varepsilon^{N{\alpha_0}}. \end{align*} Therefore if $N>6s$, we have \begin{align*}
&\left\|f_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum_{i=1}^ka_if_0(w_i)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\leq C\varepsilon^{{\frac{N+2s}{2}\alpha_0}}. \end{align*}
Moreover, if $N=6s$, we have \begin{align*}
&\int_{B(\sigma_j,\eta/2)}\left|\mathcal{P}_\varepsilon w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}^p(x) -w_{\lambda_j\varepsilon^{{\alpha_0},\sigma_j}}^p(x)\right|^\frac{2N}{N+2s}dx\\
&=p\int_{B(\sigma_j,\eta/2)}|(w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}+\theta(x)(\mathcal{P}_\varepsilon w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}(x) -w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}(x)))^{p-1}\\
&\ \ \ \ \times(\mathcal{P}_\varepsilon w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}(x) -w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}(x))|^\frac{2N}{N+2s}dx\\
&\leq C\int_{B(\sigma_j,\eta/2)}\left|\frac{\lambda_j\varepsilon^{\alpha_0}}{({\lambda_j\varepsilon^{\alpha_0}})^2+|x-\sigma_j|^2}\right|^\frac{4sN}{N+2s} (\varepsilon^{\alpha_0})^\frac{N(N-2s)}{N+2s}dx\\ &=C\int_0^\frac{1}{\varepsilon^{\alpha_0}}\left(\frac{1}{1+\rho^2}\right)^\frac{4sN}{N+2s}(\varepsilon^{\alpha_0})^{\frac{N(N-2s)}{N+2s}+N-\frac{4sN}{N+2s}}\rho^{N-1} d\rho\\
&\leq C\varepsilon^{\frac{2N}{N+2s}}|\ln\varepsilon|. \end{align*} On the other hand, if $2s<N<6s$, using the substitution $x-\sigma_j=\lambda_j\varepsilon^{\alpha_0}z$, we get \begin{align*}
&\int_{B(\sigma_j,\eta/2)}\left|\mathcal{P}_\varepsilon w_{\lambda_j\varepsilon^{\alpha_0},\sigma_j}^p(x) -w_{\lambda_j\varepsilon^{{\alpha_0},\sigma_j}}^p(x)\right|^\frac{2N}{N+2s}dx\\ &\leq C\varepsilon^{\frac{2N}{N+2s}}\int_{\mathbb R^N}\frac{1}{(1+z^2)^\frac{4Ns}{N+2s}}dz\\ &\leq C\varepsilon^{\frac{2N}{N+2s}}. \end{align*} \end{proof} \begin{lemma}\label{lemma A.6} For any $\eta>0$ and $\varepsilon_0>0$, there exists $C>0$ such that for any $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$ and for any $\varepsilon\in(0,\varepsilon_0)$ we have for $h=1,...,k$ and $j=0,1,...,N$ \begin{eqnarray}\label{Appendix A.20}
&\left\|\left[f'_0\left(\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum\limits_{i=1}^ka_if'_0(w_i)\right]\mathcal{P}_\varepsilon \psi_h^j\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\leq C\varepsilon^{\frac{N+2s}{2}\alpha_0}. \end{eqnarray} \end{lemma}
\begin{proof}
Since $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$ it holds $|\sigma_i-\sigma_j|>\eta$ for any $i\neq j(i, j=1,...,k)$, by using Lemma \ref{lemma A.4} and Lemma \ref{(M17)-Lemma A}, we have \begin{align*}
&\int_{\Omega_\varepsilon}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum_{i=1}^ka_if'_0(w_i)\right||\mathcal{P}_\varepsilon \psi_h^j|\right)^\frac{2N}{N+2s}dx\\
&\leq\int_{\Omega_\varepsilon}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-\sum_{i=1}^ka_if'_0(w_i)\right||\mathcal{P}_\varepsilon \psi_h^j-\psi_h^j|\right)^\frac{2N}{N+2s}dx\\
&\ \ \ +\int_{\Omega_\varepsilon}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum_{i=1}^ka_if'_0(w_i)\right||\psi_h^j|\right)^\frac{2N}{N+2s}dx\\
&\leq\left\|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum_{i=1}^ka_if'_0(w_i)\right\|_{L^\frac{N}{2s}(\Omega_\varepsilon)}^\frac{2N}{N+2s}\|\mathcal{P}_\varepsilon \psi_h^j-\psi_h^j\|_{L^\frac{2N}{N-2s}(\Omega_\varepsilon)}^\frac{2N}{N+2s}\\
&\ \ \ +\int_{\Omega_\varepsilon}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum_{i=1}^ka_if'_0(w_i)\right||\psi_h^j|\right)^\frac{2N}{N+2s}dx. \end{align*}
Now by using (\ref{Appendix A.3}) and (\ref{Appendix A.4}) we have \begin{align*}
&\int_{\Omega_\varepsilon}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)- \sum_{i=1}^ka_if'_0(w_i)\right||\psi_h^j|\right)^\frac{2N}{N+2s}dy\ \ \ ( \mbox{set}\ \ x=\varepsilon^{\alpha_0}y)\\
&=\varepsilon^\frac{2N\alpha_0}{N+2s}\int_\Omega\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right)- \sum_{i=1}^ka_if'_0(w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i})\right||\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|\right)^\frac{2N}{N+2s}dx\\
&\leq\varepsilon^\frac{2N\alpha_0}{N+2s}\int_{B(\sigma_h,\eta/2)}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right)- \sum_{i=1}^ka_if'_0(w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i})\right||\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|\right)^\frac{2N}{N+2s}dx\\
&\ \ \ +\varepsilon^\frac{2N\alpha_0}{N+2s}\int_{\Omega\setminus B(\sigma_h,\eta/2)}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right)- \sum_{i=1}^ka_if'_0(w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i})\right||\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|\right)^\frac{2N}{N+2s}dx.\\ \end{align*} Firstly, by Lemma \ref{lemma A.1}, we get \begin{align*}
&\int_{B(\sigma_h,\eta/2)}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right)- \sum_{i=1}^ka_if'_0(w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i})\right||\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|\right)^\frac{2N}{N+2s}dx\\
&\leq C\int_{B(\sigma_h,\eta/2)}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right)- a_hf'_0(w_{\lambda_h\varepsilon^{\alpha_0},\sigma_h})\right||\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|\right)^\frac{2N}{N+2s}dx\\
&\ \ \ +C\sum\limits_{i=1, i\neq h}^{k}\int_{B(\sigma_h,\eta/2)}|f'_0(w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i})||\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|^\frac{2N}{N+2s}dx\\
&\leq C\int_{B(\sigma_h,\eta/2)}|\mathcal{P}_\varepsilon w_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}- w_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}|^\frac{8Ns}{(N+2s)(N-2s)}|\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|^\frac{2N}{N+2s}dx\\
&\ \ \ +C\sum\limits_{i=1, i\neq h}^{k}\int_{B(\sigma_h,\eta/2)}|w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}|^\frac{8Ns}{(N+2s)(N-2s)}|\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|^\frac{2N}{N+2s}dx\\ &\leq C\varepsilon^\frac{\alpha_0N^2}{N+2s}. \end{align*} Secondly, we have \begin{align*}
&\int_{\Omega\setminus B(\sigma_h,\eta/2)}\left(\left|f'_0\left(\sum_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right)- \sum_{i=1}^ka_if'_0(w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i})\right||\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|\right)^\frac{2N}{N+2s}dx\\
&\leq\sum_{i=1}^k\int_{\Omega\setminus B(\sigma_h,\eta/2)}|w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}|^\frac{8Ns}{(N+2s)(N-2s)}|\psi_{\lambda_h\varepsilon^{\alpha_0},\sigma_h}^j|^\frac{2N}{N+2s}dx\\ &\leq C\varepsilon^\frac{\alpha_0N^2}{N+2s}. \end{align*} By the above estimations the proof is complete. \end{proof}
\begin{lemma}\label{lemma A.7} For any $\eta>0$ and $\varepsilon_0>0$ there exists $C>0$ such that for any $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$ and for any $\varepsilon\in(0,\varepsilon_0)$ we have \begin{eqnarray}\label{lemma A.7-1}
&\left\| f_\varepsilon\left(\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f_0\left(\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right\|_{L^\frac{2N}{N+2s}(\Omega_\varepsilon)}\leq C\varepsilon|\ln\varepsilon|. \end{eqnarray} \begin{eqnarray}\label{lemma A.7-2}
&\left\|f'_\varepsilon\left(\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f'_0\left(\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right\|_{L^\frac{N}{2s}(\Omega_\varepsilon)}\leq C\varepsilon|\ln\varepsilon|, \end{eqnarray} \end{lemma}
\begin{proof} Let us prove (\ref{lemma A.7-1}). The proof of (\ref{lemma A.7-2}) is similar. Since $(\mbox{\boldmath $\lambda$},\mbox{\boldmath $\sigma$})\in\mathcal{O}_\eta$ it holds $|\sigma_i-\sigma_j|>\eta$ for any $i\neq j(i, j=1,...,k)$. We have \begin{align*}
&\left\|f_\varepsilon\left(\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)-f_0\left(\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right)\right\|_{L^{\frac{2N}{N+2s}}(\Omega_\varepsilon)}^{\frac{2N}{N+2s}}\\
&=\int_{\Omega_\varepsilon}\left|\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right|^{p-\varepsilon}-\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right|^p\right|^{\frac{2N}{N+2s}}dx\\
&=\int_{\Omega_\varepsilon}\left|\varepsilon\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right|^{p-t\varepsilon}\ln\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_i\right|\right|^{\frac{2N}{N+2s}}dx\ \ \ ( \mbox{set}\ \ y=\varepsilon^{\alpha_0}x)\\
&=\varepsilon^{\frac{2N}{N+2s}(\frac{p-t\varepsilon}{2}+1)-\frac{N}{N-2s}}\int_{\Omega}\left|\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|^{p-t\varepsilon}\ln\left|\varepsilon^{\frac{(N-2s)\alpha_0}{2}}\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|\right|^{\frac{2N}{N+2s}}dy\\
&=\varepsilon^{\frac{2N}{N+2s}(\frac{p-t\varepsilon}{2}+1)-\frac{N}{N-2s}}\Bigg[\sum_{j=1}^k\int_{B(\sigma_j,\eta/2)}\left|\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|^{p-t\varepsilon}\ln\left|\varepsilon^{\frac{(N-2s)\alpha_0}{2}}\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|\right|^{\frac{2N}{N+2s}}dy\\
&\ \ \ \ +\int_{\Omega\setminus{\bigcup\limits_{j=1}^kB(\sigma_j,\eta/2)}}\left|\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|^{p-t\varepsilon}\ln\left|\varepsilon^{\frac{(N-2s)\alpha_0}{2}}\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|\right|^{\frac{2N}{N+2s}}dy\Bigg].\\ \end{align*}
Firstly, we have \begin{align*}
&\int_{\Omega\setminus{\bigcup\limits_{j=1}^kB(\sigma_j,\eta/2)}}\varepsilon^{\frac{2N}{N+2s}(\frac{p-t\varepsilon}{2}+1)-\frac{N}{N-2s}}\left|\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|^{p-t\varepsilon}\ln\left|\varepsilon^{\frac{(N-2s)\alpha_0}{2}}\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|\right|^{\frac{2N}{N+2s}}dy\\ &\leq C\varepsilon^{\frac{2N}{N+2s}(\frac{p-t\varepsilon}{2}+1)-\frac{N}{N-2s}}\sum\limits_{i=1}^k(\lambda_i\varepsilon^{\alpha_0})^{\frac{N-2s}{2}
(p-t\varepsilon)\frac{2N}{N+2s}}\left|\ln\left(\varepsilon^{\frac{(N-2s)\alpha_0}{2}}\sum
\limits_{i=1}^k(\lambda_i\varepsilon^{\alpha_0})^{\frac{N-2s}{2}}\right)\right|^{\frac{2N}{N+2s}}\\
&\leq C |\varepsilon\ln\varepsilon|^{\frac{2N}{N+2s}} \end{align*} Moreover, using the substitution $x-\sigma_i=\lambda_i\varepsilon^{\alpha_0}z$, we get \begin{align*}
&\int_{B(\sigma_j,\eta/2)}\varepsilon^{\frac{2N}{N+2s}(\frac{p-t\varepsilon}{2}+1)-\frac{N}{N-2s}}\left|\left|\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|^{p-t\varepsilon}\ln\left|\varepsilon^{\frac{(N-2s)\alpha_0}{2}}\sum\limits_{i=1}^ka_i\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|\right|^{\frac{2N}{N+2s}}dy\\ &\leq C\sum\limits_{i=1}^k\int_{B(\sigma_j,\eta/2)}\varepsilon^{\frac{2N}{N+2s}(\frac{p-t\varepsilon}{2}+1)-\frac{N}{N-2s}}
\left|w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|^{\frac{2N(p-t\varepsilon)}{N+2s}}
\left|\ln\left|\varepsilon^{\frac{(N-2s)\alpha_0}{2}}\sum\limits_{i=1}^k\mathcal{P}_\varepsilon w_{\lambda_i\varepsilon^{\alpha_0},\sigma_i}\right|\right|^{\frac{2N}{N+2s}}dy\\ &\leq C\sum\limits_{i=1}^k\int_{\mathbb R^N}\varepsilon^{\frac{2N}{N+2s}(\frac{p-t\varepsilon}{2}+1)-\frac{N}{N-2s}}(\lambda_i\varepsilon^{\alpha_0})^{N-\frac{N(N-2s)}{N+2s}(p-t\varepsilon)} \left(\frac{1}{1+z^2}\right)^{\frac{N(N-2s)}{N+2s}(p-t\varepsilon)}\\
&\ \ \ \times\left|\ln\left|\lambda_i^{-\frac{N-2s}{2}}
\Bigg[\left(\frac{1}{1+z^2}\right)^\frac{N-2s}{2}+\sum\limits_{j=1, i\neq j}^{k}\left(\frac{\lambda_i\varepsilon^{\alpha_0}\lambda_j\varepsilon^{\alpha_0}}{(\lambda_j\varepsilon^{\alpha_0})^2+|\lambda_i\varepsilon^{\alpha_0}z+\sigma_i-
\sigma_j|^2}\right)^\frac{N-2s}{2}\Bigg]\right|\right|^{\frac{2N}{N+2s}}dz\\
&\leq C |\varepsilon\ln\varepsilon|^{\frac{2N}{N+2s}}. \end{align*} \end{proof}
\end{document} | arXiv |
Mathematically modeling the biological properties of gliomas: A review
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Quantitative impact of immunomodulation versus oncolysis with cytokine-expressing virus therapeutics
2015, 12(4): 859-877. doi: 10.3934/mbe.2015.12.859
Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function
Yu Yang 1, , Shigui Ruan 2, and Dongmei Xiao 3,
School of Science and Technology, Zhejiang International Studies University, Hangzhou 310012, China
Department of Mathematics, University of Miami, Coral Gables, FL 33124-4250
Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
Received April 2014 Revised December 2014 Published April 2015
In this paper, we study an age-structured virus dynamics model with Beddington-DeAngelis infection function. An explicit formula for the basic reproductive number $\mathcal{R}_{0}$ of the model is obtained. We investigate the global behavior of the model in terms of $\mathcal{R}_{0}$: if $\mathcal{R}_{0}\leq1$, then the infection-free equilibrium is globally asymptotically stable, whereas if $\mathcal{R}_{0}>1$, then the infection equilibrium is globally asymptotically stable. Finally, some special cases, which reduce to some known HIV infection models studied by other researchers, are considered.
Keywords: Age structure, virus dynamics model, Liapunov function, infection equilibrium, global stability..
Mathematics Subject Classification: Primary: 35L60, 92C37; Secondary: 35B35, 34K2.
Citation: Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
C. L. Althaus and R. J. De Boer, Dynamics of immune escape during HIV/SIV infection,, PLoS Comput. Biol., 4 (2008). doi: 10.1371/journal.pcbi.1000103. Google Scholar
F. Brauer, Z. Shuai and P. van den Driessche, Dynamics of an age-of-infection cholera model,, Math. Biosci. Eng., 10 (2013), 1335. doi: 10.3934/mbe.2013.10.1335. Google Scholar
C. J. Browne and S. S. Pilyugin, Global analysis of age-structured within-host virus model,, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1999. doi: 10.3934/dcdsb.2013.18.1999. Google Scholar
R. V. Culshaw and S. Ruan, A delay-differential equation model of HIV infection of CD4$^+$ T-cells,, Math. Biosci., 165 (2000), 27. doi: 10.1016/S0025-5564(00)00006-7. Google Scholar
C. Cosner, D.L. DeAngelis, J. S. Ault and D. B. Olson, Effects of spatial grouping on the functional response of predators,, Theoret. Pop. Biol., 56 (1999), 65. doi: 10.1006/tpbi.1999.1414. Google Scholar
R. J. De Boer and A. S. Perelson, Target cell limited and immune control models of HIV infection: A comparison,, J. Theoret. Biol., 190 (1998), 201. Google Scholar
R. D. Demasse and A. Ducrot, An age-structured within-host model for multistrain malaria infections,, SIAM. J. Appl. Math., 73 (2013), 572. doi: 10.1137/120890351. Google Scholar
P. De Leenheer and H. L. Smith, Virus dynamics: A global analysis,, SIAM J. Appl. Math., 63 (2003), 1313. doi: 10.1137/S0036139902406905. Google Scholar
J. K. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs Vol 25, (1988). Google Scholar
J. K. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 20 (1989), 388. doi: 10.1137/0520025. Google Scholar
G. Huang, W. Ma and Y. Takeuchi, Global properties for virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 22 (2009), 1690. doi: 10.1016/j.aml.2009.06.004. Google Scholar
G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response,, Appl. Math. Lett., 24 (2011), 1199. doi: 10.1016/j.aml.2011.02.007. Google Scholar
G. Huang, Y. Takeuchi and W. Ma, Lyapunov functionals for delay differential equations model of viral infections,, SIAM J. Appl. Math., 70 (2010), 2693. doi: 10.1137/090780821. Google Scholar
G. Huang, X. Liu and Y. Takeuchi, Lyapunov functions and global stability for age-structured HIV infection model,, SIAM J. Appl. Math., 72 (2012), 25. doi: 10.1137/110826588. Google Scholar
G. Huisman and R. J. De Boer, A formal derivation of the "Beddington" functional response,, J. Theoret. Biol., 185 (1997), 389. doi: 10.1006/jtbi.1996.0318. Google Scholar
D. Kirschner and G. F. Webb, A model for treatment strategy in the chemotherapy of AIDS,, Bull. Math. Biol., 58 (1996), 367. doi: 10.1016/0092-8240(95)00345-2. Google Scholar
M. Y. Li and H. Shu, Global dynamics of an in-host viral model with intracellular delay,, Bull. Math. Biol., 72 (2010), 1492. doi: 10.1007/s11538-010-9503-x. Google Scholar
M. Y. Li and H. Shu, Impact of intracellular delays and target-cell dynamics on in vivo viral infections,, SIAM J. Appl. Math., 70 (2010), 2434. doi: 10.1137/090779322. Google Scholar
P. Magal, Compact attractors for time-periodic age-structured population models,, Electron. J. Differential Equations, 65 (2001), 1. Google Scholar
P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model,, Appl. Anal., 89 (2010), 1109. doi: 10.1080/00036810903208122. Google Scholar
P. Magal and C. C. McCluskey, Two-group infection age model including an application to nosocomial infection,, SIAM J. Appl. Math., 73 (2013), 1058. doi: 10.1137/120882056. Google Scholar
P. Magal and H. R. Thieme, Eventual compactness for semiflows generated by nonlinear age-structured models,, Commun. Pure Appl. Anal., 3 (2004), 695. doi: 10.3934/cpaa.2004.3.695. Google Scholar
C. C. McCluskey, Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes,, Math. Biosci. Eng., 9 (2012), 819. doi: 10.3934/mbe.2012.9.819. Google Scholar
P. W. Nelson, M. A. Gilchrist, D. Coombs, J. M. Hyman and A. S. Perelson, An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells,, Math. Biosci. Eng., 1 (2004), 267. doi: 10.3934/mbe.2004.1.267. Google Scholar
A. Nowak and C. R. M. Bangham, Population dynamics of immune responses to persistent viruses,, Science, 272 (1996), 74. doi: 10.1126/science.272.5258.74. Google Scholar
M. A. Nowak and R. M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology,, Oxford University Press, (2000). Google Scholar
A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo,, SIAM Rev., 41 (1999), 3. doi: 10.1137/S0036144598335107. Google Scholar
L. Rong, Z. Feng and A. S. Perelson, Mathematical analysis of age-structured HIV-1 dynamics with combination antiretroviral therapy,, SIAM. J. Appl. Math., 67 (2007), 731. doi: 10.1137/060663945. Google Scholar
H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, Differential Integral Equations, 3 (1990), 1035. Google Scholar
Hui li, Manjun Ma. Global dynamics of a virus infection model with repulsive effect. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4783-4797. doi: 10.3934/dcdsb.2019030
Yuming Chen, Junyuan Yang, Fengqin Zhang. The global stability of an SIRS model with infection age. Mathematical Biosciences & Engineering, 2014, 11 (3) : 449-469. doi: 10.3934/mbe.2014.11.449
Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva. Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1159-1186. doi: 10.3934/mbe.2017060
C. Connell McCluskey. Global stability for an $SEI$ model of infectious disease with age structure and immigration of infecteds. Mathematical Biosciences & Engineering, 2016, 13 (2) : 381-400. doi: 10.3934/mbe.2015008
Cameron Browne. Immune response in virus model structured by cell infection-age. Mathematical Biosciences & Engineering, 2016, 13 (5) : 887-909. doi: 10.3934/mbe.2016022
Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
Zhaohui Yuan, Xingfu Zou. Global threshold dynamics in an HIV virus model with nonlinear infection rate and distributed invasion and production delays. Mathematical Biosciences & Engineering, 2013, 10 (2) : 483-498. doi: 10.3934/mbe.2013.10.483
Hongying Shu, Lin Wang. Global stability and backward bifurcation of a general viral infection model with virus-driven proliferation of target cells. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1749-1768. doi: 10.3934/dcdsb.2014.19.1749
Fred Brauer, Zhisheng Shuai, P. van den Driessche. Dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1335-1349. doi: 10.3934/mbe.2013.10.1335
Jinliang Wang, Ran Zhang, Toshikazu Kuniya. A note on dynamics of an age-of-infection cholera model. Mathematical Biosciences & Engineering, 2016, 13 (1) : 227-247. doi: 10.3934/mbe.2016.13.227
Jinliang Wang, Jiying Lang, Yuming Chen. Global dynamics of an age-structured HIV infection model incorporating latency and cell-to-cell transmission. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3721-3747. doi: 10.3934/dcdsb.2017186
Bao-Zhu Guo, Li-Ming Cai. A note for the global stability of a delay differential equation of hepatitis B virus infection. Mathematical Biosciences & Engineering, 2011, 8 (3) : 689-694. doi: 10.3934/mbe.2011.8.689
Suxia Zhang, Xiaxia Xu. A mathematical model for hepatitis B with infection-age structure. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1329-1346. doi: 10.3934/dcdsb.2016.21.1329
Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999
Steffen Eikenberry, Sarah Hews, John D. Nagy, Yang Kuang. The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth. Mathematical Biosciences & Engineering, 2009, 6 (2) : 283-299. doi: 10.3934/mbe.2009.6.283
Yu Yang, Yueping Dong, Yasuhiro Takeuchi. Global dynamics of a latent HIV infection model with general incidence function and multiple delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 783-800. doi: 10.3934/dcdsb.2018207
C. Connell McCluskey. Global stability for an SEI epidemiological model with continuous age-structure in the exposed and infectious classes. Mathematical Biosciences & Engineering, 2012, 9 (4) : 819-841. doi: 10.3934/mbe.2012.9.819
Nicola Bellomo, Youshan Tao. Stabilization in a chemotaxis model for virus infection. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : 105-117. doi: 10.3934/dcdss.2020006
Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019226
Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
Yu Yang Shigui Ruan Dongmei Xiao | CommonCrawl |
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On finitely generated vector sublattices
Volume 245 / 2019
Lech Drewnowski, Witold Wnuk Studia Mathematica 245 (2019), 129-167 MSC: Primary 46B42, 28A05, 54H05. DOI: 10.4064/sm170524-23-12 Published online: 20 July 2018
We investigate various questions concerning vector sublattices, $\operatorname {vlt} S$, generated by subsets $S$ of an Archimedean vector lattice $E$. We first prove a distributivity law: $\operatorname {vlt}(X;Y,Z)=\operatorname {vlt}(X,Y)+\nobreakspace {}\operatorname {vlt}(X,Z)$ if $X,Y,Z\subset E$ and $Y\perp Z$, and derive a number of its consequences. We next show that in a topological vector lattice the dimension of the sublattice generated by an analytic set is either $\le \aleph _0$ or $2^{\aleph _0}$, and that the same is true for sublattices generated by at most countable sets in arbitrary vector lattices. In a vector lattice, we characterize those sets that generate $n$-dimensional sublattices and prove that a finite set generates a finite-dimensional sublattice if so does each pair of its elements. We also show that in a uniformly complete vector lattice every principal ideal of infinite dimension contains pairs of positive elements generating $\aleph _0$- as well as $2^{\aleph _0}$-dimensional sublattices. The special case of lattices $C(K)$ is also treated in this respect. Moreover, for a compact set $K\subset \mathbb R^n$ with a nonempty interior, it is shown that the minimal number of functions in $C(K)$ or $C(K)_+$ generating a dense sublattice is $n+1$. We also prove that every (separable) Banach lattice $C(K)$ can be embedded in a discrete (separable) Banach lattice of the same type. Finally, we prove that in a discrete and $\sigma $-Dedekind complete separable $F$-lattice one can always find a pair of positive elements generating a dense sublattice, and we use that result to show that, in general, this is far from being true even in the case of discrete separable $C(K)$ lattices.
Lech DrewnowskiFaculty of Mathematics and Computer Science
A. Mickiewicz University
Umultowska 87
Witold WnukFaculty of Mathematics and Computer Science
all title author MSCS 2010
Query phrase too short. Type at least 4 characters. | CommonCrawl |
Find the remainder when $109876543210$ is divided by $180$.
Let $N = 109876543210$. Notice that $180 = 4 \times 9 \times 5$, so by the Chinese Remainder Theorem, it suffices to evaluate the remainders when $N$ is divided by each of $4$, $9$, and $5$. We can apply the divisibility rules to find each of these. Since the last two digits of $N$ are $10$, it follows that $N \equiv 10 \equiv 2 \pmod{4}$. We know that $N$ is divisible by $5$, so $N \equiv 0 \pmod{5}$. Finally, since $N$ leaves the same residue modulo $9$ as the sum of its digits, then $$N \equiv 0 + 1 + 2 + 3 + \cdots + 9 + 1 \equiv 1+ \frac{9 \cdot 10}{2} \equiv 46 \equiv 1 \pmod{9}.$$By the Chinese Remainder Theorem and inspection, it follows that $N \equiv 10 \pmod{4 \cdot 9}$, and since $10$ is also divisible by $5$, then $N \equiv \boxed{10} \pmod{180}$. | Math Dataset |
Parallel all-pairs shortest path algorithm
A central problem in algorithmic graph theory is the shortest path problem. Hereby, the problem of finding the shortest path between every pair of nodes is known as all-pair-shortest-paths (APSP) problem. As sequential algorithms for this problem often yield long runtimes, parallelization has shown to be beneficial in this field. In this article two efficient algorithms solving this problem are introduced.
Another variation of the problem is the single-source-shortest-paths (SSSP) problem, which also has parallel approaches: Parallel single-source shortest path algorithm.
Problem definition
Let $G=(V,E,w)$ be a directed Graph with the set of nodes $V$ and the set of edges $E\subseteq V\times V$. Each edge $e\in E$ has a weight $w(e)$ assigned. The goal of the all-pair-shortest-paths problem is to find the shortest path between all pairs of nodes of the graph. For this path to be unique it is required that the graph does not contain cycles with a negative weight.
In the remainder of the article it is assumed that the graph is represented using an adjacency matrix. We expect the output of the algorithm to be a distancematrix $D$. In $D$, every entry $d-{i,j}$ is the weight of the shortest path in $G$ from node $i$ to node $j$.
The Floyd algorithm presented later can handle negative edge weights, whereas the Dijkstra algorithm requires all edges to have a positive weight.
Dijkstra algorithm
The Dijkstra algorithm originally was proposed as a solver for the single-source-shortest-paths problem. However, the algorithm can easily be used for solving the All-Pair-Shortest-Paths problem by executing the Single-Source variant with each node in the role of the root node.
In pseudocode such an implementation could look as follows:
1 func DijkstraSSSP(G,v) {
2 ... //standard SSSP-implementation here
3 return dv;
4 }
5
6 func DijkstraAPSP(G) {
7 D := |V|x|V|-Matrix
8 for i from 1 to |V| {
9 //D[v] denotes the v-th row of D
10 D[v] := DijkstraSSP(G,i)
11 }
12 }
In this example we assume that DijkstraSSSP takes the graph $G$ and the root node $v$ as input. The result of the execution in turn is the distancelist $d_{v}$. In $d_{v}$, the $i$-th element stores the distance from the root node $v$ to the node $i$. Therefore the list $d_{v}$ corresponds exactly to the $v$-th row of the APSP distancematrix $D$. For this reason, DijkstraAPSP iterates over all nodes of the graph $G$ and executes DijkstraSSSP with each as root node while storing the results in $D$.
The runtime of DijkstraSSSP is $O(|V|^{2})$ as we expect the graph to be represented using an adjacency matrix. Therefore DijkstraAPSP has a total sequential runtime of $O(|V|^{3})$.
Parallelization for up to |V| processors
A trivial parallelization can be obtained by parallelizing the loop of DijkstraAPSP in line8. However, when using the sequential DijkstraSSSP this limits the number of processors to be used by the number of iterations executed in the loop. Therefore, for this trivial parallelization $|V|$ is an upper bound for the number of processors.
For example, let the number of processors $p$ be equal to the number of nodes $|V|$. This results in each processor executing DijkstraSSSP exactly once in parallel. However, when there are only for example $p={\frac {|V|}{2}}$ processors available, each processor has to execute DijkstraSSSP twice.
In total this yields a runtime of $O(|V|^{2}\cdot {\frac {|V|}{p}})$, when $|V|$ is a multiple of $p$. Consequently, the efficiency of this parallelization is perfect: Employing $p$ processors reduces the runtime by the factor $p$.
Another benefit of this parallelization is that no communication between the processors is required. However, it is required that every processor has enough local memory to store the entire adjacency matrix of the graph.
Parallelization for more than |V| processors
If more than $|V|$ processors shall be used for the parallelization, it is required that multiple processors take part of the DijkstraSSSP computation. For this reason, the parallelization is split across into two levels.
For the first level the processors are split into $|V|$ partitions. Each partition is responsible for the computation of a single row of the distancematrix $D$. This means each partition has to evaluate one DijkstraSSSP execution with a fixed root node. With this definition each partition has a size of $k={\frac {p}{|V|}}$ processors. The partitions can perform their computations in parallel as the results of each are independent of each other. Therefore, the parallelization presented in the previous section corresponds to a partition size of 1 with $p=|V|$ processors.
The main difficulty is the parallelization of multiple processors executing DijkstraSSSP for a single root node. The idea for this parallelization is to distribute the management of the distancelist $d_{v}$ in DijkstraSSSP within the partition. Each processor in the partition therefore is exclusively responsible for ${\frac {|V|}{k}}$ elements of $d_{v}$. For example, consider $|V|=4$ and $p=8$: this yields a partition size of $k=2$. In this case, the first processor of each partition is responsible for $d_{v,1}$, $d_{v,2}$ and the second processor is responsible for $d_{v,3}$ and $d_{v,4}$. Hereby, the total distance lists is $d_{v}=[d_{v,1},d_{v,2},d_{v,3},d_{v,4}]$.
The DijkstraSSSP algorithm mainly consists of the repetition of two steps: First, the nearest node $x$ in the distancelist $d_{v}$ has to be found. For this node the shortest path already has been found. Afterwards the distance of all neighbors of $x$ has to be adjusted in $d_{v}$.
These steps have to be altered as follows because for the parallelization $d_{v}$ has been distributed across the partition:
1. Find the node $x$ with the shortest distance in $d_{v}$.
• Each processor owns a part of $d_{v}$: Each processor scans for the local minimum ${\tilde {x}}$ in his part, for example using linear search.
• Compute the global minimum $x$ in $d_{v}$ by performing a reduce-operation across all ${\tilde {x}}$.
• Broadcast the global minimum $x$ to all nodes in the partition.
2. Adjust the distance of all neighbors of $x$ in $d_{v}$
• Every processors now knows the global nearest node $x$ and its distance. Based on this information, adjust the neighbors of $x$ in $d_{v}$ which are managed by the corresponding processor.
The total runtime of such an iteration of DijkstraSSSP performed by a partition of size $k$ can be derived based on the performed subtasks:
• The linear search for ${\tilde {x}}$: $O({\frac {|V|}{k}})$
• Broadcast- and Reduce-operations: These can be implemented efficiently for example using binonmialtrees. This yields a communication overhead of $O(\log k)$.
For $|V|$-iterations this results in a total runtime of $O(|V|({\frac {|V|}{k}}+\log k))$. After substituting the definition of $k$ this yields the total runtime for DijkstraAPSP: $O({\frac {|V|^{3}}{p}}+\log p)$.
The main benefit of this parallelization is that it is not required anymore that every processor stores the entire adjacency matrix. Instead, it is sufficient when each processor within a partition only stores the columns of the adjacency matrix of the nodes for which he is responsible. Given a partition size of $k$, each processor only has to store ${\frac {|V|}{k}}$ columns of the adjacency matrix. A downside, however, is that this parallelization comes with a communication overhead due to the reduce- and broadcast-operations.
Example
The graph used in this example is the one presented in the image with four nodes.
The goal is to compute the distancematrix with $p=8$ processors. For this reason, the processors are divided into four partitions with two processors each. For the illustration we focus on the partition which is responsible for the computation of the shortest paths from node A to all other nodes. Let the processors of this partition be named p1 and p2.
The computation of the distancelist across the different iterations is visualized in the second image.
The top row in the image corresponds to $d_{A}$ after the initialization, the bottom one to $d_{A}$ after the termination of the algorithm. The nodes are distributed in a way that p1 is responsible for the nodes A and B, while p2 is responsible for C and D. The distancelist $d_{A}$ is distributed according to this. For the second iteration the subtasks executed are shown explicitly in the image:
1. Computation of the local minimum node in $d_{A}$
2. Computation of the globalminimum node in $d_{A}$ through a reduce operation
3. Broadcast of the global minimum node in $d_{A}$
4. Marking of the global nearest node as "finished" and adjusting the distance of its neighbors
Floyd algorithm
The Floyd algorithm solves the All-Pair-Shortest-Paths problem for directed graphs. With the adjacency matrix of a graph as input, it calculates shorter paths iterative. After |V| iterations the distance-matrix contains all the shortest paths. The following describes a sequential version of the algorithm in pseudo code:
1 func Floyd_All_Pairs_SP(A) {
2 $D^{(0)}$ = A;
3 for k := 1 to n do
4 for i := 1 to n do
5 for j := 1 to n do
6 $d_{i,j}^{(k)}:=\min(d_{i,j}^{(k-1)},d_{i,k}^{(k-1)}+d_{k,j}^{(k-1)})$
7 }
Where A is the adjacency matrix, n = |V| the number of nodes and D the distance matrix. For a more detailed description of the sequential algorithm look up Floyd–Warshall algorithm.
Parallelization
The basic idea to parallelize the algorithm is to partition the matrix and split the computation between the processes. Each process is assigned to a specific part of the matrix. A common way to achieve this is 2-D Block Mapping. Here the matrix is partitioned into squares of the same size and each square gets assigned to a process. For an $n\times n$-matrix and p processes each process calculates a $n/{\sqrt {p}}\times n/{\sqrt {p}}$ sized part of the distance matrix. For $p=n^{2}$ processes each would get assigned to exactly one element of the matrix. Because of that the parallelization only scales to a maximum of $n^{2}$ processes. In the following we refer with $p_{i,j}$ to the process that is assigned to the square in the i-th row and the j-th column.
As the calculation of the parts of the distance matrix is dependent on results from other parts the processes have to communicate between each other and exchange data. In the following we refer with $d_{i,j}^{(k)}$ to the element of the i-th row and j-th column of the distance matrix after the k-th iteration. To calculate $d_{i,j}^{(k)}$ we need the elements $d_{i,j}^{(k-1)}$, $d_{i,k}^{(k-1)}$ and $d_{k,j}^{(k-1)}$ as specified in line 6 of the algorithm. $d_{i,j}^{(k-1)}$ is available to each process as it was calculated by itself in the previous iteration.
Additionally each process needs a part of the k-th row and the k-th column of the $D^{k-1}$ matrix. The $d_{i,k}^{(k-1)}$ element holds a process in the same row and the $d_{k,j}^{(k-1)}$ element holds a process in the same column as the process that wants to compute $d_{i,j}^{(k)}$. Each process that calculated a part of the k-th row in the $D^{k-1}$ matrix has to send this part to all processes in its column. Each process that calculated a part of the k-th column in the $D^{k-1}$ matrix has to send this part to all processes in its row. All this processes have to do a one-to-all-broadcast operation along the row or the column. The data dependencies are illustrated in the image below.
For the 2-D block mapping we have to modify the algorithm as follows:
1 func Floyd_All_Pairs_Parallel($D^{(0)}$) {
2 for k := 1 to n do{
3 Each process $p_{i,j}$ that has a segment of the k-th row of $D^{(k-1)}$,
broadcasts it to the $p_{*,j}$ processes;
4 Each process $p_{i,j}$ that has a segment of the k-th column of $D^{(k-1)}$,
broadcasts it to the $p_{i,*}$ processes;
5 Each process waits to receive the needed segments;
6 Each process computes its part of the $D^{(k)}$ matrix;
7 }
8 }
In line 5 of the algorithm we have a synchronisation step to ensure that all processes have the data necessary to compute the next iteration. To improve the runtime of the algorithm we can remove the synchronisation step without affecting the correctness of the algorithm. To achieve that each process starts the computation as soon as it has the data necessary to compute its part of the matrix. This version of the algorithm is called pipelined 2-D block mapping.
Runtime
The runtime of the sequential algorithm is determined by the triple nested for loop. The computation in line 6 can be done in constant time ($O(1)$). Therefore, the runtime of the sequential algorithm is $O(n^{3})$.
2-D block mapping
The runtime of the parallelized algorithm consists of two parts. The time for the computation and the part for communication and data transfer between the processes.
As there is no additional computation in the algorithm and the computation is split equally among the p processes, we have a runtime of $O(n^{3}/p)$ for the computational part.
In each iteration of the algorithm there is a one-to-all broadcast operation performed along the row and column of the processes. There are $n/{\sqrt {p}}$ elements broadcast. Afterwards there is a synchronisation step performed. How much time these operations take is highly dependent on the architecture of the parallel system used. Therefore, the time needed for communication and data transfer in the algorithm is $T_{\text{comm}}=n(T_{\text{synch}}+T_{\text{broadcast}})$.
For the whole algorithm we have the following runtime:
$T=O\left({\frac {n^{3}}{p}}\right)+n(T_{\text{synch}}+T_{\text{broadcast}})$
Pipelined 2-D block mapping
For the runtime of the data transfer between the processes in the pipelined version of the algorithm we assume that a process can transfer k elements to a neighbouring process in $O(k)$ time. In every step there are $n/{\sqrt {p}}$ elements of a row or a column send to a neighbouring process. Such a step takes $O(n/{\sqrt {p}})$ time. After ${\sqrt {p}}$ steps the relevant data of the first row and column arrive at process $p_{{\sqrt {p}},{\sqrt {p}}}$ (in $O(n)$ time).
The values of successive rows and columns follow after time $O(n^{2}/p)$ in a pipelined mode. Process $p_{{\sqrt {p}},{\sqrt {p}}}$ finishes its last computation after O($n^{3}/p$) + O($n$) time. Therefore, the additional time needed for communication in the pipelined version is $O(n)$.
The overall runtime for the pipelined version of the algorithm is:
$T=O\left({\frac {n^{3}}{p}}\right)+O(n)$
References
Bibliography
• Grama, A.: Introduction to parallel computing. Pearson Education, 2003.
• Kumar, V.: Scalability of Parallel Algorithms for the All-Pairs Shortest-Path Problem. Journal of Parallel and Distributed Programming 13, 1991.
• Foster, I.: Designing and Building Parallel Programs (Online).
• Bindell, Fall: Parallel All-Pairs Shortest Paths Applications of Parallel Computers, 2011.
| Wikipedia |
What if, for a body moving in a circle, the centripetal force is not equal to $mv^2/r$?
For example, I don't understand why the speed of a satellite moving in an orbit of radius 'a' around the Earth must be equal to $\sqrt{GM/a}$ If I release a particle in outer space with a velocity perpendicular to the line joining the particle and Earth but magnitude not equal to $\sqrt{GM/a}$ what will happen? For al I know, the acceleration will still be perpendicular to its velocity at all instants. And, an always perpendicular acceleration is only capable of changing the direction of velocity continuously. Then, wouldn't the particle still rotate even if its speed not equal to the one given by F=$mv^2/r$?
newtonian-mechanics newtonian-gravity centripetal-force
DoveDove
$\begingroup$ Acceleration will only be perpendicular to the velocity only at the moment you released the particle (if the velocity is not equal to $\sqrt {GM/a} $). $\endgroup$
– cobra121
$\begingroup$ Can you please explain why the acceleration won't be perpendicular at all other instants? $\endgroup$
– Dove
$\begingroup$ For a particle having a velocity $v$ , the condition for it to move in a circle of radius $r $ is the force on it should be perpendicular to it and equal to $mv^2/r $ $\endgroup$
$\begingroup$ Lets assume the velocity is $x\sqrt {GM/a} $ where x is not 1. Here the gravitational force is $GMm/a^2$ and the force required is $mv^2/a= x^2GMm/a^2$ which are not equal. $\endgroup$
$\begingroup$ If $x>1$ then this force is not enough to curl the particle to move in a circle. Thus it will leave the circular orbit the moment you release it (not tangentially but some trajectory between the tangent and the circular path). In that trajectory the force will still be directed towards the earth, but the velocity will have a component in the line joining the particle and the earth. You can similarly reason for $x <1$. The force is more than what is required and the particle will curl more than required for a circle. $\endgroup$
You agree that a satellite with speed $\sqrt{\frac{GM}{a}}$ will undergo a circular orbit of radius $a$.
Now suppose that you take a satellite up to that distance $a$ and give it a tangential speed greater than $\sqrt{\frac{GM}{a}}$.
The satellite will start executing a curved path of smaller curvature, larger radius, than if the speed had been $\sqrt{\frac{GM}{a}}$ because the gravitational attraction of the Earth on the satellite is not large enough for the path to be of radius $a$.
The satellite would therefore move further from the Earth gaining gravitational potential energy but at the same time losing kinetic energy i.e. moving at a slower speed.
Remember that the satellite has a mass and if no force was acting on it the satellite would travel in a straight line.
With the higher speed the gravitational attraction of the Earth cannot pull the satellite enough to make the satellite execute a circular path so it goes along a less curved, elliptical path.
What happens next depends on the speed that you give the satellite.
Below a certain value the satellite would execute an elliptical orbit about the Earth i.e. the distance between the Earth and the satellite would vary as would the speed of the satellite.
At a particular speed, the escape speed, the satellite would execute a parabolic path and escape from the Earth.
Above the escape speed the path of the satellite would be hyperbolic.
FarcherFarcher
When you have a particle orbiting circularly around Earth, you can easily write the motion laws for two directions: radial direction and tangential direction. In tangential direction, you have uniform motion with constant speed. In radial direction, due to the motion of the particle, in order to have a circular motion, you should have a radial (i.e., centripetal) force with modulus: $$F=\frac{m v^2}{ R}$$ with $m$ mass of the particle, $v$ its speed and $R$ radius of the circular orbit. In this case, this force is gravity and, from the law of Gravitational attraction, you can get: $$ F = \frac{G mM}{ R^2} $$ with $G$ constant, $M$ mass of the Earth and $R$ distance between them. If you put these two expressions equal, you can get the right speed for a satellite circularly orbiting the Earth. $$ v =\sqrt{\frac{GM}{R}} $$ If you have a satellite's speed greater than the one calculated for a circular motion, you can see that in the radial direction the required centripetal force is greater than the actual Gravitational one. This means that gravity cannot keep the satellite orbiting around the Earth, you have a force term in radial direction and it will cause the satellite to go further from the Earth while it is rotating around it. On the contrary, if the speed is less than required, you still have a force in radial direction and it will cause the satellite to fall toward Earth's surface, while orbiting around it, and the orbit will become elliptical.
Bill N
JackIJackI
$\begingroup$ Nice answer. I hope you don't mind that I inserted a few clarifications. $\endgroup$
– Bill N
$\begingroup$ @BillN Thank you for your help in providing a good answer!!! $\endgroup$
– JackI
For example, I don't understand why the speed of a satellite moving in an orbit of radius 'a' around the Earth must be equal to $\sqrt{GM/a}.$
For the satellite to orbit at radius $a$, the Earth's gravitational field must exert a centripetal force $F_c$:
$$F_c=\frac{mv^2}{a}$$
This force is the gravitational force, so:
$$F_c=G\frac{mM}{a^2}$$
$$\frac{mv^2}{a}=G\frac{mM}{a^2}$$ $$\implies v=\sqrt\frac{GM}{a}\tag{1}$$
If the 'launch speed' (as you defined it) is higher, the satellite will move to a different orbit, until $(1)$ is satisfied again.
So here we assume, as per the OP, that $v_0 \neq \sqrt\frac{GM}{a}$.
The 'final orbit' $r$ is found from energy conservation (we assume gravity to be the only external force). At launch the total energy $T$ is ($v_0$ is launch velocity):
$$T=\frac{mv_0^2}{2}-\frac{GMm}{a}$$
In final orbit the total energy $T$ is:
$$T=\frac{GmM}{2r}-\frac{GmM}{r}=-\frac{GmM}{2r}$$
From the identity, $r$ can be calculated:
$$-\frac{GM}{2r}=\frac{v_0^2}{2}-\frac{GM}{a}$$ $$\frac{GM}{r}=\frac{2GM-av_0^2}{a}$$ $$\boxed{r=\frac{aGM}{2GM-av_0^2}}$$
If $v_0 > \sqrt\frac{GM}{a}$, then $r>a$, so the satellite moves to a higher orbit. It slows down because some of its kinetic energy is converted to potential energy.
If $v_0 < \sqrt\frac{GM}{a}$, then $r<a$, so the satellite moves to a higher orbit. It will speed up because some of its potential energy is converted to kinetic energy.
GertGert
$\begingroup$ First, you said that if the launch speed is higher, the satellite will move to a higher orbit. But, according to the equation when v is more then a should be less. Second, if I provide the satellite an initial velocity perpendicular to the gravitational force, then what would provide the satelite the force requored to move to a higher orbit. There is no radial component of initial velocity and the only force on the satelite os towards the center. Then, how will it move to a higher orbit? $\endgroup$
$\begingroup$ And,I'm not talking about launching the particle from earth with some initial velocity. I'm talking about taking it to some orbit of radius 'a', then providing it an initial velocity perpendicular to the radius but of magnitude not equal to $\sqrt{GM/a}$. Then, what will provide the satellite the force necessary to move to an orbit whose radi $\endgroup$
$\begingroup$ I've edited the post slightly. $\endgroup$
– Gert
I will compare circular motion in the case of a rotating frame and with the case of the Schwarzshild metric. In this general relativistic context we can examine what is meant by the centripetal force in this context for a particle in a rotating frame and a particle in a circular orbit around a gravitating mass. We can compare the two and see what is modified by general relativity, and what is interpreted by centripetal acceleration.
The metric for a rotating coordinate system with $d\phi~\rightarrow~d\phi~+~\omega dt$ is $$ ds^2~=~A(\omega,~r)\left(cdt~-~\frac{\omega r^2}{c^2A(\omega,~r)}d\phi\right)^2~-~dr^2~-~\frac{r^2}{A(\omega,~r)}d\phi^2~-~dz^2, $$ $$ A(\omega,~r)~=~(1~-~\omega^2r^2/c^2) $$ gives the Christoffel symbols $$ \Gamma^r_{tt}~=~\omega^2 r,~\Gamma^\phi_{rr}~=~-\frac{\omega}{r},~\Gamma^\phi_{r\phi}~=~-\frac{1}{r},~\Gamma^r_{\phi\phi}~=~r. $$ For circular motion we can set $\Gamma^\phi_{rr}~=~\Gamma^\phi_{r\phi}~=~0$. The geodesic equation of interest is of the form $$ \frac{d^2r}{ds^2}~+~\Gamma^r_{\phi\phi}\left(\frac{d\phi}{ds}\right)^2~=~0, $$ or equivalently $$ \frac{d^2r}{ds^2}~+~\omega^2 r\left(\frac{dt}{ds}\right)^2~+~r\left(\frac{d\phi}{ds}\right)^2~=~0. $$ This is similar to centripetal accleration.
To make the connection to Newtonian physics let us transform this to acceleration in the standard coordinates of an observer. We then have $$ \frac{d^2r}{ds^2}~=~\left(\frac{d^2r}{dt^2}\right)\left(\frac{dt}{ds}\right)^2, $$ which employs $dr/dt~=~dz/dt~=~0$ for circular motion. We use the metric with $dr~=~0$ $$ ds^2~=~A(\omega,~r)\left(cdt~-~\frac{\omega r^2}{c^2A(\omega,~r)}d\phi\right)^2~-~\frac{r^2}{A(\omega,~r)}d\phi^2, $$ so the term $dt/ds$ is seen in $$ \left(\frac{ds}{dt}\right)^2~=~A(\omega,~r)\left(1~-~\frac{\omega^2r^2}{A(\omega,~r)}\right)~-~\frac{\omega^2r^2}{A(\omega,~r)}, $$ so that $dt/ds$ is a form of Lorentz gamma factor $$ \gamma(\omega)~\dot=~\frac{dt}{ds}~=~\frac{1}{\sqrt{A(\omega,~r)\left(1~-~\frac{\omega^2r^2}{A(\omega,~r)}\right)~-~\frac{\omega^2r^2}{A(\omega,~r)}}}. $$ This then gives us a gamma factor modified form of the centripetal acceleration. Since it is evident the modified gamma factor divides out on both side this is centripetal acceleration for a particle fixed to a rotating frame. The question is then whether this applies to gravitation.
For gravitation we turn to the Schwarzschild metric $$ ds^2~=~(1~-~2m/r)dt^2~-~(1~-~2m/r)^{-1}dr^2~-~r^2(d\theta^2~+~sin^2\theta d\phi^2)~with~m~=~GM/c^2, $$ where for a circular orbit we have $dr~=~d\theta=-~0$ and $\theta~=~\pi/2$ so that $$ ds^2~=~(1~-~2m/r)dt^2~-~r^2d\phi^2. $$ Dividing through by $dt^2$ and letting $\omega~=~d\phi/dt$ gives $$ ds^2~=~[(1~-~2m/r)~-~r^2\omega^2]dt^2, $$ which gives a similar gamma factor $$ \gamma_m(\omega)~=~\frac{1}{\sqrt{1~-~2m/r~-~r^2\omega^2}}. $$ The Christoffel symbol relevant for calculation is $$ \Gamma^r_{tt}~=~\frac{m(r~-~2m)}{r^3}, $$ so that $$ \frac{d^2r}{ds^2}~+~\frac{m(r~-~2m)}{r^3}\left(\frac{dt}{ds}\right)^2~=~0. $$ It is evident that the $\gamma_m(\omega)$ divides out and this leaves the dynamical equation $$ \frac{d^2r}{dt^2}~+~\frac{m(r~-~2m)}{r^3}~=~0. $$ For $r~>>~2m$ this recovers Newton's second law with gravity.
It appears that in standard coordinates centripetal acceleration is the same. What is modified by general relativity is the nature of gravitation as a force interpreted in standard coordinates.
Lawrence B. CrowellLawrence B. Crowell
For al I know, the acceleration will still be perpendicular to its velocity at all instants. And, an always perpendicular acceleration is only capable of changing the direction of velocity continuously.
Well, let's see. For convenience, stipulate that at $t=0$, the particle is located at $r=R, \theta=\pi/2, \phi=0$ with velocity
$$\vec v(0)=v\;\hat{\boldsymbol{y}}$$
and acceleration
$$\vec a(0)=-\frac{GM}{R^2}\;\hat{\boldsymbol{x}}$$
The velocity at the next instant is then $$\vec v(0 + dt)=-\frac{GM}{R^2}dt\;\hat{\boldsymbol{x}} + v\;\hat{\boldsymbol{y}}$$
while the acceleration is
$$\vec a(0 + dt)=-\frac{GM}{R^2}\left(\hat{\boldsymbol{x}} + \frac{v}{R}\,dt\;\hat{\boldsymbol{y}}\right)$$
and so the dot product of these velocity and acceleration vectors is
$$\vec v(0 + dt)\cdot \vec a(0 + dt)=-\frac{GM}{R^2}\left[-\frac{GM}{R^2}+\frac{v^2}{R}\right]dt$$
which is zero only if
$$\frac{v^2}{R}=\frac{GM}{R^2}$$
$$v=\sqrt{\frac{GM}{R}}$$
Thus, it isn't the case that the acceleration will still be perpendicular to the velocity at all instants for arbitrary $v$.
Alfred CentauriAlfred Centauri
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research journal impact factor
In den Rahmen der finalen Bewertung zählt viele Eigenarten, damit ein möglichst gutes Ergebniss entsteht. Emanuela Reale. Clinical Research Journals Impact Factors List; Journal Name Journal Impact Factor* Citations Report; Clinical Investigation : 1.08 . It is always advised to submit your articles into a journal Journal impact factors are a metric that reflects that years' average number of citations for articles published in the last two years. Basic Definitions 2. About the Journal. All Rights Reserved. Journal Impact Factor List 2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2010 | 2009 | 2008 In der Pr… to find out other top journals and conference opportunities where you can submit your research paper or article to showcase the quality of your work. Impact factors measure the impact of a journal, not the impact of individual articles. 49 ACCOUNTS OF CHEMICAL RESEARCH 0001-4842 20.268 50 Nature Reviews Neurology 1759-4758 20.257. ASSOCIATE CHIEF EDITOR Jean-Marc SABATIER Chief Scientific Officer and Head of a Research Group France. Impact factor of journals 1. Publications2017 = 40 We cover the impact and detailed analysis of almost The Journal Impact Quartile of Review of Educational Research is Q1. $IF_x = \frac{Citations_{x-1} + Citations_{x-2}}{Publications_{x-1} + Publications_{x-2}}$. Geology (2 Year) metric. The higher the IF of the journal, the better it is ranked. We also present here the highest (best) and lowest (worst) impact Cancer Discovery. Pediatrics JCR® data may also serve advertisers interested in evaluating the potential of a specific journal. In market research, the impact factor provides quantitative evidence for editors and publishers for positioning their journals in relation to the competition–especially others in the same subject category, in a vertical rather than a horizontal or interdisciplinary comparison. All these details will be helpful when you want to select a journal or assess the quality of a journal. Daher macht es sich diese geniale Biologie Ihres Körpers zum … Cancer Research Journal Impact Factors | Springer Journal Metrics Reports 2019 Announcement of the latest impact factors from the Journal Citation Reports Researchers consider a number of factors in deciding where to publish their research, such as journal reputation, readership and community, speed of publication, and citations. Author can get information about international journal impact factor, proceedings (research papers) and information on upcoming events. Advertisement. It is often used to compare journals of the same category. Journal title Impact factor 5-yr impact factor; RSC Advances: 3.119: 3.098: Chemical Science: 9.346: 8.945: Nanoscale Advances: Not available : Not available: RSC Chemical Biology: Not available: Not available: Sign up to e-alerts for our Open Access products. Journal Impact Factor: 1.65 * ™ Frequency: Quarterly. They have been introduced as official research evaluation tools in several countries. 19+ million members; 135+ million publications ; 700k+ research projects; Join for free. About the Journal Impact Factor: SJIF 2019: 7.583. International Journal of Science and Research (IJSR) is a Open Access, Fully Refereed and Peer Reviewed International Journal. Questions (664) The JCR analysis released in 2020 is based on citations in 2019 to articles published in 2017 and 2018. The impact factor of a journal is calculated by dividing the number of current year citations to the source items published in that journal during the previous two years. Citations2017 = 60 Greenwood, D. C. (2007). 5 year Impact Factor. F1000 Research is not a journal, and doesn't want to be mistaken for a journal. Otorhinolaryngology Ecology, Evolution, Behavior and Systematics Submission; Archive ; Subscription; Advertise; Awards . 01. Scientific Education. CURRENT ISSUE. Impact factor (IF) is a measure of the number of times an average paper in a journal is cited, during a year. Citation Impact 1.835 - 2-year Impact Factor 2.179 - 5-year Impact Factor 1.098 - Source Normalized Impact per Paper (SNIP) 0.787 - SCImago Journal Rank (SJR) Usage 1,454,014 downloads Social Media Impact 1348 Altmetric mentions Show More. Publications2018 = 30 Its subject matter is the evaluation of activities concerned with scientific research, technological development and innovation … Find out more. In 2017, the Journal of Advertising Research achieved an Impact Factor of 2.328, demonstrating the journal's authority and importance within the industry.. Mathematics (miscellaneous) The aggregate journal impact factor for a subject category is calculated using the same method as the journal impact factor for a journal, but it also includes the number of citations for all journals in the category and the number of articles from all journals in the category. Der Impact Factor (IF) oder genauer der Journal Impact Factor (JIF), deutsch Impact-Faktor, ist eine errechnete Zahl, deren Höhe den Einfluss einer wissenschaftlichen Fachzeitschrift wiedergibt. SCImago Journal Rank (SJR) … Factor Full Journal Title 1 - 0007-9235 - 292.278 - CA-A CANCER JOURNAL FOR CLINICIANS 2 - 0028-4793 - 74.699 - NEW ENGLAND JOURNAL OF MEDICINE impact data over years, we compute and show the standard deviation. The Journal Impact Factor is the average number of citations received in the last year to articles published in the previous two years. We perform statistical impact analysis for various journals and conference to evaluate their impact trends. Journal ranking is widely used in academic circles in the evaluation of an academic journal's impact and quality. It cannot be computed until all publications in the previous year of 2018 are processed by the indexing agency. What is Impact Factor? Normally, journals display their Impact Factor and CiteScore on their About Page. ONLINE PAYPAL PAYMENT. Environmental Chemistry However, very few journals with high impact factor journals highly recognized by the scientific community. Ranked within the top 10 Business/Finance journals in the world, the Journal of Accounting Research is pleased to announce a 2018 Impact Factor of 4.481, and a five-year Impact Factor of 6.056. Data in the following tables were calculated by Clarivate Analytics and are collected and updated annually for full calendar years. Er dient zum bibliometrischen Vergleich verschiedener Zeitschriften. Citation Impact 3.924 - 2-year Impact Factor 4.043 - 5-year Impact Factor 1.137 - Source Normalized Impact per Paper (SNIP) 1.436 - SCImago Journal Rank (SJR) Usage 1,136,622 Downloads 2,375 Altmetric mentions Impact factor gives the approximate idea about how prestigious a particular journal is in its field. Therefore, it is always recommended to cite or refer the articles from the top journals (Which are basically the one having high impact factor), $IF_{2019}= \frac{Citations_{2018} + Citations_{2017}}{Publications_{2018} + Publications_{2017}}$, Ecology, Evolution, Behavior and Systematics, Neuropsychology and Physiological Psychology, Agricultural and Biological Sciences (miscellaneous), Immunology and Microbiology (miscellaneous), Earth and Planetary Sciences (miscellaneous), Electronic, Optical and Magnetic Materials, Political Science and International Relations, Pediatrics, Perinatology and Child Health, Business, Management and Accounting (miscellaneous), Management Science and Operations Research, Economics, Econometrics and Finance (miscellaneous), Public Health, Environmental and Occupational Health, Geotechnical Engineering and Engineering Geology, Computer Graphics and Computer-Aided Design, Physical Therapy, Sports Therapy and Rehabilitation, Renewable Energy, Sustainability and the Environment, Pharmacology, Toxicology and Pharmaceutics (miscellaneous), Tourism, Leisure and Hospitality Management, Organizational Behavior and Human Resource Management, Biochemistry, Genetics and Molecular Biology (miscellaneous), Critical Care and Intensive Care Medicine. Resources . It is frequently used as a Metric for the relative importance of a journal within its field; journals with higher Journal Impact are often deemed to be more important than … Impact Factor: The impact factor is a measure of the frequency in which the average article in a journal is cited in a particular year. The Journal Impact of an academic journal is a scientometric Metric that reflects the yearly average number of citations that recent articles published in a given journal received. For Author For Referee Open Access. Guidelines . The Journal Impact 2019-2020 of Cancer Research is 7.070, which is just updated in 2020.Compared with historical Journal Impact data, the Metric 2019 of Cancer Research dropped by 9.82 %.The Journal Impact Quartile of Cancer Research is Q1.The Journal Impact of an academic journal is a scientometric Metric that reflects the yearly average number of citations that recent articles published … The above metrics help you to better correlate and judge the all the major areas and disciplines. For a given year, the Journal's IF is computed as the number of cites, received in that throughout the year, of the scientific articles or papers published in that particular journal during the two previous years, 29.497. It should be used to compare the journals withing a single discipline. It is determined by calculating an average number of citations received by the selected articles in that journal within the last few years. Bioxbio.com includes the journal factors, ISSN, number of articles and other detailed information of over 11000 journals in different fields. Of 12,298 journals, only 239 titles, or 1.9% of the journals tracked by JCR, have a 2017 impact factor of 10 or higher. Topics include etiology, pathogenesis, management, and prevention of diabetes, as well as associated complications such as nephropathy. This factor is used for evaluating the prestige of journals. The news suggests that the Journal Citation Reports (JCR) database tracked all impact factors for more than 12000 journals. It is measured each year by the Web of Science Group and reported in the Journal Citation Reports (JCR). 1.57 (5 Yr Journal Impact Factor) Citations Report: Archives of Clinical Microbiology : 0.72 . Pharmaceutical Science About this journal. Journal Institute Society CR@NPG Editorial Office . Neuropsychology and Physiological Psychology Higher the Impact factor, higher is the ranking of the journal. (2009). Tools to Measure Journal Impact (Impact Factor) Journal Citation Reports . Discover the world's research. Journal impact search engine. The higher the JIF, the better it is ranked. Announcement of the latest impact factors from the Journal Citation Reports. The paper copy of the journal occasionally includes this data and will always provide current contact information. This will help you Thed van Leeuwen . It is proposed by the founder of the Institute for Scientific Information, Eugene Eli Garfield and is being regularly calculated beginning from 1975 for all the journals registered in the The acceptance and rejection rates of journals can be a determining factor. By using this metric you can basically evaluate and compare the journals in similar subject categories to identify their importance. Die betreffende Auswirkung von journal impact factor hormone and metabolic research kommt natürlicherweise durch jenes besondere Zusammenspiel der einzelnen Inhaltsstoffe zu Stande. Researchers consider a number of factors in deciding where to publish their research, such as journal reputation, readership and community, speed of publication, and citations. For example, an Artificial Intelligence journal's Impact Factor cannot be compared with a journal from the Management domain. Reliability of journal impact factor rankings. Many countries may, for example, not want to have details about their scientists and projects listed. We perform various analysis on the data produced by SCImago. average impact index for the last three and five years. The Journal Citation Report provides Quartile rankings derived for each journal in each of its subject categories according to which quartile of the Impact Factor (IF) distribution the journal occupies for that subject category. This journal is a member of and subscribes to the principles of the Committee on Publication Ethics (COPE). Sanofi-Cell Research Outstanding Original Paper award CSH Asia Poster Award. All Engineering Fields Applied sciences, Business and Management Physics, Chemistry, mathematics, Biology, Humanities & Social Science, Medical Sciences. It is important to note here that the 2018's IF is published in the year 2019. Computer Vision and Pattern Recognition impact of any particular journal/conference. However, you have to be careful because predatory journals post fake Impact Factors too. ISSN: 1001-0602 EISSN: 1748-7838 2018 2019 impact factor 20.507* (Clarivate Analytics, 2020) About the cover. Dabei handelt es sich um nach verschiedenen Kriterien ausgewählte, meist international orientierte Zeitschriften, die nur Artikel veröffentlichen, welche zuvor das Peer-Review-Verfahren durchla… Emergency Nursing Julia Melkers. The Journal Impact 2019-2020 of Environmental Research is 5.200, which is just updated in 2020.Compared with historical Journal Impact data, the Metric 2019 of Environmental Research grew by 2.77 %.The Journal Impact Quartile of Environmental Research is Q1.The Journal Impact of an academic journal is a scientometric Metric that reflects the yearly average number of citations that … Zeitschriften müssen aber mindestens drei Jahre im Web of Science vertreten sein, damit ein Journal Impact Factor berechnet werden kann. Impact Factor. Journal of Surgery and Research journals have high impact factor in 2018, 2019 and 2020. International Scientific Journal & Country Ranking. Learn about publishing Open Access with us Journal metrics 2.507 (2019) Impact factor 2.643 (2019) Five year impact factor 43 days Submission to first decision 225 days Submission to acceptance 92,964 (2019) Downloads You can also check in detail analysis (like five years average, highest impact in the last five years, etc.) Journals. The top 5% of journals have impact factors approximately equal to or greater than 6 (610 journals or 4.9% of the journals tracked by JCR). However, it is important to note that we respects the various scientific communities submitting their information by allowing those communities to determine what they want to have displayed globally and what they want to keep confidential. Editors. To search the impact factor of any Journal or Conference, you can query by its title or ISSN. For an example, to find the impact factor of a "ABC" journal for the year 2019, we would compute: For example, if the Journal have the following citations and publications value: Publication Fees. You can find the impact factor of thousands of journals on this website. of a particular item, by clicking on the same. Impact Factor: 2019: 2.730 The Impact Factor measures the average number of citations received in a particular year by papers published in the journal during the two preceding years. Journal Impact Factor List 2016 | 2015 | 2014 | 2013 | 2012 | 2011 | 2010 | 2009 | 2008 4.344. About This Journal. It was found that approximately only 1.9% of the journals had a 2017 impact factor of 10 or higher. Monthly archive. 01 . So far, we have covered our impact factor analysis for the years 2018, 2017, 2016, 2015, 2014, and 2013. Research Impact Factors An Impact Factor is one measure of the relative importance of a journal, individual publication, or researcher to literature and research. Timing ... Review articles are often more highly cited than original research articles: consider a journal's source data by document type. Low acceptance rate, high rejection rate journals are considered the best and most prestigious journals as the acceptance criteria is of high quality standard. Bitte beachten Sie: Bei Open-Access-Zeitschriften handelt es sich häufig um Neugründungen. Management of Technology and Innovation The European Educational Research Journal (EERJ) is a peer-reviewed scientific journal interested in the changing landscape of education research across Europe. We covered the following categories: Endocrinology, Diabetes and Metabolism Learn more about journal metrics: Research metrics: The Journal Impact Factor Journal. Volume 31, No 1, Jan 2021. The Journal publishes original articles, review articles, editorials, letters to the editor, short communication, case reports in all the field of surgery. Der Impact-Faktor ist kein Maß für die Qualität der Artikel einer Zeitschrift, sondern gibt Auskunft darüber, wie oft die Artikel einer bestimmten Zeitschrift in anderen wissenschaftlichen Publikationen durchschnittlich pro Jahr zitiertwerden. When it comes to journal publications, many publications are available in the area of AI and Machine Learning(ML). All the journal pages have pointers to Web pages of the publishers which are integrated into the CiteFactor stream pages. Journal impact search engine. In recognition of this, enjoy free access to top cited research that contributes to both the 2018 Impact Factor, and five-year Impact Factor. Journal Citation Reports (JCR). The data from the Scopus® database can also be found at resurchify.com. Cancer Research Print ISSN: 0008-5472 Journal of Cancer Research ISSN: 0099-7013 American Journal of Cancer ISSN: 0099-7374. Top 100 Impact Factor Journals of Science 2016 CIIT, Library Information Services, Islamabad. CHIEF EDITOR. 1.75 (5 Yr Journal Impact Factor) Citations Report: Journal of Clinical & Experimental Pathology : 0.50 . Journal of Current Research in Food Science, International Journal of Multidisciplinary Trends, International Journal of Advanced Research in Medicine, International Journal of Medical Ophthalmology. In recognition of this, enjoy free access to top cited research that contributes to both the 2018 Impact Factor, and five-year Impact Factor. To show the variation of Research Evaluation is an interdisciplinary peer-reviewed, international journal. Ranked within the top 10 Business/Finance journals in the world, the Journal of Accounting Research is pleased to announce a 2018 Impact Factor of 4.481, and a five-year Impact Factor of 6.056. Impact Factor is a measure of the importance of a journal. Pharmacy Researchers consider a number of factors in deciding where to publish their research, such as journal reputation, readership and community, speed of publication, and citations. In market research, the impact factor provides quantitative evidence for editors and publishers for positioning their journals in relation to the competition—especially others in the same subject category, in a vertical rather than a horizontal or intradisciplinary comparison. Original research, reviews, symposium reports, hypothesis formation and commentaries are all considered for publication. Journal Impact factor 2020. Howard, J. dividing it by the number of articles published in that journal in the previous two years.years. We perform all the analysis on Cites/Doc. While the standard Impact Factor measures citations to journal articles published within a 2-year period, the 5-year Impact Factor evaluates citations to journal articles published within a 5-year period to provide a measure of a journal's longer-term influence. for the last few years. In this article, ilovephd provides the top High Impact Factor Artificial Intelligence(AI) Journals with H5 index detail. Journal name: Research Inventy Publisher: RI publishers Online ISSN: 2278-4721 Print ISSN: 2319-6483 Publication frequency: 12 Issue per year Publication Since: 2012. The EERJ publishes education research papers and special issues which include a reflection on how the European context and other related global or regional dynamics shape their educational research … 2.571. The impact factor (IF), also named as Journal Impact Factor (JIF) is a metric used to evaluate the importance of a Journal. Journal Citation Reports (Clarivate Analytics, 2020) Time to first decision: 2.7 weeks Citations2018 = 80 Copyright © 2015 - 2021. Many journals and societies have web pages that give publication data and style requirements and often includes acceptance/rejection rates. Whether a journal is indexed in the major indexing/abstracting service in the field is another criteria that can be used to assess the worth and quality of a journal. Editorial Board. Cancer Immunology Research. Oncology (nursing) Berücksichtigt werden alle Zitationen aus Zeitschriften, die selbst im Web of Science enthalten sind. Cancer Epidemiology, Biomarkers & Prevention. However, we will update soon all our analysis for the year 2019 also, once the data is available. It is denoted as a ratio between citations and recent citable items published. Rosane Cavalcante Fragoso, Brasil. This is to show the credibility and worthiness of your research articles and your work. The Chronicle of Higher Education, 55(19), A1. J. Christopher Westland; Publishing model Hybrid (Transformative Journal). The CiteFactor server provides indexing of major international journals and proceedings. We perform the in-depth analysis method. Typically, journals with more review articles or papers are able to achieve higher JIF. Inclusion in our database through which all journals can be registered, not just those deemed appropriate by an arbitrary group of individuals. articles based on the referenced journal articles. The Indian Journal of Medical Research, 127(1), 4-6. You can find here the Journal of International Medical Research is a peer-reviewed open access journal which focuses on original clinical and preclinical research, systematic and perspective reviews, meta-analyses, pilot studies and case reports, with every article accepted by peer review given a full technical edit to make all papers highly accessible to the international medical community. Complementary and Manual Therapy Journal of Obesity & Weight Loss Therapy is an open access journal and aims to publish most complete and reliable source of information on the discoveries and current developments in the fields of Classical Childhood obesity, Obesity causes and disorders Morbid Obesity, Obesity Diabetes, Endocrine disorders, Obesity and weight loss, Obesity adolescence, Metabolic syndrome, etc. Country Amount … Journal Impact Factor - Science topic A quantitative measure of the frequency on average with which articles in a journal have been cited in a given period of time. Humanities journals confront identity crisis. Numerous critiques have been made regarding the use of impact factors. 3.406. Bioxbio.com includes the journal factors, ISSN, number of articles and other detailed information of over 11000 journals in different fields. Announcement of the latest impact factors from the Journal Citation Reports. BMC Medical Research Methodology, 7(48), 48. Gerontology Impact Factor: 2019: 5.715 The Impact Factor measures the average number of citations received in a particular year by papers published in the journal during the two preceding years. JCR data may also serve advertisers interested in evaluating the potential of a specific journal. About the journal. There is also a more general debate on the validity of the impact factor as a measure of journal importance and the effect of policies that editors may adopt to boost their impact factor (perhaps to the detriment of readers and wri… 3 Rank Journal Title ISSN Impact Factor 51 EUROPEAN HEART JOURNAL 0195-668X 20.212 52 NATURE CELL BIOLOGY 1465-7392 20.060 53 Cancer Discovery 2159-8274 20.011 54 … Pulmonary and Respiratory Medicine Impact Factor: Just released – Clarivate Web of Science Journal Citation Reports™ (JCR) 2020 Impact Factors for the AACR Journals*: Journal. The top 5% of journals had impact factors approximately equal to or greater than 6. Journal Impact Factor: 2.386 (5 year impact 2.695) Indexed online: PubMed and PubMed Central (J Pain Res) December 2020; November 2020; October 2020; September 2020; … A 2007 study noted that the most fundamental flaw is that impact factors present the mean of data that are not normally distributed, and suggested that it would be more appropriate to present the medianof these data. The impact factor (IF) is a measure of the yearly average number of citations to recent articles published in that journal. Small Animals Health Professions (miscellaneous) It helps to measure the relative importance of journals within particular areas and to compare the journals within the same areas. Scopus database includes the information of more than 15,000 journals from different fields from around 4,000 publishers and also covers around 1000 open access journals. Subjects Category. Eine Zusammenfassung unserer favoritisierten Journal of labor research impact factor. Also known as impact index. Only Open Access Journals Only SciELO Journals Only WoS Journals IF2016= (80 + 60) / (30 + 40) = 2 The calculation of IF is based on a period of two years and computes as dividing the number of times articles published in that journal were cited by the total number of articles which are citable. You can either refer to the Journal Citation Reports (JCR) or the Scopus® database to find the impact factor of the journal. Impact Factor Calculations IF 2018 = (Citation in 2017 + Citations in 2016)/(Papers Published in 2017+ Papers Published in 2016) Journal Impact Factor List 2020 ID Print-ISSN J. I. Fast Publication/Impact factor Journal (Click) IJMCE RECOMMENDATION. Der Journal Impact Factor wird auf Grundlage der im Web of Science (einem kommerziellen Produkt von Thomson Scientific) enthaltenen Zeitschriften berechnet und jährlich in den Journal Citation Reports veröffentlicht. 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Transition to turbulence in the rotating-disk boundary-layer flow with stationary vortices
E. Appelquist, P. Schlatter, P. H. Alfredsson, R. J. Lingwood
Journal: Journal of Fluid Mechanics / Volume 836 / 10 February 2018
Print publication: 10 February 2018
This paper proposes a resolution to the conundrum of the roles of convective and absolute instability in transition of the rotating-disk boundary layer. It also draws some comparison with swept-wing flows. Direct numerical simulations based on the incompressible Navier–Stokes equations of the flow over the surface of a rotating disk with modelled roughness elements are presented. The rotating-disk flow has been of particular interest for stability and transition research since the work by Lingwood (J. Fluid Mech., vol. 299, 1995, pp. 17–33) where an absolute instability was found. Here stationary disturbances develop from roughness elements on the disk and are followed from the linear stage, growing to saturation and finally transitioning to turbulence. Several simulations are presented with varying disturbance amplitudes. The lowest amplitude corresponds approximately to the experiment by Imayama et al. (J. Fluid Mech., vol. 745, 2014a, pp. 132–163). For all cases, the primary instability was found to be convectively unstable, and secondary modes were found to be triggered spontaneously while the flow was developing. The secondary modes further stayed within the domain, and an explanation for this is a proposed globally unstable secondary instability. For the low-amplitude roughness cases, the disturbances propagate beyond the threshold for secondary global instability before becoming turbulent, and for the high-amplitude roughness cases the transition scenario gives a turbulent flow directly at the critical Reynolds number for the secondary global instability. These results correspond to the theory of Pier (J. Engng Maths, vol. 57, 2007, pp. 237–251) predicting a secondary absolute instability. In our simulations, high temporal frequencies were found to grow with a large amplification rate where the secondary global instability occurred. For smaller radial positions, low-frequency secondary instabilities were observed, tripped by the global instability.
Job strain as a risk factor for clinical depression: systematic review and meta-analysis with additional individual participant data
I. E. H. Madsen, S. T. Nyberg, L. L. Magnusson Hanson, J. E. Ferrie, K. Ahola, L. Alfredsson, G. D. Batty, J. B. Bjorner, M. Borritz, H. Burr, J.-F. Chastang, R. de Graaf, N. Dragano, M. Hamer, M. Jokela, A. Knutsson, M. Koskenvuo, A. Koskinen, C. Leineweber, I. Niedhammer, M. L. Nielsen, M. Nordin, T. Oksanen, J. H. Pejtersen, J. Pentti, I. Plaisier, P. Salo, A. Singh-Manoux, S. Suominen, M. ten Have, T. Theorell, S. Toppinen-Tanner, J. Vahtera, A. Väänänen, P. J. M. Westerholm, H. Westerlund, E. I. Fransson, K. Heikkilä, M. Virtanen, R. Rugulies, M. Kivimäki, for the IPD-Work Consortium
Journal: Psychological Medicine / Volume 47 / Issue 8 / June 2017
Adverse psychosocial working environments characterized by job strain (the combination of high demands and low control at work) are associated with an increased risk of depressive symptoms among employees, but evidence on clinically diagnosed depression is scarce. We examined job strain as a risk factor for clinical depression.
We identified published cohort studies from a systematic literature search in PubMed and PsycNET and obtained 14 cohort studies with unpublished individual-level data from the Individual-Participant-Data Meta-analysis in Working Populations (IPD-Work) Consortium. Summary estimates of the association were obtained using random-effects models. Individual-level data analyses were based on a pre-published study protocol.
We included six published studies with a total of 27 461 individuals and 914 incident cases of clinical depression. From unpublished datasets we included 120 221 individuals and 982 first episodes of hospital-treated clinical depression. Job strain was associated with an increased risk of clinical depression in both published [relative risk (RR) = 1.77, 95% confidence interval (CI) 1.47–2.13] and unpublished datasets (RR = 1.27, 95% CI 1.04–1.55). Further individual participant analyses showed a similar association across sociodemographic subgroups and after excluding individuals with baseline somatic disease. The association was unchanged when excluding individuals with baseline depressive symptoms (RR = 1.25, 95% CI 0.94–1.65), but attenuated on adjustment for a continuous depressive symptoms score (RR = 1.03, 95% CI 0.81–1.32).
Job strain may precipitate clinical depression among employees. Future intervention studies should test whether job strain is a modifiable risk factor for depression.
On the global nonlinear instability of the rotating-disk flow over a finite domain
Journal: Journal of Fluid Mechanics / Volume 803 / 25 September 2016
Print publication: 25 September 2016
Direct numerical simulations based on the incompressible nonlinear Navier–Stokes equations of the flow over the surface of a rotating disk have been conducted. An impulsive disturbance was introduced and its development as it travelled radially outwards and ultimately transitioned to turbulence has been analysed. Of particular interest was whether the nonlinear stability is related to the linear stability properties. Specifically three disk-edge conditions were considered; (i) a sponge region forcing the flow back to laminar flow, (ii) a disk edge, where the disk was assumed to be infinitely thin and (iii) a physically realistic disk edge of finite thickness. This work expands on the linear simulations presented by Appelquist et al. (J. Fluid. Mech., vol. 765, 2015, pp. 612–631), where, for case (i), this configuration was shown to be globally linearly unstable when the sponge region effectively models the influence of the turbulence on the flow field. In contrast, case (ii) was mentioned there to be linearly globally stable, and here, where nonlinearity is included, it is shown that both cases (ii) and (iii) are nonlinearly globally unstable. The simulations show that the flow can be globally linearly stable if the linear wavepacket has a positive front velocity. However, in the same flow field, a nonlinear global instability can emerge, which is shown to depend on the outer turbulent region generating a linear inward-travelling mode that sustains a transition front within the domain. The results show that the front position does not approach the critical Reynolds number for the local absolute instability, $R=507$ . Instead, the front approaches $R=583$ and both the temporal frequency and spatial growth rate correspond to a global mode originating at this position.
Global linear instability of the rotating-disk flow investigated through simulations
Numerical simulations of the flow developing on the surface of a rotating disk are presented based on the linearized incompressible Navier–Stokes equations. The boundary-layer flow is perturbed by an impulsive disturbance within a linear global framework, and the effect of downstream turbulence is modelled by a damping region further downstream. In addition to the outward-travelling modes, inward-travelling disturbances excited at the radial end of the simulated linear region, $r_{end}$ , by the modelled turbulence are included within the simulations, potentially allowing absolute instability to develop. During early times the flow shows traditional convective behaviour, with the total energy slowly decaying in time. However, after the disturbances have reached $r_{end}$ , the energy evolution reaches a turning point and, if the location of $r_{end}$ is at a Reynolds number larger than approximately $R=594$ (radius non-dimensionalized by $\sqrt{{\it\nu}/{\rm\Omega}^{\ast }}$ , where ${\it\nu}$ is the kinematic viscosity and ${\rm\Omega}^{\ast }$ is the rotation rate of the disk), there will be global temporal growth. The global frequency and mode shape are clearly imposed by the conditions at $r_{end}$ . Our results suggest that the linearized Ginzburg–Landau model by Healey (J. Fluid Mech., vol. 663, 2010, pp. 148–159) captures the (linear) physics of the developing rotating-disk flow, showing that there is linear global instability provided the Reynolds number of $r_{end}$ is sufficiently larger than the critical Reynolds number for the onset of absolute instability.
Obtaining accurate mean velocity measurements in high Reynolds number turbulent boundary layers using Pitot tubes
S. C. C. Bailey, M. Hultmark, J. P. Monty, P. H. Alfredsson, M. S. Chong, R. D. Duncan, J. H. M. Fransson, N. Hutchins, I. Marusic, B. J. McKeon, H. M. Nagib, R. Örlü, A. Segalini, A. J. Smits, R. Vinuesa
Journal: Journal of Fluid Mechanics / Volume 715 / 25 January 2013
Print publication: 25 January 2013
This article reports on one component of a larger study on measurement of the zero-pressure-gradient turbulent flat plate boundary layer, in which a detailed investigation was conducted of the suite of corrections required for mean velocity measurements performed using Pitot tubes. In particular, the corrections for velocity shear across the tube and for blockage effects which occur when the tube is in close proximity to the wall were investigated using measurements from Pitot tubes of five different diameters, in two different facilities, and at five different Reynolds numbers ranging from ${\mathit{Re}}_{\theta } = 11\hspace{0.167em} 100$ to 67 000. Only small differences were found amongst commonly used corrections for velocity shear, but improvements were found for existing near-wall proximity corrections. Corrections for the nonlinear averaging of the velocity fluctuations were also investigated, and the results compared to hot-wire data taken as part of the same measurement campaign. The streamwise turbulence-intensity correction was found to be of comparable magnitude to that of the shear correction, and found to bring the hot-wire and Pitot results into closer agreement when applied to the data, along with the other corrections discussed and refined here.
Flow regimes in a plane Couette flow with system rotation
T. TSUKAHARA, N. TILLMARK, P. H. ALFREDSSON
Journal: Journal of Fluid Mechanics / Volume 648 / 10 April 2010
Published online by Cambridge University Press: 07 April 2010, pp. 5-33
Print publication: 10 April 2010
Flow states in plane Couette flow in a spanwise rotating frame of reference have been mapped experimentally in the parameter space spanned by the Reynolds number and rotation rate. Depending on the direction of rotation, the flow is either stabilized or destabilized. The experiments were made through flow visualization in a Couette flow apparatus mounted on a rotating table, where reflected flakes are mixed with the water to visualize the flow. Both short- and long-time exposures have been used: the short-time exposure gives an instantaneous picture of the turbulent flow field, whereas the long-time exposure averages the small, rapidly varying scales and gives a clearer representation of the large scales. A correlation technique involving the light intensity of the photographs made it possible to obtain, in an objective manner, both the spanwise and streamwise wavelengths of the flow structures. During these experiments 17 different flow regimes have been identified, both laminar and turbulent with and without roll cells, as well as states that can be described as transitional, i.e. states that contain both laminar and turbulent regions at the same time. Many of these flow states seem to be similar to those observed in Taylor–Couette flow.
Streamwise evolution of longitudinal vortices in a turbulent boundary layer
OLA LÖGDBERG, JENS H. M. FRANSSON, P. HENRIK ALFREDSSON
Journal: Journal of Fluid Mechanics / Volume 623 / 25 March 2009
Published online by Cambridge University Press: 06 March 2009, pp. 27-58
Print publication: 25 March 2009
In this experimental study both smoke visualization and three-component hot-wire measurements have been performed in order to characterize the streamwise evolution of longitudinal counter-rotating vortices in a turbulent boundary layer. The vortices were generated by means of vortex generators (VGs) in different configurations. Both single pairs and arrays in a natural setting as well as in yaw have been considered. Moreover three different vortex blade heights h, with the spacing d and the distance to the neighbouring vortex pair D for the array configuration, were studied keeping the same d/h and D/h ratios. It is shown that the vortex core paths scale with h in the streamwise direction and with D and h in the spanwise and wall-normal directions, respectively. A new peculiar 'hooklike' vortex core motion, seen in the cross-flow plane, has been identified in the far region, starting around 200h and 50h for the pair and the array configuration, respectively. This behaviour is explained in the paper. Furthermore the experimental data indicate that the vortex paths asymptote to a prescribed location in the cross-flow plane, which first was stated as a hypothesis and later verified. This observation goes against previously reported numerical results based on inviscid theory. An account for the important viscous effects is taken in a pseudo-viscous vortex model which is able to capture the streamwise core evolution throughout the measurement region down to 450h. Finally, the effect of yawing is reported, and it is shown that spanwise-averaged quantities such as the shape factor and the circulation are hardly perceptible. However, the evolution of the vortex cores are different both between the pair and the array configuration and in the natural setting versus the case with yaw. From a general point of view the present paper reports on fundamental results concerning the vortex evolution in a fully developed turbulent boundary layer.
Pressure statistics and their scaling in high-Reynolds-number turbulent boundary layers
Y. TSUJI, J. H. M. FRANSSON, P. H. ALFREDSSON, A. V. JOHANSSON
Journal: Journal of Fluid Mechanics / Volume 585 / 25 August 2007
Published online by Cambridge University Press: 07 August 2007, pp. 1-40
Print publication: 25 August 2007
Pressure fluctuations are an important ingredient in turbulence, e.g. in the pressure strain terms which redistribute turbulence among the different fluctuating velocity components. The variation of the pressure fluctuations inside a turbulent boundary layer has hitherto been out of reach of experimental determination. The mechanisms of non-local pressure-related coupling between the different regions of the boundary layer have therefore remained poorly understood. One reason for this is the difficulty inherent in measuring the fluctuating pressure. We have developed a new technique to measure pressure fluctuations. In the present study, both mean and fluctuating pressure, wall pressure, and streamwise velocity have been measured simultaneously in turbulent boundary layers up to Reynolds numbers based on the momentum thickness Rθ ≃ 20000. Results on mean and fluctuation distributions, spectra, Reynolds number dependence, and correlation functions are reported. Also, an attempt is made to test, for the first time, the existence of Kolmogorov's -7/3 power-law scaling of the pressure spectrum in the limit of high Reynolds numbers in a turbulent boundary layer.
Transition induced by free-stream turbulence
J. H. M. FRANSSON, M. MATSUBARA, P. H. ALFREDSSON
Published online by Cambridge University Press: 09 March 2005, pp. 1-25
Free-stream turbulence (FST) is perhaps the most important source inducing by-pass transition in boundary layer flows. The present study describes the initial energy growth of streamwise-oriented disturbances in the boundary layer originating from the presence of FST with intensities between 1.4% and 6.7%, but the study is mainly focused on the modelling of the transition zone. For this study three passive and one active turbulence-generating grids were used. The active grid was used in order to vary the turbulence intensity ($Tu$) without changing the setup in the test section. It is shown that the initial disturbance energy in the boundary layer is proportional to $Tu^2$. The experiments also show that the energy grows in linear proportion to the Reynolds number based on the downstream distance. Furthermore the transitional Reynolds number is shown to be inversely proportional to $Tu^2$ for the whole range of $Tu$ studied. The intermittency in the transitional zone was determined and it was shown that the intermittency function has a universal shape if the downstream distance is scaled with the length of the transition zone. The Reynolds number based on this transition zone length was found to increase linearly with the transition Reynolds number; however it was also noted that this non-dimensional length has a minimum value. With these results we were able to formulate an expression for the spot production rate which has a better physical base than previous models.
On the disturbance growth in an asymptotic suction boundary layer
J. H. M. FRANSSON, P. H. ALFREDSSON
Journal: Journal of Fluid Mechanics / Volume 482 / 10 May 2003
Published online by Cambridge University Press: 13 May 2003, pp. 51-90
Print publication: 10 May 2003
An experimental and theoretical study on the effect of boundary layer suction on the laminar–turbulent transition process has been carried out. In the study an asymptotic suction boundary layer was established in a wind tunnel with a free-stream velocity of 5.0 m s$^{-1}$. Wall-normal suction (suction velocity 1.44 cm s$^{-1}$) was applied over a large area and the boundary layer was nearly constant over a length of 1800 mm. Measurements were made both with and without suction so comparisons between the two cases could easily be made. Measurements of the development of the mean velocity distribution showed good agreement with theory. The Reynolds number based on the displacement thickness for the suction boundary layer was 347. Experiments on both the development of forced Tollmien–Schlichting (TS) waves and boundary layer disturbances introduced by free-stream turbulence were carried out. Spatial linear stability calculations for TS-waves, where the wall-normal velocity component is accounted for, were carried out for comparison with the experiments. This comparison shows satisfactory agreement even though the stability of the asymptotic suction profile is somewhat overpredicted by the theory. Free-stream turbulence (FST) was generated by three different grids, giving turbulence intensities at the leading edge of the plate between 1.4% and 4.0%. The FST induces disturbances in the boundary layer and it was shown that for the present suction rate the disturbance level inside the boundary layer is constant and becomes proportional to the FST intensity. In all cases transition was prevented when suction was applied whereas without suction the two highest levels of grid turbulence gave rise to transition. Despite a twofold reduction in the boundary layer thickness in the suction case compared to the no suction case the spanwise scale of the streaky structures was almost constant.
Disturbance growth in boundary layers subjected to free-stream turbulence
M. MATSUBARA, P. H. ALFREDSSON
This paper aims at a description of boundary-layer flow which is subjected to free-stream turbulence in the range from 1–6% and is based on both flow visualization results and extensive hot-wire measurements. Such flows develop streamwise elongated regions of high and low streamwise velocity which seem to lead to secondary instability and breakdown to turbulence. The initial growth of the streaky structures is found to be closely related to algebraic or transient growth theory. The data have been used to determine streamwise and spanwise scales of the streaky structures. Both the flow visualization and the hot-wire measurements show that close to the leading edge the spanwise scale is large as compared to the boundary-layer thickness, but further downstream the spanwise scale approaches the boundary-layer thickness. Wavenumber spectra in both the streamwise and the spanwise directions were calculated. A scaling for the streamwise structure of the disturbance was found, which allows us to collapse the spectra from different downstream positions. The scaling combines the facts that the streaky structures increase their streamwise length in the downstream direction which becomes proportional to the boundary-layer thickness and that the energy growth is algebraic, close to proportional to the downstream distance.
Secondary instability in rotating channel flow
Published online by Cambridge University Press: 10 August 1998, pp. 27-50
Experiments on rotating channel flow, where both the primary (induced by a Coriolis instability) and the secondary instability are triggered independently, are described, focusing on the development of a secondary instability consisting of high-frequency travelling waves and their subsequent breakdown. Detailed hot-wire velocity measurements of the secondary disturbance are made and the phase speed and growth rate for various frequencies are determined. It is shown that the frequency of highest growth rate is close to that which is observed for naturally developing flow. Some information on the later stages in the transition process is obtained from frequency spectra, which show that interaction between various modes gives rise to stochastic low-frequency disturbances, which may play an important role in the transition process. A theoretical model of the disturbance structure is described which is used to explain some of the measured results and also allows the determination of the disturbance cross-stream flow field from only streamwise velocity measurements.
An investigation of turbulent plane Couette flow at low Reynolds numbers
Knut H. Bech, Nils Tillmark, P. Henrik Alfredsson, Helge I. Andersson
The turbulent structure in plane Couette flow at low Reynolds numbers is studied using data obtained both from numerical simulation and physical experiments. It is shown that the near-wall turbulence structure is quite similar to what has earlier been found in plane Poiseuille flow; however, there are also some large differences especially regarding Reynolds stress production. The commonly held view that the maximum in Reynolds stress close to the wall in Poiseuille and boundary layer flows is due to the turbulence-generating events must be modified as plane Couette flow does not exhibit such a maximum, although the near-wall coherent structures are quite similar. For two-dimensional mean flow, turbulence production occurs only for the streamwise fluctuations, and the present study shows the importance of the pressure—strain redistribution in connection with the near-wall coherent events.
Experiments in a boundary layer subjected to free stream turbulence. Part 1. Boundary layer structure and receptivity
K. J. A. Westin, A. V. Boiko, B. G. B. Klingmann, V. V. Kozlov, P. H. Alfredsson
Journal: Journal of Fluid Mechanics / Volume 281 / 25 December 1994
Print publication: 25 December 1994
The modification of the mean and fluctuating characteristics of a flat-plate boundary layer subjected to nearly isotropic free stream turbulence (FST) is studied experimentally using hot-wire anemometry. The study is focussed on the region upstream of the transition onset, where the fluctuations inside the boundary layer are dominated by elongated flow structures which grow downstream both in amplitude and length. Their downstream development and scaling are investigated and the results are compared with those obtained by previous authors. This allows some conclusions about the parameters which are relevant for the modelling of the transition process. The mechanisms underlying the transition process and the relative importance of the Tollmien–Schlichting wave instability in this flow are treated in an accompanying paper (part 2 of the present report).
Experiments in a boundary layer subjected to free stream turbulence. Part 2. The role of TS-waves in the transition process
A. V. Boiko, K. J. A. Westin, B. G. B. Klingmann, V. V. Kozlov, P. H. Alfredsson
The natural occurrence of Tollmien-Schlichting (TS) waves has so far only been observed in boundary layers subjected to moderate levels of free stream turbulence (Tu < 1%), owing to the difficulty in detecting small-amplitude waves in highly perturbed boundary layers. By introducing controlled oscillations with a vibrating ribbon, it is possible to study small-amplitude waves using phase-selective filtering techniques. In the present work, the effect of TS-waves on the transition is studied at Tu = 1.5%. It is demonstrated that TS-waves can exist and develop in a similar way as in an undisturbed boundary layer. It is also found that TS-waves with quite small amplitudes are involved in nonlinear interactions which lead to a regeneration of TS-waves in the whole unstable frequency band. This results in a significant increase in the number of turbulent spots, which promote the onset of turbulence. | CommonCrawl |
\begin{document}
\title{Decoherence in the three-state quantum walk}
\author{Luísa Toledo Tude} \affiliation{Instituto de F\'\i sica ``Gleb Wataghin'', Universidade Estadual de Campinas, Campinas, SP, Brazil} \affiliation{School of Physics, Trinity College Dublin, Ireland} \author{Marcos C. de Oliveira } \email{[email protected]} \affiliation{Instituto de F\'\i sica ``Gleb Wataghin'', Universidade Estadual de Campinas, Campinas, SP, Brazil}
\date{\today}
\begin{abstract}
Quantum walks are dynamic systems with a wide range of applications in quantum computation and quantum simulation of analog systems, therefore it is of common interest to understand what changes from an isolated process to one embedded in an environment. In the present work, we analyze the decoherence in a three-state uni-dimensional quantum walk. The approaches taken into consideration to account for the environment effects are phase and amplitude damping Kraus operators, unitary noise on the coin space, and broken links.
\end{abstract}
\pacs{}
\maketitle
\section{Introduction} Quantum walks were initially developed to be the quantum counterpart of the random walks, but soon it was realized that they can be used as a tool to develop faster quantum search algorithms\footnote{For a complete description on quantum walks and search algorithms, see \cite{PortugalBook} and references therein}. This is basically due to the fact that its standard deviation scales as $\sigma \propto t$ which is quadratically faster than the classical case, where $\sigma \propto \sqrt{t}$. Quantum walks can be subdivided in two main groups, discrete and continuous time. Here we focus our attention on the discrete-time quantum walk (DTQW). The key element that differentiates the DTQW in the line from the simple random walk is that a quantum version of the coin is taken into account. In other words, this means that while in the random walk a coin is thrown to decide if the next step of a walker will be to the right or to the left, in the quantum case one considers a "coin" --- typically described by the Hadamard gate --- that can be in a superposition of heads and tails, causing the walker to displace in a superposition of right and left steps. The three-state quantum walk is similar to the regular one-dimension walk, but accounting for an additional probability that the walker will remain in the same site during the time step. A typical signature of this extra conditional assignment is the possibility of localization of the walker, contrasting with the standard two-sided coin conditioned evolution \cite{3QW1,3QW3e4,3QWmatriz}.
With the increase of interest in the field of quantum computation, the theoretical and experimental domain of quantum walks \cite{Gong948} became a key ingredient to the performance of quantum search algorithms. Since quantum computers are physical objects, they are always subjected to some level of noise and dissipation. Therefore, dealing with decoherence is inevitable to build quantum computers that will perform quantum search outperforming classical search algorithms. Decoherence is a key element to understand the limit between classical and quantum phenomena, and there is already an expressive literature on that for two-state quantum walks (See, e.g., \cite{DecoherenceCanBeUsefulinQW}). A three-state quantum walk has a richer dynamical structure, which could be explored for quantum simulation of several systems in both condensed matter \cite{AndersonLoc}, and high energy physics \cite{PhysRevA.81.062340}. Since those systems are generally not isolated, an investigation of the effects of decoherence in the three-state quantum walk is in order. The evolution of a system under decoherence is not necessarily described by unitary operators, therefore one could use external interactions to control a new class of evolutions that lead to different behaviors of the walk. In this paper, we investigate the effects of decoherence in the three-state quantum walk. Since it can be physically introduced in the system by many different phenomena, depending of the actual physical implementation, we give a general account for such decoherence effects by introducing different mathematical approaches.
This work is subdivided as follows. In Sec. II we present a brief overview of the three-state quantum walk in an infinite line. In each of the subsequent three sections, we consider a distinct method of accounting for decoherence. In Sec. \ref{sec:Kraus}, we introduce the Kraus operator that can be used to model phase and amplitude damping decoherence in qutrits. In Sec. \ref{sec:Unise} we investigate decoherence by unitary noise, and at last, in Sec. \ref{sec:BrokenLinks} we analyze decoherence by broken links. Sec. VI is dedicated to the final remarks and conclusions. It is important to point out that those are not the only methods for simulating decoherence, for more information about other methods to implement decoherence in discrete and continuous-time quantum walks we refer to \cite{reviewDecoherence}.
\section{Three-state quantum walk}\label{sec:3QW} The two-state quantum walk on the line is the quantum version of the simple random walk. Its dynamics can be described by two operators -- one representing the coin toss and the other, the shift of the walker on the line. The difference between the random walk and its quantum version is that in the second case the coin toss does not give an exclusive classical result such as heads or tails, but a superposition of both. In that way, instead of taking a step to the right or to the left, the walker step is a superposition of both directions.
The three-state quantum walk, also known as lazy quantum walk, is analogous to the two-state quantum walk, but with an additional degree of freedom on the "coin" space that accounts for the possibility that the walker does not move in a time step. At each time step the walker flips a "three-sided quantum coin" that falls in a superposition of its three possible states, making the walker's position state, $\ket{n}$, evolve to a superposition of three possible states, $\ket{n-1}$, $\ket{n}$, and $\ket{n+1}$. Figure \ref{fig:3QWdiagrama} illustrates the possible steps of a walker that occupies the nth site of the lattice. \begin{figure}
\caption{Diagram of three-state quantum walk.The coefficients $a_n$, $b_n$ and $c_n$ correspond to the left ($L$), no movement ($S$) and right($R$) chiralities, respectively.}
\label{fig:3QWdiagrama}
\end{figure}
The system is composed by a coin and a walker, therefore its Hilbert space is written as $\mathcal{H} = \mathcal{H}_C \otimes \mathcal{H}_P$, where $\mathcal{H}_C$ is the "coin" Hilbert space and $\mathcal{H}_P$ the Hilbert Space associated with the positions of the walker in the one-dimensional infinite lattice. The state of the system at anytime can be described as a spinor \begin{equation}
\ket{\psi(t)} = \sum^{\infty}_{n = -\infty}
\begin{pmatrix}
a_n (t)\\
b_n (t)\\
c_n (t)
\end{pmatrix}
\ket{n},\label{comp} \end{equation} where $a_n$, $b_n$, and $c_n$ are the wave components correspondent to the three possibles states of the "coin". Each time step of the Quantum Walk dynamics is composed by two unitary operations. A rotation in the coin (chirality) space ($C$), followed by a shift (Sh) operation. The operator we will consider here to represent the action of the "coin" is \begin{equation}
C = \frac{1}{3} \begin{pmatrix} -1&2&2\\ 2&-1&2\\ 2&2&-1 \end{pmatrix}, \end{equation} known as Grover coin, and the shift operator is \begin{equation}
\begin{aligned}
\text{Sh} &= \sum^{\infty}_{n = - \infty}\ket{n-1}\bra{n}\otimes \ket{L}\bra{L}\\
&+\sum^{\infty}_{n = - \infty}\ket{n}\bra{n}\otimes \ket{S}\bra{S}\\
&+\sum^{\infty}_{n = - \infty}\ket{n+1}\bra{n}\otimes \ket{R}\bra{R}.
\end{aligned} \end{equation} Therefore, using these two unitary operators, the dynamics can be summarized to
\begin{equation}
\ket{\psi(t)} = (\text{Sh} (C \otimes \mathbb{I}))^t \ket{\psi(0)} = U^t\ket{\psi(0)},\label{eq:dynamicsQW}
\end{equation}
where $\mathbb{I}$ stands for the identity in position space, and $t$ is the time parametrized as the number of time steps. Using the density matrix notation, equation \ref{eq:dynamicsQW} is equivalent to
\begin{equation}
\rho(t) = U^t \rho_0 (U^{\dagger})^t,
\end{equation}
where $\rho_0 = \ket{\psi(0)} \bra{\psi(0)}$. The interference between the states will generate a probability distribution of position completely different from the classical. This can be clearly seen as we write the evolution of the global chirality distribution (GCP) \cite{GCP, tude2020temperature},
\begin{eqnarray}
\begin{pmatrix}
P_L (t+1)\\
P_S (t+1)\\
P_R (t+1)
\end{pmatrix}
&=&\frac{1}{9}\begin{pmatrix}
1 &4&4\\
4&1&4\\
4&4&1
\end{pmatrix}
\begin{pmatrix}
P_L (t)\\
P_S (t)\\
P_R (t)
\end{pmatrix}- \frac{\mathbb{R}[Q_1(t)]}{9}
\begin{pmatrix}
4\\
4\\
-8
\end{pmatrix}\nonumber\\
&&- \frac{\mathbb{R}[Q_2(t)]}{9}
\begin{pmatrix}
4\\
-8\\
4
\end{pmatrix}
- \frac{\mathbb{R}[Q_3(t)]}{9}
\begin{pmatrix}
8\\
4\\
4
\end{pmatrix},\label{eq:evGCP} \end{eqnarray} where the GCPs are the total probabilities of having the coin in each state --- right, left, or stay still---, independently of the position,
\begin{equation}
\begin{aligned}
P_L(t) =& \sum^{\infty}_{n = -\infty} |a_n (t)|^2, \\
P_S(t) =& \sum^{\infty}_{n = -\infty} |b_n (t)|^2,\\
P_R(t) =& \sum^{\infty}_{n = -\infty} |c_n (t)|^2,\label{eq:GCD3}
\end{aligned}
\end{equation} and the terms $Q_1(t)$, $Q_2(t)$ and $Q_3(t)$, given by
\begin{equation}
\begin{aligned}
Q_1(t) &= \sum^{\infty}_{n = -\infty} a_n(t) b_n^{*}(t);\\
Q_2(t) &= \sum^{\infty}_{n = -\infty} a_n(t) c_n^{*}(t);\\
Q_3(t) &= \sum^{\infty}_{n = -\infty} b_n(t) c_n^{*}(t),
\end{aligned}
\end{equation} are responsible for the interference effects of the walk. If the presence of decoherence causes them to completely vanish, equation (\ref{eq:evGCP}) becomes \begin{equation}
\begin{pmatrix}
P_L (t+1)\\
P_S (t+1)\\
P_R (t+1)
\end{pmatrix}
=\frac{1}{9}\begin{pmatrix}
1 &4&4\\
4&1&4\\
4&4&1
\end{pmatrix}
\begin{pmatrix}
P_L (t)\\
P_S (t)\\
P_R (t)
\end{pmatrix}, \end{equation} that describes the evolution of the GCPs as a classical Markovian process. The asymptotic limit of this process gives equal probabilities for each chirality independent of the initial conditions, i.e, \begin{equation}
\lim_{n\rightarrow \infty}\begin{pmatrix}
P_L (n t)\\
P_S (n t)\\
P_R (n t)
\end{pmatrix}
=\begin{pmatrix}
1/3\\
1/3\\
1/3
\end{pmatrix}. \end{equation}
In the next sections we will describe the effects of decoherence in the final probability distribution of positions through different models and analyze how fast they approach the classical limit.
One of the main differences between the two and three-state quantum walk is that, for some initial conditions, the three state quantum walk can exhibit localization, i.e, a peak on the position distribution around the initial location of the walker. Figure \ref{fig:3QWpos} shows the distribution of displacements of a walker that was initial at the site $0$, with two different initial states of the "coin". The initial conditions were chosen so that we could see the behavior of a walk with and without localization. Those same initial conditions will be used in the rest of the paper, so when we refer to the initial condition that generates localization we will be referring to \begin{equation}\label{loc}
\ket{\psi_0} = \dfrac{1}{\sqrt{2}} \begin{pmatrix}i\\0\\1\end{pmatrix} \ket{0}, \end{equation} and when we refer to the initial condition that does not exhibit localization, we mean \begin{equation}\label{unloc}
\ket{\psi_0} = \dfrac{1}{\sqrt{6}} \begin{pmatrix}1\\-2\\1\end{pmatrix} \ket{0}. \end{equation} Note that both states are normalized.
\begin{figure}
\caption{Distribution after $100$ time steps. Two different initial conditions were considered one that generates localization, and one that does not, according to Eqs. (\ref{loc}) and (\ref{unloc}).}
\label{fig:3QWpos}
\end{figure}
\section{Kraus Operators}\label{sec:Kraus} The simplest and direct way to introduce decoherence on the walk is by adding extra nonunitary operators, in the form of Kraus operators, $K_j$, to describe the effects of noise and other external effects. Hence, the recurrence relation in respect to the evolution of the system becomes \begin{equation} \rho(t+1) = \sum_{j}K_{j} U \rho(t) U^{\dagger} K_{j}^{\dagger}.\label{eq:KrausEvolution} \end{equation} This expression is completely equivalent to considering a unitary evolution on the total space composed by the system and the environment and taking the partial trace of the environment, \cite{Nielsen}.Kraus operators can account for different effects -- here we explore the operators associated with phase and amplitude damping on the coin space, \cite{DoriguelloDiniz2016,Nielsen}. Phase damping is modeled by the Kraus operators of the form $K_{j} = \mathbb{I} \otimes E_j$, where $j = 0,1$, $\mathbb{I}$ is the identity in the position space and the chirality components of the Kraus operators are given by, \cite{KrausOp3QW}, \begin{align}
E_{0} = \sqrt{1-\gamma} \begin{pmatrix}1&0&0\\0&1&0\\0&0&1
\end{pmatrix};&&
E_{1} =\sqrt{\gamma} \begin{pmatrix}1&0&0\\0&\omega&0\\0&0&\omega^2 \end{pmatrix}.\label{eq:Kraus3QWphase} \end{align} The parameter $\gamma \in [0,1]$ is the strength of the channel, here called the relaxation parameter, and $\omega = e^{2 \pi i/3}$. This noise process describes the quantum loss of information without loss of energy. Hence, as the system evolves, the information about the phase between the energy eigenstates is lost.
The walks with and without localization were simulated considering the phase damping operators, and the result is presented in figure \ref{fig:Kraus3QWPhase}. In both cases the effect of decoherence is similar to the effect in the two-state walk in the sense that there is a transition from quantum to classical (Normal distribution) behavior, \cite{DoriguelloDiniz2016}. The main difference in the case of the three-state walk is that the transition occurs as the relaxation parameter, $\gamma$, increases, but the maximum decoherence is achieved for $\gamma = 0.5$, instead of $1$ and for $\gamma > 0.5$ the distribution starts to transit back to the quantum behavior. Also, note that the dephasing acts equally in the localized and non-localized regimes, and although the normal distribution of the dephased regime, with localized initial condition is around the initial site $0$, the typical localization rapidly disappears. This is expected since any quantum behavior is suppressed equally by the dephasing. Even though localization may occur in any wave-like propagation, here the wave-like behavior is due to the presence of quantum features, in the form of superposition and entanglement of states.
\begin{figure}\label{fig:Kraus3QWPhase}
\end{figure}
Decoherence is always associated with some kind of informational loss, while the Kraus operators for phase damping introduce loss of phase information, the amplitude damping channel introduces loss of information regarding amplitude, as well as coherence of the eigenstates. The Kraus operators in this case have a similar format as the ones presented for phase damping decoherence ($K_{j} = \mathbb{I} \otimes E_j$), however in this case, \cite{KrausOp3QW}, \begin{eqnarray}
E_{0} &=& \begin{pmatrix}1&0&0\\0& \sqrt{1-\gamma}&0\\0&0& \sqrt{1-\gamma}
\end{pmatrix};\;\;
E_{1} = \begin{pmatrix}0&\sqrt{\gamma}&0\\0&0&0\\0&0&0 \end{pmatrix};\nonumber\\
E_{2} &=& \begin{pmatrix}0&0&\sqrt{\gamma}\\0&0&0\\0&0&0 \end{pmatrix}.\label{eq:Kraus3QWamplitude} \end{eqnarray}
Figure \ref{fig:Kraus3QWAmplitude} shows the resultant distribution of the three-state quantum walk with amplitude damping for the two initial conditions we are considering. The difference between phase and amplitude damping channel effects is in how the transition from quantum to classical behavior occurs. While in the case of phase damping the transitions occur symmetrically, in the amplitude damping the transition is not symmetric. The walk with strength $\gamma = 0$ represents the walk with no decoherence and as $\gamma$ increases the classical distribution is recovered, but with a shift in the position of the lattice. This is due to the fact that when the decoherence affects the relative phase between the eigenstates, the interference between them is lost, causing the distribution to behave classically, i.e, respecting the central limit theorem. On the other hand, the amplitude damping channel not only affects the interference, but also the information regarding the mean final position. In any case, despite the asymmetry, the effect is similar to the phase damping, regarding the localization, which is rapidly suppressed with the amplitude damping.
\begin{figure}\label{fig:Kraus3QWAmplitude}
\end{figure}
\section{Unitary Noise}\label{sec:Unise} There is an additional decoherence effect that may occur in quantum walks through unitary processes. Decoherence described by unitary operators can be caused by fluctuations and drifts in parameters (usually associated with phase) of the system Hamiltonian. To consider this type of decoherence in two-state quantum walks, a method was developed in Ref. \cite{UnitaryNoise}. Here we extend the method for three-state quantum walks. It consists in changing the evolution operator $U$ in eq. (\ref{eq:dynamicsQW}) to a unitary operator with a stochastic part. This can be interpreted as a random rotation on the "coin" space at each time step. The dynamics of the system is then given by
\begin{equation}
\ket{\psi(t+1)} = \text{Sh}(C e^{i a(t)}\otimes\mathbb{I})\ket{\psi(t)} = Q(t) \ket{\psi(t)}.
\end{equation} The operator $a(t)$ is a stochastic and Hermitian operator that acts on the "coin" space. Hence, the new evolution operator, $Q(t)$, is stochastic, but remains unitary. Since the Gell-Mann matrices, ${\bf \lambda} = (\lambda_1, \lambda_2, \lambda_3, \lambda_4, \lambda_5, \lambda_6, \lambda_7, \lambda_8)$, together with the identity form a basis of the chirality space, we can write \begin{equation}
a(t) = \sum_{k = 1}^{8} \alpha_k (t) \lambda_k, \end{equation} with $\alpha_k (t)$ being real components of the expansion, that are chosen randomly at each time step. In this case, the identity does not need to be taken into account because it would only add a global phase to the state. Before simulating the effects of this decoherence on the quantum walk we made the following assumptions on the stochastic operator components $a(t)$, i.e, \begin{equation}
\begin{cases}
\langle \alpha_{k}(t) \alpha_{k^\prime}(t^\prime)\rangle = \delta_{k,k^\prime} \delta_{t,t^\prime}\alpha^2\\
\langle \alpha_{k}(t) \rangle = 0\ \ \ \ \ \ ;\text{k = x, y, z.}
\end{cases} \end{equation}
This means that there is no correlation between different components of the operator and between different times and that the probability distributions of $\alpha$ are isotropic. In the simulation, we considered a sampling through a Gaussian distribution with a standard deviation, $\sigma_{a}$, which is varied for each data set. Figure \ref{fig:Unoise3QW} shows the result of the simulations for 4 fixed values of $\sigma_a$. To obtain these distributions, the simulation ran $400$ times and we took the mean of the results. The line that corresponds to $\sigma_{a}=0$ is the limit of the coherent quantum walk, and although the distribution do not seem to reach a Gaussian shape, as $\sigma_{a}$ increases we see a tendency of accumulation on the initial state of the distribution. Therefore, as expected, the stochasticity added to the process diminished the interference effect. We see that, although the random phase affects less drastically the localization for $\sigma_a=0.1$, it affects equally the localized and the non-localized solutions for larger standard deviations of the Gaussian random phase distribution.
\begin{figure}\label{fig:Unoise3QW}
\end{figure}
\section{Broken Links}\label{sec:BrokenLinks} When the links between any two sites of the walk have a non-null probability of being broken at each time step, an alternative decoherence source is established \cite{PeriodicMeBrokewnLinks}. If, in a time step, the link is open, the particle cannot move to the neighbor vertices, of the graph where the walk is embedded. In this situation a careful analysis of the recurrence relation for the wave components of the walk, as defined in Eq. (\ref{comp}), must be taken.
The recurrence relation for the regular walk --- i.e, with no broken links --- is \begin{equation}
\begin{aligned}
a_{n}(t+1) &=\frac{1}{3}(-a_{n+1}(t) +2 b_{n+1}(t) + 2 c_{n+1}(t));
\\
b_{n}(t+1) &= \frac{1}{3}(2 a_{n}(t) - b_{n}(t) + 2 c_{n}(t)); \\
c_{n}(t+1) &= \frac{1}{3}(2 a_{n-1}(t) + 2 b_{n-1}(t) - c_{n-1}(t)).
\end{aligned} \end{equation} If the link on the left side of position $n$ is broken, then the upper component of the spinor at $n$ receives a probability flux from $n+1$. To conserve the probability flux, the outgoing flux must be passed to component $c$ at the same site. The resultant expressions are \begin{equation}
\begin{aligned}
a_{n}(t+1) &=\frac{1}{3}(-a_{n+1}(t) +2 b_{n+1}(t) + 2 c_{n+1}(t));
\\
b_{n}(t+1) &= \frac{1}{3}(2 a_{n}(t) - b_{n}(t) + 2 c_{n}(t)); \\
c_{n}(t+1) &= \frac{1}{3}(- a_{n}(t) + 2 b_{n}(t) + 2 c_{n}(t)).
\end{aligned} \end{equation} Analogously the recurrences relations in the case that there is a broken link to the right of the site $n$ are \begin{equation}
\begin{aligned}
a_{n}(t+1) &=\frac{1}{3}(2 a_{n}(t) +2 b_{n}(t) - c_{n}(t));
\\
b_{n}(t+1) &= \frac{1}{3}(2 a_{n}(t) - b_{n}(t) + 2 c_{n}(t)); \\
c_{n}(t+1) &= \frac{1}{3}(2 a_{n-1}(t) + 2 b_{n-1}(t) - c_{n-1}(t)).
\end{aligned} \end{equation} Finally, if both links that connect site $n$ with its neighbors are broken, the relations become \begin{equation}
\begin{aligned}
a_{n}(t+1) &=\frac{1}{3}(2 a_{n}(t) +2 b_{n}(t) - c_{n}(t));
\\
b_{n}(t+1) &= \frac{1}{3}(2 a_{n}(t) - b_{n}(t) + 2 c_{n}(t)); \\
c_{n}(t+1) &= \frac{1}{3}(- a_{n}(t) + 2 b_{n}(t) + 2 c_{n}(t)).
\end{aligned} \end{equation}
At each time step some links are randomly chosen (with probability p) to be broken. Then, the process evolves following a different recurrence relation to each position, depending if its neighbor links are broken or not. Figure \ref{fig:3QWBrokendiagrama} illustrates the possible steps of a walker on a lattice with some broken links.
\begin{figure}
\caption{Diagram of three-state quantum walk with broken links. At every time step of the process a new set o broken links is randomly generated.}
\label{fig:3QWBrokendiagrama}
\end{figure}
The evolution proceeds through unitary operations, however, the operators change randomly according to the topology of the graph. Note that the decoherence comes from a stochastic process --- i.e, the changes on the links of the graph --- that changes the evolution operator, $U$, to another unitary operator, just like in the case of the unitary noise model. The main difference between both cases is that here the noise directly affects the walker space, while in section \ref{sec:Unise} it accounted for decoherences in the coin space.
Using the recurrence relations derived above, we simulate the three-state quantum walk with broken links after $50$ and $200$ time steps and for different probabilities of broken links. Figures \ref{fig:BrokenLinks3QWLoc} and \ref{fig:BrokenLinks3QWNLoc} show the mean result of $1000$ simulation runs for the two initial conditions.
\begin{figure}\label{fig:BrokenLinks3QWLoc}
\end{figure} As in the processes analyzed in the previous sections, we see a transition from the quantum to a Gaussian-like distribution, however in this case and interesting feature differentiates the effect of the decoherence. When the initial condition generates localization, the broken links preserve it, changing only the other regions of the distribution. In figure \ref{fig:BrokenLinks3QWLoc} we can see clearly that outside the localization region the blue and orange curves approach a Gaussian shape and in the central region the three curves present the localized shape. This is due to the fact that the disorder introduced by the random choice of broken links, in fact contributes for localization. However it is not strong enough to imprint a localization profile for arbitrary initial conditions, only for the favorable localized initial condition is that the localization is reinforced.
\begin{figure}\label{fig:BrokenLinks3QWNLoc}
\end{figure}
There is a characteristic time, $t_c$, associated with the transition between quantum to classical behavior that depends on the probability of broken links $p$, \cite{PeriodicMeBrokewnLinks}. At the initial time, the walker is at position $0$, therefore there are only two relevant links to the walk -- the ones connecting position $0$ with $\pm 1$. As the time evolves the wave function spreads through the line covering a range of $\alpha t$ . Hence, the mean number of broken links per time step is proportional to the time, $p \alpha t$. The classical behavior starts to emerge when the mean number of broken links per time step is of order one, so $t_c = \dfrac{1}{p \alpha}$ and for $t>>t_c$ the distribution tends to a Gaussian. The transition is also reflected on the standard deviation, the spread for early times is ballistic, and for $t>>t_c$ the classical spread is dominant. This can be observed in figure \ref{fig:sigma}. All three curves start looking like a straight line but as time grows, the ballistic feature stops to be the dominant behavior. This happens first to the walk with a higher probability of broken link. The green curve remains with quantum behavior through all time accounted.
\begin{figure}\label{fig:sigma}
\end{figure}
\section{Conclusions}\label{sec:C}
Despite being considered as the quantum counterpart of random walks, quantum walks are not stochastic processes in the classical sense. That is, randomness plays a clear role at each time step of the random walk, in the sense that the result of the coin toss is not predictable and the system's dynamics is irreversible. On the other hand, in the quantum walk, the position of the walker is unknown, but the state of the system is always known. The result of the coin toss is perfectly predictable and the dynamics of the system is governed by a unitary evolution, which means that if the initial state is pure it will remain pure. The randomness of the quantum walk is uniquely due to the measurement process.
However, when the quantum walk is not completely isolated, the picture changes and the lack of knowledge about the environment that contains the system can add randomness to the system. Those decoherence effects are sometimes inevitable with the available technologies nowadays, which makes extremely important to understand what different types of interactions between system and environment have in the process we want to control. In that spirit, this paper is devoted to the analyses of the decoherence effect in the three-state quantum walk. We analyzed the behavior of the walk for different degrees of relaxation in the case of phase and amplitude damping and observed a Gaussian behavior emerging in both cases, but with the difference that the amplitude damping also generates a lost symmetry in the probability of position. In sections \ref{sec:Unise} and \ref{sec:BrokenLinks} we analyzed a decoherence that causes a random factor in the Hamiltonian. In the first case this randomness is related to the chirality space while in the second it affects the position space. In the case of decoherence by broken links we also notice that the decoherence preserves the localization of the distribution.
Besides its experimental applications, this work also presents a simple way of understanding the quantum to classical limits. As expected, our results suggest that the decoherence attenuates the quantum interference effects that are responsible for the wavy shape of the position distributions and for the linear grow of the the standard deviation of the distribution in time. \begin{acknowledgments}
This work was partially supported by CNPq (Brazil). \end{acknowledgments}
\end{document} | arXiv |
\begin{document}
\title{Some linear SPDEs driven by a fractional noise with Hurst index greater than 1/2}
\author{Raluca M. Balan\footnote{Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, ON, K1N 6N5, Canada. E-mail address: [email protected]} \footnote{Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.}}
\date{February 18, 2011}
\maketitle
\begin{abstract} \noindent In this article, we identify the necessary and sufficient conditions for the existence of a random field solution for some linear s.p.d.e.'s of parabolic and hyperbolic type. These equations rely on a spatial operator $\mathcal{L}$ given by the $L^2$-generator of a $d$-dimensional L\'evy process $X=(X_t)_{t \geq 0}$, and are driven by a spatially-homogeneous Gaussian noise, which is fractional in time with Hurst index $H>1/2$. As an application, we consider the case when $X$ is a $\beta$-stable process, with $\beta \in (0,2]$. In the parabolic case, we develop a connection with the potential theory of the Markov process $\bar{X}$ (defined as the symmetrization of $X$), and we show that the existence of the solution is related to the existence of a ``weighted'' intersection local time of two independent copies of $\bar{X}$. \end{abstract}
\noindent {\em MSC 2000 subject classification:} Primary 60H15; secondary 60H05, 60G51
\noindent {\em Keywords and phrases:} stochastic partial differential equations, fractional Brownian motion, spatially homogeneous Gaussian noise, L\'evy processes
\section{Introduction}
In 1944, in his seminal article \cite{ito44}, It\^o introduced the stochastic integral with respect to the Brownian motion, which turned out to be one of the most fruitful ideas in mathematics in the 20th century. This lead to the theory of diffusions (whose origins can be traced back to \cite{ito51}), and the development of the stochastic calculus with respect to martingales (initiated in \cite{kunita-watanabe67}). These ideas have grown into a solid branch of probability theory called stochastic analysis, which includes the study of stochastic partial differential equations (s.p.d.e.'s)
Traditionally, there have been several approaches for the study of s.p.d.e.'s. The most important are: the Walsh approach which relies on stochastic integrals with respect to martingale-measures (see \cite{walsh86}), the Da Prato and Zabczyk approach which uses stochastic integrals with respect to Hilbert-space-valued Wiener processes (see \cite{daprato-zabczyk92}), and the Krylov approach which uses the concept of function-space-valued solution (see \cite{krylov99}). These approaches have been developed at the same time, and nowadays a lot of effort is dedicated to unify them (see the recent survey \cite{dalang-quer10} and the references therein).
The fractional Brownian motion (fBm) was introduced by Kolmogorov in \cite{kolmogorov40}, who called it the ``Wiener spiral'', and is defined as a zero-mean Gaussian process $(B_t)_{t \geq 0}$ with covariance:
$$R_{H}(t,s)=E(B_t B_s)=\frac{1}{2}(t^{2H}+s^{2H}-|t-s|^{2H}).$$ The parameter $H$ lies in $(0,1)$, and is called the Hurst index (due to \cite{hurst51}). The case $H=1/2$ corresponds to the Brownian motion, whereas the cases $H>1/2$ and $H<1/2$ have many contrasting properties and cannot be handled simultaneously. The representation of the fBm as a stochastic integral with respect to the Brownian motion on $\mathbb{R}$ was obtained as early as 1968 (see \cite{mandelbrot-vanness68}), but the fBm began to be used intensively in stochastic analysis only in the late 1990's. It is the flexibility which stems from the choice of the parameter $H$ that makes the fBm a much more attractive model for the noise than the Brownian motion (and its infinite-dimensional counterparts).
Among the fBm's remarkable properties is the fact that it is {\em not} a semimartingale. Consequently, It\^o calculus cannot be used in this case. A stochastic calculus with respect to the fBm was developed for the first time in \cite{decreusefond-ustunel98}. Subsequent important contributions were made in \cite{alos-mazet-nualart01}, \cite{carmona-coutin-montseny03} and \cite{duncan-hu-pasik00}. The stochastic integral used by these authors is an extension of the It\^{o} integral introduced by Hitsuda in \cite{hitsuda72} (and refined in \cite{kabanov75} and \cite{skorohod75}), and coincides with the divergence operator. These techniques are based on Malliavin calculus. Alternative methods for defining a stochastic integral with respect to the fBm exploit the H\"older continuity property of its sample paths and are based on generalized Stieltjes integrals. We refer the reader to Chapter 6 of the monograph \cite{nualart06} for more details.
In the present article, we consider the parabolic Cauchy problem \begin{eqnarray} \label{parabolic-eq} \frac{\partial u}{\partial t}(t,x)&=&\mathcal{L} u(t,x)+\dot W(t,x), \quad t>0, x \in \mathbb{R}^d \\ \nonumber u(0,x) &=& 0, \quad x \in \mathbb{R}^d, \end{eqnarray} and the hyperbolic Cauchy problem
\begin{eqnarray} \label{hyperbolic-eq} \frac{\partial^2 u}{\partial t^2}(t,x)&=&\mathcal{L} u(t,x)+\dot W(t,x), \quad t>0, x \in \mathbb{R}^d \\ \nonumber u(0,x) &=& 0, \quad x \in \mathbb{R}^d \\ \nonumber
\frac{\partial u}{\partial t}(0,x) &=& 0, \quad x \in \mathbb{R}^d, \end{eqnarray} where $\mathcal{L}$ is a ``spatial operator'' (i.e. it acts only on the $x$ variable) given by the $L^2(\mathbb{R}^d)$-generator of a $d$-dimensional L\'evy process $X=(X_t)_{t \geq 0}$, and $W$ is a Gaussian noise whose covariance is written formally as:
$$E[W(t,x)W(s,y)]=|t-s|^{2H-2}f(x-y),$$ for some index $H>1/2$ and some kernel $f$ (to be defined below).
The rigorous definition of the noise $\dot W$ is given in Section \ref{prelim-section}. At this point, we should just mention that the covariance structure of the noise has two components: a spatially-homogeneous component specified by the kernel $f$ (the example that we have in mind being the Riesz kernel
$f(x)=|x|^{-(d-\alpha)}$, with $0<\alpha<d$), and a temporal component inherited from the fBm. This becomes clear once we realize that {\em if} $H>1/2$, $R_H(t,s)$ can be written as:
$$R_{H}(t,s)=\alpha_H\int_0^t \int_0^s |u-v|^{2H-2}du dv, \quad \mbox{with} \quad \alpha_H=H(2H-1).$$ (The case $H<1/2$ has to be treated differently and is not discussed here.)
The solution to problem (\ref{parabolic-eq}) (or (\ref{hyperbolic-eq})) is understood in the mild-sense, and one of the goals of the present article is to give a necessary and sufficient condition for the existence of this solution, in terms of the parameters $(H,f)$ of the noise, and the spatial operator $\mathcal{L}$. A similar problem has been considered in \cite{FKN09} and \cite{FK10} in the case $H=1/2$ (which corresponds to the white noise in time). This motivated us to examine the case $H>1/2$.
The case of the hyperbolic equation with spatial operator $\mathcal{L}=-(-\Delta)^{-\beta/2}, \linebreak \beta>0$, driven by a white noise in time was examined in \cite{dalang-mueller03} and \cite{dalang-sanzsole05}. In fact, these authors consider the much more difficult case of the non-linear equation $\partial_{tt} u=\mathcal{L} u+\sigma(u)\dot{W}+b(u)$ with arbitrary initial conditions, and Lipschitz continuous functions $\sigma$ and $b$. For the linear equation, it turns out that the necessary and sufficient condition for the existence of the solution is: \begin{equation} \label{Dalang-Mueller-cond}
\int_{\mathbb{R}^d} \frac{1}{1+|\xi|^{\beta}}\mu(d\xi)<\infty, \end{equation} where the measure $\mu$ is the inverse Fourier transform of $f$ in $\mathcal{S}'(\mathbb{R}^d)$.
In the case of equations (\ref{parabolic-eq}) and (\ref{hyperbolic-eq}) driven by a space-time white noise (i.e. $H=1/2$ and $f=\delta_0$), the authors of \cite{FKN09} have shown that the necessary and sufficient condition for the existence of a random field solution is: \begin{equation} \label{FKN-cond} \int_{\mathbb{R}^d} \frac{1}{1+{\rm Re} \Psi(\xi)}\mu(d\xi)<\infty, \end{equation} where $\Psi(\xi)$ is the characteristic exponent of the underlying L\'evy process $X$. An important observation of \cite{FKN09} is that condition (\ref{FKN-cond}) can be extrapolated in a different context, being the condition which guarantees the existence of a local time $\int_0^t \delta_0(\bar{X}_s)ds$ of the symmetrization $\bar{X}$ of $X$. This line of investigation was continued in \cite{FK10} in the case of the parabolic equation (\ref{parabolic-eq}) with white noise in time, but covariance kernel $f$ in space. Surprisingly, it is shown there that condition (\ref{FKN-cond}) is related not only to the existence of the ``occupation'' local time $L_t(f)=\int_{0}^{t}f(\bar{X}_s)ds$, but also to the potential theory of the process $\bar{X}$, when viewed as a Markov process.
In the present article, we carry out a similar program in the case of the fractional noise in time. More precisely, after the introduction of some background material in Section \ref{prelim-section}, the article is split between the two problems: Section \ref{parabolic-section} is dedicated to the parabolic problem (\ref{parabolic-eq}), while Section \ref{hyperbolic-section} treats the hyperbolic problem (\ref{hyperbolic-eq}). For the parabolic problem, we discuss three things: (i) the existence of a random field solution (Section 3.1); (ii) the maximal principle which gives the connection with the potential theory of Markov processes (Section 3.2); (iii) the relationship with the ``weighted'' intersection local time $L_{t,H}(f)$, defined by
$$L_{t,H}(f)=\alpha_H \int_0^t \int_0^t |r-s|^{2H-2} f(\bar{X}_r^1-\bar{X}_s^2)dr ds,$$ where $\bar{X}^1$ and $\bar{X}^2$ are two independent copies of $\bar{X}$ (Section 3.3). For the hyperbolic problem, we only discuss the existence of a random field solution in the case when $\Psi(\xi)$ is real-valued (i.e. $X$ is symmetric).
Unlike the case of the white noise in time, it turns out that for the fractional noise, the conditions for the existence of the solution are different for the parabolic and hyperbolic problems. These conditions are: \begin{equation} \label{parabolic-cond} \int_{\mathbb{R}^d} \left(\frac{1}{1+{\rm Re} \Psi(\xi)}\right)^{2H}\mu(d\xi)<\infty, \end{equation} in the parabolic case, respectively, \begin{equation} \label{cond-hyp-eq} \int_{\mathbb{R}^d} \left(\frac{1}{1+{\rm Re}\Psi(\xi)} \right)^{H+1/2}\mu(d\xi)<\infty, \end{equation} in the hyperbolic case. This phenomenon was observed for the first time in \cite{BT10-SPA} for the wave and heat equations.
As an application, we discuss the case when $X$ is a $\beta$-stable process with $\beta \in (0,2]$, and hence $\Psi(\xi)=c_{\beta}|\xi|^{\beta}$. In this case, conditions (\ref{parabolic-cond}) and (\ref{cond-hyp-eq}) turn out to be generalizations of (\ref{Dalang-Mueller-cond}).
The fact that the fractional noise induces a connection with the weighted intersection local time was also used in \cite{hu-nualart09}, in the case when $f=\delta_0$. In \cite{BT09-JTP}, it is shown that the existence of the exponential moment of $L_{t,H}(f)$ is closely related to the existence of the (mild) solution of the heat equation with multiplicative noise.
We now introduce the notation used in the present article. The Fourier transform of a function $\varphi \in L^1(\mathbb{R}^d)$ is defined by: $$\mathcal{F} \varphi (\xi)=\int_{\mathbb{R}^d} e^{-i \xi \cdot x} \varphi(x) dx.$$ It is known that the Fourier transform can be extended to $L^2(\mathbb{R}^d)$ (see e.g. \cite{folland92}). Plancherel theorem says that for any $\varphi, \psi \in L^2(\mathbb{R}^d)$, $$\int_{\mathbb{R}^d} \varphi(x) \psi(x) dx= \frac{1}{(2\pi)^{d}} \int_{\mathbb{R}^d} \mathcal{F} \varphi(\xi) \overline{\mathcal{F} \psi(\xi)}d\xi.$$
Let $\mathcal{S}(\mathbb{R}^d)$ be the Schwartz space of rapidly decreasing infinitely differentiable functions on $\mathbb{R}^d$. A continuous linear functional on $\mathcal{S}(\mathbb{R}^d)$ is called a tempered
distribution. Let $\mathcal{S}'(\mathbb{R}^d)$ be the space of
tempered distributions. The Fourier transform $\mathcal{F} S$
of a functional $S \in \mathcal{S}'(\mathbb{R}^d)$ is defined by: $$(\mathcal{F} S, \varphi)=(S, \mathcal{F} \varphi), \quad \forall \varphi \in \mathcal{S}(\mathbb{R}^d).$$
We refer the reader to \cite{schwartz66} for more details about the space $\mathcal{S}'(\mathbb{R}^d)$.
\section{Preliminaries} \label{prelim-section}
In this section, we introduce some background material about the Gaussian noise $W$ and the L\'evy process $X$.
\subsection{The Gaussian noise}
As in \cite{dalang99}, we let $f:\mathbb{R}^d \to [0,\infty]$ be a measurable locally integrable function (or a {\em kernel}). We assume that $f$ is the Fourier transform in $\mathcal{S}'(\mathbb{R}^d)$ of a tempered measure $\mu$, i.e. \begin{equation} \label{def-Fourier-mu} \int_{\mathbb{R}^d}f(x)\varphi(x)dx=\int_{\mathbb{R}^d} \mathcal{F} \varphi(\xi) \mu(d\xi), \quad \forall \varphi \in \mathcal{S}(\mathbb{R}^d). \end{equation}
It follows that for any $\varphi,\psi \in \mathcal{S}(\mathbb{R}^d)$, \begin{equation} \label{conseq-def-mu} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \varphi(x)\psi(y)f(x-y)dx dy=\int_{\mathbb{R}^d} \mathcal{F} \varphi(\xi) \overline{\psi(\xi)}\mu(d\xi). \end{equation}
Similarly to \cite{BT10-SPA}, we let $\mathcal{E}$ be the set of elementary functions of the form $$h(t,x)= \phi(t)\psi(x), \quad t \geq 0, \ x \in \mathbb{R}^d,$$ where $\phi$ is a linear combination of indicator functions $1_{[0,a]}$ with $a>0$, and $\psi \in \mathcal{S}(\mathbb{R}^d)$. We endow $\mathcal{E}$ with the inner product: $$\langle h_1, h_2 \rangle_{\mathcal{H} \mathcal{P}}=\alpha_H \int_{0}^{\infty}
\int_{0}^{\infty} \int_{\mathbb{R}^{d}} \int_{\mathbb{R}^d}|t-s|^{2H-2} f(x-y)h_1(t,x)h_2(s,y)dx dy dr ds.$$ Let $\mathcal{H} \mathcal{P}$ be the completion of $\mathcal{E}$ with respect to the inner product $\langle \cdot , \cdot \rangle_{\mathcal{H} \mathcal{P}}$. We note that the space $\mathcal{H} \mathcal{P}$ may contain distributions in both $t$ and $x$ variables.
We consider a zero-mean Gaussian process $\{W(h); h \in \mathcal{E}\}$ with covariance $$E(W(h_1)W(h_2))=\langle h_1, h_2 \rangle_{\mathcal{H} \mathcal{P}}.$$
The map $h \mapsto W(h)$ is an isometry between $\mathcal{E}$ and the Gaussian space of $W$, which can be extended to $\mathcal{H} \mathcal{P}$. This extension defines an isonormal Gaussian process $W=\{W(h);h \in \mathcal{H} \mathcal{P}\}$. We write $$ W(h)=\int_0^{\infty}\int_{\mathbb{R}^d} h(t,x)W(dt,dx), \quad \mbox{for any} \ h \in \mathcal{H} \mathcal{P}.$$ This defines the stochastic integral of an element $h \in \mathcal{H} \mathcal{P}$ with respect to the noise $W$.
\subsection{The L\'evy process}
As in \cite{FK10}, we let $X=(X_t)_{t \geq 0}$ be a $d$-dimensional L\'evy process with characteristic exponent $\Psi(\xi)$. Hence, $X_0=0$ and for any $t>0$, \begin{equation} \label{Fourier-transform-Xt} E(e^{-i \xi \cdot X_t})=e^{-t \Psi(\xi)}, \quad \mbox{for all} \ \xi \in \mathbb{R}^d. \end{equation}
By the L\'evy-Khintchine formula (see e.g Theorem 8.1 of \cite{sato99}),
$$\Psi(\xi)=i\gamma \cdot \xi +\xi^T A \xi -\int_{\mathbb{R}^d}(e^{-i \xi \cdot x}-1+i\xi \cdot x 1_{\{|x| \leq 1\}} )\nu(dx),$$ where $(\gamma,A,\nu)$ is the generating triplet of $X$. We observe that for any $\xi \in \mathbb{R}^d$, \begin{eqnarray*} {\rm Re} \Psi (\xi)&=& \xi^T A \xi +\int_{\mathbb{R}^d} [1-\cos (\xi \cdot x)] \nu(dx) \geq 0, \quad \mbox{and} \\ {\rm Im} \Psi (\xi)&=& \gamma \cdot \xi + \int_{\mathbb{R}^d}[\sin (\xi
\cdot x)-\xi \cdot x 1_{\{|x| \leq 1\}}] \nu(dx). \end{eqnarray*}
$X$ is a homogenous Markov process with transition probabilities:
$$Q_t(x;B)=P(X_{s+t} \in B|X_s=x)=P(X_t \in B-x),$$ for any $x \in \mathbb{R}^d$ and Borel set $B \subset \mathbb{R}^d$. Let $(P_t)_{t \geq 0}$ be the associated semigroup, defined by: $$(P_t \phi) (x)=\int_{\mathbb{R}^d}\phi(y)Q_t(x;dy)=E[\phi(x+X_t)],$$ for any bounded (or non-negative) measurable function $\phi: \mathbb{R}^d \to \mathbb{R}$.
Let $\mathcal{L}$ be the $L^2(\mathbb{R}^d)$-generator of $(P_t)_{t \geq 0}$, defined by: $$\mathcal{L} \phi=\lim_{t \to 0}\frac{P_t \phi-\phi}{t} \quad \mbox{in} \ L^2(\mathbb{R}^d) \quad\mbox{(if it exists)}.$$ Note that $\mathcal{L} \phi$ exists if and only if $\phi \in {\rm Dom}(\mathcal{L})$, where $${\rm Dom}(\mathcal{L})=\{\phi \in L^2(\mathbb{R}^d); (\mathcal{F} \phi)\Psi \in L^2(\mathbb{R}^d)\}$$ (see p.16 of \cite{FK10}). Moreover, $\mathcal{L}$ can be viewed as a convolution operator with Fourier multiplier
$\mathcal{F} \mathcal{L}=-\Psi$, i.e. for any $\phi \in {\rm Dom}(\mathcal{L})$ and $\xi \in \mathbb{R}^d$,
$$\mathcal{F}(\mathcal{L} \phi)(\xi)=-\Psi(\xi)\mathcal{F} \phi(\xi).$$
\section{The Parabolic Equation} \label{parabolic-section}
In this section, we assume that the law of $X_t$ has a density denoted by $p_t$. This assumption allows us to identify the fundamental solution of $\partial_t u-\mathcal{L} u=0$.
To see this, note that $Q_t(x; \cdot)$ has density $p_t( \cdot -x)$, and $P_t \phi=\phi * \tilde p_t$, where $\tilde p_t(x)=p_t(-x)$. Since the solution of the Kolmogorov's equation $\partial_t u(t,x)=\mathcal{L} u(t,x)$ with initial condition $u(0,x)=u_0(x)$ is $$u(t,x)=(P_t u_0)(x)=\int_{\mathbb{R}^d} u_0(y)p_t(y-x)dy,$$
it follows that the fundamental solution of $\partial_t u-\mathcal{L} u=0$ is the function: $$G(t,x)=p_t(-x), \quad t>0, x \in \mathbb{R}^d.$$ From (\ref{Fourier-transform-Xt}), we obtain that: \begin{equation} \label{Fourier-G} {\mathcal{F}} G(t,\cdot)(\xi)= \int_{\mathbb{R}^d}e^{i \xi \cdot x} p_t(x)dx=E(e^{i \xi \cdot X_t})=e^{-t \overline{\Psi(\xi)}}. \end{equation}
\subsection{Existence of the Random-Field Solution}
There are two equivalent ways of defining a random field solution for problem (\ref{parabolic-eq}). Similarly to \cite{BT10-SPA}, one can say that the process $\{u(t,x); t \geq 0,x \in \mathbb{R}^d\}$ defined by: $$u(t,x)=\int_0^t \int_{\mathbb{R}^d}G(t-s,x-y)W(ds,dy)$$ is a {\em random field solution} of (\ref{parabolic-eq}), provided that the stochastic integral above is well-defined, i.e. the integrand \begin{equation} \label{gtx-in-H} \mathbb{R}_{+} \times \mathbb{R}^d \ni (s,y)\mapsto g_{tx}(s,y)=1_{[0,t]}(s)G(t-s,x-y) \quad \mbox{belongs to} \quad \mathcal{H} \mathcal{P}. \end{equation}
Since $g_{tx}$ satisfies conditions (i)-(iii) of Theorem 2.1 of \cite{BT10-SPA}, to check that $g_{tx} \in \mathcal{H} \mathcal{P}$, it suffices to prove that:
$$I_t:=\alpha_H \int_{\mathbb{R}^d} \int_{0}^{\infty} \int_{0}^{\infty} \mathcal{F} g_{tx}(r,\cdot)(\xi) \overline{\mathcal{F} g_{tx}(s,\cdot)(\xi)}|r-s|^{2H-2} dr ds \mu(d\xi)<\infty.$$ In this case,
$$E|u(t,x)|^2=E|W(g_{tx})|^2=
\|g_{tx}\|_{\mathcal{H} \mathcal{P}}^2=I_t.$$
Note that by (\ref{Fourier-G}), $I_t=\int_{\mathbb{R}^d}N_t(\xi)\mu(d \xi)$, where
$$N_t(\xi)=\alpha_H \int_0^t \int_0^t e^{-r \Psi(\xi)} e^{-s \overline{\Psi(\xi)}}|r-s|^{2H-2}dr ds.$$ Therefore, the question about the existence of a random field solution of (\ref{parabolic-eq}) reduces to finding suitable upper and
lower bounds for $N_t(\xi)$.
Alternatively, the authors of \cite{FKN09} suggest a different method for defining the random field solution of (\ref{parabolic-eq}), which has the advantage that can be applied also to the hyperbolic problem (\ref{hyperbolic-eq}) (for which one cannot identify the fundamental solution $G$). Since this is the method that we use in the present article, we explain it below.
We say that $\{u(t,\varphi);t \geq 0, \varphi \in \mathcal{S}(\mathbb{R}^d)\}$ is a {\em the weak solution} of (\ref{parabolic-eq}) if $$u(t,\varphi)=\int_0^t \int_{\mathbb{R}^d} \left(\int_{\mathbb{R}^d}G(t-s,x-y)\varphi(x) dx \right) W(ds,dy).$$ Note that the stochastic integral above is well-defined (as a random variable in $L^2(\Omega)$) if and only if the integrand $$\mathbb{R}_{+} \times \mathbb{R}^d \ni (s,y) \mapsto h_{t,\varphi}(s,y)=1_{[0,t]}(s)(\varphi * p_{t-s})(y) \quad \mbox{belongs to $\mathcal{H} \mathcal{P}$}.$$
Since $h_{t,\varphi}$ satisfies the conditions (i)-(iii) of Theorem 2.1 of \cite{BT10-SPA}, to check that $h_{t,\varphi} \in \mathcal{H} \mathcal{P}$ it suffices to show that
$$I_{t,\varphi}:=\alpha_H \int_{\mathbb{R}^d} \int_0^{\infty} \int_0^{\infty}|r-s|^{2H-2} \mathcal{F} h_{t,\varphi}(r,\cdot)(\xi) \overline{\mathcal{F} h_{t,\varphi}(s,\cdot)(\xi)}dr ds \mu(d\xi)<\infty.$$ In this case, we have
$$E|u(t,\varphi)|^2=E|W(h_{t,\varphi})|^2=
\|h_{t,\varphi}\|_{\mathcal{H} \mathcal{P}}^2=I_{t,\varphi}.$$
Since both $\varphi$ and $p_{t-s}$ are in $L^1(\mathbb{R}^d)$, $$\mathcal{F}(\varphi * p_{t-s})(\xi)=\mathcal{F} \varphi(\xi) \mathcal{F} p_{t-s}(\xi)=\mathcal{F} \varphi(\xi) e^{-(t-s) \Psi(\xi)},$$ and hence
$$I_{t,\varphi}=\int_{\mathbb{R}^d}N_t(\xi)|\mathcal{F} \varphi(\xi)|^2 \mu(d\xi).$$
Using the trivial bound $N_t(\xi) \leq t^{2H}$ (which is obtained using the fact that $|e^{-s \Psi(\xi)}|=e^{-s {\rm Re} \Psi(\xi)} \leq 1$ for all $s>0$ and $\xi \in \mathbb{R}^d$), we get: $$I_{t,\varphi}\leq t^{2H} \int_{\mathbb{R}^d}|\mathcal{F} \varphi(\xi)|^2 \mu(d\xi)<\infty \quad \mbox{for all} \ \varphi \in \mathcal{S}(\mathbb{R}^d).$$ This shows that $u(t,\varphi)$ is a well-defined random variable in $L^2(\Omega)$.
We continue to explain the method of \cite{FKN09}. We endow $\mathcal{S}(\mathbb{R}^d)$ with the inner product: $$\langle \varphi,\psi \rangle_t=E(u(t,\varphi)u(t,\psi)).$$
We denote by $\| \cdot \|_t$ the norm induced by the inner product $\langle \cdot, \cdot \rangle_t$, i.e. \begin{equation} \label{def-norm-2bars}
\|\varphi \|_t^2=E|u(t,\varphi)|^2 =\int_{\mathbb{R}^d} N_t(\xi)|\mathcal{F} \varphi(\xi)|^2 \mu(d\xi). \end{equation}
Let $M_t$ be the completion of $\mathcal{S}(\mathbb{R}^d)$ with respect to $\langle \cdot, \cdot \rangle_t$ and $M=\cap_{t>0}M_t$. We say that (\ref{parabolic-eq}) has a {\em random field solution} if and only if \begin{equation} \label{delta-x-in-M} \delta_x \in M \quad \mbox{for all} \ x \in \mathbb{R}^d. \end{equation} The random field solution is defined by $\{u(t,x)=u(t,\delta_x); t \geq 0,x \in \mathbb{R}^d\}$.
To prove (\ref{delta-x-in-M}), we introduce the space $\mathcal{Z}=\cap_{t>0}\mathcal{Z}_t$, where $\mathcal{Z}_t$ is the completion of $\mathcal{S}(\mathbb{R}^d)$ with respect to the inner product $[\cdot,\cdot]_t$ defined by: $$[\varphi, \psi]_t= \int_{\mathbb{R}^d} \left(\frac{1}{1/t+ {\rm Re} \Psi(\xi)} \right)^{2H} \mathcal{F} \varphi(\xi) \overline{\mathcal{F} \psi(\xi)}\mu(d\xi)=:\mathcal{E}(t; \varphi,\psi)$$
We denote by $|\| \cdot \||_t$ the norm induced by the inner product $[\cdot, \cdot]_t$, i.e. \begin{equation} \label{def-norm-3bars}
|\| \varphi \||_t =\int_{\mathbb{R}^d} \left(\frac{1}{1/t + {\rm Re} \Psi(\xi)} \right)^{2H} |\mathcal{F} \varphi(\xi)|^2\mu(d\xi)=:\mathcal{E}(t; \varphi). \end{equation}
By Lemma \ref{ineq-cE} (Appendix A), for any $s,t>0$, there exist some positive constants $c_1(s,t)$ and $c_2(s,t)$ such that for any $\varphi \in \mathcal{S}(\mathbb{R}^d)$, $$c_1(s,t)^{2H} \mathcal{E}(s;\varphi) \leq \mathcal{E}(t; \varphi) \leq c_2(s,t)^{2H} \mathcal{E}(s; \varphi).$$
Therefore, the norms $|\| \cdot \||_t$ and $|\| \cdot \||_s$ are equivalent and $\mathcal{Z}_t =\mathcal{Z}_s=\mathcal{Z}$.
The idea for proving (\ref{delta-x-in-M}) is to show that any norm $\| \cdot \|_t$ is equivalent to a norm $|\| \cdot \||_{\rho(t)}$, for a certain bijective function $\rho:\mathbb{R}_{+} \to \mathbb{R}_{+}$. From this, one infers that $M_t=\mathcal{Z}_{\rho(t)}$ for any $t>0$, and hence $M=\mathcal{Z}=\mathcal{Z}_1$. Condition (\ref{delta-x-in-M}) becomes $\delta_x \in \mathcal{Z}_1$ for all $x \in \mathbb{R}^d$, for which one can find a natural necessary and sufficient condition (see Theorem \ref{nec-suf-delta-in-Z1} below). In the case of the parabolic problem (\ref{parabolic-eq}), it turns out that $\rho(t)=t$. (We will see in Section \ref{hyperbolic-section} that for the hyperbolic problem (\ref{hyperbolic-eq}), $\rho(t)=t^2$.)
The next theorem is the main result of the present section, and gives the desired upper and lower bounds for $N_t(\xi)$. Unfortunately, for the lower bound, we had to introduce an additional condition of boundedness on the ratio between the imaginary part and the real part of the characteristic exponent $\Psi(\xi)$. A similar difficulty has been encountered in \cite{khoshnevisan-xiao09} for obtaining a lower bound for the ``sojourn operator''. Our condition (\ref{cond-Im-Re}) is similar to condition (3.3) of \cite{khoshnevisan-xiao09}, and is trivially
satisfied when $\Psi$ is real-valued.
We use the following inequality: there exists a constant $b_H>0$, such that \begin{equation} \label{MMV-ineq} \alpha_H \int_{\mathbb{R}}
\int_{\mathbb{R}} |\varphi(r)| |\varphi(s)||r-s|^{2H-2}dr ds \leq b_H^2 \left(
\int_{\mathbb{R}} |\varphi(s)|^{1/H}ds \right)^{2H} \end{equation} for any $\varphi \in L^{1/H}(\mathbb{R})$. This inequality was proved in \cite{MMV01} and is a consequence of the Littlewood-Hardy inequality.
For complex-valued functions $\varphi$, we define:
$$ \|\varphi \|_{\mathcal{H}(0,t)}^2:=\alpha_H \int_0^t
\int_0^t \varphi(r) \overline{\varphi(s)}|r-s|^{2H-2}dr ds =\|{\rm Re} \varphi \|_{\mathcal{H}(0,t)}^2+ \|{\rm Im} \varphi\|_{\mathcal{H}(0,t)}^{2}.$$
\begin{theorem} \label{bounds-Nt-xi} For any $t>0$ and $\xi \in \mathbb{R}^d$, $$N_t(\xi) \leq C_H \left(\frac{1}{1/t+ {\rm Re} \Psi(\xi)} \right)^{2H},$$ where $C_H =H^{2H} b_H^2 e^{2}$. If in addition, there exists a constant $K>0$ such that: \begin{equation} \label{cond-Im-Re}
|{\rm Im} \Psi (\xi)| \leq K {\rm Re} \Psi(\xi), \quad \forall \xi \in \mathbb{R}^d. \end{equation} then, for any $t>0$ and $\xi \in \mathbb{R}^d$, $$N_t(\xi) \geq C_{H,K} \left(\frac{1}{1/t+ {\rm Re} \Psi(\xi)} \right)^{2H},$$ where $C_{H,K}$ is a positive constant depending on $H$ and $K$. \end{theorem}
\noindent {\bf Proof:} For the upper bound, we note that $N_t(\xi)$ can be written as
$$N_t(\xi)=\alpha_H \int_0^{t} \int_0^t e^{-r {\rm Re} \Psi(\xi)} e^{-s {\rm Re} \Psi(\xi)} |r-s|^{2H-2}\cos[(r-s){\rm Im} \Psi(\xi)] dr ds.$$
Using the fact that $|\cos x| \leq 1$ and $e^{-r {\rm Re} \Psi(\xi)} \leq e^{t/\lambda} e^{-r(1/\lambda+{\rm Re} \Psi(\xi))}$ for any $r \in [0,t]$, we get:
$$N_t(\xi) \leq e^{2t/\lambda} \alpha_H \int_0^{t} \int_0^t e^{-r(1/\lambda +{\rm Re} \Psi(\xi))} e^{-s(1/\lambda +{\rm Re} \Psi(\xi))} |r-s|^{2H-2}dr ds $$
By (\ref{MMV-ineq}), it follows that: $$N_t(\xi) \leq e^{2t/\lambda} b_H^2 \left(\int_0^t e^{-r(1/\lambda+{\rm Re} \Psi(\xi))/H}dr \right)^{2H} \leq b_H^2 e^{2t/\lambda}\left(\frac{H}{1/\lambda+ {\rm Re} \Psi(\xi)}\right)^{2H}.$$ The conclusion follows by taking $\lambda=t$.
For the lower bound, suppose first that $t {\rm Re} \Psi(\xi) \leq a$, for some constant $a=a_K \in (0,1)$
such that $Ka < \pi/2$. By (\ref{cond-Im-Re}), $$t|{\rm Im} \Psi(\xi)| \leq K t {\rm Re} \Psi (\xi) \leq Ka <\frac{\pi}{2}.$$
Using the fact that $e^{-x} \geq 1-x$ for $x>0$, we obtain: for any $r \in [0,t]$, $$e^{-r {\rm Re} \Psi(\xi)} \geq 1-r {\rm Re} \Psi (\xi)\geq 1-a.$$
Since $\cos$ is decreasing on the interval $[0,\frac{\pi}{2}]$, for any $0<s<r<t$, $$\cos[(r-s)|{\rm Im} \Psi(\xi)|] \geq \cos [t|{\rm Im} \Psi(\xi)|] \geq \cos (Ka)>0.$$ Therefore, \begin{eqnarray*}
N_t(\xi)&=&2\alpha_H\int_0^{t} \int_0^r e^{-r {\rm Re} \Psi(\xi)} e^{-s {\rm Re} \Psi(\xi)} (r-s)^{2H-2}\cos[(r-s)|{\rm Im} \Psi(\xi)|] ds dr \\ & \geq & (1-a)^2 \cos (Ka) 2\alpha_H \int_0^{t} \int_0^r (r-s)^{2H-2} ds dr \\ &=& (1-a)^2 \cos (Ka) t^{2H} \geq (1-a)^2 \cos (Ka) \left(\frac{1}{1/t+{\rm Re} \Psi(\xi)} \right)^{2H}, \end{eqnarray*} where for the last inequality we used the fact that $t \geq \frac{1}{t^{-1}+{\rm Re} \Psi(\xi)}$.
Suppose next that $t {\rm Re} \Psi(\xi) \geq a$. Note that
$$N_t(\xi)=\|e^{- \cdot \Psi(\xi)} \|_{\mathcal{H}(0,t)}^2.$$
Using Lemma \ref{H-norm-exp} (Appendix B) for expressing the $\mathcal{H}(0,t)$-norm of the exponential function in the spectral domain, we obtain:
$$N_t(\xi)=c_{H} \int_{\mathbb{R}} \frac{\sin^2[(\tau+ {\rm Im} \Psi(\xi))t]+\{e^{-t {\rm Re} \Psi(\xi)}-\cos[(\tau+{\rm Im} \Psi(\xi))t]\}^2}{[{\rm Re} \Psi(\xi)]^2 + [\tau+ {\rm Im} \Psi(\xi)]^2} |\tau|^{-(2H-1)} d\tau.$$
We denote $$T=t {\rm Re} \Psi(\xi) \quad \mbox{and} \quad b=\frac{{\rm Im} \Psi(\xi)}{{\rm Re} \Psi(\xi)}.$$ Using the change of variable $\tau'=\tau/{\rm Re} \Psi(\xi)$, we obtain that:
\begin{equation} \label{new-form-Ntxi-par}
N_t(\xi)=\frac{c_H}{[{\rm Re} \Psi(\xi)]^{2H}} \int_{\mathbb{R}}\frac{|\tau|^{-(2H-1)}}{1+(\tau+b)^2}[f_T^2 (\tau)+g_T^{2}(\tau)] d\tau, \end{equation} where $f_T(\tau)=\sin [(\tau+b)T]$ and $g_T(\tau)=e^{-T}-\cos[(\tau+b)T]$.
From the proof of Lemma \ref{H-norm-exp} (Appendix B), we know that:
$$\frac{1}{1+(\tau+b)^2}[f_T^2 (\tau)+g_T^{2}(\tau)]=|\mathcal{F}_{0,T} \varphi(\tau)|^2,$$ where $\varphi(x)=e^{-x(1+ib)}$.
Let $\rho>K$ be positive constant whose value will be specified later. Since the integrand of (\ref{new-form-Ntxi-par}) is non-negative, the integral can be bounded below by the integral over the region $|\tau| \leq \rho$. In this region, $|\tau|^{-(2H-1)} \geq \rho^{-(2H-1)}$. We obtain: \begin{equation} \label{LB-Nt-heat-step1} N_t(\xi) \geq \frac{c_H \rho^{-(2H-1)}}{[{\rm Re} \Psi(\xi)]^{2H}}
\left(I(T) - \int_{|\tau| \geq \rho}\frac{1}{1+(\tau+b)^2}[f_T^2 (\tau)+g_T^{2}(\tau)] d\tau \right), \end{equation} where \begin{equation} \label{Planch-id-heat}
I(T):=\int_{\mathbb{R}}\frac{1}{1+(\tau+b)^2}[f_T^2 (\tau)+g_T^{2}(\tau)] d\tau=2 \pi \int_0^T |e^{-x(1+ib)}|^2 dx=\pi(1-e^{-2T}), \end{equation} by Plancherel's theorem.
Using (\ref{Planch-id-heat}), we obtain the lower bound: \begin{equation} \label{LB-Nt-heat-step2} I(T) \geq \pi(1-e^{-2a}), \quad \mbox{since} \quad T \geq a. \end{equation}
To find an upper bound for the second integral on the right-hand side of (\ref{LB-Nt-heat-step1}), we use the fact that: $$f_T^2(\tau)+g_T^2(\tau) \leq 5, \quad \forall \tau \in \mathbb{R}.$$
It follows that: \begin{equation} \label{LB-Nt-heat-step3}
\int_{|\tau| \geq \rho} \frac{f_T^{2}(\tau)+g_T^{2}(\tau)}{1+(\tau+b)^2}d\tau \leq \int_{|\tau| \geq \rho}\frac{5}{(\tau+b)^2}d\tau =
\frac{10\rho}{\rho^2-b^2} \leq \frac{10\rho}{\rho^2-K^2}, \end{equation}
since $|b| \leq K$ (by (\ref{cond-Im-Re})). We choose $\rho=\rho_K$ large enough such that $$C_K:=\pi(1-e^{-2a})-\frac{10\rho}{\rho^2-K^2}>0.$$
Using (\ref{LB-Nt-heat-step1}), (\ref{LB-Nt-heat-step2}) and (\ref{LB-Nt-heat-step3}), we obtain: $$N_t(\xi) \geq C_K \frac{c_H \rho^{-(2H-1)}}{[{\rm Re} \Psi(\xi)]^{2H}} \geq C_{K} c_H \rho^{-(2H-1)} \left(\frac{1}{1/t+{\rm Re \Psi(\xi)}} \right)^{2H}.$$
The conclusion follows, letting $$C_{H,K}=\min\left\{(1-a)^2 \cos (Ka), C_{K} c_H \rho^{-(2H-1)} \right\}.$$
$\Box$
The following result is an immediate consequence of Theorem \ref{bounds-Nt-xi}.
\begin{corollary} \label{corol-bounds-Nt-xi} a) For any $t>0, \varphi \in \mathcal{S}(\mathbb{R}^d)$,
$$E|u(t,\varphi)|^2 \leq C_H \mathcal{E}(t;\varphi),$$ where $C_H=H^{2H} b_H^2 e^{2}$. Hence, $M_t \supset \mathcal{Z}_t$ for all $t>0$, and $M \supset \mathcal{Z}$.
b) If (\ref{cond-Im-Re}) holds, then for any $t>0, \varphi \in \mathcal{S}(\mathbb{R}^d)$,
$$E|u(t,\varphi)|^2 \geq C_{H,K} \mathcal{E}(t;\varphi),$$ where $C_{H,K}$ is a positive constant depending on $H$ and $K$. Hence, $M_t=\mathcal{Z}_t$ for all $t>0$, and $M=\mathcal{Z}$. \end{corollary}
\noindent {\bf Proof:} We use Theorem \ref{bounds-Nt-xi} and the definitions (\ref{def-norm-2bars}) and (\ref{def-norm-3bars}) of the norms $\|\cdot \|_t$, respectively $|\| \cdot \||_t$. $\Box$
The next result gives the necessary and sufficient condition for $\delta_x \in \mathcal{Z}_1$ for all $x \in \mathbb{R}^d$.
\begin{theorem} \label{nec-suf-delta-in-Z1} In order that $\delta_x \in \mathcal{Z}_1$ for all $x \in \mathbb{R}^d$, it is necessary and sufficient that condition (\ref{parabolic-cond}) holds. \end{theorem}
\noindent {\bf Proof:} Suppose first that (\ref{parabolic-cond}) holds. To show that $\delta_x \in \mathcal{Z}_1$ for all $x \in \mathbb{R}^d$, we use an argument similar to the proof of Theorem 2 of \cite{dalang99}.
Let $\mathcal{Z}_0$ be the set Schwartz distributions $\varphi$ such that $\mathcal{F} \varphi$ is a function and
$$|\|\varphi \||_1:=\int_{\mathbb{R}^d} \left(\frac{1}{1+{\rm Re} \Psi(\xi)} \right)^{2H}|\mathcal{F} \varphi(\xi)|^2 \mu(d\xi)<\infty.$$
Note that $\mathcal{S}(\mathbb{R}^d) \subset \mathcal{Z}_0$ and the definition of $|\| \cdot \| |_1$ agrees on $\mathcal{S}(\mathbb{R}^d)$ with the one given by (\ref{def-norm-3bars}). Therefore, to show that a distribution $\varphi \in \mathcal{Z}_0$ is in $\mathcal{Z}_1$, it suffices to show that there exists a sequence $(\varphi_n)_{n \geq 1} \subset \mathcal{S}(\mathbb{R}^d)$ such that $|\|\varphi_n-\varphi \||_1 \to 0$. We apply this to $\varphi=\delta_x$. In this case, $\mathcal{F} \varphi(\xi)=e^{-i \xi \cdot x}$, $|\mathcal{F} \varphi (\xi)|=1$ for all $\xi \in \mathbb{R}^d$, and
$|\| \varphi \||_1$ coincides with the integral of (\ref{parabolic-cond}).
Let $\varphi_n=\varphi * \phi_n \in \mathcal{S}(\mathbb{R}^d)$, where $\phi_n(x)=n^d \phi(nx)$ and $\phi \in \mathcal{S}(\mathbb{R}^d)$ is such that $\phi \geq 0$ and $\int_{\mathbb{R}^d}\phi(x)dx=1$. Then $\mathcal{F} \varphi_n(\xi)=\mathcal{F} \varphi (\xi) \mathcal{F} \phi_n(\xi)$ and \begin{eqnarray*}
|\|\varphi_n-\varphi \||_1 &=& \int_{\mathbb{R}^d}\left(\frac{1}{1+ {\rm Re} \Psi(\xi)} \right)^{2H} |\mathcal{F} \varphi_n (\xi) -\mathcal{F} \varphi (\xi)|^2 \mu(d\xi) \\
&=& \int_{\mathbb{R}^d}\left(\frac{1}{1+ {\rm Re} \Psi(\xi)} \right)^{2H} |\mathcal{F} \phi_n (\xi) -1|^2 \mu(d\xi) \to 0, \end{eqnarray*}
by the Dominated Convergence Theorem, since $|\mathcal{F} \phi_n(\xi)| \leq 1$ for all $\xi \in \mathbb{R}^d$.
For the reverse implication, suppose that $\delta_x \in \mathcal{Z}_1$ for all $x \in \mathbb{R}^d$. To show that (\ref{parabolic-cond}) holds, one can use the same argument as in the proof of Lemma 4.2 of \cite{FK10}. We omit the details. $\Box$
The following result concludes our discussion about the existence of a random-field solution.
\begin{theorem} \label{th-existence-parabolic} {\rm (Existence of Solution in the Parabolic Case)}
a) If (\ref{parabolic-cond}) holds, then equation (\ref{parabolic-eq}) has a random-field solution.
b) Suppose that (\ref{cond-Im-Re}) holds. If (\ref{parabolic-eq}) has a random-field solution, then (\ref{parabolic-cond}) holds. \end{theorem}
\noindent {\bf Proof:} a) Suppose that (\ref{parabolic-cond}) holds. By Theorem \ref{nec-suf-delta-in-Z1}, $\delta_x \in \mathcal{Z}_1$ for all $x \in \mathbb{R}^d$. By Corollary \ref{corol-bounds-Nt-xi}.a), $\mathcal{Z}_1=\mathcal{Z} \subset M$. Hence (\ref{delta-x-in-M}) holds.
b) Suppose that (\ref{delta-x-in-M}) holds. By Corollary \ref{corol-bounds-Nt-xi}.b), $M=\mathcal{Z}=\mathcal{Z}_1$. Hence $\delta_x \in \mathcal{Z}_1$ for all $x \in \mathbb{R}^d$. By Theorem \ref{nec-suf-delta-in-Z1}, (\ref{parabolic-cond}) holds. $\Box$
\begin{example} \label{stable-ex}{\rm (Stable processes)}
{\rm Suppose that $\mathcal{L}=-(-\Delta)^{\beta/2}$ for $\beta \in (0,2]$. Then $(X_t)_{t \geq 0}$ is a rotation invariant strictly $\beta$-stable process on $\mathbb{R}^d$, and $\Psi(\xi)=c_{\beta}|\xi|^{\beta}$ (see Theorem 14.14 in \cite{sato99} and Example 30.6 in \cite{sato99}). It can be shown that $(X_t)_{t \geq 0}$ is subordinate to the Brownian motion on $\mathbb{R}^d$ by a strictly ($\beta/2$)-stable subordinator (see Example 32.7 of \cite{sato99}).
In this case, condition (\ref{parabolic-cond}) becomes: \begin{equation} \label{cond-stable-proc}
\int_{\mathbb{R}^d} \left(\frac{1}{1+|\xi|^{\beta}} \right)^{2H}\mu(d\xi)<\infty. \end{equation}
We consider two kernels:
(i) $f(x)=c_{\alpha,d}|x|^{-(d-\alpha)}$ for $0<\alpha<d$. In this case, $\mu(d\xi)=|\xi|^{-\alpha}d\xi$ (see p.117 of \cite{stein70}), and condition (\ref{cond-stable-proc}) is equivalent to \begin{equation} \label{cond-H-alpha-beta} 2H \beta>d-\alpha. \end{equation}
(ii) $f(x)=\prod_{i=1}^{d}(\alpha_{H_i}|x_i|^{2H_i-2})$. In this case, $\mu(d\xi)=\prod_{i=1}^{n}c_{H_i}|\xi_i|^{-(2H_i-1)}$, and condition (\ref{cond-stable-proc}) is equivalent to $$2H \beta >d-\sum_{i=1}^{d}(2H_i-1).$$ } \end{example}
\begin{remark} {\rm (Fractional Powers of the Laplacian)} \label{stable-rem} {\rm As in \cite{dalang-mueller03} and \cite{dalang-sanzsole05}, we can consider also the case $\mathcal{L}=-(-\Delta)^{\beta/2}$ for arbitrary $\beta>0$, even if there is no corresponding L\'evy process whose generator is $\mathcal{L}$. Note that the fundamental solution $G$ of $\partial_t u- \mathcal{L} u=0$ exists and satisfies:
$$\mathcal{F} G (t,\xi)=\exp(-c_{\beta}t|\xi|^{\beta}).$$
Using Theorem 2.1 of \cite{BT10-SPA} and estimates similar to those given by Theorem \ref{bounds-Nt-xi} above, one can show that a random field solution of (\ref{parabolic-eq}) (in the sense of \cite{BT10-SPA}) exists if and only if (\ref{cond-stable-proc}) holds. } \end{remark}
\subsection{A Maximum Principle}
Throughout this section, we assume that $p_t \in L^2(\mathbb{R}^d)$ for all $t>0$, and \begin{equation} \label{mu-has-density} \mbox{$\mu$ has a (non-negative) density $g$,} \end{equation} i.e. $f$ is a kernel of positive type (see Definition 5.1 of \cite{khoshnevisan-xiao09}).
We consider the symmetric L\'evy process $\bar{X}=(\bar{X}_t)_{t \geq 0}$ defined by: $$\bar{X}_t:=X_t-\tilde{X}_t,$$ where $(\tilde{X}_t)_{t \geq 0}$ is an independent copy of $(X_t)_{t \geq 0}$. We denote by $(\bar{P}_t)_{t \geq 0}$ the semigroup of $(\bar{X}_t)_{t \geq 0}$, i.e. $$(\bar{P}_t \phi)(x)=\int_{\mathbb{R}^d}\phi(y)\bar{p}_t(x-y)dy,$$ where $\bar{p}_t=p_t*\tilde{p}_t$. From (\ref{Fourier-G}), it follows that $\mathcal{F} \bar{p}_{t}(\xi)=e^{-2t {\rm Re} \Psi(\xi)}$.
Let $(\bar{R}_{\alpha})_{\alpha>0}$ be the resolvent of $(\bar{P}_t)_{t \geq 0}$, i.e. $$(\bar{R}_{\alpha}\phi)(x)=\int_{0}^{\infty}e^{-\alpha s} (\bar{P}_s \phi)(x)ds.$$
The following maximum principle has been obtained recently in \cite{FK10}: \begin{equation} \label{max-principle-FK} (\bar{R}_{\alpha}f)(0)=\sup_{x \in \mathbb{R}^d}(\bar{R}_{\alpha}f)(x)=\Upsilon(\alpha):=\int_{\mathbb{R}^d} \frac{1}{\alpha+2{\rm Re} \Psi(\xi)}\mu(d\xi). \end{equation}
\begin{remark} \label{remark-kernel-of-positive-type} {\rm Recall that $f=\mathcal{F} g$ in $\mathcal{S}'(\mathbb{R}^d)$. The authors of \cite{FK10} work with the Fourier transform $\mathcal{F} f$ instead of $g$, which introduces an additional factor $(2\pi)^{-d}$. To see this, note that by the Fourier inversion theorem on $\mathcal{S}(\mathbb{R}^d)$, relation (\ref{def-Fourier-mu}) becomes: for any $\phi \in \mathcal{S}(\mathbb{R}^d)$, $$\int_{\mathbb{R}^d} \phi(\xi) g(\xi) d\xi=\frac{1}{(2\pi)^d} \int_{\mathbb{R}^d}f(x)\mathcal{F} \phi(x)dx.$$ This shows that $g=(2\pi)^{-d} \mathcal{F} f$ in $\mathcal{S}'(\mathbb{R}^d)$.
} \end{remark}
Note that $\Upsilon(\alpha)<\infty$ for all $\alpha>0$ if and only if $\Upsilon(\alpha)<\infty$ for some $\alpha>0$. An important consequence of (\ref{max-principle-FK}) (combined with the results of \cite{dalang99}) is that the potential-theoretic condition: $$(\bar{R}_{\alpha}f)(0)<\infty \quad \mbox{for all} \quad \alpha>0$$ is necessary and sufficient for the existence of a random field solution of (\ref{parabolic-eq}), when the Gaussian noise $W$ is white in time (i.e. $H=1/2$).
In the present article, we develop a maximal principle similar to (\ref{max-principle-FK}), which has a connection with the existence of a random field solution of (\ref{parabolic-eq}), when the noise $W$ is fractional in time.
We define the following ``fractional analogue'' of the resolvent operator:
$$(\bar{R}_{\alpha,H} \phi)(x)=\alpha_H \int_0^{\infty} \int_0^{\infty}|r-s|^{2H-2}e^{-\alpha (r+s)} (\bar{P}_{r+s}\phi)(x)dr ds,$$ and we let
$$\Upsilon_{H}(\alpha):=\alpha_H \int_{\mathbb{R}^d} \int_0^{\infty} \int_0^{\infty} |r-s|^{2H-2} e^{-(\alpha+ 2{\rm Re}\Psi(\xi)) (r+s)} dr ds \mu(d\xi).$$
As in \cite{khoshnevisan-xiao09}, we assume that $f$ satisfies the following condition: \begin{equation} \label{cond-on-f} \mbox{$f(x)<\infty$ if and only if $x \not =0$.} \end{equation} Under this condition, the following harmonic-analysis result holds: \begin{equation} \label{harmonic-result} \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \varphi(x)\psi(y)f(x-y)dx dy=\int_{\mathbb{R}^d} \mathcal{F} \varphi(\xi)\overline{\mathcal{F} \psi(\xi)}g(\xi)d\xi, \end{equation} for any non-negative functions $\varphi,\psi \in L^1(\mathbb{R}^d)$ (see Lemma 5.6 of \cite{khoshnevisan-xiao09}).
\begin{theorem}{\rm (A maximum principle)} \label{max-principle} If (\ref{cond-on-f}) holds, then for any $\alpha>0$, $$(\bar{R}_{\alpha,H}f)(0)=\sup_{x \in \mathbb{R}^d} (\bar{R}_{\alpha,H}f)(x)=\Upsilon_{H}(\alpha).$$ \end{theorem}
The proof of Theorem \ref{max-principle} follows from Lemma \ref{max-principle-step3} and Lemma \ref{max-principle-step4} below. Before this, we need some intermediate results.
Let $C_0(\mathbb{R}^d)$ be the space of continuous functions which vanish at infinity.
\begin{lemma} \label{max-principle-step1} For any $\phi \in \mathcal{S}(\mathbb{R}^d)$, we have: \\ a) $f*\phi \in C_0(\mathbb{R}^d)$ and $\bar{R}_{\alpha,H}(f*\phi) \in C_0(\mathbb{R}^d)$ for any $\alpha>0$;\\ b) $f*\phi \in L^2(\mathbb{R}^d)$ and $\mathcal{F} (f*\phi)(\xi)=(2\pi)^d \mathcal{F} \phi(\xi) g(\xi)$. \end{lemma}
\noindent {\bf Proof:} a) Since $f$ is tempered (i.e. $f(x) \leq C (1+|x|)^k$ for all $x \in \mathbb{R}^d$, for some $k \geq 0,C>0$), the function $f*\phi$ is well-defined. By (\ref{def-Fourier-mu}) and (\ref{mu-has-density}), \begin{equation} \label{formula-f*phi} (f*\phi)(x)=\int_{\mathbb{R}^d}f(y)\phi(x-y)dy=\int_{\mathbb{R}^d}e^{-i \xi \cdot x} \overline{\mathcal{F} \phi(\xi)}g(\xi)d\xi. \end{equation} Since $g$ is tempered, $(\overline{\mathcal{F} \phi}) g \in L^p(\mathbb{R}^d)$ for any $p \geq 1$. By Riemann-Lebesgue lemma, $f*\phi \in C_0(\mathbb{R}^d)$. Finally, we note that $\bar{R}_{\alpha,H} :C_0(\mathbb{R}^d) \to C_0(\mathbb{R}^d)$.
b) By (\ref{formula-f*phi}), $f*\phi=\mathcal{F} h$, where $h:=(\overline{\mathcal{F} \phi}) g \in L^2(\mathbb{R}^d)$. Hence, $f*\phi \in L^2(\mathbb{R}^d)$. By the Fourier inversion formula in $L^2(\mathbb{R}^d)$ (see e.g. p.222 of \cite{folland92}), $$\mathcal{F} (f*\phi)(\xi)=(2\pi)^{d}\overline{h(\xi)}=(2\pi)^d {\mathcal{F} \phi}(\xi)g(\xi).$$ $\Box$
\begin{lemma} \label{max-principle-step2} For any $\phi \in \mathcal{S}(\mathbb{R}^d)$ and $x \in \mathbb{R}^d$,
$$(\bar{R}_{\alpha,H}(f*\phi))(x)=\alpha_H \int_{\mathbb{R}^d} e^{-i\xi \cdot x}\overline{\mathcal{F} \phi(\xi)} \int_{\mathbb{R}_{+}^2} |r-s|^{2H-2} e^{-(\alpha+2{\rm Re} \Psi(\xi))(r+s)}dr ds \mu(d\xi).$$
Consequently, for any $\phi \in \mathcal{S}(\mathbb{R}^d)$ with $\|\phi\|_1=1$,
$$|(\bar{R}_{\alpha,H}(f*\phi))(x)| \leq \Upsilon_{H}(\alpha), \quad \forall x \in \mathbb{R}^d.$$ \end{lemma}
\noindent {\bf Proof:} Since $\bar{P}_{r+s}=\bar{P}_r \bar{P}_s$, we have: \begin{equation} \label{calcul-P-st} (\bar{P}_{r+s}(f*\phi))(x)=\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} (f*\phi)(y-z)\bar{p}_r(x-y) \bar{p}_s(z) dy dz. \end{equation} Using Lemma \ref{version-Plancherel} (Appendix C) with $\varphi=f*\phi$, $\psi_1=\bar{p}_r(x-\cdot)$ and $\psi_2=\bar{p}_s$, $$(\bar{P}_{r+s}(f*\phi))(x)=\frac{1}{(2\pi)^d} \int_{\mathbb{R}^d} e^{-i \xi \cdot x} \overline{\mathcal{F} \bar{p}_r(\xi)} \ \overline{\mathcal{F} \bar{p}_s(\xi)} \ \overline{\mathcal{F}(f*\phi)(\xi)}d\xi.$$ The result follows using Lemma \ref{max-principle-step1}.b), the fact that $\mathcal{F} \bar{p}_{r}(\xi)=e^{-2r {\rm Re} \Psi(\xi)}$, and Fubini's theorem. $\Box$
\begin{lemma} \label{max-principle-step3} For any $\alpha>0$, $$\Upsilon_{H}(\alpha)=\sup_{x \in \mathbb{R}^d}(\bar{R}_{\alpha,H}f)(x)=\limsup_{x \to 0} (\bar{R}_{\alpha,H}f)(x).$$ \end{lemma}
\noindent {\bf Proof:} The proof is similar to Proposition 3.5 of \cite{FK10}.
By Fatou's lemma and Lemma \ref{max-principle-step2}, for any $x \in \mathbb{R}^d$, $$(\bar{R}_{\alpha,H}f)(x) \leq \liminf_{n \to \infty}(\bar{R}_{\alpha,H}(f*\phi_n))(x) \leq \Upsilon_H(\alpha),$$ where $(\phi_n)_{n \geq 1}$ is a sequence of approximations to the identity, consisting of probability density functions in $\mathcal{S}(\mathbb{R}^d)$. Hence, $$\sup_{x \in \mathbb{R}^d}(\bar{R}_{\alpha,H}f)(x) \leq \Upsilon_{H}(\alpha).$$
For the reverse inequality, we let $\phi_n(x)=(2\pi)^{-d/2}n^{d/2}\exp(-n|x|^2/2)$. By Lemma \ref{max-principle-step2},
$$(\bar{R}_{\alpha,H}(f*\phi_n))(0)=\alpha_H \int_{\mathbb{R}^d} e^{-|\xi|^2/(2n)}\int_{\mathbb{R}_{+}^2}|r-s|^{2H-2} e^{-(\alpha+2{\rm Re}\Psi(\xi))(r+s)} dr ds \mu(d\xi),$$ and therefore, by applying the monotone convergence theorem, \begin{equation} \label{limit-R-n-0} \lim_{n \to \infty}(\bar{R}_{\alpha,H}(f*\phi_n))(0)=\Upsilon_{H}(\alpha). \end{equation}
Using (\ref{calcul-P-st}) and the symmetry of the function $\phi_n$, we obtain: $$(\bar{P}_{r+s}(f*\phi_n))(0)=\int_{\mathbb{R}^d}(\bar{P}_{r+s}f)(x)\phi_n(x) dx.$$ Therefore, \begin{equation} \label{calcul-R-n-0} (\bar{R}_{\alpha,H}(f*\phi_n))(0)=\int_{\mathbb{R}^d}(\bar{R}_{\alpha,H}f)(x)\phi_n(x) dx \leq \sup_{x \in \mathbb{R}^d} (\bar{R}_{\alpha,H}f)(x). \end{equation} From (\ref{limit-R-n-0}) and (\ref{calcul-R-n-0}), we infer that $\Upsilon_{H}(\alpha) \leq \sup_{x \in \mathbb{R}^d} (\bar{R}_{\alpha,H}f)(x)$.
The last assertion follows by taking $\phi_n$ with the support in the ball of radius $1/n$ and center $0$. $\Box$
\begin{lemma} \label{max-principle-step4} If $f$ satisfies (\ref{cond-on-f}), then for any $\alpha>0$, $$(\bar{R}_{\alpha,H}f)(0)=\Upsilon_{H}(\alpha).$$ \end{lemma}
\noindent {\bf Proof:}
Using (\ref{harmonic-result}), we have: $$(\bar{P}_{r+s}f)(0)=\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\bar{p}_r(x)\bar{p}_{s}(y)f(x-y)dx dy=\int_{\mathbb{R}^d}e^{-2(r+s){\rm Re} \Psi(\xi)}g(\xi)d\xi.$$ The conclusion follows from the definitions of $(\bar R_{\alpha,H}f)(0)$ and $\Upsilon_{H}(\alpha)$.
$\Box$
To investigate the connection with the parabolic problem (\ref{parabolic-eq}), we let $$\Upsilon_H^{*}(\alpha)=\int_{\mathbb{R}^d} \left(\frac{1}{\alpha+2{\rm Re}\Psi(\xi)} \right)^{2H}\mu(d\xi).$$ By Lemma \ref{lemmaA} (Appendix A), $\Upsilon_H^{*}(\alpha)<\infty$ for all $\alpha>0$ if and only if $\Upsilon_H^{*}(\alpha)<\infty$ for some $\alpha>0$.
The following result gives the relationship between $\Upsilon_H(\alpha)$ and $\Upsilon_H^*(\alpha)$.
\begin{lemma} \label{ineq-Upsilon} For any $\alpha>0$, $$c_{\alpha,H} \Upsilon_H^*(\alpha) \leq \Upsilon_{H}(\alpha) \leq b_H^2 H^{2H} \Upsilon_{H}^{*}(\alpha),$$ where $c_{\alpha,H}=2^{-(2H+2)}[(\alpha \wedge 1)/(\alpha+3/2)]^{2H}$. \end{lemma}
\noindent {\bf Proof:} The second inequality follows by (\ref{MMV-ineq}). For the first inequality, we note that, since the integrand from the definition of $\Upsilon_H(\alpha)$ is non-negative, the integral $dr ds$ over $[0,\infty)^2$ can be bounded below by the integral over $[0,1]^2$. By Proposition 4.3 of \cite{BT10-SPA}, for any $t>0$ and $\lambda \geq 0$
$$\alpha_{H} \int_0^t \int_0^t |r-s|^{2H-2} e^{-\lambda (r+s)} dr ds \geq \frac{1}{4} (t^{2H} \wedge 1) \left( \frac{1}{2}\right)^{2H}\left( \frac{1}{1/2+\lambda}\right)^{2H}.$$ Applying this result for $t=1$ and $\lambda=\alpha+2{\rm Re} \Psi(\xi)$, we obtain:
$$\int_0^1 \int_0^1 |r-s|^{2H-2} e^{-(\alpha+2{\rm Re} \Psi(\xi))(r+s)} dr ds \geq \left( \frac{1}{2}\right)^{2H+2} \left(\frac{1}{1/2+\alpha+2{\rm Re} \Psi(\xi)} \right)^{2H}.$$ Hence, $$\Upsilon_H(\alpha) \geq \left(\frac{1}{2}\right)^{2H+2} \Upsilon_H^*(\alpha+1/2) \geq c_{\alpha,H} \Upsilon_H^*(\alpha),$$ where we used Lemma \ref{lemmaA} (Appendix A) for the second inequality. $\Box$
Recall that by Theorem \ref{th-existence-parabolic}, condition (\ref{parabolic-cond}) is the necessary and sufficient for problem (\ref{parabolic-eq}) to have a random field solution. As a consequence of the maximum principle, we obtain the following result.
\begin{corollary} \label{resolvent-cond-equiv} Suppose that $f$ satisfies (\ref{cond-on-f}). Then (\ref{parabolic-cond}) is equivalent to \begin{equation} \label{resolvent-cond} (\bar{R}_{\alpha,H}f)(0)<\infty \quad \mbox{for any} \ \alpha>0. \end{equation} \end{corollary}
\noindent {\bf Proof:} By Theorem \ref{max-principle}, (\ref{resolvent-cond}) holds if and only if $\Upsilon_H(\alpha)<\infty$ for any $\alpha>0$. By Lemma \ref{ineq-Upsilon}, this is equivalent to $\Upsilon_H^*(\alpha)<\infty$ for any $\alpha >0$, which in turn, is equivalent to (\ref{parabolic-cond}) (i.e. $\Upsilon_H^*(2)<\infty$), by Lemma \ref{lemmaA} (Appendix A). $\Box$
\subsection{Connection with the Intersection Local Time}
When the noise $W$ is white in time, the authors of \cite{FKN09} and \cite{FK10} noticed an interesting connection between the existence of a random field solution of problem (\ref{parabolic-eq}) and the existence of the occupation time $$L_t(f)=\int_0^t f(\bar{X}_s)ds.$$
In this section, we develop a similar connection in the case of the fractional noise, by considering the ``weighted'' intersection local time:
$$L_{t,H}(f)=\alpha_H \int_0^t \int_0^t |r-s|^{2H-2}f(\bar{X}_r^1-\bar{X}_s^2)dr ds,$$ where $(\bar{X}_t^1)_{t \geq 0}$ and $(\bar{X}_t^2)_{t \geq 0}$ are two independent copies of $(\bar{X}_t)_{t \geq 0}$.
Clearly for any fixed $t>0$, $E[L_{t,H}(f)]<\infty$ is a sufficient condition for $L_{t,H}(f)<\infty$ a.s., but the negligible set depends on $t$. Our result will show that under condition (\ref{resolvent-cond}), $L_{t,H}(f) <\infty$ for all $t>0$ a.s. To motivate this result, we consider first an example, in which we proceed to the calculation of $E[L_{t,H}(f)]$ in a particular case.
Note that $\bar{X}_r^1-\bar{X}_s^2 \stackrel{d}{=}\bar{X}_{r+s}$ for any $r,s \in [0,t]$,
and therefore, \begin{equation} \label{mean-Lt(f)}
E[L_{t,H}(f)]=\alpha_H \int_0^t \int_0^t |r-s|^{2H-2}E[f(\bar{X}_{r+s})]dr ds. \end{equation}
\begin{example}{\rm (Stable processes)} {\rm Refer to Example \ref{stable-ex}. Since $(\bar{X}_t)_{t \geq 0}$ is self-similar with exponent $1/\beta$ (see Theorem 13.5 of \cite{sato99}), $$\bar{X}_{r+s}\stackrel{d}{=} (r+s)^{1/\beta}\bar{X}_{1}.$$
Suppose in addition that $f(x)=|x|^{-(d-\alpha)}$ for $0<\alpha<d$. Then $E[f(\bar{X}_{r+s})]=E|\bar{X}_{r+s}|^{-(d-\alpha)}=c_{\alpha,d}(r+s)^{-(d-\alpha)/\beta}$, where
$c_{\alpha,d}=E|\bar{X}_1|^{-(d-\alpha)}$. By (\ref{mean-Lt(f)}),
$$E[L_{t,H}(f)]=\alpha_H c_{\alpha,d}\int_0^t \int_0^t |r-s|^{2H-2}(r+s)^{-(d-\alpha)/\beta}dr ds.$$ One can see that $E[L_{t,H}(f)]<\infty$ for any $t>0$ if and only if (\ref{cond-H-alpha-beta}) (or equivalently, (\ref{parabolic-cond})) holds. But by Corollary \ref{resolvent-cond-equiv}, (\ref{parabolic-cond}) is equivalent to (\ref{resolvent-cond}). } \end{example}
The previous example shows that the existence of $L_{t,H}(f)$ is related to the potential-theoretic condition (\ref{resolvent-cond}), which is in turn the necessary and sufficient condition for the existence of a random field solution to problem (\ref{parabolic-eq}) (by Theorem \ref{th-existence-parabolic} and Corollary \ref{resolvent-cond-equiv}). We will see below that this is a general phenomenon. For this, suppose that $$\bar{X}_0^1=x_1, \quad \bar{X}_0^2=x_2,$$ and let $P_{x_i}$ be the law of $\bar{X}^i$ for $i=1,2$. Then $P_{x_1,x_2}=P_{x_1} \times P_{x_2}$ is the law of $(\bar{X}^1,\bar{X}^2)$. We denote by $E_{x_1,x_2}$ the expectation under $P_{x_1,x_2}$.
The next result shows the existence of the intersection local time $L_{t,H}(f)$ under condition (\ref{resolvent-cond}).
\begin{theorem} {\rm (Connection with the Local Time)} Suppose that $f$ satisfies (\ref{cond-on-f}). If (\ref{resolvent-cond}) holds, then for any $x_1,x_2 \in \mathbb{R}^d$, $$P_{x_1,x_2}(L_{t,H}(f)<\infty \quad \mbox{for all} \ t>0)=1$$ $$P_{x_1,x_2}\left(\limsup_{t \to \infty}\frac{\log L_{t,H}(f)}{t} \leq 0\right)=1.$$ \end{theorem}
\noindent {\bf Proof:} We follow the lines of the proof of Theorem 3.13 of \cite{FK10}. Since $f$ is non-negative, it follows that for any $t>0$, \begin{equation} \label{bound-Lt-e}
e^{-2\alpha t} L_{t,H}(f) \leq \alpha_H \int_0^{\infty} \int_0^{\infty} e^{-\alpha (r+s)}|r-s|^{2H-2}f(\bar{X}_r^1-\bar{X}_s^2)drds. \end{equation}
Note that $$E_{x_1,x_2}[f(\bar{X}_r^1-\bar{X}_s^2)]=\int_{\mathbb{R}^d} \int_{\mathbb{R}^d}f(y-z)\bar{p}_{r}(x_1-y)\bar{p}_{s}(x_2-z)dy dz=(\bar{P}_{r+s}f)(x_1-x_2).$$ Taking supremum over $t$, and expectation with respect to $P_{x_1,x_2}$ in (\ref{bound-Lt-e}), we obtain: $$E_{x_1,x_2}[\sup_{t>0}(e^{-2\alpha t} L_{t,H}(f))] \leq (\bar{R}_{\alpha,H}f)(x_1-x_2).$$ From here, using Theorem \ref{max-principle} and condition (\ref{resolvent-cond}), we infer that: $$\sup_{x_1,x_2 \in \mathbb{R}^d}E_{x_1,x_2}[\sup_{t>0}(e^{-2\alpha t} L_{t,H}(f))] \leq \sup_{x \in \mathbb{R}^d} (\bar{R}_{\alpha,H}f)(x)=(\bar{R}_{\alpha,H}f)(0)<\infty.$$ The result follows. $\Box$
\section{The Hyperbolic Equation} \label{hyperbolic-section}
In this section we consider the hyperbolic problem (\ref{hyperbolic-eq}). Throughout this section, we assume that $X$ is symmetric, i.e. \begin{equation} \label{imaginary-zero} {\rm Im} \Psi(\xi)=0, \quad \mbox{for all} \ \xi \in \mathbb{R}^d. \end{equation} Since ${\rm Re} \Psi(\xi)=\Psi (\xi)$, we use the notation $\Psi(\xi)$ to simplify the writing.
To define the weak solution, we cannot use the same method as in the parabolic case, since in general, we may not be able to identify the fundamental solution $G$ of $\partial_{tt}u-\mathcal{L} u=0$. Note that in some particular cases, we are able to identify $G$ (see Remark \ref{stable-rem-hyp} below).
To circumvent this difficulty, we use the method of \cite{FKN09}, whose salient features we recall briefly below. Consider first the deterministic equation: \begin{equation} \label{deterministic-wave} \frac{\partial^2 u}{\partial t^2}(t,x)=\mathcal{L} u(t,x)+F(t,x), \end{equation} with zero initial conditions, where $F$ is a smooth function. By taking formally the Fourier transform in the $x$ variable, and using the fact that $\mathcal{F} {\mathcal{L}}=-\Psi$, we obtain that $\mathcal{F} u$ satisfies the following equation: \begin{equation} \label{deterministic-wave-Fourier} \frac{\partial^2 (\mathcal{F} u)}{\partial t^2}(t,\xi)=-\Psi(\xi)\mathcal{F} u(t,\xi)+\mathcal{F} F(t,\xi), \end{equation} with zero initial conditions. Equation (\ref{deterministic-wave-Fourier}) can be solved using Duhamel's principle. We obtain: $$\mathcal{F} u(t,\xi)=\frac{1}{\sqrt{\Psi(\xi)}} \int_0^t \sin (\sqrt{\Psi(\xi)}(t-s)) \mathcal{F} F(s,\xi)ds.$$ We apply formally the Fourier inversion formula. Multiplying by $\varphi \in \mathcal{S}(\mathbb{R}^d)$, and integrating $dx$, we arrive to the following (formal) definition of a weak solution of (\ref{deterministic-wave}): \begin{equation} \label{formal-solution-wave} u(t,\varphi)=\frac{1}{(2\pi)^{d}} \int_{0}^{t}\int_{\mathbb{R}^d}\frac{\sin (\sqrt{\Psi(\xi)}(t-s))}{\sqrt{\Psi(\xi)}}\overline{\mathcal{F} \varphi(\xi)} \mathcal{F} F(s,\xi) d\xi ds. \end{equation}
If instead of the smooth function $F$ we consider the random noise $\dot W$,
the integral above is replaced by a stochastic integral $\mathcal{F} W(ds,d\xi)$, where $\mathcal{F} W$ is a Gaussian process which we define below.
As in \cite{balan-tudor08}, we let $\mathcal{P}(\mathbb{R}^d)$ be the completion of $\mathcal{S}(\mathbb{R}^d)$ with respect to the inner product: \begin{eqnarray*} \langle \varphi_1, \varphi_2 \rangle_{\mathcal{P}(\mathbb{R}^d)} &:=&\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \varphi_1(x)\overline{\varphi_2(y)}f(x-y)dx dy \\ &=& \int_{\mathbb{R}^d} \mathcal{F} \varphi_1(\xi) \overline{\mathcal{F} \varphi_2(\xi)}\mu(d\xi). \end{eqnarray*}
Let $\widehat{\mathcal{P}(\mathbb{R}^d)}$ be the completion of $\mathcal{S}(\mathbb{R}^d)$ with respect to the inner product: \begin{equation} \label{def-product-hat-P} \langle \psi_1,\psi_2 \rangle_{\widehat{\mathcal{P}(\mathbb{R}^d)}}:= \langle \mathcal{F} \psi_1, \mathcal{F} \psi_2 \rangle_{\mathcal{P}(\mathbb{R}^d)}. \end{equation}
Note that if the noise $W$ is white in space, then by Plancherel theorem, $\langle \psi_1,\psi_2 \rangle_{\widehat{\mathcal{P}(\mathbb{R}^d)}}=(2\pi)^d\langle \psi_1,\psi_2 \rangle_{L^2(\mathbb{R}^d)}$ and $\widehat{\mathcal{P}(\mathbb{R}^d)}=\mathcal{P}(\mathbb{R}^d)=L^2(\mathbb{R}^d)$.
The following lemma gives a more direct way of calculating $\langle \psi_1,\psi_2 \rangle_{\widehat{\mathcal{P}(\mathbb{R}^d)}}$.
\begin{lemma} \label{direct-calcul-norm-hatHP} For any $\psi_1,\psi_2 \in \mathcal{S}(\mathbb{R}^d)$, $$\langle \psi_1,\psi_2 \rangle_{\widehat{\mathcal{P}(\mathbb{R}^d)}}=(2\pi)^{2d} \int_{\mathbb{R}^d} \psi_1(\xi)\overline{\psi_2(\xi)}\mu(d\xi).$$ \end{lemma}
\noindent {\bf Proof:} Note that for any $\varphi \in L^1(\mathbb{R}^d)$, $\overline{\mathcal{F} \varphi(\xi)}=\mathcal{F}^{-1}\overline{\varphi}(\xi)$, where $$\mathcal{F}^{-1} \varphi(\xi):=\int_{\mathbb{R}^d} e^{i \xi \cdot x}\varphi(x)dx, \quad \forall \xi \in \mathbb{R}^d.$$
We denote $\varphi_i:=\overline{\mathcal{F} \psi_i} \in \mathcal{S}(\mathbb{R}^d)$ for $i=1,2$. We obtain: \begin{eqnarray*} \langle \psi_1,\psi_2 \rangle_{\widehat{\mathcal{P}(\mathbb{R}^d)}} &=&
\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \overline{\varphi_1(x)}\varphi_2(y)f(x-y)dx dy \\ &=& \int_{\mathbb{R}^d} \overline{\mathcal{F} \varphi_1(\xi)}\mathcal{F} \varphi_2(\xi) \mu(d\xi) \\ &=& \int_{\mathbb{R}^d} \mathcal{F}^{-1} (\mathcal{F} \psi_1)(\xi) \overline{\mathcal{F}^{-1}(\mathcal{F} \psi_2)(\xi)} \mu(d\xi). \end{eqnarray*} By the Fourier inversion theorem, $\mathcal{F}^{-1}(\mathcal{F} \psi_i)=(2\pi)^d \psi_i$ for $i=1,2$. The result follows. $\Box$
We endow the space $\mathcal{E}$ with the inner product: $$\langle h_1,h_2 \rangle_{\widehat{\mathcal{H} \mathcal{P}}}:= \langle \mathcal{F} h_1, \mathcal{F} h_2 \rangle_{\mathcal{H} \mathcal{P}},$$ where $\mathcal{F}$ denotes the Fourier transform in the $x$ variable.
Note that by Lemma \ref{direct-calcul-norm-hatHP}, for any $h_1, h_2 \in \mathcal{E}$,
$$\langle h_1,h_2 \rangle_{\widehat{\mathcal{H} \mathcal{P}}}= \alpha_H (2\pi)^{2d} \int_{\mathbb{R}^d} \int_{0}^{\infty} \int_{0}^{\infty}|r-s|^{2H-2} h_1(r,\xi) \overline{h_2(s,\xi)} dr ds\mu(d\xi).$$
For any $h \in \mathcal{E}$, we define: $$\mathcal{F} W(h):= W(\mathcal{F} h).$$ By the isometry property of $W$, for any $h \in \mathcal{E}$,
$$E|\mathcal{F} W(h)|^2=E|W(\mathcal{F} h)|^2= \| \mathcal{F} h \|_{\mathcal{H} \mathcal{P}}^2=\|h\|_{\widehat{\mathcal{H} \mathcal{P}}}^2.$$
Let $\widehat{\mathcal{H} \mathcal{P}}$ be the completion of $\mathcal{E}$ with respect to the inner product $\langle \cdot, \cdot \rangle_{\widehat{\mathcal{H} \mathcal{P}}}$. The map $\mathcal{E} \ni h \mapsto \mathcal{F} W(h) \in L^2(\Omega)$ is an isometry which can be extended to $\widehat{\mathcal{H} \mathcal{P}}$. We denote this extension by: $$h \mapsto \int_0^{\infty}\int_{\mathbb{R}^d}h(t,\xi)\mathcal{F} W (dt,d\xi)=:\mathcal{F} W(h).$$
This gives the rigorous construction of the isonormal Gaussian process $\mathcal{F} W=\{\mathcal{F} W(h); h \in \widehat{\mathcal{H} \mathcal{P}}\}$ which was mentioned formally above.
The following result gives a criterion for a function $h$ to be in $\widehat{\mathcal{H} \mathcal{P}}$.
\begin{lemma} \label{criterion-phi-in-hatHP} Let $h: \mathbb{R}_{+} \times \mathbb{R}^d \to \mathbb{C}$ be a deterministic function such that $h(t,\cdot)=0$ if $t>T$. Suppose that $h$ satisfies the following conditions: \\ (i) $h(t, \cdot) \in L^2(\mathbb{R}^d)$ for all $t \in [0,T]$;\\ (ii) $\mathcal{F} h \in \mathcal{H} \mathcal{P}$, where $\mathcal{F} h$ denotes the Fourier transform in the $x$ variable. \\ Then $h \in \widehat{\mathcal{H} \mathcal{P}}$ and
$$\|h\|_{\widehat{\mathcal{H} \mathcal{P}}}^2= \alpha_H (2\pi)^{2d} \int_{\mathbb{R}^d} \int_{0}^{T} \int_{0}^{T}|r-s|^{2H-2} h(r,\xi) \overline{h(s,\xi)}dr ds \mu(d\xi).$$ In particular, the stochastic integral of $h$ with respect to the noise $\mathcal{F} W$ is well-defined. \end{lemma}
\noindent {\bf Proof:} Let $g=\mathcal{F} h$. By the Fourier inversion formula on $L^2(\mathbb{R}^d)$, $h(s,\xi)=(2\pi)^{-d}\overline{\mathcal{F}(s,\xi)}$, and hence, \begin{equation} \label{F-inversion-g} \mathcal{F} g(s,\xi)=(2\pi)^d \overline{h(s,\xi)}. \end{equation}
Since $g \in \mathcal{H} \mathcal{P}$, there exists a sequence $(g_n)_{n \geq 1}$ of the form $g_n(s,x)=\phi_n(s)\gamma_n(x)$, where $\phi_n$ is a linear combination of indicator functions $1_{[0,a]}, a \in [0,T]$ and $\gamma_n \in \mathcal{S}(\mathbb{R}^d)$, such that $\|g_n-g\|_{\mathcal{H} \mathcal{P}} \to 0$ (see \cite{BT10-SPA}).
Let $\psi_n:=(2\pi)^{-d}\overline{\mathcal{F} \gamma_n} \in \mathcal{S}(\mathbb{R}^d)$ and $h_n(s,\xi)=\phi_n(s)\psi_n(\xi)$. Then \begin{equation} \label{F-inversion-gn}\mathcal{F} g_n(s,\xi)=\phi_n(s) \mathcal{F} \gamma_n(\xi)=(2\pi)^d \phi_n(s) \overline{\psi_n(\xi)}=(2\pi)^d \overline{h_n(s,\xi)}. \end{equation}
Using (\ref{F-inversion-g}) and (\ref{F-inversion-gn}), we obtain that \begin{eqnarray*}
\lefteqn{\alpha_H (2\pi)^{2d} \int_{\mathbb{R}^d} \int_0^T \int_0^T |r-s|^{2H-2} (h_n-h)(r,\xi) \overline{ (h_n-h)(s,\xi)}dr ds \mu(d\xi) } \\
& & = \alpha_H \int_{\mathbb{R}^d} \int_0^T \int_0^T |r-s|^{2H-2} \overline{(\mathcal{F} g_n-\mathcal{F} g)(r,\xi)} (\mathcal{F} g_n-\mathcal{F} g)(s,\xi) dr ds \mu(d\xi)\\
& & =\|g_n-g\|_{\mathcal{H} \mathcal{P}}^2 \to 0. \end{eqnarray*} The conclusion follows. $\Box$
We now return to equation (\ref{hyperbolic-eq}). By analogy with (\ref{formal-solution-wave}), we say that the process $\{u(t,\varphi); t \geq 0,\varphi \in \mathcal{S}(\mathbb{R}^d)\}$ defined by: $$u(t,\varphi)=\frac{1}{(2\pi)^{d}} \int_{0}^{t} \int_{\mathbb{R}^d}\frac{\sin (\sqrt{\Psi(\xi)}(t-s))}{\sqrt{\Psi(\xi)}}\overline{\mathcal{F} \varphi(\xi)} \mathcal{F} W(ds,d\xi),$$ is a {\em weak solution} of (\ref{hyperbolic-eq}). The stochastic integral above is well-defined if and only if the integrand $$(s,\xi) \mapsto h_{t,\varphi}(s,\xi)=\frac{1}{(2\pi)^d} 1_{[0,t]}(s) \frac{\sin (\sqrt{\Psi(\xi)}(t-s))}{\sqrt{\Psi(\xi)}}\overline{\mathcal{F} \varphi(\xi)} \quad \mbox{belongs to $\widehat{\mathcal{H} \mathcal{P}}$}.$$
To check that $h_{t,\varphi} \in
\widehat{\mathcal{H} \mathcal{P}}$, it suffices to show that $h_{t,\varphi}$ satisfies conditions (i) and (ii) of Lemma \ref{criterion-phi-in-hatHP}. Condition (i) holds since $|\sin x| \leq |x|$ for any $x$. For (ii), we have to show that $g_{t,\varphi}:=\mathcal{F} h_{t,\varphi} \in \mathcal{H} \mathcal{P}$. For this we apply Theorem 2.1 of \cite{BT10-SPA}. Note that the function $(s,\xi) \mapsto \mathcal{F} g_{t,\varphi}(s,\xi)=(2\pi)^d \overline{h_{t,\varphi}(s,\xi)}$ satisfies conditions (i)-(iii) of this theorem. So, if suffices to show that: $$I_{t,\varphi}:=\alpha_H (2\pi)^{2d}\int_{\mathbb{R}^d} \int_0^{\infty}
\int_0^{\infty}|r-s|^{2H-2} \overline{h_{t,\varphi}(r,\xi)} h_{t,\varphi}(s,\xi) dr ds \mu(d\xi)<\infty.$$
Note that $$I_{t,\varphi}=
\int_{\mathbb{R}^d}N_t(\xi)|\mathcal{F} \varphi(\xi)|^2 \mu(d\xi),$$ where
$$N_t(\xi)=\frac{\alpha_H}{\Psi(\xi)} \int_0^t \int_0^t \sin (r\sqrt{\Psi(\xi)})\sin (s\sqrt{\Psi(\xi)})|r-s|^{2H-2} dr ds.$$
Using the fact that $|\sin x| \leq |x|$ for any $x$, it follows that $$N_t (\xi) \leq \alpha_H \int_0^t \int_0^t rs |r-s|^{2H-2}drds \leq t^{2H+2},$$ and hence $I_{t,\varphi} \leq
t^{2H+2} \int_{\mathbb{R}^d}|\mathcal{F} \varphi(\xi)|^2 \mu(d\xi)<\infty$. This proves that $u(t,\varphi)$ is a well-defined random variable in $L^2( \Omega)$, for any $t>0$ and $\varphi \in \mathcal{S}(\mathbb{R}^d)$. Moreover,
$$E|u(t,\varphi)|^2=E|\mathcal{F} W(h_{t,\varphi})|^2=\|h_{t,\varphi}\|_{\widehat{\mathcal{H} \mathcal{P}}}^2=I_{t,\varphi}.$$
To define the random field solution of (\ref{hyperbolic-eq}), we proceed as in the case of the parabolic equation. We define the norms: \begin{eqnarray*}
\|\varphi\|_t^2 &:=& E|u(t,\varphi)|^2 =\int_{\mathbb{R}^d} N_t(\xi) |\mathcal{F}
\varphi(\xi)|^2 \mu(d\xi)\\
|\|\varphi \||_{t}^2 &:=& \mathcal{E}(t;\varphi)=\int_{\mathbb{R}^d}
\left(\frac{1}{1/t+\Psi(\xi)} \right)^{H+1/2}|\mathcal{F}
\varphi(\xi)|^2\mu(d\xi). \end{eqnarray*}
Let $M_t$ and $\mathcal{Z}_t$ be the completions of $\mathcal{S}(\mathbb{R}^d)$ with respect to the norms $\|\cdot\|_t$, respectively $|\|\cdot \||_t$. Let $M=\cap_{t>0}M_t$ and $\mathcal{Z}=\cap_{t>0}\mathcal{Z}_t$. By Lemma \ref{lemmaA} (Appendix A), $\mathcal{Z}_t=\mathcal{Z}_s=\mathcal{Z}$ for any $s,t>0$.
We say that equation (\ref{hyperbolic-eq}) has a {\em random field solution} if $\delta_x \in M$ for any $x \in \mathbb{R}^d$. In this case, the random field solution is defined by $\{u(t,x)=u(t,\delta_x); t \geq 0,x \in \mathbb{R}^d\}$.
The following result gives some upper and lower bounds for $N_t(\xi)$. For the upper bound, we use an argument similar to Proposition 3.7 of \cite{BT10-SPA}. For the lower bound, we use a new argument.
\begin{theorem} \label{bounds-for-Ntxi-hyp} For any $t>0$ and $\xi \in \mathbb{R}^d$, $$D_H^{(2)} t \left(\frac{1}{1/t^2+\Psi(\xi)} \right)^{H+1/2} \leq N_t(\xi) \leq D_H^{(1)} t \left(\frac{1}{1/t^2+\Psi(\xi)} \right)^{H+1/2},$$ where $D_H^{(1)}$ and $D_H^{(2)}$ are some positive constants depending only on $H$. \end{theorem}
\noindent {\bf Proof:} We first prove the upper bound. Suppose that $t^2 \Psi(\xi) \leq 1$. Using (\ref{MMV-ineq}), the fact that
$\|\varphi\|_{L^{1/H}(0,t)}^2 \leq t^{2H-1}\|\varphi\|_{L^2(0,t)}^2$
and $|\sin x| \leq x$ for all $x>0$, we obtain: \begin{eqnarray*} N_t(\xi) & \leq & b_H^2 t^{2H-1}\frac{1}{\Psi(\xi)} \int_0^t \sin^2 (r \sqrt{\Psi(\xi)}) dr \leq b_H^2 t^{2H-1} \int_0^t r^2 dr \\ &=& \frac{1}{3} b_H^2 t^{2H+2} \leq \frac{1}{3} b_H^2 2^{H+1/2}t \left(\frac{1}{1/t^2+\Psi(\xi)} \right)^{H+1/2}, \end{eqnarray*} where for the last inequality, we used the fact that $\frac{t^2}{2} \leq \frac{1}{t^{-2}+\Psi(\xi)}$ if $t^2 \Psi(\xi) \leq 1$.
Suppose next that $t^2 \Psi(\xi) \geq 1$. We denote $$T=t \sqrt{\Psi(\xi)}.$$ Using the change of variable $r'=r\sqrt{\Psi(\xi)}$ and $s'=s \sqrt{\Psi(\xi)}$, we obtain:
$$N_t(\xi)=\frac{1}{\Psi(\xi)^{H+1}} \| \sin (\cdot)\|_{\mathcal{H}(0,T)}^2.$$
We now use Lemma B.1 of \cite{BT10-SPA} for expressing the $\mathcal{H}(0,T)$-norm of the sinus function in the spectral domain. We obtain that: \begin{equation} \label{Nt-wave-spectral} N_t(\xi)=\frac{c_H}{\Psi(\xi)^{H+1}}
\int_{\mathbb{R}} \frac{|\tau|^{-(2H-1)}}{(\tau^2-1)^2} [f_{T}^{2}(\tau)+g_T^2 (\tau)]d\tau, \end{equation} where $$f_T(\tau)=\sin(\tau T)-\tau \sin T \quad \mbox{and} \quad g_T(\tau)=\cos(\tau T)- \cos T.$$
Letting $\varphi(x)=\sin x$, we have: (see the proof of Lemma B.1 of \cite{BT10-SPA}) $$|\mathcal{F}_{0,T}\varphi(\tau)|^2=\frac{1}{(\tau^2-1)^2}[f_{T}^2(\tau)+g_T^2(\tau)].$$
We split the integral in (\ref{Nt-wave-spectral}) into the regions $|\tau| \leq 1/2$ and
$|\tau| \geq 1/2$, and denote the two integrals by $N_t^{(1)}(\xi)$ and $N_t^{(2)}(\xi)$. Using the same argument as in the proof of Proposition 3.7 of \cite{BT10-SPA}, we get: \begin{equation} \label{estimate-Nt1} N_t^{(1)}(\xi) \leq C \frac{c_H}{\Psi(\xi)^{H+1}} \cdot \frac{2^{2H-2}}{1-H}\leq C \frac{c_H }{1-H} 2^{2H-2} t \left( \frac{2}{1/t^2+\Psi(\xi)}\right)^{H+1/2}, \end{equation} where $C=11.11$, and for the second inequality we used the fact that $\frac{1}{\Psi(\xi)^{1/2}} \leq t$ and $\frac{1}{\Psi(\xi)} \leq \frac{2}{t^{-2}+\Psi(\xi)}$ if $t^2 \Psi(\xi) \geq 1$.
On the other hand, $$N_t^{(2)}(\xi) \leq c_H 2^{2H-1} \frac{1}{\Psi(\xi)^{H+1}} I(T),$$ where \begin{equation} \label{Planch-id-wave} I(T):=\int_{\mathbb{R}} \frac{f_T^2(\tau)+g_T^2(\tau)}{(\tau^2-1)^2}d\tau =2\pi \int_0^T \sin^2 x dx=\pi T \left[1-\frac{\sin(2 T)}{2 T} \right]. \end{equation} by Plancherel's theorem. This yields the estimate $I(T) \leq 2\pi T$. We obtain: \begin{equation} \label{estimate-Nt2} N_t^{(2)}(\xi) \leq c_H 2^{2H-1} 2\pi t \left(\frac{1}{\Psi(\xi)}\right)^{H+1/2} \leq c_H 2^{2H-1} 2 \pi t \left( \frac{2}{1/t^2+\Psi(\xi)} \right)^{H+1/2}, \end{equation} where for the second inequality we used the fact that $t^2 \Psi(\xi) \geq 1$.
Combining (\ref{estimate-Nt1}) and (\ref{estimate-Nt2}), we conclude that: $$N_t(\xi) \leq C \frac{c_H}{1-H} 2^{3H-1/2} t\left( \frac{1}{1/t^2+\Psi(\xi)} \right)^{H+1/2}.$$ The upper bound follows, letting $$D_H^{(1)}=\max \left\{ \frac{1}{3} b_H^2 2^{H+1/2}, C \frac{c_H}{1-H} 2^{3H-1/2} \right\}.$$
We now treat the lower bound. Suppose first that $t^2 \Psi(\xi) \leq 1$. Using the fact that $\sin x \geq x \sin 1$ for all $x \in [0,1]$, we obtain: \begin{eqnarray*} N_t(\xi) & \geq & \alpha_H \sin^2 1 \int_0^t \int_0^t rs
|r-s|^{2H-2}dr ds= \alpha_H \sin^2 1 \frac{\beta(2,2H-1)}{H+1} t^{2H+2} \\ & \geq & \alpha_H \sin^2 1 \frac{\beta(2,2H-1)}{H+1} t \left( \frac{1}{1/t^2+\Psi(\xi)}\right)^{H+1/2}, \end{eqnarray*} where $\beta$ denotes the Beta function and we used the fact that $t^2 \geq \frac{1}{t^{-2}+\Psi(\xi)}$.
Suppose next that $T^2=t^2 \Psi(\xi) \geq 1$. Let $\rho>1$ be a constant which will be specified below. We use (\ref{Nt-wave-spectral}). Since the integrand is non-negative, $N_t(\xi)$ is bounded below by the integral over the region $|\tau| \leq \rho$. In that region,
$|\tau|^{-(2H-1)} \geq \rho^{-(2H-1)}$, and hence \begin{equation} \label{LB-Nt-wave-step1} N_t(\xi) \geq \frac{c_H \rho^{-(2H-1)}}{\Psi(\xi)^{H+1}} \left(
I(T)-\int_{|\tau| \geq \rho} \frac{f_T^2(\tau)+g_T^2(\tau)}{(\tau^2-1)^2}d\tau\right). \end{equation}
Using (\ref{Planch-id-wave}) and the inequality $1-(\sin x)/x \geq 1/2$ for any $x \geq 2$, we get the lower bound: \begin{equation} \label{LB-Nt-wave-step2} I(T) \geq \frac{\pi }{2} T , \quad \mbox{since} \quad T \geq 1. \end{equation}
To find an upper bound for the second integral in the right-hand side of (\ref{LB-Nt-wave-step1}), we use the fact that: $$f_T^2(\tau)+g_T^2(\tau) \leq 2T
(1+|\tau|)^2, \quad \forall \tau \in \mathbb{R}.$$
(To see this, note that $|f_T(\tau)| \leq 1+|\tau|$ and
$|f_T(\tau)| \leq 2 T|\tau|$, since $|\sin x| \leq |x|$. Hence, $f_T^2(\tau) \leq 2 T |\tau|(1+|\tau|)$. Similarly, $|g_T(\tau)| \leq 2$ and $|g_T(\tau)| \leq T(1+|\tau|)$, since $|1-\cos x| \leq |x|$. Hence,
$g_T^2(\tau) \leq 2T (1+|\tau|)$.)
It follows that: \begin{equation}
\label{LB-Nt-wave-step3} \int_{|\tau| \geq \rho} \frac{f_T^2(\tau)+g_T^2(\tau)}{(\tau^2-1)^2}d\tau \leq C_{\rho}T, \end{equation}
where $C_{\rho}=2
\int_{|\tau| \geq \rho} \frac{(1+|\tau|)^2}{(\tau^2-1)^2}d \tau$. Using (\ref{LB-Nt-wave-step1}), (\ref{LB-Nt-wave-step2}) and (\ref{LB-Nt-wave-step3}), we obtain that: $$N_t(\xi) \geq \frac{c_H \rho^{-(2H-1)}}{\Psi(\xi)^{H+1}} \left(\frac{\pi}{2}-C_{\rho} \right) t \sqrt{\Psi(\xi)}.$$ Choose $\rho$ large enough such that $C_{\rho}<\pi/2$, e.g. $\rho=4$, for which $C_{\rho}<4/3$. Using the fact that $\frac{1}{\Psi(\xi)} \geq \frac{1}{t^{-2}+\Psi(\xi)}$, we get $$N_t(\xi) \geq c_H 4^{-(2H-1)} \left(\frac{\pi}{2}-\frac{4}{3} \right) t \left(\frac{1}{1/t^2+\Psi(\xi)} \right)^{H+1/2}.$$ The lower bound follows, letting $$D_H^{(2)}=\min \left\{\alpha_H \sin^2 1 \frac{\beta(2,2H-1)}{H+1}, c_H 4^{-(2H-1)} \left(\frac{\pi}{2}-\frac{4}{3} \right) \right\}.$$
$\Box$
A consequence of the previous result is that the norms $\|\cdot
\|_t$ and $|\| \cdot\||_{t^2}$ are equivalent, for any $t>0$.
\begin{corollary} For any $t>0,\varphi \in \mathcal{S}(\mathbb{R}^d)$,
$$d_H t\mathcal{E}(t^2;\varphi) \leq E|u(t,\varphi)|^2 \leq D_H t\mathcal{E}(t^2;\varphi).$$ Hence $M_{t}=\mathcal{Z}_{t^2}$ for any $t>0$, and $M=\mathcal{Z}$. \end{corollary}
Below is the main result of this section.
\begin{theorem} {\rm (Existence of Solution in the Hyperbolic Case)} Assume that (\ref{imaginary-zero}) holds. Then (\ref{hyperbolic-eq}) has a random field solution if and only if (\ref{cond-hyp-eq}) holds. \end{theorem}
\noindent {\bf Proof:} As in the proof of Theorem \ref{nec-suf-delta-in-Z1}, one can show that (\ref{cond-hyp-eq}) is a necessary and sufficient condition for $\delta_x \in \mathcal{Z}_1=M$ for all $x \in \mathbb{R}^d$. We omit the details. $\Box$.
\begin{example} {\rm (Stable processes)}
{\rm As in Example \ref{stable-ex}, let $\mathcal{L}=-(-\Delta)^{\beta/2}$ for $\beta \in (0,2]$. Then $\Psi(\xi)=c_{\beta}|\xi|^{\beta}$ and (\ref{cond-hyp-eq}) becomes: \begin{equation} \label{cond-stable-proc-hyp}
\int_{\mathbb{R}^d} \left(\frac{1}{1+|\xi|^{\beta}} \right)^{H+1/2}\mu(d\xi)<\infty. \end{equation}
\noindent When $f(x)=c_{\alpha,d}|x|^{-(d-\alpha)}$ with $0<\alpha<d$, (\ref{cond-stable-proc-hyp}) is equivalent to $$\left(H+\frac{1}{2}\right) \beta>d-\alpha,$$
whereas for $f(x)=\prod_{i=1}^{d}(\alpha_{H_i}|x_i|^{2H_i-2})$, (\ref{cond-stable-proc-hyp}) is equivalent to $$\left(H+\frac{1}{2}\right) \beta >d-\sum_{i=1}^{d}(2H_i-1).$$ } \end{example}
\begin{remark} \label{stable-rem-hyp} {\rm (Fractional Powers of the Laplacian)} {\rm As in Remark \ref{stable-rem}, we can consider the case $\mathcal{L}=-(-\Delta)^{\beta/2}$ for arbitrary $\beta>0$. Note that the fundamental solution $G$ of $\partial_{tt} u- \mathcal{L} u=0$ exists and satisfies:
$$\mathcal{F} G (t,\xi)=\frac{\sin(t|\xi|^{\beta/2})}{|\xi|^{\beta/2}}.$$
(see p.11 of \cite{dalang-sanzsole05}). Using Theorem 2.1 of \cite{BT10-SPA} and estimates similar to those given by Theorem \ref{bounds-for-Ntxi-hyp} above, one can show that a random field solution of (\ref{hyperbolic-eq}) (in the sense of \cite{BT10-SPA}) exists if and only if (\ref{cond-stable-proc-hyp}) holds. } \end{remark}
\appendix
\section{Some elementary inequalities}
\begin{lemma} \label{lemmaA} For any $\alpha,\beta>0$, $$c_1(\alpha,\beta)^{2H}\Upsilon_{H}^*(\beta) \leq \Upsilon_H^*(\alpha) \leq c_2(\alpha,\beta)^{2H}\Upsilon_H^*(\beta),$$ where $c_1(\alpha,\beta)=(\beta \wedge 1)/(\alpha+1)$ and $c_2(\alpha,\beta)=(\beta+1)[(1/\alpha) \vee 1]$. \end{lemma}
\noindent {\bf Proof:} We denote by $\Upsilon_{H,1}^{*}(\alpha)$ and $\Upsilon_{H,2}^{*}(\alpha)$ the integrals over the regions $\{2{\rm Re} \Psi(\xi) \leq 1\}$, respectively $\{2{\rm Re} \Psi(\xi) \geq 1\}$. Using the inequality $(\alpha+1)^{-1} \leq [1+2{\rm Re} \Psi(\xi)]^{-1} \leq \alpha^{-1}$ if $2{\rm Re} \Psi(\xi) \leq 1$, we obtain that: $$\left(\frac{1}{\alpha+1}\right)^{2H} \int_{2{\rm Re} \Psi(\xi) \leq 1}\mu(d\xi) \leq \Upsilon_{H,1}^{*}(\alpha) \leq \left(\frac{1}{\alpha}\right)^{2H} \int_{2{\rm Re} \Psi(\xi) \leq 1}\mu(d\xi).$$ Combining this with the similar inequality for $\Upsilon_{H,1}^{*}(\beta)$, we get: \begin{equation} \label{elem-ineq-upsilon1} \left(\frac{\beta}{\alpha+1}\right)^{2H} \Upsilon_{H,1}^{*}(\beta) \leq \Upsilon_{H,1}^{*}(\alpha) \leq \left(\frac{\beta+1}{\alpha}\right)^{2H} \Upsilon_{H,1}^{*}(\beta). \end{equation}
A similar argument works for $\Upsilon_{H,2}^{*}(\alpha)$. We obtain:
\begin{equation} \label{elem-ineq-upsilon2} \left(\frac{1}{\alpha+1}\right)^{2H} \Upsilon_{H,2}^{*}(\beta) \leq \Upsilon_{H,2}^{*}(\alpha) \leq (\beta+1)^{2H} \Upsilon_{H,2}^{*}(\beta). \end{equation} The result follows by taking the sum of (\ref{elem-ineq-upsilon1}) and (\ref{elem-ineq-upsilon2}). $\Box$
We recall the definitions of the functionals $\mathcal{E}(t;\varphi)$ introduced in Section \ref{parabolic-section}, respectively Section \ref{hyperbolic-section}. To make a distinction between these functionals in the two cases, we use the index ``par'' for parabolic, and ``hyp'' for hyperbolic: \begin{eqnarray*}
\mathcal{E}_{{\rm par}}(t;\varphi) &=& \int_{\mathbb{R}^d} \left(\frac{1}{1/t+ {\rm Re} \Psi(\xi)} \right)^{2H}|\mathcal{F} \varphi(\xi)|^2 \mu(d\xi)\\
\mathcal{E}_{{\rm hyp}}(t;\varphi) &=& \int_{\mathbb{R}^d} \left(\frac{1}{1/t+ {\rm Re }\Psi(\xi)} \right)^{H+1/2}|\mathcal{F} \varphi(\xi)|^2 \mu(d\xi). \end{eqnarray*}
\begin{lemma} \label{ineq-cE} For any $s>0,t>0$ and $\varphi \in \mathcal{S}(\mathbb{R}^d)$, $$c_1(s,t)^{2H} \mathcal{E}_{{\rm par}}(s;\varphi) \leq \mathcal{E}_{{\rm par}}(t; \varphi) \leq c_2(s,t)^{2H} \mathcal{E}_{{\rm par}}(s; \varphi)$$ $$c_1(s,t)^{H+1/2} \mathcal{E}_{{\rm hyp}}(s;\varphi) \leq \mathcal{E}_{{\rm hyp}}(t; \varphi) \leq c_2(s,t)^{H+1/2} \mathcal{E}_{{\rm hyp}}(s; \varphi),$$ where $c_1(s,t)=(s^{-1} \wedge 1)/(t^{-1}+1)$ and $c_2(t)=(s^{-1}+1)(t \vee 1)$. \end{lemma}
\noindent {\bf Proof:} The argument is similar to the proof of Lemma \ref{lemmaA}. $\Box$
\section{The $\mathcal{H}(0,T)$-norm of the exponential}
The next result gives the expression of the $\mathcal{H}(0,T)$-norm of the complex-valued exponential function in the spectral domain, which is needed in the proof of Theorem \ref{bounds-Nt-xi}.
\begin{lemma} \label{H-norm-exp} Let $\varphi(x)=e^{-x(a+ib)}$ for $x \in (0,T)$, where $a,b \in \mathbb{R}$. Then
$$\|\varphi\|_{\mathcal{H}(0,T)}^2=c_H \int_{\mathbb{R}}\frac{\sin^2[(\tau+b)T]+ \{e^{-at}-\cos[(\tau+b)T]\}^2}{a^2+(\tau+b)^2}
|\tau|^{-(2H-1)}d\tau,$$ where $c_H=\Gamma(2H+1) \sin(\pi H)/(2H)$. \end{lemma}
\noindent {\bf Proof:} For complex-valued functions $\varphi \in L^2(0,t)$, we can apply Lemma A.1 of \cite{BT10-SPA} to ${\rm Re}\varphi$ and ${\rm Im}\varphi$ to obtain that: \begin{equation} \label{H-norm-spec-domain}
\|\varphi\|_{\mathcal{H}(0,T)}^2=c_H \int_{\mathbb{R}} |\mathcal{F}_{0,T}\varphi(\tau)|^2 |\tau|^{-(2H-1)}d\tau, \end{equation} where $\mathcal{F}_{0,T}\varphi (\tau):=\int_0^T e^{-i \tau x}\varphi(x)dx$.
An elementary calculation shows that for $\varphi(x)=e^{-x(a+ib)}$,
$$|\mathcal{F}_{0,T} \varphi(\tau)|^2=\frac{1}{a^2+(\tau+b)^2}[f_T^2 (\tau)+g_T^{2}(\tau)],$$ where $f_T(\tau)=\sin [(\tau+b)T]$ and $g_T(\tau)=e^{-at}-\cos[(\tau+b)T]$. The result follows by (\ref{H-norm-spec-domain}). $\Box$
\section{A version of Plancherel theorem}
The following result is a version of Plancherel theorem needed for the calculation of $(\bar{P}_{r+s} (f*\phi))(x)$ in the proof of Lemma \ref{max-principle-step2}.
\begin{lemma} \label{version-Plancherel} For any $\varphi \in L^2(\mathbb{R}^d)$, $\psi_1 \in L^2(\mathbb{R}^d)$ and $\psi_2 \in L^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d)$, $$\int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\psi_1(x)\psi_2(y)\varphi(x-y)dydx=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d} \mathcal{F} \psi_1(\xi) \overline{\mathcal{F} \psi_2(\xi)} \ \overline{\mathcal{F} \varphi(\xi)}d\xi.$$ \end{lemma}
\noindent {\bf Proof:} By Young's inequality, $\|\varphi * \psi_2\|_{2} \leq \|\varphi\|_2 \|\psi_2\|_{1}$ and $\varphi*\psi_2 \in L^2(\mathbb{R}^d)$. The result follows by Plancherel theorem, since $$\int_{\mathbb{R}^d} \psi_1(x) (\varphi*\psi_2)(x)dx=\frac{1}{(2\pi)^d}\int_{\mathbb{R}^d}\mathcal{F} \psi_1(\xi) \overline{\mathcal{F}(\varphi*\psi_2)(\xi)}d\xi.$$ $\Box$
\end{document} | arXiv |
von Mises distribution
In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle $\theta $ on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic potential, i.e. with a preferred orientation.[1] The von Mises distribution is the maximum entropy distribution for circular data when the real and imaginary parts of the first circular moment are specified. The von Mises distribution is a special case of the von Mises–Fisher distribution on the N-dimensional sphere.
von Mises
Probability density function
The support is chosen to be [−π,π] with μ = 0
Cumulative distribution function
The support is chosen to be [−π,π] with μ = 0
Parameters $\mu $ real
$\kappa >0$
Support $x\in $ any interval of length 2π
PDF ${\frac {e^{\kappa \cos(x-\mu )}}{2\pi I_{0}(\kappa )}}$
CDF (not analytic – see text)
Mean $\mu $
Median $\mu $
Mode $\mu $
Variance ${\textrm {var}}(x)=1-I_{1}(\kappa )/I_{0}(\kappa )$ (circular)
Entropy $-\kappa {\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}+\ln[2\pi I_{0}(\kappa )]$ (differential)
CF ${\frac {I_{|t|}(\kappa )}{I_{0}(\kappa )}}e^{it\mu }$
Definition
The von Mises probability density function for the angle x is given by:[2]
$f(x\mid \mu ,\kappa )={\frac {\exp(\kappa \cos(x-\mu ))}{2\pi I_{0}(\kappa )}}$
where I0($\kappa $) is the modified Bessel function of the first kind of order 0, with this scaling constant chosen so that the distribution sums to unity: $ \int _{-\pi }^{\pi }\exp(\kappa \cos x)dx={2\pi I_{0}(\kappa )}.$
The parameters μ and 1/$\kappa $ are analogous to μ and σ2 (the mean and variance) in the normal distribution:
• μ is a measure of location (the distribution is clustered around μ), and
• $\kappa $ is a measure of concentration (a reciprocal measure of dispersion, so 1/$\kappa $ is analogous to σ2).
• If $\kappa $ is zero, the distribution is uniform, and for small $\kappa $, it is close to uniform.
• If $\kappa $ is large, the distribution becomes very concentrated about the angle μ with $\kappa $ being a measure of the concentration. In fact, as $\kappa $ increases, the distribution approaches a normal distribution in x with mean μ and variance 1/$\kappa $.
The probability density can be expressed as a series of Bessel functions[3]
$f(x\mid \mu ,\kappa )={\frac {1}{2\pi }}\left(1+{\frac {2}{I_{0}(\kappa )}}\sum _{j=1}^{\infty }I_{j}(\kappa )\cos[j(x-\mu )]\right)$
where Ij(x) is the modified Bessel function of order j.
The cumulative distribution function is not analytic and is best found by integrating the above series. The indefinite integral of the probability density is:
$\Phi (x\mid \mu ,\kappa )=\int f(t\mid \mu ,\kappa )\,dt={\frac {1}{2\pi }}\left(x+{\frac {2}{I_{0}(\kappa )}}\sum _{j=1}^{\infty }I_{j}(\kappa ){\frac {\sin[j(x-\mu )]}{j}}\right).$
The cumulative distribution function will be a function of the lower limit of integration x0:
$F(x\mid \mu ,\kappa )=\Phi (x\mid \mu ,\kappa )-\Phi (x_{0}\mid \mu ,\kappa ).\,$
Moments
The moments of the von Mises distribution are usually calculated as the moments of the complex exponential z = eix rather than the angle x itself. These moments are referred to as circular moments. The variance calculated from these moments is referred to as the circular variance. The one exception to this is that the "mean" usually refers to the argument of the complex mean.
The nth raw moment of z is:
$m_{n}=\langle z^{n}\rangle =\int _{\Gamma }z^{n}\,f(x|\mu ,\kappa )\,dx$
$={\frac {I_{|n|}(\kappa )}{I_{0}(\kappa )}}e^{in\mu }$
where the integral is over any interval $\Gamma $ of length 2π. In calculating the above integral, we use the fact that zn = cos(nx) + i sin(nx) and the Bessel function identity:[4]
$I_{n}(\kappa )={\frac {1}{\pi }}\int _{0}^{\pi }e^{\kappa \cos(x)}\cos(nx)\,dx.$
The mean of the complex exponential z is then just
$m_{1}={\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}e^{i\mu }$
and the circular mean value of the angle x is then taken to be the argument μ. This is the expected or preferred direction of the angular random variables. The variance of z, or the circular variance of x is:
${\textrm {var}}(x)=1-E[\cos(x-\mu )]=1-{\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}.$
Limiting behavior
When $\kappa $ is large, the distribution resembles a normal distribution. More specifically, for large positive real numbers $\kappa $,
$f(x\mid \mu ,\kappa )\approx {\frac {1}{\sigma {\sqrt {2\pi }}}}\exp \left[{\dfrac {-(x-\mu )^{2}}{2\sigma ^{2}}}\right]$
where σ2 = 1/$\kappa $ and the difference between the left hand side and the right hand side of the approximation converges uniformly to zero as $\kappa $ goes to infinity. Also, when $\kappa $ is small, the probability density function resembles a uniform distribution:
$\lim _{\kappa \rightarrow 0}f(x\mid \mu ,\kappa )=\mathrm {U} (x)$
where the interval for the uniform distribution $\mathrm {U} (x)$ is the chosen interval of length $2\pi $ (i.e. $\mathrm {U} (x)=1/(2\pi )$ when $x$ is in the interval and $\mathrm {U} (x)=0$ when $x$ is not in the interval).
Estimation of parameters
A series of N measurements $z_{n}=e^{i\theta _{n}}$ drawn from a von Mises distribution may be used to estimate certain parameters of the distribution.[5] The average of the series ${\overline {z}}$ is defined as
${\overline {z}}={\frac {1}{N}}\sum _{n=1}^{N}z_{n}$
and its expectation value will be just the first moment:
$\langle {\overline {z}}\rangle ={\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}e^{i\mu }.$
In other words, ${\overline {z}}$ is an unbiased estimator of the first moment. If we assume that the mean $\mu $ lies in the interval $[-\pi ,\pi ]$, then Arg$({\overline {z}})$ will be a (biased) estimator of the mean $\mu $.
Viewing the $z_{n}$ as a set of vectors in the complex plane, the ${\bar {R}}^{2}$ statistic is the square of the length of the averaged vector:
${\bar {R}}^{2}={\overline {z}}\,{\overline {z^{*}}}=\left({\frac {1}{N}}\sum _{n=1}^{N}\cos \theta _{n}\right)^{2}+\left({\frac {1}{N}}\sum _{n=1}^{N}\sin \theta _{n}\right)^{2}$
and its expectation value is [6]
$\langle {\bar {R}}^{2}\rangle ={\frac {1}{N}}+{\frac {N-1}{N}}\,{\frac {I_{1}(\kappa )^{2}}{I_{0}(\kappa )^{2}}}.$
In other words, the statistic
$R_{e}^{2}={\frac {N}{N-1}}\left({\bar {R}}^{2}-{\frac {1}{N}}\right)$
will be an unbiased estimator of ${\frac {I_{1}(\kappa )^{2}}{I_{0}(\kappa )^{2}}}\,$ and solving the equation $R_{e}={\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}\,$ for $\kappa \,$ will yield a (biased) estimator of $\kappa \,$. In analogy to the linear case, the solution to the equation ${\bar {R}}={\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}\,$ will yield the maximum likelihood estimate of $\kappa \,$ and both will be equal in the limit of large N. For approximate solution to $\kappa \,$ refer to von Mises–Fisher distribution.
Distribution of the mean
The distribution of the sample mean ${\overline {z}}={\bar {R}}e^{i{\overline {\theta }}}$ for the von Mises distribution is given by:[7]
$P({\bar {R}},{\bar {\theta }})\,d{\bar {R}}\,d{\bar {\theta }}={\frac {1}{(2\pi I_{0}(\kappa ))^{N}}}\int _{\Gamma }\prod _{n=1}^{N}\left(e^{\kappa \cos(\theta _{n}-\mu )}d\theta _{n}\right)={\frac {e^{\kappa N{\bar {R}}\cos({\bar {\theta }}-\mu )}}{I_{0}(\kappa )^{N}}}\left({\frac {1}{(2\pi )^{N}}}\int _{\Gamma }\prod _{n=1}^{N}d\theta _{n}\right)$
where N is the number of measurements and $\Gamma \,$ consists of intervals of $2\pi $ in the variables, subject to the constraint that ${\bar {R}}$ and ${\bar {\theta }}$ are constant, where ${\bar {R}}$ is the mean resultant:
${\bar {R}}^{2}=|{\bar {z}}|^{2}=\left({\frac {1}{N}}\sum _{n=1}^{N}\cos(\theta _{n})\right)^{2}+\left({\frac {1}{N}}\sum _{n=1}^{N}\sin(\theta _{n})\right)^{2}$
and ${\overline {\theta }}$ is the mean angle:
${\overline {\theta }}=\mathrm {Arg} ({\overline {z}}).\,$
Note that product term in parentheses is just the distribution of the mean for a circular uniform distribution.[7]
This means that the distribution of the mean direction $\mu $ of a von Mises distribution $VM(\mu ,\kappa )$ is a von Mises distribution $VM(\mu ,{\bar {R}}N\kappa )$, or, equivalently, $VM(\mu ,R\kappa )$.
Entropy
By definition, the information entropy of the von Mises distribution is[2]
$H=-\int _{\Gamma }f(\theta ;\mu ,\kappa )\,\ln(f(\theta ;\mu ,\kappa ))\,d\theta \,$ ;\mu ,\kappa )\,\ln(f(\theta ;\mu ,\kappa ))\,d\theta \,}
where $\Gamma $ is any interval of length $2\pi $. The logarithm of the density of the Von Mises distribution is straightforward:
$\ln(f(\theta ;\mu ,\kappa ))=-\ln(2\pi I_{0}(\kappa ))+\kappa \cos(\theta )\,$ ;\mu ,\kappa ))=-\ln(2\pi I_{0}(\kappa ))+\kappa \cos(\theta )\,}
The characteristic function representation for the Von Mises distribution is:
$f(\theta ;\mu ,\kappa )={\frac {1}{2\pi }}\left(1+2\sum _{n=1}^{\infty }\phi _{n}\cos(n\theta )\right)$ ;\mu ,\kappa )={\frac {1}{2\pi }}\left(1+2\sum _{n=1}^{\infty }\phi _{n}\cos(n\theta )\right)}
where $\phi _{n}=I_{|n|}(\kappa )/I_{0}(\kappa )$. Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written:
$H=\ln(2\pi I_{0}(\kappa ))-\kappa \phi _{1}=\ln(2\pi I_{0}(\kappa ))-\kappa {\frac {I_{1}(\kappa )}{I_{0}(\kappa )}}$
For $\kappa =0$, the von Mises distribution becomes the circular uniform distribution and the entropy attains its maximum value of $\ln(2\pi )$.
Notice that the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified[8] or, equivalently, the circular mean and circular variance are specified.
See also
• Bivariate von Mises distribution
• Directional statistics
• Von Mises–Fisher distribution
• Kent distribution
References
1. Risken, H. (1989). The Fokker–Planck Equation. Springer. ISBN 978-3-540-61530-9.
2. Mardia, Kantilal; Jupp, Peter E. (1999). Directional Statistics. Wiley. ISBN 978-0-471-95333-3.
3. see Abramowitz and Stegun §9.6.34
4. See Abramowitz and Stegun §9.6.19
5. Borradaile, G. J. (2003). Statistics of earth science data : their distribution in time, space, and orientation. Springer. ISBN 978-3-662-05223-5.
6. Kutil, Rade (August 2012). "Biased and unbiased estimation of the circular mean resultant length and its variance". Statistics: A Journal of Theoretical and Applied Statistics. 46 (4): 549–561. doi:10.1080/02331888.2010.543463. S2CID 7045090.
7. Jammalamadaka, S. Rao; Sengupta, A. (2001). Topics in Circular Statistics. World Scientific Publishing Company. ISBN 978-981-02-3778-3.
8. Jammalamadaka, S. Rao; SenGupta, A. (2001). Topics in circular statistics. New Jersey: World Scientific. ISBN 981-02-3778-2. Retrieved 2011-05-15.
Further reading
• Abramowitz, M. and Stegun, I. A. (ed.), Handbook of Mathematical Functions, National Bureau of Standards, 1964; reprinted Dover Publications, 1965. ISBN 0-486-61272-4
• "Algorithm AS 86: The von Mises Distribution Function", Mardia, Applied Statistics, 24, 1975 (pp. 268–272).
• "Algorithm 518, Incomplete Bessel Function I0: The von Mises Distribution", Hill, ACM Transactions on Mathematical Software, Vol. 3, No. 3, September 1977, Pages 279–284.
• Best, D. and Fisher, N. (1979). Efficient simulation of the von Mises distribution. Applied Statistics, 28, 152–157.
• Evans, M., Hastings, N., and Peacock, B., "von Mises Distribution". Ch. 41 in Statistical Distributions, 3rd ed. New York. Wiley 2000.
• Fisher, Nicholas I., Statistical Analysis of Circular Data. New York. Cambridge 1993.
• "Statistical Distributions", 2nd. Edition, Evans, Hastings, and Peacock, John Wiley and Sons, 1993, (chapter 39). ISBN 0-471-55951-2
• Borradaile, Graham (2003). Statistics of Earth Science Data. Springer. ISBN 978-3-540-43603-4. Retrieved 31 Dec 2009.
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| Wikipedia |
\begin{document}
\hbox{Preprint}
\title{On Practical Numbers of Some Special Forms}
\author{Li-Yuan Wang} \address{Department of Mathematics, Nanjing University\\ Nanjing, 210093, People's Republic of China} \email{[email protected]}
\author{Zhi-Wei Sun } \address{Department of Mathematics, Nanjing University\\ Nanjing, 210093, People's Republic of China} \email{[email protected]}
\begin{abstract} In this paper we study practical numbers of some special forms. For any integers $b\gs0$ and $c>0$, we show that if $n^2+bn+c$ is practical for some integer $n>1$, then there are infinitely many nonnegative integers $n$ with $n^2+bn+c$ practical. We also prove that there are infinitely many practical numbers of the form $q^4+2$ with $q$ practical, and that there are infinitely many practical Pythagorean triples $(a,b,c)$ with $\gcd(a,b,c)=6$ (or $\gcd(a,b,c)=4$). \end{abstract}
\thanks{2010 {\it Mathematics Subject Classification}. Primary 11B83; Secondary 11D09. \newline\indent {\it Keywords}. Practical numbers, cyclotomic polynomials, Pythagorean triples. \newline \indent The second author is the corresponding author, and supported by the National Natural Science Foundation of China (grant 11571162).}
\maketitle \section{Introduction} \setcounter{lemma}{0} \setcounter{theorem}{0} \setcounter{corollary}{0} \setcounter{remark}{0} \setcounter{equation}{0} \setcounter{conjecture}{0}
A positive integer $m$ is called a {\it practical number} if each $n=1,\ldots,m$ can be written as the sum of some distinct divisors of $n$. This concept was introduced by Srinivasan \cite{name-of-practical} who noted that any practical number greater than $2$ must be divisible by $4$ or $6$. In 1954, Stewart \cite{structure-theorem} obtained the following structure theorem for practical numbers. \begin{theorem} Let $p_1<\ldots<p_k$ be distinct primes and let $a_1,\ldots,a_k\in\Bbb Z^+=\{1,2,3,\ldots\}$. Then $m=p_{1}^{a_{1}}p_{2}^{a_{2}} \cdots p_{k}^{a_{k}}$ is practical if and only if $p_{1}=2$ and $$p_{j}-1\leqslant \sigma (p_{1}^{a_{1}}p_{2}^{a_{2}} \cdots p_{j-1}^{a_{j-1}})\ \ \text{for all}\ 1<j\leqslant k,$$ where $\sigma(n)$ denotes the sum of the positive divisors of $n$. \label{structure} \end{theorem}
It is interesting to compare practical numbers with primes. All practical numbers are even except 1 while all primes are odd except 2. Moreover, if $P(x)$ denotes the number of practical numbers not exceeding $x$, then there is a positive constant $c$ such that \begin{equation}
P(x)\sim \frac{cx}{\log x}\quad\text{as}\ x\to\infty,
\label{zero}
\end{equation} which was established in \cite{distribution}. This is quite similar to the Prime Number Theorem.
Inspired by the famous Goldbach conjecture and the twin prime conjecture, Margenstern \cite{french} conjectured that every positive even integer is the sum of two practical numbers and that there are infinitely many practical numbers $m$ with $m-2$ and $m+2$ also practical. Both conjectures were confirmed by Melfi \cite{two-conjecture} in 1996. Guo and Weingartner \cite{GW} proved in 2018 that for any odd integer $a$ there are infinitely many primes $p$ with $p+a$ practical.
An open conjecture of Sun \cite[Conjecture 3.38] {S17} states that any odd integer greater than one can be written as the sum of a prime and a practical number.
Whether there are infinitely many primes of the form $x^2+1$ with $x\in\Bbb Z$ is a famous unsolved problem in number theory. Motivated by this, in 2017 Sun \cite[A294225]{S} conjectured that there are infinitely many positive integers $q$
such that $q$, $q+2$ and $q^2+2$ are all practical, which looks quite challenging. Thus, it is natural to study for what $a,b,c\in\Bbb Z^+$ there are infinitely many
practical numbers of the form $an^2+bn+c$. Note that if $a\equiv b\pmod 2$ and $2\nmid c$ then $an^2+bn+c$ is odd for any $n\in\Bbb N=\{0,1,2,\ldots\}$ and hence
$an^2+bn+c$ cannot take practical values for infinitely many $n\in\Bbb N$.
Based on our computation we formulate the following conjecture.
\begin{conjecture} Let $a,b,c$ be positive integers with $2\nmid ab$ and $2\mid c$. Then there are infinitely many $n\in\Bbb N$ with $an^2+bn+c$ practical. Moreover, in the case $a=1$, there is an integer $n$ with $1<n\leqslant\max\{b,c\}$ such that $n^2+bn+c$ is practical. \end{conjecture}
Though we are unable to prove this conjecture fully, we make the following progress.
\begin{theorem}\label{Th1.1} Let $b\in\Bbb N$ and $c\in\Bbb Z^+$. If $n^2+bn+c$ is practical for some integer $n>1$, then there are infinitely many $n\in\Bbb N$ with $n^2+bn+c$ practical. \label{quadratic} \end{theorem}
If $1\leqslant b\leqslant 100$ and $1\leqslant c\ls100$ with $2\nmid b$ and $2\mid c$, then we can easily find $1<n\leqslant \max\{b,c\}$ with $n^2+bn+c$ practical. For example, $n^2+n+2$ with $n=2$ is practical. For each positive even number $b\leqslant 20$ we make the set $$S_b:=\{1\leqslant c\leqslant 100:\ n^2+bn+c\ \text{is practical for some}\ n=2,\ldots,20000\}$$ explicit: \begin{align*}S_0=&\{1\leqslant c\leqslant 100:\ c\not\equiv 1,10\pmod{12}\ \text{and}\ c\not=43,67,93\}, \\S_2=&\{1\leqslant c\leqslant 100:\ c\not\eq2,11\pmod{12}\ \text{and}\ c\not=44,68,94\}, \\S_4=&\{1\leqslant c\leqslant 100:\ c\not\equiv 2,5\pmod{12}\ \text{and}\ c\not=47,71,97\}, \\S_6=&\{1\leqslant c\leqslant 100:\ c\not\eq7,10\pmod{12}\ \text{and}\ c\not=52,76\}, \\S_8=&\{1\leqslant c\leqslant 100:\ c\not\eq2,5\pmod{12}\ \text{and}\ c\not=59,83\}, \\S_{10}=&\{1\leqslant c\leqslant 100:\ c\not\equiv 2,11\pmod{12}\ \text{and}\ c\not=68,92\}, \\S_{12}=&\{1\leqslant c\leqslant 100:\ c\not\eq1,10\pmod{12}\ \text{and}\ c\not=79\}, \\S_{14}=&\{1\leqslant c\leqslant 100:\ c\not\eq2,11\pmod{12}\ \text{and}\ c\not=92\}, \\S_{16}=&\{1\leqslant c\leqslant 100:\ c\not\equiv 2,5\pmod{12}\}, \\S_{18}=&\{1\leqslant c\leqslant 100:\ c\not\eq7,10\pmod{12}\}, \\S_{20}=&\{1\leqslant c\leqslant 100:\ c\not\eq2,5\pmod{12}\}. \end{align*} For example, applying Theorem \ref{Th1.1} with $b=20$, we see that for any $c=1,\ldots,100$ with $c\not\eq2,5\pmod{12}$ there are infinitely many $n\in\Bbb N$ with $n^2+20n+c$ practical. It is easy to see that if $c$ is congruent to $2$ or $5$ modulo $12$ then $n^2+20n+c$ is not practical for any integer $n\gs2$.
By Theorem \ref{Th1.1} and the fact $2\in S_0$, there are infinitely many $n\in\Bbb N$ with $n^2+2$ practical. Moreover, we have the following stronger result.
\begin{theorem}\label{Th1.2} $2^{35\times3^k+1}+2$ is practical for every $k=0,1,2,\ldots$. Hence there are infinitely many practical numbers $q$ with $q^4+2$ also practical. \label{infinity} \end{theorem}
We prove Theorem \ref{Th1.2} by modifying Melfi's cyclotomic method in \cite{two-conjecture}.
We now turn to Pythagorean triples involving practical numbers, and call a Pythagorean triple $(a,b,c)$ with $a,b,c$ all practical a {\it practical Pythagorean triple}. Obviously, there are infinitely many practical Pythagorean triples. In fact, if $a^2+b^2=c^2$ with $a,b,c$ positive integers then $({2^ka})^2+({2^kb})^2=({2^kc})^2$ for all $k=0,1,2,\ldots$. By Theorem \ref{structure}, $2^ka$, $2^kb$ and $2^kc$ are all practical if $k$ is large enough.
Our following theorem was originally conjectured by Sun \cite[A294112]{S}.
\begin{theorem}\label{Th1.3} Let $d$ be $4$ or $6$. Then there are infinitely many practical Pythagorean triples $(a,b,c)$ with $\gcd(a,b,c)=d$. \label{Pythagorean} \end{theorem}
We are going to show Theorems \ref{Th1.1}-\ref{Th1.3} in the next section. \section{Proofs of Theorems 1.2-1.4}
\begin{lemma} \label{Lem2.1} Let $m$ be any practical number. Then $mn$ is practical for every $n=1,\ldots,\sigma(m)+1$.
In particular, $mn$ is practical for every $1\leqslant n\leqslant 2m$. \label{twotimes} \end{lemma} This lemma follows easily from Theorem \ref{structure}; see \cite{two-conjecture} for details. Note that if $m>1$ is practical then $m-1$ can be written as the sum of some divisors of $m$ and hence $(m-1)+m\leqslant\sigma(m)$.
\noindent{\it Proof of Theorem \ref{Th1.1}}. Set $f(n)=n^2+bn+c$. It is easy to verify that $$f(n+f(n))=f(n)(f(n)+2n+b+1).$$ Note that $$f(n)-(2n+b+1)=n(n-2)+b(n-1)+c-1\gs0.$$ If $n\gs2$ is an integer with $f(n)$ practical, then $f(n+f(n))=f(n)(f(n)+2n+b+1)$ is also practical by Lemma \ref{Lem2.1} and the inequality $$f(n)+2n+b+1\leqslant 2f(n).$$ So the desired result follows. \qed
For a positive integer $m$, the cyclotomic polynomial $\Phi_m(x)$ is defined by $$\Phi_m(x):=\prod_{a=1\atop\gcd(a,m)=1}^m\left(x-e^{2\pi ia/m}\right).$$ Clearly, \begin{equation}\label{cyc}x^n-1=\prod_{d\mid n}\Phi_{d}(x)\quad\text{for all}\ n=1,2,3,\ldots. \end{equation}
\noindent{\it Proof of Theorem \ref{infinity}}. Write $m_k=2^{35\times3^k+1}+2$ for $k=0,1,2,\ldots$. Note that $m_{2k}=q_k^4+2$ with $q_k=2^{(35\times9^k+1)/4}$ practical. So it suffices to prove that $m_k$ is practical for every $k=0,1,2,\ldots$.
Via a computer we find that $$m_{0}=2^{36}+2,\ m_{1}=2^{106}+2,\ m_{2}=2^{316}+2$$
are all practical.
Now assume that $m_{k}$ is practical for a fixed integer $k\geqslant 2$. For convenience, we write $x$ for $2^{3^k}$. Then $$x\geqslant 2^9=512,\ m_k=2(x^{35}+1)\ \text{and}\ m_{k+1}=2(x^{105}+1).$$
In view of \eqref{cyc}, \begin{equation}\label{105} \frac{x^{210}-1}{x^{105}-1}=\frac{x^{70}-1}{x^{35}-1}\Phi_{6}(x)\Phi_{30}(x)\Phi_{42}(x)\Phi_{210}(x). \end{equation} Since $x\gs512$, we have \begin{equation}
\frac{x^2}{2}< \Phi_{6}(x)=x^2-x+1<x^2.
\label{phi6} \end{equation} Clearly, $$x^7>x^3\frac{x^3-1}{x-1}=x^5+x^4+x^3$$ and $$x^8>2x^7\geqslant x^7+x+1.$$ Thus \begin{equation}
x^8< \Phi_{30}(x)=x^8+x^7-x^5-x^4-x^3+x+1<2x^8
\label{phi30} \end{equation} Similarly, for $$\Phi_{42}(x) =x^{12}+x^{11}-x^9-x^8+x^6-x^4-x^3+x+1$$ and \begin{align*} \Phi_{210}(x)=&x^{48}-x^{47}+x^{46}+x^{43}-x^{42}+2 x^{41}-x^{40}+x^{39}+x^{36} \\&-x^{35}+x^{34}-x^{33}+x^{32}-x^{31}-x^{28}-x^{26}-x^{24}-x^{22} \\&-x^{20}-x^{17}+x^{16}-x^{15}+x^{14}-x^{13}+x^{12}+x^9-x^8 \\&+2x^7-x^6+x^5+x^2-x+1, \end{align*} we can prove that \begin{equation}\label{phi42}
x^{12}< \Phi_{42}(x)<2x^{12}\ \text{and}\
\Phi_{210}(x)<x^{48}. \end{equation} Combining (\ref{phi6}), (\ref{phi30}) and (\ref{phi42}), we get \begin{equation} \frac{x^{22}}{2}<\Phi_{6}(x)\Phi_{30}(x)\Phi_{42}(x)<4x^{22} \label{inequality} \end{equation} and hence $\Phi_{6}(x)\Phi_{30}(x)\Phi_{42}(x)<4(x^{35}+1)$. Thus, by Lemma \ref{twotimes} and the induction hypothesis we obtain that $$2(x^{35}+1)\Phi_{6}(x)\Phi_{30}(x)\Phi_{42}(x)$$ is practical.
By (\ref{inequality}), $$2(x^{35}+1)\Phi_{6}(x)\Phi_{30}(x)\Phi_{42}(x)>x^{57}>x^{48}.$$ So, applying (\ref{phi42}) and Lemma \ref{twotimes}, we conclude that $$2(x^{35}+1)\Phi_{6}(x)\Phi_{30}(x)\Phi_{42}(x)\Phi_{210}(x)$$ is practical. In view of \eqref{105}, this indicates that $m_{k+1}$ is practical. This completes the proof. \qed
\begin{lemma} {\rm (Melfi \cite{two-conjecture})} For every $k \in \Bbb N$, both $ 2(3^{3^k\cdot 70}-1) $ and $ 2(3^{3^k\cdot 70}+1)$ are practical numbers. \label{bothpractical} \end{lemma}
\noindent{\it Proof of Theorem} 1.4. (i) We first consider the case $d=4$. For each $k=0,1,2,\ldots$, define $$a_k=2(3^{3^k\cdot 70}-1),\ b_k=4\cdot 3^{3^k\cdot 35},\ \text{and}\ c_k=2(3^{3^k\cdot 70}+1).$$ It is easy to see that $a_k^2+b_k^2=c_k^2$ and $\gcd(a_k,b_k,c_k)=4$. By Lemma \ref{bothpractical}, $a_k$ and $c_k$ are both practical. Theorem \ref{twotimes} implies that $b_k$ is practical. This proves Theorem 1.4 for $d=4$.
(ii) Now we handle the case $d=6$. For any $k=0,1,2,\ldots$, define $$x_k=3(3^{3^k\cdot 70}-1),\ y_k=6\cdot 3^{3^k\cdot 35},\ \text{and}\ z_k=3(3^{3^k\cdot 70}+1).$$
Then $x_k^2+y_k^2=z_k^2$ and $\gcd(x_k,y_k,z_k)=6$. Note that
$y_k$ is practical for any $k=0,1,2,\ldots$ by Theorem \ref{twotimes}.
Now it remains to show by induction that $x_k$ and $z_k$ are practical for all $k=0,1,2,\ldots$.
Via a computer, we see that $x_0=3^{71}-3$ and $z_0=3^{71}+3$ are practical numbers.
Suppose that $x_k$ and $z_k$ are practical for some nonnegative integer $k$. Then
\begin{equation}\label{2.8}
x_{k+1}=3(3^{3^{k+1}\cdot 70}-1)=x_k(3^{3^k\cdot 70}-3^{3^k\cdot 35}+1)(3^{3^k\cdot 70}+3^{3^k\cdot 35}+1)
\end{equation}
and
\begin{equation}\label{2.9} z_{k+1}=3(3^{3^{k+1}\cdot 70}+1)= z_k\Phi_{12}(3^{3^k})\Phi_{60}(3^{3^k})\Phi_{84}(3^{3^k})\Phi_{420}(3^{3^k}).
\end{equation} In view of (\ref{2.8}), by applying Lemma \ref{twotimes} twice, we see that $x_{k+1}$ is practical. It is easy to check that \begin{gather*} \Phi_{12}(3^{3^k})\leqslant 2 z_k,\ \Phi_{60}(3^{3^k})\leqslant 2z_k\Phi_{12}(3^{3^k}), \\\Phi_{84}(3^{3^k})\leqslant 2z_k\Phi_{12}(3^{3^k})\Phi_{60}(3^{3^k}),\ \Phi_{420}(3^{3^k})\leqslant 2 z_k\Phi_{12}(3^{3^k})\Phi_{60}(3^{3^k})\Phi_{84}(3^{3^k}). \end{gather*} In light of these and \eqref{2.9}, by applying Lemma \ref{twotimes} four times, we see that $z_{k+1}$ is practical. This concludes the induction step.
The proof of Theorem 1.4 is now complete. \qed
\end{document} | arXiv |
Restricted until 1 September 2020.
Möbius transformations have been thoughtfully studied over the field of complex numbers. In this thesis, we investigate Möbius transformations over two rings which are not fields: the ring of double numbers $\mathbb O$ and the ring of dual numbers $\mathbb D$. We will see certain similarity between the cases of fields and rings along with some significant distinctions. After the introduction and necessary background material, given in the first two chapters, I introduce general linear groups, projective lines and Möbius transformations over several rings such us the ring of integer numbers, the Cartesian product ring and the two rings $\mathbb O$ and $\mathbb D.$ In the following chapters, we consider in details metrics, classification of Möbius maps based on the number of fixed points, connected continuous one-parameter subgroups and an application of Möbius maps. | CommonCrawl |
\begin{document}
\title*{A Solution of polynomial equations} \titlerunning{A Solution of polynomial equations} \author{N. Tsirivas}
\authorrunning{N.~Tsirivas } \institute{N.~Tsirivas \at Diovouniotou 30-32, T.K. 11741, Athens, Greece\\
\email{[email protected]} } \maketitle
\abstract{We present a method for the solution of polynomial equations. We do not intend to present one more method among several others, because today there are many excellent methods. Our main aim is educational. Here we attempt to present a method with elementary tools in order to be understood and useful by students and educators. For this reason, we provide a self contained approach. Our method is a variation of the well known method of resultant, that has its origin back to Euler. Our goal, in the present paper, is in the spirit of calculus and secondary school mathematics. An extensive discussion of the theory of zeros of polynomials and extremal problems for polynomials the reader can find in the books \cite{10} and \cite{13}.}\vspace*{0.2cm}
\noindent
{\em MSC (2010)}: 65H04\\ {\em Keywords}: Polynomial equation, resultant, Gr\"{o}bner bases.
\\
\noindent
{\bf Introduction}
It is well known that many problems in Physics, Chemistry and Science lead generally to a polynomial equation.
In pure mathematics also, there are classical problems that lead to a polynomial equation.
Let us give two examples:
1) If we are to compute the integral $\displaystyle\int^\bi_\al\dfrac{p(x)}{q(x)}dx$, where $\al,\bi\in{\mathbb{R}}$, $\al<\bi$, and $p(x),q(x)$ are two real polynomials of one variable, and $q(x)$ is a non-zero polynomial that does not have any root in the interval $[\al,\bi]$, then we are led to the problem of finding the real roots of $q(x)$.
2) Let $n\in{\Bbb N}$, $a_i\in{\mathbb{R}}$ for $i=1,\ld,n$, where ${\Bbb N},{\mathbb{R}}$ are the sets of natural and real numbers respectively.
We can consider the differential equation
\[ a_ny^{(n)}+a_{n-1}y^{(n-1)}+\cdots+a_1y+a_0=0, \]
where $y$ is the unknown function.
In order to solve this simple equation we have to find all the roots of the polynomial
\[ p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0. \]
So, the utility to solve a polynomial equation, or in other words to find the roots of a polynomial is undoubted. This problem is a very old, classical problem in mathematics and Numerical Analysis, especially. For this reason, there exist many methods that solve it.
However, if a scientist wants to solve an equation for his work, it is sufficient to use programs as ``mathematica'' and ``maple'', nowadays. So, the utility of the problem has an other direction, which is the finding of better algorithms and programs. This is the main line of research in the area experts, nowadays.
We are moving in an other direction in this paper.
Our main aim is mainly educational.
In this paper we present a method of solving a polynomial equation with full details for educational reasons so that a student of positive sciences can improve the level of knowledge in the subject. First of all, let us state our problem. We denote ${\Bbb C}$ as the set of complex numbers. Let $n\in{\Bbb N}$ and $a_i\in{\Bbb C}$ for $i=0,1,\ld,n$. We then consider the polynomial
\[ p(z)=a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0, \]
that is a polynomial of one complex variable $z$ with complex coefficients. We suppose that $a_n\neq0$. The natural number $n$ is called the degree of $p(z)$ and it is denoted by $degp(z)=n$. The number $a\in{\Bbb C}$ is a root of $p(z)$ when it is applicable: $p(a)=0$. Our problem is to find all the roots of $p(z)$, or in other words to solve the equation, $p(z)=0$. Polynomials are simple and specific functions that have the following fundamental property:
\\
\noindent
{\bf Fundamental Theorem of Algebra}. Every polynomial of one complex variable with complex coefficients and a degree greater or equal to one has at least one root in ${\Bbb C}$.
This result is central. It is the basis of our method.
However, even if this theorem is fundamental, its proof is not trivial. Its simplest proof comes from complex analysis that many students do not learn in university. In the appendix we give one of the simplest proofs of the fundamental Theorem.
Many of the best methods of our problem are iterative. They are based on the construction of specific sequences that approach to the roots of the supposed polynomial. Our method here uses algebra as much as possible, and when algebra cannot go further, analysis takes its role in solving the problem. Here we do not deal with the problem of speed of convergence. We use numerical analysis as little as we can. It is sufficient for us to use the simplest method in order to find a root in a specific real open interval, the bisection method.
Most of the books on numerical analysis describe the bisection method with details. For example see (\cite{8}, \cite{11}).
There are some formulas that provide bounds of the roots of a polynomial. A. Cauchy had given such a bound, see \cite{8}. In the frame of our method we provide such a bound.
There are some results that give information about the number of positive or real roots, for example Descart's law of signs and Sturm's sequence \cite{11}. A basic problem is to find disjoint real intervals so that every one of them contains one root exactly. There are, also, many methods for this.
Let us describe now, roughly, the stages of our method.
1) In the first stage we find all the real roots of a polynomial. For this reason we are based on two results. First of all the bisection method and secondly by the following result:
If we have a polynomial $p$ of one real variable with real coefficients with a degree greater or equal to one for which we know the roots of $p'$, then using the bisection method we can find all the real roots of $p$.
The first stage is simple. It uses only elementary knowledge and it is also convenient for students of secondary school!
We think that it is very useful for students of secondary school to know a method that find all the real roots of an arbitrary real polynomial with their knowledge base.
2) In the second stage, we provide a method that gives all the real roots of a system of the form:
\[ \left\{\begin{array}{c}
p(x,y)=0 \\ [2ex]
q(x,y)=0
\end{array}\right. \eqno{\mbox{(A)}} \]
where $p(x,y),q(x,y)$ are polynomials of two real variables $x$ and $y$ with real coefficients. Our method here is a variation of the well known method of resultant (see \cite{6}, \cite{13}), that has its origin in Euler. With this method the solution of the above system $A$ is reduced to the first stage. As in the first stage, the second stage is also convenient for students of secondary school, (except for Theorem 3.17 in our prerequisites).
3) In the third stage we show that the solution of our problem is reduced to the second stage.
So, roughly speaking, our main aim in this paper is to present a method that is in the frame of the usual lessons of calculus in secondary school or in university and present it with all the necessary details in order for it to be understand by students.
As for the notation. Let $p(x,y)$ be a polynomial of two real variables $x$ and $y$, with real coefficients. We denote $deg_xp(x,y)$ the greatest degree of $p(x,y)$ with respect to $x$ and $deg_yp(x,y)$ the greatest degree of $p(x,y)$ with respect to $y$. If $deg_xp(x,y)\ge1$ and $deg_yp(x,y)\ge1$, we call the polynomial $p(x,y)$ a pure polynomial. If $p(z)$, $q(z)$ are two complex polynomials, we write $p(z)\equiv q(z)$, when they are equal by identity. We also write $p(z)\equiv0$, when $p(z)$ is equal to zero polynomial by identity. We write $p(z)\not\equiv q(z)$ when, polynomials $p(z)$, $q(z)$, are not equal by identity and $p(z)\not\equiv0$, when $p(z)$ is not the zero polynomial.
There are many methods and algorithms to the solution of polynomial equations. Some of them are very old like the methods of Horner, Graeffe and Bernoulli, whereas today there are some others like the methods of Rutishauser, Lehmer, Lin, Bairstow, Bareiss and many others. Another method, similar to Bernoulli method is the QD method. A classical and popular method today is that of Muller. It is a general method, not only for polynomials.
The interested reader can find the details of some of the above methods in the books of our references, see \cite{1}, \cite{3}, \cite{4}, \cite{7}, \cite{9}, \cite{10}, \cite{11} and \cite{12}. As we said there exist many algorithms and programs to our problem.
One of the best is the subroutine ZEROIN. One can find the details of this program in \cite{4}.
As we said formerly, the basis of our method is the resultant (or eliminent). With this method we can convert a system of polynomial equations in one equation with only one unknown!
Theoretically, we can succeed in that, but the complexity of calculations is enormous, so its value today is only for polynomial equations with a low degree, and is used as a theoretical tool. For details of the resultant see \cite{6}, \cite{12}. Apart from this there are some cases where the resultant fails. This can happen, for example, when we have to solve a system of two equations with two unknowns and one of the two equations is a multiple of the other, and the system has a finite number of solutions. See, for example, the equation: $(x^2-1)^2+(y^2-2)^2=0$, that has the set of solutions
\[ L=\{(1,\sqrt{2}), (1,-\sqrt{2}), (-1,\sqrt{2}), (-1,-\sqrt{2})\}. \]
We describe with details how we handle these cases in our method here. An alternate method for our problem is to solve it with Gr\"{o}bner bases. Gr\"{o}bner bases is a method that was developed in 1960 for the division of polynomials with more than one variables. With Gr\"{o}bner bases we can also convert a system of equations in an equation with only one unknown, as the resultant does. This is the main application of Gr\"{o}bner bases. This can be done in most cases.
However, there are some cases where Gr\"{o}bner bases fail to succeed in the above, like the above case.
For Gr\"{o}bner bases, see \cite{2}. Many books of secondary school contain the elementary theory of polynomials and Euclidean division that we refer to in our prerequisites.
The structure of our paper is as follows:
In the first paragraph we give a roughly description of our method. In the second paragraph we give the complete description of our method. In the third paragraph we collect all the prerequisites tools of our method from Algebra and Analysis and we present them with all the necessary proofs, especially for results that someone cannot find easily in books.
Finally, in the last paragraph 4 (Appendix) we give one of the simplest proofs of the fundamental Theorem of Algebra that one cannot find easily in books.
We, also, give a short description of the solution of binomial equation: $x^n=a$, where $n\in{\Bbb N}$, $n\ge2$ and $a$ is a positive number.
\section{General description of the solution of our problem} \label{sec1}
\noindent
For methodological reasons, we divide the solution of our problem in three stages.
\subsection{First stage} In this stage we find all the real roots of the polynomial equation
\[ a_vx^v+a_{v-1}x^{v-1}+\cdots+a_1x+a_0=0, \]
where $a_i\in{\mathbb{R}}$, for every $i=0,1,\ld,v$, where $v\in{\Bbb N}$.
\subsection{Second stage} Let $p_1(x,y)$, $p_2(x,y)$ be two polynomials of two real variables $x$ and $y$ whose coefficients are in ${\mathbb{R}}$. We consider the system of equations.
\[ \left\{\begin{array}{cc}
p_1(x,y)=0 & (1) \\[2ex]
p_2(x,y)=0 & (2)
\end{array}\right. \eqno{\mbox{(A)}} \]
Let $L_A$ be the set of solutions of the above system (A), in ${\mathbb{R}}^2$. That is, we consider the set
\[
L_A:=\{(x,y)\in{\mathbb{R}}^2|p_1(x,y)=0 \ \ \text{and} \ \ p_2(x,y)=0\}, \]
of solutions of the above system (A), in ${\mathbb{R}}^2$. In the second stage we find the set $L_A$ under the following supposition (S)
\\
(S): {\bf Supposition}: We suppose that the set $L_A$ is finite.
That is, we solve the above system (A), in ${\mathbb{R}}^2$, in the case of supposition (S) holds. We note, that we succeed the second stage using first stage.
\subsection{Third stage} In the third stage we completely solve our initial problem of finding all the roots of the polynomial equation $a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0=0$, where $n\in{\Bbb N}$,\linebreak $a_i\in{\Bbb C}$, for every $i=0,1,\ld,n$, using the previous two stages.
The first stage is the analytical part, whereas the second and third stages are the algebraic parts of our method. The prerequisites of our method are few. Elementary calculus and the elementary linear algebra of secondary school are enough, except only for a specific case, where we use Theorem 3.17 from our prerequisites, (a very well-known result from calculus of several variables).
In the following paragraph, we give the complete description of our method.
\section{Complete description of our method}\label{sec2}
\subsection{First Stage} Let $a_i\in{\mathbb{R}}$, for $i=0,1,\ld,v$, $v\in{\Bbb N}$ and a polynomial
\[ p(x)=a_vx^v+a_{v-1}x^{v-1}+\cdots+a_1x+a_0, \]
where $a_v\neq0$, so $deg p(x)=v$.
Here we find all the real roots of $p(x)$. If $v=1$, or $v=2$ we know how to find the real roots of $p(x)$ from secondary school. Let us suppose that $v\ge 3$. We find all the real roots of $p'$ (if any) and then we find the roots of $p$ by applying basic Lemma 3.8 or Corollaries 3.9, 3.10.
More generally, we suppose that $p$ has degree $v\in{\Bbb N}$, $v\ge3$. We consider polynomials $p',p'',\ld,p^{(v-3)}$ $p^{(v-2)}$. Polynomial $p'$ has degree $v-1$, $p''$ has degree $v-2$, polynomial $p^{(v-2)}$ has degree 2.
We find the roots of $p^{(v-2)}$ (if any). After using basic Lemma 3.8, or Corollaries 3.9, 3.10 we find the roots of $p^{(v-3)}$ and going inductively, after a finite number of steps, we find the roots of $p'$ and finally in the same way the roots of $p$, and thus we complete our first stage.
\subsection{Second stage} We will now consider the system of two polynomials $p_1(x,y),p_2(x,y)$ of two real variables $x$ and $y$ with coefficients in ${\mathbb{R}}$. We solve the system (A) where
\[ \left\{\begin{array}{cc}
p_1(x,y)=0 & (1) \\[2ex]
p_2(x,y)=0 & (2)
\end{array}\right. \eqno{\mbox{(A)}} \]
We solve system (A) with the following
\\
{\bf Supposition}: We suppose that system (A) has a finite number of solutions, that is, the set
\[
L_A=\{(x,y)\in{\mathbb{R}}^2|p_1(x,y)=p_2(x,y)=0\} \]
is non-void and finite.
Firstly, we notice that one of the polynomials $p_1(x,y)$, $p_2(x,y)$, at least, is non zero, or else if $p_1(x,y)\equiv p_2(x,y)\equiv0$ for every $(x,y)\in{\mathbb{R}}$, then we have $L_A={\mathbb{R}}^2$, that is false because the set $L_A$ is finite. We will examine some cases.
First of all, we suppose that at least one of the polynomials is of one variable only. We can distinguish some cases here. Let $p_1(x,y)\equiv q_1(x)$, $p_2(x,y)\equiv q_2(x)$. Then, we solve the equations $q_1(x)=0$ and $q_2(x)=0$ with the method of the first stage and after we conclude that set $L_A$ is the set of all $(x,y)$, where $x$ is one of the common solutions of equations $q_1(x)=0$ and $q_2(x)=0$ and $y\in{\mathbb{R}}$, that is $L_A$ is an infinite set, which is false by our supposition. So this case itself cannot occur. Similarly, we can't have the case where $p_1(x,y)\equiv r_1(y)$ and $p_2(x,y)\equiv r_2(y)$. Now we consider the case where:
\[ p_1(x,y)\equiv q_1(x) \ \ \text{and} \ \ p_2(x,y)\equiv q_2(y). \]
Then, we can solve the equations $q_1(x)=0$ and $q_2(y)=0$ with the method of the first stage, and we find the finite sets $A_1=\{\rho_1,\rho_2,\ld,\rho_v\}$ and $B_1=\{\la_1,\la_2,\ld,\la_m\}$, $A_1\cup B_1\subseteq{\mathbb{R}}$ where $A_1$ is the set of roots of $q_1$ and $B_1$ is the set of roots of $q_2$, $v,m\in{\Bbb N}$. Then, we have $L_A=\{(\rho_i,\la_j), i=1,\ld,v, j=,\ld,m\}$. In a similar way, we can solve the system A, when $p_1(x,y)=r_1(y)$ and $p_2(x,y)=r_2(x)$, for some polynomials $r_1(y), r_2(x)$.
Now, we consider the case where $p_1(x,y),p_2(x,y)$ is two pure polynomials.
(i) The simplest case is when $deg_yp_1(x,y)=deg_yp_2(x,y)=1$. Then we have:
\[ \begin{array}{lc}
p_1(x,y)=\al_1(x)y+\al_2(x) & \text{and} \\ [2ex]
p_2(x,y)=\bi_1(x)y+\bi_2(x), & \end{array} \]
where $\al_1(x),\al_2(x),\bi_1(x),\bi_2(x)$ are some polynomials of real variable $x$ only and $\al_1(x)\not\equiv0$ an $\bi_1(x)\not\equiv0$, because $p_1(x,y)$, $p_2(x,y)$ are pure polynomials.
So we have to solve the system:
\[ \left\{\begin{array}{cc}
\al_1(x)y+\al_2(x)=0 & (3) \\[2ex]
\bi_1(x)y+\bi_2(x)=0 & (4)
\end{array}\right. \]
We can distinguish some cases here. There exists a $(x_0,y_0)\in L_A$, so that:
\begin{enumerate} \item[1)] $\al_1(x_0)=\bi_1(x_0)=0$. Then with (3) and (4), we get: $\al_2(x_0)=\bi_2(x_0)=0$. We get $(x_0,y)\in L_A$ for every $y\in{\mathbb{R}}$, that is false because $L_A$ is finite. So, this case can not occur.
\item[2)] $\al_1(x_0)=0$ and $\bi_1(x_0)\neq0$. Then with (4) we take:
$y_0=-\dfrac{\bi_2(x_0)}{\bi_1(x_0)}$ (5). With (3) we have: $\al_2(x_0)=0$.
So, in this case we find the common roots of polynomials $\al_1(x)$ and $\al_2(x)$, and for every common root $x_0$ of $\al_1(x)$ and $\al_2(x)$, so that $\bi_1(x_0)\neq0$, the couple $(x_0,y_0)\in L_A$, where $y_0$ is given from (5). Of course we find the real roots of polynomials $\al_1(x)$ and $\al_2(x)$ with the method of our first stage. In a similar way we find the roots $(x_0,y_0)\in L_A$, so that, $\al_1(x_0)\neq0$ and $\bi_1(x_0)=0$.
\item[3)] $\al_1(x_0)\neq0$ and $\bi_1(x_0)\neq0$.
Here, we have some cases:
\begin{itemize} \item[(i)] $\al_2(x_0)=\bi_2(x_0)=0$. Then with (3), (4) and our supposition, we get: $y=0$. So, in this case we find the common roots of polynomials $\al_2(x)$ and $\bi_2(x)$, so that they are not roots of polynomials $\al_1(x)$ and $\bi_1(x)$, with the method of the first stage. If $x_0$ is such a root, that is: $\al_2(x_0)=\bi_2(x_0)=0$ and $\al_1(x_0)\neq0$ and $\bi_2(x_0)\neq0$, then: $(x_0,0)\in L_A$.
\item[(ii)]\ \ $\al_2(x_0)=0$ and $\bi_2(x_0)\neq0$.
Then, with (3), and the facts $\al_2(x_0)=0$ and $\al_1(x_0)\neq0$, we get: $y=0$. Then, because $y=0$, by (4) we get $\bi_2(x_0)=0$, that is a contradiction by our supposition. So, this case cannot occur.
\item[(iii)]\ \ $\al_2(x_0)\neq0$ and $\bi_2(x_0)=0$. As in the previous case (ii), this case cannot occur.
\item[(iv)] \ \ $\al_2(x_0)\neq0$ and $\bi_2(x_0)\neq0$.
Then, with (3) and (4) we get:
$y_0=-\dfrac{\al_2(x_0)}{\al_1(x_0)}$ (6) and $y_0=-\dfrac{\bi_2(x_0)}{\bi_1(x_0)}$ (7).
With (6) and (7) we get:
$-\dfrac{\al_2(x_0)}{\al_1(x_0)}=-\dfrac{\bi_2(x_0)}{\bi_1(x_0)}\Leftrightarrow \al_2(x_0)\bi_1(x_0)-\al_1(x_0)\bi_2(x_0)=0$ (8) \end{itemize}
\end{enumerate}
Now, we can consider two systems of relations (A1) and (A2) as follows:
\[ \left\{\begin{array}{l}
\al_1(x)y+\al_2(x)=0, \quad (i) \\ [1.5ex]
\bi_1(x)y+\bi_2(x)=0,\quad (ii) \\ [1.5ex]
\al_1(x)\neq0, \al_2(x)\neq0, \bi_1(x)\neq0, \bi_2(x)\neq0
\end{array}\right. \eqno{\mbox{(A$_1$)}} \]
and
\[ \left\{\begin{array}{l}
\al_2(x)\bi_1(x)-\al_1(x)\bi_2(x)=0, \quad (i) \\ [1.5ex]
y=-\dfrac{\al_2(x)}{\al_1(x)},\quad (ii) \\ [1.5ex]
\al_1(x)\neq0, \al_2(x)\neq0, \bi_1(x)\neq0, \bi_2(x)\neq0
\end{array}\right. \eqno{\mbox{(A$_2$)}} \]
Let $L_{A_1}$, $L_{A_2}$ be the two set of solutions of systems $A_1$ and $A_2$ respectively. We prove that $L_{A_1}=L_{A_2}$.
By previous procedure and equalities (6) and (8) we get:
\[ L_{A_1}\subseteq L_{A_2} \quad (9) \]
Now let $(x,y)\in L_{A_2}$. Then equality ii) of $A_2$ gives equality ii) of $A_1$. By equality (i) of $(A_2)$ we get: $\al_2(x)\bi_1(x)=\al_1(x)\bi_2(x)$ and by the fact that $\al_1(x)\neq0$ and $\bi_1(x)\neq0$, we get:
\[ -\frac{\al_2(x)}{\al_1(x)}=-\frac{\bi_2(x)}{\bi_1(x)}. \quad (10) \]
Through the equality (ii) of $(A_2)$ and (10) we get:
\[ y=-\frac{\bi_2(x)}{\bi_1(x)}. \quad (11) \]
Equality (11) gives equality (ii) of $(A_1)$. So we have $(x,y)\in L_{A_1}$, that is $L_{A_2}\subseteq L_{A_1}$ \ \ (12).
By (9) and (12) we get: $L_{A_1}=L_{A_2}$.
So, we proved that in order to solve system $(A_1)$ it suffices to solve system $(A_2)$. Thus, we solve equation (i) of $(A_2)$ with the method of the first stage, and for every root $x$ of polynomial $\al_2(x)\bi_1(x)-\al_1(x)\bi_2(x)$ so that $\al_1(x)\neq0$, $\al_2(x)\neq0$,\linebreak $\bi_1(x)\neq0$, $\bi_2(x)\neq0$, we get the respective $y$ from equality (ii) of $(A_2)$.
So far we have completely solved the system (A), in the case of\\ $deg_yp_1(x,y)=deg_yp_2(x,y)=1$.
For the sequel we solve the case ii) where $deg_yp_1(x,y)\le2$ and $deg_yp_2(x,y)\le2$ and $p_1(x,y),p_2(x,y)$ are two pure polynomials. Of course, we have $deg_yp_1(x,y)\ge1$ and $deg_yp_2(x,y)\ge1$, because $p_1(x,y),p_2(x,y)$ are pure polynomials.
We have already examined the case $deg_yp_1(x,y)=deg_yp_2(x,y)=1$.
So we, here, examine the case where at least one of natural numbers $deg_yp(x,y)$, $deg_yp_2(x,y)$ are equal to 2.
We examine, firstly, the case where:
\[ deg_yp_1(x,y)=2 \ \ \text{and} \ \ deg_yp_1(x,y)=1. \]
Then, we can write the system (A) as follows:
\[ \left\{\begin{array}{rc}
\al_2(x)y^2+\al_1(x)y+\al_0(x)=0 & (13) \\ [2ex]
\bi_1(x)y+\bi_0(x)=0 & (14) \\
\end{array}\right. \eqno{\mbox{(A)}} \]
If $\al_2(x)\equiv0$, we have the previous system. So we suppose that $\al_2(x)\not\equiv0$.
Now let some $(x_0,y_0)\in L_A$ as above. We distinguish some cases:
\begin{enumerate} \item[1)] $\al_2(x_0)=0$. Then, we solve the system $\left\{\begin{array}{l} \al_1(x)y+\al_0(x)=0 \\ \bi_1(x)y+\bi_0(x)=0 \end{array}\right.$ as previously and we take only the solutions $(x,y)$ of this system so that $\al_2(x)=0$ holds, solving the equation $\al_2(x)=0$ with the method of the first stage.
\item[2)] $\al_2(x_0)\neq0$. We distinguish some cases.
\begin{itemize} \item[(i)] \ \ $\al_1(x_0)=\bi_1(x_0)=0$. Then we have to solve the system
\[ \left\{\begin{array}{rc}
\al_2(x)y^2+\al_2(x)=0 & (15) \\ [2ex]
\bi_0(x)=0 & (16) \end{array}\right.. \eqno{\mbox{(B$_1$)}} \]
By (15) we take:
\[ y^2=-\frac{\al_0(x)}{\al_2(x)}. \quad (17) \]
So, in order to solve this system we do the following:
First of all we find all the common roots $x$ of three polynomials $\al_1(x)$, $\bi_1(x)$, $\bi_0(x)$ that are not roots of polynomial $\al_2(x)$.
If $x\in{\mathbb{R}}$ and $\al_1(x)=\bi_1(x)=\bi_0(x)=0$ and $\al_2(x)\neq0$, we consider the number $-\dfrac{\al_0(x)}{\al_2(x)}$. If $-\dfrac{\al_0(x)}{\al_2(x)}\ge0$, then we set
$\left(y_1=\sqrt{-\dfrac{\al_0(x)}{\al_2(x)}}\right.$ and $y_2=-\sqrt{-\dfrac{\al_0(x)}{\al_2(x)}}$, if $\left.-\dfrac{\al_0(x)}{\al_2(x)}>0\right)$ and\\ $(y=0$ if $\al_0(x)=0)$, and then under the above conditions $(x,y)\in L_A$.
We find the roots of polynomials $\al_1(x)$, $\bi_1(x)$, $\bi_0(x)$ with the method of the first stage.
Of course, if we cannot find couples $(x,y)\in{\mathbb{R}}^2$ so that all the above conditions hold, this means, that we do not have solutions to this case.
\item[(ii)] \ \ $\al_1(x_0)=0$, $\bi_1(x_0)\neq0$.
We consider the system:
\[ \left\{\begin{array}{rc}
\al_2(x)y^2+\al_0(x)=0 & (17) \\ [2ex]
\bi_1(x)y+\bi_0(x)=0 & (18)
\end{array}\right.. \eqno{\mbox{(B$_1$)}} \]
Through (17) and (18) we get:
\[ y^2=-\frac{\al_2(x)}{\al_2(x)} \quad (19) \]
\[ y=-\frac{\bi_0(x)}{\bi_1(x)} \quad (20) \ \ \Rightarrow \ \ y^2=\bigg(\frac{\bi_0(x)}{\bi_1(x)}\bigg)^2 \quad (21) \]
Through (19) and (21) we get:
\[ -\frac{\al_0(x)}{\al_2(x)}=\bigg(\frac{\bi_0(x)}{\bi_1(x)}\bigg)^2\Leftrightarrow\al_2(x) \bi_0(x)^2+\al_0(x)\bi_1(x)^2=0. \quad (22) \]
From the above, in order to find a solution of system $(B_2)$ we do the following:
We find all the common roots of two polynomials $\al_2(x)\bi_0(x)^2+\al_0(x)\bi_1(x)^2$ and $\al_1(x)$, that are not roots of polynomials $\al_2(x)$ and $\bi_1(x)$ (if any). Let $x$ be such a root. We set $y=-\dfrac{\bi_0(x)}{\bi_1(x)}$, and then $(x,y)$ is a solution of $(B_2)$ and we get all the other solutions of $(B_2)$ in the same way.
\item[(iii)] \ \ $\al_1(x_0)\neq0$, $\bi_1(x_0)=0$.
Then, through (14) we get $\bi_0(x_0)=0$. So, in order to solve system (A) in this case, we do the following.
We find all the common roots (if any) $x$ of polynomials $\bi_1(x)$, $\bi_0(x)$, so that $\al_2(x)\neq0$ and $\al_1(x)\neq0$. Of course this is a finite set of numbers $x$.
For such a root $x_0$ we solve the equation $\ \al_2(x_0)y^2+\al_1(x_0)y+\al_0(x)=0$ and we find the respective number $y$ (if any).
All these couples $(x,y)\in{\mathbb{R}}^2$ (if any) are the set of solutions of system (A) in this case.
\item[(iv)] \ \ $\al_1(x_0)\neq0$, $\bi_1(x_0)\neq0$.
We leave this case for later. In a similar way we examine the case where $deg_yp_1(x,y)=1$ and $deg_yp_2(x,y)=2$.
\end{itemize}
\end{enumerate}
Now, we examine the case where:
\[ deg_yp_1(x,y)=deg_yp_2(x,y)=2. \]
We have the system:
\[ \left\{\begin{array}{cc}
\al_2(x)y^2+\al_1(x)y+\al_0(x)=0 & (23) \\ [2ex]
\bi_2(x)y^2+\bi_1(x)y+\bi_0(x)=0 & (24)
\end{array}\right.\eqno{\mbox{(B$_3$)}} \]
Here we examine some cases:
\begin{enumerate} \item[1)] Let $(x_0,y_0)\in L_{B_3}$.
If $\al_2(x_0)=0$, or $\bi_2(x_0)=0$ we have a system as in the previous case.
So, we suppose that:
\[ \al_2(x_0)\neq0 \ \ \text{and} \ \ \bi_2(x_0)\neq0. \]
Now, we can distinguish some cases:
\begin{itemize} \item[i)] \ \ $\al_1(x_0)=\bi_1(x_0)=0$.
So, we are to solve the system:
\[ \begin{array}{rl}
\al_2(x_0)y^2+\al_0(x_0)=0 & (25) \ \ \text{and} \\[1.5ex]
\bi_2(x_0)y^2+\bi_0(x_0)=0 & (26). \end{array} \]
Through (25) we have $y^2=-\dfrac{\al_0(x_0)}{\al_2(x_0)}$ (27) and by (26) we get
\[ y^2=-\frac{\bi_0(x_0)}{\bi_2(x_0)}. \eqno{\mbox{(28)}} \]
Through (27) and (28) we get:
\[ -\frac{\al_0(x_0)}{\al_2(x_0)}=-\frac{\bi_0(x_0)}{\bi_2(x_0)}\Leftrightarrow \al_0(x_0)\bi_2(x_0)-\bi_0(x_0)\al_2(x_0)=0. \eqno{\mbox{(29)}} \]
From the above, we have the following solution:
We find the common roots of polynomials\\ $\al_1(x),\bi_1(x),\al_0(x)\bi_2(x)-\bi_0(x)\al_2(x)$ so that $\al_2(x)\neq0$ and $\bi_2(x)\neq0$.
We suppose that there exists such a root $x_0$. If $\al_0(x_0)=0$, we get $y_0=0$.\\ If $\dfrac{\al_0(x_0)}{\al_2(x_0)}<0$, we consider
\[ y_1=\sqrt{-\frac{\al_0(x_0)}{\al_2(x_0)}}, \ \ y_2=-\sqrt{-\frac{\al_0(x_0)}{\al_2(x_0)}}, \]
and $(x_0,y_1),(x_0,y_2)\in L_{B_3}$. We get all the other solutions of $B_3$ in the same way. Of course, if $(x,y)$ does not exist with the above conditions, we do not have solutions of $B_3$ in this case.
\item[ii)] \ \ $\al_1(x_0)=0$ and $\bi_1(x_0)\neq0$.
We will postpone this case for later.
\item[iii)] \ \ $\al_1(x_0)\neq0$ and $\bi_1(x_0)=0$.
We will also postpone this case for later.
iv) \ \ $\al_1(x_0)\neq0$ and $\bi_1(x_0)\neq0$.
This is the central case of system $(B_3)$. \end{itemize}
\end{enumerate}
We consider the number:
\[
D=\al_2(x_0)\bi_1(x_0)-\al_1(x_0)\bi_2(x_0)=\left|\begin{array}{ccc}
\al_2(x_0) & & \al_1(x_0) \\ [2ex]
\bi_2(x_0) & & \bi_1(x_0)
\end{array}\right| \]
that we call it: the determinant of system $(B_3)$.
We distinguish two cases:
\\
a) $ D\neq0$.\\
We consider the linear system:
\[ \left\{\begin{array}{cl}
\al_2(x_0)z+\al_1(x_0)\oo=-\al_0(x_0) & (30) \\ [2ex]
\bi_2(x_0)z+\bi_1(x_0)\oo=-\bi_0(x_0)& (31)
\end{array}\right.\eqno{\mbox{(B$_4$)}} \]
This linear system has determinant $D\neq0$, so, it has exactly one solution.
We set:
\[
D_1=\left|\begin{array}{ccc}
-\al_0(x_0) &&\al_1(x_0) \\ [2ex]
-\bi_0(x_0) && \bi_1(x_0)
\end{array} \right| =\al_1(x_0)\bi_0(x_0)-\al_0(x_0)\bi_1(x_0) \eqno{\mbox{(32)}} \]
and
\[
D_2=\left|\begin{array}{ccc}
\al_2(x_0) &&-\al_0(x_0) \\ [2ex]
\bi_2(x_0) && -\bi_0(x_0)
\end{array} \right| =\al_0(x_0)\bi_2(x_0)-\al_2(x_0)\bi_2(x_0) \eqno{\mbox{(33)}} \]
Through Cramer's law of linear algebra we get the unique solution $(z,\oo)$ of system $B_4$, that is
\[ z=\frac{D_1}{D} \ \ \text{and} \ \ \oo=\frac{D_2}{D}. \]
From our supposition the couple $(x_0,y_0)\in L_{B_3}$. This means that the numbers $y^2_0$ and $y_0$ satisfy equations (23) and (24) of $(B_3)$, or differently, in other words the couple $(y^2_0,y_0)$ is a solution of the linear system $(B_4)$. But because of our supposition $D\neq0$, the couple $(z,\oo)$, where $z=\dfrac{D_1}{D}$ \ \ (34) and $\oo=\dfrac{D_2}{D}$ \ \ (35), is the unique solution of system $(B_4)$, as it is well known in linear algebra. So, we have $z=y^2_0$ and $\oo=y_0$, and by (34) and (35) we get:
\[ y^2_0=\frac{D_1}{D} \quad (36) \ \ \text{and} \ \ y_0=\frac{D_2}{D} \quad (37) \]
Now, we exploit the inner relation that numbers $y_0$ and $y^2_0$ have, that is:
\[ y^2_0=y_0\cdot y_0. \eqno{\mbox{(38)}} \]
Replacing (36) and (37) in relation (38), we get:
\[ \frac{D_1}{D}=y_0\cdot\frac{D_2}{D}\Rightarrow D_1-y_0D_2=0. \eqno{\mbox{(39)}} \]
By (37) we have
\[ Dy_0-D_2=0\eqno{\mbox{(40)}} \]
So, the couple $(x_0,y_0)\in L_{B_3}$ also satisfies the system:
\[ \left\{\begin{array}{rcl}
D_1-yD_2=0 && (39) \\ [2ex]
Dy-D_2=0 && (40)
\end{array}\right. \]
From the above, we have the two systems:
\[ \left\{\begin{array}{l}
\al_2(x)y^2+\al_1(x)y+\al_0(x)=0 \\ [1.5ex]
\bi_2(x)y^2+\bi_1(x)y+\bi_0(x)=0 \\ [1.5ex]
\al_2(x)\neq0,\bi_2(x)\neq0,\;\al_1(x)\neq,\;\bi_1(x)\neq0, \\ [1.5ex]
D=\left|\begin{array}{cc}
\al_2(x) & \al_1(x) \\ [2ex]
\bi_2((x) & \bi_1(x)
\end{array}\right|\neq0
\end{array}\right. \eqno{\mbox{(B$_5$)}} \]
and
\[ \left\{\begin{array}{l}
D_1=yD_2=0 \\ [1.5ex]
Dy-D_2=0 \\ [1.5ex]
\al_2(x)\neq0,\;\bi_2(x)\neq0,\;\al_1(x)\neq0,\;\bi_1(x)\neq0, \\ [1.5ex]
D\neq0
\end{array}\right.. \eqno{\mbox{(B$_6$)}} \]
Let $L_{B_5}$, $L_{B_6}$ be the set of solutions of systems $(B_5)$ and $(B_6)$. We now show that $L_{B_5}=L_{B_6}$. Of course we have $L_{B_5}\subseteq L_{B_6}$ from the previous procedure, because we obtained equalities (39) and (40) of system $(B_6)$ from system $B_5$.
Reversely, let $(x_0,y_0)\in L_{B_6}$. From the first two equalities of $(B_6)$ we get:
\[ y_0=\frac{D_1}{D_2} \ \ \text{and} \ \ y_0=\frac{D_2}{D} \eqno{\mbox{(37)}} \]
We multiply these equalities and we take: $y^2_0=\dfrac{D_1}{D}$ (36).
Now, we consider the linear system $(B_4)$. Because $D\neq0$ (by our supposition), this system has a unique solution $(z,\oo)=\Big(\dfrac{D_1}{D},\dfrac{D_2}{D}\Big)$, (41) as it is well known, in Cramer's law.
From (36), (37) and (41) we have: $z=y^2_0$ (42) and $\oo=y_0$ (43).
Replacing (42) and (43) in (30) and (31) of $(B_4)$ we get the first two equalities of $(B_5)$, that is $(x_0,y_0)\in L_{B_5}$. So, we have: $L_{B_6}\subseteq L_{B_5}$.\\ From the above we have $L_{B_5}=L_{B_6}$. So, we are led to solve system $B_6$, which we have examined previously, in the system: $\left.\begin{array}{l}
p_1(x,y)=0 \\[1.5ex]
p_2(x,y)=0
\end{array}\right\}$, where $\begin{array}{l}
deg_yp_1(x,y)\le1 \ \ \text{and} \\ [1.5ex]
deg_yp_2(x,y)\le1.
\end{array}$
\\
b) $D=0$\\
The solution of these cases is taken as follows:
We take the roots of polynomial $D$ $x_1$, that is $\al_2(x_1)\bi_1(x_1)-\al_1(x_1)\bi_2(x_1)=0$ so that $\al_2(x_1)\neq0$, $\bi_2(x_1)\neq0$, $\al_1(x_1)\neq0$ and $\bi_1(x_1)\neq0$. We get $y$ that satisfies one of the equations (23), or (24) of $(B_3)$, that is:
\[ \al_2(x_1)y^2+\al_1(x_1)y+\al_0(x_1)=0. \]
This holds because the two equations (23) and (24) are equivalent, (as we have shown in prerequisites of linear algebra), and each of them is a multiple of the other.\vspace*{0.2cm}\\
\noindent
{\bf Remark 2.2.1.} {\em We note that the three remaining cases we have left are similar to case a) above where $D\neq0$}.
So far, we have examined system (A), where $deg_yp_1(x,y)\le2$ and\linebreak $deg_yp_2(x,y)\le2$. We set
\[ m_0:=\max\{deg_yp_1(x,y),deg_yp_2(x,y)\}. \]
We solve system (A) in the general case with induction above the number $m_0$. We have examined the cases where $m_0=1$ or $m_0=2$.
We suppose that for $k_0\in{\Bbb N}$, $k_0\ge3$, we have solved system (A) for every system so that $m_0\le k_1-1$. We now solve system (A) when $m_0=k_0$.
We can write polynomials $p_1(x,y),p_2(x,y)$ as follows:
\begin{align*} \al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)&=p_1(x,y) \ \ \text{and} \ \ \bi_{n_0}(x)y^{n_0}+\bi_{\mi_0}(x)y^{\mi_0}+q_2(x,y)\\ &=p_2(x,y), \end{align*}
where $v_0<m_0$, $v_0,m_0\in{\Bbb N}$, $deg_yq_1(x,y)<v_0$ and $\mi_0<n_0$, $\mi_0,n_0\in{\Bbb N}$, $n_0\le m_0$, $deg_yq_2(x,y)<\mi_0$.
So, the initial system can be written as follows:
\[ \left\{\begin{array}{rl}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 & (1) \\ [2ex]
\bi_{n_0}(x)y^{n_0}+\bi_{\mi_0}(x)y^{\mi_0}+q_2(x,y)=0 & (2)
\end{array}\right. \eqno{\mbox{(A)}} \]
where $\al_{m_0}(x),q_{v_0}(x),\bi_{n_0}(x),\bi_{\mi_0}(x)$ are polynomials of the real variable $x$ only and $q_1(x,y),q_2(x,y)$ be polynomials of real variables $x$ and $y$.
We, also, suppose that $a_{m_0}(x)\not\equiv0$. We can distinguish some cases as previously:
1) Let $\al_{v_0}(x)\equiv q_1(x,y)\equiv\bi_{n_0}(x)\equiv\bi_{\mi_0}(x)\equiv q_2(x,y)\equiv0$. Then we have the system: $\al_{m_0}(x)y^{m_0}=0$. If $m_0=0$, and $\al_{m_0}(x)=c\in{\mathbb{R}}$ then every couple $(x,0)\in L_A$ and the system has an infinite set of solutions, which is false. So, this case cannot occur.
2) $\bi_{n_0}(x)\equiv\bi_{\mi_0}(x)\equiv q_2(x,y)\equiv0$.
We will examine this case later.
3) $\bi_{n_0}(x)\not\equiv0$, $\al_{v_0}(x)\equiv q_1(x,y)\equiv\bi_{\mi_0}(x)\equiv q_2(x,y)\equiv0$.
We have the system
\[ \left\{\begin{array}{c}
\al_{m_0}(x)y^{m_0}=0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}=0
\end{array}\right. \eqno{\mbox{(A)}} \]
If $deg\al_{m_0}(x)=deg\bi_{n_0}(x)=0$, then any couple $(x,0)\in L_A$ and the set of solutions is infinite, which is false. So, this case cannot occur.
4) $\bi_{n_0}(x)\not\equiv0$, $\al_{v_0}(x)\equiv\bi_{\mi_0}(x)\equiv0\equiv q_1(x,y)$ and $q_2(x,y)\equiv r(x)\not\equiv0$.
So, we have the system:
\[ \left\{\begin{array}{rl}
\al_{m_0}(x)y^{m_0}=0 & (3) \\ [2ex]
\bi_{n_0}(x)y^{n_0}+r(x)=0 & (4) \end{array}\right. \eqno{\mbox{(A)}} \]
We can distinguish some cases here.
Firstly, we suppose that (A) has a solution $(x_0,y_0)\in L_A$.
\begin{enumerate} \item[i)] $\al_{m_0}(x_0)=0=\bi_{n_0}(x_0)$. Then, of course $r(x_0)=0$. So, if the polynomials $\al_{m_0}(x), \bi_{n_0}(x),r(x)$, has a common root $x_0$, then any couple $(x_0,y)\in L_A$, for every $y\in{\mathbb{R}}$, which is false of course.
So, this case cannot happen.
\item[ii)] $\al_{m_0}(x_0)=0$, $\bi_{n_0}(x_0)\neq0$.
By (4), we take $y_0=-\dfrac{r(x_0)}{\bi_{n_0}(x_0)}$.
Thus, in this case we solve the system as follows:
We find the roots $x$ of $\al_{m_0}(x)$, so that $\bi_{n_0}(x)\neq0$. For every such root the couple $(x,y)=\Big(x,-\dfrac{r(x)}{\bi_{n_0}(x)}\Big)\in L_A$.
We get all the other solutions of this system in the same way.
\item[iii)] $\al_{m_0}(x_0)\neq0$ and $\bi_{n_0}(x_0)=0$.
Through (3), we get $y=0$. By (4), we take $r(x_0)=0$.
So, in this case we can solve the system as follows: We find the roots of $r(x)$ such that $\al_{m_0}(x)\neq0$ and $\bi_{n_0}(x)=0$. For every such root $x$, the couple $(x,0)\in L_A$.
\item[iv)] $\al_{m_0}(x_0)\neq0$ and $\bi_{n_0}(x_0)\neq0$.
By (3) we get $y_0=0$, and by (4), for $y_0=0$ we get $r(x_0)=0$.
So, in this case we can solve the system as follows: We find the roots $x$ of $r(x)$ so that $\al_{m_0}(x)\neq0$ and $\bi_{n_0}\neq0$. Then the couple $(x,0)$ is a solution of (A).
\item[v)] In a similar way we can solve a system of the form:
\[ \left\{\begin{array}{r}
\al_{m_0}(x)y^{m_0}+r(x)=0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}=0.
\end{array}\right. \]
\end{enumerate}
5) $\bi_{n_0}(x)\not\equiv0$, $\al_{v_0}(x)\equiv\bi_{\mi_0}(x)\equiv0$, $q_1(x,y)\equiv r_v(x)\not\equiv0$, $q_2(x,y)\equiv r_2(x)\not\equiv0$.
So, we have the system:
\[ \left\{\begin{array}{c}
\al_{m_0}(x)y^{m_0}+r_1(x)=0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}+r_2(x)=0
\end{array}\right. \eqno{\mbox{(A)}} \]
Let $(x_0,y_0)\in L_A$.
If $r_1(x_0)=0$ or $r_2(x_0)=0$, then we have the system of the previous case 4. So, we suppose that: $r_1(x_0)\neq0$ and $r_2(x_0)\neq0$. We can distinguish some cases:
\begin{enumerate} \item[(i)] $\al_{m_0}(x_0)=\bi_{n_0}(x_0)=0$
So, we have the system:
\[ \left\{\begin{array}{c}
r_1(x)=0 \\ [2ex]
r_2(x)=0
\end{array}\right.. \eqno{\mbox{(A)}} \]
Let $x_0$ be a common root of $r_1(x),r_2(x)$. Then, we have $(x_0,y)\in L_A$, for every $y\in{\mathbb{R}}$, which is false of course, because the set $L_A$ is finite.
So, this case cannot occur.
\item[(ii)] $\al_{m_0}(x_0)=0$ and $\bi_{n_0}(x_0)\neq0$. Then, we get $r_1(x_0)=0$ and $y^{n_0}_0=-\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}$, and if $n_0$ is odd we have $y_0=\sqrt[n_0]{-\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}}$ if $\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}\le0$ and $y_0=-\sqrt[n_0]{\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}}$\\ if $\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}>0$.
If $n_0$ is even and $\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}\le0$, we have:
\[ y_1=\sqrt[n_0]{-\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}} \ \ \text{and} \ \ y_2=-\sqrt[n_0]{-\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}}. \]
So, in this case we solve the system as follows:
We find the common roots of polynomials $\al_{m_0}(x)$ and $r_1(x_0)$, such that $\bi_{n_0}(x)\neq0$. Let $x_0$ such a root.
Then, if $n_0$ is odd and $\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}\le0$, then the couple $(x_0,y_0)\in L_A$, where
\[ y_0=\sqrt[n_0]{-\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}}, \]
whereas if $\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}>0$, then the couple $(x_0,y_0)\in L_A$, where
\[ y_0=-\sqrt[n_0]{\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}}. \]
If $n_0$ is even, we set
\[ y_1=\sqrt[n_0]{-\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}} \ \ \text{and} \ \ y_2=-\sqrt[n_0]{-\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}}, \]
and the couples $(x_0,y_1),(x_0,y_2)\in L_A$, where $\dfrac{r_2(x_0)}{\bi_{n_0}(x_0)}\le0$. This case can happen if the above conditions hold, of course.
\item[(iii)] $\al_{m_0}(x_0)\neq0$, $\bi_{n_0}(x_0)=0$.
This case is similar to the previous case (ii).
\item[(iv)] $\al_{m_0}(x_0)\neq0$ and $\bi_{n_0}(x_0)\neq0$.
Through the equations of $(A)$ we get:
\[ y^{m_0}_0=-\frac{r_1(x_0)}{\al_{m_0}(x_0)} \ \ \text{and} \ \ y^{n_0}_0=-\frac{r_2(x_0)}{\bi_{n_0}(x_0)}. \]
Through these equations we get:
\[ y^{n_0m_0}_0=(-1)^{n_0}\bigg(\frac{r_1(x_0)}{\al_{m_0}(x_0)}\bigg)^{n_0} \ \ \text{and} \ \ y^{n_0m_0}_0=(-1)^{m_0}\bigg(\frac{r_2(x_0)}{\bi_{n_0}(x_0)}\bigg)^{m_0} \]
and by these equations we get:
\begin{align*} (-1)^{n_0}\bigg(\frac{r_1(x_0)}{\al_{m_0}(x_0)}\bigg)^{n_0}&=(-1)^{m_0} \bigg(\frac{r_2(x_0)}{\bi_{n_0}(x_0)}\bigg)^{m_0}\\ &\Leftrightarrow(-1)^{m_0-n_0}r_2(x_0)^{m_0}\al_{m_0} (x_0)^{n_0}-r_1(x_0)^{n_0}\bi_{n_0}(x_0)^{m_0}=0. \end{align*}
So, we solve this case as follows:
We find the real roots of polynomial $(-1)^{m_0-n_0}r_2(x)^{m_0}\al_{m_0}(x)^{n_0}-r_1(x)^{n_0}\bi_{n_0}(x)^{m_0}$, so that $\al_{m_0}(x)\neq0$ and $\bi_{n_0}(x)\neq0$.
For every $x_0$, we get $y_0$ so that $y^{m_0}_0=-\dfrac{r_1(x_0)}{\al_{m_0}(x_0)}$ as in the previous case (ii). \end{enumerate}
6) $\bi_{n_0}(x)\not\equiv0$, $\al_{v_0}(x)\equiv\bi_{\mi_0}(x)\equiv0$, $q_1(x,y),q_2(x,y)$ are two pure polynomials. So, we have to solve the system:
\[ \left\{\begin{array}{r}
\al_{m_0}(x)y^{m_0}+q_1(x,y)=0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}+q_2(x,y)=0
\end{array}\right. \eqno{\mbox{(A)}} \]
where $deg_yq_1(x,y)\ge1$, $deg_yq_2(x,y)\ge1$, and $q_1(x,y),q_2(x,y)$ are two monomials. So, in this case we have the system:
\[ \left\{\begin{array}{r}
\al_{m_0}(x)y^{m_0}+\al_0(x)y^{\la_1}=0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}+\bi_0(x)y^{\la_2}=0
\end{array}\right., \]
where $\al_0(x),\bi_0(x)$ are two polynomials so that one of them (at least) is non-zero and $\la_1,\la_2\in{\Bbb N}$, so that $\la_1<m_0$ and $\la_2<n_0$. We can write the system as follows:
\[ \left\{\begin{array}{r}
y^{\la_1}(\al_{m_0}(x)y^{m_0-\la_1}+\al_0(x))=0 \\ [2ex]
y^{\la_2}(\bi_{n_0}(x)^{n_0-\la_2}+\bi_0(x))=0
\end{array}\right., \eqno{\mbox{(A)}} \]
so for every $x\in{\mathbb{R}}$, the couple $(x,0)\in L_A$, which is false, because $L_A$ is finite. Thus, this cannot occur.
7) We suppose that $\bi_{n_0}(x)\not\equiv0$ $q_1(x,y)\equiv r_1(x)$, $\al_{v_0}(x)\not\equiv0$ or $\bi_{\mi_0}(x)\not\equiv0$, $q_2(x,y)\equiv r_2(x)$, where $r_1(x)\not\equiv0$ or $r_2(x)\not\equiv0$. So, we get the system:
\[ \left\{\begin{array}{r}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+r_1(x)=0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}+\bi_{\mi_0}(x)y^{\mi_0}+r_2(x)=0 \end{array}\right. \eqno{\mbox{(A)}} \]
We distinguish some cases:
\begin{enumerate} \item[(i)] $r_1(x)\equiv0$ and $r_2(x)\not\equiv0$. So, we get the system:
\[ \left\{\begin{array}{lr}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}& =0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}+\bi_{\mi_0}(x)y^{\mi_0}+r_2(x)& =0
\end{array}\right. \eqno{\mbox{(A)}} \]
Let $(x_0,y_0)\in L_A$.
Through the first equation we get
\[ y^{v_0}(\al_{m_0}(x)y^{m_0-v_0}+\al_{v_0}(x))=0 \]
If $y=0$, by the second equation we get $r_2(x_0)=0$. So, if $x_0$ is a root of $r_2(x)$, then the couple $(x_0,0)\in L_A$.
Let $y_0\neq0$. Then we get:
\[ \al_{m_0}(x_0)y^{m_0-v_0}_0+\al_{v_0}(x_0)=0 \ \ \text{and} \ \ \bi_{n_0}(x_0)y^{n_0}_0+\bi_{\mi_0}(x_0)y^{\mi_0}_0+r_2(x_0)=0. \]
If $r_2(x_0)=0$ we have some of the previous cases that we have already examined. So, we suppose that $r_2(x_0)\neq0$. If $n_0<m_0$ we have a system that we supposedly can solve through the induction step. Thus, we suppose that $n_0=m_0$. So we have the system:
\[ \left\{\begin{array}{lr}
\al_{m_0}(x_0)y^{m_0-v_0}_0+\al_{v_0}(x_0) & =0 \\ [2ex]
\bi_{n_0}(x_0)y^{m_0}_0+\bi_{\mi_0}(x_0)y^{\mi_0}+r_2(x_0) & =0
\end{array}\right. \]
We will postpone this case, because we will examine a more general case later, that covers this case.
\item[(ii)] $r_1(x)\not\equiv0$ and $r_2(x)\equiv0$.
This case is similar to the previous.
\item[(iii)] $r_1(x)\not\equiv0$ and $r_2(x)\not\equiv0$. \end{enumerate}
Let $(x_0,y_0)\in L_A$. If $r_1(x_0)=0$, or $r_2(x_0)=0$ we have some of the previous cases. So, we suppose that $r_1(x_0)\neq0$ and $r_2(x_0)\neq0$. We have some cases:
\begin{enumerate} \item[(i)] $\al_{v_0}(x)\not\equiv0$ and $\bi_{\mi_0}(x)\equiv0$.
So, we have the system
\[ \left\{\begin{array}{lr}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0} & +r_1(x)=0 \\ [2ex]
\bi_{n_0}(x)y^{m_0} & +r_2(x)=0
\end{array}\right.. \eqno{\mbox{(A)}} \]
We consider some cases:
\begin{itemize} \item[(a)]\ \ Let $(x_0,y_0)\in L_A$, $\al_{m_0}(x_0)=\bi_{n_0}(x_0)=0$. We have examined this case previously.
\item[(b)] \ \ $\al_{m_0}(x_0)=0$ and $\bi_{n_0}(x_0)\neq0$. We have examined this case previously.
\item[(c)] \ \ $\al_{m_0}(x_0)\neq0$ and $\bi_{n_0}(x_0)=0$.
If $\al_{v_0}(x_0)=0$ we have examined this case previously. So, we suppose that $\al_{v_0}(x_0)\neq0$. Then we get $r_2(x_0)=0$ by the second equation of (A) because $\bi_{n_0}(x_0)=0$, that is false by our supposition. So, this case cannot occur.
\item[(d)] \ \ $\al_{m_0}(x_0)\neq0$ and $\bi_{n_0}(x_0)\neq0$. \end{itemize}
If $\al_{v_0}(x_0)=0$, we have examined this case previously. So, we suppose that $\al_{v_0}(x_0)\neq0$. We will examine this case later.
\item[(ii)] $\al_{v_0}(x)\equiv0$ and $\bi_{\mi_0}(x)\not\equiv0$.
This case is similar to the previous:
\item[(iii)] $\al_{v_0}(x)\not\equiv0$ and $\bi_{\mi_0}(x)\not\equiv0$.
We have some cases here:
\begin{itemize} \item[(a)] \ \ We suppose that $\bi_{n_0}(x_0)=\bi_{\mi_0}(x_0)=0$, and $\al_{m_0}(x_0)\neq0$, $\al_{v_0}(x_0)\neq0$. So, we have the system:
\[ \left\{\begin{array}{r}
\al_{m_0}(x_0)y^{m_0}+\al_{v_0}(x_0)y^{v_0}+r_1(x_0)=0 \\ [2ex]
r_2(x_0)=0
\end{array}\right.. \eqno{\mbox{(A)}} \]
This cannot happen because $r_2(x_0)\neq0$ by our supposition.
\item[(b)] \ \ We suppose that $\al_{m_0}(x_0)=\al_{v_0}(x_0)=0$. Then we get that $r_1(x_0)=0$, which is false by our supposition. Thus, this case cannot occur.
\item[(c)] \ \ If $\al_{m_0}(x_0)=0$, or $\al_{v_0}(x_0)=0$, or $\bi_{n_0}(x_0)=0$, or $\bi_{\mi_0}(x_0)=0$, then we get some of the previous cases.
\item[(d)] $\al_{m_0}(x_0)\neq0$ and $\al_{v_0}(x_0)\neq0$ and $\bi_{n_0}(x_0)\neq0$ and $\bi_{\mi_0}(x_0)\neq0$.
We have to solve the system:
\[ \left\{\begin{array}{r}
\al_{m_0}(x_0)y^{m_0}+\al_{v_0}(x_0)y^{v_0}+r_1(x_0)=0 \\ [2ex]
\bi_{n_0}(x_0)y^{m_0}+\bi_{\mi_0}(x_0)y^{\mi_0}+r_2(x_0)=0 \end{array}\right.. \eqno{\mbox{(A)}} \]
We have here the basic case of this system.
We will examine this case later in a more general case. \end{itemize}
\end{enumerate}
Now, we will examine the system:
\[ \left\{\begin{array}{r}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}+\bi_{\mi_0}(x)y^{\mi_0}+q_2(x,y)=0 \end{array}\right., \]
where $\al_{m_0}(x)\not\equiv0$, $m_0\ge3$, $v_0<m_0$, $n_0\le m_0$ $n_0>\mi_0$, $q_1(x,y),q_2(x,y)$ are two pure polynomials, $\al_{v_0}(x)\not\equiv0$, $\bi_{\mi_0}(x)\not\equiv0$.
We candistinguish the following cases:
1) $\bi_{n_0}(x)\equiv0$.
We get $(x_0,y_0)\in L_A$.
If $\al_{m_0}(x_0)=0$, then we have a system from the induction step. So, we suppose $\al_{m_0}(x_0)\neq0$.
We have some cases:
\begin{enumerate} \item[(i)] $\al_{v_0}(x_0)=\bi_{\mi_0}(x_0)=0$.
Then, we analyze polynomials $q_1(x,y),q_2(x,y)$ and we reach a system of the following form
\[ \left\{\begin{array}{r}
\al_{m_0}(x_0)y^{m_0}+\al_{v_1}(x_0)y^{v_1}+q_3(x_0,y_0)=0 \\ [2ex] \bi_{m_1}(x_0)y^{\mi_1}+q_3(x_0,y_0)=0 \end{array}\right., \eqno{\mbox{(A)}} \]
where $\al_{v_1}(x_0)\neq0$, $\bi_{\mi_1}(x_0)\neq0$, $v_1<m_0$, $\mi_1\in{\Bbb N}$, $deg_yq_3(x,y)<v_1$,\\ $deg_yq_3(x,y)<\mi_1$.
We will see later how we can solve such a system.
\item[(ii)] $\al_{v_0}(x_0)\neq0$ and $\bi_{\mi_0}(x_0)=0$.
Then, we analyze polynomial $q_2(x,y)$ and we reach a system of the following form:
\[ \left\{\begin{array}{rr}
\al_{v_0}(x_0)y^{m_0}_0+\al_{v_0}(x_0)y^{v_0}_0+q_1(x_0,y_0)=0 & (1)\\ [2ex]
\bi_{\mi_1}(x_0)y^{\mi_1}_0+r_2(x_0)=0 & (2)
\end{array} \right., \eqno{\mbox{(B)}} \]
If $\bi_{\mi_1}(x_0)=0$, we get from (2) that: $r_2(x_0)=0$.
In this case we solve system (B) as follows:
We take $x_0$ that is a common root of polynomials in the second equation (B) so that $\al_{m_0}(x_0)\neq0$ and $\al_{v_0}(x_0)\neq0$, and we find $y_0$ from the first equation of (B).
\item[(iii)] $\al_{v_0}(x_0)\neq0$ and $\bi_{\mi_0}(x_0)\neq0$. We will see later how we solve this system.
{\bf After all the above cases we reach now in the most important case}.
We have the system:
\[ \left\{\begin{array}{r}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 \\ [2ex]
\bi_{n_0}(x)y^{n_0}+\bi_{\mi_0}(x)y^{\mi_0}+q_2(x,y)=0 \end{array}\right. \eqno{\mbox{(A)}} \]
where we have:
\[ \begin{array}{l}
m_0>v_0>deg_yq_1(x,y), \\ [2ex]
n_0>\mi_0>deg_yq_2(x,y), \end{array} \] $\al_{m_0}(x)\not\equiv0$, $\al_{v_0}(x)\not\equiv0$, $\bi_{n_0}(x)\not\equiv0$, $\bi_{\mi_0}(x)\not\equiv0$, $q_1(x,y)$, $q_2(x,y)$ be two pure polynomials.
We distinguish some cases:
\item[(i)] $m_0=n_0$.
We also have some cases here.
\begin{itemize} \item[(a)] \ \ $v_0=\mi_0$.
So, we have the system:
\[ \left\{\begin{array}{r}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 \\ [2ex]
\bi_{m_0}(x)y^{m_0}+\bi_{v_0}(x)y^{v_0}+q_2(x,y)=0 \end{array}\right. \eqno{\mbox{(A)}} \]
Firstly, we examine the case where:
\[ (x_0,y_0)\in L_A \ \ \text{and} \ \ \al_{m_0}(x_0)\cdot\al_{v_0}(x_0)\cdot\bi_{m_0}(x_0)\cdot\bi_{m_0}(x_0) \cdot\bi_{v_0}(x_0)\neq0. \]
Let
\[
D=\left|\begin{array}{cc}
\al_{m_0}(x_0) & \al_{v_0}(x_0) \\ [2ex]
\bi_{m_0}(x_0) & \bi_{v_0}(x_0)
\end{array}\right|=\al_{m_0}(x_0)\bi_{v_0}(x_0)-\al_{v_0}(x_0)
\bi_{m_0}(x_0). \]
We suppose that $D\neq0$.
This exactly {\bf is the first basic case}. We will study three basic cases overall. We consider the linear system:
\[ \left\{\begin{array}{cc}
\al_{m_0}(x_0)z+\al_{v_0}(x_0)\oo=-q_1(x_0,y_0) & (1) \\ [2ex]
\bi_{m_0}(x_0)z+\bi_{v_0}(x_0)\oo=-q_2(x_0,y_0) & (2)
\end{array}\right. \eqno{\mbox{(B)}} \]
We set:
\[
D_1=\left|\begin{array}{lc}
-\al_1(x_0,y_0) & \al_{v_0}(x_0) \\ [2ex]
-q_2(x_0,y_0) & \bi_{v_0}(x_0)
\end{array}\right| \ \ \text{and} \ \
D_2=\left|\begin{array}{lc}
\al_{m_0}(x_0) & -q_1(x_0,y_0) \\ [2ex]
\bi_{m_0}(x_0) & -q_2(x_0,y_0)
\end{array}\right|. \]
That is, we have:
\[ \begin{array}{l}
D_1=\al_{v_0}(x_0)q_2(x_0,y_0)-\bi_{v_0}(x_0)q_1(x_0,y_0) \ \ \text{and} \\ [2ex]
D_2=\bi_{m_0}(x_0)q_1(x_0,y_0)-\al_{m_0}(x_0)q_2(x_0,y_0). \end{array} \]
\end{itemize}
\end{enumerate}
Because $D\neq0$, by our supposition, we take it that system (B) has only one solution $(z_0,\oo_0)$, where $z_0=\dfrac{D_1}{D}$ (3) and $\oo_0=\dfrac{D_2}{D}$ (4), as it is well known by linear algebra by Cramer's law.
Because $(x_0,y_0)\in L_A$, (by our supposition) this means that the couple $(y^{m_0}_0,y^{v_0}_0)$ is a solution of system (B).
But, $(z_0,\oo_0)$ is the unique solution of system (B). So, we have: $(z_0,\oo_0)=(y^{m_0}_0,y^{v_0}_0)\Leftrightarrow z_0=y^{m_0}_0$ (5) and $\oo_0=y^{v_0}_0$ (6). By (3), (4), (5) and (6), we get:
\[ y^{m_0}_0=\frac{D_1}{D} \ \ \text{(7)} \ \ \text{and} \ \ y^{v_0}_0 =\frac{D_2}{D}. \ \ \text{(8)} \]
Now, we use the obvious relation of numbers $y^{m_0}_0$ and $y^{v_0}_0$, that is: $y^{m_0}_0=y^{m_0-v_0}_0\cdot y^{v_0}_0$ (9), where $v_0<m_0$, by our supposition.
Replacing by (7) and (8) in (9), we get:
\[ \frac{D_1}{D}=y^{m_0-v_0}_0\cdot\frac{D_2}{D}\Leftrightarrow D_2y^{m_0-v_0}_0-D_1=0. \eqno{\mbox{(10)}} \]
From the above we see that $(x_0,y_0)$ satisfies the two equations:
\[ \left\{\begin{array}{rc}
Dy^{v_0}_0-D_2=0 & (11) \\ [2ex]
D_2y^{m_0-v_0}_0-D_1=0 & (12)
\end{array}\right. \eqno{\mbox{(C)}} \]
We notice that polynomials in (11) and (12) have degree with respect to $y$ lower than $m_0$.\\
Let us consider now the following systems
\[ \left\{\begin{array}{lc}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 & (13) \\ [1.5ex]
\bi_{m_0}(x)y^{m_0}+\bi_{v_0}(x)y^{v_0}+q_2(x,y)=0 & (14) \\[1.5ex]
y\cdot\al_{m_0}(x)\cdot\al_{v_0}(x)\bi_{m_0}(x)\cdot\bi_{v_0}(x)D\neq0 &
\end{array}\right. \eqno{\mbox{(A)}} \]
\[ \left\{\begin{array}{lc}
Dy^{v_0}-D_2=0 & (15) \\ [1.5ex]
D_2y^{m_0-v_0}-D_1=0 & (16) \\ [1.5ex]
y\cdot\al_{m_0}(x)\cdot\al_{v_0}(x)\cdot\bi_{m_0}(x)\cdot\bi_{v_0}(x)
\cdot D\neq0 & \end{array}\right. \eqno{\mbox{(B)}} \]
where $D=\al_{m_0}(x)\bi_{v_0}(x)-\al_{v_0}(x)\bi_{m_0}(x)$.
\[ \begin{array}{l}
D_1=\al_{v_0}(x)q_2(x,y)-\bi_{v_0}(x)q_1(x,y), \\ [2ex]
D_2=\bi_{m_0}(x)q_1(x,y)-\al_{m_0}(x)q_2(x,y). \end{array} \]
It is obvious that $deg_y(Dy^{v_0})=v_0<m_0$, because $D\neq0$ and $deg_y(D_2y^{m_0-v_0})<m_0$, because $deg_yD_2<v_0$, by our suppositions. We will prove now that $L_A=L_B$. It is obvious that $L_A\subseteq L_B$ (17) from the previous procedure, because we got equations (11) and (12) of system ($\Ga$) from equations of system A.
Now, let $(x_0,y_0)\in L_B$.
Through equations (15), (16) of (B) and the fact that $y_0\neq0$, we get:
\[ y^{v_0}_0=\frac{D_2}{D} \ \ \text{(18)} \ \ \text{and} \ \ y^{m_0-v_0}_0=\frac{D_1}{D_2} \ \ \text{(19)} \]
Through equations (18) and (19) we get: $y^{m_0}_0=\dfrac{D_1}{D}$ (20). Now, we consider system (B). Because $D\neq0$ this system has unique solution $(z_0,\oo_0)=\Big(\dfrac{D_1}{D},\dfrac{D_2}{D}\Big)$ (21), from Cramer's Law. Through (18), (20) and (21) we get $z_0=y^{m_0}_0$ (22) and $\oo_0=y^{v_0}_0$ (23).
Replacing (22) and (23) in equations of (B) we take it that $(x_0,y_0)\in L_A$, so $L_B\subseteq L_A$ (24). By (17) and (24), we have $L_A=L_B$ (25). The equality (25) means that: in order to solve system (A), it suffices to solve system (B), whose degree with respect to $y$ is smaller than $m_0$, that is the degree of system (A) with respect to $y$. But with the induction step, we can solve a system whose degree with respect to $y$ is smaller than $m_0$, and thus we complete this case.
{\bf The second basic case is the following} $D\not\equiv0$, but
\[ D(x_0)=\al_{m_0}(x_0)\bi_{v_0}(x_0)-\al_{v_0}(x_0)\bi_{m_0}(x_0)=0. \]
In this case the two equations of system (A) are equivalent to those of linear algebra, as we have shown in prerequisites.
So, we can solve this case as follows:
We find the roots of polynomial $D=\al_{m_0}(x)\bi_{v_0}(x)-\al_{v_0}(x)\bi_{m_0}(x)$, so that: $\al_{m_0}(x)\cdot\al_{v_0}(x)\cdot\bi_{n_0}(x)\cdot\bi_{\mi_0}(x)$.
For every such root $x_0$ we find $y_0$ from one of the equations of (A) that are equivalent. We can complete this case by finding the solutions of the form $(x,0)$ (if any).
{\bf Third basic case (singular case)}
We suppose that $D\equiv0\equiv\al_{m_0}(x)\bi_{v_0}(x)-\al_{v_0}(x)\bi_{m_0}(x)$. We call this case {\bf the singular case}.
We consider the system:
\[ \left\{\begin{array}{lc}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 & (1) \\ [1.5ex]
\bi_{m_0}(x)y^{m_0}+\bi_{v_0}(x)y^{v_0}+q_2(x,y)=0 & (2) \\ [1.5ex]
\al_{m_0}(x)\al_{v_0}(x)\bi_{m_0}(x)\bi_{v_0}(x)\neq0 &
\end{array}\right.. \eqno{\mbox{(A)}} \]
We consider our general supposition. That is, we suppose $L_A\neq\emptyset$. Let $(x_0,y_0)\in L_A$. Then we get:
\[ \left\{\begin{array}{cc}
\al_{m_0}(x_0)y^{m_0}_0+\al_{v_0}(x_0)y^{v_0}_0=-q_1(x_0,y_0) & (3) \\ [2ex]
\bi_{m_0}(x_0)y^{m_0}_0+\bi_{v_0}(x_0)y^{v_0}_0=-q_2(x_0,y_0) & (4) \end{array}\right.. \eqno{\mbox{(B)}} \]
We get
\[ D(x_0)=\al_{m_0}(x_0)\bi_{v_0}(x_0)-\al_{v_0}(x_0)\bi_{m_0}(x_0)=0. \]
Let
\[ D_1(x_0,y_0)=\al_{v_0}(x_0)q_2(x_0,y_0)-\bi_{v_0}(x_0)q_1(x_0.y_0) \]
\[ D_2(x_0,y_0)=\bi_{m_0}(x_0)q_1(x_0,y_0)-\al_{m_0}(x_0)q_2(x_0,y_0). \]
We now consider the following system
\[ \left\{\begin{array}{cc}
\al_{m_0}(x_0)z+\al_{v_0}(x_0)\oo=-q_1(x_0,y_0) & (5) \\ [2ex]
\bi_{m_0}(x_0)z+\bi_{v_0}(x_0)\oo=-q_2(x_0,y_0) & (6) \end{array}\right. \eqno{\mbox{($\Ga$)}} \]
Through the previous system (B) we have that $(y^{m_0}_0,y^{v_0}_0)$ is a solution of ($\Ga$). That is, ($\Ga$) is a linear system that has a solution and $D(x_0)=0$. So, we have that\linebreak $D_1(x_0,y_0)=0$ through linear algebra.
We consider now the following two systems: (A) and the following:
\[ \left\{\begin{array}{lc}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 & (7) \\ [1.5ex]
\al_{v_0}(x)q_2(x,y)-\bi_{v_0}(x)q_1(x,y)=0 & (8) \\[1.5ex]
y\cdot\al_{m_0}(x)\al_{v_0}(x)\bi_{m_0}(x)\bi_{v_0}(x)\neq0 &
\end{array}\right. \eqno{\mbox{($\De$)}} \]
From the above we have $L_A\subseteq L_\De$ (9).
We can now prove the reverse inclusion of (9).
Let $(x_0,y_0)\in L_\De$. Of course $(x_0,y_0)$ satisfies equation (1) of (A). We distinguish two cases:
(i) $q_1(x_0,y_0)\neq0$.
We can consider linear system ($\Ga$). This system has $D=D_1=0$ by our supposition $D\equiv0$ and $D_1=0$ because (8) holds for $(x_0,y_0)$, that is $D_1(x_0,y_0)=0$. So, system ($\Ga$) has an infinity of solutions and $D_2(x_0,y_0)=0$, because $D=D_1(x_0,y_0)=0$. Of course we have: $\al_{m_0}(x_0)q_{v_0}(x_0)\bi_{m_0}(x_0)\bi_{v_0}(x_0)\neq0$, by our supposition.
By relation $D(x_0)=0$ we take:
\[ \al_{m_0}(x_0)\bi_{v_0}(x_0)-\al_{v_0}(x_0)\bi_{m_0}(x_0)=0\Leftrightarrow \frac{\al_{m_0}(x_0)}{\bi_{m_0}(x_0)}=\frac{\al_{v_0}(x_0)}{\bi_{v_0}(x_0)}. \eqno{\mbox{(10)}} \]
By equation $D_1(x_0,y_0)=0$ we take:
\[ \al_{v_0}(x_0)q_2(x_0,y_0)=\bi_{ v_0}(x_0)q_1(x_0,y_0)\Rightarrow \frac{\al_{v_0}(x_0)}{\bi_{v_0}(x_0)}=\frac{q_1(x_0,y_0)}{q_2(x_0,y_0)} \eqno{\mbox{(11)}} \]
We have $q_2(x_0,y_0)\neq0$ or else if $q_2(x_0,y_0)=0\Rightarrow q_1(x_0,y_0)=0$, that is false by our supposition. So, (11) holds. By (10) and (11) we set
\[ 0\neq\la=\frac{\al_{m_0}(x_0)}{\bi_{m_0}(x_0)}=\frac{\al_{v_0}(x_0)}{\bi_{v_0}(x_0)}= \frac{q_1(x_0,y_0)}{q_2(x_0,y_0)}\Rightarrow\bi_{m_0}(x_0)=\frac{1}{\la}\al_{m_0}(x_0), \eqno{\mbox{(12)}} \]
\[ \bi_{v_0}(x_0)=\frac{1}{\la}\al_{v_0}(x_0), \eqno{\mbox{(13)}} \]
\[ q_2(x_0,y_0)=\frac{1}{\la}q_1(x_0,y_0). \eqno{\mbox{(14)}} \]
By (12), (13) and (14) we get:
\begin{align*} \bi_{m_0}(x_0)y^{m_0}_0+\bi_{v_0}(x_0)y^{v_0}_0+q_2(x_0,y_0)&= \frac{1}{\la}\al_{m_0}(x_0)y^{m_0}_0+\frac{1}{\la}\al_{v_0}(x_0)y^{v_0}_0+\frac{1}{\la} q_1(x_0,y_0)\\ &=\frac{1}{\la}(\al_{m_0}(x_0)y^{m_0}_0+\al_{v_0}(x_0)y^{v_0}_0+q_1(x_0,y_0)\\ &=\frac{1}{\la}\cdot0=0, \end{align*}
because $(x_0,y_0)\in L_\De$, which means that $(x_0,y_0)$ satisfies equality (7). So, we proved that if $(x_0,y_0)\in L_\De$ and $q_1(x_0,y_0)\neq0$, then $(x_0,y_0)\in L_A$.
(ii) $q_1(x_0,y_0)=0$.
Then, because $(x_0,y_0)\in L_\De$, through equality (8) we get: $q_2(x_0,y_0)=0$, because $\al_{v_0}(x_0)\neq0$, by our supposition.
As previously, because $D(x_0)=0$ and $\bi_{m_0}(x_0)\bi_{v_0}(x_0)\neq0$, we take it that: (12), and (13) holds, so
\begin{align*} \bi_{m_0}(x_0)y^{m_0}_0+\bi_{v_0}(x_0)y^{v_0}_0+q_2(x_0,y_0) &=\frac{1}{\la}\al_{m_0}(x_0)y^{m_0}_0+\frac{1}{\la}\al_{v_0}(x_0)y^{v_0}_0+0\\ &=\frac{1}{\la}(\al_{m_0}(x_0)y^{m_0}_0+\al_{v_0}(x_0)y^v_0+q_1(x_0,y_0)=0, \end{align*}
by equality (7) of $(\De)$ because $(x_0,y_0)\in L_\De$ by our supposition.
So, equality (2) of (A) holds, that is $(x_0,y_0)\in L_A$. So, we have $L_\De\subseteq L_A$ (15). Through (9) and (15), we get: $L_A=L_\De$. So, in order to solve system (A), it suffices to solve system $(\De)$. What is the profit from system $(\De)$. The profit is that polynomial in equation (8) of $(\De)$, that is $D_1$, has $deg_yD_1(x,y)<v_0$ or $D_1(x,y)\equiv0$. We examine now how we exploit these facts.
We leave the case $D_1(x,y)\equiv0$, for the end.
We examine now the case where $D_1(x,y)\not\equiv0$. We can write $D_1(x,y)$ in the following form:
\[ D_1(x,y)=\al_{v_1}(x)y^{v_1}+\al_{v_2}(x)y^{v_2}+q_3(x,y), \]
where $v_0>v_1>v_2$, $deg_yq_3(x,y)<v_2$, or $q_3(x,y)\equiv0$. This is the general case.
We suppose, also, that $\al_{v_1}(x)\not\equiv0$, $\al_{v_2}(x)\not\equiv0$ and $q_3(x,y)$ is a pure polynomial.
We get:
\[ y^{m_0-v_1}D_1(x,y)=\al_{v_1}(x)y^{m_0}+\al_{v_2}(x)y^{m_0-v_1+v_2}+y^{m_0-v_1}q_3(x,y), \]
where $deg_y(y^{m_0-v_1}q_3(x,y))<m_0-v_1+v_2$ because $deg_yq_3(x,y)<v_2$, by our supposition.
We consider the system:
\[ \left\{\begin{array}{l}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 \\ [1.5ex]
\al_{v_1}(x)y^{m_0}+\al_{v_2}(x)y^{m_0-v_1+v_2}+y^{m_0-v_1}q_3(x,y)=0 \\ [1.5ex]
y\al_{m_0}(x)\al_{v_0}(x)\bi_{m_0}(x)\bi_{v_0}(x)\neq0
\end{array}\right.. \eqno{\mbox{(E)}} \]
If $\al_{v_1}(x)=0$, or $\al_{v_2}(x)=0$ for $x\in{\mathbb{R}}$ we examine whether system (E) has a root of $\al_{v_1}(x)$ or $\al_{v_2}(x)$ that satisfies system (E). So, we examine the case where\linebreak $\al_{v_1}(x)\cdot\al_{v_2}(x)\neq0$.
Let
\[
D=\left|\begin{array}{cc}
\al_{m_0}(x) & \al_{v_0}(x) \\ [2ex]
\al_{v_1}(x) & \al_{v_2}(x)
\end{array}\right|=\al_{m_0}(x)\al_{v_2}(x)-\al_{v_0}(x)\al_{v_1}(x). \]
Then, system (E) is a system similar to system (A).
So, we examine the similar cases with the same way.
Here, we examine only the case where $D=\al_{m_0}(x)\al_{v_2}(x)-\al_{v_0}(x)\al_{v_1}(x)\equiv0$.\\ In this case we again reach a system similar to $(\De)$, so that the respective equation (8) of the new system has $D_1(x,y)\not\equiv0$.
We handle this case as follows:
In system (A) of page 23, we can take any of two equations in order to get an equivalent system as $(\De)$. So it helps us to take the equation in which the respective pure polynomial $q_1(x,y)$ or $q_2(x,y)$ has the smallest number of terms. For this reason in system (E) (that is similar to A) we take as a first equation (of system $(\De)$) the second equation because this polynomial $y^{m_0-v_1}q_3(x,y)$ has at most $v_2$ terms with respect to $y$ (by its definition), where $v_2<v_1<v_0\Rightarrow v_2\le v_0-2$.
In the new system $(\De)$ we take that the respective $D_1(x,y)$ polynomial of $(\De)$ has $deg_yD_1(x,y)<v_0$, so, if we write this polynomial again in the form:
\[ D_1(x,y)=\al_{v_1}(x)y^{v_1}+\al_{v_2}(x)y^{v_2}+q_4(x,y), \]
the new polynomial $q_4(x,y)$, has at most $v_2\le v_0-2$ terms.
So, the profit, is that the new polynomials $q_1(x,y)$, $q_2(x,y)$ of the new system $(\De)$ will have at most $v_0-2$ terms each one of them and the respective new polynomial $D_1(x,y)$ also. So, the profit is the following:
In system $(\De)$ polynomial $D_1(x,y)$ has at most $v_0$ terms with respect to $y$, whereas in a new system like $(\De)$ in a following stage the respective polynomial $D_1(x,y)$ of the new system $(\De)$ will have at most $v_0-2$ terms with respect to $y$.
With the same procedure we can see that the terms of the respective polynomials $D_1(x,y)$ are decreasing, so that after a finite number of steps we reach a polynomial $D_1(x,y)\equiv0$ or $D_1(x,y)\equiv r(x)$ for polynomial $r(x)\not\equiv0$. If $D_1(x,y)\equiv0$ we solve this case in the final step, or else if $r(x)\not\equiv0$, it suffices to find the roots of polynomial $r(x)$, otherwise we have some of the previous cases that we have already examined.
Now we will examine the remaining case. In system (A), page 21. If $v_0\neq\mi_0$ and $n_0=m_0$ we have the first basic case where $D(x_0)\neq0$.
Now, let $m_0\neq n_0$, that is $n_0<m_0$. If $n_0\ge v_0$, we have the first basic case. So, we can examine the case $v_0>n_0$. In this case we have:
\begin{align*} &y^{m_0-n_0}(\bi_{n_0}(x)y^{n_0}+\bi_{\mi_0}(x)y^{\mi_0}+q_2(x,y))=0\\ &\Leftrightarrow\bi_{n_0}(x)y^{m_0}+\bi_{\mi_0}(x)y^{m_0-n_0+\mi_0}+y^{m_0-n_0} q_2(x,y)=0 \end{align*}
and instead of (A) we examine the system:
\[ \left\{\begin{array}{l}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 \\ [2ex]
\bi_{n_0}(x)y^{m_0}+\bi_{\mi_0}(x)y^{m_0-n_0+\mi_0}+y^{m_0-n_0}q_2(x,y)=0 \end{array}\right. \]
This system is of the case where $m_0=n_0$, which we have already examined. So, up to now, we have examined all the possible cases of the initial system except one only, that we will examine now.
In the third basic case we will examine now the case where $D_1(x,y)\equiv0$. Then, as in pages 23, 24 we take it that $D_2(x,y)\equiv0$, also that for every $(x,y)\in{\mathbb{R}}^2$ there exists $c\in{\mathbb{R}}$, such that
\[ \al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=c\cdot(\bi_{m_0}(x)y^{m_0}+\bi_{v_0}(x) y^{v_0}+q_2(x,y)) \eqno{\mbox{($\ast$)}} \]
and $c\neq0$. The number $c$ depends on the couple $(x,y)$, so it is better to write $c(x,y)$, instead of $c$.
Now, we will consider system (A$^\ast$)
\[ \left\{\begin{array}{l}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 \\ [2ex]
\al_{m_0}(x)\al_{v_0}(x)\bi_{m_0}(x)\bi_{v_0}(x)\neq0
\end{array}\right.. \eqno{\mbox{(A$^\ast$)}} \]
Equality $(\ast)$ gives us that
\[ L_A=L_{A^\ast}. \]
So, in order to solve system (A) it suffices to solve the ``simpler'' system (A$^\ast$) that has only one equation.
Now, it is the time to exploit the unique supposition that we have not used up to now.
That is: The set $L_A=L_{A^\ast}$ is finite. As we have seen in the prerequisites there are polynomials $p(x,y)$ of two real variables that have a finite set of roots only. \\
For example let:
\[ p(x,y)=(x^2-4)^2+(y^2-9)^2. \]
It is easy to see that
\[ L_{p(x,y)}=\{(2,3),(2,-3),(-2,3),(-2,-3)\}. \]
We denote:
\[ R(x,y)=\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y), \]
for simplicity.
So, we solve the system:
\[ \left\{\begin{array}{l}
R(x,y)=0 \\ [2ex]
\al_{m_0}(x)\al_{v_0}(x)\bi_{m_0}(x)\bi_{v_0}(x)\neq0
\end{array}\right.. \eqno{\mbox{(A$^\ast$)}} \]
Of course, we get $R(x,y)\not\equiv 0$, because $\al_{m_0}\not\equiv0$.
Now, it is the time to use the results of our prerequisites.
By the suppositions of the third case we get $\al_{m_0}(x)\not\equiv0$ and $m_0>1$, that gives that $R(x,y)$ is a pure polynomial, that has a finite set of roots, non empty.
We apply Corollary 3.16 by our prerequisites and we take it that 0 is the global maximum or minimum of $R(x,y)$.
Without loss of generality we suppose that 0 is the global minimum of $R(x,y)$. This means that if we consider the function $F:U{\rightarrow}{\mathbb{R}}$ (where\\ $U=\{(x,y)\in{\mathbb{R}}^2\mid\al_{m_0}(x)\al_{v_0}(x)\bi_{m_0}(x)\bi_{v_0}(x)\neq0\}$ is an open subset of ${\mathbb{R}}^2$).\linebreak $F((x,y))=R(x,y)$ for every $(x,y)\in U$, then it holds $F((x,y))\ge0$ for every $(x,y)\in U$, and there exists $(x_0,y_0)\in U$ so that $F((x_0,y_0))=0$.
Let $(x_0,y_0)\in{\mathbb{R}}^2$ so that $(x_0,y_0)\in L_{A^\ast}$. Then, we have $F((x_0,y_0))=0$ and function $F$ has a global minimum in $(x_0,y_0)$. Then, by Theorem 3.17 we get $\nabla F(x_0,y_0)=(0,0)$. So we have: $\dfrac{\partial F}{\partial y}((x_0,y_0))=0$.
We can now consider the system:
\[ \left\{\begin{array}{l}
F((x,y))=0 \\ [1.5ex]
\dfrac{\partial F}{\partial y}((x,y))=0 \\ [1.5ex]
\al_{m_0}(x)\al_{v_0}(x)\bi_{m_0}(x)\bi_{v_0}(x)\neq0
\end{array}\right.. \eqno{\mbox{((A$_1$)}} \]
Of course we get $L_{A_1}\subseteq L_{A^\ast}=L_A$ and by the above we also get: $L_A^\ast\subseteq L_{A_1}$. So we get:
\[ L_{A_1}=L_A. \]
So, in order to solve system (A$^\ast$) it suffices to solve system (A$_1$). We need to write a more analytic system (A$_1$). We get:
\[ \left\{\begin{array}{l}
\al_{m_0}(x)y^{m_0}+\al_{v_0}(x)y^{v_0}+q_1(x,y)=0 \\ [1.5ex]
m_0\al_{m_0}(x)y^{m_0-1}+v_0\al_{v_0}(x)y^{v_0-1}+
\dfrac{\partial q_1}{\partial y}(x,y)=0 \\ [1.5ex]
\al_{m_0}(x)\al_{v_0}(x)\bi_{m_0}(x)\bi_{v_0}(x)\neq0
\end{array}\right. \eqno{\mbox{(A$_1$)}} \]
Because $m_0>v_0\Rightarrow m_0-1\ge v_0$. This shows that system (A$_1$) is the first basic case, and so we can transfer system (A$_1$) to a system that has smaller than $m_0$ degree with respect to $y$ that we can solve with the induction step. So, inductively we have managed to solve the initial system in any case. So, we have completed our second stage.
\subsection{Third stage} Let a polynomial
\[ p(z)=\al_0+\al_1z+\cdots+\al_{v-1}z^{v-1}+\al_vz^v, \]
for $v\in{\Bbb N}$, $\al_1\in{\Bbb C}$, for $i=0,1,\ld,v$, $\al_v\neq0$, of one complex variable.
We are now ready to solve completely the equation $p(z)=0$, or in other words to find the roots of polynomial $p(z)$ with degree $v$.
We distinguish two cases:
(i) $\al_i\in{\mathbb{R}}$ for every $i=0,1,\ld,v$, and (ii) $\al_i\in{\Bbb C}$, $i=0,1,\ld,v$. Firstly, we prove the following lemma:
\setcounter{lem}{3} \begin{lem}\label{lem2.4} (A well known lemma).
Let $p(z)$, be a polynomial as above with degree $v=deg p(z)\in{\Bbb N}$. Then, there exist two polynomials $p_1(x,y)$, $p_2(x,y)$ of two real variables with real coefficients, so that it holds:
\[ p(x+yi)=p_1(x,y)+ip_2(x,y) \]
for every $(x,y)\in{\mathbb{R}}^2$. \end{lem}
\begin{proof} We can prove this lemma with induction above the degree $v$ of $p(z)$. Let $p(z)=\al_0+\al_1z$, $\al_0,\al_1\in{\mathbb{R}}$, $\al_1\neq0$. Let $(x,y)\in{\mathbb{R}}^2$. We get:
\[ p(x+yi)=\al_0+\al_1(x+yi)=(\al_0+\al_1x)+\al_1yi, \ \ \text{for} \ \ v=1 \]
so for $p_1(x,y)=\al_0+\al_1x$ and $p_2(x,y)=\al_1y$, the result holds.
For $v=2$.
\\
Let $p(z)=\al_0+\al_1z+\al_2z^2$, where $\al_0,\al_1,\al_2\in{\mathbb{R}}$, $\al_2\neq0$.
Let $z=x+yi$, $(x,y)\in{\mathbb{R}}^2$. We get:
\begin{align*} p(z)&=p(x+yi)=\al_0+\al_1(x+yi)+\al_2(x+yi)^2\\ &=(\al_0+\al_1+\al_2x^2-\al_2y^2)+(\al_1y+2\al_2xy)i, \end{align*}
so for $p_1(x,y)=\al_0+\al_1x+\al_2x^2-\al_2y^2$ and $p_2(x,y)=\al_1y+2\al xy$, the result holds. We suppose now, that the result holds for any $1\le i\le k_0\in{\Bbb N}$. We can prove that result holds for $k_0+1$.
Let
\[ p(z)=\al_0+\al_1z+\cdots+\al_{n_0}z^{k_0}+\al_{k_0+1}z^{k_0+1}, \]
be a polynomial with $\al_{k_0+1}\neq0$, $\al_i\in{\mathbb{R}}$, for every $i=0,1,\ld,k_0+1$.
Let $(x,y)\in{\mathbb{R}}^2$. We have:
\[ p(z)=q(z)+\al_{k_0+1}z^{k_0+1}, \]
we distinguish two cases:
(a) $q(z)\not\equiv0$. Then, through the induction step we can show that there exist two polynomials $p_1(x,y)$, $p_2(x,y)$ of two real variables $x$ and $y$ with real coefficients, so that:
\[ q(x+y_i)=p_1(x,y)+p_2(x,y)i \ \ \text{for every} \ \ (x,y)\in{\mathbb{R}}^2. \eqno{\mbox{(1)}} \]
We get:
\begin{align*} \al_{k_0+1}z^{k_0+1}=&\,\al_{k_0+1}(x+yi)^{k_0+1}=\al_{k_0+1}\sum^{k_0+1}_{j=0} \binom{k_0+1}{j}x^j\cdot(yi)^{k_0+1-j}\\ =&\,\al_{k_0+1}\sum^{k_0+1}_{j=0}\binom{k_0+1}{j}x^jy^{k_0+1-j}i^{k_0+1-j} \\ =&\sum_{\scriptsize{\begin{array}{c}
k_0+1-j=2\rho \\
\rho\in {\Bbb N} \\
\al<j\le k_0+1
\end{array}}}\al_{k_0+1}\binom{k_0+1}{j}x^jy^{k_0+1-j}(-1)^{(n_0+1-j)/2}\\
&+\sum_{\scriptsize{\begin{array}{c}
k_0+1-j=2\rho+1 \\
\rho\in {\Bbb N} \\
0\le j\le k_0+1
\end{array}}}\al_{k_0+1}x^jy^{k_0+1-j}i^{k_0+1-j}\\ =&\,q_1(x,y)+iq_2(x,y) \hspace*{6.5cm} {\mbox{(2)}} \end{align*}
where
\[ q_1(x,y)=\sum_{\scriptsize{\begin{array}{c}
k_0+1-j=2\rho \\
\rho\in {\Bbb N} \\
0\le j\le k_0+1
\end{array}}}\al_{k_0+1}\binom{k_0+1}{j}x^jy^{k_0+1-j}(-1)^{(n_0+1-j)/2} \ \ \text{and} \]
\[ iq_2(x,y)=\sum_{\scriptsize{\begin{array}{c}
k_0+1-j=2\rho+1 \\
\rho\in {\Bbb N} \\
0\le j\le k_0+1
\end{array}}}\al_{k_0+1}x^jy^{k_0+1-j}i^{k_0+1-j} \]
where $q_1(x,y)$, $q_2(x,y)$ are two polynomials of the two real variables with real coefficients because $i^{2v+1}=i$ or $-i$, $v\in{\Bbb N}$.
So, we get: $\al_{k_0+1}z^{k_0+1}=q_1(x,y)+iq_2(x,y)$. So, we get: by (1) and (2)
\begin{align*} p(z)&=q(z)+\al_{k_0+1}z^{k_0+1}=(p_1(x,y))+p_2(x,y)i)+(q_1(x,y)+q_2(x,y)i) \\ &=(p_1(x)y)+q_1(x,y))+(p_2(x,y)+q_2(x,y))i \end{align*}
and the result also holds for every $(x,y)\in{\mathbb{R}}^2$.
(b) $q(z)\equiv0$. Then, with the above equality (2) we get\\ $p(z)=\al_{k_0+1}z^{k_0+1}=q_1(x,y)+iq_2(x,y)$ for every $(x,y)\in{\mathbb{R}}^2$ and the result also holds. So, by induction we see that the result holds in this case.
Now, we suppose that $\al_i\in{\Bbb C}$, for every $i=0,1,\ld,v$.
Let $\al_j=\bi_j+\ga_ji$ for every $j=0,1,\ld,v$, where $\bi_j,\ga_j\in{\mathbb{R}}$ for every $j=0,1,\ld,v$. Let $z=x+yi\in{\Bbb C}$, $(x,y)\in{\mathbb{R}}^2$. We get:
\begin{align*} p(z)&=p(x+yi)=\al_0+\al_1z+\cdots+\al_{v-1}z^{v-1}+\al_vz^v \\ &=(\bi_0+\ga_0i)+(\bi_1+\ga_1i)z+\cdots+(\bi_{v-1}+\ga_{v-1}i)z^{v-1} +(\bi_v+\ga_vi)z^v \\ &=(\bi_0+\bi_1z+\cdots+\bi_{v-1}z^{v-1}+\bi_vz^v)+(\ga_0+\ga_1z+\cdots+ \ga_{v-1}z^{v-1}+\ga_vz^v)i. \hspace*{0.2cm} (3) \end{align*}
In the previous case (i) we see that there exist polynomials $p_1(x,y)$, $p_2(x,y)$, $q_1(x,y)$, $q_2(x,y)$ of the two real variables $x$ and $y$ with real coefficients, so that
\[ \bi_0+\bi_1z+\cdots+\bi_{v-1}z^{v-1}+\bi_vz^v=p_1(x,y)+p_2(x,y)i \ \eqno{\mbox{(4)}} \]
and
\[ \ga_0+\ga_1z+\cdots+\ga_{v-1}z^{v-1}+\ga_vz^v=q_1(x,y)+q_2(x,y)i \eqno{\mbox{(5)}} \]
for every $(x,y)\in{\mathbb{R}}^2$.
By (3), (4) and (5) we get:
\begin{align*} p(z)&=(p_1(x,y)+p_2(x,y)i)+(q_1(x,y)+q_2(x,y)i)i \\ &=(p_1(x,y)-q_2(x,y))+(p_2(x,y)+q_1(x,y))i \end{align*}
and the result holds also. $\quad\blacksquare$
With the help of this Lemma we can now solve the equation $p(z)=0$ as follows:
We examine the general case where
\[ p(z)=\al_0+\al_1z+\cdots+\al_{v-1}z^{v-1}+\al_vz^v, \ \ v\in{\Bbb N}, \ \ \al_v\neq0, \ \ \al_i\in{\Bbb C}, \]
for every $i=0,1,\ld,v$.
With the help of the above lemma we write:
\[ p(x+yi)=q_1(x,y)+q_2(x,y)i \eqno{\mbox{($\ast$))}} \]
for every $(x,y)\in{\mathbb{R}}^2$, where $q_1(x,y),q_2(x,y)$ are two polynomials of two real variables $x$ and $y$ with real coefficients.
Let $A$ be the set of roots of $p(z)$. We consider the system:
\[ \left\{\begin{array}{l}
q_1(x,y)=0 \\ [2ex]
q_2(x,y)=0
\end{array}\right.. \eqno{\mbox{(B)}} \]
It is obvious from the above equality $(\ast)$ that $A=L_B$. So, in order to find all the roots of A, it suffices to find all the real roots of system (B).
So, we solve system B with the method we have developed in the second stage, and thus we find all the roots of polynomial $p(z)$.
Our method has been completed now because our supposition (S) (that system (B) has a solution) is satisfied because the same holds for (A). So, in all the cases we can reduce our initial system to a system in which the two polynomials have a lower degree than that of the polynomials of the initial system. Thus, we apply the induction step and the system is solved inductively. \end{proof}
\section{Prerequisites}\label{sec3}
\noindent
a) Prerequisites from Algebra.
We use some basic tools and results from theory of polynomials.
We denote ${\Bbb C}[z]$ as the set of complex polynomials. We denote ${\mathbb{R}}[x]$ as the set of real polynomials, that is the set of polynomials of one real variable with coefficients in the set of real numbers ${\mathbb{R}}$.
We begin with the following basic result, that is a simple implication of the algorithm of Euclidean division.
\begin{prop}\label{prop3.1} Let $p(z)\in{\Bbb C}[z]$, $deg p(z)\ge1$. The number $r\in{\Bbb C}$ is a root of $p(z)$ if and only if there exists a unique polynomial $q(z)\in{\Bbb C}[z]$ so that:
\[ p(z)=(z-r)q(z). \quad \blacksquare \]
\end{prop}
We need the definition of multiplicity of a root of a polynomial.
\begin{Def}\label{def3.1} Let $p(z)\in{\Bbb C}[z]$. Let $\rho\in{\Bbb C}$ be a root of $p(z)$. The natural number $m$ is a multiplicity of the root $\rho$ of $p(z)$ if polynomial $(z-\rho)^m$ divides $p(z)$, whereas polynomial $(z-\rho)^{m+1}$ does not divide $p(z)$. \end{Def}
As consequence of Proposition 3.1 there is the following proposition:
\begin{prop}\label{prop3.3} Every root of a polynomial $p(z)\in{\Bbb C}[z]$ has a multiplicity, that is unique. $\quad\blacksquare$ \end{prop}
We state now the fundamental Theorem of algebra, whose proof is not simple and needs some tools from analysis.
\begin{thm}\label{thm3.4} Every complex polynomial $p(z)$, with $deg p(z)\ge1$ has at least one root. $\quad\blacksquare$ \end{thm}
From Theorem 3.4 and Proposition 3.1 we get the following fundamental result:
\begin{thm}\label{thm3.5} Let $p(z)\in{\Bbb C}[z]$ be a complex polynomial with $deg p(z)\ge1$. Then $p(z)$ has a finite number of different roots.
Let $\rho_1,\rho_2,\ld,\rho_v$ be the different roots of $p(z)$ with respective to multiplicities $m_1,m_2,\ld,m_v$. Then, the following formula holds:
\[ p(z)=\al\cdot(z-\rho_1)^{m_1}(z-\rho_2)^{m_2}\cdots (z-\rho_v)^{m_v}, \]
where $\al\neq0$ and $\al$ is the coefficient of the monomial of greater grade $m_0=deg p(z)$, and $m_0=m_1+m_2+\cdots+m_v$. $\quad\blacksquare$ \end{thm}
Now, we describe a simple algorithm in order to find the multiplicity of a root of a complex polynomial.
\\
{\bf 3.6. An algorithm for the multiplicity of a root}
Let $p(z)\in{\Bbb C}[z]$ be a complex polynomial of degree $deg p(z)\ge1$.
By Theorem \ref{thm3.5} polynomial $p(z)$ has a finite number of roots. Let $\rho$ be a root of $p(z)$. We describe with details a way in order to find the multiplicity of $\rho$.
By Proposition 3.1 there exists a unique polynomial $q(z)$ so that:
\[ p(z)=(z-\rho)q(z). \eqno{\mbox{(1)}} \]
We find the polynomial $q(z)$ through the algorithm of Euclidean division, for example using Horner's scheme.
Afterwards, we compute the number $q(\rho)$, for example with Horner's scheme.\\ If $q(\rho)\neq0$, then the root $\rho$ has multiplicity 1. In order to prove this we suppose that the root $\rho$ does not have multiplicity 1. By Proposition 3.3 the root $\rho$ has a unique multiplicity, $m\in{\Bbb N}$ (see Definition 3.2). Because of $m\neq1$, we have that $m\ge2$. By the definition of multiplicity we have that polynomial $(z-\rho)^m$ divides $p(z)$. This means (by the definition of division) that there exists a polynomial $R(z)\in{\Bbb C}[z]$ sο that:
\[ p(z)=(z-\rho)^mR(z). \eqno{\mbox{(2)}} \]
By relations (1) and (2) we get:
\[ (z-\rho)q(z)=(z-\rho)^mR(z)\Leftrightarrow(z-\rho)(q(z)-(z-\rho)^{m-1}R(z)=0. \eqno{\mbox{(3)}} \]
The expressions $z-\rho$ and $q(z)-(z-\rho)^{m-1}R(z)$ are polynomials in ${\Bbb C}[z]$ of course, because $m\ge2$ (as we have seen). Because of $z-\rho\not\equiv0$, we take it that
\[ q(z)-(z-\rho)^{m-1}R(z)=0, \eqno{\mbox{(4)}} \]
because the Ring of polynomials ${\Bbb C}[z]$ is an integer neighbourhood, as is well known from Algebra. Relation (4) gives $q(\rho)=0$ (because $m\ge2$), which is false because we have supposed that $q(\rho)\neq0$. So, if $q(\rho)\neq0$, then root $\rho$ has multiplicity 1.
Whereas if $q(\rho)=0$, then through Proposition \ref{prop3.1}, we take it that there exists a polynomial $q_1(z)\in{\Bbb C}[z]$, so that:
\[ q(z)=(z-\rho)q_1(z). \eqno{\mbox{(5)}} \]
By (1) and (5) we take that
\[ p(z)=(z-\rho)^2q_1(z). \eqno{\mbox{(6)}} \]
Relation (6) tells us that polynomial $(z-\rho)^2$ divides $p(z)$. Afterwards, we find polynomial $q_1(z)$ by (5) with the Euclidean Algorithm, for example from Horner's scheme, because we have found polynomial $q(z)$ previously. After that, we compute number $q_1(\rho)$, for example with Horner's scheme. If $q_1(\rho)\neq0$, then the multiplicity of $\rho$ is 2, with a proof similar to what we had found previously. Or otherwise if $q_1(\rho)=0$, then again through Proposition \ref{prop3.1} there exists a polynomial $q_2(z)\in{\Bbb C}[z]$ so that
\[ q_1(z)=(z-\rho)q_2(z). \eqno{\mbox{(7)}} \]
Through (6) and (7) we take it:
\[ p(z)=(z-\rho)^3q_2(z). \eqno{\mbox{(8)}} \]
We inductively continue this procedure of finding a sequence of polynomials \[ q_j(z)\in{\Bbb C}[z], \]
for $j=1,2,\ld$, where $q_j(z)=(z-\rho)q_{j+1}(z)$, for $j=1,2,\ld$, and
\[ p(z)=(z-\rho)^{j+1}q_j(z). \]
If $p(z)=(z-\rho)^{j+1}q_j(z)$ for $j\in{\Bbb N}$, (where $q_j(\rho)\neq0)$, then the multiplicity of $\rho$ is $j+1$, with a proof similar to what we have shown previously. This procedure stops if some natural number $j\in{\Bbb N}$, or if $deg p(z)=v_0\in{\Bbb N}$, then we take it that $p(z)=(z-\rho)^{v_0+1}q_{v_0}(z)$, where $q_{v_0}(z)\neq0$ (or else $p(z)=0$, which is false because $deg p(r)\ge1$, by supposition), so $deg((z-\rho)^{v_0+1}q_{v_0}(z))\ge v_0+1$, which is false of course because $deg p(z)=v_0$.
That is we take it that $p(z)=(z-\rho)^{j+1}q_j(z)$ for some $j\in{\Bbb N}$, $j<v_0-1$, and $q_j(\rho)\neq0$, that gives that multiplicity of $\rho$ is $j+1<v_0$ otherwise we take it that
\[ p(z)=(z-\rho)^{v_0}q_{v_0-1}(z). \eqno{\mbox{(9)}} \]
Relation (9) gives that $q_{v_0-1}(z)\neq0$, (or else $p(z)=0$ which is false of course) and by relation (9) we take it also that $q_{v_0-1}(z)$ is a constant polynomial with value say $c_0$. That is $p(z)=(z-\rho)^{v_0}c_0$. Of course polynomial $(z-\rho)^{v_0+1}$ cannot divide $p(z)$, because this polynomial has a degree $deg((z-\rho)^{v_0+1})>v_0=deg p(z)$, that gives that multiplicity of $\rho$ is $v_0$ (by the definition of multiplicity). So we have described a complete algorithm that gives us the multiplicity of a root of a complex\linebreak polynomial. $\quad\blacksquare$
\\
\noindent
{\bf Remark 3.7} {\em We can combine Proposition \ref{prop3.1} with Theorem 3.4 and the previous algorithm (and of course Proposition \ref{prop3.3}), in order to prove Theorem \ref{thm3.5}. We leave it as an easy exercise for the reader. So far we have developed all we need from polynomials of one complex variable. We also obtain some basic results from Linear Algebra. Here we will now consider the following linear system of two equations
\[ \left\{\begin{array}{lc}
\al_1x+\bi_1y=\ga_1 & (1) \\ [2ex]
\al_2x+\bi_2y=\ga_2 & (2)
\end{array}\right. \eqno{\mbox{(A)}} \]
where $\al_i,\bi_i,\ga_i\in{\Bbb C}$ for $i=1,2$. We consider the determinants $D,D_x,D_y$ where
\[ \begin{array}{l}
D=\left|\begin{array}{cc}
\al_1 & \bi_1 \\ [1.5ex]
\al_2 & \bi_2
\end{array}\right|=\al_1\bi_2-\al_2\bi_1,
\\ [4ex]
D_x=\left|\begin{array}{cc}
\ga_1 & \bi_1 \\[1.5ex]
\ga_2 & \bi_2
\end{array}\right|=\ga_1\bi_2-\bi_1\ga_2,D_y
=\left|\begin{array}{cc} \al_1 & \ga_1 \\ [1.5ex] \al_2 & \ga_2
\end{array}\right|=\al_1\ga_2-\al_2\ga_1. \end{array} \]
When $D\neq0$, then system (A) has only one solution $(x_0,y_0)$, where $x_0=\dfrac{D_x}{D}$, $y_0=\dfrac{D_y}{D}$. When $D=0$ and $D_x\neq0$, or $D_y\neq0$, then system (A) does not have any solution, whereas when $D=D_x=D_y=0$, then system (A) has an infinite number of solutions except only in the case where $\al_1=\al_2=\bi_1=\bi_2=0$ and only one of the numbers $\ga_1,\ga_2$ is non zero. We need the case where $D\neq0$ and the case where $D=D_x=D_y=0$. We consider the case where $D=D_x=D_y=0$. We suppose that system (A) is a pure system of two variables $x$ and $y$, that is, we suppose that at least one of the numbers $\al_1,\al_2$ is non-zero also. That is $\al_1\neq0$ or $\al_2\neq0$ and $\bi_1\neq0$ or $\bi_2\neq0$, otherwise we do not have a system of equations of two different variables}.
We have two cases:
(i) One from the six numbers $\al_i,\bi_i,\ga_i$, $i=1,2$ is zero
Let $\al_1=0$ (3). We have $D=0$, that is $\al_1\bi_2-\al_2\bi_1=0\Rightarrow\al_1\bi_2=\al_2\bi_1$ (4). Through (3) and (4) we have $\al_2\bi_1=0$ (5). Because of $\al_1=0$ and our hypothesis we have $\al\neq0$ (6). By (5) and (6) we get $\bi_1=0$ (7).
Through (3) and the fact that $D_y=0$ we get in a similar way that $\ga_1=0$. That is the equation (1) is the equation $0\cdot x+0\cdot y=0$, with set of solutions the set ${\mathbb{R}}^2$. This means that system (A) is equivalent to the equation (2) of (A) only. If $\bi_1=0$, or $\ga_1=0$, we get in a similar way that $\al_1=\bi_1=\ga_1=0$ and we have similarly the same implication, that is system (A) is equivalent with equation (2) of (A) only. If $\al_2=0$, or $\bi_2=0$, or $\ga_2=0$, we take it that $\al_2=\bi_2=\ga_2=0$ in an analogous way and finally system (A) is equivalent with the equation (1) of (A) only.
Now, we suppose that $\al_1\bi_1\ga_1\al_2\bi_2\ga_2\neq0$, that is, non of six numbers $\al_i,\bi_i,\ga_i$, $i=1,2,$ is zero. We have $D=0\Leftrightarrow\al_1\bi_2-\al_2\bi_1=0\Leftrightarrow\dfrac{\al_2}{\al_1} =\dfrac{\bi_2}{\bi_1}$ $(\al_1\neq0$, $\bi_1\neq0$). We set $\la=\dfrac{\al_2}{\al_1}=\dfrac{\bi_2}{\bi_1}$ (8).
We have $D_x=0\Leftrightarrow\ga_1\bi_2-\bi_1\ga_2=0\Leftrightarrow\la=\dfrac{\bi_2}{\bi_1} =\dfrac{\ga_2}{\ga_1}$ (9). With (8) and (9) we get: $\al_2=\la\al_1$, $\bi_2=\la\bi_1$, $\la_2=\la\ga_1$, that is $\al_2x+\bi_2y=\ga_2\Leftrightarrow(\la\al_1)x+(\la\bi_1)y=(\la\ga_1)\Leftrightarrow \la\cdot(\al_1x+\bi_1y)=\la\ga_1\overset{\la\neq0}{\Leftrightarrow}\al_1x+\bi_y=\ga_1$, that is equations (1) and (2) of (A) are equivalent, that is they have the same set of solutions, which means that system (A) is equivalent to one only from the equations (1) and (2), whichever of the two).
So, we have proved that in the case of $D=D_x=D_y=0$, system (A) has an infinite number of solutions and it is equivalent with only one from the equations (1) and (2). So we have stated our prerequisites from Algebra.
\\
{\bf b) Prerequisites from Analysis}
As it is well known, by Galois theory, there are no formulas that give the roots of an arbitrary polynomial as a function of its coefficients with radicals. So, for an arbitrary polynomial the only way to find its roots is to approximate them with a numerical method. Perhaps, the simplest numerical method for algebraic equations is the bisection method, which is presented in all classical books of Numerical Analysis:
It is a simple method, and here we have based in it in our problem. The bisection method has very weak suppositions, and it is convenient for secondary students also.
Let $\al,\bi\in{\mathbb{R}}$, $\al<\bi$, and $f:[\al,\bi]{\rightarrow}{\mathbb{R}}$ be a continuous function. We suppose that $f(\al)\cdot f(\bi)<0$. Then, function $f$ has a root, at least in the interval $(\al,\bi)$, and bisection method approximates a root of $f$ in $(\al,\bi)$, as closely as we want with a specific minor error.
There are many different numerical methods that find the roots in a specific interval. We will not discuss this subject. This is a vast subject in Numerical Analysis. In this text, it is enough for us to find only one root in a specific interval and approximate it using bisection method.
The solution to all the real roots of a polynomial will be based on the following basic lemma.
\\
\noindent
{\bf Basic Lemma 3.8}. {\em Let $v\in{\Bbb N}$, $v\ge3$, $p(x)=\al_vx^v+\al_{v-1}x^{v-1}+\cdots+\al_1x+\al_0$, be a polynomial $p(x)\in{\mathbb{R}}[x]$, with degree $deg p(x)=v$.
Let $\rho_1,\rho_2,\ld,\rho_k$ be all the different real roots of polynomial $p'(x)$, $k\in{\Bbb N}$, $k\ge2$, $\rho_i\neq\rho_j$, for all $i,j\in\{1,2,\ld,k\}$, $i\neq j$.
Then, we can find, with an algorithm, all the real roots of $p(x)$, with their multiplicities.}
\begin{proof} Let $L=\{\rho_1,\rho_2,\ld,\rho_k\}$, be the set of all real roots of $p'(x)$. We suppose, also, without loss of generality that $\rho_1<\rho_2<\cdots<\rho_k$.
Let $i_0\in\{1,\ld,k-1\}$. Then $p'(x)>0$ for every $x\in(\rho_{i_0},\rho_{i_0+1})$ or $p'(x)<0$ for every $x\in(\rho_{i_0},\rho_{i_0+1})$. This gives that $p$ is a strictly decreasing or strictly increasing function on $[\rho_{i_0},\rho_{i_0+1}]$. If $p(\rho_{i_0})=0$, then $\rho_{i_0}$ is the unique root of $p$ in $[\rho_{i_0},\rho_{i_0+1}]$. The same holds if $p(\rho_{i_0+1})=0$, that is $\rho_{i_0+1}$ is the unique root of $p$ in $[\rho_{i_0},\rho_{i_0+1}]$, if $p(\rho_{i_0+1})=0$.
Of course polynomial $p$ can't have the numbers $\rho_{i_0}$ and $\rho_{i_0+1}$ as roots simultaneously, by its monotonicity. We suppose now that $p(\rho_{i_0})\cdot p(\rho_{i_0+1})\neq0$. Then, if $p(\rho_{i_0})\cdot p(\rho_{i_0+1})>0$, polynomial $p$ does not have any root in $[\rho_{i_0},\rho_{i_0+1}]$. If $p(\rho_{i_0})\cdot p(\rho_{i_0+1})<0$, then $p$ has one root exactly in the interval $[\rho_{i_0},\rho_{i_0+1}]$, and more specifically this root belongs in $(\rho_{i_0},\rho_{i_0+1})$.
Applying the bisection method we find this root, because the suppositions of bisection method are satisfied now. We do the same in every interval $[\rho_i,\rho_{i+1}]$.
So we find all the roots of $p$ in the interval $[\rho_1,\rho_k]$. We examine the roots in $[\rho_k,+\infty)$. Because $\al_v\neq0$, we have two cases:
\begin{enumerate} \item[i)] If $\al_v>0$, then $\lim\limits_{x{\rightarrow}+\infty}p(x)=+\infty$.
Then $p$ is a strictly increasing function in $[\rho_k,+\infty)$.
\begin{itemize} \item[a)] $p(\rho_k)=0$, then $\rho_k$ is the unique root of $p$ in $[\rho_k,+\infty)$.
\item[b)] If $p(\rho_k)>0$, then $p$ does not have any root in $[\rho_k,+\infty)$.
\item[c)] If $p(\rho_k)<0$, then $p$ has one root exactly, (say $\rho_{k+1})$ in $[\rho_k,+\infty)$ and more specifically $\rho_{k+1}\in(\rho_k,+\infty)$. \end{itemize}
Because $\lim\limits_{x{\rightarrow}+\infty}p(x)=+\infty$, there exists some $x_0\in{\mathbb{R}}$, $x_0>\rho_k$, so that $p(x_0)>0$. Then $p(\rho_k)\cdot p(x_0)<0$ and thus $\rho_{k+1}\in(\rho_k,x_0)$.
Applying bisection method in $[\rho_{k+1},x_0]$, we approximate the root $\rho_{k+1}$. Later, we will see how we compute a number like $x_0$, in order to apply bisection method.
\item[ii)] If $\al_v<0$, then $\lim\limits_{x{\rightarrow}+\infty}p(x)=-\infty$. Polynomial $p$ is a strictly decreasing function in $[\rho_k,+\infty)$.
\begin{itemize} \item[a)] If $p(\rho_k)=0$, then $\rho_k$ is the unique root of $p$ in $[\rho_k,+\infty)$.
\item[b)] If $p(\rho_k)<0$, then $p$ does not have any root in $[\rho_k,+\infty)$.
\item[c)] If $p(\rho_k)>0$, then $p$ has unique one root in $[\rho_k,+\infty)$ (say $\rho_{k+1})$ and more specifically $\rho_{k+1}\in(\rho_k,+\infty)$. \end{itemize}
Because $\lim\limits_{x{\rightarrow}+\infty}p(x)=-\infty$, there exists some $x_0\in(\rho_k,+\infty)$, so that $p(x_0)<0$.
Then, $p(\rho_k)\cdot p(x_0)<0$, and $\rho_{k+1}\in(\rho_k,x_0)$, and applying bisection method we approximate the unique root $\rho_{k+1}$ in $(\rho_k,x_0)$. Now we examine the roots in $(-\infty,\rho_1]$. Whether $p(\rho_1)=0$, then $\rho_1$ is the unique root of $p$ in $(-\infty,\rho_1]$.
Now we suppose that $p(\rho_1)\neq0$. We examine two cases:
\item[i)] $\lim\limits_{x{\rightarrow}-\infty}p(x)=+\infty$.
This happens when $v$ is even and $\al_v>0$, or $v$ is odd and $\al_v<0$. Then $p$ is a strictly decreasing function in $(-\infty,\rho_1]$.
i), 1) If $p(\rho_1)>0$, then $p$ does not have any root in $(-\infty,\rho_1]$.
i), 2) If $p(\rho_1)<0$, then $p$ has a unique root in $(-\infty,\rho_1]$ (say $\rho_{k+2})$ and more specifically $\rho_{k+2}\in(-\infty,\rho_1)$. Because $\lim\limits_{x{\rightarrow}-\infty}p(x)=+\infty$, there exists some\linebreak $x_0<\rho_1$, so that $p(x_0)>0$. Then $\rho_{k+2}\in(x_0,\rho_1)$ and applying bisection method in $[x_0,\rho_1]$, we approximate root $\rho_{k+2}$.
\item[ii)] $\lim\limits_{x{\rightarrow}-\infty}p(x)=-\infty$. This is happened when $v$ is even and $\al_v<0$, or $v$ is odd and $\al_v>0$. Then $p$ is a strictly increasing function in $(-\infty,\rho_1]$.
We have two cases:
ii), 1)$p(\rho_1)<0$. Then, $p$ does not have any root in $(-\infty,\rho_1]$.
ii), 2) $p(\rho_1)>0$. Then $p$ has unique root in $(-\infty,\rho_1]$ (say $\rho_{k+2})$ and more specifically $\rho_{k+2}\in(-\infty,\rho_1)$. Because $\lim\limits_{x{\rightarrow}-\infty}p(x)=-\infty$, there exists some $x_0<\rho_1$, such that $p(x_0)<0$. Then $p(x_0)\cdot p(\rho_1)<0$ and $\rho_{k+2}\in(x_0,\rho_1)$.
Applying bisection method in $[x_0,\rho_1]$ we approximate root $\rho_{k+2}$. All the implications of this lemma are easy to prove and are left as an easy exercise for the interested reader. The proofs are of secondary school. \end{enumerate} \end{proof}
\noindent
{\bf Corollary 3.9.} {\em Basic Lemma 3.8 holds again, in the case when polynomial $p'$ has only one root}.
\begin{proof} The proof is similar to that of basic lemma for the intervals $(-\infty,\rho_1]$ and $[\rho_1,+\infty)$, where $p'(\rho_1)=0$. $\quad\blacksquare$ \end{proof}
\noindent
{\bf Corollary 3.10.} {\em Let $v\in{\Bbb N}$, $v\ge3$, $p(x)=\al_vx^v+\al_{v-1}x^{v-1}+\cdots+\al_1x+\al_0$, be a polynomial $p(x)\in{\mathbb{R}}[x]$, with degree $deg p(x)=v$.
We suppose that $p'$ does not have any root. Then $p$ has unique real root and we can construct an algorithm in order to find it}.
\begin{proof} Of course $p'$ is a polynomial of even degree $deg p'=v-1$, so $p$ is a polynomial of odd degree. Thus $p$ has, at least, one real root. Because $p'$ does not have any root, we have $p'(x)\neq0$, for every $x\in{\mathbb{R}}$. Thus, $p'(x)>0$ for every $x\in{\mathbb{R}}$, or $p'(x)<0$ for every $x\in{\mathbb{R}}$, or else if there exist $\al,\bi\in{\mathbb{R}}$, so that $p'(\al)<0$ and $p'(\bi)>0$, (of course $\al\neq\bi$), then because $p'$ is a continuous function (as a polynomial) and $p'(\al)\cdot p'(\bi)<0$, we take it that there exists $\ga\in(\al,\bi)$ (if $\al<\bi$) or $\ga\in(\bi,\al)$ (if $\bi<\al$) so that $p'(\ga)=0$, that is a contradiction because $p'(x)\neq0$ for every $x\in{\mathbb{R}}$. Thus, $p$ is a strictly increasing function in ${\mathbb{R}}$, if $p'(x)>0$, for every $x\in{\mathbb{R}}$, or else $p$ is a strictly decreasing function in ${\mathbb{R}}$ if $p'(x)<0$ for every $x\in{\mathbb{R}}$. If $p$ is a strictly increasing function, then $\lim\limits_{x{\rightarrow}+\infty}p(x)=+\infty$ and $\lim\limits_{x{\rightarrow}-\infty}p(x)=-\infty$, or else if $p$ is a strictly decreasing function in ${\mathbb{R}}$, then $\lim\limits_{x{\rightarrow}+\infty}p(x)=-\infty$ and $\lim\limits_{x{\rightarrow}-\infty}p(x)=+\infty$.
Polynomial $p$ is a strictly increasing function if $\al_v>0$, or else if $\al_v<0$, then $p$ is a strictly decreasing function.
If $\al_v>0$, then because $\lim\limits_{x{\rightarrow}+\infty}p(x)=+\infty$, there exists $y_0\in{\mathbb{R}}$, so that\linebreak $p(y_0)>0$, and because $\lim\limits_{x{\rightarrow}-\infty}=-\infty$, there exists $x_0\in{\mathbb{R}}$, $x_0<y_0$, so that $p(x_0)<0$. So $p(x_0)\cdot p(y_0)<0$ and $p$ has unique root in ${\mathbb{R}}$, (say $\rho$) so that $\rho\in(x_0,y_0)$.
If $\al_v<0$, then because $\lim\limits_{x{\rightarrow}+\infty}p(x)=-\infty$ there exists $y_0\in{\mathbb{R}}$, so that $p(y_0)<0$. Because $\lim\limits_{x{\rightarrow}-\infty}p(x)=+\infty$, there exists $x_0\in{\mathbb{R}}$, $x_0<y_0$ so that $p(x_0)>0$. Thus $p(x_0)\cdot p(y_0)<0$, and $p$ has unique root in ${\mathbb{R}}$ (say $\rho$) so that $\rho\in(x_0,y_0)$.
We sill see later how we compute numbers $x_0,y_0$ as above.
In any of the cases above we apply the bisection method in the interval $[x_0,y_0]$, to find the unique real root of $p$. $\quad\blacksquare$ \end{proof}
\noindent
{\bf Remark 3.11} {\em The multiplicity of a root is found with the algebraic algorithm 3.6.
However, we can find the multiplicity of a root in an analytic way.
More specifically:\\
Let $p(x)\in{\Bbb C}[x]$ be a polynomial and $\rho$ be a root of $p$, where $deg p(x)=v\in{\Bbb N}$.
Then, there exists a unique natural number $k\in{\Bbb N}\cup\{0\}$ $k\le v-1$, so that:\\
$p(\rho)=0$, $p'(\rho)=0,\ld$, $p^{(k)}(\rho)=0$ and $p^{(k+1)}(\rho)\neq0$, that is $p^{(i)}(\rho)=0$, for all $i=0,1,\ld,k$ and $p^{(k+1)}(\rho)=0$, where $p^{(0)}(\rho)=p(\rho)$.
The natural number $k+1$ is the multiplicity of root $\rho$ of $p$. (Of course we have always $p^{(v)}(\rho)\neq0$)\\
This is a classical result in calculus, that is proven easily.
Now, we cover the gap from basic Lemma 3.8, computing a number like $x_0$ in this lemma}.
\\
\noindent
{\bf Remark 3.12.} {\em Let $p(x)\in{\mathbb{R}}[x]$ be a real polynomial,
\[ p(x)=\al_0+\al_1x+\cdots+\al_{v-1}x^{v-1}+\al_vx^v,\ \ v=deg p(x),\ \ v\ge3. \]
We suppose that $\al_v>0$, and that $p'(x)$ has real roots}.
\begin{proof} Let $\rho$ be the greater real root of $p'(x)$. We suppose that $p(\rho)<0$. We consider an arbitrary real number $x_0$, so that $x_0>\rho$, $x_0>1$ and\\
$x_0>\dfrac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}$. We prove that $p(x_0)>0$.
We have of course $-|y|\le y$ for every $y\in{\mathbb{R}}$ (1). We apply (1) for
\[ y=\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x}\eqno{\mbox{(1)}} \]
for some $x\in{\mathbb{R}}-\{0\}$ and we have:
\[
-\bigg|\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x}\bigg| \le\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x}. \eqno{\mbox{(2)}} \]
Adding the number $\al_v$ in two members of (2) we take:
\[
\al_v-\bigg|\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x}\bigg| \le\al_v+\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{a_{v-1}}{x}. \eqno{\mbox{(3)}} \]
By the triangle inequality we take for $x>0$:
\begin{align*}
\bigg|\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x}\bigg|
&\le\bigg|\frac{\al_0}{x^v}\bigg|+\bigg|\frac{\al_1}{x^{v-1}}\bigg|+\cdots+
\bigg|\frac{\al_{v-1}}{x}\bigg|\\
&\Leftrightarrow-\bigg(\frac{|\al_0|}{x^v}+\frac{|\al_v|}
{x^{v-1}}+\cdots+\frac{|\al_{v-1}|}{x}\bigg)\\
&\le-\bigg|\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x}\bigg| \hspace*{3cm} {\mbox{(4)}} \end{align*}
Adding the number $\al_v$ in two members of (4) we take
\[
\al_v-\bigg(\frac{|\al_0|}{x^v}+\frac{|\al_1|}{x^{v-1}}+\cdots+\frac{|\al_{v-1}|}{x}\bigg)
\le\al_v-\bigg|\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x}\bigg|, \ \ \text{for} \ \ x>0. \eqno{\mbox{(5)}} \]
Let some $x>1$. Then we have:
\begin{align*} &x\ge x,x^2\ge x,\ld,x^v\ge x\Rightarrow\frac{1}{x}\le\frac{1}{x}, \frac{1}{x^2}<\frac{1}{x},\ld,\frac{1}{x^v}<\frac{1}{x}\\
&\Rightarrow\frac{|\al_{v-1}|}{x}\le\frac{|\al_{v-1}|}{x},\frac{|\al_{v-2}|}{x^2}\le, \frac{|\al_{v-2}|}{x},\ld,
\frac{|\al_1|}{x^{v-1}}\le\frac{|\al_1|}{x},\frac{|\al_0|}{x^v}\le\frac{|\al_0|}{x}. \end{align*}
Adding the above inequalities in pairs we take:
\begin{align*}
&\frac{|\al_{v-1}|}{x}+\cdots+\frac{|\al_1|}{x^{v-1}}+\frac{|\al_0|}{x^v}\le
\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{x}\Rightarrow\\
&-\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{x}\le-\bigg(\frac{|\al_{v-1}|}{x}+\cdots
+\frac{|\al_1|}{x^{v-1}}+\frac{|\al_0|}{x^v}\bigg)\Rightarrow \\
&\al_v-\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{x}\le\al_v-\bigg(\frac{|\al_{v-1}|}{x}
+\cdots+\frac{|\al_1|}{x^{v-1}}+\frac{|\al_0|}{x^v}\bigg). \hspace*{1.6cm} {\mbox{(6)}} \end{align*}
Through inequalities (3), (5) and (6), we get:
\[
\al_v-\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{x}\le\al_v+\frac{\al_0}{x^v}+ \frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x} \ \ \text{for} \ \ x>1. \eqno{\mbox{(7)}} \]
Now, for every $x>1$, $x>\dfrac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}$, we take
\[
\al_v>\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{x}\Rightarrow\al_v-\frac{|\al_0|
+|\al_1|+\cdots+|\al_{v-1}|}{x}>0. \eqno{\mbox{(8)}} \]
Through (7) and (8), we take it that for every $x>1$, $x>\dfrac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}$ we get
\[ \al_v+\dfrac{\al_{v-1}}{x}+\cdots+\frac{\al_1}{x^{v-1}}+\frac{\al_0}{x^v}>0. \eqno{\mbox{($\ast$)}} \]
This gives
\[ x^v\cdot\bigg(\al_v+\frac{\al_v-1}{x}+\cdots+\frac{\al_1}{x^{v-1}}+\frac{\al_0}{x^v}\bigg)>0\Leftrightarrow p(x)>0 \ \ \text{(by the definition of $p(x)$)} \eqno{\mbox{(9)}} \]
We apply (9) for the number $x_0\in{\mathbb{R}}$ so that $x_0>\rho$, $x_0>1$ and\\
$x_0>\dfrac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}$ and we take it that $p(x_0)>0$.
Thus, we have $p(\rho)\cdot p(x_0)<0$. This means that the unique real root of $p(x)$ in $[\rho,+\infty)$ belongs in $(\rho,c_0)$. Applying the bisection method in the interval $[\rho,x_0]$, we compute the unique real root $x_0^\ast$ of $p(x)$ in $[\rho,+\infty)$ that is $x^\ast_0\in(\rho,x_0)$. Of course if $p(\rho)>0$, polynomial $p$ does not have any real root in $[\rho,+\infty)$ as we have seen in basic Lemma 3.8 and if $p(\rho)=0$, then $\rho$ is the unique real root of $p$ in $[\rho,+\infty)$.
Now, we suppose that $\al_v<0$, and that $p'(x)$ has real roots.
Let $\rho$ be the greatest real root of $p'(x)$. We suppose that $p(\rho)>0$. We consider an arbitrary real number $x_0$, so that $x_0>\rho$, $x_0>1$ and
\[
x_0>\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{-\al_v}=\frac{|\al_0|+|\al_1|
+\cdots+|\al_{v-1}|}{|\al_v|}. \eqno{\mbox{(10)}} \]
By (10) we get (because $|\al_v|>0$ and $x>0$)
\[
|\al_v|>\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{x_0}\Rightarrow
\al_v+\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{x_0}<0. \eqno{\mbox{(11)}} \]
Let $x>1$. Because of $x>1$ we get
\begin{align*} &x\ge x,x^2\ge x,\ld,x^{v-1}\ge x,x^v\ge x\Rightarrow\frac{1}{x^v}\le\frac{1}{x},\frac{1}{x^{v-1}}\le\frac{1}{x},\ld, \frac{1}{x}\le\frac{1}{x}\Rightarrow \\
&\frac{|\al_0|}{x^v}\le\frac{|\al_0|}{x},\frac{|\al_1|}{x^{v-1}}\le\frac{|\al_1|}
{x^{v-1}},\ld,\frac{|\al_{v-1}|}{x}\le\frac{|\al_{v-1}|}{x}. \end{align*}
Adding by pairs the previous inequalities we get:
\begin{align*}
&\frac{|\al_0|}{x^v}+\frac{|\al_1|}{x^{v-1}}+\cdots+\frac{|\al_{v-1}|}{x}\le
\frac{|\al_0|}{x}+\frac{|\al_1|}{x}+\cdots+\frac{\al_{v-1}|}{x}\Rightarrow \\
&\al_v+\frac{|\al_0|}{x^v}+\frac{\al_v|}{x^{v-1}}+\cdots+\frac{|\al_0-1|}{x}\le\al_v+
\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{x}. \hspace*{1.6cm} {\mbox{(12)}} \end{align*}
Of course we get:
\begin{align*}
\al_0\le|\al_0|,\al_1\le|\al_1|,\ld,\al_{v-1}&\le|\al_{v-1}|\overset{x>0}{\Longrightarrow}
\frac{\al_0}{x^v}\le\frac{|\al_0|}{x^v},\frac{\al_1}{x^{v-1}}\\
&\le\frac{|\al_1|}{x^{v-1}}
,\ld,\frac{\al_{v-1}}{x}\le\frac{|\al_{v-1}|}{x} \end{align*}
and adding by pairs the previous inequalities we get:
\begin{align*}
&\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}\cdots+\frac{\al_{v-1}}{x}\le\frac{|\al_0|}{x^v}
+\frac{|\al_1|}{x^{v-1}}+\cdots+\frac{|\al_{v-1}|}{x}\Rightarrow \\ &\al_v+\frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_{v-1}}{x}\le
\al_v+\frac{|\al_0|}{x^v}+\frac{|\al_1|}{x^{v-1}}+\cdots+\frac{|\al_{v-1}|}{x}.
\hspace*{1.7cm} {\mbox{(13)}} \end{align*}
Of course as we have seen in basic Lemma 3.8 if $p(\rho)<0$, then $p$ does not have any root in $[\rho,+\infty)$, and if $p(\rho)=0$, number $\rho$ is the unique real root of $p$ in $[\rho,+\infty)$. So far we have seen how we compute the unique real root of $p$ (if any) in $[\rho,+\infty)$, when $\rho$ is the greatest real root of $p'$.
In a similar way we compute the unique real root of $p$ in $(-\infty,\rho^\ast]$ (if any), where $\rho^\ast$ is the smallest real root of $p'$. It suffices to observe the following:
We simply write $p(x)=p(-(-x))$ and we find easily a polynomial $q\in{\mathbb{R}}[x]$, such that $p(x)=q(-x)$ (it is trivial to find such a polynomial $q$).
Now, for $x_0>1$, $x_0>\rho$ and $x_0>\dfrac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{|\al_v|}$ the previous inequalities (11), (12) and (13) hold simultaneously for $x=x_0$, thus we get
\begin{align*} &\al_v+\frac{\al_{v-1}}{x_0}+\cdots+\frac{\al_1}{x^{v-1}_0}+\frac{\al_0}{a^v_0}<0\Rightarrow \\ &x^v_0\bigg(\al_v+\frac{\al_{v-1}}{x_0}+\cdots+\frac{\al_1}{x^{v-1}_0}+\frac{\al_0}{x^v_0}\bigg) <0\Rightarrow p(x_0)<0 \ \ \text{(by the definition of $p$)}. \end{align*}
So, we get $p(\rho)\cdot p(x_0)<0$ and $p$ has a root exactly in $(\rho,x_0)$, (say $x^\ast_0$), where $x^\ast_0$ is the unique real root of $p$ in $[\rho,+\infty)$.
We apply the bisection method in the interval $[\rho,x_0]$ and we compute the unique real root $x^\ast_0\in(\rho,x_0)$ of $p$ in $[\rho,+\infty)$.
Now we suppose that $p$ has real roots, and let $\rho$ be the smallest real root of $p$. Let $r$ be an arbitrary real root of $p$. Then $p(r)=0$ and by (14) we take that $q(-r)=0$, that is $-r$ is a real root of $q$. This gives that $-\rho$ is the biggest real root of $q$.
By (14) we also get that $p'(x)=-q'(-x)$, which gives that if $\al$ is a real root of $p'$, then $-\al$ is a real root of $q'$. Thus, because by supposition $p'$ has real roots, the same holds for $q'$. Then, we apply the previous procedure and we compute the greatest real root of $q$, (say $-\rho$), which means that $\rho$ is the smallest real root of $p$. Thus, we compute the smallest real root of $p$ also in any case. Now, we consider a polynomial $p\in{\mathbb{R}}[x]$, with $deg p(x)=v\in{\Bbb N}$, $v\ge3$, so that polynomial $p'$ does not have any real root. Because of $p'$ does not have any root and $deg p'(x)\ge2$ (because $deg p(x)\ge3$), we conclude that $p'$ is a polynomial of even degree (because any polynomial of odd degree has a real root at least). This means that $p$ is a polynomial of odd degree that has one real root at least. Because $p'(x)\neq0$ for every $x\in{\mathbb{R}}$, we take it that $p'(x)>0$ for every $x\in{\mathbb{R}}$ or $p'(x)<0$ for every $x\in{\mathbb{R}}$. If $\al_v>0$, then $p'(x)>0$ for every $x\in{\mathbb{R}}$ and $p$ is a strictly increasing function in ${\mathbb{R}}$, such that $\lim\limits_{x{\rightarrow}+\infty}p(x)=+\infty$ and $\lim\limits_{x{\rightarrow}-\infty}p(x)=-\infty$. Thus, there exists $x_0<0$ so that $p(x_0)<0$ and $y_0>0$ so that $p(y_0)>0$, which gives that $p$ has a real root in $(x_0,y_0)$, say $\rho$. Because $p$ is a strictly monotonous function, we take it that the root $\rho$ is the unique real root of $p$.
We can now compute some numbers $x_0,y_0$ with the above properties. By inequality $(\ast)$ in page 40 we take it that if $x\in{\mathbb{R}}$, so that $x>1$ and $x>\dfrac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}$, then we get $\al_v+\dfrac{\al_v-1}{x}+\cdots+\dfrac{\al_1}{x^{v-1}}+\dfrac{\al_0}{x^v}>0$. We choose some $y_0>1$, so that $y_0>\dfrac{|\al_0|+|\al_1|+\cdots+|\al_v-1|}{\al_v}$, then by inequality $(\ast)$ in page 40 we get
\begin{align*} &\al_v+\frac{\al_v-1}{y_0}+\cdots+\frac{\al_1}{y^{v-1}_0}+\frac{\al_0}{y^v_0}>0\Rightarrow\\ &y^v_0\bigg(\al_v+\frac{\al_v-1}{y_0}+\cdots+\frac{\al_1}{y^{v-1}_0}+\cdots+\frac{\al_0}{y^v_0}\bigg)>0\Leftrightarrow p(y_0)>0. \hspace*{2.5cm} {\mbox{(15)}} \end{align*}
Now, let some $x<-1$. Then we have $(x\neq0)$
\[
\frac{\al_0}{x^v}\ge-\bigg|\frac{\al_0}{x^v}\bigg|,\ \ \frac{\al_1}{x^{v-1}}\ge-\bigg|\frac{\al_1}{x^{v-1}}\bigg|,\ld,\frac{\al_{v-1}}{x}\ge-\bigg|
\frac{\al_{v-1}}{x}\bigg|. \]
Adding these inequalities we get:
\[ \frac{\al_0}{x^v}+\frac{\al_1}{x^{v-1}}+\cdots+\frac{\al_v-1}{x}\ge-
\bigg(\bigg|\frac{\al_0}{x^v}\bigg|+\bigg|\frac{\al_1}{x^{v-1}}\bigg|+\cdots+\bigg|
\frac{\al_{v-1}}{x}\bigg|\bigg) \eqno{\mbox{(16)}} \]
We get also
\begin{align*}
|x|>1&\Rightarrow|x|\ge|x|,|x^2|>|x|,\ld,|x^{v-1}|>|x|,|x^v|>|x|\\
&\Rightarrow\frac{1}{|x|}\le\frac{1}{|x|},\ld,\frac{1}{|x^{v-1}|}<
\frac{1}{|x|},\frac{1}{|x^v|}<\frac{1}{|x|} \\
&\Rightarrow\bigg|\frac{\al_{v-1}}{x}\bigg|\le\bigg|\frac{\al_{v-1}|}{|x|},\ld,
\bigg|\frac{\al_1}{x^{v-1}}\bigg|<\frac{|\al_1|}{|x|},\bigg|\frac{\al_0}{x^v}\bigg|<\bigg|\frac{\al_0}{x}\bigg|\\
&\Rightarrow-\bigg|\frac{\al_{v-1}}{x}\bigg|\ge-\bigg|\frac{\al_{v-1}}{x}\bigg|,\ld,-
\bigg|\frac{\al_1}{x^{v-1}}\bigg|>-\frac{|\al_1|}{|x|},-\bigg|\frac{\al_0}{x^v}\bigg|>-
\bigg|\frac{\al_0}{x}\bigg|. \end{align*}
Adding by pairs the previous inequalities we get:
\[
-\bigg(\bigg|\frac{\al_0}{x^v}\bigg|+\bigg|\frac{\al_1}{x^{v-1}}\bigg|+\cdots+
\bigg|\frac{\al_{v-1}}{x}\bigg|\bigg)\ge-\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{|x|}. \eqno{\mbox{(17)}} \]
By (16) and (17) we get
\[ \al_v+\frac{\al_{v-1}}{x}+\cdots+\frac{\al_v}{x^{v-1}}+\frac{\al_0}{x^v}\ge\al_v+
\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{|x|} \eqno{\mbox{(18)}} \]
for every $x\in{\mathbb{R}}$, $x<-1$.
Now we get $x_0\in{\mathbb{R}}$, so that
\[
x_0<-1 \ \ \text{and} \ \ x_0<-\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}. \eqno{\mbox{(19)}} \]
Then by (19) we get:
\begin{align*}
&-x_0>\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}>0 \ \ \text{(because $\al_v>0$)} \\
&\Rightarrow|x_0|>\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v} \\
&\Rightarrow\al_v-\frac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{|x_0|}>0. \hspace*{5.5cm} {\mbox{(20)}} \end{align*}
So, for $x<-1$, $x_0<-\dfrac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}$ we get from (20), that
\[ \al_v+\frac{\al_{v-1}}{x_0}+\cdots+\frac{\al_1}{x^{v-1}_0}+\frac{\al_0}{x^v_0}>0\Rightarrow \]
(because $x_0<0$ and $v$ is odd $x^v_0<0$)
\[ x^v_0\cdots\bigg(\al_v+\frac{\al_{v-1}}{x_0}+\cdots+\frac{\al_1}{x^{v-1}_0}+ \frac{\al_0}{x^v_0}\bigg)<0\Leftrightarrow p(x_0)<0. \]
Thus, for some $x_0<-1$, $x_0<-\dfrac{|\al_0|+|\al_1|+\cdots+|\al_{v-1}|}{\al_v}$ we get $p(x_0)<0$ (21).
So with (15) and (12) we get $p(x_0)\cdot p(y_0)<0$ and by applying the bisection method we can compute the unique real root $\rho$ of $p$, so that $\rho\in(x_0,y_0)$.
Finally in the case of $\al_v<0$, we consider the polynomial $-p(x)$. Then
\[ (-p)'(x)=-p'(x)\neq0\ \ \text{for every} \ \ x\in{\mathbb{R}} \]
and the coefficient of the monomial of greater degree of $-p$ is positive now. We apply the previous for $-p$ and we compute the unique real root $\rho$ of $-p$, that is the unique real root of $p$ also.
So in this remark we have covered the gap, we have left from basic Lemma 3.8 and Corollaries 3.9 and 3.10 and we have computed specific numbers $x_0,y_0$. For the sequel we also need some tools from real polynomials of two real variables.
Before this, let us give some specific examples for the roots of polynomials.
As it is well known, from elementary calculus, any polynomial of odd degree has a real root at least.
On the other hand, there are many polynomials of any even degree that do not have any real root. For example, let $p(x)$ be any polynomial that is non constant, let $k\in{\Bbb N}$, and $\thi>0$. Then polynomial $q(x)=p(x)^{2k}+\thi$ does not have any real root, as we can easily see, and has degree $2k\cdot v$, where $v=deg p(x)$.
Of course for every finite set of real numbers $A=\{\rho_1,\rho_2,\ld,\rho_v\}$, $v\in{\Bbb N}$, $\rho_i\neq\rho_j$, for $i,j\in\{1,2,\ld,v\}$, $i\neq j$, polynomial $p(x)=(x-\rho_1)(x-\rho_2)\ld(x-\rho_v)$ has roots the numbers $\rho_i$, $i=1,\ld,v$, and polynomial
\[ q(x)=((x-\rho_1)(x-\rho_2)\ld(x-\rho_v))^{2k}=p(x)^{2k} \]
is a polynomial of even degree $deg q(x)=2kv$, with roots the numbers $\rho_i$, $i=1,2,\ld,v$, also. Now we consider polynomials of two real variables with real coefficients, that is we consider the set
\[ {\mathbb{R}}^2[x,y]=\{p(x,y):p(x,y) \]
is a polynomial of two real variables $x$ and $y$ with coefficients in ${\mathbb{R}}\}$.
Let $p(x,y)\in{\mathbb{R}}^2[x,y]$. We say that polynomial $p(x,y)$ is a pure polynomial, when $deg_x p(x,y)\ge1$ and $deg_yp(x,y)\ge1$ where $deg_xp(x,y)$, $deg_yp(x,y)$ are the greatest degree of its monomials with respect to $x$ (or $y$ respectively).
The set of roots of $p(x,y)$ is the set
\[ L_p(x,y)=\{(x,y)\in{\mathbb{R}}^2\mid p(x,y)=0\}. \]
As in polynomials of one real variable, we can easily see that there are many pure polynomials $p(x,y)\in{\mathbb{R}}^2[x,y]$, that do not have any roots.
For example, let $p(x,y)$ be any pure polynomial. Then polynomial\linebreak $q(x,y)=p(x,y)^{2k}\!+\!\thi$, where $k\!\in\!{\Bbb N}$, $\thi\!>\!0$, is a pure polynomial that does not have any real root, as we easily see. Of course these polynomials are of even degree for $x$ and $y$.
On the other hand let $A=\{\al_1,\al_2,\ld,\al_v\}$, $B=\{\bi_1,\bi_2,\ld,\bi_m\}$, $A\cup B\subseteq{\mathbb{R}}$, $v,m\in{\Bbb N}$, $\al_i\neq\al_j$ for every $i,j\in\{1,\ld,v\}$ $i\neq j$ and $\bi_i\neq\bi_j$ for every $i,j\in\{1,2,\ld,m\}$, $i\neq j$. Let Also $k\in{\Bbb N}$. We consider the pure polynomial
\[ p(x,y)=((x-\al_1)(x-\al_2)\ld(x-\al_v))((y-\bi_1)(y-\bi_2)\ld(y-\bi_m))^{2k}. \]
Then $p$ is a pure polynomial of even degree with respect to $x$ and $y$ such that
\[ L_0=\{(\al_i,\bi_j),i\in\{1,\ld,v\},j\in\{1,2,\ld,m\}\}\subsetneqq L_p(x,y). \]
We remark also that for ever $y\in{\mathbb{R}}$, the couple $(\al_i,y)$ is a root of $p(x,y)$. This fact differentiates pure polynomials $p(x,y)$ from polynomials of one variable.
That is, there exist uncountable pure polynomials, each one having uncountable set of real roots. Especially, this holds for pure polynomials of an odd degree with respect to $x$ and $y$. We have the following proposition. \end{proof}
\noindent
{\bf Proposition 3.13.} {\em Let $p(x,y)$ be a pure polynomial such that $deg_x(x,y)=v$ is odd or $deg p_y(x,y)$ is odd. Then for every $r\in{\mathbb{R}}$ the set $L_r=\{(x,y)\in{\mathbb{R}}^2:p(x,y)=r\}$ is uncountable}.
\begin{proof} We suppose, without loss of generality, that number $v=deg p_x(x,y)$ is odd. Then, as we can see easily, we can write polynomial $p(x,y)$ as follows:
\[ p(x,y)=\al_v(y)x^v+\al_{v-1}(y)x^{v-1}+\cdots+\al_1(y)x+\al_0(y), \]
where $\al_i(y)\in{\mathbb{R}}[y]$ for every $i=0,1,\ld,v$, and $\al_v(y)\neq0$, because $v=deg p_x(x,y)$. Because $\al_v(y)\neq0$, polynomial $\al_v(y)$ has a finite set of roots. Let $A_v$ be the set of roots of $\al_v(y)$, that is $A_v=\{y\in{\mathbb{R}}\mid\al_v(y)=0\}$. Let $y_0\in{\mathbb{R}}\sm A_v$. Then $\al_v(y_0)\neq0$. Also let $r\in{\mathbb{R}}$. We consider the polynomial
\[ p_r(x)=\al_v(y_0)x^v+\al_{v-1}(y_0)x^{v-1}+\cdots+\al_1(y_0)x+\al_0(y_0)-r. \]
Then, $p_r(x)$ is a polynomial of odd degree $deg p_r(x)=v$, thus polynomial $p_r(x)$ has a real root, say $x_0$, at least, that is we have:
\[ \begin{array}{l}
p_r(x_0)=0\Rightarrow\al_v(y_0)x^v_0+\al_{v-1}(y_0)x^{v-1}_0+\cdots+ \al_1(y_0)x_0+\al_0(y_0)=r\Leftrightarrow \\ [1.5ex] p(x_0,y_0)=r\Rightarrow(x_0,y_0)\in L_r. \end{array} \]
That is we proved that for every $y\in{\mathbb{R}}\sm A_v$, we have that there exists some $x\in{\mathbb{R}}$, such that $(x,y)\in L_r$. Of course, if $y_1,y_2\in{\mathbb{R}}\sm A_v$, $y_1\neq y_2$ and $(x_1,y_1),(x_2,y_2)\in L_r$, we have $(x_1,y_1)\neq(x_2,y_2)$ so the set $L_r$ is uncountable, and the proof of this proposition is complete. $\quad\blacksquare$ \end{proof}
\noindent
{\bf Corollary 3.14.} {\em Let $p(x,y)$ be a pure polynomial such that $deg_xp(x,y)$ or $deg_yp(x,y)$ is odd. Then the set of real roots of $p(x,y)$ is uncountable}.
\begin{proof} It is a simple application of the previous Proposition 3.13 for $r=0$. $\quad\blacksquare$ \end{proof}
As we have noticed previously there are also such pure polynomials that the numbers $deg_xp(x,y)$ and $deg_yp(x,y)$ are even whose set of real roots is uncountable, as well as there being polynomials that do not have any roots.
Of course here we have the natural question: Are there pure polynomials $p(x,y)$ whose set of real roots $p(x,y)$ is non empty and finite? Of course, let us give a simple example:
We consider the polynomial: $p(x,y)=(x^2-1)^2+(y^2-4)^2$. It is easy to see that
\[ L_p(x,y)=\{(1,2),(1,-2),(-1,2),(-1,-2)\}. \]
More generally, let $p_1(x)$ be a polynomial with real roots $\al_1,\al_2,\ld,\al_v$, and $p_2(y)$ be a polynomial with real roots $\bi_1,\bi_2,\ld,\bi_m$.
We consider the pure polynomial $p(x,y)=p_1(x)^{2k_1}+p_2(y)^{2k_2}$, where $k_1,k_2,\in{\Bbb N}$. Then, it is easy to see that:
\[ L_p(x,y)=\{(\al_i,\bi_j),i\in\{1,\ld,v\},j\in\{1,\ld,m\}\}. \]
Of course by Corollary 3.14 only pure polynomials whose numbers $deg_xp(x,y)$, $deg_yp(x,y)$ are even can have finite set of roots, as in the previous examples. From the previous results we also have a significant observation.
These polynomials have the number zero as a global minimum!
For the sequel, we have to concentrate our attention to pure polynomials $p(x,y)$ that have a finite set of roots. So, from the previous observation we are led to ask whether the reverse result holds. That is, does any pure polynomial that has a global minimum have a finite set of real roots? The answer is no, and we can give a simple example. We consider the pure polynomial $p(x,y)=((x-1)(y-2))^2-7$. It is easy to check that polynomial $p(x,y)$ has the number $-7$ as a global minimum. For this polynomial we get:
\[ p(x,3)=(x-1)^2-7 \ \ \text{for every} \ \ x\in{\mathbb{R}} \]
so $p(1,3)=-7<0$ and $p(4,1)=2>0$, thus there exists $x_0\in(1,4)$ so that $p(x_0,3)=0$. Similarly take any real number $y_0\in(3,4)$. That is $3<y_0<4\Rightarrow1<y_0-2<2\Rightarrow1<(y_0-2)^2<4$ (1).
We get:
\[ \begin{array}{l}
p(1,y_0)=-7<0 \\ [1.5ex]
p(8,y_0)=7^2(y_0-2)^2>7^2>0 \end{array} \]
by (1), so there exists $x_1\in(1,8)$ such that $p(x_1,y_0)=0$.
Thus, for every $y\in(3,4)$, there exists $x\in{\mathbb{R}}$, so that $p(x,y)=0$, and of course if $y_1,y_2\in(3,4)$, $x_1,x_2\in{\mathbb{R}}$, $p(x_1,y_1)=p(x_2,y_2)=0$ and $y_1\neq y_2$ we have $(x_1,y_1)\neq(x_2,y_2)$, so the set of real roots of $p$ is uncountable even if polynomial $p(x,y)$ has a global minimum. However, the property of a pure polynomial to have a global minimum (or maximum also) is a crucial property that have all pure polynomials that have a finite number of roots, as we will prove now with the following proposition.
\\
\noindent
{\bf Proposition 3.15.} {\em (topological lemma). Let $p(x,y)$ be a pure polynomial. We suppose that there exist two couples $(x_1,y_1),(x_2,y_2)\in{\mathbb{R}}^2$ such that\linebreak $p(x_1,y_1)\cdot p(x_2,y_2)<0$. Then, set $L_p(x,y)$ is uncountable}.
\begin{proof}
We set $A=(x_1,y_1)$, $B=(x_2,y_2)$. We get $A\neq B$, or otherwise we have $A=B$ and $p(x_1,y_1)\cdot p(x_2,y_2)=p(A)\cdot p(B)=p(A)^2\ge0$, which is false. So we get $A\neq B$. We consider the midperpendicular $\el$ of segment $[A,B]$. For every point $\Ga\in\el$, we consider the union of two segments $[A,\Ga]\cup[\Ga,B]$. We write $A\Ga B=[A,\Ga]\cup[\Ga,B]$ for simplicity. Of course $A\Ga B\subseteq{\mathbb{R}}^2$. We consider the restriction $p|_{A\Ga B}$ for simplicity, and we write $p=p|_{A\Ga B}$ also for simplicity.
Of course the set $A\Ga B$ is a compact and connected subset of ${\mathbb{R}}^2$. So the set $p(A\Ga B)$ os a closed interval of ${\mathbb{R}}$.
We suppose that $p(A)<0$ and $p(B)>0$, without loss of generality. So $p(A),p(B)\in p(A\Ga B)$ and gives that $0\in p(A\Ga B)$, that is there exists some point $\De\in A\Ga B$ so that $p(\De)=0$. Of course $\De\neq A$ and $\De\neq B$. So, for every $\Ga\in\el$ and every curve $A\Ga B$, there exists some $\De\in A\Ga B$, $\De\neq A$, $\De\neq B$, such that $p(\De)=0$.
Because the set ${\cal{A}}=\{A\Ga B,\Ga\in\el\}$ is an uncountable supset of $P({\mathbb{R}}^2)$ (the powerset of ${\mathbb{R}}^2$), and for every $\Ga_1,\Ga_2\in\el,\Ga_1\neq\Ga_2$, we have that $A\Ga_1B\cap A\Ga_2B=\{A,B\}$ this means that the set
\[ {\cal{B}}=\{\De\in A\Ga B\mid\Ga\in\el \ \ \text{and} \ \ p(\De)=0\} \]
is uncountable, that gives that the set $Lp(x,y)$ of roots of $p(x,y)$ is uncountable and the proof of this proposition is complete. $\quad\blacksquare$ \end{proof}
\noindent
{\bf Corollary 3.16.} {\em Let $p(x,y)$ be a pure polynomial that has a finite set of roots, non empty. Then, number 0 is the global minimum or maximum of $p(x,y)$, or in other words polynomial $p(x,y)$ has a global maximum or minimum, and when this holds, then this global maximum or minimum is number 0}.
\begin{proof} There exists no two points $(x_1,y_1),(x_2,y_2)\in{\mathbb{R}}^2$ so that:
\[ p(x_1,y_1)\cdot p(x_2,y_2)<0. \]
Or else, if there exist two points $(x_1,y_1),(x_2,y_2)\!\in\!{\mathbb{R}}^2$, so that $p(x_1,y_1)\!\cdot\! p(x_2,y_2)<0$, then set $L_p(x,y)$ is uncountable (by the previous Proposition 3.13), which is false by our supposition.
This means that we have:
i) $p(x,y)\ge0$ for every $(x,y)\in{\mathbb{R}}^2$, or
ii) $p(x,y)\le0$ for every $(x,y)\in{\mathbb{R}}^2$.
We suppose that i) holds. Because set $Lp(x,y)$ is non-empty, this means that there exists $(x_0,y_0)\in{\mathbb{R}}^2$ so that $p(x_0,y_0)=0$, so we get $p(x,y)\ge p(x_0,y_0)$ for every $(x,y)\in{\mathbb{R}}^2$. So, polynomial $p(x,y)$ has in point $(x_0,y_0)$ its global minimum the number 0, because $p(x_0,y_0)=0$. If ii) holds, then we take with a similar way that $p$ has global maximum the number 0 in a point, and the proof of corollary is complete. $\quad\blacksquare$ \end{proof}
The above corollary is a basic result that we use in the second stage of our method.
Finally, we refer here the most advanced result, that we use in our method.
This result is called many times as Fermat's Theorem in calculus of several variables.
\\
\noindent
{\bf Theorem 3.17.} {\em Let $U\subseteq{\mathbb{R}}^2$, $U$ open and $f:U{\rightarrow}{\mathbb{R}}$ be a differentiable function in $x_0\in U$, where $x_0$ is a point of local maximum or local minimum of $f$. Then the following holds: $\bigtriangledown f(x_0)=0$, that is $x_0$ is a crucial point of $f$, where $\nabla f(x_0)$ is the gradient of $f$ in $x_0$}.
\section{Appendix}
\noindent
Fundamental Theorem of Algebra is a powerful and basic result in the theory of polynomials, especially in polynomial equations.
Gauss gave the first complete proof of this result in his Ph.D. There are many proofs for this important Theorem but none of them is trivial in order to be presented in books of secondary school.
Its simplest proof comes from complex analysis and uses an advanced Theorem of complex analysis, Liouville's Theorem. Here we give a proof that uses the most elementary tools that an undergraduate student learns.
We think that it is difficult for an undergraduate student to find this proof in books, so we try to present it with details for educational reasons.
For this reason we give firstly some elementary lemmas.
\\
\noindent
{\bf Lemma 4.1} {\em Let $p(z)\in{\Bbb C}[z]$ be a complex polynomial, and $z_0\in{\Bbb C}$. We consider polynomial $Q(z)=p(r+z_0)$, $z\in{\Bbb C}$. If $p(z)\equiv0$, then of course $Q(z)\equiv0$. If $p(z)\not\equiv0$, then $deg Q(z)=deg p(z)$}.
\begin{proof} If $deg p(z)=0$, then the result is obvious. Let $deg p(z)=n\in{\Bbb N}$, $n\ge1$. We suppose that $n=1$, so we get $p(z)=az+b$, where $a,b\in{\Bbb C}$, $a\neq0$. We get: $Q(z)=p(z+z_0)=a(r+z_0)+b=az+(az_0+b)$, and $deg Q(z)=1$, because $a\neq0$. So, the result holds for $n=1$. We prove the result inductively. We suppose $z_0\neq0$.
For $n=1$, the result holds.
We suppose that result holds for $k\in{\Bbb N}$, $k\ge1$ and for every $j\in{\Bbb N}$, $1\le j\le k$. We prove that result holds for $k+1$.
We suppose that
\[ p(z)=a_0+a_1z+\cdots+a_kz+a_{k+1}z^{k+1} \ \ \text{and} \ \ a_{k+1}\neq0, \ \ \text{so} \ \ deg p(z)=k+1. \]
We distinguish two cases:
(i) $q(z)=a_0+a_1z+\cdots+a_kz^k\not\equiv0$. \\
Then we have: $p(z)=q(z)+a_{k+1}z^{k+1}$.
We have
\[ Q(z)=p(z+z_0)=q(z+z_0)+a_{k+1}(z+z_0)^{k+1}. \eqno{\mbox{(1)}} \]
We set $r(z)=q(z+z_0)$, $z\in{\Bbb C}$. Because $q(z)\not\equiv0$, by induction step we have $deg r(z)=deg q(z)\le k$ (2).
We have by Newton's binomial
\[ a_{k+1}(z+z_0)^{k+1}=a_{k+1}\sum^{k+1}_{j=0}z^{k+1-j}z^j_0 =\sum^{k+1}_{j=0}a_{n+1}z^j_0z^{k+1-j}. \eqno{\mbox{(3)}} \]
Because $z_0\neq0$ (by our supposition) and $a_{k+1}\neq0$ we have $deg a_{k+1}(z+z_0)^{k+1}=k+1$ (4), by equality (3).
By (1), (2) and (4), we get $deg Q(z)=k+1$ and the result holds.
(ii) $q(z)=a_0+a_1z+\cdots+a_kz^k\equiv0$. The proof is similar to case (i), so the result holds by induction.
Of course if $z_0=0$ the result is obvious, because $Q(z)=p(z)$, so $deg Q(z)=deg p(z)$. \end{proof}
\noindent
{\bf Lemma 4.2.} {\em We consider polynomial $p(z)\in{\Bbb C}[z]$. Of course we have $|p(z)|\ge0$ for every $z\in{\Bbb C}$. So the set
\[
A=\{x\in{\mathbb{R}}\mid\exists\;z\in{\Bbb C}:x=|p(z)|\} \]
is low bounded by 0.
We set $m=\inf(A)$. Then there exist $R>0$ so that:
\[
m=\inf(\{x\in{\mathbb{R}}\mid\exists z\in\overline{D(0,R)}:x=|p(z)|\}) \]
where $\overline{D(0,R)}=\{z\in{\Bbb C}:|z|\le R\}$}.
\begin{proof}
We set $B_R=\{x\in{\mathbb{R}}\mid\exists z\in D(0,R):x=|p(z)|\}$ for some $R>0$.
The result is obvious when $p(z)\equiv0$, so we suppose that $p(z)\not\equiv0$. It is obvious that $B_R\subseteq A$ by definitions of sets $A$ and $B_R$ for $R>0$. Let $x\in B_R$ for some $R>0$. Then $x\in A$. So: $m\le x$, because $m$ is a lower bound of $A$. So we have: $m\le x$ for every $x\in B_R$. This means that $m$ is a lower bound of $B_R$, so: $m\le m^\ast_R$ (1), where $m^\ast_R=\inf(B_R)$. That is we get: $m\le m^\ast_R$ for every $R>0$.
We suppose that:\\
$p(z)=a_0+a_1z+\cdots+a_nz^n$, where $n\in{\Bbb N}\cup\{0\}$, $a_n\neq0$. When $n=0$, we get of course $m^\ast_R=m=|p(z)|$ for every $z\in{\Bbb C}$ and every $R>0$, and the result is obvious of course. So we suppose that $n\ge1$.
Then for every $z\in{\Bbb C}\sm\{0\}$ we get
\[ p(z)=z^n\cdot\bigg(\frac{a_0}{z^n}+\frac{a_1}{z^{n-1}}+\cdots+\frac{a_{n-1}}{z}+a_n\bigg). \]
By calculus of the elementary limits in complex analysis we have:\\
$\lim\limits_{z{\rightarrow}\infty}\dfrac{a_0}{z^n}=\lim\limits_{z{\rightarrow}\infty}\dfrac{a_1}{z^{n-1}}=
\cdots=\lim\limits_{z{\rightarrow}\infty}\dfrac{a_{n-1}}{z}=0$ and $\lim\limits_{z{\rightarrow}\infty}z^n=\infty$, so we have: $(a_n\neq0)$ $\lim\limits_{z{\rightarrow}\infty}p(z)=\infty$. By definition of $\lim\limits_{z{\rightarrow}\infty}p(z)$, this means that: for $m+1$, there exists $R_0>0$, so that: $|p(z)|>m+1$ for every $z\in{\Bbb C}$, $|z|>R_0^{(\ast)}$.
From (1) we have of course $m\le m^\ast_{R_0}$ (2). Take $w\in{\Bbb C}$: $|w|>R_0$. Then, by the above we have $|p(w)|>m+1$ (3).
Now there exists $z_1\in{\Bbb C}$ so that $|p(z_1)|<m+1$ (4), or otherwise we have\linebreak $|p(z)|\ge m+1$ for every $z\in{\Bbb C}$, so $m+1$ is a lower bound of $A$, that is\linebreak $m=\inf(A)\ge m+1$, which is false. Of course $z_1\in\overline{D(0,R_0)}$ by implication $(\ast)$, or else $|z_1|>R_0$ that means $|p(z_1)|>m+1$ (5) that is false by the above inequalities (4) and (5). So we have $m^\ast_{R_0}\le|p(z_1)|<m+1<|p(w)|\Rightarrow m^\ast_{R_0}\le|p(w)|$.
So we get: $m^\ast_{R_0}\le|p(z)|$ for every $z\in{\Bbb C}:|z|>R_0$. Of course we have also\linebreak $m^\ast_{R_0}\le|p(z)|$ for every $z\in\overline{D(0,R_0)}$ by definition of $m^\ast_{R_0}$. So we get $m^\ast_{R_0}\le|p(z)|$ for every $z\in{\Bbb C}$, that means that $m^\ast_{R_0}$ is a lower bound of $A$, that is $m^\ast_{R_0}\le m$ (6). From (2) and (6) we get $m=m^\ast_{R_0}$, that is Lemma 4.2 has been proven. $\quad\blacksquare$ \end{proof}
\noindent
{\bf Remark 4.3.} {\em {\bf De Moivre Theorem:} We remind here the following result.\\
Let $n\in{\Bbb N}$, $n\ge2$. Then every non-zero complex number has exactly $n$ roots, that is if $w\in{\Bbb C}$, $w\neq0$, then equation $z^n=w$ has exactly $n$ solutions: This result is proven easily by elementary properties of complex numbers and it is well known as De Moivre's Theorem, using properties of functions sine and cosine. We also need a topological Theorem.}
\\
\noindent
{\bf Theorem 4.4} {\em Let $K\subseteq{\Bbb C}$ be compact and $f:K{\rightarrow}{\mathbb{R}}$ be continuous. Then $f$ attains its supremum and its infimum and both are finite. For this theorem see \cite{5}. After the above, we are now ready to give the proof of fundamental Theorem of Algebra}.
\\
\noindent
{\bf Fundamental Theorem of Algebra 4.5}
\begin{proof} We consider polynomial
\[ p(z)=a_0+a_1z+\cdots+a_nz^n, \ \ a_i\in{\Bbb C} \ \ \text{for every} \ \ i=0,1,\ld,n, \ \ n\in{\Bbb N}, \ \ a_n\neq0, \ \ n\ge1. \]
We prove that $p(z)$ has a root that is there exists $z_0\in{\Bbb C}$, so that $p(z_0)=0$. First of all we examine the case of $a_n=1$.
Of course we have $|p(z)|\ge0$ for every $z\in{\Bbb C}$. We set
\[
A=\{x\in{\mathbb{R}}\mid\exists z\in{\Bbb C}:x=|p(z)|\}. \]
Set $A$ is low bounded by 0. We set $m=\inf(A)$. Of course $m\ge0$. For every $R>0$ we set:
\[
B_R=\{x\in{\mathbb{R}}\mid\exists z\in\overline{D(0,R)}:x=|p(z)|\}, \ \ \text{and} \]
\[
m^\ast_R=\inf(B_R),\overline{D(0,R)}=\{z\in{\Bbb C}:|z|\le R\}. \]
Applying Lemma 4.2 we take that there exists $R_0>0$ so that: $m=m^\ast_{R_0}$ (1).
By Theorem 4.3, page 233 [5], Ball $\overline{D(0,R_o)}=\{z\in{\Bbb C}:|z|\le R_0\}$ is a compact set as a set closed and bounded. Polynomial $p$ is a continuous function in ${\Bbb C}$. This is a well known result in elementary Complex analysis.Usual norm $|\;|:{\Bbb C}{\rightarrow}{\mathbb{R}}$ is a continuous function also in ${\Bbb C}$, by elementary complex analysis. So, the composition function $F:{\Bbb C}{\rightarrow}{\mathbb{R}}$, $F=|\cdot|\circ p$, where $p:{\Bbb C}{\rightarrow}{\Bbb C}$, $|\cdot|:{\Bbb C}{\rightarrow}{\mathbb{R}}$ with formula\linebreak $F(z)=(|\cdot|\circ p)(z)=|p(z)|$ for every $z\in{\Bbb C}$ is a continuous functions as the composition of continuous functions $|\cdot|$ and $p$. Applying now Theorem 4.4 for $K=\overline{D(0,R_0)}$ and $f=F$ we take it that function $F$ attains its infimum is some point $z_0\in\overline{D(0,R_0)}$. This means that $|p(z_0)|=m^\ast_{R_0}$ (2). By (1) and (2) we have $m=|p(z_0)|$ (3). We argue that $m=0$. To take a contradiction we suppose that $m>0$. Because $|p(z_0)|=m>0$, we see that $p(z_0)\neq0$.
We consider polynomial $Q(z)=\dfrac{p(z+z_0)}{p(z_0)}$, that is defined well because $p(z_0)\neq0$.
Applying Lemma 4.1 we see that $deg p(z+z_0)=deg p(z)=n$, and by definition of $Q(z)$, we get: $deg Q(z)=n$. We have $Q(0)=\dfrac{p(0+z_0)}{p(z_0)}=1$, so polynomial $Q(z)$ has constant term equal to 1.
Let
\[ Q(z)=1+c_kz^k+\cdots+c_nz^n, \ \ c_n\neq0, \ \ \text{for every} \ \ z\in{\Bbb C}, \ \ \text{where} \ \ k\in{\Bbb N}, \ \ 1\le k\le n \]
and $k$ be the smallest natural number such that $c_k\neq0$, (maybe $k=n$ of course).
So, we get: $-|c_k|/c_k\neq0$. From Remark 4.3 there exists $j\in{\Bbb C}$, so that\linebreak$j^k=-|c_k|/c_k$ (4). (Of course there are $k$ different complex numbers such that (4) holds). By (4) we take $|j^k|=|-|c_k|/c_k|=1\Rightarrow|j|=1$ (5).
By choice of $j$ we have: for $r\in{\Bbb C}$ $|1+c_kr^kj^k|\overset{(4)}{=}|1+c_kr^k\cdot(-|c_k|/c_k)=1-|c_k|r^k$ (6).
By definition of $Q(z)$ we compute for $z=rj$ for $r\in{\Bbb C}$:
\begin{align*} Q(z)&=Q(rj)=1+c_k(rj)^k+\cdots+c_n(rj)^n\\ &=1+c_kr^kj^k+c_{k+1}r^{k+1}j^{k+1}+\cdots+ c_nr^nj^n. \hspace*{4.4cm} (7) \end{align*}
By (7) and triangle inequality we get:
\[
|Q(rj)|\le|1+c_kr^kj^k|+|c_{k+1}r^{k+1}j^{k+1}|+\cdots+|c_nr^nj^n|. \eqno{\mbox{(8)}} \]
Applying (6) we get by (8)
\begin{align*}
|Q(rj)|&\le1-|c_k||r^k|+|c_{k+1}||r|^{k+1}+\cdots+|c_n||r^n| \\
&=1-|r^k|(|c_k|-|c_{k+1}||r|-\cdots-|c_n||r|^{n-k}), \ \ \text{for every} \ \ r\in{\Bbb C}\hspace*{1.3cm} (10) \end{align*}
By definition of $m$ we get:
\[
m\le|p(z+z_0)| \ \ \text{for every} \ \ z\in{\Bbb C}. \eqno{\mbox{(11)}} \]
By (3) and (11) we get:
\begin{align*}
|p(z_0)|\le|p(z+z_0)| \ \ &\text{for every} \ \ z\in{\Bbb C} \\
\Rightarrow \bigg|\frac{p(z+z_0)}{p(z_0)}\bigg|\ge1 \ \ &\text{for every} \ \ z\in{\Bbb C}\\
\Rightarrow |Q(z)|\ge1 \ \ &\text{for every} \ \ z\in{\Bbb C} \ \ \text{(by definition of $Q$)} \hspace*{3cm} (12) \end{align*}
Now, we distinguish two cases:
(i) $k=n$. Then, from (10) we get:
\[
|Q(rj)|\le1-|r|^k|c_k|, \eqno{\mbox{(11)}} \]
So, for every $r\neq0$, we get by (12)
\[
|Q(rj)|\ge 1 \ \ \text{and} \eqno{\mbox{(13)}} \]
\[
|Q(rj)|\le1-|r^k||c_k|<1 \eqno{\mbox{(14)}} \]
and we take a contradiction from (13) and (14).
(ii) $k<n$.
By properties of complex limits we get:
\[
\lim_{r\ra0}(|c_k|-|c_{k+1}||r|-\cdots-|c_n|r|^{n-k})=|c_k|>0. \]
This limit shows us that: there exists some small $r_0$ so that
\[
|c_k|-|c_{k+1}||r_0|-\cdots-|c_n||r_0|^{n-k}>0. \eqno{\mbox{(15)}} \]
We set $\thi:=|c_k|-|c_{k+1}||r_0|-\cdots-|c_n||r_0|^{n-k}$. So we have $\thi_0>0$. From (10) and (15) we get:
\[
|Q(r_0j)|\le1-|r_0|^k\thi_0<1. \eqno{\mbox{(16)}} \]
From (12) we get: $|Q(r_0j)|\ge1$ (17). By (16) and (17) we get a contradiction. So, our supposition that $m>0$ is false. So we have $m=0$ and from (3) we get $0=p(z_0)$, that is polynomial $p$ has, as a root number $z_0$. If $a_n\neq1$ we write: \\ $\dfrac{1}{a_n}p(z)=\dfrac{a_0}{a_n}+\dfrac{a_1}{a_n}z+\cdots+z^n$, and applying the previous result we take it that there exists some $w\in{\Bbb C}$ such that $\dfrac{1}{a_n}p(w)=0\Leftrightarrow p(w)=0$, so polynomial $p$ has a root again. The proof of fundamental Theorem has completed now. $\quad\blacksquare$ \end{proof}
\noindent
{\bf Remark 4.6.} {\em Inside our work we have used the well known binomial equation\linebreak $x^n=a$, where $a>0$. We remind how we solve this equation here, for $n\ge2$, $n\in{\Bbb N}$. We will distinguish two cases:
(i) $a>1$. We consider function $f:[1,a]{\rightarrow}{\mathbb{R}}$, with the formula $f(x)=x^n-a$ for every $x\in[1,a]$. We get $f(1)=1^n-a<1$, from our supposition and\\ $f(a)=a^n-a=a(a^{n-1}-1)>0$. So we have $f(1)\cdot f(a)<0$ and because $f$ is continuous we understand from Bolzano Theorem that there exists $x_0\in(1,a)$ so that: $f(x_0)=0\Leftrightarrow x^n_0-a=0\Leftrightarrow x_0=\sqrt[n]{a}$. Because $f$ is strictly increasing in $[1,a]$, (because $f'(x)=nx^{n-1}>0$ for every $x\in[1,a]$) equation $f(x)=0$ has unique root in $[1,a]$, that is number $\sqrt[n]{a}$. Applying bisection method we approximate number $\sqrt[n]{a}$, or in other words we solve the equation $x^n=a$.
(ii) $a\in(0,1)$. Then we apply the above procedure similarly to the function\linebreak $g:[0,1]{\rightarrow}{\mathbb{R}}$ with the formula $g(x)=x^n-a$, for every $x\in[0,1]$}.\vspace*{0.5cm} \\
\noindent
{\bf Acknowledgements:} Many thanks to Vasilli Karali for his contribution in the presentation of this paper.
\vspace*{1.5cm}
\noindent Nikos Tsirivas,\\
Department of Mathematics, University of Patras and\\
Department of Marine Engineering, University of West Attica.
\end{document} | arXiv |
2012 CMS Summer Meeting
Regina Inn and Ramada Hotels (Regina~Saskatchewan), June 2 - 4, 2012 www.cms.math.ca//Events/summer12
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Plenary, Prize and Public Lectures
Contributed Papers Session
Student Poster Session
By speaker
Overview Plenary, Prize and Public Lectures By session By speaker Posters
Homotopy Theory
Org: Kristine Bauer (Calgary) and Marcy Robertson (Western)
JULIE BERGNER, University of California, Riverside
Homotopy operads as diagrams [PDF]
Many algebraic structures on spaces can be encoded via product-preserving functors from an algebraic theory to the category of spaces. For some structures, the algebraic theory can be replaced by a simpler diagram. For example, simplicial monoids are known to be equivalent to Segal monoids, given by certain $\Delta^{op}$-diagrams. In joint work with Philip Hackney, we establish an equivalence of model categories between simplicial operads and certain $\Omega^{op}$-diagrams, where $\Omega$ is the Moerdijk-Weiss dendroidal category. Furthermore, we extend this result to a diagrammatic description of simplicial operads with a group action.
MICHAEL CHING, Amherst College
A classification of Taylor towers [PDF]
Goodwillie's homotopy calculus provides a systematic way to approximate a functor F between the categories of based spaces and/or spectra with a `Taylor tower' of polynomial functors. The layers in this tower can be described by a sequence of spectra which play the role of the derivatives of F (at the one-point object). The goal of this talk is to describe additional structure on these derivatives that specifies the extensions in the Taylor tower. This allows us to describe the polynomial approximations as derived mapping objects for coalgebras over certain comonads. I'll connect the structure of these comonads to that of right modules over various operads. This is all joint work with Greg Arone.
MARTIN FRANKLAND, University of Illinois at Urbana-Champaign
Non-realizable 2-stage $\Pi$-algebras [PDF]
It is a classic fact that Eilenberg-MacLane spaces exist and are unique up to weak equivalence. However, one cannot always find a space with two non-zero homotopy groups and prescribed primary homotopy operations. Using work of Baues and Goerss, we will present examples of non-realizable 2-stage $\Pi$-algebras, focusing on the stable range.
PHIL HACKNEY, University of California, Riverside
Homotopy theory of props [PDF]
Props have the capability to control algebraic structures more general than those described by operads; for example, there is a prop governing Hopf algebras and a prop governing conformal field theories. We study the category consisting of all (colored, simplicial) props. We show that this category is a closed symmetric monoidal category with tensor product closely related to the Boardman-Vogt tensor product of operads. Furthermore, this category admits a Quillen model structure which restricts to the model structure for (colored) operads developed by Robertson and to the Bergner model structure for simplicial categories. (joint with Marcy Robertson)
RICK JARDINE, University of Western Ontario
Cosimplicial spaces and cocycles [PDF]
Cosimplicial spaces were introduced by Bousfield and Kan in the early 1970s as a technical device in their theory of homology completions. These objects have since become fundamental tools in much of homotopy theory, but the original theory remains rather mysterious. The point of this talk is that cosimplicial spaces are quite amenable to study with modern methods of sheaf theoretic homotopy theory and cocycle categories. Non-abelian cohomology theory has a particularly interesting and useful interpretation in this context.
BRENDA JOHNSON, Union College
Models for Taylor Polynomials of Functors [PDF]
Let ${\mathcal C}$ and ${\mathcal D}$ be simplicial model categories. Let $f:A\rightarrow B$ be a fixed morphism in ${\mathcal C}$ and ${\mathcal C}_f$ be the category whose objects are pairs of morphisms $A\rightarrow X\rightarrow B$ in ${\mathcal C}$ that factor $f$. Using a generalization of Eilenberg and Mac Lane's notion of cross effect functors (originally defined for functors of abelian categories) to functors from ${\mathcal C}_f$ to ${\mathcal D}$, we produce a tower of functors, $\dots\rightarrow \Gamma _n^f F\rightarrow \Gamma _{n-1}^fF\rightarrow \dots \rightarrow \Gamma _0^fF$, that acts like a Taylor series for the functor $F$. We compare this to the Taylor tower for $F$ produced by Tom Goodwillie's calculus of homotopy functors, and use it to better understand the roles of the initial and final objects, $A$ and $B$, in the calculus of homotopy functors. This is joint work with Kristine Bauer, Rosona Eldred, and Randy McCarthy.
KEITH JOHNSON, Dalhousie University
Homogeneous integer valued polynomials and the stable homotopy of BU [PDF]
The use of homogeneous integer valued multivariable polynomials to detect elements in the stable homotopy groups of BU originated with work of Baker, Clarke, Ray and Schwartz (Trans. AMS 316(1989)). In this talk we will demonstrate some new constructions of rings of such polynomials and study their topological consequences.
DAN LIOR, University of Illinois, Urbana
The use of labelled trees in the Goodwillie-Taylor tower of discrete modules [PDF]
A discrete module is a functor of finite pointed sets taking values in chain complexes of abelian groups. For an arbitrary discrete module F, McCarthy, Johnson and Intermont described the first homogeneous layer $D_1F$ of the Goodwillie-Taylor tower of F in terms of the cross effects of F and the multilinear parts of finitely generated free Lie algebras. I will describe a category of trees which illustrates this connection and extends it to the rest of the layers $D_nF$ of the Goodwillie-Taylor tower of F.
PARKER LOWREY, University of Western Ontario
The derived motivic Hall algebra associated to a projective variety. [PDF]
We discuss how to associate a locally geometric derived moduli stack classifying objects in the bounded derived category associated to any projective variety. This is the main ingredient needed in defining a Hall algebra for this triangulated category. It extends the work of To\"en, Kontsevich, and Soibelman to the singular case and is the first step in applying Donaldson-Thomas theory (and Joycey's extensions) to these homologically unwieldy categories. The talk will contain a good deal of algebro-geometric and homotopy theoretic material.
HUGO RODRIGUEZ ORDONEZ, Universidad Autónoma de Aguascalientes
Dimensional restrictions upon counterexamples to Ganea's conjecture [PDF]
The long standing conjecture by Ganea on the Lusternik-Schnirelmann category was disproved in the late 1990s by means of a family of counterexamples whose least dimensional element has dimension 10. In a previous work, the authors proved that there is a 7-dimensional counterexample. In this work, we present a proof that there is no counterexample to this conjecture with dimension 6 or less. This is joint work with Don Stanley.
SIMONA PAOLI, University of Leicester
n-fold groupoids and n-types [PDF]
Most homotopy invariants of topological spaces are filtered by dimension, so it is useful to have finite dimensional approximations to homotopy theories. We describe an algebraic model for the latter, which we call n-track categories. An appropriate algebraic model of n-types is developed for this purpose, with a class of n-fold groupoids which we call n-typical. This model leads to an explicit connection between homotopy types and iterated loop spaces and exhibit other useful properties. This is joint work with David Blanc.
DORETTE PRONK, Dalhousie University
Bredon Cohomology with Local Coefficients [PDF]
Bredon [1] defined his version of equivariant cohomology with constant coefficients for spaces with an action of a discrete group $G$. This was generalized to arbitrary topological groups by Illman [2]. This definition was then extended to local coefficient systems independently by Moerdijk and Svensson [3] and by the Mukherjees [4]. Moerdijk and Svensson's approach was only applicable to discrete groups and used the cohomology of a category constructed to represent the $G$-space. The Mukherjees' approach was closer to the work by Illman. Mukherjee and Pandey [5] showed that the two definitions agree when the group $G$ is discrete.
Laura Scull and I have generalized the construction of the category given by Moerdijk and Svensson to $G$-spaces for an arbitrary toplogical group $G$. We will show that the resulting definition of Bredon cohomology agrees with the one given by the Mukherjees. As an application we get the Serre spectral sequence in the more general setting of a topological group $G$.
[1] G.E. Bredon, {\em Introduction to Compact Transformation Groups}, Academic Press (1972).
[2] S. Illman, Equivariant Singular Homology and Cohomology, Bull. AMS 79 (1973) pp. 188–192.
[3] I. Moerdijk, J.-A. Svensson, The equivariant Serre spectral sequence, {\em Proceedings of the AMS} 118 (1993), pp. 263–278.
[4] A. Mukherjee, G. Mukherjee, Bredon-Illman cohomology with local coefficients, {\em Quart. J. Math. Oxford} 47 (1996), pp. 199-219.
[5] Goutam Mukherjee, Neeta Pandey, Equivariant cohomology with local coefficients, {\em Proceedings of the AMS} 130 (2002), pp. 227-232.
MARCY ROBERTSON, University of Western Ontario
On Topological Triangulated Orbit Categories [PDF]
In 2005, Keller showed that the orbit category associated to the bounded derived category of a hereditary category under an auto equivalence is triangulated. As an application he proved that the cluster category is triangulated. We show that this theorem generalizes to triangulated categories with topological origin (i.e. the homotopy category of a stable model category). As an application we construct a topological triangulated category which models the cluster category. This is joint work with Andrew Salch.
DON STANLEY, University of Regina
Homotopy invariance of configuration spaces [PDF]
Given a closed manifold $M$, the configuration space of $n$ points in $M$, $F(M,k)$ is the set $k$ distinct points in $M$. Levitt showed that if $M$ is $2$-connected then $F(M,2)$ only depends on the homotopy type of $M$. When $M$ is a smooth projective variety, Kriz constructed a model for the rational homotopy type of $F(M,k)$. In this talk we show that a variant of the Kriz model works for any sufficiently connected closed manifold, and discuss the related problem of the homotopy invariance of $F(M,3)$.
SEAN TILSON, Wayne State University
Power Operations and the Kunneth Spectral Sequence [PDF]
Power operations have been constructed and successfully utilized in the Adams and Homological Homotopy Fixed Point Spectral Sequences by Bruner and Bruner-Rognes. It was thought that such results were not specific to the spectral sequence, but rather that they arose because highly structured ring spectra are involved. In this talk, we show that while the Kunneth Spectral Sequence enjoys some nice multiplicative properties, there are no non-zero algebraic operations in $E_2$ (other than the square). Despite the negative results we are able to use old computations of Steinberger's with our current work to compute operations in the homotopy of some relative smash products.
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Hilbert–Schmidt theorem
In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems.
Statement of the theorem
Let (H, ⟨ , ⟩) be a real or complex Hilbert space and let A : H → H be a bounded, compact, self-adjoint operator. Then there is a sequence of non-zero real eigenvalues λi, i = 1, …, N, with N equal to the rank of A, such that |λi| is monotonically non-increasing and, if N = +∞,
$\lim _{i\to +\infty }\lambda _{i}=0.$
Furthermore, if each eigenvalue of A is repeated in the sequence according to its multiplicity, then there exists an orthonormal set φi, i = 1, …, N, of corresponding eigenfunctions, i.e.,
$A\varphi _{i}=\lambda _{i}\varphi _{i}{\mbox{ for }}i=1,\dots ,N.$
Moreover, the functions φi form an orthonormal basis for the range of A and A can be written as
$Au=\sum _{i=1}^{N}\lambda _{i}\langle \varphi _{i},u\rangle \varphi _{i}{\mbox{ for all }}u\in H.$
References
• Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. pp. 356. ISBN 0-387-00444-0. (Theorem 8.94)
• Royden, Halsey; Fitzpatrick, Patrick (2017). Real Analysis (Fourth ed.). New York: MacMillan. ISBN 0134689496. (Section 16.6)
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| Wikipedia |
Minimum Cost Path Graph
Let S be the set of vertices whose minimum distance from the source vertex has been found. The following table lists the Port Cost value for different bandwidths. The shortest path computed in the reduced cost graph is the same as the shortest path in the original graph. Steps for finding a minimum-cost spanning tree using _____ Algorithm: Add edges in order of cheapest cost so that no circuits form. The next lowest cost. I am looking to run MST on some vectors with geometries and cost. In this graph, cost of an edge (i, j) is represented by c(i, j). Your program will either return a sequence of nodes for a minimum-cost path or indicate that no solution exists. Minimum weight perfect matching problem: Given a cost c ij for all (i,j) ∈ E, find a perfect matching of minimum cost where the cost of a matchinPg M is given by c(M) = (i,j)∈M c ij. A minimum-cost spanning tree is one which has the smallest possible total weight (where weight represents cost or distance). This problem is concerned with finding the cheapest path between verticesa and b in a graph G = (V,E). the cost-minimized pathfrom the seed point to the goal point. Dynamic Graph Clustering Using Minimum-Cut Trees 1 Introduction Graph clustering has become a central tool for the analysis of networks in gen-eral, with applications ranging from the eld of social sciences to biology and to the growing eld of complex systems. Given a weighted graph, find the maximum cost path from given source to destination that is greater than a given integer x. for example, if your answer is 1 write 1 without decimal points. Initially, this quantity is infinity (i. the total intuitionistic fuzzy cost for traveling through the shortest path. $\begingroup$ Hint: both shortest path and min-cost flow determine the minimum of a sum. We will use Prim's algorithm to find the minimum spanning tree. Version 05/03/2011 Minimum Spanning Trees & Shortest Path—Graph Theory ©2013 North Carolina State University Chapter 8 – Page 1 Section 8. Given a path state state of type AbstractPathState, return a vector (indexed by vertex) of the paths between the source vertex used to compute the path state and a single destination vertex, a list of destination vertices, or the entire graph. NOTE: This algorithm really does always give us the minimum-cost spanning tree. This example. The distance from a vertex v i to a vertex v j in G is the minimum cost over all paths from v i to v j in G denoted by d∗ ij. a) Suppose that each edge in the graph has a weight of zero. particular, this package provides solving tools for minimum cost spanning tree problems, minimum cost arborescence problems, shortest path tree problems and minimum cut tree problem. can u much detail abt this…its very helpful to me…. Tarjan Princeton and HP Labs Abstract Consider a bipartite graph G= (X;Y;E) with real-valued weights on its edges, and suppose that Gis balanced, with jXj= jYj. In this case, we start with single edge of graph and we add edges to it and finally we get minimum cost tree. along path p; and (2) path p has the minimum cost (toll fee) among all the paths satisfying the condition (1). Verify that your result is a maximum or minimum value using the first or second derivative test for extrema. Two vertices are adjacent when they are both incident to a common edge. Dijkstra's algorithm is a graph search algorithm that can solve the single-source shortest path problem for a graph with non- negative edge path cost, outputting a shortest path tree. A number of algorithms have been proposed to enumerate all spanning trees of an undirected graph. Figure 1: Example of a shortest path problem and its mapping to the minimum cost ow model 1. path scheduling with all activity durations assumed to be at minimum cost. A) $2100 B) $2400 C) $2900 D) $6200. Find a minimum cost spanning tree on the graph below using Kruskal's algorithm. The cost is determined depending upon the criteria to be optimized. The following will run the k-maximum spanning tree algorithm and write back results: MATCH (n:Place{id:"D"}) CALL algo. The Prim's algorithm operates on two disjoint sets of edges in the graph. This paper is organized as follows: In Section 2, we provide background ma- terial on network flow algorithms and present some preliminary results. We can, therefore define function for any V,. Long-run average cost (LRAC) curve is a graph that plots average cost of a firm in the long-run when all inputs can be changed. Removes the connection between the specified origin node and the specified destination node Keep in mind that this only removes the connection in one direction, for undirected graphs, the function must be called again with the destination node as the origin. Given a vertex s in graph G, find the shortest path from s to every other vertex in G A C B E 10 3 15 5 2 11 D 20 Closely related problem is to find the shortest (minimum cost) path between two nodes of a graph. The well-known basic problem concerns finding the shortest paths in graphs given any set of journeys and a weighted, connected graph. We analyze the problem of finding a minimum cost path between two given vertices such that the vector sum of all edges in the path equals a given target vector m. 2) Areas less than the minimum core habitat percentage times the area of the foraging radius are eliminated 3) A cost surface is created from the habitat quality raster, cells of high quality have a low cost and vise versa 4) The remaining patches are grown outwards across the cost surface to a distance equal to the foraging radius. of Gsuch that the undirected version of T is a tree and T contains a directed path from rto any other vertex in V. hk Ruifeng Liu Chinese University of Hong Kong rfl[email protected] The minimum cost spanning tree (MST) Spanning tree: is a free tree that connects all the vertices in V • cost of a spanning tree = sum of the costs of the edges in the tree Minimum spanning tree property: • G = ( V, E): a connected graph with a cost function defined on the edges; U ⊆V. Part 3: Remember that we are suppose to find the point (x,y) on the graph of the parabola, y = x 2 + 1, that minimizes d. Optimal Discounted Cost in Weighted Graphs Ashutosh Trivedi = Cost(ˇ): | Now consider the path ˇ we write DCost(v) as minimum discounted cost of all in nite. A connected graph with no circuits, called trees, are also discussed in this chapter. The idea is to start with an empty graph and try to add edges one at a time, always making sure that what is built remainsacyclic. In this way, each distant node influences the cen-ter node through a path connecting the two with minimum cost, providing a robust estimation and intact exploration of the graph structure. As A* traverses the graph, it follows a path of the lowest known cost, Keeping a sorted priority queue of alternate path segments along the way. Here is my Graph class that implements a graph and has nice a method to generate its spanning tree using Kruskal's algorithm. And here comes the definition of an AI agent. graph find a minimum cost to find the shortest path between two points. Total cost of a path to reach (m, n) is sum of all the costs on that path (including both source and destination). Specifically distance[v] stores the minimum distance so far from the source vertex s to some other vertex v. min_cost_flow (G[, demand, capacity, weight]) Return a minimum cost flow satisfying all demands in digraph G. In essence, the planner develops a list of activities on the critical path ranked with their cost slopes. The weight of a shortest path tree. The path is (0, 0) –> (0, 1) –> (1, 2) –> (2, 2). {Each node has a value b(v). For a given source vertex (node) in the graph, the algorithm finds the path with lowest cost (i. satisfaction) problems with costs. [Tree, pred] = graphminspantree(G) finds an acyclic subset of edges that connects all the nodes in the undirected graph G and for which the total weight is minimized. Eulerization of a graph is the process of finding an Euler circuit for that graph. It should return and integer that represents the minimum weight to connect all nodes in the graph provided. Initially, this quantity is infinity (i. the distance of the right path (between robot 3's vertex and 2's goal) be x 1. A simple graph with (a) a face-spanning subgraph of cost 11 and (b) another face-spanning subgraph of cost 13. To find the path, the image is first modeled as a graph. Such a route is easily obtained by a breadth first search method. It adds one more node in each iteration to the minimum cost spanning tree. Steps for finding a minimum-cost spanning tree using _____ Algorithm: Add edges in order of cheapest cost so that no circuits form. Describe and analyze an e cient algorithm for nding a minimum-cost monotone path in such a graph, G. An edge is the line segment connecting two nodes and has the same length in either direction. Prim's Algorithm. Proof: Consider any path from sto some node t. Tarjan Princeton and HP Labs Abstract Consider a bipartite graph G= (X;Y;E) with real-valued weights on its edges, and suppose that Gis balanced, with jXj= jYj. minimum cost on the section from s to t, which makes the max-flow also min-cost. We then will see how the basic approach of this algorithm can be used to solve other problems including finding maximum bottleneck paths and the minimum spanning tree (MST) problem. In the maze defined, True are the open ways. Both search methods can be used to obtain a spanning tree of the graph, though if I recall correctly, BFS can also be used in a weighted graph to generate a minimum cost spanning tree. • Route signals along minimum cost path • If congestion/overuse – assign higher cost to congested resources • Makes problem a shortest path search • Allows us to adapt costs/search to problem • Repeat until done Penn ESE 535 Spring 2015 -- DeHon 26 Key Idea • Congested paths/resources become expensive. Let G = (V; E) be an (undirected) graph with. Use Kruskal's algorithm for minimum-cost spanning trees on the graph below. Suppose you are given a connected graph G, with edge costs that are all distinct. Finding minimum cost to visit all vertices in a graph and returning back. path scheduling with all activity durations assumed to be at minimum cost. Before increasing the edge weights, shortest path from vertex 1 to 4 was through 2 and 3 but after increasing Figure 1: Counterexample for Shortest Path Tree the edge weights shortest path to 4 is from vertex 1. the total intuitionistic fuzzy cost for traveling through the shortest path. ) In this context, given an input graph G, one seeks a homomorphism f of G to H with minimum cost, i. Each cell of the matrix represents a cost to traverse through that cell. A minimum spanning tree of an undirected graph can be easily obtained using classical algorithms by Prim or Kruskal. Now, let's ask: what's the shortest path cost to, say, Ben and Jerry's? * To move to weighted graphs, we appeal to the mighty power of Dijkstra. Using this answer, by finding the minimum cost closed walk (or just it's cost) of an arbitrary 4-regular planar graph, with weights 1, we can decide whether it has a Hamiltonian Path, but this problem is NP-complete. As I stand now I'm using DFS, and it's pretty slow (high number of nodes and maximum length too). For positive edge weight graphs. NOTE: This algorithm really does always give us the minimum-cost spanning tree. The cost is determined depending upon the criteria to be optimized. This is the single-source minimum-cost paths problem. (eds) Graph Based Representations in Pattern Recognition. ° Among all the spanning trees of a weighted and connected graph, the one (possibly more) with the least total weight is called a minimum spanning tree (MST). shortest path finding as a complete graph G (V, E). The multistage graph problem is finding the path with minimum cost from source s to sink t. A spanning tree of a graph is a tree that has all the vertices of the graph connected by some edges. Dijkstra solves the shortest path problem (from a specified node), while Kruskal and Prim finds a minimum-cost spanning tree. Now any positive value (>0) may exist on each edge. Suppose we have a weighted graph G = (V, E, c), where V is the set of vertices, E is the set of arcs, and. along path p; and (2) path p has the minimum cost (toll fee) among all the paths satisfying the condition (1). There can be many spanning trees. A minimum directed spanning tree (MDST) rooted at ris a directed spanning tree rooted at rof minimum cost. Next, the planner can examine activities on the critical path and reduce the scheduled duration of activities which have the lowest resulting increase in costs. replacing the edge weights with ((LCM of all edges)/(weight of the edge)) makes the longest edge as smallest and smallest edge as longest. Minimum Spanning Tree Problem We are given a undirected graph (V,E) with the node set V and the edge set E. A number of problems from graph theory are called Minimum spanning tree. A minimum directed spanning tree (MDST) rooted at ris a directed spanning tree rooted at rof minimum cost. // Indexed in order of stages E is a set of edges. Assuming that you don't expect the paths to be more than 1000 steps long, you can choose p = 1/1000. There also can be many minimum spanning trees. Minimum Cost flow problem is a way of minimizing the cost required to deliver maximum amount of flow possible in the network. As mentioned there, grid problem reduces to smaller sub-problems once choice at the cell is made, but here move will be in reverse direction. ing tree, or a shortest path. Each unit of flow on each arc corresponds to onepathgoingthroughthatnode. A circuit that uses every edge exactly once is an Euler circuit. The cost w(T) of a directed spanning tree Tis the sum of the costs of its edges, i. The program must be possible to open these files (check the format). However, when we add a 2 to this graph, we penalize longer paths, so the shortest path from a to d is a !b !d. Version 05/03/2011 Minimum Spanning Trees & Shortest Path—Graph Theory ©2013 North Carolina State University Chapter 8 – Page 1 Section 8. The shortest path problem can be defined for graphs whether undirected, directed, or mixed. Successive Shortest Path Algorithm for the Minimum Cost Flow Problem in Dynamic Graphs MathildeVernet1,MaciejDrozdowski2,YoannPigné1,EricSanlaville1 1Normandie Univ, UNIHAVRE, UNIROUEN, INSA Rouen, LITIS, 76600 Le Havre, France. Graph Magics - an ultimate software for graph theory, having many very useful things, among which a strong graph generator and more than 15 different algorithms that one may apply to graphs (ex. An Extended Path Following Algorithm for Graph-Matching Problem Zhi-Yong Liu, Hong Qiao,Senior Member, IEEE, and Lei Xu,Fellow, IEEE Abstract—The path following algorithm was proposed recently to approximately solve the matching problems on undirected graph models and exhibited a state-of-the-art performance on matching accuracy. I fear you will have to work a little and make a program that check every possible path until you find the one with minimum cost. Metanet is a toolbox of Scilab for graphs and networks computations. On a graph, transportation problems can be used to express challenging tasks involving matching supply to demand with minimal shipment expense; in discrete language, these become minimum-cost network ow problems. This contradicts the maximality of M. A path that uses every edge exactly once is an Euler path. Connected graph: a path exists between every pair of find lowest cost set of roads to repair so that all cities are connected This is a minimum spanning tree. The path to reach (m, n) must be through one of the 3 cells: (m-1, n-1) or (m-1, n) or (m, n-1). applying IG-ÿskal's algorithm for finding a minimum-cost spanning tree for a graph. It adds one more node in each iteration to the minimum cost spanning tree. Abstract: We design and analyse approximation algorithms for the minimum-cost connected T-join problem: given an undirected graph G = (V;E) with nonnegative costs on the edges, and a subset of nodes T, find (if it exists) a spanning connected subgraph H of minimum cost such that every node in T has odd degree and every node not in T has even degree; H may have multiple copies of any edge of G. Minimum weight perfect matching problem: Given a cost c ij for all (i,j) ∈ E, find a perfect matching of minimum cost where the cost of a matchinPg M is given by c(M) = (i,j)∈M c ij. Shortest Distance Problems in Graphs Using History-Dependent Transition Costs with Application to Kinodynamic Path Planning Raghvendra V. Fig 1: This graph shows the shortest path from node "a" or "1" to node "b" or "5" using Dijkstras Algorithm. This paper involved in illustrating the best way to travel between two points and in doing so, the shortest path algorithm was created. If there is more than one minimum cost path from v to w, will Dijkstra's algorithm always find the path with the fewest edges? If not, explain in a few sentences how to modify Dijkstra's algorithm so that if there is more than one minimum path from v to w, a path with the fewest edges is chosen. @ inding an Euler circuit on a graph. Our results are based on a new approach to speed-up augmenting path based matching algorithms, which we describe next. Minimum Spanning Tree Problem We are given a undirected graph (V,E) with the node set V and the edge set E. Dijkstra's algorithm Like BFS for weighted graphs. This model consid-ers the phenomenon that some vehicles may choose to stop at some places to avoid tra c jams. Graph search algorithms explore a graph either for general discovery or explicit search. Similar is story for vertex C. Can you move some of the vertices or bend. 8 If the graph is directed it is possible for a tree of shortest paths from s and a minimum spanning tree in G. Each cell of the matrix represents a cost to traverse through that cell. Handout MS2: Midterm 2 Solutions 2 eb, we obtain a new spanning tree for the original graph with lower cost than T, since the ordering of edge weights is preserved when we add 1 to each edge weight. In this case, as well, we have n-1 edges when number of nodes in graph are n. Consider an undirected graph containing nodes and edges. Using this answer, by finding the minimum cost closed walk (or just it's cost) of an arbitrary 4-regular planar graph, with weights 1, we can decide whether it has a Hamiltonian Path, but this problem is NP-complete. In this Java Program first we input the number of nodes and cost matrix weights for the graph ,then we input the source vertex. The Dijkstra's algorithm gradually builds a short path tree using links in the network. This week we finish our look at pathfinding and graph search algorithms, with a focus on the Minimum Weight Spanning Tree algorithm, which calculates the paths along a connected tree structure with the smallest value (weight of the relationship such as cost, time or capacity) associated with visiting all nodes in the tree. Additionally, the graph is expected to have very few edges, so the average degree is very small. BANSAL Department of Mathematics, A. Need the graph to be connected, and minimize the cost of laying the cables. Unlike the situation where you're trying to find a minimum cost Hamilton circuit, there is an algorithm. of biconnected graph a linear cost function on the face cycles. Steiner tree problem or so called Steiner Problem in Graphs (SPG) is a classic combinatorial optimization problem. The minimum-cost spanning tree produced by applying Kruskal's algorithm will always contain the lowest cost edge of the graph. Minimum spanning tree is a tree in a graph that spans all the vertices and total weight of a tree is minimal. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest paths in G. By Ion Cozac. i have an adjacency list representation of a graph for the problem, now i am trying to implement dijkstra's algorithm to find the minimum cost paths for the 'interesting cities' as suggested by @Kolmar. cost a configuration that is a Nash Equilibrium can get in total cost to the minimum solution. In order to be able to run this solution, you will need. ) In this context, given an input graph G, one seeks a homomorphism f of G to H with minimum cost, i. e the Global Processing Via Graph Theoretic technique and comes in sem 7 exams. This paper contains two similar theorems giving con-ditions for a minimum cover and a maximum matching of a graph. If all edge lengths are equal, then the Shortest Path algorithm is equivalent to the breadth-first search algorithm. The Minimum Weight Spanning Tree excludes the relationship with cost 6 from D to E, and the one with cost 3 from B to C. Kruskal's algorithm is used for finding a minimum cost spanning tree. {positive b(v) is a supply {negative b(v) is a demand. Hence, the cost of path from source s to sink t is the sum of costs of each edges in this path. A minimum spanning tree of an undirected graph can be easily obtained using classical algorithms by Prim or Kruskal. We transform an dependency attack graph into a Boolean formula and assign cost metrics to attack variables in the formula, based on the severity metrics. We have to go from A to B. This is the single-source shortest paths problem. A number of algorithms have been proposed to enumerate all spanning trees of an undirected graph. Given an n-d costs array, this class can be used to find the minimum-cost path through that array from any set of points to any other set of points. To find minimum cost at cell (i,j), first find the minimum cost to the cell (i-1, j) and cell (i, j-1). Nota: Java knows the length of arrays, in. [costs] is an LxM matrix of minimum cost values for the minimal paths [paths] is an LxM cell containing the shortest path arrays [showWaitbar] (optional) a scalar logical that initializes a waitbar if nonzero. hey, I am trying to find the cost or the length of the path by the following code. The cost reduction strategy converts an existing ow into a ow of lower cost by nding negative cost cycles in the residual graph and adding ow to those cycles. The goal is to obtain an Eulerian Path that has a minimal total cost. – fsociety May 11 '15 at 7:43. , there exist a path from ) •Breadth-first search •Depth-first search •Searching a graph -Systematically follow the edges of a graph to visit the vertices of the graph. the distance of the right path (between robot 3's vertex and 2's goal) be x 1. Assume that $ C_v = 1 $ for all vertices $ 1 \leq v \leq n $ (i. particular, this package provides solving tools for minimum cost spanning tree problems, minimum cost arborescence problems, shortest path tree problems and minimum cut tree problem. [1] There are some theorems that can be used in specific circumstances, such as Dirac's theorem, which says that a Hamiltonian circuit must exist on a graph with n vertices if each vertex has degree n /2 or greater. Finally, we open the black box in order to generalize a recent linear-time algorithm for multiple-source shortest paths in unweighted undirected planar graphs to work in arbitrary orientable surfaces. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest paths in G. Continue until every vertex is on some-edge you have chosen. acyclic graphs (DAGs) and propose structured sparsity penalties over paths on a DAG (called "path coding" penalties). We propose search several fast algorithms, which allow us to define minimal time cost path and minimal cost path. 2-vertex-connected subgraphs of low cost; no previous approximation algorithm was known for either problem. hk ABSTRACT The computation of Minimum Spanning Trees (MSTs) is a funda-mental graph problem with. Minimum spanning tree Given a connected graph G = (V, E) with edge weights c e, an MST is a subset of the edges T ⊆ E such that T is a spanning tree whose sum of edge weights is minimized. Find a minimum cost spanning tree on the graph below using Kruskal's algorithm. Conclusion We have to study the minimum cost spanning tree using the Prim's algorithm and find the minimum cost is 99 so the final path of minimum cost of spanning is {1, 6}, {6, 5}, {5, 4}, {4, 3}, {3, 2}, {2, 7}. Now I relax all edges leaving B,and set path cost of C as 3, and path cost to e as -2. In this chapter, we consider four specific network models—shortest-path prob-lems, maximum-flow problems, CPM-PERT project-scheduling models, and minimum-spanning. In this paper, we address this issue for directed acyclic graphs (DAGs) and propose structured sparsity penalties over paths on a DAG (called "path coding" penalties). Algorithms in graphs include finding a path between two nodes, finding the shortest path between two nodes, determining cycles in the graph (a cycle is a non-empty path from a node to itself), finding a path that reaches all nodes (the famous "traveling salesman problem"), and so on. Lecture notes on bipartite matching 3 Theorem 2 A matching M is maximum if and only if there are no augmenting paths with respect to M. Repeatedly augment along a minimum -cost augmenting path. Minimum spanning tree. graph find a minimum cost to find the shortest path between two points. Part 3: Remember that we are suppose to find the point (x,y) on the graph of the parabola, y = x 2 + 1, that minimizes d. A logarithmic algorithm for the minimum path problem in Knödel graphs is an open problem despite the fact that they are bipartite and highly symmetric. MCP ¶ class skimage. In this graph, cost of an edge (i, j) is represented by c(i, j). A tax of 1 cent per mile on commercial trucks' travel would have raised $2. Weighted Shortest Path Problem Single-source shortest-path problem: Given as input a weighted graph, G = ( V, E ), and a distinguished starting vertex, s, find the shortest weighted path from s to every other vertex in G. • Total cost: C = C(v, w, q) Minimum Total Cost is a function of input prices and output quantity. i have an adjacency list representation of a graph for the problem, now i am trying to implement dijkstra's algorithm to find the minimum cost paths for the 'interesting cities' as suggested by @Kolmar. First, we identify optimality conditions, which tell us when a given perfect matching is in fact minimum. One classical model that has resurfaced in many multi-assembly methods (e. This is not a trivial problem, because the shortest path may not be along the edge (if any) connecting two vertices, but rather may be along a path involving one or more intermediate vertices. Must Read: C Program To Implement Kruskal's Algorithm Every vertex is labelled with pathLength and predecessor. true Suppose a veteran is planning a visit to all the war memorials in Washington, D. In this section we shall show how to find a minimum-cost spanning tree for G. The multistage graph problem is finding the path with minimum cost from source s to sink t. A typical application for minimum-cost spanning trees occurs in the design of communications networks. Suppose that each edge in the graph has a weight of zero (while non-edges have a cost of $ \infty $ ). The output is either a single float (when a single vertex is provided) or a vector of floats corresponding to the vertex vector. In this paper, we implemented two graph theory methods that extend the least-cost path approach: the Conditional Minimum Transit Cost (CMTC) tool and the Multiple Shortest Paths (MSPs) tool. {Find ow which satis es supplies and demands and has minimum total cost. It can be said as an extension of maximum flow problem with an added constraint on cost(per unit flow) of flow for each edge. // Indexed in order of stages E is a set of edges. Shortest Path using Dijkstra's Algorithm is used to find Single Source shortest Paths to all vertices of graph in case the graph doesn't have negative edges. As you can probably imagine, larger graphs have more nodes and many more possibilities for subgraphs. Generic approach: A tree is an acyclic graph. Consider a connected undirected graph G with not necessarily distinct edge costs. The cost of this spanning tree is (5 + 7 + 3 + 3 + 5 + 8 + 3 + 4) = 38. In case it can, what would be the minimum cost of such path?. Kruskal's algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. Prim's Algorithm. scrolling computer game in terms of nding a minimum-cost monotone path in the graph, G, that represents this game. Minimum Spanning Tree Problem Find a minimum-cost set of edges that connect all vertices of a graph at lowest total cost Applications Connecting "nodes" with a minimum of "wire" Networking Circuit design Collecting nearby nodes Clustering, taxonomy construction Approximating graphs Most graph algorithms are faster on trees. A tax of 1 cent per mile on commercial trucks' travel would have raised $2. I want to: Make it pythonic Improve readability Improve the abstracti. A third is in the process of obtaining a subset of the overall graph, called a Spanning Tree which connects every desired node with a path, but has no paths which can start and end on the same node (such a path is called a cycle). 3: Computing the Single Source Shortest Path in a graph. of biconnected graph a linear cost function on the face cycles. It's important to be acquainted with all of these algorithms - the motivation behind them, their implementations and applications. G is usually assumed to be a weighted graph. Both of these conditions depend on the concept of an alternating path, due to Petersen [2]. These fun activities will help students learn how to read and gather data and use tables to create. Looks at the successors of the current lowest cost vertex in the wavefront. If G is a weighted graph, then the minimum spanning tree Span(G) is the spanning tree over G with minimum weight. if there are multiple edges, keep the lowest cost one Iedge weights are g(x t;u t) Iadd additional target vertex z with an edge from each x 2X T with weight g T (x) Ia sequence of actions is a path through the unrolled graph from x 0 to z Iassociated objective is total, weighted path length 4. In this case, we start with single edge of graph and we add edges to it and finally we get minimum cost tree. To find minimum cost at cell (i,j), first find the minimum cost to the cell (i-1, j) and cell (i, j-1). Minimum spanning tree is the spanning tree where the cost is minimum among all the spanning trees. This model consid-ers the phenomenon that some vehicles may choose to stop at some places to avoid tra c jams. 32 to just. The first distinction is that Dijkstra's algorithm solves a different problem than Kruskal and Prim. Dijkstra's algorithm (also called uniform cost search) - Use a priority queue in general search/traversal. Conclusion We have to study the minimum cost spanning tree using the Prim's algorithm and find the minimum cost is 99 so the final path of minimum cost of spanning is {1, 6}, {6, 5}, {5, 4}, {4, 3}, {3, 2}, {2, 7}. We transform an dependency attack graph into a Boolean formula and assign cost metrics to attack variables in the formula, based on the severity metrics. Given a graph, the start node, and the goal node, your program will search the graph for a minimum-cost path from the start to the goal. The cost of a path is the sum of the costs of the edges and vertices encountered on the path. Weighted Graphs Data Structures & Algorithms 3 [email protected] ©2000-2009 McQuain Dijkstra's SSAD Algorithm* We assume that there is a path from the source vertex s to every other vertex in the graph. The limitation of this type of analysis is that only a single path is identified, even though alternative paths with comparable costs might exist. It works for non-loopy mazes which was already my goal. The cost of the spanning tree is the sum of the weights of all the edges in the tree. Repeatedly augment along a minimum -cost augmenting path. In this paper, the time dependent graph is presented. MCP(costs, offsets=None, fully_connected=True)¶. A heuristic is admissible if for any node, n, in the graph, the heuristic estimate of the cost of the path from n to t is less than or equal to the true cost of that path. There are nn–2 spanning trees of K n. (2)Then I process vertex b, and it is now included in S as it's shortest path from source is determined. The goal of the proposal is to obtain an optimal path with the same cost as the path returned by Dijkstra's algorithm, for the same origin and destination, but using a reduced graph. Maintains a cost to visit every vertex. 4 Problem 5. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Dijkstra's Algorithm [2] successfully finds the lowest cost path for each journey. [1] There are some theorems that can be used in specific circumstances, such as Dirac's theorem, which says that a Hamiltonian circuit must exist on a graph with n vertices if each vertex has degree n /2 or greater. Total cost of a path to reach (m, n) is sum of all the costs on that path (including both source and destination). The problem is solved by using the Minimal Spanning Tree Algorithm. Starting from node , we select the lower weight path, i. Before increasing the edge weights, shortest path from vertex 1 to 4 was through 2 and 3 but after increasing Figure 1: Counterexample for Shortest Path Tree the edge weights shortest path to 4 is from vertex 1. Note : It is assumed that negative cost cycles do not exist in input matrix. The path is (0, 0) –> (0, 1) –> (1, 2) –> (2, 2). acyclic graphs (DAGs) and propose structured sparsity penalties over paths on a DAG (called "path coding" penalties). Determining a minimum cost path between two given nodes of this graph can take O(mlogn) time, where n = jV j and m = jEj: If this graph is huge, say n … 700000 and m. [Tree, pred] = graphminspantree(G) finds an acyclic subset of edges that connects all the nodes in the undirected graph G and for which the total weight is minimized. Given a square grid of size N, each cell of which contains integer cost which represents a cost to traverse through that cell, we need to find a path from top left cell to bottom right cell by which total cost incurred is minimum. [1] There are some theorems that can be used in specific circumstances, such as Dirac's theorem, which says that a Hamiltonian circuit must exist on a graph with n vertices if each vertex has degree n /2 or greater. A minimum-cost spanning tree is one which has the smallest possible total weight (where weight represents cost or distance). Computing a planarorthogonaldrawingofa planar graphwith the minimum number of bends over all possible embeddings is in general NP-hard [17,18]. A typical application of this problem involves finding the best delivery route from a factory to a warehouse where the road network has some capacity and cost associated. min_paths(+Vertex, +WeightedGraph, -Tree) Tree is a tree of all the minimum-cost paths from Vertex to every other vertex in WeightedGraph. , there exist a path from ) •Breadth-first search •Depth-first search •Searching a graph -Systematically follow the edges of a graph to visit the vertices of the graph. Minimum Cost Flow Problem • Objective: determine the least cost movement of a commodity through a network in order to satisfy demands at certain nodes from available supplies at other nodes. Kruskal's algorithm is a minimum-spanning-tree algorithm which finds an edge of the least possible weight that connects any two trees in the forest. 1 Shortest path problem The shortest path problem is one of the simplest of all network ow problems. In kruskal's algorithm, edges are added to the spanning tree in increasing order of cost. Minimum spanning tree. G is usually assumed to be a weighted graph. ] Each set V i is called a stage in the graph. A spanning tree in a given graph is a tree built using all the vertices of the graph and just enough of its edges to obtain a tree. Assuming that you don't expect the paths to be more than 1000 steps long, you can choose p = 1/1000. Therefore, we can use the same reduction to also compute a minimum-cost maximum cardinality matching in O~(mn2=5) time. Operations Research Methods 8. • The edges for shortest path are , , and is the minimum cost of path 1 to 4 is "4", cost of path 1 to 2 is "6", and cost of path 4 to 3 is "5" of the given graph 6 + 4 + 5 = 15. This problem is similar to Finding possible paths in grid. 412 G orke et al. These algorithms carve paths through the graph, but there is no expectation that those paths are computationally optimal. This contradicts the maximality of M. We can, therefore define function for any V,. Investigate ideas such as planar graphs, complete graphs, minimum-cost spanning trees, and Euler and Hamiltonian paths. The cost of a path from s to t is the sum of costs of the edges on the path. i have an adjacency list representation of a graph for the problem, now i am trying to implement dijkstra's algorithm to find the minimum cost paths for the 'interesting cities' as suggested by @Kolmar. An expansion path provides a long-run view of a firm's production decision and can be used to create its long-run cost curves. It is expanded, yielding nodes B, C, D. In any graph G, the shortest path from a source vertex to a destination vertex can be calculated using Dijkstra Algorithm. Must Read: C Program To Implement Kruskal's Algorithm Every vertex is labelled with pathLength and predecessor. the original graph. As it turns out, the minimum cost flow problem is equivalent to minimum cost circulation problem and transshipment problem in the sense that they can be reduce to each other while blowing up the input size by a constant factor. Describe and analyze an e cient algorithm for nding a minimum-cost monotone path in such a graph, G. This problem is also called the assignment problem. 2) Areas less than the minimum core habitat percentage times the area of the foraging radius are eliminated 3) A cost surface is created from the habitat quality raster, cells of high quality have a low cost and vise versa 4) The remaining patches are grown outwards across the cost surface to a distance equal to the foraging radius. Shortest Path, Network Flows, Minimum Cut, Maximum Clique, Chinese Postman Problem, Graph Center, Graph Median etc. Negative Edge Costs Single-Source Shortest-Path Problem Problem Given as input a weighted graph, G = (V,E), and a distinguished vertex, s, find the shortest weighted path from s to every other vertex in G. cost(e), but cannot be shared by more than cap(e) paths even if we pay the cost of e. Can you move some of the vertices or bend. Given a graph, the start node, and the goal node, your program will search the graph for a minimum-cost path from the start to the goal. An Extended Path Following Algorithm for Graph-Matching Problem Zhi-Yong Liu, Hong Qiao,Senior Member, IEEE, and Lei Xu,Fellow, IEEE Abstract—The path following algorithm was proposed recently to approximately solve the matching problems on undirected graph models and exhibited a state-of-the-art performance on matching accuracy. For example, consider below graph. Abstract: Let G be an edge-weighted directed graph with n vertices embedded on a surface of genus g. | CommonCrawl |
A ball is dropped straight down from a height of 16 feet. If it bounces back each time to a height one-half the height from which it last fell, how far will the ball have traveled when it hits the floor for the sixth time, in feet?
The ball first drops 16 feet. It then travels up 8 feet and down 8 feet. When it hits the floor for the sixth time, it will have traveled $16 + 8 + 8 + 4 + 4 + 2 + 2 + 1 + 1 + 1/2 + 1/2 = \boxed{47}$ feet. | Math Dataset |
Does galvanic cathodic protection by aluminum anodes impact marine organisms?
Anna Maria Bell1,
Marcus von der Au2,4,
Julia Regnery1,
Matthias Schmid3,
Björn Meermann2,4,
Georg Reifferscheid1,
Thomas Ternes2 &
Sebastian Buchinger ORCID: orcid.org/0000-0003-4859-89721
Environmental Sciences Europe volume 32, Article number: 157 (2020) Cite this article
Cathodic protection by sacrificial anodes composed of aluminum-zinc-indium alloys is often applied to protect offshore support structures of wind turbines from corrosion. Given the considerable growth of renewable energies and thus offshore wind farms in Germany over the last decade, increasing levels of aluminum, indium and zinc are released to the marine environment. Although these metals are ecotoxicologically well-studied, data regarding their impact on marine organisms, especially sediment-dwelling species, as well as possible ecotoxicological effects of galvanic anodes are scarce. To investigate possible ecotoxicological effects to the marine environment, the diatom Phaedactylum tricornutum, the bacterium Aliivibrio fischeri and the amphipod Corophium volutator were exposed to dissolved galvanic anodes and solutions of aluminum and zinc, respectively, in standardized laboratory tests using natural seawater. In addition to acute toxicological effects, the uptake of these elements by C. volutator was investigated.
The investigated anode material caused no acute toxicity to the tested bacteria and only weak but significant effects on algal growth. In case of the amphipods, the single elements Al and Zn showed significant effects only at the highest tested concentrations. Moreover, an accumulation of Al and In was observed in the crustacea species.
Overall, the findings of this study indicated no direct environmental impact on the tested marine organisms by the use of galvanic anodes for cathodic protection. However, the accumulation of metals in, e.g., crustaceans might enhance their trophic transfer within the marine food web.
Offshore installations, such as wind turbines, are founded on support structures that mainly consist of steel. In order to ensure sufficient durability of submerged substructures, appropriate corrosion protection strategies are required. Because of its comparatively low weight and high electrochemical capacity, aluminum-zinc-indium alloys are commonly applied as sacrificial (i.e., galvanic) anodes for cathodic corrosion protection of steel in the marine environment [1]. The galvanic anode and the steel are combined to an electrochemical local element resulting in a continuous oxidation of the galvanic anode and thus transfer of electrons to the steel that exhibits a higher standard electrode potential [2, 3]. As numbers of offshore wind farms increase to meet the goals of the European strategy for a "prosperous, modern, competitive and climate neutral economy" [4], an increasing amount of galvanic anodes is expected to be used in the marine environment. More than 5,000 offshore wind turbines have been installed across 12 European countries until 2019, which corresponds to a total capacity of around 22 GW. The majority (77%) of this capacity is located in the North Sea [5]. The necessary amount of a certain anode material mainly depends on its desired life time and the surface area (i.e. foundation type) that requires protection against corrosion [6]. A conservative estimate based on calculations from Kirchgeorg et al. [7] reveals a current annual release of around 1900 t aluminum and 90 t zinc solely to the North Sea. This evaluation assumes the exclusive installation of aluminum anodes (Al-Zn-In alloy) with a life time of 25 years on uncoated monopile foundations without considering offshore substations or converter platforms. By the end of 2030 the wind capacity in Europe is expected to grow to 76 GW [5], which more than triples the need of corrosion protection as well.
The release of metals into the marine environment due to the use of galvanic anodes raises questions regarding potential environmental effects on marine organisms either directly or along the food chain across trophic levels [8]. In laboratory experiments, Deborde et al. demonstrated a significant increase of suspended and dissolved metal fractions in the water phase. The total concentrations reached up to 7280 µg/l Al and 360 µg/l Zn during the activation period of the galvanic anode [9]. A field study in a French harbor reported elevated Al concentrations up to 32.5 g/kg in sediments sampled close to galvanic anodes. Furthermore, up to 380 mg/kg acid-soluble Al was detected [10]. This increased proportion of a labile Al fraction might indicate a higher bioavailability of Al released by galvanic anodes. Leleyter et al. observed a total Al concentration of 29 g/kg in natural sediments contaminated by aluminum anodes [11]. Moreover, galvanic anodes caused local elevations of Zn concentrations, both in sediment and dissolved in seawater. Up to 23 µg/l dissolved and 300 µg/g sediment associated Zn was measured in enclosed marinas on the south coast of England [12]. During a monitoring campaign at the Port of Calais, Caplat et al. found elevated levels of Zn, Cu, Al and In due to the use of galvanic anodes. At undisturbed—e.g. no dredging activities—sampling points near the anodes up to 371 mg/kg Zn, 56 mg/kg Cu, 49501 mg/kg Al and 12.6 mg/kg In were detected in the sediment [8].
Despite the ubiquitous application of galvanic anodes and the subsequent release of metals, little is known about the consequences for benthic organisms in marine environments. Adverse effects of aluminum anodes were first observed in sea urchin Paracentrotus lividus causing changed fertilization rates and the induction of mitotic abnormalities at concentrations of 3 µM and 0.1 µM anode-derived Al3+, respectively [13]. In the digestive glands of the mussel Mytilus edulis, an accumulation of up to 1706 mg/kg Al released from galvanic anodes was observed that was reversible after the transfer of organisms to uncontaminated water [14]. A number of ecotoxicological studies on Al characterize the impact of acidification on the solubility and release of Al from e.g. insoluble hydroxides, which might lead to a deterioration of soil and freshwater quality. In this context, a variety of plants and freshwater organisms were examined and showed a broad range of sensitivities depending on pH and Al speciation [15, 16]. In this context, compared to invertebrates, fish have been proven to be particularly sensitive to Al exposition, in neutral to basic water a no effect concentration of 100 µg/l Al was reported for trouts. In contrast, marine studies on the ecotoxicological impact of galvanic anodes are scarce [17]. So far, very little attention has been paid to studying the toxicity and accumulation of Al in benthic organisms. Insoluble and particle-reactive hydroxide (Al(OH)3) and aluminate ([Al(OH)4]−) species dominate in natural seawater at its common pH around 8 [18] and could particularly cause a threat to marine sediments and sediment-dwelling organisms. The partitioning of Al in sediments is dominated by mineral-bound species such as aluminosilicates in feldspars, mica or clays. Leleyter et al. reported a mineral associated content of Al of 91 to 96% in various natural sediments and suggested that the labile fraction in turn mainly consists of Al oxides (95%) [11, 19].
Unlike Al, Zn–as the second main component of galvanic anodes–serves as an essential micronutrient, which could become toxic at higher concentrations. Several studies observed toxic effects of Zn to a variety of organisms such as plants, fish and microorganisms [20,21,22]. Most recent research pays attention to the toxicity of nanoscale zinc oxide. In this context Wong et al. showed that marine crustaceans were more affected than diatoms and the majority of effects was related to dissolved Zn2+ ions. For the amphipod Elasmopus rapax a half maximal effect concentration of 0.80 mg/l was determined, whereas the growth of the marine diatom Thalassiosia pseudonana was inhibited to 50% at 3.48 mg/l dissolved Zn [23]. At pH values around 8, the free ion (Zn2+) and the mononuclear hydroxide ([ZnOH]+) are prevalent in the aqueous phase [24]. If the maximum soluble concentration is exceeded, Zn can also precipitate as Zn(OH)2 [25]. However, adverse effects of Zn released from aluminum anodes in the marine environment are mainly expected due to its dissolved species, as Zn is a minor component in these anodes and local saturation is most likely not reached.
More research is needed to adequately assess the environmental impact of galvanic anodes on the marine environment, in particular with regard to sediment-dwelling marine organisms. Therefore, the main objective of this study was to investigate the acute toxicity of metals released from galvanic aluminum anodes on various trophic levels in the marine environment using the following marine standard test organisms: the sediment-dwelling mud shrimp Corophium volutator, the marine diatom Phaeodactylum tricornutum and the luminescent bacterium Aliivibrio fischeri. Besides the acute toxicity testing, a basic uptake experiment using dissolved material from a galvanic anode was carried out at laboratory scale to mimic worst case exposure conditions of sediment-dwelling organisms near offshore installations. Due to the pH-dependent precipitation of aluminum hydroxide after the release of Al from anodes in seawater, this batch experiment focused on sediment toxicity and a potential uptake of Al, Zn and In by C. volutator.
Rationale of exposure scenarios
Different exposure scenarios for bacteria, algae and amphipods were used because of the following rationale. Bacteria and algae belong to the marine plankton and are most likely exposed to released metal ions that are present in dissolved form under natural conditions. Therefore, only the soluble fraction of the dissolved anode material was used for the exposure of bacteria and algae. Similarly, only the soluble fractions of the ionic Al and Zn standards were used for these tests. In case of the dissolved anode material and Al, the assays were performed as limit tests, i.e. exposure at saturation concentration, due to the low solubility of Al at the pH of natural sea water. In case of C. volutator, the testing was done with the total amount of metal ions, because the sediment dwelling C. volutator is likely to be exposed to both, the dissolved and the precipitated fraction of the metal ions that sediment on the seabed.
Preparation of test solutions
The natural seawater used in all experiments originated from the amphipod collecting site mentioned in Sect. "Amphipoda". For the testing a stock solution (10 g/l) of a commercial galvanic aluminum anode (Al-Zn-In alloy, casted in terms to VG 81257:2009 by Raguse + Voss Metallgießerei GmbH, Germany; 95 ± 1·104 mg/kg Al; 5 ± 1·104 mg/kg Zn, 203 ± 1 mg/kg In) was prepared by dissolving filings of the anode in 30% NaOH (w/v) (Merck Suprapur). Al3+ and Zn2+ were used as single element standards (Alfa Aesar Plasma Standard Specpure, 10 g/l Al in HCl; SCP Science PlasmaCAL Custom Standard, 1 g/l Zn in 5% HCl) in the various bioassays for the testing of single elements. Prior to the start of each bioassay, aliquots of the respective metal stock solutions were adjusted to pH 8 with either 30% NaOH (w/v) (Merck Suprapur) or 30% HCl (w/v) (Merck Suprapur, subboiled), resulting in the formation of precipitates. For experiments with the sediment-dwelling amphipods, the saturated solutions with the precipitate were dosed to filtered (Pall VacuCap 90 Filter Unit, 0.45 µm) natural seawater at the desired total bioassay concentration. In contrast, toxicity testing of Al and the anode material to algae and luminescent bacteria was designed as a limit test using the maximal seawater soluble metal concentrations at natural pH. For this purpose, the pH adjusted stock solutions were added to natural seawater in the ratio of 1 to 1000 and stirred for 24 h until an equilibrium of Al in the medium was reached (Additional file 1: Fig SI1 and Table SI1). Subsequently, the filtered solution (Nalgene Rapid-Flow Bottle Top Filter with SFCA Membrane, 0.45 µm) was used for the respective limit test. ZnCl2 (Merck Emsure) with a maximal concentration of 100 mg/l Zn2+ in natural seawater was used to characterize the dose–response relationship of Zn for algae toxicity. For the testing of bacteria and algae, Al and Zn concentrations were measured before the addition of test organisms as well as at the beginning and the end of toxicity testing by ICP-MS.
Bioassays
Luminescent bacteria
The bioluminescence inhibition assay was performed according to DIN EN ISO 11348-2 [26] with liquid-dried bacteria (LCK 482, Hach Lange). This standardized test was conducted in glass cuvettes after the reconstitution of the bacteria and measured using a LUMIStox 300 instrument (Hach Lange, Germany). The assay utilizes the bioluminescence of the marine bacterium Aliivibrio fischeri and quantifies the inhibition of the bacterial light emission after exposure to the test sample as a measure for the acute bacterial toxicity. The addition of bacteria to the filtered test solutions reduced the maximal soluble metal concentration in the assay to 80% of the previously dissolved concentration due to dilution. The initial Al and Zn concentrations were analyzed by inductively coupled plasma-mass spectrometry (ICP-MS) as described in Sect. "Chemical Analyses". Each limit test was performed with four replicates for the dissolved aluminum anode as well as the aluminum standard. Every experiment was repeated three times in total. Natural seawater was used as negative control and 3,5-dichlorophenol (4.5 mg/l) served as the positive control.
The growth inhibition test using the marine algae P. tricornutum (SAG1090-1a) was based on DIN EN ISO 10253 [27]. The incubation was performed in an incubation shaker (Infors Multitron Pro) at 20 °C and 120 rpm in polycarbonate Erlenmeyer flasks with a testing volume of 50 ml. Test vessels were continuously illuminated with 60–120 µmol photons/(m2·s). To simulate the natural speciation and bioavailability of the investigated metals, tests were conducted in natural seawater at pH 8.1 without the addition of chelating agents or trace metals. Merely 15 mg NO3−/l and 1.5 mg PO43−/l were added to the growth medium to prevent nutrient limitation of algae growth. An exponentially growing pre-culture (up to 104 cells/ml) was measured with a fluorescence spectrophotometer (Hitachi F-2500) and adjusted to 5 × 103 cells/ml. The inoculation of the test solutions diluted the metal concentration in the assay to 80% of the previously dissolved concentration. Al and Zn concentrations were measured at the beginning and the end of the exposure by ICP-MS. Single concentrations of the dissolved aluminum anode and aluminum standard or dilution series of Zn (test concentration 10 to 1 mg/l) were investigated as triplicates. Every experiment was repeated three times in total. Natural seawater was used as negative control and 3,5-dichlorophenol (2.5 mg/l) served as the positive control. The chlorophyll fluorescence of each sample was determined daily (excitation wavelength 435 nm; emission wavelength 685 nm, plate reader Tecan Infinite M200 Pro). The evaluation of the growth inhibition was performed after 72 h.
The DIN EN ISO 16712 standard [28] for the determination of acute toxicity of sediment to amphipods was adapted to study the selected metals. In addition to the classic exposure for 10 days in the presence of sediment, this test was performed according to procedures used for positive controls, i.e. water-only exposures for 3 days, as a worst case scenario reflecting a direct contact of the organisms to the precipitated Al. This exposure was performed under identical conditions and with the same procedural steps except that no sediment was added to the test vessels so that the organisms cannot avoid the exposure to the precipitated metals by digging into the sediment. The test species Corophium volutator had been collected from uncontaminated reference sites at Sylt and Norderney, Germany, respectively. Corresponding sediment and seawater for exposure and control experiments originated from the same collection site (physicochemical characterization ensured consistent sediment and water quality, Additional file 1: Table SI2). The background concentrations in the sediment fractions < 20 µm ranged from 48,900 to 60,500 mg/kg Al and from 150 to 221 mg/kg Zn. The dissolved amount of Al and Zn in seawater was always below the limit of quantification (0.02 mg/l Al and 0.11 mg/l Zn). The sediment was sieved through a 1 mm screen prior usage to remove native macroorganisms. Accordingly, the seawater was filtered to 0.45 µm (Pall VacuCap 90 Filter Unit). After acclimatization to laboratory conditions, 20 individuals were added to 1 l beakers filled with 300 g sediment (approx. 2 cm depth) and 700 g test solution or 900 g test solution without sediment. The incubation was performed under constant aeration and lighting at 15 °C. At the end of respective testing periods, the content of test vessels was sieved and living organisms were counted. Al and the dissolved aluminum anode were investigated at nominal concentrations of 1, 10 and 100 mg/kg seawater. The toxicity of Zn was characterized at nominal concentrations of 0.1, 1 and 10 mg/kg seawater. Nominally dosed concentrations in this bioassay were not accompanied by chemical analyses of seawater. Two to six independent replicates were tested for each concentration level and exposure type (with and without sediment). Each experiment accompanied by negative controls with only seawater (with and without sediment). To assess the sensitivity of the test populations, ammonium chloride (30–150 mg/l NH4+) served as reference toxicant (without sediment).
Uptake of metals by Corophium volutator
The possible uptake of metals from galvanic anodes was investigated using C. volutator. As toxicity tests with C. volutator according to DIN EN ISO 16712 [28] provided not enough sample material for chemical analysis, the experimental set up was scaled up from 1 l beakers to 5 l aquariums with about 200 individuals per tested exposure concentration. All other test parameters, such as concentration levels and exposure time, remained the same. More specifically, the investigations were carried out in presence and absence of sediment and at nominal concentrations of 1, 10 and 100 mg dissolved aluminum anode per kg seawater. Due to the high amount of mud shrimp required for analysis in this basic uptake experiment, exposure was not carried out in replicate. At the end of the testing period, test organisms were kept for at least 1 h in fresh seawater for gut purging and were subsequently anaesthetized by aeration with CO2. The gut passage time of sediment for C. volutator was previously verified on multiple individuals by the observation under a stereomicroscope. Preliminary tests showed that precipitates adhere to plumose setae (i.e., the hair-like structures of the pereiopods and the pleopods of the test organisms). Thus, attached particles had to be removed prior to the determination of metal accumulation by ICP-MS to prevent an overestimation of metal contents. After wash conditions had been optimized (Additional file 1: Fig SI2 and Table SI3), exposed organisms were washed with 0.5% HNO3 under agitation at 250 rpm for 30 min to ensure sufficient removal of attached material. Subsequently, organisms were rinsed with ultra-pure water, shock frozen in liquid nitrogen, freeze-dried and homogenized for 1 min at 20 Hz with a mixer mill (Retsch M400).
The statistical analysis of bioassay data was performed with the open source software R using the core distribution, version 3.4.3 and R-Studio, version 1.1.383.
After checking for normal distribution with Shapiro–Wilk's test and homoscedasticity with F-test, an unpaired, two-sided t-test (normally distributed, homoscedastic data), Welch's t-test (normally distributed, not homoscedastic data) or Mann–Whitney U test (data not normally distributed) was used to determine statistical differences in ecotoxicological effects. In case of statistically significant differences in t-test, the strength was calculated. All tests were performed at the significance level α = 0.05.
The fit of concentration–response relationships and the estimate of the half maximal effective concentration (EC50) were calculated by a five-parameter log-logistic function (Eq. 1) by means of the extension package drc (version 3.0.1) [29].
$$\varvec{f}\left( \varvec{x} \right) = \varvec{c} + \frac{{\varvec{d} - \varvec{c}}}{{\left( {1 + \exp \left( {\varvec{b} \times \left( {\log \left( \varvec{x} \right) - \log \left( \varvec{e} \right)} \right)} \right)} \right)^{\varvec{f}} }}.$$
The response is evaluated as a function of the concentration x with the parameters c and d as lower and upper response limits, respectively. The parameter e is defined as inflection point, parameter b denotes the relative slope and the parameter f describes the asymmetry of the curve.
Chemical analyses
Al, Zn and In concentrations in seawater samples were analyzed by ICP-MS (7700 ICP-MS, Agilent Technologies, Inc., USA) according to Düster et al. [30]. Samples were diluted at least 1:10 to reduce salt concentrations and thus matrix load. If analyte concentrations exceeded the linear range of the calibrated system (i.e., > 200 µg/l), samples were diluted. All samples were analyzed as triplicates and mean values were provided for statistical analysis. The following reference materials were used for method validation: SPS-SW1, SPS-SW2 (Spectrapure Standards, Norway) and SRM 1640a (NIST, USA).
Prior to the analysis of Al, Zn and In in the bulk of C. volutator by ICP-MS, amphipods were dissolved using microwave pressure digestion (turboWave, MLS GmbH, Germany) as summarized in Additional file 1: Table SI4. For the digestion 50 mg of freeze-dried, ground and homogenized amphipods, 0.6 ml HNO3 (65%, Suprapur, subboiled, Merck KGaA, Germany) and 0.4 ml HCl (30%, Suprapur, subboiled, Merck KGaA, Germany) were used. All samples were digested in triplicate and analyzed as described above. In addition, NCS ZC73034 (China National Analysis Center for Iron & Steel, China), a prawn standard, and SRM 2976 (NIST, USA), a mussel tissue standard, were digested and measured as reference materials. All procedures were optimized to obtain a recovery of 100 ± 10% for the analytes in the reference material.
In case of testing with A. fischeri, no significant effects were detected for any of the tested materials (Additional file 1: Table SI5). In case of P. tricornutum, the dissolved anode and Al at saturation concentration at pH = 8.1 caused a comparable and statistically significant decrease of the growth rate (Fig. 1 and Additional file 1: Table SI6) with an average growth inhibition of 28.3 ± 6.3% and 26.0 ± 2.6%, respectively. The mean exposure concentration of Al from the dissolved anode material was 818 ± 19 µg/l (824 µg/l at the beginning and 817 µg/l at the end of exposure), while the determined concentrations of zinc and indium were below their respective detection limits of 32 µg/l and 29 µg/l in seawater. When tested as a single element, the mean exposure concentration of Al was 880 ± 24 µg/l (899 µg/l at the beginning and 874 µg/l at the end of exposure). Additionally, the effect of Zn on algal growth was characterized as a concentration–response curve at dissolved concentrations from 0.9 up to 8 mg/l (Fig. 2 and Additional file 1: Table SI8). Applied metal concentrations quantified by ICP-MS are summarized in Additional file 1: Tables SI7 and SI9. Higher concentrations could not be tested without decreasing the pH value of the medium. Under pH conditions of the investigated natural seawater, the EC50 of Zn2+ for P. tricornutum was determined as 5.70 ± 0.33 mg/l (with a fixed maximum growth inhibition of 100%).
Average growth rate of P. tricornutum during limit testing of aluminum and the aluminum anode. Diatoms were exposed for 72 h to maximum soluble metal concentrations (n = 3, error bars indicate SE) under natural seawater pH conditions. Significantly different means are marked with different letters (two-tailed unpaired t-test, p < 0.05)
Concentration-dependent growth inhibition of P. tricornutum during zinc exposure for 72 h (mean ± SE, n = 3). The dashed line indicates 20% effect level and dotted line shows 5-parametric log-logistic fit of data (b = −10.3, c = 6.8, d = 100, e = 7.99, f = 0.20; Eq. 1)
In case of C. volutator, the investigated anode material caused no acute toxicity. However, the single elements Al and Zn showed significant effects at the highest tested concentrations (100 mg/kg aluminum, 10 mg/kg zinc) as illustrated in Fig. 3 and Additional file 1: Table SI10. In case of Al, experiments in the absence of sediments exhibited higher effects on C. volutator mortality (max. mortality of 17.5 ± 4.2%) compared to exposure in the presence of sediment. On the contrary, Zn caused the highest mortality of C. volutator (52.5 ± 7.5%) in the presence of sediment. In the absence of sediment, the mortality was 30 ± 5.4%. Statistically significant differences between exposures with and without sediments were only detected in case of the exposition with 100 mg/kg aluminum (two-tailed, unpaired Mann–Whitney U test, p = 0.017).
Average toxicity (± SE) caused by the dissolved galvanic anode and its main components aluminum and zinc to C. volutator. Test organisms were exposed to the different metal concentrations for ten days with (w/) sediment and 3 days without (w/o) sediment. The numbers below the bars represent the number of replicates of each experiment. Experiments with statistically significant differences to the corresponding negative control are marked with an asterisk (two-tailed unpaired t-test, p < 0.01). See Additional file 1: Table SI11 for complete data of statistical evaluation
Uptake of metals
To study the potential uptake of the three components Al, Zn and In of galvanic aluminum anodes, concentrations of these metals were determined in C. volutator after exposure to dissolved galvanic anode material in the presence and absence of sediment, respectively. The results are shown in Figs. 4, 5 and Additional file 1: Table SI12 and Table SI13. No increase of residual metal content compared to the control group was observed in C. volutator in the experiments with sediment. In contrast, all investigated elements revealed increased concentrations in C. volutator after exposure without sediment. The extent of accumulation indicated differences among the studied metals. As expected, given its prevalence in the dissolved galvanic anode solution, Al showed the highest concentrations in the mud shrimp after exposure. Concentration increased up to 2,200 mg/kg dry weight, which resulted in Al levels five times above those in the unexposed control group.
Average Al, Zn and In concentrations (± SE of technical triplicates) in C. volutator. The exposure with the dissolved aluminum anode proceeded for 10 days with sediment and three days without sediment. At the end of the testing period, the organisms were washed with 0.5% HNO3 for 30 min to remove adherent metal precipitates from exoskeleton, freeze-dried, homogenized, digested and subsequently analyzed by ICP-MS
Enrichment factor in C. volutator after exposure to dissolved aluminum anode material. Results are shown for the exposure with three different concentrations for 3 days in the absence of sediment. Analyte residue concentrations are based on dry weight of organisms. The enrichment factor represents the ratio of analyte residues detected in exposed compared to non-exposed organisms
Although In represents a minor constituent of the tested galvanic anode, it showed the highest relative enrichment in mud shrimp. At the highest exposure level, the measured In concentration in C. volutator increased 136-fold compared to the negative control. The lowest enrichment was found for Zn. The Zn concentration in the test organisms was elevated by approximately 28% at the highest exposure level compared to the negative control. In summary, dissolved galvanic anode concentrations in seawater showed a positive correlation with residual metal concentration in biota. However, enrichment expressed no linear relationship in terms of applied test concentration of the dissolved galvanic anode. Nevertheless, the highest enrichment was observed at the highest exposure concentration (Fig. 5).
In order to assess potential ecotoxicological risks, a worst case exposure of three marine organisms from different trophic levels, i.e. the marine algae P. tricornutum, the bacterium A. fischeri and the amphipod C. volutator was performed for aluminum, zinc and the dissolved galvanic anode. For P. tricornutum, an EC50 value for zinc of about 42 mg/l for exponentially growing algae in f/2 medium was reported [31]. The respective EC50 value of 5.34 mg/l determined in the present study is eightfold lower. In contrast to the EC50 previously published, no further compounds such as vitamins and other trace metals that are present in the f/2 medium were added to the current experiments to achieve effect data under most realistic conditions. These differences might explain the higher Zn sensitivity for P. tricornutum reported here in comparison to previous studies. Deborde et al. reported a maximum increase of dissolved Zn content to around 200 µg/l during the anode activation period in a tank experiment [9]. Based on the 20% effect concentration (EC20, around 3 mg/l) as an estimate for the boundary between the lowest and no observed effect concentration (LOEC and NOEC), the resulting risk quotient is about 0.07, which indicates that there is no acute risk for P. tricornutum from the use of galvanic anodes in marine environments. In comparison to the freshwater green algae Pseudokirchneriella subcapitata, the marine algae P. tricornutum seems to be more tolerant for Zn2+. The reported EC50 for growth inhibition of the freshwater algae at pH = 7.5 is 16.3 µg/l [32].
The observed growth inhibition of P. tricornutum at Al concentrations around 850 µg/l is in agreement with Gillmore et al., who reported an EC10 of 920 µg/l for growth inhibition of the same algae under similar conditions [33]. During the anodic dissolution in a tank experiment, Deborde et al. measured dissolved aluminum concentrations of 0.2 to 0.5 µg/l, which was in the range of the natural background concentration [9]. Thus, effects on growth of marine algae caused by Al seem unlikely. The same conclusion was drawn by Zhou et al., who reviewed the aluminum toxicity to marine phytoplankton [17]. However, Gillmore et al. showed that diatoms with higher silicon content in cell walls are more sensitive [33]. For Ceratoneis closterium and Minutocellus polymorphus a 10% inhibition of the growth rate was observed already at Al concentrations of 69 and 440 µg/l, respectively. Based on these numbers, a growth inhibition of C. closterium might occur in close proximity to the galvanic anodes when the released aluminum is not yet further diluted by the surrounding water. The presented results indicate furthermore that the inhibition of algae growth by the dissolved galvanic anode reported in the current study is mainly caused by aluminum, as the observed effects do not differ significantly between the exposure to the dissolved galvanic anode and aluminum alone. Thus, possible mixture effects of metals released from a galvanic anode on the growth of P. tricornutum are unlikely.
No acute toxicity for any of the tested materials was observed for A. fischeri, indicating a lower toxicity of metals released from galvanic anodes for marine bacteria compared to algae. However, due to the scarcity of literature about Al toxicity on bacteria, it is challenging to extrapolate this finding to the field. A varying sensitivity for other marine bacteria is to be expected. In this respect, it is interesting that recent studies have shown that aluminum can even have a stimulative effect on the growth of different cyanobacteria [34, 35]. Nevertheless, it can be concluded that bacterial toxicity is unlikely to be the main driver for an environmental risk assessment of galvanic anodes because species from other trophic levels showed higher sensitivities.
Corophium volutator showed only a significantly increased mortality at nominal Al concentrations of 100 mg/l in the exposure without sediment. Although not directly indicated by a comparison of the mortality in the controls with and without sediment, the toxicity observed in the absence of sediment might be caused by a higher general stress level of the organisms when exposed solely in water. A further explanation might be the possibility for the organisms to avoid the exposure to the precipitated Al(OH)3 after digging into the sediment or absorption of the element to the sediment. This observation suggests a protective function of the sediment for benthic organisms. Furthermore, toxic Al3+ ions might be sequestered by the sediment lowering the bioavailability of Al for the exposed organisms. On the other hand a saturated Al concentration might be expected because of the precipitated Al(OH)3 that can replenish the Al aquo complexes in solution. The formation of colloidal Al(OH)3 and various polymeric Al(III) cations in dependence of the surrounding pH is challenging for the assessment of the bioavailability of Al species. The chronic toxicity of aluminum to freshwater organisms was successfully predicted by multiple linear regression [36] and a biotic ligand model [37]. However, a valid model is currently not available for marine ecosystems.
In case of Zn, a reduction of toxicity due to the presence of sediment was reported by Bat et al., the EC50 values for Zn-induced mortality of C. volutator was 14.12 mg/l in the presence and 9.79 mg/l in the absence of sediment [38]. Although exposure times differed between the published (four days for both setups) and the current experiments, the values for the observed mortality are in good agreement with the previous study (about 50% mortality at 10 mg/l zinc). The comparatively higher toxicity for exposure in the presence of sediment in the current work might be explained by the longer exposure times for this setup compared to the earlier report. Concentrations of total Zn during the anode activation period in a tank experiment ranged from 28.3 to 360.0 µg/l [9]. Under consideration of this maximum release and a roughly estimated NOEC of 1 mg/l, the risk quotient for zinc ranges from 0.03 to 0.36. Conradi & Depledge reported values for chronic Zn exposure in sediment over up to 107 days [39]. They found a significantly decreased survival at Zn concentrations of 800 µg/l, which is more than two times above the highest Zn concentration determined during the tank experiments of Deborde et al. [9].
To assess the potential uptake of the metals by C. volutator, the metal composition with respect to aluminum, zinc and indium of this organism was analyzed before and after exposure to the three different concentrations of the dissolved galvanic anode. Interestingly, the maximum enrichment factor of the three metals in biota reflects the ratio between the metal content in the galvanic anode and the natural biota-level of the metals in the mud shrimp. If normalized to Zn = 1, the ratio between metal concentrations in the galvanic anode and the non-exposed shrimp was 3.4 for aluminum and 170 for indium. This corresponds well to the normalized maximum enrichment factor of 3.8 for aluminum and about 105 for indium. In other words, the metal composition in biota after the exposure reflects the metal composition of the galvanic anode indicating that there is no specific mechanism in C. volutator that would favor the uptake of one of the investigated elements over the other. The uptake of potentially toxic metals by aquatic organisms can take place in general across the integument, through the respiratory surfaces (e.g. gills), or the gut after ingestion of contaminated food [40]. The mud shrimp C. volutator, due to its filter and deposit feeding activity, is exposed to the particular risk of particle-associated contaminants and therefore the uptake of metals from precipitates in the current experiments. Analysis of the exposed organisms provided valuable additional information but raised the question, where the enriched elements are located, i.e., in or at the organism. As Rainbow & Luoma described, metal toxicity is not necessarily related to the total accumulated amount in the organism, because the metals are frequently stored in detoxified forms [41]. The removal of metals from cytoplasm by complexation and accumulation in spherical precipitates in membrane bound vesicles or vacuoles is a well-known detoxification strategy of crustaceans [40]. The same applies to C. volutator, where e.g. zinc-rich granules could be observed in hepatopancreas [42]. Alternatively, the elements released by the galvanic anode might be absorbed at the shell of the crustacea. In fact, the biosorption of various metal ions to chitin is used as an approach to purify water from e.g. mining sites [43, 44]. It has been demonstrated that chitin can remove metal ions from aqueous solutions by adsorption to functional groups such as hydroxyl- and amino-residues [45, 46]. Based on modeling results, Tarpani et al. proposed chemisorption of Al3+ to chitin [44]. A strong binding of metal ions to chitin might explain why the elements were detected in elevated levels despite the acidic washing step. However, the bulk analysis of exposed organisms alone provides no further information. To elucidate the localization of the accumulated metals, methods allowing a special analysis such as laser ablation ICP-MS [47,48,49] should be applied. The reported enrichment of metal ions by C. volutator—either due to uptake or adsorption—might promote a trophic transfer in the marine environment. Benthic invertebrates usually serve as food for larger invertebrates, fish and also birds. As previously reported, aluminum in freshwater can be transferred from a primary to a secondary consumer [50]. In particular with respect to the potential use of areas surrounding offshore wind farms for aquaculture [51, 52], further studies are required to investigate the direct and indirect uptake of metals released from galvanic anodes by marine organisms for a risk assessment on environment and human health.
Based on the assessment of the acute toxicity to three organisms of different trophic levels, a direct environmental threat by the use of galvanic anodes for cathodic protection of wind turbine support structures in the marine environment was not indicated. Toxicity thresholds for C. volutator, A. fischeri and P. tricornutum determined in worst case exposure scenarios were in most cases at least one order of magnitude higher compared to the concentrations of aluminum and zinc that are expected to be released during cathodic protection. However, a growth inhibition of marine algae in close proximity to an operating galvanic anode might occur if the dilution of the released metal ions occurs by diffusion only. The observed accumulation of aluminum and indium in crustacea might facilitate their enhanced entry into the food chain. To develop a broader picture of the environmental impact of galvanic anodes in marine environments, future investigations should be extended to higher trophic levels, the potential trophic transfer of respective metals and possibly more sensitive chronic endpoints. If sublethal or developmental assays resulted in more significant effects, also the investigation of mixture toxicity could be useful extension of the presented work.
All data generated or analyzed during this study are included in this published article and its supplementary information file.
EC20:
20% effect concentration
Half maximal effective concentration
ICP-MS:
Inductively coupled plasma-mass spectrometry
LOEC:
Lowest observed effect concentration
NOEC:
No observed effect concentration
Zn:
DNV (2017) Recommended practice DNVGL-RP-B401: Cathodic protection design
DIN (2014) Galvanic Anodes for cathodic protection in seawater and saline mud. https://doi.org/10.31030/2088472
Szabo S, Bakos I (2006) Cathodic protection with sacrificial anodes. Corros Rev 24(3–4):231–280
European Commission (2018). A Clean Planet for all - A European strategic long-term vision for a prosperous, modern, competitive and climate neutral economy
WindEurope (2019) Offshore Wind in Europe - Key trends and Statistics. https://windeurope.org/wp-content/uploads/files/about-wind/statistics/WindEurope-Annual-Offshore-Statistics-2019.pdf
HTG (2009) Kathodischer Korrosionsschutz im Wasserbau. https://www.htg-online.de/fileadmin/dateien/FA/FA_Korrosionsfragen/HTG-Handbuch_Dezember_2009_Korrektur_mit_Titelseite.pdf
Kirchgeorg T, Weinberg I, Hörnig M, Baier R, Schmid M, Brockmeyer B (2018) Emissions from corrosion protection systems of offshore wind farms: evaluation of the potential impact on the marine environment. Mar Pollut Bull 136:257–268. https://doi.org/10.1016/j.marpolbul.2018.08.058
Caplat C, Basuyaux O, Pineau S, Deborde J, Grolleau AM, Leglatin S, Mahaut ML (2019) Transfer of elements released by aluminum galvanic anodes in a marine sedimentary compartment after long-term monitoring in harbor and laboratory environments. Chemosphere 239:124720. https://doi.org/10.1016/j.chemosphere.2019.124720
Deborde J, Refait P, Bustamante P, Caplat C, Basuyaux O, Grolleau AM, Mahaut ML, Brach-Papa C, Gonzalez JL, Pineau S (2015) Impact of galvanic anode dissolution on metal trace element concentrations in marine waters. Water Air Soil Pollut 226(12):14. https://doi.org/10.1007/s11270-015-2694-x
Gabelle C, Baraud F, Biree L, Gouali S, Hamdoun H, Rousseau C, van Veen E, Leleyter L (2012) The impact of aluminium sacrificial anodes on the marine environment: a case study. Appl Geochem 27(10):2088–2095. https://doi.org/10.1016/j.apgeochem.2012.07.001
Leleyter L, Baraud F, Reinert T, Gouali S, Lemoine M, Gil O (2018) Fate of aluminium released by sacrificial anodes - Contamination of marine sediments by environmentally available compounds. C R Geosci 350(5):195–201. https://doi.org/10.1016/j.crte.2018.05.003
Bird P, Comber SDW, Gardner MJ, Ravenscroft JE (1996) Zinc inputs to coastal waters from sacrificial anodes. Sci Total Environ 181(3):257–264. https://doi.org/10.1016/0048-9697(95)05025-6
Caplat C, Oral R, Mahaut ML, Mao A, Barillier D, Guida M, Della Rocca C, Pagano G (2010) Comparative toxicities of aluminum and zinc from sacrificial anodes or from sulfate salt in sea urchin embryos and sperm. Ecotoxicol Environ Saf 73(6):1138–1143. https://doi.org/10.1016/j.ecoenv.2010.06.024
Mao A, Mahaut M-L, Pineau S, Barillier D, Caplat C (2011) Assessment of sacrificial anode impact by aluminum accumulation in mussel Mytilus edulis: a large-scale laboratory test. Mar Pollut Bull 62(12):2707–2713. https://doi.org/10.1016/j.marpolbul.2011.09.017
Gensemer RW, Playle RC (1999) The bioavailability and toxicity of aluminum in aquatic environments. Critical Rev Environmental Sci Technol 29(4):315–450. https://doi.org/10.1080/10643389991259245
Singh S, Tripathi DK, Singh S, Sharma S, Dubey NK, Chauhan DK, Vaculik M (2017) Toxicity of aluminium on various levels of plant cells and organism: a review. Environ Exp Bot 137:177–193. https://doi.org/10.1016/j.envexpbot.2017.01.005
Zhou LB, Tan YH, Huang LM, Fortin C, Campbell PGC (2018) Aluminum effects on marine phytoplankton: implications for a revised Iron Hypothesis (Iron-Aluminum Hypothesis). Biogeochemistry 139(2):123–137. https://doi.org/10.1007/s10533-018-0458-6
Tria J, Butler ECV, Haddad PR, Bowie AR (2007) Determination of aluminium in natural water samples. Anal Chim Acta 588(2):153–165. https://doi.org/10.1016/j.aca.200'1.02.048
Leleyter L, Probst JL (1999) A new sequential extraction procedure for the speciation of particulate trace elements in river sediments. Int J Environ Anal Chem 73(2):109–128. https://doi.org/10.1080/03067319908032656
Admiraal W, Blanck H, Buckert-De Jong M, Guasch H, Ivorra N, Lehmann V, Nystrom BAH, Paulsson M, Sabater S (1999) Short-term toxicity of zinc to microbenthic algae and bacteria in a metal polluted stream. Water Res 33(9):1989–1996. https://doi.org/10.1016/s0043-1354(98)00426-6
De Schamphelaere KAC, Janssen CR (2004) Bioavailability and chronic toxicity of zinc to juvenile rainbow trout (Oncorhynchus mykiss): comparison with other fish species and development of a biotic ligand model. Environ Sci Technol 38(23):6201–6209. https://doi.org/10.1021/es049720m
Rout GR, Das P (2003) Effect of metal toxicity on plant growth and metabolism: i. Zinc. Agronomie 23(1):3–11. https://doi.org/10.1051/agro:2002073
Wong SWY, Leung PTY, Djurisic AB, Leung KMY (2010) Toxicities of nano zinc oxide to five marine organisms: influences of aggregate size and ion solubility. Anal Bioanal Chem 396(2):609–618. https://doi.org/10.1007/s00216-009-3249-z
Reichle RA, McCurdy KG, Hepler LG (1975) Zinc hydroxide: solubility product and hydroxy-complex stability constants from 12.5–75 C. Canadian Journal of Chemistry 53 (24):3841-3845
Jelmert A, van Leeuwen J (2000) Harming local species or preventing the transfer of exotics? Possible negative and positive effects of using zinc anodes for corrosion protection of ballast water tanks. Water Res 34(6):1937–1940. https://doi.org/10.1016/s0043-1354(99)00416-9
ISO (2007) Water quality - Determination of the inhibitory effect of water samples on the light emission of Vibrio fischeri (Luminescent bacteria test) - Part 2: Method using liquid-dried bacteria
ISO (2018) Water quality - Marine algal growth inhibition test with Skeletonema costatum and Phaeodactylum tricomutum
ISO (2007) Water quality - Determination of acute toxicity of marine or estuarine sediment to amphipods
Ritz C, Baty F, Streibig JC, Gerhard D (2015) Dose-Response Analysis Using R. PLoS ONE 10(12):13. https://doi.org/10.1371/journal.pone.0146021
Duester L, Wahrendorf DS, Brinkmann C, Fabricius AL, Meermann B, Pelzer J, Ecker D, Renner M, Schmid H, Ternes TA, Heininger P (2017) A framework to evaluate the impact of armourstones on the chemical quality of surface water. PLoS ONE 12(1):12. https://doi.org/10.1371/journal.pone.0168926
Horvatic J, Persic V (2007) The effect of Ni2 + , Co2 + , Zn2 + , Cd2 + and Hg2 + on the growth rate of marine diatom Phaeodactylum tricornutum bohlin: microplate growth inhibition test. Bull Environ Contam Toxicol 79(5):494–498. https://doi.org/10.1007/s00128-007-9291-7
Heijerick DG, De Schamphelaere KAC, Janssen CR (2002) Biotic ligand model development predicting Zn toxicity to the alga Pseudokirchneriella subcapitata: possibilities and limitations. Comparative Biochemistry and Physiology C-Toxicology & Pharmacology 133(1–2):207–218. https://doi.org/10.1016/s1532-0456(02)00077-7
Gillmore ML, Golding LA, Angel BM, Adams MS, Jolley DF (2016) Toxicity of dissolved and precipitated aluminium to marine diatoms. Aquat Toxicol 174:82–91. https://doi.org/10.1016/j.aquatox.2016.02.004
Liu JX, Zhou LB, Ke ZX, Li G, Shi RJ, Tan YH (2018) Beneficial effects of aluminum enrichment on nitrogen-fixing cyanobacteria in the South China Sea. Mar Pollut Bull 129(1):142–150. https://doi.org/10.1016/j.marpolbul.2018.02.011
Shi RJ, Li G, Zhou LB, Liu JX, Tan YH (2015) The increasing aluminum content affects the growth, cellular chlorophyll a and oxidation stress of cyanobacteria Synechococcus sp WH7803. Oceanol Hydrobiol Stud 44(3):343–351. https://doi.org/10.1515/ohs-2015-0033
DeForest DK, Brix KV, Tear LM, Cardwell AS, Stubblefield WA, Nordheim E, Adams WJ (2020) Updated multiple linear regression models for predicting chronic aluminum toxicity to freshwater aquatic organisms and developing water quality guidelines. Environ Toxicol Chem 39(9):1724–1736. https://doi.org/10.1002/etc.4796
Santore RC, Ryan AC, Kroglund F, Rodriguez PH, Stubblefield WA, Cardwell AS, Adams WJ, Nordheim E (2018) Development and application of a biotic ligand model for predicting the chronic toxicity of dissolved and precipitated aluminum to aquatic organisms. Environ Toxicol Chem 37(1):70–79. https://doi.org/10.1002/etc.4020
Bat L, Raffaelli D, Marr IL (1998) The accumulation of copper, zinc and cadmium by the amphipod Corophium volutator (Pallas). J Exp Mar Biol Ecol 223(2):167–184. https://doi.org/10.1016/s0022-0981(97)00162-7
Conradi M, Depledge MH (1999) Effects of zinc on the life-cycle, growth and reproduction of the marine amphipod Corophium volutator. Mar Ecol Prog Ser 176:131–138. https://doi.org/10.3354/meps176131
Ahearn GA, Mandal PK, Mandal A (2004) Mechanisms of heavy-metal sequestration and detoxification in crustaceans: a review. J Comp Physiol B-Biochem Syst Environ Physiol 174(6):439–452. https://doi.org/10.1007/s00360-004-0438-0
Rainbow PS, Luoma SN (2011) Metal toxicity, uptake and bioaccumulation in aquatic invertebrates-Modelling zinc in crustaceans. Aquat Toxicol 105(3–4):455–465. https://doi.org/10.1016/j.aquatox.2011.08.001
Fabrega J, Tantra R, Amer A, Stolpe B, Tomkins J, Fry T, Lead JR, Tyler CR, Galloway TS (2012) Sequestration of zinc from zinc oxide nanoparticles and life cycle effects in the sediment dweller amphipod corophium volutator. Environ Sci Technol 46(2):1128–1135. https://doi.org/10.1021/es202570g
McAfee BJ, Gould WD, Nadeau JC, da Costa ACA (2001) Biosorption of metal ions using chitosan, chitin, and biomass of Rhizopus oryzae. Sep Sci Technol 36(14):3207–3222. https://doi.org/10.1081/ss-100107768
Tarpani RRZ, Lapolli FR, Lobo-Recio MA (2015) Removal of aluminum from synthetic solutions and well water by chitin: batch and continuous experiments. Desalin Water Treat 53(13):3531–3542. https://doi.org/10.1080/19443994.2013.873741
Anastopoulos I, Bhatnagar A, Bikiaris DN, Kyzas GZ (2017) Chitin Adsorbents for Toxic Metals: a Review. Int J Mol Sci 18(1):11. https://doi.org/10.3390/ijms18010114
Lobo-Recio MA, Lapolli FR, Belli TJ, Folzke CT, Tarpani RRZ (2013) Study of the removal of residual aluminum through the biopolymers carboxymethylcellulose, chitin, and chitosan. Desalin Water Treat 51(7–9):1735–1743. https://doi.org/10.1080/19443994.2012.715133
Becker JS, Zoriy M, Matusch A, Wu B, Salber D, Palm C, Becker JS (2010) Bioimaging of metals by laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS). Mass Spectrom Rev 29(1):156–175. https://doi.org/10.1002/mas.20239
Pozebon D, Scheffler GL, Dressler VL (2017) Recent applications of laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) for biological sample analysis: a follow-up review. J Anal At Spectrom 32(5):890–919. https://doi.org/10.1039/c7ja00026j
Pozebon D, Scheffler GL, Dressler VL, Nunes MAG (2014) Review of the applications of laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) to the analysis of biological samples. J Anal At Spectrom 29(12):2204–2228. https://doi.org/10.1039/c4ja00250d
Walton RC, McCrohan CR, Livens F, White KN (2010) Trophic transfer of aluminium through an aquatic grazer-omnivore food chain. Aquat Toxicol 99(1):93–99. https://doi.org/10.1016/j.aquatox.2010.04.003
Buck BH, Krause G, Michler-Cieluch T, Brenner M, Buchholz CM, Busch JA, Fisch R, Geisen M, Zielinski O (2008) Meeting the quest for spatial efficiency: progress and prospects of extensive aquaculture within offshore wind farms. Helgoland Mar Res 62(3):269–281. https://doi.org/10.1007/s10152-008-0115-x
Di Tullio GR, Mariani P, Benassai G, Di Luccio D, Grieco L (2018) Sustainable use of marine resources through offshore wind and mussel farm co-location. Ecol Model 367:34–41. https://doi.org/10.1016/j.ecolmodel.2017.10.012
The authors thank Emily Fischer for technical assistance with ICP-MS analyses.
Open Access funding enabled and organized by Projekt DEAL. The project was funded within the governmental research and development-budget of the German Federal Ministry of Transport and Digital Infrastructure (BMVI).
Department G3 - Biochemistry, Federal Institute of Hydrology, Ecotoxicology, Am Mainzer Tor 1, Koblenz, 56068, Germany
Anna Maria Bell, Julia Regnery, Georg Reifferscheid & Sebastian Buchinger
Department G2 - Aquatic Chemistry, Federal Institute of Hydrology, Am Mainzer Tor 1, Koblenz, 56068, Germany
Marcus von der Au, Björn Meermann & Thomas Ternes
Section B2 - Steel Structures and Corrosion Protection, Federal Waterways Engineering and Research Institute, Kußmaulstr. 17, Karlsruhe, 76187, Germany
Division 1.1 - Inorganic Trace Analysis, Federal Institute for Materials Research and Testing, Richard-Willstätter-Str. 11, Berlin, 12489, Germany
Marcus von der Au & Björn Meermann
Anna Maria Bell
Marcus von der Au
Julia Regnery
Björn Meermann
Georg Reifferscheid
Thomas Ternes
Sebastian Buchinger
BM and TT formulated the overarching research goals and aims. AMB and SB conceived and designed this study. AMB and MvdA coordinated and conducted the experiments. The data analysis and visualization were performed by AMB and SB. All authors contributed to the discussion and interpretation of results. The manuscript was drafted by AMB, MvdA and SB and revised by all authors. All authors read and approved the final manuscript.
Correspondence to Sebastian Buchinger.
Tables and Figures.
Bell, A.M., von der Au, M., Regnery, J. et al. Does galvanic cathodic protection by aluminum anodes impact marine organisms?. Environ Sci Eur 32, 157 (2020). https://doi.org/10.1186/s12302-020-00441-3
Galvanic anodes
Metal uptake
Corophium volutator | CommonCrawl |
\begin{document}
\title{Bound state nodal solutions for the non-autonomous Schr\"{o} dinger--Poisson system in $\mathbb{R}^{3}$} \author{Juntao Sun$^{a}$\thanks{ E-mail address: [email protected](J. Sun)}, Tsung-fang Wu$^{b}$\thanks{ E-mail address: [email protected] (T.-F. Wu)} \\
{\footnotesize $^a$\emph{School of Mathematics and Statistics, Shandong University of Technology, Zibo, 255049, P.R. China }}\\ {\footnotesize $^b$\emph{Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan }}} \date{} \maketitle
\begin{abstract} In this paper, we study the existence of nodal solutions for the non-autonomous Schr\"{o}dinger--Poisson system: \begin{equation*} \left\{ \begin{array}{ll}
-\Delta u+u+\lambda K(x) \phi u=f(x) |u|^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2} & \text{ in }\mathbb{R}^{3}, \end{array} \right. \end{equation*} where $\lambda >0$ is a parameter and $2<p<4$. Under some proper assumptions on the nonnegative functions $K(x)$ and $f(x)$, but not requiring any symmetry property, when $\lambda$ is sufficiently small, we find a bounded nodal solution for the above problem by proposing a new approach, which changes sign exactly once in $\mathbb{R}^{3}$. In particular, the existence of a least energy nodal solution is concerned as well. \end{abstract}
\section{Introduction}
Consider the non-autonomous Schr\"{o}dinger--Poisson system in the form: \begin{equation} \left\{ \begin{array}{ll}
-\Delta u+u+\lambda K(x)\phi u=f\left( x\right) |u|^{p-2}u & \text{ in } \mathbb{R}^{3}, \\ -\Delta \phi =K(x)u^{2} & \text{ in }\mathbb{R}^{3}, \end{array} \right. \tag{$SP_{\lambda }$} \end{equation} where $\lambda >0,$ $2<p<4$ and the functions $f(x)$ and $K(x)$ satisfy the following assumptions:
\begin{itemize} \item[$\left( F1\right) $] $f(x)$ is a positive continuous function on $ \mathbb{R}^{3}$ such that \begin{equation*} \lim_{\left\vert x\right\vert \rightarrow \infty }f\left( x\right) =f_{\infty }>0\text{ uniformly on}\ \mathbb{R}^{3}, \end{equation*} and \begin{equation*} f_{\max }:=\sup_{x\in \mathbb{R}^{3}}f\left( x\right) <\frac{f_{\infty }}{ A\left( p\right) ^{\frac{p-2}{2}}}, \end{equation*} where \begin{equation*} A\left( p\right) =\left\{ \begin{array}{ll} \left( \frac{4-p}{2}\right) ^{\frac{1}{p-2}}, & \text{ if }2<p\leq 3, \\ \frac{1}{2}, & \text{ if }3<p<4. \end{array} \right. \end{equation*}
\item[$\left( K1\right) $] $K(x)\in L^{\infty }(\mathbb{R}^{3})\backslash \left\{ 0\right\} $ is a nonnegative function on $\mathbb{R}^{3}.$ \end{itemize}
In quantum mechanics, Schr\"{o}dinger--Poisson systems (SP systems for short), of the form similar to system $(SP_{\lambda }),$ can be used to describe the interaction of a charged particle with the electrostatic field. Indeed, the unknowns $u$ and $\phi $ represent the wave functions associated with the particle and the electric potentials, respectively. The function $ K(x)$ denotes a nonnegative density charge, and the local nonlinearity $
f\left( x\right) |u|^{p-2}u$ (or, more generally, $g(x,u)$) simulates the interaction effect among many particles. For more details about its physical meaning, we refer the reader to \cite{BF} and the references therein.
It is well-known that SP systems can be transformed into the nonlinear Schr \"{o}dinger equations with a non-local term \cite{BF,CG,R}. Using system $ (SP_{\lambda })$ as an example, it becomes the following equation \begin{equation} \begin{array}{ll} -\Delta u+u+\lambda K\left( x\right) \phi _{K,u}u=f\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{3}, \end{array} \tag*{$\left( E_{\lambda }\right) $} \end{equation} where $\phi _{K,u}(x)=\frac{1}{4\pi }\int_{\mathbb{R}^{3}}\frac{K(y)}{ \left\vert x-y\right\vert }u^{2}(y)dy.$ Eq. $\left( E_{\lambda }\right) $ is variational, and its solutions are the critical points of the energy functional $I_{\lambda }\left( u\right) $ defined in $H^{1}(\mathbb{R}^{3})$ by \begin{equation*} I_{\lambda }\left( u\right) =\frac{1}{2}\left\Vert u\right\Vert _{H^{1}}^{2}+ \frac{\lambda }{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx-\frac{1}{p} \int_{\mathbb{R}^{3}}f(x)\left\vert u\right\vert ^{p}dx, \end{equation*} where $\left\Vert u\right\Vert _{H^{1}}=\left[ \int_{\mathbb{R}^{3}}\left( \left\vert \nabla u\right\vert ^{2}+u^{2}\right) dx\right] ^{1/2}$ is the standard norm in $H^{1}(\mathbb{R}^{3}).$ In view of this, variational methods have been effective tools in finding nontrivial solutions of SP systems.
In recent years, there has been much attention to SP systems like system $ (SP_{\lambda })$ on the existence of positive solutions, ground states, radial solutions and semiclassical states. We refer the reader to \cite {AR,AP,CM,CV,CKW,IR,MT,M,R,SCN,SWF,SWF1,ZZ}. More precisely, Ruiz \cite{R} studied the autonomous SP system \begin{equation} \left\{ \begin{array}{ll}
-\Delta u+u+\lambda \phi u=|u|^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =u^{2} & \text{ in }\mathbb{R}^{3}. \end{array} \right. \label{1-4} \end{equation} In order to find nontrivial solutions of system $(\ref{1-4})$ with $2<p<6,$ a Nehari-Pohozaev manifold is constructed, with the aid of the Pohozaev identity corresponding to system $(\ref{1-4}).$ As a consequence, for $ \lambda >0$ sufficiently small, two positive radial solutions and one positive radial solution have been obtained when $2<p<3$ and $3\leq p<6,$ respectively. Moreover, when $\lambda \geq \frac{1}{4},$ it has been shown that $p=3$ is a critical value for the existence of nontrivial solutions. The corresponding results have been further improved by Azzollini-Pomponio \cite{AP} by showing the existence of ground state solutions when $\lambda >0 $ and $3<p<6.$
Cerami and Varia \cite{CV} studied a class of non-autonomous SP systems without any symmetry assumptions, i.e., system $(SP_{\lambda })$ with $ \lambda =1.$ By establishing the compactness lemma and using the Nehari manifold, when $K(x)$ and $f(x)$ satisfy some suitable assumptions, the existence of positive ground state and bound state solutions have been proved for $4<p<6.$ Later, when the mass term $u$ is replaced by $V(x)u$ in system $(SP_{\lambda }),$ by assuming the decay rate of the coefficients $ V(x),K(x)$ and $f(x)$, Cerami and Molle \cite{CM} obtained the existence of positive bound state solution for system $(SP_{\lambda })$ with $\lambda =1$ and $4<p<6$ via the Nehari manifold, which complements the result in \cite {CV} in some sense.
Very recently, we \cite{SWF1} investigated the existence of a positive solution for system $(SP_{\lambda })$ with $2<p<4$ when $\lambda $ is sufficiently small. Distinguishing from the case of $4\leq p<6,$ we notice that in this case the (PS)--sequences for the energy functional $I_{\lambda } $ may not be bounded and $I_{\lambda }(tu)\rightarrow \infty $ as $ t\rightarrow \infty $ for each $u\in H^{1}(\mathbb{R}^{3})\backslash \left\{ 0\right\} $. So variational methods cannot be applied in a standard way, even restricting $I_{\lambda }$ on the Nehari manifold. Moreover, the Nehari-Pohozaev manifold presented by Ruiz is also not a ideal choice for the non-autonomous system like system $(SP_{\lambda })$, since the Pohozaev identity corresponding to system $(SP_{\lambda })$ is extremely complicated. For these reasons, in \cite{SWF1} we introduced a filtration of the Nehari manifold $\mathbf{M}_{\lambda }$ as follows \begin{equation*} \mathbf{M}_{\lambda }(c)=\{u\in \mathbf{M}_{\lambda }:I_{\lambda }(u)<c\} \text{ for some }c>0, \end{equation*} and showed that this set $\mathbf{M}_{\lambda }(c)$ under the given assumptions is the union of two disjoint nonempty sets, namely, \begin{equation*} \mathbf{M}_{\lambda }(c)=\mathbf{M}_{\lambda }^{(1)}\cup \mathbf{M}_{\lambda }^{(2)}, \end{equation*} which are both $C^{1}$ sub-manifolds of $\mathbf{M}_{\lambda }$ and natural constraints of $I_{\lambda }.$ Moreover, $\mathbf{M}_{\lambda }^{(1)}$ is bounded such that $I_{\lambda }$ is coercive and bounded below on it, whereas $I_{\lambda }$ is unbounded below on $\mathbf{M}_{\lambda }^{(2)}.$ In fact, $\mathbf{M}_{\lambda }^{(2)}$ may not contain any non-zero critical point of $I_{\lambda }$ for $\frac{1+\sqrt{73}}{3}<p<4$ (see \cite[Theorem 1.6]{SWF1}). Thus, our approach is seeking a minimizer of $I_{\lambda }$ on the constraint $\mathbf{M}_{\lambda }^{(1)}.$
Another topic which has received increasingly interest of late years is the existence of nodal (or sign-changing) solutions for SP systems, see, for example, \cite{AS,ASS,BF1,CT,I,KS,LXZ,LWZ,SW,WZ}. Recall that a solution $ (u,\phi )$ to SP systems is called a nodal solution if $u$ changes sign, i.e., $u^{\pm }\not\equiv 0,$ where \begin{equation*} u^{+}(x)=\max \{u(x),0\}\text{ and }u^{-}(x)=\min \{u(x),0\}. \end{equation*} By using the Nehari manifold and gluing solution pieces together, Kim and Seok \cite{KS} proved the existence of a radial nodal solution with prescribed numbers of nodal domains for system $(\ref{1-4})$ with $\lambda >0 $ and $4<p<6$. Almost simultaneously, a similar result to \cite{KS} for $ 4\leq p<6$ has been established by Ianni \cite{I} via a dynamical approach together with a limit procedure. Of particular note is that all nodal solutions found in \cite{I,KS} have certain types of symmetries, and thus the system is required to have a certain group invariance.
In \cite{WZ}, Wang and Zhou studied the following non-autonomous SP system without any symmetry \begin{equation} \left\{ \begin{array}{ll} -\Delta u+V(x)u+\phi u=\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R }^{3}, \\ -\Delta \phi =u^{2} & \ \text{in }\mathbb{R}^{3}. \end{array} \right. \label{1-6} \end{equation} By using the nodal Nehari manifold \begin{equation*}
\mathbf{N}=\left\{ u\in H\text{ }|\text{ }\left\langle I^{\prime }(u),u^{+}\right\rangle =\left\langle I^{\prime }(u),u^{-}\right\rangle =0 \text{ and }u^{\pm }\neq 0\right\} \end{equation*} as well as the Brouwer degree theory, the existence of a least energy nodal solution for system $(\ref{1-6})$ with $4<p<6$ has been proved when either $ V(x)$ is a positive constant or $V(x)\in C(\mathbb{R}^{3},\mathbb{R}^{+})$ such that $H\subset H^{1}(\mathbb{R}^{3})$ and the embedding $ H\hookrightarrow L^{q}(\mathbb{R}^{3})(2<p<6)$ is compact. Applying the same approach, some similar results to \cite{WZ} have been obtained in \cite {AS,ASS,BF1,CT,LXZ,SW} when the nonlinearity is either $g(x,u)$ or $ f(x)\left\vert u\right\vert ^{p-2}u(4\leq p<6)$. Note that such a $g(x,u)$ is merely a general form of $f(x)\left\vert u\right\vert ^{p-2}u(4\leq p<6)$ , not covering the case of $2<p<4.$
In \cite{LWZ}, Liu, Wang and Zhang proved the existence of infinitely many nodal solutions for system $(\ref{1-6})$ with $3<p<6$ when $V(x)$ is coercive in $\mathbb{R}^{3}$ for recovering the compactness. The proof is mainly based on the method of invariant sets of descending flow. Furthermore, in the case of $3<p<4,$ a perturbation approach is also used by constructing an auxiliary system and passing the limit to the original one.
To the best of our knowledge, there seems no result in the existing literature on nodal solutions of SP systems in the case of $2<p<4$, except \cite{LWZ}. Inspired by this fact, in the present paper we are interested in the existence of a nodal solution for a class of non-autonomous SP systems when the nonlinearity is like $f(x)\left\vert u\right\vert ^{p-2}u(2<p<4)$, i.e., system $(SP_{\lambda })$ with $2<p<4.$ It is worth emphasizing that in this case the existence of a least energy nodal solution is concerned as well.
We wish to point out that the approaches in \cite {AS,ASS,BF1,CT,I,KS,LXZ,SW,WZ} are only valid for the case of $4\leq p<6,$ and that the approach in \cite{LWZ} can only solve the case of $3<p<6.$
In this study, following a part of the idea in our recent paper \cite{SWF1}, we propose a new approach to seek nodal solutions of system $(SP_{\lambda })$ with $2<p<4.$ That is, we construct a nonempty nodal set $\mathbf{N} _{\lambda }^{(1)}$ in the bounded set $\mathbf{M}_{\lambda }^{(1)}$ introduced in \cite{SWF1}, where $I_{\lambda }$ is coercive and bounded below, and then minimize $I_{\lambda }$ on it, not on the nodal Nehari manifold $\mathbf{N.}$ In fact, such a $\mathbf{N}_{\lambda }^{(1)}$ is a subset of $\mathbf{N}.$
In analysis, we have to face several challenges. First of all, note that the nodal set\ $\mathbf{N}_{\lambda }^{(1)}$ is not manifold. Then one cannot talk about vector fields on $\mathbf{N}_{\lambda }^{(1)}$ and one cannot easily construct deformations on $\mathbf{N}_{\lambda }^{(1)}.$ As a consequence, min-max values for $I_{\lambda }$ on $\mathbf{N}_{\lambda }^{(1)}$ are not automatically critical points of $I_{\lambda }.$ In fact, $ \mathbf{N}_{\lambda }^{(1)}\cap H^{2}(\mathbb{R}^{3})$ are codimension $2$ submanifolds of $H^{2}(\mathbb{R}^{3})$ (see \cite{BW2,BW}). Secondly, since $\mathbf{N}_{\lambda }^{(1)}$ is just a subset of the nodal Nehari manifold $ \mathbf{N}$, it seems not easy to show that $\mathbf{N}_{\lambda }^{(1)}\neq \emptyset ,$ which has never been involved before. Thirdly, for each $u\in H^{1}(\mathbb{R}^{3})$ with $u^{\pm }\not\equiv 0,$ the function $\widetilde{ h}(s,t)=I_{\lambda }\left( su^{+}+tu^{-}\right) $ is not strictly concave on $(0,\infty )\times (0,\infty )$ when $2<p<4,$ which is totally different from the case of $4\leq p<6$. Finally, we notice that the decomposition \begin{equation*} \int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx=\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}\left( u^{+}\right) ^{2}dx+\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}\left( u^{-}\right) ^{2}dx \end{equation*} does not hold in general, making the problem more complicated. In order to overcome these difficulties, in this paper some new ideas are introduced and some new estimates are established.
\begin{definition} $(u,\phi )$ is called a least energy nodal solution of system $(SP_{\lambda })$, if $(u,\phi )$ is a solution of system $(SP_{\lambda })$ which has the least energy among all nodal solutions of system $(SP_{\lambda })$. \end{definition}
We now summarize our main results as follows.
\begin{theorem} \label{t2}Suppose that $2<p<4,$ and conditions $(F1)$ and $(K1)$ hold. In addition, we assume that
\begin{itemize} \item[$(F2)$] there exists $0<r_{f}<1$ such that $f\left( x\right) \geq f_{\infty }+d_{0}\exp \left( -\left\vert x\right\vert ^{r_{f}}\right) $ for some $d_{0}>0$ and for all $x\in \mathbb{R}^{3};$
\item[$(K2)$] $K\left( x\right) \lneqq K_{\infty }$ for all $x\in \mathbb{R} ^{3}$ and $\lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) =K_{\infty }>0\ $uniformly on$\ \mathbb{R}^{3}.$ \end{itemize}
Then there exists $\lambda ^{\ast }>0$ such that for every $0<\lambda <\lambda ^{\ast },$ system $(SP_{\lambda })$ admits a nodal solution $ (u_{\lambda },\phi _{K,u_{\lambda }})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}( \mathbb{R}^{3}),$ which changes sign exactly once in $\mathbb{R}^{3}.$ Furthermore, there holds \begin{equation*} \left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{1}{p-2}}\leq \left\Vert u_{\lambda }^{\pm }\right\Vert _{H^{1}}<\left( \frac{2S_{p}^{p}}{\left( 4-p\right) f_{\max }}\right) ^{\frac{1}{p-2}}, \end{equation*} and \begin{equation*} \left\Vert \phi _{K,u_{\lambda}}\right\Vert _{D^{1,2}}\leq\overline{S} ^{-1}S_{12/5}^{-2}K_{\max }\left( \frac{2S_{p}^{p}}{\left( 4-p\right) f_{\max }}\right) ^{\frac{2}{p-2}}, \end{equation*} where $S_{p}$ is the best constant for the embedding of $H^{1}(\mathbb{R} ^{3})$ in $L^{p}(\mathbb{R}^{3})$ with $2<p<4,$ $\overline{S}$ is the best constant for the embedding of $D^{1,2}(\mathbb{R}^{3})$ in $L^{6}(\mathbb{R} ^{3}),$ and $S_{12/5}=S_{p}$ with $p=12/5$. \end{theorem}
\begin{theorem} \label{t3}Suppose that $2<p<4$ and conditions ${(F1)}-{(F2)}$ and $\left( K1\right) $ hold. In addition, we assume that
\begin{itemize} \item[$\left( K3\right) $] $K(x)\in L^{2}(\mathbb{R}^{3})$ and $ \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) =0.$ \end{itemize}
Then there exists $\overline{\lambda }^{\ast }>0$ such that for each $ 0<\lambda <\overline{\lambda }^{\ast },$ system $(SP_{\lambda })$ admits a nodal solution $(u_{\lambda },\phi _{K,u_{\lambda }})\in H^{1}(\mathbb{R} ^{3})\times D^{1,2}(\mathbb{R}^{3}),$ which changes sign exactly once in $ \mathbb{R}^{3}.$ Furthermore, there holds \begin{equation*} \left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{1}{p-2}}\leq \left\Vert u_{\lambda }^{\pm }\right\Vert _{H^{1}}<\left( \frac{2S_{p}^{p}}{\left( 4-p\right) f_{\max }}\right) ^{\frac{1}{p-2}}, \end{equation*} and \begin{equation*} \left\Vert \phi _{K,u_{\lambda}}\right\Vert _{D^{1,2}}\leq\overline{S} ^{-1}S_{12/5}^{-2}K_{\max }\left( \frac{2S_{p}^{p}}{\left( 4-p\right) f_{\max }}\right) ^{\frac{2}{p-2}}. \end{equation*} \end{theorem}
According to \cite[Theorem 1.6]{SWF1}, we have the following theorem on the existence of a least energy nodal solution.
\begin{theorem} \label{t4}Suppose that $\frac{1+\sqrt{73}}{3}<p<4,$ and conditions ${(F1)}$ and ${(K1)}$ hold. In addition, we assume that
\begin{itemize} \item[$\left( D_{K,f}\right) $] the functions $f(x),K(x)\in C^{1}(\mathbb{R} ^{3})$ satisfy $\langle \nabla f(x),x\rangle \leq 0$ and \begin{equation*} \frac{3p^{2}-2p-24}{2(6-p)}K(x)+\frac{p-2}{2}\langle \nabla K(x),x\rangle \geq 0. \end{equation*} \end{itemize}
If $(u_{\lambda },\phi _{K,u_{\lambda }})$ is the nodal solution as described in Theorem \ref{t2} or \ref{t3}, then $(u_{\lambda },\phi _{K,u_{\lambda }})$ is a least energy nodal solution of system $(SP_{\lambda }).$ \end{theorem}
This paper is organized as follows. After introducing various preliminaries in Section 2, we give the estimates of energy and construct the Palais--Smale sequences in Sections 3 and 4, respectively. In Sections 5 and 6, we prove Theorems \ref{t2} and \ref{t3}, respectively.
\section{Preliminaries}
As pointed out in the section of Introduction, system $(SP_{\lambda })$ can be transferred into a nonlocal Schr\"{o}dinger equation, i.e., Eq. $\left( E_{\lambda }\right) $, and its corresponding energy functional is $ I_{\lambda }(u)$. It is not difficult to prove that $I_{\lambda }$ is a $ C^{1}$ functional with the derivative given by \begin{equation*} \left\langle I_{\lambda }^{\prime }(u),\varphi \right\rangle =\int_{\mathbb{R
}^{3}}\left( \nabla u\nabla \varphi +u\varphi +\lambda K(x)\phi _{K,u}u\varphi -f(x)|u|^{p-2}u\varphi \right) dx \end{equation*} for all $\varphi \in H^{1}(\mathbb{R}^{3}),$ where $I_{\lambda }^{\prime }$ is the Fr\'{e}chet derivative of $I_{\lambda }.$ Note that $(u,\phi )\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ is a solution of system $\left( SP_{\lambda }\right) $ if and only if $u$ is a critical point of $ I_{\lambda }$ and $\phi =\phi _{K,u}$.
Next, we give a characterization of the weak convergence for the Poisson term. The proof can be made in a similar argument as in \cite{IR}.
\begin{lemma} \label{h2}Suppose that condition $(K1)$ holds. Define the operator $\Pi : \left[ H^{1}(\mathbb{R}^{3})\right] ^{4}\rightarrow \mathbb{R}$ by \begin{equation*} \Pi \left( u,v,w,z\right) =\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{ K(x)K\left( y\right) }{\left\vert x-y\right\vert }u\left( x\right) v\left( x\right) w\left( y\right) z\left( y\right) dxdy \end{equation*} for all $\left( u,v,w,z\right) \in \left[ H^{1}(\mathbb{R}^{3})\right] ^{4}.$ Then for all $\{u_{n}\},\{v_{n}\},\{w_{n}\}\subset H^{1}(\mathbb{R}^{3})$ satisfying $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{3}),v_{n} \rightharpoonup v$ in $H^{1}(\mathbb{R}^{3}),w_{n}\rightharpoonup w$ in $ H^{1}(\mathbb{R}^{3})$ and for all $z\in H^{1}(\mathbb{R}^{3}),$ there holds \begin{equation*} \Pi \left( u_{n},v_{n},w_{n},z\right) \rightarrow \Pi \left( u,v,w,z\right) . \end{equation*} \end{lemma}
In the following lemma we summarize some useful properties on the function $ \phi _{K,u}$, which have been obtained in \cite{AP,CV}.
\begin{lemma} \label{h1}Suppose that condition $(K1)$ holds. Then for each $u\in H^{1}( \mathbb{R}^{3})$, we have the following statements.\newline $\left( i\right) $ $\left\Vert \phi _{K,u}\right\Vert _{D^{1,2}}\leq \overline{S}^{-1}S_{12/5}^{-2}K_{\max }\left\Vert u\right\Vert _{H^{1}}^{2}$ holds. As a consequence, there holds \begin{equation*} \int_{\mathbb{R}^{3}}K(x)\phi _{K,v}u^{2}dx\leq \overline{S} ^{-2}S_{12/5}^{-4}K_{\max }^{2}\Vert v\Vert _{H^{1}}^{2}\Vert u\Vert _{H^{1}}^{2}; \end{equation*} $\left( ii\right)$ Both $\phi _{K,u}\geq 0$ and $\phi _{K,u}\left( x\right) = \frac{1}{4\pi }\int_{\mathbb{R}^{3}}\frac{K\left( y\right) u^{2}\left( y\right) }{\left\vert x-y\right\vert }dy$ hold;\newline $\left( iii\right) $ For any $t>0,\ \phi _{K,tu}=t^{2}\phi _{K,u}$ holds; \newline $\left( iv\right) $ If $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{3}),$ then $\Phi \left[ u_{n}\right] \rightharpoonup \Phi \left[ u\right] $ in $ D^{1,2}(\mathbb{R}^{3}),$ where the operator $\Phi :H^{1}(\mathbb{R} ^{3})\rightarrow D^{1,2}(\mathbb{R}^{3})$ as $\Phi \left[ u\right] =\phi _{K,u};$\newline $\left( v\right) $ If we, in addition, assume that condition $\left( K3\right) $ holds, then \begin{equation*} \int_{\mathbb{R}^{3}}K(x)\phi _{K,u_{n}}u_{n}^{2}dx\rightarrow \int_{\mathbb{ R}^{3}}K(x)\phi _{K,u}u^{2}dx\text{ as }n\rightarrow \infty , \end{equation*} when $u_{n}\rightharpoonup u$ in $H^{1}(\mathbb{R}^{3}).$ \end{lemma}
Define the Nehari manifold \begin{equation*} \mathbf{M}_{\lambda }=\left\{ u\in H^{1}(\mathbb{R}^{3})\backslash \{0\}
\text{ }|\text{ }\left\langle I_{\lambda }^{\prime }\left( u\right) ,u\right\rangle =0\right\} . \end{equation*} Then $u\in \mathbf{M}_{\lambda }$ if and only if \begin{equation*}
\left\Vert u\right\Vert _{H^{1}}^{2}+\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx-\int_{\mathbb{R}^{3}}f(x)|u|^{p}dx=0. \end{equation*} Moreover, it follows from the Sobolev inequality that \begin{eqnarray*} \left\Vert u\right\Vert _{H^{1}}^{2} &\leq &\left\Vert u\right\Vert _{H^{1}}^{2}+\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx \\ &=&\int_{\mathbb{R}^{3}}f(x)\left\vert u\right\vert ^{p}dx\leq S_{p}^{-p}f_{\max }\left\Vert u\right\Vert _{H^{1}}^{p}\text{ for all }u\in \mathbf{M}_{\lambda }, \end{eqnarray*} this implies\ that \begin{equation} \left\Vert u\right\Vert _{H^{1}}\geq \left( \frac{S_{p}^{p}}{f_{\max }} \right) ^{\frac{1}{p-2}}\text{ for all }u\in \mathbf{M}_{\lambda }, \label{2} \end{equation} where $S_{p}$ is the best Sobolev constant for the embedding of $H^{1}( \mathbb{R}^{3})$ in $L^{p}(\mathbb{R}^{3}).$
As we know, the Nehari manifold $\mathbf{M}_{\lambda }$ is closely related to the behavior of the function $h_{u}:t\rightarrow I_{\lambda }\left( tu\right) $ for $t>0.$ Such map is known as fibering map. About its theory and application, we refer the reader to \cite{BB,BZ,DP,P1,P2}. For $u\in H^{1}(\mathbb{R}^{3}),$ we have \begin{equation*} h_{u}\left( t\right) =\frac{t^{2}}{2}\left\Vert u\right\Vert _{H^{1}}^{2}+ \frac{\lambda t^{4}}{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx-\frac{ t^{p}}{p}\int_{\mathbb{R}^{3}}f(x)\left\vert u\right\vert ^{p}dx. \end{equation*} By a calculation on the first and second derivatives, we find \begin{eqnarray*} h_{u}^{\prime }\left( t\right) &=&t\left\Vert u\right\Vert _{H^{1}}^{2}+\lambda t^{3}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx-t^{p-1}\int_{\mathbb{R}^{3}}f(x)\left\vert u\right\vert ^{p}dx, \\ h_{u}^{\prime \prime }\left( t\right) &=&\left\Vert u\right\Vert _{H^{1}}^{2}+3\lambda t^{2}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx-\left( p-1\right) t^{p-2}\int_{\mathbb{R}^{3}}f(x)\left\vert u\right\vert ^{p}dx \end{eqnarray*} and \begin{equation*} th_{u}^{\prime }\left( t\right) =\left\Vert tu\right\Vert _{H^{1}}^{2}+\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,tu}\left( tu\right) ^{2}dx-\int_{\mathbb{R}^{3}}f(x)\left\vert tu\right\vert ^{p}dx. \end{equation*} Thus, for any $u\in H^{1}(\mathbb{R}^{3})\backslash \left\{ 0\right\} $ and $ t>0,$ $h_{u}^{\prime }\left( t\right) =0$ holds if and only if $tu\in \mathbf{M}_{\lambda }$. In particular, $h_{u}^{\prime }\left( 1\right) =0$ if and only if $u\in \mathbf{M}_{\lambda }.$ It is natural to split $\mathbf{ M}_{\lambda }$ into three parts corresponding to local minima, local maxima and points of inflection. Accordingly, following \cite{T}, we define \begin{eqnarray*}
\mathbf{M}_{\lambda }^{+} &=&\{u\in \mathbf{M}_{\lambda }\text{ }|\text{ } h_{u}^{\prime \prime }\left( 1\right) >0\}; \\
\mathbf{M}_{\lambda }^{0} &=&\{u\in \mathbf{M}_{\lambda }\text{ }|\text{ } h_{u}^{\prime \prime }\left( 1\right) =0\}; \\
\mathbf{M}_{\lambda }^{-} &=&\{u\in \mathbf{M}_{\lambda }\text{ }|\text{ } h_{u}^{\prime \prime }\left( 1\right) <0\}. \end{eqnarray*}
In order to look for nodal solutions of system $(SP)_{\lambda },$ we define the nodal Nehari manifold by \begin{equation*}
\mathbf{N}_{\lambda }=\left\{ u\in H^{1}(\mathbb{R}^{3})\text{ }|\text{ } \left\langle I_{\lambda }^{\prime }\left( u\right) ,u^{+}\right\rangle =\left\langle I_{\lambda }^{\prime }\left( u\right) ,u^{-}\right\rangle =0 \text{ and }u^{\pm }\neq 0\right\} , \end{equation*} which is a subset of the Nehari manifold $\mathbf{M}_{\lambda }.$ Clearly, $ u\in \mathbf{N}_{\lambda }$ if and only if \begin{equation*} \left\langle I_{\lambda }^{\prime }\left( u\right) ,u^{+}\right\rangle =\left\langle I_{\lambda }^{\prime }\left( u^{+}\right) ,u^{+}\right\rangle +\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}\left( u^{+}\right) ^{2}dx=0 \end{equation*} and \begin{equation*} \left\langle I_{\lambda }^{\prime }\left( u\right) ,u^{-}\right\rangle =\left\langle I_{\lambda }^{\prime }\left( u^{-}\right) ,u^{-}\right\rangle +\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}\left( u^{-}\right) ^{2}dx=0. \end{equation*} Moreover, by virtue of Lemma \ref{h1} $(ii),$ it is easy to verify that \begin{equation*} \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}\left( u^{+}\right) ^{2}dx=\int_{ \mathbb{R}^{3}}K(x)\phi _{K,u^{+}}\left( u^{-}\right) ^{2}dx. \end{equation*} For each $u\in \mathbf{M}_{\lambda },$ there holds \begin{eqnarray} h_{u}^{\prime \prime }\left( 1\right) &=&\left\Vert u\right\Vert _{H^{1}}^{2}+3\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx-\left( p-1\right) \int_{\mathbb{R}^{3}}f(x)\left\vert u\right\vert ^{p}dx \notag \\ &=&-\left( p-2\right) \left\Vert u\right\Vert _{H^{1}}^{2}+\lambda \left( 4-p\right) \int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx \label{2-1} \\ &=&-2\left\Vert u\right\Vert _{H^{1}}^{2}+\left( 4-p\right) \int_{\mathbb{R} ^{3}}f(x)\left\vert u\right\vert ^{p}dx \label{2-2} \\ &\leq &-2\left\Vert u\right\Vert _{H^{1}}^{2}+\left( 4-p\right) S_{p}^{-p}f_{\max }\Vert u\Vert _{H^{1}}^{p}. \notag \end{eqnarray} By $(\ref{2})$ and $(\ref{2-2}),$ we have \begin{eqnarray*} I_{\lambda }(u) &=&\frac{1}{4}\left\Vert u\right\Vert _{H^{1}}^{2}-\frac{4-p
}{4p}\int_{\mathbb{R}^{3}}f(x)|u|^{p}dx \\ &>&\frac{p-2}{4p}\left\Vert u\right\Vert _{H^{1}}^{2} \\ &\geq &\frac{p-2}{4p}\left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{2}{p-2 }}>0\text{ for all }u\in \mathbf{M}_{\lambda }^{-}, \end{eqnarray*} which indicates that $I_{\lambda }$ is coercive and bounded below on $ \mathbf{M}_{\lambda }^{-}.$
Let \begin{equation*} C\left( p\right) =\frac{A\left( p\right) \left( p-2\right) }{2p}\left( \frac{ 2}{4-p}\right) ^{\frac{2}{p-2}}\text{ for }2<p<4. \end{equation*} It is not difficult to verify that $C\left( p\right) $ is increasing on $ 2<p<4$ and that \begin{equation*} C\left( p\right) >\left\{ \begin{array}{ll} \frac{\sqrt{e}\left( p-2\right) }{p}, & \text{ if }2<p\leq 3, \\ \frac{e\left( p-2\right) }{2p}, & \text{ if }3<p<4. \end{array} \right. \end{equation*} Following \cite{SWF1}, for any $u\in \mathbf{M}_{\lambda }$ with $I_{\lambda }\left( u\right) <C\left( p\right) \left( \frac{S_{p}^{p}}{f_{\infty }} \right) ^{\frac{2}{p-2}},$ we deduce that \begin{eqnarray} C\left( p\right) \left( \frac{S_{p}^{p}}{f_{\infty }}\right) ^{\frac{2}{p-2} } &>&I_{\lambda }(u) \notag \\ &=&\frac{1}{2}\left\Vert u\right\Vert _{H^{1}}^{2}+\frac{\lambda }{4}\int_{ \mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx-\frac{1}{p}\int_{\mathbb{R}
^{3}}f(x)|u|^{p}dx \notag \\ &=&\frac{p-2}{2p}\left\Vert u\right\Vert _{H^{1}}^{2}-\frac{\lambda (4-p)}{4p }\int_{\mathbb{R}^{3}}K(x)\phi _{K,u}u^{2}dx \label{2-4} \\ &\geq &\frac{p-2}{2p}\left\Vert u\right\Vert _{H^{1}}^{2}-\lambda \overline{S }^{-2}S_{12/5}^{-4}K_{\max }^{2}\left( \frac{4-p}{4p}\right) \Vert u\Vert _{H^{1}}^{4}. \label{2-5} \end{eqnarray} It follows from (\ref{2-5}) that for $2<p<4$ and $0<\lambda <\lambda _{0},$ there exist two positive numbers $D_{1}$ and $D_{2}$ satisfying \begin{equation*} \sqrt{A\left( p\right) }\left( \frac{2S_{p}^{p}}{f_{\infty }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}<D_{1}<\left( \frac{2S_{p}^{p}}{f_{\max }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}<\sqrt{2}\left( \frac{2S_{p}^{p} }{f_{\infty }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}<D_{2} \end{equation*} such that \begin{equation*} \left\Vert u\right\Vert _{H^{1}}<D_{1}\text{ or }\left\Vert u\right\Vert _{H^{1}}>D_{2}, \end{equation*} where \begin{equation*} \lambda _{0}:=\frac{p-2}{2(4-p)}\left[ 1-A\left( p\right) \left( \frac{ f_{\max }}{f_{\infty }}\right) ^{\frac{2}{p-2}}\right] \left( \frac{ f_{\infty }(4-p)}{pS_{p}^{p}}\right) ^{\frac{2}{p-2}}\overline{S} ^{2}S_{12/5}^{4}K_{\max }^{-2}>0. \end{equation*} Note that $D_{1}\rightarrow \infty $ as $p\rightarrow 4^{-}.$ Thus, there holds \begin{eqnarray*} \mathbf{M}_{\lambda }\left( C\left( p\right) \left( \frac{S_{p}^{p}}{ f_{\infty }}\right) ^{\frac{2}{p-2}}\right) &=&\left\{ u\in \mathbf{M} _{\lambda }:J_{\lambda }\left( u\right) <C\left( p\right) \left( \frac{ S_{p}^{p}}{f_{\infty }}\right) ^{\frac{2}{p-2}}\right\} \\ &=&\mathbf{M}_{\lambda }^{(1)}\cup \mathbf{M}_{\lambda }^{(2)}, \end{eqnarray*} where \begin{equation*} \mathbf{M}_{\lambda }^{(1)}:=\left\{ u\in \mathbf{M}_{\lambda }\left( C\left( p\right) \left( \frac{S_{p}^{p}}{f_{\infty }}\right) ^{\frac{2}{p-2} }\right) :\left\Vert u\right\Vert _{H^{1}}<D_{1}\right\} \end{equation*} and \begin{equation*} \mathbf{M}_{\lambda }^{(2)}:=\left\{ u\in \mathbf{M}_{\lambda }\left( C\left( p\right) \left( \frac{S_{p}^{p}}{f_{\infty }}\right) ^{\frac{2}{p-2} }\right) :\left\Vert u\right\Vert _{H^{1}}>D_{2}\right\} . \end{equation*} For $2<p<4$ and $0<\lambda <\lambda _{0},$ we further have \begin{equation} \left\Vert u\right\Vert _{H^{1}}<D_{1}<\left( \frac{2S_{p}^{p}}{f_{\max }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}\text{ for all }u\in \mathbf{M} _{\lambda }^{(1)} \label{4-5} \end{equation} and \begin{equation} \left\Vert u\right\Vert _{H^{1}}>D_{2}>\sqrt{2}\left( \frac{2S_{p}^{p}}{ f_{\infty }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}\text{ for all }u\in \mathbf{M}_{\lambda }^{(2)}. \label{4-6} \end{equation} From $(\ref{2-2}),(\ref{4-5})$ and the Sobolev inequality it follows that \begin{equation*} h_{\lambda ,u}^{\prime \prime }\left( 1\right) \leq -2\left\Vert u\right\Vert _{H^{1}}^{2}+\left( 4-p\right) S_{p}^{-p}f_{\max }\left\Vert u\right\Vert _{H^{1}}^{p}<0\text{ for all }u\in \mathbf{M}_{\lambda }^{(1)}. \end{equation*} Using $\left( \ref{4-6}\right) $ leads to \begin{eqnarray*} \frac{1}{4}\left\Vert u\right\Vert _{H^{1}}^{2}-\frac{4-p}{4p}\int_{\mathbb{R
}^{3}}f(x)|u|^{p}dx &=&J_{\lambda }\left( u\right) \\ &<&\frac{A\left( p\right) (p-2)}{2p}\left( \frac{2S_{p}^{p}}{f_{\infty }\left( 4-p\right) }\right) ^{\frac{2}{p-2}} \\ &<&\frac{p-2}{2p}\left( \frac{2S_{p}^{p}}{f_{\infty }\left( 4-p\right) } \right) ^{\frac{2}{p-2}} \\ &<&\frac{p-2}{4p}\left\Vert u\right\Vert _{H^{1}}^{2}\text{ for all }u\in \mathbf{M}_{\lambda }^{(2)}. \end{eqnarray*} This implies that \begin{equation*} 2\left\Vert u\right\Vert _{H^{1}}^{2}<\left( 4-p\right) \int_{\mathbb{R}
^{3}}f(x)|u|^{p}dx\text{ for all }u\in \mathbf{M}_{\lambda }^{(2)}. \end{equation*} Combining the above inequality with $(\ref{2-2})$ gives \begin{equation*} h_{\lambda ,u}^{\prime \prime }\left( 1\right) >0\text{ for all }u\in \mathbf{M}_{\lambda }^{(2)}. \end{equation*}
Set \begin{equation*} \mathbf{N}_{\lambda }^{\left( 1\right) }=\left\{ u\in \mathbf{M}_{\lambda
}^{(1)}\text{ }|\text{ }\left\langle I_{\lambda }^{\prime }\left( u\right) ,u^{+}\right\rangle =\left\langle I_{\lambda }^{\prime }\left( u\right) ,u^{-}\right\rangle =0\text{ and }u^{\pm }\not\equiv 0\right\} . \label{2-15} \end{equation*} Clearly, $\mathbf{N}_{\lambda }^{\left( 1\right) }$ is a subset of $\mathbf{M }_{\lambda }^{(1)},$ and also of $\mathbf{N}_{\lambda }.$ Moreover, for $ u\in \mathbf{N}_{\lambda }^{\left( 1\right) },$ there holds \begin{equation} \left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{1}{p-2}}\leq \left\Vert u^{\pm }\right\Vert _{H^{1}}<D_{1}<\left( \frac{2S_{p}^{p}}{\left( 4-p\right) f_{\max }}\right) ^{\frac{1}{p-2}}. \label{2-20} \end{equation}
Let \begin{eqnarray} J_{\lambda }^{+}\left( u^{+},u^{-}\right) &=&\frac{1}{2}\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\frac{\lambda }{4}\left( \int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx+\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx\right) \notag \\ &&-\frac{1}{p}\int_{\mathbb{R}^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx \label{2-21} \end{eqnarray} and \begin{eqnarray} J_{\lambda }^{-}\left( u^{+},u^{-}\right) &=&\frac{1}{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\frac{\lambda }{4}\left( \int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{+}}(u^{-})^{2}dx+\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{-})^{2}dx\right) \notag \\ &&-\frac{1}{p}\int_{\mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx. \label{2-25} \end{eqnarray} Now we denote the function $\widetilde{h}\left( s,t\right) $ by \begin{equation} \widetilde{h}\left( s,t\right) =J_{\lambda }^{+}\left( su^{+},tu^{-}\right) +J_{\lambda }^{-}\left( su^{+},tu^{-}\right) \text{ for }s,t>0. \label{2-28} \end{equation} Clearly, $\widetilde{h}\left( s,t\right) =I_{\lambda }(su^{+}+tu^{-}).$ Moreover, a direct calculation shows that \begin{eqnarray*} \frac{\partial }{\partial s}\widetilde{h}\left( s,t\right) &=&s\left\Vert u^{^{+}}\right\Vert _{H^{1}}^{2}+\lambda st^{2}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx+\lambda s^{3}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{^{+}}}(u^{+})^{2}dx \\ &&-s^{p-1}\int_{\mathbb{R}^{3}}f(x)\left\vert u^{^{+}}\right\vert ^{p}dx, \\ \frac{\partial }{\partial t}\widetilde{h}\left( s,t\right) &=&t\left\Vert u^{^{-}}\right\Vert _{H^{1}}^{2}+\lambda s^{2}t\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{-})^{2}dx+\lambda t^{3}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{^{-}}}(u^{-})^{2}dx \\ &&-t^{p-1}\int_{\mathbb{R}^{3}}f(x)\left\vert u^{^{-}}\right\vert ^{p}dx \end{eqnarray*} and \begin{eqnarray*} \frac{\partial ^{2}}{\partial s^{2}}\widetilde{h}\left( s,t\right) &=&\left\Vert u^{^{+}}\right\Vert _{H^{1}}^{2}+\lambda t^{2}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx+3\lambda s^{2}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{^{+}}}(u^{+})^{2}dx \\ &&-\left( p-1\right) s^{p-2}\int_{\mathbb{R}^{3}}f(x)\left\vert u^{^{+}}\right\vert ^{p}dx, \\ \frac{\partial ^{2}}{\partial t^{2}}\widetilde{h}\left( s,t\right) &=&\left\Vert u^{^{-}}\right\Vert _{H^{1}}^{2}+\lambda s^{2}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{+}}(u^{-})^{2}dx+3\lambda t^{2}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{^{-}}}(u^{-})^{2}dx \\ &&-\left( p-1\right) t^{p-2}\int_{\mathbb{R}^{3}}f(x)\left\vert u^{^{-}}\right\vert ^{p}dx. \end{eqnarray*} If $u\in \mathbf{N}_{\lambda }^{\left( 1\right) },$ then $\frac{\partial }{ \partial s}\widetilde{h}\left( 1,1\right) =\frac{\partial }{\partial t} \widetilde{h}\left( 1,1\right) =0,$ \begin{eqnarray*} \frac{\partial ^{2}}{\partial s^{2}}\widetilde{h}\left( 1,1\right) &=&2\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx-\left( p-2\right) \int_{\mathbb{R}^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx \\ &=&-\left( p-2\right) \left( \left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx\right) \\ &&+\left( 4-p\right) \lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx \\ &=&-2\left( \left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\lambda \int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx\right) +\left( 4-p\right) \int_{ \mathbb{R}^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx \\ &\leq &-2\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\left( 4-p\right) f_{\max }S_{p}^{-p}\Vert u^{+}\Vert _{H^{1}}^{p} \\ &=&\left( 4-p\right) \left\Vert u^{+}\right\Vert _{H^{1}}^{2}\left( f_{\max }S_{p}^{-p}\Vert u^{+}\Vert _{H^{1}}^{p-2}-\frac{2}{4-p}\right) <0 \end{eqnarray*} and \begin{eqnarray*} \frac{\partial ^{2}}{\partial t^{2}}\widetilde{h}\left( 1,1\right) &=&2\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{-})^{2}dx-\left( p-2\right) \int_{\mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx \\ &=&-\left( p-2\right) \left( \left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{-})^{2}dx\right) \\ &&+\left( 4-p\right) \lambda \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{-})^{2}dx \\ &=&-2\left( \left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\lambda \int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{+}}(u^{-})^{2}dx\right) +\left( 4-p\right) \int_{ \mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx \\ &\leq &-2\left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\left( 4-p\right) f_{\max }S_{p}^{-p}\Vert u^{-}\Vert _{H^{1}}^{p} \\ &=&\left( 4-p\right) \left\Vert u^{-}\right\Vert _{H^{1}}^{2}\left( f_{\max }S_{p}^{-p}\Vert u^{-}\Vert _{H^{1}}^{p-2}-\frac{2}{4-p}\right) <0. \end{eqnarray*} Furthermore, we have the following result.
\begin{lemma} \label{h3-3}Suppose that $2<p<4$ and conditions $\left( F1\right) $ and $ \left( K1\right) $ hold. Then there exists a positive number $\widetilde{ \lambda }\leq \lambda _{0}$ such that for every $0<\lambda <\widetilde{ \lambda }$ and $u\in \mathbf{N}_{\lambda }^{\left( 1\right) },$ there exist $ \left( \frac{p}{2}\right) ^{\frac{1}{p-2}}<\widetilde{s}_{\lambda }, \widetilde{t}_{\lambda }\leq \left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}$ such that $I_{\lambda }\left( \widetilde{s}_{\lambda }u^{+}+\widetilde{t} _{\lambda }u^{-}\right) <0.$ Furthermore, there holds \begin{equation*} I_{\lambda }\left( u^{+}+u^{-}\right) =\sup_{\left( s,t\right) \in \left[ 0, \widetilde{s}_{\lambda }\right] \times \left[ 0,\widetilde{t}_{\lambda } \right] }I_{\lambda }(su^{+}+tu^{-}). \end{equation*} \end{lemma}
\begin{proof} By Lemma \ref{h1} $(i)$ and Young's inequality, \begin{equation*} s^{2}t^{2}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx\leq \frac{ s^{4}}{2}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{+}\right\Vert _{H^{1}}^{4}+\frac{t^{4}}{2}\overline{S} ^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{4}. \end{equation*} Using the above inequality, together with $(\ref{2-21})-(\ref{2-28})$ leads to \begin{eqnarray*} I_{\lambda }\left( su^{+}+tu^{-}\right) &=&\widetilde{h}\left( s,t\right) \\ &\leq &\frac{s^{2}}{2}\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\frac{\lambda s^{4}}{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx-\frac{s^{p}}{p }\int_{\mathbb{R}^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx \\ &&+\frac{t^{2}}{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\frac{\lambda t^{4}}{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{-})^{2}dx-\frac{t^{p}}{p }\int_{\mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx \\ &&+\frac{\lambda s^{4}}{4}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{+}\right\Vert _{H^{1}}^{4}+\frac{\lambda t^{4}}{4} \overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{4} \\ &\leq &g^{+}(s)+g^{-}(t), \end{eqnarray*} where \begin{equation*} g^{+}\left( s\right) =\frac{s^{2}}{2}\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\frac{\lambda s^{4}}{2}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{+}\right\Vert _{H^{1}}^{4}-\frac{s^{p}}{p}\int_{\mathbb{R} ^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx \end{equation*} and \begin{equation*} g^{-}\left( t\right) =\frac{t^{2}}{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\frac{\lambda t^{4}}{2}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{4}-\frac{t^{p}}{p}\int_{\mathbb{R} ^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx. \end{equation*} In order to arrive at the conclusion, we only need to show that there exist $ \widetilde{s}_{\lambda },\widetilde{t}_{\lambda }>0$ such that $g^{+}\left( \widetilde{s}_{\lambda }\right) ,g^{-}\left( \widetilde{t}_{\lambda }\right) <0.$
Let \begin{equation*} \widetilde{g}(t)=\frac{t^{-2}}{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{2}- \frac{t^{p-4}}{p}\int_{\mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx \text{ for }t>0. \end{equation*} A straightforward calculation gives \begin{equation*} \widetilde{g}(t_{\lambda })=0,\ \lim_{t\rightarrow 0^{+}}\widetilde{g} (t)=\infty \ \text{and}\ \lim_{t\rightarrow \infty }\widetilde{g}(t)=0, \end{equation*} where \begin{equation*} t_{\lambda }:=\left( \frac{p\left\Vert u^{-}\right\Vert _{H^{1}}^{2}}{2\int_{ \mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx}\right) ^{\frac{1}{p-2} }. \end{equation*} By the fact of $\frac{\partial }{\partial t}\widetilde{h}\left( 1,1\right) =0 $ and $(\ref{2-20})$ one has \begin{equation} \left( \frac{p(4-p)}{4}\right) ^{\frac{1}{p-2}}<t_{\lambda }\leq \left( \frac{p}{2}\right) ^{\frac{1}{p-2}}. \label{2-17} \end{equation} By calculating the derivative of $\widetilde{g}(t),$ we find \begin{eqnarray*} \widetilde{g}^{\prime }\left( t\right) &=&-t^{-3}\left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\frac{\left( 4-p\right) t^{p-5}}{p}\int_{ \mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx \\ &=&t^{-3}\left[ -\left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\frac{\left( 4-p\right) t^{p-2}}{p}\int_{\mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx\right] , \end{eqnarray*} which indicates that there exists $\widetilde{t}_{\lambda }=\left( \frac{2}{ 4-p}\right) ^{\frac{1}{p-2}}t_{\lambda }$ such that $\widetilde{g}\left( t\right) $ is decreasing when $0<t<\widetilde{t}_{\lambda }$ and is increasing when $t>\widetilde{t}_{\lambda }.$ Moreover, using $\left( \ref {2-17}\right) $ gives \begin{equation} 1<\left( \frac{p}{2}\right) ^{\frac{1}{p-2}}<\widetilde{t}_{\lambda }\leq \left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}. \label{2-18} \end{equation} Thus, by virtue of $\left( \ref{2-20}\right) $ and $(\ref{2-18}),$ we have \begin{eqnarray} \inf_{t>0}\widetilde{g}\left( t\right) &=&\widetilde{g}\left( \widetilde{t} _{\lambda }\right) \notag \\ &=&-\frac{p-2}{2(4-p)}\left[ \frac{p\left\Vert u^{-}\right\Vert _{H^{1}}^{2} }{(4-p)\int_{\mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx}\right] ^{-\frac{2}{p-2}}\left\Vert u^{-}\right\Vert _{H^{1}}^{2} \notag \\ &\leq &-\frac{p-2}{2(4-p)}\left( \frac{p}{4-p}\right) ^{-\frac{2}{p-2} }\left\Vert u^{-}\right\Vert _{H^{1}}^{2} \notag \\ &\leq &-\frac{p-2}{2(4-p)}\left( \frac{S_{p}^{p}(4-p)}{pf_{\max }}\right) ^{ \frac{2}{p-2}}, \notag \end{eqnarray} which implies that there exists a positive constant $\lambda _{1}\leq \lambda _{0}$ such that for every $0<\lambda <\lambda _{1},$ \begin{equation*} \inf_{t>0}\widetilde{g}\left( t\right) =\widetilde{g}\left( \widetilde{t} _{\lambda }\right) <-\frac{\lambda }{2}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{4}. \end{equation*} This indicates that \begin{eqnarray} g^{-}\left( \widetilde{t}_{\lambda }\right) &=&\frac{\widetilde{t}_{\lambda }^{2}}{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\frac{\lambda \widetilde{t} _{\lambda }^{4}}{2}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{4}-\frac{\widetilde{t}_{\lambda }^{p}}{p}\int_{ \mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx \notag \\ &=&\widetilde{t}_{\lambda }^{4}\left( \widetilde{g}\left( \widetilde{t} _{\lambda }\right) +\frac{\lambda }{2}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{-}\right\Vert _{H^{1}}^{4}\right) <0. \label{2-12} \end{eqnarray} Similarly, we also obtain that there exists \begin{equation*} 1<\left( \frac{p}{2}\right) ^{\frac{1}{p-2}}<\widetilde{s}_{\lambda }\leq \left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}} \end{equation*} such that \begin{equation} g^{+}\left( \widetilde{s}_{\lambda }\right) =\frac{\widetilde{s}_{\lambda }^{2}}{2}\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\frac{\lambda \widetilde{s} _{\lambda }^{4}}{2}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }^{2}\left\Vert u^{+}\right\Vert _{H^{1}}^{4}-\frac{\widetilde{s}_{\lambda }^{p}}{p}\int_{ \mathbb{R}^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx<0. \label{2-24} \end{equation} Thus, by $(\ref{2-12})$ and $(\ref{2-24}),$ for every $u\in \mathbf{N} _{\lambda }^{\left( 1\right) }$ there holds \begin{equation*} I_{\lambda }\left( \widetilde{s}_{\lambda }u^{+}+\widetilde{t}_{\lambda }u^{-}\right) =\widetilde{h}\left( \widetilde{s}_{\lambda },\widetilde{t} _{\lambda }\right) <0\text{ for all }0<\lambda <\lambda _{1}. \end{equation*} Next, we show that \begin{equation*} \sup_{\left( s,t\right) \in \left[ 0,\widetilde{s}_{\lambda }\right] \times \left[ 0,\widetilde{t}_{\lambda }\right] }I_{\lambda }\left( su^{+}+tu^{-}\right) =I_{\lambda }(u^{+}+u^{-}). \end{equation*} Set $Q_{\lambda }=\left[ 0,\widetilde{s}_{\lambda }\right] \times \left[ 0, \widetilde{t}_{\lambda }\right] .$ First, we claim that \begin{equation*} \sup_{\left( s,t\right) \in \partial Q_{\lambda }}I_{\lambda }\left( su^{+}+tu^{-}\right) <\sup_{\left( s,t\right) \in Q_{\lambda }}I_{\lambda }(su^{+}+tu^{-}). \end{equation*} Let us define \begin{eqnarray*} A_{1} &=&\left\Vert u^{+}\right\Vert _{H^{1}}^{2},A_{2}=\int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx,A_{3}=\int_{\mathbb{R} ^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx, \\ B_{1} &=&\left\Vert u^{-}\right\Vert _{H^{1}}^{2},B_{2}=\int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{-}}(u^{-})^{2}dx,B_{3}=\int_{\mathbb{R} ^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx \end{eqnarray*} and \begin{equation*} C=\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx=\int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{+}}(u^{-})^{2}dx. \end{equation*} Then \begin{equation*} \widetilde{h}\left( s,t\right) =\frac{s^{2}}{2}A_{1}+\frac{t^{2}}{2} B_{1}+\lambda \frac{s^{4}}{4}A_{2}+\lambda \frac{t^{4}}{4}B_{2}+\lambda \frac{s^{2}t^{2}}{2}C-\frac{s^{p}}{p}A_{3}-\frac{t^{p}}{p}B_{3}. \end{equation*} Clearly, there holds \begin{eqnarray*} \frac{\partial }{\partial s}\widetilde{h}\left( s,t\right) &=&s\left( A_{1}+\lambda s^{2}A_{2}+\lambda t^{2}C-s^{p-2}A_{3}\right) ; \\ \frac{\partial }{\partial t}\widetilde{h}\left( s,t\right) &=&t\left( B_{1}+\lambda t^{2}B_{2}+\lambda s^{2}C-t^{p-2}B_{3}\right) . \end{eqnarray*} It is not difficult to obtain that there exist $s_{0},t_{0}>0$ sufficiently small such that \begin{equation} \frac{\partial }{\partial s}\widetilde{h}\left( s,t\right) >0\text{ for all } \left( s,t\right) \in \left( 0,s_{0}\right) \times \left[ 0,\widetilde{t} _{\lambda }\right] \label{3-2} \end{equation} and \begin{equation} \frac{\partial }{\partial t}\widetilde{h}\left( s,t\right) >0\text{ for all } \left( s,t\right) \in \left[ 0,\widetilde{s}_{\lambda }\right] \times (0,t_{0}). \label{3-3} \end{equation} Note that $A_{1}<A_{3},B_{1}<B_{3}$ and $\widetilde{s}_{\lambda },\widetilde{ t}_{\lambda }>\left( \frac{p}{2}\right) ^{\frac{1}{p-2}}>1.$ Then there exists a positive constant $\lambda _{2}\leq \lambda _{1}$ such that for every $0<\lambda <\lambda _{2},$ \begin{equation} \frac{\partial }{\partial s}\widetilde{h}\left( s,t\right) <0\text{ for all } \left( s,t\right) \in \left\{ \widetilde{s}_{\lambda }\right\} \times \left[ 0,\widetilde{t}_{\lambda }\right] \label{3-4} \end{equation} and \begin{equation} \frac{\partial }{\partial t}\widetilde{h}\left( s,t\right) <0\text{ for all } \left( s,t\right) \in \left[ 0,\widetilde{s}_{\lambda }\right] \times \left\{ \widetilde{t}_{\lambda }\right\} . \label{3-5} \end{equation} By $(\ref{3-2})-(\ref{3-5}),$ we can conclude that \begin{equation*} \sup_{\left( s,t\right) \in \partial Q_{\lambda }}I_{\lambda }(su^{+}+tu^{-})<\sup_{\left( s,t\right) \in Q_{\lambda }}I_{\lambda }(su^{+}+tu^{-}). \end{equation*} Second, we prove that $I_{\lambda }\left( u^{+}+u^{-}\right) =\sup_{\left( s,t\right) \in Q_{\lambda }}I_{\lambda }(su^{+}+tu^{-}).$ Since $\frac{ \partial }{\partial s}\widetilde{h}\left( 1,1\right) =\frac{\partial }{ \partial t}\widetilde{h}\left( 1,1\right) =0,$ we have $\left( 1,1\right) $ is a critical point of $\widetilde{h}\left( s,t\right) $ for all $\lambda >0. $ By a calculation, we deduce that \begin{eqnarray*} \frac{\partial ^{2}}{\partial s^{2}}\widetilde{h}\left( s,t\right) &=&A_{1}+3\lambda s^{2}A_{2}+\lambda t^{2}C-\left( p-1\right) s^{p-2}A_{3}; \\ \frac{\partial ^{2}}{\partial t^{2}}\widetilde{h}\left( s,t\right) &=&B_{1}+3\lambda t^{2}B_{2}+\lambda s^{2}C-\left( p-1\right) t^{p-2}B_{3}; \\ \frac{\partial ^{2}}{\partial s\partial t}\widetilde{h}\left( s,t\right) &=&2\lambda stC. \end{eqnarray*} Then the Hessian matric of $\widetilde{h}$ at $\left( 1,1\right) $ is \begin{eqnarray*} H_{\lambda } &=&\left[ \begin{array}{cc} A_{1}+3\lambda A_{2}+\lambda C-\left( p-1\right) A_{3} & 2\lambda C \\ 2\lambda C & B_{1}+3\lambda B_{2}+\lambda C-\left( p-1\right) B_{3} \end{array} \right] \\ &=&\left[ \begin{array}{cc} A_{1}-\left( p-1\right) A_{3} & 0 \\ 0 & B_{1}-\left( p-1\right) B_{3} \end{array} \right] +\lambda \left[ \begin{array}{cc} 3A_{2}+C & 2C \\ 2C & 3B_{2}+C \end{array} \right] \end{eqnarray*} for all $\lambda >0.$ We notice that the matrix \begin{equation*} -\left[ \begin{array}{cc} A_{1}-\left( p-1\right) A_{3} & 0 \\ 0 & B_{1}-\left( p-1\right) B_{3} \end{array} \right] \end{equation*} is positive definite, since $0<A_{1}<A_{3},0<B_{1}<B_{3}$ and $2<p<4.$ Using this, together with the fact that $A_{2},B_{2},C$ are uniformly bounded for all $\lambda >0$, we get $-H_{\lambda }$ is positive definite for $\lambda >0 $ sufficiently small. This implies that there exists $r_{0}>0$ sufficiently small, independent of $\lambda $ such that $\widetilde{h}\left( 1,1\right) $ is a unique global maximum point on \begin{equation*} B_{r_{0}}\left( \left( 1,1\right) \right) =\left\{ \left( s,t\right) :s,t>0 \text{ and }\left\vert \left( s,t\right) -\left( 1,1\right) \right\vert <r_{0}\right\} \subset Q_{\lambda }. \end{equation*} Next, we show that $\widetilde{h}\left( 1,1\right) $ is a unique global maximum on $Q_{\lambda }$ for $\lambda >0$ sufficiently small$.$ If not, there exist a sequence $\left\{ \lambda _{n}\right\} \subset \mathbb{R}^{+}$ with $\lambda _{n}\rightarrow 0$ as $n\rightarrow \infty $ and points $ \left( s_{\lambda _{n}},t_{\lambda _{n}}\right) \in Q_{\lambda _{n}}\backslash B_{r_{0}}\left( \left( 1,1\right) \right) $ such that \begin{equation*} \widetilde{h}\left( s_{\lambda _{n}},t_{\lambda _{n}}\right) =\sup_{\left( s,t\right) \in Q_{\lambda _{n}}\backslash B_{r_{0}}\left( \left( 1,1\right) \right) }\widetilde{h}(s,t). \end{equation*} Since \begin{equation*} Q_{\lambda _{n}}\subset \left[ 0,\left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}} \right] \times \left[ 0,\left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}\right] , \end{equation*} we have $\left\{ \left( s_{\lambda _{n}},t_{\lambda _{n}}\right) \right\} $ is a bounded sequence. Then there exist a subsequence $\left\{ \left( s_{\lambda _{n}},t_{\lambda _{n}}\right) \right\} $ and \begin{equation*} \left( s_{0},t_{0}\right) \in \left[ 0,\left( \frac{p}{4-p}\right) ^{\frac{1 }{p-2}}\right] \times \left[ 0,\left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}} \right] \backslash B_{r_{0}}\left( \left( 1,1\right) \right) \end{equation*} such that \begin{equation*} \left( s_{\lambda _{n}},t_{\lambda _{n}}\right) \rightarrow \left( s_{0},t_{0}\right) \text{ as }n\rightarrow \infty \end{equation*} and \begin{equation*} \widetilde{h}\left( s_{0},t_{0}\right) \geq \widetilde{h}(1,1), \end{equation*} which contradicts to the fact that $\left( 1,1\right) $ is a unique global maximum point of $\widetilde{h}$ for $\lambda =0.$ Therefore, there exists a positive constant $\widetilde{\lambda }\leq \lambda _{2}$ such that for every $0<\lambda <\widetilde{\lambda },$ \begin{equation*} I_{\lambda }\left( u^{+}+u^{-}\right) =\sup_{\left( s,t\right) \in \left[ 0, \widetilde{s}_{\lambda }\right] \times \left[ 0,\widetilde{t}_{\lambda } \right] }I_{\lambda }\left( su^{+}+tu^{-}\right) . \end{equation*} This completes the proof. \end{proof}
Similar to the argument in Lemma \ref{h3-3}, we obtain that for every $ 0<\lambda <\widetilde{\lambda }$ and $u\in \mathbf{N}_{\lambda }^{\left( 1\right) }$ there exist $\left( \frac{p}{2}\right) ^{\frac{1}{p-2} }<s_{\lambda }^{+},t_{\lambda }^{-}\leq \left( \frac{p}{4-p}\right) ^{\frac{1 }{p-2}}$ such that $J_{\lambda }^{+}\left( s_{\lambda }^{+}u^{+},u^{-}\right) <0,J_{\lambda }^{-}\left( u^{+},t_{\lambda }^{-}u^{-}\right) <0$ and \begin{equation*} J_{\lambda }^{+}\left( u^{+},u^{-}\right) =\sup_{s\in \left[ 0,s_{\lambda }^{+}\right] }J_{\lambda }^{+}\left( su^{+},u^{-}\right) ;\text{ }J_{\lambda }^{-}\left( u^{+},u^{-}\right) =\sup_{t\in \left[ 0,t_{\lambda }^{-}\right] }J_{\lambda }^{-}(u^{+},tu^{-}). \end{equation*} Furthermore, we have the following result.
\begin{lemma} \label{h3-5}Suppose that $2<p<4,$ and conditions $\left( F1\right) $ and $ \left( K1\right) $ hold. Let $\widetilde{\lambda }>0$ be as in Lemma \ref {h3-3}. Then for every $0<\lambda <\widetilde{\lambda }$ and $u\in \mathbf{N} _{\lambda }^{\left( 1\right) }$ there exist $0<s_{0}^{+},t_{0}^{-}\leq 1$ such that $s_{0}^{+}u^{+},t_{0}^{-}u^{-}\in \mathbf{M}_{\lambda }^{-}$ and \begin{equation*} \sup_{s\in \left[ 0,s_{\lambda }^{+}\right] }I_{\lambda }\left( su^{+}\right) =I_{\lambda }\left( s_{0}^{+}u^{+}\right) \geq \alpha _{\lambda }^{-};\text{ }\sup_{t\in \left[ 0,t_{\lambda }^{-}\right] }I_{\lambda }\left( tu^{-}\right) =I_{\lambda }\left( t_{0}^{-}u^{-}\right) \geq \alpha _{\lambda }^{-}, \end{equation*} where $\alpha _{\lambda }^{-}=\inf_{u\in \mathbf{M}_{\lambda }^{-}}I_{\lambda }(u).$ In particular, \begin{equation*} J_{\lambda }^{\pm }\left( u^{+},u^{-}\right) \geq \alpha _{\lambda }^{-} \text{ for all }u\in \mathbf{N}_{\lambda }^{\left( 1\right) }. \end{equation*} \end{lemma}
\begin{proof} We only prove the case of $"+",$ since the case of $"-"$ is analogous. Let \begin{eqnarray*} \widehat{h}_{u^{+}}\left( s\right) &=&I_{\lambda }\left( su^{+}\right) \\ &=&\frac{1}{2}\left\Vert su^{+}\right\Vert _{H^{1}}^{2}+\frac{\lambda }{4} \int_{\mathbb{R}^{3}}K(x)\phi _{K,su^{+}}(su^{+})^{2}dx-\frac{1}{p}\int_{
\mathbb{R}^{3}}f(x)|su^{+}|^{p}dx \end{eqnarray*} for $s>0.$ Clearly, \begin{eqnarray*}
\widehat{h}_{u^{+}}^{\prime }\left( s\right) &=&s\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\lambda s^{3}\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx-s^{p-1}\int_{\mathbb{R}^{3}}f(x)|u^{+}|^{p}dx \\ &=&s^{3}\left( \widehat{g}_{u^{+}}\left( s\right) +\lambda \int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx\right) , \end{eqnarray*} where \begin{equation*}
\widehat{g}_{u^{+}}\left( s\right) =s^{-2}\left\Vert u^{+}\right\Vert _{H^{1}}^{2}-s^{p-4}\int_{\mathbb{R}^{3}}f(x)|u^{+}|^{p}dx. \end{equation*} Analyzing the functions $\widehat{g}_{u^{+}}$ leads to \begin{equation*} \widehat{g}_{u^{+}}\left( \widehat{s}\right) =0,\ \lim_{s\rightarrow 0^{+}} \widehat{g}_{u^{+}}(s)=\infty \ \text{and}\ \lim_{s\rightarrow \infty } \widehat{g}_{u^{+}}(s)=0, \end{equation*} where \begin{equation*} \left( \frac{4-p}{2}\right) ^{\frac{1}{p-2}}<\widehat{s}:=\left( \frac{ \left\Vert u^{+}\right\Vert _{H^{1}}^{2}}{\int_{\mathbb{R}
^{3}}f(x)|u^{+}|^{p}dx}\right) ^{\frac{1}{p-2}}\leq 1. \end{equation*} Moreover, the derivative of $\widehat{g}_{u^{+}}\left( s\right) \ $is the following \begin{eqnarray*} \widehat{g}_{u^{+}}^{\prime }\left( s\right) &=&-2s^{-3}\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\left( 4-p\right) s^{p-5}\int_{\mathbb{R}
^{3}}f(x)|u^{+}|^{p}dx \\ &=&s^{-3}\left( -2\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\left( 4-p\right)
s^{p-2}\int_{\mathbb{R}^{3}}f(x)|u^{+}|^{p}dx\right) , \end{eqnarray*} which indicates that $\widehat{g}_{u^{+}}\left( s\right) $ is decreasing when $0<s<\left( \frac{2}{4-p}\right) ^{\frac{1}{p-2}}\widehat{s}$ and is increasing when $s>\left( \frac{2}{4-p}\right) ^{\frac{1}{p-2}}\widehat{s}$ and \begin{eqnarray*} \inf_{s>0}\widehat{g}_{u^{+}}\left( s\right) &=&\widehat{g}_{u^{+}}\left( \left( \frac{2}{4-p}\right) ^{\frac{1}{p-2}}\widehat{s}\right) \\ &=&\frac{2-p}{4-p}\left( \frac{\left\Vert u^{+}\right\Vert _{H^{1}}^{2}}{ \int_{\mathbb{R}^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx}\right) ^{-\frac{ 2}{p-2}}\left( \frac{2}{4-p}\right) ^{-\frac{2}{p-2}}\left\Vert u^{+}\right\Vert _{H^{1}}^{2} \\ &=&-\frac{p-2}{4-p}\left( \frac{4-p}{2}\right) ^{\frac{2}{p-2}}\widehat{s} ^{-2}\Vert u^{+}\Vert _{H^{1}}^{2} \\ &<&-\frac{p-2}{4-p}\left( \frac{4-p}{2}\right) ^{\frac{2}{p-2}}\Vert u^{+}\Vert _{H^{1}}^{2}. \end{eqnarray*} Note that \begin{equation*} \left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{1}{p-2}}\leq \left\Vert u^{+}\right\Vert _{H^{1}}<\left( \frac{2S_{p}^{p}}{f_{\max }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}. \end{equation*} Similar to the argument in Lemma \ref{h3-3}, we obtain that for $0<\lambda < \widetilde{\lambda },$ \begin{equation*} \inf_{s>0}\widehat{g}_{u^{+}}\left( s\right) <-\lambda \int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx. \end{equation*} Then there are two numbers $s_{0}^{+}$ and $\widehat{s}_{0}^{+}$ satisfying \begin{equation*} \widehat{s}<s_{0}^{+}<\left( \frac{2}{4-p}\right) ^{\frac{1}{p-2}}\widehat{s} <\widehat{s}_{0}^{+} \end{equation*} such that \begin{equation*} \widehat{g}_{u^{+}}\left( s_{0}^{+}\right) +\lambda \int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx=0 \end{equation*} and \begin{equation*} \widehat{g}_{u^{+}}\left( \widehat{s}_{0}^{+}\right) +\lambda \int_{\mathbb{R }^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx=0. \end{equation*} Moreover, $\widehat{h}_{u^{+}}\left( s\right) $ is increasing when $s\in \left( 0,s_{0}^{+}\right) \cup \left( \widehat{s}_{0}^{+},\infty \right) $ and is decreasing when $s_{0}^{+}<s<\widehat{s}_{0}^{+}.$ Note that $ \widehat{h}_{u^{+}}\left( s_{\lambda }^{+}\right) =I_{\lambda }\left( s_{\lambda }^{+}u^{+}\right) <0$ by the fact of $J_{\lambda }^{+}\left( s_{\lambda }^{+}u^{+},u^{-}\right) <0.$ Thus, there holds $s_{0}^{+}u^{+}\in \mathbf{M}_{\lambda }^{-}$ and \begin{equation*} \sup_{s\in \left[ 0,s_{\lambda }^{+}\right] }I_{\lambda }\left( su^{+}\right) =I_{\lambda }\left( s_{0}^{+}u^{+}\right) \geq \alpha _{\lambda }^{-}. \end{equation*} Since $u\in \mathbf{N}_{\lambda }^{\left( 1\right) },$ we have \begin{equation*} \left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\lambda \left( \int_{\mathbb{R}
^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx+\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx\right) -\int_{\mathbb{R}^{3}}f(x)|u^{+}|^{p}dx=0, \end{equation*} which implies that \begin{equation*} \widehat{h}_{u^{+}}^{\prime }\left( 1\right) =-\lambda \int_{\mathbb{R} ^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx\leq 0. \end{equation*} This indicates that $0<s_{0}^{+}\leq 1.$ Finally, we obtain \begin{eqnarray*} J_{\lambda }^{+}\left( u^{+},u^{-}\right) &=&\sup_{s\in \left[ 0,s_{\lambda }^{+}\right] }J_{\lambda }^{+}\left( su^{+},u^{-}\right) \geq \sup_{s\in \left[ 0,s_{\lambda }^{+}\right] }I_{\lambda }\left( su^{+}\right) \\ &=&I_{\lambda }\left( s_{0}^{+}u^{+}\right) \geq \alpha _{\lambda }^{-}. \end{eqnarray*} This completes the proof. \end{proof}
\section{Estimates of energy}
Consider the following autonomous Schr\"{o}dinger-Poisson systems: \begin{equation} \left\{ \begin{array}{ll} -\Delta u+u+\lambda K_{\infty }\phi u=f_{\infty }\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R}^{3}, \\ -\Delta \phi =K_{\infty }u^{2} & \ \text{in }\mathbb{R}^{3}, \end{array} \right. \tag{$SP_{\lambda }^{\infty }$} \end{equation} where $\lambda >0$ and $2<p<4.$ By \cite[Theorem 1.3]{SWF1}, there exists $ \Lambda >0$ such that for each $0<\lambda <\Lambda ,$ system $\left( SP_{\lambda }^{\infty }\right) $ admits a positive solution $(w_{\lambda }^{\infty },\phi _{K_{\infty },w_{\lambda }^{\infty }})\in H^{1}(\mathbb{R} ^{3})\times D^{1,2}(\mathbb{R}^{3})$ satisfying \begin{equation} \alpha _{\lambda }^{\infty }:=I_{\lambda }^{\infty }\left( w_{\lambda }^{\infty }\right) >\alpha _{0}^{\infty }:=\frac{p-2}{2p}\left( \frac{ S_{p}^{p}}{f_{\infty }}\right) ^{\frac{2}{p-2}} \label{3-7} \end{equation} and \begin{equation} \alpha _{\lambda }^{\infty }\rightarrow \alpha _{0}^{\infty }=\frac{p-2}{2p} \left( \frac{S_{p}^{p}}{f_{\infty }}\right) ^{\frac{2}{p-2}}\text{ as } \lambda \rightarrow 0^{+}, \label{3-9} \end{equation} where $I_{\lambda }^{\infty }$ is the energy functional of system $\left( SP_{\lambda }^{\infty }\right) .$ Note that conditions $\left( F1\right) -\left( F2\right) $ and $\left( K1\right) -\left( K2\right) $ satisfy conditions $\left( D1\right) -\left( D3\right) $ in \cite[Theorem 1.4]{SWF1} , and thus for each $0<\lambda <\Lambda ,$ system $\left( SP_{\lambda }\right) $ admits a positive solution $(v_{\lambda },\phi _{K,v_{\lambda }})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ satisfying \begin{equation} \frac{p-2}{4p}\left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{2}{p-2} }<\alpha _{\lambda }^{-}:=I_{\lambda }\left( v_{\lambda }\right) <\alpha _{\lambda }^{\infty }. \label{3-10} \end{equation} Moreover, by using the Moser's iteration and the De Giorgi's iteration (or see \cite[Proposition 1]{MT}), we can easily prove that both $w_{\lambda }^{\infty }$ and $v_{\lambda }$ have exponential decay, and so, both $\phi _{K_{\infty },w_{\lambda }^{\infty }}$ and $\phi _{K,v_{\lambda }}$ have the same behavior. That is, for each $0<\varepsilon <1$ there exists $ C_{\varepsilon }>0$ such that \begin{equation*} v_{\lambda }\left( x\right) ,w_{\lambda }^{\infty }\left( x\right) ,\phi _{K_{\infty },w_{\lambda }^{\infty }}\left( x\right) ,\phi _{K,v_{\lambda }}\leq \left( x\right) C_{\varepsilon }\exp \left( -2^{\frac{\varepsilon }{ 1-\varepsilon }}\left( 1+\left\vert x\right\vert \right) ^{1-\varepsilon }\right) . \end{equation*} Note that \begin{equation*} \left( 1+\left\vert x\right\vert \right) ^{1-\varepsilon }\geq 2^{\frac{ -\varepsilon }{1-\varepsilon }}\left( 1+\left\vert x\right\vert ^{1-\varepsilon }\right) . \end{equation*} Then, we have \begin{equation} v_{\lambda }\left( x\right) ,w_{\lambda }^{\infty }\left( x\right) ,\phi _{K_{\infty },w_{\lambda }^{\infty }}\left( x\right) ,\phi _{K,v_{\lambda }}\left( x\right) \leq C_{\varepsilon }\exp \left( -\left\vert x\right\vert ^{1-\varepsilon }\right) . \label{3-1} \end{equation} For $n\in \mathbb{N},$ we define the sequence $\{w_{n}\}$ by \begin{equation} w_{n}\left( x\right) =w_{\lambda }^{\infty }(x-ne_{1}), \label{3-8} \end{equation} where $e_{1}=\left( 1,0,0\right) $. Clearly, $I_{\lambda }^{\infty }\left( w_{n}\right) =I_{\lambda }^{\infty }\left( w_{\lambda }^{\infty }\right) $ for all $n\in \mathbb{N},$ and by $(\ref{3-1})$ one has \begin{equation} w_{n}\left( x\right) =w_{\lambda }^{\infty }\left( x-ne_{1}\right) \leq C_{\varepsilon }\exp \left( \left\vert x\right\vert ^{1-\varepsilon }-n^{1-\varepsilon }\right) . \label{3-11} \end{equation} Then following \cite{ASS}, we have the following result.
\begin{lemma} \label{h-1}Suppose that conditions $\left( F1\right) -\left( F2\right) $ and $\left( K1\right) -\left( K2\right) $ hold. Then for each $0<\varepsilon <1$ there exists $C_{\varepsilon }>0$ such that\newline $(i) $ $\int_{\mathbb{R}^{3}}K(x)\phi _{K,w_{n}}v_{\lambda }^{2}dx=\int_{ \mathbb{R}^{3}}K(x)\phi _{K,v_{\lambda }}w_{n}^{2}dx\leq K_{\max }^{2}C_{\varepsilon }^{2}\exp \left( -2n^{1-\varepsilon }\right) ;$\newline $(ii) $ $\left\vert \int_{\mathbb{R}^{3}}K(x)\phi _{K,v_{\lambda }-w_{n}}\left( v_{\lambda }-w_{n}\right) ^{2}dx-\int_{\mathbb{R} ^{3}}K(x)\phi _{K,v_{\lambda }}v_{\lambda }^{2}dx-\int_{\mathbb{R} ^{3}}K(x)\phi _{K,w_{n}}w_{n}^{2}dx\right\vert \leq C_{\varepsilon }^{2}\exp \left( -2n^{1-\varepsilon }\right) .$ \end{lemma}
\begin{proof} $\left( i\right) $ By Fubini's Theorem, we have \begin{equation*} \int_{\mathbb{R}^{3}}K(x)\phi _{K,w_{n}}v_{\lambda }^{2}dx=\int_{\mathbb{R} ^{3}}K(x)\phi _{K,v_{\lambda }}w_{n}^{2}dx. \end{equation*} Then it follows from $(\ref{3-1})$ and $(\ref{3-11})$ that \begin{eqnarray*} \int_{\mathbb{R}^{3}}K(x)\phi _{K,w_{n}}v_{\lambda }^{2}dx &=&\int_{\mathbb{R }^{3}}K(x)\phi _{K,v_{\lambda }}w_{n}^{2}dx \\ &\leq &K_{\max }^{2}C_{\varepsilon }^{2}\int_{\mathbb{R}^{3}}\exp \left( -2\left\vert x\right\vert ^{1-\varepsilon }\right) \exp \left( -2\left\vert x-e_{1}\right\vert ^{1-\varepsilon }\right) dx \\ &\leq &K_{\max }^{2}C_{\varepsilon }^{2}\exp \left( -2n^{1-\varepsilon }\right) . \end{eqnarray*} $\left( ii\right) $ By part $\left( i\right) ,$ we easily arrive at the conclusion. \end{proof}
\begin{lemma} \label{h-2}Suppose that $2<p<4,$ and conditions $\left( F1\right) -\left( F2\right) $ and $\left( K1\right) -\left( K2\right) $ hold. Then there exists a positive number $\lambda _{3}\leq \min \{\widetilde{\lambda } ,\Lambda \}$ such that for each $0<\lambda <\lambda _{3},$ there exist two numbers $s_{\lambda }^{\left( 1\right) }$ and $s_{\lambda }^{\left( 2\right) }$ satisfying \begin{equation*} s_{\lambda }<1=s_{\lambda }^{\left( 1\right) }<\left( \frac{2}{4-p}\right) ^{ \frac{1}{p-2}}s_{\lambda }<s_{\lambda }^{\left( 2\right) } \end{equation*} and $s_{\lambda }^{\left( j\right) }v_{\lambda }\in \mathbf{M}_{\lambda }^{(j)}$ for $j=1,2,$ where \begin{equation} s_{\lambda }=\left( \frac{\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}}{ \int_{\mathbb{R}^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx}\right) ^{ \frac{1}{p-2}}. \label{2-9} \end{equation} Furhtermore, we have \begin{equation*} I_{\lambda }\left( v_{\lambda }\right) =\sup_{0\leq t\leq \widehat{s} _{\lambda }}I_{\lambda }(sv_{\lambda }) \end{equation*} and \begin{equation*} I_{\lambda }\left( s_{\lambda }^{\left( 2\right) }v_{\lambda }\right) =\inf_{t\geq 0}I_{\lambda }\left( sv_{\lambda }\right) <I_{\lambda }\left( \widehat{s}_{\lambda }v_{\lambda }\right) <0, \end{equation*} where $\left( \frac{p}{2}\right) ^{\frac{1}{p-2}}<\widehat{s}_{\lambda }=\left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}s_{\lambda }<\left( \frac{p}{ 4-p}\right) ^{\frac{1}{p-2}}.$ \end{lemma}
\begin{proof} Similar to the argument of Lemma \ref{h3-5}, one can easily prove that there exist two numbers $s_{\lambda }^{\left( 1\right) }$ and $s_{\lambda }^{\left( 2\right) }$ satisfying \begin{equation*} s_{\lambda }<1=s_{\lambda }^{\left( 1\right) }<\left( \frac{2}{4-p}\right) ^{ \frac{1}{p-2}}s_{\lambda }<s_{\lambda }^{\left( 2\right) } \end{equation*} such that $s_{\lambda }^{\left( j\right) }v_{\lambda }\in \mathbf{M} _{\lambda }^{(j)}$ for $j=1,2,$ and $I_{\lambda }\left( s_{\lambda }^{\left( 2\right) }v_{\lambda }\right) =\inf_{t\geq 0}I_{\lambda }(sv_{\lambda }),$ where $s_{\lambda }$ is defined as $(\ref{2-9}).$
Note that \begin{eqnarray*} I_{\lambda }\left( sv_{\lambda }\right) &=&\frac{s^{2}}{2}\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}+\frac{\lambda s^{4}}{4}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,v_{\lambda }}v_{\lambda }^{2}dx-\frac{s^{p}}{p}\int_{ \mathbb{R}^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx \\ &=&s^{4}\left[ \overline{g}_{v_{\lambda }}\left( s\right) +\frac{\lambda }{4} \int_{\mathbb{R}^{3}}K(x)\phi _{K,v_{\lambda }}v_{\lambda }^{2}dx\right] , \end{eqnarray*} where \begin{equation*} \overline{g}_{v_{\lambda }}\left( s\right) =\frac{s^{-2}}{2}\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}-\frac{s^{p-4}}{p}\int_{\mathbb{R} ^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx. \end{equation*} Clearly, $I_{\lambda }\left( sv_{\lambda }\right) =0$ if and only if \begin{equation*} \overline{g}_{v_{\lambda }}\left( s\right) +\frac{\lambda }{4}\int_{\mathbb{R }^{3}}K(x)\phi _{K,v_{\lambda }}v_{\lambda }^{2}dx=0. \end{equation*} By analyzing the functions $\overline{g}_{v_{\lambda }},$ one has \begin{equation*} \overline{g}_{v_{\lambda }}\left( \overline{s}_{\lambda }\right) =0,\ \lim_{s\rightarrow 0^{+}}g_{\lambda }(s)=\infty \text{ and}\ \lim_{s\rightarrow \infty }g_{\lambda }(s)=0, \end{equation*} where \begin{equation} \overline{s}_{\lambda }=\left( \frac{p\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}}{2\int_{\mathbb{R}^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx}\right) ^{\frac{1}{p-2}}. \label{2-8} \end{equation} Moreover, it is easy to see that \begin{eqnarray*} \overline{g}_{v_{\lambda }}^{\prime }\left( s\right) &=&-s^{-3}\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}+\frac{4-p}{p}s^{p-5}\int_{\mathbb{R} ^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx \\ &=&s^{-3}\left( -\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}+\frac{4-p}{p }s^{p-2}\int_{\mathbb{R}^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx\right) . \end{eqnarray*} This indicates that $\overline{g}_{v_{\lambda }}\left( s\right) $ is decreasing when $0<s<\widehat{s}_{\lambda }$ and is increasing when $s> \widehat{s}_{\lambda },$ where \begin{equation*} \left( \frac{p}{2}\right) ^{\frac{1}{p-2}}<\widehat{s}_{\lambda }:=\left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}s_{\lambda }<\left( \frac{p}{4-p} \right) ^{\frac{1}{p-2}}. \end{equation*} Moreover, there holds \begin{eqnarray*} \inf_{s>0}\overline{g}_{v_{\lambda }}\left( s\right) &=&\overline{g} _{v_{\lambda }}\left( \widehat{s}_{\lambda }\right) =-\frac{p-2}{2\left( 4-p\right) }\left( \frac{4-p}{p}\right) ^{\frac{2}{p-2}}s_{\lambda }^{-2}\Vert v_{\lambda }\Vert _{H^{1}}^{2} \\ &<&-\frac{p-2}{2\left( 4-p\right) }\left( \frac{4-p}{p}\right) ^{\frac{2}{p-2 }}\Vert v_{\lambda }\Vert _{H^{1}}^{2}. \end{eqnarray*} Note that \begin{equation*} \left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{1}{p-2}}\leq \left\Vert v_{\lambda }\right\Vert _{H^{1}}<\left( \frac{2S_{p}^{p}}{f_{\max }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}. \end{equation*} Then there exists a positive constant $\lambda _{3}\leq \min \{\widetilde{ \lambda },\Lambda \}$ such that for each $0<\lambda <\lambda _{3},$ \begin{equation} \inf_{s>0}\overline{g}_{v_{\lambda }}\left( s\right) =\overline{g} _{v_{\lambda }}\left( \widehat{s}_{\lambda }\right) <-\frac{\lambda }{4} \int_{\mathbb{R}^{3}}K(x)\phi _{K,v_{\lambda }}v_{\lambda }^{2}dx. \label{2-13} \end{equation} Thus, there exist two numbers $\overline{s}_{\lambda }^{(j)}(j=1,2)$ satisfying \begin{equation*} \overline{s}_{\lambda }<\overline{s}_{\lambda }^{(1)}<\left( \frac{p}{4-p} \right) ^{\frac{1}{p-2}}s_{\lambda }<\overline{s}_{\lambda }^{(2)} \end{equation*} such that \begin{equation*} \overline{g}_{v_{\lambda }}\left( \overline{s}_{\lambda }^{(j)}\right) +\lambda \int_{\mathbb{R}^{3}}K(x)\phi _{v_{\lambda }}v_{\lambda }^{2}dx=0, \end{equation*} namely, $I_{\lambda }\left( \overline{s}_{\lambda }^{(j)}v_{\lambda }\right) =0,$ where $s_{\lambda }$ and $\overline{s}_{\lambda }$ are defined as $(\ref {2-9})$ and $(\ref{2-8})$, respectively. It follows from $(\ref{2-13})$ that \begin{eqnarray*} I_{\lambda }\left( \widehat{s}_{\lambda }v_{\lambda }\right) &=&I_{\lambda }\left( \left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}s_{\lambda }v_{\lambda }\right) \\ &=&\left( \left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}s_{\lambda }\right) ^{4}\left[ g_{\lambda }\left( \left( \frac{p}{4-p}\right) ^{\frac{1}{p-2} }s_{\lambda }\right) +\frac{\lambda }{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,v_{\lambda }}v_{\lambda }^{2}dx\right] \\ &<&0, \end{eqnarray*} which leads to \begin{equation*} \inf_{t\geq 0}I_{\lambda }\left( tv_{\lambda }\right) <I_{\lambda }\left( \widehat{s}_{\lambda }v_{\lambda }\right) <0, \end{equation*} and \begin{equation*} I_{\lambda }\left( v_{\lambda }\right) =I_{\lambda }\left( s_{\lambda }^{\left( 1\right) }v_{\lambda }\right) =\sup_{0\leq t\leq \widehat{s} _{\lambda }}I_{\lambda }(sv_{\lambda }). \end{equation*} This completes the proof. \end{proof}
\begin{lemma} \label{h-3}Suppose that $2<p<4$ and conditions $\left( F1\right) ,\left( F2\right) ,$ $\left( K1\right) $ and $\left( K2\right) $ hold. Then there exist two positive number $\lambda _{4}\leq \min \{\widetilde{\lambda } ,\Lambda \}$ and $n_{0}\in \mathbb{N}$ such that for every $0<\lambda <\lambda _{4}$ and $n\geq n_{0},$ there exist two numbers $t_{n}^{\left( 1\right) }$ and $t_{n}^{\left( 2\right) }$ satisfying \begin{equation*} t_{n}^{\infty }<1=t_{n}^{\left( 1\right) }<\left( \frac{2}{4-p}\right) ^{ \frac{1}{p-2}}t_{n}^{\infty }<t_{n}^{\left( 2\right) } \end{equation*} and $t_{n}^{\left( j\right) }w_{n}\in \mathbf{M}_{\lambda }^{(j)}$ for $ j=1,2,$ where \begin{equation} t_{n}^{\infty }=\left( \frac{\left\Vert w_{n}\right\Vert _{H^{1}}^{2}}{\int_{ \mathbb{R}^{3}}f_{\infty }\left\vert w_{n}\right\vert ^{p}dx}\right) ^{\frac{ 1}{p-2}}. \label{5-1} \end{equation} Furthermore, we have \begin{equation*} I_{\lambda }\left( w_{n}\right) =\sup_{0\leq t\leq \widehat{t} _{n}}I_{\lambda }(tw_{n}), \end{equation*} and \begin{equation*} I_{\lambda }\left( t_{n}^{\left( 2\right) }w_{n}\right) =\inf_{t\geq 0}I_{\lambda }\left( tw_{n}\right) <I_{\lambda }\left( \widehat{t} _{n}w_{n}\right) <0, \end{equation*} where $\left( \frac{p}{2}\right) ^{\frac{1}{p-2}}<\widehat{t}_{n}<\left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}.$ \end{lemma}
\begin{proof} Similar to the argument of Lemma \ref{h3-5}, it is easy to prove that there exist two numbers $t_{n}^{\left( 1\right) }$ and $t_{n}^{\left( 2\right) }$ satisfying \begin{equation*} t_{n}^{\infty }<1=t_{n}^{\left( 1\right) }<\left( \frac{2}{4-p}\right) ^{ \frac{1}{p-2}}t_{n}^{\infty }<t_{n}^{\left( 2\right) } \end{equation*} and $t_{n}^{\left( j\right) }w_{n}\in \mathbf{M}_{\lambda }^{(j)}$ for $ j=1,2,$ and $I_{\lambda }\left( t_{n}^{\left( 2\right) }w_{n}\right) =\inf_{t\geq 0}I_{\lambda }\left( tw_{n}\right) ,$ where $t_{n}^{\infty }$ is defined as $(\ref{5-1})$ satisfying \begin{equation} \left( \frac{4-p}{2}\right) ^{\frac{1}{p-2}}<t_{n}^{\infty }<1. \label{5-13} \end{equation}
Note that \begin{equation*} \int_{\mathbb{R}^{3}}\left( f(x)-f_{\infty }\right) \left\vert w_{n}\right\vert ^{p}dx=o\left( 1\right) \end{equation*} and \begin{equation*} \int_{\mathbb{R}^{3}}K(x)\phi _{K,w_{n}}w_{n}^{2}dx-\int_{\mathbb{R} ^{3}}K_{\infty }\phi _{K_{\infty },w_{n}}w_{n}^{2}dx=o(1). \end{equation*} Then it follows from $\left( \ref{5-1}\right) -\left( \ref{5-13}\right) $ that there exists $n_{0}\in \mathbb{N}$ such that for any $n\geq n_{0},$ \begin{equation*} \left( \frac{4-p}{2}\right) ^{\frac{1}{p-2}}<t_{n}:=\left( \frac{\left\Vert w_{n}\right\Vert _{H^{1}}^{2}}{\int_{\mathbb{R}^{3}}f(x)\left\vert w_{n}\right\vert ^{p}dx}\right) ^{\frac{1}{p-2}}<1. \end{equation*} Moreover, by $(\ref{3-7}),$ one has $I_{\lambda }\left( w_{n}\right) =\alpha _{\lambda }^{\infty }$ for any $n\geq n_{0}.$ It is easy to see that \begin{eqnarray} I_{\lambda }\left( tw_{n}\right) &=&\frac{t^{2}}{2}\left\Vert w_{n}\right\Vert _{H^{1}}^{2}+\frac{\lambda t^{4}}{4}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,w_{n}}w_{n}^{2}dx-\frac{t^{p}}{p}\int_{\mathbb{R} ^{3}}f(x)\left\vert w_{n}\right\vert ^{p}dx \notag \\ &=&t^{4}\left[ g_{w_{n}}\left( t\right) +\frac{\lambda }{4}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,w_{n}}w_{n}^{2}dx\right] , \label{5-14} \end{eqnarray} where \begin{equation*} g_{w_{n}}\left( t\right) =\frac{t^{-2}}{2}\left\Vert w_{n}\right\Vert _{H^{1}}^{2}-\frac{t^{p-4}}{p}\int_{\mathbb{R}^{3}}f(x)\left\vert w_{n}\right\vert ^{p}dx. \end{equation*} Clearly, $I_{\lambda }\left( tw_{n}\right) =0$ if and only if \begin{equation*} g_{w_{n}}\left( t\right) +\frac{\lambda }{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,w_{n}}w_{n}^{2}dx=0. \end{equation*} By analyzing the functions $g_{w_{n}}$ one has \begin{equation*} g_{w_{n}}\left( \widetilde{t}_{n}\right) =0,\ \lim_{t\rightarrow 0^{+}}g_{w_{n}}(t)=\infty \text{ and}\ \lim_{t\rightarrow \infty }g_{w_{n}}(t)=0, \end{equation*} where \begin{equation} \widetilde{t}_{n}=\left( \frac{p\left\Vert w_{n}\right\Vert _{H^{1}}^{2}}{ 2\int_{\mathbb{R}^{3}}f(x)\left\vert w_{n}\right\vert ^{p}dx}\right) ^{\frac{ 1}{p-2}}. \label{5-4} \end{equation} A direct calculation shows that \begin{equation*} g_{w_{n}}^{\prime }\left( t\right) =t^{-3}\left( -\left\Vert w_{n}\right\Vert _{H^{1}}^{2}+\frac{4-p}{p}t^{p-2}\int_{\mathbb{R} ^{3}}f(x)\left\vert w_{n}\right\vert ^{p}dx\right) . \end{equation*} This implies that $g_{w_{n}}\left( t\right) $ is decreasing when $0<t< \widehat{t}_{n}$ and is increasing when $t>\widehat{t}_{n},$ where \begin{equation} \left( \frac{p}{2}\right) ^{\frac{1}{p-2}}<\widehat{t}_{n}:=\left( \frac{p}{ 4-p}\right) ^{\frac{1}{p-2}}t_{n}<\left( \frac{p}{4-p}\right) ^{\frac{1}{p-2} }. \label{5-15} \end{equation} Moreover, there holds \begin{eqnarray*} \inf_{t>0}g_{w_{n}}\left( t\right) &=&g_{w_{n}}\left( \widehat{t} _{n}\right) =-\frac{p-2}{2\left( 4-p\right) }\left( \frac{4-p}{p}\right) ^{ \frac{2}{p-2}}t_{n}^{-2}\Vert w_{n}\Vert _{H^{1}}^{2} \\ &<&-\frac{p-2}{2\left( 4-p\right) }\left( \frac{4-p}{p}\right) ^{\frac{2}{p-2 }}\Vert w_{n}\Vert _{H^{1}}^{2}. \end{eqnarray*} Since \begin{equation*} \left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{1}{p-2}}\leq \left\Vert w_{n}\right\Vert _{H^{1}}<\left( \frac{2S_{p}^{p}}{f_{\max }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}, \end{equation*} there exists a positive number $\lambda _{4}\leq \min \{\widetilde{\lambda } ,\Lambda \}$ such that \begin{equation*} \inf_{t>0}g_{w_{n}}\left( t\right) =g_{w_{n}}\left( \widehat{t}_{n}\right) <- \frac{\lambda }{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,w_{n}}w_{n}^{2}dx \end{equation*} for all $0<\lambda <\lambda _{4}.$ Thus, there are two numbers $\widehat{t} _{n}^{(1)}$ and $\widehat{t}_{n}^{\left( 2\right) }$ satisfying $\widetilde{t }_{n}<\widehat{t}_{n}^{\left( 1\right) }<\widehat{t}_{n}<\widehat{t} _{n}^{\left( 2\right) }$ such that \begin{equation*} g_{w_{n}}\left( \widehat{t}_{n}^{\left( j\right) }\right) +\frac{\lambda }{4} \int_{\mathbb{R}^{3}}K(x)\phi _{w_{n}}w_{n}^{2}dx=0\text{ for }j=1,2, \end{equation*} i.e., $I_{\lambda }\left( \widehat{t}_{n}^{\left( j\right) }w_{n}\right) =0,$ where $\widetilde{t}_{n}$ and $\widehat{t}_{n}$ are defined as $(\ref{5-4})$ and $(\ref{5-15})$, respectively. It follows from $(\ref{5-14})$ that \begin{eqnarray*} I_{\lambda }\left( \widehat{t}_{n}w_{n}\right) &=&\left( \left( \frac{p}{4-p }\right) ^{\frac{1}{p-2}}t_{n}\right) ^{4}\left[ g_{w_{n}}\left( \left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}t_{n}\right) +\frac{\lambda }{4}\int_{ \mathbb{R}^{3}}K(x)\phi _{K,w_{n}}w_{n}^{2}dx\right] \\ &<&0, \end{eqnarray*} which gives $\inf_{t\geq 0}I_{\lambda }\left( tw_{n}\right) <I_{\lambda }\left( \widehat{t}_{n}w_{n}\right) <0$ and $I_{\lambda }\left( w_{n}\right) =\sup_{0\leq t\leq \widehat{t}_{n}}I_{\lambda }(tw_{n}).$ This completes the proof. \end{proof}
\begin{lemma} \label{h-4}Suppose that $2<p<4,$ and conditions $\left( F1\right) -\left( F2\right) $ and $\left( K1\right) -\left( K2\right) $ hold. Then for any $ 0<\lambda <\min \{\lambda _{3},\lambda _{4}\}$ and $n\geq n_{0},$ there holds \begin{equation*} I_{\lambda }(v_{\lambda }-w_{n})>\sup_{\left( s,t\right) \in \partial \left\{ \left[ 0,\widehat{s}_{\lambda }\right] \times \left[ 0,\widehat{t} _{n}\right] \right\} }I_{\lambda }(sv_{\lambda }-tw_{n}), \end{equation*} where $\widehat{s}_{\lambda }$ and $\widehat{t}_{n}$ are defined in Lemmas \ref{h-2} and \ref{h-3}, respectively. Furthermore, we have \begin{equation*} \lim_{n\rightarrow \infty }I_{\lambda }(v_{\lambda }-w_{n})=\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }. \end{equation*} \end{lemma}
\begin{proof} Note that $1<\left( \frac{p}{2}\right) ^{\frac{1}{p-2}}<\widehat{s}_{\lambda },\widehat{t}_{n}<\left( \frac{p}{4-p}\right) ^{\frac{1}{p-2}}$ for all $ 0<\lambda <\min \{\lambda _{3},\lambda _{4}\}$ and $n\geq n_{0}$ by Lemmas \ref{h-2} and \ref{h-3}. Then for all $\left( s,t\right) \in \left[ 0, \widehat{s}_{\lambda }\right] \times \left[ 0,\widehat{t}_{n}\right] ,$ by virtue of Lemma \ref{h-1} and $(\ref{3-8}),$ we have \begin{eqnarray} &&\int_{\mathbb{R}^{3}}K(x)\phi _{K,\left( sv_{\lambda }-tw_{n}\right) }\left( sv_{\lambda }-tw_{n}\right) ^{2}dx \notag \\ &=&s^{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,v_{\lambda }}v_{\lambda }^{2}dx+t^{4}\int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty },w_{\lambda }^{\infty }}\left( w_{\lambda }^{\infty }\right) ^{2}dx+o(1). \label{5-8} \end{eqnarray} Moreover, using the fact of $w_{n}\rightarrow 0$ a.e. in $\mathbb{R}^{3}$ and \cite[Brezis-Lieb Lemma]{BLi} gives \begin{equation} \left\Vert sv_{\lambda }-tw_{n}\right\Vert _{H^{1}}^{2}=s^{2}\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}+t^{2}\left\Vert w_{n}\right\Vert _{H^{1}}^{2}+o(1) \label{5-2} \end{equation} and \begin{equation} \int_{\mathbb{R}^{3}}f(x)\left\vert sv_{\lambda }-tw_{n}\right\vert ^{p}dx=s^{p}\int_{\mathbb{R}^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx+t^{p}\int_{\mathbb{R}^{3}}f_{\infty }\left\vert w_{\lambda }^{\infty }\right\vert ^{p}dx+o(1). \label{5-9} \end{equation} It follows from $\left( \ref{5-8}\right) -\left( \ref{5-9}\right) $ that \begin{eqnarray*} I_{\lambda }\left( v_{\lambda }-w_{n}\right) &=&\frac{1}{2}\left\Vert v_{\lambda }-w_{n}\right\Vert _{H^{1}}^{2}+\frac{\lambda }{4}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,\left( v_{\lambda }-w_{n}\right) }\left( v_{\lambda }-w_{n}\right) ^{2}dx \\ &&-\frac{1}{p}\int_{\mathbb{R}^{3}}f(x)\left\vert v_{\lambda }-w_{n}\right\vert ^{p}dx \\ &=&\frac{1}{2}\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}+\frac{\lambda }{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,v_{\lambda }}v_{\lambda }^{2}dx-\frac{1 }{p}\int_{\mathbb{R}^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx \\ &&+\frac{1}{2}\left\Vert w_{\lambda }^{\infty }\right\Vert _{H^{1}}^{2}+ \frac{\lambda }{4}\int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty },w_{\lambda }^{\infty }}\left( w_{\lambda }^{\infty }\right) ^{2}dx-\frac{1 }{p}\int_{\mathbb{R}^{3}}f_{\infty }\left\vert w_{\lambda }^{\infty }\right\vert ^{p}dx+o\left( 1\right) \\ &=&\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }+o(1). \end{eqnarray*} Thus, by Lemmas \ref{h-2} and \ref{h-3}, for all $0<\lambda <\min \{\lambda _{3},\lambda _{4}\}$ and $n\geq n_{0},$ there holds \begin{equation*} I_{\lambda }(v_{\lambda }-w_{n})>\sup_{\left( s,t\right) \in D}I_{\lambda }(sv_{\lambda }-tw_{n}), \end{equation*} where $D=\left( \left[ 0,\widehat{s}_{\lambda }\right] \times \left\{ 0\right\} \right) \cup \left( \left\{ 0\right\} \times \left[ 0,\widehat{t} _{n}\right] \right) .$ Similarly, we also get \begin{equation*} I_{\lambda }(v_{\lambda }-w_{n})>\sup_{t\in \left[ 0,\widehat{t}_{n}\right] }I_{\lambda }(\widehat{s}_{\lambda }v_{\lambda }-tw_{n}) \end{equation*} and \begin{equation*} I_{\lambda }(v_{n}-w_{n})>\sup_{s\in \left[ 0,\widehat{s}_{\lambda }\right] }I_{\lambda }\left( sv_{n}-\widehat{t}_{n}w_{n}\right) . \end{equation*} These imply that \begin{equation*} I_{\lambda }\left( v_{\lambda }-w_{n}\right) >\sup_{\left( s,t\right) \in \partial \left\{ \left[ 0,\widehat{s}_{\lambda }\right] \times \left[ 0, \widehat{t}_{n}\right] \right\} }I_{\lambda }(sv_{n}-tw_{n}). \end{equation*} This completes the proof. \end{proof}
For all $\left( s,t\right) \in \left[ 0,\widehat{s}_{\lambda }\right] \times \left[ 0,\widehat{t}_{n}\right] ,$ we define \begin{eqnarray*} \widehat{h}\left( s,t\right) &=&I_{\lambda }\left( sv_{\lambda }-tw_{n}\right) \\ &=&\frac{1}{2}\left\Vert sv_{\lambda }-tw_{n}\right\Vert _{H^{1}}^{2}+\frac{ \lambda }{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,\left( sv_{\lambda }-tw_{n}\right) }\left( sv_{\lambda }-tw_{n}\right) ^{2}dx \\ &&-\frac{1}{p}\int_{\mathbb{R}^{3}}f(x)\left\vert sv_{\lambda }-tw_{n}\right\vert ^{p}dx. \end{eqnarray*} A direct calculation shows that \begin{eqnarray*} \frac{\partial }{\partial s}\widehat{h}\left( s,t\right) &=&\left\langle I_{\lambda }^{\prime }\left( sv_{\lambda }-tw_{n}\right) ,v_{\lambda }\right\rangle \\ &=&s\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}+\lambda st^{2}\int_{ \mathbb{R}^{3}}K(x)\phi _{K,w_{n}}v_{\lambda }^{2}dx+\lambda s^{3}\int_{ \mathbb{R}^{3}}K(x)\phi _{K,v_{n}}v_{\lambda }^{2}dx \\ &&-s^{p-1}\int_{\mathbb{R}^{3}}f(x)\left\vert v_{\lambda }\right\vert ^{p}dx+o(1) \end{eqnarray*} and \begin{eqnarray*} \frac{\partial }{\partial t}\widehat{h}\left( s,t\right) &=&\left\langle I_{\lambda }^{\prime }\left( sv_{\lambda }-tw_{n}\right) ,-w_{n}\right\rangle \\ &=&t\left\Vert w_{n}\right\Vert _{H^{1}}^{2}+\lambda s^{2}t\int_{\mathbb{R} ^{3}}K(x)\phi _{K,v_{\lambda }}w_{n}^{2}dx+\lambda t^{3}\int_{\mathbb{R} ^{3}}K(x)\phi _{K,w_{n}}w_{n}^{2}dx \\ &&-t^{p-1}\int_{\mathbb{R}^{3}}f(x)\left\vert w_{n}\right\vert ^{p}dx+o(1). \end{eqnarray*} Then we have the following result.
\begin{proposition} \label{h-5}Suppose that $2<p<4,$ and conditions $\left( F1\right) -\left( F2\right) $ and $\left( K1\right) -\left( K2\right) $ hold. Then there exist two positive numbers $\lambda ^{\ast }\leq \min \{\lambda _{3},\lambda _{4}\} $ and $n^{\ast }\in \mathbb{N}$ such that for every $0<\lambda <\lambda ^{\ast }$ and $n\geq n^{\ast },$ there exists $\left( s_{\lambda }^{\ast },t_{n}^{\ast }\right) \in \left( 0,\widehat{s}_{\lambda }\right) \times \left( 0,\widehat{t}_{n}\right) $ such that \begin{equation*} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) =\sup_{\left( s,t\right) \in \left[ 0,\widehat{s}_{\lambda } \right] \times \left[ 0,\widehat{t}_{n}\right] }I_{\lambda }\left( sv_{\lambda }-tw_{n}\right) <\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }, \end{equation*} and $s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\in \mathbf{N} _{\lambda }^{(1)}.$ \end{proposition}
\begin{proof} It follows from Lemmas \ref{h-2} and \ref{h-3} that $1<\left( \frac{p}{2} \right) ^{\frac{1}{p-2}}<\widehat{s}_{\lambda },\widehat{t}_{n}<\left( \frac{ p}{4-p}\right) ^{\frac{1}{p-2}}$ for all $0<\lambda <\min \{\lambda _{3},\lambda _{4}\}$ and $n\geq n_{0}.$ By Lemma \ref{h-4}, there exists $ \left( s_{\lambda }^{\ast },t_{n}^{\ast }\right) \in \left( 0,\widehat{s} _{\lambda }\right) \times \left( 0,\widehat{t}_{n}\right) $ such that \begin{equation*} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) =\sup_{\left( s,t\right) \in \left[ 0,\widehat{s}_{\lambda } \right] \times \left[ 0,\widehat{t}_{n}\right] }I_{\lambda }(sv_{\lambda }-tw_{n})\geq I_{\lambda }(v_{\lambda }-w_{n}), \end{equation*} and \begin{eqnarray} \frac{\partial }{\partial s}\widehat{h}\left( s_{\lambda }^{\ast },t_{n}^{\ast }\right) &=&\left\langle I_{\lambda }^{\prime }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) ,v_{\lambda }\right\rangle =0, \label{5-17} \\ \frac{\partial }{\partial t}\widehat{h}\left( s_{\lambda }^{\ast },t_{n}^{\ast }\right) &=&\left\langle I_{\lambda }^{\prime }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) ,-w_{n}\right\rangle =0. \label{5-18} \end{eqnarray} This implies that \begin{equation*} \left\langle I_{\lambda }^{\prime }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) ,s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right\rangle =0, \end{equation*} i.e., $s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\in \mathbf{M} _{\lambda }.$
Next, we show that \begin{equation} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) =\sup_{\left( s,t\right) \in \left[ 0,\widehat{s}_{\lambda } \right] \times \left[ 0,\widehat{t}_{n}\right] }I_{\lambda }\left( sv_{\lambda }-tw_{n}\right) <\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }. \label{5-20} \end{equation} By virtue of Lemma \ref{h-1}, one has \begin{eqnarray} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) &=&\frac{1}{2}\left\Vert s_{\lambda }^{\ast }v_{\lambda }-t^{\ast }w_{n}\right\Vert _{H^{1}}^{2}-\frac{1}{p}\int_{\mathbb{R} ^{3}}f(x)\left\vert s_{\lambda }^{\ast }v_{\lambda }-t^{\ast }w_{n}\right\vert ^{p}dx \notag \\ &&+\frac{\lambda }{4}\int_{\mathbb{R}^{3}}K(x)\phi _{K,\left( s_{\lambda }^{\ast }v_{\lambda }-t^{\ast }w_{n}\right) }\left( s_{\lambda }^{\ast }v_{\lambda }-t^{\ast }w_{n}\right) ^{2}dx \notag \\ &\leq &I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }\right) +I_{\lambda }^{\infty }\left( t_{n}^{\ast }w_{n}\right) +\frac{\lambda }{4} C_{\varepsilon }^{2}\exp \left( -2n^{1-\varepsilon }\right) \notag \\ &&-\frac{\left( t_{n}^{\ast }\right) ^{p}}{p}\int_{\mathbb{R}^{3}}(f\left( x\right) -f_{\infty })\left\vert w_{n}\right\vert ^{p}dx \notag \\ &&-\frac{1}{p}\int_{\mathbb{R}^{3}}f(x)\left( \left\vert s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right\vert ^{p}-\left\vert s_{\lambda }^{\ast }v_{\lambda }\right\vert ^{p}-\left\vert t_{n}^{\ast }w_{n}\right\vert ^{p}\right) dx. \label{5-3} \end{eqnarray} Since $I_{\lambda }(v_{\lambda })=\sup_{0\leq t\leq \widehat{s}_{\lambda }}I_{\lambda }(sv_{\lambda })$ and $I_{\lambda }^{\infty }(w_{n})=\sup_{0\leq t\leq \widehat{t}_{n}}I_{\lambda }^{\infty }(tw_{n}),$ we have \begin{equation} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }\right) \leq \alpha _{\lambda }^{-}\text{ and }I_{\lambda }^{\infty }\left( t_{n}^{\ast }w_{n}\right) \leq \alpha _{\lambda }^{\infty }. \label{5-16} \end{equation} Using the inequality \begin{equation*} \left\vert c-d\right\vert ^{p}>c^{p}+d^{p}-C_{\ast }\left( p\right) \left( c^{p-1}d+cd^{p-1}\right) \end{equation*} for all $c,d>0$ and for some constant $C^{\ast }\left( p\right) >0,$ together with $(\ref{5-3})-(\ref{5-16}),$ leads to \begin{eqnarray} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) &\leq &\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }+ \frac{\lambda }{4}C_{\varepsilon }^{2}\exp \left( -2n^{1-\varepsilon }\right) -\frac{\left( t_{n}^{\ast }\right) ^{p}}{p}\int_{\mathbb{R} ^{3}}(f\left( x\right) -f_{\infty })\left\vert w_{n}\right\vert ^{p}dx \notag \\ &&+\frac{C^{\ast }\left( p\right) }{p}\int_{\mathbb{R}^{3}}f\left( x\right) \left( \left\vert s_{\lambda }^{\ast }v_{\lambda }\right\vert ^{p-1}t_{n}^{\ast }w_{n}+s_{\lambda }^{\ast }v_{\lambda }\left\vert t_{n}^{\ast }w_{n}\right\vert ^{p-1}\right) dx \notag \\ &\leq &\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }+\frac{\lambda }{4} C_{\varepsilon }^{2}\exp \left( -2n^{1-\varepsilon }\right) -\frac{\left( t_{n}^{\ast }\right) ^{p}}{p}\int_{\mathbb{R}^{3}}(f\left( x\right) -f_{\infty })\left\vert w_{n}\right\vert ^{p}dx \notag \\ &&+\frac{C^{\ast }\left( p\right) }{p}\left( \frac{p}{4-p}\right) ^{\frac{p}{ p-2}}\int_{\mathbb{R}^{3}}f\left( x\right) \left( \left\vert v_{\lambda }\right\vert ^{p-1}w_{n}+v_{\lambda }\left\vert w_{n}\right\vert ^{p-1}\right) dx. \label{5-19} \end{eqnarray} By condition $(F2),$ one has \begin{eqnarray} \int_{\mathbb{R}^{3}}(f\left( x\right) -f_{\infty })\left\vert w_{n}\right\vert ^{p}dx &\geq &d_{0}\int_{\mathbb{R}^{3}}\exp \left( -\left\vert x+ne_{1}\right\vert ^{r_{f}}\right) (w_{\lambda }^{\infty })^{p}\left( x\right) dx \notag \\ &\geq &\left( \min_{x\in B_{1}\left( 0\right) }\left( w_{\lambda }^{\infty }\right) ^{p}\right) \int_{B_{1}\left( 0\right) }\exp \left( -\left\vert x+ne_{1}\right\vert ^{r_{f}}\right) dx \notag \\ &\geq &\left( \min_{x\in B_{1}\left( 0\right) }\left( w_{\lambda }^{\infty }\right) ^{p}\right) d_{0}\int_{B_{1}\left( 0\right) }\exp \left(
-\left\vert x\right\vert ^{r_{f}}-|e_{1}|n^{r_{f}}\right) dx \notag \\ &=&\left( \min_{x\in B_{1}\left( 0\right) }\left( w_{\lambda }^{\infty }\right) ^{p}\right) D_{0}\exp \left( -n^{r_{f}}\right) . \label{5-10} \end{eqnarray} Moreover, by \cite[Lemma 4.6]{Lin}, there exists $n_{1}>0$ such that for all $n>n_{1},$ \begin{eqnarray} \int_{\mathbb{R}^{3}}f\left( x\right) \left\vert v_{\lambda }\right\vert ^{p-1}w_{n}dx &\leq &f_{\max }\int_{\mathbb{R}^{3}}\exp \left( -\left(
p-1\right) |x|^{1-\varepsilon }\right) \exp \left(
-|x-e_{1}n|^{1-\varepsilon }\right) dx \notag \\ &\leq &\overline{C}_{\varepsilon ,1}\exp \left( -n^{1-\varepsilon }\right) \text{ for some }\overline{C}_{\varepsilon ,1}>0. \label{5-11} \end{eqnarray} Similarly, we also obtain that there exists $n_{2}>0$ such that for all $ n>n_{2},$ \begin{equation} \int_{\mathbb{R}^{3}}f\left( x\right) v_{\lambda }\left\vert w_{n}\right\vert ^{p-1}dx\leq \overline{C}_{\varepsilon ,2}\exp \left( -n^{1-\varepsilon }\right) \text{ for some }\overline{C}_{\varepsilon ,2}>0. \label{5-12} \end{equation} Hence, by $(\ref{5-19})-(\ref{5-12}),$ we may take $0<\varepsilon <1-r_{f}$ and $n^{\ast }\geq \max \left\{ n_{0},n_{1},n_{2}\right\} $ such that for every $0<\lambda <\min \{\lambda _{3},\lambda _{4}\}$ and $n\geq n^{\ast },$ there holds \begin{eqnarray*} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) &\leq &\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }+ \overline{C}_{\varepsilon }\exp \left( -n^{1-\varepsilon }\right) -C_{0}\exp \left( -n^{r_{f}}\right) \\ &<&\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }, \end{eqnarray*} where $\overline{C}_{\varepsilon }$ and $C_{0}$ are two positive constants.
Finally, we claim that $s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\in \mathbf{N}_{\lambda }^{(1)}.$ Note that \begin{equation*} C\left( p\right) >\left\{ \begin{array}{ll} \frac{\sqrt{e}\left( p-2\right) }{p}, & \text{ if }2<p\leq 3, \\ \frac{e\left( p-2\right) }{2p}, & \text{ if }3<p<4. \end{array} \right. \end{equation*} Then from $(\ref{3-9})-(\ref{3-10})$ and $(\ref{5-20})$ it follows that for $ \lambda >0$ sufficiently small, \begin{equation*} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) <\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }<2\alpha _{\lambda }^{\infty }<C\left( p\right) \left( \frac{S_{p}^{p}}{f_{\infty }} \right) ^{\frac{2}{p-2}}, \end{equation*} and so we can conclude that either $s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\in \mathbf{M}_{\lambda }^{(1)}$ or $s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\in \mathbf{M}_{\lambda }^{(2)}.$ If $ s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\in \mathbf{M}_{\lambda }^{(2)},$ then by $(\ref{2-1})$ and $(\ref{2-4}),$ we have \begin{eqnarray} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) &=&\frac{p-2}{2p}\left\Vert s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right\Vert _{H^{1}}^{2} \notag \\ &&-\frac{\lambda (4-p)}{4p}\int_{\mathbb{R}^{3}}K\left( x\right) \phi _{K,s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}}(s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n})^{2}dx \notag \\ &<&\lambda \left( \frac{4-p}{2p}-\frac{4-p}{4p}\right) \int_{\mathbb{R} ^{3}}K\left( x\right) \phi _{K,s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}}(s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n})^{2}dx \notag \\ &=&\frac{\lambda (4-p)}{4p}\int_{\mathbb{R}^{3}}K(x)\phi _{K,s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}}(s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n})^{2}dx \notag \\ &\leq &\frac{\lambda (4-p)}{4p}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }\left\Vert s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right\Vert _{H^{1}}^{4}. \label{5-21} \end{eqnarray} Moreover, using Lemmas $\ref{h-2}$ and $\ref{h-4}$, leads to \begin{equation} I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) \geq I_{\lambda }\left( v_{\lambda }-w_{n}\right) =\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }+o(1) \label{5-22} \end{equation} and \begin{equation} \left\Vert s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right\Vert _{H^{1}}^{2}=\left( s_{\lambda }^{\ast }\right) ^{2}\left\Vert v_{\lambda }\right\Vert _{H^{1}}^{2}+\left( t_{n}^{\ast }\right) ^{2}\left\Vert w_{n}\right\Vert _{H^{1}}^{2}-2s_{\lambda }^{\ast }t_{n}^{\ast }\left\langle v_{\lambda },w_{n}\right\rangle \leq C_{0}. \label{5-23} \end{equation} Thus, by $(\ref{5-21})-(\ref{5-23}),$ we can conclude that for $\lambda >0$ sufficiently small and $n\geq n^{\ast },$ there holds \begin{eqnarray*} \alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty }+o\left( 1\right) &<&I_{\lambda }\left( s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right) \\ &<&\frac{\lambda (4-p)}{4p}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }\left\Vert s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\right\Vert _{H^{1}}^{4} \\ &\leq &\frac{\lambda (4-p)}{4p}\overline{S}^{-2}S_{12/5}^{-4}K_{\max }C_{0}^{2} \\ &<&\alpha _{\lambda }^{\infty }, \end{eqnarray*} which a contradiction. This indicates that there exists a positive number $ \lambda ^{\ast }\leq \min \{\lambda _{3},\lambda _{4}\}$ such that $ s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\in \mathbf{M}_{\lambda }^{(1)}$ for all $0<\lambda <\lambda ^{\ast }$ and $n\geq n^{\ast }.$ Combining $(\ref{5-17})$ and $(\ref{5-18})$ gives $s_{\lambda }^{\ast }v_{\lambda }-t_{n}^{\ast }w_{n}\in \mathbf{N}_{\lambda }^{(1)}.$ This completes the proof. \end{proof}
Let $w_{0}$ be the unique positive solution of the following Schr\"{o}dinger equation \begin{equation} \begin{array}{ll} -\Delta u+u=f_{\infty }\left\vert u\right\vert ^{p-2}u & \text{ in }\mathbb{R }^{3}. \end{array} \tag*{$\left( E_{0}^{\infty }\right) $} \end{equation} From \cite{K}, one can see that \begin{equation*} I_{0}^{\infty }\left( w_{0}\right) =\alpha _{0}^{\infty }:=\frac{p-2}{2p} \left( \frac{S_{p}^{p}}{f_{\infty }}\right) ^{\frac{2}{p-2}}, \end{equation*} where $I_{0}^{\infty }$ is the energy functional of Eq. $(E_{0}^{\infty })$ in $H^{1}(\mathbb{R}^{3})$ in the form \begin{equation*} I_{0}^{\infty }\left( u\right) =\frac{1}{2}\left\Vert u\right\Vert _{H^{1}}^{2}-\frac{1}{p}\int_{\mathbb{R}^{3}}f_{\infty }\left\vert u\right\vert ^{p}dx. \end{equation*} Moreover, by \cite{GNN}, for any $\varepsilon >0,$ there exist positive numbers $A_{\varepsilon }$ and $B_{0}$ such that \begin{equation} A_{\varepsilon }\exp \left( -\left( 1+\varepsilon \right) \left\vert x\right\vert \right) \leq w_{0}\leq B_{0}\exp \left( -\left\vert x\right\vert \right) \text{ for all }x\in \mathbb{R}^{N}. \label{45} \end{equation} For $n\in \mathbb{N},$ we define the sequence \begin{equation*} \overline{w}_{n}(x)=w_{0}(x-ne_{1}). \end{equation*} Clearly, $I_{0}^{\infty }\left( \overline{w}_{n}\right) =I_{0}^{\infty }\left( w_{0}\right) $ for all $n\in \mathbb{N},$ where $I_{0}^{\infty }$ is the energy functional of Eq. $(E_{0}^{\infty }).$ Moreover, by $(\ref{45})$ one has \begin{equation*} \overline{w}_{n}\left( x\right) =w_{0}\left( x-ne_{1}\right) \leq C_{\varepsilon }\exp (\left\vert x\right\vert -n). \end{equation*} Note that conditions $(F1),(F2),(K1)$ and $(K3)$ satisfy conditions $ (D1),(D2)$ and $(D4)$ in \cite[Theorem 1.5]{SWF1}. Then from \cite[Theorem 1.5]{SWF1}, we obtain that there exists $\overline{\Lambda }>0$ such that for every $0<\lambda <\overline{\Lambda },$ system $\left( SP_{\lambda }\right) $ admits a positive solution $(\overline{v}_{\lambda },\phi _{K, \overline{v}_{\lambda }})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R} ^{3})$ satisfying \begin{equation*} \frac{p-2}{4p}\left( \frac{S_{p}^{p}}{f_{\max }}\right) ^{\frac{2}{p-2} }<I_{\lambda }\left( \overline{v}_{\lambda }\right) =\alpha _{\lambda }^{-}<\alpha _{0}^{\infty }, \end{equation*} and $\overline{v}_{\lambda }$ has also exponential decay like $(\ref{3-1}).$ Moreover, similar to Lemma \ref{h-1} and Proposition \ref{h-5}, we have the following two conclusions.
\begin{lemma} \label{h-7}Suppose that conditions ${(F1)},{(F2)},\left( K1\right) $ and $ \left( K3\right) $ hold. Then for each $0<\varepsilon <1$ there exists $ C_{\varepsilon }>0$ such that\newline $\left( i\right) $ $\int_{\mathbb{R}^{3}}K(x)\phi _{K,\overline{w}_{n}} \overline{v}_{\lambda }^{2}dx=\int_{\mathbb{R}^{3}}K(x)\phi _{K,\overline{v} _{\lambda }}\overline{w}_{n}^{2}dx\leq C_{\varepsilon }e^{-n^{1-\varepsilon }};$\newline $\left( ii\right) $ $\left\vert \int_{\mathbb{R}^{3}}K(x)\phi _{K,\overline{v }_{\lambda }-\overline{w}_{n}}\left( \overline{v}_{\lambda }-\overline{w} _{n}\right) ^{2}dx-\int_{\mathbb{R}^{3}}K(x)\phi _{K,\overline{v}_{\lambda }} \overline{v}_{\lambda }^{2}dx-\int_{\mathbb{R}^{3}}K(x)\phi _{K,\overline{w} _{n}}\overline{w}_{n}^{2}dx\right\vert \leq C_{\varepsilon }e^{-n^{1-\varepsilon }}.$ \end{lemma}
\begin{proposition} \label{h-6}Suppose that $2<p<4,$ and conditions ${(F1)},{(F2)},\left( K1\right) $ and $\left( K3\right) $ hold. Then there exist two positive numbers $\overline{\lambda }^{\ast }\leq \min \{\widetilde{\lambda }, \overline{\Lambda }\}$ and $\overline{n}^{\ast }\in \mathbb{N}$ such that for every $0<\lambda <\overline{\lambda }^{\ast }$ and $n\geq \overline{n} ^{\ast },$ there exists $\left( \overline{s}_{\lambda }^{\ast },\overline{t} _{n}^{\ast }\right) \in \left( 0,\infty \right) \times \left( 0,\infty \right) $ such that $\overline{s}_{\lambda }^{\ast }\overline{v}_{\lambda }- \overline{t}_{n}^{\ast }\overline{w}_{n}\in \mathbf{N}_{\lambda }^{(1)}$ and \begin{equation*} I_{\lambda }\left( \overline{s}_{\lambda }^{\ast }\overline{v}_{\lambda }- \overline{t}_{n}^{\ast }\overline{w}_{n}\right) <\alpha _{\lambda }^{-}+\alpha _{0}^{\infty }. \end{equation*} \end{proposition}
The proofs of the two results above are analogous to those of Lemma \ref{h-1} and Proposition \ref{h-5}, respectively, and so we omit here.
\section{Palais--Smale Sequences}
Define \begin{equation*} \theta _{\lambda }^{-}=\inf_{u\in \mathbf{N}_{\lambda }^{(1)}}I_{\lambda }\left( u\right) . \end{equation*} Then by Lemma \ref{h3-5} and Proposition \ref{h-5} or \ref{h-6}, we have \begin{equation} 2\alpha _{\lambda }^{-}\leq \theta _{\lambda }^{-}<\alpha _{\lambda }^{-}+\alpha _{\lambda }^{\infty } \label{18-0} \end{equation} or \begin{equation*} 2\alpha _{\lambda }^{-}\leq \theta _{\lambda }^{-}<\alpha _{\lambda }^{-}+\alpha _{0}^{\infty }. \end{equation*} Next, we define \begin{equation*} \Phi _{\lambda }^{+}\left( u\right) =\frac{\left\Vert u^{+}\right\Vert _{H^{1}}^{2}+\lambda \left( \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{+})^{2}dx+\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{+})^{2}dx\right) }{\int_{\mathbb{R}^{3}}f(x)\left\vert u^{+}\right\vert ^{p}dx} \end{equation*} and \begin{equation*} \Phi _{\lambda }^{-}\left( u\right) =\frac{\left\Vert u^{-}\right\Vert _{H^{1}}^{2}+\lambda \left( \int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{+}}(u^{-})^{2}dx+\int_{\mathbb{R}^{3}}K(x)\phi _{K,u^{-}}(u^{-})^{2}dx\right) }{\int_{\mathbb{R}^{3}}f(x)\left\vert u^{-}\right\vert ^{p}dx}. \end{equation*} Then for each $u\in \mathbf{N}_{\lambda }^{(1)},$ there holds $\Phi _{\lambda }^{+}\left( u\right) =\Phi _{\lambda }^{-}\left( u\right) =1.$ Furthermore, we have the following results.
\begin{lemma} \label{f4-1}For each $\epsilon >0$ there exists $\mu (\epsilon )>0$ such that for every $v\in \mathbf{N}_{\lambda }^{(1)}$ and $u\in H^{1}(\mathbb{R} ^{3})$ with $\left\Vert v-u\right\Vert _{H^{1}}<\mu (\epsilon ),$ there holds $\left\vert \Phi _{\lambda }^{+}\left( u\right) -1\right\vert +\left\vert \Phi _{\lambda }^{-}\left( u\right) -1\right\vert <\epsilon .$ \end{lemma}
\begin{lemma} \label{f4-2}Suppose that $2<p<4.$ Then for each $v_{0}\in \mathbf{N} _{\lambda }^{(1)},$ there exists a map $\phi _{\lambda }:H^{1}(\mathbb{R} ^{3})\rightarrow \mathbb{R}^{2}$ such that\newline $\left( i\right) \ \phi _{\lambda }\left( s_{1}v_{0}^{+}+s_{2}v_{0}^{-}\right) =\left( s_{1},s_{2}\right) $ for $ \left( s_{1},s_{2}\right) \in \left[ 0,\widetilde{s}_{\lambda }\right] \times \left[ 0,\widetilde{t}_{\lambda }\right] ;$\newline $\left( ii\right) \ \phi _{\lambda }\left( u\right) =\left( 1,1\right) $ if and only if $u\in \mathbf{N}_{\lambda }^{(1)}.$ \end{lemma}
The proofs of Lemmas \ref{f4-1} and \ref{f4-2} are almost the same as those in Clapp and Weth \cite[Lemma 13]{CW2} and we omit them here.
\begin{proposition} \label{f5-1}Let $\epsilon ,\mu (\epsilon )>0$ be as in Lemma \ref{f4-1}. Then for each \begin{equation*} 0<\eta <C\left( p\right) \left( \frac{S_{p}^{p}}{f_{\infty }}\right) ^{\frac{ 2}{p-2}}-\theta _{\lambda }^{-} \end{equation*} and $\mu \in (0,\mu (\epsilon )),$ there exists $u_{0}\in H^{1}(\mathbb{R} ^{3})$ such that for every $0<\lambda <\widetilde{\lambda },$\newline $\left( i\right) \ dist\left( u_{0},\mathbf{N}_{\lambda }^{(1)}\right) \leq \mu ;$\newline $\left( ii\right) \ I_{\lambda }\left( u_{0}\right) \in \lbrack \theta _{\lambda }^{-},\theta _{\lambda }^{-}+\eta );$\newline $\left( iii\right) \ \left\Vert I_{\lambda }^{\prime }\left( u_{0}\right) \right\Vert _{H^{-1}}\leq \max \left\{ \sqrt{\eta },\frac{\eta }{\mu } \right\} ;$\newline $\left( iv\right) \ \left\vert \Phi _{\lambda }^{+}\left( u\right) -1\right\vert +\left\vert \Phi _{\lambda }^{-}\left( u\right) -1\right\vert <\epsilon .$ \end{proposition}
\begin{proof} Let us fix $v_{0}\in \mathbf{N}_{\lambda }^{(1)}$ such that $I_{\lambda }\left( v_{0}\right) <\theta _{\lambda }^{-}+\eta ,$ and fix $\widetilde{s} _{\lambda },\widetilde{t}_{\lambda }>1$ as in Lemma \ref{h3-3} such that $ I_{\lambda }\left( \widetilde{s}_{\lambda }v_{0}^{+}+\widetilde{t}_{\lambda }v_{0}^{-}\right) \leq 0.$ Let $\phi _{\lambda }:H^{1}(\mathbb{R} ^{3})\rightarrow \mathbb{R}^{2}$ as in Lemma \ref{f4-2}. We define a map $ \beta _{\lambda }:Q_{\lambda }\rightarrow H^{1}(\mathbb{R}^{3})$ by \begin{equation*} \beta _{\lambda }\left( s_{1},s_{2}\right) =s_{1}v_{0}^{+}+s_{2}v_{0}^{-}, \end{equation*} where $Q_{\lambda }=\left[ 0,\widetilde{s}_{\lambda }\right] \times \left[ 0, \widetilde{t}_{\lambda }\right] .$ Then $\phi _{\lambda }\circ \beta _{\lambda }=id:Q_{\lambda }\rightarrow Q_{\lambda }.$ In particular, there holds \begin{equation} \deg \left( \phi _{\lambda }\circ \beta _{\lambda },Q_{\lambda },\left( 1,1\right) \right) =1. \label{20} \end{equation} Moreover, we also have \begin{equation} I_{\lambda }\left( \beta _{\lambda }\left( s_{1},s_{2}\right) \right) \leq I_{\lambda }\left( v_{0}\right) <\theta _{\lambda }^{-}+\eta \text{ for all } \left( s_{1},s_{2}\right) \in Q_{\lambda }. \label{21} \end{equation} Now we choose a Lipschitz continuous function $\chi :\mathbb{R}\rightarrow \mathbb{R}$ such that $0\leq \chi \leq 1,\chi \left( s\right) =1$ for $s\geq 0$ and $\chi \left( s\right) =0$ for $s\leq -1.$ Since $I_{\lambda }\in C^{2}(H^{1}(\mathbb{R}^{3}),\mathbb{R)},$ there is a semiflow $\varphi :[0,\infty )\times H^{1}(\mathbb{R}^{3})\rightarrow H^{1}(\mathbb{R}^{3})$ satisfying \begin{equation*} \left\{ \begin{array}{l} \frac{\partial }{\partial t}\varphi \left( t,u\right) =-\chi \left( I_{\lambda }\left( \varphi \left( t,u\right) \right) \right) I_{\lambda }^{\prime }\left( \varphi \left( t,u\right) \right) , \\ \varphi \left( 0,u\right) =u. \end{array} \right. \end{equation*} For convenience, we always write $\varphi \left( t,\cdot \right) $ by $ \varphi ^{t}$ in the sequel. Since $\max \left\{ I_{\lambda }\left( \widetilde{s}_{\lambda }v_{0}^{+}\right) ,I_{\lambda }\left( \widetilde{t} _{\lambda }v_{0}^{-}\right) \right\} <0,$ similar to the argument in Lemma \ref{h3-3}, we have \begin{equation*} \sup I_{\lambda }\left( \beta _{\lambda }\left( \partial Q_{\lambda }\right) \right) <2\alpha _{\lambda }^{-}. \end{equation*} Hence, \begin{equation*} \left( \varphi ^{t}\circ \beta _{\lambda }\right) \left( \partial Q_{\lambda }\right) \cap \mathbf{N}_{\lambda }^{(1)}=\emptyset \text{ for all }t\geq 0. \end{equation*} Using Lemma \ref{f4-2} gives \begin{equation*} \left( \phi _{\lambda }\circ \varphi ^{t}\circ \beta _{\lambda }\right) \left( y\right) \neq \left( 1,1\right) \text{ for all }y\in \partial Q_{\lambda }\text{ and }t\geq 0. \end{equation*} By $(\ref{20})$ and the global continuation principle of Leray-Schauder (see e.g. Zeider \cite[p.629]{Za}), we obtain that there exists a connected subset $Z\subset Q_{\lambda }\times \left[ 0,1\right] $ such that \begin{equation*} \begin{array}{l} \left( 1,1,0\right) \in Z, \\ \varphi ^{t}\left( \beta _{\lambda }\left( s_{1},s_{2}\right) \right) \in \mathbf{N}_{\lambda }^{(1)}\text{ for all }\left( s_{1},s_{2},t\right) \in Z, \\ Z\cap \left( Q_{\lambda }\times \left\{ 1\right\} \right) \neq \emptyset . \end{array} \end{equation*} Set \begin{equation*} \Gamma =\left\{ \varphi ^{t}\left( \beta _{\lambda }\left( s_{1},s_{2}\right) \right) \in \mathbf{N}_{\lambda }^{(1)}:\left( s_{1},s_{2},t\right) \in Z\right\} . \end{equation*} From $(\ref{21})$ it follows that \begin{equation*} \sup_{u\in \Gamma }I_{\lambda }\left( u\right) <\theta _{\lambda }^{-}+\eta , \end{equation*} which implies that $\Gamma \subset \mathbf{N}_{\lambda }^{(1)},$ since $Z$ is connected. Now we pick $\left( \bar{s}_{1},\bar{s}_{2},1\right) \in Z\cap \left( Q_{\lambda }\times \left\{ 1\right\} \right) $ and set \begin{equation*} v_{1}:=\psi _{\lambda }\left( \bar{s}_{1},\bar{s}_{2}\right) \text{ and } v_{2}:=\varphi ^{1}(v_{1}). \end{equation*} Clearly, $v_{2}\in \Gamma \subset \mathbf{N}_{\lambda }^{(1)}$ and $\Phi _{\lambda }^{+}\left( v_{2}\right) =\Phi _{\lambda }^{-}\left( v_{2}\right) =1.$ We distinguish two cases as follows:\newline Case $(i):\left\Vert \varphi ^{t}\left( v_{1}\right) -v_{2}\right\Vert _{H^{1}}\leq \mu $ for all $t\in \left[ 0,1\right] .$ By Lemma \ref{f4-1} one has \begin{equation*} \left\vert \Phi _{\lambda }^{+}\left( \varphi ^{t}\left( v_{1}\right) \right) -1\right\vert +\left\vert \Phi _{\lambda }^{-}\left( \varphi ^{t}\left( v_{1}\right) \right) -1\right\vert <\epsilon \text{ for all }t\in \left[ 0,1\right] . \end{equation*} Choosing $t_{0}\in \left[ 0,1\right] $ with \begin{equation*} \left\Vert I_{\lambda }^{\prime }\left( \varphi ^{t_{0}}\left( v_{1}\right) \right) \right\Vert _{H^{-1}}=\min_{0\leq t\leq 1}\left\Vert I_{\lambda }^{\prime }\left( \varphi ^{t}\left( v_{1}\right) \right) \right\Vert _{H^{-1}} \end{equation*} and setting $u_{0}=\varphi ^{t_{0}}(v_{1}).$ Then, we have \begin{eqnarray*} \eta &\geq &I_{\lambda }\left( v_{1}\right) -I_{\lambda }\left( v_{2}\right) =-\int_{0}^{1}\frac{\partial }{\partial t}I_{\lambda }\left( \varphi ^{t}\left( v_{1}\right) \right) dt \\ &=&\int_{0}^{1}\left\Vert I_{\lambda }^{\prime }\left( \varphi ^{t_{0}}\left( v_{1}\right) \right) \right\Vert _{H^{-1}}^{2}dt\geq \left\Vert I_{\lambda }^{\prime }\left( u_{0}\right) \right\Vert _{H^{-1}}^{2}. \end{eqnarray*} Therefore, $u_{0}$ satisfies the desired properties.\newline Case $(ii):$ There exists $\bar{t}\in \left[ 0,1\right] $ such that $ \left\Vert \varphi ^{\bar{t}}\left( v_{1}\right) -v_{2}\right\Vert _{H^{1}}>\mu .$ Let \begin{equation*}
t_{1}=\sup \left\{ t\geq \bar{t}\ |\ \left\Vert \varphi ^{t}\left( v_{1}\right) -v_{2}\right\Vert >\mu \right\} . \end{equation*} Then by Lemma \ref{f4-1}, we have \begin{equation*} \left\vert \Phi _{\lambda }^{+}\left( \varphi ^{t}\left( v_{1}\right) \right) -1\right\vert +\left\vert \Phi _{\lambda }^{-}\left( \varphi ^{t}\left( v_{1}\right) \right) -1\right\vert <\epsilon \end{equation*} for all $t\in \lbrack t_{1},1].$ Choosing $t_{0}\in \left[ t_{1},1\right] $ with \begin{equation*} \left\Vert I_{\lambda }^{\prime }\left( \varphi ^{t_{0}}\left( v_{1}\right) \right) \right\Vert _{H^{-1}}=\min_{t_{1}\leq t\leq 1}\left\Vert I_{\lambda }^{\prime }\left( \varphi ^{t}\left( v_{1}\right) \right) \right\Vert _{H^{-1}} \end{equation*} and setting $u_{0}=\varphi ^{t_{0}}(v_{1}).$ Then there holds \begin{equation*} \mu \leq \int_{t_{1}}^{1}\left\Vert \frac{\partial }{\partial t}\varphi ^{t}\left( v_{1}\right) \right\Vert _{H^{1}}dt\leq \int_{t_{1}}^{1}\left\Vert I_{\lambda }^{\prime }\left( \varphi ^{t}\left( v_{1}\right) \right) \right\Vert _{H^{-1}}dt \end{equation*} and \begin{eqnarray*} \eta &\geq &I_{\lambda }\left( \varphi ^{t_{1}}\left( v_{1}\right) \right) -I_{\lambda }\left( v_{2}\right) =\int_{t_{1}}^{1}\left\Vert I_{\lambda }^{\prime }\left( \varphi ^{t}v_{1}\right) \right\Vert _{H^{-1}}^{2}dt \\ &\geq &\left\Vert I_{\lambda }^{\prime }\left( u_{0}\right) \right\Vert _{H^{-1}}\int_{t_{1}}^{1}\left\Vert I_{\lambda }^{\prime }\left( \varphi ^{t}\left( v_{1}\right) \right) \right\Vert _{H^{-1}}dt, \end{eqnarray*} which implies that $\left\Vert I_{\lambda }^{\prime }\left( u_{0}\right) \right\Vert _{H^{-1}}\leq \frac{\eta }{\mu }.$ Therefore, $u_{0}$ satisfies the desired properties. The proof is complete. \end{proof}
\begin{corollary} \label{f5-2}For each $0<\lambda <\widetilde{\lambda },$ there exists a sequence $\left\{ u_{n}\right\} \subset H^{1}(\mathbb{R}^{3})$ such that \newline $\left( i\right) \ dist\left( u_{n},\mathbf{N}_{\lambda }^{(1)}\right) \rightarrow 0;$\newline $\left( ii\right) \ I_{\lambda }\left( u_{n}\right) \rightarrow \theta _{\lambda }^{-};$\newline $\left( iii\right) \ I_{\lambda }^{\prime }(u_{n})=o(1)$ strongly in $H^{-1}( \mathbb{R}^{3});$\newline $\left( iv\right) $ $\left\vert \Phi _{\lambda }^{+}\left( u_{n}\right) -1\right\vert +\left\vert \Phi _{\lambda }^{-}\left( u_{n}\right) -1\right\vert \rightarrow 0.$ \end{corollary}
\section{Proof of Theorem \protect\ref{t2}}
Before proving Theorem \ref{t2}, we first give a precise description of the Palais--Smale sequence for $I_{\lambda }$ in this section.
\begin{proposition} \label{d1}Suppose that $2<p<4,$ and conditions $(F1)-\left( F2\right) $ and $ (K1)-\left( K2\right) $ hold. Let $\left\{ u_{n}\right\} \subset H^{1}( \mathbb{R}^{3})$ be a sequence satisfying\newline $\left( i\right) \ dist\left( u_{n},\mathbf{N}_{\lambda }^{(1)}\right) \rightarrow 0;$\newline $\left( ii\right) \ I_{\lambda }\left( u_{n}\right) \rightarrow \theta _{\lambda }^{-};$\newline $\left( iii\right) \ I_{\lambda }^{\prime }(u_{n})=o(1)$ strongly in $H^{-1}( \mathbb{R}^{3});$\newline $\left( iv\right) $ $\left\vert \Phi _{\lambda }^{+}\left( u_{n}\right) -1\right\vert +\left\vert \Phi _{\lambda }^{-}\left( u_{n}\right) -1\right\vert \rightarrow 0.$\newline Then there exist a subsequence $\left\{ u_{n}\right\} $ and $u_{\lambda }\in \mathbf{N}_{\lambda }^{(1)}$ such that $u_{n}\rightarrow u_{\lambda }$ strongly in $H^{1}(\mathbb{R}^{3})$ for each $0<\lambda <\lambda ^{\ast }.$ \end{proposition}
\begin{proof} Since $\left\{ u_{n}\right\} $ is bounded in $H^{1}(\mathbb{R}^{3}),$ we can assume that there exists $u_{\lambda }\in H^{1}(\mathbb{R}^{3})$ such that \begin{eqnarray} u_{n} &\rightharpoonup &u_{\lambda }\text{ and }u_{n}^{\pm }\rightharpoonup u_{\lambda }^{\pm }\text{ weakly in }H^{1}(\mathbb{R}^{3}), \notag \\ u_{n} &\rightarrow &u_{\lambda }\text{ and }u_{n}^{\pm }\rightarrow u_{\lambda }^{\pm }\text{ strongly in }L_{loc}^{r}(\mathbb{R}^{3})\text{ for }1\leq r<2^{\ast }, \label{18-2} \\ u_{n} &\rightarrow &u_{\lambda }\text{ and }u_{n}^{\pm }\rightarrow u_{\lambda }^{\pm }\text{ a.e. in }\mathbb{R}^{3}. \notag \end{eqnarray}
First, we claim that$\ u_{\lambda }^{\pm }\not\equiv 0.$ Suppose on the contrary. Then we can assume without loss of generality that $u_{\lambda }^{+}\equiv 0.$ Since$\ dist\left( u_{n},\mathbf{N}_{\lambda }^{(1)}\right) \rightarrow 0$ as $n\rightarrow \infty $ and $2\alpha _{\lambda }^{-}\leq \theta _{\lambda }^{-}<\alpha _{\lambda }^{\infty }+\alpha _{\lambda }^{-},$ we deduce from the Sobolev imbedding theorem that $\left\Vert u_{n}^{+}\right\Vert _{H^{1}}>\nu >0$ for some constant $\nu $ and for all $ n>0.$ Applying the concentration-compactness principle of P.L. Lions \cite {Li}, there exist positive constants $R,d$ and a sequence $\left\{ x_{n}\right\} \subset \mathbb{R}^{3}$ such that \begin{equation} \int_{B_{R}\left( 0\right) }\left\vert u_{n}^{+}\left( x+x_{n}\right) \right\vert ^{p}dx\geq d\text{ for }n\text{ sufficiently large.} \label{23} \end{equation} We will show that $\left\{ x_{n}\right\} $ is a unbounded sequence in $ \mathbb{R}^{3}.$ Suppose otherwise, we can assume that $x_{n}\rightarrow x_{0}$ for some $x_{0}\in \mathbb{R}^{3}.$ It follows from $(\ref{18-2})$ and $(\ref{23})$ that \begin{equation*} \int_{B_{R}\left( x_{0}\right) }\left\vert u_{\lambda }^{+}\right\vert ^{p}dx\geq d, \end{equation*} which contradicts with $u_{\lambda }^{+}\equiv 0.$ Thus, $\left\{ x_{n}\right\} $ is a unbounded sequence in $\mathbb{R}^{3}.$ Set $\widetilde{ u}_{n}\left( x\right) =u_{n}\left( x+x_{n}\right) .$ Clearly, $\left\{ \widetilde{u}_{n}\right\} $ is also bounded in $H^{1}(\mathbb{R}^{3}).$ Then we may assume that there exists $\widetilde{u}_{0}\in H^{1}(\mathbb{R}^{3})$ such that \begin{eqnarray} \widetilde{u}_{n} &\rightharpoonup &\widetilde{u}_{\lambda }\text{ and } \widetilde{u}_{n}^{\pm }\rightharpoonup \widetilde{u}_{\lambda }^{\pm }\text{ weakly in }H^{1}(\mathbb{R}^{3}), \label{24} \\ \widetilde{u}_{n} &\rightarrow &\widetilde{u}_{\lambda }\text{ and } \widetilde{u}_{n}^{\pm }\rightarrow \widetilde{u}_{\lambda }^{\pm }\text{ a.e. in }\mathbb{R}^{3}. \notag \end{eqnarray} By $(\ref{23}),$ we have $\widetilde{u}_{\lambda }^{+}\not\equiv 0$ in $ \mathbb{R}^{3}.$ Note that \begin{equation} \left\Vert \widetilde{u}_{n}^{\pm }\right\Vert _{H^{1}}=\left\Vert u_{n}^{\pm }\right\Vert _{H^{1}}<D_{1}<\left( \frac{2S_{p}^{p}}{f_{\max }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}, \label{18-8} \end{equation} it follows from Fatou's Lemma that \begin{equation*} \left\Vert \widetilde{u}_{\lambda }^{+}\right\Vert _{H^{1}}\leq \lim \inf_{n\rightarrow \infty }\left\Vert \widetilde{u}_{n}^{+}\right\Vert _{H^{1}}\leq D_{1}<\left( \frac{2S_{p}^{p}}{f_{\max }\left( 4-p\right) } \right) ^{\frac{1}{p-2}}. \end{equation*} By conditions $(F1),(K1)$ and$(K2),$ we have $K\left( x-x_{n}\right) \rightarrow K_{\infty }$ and $f\left( x-x_{n}\right) \rightarrow f_{\infty }$ as $n\rightarrow \infty .$ Thus, from Lemma \ref{h2} and the fact of $ I_{\lambda }^{\prime }(u_{n})\rightarrow 0$ on $H^{-1}(\mathbb{R}^{3})$ it follows that \begin{eqnarray} &&\left\Vert \widetilde{u}_{\lambda }^{+}\right\Vert _{H^{1}}^{2}+\lambda \left( \int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty },\widetilde{u} _{\lambda }^{+}}(\widetilde{u}_{\lambda }^{+})^{2}dx+\int_{\mathbb{R} ^{3}}K_{\infty }\phi _{K_{\infty },\widetilde{u}_{\lambda }^{-}}(\widetilde{u }_{\lambda }^{+})^{2}dx\right) \notag \\ &=&\int_{\mathbb{R}^{3}}f_{\infty }\left\vert \widetilde{u}_{\lambda }^{+}\right\vert ^{p}dx, \label{18-6} \end{eqnarray} and \begin{eqnarray} &&\left\Vert \widetilde{u}_{n}^{\pm }\right\Vert _{H^{1}}^{2}+\lambda \left( \int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty },\widetilde{u}_{n}^{\pm }}( \widetilde{u}_{n}^{\pm })^{2}dx+\int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty },\widetilde{u}_{n}^{\mp }}^{2}(\widetilde{u}_{n}^{\pm })^{2}dx\right) \notag \\ &=&\int_{\mathbb{R}^{3}}f_{\infty }\left\vert \widetilde{u}_{n}^{\pm }\right\vert ^{p}dx+o(1). \label{18-4} \end{eqnarray} Set $v_{n}=\widetilde{u}_{n}^{+}-\widetilde{u}_{\lambda }^{+}.$ We distinguish two cases as follows:\newline Case $I:\left\Vert v_{n}\right\Vert _{H^{1}}\rightarrow 0$ as $n\rightarrow \infty .$ Since $dist\left( u_{n},\mathbf{N}_{\lambda }^{(1)}\right) \rightarrow 0,$ it follows from $(\ref{18-6})$ and Lemma \ref{h3-5} that \begin{eqnarray*} I_{\lambda }(u_{n}) &=&J_{\lambda }^{+}(u_{n}^{+},u_{n}^{-})+J_{\lambda }^{-}(u_{n}^{+},u_{n}^{-}) \\ &=&J_{\lambda }^{+}\left( \widetilde{u}_{n}^{+},\widetilde{u}_{n}^{-}\right) +J_{\lambda }^{-}\left( u_{n}^{+},u_{n}^{-}\right) \\ &=&(J_{\lambda }^{+})^{\infty }\left( \widetilde{u}_{\lambda }^{+}, \widetilde{u}_{\lambda }^{-}\right) +J_{\lambda }^{-}\left( u_{n}^{+},u_{n}^{-}\right) +o\left( 1\right) \\ &\geq &\alpha _{\lambda }^{\infty }+\alpha _{\lambda }^{-}+o(1), \end{eqnarray*} where $(J_{\lambda }^{+})^{\infty }=J_{\lambda }^{+}$ with $K(x)\equiv K_{\infty }$ and $f(x)\equiv f_{\infty }$. Thus, $\theta _{\lambda }^{-}\geq \alpha _{\lambda }^{\infty }+\alpha _{\lambda }^{-},$ which contradicts to $ \theta _{\lambda }^{-}<\alpha _{\lambda }^{\infty }+\alpha _{\lambda }^{-}.$ \newline Case $II:\left\Vert v_{n}\right\Vert _{H^{1}}\geq c_{0}$ for large $n$ and for some constant $c_{0}>0.$ Following Brezis-Lieb Lemma \cite{BLi} and \cite [Lemma 2.2]{ZZ}, together with $(\ref{18-6})$ and $(\ref{18-4}),$ we have \begin{equation} \left\Vert v_{n}\right\Vert _{H^{1}}^{2}+\lambda \left( \int_{\mathbb{R} ^{3}}K_{\infty }\phi _{K_{\infty },v_{n}}v_{n}^{2}dx+\int_{\mathbb{R} ^{3}}K_{\infty }\phi _{K_{\infty },\widetilde{u}_{n}^{-}}v_{n}^{2}dx\right) -\int_{\mathbb{R}^{3}}f_{\infty }\left\vert v_{n}\right\vert ^{p}dx=o(1). \label{18-9} \end{equation} Note that $\Vert \widetilde{u}_{\lambda }^{+}\Vert _{H^{1}}\geq \left( \frac{ S_{p}^{p}}{f_{\max }}\right) ^{\frac{1}{p-2}}$ and $\left\Vert v_{n}\right\Vert _{H^{1}}^{2}=\left\Vert \widetilde{u}_{n}^{+}\right\Vert _{H^{1}}^{2}-\Vert \widetilde{u}_{\lambda }^{+}\Vert _{H^{1}}^{2}+o(1).$ Then it follows from $\left( \ref{24}\right) $ and $\left( \ref{18-8}\right) $ that \begin{equation} \left\Vert v_{n}\right\Vert _{H^{1}}<D_{1}<\left( \frac{2S_{p}^{p}}{f_{\max }\left( 4-p\right) }\right) ^{\frac{1}{p-2}}\text{ for sufficiently large }n. \label{18-10} \end{equation} By $(\ref{18-9}),(\ref{18-10})$ and the fact of $\left\Vert v_{n}\right\Vert _{H^{1}}\geq c_{0}$ for sufficiently large $n,$ it is straightforward to find a sequence $\left\{ s_{n}\right\} \subset \mathbb{R}^{+}$ with $ s_{n}\rightarrow 1$ as $n\rightarrow \infty $ such that \begin{eqnarray*} &&\left\Vert s_{n}v_{n}\right\Vert _{H^{1}}^{2}+\lambda \left( \int_{\mathbb{ R}^{3}}K_{\infty }\phi _{K_{\infty },s_{n}v_{n}}(s_{n}v_{n})^{2}dx+\int_{ \mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty },\widetilde{u} _{n}^{-}}(s_{n}v_{n})^{2}dx\right) \\ &=&\int_{\mathbb{R}^{3}}f_{\infty }\left\vert s_{n}v_{n}\right\vert ^{p}dx. \end{eqnarray*} Thus, similar to the argument in Lemma \ref{h3-5}, we obtain \begin{eqnarray*} &&\frac{1}{2}\left\Vert v_{n}\right\Vert _{H^{1}}^{2}+\frac{\lambda }{4} \left( \int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty },v_{n}}v_{n}^{2}dx+\int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty }, \widetilde{u}_{n}^{-}}v_{n}^{2}dx\right) -\frac{1}{p}\int_{\mathbb{R} ^{3}}f_{\infty }\left\vert v_{n}\right\vert ^{p}dx \\ &\geq &\alpha _{\lambda }^{\infty }+o(1), \end{eqnarray*} where we have used the fact of $s_{n}\rightarrow 1.$ It follows from Lemma \ref{h3-5}, Brezis-Lieb Lemma \cite{BLi} and \cite[Lemma 2.2]{ZZ} that \begin{eqnarray*} I_{\lambda }\left( u_{n}\right) &=&J_{\lambda }^{+}\left( u_{n}^{+},u_{n}^{-}\right) +J_{\lambda }^{-}\left( u_{n}^{+},u_{n}^{-}\right) \\ &=&(J_{\lambda }^{+})^{\infty }\left( \widetilde{u}_{n}^{+},\widetilde{u} _{n}^{-}\right) +J_{\lambda }^{-}\left( u_{n}^{+},u_{n}^{-}\right) +o\left( 1\right) \\ &=&\frac{1}{2}\left\Vert v_{n}\right\Vert _{H^{1}}^{2}+\frac{\lambda }{4} \left( \int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty },v_{n}}v_{n}^{2}dx+\int_{\mathbb{R}^{3}}K_{\infty }\phi _{K_{\infty }, \widetilde{u}_{n}^{-}}v_{n}^{2}dx\right) \\ &&-\frac{1}{p}\int_{\mathbb{R}^{3}}f_{\infty }\left\vert v_{n}\right\vert ^{p}dx+(J_{\lambda }^{+})^{\infty }\left( \widetilde{u}_{\lambda }^{+}, \widetilde{u}_{\lambda }^{-}\right) +J_{\lambda }^{-}\left( u_{n}^{+},u_{n}^{-}\right) +o\left( 1\right) \\ &\geq &2\alpha _{\lambda }^{\infty }+\alpha _{\lambda }^{-}+o(1), \end{eqnarray*} which implies that \begin{equation*} \lim_{n\rightarrow \infty }I_{\lambda }(u_{n})=\theta _{\lambda }^{-}\geq 2\alpha _{\lambda }^{\infty }+\alpha _{\lambda }^{-}. \end{equation*} This contradicts to $(\ref{18-0}).$ Hence, $u_{\lambda }^{+}\not\equiv 0.$ Similarly, we also obtain $u_{\lambda }^{-}\not\equiv 0.$
Next, we show that $u_{n}\rightarrow u_{0}$ strongly in $H^{1}(\mathbb{R} ^{3}).$ Similar to the argument of Case $II,$ we can easily arrive at the conclusion. Moreover, we have $u_{\lambda }\in \mathbf{N}_{\lambda }^{(1)}$ and $I_{\lambda }(u_{\lambda })=\theta _{\lambda }^{-}.$ This indicates that $u_{\lambda }$ is a nodal solution for each $0<\lambda <\lambda ^{\ast }$. The proof is complete. \end{proof}
\textbf{We are ready to prove Theorem \ref{t2}:} By Corollary \ref{f5-2} and Proposition \ref{d1}, for each $0<\lambda <\lambda ^{\ast },$ Eq. $\left( E_{\lambda }\right) $ has a nodal solution $u_{\lambda }$ such that $ I_{\lambda }\left( u_{\lambda }\right) =\theta _{\lambda }^{-}.$ Moreover, similar to the argument in \cite[Theorem 1.3]{AS}, $u_{\lambda }$ changes sign exactly once in $\mathbb{R}^{3}.$ Consequently, system $(SP_{\lambda })$ admits a nodal solution $(u_{\lambda },\phi _{K,u_{\lambda }})\in H^{1}( \mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ for each $0<\lambda <\lambda ^{\ast },$ which changes sign exactly once in $\mathbb{R}^{3}.$
\section{Proof of Theorem \protect\ref{t3}}
As in Section 5, we also give a precise description of the Palais--Smale sequence for $I_{\lambda }$ at the beginning of this section.
\begin{proposition} \label{d2}Suppose that $2<p<4,$ and conditions ${(F1)}-{(F2)},(K1)$ and $ (K3) $ hold. Let $\left\{ \overline{u}_{n}\right\} \subset H^{1}(\mathbb{R} ^{3})$ be a sequence satisfying\newline $\left( i\right) \ dist\left( \overline{u}_{n},\mathbf{N}_{\lambda }^{(1)}\right) \rightarrow 0;$\newline $\left( ii\right) \ I_{\lambda }(\overline{u}_{n})\rightarrow \theta _{\lambda }^{-};$\newline $\left( iii\right) \ I_{\lambda }^{\prime }(\overline{u}_{n})=o(1)$ strongly in $H^{-1}(\mathbb{R}^{3});$\newline $\left( iv\right) $ $\left\vert \Phi _{\lambda }^{+}\left( \overline{u} _{n}\right) -1\right\vert +\left\vert \Phi _{\lambda }^{-}\left( \overline{u} _{n}\right) -1\right\vert \rightarrow 0.$\newline Then there exist a subsequence $\{\overline{u}_{n}\}$ and $u_{\lambda }\in \mathbf{N}_{\lambda }^{(1)}$ such that $\overline{u}_{n}\rightarrow u_{\lambda }$ strongly in $H^{1}(\mathbb{R}^{3})$ for each $0<\lambda < \overline{\lambda }^{\ast }.$ \end{proposition}
\begin{proof} The proof is analogous to that of Proposition \ref{d1}, and we omit it here. \end{proof}
\textbf{We now begin to prove Theorem \ref{t3}:} By Corollary \ref{f5-2} and Proposition \ref{d2}, for each $0<\lambda <\overline{\lambda }^{\ast },$ Eq. $\left( E_{\lambda }\right) $ admits a nodal solution $u_{\lambda }$ such that $I_{\lambda }\left( u_{\lambda }\right) =\theta _{\lambda }^{-}.$ Moreover, similar to the argument in \cite[Theorem 1.3]{AS}, $u_{\lambda }$ changes sign exactly once in $\mathbb{R}^{3}.$ Consequently, system $ (SP_{\lambda })$ admits a nodal solution $(u_{\lambda },\phi _{K,u_{\lambda }})\in H^{1}(\mathbb{R}^{3})\times D^{1,2}(\mathbb{R}^{3})$ for each $ 0<\lambda <\overline{\lambda }^{\ast },$ which changes sign exactly once in $ \mathbb{R}^{3}.$
\section{Acknowledgments}
J. Sun is supported by the National Natural Science Foundation of China (Grant No. 11671236). T.F. Wu is supported in part by the Ministry of Science and Technology, Taiwan (Grant 106-2115-M-390-002-MY2), the Mathematics Research Promotion Center, Taiwan and the National Center for Theoretical Sciences, Taiwan.
\end{document} | arXiv |
Achille Sannia
Achille Sannia (14 April 1822 – 2 August 1892) was an Italian mathematician and politician.
Achille Sannia
Born
Campobasso
Died
Naples
CitizenshipItalian
Known forGeometry and Politics
AwardsGreat official of the Order of the Crown of Italy, Commendatore of the Order of Saints Maurizio and Lazzaro
Scientific career
FieldsMathematician
Biography
Achille Sannia was a senator of the Kingdom of Italy.[1] He was born in Campobasso and later moved from Molise to Naples to continue his studies together with Enrico D'Ovidio. He first taught in a private school before moving to a University in 1865 as a professor of geometry.
In 1871, he created a school of Electrical engineering. He wrote two important treatises, one concerning projective geometry and the other elementary geometry. He was a member of the Academy of Sciences.[2]
He had a son, Gustavo Sannia who was also a mathematician.[3]
Works
• Planimetry, with Enrico D'Ovidio, Stab. typ. of the fine arts, Naples 1869, II ed. 1871
• Geometry elements, with Enrico D'Ovidio, (14 editions), Naples 1868-69, 12th edition, B. Pellerano LC Scientific and Industrial Library successor, Naples 1906
• Projection Geometry Lessons dictated in the Royal University of Naples by Prof. Achille Sannia- Ed. Pellerano Naples 1891 of 691pagg.
References
1. "Scheda senatore SANNIA Achille". notes9.senato.it. Archived from the original on 2016-03-04. Retrieved 2019-01-19.
2. "Achille SANNIA". www.dm.unito.it. Archived from the original on 29 February 2012. Retrieved 19 January 2019.
3. "Gustavo Sannia | MATEpristem". matematica.unibocconi.it. Archived from the original on 2015-10-19. Retrieved 2019-01-19.
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Search for a heavy pseudoscalar boson decaying to a Z and a Higgs boson at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te
A. M. Sirunyan, A. Tumasyan, W. Adam, F. Ambrogi, more
The European Physical Journal C > 2019 > 79 > 7 > 1-27
A search is presented for a heavy pseudoscalar boson $$\text {A}$$ A decaying to a Z boson and a Higgs boson with mass of 125$$\,\text {GeV}$$ GeV . In the final state considered, the Higgs boson decays to a bottom quark and antiquark, and the Z boson decays either into a pair of electrons, muons, or neutrinos. The analysis is performed using a data sample corresponding to an integrated luminosity...
Search for supersymmetry in final states with photons and missing transverse momentum in proton-proton collisions at 13 TeV
The CMS collaboration, A. M. Sirunyan, A. Tumasyan, W. Adam, more
Journal of High Energy Physics > 2019 > 2019 > 6 > 1-34
Abstract Results are reported of a search for supersymmetry in final states with photons and missing transverse momentum in proton-proton collisions at the LHC. The data sample corresponds to an integrated luminosity of 35.9 fb−1 collected at a center-of-mass energy of 13 TeV using the CMS detector. The results are interpreted in the context of models of gauge-mediated supersymmetry breaking. Production...
Search for the associated production of the Higgs boson and a vector boson in proton-proton collisions at s $$ \sqrt{s} $$ = 13 TeV via Higgs boson decays to τ leptons
Abstract A search for the standard model Higgs boson produced in association with a W or a Z boson and decaying to a pair of τ leptons is performed. A data sample of proton-proton collisions collected at s $$ \sqrt{s} $$ = 13 TeV by the CMS experiment at the CERN LHC is used, corresponding to an integrated luminosity of 35.9 fb−1. The signal strength is measured relative to the expectation...
Search for a low-mass τ−τ+ resonance in association with a bottom quark in proton-proton collisions at s $$ \sqrt{s} $$ = 13 TeV
Abstract A general search is presented for a low-mass τ−τ+ resonance produced in association with a bottom quark. The search is based on proton-proton collision data at a center-of-mass energy of 13 TeV collected by the CMS experiment at the LHC, corresponding to an integrated luminosity of 35.9 fb−1. The data are consistent with the standard model expectation. Upper limits at 95% confidence level...
Search for supersymmetry in events with a photon, jets, $$\mathrm {b}$$ b -jets, and missing transverse momentum in proton–proton collisions at 13$$\,\text {Te}\text {V}$$ Te
A search for supersymmetry is presented based on events with at least one photon, jets, and large missing transverse momentum produced in proton–proton collisions at a center-of-mass energy of 13$$\,\text {Te}\text {V}$$ Te . The data correspond to an integrated luminosity of 35.9$$\,\text {fb}^{-1}$$ fb-1 and were recorded at the LHC with the CMS detector in 2016. The analysis characterizes signal-like...
Combined measurements of Higgs boson couplings in proton–proton collisions at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te
Combined measurements of the production and decay rates of the Higgs boson, as well as its couplings to vector bosons and fermions, are presented. The analysis uses the LHC proton–proton collision data set recorded with the CMS detector in 2016 at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te , corresponding to an integrated luminosity of 35.9$${\,\text {fb}^{-1}} $$ fb-1 . The combination is based...
Combinations of single-top-quark production cross-section measurements and |fLVVtb| determinations at s $$ \sqrt{s} $$ = 7 and 8 TeV with the ATLAS and CMS experiments
The ATLAS collaboration, M. Aaboud, G. Aad, B. Abbott, more
Abstract This paper presents the combinations of single-top-quark production cross-section measurements by the ATLAS and CMS Collaborations, using data from LHC proton-proton collisions at s $$ \sqrt{s} $$ = 7 and 8 TeV corresponding to integrated luminosities of 1.17 to 5.1 fb−1 at s $$ \sqrt{s} $$ = 7 TeV and 12.2 to 20.3 fb−1 at s $$ \sqrt{s} $$ = 8 TeV. These combinations...
Measurement of inclusive very forward jet cross sections in proton-lead collisions at s N N $$ \sqrt{s_{\mathrm{NN}}} $$ = 5.02 TeV
Abstract Measurements of differential cross sections for inclusive very forward jet production in proton-lead collisions as a function of jet energy are presented. The data were collected with the CMS experiment at the LHC in the laboratory pseudorapidity range −6.6 < η < −5.2. Asymmetric beam energies of 4 TeV for protons and 1.58 TeV per nucleon for Pb nuclei were used, corresponding to a...
Measurement of the energy density as a function of pseudorapidity in proton–proton collisions at $$\sqrt{s} =13\,\text {TeV} $$ s=13TeV
A measurement of the energy density in proton–proton collisions at a centre-of-mass energy of $$\sqrt{s} =13$$ s=13 $$\,\text {TeV}$$ TeV is presented. The data have been recorded with the CMS experiment at the LHC during low luminosity operations in 2015. The energy density is studied as a function of pseudorapidity in the ranges $$-\,6.6<\eta <-\,5.2$$ -6.6<η<-5.2 and $$3.15<|\eta...
Measurement of the $${\mathrm {t}\overline{\mathrm {t}}}$$ tt¯ production cross section, the top quark mass, and the strong coupling constant using dilepton events in pp collisions at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te
A measurement of the top quark–antiquark pair production cross section $$\sigma _{\mathrm {t}\overline{\mathrm {t}}} $$ σtt¯ in proton–proton collisions at a centre-of-mass energy of 13$$\,\text {Te}\text {V}$$ Te is presented. The data correspond to an integrated luminosity of $$35.9{\,\text {fb}^{-1}} $$ 35.9fb-1 , recorded by the CMS experiment at the CERN LHC in 2016. Dilepton events ($$\mathrm...
Search for vector-like quarks in events with two oppositely charged leptons and jets in proton–proton collisions at $$\sqrt{s} = 13\,\text {Te}\text {V} $$ s=13Te
A search for the pair production of heavy vector-like partners $$\mathrm {T}$$ T and $$\mathrm {B}$$ B of the top and bottom quarks has been performed by the CMS experiment at the CERN LHC using proton–proton collisions at $$\sqrt{s} = 13\,\text {Te}\text {V} $$ s=13Te . The data sample was collected in 2016 and corresponds to an integrated luminosity of 35.9$$\,\text {fb}^{-1}$$ fb-1 . Final states...
Measurements of the pp → WZ inclusive and differential production cross sections and constraints on charged anomalous triple gauge couplings at s $$ \sqrt{s} $$ = 13 TeV
Abstract The WZ production cross section is measured in proton-proton collisions at a centre-of-mass energy s $$ \sqrt{s} $$ = 13 TeV using data collected with the CMS detector, corresponding to an integrated luminosity of 35.9 fb−1. The inclusive cross section is measured to be σtot(pp → WZ) = 48.09 − 0.96+ 1.00 (stat) − 0.37+ 0.44 (theo) − 2.17+ 2.39 (syst) ± 1.39(lum) pb, resulting in...
Search for nonresonant Higgs boson pair production in the b b ¯ b b ¯ $$ \mathrm{b}\overline{\mathrm{b}}\mathrm{b}\overline{\mathrm{b}} $$ final state at s $$ \sqrt{s} $$ = 13 TeV
Abstract Results of a search for nonresonant production of Higgs boson pairs, with each Higgs boson decaying to a b b ¯ $$ \mathrm{b}\overline{\mathrm{b}} $$ pair, are presented. This search uses data from proton-proton collisions at a centre-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 35.9 fb−1, collected by the CMS detector at the LHC. No signal is observed, and...
Search for contact interactions and large extra dimensions in the dilepton mass spectra from proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV
Abstract A search for nonresonant excesses in the invariant mass spectra of electron and muon pairs is presented. The analysis is based on data from proton-proton collisions at a center-of-mass energy of 13 TeV recorded by the CMS experiment in 2016, corresponding to a total integrated luminosity of 36 fb−1. No significant deviation from the standard model is observed. Limits are set at 95% confidence...
Measurement of the top quark mass in the all-jets final state at $$\sqrt{s}=13\,\text {TeV} $$ s=13TeV and combination with the lepton+jets channel
A top quark mass measurement is performed using $$35.9{\,\text {fb}^{-1}} $$ 35.9fb-1 of LHC proton–proton collision data collected with the CMS detector at $$\sqrt{s}=13\,\text {TeV} $$ s=13TeV . The measurement uses the $${\mathrm {t}\overline{\mathrm {t}}}$$ tt¯ all-jets final state. A kinematic fit is performed to reconstruct the decay of the $${\mathrm {t}\overline{\mathrm {t}}}$$ tt¯ system...
Search for resonant production of second-generation sleptons with same-sign dimuon events in proton–proton collisions at $$\sqrt{s} = 13\,\text {TeV} $$ s=13TeV
A search is presented for resonant production of second-generation sleptons ($$\widetilde{\mu } _{\mathrm {L}}$$ μ~L , $$\widetilde{\nu }_{\mu }$$ ν~μ ) via the R-parity-violating coupling $${\lambda ^{\prime }_{211}}$$ λ211′ to quarks, in events with two same-sign muons and at least two jets in the final state. The smuon (muon sneutrino) is expected to decay into a muon and a neutralino (chargino),...
Search for resonant t t ¯ $$ \mathrm{t}\overline{\mathrm{t}} $$ production in proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV
Abstract A search for a heavy resonance decaying into a top quark and antiquark t t ¯ $$ \left(\mathrm{t}\overline{\mathrm{t}}\right) $$ pair is performed using proton-proton collisions at s = 13 $$ \sqrt{s}=13 $$ TeV. The search uses the data set collected with the CMS detector in 2016, which corresponds to an integrated luminosity of 35.9 fb−1. The analysis considers three exclusive...
Search for excited leptons in ℓℓγ final states in proton-proton collisions at s = 13 $$ \sqrt{\mathrm{s}}=13 $$ TeV
Abstract A search is presented for excited electrons and muons in ℓℓγ final states at the LHC. The search is based on a data sample corresponding to an integrated luminosity of 35.9 fb−1 of proton-proton collisions at a center-of-mass energy of 13 TeV, collected with the CMS detector in 2016. This is the first search for excited leptons at s $$ \sqrt{s} $$ = 13 TeV. The observation is consistent...
Search for dark matter produced in association with a Higgs boson decaying to a pair of bottom quarks in proton–proton collisions at $$\sqrt{s}=13\,\text {Te}\text {V} $$ s=13Te
A search for dark matter produced in association with a Higgs boson decaying to a pair of bottom quarks is performed in proton–proton collisions at a center-of-mass energy of 13$$\,\text {Te}\text {V}$$ Te collected with the CMS detector at the LHC. The analyzed data sample corresponds to an integrated luminosity of 35.9$$\,\text {fb}^{-1}$$ fb-1 . The signal is characterized by a large missing transverse...
Measurement of exclusive $$\mathrm {\Upsilon }$$ Υ photoproduction from protons in $$\mathrm {p}$$ p Pb collisions at $$\sqrt{\smash [b]{s_{_{\mathrm {NN}}}}} = 5.02\,\text {TeV} $$ sNN=5.02TeV
The exclusive photoproduction of $$\mathrm {\Upsilon }\mathrm {(nS)} $$ Υ(nS) meson states from protons, $$\gamma \mathrm {p} \rightarrow \mathrm {\Upsilon }\mathrm {(nS)} \,\mathrm {p}$$ γp→Υ(nS)p (with $$\mathrm {n}=1,2,3$$ n=1,2,3 ), is studied in ultraperipheral $$\mathrm {p}$$ p Pb collisions at a centre-of-mass energy per nucleon pair of $$\sqrt{\smash [b]{s_{_{\mathrm {NN}}}}} = 5.02\,\text...
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\begin{document}
\renewcommand{References}{References} \renewcommand{Abstract}{Abstract} \title[Obstacle problem for the PME]{On two notions of solutions to
the obstacle problem for the singular porous medium equation} \author[K. Moring]{Kristian Moring} \address{Kristian Moring\\
Fakult\"at f\"ur Mathematik, Universit\"at Duisburg-Essen\\
Thea-Leymann-Str. 9, 45127 Essen, Germany} \email{[email protected]}
\author[C. Scheven]{Christoph Scheven} \address{Christoph Scheven\\
Fakult\"at f\"ur Mathematik, Universit\"at Duisburg-Essen\\
Thea-Leymann-Str. 9, 45127 Essen, Germany} \email{[email protected]}
\begin{abstract} We show that two different notions of solutions to the obstacle problem for the porous medium equation, a potential theoretic notion and a notion based on a variational inequality, coincide for regular enough compactly supported obstacles. \end{abstract} \makeatother
\maketitle
\section{Introduction}
In this paper we consider the porous medium equation (PME for short), which can be written as \begin{equation}\label{evo_eqn} \partial_t u -\Delta( u^m)=0, \end{equation} for $0<m<\infty$ and nonnegative $u$. Here we focus on the singular range $0 < m < 1$. For the standard theory of this equation we refer to the monographs~\cite{Vazquez,Vazquez2,DK}.
In the obstacle problem, the objective roughly is to find a solution to the equation with an additional constraint. The constraint here is that the solution must stay above the given obstacle in the whole domain. Besides being interesting in its own right, the obstacle problem plays a role as a standard tool in nonlinear potential theory for example (see e.g.~\cite{HKM,KinnunenLindqvist2006,KinnunenLindqvist_crelle}). In this paper, we consider two different notions of solutions to the obstacle problem: weak solution and the minimal supersolution above the given obstacle.
For the porous medium equation, the existence of weak (or variational) solutions has been studied in~\cite{AL,BLS,Schaetzler1,Schaetzler2}, and regularity questions are addressed in~\cite{MS,CS-obstacle,BLS-holder,MSb}. Typically, existence proofs for weak solutions require sufficient smoothness of the obstacle; in particular, regularity in time is a frequent presumption~\cite{BDM,AL,BLS,Schaetzler1,Schaetzler2}. On the contrary, existence and uniqueness of the minimal supersolution above the given obstacle are rather immediate consequences due to the concept of balayage from potential theory without any differentiability assumptions on the obstacle. It is also worth to mention that a more explicit construction of minimal supersolutions above the obstacle based on Schwartz alternating method has been proposed, see e.g.~\cite{korte_lscobstacle,kokusi}.
Main objective of the paper is to show a connection between the notions of weak solutions to the obstacle problem and minimal supersolutions above the given obstacle. In the case of the parabolic $p$-Laplace equation, this direction has been studied in~\cite{LP}. For the porous medium equation in the slow diffusion case partial results were given in~\cite{AvLu}, where the authors showed that the minimal supersolution is a weak solution to the obstacle problem. In this paper we are able to show that in the fast diffusion case these notions actually coincide under appropriate assumptions. More precisely, we suppose that the obstacle and its first weak derivatives exist with appropriate integrability conditions such that the existence of a weak solution is guaranteed (\hspace{1sp}\cite{BLS,Schaetzler2}). Furthermore, we require that the obstacle is H\"older continuous to ensure that weak solutions are continuous (see~\cite{MS,CS-obstacle} and also~\cite{KMN,kokusi} for the parabolic $p$-Laplace equation).
In Section~\ref{sec:varsol} we introduce weak solutions to the obstacle problem and prove some properties that are beneficial later on. More specifically, we show that outside the contact set a weak solution to the obstacle problem is a weak solution to the obstacle free problem (Lemma~\ref{l.variational-solution-noncontactset}). In Section~\ref{sec:minimal} we consider minimal supersolutions above the given obstacle, which we define via the concept of balayage. The definition is based on the notion of supercaloric functions that have been studied for the porous medium equation e.g. in~\cite{KinnunenLindqvist_crelle,KLLP-supercal,MSch}. Many useful results on supercaloric functions and their connection to weak supersolutions in the fast diffusion case were established in~\cite{MSch}. In particular, it was shown that in each connected component of the domain the positivity set of a supercaloric function can be expressed as a countable union of time intervals. Our result on the connection of the two notions for obstacle problems relies heavily on this property.
The connection between the two notions is established in Section~\ref{sec:connection}. The precise statement is given in Theorem~\ref{t.weakvar-is-minimal}. The proof is closely connected to a suitable comparison principle (as in~\cite{AvLu}) together with results on positivity sets of supercaloric functions in the fast diffusion case established in~\cite{MSch}. The main idea is to show that a weak solution to the obstacle problem is below the minimal supersolution in the domain, after which the conclusion is rather immediate by the minimality property of the latter.
In the linear case, the comparison principle for general open sets is a rather straightforward consequence once the comparison principle in finite unions of space-time cylinders is at disposal. This is mainly due to the fact that the class(es) of solutions is closed under addition of constants. The porous medium equation does not enjoy this property, which makes such a comparison principle more difficult to verify. The idea proposed in~\cite{AvLu} was, instead of adding a small constant to solution, to multiply the solution with a constant close to one. While the resulting function is not a weak solution in the original sense exactly, it still satisfies the equation with a certain source term, which disappears when the constant tends to one. In~\cite{AvLu}, this observation allowed to deduce a comparison principle between solutions and supersolutions in more general sets than space-time cylinders. More precisely, the result was shown in cylinders from which certain compact set can be removed. A crucial assumption that was made in the proof -- due to multiplication instead of addition of a constant -- was that the supersolution stays strictly positive in this compact set.
In order to adapt this comparison principle to the proof that weak solutions to the obstacle problem and minimal supercaloric functions coincide (Theorem~\ref{t.weakvar-is-minimal}), we rely on the fact that in the fast diffusion case positivity sets of supercaloric functions depend only on the time variable (\hspace{1sp}\cite{MSch}). Due to this property, we are able to circumvent the additional positivity assumption on the supersolution (minimal supersolution above the obstacle) and finally conclude the result.
\noindent {\bf Acknowledgments.} K.~Moring has been supported by the Magnus Ehrnrooth Foundation.
\section{Preliminaries}
Let $\Omega \subset \ensuremath{\mathbb{R}}^n$ be an open and bounded set. For $T>0$ we denote by $\Omega_T := \Omega \times (0,T)$ a space-time cylinder in $\ensuremath{\mathbb{R}}^{n+1}$. The parabolic boundary of $\Omega_T$ is defined as $\partial_p \Omega_T := \left( \Omega \times \{0\} \right) \cup \left( \partial \Omega \times [0,T) \right)$. Up next we define weak (super/sub)solutions.
\begin{definition} \label{d.weak_sol} A measurable function $u: \Omega_T \to [0,\infty]$ satisfying $$ u^m \in L_{\loc}^2(0,T;H^{1}_{\loc}(\Omega)) \cap L^\frac{1}{m}_{\loc}(\Omega_T) $$ is called a weak solution to the PME~\eqref{evo_eqn} if and only if $u$ satisfies the integral equality \begin{equation} \label{eq:wsol} \iint_{\Omega_T} \left(-u \partial_t \varphi + \nabla u^m \cdot \nabla \varphi \right)\:\! \mathrm{d} x\:\! \mathrm{d} t =0 \end{equation} for every $\varphi \in C^\infty_0(\Omega_T)$. Further, we say that $u$ is a weak supersolution if the integral above is nonnegative for all nonnegative test functions $\varphi \in C^{\infty}_0(\Omega_T)$. If the integral is nonpositive for such test functions, we call $u$ a weak subsolution.
Furthermore, we say that $u$ is a global weak solution if $u$ satisfies~\eqref{eq:wsol} and $$ u^m \in L^2(0,T;H^1(\Omega)) \cap L^\frac{1}{m}(\Omega_T). $$ \end{definition}
We recall an existence and comparison result for continuous, global weak solutions, see~\cite{Abdulla1,Abdulla2,Bjorns_boundary,MSch}.
\begin{theorem} \label{t.existence} Let $0<m<1$ and $\Omega_T$ be a $C^{1,\alpha}$-cylinder with $\alpha >0$. Suppose that the function $g \in C(\overline{\Omega_T})$ satisfies $g^m \in L^2(0,T;H^1(\Omega))$ and $\partial_t g^m \in L^\frac{m+1}{m}(\Omega_T)$. Then, there exists a unique global weak solution $u$ to~\eqref{evo_eqn} such that $u \in C(\overline{\Omega_T})$, $u$ is locally H\"older continuous and $u = g$ on $\partial_p \Omega_T$. Moreover, if $g'$ satisfies conditions above and $g\leq g'$ on $\partial_p \Omega$ and $h' \in C(\overline {\Omega_T})$ is a global weak solution with boundary values $g'$ on $\partial_p \Omega_T$, then $h \leq h'$ in $\Omega_T$. \end{theorem}
The next result states that a weak supersolution is lower semicontinuous after possible redefinition in a set of measure zero, see~\cite{Naian}.
\begin{theorem} \label{t.super_lsc} Suppose that $m > 0$. Let $u$ be a weak supersolution according to Definition~\ref{d.weak_sol}. Then, there exists a lower semicontinuous function $u_*$ such that $u_*(x,t) = u(x,t)$ for a.e. $(x,t) \in \Omega_T$. Moreover, $u_*$ satisfies \begin{equation}\label{def:u-star} u_*(x,t) = \essliminf_{\substack{(y,s) \to (x,t) \\ s<t}} u(y,s) = \lim_{\varrho\to0}\left( \essinf_{B_\varrho(x)\times (t-\varrho^2,t)} u \right). \end{equation} \end{theorem}
The following Caccioppoli inequality holds for bounded weak supersolutions, see~\cite[Lemma 2.15]{KinnunenLindqvist_crelle}.
\begin{lemma} \label{l.bounded_caccioppoli} Suppose that $m>0$ and $u \leq M$ is a weak supersolution in $\Omega_T$. Then, there exists a numerical constant $c > 0$ such that $$
\int_{0}^{T}\int_{\Omega} \xi^2 \left| \nabla u^m \right|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \leq c M^{2m} T \int_\Omega \left| \nabla \xi \right|^2 \, \:\! \mathrm{d} x + c M^{m+1} \int_\Omega \xi^2 \, \:\! \mathrm{d} x $$ for every $\xi = \xi(x) \in C_0^\infty (\Omega)$ with $\xi \geq 0$. \end{lemma}
Then we recall the definition of super- and subcaloric functions.
\begin{definition} \label{d.supercal} We call $u \colon \Omega_T \to [0,\infty]$ a supercaloric function, if the following properties hold: \begin{itemize} \item[(i)] $u$ is lower semicontinuous, \item[(ii)] $u$ is finite in a dense subset, \item[(iii)] $u$ satisfies the comparison principle in every subcylinder $Q_{t_1,t_2}=Q \times (t_1,t_2) \Subset \Omega_T$: if $h\in C(\overline{Q}_{t_1,t_2})$ is a weak solution in $Q_{t_1,t_2}$ and if $h \leq u$ on the parabolic boundary of $Q_{t_1,t_2}$, then $ h\leq u$ in $Q_{t_1,t_2}$. \end{itemize}
A function $v: \Omega_T \to [0,\infty)$ is called subcaloric function if the conditions (i), (ii) and (iii) above hold with (i) replaced by upper semicontinuity, and the inequalities in (iii) by $\geq$. \end{definition}
\begin{remark} In~\cite{MSch} the authors considered so called quasi-super(sub)caloric functions, for which the comparison principle (iii) in Definition~\ref{d.supercal} is required only in $C^{2,\alpha}$-subcylinders of $\Omega_T$. It was shown that the classes of quasi-super(sub)caloric and super(sub)caloric functions coincide, which implies that it is sufficient to require that the comparison principle in item (iii) holds in all $C^{2,\alpha}$-subcylinders. \end{remark}
Supercaloric functions are closed under increasing limits provided that the limit function is finite in a dense set, see~\cite[Proposition 4.6]{Bjorns_boundary}.
\begin{lemma} \label{l.superc_increasing_lim} Let $m>0$ and $u_k$ be a nondecreasing sequence of supercaloric functions in $\Omega_T$. If $u := \lim_{k\to \infty} u_k$ is finite in a dense subset of $\Omega_T$, then $u$ is supercaloric in $\Omega_T$. \end{lemma}
Furthermore, we can extend supercaloric functions by zero to the past, see~\cite{MSch}.
\begin{lemma} \label{l.zero-past-extension} Let $0<m<1$ and $v : \Omega_T \to [0,\infty]$ be a supercaloric function in $\Omega_T$. Then \[ u = \begin{cases} v\quad &\text {in } \Omega \times (0,T), \\ 0\quad &\text {in } \Omega \times (-\infty,0], \end{cases} \] is a supercaloric function in $\Omega \times (-\infty,T)$. \end{lemma}
In the fast diffusion case, on a fixed time slice a weak supersolution is either positive or zero everywhere provided that the domain is connected, see~\cite{MSch}.
\begin{lemma}\label{lem:alternatives}
Let $0<m<1$ and assume that $u$ is a supercaloric function to the PME in $\Omega_T$,
where $\Omega\subset\ensuremath{\mathbb{R}}^n$ is open and
connected. Then, for any time $t\in(0,T)$ either
$u$ is positive on the whole time slice $\Omega\times\{t\}$ or $u$
vanishes on the whole time slice.
In particular, the positivity set of $u$ can be written as the union
of cylinders $\Omega\times\Lambda_i$ for at most countably many open intervals
$\Lambda_i\subset(0,T)$. \end{lemma}
We also have the following connections of supercaloric functions and weak supersolutions, see~\cite{MSch}.
\begin{proposition} \label{p.bdd-supercal-supersol} Let $0<m<1$ and suppose that $u$ is a locally bounded supercaloric function to the porous medium equation in $\Omega_T$,
where $\Omega\subset\ensuremath{\mathbb{R}}^n$ is an open set. Then, $u$ is a weak supersolution. \end{proposition}
In connection to the next result, see Theorem~\ref{t.super_lsc}. For the proof we refer to~\cite{MSch}. \begin{lemma} \label{l.weasuper-is-supercal} Let $0<m<1$ and suppose that $u$ is a weak supersolution in $\Omega_T$. Then, $u_*$ is a supercaloric function in $\Omega_T$. \end{lemma}
For the proof of the next lemma we refer to~\cite{MSch}, see also~\cite[Theorem 3.6]{Bjorns_boundary} and~\cite[Theorem 3.3]{KLL}.
\begin{lemma} \label{l.supersubcal-cylinder-comparison} Let $m>0$ and $U_{t_1,t_2} \Subset \ensuremath{\mathbb{R}}^{n+1}$ be a cylinder. Suppose that $u$ is a supercaloric and $v$ is a subcaloric function in $U_{t_1,t_2}$. If $$ \infty \neq \limsup_{U_{t_1,t_2} \ni (y,s) \to (x,t)} v(y,s) \leq \liminf_{U_{t_1,t_2} \ni (y,s) \to (x,t)} u(y,s) $$ for every $(x,t) \in \partial_p U_{t_1,t_2}$, then $v \leq u$ in $U_{t_1,t_2}$. \end{lemma}
For supercaloric functions the following result holds true by~\cite{MSch}.
\begin{theorem} \label{t.supercal-essliminf} Let $0 < m< 1$ and $u : \Omega_T \to [0,\infty]$ be a supercaloric function in $\Omega_T$. Then, $$ u(x,t) = \essliminf_{\substack{(y,s) \to (x,t) \\ s<t}} u(y,s)\quad \text{for every } (x,t) \in \Omega_T. $$ \end{theorem}
\begin{definition} For $v \in L^1(\Omega_T)$, $v_o \in L^1(\Omega)$ and $h>0$, we define a mollification in time by \begin{equation} \label{eq:time-mollif}
\mollifytime{v}{h}(x,t)
:=
\mollifytime{v}{h,v_o}(x,t)
:= e^{-\frac{t}{h}}v_o(x) + \tfrac1h \int_0^t e^{\frac{s-t}{h}}
v(x,s)\, \mathrm{d}s
\end{equation}
for $(x,t)\in \Omega\times[0,T]$. \end{definition}
We state some useful properties of the mollification in the following lemma, see~\cite[Lemma 2.2]{KinnunenLindqvist2006}.
\begin{lemma} \label{lem:mollifier}
Let $\mollifytime{v}{h}$ be defined as in~\eqref{eq:time-mollif}. Then the following properties hold:
\begin{enumerate}[(i)]
\itemsep2mm
\item Let $p\ge 1$ and $X\in \{L^p(\Omega),W^{1,p}(\Omega),W^{1,p}_0(\Omega)\}$. If $v \in L^p(0,T;X)$ and $v_o \in X$,
then $\mollifytime{v}{h}\in C([0,T];X)$ and $\mollifytime{v}{h}(\cdot,0)=v_o$.
Furthermore $\mollifytime{v}{h} \xrightarrow{h\to 0} v$ in $L^p(0,T;X)$ . \item If $v \in C(\overline{\Omega_T})$, $v_o = v(\cdot,0)$ and $\Omega \subset \ensuremath{\mathbb{R}}^n$ is a bounded set, then $$ \mollifytime{v}{h} \xrightarrow{h\to 0} v\quad \text{ uniformly in } \Omega_T. $$ \item If $v\in L^p(\Omega_T)$ for some $p\ge1$,
the weak time derivative $\partial_t \mollifytime{v}{h}$ exists in
$L^p(\Omega_T)$ and is given by the formula \begin{equation*}
\partial_t \mollifytime{v}{h} = \frac{1}{h} ( v - \mollifytime{v}{h} ). \end{equation*} \end{enumerate} \end{lemma}
\section{Weak solution to the obstacle problem} \label{sec:varsol}
We consider an obstacle function $\psi:\Omega\times[0,T]\to\ensuremath{\mathbb{R}}_{\ge0}$ and abbreviate $\psi_o:=\psi(\cdot,0)$. Now we define the notion of weak solution to the obstacle problem. Let $$ K_\psi(\Omega_T) := \{ v \colon v^m \in L^2(0,T;H^{1}(\Omega)),\, u\in C([0,T];L^{m+1}(\Omega)), v \geq \psi \text{ a.e. in } \Omega_T \} $$ and $$ K'_\psi(\Omega_T) := \{ v \in K_\psi(\Omega_T) : \partial_t (v^m) \in L^\frac{m+1}{m}(\Omega_T) \}. $$ For cutoff functions $\eta\in C^1_0(\Omega, \ensuremath{\mathbb{R}}_{\geq 0})$ and $\alpha \in W^{1,\infty}_0((0,T);\ensuremath{\mathbb{R}}_{\geq 0})$ we define \begin{align*} \llangle \partial_t u, \alpha \eta (v^m - u^m) \rrangle &:= \iint_{\Omega_T} \eta \left[ \alpha' \left( \tfrac{1}{m+1} u^{m+1} - uv^m \right) -\alpha u \partial_t v^m \right] \, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align*}
\begin{definition} \label{d.local-weak-obstacle} A function $u \in K_\psi(\Omega_T)$ is a local weak solution to the obstacle problem for the PME if \begin{equation} \label{e.local_var_eq} \llangle \partial_t u, \alpha \eta (v^m - u^m) \rrangle + \iint_{\Omega_T} \alpha \nabla u^m \cdot \nabla \left(\eta( v^m - u^m) \right) \:\! \mathrm{d} x \:\! \mathrm{d} t \geq 0 \end{equation} holds true for all comparison maps $v \in K'_\psi(\Omega_T)$, every $\eta\in C^1_0(\Omega,\ensuremath{\mathbb{R}}_{\ge0})$ and $\alpha \in W^{1,\infty}_0((0,T);\ensuremath{\mathbb{R}}_{\geq 0})$. \end{definition}
For a weak solution, we additionally require that it agrees with the obstacle on the parabolic boundary.
\begin{definition} \label{d.variational-obstacle} A function $u \in K_{\psi}(\Omega_T)$ is a weak solution to the obstacle problem for the PME with $u=\psi$ on $\partial_p\Omega_T$ if it is a local weak solution in the sense of Definition~\ref{d.local-weak-obstacle}, it attains the initial values $u(\cdot,0)=\psi_o$ a.e. in $\Omega$ and $u^m - \psi^m \in L^2(0,T;H^1_0(\Omega))$. \end{definition}
We will rely on the following results on existence and regularity of weak solutions to the obstacle problem. For the existence, we refer to \cite{BLS,Schaetzler2}, see also Appendix~\ref{appendix-a}. Regularity follows from~\cite{CS-obstacle,MS}. \begin{theorem}\label{t.cont-exist}
Let $\Omega \subset \ensuremath{\mathbb{R}}^n$ be a bounded Lipschitz domain and $\psi^m\in
L^2(0,T;H^1(\Omega))\cap C([0,T];L^{\frac{m+1}{m}}(\Omega))$ with $\partial_t(\psi^m)\in
L^{\frac{m+1}{m}}(\Omega_T)$ and $\psi_o^m \in
L^{\frac{m+1}{m}}(\Omega) \cap H^1(\Omega)$. Then, there exists a weak
solution $u$ to the obstacle problem with $u=\psi$ on
$\partial_p\Omega_T$ in the sense of Definition~\ref{d.variational-obstacle}.
If the obstacle additionally satisfies $\psi\in C^{0;\beta,\frac{\beta}{2}}(\Omega_T)$ for some $\beta \in (0,1)$,
then every locally bounded local weak solution to the obstacle problem is locally H\"older continuous in $\Omega_T$. \end{theorem}
We show that every local weak solution to the obstacle problem is a weak supersolution to the obstacle free problem.
\begin{lemma} \label{l.variationalsol-is-supersol} Suppose that $\Omega \Subset \ensuremath{\mathbb{R}}^n$ and let $u \in K_\psi(\Omega_T)$ be a local weak solution according to Definition~\ref{d.local-weak-obstacle} to the obstacle problem with obstacle $\psi \in C(\overline{\Omega_T})$. Then $u$ is a weak supersolution. \end{lemma}
\begin{remark} One could alternatively assume that $\psi^m \in L^2(0,T;H^1(\Omega))$ and $\partial_t \psi^m \in L^\frac{m+1}{m}(\Omega_T)$ instead of continuity of $\psi$. \end{remark}
\begin{proof} Let $\varphi \in C^\infty_0(\Omega_T, \ensuremath{\mathbb{R}}_{\geq 0})$. In the definition of local weak solution, we consider cutoff functions $\eta\in C^1_0(\Omega,\ensuremath{\mathbb{R}}_{\ge0})$ and $\alpha \in W^{1,\infty}([0,T];\ensuremath{\mathbb{R}}_{\geq 0})$ with $\eta \alpha \equiv 1$ in $\operatorname{spt} (\varphi)$ such that $\alpha$ is compactly supported in $(0,T)$. Let $R \geq 1$ such that $B_R(0) \supset \Omega$, and extend $\psi(\cdot,0)$ as continuous function to $B_{2R} (0) \setminus \overline{\Omega}$ and as zero to $\ensuremath{\mathbb{R}}^n \setminus B_{2R}(0)$. Denote this extension by $\psi_o$. We will use time mollifications $\mollifytime{u^m}{h,(\psi_o^m)_\delta}$ and $\mollifytime{\psi^m}{h,(\psi_o^m)_\delta}$. Observe that for each sequence $\varepsilon_i \xrightarrow{i\to \infty} 0$ there exists a sequence $\delta_i \xrightarrow{i \to \infty} 0$ such that $$
\|\mollifytime{\psi^m}{h,(\psi_o^m)_{\delta}} -
\mollifytime{\psi^m}{h,\psi_o^m}\|_{L^\infty(\Omega_T)} =
\|(\psi_o^m)_{\delta} -\psi_o^m\|_{L^\infty(\Omega)} < \varepsilon_i $$ for every $\delta \in (0,\delta_i]$, uniformly in $h>0$, and by Lemma~\ref{lem:mollifier}\,(ii) there is a sequence $\hat h_i \xrightarrow{i \to \infty}0$ such that $$
\|\mollifytime{\psi^m}{h,\psi_o^m} - \psi^m\|_{L^\infty(\Omega_T)} <
\varepsilon_i $$ for every $h \in (0,\hat h_i]$. Furthermore, for $\varepsilon_i$ and $\delta_i$ constructed above, Lemma~\ref{lem:mollifier}\,(i) yields a sequence $\tilde h_i \xrightarrow{i \to \infty} 0$ such that $$
\max\{\| \mollifytime{u^m}{h,(\psi_o^m)_{\delta_i}} - u^m
\|_{L^\frac{m+1}{m}(\Omega_T)},
\|\mollifytime{u^m}{h,(\psi_o^m)_{\delta_i}} -
u^m\|_{L^2(0,T;H^1(\Omega))} \} < \varepsilon_i $$ for each $h \in (0,\tilde h_i]$. By choosing $h_i := \min\{\hat h_i, \tilde h_i\}$ we have that \begin{align*}
\begin{array}{rl}
\mollifytime{\psi^m}{i} :=
\mollifytime{\psi^m}{h_i,(\psi_o^m)_{\delta_i}} \xrightarrow{i \to
\infty } \psi^m&\mbox{ uniformly in $\Omega_T$ and}\\[0.8ex]
\mollifytime{u^m}{i} := \mollifytime{u^m}{h_i,(\psi_o^m)_{\delta_i}} \xrightarrow{i \to \infty } u\phantom{^m}&\mbox{ in $L^2(0,T;H^1(\Omega))$ and in $L^\frac{m+1}{m}(\Omega_T)$}.
\end{array} \end{align*} Define a comparison map $$
v^m_i = \mollifytime{u^m}{i} + \varphi + \|\psi^m - \mollifytime{\psi^m}{i}\|_\infty. $$ It follows that $v^m_i \in K'_\psi(\Omega_T)$ and in particular, $$ v^m_i \geq \mollifytime{u^m}{i} + \psi^m - \mollifytime{\psi^m}{i} \geq \psi^m \quad \text{ a.e. in } \Omega_T, $$ since $\varphi \geq 0$ and $u \geq \psi$ a.e. in $\Omega_T$.
For the parabolic part in~\eqref{e.local_var_eq} we have \begin{align*}
&\llangle \partial_t u , \alpha \eta (v^m_i -u^m) \rrangle \\
&\qquad= \iint_{\Omega_T} \eta \alpha' \left(\tfrac{1}{m+1}u^{m+1} - u (\mollifytime{u^m}{i} + \varphi + \|\psi^m - \mollifytime{\psi^m}{i}\|_\infty )\right) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\qquad\phantom{+} - \iint_{\Omega_T} \alpha \eta u \partial_t \left(
\mollifytime{u^m}{i} + \varphi + \|\psi^m - \mollifytime{\psi^m}{i}\|_\infty \right) \, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align*} Since Lemma~\ref{lem:mollifier} (iii) implies
\begin{align}\label{time-deriv-moll}
u\partial_t\mollifytime{u^m}{i}
&=
\big(u-\mollifytime{u^m}{i}^{\frac1m}\big)\partial_t\mollifytime{u^m}{i}
+
\mollifytime{u^m}{i}^{\frac1m}\partial_t\mollifytime{u^m}{i}\\\nonumber
&\ge
\mollifytime{u^m}{i}^{\frac1m}\partial_t\mollifytime{u^m}{i}
=
\tfrac{m}{m+1}\partial_t\mollifytime{u^m}{i}^{\frac{m+1}{m}},
\end{align} we may estimate \begin{align*} - \iint_{\Omega_T} \alpha \eta u \partial_t \mollifytime{u^m}{i} \, \:\! \mathrm{d} x \:\! \mathrm{d} t &\leq
\tfrac{m}{m+1}\iint_{\Omega_T} \alpha' \eta \mollifytime{u^m}{i}^\frac{m+1}{m} \, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align*} By combining the estimates above, and using Lemma~\ref{lem:mollifier} (i), (ii) together with the fact $\alpha\eta \equiv 1$ in $\operatorname{spt}(\varphi)$, we have \begin{align*} \limsup_{i\to\infty}\llangle \partial_t u , \alpha \eta (v_i^m -u^m) \rrangle &\leq - \iint_{\Omega_T} \eta u( \alpha' \varphi + \alpha \partial_t \varphi) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &= -\iint_{\Omega_T} u \partial_t \varphi \, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align*} For the divergence part we obtain \begin{align*} \iint_{\Omega_T} &\alpha \nabla u^m \cdot \nabla(\eta (v^m_i - u^m) ) \, \:\! \mathrm{d} x \:\! \mathrm{d} t\\ &= \iint_{\Omega_T} \alpha (\mollifytime{u^m}{i} - u^m ) \nabla u^m \cdot \nabla\eta \, \:\! \mathrm{d} x \:\! \mathrm{d} t +\iint_{\Omega_T} \alpha \varphi \nabla u^m \cdot \nabla\eta \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\
&\phantom{+}+\iint_{\Omega_T} \alpha \|\psi^m - \mollifytime{\psi^m}{i}\|_\infty \nabla u^m \cdot \nabla \eta\, \:\! \mathrm{d} x\:\! \mathrm{d} t \\ &\phantom{+}+ \iint_{\Omega_T} \alpha \eta \nabla u^m \cdot \nabla(\mollifytime{u^m}{i} - u^m ) \, \:\! \mathrm{d} x \:\! \mathrm{d} t+ \iint_{\Omega_T} \alpha \eta \nabla u^m \cdot \nabla \varphi \, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align*} Again, by using the fact $\alpha \eta \equiv 1$ in $\operatorname{spt}(\varphi)$ and Lemma~\ref{lem:mollifier} (i), (ii) we arrive at $$ \lim_{i\to\infty} \iint_{\Omega_T} \alpha \nabla u^m \cdot \nabla(\eta (v^m_i - u^m) ) \, \:\! \mathrm{d} x \:\! \mathrm{d} t = \iint_{\Omega_T} \nabla u^m \cdot \nabla \varphi \, \:\! \mathrm{d} x \:\! \mathrm{d} t. $$ By combining the results, we obtain \begin{equation*}
\iint_{\Omega_T} (-u\partial_t\varphi+\nabla u^m \cdot \nabla \varphi) \, \:\! \mathrm{d} x \:\! \mathrm{d} t\ge0, \end{equation*} and the claim follows. \end{proof}
The next lemma shows that local weak solutions to the obstacle problem are weak solutions to the PME outside of the contact set.
\begin{lemma} \label{l.variational-solution-noncontactset}
Let $\psi\in C(\overline{\Omega_T})$, and
$u\in K_\psi(\Omega_T)$ be a continuous, local weak solution to the obstacle problem
for the porous medium equation in the sense of
Definition~\ref{d.local-weak-obstacle}.
Then, $u$ is a weak solution to the
porous medium equation
in the set $\{(x,t)\in\Omega_T\colon u(x,t)>\psi(x,t)\}$. \end{lemma}
\begin{proof}
It suffices to prove that $u$ is a weak solution in any parabolic
cylinder $Q=B\times(t_o,t_1)\Subset\{u>\psi\}$.
Without loss generality, we may assume that $u^m(\cdot,t_o)\in
H^1(B)$.
We consider an arbitrary test function $\varphi\in
C^\infty_0(Q)$. In~\eqref{e.local_var_eq}, we choose cutoff functions
$\alpha\in W^{1,\infty}_0((t_o,t_1),[0,1])$ and $\eta\in
C^1_0(B,[0,1])$ with $\alpha\eta\equiv1$ in $\operatorname{spt}(\varphi)$ and
$\operatorname{spt}(\alpha\eta)\Subset Q$.
We use the time mollification $\mollifytime{u^m}{h}$
introduced in~\eqref{eq:time-mollif}
on the domain $Q$ instead of $\Omega_T$, with initial values
$u^m(\cdot,t_o)\in H^1(B)\cap L^{\frac{m+1}{m}}(B)$.
Then we define a comparison map as
\begin{equation*}
v^m_h:= \mollifytime{u^m}{h}+\varepsilon\varphi
\end{equation*}
with appropriately chosen $\varepsilon>0$. Since $u\in C(\overline Q)$, we have $\mollifytime{u^m}{h}\xrightarrow{h\downarrow0} u^m$ uniformly in
$Q$ by Lemma~\ref{lem:mollifier} (ii), so that we may assume
\begin{equation*}
\mollifytime{u^m}{h}>u^m-d \mbox{\quad in $Q$, \qquad with }
d:=\tfrac12\inf_{Q}(u^m-\psi^m)>0,
\end{equation*}
by choosing $h>0$ sufficiently small. If $\varphi\ge0$ in $Q$, we
consider an arbitrary $\varepsilon>0$. Otherwise, we restrict ourselves to
parameters
\begin{equation*}
0<\varepsilon<\frac{d}{-\inf\varphi}.
\end{equation*}
In any case, we obtain $v^m_h>u^m-2d\ge\psi^m$ in $Q$. We note that by our choices of the cutoff
functions, it is sufficient to ensure the obstacle constraint in
$Q$. In fact, instead of $v_h^m$ we can use the comparison map
\begin{equation*}
v^m:=\zeta v_h^m+(1-\zeta)\sup_{\Omega_T}\psi^m,
\end{equation*}
with a cutoff function $\zeta\in C^\infty_0(Q,[0,1])$ satisfying
$\zeta\equiv1$ on $\operatorname{spt}(\alpha\eta)$. Since $v_h\ge\psi$ a.e. in
$Q$, we have $v\ge\psi$ a.e. in $\Omega_T$. Therefore, we can use
$v$ in the variational inequality~\eqref{e.local_var_eq}. By our
choice of $\zeta$, this gives the same result as plugging in
$v_h$, i.e.
\begin{align}\label{var-ineq}
0&\le
\langle\!\langle\partial_tu,\alpha\eta(v^m_h-u^m)\rangle\!\rangle
+
\iint_{\Omega_T}\alpha \nabla u^m\cdot
\nabla\big(\eta(v^m_h-u^m)\big) \, \mathrm{d}x\mathrm{d}t\\\nonumber
&=
\iint_{\Omega_T} \eta \big( \alpha'\big(\tfrac1{m+1}u^{m+1}-u\mollifytime{u^m}{h}\big)
-\alpha u\partial_t\mollifytime{u^m}{h} \big) \, \mathrm{d}x\mathrm{d}t\\\nonumber
&\qquad
-\varepsilon\iint_{\Omega_T}\eta\big( \alpha'u\varphi+\alpha
u\partial_t\varphi\big) \, \mathrm{d}x\mathrm{d}t\\\nonumber
&\qquad+
\iint_{\Omega_T}\alpha \nabla u^m\cdot
\nabla\big(\eta(\mollifytime{u^m}{h}-u^m+\varepsilon\varphi
)\big) \, \mathrm{d}x\mathrm{d}t\\\nonumber
&=:\mathrm{I}+\mathrm{II}+\mathrm{III}.
\end{align}
Using estimate~\eqref{time-deriv-moll},
we can bound the first term by
\begin{align*}
\mathrm{I}
&\le
\iint_{\Omega_T} \eta \big( \alpha'\big(\tfrac1{m+1}u^{m+1}-u\mollifytime{u^m}{h}\big)
-\alpha \tfrac{m}{m+1}\partial_t\mollifytime{u^m}{h}^{\frac{m+1}{m}}
\big)\mathrm{d}x\mathrm{d}t\\
&=
\iint_{\Omega_T} \eta \alpha'\big(\tfrac1{m+1}u^{m+1}-u\mollifytime{u^m}{h}
+\tfrac{m}{m+1}\mollifytime{u^m}{h}^{\frac{m+1}{m}}\big)\mathrm{d}x\mathrm{d}t\\
&\xrightarrow{h\downarrow0} 0.
\end{align*}
The last convergence follows from Lemma~\ref{lem:mollifier}\,(i).
Moreover, since $\alpha\eta\equiv1$ on
$\operatorname{spt}(\varphi)$, we have
\begin{align*}
\mathrm{II}
=
-\varepsilon\iint_{\Omega_T}u\partial_t\varphi \, \mathrm{d}x\mathrm{d}t,
\end{align*}
and Lemma~\ref{lem:mollifier}\,(i) implies
\begin{align*}
\mathrm{III}
\xrightarrow{h\downarrow0}
\varepsilon\iint_{\Omega_T}\alpha\nabla u^m\cdot\nabla(\eta\varphi)\,\mathrm{d}x\mathrm{d}t
=
\varepsilon\iint_{\Omega_T}\nabla u^m\cdot\nabla\varphi \, \mathrm{d}x\mathrm{d}t.
\end{align*}
Therefore, by letting $h\downarrow0$
in~\eqref{var-ineq}, we infer
\begin{equation*}
0\le\varepsilon\iint_{\Omega_T}\big(-u\partial_t\varphi+\nabla
u^m\cdot\nabla\varphi\big) \, \mathrm{d}x\mathrm{d}t.
\end{equation*}
Since the same inequality holds for $-\varphi$ in place of
$\varphi$, we actually have equality. After dividing by $\varepsilon>0$, we
thus obtain
\begin{equation*}
\iint_{\Omega_T}\big(-u\partial_t\varphi+\nabla u^m\cdot\nabla\varphi\big) \, \mathrm{d}x\mathrm{d}t=0
\end{equation*}
for every $\varphi\in C^\infty_0(Q)$, for an arbitrary cylinder
$Q\Subset\{u>\psi\}$. A partition of unity argument now implies that
$u$ is a weak solution to the porous medium equation in the set
$\{u>\psi\}$. \end{proof}
\section{Minimal supersolution above the obstacle} \label{sec:minimal}
We define the minimal supersolution above the given obstacle as follows.
\begin{definition} \label{d.minimal-supercal} Let $\psi \in C(\Omega_T) \cap L^\infty(\Omega_T)$. We say that $u: \Omega_T \to [0,\infty)$ is a minimal supersolution above the obstacle $\psi$ if the following properties hold true: \begin{itemize} \item[(i)] $u(x,t) \geq \psi(x,t)$ for every $(x,t) \in \Omega_T$; \item[(ii)] $u$ is a supercaloric function in $\Omega_T$; \item[(iii)] $u$ is the smallest supercaloric function in $\Omega_T$ which lies above $\psi$, i.e., if $v$ is a supercaloric function with $v(x,t)\geq \psi(x,t)$ for all $(x,t) \in \Omega_T$, then $v(x,t)\geq u(x,t)$ for all $(x,t) \in \Omega_T$; \item[(iv)] $u$ is a weak solution to~\eqref{evo_eqn} in the set $\{(x,t) \in \Omega_T : u(x,t) > \psi (x,t) \}$. \end{itemize} \end{definition}
Next we define the balayage of the given obstacle $\psi$, which will be our candidate for the solution in Definition~\ref{d.minimal-supercal}.
\begin{definition} Let $\psi: \Omega_T \to [0,\infty]$ be a function, and denote $$ \mathcal U_\psi := \{ v \text{ is a supercaloric function in } \Omega_T : v(x,t) \geq \psi(x,t)\, \text{ for every } (x,t) \in \Omega_T \}. $$ We define the r\'eduite of $\psi$ as $$ R_\psi(x,t) = \inf \{ v(x,t): v \in \mathcal U_\psi \}, $$ and the balayage of $\psi$ as its lower semicontinuous regularization $$ \hat R_\psi(x,t) := \lim_{\varrho \to 0} \left( \inf_{B_\varrho(x) \times (t-\varrho^2,t+\varrho^2)} R_\psi \right). $$ \end{definition} From the definition it clearly follows that $\hat R_\psi$ is a lower semicontinuous function. Observe that, in general, $\hat R_\psi$ is not necessarily above $\psi$ at every point. However, for example if $\psi$ is lower semicontinuous, then $\hat R_\psi \geq \psi$ everywhere in $\Omega_T$.
\begin{remark} By the argument in~\cite[Lemma 2.7]{LP} together with Theorem~\ref{t.supercal-essliminf} we have that $\hat R_\psi = (R_\psi)_*$ everywhere in $\Omega_T$, where $(R_\psi)_*$ denotes the $\essliminf$-regularization of $R_\psi$ as introduced in~\eqref{def:u-star}. Thus it does not matter which regularization we use for the r\'eduite $R_\psi$. Moreover, if $\psi$ is a bounded function, then it follows that $\hat R_\psi = R_\psi$ a.e. in $\Omega_T$, see~\cite[Theorem 2.8]{LP} in connection with Proposition~\ref{p.bdd-supercal-supersol}, Theorem~\ref{t.super_lsc} and Lemma~\ref{l.bounded_caccioppoli}. \end{remark}
\begin{lemma} \label{l.balayage-supercal} Let $\psi : \Omega_T \to [0,\infty)$ be a bounded, lower semicontinuous function in $\Omega_T$. Then $\hat R_\psi$ exists, it is bounded and satisfies properties (i), (ii) and (iii) in Definition~\ref{d.minimal-supercal}. \end{lemma}
\begin{proof} Since $\sup_{\Omega_T} \psi < \infty$ is a supercaloric function as a constant, it follows that $\sup_{\Omega_T} \psi \in \mathcal U_\psi$. Hence, the set $\mathcal U_\psi$ is nonempty and $\hat R_\psi \leq R_\psi \leq \sup_{\Omega_T} \psi < \infty$, which implies existence and boundedness of $\hat R_\psi$.
Clearly $R_\psi(x,t) \geq \psi(x,t)$ for every $(x,t) \in \Omega_T$ and since $\psi$ is lower semicontinuous, also $\hat R_\psi(x,t) \geq \psi(x,t)$ for every $(x,t) \in \Omega_T$. This shows property (i).
To show that $\hat R_\psi$ is supercaloric, we are left to ensure the comparison principle, item (iii) in Definition~\ref{d.supercal}. Let $Q \Subset \Omega_T$ be a cylinder and $h \in C(\overline{Q})$ be a weak solution in $Q$ such that $h \leq \hat R_\psi $ on $\partial_p Q$. This implies $h \leq v$ on $\partial_p Q$ for any $v\in \mathcal U_\psi$, and since $v$ is supercaloric, we deduce $h \leq v$ in $Q$. Since $v\in \mathcal U_\psi$ is arbitrary, this implies that $h \leq R_\psi$ in $Q$. Since $h$ is continuous, it follows that $h \leq \hat R_\psi$ in $Q$ completing the proof of property (ii).
To show item (iii) in Definition~\ref{d.minimal-supercal}, we observe that for every $v\in\mathcal U_\psi$, we have $\psi\le\hat R_\psi\le R_\psi\le v$ everywhere in $\Omega_T$. Hence, $\hat R_\psi$ is the smallest supercaloric function above $\psi$. \end{proof}
\begin{proposition}\label{p.balayage-supercal} Let $\psi \in C(\Omega_T) \cap L^\infty(\Omega_T)$. Then, $\hat R_\psi$ is the minimal supersolution above the obstacle $\psi$ according to Definition~\ref{d.minimal-supercal}. \end{proposition} \begin{proof} Properties (i), (ii) and (iii) in Definition~\ref{d.minimal-supercal} follow directly from Lemma~\ref{l.balayage-supercal}. Thus we are left to prove the property (iv).
We denote $u:= \hat R_\psi$. Clearly, the set $\{u>\psi\}$ is open by lower semicontinuity of $u$ and continuity of $\psi$. Thus for fixed $(x_o,t_o) \in \{u>\psi\}$ we can find $\lambda > 0$, $\delta > 0$ and a $C^{2,\alpha}$-cylinder $U_{t_1,t_2} \Subset \Omega_T$ with $(x_o,t_o)\in U_{t_1,t_2}$ such that $$ u>\lambda > \psi\quad \text{ in } \overline{U_{t_1,t_2+\delta}}. $$ Let $\phi_j$ be a sequence of nonnegative Lipschitz functions in $\overline{U_{t_1,t_2+\delta}}$ with the property $\lambda^m \leq \phi_j \leq \phi_{j+1}\leq u^m$, and $\lim_{j\to \infty} \phi_j(x,t) = u^m(x,t)$ for all $(x,t) \in \overline{U_{t_1,t_2+\delta}}$. Let $h_j$ be a global weak solution in $U_{t_1,t_2+\delta}$ taking continuously boundary values $(\phi_j)^\frac{1}{m}$ on $\partial_p U_{t_1,t_2+\delta}$. This solution exists according to Theorem~\ref{t.existence}. We use it to define a Poisson modification \[ u_P^j := \begin{cases} h_j\quad &\text{ in } U \times(t_1,t_2] \\ u\quad &\text{ in } \Omega_T \setminus ( U \times (t_1,t_2]). \end{cases} \] Observe that since the boundary values $(\phi_j)^\frac{1}{m}$ are increasing, Theorem~\ref{t.existence} implies that the sequence $h_j$ is increasing and thus $u_P^j$ is increasing. Hence, we may define $$ u_P := \lim_{j\to \infty} u_P^j. $$ Since $u$ is bounded, it follows that $h_j$ is uniformly bounded in $\overline{U_{t_1,t_2+ \delta}}$. Since the sequence $\{h_j\}$ is also increasing, we may conclude that the limit $h = \lim_{j\to \infty} h_j$ is a continuous weak solution in $U_{t_1,t_2 + \delta}$ by~\cite[Theorem 18.1, Chapter 6]{DGV} together with Lemma~\ref{l.bounded_caccioppoli}. We claim that $u_P$ is a supercaloric function. Clearly $u_P$ is lower semicontinuous in $ \Omega_T \setminus \partial U_{t_1,t_2}$ by continuity of $h$ in $U_{t_1,t_2}$ and lower semicontinuity of $u$. Moreover, $u_P$ is lower semicontinuous on $\partial_p U_{t_1,t_2}$ since $h$ is lower semicontinuous in $\overline U_{t_1,t_2}$ as an increasing limit of continuous functions and on $\overline {\partial_p U_{t_1,t_2}}$ the limit is $u$. On the slice $U \times \{t_2\}$ we have $h \leq u$, which together with lower semicontinuity of $u$ concludes that $u_P$ is lower semicontinuous in $\Omega_T$. Next we show the comparison principle.
Let $Q \Subset \Omega_T$ such that $Q \cap U_{t_1,t_2} \neq \varnothing$ and $Q \setminus U_{t_1,t_2} \neq \varnothing$, and let $\bar h \in C(\overline{Q})$ be a weak solution in $Q$ with $\bar h \leq u_P$ on $\partial_p Q$. Since $u_P\le u$ and $u$ is supercaloric, this immediately implies that $\bar h \leq u$ in $Q$. Now we are left to show that $\bar h \leq u_P = h$ in $Q \cap U_{t_1,t_2}$ and on the slice $Q \cap (U \times \{t_2\})$. In the former set, $h$ is a supercaloric function as a continuous weak solution, see Lemma~\ref{l.weasuper-is-supercal}, and similarly, $\bar h$ is subcaloric. Now we are able to use Lemma~\ref{l.supersubcal-cylinder-comparison} to conclude that also $\bar h \leq h$ in $Q\cap U_{t_1,t_2}$. Continuity of $h$ and $\bar h$ imply that this also holds on the slice $Q \cap (U \times \{t_2\})$. Altogether we have shown $\bar h\le u_P$ in $Q$. Thus $u_P$ is a supercaloric function in $\Omega_T$.
Since $u_P^j \leq u$ in $\Omega_T$ for every $j \in \ensuremath{\mathbb{N}}$, it follows that $u_P \leq u$. Moreover, since $u_P$ satisfies $u_P > \lambda$ on $\partial_p U_{t_1,t_2}$, we may use the comparison principle in the definition of supercaloric functions to conclude that $u_P > \lambda$ in $U_{t_1,t_2}$. Therefore, we have that $$ u \geq u_P > \lambda > \psi \quad \text{ in } U_{t_1,t_2}. $$ Since $u = \hat R_\psi \leq v$ in $\Omega_T$ for any $v \in \mathcal U_\psi$, it follows that in particular $u \leq u_P$ in $\Omega_T$. This implies that $u = u_P$, and further that $u$ is a weak solution in $U_{t_1,t_2}$ since $u_P$ is. Since $(x_o,t_o)\in \{u>\psi\}$ was arbitrary, this holds in a neighbourhood of any point in $\{u > \psi\}$, proving property (iv). \end{proof}
Next we show that for a bounded, compactly supported obstacle the minimal supercaloric function has zero boundary values. In particular, it belongs to the appropriate Sobolev space. Here we will assume that $\Omega$ is a $C^{1,\alpha}$-domain. This is used to ensure the existence of the Poisson modification by applying Theorem~\ref{t.existence}.
The following proof is partly based on the ideas in~\cite{LP}.
\begin{lemma} \label{l.minimal-supercal-zerobv} Let $\Omega$ be a $C^{1,\alpha}$-domain. Let $\psi$ be a bounded, lower semicontinuous obstacle with compact support in $\Omega_T$. Then, $\hat R_\psi^m \in L^2(0,T; H^1_0(\Omega))$. Furthermore, there exists $\delta >0$ such that $\hat R_\psi(\cdot,t) \equiv 0$ for every $t \in (0,\delta]$. \end{lemma}
\begin{proof} Let $\delta >0$ such that $\psi(\cdot,t) \equiv 0$ for every $t \in (0,\delta]$. Such $\delta$ exists since $\psi$ has compact support in $\Omega_T$. Let $u$ denote the zero extension of $\hat R_\psi$ to the times $(0,\delta]$, i.e., $u = \hat R_\psi$ whenever $t \in (\delta, T)$ and $u \equiv 0$ when $t \in (0,\delta]$. This implies $u \leq \hat R_\psi$. Furthermore, $u$ is supercaloric in $\Omega_T$ by Lemma~\ref{l.zero-past-extension} and clearly $u \geq \psi$ everywhere in $\Omega_T$, which implies $\hat R_\psi \leq u$. Thus, $u = \hat R_\psi$.
We know that $\hat R_\psi$ is a weak supersolution by Proposition~\ref{p.bdd-supercal-supersol} since $\hat R_\psi \leq \sup_{\Omega_T} \psi < \infty$ everywhere in $\Omega_T$. This implies $\hat R_\psi^m \in L^2_{\loc}(0,T; H^1_{\loc}(\Omega) )$ and further $\hat R_\psi^m \in L^2(0,T; H^1_{\loc}(\Omega) )$ by the Caccioppoli inequality for bounded weak supersolutions, Lemma~\ref{l.bounded_caccioppoli}. Up next, we show $\hat R_\psi^m \in L^2(0,T; H^1_0(\Omega) )$.
We prove this by using a Poisson modification, and denote $u:=\hat R_\psi$. Let $D\Subset\Omega$ be a $C^{1,\alpha}$-domain such that $\operatorname{spt}(\psi) \subset D \times (0,T)$ and $(\Omega\setminus \overline D)\times(0,T)$ is a $C^{1,\alpha}$-cylinder. Furthermore, let $U \subset \ensuremath{\mathbb{R}}^n$ be a $C^{1,\alpha}$-domain such that $D \Subset U \Subset \Omega$. Let $\phi_j $ be a sequence of nonnegative Lipschitz functions such that $\phi_j \leq \phi_{j+1} \leq u^m \chi_{U \times (0,T)}$ and $\phi_j \xrightarrow{j \to
\infty} u^m$ pointwise everywhere in $U \times (0,T)$. Define Poisson modifications \[ u_P^j := \begin{cases} u & \text{ in } \overline{D} \times (0,T),\\ h_j & \text{ in } (\Omega \setminus \overline{D}) \times (0,T), \end{cases} \] in which $h_j$ is a continuous weak solution with boundary values \[ h_j := \begin{cases} 0 (=\phi_j^\frac{1}{m}) & \text{ on } \partial \Omega \times (0,T)\\ 0 (=\phi_j^\frac{1}{m}) & \text{ on } (\overline{\Omega} \setminus D ) \times \{0\} \\ \phi_j^\frac{1}{m} & \text{ on } \partial D \times (0,T). \end{cases} \] We claim that $u_P := \lim_{j\to \infty} u_P^j$ is a supercaloric function with $u_P \leq u$ and $u_P^m \in L^2(0,T; H^1_0(\Omega))$. Observe that by Theorem~\ref{t.existence} the sequence $h_j$ is increasing so that the limit $\lim_{j\to \infty} h_j$ exists.
Clearly $u_P \leq u$ in $\overline{D} \times (0,T)$ by definition. By Lemma~\ref{l.supersubcal-cylinder-comparison} we have that $u_P^j \leq u$ in $(\Omega \setminus \overline{D}) \times (0,T)$ for every $j\in \ensuremath{\mathbb{N}}$. This also holds in the limit $j \to \infty$, which implies $u_P \leq u$ everywhere.
To show that $u_P$ is supercaloric, fix a $C^{2,\alpha}$-cylinder $Q_{t_1,t_2} \Subset \Omega_T$ (and suppose it intersects both regions where $U_P$ is defined differently) and let $\bar h \in C(\overline{Q_{t_1,t_2}})$ be a weak solution in $Q_{t_1,t_2}$ such that $\bar h \leq u_P$ on $\partial_p Q_{t_1,t_2}$. This immediately implies that $\bar h \leq u$ in $Q_{t_1,t_2}$ since $u$ is supercaloric, which takes care of the part $Q_{t_1,t_2} \cap [\overline{D} \times (0,T)]$. Then consider $Q_{t_1,t_2} \setminus [\overline{D} \times (0,T)]$. Since we know that $h= \lim_j h_j$ is a continuous weak solution in $(\Omega \setminus \overline{D} ) \times (0,T)$, we have that $\bar h \leq h$ on $\partial_p Q_{t_1,t_2} \cap [(\Omega \setminus \overline{D} ) \times (0,T)]$. Since $h$ is an increasing limit of continuous functions in $(\overline{\Omega} \setminus D ) \times [0,T)$, we have that the limit is lower semicontinuous (and continuous in the interior). This implies that for any point $(x,t) \in \partial D \times [t_1,t_2)$ we have $$ \liminf_{ (\Omega \setminus \overline{D}) \times (t_1,t_2) \ni (y,s) \to (x,t)} h(y,s) \geq h(x,t) = u(x,t) \geq \bar h(x,t), $$ which implies that we can use Lemma~\ref{l.supersubcal-cylinder-comparison} to conclude that also $\bar h \leq h$ in $Q_{t_1,t_2} \setminus [\overline{D} \times (0,T)]$. In total, we have now that $u_P \leq u$ and $u_P$ is a supercaloric function in $\Omega_T$. Also, $u_P \geq \psi$ clearly, which implies $u = u_P$ by minimality of $u$. Now $u_P$ is a weak solution in $(\Omega\setminus\overline D)\times(0,T)$.
We are left to show that $u^m \in L^2(0,T; H^1_0(\Omega))$. We know that $h_j^m(\cdot,t)$ has zero boundary values in Sobolev sense on $\partial \Omega$ for a.e. $t \in (0,T)$ and $h_j(\cdot,0) \equiv 0$ for every $j \in \ensuremath{\mathbb{N}}$.
Since $h_j \leq \| h\|_\infty < \infty$ for all $j \in \ensuremath{\mathbb{N}}$, we can derive a Caccioppoli inequality for weak solutions $h_j$ in the form \begin{align}\label{eq:caccio-weak}
\iint_{(\Omega \setminus D)\times (0,T)} &\eta^2 |\nabla h_j^m|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \leq c
\|h\|_{\infty}^{2m} T \int_{\Omega \setminus D} |\nabla \eta|^2 \, \:\! \mathrm{d} x \end{align} with a numerical constant $c > 0$ and for any $\eta \in C^\infty(\overline{\Omega} \setminus \overline{D})$ which vanishes in a neighbourhood of $\partial D$. This can be obtained by testing the mollified weak formulation of $h_j$ with test function $- \alpha(t) \eta^2(x) h_j^m(x,t)$, where $\alpha$ is a cutoff function in time with $\alpha(T) = 0$ (cf. Lemma~\ref{l.bounded_caccioppoli}). We define $U_\lambda = \{x \in \overline{\Omega} \setminus \overline{D} : \text{dist} (x, \partial D) > \lambda \}$ for $\lambda\in(0,\frac14\dist(D,\partial\Omega))$. We apply~\eqref{eq:caccio-weak} with a function $\eta_\lambda$ such that $\eta_\lambda \equiv 1$ in $U_\lambda$ and $\eta_\lambda \equiv 0$ in $(\Omega \setminus D) \setminus U_{\lambda/2}$. With this choice, estimate~\eqref{eq:caccio-weak}
implies that $|\nabla h_j^m|$ is uniformly bounded in $L^2(U_\lambda \times (0,T))$, which in turn implies that there exists a subsequence converging weakly, i.e., $\nabla h_j^m \rightharpoonup \nabla h^m$ weakly in $L^2(U_\lambda \times (0,T))$. Furthermore $\eta_{4\lambda} h_j^m \rightharpoonup \eta_{4\lambda} h^m$ weakly in $L^2(0,T; H^1(U_\lambda))$. Since $\eta_{4\lambda} h_j^m \in L^2(0,T;H^1_0(U_\lambda))$, by Hahn-Banach theorem it follows that also the weak limit satisfies $\eta_{4\lambda} h^m \in L^2(0,T;H^1_0(U_\lambda))$. Since $\eta_{4\lambda} \equiv 1$ in $U_{4\lambda}$, this implies that $u^m = u_P^m \in L^2(0,T;H^1_0(\Omega))$.
\end{proof}
\section{Connection between the two notions} \label{sec:connection}
Throughout this section we will assume that $\Omega$ is a $C^{2,\alpha}$-domain. This assumption is made to be able to apply the duality type proof in the comparison principles used in Theorem~\ref{t.weakvar-is-minimal}.
We will use a comparison principle in a union of space time cylinders. Let $\left \{U^i_{t_1^i,t_2^i}\right\}_i$ be a finite collection of open space time cylinders. The lateral boundary of the union of such cylinders is denoted by $$ \mathcal S \big(\cup_i U^i_{t_1^i,t_2^i} \big) = \Big(\! \cup_i
\partial U^i\times [t_1^i,t_2^i] \Big) \setminus \left( \cup_i U^i_{t_1^i,t_2^i} \right). $$ The tops of the union we denote by $$ \mathcal T \big( \cup_i U^i_{t_1^i,t_2^i} \big) = \left( \cup_i \overline{U^i} \times \{t_2^i\}\right) \setminus \left( \cup_i U^i_{t_1^i,t_2^i} \right), $$ and the bottoms by $$ \mathcal B \big( \cup_i U^i_{t_1^i,t_2^i} \big) = \left( \cup_i \overline{U^i} \times \{t_1^i\}\right) \setminus \left( \cup_i U^i_{t_1^i,t_2^i} \right). $$
We start with a technical covering lemma. \begin{lemma} \label{l.covering}
Let $\Omega\subset\ensuremath{\mathbb{R}}^n$ be a $C^{2,\alpha}$-domain and
$S<T$. Consider two sets
$K,D\subset\Omega\times(S,T)$ such that $K$ has positive
distance to $D$ and to the lateral boundary
$\partial\Omega\times[S,T]$. Then there exist $C^{2,\alpha}$-domains
$U_k\subset\Omega$, $k=1,\ldots,m$, and times
$S=t_0<t_1<\ldots<t_m=T$ such that
\begin{equation}\label{cover-D}
D\subset \bigcup_{k=1}^m U_k\times(t_{k-1},t_k] =:D^F
\qquad\mbox{and}\qquad \dist(D^F,K)>0.
\end{equation}
If $D_1\subset D_2\subset\ldots\subset\Omega\times(S,T)$
is an increasing sequence of sets
with $\dist(K,D_\ell)>0$ for each $\ell\in\ensuremath{\mathbb{N}}$, then the
corresponding coverings can be chosen as an increasing sequence as
well, i.e. $D_1^F\subset D_2^F\subset\ldots$. \end{lemma}
\begin{proof}
We abbreviate $d:=\tfrac13\dist(K,D\cup(\partial\Omega\times[S,T]))>0$.
For $m:=\lceil \frac{T-S}{d}\rceil\in\ensuremath{\mathbb{N}}$ we let $t_k:=S+kd$ for
$k=0,\ldots,m-1$ and $t_m:= T$. Then we define
\begin{equation*}
\widetilde U_k
:=
\{x\in\Omega\colon (x,s)\in D\mbox{ for some }s\in(t_{k-1},t_k]\},
\end{equation*}
for $k=1,\ldots,m$. We claim that
\begin{equation}\label{pos-dist-U-tilde}
\dist\big(K,\widetilde U_k\times(t_{k-1},t_k]\big)\ge 2d>0.
\end{equation}
For the proof of this estimate, we consider a point $(x,t)\in
\widetilde U_k\times(t_{k-1},t_k]$. By
definition of $\widetilde U_k$, there exists a time $s\in (t_{k-1},t_k]$
with $(x,s)\in D$. Therefore, for any $z\in K$, we can estimate
\begin{equation*}
|(x,t)-z|\ge |(x,s)-z|-|t-s|\ge \dist(K,D)-(t_k-t_{k-1})\ge 3d-d=2d,
\end{equation*}
which proves \eqref{pos-dist-U-tilde}.
Next, we claim that there exist $C^{2,\alpha}$-domains $U_k$ with
$\widetilde U_k\subset U_k\subset\Omega$ and
\begin{equation}\label{pos-dist-U}
\dist\big(K,U_k\times(t_{k-1},t_k]\big)\ge d>0.
\end{equation}
For the construction of these sets, we consider a mollification $g_\ast\in C^\infty(\Omega)$ of
$g(x):=\dist(x,\widetilde U_k\cup\partial\Omega)$ with
$\|g-g_\ast\|_{C^0}<\frac d4$. By Sard's theorem~\cite[Theorem 10.1, Chapter 2]{CH-book}, the critical
values of $g_\ast$ form a zero set. Therefore, for a.e. $r\in(\frac
d4,\frac{3d}{4})$, the level sets $\{x\in\Omega\colon g_\ast(x)=r\}$
are smooth submanifolds. We fix a value $r\in(\frac
d4,\frac{3d}{4})$ with this property and define
\begin{equation*}
U_k:=\big\{x\in\Omega\colon g_\ast(x)<r\big\}.
\end{equation*}
The boundary $\partial U_k$ is the disjoint union
of the smooth submanifold $\{x\in\Omega\colon g_\ast(x)=r\}$
and $\partial\Omega$. Since $\Omega$ is a $C^{2,\alpha}$-domain, the
sets $U_k$ are $C^{2,\alpha}$-domains as well.
Moreover, by definition the set $U_k$ is
contained in the $d$-neighbourhood of $\widetilde U_k\cup\partial\Omega$. Therefore,
estimate~\eqref{pos-dist-U-tilde} and the definition of $d$
imply~\eqref{pos-dist-U}.
This proves claim~\eqref{cover-D} for a single set $D$.
In the case of an increasing sequence $D_1\subset D_2\subset\ldots$,
we start by constructing the covering $D_1^F\supset D_1$ in the same
way as above. Assuming that the coverings $D_1^F\subset
D_2^F\subset\ldots\subset D_{\ell}^F$ have already been
chosen for some $\ell\in\ensuremath{\mathbb{N}}$, we apply the preceding construction to the set $D_{\ell+1}\cup
D_{\ell}^F$ instead of $D$ to construct a covering
$D_{\ell+1}^F\supset D_{\ell+1}\cup D_{\ell}^F$ with the asserted
properties. This procedure leads to an increasing sequence of
coverings $D_1^F\subset D_2^F\subset\ldots$ such
that~\eqref{cover-D} is satisfied for $D_\ell^F$ in place of $D^F$. \end{proof}
In the following we suppose that $\psi$ is a nonnegative function satisfying \begin{equation} \label{a.obstacle_weak} \psi^m \in L^2(0,T;H^1_0(\Omega)),\quad \partial_t(\psi^m) \in L^\frac{m+1}{m}(\Omega_T), \end{equation} and furthermore, $\psi$ is H\"older continuous with compact support in the following sense. \begin{equation} \label{e.holder-obstacle} \psi \in C^{0;\beta,\frac{\beta}{2}}_0(\Omega_T)\quad \text{ for some } \beta \in (0,1). \end{equation}
Assumption \eqref{a.obstacle_weak} guarantees existence and~\eqref{e.holder-obstacle} continuity of a weak solution to the obstacle problem with $u=0$ on $\partial_p\Omega_T$ by Theorem~\ref{t.cont-exist}. Observe that $u$ is bounded in $\Omega_T$ under these assumptions by~\cite{MSb}. Existence of the minimal supersolution above the obstacle is guaranteed by Proposition~\ref{p.balayage-supercal}, and its uniqueness is clear by definition. Then we show that if the obstacle satisfies conditions above, weak solutions to the obstacle problem and minimal supersolutions above the obstacle coincide.
\begin{theorem} \label{t.weakvar-is-minimal} Let $0<m<1$, $\Omega \Subset \ensuremath{\mathbb{R}}^n$ be a $C^{2,\alpha}$-domain and $\psi$ an obstacle satisfying~\eqref{a.obstacle_weak} and~\eqref{e.holder-obstacle}. Then, a weak solution $u$ to the obstacle problem with $u=0$ on $\partial_p\Omega_T$ in the sense of Definition~\ref{d.variational-obstacle} and the minimal supersolution $v$ above the obstacle in the sense of Definition~\ref{d.minimal-supercal} coincide, i.e., $u = v$ in $\Omega_T$. \end{theorem}
\begin{remark} In principle, instead of~\eqref{a.obstacle_weak} and~\eqref{e.holder-obstacle} one can merely assume that $\psi \in C_0(\Omega_T)$. Then, the result holds whenever a weak solution $u\in C(\Omega_T)$ to the obstacle problem with zero boundary values and such obstacle exists. \end{remark}
\begin{proof} We start by proving the result by using a similar approach as in the proof of~\cite[Theorem 3.1]{AvLu}. Because of our assumptions~\eqref{a.obstacle_weak} and~\eqref{e.holder-obstacle}, Theorem~\ref{t.cont-exist} implies that the weak solution $u$ to the obstacle problem is continuous. Moreover, it is a weak supersolution by Lemma~\ref{l.variationalsol-is-supersol} and a supercaloric function by
Lemma~\ref{l.weasuper-is-supercal}. Therefore, and since we are
considering the fast diffusion range $0<m<1$,
Lemma~\ref{lem:alternatives} guarantees that the positivity set of
$u$ is the union of at most countably many sets
of the form $\Omega^j\times\Lambda_i^j$ for open time intervals $\Lambda_i^j\subset(0,T)$ for each connected component $\Omega^j$ of $\Omega$. We consider the connected components $\Omega^j$ separately and for simplicity we omit $j$ from the superscript. Let $(t_1,t_2) \subset (0,T)$ be one of these positivity intervals, such that $u>0$ in $\Omega \times (t_1,t_2)$ and $u(\cdot,t_1) \equiv u(\cdot,t_2) \equiv0$, unless $t_2 = T$. Observe that this implies that $\psi(\cdot,t_1) \equiv \psi(\cdot,t_2) \equiv 0$.
By Proposition~\ref{p.balayage-supercal}, the minimal supersolution above the obstacle is given by $v = \hat R_\psi$, which is bounded since $v \leq \sup_{\Omega_T} \psi < \infty$, and $v^m \in L^2(0,T;H^1_0(\Omega))$ by Lemma~\ref{l.minimal-supercal-zerobv}. For $s \in(t_1,t_2)$, let $\tau_1^i$ be a decreasing sequence such that $s>\tau_1^i \to t_1$ as $i \to \infty$. Let us consider $Q_i = \Omega \times (\tau_1^i, s)$ and $$ D_{\varepsilon,s}^i = \left\{(x,t)\in Q_i : \frac{u(x,t)}{1+\varepsilon} \geq v(x,t) \right\}, $$ for any fixed $\varepsilon>0$. Observe that $Q_i \subset Q_{i+1}$ and $D_{\varepsilon,s}^i \subset D_{\varepsilon,s}^{i+1}$. Define $K_i := \{(x,t) \in Q_i : u(x,t) = \psi(x,t) \}$. Observe that in the set $K_i$ we have $0 < u = \psi \leq v$ and $u$ is a weak solution in $Q_i \setminus K_i$ by Lemma~\ref{l.variational-solution-noncontactset}. Let us denote $u_\varepsilon := u/(1+\varepsilon)$. Now the function $u_\varepsilon - v$ is upper semicontinuous, which implies that $D_{\varepsilon,s}^i$ is closed in $Q_i$, and there is a positive distance between $D_{\varepsilon,s}^i$ and $K_i$, as well as between $\mathcal S (\Omega_T)$ and $K_i$. Therefore, for each fixed $\varepsilon>0$ and $i \in \ensuremath{\mathbb{N}}$, by Lemma~\ref{l.covering} we find a collection of a finite number of $C^{2,\alpha}$-cylinders, denoted by $D_{\varepsilon,s}^{F,i}$, that covers $D_{\varepsilon,s}^i$ and does not intersect $K_i$. We fix an arbitrary $\varepsilon_o>0$ and a
parameter $\varepsilon\in(0,\varepsilon_o]$. For $i\in\ensuremath{\mathbb{N}}$, we can choose the coverings
$D_{\varepsilon,s}^{F,i}$ such that $D^{F,1}_{\varepsilon_o,s} \subset
D^{F,i}_{\varepsilon,s}\subset D^{F,i+1}_{\varepsilon,s}$ for any $i\in\ensuremath{\mathbb{N}}$.
Since $u$ is a weak solution to the PME in $Q_i \setminus K_i$, we have \begin{align*} \iint_{Q_i \setminus K_i} -u_{\varepsilon}\partial_t \varphi + \nabla u_{\varepsilon}^m\cdot \nabla \varphi \, \:\! \mathrm{d} x\:\! \mathrm{d} t &= \iint_{Q_i \setminus K_i} - \tfrac{1}{1+\varepsilon} u\partial_t \varphi + \tfrac{1}{(1+\varepsilon)^m} \nabla u^m\cdot \nabla \varphi \, \:\! \mathrm{d} x\:\! \mathrm{d} t \\ &= \tfrac{1-(1+\varepsilon)^{m-1}}{(1+ \varepsilon)^m} \iint_{Q_i \setminus K_i} \nabla u^m\cdot \nabla \varphi \, \:\! \mathrm{d} x\:\! \mathrm{d} t \\ &= \iint_{Q_i \setminus K_i} \nabla f\cdot \nabla \varphi \, \:\! \mathrm{d} x\:\! \mathrm{d} t \end{align*} for every $\varphi \in C^\infty_0(Q_i \setminus K_i)$, in which we denoted $f = \tfrac{1-(1+\varepsilon)^{m-1}}{(1+ \varepsilon)^m} u^m$. We consider an arbitrary nonnegative test function \begin{equation}\label{choice-test-function}
\phi\in C^\infty_0\Big(\overline{D_{\varepsilon_o,s}^{F,1}}\cap (\Omega\times\{s\}),\ensuremath{\mathbb{R}}_{\ge0}\Big), \end{equation} which we extend to a function $\phi\in C^\infty(\Omega\times(0,s],\ensuremath{\mathbb{R}}_{\ge0})$ with $\phi=0$ on $\Omega\times(0,s]\setminus \overline{D_{\varepsilon_o,s}^{F,1}}$. Since $D_{\varepsilon_o,s}^{F,1}\subset D_{\varepsilon,s}^{F,i}$ for any $i\in\ensuremath{\mathbb{N}}$, this choice implies in particular $\phi=0$ on $\partial D_{\varepsilon,s}^{F,i}\setminus(\Omega\times\{s\})$. From now on, we omit the index $i$ in the superscript. We consider a function $h = C^{2,1}(\overline{D_{\varepsilon,s}^F},\ensuremath{\mathbb{R}}_{\geq 0})$ with the boundary values \begin{align*}
\left\{
\begin{array}{cl}
h=\phi&\qquad\mbox{on $\mathcal T(D_{\varepsilon,s}^F)$},\\[1ex]
h=0&\qquad\mbox{on $\mathcal S(D_{\varepsilon,s}^F)$},
\end{array}
\right. \end{align*} which will later be chosen as the solution of a dual problem. Note that as a consequence of the boundary condition and the fact $h\ge0$, the outer normal derivative satisfies $\partial_n h \leq 0$ on $\mathcal S(D_{\varepsilon,s}^F)$.
Denote $b = u_\varepsilon - v$. We have that $b \leq 0$ on $\partial D_{\varepsilon,s}^F \cap Q_i$. On $\partial \Omega \times (\tau_1^i, s)$ we have $0 = u = u_\varepsilon = v$, i.e., $u_\varepsilon^m - v^m = 0$ in the sense of traces. On $\Omega \times \{\tau_1^i\}$ and $\Omega \times \{ s \}$ we do not have information on the sign of $b$, only that $u,u_\varepsilon > 0$ and $v \geq 0$ on these sets. Since $u_\varepsilon$ is a weak solution with source term $f$ and $v$ is a weak supersolution in $D_{\varepsilon,s}^F$, we can subtract the (very) weak formulations and obtain \begin{align*} \iint_{D_{\varepsilon,s}^F} \nabla f \cdot \nabla h \, \:\! \mathrm{d} x \:\! \mathrm{d} t &\geq \int_{\mathcal T(D_{\varepsilon,s}^F)} b \phi \, \:\! \mathrm{d} x - \int_{\mathcal B(D_{\varepsilon,s}^F)} bh \, \:\! \mathrm{d} x - \iint_{D_{\varepsilon,s}^F} b h_t \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} - \iint_{D_{\varepsilon,s}^F} (u_\varepsilon^m - v^m) \Delta h \, \:\! \mathrm{d} x \:\! \mathrm{d} t + \iint_{\mathcal S(D_{\varepsilon,s}^F)} (u_\varepsilon^m - v^m) \partial_n h \, \:\! \mathrm{d} \sigma \:\! \mathrm{d} t \\ &\geq \int_{\mathcal T(D_{\varepsilon,s}^F)} b \phi \, \:\! \mathrm{d} x - \int_{\mathcal B(D_{\varepsilon,s}^F)} bh \, \:\! \mathrm{d} x - \iint_{D_{\varepsilon,s}^F} b h_t \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} - \iint_{D_{\varepsilon,s}^F} (u_\varepsilon^m - v^m) \Delta h \, \:\! \mathrm{d} x \:\! \mathrm{d} t, \end{align*} which can be written as \begin{align} \label{e.comparison-est} \int_{\mathcal T(D_{\varepsilon,s}^F)} b \phi \, \:\! \mathrm{d} x &\leq \iint_{D_{\varepsilon,s}^F} b(h_t + a \Delta h) \, \:\! \mathrm{d} x \:\! \mathrm{d} t + \iint_{D_{\varepsilon,s}^F} \nabla f \cdot \nabla h \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} + \int_{\mathcal B(D_{\varepsilon,s}^F)} bh \, \:\! \mathrm{d} x \nonumber \end{align} with \[ a = \begin{cases}
\frac{u_\varepsilon^m - v^m}{u_\varepsilon-v}, & \text{ if } u_\varepsilon \neq v, \\
0, & \text{ if } u_\varepsilon = v.
\end{cases} \] By defining $a_{k} = \min \{ k, \max\{ a, \frac{1}{k} \} \}$ we have $0< 1/k \leq a_{k} \leq k < \infty $, and by $(a_k)_\delta$ we denote the standard mollification of $a_k$, for a sufficiently small parameter $\delta=\delta(k)>0$. In this case, to be able to define $(a_k)_\delta$ up to the boundary $\partial_p \Omega_T$, we set $a = 0$ in $\ensuremath{\mathbb{R}}^{n+1}\setminus \Omega_T$. Let $h_k$ be the solution to a backward in time boundary value problem \begin{equation}\label{eq:dual-problem}
\left\{
\begin{array}{ll}
u_t + (a_k)_\delta \Delta u = 0 & \text{ in } D_{\varepsilon,s}^F, \\
u = \phi & \text{ on } \mathcal T(D_{\varepsilon,s}^F), \\
u = 0 & \text{ on } \mathcal S(D_{\varepsilon,s}^F).
\end{array}
\right. \end{equation} By the linear theory and the fact that $D_{\varepsilon,s}^F$ can be expressed as in~\eqref{cover-D}, the solution satisfies $h_k \in C^{2,1}(\overline{D_{\varepsilon,s}^F})$. By Hölder's inequality we obtain \begin{align} \label{e.bp-term} \iint_{D_{\varepsilon,s}^F} &b((h_k)_t + a \Delta h_k) \, \:\! \mathrm{d} x \:\! \mathrm{d} t = \iint_{D_{\varepsilon,s}^F} b[ a-(a_k)_\delta ]\Delta h_k \, \:\! \mathrm{d} x \:\! \mathrm{d} t \nonumber \\ &\leq \left( \iint_{D_{\varepsilon,s}^F} b^2 \frac{[(a_k)_\delta - a ]^2}{(a_k)_\delta}\, \:\! \mathrm{d} x \:\! \mathrm{d} t \right)^\frac{1}{2} \left( \iint_{D_{\varepsilon,s}^F} (a_k)_\delta \left( \Delta h_k \right)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \right)^\frac{1}{2}. \end{align} For the first term on the right-hand side we have \begin{align*}
\left\|\frac{b[(a_k)_\delta - a]}{\sqrt{(a_k)_\delta}} \right\|_{L^2(D_{\varepsilon,s}^F)} &\leq \left\|\frac{b[(a_k)_\delta - a_k]}{\sqrt{(a_k)_\delta}} \right\|_{L^2(D_{\varepsilon,s}^F)} + \left\|\frac{b[a_k -a ]}{\sqrt{(a_k)_\delta}} \right\|_{L^2(D_{\varepsilon,s}^F)} \\
&\xrightarrow{\delta \to 0}\left\|\frac{b[a_k -a ]}{\sqrt{a_k}} \right\|_{L^2(D_{\varepsilon,s}^F)}, \end{align*} since $(a_k)_\delta \to a_k$ a.e., $\frac{1}{k} \leq (a_k)_\delta \leq k$, $b\in L^2(D_{\varepsilon,s}^F)$ and by using the dominated convergence theorem. We further estimate \begin{align*} \iint_{D_{\varepsilon,s}^F} b^2 \frac{(a_k - a )^2}{a_k}\, \:\! \mathrm{d} x \:\! \mathrm{d} t &= \iint_{D_{\varepsilon,s}^F \cap \{a < \frac{1}{k}\}} b^2 \frac{(\tfrac{1}{k} - a )^2}{\tfrac{1}{k}}\, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} + \iint_{D_{\varepsilon,s}^F \cap \{a > k\}} b^2 \frac{(a-k )^2}{k}\, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\leq \tfrac{1}{k} \iint_{D_{\varepsilon,s}^F \cap \{a < \frac{1}{k}\}} b^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} + \tfrac{1}{k} \iint_{D_{\varepsilon,s}^F \cap \{a > k\}} (u_\varepsilon^m - v^m)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\le \frac{c_o}{k} \end{align*}
for a constant $c_o = c_o(m,\|u\|_{L^2(\Omega_T)},\|v\|_{L^2(\Omega_T)})>0$. Because of the two preceding estimates, by choosing $\delta>0$ small enough in dependence on $k$, we achieve \begin{equation*} \iint_{D_{\varepsilon,s}^F} b^2 \frac{[(a_k)_\delta - a ]^2}{(a_k)_\delta}\, \:\! \mathrm{d} x \:\! \mathrm{d} t \le \frac{2c_o}{k}. \end{equation*} By using the property that $h_k$ solves the boundary value problem~\eqref{eq:dual-problem}, we have \begin{align*} \iint_{D_{\varepsilon,s}^F} &(a_k)_\delta (\Delta h_k)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t = - \iint_{D_{\varepsilon,s}^F} (h_k)_t \Delta h_k \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &= \iint_{D_{\varepsilon,s}^F} h_k (\Delta h_k)_t \, \:\! \mathrm{d} x \:\! \mathrm{d} t - \int_{\mathcal T(D_{\varepsilon,s}^F)} \phi \Delta \phi \, \:\! \mathrm{d} x + \int_{\mathcal B(D_{\varepsilon,s}^F)} h_k \Delta h_k \, \:\! \mathrm{d} x \\
&= \iint_{D_{\varepsilon,s}^F} h_k \Delta (h_k)_t \, \:\! \mathrm{d} x \:\! \mathrm{d} t + \int_{\mathcal T(D_{\varepsilon,s}^F)} |\nabla \phi|^2 \, \:\! \mathrm{d} x - \int_{\mathcal B(D_{\varepsilon,s}^F)} |\nabla h_k|^2 \, \:\! \mathrm{d} x \\
&\leq \iint_{D_{\varepsilon,s}^F} (h_k)_t \Delta h_k \, \:\! \mathrm{d} x \:\! \mathrm{d} t - \iint_{\mathcal S(D_{\varepsilon,s}^F)} (h_k)_t \partial_n h_k \, \:\! \mathrm{d} \sigma \:\! \mathrm{d} t + \int_{\mathcal T(D_{\varepsilon,s}^F)} |\nabla \phi|^2 \, \:\! \mathrm{d} x \\
&= - \iint_{D_{\varepsilon,s}^F} (a_k)_\delta (\Delta h_k)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t + \int_{\mathcal T(D_{\varepsilon,s}^F)} |\nabla \phi|^2 \, \:\! \mathrm{d} x \end{align*} by integrating by parts in time and space, and using the fact that $h_k \in C^{2,1}(\overline{D_{\varepsilon,s}^F})$ together with the boundary conditions. This implies $$ \iint_{D_{\varepsilon,s}^F} (a_k)_\delta (\Delta h_k)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \leq
\frac12 \int_{\mathcal T(D_{\varepsilon,s}^F)} |\nabla \phi|^2 \, \:\! \mathrm{d} x =
\frac12 \int_{\mathcal{T}(D_{\varepsilon_o,s}^{F,1})\cap(\Omega\times\{s\})} |\nabla \phi|^2 \, \:\! \mathrm{d} x, $$ by the choice of $\phi$. Collecting the previous results in~\eqref{e.bp-term}, we deduce \begin{equation*}
\iint_{D_{\varepsilon,s}^F} b((h_k)_t + a \Delta h_k) \, \:\! \mathrm{d} x \:\! \mathrm{d} t
\le
\frac{c_1}{\sqrt{k}}, \end{equation*}
for a constant $c_1 = c_1(m,\|u\|_{L^2(\Omega_T)},\|v\|_{L^2(\Omega_T)}, \|\nabla \phi\|_{L^2\left(\Omega \times \{s\} \right)}) > 0$. For the term with $f$ in~\eqref{e.comparison-est}, we estimate $$
\iint_{D_{\varepsilon,s}^F} \nabla f \cdot \nabla h_k \, \:\! \mathrm{d} x \:\! \mathrm{d} t \leq \frac{ 1 -(1+\varepsilon)^{m-1} }{(1+\varepsilon)^m} \|\nabla u^m\|_{L^2(\Omega_T)} \left( \iint_{D_{\varepsilon,s}^F} |\nabla h_k|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \right)^\frac{1}{2}. $$ By using the fact that $h_k$ is a solution to the linear boundary value problem, with a cutoff-function $\alpha= \alpha (t)$ with properties $\alpha(0) = 1/2$, $\alpha(s) = 1$ and $\alpha' = \frac{1}{2s}$ we obtain \begin{align*} 0 &= \iint_{D_{\varepsilon,s}^F} \alpha (h_k)_t \Delta h_k \, \:\! \mathrm{d} x \:\! \mathrm{d} t + \iint_{D_{\varepsilon,s}^F} \alpha (a_k)_\delta (\Delta h_k)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &= - \iint_{D_{\varepsilon,s}^F} \alpha \nabla (h_k)_t \cdot \nabla h_k \, \:\! \mathrm{d} x \:\! \mathrm{d} t + \iint_{D_{\varepsilon,s}^F} \alpha (a_k)_\delta (\Delta h_k)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\
&= - \frac{1}{2} \iint_{D_{\varepsilon,s}^F} \alpha (|\nabla h_k|^2)_t \, \:\! \mathrm{d} x \:\! \mathrm{d} t + \iint_{D_{\varepsilon,s}^F} \alpha (a_k)_\delta (\Delta h_k)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\
&= \frac{1}{2} \iint_{D_{\varepsilon,s}^F} \alpha' |\nabla h_k|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t - \frac12 \int_{\mathcal T(D_{\varepsilon,s}^F)} \alpha |\nabla h_k|^2 \, \:\! \mathrm{d} x \\
&\phantom{+} + \frac12 \int_{\mathcal B(D_{\varepsilon,s}^F)} \alpha |\nabla h_k|^2 \, \:\! \mathrm{d} x + \iint_{D_{\varepsilon,s}^F} \alpha (a_k)_\delta (\Delta h_k)^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\
&\geq \frac{1}{4s} \iint_{D_{\varepsilon,s}^F} |\nabla h_k|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t - \frac12 \int_{\mathcal T(D_{\varepsilon,s}^F)} |\nabla\phi|^2 \, \:\! \mathrm{d} x, \end{align*} which implies $$
\left( \iint_{D_{\varepsilon,s}^F} |\nabla h_k|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \right)^\frac12 \leq (2s)^\frac12 \left( \int_{\mathcal T(D_{\varepsilon,s}^F)} |\nabla\phi|^2 \, \:\! \mathrm{d} x \right)^\frac12. $$ Therefore, we arrive at the bound \begin{align*}
\iint_{D_{\varepsilon,s}^F} \nabla f \cdot \nabla h_k \, \:\! \mathrm{d} x \:\! \mathrm{d} t \leq (2s)^\frac12 \frac{1-(1+\varepsilon)^{m-1}}{(1+\varepsilon)^m} \|\nabla u^m\|_{L^2(\Omega_T)} \left( \int_{\mathcal T(D_{\varepsilon,s}^F)} |\nabla \phi|^2 \, \:\! \mathrm{d} x \right)^\frac12. \end{align*} Now, taking also the indices $i$ into account and recalling the fact $b \leq 0$ on $\partial D_{\varepsilon,s}^F \cap Q_i$, we can write~\eqref{e.comparison-est} in the form \begin{align*}
\int_{\mathcal T(D_{\varepsilon,s}^{F,i})} (u_\varepsilon -v)\phi \, \:\! \mathrm{d} x &\leq \frac{c_1}{\sqrt{k}} + (2s)^\frac12 \frac{1-(1+\varepsilon)^{m-1}}{(1+\varepsilon)^m} \left( \int_{\mathcal T(D_{\varepsilon,s}^{F,i})} |\nabla \phi|^2 \, \:\! \mathrm{d} x \right)^\frac12 \\ &\phantom{+} + \int_{\mathcal B(D_{\varepsilon,s}^{F,i})\cap (\Omega \times \{\tau_1^i\})} (u_\varepsilon - v)h_k \, \:\! \mathrm{d} x. \end{align*} Observe that \begin{align*} \int_{\mathcal B(D_{\varepsilon,s}^{F,i})\cap (\Omega \times \{\tau_1^i\})} (u_\varepsilon - v)h_k \, \:\! \mathrm{d} x &\leq \int_{\mathcal B(D_{\varepsilon,s}^{F,i})\cap (\Omega \times \{\tau_1^i\})} (u_\varepsilon-v)_+ h_k \, \:\! \mathrm{d} x \\
&\leq \|\phi\|_{L^\infty(\mathcal T(D_{\varepsilon,s}^{F,i}))} \int_{\Omega \times \{\tau_1^i\}} (u_\varepsilon-v)_+ \, \:\! \mathrm{d} x \end{align*} by using the comparison principle for $h_k$. Recall that we chose $\phi$ with the property $\phi=0$ in $\Omega\times(0,s]\setminus \overline{D_{\varepsilon_o,s}^{F,1}}$. Therefore, after passing to the limit $k \to \infty$ we infer the bound \begin{align*}
\int_{\overline{D_{\varepsilon_o,s}^{F,1}}\times(\Omega\times\{s\})} (u_\varepsilon -v)\phi \, \:\! \mathrm{d} x &\leq (2s)^\frac12 \frac{1-(1+\varepsilon)^{m-1}}{(1+\varepsilon)^m} \left( \int_{\mathcal T(D_{\varepsilon_o,s}^{F,1})} |\nabla \phi|^2 \, \:\! \mathrm{d} x \right)^\frac12 \\
&\phantom{+} + \|\phi\|_{L^\infty(\mathcal T(D_{\varepsilon_o,s}^{F,1}))} \int_{\Omega \times \{\tau_1^i\}} (u-v)_+ \, \:\! \mathrm{d} x. \end{align*} By passing to the limit $i \to \infty$, the last term vanishes, since $u \in C([0,T];L^{m+1}(\Omega))$ and $(u-v)_+(\cdot,\tau_1^i)\le u(\cdot,\tau_1^i)\to 0$ as $\tau_1^i\to t_1$. In the resulting inequality, we let $\varepsilon\downarrow0$ and conclude $$ \int_{\overline{D_{\varepsilon_o,s}^{F,1}} \cap (\Omega \times \{s\})} (u-v)\phi \, \:\! \mathrm{d} x \leq 0 $$ for every test function $\phi$ as in~\eqref{choice-test-function}, i.e. that $u \leq v$ a.e. on $\overline{D_{\varepsilon_o,s}^{F,1}} \cap (\Omega \times \{s\})$. In particular, on the set $\overline{D_{\varepsilon_o,s}^{1}} \cap (\Omega \times \{s\})$ we have $0<u\le v\le \frac{u}{1+\varepsilon_o}$, which is not possible. Therefore, this set must be empty for every $\varepsilon_o>0$, which implies $u\le v$ a.e. on $\Omega\times\{s\}$. Since $s\in (t_1,t_2)$ is arbitrary, we obtain $u \leq v$ a.e in $\Omega \times (t_1,t_2)$. Since $u$ is continuous and $v = \hat R_\psi$ is a supercaloric function, Theorem~\ref{t.supercal-essliminf} implies that $u \leq v$ everywhere in $\Omega \times (t_1,t_2)$. This can be deduced in any connected component $\Omega \times \Lambda_i$ of the positivity set of $u$. Outside these sets we trivially have $u = 0 \leq v$, which implies that $u \leq v= \hat R_\psi$ everywhere in $\Omega_T$. Since $u$ itself is a continuous weak supersolution, and thus a supercaloric function by Lemma~\ref{l.weasuper-is-supercal} satisfying $u \geq \psi$, it follows that $R_\psi \leq u$ everywhere. Now we have $u \leq \hat R_\psi \leq R_\psi \leq u$ everywhere, which concludes the result for each connected component of $\Omega$. Thus the claim follows. \end{proof}
From the proof of Theorem~\ref{t.weakvar-is-minimal} we can extract the following comparison result, cf.~\cite[Theorem 3.1]{AvLu}.
\begin{proposition} Suppose that $0<m<1$. Let $K \Subset \Omega_T$ be a compact set and $\Omega \Subset \ensuremath{\mathbb{R}}^n$ be a $C^{2,\alpha}$-domain. Suppose that $u \in C(\Omega_T)$ is a supercaloric function in $\Omega_T$ such that $u$ is a weak solution in $\Omega_T \setminus K$ with $u^m \in L^2(0,T;H^1(\Omega)) \cap L^\frac{2}{m}(\Omega_T)$ and taking initial values $u_o$ in $L^1$-sense. Let $v$ be a supercaloric function in $\Omega_T$ satisfying $v^m\in L^2(0,T;H^1(\Omega)) \cap L^\frac{2}{m}(\Omega)$ and taking the initial values $v_o$ in $L^1$-sense. If $u_o \leq v_o$ a.e. on $\Omega$, $u^m(\cdot,t) \leq v^m(\cdot,t)$ on $\partial \Omega$ in the sense of $H^1$-trace for a.e. $t \in (0,T)$ and $u \leq v$ on $K´$, then $u \leq v$ everywhere in $\Omega_T$. \end{proposition}
As an immediate consequence of Theorem~\ref{t.weakvar-is-minimal} we can conclude the following comparison principle for weak solutions to the obstacle problem.
\begin{corollary} Let $\Omega$ be a $C^{2,\alpha}$-domain and $\psi_1$ and $\psi_2$ be obstacles with compact support in $\Omega_T$ satisfying~\eqref{a.obstacle_weak} and~\eqref{e.holder-obstacle}, and let $u_1$ and $u_2$ be corresponding weak solutions to the obstacle problems with zero boundary values on $\partial_p \Omega_T$ in the sense of Definition~\ref{d.variational-obstacle}. If $\psi_1 \leq \psi_2$, then $u_1 \leq u_2$. \end{corollary}
\appendix
\section{On the notions of variational solution and weak solution to the obstacle problem} \label{appendix-a}
In Theorem~\ref{t.existence} we use an existence result for the obstacle problem. In order to cover the full fast diffusion range (i.e., also the subcritical range) we apply the result given in~\cite{Schaetzler2}. The aforementioned existence result is established for the notion of variational solution (see Definition~\ref{d.obstacle-varsol}). Here we show that a variational solution is also a weak solution to the obstacle problem according to Definition~\ref{d.variational-obstacle} provided that the obstacle is regular enough.
Suppose that the obstacle $\psi$ satisfies \begin{equation} \label{eq:g-cond} \begin{aligned} &\psi^m \in L^2(0,T;H^1(\Omega)),\quad \partial_t \psi^m \in L^\frac{m+1}{m}(\Omega_T), \\ &\psi \in C([0,T];L^{m+1}(\Omega)),\quad \text{ and } \quad \psi_o^m := \psi^m(\cdot,0) \in L^{\frac{m+1}{m}}(\Omega) \cap H^1(\Omega). \end{aligned} \end{equation} For a variational solution $u$, suppose \begin{equation} \label{eq:u-class} u \in L^\infty(0,T; L^{m+1}(\Omega)), \quad u^m \in L^2(0,T;H^1(\Omega)),\quad u\geq \psi \, \text{ a.e. in } \Omega_T, \end{equation} and that it attains the same boundary and initial values as $\psi$ in the sense \begin{equation} \label{eq:boundary-data} u^m - \psi^m \in L^2(0,T;H^1_0(\Omega)), \quad
\lim_{h\to 0} \Xint-_0^h \int_\Omega |u(x,t) - \psi_o|^{m+1} \, \:\! \mathrm{d} x \:\! \mathrm{d} t = 0. \end{equation}
Denote \begin{align*} I(u,v) &:= \tfrac{1}{m+1}\left( u^{m+1} - v^{m+1} \right) - v^m (u-v) \\ &= \tfrac{1}{m+1} u^{m+1} + \tfrac{m}{m+1} v^{m+1} - v^mu. \end{align*} Now we recall the notion of solution used in \cite{Schaetzler2}. \begin{definition}\label{d.obstacle-varsol} Let $\psi$ satisfy~\eqref{eq:g-cond}. A measurable map $u \colon \Omega_T \to \ensuremath{\mathbb{R}}_{\geq0}$ satisfying~\eqref{eq:u-class} and~\eqref{eq:boundary-data} is called a variational solution to the obstacle problem for the PME if \begin{align}\label{var-ineq-appendix}
\frac12 \iint_{\Omega_\tau} |\nabla u^m|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t &\leq
\iint_{\Omega_\tau} \partial_t v^m(v - u) \, \:\! \mathrm{d} x \:\! \mathrm{d} t + \frac12 \iint_{\Omega_\tau}|\nabla v^m|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\\nonumber &\phantom{+} - \int_{\Omega} I(u(\tau),v(\tau))\, \:\! \mathrm{d} x +\int_{\Omega} I (\psi_o,v(0)) \, \:\! \mathrm{d} x \nonumber \end{align} for a.e. $\tau \in [0,T]$ and every comparison map $v$ satisfying $v^m \in \psi^m + L^2(0,T; H^1_0(\Omega))$ with $\partial_t v^m \in L^1(0,T;L^\frac{m+1}{m}(\Omega))$, $v(0) \in L^{m+1}(\Omega)$ and $v \geq \psi$ a.e. in $\Omega_T$. \end{definition}
As an intermediate step, we will prove that every variational solution is a solution in the following sense.
\begin{definition} \label{d.obstacle-wsol} Let $\psi$ satisfy~\eqref{eq:g-cond}. A measurable map $u \colon \Omega_T \to \ensuremath{\mathbb{R}}_{\geq0}$ satisfying~\eqref{eq:u-class} and~\eqref{eq:boundary-data}$_1$ is called a weak solution to the obstacle problem for the PME if \begin{equation}\label{global-var-ineq} \llangle \partial_t u , \alpha (v^m-u^m) \rrangle_{\psi_o} + \iint_{\Omega_T} \alpha \nabla u \cdot \nabla \left(v-u \right) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \geq 0 \end{equation} for every cutoff function $\alpha\in C^{0,1}([0,T]; \ensuremath{\mathbb{R}}_{\geq 0})$ with $\alpha(T) = 0$ and all comparison maps $v$ satisfying $v^m \in \psi^m + L^2(0,T; H^1_0(\Omega))$ with $\partial_t v^m \in L^\frac{m+1}{m}(\Omega_T)$, $v \in C([0,T];L^{m+1}(\Omega))$ and $v \geq \psi$ a.e. in $\Omega_T$. We denoted
\begin{align*} \llangle \partial_t u, \alpha (v^m-u^m) \rrangle_{\psi_o} := &\iint_{\Omega_T} \left[ \alpha' \left( \tfrac{1}{m+1}u^{m+1} - u v^m \right) - \alpha u \partial_t v^m \right] \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{=} + \alpha(0) \int_\Omega \left[ \tfrac{1}{m+1} \psi_o^{m+1} - \psi_o v^m(\cdot,0) \right] \, \:\! \mathrm{d} x . \end{align*} \end{definition}
By a simple change of variables ($q = \frac{1}{m}$)~\cite[Theorem 1.2]{Schaetzler2} implies that solutions according to Definition~\ref{d.obstacle-varsol} exists. Then, we show that the solution for which the existence is guaranteed, is actually a solution according to Definition~\ref{d.obstacle-wsol} and finally, according to Definition~\ref{d.variational-obstacle}.
\begin{lemma} \label{lem:varsol-is-weaksol} Let $u$ be a variational solution according to Definition~\ref{d.obstacle-varsol}. Then $u$ is a weak solution to the obstacle problem according to Definition~\ref{d.obstacle-wsol}. \end{lemma}
\begin{proof} Let $v$ be a comparison map and $\eta,\alpha$ cutoff functions satisfying the assumptions given in Definition~\ref{d.obstacle-wsol}. Without loss of generality we may assume that $0 \leq \alpha\leq 1$.
Let us use a comparison map $$ v_h^m = s v^m + (1-s) (\mollifytime{u^m}{h,\psi_o} - \mollifytime{\psi^m}{h,\psi_o} + \psi^m), $$ where we denote $s = s(t) := \varepsilon \alpha(t)$ with $\varepsilon \in (0,1)$, and where we used the time mollification introduced in~\eqref{eq:time-mollif}. Observe that $v_h$ is an admissible comparison map. In particular $v_h \geq \psi$ as convex combination of functions enjoying that property and $v_h^m -\psi^m \in L^2(0,T;H^1_0(\Omega))$.
First, suppose that $\alpha$ vanishes in a neighbourhood of $T$ and let $\tau \in (0,T)$ be so large that $\alpha(t) = 0$ for all $t \in [\tau,T]$. For the parabolic term in~\eqref{var-ineq-appendix} we have \begin{align*} \iint_{\Omega_\tau} &\partial_t v_h^m (v_h - u) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &=
\tfrac{m}{m+1} \int_{\Omega} v_h^{m+1}\, \:\! \mathrm{d} x \bigg|_0^\tau \\ &\phantom{+} - \iint_{\Omega_\tau}u \left( \partial_t(s v^m) + \partial_t [(1-s) \mollifytime{u^m}{h,\psi_o}] + \partial_t [ (1-s)(\psi^m - \mollifytime{\psi^m}{h,\psi_o} )] \right) \, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align*} A part of the second term on the right hand side can be estimated by \begin{align*} - \iint_{\Omega_\tau} u (1-s) \partial_t \mollifytime{u^m}{h,\psi_o} \, \:\! \mathrm{d} x \:\! \mathrm{d} t &\leq - \tfrac{m}{m+1} \iint_{\Omega_\tau}(1-s) \partial_t \mollifytime{u^m}{h,\psi_o}^\frac{m+1}{m} \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\
&= - \tfrac{m}{m+1} \int_{\Omega}(1-s) \mollifytime{u^m}{h,\psi_o}^\frac{m+1}{m} \, \:\! \mathrm{d} x \bigg|_0^\tau \\ &\phantom{+} - \tfrac{m}{m+1} \varepsilon \iint_{\Omega_\tau} \alpha' \mollifytime{u^m}{h,\psi_o}^\frac{m+1}{m}\, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align*} Since according to Lemma~\ref{lem:mollifier}\,(i), we have $$
\int_0^T \|\mollifytime{u^m}{h,\psi_o}(t) - u^m(t)\|^\frac{m+1}{m}_{L^\frac{m+1}{m}(\Omega)}\, \:\! \mathrm{d} t \xrightarrow{h\to 0} 0, $$
it follows that along a subsequence $\|\mollifytime{u^m}{h,\psi_o}(t)
- u^m(t)\|_{L^\frac{m+1}{m}(\Omega)} \xrightarrow{h\to
0} 0$ for a.e. $t \in (0,T)$. Let $\tau$ be an instant of time at which this convergence holds. Then, by passing to the limit $h\to 0$ (along the aforementioned subsequence) we obtain \begin{align*} \limsup_{h\to0} &\left( \iint_{\Omega_\tau} \partial_t v_h^m(v_h - u) \, \:\! \mathrm{d} x \:\! \mathrm{d} t - \int_{\Omega} I(u(\tau),v_h(\tau))\, \:\! \mathrm{d} x +\int_{\Omega} I (\psi_o,v_h(0)) \, \:\! \mathrm{d} x \right) \\ &\leq \varepsilon \iint_{\Omega_\tau} \left[ \alpha' \left( \tfrac{1}{m+1}u^{m+1} - u v^m \right) - \alpha u \partial_t v^m \right] \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{=} + \varepsilon \alpha(0) \int_\Omega \left[ \tfrac{1}{m+1} \psi_o^{m+1} - \psi_o v^m(\cdot,0) \right] \, \:\! \mathrm{d} x . \end{align*}
We also have that \begin{align*}
\iint_{\Omega_\tau} |\nabla v_h^m|^2 \, \:\! \mathrm{d} x \:\! \mathrm{d} t \xrightarrow{h\to 0} \iint_{\Omega_\tau} |\nabla u^m + \varepsilon\alpha \nabla(v^m-u^m) |^2 \, \:\! \mathrm{d} x\:\! \mathrm{d} t. \end{align*} By dividing every term in~\eqref{var-ineq-appendix} by $\varepsilon> 0$, for the gradient terms we have \begin{align*}
\iint_{\Omega_\tau} &\frac{1}{2\varepsilon}(|\nabla u^m + \varepsilon\alpha \nabla(v^m-u^m)|^2 -|\nabla u^m|^2) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\xrightarrow{\varepsilon \to 0} \iint_{\Omega_\tau} \alpha \nabla u^m \cdot \nabla(v^m-u^m)\, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align*} By combining all the estimates and passing to the limit $\tau \to T$, the claim follows with $\alpha$ vanishing in a neighbourhood of $T$. For $\alpha$ that is required to satisfy only $\alpha(T) = 0$ it is deduced by a standard cutoff argument. \end{proof}
\begin{lemma} \label{lem:global-wsol-is-local} Suppose that $u$ is a weak solution to the obstacle problem according to Definition~\ref{d.obstacle-wsol}, then $u$ is a weak solution to the obstacle problem with $u=\psi$ on $\partial_p\Omega_T$ according to Definition~\ref{d.variational-obstacle}. \end{lemma}
\begin{proof} Observe that~\cite[Lemma 5.2]{BLS} implies $u \in C([0,T];L^{m+1}(\Omega))$ and $u(\cdot,0) = \psi_o$ a.e. in $\Omega$. Therefore, it only remains to prove the variational inequality~\eqref{e.local_var_eq}.
Let $v\in K'_\psi(\Omega_T)$ and $\eta \in C_0^1(\Omega,\ensuremath{\mathbb{R}}_{\geq0})$. Without loss of generality, we may assume $0 \leq \eta \leq 1$. In the variational inequality~\eqref{global-var-ineq}, we use a test function $$ v_h^m = \eta v^m + (1-\eta) (\mollifytime{u^m}{h,\psi_o} - \mollifytime{\psi^m}{h,\psi_o} + \psi^m), $$ which satisfies the assumptions from Definition~\ref{d.obstacle-wsol}. For the divergence part we have \begin{align}\label{eq:conv-div-part} \iint_{\Omega_T}& \alpha \nabla u^m \cdot \nabla (v_h^m - u^m) \, \:\! \mathrm{d} x
\:\! \mathrm{d} t\\\nonumber
&= \iint_{\Omega_T} \alpha \nabla u^m \cdot \nabla (\eta (v^m-\mollifytime{u^m}{h,\psi_o})) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\\nonumber &\phantom{+}+ \iint_{\Omega_T} \alpha \nabla u^m\cdot \nabla ( \mollifytime{u^m}{h,\psi_o} - u^m) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\\nonumber &\phantom{+} + \iint_{\Omega_T} \alpha \nabla u^m\cdot \nabla ((1-\eta) (\psi^m - \mollifytime{\psi^m}{h,\psi_o})) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\\nonumber &\xrightarrow{h\to 0} \iint_{\Omega_T} \alpha \nabla u^m\cdot \nabla (\eta (v^m - u^m)) \, \:\! \mathrm{d} x \:\! \mathrm{d} t. \end{align} Here we used Lemma~\ref{lem:mollifier}\,(i) to pass to the limit. Moreover, we may estimate \begin{align*} \iint_{\Omega_T} - \alpha u \partial_t v_h^m \, \:\! \mathrm{d} x\:\! \mathrm{d} t &= \iint_{\Omega_T}-\alpha \eta u \partial_t v^m \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} +\iint_{\Omega_T} - \alpha (1-\eta) u \partial_t \mollifytime{u^m}{h,\psi_o} \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} + \iint_{\Omega_T} -\alpha (1-\eta) u \partial_t (\psi^m - \mollifytime{\psi^m}{h,\psi_o} ) \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\leq \iint_{\Omega_T}-\alpha \eta u \partial_t v^m \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} + \tfrac{m}{m+1}\iint_{\Omega_T} \alpha' (1-\eta) \mollifytime{u^m}{h,\psi_o}^\frac{m+1}{m} \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{+} + \tfrac{m}{m+1} \alpha(0) \int_{\Omega} (1-\eta) \psi_o^{m+1} \, \:\! \mathrm{d} x \\ &\phantom{+} + \iint_{\Omega_T} -\alpha (1-\eta) u \partial_t (\psi^m - \mollifytime{\psi^m}{h,\psi_o} ) \, \:\! \mathrm{d} x \:\! \mathrm{d} t . \end{align*} Observe that also $$ -\psi_o v_h^m(\cdot,0) = -\eta \psi_o v^m(\cdot,0) - (1-\eta) \psi_o^{m+1}. $$ Now by combining the estimates and passing to the limit $h \to0$, we obtain \begin{align*} \limsup_{h\to 0} \llangle \partial_t u, \alpha (v_h^m-u^m) \rrangle_{\psi_o} &\leq \iint_{\Omega_T} \eta \left[ \alpha' \left( \tfrac{1}{m+1}u^{m+1} - u v^m \right) - \alpha u \partial_t v^m \right] \, \:\! \mathrm{d} x \:\! \mathrm{d} t \\ &\phantom{=} + \alpha(0) \int_\Omega \eta \left[ \tfrac{1}{m+1} \psi_o^{m+1} - \psi_o v^m(\cdot,0) \right] \, \:\! \mathrm{d} x. \end{align*} For cutoff functions $\alpha$ with compact support in $(0,T)$, as they are considered in~\eqref{e.local_var_eq}, the last integral vanishes. Consequently, for such cutoff functions the last formula becomes \begin{equation*}
\limsup_{h\to 0} \llangle \partial_t u, \alpha (v_h^m-u^m)
\rrangle_{\psi_o}
\le
\llangle \partial_t u, \alpha\eta (v_h^m-u^m) \rrangle. \end{equation*} In combination with~\eqref{eq:conv-div-part}, this yields the variational inequality~\eqref{e.local_var_eq} and concludes the proof. \end{proof}
\end{document} | arXiv |
The Journal of Korean Academy of Dental Technology (대한치과기공학회지)
Korean Academy of Dental Technology (대한치과기공학회)
Health Sciences > Dental Science
학회지 발행목적 본 학회지는 대한치과기공학회에서 발행하는 학술지로서 1997년 창간되어, 국민의 구강보건향상을 위하여 치과기공 분야의 학술연마를 위해 발간한다. 학회 및 학회지의 영문약자 대한치과기공학회의 약자는 KADT이다. 대한치과기공학회지의 약자는 J Kor Aca Den Tec(Journal of Korean Academy of Dental Technology)이다. 접수원고 원고의 종류는 실험논문, 관찰논문 등으로 한다. 그리고 본 학회에서 필요로 하는 청탁논문이나 리뷰논문, 임상 기술논문 등이 포함될 수 있다. 학술지 발행 본 학술지는 2011년 1월 1일부터 연4회 발간한다(발행일 : 3월 30일, 6월 30일, 9월 30일, 12월 30일).
A Study on improvements in manufatured technique of all Ceramic Crown
Shin, Moo-Hak;Kim, Yeon-Soo;Choi, Un-Jea;Chung, Hee-Sun 9
A new kind of 'All Ceramic Crown' could be manufactured by making improvements in the manufacturing technique for the current 'All Ceramic Crown' which does not use a special ceramic but rather a general one as a substitute. If we use the manufacturing technique for the 'All Ceramic Crown', metal coping and core are not produced. The effects of the new manufacturing technique for the 'All Ceramic Crown' are as follows: First. We do not need to use new material or special machinery or tools. Second. We can use general machinery and tools. Third. Using the basic 'All Ceramic Technique', we anticipate improvement in learning in our students. Forth. We can save effort, materials and time. Fifth. The technique also has advantages for esthetic 'temporary crown'.
A Study on The Dept. of Dental Laboratory Technology Curricula by Term in the Nation
Kwon, Soon-Suk 17
The purpose of this study was to examine the 2001 curricula in 17 departments of dental technology across the nation in an attempt to find out the educational realities of the departments by term and school year and serve as a basis for the development of more advanced, efficient dental technology curriculum and common educational objectives. For that purpose, the 2001 curricula of the three-year dental laboratory technology departments were analyzed by school year and term to calculate the amount of required credit, the number of subjects, and the weekly classes for electives and major. The findings of this study could be listed as below: 1. The departments of dental laboratory technology nationwide investigated require students to get 120 to 135 credits in total. Out of the credits, 10 to 25 credits are assigned to the electives, and 106 to 11 8 credits are given to the major. 2. There are 50 to 68 subjects in the departments of dental technology. 5 to 16 subjects are the electives, and 41 to 59 are the major. 3. There are 150 to 196 classes per week, which consist of 10 to 30 ones for the electives and 137 to In for the major. 4. The curricula for the first semester of the first year are as follows: 1) 20 to 24 credits are required. 4 to 11 credits are alloted to the electives, and 9 to 19 credits are assigned to the major. 2) The number of subject is 9 to 13, which are composed of 2 to 7 for the electives and 4 to 9 for the major. 3) The weekly classes are 22 to 29. The classes for the electives range from 4 to 14 per week, and 10 to 20 classes a week are for the major. 5. The curricula for the second semester of the first year are as below: 1) There are 20 to 25 credits. 3 to 12 credits are assigned to the electives, and 12 to 19 credits are for the major. 2) The number of subject is 10 to 14, which consist of 2 to 6 for the electives and 6 to 10 for the major. 3) The weekly classes are 22 to 29. and 3 to 12 classes a week are for the electives, and 15 to 24 classes are for the major. 6. The curricula for the first semester of the second year are as below: 1) The number of credits ranges from 20 to 24. Only six colleges offer 2 credits for the electives and the major account for 18 to 24 ones. 2) There are 8 to 12 subjects. Only six colleges offer one or two electives, and 8 to 12 are the major. 3) The weekly classes are 23 to 33. Only six colleges offer 2 or 3 classes a week for the electives, and 21 to 33 classes are for the major. 7. The curricula for the second semester of the second year are as below: 1) The number of credits ranges from 19 to 24. Only two colleges offer 2 credits for the electives and the major account for 18 to 24 ones. 2) There are 7 to 12 subjects. Only two colleges offer one or two electives, and 8 to 12 are the major. 3) The weekly classes are 24 to 36. Only two colleges offer 2 classes a week for the electives, and 24 to 36 classes are for the major. 8. The curricula for the first semester Of the third year are as below: 1) There are 16 to 24 credits. Just a college assigns 2 credits to the electives, and 16 to 24 credits are given to the major. 2) The number of subject is 5 to 12. Only a college offers one elective for optional course, and 5 to 12 are the major. 3) The weekly classes range from 18 to 39. Just a college offer 2 classes a week for the electives, and 18 to 39 classes are for the major. 9. The curricula for the second semester of the third year are as below: 1) There are 16 to 23 credits. Just a college assigns 2 credits to the electives, and 16 to 23 credits are given to the major. 2) The number of subject is 5 to 12. Only a college offers one elective for optional course, and 5 to 12 are the major. 3) The weekly classes range from 18 to 39. Just a college offer 2 classes a week for the electives, and 18 to 39 classes are for the major.
A Study on new Photoinitiator of Visible Light Dental Composite Resin
Choi, Yong-Seok;Sun, Gum-Ju 49
The photopolymerization efficiency and surface hardness of composite resin containing 1,2-phenylpropanedione (PD) and diacetyl (DA) as photoinitiators were studied by IR and Vickers hardness and the results were compared with that of camphorquinone (CQ). Relative photopolymerization efficiency of the photoinitiators increased in the order of DA < CQ < PD. Vickers hardness of composite resin containing the photoinitiators increased in the order of CQ < PD < DA. Thus, PD is a new visible light photoinitiator for dental composite resin with higher photopolymerization efficiency and surface hardness than that of CQ. Mechanical properties such as Vickers hardness, diametral tensile strength, and flexural strength of the experimental resin composite prepared by addition of the photosensitizer into a resin of bis-GMA improved with increasing the photosensitizer content and irradiation time. The resin composite of bis-GMA containing DA or PD shows better mechanical properties than that of CQ.
A Patient's Satisfaction with Denture in the Old People
Lee, In-Kyu;Song, Yun-Hee 61
This experiment was to find out the complacency of wearing dentures and the linkage to the quality of life style of oldsters 60 years old and over. 122 numbers of oldsters who came to aid to the health center were put to survey. This was taken place within the areas of Deajun, Koonsan, Mooju and Jinahn, thus procured the following result. 1. The length of edentulous period of 1-6years of oldsters aged around 60s showed 26.0%. The oldsters with edentulous period of over 7years aged in the 70s showed 26.1 % and 56.0% on oldsters aged in the 80s. This shows that as the age increases the edentulous period lengthens. (P<0.05) The length of time of using the denture shows. llyears or over on women 41.9%, less than 6years on men 71.4% as the highest rate. 11 years or over on towns/subcounty show 57.5%, small and medium cities more than 1 year 63.6%, less than six years also 63.6% and Kwangyuk city 47.6%. 2. The complacency on medical treatment of dentures was highest in Kwangyuk city of 61.3%, compared to towns/subcounty of 50.8% and small and medium cities of 33.3%. (P<0.05) 3. The complacency on mastication and pronunciation appears, 2.74% in Kwangyuk city, 3.10% in towns/ subcounty which is higher than the small and medium cities showing 1.09% on average. Satisfaction rate tends to be higher as the length of time of using the denture is longer. 4. Inconvenience on eating habits caused by dentures were felt by women. Wanting to get a new denture was 25.6% by women showing much higher rate than that of men which is 2.8% by men. (P<0.05) 5. The complacency of change in their life style after wearing the dentures were higher in Kwangyuk city of 64.5% whereas it showed 27.0% in towns! subcounty and 16.7%in small and medium cities. (P<0.05)
Research for the Standard Model of the Items of the National Qualification Examination for the Dental Technician
Lee, Duck-Hye;Chung, In-Sung;Han, Chang-Sik 75
This research was preformed for the purpose of preparing the items of standard model of the national dental technician test base on the duty analysis of the dental technician. The results of the duty analysis for the dental technician follows. 1. The dental technician is a profession to make the oral function smooth through the dental supplement and equipment in a scientific method and the skilled technique. 2. The duty of the dental technician are determined as A. preparation for manufacture B. manufacture C. management of the place of the dental technology D. self-development. A. The field of "the preparation for manufacture" are determined as 1. to confirm work authorization 2. To confirm the working model, B. The field of "In manufacture" are determined as 1. to manufacture the temporary crown 2. to manufacture the inlay and crown & bridge prosthesis 3. to manufacture the porcelain fused metal crown prosthesis 4. to manufacture the all ceramic crown prosthesis 5. to manufacture the temporary denture prosthesis 6. to manufacture the partial denture prosthesis 7. to manufacture the complete denture prosthesis 8. to manufacture the attachment prosthesis 9. to manufacture implant prosthesis 10. to manufacture the removable orthodontic device, 11. to manufacture the fixed orthodontic device, 12. to manufacture the orthodontic study cast C. The field of "in management of the dental lab." are determined as 1. management 2. to control the dental lab. D. The field of "In the self-development" are determined as 1. to improve the professionalism 2. self-control. 3. The developing items selected under the duty evaluation of the dental technician are l7s in the manufacture preparation, 1,011s in the manufacture, 7s in the management for the dental technology, 5s in self-development, and in all together 1,040s
Research for The Comparing Test of the Fracture Strength According to the Heat Curing Method in the Denture Base Resin
Han, Min-Soo 95
For this study, self curing resin and heat curing resin used for existing usual resin denture base in the denture industry were chosen by manufacturer. Curing tests for 30-minute, I-hour, 2-hour and 3-hour were conducted to know the strength of the resins and conduct analysis to get other necessary information. The results obtained are as follows: 1. Heat curing resins show a little differences among the manufacturers. However 30-minute curing resin shows great difference as shown in the fracture strength test. 2. The effect from the granularity of the resins on the fracture strength was found insignificant which means there is no difference between coherence and strength. 3. To summarize the results from each time level, the longer the time is, the more the minute cracks on the surface, which is the cause of reduced strength. From this test, it was identified that in making the denture base for patients in dental clinics, 30-minute curing is most efficient and effective in reducing discoloration and monomers, although long-time curing has been considered to be the principal.
A Study on the Work Stress of Dental Technicians
Kwon, Eun-Ja;Kim, Ji-Hwan 105
This study was designed to grasp the degree of job stress affecting dental technicians and the degree of the symptoms of their job stress. With this in mind, the researcher selected a total of 170 dental technicians living in Seoul and Incheon, conducting a research in a period ranging from August 1,2001, to August 20, 2001. The researcher made use of a structured questionnaire whose reliability and feasibility are proved. The questionnaire is composed of a total of 55 questions: thirteen questions related to the subjects' general characteristics; 28 questions connected to the measurement of job stress and fourteen questions linked with the measurement of job-stress symptoms. The researcher analyzed the findings with the aid of SPSS(Statistical Package for Social Sciences). The research came to draw the following conclusions on the basis of frequency, percentage, T-test, and F-test, multiple regression. I. The analysis into the job stress of the subjects indicates that there is a significant difference in difficult questions among their work places, working hours, academic background, job satisfaction and jobs(P<.05). The job stress stands at 3.48 on the average, and the area of conflict among too much work and job performance turns out to be highly perceived. 2. The analysis into the degree of the symptoms of the job stress of the subjects shows that there is a significant difference in work place, working hours, job satisfaction and the continual maintenance of job(P<.05). The symptoms of job stress accounts for 2.65 on the average. Physical symptoms turn out to be highly perceived; The response 'My arms and legs are killing me' proves to stand for 3.03. 3. The correlation between job stress and the symptoms of job stress turns out to be significant(r=0.519, P<0.001), and there is a significant correlation between the average points of each job-stress area and the average points of the symptoms of job stress. All in all, it is necessary that dental technicians themselves should make positive efforts to control and relieve stress and that more efficient programs should be implemented with a view to dealing with stress.
A Study on Problems and Solution Activities of College Students at the Departments of Dental Technology
Kim, Youn-Su;Shin, Moo-Hak;Song, Yun-Hee;Chung, Hee-Sun 121
This Study aims at investigating the tendency of individual problems that the students at the departments of dental technology are faced with, their efforts and types of activities to solve the problems. Surveyed for this purpose were 700 freshmen to juniors enrolled in the departments of dental technology at the universities across the country and they were analyzed by area, grade and sex. 1. Tendency of Problems The areas of problems the students at the departments of dental technology considered as most serious were health, human relation, view of value and economy in sequence, while the areas they considered as least serious were future course and employment. In general, there were tendencies as follows: divided by grade, the higher the grades of the students are, the higher scores of problems they get(F=12.14, P=.000) : in the areas of health and human relation, as the grade gets higher, the scores of their problems also increase, accordingly(F= 17.58, P=.000)(F=8.39, P=.000) : in the areas of economy, parents and home, freshmen and juniors are found to have higher scores of problems compared to sophomores,:F=7.89, P=.000)(F=11.63, P=.000) : divided by sex, female students recorded higher scores of problems than their counterparts(F=31.85, P=.000) 2. Efforts for Problems View of value, human relation, parents and home appeared to be the areas for which the students made the highest rate of efforts in sequence while the areas for which they paid lowest grade of efforts were health and economy. In other words, the students gave higher scores for human relation and view of value in terms of tendency as well as efforts of problems, while they rated high for health and economy in the tendency but low in the efforts of problems. Divided by grade, freshmen and juniors appeared to score higher points than sophomores in their efforts to solve problems, even though juniors made relatively higher efforts than freshmen(F=6.53, P=.002). Divided by sex, female students scored higher points in their efforts than males(F=15.98, P=.000) 3. Difference of Efforts to Solve Problems. In the analysis into the scores of difference showing the disagreement between the tendency of problems and the efforts to solve problems, the area of health was rated as highest, followed by economy. There was little difference of scores between human relation and view of value, an indicator of agreement between the scores of problems and efforts. The difference of scores between future course and employment, between liberal arts and recreation, between characters/parents and home/school work turned out to be negative in the listed sequence, leading to a conclusion that the students enrolled in the departments of dental technology are making more efforts to solve problems than actually needed in most areas. By grade, there was a significant difference only in the area of health(F=3.00, P=.050). This difference itself was widening as the students come to higher grades. By sex, females showed bigger difference than males. 4. Types of Activities to Solve Problems. The analysis into types of activities to solve problems revealed that the surveyed students seemed to actively cope with the problems in all areas, relying at the same time on personal relationship. Especially in the areas of health, economy, parents, home, human relation, future course and employment, the students appeared to deal with the problems actively as well as emotionally.
Influences of Sprue joint Forms on Castability in dental Restorations
Choi, Un-Jea;So, Jung-Mo 151
The purpose of this study was to evaluate the influence of sprue joint form on castability in dental prosthetics. The researcher carried out experimental study according to the type was divided by sprue joint form. Therefore type A was made thicken than sprue diameter of type B and type C was made it conversely. The results were as follows: 1. In the order of castability, type C(87.8%) was first and then type A(87.0%) and type B(79.2%) was last. However there was no significant difference statistically. 2. If reservior was located at the correct position and in sufficient size, internal shrinkage porosity in the casting bodies were not generated in the type of all. 3. At the reservoir, internal shrinkage porosity was generated first in type A, followed by type Band type C in the order named. Although it gives no impact on casting body, porosity seems to appear the least when it's produced in the form of type C.
A Study on Improvement of Metal-Ceramic Bonding Strength by Addition of Aluminum to Casting Metal Alloy
Lee, Jae-Won;Min, Byong-Kuk;Han, Min-Soo 161
The Purpose of this study was to investigate the chemically improvement of metal-ceramics bond strength in the course of recasting Ni-Cr metal composite system with 10wt.%, 20wt.% and 30wt.% aluminum respectively. We have tested the bond strength, micro-structure, chemical composition of each metal composites and metal- ceramic bond interfaces by 3-point bending strength tester, SEM and EDS. We have made the conclusions through this study as follow: 1. The most suitable amount of aluminum to the Ni-Cr metal composite recasting is 20wt. % for improving metal-ceramics bond strength with debonding strength value of 49.54 kgf/mm2. 2. The aluminum must be changed to small spread alumina like phases and second aluminum-metal composites phases in the morphology of Ni-Cr metal composite system by adding during it's casting. These second phases have inclined functional oxide phases mixed with metal elements and they must take roll to improvement of metal-ceramics bond strength. 3. In the case of 30wt.% aluminum appended to Ni-Cr metal composite system, an excess of second inclined functional oxide phases produce cracks and spalling of them apart from it's base material. It must be a important factor of reduction of metal-ceramics bond strength.
Force per unit Displacement according to the Shape of a Clasp Arm and Flexibility of the Material
Lim, Dong-Chun 171
The purpose of this study is to evaluate force per unit displacements according to the shape of a clasp arm and flexibility of the material. Effect of four shape parameters of a clasp, base width and thickness and tip width and thickness, on tip displacement and force per unit displacement was investigated to get the fact that displacement and force per unit displacement at the tip increase as thickness and width of clasp arm increase just as expected. But force per unit displacement is much more affected by the change in thickness than by change in width. So it is effective to increase the thickness rather than width in order to increase the force at the tip using the same amount of the material.
An Experimental Study on Margin Consistency of Prosthesis According to the Continued Time of Casting-Ring in the course of the Casting of Dental Alloy
Hwang, Seong-Sig;Lee, Sang-Hyeok 179
With regard to the manufacture of dental prosthesis, all the dental mechanism is of vital significance at the aspect of activating its function by fixing the prosthesis to Patient's oral cavity. However, if there we will take our immediate action without the discretion about its process none the less for the importance of dental mechanism, then we might have a serious problem. Accordingly, there need to pay attention to the dilatability makes up for the shrinkage state occurring by the feature of metal materials and manufacturing process which appeared in the process of dental mechanism, which eventually is expected to playa very important role in casting a dental prosthesis appropriate to one's oral tissue. This study was designed to take into account of the effects on margin consistency of prosthesis according to the continued time of casting-ring in the course of the casting of dental alloy. For this, the researcher made an experiment on the casting of dental alloy, its dilatability, and the change of phase. The results of this study were as follows: First, the researcher could see that the sample which was cast under the condition of $650^{\circ}C/20$ Minutes(the continued time) was far superior to others at the aspect of margin consistency. Second, according to the measurement of expansion coefficient by Dilatometer, the researcher perceived the fact that the expansion-coefficient showed a maximum of $37.1{\mu}m$ considering the sample's length which was cast with ordinary temperature under the condition of $650^{\circ}C/20$ Minutes. Third, from the result of X-ray diffraction under the condition of $650^{\circ}C/20$ Minutes(the continued time), the researcher could find that there's no difference between the change of phase and its intensity. As mentioned above, the researcher could ascertain the fact that its contraction don't give rise to the change of phase.
A study on reflective spectrum between In-Ceram alumina core and IPS Empress 2 core
Park, Myung-Ja;Kim, Joo-Won 189
A study on reflective spectrum between In-Ceram alumina plate(IAP) and IPS empress 2 plate were carried out in order to determine the difference of reflective spectrum rates between materials and between thicknesses of materials(0.8mm, 1.0mm, 1.2mm) by visible wave. The rates are measured by spectrophotometer(Top scan model TC-1800). And an analysis of ANOVA and paired sample t-test were carried out. The results are summarized as follows. 1. The reflective rates of IAP and IPS increased slowly as visible wave spectrum increased. 2. The difference of reflective rates of IAP between 0.8mm and 1.0mm is significant but a little(less than 1%). The differences of reflective rates of IAP between 0.8mm and l.2mm and between 1.0mm and l.2mm are significant(8%, 10% ). 3. The differences of reflective rates of IPS between 0.8mm and 1.0mm and between 0.8mm and 1.2mm and between 1.0mm and 1.2mm are significant(0.06 %, 0.01 %, 2 %). 4. The differences of reflective rates of 0.8mm, 1.0mm between IAP and IPS are significant but a little(less than 0.9%, 0.5%). The difference of reflective rates of l.2mm between IAP and IPS is significant and larger than the other thicknesses relatively(7 %).
The Effect of Cooling method on the Surface Reaction Zone of CP Titanium Casting Body
Moom, Soo;Choi, Seog-Soon;Moon, Il 203
This test is to conduct applied research the reaction area of the Ti-cast metal body which is made use of Dental Phosphate-silica alumina bonded investment material selling at a market, and the cooling method is how to effect on the acicular. The experimentation is as followings, 1. Experimental specimens After invest with Dental Phosphate-silica alumina bonded investment material, the $10{\times}10{\times}1.0mm^3$ wax pattern was casted by Dental high vacuum argon centrifugal casting machine. 2. Test We can analyze SEM/EDS, XRD utilize the fractography(an optical microscope). 3. Conclusion The pure cast metal body constituted of reaction products layer, stability layer and contamination layer. This pure cast have no connection with the cooling condition. The pure Titanium shows difference in a component distribution according to the cooling condition. Through this experimentation we can establish that acicular in the pure Ti-cast metal is consist of Hexagonal structure a=2.9505$\AA$, c=4.6826$\AA$.
A Study on Opaque Porcelain for P.F.M Crown - Focused on Paste Opaque -
Kim, Sa-Hak;Ko, Dae-Jin;Lee, Yong-Keun;Kim, Kwang-Mahn;Kim, Kyoung-Nam 211
The purpose of this study was to analyze the commercial paste opaque products currently available in domestic of foreign as well as domestic, such as Duceram Plus(DU; Ducera Dental GmbH, Germany), VMK 95(VM; Vita Co., Germany), Noritake EX-3(EX; Noritake Co., Japan) and Ceramax(CE; Alphadent Co., Korea). They were characterized in thermal expansion coefficient, particle size distribution, viscosity and solvent using thermomechanical analyzer, particle size analyzer, rheometer and infrared spectrophotometer. Experimental results are as follows; Firstly, thermal expansion coefficients were determined $13.9{\times}10-6/^{\circ}C$ for DU, $14.3{\times}10-6/^{\circ}C$ for VM, $13.3{\times}10-6/^{\circ}C$ for EX, and $14.0{\times}10-6/^{\circ}C$ for CE. Secondly, percent of partice size below $1{\mu}m$ were 12% for DU, VM and CE, and 13% for EX, percent between $1{\mu}m$ and $5{\mu}m$ were 42% for DU, 42% for VM, 38% for EX, and 61 % for CE, percent between $5{\mu}m$ and $10{\mu}m$ were 21 % for DU, 24% for VM, 20% for EX, and 18% for CE, and over $10{\mu}m$ were 25% for DU, 22% for VM, 29% for EX, 9% for CE. Thirdly, the basic composition of the solvent in all of the commercial paste opaques were determined as ethylene glycol from FT-IR investigation. Lastly, measured viscosities were 1798 cp for DU, 536 cp for VM, 1110 cp for EX, and 721 cp for CE.
A study on microstruture and corrosion resistance of Ti-Nb alloys by hot rolling
Park, Hyo-Byung 223
Pure titanium and Ti6Al4V alloy have been mainly used as implant materials but the cytotoxicity of V, neurotoxicity of Al resulting in Alzheimer disease had been reported. This paper was described the influence of composition of Ti-Nb alloys with 3 wt%Nb, 20 wt%Nb on the microstructure and corrosion resistance. Specimens of Ti alloys were melted in vacuum arc furnace and homogenized at $1000^{\circ}C$ for 24hr. The alloys were rolled in $\beta$ and ${\alpha}+{\beta}$ regions. The corrosion resistance of Ti alloys were evaluated by potentiodymic polarization test in 0.9% NaCl and 5% HCl solutions. The results can be summarized as follows: 1. The microstructure was transformed from $\alpha$ phase to ${\alpha}+{\beta}$ phase by adding Nb 2. The hardness of Ti-20Nb alloy was greater than Cp- Ti, Ti-3Nb alloy. 3. The corrosion resistance of Ti-20Nb alloy was better than that of Cp-Ti, Ti-3Nb alloy in 0.9%NaCl and 5%HCl solutions. | CommonCrawl |
Simulations of in situ X-ray diffraction from uniaxially compressed highly textured polycrystalline targets
McGonegle, David, Milathianaki, Despina, Remington, Bruce A. et al. (2 more authors) (2015) Simulations of in situ X-ray diffraction from uniaxially compressed highly textured polycrystalline targets. Journal of Applied Physics. 065902. ISSN 0021-8979
A growing number of shock compression experiments, especially those involving laser compression, are taking advantage of in situ x-ray diffraction as a tool to interrogate structure and microstructure evolution. Although these experiments are becoming increasingly sophisticated, there has been little work on exploiting the textured nature of polycrystalline targets to gain information on sample response. Here, we describe how to generate simulated x-ray diffraction patterns from materials with an arbitrary texture function subject to a general deformation gradient. We will present simulations of Debye-Scherrer x-ray diffraction from highly textured polycrystalline targets that have been subjected to uniaxial compression, as may occur under planar shock conditions. In particular, we study samples with a fibre texture, and find that the azimuthal dependence of the diffraction patterns contains information that, in principle, affords discrimination between a number of similar shock-deformation mechanisms. For certain cases we compare our method with results obtained by taking the Fourier Transform of the atomic positions calculated by classical molecular dynamics simulations. Illustrative results are presented for the shock-induced $\alpha$-$\epsilon$ phase transition in iron, the $\alpha$-$\omega$ transition in titanium and deformation due to twinning in tantalum that is initially preferentially textured along [001] and [011]. The simulations are relevant to experiments that can now be performed using 4th generation light sources, where single-shot x-ray diffraction patterns from crystals compressed via laser-ablation can be obtained on timescales shorter than a phonon period.
McGonegle, David
Milathianaki, Despina
Remington, Bruce A.
Wark, Justin S.
Higginbotham, Andrew https://orcid.org/0000-0001-5211-9933
10 pages, 4 figures; typo corrected in Fig. 1 caption. © AIP Publishing 2015. This is an author produced version of a paper accepted for publication in Journal of Applied Physics. Uploaded in accordance with the publisher's self-archiving policy.
cond-mat.mtrl-sci,Physics and Astronomy
The University of York > Faculty of Sciences (York) > Physics (York)
http://arxiv.org/abs/1501.05474
Submitted Version
Filename: 150105474v2.pdf
Description: 150105474v2 | CommonCrawl |
E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
• Normal subgroup
• Quotient group
• (Semi-)direct product
Group homomorphisms
• kernel
• image
• direct sum
• wreath product
• simple
• finite
• infinite
• continuous
• multiplicative
• additive
• cyclic
• abelian
• dihedral
• nilpotent
• solvable
• action
• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
• Symmetric group Sn
• Alternating group An
• Dihedral group Dn
• Quaternion group Q
• Cauchy's theorem
• Lagrange's theorem
• Sylow theorems
• Hall's theorem
• p-group
• Elementary abelian group
• Frobenius group
• Schur multiplier
Classification of finite simple groups
• cyclic
• alternating
• Lie type
• sporadic
• Discrete groups
• Lattices
• Integers ($\mathbb {Z} $)
• Free group
Modular groups
• PSL(2, $\mathbb {Z} $)
• SL(2, $\mathbb {Z} $)
• Arithmetic group
• Lattice
• Hyperbolic group
Topological and Lie groups
• Solenoid
• Circle
• General linear GL(n)
• Special linear SL(n)
• Orthogonal O(n)
• Euclidean E(n)
• Special orthogonal SO(n)
• Unitary U(n)
• Special unitary SU(n)
• Symplectic Sp(n)
• G2
• F4
• E6
• E7
• E8
• Lorentz
• Poincaré
• Conformal
• Diffeomorphism
• Loop
Infinite dimensional Lie group
• O(∞)
• SU(∞)
• Sp(∞)
Algebraic groups
• Linear algebraic group
• Reductive group
• Abelian variety
• Elliptic curve
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
Basic description
The Lie group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is eight.
Therefore, the vectors of the root system are in eight-dimensional Euclidean space: they are described explicitly later in this article. The Weyl group of E8, which is the group of symmetries of the maximal torus which are induced by conjugations in the whole group, has order 214 35 52 7 = 696729600.
The compact group E8 is unique among simple compact Lie groups in that its non-trivial representation of smallest dimension is the adjoint representation (of dimension 248) acting on the Lie algebra E8 itself; it is also the unique one which has the following four properties: trivial center, compact, simply connected, and simply laced (all roots have the same length).
There is a Lie algebra Ek for every integer k ≥ 3. The largest value of k for which Ek is finite-dimensional is k = 8, that is, Ek is infinite-dimensional for any k > 8.
Real and complex forms
There is a unique complex Lie algebra of type E8, corresponding to a complex group of complex dimension 248. The complex Lie group E8 of complex dimension 248 can be considered as a simple real Lie group of real dimension 496. This is simply connected, has maximal compact subgroup the compact form (see below) of E8, and has an outer automorphism group of order 2 generated by complex conjugation.
As well as the complex Lie group of type E8, there are three real forms of the Lie algebra, three real forms of the group with trivial center (two of which have non-algebraic double covers, giving two further real forms), all of real dimension 248, as follows:
• The compact form (which is usually the one meant if no other information is given), which is simply connected and has trivial outer automorphism group.
• The split form, EVIII (or E8(8)), which has maximal compact subgroup Spin(16)/(Z/2Z), fundamental group of order 2 (implying that it has a double cover, which is a simply connected Lie real group but is not algebraic, see below) and has trivial outer automorphism group.
• EIX (or E8(−24)), which has maximal compact subgroup E7×SU(2)/(−1,−1), fundamental group of order 2 (again implying a double cover, which is not algebraic) and has trivial outer automorphism group.
For a complete list of real forms of simple Lie algebras, see the list of simple Lie groups.
E8 as an algebraic group
By means of a Chevalley basis for the Lie algebra, one can define E8 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as “untwisted”) form of E8. Over an algebraically closed field, this is the only form; however, over other fields, there are often many other forms, or “twists” of E8, which are classified in the general framework of Galois cohomology (over a perfect field k) by the set H1(k,Aut(E8)) which, because the Dynkin diagram of E8 (see below) has no automorphisms, coincides with H1(k,E8).[1]
Over R, the real connected component of the identity of these algebraically twisted forms of E8 coincide with the three real Lie groups mentioned above, but with a subtlety concerning the fundamental group: all forms of E8 are simply connected in the sense of algebraic geometry, meaning that they admit no non-trivial algebraic coverings; the non-compact and simply connected real Lie group forms of E8 are therefore not algebraic and admit no faithful finite-dimensional representations.
Over finite fields, the Lang–Steinberg theorem implies that H1(k,E8)=0, meaning that E8 has no twisted forms: see below.
The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are (sequence A121732 in the OEIS):
1, 248, 3875, 27000, 30380, 147250, 779247, 1763125, 2450240, 4096000, 4881384, 6696000, 26411008, 70680000, 76271625, 79143000, 146325270, 203205000, 281545875, 301694976, 344452500, 820260000, 1094951000, 2172667860, 2275896000, 2642777280, 2903770000, 3929713760, 4076399250, 4825673125, 6899079264, 8634368000 (twice), 12692520960…
The 248-dimensional representation is the adjoint representation. There are two non-isomorphic irreducible representations of dimension 8634368000 (it is not unique; however, the next integer with this property is 175898504162692612600853299200000 (sequence A181746 in the OEIS)). The fundamental representations are those with dimensions 3875, 6696000, 6899079264, 146325270, 2450240, 30380, 248 and 147250 (corresponding to the eight nodes in the Dynkin diagram in the order chosen for the Cartan matrix below, i.e., the nodes are read in the seven-node chain first, with the last node being connected to the third).
The coefficients of the character formulas for infinite dimensional irreducible representations of E8 depend on some large square matrices consisting of polynomials, the Lusztig–Vogan polynomials, an analogue of Kazhdan–Lusztig polynomials introduced for reductive groups in general by George Lusztig and David Kazhdan (1983). The values at 1 of the Lusztig–Vogan polynomials give the coefficients of the matrices relating the standard representations (whose characters are easy to describe) with the irreducible representations.
These matrices were computed after four years of collaboration by a group of 18 mathematicians and computer scientists, led by Jeffrey Adams, with much of the programming done by Fokko du Cloux. The most difficult case (for exceptional groups) is the split real form of E8 (see above), where the largest matrix is of size 453060×453060. The Lusztig–Vogan polynomials for all other exceptional simple groups have been known for some time; the calculation for the split form of E8 is far longer than any other case. The announcement of the result in March 2007 received extraordinary attention from the media (see the external links), to the surprise of the mathematicians working on it.
The representations of the E8 groups over finite fields are given by Deligne–Lusztig theory.
Constructions
One can construct the (compact form of the) E8 group as the automorphism group of the corresponding e8 Lie algebra. This algebra has a 120-dimensional subalgebra so(16) generated by Jij as well as 128 new generators Qa that transform as a Weyl–Majorana spinor of spin(16). These statements determine the commutators
$\left[J_{ij},J_{k\ell }\right]=\delta _{jk}J_{i\ell }-\delta _{j\ell }J_{ik}-\delta _{ik}J_{j\ell }+\delta _{i\ell }J_{jk}$
as well as
$\left[J_{ij},Q_{a}\right]={\frac {1}{4}}\left(\gamma _{i}\gamma _{j}-\gamma _{j}\gamma _{i}\right)_{ab}Q_{b},$
while the remaining commutators (not anticommutators!) between the spinor generators are defined as
$\left[Q_{a},Q_{b}\right]=\gamma _{ac}^{[i}\gamma _{cb}^{j]}J_{ij}.$
It is then possible to check that the Jacobi identity is satisfied.
Geometry
The compact real form of E8 is the isometry group of the 128-dimensional exceptional compact Riemannian symmetric space EVIII (in Cartan's classification). It is known informally as the "octooctonionic projective plane" because it can be built using an algebra that is the tensor product of the octonions with themselves, and is also known as a Rosenfeld projective plane, though it does not obey the usual axioms of a projective plane. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits (Landsberg & Manivel 2001).
E8 root system
A root system of rank r is a particular finite configuration of vectors, called roots, which span an r-dimensional Euclidean space and satisfy certain geometrical properties. In particular, the root system must be invariant under reflection through the hyperplane perpendicular to any root.
The E8 root system is a rank 8 root system containing 240 root vectors spanning R8. It is irreducible in the sense that it cannot be built from root systems of smaller rank. All the root vectors in E8 have the same length. It is convenient for a number of purposes to normalize them to have length √2. These 240 vectors are the vertices of a semi-regular polytope discovered by Thorold Gosset in 1900, sometimes known as the 421 polytope.
Construction
In the so-called even coordinate system, E8 is given as the set of all vectors in R8 with length squared equal to 2 such that coordinates are either all integers or all half-integers and the sum of the coordinates is even.
Explicitly, there are 112 roots with integer entries obtained from
$\left(\pm 1,\pm 1,0,0,0,0,0,0\right)\,$
by taking an arbitrary combination of signs and an arbitrary permutation of coordinates, and 128 roots with half-integer entries obtained from
$\left(\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}},\pm {\tfrac {1}{2}}\right)\,$
by taking an even number of minus signs (or, equivalently, requiring that the sum of all the eight coordinates be even). There are 240 roots in all.
The 112 roots with integer entries form a D8 root system. The E8 root system also contains a copy of A8 (which has 72 roots) as well as E6 and E7 (in fact, the latter two are usually defined as subsets of E8).
In the odd coordinate system, E8 is given by taking the roots in the even coordinate system and changing the sign of any one coordinate. The roots with integer entries are the same while those with half-integer entries have an odd number of minus signs rather than an even number.
Dynkin diagram
The Dynkin diagram for E8 is given by .
This diagram gives a concise visual summary of the root structure. Each node of this diagram represents a simple root. A line joining two simple roots indicates that they are at an angle of 120° to each other. Two simple roots which are not joined by a line are orthogonal.
Cartan matrix
The Cartan matrix of a rank r root system is an r × r matrix whose entries are derived from the simple roots. Specifically, the entries of the Cartan matrix are given by
$A_{ij}=2{\frac {\left(\alpha _{i},\alpha _{j}\right)}{\left(\alpha _{i},\alpha _{i}\right)}}$
where ( , ) is the Euclidean inner product and αi are the simple roots. The entries are independent of the choice of simple roots (up to ordering).
The Cartan matrix for E8 is given by
$\left[{\begin{array}{rr}2&-1&0&0&0&0&0&0\\-1&2&-1&0&0&0&0&0\\0&-1&2&-1&0&0&0&0\\0&0&-1&2&-1&0&0&0\\0&0&0&-1&2&-1&0&-1\\0&0&0&0&-1&2&-1&0\\0&0&0&0&0&-1&2&0\\0&0&0&0&-1&0&0&2\end{array}}\right].$
The determinant of this matrix is equal to 1.
Simple roots
A set of simple roots for a root system Φ is a set of roots that form a basis for the Euclidean space spanned by Φ with the special property that each root has components with respect to this basis that are either all nonnegative or all nonpositive.
Given the E8 Cartan matrix (above) and a Dynkin diagram node ordering of:
One choice of simple roots is given by the rows of the following matrix:
$\left[{\begin{array}{rr}1&-1&0&0&0&0&0&0\\0&1&-1&0&0&0&0&0\\0&0&1&-1&0&0&0&0\\0&0&0&1&-1&0&0&0\\0&0&0&0&1&-1&0&0\\0&0&0&0&0&1&1&0\\-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}&-{\frac {1}{2}}\\0&0&0&0&0&1&-1&0\\\end{array}}\right].$
Weyl group
The Weyl group of E8 is of order 696729600, and can be described as O+
8
(2): it is of the form 2.G.2 (that is, a stem extension by the cyclic group of order 2 of an extension of the cyclic group of order 2 by a group G) where G is the unique simple group of order 174182400 (which can be described as PSΩ8+(2)).[3]
E8 root lattice
Main article: E8 lattice
The integral span of the E8 root system forms a lattice in R8 naturally called the E8 root lattice. This lattice is rather remarkable in that it is the only (nontrivial) even, unimodular lattice with rank less than 16.
Simple subalgebras of E8
The Lie algebra E8 contains as subalgebras all the exceptional Lie algebras as well as many other important Lie algebras in mathematics and physics. The height of the Lie algebra on the diagram approximately corresponds to the rank of the algebra. A line from an algebra down to a lower algebra indicates that the lower algebra is a subalgebra of the higher algebra.
Chevalley groups of type E8
Chevalley (1955) showed that the points of the (split) algebraic group E8 (see above) over a finite field with q elements form a finite Chevalley group, generally written E8(q), which is simple for any q,[4][5] and constitutes one of the infinite families addressed by the classification of finite simple groups. Its number of elements is given by the formula (sequence A008868 in the OEIS):
$q^{120}\left(q^{30}-1\right)\left(q^{24}-1\right)\left(q^{20}-1\right)\left(q^{18}-1\right)\left(q^{14}-1\right)\left(q^{12}-1\right)\left(q^{8}-1\right)\left(q^{2}-1\right)$
The first term in this sequence, the order of E8(2), namely 337804753143634806261388190614085595079991692242467651576160959909068800000 ≈ 3.38×1074, is already larger than the size of the Monster group. This group E8(2) is the last one described (but without its character table) in the ATLAS of Finite Groups.[6]
The Schur multiplier of E8(q) is trivial, and its outer automorphism group is that of field automorphisms (i.e., cyclic of order f if q=pf where p is prime).
Lusztig (1979) described the unipotent representations of finite groups of type E8.
Subgroups
The smaller exceptional groups E7 and E6 sit inside E8. In the compact group, both E7×SU(2)/(−1,−1) and E6×SU(3)/(Z/3Z) are maximal subgroups of E8.
The 248-dimensional adjoint representation of E8 may be considered in terms of its restricted representation to the first of these subgroups. It transforms under E7×SU(2) as a sum of tensor product representations, which may be labelled as a pair of dimensions as (3,1) + (1,133) + (2,56) (since there is a quotient in the product, these notations may strictly be taken as indicating the infinitesimal (Lie algebra) representations). Since the adjoint representation can be described by the roots together with the generators in the Cartan subalgebra, we may see that decomposition by looking at these. In this description,
• (3,1) consists of the roots (0,0,0,0,0,0,1,−1), (0,0,0,0,0,0,−1,1) and the Cartan generator corresponding to the last dimension;
• (1,133) consists of all roots with (1,1), (−1,−1), (0,0), (−1⁄2,−1⁄2) or (1⁄2,1⁄2) in the last two dimensions, together with the Cartan generators corresponding to the first seven dimensions;
• (2,56) consists of all roots with permutations of (1,0), (−1,0) or (1⁄2,−1⁄2) in the last two dimensions.
The 248-dimensional adjoint representation of E8, when similarly restricted, transforms under E6×SU(3) as: (8,1) + (1,78) + (3,27) + (3,27). We may again see the decomposition by looking at the roots together with the generators in the Cartan subalgebra. In this description,
• (8,1) consists of the roots with permutations of (1,−1,0) in the last three dimensions, together with the Cartan generator corresponding to the last two dimensions;
• (1,78) consists of all roots with (0,0,0), (−1⁄2,−1⁄2,−1⁄2) or (1⁄2,1⁄2,1⁄2) in the last three dimensions, together with the Cartan generators corresponding to the first six dimensions;
• (3,27) consists of all roots with permutations of (1,0,0), (1,1,0) or (−1⁄2,1⁄2,1⁄2) in the last three dimensions.
• (3,27) consists of all roots with permutations of (−1,0,0), (−1,−1,0) or (1⁄2,−1⁄2,−1⁄2) in the last three dimensions.
The finite quasisimple groups that can embed in (the compact form of) E8 were found by Griess & Ryba (1999).
The Dempwolff group is a subgroup of (the compact form of) E8. It is contained in the Thompson sporadic group, which acts on the underlying vector space of the Lie group E8 but does not preserve the Lie bracket. The Thompson group fixes a lattice and does preserve the Lie bracket of this lattice mod 3, giving an embedding of the Thompson group into E8(F3).
Applications
The E8 Lie group has applications in theoretical physics and especially in string theory and supergravity. E8×E8 is the gauge group of one of the two types of heterotic string and is one of two anomaly-free gauge groups that can be coupled to the N = 1 supergravity in ten dimensions. E8 is the U-duality group of supergravity on an eight-torus (in its split form).
One way to incorporate the standard model of particle physics into heterotic string theory is the symmetry breaking of E8 to its maximal subalgebra SU(3)×E6.
In 1982, Michael Freedman used the E8 lattice to construct an example of a topological 4-manifold, the E8 manifold, which has no smooth structure.
Antony Garrett Lisi's incomplete "An Exceptionally Simple Theory of Everything" attempts to describe all known fundamental interactions in physics as part of the E8 Lie algebra.[7][8]
R. Coldea, D. A. Tennant, and E. M. Wheeler et al. (2010) reported an experiment where the electron spins of a cobalt-niobium crystal exhibited, under certain conditions, two of the eight peaks related to E8 that were predicted by Zamolodchikov (1989).[9][10]
History
Wilhelm Killing (1888a, 1888b, 1889, 1890) discovered the complex Lie algebra E8 during his classification of simple compact Lie algebras, though he did not prove its existence, which was first shown by Élie Cartan. Cartan determined that a complex simple Lie algebra of type E8 admits three real forms. Each of them gives rise to a simple Lie group of dimension 248, exactly one of which (as for any complex simple Lie algebra) is compact. Chevalley (1955) introduced algebraic groups and Lie algebras of type E8 over other fields: for example, in the case of finite fields they lead to an infinite family of finite simple groups of Lie type. E8 continues to be an area of active basic research by Atlas of Lie Groups and Representations, which aims to determine the unitary representations of all the Lie groups.[11]
See also
• En
Footnotes
1. Платонов, Владимир П.; Рапинчук, Андрей С. (1991), Алгебраические группы и теория чисел, Наука, ISBN 5-02-014191-7 (English translation: Platonov, Vladimir P.; Rapinchuk, Andrei S. (1994), Algebraic groups and number theory, Academic Press, ISBN 0-12-558180-7), §2.2.4
2. "The 600-Cell (Part 1)". December 16, 2017.
3. Conway, John Horton; Curtis, Robert Turner; Norton, Simon Phillips; Parker, Richard A; Wilson, Robert Arnott (1985), Atlas of Finite Groups: Maximal Subgroups and Ordinary Characters for Simple Groups, Oxford University Press, p. 85, ISBN 0-19-853199-0
4. Carter, Roger W. (1989), Simple Groups of Lie Type, Wiley Classics Library, John Wiley & Sons, ISBN 0-471-50683-4
5. Wilson, Robert A. (2009), The Finite Simple Groups, Graduate Texts in Mathematics, vol. 251, Springer-Verlag, ISBN 978-1-84800-987-5
6. Conway &al, op. cit., p. 235.
7. A. G. Lisi; J. O. Weatherall (2010). "A Geometric Theory of Everything". Scientific American. 303 (6): 54–61. Bibcode:2010SciAm.303f..54L. doi:10.1038/scientificamerican1210-54. PMID 21141358.
8. Greg Boustead (2008-11-17). "Garrett Lisi's Exceptional Approach to Everything". SEED Magazine. Archived from the original on 2009-02-02.{{cite news}}: CS1 maint: unfit URL (link)
9. Shiga, David (2010-01-07). "'Most beautiful' math structure appears in lab for first time". New Scientist. Retrieved 2023-02-01.
10. Did a 1-dimensional magnet detect a 248-dimensional Lie algebra?, Notices of the American Mathematical Society, September 2011.
11. "AIM math: Representations of E8". aimath.org.
References
• Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 978-0-226-00526-3, MR 1428422
• Baez, John C. (2002), "The octonions", Bulletin of the American Mathematical Society, New Series, 39 (2): 145–205, arXiv:math/0105155, doi:10.1090/S0273-0979-01-00934-X, MR 1886087, S2CID 586512
• Chevalley, Claude (1955), "Sur certains groupes simples", The Tohoku Mathematical Journal, Second Series, 7 (1–2): 14–66, doi:10.2748/tmj/1178245104, ISSN 0040-8735, MR 0073602
• Coldea, R.; Tennant, D. A.; Wheeler, E. M.; Wawrzynska, E.; Prabhakaran, D.; Telling, M.; Habicht, K.; Smeibidl, P.; Kiefer, K. (2010), "Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8 Symmetry", Science, 327 (5962): 177–180, arXiv:1103.3694, Bibcode:2010Sci...327..177C, doi:10.1126/science.1180085, PMID 20056884, S2CID 206522808
• Garibaldi, Skip (2016), "E8, the most exceptional group", Bulletin of the American Mathematical Society, 53 (4): 643–671, arXiv:1605.01721, doi:10.1090/bull/1540, S2CID 15810796
• Griess, Robert L.; Ryba, A. J. E. (1999), "Finite simple groups which projectively embed in an exceptional Lie group are classified!", Bulletin of the American Mathematical Society, New Series, 36 (1): 75–93, doi:10.1090/S0273-0979-99-00771-5, MR 1653177
• Killing, Wilhelm (1888a), "Die Zusammensetzung der stetigen endlichen Transformationsgruppen", Mathematische Annalen, 31 (2): 252–290, doi:10.1007/BF01211904, S2CID 120501356
• Killing, Wilhelm (1888b), "Die Zusammensetzung der stetigen endlichen Transformationsgruppen", Mathematische Annalen, 33 (1): 1–48, doi:10.1007/BF01444109, S2CID 124198118
• Killing, Wilhelm (1889), "Die Zusammensetzung der stetigen endlichen Transformationsgruppen", Mathematische Annalen, 34 (1): 57–122, doi:10.1007/BF01446792, S2CID 179177899, archived from the original on 2015-02-21, retrieved 2013-09-12
• Killing, Wilhelm (1890), "Die Zusammensetzung der stetigen endlichen Transformationsgruppen", Mathematische Annalen, 36 (2): 161–189, doi:10.1007/BF01207837, S2CID 179178061
• Landsberg, Joseph M.; Manivel, Laurent (2001), "The projective geometry of Freudenthal's magic square", Journal of Algebra, 239 (2): 477–512, arXiv:math/9908039, doi:10.1006/jabr.2000.8697, MR 1832903, S2CID 16320642
• Lusztig, George (1979), "Unipotent representations of a finite Chevalley group of type E8", The Quarterly Journal of Mathematics, Second Series, 30 (3): 315–338, doi:10.1093/qmath/30.3.301, ISSN 0033-5606, MR 0545068
• Lusztig, George; Vogan, David (1983), "Singularities of closures of K-orbits on flag manifolds", Inventiones Mathematicae, Springer-Verlag, 71 (2): 365–379, Bibcode:1983InMat..71..365L, doi:10.1007/BF01389103, S2CID 120917588
• Zamolodchikov, A. B. (1989), "Integrals of motion and S-matrix of the (scaled) T=Tc Ising model with magnetic field", International Journal of Modern Physics A, 4 (16): 4235–4248, Bibcode:1989IJMPA...4.4235Z, doi:10.1142/S0217751X8900176X, MR 1017357
External links
Lusztig–Vogan polynomial calculation
• Atlas of Lie groups
• Kazhdan–Lusztig–Vogan Polynomials for E8
• Narrative of the Project to compute Kazhdan–Lusztig Polynomials for E8
• American Institute of Mathematics (March 2007), Mathematicians Map E8
• The n-Category Café, a University of Texas blog posting by John Baez on E8.
Other links
• Graphic representation of E8 root system.
• The list of dimensions of irreducible representations of the complex form of E8 is sequence A121732 in the OEIS.
Exceptional Lie groups
• G2
• F4
• E6
• E7
• E8
String theory
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| Wikipedia |
Chapter 3: Kan Complexes
Section 3.2: Homotopy Groups
Subsection 3.2.5: The Long Exact Sequence of a Fibration (cite)
3.2.5 The Long Exact Sequence of a Fibration
If $(X,x)$ is a pointed Kan complex, then we regard each $\pi _{n}(X,x)$ as a pointed set, with base point given by the homotopy class of the constant map $\Delta ^{n} \rightarrow \{ x\} \subseteq X$ (if $n \geq 1$, then this is the identity element with respect to the group structure on $\pi _{n}(X,x)$). Recall that a diagram of pointed sets
\[ \cdots \rightarrow ( G_{n+1}, e_{n+1}) \xrightarrow {f_ n} ( G_ n, e_ n ) \xrightarrow { f_{n-1} } (G_{n-1}, e_{n-1} ) \rightarrow \cdots \]
is said to be exact if the image of each $f_{n}$ is equal to the fiber $f_{n-1}^{-1} \{ e_{n-1} \} = \{ g \in G_ n: f_{n-1}(g) = e_{n-1} \} $. Our goal in this section is to prove the following:
Theorem 3.2.5.1. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes. Then the sequence of pointed sets
\[ \cdots \rightarrow \pi _{2}(S,s) \xrightarrow {\partial } \pi _{1}(X_ s, x) \rightarrow \pi _{1}( X, x) \rightarrow \pi _1(S,s) \xrightarrow {\partial } \pi _{0}(X_ s, x) \rightarrow \pi _0( X,x) \rightarrow \pi _0(S,s) \]
is exact; here $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ denotes the connecting homomorphism of Construction 3.2.4.3.
Theorem 3.2.5.1 really amounts to three separate assertions, which we will formulate and prove individually (Propositions 3.2.5.2, 3.2.5.4, and 3.2.5.6).
Proposition 3.2.5.2. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes and let $n \geq 0$ be an integer. Then the sequence of pointed sets
\[ \pi _{n}(X_ s, x) \rightarrow \pi _{n}(X,x) \rightarrow \pi _{n}(S,s) \]
is exact.
In the special case $n = 0$, the content of Proposition 3.2.5.2 can be formulated without reference to the base point $x \in X$:
Corollary 3.2.5.3. Let $f: X \rightarrow S$ be a Kan fibration between Kan complexes, let $s$ be a vertex of $S$, and set $X_{s} = \{ s\} \times _{S} X$. Then the image of the map $\pi _0(X_{s} ) \rightarrow \pi _0(X)$ is equal to the fiber of the map $\pi _0(f): \pi _0(X) \rightarrow \pi _0(S)$ over the connected component $[s] \in \pi _0(S)$ determined by the vertex $s$. In other words, a vertex $x \in X$ satisfies $[ f(x) ] = [s]$ in $\pi _0(S)$ if and only if the connected component of $x$ has nonempty intersection with the fiber $X_{s}$.
Proof of Proposition 3.2.5.2. Fix an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ is the constant map carrying $\operatorname{\partial \Delta }^{n}$ to the base point $x \in X$. We wish to show that the homotopy class $[\sigma ]$ belongs to the image of the map $\pi _{n}(X_ s, x) \rightarrow \pi _{n}(X,x)$ if and only if the image $[f(\sigma )]$ is equal to the base point of $\pi _{n}(S,s)$. The "only if" direction is clear, since the composite map $X_{s} \hookrightarrow X \xrightarrow {f} S$ is equal to the constant map taking the value $s$. For the converse, suppose that $[ f(\sigma ) ]$ is the base point of $\pi _{n}(S,s)$. Then there exists a homotopy $h: \Delta ^{1} \times \Delta ^{n} \rightarrow S$ from $f(\sigma )$ to the constant map $\sigma '_0: \Delta ^{n} \rightarrow \{ s\} \subseteq S$, which is constant when restricted to the boundary $\operatorname{\partial \Delta }^ n$. Since $f$ is a Kan fibration, we can lift $h$ to a homotopy $\widetilde{h}: \Delta ^{1} \times \Delta ^{n} \rightarrow X$ from $\sigma $ to another $n$-simplex $\sigma ': \Delta ^{n} \rightarrow X$, where $\widetilde{h}$ is constant along the boundary $\operatorname{\partial \Delta }^{n}$ and $f( \sigma ') = \sigma '_0$ (Remark 3.1.5.3). Then $\sigma '$ represents a homotopy class $[\sigma '] \in \pi _{n}(X_ s, x)$, and the homotopy $\widetilde{h}$ witnesses that $[\sigma ]$ is equal to the image of $[\sigma ']$ in $\pi _{n}(X,x)$. $\square$
Proposition 3.2.5.4. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes and let $n \geq 0$ be an integer. Then the sequence of pointed sets $\pi _{n+1}(S,s) \xrightarrow { \partial } \pi _{n}(X_ s, x) \rightarrow \pi _{n}(X,x)$ is exact, where $\partial $ is the connecting homomorphism of Construction 3.2.4.3.
In the special case $n=0$, Proposition 3.2.5.4 can also be formulated without reference to the base point $x \in X$.
Corollary 3.2.5.5. Let $f: X \rightarrow S$ be a Kan fibration between Kan complexes, let $s$ be a vertex of $S$, and set $X_{s} = \{ s\} \times _{S} X$. Then two elements of $\pi _0( X_ s )$ have the same image in $\pi _0( X )$ if and only if they belong to the same orbit of the action of the fundamental group $\pi _{1}(S,s)$ (see Variant 3.2.4.5). In other words, the inclusion of Kan complexes $X_{s} \hookrightarrow X$ induces a monomorphism of sets $(\pi _{1}(S,s) \backslash \pi _{0}(X_ s)) \hookrightarrow \pi _0(X)$.
Proof. Combine Variant 3.2.4.5 with Proposition 3.2.5.4. $\square$
Proof of Proposition 3.2.5.4. Fix an $n$-simplex $\sigma : \Delta ^{n} \rightarrow X_ s$ such that $\sigma |_{ \operatorname{\partial \Delta }^{n} }$ is the constant map carrying $\operatorname{\partial \Delta }^{n}$ to the base point $x \in X_ s$. By construction, the homotopy class $[ \sigma ] \in \pi _{n}(X_ s, x)$ belongs to the image of the connecting homomorphism $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ if and only if there exists an $(n+1)$-simplex $\tau : \Delta ^{n+1} \rightarrow S$ such that $\tau |_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $s$ and $\sigma $ is incident to $\tau $, in the sense of Definition 3.2.4.1. This condition is equivalent to the existence of an $(n+1)$-simplex $\widetilde{\tau }: \Delta ^{n+1} \rightarrow X$ satisfying $d_0( \widetilde{\tau } ) = \sigma $ and $d_ i( \widetilde{\tau } )$ is equal to the constant map $e: \Delta ^{n} \rightarrow \{ x\} \subseteq X$ for $1 \leq i \leq n+1$. In other words, it is equivalent to the assertion that the tuple of $n$-simplices of $X$ $( \sigma , e, e, \ldots , e)$ bounds, in the sense of Notation 3.2.3.1. For $n \geq 1$, this is equivalent to the vanishing of the image of $[\sigma ]$ in the homotopy group $\pi _{n}(X,x)$ (Theorem 3.2.2.10). When $n=0$, it is equivalent to the equality $[\sigma ] = [x]$ in $\pi _0(X)$ by virtue of Remark 1.3.6.13. $\square$
Proposition 3.2.5.6. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes and let $n \geq 0$ be an integer. Then the sequence of pointed sets $\pi _{n+1}(X,x) \xrightarrow { \pi _{n+1}(f)} \pi _{n+1}(S,s) \xrightarrow {\partial } \pi _{n}(X_ s,x)$ is exact, where $\partial $ is the connecting homomorphism of Construction 3.2.4.3.
Corollary 3.2.5.7. Let $f: (X,x) \rightarrow (S,s)$ be a Kan fibration between pointed Kan complexes. Then the image of the induced map $\pi _{1}(f): \pi _1(X,x) \rightarrow \pi _{1}(S,s)$ is equal to the stabilizer of $[x] \in \pi _0(X_ s)$ (with respect to the action of $\pi _{1}(S,s)$ on $\pi _0(X_ s)$ supplied by Variant 3.2.4.5.
Proof of Proposition 3.2.5.6. Fix an $(n+1)$-simplex $\tau : \Delta ^{n+1} \rightarrow S$ for which $\tau |_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $s$. By construction, the connecting homomorphism $\partial : \pi _{n+1}(S,s) \rightarrow \pi _{n}(X_ s, x)$ carries $[ \tau ]$ to the base point of $\pi _{n}(X_ s,x)$ if and only if the constant map $e: \Delta ^{n} \rightarrow \{ x\} \hookrightarrow X_{s}$ is incident to $\tau $, in the sense of Definition 3.2.4.1. This is equivalent to the requirement that $\tau $ can be lifted to a map $\widetilde{\tau }: \Delta ^{n+1} \rightarrow X$ for which $\widetilde{\tau }|_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $x$, which clearly implies that that $[\tau ]$ belongs to the image of the map $\pi _{n+1}(f): \pi _{n+1}(X,x) \rightarrow \pi _{n+1}(S,s)$. To prove the reverse implication, suppose that $[\tau ]$ belongs to the image of $\pi _{n+1}(f)$, so that we can write $[ \tau ] = [ f( \widetilde{\tau }' ) ]$ for some map $\widetilde{\tau }': \Delta ^{n+1} \rightarrow X$ for which $\widetilde{\tau }'|_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $x$. It follows that there is a homotopy $h: \Delta ^{1} \times \Delta ^{n+1} \rightarrow S$ from $f( \widetilde{\tau }' )$ to $\tau $ which is constant along the boundary $\operatorname{\partial \Delta }^{n+1}$. Since $f$ is a Kan fibration, we can lift $h$ to a map $\widetilde{h}: \Delta ^{1} \times \Delta ^{n+1} \rightarrow X$ such that $h|_{ \{ 0\} \times \Delta ^{n+1} } = \widetilde{\tau }'$ and $h|_{ \Delta ^{1} \times \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $x$ (Remark 3.1.5.3). The restriction $\widetilde{\tau } = h|_{ \{ 1\} \times \Delta ^{n+1} }$ then satisfies $f( \widetilde{\tau } ) = \tau $ and $\widetilde{\tau }|_{ \operatorname{\partial \Delta }^{n+1} }$ is the constant map taking the value $x$. $\square$ | CommonCrawl |
\begin{document}
\title{Geodesic and Contour Optimization Using Conformal Mapping}
\author{Ricky Fok \and
Aijun An \and
Xiaogong Wang }
\institute{Ricky Fok \at
Department of Computer Science, York University,
4700 Keele Street, Toronto, M3J 1P3, Canada. \\
\email{[email protected]}
\and
Aijun An \at
Department of Computer Science, York University,
4700 Keele Street, Toronto, M3J 1P3, Canada. \\
\email{[email protected]}
\and
Xiaogong Wang \at
Department of Mathematics and Statistics, York University,
4700 Keele Street, Toronto, M3J 1P3, Canada. \\
\email{[email protected]} }
\date{Received: date / Accepted: date}
\maketitle
\begin{abstract} We propose a novel optimization algorithm for continuous functions using geodesics and contours under conformal mapping.
The algorithm can find multiple optima by first following a geodesic curve to a local optimum then traveling to the next search area by following a contour curve. To improve the efficiency, Newton-Raphson algorithm is also employed in local search steps. A proposed jumping mechanism based on realized geodesics enables the algorithm to jump to a nearby region and consequently avoid trapping in local optima.
Conformal mapping is used to resolve numerical instability associated with solving the classical geodesic equations.
Geodesic flows under conformal mapping are constructed numerically by using local quadratic approximation.
The parameters in the algorithm are adaptively chosen to reflect local geometric features of the objective function.
Comparisons with many commonly used optimization algorithms including gradient, trust region, genetic algorithm and global search methods have shown
that the proposed algorithm outperforms most widely used methods in almost all test cases with only a couple of exceptions.\end{abstract}
\keywords{Contour, geodesic, gradient, jumping mechanism}
\section{Introduction}
\parindent 20pt
Optimization is an essential process in scientific investigation. There are many effective and efficient methods proposed in the literature, see for example \cite{Polak1997}, \cite{MN2002}, \cite{BL2004} and \cite{GS2007}. We propose a new algorithm that solves optimization problems from a point of view that is different than most of the major methods in the literature. The proposed method builds one dimensional paths or curves and travels on it with a constant speed to search for the global optimum.
The main idea of the proposed optimization algorithm is built upon geodesics which are a generalization of straight lines in Euclidean space that minimizes the non-Euclidean distance between two points on a given manifold defined by the objective function. The paths of optimization are constructed numerically by using a local quadratic approximation since there is no analytical solution to the geodesic equation in a general manifold. To avoid numerical instabilities in converting ill-conditioned matrices, the geodesics are constructed on a manifold under conformal mapping which preserves intrinsic geometrical features of the objective function. The algorithm will then follow either the geodesics or contours under conformal mapping to search for the optimum. Each contour curve could provide a bridge to a new search area and the constant speed enforced by the algorithm ensures that the search never stops or be trapped at a stationary point. Even with carefully constructed geodesics, the proposal algorithm can still be trapped within one region. In order to search another promising region, we also build a jumping mechanism by examining the values of the objective function along the geodesics to detect any potential and hidden influence to the geodesic flow from an nearby optimum.
The algorithm can be further improved by integrating with other search methods. For example, one can use a few points along the geodesic as starting points to a Quasi-Newton algorithm to improve computational efficiency. From the Quasi-Newton outputs the algorithm is able to change its parameters adaptively for oscillating and smooth objective functions. We found that the resulting algorithm performs well in both types of functions in moderately high dimensions. Furthermore, we built a stopping criterion for the algorithm using Quasi-Newton methods by setting a threshold on the number of the maximum found within tolerance.
The remaining part of the paper is organized as follows. In Section \ref{DiffGeo} we give an introductory review on differential geometry. Theoretical properties of the proposed algorithm are established in Section \ref{theo}. In Section \ref{Alg}, we give a general description of the algorithm and we also describe the method of choosing the parameters adaptively. Numerical results comparing the proposed algorithm with the Quasi-Newton, genetic algorithm, wedge trust region methods and the global search function in Matlab's global optimization package are provided in Section \ref{results}. Finally, the conclusion is given in Section \ref{conclusion}.
\section{The Main Idea} \label{DiffGeo} We generalize the line search method with geodesics in order to discover multiple maxima on a manifold conformally related to $\mathbb{R}^n$.
\subsection{ Geodesics and Geodesic Equations}
We consider a topological manifold that is a second countable and locally compact Hausdorff space. It is also connected and completely regular. Detailed discussions can be found in \cite{Boothby2003} and \cite{Lee2010}. A Riemannian metric on a smooth and differentiable manifold $M$ is a 2-tensor field ${\cal T}^2(M)$ that is symmetric and positive definite. A Riemannian metric thus determines an inner product on each tangent space $T_p(M)$, which is typically written as $g(U, V)$ for $U,V \in T_p(M)$. For an Euclidean space, the metric matrix (or just metric henceforth) in component form is the Kronecka delta, i.e $g_{ij}=\delta_{ij}$. The inner product $g(U, V)$ in Euclidean reduces to the dot product, $\sum_{ij} \delta_{ij}U^i V^j$, where the sum is over all dimensions. In the Einstein summation convention, it is understood that repeated indices are summed over and the inner product is expressed as $g_{ij}U^i V^j$.
A geodesic is defined to be the path of minimum length for two given distinct points in a connected manifold. It is simply a straight line in Euclidean space. In a non-flat manifold, however, it is a curve and no longer a straight line.
Let $X^i(t)$ denote the local coordinate for the $i$-th dimension for a parameter $t$ which is a time step in our case.
The geodesic is then characterized by a set of partial differential equations, using the Einstein summation convention: \begin{equation} \frac{d^2 X^i(t)}{dt^2} + \Gamma^i_{jk} \frac{d X^j(t)}{dt} \frac{d X^k(t)}{dt} = 0, \label{Geodesic} \end{equation} where $\Gamma^i_{jk}$ are Christoffel symbols defined to be \[ \Gamma^i_{jk} = \frac{1}{2} g^{im}\bigg( \frac{\partial g_{mj}}{\partial x^k} + \frac{\partial g_{mk}}{\partial x^j} - \frac{\partial g_{jk}}{\partial x^m} \bigg), \] $g_{ij}$ is the metric and $g^{ij}$ is the inverse metric.
There exists a unique vector field on the tangent bundle of manifold, denoted as $TM$, whose trajectories are of the form $(\gamma(t), \gamma^{'}(t))$ where $\gamma$ is the geodesic. Geodesics play an important role in General Relativity, see \cite{Foster2006}, where the path of a planet orbiting around a star is the projection of a geodesic of the curved 4-D spacetime geometry around the star onto a 3-D Euclidean space.
\subsection{ Conformal Mapping}
Numerical calculations of the Christoffel symbols involve the inversion of the metric and can be unstable and computationally costly. One strategy to avoid such complications is to calculate the Christoffel symbols in a manifold where the metric is easily inverted and then map the results to the manifold desired. We consider the case where the manifold containing information about the objective function is mapped from $\mathbb{R}^n$, where the metric and the inverse metric is the Kronecka delta. In such a case an analytic expression for the Christoffel symbols is available and the costly matrix inversion is avoided.
Each local neighborhood in the new manifold is holomorphic to $\mathbb{R}^n$. The resulting metric under the conformal mapping is said to be conformally related to the Euclidean metric
\[ g_{ij} = \Psi(x)^2 \delta_{ij}, \] where the scale factor $\Psi(x) = e^{\phi(x)}$ and $\phi(x)$ is a real valued objective function. Trivially, the manifolds obtained this way are Riemannian as the metric tensors are both symmetric and positive definite. The existence of such mappings is trivial since we assume the existence of $\phi(x)$ to begin with.
\section{Theoretical Properties of the Geodesic in a Conformally Flat Metric} \label{theo}
This section investigates the path of the geodesic by considering the direction of its tangent vector and the jumping mechanism used in the algorithm. Unless otherwise stated, the Einstein summation convention is used on all quantities in component form.
\subsection{The Geodesics under Conformal Mapping} \label{attractor} \paragraph{Theorem 3.1} {\bf The level curves and the gradient of the objective function $\phi$ are the attractors of the geodesics on any manifold conformally related to Euclidean space.}
\subsection{Jumping Mechanism} Occasionally, the geodesic can be confined in the neighborhood of a local maximum. Here we discuss a method to estimate the direction of a neighboring maximum from the local maximum using a geodesic so that a jump can be implemented to restart the geodesic along that direction.
Let $l$ be the length of the geodesic, $\gamma$. We define the jumping direction to be along the vector \[ \Delta \mathbf{x} := \frac{1}{l} \int_{\gamma} \widehat{\phi}(\mathbf{x}) \mathbf{x} \ d \mathbf{x} - \frac{1}{l} \int_{\gamma} \mathbf{x} \ d \mathbf{x}, \] where the integral is over the geodesic and $\widehat{\phi}$ being the normalized $\phi$ over $t$, . In practice, this is approximated by the sum over all steps along the geodesic \[ \Delta \mathbf{x} \simeq \frac{1}{T} \sum_{t=1}^{T} [ \widehat{\phi}(\mathbf{x}_t) \mathbf{x}_t - \mathbf{x}_t], \] where $T$ is the total number of steps and $\mathbf{x}_t \in \gamma$. This is just the difference of the weighted mean and the mean position vectors along the geodesic. Suppose that a neighboring maximum exists and the geodesic is symmetric about a local maximum (as it would usually be the case if the geodesic is trapped, for instance, as in Figure \ref{ST}). Then the weighted mean would be slightly biased towards the neighboring maximum. And so $\Delta \mathbf{x}$ would be pointed towards the neighboring maximum. We used a decreasing jump distance for each jump. This is by no means an accurate estimate of the direction to the next maximum. However it has been proven to be sufficient for our algorithm to discover the global maximum in many objective functions of multiple maxima.
\subsection{Solving the Geodesic Equation with a Quadratic Approximation} \label{stepsizes} In this subsection we discuss the quadratic approximation used to solve the geodesic equation iteratively. We give an estimation of the adaptive step sizes to ensure that the approximation is valid. The geodesic equation is \begin{equation} \frac{d^2 x^i(t)}{dt^2} + \Gamma^i_{jk} \frac{d x^j(t)}{dt} \frac{d x^k(t)}{dt} = 0. \label{Geq} \end{equation} In the neighborhood of $\mathbf{x}_t$, the (discretized) approximation to the solution of the geodesic equation is \begin{equation} \label{soln} \mathbf{x}_{t+1} = \mathbf{x}_t + \mathbf{v}_t \delta t + \mathbf{c}_t (\delta t)^2, \end{equation} where $\mathbf{v}_t$ is the unit tangent vector of the geodesic at $\mathbf{x}_t$, $\delta t$ is the step size, and \begin{equation} \mathbf{c}_t = \frac{1}{2} \frac{d \mathbf{v}}{dt} = \frac{1}{2}[\nabla \phi(\mathbf{x}_t) - 2(\mathbf{v}_t \cdot \nabla \phi(\mathbf{x}_t)) \mathbf{v}_t]. \end{equation} The tangent vector is estimated by the (normalized) difference $\mathbf{x}_{t} - \mathbf{x}_{t-1}$ and we set the initial tangent vector to be the gradient, $\mathbf{v}_{t=1} = \nabla \phi(\mathbf{x}_{t=1})$. Note that the quadratic approximation (\ref{soln}) is not valid when there exist a component $i$ such that $O(v_{ti} \delta t) \gg O(c_{ti}(\delta t)^2)$, as the approximation is only accurate when the quadratic term is small. The value of $\delta t$ when the linear term equals to the quadratic term in magnitude is $\delta t = t_C$, where \[
t_C = \min_i \bigg|\frac{v_{ti}}{c_{ti}}\bigg|. \] If the quadratic term has opposite sign to the linear term, then $x^i_{t+1} = x^i_{t}$ when $\delta t = t_C$ for some component $i$. Also, when $\delta t = \frac{1}{2} t_C$, $x^i_{t+1} - x^i_t$ is maximized. Therefore, at every time step, the algorithm chooses a step size of $\delta t = \frac{1}{2} t_C$, if it is not smaller than the specified lower bound on $t$ (to be explained in the algorithm section).
Since the geodesic aligns itself with the gradient (or the level curves) as shown in the Section \ref{attractor}, in both cases, the step sizes are \[
||\mathbf{x}_{t+1} - \mathbf{x}_t || = \frac{3}{4} \frac{1}{| \nabla \phi| }. \] It can be found by substituting $\delta t = \frac{1}{2} t_C$ into Equation \ref{soln} and setting $v_i = 1$ in the component parallel to the gradient (or the level curves). Furthermore, as the geodesic travels towards a maximum following the gradient, it has a linear rate of convergence similar to gradient descent.
\section{Algorithm}
\label{Alg}
The algorithm has two parts. The first is a geodesic guided optimization (GEO) algorithm. It estimates the geodesic using the quadratic approximation for a total of $T$ steps. The step size is adaptive and bounded by the validity of the quadratic approximation. Quasi-Newton (QN) optimization may be performed, using points along the geodesic as the starting points. Figure \ref{cam} shows the estimated geodesic moving through three local maxima. The algorithm returns the location of the maximum and its function value along the geodesic, or the one obtained by QN, whichever is the largest.
\begin{figure}
\caption{Geodesic traversing through multiple local maxima in the search space.}
\label{cam}
\end{figure}
\subsection{Choosing Parameters} There are cases where the geodesic fails to reach multiple maxima, for instance, as in Figure \ref{ST}. When the objective function is highly oscillatory, the global maximum is less likely to be found by the geodesic. Furthermore, a lower bound on the step size $\delta t_{LB}$ must be specified as an input parameter to prevent the step size to become impractically small in regions of large gradient, but the choice of an appropriate lower bound for any objective function is difficult (if not impossible) to determine. Intuitively, a large $\delta t_{LB}$ would allow the geodesic to escape from local fluctuations. On the other hand, it may prevent the geodesic from visiting the global maximum.
The second part of the algorithm, Sequential GEO (SGEO), is developed to overcome these difficulties. Information from the geodesic is obtained and passed to SGEO. It includes an estimate of the direction of a neighboring local maximum, $\widehat{\Delta \mathbf{x}}$, an indicator, $k$, to denote whether the geodesic is trapped in a local maximum, and the average distance between the starting points and the end points of QN, $\bar{R}$, to determine whether the objective function is oscillatory. \begin{figure}
\caption{Geodesic trapped in a local maximum.}
\label{ST}
\end{figure}
SGEO calls GEO sequentially with decreasing $\delta t_{LB}$ for $N$ times. In each subsequent run, $\delta t_{LB}$ is reduced by a factor of $\alpha$, determined by requiring that $\delta t_{LB}$ in the last run to be 1000 times smaller than that in the first run. The next GEO run starts from a position obtained by translating $\mathbf{x^*}$ along $\widehat{\Delta \mathbf{x}}$, with the magnitude and method of the jump determined by $k$. The two mechanisms described above assist in the escape from local maxima. In the case of oscillatory functions, QN is not performed, allowing for a larger number of GEO runs. The algorithm first assumes a non-oscillatory function, and adaptively adjusts its parameters suitable for an oscillatory function by setting a threshold on $\bar{R}$. Finally, we impose a stopping criterion to improve its computational cost. The technical details are described in the follow subsections. The only inputs needed are the number of GEO runs, $N$, total number of steps $N_T$, the stopping threshold, $N_{th}$, which only depends on the dimensionality, and the initial $\delta t_{LB}$ which is only dependent on the dimensionality and the size of the search region.
\subsection{The first component, GEO} GEO estimates the geodesic corresponding to a conformally Euclidean metric with the conformal factor given by the objective function up to $T$ steps and evaluates the objective function $\phi(\mathbf{x}_t)$ at every time step, $t$, along the geodesic.
At each $t$, the normalized tangent vector is used to evaluate $\mathbf{x}_{t+1}$ in Equation \ref{soln}. Since the step size is at most of order $O(1 / \nabla_i \phi)$ as discussed in Section \ref{stepsizes}. The geodesic tends to get trapped in regions of large gradient. To avoid this problem we introduce a lower bound on the step size, $\delta t_{LB}$, so that the step size is $\delta t = \max \{0.5 t_C, \delta t_{LB}\}$. The lower bound also ensures that the geodesic has length of at least $T \delta t$.
Now, consider the case where $t_{LB} \gg t_C$. The trajectory is dominated by the quadratic term $\mathbf{c}_t (\delta t)^2$. But at $t=1$, the tangent vector is the unit gradient and so $\mathbf{c}_{t=1} = - \nabla \phi /2$, the geodesic moves against the gradient. An additional minus sign is introduced to $\mathbf{c}_t$ whenever $t_{LB} > 0.5 t_C$ at $t=1$. A backward geodesic that initially moves against the gradient is also estimated by using the initial condition $\mathbf{v}_{t=1} = - \nabla \phi$.
For both the forward and backward geodesics, Quasi-Newton optimization can be performed at every $T_{QN}$ steps. Whenever $\mathbf{x}_{t+1}$ is outside the search region, the algorithm sets $\mathbf{x}_{t+1}$ to be a random point sampled uniformly within the search region. The following quantities are also evaluated to pass to SGEO, the mean distance between the Quasi-Newton initial position and the solution $\bar{R}$, the normalized mean of $\phi(\mathbf{x}_t)\mathbf{x}_t - \mathbf{x}_t$ over all $t$, and an integer $k \in \{0 , 1 ,2\}$ which characterizes the degree of locality of the geodesics, \[ k = \begin{cases} 0 & \mbox{if $\phi^*_t$ is not unique in the forward geodesic within tolerance $\forall t$. } \\ 1 & \mbox{if $\phi^*_t$ is unique in only the forward geodesic within tolerance $\forall t$.} \\ 2 & \mbox{if $\phi^*_t$ is unique in both geodesics within tolerance $\forall t$,} \end{cases} \] where $\phi^*_t$ is the larger of $\phi(\mathbf{x}_t)$ and the optimized value with QN. The algorithm returns the maximum objective function value and its position along the geodesics as well as the information needed to pass onto SGEO.
\subsection{Algorithm 1: GEO, Geodesic Guided Optimization} Matlab's fminunc() function is used for the Quasi-Newton optimization. Its input parameters are set as $MaxIter = 200, Tol_{f} = 0.05, Tol_X = 0.01$, where the last two quantities are the tolerances in $\phi$ and $\mathbf{x}$, respectively.
The set of input parameters for GEO is
\begin{itemize} \item $T$, the number of time steps along the geodesic, \item $T_{QN}$, the number of time steps between each QN call. \item $\mathbf{x}_0$, a vector the initial point of the geodesic, \item $\phi(\cdot)$, the objective function, \item $\nabla \phi(\cdot)$, the gradient of the objective function, \item $\mathbf{L}$, a vector containing the lower bounds of the search space, \item $\mathbf{U}$, a vector containing the upper bounds of the search space, \item $\delta t_{LB}$, the lower bound on the step size, \item $s_{QN}$, boolean variable denoting whether QN is performed. \end{itemize}
The algorithm is as follows
\begin{enumerate} \item For $t = 1:T$
\begin{enumerate}
\item Calculate $\phi(\mathbf{x}_t)$ and $\nabla \phi(\mathbf{x}_t)$.
\item If $t = 1$, set $\mathbf{x}_t := \mathbf{x}_0$, $k := 1$ and $\delta \mathbf{x}_t := \nabla \phi(\mathbf{x}_t)$,
\begin{enumerate}
\item else, set $\delta \mathbf{x}_t := \mathbf{x}_t - \mathbf{x}_{t-1}$.
\end{enumerate}
\item If $mod(t, T_{QN}) = 0$ and $s_{QN} = 1$,
\begin{enumerate}
\item call $QN(\phi(\cdot), \mathbf{x}_t)$ and obtain $\{\phi^{*(1)}_t, \mathbf{x}^*\}$,
\item set $R^{(1)}_{m} := ||\mathbf{x}^* - \mathbf{x}_t||$ and set $ m := m+1$,
\item else set $\phi^{*(1)}_t := \phi(\mathbf{x}_t)$.
\end{enumerate}
\item Calculate the normalized tangent vector $\mathbf{v}_t := \delta \mathbf{x}_t / ||\delta \mathbf{x}_t||$.
\item Calculate $\mathbf{c}_t := \frac{1}{2}[\nabla \phi(\mathbf{x}_t) - 2(\mathbf{v}_t \cdot \nabla \phi(\mathbf{x}_t))\mathbf{v}_t]$.
\item Set $t_C := \min_i |v_{ti}/c_{ti}|$, where $i \in \{1,\ldots, D\}$ denotes the $i$-th component.
\item Set the step size $\delta t := \max \{\frac{1}{2} t_C, \delta t_{LB}\}$.
\item If $\delta t = \delta t_{LB}$ and $t=1$, change the sign of $\mathbf{c}_t := -\mathbf{c}_t$.
\item Calculate $x_{t+1} := \mathbf{x}_t + \mathbf{v}_t \delta t + \mathbf{c}_t (\delta t)^2$.
\item If $x_{(t+1)i} < L_i$ or $x_{(t+1)i} > U_i$ for any component $i$, sample $\mathbf{x}_{(t+1)}$ from a uniform distribution in $[\mathbf{L}, \mathbf{U}]$.
\item Calculate $\Delta \mathbf{x}^{(1)} := \frac{1}{T}[\sum_{t = 1}^T (\mathbf{x}_t \widehat{\phi}^{*(1)}_t - \mathbf{x}_t)] $, where $\widehat{\phi}^{*(1)}_t = \phi^{*(1)}_t / \sum \phi^{*(1)}_t$
\item Set $\phi^{*(1)} = \max \phi_t^{*(1)}$.
\item If $| \phi^{*(1)} - \phi_t^{*(1)}| < Tol_{f} \phi^{*(1)}$ for all $t$, then set $k =1$. Else set $k = 0 $.
\end{enumerate}
\item For the backward geodesic, step (1) is repeated with the following adjustments,
\begin{enumerate}
\item $\delta \mathbf{x}_t$ is defined as $\delta \mathbf{x}_t := -\nabla \phi(\mathbf{x}_t)$, in step (1b),
\item $\phi^{*(2)}_t := QN(\phi(\cdot), \mathbf{x}_t)$ in step (1c(i)),
\item $R^{(2)}_{m} := ||\mathbf{x}^* - \mathbf{x}_t||$ in step (1c(iii)), and
\item omitting step (1h).
\item $\Delta \mathbf{x}^{(2)} := \frac{1}{T}[\sum_{t = 1}^T (\mathbf{x}_t \widehat{\phi}^{*(2)}_t - \mathbf{x}_t)] $ in step (1k).
\item Set $\phi^{*(2)} = \max \phi_t^{*(2)}$ in step (1l).
\item If $| \phi^{*(2)} - \phi^{*(1)}| < Tol_{f} \phi^{*(2)}$ for all $t$ and $k =1$, set $k = 2 $.
\end{enumerate} \item Set $\Delta \mathbf{x} := \frac{1}{2}[\Delta \mathbf{x}^{(1)} + \Delta \mathbf{x}^{(2)}]$.
\item Return $\phi^* := \max \{\phi^{*(1)}, \phi^{*(2)}\}, \mathbf{x}^* := \{ \mathbf{x} | \phi(\mathbf{x}) = \phi^*\}, \bar{R} := \textrm{mean} \{ \mathbf{R}^{(1)},\mathbf{R}^{(2)}\}$, $\widehat{\Delta \mathbf{x}} := \Delta \mathbf{x} / || \Delta \mathbf{x} ||$ and $k$. \end{enumerate}
\subsection{The second component, SGEO} SGEO runs GEO sequentially with different parameters. Let $\mathbf{U}$ and $\mathbf{L}$ be vectors denoting the upper and lower bounds of the search space and let $\Lambda = \min (\mathbf{U} - \mathbf{L})$. This algorithm checks whether the objective function is highly oscillating. We use the following criterion that an oscillatory function must satisfy: $\bar{R} < 0.1 \Lambda \sqrt{D}$ in any the first two GEO calls. The reason for limiting to just the first two GEO runs is that $\delta t_{LB}$ gets smaller after each consecutive runs and it is more likely for $\bar{R}$ to be small even for non-oscillatory functions. In the case of a high dimensional ($D > 10$) oscillatory function, no Quasi-Newton optimization is performed to allow for a higher number of GEO runs. Both of which are crucial in locating the global optimum of highly oscillating functions.
The algorithm uses a procedure similar to annealing to reduce $\delta t_{LB}$ for each GEO run. Initially, $\delta t_{LB}^{(n=0)}$ is set to be $\Lambda \sqrt{D} / 100$, where $n$ denotes the $n$-th GEO run. Then the lower bound on $\delta t$ is lowered such that $\delta t_{LB}^{n} = \alpha^n \delta t_{LB}^{(n=0)}, \alpha \in (0,1)$. The factor $\alpha$ is chosen such that $\delta t_{LB}^{(n=N)}/ \delta t_{LB}^{(n=0)} = 10^{-3}$.
After each GEO call, the initial value, $\mathbf{x}_0^{(n)}$, for the next GEO call is estimated depending on the value of $k$ passed from GEO. Intuitively, $\Delta \mathbf{x}$ would be a vector pointing roughly towards a neighboring maximum. For $k = 0$, the local geodesic is not trapped, \[ \mathbf{x}_0^{(n+1)} = \mathbf{x}^{*(n)} + (\alpha^{n} \Lambda) \widehat{\Delta \mathbf{x}}^{(n)}. \] For $k=1$, the forward geodesic is trapped and we set $\mathbf{x}_0^{(n+1)}$ to be further away from $\mathbf{x}_0^{(n)}$, \[ \mathbf{x}_0^{(n+1)} = \mathbf{x}^{*(n)} + (\alpha \Lambda) \widehat{\Delta \mathbf{x}}^{(n)}. \] Finally for $k=2$, when both the backward and forward geodesics are trapped, the method using $\Delta \mathbf{x}$ becomes ineffective as the objective function has similar values along the geodesics - $\Delta \mathbf{x}^{(n)}$ points in the same direction as $\mathbf{x}^{(n)}$. Therefore we simply set $\mathbf{x}_0^{(n+1)}$ to be a point reflected across the midpoint of the search space from $\mathbf{x}_0^{(n)}$, \[ \mathbf{x}^{(n+1)}_0 := \frac{\mathbf{L} + \mathbf{U}}{2} + \alpha^n \Lambda(\frac{\mathbf{L} + \mathbf{U}}{2}- \mathbf{x}^{(n)}_0). \]
A stopping criterion is imposed to reduce the computational cost. Let $\Phi^{(n)} = \{\phi^{*(n=1)}, \ldots, \phi^{*(n)} \}$ be a series of maxima found up to the $n$-th GEO run, $\phi^* = \max \Phi^{(n)}$ and $N^*$ be the number of elements in $\Phi^{(n)}$ that are within tolerance of $\phi^*$. The algorithm is stopped if $N^* > N_{th}(D)$, where \[ N_{th}(D) = \begin{cases} 5 & D < 10 \\ 10 & 10 \leq D < 20 \\ 20 & 20 \leq D \leq 50. \end{cases} \]
The algorithm returns $\phi^*$ and $x^*= \{ \mathbf{x} |\phi( \mathbf{x}) = \phi^*\}$.
\subsection{Algorithm 2: SGEO, Sequential Geodesic Optimization} Let $\Lambda= \min (\mathbf{U} - \mathbf{L})$, the set of input parameters is
\begin{itemize} \item $N$, the number of GEO runs, \item $N_T$, total step number, \item $\delta t_{LB}^{(n=0)}$, starting lower bound on the step size,
\item $N_{th}(D)$. \end{itemize}
The algorithm is \begin{enumerate} \item Set $\Lambda := \min (\mathbf{U}-\mathbf{L})$, then set $\{ N, \alpha, N_T, \delta t_{LB}^{(n=0)}, T_{QN},s_{QN}\} = \{ 20, 0.7, 500, \Lambda \sqrt{D} / 100, 10,1\}$. \item Calculate $T := \lfloor \frac{N_T}{N} \rfloor$ and sample $\mathbf{x}_0^{(n=1)}$ uniformly in $[\mathbf{L}, \mathbf{U}]$. \item For $n = 1:N$
\begin{enumerate}
\item Calculate $\delta t^{(n)}_{LB} := \alpha \delta t_{LB}^{(n-1)}$.
\item Obtain $\{\phi^{*(n)}, \mathbf{x}^{*(n)}, \bar{R}, \widehat{\Delta \mathbf{x}}^{(n)},k \} $ by calling GEO($\mathbf{x}^{(n)}_0,\delta t^{(n)}_{LB},s_{QN}, T, T_{QN}$).
\item If $D > 10$, $\bar{R} < 0.1 \Lambda \sqrt{D}$, $n \leq 2$ and $s_{QN} = 1$,
\begin{enumerate}
\item set $\{ N, \alpha, N_T,s_{QN}\} := \{ 400, 0.98, 4000,0\}$ and
\item break and restart current loop with the parameters in the above step in place of those in step 1.
\end{enumerate}
\item If $k = 0$, set $\mathbf{x}^{(n+1)}_0 := \mathbf{x}^{*(n)} + (\alpha^n \Lambda) \widehat{\Delta \mathbf{x}}^{(n)}$. Else if $k = 1$, set $\mathbf{x}^{(n+1)}_0 := \mathbf{x}^{*(n)} + (\alpha \Lambda) \widehat{\Delta \mathbf{x}}^{(n)}$. Else set $\mathbf{x}^{(n+1)}_0 := \frac{\mathbf{L} + \mathbf{U}}{2} + \alpha^n \Lambda(\frac{\mathbf{L} + \mathbf{U}}{2}- \mathbf{x}^{(n)}_0)$.
\item If $n = 1$, set $\phi^* = \phi^{*(n=1)}$. Else set $\phi^* := \max \{ \phi^*, \phi^{*(n)}\}$.
\item Find $N^*$, the number of instances such that $|\phi^* - \phi^{*(n')}| < Tol_{f} \phi^*$, $n' \in \{1, \ldots, n\}$.
\item If $N^* \geq N_{th}(D)$, exit loop.
\end{enumerate}
\item Return $\phi^*$ and $\mathbf{x}^* := \{ \mathbf{x} |\phi( \mathbf{x}) = \phi^*\}$.
\end{enumerate}
\section{Numerical Experiments} \label{results}
In this section we compare SGEO with other algorithms on test functions commonly used in the literature. The objective functions\footnote{These are obtained from http://www.sfu.ca/$\sim$ssurjano/optimization.html. We have used the log of the Hartman and Goldstein-Price functions as the values of these functions vary across five orders of magnitude within the search space.} used have dimensions ranging from 2 to 50. Sixteen of which are reasonably smooth, the other twelve are oscillatory. All calculations are performed in Matlab.
Table \ref{tab:1} shows the number of failures in locating the global maximum for existing methods. A success is defined when the estimated function value is within 5\% of the maximum value. If the maximum value is zero, a success corresponds to finding a function value less than 0.05. As it can be seen in Table \ref{tab:1}, the Global Search (GS) method in Matlab's global optimization toolbox outperforms Quasi-Newton (QN), Trust Region (TR), and Genetic Algorithm (GA).
In Table \ref{tab:2} we justify the use of QN and the jumping mechanism in SGEO. The algorithm without QN fails in high dimensions whereas without jumping the algorithm fails in oscillatory cases.
For finding a global maximum from the chosen test functions, we found that the global search method is the best among all commonly used optimization method. We therefore did an extensive comparison with the global search method in Table \ref{tab:3}. We found that SGEO can discover the true global maximum with a higher chance than GS in the objective functions tested. Our method is more accurate than the global search in many test functions. There is only one function that our method is not as good as the global search. At the same time, the computational cost represented by the number of function calls and the computational time remains similar in most cases.
\begin{table}[ht]
\caption{Number of failures over 100 runs for Quasi-Newton (QN), Trust Region (TR), Genetic Algorithm (GA), and Global Search (GS).} \label{tab:1}
\end{table}
\begin{table}[ht]
\caption{Number of failures over 50 runs for SGEO, without Quasi-Newton (with $T=200$), and without jumping.} \label{tab:2}
\end{table}
\begin{table}[ht]
\caption{The number of failures ($N_{failure}$), computational time, and the number of function calls ($N_{call}$) over 50 runs of SGEO and GS.} \label{tab:3}
\end{table}
\section{Conclusion} \label{conclusion}
A new algorithm is proposed in order to find multiple optima of a continuous objective function. The path constructed by the algorithm follows either a geodesic or a contour line. Conformal mapping and the Newton Raphston algorithm are employed to enhance computational efficiency. A built-in jumping mechanism also directs the proposed algorithm to a more promising search area. A stopping criterion is implemented if the same maximum is found too many times. We are extending this algorithm to handle optimization in high dimensions with contestation.
\section{Acknoledgements} This research is supported by the Discovery Grants and Discovery Accelerator Supplement from Natural Sciences and Engineering Research Council of Canada (NSERC).
\vskip 0.2in
\end{document} | arXiv |
# Lecture Notes for MATH 2040 <br> Mathematical Logic I
Semester 1, 2009/10
Michael Rathjen
## Chapter 0. Introduction
Maybe not all areas of human endeavour, but certainly the sciences presuppose an underlying acceptance of basic principles of logic. They may not have much in common in the way of subject matter or methodology but what they have in common is a certain standard of rationality. It is assumed that the participants can differentiate between rational argumentation based on accepted principles, and wild speculation or total nonsequiturs.
The importance of logic has been recognized since antiquity. Logical principles principles connecting the syntactic structure of sentences with their truth and falsity, their meaning, or the validity of arguments in which they figure - can be found in scattered locations in the work of Plato (428-348 B.C.). The Stoic school of logic was founded some 300 years B.C. by Zeno of Citium (not to be confused with Zeno of Elea). After Zeno's death in 264 B.C., the school was led by Cleanthes, who was followed by Chrysippus. It was largely through the copious writings of Chrysippus that the Stoic school became established, though many of these writings have been lost.
The patterns of reasoning described by Stoic logic are the patterns of interconnection between propositions that are completely independent of what those propositions say. Thus, in Stoic logic, propositions are treated the way atoms are treated in present-day chemistry, where the focus is on the way atoms fit together to form molecules, rather than on the internal structure of atoms. The Stoics commenced their analysis by examining a number of ways in which two propositions can be combined to give a third, more complicated proposition. The operations of forming complex propositions from given propositions are called connectives. The "if ... , then ..." and "or" operations are examples of connectives. The Stoics studied the logic of connectives, also known as propositional logic. An example of a logically valid argument involving the connectives "or" and "not" is
$$
\begin{aligned}
& A \text { or } B . \\
& \text { Not } B
\end{aligned}
$$
## A
The idea is that the third assertion, the one below the line, follows logically from the previous two.
The first known systematic study of logic which involved quantifiers, components such as "for all" and "some", was carried out by Aristotle (384-322 B.C.) whose work was assembled by his students after his death as a treatise called the Organon, the first systematic treatise on logic.
Aristotle tried to analyze logical thinking in terms of simple inference rules called syllogisms. These are rules for deducing one assertion from exactly two others. An example of a syllogism is:
P1. All men are mortal.
$P 2$. Socrates is a man.
## Socrates is mortal.
Again, the idea is that the third assertion follows logically from the previous two. In the case of this simple example, this deduction certainly seems correct enough, albeit pretty obvious. But as with any scientific explanation, the idea is to uncover general patterns that apply in many instances. In the case of the above syllogism, it is obvious that there is a general pattern, namely:
$$
\begin{aligned}
& P 1 . \quad \text { All } M \text { are } P \text {. } \\
& P 2 . \quad S \text { is a } M .
\end{aligned}
$$
$$
\text { C. } S \text { is } P \text {. }
$$
The general deduction rule is true whatever $M, P$ and $S$ may be.
Some of the other syllogisms Aristotle formulated are less obvious. For example the following rule is valid, but you have to think a bit to convince yourself that this is so:
## $P 1$. No $M$ is $P$. <br> $P 2$. Some $S$ is $M$.
## $C$. Some $S$ is not $P$.
Aristotle's proposal was that any logical argument can, in principle, be broken down into a series of applications of a small number of syllogisms involved in logical reasoning. He listed a total of 19, though some on his list were subsequently shown to rely on a tacit assumption to be valid.
As it turned out, the syllogism was found to be too restrictive. There are logically valid arguments that cannot be broken down into a series of syllogisms, now matter how finely those arguments are analyzed. Nevertheless, Aristotle's attempt was one of the first to analyze logical thought, and for that alone his prominent place in history is well deserved. For almost 2000 years Aristotle was revered as the ultimate authority on logical matters.
Bachelors and Masters of arts who do not follow Aristotle's philosophy are subject to a fine of five shillings for each point of divergence, as well as for infractions of the rules of the ORGANON.
- Statuses of the University of Oxford, fourteenth century. However, Aristotle's logic was very weak by modern standards. In particular it is too weak to account for most of the inferences employed in mathematics. In the centuries following Aristotle, various people tried to extend his theory. However, major advances in logic were not achieved until the seventeenth century. The ideas of creating an artificial formal language patterned on mathematical notation in order to clarify logical relationships - called characteristica universalis and of reducing logical inference to a mechanical reasoning process in a purely formal language - called calculus rationatur - were due to Gottfried Wilhelm Leibniz (1646-1716). But logic as we know it today has only emerged over the past 120 years. The name chiefly associated with this emergence is Gottlob Frege (1848 - 1925) who in his Begriffsschrift 1879 (Concept Script) invented the first programming language. His Begriffsschrift marked a turning point in the history of logic. It broke new ground, including a rigorous treatment of quantifiers and the ideas of functions and variables. Frege wanted to show that mathematics grew out of logic, but in so doing devised techniques that took him far beyond the Aristotelian syllogistic and Stoic propositional logic that had come down to him in the logical tradition.
One reason for using a precise mathematical language and grammar is that the statements of mathematics are supposed to be completely precise, and it is not possible to achieve complete precision unless the language one uses is free of many of the vaguenesses and ambiguities of ordinary speech. Mathematical sentences can also be highly complex: if the parts that made them up were not clear and simple, then the unclarities would rapidly accumulate and render the sentences unintelligible. An ambiguity in the English language is exploited by the following old joke that suggests that traditional grammar, with its structuring principle of parts of speech into subjects, predicates, and objects, needs to be radically rethought.
Nothing is better than lifelong happiness.
But a cheese sandwich is better than nothing.
Therefore, a cheese sandwich is better than lifelong happiness.
Let us try to be precise about how this play on words works. It hinges on the word "nothing", which is used in two different ways. The first sentence means "There is no single thing that is better than lifelong happiness", whereas the second means "It is better to have a cheese sandwich than nothing at all." In other words, in the second sentence "nothing" stands for what one might call the null option, the option of having nothing, whereas in the first it does not (to have nothing is not better than to have lifelong happiness).
Words like "all", "some", "any", "every", and "nothing" are examples of quantifiers, and in the English language they are highly prone to this kind of ambiguity.
## The 20th Century
Over the past century the study of logic has undergone rapid and important advances. Spurred on by logical problems in that most deductive discipline, mathematics, it developed into a discipline in its own right, with its own concepts, methods, techniques, and language.
More recently the study of logic has played a major role in the development of modern day computers and programming languages. Logic continues to play an im- portant part in computer science. Some of the central questions of mathematical logic are:
- What is a mathematical proof?
- How can axioms and proofs be justified? (The foundations of mathematics)
- Are there limitations to provability?
- To what extent can machines carry out mathematical proofs?
Only in the last century has there been success in obtaining substantial and satisfactory answers.
## Chapter I. Sets, Functions, Countable sets, Mathematical Induction
Up to a point, one can do and speak mathematics without knowing how to classify the different sorts of words one is using, but many of the sentences of advanced mathematics have a complicated structure that is much easier to understand if one knows a few basic terms from set theory and logic. It is a surprising fact that a small number of set-theoretic concepts and logical terms can be used to provide a precise language that is versatile enough to express all the statements of ordinary mathematics.
### A bit of Naive Set Theory
Goal. Get used to notation, practice some basic proof writing, see the connection with propositional logic.
We introduce naive set theory - the point here is that we do not give formal axioms. Part of the idea is to get used to set theory notation, and to writing proofs - both essential later in the module.
Set theory is the creation of the German mathematician Georg Cantor (1845-1918). What is a set?
1.1 (Naive) Definition A set is a collection of distinct objects, called its elements or members.
If $x$ is an element of the set $A$, we write $x \in A$.
We write $x \notin A$, if $x$ is not an element of $A$.
Remark. Two sets $A$ and $B$ are equal, written $A=B$, if they have the same elements; for example $\{x \in \mathbb{R}: x \geq 0\}=\left\{x \in \mathbb{R}\right.$ : there is a $y \in \mathbb{R}$ with $\left.y^{2}=x\right\}$.
Why did I call the above definition 'naive'?
It can lead to contradictions:
1.2 Russell's Paradox (1901). Some sets are members of themselves. For example, let
$A=$ the set consisting of all sets describable in ten words.
The sentence on the right has ten words, so $A \in A$.
Consider the collection $X$ of sets which are not elements of themselves, i.e. $X=$ $\{x: x$ is a set and $x \notin x\}$. Then consider the question: Is $X \in X$ ?
If $X \in X$, then $X \notin X$ by definition of $X$.
If $X \notin X$, then $X \in X$ by definition of $X$.
So the question can't be answered. That is, we have a contradiction.
This paradox comes in many different guises, for example:
There is a barber who shaves all and only those people who don't shave themselves. Decide whether he shaves himself or not!
These paradoxes showed that set theory had to be put on a solid, axiomatic foundation. This was carried out by Ernst Zermelo in 1908. His axiomatization was later modified by Abraham Fraenkel into what we now call ZF set-theory. Ultimately every mathematical object can be thought of (more precisely 'is') a set, built from the empty set $\emptyset$. For example each natural number is a set as follows:
$$
\begin{aligned}
& 0=\emptyset \\
& 1=\{\emptyset\} \\
& 2=\{\emptyset,\{\emptyset\}\} \\
& 3=\{\emptyset,\{\emptyset\},\{\emptyset,\{\emptyset\}\}\}=\{0,1,2\} \\
& \ldots
\end{aligned}
$$
But not every collection of objects we can think of, can be allowed to be a set,because of Russell's Paradox..
Since this course is not about set theory, we will take the 'naive' approach and work with Definition 1.
1.3 Definition. (a) $A$ is a subset of $B$, written, $A \subseteq B$, if every element in $A$ belongs to $B$. We also say $A$ is contained in $B$. Note $A \subseteq A$.
(b) $A$ is a proper subset of $B$, written $A \subset B$, or $A \nsubseteq B$, if $A \subseteq B$ and $A \neq B$. We also say that $A$ is strictly contained in $B$.
### Examples.
(a) $\emptyset$ is the empty set. Note: $\emptyset \subseteq A$ for every set $A$.
(b) Sets of numbers:
$\mathbb{N}=\{0,1,2,3,4,5 \ldots\}$, the set of natural numbers.
$\mathbb{Z}=\{\ldots,-4,-3,-2,-1,0,1,2,3,4,5, \ldots\}$, the set of integers.
$\mathbb{Q}=\left\{\frac{a}{b}: a, b \in \mathbb{Z}, b \neq 0\right\}$, the set of rational numbers.
$\mathbb{R}=$ 'the set of all decimal numbers' (= the completion of $\mathbb{Q})$.
$\mathbb{C}=$ 'the complex numbers'
(c) The set of all even integers.
(d) The set of all functions from $\mathbb{Q}$ to $\mathbb{R}$, also written as $\mathbb{R}^{\mathbb{Q}}$.
(e) For $n \in \mathbb{N}$, the set $[n]=\{0,1,2,3, \ldots, n-1\}$.
(f) If $A$ is a set, and $P$ is some condition or property then
$$
\{x \in A: x \text { has property } P\}
$$
is a set. For example, $\left\{x \in \mathbb{R}: x^{2}=-1\right\}=\emptyset$.
How do we show that two sets are the same? For example we can use that $A=B$ if and only if $A \subseteq B$ and $B \subseteq A$, i.e. every element in $A$ is in $B$, and every element in $B$ is in $A$.
## Boolean Operations on Sets.
### Definition. Let $A$ and $B$ be sets.
(1) The union $A \cup B=\{x: x \in A$ or $x \in B\}$.
(2) The intersection $A \cap B=\{x: x \in A$ and $x \in B\}$.
(3) The difference $A \backslash B=\{x: x \in A$ and $x \notin B\}$.
(4) $A$ and $B$ are called disjoint, if $A \cap B=\emptyset$. (5) The complement of $A$ in some fixed universe $U$ is $A^{c}=\bar{A}=-A=U-A=$ $\{x \in U: x \notin A\}$.
Visualize by Venn-Diagrams.
1.6 Lemma. Let $A$ and $B$ be sets. Then
(1) $A \cup B=B \cup A$.
(2) $A \cap B=B \cap A$.
(3) $(A \backslash B) \cap(B \backslash A)=\emptyset$.
(4) $(A \backslash B) \cup(B \backslash A)=(A \cup B) \backslash(A \cap B)$.
(5) $A \cup B=(A \cap B) \cup(A \backslash B) \cup(B \backslash A)$.
(6) $A \cup \emptyset=A, A \cap \emptyset=\emptyset$.
(7) De Morgan's Laws:
(i) $(A \cap B)^{c}=A^{c} \cup B^{c}$.
(ii) $(A \cup B)^{c}=A^{c} \cap B^{c}$
Proof. In class and homework.
1.7 Definition. The power set $\mathcal{P}(A)$ of $A$ is the set of all subsets of $A$.
For example, for $A=\{1,2\}, \mathcal{P}(A)=\{\emptyset,\{1\},\{2\},\{1,2\}\}$.
Question: If $A$ has $n$ elements then $\mathcal{P}(A)$ has how many elements?
We will come back to this question later.
If $A$ is a finite set, we will write $|A|=n$ if $A$ has $n$ elements.
1.8 Definition. Let $A$ and $B$ be sets.
(a) The cartesian product of $A$ and $B$, written $A \times B$ is the set
$$
\{(x, y): x \in A, y \in B\} .
$$
The elements $(x, y)$ are called ordered pairs or 2-tuples.
Remark: $(x, y) \neq(y, x)$, unless $x=y$. However: $\{x, y\}=\{y, x\}$.
(b) The set of $k$-tuples over $A$ is $A^{k}=\left\{\left(x_{1}, \ldots, x_{k}\right): x_{i} \in A\right\}$.
Note $A^{2}=A \times A$.
For $A$ and $B$ finite sets, $|A \times B|=|A| \cdot|B|$ and $\left|A^{k}\right|=|A|^{k}$.
1.9 Example. Let $A=\{1,3,5\}$.
The set of ordered pairs over $A$ is
$$
A \times A=\{(1,1),(1,3),(1,5),(3,1),(3,3),(3,5),(5,1),(5,3),(5,5)\} .
$$
The set of 2-element subsets of $A$ is $\{\{1,3\},\{1,5\},\{3,5\}\}$.
## Functions
1.10 (Vague) Definition. A function from a set $A$ to a set $B$ is a rule which assigns to each element $a \in A$ a unique element $b \in B$. Write $f: A \rightarrow B$, with $x \mapsto f(x)$.
But what is a 'rule'?
1.11 (Precise) Definition. A function $f$ from a set $A$ to a set $B$ is a non-empty subset of $A \times B$ such that for each element $a \in A$ there is one and only one pair $(x, y) \in f$ with $x=a$, i.e. for all $a \in A$ there is $b \in B$ with $(a, b) \in f$, and for all $a \in A$, and $b, c \in B$, if $(a, b) \in f$ and $(a, c) \in f$ then $b=c$.
(This set $f$ is also referred to as the graph of $f$. So we identify $f$ with its graph; i.e. writing $f(x)=y$ really means $(x, y) \in f$.
$A$ is called the domain of $f, B$ is called the codomain of $f$.
If $(a, b) \in f$, then we say that $b$ is the image of a under $f$ or the value of $f$ at $a$, and write $b=f(a)$.
The set $f(A)=\{f(a): a \in A\}=\{b \in B:$ there is $a \in A$ with $(a, b) \in f\}$ is called the image of $A$ under $f$, the image of $f$, or the range of $f$.
Example. Consider $f: \mathbb{Z} \rightarrow \mathbb{N}$ with $f(n)=n^{2}$. Then, formally,
$$
f=\{(0,0),(1,1),(-1,1),(2,4),(-2,4),(3,9),(-3,9), \ldots\} .
$$
The domain of $f$ is $\mathbb{Z}$, and the codomain is $\mathbb{N}$. Then the range of $f$ is a proper subset of $\mathbb{N}$, namely Range $(f)=\{0,1,4,9, \ldots\}$.
For $g: \mathbb{R} \rightarrow \mathbb{R}$ with $x \mapsto x^{3}$, the range of $f$ equals the codomain.
## Injections, Surjections, Bijections.
1.12 Definition. A function $f$ from $A$ to $B$ is called injective (an injection, or one-to-one) if for every $b \in B$ there is at most one $a \in A$ with $f(a)=b$.
Equivalently: $f: A \rightarrow B$ is injective if for all $x_{1}, x_{2} \in A, x_{1} \neq x_{2}$ implies $f\left(x_{1}\right) \neq$ $f\left(x_{2}\right)$.
Equivalently: $f: A \rightarrow B$ is injective if for all $x_{1}, x_{2} \in A, f\left(x_{1}\right)=f\left(x_{2}\right)$ implies $x_{1}=x_{2}$. (This last is generally the most convenient form to work with.)
1.13 Examples. Which of the following functions is injective, which is not?
(i) $f: \mathbb{R} \rightarrow \mathbb{R}$, with $f(x)=x^{3}$.
(ii) $g: \mathbb{R} \rightarrow \mathbb{R}$ with $g(x)=x^{2}$
(iii) $h: \mathbb{N} \rightarrow \mathbb{Z}$ with $h(x)=x^{2}$.
(iv) $j: \mathbb{Z} \rightarrow \mathbb{Z}$ with $j(x)=x^{2}$,
(v) $k: \mathbb{N} \rightarrow\{0,1\}$ with $f(n)= \begin{cases}0 & , \text { if } n \text { is even } \\ 1 & , \text { if } n \text { is odd }\end{cases}$
(vi) $l: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$, where $f(n, m)=n+m$.
Ans: Only (i), (iii) are injective. 1.14 Definition. A function $f: A \rightarrow B$ is called surjective (a surjection, or onto) if for all $b \in B$ there is at least one $a \in A$ with $f(a)=b$.
Equivalently: $f: A \rightarrow B$ is surjective if all elements in $B$ are images.
Examples Which of the functions in Examples 1.13 are surjective? Answer: (i), (v), (vi).
1.15 Definition. If $f: A \rightarrow B$ and $g: B \rightarrow C$ are functions, then the composition $h:=g \circ f: A \rightarrow C$ is defined by: $h(x)=g(f(x))$ for all $x \in A$.
1.16 Exercise. In 1.15, if $f, g$ are injective, so is $g \circ f$. If $f, g$ are both surjective, so is $g \circ f$. Prove these facts!
Combining both conditions:
1.17 Definition. A function $f: A \rightarrow B$ is called bijective (a bijection, or a one-to-one correspondence) if $f$ is injective and surjective.
Equivalently: $f: A \rightarrow B$ is bijective if for all $b$ in $B$ there is exactly one $a$ in $A$ with $f(a)=b$.
So a bijection $f: A \rightarrow B$ establishes a one-to-one correspondence between the elements of $A$ and the elements of $B$.
Note. If $f: A \rightarrow B$ is a bijection, then $f^{-1}: B \rightarrow A$ is also a bijection.
1.18 Examples. Each of the following are bijections. Write down their inverses.
(i) Most trivial example: For any set $A$, the function $f: A \rightarrow A$ with $f(a)=a$ for all $a$ is a bijection.
(ii) $f: \mathbb{Z} \rightarrow \mathbb{Z}$ with $f(x)=x+1$.
(iii) $f: \mathbb{Q} \rightarrow \mathbb{Q}$ with $f(x)=2 x$.
(iv) $f:(0,1] \rightarrow[1, \infty)$ with $f(x)=\frac{1}{x}$.
### Cardinality - the sizes of sets
How can we find a mathematically precise way for measuring the 'size' of a set, particularly of an infinite set? Do all infinite sets have the same 'size'?
In this module we will not formally define the size/cardinality $|A|$ of a set $A$. We will only be able to compare sizes of sets, we give a meaning to the expression $|A| \leq|B|$.
For finite sets, it's pretty obvious that we just have to count the number of elements. More formally:
### Definition.
(1) A set $A$ is finite if $A=\emptyset$ or there is a bijection from $A$ to $[n]=\{0,1,2, \ldots, n-1\}$ for some $n \geq 1$. Then we say that $A$ has $n$ elements, and we write $|A|=n$ $(|A|=0$ for $A=\emptyset)$. (2) We say that $A$ is infinite if $A$ is not finite.
(3) Two sets $A$ and $B$ are said to have the same cardinality (the same size) if there is a bijection from $A$ to $B$ (or equivalently from $B$ to $A$ ). In this case, we write $|A|=|B|$ or also $A={ }_{c} B$.
(4) With $|A| \leq|B|$ (or $A \leq_{c} B$ ) we convey that there is an injection from $A$ to $B$.
1.20 Lemma. (i) $|A| \leq|B|$ if and only if $A=\emptyset$ or there is an surjection from $B$ to A.
(ii) $|A| \leq|B|$ and $|B| \leq|C|$ imply $|A| \leq|C|$. Likewise $|A|=|B|$ and $|B|=|C|$ imply $|A|=|C|$.
Proof. For (i) see HW2, Q4. (ii) follows from Exercise 1.16.
1.21 Theorem (Schröder-Bernstein). If $|A| \leq|B|$ and $|B| \leq|A|$ then $|A|=|B|$. This is not obvious: you have to stitch together two injections (or two surjections) to create a bijection. This is beyond this module, but a proof can be found in many set theory books.
Note: We have not defined what the cardinality of $A$ is!
Let's first look at the size of some finite sets:
Question. How many different functions are there from $A=\{a, b, c\}$ to $B=\{0,1\}$ ?
Answer. For each element of $A$ there are 2 possibilities for its image. So there are $2 \cdot 2 \cdot 2=8$ different such functions.
1.22 Lemma. (i) In general, if $|A|=m$ and $|B|=n(m, n \geq 1)$, there are $|B|^{|A|}=n^{m}$ different functions from $A$ to $B$.
The set of all functions from $A$ to $B$ is often denoted by $B^{A}$, and for $A$ and $B$ finite we have $\left|B^{A}\right|=|B|^{|A|}$. See Problem Sheet 2, Q2.
(ii) For $|A|=n,|\mathcal{P}(A)|=2^{n}$.
Proof. (i) For every element in $A$ there are $n$ possibilities for its image.
(ii) By HW2 Q3, there is a bijection
$$
\begin{array}{rlr}
G: \mathcal{P}(A) & \rightarrow\{0,1\}^{A} \\
S & \mapsto f_{S}
\end{array}
$$
Therefore, $|\mathcal{P}(A)|=\left|\{0,1\}^{A}\right|=2^{|A|}=2^{n}$.
Here is a complete proof:
Any subset $S$ of $A$ can be identified with the following function
$$
f_{S}: A \rightarrow\{0,1\} \text { with } f_{S}(a)= \begin{cases}0 & , \text { if } a \notin S \\ 1 & , \text { if } a \in S\end{cases}
$$
The function $f_{S}$ is called the characteristic function of $S$.
In fact, we have a bijection:
$$
\begin{aligned}
G: \mathcal{P}(A) & \rightarrow\{0,1\}^{A} \\
S & \mapsto f_{S}
\end{aligned}
$$
Therefore, $|\mathcal{P}(A)|=\left|\{0,1\}^{A}\right|=2^{|A|}=2^{n}$.
1.23 Question. For $A$ and $B$ finite sets, how many bijections are there from $A$ to $B$ ?
Answer. If $|A| \neq|B|$, there are none. If $|A|=|B|=n \geq 1$, there are $n$ ! bijections from $A$ to $B$. Why?
## How 'big' are infinite sets?
1.24 Definition. A set $S$ is called countably infinite (or has cardinality $\aleph_{0}$, read 'aleph zero' or 'aleph nought') if it has the same cardinality as $\mathbb{N}$, i.e. if there is a bijection from $\mathbb{N}$ to $S$ (or from $S$ to $\mathbb{N}$ ).
A set $S$ is called countable, if $S$ is finite or countably infinite.
### Examples.
(i) Let $E \subseteq \mathbb{N}$ be the set of all even natural numbers, $E=\{0,2,4,6, \ldots\}$. Then $E$ is countably infinite since $f: \mathbb{N} \rightarrow E$ with $f(n)=2 n$ is a bijection.
(ii) Claim. $\mathbb{Z}$, the set of integers, is countably infinite.
Proof. Define $f: \mathbb{N} \rightarrow \mathbb{Z}$ as follows:
$$
f(n)=\left\{\begin{array}{cl}
m & , \text { if } n=2 m \\
-m & , \text { if } n=2 m-1
\end{array}\right.
$$
Describe this function? How are the elements of $\mathbb{Z}$ 'listed'? Show that $f$ is a bijection. $f(0)=0, f(1)=-1, f(2)=1, f(3)=-2, f(4)=2$, etc.
1.26 Theorem. $\mathbb{N} \times \mathbb{N}$ is countably infinite.
Proof. Idea:
$$
\begin{array}{ccccc}
{ }^{j} & 0 & 1 & 2 & \ldots \\
0 & (0,0) & (0,1) & (0,2) & \ldots \\
1 & (1,0) & (1,1) & (1,2) & \ldots \\
2 & (2,0) & (2,1) & (2,2) & \ldots
\end{array}
$$
Note:
1. On every diagonal, the sum of the two coordinates $i+j$ is constant.
2. $1+2+3+\cdots+(i+j)=\frac{(i+j)(i+j+1)}{2}$
So define $f: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ by
$$
f((i, j))=\frac{(i+j)(i+j+1)}{2}+j
$$
Note: Another bijection $f$ from $\mathbb{N} \times \mathbb{N}$ to $\mathbb{N}$ is given by $f(i, j)=2^{i}(2 j+1)-1$. Prove that this works!
1.27 Corollary. $\mathbb{N}^{k}$ has cardinality $\aleph_{0}$ for all $k \geq 1$.
Proof. Idea:
$k=1: \mathbb{N}$ is countably infinite.
$k=2: \mathbb{N} \times \mathbb{N}$ is countably infinite by the previous theorem.
$k=3$ : For $\mathbb{N} \times \mathbb{N} \times \mathbb{N}=(\mathbb{N} \times \mathbb{N}) \times \mathbb{N}$ use the function $f$ from the proof of the previous theorem to define a bijection $g$ from $(\mathbb{N} \times \mathbb{N}) \times \mathbb{N}$ to $\mathbb{N}$ as follows:
$$
g((i, j, k))=f(f(i, j), k)
$$
$k=4$ : keep going.
This proof requires induction on $k$. This will be discussed later.
How about
$$
\mathbb{N} \cup \mathbb{N}^{2} \cup \mathbb{N}^{3} \cup \ldots=\bigcup_{k \geq 1} \mathbb{N}^{k} ?
$$
This is the set of all finite tuples (or sequences) over $\mathbb{N}$. It is a countably infinite union of countably infinite sets.
(Note: Usually one includes the empty sequence as an element in this set and looks at $\bigcup_{k \geq 0} \mathbb{N}^{k}$.)
We will prove that a countable union of countable sets is countable.
We first need two lemmas:
1.28 Lemma. Any subset $A$ of a countable set $S$ is also countable.
Proof. If $A$ is finite, there is nothing to show.
So suppose $A$ is infinite. Then $S$ is infinite and, since $S$ is countable we can write $S=\left\{b_{0}, b_{1}, b_{2}, b_{3}, \ldots\right\}$. Then define the function $f: \mathbb{N} \rightarrow A$ inductively as follows.
$f(0)=$ the element $b_{i}$ with smallest index $i$ with $b_{i} \in A$, and
$f(n+1)=$ the element $b_{i}$ with smallest index $i$ with $b_{i} \in A \backslash\{f(0), \ldots, f(n)\}$. "Clearly" (why?) $f$ defines a bijection from $\mathbb{N}$ to $A$.
1.29 Lemma. Let $A \neq \emptyset$. The following are equivalent:
(i) $A$ is countable.
(ii) There is a surjection from $\mathbb{N}$ to $A$.
(iii) There is a surjection from a countable set $B$ onto $A$.
Proof. The proofs that (i) implies (ii), and that (ii) implies (iii) are trivial. To prove that (iii) implies (i), let $f: B \rightarrow A$ be a surjection with $B$ countable. So $B=\left\{b_{0}, b_{1}, b_{2}, \ldots, b_{n}\right\}$ if $B$ is finite, or $B=\left\{b_{0}, b_{1}, b_{2}, b_{3}, \ldots\right\}$ if $B$ infinite. Define $g: A \rightarrow B$ by letting
$$
g(a)=b_{i}
$$
where $f(a)=b_{i}$ and for all $j<i, f(a) \neq b_{j}$. Clearly $g$ is an injection from $A$ to $B$, and hence $A={ }_{c} g(A)$. Since $g(A)$ is countable by the previous lemma, $A$ is countable.
1.30 Theorem. A countable union of countable sets is countable.
Proof. (Possibly a homework problem.) Let $A_{i}, i \in \mathbb{N}$, each be countable. We want to show that $A=\bigcup_{i \in \mathbb{N}} A_{i}$ (which means $A_{0} \cup A_{1} \cup A_{2} \cup \ldots$ ) is countable. (Note: if this is a finite union many of the $A_{i}$ are the same.) define
By the previous lemma, for each $i \in \mathbb{N}$ there is a surjection $f_{i}: \mathbb{N} \rightarrow A_{i}$. Then
$$
f: \mathbb{N} \times \mathbb{N} \rightarrow A \text { with } f(i, j)=f_{i}(j) .
$$
Clearly $f$ is surjective and $\mathbb{N} \times \mathbb{N}$ is countable, so by the previous lemma, $A$ is countable.
1.31 Theorem. (Cantor) $\mathbb{Q}$ is countable.
Proof. $\mathbb{Q}=\left\{\frac{a}{b}: a, b \in \mathbb{Z}, b \neq 0\right\}$. Define
$$
\begin{array}{ccc}
f: & \mathbb{Z} \times \mathbb{Z} & \rightarrow \mathbb{Q} \\
\text { with } & (a, b) & \mapsto \begin{cases}0 & \text { if } b=0 \\
\frac{a}{b} & \text { if } b \neq 0\end{cases}
\end{array}
$$
Clearly $f$ is onto. Since $\mathbb{Z}$ is countable, $\mathbb{Z} \times \mathbb{Z}$ is countable (Why?). So by Lemma $1.29 \mathbb{Q}$ is countable.
Note: This theorem says that one can list the rational numbers. Note however that it is impossible to list them in their natural order, since between any two rational numbers there are infinitely many other rational numbers. The ordering of this list is very different from the usual ordering.
Question. Are there sets which are not countable?
Answer. Yes, lots of them. First we will to show that the set of real numbers $\mathbb{R}$ is not countable.
Decimal Expansions. One way of representing the real numbers is via decimal expansions:
Every real number can be written in the form $k \cdot d_{1} d_{2} d_{3} \ldots$ where $k$ is an integer and $d_{i} \in\{0,1,2, \ldots, 9\}$.
This representation is unique except for those decimal expansions which are constantly 9 from some point onward (i.e end with recurring 9s).
Example: $0.999 \ldots=0 . \overline{9}=1=1.000 \ldots$
$-5.12999 \ldots=-5.13=-5.13000 \ldots$
We will use only representations which are not constantly 9 from some point onward. So we can assume that the representation is unique.
1.32 Theorem. $\mathbb{R}$ is not countable. (We say $\mathbb{R}$ is uncountable.)
Note that this means that there is no way to make a '(countably) infinitely long list' of all the real numbers.
Proof. Suppose $\mathbb{R}$ is countable. Then $[0,1) \subset \mathbb{R}$ is countable. Then there is a bijection $f: \mathbb{N} \rightarrow[0,1)$. We will show that $f$ is not onto (which is a contradiction): We write each real $f(0), f(1), f(2), \ldots$ in the form
$$
f(i)=0 . d_{0}^{i} d_{1}^{i} d_{2}^{i} \cdots
$$
Let
$$
\begin{aligned}
& f(0)=0 . d_{0}^{0} d_{1}^{0} d_{2}^{0} d_{3}^{0} \cdots \\
& f(1)=0 . d_{0}^{1} d_{1}^{1} d_{2}^{1} d_{3}^{1} \cdots \\
& f(2)=0 . d_{0}^{2} d_{1}^{2} d_{2}^{2} d_{3}^{2} \cdots \\
& f(3)=0 . d_{0}^{3} d_{1}^{3} d_{2}^{3} d_{3}^{3} \cdots
\end{aligned}
$$
using decimal expansions described above without recurring 9s.
Define $r=0 . d_{0}^{*} d_{1}^{*} d_{2}^{*} d_{3}^{*} \ldots$
$$
\text { by } d_{i}^{*}=\left\{\begin{array}{ll}
4 & \text { if } d_{i}^{i} \neq 4 \\
5 & \text { if } d_{i}^{i}=4
\end{array} \text { for all } i \geq 0\right. \text {. }
$$
Example: If
$$
\begin{aligned}
f(0) & =0.132978113 \ldots \\
f(1) & =0.0 \underline{1} 9620013 \ldots \\
f(2) & =0.00 \underline{0} 000000 \ldots \\
f(3) & =0.223 \underline{4} 44444 \ldots \\
f(4) & =0.4239 \underline{2} 8776 \ldots
\end{aligned}
$$
then $r=0.44454 \ldots$
So $r \neq f(i)$ for all $i \in \mathbb{N}$ since $r$ differs from $f(i)$ in the $i$ 'th place. Thus, $r \in[0,1)$, but $r$ is not in the range of $f$. So $f$ is not surjective, which is a contradiction. So our assumption that $\mathbb{R}$ is countable was wrong.
Remark. This type of proof is called a diagonal argument. The proof of the following theorem also uses a diagonal argument (in a more abstract form). 1.33 Theorem (Cantor). For any set $X,|\mathcal{P}(X)| \neq|X|$.
Note: Clearly $|X| \leq|\mathcal{P}(X)|$, since there is an injection $f: X \rightarrow \mathcal{P}(X)$ given by $f(x)=\{x\}$ for all $x \in X$. Thus, the theorem means that the power set of a set $X$ is strictly larger than $X$, i.e. there cannot be a bijection from $X$ to $\mathcal{P}(X)$. So there are arbitrarily large infinite sets, and
$$
|\mathbb{N}|<|\mathcal{P}(\mathbb{N})|<|\mathcal{P}(\mathcal{P}(\mathbb{N}))|<\ldots
$$
Proof of Theorem 1.33. If $X=\emptyset$ the statement is true since $|X|=|\emptyset|=0$ and $|\mathcal{P}(X)|=1($ as $\mathcal{P}(\emptyset)=\{\emptyset\}-$ why?).
So suppose $X \neq \emptyset$, and let $f: X \rightarrow \mathcal{P}(X)$ be a function. We will show that $f$ cannot be surjective.
Define
$$
S=\{x \in X: x \notin f(x)\} .
$$
$S$ is different from all $f(y)$ since they differ on the element $y$. (More precisely, for each $y$,
$$
y \in f(y) \Leftrightarrow y \notin S
$$
so $y \neq f(S)$.)
So $S$ is not in the range of $f$, so $f$ is not surjective. This completes the proof.
1.34 Exercise. Why is the following not a proof of this theorem? The function $f: X \rightarrow \mathcal{P}(X)$ with $f(x)=\{x\}$ for each $x \in X$ is not a surjection. So $|X| \neq|\mathcal{P}(X)|$.
1.35 Question. Where does $|\mathbb{R}|$ fit into the hierarchy
$$
|\mathbb{N}|<|\mathcal{P}(\mathbb{N})|<|\mathcal{P}(\mathcal{P}(\mathbb{N}))|<\ldots ?
$$
1.36 Remarks. (1) For any set $X,|\mathcal{P}(X)|=\left|[2]^{X}\right|$. (Recall: as $[2]:=\{0,1\},[2]^{X}$ is the set of all functions from $X$ to $\{0,1\}$, also denoted by $2^{X}$.)
Proof. For every subset $S$ of $X$ define the function
$$
f_{S}: X \rightarrow\{0,1\} \text { with } f_{S}(x)= \begin{cases}1 & \text { if } x \in S \\ 0 & \text { if } x \notin S\end{cases}
$$
( $f_{S}$ is the characteristic function of $S$ which we defined earlier.)
Then the function $F: \mathcal{P}(X) \rightarrow 2^{X}$ with $F(S)=f_{S}$ is a bijection. Therefore $|\mathcal{P}(X)|=$ $\left|2^{X}\right|$.
(2) One can show that $|\mathbb{R}|=\left|2^{\mathbb{N}}\right|$ by using the fact that every real number has a binary representation (using 0 and 1 ). So $|\mathbb{R}|=|\mathcal{P}(\mathbb{N})|$.
## Mathematical Induction
I will assume some familiarity with mathematical induction.
### Principle of Mathematical Induction.
Let $P(n)$ be a statement about the natural number $n$.
(1) Induction base: $P(0)$ is true, and
(2) Induction step: for all $k \geq 0, P(k)$ implies $P(k+1)$,
## then
$P(n)$ is true for all $n \in \mathbb{N}$.
### Examples.
(i) $\sum_{i=1}^{n} i=\frac{n(n+1)}{2}$. Prove it using induction!
(ii) $|P([n])|=2^{n}$.
We proved this before without induction. Here is the hint for a proof using induction:
$$
\mathcal{P}([n+1])=\mathcal{P}([n]) \cup\{S \cup\{n\}: S \in \mathcal{P}([n])\} .
$$
(iii) The sum of the first $n$ odd numbers is $n^{2}$. That is,
$$
1+3+5+\ldots+(2 n-1)=n^{2}
$$
1.39 Lemma We can start mathematical induction at any natural number:
Let $P(n)$ be a statement about the natural number $n$. Let $n_{0}$ be a fixed natural number. If
(1) Induction base: $P\left(n_{0}\right)$ is true, and
(2) Induction step: for all $k \geq n_{0}, P(k)$ implies $P(k+1)$,
## then
$P(n)$ is true for all $n \geq n_{0}$.
1.40 Example. $2^{n}>n^{3}$ for all $n \geq 10$. Indeed, $2^{10} \geq 10^{3}$ (as $1028>1000$ ); and suppose $2^{n} \geq n^{3}$, with $n \geq 10$. Then $2^{n+1}=2 \times 2^{n} \geq 2 n^{3} \geq n^{3}+3 n^{2}+2 n+1=(n+1)^{3}$. The last inequality holds because $n^{3} \geq 7 n^{2} \geq 3 n^{2}+3 n+1$ for $n \geq 10$.
1.41 Exercise. Show that all people have the same gender.
Proof. Let $n$ be the number of people in the world. If $n=1$, the statement is true. Now let $S=\left\{a_{1}, \ldots, a_{n+1}\right\}$ be the set of people. Since $\left|S \backslash\left\{a_{1}\right\}\right|=n$, by induction hypothesis all people in $S \backslash\left\{a_{1}\right\}$ have the same gender, the same gender as $a_{2}$. Also, since $\left|S \backslash\left\{a_{n+1}\right\}\right|=n$, by induction hypothesis all people in $S \backslash\left\{a_{n+1}\right\}$ have the same gender, again the same gender as $a_{2}$. So all people in $S$ have the same gender as $a_{2}$.
What is wrong with this proof?
The argument does not work when $n=1$. Moral of the story: Make sure that the induction step holds for all relevant $k$.
There are certain situations (frequently in logic, but sometimes also in algebra) where we need
### The Principle of Strong Induction:
Let $P(n)$ be a statement about the natural number $n$.
If
(1) Induction base: $P(0)$ is true, and
(2) Induction step: for all $k \geq 0$, if $P(i)$ is true for all $0 \leq i \leq k$, then $P(k+1)$ is true,
then
$P(n)$ is true for all $n \in \mathbb{N}$.
Note. Though I have formulated all these induction principles so that the inductive step involves an assumption on $k$, in practice we often don't introduce a symbol $k$, but just work directly with $n$.
1.43 Example. Every natural number is a product of prime numbers. (Here, $p \in \mathbb{N}$ is prime if $p \neq 0,1$, and for any expression $p=a b$ where $a, b \in \mathbb{N}$, we have $a=1$ or $b=1)$.
Proof. Let $P(k)$ be the statement that the natural number $k$ is a product of primes. For $n=0$ and $n=1$ the statement is clearly true. Suppose $n \geq 2$ and $P(k)$ holds for all $k<n$. If $n$ is prime, then $P(n)$ holds. Otherwise, there are $a, b \in \mathbb{N}$ with $a, b<n$ and $a b=n$. Since $P(a)$ and $P(b)$ hold, $a$ and $b$ are each a product of primes, and hence (multiplying all these primes together), so is $n$.
## Chapter 2. Propositional Logic
How can we formalize the following arguments?
If demand rises, then companies expand.
If companies expand, then they hire workers.
If demand rises, then companies hire workers
and
This computer program has a bug, or the input is erroneous.
The input is not erroneous.
This computer program has a bug.
If the premises are true, then we cannot avoid the conclusions. The type of the argument in the first example is
$$
\begin{aligned}
& \text { If } A \text {, then } B \text {. } \\
& \text { If } B \text {, then } C \text {. }
\end{aligned}
$$
## If $A$, then $C$.
It doesn't matter what the propositions $A, B$, and $C$ stand for.
The form of the argument in the second example is
$A$ or $B$.
Not $A$.
$B$.
In Propositional Logic, rather than analyzing the meaning of 'atomic' sentences $A, B, C, \ldots$, we explore how more complicated statements are built from $A, B, C, \ldots$, and how their truth depends on that of $A, B, C, \ldots$
Definition 2.1. (Syntax of propositional logic, PROP)
Alphabet and Symbols. The alphabet of the language consists of $A_{1}, A_{2}, A_{3}, \ldots$, called the atomic formulas. We use parentheses '(' and ')', and symbols $\neg, \wedge, \vee, \rightarrow$, which are called logical connectives.
Formulas are defined inductively as follows:
1. All atomic formulas $A_{1}, A_{2}, A_{3}, \ldots$ (or sometimes $A, B, C$ ) are formulas.
2. If $F$ is a formula, then $(\neg F)$ is a formula, called the negation of $F$.
3. If $F$ and $G$ are formulas, then
(i) $(F \vee G)$ is a formula, read as ' $F$ or $G$ ', the disjunction of $F$ and $G$. (ii) $(F \wedge G)$ is a formula, read as ' $F$ and $G$ ', the conjunction of $F$ and $G$.
(iii) $(F \rightarrow G)$ is a formula, read as ' $F$ implies $G$ ', or 'If $F$ then $G$ ', called implication.
The parentheses are needed to make it unambiguous how a formula was formed. When it is unambiguous, we sometimes leave out some of the parenthesis. There adopt certain conventions about omitting parentheses, for example $\neg F \vee G$ means $(\neg F) \vee G$, not $\neg(F \vee G)$.
$(F \leftrightarrow G)$ is an abbreviation for $((F \rightarrow G) \wedge(G \rightarrow F))$.
The set of all formulas is denoted by FOM.
Note 2. To every formula is associated a tree, its parsing tree, which displays how the formula is generated:
For example the formula $\left(\left(\left(\neg A_{2}\right) \rightarrow\left(A_{4} \wedge A_{5}\right)\right) \vee A_{1}\right)$ has the following tree:
$$
\begin{aligned}
& \left(\left(\neg A_{2}\right) \rightarrow\left(A_{4} \wedge A_{5}\right)\right) \vee A_{1} \\
& \left(\neg A_{2}\right) \rightarrow\left(A_{4} \wedge A_{5}\right) \\
& \neg A_{2} \quad A_{4} \wedge A_{5} \\
& \begin{array}{llll}
A_{2} & A_{4} & A_{5} & A_{1}
\end{array}
\end{aligned}
$$
Important: The letters $A_{1}, A_{2}, A_{3}, \ldots$ and $A, B, C, \ldots$ always denote atomic formulas, whereas $F_{1}, F_{2}, F_{3}, \ldots$ and $F, G, H, \ldots$ denote arbitrary formulas.
### Semantics for Propositional Logic.
In formal logic, we keep a clear distinction between:
- syntax: formal languages, formulas, proofs, strings of symbols, etc.
on the one hand and
- semantics: truth values, models, interpretation, meaning, content, etc.
on the other hand.
At this point, formulas are simply strings of symbols with no meaning assigned to them. The following definition associates them with the notion of truth:
Definition 2.3. (Semantics of Propositional Logic) A truth assignment is a function
$$
\mu: F O M \rightarrow\{0,1\}
$$
satisfying the following conditions for all formulas $F, G \in \mathcal{F}$ :
(1) $\mu((\neg F))=1-\mu(F)$. (2) $\mu((F \vee G))=\max \{\mu(F), \mu(G)\}$.
(3) $\mu((F \wedge G))=\mu(F) \cdot \mu(G)$.
(4) $\mu((F \rightarrow G))=\max \{1-\mu(F), \mu(G)\}$.
Here, 0 and 1 are called the truth values, where $\mu(F)=0$ is interpreted as meaning ' $F$ is false' and $\mu(F)=1$ as ' $F$ is true'.
Notice:
$(1)^{\prime} \mu((\neg F))=1$ iff $\mu(F)=0$.
$(2)^{\prime} \mu((F \vee G))=1$ iff $\mu(F)=1$ or $\mu(G)=1$. ('or' always mean 'one or the other or both'.)
$(3)^{\prime} \mu((F \wedge G))=1$ iff $\mu(F)=1$ and $\mu(G)=1$.
$(4)^{\prime} \mu((F \rightarrow G))=1$ iff $\mu(F)=0$ or $\mu(G)=1$, also $(\mu(F \rightarrow G))=0$ iff $\mu(F)=1$ and $\mu(G)=0$.
Example. Let $A, B, C$ be atomic formulas with $\mu(A)=1, \mu(B)=1$, and $\mu(C)=0$. What is $\mu(F)$, where $F=\neg((A \wedge B) \vee C)$ ? (Use the tree associated to this formula. You should find $\mu(F)=0$.)
Remark 2.4. The truth table for implication " $\rightarrow$ " coincides with the meaning of implication in the following scenario, where we make make a promise to a small child.
## If the sun shines on Sunday, we go to the zoo.
Letting $A$ stand for "the sun shines on Sunday" and B for "we go to the zoo", there is only one situation when the child could justifiably complain, namely when the sun shines on Sunday but we don't go to the zoo. This is when $A$ is true but $B$ is false. In all other cases, $A \rightarrow B$ is true, which is in keeping with not breaking our promise.
Lemma 2.5. If $\mu^{\prime}$ is a function from $\left\{A_{1}, A_{2}, \ldots\right\}$ to $\{0,1\}$, then $\mu^{\prime}$ extends uniquely to a truth assignment $\mu$. (So every truth assignment is uniquely determined by its values on $A_{1}, A_{2}, A_{3}, \ldots$ )
This is pretty obvious. But we will write down a precise proof, using induction on the length of the formulas. This is a standard method of proof in logic.
So let's first define the length of a formula.
Definition 2.6. The length $l(F)$ of a formula $F$ is the number of symbols in $F$, counting atomic formulas, logical connectives and parenthesis.
Proof of Lemma 2.5. We have to show:
(a) Existence. If $\mu^{\prime}$ is an arbitrary function from $\left\{A_{1}, A_{2}, \ldots\right\}$ to $\{0,1\}$, then $\mu^{\prime}$ extends to a truth assignment $\mu$.
Prove this by induction on the length of a formula. The details are omitted.
(b) (Uniqueness. If $\mu_{1}$ and $\mu_{2}$ are two truth assignments with $\mu_{1}\left(A_{i}\right)=\mu_{2}\left(A_{i}\right)$ for all $i \geq 1$, then $\mu_{1}=\mu_{2}$. We prove part (b) by induction on the length of formulas. We show that for all $F \in \mathcal{F}$, $\mu_{1}(F)=\mu_{2}(F)$ :
Induction base. $l(F)=1$ means $F=A_{i}$. So $\mu_{1}\left(A_{i}\right)=\mu_{2}\left(A_{i}\right)$ as given.
Induction hypothesis. Let $n \geq 1$. Assume that for all formulas $F$ with $l(F) \leq n$, $\mu_{1}(F)=\mu_{2}(F)$.
Induction step. Let $l(F)=n+1$. Then $F$ is built from simpler formulas (of smaller length) in one of four ways.
(i): $F=(\neg G)$. Since $l(G) \leq n, \mu_{1}(G)=\mu_{2}(G)$. So $\mu_{1}(F)=1-\mu_{1}(G)=1-\mu_{2}(G)=$ $\mu_{2}(F)$.
(ii): $F=(G \vee H)$. Since $l(G) \leq n$ and $l(H) \leq n, \mu_{1}(G)=\mu_{2}(G)$, and $\mu_{1}(H)=\mu_{2}(H)$. So
$$
\mu_{1}(F)=\max \left\{\mu_{1}(G), \mu_{1}(H)\right\}=\max \left\{\mu_{2}(G), \mu_{2}(H)\right\}=\mu_{2}(F) .
$$
(iii): $F=(G \wedge H)$. Fill in yourself.
(iv): $F=(G \rightarrow H)$. Fill in yourself.
Q: Did we use induction or strong induction?
2.7. Truth Tables. This is a way of describing how the truth value of a formula depends on those of its atomic subformulas. First, we give truth tables for the connectives.
| $\mu(A)$ | $\mu(B)$ | $\mu(\neg A)$ | $\mu(A \vee B)$ | $\mu(A \wedge B)$ | $\mu(A \rightarrow B)$ | $\mu(A \leftrightarrow B)$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 0 | 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 1 | 0 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 | 1 |
Note that each line in this truth table represents a different truth assignment.
You could introduce other connectives, like ' $A$ iff $\neg B$ '. Is there any point?
We can write down truth tables for any formula, such as the following one for $(A \wedge B \rightarrow \neg C)$.
| $A$ | $B$ | $C$ | $A \wedge B$ | $\neg C$ | $(A \wedge B) \rightarrow \neg C$ |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 1 | 1 | 1 | 0 | 0 |
| 1 | 1 | 0 | 1 | 1 | 1 |
| 1 | 0 | 1 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 1 | 1 |
| 0 | 1 | 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 1 | 1 |
| 0 | 0 | 1 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 1 |
Write down the truth tables of the following formulas:
(1) $(\neg A) \vee B$
(2) $(\neg A) \rightarrow B$
(3) $(\neg B) \rightarrow(\neg A)$ (4) $\neg(\neg A)$
What do you notice? $(A \rightarrow B) \equiv(1) \equiv(3), A \vee B \equiv(2), A \equiv(4)$.
This motivates the following definition:
Definition 2.8. Two formulas $F$ and $G$ are (semantically) equivalent, if for every truth assignment $\mu, \mu(F)=\mu(G)$. We denote this by $F \equiv G$.
Note. The symbol $\leftrightarrow$ is read as (syntactically) equivalent, whereas $\equiv$ is semantic.
Question. Can formulas containing different atomic formulas be semantically equivalent?
Adequate Sets of Connectives. I raised earlier the question whether $\neg, \wedge, \vee, \rightarrow$ are 'enough'.
Definition 2.9. A set of connectives $\mathcal{C}$ is called adequate (or functionally complete), if every connective can be expressed with connectives from $\mathcal{C}$; more precisely, $\mathcal{C}$ is adequate if for every formula $F$ in $\mathcal{F}$ there is a formula $G$ just using connectives from $\mathcal{C}$ such that $F \equiv G$.
Lemma 2.10. $\{\neg, \wedge\}$ is adequate.
Proof. This is by induction on the length of a formula. The base case is obvious - if $l(F)=1$ then $F$ is atomic so has no connectives. We assume that for all $F \in \mathcal{F}$ with $l(F) \leq n$ there is $G$ just using $\neg, \wedge$ such that $F \equiv G$. Suppose $l(F)=n+1$.
Case (i). $F$ is $\neg F^{\prime}$. Now $l\left(F^{\prime}\right) \leq n$, so by induction there is $G^{\prime}$ just using $\neg, \wedge$ with $F^{\prime} \equiv G^{\prime}$. Then $F \equiv \neg G^{\prime}$, and $\neg G^{\prime}$ just uses $\wedge, \neg$.
Case (ii). $F$ is $F_{1} \wedge F_{2}$ : Exercise.
Case (iii). $F$ is $F_{1} \vee F_{2}$. By induction there are $G_{1}, G_{2}$ just using $\wedge, \neg$ with $F_{1} \equiv G_{1}$ and $F_{2} \equiv G_{2}$. Then $F$ is $F_{1} \vee F_{2} \equiv \neg\left(\neg F_{1} \wedge \neg F_{2}\right) \equiv \neg\left(\neg G_{1} \wedge \neg G_{2}\right)$.
Case (iv). $F$ is $F_{1} \rightarrow F_{2}$. Find $G_{1}, G_{2}$ as in case (iii). Now $F$ is $F_{1} \rightarrow F_{2} \equiv$ $\neg\left(F_{1} \wedge \neg F_{2}\right) \equiv \neg\left(G_{1} \wedge \neg G_{2}\right)$.
Corollary 2.11. $\{\neg, \vee\}$ is adequate.
Proof. Just note $(F \wedge G) \equiv \neg(\neg F \vee \neg G)$.
Question. Is there a single connective which is adequate?
Definition 2.12. (a) The connective $\downarrow$ ('joint denial', 'not or', 'nor') is defined as follows:
$$
\mu(F \downarrow G)=1-\max \{\mu(F), \mu(G)\} .
$$
So its truth table looks like this:
| $F$ | $G$ | $F \downarrow G$ |
| :---: | :---: | :---: |
| 1 | 1 | 0 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
(b) The connective | ('alternative denial', 'not and') is defined as follows:
$$
\mu(F \mid G)=1-\mu(F) \cdot \mu(G) .
$$
So its truth table looks like this:
| $F$ | $G$ | $F \downarrow G$ |
| :---: | :---: | :---: |
| 1 | 1 | 0 |
| 1 | 0 | 1 |
| 0 | 1 | 1 |
| 0 | 0 | 1 |
Proposition 2.13. (a) $\{\downarrow\}$ is adequate.
(b) $\{\mid\}$ is adequate.
Proof. Homework. Note that it suffices to show that $\neg$ and $\wedge$ (or $\neg$ and $\vee$ ) can be expressed using just $\downarrow$ (same for $\mid$ ).
Remark. One can show that $\downarrow$ and $\mid$ are the only connectives which by themselves are adequate.
## Definition 2.14.
(1) A formula $F$ is called satisfiable if there is a truth assignment $\mu$ with $\mu(F)=1$.
(2) If a formula $F$ is not satisfiable, i.e. $\mu(F)=0$ for all truth assignments $\mu$, then $F$ is called unsatisfiable, or contradictory.
(3) A formula $F$ is called valid or a tautology if for every truth assignment $\mu$, we have $\mu(F)=1$.
Examples. $A$ is satisfiable.
$A \wedge \neg A$ is contradictory.
$A \vee \neg A$ is valid.
This can be seen by the truth table below.
| $A$ | $\neg A$ | $A \wedge \neg A$ | $A \vee \neg A$ |
| :---: | :---: | :---: | :---: |
| 1 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 |
Note.
- If $F$ is valid then $F$ is satisfiable. (Prove it!)
- $F$ is a tautology iff $\neg F$ is unsatisfiable. (Prove it!) - It does not make sense to talk about a formula being true, but rather it having truth value 1 under a truth assignment.
Exercise 2.15. Decide whether the following formulas are satisfiable, contradictory, tautologies:
(i) $F_{1}=\neg A \rightarrow(A \rightarrow B)$ (tautology)
(ii) $F_{2}=(A \rightarrow B) \rightarrow(B \rightarrow A)$ (satisfiable, but not valid)
(iii) $F_{3}=(\neg(A \rightarrow B)) \wedge(\neg A \vee B)$ (contradictory)
We check these three through the following truth table.
| $A$ | $B$ | $\neg A$ | $A \rightarrow B$ | $F_{1}$ | $B \rightarrow A$ | $F_{2}$ | $\neg(A \rightarrow B)$ | $\neg A \vee B$ | $F_{3}$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | 1 | 0 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 0 | 0 |
| 0 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 0 |
The truth table checks the truth value of $F_{1}, F_{2}, F_{3}$ under all truth assignments. Since the $F_{1}$ column is all-1, $F_{1}$ is a tautology. Since the $F_{2}$ column has a $1, F_{2}$ is satisfiable. Since the $F_{3}$ column is all- $0, F_{3}$ is contradictory.
(iv) True or false: If $F \rightarrow G$ is satisfiable, and $F$ is satisfiable, then $G$ is satisfiable.
This is false: put $F=A_{1}, G=A_{2} \wedge \neg A_{2}$. Then $F \rightarrow G$ is satisfiable (let $\mu\left(A_{1}\right)=0$ ), and $F$ is satisfiable (let $\mu\left(A_{1}\right)=1$ ), but $G$ is not satisfiable.
(v) True or false: If $F \rightarrow G$ is valid, and $F$ is valid, then $G$ is valid.
This is true. Let $\mu$ be any truth assignment. Then $\mu(F \rightarrow G)=1$ and $\mu(F)=1$ (as these are valid), so $\mu(G)=1$.
Question 2.16. (1) How many formulas are there just using the three atomic formulas $A, B, C$ ?
Ans.: there are countably infinitely many (consider $A, A \wedge A, A \wedge(A \wedge A)$, etc.
(2) How many (semantically) non-equivalent such formulas are there?
This really breaks up into two questions.
(i) How many truth tables are there?
For formulas with $n$ atomic formulas, there are $2^{n}$ possible assignments of truth values to the atomic formulas, and a truth table assigns 0 or 1 to each of these $2^{n}$ choices. So a truth table is really a function $\{0,1\}^{n} \rightarrow\{0,1\}$, so there are $2^{2^{n}}$ truth tables with $n$ atomic subformulas. For example, for $n=4$, there are 65,536 truth tables!
(ii) Is every such truth table (i.e. every function $\{0,1\}^{n} \rightarrow\{0,1\}$ expressible by a propositional formula from $\mathcal{F}$ ? For $n=2$, this is easily checked - there are $2^{2^{2}}=16$ functions. In general, the answer is yes.
I describe now how to obtain a formula, given a truth table. Consider the truth table below.
| $A_{1}$ | $A_{2}$ | $A_{3}$ | $\phi$ |
| :--- | :--- | :--- | :--- |
| 1 | 1 | 1 | 0 |
| 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 1 | 1 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
The formula $\phi$ is true if and only if $A_{1}$ and $A_{2}$ are true and $A_{3}$ is false (line 2), or $A_{1}$ is false and $A_{2}, A_{3}$ true (1.5) or all the $A_{i}$ are false (1.8). So the formula $\phi$ is equivalent to
$$
\left(A_{1} \wedge A_{2} \wedge \neg A_{3}\right) \vee\left(\neg A_{1} \wedge A_{2} \wedge A_{3}\right) \vee\left(\neg A_{1} \wedge \neg A_{2} \wedge \neg A_{3}\right) .
$$
## Normal Forms
Definition 2.17 (a) A literal is an atomic formula (positive literal) or the negation of an atomic formula (negative literal).
(b) A formula $F$ is in conjunctive normal form (CNF) if it is a conjunction of disjunctions of literals, i.e.
$F=\left(L_{11} \vee L_{12} \vee \ldots \vee L_{1 m_{1}}\right) \wedge\left(L_{21} \vee L_{22} \vee \ldots \vee L_{2 m_{2}}\right) \wedge \ldots \wedge\left(L_{n 1} \vee L_{n 2} \vee \ldots \vee L_{n m_{n}}\right)$, for short:
$$
F=\bigwedge_{i=1}^{n}\left(\bigvee_{j=1}^{m_{i}} L_{i, j}\right),
$$
where $L_{i, j} \in\left\{A_{1}, A_{2}, \ldots\right\} \cup\left\{\neg A_{1}, \neg A_{2}, \ldots\right\}$.
(c) A formula $F$ is in disjunctive normal form (DNF) if it is a disjunction of conjunctions of literals, i.e.
$$
F=\bigvee_{i=1}^{n}\left(\bigwedge_{j=1}^{m_{i}} L_{i, j}\right),
$$
where $L_{i, j} \in\left\{A_{1}, A_{2}, \ldots\right\} \cup\left\{\neg A_{1}, \neg A_{2}, \ldots\right\}$.
Note that $\wedge$ and $\vee$ are associative, i.e. $(F \wedge G) \wedge H) \equiv F \wedge(G \wedge H)$ and $(F \vee G) \vee H \equiv$ $F \vee(G \vee H)$, so the expressions above are not ambiguous.
Theorem 2.18. For every formula $F$ there is a formula $C$ in conjunctive normal form and a formula $D$ in disjunctive normal form with $C \equiv F \equiv D$.
Proof. We will use two different methods to prove this:
(1) The truth table method, discussed above, yields a formula in DNF. It can be adapted to get a formula in CNF. For example, consider the formula
$$
\left.\left.\left(A_{1} \wedge A_{2} \wedge \neg A_{3}\right) \vee\left(\neg A_{1}\right) \wedge A_{2} \wedge A_{3}\right) \vee\left(\neg A_{1}\right) \wedge \neg A_{2} \wedge \neg A_{3}\right)
$$
arising from the last truth table considered above. By considering the same truth table, the formula $\neg \phi$ is equivalent to the following one in DNF (obtained by considering the rows where $\phi$ has a 0 :
$\left(A_{1} \wedge A_{2} \wedge A_{3}\right) \vee\left(A_{1} \wedge \neg A_{2} \wedge A_{3}\right) \vee\left(A_{1} \wedge \neg A_{2} \wedge \neg A_{3}\right) \vee\left(\neg A_{1} \wedge A_{2} \wedge \neg A_{3}\right) \vee\left(\neg A_{1} \wedge \neg A_{2} \wedge A_{3}\right)$
Thus, applying De Morgan's Laws to the last formula, $\phi$ is equivalent to the following one:
$\left(\neg A_{1} \vee \neg A_{2} \vee \neg A_{3}\right) \wedge\left(\neg A_{1} \vee A_{2} \vee \neg A_{3}\right) \wedge\left(\neg A_{1} \vee A_{2} \vee A_{3}\right) \wedge\left(A_{1} \vee \neg A_{2} \vee A_{3}\right) \wedge\left(A_{1} \vee A_{2} \vee \neg A_{3}\right)$
(2) Now we will discuss a method for finding $C$ and $D$ without going through the truth table of the formula. We prove that $C$ and $D$ exist, but our proof also gives an algorithm for finding $C$ and $D$.
We need the following facts:
- Let $F$ be a formula, $G$ a subformula of $F$ (i.e. a formula occurring in the tree of $F$ ). If $G_{1} \equiv G$ and $F_{1}$ is the formula obtained from $F$ by replacing $G$ by $G_{1}$, then $F \equiv F_{1}$.
- $\neg(\neg F) \equiv F$
- De Morgan's Laws:
$\neg(F \wedge G) \equiv \neg F \vee \neg G$ (see homework).
$\neg(F \vee G) \equiv \neg F \wedge \neg G$
In general:
$\neg\left(\bigwedge_{i=1}^{n} F_{i}\right) \equiv \bigvee_{i=1}^{n}\left(\neg F_{i}\right)$
$\neg\left(\bigvee_{i=1}^{n} F_{i}\right) \equiv \bigwedge_{i=1}^{n}\left(\neg F_{i}\right)$
(Prove these by induction on $n$ using the equivalences above.)
- Distributive Laws:
$F \wedge(G \vee H) \equiv(F \wedge G) \vee(F \wedge H)$
$F \vee(G \wedge H) \equiv(F \vee G) \wedge(F \vee H)$
These generalize to:
$F \wedge\left(\bigvee G_{i}\right) \equiv \bigvee\left(F \wedge G_{i}\right)$
$F \vee\left(\bigwedge G_{i}\right) \equiv \bigwedge\left(F \vee G_{i}\right)$
- Associative Laws: (which we are using all along when using $\wedge$ and $\bigvee$ )
$F \wedge(G \wedge H) \equiv(F \wedge G) \wedge H$
$F \vee(G \vee H) \equiv(F \vee G) \vee H$ We know that $\{\neg, \vee\}$ is an adequate set of connectives. So we can assume that $F$ only contains these connectives by replacing occurrences of the form $G \rightarrow H$ by $\neg G \vee H$, and occurrences of $G \wedge H$ by $\neg(\neg G \vee \neg H)$. For example, if $F$ is $(A \leftrightarrow B)$, we get
$A \leftrightarrow B \equiv(A \rightarrow B) \wedge(B \rightarrow A) \equiv(\neg A \vee B) \wedge(\neg B \vee A) \equiv \neg(\neg(\neg A \vee B) \vee \neg(\neg B \vee A))$ $F$ :
The proof of Theorem 2.18 is now by induction on the length $l(F)$ of the formula
Induction base. If $F$ is an atomic formula, $F$ is already in CNF and DNF.
Induction step. (1) Let $F=\neg G$. By induction hypothesis, as $l(G)<l(F)$, there are $G_{c}, G_{d}$ so that $G \equiv G_{c}$ with $G_{c}=\bigwedge_{i} \bigvee_{j} L_{i j}$ in CNF, and $G \equiv G_{d}$ with $G_{d}=\bigvee_{i} \bigwedge_{j} K_{i j}$ in DNF. Then
$$
F=\neg G \equiv \neg G_{c}=\neg\left(\bigwedge_{i} \bigvee_{j} L_{i j}\right) \equiv \bigvee_{i} \neg\left(\bigvee_{j} L_{i j}\right) \equiv \bigvee_{i}\left(\bigwedge_{j} \neg L_{i j}\right) \equiv \bigvee_{i} \bigwedge_{j}\left(\overline{L_{i j}}\right)
$$
where $\overline{L_{i j}}=\left\{\begin{array}{ll}\neg A_{k} & \text { if } L_{i j}=A_{k}, \text { a positive literal } \\ A_{k} & \text { if } L_{i j}=\neg A_{k}, \text { a negative literal }\end{array}\right.$.
So $\bigvee_{i} \bigwedge_{j}\left(\overline{L_{i j}}\right)$ is in DNF.
Using $F \equiv \neg G_{d}$ yields a formula in CNF equivalent to $F$. (Fill the details in yourself!)
(2) For $F=G \vee H$. Again, as $l(G), l(H)<l(F)$, there are $G_{c}, G_{d}, H_{c}, H_{d}$ so that $G \equiv G_{c}$ with $G_{c}=\bigwedge_{i} \bigvee_{j} L_{i j}^{1}$ in CNF, and $G \equiv G_{d}$ with $G_{d}=\bigvee_{i} \bigwedge_{j} K_{i j}^{1}$ in DNF, and $H \equiv H_{c}$ with $H_{c}=\bigwedge_{n} \bigvee_{m} L_{n m}^{2}$ in CNF, and $H \equiv H_{d}$ with $H_{d}=\bigvee_{n} \bigwedge_{m} K_{n m}^{2}$ in DNF , we have
$$
F \equiv G_{c} \vee H_{c}=\bigwedge_{i} \bigvee_{j} L_{i j}^{1} \vee \bigwedge_{n} \bigvee_{m} L_{n m}^{2} \equiv \bigwedge_{i, n}\left(\bigvee_{j} L_{i j}^{1} \vee \bigvee_{m} L_{n m}^{2}\right) \text { (distributive law) }
$$
which is in CNF.
Also
$$
F \equiv G_{d} \vee H_{d}=\bigvee_{i} \bigwedge_{j} K_{i j}^{1} \vee \bigvee_{n} \bigwedge_{m} K_{n m}^{2}
$$
which is in DNF.
Example. Put $(\neg A \rightarrow B) \wedge((A \wedge \neg C) \leftrightarrow B)$ in CNF .
Here,
$$
\begin{gathered}
(\neg A \rightarrow B) \wedge((A \wedge \neg C) \leftrightarrow B) \\
\equiv(\neg \neg A \vee B) \wedge[((A \wedge \neg C) \rightarrow B) \wedge(B \rightarrow(A \wedge \neg C))] \\
\equiv(A \vee B) \wedge[(\neg(A \wedge \neg C) \vee B) \wedge(\neg B \vee(A \wedge \neg C))] \\
\equiv(A \vee B) \wedge[((\neg A \vee C) \vee B) \wedge((\neg B \vee A) \wedge(\neg B \vee \neg C))]
\end{gathered}
$$
$$
\equiv(A \vee B) \wedge(\neg A \vee C \vee B) \wedge(\neg B \vee A) \wedge(\neg B \vee \neg C),
$$
which is in CNF.
As another example, the formula $\psi=\left(\neg A_{1} \vee A_{2}\right) \wedge\left(\neg A_{3} \vee A_{4} \vee A_{5}\right) \wedge A_{6}$ which is already in CNF, is equivalent to the following formula in DNF, found using distributivity.
$$
\begin{gathered}
\left(\neg A_{1} \wedge \neg A_{3} \wedge A_{6}\right) \vee\left(\neg A_{1} \wedge A_{4} \wedge A_{6}\right) \vee\left(\neg A_{1} \wedge A_{5} \wedge A_{6}\right) \vee \\
\vee\left(A_{2} \wedge \neg A_{3} \wedge A_{6}\right) \vee\left(A_{2} \wedge A_{4} \wedge A_{6}\right) \vee\left(A_{2} \wedge A_{5} \wedge A_{6}\right) .
\end{gathered}
$$
| Textbooks |
Optimal control problems for a neutral integro-differential system with infinite delay
EECT Home
Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory"
February 2022, 11(1): 169-175. doi: 10.3934/eect.2020106
Some results on the behaviour of transfer functions at the right half plane
Tahir Aliyev Azeroğlu 1, , Bülent Nafi Örnek 2,, and Timur Düzenli 2,3,
Department of Mathematics, Gebze Technical University, Gebze, Kocaeli, Turkey
Amasya University, Technology Faculty, Department of Computer Engineering
Amasya University, Technology Faculty, Department of Electrical and Electronics Engineering, Amasya, Turkey
* Corresponding author: Bülent Nafi Örnek
Received February 2020 Revised September 2020 Published February 2022 Early access December 2020
In this paper, an inequality for a transfer function is obtained assuming that its residues at the poles located on the imaginary axis in the right half plane. In addition, the extremal function of the proposed inequality is obtained by performing sharpness analysis. To interpret the results of analyses in terms of control theory, root-locus curves are plotted. According to the results, marginally and asymptotically stable transfer functions can be determined using the obtained extremal function in the proposed theorem.
Keywords: Positive real function, control system, extremal function, sharpness analysis, marginal stability, transfer function.
Mathematics Subject Classification: Primary: 32A10, 32A05.
Citation: Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2022, 11 (1) : 169-175. doi: 10.3934/eect.2020106
M. Corless, E. Zeheb and R. Shorten, On the SPRification of linear descriptor systems via output feedback, IEEE Transactions on Automatic Control, 64 (2019), 1535-1549. doi: 10.1109/TAC.2018.2849613. Google Scholar
G. Fernández-Anaya, J.-J. Flores-Godoy and J. Álvarez-Ramírez, Preservation of properties in discrete-time systems under substitutions, Asian Journal of Control, 11 (2009), 367-375. doi: 10.1002/asjc.114. Google Scholar
J.-S. Hu and M.-C. Tsai, Robustness analysis of a practical impedance control system, IFAC Proceedings Volumes, 37 (2004), 725-730. doi: 10.1016/S1474-6670(17)31695-6. Google Scholar
S. S. Khilari, Transfer Function and Impulse Response Synthesis Using Classical Techniques, Master Thesis, University of Massachusetts Amherst, 2007. Google Scholar
E. Landau and G. Valiron, A deduction from Schwarz's lemma, Journal of the London Mathematical Society 4, (1929), 162–163. doi: 10.1112/jlms/s1-4.3.162. Google Scholar
M. Liu and et al., On positive realness, negative imaginariness, and h-$\inf$ control of state-space symmetric systems, Automatica J. IFAC, 101 (2019), 190-196. doi: 10.1016/j.automatica.2018.11.031. Google Scholar
F. Mukhtar, Y. Kuznetsov and P. Russer, Network modelling with Brune's synthesis, Advances in Radio Science, 9 (2011), 91-94. doi: 10.5194/ars-9-91-2011. Google Scholar
R. H. Nevanlinna, Eindeutige Analytische Funktionen, Springer-Verlag, Berlin, 1953. Google Scholar
A. Ochoa, Driving point impedance and signal flow graph basics: a systematic approach to circuit analysis, in Feedback in Analog Circuits, Springer, Cham, 2016, 13–34. Google Scholar
Y. Pan, M. J. Er, R. Chen and H. Yu, Output feedback adaptive neural control without seeking spr condition, Asian Journal of Control, 17 (2015), 1620-1630. doi: 10.1002/asjc.966. Google Scholar
F. Reza, A bound for the derivative of positive real functions, SIAM Review, 4 (1962), 40-42. doi: 10.1137/1004005. Google Scholar
M. Şengül, Foster impedance data modeling via singly terminated LC ladder networks, Turkish Journal of Electrical Engineering & Computer Sciences, 21 (2013), 785-792. Google Scholar
A. Sharma and T. Soni, A review on passive network synthesis using cauer form, World J. Wireless Devices Eng., 1 (2017), 39-46. Google Scholar
W. Sun, P. P. Khargonekar and and D. Shim, Solution to the positive real control problem for linear time-invariant systems, IEEE Transactions on Automatic Control, 39 (1994), 2034-2046. doi: 10.1109/9.328822. Google Scholar
M. S. Tavazoei, Passively realisable impedance functions by using two fractional elements and some resistors, IET Circuits, Devices & Systems, 12 (2017), 280-285. doi: 10.1049/iet-cds.2017.0342. Google Scholar
A. D. Wunsch and S.-P. Hu, A closed-form expression for the driving-point impedance of the small inverted L antenna, IEEE Transactions on Antennas and Propagation, 44 (1996), 236-242. doi: 10.1109/8.481653. Google Scholar
C. Xiao and D. J. Hill, Concepts of strict positive realness and the absolute stability problem of continuous-time systems, Automatica J. IFAC, 34 (1998), 1071-1082. doi: 10.1016/S0005-1098(98)00049-1. Google Scholar
Figure 1. Root-locus curves for the transfer function $ H(s) = \sum\limits_{i = 1}^{n}\frac{\alpha _{i}}{s-s_{i}}+i\beta $. It is assumed that $ \alpha_{i} $'s equal to 1 and $ \beta $ is zero. The figures are presented for different $ n $ values: (a) $ n = 1 $, (b) $ n = 2 $, (c) $ n = 3 $, (d) $ n = 4 $
Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321
Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795
Harry L. Johnson, David Russell. Transfer function approach to output specification in certain linear distributed parameter systems. Conference Publications, 2003, 2003 (Special) : 449-458. doi: 10.3934/proc.2003.2003.449
Michal Málek, Peter Raith. Stability of the distribution function for piecewise monotonic maps on the interval. Discrete & Continuous Dynamical Systems, 2018, 38 (5) : 2527-2539. doi: 10.3934/dcds.2018105
Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
Yipeng Chen, Yicheng Liu, Xiao Wang. Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021023
Yuri Latushkin, Alim Sukhtayev. The Evans function and the Weyl-Titchmarsh function. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 939-970. doi: 10.3934/dcdss.2012.5.939
Guozhen Lu, Yunyan Yang. Sharp constant and extremal function for the improved Moser-Trudinger inequality involving $L^p$ norm in two dimension. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 963-979. doi: 10.3934/dcds.2009.25.963
J. William Hoffman. Remarks on the zeta function of a graph. Conference Publications, 2003, 2003 (Special) : 413-422. doi: 10.3934/proc.2003.2003.413
H. N. Mhaskar, T. Poggio. Function approximation by deep networks. Communications on Pure & Applied Analysis, 2020, 19 (8) : 4085-4095. doi: 10.3934/cpaa.2020181
Hassan Emamirad, Philippe Rogeon. Semiclassical limit of Husimi function. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 669-676. doi: 10.3934/dcdss.2013.6.669
Ken Ono. Parity of the partition function. Electronic Research Announcements, 1995, 1: 35-42.
Tomasz Downarowicz, Yonatan Gutman, Dawid Huczek. Rank as a function of measure. Discrete & Continuous Dynamical Systems, 2014, 34 (7) : 2741-2750. doi: 10.3934/dcds.2014.34.2741
Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017
Ariela Briani, Hasnaa Zidani. Characterization of the value function of final state constrained control problems with BV trajectories. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1567-1587. doi: 10.3934/cpaa.2011.10.1567
Nguyen Huy Chieu, Jen-Chih Yao. Subgradients of the optimal value function in a parametric discrete optimal control problem. Journal of Industrial & Management Optimization, 2010, 6 (2) : 401-410. doi: 10.3934/jimo.2010.6.401
Karam Allali, Sanaa Harroudi, Delfim F. M. Torres. Optimal control of an HIV model with a trilinear antibody growth function. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021148
Peter Howard, K. Zumbrun. The Evans function and stability criteria for degenerate viscous shock waves. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 837-855. doi: 10.3934/dcds.2004.10.837
Cruz Vargas-De-León, Alberto d'Onofrio. Global stability of infectious disease models with contact rate as a function of prevalence index. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1019-1033. doi: 10.3934/mbe.2017053
Ramon Plaza, K. Zumbrun. An Evans function approach to spectral stability of small-amplitude shock profiles. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 885-924. doi: 10.3934/dcds.2004.10.885
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Tahir Aliyev Azeroğlu Bülent Nafi Örnek Timur Düzenli | CommonCrawl |
Density-functional fluctuation theory of crowds
J. Felipe Méndez-Valderrama ORCID: orcid.org/0000-0003-3026-89401 na1,
Yunus A. Kinkhabwala ORCID: orcid.org/0000-0003-2320-11822 na1,
Jeffrey Silver ORCID: orcid.org/0000-0002-8453-10303,
Itai Cohen4 &
T. A. Arias ORCID: orcid.org/0000-0001-5880-02604
103 Altmetric
Computational biophysics
Computer modelling
Statistical physics
A primary goal of collective population behavior studies is to determine the rules governing crowd distributions in order to predict future behaviors in new environments. Current top-down modeling approaches describe, instead of predict, specific emergent behaviors, whereas bottom-up approaches must postulate, instead of directly determine, rules for individual behaviors. Here, we employ classical density functional theory (DFT) to quantify, directly from observations of local crowd density, the rules that predict mass behaviors under new circumstances. To demonstrate our theory-based, data-driven approach, we use a model crowd consisting of walking fruit flies and extract two functions that separately describe spatial and social preferences. The resulting theory accurately predicts experimental fly distributions in new environments and provides quantification of the crowd "mood". Should this approach generalize beyond milling crowds, it may find powerful applications in fields ranging from spatial ecology and active matter to demography and economics.
Identifying the role of social interactions and environmental influences on living systems has been the goal of many recent studies of collective population behavior1,2,3,4,5,6,7,8,9,10,11,12,13,14,15. Current agent-based models of crowds can reproduce many emergent behaviors, ranging from random milling to swarming, but often must postulate preconceived rules for individual agent interactions with each other and their environment1,2,3,4,5,6,7,8,9,10. In contrast to such bottom-up approaches, some studies have inferred interaction rules from observations of individual motions within a crowd for a few species of fish11,12, birds13, and insects14,15, but these studies have largely been limited to specific behaviors and have not been developed for making predictions under new circumstances. To date, a general predictive approach to emergent collective behavior in living systems has been lacking.
Such approaches, however, have been developed successfully for large collections of interacting atoms and molecules in the field of statistical physics. One of the central tenants of statistical physics is that generic thermodynamic behaviors emerge from underlying interaction rules among large numbers of particles16,17. Remarkably, these emergent behaviors are often insensitive to the detailed nature of the underlying interactions. Here, we pursue the hypothesis that a similar scenario emerges in the study of large crowds18,19,20,21,22 so that behaviors arising from generic agent-based models can be predicted using a top-down approach. Accordingly, our strategy is to begin with a family of models that roughly capture the "microscopic" behaviors of individuals as they rearrange within a crowd. We do this, not because we are interested directly in individual behaviors, but rather because we are interested in the generic "macroscopic" behaviors that emerge in crowds en masse. This tack is not a priori obvious since active systems do not possess a fixed energy, their temperature is ill-defined, and there are no obvious equilibrium states23. Nonetheless, we show here that mathematical equivalents of free energy, the Hamiltonian, and equilibrium states arise naturally from plausible models of crowd behavior.
In this work, we present the following results. We introduce a general class of plausible agent-based models in which two different functions,"vexation" and "frustration," quantify location and social preferences, respectively. For this class of models, we develop a coarse-grained approach stemming from classical density-functional theory (DFT) that allows us to determine the general mathematical form of the probability distributions describing a crowd. We then discuss the conditions a system must possess to be describable by our theory and test our approach using a living system consisting of walking fruit flies (Drosophila melanogaster), which we confine to a variety of two-dimensional environments. For this fruit-fly system, we successfully extract the vexation and frustration functions corresponding to a variety of different physical settings. Furthermore, these functions are sufficiently stable that, by mixing and matching functions from different experiments, we accurately predict crowd distributions in new environments. Finally, by exposing the fly system to conditions that elicit distinct social motivations, we are able to identify changes in the overall behavior of the crowd, i.e., its "mood," by tracking the evolution of the social preference function.
General mathematical form of crowd-density distributions
Consider, as an example, a crowd at a political rally (Fig. 1a). Under such circumstances, individuals will seek the best locations—presumably closest to the stage—while avoiding overcrowded areas where there is insufficient "personal space." Moreover, individuals will, from time to time, move to new, better locations that become available.
Resulting density-functional approach. a Schematic of crowd in which agents attempt to get as close to the stage as possible while avoiding overcrowding. b In the absence of interactions, the mean of each probability distribution (vertical dashed line) indicates location preference, from which we can extract a bin-dependent vexation functional, vb. c Resulting bin-dependent vexations. d–f Crowds in environments with uniform vexation but with neutral, repulsive, or attractive interactions. For neutral interactions, we expect complete spatial randomness leading to Poisson distributed counts within each bin. The repulsive and attractive interactions are thus reflected in the deviation of the probability distribution from the Poisson form26. From these deviations we can extract a bin-independent frustration functional, fN, whose curvature indicates the nature and intensity of the interaction
A plausible agent-based model of this behavior would assign an intrinsic desirability of each location x through a "vexation" function V(x) that takes its minimum value at the most ideal location near the stage. In addition, it would account for crowding effects through the local crowd areal density n(x) by introducing a "frustration" function f′(n), so that the relative preferablity of location x is actually the sum of vexation and frustration effects, V(x) + f′(n(x)). Finally, this model would include a behavioral rule to account for the tendency for individuals to seek improved locations. When an agent considers a move from location x to x′, the change in the agent's dissatisfaction is ΔH ≡ (V(x′) + f′(n(x′)) − (V(x) + f′(n(x)). A rule where each agent executes such moves with probability 1/(eΔH + 1) captures the intuition that moves that increase the dissatisfaction ΔH > 0 are unlikely, and moves that decrease the dissatisfaction ΔH < 0 are likely, while moves where ΔH = 0 occur with 50% probability. The disadvantage of such an agent based modeling approach is that the rules for each agent are postulated and comparison with experiment requires gathering statistics from repeated simulations, each of which scales as the number of agents or worse. Again, our purpose here is not to develop such a model in detail, but rather to explore the top-level, global behaviors that emerge from this class of models, which we conjecture should apply to crowds more generally.
To extract such global behaviors, we develop a top-down approach by considering the system as a whole and summing the changes in the individual agent dissatisfactions ΔH to obtain a net global population dissatisfaction functional H[n(x)] (Methods). Integrating over dn and area element dA yields
$$H[n({\mathbf{x}})] \equiv F[n({\mathbf{x}})] + {\int} V({\mathbf{x}})n({\mathbf{x}})dA,$$
where the net frustration effect at location x is described by \(f(n) = {\int} f\prime (n){\kern 1pt} dn\), and a local density approximation24,25 \(F[n({\mathbf{x}})] \equiv {\int} f(n({\mathbf{x}})){\kern 1pt} dA\) is in this case sufficient for capturing the crowd behavior. This global functional H[n(x)] and the model described above then lead mathematically to the prediction (Methods) that the probability for observing a crowd arrangement with density n(x) will be given by the probability density functional
$$P[n({\mathbf{x}})] = Z^{ - 1}{\mathrm{exp}}( - H[n({\mathbf{x}})]),$$
where Z is an overall normalization constant. Since we cannot measure the function n(x) directly in experimental crowds, we instead consider discrete counts of individuals within equal area bins (quadrats)26. Thus, to make contact with experiments we discretize Eq. 1 as \(H = \mathop {\sum}\nolimits_b \left( {f_{N_b} + v_bN_b} \right)\), where vb is the average value of the vexation V(x) over bin b, and \(f_{N_b} \equiv f(N_b/A)A\) approximates the total frustration contribution of bin b of area A (Methods). Substituting this discretization into Eq. 2, the overall probability factors into independent distributions for each bin of the form
$$P_b(N) = z_b^{ - 1}\frac{1}{{N!}}\left( {e^{ - v_b}} \right)^Ne^{ - f_N},$$
where zb is a bin-dependent normalization constant and N! accounts for equivalent configurations among the bins (Methods). Thus, we predict that the fluctuations of the bin counts will be statistically independent and follow a modified Poisson form for each bin. This formulation dramatically reduces the complexity of the system description from tracking each individual to tracking the local density in each bin. Additionally, instead of rules with potentially complex interactions for each agent, the global system behavior of the density is determined by just two functions, vb and a bin-independent fN. Because this reduction in the number of variables is the result of transitioning to a local-density description as in classical density-functional theory, but now with the modification that interactions are inferred from density fluctuations, we call our approach density-functional fluctuation theory (DFFT).
Remarkably, rather than postulating these functions, they can be extracted directly from measurements of density distributions in each bin. In particular, in the case of neutral interactions (fN = 0), the bin counts will be single-parameter Poisson distributed, as expected for an experiment counting so-called completely spatially random events26. From the mean of these distributions one can extract an effective vb (Fig. 1b, c), or logarithm of the so-called intensity26, that can arise either from actual preferences for particular locations or from other kinetic interactions with the environment, such as slowing down near barriers27. In the case of interactions, such probability distributions can vary substantially from their non-interacting form (Fig. 1d) when the interactions are included (Fig. 1e, f). For example, so-called contagious distributions, which correspond to attractive interactions and show increased variance-to-mean ratios, have been observed26,28,29. If the interaction is strongly attractive, groups will form, resulting in a bimodal bin probability distribution corresponding to low and high density regions (Fig. 1e), with the high density region constrained by the packing limit. In contrast, highly repulsive interactions (Fig. 1f) lead to more uniform distribution of individuals in the crowd26 and will narrow the bin probability distribution. Finally, from distortions off of the Poisson form, we can determine an effective frustration function fN, without assuming any particular functional form, that describes any local interaction, attractive or repulsive. This formulation holds whether the interaction is directly related to density or to more complex factors such as orientation distributions, as well as higher-order many body interactions (Methods). The power of this approach is that, since vb is tied to the interactions with the environment and fN is tied to inter-agent interactions, it may be possible to combine vexations and frustrations from previous measurements to predict future crowd behaviors.
Several conditions must be met when applying this methodology to crowds under realistic circumstances. For example, the system must be sufficiently ergodic. Thus, the time scales for measurements must be longer than the system decorrelation time. In addition, the agent interactions with their environment should be sufficiently independent of the agent density, the agent interactions should be sufficiently independent of location, and both should be stable over the measurement time. Finally, bin sizes must be appropriately chosen. The bins must be large enough to yield reliable estimates of density, as well as to avoid trivial correlations in neighboring bins, yet small enough that the underlying vexation and local density are nearly constant across each bin.
Extraction of functionals for model system of walking flies
To test whether this approach applies to actual populations, we consider a model crowd consisting of wild-type male Drosophila melanogaster from an out-bred laboratory stock. It is well know that flies exhibit complex spatial preferences30,31 and social behaviors32,33. Here we seek to determine whether a large crowd of individuals with such complex behaviors indeed can be described within our vexation and frustration framework. The flies are confined in 1.5 mm tall transparent chambers where they can walk freely but cannot fly or climb on top of each other. We record overhead videos of the flies, bin the arena, and use custom Matlab-based tracking algorithms (Methods) to measure the individual bin counts Nb in each video frame. To explore a variety of behaviors, we use arenas of different shapes30 and apply heat gradients34 across the arenas to generate different spatial preferences. We find that the flies fully adjust to such changes in their environments after 5 min. We also find that the behavior of the flies changes slowly over a time scale of hours (Methods). We thus take care to make our observations over 10 minute windows during time periods where the behavior is stable.
A top down image of 65 flies in a quasi 1D arena that is uncomfortably heated on the right is shown in Fig. 2a. We find that a bin size of 0.15 cm2, corresponding to the area of approximately 7 flies, ensures that the counts are spatially independent (Fig. 2b) and that the density does not vary substantially over each bin. We also find that the decorrelation time for Nb is about 5 s (Fig. 2c) indicating the system is sufficiently ergodic over the time scale of our observation windows. We show representative probability distributions Pb(N) for a high and a low density bin in Fig. 2d, e, respectively. We find that the distribution peaks are centered at higher N near the left side of the chamber suggesting lower vexation there. Additionally, the high density probability distribution is significantly narrower than the fitted Poisson distribution, hinting that there are repulsive interactions among the flies.
Statistical analysis and extraction of functionals for walking fruit fly experiments. a Single frame of 65 flies walking in a quasi 1D chamber of dimensions 10 cm × 0.8 cm divided into 48 bins with approximate area 0.15 cm2. Heat is applied on the right side of the chamber so that the temperature varies from 35 °C on the left to 50 °C on the right. b Averaged spatial correlation function. c Averaged temporal correlation function. d–e Probability distributions of the number of flies in the two bins outlined in a in red and magenta, respectively. f The "pseudo-free energy," −ln(N!Pb(N)), for eight representative bins. The observed positive curvature indicates deviations from the Poisson form and repulsive interactions. g Frustration functional, fN, obtained from collapse of the pseudo-free energies for all 48 bins upon removal of the Poisson contributions. h Vexation for each bin as measured from the Poisson contributions to the pseudo-free energies. S.d. error bars in d–f computed from Bayesian posterior distribution assuming a Dirichlet prior. S.d. errors bars in g computed from linear propagation of errors displayed in f
To validate our description and quantify the vexations and frustrations, we plot what we call as a mnemonic the "pseudo-free energy" −ln(N!Pb(N)) = (vbN +ln zb) + fN versus N in Fig. 2f. To determine whether the frustration fN is indeed universal, we subtract a linear term corresponding to a bin-dependent vexation and normalization constant, vbN +ln zb, from each curve. Remarkably, the resulting curves can be made to collapse, indicating that a single, universal frustration function fN applies equally well to all bins (Fig. 2g). The positive curvature indicates that higher densities are less preferable than expected from non-interacting populations, and thus indicates repulsive interactions. We also show the bin-dependent vexation values vb used to collapse the curves in Fig. 2h. Finally, as an indicator of the strength of the collapse, we find that modifying the best least-squared fit Poisson distributions by including just eight universal frustration values (f0 through f7) decreases our reduced χ2 value for 166 degrees of freedom from 8.1 to 0.95. Additionally, our DFFT model is favored by the likelihood ratio test with probability p < 0.001 for accepting the hypothesis that the frustration values should be taken to be zero and a vexation-only model be used. This latter test confirms that the aforementioned reduction in χ2 is not a result of overfitting (Methods).
Predictions of crowd density under new circumstances
An important consequence of the physical independence of fN from vb is that it should be possible to use the frustrations extracted from the quasi-1D chamber to predict fly distributions in distinct vexations (Fig. 3). We demonstrate this capability by predicting the measured density distributions for large numbers of flies (on the order of 100) in three distinct geometries and temperature gradients (Fig. 3a). Using measurements of just a few flies in each chamber, we extract density distributions and determine the corresponding vexation vb. Combining this few-fly vexation for each environment with the many-fly frustration fN extracted from the quasi 1D geometry, we predict the fly distributions under dense conditions. Fig. 3b shows this procedure for the stair-case geometry. We find that the individual fly probability distributions (density normalized by total number of flies) for low and high densities are significantly different (Fig. 3c). In contrast, including the interactions through our DFFT approach predicts a more homogeneous population that matches the observed distribution (Fig. 3d). These results demonstrate that, using our DFFT analysis, it is indeed possible to make accurate predictions by combining vexations from low-density experiments in different environments with a frustration that corresponds to a particular behavior ("mood").
Predictions of large crowd distributions in three new environments. a Experimental observations of dense crowds (124, 219, and 189 flies) in three chambers with different geometries, two with applications of heat creating temperature differences of up to 20 °C. b Measured single-fly probability distributions, NAve/NTot. c, d DFFT protocol applied to the stair-case geometry. c Measurement of the density for 3 flies is used to determine the vexation, vb. d Combining this vexation with the extracted quasi 1D frustration from Fig. 2 leads to the high density DFFT prediction. e Comparison of single-fly probabilities for the sparse and dense populations shows significant population shifts as indicated by a correlation coefficient r = 0.73 and a σmean = 3.8. f DFFT analysis that incorporates interactions predicts the measured dense population distribution within statistical uncertainty (r = 0.96 with a σmean = 1.0). Vertical error bars correspond to s.d. of bin-occupation distributions and horizontal error bars correspond to s.e.m. of the observed density within a given bin
Frustration used to quantify the "mood" of a crowd
Conversely, by keeping the environmental conditions fixed and analyzing different time points in the experiments or changing the ratio of male to female flies, the resulting change in "mood" can be quantified by extracting the corresponding functionals. For example, after spending about six hours in the chamber without food or water, the flies exhibit transient groups or clusters of about 10-20 individuals. This change in behavior is quantified by the different curvatures for the frustrations fN characterizing the initial (blue curve) and deprived states (red curve) in Fig. 4. The nearly flat frustration associated with this behavior indicates that male flies are willing to surmount their natural repulsion and form higher density groups under deprivation conditions, a previously undocumented spontaneous self-organized change in collective behavior31,32,35. Attraction between individuals can be induced by introducing female flies. For groups of flies with equal numbers of males and females which have been separated for several days, we find pair formation (yellow ellipses). This behavior is characterized by a sharp downward curvature in the frustration at low N (yellow curve). Exposing this population to similar deprivation conditions drives formation of larger groups (purple circle) at the expense of pair formation. This behavior is captured by the shift of downward curvature in the frustration to larger bin occupations of N≈7 (purple curve). These data establish that the DFFT approach has the power to detect and quantify changes in social behaviors.
Extracting frustrations to quantify changing behavior. Frustrations measured for flies in a 4 cm square chamber. The experiment duration was seven hours. The frustrations were extracted from two different 10 minute intervals corresponding to the initial and final stages of experiments on two different populations. The blue curve (90♂) exhibits a positive curvature at all occupancies, indicating an aversion to crowding at all densities. The red curve characterizes interactions for the same population 6 hours later. The lower curvature indicates significantly reduced aversion to grouping. The yellow curve (30♂ + 25♀) exhibits a downward curvature at low occupations, reflecting mating interactions between pairs of flies (yellow ellipses). At higher occupancies, the lack of curvature indicates a more neutral response to changes in occupation number. Finally, the purple curve characterizes interactions for the same mixed-sex population 6 hours later. The downward curvature shifts to higher occupancies and is followed by a region of positive curvature. The corresponding inflection point indicates a preference for group formation with a density of about eight flies per bin. S.d. error bars calculated from the maximum likelihood (ML) covariance matrix of DFFT distribution in Eq. 3
Collectively, these results demonstrate that top-down approaches are a promising method for predicting crowd distributions and quantifying crowd behaviors. The DFFT analysis that we present is particularly powerful because it separates the influence of the environment on agents from interactions among those agents. This separation then enables predictions of crowd distributions in new situations through mixing and matching of the vexations and frustrations from previous observations in different scenarios. In addition, the real-time quantification of frustrations opens the door to tracking behavioral changes and potentially extrapolating the time evolution of frustrations to anticipate future behaviors.
There are a number of directions in which the formal framework suggested here can be extended, paralleling developments from the traditional density-functional theory literature. Extensions to time-dependent DFT methods (TDDFT)36,37 would enable the prediction of situations in which crowds gather and disperse in response to changes in the environment. This approach would also apply to situations in which the center of mass of the entire group is moving as whole, such as in herd migration and bacterial and insect swarming. Moreover, by including the local current density ("flow") in the functional, such approaches may even be able to describe crowds where correlated subgroups move with different local velocities, such as in flocks of birds. Likewise, extensions to multicomponent DFT38 would enable corresponding predictions and observations in crowds composed of distinct groups exhibiting interactions such as inter-group conflict, predator-prey relations, or mating behavior.
Should these results extend to human populations, the implications are profound. From publicly available video data of people milling in public spaces, this approach could predict how people would distribute themselves under extreme crowding. Additionally, a simple application running on a hand-held device could easily measure density fluctuations and extract functionals that are indicative of the current behavioral state or mood of the crowd. Through comparison with a library of functionals measured from past events, such an application could provide early warning as a crowd evolves towards a dangerous behavior. Finally, given the recent proliferation of newly available cell-phone and census data39,40 these approaches may also extend to population flows on larger scales, such as migration. Here, vexations could correspond to political or environmental drivers and frustrations to population pressures. The resulting predictions of migration during acute events would enable better planning by all levels of government officials, from local municipalities to international bodies40,41, with the potential to save millions of human lives.
Global dissatisfaction functional H[n(x)]
The main text describes a net global population dissatisfaction functional H[n(x)]. To derive this functional, we begin by considering a deterministic model, in which agents reject or accept potential moves with unit probability according to whether ΔH ≡ (V(x′) + f′(n(x′)) − (V(x) + f′(n(x)) is positive or negative, respectively. In such a model, it is clear that equilibrium is attained and all motion ceases when ΔH = 0 for all pairs of points x and x′. This statement is equivalent to the combination V(x) + f′(n(x)) attaining some constant value μ across the system,
$$V({\mathbf{x}}) + f\prime (n({\mathbf{x}})) = \mu .$$
This equation corresponds precisely to the Lagrange-multiplier equation for minimization of the functional
$$H[n({\mathbf{x}})] \equiv {\int} f(n({\mathbf{x}})){\kern 1pt} dA + {\int} V({\mathbf{x}})n({\mathbf{x}}){\kern 1pt} dA,$$
subject to the constraint of fixed number of agents \(N = {\int} n({\mathbf{x}})\,dA\), with μ being the corresponding Lagrange-multiplier. Here, μ plays an analogous role to the "chemical potential" from Statistical Physics.
Probability density functional P[n(x)]
To make the transition to the probability functional P[n(x)], we note that the stochastic model described in the text maps directly onto a particular Markov chain. Each step on this chain corresponds to a three-stage process. First, (a) an agent is selected at random to consider a possible move from current location x. Selecting a random agent at each time step allows agents to adjust their locations at equal rates. In this approach, choosing the physical time interval between Markov steps to be inversely related to the number of agents preserves the time scale of the overall crowd dynamics. Second, (b) a location x′ nearby x is selected at random as a move to be considered by the given agent. We note that for this work, we assume that the new location x′ is selected in a symmetric way so that that agents at x contemplate moves to x′ with the same probability that agents at x′ contemplate moves to x. This assumption seems most plausible given the systems we consider here. Other selection criteria, however, are possible and would modify the distribution below. Finally, (c) the contemplated move is accepted or rejected according to the probability 1/(eΔH + 1), where ΔH is defined specifically as the change in the value of the functional described in Eq. 5 as a result of the move.
There are two critical things to note about this Markov chain. The first is that it gives a very natural description of agent behavior. The second is that it corresponds precisely to the standard Metropolis-Barker algorithm42,43 for drawing random samples from the Boltzmann distribution P ∝ exp(−H) for a Hamiltonian H. Thus, under our proposed motion model, the population itself naturally samples from the distribution quoted in the text,
$$P[n({\mathbf{x}})] = Z^{ - 1}{\mathrm{exp}}( - H[n({\mathbf{x}})]).$$
Discretization H=Σb(\({\boldsymbol{ f}}_{{\boldsymbol{N}}_{\boldsymbol{b}}} + {\boldsymbol{v}}_{\boldsymbol{b}} {\boldsymbol{N}}_{\boldsymbol{b}}\))
To arrive at the discretization described in the text, it is important to note that the density n(x) appearing in the probability functional P[n(x)] corresponds to the fluctuating crowd density, as opposed to the average density nave(x). As such, in practice, this density must be described in terms of the discrete locations xa of all agents a in the crowd at any give time. The most natural description for the associated density operator is
$$n({\mathbf{x}}) = \mathop {\sum}\limits_a \delta ^{(\sigma )}({\mathbf{x}},{\mathbf{x}}_a),$$
where δ(σ)(x, xa) is a function describing the range over which the presence of an agent at xa contributes to the density n(x) at point x. To conserve number of agents, this function must integrate to unity. The analysis carried out in the text divides space into bins b of area Ab, and estimates the density in each bin as n = Nb/Ab where Nb corresponds to the total number of agents in bin b. This definition sets the range function as
$${\delta}^{(\sigma )}({\mathbf{x}},{\mathbf{x}}_{a}) \equiv \left\{ {\begin{array}{*{20}{l}} {\frac{1}{A_{b}}} \hfill & {{\mathrm{if}}\,{\mathbf{x}}\,{\mathrm{and}}\,{\mathbf{x}}_{a}\,{\mathrm{are}}\,{\mathrm{in}}\,{\mathrm{the}}\,{\mathrm{same}}\,{\mathrm{bin}}\,b{\kern 1pt} } \hfill \\ 0 \hfill & {{\mathrm{otherwise}}{\kern 1pt} } \hfill \end{array}} \right.$$
To capture relevant variations in vexation and density, the bins cannot be selected so large that these quantities vary significantly across each bin. Alternately, to avoid missing the effects of nearby agents, the bins cannot be selected to be smaller than the agent's interaction range.
Finally, combining equations 5, 7, and 8, yields
$$H[n({\mathbf{x}})] = \mathop {\sum}\limits_b f_{N_b} + \mathop {\sum}\limits_b v_bN_b,$$
where \(f_{N_b} \equiv f(N_b/A_b)A_b\) and \(v_b \equiv {\int}_b V({\mathbf{x}}){\kern 1pt} dA/A_b\).
Bin occupation probability distributions P b(N)
To arrive at the final discrete probability expression in the text, there are now two routes. One can directly insert Eq. 9 above into Eq. 2 from the main text, or one can employ Eq. 9 directly to compute ΔH to determine the probabilities for moves. In the latter case, the predicted probability distribution becomes exact so long as we interpret f′(n) in the main text at points x′ and x to represent forward and reverse finite difference derivatives \(f_{+}^{\prime} (n({\mathbf{x}}\prime )) = (f(n({\mathbf{x}}\prime ) + {\mathrm{\Delta }}) - f(n({\mathbf{x}}\prime )))/{\mathrm{\Delta }}\) and \(f_{-}^\prime \left( {n\left( {\mathbf{x}} \right)} \right) = \left( {f\left( {n\left( {\mathbf{x}} \right)} \right) - f\left( {n\left( {\mathbf{x}} \right) - {\mathrm{\Delta }}} \right)} \right)/{\mathrm{\Delta }}\), respectively, where Δ ≡ 1/Ab. Finally, because the Boltzmann factor above gives probabilities for individual arrangements of agents among bins, we must account for the multiple ways to realize a set of bin counts {Nb} by permuting individuals among the bins. Multiplying by the combinatorial factor Ntot!/(N1!…Nb!…), we find
$$P(\{ N_b\} ) = \frac{{N_{{\mathrm{tot}}}!}}{Z}\mathop {\prod}\limits_b \frac{{e^{ - f_{N_b} - v_bN_b}}}{{N_b!}},$$
where Z is a normalization factor.
As described in the text, we note that the form of the joint probability distribution above predicts the occupations of different bins to be very nearly statistically independent. The only deviation from complete statistical independence comes from the constraint of a fixed total number of agents \(N_{{\mathrm{tot}}} = \mathop {\sum}\nolimits_b N_b\). Due to this constraint, the probability distribution is difficult to use in making predictions. We can overcome this difficulty using a standard technique from statistical physics. Specifically, introducing a factor \(e^{\mu N_{{\mathrm{tot}}}}\) removes the constraint without significantly affecting the calculated local distributions. As a result, the individual bin distributions then become statistically independent and of the form
$$P_b(N) = z_b^{ - 1}\frac{1}{{N!}}\left( {e^{ - (v_b - \mu )}} \right)^Ne^{ - f_N}.$$
In statistical physics this mathematical transformation corresponds to using a Grand Canonical Ensemble44 to simplify statistical calculations. Physically, this approach corresponds to relaxing the constraint of a fixed number of agents by allowing exchanges between the system being considered and a large reservoir whose vexation is controlled by μ. Mathematically, we can add and subtract a constant within the exponent, (vb − c − (μ − c)) without affecting the distribution. Accordingly, we redefine vb and μ with a constant shift such that vb ← vb − c and μ ← μ − c and, further, choose c so that μ = 0, resulting in Eq. 3 in the text. Note that motion between bins is controlled only by differences in vexations, so that none of this affects the dynamics represented in our analysis. When considering a different number of agents in the same chamber, however, μ will take on a different value and so μ − c can no longer be set to zero. Accordingly, to predict distributions for new numbers of flies, we employ Eq. 11 above and adjust μ so that the vexation of the associated reservoir fixes the new total number of flies.
Orientation and higher-order many-body interactions
Remarkably, our conclusions hold also for plausible models in which the inter-agent interactions are not explicitly expressed in terms of the local density n(x). To see this, we can consider the same behavioral rule of moves accepted according to probability 1/(eΔH + 1), but with H now defined as a sum of two parts,
$$H \equiv U({\mathbf{x}}_a) + \mathop {\sum}\limits_a V({\mathbf{x}}_a),$$
where V(x) is the usual vexation function for the individual agents, and now U(xa) is some potentially complex many-body interaction of finite range depending explicitly on the locations of all of the agents xa.
As above, the form of the Markov chain associated with the move model leads directly to the Boltzmann distribution P(xa) = Z−1e−H. To recover the frustration-vexation probability form analyzed throughout the text, we now follow the standard Statistical Mechanics approach of defining an pseudo-free-energy functional by integrating out internal degrees of freedom. Specifically, we will keep the bin occupancies constant while integrating over all arrangements of agents consistent with these occupancies. For sufficiently small bins in which vexation does not vary significantly, we again find to a good approximation \(\mathop {\sum}\nolimits_a V({\mathbf{x}}_a) = \mathop {\sum}\nolimits_b v_bN_b\), so that vexation simply gives a constant factor. Next, for sufficiently large bins, the net contributions to U(xa) from interactions occurring within the bins will be large compared to the boundary effects from contributions from interactions crossing bin boundaries. Thus, we can imagine decomposing the overall interaction into a sum over the bins of the interactions just among agents a within each bin b, \(U({\mathbf{x}}_a) = \mathop {\sum}\nolimits_b U(\{{\mathbf{x}}_a\} _{a\, \in \,b})\), where we can improve accuracy by repeating the same agent locations {xa}a∈b in neighboring bins (so-called periodic boundary conditions).
Combining these approximations, and summing over all ways to assign agents to bins with counts {Nb} and over all possible locations for the agents within each bin, yields the same frustration-vexation form considered throughout the text,
$$P\left( {\left\{ {N_b} \right\}} \right) = Z^{ - 1}\left( {\begin{array}{*{20}{c}} N \\ {N_1! \ldots N_B!} \end{array}} \right)\left( {\mathop {\prod}\limits_b e^{ - f_{N_b}}} \right)e^{ - \mathop {\sum}\limits_b v_bN_b},$$
where B is the total number of bins, and
$$e^{ - f_N} \equiv {\int}_A \ldots {\int}_A e^{ - U\left( {{\mathbf{x}}_1, \ldots ,{\mathbf{x}}_N} \right)}{\kern 1pt} dA_1 \ldots dA_N$$
defines the effective bin-frustration functional fN as an N-dimensional integral over the area of a single bin (with periodic boundary conditions applied to the interactions). Finally, we note that the above generalizes naturally to orientation-dependent interactions by considering the coordinates {xa} to include orientation, as well as spatial coordinates. If the vexation is orientation-independent, we recover precisely the form above. Otherwise, the entire framework generalizes naturally to consideration of joint location-orientation densities n(x,θ).
All experiments were performed 3–15 days post-eclosion using common fruit flies (D. melanogaster) from an out-bred laboratory stock reared at room temperature on a 12 h/12h day-night cycle. Flies are anesthetized using CO2 and sorted within a few days post-eclosure. We wait for 24 h after sorting before running experiments. Most observations started between 1–5 h after the light was turned on. The experiment chambers are constructed by sandwiching a 1.5 mm thick aluminum frame between two transparent acrylic sheets. The chamber is suspended above an LED light table. Holes in the upper acrylic sheet allow for the introducing flies via aspiration from above. To heat the chambers, 2 Ω high-power resistors are adhered using JB Weld to the aluminum sheet and powered by a variable power supply. On the opposite side of the sheet, a beaker of ice water is used as a heat sink. Chamber temperature is measured for two locations using a contact thermometer to ensure no more than 2 degrees Celsius drift and consistent temperature gradients between trials. We heat one side of the chamber to temperatures between 40–50 degrees Celsius34. The opposing side of the chamber is connected to a heat sink and kept at temperatures between 25–35 degrees Celsius. We find that the resulting temperature gradient drives a strong avoidance behavior for the hotter wall while avoiding fly death as the flies avoid the high-temperature region. A video camera (AVT Marlin, Andover, MA) records overhead images of flies at frame rates around 30 fps and relays these images to a computer where they are analyzed by a custom MATLAB program in real-time. The entire apparatus was enclosed in a black box to prevent biases introduced by ambient light or additional visual cues.
To label fly centroids, images were thresholded to find fly silhouettes. For high density experiments, large groups become common and a more sophisticated approach is necessary to separate clusters, which may be as large at 10 flies. First, the images of several individual flies are combined to make a single, averaged fly mask. This mask is then convolved with images of fly groups. The best fits for these convolutions are used to approximate the locations of flies whose silhouettes overlap. (For additional details, see code provided under Code Availability statement below.) Labeling is then manually checked and we find this technique robust enough to label male flies with 0.25 % error or 1 in 400 flies mislabeled. The mating flies required extensive manual corrections due to changes in the fly postures and the polydispersity of fly sizes, since females are larger than males. For the analysis in this paper we sampled these positions at intervals of 1 s.
Due to wall-exclusion effects, the area of a chamber is different from the area accessible by the centroid of a fly. We thus exclude the outer area of the chamber that corresponds to approximately half the width of a fly. Areas of the bins are then extracted using images from the experiment.
To demonstrate another method for tracking flies that only measures local densities, a simpler method was used for counting flies in the "C" shaped chamber. After thresholding, the number of pixels corresponding to a fly were summed in each bin and then a discrete fly density was assigned to each bin using knowledge of the total number of flies in the chamber. This method has the advantage of computational speed, but weights larger flies more heavily and requires reanalysis for different bin sizes.
Measurement timing and thermal ramp protocol
Observations for Fig. 3 were conducted using time intervals from approximately 5–15 min after being introduced into the chamber so that the flies could explore their new chamber and adjust to a steady state. To measure the vexation of the square experiment, we performed 12 separate single fly measurements each lasting 10 min. Similar results are obtained if three flies are used over a single 10 min period. Thus, measurements of vexation in the "C" and stair shaped chambers used two and three concurrent flies and only needed a single ten minute observation to measure the vexation.
To probe the changing fly behaviors shown in Fig. 4, we track the flies for up to 9 h before flies begin to die from deprivation45,46. To test whether fly behavior is changing over our standard 10 minute time windows, we compare the probabilities, Pb(N), from the first 5 min of the window with the last 5 min and find that they are consistent. The only exception to this is during the very first 5 min after the flies are introduced into the chamber as they become oriented to their new environment that we do not include in our analysis. To elicit different behaviors and location preferences with the same population of flies, we apply a heat gradient to generate an avoidance behavior34 starting at 20 min after being introduced to the chamber. By minute 30, the chamber has reached a steady temperature and we observe that the flies exhibit an approximately constant average distribution. At minute 40, we turn the heat off and let it adjust to room temperature for the remainder of the experiment. Throughout these observations, we qualitatively observe several different behaviors. For the first 5 min, flies are most active and their frustration has a slightly higher positive curvature than the frustration for the 5–15 min period. When the chamber is heated, the frustration stays approximately the same despite the drastic change in the vexation. After the chamber cools down, flies enter a readjustment phase where they are much less active. After this readjustment phase, however, flies again exhibit behavior similar to that from the 5–15 min interval. By 6 h, flies in all the experiments switch to a grouping behavior as shown in Fig. 4.
Validation of assumptions underlying theoretical analysis
As mentioned above, we made some general assumptions developing our theory which we now validate for the walking fly system. First, to verify attainment of equilibrium and sufficient ergodicity, we consider the normalized autocorrelation function
$$c_{\mathrm{T}}({\mathrm{\Delta }}t) \equiv \frac{{\left\langle {\mathop {\sum}\nolimits_b {N_b(t)N_b(t + {\mathrm{\Delta }}t)} } \right\rangle _t}}{{\left\langle {\mathop {\sum}\nolimits_b {N_b(t)N_b(t)} } \right\rangle _t}},$$
where <…>t indicates average over all times. This function shows the expected rapid exponential decay (Fig. 2c), and has an integral which gives the decorrelation time τ = 0.92 s. Indeed, we find this time to be quite short, typically on the order of a few seconds, for all of our experimental runs. This decay time is two orders of magnitude faster than the typical run time and does not vary significantly when computed in different time sub-windows, strongly suggesting rapid mixing and stationarity of the random process, thereby allowing the interchange of time and ensemble averages, and establishing the existence of equilibrium in the timescales under study. Our videos thus represent hundreds of independent samples drawn from the equilibrium ensemble underlying our analysis.
We next consider whether the bins are truly independently distributed as expected in Eq. 3. Accordingly, we consider the normalized time-averaged spatial-correlation function
$$c_{\mathrm{S}}({\mathrm{\Delta }}) \equiv \frac{{\left\langle {\mathop {\sum}\nolimits_b {N_b(t)N_{b + {\mathrm{\Delta }}}(t)} } \right\rangle _{b,t}}}{{\left\langle {N_b(t)N_b(t)} \right\rangle _{b,t}}},$$
where <…>b,t indicates average all times and bins, and Δ is the two-dimensional vector displacement between bins (Fig. 2b)). The data show essentially no correlation between bins, thereby verifying the product form of the global bin distribution function in Eq. 3 in the main text. This confirms not only that we have chosen appropriately sized bins but also, more fundamentally, establishes that there are little or no fly–fly interaction effects between bins, so that the local density approximation (LDA) form for the frustration, \(F[n({\mathbf{x}})] = {\int} f(n({\mathbf{x}}))dA\), indeed gives a good representation of the behavior of the fly populations at scales greater than 0.15 cm2.
To estimate the frustration and vexation for the crowds in our experiments, we start by constructing the posterior function P(fN, vb|Nb(t)), which represents the relative likelihood of different parameter choices for our model given the data (number counts within each bin) that has actually been observed. Then, to find the a posteriori estimate of the parameters, we maximize this likelihood by performing a numerical gradient minimization of
$$\begin{array}{l} - {\mathrm{ln}}P(f_N,v_b|N_b(t)) = C + TB\left( {\left\langle {{\mathrm{ln}}z_b} \right\rangle _b + \left\langle {v_bN_b(t) + {\mathrm{ln}}N_b(t)! + f_{N_b(t)}} \right\rangle _{b,t}} \right)\\ + \mathop {\sum}\limits_N \frac{{f_N^2}}{{2\sigma ^2}} + \mathop {\sum}\limits_b \frac{{v_b^2}}{{2\sigma ^2}},\end{array}$$
where C is an irrelevant normalization constant, B corresponds to the total number of bins in the system, T the total number of independent time samples employed, and 〈…〉b and 〈…〉b,t represent averages over either all bins or bins and times, respectively. Finally, for the last two terms, σ accounts for the range about zero of a Gaussian prior distribution on the frustration and vexation parameters. This Gaussian prior distribution reflects the fact that the frustration and vexation parameters vb and fN can in principle take any real value, but in practice generally fall in a range on the order of from about −15 to 15 because these parameters enter as exponentials in our probability models. Because the amount of data that we handle is on the order of tens of thousands of frames, the likelihood peaks strongly around its maximum, and the precise form of the Gaussian prior is largely irrelevant. Indeed, changing the value of σ from a reasonable value of 15 to an unreasonably small value of 1, only changes our final results for the frustration by 11.4%. Throughout the rest of our work, we take σ = 15.
Uncertainty in parameter estimation
The sharp peaks associated with the large amount of data ensure the accuracy of the asymptotic Gaussian approximation, in which the joint probability distribution representing the range of parameters supported by the data is a multivariate Gaussian distribution. As a result, the associated covariance matrix of uncertainties in the parameters is the inverse of the Fisher information matrix I (i.e., the second derivative of −lnP evaluated at the location of its maximum). The matrices of parameter uncertainties and cross-correlations among them are computed as follows. For our full DFFT model, with vexation and frustration, and the simple Poisson model, with vexation only, we calculate the inverses of the following matrices, respectively,
$$I_{{\mathrm{DFFT}}}(\{ f_N\} ,\{ v_b\} ) = \left( {\begin{array}{*{20}{c}} {\left[ {I_{ff}} \right]_{N_{{\mathrm{max}}} \times N_{{\mathrm{max}}}}} & {\left[ {I_{fv}} \right]_{N_{{\mathrm{max}}} \times B}} \\ {\left[ {I_{fv}^T} \right]_{N_{{\mathrm{max}}} \times B}} & {\left[ {I_{vv}} \right]_{B \times B}} \end{array}} \right),$$
$$I_{{\mathrm{Poisson}}}(\{ v_b\} ) = \left[ {I_{vv}} \right]_{B \times B},$$
where the matrix elements of each block are
$$\left[ {I_{ff}} \right]_{N,N^\prime } = T\delta _{NN^\prime }\left( {\mathop {\sum}\limits_{\bar{b}} P_{\,\bar{b}}\left( N \right) - \mathop {\sum}\limits_{\bar{b}} \left({P_{\,\bar{b}}\left( N \right)P_{\,\bar{b}}\left( {N^\prime } \right)} \right)} \right)$$
$$\left[ {I_{fv}} \right]_{N,b} = TP_b\left( N \right)\left( {N - \mathop {\sum}\limits_{\widetilde{N}} \widetilde{N}P_b\left( {\widetilde{N}} \right)} \right)$$
$$\left[ {I_{vv}} \right]_{b,b\prime } = T\delta _{bb\prime }\left( {\mathop {\sum}\limits_{\widetilde{N}} \widetilde{N}^{2} P_b\left( {\widetilde{N}} \right) - \left( {\mathop {\sum}\limits_{\widetilde{N}} \widetilde{N} P_b \left( {\widetilde{N}} \right) } \right)^{2}} \right).$$
Here, Pb(N) is defined as in Eq. 3 in the main text, T again represents the total number of independent time frames, and the " ~ " indicates internal summation indices.
Finally, a subtle, but important, ambiguity arises in the extraction of frustrations and vexations. Specifically, because the exponent in the observed probabilities for each bin takes the form (ln zb + vbN + fN), making the replacements (vb → vb − α; zb → zb − β; fN → fN + β + αN;) leaves the predictions of the model unchanged, and any choice of parameters corresponding to these replacements represents the data equally well. As a result, the Fisher matrices described above are singular. To resolve this "gauge invariance" and remove the singularity, we must break the symmetry among equivalent models by adding two constraints (one for α and one for β) to our choice of fN. Here, we do this by enforcing the natural choice that f0 ≡ 0 and f1 ≡ 0, corresponding to the convention that that the frustration does not affect the probability for bins with either N = 0 or N = 1 flies. Finally, in terms of the information matrices above, implementing this constraint corresponds to dropping the first two rows and columns associated with these parameters from the IDFFT matrix.
Uncertainty in predictions of average occupations
With the uncertainties in the extraction of the vexation and frustration parameters from above, we next determined the uncertainties in our predictions of the average bin occupations for large populations in new arenas. The predicted mean densities are
$$\bar N_b = \mathop {\sum}\limits_{N = 0}^{N_{{\mathrm{max}}}} NP_b(N) = \frac{1}{{z_b}}\mathop {\sum}\limits_{N = 0}^{N_{{\mathrm{max}}}} N\frac{{e^{ - (v_b - \mu )N - f_N}}}{{N!}},$$
where the normalization is
$$z_b = \mathop {\sum}\limits_{N = 0}^{N_{{\mathrm{max}}}} \frac{{e^{ - (v_b - \mu )N - f_N}}}{{N!}},$$
where Pb(N) is the probability of having N flies in bin b, vb is the vexation in bin b, and fN is the frustration associated with having N flies in a bin. We accordingly computed the associated uncertainties using standard linearized error propagation as
$$\sigma (\bar N_b) = \sqrt {\left( {\frac{{\partial \bar N_b}}{{\partial v_b}}} \right)^2{\mathrm{var}}(v_b) + \mathop {\sum}\limits_{N,N^\prime = 2}^{N_{{\mathrm{max}}}} \frac{{\partial \bar N_b}}{{\partial f_N}}\frac{{\partial \bar N_b}}{{\partial f_{N^{\prime}}}}{\mathrm{covar}}(f_N,f_{N^\prime })} ,$$
where var(X) and covar(X, Y) represent the variance of random variable X and covariance between X and Y, respectively, as determined by the inverse of the Fisher information matrix as discussed above. Finally, the derivatives needed in Eq. 25 are
$$\frac{{\partial \bar N_b}}{{\partial v_b}} = - \left( {\left\langle {N_b^2} \right\rangle - \bar N_b^2} \right),$$
$$\frac{{\partial \bar N_b}}{{\partial f_N}} = - \left( {\frac{{N_b - \bar N_b}}{Z}} \right)\frac{{e^{ - \,N_b(v_b - \mu ) - f_{N_b}}}}{{N_b!}},$$
where \(\left\langle {N_b^2} \right\rangle \equiv \mathop {\sum}\nolimits_N N^2P_b(N)\) with Pb(N) as defined above.
A few technical notes are in order to understand the terms present in Eq. 25. First, note that cross-correlations between vexations in different bins are not relevant because \(\bar N_b\) depends solely on vb and not on vexations from other bins. Also, cross-correlations between extracted vexations vb and frustrations fN are zero in our case because we extract the vexations and frustrations from different, and thus independent, experiments when making our predictions for average occupations. Finally, the uncertainties in f0 and f1 are not included because these uncertainties are zero due to the gauge choice discussed in the section above.
Uncertainty in experimentally measured bin statistics
For each independent bin, we obtain from the experiment a sequence of length NT with elements each corresponding to a bin occupation that can range from zero to the maximum packing of files, N = 0,…,Nmax. From this data, we hope to extract probability parameters pN describing the bin occupation distributions studied in the main text. For simplicity of notation, we here use lower case p to denote experimentally measured probabilities.
To account for time-correlations in bin occupancies, particularly at high frame rates, we down-sample at intervals given by the decorrelation time τ and actually consider uncorrelated sequences of length T = NT/τ. The data then correspond to the result of a random process of making T independent selections among Nmax + 1 possible bin occupations. Thus, for each bin, the probability of observing a given data sequence becomes the multinomial distribution,
$$\left( {\begin{array}{*{20}{c}} T \\ {h_0 \cdots h_{N_{{\mathrm{max}}}}} \end{array}} \right)p_0^{h_0} \cdots p_{N_{{\mathrm{max}}}}^{h_{N_{{\mathrm{max}}}}},$$
where hN represents the number of times ("hits") we observe each of the possible occupancies N.
To extract the underlying uncertainties, we note that Bayes' theorem gives the following distribution for the probability parameters to take the values {pN} given the actually observed counts {hN},
$$P(\{ p_N\} |\{ h_N\} ) = \frac{{P\left( {\left. {\{ h_N\} } \right|\{ p_N\} } \right)P\left( {\{ p_N\} } \right)}}{{P(\{ h_N\} )}} \propto \left( {\mathop {\prod}\limits_{n = 0}^{N_{{\mathrm{max}}}} \frac{{p_n^{h_n}}}{{h_n!}}} \right)P(\{ p_N\} ).$$
This posterior probability is proportional to an undetermined prior probability P({pN}) describing our a priori expectations for the values of the {pN} parameters. However, as per our discussion surrounding Eq. 17 above, in the large T limit, the Poisson-like product factor in Eq. 29 above will be highly peaked, and the unknown prior P({pN}) will not have a substantial effect on the posterior distribution.
To completely eliminate the effects of unwarranted assumptions entering through our choice of prior, we assume an uninformative prior distribution that is consistent with the invariance of the probability values under the inclusion of new samples, and choose the multivariate generalization of Haldane's uninformative improper prior distribution47,
$$P(\{ p_N\} ) = \frac{1}{{\mathop {\prod}\nolimits_{n = 0}^{N_{{\mathrm{max}}}} {p_n} }}.$$
With this choice, upon normalization, Eq. 29 becomes the Dirichlet distribution,
$$P(\{ p_N\} |\{ h_N\} ) = \Gamma \left( {\mathop {\sum}\limits_{n\prime = 0}^{N_{{\mathrm{max}}}} h_{N\prime }} \right)\mathop {\prod}\limits_{N = 0}^{N_{{\mathrm{max}}}} \frac{{p_N^{h_N - 1}}}{{\Gamma (h_N)}},$$
where Γ(x) is the Gamma function. This distribution yields expected values for the probabilities equal precisely to the observed frequencies \(\bar p_N = h_N/T\). The variances of this distribution, then give our desired uncertainties,
$$\sigma (p_N) = \sqrt {\frac{{h_N\left( {T - h_N} \right)}}{{T^2(T + 1)}}} = \sqrt {\frac{{\bar p_N(1 - \bar p_N)}}{{T + 1}}} .$$
Note that when T is large and \(\bar p_N \ll 1\), the uncertainties correspond to what we would naïvely expect from Poisson counting, namely an uncertainty of \(\sqrt {h_N}\) in the counts, corresponding to an uncertainty of \(\sqrt {h_N} /T = \sqrt {\overline{p}_N/T}\) in the extracted probabilities. Such an analysis, however, misses the important factor of \(\sqrt {1 - \overline{p}_N}\) and leads to significant errors in our case.
Finally, for the uncertainty in the experimental average occupation \(\bar N_{{\mathrm{exp}}t} = \mathop {\sum}\nolimits_N Np_N\), the corresponding variance is
$${\mathrm{var}}\left( {\bar N_{{\mathrm{expt}}}} \right) = \mathop {\sum}\limits_{N \ne N^{\prime}} NN^{\prime}{\mathrm{covar}}\left( {p_N,p_{N^\prime }} \right) + \mathop {\sum}\limits_N N^2\sigma (p_N)^2,$$
where the needed covariances of the Dirichlet distribution are
$${\mathrm{covar}}(p_N,p_{N^\prime }) = \frac{{ - h_Nh_{N^\prime }}}{{T^2(T + 1)}} = \frac{{ - \bar p_N\bar p_{N^\prime }}}{{T + 1}}$$
Readers can access the code related to parameter estimation and crowd density predictions by going to (https://github.com/MendezV/DFFT) or to (https://doi.org/10.5281/zenodo.1285931). Readers can also access code related to image analysis procedures by visitng (https://github.com/yunuskink/Fitfly-fly-tracking) or (https://doi.org/10.5281/zenodo.1304326). There are no access restrictions to this software.
The fly density data that support the findings of this study are available in the Open Science Framework database at (https://doi.org/10.17605/OSF.IO/7UBZ2).
Silverberg, J. L., Bierbaum, M., Sethna, J. P. & Cohen, I. Collective motion of humans in mosh and circle pits at heavy metal concerts. Phys. Rev. Lett. 110, 228701 (2013).
ADS Article PubMed CAS Google Scholar
Helbing, D., Farkas, I. & Vicsek, T. Simulating dynamical features of escape panic. Nature 407, 487 (2000).
Bellomo, N. & Dogbe, C. On the modeling of traffic and crowds: a survey of models, speculations, and perspectives. SIAM Rev. 53, 409–463 (2011).
MathSciNet Article MATH Google Scholar
Leonard, N. E. & Fiorelli, E. Virtual leaders, artificial potentials and coordinated control of groups. Decision and Control, 2001. In Proceedings of the 40th IEEE Conference on, Vol. 3, 2968–2973. (IEEE, 2001).
Couzin, I. D., Krause, J., Franks, N. R. & Levin, S. A. Effective leadership and decision-making in animal groups on the move. Nature 433, 513 (2005).
Vicsek, T., Czirók, A., Ben-Jacob, E., Cohen, I. & Shochet, O. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett. 75, 1226 (1995).
ADS MathSciNet Article PubMed CAS Google Scholar
Moussad, M., Helbing, D. & Theraulaz, G. How simple rules determine pedestrian behavior and crowd disasters. Proc. Natl Acad. Sci. USA 108, 6884–6888 (2011).
An, L. Modeling human decisions in coupled human and natural systems: review of agent-based models. Ecol. Model. 229, 25–36 (2012).
Zheng, X., Zhong, T. & Liu, M. Modeling crowd evacuation of a building based on seven methodological approaches. Build. Environ. 44, 437–445 (2009).
Treuille, A., Cooper, S. & Popović, Z. Continuum crowds. ACM Trans. Graph. 25, 1160–1168 (2006).
Hinz, R. C. & de Polavieja, G. G. Ontogeny of collective behavior reveals a simple attraction rule. Proc. Natl Acad. Sci. USA 114, 2295–2300 (2017).
Katz, Y., Tunstrøm, K., Ioannou, C. C., Huepe, C. & Couzin, I. D. Inferring the structure and dynamics of interactions in schooling fish. Proc. Natl Acad. Sci. USA 108, 18720–18725 (2011).
Ballerini, M. et al. Interaction ruling animal collective behavior depends on topological rather than metric distance: evidence from a field study. Proc. Natl Acad. Sci. USA 105, 1232–1237 (2008).
ADS Article PubMed Google Scholar
Berman, G. J., Choi, D. M., Bialek, W. & Shaevitz, J. W. Mapping the stereotyped behaviour of freely moving fruit flies. J. R. Soc. Interface 11, 20140672 (2014).
Kelley, D. H. & Ouellette, N. T. Emergent dynamics of laboratory insect swarms. Sci. Rep., 3, 1073 (2013).
Boltzmann, L. Über die mechanische Bedeutung des zweiten Hauptsatzes der Wärmetheorie:(vorgelegt in der Sitzung am 8 February 1866). (Staatsdruckerei, 1866).
Diderik Van der Waals, J. Over de Continuiteit van den Gas-en Vloeistoftoestand, volume 1. Sijthoff, 1873.
Toner, J. & Tu, Y. Flocks, herds, and schools: a quantitative theory of flocking. Phys. Rev. E 58, 4828 (1998).
ADS MathSciNet Article CAS Google Scholar
Mora, T. & Bialek, W. Are biological systems poised at criticality? J. Stat. Phys. 144, 268–302 (2011).
ADS MathSciNet Article MATH Google Scholar
Cavagna, A. et al. Dynamic scaling in natural swarms. Nat. Phys. 13, 914 (2017).
Tennenbaum, M., Liu, Z., Hu, D. & Fernandez-Nieves, A. Mechanics of fire ant aggregations. Nat. Mater. 15, 54 (2016).
Sinhuber, Ml & Ouellette, N. T. Phase coexistence in insect swarms. Phys. Rev. Lett. 119, 178003 (2017).
Solon, A. P. et al. Pressure is not a state function for generic active fluids. Nat. Phys. 11, 673 (2015).
Sham, L. J. & Kohn, W. One-particle properties of an inhomogeneous interacting electron gas. Phys. Rev. 145, 561 (1966).
ADS Article CAS Google Scholar
Hohenberg, P. & Kohn, W. Inhomogeneous electron gas. Phys. Rev. 136(3B), B864 (1964).
ADS MathSciNet Article Google Scholar
Diggle, P. J. Statistical Analysis of Spatial and Spatio-Temporal Point Patterns, Third Edition. Chapman & Hall/CRC Monographs on Statistics and Applied Probability. (CRC Press, 2013).
Kim, I. S. & Dickinson, M. H. Idiothetic path integration in the fruit fly Drosophila melanogaster. Curr. Biol. 27, 2227–2238 (2017).
Neyman, J. On a new class of contagious distributions, applicable in entomology and bacteriology. Ann. Math. Statist. 10, 35–57 (1939).
Article MATH Google Scholar
Taylor, L. R. Aggregation, variance and the mean. Nature 189, 732–735 (1961).
Besson, M. & Martin, J.-R. Centrophobism/thigmotaxis, a new role for the mushroom bodies in Drosophila. Dev. Neurobiol. 62, 386–396 (2005).
Ramdya, P. et al. Mechanosensory interactions drive collective behaviour in Drosophila. Nature 519, 233 (2015).
Schneider, J., Atallah, J. & Levine, J. D. 3 one, two, and many—a perspective on what groups of Drosophila melanogaster can tell us about social dynamics. Adv. Genet. 77, 59 (2012).
Schneider, J., Dickinson, M. H. & Levine, J. D. Social structures depend on innate determinants and chemosensory processing in Drosophila. Proc. Natl Acad. Sci. USA 109, 17174–17179 (2012).
Zars, M. & Zars, T. High and low temperatures have unequal reinforcing properties in Drosophila spatial learning. J. Comp. Physiol. A. 192, 727 (2006).
Ramdya, P., Schneider, J. & Levine, J. D. The neurogenetics of group behavior in Drosophila melanogaster. J. Exp. Biol. 220, 35–41 (2017).
Runge, E. & Gross, E. K. U. Density-functional theory for time-dependent systems. Phys. Rev. Lett. 52, 997 (1984).
Chan, G. K.-L. & Finken, R. Time-dependent density functional theory of classical fluids. Phys. Rev. Lett. 94, 183001 (2005).
Capitani, J. F., Nalewajski, R. F. & Parr, R. G. Non-Born–Oppenheimer density functional theory of molecular systems. J. Chem. Phys. 76, 568–573 (1982).
Wesolowski, A. et al. Quantifying the impact of human mobility on malaria. Science 338, 267–270 (2012).
ADS Article PubMed PubMed Central CAS Google Scholar
Hauer, M. E. Migration induced by sea-level rise could reshape the US population landscape. Nat. Clim. Change 7, 321–325 (2017).
Clark, P. U. et al. Consequences of twenty-first-century policy for multi-millennial climate and sea-level change. Nat. Clim. Chang. 6, 360–369 (2016).
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. & Teller, E. Equation of state calculations by fast computing machines. J. Chem. Phys. 21, 1087–1092 (1953).
Barker, A. A. Monte Carlo calculations of the radial distribution functions for a proton-electron plasma. Aust. J. Phys. 18, 119–134 (1965).
Reif, F. Chapter 8, Fundamentals of Statistical and Thermal Physics. (McGraw Hill, New York, 1965).
Ji, F. & Zhu, Y. A novel assay reveals hygrotactic behavior in Drosophila. PLoS. One. 10, e0119162 (2015).
Bell, W. J., Cathy, T., Roggero, R. J., Kipp, L. R. & Tobin, T. R. Sucrose-stimulated searching behaviour of Drosophila melanogaster in a uniform habitat: modulation by period of deprivation. Anim. Behav. 33, 436–448 (1985).
Haldane, J. A note on inverse probability. Math. Proc. Camb. Philos. Soc. 28, 55–61 (1932).
The authors thank Xiaoning Wang, Marc-Antoine Bouvattier, Alonso Botero, Tom Corwin, Tom Mifflin, Greg Godfrey, and Nathan Sitaraman for their help with the initial stages of the project. We further thank the Cohen and Arias groups for discussions throughout this work. The work was primarily funded by the Army Research Office Army-ARO W911NF-16-1-0433. J.F.M. was also supported in part by an Office of the Vice President for Research at the University of Los Andes. Y.K. was also supported in part by funding from the National Science Foundation Graduate Research Fellowship Award No. DGE-1650441.
These authors contributed equally: J. Felipe Méndez-Valderrama, Yunus A. Kinkhabwala
Department of Physics, Universidad de Los Andes, Bogotá, 111711, Colombia
J. Felipe Méndez-Valderrama
Department of Applied and Engineering Physics, Cornell University, Ithaca, NY, 14853, USA
Yunus A. Kinkhabwala
Metron Inc., Scientific Solutions, Reston, VA, 2019, USA
Jeffrey Silver
Department of Physics, Cornell University, Ithaca, NY, 14853, USA
Itai Cohen & T. A. Arias
Itai Cohen
T. A. Arias
J.F.M.V.: Development and implementation of analyses to extract vexations, frustrations, and predicted mean occupations. Analysis of statistical uncertainties in all of these quantities and also in the extraction of bin-occupancy distributions from experimental data. Final display format for cross-correlating predicted mean occupancies with predictions. Theoretical parts of the Methods Section. Significant contributions to main text. Y.A.K.: Design and implementation of experiments along with development of image analysis techniques. Running data through analyses provided by Méndez Valderrama. Experimental parts of the Methods Section, design, and implementation of figures, and an early draft of the manuscript. Significant contributions to main text. Y.A.K. and J.F.M.V. contributed equally to this work. J.S.: Co-development of underlying Markov chain, proper accounting for degeneracy factor, identification of multinomial and Poisson distributions for the non-interacting case, initiation of use of maximum-likelihood estimation and Bayesian uncertainty techniques. I.C.: Significant input into design of experiments, and primary responsibility for main text. T.A.A.: Development of underlying motion model, density-functional theory analysis, and prediction of form of population fluctuations. Co-development of underlying Markov chain. Complete early draft of manuscript, and significant contributions to main text.
Correspondence to T. A. Arias.
Méndez-Valderrama, J.F., Kinkhabwala, Y.A., Silver, J. et al. Density-functional fluctuation theory of crowds. Nat Commun 9, 3538 (2018). https://doi.org/10.1038/s41467-018-05750-z
DOI: https://doi.org/10.1038/s41467-018-05750-z
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The Demographic Window of Opportunity and Economic Growth at Sub-National Level in 91 Developing Countries
Lamar Crombach ORCID: orcid.org/0000-0002-7634-93431 &
Jeroen Smits ORCID: orcid.org/0000-0002-5802-975X2
Social Indicators Research volume 161, pages 171–189 (2022)Cite this article
Data for low- and middle- income countries (LMICs) are used to investigate the effect of the demographic transition on economic growth at sub-national level. We introduce a detailed classification of demographic window phases, determine how these phases are distributed among and within LMICs, and analyze the relationship between the demographic window of opportunity (DWO) and economic growth for 1921 urban and rural areas of sub-national regions within 91 LMICs. Many areas in Asia, Latin America and the Middle East have entered the window, but most of Sub-Saharan Africa is still in the traditional or pre-window phase. Our analyses reveal higher growth rates in areas passing through the DWO. Positive growth effects are particularly strong in rural and more educated regions and in countries with lower levels of corruption. Policy measures aimed at effectively using the DW for achieving growth should combine investments in education and rural development with better governance.
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Avoid the common mistakes
The role of the population age structure as a potential source for economic growth is gaining importance in the economic literature (e.g. Canning et al., 2015; Groth & May, 2017; Kelley & Schmidt, 2005). Fertility reduction may potentially enable a region to have a period of rapid economic growth during the Demographic Window of Opportunity (DWO). The DWO is a period of several decades that countries go through when moving from a situation of high fertility and mortality to low fertility and mortality. This transition is generally characterized by a baby boom period, in which (child) mortality is already reduced, but fertility levels are still high. When later fertility also decreases, the baby boom generation moves up in the age distribution and after some time enters the working age population. During this period, more women may enter the labor force as less time needs to be spent on children (Aaronson et al., 2021; Bloom et al., 2009; Cristia, 2008). Given that there are still few elderly—as their generation suffered from higher mortality rates in the past—the region experiences a period in which the working age population is large and the dependent population—the green and the grey—is small. Hence, there is high potential for economic growth, which, if realized, is called the "demographic dividend" (Bloom et al., 2003, 2009; Groth & May, 2017)
The demographic dividend is often used as an explanation for the rapid economic growth that occurred in East Asian economies in the end of the twentieth century, accounting for as much as one third of the observed national economic growth (Bloom & Williamson, 1998). The size of this demographic dividend, however, is found to depend on policies regarding labor markets, financial markets, education and health (Bloom et al., 2017; Bloom & Williamson, 1998; Groth & May, 2017). Most research tends to focus on the level or growth rate of the dependency ratio, which is measured as the ratio of the share of the dependent population (those aged below 15 and above 65) to the working-age population (those aged 15–65). A low or declining dependency ratio indicates that an area is passing through the DWO.
Due to issues of data availability, the current literature has not yet extensively analyzed the effects of the DWO on economic growth in low-income countries. These countries are often thought of as being stuck in a high-fertility high-mortality situation, i.e. not to have entered the DWO yet (Bloom et al., 2003). Consequently, the empirical basis of the literature has been heavily biased towards growth in middle- income countries with a well-educated workforce and good infrastructure—such as the East Asian Tigers—where the potential for economic growth may have been better than in many of the current low-income countries (Bloom & Williamson, 1998; Radelet et al., 2001; Williamson, 2013).
The empirical literature has also been very focused on the national level, whereas fertility decline and economic growth may differ substantially between different areas within a country. For instance, cities may show entirely different patterns than rural areas (Sander & Charles-Edwards, 2017; Williamson, 2013). Furthermore, the national-level fertility transition has been shown to lead to within-country increases in fertility inequality (Eloundou-Enyegue et al., 2017). By analyzing data at a sub-national level—as is done in the current study—the added variation allows for a more refined analysis of the variables at hand. Indeed, a simple multilevel model of our dependent variable reveals that about 71% of variation occurs at the sub-national level, while 29% occurs at the national level.
There are a few published papers that have analyzed the effects of demographic variables at a sub-national level, but they tend to be restricted to regions within a single country. Firstly, Wei and Hao (2010) analyze the impacts of the growth rate and level of the dependency ratio, the total population, and population density, on economic growth in Chinese regions. They also include interactions to analyze under which conditions the dependency ratio is more or less important. Their analyses do not reveal effects of the level of population, urbanization, or the growth rate of the dependency ratio, but they do find significant negative effects of the dependency ratio and population density. Secondly, Baerlocher et al. (2019) estimate the impacts of the participation rate, the working age population, the growth rates of the (working age) population, and the level of and change in the mean years of schooling in micro-regions in Brazil. They find that demographic variables do matter for economic growth, though predominantly through their effect on human capital accumulation. Lastly, Kumar (2013) analyzes the effects of the level of, and growth in, the working age share of the population on state-level net GDP per capita in 16 Indian states for the period 1971–2001 and finds significant positive effects of both level and growth, thus pointing towards a favorable effect of the DWO on growth.
The aim of the current paper is threefold. First, we want to find out how the DWO has spread across subnational regions of low and middle-income countries. Second, we aim to determine whether the positive effect of the DWO on economic growth—which has been found for middle-income countries—is also present in low-income countries and in particular at the level of sub-national regions. Third, we want to determine under which circumstances the effect of the DWO is strongest to help policy makers develop contextualized measures that facilitate reaping the demographic dividend in specific areas that have already entered the DWO. Therefore, our central research questions sound:
What is the variation in DWO phases at the level of sub-national regions across the developing world?
To what extent does the DWO foster economic growth at the sub-national level
Which factors are associated with a more effective use of the DWO in terms of economic growth
Our study contributes to the literature in several important ways. It is the first to compare sub-national regions across the developing world in terms of the impact of the DWO on economic growth. We use a database with information for two points in time for 1921 sub-national areas, both urban and rural, spread across 91 different LMICs. Second, a dataset with such richness offers more possibilities for analyzing the role of context factors than the national data on—at most—some 200 countries that are usually used for growth models. Third, besides average effects, we also study interaction effects. Interaction effects help us to understand how the effect of the DWO depends on specific circumstances. Such insights allow policymakers to implement better-targeted policies. Fourth, economic growth at the sub-national level is indicated by the International Wealth Index (IWI), an asset-based wealth index that is comparable over time and across countries (Smits & Steendijk, 2015). Analyzing the effects of demographics on the IWI will provide valuable new insights, as IWI measures a household's possession of durables, housing quality and access to basic services and is as such a broader indicator than income or expenditure. As an outcome-based measure, IWI does not need arbitrary adjustments in terms of purchasing power parities, common baskets of goods, or inflation, as is the case with GDP per capita.
The remainder of this paper is structured as follows. The next section will outline the theoretical model, our measure of economic growth and the context factors that are potential determinants of DWO effectiveness. The third section describes the data, the methodology and the control factors. The fourth section outlines the results, and the fifth section will discuss and conclude.
The Theoretical & Empirical Model
We begin by defining a standard Cobb–Douglas aggregate production function for region i in country j in the year T2:
$${Y}_{ij}^{T2}={A}_{ij}^{T2}{{K}_{ij}^{T2}}^{\alpha }{{H}_{ij}^{T2}}^{1-\alpha }$$
where \({Y}_{ij}\) is GDP, \({A}_{ij}\) is Total Factor Productivity (TFP), \({K}_{ij}\) is the capital stock, and \({H}_{ij}\) is the stock of efficient labor, which consists of the labor force, \({L}_{ij}\), multiplied by the human capital per worker, \({h}_{ij}\). Human capital per worker is determined by the years of schooling, \({s}_{ij}\), and the return to schooling, \(\theta \), which is assumed to be constant, in the following functional format:
$${h}_{ij}^{T2}={e}^{{s}_{ij}^{T2}\theta }$$
Dividing output by the labor force allows us to rewrite the production function in per worker terms:
where \({y}_{ij}\) is GDP per worker, and \(k_{ij}\) is the capital stock per worker. To simplify notation, the growth rate is assumed to refer to the growth in period T1-T2. Substituting Eq. 2 into Eq. 3, taking logs and the first difference gives us the following growth rate equation:
$$ g_{{ij}}^{y} = g_{{ij}}^{A} + \left( {1 - \alpha } \right)\theta \Delta s_{{ij}} + \alpha g_{{ij}}^{k} $$
We follow Baerlocher et al. (2019) by allowing TFP growth to depend on the level of mean years of schooling, with the argument that countries with higher levels of schooling are better able to innovate and generate economic growth. Additionally, countries that are closer to the technological frontier, as proxied by GDP per worker in period T1, will have lower TFP growth (Barro, 1991):
$${g}_{ij}^{A}=\delta +\psi {s}_{ij}^{T1}-\mu \mathrm{ln}\left({y}_{ij}^{T1}\right)$$
In our analysis, we will control for capital stock per worker growth at the national level by using a fixed effects country dummy model. Substituting TFP into the GDP per worker growth equation gives:
$$ g_{{ij}}^{y} = \delta + \psi s_{{ij}}^{{T1}} - \mu {\text{ln}}\left( {y_{{ij}}^{{T1}} } \right) + \left( {1 - \alpha } \right)\theta \Delta s_{{ij}} + \alpha g_{{ij}}^{k} $$
To translate the model from GDP per worker to GDP per capita terms, we must adjust the neoclassical model and recognize that a population does not merely consist of workers, but also of dependents:
$${\left(\frac{Y}{N}\right)}_{ij}={\left(\frac{Y}{L}\right)}_{ij}{\left(\frac{L}{N}\right)}_{ij}={\left(\frac{Y}{WA}\right)}_{ij}{\left(\frac{WA}{N}\right)}_{ij}$$
where \({N}_{ij}\) is the total population, and \(WA_{ij}\) is the working age share of the population, i.e., those aged between 15 and 65.
Thus, GDP per capita consists of a productivity component, \(\frac{Y}{WA}=y\), and a demographic component: \(\frac{WA}{N}\). Additionally, for reasons of data availability, we assume \({L}_{ij}=W{A}_{ij}\), i.e., full employment. Moreover, as in Wei and Hao (2010), we express the demographic component in terms of the dependency ratio, \({D}_{ij}\), as opposed to share of the working age population to the total population: \(D_{ij}=\frac{N_{ij}-WA_{ij}}{WA_{ij}}\), which gives:
$${\left(\frac{Y}{N}\right)}_{ij}={\overline{y} }_{ij}=\frac{{y}_{ij}}{1+D_{ij}}$$
where \({\overline{y} }_{ij}\) is GDP per capita.
As before, we take logs and the first difference to obtain growth rates. Additionally, we substitute the TFP growth equation into the growth rate of GDP per capita equation. Moreover, note that TFP growth depends on the initial level of income per worker, which we now translate to the initial level of income per capita as well:
$$ \begin{gathered} g_{ij}^{{\overline{y}}} = \delta + \psi s_{ij}^{T1} - \mu \ln \left( {\overline{y}_{ij}^{T1} } \right) - \mu \ln \left( {1 + D_{ij}^{T1} } \right) + \alpha g_{ij}^{k} + \hfill \\ \quad \quad \quad \left( {1 - \alpha } \right)\theta \Delta s_{ij} - \Delta \ln \left( {1 + D_{ij} } \right). \hfill \\ \end{gathered} $$
Transforming the above equation into an empirical specification with country-specific fixed effects (which absorb national capital stock per worker growth, \({g}_{ij}^{k}\)), \({\upgamma }_{j}\), and a random error term at the sub-national level, \({\epsilon }_{ij}\), gives:
$$ \begin{gathered} g_{ij}^{{\overline{y}}} = \beta_{0} + \beta_{1} s_{ij}^{T1} + \beta_{2} \ln \left( {\overline{y}_{ij}^{T1} } \right) + \beta_{3} \ln \left( {1 + D_{ij}^{T1} } \right) + \hfill \\ \quad \quad \beta_{4} {\Delta }s_{ij} + \beta_{5} {\text{g}}^{{1 + {\text{D}}_{{{\text{ij}}}} }} + \gamma_{j} + \in_{ij} . \hfill \\ \end{gathered} $$
Economic Growth at Sub-National Level
A key issue to overcome is that for most LMICs data on (changes in) GDP per capita at sub-national level is not available. Therefore, we will use (changes in) standard of living of households, as indicated by the International Wealth Index (IWI) (Smits & Steendijk, 2015) as an alternative. IWI is an asset-based wealth index that measures household wealth in LMICs on the basis of ownership of consumer durables (TV, fridge, phone, car, etc.), quality of housing and access to public services (clean water, electricity). IWI ranks households from 0 to 100, with 0 meaning none of the durables, bad quality housing and no access to services and 100 meaning all durables, good quality housing and access to high quality services.
Conveniently, one can aggregate IWI from the household level to higher levels, such as the level of sub-national areas and the country level. It then indicates the mean wealth level of households in terms of asset ownership in the sub-national areas or country, while increases in the IWI level of an area over a period of time mean that households in the area were able to increase their stock of assets during that period. An advantage compared to indices based on household income in the context of LMICs is that IWI is less volatile, as it is not very sensitive to income changes due to unstable employment or economic shocks, which are rather prevalent in poor regions (Loayza et al., 2007).
Testing the performance of IWI as indicator of economic growth at the subnational level is not possible, as there is no other indicator available at that level for comparison. However, as Figure S1 in the Supplementary Information shows, at the national level IWI is highly (0.86) correlated with the logarithm of GDP per capita and even more highly (0.92) with the Human Development Index, which suggests that also at subnational level it might be a reasonable indicator of the level of development.
Thus, \(IW{I}_{ijt}\approx \mathrm{ln}\left({y}_{ijt}\right)\), i.e., we assume IWI to be an appropriate proxy for the logarithm of GDP per Capita. This changes the specification as follows:
$$ \begin{gathered} g_{ij}^{y} = \ln \left( {y_{ij}^{T2} } \right) - \ln \left( {y_{ij}^{T1} } \right) \approx {\Delta }IWI_{ij} = \beta_{0} + \beta_{1} s_{ij}^{T1} + \hfill \\ \beta_{2} IWI_{ij}^{T1} + \beta_{3} \ln \left( {1 + D_{ij}^{T1} } \right) + \beta_{4} {\Delta }s_{ij} + \beta_{5} {\text{g}}^{{1 + {\text{D}}_{{{\text{ij}}}} }} + \gamma_{j} + \in_{ij} . \hfill \\ \end{gathered} $$
The Role of the Context
The effect of the DWO on economic growth is not expected to be the same everywhere, but may depend on characteristics of the context in which it takes place (Bloom et al., 2017; Bloom & Williamson, 1998; Groth & May, 2017). According to Zuber et al. (2017), DWO effectiveness is influenced by three major factors: (i) job creation, (ii) human capital building and (iii) good governance. A relatively high working age share of the population can only translate into growth if the workers have sufficient employment opportunities. To achieve these opportunities prudent macroeconomic policies are supposed to be required, including a high level of financial market development, a high degree of economic openness and low levels of positive inflation (Collier & Dollar, 2001; Turbat, 2017). However empirical evidence is mixed. Whereas Easterly (2005) found the effect of good policies on growth to be small and not robust to different econometric specifications, Wei and Hao (2010) present data for China showing the effect of the DWO to be stronger in regions with higher levels of market openness. Regarding the role of human capital building there is not yet much empirical support available. Whereas Kelley and Schmidt (2005) found little effect of education on the size of the DWO effect, other studies (Baerlocher et al., 2019; Crespo Cuaresma et al., 2014) suggest that most gains of the DWO are related to education decisions that are the result of the demographic changes.
The third factor is good governance. We follow Kaufmann et al. (2011) in defining governance as "the traditions and institutions by which authority in a country is exercised". We see governance as good when it is participatory, consensus-oriented, accountable, transparent, responsive, effective and efficient, equitable and inclusive and follows the rule of the law (Zuber et al., 2017). Good governance implies that the government offers high-quality and affordable education and health care to everyone, both of which increase human capital building. Further, the increase in tax income resulting from the relative increase in workers must be used productively. We therefore expect that if the additional resources are wasted on corruption and inefficiencies, the demographic dividend may be substantially harmed.
A subnational indicator database was constructed from data of the Database Developing World of the Global Data Lab (GDL) (www.globaldatalab.org), which contains harmonized data for over 30 million persons in 130 + LMICs. The data is derived from Demographic and Health Surveys (DHS, www.dhsprogram.com), Unicef MICS Surveys (mics.unicef.org), IPUMS census data (international.ipums.org), Afrobarometer surveys (www.afrobarometer.org) and several stand-alone surveys (Smits, 2016). Variables obtained from these surveys were aggregated to the level of urban and rural areas of sub-national regions using the regional/provincial codes available in the datasets. In this way, a dataset was created with information for two points in time (T1 and T2) for 1921 urban and rural sub-national areas within 91 countries. For each country, data were selected from the last two available surveys that were at least four years apart. Because the surveys were held in different years for the different countries, the time period between the two surveys could vary between four and 16 years. To obtain a simple and comparative measure of economic growth, the average annual change in IWI between the two survey years (T1 and T2) was used. The other change variables in the model reflect the average annual change between times T2 and T2 as well. In this way a dataset was obtained with for each of the 1921 regions the value of the regional characteristics in the first survey (T1) and the annual changes in these characteristics between the first and the second survey (T2).
To address the fact that the 1921 sub-national regions, which are the units of analysis, are nested within the 91 countries, our regression model included the complete set of fixed effects country dummies. In this way the direct effects of all (measured and unmeasured) country-specific characteristics—including number of sub-national regions and variation in number of years between T1 and T2—are accounted for, while keeping the possibility to study cross-level interactions with the country-level explanatory variables. Given that within the regions only data for two years was used, a two-period panel data analysis could be performed (Wooldridge, 2013) and no further control for multiple observations within the regions was needed.
The main independent variables are the dependency ratio and the growth rate of the dependency ratio. The dependency ratio is measured as the ratio of the share of the working age population to the share of the dependent population. We define the working age population as all individuals aged between 15 and 64. We consider all individuals aged below 15 or above 64 as the dependent population.
Besides the dependency ratios, the models contained independent variables at subnational and national level. At subnational level, these variables included the T1 value of IWI, education, the gender difference in education and urbanization. Education was measured as the mean years of schooling of male individuals aged 20 and above in the region. We choose education for the 20 + population because the starting age of schooling is often high in LMICs (Huisman & Smits, 2009) and individuals under 20 might not yet have completed their education. Gender inequality was measured by the difference between the mean years of schooling for men and for women (education men minus education women). Urbanization was measured by a dummy variable with value zero in rural areas and value one in urban areas. The dummy's value depends on the designation (urban or rural) given by the source surveys, which are based on the official national definition used in the country (United Nations, 2018). Given this rather crude division in only two categories, it may be possible that there are households living in urban areas (e.g., slums) which are less well-off than rural households. The level of IWI at T1 was included as explanatory variable to control for the distance to the technological frontier (convergence) (Barro, 1991).
At the national level, quality of governance, inflation, market openness and financial development were included. To test for good governance, a multidimensional governance index was used, aggregated from the six World Bank worldwide governance indicators (Kaufmann et al., 2011): control of corruption, rule of law, political stability and violence, voice and accountability, government effectiveness and regulatory quality. The scores for these dimensions were all obtained through expert surveys. Each dimension, as well as the aggregated governance index, ranges from −2.5 (low) to + 2.5 (high). Inflation was taken from the World Development Indicators (World Bank, 2021) and measured as the annual %-change in consumer prices. For market openness, the KOF economic globalization index is used, which ranges from 0 (closed) to 100 (open) and measures the degree of financial and trade openness (Dreher, 2006; Dreher et al., 2008; Gygli et al., 2019). For financial development, the IMF financial development index is used, which ranges from 0 (low development) to 1 (high development) (Svirydzenka, 2016). Missing values on the national variables were addressed using dummy variable adjustment, whereby the mean of the valid cases was imputed and a dummy was added to nullify them (compare Allison, 2002).The national variables were only used to study their interactions with the dependency ratios, as the direct effects of these variables on economic growth are completely covered by the fixed effects approach.
Although the risk of endogeneity bias was restricted because the current growth of the working age population is based on fertility decisions taken in the past, it remains possible that an omitted variable bias in our model was caused by migration. The classic push–pull migration model (Lee, 1966) postulates that migration is caused by the uneven processes of development. Economic opportunities in the form of wage differentials may drive migration streams. Additionally, the demographic transition will lead to a large labor supply (Preston et al., 1989), and mostly in urban areas (Sander & Charles-Edwards, 2017; Williamson, 2013), which may lead to immigration/emigration if there are many/few opportunities available in the area. Given that migration in low- and middle income areas consists mostly of young adults, as they are the most mobile group (Sander & Charles-Edwards, 2017), this may lead to a positive relationship between economic development and immigration. As a robustness test, we therefore repeated our analysis with an indicator for migration in the study period. Given that such a measure is not readily available at the subnational level, we developed a proxy of migration as follows:
$$Migratio{n}_{ij}=\frac{(100+Actual\, change\, working\, age \,share)}{(100+Expected\, change\, working\, age\, share)}$$
Substituting for the actual and expected changes in the working age share gives us:
$$Migration=\frac{100+\sum_{k=15}^{64}\Delta {\mathrm{share}}_{\mathrm{k}}}{100+\sum_{m=15-T}^{15}shar{e}_{m}^{T1}-\sum_{n=64-T}^{64}shar{e}_{n}^{T1}-\sum_{p=0}^{T}\Delta shar{e}_{p}\times \frac{WorkAg{e}^{T1}}{Elderl{y}^{T1}}}$$
where share is the share of the population of a certain age, T is the total amount of years in the time period of the subnational region, T1 indicates that the value is from the start of the period, WorkAge is the share of the population aged 15–64, and Elderly is the share of the population aged 65 or over. Simply put, the migration measure is the ratio of the actual changes of the 15–64 population shares, divided by the expected changes of the 15–64 population shares based on the T1 population structure, with an adjustment for fertility. This measure is expected to provide a reasonable indication of the role of migration, as the population dynamics in the younger age groups are appropriately included and labor migration is a process that involves mostly young people.
As a second robustness test, we repeated the main analysis, but using data from the period before the T1–T2 setup, i.e., the T0–T1 period. This enables the reader to see to what extent the results depended on the used sample. In a third robustness test we perform the same analysis using data at the national level, but split up into a rural and urban part to maintain comparability with the other analyses. This setup may highlight the importance of using sub-national data to capture context factors that are important during the DWO.
Descriptive Analyses
Before the discussion of the regression results, this section offers a descriptive perspective on the status of the DWO across the developing world. Because a detailed classification scheme for the DWO phase of a region was lacking, we use a new one that takes a simpler scheme used by the United Nations (UN) Population Division (UN, 2004, p. 70) as a starting point (compare Smits, 2016). The UN scheme distinguishes three DW phases: a pre-window phase with 30 or more percent of the population under 15 years old, a window phase with less than 30 percent under 15 years old and less than 15 percent above 64 years old, and a post-window phase with 15 or more percent above 64 years old. To get a more refined picture of the window, the GDL added a traditional phase, for countries that show hardly any sign of fertility reduction, and further subdivided the first two phases of the UN classification to obtain the following scheme:
1. Traditional phase (> 40% under 15 and < 15% over 64),
2. Pre-window phase (30–40% under 15 and < 15% over 64),
3. Early-window phase (25–30% under 15 and < 15% over 64),
4. Mid-window phase (20–25% under 15 and < 15% over 64),
5. Late-window phase (< 20% under 15 and < 15% over 64),
6. Post-window phase (> 15% over 64).
Figure 1 displays the DWO phase for 1921 subnational regions in 91 LMICs. The map reveals a substantial amount of variation in terms of DW phases both between and within LMICs. China is the only country where some sub-national regions are already in the post-window phase, while its northern neighbor, Mongolia, is mostly in the pre-window phase. India, on the other hand, has a clear north–south divide. While most of the south is in the mid-window phase, the north is still mostly in the pre-window phase. The MENA region shows a diverse pattern, with many regions in Turkey, Tunisia, and Iran already in the early-window phase, while other countries are still in the pre-window or traditional phases. In South America, countries such as Brazil and Argentina are in the lead in terms of the demographic transition.
Source: Global Data Lab
Demographic Window Phases across the developing world.
In Sub-Saharan Africa (SSA), the picture is rather different from the other continents, with much less variation between countries and regions. Although a few areas, mostly in the south, appear to be past the traditional or pre-window phases, the majority of the central African region remains in the traditional phase. In Fig. 2, which zooms in on the SSA region, the picture remains more or less the same as well, with only a limited number of areas in the South, around the Gulf of Guinea and at a few other places that are in a later stage. These areas largely coincide with the countries called 'vanguard countries' by Eloundou-Enyegue and Hirschl (2017), i.e. those countries that started the earliest with the DWO as measured by having a Total Fertility Rate (TFR) of less than 3.5.
Demographic Window Phases across the African continent.
To have an even more refined view of the demographics in SSA, Table 1 displays the SSA countries that, at the national level, are not in the traditional phase anymore. The tourist destinations Mauritius (Phase 5) and Cape Verde (Phase 3) turn out to be the most developed in this respect, while Gabon and the five most southern countries are in phase 2. Thus, in SSA, only eight countries are not in the traditional phase when countries as a whole are considered.
Table 1 Demographic window phases of SSA countries with national or urban areas that are not in the traditional demographic window phase
However, demographic developments are generally not homogeneously spread within countries. As was already clear from Fig. 1, in many countries there is substantial sub-national variation in terms of DW phases. Given that fertility reduction tends to start earlier in urban than in rural areas (Easterlin, 1971), we would expect the first signs of an emerging window to be found in the cities (Williamson, 2013). For the SSA region, variation in fertility has already been observed for countries like Nigeria and the Democratic Republic of Congo, where fertility is substantially lower in more urban and capital areas than in rural and remote areas of the countries (Jimenez & Pate, 2017; Shapiro et al., 2017). To see whether more of such signs can be discerned in the SSA region, Table 1 also displays the DW phases of the SSA countries with an urban DW phase above phase one (excluding Mauritius and Cape Verde for which no sub-national DW data were available). In 24 countries, the urban areas are in phase two, and in two countries, Lesotho, and South Africa, already in phase three.
To look even more in depth, Table 2 displays the urban areas of sub-national regions that have a DW phase of three or over (i.e., the urban areas that have surpassed the pre-window phase and have thus actually entered the window). It shows that the DW is opening in more places than one would expect based on Table 1. For instance, while Botswana as a whole is only in the second phase of the DWO and its urban areas as well, the urban area of one of its regions—South-East—is already in the fourth phase, i.e., the mid-window phase. Lastly, while Ethiopia as a whole is in phase one and its urban areas on average in phase two, the country's capital Addis Ababa is already in phase four.
Table 2 Demographic window phases of urban areas of subnational regions in SSA that have entered the DWO (phase 3 or higher)
The different countries and the time periods used in the analyses are shown in the online Supplementary Information (SI). In terms of the descriptive statistics, Table 3 shows there is substantial variation in terms of all variables. T1 IWI, i.e., the value for IWI at the start of the analyzed period, ranges from 1.82 to 94.99. Thus, there are regions where the average household in the first year owned almost none of the assets that are included in the index, while there are also regions in which households owned almost all assets in the first year. In terms of economic growth, there is also substantial variation, as the lowest average yearly change between T1 and T2 in IWI was −3.96 while the largest average yearly change was 5.72. In terms of male education, average mean years of schooling ranged from 0.23 years to 13.40 years. Regarding gender inequality in education, there are regions where men, on average, went to school 5.77 years longer than women, while there are also regions where women, on average, went to school 2.08 years longer than men.
Table 3 Sample Descriptive Statistics
Tables 4 and 5 show the results of our multivariate analyses. Model 1 in Table 4 includes all sub-national determinants of the change in IWI. The demographic window effect is clearly present, as the results show that both a lower T1 dependency ratio and a lower growth rate of the dependency ratio are associated with a significant increase in IWI growth.
Table 4 Coefficients of fixed effects regression models of selected independent variables on economic growth in subnational regions of LMICs
Table 5 Fixed effects regression models of effects of the demographic window phases on economic growth in subnational regions of LMICs
Regarding the other factors in the model, we observe the expected negative effect of IWI at T1 on the growth in IWI. IWI growth is also higher in urban regions, in regions with higher levels of male schooling, in regions with a higher growth in male schooling, in regions with a smaller gender difference in schooling and in regions where the gender difference in schooling is decreasing.
Model 2 in Table 4 introduces all significant interaction terms between the independent variables and the dependency ratios and between the subnational control factors themselves. Regarding the main effects of the independent variable, we observe little change. The coefficients of the T1 dependency ratio and of the growth of the dependency ratio remain highly significant and negative, thus again confirming that both a lower dependency ratio and a decreasing dependency ratio are associated with significantly more economic growth.
The interaction coefficients show that the effect of the dependency ratio is conditional on the level of urbanization of the region. Both the T1 dependency ratio and the growth rate of the dependency ratio have a significantly weaker negative effect in urban regions. Hence the DWO effect is, on average, stronger in rural areas. Regarding education, we observe that declines in the T1 dependency ratio are more effective in increasing growth when the region has high levels of initial male education (human capital stock). In terms of the growth rate of the dependency ratio, we find that its effect is larger in countries with a stronger institutional environment. In unreported analyses, we find that the governance effect predominantly occurs due to variation in the control of corruption component of the Worldwide Governance Indicators and that no interaction of the other components is significant if control of corruption is included. The interaction coefficient of control of corruption is −0.165 and has a t-value of −4.02. Somewhat surprisingly, we find that the DWO is more effective in regions with less financial development. This suggests that the DWO might be able to create growth even in regions where factors usually associated with job creation (inflation reduction, financial market development, economic openness) are not (yet) well developed.
Our effects are quantitatively important. For instance, a one standard deviation decrease in the T1 dependency ratio or the growth in the dependency ratio, leads to a 0.36 or 0.15 standard deviation increase in IWI growth for an average region, respectively. The effect of the T1 dependency ratio is 50% stronger than the average effect in regions with one standard deviation of T1 male education above the average, and 28% weaker than the average effect in urban regions. Further, the effect of growth in the dependency ratio is 47% stronger in regions with one standard deviation of governance above the average.
The negative effect of IWI at T1 remains in the interaction model, but it is now non-linear, and conditional on the level and change in (male) schooling. The larger the level of or increase in schooling, the stronger the T1 IWI effect. Conversely, one could argue that (faster) increases in the level of schooling have a stronger effect in regions with an initially lower level of development.
To get a more detailed picture of the relationship between the DWO and economic growth, in Table 5 the dependency ratios are replaced by the T1 DW phases. With regard to the other independent variables, the model is equal to Model 1 of Table 4. The table shows that regions in the second, third and fourth DW phase have significant higher growth rates than the traditional DW phase. This effect is still positive but not significant in the late window phase. In the post-window phase, it has disappeared completely. As such, the results are in line with the idea that the DWO is a temporary period of higher growth. However, given that the effect is already positive in the pre-window phase and not significant anymore in the late window phase, it seems that the positive effects of a decrease of the dependency ratio can already be felt earlier than has been assumed before (e.g., UN, 2004).
Robustness Tests
The models presented in Table S1 in the Supplementary Materials use the same strategy as those in Table 4, but with different data, estimation techniques or variables. The first column shows the coefficients of the model that controls for migration flows within and between regions. Although the coefficient of the migration variable has the expected positive sign, it is not significant at the conventional confidence level of 95%. All other effects are robust to the inclusion of the proxy for migratory flows. In the T0–T1 column, we use the same strategy as before, but we go back one period in time for all countries where another period of at least four years is available. Thus, instead of explaining the variation in changes in the period between times T1 and T2, we now analyze the variation in changes in the period between times T0 and T1. This is possible for 1254 regions in 55 countries. The effects of the main variables—dependency ratio and change in dependency ratio—remain significant. We also note that the main effects of the DWO increase in size, while only the interaction between the change in the dependency ratio and T1 governance remains significant. Of the other variables, we observe that the T1 difference between male and female schooling, and some interactions, lose their significance. The third column shows the results we would have obtained if we only had data at the national level, but with a split between urban/rural regions. This analysis is based on data for 91 countries and 182 observations in total. In this setup, the T1 dependency ratio and the change in the dependency ratio are no longer significant. In addition, many of the control factors and all interactions lose their significance. This highlights the added value of our sub-national approach.
Overall, we conclude that our main findings regarding DWO effectiveness are robust to the inclusion of migratory flows and the use of older survey data. Nevertheless, T1 Governance is the only significant determinants of DWO effectiveness in the T0-T1 setup. Further, we highlight that sub-national data is better able to capture context-specific effects than national data.
Discussion and Conclusion
In this study, we aim to contribute to the existing literature on the effects of the DWO on economic growth, by studying this relationship at the level of sub-national regions within LMICs. Our data reveal that there is substantial variation in terms of the DWO at the sub-national level, not only in middle-income countries, but also in low-income countries. As such, significant informative value can be gained from analyzing the effect of population age-structures on economic growth at a sub-national level in LMICs.
To achieve this, we created a dataset of 1921 regions in 91 LMICs and we used a fixed effects country dummy model to explain the variation in growth (measured by the International Wealth Index) among these regions. The results indicate that the DWO is indeed a statistical reality. Economic growth was largest in regions with a lower dependency ratio, as well as in regions where the dependency ratio decreased during the study period. A more detailed analysis in which economic growth was compared between regions in different phases of the DWO revealed that economic growth was largest in regions that are in the second to fourth phase of the DWO, called here the pre, early and mid-window phases.
Our interaction analysis showed that the effects of the DWO on economic growth are not everywhere the same. The positive associations between a lower dependency ratio and economic growth, as well as of a reduction of the dependency ratio and economic growth, are stronger in rural areas. This indicates that these areas can profit more of the DWO than more urban areas. Also, investments in education have the expected effect with a more effective DWO in areas with a higher educational level. Together, these finding suggest that rural areas, which have invested in reasonable educational facilities, have the highest propensity to reap the demographic dividend.
The positive effect of a reduction in the dependency ratio on economic growth is strengthened in regions with higher levels of development, in regions with good governance and in regions with lower levels of financial development. The first finding makes clear that for very poor regions it is more difficult to make use of the window, which sets sub-Saharan Africa at a disadvantage compared to LMICs in other regions. Regarding governance, corruption seems the decisive factor, with areas within countries with higher levels of corruption being less able to turn a decreasing dependency ratio into growth.
A low level of financial development, on the other hand, does not seem problematic for reaping the DWO. This might have to do with the way growth is measured: on the basis of household wealth instead of GDP. Whereas GDP growth might depend on business activities that require a well-developed financial infrastructure, this is less the case for increases in household wealth (as measured by the IWI). Our data show that in countries with weak financial institutions households are even better able to translate the extra income due to reduction of the number (and hence costs) of children into asset ownership and improved housing quality. This may have to do with a greater importance of informal social networks in these countries, including local savings and credit associations (Anderson & Baland, 2002; Besley et al., 1993). There are indications that informal institutions may become less effective under stricter formal regulation (Williamson, 2009).
Important is also our finding that already in the second phase of the DWO, called pre-window phase, a significant increase in economic growth is observed. Hence, regions in an early phase of fertility reduction might already experience benefits in terms of economic growth. On the other hand, regions that are in the late window phase (with less than 20% under 15 and less than 15% over 65) seem to experience less advantages of their demographic situation than countries in an earlier phase. This might mean that in those countries growth chances already start to be affected by the growing number of elderly, and that the effect of the old-age dependency ratio on growth is relatively strong.
In terms of policy implications, we urge policymakers to invest in education and particularly so in rural areas. Control of corruption should be another important policy concern, as the positive effects of good governance that are observed are completely due to improvements made in its corruption dimension. Moreover, we find that investing in education during the DWO leads to a much lower demographic dividend than investing in education early on.
In conclusion, our paper contributes to the existing literature by analyzing the effect of the DWO at subnational level for a large sample of LMICs. Population age-structures matter for economic growth, not only at the national level but also for specific areas within countries. Additionally, our interaction analysis confirms that the relationship between demography and economic growth is not a simple one but depends to a substantial extent on characteristics of the context in which the changes are taking place. Consequently, the implementation of policies such as family planning, labor market policies and education policies should take the unique characteristics of a specific area into account.
All data will be made available in case of acceptance.
Aaronson, D., Dehejia, R., Jordan, A., Pop-Eleches, C., Samii, C., & Schulze, K. (2021). The effect of fertility on mothers' labor supply over the last two centuries. The Economic Journal, 131(633), 1–32. https://doi.org/10.1093/ej/ueaa100
Allison, P. D. (2002). Missing data. Sage Publications.
Anderson, S., & Baland, J. M. (2002). The economics of roscas and intrahousehold resource allocation. The Quarterly Journal of Economics, 117(3), 963–995. https://doi.org/10.1162/003355302760193931.
Baerlocher, D., Parente, S., & L., and Rios-Neto, E. . (2019). Economic effects of demographic dividend in Brazilian regions. The Journal of the Economics of Ageing. https://doi.org/10.1016/j.jeoa.2019.100198
Barro, R. J. (1991). Economic growth in a cross section of countries. The Quarterly Journal of Economics, 106(2), 407–443. https://doi.org/10.2307/2937943
Besley, T., Coate, S., & Loury, G. (1993). The economics of rotating savings and credit associations. The American Economic Review, 792–810. https://www.jstor.org/stable/2117579.
Bloom, D., & Canning, D. (2008). Global demographic change: Dimensions and economic significance. Population and Development Review, 34, 17–51.
Bloom, D., Canning, D., Fink, G., & Finlay, J. E. (2009). Fertility, female labor force participation, and the demographic dividend. Journal of Economic Growth, 14(2), 79–101. https://doi.org/10.1007/s10887-009-9039-9
Bloom, D., Canning, D., & Sevilla, J. (2003). The demographic dividend: A new perspective on the economic consequences of population change. RAND Corporation. https://doi.org/10.7249/MR1274
Bloom, D., Kuhn, M., & Prettner, K. (2017). Africa's prospects for enjoying a demographic dividend. Journal of Demographic Economics, 83(1), 63–76. https://doi.org/10.1017/dem.2016.19
Bloom, D., & Williamson, J. G. (1998). Demographic transitions and economic miracles in emerging Asia. The World Bank Economic Review, 12(3), 419–455. https://doi.org/10.1093/wber/12.3.419
Canning, D., Raja, S., and Yazbeck, A. (Eds.). (2015). Africa's demographic transition: Dividend or disaster? The World Bank.
Collier, P., & Dollar, D. (2001). Can the world cut poverty in half? How policy reform and effective aid can meet international development goals. World Development, 29(11), 1787–1802. https://doi.org/10.1016/S0305-750X(01)00076-6
Crespo Cuaresma, J., Lutz, W., & Sanderson, W. (2014). Is the demographic dividend an education dividend? Demography, 51(1), 299–315. https://doi.org/10.1007/s13524-013-0245-x
Cristia, J. P. (2008). The effect of a first child on female labor supply: Evidence from women seeking fertility services. Journal of Human Resources, 43(3), 487–510. https://doi.org/10.3368/jhr.43.3.487
Dreher, A. (2006). Does globalization affect growth? Evidence from a new index of globalization. Applied Economics, 38(10), 1091–1110. https://doi.org/10.1080/00036840500392078
Dreher, A., Gaston, N., and Martens, P. (2008). Measuring Globalization: Gauging its Consequences. Springer.
Easterlin, R. A. (1971). Does human fertility adjust to the environment? The American Economic Review, 61(2), 399–407.
Eloundou-Enyegue, P. M., Giroux, S., & Tenikue, M. (2017). African transitions and fertility inequality: A demographic Kuznets hypothesis. Population and Development Review, 43, 59–83. https://doi.org/10.1111/padr.12034
Eloundou-Enyegue, P. M., and Hirschl, N. (2017). Fertility transitions and schooling dividends in Sub-Saharan Africa: The experience of Vanguard Countries. In H. Groth and J. F. May (Eds.), Africa's Population: In Search of a Demographic Dividend (pp. 101–111). Springer International Publishing. https://doi.org/10.1007/978-3-319-46889-1_7
Groth, H., and May, J. F. (Eds.). (2017). Africa's Population: In Search of a Demographic Dividend (1st ed. 2017). Springer International Publishing : Imprint: Springer. https://doi.org/10.1007/978-3-319-46889-1
Gygli, S., Haelg, F., Potrafke, N., & Sturm, J.-E. (2019). The KOF globalisation index—revisited. The Review of International Organizations, 14, 543–574. https://doi.org/10.1007/s11558-019-09357-x
Huisman, J., & Smits, J. (2009). Effects of household-and district-level factors on primary school enrollment in 30 developing countries. World Development, 37(1), 179–193. https://doi.org/10.1016/j.worlddev.2008.01.007.
Jimenez, E., and Pate, M. A. (2017). Reaping a demographic dividend in Africa's largest Country: Nigeria. In H. Groth and J. F. May (Eds.), Africa's Population: In Search of a Demographic Dividend (pp. 33–51). Springer International Publishing. https://doi.org/10.1007/978-3-319-46889-1_3
Kaufmann, D., Kraay, A., & Mastruzzi, M. (2011). The worldwide governance indicators: Methodology and analytical issues. Hague Journal on the Rule of Law, 3(2), 220–246. https://doi.org/10.1017/S1876404511200046
Kelley, A. C., & Schmidt, R. M. (2005). Evolution of recent economic-demographic modeling: A synthesis. Journal of Population Economics, 18, 275–300. https://doi.org/10.1007/s00148-005-0222-9
Kumar, U. (2013). India's demographic transition: Boon or bane? Asia & the Pacific Policy Studies, 1(1), 186–203. https://doi.org/10.1002/app5.9
Lee, E. S. (1966). A theory of migration. Demography, 3(1), 47. https://doi.org/10.2307/2060063
Loayza, N. V., Rancière, R., Servén, L., & Ventura, J. (2007). Macroeconomic volatility and welfare in developing countries: An introduction. The World Bank Economic Review, 21(3), 343–357. https://doi.org/10.1093/wber/lhm017
Preston, S. H., Himes, C., & Eggers, M. (1989). Demographic conditions responsible for population aging. Demography, 26(4), 691. https://doi.org/10.2307/2061266
Radelet, S., Sachs, J., & Jong-Wha, L. (2001). The determinants and prospects of economic growth in Asia. International Economic Journal, 15(3), 1–29. https://doi.org/10.1080/10168730100000041
Sander, N., and Charles-Edwards, E. (2017). Internal and International Migration. In H. Groth and J. F. May (Eds.), Africa's Population: In Search of a Demographic Dividend (pp. 333–349). Springer International Publishing. https://doi.org/10.1007/978-3-319-46889-1_21
Shapiro, D., Tambashe, B. O., and Romaniuk, A. (2017). The Third Biggest African Country: The Democratic Republic of the Congo. In H. Groth and J. F. May (Eds.), Africa's Population: In Search of a Demographic Dividend (pp. 71–86). Springer International Publishing. https://doi.org/10.1007/978-3-319-46889-1_5
Smits, J. (2016). GDL Area Database. Sub-national development indicators for research and policy-making. GDL Working Paper, 16–101.
Smits, J., & Steendijk, R. (2015). The international wealth index (IWI). Social Indicators Research, 122, 65–85. https://doi.org/10.1007/s11205-014-0683-x
Svirydzenka, K. (2016). Introducing a New Broad-Based Index of Financial Development. IMF Working Paper No. 16/5, 1–44.
Turbat, V. (2017). Economic Growth and Public and Private Investments. In H. Groth and J. F. May (Eds.), Africa's Population: In Search of a Demographic Dividend (pp. 353–365). Springer International Publishing. https://doi.org/10.1007/978-3-319-46889-1_22
United Nations. (2004). World Population to 2300. United Nations Population Division.
United Nations. (2018). World Urbanization Prospects: The 2018 Revision. Department of Economic and Social Affairs, Population Division.
Wei, Z., & Hao, R. (2010). Demographic structure and economic growth: Evidence from China. Journal of Comparative Economics, 38(4), 472–491. https://doi.org/10.1016/j.jce.2010.08.002
Williamson, C. R. (2009). Informal institutions rule: Institutional arrangements and economic performance. Public Choice, 139(3–4), 371–387. https://doi.org/10.1007/s11127-009-9399-x
Williamson, J. G. (2013). Demographic dividends revisited. Asian Development Review, 30(2), 1–25.
Wooldridge, J. M. (2013). Introductory econometrics: A modern approach (5th ed). South-Western Cengage Learning.
World Bank. (2021). World Development Indicators. Inflation, consumer prices (annual %). https://databank.worldbank.org/home.aspx
Zuber, A., Blickenstorfer, C., and Groth, H. (2017). Governance, Transparency, and the Rule of Law. In H. Groth and J. F. May (Eds.), Africa's Population: In Search of a Demographic Dividend (pp. 367–384). Springer International Publishing. https://doi.org/10.1007/978-3-319-46889-1_23
We gratefully acknowledge the comments and suggestions made by participants of the 8th African Population Conference held in Entebbe, Uganda in November 2019.
Open Access funding provided by ETH Zurich. There was no project-specific funding.
ETH Zürich, KOF Swiss Economic Institute, Konjunkturforschungsstelle, LEE G224 Leonhardstrasse 21, 8092, Zürich, Switzerland
Lamar Crombach
Institute for Management Research, Global Data Lab, Radboud University, PO.Box 9108, 6500HK, Nijmegen, The Netherlands
Jeroen Smits
Correspondence to Lamar Crombach.
Below is the link to the electronic supplementary material.
Supplementary file1 (DOCX 141 kb)
Crombach, L., Smits, J. The Demographic Window of Opportunity and Economic Growth at Sub-National Level in 91 Developing Countries. Soc Indic Res 161, 171–189 (2022). https://doi.org/10.1007/s11205-021-02802-8
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Elliptical orbit of revolution of earth [duplicate]
How did planets have an elliptical orbit in the first place? [duplicate] 5 answers
Why don't planets have Circular orbits? 6 answers
Well known Kepler laws state that earth revolve around sun in an elliptical path with sun at one of the focii. My question is rather simple. Why so?
I mean for an equilibrium of earth that is it is not accelerated towards or away from sun its gravitational force must be balanced by centrifugal force (pseudo force as frame corresponding to earth)
But following an elliptical path it distance from sun won't be same and hence the gravitational force and hence how is this?
I suspect may be it is due to presence of other celestial bodies but that seems too vague and escaping from the truth
Also maximum of derivations in gravitation are done by assuming circular path of earth around sun. (atleast at my level)
Please give me the real physics behind it?
newtonian-mechanics newtonian-gravity orbital-motion celestial-mechanics
Pranjal RanaPranjal Rana
marked as duplicate by Emilio Pisanty, stafusa, M. Enns, John Rennie newtonian-mechanics Users with the newtonian-mechanics badge can single-handedly close newtonian-mechanics questions as duplicates and reopen them as needed. Sep 30 '17 at 4:46
$\begingroup$ What do you mean by "maximum of derivations in gravitation are done by assuming circular path of earth"? $\endgroup$ – Floris Sep 29 '17 at 14:43
$\begingroup$ Many of them like proof of time and orbit relationship of kepler $\endgroup$ – Pranjal Rana Sep 29 '17 at 14:45
$\begingroup$ There's a story that when asked why planets move in ellipses, to answer that question, Newton invented Calculus. Kepler didn't know why, he just worked out what the orbits did. Kepler wasn't even a fan of ellipses cause they seemed as counter-intuitive to him as everyone else in his day. He just worked out and published the math. Newton worked out why. futurism.com/… $\endgroup$ – userLTK Sep 29 '17 at 15:16
$\begingroup$ Given that there has to be an orbit even the smallest perturbation to a circular orbit will result in an elliptical orbit so the chance of an orbit being exactly circular is negligible. $\endgroup$ – Farcher Sep 29 '17 at 15:37
$\begingroup$ Why do you think the earth isn't accelerating towards the sun? It is constantly accelerating towards the sun. The direction of the acceleration and the direction of the movement need not be the same; I suspect you are confusing them. $\endgroup$ – Eric Lippert Sep 29 '17 at 17:15
What you are missing is that Earth speed is varying along its orbit, contrary to a circular one.
More precisely, at any point of the orbit, the acceleration of the Earth has a component tangent to the orbit, and a component perpendicular to the orbit. For a circular orbit, the former is always zero, and the latter is therefore exactly equal to the gravitational force, i.e. a centripetal acceleration. That was in an inertial frame, so now if we take a frame rotating along with the Earth, that's the picture you had in mind: the centrifugal force, which is opposite to the centripetal acceleration in the inertial frame, compensates the gravitational force.
But for an elliptic orbit, the component of the acceleration tangent to the orbit is only zero at the apogee and perigee. Since the gravitational force is equal to the sum of the tangent and normal component of the acceleration, we can say that part of the gravitational force bends the trajectory toward the Sun, and part accelerate the Earth along the trajectory (or decelerate it, depending on the position on the orbit).
There are many mathematical derivations of this online; let me give you the intuitive explanation that I think you are looking for.
I will look at two points on the ellipse: when the Earth is closest to the sun (perihelion) and when it is furthest (aphelion). Throughout, we assume conservation of angular momentum - that is, the product of the orbital velocity and the distance are constant.
When the Earth is at aphelion, we will assume velocity $v_a$, and distance to the Sun $r_a$. We know the force of gravity will be proportional to $\frac{1}{r_a^2}$. The centripetal force needed for a circular orbit is $$F_c = \frac{mv_a^2}{r_a}$$
If we express this in terms of the angular momentum $L=mv_ar_a$ (which is constant) we get
$$F_c = \frac{L^2}{mr_a^3}$$
But we know the force of gravity scales with the inverse square, not the inverse cube: so when we are far away, the force will be "stronger than is needed" and the Earth will "fall towards the Sun". When we are close, the force is not strong enough and we "don't fall fast enough for a circular orbit".
The result is an elliptical orbit. This was already proven by Newton (incidentally, the elliptical result only follows if the power law is inverse square - that makes for an interesting large scale proof that gravity is indeed following that law at large scales).
FlorisFloris
$\begingroup$ Fantastic job Floris, I am agreed but one small doubt flickers..... If the orbit is circular then what is the use of aphelion and perhelion. $\endgroup$ – Pranjal Rana Sep 29 '17 at 15:15
$\begingroup$ When the orbit is circular, the velocity is just right and there is no "closest" or "furthest" point... $\endgroup$ – Floris Sep 29 '17 at 15:19
There are essentially five solutions to the two body problem, where the bodies are attracted to each other by gravity.
A hyperbolic path - the two bodies are moving away from each other too quickly for gravity to bring them into a closed orbit.
A parabolic path - the two bodies are moving away from each other just quickly enough for gravity not to be able to bring them into a closed orbit. This is a kind of "boundary case" between the hyperbolic path and the elliptical orbit.
An elliptical orbit - the general case when gravity between the two bodies limits the distance between them.
A circular orbit - a very special variant of the elliptical orbit, where the direction of the bodies' velocity away from each other is exactly perpendicular to the displacement between them, and the magnitude of that velocity is exactly the right amount to balance out the gravitational attraction.
A collision course where the bodies move directly towards each other, although they may start off moving away from each other.
For the parabolic solution or the circle solution, conditions need to be exactly right. That is, the initial relative velocity of the bodies needs to be a precisely calculated magnitude, in a precisely calculated direction. This would require an enormous coincidence. That's why the other three solutions (hyperbola, ellipse or collision course) are the only ones that every happen in nature.
In particular, if there were ever a satellite or a planet orbiting a planet or a star in an exactly circular orbit, it would only take a very slight tug from a third body to perturb it into an elliptical orbit.
Dawood ibn KareemDawood ibn Kareem
Not the answer you're looking for? Browse other questions tagged newtonian-mechanics newtonian-gravity orbital-motion celestial-mechanics or ask your own question.
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\begin{document}
\title{Enhanced the thermal Entanglement in Anisotropy Heisenberg $XYZ$ Chain} \author{L. Zhou, H. S. Song, Y. Q. Guo and C. Li} \address{{\small Department of Physics, Dalian University of Technology,}\\ Dalian, 116024, P. R. China} \maketitle
\begin{abstract} The thermal entanglement in Heisenberg $XYZ$ chain is investigated in the presence of external magnetic field $B$. In the two-qubit system, the critical magnetic field $B_c$ is increased because of introducing the interaction of the z-component of two neighboring spins $J_z$. This interaction not only improves the critical temperature $T_c,$ but also enhances the entanglement for particular fixed $B$. We also analyze the pairwise entanglement between nearest neighbors in three qubits. The pairwise entanglement, for a fixed $T$, can be strong by controlling $B$ and $J_z$.
PACS: 03.65. Ud, 03.67. -a, 75.10. Jm \end{abstract}
\section{Introduction}
Entanglement is an important resource in quantum information \cite {bennett}. The ideal case which Quantum computing and quantum communication \ are put into use \ is to find entanglement resource in solid system at a finite temperature. The Heisenberg model is a simple but realistic and extensively studied solid state system \cite{PRH}\cite{SE}. Recently, Heisenberg interaction is not localized in spin system. It can be realized in quantum dots \cite{DL}, nuclear spins \cite{BEK}, cavity QED \cite{AI} \cite{SBZ}. This effect Hamiltonian can be used for quantum computation \cite {DAL} and controlled NOT gate \cite{SBZ}. The thermal entanglement in isotropic Heisenberg spin chain has been studied in the absence [9,10,15] and in the presence of an external magnetic $B$ [9,10,14]. The entanglement of two-qubit isotropic Heisenberg system decreases with the increasing $T$ and vanishes beyond a critical value $T_c$ [9,10], which is independent of $ B.$ Pairwise entanglement in $N$-qubit isotropic Heisenberg system in certain degree can be increased by increasing the temperature or the external field $B$ \cite{MC}. Anisotropic Heisenberg spin chain has been investigated in the case of $B=0$ \cite{wxg} and $B\neq 0$ \cite{GL}. For a two-qubit anisotropic Heisenberg $XY$ chain, one is able to produce entanglement for finite $T$ by adjusting the magnetic field strength \cite {GL}. However, the entanglement by increasing $T$ or $B$, in two-qubit anisotropic Heisenberg $XY$ chain \cite{GL} or in $N$-qubit isotropic Heisenberg chain \cite{MC}, is very weak. How to produce strong entanglement is worthy to study.
On the other hand, we have not find the work about two-qubit or the $N$ -qubit anisotropic $XYZ$ Heisenberg chain in the presence of magnetic field. Although the $N$-qubit Heisenberg chain has been studied \cite{KMO} \cite{MC} , in Ref. \cite{KMO} the authors studied the maximum possible nearest neighbor entanglement for ground state in a ring of $N$ qubits, and in \cite {MC} they just investigated the case of isotropy $N$ qubits Heisenberg chain. In this paper, we study the entanglement of two-qubit anisotropic Heisenberg $XYZ$ chain and the pairwise entanglement of three-qubit anisotropic Heisenberg $XYZ$ chain. Introducing the interaction of the z-component of two neighboring spins not only improve the critical temperature $T_c$ but also enhance the entanglement for fixed $B$ and $T$ in particular regions. In the case of anisotropic three-qubit Heisenberg $XYZ$ chain, the effect of partial anisotropy $\gamma $ make the revival phenomenon more apparent than in two-qubit chain; for a fixed $T$, one can obtain a robust entanglement by controlling $B$ and $J_z$.
The Hamiltonian of $N$-qubit anisotropic Heisenberg $XYZ$ model in an external magnetic field $B$ is \cite{GL} \begin{equation} H=\frac 12\sum_{i=1}^N[J_x\sigma _i^x\sigma _{i+1}^x+J_y\sigma _i^y\sigma _{i+1}^y+J_z\sigma _i^z\sigma _{i+1}^z+B(\sigma _i^z+\sigma _{i+1}^z)], \end{equation} where $\overrightarrow{\sigma }_j=(\sigma _j^x,\sigma _j^y,\sigma _j^z)$ is the vector of Pauli matrices and $J_i(i=x,y,z)$ is real coupling coefficient. The coupling coefficient $J_i$ of arbitrary nearest neighbor two qubits is equal in value. For the spin interaction, the chain is said to be antiferromagnetic for $J_i>0$ and ferromagnetic for $J_i$ $<0$.
For a system in equilibrium at temperature $T$ , the density operator is $ \rho =Z^{-1}\exp (-H/k_BT)$, where $Z=Tr[\exp (-H/k_BT)]$ is the partition function and $k_B$ is Boltzmann's constant. For simplicity we write $k_B=1$. Entanglement of two qubits can be measured by concurrence $C$ which is written as $C=\max (0,2\max \{\lambda _i\}-\sum_{i=1}^4\lambda _i)$ \cite {CHB}\cite{sh}\cite{ca}, where $\lambda _i$ is the square roots of the eigenvalues of the matrix $R=\rho S\rho ^{*}S$, $\rho $ is the density matrix, $S=\sigma _1^y\otimes \sigma _2^y$ and $*$ stand for complex conjugate. The concurrence is available no matter what $\rho $ is pure or mixed.
\
\section{Two-qubit Heisenberg $XYZ$ chain}
Now, we consider the Hamiltonian for anisotropic two-qubit Heisenberg $XYZ$ chain in an external magnetic field $B$. The Hamiltonian can be expressed as \begin{equation} H=J(\sigma _1^{+}\sigma _2^{-}+\sigma _1^{-}\sigma _2^{+})+J\gamma (\sigma _1^{+}\sigma _2^{+}+\sigma _1^{-}\sigma _2^{-})+\frac{J_z}2\sigma _1^z\sigma _2^z+\frac B2(\sigma _1^z+\sigma _2^z) \end{equation} where $\sigma ^{\pm }=\frac 12(\sigma ^x\pm i\sigma ^y)$ is raising and lowering operator respectively, and $J=\frac{J_x+J_y}2,$ $\gamma =\frac{
J_x-J_y}{J_x+J_y}$. The parameter $\gamma $ $(0<\gamma <1)$ measure the anisotropy (partial anisotropy) in $XY$ plane. When the Hamiltonian of the system has the form of Eq.(2), in the standard basis $\{|00\rangle
,|01\rangle ,|10\rangle ,|11\rangle \}$, the density matrix of the system can be written as \begin{equation} \rho _{12}=\left( \begin{array}{llll} u_1 & 0 & 0 & v \\ 0 & w & z & 0 \\ 0 & z & w & 0 \\ v & 0 & 0 & u_2 \end{array} \right) . \end{equation} These nonzero matrix element can be calculated through \begin{eqnarray}
u_1 &=&Tr(|00\rangle \langle 00|\rho ),u_2=Tr(|11\rangle \langle 11|\rho ), \nonumber \\
w &=&Tr(|01\rangle \langle 01|\rho ),v=Tr(|00\rangle \langle 11|\rho
),z=Tr(|01\rangle \langle 10|\rho ). \end{eqnarray}
The square roots of the eigenvalues of the matrix $R$ are $\lambda _{1,2}=|w\pm z|,\lambda _{3,4}=|\sqrt{u_1u_2}\pm v|$. Therefore, we can calculate the concurrence.
The eigenvalues and eigenstates of $H$ are easily obtained as $H|\Psi ^{\pm
}\rangle =(-\frac{J_z}2\pm J)|\Psi ^{\pm }\rangle $, $H|\Sigma ^{\pm
}\rangle =(\frac{J_z}2\pm \eta )|\Sigma ^{\pm }\rangle ,$with the eigenstates $|\Psi ^{\pm }\rangle =\frac 1{\sqrt{2}}(|01\rangle \pm
|10\rangle )$, $|\Sigma ^{\pm }\rangle =\frac 1{\sqrt{2\eta (\eta \mp B)}}
[(\eta \mp B)|00\rangle \pm J\gamma |11\rangle ]$, where $\eta =\sqrt{
B^2+(J\gamma )^2}$. One can notice that the eigenstates are the same as the case of $J_z=0$\cite{GL}. Because the basises $|01\rangle $ and $|10\rangle $ are the two degenerate eigenstates of $\sigma _1^z\sigma _2^z$ with eigenvalue $-1$, hence the superposition of the two degenerate states $
|01\rangle $ and $|10\rangle $ still is the eigenstate of $\sigma _1^z\sigma _2^z$, that is, $|\Psi ^{\pm }\rangle $ is the eigenstate of $J_z=0$ as well as that of $J_z\neq 0$. The same reason account for $|\Sigma ^{\pm }\rangle $ both as an eigenstate of Eq.(2) and as that of the case of $J_z=0$ . From Eq.(4), tracing on the eigenstates, we obtain the square roots of the eigenvalues of the matrix $R$ \begin{eqnarray} \lambda _{1,2} &=&Z^{-1}e^{\frac{\beta J_z}2}e^{\pm \beta J}, \nonumber \\
\lambda _{3,4} &=&Z^{-1}e^{-\frac{\beta J_z}2}|\sqrt{1+(\frac{J\gamma }\eta
\sinh \beta \eta )^2}\mp \frac{J\gamma }\eta \sinh \beta \eta |, \end{eqnarray} where the partition function $Z=2(e^{-\frac{J_z}{2T}}\cosh \beta \eta +e^{ \frac{\beta J_z}2}\cosh \beta J)$. Because the concurrence is invariant under the substitutions $J\rightarrow -J$ and $\gamma \rightarrow -\gamma $ \cite{GL}, we will consider the case $J>0$ and $0<\gamma <1$. But with substitution $J_z\rightarrow -J_z$ the the concurrence is variant. We choose $J_z>0$, and we will state the reason later.
We first review the circumstance of anisotropic Heisenberg $XY$ chain, which is analyzed in \cite{GL}. At $T=0,$ exist a critical magnetic field $B_c$. As $B$ cross $B_c$, the concurrence $C$ drops suddenly then undergoes a ''revival'' for sufficient large $\gamma .$ However, we noticed that $B_c$ decrease with the increasing of the anisotropic parameter $\gamma $. Although with $\gamma $ increasing the critical temperature $T_c$ is improved, the entanglement, when temperature is in the revival region, is very weak.
With $\gamma =0.3$, we show the concurrence as a function of $B$ and $T$ for two values of $J_z$ in Fig. 1. For $J_z=0$ (Fig.1a) corresponding to the circumstance of anisotropic Heisenberg $XY$ chain \cite{GL}, one can observe a revival phenomenon and the weak entanglement in revival region. For the convenience of representation, we define the main region in which concurrence $C$ keeping its constant and maximal value. Comparing Fig. 1 (a) with (b), we find that with the increasing of $J_z,$ the main region is extended in terms of $B$ and $T$, i.e., the critical magnetic field $B_c$ is broadened and the critical temperature $T_c$ in main region is improved. That is to say, the range of concurrence $C$ keeping its constant and maximal is extended in terms of $B$ and $T,$ so we can obtain strong entanglement in the extended range.
We can understand the effect of $J_z$ on $B_c$ from the case of $T=0$. For $ T=0$ under the condition of $J_z\leq J$, $C$ can be written analytically as \begin{equation} C(T=0)=\left\{ \begin{array}{c} 1\text{ \qquad \qquad \quad \quad for }\eta <J+J_z \\ (1-J\gamma /\eta )/2\text{ \quad for }\eta =J+J_z \\ J\gamma /\eta \text{ \quad \quad \quad \quad for }\eta >J+J_z \end{array} \right. \end{equation} The parameters $J$, $\eta $ and $\gamma $ are independent of $J_z$ in the case of two interacting qubits. Comparing Eq.(6) with Eq.(6) of Ref.\cite{GL} , we can see clearly that if $J_z$ is positive, $J_z$ makes the intersection points of piecewise function shift. In this paper, we consider the case of $ J_z>0$. Fig. 2 shows the concurrence at $T=0$ for three values of positive $ J_z$. It show clearly that concurrence drops sharply at a finite value of magnetic field $B$, which is called critical magnetic field $B_c$, at which the quantum phase transition occurs\cite{GL}. But with the increasing of $ J_z $, $B_c$ is increased. The interaction of the z-component of two neighboring spins $J_z$ causes a shift in the locations of the phase transitions. Namely, the presence of positive Jz increases the region over which the concurrence C attains its maximum value.This result means that in larger region of $B$ and $T$ we can obtain stronger entanglement. The effect of $J_z $ is different with that of $\gamma $ on changing $B_c$. In the case of $J_z=0$ \cite{GL}, although with the increasing of $\gamma $ the critical temperature $T_c$ is increased, the larger the values of $\gamma $, the smaller the critical magnetic field $B_c$. Here, introducing the z-component interaction of two neighboring spins not only extends critical magnetic field $B_c$ but also improves critical temperature $T_c$ and the entanglement (we will further show it in Fig. 3).
Let us consider concurrence changing with temperature for different values of $J_z$ in a fixed $B$ ($B=1.1$). We plot it in Fig. 3 with $\gamma =0.3$. We notice that existing a critical temperature $T_c$ at which the entanglement vanishes. Obviously, $T_c$ is improved monotonously with increasing of $J_z$. Under the condition $J_z=0$ (corresponding to $XY$ model \cite{GL}), the concurrence exhibit a revival phenomenon, but the maximal values of entanglement in both area are small. If introducing the $ J_z$, the critical external magnetic field $B_c$ become larger so that $ B=1.1 $ is less than $B_c$ (the critical magnetic when $J_z=0.2,0.5$ or $ J_z=0.9$), thus we observe the maximal value of entanglement 1. In the temperature range $0<T<1.725$ , the larger $J_z$ the stronger entanglement. Therefore, $J_z$ not only improve the critical temperature $T_c,$ but also enhance the entanglement for particular fixed $B$ and $\gamma .$
\section{The pairwise entanglement in three qubits}
The calculation of pairwise entanglement in $N$ qubits is very complicated due to the anisotropy in Heisenberg $XYZ$ chain. Here we just calculate the pairwise entanglement in three qubits to show the effects of $J_z$ . We now solve the eigenvalue problems of the three-qubit $XYZ$ Hamiltonian. We list the eigenvalues and the corresponding eigenvectors as follow
\begin{eqnarray}
E_{1,2} &=&-J-\frac{J_z}2+B:|\Phi _{1,2}\rangle =\pm \frac 12(1\mp \frac 1{
\sqrt{3}})|110\rangle +\frac 1{\sqrt{3}}|101\rangle \mp \frac 12(1\pm \frac 1
{\sqrt{3}})|011\rangle ), \nonumber \\
E_{3,4} &=&J+\frac{J_z}2-B\pm \eta _{-}:|\Phi _{3,4}\rangle =\frac 1{\sqrt{
2\eta _{-}[\eta _{-}\pm (J_z-2B-J)]}}[(J_z-2B-J\pm \eta _{-})|000\rangle
+J\gamma \sum_{n=0}^2\Upsilon ^n|110\rangle ]; \nonumber \\
E_{5,6} &=&-J-\frac{J_z}2-B:|\Phi _{5,6}\rangle =\pm \frac 12(1\mp \frac 1{
\sqrt{3}})|010\rangle +\frac 1{\sqrt{3}}|100\rangle \mp \frac 12(1\pm \frac 1
{\sqrt{3}})|001\rangle ); \nonumber \\
E_{7,8} &=&J+\frac{J_z}2+B\pm \eta _{+}:|\Phi _{7,8}\rangle =\frac 1{\sqrt{
2\eta _{+}[\eta _{+}\pm (J_z+2B-J)]}}[(J_z+2B-J\pm \eta _{+})|111\rangle
+J\gamma \sum_{n=0}^2\Upsilon ^n|010\rangle ]. \end{eqnarray} where $\eta _{\pm }=\sqrt{(J_z-J\pm 2B)^2+3(J\gamma )^2},\Upsilon $ is the cyclic right shift operator \cite{wxg3}. The reduced density matrix of two nearest-neighbor qubits in $N$ qubits system also has the form of Eq.(3). Employing Eq.(4) and tracing on the basis of eigenstates shown in Eq. (7), one can get the density matrix $\mu _1$, $\mu _2,w$, $z,v$, then further obtain the concurrence. Here we do not write the expressions of $\lambda _i$ because it is very long. We will directly plot some curves to show the effect of $J_z$ on enhancing entanglement.
Fig.4 show concurrence as a function of $B$ and $T$ with $\gamma =0.3$, $ J_z=0.9$ and $J=1.0$ in three-qubit $XYZ$ Heisenberg chain. We see that with the same $\gamma =0.3,$ the effect of partial anisotropy $\gamma $ make the revival phenomenon more apparent than in two-qubit chain. When $B=4$ in Fig. 1, the largest critical temperature $T_c$ produced by $\gamma $ is about $ 1.0 $ (Fig.1a); due to the restrain of $J_z$ the maximum temperature only caused by $\gamma $ is below 0.8(Fig.1b). However, in three-qubit system if $ B=4$ with the same set of parameters, comparing Fig.1b with Fig.4, the critical temperature $T_c$ in revival region almost equal to $1.8$. The stronger effect of $\gamma $ implies that if we aim to obtain strong entanglement we can decrease $\gamma $ properly and increase $J_z$, otherwise increasing $\gamma $ can make the revival phenomenon more evident. Of course, the coupling constant $J_z$ also increase magnetic field $B_c$ and expend the region of concurrence keeping constant in terms of $B$ and $T$ as it do in two-qubit (for the limited of the page,we do not plotted here) .
For $T=0.6$, Fig.5 show concurrence as function of $B$ and $J_z$. There is no entanglement for $B=0$, which corresponds with Fig.4. If $J_z$ is below a certain value, in case of Fig. 5 the value
is about $0.2$, the entanglement appears in one area corresponding to the ''revival''\cite{GL} on condition that the magnetic field is larger than a certain value, and the certain value of $B$ is increased with the enhanced of $J_z$. But, if $J_z$ is larger than $0.2$, there are two areas appearing entanglement, and the entanglement appearing in the lower range of $B$ can be much stronger than that in higher magnetic field. In the lower range of $B$, for a certain $B$, the large $J_z$ the large concurrence. Thus, in the $N$-qubits $XYZ$ system, for a fixed $T$, one can obtain a robust entanglement by controlling $B$ and $J_z$.
\section{Conclusion}
The thermal entanglement in anisotropic $XYZ$ Heisenberg chain is investigated. Through analyzing the two-qubit system, we find that with the increasing of $J_z,$ the critical magnetic field $B_c$ is increased; the coupling along $Z$ not only improves the critical temperature $T_c$, but also enhances the entanglement for certain fixed $B$. We also analyze the entanglement between two nearest neighbors in three qubits and find that the effect of partial anisotropy is more evident than it do in two-qubit system. The pairwise entanglement exhibit a interesting phenomenon. For certain fixed $B$ , if the coupling constant $J_z$ is small, the pairwise entanglement only exists in relative strong magnetic field $B$ and the entanglement is weak. By increasing $J_z,$ in lower range of $B$ , one can obtain a strong entanglement. Therefore, interaction constant of the z-component of two neighboring spins $J_z$ play important role in enhancing entanglement and in improving the critical temperature.
This work was supported by Ministry of Science and Technology of China under Grant No.2100CCA00700
The captions of the figure:
Fig. 1 Concurrence in two-qubit Heisenberg $XYZ$ chain is plotted vs $T$ and $B$, where (a): $J_z=0$, (b): $J_z=0.9$. For all plotted $J=1.0$, $\gamma =0.3$.
Fig. 2 Concurrence in two-qubit Heisenberg $XYZ$ chain vs $B$ at zero temperature for various values of $J_z$ with $\gamma =0.3$ and $J=1.0$. From left to right $J_z$ equal to 0, 0.5, 0.9, respectively.
Fig.3 Concurrence in two qubits Heisenberg $XYZ$ chain is plotted vs $T$ . For all plotted $J=1.0$, $B=1.1$,$\gamma =0.3$.From top to bottom $J_z$ equal to 0.9, 0.5, 0.2, 0, respectively.
Fig.4 Pairwise entanglement in three-qubit Heisenberg $XYZ$ chain is plotted as a function of $T$ and $B$, where $\gamma =0.3$, $J=1.0$, $J_z=0.9$.
Fig.5 Pairwise entanglement is plotted as a function of $B$ and $J_z$, where $T=0.6$, $J=1.0$, $\gamma =0.3$ .
\begin{references} \bibitem{bennett} C. H. Bennett and D. P. DiVincenze, Nature {\bf 404}, 247 (2000)
\bibitem{PRH} P. R. Hammar et al, Phys. Rev. B 59, 1008 (1999)
\bibitem{SE} S. Eggert, I. Affleck and M. Takahashi, Phys. Rev. Lett.,73, 332 (1994)
\bibitem{DL} D. Loss and D. P. DiVincenzo, Phys. Rev. A, 57, 120 (1998);G. Burkard, D. Loss and D. P. DiVincenzo, Phys. Rev. B, 59, 2070 (1999)
\bibitem{BEK} B. E. Kane, Nature 393, 133 (1998)
\bibitem{AI} A. Imamoglu, D. D. Awschalom, G. Burkard, D. P. DiVincenzo, D. Loss, M. Sherwin, A. Small, Phys. Rev. Lett. 83, 4204 (1999)
\bibitem{SBZ} S. B. Zheng, G. C. Guo, Phys. Rev. Lett. 85, 2392 (2000)
\bibitem{DAL} D. A. Lidar, D. Bacon, and K. B. Whaley, Phys. Rev. Lett. 82, 4556 (1999)
\bibitem{MC} M. C. Arnesen, S. Bose and V. Vedral, Phys. Rev. Lett. 87, 017901 (2001)
\bibitem{wxg} X. Wang, Phys. Rev. A 64, 012313 (2001)
\bibitem{GL} G. L. Kamta and A. F. Starace, Phys. Rev. Lett. 88, 107901(2002)
\bibitem{KMO} K. M. OConnor and W. K.Wootters Phys. Rev. A 63, 052302 (2001)
\bibitem{ca} C. Anteneodo and M. C. Souza, J of Opt. B 5, 73 (2003)
\bibitem{wxg2} X. Wang, Phys. Rev. A 66,034302 (2002)
\bibitem{wxg3} X. Wang, Phys. Rev. A 66,044305 (2002)
\bibitem{CHB} C. H. Bennett, D. P. DiVincenzo, J. A. Smolin and W. K. Wootters Phys. Rev. A 54, 3824 (1996)
\bibitem{sh} S. Hill and W. K. Wootters Phys. Rev. Lett. 78, 5022 (1997) \end{references}
\end{document} | arXiv |
\begin{document}
\begin{titlepage}
\begin{center}
\vspace*{1cm}
\Huge \textbf{The Freyd-Mitchell Embedding Theorem}
\Large
\textbf{Arnold Tan Junhan}
\Large
\textbf{Michaelmas 2018 Mini Projects: Homological Algebra}
\Large
University of Oxford
\end{center} \end{titlepage} \tableofcontents
\section{Abstract}
Given a small abelian category $\mathcal{A}$, the \textit{Freyd-Mitchell embedding theorem} states the existence of a ring $R$ and an exact full embedding $\mathcal{A} \rightarrow R$-Mod, $R$-Mod being the category of left modules over $R$. This theorem is useful as it allows one to prove general results about abelian categories within the context of $R$-modules. The goal of this report is to flesh out the proof of the embedding theorem. \\
\noindent We shall follow closely the material and approach presented in Freyd (1964). \\ This means we will encounter such concepts as projective generators, injective cogenerators, the Yoneda embedding, injective envelopes, Grothendieck categories, subcategories of mono objects and subcategories of absolutely pure objects. This approach is summarised as follows: \begin{itemize}
\item the functor category $[\mathcal{A}, Ab]$ is abelian and has injective envelopes.
\item in fact, the same holds for the full subcategory $\mathcal{L}(\mathcal{A})$ of left-exact functors.
\item $\mathcal{L}(\mathcal{A})^{op}$ has some nice properties: it is cocomplete and has a projective generator.
\item such a category embeds into $R$-Mod for some ring $R$.
\item in turn, $\mathcal{A}$ embeds into such a category. \end{itemize} \section{Basics on abelian categories}
Fix some category $\mathcal{C}$. Let us say that a monic $A \rightarrow B$ is \textbf{contained} in another monic $A' \rightarrow B$ if there is a map $A \rightarrow A'$ making the diagram
\[
\begin{tikzcd}[row sep=0.4em,column sep=2em]
A \arrow[dd] \arrow[dr] \\
& B & \text{commute.}\\
A' \arrow[ur] & \\
\end{tikzcd}
\]
\noindent We declare two monics $A \rightarrow B$ and $A' \rightarrow B$ to be \textbf{equivalent} if each is contained in the other. In this case $A$ and $A'$ are isomorphic. \\ A \textbf{subobject} of $B$ is an equivalence class of monics into $B$. The relation of containment gives a partial ordering on subobjects. \\
\noindent Dually, let us declare two epics $B \rightarrow C$ and $B \rightarrow C'$ in $\mathcal{C}$ to be \textbf{equivalent} if there are maps $C \rightarrow C'$ and $C' \rightarrow C$ such that
\[
\begin{tikzcd}[row sep=0.4em,column sep=2em]
& C \arrow[dd] &&& C' \arrow[dd] \\
B \arrow[ur] \arrow[dr] &&\text{and}& B \arrow[ur] \arrow[dr]&& \text{commute.}\\
& C' &&& C \\
\end{tikzcd}
\]
\noindent A \textbf{quotient object} of $B$ is an equivalence class of epics out of $B$, and we say the quotient object represented by $B \rightarrow C$ is \textbf{smaller} than that represented by $B \rightarrow C'$ if we have just the right diagram above.
\noindent If two quotient objects $B \rightarrow C$, $B \rightarrow C'$ are equivalent, then $C \cong C'$. \\
\noindent When the context is clear, we will often just say \textit{$A$ is a subobject of $B$}, or \textit{$C$ is a quotient object of $B$}. \\
\begin{defn} A category is \textbf{complete} if every pair of maps has an equaliser, and every indexed set of objects has a product. Dually, a category is \textbf{cocomplete} if every pair of maps has a coequaliser, and every indexed set of objects has a sum. \\ A category is \textbf{bicomplete} if it is both complete and cocomplete. \\ \end{defn}
\begin{defn} A category $\mathcal{A}$ is \textbf{abelian} if
\begin{itemize}
\item[A0.] $\mathcal{A}$ has a zero object.
\\
\item[A1.] For every pair of objects there is a product and
\item[A1*.] a sum.
\\
\item[A2.] Every map has a kernel and
\item[A2*.] a cokernel.
\\
\item[A3.] Every monomorphism is a kernel of a map.
\item[A3*.] Every epimorphism is a cokernel of a map.
\\
\\ \end{itemize} \end{defn}
\noindent Let $A$ be an object in abelian category $\mathcal{A}$. Let $\mathcal{S}$ and $\mathcal{Q}$ be the families of subobjects and quotient objects of $A$, respectively. Define two functions $Cok: \mathcal{S} \rightarrow \mathcal{Q}$ and $Ker: \mathcal{Q} \rightarrow \mathcal{S}$, where $Cok$ assigns to each subobject its cokernel, and $Ker$ assigns to each quotient object its kernel. Note that these are order-reversing functions. For instance, if the monic $A \rightarrow B$ is contained in the monic $A' \rightarrow B$, then the epic $Cok(A' \rightarrow B)$ is smaller than the epic $Cok(A \rightarrow B)$. \\
\begin{thm} \label{2.11} For each $A$ in an abelian category $\mathcal{A}$, $Ker$ and $Cok$ are mutually inverse functions. \end{thm} \begin{proof} We note in passing that $Ker$ and $Cok$ really are well-defined, as a kernel is always monic and a cokernel always epic. \\ Now let $A' \rightarrow A$ be a monic. Let $A \rightarrow F$ be the cokernel of $A' \rightarrow A$, and $K \rightarrow A$ the kernel of $A \rightarrow F$. We must show that $K \rightarrow A$ is the same subobject as $A' \rightarrow A$. \\ By Axiom A3 $A' \rightarrow A$ is the kernel of some $A \rightarrow B$. \begin{itemize}
\item $A' \rightarrow A \rightarrow F = 0$, so $A' \rightarrow A$ factors through the kernel of $A \rightarrow F$ as $A' \rightarrow K \rightarrow F$.
\item On the other hand, $A' \rightarrow A \rightarrow B = 0$ so $A \rightarrow B$ factors through the cokernel of $A' \rightarrow A$ as $A \rightarrow F \rightarrow B$. \\ Therefore $K \rightarrow A \rightarrow B =K \rightarrow A \rightarrow F \rightarrow B = 0$, and $K \rightarrow A$ factors through the kernel of $A \rightarrow B$ as $K \rightarrow A' \rightarrow B$. \end{itemize} We have shown that $KerCok$ is identity. Dually, $CokKer$ is identity. \\ \end{proof}
\begin{thm}[Abelian categories are balanced] In an abelian category, every monic epic map is an isomorphism. \\ \end{thm}
\begin{proof} Let $A \xrightarrow{x} B$ be monic and epic. \\ Obviously, $B \rightarrow 0 = Cok(x)$, so by the result above, $x = Ker(B \rightarrow 0)$. \\ $B \xrightarrow{1_B} B$ factors through the kernel $x$ of $B \rightarrow 0$: there is $B \xrightarrow{y} A$ with $xy = B \xrightarrow{1_B} B$. ($x$ is \textit{split epic}.) \noindent Then $xyx = 1_Bx=x1_A$, and since $x$ is monic, $yx=1_A$. \\ \end{proof}
\noindent The \textbf{intersection} of two subobjects of $\mathcal{A}$ is their greatest lower bound in the family of subobjects of $A$, with respect to containment.
\begin{thm} In an abelian category, every pair of subobjects has an intersection. \end{thm}
\begin{proof} Let $A_1 \xrightarrow{f_1} A$ and $A_2 \xrightarrow{f_2} A$ be monics, $A \xrightarrow{c} F$ a cokernel of $A_1 \xrightarrow{f_1} A$, and $A_{12} \xrightarrow{k} A_2$ a kernel of $A_2 \xrightarrow{f_2} A \xrightarrow{c} F$. \\ By definition of $k$, we have $A_{12}\xrightarrow{f_2k} A \xrightarrow{c} F = 0$. Since $A_1 \xrightarrow{f_1} A$ is a kernel of $A \xrightarrow{c} F$, $f_2k$ factors as $A_{12} \xrightarrow{m} A_1 \xrightarrow{f_1} A$. $m$ must be monic, since $f_1m=f_2k$ is monic as the composition of two monics. \\ We therefore have a commutative diagram \[ \begin{tikzcd} A_{12} \arrow{r}{k} \arrow[swap]{d}{m} & A_2 \arrow{d}{f_2} \\ A_1 \arrow{r}{f_1} & A \end{tikzcd} \] which we claim is actually a pullback square: for each commutative diagram \[ \begin{tikzcd} X \arrow{r}{x_2} \arrow[swap]{d}{x_1} & A_2 \arrow{d}{f_2} \\ A_1 \arrow{r}{f_1} & A \end{tikzcd} \] there exists a unique $X \xrightarrow{x} A_{12}$ such that $mx=x_1$ and $kx=x_2$. \\
\noindent Well, we have $(cf_2)x_2 = cf_1x_1=0$, so $x_2$ factors uniquely through the kernel $A_{12} \xrightarrow{k} A_2$ of $cf_2$: $$X \xrightarrow{x_2} A_2 = X \xrightarrow{x} A_{12} \xrightarrow{k} A_2,$$ where $x$ is unique such that $kx = x_2$. It only remains to see that $mx=x_1$. This is because $$f_1mx=f_2kx=f_2x_2=f_1x_1,$$ and $f_1$ is monic. \\
\noindent In particular, $A_{12} \xrightarrow{f_1m \ = \ f_2k} A$ is the intersection of $A_1 \xrightarrow{f_1} A$ and $A_2 \xrightarrow{f_2} A$, since when $X$ is a subobject contained in $A_1$ and $A_2$, $X$ will also be contained in $A_{12}$. \end{proof}
\noindent Dually, any two quotient objects have a greatest lower bound. Since $Ker$ and $Cok$ are order-reversing and mutually inverse, every pair of subobjects has a least upper bound: for maps $A_i \rightarrow B$, $(i=1,2)$, their cokernels $B \rightarrow C_i$ have a least upper bound $B \rightarrow C_0$. Take the kernels:
\[
\begin{tikzcd}[row sep=0.8em,column sep=2em]
C_1 \arrow[d] &&& A_1 \arrow[d] \arrow[dr] \\
C_0 & B \arrow{l} \arrow{ul} \arrow{dl} &\text{$\xrightarrow[\text{kernels}]{\text{take}}$}& A_0 \arrow{r} & B & \\
C_2 \arrow[u] &&& A_1\arrow[ur] \arrow{u}\\
\end{tikzcd}
\] \noindent Hence the family of subobjects of $A$ is a lattice. We write $\bigcap$ for the greatest lower bound operation (\textit{meet}) and $\bigcup$ for the least upper bound operation (\textit{join}). \\
\begin{fact} Abelian categories have all equalisers and all pullbacks. Dually, abelian categories have all coequalisers and all pushouts. \end{fact}
\noindent For instance, then, when we want to show that an abelian category is complete, we just need to check that it has all products. \\ \begin{defn} The \textbf{image} $Im(A \rightarrow B)$ of a map $A \rightarrow B$ is the smallest subobject of $B$ such that $A \rightarrow B$ factors through the representing monics. \\ Dually, the \textbf{coimage} $Coim(A \rightarrow B)$ of $A \rightarrow B$ is the smallest quotient object of $A$ through which $A \rightarrow B$ factors. \end{defn}
\noindent Recall that $Ker$ and $Cok$ were mutually inverse on subobjects and quotients, but we may of course take the $Ker$ and $Cok$ of any map, then $KerCok$ and $CokKer$ need not be identity. In fact:
\begin{fact} In an abelian category, \begin{itemize}
\item $A \rightarrow B$ has an image, namely, $KerCok(A \rightarrow B)$.
\item $A \xrightarrow{x} B$ is epic iff $Im(x)=B$, and hence, iff $Cok(x) = 0$.
\item $A \xrightarrow{x} Im(x)$ is epic. \end{itemize} Dually, \begin{itemize}
\item $A \rightarrow B$ has a coimage, namely, $CokKer(A \rightarrow B)$.
\item $A \xrightarrow{x} B$ is monic iff $Coim(x)=A$, and hence, iff $Ker(x) = 0$.
\item $Coim \xrightarrow{x} B$ is monic.
\\ \end{itemize} \end{fact}
\noindent Next, we state a couple of lemmas for abelian categories:
\begin{lem} \label{2.64} Suppose we have exact columns and exact middle row in the following commutative diagram: \[ \begin{tikzcd}[row sep=0.8em,column sep=1em]
&0 \arrow{d} &0 \arrow{d} &0 \arrow{d}\\ 0 \arrow{r} & B_{11} \arrow{r} \arrow[swap]{d} & B_{12} \arrow{r} \arrow{d} & B_{13} \arrow{d} \\ 0 \arrow{r} & B_{21} \arrow{r} \arrow[swap]{d} & B_{22} \arrow{r} \arrow{d} & B_{23} \\ 0 \arrow{r} & B_{31} \arrow{r} \arrow{d} & B_{32} \arrow{d} \\ &0 &0 \\ \end{tikzcd} \] Then the bottom row is exact iff the top row is exact. \end{lem}
\begin{proof} First, we prove the forward direction. \begin{itemize}
\item $Ker(B_{11} \rightarrow B_{12}) = 0:$
\\ Let $A \rightarrow B_{11} \rightarrow B_{12}=0$. Then $A \rightarrow B_{11} \rightarrow B_{21} \rightarrow B_{22} = A \rightarrow B_{11} \rightarrow B_{12} \rightarrow B_{22}=0$.
\\ Hence $A \rightarrow B_{11} \rightarrow B_{21}$ factors through $Ker(B_{21} \rightarrow B_{22})=0$. Hence $A \rightarrow B_{11}$ factors through $Ker(B_{11} \rightarrow B_{21})=0$, hence $A \rightarrow B_{11} = 0$.
\item $Im(B_{11} \rightarrow B_{12}) \subset Ker(B_{12} \rightarrow B_{13}) :$
\\ It is enough to see that $B_{11} \rightarrow B_{12}$ factors through $Ker(B_{12} \rightarrow B_{13})$, i.e., $B_{11} \rightarrow B_{12} \rightarrow B_{13} = 0$.
\\ This follows because $B_{11} \rightarrow B_{12} \rightarrow B_{13} \rightarrow B_{23} =B_{11} \rightarrow B_{21} \rightarrow B_{22} \rightarrow B_{23}= 0$, and $B_{13} \rightarrow B_{23}$ is monic.
\item $ Ker(B_{12} \rightarrow B_{13}) \subset Im(B_{11} \rightarrow B_{12}) :$
\\ We show that whenever $L \rightarrow B_{12} \rightarrow B_{13} =0$, $L \rightarrow B_{12}$ factors through $B_{11} \rightarrow B_{12}$.
\\ Well, $0 = L \rightarrow B_{12} \rightarrow B_{13} \rightarrow B_{23} = L \rightarrow B_{12} \rightarrow B_{22} \rightarrow B_{23}$,
\\ so $L \rightarrow B_{12} \rightarrow B_{22} \subset Ker(B_{22} \rightarrow B_{23}) = B_{21} \rightarrow B_{22}$.
\\ That is, there is some $L \rightarrow B_{21}$ such that $L \rightarrow B_{12} \rightarrow B_{22} = L \rightarrow B_{21} \rightarrow B_{22}$.
\\ Next, we have $L \rightarrow B_{21} \rightarrow B_{31} \rightarrow B_{32} =L \rightarrow B_{12} \rightarrow B_{22} \rightarrow B_{32} =0$, so $L \rightarrow B_{21} \rightarrow B_{31} = 0$.
\\ Hence $L \rightarrow B_{21} \subset Ker(B_{21} \rightarrow B_{31}) = B_{11} \rightarrow B_{21}$, and we factor $L \rightarrow B_{21} = L \rightarrow B_{11} \rightarrow B_{21}$.
\\ Then $L \rightarrow B_{11} \rightarrow B_{12} \rightarrow B_{22} = L \rightarrow B_{11} \rightarrow B_{21} \rightarrow B_{22} = L \rightarrow B_{21} \rightarrow B_{22} = L \rightarrow B_{12} \rightarrow B_{22},$
\\ and we are done since $B_{12} \rightarrow B_{22}$ is monic. \end{itemize} For the other direction, we need only show that $Ker(B_{31} \rightarrow B_{32}) = 0$. This can be chased similarly, but we note it follows immediately by the snake lemma applied to the top two rows (after replacing $B_{13}$ with $I$, where $I \rightarrow B_{13} = Im(B_{12} \rightarrow B_{13})$). \end{proof}
\begin{lem}[Nine Lemma] \label{2.65} Suppose we have exact columns and exact middle row in the commutative diagram: \[ \begin{tikzcd}[row sep=0.8em,column sep=1em]
&0 \arrow{d} &0 \arrow{d} &0 \arrow{d}\\ 0 \arrow{r} & B_{11} \arrow{r} \arrow[swap]{d} & B_{12} \arrow{r} \arrow{d} & B_{13} \arrow{d} \arrow{r} & 0 \\ 0 \arrow{r} & B_{21} \arrow{r} \arrow[swap]{d} & B_{22} \arrow{r} \arrow{d} & B_{23} \arrow{r} \arrow{d} & 0 \\ 0 \arrow{r} & B_{31} \arrow{r} \arrow{d} & B_{32} \arrow{d} \arrow{r} & B_{33} \arrow{d} \arrow{r} &0\\ &0 &0 &0\\ \end{tikzcd} \] Then the bottom row is exact iff the top row is exact. \end{lem}
\begin{proof} This follows immediately from Lemma \ref{2.64} and its dual.
\\ \end{proof}
\noindent Recall that the direct sum $A \oplus B$ plays the role of the (binary) categorical sum and product in an abelian category: \begin{itemize}
\item We have projection maps $A \oplus B \xrightarrow{\pi_1} A$, $A \oplus B \xrightarrow{\pi_2} B$. Two maps $C \xrightarrow{f_1} A$, $C \xrightarrow{f_2} B$ define a unique map $C \xrightarrow{\langle f_1,f_2 \rangle} A \oplus B$ such that $\pi_j \circ \langle f_1,f_2 \rangle = f_j$.
\item We have inclusion maps $A \xrightarrow{\iota_1} A \oplus B$, $B \xrightarrow{\iota_2} A \oplus B$. Two maps $A \xrightarrow{g_1} C$, $B \xrightarrow{g_2} C$ define a unique map $A \oplus B \xrightarrow{[ g_1,g_2]} C$ such that $[ g_1,g_2 ] \circ \iota_j = g_j$.
\\ \end{itemize} \noindent We may add two maps $f,g:A \rightarrow B$ by defining $f+g: A \rightarrow B$ to be the map $A \xrightarrow{\Delta = \langle 1,1 \rangle} A \oplus A \xrightarrow{[f,g]} B$. Alternatively, we could define it as $A \xrightarrow{ \langle f,g \rangle} B \oplus B \xrightarrow{\Sigma = [1,1]} B$. Both have the zero map $A \xrightarrow{0} B$ as a unit, so an Eckmann-Hilton argument shows that the two operations are the same, and in fact associative and commutative. In fact:
\begin{thm} The set $Hom(A,B)$ with the operation $+$ is an abelian group. \end{thm}
\begin{proof} It remains to exhibit an inverse $A \xrightarrow{-x} B$ for $A \xrightarrow{x} B$. \\ It is convenient to introduce matrix notations for maps to/from the direct sum: \\ Write $\begin{psmallmatrix} w \\y\end{psmallmatrix}$ for $[w,y]$, $\begin{psmallmatrix} w & x\end{psmallmatrix}$ for $\langle w, x \rangle$, and $\begin{psmallmatrix}w & x\\y & z\end{psmallmatrix}$ for $[\langle w,x \rangle, \langle y,z \rangle] = \langle [w,y], [x,z] \rangle$. Then a map $\begin{psmallmatrix}w & x\\y & z\end{psmallmatrix} \circ \begin{psmallmatrix} p & q\\r & s \end{psmallmatrix}$ is computed as the matrix product $\begin{psmallmatrix}p & q \\ r & s \end{psmallmatrix} \begin{psmallmatrix}w & x\\y & z\end{psmallmatrix}$. \\
\noindent Define a map $A \oplus B \xrightarrow{\begin{psmallmatrix}1 & x\\0 & 1\end{psmallmatrix}} A \oplus B$. \\ The kernel of $\begin{psmallmatrix}1 & x\\0 & 1\end{psmallmatrix}$ is $K \xrightarrow{\begin{psmallmatrix}a & b\end{psmallmatrix}} A \oplus B$ where $0 = K \xrightarrow{\begin{psmallmatrix}a & b\end{psmallmatrix}} A \oplus B \xrightarrow{\begin{psmallmatrix}1 & x\\0 & 1\end{psmallmatrix}} A \oplus B = K \xrightarrow{\begin{psmallmatrix}a & xa+b \end{psmallmatrix}} A \oplus B$, so $a=b=0$. \\ This shows that $\begin{psmallmatrix}1 & 0\\0 & 1\end{psmallmatrix}$ is monic. Dually, it is epic, hence it has an inverse map $\begin{psmallmatrix}p & q\\r & s\end{psmallmatrix}$. \\ Since $\begin{psmallmatrix}1 & x\\0 & 1\end{psmallmatrix} \begin{psmallmatrix}p & q\\r & s\end{psmallmatrix} = \begin{psmallmatrix}1 & 0\\0 & 1\end{psmallmatrix}$, we conclude in particular that $q+x = 0$. \\ \end{proof}
\noindent This upgrades the representable $Hom(A,-)$ (for each $A \in \mathcal{A}$) from a functor $\mathcal{A} \rightarrow Set$ to a functor $\mathcal{A} \rightarrow Ab$, where $Ab$ is the category of abelian groups. \\
\begin{remark} \label{2.42} We know that the direct sum $A \oplus B$ is unique up to isomorphism, and may be characterised as a system $A \overset{\iota_1}{\underset{\pi_1}\rightleftarrows} X \overset{\iota_2}{\underset{\pi_2}\leftrightarrows} B $ where $\pi_1 \iota_1 = 1_A, \pi_2 \iota_2 = 1_B, \pi_1 \iota_2=\pi_2 \iota_1=0$, and $\iota_1 \pi_1 + \iota_2 \pi_2 = 1_X$. \\ Equivalently, it is a system $A \overset{\iota_1}{\underset{\pi_1}\rightleftarrows} X \overset{\iota_2}{\underset{\pi_2}\leftrightarrows} B $ where $\pi_1 \iota_1 = 1_A, \pi_2 \iota_2 = 1_B$, and $A \xrightarrow{\iota_1} X \xrightarrow{\pi_2} B$, $B \xrightarrow{\iota_2} X \xrightarrow{\pi_1} A$ are exact. \end{remark}
\section{Additives and representables}
To any functor $F: \mathcal{A} \rightarrow \mathcal{B}$ is associated a function $Hom(A_1,A_2) \rightarrow Hom(FA_1,FA_2)$. \\ If $\mathcal{A}$ and $\mathcal{B}$ are abelian categories, we say $F$ is \textbf{additive} if this function is a group homomorphism (with respect to $+$) for every $A_1, A_2 \in \mathcal{A}$. \\ The functors $Hom(A,-) : \mathcal{A} \rightarrow Ab$ and $Hom(-,A): \mathcal{A}^{op} \rightarrow Ab$ are additive, because they are left-exact (see Corollary \ref{3.12}). \\
\begin{thm} A functor between abelian categories is additive iff it carries direct sums into direct sums. \end{thm}
\begin{proof} Suppose $A \overset{\iota_1}{\underset{\pi_1}\rightleftarrows} X \overset{\iota_2}{\underset{\pi_2}\leftrightarrows} B $ is a direct sum system in $\mathcal{A}$ (so $\pi_1 \iota_1 = 1_A, \pi_2 \iota_2 = 1_B, \pi_1 \iota_2=\pi_2 \iota_1=0$, and $\iota_1 \pi_1 + \iota_2 \pi_2 = 1_X$). \\
\noindent Applying a functor $F: \mathcal{A} \rightarrow \mathcal{B}$ yields a direct sum system in $\mathcal{B}$, if $F$ is additive. \\ Conversely, suppose applying $F: \mathcal{A} \rightarrow \mathcal{B}$ yields a direct sum system in $\mathcal{B}$ . Let us show that $F(x+y)=F(x)+F(y)$ for any $x,y: A \rightarrow B$. \\ By definition, $A \xrightarrow{x+y} B = A \xrightarrow{\begin{psmallmatrix}1 & 1\end{psmallmatrix}} A \oplus A \xrightarrow{\begin{psmallmatrix} x\\y\end{psmallmatrix}} B$, so $$F(A \xrightarrow{x+y} B) = FA \xrightarrow{F\begin{psmallmatrix}1 & 1\end{psmallmatrix}} F(A \oplus A) \xrightarrow{F\begin{psmallmatrix} x\\y\end{psmallmatrix}} FB = FA \xrightarrow{\begin{psmallmatrix}1 & 1\end{psmallmatrix}} F(A \oplus A) \xrightarrow{\begin{psmallmatrix} Fx\\Fy\end{psmallmatrix}} FB = FA \xrightarrow{Fx+Fy} FB .$$ \end{proof}
\noindent Working over an abelian category $\mathcal{A}$, let us call a sequence $\cdots \rightarrow A_1 \rightarrow A_2 \rightarrow A_3 \rightarrow \cdots$ \textbf{exact} if for each $i$, the kernel of $A_i \rightarrow A_{i+1}$ equals the image of $A_{i-1} \rightarrow A_i$ as subobjects of $A_i$. \\ An exact sequence of the form $0 \rightarrow A' \rightarrow A \rightarrow A''$ is \textbf{left-exact}, and one of the form $A' \rightarrow A \rightarrow A'' \rightarrow 0$ is \textbf{right-exact}. \\ We say a functor between abelian categories is \textbf{left-exact} if it carries left-exact sequences into left-exact sequences, \textbf{right-exact} if it carries right-exact sequences into right-exact sequences, and \textbf{exact} if it is both.
\begin{cor} \label{3.12} Any left-exact or right-exact functor is additive. \\ \end{cor}
\begin{proof} If $A \overset{\iota_1}{\underset{\pi_1}\rightleftarrows} X \overset{\iota_2}{\underset{\pi_2}\leftrightarrows} B $ is a direct sum system in $\mathcal{A}$ (so $\pi_1 \iota_1 = 1_A, \pi_2 \iota_2 = 1_B$, and $A \xrightarrow{\iota_1} X \xrightarrow{\pi_2} B$, $B \xrightarrow{\iota_2} X \xrightarrow{\pi_1} A$ are exact), then these conditions are preserved by left-exact or right-exact functors. \end{proof}
\noindent Let us say that a functor $F: \mathcal{A} \rightarrow \mathcal{B}$ is \textbf{faithful}, or an \textbf{embedding}, if for any $A_1, A_2 \in \mathcal{A}$ we have that the function $Hom(A_1,A_2) \rightarrow Hom(FA_1,FA_2)$ is injective. \\
\begin{lem} \label{2.21} For $A \rightarrow B \rightarrow C$ the following conditions are equivalent: \begin{enumerate}
\item $Im(A \rightarrow B) = Ker(B \rightarrow C)$;
\item $Cok(A \rightarrow B) = Coim(B \rightarrow C)$;
\item $A \rightarrow B \rightarrow C = 0$ and $K \rightarrow B \rightarrow F =0$, \end{enumerate} where $K \rightarrow B$ is a kernel of $B \rightarrow C$, and $B \rightarrow F$ is a cokernel of $A \rightarrow B$. \end{lem}
\begin{proof} We prove equivalence of the first and third items; equivalence of the second and third is proven dually. \begin{itemize}
\item The first item implies the third:
\\ $A \rightarrow B \rightarrow C = A \rightarrow Im(A \rightarrow B) \rightarrow B \rightarrow C = A \rightarrow Ker(B \rightarrow C) \rightarrow B \rightarrow C =0$.
\\ $K \rightarrow B \rightarrow F = 0$ because $K \rightarrow B$ is a kernel of $B \rightarrow F$:
$$K \rightarrow B = Ker(B \rightarrow C) = Im(A \rightarrow B) = KerCok(A \rightarrow B) = Ker(A \rightarrow B).$$
\item The third item implies the first:
\\ Since $A \rightarrow B \rightarrow C =0$, $A \rightarrow B$ factors through $Ker(B \rightarrow C)$. \\ Therefore, by definition of image, $Im(A \rightarrow B) \subset Ker(B \rightarrow C)$.
\\ On the other hand, since $K \rightarrow B \rightarrow F =0$, $K \rightarrow B$ factors through the kernel of $B \rightarrow F$:
$$Ker(B \rightarrow C) = K \rightarrow B \subset Ker(B \rightarrow F) = KerCok(A \rightarrow B) = Im(A\rightarrow B).$$ \end{itemize} \end{proof}
\begin{thm} \label{3.21} Let $F: \mathcal{A} \rightarrow \mathcal{B}$ be an additive functor between abelian categories. The following are equivalent: \begin{itemize}
\item[(a)] F is an embedding.
\item[(b)] F carries noncommutative diagrams into noncommutative diagrams.
\item[(c)] F carries nonexact sequences into nonexact sequences. \end{itemize} \end{thm}
\begin{proof} \begin{itemize}
\item The first two statements are trivially equivalent.
\item The third implies the first:
Let $A' \xrightarrow{x} A \neq 0$. Then $A' \xrightarrow{1} A' \xrightarrow{x} A$ is not exact, so neither is $FA' \xrightarrow{1} FA' \xrightarrow{Fx} FA$, hence $Fx \neq 0$.
\item The first implies the third:
\\ Let $A' \rightarrow A \rightarrow A''$ be a nonexact sequence in $\mathcal{A}$. Let $0 \rightarrow K \rightarrow A \rightarrow A''$ and $A' \rightarrow A \rightarrow G \rightarrow 0$ be exact.
By Lemma \ref{2.21}, either $A' \rightarrow A \rightarrow A'' \neq 0$ or $K \rightarrow A \rightarrow G \neq 0$. By assumption, $F$ applied to a nonzero map is nonzero, so we have two cases:
\begin{enumerate}
\item If $FA' \rightarrow FA \rightarrow FA'' \neq 0$ then by Lemma \ref{2.21} $FA' \rightarrow FA \rightarrow FA''$ is nonexact.
\item If $FK \rightarrow FA \rightarrow FG \neq 0$, let $0 \rightarrow L \rightarrow FA \rightarrow FA''$ and $FA' \rightarrow FA \rightarrow H \rightarrow 0$ be exact in $\mathcal{B}$.
\\ Since $FK \rightarrow FA \rightarrow FA'' = 0$, $FK \rightarrow FA$ factors through the kernel as $FK \rightarrow L \rightarrow FA$.
\\ Since $FA' \rightarrow FA \rightarrow FG = 0$, $FA \rightarrow FG$ factors through the cokernel as $FA \rightarrow H \rightarrow FG$.
\\ We see that $FA' \rightarrow FA \rightarrow FA''$ cannot be exact, otherwise Lemma \ref{2.21} would imply \\ $L \rightarrow FA \rightarrow H = 0$, then $$FK \rightarrow FA \rightarrow FG =FK \rightarrow L \rightarrow FA \rightarrow H \rightarrow FG= 0,$$ contradicting our assumption.
\end{enumerate} \end{itemize} \end{proof}
\begin{cor} \label{exemb} If a functor $F: \mathcal{A} \rightarrow \mathcal{B}$ between abelian categories is an exact embedding, then the exactness (resp. commutativity) of a diagram in $\mathcal{A}$ is equivalent to the exactness (resp. commutativity) of the $F$-image of the diagram. \\ \end{cor}
\noindent Let us say an object $P$ in an abelian category $\mathcal{A}$ is \textbf{projective} if the functor $Hom(P,-): \mathcal{A} \rightarrow Ab$ is exact. \\ Of course, $Hom(A,-)$ is left-exact for any $A \in \mathcal{A}$, so we may equally just demand right-exactness in this definition. \\ Unpacking the definition, we see that $P$ is projective iff for any map $P \xrightarrow{p} B$ and epic $A \xrightarrow{e} B$, there is a map $P \xrightarrow{\tilde{p}} A$ (a \textit{lift} of $p$) such that $e \circ \tilde{p} = p$. \\
\begin{prop} \label{3.32} If $\{P_j\}$ is a family of projectives in an abelian category, then the direct sum $\Sigma_j P_j$ (if it exists) is projective. \end{prop}
\begin{proof} A map $\Sigma_j P_j \xrightarrow{p} B$ is given by individual maps $P_i \xrightarrow{p_j} B$. If we have an epic $A \xrightarrow{e} B$, there are componentwise lifts a map $P_j \xrightarrow{\tilde{p_j}} A$. That is, for each $i$, $e \circ \tilde{p_j} = p_j$. These collect into a map $\Sigma_j P_j \xrightarrow{\tilde{p}} A$ which lifts $p$: $e \circ \tilde{p} = p$, because these maps agree on each $P_j$: $$(e \circ \tilde{p}) \circ \iota_j = e \circ (\tilde{p} \circ \iota_j)= e \circ (\tilde{p_j}) = p_j = p \circ \iota_j,$$ where $\iota_j$ is the $j$th inclusion into the sum. \\ \end{proof}
\noindent Let us say an object $G \in \mathcal{A}$ is a \textbf{generator} if the functor $Hom(G,-): \mathcal{A} \rightarrow Ab$ is an embedding. \\
\begin{prop} \label{3.33} The following are equivalent: \begin{itemize}
\item $G$ is a generator.
\item For every $A \rightarrow B \neq 0$ there is a map $G \rightarrow A$ such that $G \rightarrow A \rightarrow B \neq 0$.
\item For every proper subobject of $A$ there is a map $G \rightarrow A$ whose image is not contained in the given subobject. \end{itemize} \end{prop}
\begin{proof} \begin{itemize}
\item Unpacking the definition, $G$ is a generator if and only if the function $$Hom(A,B) \rightarrow Hom(Hom(G,A),Hom(G,B)), f \mapsto f \circ -$$ is injective. \\ This is if and only if for any nonzero $f \in Hom(A,B)$, the map $f \circ -$ is nonzero, meaning there is some $g \in Hom(G,A)$ with $f \circ g$ nonzero. \\ Hence the first two statements are equivalent.
\item The second statement implies the third.
\\ Let $C \xrightarrow{s} A$ be a proper subobject. In particular $s$ is not epic, otherwise it would be an isomorphism as abelian categories are balanced. Take its cokernel $A \xrightarrow{c} B \neq 0$. There is some $G \xrightarrow{g} A$ with $cg \neq 0$. $Im(g)= I \xrightarrow{i} A$ is not contained in $C$. If it were, then by definition there would be a map $I \xrightarrow{f} C$ with $s \circ f = i $. Then
$$cg = G \rightarrow I \xrightarrow{i} A \xrightarrow{c} B = G \rightarrow I \xrightarrow{f} C \xrightarrow{s} A \xrightarrow{c} B = 0,$$
since $c$ was a cokernel of $s$. This is a contradiction.
\item The third statement implies the second.
\\ Given $A \xrightarrow{c} B \neq 0$, its kernel $K$ is a proper subobject of $A$, so there is some $G \xrightarrow{g} A$ whose image is not contained in $K$. In particular $cg \neq 0$.
\\ \end{itemize} \end{proof}
\begin{prop} If $P$ is projective then it is a generator iff $Hom(P,A)$ is nontrivial for all nontrivial $A$. \end{prop}
\begin{proof} \begin{itemize}
\item Let $P$ be a generator, and $A \neq 0$. Then $A \xrightarrow{1} A \neq 0$, and by the result above, there is some $P \xrightarrow{g} A$ with $g=1g \neq 0$.
\item Let $P$ be projective, but not a generator. There is some $A \xrightarrow{c} B \neq 0$ such that for every $P \xrightarrow{g} A$, $cg=0$.
\\ $c$ factors through $Im(c) = I \xrightarrow{i} B$ as $c= A \rightarrow I \xrightarrow{i} B$, where $A \rightarrow I$ is epic.
\\ Then $I$ is nontrivial with trivial $Hom(P,I)$. \end{itemize} \end{proof}
\noindent Say a category is \textbf{well-powered} if the family of subobjects of any object is a set.
\begin{prop} \label{3.35} An abelian category that has a generator is well-powered. \end{prop}
\begin{proof} Let $G$ be a generator, and $A$ any object. Then a subobject $A' \rightarrow A$ is distinguished by the subset $Hom(G,A') \subset Hom(G,A)$. (We have identified $Hom(G,A')$ with its image under $Hom(G,-)(A' \rightarrow A)$. In other words, there are no more subobjects of $A$ than subsets of $Hom(G,A)$.) \end{proof}
\begin{prop}\label{3.36} $G$ is a generator in a cocomplete abelian category $\mathcal{A}$ iff for every $A \in \mathcal{A}$ the obvious map $\Sigma_{Hom(G,A)}G \rightarrow A$ is epic. \end{prop}
\begin{proof} Let $G$ be a generator. Suppose for a contradiction there is some $A \in \mathcal{A}$ with $$A \rightarrow B := Cok(\Sigma_{Hom(G,A)}G \rightarrow A) \neq 0.$$ Then there is a map $G \rightarrow A$ with $G \rightarrow A \rightarrow B \neq 0$, but this contradicts that $$\Sigma_{Hom(G,A)}G \rightarrow A \rightarrow B = 0.$$ \\
\noindent Conversely, suppose $\Sigma_{Hom(G,A)}G \rightarrow A$ is epic, so its cokernel is zero. Suppose for a contradiction there is some $A \rightarrow B \neq 0$ such that every $G \rightarrow A$ has $G \rightarrow A \rightarrow B =0$. Then we have $\Sigma_{Hom(G,A)}G \rightarrow A \rightarrow B = 0$, so $A \rightarrow B$ factors through $Cok(\Sigma_{Hom(G,A)}G \rightarrow A)=0$. Then $A \rightarrow B =0$. \end{proof}
\noindent The dual notions are as follows: \\ An object $Q$ is \textbf{injective} if the functor $Hom(-,Q): \mathcal{A}^{op} \rightarrow Ab$ is exact. \\ An object $C$ is a \textbf{cogenerator} if the functor $Hom(-,C): \mathcal{A}^{op} \rightarrow Ab$ is an embedding. \\
\noindent Note that $Q$ is injective in $\mathcal{A}$ iff it is projective in $\mathcal{A}^{op}$, and $C$ is a cogenerator for $\mathcal{A}$ iff it is a generator for $\mathcal{A}^{op}$.
\begin{prop} \label{3.37} Let $\mathcal{A}$ be a complete abelian category with a generator. \\ There is, out of every object in $\mathcal{A}$, a monic to an injective object iff $\mathcal{A}$ has an injective cogenerator. \end{prop}
\begin{proof} \begin{itemize}
\item Let $C$ be an injective cogenerator for $\mathcal{A}$, and $A \in \mathcal{A}$. The obvious map $A \rightarrow \Pi_{Hom(A,C)}C$ is monic, and $\Pi_{Hom(A,C)}C$ is injective. (We are using the duals of Propositions \ref{3.32} and \ref{3.36}.)
\item Let $G$ be a generator for $\mathcal{A}$.
\\ By Proposition \ref{3.35}, the class of quotient objects of $G$ is a set. (The class of subobjects of $G$ is a set, but this is in bijection with the class of quotient objects by Theorem {2.11}.)
\\ Therefore we may define $P = \Pi_{\{ \text{quotient objects $Q$ of }G \} } Q$.
\\
By assumption we have a monic $P \rightarrow E$ where $E$ is injective. We claim $E$ is a cogenerator.
\\Let $A \xrightarrow{c} B \neq 0$.
\\ It is enough, by the dual of Proposition \ref{3.33}, to name some $B \rightarrow E $ such that $A \rightarrow B \rightarrow E \neq 0$.
\\ Well, since $G$ is a generator, there is $G \xrightarrow{g} A$ with $cg \neq 0$.
\\ Let $I \xrightarrow{i} B$ be the image of $G \xrightarrow{g} A \xrightarrow{c} B$, so $G \xrightarrow{cg} B = G \xrightarrow{cg} I \xrightarrow{i} B $.
\\ Let $I \rightarrow P \rightarrow E$ be a monic $m$. (Since $I$ is a quotient object of $G$, it appears as a factor in $P$, so we may just take $I \rightarrow P$ to consist of the identity $I \rightarrow I$ and zero maps from $I$ to any other factor of $P$. This is monic because that identity component $I \rightarrow I$ is monic.)
\\ Since $E$ is injective and $I \xrightarrow{i} B$ is monic, there is some $B \xrightarrow{b} E$ such that $bi = I \xrightarrow{m} E$.
\\ We indeed have $A \xrightarrow{c} B \xrightarrow{b} E \neq 0$, since
$$bcg = bicg = mcg \neq 0.$$
(The last step is because $cg \neq 0$ and $m$ is monic.)
\\ \end{itemize} \end{proof}
\noindent Recall that a \textbf{subcategory} $\mathcal{A'}$ of the category $\mathcal{A}$ is just a subclass of the objects of $\mathcal{A}$, with, for any two objects $A', A$ in this subclass, a subclass of $Hom(A', A)$ closed under composition and identities. $\mathcal{A'}$ is, of course, a category, and there is an obvious inclusion functor $\mathcal{A}' \rightarrow \mathcal{A}$. \\
\noindent Let $\mathcal{A'}$ be a subcategory of abelian category $\mathcal{A}$. We say $\mathcal{A}'$ is \textbf{exact} if $\mathcal{A'}$ is abelian and the inclusion functor is exact. The inclusion functor is automatically an embedding, so in this situation a diagram in $\mathcal{A}'$ is exact iff it is exact in $\mathcal{A}$ --- this was Corollary \ref{exemb}. \\
\noindent Let us say that a functor $F: \mathcal{A} \rightarrow \mathcal{B}$ is \textbf{full} if for any $A_1, A_2 \in \mathcal{A}$ we have that the function $Hom(A_1,A_2) \rightarrow Hom(FA_1,FA_2)$ is surjective. \\ \noindent A subcategory is \textbf{full} if the inclusion functor is full. A full subcategory of $\mathcal{A}$ can be specified simply by naming a subclass of the objects of $\mathcal{A}$. \\
\noindent We also remark that any functor $F: \mathcal{A} \rightarrow \mathcal{B}$ restricts in the obvious way to a functor $F \ _{\mkern 1mu \vrule height 2ex\mkern2mu \mathcal{A'}}: \mathcal{A'} \rightarrow \mathcal{B}$ on any subcategory $\mathcal{A'}$ of $\mathcal{A}$. \\ When $F$ is exact, full, or an embedding, then the restriction $F \ _{\mkern 1mu \vrule height 2ex\mkern2mu \mathcal{A'}}$ will respectively be exact, full, or an embedding. \section{A special case of Freyd-Mitchell}
An abelian category $\mathcal{A}$ is \textbf{fully abelian} if for every full small exact subcategory $\mathcal{A'}$ of $\mathcal{A}$ there is a ring $R$ and a full exact embedding of $\mathcal{A}'$ into $R$-Mod. \\
\noindent We shall now state a special case of the Freyd-Mitchell embedding theorem, that is easy to prove.
\begin{thm}[Mitchell] \label{Mitch} A cocomplete abelian category with a projective generator is fully abelian. \end{thm}
\begin{proof} Let $\mathcal{A}'$ be a small full exact subcategory of a cocomplete category $\mathcal{A}$. Let $P'$ be a projective generator for $\mathcal{A}$. We wish to give a full exact embedding of $\mathcal{A}'$ into $R$-Mod, for some ring $R$. \\
\noindent First of all, let us slightly modify $P'$. \\ For each $A \in \mathcal{A}'$ consider the epic $\Sigma_{Hom(P',A)}P' \rightarrow A$ from Proposition \ref{3.36}. \\ Let $I = \bigcup_{A \in \mathcal{A}'} Hom(P',A)$. Define $P=\Sigma_I P'$. \\ By Proposition \ref{3.32}, $P$ is still a projective generator, but now we have an additional property: for each $A \in \mathcal{A}'$ there is an epic $P \rightarrow A$. \\ (For instance, define $P \rightarrow A$ as the epic $\Sigma_{Hom(P',A)}P' \rightarrow A$ on the summands indexed over by $Hom(P',A)$, and as zero on all other summands.) \\
\noindent Let $R$ be the ring $End(P)$ of endomorphisms on $P$. \\ We had previously upgraded the functor $Hom(P,-): \mathcal{A} \rightarrow Set$ to a functor $Hom(P,-): \mathcal{A} \rightarrow Ab$, but now let us upgrade it further to a functor $Hom(P,-): \mathcal{A} \rightarrow R$-Mod. \begin{itemize}
\item For every $A \in \mathcal{A}$, the abelian group $Hom(P,A)$ has a canonical $R$-module structure:
\\ given $P \xrightarrow{x} A \in Hom(P,A)$ and $P \xrightarrow{r} P \in R$, define $$r \cdot x = x \circ r \in Hom(P,A).$$
\item For every map $A \xrightarrow{y} B$ in $\mathcal{A}$, the induced map $Hom(P,A) \xrightarrow{y \circ -} Hom(P,B)$ is $R$-linear:
$$(y \circ -)(r \cdot x) = y \circ (r \cdot x) = y \circ (x \circ r) = (y \circ x) \circ r = r \cdot(y \circ x) = r \cdot ((y \circ -) x).$$ \end{itemize} Hence we do get a functor $F=Hom(P,-): \mathcal{A} \rightarrow R$-Mod. \\ $F$ is an exact embedding since $P$ is a projective generator. \\ (To be slightly pedantic, the functor $Hom(P,-): \mathcal{A} \rightarrow Ab$ is an exact embedding by definition of $P$ as a projective generator, but $R$-Mod is an exact subcategory of $Ab$ --- the forgetful inclusion functor $R\text{-Mod} \rightarrow Ab$ has left and right adjoints, so it preserves finite limits and colimits, so it is exact.) \\ The restriction $F \ _{\mkern 1mu \vrule height 2ex\mkern2mu \mathcal{A'}}$ is therefore an exact embedding; it only remains to show it is full. \\
\noindent Suppose we have $A, B \in \mathcal{A'}$ and a map $FA \xrightarrow{\tilde{y}} FB$ in $R$-Mod. We must exhibit a map $A \xrightarrow{y} B$ in $\mathcal{A'}$ such that $Fy = \tilde{y}$, where $Fy = y \circ -$. \\ Since $A,B \in \mathcal{A}'$, we have exact sequences $0 \rightarrow K \rightarrow P \rightarrow A \rightarrow 0$ and $P \rightarrow B \rightarrow 0$ in $\mathcal{A}$ coming from the epics $P \rightarrow A$ and $P \rightarrow B$. (Just take $K \rightarrow P = Ker(P \rightarrow A)$.) \\ Since $FP=R$, taking $F$ gives us the following commutative diagram in $R$-Mod:
\[ \begin{tikzcd} 0 \arrow{r} &FK \arrow{r} &R \arrow{d}{f} \arrow{r} &FA \arrow{d}{\tilde{y}} \arrow{r} &0 \\ & &R \arrow{r} &FB \arrow{r} &0 \end{tikzcd} \] where $f$ is a lift of $R \rightarrow FA \xrightarrow{\tilde{y}} FB$. ($R$ is projective and $R \rightarrow FB$ is epic.) \\
\noindent Since $R$ is a ring, we have $End(R) \cong R^{op}$ --- in other words, any endomorphism on $R$ is given by multiplication \textit{on the right} by some $R$-element. Hence, write $f(s)=sr = s \circ r$ for all $s \in R$, where $P \xrightarrow{r} P \in R$. \\
\noindent Return to $\mathcal{A}$: in the diagram \[ \begin{tikzcd} 0 \arrow{r} &K \arrow{r} &P \arrow{d}{r} \arrow{r} &A \arrow{r} &0 \\ & &P \arrow{r} &B \arrow{r} &0 \end{tikzcd} \] we have that $K \rightarrow P \xrightarrow{r} P \rightarrow B = 0$, as $FK \rightarrow R \xrightarrow{f} R \rightarrow FB = 0$ and $F$ is an embedding. \\ Hence $P \xrightarrow{r} P \rightarrow B$ factors through the cokernel $P \rightarrow A$ --- there is $A \xrightarrow{y} B$ such that
\[ \begin{tikzcd} P \arrow{r} \arrow{d}{r} & A \arrow{d}{y} \\ P \arrow{r} & B & \text{commutes.} \end{tikzcd} \] Hence
\[ \begin{tikzcd} R \arrow{r} \arrow{d}{f} & F A \arrow{d}{Fy} \\ R \arrow{r} & FB & \text{commutes.} \end{tikzcd} \] \\
\noindent Therefore $R \rightarrow FA \xrightarrow{Fy} FB = R \xrightarrow{f} R \rightarrow FB = R \rightarrow FA \xrightarrow{\tilde{y}} FB$. \\ Since $R \rightarrow FA$ is epic, $Fy = \tilde{y}$. \\ \\ \end{proof}
\noindent The full statement of the \textbf{Freyd-Mitchell embedding theorem} is: \textit{Every abelian category is fully abelian.} \\ We have just shown that this is true if our category is cocomplete with a projective generator. \\ Therefore, if we want to show that every abelian category is fully abelian, it is enough to solve the following problem: Given a small abelian category $\mathcal{A}$, find a cocomplete abelian category $\mathcal{L}$ with a projective generator and an exact full embedding $\mathcal{A} \rightarrow \mathcal{L}$. (The composition of two full exact embeddings is again a full exact embedding.)
\section{Functor categories}
Let $\mathcal{A}$ be a small abelian category. Let $[\mathcal{A}, Ab]$ denote the category of additive functors from $\mathcal{A}$ to $Ab$. Its objects are functors, and its maps are natural transformations. \\ \begin{thm} $[\mathcal{A}, Ab]$ is an abelian category. \end{thm}
\begin{proof} We briefly run through the axioms. \begin{itemize}
\item[A0.] The constantly zero functor is a zero object.
\item[A1, A1*.] (Binary) sums and products are computed pointwise. Given $F_1, F_2 \in [\mathcal{A}, Ab]$, define a functor $F_1 \oplus F_2$ on objects as $( F_1 \oplus F_2 )(A) = F_1(A) \oplus F_2(A)$ and on maps as
$$( F_1 \oplus F_2 )(x) = \begin{pmatrix}
F_1(x) & 0 \\
0 & F_2(x) \end{pmatrix}. $$ This plays the role of binary sum and product.
\item[A2.] Let $F_1 \rightarrow F_2$ in $[\mathcal{A}, Ab]$. We construct a kernel $K \rightarrow F_1$.
\\ For each $A \in \mathcal{A}$, let $K(A):=Ker(F_1A \rightarrow F_2A)$.
\\ Given $A \xrightarrow{x} B$ in $\mathcal{A}$ there is a unique map $K(x): K(A) \rightarrow K(B)$ such that
\[ \begin{tikzcd} K(A) \arrow{r} \arrow{d}{K(x)} & F_1(A) \arrow{d}{F_1(x)} \\ K(B) \arrow{r} & F_1(B) & \text{commutes.} \end{tikzcd} \]
The uniqueness forces $K$ to be a functor, and $K \rightarrow F_1$ is a natural transformation. (The diagram above is a naturality square.)
\item[A2*.] Dually to A2, we construct a cokernel $F_2 \rightarrow C$ for each $F_1 \rightarrow F_2$ pointwise.
\item[A3.] The construction in A2 shows that a natural transformation $F_1 \rightarrow F_2$ is monic in $[\mathcal{A}, Ab]$ iff $F_1A \rightarrow F_2A$ is monic in $\mathcal{A}$ for each $A$. The construction for A2* shows that if $F_1 \rightarrow F_2$ is monic, then it is a kernel of its cokernel.
\item[A3*.] Dual to A3. \end{itemize} \end{proof}
\noindent These constructions indicate that a sequence $F' \rightarrow F \rightarrow F''$ is exact in $[\mathcal{A}, Ab]$ iff the sequences \\ $F'A \rightarrow FA \rightarrow F''A$ are exact in $\mathcal{A}$ for all $A \in \mathcal{A}$. \\ More formally, the \textbf{evaluation functor} $E_A: [\mathcal{A},Ab] \rightarrow Ab$ defined by $E_A(F_1 \xrightarrow{\eta} F_2) = F_1A \xrightarrow{\eta(A)} F_2A$ is an exact functor for each $A \in \mathcal{A}$. \\ The product $(\Pi_{\mathcal{A}}E_A): [\mathcal{A}, Ab] \rightarrow Ab$ defined by $(\Pi_{\mathcal{A}}E_A)(F) = \Pi_{\mathcal{A}}E_A(F) = \Pi_{\mathcal{A}}FA$ is an exact embedding.
\\ \begin{prop} $[\mathcal{A}, Ab]$ is a bicomplete abelian category. \end{prop}
\begin{proof} Let $\{F_i\}_I$ be a (small) collection of functors in $[\mathcal{A}, Ab]$. \\ We construct $\Pi_IF_i$ and $\Sigma_IF_i$ pointwise, just as we did finite direct sums: $$(\Pi_IF_i)(A)=\Pi_IF_iA \qquad \text{and} \qquad (\Sigma_IF_i)(A)=\Sigma_IF_iA.$$ \end{proof}
\noindent The next definition generalises a property that is possessed by categories like $Ab$ and $R$-Mod, where $R$ is a ring. \begin{defn} Let $\mathcal{A}$ be a bicomplete well-powered abelian category. \\ We say $\mathcal{A}$ is a \textbf{Grothendieck} category if for each chain $\{S_i\}_I$ in the lattice of subobjects of an object $S$, and $T$ is any subobject of $S$, then we have $$T \cap \bigcup S_i = \bigcup (T \cap S_i).$$ \end{defn}
\noindent That $R$-Mod satisfies this property really is quite trivial, because the union and the intersection are just set-theoretic union and intersection. It was important, then, that we demanded the family of subobjects to be a chain --- this guarantees that the set-theoretic union is again a module. \\ \begin{prop} $[\mathcal{A}, Ab]$ is a Grothendieck category. \end{prop}
\begin{proof} $[\mathcal{A}, Ab]$ is certainly well-powered (Proposition \ref{3.35} and Theorem \ref{5.35}, for instance). \\ Note that given a collection $\{F_i\}_I$ of subfunctors of $F$, their union and intersection are constructed pointwise: $$(\bigcup F_i)(A) = \bigcup (F_iA) \subset FA,$$ since we know that $F_i \rightarrow F$ is monic only if each component is monic. \\ Hence, given a chain $\{F_i\}$ and subfunctor $H \subset F$, we have $$(H \cap \bigcup F_i)(A) = HA \cap \bigcup F_iA = \bigcup (HA \cap F_iA) = (\bigcup (H \cap F_i))(A),$$ where the second equality uses that $Ab$ is Grothendieck. \\ \end{proof}
\noindent Recall that the \textbf{(co)-Yoneda embedding} is the functor $H: \mathcal{A}^{op} \rightarrow [\mathcal{A}, Ab]$ given on objects by \\ $H(A) = Hom(A, -)$, and on maps by $H(A \xrightarrow{x} B) = Hom(B, -) \xrightarrow{ (x , -)} Hom(A,-)$. \\ If we denote $Hom(A,-)$ by $H^A$, then we may as well denote $H(A \xrightarrow{x} B)$ by $H^B \xrightarrow{H^x} H^A$. \\
\begin{thm} The Yoneda embedding $H: \mathcal{A}^{op} \rightarrow [\mathcal{A}, Ab]$ is left-exact. \end{thm}
\begin{proof} Let $0 \rightarrow A'\rightarrow A \rightarrow A''$ be exact in $\mathcal{A}$. We show $H^{A''} \rightarrow H^A \rightarrow H^{A'} \rightarrow H^0$ is exact in $[\mathcal{A}, Ab]$. \\ We know this holds iff $Hom(0,B) \rightarrow Hom(A',B) \rightarrow Hom(A,B) \rightarrow Hom(A'',B)$ is exact in $Ab$ for each $B \in \mathcal{A}$, but this holds because $Hom(-,B): \mathcal{A}^{op} \rightarrow Ab$ is left exact. \\ \end{proof}
\noindent We recall the following famous lemma from category theory.
\begin{lem}[Yoneda Lemma] $Hom(H^A,F)$ is naturally isomorphic to $F(A)$ in $A \in \mathcal{A}^{op}$ and $F \in [\mathcal{A}, Ab]$. \\ \end{lem}
\begin{thm} \label{5.35} $\Sigma_\mathcal{A} H^A$ is a projective generator for $[\mathcal{A}, Ab]$. \end{thm}
\begin{proof} Let us be more specific about what the Yoneda Lemma says. \\ There are functors $D, E : \mathcal{A}^{op} \times [\mathcal{A}, Ab] \rightarrow Ab$ defined by $$D = \mathcal{A}^{op} \times [\mathcal{A}, Ab] \xrightarrow{H \times 1} [\mathcal{A}, Ab] \times [\mathcal{A}, Ab] \xrightarrow{Hom_{[\mathcal{A}, Ab]}} Ab,$$ (so $D(A,F) = Hom (H^A, F)$,) \\ and the \textit{evaluating functor} $$E(A,F) = F(A); \quad E(A,F_1 \xrightarrow{\eta} F_2) = F_1(A) \xrightarrow{\eta_A} F_2(A), \quad E(A_1 \xrightarrow{x} A_2, F) = F(A_1) \xrightarrow{F(x)} F(A_2).$$ The Yoneda Lemma says $D$ is naturally isomorphic to $E$. \\ Hence, as functors $[\mathcal{A}, Ab] \rightarrow Ab$, we have that $Hom(\Sigma_\mathcal{A} H^A, - )$ is naturally isomorphic to $(\Pi E_A)$: $$Hom(\Sigma_\mathcal{A} H^A, - ) = \Pi_\mathcal{A} Hom( H^A, - ) =\Pi_\mathcal{A} D( A, -) \cong \Pi_\mathcal{A} E( A, -) = (\Pi_\mathcal{A} E_A), $$ The latter is an exact embedding. \end{proof}
\begin{thm} \label{5.36} The Yoneda embedding $H: \mathcal{A}^{op} \rightarrow [\mathcal{A}, Ab]$ is a full embedding. \end{thm}
\begin{proof} This follows immediately from setting $F = H^B$ in the Yoneda Lemma: $$Hom_{[\mathcal{A},Ab]}(H^A,H^B) \cong H^B(A)=Hom_\mathcal{A}(B,A)=Hom_{\mathcal{A}^{op}}(A,B).$$ \end{proof} \section{Injective Envelopes}
The key result of this section will be the following: In a Grothendieck category that has a generator, every object has an injective envelope. \\
\noindent In particular this applies to $[\mathcal{A}, Ab]$, and will be very useful in the next section. \\
\noindent Throughout let $\mathcal{A}$ be an abelian category. Given an object $A \in \mathcal{A}$, an \textbf{extension} of $A$ is simply a monic $A \rightarrow B$ out of $A$. Sometimes we will call $B$ itself an extension of $A$. \\ A \textbf{trivial} extension of $A$ is a \textbf{split} monic --- a monic $A \xrightarrow{x} B$ for which there is some $B \xrightarrow{y} A$ with $yx=1_A$. Equivalently, $a \rightarrow B$ is a trivial extension if there is an object $C$ with $B = A \oplus C$, and $A \rightarrow B$ is the inclusion $A \xrightarrow{i_1} A \oplus C$. ($C$ must then be the cokernel of $A \rightarrow B$.)
\begin{prop} An object in $\mathcal{A}$ is injective iff it has only trivial extensions. \end{prop}
\begin{proof} The forward direction is clear: if $I$ is injective and $I \xrightarrow{x} B$ is monic, then $I\xrightarrow{1}I$ extends to a map $B \xrightarrow{y} I$, meaning $yx = 1_I$. \\
\noindent For the reverse direction, suppose $E$ has only trivial extensions. Let $A \xrightarrow{x} B$ be monic, and $A \xrightarrow{a}E$ be any map. We find a map $B \xrightarrow{y} E$ with $yx=a$. \\ Make a pushout diagram \[ \begin{tikzcd} A \arrow{r}{x} \arrow[swap]{d}{a} & B \arrow{d}{b} \\ E \arrow{r}{e} & P \end{tikzcd} \] and observe that since $x$ is monic, so is $e$. By assumption, $P$ must be a trivial extension of $E$, meaning there is $P \xrightarrow{f} E$ with $fe=1_E$. Put $y=B \xrightarrow{fb} E$; then $yx=fbx=fea=1_Ea=a$. \\ \end{proof}
\noindent An \textbf{essential extension} is a monic $A \rightarrow B$ such that for every nonzero monic $B' \rightarrow B$, ther intersections (of the images) of $A \rightarrow B$ and $B' \rightarrow B$ are nonzero. \\
\begin{prop} An extension $A \rightarrow B$ is essential if for every $B \rightarrow F$ such that $A \rightarrow B \rightarrow F$ is monic, we have that $B \rightarrow F$ is monic. \end{prop}
\iffalse Note that we never have an essential extension $0 \rightarrow B$ if $B \neq 0$. Indeed, the intersection of $0 \rightarrow B$ and $B \xrightarrow{1} B$ is zero. \fi \begin{proof} \begin{itemize}
\item Let $A \rightarrow B$ be essential, and $B \rightarrow F$ be such that $A \rightarrow B \rightarrow F$ is monic. We claim $B \rightarrow F$ is monic. Suppose not, then $B' \rightarrow B := Ker (B \rightarrow F) \neq 0$ is monic, so by assumption $$(A \rightarrow B) \cap (B' \rightarrow B) \neq 0.$$
On the other hand, we show the intersection is zero, for a contradiction. Suppose the monic $C \rightarrow B$ is contained in the intersection, so $C \rightarrow B$ factors as $C \rightarrow A \rightarrow B$, and also factors through the kernel of $B \rightarrow F$. In particular, $$C \rightarrow A \rightarrow B \rightarrow F=C \rightarrow B\rightarrow F= 0.$$
Since $C \rightarrow A$ is monic, we conclude $A \rightarrow B \rightarrow F = 0$, but this was a monic, so $A = 0$. Then $A \rightarrow B = 0$, so the intersection has to be zero.
\item Conversely, suppose $B' \rightarrow B$ is a nonzero monic with $ (A \rightarrow B) \cap (B' \rightarrow B) = 0$. Set $B \rightarrow F := Cok(B' \rightarrow B)$. \\ We see that $B \rightarrow F$ is not monic --- otherwise $0 = Ker(B \rightarrow F) = B' \rightarrow B$. \\ On the other hand, $Ker(A \rightarrow B\rightarrow F) = 0$:
Suppose $A' \rightarrow A$ is such that $A' \rightarrow A \rightarrow B \rightarrow F = 0$. We must show it is zero.
\\ Consider the monic $Im(A' \rightarrow A \rightarrow B) = I \rightarrow B$. By definition of image, this factors through $A \rightarrow B$. It also factors through $B' \rightarrow B = Ker(B \rightarrow F)$, since $A' \rightarrow I$ is epic and
$$A' \rightarrow I \rightarrow B \rightarrow F = A' \rightarrow A \rightarrow B \rightarrow F = 0.$$
Hence it lies in the intersection $(A \rightarrow B) \cap (B' \rightarrow B) = 0$ as required. \end{itemize} \end{proof}
\begin{thm} \label{6.13} In a Grothendieck category, an object is injective iff it has no proper essential extensions. \end{thm}
\begin{proof} Certainly if $E$ is injective, then its only proper extensions are trivial, $E \xrightarrow{i_1} E \oplus B$, $B \neq 0$. \\ Then $E \oplus B \xrightarrow{\pi_1} E$ is not monic (it is epic but not an isomorphism); however $\pi_1 i_1=1_E$ is monic. By definition this is not essential. \\
\noindent Conversely, let $E$ have no proper essential extensions. Let $E \rightarrow B$ be any extension; we show it must be trivial. \\ Let $\mathcal{F}$ be the poset (ordered by inclusion) of subobjects of $B$ which have zero intersections with (the image of) $E \rightarrow B$. \begin{itemize}
\item[\textbf{Claim:}] If $\{B_i\}_I$ is an ascending chain in $\mathcal{F}$ then $\bigcup B_i \in \mathcal{F}.$
\item[\textbf{Proof:}] $\bigcup B_i$ exists as a subobject of $B$. We show it has zero intersection with $Im(E \rightarrow B) = I \rightarrow B$:
$$I \cap \bigcup B_i = \bigcup (I \cap B_i) = \bigcup 0 = 0.$$
The claim is proven. \end{itemize} \noindent Hence, Zorn's Lemma guarantees us a maximal element $B' \subset B$ of $\mathcal{F}$. \\ Let us switch perspectives by taking cokernels, to get a corresponding family $\tilde{\mathcal{F}}$ of quotient objects of $B$, where $$B \rightarrow F \in \tilde{\mathcal{F}} \text{ iff } E \rightarrow B \rightarrow \text{ F is monic.}$$ This must have a minimal element $B \rightarrow B''$ (corresponding to $B' \subset B$).
\\ Certainly $E \rightarrow B \rightarrow B''$ is monic; let us show it is essential. \\ Suppose $B'' \rightarrow F$ is such that $E \rightarrow B \rightarrow B'' \rightarrow F$ is monic, then by definition, the coimage of $B \rightarrow B'' \rightarrow F$ is smaller than $B \rightarrow B''$. By minimality of $B''$, it must be equal to this coimage, and in particular is monic. \\
\noindent By hypothesis, the essential extension $E \rightarrow B \rightarrow B''$ cannot be proper, so it is an isomorphism, and $E \rightarrow B$ is trivial. (Writing $\psi = E \xrightarrow{j} B \xrightarrow{k} B''$, $\psi^{-1}k:B \rightarrow E$ is such that $\psi^{-1}k j = (kj)^{-1}kj=1_E$.) \\ \end{proof}
\noindent The following falls out easily as a corollary. We include it because the key result of this section is proven similarly. \begin{thm}[Baer's Criterion] Let $R$ be a ring, and $A$ be a left $R$-module. \\ If for every left ideal $I \subset R$ we have that $Hom(R,A) \rightarrow Hom(I,A)$ is epic, then $A$ is injective in $R$-Mod. \end{thm}
\begin{proof} By the theorem above, it suffices to show that $A$ has no proper essential extensions. \\ Let $A \subset B$, $x \in B \backslash A$. We show $A \subset B$ is not essential. \\ Let $R \xrightarrow{x} B$ be the map sending $1 \mapsto x$. Make a pullback diagram:
\[ \begin{tikzcd} I \arrow{r}{i_1} \arrow[swap]{d}{i_2} & R \arrow{d}{x} \\ A \arrow{r}{j} & B \end{tikzcd} \] $I = \{(a,r): a=rx\} = \{(rx,x): rx \in A\}$ may be identified with the ideal $\{r \in R: rx \in A\}$, \\ so by assumption $I \xrightarrow{i_2} A$ extends to a map $R \rightarrow A$: there is some $y \in A$ with $I \xrightarrow{i_1} R \xrightarrow{y} A = I \xrightarrow{i_2} A$. \\
\noindent We have $x-y \neq 0$ since $x \notin A \ni y$. On the other hand, the submodule $M = \{r(x-y): r \in R\}$ of $B$ generated by $x-y$ meets $A$ only trivially. \\ In other words, consider the nonzero monic $M \subset B$. $B$ is not essential, because the intersection of the images of $A \subset B$ and $M \subset B$ is zero --- given $r(x-y) \in A$ where $r \in R$, then $rx=r(x-y)+ry \in A$, so $r \in I$, so $r(x-y) = 0$ because $$rx= xi_1(r)= ji_2(r)=jyi_1(r) = y(r) = ry.$$ \\ \end{proof}
\begin{defn} An \textbf{injective envelope} of $A$ is an injective essential extension. \end{defn}
\noindent An injective envelope is a maximal essential extension and a minimal injection extension. \\
\begin{lem} An essential extension of an essential extension is essential. \end{lem}
\begin{proof} Let $A \xrightarrow{a}B$, $B \xrightarrow{b}C$ be essential extensions. We show the extension $A \xrightarrow{ba} C$ is essential. \\ Let $C \xrightarrow{c} F$ be such that $cba$ is monic. \\ Since $a$ is essential, $cb$ is monic. Since $b$ is essential, $c$ is monic. \\ \end{proof}
\begin{lem} \label{6.22} Let $A \rightarrow E$ be an extension of $A$ in a Grothendieck category, and $\{E_i\}$ an ascending chain of subobjects between (the image of) $A$ and $E$. If $E_i$ is an essential extension of $A$ for each $i$, then $\bigcup E_i$ is an essential extension of $A$. \end{lem}
\begin{proof} Let $S$ be any nonzero subobject of $\bigcup E_i$. \\ Then $S = S \cap \bigcup E_i = \bigcup (S \cap E_i)$, hence $S \cap E_i \neq 0$ for some $i$. \\ Since $E_i$ is essential, we have $S \cap A = (S \cap E_i) \cap A \neq 0$. \\ \end{proof}
\noindent Although $E$ does not appear explicitly in the proof above, the proof really does hinge on the fact that $A$ and the $E_i$ are contained in $E$; otherwise we could not even speak of $\bigcup E_i$.
\noindent It is the next lemma that asserts that every ascending chain of extensions may indeed be embedded in a common extension $E$, and therefore, the lemma above becomes the statement that every ascending chain of essential extensions is bounded by an essential extension. \\
\begin{thm} \label{6.23} Let $\mathcal{B}$ be a Grothendieck category, $J$ an ordered set, and $\{E_j \rightarrow E_k\}_{j < k}$ a family of monics such that whenever $j < k< l$, $E_j \rightarrow E_k \rightarrow E_l = E_j \rightarrow E_l$. \\ Then there is an object $E \in \mathcal{B}$ such that whenever $j < k$, $$E_j \rightarrow E_k \rightarrow E = E_j \rightarrow E$$. \end{thm}
\begin{proof} Let $S = \Sigma_J E_j$. For each $j \in J$ let $E_j \xrightarrow{\iota_j} S$ be the $j$th inclusion. For each $j \in J$, define a map $S \xrightarrow{h_j} S$ on the component $E_k$ as
$$E_k \xrightarrow{\iota_k} S \xrightarrow{h_j} S = \left\{
\begin{array}{ll}
E_k \rightarrow E_j \xrightarrow{\iota_j} S & \mbox{if } k \leq j \\
E_k \xrightarrow{\iota_k} S & \mbox{if } j \leq k.
\end{array} \right. $$
\noindent Let $S \xrightarrow{h} E$ be an epic such that $Ker(h) = \bigcup_k Ker(h_k)$. (Just take the cokernel of the subobject $\bigcup_k Ker(h_k)$ of $S$.) \\ Note that $\{Ker(h_k)\}$ is an ascending family, because for $k \leq {k'}$ we have $$S \xrightarrow{h_{k'}} S = S \xrightarrow{h_k} S \xrightarrow{h_{k'}} S.$$
\\ It remains to see that each $E_j \xrightarrow{\iota_j} S \xrightarrow{h} E$ is monic. \\ For this, it suffices to show that $Im(E_j \rightarrow S) \cap ( \bigcup_k Ker(h_k)) =0$.
\\ We know each $E_j \rightarrow S \xrightarrow{h_k} S$ is monic, so $Im(E_j \rightarrow S) \cap ( Ker(h_k)) =0$ for each $k$, and we are done by the Grothendieck axiom. \\ \end{proof}
\noindent Recall our goal for this section: to prove that in a Grothendieck category with a generator, every object has an injective envelope. \\ Let $\mathcal{B}$ be a Grothendieck category. By Theorem \ref{6.13} we may choose for each non-injective object $A \in \mathcal{B}$ a proper essential extension $E(A):= (A \rightarrow B)$. If $A \in \mathcal{B}$ is injective, setting $E(A) := A \rightarrow A$ already gives us an injective envelope. \\Define $E^\gamma (A)$ by transfinite recursion, as follows: \begin{itemize}
\item on zero: $E^0 (A)=E(A)$;
\item on successor ordinals: $E^{\gamma + 1} (A) = E \rightarrow E^\gamma (A) \rightarrow E(E^{\gamma} (A))$;
\item on limit ordinals: $E^\alpha(A)$ is a minimal essential extension that bounds $E^\gamma (A)$ for all $\gamma < \alpha$. (Such an extension exists by Theorem \ref{6.23}.) \end{itemize} \noindent Then the sequence $\{ E^\gamma (A) \}$ becomes stationary precisely when it reaches an injective essential extension, i.e., an injective envelope of $A$. We show that this does happen, when $\mathcal{B}$ has a generator.
\begin{thm} If $\mathcal{B}$ is a Grothendieck category with a generator $G$ then every object has an injective envelope. \end{thm}
\begin{proof} We start out similarly to the proof of Theorem \ref{Mitch}: Let $R = End (G)$; there is a functor $F: \mathcal{F} \rightarrow R$-Mod sending $\mathcal{B} \ni B \mapsto Hom(G,B) \in R$-Mod. \begin{itemize}
\item[\textbf{Claim}:] If $A \rightarrow E$ is an essential extension in $\mathcal{B}$, then $FA \rightarrow FE$ is an essential extension in $R$-Mod.
\item[\textbf{Proof:}] $FA \rightarrow FE$ is an extension because $F=Hom(G,-)$ is left-exact.
\\ Let $M \subset FE$ be a nonzero submodule, so there is $x \in M$. We need to construct a nonzero element in $M \cap Im(FA \rightarrow FE)$.
\\ $x \in M \subset FE = Hom(G,E)$, so take a pullback diagram: \[ \begin{tikzcd} P \arrow{r} \arrow[swap]{d} & G \arrow{d}{x} \\ A \arrow{r} & E \end{tikzcd} \]
$A \rightarrow E$ was essential and $x \neq 0$, so $P \neq 0$, and $G \xrightarrow{1} G$ factors as $G \rightarrow P \rightarrow G$.
\\ Now $0 \neq G \rightarrow P \rightarrow G \xrightarrow{x} E$ is an element of $M$, and is contained in $Im(F(A \rightarrow E))$.
\end{itemize} \noindent Now, we use the fact that for any ring $R$, $R$-Mod \textit{has enough injectives}: there is an injective extension out of every $R$-module. In particular there is an injective extension $FA \rightarrow Q$, which factors by injectivity of $Q$ as $FA \rightarrow FE \rightarrow Q$. Further, we have that $FE$ is isomorphic to a subobject of $Q$. \\ The above holds for \textit{any} essential extension $E$ of $A$, so, simply take any ordinal $\Omega$ whose cardinality is larger than that of the set of subobjects of $Q$. Since $F$ is an embedding, any sequence of proper essential extensions of $A$ must terminate before $\Omega$. (There are no more essential extensions $A \rightarrow E$ than the extensions $FA \rightarrow FE$, but there are no more of \textit{these} than subobjects of $Q$.)
\end{proof} \section{The Embedding Theorem}
\begin{prop} \label{7.11} If an object $E \in [\mathcal{A},Ab]$ is injective, then it is a right-exact functor. \end{prop}
\begin{proof} Let $A' \rightarrow A \rightarrow A'' \rightarrow 0$ be an exact sequence in $\mathcal{A}$. Applying the Yoneda embedding $H$, we obtain in $[\mathcal{A},Ab]$ an exact sequence $$0 \rightarrow H^{A''} \rightarrow H^{A} \rightarrow H^{A'}.$$ By definition of $E$ being injective, the functor $Hom(-,E)$ is exact. Therefore we obtain in $Ab$ an exact sequence $$ Hom(H^{A'},E) \rightarrow Hom(H^{A},E) \rightarrow Hom(H^{A''},E) \rightarrow 0.$$ By the Yoneda Lemma, this sequence is isomorphic to $$ E(A') \rightarrow E(A) \rightarrow E(A'') \rightarrow 0,$$ so $E$ is right-exact. \\ \end{proof}
\noindent A functor is \textbf{mono} if it preserves monics. In particular a right-exact functor is exact iff it is mono, so an injective mono functor is exact. \\ The injective envelope of a mono functor is an exact functor: \\
\begin{lem} \label{7.12} Let $M \rightarrow E$ be an essential extension in $[\mathcal{A}, Ab]$. If $M$ is a mono functor, then so is $E$. \end{lem}
\begin{proof} Suppose $E$ is not mono, so there is a monic $A' \rightarrow A$ in $\mathcal{A}$ such that $EA' \rightarrow EA$ is not monic in $Ab$. There is $0 \neq x \in EA'$ with $(EA' \rightarrow EA)(x) =0$; we construct the subfunctor $F \subset E$ \textit{generated by} $x$ as follows. \\ Define it on objects as $F(B) = \{ y \in EB: \text{ there exists } A' \rightarrow B \text{ in } \mathcal{A} \text{ such that } (EA' \rightarrow EB)(x) = y\}$, from which it follows that $$(EB' \rightarrow EB)(FB') \subset FB$$ for $B' \rightarrow B$: If $y \in FB'$ then there is $A' \rightarrow B'$ in $\mathcal{A}$ with $(EA' \rightarrow EB')(x)=y$. Then $A' \rightarrow B' \rightarrow B$ witnesses that $(EB' \rightarrow EB)(y) \in FB$. \\ Hence we may define $F(B' \rightarrow B)$ by restriction: $$F(B' \rightarrow B) = FB' \rightarrow FB, y \mapsto (EB' \rightarrow EB)(y).$$ (Functoriality is then tautological.) \\ $F$ is still a set-valued functor. We would like to upgrade it to a functor $\mathcal{A} \rightarrow Ab$, and we do this by observing that $FB$ is a subgroup of $EB$: \begin{itemize}
\item $0 \in FB$, since the zero map sends $x$ to it.
\item if $y,z \in FB$ then there are $f,g: A' \rightarrow B$ with $(Ef)(x)=y, (Eg)(x)=z$.
\\ Then $(E(f-g))(x)=(Ef-Eg)(x)=(Ef)(x)-(Eg)(x)=y-z$, so $y-z \in FB$,
\\ where the first equality uses Proposition \ref{7.11} and Corollary \ref{3.12}. \end{itemize}
\noindent Since $x \in FA' \subset EA'$, we have $F \neq 0$. Since $M \rightarrow E$ is essential, $F \cap M \neq 0$, so there is some $B$ with $FB \cap MB \neq 0$, so there is $0 \neq y \in FB \cap MB$. \\ Since $y \in FB$, there is $A' \rightarrow B$ with $y=(EA' \rightarrow EB)(x)$. Let \[ \begin{tikzcd} A' \arrow{d} \arrow{r} &A \arrow{d}\\ B \arrow{r} &P \end{tikzcd} \] be a pushout diagram. Since $A' \rightarrow A$ was monic, so will be $B \rightarrow P$. Hence so too is $MB \rightarrow MP$ (since $M$ was mono), therefore $MB \rightarrow MP \neq 0$. \\ Therefore, $$0 \neq (EB \rightarrow EP)(y) = (EB \rightarrow EP)(EA' \rightarrow EB)(x) = (EA' \rightarrow EP)(x)= (EA \rightarrow EP)(EA' \rightarrow EA)(x)=0,$$ a contradiction. The first step is because $MB \rightarrow EB \rightarrow EP = MB \rightarrow MP \rightarrow EP$ (a naturality square), and $MP \rightarrow EP$ is monic. \\ \end{proof}
\noindent Let $\mathcal{M}(\mathcal{A})$ be the full subcategory of $[\mathcal{A}, Ab]$ whose objects are the mono functors. \\ $\mathcal{M}(\mathcal{A})$ is closed under taking subobjects, products, and essential extensions: \begin{itemize}
\item Let $E$ be a subfunctor of $F$ mono, so each component $EA \rightarrow FA$ is monic.
\\ If $A \rightarrow A''$ is monic in $\mathcal{A}$, then $FA \rightarrow FA''$ is monic in $Ab$. Since $E \rightarrow F$ is a natural transformation, $$EA \rightarrow EA'' \rightarrow FA'' = EA \rightarrow FA \rightarrow FA'', \text{ which is monic.}$$
\item Closure under products is easy: we prove a similar result in Theorem \ref{7.27}. (There we show the full subcategory of left-exact functors is closed under products.)
\item We have just proven closure under essential extensions.
\\ \end{itemize}
\noindent To generalise the situation, let $\mathcal{B}$ be a Grothendieck category with injective extensions, and let $\mathcal{M}$ be a full subcategory closed under taking subobjects, products, and essential extensions. Let us call the objects in $\mathcal{M}$ \textit{mono objects}. \\ As an example, if $R$ is an integral domain, then $\mathcal{B} = R$-Mod is Grothendieck, and the subcategory $\mathcal{M}$ of torsion-free modules is closed under these three operations.
\begin{prop} Every object $B \in \mathcal{B}$ has a maximal quotient object $B \rightarrow \mathcal{M} (B)$ in $\mathcal{M}$. \end{prop}
\begin{proof} Let $\mathcal{F}$ be the set of mono quotients of $\mathcal{B}$. Define $M(B)$ as the coimage of $B \xrightarrow{h} \Pi_{B' \in \mathcal{F}} B'$, where each component of $h$ is the obvious epic. \\ Since $\mathcal{M}$ is closed under products and subobjects, and a coimage is a subobject of its codomain, $M (B) \in \mathcal{M}$. By definition, the coimage is a quotient object of $B$, so it remains to see that it is maximal in $\mathcal{M}$. \\ Given an epic $B \rightarrow B''$ with $B'' \in \mathcal{M}$, there is a map $M(B) \rightarrow B''$ such that
\[
\begin{tikzcd}[row sep=0.4em,column sep=2em]
& M(B) \arrow{dd} \\
B \arrow[ur] \arrow[dr] && \text{commutes.}\\
& B'' \\
\end{tikzcd}
\] Indeed, since $B''$ appears as a factor in the product $\Pi B'$, we have a projection map $\Pi B_i \xrightarrow{\pi} B''$. Let us take $$M(B) \rightarrow B'' = M(B) \rightarrow \Pi B' \xrightarrow{\pi} B''.$$ Then, as desired, $$B \rightarrow M(B) \rightarrow B'' = B \rightarrow M(B) \rightarrow \Pi B' \xrightarrow{\pi} B'' = B \rightarrow \Pi B' \xrightarrow{\pi} B'' = B \rightarrow B''.$$ \end{proof}
\begin{prop} \label{7.22} Let $B \rightarrow M$ be any map, where $B \in \mathcal{B}, M \in \mathcal{M}$. \\ There is a unique map $M(B) \rightarrow M$ such that
\[
\begin{tikzcd}[row sep=0.4em,column sep=2em]
& M(B) \arrow{dd} \\
B \arrow[ur] \arrow[dr] && \text{commutes.}\\
& M \\
\end{tikzcd}
\]
(This says that $B \rightarrow M(B)$ is a \textbf{reflection} of $B$ in $\mathcal{M}$.) \end{prop}
\begin{proof} Let $B \xrightarrow{c} B''$ be the coimage of $B \xrightarrow{b} M$, so $b$ factors as $B \xrightarrow{c} B'' \xrightarrow{d} M$. \\ $B''$ is mono since $\mathcal{M}$ is closed under subobjects. By maximality of $M(B)$ among mono quotients, $B \xrightarrow{c} B''$ factors as $B \xrightarrow{q} M(B) \xrightarrow{u} B''$. \\ Let us define $M(B) \xrightarrow{x} M$ as $du$. \\ We clearly have $b = dc = duq = xq$. \\ $x$ is the unique such map, because $q$ is epic: if also $b = x'q$, then $x'q = b = xq$ implies $x=x'$. \\ \end{proof}
\noindent Given any $B' \rightarrow B$ in $\mathcal{B}$, we get a unique map $M(B') \rightarrow M(B)$ such that
\[ \begin{tikzcd} B' \arrow{r} \arrow[swap]{d} & M(B') \arrow{d} \\ B \arrow{r} & M(B) & \text{commutes,} \end{tikzcd} \]
\noindent by taking $B' \rightarrow M(B)$ to be the composite $B' \rightarrow B \rightarrow M(B)$ in
\[
\begin{tikzcd}[row sep=0.3em,column sep=2em]
& M(B') \arrow{dd}{\exists !} \\
B' \arrow[ur] \arrow{dr}\\
& M(B) \ . \\
\end{tikzcd}
\]
\noindent The uniqueness forces $M$ to be an additive functor $\mathcal{B} \rightarrow \mathcal{M}$. \begin{itemize}
\item $M(1_B) = 1_M(B)$, because $1_{M(B)}$ makes the diagram \[ \begin{tikzcd} B \arrow{r} \arrow[swap]{d}{1_B} & M(B) \arrow{d}{1_{M(B)}} \\ B \arrow{r} & M(B) & \text{commute.} \end{tikzcd} \]
\item $M(B' \xrightarrow{f} B \xrightarrow{g} B'') = M(g) \circ M(f)$, because $M(g) \circ M(f)$ makes the diagram \[ \begin{tikzcd} B' \arrow{r} \arrow[swap]{d}{g \circ f} & M(B') \arrow{d}{M(g) \circ M(f)} \\ B'' \arrow{r} & M(B'') & \text{commute. (The diagram expands as two commutative squares, one above the other.)} \end{tikzcd} \]
\item $M$ is additive -- each square in \[ \begin{tikzcd} B' \arrow{r}{q'} \arrow[swap]{d}{\Delta = \langle 1,1 \rangle } & M(B') \arrow{d}{\Delta = \langle 1,1 \rangle } \\ B' \oplus B' \quad \arrow{r} \arrow[swap]{d}{[ f, g ]} &\quad M(B') \oplus M(B') \arrow{d}{[ Mf, Mg ] } \\ B \arrow{r}{q} & M(B') & \text{commutes, so the ``outer'' square commutes:} \end{tikzcd} \] $$[ Mf, Mg ] \circ [i_1 \circ q', i_2 \circ q'] = [[Mf, Mg] \circ i_1 \circ q', [ Mf, Mg ] \circ i_2 \circ q'] = [Mf \circ q', Mg \circ q'] = [q \circ f, q \circ g] = q \circ [f,g].$$ $$\langle q' \circ \pi_1, q' \circ \pi_2 \rangle \circ \langle 1,1 \rangle = \langle q' \circ \pi_1 \circ \langle 1,1 \rangle \, q' \circ \pi_2 \circ \langle 1,1 \rangle \rangle = \langle q' \circ 1,q' \circ 1 \rangle = \langle 1 \circ q',1 \circ q' \rangle = \langle 1,1 \rangle q'.$$ \end{itemize}
\noindent (We have shown that $\mathcal{M}$ is a \textbf{reflective} subcategory of $\mathcal{B}$, and the functor $M: \mathcal{B} \rightarrow \mathcal{M}$ is a \textbf{reflector}.) \\ \\
\noindent Let us call $T \in \mathcal{B}$ a \textbf{torsion} object if $Hom(T,N) = 0$ for each $N \in \mathcal{M}$.
\begin{prop} $T$ is torsion iff $M(T) = 0$. \end{prop}
\begin{proof} Suppose $M(T) = 0$. Let $T \xrightarrow{k} N$ be any map, where $N \in \mathcal{M}$. By Proposition \ref{7.22} this map factors as $k = T \rightarrow 0 \rightarrow N = 0$. \\
\noindent In the other direction, $Hom(T, M(T)) = 0$ means the obvious epic $T \rightarrow M(T)$ is zero, so $M(T) = 0$. \\ (For instance, any map out of $M(T)$ must be zero, since $T \xrightarrow{0} M(T)$ is epic. This shows $M(T)$ is initial.) \\ \end{proof}
\begin{prop} \label{7.24} $Ker(B \rightarrow M(B))$ is the maximal torsion subobject of $B$. \end{prop}
\begin{proof} For any torsion object $T$ and map $T \rightarrow B$, the image of $T \rightarrow B$ is contained in $Ker(B \rightarrow M(B))$. ($T \rightarrow B \rightarrow M(B) = 0$, because Proposition \ref{7.22} says this map factors as $T \rightarrow M(T) \rightarrow M(B)$, and $M(T) = 0$.) \\ Hence, as soon as we that $K=Ker(B \rightarrow M(B))$ is torsion, we are done: it is maximal as such. \\
\noindent Let $B'' \in \mathcal{M}$. We show that any map $K \rightarrow B''$ is zero. \\ We have an exact sequence $0 \rightarrow K \rightarrow B \rightarrow M(B) \rightarrow 0$. \\ Form an injective envelope $B'' \rightarrow E$. $\mathcal{M}$ is closed under this operation, so $E \in \mathcal{M}$. \\ Since $K \rightarrow B$ is monic and $E$ is injective, $K \rightarrow B'' \rightarrow E$ extends to a map $B \rightarrow E$. \\ We obtain the following commutative diagram:
\[ \begin{tikzcd} 0 \arrow{r} &K \arrow{d} \arrow{r} &B \arrow{d} \arrow{r} &M(B) \arrow{dl} \arrow{r} &0 \\ & B'' \arrow{r} &E \end{tikzcd} \] where $M(B) \rightarrow E$ is a map as in Proposition \ref{7.22}. \\By commutativity of the diagram and exactness of the upper row, we have $$K \rightarrow B'' \rightarrow E = K \rightarrow B \rightarrow M(B) \rightarrow E = 0,$$ so $K \rightarrow B''$ is also zero since $ B'' \rightarrow E$ is monic. \\ \end{proof}
\noindent In general, although $\mathcal{B}$ was abelian, $\mathcal{M}$ need not be: not every monic in $\mathbb{M}$ is realised as a kernel of a map in $\mathcal{M}$. For instance, in the situation where $R = \mathbb{Z}$, $\mathcal{B} = Ab$ and $\mathcal{M}$ is the subcategory of torsion-free abelian groups, $\mathcal{M}$ is not abelian because the monic $\mathbb{Z} \xrightarrow{2} \mathbb{Z}$ is not a kernel. \\ (If it was a kernel of some $\mathbb{Z} \xrightarrow{f} B$, then $2f(1)=f(2)=0$, which forces $f(1)=0$ since $B$ is torsion free. So $f=0$. Then $\mathbb{Z} \xrightarrow{1} \mathbb{Z}$ would also factor through the kernel of $f$, which implies that $1$ is even: a contradiction.) \\
\noindent What we \textit{can} do is go one level deeper to define a full subcategory $\mathcal{L}$ of $\mathcal{M}$ that will turn out to be abelian. In the case when our Grothendieck category is $[\mathcal{A}, Ab]$, $\mathcal{L}$ will be our key to proving the Mitchell embedding theorem. \\
\noindent Let us call a subobject $M' \subset M \in \mathcal{M}$ to be \textbf{pure} if $M/M' \in \mathcal{M}$, where $M \rightarrow M/M'$ is the cokernel of $M' \rightarrow M$. Let us call a mono object \textbf{absolutely pure} if whenever it appears as a subobject of a mono object, it is a pure subobject. \\ Define $\mathcal{L}$ to be the full subcategory of absolutely pure objects.
\begin{lem} \label{inj abspure} All injective mono objects are absolutely pure. \end{lem}
\begin{proof} Let $E \in \mathcal{M}$ be injective. Let $E \rightarrow F$ be monic, where $F \in \mathcal{M}$. We must show that $F/E \in \mathcal{M}$. Well, the extension $E \rightarrow F$ must be split, so $F$ is the direct sum of $E$ and $F/E$. In particular $F/E$ is a subobject of $F \in \mathcal{M}$, so $F/E \in \mathcal{M}$. \\ \end{proof}
\begin{lem} If $0 \rightarrow M_1 \rightarrow B \rightarrow M_2 \rightarrow 0$ is exact in $\mathcal{B}$ and $M_1, M_2 \in \mathcal{M}$, then $B \in \mathcal{M}$. \end{lem}
\begin{proof} Let $M_1 \rightarrow E$ be an injective envelope. \\ We have $E \in \mathcal{M}$, hence $E \oplus M_2 \in \mathcal{M}$. \\ Since $E$ is injective, $B \rightarrow E$ extends to a map $M_1 \rightarrow E$. Once we see that the obvious map $B \xrightarrow{m} E \oplus M_2$ is monic, we will be done. \\ Suppose $f,g$ are maps $A \rightarrow B$ with $mf=mg$. It is enough to see that $d:=f-g = 0$. \\ Since $md=0$, $A \xrightarrow{d} B \rightarrow M_2 =A \xrightarrow{d} B \rightarrow E = 0$. In particular, $d$ factors through the kernel of $B \rightarrow M_2$ as $d = A \rightarrow M_1 \rightarrow B$. Now $$A \rightarrow M_1 \rightarrow E = A \rightarrow M_1 \rightarrow B \rightarrow E = A \xrightarrow{d} B \rightarrow E = 0,$$ but $M_1 \rightarrow E$ is monic, so $A \rightarrow M_1 = 0$, and finally $d = A \rightarrow M_1 \rightarrow B= 0$. \\ \end{proof}
\begin{lem} \label{7.26} A pure subobject of an absolutely pure object is absolutely pure. \end{lem}
\begin{proof} Let $A$ be absolutely pure, $P \rightarrow A$ a pure subobject, and $P \rightarrow M$ a monic, where $M \in \mathcal{M}$. We must show that $M/P \in \mathcal{M}$. \\ Make a pushout diagram \[ \begin{tikzcd}[row sep=0.8em,column sep=1em] P \arrow{r} \arrow[swap]{d} & A \arrow{d} \\ M \arrow{r} & R \end{tikzcd} \] and extend it to an exact commutative diagram \[ \begin{tikzcd}[row sep=0.8em,column sep=1em]
&0 \arrow{d} &0 \arrow{d} &0 \arrow{d}\\ 0 \arrow{r} & P \arrow{r} \arrow[swap]{d} & A \arrow{r} \arrow{d} &A/P \arrow{d} \arrow{r} & 0 \\ 0 \arrow{r} & M \arrow{r} \arrow[swap]{d} & R \arrow{r} \arrow{d} &R/M \arrow{d} \arrow{r} & 0 \\ 0 \arrow{r} & M/P \arrow{d} \arrow{r} & R/A \arrow{d} \arrow{r} & 0 \\ &0 &0 \\ \end{tikzcd} \] by noting that $M/P \rightarrow R/A$ and $A/P \rightarrow R/M$ are both isomorphisms. \\
\noindent Since $M$ and $R/M \cong A/P$ are mono, $R$ is mono. Hence $R/A$ is mono, hence $M/P$ is mono, as required. \\ \end{proof}
\begin{thm} \label{7.27} A mono functor $M \in [\mathcal{A}, Ab]$ is absolutely pure iff it is left-exact. \end{thm}
\begin{proof} First, we prove a claim. \begin{itemize}
\item[\textbf{Claim:}] A subfunctor of a left-exact functor is pure iff it is left-exact.
\item[\textbf{Proof:}]
Let $0 \rightarrow M \rightarrow E \rightarrow F \rightarrow 0$ be exact in $[\mathcal{A}, Ab]$, where $E$ is left-exact. We must show $M$ is left-exact. \\ Let $0 \rightarrow A' \rightarrow A \rightarrow A''$ be exact in $\mathcal{A}$. We have a commutative diagram
\[ \begin{tikzcd}[row sep=0.8em,column sep=1em]
&0 \arrow{d} &0 \arrow{d} &0 \arrow{d}\\ 0 \arrow{r} & MA' \arrow{r} \arrow[swap]{d} & MA \arrow{r} \arrow{d} &MA'' \arrow{d} \\ 0 \arrow{r} & EA' \arrow{r} \arrow[swap]{d} & EA \arrow{r} \arrow{d} &EA'' \\ 0 \arrow{r} & FA' \arrow{d} \arrow{r} & FA \arrow{d} \\ &0 &0 \\ \end{tikzcd} \] whose columns are exact since the evaluation functor $[\mathcal{A}, Ab] \rightarrow Ab$ for each of $A',A, \text{ and } A''$ is exact. \\ The middle row is exact since $E$ is left-exact, so the hypothesis of Lemma \ref{2.64} is satisfied. \\ Hence, $F$ is mono iff $M$ is left-exact. The claim is proven. \end{itemize}
\noindent Now, suppose we have a mono functor $M$. Take an injective envelope $M \rightarrow E$. We know $E$ is left-exact (it is injective and mono, hence exact) and absolutely pure (by Lemma \ref{inj abspure}). \\ If $M$ is absolutely pure, then $M \rightarrow E$ is pure, so the claim implies $M$ is left-exact. \\ Conversely, if $M$ is left-exact, the claim implies $M \rightarrow E$ is pure, so we finish by Lemma \ref{7.26}. \end{proof}
\noindent Recall that in the general setting we have a Grothendieck category $\mathcal{B}$, a full subcategory $\mathcal{M}$ of $\mathcal{B}$ closed under taking subobjects, products, and essential extensions, and a full subcategory $\mathcal{L}$ of $\mathcal{M}$ consisting of the absolutely pure objects. \\
\noindent Given $M \in \mathcal{M}$ and $R \in \mathcal{L}$, a map $M \rightarrow R$ is a \textbf{reflection} of $M$ in $\mathcal{L}$ if for every map $M \rightarrow L$ where $L \in \mathcal{L}$, there is a unique map $R \rightarrow L$ such that
\[
\begin{tikzcd}[row sep=0.4em,column sep=2em]
& R \arrow{dd} \\
M \arrow[ur] \arrow[dr] && \text{commutes.}\\
& L \\
\end{tikzcd}
\]
\begin{thm}[Recognition Theorem] \label{7.28} If the sequence $0 \rightarrow M \rightarrow R \rightarrow T \rightarrow 0$ is exact in $\mathcal{B}$ for $M$ mono, $R$ absolutely pure, and $T$ torsion, then $M \rightarrow R$ is a reflection of $M$ in $\mathcal{L}$. \end{thm}
\begin{proof} Given $L \in \mathcal{L}$ and $M \xrightarrow{m} L$, let $L\xrightarrow{l} E$ be an injective envelope and $E \xrightarrow{c} F = Cok(L\xrightarrow{l} E)$. \\ We obtain a commutative diagram with exact rows
\[ \begin{tikzcd}[row sep=1.1em,column sep=1.5em] 0 \arrow{r} &M \arrow{d}{m} \arrow{r}{i} &R \arrow{d}{r} \arrow{r} &T \arrow{d} \arrow{r} &0 \\ 0 \arrow{r} &L \arrow{r}{l} &E \arrow{r}{c} &F \arrow{r} &0 \\ \end{tikzcd} \] as follows: \\ We already have exact rows, and the vertical map $M \xrightarrow{m} L$. Since $M \xrightarrow{i} R$ is monic and $E$ is injective, $M \rightarrow L \rightarrow E$ extends to a map $R \xrightarrow{r} E$. Finally, we get a map $T \rightarrow F$ because the map $R \xrightarrow{r} E \xrightarrow{c} F$ factors through the cokernel $R \rightarrow T$ of $M \xrightarrow{i} R$. ($cri = clm = 0m = 0$.) \\
\noindent $E$ is mono by Lemma \ref{7.12}, so $F$ is mono by absolute purity of $L$. Hence $T \rightarrow F = 0$, by definition of $T$ torsion. Then $R \xrightarrow{r} E \xrightarrow{c} F = 0$, so $r$ factors through the kernel $L$ of $c$: there is some $R \xrightarrow{u} L$ with $lu=r$. \\ Then $lui=ri=lm$, but $l$ is monic so we have found that
\[
\begin{tikzcd}[row sep=0.4em,column sep=2em]
& R \arrow{dd}{u} \\
M \arrow{ur}{i} \arrow{dr}{m} && \text{commutes.}\\
& L \\
\end{tikzcd}
\]
It remains to see that $u$ is the unique such map.
\\ If we have $u, u' : R \rightarrow L$ with $ui=m=u'i$, then $d:=u-u'$ factors through the cokernel $R \rightarrow T$ of $i$,
$$R \xrightarrow{d} L = R \rightarrow T \rightarrow L = 0,$$ where the last equality follows since $T$ is torsion. \\ \end{proof}
\begin{thm}[Construction Theorem] \label{7.29} For every $M \in \mathcal{M}$ there is a monic $M \rightarrow R$ which is a reflection of $M$ in $\mathcal{L}$. \end{thm}
\begin{proof} Let $M \rightarrow E$ be an injective envelope. In particular $E$ is absolutely pure. \\ Construct an exact commutative diagram \[ \begin{tikzcd}[row sep=0.8em,column sep=1em]
&0 \arrow{d} &0 \arrow{d} &0 \arrow{d}\\ 0 \arrow{r} & M \arrow{r} \arrow[swap]{d} & R \arrow{r} \arrow{d} &T \arrow{d} \arrow{r} &0 \\ 0 \arrow{r} & M \arrow{r} \arrow[swap]{d} & E \arrow{r} \arrow{d} &F \arrow{r} \arrow{d} &0 \\
& 0 \arrow{r} & M(F) \arrow{d} \arrow{r} &M(F) \arrow{d} \arrow{r} &0 \\ &&0 &0 \\ \end{tikzcd} \] by starting with the middle row (constructed from the monic $M \rightarrow E$), then the right column (constructed from the epic $F \rightarrow M(F)$), the the bottom row, then the middle column (constructed from $E \rightarrow M(F)$, epic as the composition of two epics), then the top row. The top row is the only part that is not exact by construction; the map $M \rightarrow R$ exists because $M \rightarrow E$ factors through the kernel $R \rightarrow E$ of $E \rightarrow M(F)$. \\ The top row is exact by Lemma \ref{2.65}. \\
\noindent We finish simply by applying Theorem \ref{7.28} to the top row, since $M$ is mono, $T$ is torsion by Proposition \ref{7.24}, and $R$ is absolutely pure ($R \rightarrow E$ is pure since $M(F) \in \mathcal{M}$, and $E$ is absolutely pure). \\ \end{proof}
\noindent $\mathcal{L}$ is seen to be a reflective subcategory of $\mathcal{M}$, in exactly the same way we showed that $\mathcal{M}$ was a reflective subcategory of $\mathcal{B}$: choosing a reflection $M \rightarrow R(M)$ in $\mathcal{L}$ for each $M \in \mathcal{M}$ yields an additive functor $R: \mathcal{M} \rightarrow \mathcal{L}$. \\ (Reflections are unique up to isomorphism.) \\
\begin{thm} \label{7.31} $\mathcal{L}$ is abelian, and every object has an injective envelope. \end{thm}
\begin{proof} We check the axioms. \begin{itemize}
\item[A0.] The constantly zero functor is a zero object.
\item[A1, A1*.] For $M \in \mathcal{M}$, we have $M \in \mathcal{L}$ iff $M \rightarrow R(M)$ is an isomorphism.
\\ (If $M \in \mathcal{L}$ then $M \xrightarrow{1} M$ is a reflector; conversely if $M \cong R(M) \in \mathcal{L}$ then $M \in \mathcal{L}$.)
\\ Since $R$ is an additive functor, it preserves direct sums: given $N,N' \in \mathcal{L}$, we have
$$R(N \oplus N') \cong R(N) \oplus R(N') \cong N \oplus N',$$
so $N \oplus N' \in \mathcal{L}$.
\item[A2.] By Lemma \ref{7.26}, the $\mathcal{B}$-kernel of an $\mathcal{L}$-map $L \rightarrow L'$ is in $\mathcal{L}$, so $\mathcal{L}$ has kernels. Indeed, write $K \rightarrow L$ for the $\mathcal{B}$-kernel. Then $K \in \mathcal{L}$, since $L \in \mathcal{L}$ and $K \rightarrow L$ is a pure subobject (as K/L is just $L' \in \mathcal{L}$).
\\ In fact, an $\mathcal{L}$-map s an $\mathcal{L}$-monic iff it is a $\mathcal{B}$-monic. (Both conditions are equivalent to the kernel being zero.)
\item[A3.] Let $L \rightarrow L'$ be an $\mathcal{L}$-monic, and let $L' \rightarrow L'/L$ be its $\mathcal{B}$-cokernel. Since $L$ is absolutely pure and $L' \in \mathcal{M}$, $M:=L'/L \in \mathcal{M}$. Therefore it has a reflection $M \rightarrow R(M)$ in $\mathcal{L}$.
\\ Now $L \rightarrow L'$ is the $\mathcal{B}$-kernel of $L' \rightarrow M$, and hence the $\mathcal{B}$-kernel of $L' \rightarrow R(M) = L' \rightarrow M \rightarrow R(M)$, since $M \rightarrow R(M)$ is monic.
\\ Of course, the $\mathcal{B}$-kernel of an $\mathcal{L}$-map \textit{is} its $\mathcal{L}$-kernel, so $L \rightarrow L'$ the $\mathcal{L}$-kernel of $L' \rightarrow R(M))$.
\item[A2*.] Let $L \rightarrow L'$ be an $\mathcal{L}$-map. Take the $\mathcal{B}$-cokernel $L' \rightarrow F$.
\\ Then $L' \rightarrow F \rightarrow M(F) \rightarrow R(M(F))$ is an $\mathcal{L}$-cokernel:
\\ Certainly $L \rightarrow L' \rightarrow F \rightarrow M(F) \rightarrow R(M(F)) =0$, since $L \rightarrow L' \rightarrow F =0$. Now suppose $L \rightarrow L' \rightarrow N =0$. Then $L' \rightarrow N$ factors uniquely through the cokernel as $L' \rightarrow F \xrightarrow{\exists !} N$. In turn, $F \rightarrow N$ factors uniquely through $M(F)$ as $F \rightarrow M(F) \xrightarrow{\exists !} N$, and in turn still, $M(F) \rightarrow N$ factors uniquely through $R(M(F))$ as $M(F) \rightarrow R(M(F)) \xrightarrow{\exists !} N$.
\\ Taken altogether, $L' \rightarrow N$ factors uniquely through $L' \rightarrow F \rightarrow M(F) \rightarrow R(M(F))$.
\item[A3*.] The above shows that an $\mathcal{L}$-map $L \rightarrow L'$ is an $\mathcal{L}$-epic iff the $\mathcal{B}$-cokernel of $L \rightarrow L'$ is torsion.
\\ Let $L \rightarrow L'$ be an $\mathcal{L}$-epic. Take its $\mathcal{B}$-image $M \rightarrow L'$ to get an exact sequence $ 0 \rightarrow M \rightarrow L' \rightarrow T \rightarrow 0$ in $\mathcal{B}$. We have just remarked that $T$ must be torsion; furthermore $M$ is mono as a subobject of $L' \in \mathcal{M}$, so we may apply Theorem \ref{7.28} to this sequence to deduce that $L' \cong R(M)$.
\\ Therefore, write $K \rightarrow L = Ker(L \rightarrow M)$; then the $\mathcal{B}$-cokernel of $K \rightarrow L$ is $L \rightarrow M$.
\\ We know that the $\mathcal{L}$-cokernel of $K \rightarrow L$ must be the $\mathcal{B}$-cokernel postcomposed with a reflection down to $\mathcal{M}$ and then another reflection down to $\mathcal{L}$, but this is just $L \rightarrow R(M) = L \rightarrow L'$.
\\ We have exhibited $L \rightarrow L'$ as an $\mathcal{L}$-cokernel! \end{itemize} \noindent Therefore $\mathcal{L}$ is abelian. \\Let us see that every object in $\mathcal{L}$ has an injective envelope. Since monics are the same in $\mathcal{B}$ and in $\mathcal{L}$, if $E$ is a $\mathcal{B}$-injective envelope of an $\mathcal{L}$-object, then it is injective in $\mathcal{L}$. \\ (To spell this out: take an injective envelope $L \rightarrow E$ in $\mathcal{B}$. $E \in \mathcal{B}$ is injective and mono, hence absolutely pure. $L \rightarrow E$ is still an injective essential extension in $\mathcal{L}$.) \\ \end{proof}
\noindent Finally, let us return to the case of the Grothendieck category $[\mathcal{A}, Ab]$. Just as $\mathcal{M}(\mathcal{A})$ was the full subcategory of mono functors, let us define $\mathcal{L}(\mathcal{A})$ to be the full subcategory of left-exact functors. \\ Theorems \ref{7.27} and \ref{7.31} say that $\mathcal{L}(\mathcal{A})$ is an abelian category wtih injective envelopes. The Yoneda embedding $H: \mathcal{A} \rightarrow [\mathcal{A}, Ab]$ factors through $\mathcal{L}(\mathcal{A})$, precisely because each $H^A=Hom(A,-)$ is left-exact. \\
\begin{thm} \label{7.32} $\mathcal{L}(\mathcal{A})$ is complete and has an injective cogenerator. \end{thm}
\begin{proof} Products in $\mathcal{L}(\mathcal{A})$ are just products in $[\mathcal{A},Ab]$, because the product of left-exact functors of $[\mathcal{A},Ab]$ is left-exact: Suppose we have a family $\{F_i\}_I$ in $\mathcal{L}(\mathcal{A})$. Let $0 \rightarrow A' \rightarrow A \rightarrow A$ be exact in $\mathcal{A}$. Then $0 \rightarrow F_iA' \rightarrow F_iA \rightarrow F_A''$ is exact for each $i$. \\Taking the product of these sequences in $Ab$ yields $0 \rightarrow \Pi (F_iA') \rightarrow \Pi(F_iA) \rightarrow \Pi(F_i A'')$, exact in $Ab$, but of course this last sequence is just $0 \rightarrow (\Pi F_i)A' \rightarrow (\Pi F_i)A \rightarrow (\Pi F_i)A''$. \\
\noindent In particular the product of all the representables $\{H^A\}_{A \in \mathcal{A}}$ is also left-exact, and since this was a generator for $[\mathcal{A}, Ab]$ (Theorem \ref{5.35}), it is a generator for $\mathcal{L}(\mathcal{A})$. \\ By Proposition \ref{3.37}, $\mathcal{L}(\mathcal{A})$ has an injective cogenerator.
\\ \end{proof}
\begin{thm} $H: \mathcal{A}^{op} \rightarrow \mathcal{L}(\mathcal{A})$ is an exact full embedding. \end{thm}
\begin{proof} We know $H$ is a full embedding (Theorem \ref{5.36}); it remains to show $H$ is exact. \\ Let $0 \rightarrow A' \rightarrow A \rightarrow A'' \rightarrow 0$ be exact in $\mathcal{A}$. We must show $0 \rightarrow H^{A''} \rightarrow H^{A} \rightarrow H^{A'} \rightarrow 0$ is exact in $\mathcal{L}(\mathcal{A})$. \\ This is the case iff the sequence $0 \rightarrow Hom(H^{A'},E) \rightarrow Hom(H^{A},E) \rightarrow Hom(H^{A''},E) \rightarrow 0$ is exact in $Ab$ for an injective cogenerator $E$ in $\mathcal{L}(\mathcal{A})$. \\ ($E$ is injective, so $Hom(-,E)$ is exact; $E$ is a cogenerator, so $Hom(-,E)$ is an embedding; apply Corollary \ref{exemb}.)\\ \noindent That last sequence is isomorphic by the Yoneda Lemma to $0 \rightarrow EA' \rightarrow EA \rightarrow EA'' \rightarrow 0$, and this sequence is always exact iff $E$ is an exact functor. \\ This is indeed the case: $E$ is right-exact by Lemma \ref{7.11}, and left-exact since it lies in $\mathcal{L}(\mathcal{A})$. \\ \end{proof}
\begin{thm}[Freyd-Mitchell] Every abelian category is fully abelian. \end{thm}
\begin{proof} The Yoneda embedding $H: \mathcal{A}^{op} \rightarrow \mathcal{L}(\mathcal{A})$ provides an exact full embedding into a complete abelian category with an injective cogenerator. \\ We may of course view this as a functor $H: \mathcal{A} \rightarrow \mathcal{L}(\mathcal{A})^{op}$. This is an exact full embedding into a cocomplete abelian category with a projective generator. Now apply Theorem \ref{Mitch}. \end{proof}
\begin{cor} For every small abelian category $\mathcal{A}$ there is a ring $R$ and an exact full embedding \\ $\mathcal{A} \rightarrow R$-Mod. \end{cor}
\end{document} | arXiv |
\begin{document}
\title[FLLN for spatially dense non-Markovian epidemic models ]{\ttl}
\author[Guodong \ Pang]{Guodong Pang$^*$} \address{$^*$Department of Computational Applied Mathematics and Operations Research, George R. Brown School of Engineering, Rice University, Houston, TX 77005} \email{[email protected]}
\author[{\'E}tienne \ Pardoux]{{\'E}tienne Pardoux$^\dag$} \address{$^\dag$Aix Marseille Univ, CNRS, I2M, Marseille, France} \email{[email protected]}
\begin{abstract} We study an individual-based stochastic spatial epidemic model where the number of locations and the number of individuals at each location both grow to infinity. Each individual is associated with a random infection-age dependent infectivity function.
Individuals are infected through interactions across the locations with heterogeneous effects. The epidemic dynamics can be described using a time-space representation for the the total force of infection, the number of susceptible individuals, the number of infected individuals that are infected at each time and have been infected for a certain amount of time, as well as the number of recovered individuals.
We prove a functional law of large numbers for these time-space processes, and in the limit, we obtain a set of time-space integral equations. We then derive the PDE models from the limiting time-space integral equations, in particular, the density (with respect to the infection age) of the time-age-space integral equation for the number of infected individuals tracking the age of infection satisfies a linear PDE in time and age with an integral boundary condition. These integral equation and PDE limits can be regarded as dynamics on graphon under certain conditions.
\end{abstract}
\keywords{spatially epidemic model, infection-age dependent infectivity, growing number of locations and population size, functional law of large numbers, time-space integral equations, PDE model}
\maketitle
\allowdisplaybreaks
\section{Introduction}
In order to capture the geographic heterogeneity, spatial epidemic models have been well developed, both in discrete and continuous spaces.
In discrete space, multi-patch epidemic models have been studied in \cite{sattenspiel1995structured,arrigoni2002limits,allen2007asymptotic,xiao2014transmission,bichara2018multi,nzipardouxyeo} and recently by the authors \cite{PP-2020b}, where each patch represents a geographic location, and infection may occur within each patch and from the distance (for example, due to short travels). See also the multi-patch multi-type epidemic models in \cite{bichara2018multi,FPP2021-MPMG}, as well as relevant models in \cite{ball2008network,magal2016final,magal2018final}. Some of these studies assume migration of individuals among different patches \cite{sattenspiel1995structured,allen2007asymptotic,nzipardouxyeo,PP-2020b,FPP2021-MPMG}, while others do not but assume interactions between patches to induce infection \cite{arrigoni2002limits,bichara2018multi,xiao2014transmission,magal2016final,magal2018final}. In continuous space, various PDE models have been developed (see the monographs \cite{RR2003book,martcheva2015,BCF-2019} and a survey \cite{ruan2007spatial}). There are two well--known models without spatial movement: Kendall's spatial model \cite{kendall1957, kendall1965} and Diekmann-Thieme's PDE model \cite{diekmann1978thresholds,diekmann1979run,thieme1977asymptotic,thieme1977model}. Kendall's spatial model is a system of ODEs with a spatial parameter (without spatial partial derivative).
It was proved to be the FLLN limit of the multitype Markovian SIR model by Andersson and Djehiche \cite{andersson1995limit}, where both the number of types and the population size go to infinity. Diekmann-Thieme's spatial PDE model (with partial derivatives with respect to time and infection-age) has the infection rate depending on the age of infection, as in the PDE model first proposed by Kermack and McKendrick in their 1932 paper \cite{KM32}. Similar to Kendall's spatial model, there is no partial derivative with respect to the spatial parameter, since there is no movement in space. The Diekmann-Thieme PDE model was not yet proved to be the FLLN limit of a stochastic epidemic model, and can be seen as is a special case of our FLLN limit, which is new. We should also mention the spatial models in continuous space in \cite{bowongemakouapardoux} and \cite{vuong2021conditional}, where the stochastic model starts with a continuous process for the movement of individuals, in particular, it is assumed that individual movements follow an It{\^o} diffusion process, and the epidemic models are Markovian.
In this paper, we start with an individual-based stochastic epidemic model at a finite number of locations. Each individual at every location may be infected from his or her own location or from other locations (see the infection rate function in equation \eqref{eqn-upsilon}). Note that individuals do not migrate from one location to another in our model. Each individual is associated with a random infectivity function/process, independent from any other individual but having the same law as all the other individuals. This random infectivity function also determines the law of the infectious duration of each individual. Those random functions are i.i.d. for all individuals. For each individual, we track the age of infection, that is, the elapsed time since the individual was infected (for the initially infected individuals, this means we also know their infection times before time zero). To describe the epidemic dynamics at each location, we use the aggregate infectivity process of the population and a two-parameter (equivalently, measure-valued) process tracking the number of individuals that have been infected for less than or equal to a certain amount time as well as the numbers of susceptible and recovered individuals. Such an individual-based stochastic model with only one location has been studied by the authors in \cite{PP-2021}, where an FLLN is established and the associated PDE model for the limit is derived. In our previous works of large population scaling limits for stochastic epidemic models (see the survey \cite{FPP-survey}), most models consider a homogeneous population with the two exceptions of a multi-patch (discrete space) model \cite{PP-2020b,FPP2021-MPMG}. Our model in this paper starts from a dense discrete space model, while the limit as both the size of the population and the number of patches/locations tend to infinity simultaneously is a deterministic spatial model in continuous space. In particular, the PDE model includes the Diekmann-Thieme spatial model as a special case (see Remarks \ref{rem-special} and \ref{rem-Diekman-Kendall}).
We consider this stochastic epidemic model in a spatially dense setting, where the number of locations increases to infinity while the number of individuals in each location (and the total population) also goes to infinity. This has the same flavor as the asymptotic regime in \cite{andersson1995limit} for the multitype Markovian SIR model where the number of types goes to infinity while the population in each type also go to infinity. It is worth mentioning the paper \cite{arrigoni2002limits} in which a measure-valued limit is proved for a multi-patch Markovian SIS epidemic model without migration in the asymptotic regime with both the number of patches and the number of individuals in each patch going to infinity. This is also in a similar fashion as the asymptotic regime of the Markovian SIR epidemic model with migration on a refining spatial grid in $\R^d$ ($d=1,2,3$), recently studied in \cite{nzipardouxyeo}, where the mesh of the grid goes to zero and the population size at each site also goes to infinity. In the limit of that model, a Laplace operator describes the spatial movement in the time-space dynamics. Although our model does not have migration, our model is non-Markovian and the infection process with the infection-age dependent infectivity is much more complicated than these previous relevant works.
For this model, it is convenient to describe the epidemic dynamics at all locations using a time-space representation of the vector-valued processes (for the number of infected individuals tracking the age of infection, this in fact becomes a time-age-space process). We treat the time-space processes in the functional spaces ${\mathbf D}$ and ${\mathbf D}_{\mathbf D}$ given the spatial component, while choosing the $L^1$ norm on the spatial component. We prove an FLLN (Theorem \ref{thm-FLLN}) for the scaled time-space processes under a set of regularity conditions on the initial conditions, infection contact rates and and random infectivity functions (Assumptions \ref{AS-LLN-1}, \ref{AS-LLN-2} and \ref{AS-lambda}). The limits in the FLLN are described by a set of time-space integral equations. It is worth highlighting that the heterogeneity of interaction effects between different locations is represented by a function $\beta(x,y)$ for $x,y\in [0,1]$ (which resembles the kernel function of a graphon, see further discussions below).
For the weak convergence of the time-space processes, we introduce new weak convergence criteria for these time-space processes (Theorems \ref{thm-D-conv-x} and \ref{thm-DD-conv-x}), which involves the $L_1$ norm for the spatial component. To verify these criteria, we establish moment estimates for the increments of these processes, which is challenging due to the interactions among the individuals at the different locations. In particular, the interactions introduce nontrivial dependence in various components of the time-space processes. We first study the joint time-space dynamics of the susceptible population and total force of infection (Section \ref{sec-proof-conv-S-F}). This involves the existence and uniqueness of solution to a set of time-space Volterra-type integral equations (see equations \eqref{eq:SF}-\eqref{eqn-bar-mfF-0}), and the moment estimates associated with the increments involving the varying infectivity functions together with their interactions (in order to use Theorem \ref{thm-D-conv-x}). Given this convergence, we then establish the convergence of the time-age-space process tracking the infection ages of individuals (Section \ref{sec-proof-conv-I}). In order to employ Theorem \ref{thm-DD-conv-x}, we need to establish the moment estimates for the increments with respect to both time and infection-age parameters, for which the dependence due to interactions also brings additional challenges.
From the limit tracking the rescaled number of infected individuals with a given age of infection, we derive a PDE model with partial derivatives with respect to time and the age of infection (not with respect to the spatial variable, since there is no migration among locations). It is a linear PDE model with an integral boundary condition. It may be seen as an extension of the PDE models in \cite{PP-2021}, with the addition of a spatial component. We then discuss how the PDE model is related to the well-known Diekmann-Thieme PDE model and how it reduces to Kendall's PDE model in the Markovian case (see Remarks \ref{rem-special} and \ref{rem-Diekman-Kendall}). Note that our PDE model is more general since we do not require any condition on the distribution function of the infectious periods.
Our work also contributes to the recent studies of stochastic dynamics on graphon. Keliger et al. \cite{keliger2022local} consider a finite-state Markov chain with local density-dependence on a discretized graph of a graphon, and then prove an FLLN for the Markovian time-space dynamics. Their model includes a Markovian SIS model on graphon, and since each individual is a node on the sampled graph and naturally there is no spatial movement, the limit is in fact a system of ODEs without spatial partial derivative. There is some resemblance between that limit and our PDE model for the Markovian SIS model, see further discussions in Remark \ref{rem-SIS-PDE}, although it is important to note that in our stochastic multi-patch model, the number of individuals in each patch also goes to infinity while in the stochastic model on the sampled graph from a graphon in \cite{keliger2022local}, there is only one individual in each node of the graph. Petit et al. \cite{petit2021random} consider a random walk on graphon and prove an LLN limit for the Markovian time-space dynamics, which is again a system of ODEs without spatial partial derivative. However, we start with a non-Markovian multi-patch epidemic dynamics, and the limiting integral equations in Theorem \ref{thm-FLLN} and the PDE models in Proposition \ref{prop-PDE-g} and Corollaries \ref{coro-PDE-ac} and \ref{coro-PDE-det} can be regarded as dynamics on graphon, when the kernel function $\beta(x,y)$ is symmetric and takes values in $[0,1]$ (see further discussions in Remark \ref{rem-graphon}).
\subsection{Organization of the paper} The paper is organized as follows. In Section \ref{sec-model}, we provide the detailed model description. We then present the scaled processes and assumptions and state the FLLN result in Section \ref{sec-FLLN}. We derive the PDE models from the FLLN limits and discuss how they are related to the already known spatial PDE models in Section \ref{sec-PDE}. The proofs of the FLLN are given in Sections \ref{sec-proof-conv-S-F} and \ref{sec-proof-conv-I} after some technical preliminaries in Section \ref{sec-technical}.
\subsection{Notation}
All random variables and processes are defined on a common complete probability space $(\Omega, {\mathcal{F}}, \mathbb{P})$. Throughout the paper, ${\mathbb N}$ denotes the set of natural numbers, and ${\mathbb R}^k ({\mathbb R}^k_+)$ denotes the space of $k$-dimensional vectors with real (nonnegative) coordinates, with ${\mathbb R} ({\mathbb R}_+)$ for $k=1$. Let ${\mathbf D}={\mathbf D}({\mathbb R}_+;{\mathbb R})$ denote the space of ${\mathbb R}$--valued c{\`a}dl{\`a}g functions defined on ${\mathbb R}_+$. Here, convergence in ${\mathbf D}$ means convergence in the Skorohod $J_1$ topology, see Chapter 3 of \cite{billingsley1999convergence}.
Let ${\mathbf C}$ be the subset of ${\mathbf D}$ consisting of continuous functions.
Let ${\mathbf D}_{\mathbf D}= {\mathbf D}({\mathbb R}_+; {\mathbf D}({\mathbb R}_+;{\mathbb R}))$ be the ${\mathbf D}$-valued ${\mathbf D}$ space, and the convergence in the space ${\mathbf D}_{\mathbf D}$ means that both ${\mathbf D}$ spaces are endowed with the Skorohod $J_1$ topology. For any increasing c{\`a}dl{\`a}g function $F(\cdot): \R_+\to \R_+$, abusing notation, we write $F(dx)$ by treating $F(\cdot)$ as the positive (finite) measure on $\R_+$ whose distribution function is $F$. For any ${\mathbb R}$--valued c{\`a}dl{\`a}g function $\phi(\cdot)$ on $\R_+$, the integral $\int_{a}^b \phi(x)F(dx)$ represents $\int_{(a,b]} \phi(x) F(dx)$ for $a<b$. We use ${\mathbf 1}_{\{\cdot\}}$ for the indicator function. For $x,y \in{\mathbb R}$, we denote $x\wedge y = \min\{x,y\}$ and $x\vee y = \max\{x,y\}$.
We use $\|\cdot\|_1$ to denote the $L^1([0,1])$ norm. For time-space processes $Z(t,x)$ and $Z(t,s,x)$, for each $x$, we regard them in the spaces ${\mathbf D}$ and ${\mathbf D}_{\mathbf D}$, respectively. For the weak convergence of the time-space processes $Z^N(t,x)$ to $Z(t,x)$ as $N\to\infty$, we use the Skorohod topology for the processes in ${\mathbf D}$ with the $L^1([0,1])$ norm with respect to $x$. Similarly, for the weak convergence of the time-space processes $Z^N(t,s,x)$ to $Z(t,s,x)$ as $N\to\infty$, we use the Skorohod topology for the processes in ${\mathbf D}_{\mathbf D}$ with the $L^1([0,1])$ norm with respect to $x$. We write these spaces as ${\mathbf D}(\R_+, L^1([0,1]))$ and ${\mathbf D}(\R_+, {\mathbf D}(\R_+, L^1([0,1]))$, or ${\mathbf D}(\R_+, L^1)$ and ${\mathbf D}(\R_+, {\mathbf D}(\R_+, L^1))$ for short. See the weak convergence criteria in Theorems \ref{thm-D-conv-x} and \ref{thm-DD-conv-x}.
\section{Model and FLLN} \label{sec-model-FLLN}
\subsection{Model Description} \label{sec-model}
We consider a population of fixed size $N$ distributed in $K$ locations in some bounded domain ${\mathcal{S}}$ in ${\mathbb R}^d$ ($d\ge1$). To be specific, we choose ${\mathcal{S}}=[0,1]$. The arguments in the paper would remain the same for any such a domain ${\mathcal{S}}$ since we do not consider migration among locations.
Also let $K$ depend on $N$, denoted as $K^N$. Let the $K^N$ locations be at $x^N_k,\, k=1,\dots,K^N$ in $[0,1]$ such that $0 \le x^N_1< x^N_2<\cdots<x^N_{K^N} \le 1$. For notational convenience, let $\mathtt{I}^N_k, k=1,\dots,K^N$
be a partition of $[0,1]$ such that $x^N_k\in \mathtt{I}^N_k$ and $|\mathtt{I}^N_k|=(K^N)^{-1}$ for all $1\le k\le K^N$. In each location, individuals are categorized into three groups: susceptible, infected (possibly including both exposed and infectious) and recovered. We assume that individuals do not move among the different locations, and susceptible individuals in each location can be infected from their own location as well as from other locations (as explained below). Suppose that there are $B^N_k$ individuals at location $k$, such that $B^N_1+\cdots+B^N_{K^N}=N$. (For example, there is an equal number of individuals in each path, that is, $B^N_k = N/K^N$ for all $k$.) We assume that \begin{align} \label{eqn-KB-condition} \text{both } K^N\to\infty& \quad\mbox{and}\quad \frac{N}{K^N}\to\infty, \quad\mbox{as}\quad N\to\infty\,. \end{align}
Notation: Whenever not causing any confusion, we drop the superscript $N$ in $x^N_k$, $\mathtt{I}^N_k$, $K^N$ and $B^N_k$. For any vector ${\bf z}=(z_1,\dots,z_K)$, we write $z(x)=\sum_{k=1}^K z_k {\bf 1}_{\mathtt{I}^N_k}(x)$ where ${\bf 1}_{\mathtt{I}^N_k}(\cdot)$ denotes the indicator function of the set $\mathtt{I}^N_k$. For a process ${\bf Z}(t)=(Z_1(t),\dots,Z_K(t))$, we write $Z(t,x) = \sum_{k=1}^K Z_k(t) {\bf 1}_{\mathtt{I}^N_k}(x)$ for $t \ge 0, x \in [0,1]$.
Let $S^N_k(t)$, $I^N_k(t)$ and $R^N_k(t)$ be the numbers of susceptible, infected and recovered individuals in location $x_k$ at time $t$. We can also write the vectors ${\bf S}^N(t)=(S^N_1(t),\dots,S^N_K(t))$, ${\bf I}^N(t)=(I^N_1(t),\dots,I^N_K(t))$ and ${\bf R}^N(t)=(R^N_1(t),\dots,R^N_K(t))$, as the following time-space processes $S^N(t,x) = \sum_{k=1}^K S^N_k(t){\bf 1}_{\mathtt{I}_k}(x)$, $I^N(t,x) = \sum_{k=1}^K I^N_k(t){\bf 1}_{\mathtt{I}_k}(x)$ and $R^N(t,x) = \sum_{k=1}^K R^N_k(t){\bf 1}_{\mathtt{I}_k}(x)$, respectively. Note that $S^N_k(t) = S^N(t,x_k)$, and so on.
To each infected individual is attached a random infectivity function. Individual $j$ in location $k$ has a random infectivity function $\lambda_{j,k}(\cdot)$. The initially infected individual $j$ from location $k$ gets infected at time $\tau^N_{j,k}<0$, for $j = - I^N_k(0), \dots, -1$, and
has at time $t\ge 0$ the infectivity $\lambda_{j,k}(\tilde{\tau}^N_{-j,k}+t)$ where $\tilde{\tau}^N_{-j,k} = -\tau^N_{j,k}$. The initially susceptible individual $j$ that gets infected at time $\tau^N_{j,k}$ has the infectivity $\lambda_{j,k}(t-\tau^N_{j,k})$ at time $t\ge 0$ for each $j\ge 1$. We assume that the sequence $\{\lambda_{j,k}: j \in {\mathbb Z} \backslash \{0\},k =1,\dots,K \}$ is i.i.d. (Since we are concerned about the same disease, it is reasonable to require all the individuals at all the locations have the same law of infectivity and recovery, that is, homogeneous over locations.) Also, let $\bar{\lambda}(t) = \mathbb{E}[\lambda_{j,k}(t)]$ for $ j \in {\mathbb Z} \backslash \{0\} $ and $k=1,\dots,K$, for each $t\ge 0$.
We assume that
$ \lambda_{j, k}(t) =0$ a.s. for $t<0$,
for all $j\in{\mathbb Z} \backslash\{0\}$, $ k=1,\dots, K$, and that each $\lambda_{j,k}$ has paths in ${\mathbf D}$. Define $\eta_{j,k} = \sup\{t>0: \lambda_{j,k}(t)>0\}$, which represents the duration of the infected period for individual $j$. Note that this may include both the exposed and infectious periods. Under the above assumption on $\{\lambda_{j,k}\}$, the variables $\{\eta_{j,k} \}$ are also i.i.d. Let $F(t) = \mathbb{P}(\eta_{j,k} \le t)$ for $j \in {\mathbb Z} \backslash \{0\}$ and $k =1,\dots, K$, representing the cumulative distribution function for the newly infected individuals.
For each $j\le -1$, let $\eta^0_{j,k} =\inf\{t>0: \lambda_j(\tilde{\tau}_{j,k}^N+r)=0,\, \forall r \ge t\}$ be the remaining infected period, which depends on the elapsed infection time $\tilde{\tau}_{j,k}^N$, but is independent of the elapsed infection times of other initially infected individuals. In particular, for $j \le -1$, the conditional distribution of $\eta^0_{j,k}$ given that $\tilde{\tau}_{j,k}^N=s>0$ is given by \begin{align} \label{enq-eta0-age}
\mathbb{P}(\eta^0_{j,k}> t | \tilde{\tau}_{j,k}^N=s) = \frac{F^c(t+s)}{F^c(s)}, \quad\mbox{for}\quad t, s >0. \end{align} Note that the $\eta^0_{j,k}$'s are independent but not identically distributed.
The total force of infection of the infected individuals in location $k$ is given by \begin{equation} \label{eqn-mfk} \mathfrak{F}_k^N(t) = \sum_{j=1}^{I^N_k(0)} \lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) + \sum_{j=1}^{S^N_k(0)} \lambda_{j,k} (t-\tau^N_{j,k}), \quad t \ge 0. \end{equation} We similarly write the time-space process for the total force of infection in the population: $$\mathfrak{F}^N(t,x) = \sum_{k=1}^{K^N} \mathfrak{F}_k^N(t){\bf 1}_{\mathtt{I}_k}(x).$$
The rate of infection for individuals in location $k$ is given by \begin{equation} \label{eqn-upsilon} \Upsilon^N_k(t) = \frac{S^N_k(t)}{B^N_k} \frac{1}{K^N} \sum_{k'=1}^{K^N}\beta^N_{k,k'} \mathfrak{F}_{k'}^N(t), \quad t \ge 0. \end{equation} Here the factor $\beta^N_{k,k'}$ reflects the effect of infection of individuals from location $k'$ upon those from location $k$. It also represents the heterogeneity of the effects of the interactions among different locations.
The number of newly infected individuals in location $k$ by time $t$ is given by \begin{equation}\label{eqn-An-k-rep} A^N_k(t) = \int_0^t \int_0^\infty {\bf 1}_{u \le \Upsilon^N_k(s) } Q_{k}(ds, d u), \end{equation} where $\{Q_k(ds,du),\ 1\le k\le K\}$ are mutually independent standard (i.e., with mean measure the Lebesgue measure) Poisson random measures (PRMs) on ${\mathbb R}^2_+$. The counting process $A^N_k$ has the event times $\{\tau^N_{j,k}, j \ge 1\}$.
Let ${\mathfrak{I}}^N_k(t,\mathfrak{a})$ be the number of infected individuals in location $k$ that are infected at time $t$ and have been infected for less than or equal to $\mathfrak{a}$. Then we can write \begin{align} \label{eqn-In-k-rep} {\mathfrak{I}}^N_k(t,\mathfrak{a}) = \sum_{j=1}^{I^N_k(0)} {\bf1}_{\eta^0_{-j,k} >t} {\bf 1}_{ \tilde{\tau}^N_{-j,k} \le (\mathfrak{a}-t)^+} + \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t}\,\,. \end{align} We suppose that there exists $\bar{\mathfrak{a}} \in [0, \infty)$ such that $I^N_k(0)= {\mathfrak{I}}^N_k(0,\bar{\mathfrak{a}})$. It is also clear that for all $t\ge0$, $$ I^N_k(t) = {\mathfrak{I}}^N_k(t, \infty). $$
To account for the location, we also write the time-age-space process $$ {\mathfrak{I}}^N(t,\mathfrak{a},x) = \sum_{k=1}^{K^N} {\mathfrak{I}}^N_k(t,\mathfrak{a}) {\bf 1}_{\mathtt{I}_k}(x). $$ Note that for each $x$, the process ${\mathfrak{I}}^N(t,\mathfrak{a},x) $ has paths in ${\mathbf D}_{\mathbf D}$.
The dynamics of $S^N_k(t)$, $I^N_k(t)$ and $R^N_k(t)$ can be expressed as \begin{align*} S^N_k(t) &= S^N_k(0) - A^N_k(t), \\ I^N_k(t) &= \sum_{j=1}^{I^N_k(0)} {\bf 1}_{\eta^0_{-j,k} >t} + \sum_{j=1}^{A^N_k(t)} {\bf 1}_{\tau^N_{j,k} + \eta_{j,k} >t}\,, \\ R^N_k(t) &= R^N_k(0)+ \sum_{j=1}^{I^N_k(0)} {\bf 1}_{\eta^0_{-j,k} \le t} + \sum_{j=1}^{A^N_k(t)} {\bf 1}_{\tau^N_{j,k} + \eta_{j,k} \le t}\,. \end{align*}
\subsection{FLLN} \label{sec-FLLN} We recall that \begin{equation} \label{eqn-N-B-1} N = \sum_{k=1}^{K^N} (S^N_k(t)+ I^N_k(t) + R^N_k(t) ) = \sum_{k=1}^{K^N} B^N_k \,, \end{equation} and observe that \begin{equation} \label{eqn-N-B-2} \int_0^1 (S^N(t,x)+ I^N(t,x) + R^N(t,x) ) dx = \frac{1}{K^N} \sum_{k=1}^{K^N} (S^N_k(t)+ I^N_k(t) + R^N_k(t) )= \frac{N}{K^N} \,. \end{equation} It is then reasonable to introduce the scaling of the processes by $N/K^N$, that is, for any process $Z^N_k = \mathfrak{F}_k^N, {\mathfrak{I}}^N_k, \Upsilon^N_k, A^N_k, S^N_k, I^N_k, R^N_k$, we define $\bar{Z}^N_k= (N/K^N)^{-1} Z^N_k$. We then define the scaled time-space processes \[
\bar{Z}^N(t,x)= \sum_{k=1}^{K^N} \bar{Z}^N_k(t) {\bf 1}_{\mathtt{I}_k}(x), \quad Z^N_k = \mathfrak{F}_k^N, \Upsilon^N_k, A^N_k, S^N_k, I^N_k, R^N_k \] and \[
\bar{{\mathfrak{I}}}^N(t,\mathfrak{a},x)= \sum_{k=1}^{K^N}\bar{{\mathfrak{I}}}_k^N (t,\mathfrak{a}){\bf 1}_{\mathtt{I}_k}(x)\,. \]
In addition, define the scaled population size at each location \[\bar{B}^N(x) = \sum_{k=1}^{K^N}\bar{B}_k^N {\bf 1}_{\mathtt{I}_k}(x), \quad \text{with} \quad \bar{B}_k^N = (N/K^N)^{-1} B^N_k.\] Hence, from \eqref{eqn-N-B-2} and the scaling, we obtain \[ \int_0^1 (\bar{S}^N(t,x)+ \bar{I}^N(t,x) + \bar{R}^N(t,x) ) dx= \int_0^1 \bar{B}^N(x) dx=1\,. \]
We make the following assumption on the initial condition.
\begin{assumption} \label{AS-LLN-1} There exist nonnegative deterministic functions $(\bar{S}(0,x), \bar{{\mathfrak{I}}}(0,\mathfrak{a}, x), \bar{R}(0,x))$ such that for each $x$, $\bar{{\mathfrak{I}}}(0,\cdot, x)$ is in ${\mathbf C}$, and
for each $\mathfrak{a} \in [0, \bar\mathfrak{a}]$, \begin{align}\label{eqn-initial-L1conv}
\|\bar{S}^N(0,\cdot) - \bar{S}(0,\cdot)\|_{1} \to 0, \quad \ \|\bar{{\mathfrak{I}}}^N(0,\mathfrak{a}, \cdot) - \bar{{\mathfrak{I}}}(0,\mathfrak{a}, \cdot)\|_{1} \to 0, \quad \|\bar{R}^N(0,\cdot) - \bar{R}(0,\cdot)\|_{1} \to 0 \end{align} in probability as $N\to\infty$, where letting $\bar{I}(0,x) = \bar{{\mathfrak{I}}}(0,\infty, x)$, we have \begin{equation} \label{eqn-initial-integral} \int_0^1 (\bar{S}(0,x)+\bar{I}(0,x)+\bar{R}(0,x)) dx =1. \end{equation} In addition, there exists $\bar{B}(x)$ such that \begin{equation}\label{cvinfty}
\|\bar{B}^N(\cdot) - \bar{B}(\cdot)\|_{\infty} = \sup_{x\in [0,1]} |\bar{B}^N(x) - \bar{B}(x) | \to 0, \end{equation} where for some constants $0<c_B<C_B<\infty$, \begin{equation} \label{eqn-barB-condition} \bar{B}(x) \in [c_B, C_B] \quad \forall x\in[0,1], \end{equation} and \[
\int_0^1 \bar{B}(x) dx =1\,. \]
\end{assumption} Note that, thanks to \eqref{cvinfty} and \eqref{eqn-barB-condition}, we may and do assume that $c_B$ and $C_B$ have been chosen in such a way that \begin{equation}\label{eqn-CB} \bar{B}^N(x) \in [c_B, C_B] \quad \forall N\ge1, \, x\in[0,1]. \end{equation}
Under the assumption in \eqref{eqn-initial-L1conv}, it follows that \begin{align*}
\|\bar{I}^N (0,\cdot) - \bar{I}(0,\cdot) \|_{1} \to 0
\end{align*} in probability as $N\to \infty$.
We introduce for each $x,x'\in [0,1]$, \begin{equation} \label{eqn-betaN-xx'} \beta^N(x,x') = \sum_{k,k'} \beta^N_{k,k'} {\bf 1}_{\mathtt{I}_k}(x) {\bf 1}_{\mathtt{I}_{k'}}(x')\,. \end{equation}
\begin{assumption} \label{AS-LLN-2} There exists a constant $C_\beta>0$ such that for all $N\ge1$, $x \in[0,1]$, \begin{equation}\label{eqn-C-beta} \int_0^1\beta^N(x,y)dy \vee\int_0^1\beta^N(y,x)dy\le C_\beta\, . \end{equation} There exists a function $\beta: [0,1]\times [0,1]\mapsto \R_+$ such that for any bounded measurable function $\phi: [0,1]\mapsto {\mathbb R}$, \begin{align}\label{conv-beta}
\left\| \int_0^1[\beta^N(\cdot,y)-\beta(\cdot,y)]\phi(y)dy\right\|_1\to0\,. \end{align} \end{assumption}
\begin{remark} Concerning condition \eqref{eqn-C-beta}, let us first note that, if $\beta^N_{k,k'}=\beta^N_{k',k}$ (symmetric) for all $N\ge1,\ 1\le k,k'\le K$, the boundedness of $\int_0^1\beta^N(x,y)dy$ is equivalent to that of
$\int_0^1\beta^N(y,x)dy$. Clearly \eqref{conv-beta} implies that \eqref{eqn-C-beta} is satisfied with $\beta^N$ replaced by $\beta$. We note that this assumption allows in particular $\beta(x,y)$ to explode on the diagonal $x=y$, for example, $\beta(x,y) = \frac{c}{\sqrt{|x-y|}}$ for some $c>0$, meaning that infectious interactions between ``close by" individuals are much more frequent than between distant ones. See further discussions in Remark \ref{rem-graphon}. \end{remark}
We make the following assumption on the random function $\lambda$.
\begin{assumption} \label{AS-lambda}
Let $\lambda(\cdot)$ be a process having the same law of $\{\lambda_j^0(\cdot)\}_j$ and $\{\lambda_i(\cdot)\}_i$. Assume that there exists a constant $\lambda^*$ such that for each $0<T<\infty$, $\sup_{t\in [0,T]} \lambda(t) \le \lambda^*$ almost surely. Assume that there exist an integer $\kappa$, a random sequence $0=\zeta^0 \le \zeta^1 \le \cdots \le \zeta^\kappa $ and associated random functions $\lambda^\ell \in {\mathbf C}({\mathbb R}_+;[0,\lambda^\ast])$, $1\le\ell \le \kappa$, such that \begin{align} \label{eqn-lambda-assump} \lambda(t) = \sum_{\ell=1}^\kappa \lambda^\ell(t) {\mathbf 1}_{[\zeta^{\ell-1},\zeta^\ell)}(t). \end{align} We write $F_\ell$ for the c.d.f. of $\zeta^\ell$, $\ell =1, \dots,\kappa$. In addition,
we assume that there exists a deterministic nondecreasing function $\varphi \in {\mathbf C}({\mathbb R}_+;{\mathbb R}_+)$ with $\varphi(0)=0$ such that $|\lambda^\ell(t) - \lambda^\ell(s)| \le \varphi(t-s)$ almost surely for all $t,s \ge 0$ and for all $\ell\ge 1$. Let $\bar{\lambda}(t) = \mathbb{E}[\lambda_i(t)] =\mathbb{E}[\lambda^0_j(t)]$ and $v(t) =\text{\rm Var}(\lambda(t)) = \mathbb{E}\big[\big(\lambda(t) - \bar\lambda(t)\big)^2\big]$ for $t\ge 0$. \end{assumption}
\begin{theorem} \label{thm-FLLN} Under Assumptions \ref{AS-LLN-1}, \ref{AS-LLN-2} and \ref{AS-lambda}, \begin{align} \label{eqn-LLN-conv}
&\|\bar{\mathfrak{F}}^N(t,\cdot) - \bar{\mathfrak{F}}(t,\cdot)\|_{1} \to 0, \quad \| \bar{S}^N(t,\cdot) -\bar{S}(t,\cdot)\|_{1} \to 0, \quad \quad \| \bar{R}^N(t,\cdot) -\bar{R}(t,\cdot)\|_{1} \to 0, \nonumber \\
& \| \bar{{\mathfrak{I}}}^N(t,\mathfrak{a}, \cdot) - \bar{{\mathfrak{I}}}^N(t,\mathfrak{a}, \cdot) \|_{1} \to 0 \end{align}
in probability as $N\to \infty$, locally uniformly in $t$ and $\mathfrak{a}$, where the limits are given by the unique solution to the following set of integral equations. The limit $(\bar{S}(t,x),\bar{\mathfrak{F}}(t,x))$ is a unique solution to the system of integral equations: for $t\ge 0$ and $x\in [0,1]$, \begin{align} \bar{S}(t,x) &= \bar{S}(0,x) - \int_0^t \bar\Upsilon(s,x) ds\,, \label{eqn-barS-tx} \\ \bar{\mathfrak{F}}(t,x) &=\int_0^\infty \bar{\lambda}(\mathfrak{a}+t) \bar{{\mathfrak{I}}}(0,d \mathfrak{a}, x) + \int_0^t \bar{\lambda}(t-s) \bar\Upsilon(s,x) ds\,, \label{eqn-barmfF-tx} \end{align} where \begin{align} \label{eqn-barUpsilon-tx} \bar\Upsilon(t,x) =\frac{ \bar{S}(t,x)}{\bar{B}(x)} \int_0^1 \beta(x,x') \bar{\mathfrak{F}}(t,x')dx' = \bar{{\mathfrak{I}}}_\mathfrak{a}(t,0, x) \,. \end{align} Given $\bar{S}(t,x)$ and $\bar{\mathfrak{F}}(t,x)$, the limits $\bar{{\mathfrak{I}}}(t,\mathfrak{a}, x) $ and $\bar{R}(t,x) $ are given by \begin{align} \bar{{\mathfrak{I}}}(t,\mathfrak{a}, x) &=\int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}', x) + \int_{(t-\mathfrak{a})^+}^t F^c(t-s) \bar\Upsilon(s,x) ds \,, \label{eqn-barsI-tax}\\ \bar{R}(t,x) &= \bar{R}(0,x) + \int_0^{\infty}\Big(1- \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big) \bar{{\mathfrak{I}}}(0,d \mathfrak{a}', x) + \int_0^t F(t-s) \bar\Upsilon(s,x) ds \,. \label{eqn-barR-tx} \end{align} In addition, \begin{align*}
\| \bar{I}^N(t,\cdot) -\bar{I}(t,\cdot)\|_{1} \to 0 \end{align*} locally uniformly in $t$ in probability as $N\to \infty$, where \begin{align} \label{eqn-barI-tx} \bar{I}(t,x) =\int_0^{\infty} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}', x) + \int_{0}^t F^c(t-s) \bar\Upsilon(s,x) ds\,. \end{align} For each $x$, the limits $\bar{S}(t,x)$, $\bar{\mathfrak{F}}(t,x)$, $\bar{{\mathfrak{I}}}(t,\mathfrak{a}, x)$, $\bar{I}(t,x)$ and $\bar{R}(t,x) $ are continuous in $t$ and $\mathfrak{a}$.
\end{theorem}
\begin{remark} \label{rem-graphon} Our model can be regarded in some sense as non-Markovian epidemics dynamics on graphon. In particular, the function $\beta(x,x')$ can be regarded as the graphon kernel function, representing the inhomogeneity in the connectivity. However, the kernel function is often assumed to take values in $[0,1]$ and to be symmetric in the graphon literature. In our model, $\beta(x,x')$ does not necessarily take values in $[0,1]$ although it can be rescaled to $[0,1]$ in case it is bounded, and the function $\beta(x,x')$ may not be necessarily symmetric. In the prelimit (the $N^{\rm th}$ system), the locations $\{\mathtt{I}^N_k\}_k$ can be regarded as a discretization of the unit interval $[0,1]$ and the infection rate functions between different locations $\beta^N_{k,k'}$ in \eqref{eqn-betaN-xx'} can then be regarded as the corresponding discretization of the function $\beta(x,x')$. We refer the readers to \cite{keliger2022local} and \cite{petit2021random} for Markov dynamics on graphon and the corresponding ODE approximations with a spatial parameter (no spatial partial derivative). See also Remark \ref{rem-SIS-PDE} for further discussions on how our PDE model relates to the ODE limit with a spatial parameter for the Markovian SIS model on graphon in \cite{keliger2022local}. \end{remark}
\begin{remark} \label{rem-SIS} For the spatial SIS model, we have the identity $\sum_{k=1}^{K^N} (S^N_k(t)+ I^N_k(t))=N$
and $\int_0^1 (\bar{S}(t,x) + \bar{I}(t,x)) dx=1$.
We use two processes $\bar{\mathfrak{F}}^N(t,x)$ and $\bar{{\mathfrak{I}}}^N(t,\mathfrak{a}, x) $ to describe the epidemic dynamics, and can show that $\|\bar{\mathfrak{F}}^N(t,\cdot)-\bar{\mathfrak{F}}(t,\cdot)\|_1 \to 0$ and $ \|\bar{{\mathfrak{I}}}^N(t,\mathfrak{a}, \cdot)- \bar{{\mathfrak{I}}}(t,\mathfrak{a}, x)\|_1 \to 0$ in probability locally uniformly in $t$ and $\mathfrak{a}$ as $N\to\infty$, where \begin{align}\label{eqn-bar-mfF-SIS} \bar{\mathfrak{F}}(t,x) &=\int_0^\infty \bar{\lambda}(\mathfrak{a}+t) \bar{{\mathfrak{I}}}(0,d \mathfrak{a}, x) + \int_0^t \bar{\lambda}(t-s) \bar{S}(s,x) \int_0^1 \beta(x,x') \bar{\mathfrak{F}}(s,x')dx' ds\,, \end{align} and \begin{align} \label{eqn-bar-sI-SIS} \bar{{\mathfrak{I}}}(t,\mathfrak{a}, x) &=\int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}', x) + \int_{(t-\mathfrak{a})^+}^t F^c(t-s) \bar{S}(s,x) \int_0^1 \beta(x,x') \bar{\mathfrak{F}}(s,x')dx' ds \,, \end{align} with $ \bar{S}(t,x)$ satisfying \begin{equation}\label{eqn-bar-SI-sum} \int_0^1 (\bar{S}(t,x) + \bar{{\mathfrak{I}}}(t,\infty, x) ) dx=1\,. \end{equation} Using $\bar{I}(t,x)=\bar{{\mathfrak{I}}}(t,\infty, x)$, we can write the last equation as $\int_0^1 (\bar{S}(t,x) + \bar{I}(t,x)) dx=1$, and
the limit $\bar{I}(t,x)$ is given by \begin{align*} \bar{I}(t,x) =\int_0^{\infty} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}', x) + \int_{0}^t F^c(t-s) \bar{S}(s,x) \int_0^1 \beta(x,x') \bar{\mathfrak{F}}(s,x')dx' ds\,. \end{align*} \end{remark}
\section{PDE Models} \label{sec-PDE}
In this section we derive the PDE models associated with the limits from the FLLN.
For each $t$, the limits $\bar{S}(t,x), \bar{\mathfrak{F}}(t,x), \bar{I}(t,x), \bar{R}(t,x)$ can be regarded as the densities of the quantities, susceptibles, aggregate infectivity, infected and recovered, distributed over the location $x \in [0,1]$, and for each $t$ and $\mathfrak{a}$, the function $\bar{\mathfrak{I}}(t,\mathfrak{a},x)$ can be also regarded as the density of the proportion of infected individuals at time $t$ with infection age less than or equal to $\mathfrak{a}$, over the location $x\in [0,1]$. In addition, for each fixed $t$ and $x$, $\bar{\mathfrak{I}}(t,\mathfrak{a},x)$ is increasing in $\mathfrak{a}$, and can be regarded as a ``distribution" over the infection ages. If $\bar{\mathfrak{I}}(t,\mathfrak{a},x)$ is absolutely continuous in $\mathfrak{a}$, we let $\bar\mathfrak{i}(t,\mathfrak{a},x) = \bar{\mathfrak{I}}_\mathfrak{a}(t,\mathfrak{a},x)$ be the density function of $\bar{{\mathfrak{I}}}(t,\mathfrak{a}, x)$ with respect to the infection age $\mathfrak{a}$.
In the following we will consider the dynamics of $\bar{S}(t,x), \bar{\mathfrak{F}}(t,x), \bar{I}(t,x), \bar{R}(t,x), \bar{{\mathfrak{I}}}(t,\mathfrak{a}, x)$ in $t$ and $\mathfrak{a}$, as a PDE model. Since there is no movement of individuals between locations, no derivative with respect to $x$ will appear. However, the interaction among individuals in different locations will be captured in these dynamics, in particular, in the expression of $\bar\Upsilon(t,x)$ in \eqref{eqn-barUpsilon-tx}.
We consider any arbitrary distribution $F$, and for notational convenience, we let $G(t) = F(t^-)$ and $G^c(t) = 1-G(t) = F^c(t^-)$, which are the left continuous versions of $F$ and $F^c$. Denote $\nu(\cdot)$ the law of $\eta$. Then $\frac{\nu(d\mathfrak{a})}{G^c(\mathfrak{a})}$ can be regarded as a generalized hazard rate function.
\begin{remark}\label{GvsF} Let us explain why we introduce here the left continuous version of $F$. Note that when $F$ is absolutely continuous, this makes no difference. The simplest example which motivates this choice is the following: $\nu=\delta_{t_0}$. In this case, $G^c(t_0)=1$, while $F^c(t_0)=0$. So, with the convention that $\frac{\nu(dt)}{G^c(t)}$ is zero outside the support of $\nu$, whatever the denominator might be, this fraction is well defined, which would not be the case if we replace $G^c(t)$ by $F^c(t)$ in the denominator. \end{remark}
\begin{prop} \label{prop-PDE-g} Suppose that for each $x$, $\bar{\mathfrak{I}}(0,\mathfrak{a},x)$ is absolutely continuous with respect to $\mathfrak{a}$ with density $\bar\mathfrak{i}(0,\mathfrak{a},x) =\bar{\mathfrak{I}}_\mathfrak{a}(0,\mathfrak{a},x)$. Then for $t,\mathfrak{a}>0$ and $x\in [0,1]$, the function $\bar{\mathfrak{I}}(t,\mathfrak{a},x)$ is absolutely continuous in $t$ and $\mathfrak{a}$, and its density $\bar{\mathfrak{i}}(t,\mathfrak{a},x) =\bar{\mathfrak{I}}_\mathfrak{a}(t,\mathfrak{a},x) $ with respect to $\mathfrak{a}$ satisfies \begin{equation} \label{eqn-mfi-PDE-g} \frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial t} + \frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial \mathfrak{a}} = -\frac{ \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{G^c(\mathfrak{a})} \nu(d\mathfrak{a}) \,, \end{equation} $(t,\mathfrak{a},x)$ in $(0,\infty)^2\times [0,1]$, with the initial condition $\bar\mathfrak{i}(0,\mathfrak{a},x)=\bar{\mathfrak{I}}_\mathfrak{a}(0,\mathfrak{a},x)$ for $(\mathfrak{a},x)\in (0,\infty) \times [0,1]$, and the boundary condition \begin{align} \label{eqn-mfi-PDE-BC1-g} \bar\mathfrak{i}(t,0,x) = \frac{\bar{S}(t,x)}{\bar{B}(x)} \int_0^1 \beta(x,x') \Bigg(\int_0^{t+\bar\mathfrak{a}} \frac{\bar\lambda(\mathfrak{a}')}{\frac{G^c(\mathfrak{a}')}{G^c(\mathfrak{a}'-t)}} \, \bar\mathfrak{i}(t,\mathfrak{a}',x') d \mathfrak{a}' \Bigg) d x' \,, \end{align} where $G^c\equiv 1$ on $\R_{-}$ and the integrand inside the second integral is set to zero whenever $G^c(\mathfrak{a})=0$.
The function $\bar{S}(t,x)$ satisfies \begin{equation}\label{eqn-barS-PDE}
\frac{\partial \bar{S}(t,x)}{ \partial t} = -\bar{\mathfrak{i}}(t,0,x)\,, \end{equation} with $\bar{S}(0,x)$ satisfying \eqref{eqn-initial-integral}.
Moreover, the PDE \eqref{eqn-mfi-PDE-g}-\eqref{eqn-mfi-PDE-BC1-g} has a unique non-negative solution which is given as follows: for $\mathfrak{a}\ge t$ and $x \in [0,1]$, \begin{equation} \label{eqn-bar-mfi-s1-g} \bar\mathfrak{i}(t,\mathfrak{a},x) = \frac{G^c(\mathfrak{a})}{G^c(\mathfrak{a}-t)} \, \bar\mathfrak{i}(0, \mathfrak{a}-t, x), \end{equation} and for $t>\mathfrak{a}$ and $x\in [0,1]$, \begin{equation}\label{eqn-bar-mfi-s2-g} \bar\mathfrak{i}(t,\mathfrak{a},x) = G^c(\mathfrak{a}) \, \bar\mathfrak{i}(t-\mathfrak{a},0, x), \end{equation} and the boundary function is the unique non-negative solution to the integral equation \begin{align}\label{eqn-mfi-PDE-BC2-g} \bar{\mathfrak{i}}(t,0,x) & = (\bar{B}(x))^{-1} \Big( \bar{S}(0,x) - \int_0^t \bar{\mathfrak{i}}(s,0,x) ds \Big) \nonumber \\ & \quad \times \int_0^1 \beta(x,x') \left( \int_0^\infty \bar{\lambda}(\mathfrak{a}+t)\, \bar{\mathfrak{i}}(0,\mathfrak{a}, x')d \mathfrak{a} + \int_0^t \bar{\lambda}(t-s) \, \bar{\mathfrak{i}}(s,0,x') ds \right) dx' \,. \end{align}
\end{prop}
Provided with the PDE solution $\bar{\mathfrak{i}}(t,\mathfrak{a},x)$ and with $\bar\Upsilon(t,x)=\bar\mathfrak{i}(t,0,x)$, the functions $\bar{I}(t,x)$ and $\bar{R}(t,x)$ are given by \begin{align*} \bar{I}(t,x) &=\int_0^{\infty} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \, \bar{\mathfrak{i}}(0, \mathfrak{a}', x)d \mathfrak{a}' + \int_{0}^t F^c(t-s) \, \bar\mathfrak{i}(s,0,x) ds\,,\\ \bar{R}(t,x) &= \bar{R}(0,x) + \int_0^{\infty}\Big(1- \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big) \, \bar{\mathfrak{i}}(0, \mathfrak{a}', x) d \mathfrak{a}' + \int_0^t F(t-s)\, \bar\mathfrak{i}(s,0,x) ds \,. \end{align*} Also, by definition, \[ \bar{I}(t,x) = \bar{{\mathfrak{I}}}(t,\infty,x) = \int_0^\infty \bar\mathfrak{i}(t,\mathfrak{a},x)d\mathfrak{a}. \]
\begin{proof} Using the expression of $ \bar\Upsilon(s,x)= \bar{{\mathfrak{I}}}_\mathfrak{a}(s,0, x)$ in \eqref{eqn-barUpsilon-tx} and with $G^c$, we can equivalently rewrite \eqref{eqn-barsI-tax} as \begin{align}\label{eqn-barsI-tax-g} \bar{{\mathfrak{I}}}(t,\mathfrak{a}, x) &=\int_0^{(\mathfrak{a}-t)^+} \frac{G^c(\mathfrak{a}'+t)}{G^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}_\mathfrak{a}(0, \mathfrak{a}', x) d \mathfrak{a}' + \int_{(t-\mathfrak{a})^+}^t G^c(t-s) \bar{{\mathfrak{I}}}_\mathfrak{a}(s,0, x) ds \,. \end{align}
Exploiting the fact that $\frac{\partial}{\partial t}+\frac{\partial}{\partial \mathfrak{a}}$ of a function of $t-\mathfrak{a}$ vanishes, we deduce from \eqref{eqn-barsI-tax-g} that \begin{align*} \bar{{\mathfrak{I}}}_t(t,\mathfrak{a}, x) +\bar{{\mathfrak{I}}}_\mathfrak{a}(t,\mathfrak{a}, x) &= - \int_0^{(\mathfrak{a}-t)^+} \frac{1}{G^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}_\mathfrak{a}(0,\mathfrak{a}', x)\nu( t+d\mathfrak{a}') \\ & \quad + \bar{{\mathfrak{I}}}_\mathfrak{a}(t,0, x) - \int_{(t-\mathfrak{a})^+}^t \bar{{\mathfrak{I}}}_\mathfrak{a}(s,0, x) \nu(t-ds) \\ &= - \int_t^{\mathfrak{a}\vee t} \frac{1}{G^c(\mathfrak{a}'-t)} \bar{{\mathfrak{I}}}_\mathfrak{a}(0,\mathfrak{a}'-t, x)\nu(d\mathfrak{a}') \\ & \quad + \bar{{\mathfrak{I}}}_\mathfrak{a}(t,0, x) - \int_{0}^{\mathfrak{a}\wedge t} \bar{{\mathfrak{I}}}_\mathfrak{a}(t-s,0, x) \nu(ds)\,. \end{align*} Here we consider the derivative with respect to $t$ in the distributional sense and use the measure $\nu(\cdot)$ associated with $G$ since we do not necessarily have differentiability of $G^c$. We then take derivative with respect to $\mathfrak{a}$ on both sides of this equation (denoting $\bar{{\mathfrak{I}}}_{t,\mathfrak{a}}(t,\mathfrak{a}, x)$ and $\bar{{\mathfrak{I}}}_{\mathfrak{a},\mathfrak{a}}(t,\mathfrak{a}, x)$ as the derivatives of $\bar{{\mathfrak{I}}}_t(t,\mathfrak{a}, x)$ and $\bar{{\mathfrak{I}}}_\mathfrak{a}(t,\mathfrak{a}, x) $ with respect to $\mathfrak{a}$) and obtain the following: \begin{align*} \bar{{\mathfrak{I}}}_{t,\mathfrak{a}}(t,\mathfrak{a}, x) +\bar{{\mathfrak{I}}}_{\mathfrak{a},\mathfrak{a}}(t,\mathfrak{a}, x) &= - {\bf1}_{\mathfrak{a} \ge t} \frac{\nu(d\mathfrak{a})}{G^c(\mathfrak{a}-t)} \bar{{\mathfrak{I}}}_\mathfrak{a}(0,\mathfrak{a}-t, x) - {\bf1}_{t>\mathfrak{a}} \nu(d\mathfrak{a}) \bar{{\mathfrak{I}}}_\mathfrak{a}(t-\mathfrak{a},0, x) \,. \end{align*} Rewriting $\frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial t} =\bar{{\mathfrak{I}}}_{\mathfrak{a},t}(t,\mathfrak{a}, x) = \bar{{\mathfrak{I}}}_{t,\mathfrak{a}}(t,\mathfrak{a}, x)$ and $\frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial \mathfrak{a}} = \bar{{\mathfrak{I}}}_{\mathfrak{a},\mathfrak{a}}(t,\mathfrak{a}, x)$, we obtain the PDE: \begin{align} \label{eqn-mfi-PDE-1-g} \frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial t} +\frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial \mathfrak{a}}
&= - {\bf1}_{\mathfrak{a} \ge t} \frac{\nu(d\mathfrak{a})}{G^c(\mathfrak{a}-t)} \, \bar{\mathfrak{i}}(0,\mathfrak{a}-t, x) - {\bf1}_{t>\mathfrak{a}} \nu(d\mathfrak{a}) \,\bar{\mathfrak{i}}(t-\mathfrak{a},0, x) \,. \end{align}
In order to see that the right hand side coincides with that in \eqref{eqn-mfi-PDE-g}, we
first establish \eqref{eqn-bar-mfi-s1-g} and \eqref{eqn-bar-mfi-s2-g}. For $\mathfrak{a} \ge t$, $0 \le s \le t$ and $x\in [0,1]$, \begin{align*} \frac{\partial \bar{\mathfrak{i}}(s,\mathfrak{a}-t+s,x)}{\partial s} = - \frac{\nu(\mathfrak{a}-t+ds)}{G^c(\mathfrak{a}-t)}\, \bar\mathfrak{i}(0,\mathfrak{a}-t,x)\,, \end{align*} and for $t>\mathfrak{a}$, $0 \le s \le \mathfrak{a}$ and $x \in [0,1]$, \begin{align*} \frac{\partial \bar{\mathfrak{i}}(t-\mathfrak{a}+s,s,x)}{\partial s} = -\nu(ds) \, \bar\mathfrak{i}(t-\mathfrak{a},0,x)\,. \end{align*} From these, by integration and simple calculations, we obtain
\eqref{eqn-bar-mfi-s1-g} and \eqref{eqn-bar-mfi-s2-g}. Now \eqref{eqn-mfi-PDE-g} follows from \eqref{eqn-mfi-PDE-1-g}, \eqref{eqn-bar-mfi-s1-g} and \eqref{eqn-bar-mfi-s2-g}.
Then using \eqref{eqn-bar-mfi-s1-g} and \eqref{eqn-bar-mfi-s2-g}, by \eqref{eqn-barmfF-tx} and the second equality in \eqref{eqn-barUpsilon-tx}, we obtain \begin{align} \label{eqn-mfF-tx-i} \bar{\mathfrak{F}}(t,x) &=\int_0^\infty \bar{\lambda}(\mathfrak{a}+t) \, \bar{\mathfrak{i}}(0,\mathfrak{a}, x)d \mathfrak{a} + \int_0^t \bar{\lambda}(t-s) \, \bar{\mathfrak{i}}(s,0,x) ds\,. \end{align} The expression for the boundary condition in \eqref{eqn-mfi-PDE-BC2-g} then follows directly from
\eqref{eqn-barUpsilon-tx} using this expression of $\bar{\mathfrak{F}}(t,x)$. Again, using \eqref{eqn-bar-mfi-s1-g} and \eqref{eqn-bar-mfi-s2-g}, we see that the boundary condition \eqref{eqn-mfi-PDE-BC2-g} is equivalent to \eqref{eqn-mfi-PDE-BC1-g}.
We now sketch the proof of existence and uniqueness of a non-negative solution to \eqref{eqn-mfi-PDE-BC2-g}. Note that, thanks to \eqref{eqn-bar-mfi-s1-g} and \eqref{eqn-bar-mfi-s2-g}, existence and uniqueness of a non-negative solution to the PDE \eqref{eqn-mfi-PDE-g}-\eqref{eqn-mfi-PDE-BC1-g} will follow from that result. First of all, let us rewrite that equation as \[ u(t,x)=(\bar{B}(x))^{-1}\left(f(x)-\int_0^t u(s,x)ds\right)\times\int_0^1\beta(x,x')\left(g(t,x')+\int_0^t\bar{\lambda}(t-s)u(s,x')ds\right)dx',\] where $0\le f(x)\le1$ and $0\le g(t,x)\le \lambda^\ast$ are given from the initial conditions. Any nonnegative solution satisfies \begin{align*} u(t,x)&\le\int_0^1\beta(x,x')\left(g(t,x')+\int_0^t\bar{\lambda}(t-s)u(s,x')ds\right)dx',\ \text{hence}\\
\|u(t,\cdot)\|_\infty&\le C_\beta\lambda^\ast\left(1+\int_0^t \|u(s,\cdot)\|_\infty ds\right)\\ &\le C_\beta\lambda^\ast e^{C_\beta\lambda^\ast t}\,. \end{align*}
Here $\|u(t,\cdot)\|_\infty = \sup_{x\in [0,1]} |u(t,x)|$. Let now $u$ and $v$ be two non negative solutions. Then, \begin{align*}
|u(t,x)-v(t,x)|&\le (\bar{B}(x))^{-1}\left(\int_0^1\beta(x,x')\left[g(t,x')+\int_0^t\bar{\lambda}(t-s) u(s,x')ds\right]\right)\int_0^t|u(s,x)-v(s,x)|ds\\&\quad
+ (\bar{B}(x))^{-1} \left(f(x)+\int_0^tv(s,x)ds\right)\int_0^1\beta(x,x')\int_0^t\bar{\lambda}(t-s) |u(s,x')-v(s,x')|dsdx'\,. \end{align*} Integrating over $dx$, exploiting the previous a priori estimate and \eqref{eqn-C-beta}, we deduce the uniqueness from Gronwall's Lemma. Finally, the existence of a nonnegative $L^1([0,1])$-valued solution can be established using a Picard iteration argument. Note that in the previous lines we have used the two distinct inequalities contained in \eqref{eqn-C-beta}. \end{proof}
If $F$ is absolutely continuous, with density $f$, we denote by $\mu(\mathfrak{a})$ the hazard function, i.e., $\mu(\mathfrak{a}) = \frac{f(\mathfrak{a})}{F^c(\mathfrak{a})}$ for all $\mathfrak{a}\ge 0$. We obtain the following corollary in this case.
\begin{coro} \label{coro-PDE-ac} Under the assumptions of Proposition \ref{prop-PDE-g}, if $F$ is absolutely continuous with density $f$, then the PDE in \eqref{eqn-mfi-PDE-g} becomes \begin{equation} \label{eqn-mfi-PDE} \frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial t} + \frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial \mathfrak{a}} = - \mu(\mathfrak{a}) \, \bar{\mathfrak{i}}(t,\mathfrak{a},x) \,, \end{equation} with the initial condition $\bar\mathfrak{i}(0,\mathfrak{a},x)=\bar{\mathfrak{I}}_\mathfrak{a}(0,\mathfrak{a},x)$ for $(\mathfrak{a},x)\in (0,\infty) \times [0,1]$ and the boundary condition \eqref{eqn-mfi-PDE-BC1-g}.
The function $\bar{S}(t,x)$ satisfies \eqref{eqn-barS-PDE}, and the PDE \eqref{eqn-mfi-PDE} has a unique solution which is given
by \eqref{eqn-bar-mfi-s1-g} and \eqref{eqn-bar-mfi-s2-g}, and the boundary function is the same as in \eqref{eqn-mfi-PDE-BC2-g}.
\end{coro}
When the infectious periods are deterministic, we obtain the following corollary.
\begin{coro} \label{coro-PDE-det} Suppose that the infectious periods are deterministic and equal to $t_i$, that is, $F(t) ={\bf1}_{t\ge t_i}$ and $G(t) = {\bf 1}_{t>t_i}$. Then the PDE in in \eqref{eqn-mfi-PDE-g} becomes \begin{equation} \label{eqn-mfi-PDE-det} \frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial t} + \frac{\partial \bar{\mathfrak{i}}(t,\mathfrak{a},x)}{\partial \mathfrak{a}} = - \delta_{t_i}(\mathfrak{a}) \, \bar{\mathfrak{i}}(t,\mathfrak{a},x) \,, \end{equation} with $\delta_{t_i}(\mathfrak{a})$ being the Dirac measure at $t_i$, with the initial condition $\bar\mathfrak{i}(0,\mathfrak{a},x)=\bar{\mathfrak{I}}_\mathfrak{a}(0,\mathfrak{a},x)$ for $\mathfrak{a}\in (0, t_i) \times [0,1]$, and the boundary condition \begin{align} \label{eqn-mfi-PDE-BC1-det} \bar\mathfrak{i}(t,0,x) = \frac{\bar{S}(t,x)}{\bar{B}(x)} \int_0^1 \beta(x,x') \Bigg(\int_0^{t_i} \bar\lambda(\mathfrak{a}') \, \bar\mathfrak{i}(t,\mathfrak{a}',x') d \mathfrak{a}' \Bigg) d x' \,, \end{align} The PDE \eqref{eqn-mfi-PDE-det} has a unique solution which is given as follows: for $t \le \mathfrak{a} < t_i$ and $x \in [0,1]$, \begin{equation} \label{eqn-bar-mfi-s1-det} \bar\mathfrak{i}(t,\mathfrak{a},x) = \bar\mathfrak{i}(0, \mathfrak{a}-t, x), \end{equation} and for $\mathfrak{a}< t \wedge t_i$ and $x\in [0,1]$, \begin{equation}\label{eqn-bar-mfi-s2-det} \bar\mathfrak{i}(t,\mathfrak{a},x) = \bar\mathfrak{i}(t-\mathfrak{a},0, x), \end{equation} and for $\mathfrak{a} \ge t_i$, $\bar\mathfrak{i}(t,\mathfrak{a},x)=0$. The boundary function is the unique solution to the integral equation: for $0 <t<t_i$, \begin{align}\label{eqn-mfi-PDE-BC2-det1} \bar{\mathfrak{i}}(t,0,x) & = \bar{B}(x)^{-1}\Big( \bar{S}(0,x) - \int_0^t \bar{\mathfrak{i}}(s,0,x) ds \Big) \nonumber \\ & \quad \times \int_0^1 \beta(x,x') \left( \int_t^{t_i} \bar{\lambda}(\mathfrak{a}) \, \bar{\mathfrak{i}}(0,\mathfrak{a}-t, x')d \mathfrak{a} + \int_0^t \bar{\lambda}(t-s) \,\bar{\mathfrak{i}}(s,0,x') ds \right) dx' \,, \end{align} and for $t\ge t_i$, \begin{align}\label{eqn-mfi-PDE-BC2-det2} \bar{\mathfrak{i}}(t,0,x) & = \bar{B}(x)^{-1} \Big( \bar{S}(0,x) - \int_0^t \bar{\mathfrak{i}}(s,0,x) ds \Big) \times \int_0^1 \beta(x,x') \int_0^{t_i} \bar{\lambda}(t-s)\, \bar{\mathfrak{i}}(s,0,x') ds dx' \,. \end{align}
\end{coro}
\begin{remark} \label{rem-special} In the special case when $\lambda_i(t) =\tilde\lambda(t){\bf1}_{t<\eta_i}$ for a deterministic function $\tilde\lambda(t)$, the boundary condition \eqref{eqn-mfi-PDE-BC1-g} \ becomes \begin{align} \label{eqn-mfi-PDE-BC1-special} \bar\mathfrak{i}(t,0,x) = \frac{\bar{S}(t,x)}{\bar{B}(x)} \int_0^1 \beta(x,x') \Bigg(\int_0^{t+\bar\mathfrak{a}} \tilde\lambda(\mathfrak{a}') \, \bar\mathfrak{i}(t,\mathfrak{a}',x') d \mathfrak{a}' \Bigg) d x' \,. \end{align} This is because
$\bar\lambda(t) = \tilde{\lambda}(t) F^c(t)$ and $\mathbb{E}[ \tilde{\lambda}(t) {\mathbf 1}_{t < \eta_0}|\tilde{\tau}_0=y] = \tilde{\lambda}(t+y) \frac{F^c(t+y)}{F^c(y)}$.
This boundary condition resembles that given in the Diekmann PDE model \cite{diekmann1978thresholds} (without $\bar{B}(x)$ in the denominator). See further discussions in Remark \ref{rem-Diekman-Kendall}. We remark that the PDE model first proposed by Kermack and McKendrick in \cite{KM32} also corresponds to this special infectivity function $\lambda_i(t) =\tilde\lambda(t){\bf1}_{t<\eta_i}$; see further discussions on the PDE models with infection-age dependent infectivity in Remarks 3.3 and 3.4 of \cite{PP-2021}.
\end{remark}
\begin{remark} By using the solution expressions in \eqref{eqn-bar-mfi-s1-g} and \eqref{eqn-bar-mfi-s2-g} together with the second identity $\bar\Upsilon(t,x) = \bar{{\mathfrak{I}}}_\mathfrak{a}(t,0, x) $ in \eqref{eqn-barUpsilon-tx}, we can rewrite $\bar{\mathfrak{F}}(t,x)$ in \eqref{eqn-barmfF-tx} as \begin{align} \label{eqn-barmfF-tx-2} \bar{\mathfrak{F}}(t,x) &=\int_0^\infty \bar{\lambda}(\mathfrak{a}+t) \, \bar{\mathfrak{i}}(0, \mathfrak{a}, x) d \mathfrak{a}+ \int_0^t \bar{\lambda}(t-s) \, \bar{\mathfrak{i}}(s,0, x) ds \nonumber\\ &= \int_0^\infty \bar{\lambda}(\mathfrak{a}+t) \frac{G^c(\mathfrak{a})}{G^c(t+\mathfrak{a})} \, \bar{\mathfrak{i}}(t, t+ \mathfrak{a}, x) d \mathfrak{a}+ \int_0^t \bar{\lambda}(\mathfrak{a}) \frac{1}{G^c(\mathfrak{a})}\, \bar{\mathfrak{i}}(t,\mathfrak{a}, x) d\mathfrak{a} \nonumber \\ &= \int_0^{t+\bar\mathfrak{a}} \frac{G^c(\mathfrak{a}-t)}{G^c(\mathfrak{a})} \bar\lambda(\mathfrak{a}) \, \bar{\mathfrak{i}}(t,\mathfrak{a}, x) d\mathfrak{a}\,, \end{align} where $G^c(\mathfrak{a}) =1$ for $\mathfrak{a} \le 0$. In the special case when $\lambda_i(t) =\tilde\lambda(t){\bf1}_{t<\eta_i}$ as described in the previous remark, we obtain \begin{align} \label{eqn-barmfF-tx-2-special} \bar{\mathfrak{F}}(t,x) = \int_0^{t+\bar\mathfrak{a}} \tilde\lambda(\mathfrak{a}) \, \bar{\mathfrak{i}}(t,\mathfrak{a}, x) d\mathfrak{a}\,, \end{align} which further gives \begin{align} \label{eqn-barUpsilon-tx-2-special} \bar\Upsilon(t,x) & = \frac{\bar{S}(t,x)}{\bar{B}(x)} \int_0^1 \beta(x,x') \int_0^{t+\bar\mathfrak{a}} \tilde\lambda(\mathfrak{a}) \, \bar{\mathfrak{i}}(t,\mathfrak{a}, x') d\mathfrak{a} dx' \nonumber \\ &=\frac{\bar{S}(t,x)}{\bar{B}(x)} \int_0^{t+\bar\mathfrak{a}} \int_0^1 \beta(x,x') \tilde\lambda(\mathfrak{a}) \, \bar{\mathfrak{i}}(t,\mathfrak{a}, x') dx'd\mathfrak{a} \,. \end{align} \end{remark}
\begin{remark} \label{rem-Diekman-Kendall} In Diekmann \cite{diekmann1978thresholds}, the spatial-temporal deterministic model is specified as follows.
The function $\bar{I}(t,x)$ is written as an integral of the function $\bar{\mathfrak{i}}(t, \mathfrak{a},x)$: \[ \bar{I}(t,x) = \int_0^\infty \bar{\mathfrak{i}}(t, \mathfrak{a},x) d\mathfrak{a}\,. \] The infectivity function is given by \begin{equation} \label{eqn-barUpsilon-Diekman} \bar\Upsilon(t,x) = \bar{S}(t,x)\int_0^\infty \int_0^1 \bar{\mathfrak{i}}(t, \mathfrak{a},x') A(\mathfrak{a}, x, x') dx' d\mathfrak{a}\,, \end{equation} where $A(\mathfrak{a}, x, x')$ is the infectivity at $x$ due to the infected individual with the infection age $\mathfrak{a}$ at $x'$. (Note the difference of $\bar\Upsilon(t,x)$ in \eqref{eqn-barUpsilon-Diekman} from our limit $\bar\Upsilon(t,x)$ in \eqref{eqn-barUpsilon-tx-2-special} with $\bar{B}(x)$ in the denominator, and abusing notation we use the same symbols in this remark).
Therefore, in order to match the model by Diekmann \cite{diekmann1978thresholds}, we can take \begin{equation} \label{eqn-A-beta} A(\mathfrak{a}, x,x') = \beta(x,x') \bar\lambda(\mathfrak{a}). \end{equation}
By \eqref{eqn-barS-PDE} and \eqref{eqn-mfi-PDE-BC2-g}, we obtain \begin{align} \label{eqn-barS-tx-D1}
\frac{\partial \bar{S}(t,x)}{ \partial t}
& = - \bar{S}(t,x) \int_0^\infty \int_0^1 \beta(x,x')
\bar{\lambda}(\mathfrak{a}+t) \, \bar{\mathfrak{i}}(0,\mathfrak{a}, x') dx' d \mathfrak{a} \nonumber\\ & \quad -\bar{S}(t,x) \int_0^t \int_0^1 \beta(x,x')
\bar{\lambda}(t-s) \, \bar{\mathfrak{i}}(s,0,x') dx' ds \nonumber \\
& = \bar{S}(t,x) \bigg( \int_0^t \int_0^1 \beta(x,x')
\bar{\lambda}(\mathfrak{a}) \frac{\partial \bar{S}(t-\mathfrak{a},x')}{ \partial t} dx' d\mathfrak{a} - h(t,x)\bigg)\,, \end{align} where \[ h(t,x) = \int_0^\infty \int_0^1 \beta(x,x') \bar{\lambda}(\mathfrak{a}+t) \, \bar{\mathfrak{i}}(0,\mathfrak{a}, x') dx' d \mathfrak{a}\,. \] Then integrating \eqref{eqn-barS-tx-D1} with respect to $t$, we also get \[ u(t,x) = - \ln \frac{\bar{S}(t,x)}{\bar{S}(0,x)} = \int_0^t \int_0^1 (1- e^{-u(t-\mathfrak{a},x')}) \bar{S}(0, x') \beta(x,x')
\bar{\lambda}(\mathfrak{a}) dx' d\mathfrak{a} + \int_0^t h(s,x) ds. \] By using \eqref{eqn-A-beta}, we obtain the specification of $u(t,x)$ in \cite{diekmann1978thresholds}.
In the special case $\lambda(t) = \tilde{\lambda}(t) {\mathbf 1}_{t < \eta}$ for some deterministic function $\tilde\lambda(t)$ as described in Remark \ref{rem-special}, given the expressions in \eqref{eqn-barmfF-tx-2-special} and \eqref{eqn-barUpsilon-tx-2-special},
to match the model by Diekmann \cite{diekmann1978thresholds}, we can take \[ A(\mathfrak{a}, x,x') = \beta(x,x') \tilde\lambda(\mathfrak{a}). \]
Moreover, if the infection rate is the constant $\lambda$ and the infectious periods are exponential of rate $\mu$, we have $\bar{\mathfrak{F}}(t,x) = \lambda \bar{I}(t,x) $, and as a result, the infectivity function of Diekmann in \eqref{eqn-barUpsilon-Diekman} becomes \begin{align} \label{eqn-barUpsilon-Kendall} \bar\Upsilon(t,x) = \bar{S}(t,x) \int_0^1 \beta(x,x') \lambda \bar{I}(t,x')dx'\,. \end{align} Because of the memoryless property of exponential periods, it is adequate to use the process $I(t,x)$ to describe the dynamics instead of ${\mathfrak{I}}(t,\mathfrak{a},x)$. In this case, we obtain Kendall's spatial model \cite{kendall1957,kendall1965}, in which given the limit $\bar\Upsilon(t,x) $ in \eqref{eqn-barUpsilon-Kendall},
\begin{align} \label{eqn-Kendall-ODE-model}
\frac{\partial \bar{S}(t,x)}{\partial t} = -\bar\Upsilon(t,x), \quad \frac{\partial \bar{I}(t,x)}{\partial t} =\bar\Upsilon(t,x) - \mu \bar{I}(t,x), \quad \frac{\partial \bar{R}(t,x)}{\partial t} = \mu \bar{I}(t,x)\,.
\end{align}
\end{remark}
\begin{remark} \label{rem-SIS-PDE}
Recall the spatial SIS model in Remark \ref{rem-SIS}. We obtain the same PDE in \eqref{eqn-mfi-PDE-g} with the boundary condition in \eqref{eqn-mfi-PDE-BC1-g}, in which $\bar{S}(t,x)$ is the solution to \eqref{eqn-barS-PDE} with $\bar{S}(0,x)$ satisfying $\int_0^1 (\bar{S}(0,x)+\bar{I}(0,x)) dx =1$. The solution to the PDE is also given by \eqref{eqn-bar-mfi-s1-g}--\eqref{eqn-bar-mfi-s2-g} with the boundary condition in \eqref{eqn-mfi-PDE-BC1-g}, in which $\bar{S}(0,x)$ satisfying $\int_0^1 (\bar{S}(0,x)+\bar{I}(0,x)) dx =1$. Similarly, we also obtain the expression of $\bar{\mathfrak{F}}(t,x)$ in \eqref{eqn-barmfF-tx-2}.
In the Markovian case with a constant infection rate $\lambda$ and recovery rate $\mu$, our model reduces to the following ODE with a spatial parameter: \begin{align} \label{eqn-SIS-Markov-ODE}
\frac{\partial \bar{I}(t,x)}{\partial t} =\lambda \bar{S}(t,x) \int_0^1 \beta(x,x') \bar{I}(t,x')dx' - \mu \bar{I}(t,x) \end{align} with $ \bar{S}(t,x)$ satisfying $\int_0^1 (\bar{S}(t,x) + \bar{I}(t,x)) dx=1$ for each $t\ge 0$. (This can be also seen from \eqref{eqn-Kendall-ODE-model} and \eqref{eqn-barUpsilon-Kendall}.) This resembles the ODE limit with a spatial parameter as established by Keliger et al. \cite{keliger2022local} for the finite-state Markov SIS model on a sampled graph from graphon (since there is only one individual at each node of the graph, $\bar{S}(t,x)$ in \eqref{eqn-SIS-Markov-ODE} is replaced by $1-\bar{I}(t,x)$, see equation (10) in that paper).
Returning to the integral limit for the spatial SIS model in Remark \ref{rem-SIS}, we assume that $\lim_{t\to\infty}\bar{{\mathfrak{I}}}(t,\mathfrak{a}, x)$ exists and the limit is denoted as $\bar{{\mathfrak{I}}}^*(\mathfrak{a}, x)$, and let $\bar{I}^*(x) =\lim_{t\to\infty} \bar{I}(t,x) =\bar{{\mathfrak{I}}}^*(\infty, x)$. Also let $\bar{S}^*(x) =\lim_{t\to\infty} \bar{S}(t,x)$. Note that \begin{equation} \label{eqn-SI-eqlm} \int_0^1 (\bar{S}^*(x) + \bar{I}^*(x) ) dx=1. \end{equation}
Let $\beta^{-1}= \int_0^\infty F^c(\mathfrak{a}) d \mathfrak{a}$ and $F_e(\mathfrak{a})= \beta \int_0^\mathfrak{a} F^c(s) ds$.
By \eqref{eqn-bar-sI-SIS} and \eqref{eqn-barmfF-tx-2}, we obtain \begin{align} \label{eqn-bar-sI-SIS-eqlm} \bar{{\mathfrak{I}}}^*(\mathfrak{a}, x)
&= \int_0^\mathfrak{a} F^c(s) ds\, \bar{S}^*(x) \int_0^1 \beta(x,x') \int_0^{\infty} \frac{1}{G^c(\mathfrak{a}')} \bar\lambda(\mathfrak{a}') \bar{\mathfrak{I}}^*(d\mathfrak{a}', x') dx' \nonumber\\ &= \beta^{-1} F_e(\mathfrak{a}) \bar{S}^*(x) \int_0^1 \beta(x,x') \int_0^{\infty} \frac{1}{G^c(\mathfrak{a}')} \bar\lambda(\mathfrak{a}') \bar{\mathfrak{I}}^*(d\mathfrak{a}', x') dx'\,. \end{align} By letting $\mathfrak{a}\to\infty$ on the both sides, we obtain \begin{align} \bar{I}^*(x) &= \beta^{-1} \bar{S}^*(x) \int_0^1 \beta(x,x') \int_0^{\infty} \frac{1}{G^c(\mathfrak{a}')} \bar\lambda(\mathfrak{a}') \bar{\mathfrak{I}}^*(d\mathfrak{a}', x') dx' \,. \end{align} This implies \[ \bar{{\mathfrak{I}}}^*(\mathfrak{a}, x) = F_e(\mathfrak{a}) \bar{I}^*(x)\,, \] which then gives \[ \frac{\partial}{\partial \mathfrak{a}}\bar{{\mathfrak{I}}}^*(\mathfrak{a}, x) = \beta F^c(\mathfrak{a}) \bar{I}^*(x) \,. \] Thus, \begin{align*} \bar{{\mathfrak{I}}}^*(\mathfrak{a}, x) &= \beta^{-1} F_e(\mathfrak{a}) \bar{S}^*(x) \int_0^1 \beta(x,x') \int_0^{\infty} \frac{1}{G^c(\mathfrak{a}')} \bar\lambda(\mathfrak{a}') \beta F^c(\mathfrak{a}') \bar{I}^*(x') d \mathfrak{a}' dx' \\ &= F_e(\mathfrak{a})\Big( \int_0^{\infty} \bar\lambda(\mathfrak{a}') d \mathfrak{a}' \Big) \bar{S}^*(x) \int_0^1 \beta(x,x') \bar{I}^*(x') dx' \,. \end{align*} Recall that $R_0= \int_0^{\infty} \bar\lambda(t) d t$. By letting $\mathfrak{a}\to\infty$ again on both sides, we obtain \begin{align*} \bar{I}^*(x) &= R_0 \bar{S}^*(x) \int_0^1 \beta(x,x') \bar{I}^*(x') dx' \,. \end{align*} This equation together with the identity \eqref{eqn-SI-eqlm} determines the values $\bar{I}^*(x)$ and $\bar{S}^*(x)$.
\end{remark}
\section{Some technical preliminaries} \label{sec-technical}
We will use the following convergence criteria for the processes: a) $X^N(t,x)$ in ${\mathbf D}(\R_+, L^1)$ and b) $X^N(t,s,x)$ in ${\mathbf D}(\R_+,{\mathbf D}(\R_+, L^1))$.
They extend the convergence criterion for the processes in ${\mathbf D}$ (the Corollary on page 83 of \cite{billingsley1999convergence}) and in ${\mathbf D}_{\mathbf D}$ (\cite[Theorem 4.1]{PP-2021}).
The proof is a straightforward extension of those results (in \cite{billingsley1999convergence} it is noted that with very little change, the theory can be extended to functions taking values in metric spaces that are separable and complete).
We remark that one may also replace the $L_1$ norm $\|\cdot \|_1$ by the $L_2$ norm in the following results.
\begin{theorem} \label{thm-D-conv-x} Let $\{X^N(t,x): N \ge 1\}$ be a sequence of random elements such that $X^N$ is in ${\mathbf D}(\R_+, L^1)$.
If the following two conditions are satisfied: for any $T>0$, \begin{itemize}
\item[(i)] for any $\epsilon>0$, $ \sup_{t \in [0,T]} \mathbb{P} \big( \|X^N(t, \cdot)\|_{1}> \epsilon \big) \to 0$ as $N\to\infty$, and \item[(ii)] for any $\epsilon>0$, as $\delta\to0$, \begin{align*}
& \limsup_N \sup_{t\in [0,T]} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{u \in [0,\delta]}\|X^N(t+u,\cdot) - X^N(t,\cdot)\|_1 > \epsilon\bigg) \to 0, \end{align*} \end{itemize}
then $\|X^N(t,\cdot) \|_1 \to 0 $ in probability, locally uniformly in $t$, as $N\to \infty$. \end{theorem}
\begin{theorem} \label{thm-DD-conv-x} Let $\{X^N: N \ge 1\}$ be a sequence of random elements such that $X^N$ is in
${\mathbf D}(\R_+,{\mathbf D}(\R_+, L^1))$.
If the following two conditions are satisfied: for any $T, S>0$, \begin{itemize}
\item[(i)] for any $\epsilon>0$, $ \sup_{t \in [0,T]}\sup_{s\in [0,S]} \mathbb{P} \big( \|X^N(t, s,\cdot)\|_1> \epsilon \big) \to 0$ as $N\to\infty$, and \item[(ii)] for any $\epsilon>0$, as $\delta\to0$, \begin{align*}
& \limsup_N \sup_{t\in [0,T]} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{u \in [0,\delta]}\sup_{s \in [0,S]} \|X^N(t+u,s,\cdot) - X^N(t,s,\cdot)\|_1 > \epsilon\bigg) \to 0, \\
& \limsup_N \sup_{s\in [0,S]} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \|X^N(t,s+v,\cdot) - X^N(t,s,\cdot)\|_1 > \epsilon\bigg) \to 0, \end{align*} \end{itemize}
then $\|X^N(t,s,\cdot)\|_1 \to 0 $ in probability, locally uniformly in $t$ and $s$, as $N\to\infty$. \end{theorem}
We shall also need the following Lemma. \begin{lemma}\label{convPort} For each $N\ge1$, let $f_N:\R_+\times[0,1]\mapsto\R_+$ be measurable and such that $t\mapsto f_N(t,x)$ is non--decreasing for each $x\in[0,1]$. Assume that there exists $f:\R_+\times[0,1]\mapsto\R_+$ such that
$t\mapsto f(t,x)$ is continuous for each $x\in[0,1]$, and for all $t\ge0$,
as $N\to\infty$, \begin{equation}\label{conv}
\|f_N(t,\cdot)-f(t,\cdot)\|_{1}\to0\, . \end{equation} Let $g\in {\mathbf D}(\R;\R_+)$ be such that there exists $C>0$ with $g(t)\le C$ for all $t\ge0$. Define \[ h_N(t,x)=\int_0^tg(s)f_N(ds,x),\quad h(t,x)=\int_0^tg(s)f(ds,x)\,.\]
Then for any $t>0$, $\|h_N(t,\cdot)- h(t,\cdot)\|_{1}\to 0$ as $N\to\infty$. In addition, $\int_0^1h_N(t,x)dx\to\int_0^1h(t,x)dx$ locally uniformly in $t$, as $N\to\infty$.
Moreover, if for each $N\ge1$, $f_N$ is random and the convergence \eqref{conv} holds in probability, then the conclusion holds in probability as well. \end{lemma} \begin{proof} Let $\{s_n,\ n\ge1\}$ be a countable dense subset of $[0,1]$. By successive extraction of subsequences we can extract a subsequence from the original sequence $\{f_N,\ N\ge1\}$, which by an abuse of notation we still denote as the original sequence, and which is such that there exists a subset $\mathcal{N}\subset[0,1]$ with zero Lebesgue measure, such that for all $n\ge1$ and $x\in[0,1]\backslash\mathcal{N}$, $f_N(s_n,x)\to f(s_n,x)$. Since for all $N$ and $x$, $s\mapsto f_N(s,x)$ is nondecreasing and $s\mapsto f(s,x)$ is continuous, we deduce that for all $s\in[0,T]$ and $x\in[0,1]\backslash\mathcal{N}$, $f_N(s,x)\to f(s,x)$. Consequently, for all $x\in[0,1]\backslash\mathcal{N}$, the sequence of measures $f_N(ds,x)$ on $[0,T]$ converges weakly to the measure $f(ds,x)$. Since the set of points of discontinuity of $g$ on $[0,T]$ is at most countable and $s\mapsto f(s,x)$ is continuous, that set is of zero $f(ds,x)$ measure. Hence a slight extension of the Portmanteau theorem (see Theorem 1.2.1 in \cite{billingsley1999convergence}) yields that for all $x\in[0,1]\backslash\mathcal{N}$, $h_N(t,x)\to h(t,x)$. Moreover, $0\le h_N(t,x)\le C f_N(t,x)$, and the upper bound converges in $L^1([0,1])$, hence the sequence $h_N(t,\cdot)$ is uniformly integrable and converges in $L^1([0,1])$ towards $h(t,x)$. Now all converging subsequences have the same limit, so the the whole sequence converges.
The ``locally uniform in $t$" convergence of the integrals follows from the second Dini theorem (see, e..g, Problem 127 on pages 81 and 270 in \cite{Polya-Szego}). Indeed the convergence $\int_0^1h_N(t,x)dx\to\int_0^1h(t,x)dx$ for each $t$ follows from the above arguments, for each $N\ge1$, $t\mapsto\int_0^1h_N(t,x)dx$ is non--decreasing and the limit $t\mapsto\int_0^1h(t,x)dx$ is continuous.
The case of random $f_N$ is treated similarly. The extraction of subsequences is done in such a way that for each $n$, $f_N(s_n,x)$ converges as $N\to\infty$ on a subset of $\Omega\times[0,1]$ of full $d\,\mathbb{P} \otimes dx$ measure. We conclude that from any subsequence of the original sequence $\{h_N(t,\cdot),\ N\ge1\}$, we can extract a further subsequence which converges a.s. in $L^1([0,1])$, hence the convergence in probability in $L^1([0,1])$, as claimed. \end{proof}
\section{Proof of the Convergence of $\bar{S}^N(t,x)$ and $\bar{\mathfrak{F}}^N(t,x)$} \label{sec-proof-conv-S-F}
In this section we prove the convergence of $\bar{S}^N(t,x)$ and $\bar{\mathfrak{F}}^N(t,x)$ to
$\bar{S}(t,x)$ and $\bar{\mathfrak{F}}(t,x)$ given by the set of equations \eqref{eqn-barS-tx} and \eqref{eqn-barmfF-tx} together with \eqref{eqn-barUpsilon-tx}. We first write $S^N_k(t) = S^N_k(0)-A^N_k(t)$ as follows by \eqref{eqn-An-k-rep}: \begin{align*} S^N_k(t) = S^N_k(0) - \int_0^t \int_0^\infty {\bf 1}_{u \le \Upsilon^N_k(s) } Q_{k}(ds, d u), \end{align*} and recall $\mathfrak{F}_k^N(t)$ in \eqref{eqn-mfk}. Then, we have \begin{align} \label{eqn-barSn-tx} \bar{S}^N(t,x) &= \bar{S}^N(0,x) - \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^t \int_0^\infty {\bf 1}_{u \le \Upsilon^N_k(s) } Q_{k}(ds, d u)\, {\bf 1}_{\mathtt{I}_k}(x) \nonumber \\
&= \bar{S}^N(0,x) - \int_0^t \bar\Upsilon^N(s,x) ds \, - \bar{M}^N_A(t,x)\,, \end{align} where $\overline{Q}_k(ds,du)= Q_k(ds,du)-dsdu$ and \begin{align} \label{eqn-barM-An-tx}
\bar{M}^N_A(t,x):= \sum_{k=1}^{K^N} \frac{K^N}{N}\int_0^t \int_0^\infty {\bf 1}_{u \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d u)\, {\bf 1}_{\mathtt{I}_k}(x)\,. \end{align} We then write
\begin{align} \label{eqn-bar-mfFn-tx} \bar{\mathfrak{F}}^N(t,x)
&= \bar{\mathfrak{F}}^N_0(t,x) + \int_0^t \bar{\lambda} (t-s) \bar{\Upsilon}^N(s,x) ds + \Delta^{N}_{1,1}(t,x)+ \Delta^{N}_{1,2}(t,x)\,, \end{align}
where
\begin{equation} \label{eqn-bar-mfFn-0} \bar{\mathfrak{F}}^N_0(t,x) = \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t){\bf 1}_{\mathtt{I}_k}(x), \end{equation} \begin{equation} \label{eqn-Delta-11}
\Delta^{N}_{1,1}(t,x)= \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} \Big( \lambda_{j,k} (t-\tau^N_{j,k}) - \bar\lambda(t-\tau^N_{j,k})\Big){\bf 1}_{\mathtt{I}_k}(x) \,, \end{equation} and \begin{align} \label{eqn-Delta-12} \Delta^{N}_{1,2}(t,x) = \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^t \bar{\lambda} (t-s) \int_0^\infty {\bf 1}_{u \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d u) \,{\bf 1}_{\mathtt{I}_k}(x)\,. \end{align} Observe that \begin{align} \label{eqn-barUpsilon-n-sx} \bar{\Upsilon}^N(s,x) & = \sum_{k=1}^{K^N} \frac{K^N}{N} \frac{S^N_k(s)}{B^N_k} \frac{1}{K^N} \sum_{k'=1}^{K^N}\beta^N_{k,k'} \mathfrak{F}_{k'}^N(s){\bf 1}_{\mathtt{I}_k}(x) \nonumber \\ & = \sum_{k=1}^{K^N} \frac{\bar{S}^N_k(s)}{\bar{B}^N_k}{\bf 1}_{\mathtt{I}_k}(x) \int_0^1\sum_{k'=1}^{K^N} \beta^N_{k,k'} \bar{\mathfrak{F}}_{k'}^N(s) {\bf 1}_{\mathtt{I}_{k'}}(x')dx' \nonumber \\ &= \frac{ \bar{S}^N(s,x)}{\bar{B}^N(x)}\int_0^1\beta^N(x,x')\bar{\mathfrak{F}}^N(s,x') dx' \, , \end{align} where $\beta^N(x,x')$ is defined in \eqref{eqn-betaN-xx'}.
Before proceeding to prove the convergence of $\bar{S}^N(t,x)$ and $\bar{\mathfrak{F}}^N(t,x)$, we describe the proof strategy as follows. In the expressions of $\bar{S}^N(t,x)$ and $\bar{\mathfrak{F}}^N(t,x)$ in \eqref{eqn-barSn-tx} and \eqref{eqn-bar-mfFn-tx}, the stochastic terms $ \bar{M}^N_A(t,x)$, $\Delta^{N}_{1,1}(t,x)$ and $ \Delta^{N}_{1,2}(t,x)$ will converge to zero in probability as $N\to \infty$, which are proved in Lemmas \ref{lem-barAn-tight} and \ref{lem-mfN1-conv}. The term $\bar{\mathfrak{F}}^N_0(t,\cdot)$ will converge to a limit $\bar{\mathfrak{F}}_0(t,\cdot)$ (in the $\|\cdot\|_1$ norm in probability), which is proved in Lemma \ref{lem-mfN0-conv}. Thus, the proof for the convergence of $\bar{S}(t,x)$ and $\bar{\mathfrak{F}}^N(t,x)$ can be carried out by studying the set of integral equations
\eqref{eqn-barSn-tx} and \eqref{eqn-bar-mfFn-tx} together with the expression of $\bar\Upsilon^N(s,x)$ above, given the convergence of the terms $\bar{S}^N(0,\cdot)$, $\bar{\mathfrak{F}}^N_0(t,\cdot)$,
$ \bar{M}^N_A(t,x)$, $\Delta^{N}_{1,1}(t,x)$ and $ \Delta^{N}_{1,2}(t,x)$.
In the following we will first provide this argument in Proposition \ref{prop-conv-S-mfF} and then provide the proofs for the convergence of the required individual terms.
The following Lemma follows readily from
\eqref{eqn-N-B-2} and \eqref{eqn-mfk}, and the conditions on $\bar{B}^N(x)$ in \eqref{eqn-CB}.
\begin{lemma}\label{aprioriN} The processes $\bar{S}^N(t,x)$ and $\bar{\mathfrak{F}}^N(t,x)$ are nonnegative and satisfy the following a priori bounds: \[ \sup_N \sup_{ t\ge0,\ x\in[0,1]}\bar{S}^N(t,x)\le C_B\quad\mbox{and}\quad \sup_N \sup_{t\ge0, \ x\in[0,1]}\bar{\mathfrak{F}}^N(t,x)\le\lambda^\ast C_B \quad \text{a.s.} \]
\end{lemma}
Next, recall the set of the limiting equations: \begin{equation}\label{eq:SF} \begin{aligned} \bar{S}(t,x)&=\bar{S}(0,x)-\int_0^t\frac{\bar{S}(s,x)}{\bar{B}(x)}\int_0^1\beta(x,y)\bar{\mathfrak{F}}(s,y)dyds,\\ \bar{\mathfrak{F}}(t,x)&=\bar{\mathfrak{F}}_0(t,x)+\int_0^t\bar{\lambda}(t-s) \frac{\bar{S}(s,x)}{\bar{B}(x)} \int_0^1\beta(x,y)\bar{\mathfrak{F}}(s,y)dyds\,, \end{aligned} \end{equation} where \begin{equation} \label{eqn-bar-mfF-0}
\bar{\mathfrak{F}}_0(t,x) := \int_0^\infty \bar{\lambda}(\mathfrak{a}+t) \bar{{\mathfrak{I}}}(0,d \mathfrak{a}, x). \end{equation}
We have the following lemmas on the solution properties to this set of equations, and also the existence and uniqueness of its solution. \begin{lemma}\label{apriori} Under Assumptions \ref{AS-LLN-1} and \ref{AS-lambda}, any $(L^\infty([0,1]))^2$--valued solution $(\bar{S}(t,x),\bar{\mathfrak{F}}(t,x))$ of equation \eqref{eq:SF} is nonnegative, and satisfies $\sup_{t\ge0}\bar{S}(t,x)\le\bar{S}(0,x)\le C_B$ and for any $T>0$, there exists $C_T>0$ such that \[ \sup_{0\le t\le T, x\in[0,1]}\bar{\mathfrak{F}}(t,x)\le C_T\,.\] \end{lemma} \begin{proof} The non--negativity of $\bar{S}$ follows from that of the initial condition and the linearity of the equation.
For the second statement, we first note that $\bar{{\mathfrak{I}}}(0,\infty, x)\le C_B$, hence from \eqref{eqn-bar-mfF-0} and Assumption \ref{AS-lambda}, $0\le \bar{\mathfrak{F}}_0(t,x)\le\lambda^\ast C_B$. Hence from the second line of \eqref{eq:SF} and \eqref{eqn-C-beta} and from the assumption that $\bar{B}(x) \ge c_B>0$ for each $x \in [0,1]$ in \eqref{eqn-barB-condition},
we obtain
\[ \|\bar{\mathfrak{F}}(t,\cdot)\|_\infty\le\lambda^\ast C_B+\frac{C_\beta}{c_B}\lambda^\ast C_B\int_0^t \|\bar{\mathfrak{F}}(s,\cdot)\|_\infty ds.\] Thus, the second statement with $C_T=\lambda^\ast C_B \exp\big(\frac{C_\beta}{c_B}\lambda^\ast C_BT \big)$ follows from Gronwall's lemma. We next show that $\bar{\mathfrak{F}}(t,x)\ge0$. Suppose that $\bar{\mathfrak{F}}(t,x)=\bar{\mathfrak{F}}_+(t,x)-\bar{\mathfrak{F}}_-(t,x)$. Then we have \[ \bar{\mathfrak{F}}_-(t,x)\le \int_0^t \bar{\lambda}(t-s) \frac{\bar{S}(s,x)}{\bar{B}(x)}\int_0^1\beta(x,y)\bar{\mathfrak{F}}_-(s,y)dyds,\]
and by a similar argument as above using Gronwall's Lemma, we deduce that $\|\bar{\mathfrak{F}}_-(t,\cdot)\|_\infty=0$, hence the result. Finally it follows readily from Assumption \ref{AS-LLN-1} that $\bar{S}(0,x)\le \sup_N\bar{S}^N(0,x)\le C_B$ for all $x$. From the first line of \eqref{eq:SF}, since $\bar{S}$ and $\bar{\mathfrak{F}}$ are nonnegative, $\bar{S}(t,x)\le \bar{S}(0,x)$, hence the first statement. \end{proof}
\begin{lemma}\label{existuniq} Under Assumptions \ref{AS-LLN-1} and \ref{AS-lambda}, equation \eqref{eq:SF} has a unique $(L^\infty([0,1]))^2$--valued solution. \end{lemma} \begin{proof} We already know that any solution is nonnegative and locally bounded. Uniqueness is then easy to deduce from the following estimate. Consider two solutions $(\bar{S},\bar{\mathfrak{F}})$ and $(\bar{S}',\bar{\mathfrak{F}}')$, and define $\bar{\Upsilon}(t,x)=\frac{\bar{S}(t,x)}{\bar{B}(x)}\int_0^1\beta(x,y)\bar{\mathfrak{F}}(t,y)dy$, $\bar{\Upsilon}'(t,x)$ similarly, replacing $(\bar{S},\bar{\mathfrak{F}})$ by $(\bar{S}',\bar{\mathfrak{F}}')$.
Since from \eqref{eqn-barB-condition} $\bar{B}(x)\ge c_B$, and from Lemma \ref{apriori} $ \bar{S}(t,x)\le C_B$ and for $0\le t\le T, x\in[0,1]$, $\bar{\mathfrak{F}}(t,x)\le C_T$, we obtain \begin{align*}
\|\bar{\Upsilon}(t,\cdot)-\bar{\Upsilon}'(t,\cdot)\|_\infty&\le\sup_{x \in [0,1]}\bigg|\frac{\bar{S}(t,x)}{\bar{B}(x)}- \frac{\bar{S}'(t,x)}{\bar{B}(x)}\bigg|\int_0^1\beta(x,y)\bar{\mathfrak{F}}(t,y)dy\\
&\quad+\sup_{x \in [0,1]} \frac{\bar{S}'(t,x)}{\bar{B}(x)}\int_0^1\beta(x,y)|\bar{\mathfrak{F}}(t,y)-\bar{\mathfrak{F}}'(t,y)|dy\\
& \le \frac{1}{c_B} \|\bar{S}(t,\cdot)-\bar{S}'(t,\cdot)\|_\infty\sup_{x \in [0,1]} \int_0^1\beta(x,y)\bar{\mathfrak{F}}(t,y)dy\\
& \quad + \frac{C_\beta C_B}{c_B} \|\bar{\mathfrak{F}}(t,\cdot)-\bar{\mathfrak{F}}'(t,\cdot)\|_\infty \\
& \le \frac{C_\beta}{c_B}C_T \|\bar{S}(t,\cdot)-\bar{S}'(t,\cdot)\|_\infty + \frac{C_\beta C_B}{c_B} \|\bar{\mathfrak{F}}(t,\cdot)-\bar{\mathfrak{F}}'(t,\cdot)\|_\infty. \end{align*}
From this inequality, we see that uniqueness follows from Gronwall's Lemma. The same estimate can be used repeatedly for proving convergence in $L^\infty([0,1])$ of the Picard iteration procedure, which establishes existence. \end{proof}
We can now prove the main result of this section. Let us first introduce a notation. We let $\mathcal{E}^N_{\mathfrak{F}}(t,x) =\Delta^{N}_{1,1}(t,x)+ \Delta^{N}_{1,2}(t,x)$ and \begin{align*}
\Psi^N(t):=\int_0^1|\bar{\mathfrak{F}}^N_0(t,x)-\bar{\mathfrak{F}}_0(t,x)|dx
+ \int_0^1|\bar{M}^N_A(t,x)|dx
+\int_0^1|\mathcal{E}^N_{\mathfrak{F}}(t,x)|dx\,. \end{align*} \begin{prop} \label{prop-conv-S-mfF}
Let $T>0$ be arbitrary. Given that $\int_0^1|\bar{S}^N(0,x)-\bar{S}(0,x)|dx\to0$ in Assumption \ref{AS-LLN-1}, and assuming that $\sup_{0\le t\le T}\Psi^N(t)\to0$ in probability as $N\to\infty$, we have
\[ \sup_{0\le t\le T}\left(\|\bar{S}^N(t,\cdot)-\bar{S}(t,\cdot)\|_1+\|\bar{\mathfrak{F}}^N(t,\cdot)-\bar{\mathfrak{F}}(t,\cdot)\|_1\right)\to0\] in probability as $N\to\infty$. \end{prop} \begin{proof} Referring to the notations in Lemmas \ref{aprioriN} and \ref{apriori}, let us assume that $\lambda^\ast\le C_T$. We first upper bound the following difference \begin{align*}
&\frac{\bar{S}(t,x)}{\bar{B}(x)}\int_0^1\beta(x,y)\bar{\mathfrak{F}}(t,y)dy- \frac{\bar{S}^N(t,x)}{\bar{B}^N(x)}\int_0^1\beta^N(x,y)\bar{\mathfrak{F}}^N(t,y)dy\\ &=\bigg( \frac{\bar{S}(t,x)}{\bar{B}(x)} - \frac{\bar{S}^N(t,x)}{\bar{B}^N(x)} \bigg)\int_0^1\beta^N(x,y)\bar{\mathfrak{F}}^N(t,y)dy\\ &\quad+\frac{\bar{S}(t,x)}{\bar{B}(x)} \left(\int_0^1\beta(x,y)\bar{\mathfrak{F}}(t,y)dy-\int_0^1\beta^N(x,y)\bar{\mathfrak{F}}^N(t,y)dy\right)\\
&\le C_\beta C_T\bigg| \frac{\bar{S}(t,x)}{\bar{B}(x)} - \frac{\bar{S}^N(t,x)}{\bar{B}^N(x)} \bigg|+\int_0^1\beta^N(x,y)(\bar{\mathfrak{F}}(t,y)-\bar{\mathfrak{F}}^N(t,y))dy \\ &\qquad +\int_0^1(\beta(x,y)-\beta^N(x,y))\bar{\mathfrak{F}}(t,y)dy \, . \end{align*} Note that by \eqref{eqn-barB-condition} and \eqref{eqn-CB}, \begin{align*}
\bigg| \frac{\bar{S}(t,x)}{\bar{B}(x)} - \frac{\bar{S}^N(t,x)}{\bar{B}^N(x)} \bigg|
&= \bigg| \frac{\bar{S}(t,x)-\bar{S}^N(t,x) }{\bar{B}(x)} + \bar{S}^N(t,x) \bigg( \frac{1}{\bar{B}(x)} - \frac{1}{\bar{B}^N(x)}\bigg) \bigg| \\
& \le c_B^{-1} |\bar{S}(t,x)-\bar{S}^N(t,x)| + c_B^{-2} C_B|\bar{B}^N(x) - \bar{B}(x) | \,. \end{align*} Consequently, \begin{align*}
\bigg\| \frac{\bar{S}(t,\cdot)}{\bar{B}(\cdot)} &\int_0^1\beta(\cdot,y)\bar{\mathfrak{F}}(t,y)dy-\frac{\bar{S}^N(t,\cdot)}{\bar{B}^N(\cdot)}\int_0^1\beta^N(\cdot,y)\bar{\mathfrak{F}}^N(t,y)dy\bigg\|_1\\
&\le C_\beta C_T c_B^{-1}\|\bar{S}(t,\cdot)-\bar{S}^N(t,\cdot)\|_1 + C_\beta C_T c_B^{-2} C_B\|\bar{B}^N(\cdot) - \bar{B}(\cdot) \|_1 \\
& \quad +\left(\sup_{N,y}\int_0^1\beta^N(x,y)dx\right)\|\bar{\mathfrak{F}}(t,\cdot)-\bar{\mathfrak{F}}^N(t,\cdot)\|_1\\
&\quad+\int_0^1\left|\int_0^1(\beta(x,y)-\beta^N(x,y))\bar{\mathfrak{F}}(t,y)dx\right|dy \,. \end{align*}
We can now estimate the norm $\|\bar{S}(t,\cdot)-\bar{S}^N(t,\cdot)\|_1$ and $\|\bar{\mathfrak{F}}(t,\cdot)-\bar{\mathfrak{F}}^N(t,\cdot)\|_1$. Let $\bar{C}:=\max\{C_\beta,C_\beta C_Tc_B^{-1}, \\ C_\beta C_Tc_B^{-2} C_B \}$. We now deduce from \eqref{eqn-barSn-tx}, \eqref{eq:SF} and the last computation that \begin{align*}
\|\bar{S}(t,\cdot)-\bar{S}^N(t,\cdot)\|_1&\le\|\bar{S}(0,\cdot)-\bar{S}^N(0,\cdot)\|_1+\|\bar{M}^N_A(t,\cdot)\|_1 \\
& \qquad +\int_0^t\int_0^1\left|\int_0^1(\beta(x,y)-\beta^N(x,y))\bar{\mathfrak{F}}(s,y)dx\right|dyds\\&\qquad+
\bar{C}\int_0^t\|\bar{S}(s,\cdot)-\bar{S}^N(s,\cdot)\|_1 ds + \bar{C} \|\bar{B}^N(\cdot) - \bar{B}(\cdot) \|_1 \\
&\qquad +\bar{C}\int_0^t\|\bar{\mathfrak{F}}(s,\cdot)-\bar{\mathfrak{F}}^N(s,\cdot)\|_1 ds. \end{align*} Next from \eqref{eqn-bar-mfFn-tx} and \eqref{eq:SF}, we get \begin{align*}
\|\bar{\mathfrak{F}}(t,\cdot)-\bar{\mathfrak{F}}^N(t,\cdot)\|_1&\le \|\bar{\mathfrak{F}}_0(t,\cdot)-\bar{\mathfrak{F}}^N_0(t,\cdot)\|_1+\|\mathcal{E}^N_{\mathfrak{F}}(t,\cdot)\|_1 \\
& \qquad +\int_0^t\int_0^1\left|\int_0^1(\beta(x,y)-\beta^N(x,y))\bar{\mathfrak{F}}(s,y)dx\right|dyds\\&\qquad+
\bar{C}\int_0^t\|\bar{S}(s,\cdot)-\bar{S}^N(s,\cdot)\|_1 ds + \bar{C} \|\bar{B}^N(\cdot) - \bar{B}(\cdot) \|_1 \\
& \qquad +\bar{C}\int_0^t\|\bar{\mathfrak{F}}(s,\cdot)-\bar{\mathfrak{F}}^N(s,\cdot)\|_1 ds\,. \end{align*} Adding those two inequalities, the result follows from our assumptions, the fact that \eqref{conv-beta} in Assumption \ref{AS-LLN-2} implies that
\[ \int_0^t\int_0^1\left|\int_0^1(\beta(x,y)-\beta^N(x,y))\bar{\mathfrak{F}}(s,y)dx\right|dyds\to0\quad \text{ as } \quad N\to\infty,\] and the following variant of Gronwall's Lemma: if $f(t)$ and $g(t)$ are nonnegative real-valued functions of $t$ and satisfy $f(t)\le g(t)+c\int_0^t f(s)ds$ for all $0\le t\le T$ and for some $c>0$, then for those $t$, $f(t)\le g(t)+c\int_0^te^{c(t-s)}g(s)ds$. \end{proof}
It remains to show that $\sup_{0\le t\le T}\Upsilon^N(t)\to0$ in probability, which follows from the
next three lemmas, where we establish the convergence of $\bar{\mathfrak{F}}^N_0(t,\cdot)$ to $ \bar{\mathfrak{F}}_0(t,x)$, and that the stochastic terms $ \bar{M}^N_A(t,x)$, $\Delta^{N}_{1,1}(t,x)$ and $ \Delta^{N}_{1,2}(t,x)$ of \eqref{eqn-barM-An-tx}, \eqref{eqn-Delta-11} and \eqref{eqn-Delta-12} tend to $0$ in probability, as $N\to\infty$.
\begin{lemma} \label{lem-mfN0-conv} Under Assumptions \ref{AS-LLN-1} and \ref{AS-lambda}, \begin{equation}
\|\bar{\mathfrak{F}}^N_0(t,\cdot) - \bar{\mathfrak{F}}_0(t,\cdot)\|_1 \to 0 \end{equation} in probability, locally uniformly in $t$, as $N \to \infty$, where $\bar{\mathfrak{F}}_0(t,x) $ is defined in \eqref{eqn-bar-mfF-0}.
\end{lemma}
\begin{proof} We apply Theorem \ref{thm-D-conv-x}. First, we have \[
\bar{\mathfrak{F}}^N_0(t,x) - \bar{\mathfrak{F}}_0(t,x) = \Delta^{N}_{0,1}(t,x) + \Delta^{N}_{0,2}(t,x) , \] where \begin{align*}
\Delta^{N}_{0,1}(t,x) &=\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big(\lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) - \bar\lambda(\tilde{\tau}^N_{-j,k}+ t) \Big){\bf 1}_{\mathtt{I}_k}(x)\,, \\
\Delta^{N}_{0,2}(t,x) &= \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \bar\lambda(\tilde{\tau}^N_{-j,k}+ t){\bf 1}_{\mathtt{I}_k}(x) - \int_0^{\bar{\mathfrak{a}}} \bar{\lambda}(\mathfrak{a}+t) \bar{{\mathfrak{I}}}(0,d \mathfrak{a}, x)\\ & = \int_0^{\bar{\mathfrak{a}}} \bar{\lambda}(\mathfrak{a}+t) [\bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}, x)-\bar{{\mathfrak{I}}}(0,d \mathfrak{a}, x)]\,.
\end{align*}
We now verify condition (i) of Theorem \ref{thm-D-conv-x}. For the first term $ \Delta^{N}_{0,1}(t,x)$, we have \begin{align*}
\| \Delta^{N}_{0,1}(t,\cdot)\|_1
& \le \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \left|\sum_{j=1}^{I^N_k(0)} \Big(\lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) - \bar\lambda(\tilde{\tau}^N_{-j,k}+ t) \Big)\right|\,. \end{align*} Here the summands over $k$ are independent, and for each $k$, conditional on $\{\tilde{\tau}^N_{-j,k}\}_j$, the summands over $j$ are also independent and centered. Using Jensen's inequality for the sum over $k$, and the conditional independence for the sum over $j$, we deduce \begin{align*}
& \mathbb{E}\left[ \left( \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \left|\sum_{j=1}^{I^N_k(0)} \Big(\lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) - \bar\lambda(\tilde{\tau}^N_{-j,k}+ t) \Big)\right| \right)^2 \right] \\
& \le \mathbb{E}\left[ \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{\bar{\mathfrak{a}}} v(\mathfrak{a}+t) \bar{{\mathfrak{I}}}_k^N(0,d \mathfrak{a})\right] \to 0 \quad\mbox{as}\quad N \to \infty, \end{align*} since under Assumption \ref{AS-LLN-1}, thanks to Lemma \ref{convPort}, \[ \frac{1}{K^N}\sum_{k=1}^{K^N} \int_0^{\bar{\mathfrak{a}}} v(\mathfrak{a}+t) \bar{{\mathfrak{I}}}_k^N(0,d \mathfrak{a}) \to \int_0^1 \int_0^{\bar{\mathfrak{a}}} v(\mathfrak{a}+t) \bar{{\mathfrak{I}}}(0,d \mathfrak{a},x)dx \] in probability and $\frac{K^N}{N} \to 0 $ as $N \to \infty$. Recall that $v(t)$ is the variance of the random function $\lambda(t)$ in Assumption \ref{AS-lambda}, which is bounded.
The fact that $\|\Delta^N_{0,2}\|_1\to0$ in probability follows again from Lemma \ref{convPort} and Assumption \ref{AS-LLN-1}.
Now to check condition (ii) of Theorem \ref{thm-D-conv-x}, we first have for $t, u >0$, \begin{align*}
& \Delta^{N}_{0,1}(t+u,x) - \Delta^{N}_{0,1}(t,x) \\
& = \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big(\lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t+u) - \lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) \Big){\bf 1}_{\mathtt{I}_k}(x) \\
& \quad - \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big( \bar\lambda(\tilde{\tau}^N_{-j,k}+ t+u) - \bar\lambda(\tilde{\tau}^N_{-j,k}+ t) \Big){\bf 1}_{\mathtt{I}_k}(x)\,.
\end{align*} Observe that \begin{align*}
& \left\|\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big(\lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t+u) - \lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) \Big){\bf 1}_{\mathtt{I}_k}(x) \right\|_1\\
&\quad \le \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big|\lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t+u) - \lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) \Big| \,, \end{align*} and similarly for the second term. Thus, \begin{align*}
\| \Delta^{N}_{0,1}(t+u,x) - \Delta^{N}_{0,1}(t,x)\|_1 & \le \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big|\lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t+u) - \lambda_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) \Big| \\
& \quad + \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big| \bar\lambda(\tilde{\tau}^N_{-j,k}+ t+u) - \bar\lambda(\tilde{\tau}^N_{-j,k}+ t) \Big| \\
&=: \Delta^{N, (1)}_{0,1}(t,u) + \Delta^{N, (2)}_{0,1}(t,u)\,.
\end{align*}
By Assumption \ref{AS-lambda}, using the expression of $\lambda(t)$ in \eqref{eqn-lambda-assump}, that is, $\lambda_{-j,k}(t) = \sum_{\ell=1}^\kappa \lambda^\ell_{-j,k}(t) {\mathbf 1}_{[\zeta_{-j,k}^{\ell-1},\zeta_{-j,k}^\ell)}(t)$, we obtain \begin{align} \label{eqn-Delta01-1-bound}
\Delta^{N, (1)}_{0,1}(t,u) &= \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \bigg|\sum_{\ell=1}^\kappa \lambda^\ell_{-j,k}(\tilde{\tau}^N_{-j,k}+ t+u) {\mathbf 1}_{[\zeta_{-j,k}^{\ell-1},\zeta_{-j,k}^\ell)}(\tilde{\tau}^N_{-j,k}+ t+u) \nonumber \\
& \qquad \qquad \qquad \qquad - \sum_{\ell=1}^\kappa \lambda^\ell_{-j,k}(\tilde{\tau}^N_{-j,k}+ t) {\mathbf 1}_{[\zeta_{-j,k}^{\ell-1},\zeta_{-j,k}^\ell)}(\tilde{\tau}^N_{-j,k}+ t) \bigg| \nonumber \\
&\le \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \sum_{\ell=1}^\kappa \Big| \lambda^\ell_{-j,k}(\tilde{\tau}^N_{-j,k}+ t+u) - \lambda^\ell_{-j,k}(\tilde{\tau}^N_{-j,k}+ t)\Big| {\mathbf 1}_{\zeta_{-j,k}^{\ell-1} \le \tilde{\tau}^N_{-j,k}+ t \le \tilde{\tau}^N_{-j,k}+ t+u \le \zeta_{-j,k}^\ell} \nonumber \\ & \quad + \lambda^* \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \sum_{\ell=1}^\kappa
{\mathbf 1}_{\tilde{\tau}^N_{-j,k}+ t \le \zeta_{-j,k}^\ell \le \tilde{\tau}^N_{-j,k}+ t+u } \nonumber \\
& \le \varphi(u) \frac{1}{K^N} \sum_{k=1}^{K^N} \bar{I}^N_k(0) + \lambda^* \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \sum_{\ell=1}^\kappa
{\mathbf 1}_{\tilde{\tau}^N_{-j,k}+ t \le \zeta_{-j,k}^\ell \le \tilde{\tau}^N_{-j,k}+ t+u }\,. \end{align} Since both terms in the right hand side are increasing in $u$, we obtain \begin{equation}\label{eqn-mfN0-conv-p0} \sup_{u \in [0,\delta]}\Delta^{N, (1)}_{0,1}(t,u) \le \varphi(\delta) \frac{1}{K^N} \sum_{k=1}^{K^N} \bar{I}^N_k(0) + \lambda^* \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)}
{\mathbf 1}_{\tilde{\tau}^N_{-j,k}+ t \le \zeta_{-j,k}^\ell \le \tilde{\tau}^N_{-j,k}+ t+\delta}\,. \end{equation} Note that \[\frac{1}{K^N} \sum_{k=1}^{K^N} \bar{I}^N_k(0) \to \int_0^1 \bar{I}(0,x)dx \quad\mbox{as}\quad N \to \infty\] under Assumption \ref{AS-LLN-1}. For the second term in \eqref{eqn-mfN0-conv-p0}, we have \begin{align}\label{eqn-mfN0-conv-p1} & \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)}
{\mathbf 1}_{\tilde{\tau}^N_{-j,k}+ t \le \zeta_{-j,k}^\ell \le \tilde{\tau}^N_{-j,k}+ t+\delta } \nonumber\\
& = \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big[ {\mathbf 1}_{\tilde{\tau}^N_{-j,k}+ t \le \zeta_{-j,k}^\ell \le \tilde{\tau}^N_{-j,k}+ t+\delta } - \Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \Big] \nonumber \\ & \quad + \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \,. \end{align} In both expressions, the summands over $k$ are independent, and in the first, for each $k$, conditional on $\{\tilde{\tau}^N_{-j,k}\}_j$, the summands over $j$ are also independent. We have \begin{align*} & \mathbb{E}\left[ \left(\frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big[ {\mathbf 1}_{\tilde{\tau}^N_{-j,k}+ t \le \zeta_{-j,k}^\ell \le \tilde{\tau}^N_{-j,k}+ t+\delta } - \Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \Big] \right)^2\right] \\ & \le \mathbb{E} \left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \Bigg( \sum_{j=1}^{I^N_k(0)} \Big[ {\mathbf 1}_{\tilde{\tau}^N_{-j,k}+ t \le \zeta_{-j,k}^\ell \le \tilde{\tau}^N_{-j,k}+ t+\delta } - \Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \Big] \Bigg)^2 \right] \\ & = \mathbb{E} \left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \sum_{j=1}^{I^N_k(0)} \Big[ {\mathbf 1}_{\tilde{\tau}^N_{-j,k}+ t \le \zeta_{-j,k}^\ell \le \tilde{\tau}^N_{-j,k}+ t+\delta } - \Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \Big]^2 \right] \\ &= \mathbb{E} \Bigg[ \frac{1}{K^N} \sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \sum_{j=1}^{I^N_k(0)} \Big[ \Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \\ & \qquad \qquad \qquad \qquad \qquad \qquad \times \Big(1-\Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \Big) \Big] \Bigg] \\ & = \mathbb{E} \left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{\bar\mathfrak{a}} \Big[ \Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \Big(1-\Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \Big) \Big] \bar{{\mathfrak{I}}}^N_k(0, d\mathfrak{a}) \right] \\ &\to 0\quad\mbox{as}\quad N \to \infty\,, \end{align*} since under Assumption \ref{AS-LLN-1}, \begin{align*} & \frac{1}{K^N} \sum_{k=1}^{K^N} \int_0^{\bar\mathfrak{a}} \Big[ \Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \Big(1-\Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \Big) \Big] \bar{{\mathfrak{I}}}^N_k(0, d\mathfrak{a}) \\ & \to \int_0^1 \int_0^{\bar\mathfrak{a}} \Big[ \Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \Big(1-\Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \Big) \Big] \bar{{\mathfrak{I}}}(0, d\mathfrak{a},x)dx\,, \end{align*} and $ \frac{K^N}{N} \to 0$ as $N\to \infty$. Hence, the first term in \eqref{eqn-mfN0-conv-p1} converges to zero in probability as $N\to\infty$. For the second term in \eqref{eqn-mfN0-conv-p1}, we have \begin{align*} & \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \\ & = \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \int_0^{\bar{\mathfrak{a}}} \Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \bar{{\mathfrak{I}}}^N_k(0, d\mathfrak{a}) \\ & \to \sum_{\ell=1}^\kappa\int_0^1 \int_0^{\bar{\mathfrak{a}}} \Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \bar{{\mathfrak{I}}}(0, d\mathfrak{a}, x)dx \,, \end{align*} in probability as $N\to \infty$.
For each $\ell=1,\dots, \kappa$, the function $\delta \to \int_0^1 \int_0^{\bar{x}} \Big(F_\ell(\mathfrak{a}+ t+\delta) -F_\ell(\mathfrak{a}+ t) \Big) \bar{{\mathfrak{I}}}(0, d\mathfrak{a}, x) dx$ is continuous and equal to zero at $\delta=0$. Thus we have shown that for any small enough $\delta>0$, \begin{align} \label{eqn-mfN0-conv-p2} \limsup_{N\to\infty}\sup_{t\in [0,T]} \frac{1}{\delta}\mathbb{P} \left( \sup_{0 \le u \le \delta} \Delta^{N, (1)}_{0,1}(t,u) > \epsilon/2\right) = 0. \end{align}
Note that \begin{align}\label{201}
\Delta^{N, (2)}_{0,1}(t,u)=\int_0^1\int_0^{\bar{\mathfrak{a}}}\left|\bar{\lambda}(\mathfrak{a}+t+u)-\bar{\lambda}(\mathfrak{a}+t)\right|\bar{{\mathfrak{I}}}^N(0,d\mathfrak{a},x)dx\,. \end{align} By similar calculations leading to \eqref{eqn-mfN0-conv-p0}, we obtain for any small enough $\delta>0$, \begin{align*} \sup_{u \in [0,\delta]}\Delta^{N, (2)}_{0,1}(t,u) & \le \varphi(\delta) \frac{1}{K^N} \sum_{k=1}^{K^N} \bar{I}^N_k(0) \\ & \qquad + \lambda^* \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} \Big(F_\ell(\tilde{\tau}^N_{-j,k}+ t+\delta) -F_\ell(\tilde{\tau}^N_{-j,k}+ t) \Big) \,. \end{align*} Thus, by the same arguments for these two terms as in the proof for \eqref{eqn-mfN0-conv-p2}, we obtain that \eqref{eqn-mfN0-conv-p2} holds for $\Delta^{N, (2)}_{0,1}(t,u)$. Thus, combining these two results, we obtain that for any $\epsilon>0$, for $\delta>0$ small enough, \begin{align} \label{eqn-mfN0-conv-p3}
\limsup_{N\to\infty}\sup_{t\in [0,T]} \frac{1}{\delta}\mathbb{P} \left( \sup_{0 \le u \le \delta} \| \Delta^{N}_{0,1}(t+u,x) - \Delta^{N}_{0,1}(t,x)\|_1 > \epsilon\right) =0\,. \end{align}
Now for $\Delta^{N}_{0,2}(t,x)$, we have for $t,u>0$,
\begin{align*}
& \|\Delta^{N}_{0,2}(t+u,x) - \Delta^{N}_{0,2}(t,x)\|_1 \\
&\le\int_0^1\int_0^{\bar{\mathfrak{a}}}|\bar{\lambda}(\mathfrak{a}+t+u)-\bar{\lambda}(\mathfrak{a}+t)|[\bar{{\mathfrak{I}}}^N(0,d\mathfrak{a},x)+\bar{{\mathfrak{I}}}(0,d\mathfrak{a},x)]dx, \end{align*} which is treated exactly as $\Delta^{N, (2)}_{0,1}(t,u)$, see formula \eqref{201}. This completes the proof of the lemma. \end{proof}
\begin{lemma} \label{lem-barAn-tight} Under Assumptions \ref{AS-LLN-1}, \ref{AS-LLN-2} and \ref{AS-lambda},
for all $T>0$,
\begin{align}\label{MAto0}
\mathbb{E}\bigg[\sup_{t \in [0,T]} \| \bar{M}^N_A(t,\cdot)\|^2_1\bigg]\to0\, , \end{align} and thus, \begin{equation}\label{eqn-barA-Upsilon-int}
\Big\| \bar{A}^N(t,\cdot) - \int_0^t \bar{\Upsilon}^N(s,\cdot) ds\Big\|_1 \to 0 \end{equation} in probability, locally uniformly in $t$.
In addition, there exists $C_T>0$ such that for all $N\ge1$, \begin{equation}\label{momentestim}
\mathbb{E}\left[\sup_{t\le T}\|\bar{A}^N(t,\cdot)\|_1\right] \le C_T\,. \end{equation}
\end{lemma}
\begin{proof} Recall the expressions of $A^N_k(t)$ in \eqref{eqn-An-k-rep} and $\Upsilon^N_k(t)$ in \eqref{eqn-upsilon}. By \eqref{eqn-mfk}, under Assumption \ref{AS-lambda} that $\lambda(t) \le \lambda^*$, under the condition on $\bar{B}(x)$ in \eqref{eqn-barB-condition}, and \eqref{eqn-CB}, we have $\bar{\mathfrak{F}}^N(t,x) \le \lambda^* C_B$ and thus, under Assumption \ref{AS-LLN-2}, $\bar\Upsilon^N(t,x)\le \lambda^* C_BC_\beta $, where we have used \eqref{eqn-C-beta}.
Hence $\| \bar\Upsilon^N(t,\cdot) \|_1\le\lambda^* C_BC_\beta$, and
\begin{align} \label{eqn-int-Upsilon-bound}
\Big\| \int_0^t \bar\Upsilon^N(r,\cdot)dr -\int_0^s \bar\Upsilon^N(r,\cdot)dr \Big\|_1 \le \lambda^* C_BC_\beta (t-s) \,. \end{align}
For each $k$, we can write \begin{equation*}\label{eqn-barAn-decomp} \bar{A}^N_k(t) = \int_0^t \bar\Upsilon^N_k(s)ds + \bar{M}^N_{A,k}(t) \end{equation*} where \[ \bar{M}^N_{A,k}(t) = \frac{K^N}{N}\int_0^t \int_0^\infty {\mathbf 1}_{u \le \Upsilon^N_k(s^-)}\bar{Q}_k(ds,du) \] with $\bar{Q}_k(ds,du) = Q_k(ds,du) - ds du$ being the compensated PRM associated with $Q_k$. Let $\bar{M}^N_A(t,x) = \sum_{k=1}^{K^N}\bar{M}^N_{A,k}(t){\bf 1}_{\mathtt{I}_k}(x)$. Then we have the time-space representation: \begin{equation} \label{eqn-barAN} \bar{A}^N(t,x)=\int_0^t \bar\Upsilon^N(r,x)dr+\bar{M}^N_A(t,x)\,.
\end{equation}
It is clear that for each $k$, $\{\bar{M}^N_{A,k}(t): t \ge 0\}$ is a square-integrable martingale with respect to the filtration ${\mathcal{F}}^N_A= \{{\mathcal{F}}^N_A(t): t\ge0\}$ where \begin{align*} {\mathcal{F}}^N_A(t) &:= \sigma\big\{ I^N_k(0), \tilde{\tau}^N_{-j,k}: j =1,\dots, I^N_k(0), k=1,\dots,K\big\} \vee \sigma \big\{\lambda_{j,k}(\cdot), \, j \in {\mathbb Z}\setminus\{0\}, k=1,\dots, K \big\} \\
&\qquad \vee \sigma\bigg\{ \int_0^{t'} \int_0^\infty {\mathbf 1}_{u \le \Upsilon^N_k(s^{-})} Q_k(ds,du): 0 \le t' \le t, \, k=1, \dots,K \bigg\}. \end{align*} and has the quadratic variation \begin{equation*} \label{eqn-MA-qv} \langle \bar{M}^N_{A,k} \rangle(t) = \frac{K^N}{N}\int_0^t \bar\Upsilon^N_k(s)ds, \quad t \ge 0. \end{equation*} Then, \begin{align}\label{estimM}
\| \bar{M}^N_A(t,\cdot)\|_1
\le\int_0^1 \bigg|\sum_{k=1}^{K^N} \bar{M}^N_{A,k}(t){\bf 1}_{\mathtt{I}_k}(x)\bigg| dx \le \frac{1}{K^N} \sum_{k=1}^{K^N} \big|\bar{M}^N_{A,k}(t) \big|\,. \end{align} By Doob's inequality for submartingales, \begin{equation*} \label{eqn-barM-supbound-conv0}
\mathbb{E}\bigg[\sup_{t \in [0,T]} \big|\bar{M}^N_{A,k}(t) \big|^2 \bigg] \le \mathbb{E}\big[\big|\bar{M}^N_{A,k}(T) \big|^2\big] = \mathbb{E}\left[ \frac{K^N}{N} \int_0^T \bar\Upsilon^N_k(s)ds\right] \le \lambda^*C_B C_\beta T \frac{K^N}{N} \,. \end{equation*} Since $ \frac{K^N}{N} \to 0$ as $N\to\infty$, the last inequality entails that as $N\to\infty$,
\[ \sup_{1\le k\le K} \mathbb{E}\bigg[\sup_{t \in [0,T]} \big|\bar{M}^N_{A,k}(t) \big|^2 \bigg]\to0.\] This combined with \eqref{estimM} implies that \eqref{MAto0} holds.
Note that the above computations, combined with \eqref{eqn-barAN} and \eqref{eqn-int-Upsilon-bound}, yield \eqref{momentestim}.
Finally \eqref{eqn-barA-Upsilon-int} follows directly from \eqref{eqn-barAN} and \eqref{MAto0}. \end{proof}
We finally show that $ \Delta^{N}_{1,1}(t,\cdot)$ and $ \Delta^{N}_{1,2}(t,\cdot)$ tend to $0$.
\begin{lemma} \label{lem-mfN1-conv} Under Assumptions \ref{AS-LLN-1}, \ref{AS-LLN-2} and \ref{AS-lambda}, as $N\to \infty$, both $ \Delta^{N}_{1,1}(t,\cdot)$ and $ \Delta^{N}_{1,2}(t,\cdot)$ defined in \eqref{eqn-Delta-11} and \eqref{eqn-Delta-12} converge to zero in $L^1([0,1])$ in probability, locally uniformly in $t$.
\end{lemma}
\begin{proof}
We apply Theorem \ref{thm-D-conv-x}. We first consider $ \Delta^{N}_{1,1}(t,x)$. To verify condition (i) of Theorem \ref{thm-D-conv-x}, we have \begin{align*}
\| \Delta^{N}_{1,1}(t,\cdot)\|_1 & \le \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \left|\sum_{j=1}^{A^N_k(t)} \left(\lambda_{j,k} (t-\tau^N_{j,k}) - \bar\lambda(t-\tau^N_{j,k})\right)\right|\,. \end{align*}
Recall the expression of $A^N_k(t)$ in \eqref{eqn-An-k-rep} and the associated $\Upsilon^N_k(t) $ in \eqref{eqn-upsilon}. It is clear that the summands over $k$ are not independent due to the interactions among individuals in different locations in the infection process.
Using first Jensen's inequality, and then the fact that for each $k$, conditional on the arrivals $\{\tau^N_{j,k}\}_j$, the summands over $j$ are independent and centered, we have \begin{align*}
& \mathbb{E} \left[ \left( \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \left|\sum_{j=1}^{A^N_k(t)} \Big( \lambda_{j,k} (t-\tau^N_{j,k}) - \bar\lambda(t-\tau^N_{j,k})\Big) \right|\right)^2 \right] \\ & \le \mathbb{E} \left[ \frac{1}{K^N}\sum_{k=1}^{K^N} \left( \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} \Big( \lambda_{j,k} (t-\tau^N_{j,k}) - \bar\lambda(t-\tau^N_{j,k})\Big) \right)^2 \right] \\
& = \mathbb{E} \left[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \sum_{j=1}^{A^N_k(t)} \Big| \lambda_{j,k} (t-\tau^N_{j,k}) - \bar\lambda(t-\tau^N_{j,k})\Big|^2 \right]\\ &= \mathbb{E} \left[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \int_0^t v(t-s) d A^N_k(s) \right] \\ & \le (\lambda^*)^2 \mathbb{E} \left[ \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \bar{A}^N_k(t) \right] \\
& = (\lambda^*)^2 \frac{K^N}{N} \mathbb{E} \left[ \|\bar{A}^N(t,\cdot)\|_{1} \right] \to0 \quad\mbox{as}\quad N \to \infty\,, \end{align*} where we used $v(t) \le (\lambda^*)^2$ under Assumption \ref{AS-lambda}, and the convergence follows from the assumption that $\frac{K^N}{N} \to 0$ as $N\to\infty$, and \eqref{momentestim} in Lemma \ref{lem-barAn-tight}.
We next check condition (ii) in Theorem \ref{thm-D-conv-x} for $ \Delta^{N}_{1,1}(t,x)$. We have \begin{align*}
\Delta^{N}_{1,1}(t+u,x) - \Delta^{N}_{1,1}(t,x) &= \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} \Big( \lambda_{j,k} (t+u-\tau^N_{j,k}) - \lambda_{j,k} (t-\tau^N_{j,k}) \Big){\bf 1}_{\mathtt{I}_k}(x) \\ & \quad - \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} \Big( \bar\lambda(t+u-\tau^N_{j,k})- \bar\lambda(t-\tau^N_{j,k})\Big){\bf 1}_{\mathtt{I}_k}(x) \\
& \quad + \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k(t)+1}^{A^N_k(t+u)} \Big( \lambda_{j,k} (t+u-\tau^N_{j,k}) - \bar\lambda(t+u-\tau^N_{j,k})\Big){\bf 1}_{\mathtt{I}_k}(x)\,, \end{align*} and \begin{align*}
\|\Delta^{N}_{1,1}(t+u,x) - \Delta^{N}_{1,1}(t,x) \|_1
& \le \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} \Big| \lambda_{j,k} (t+u-\tau^N_{j,k}) - \lambda_{j,k} (t-\tau^N_{j,k}) \Big| \\
& \quad + \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} \Big| \bar\lambda(t+u-\tau^N_{j,k})- \bar\lambda(t-\tau^N_{j,k})\Big| \\
& \quad + \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k(t)+1}^{A^N_k(t+u)} \Big| \lambda_{j,k} (t+u-\tau^N_{j,k}) - \bar\lambda(t+u-\tau^N_{j,k})\Big|\\
& =: \Delta^{N,(1)}_{1,1}(t,u) + \Delta^{N,(2)}_{1,1}(t,u) + \Delta^{N,(3)}_{1,1}(t,u)\,. \end{align*}
Similar to $\Delta^{N,(1)}_{0,1}(t,u)$ in \eqref{eqn-Delta01-1-bound}, we have \begin{align*} \sup_{u \in [0,\delta]}\Delta^{N,(1)}_{1,1}(t,u) &\le \varphi(\delta) \int_0^1\bar{A}^N(t,x)dx + \lambda^* \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} \sum_{\ell=1}^\kappa
{\mathbf 1}_{t-\tau^N_{j,k} \le \zeta_{j,k}^\ell \le t+\delta -\tau^N_{j,k} }\,. \end{align*} We note that \begin{align*} \int_0^1\bar{A}^N(t,x)dx&= \int_0^1\int_0^t\bar{\Upsilon}^N(s,x)dsdx+\int_0^1\bar{M}^N_A(t,x)dx\\
&\le\lambda^\ast C_B C_\beta t+\int_0^1\bar{M}^N_A(t,x)dx\,. \end{align*} Hence, we deduce from \eqref{MAto0} that as soon as $\delta>0$ is small enough such that $\varphi(\delta)\lambda^\ast C_B C_\beta t<\epsilon/6$, \begin{equation}\label{equals0} \limsup_{N}\frac{1}{\delta}\mathbb{P}\left(\varphi(\delta) \int_0^1\bar{A}^N(t,x)dx>\epsilon/6\right)=0\,. \end{equation}
For the second term, we have \begin{align*} & \mathbb{E}\left[\left(\sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} {\mathbf 1}_{t-\tau^N_{j,k} \le \zeta_{j,k}^\ell \le t+\delta -\tau^N_{j,k} } \right)^2\right ] \\
& \le 2 \mathbb{E}\left[\left(\sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N}
\int_0^t \int_0^\infty \int_{t-s}^{t+\delta -s} {\mathbf 1}_{r \le \Upsilon^N_k(s^-)} \overline{Q}_{k,\ell}(ds,dr,d\zeta)\right)^2\right ] \\
& \quad + 2 \mathbb{E}\left[\left(\sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N}
\int_0^t \Big(F_\ell( t+\delta-s ) - F_\ell( t-s )\Big) \Upsilon^N_k(s) ds \right)^2\right ] \end{align*} where $Q_{k,\ell}(ds,dr,d\zeta)$ is a PRM on ${\mathbb R}_+^3$ with mean measure $dsdrF_\ell(d\zeta)$ whose projection on the first two coordinates is $Q_k$, and $\overline{Q}_{k,\ell}(ds,dr,d\zeta)$ is the corresponding compensated PRM. Observe that \begin{align*} & \mathbb{E}\left[\left( \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N}
\int_0^t \int_0^\infty \int_{t-s}^{t+\delta -s} {\mathbf 1}_{r \le \Upsilon^N_k(s^-)} \overline{Q}_{k,\ell}(ds,dr,d\zeta)\right)^2\right ] \\
& \le \kappa \sum_{\ell=1}^\kappa \mathbb{E}\left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \left( \frac{K^N}{N}
\int_0^t \int_0^\infty \int_{t-s}^{t+\delta -s} {\mathbf 1}_{r \le \Upsilon^N_k(s^-)} \overline{Q}_{k,\ell}(ds,dr,d\zeta)\right)^2\right ] \\
& = \kappa \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \mathbb{E}\left[ \int_0^t \Big(F_\ell( t+\delta-s ) - F_\ell( t-s )\Big) \Upsilon^N_k(s) ds \right ] \\
& \le \lambda^* C_BC_\beta \kappa \sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N}\int_0^t \Big(F_\ell( t+\delta-s ) - F_\ell( t-s )\Big)ds \\
& \le \lambda^* C_B C_\beta \kappa^2 \delta \frac{K^N}{N} \to 0 \quad\mbox{as}\quad N \to \infty, \end{align*}
where we have used the inequality \begin{align}\label{delta} 0\le\int_0^t[F_\ell(s+\delta)-F_\ell(s)]ds\le\int_0^{t+\delta}F_\ell(s)ds-\int_0^tF_\ell(s)ds\le\delta\,, \end{align} and \begin{align*} & \mathbb{E}\left[\left(\sum_{\ell=1}^\kappa \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N}
\int_0^t \Big(F_\ell( t+\delta-s ) - F_\ell( t-s )\Big) \Upsilon^N_k(s) ds \right)^2\right ] \\ & \le\kappa \sum_{\ell=1}^\kappa\frac{1}{K^N} \sum_{k=1}^{K^N} \mathbb{E}\left[\left(\frac{K^N}{N}
\int_0^t \Big(F_\ell( t+\delta-s ) - F_\ell( t-s )\Big) \Upsilon^N_k(s) ds \right)^2\right ] \\
&\le \kappa ( \lambda^* C_B C_\beta )^2 \sum_{\ell=1}^\kappa\left(\int_0^t[F_\ell(s+\delta)-F_\ell(s)]ds\right)^2\\
&\le(\kappa\lambda^\ast C_B C_\beta \delta)^2 \,. \end{align*}
This combined with \eqref{equals0} shows that \begin{align} \label{eqn-mfN1-conv11} \limsup_{N\to\infty}\sup_{t\in [0,T]} \frac{1}{\delta}\mathbb{P} \left( \sup_{0 \le u \le \delta} \Delta^{N, (1)}_{1,1}(t,u) > \epsilon/3\right) \to 0\,\quad \text{ as } \quad \delta\to0\, . \end{align}
Next, similar to $\Delta^{N,(1)}_{0,1}(t,u)$ in \eqref{eqn-Delta01-1-bound}, we have \begin{align*} \sup_{u \in [0,\delta]} \Delta^{N,(2)}_{1,1}(t,u) &\le \varphi(\delta) \frac{1}{K^N} \sum_{k=1}^{K^N} \bar{A}^N_k(t) + \lambda^* \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{A^N_k(t)} \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta -\tau^N_{j,k} ) - F_\ell( t-\tau^N_{j,k} )\Big) \,. \end{align*} Then using the same arguments leading to \eqref{eqn-mfN1-conv11}, we obtain that \eqref{eqn-mfN1-conv11} holds for $ \Delta^{N,(2)}_{1,1}(t,u) $.
Finally, for $\Delta^{N,(3)}_{1,1}(t,u)$, we have \begin{align*} \sup_{0 \le u \le \delta}\Delta^{N,(3)}_{1,1}(t,u) & \le \lambda^* \frac{1}{K^N} \sum_{k=1}^{K^N} (\bar{A}^N_k(t+\delta) - \bar{A}^N_k(t)) \\ & = \lambda^* \int_0^1 \int_t^{t+\delta} \bar{A}^N(ds, x)dx\,. \end{align*} So \begin{align*} \mathbb{P}\left(\sup_{0 \le u \le \delta}\Delta^{N,(3)}_{1,1}(t,u)>\epsilon/3\right) &\le\frac{18(\lambda^\ast)^2}{\epsilon^2}\Bigg\{\mathbb{E}\left[\left(\int_0^1\int_t^{t+\delta}\bar{\Upsilon}^N(s,x)dsdx\right)^2\right] \\ & \qquad \qquad \qquad +
\mathbb{E}\left[\|\bar{M}_A^N(t+\delta,\cdot)-\bar{M}_A^N(t,\cdot)\|_1^2\right]\Bigg\}, \end{align*} and from \eqref{MAto0} and \eqref{eqn-int-Upsilon-bound}, \begin{align*} \limsup_{N\to\infty}\sup_{t\in [0,T]}\frac{1}{\delta}\mathbb{P}\left(\sup_{0 \le u \le \delta}\Delta^{N,(3)}_{1,1}(t,u)>\epsilon/3\right)&\le \frac{18(\lambda^\ast)^4(C_B)^2 C_\beta^2}{\epsilon^2}\delta\\ &\to0,\quad\text{ as } \quad \delta\to0\,. \end{align*} Consequently \eqref{eqn-mfN1-conv11} holds for $ \Delta^{N,(3)}_{1,1}(t,u) $.
Thus combining the three last results, we obtain \begin{align} \label{eqn-mfN11-conv}
\limsup_{N\to\infty}\sup_{t\in [0,T]} \frac{1}{\delta}\mathbb{P} \left( \sup_{0 \le u \le \delta} \| \Delta^{N}_{1,1}(t+u,x) - \Delta^{N}_{1,1}(t,x)\|_1 > \epsilon\right) \to 0, \quad\mbox{as}\quad \delta \to 0. \end{align} Thus we have shown that $\Delta^{N}_{1,1}(t,\cdot)\to 0$ in $L^1([0,1])$ in probability, locally uniformly in $t$, as $N\to\infty$.
We now consider $\Delta^{N}_{1,2}(t,x)$. To check condition (i) in Theorem \ref{thm-D-conv-x}, we have for each $t\le T$, \begin{align*}
\mathbb{E}\big[\| \Delta^{N}_{1,2}(t,\cdot)\|_1^2\big] &\le \mathbb{E}\left[ \left( \frac{1}{K^N} \sum_{k=1}^{K^N}\frac{K^N}{N} \int_0^t \int_0^\infty \bar{\lambda} (t-s) {\bf 1}_{u \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d u) \right)^2 \right] \\ & \le \mathbb{E}\left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \left(\int_0^t \int_0^\infty \bar{\lambda} (t-s) {\bf 1}_{u \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d u) \right)^2 \right] \\ & = \mathbb{E}\left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \int_0^t \bar{\lambda} (t-s)^2 \Upsilon^N_k(s) ds \right] \\ & \le (\lambda^*)^2\frac{K^N}{N} \mathbb{E}\left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \int_0^t \bar\Upsilon^N_k(s) ds \right] \\ & \le (\lambda^*)^3 C_B C_\beta T\frac{K^N}{N} \to 0 \end{align*} as $N\to \infty$.
To check condition (ii) in Theorem \ref{thm-D-conv-x}, we have \begin{align*} & \Delta^{N}_{1,2}(t+u,x) - \Delta^{N}_{1,2}(t,x)
\\&= \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+u} \int_0^\infty \big( \bar{\lambda} (t+u-s) -\bar{\lambda} (t-s)\big) {\bf 1}_{r \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d r) \,{\bf 1}_{\mathtt{I}_k}(x) \\
& \qquad + \sum_{k=1}^{K^N} \frac{K^N}{N} \int_t^{t+u} \int_0^\infty \bar{\lambda} (t-s) {\bf 1}_{r \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d r) \,{\bf 1}_{\mathtt{I}_k}(x)\,. \end{align*} Thus, \begin{align*}
& \| \Delta^{N}_{1,2}(t+u,\cdot) - \Delta^{N}_{1,2}(t,\cdot)\|_1 \\
& \le \frac{1}{K^N} \sum_{k=1}^{K^N} \bigg| \frac{K^N}{N} \int_0^{t+u} \int_0^\infty \big( \bar{\lambda} (t+u-s) -\bar{\lambda} (t-s)\big) {\bf 1}_{r \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d r) \bigg| \\
& \qquad + \frac{1}{K^N}\sum_{k=1}^{K^N} \bigg| \frac{K^N}{N} \int_t^{t+u} \int_0^\infty \bar{\lambda} (t-s) {\bf 1}_{r \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d r) \bigg| \\
& \le \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+u} \int_0^\infty \big| \bar{\lambda} (t+u-s) -\bar{\lambda} (t-s)\big| {\bf 1}_{r \le \Upsilon^N_k(s) } Q_{k}(ds, d r) \\
& \quad + \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+u} \big| \bar{\lambda} (t+u-s) -\bar{\lambda} (t-s)\big| \Upsilon^N_k(s) ds \\ & \quad + \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \int_t^{t+u} \int_0^\infty \bar{\lambda} (t-s) {\bf 1}_{r \le \Upsilon^N_k(s) } Q_{k}(ds, d r) \\ & \quad + \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \int_t^{t+u} \bar{\lambda} (t-s) \Upsilon^N_k(s) ds\,,
\end{align*} from which we obtain \begin{align*}
& \sup_{0 \le u\le \delta }\| \Delta^{N}_{1,2}(t+u,\cdot) - \Delta^{N}_{1,2}(t,\cdot)\|_1 \\
& \le \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+\delta} \int_0^\infty \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big]{\bf 1}_{r \le \Upsilon^N_k(s) } Q_{k}(ds, d r) \\ & \quad + \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+\delta} \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big] \Upsilon^N_k(s) ds \\ & \quad + \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \int_t^{t+\delta} \int_0^\infty \bar{\lambda} (t-s) {\bf 1}_{r \le \Upsilon^N_k(s) } Q_{k}(ds, d r) \\ & \quad + \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \int_t^{t+\delta} \bar{\lambda} (t-s) \Upsilon^N_k(s) ds\,. \end{align*} For the first term, we have \begin{align*} &\mathbb{E}\left[ \left(\frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+\delta} \int_0^\infty \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big]{\bf 1}_{r \le \Upsilon^N_k(s) } Q_{k}(ds, d r) \right)^2\right] \\ & \le 2 \mathbb{E}\left[ \left(\frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+\delta} \int_0^\infty \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big]{\bf 1}_{r \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d r) \right)^2\right] \\
& \quad + 2 \mathbb{E}\left[ \left( \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+\delta} \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big] \Upsilon^N_k(s) ds \right)^2\right] \\
& \le 2 \mathbb{E}\left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \left( \frac{K^N}{N} \int_0^{t+\delta} \int_0^\infty \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big]{\bf 1}_{r \le \Upsilon^N_k(s) } \overline{Q}_{k}(ds, d r) \right)^2\right] \\
& \quad + 2 \mathbb{E}\left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \left( \frac{K^N}{N}\int_0^{t+\delta} \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big] \Upsilon^N_k(s) ds \right)^2\right] \\
& \le 2 \mathbb{E}\left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \int_0^{t+\delta} \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big]^2 \bar\Upsilon^N_k(s) ds \right] \\
& \quad + 2 \mathbb{E}\left[ \frac{1}{K^N} \sum_{k=1}^{K^N} \left( \int_0^{t+\delta} \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big] \bar\Upsilon^N_k(s) ds \right)^2\right] \\
& \le 2 \frac{K^N}{N} \lambda^* C_B C_\beta \int_0^{t+\delta} \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big]^2 ds \\
& \quad + 2 (\lambda^*C_BC_\beta)^2 \left( \int_0^{t+\delta} \Big[\varphi(\delta) + \lambda^* \sum_{\ell=1}^\kappa \Big(F_\ell(t+\delta-s ) - F_\ell( t-s) \Big)
\Big] ds \right)^2 \,. \end{align*} Since the integral terms can be made arbitrarily small by choosing $\delta>0$ small enough, we have that \[\limsup_{N\to\infty}\sup_{t\in [0,T]} \mathbb{P} \left(\sup_{0 \le u\le \delta } \Delta^{N, (1)}_{1,2}(t,u) > \epsilon/4\right)=0\] for $\delta>0$ small enough. The second term is already treated above as the second component in the upper bound. The other two terms can be treated in a similar but simpler way. Thus we have shown that \begin{align} \label{eqn-mfN12-conv}
\limsup_{N\to\infty}\sup_{t\in [0,T]} \frac{1}{\delta}\mathbb{P} \left( \sup_{0 \le u \le \delta} \| \Delta^{N}_{1,2}(t+u,x) - \Delta^{N}_{1,2}(t,x)\|_1 > \epsilon\right) \to 0, \quad\mbox{as}\quad \delta \to 0. \end{align} Thus we have shown that $\Delta^{N}_{1,2}(t,\cdot)\to 0$ in $L^1([0,1])$ in probability, locally uniformly in $t$, as $N\to\infty$. The proof for the lemma is complete. \end{proof}
We now deduce the following Corollary from the results in Proposition \ref{prop-conv-S-mfF} and Lemmas \ref{lem-barAn-tight}, \ref{lem-mfN0-conv} and \ref{lem-mfN1-conv}.
\begin{coro} \label{coro-conv-A}
Under Assumptions \ref{AS-LLN-1}, \ref{AS-LLN-2} and \ref{AS-lambda}, we have that $\|\bar\Upsilon^N(t,\cdot) - \bar\Upsilon(t,\cdot)\|_1\to 0$ in probability, locally uniformly in $t$, as $N\to \infty$ where $\bar\Upsilon(t,x)$ is given in \eqref{eqn-barUpsilon-tx}, and thus,
$\|\bar{A}^N(t,\cdot) - \bar{A}(t,\cdot)\|_1\to 0$ in probability, locally uniformly in $t$, as $N\to \infty$, where \begin{equation} \label{eqn-barA-tx} \bar{A}(t,x)= \int_0^t\frac{\bar{S}(s,x)}{\bar{B}(x)}\int_0^1\beta(x,x')\bar{\mathfrak{F}}(s,x')dx'ds = \int_0^t \bar\Upsilon(s,x)ds\,. \end{equation} \end{coro}
\begin{proof}
Combining the results in Lemmas \ref{lem-barAn-tight}, \ref{lem-mfN0-conv} and \ref{lem-mfN1-conv} we have shown that $\sup_{0\le t\le T}\Psi^N(t)\to0$ in probability as $N\to\infty$. Thus by Proposition \ref{prop-conv-S-mfF}, we can conclude the convergence of $\bar{S}^N(t,\cdot)$ and $\bar{\mathfrak{F}}^N(t,\cdot)$ in $L^1([0,1])$ in probability, locally uniformly in $t$. By the expression of $\bar\Upsilon^N(t,x)$ in \eqref{eqn-barUpsilon-n-sx}, we immediately obtain the convergence of $\bar\Upsilon^N(t,\cdot)$. Then by the expression of $\bar{A}^N(t,x)$ in \eqref{eqn-barAN}, we obtain the convergence in probability of $\bar{A}^N(t,\cdot)$ to $\bar{A}(t,\cdot)$ given in \eqref{eqn-barA-tx}, as announced. The uniformity in $t$ follows from the second Dini theorem. \end{proof}
\section{Proof for the Convergence of $\bar{{\mathfrak{I}}}^N(t,\mathfrak{a},x)$} \label{sec-proof-conv-I}
In this section, we prove the convergence of $\bar{{\mathfrak{I}}}^N(t,\mathfrak{a},x)$ to $\bar{{\mathfrak{I}}}(t,\mathfrak{a},x)$ as stated in Proposition \ref{prop-sIn-conv} below. Recall ${\mathfrak{I}}^N_k(t,\mathfrak{a}) $ in \eqref{eqn-In-k-rep}. We write the two decomposed processes: \begin{equation} \label{eqn-bar-sIn-0} \bar{{\mathfrak{I}}}^N_{0}(t,\mathfrak{a},x) =\sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=1}^{I^N_k(0)} {\bf1}_{\eta^0_{-j,k} >t} {\bf 1}_{ \tilde{\tau}^N_{-j,k} \le (\mathfrak{a}-t)^+} {\bf 1}_{\mathtt{I}_k}(x) =\sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^+)} {\bf1}_{\eta^0_{-j,k} >t} {\bf 1}_{\mathtt{I}_k}(x) \,\,, \end{equation} and \begin{equation} \label{eqn-bar-sIn-1} \bar{{\mathfrak{I}}}^N_{1}(t,\mathfrak{a},x) =\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} {\bf 1}_{\mathtt{I}_k}(x)\,\,. \end{equation}
\begin{lemma} \label{lem-sIn0-conv} Under Assumptions \ref{AS-LLN-1} and \ref{AS-lambda}, \begin{equation}
\|\bar{{\mathfrak{I}}}^N_0(t,\mathfrak{a}, \cdot) - \bar{{\mathfrak{I}}}_0(t, \mathfrak{a}, \cdot)\|_1 \to 0 \end{equation} in probability, locally uniformly in $t$ and $\mathfrak{a}$, as $N \to \infty$, where \begin{equation} \label{eqn-bar-sI-0}
\bar{{\mathfrak{I}}}_0(t,\mathfrak{a}, x) := \int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}', x) \,. \end{equation} \end{lemma}
\begin{proof} We first write \[ \bar{{\mathfrak{I}}}^{N}_{0}(t,\mathfrak{a}, x) =\bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},x) + \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},x) \] where \begin{align} \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},x) &=\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^+)} \frac{F^c(\tilde{\tau}^N_{-j,k}+t)}{F^c(\tilde{\tau}^N_{-j,k})} {\bf 1}_{\mathtt{I}_k}(x) = \int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x) \,,\label{eqn-bar-sI-01} \\ \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},x) &=\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^+)} \bigg( {\bf1}_{\eta^0_{-j,k} >t} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg){\bf 1}_{\mathtt{I}_k}(x)\,.\label{eqn-bar-sI-02} \end{align}
We apply Theorem \ref{thm-DD-conv-x}. We first consider the process $\bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},x)$ and show that \begin{equation} \label{eqn-bar-sI-01-conv}
\|\bar{{\mathfrak{I}}}^N_{0,1}(t,\mathfrak{a}, \cdot) - \bar{{\mathfrak{I}}}_{0}(t, \mathfrak{a}, \cdot)\|_1 \to 0, \quad \text{in probability, locally uniformly in $t$ and $\mathfrak{a}$,} \end{equation} as $N \to \infty$. We first check condition (i) of Theorem \ref{thm-DD-conv-x}.
we have \begin{align*} \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},x) - \bar{{\mathfrak{I}}}_{0}(t,\mathfrak{a},x) = \int_0^{(\mathfrak{a}-t)^+}\frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')}[\bar{{\mathfrak{I}}}^N(0,d\mathfrak{a}',x)-\bar{{\mathfrak{I}}}(0,d\mathfrak{a}',x)]\,. \end{align*}
Condition (i) of Theorem \ref{thm-DD-conv-x} follows from Lemma \ref{convPort} and Assumption \ref{AS-LLN-1}.
Next, we check condition (ii) of Theorem \ref{thm-DD-conv-x} for the processes $\bar{{\mathfrak{I}}}^{N}_{0,1}(t, \mathfrak{a},x) - \bar{{\mathfrak{I}}}_{0}(t,\mathfrak{a},x)$. We verify the condition for $\bar{{\mathfrak{I}}}^{N}_{0,1}(t, \mathfrak{a},x) $ in detail below, since the similar calculations can be done for $\bar{{\mathfrak{I}}}_{0}(t, \mathfrak{a},x) $. Namely, we show that
for any $\epsilon>0$, and for any $T, \bar{\mathfrak{a}}'>0$, as $\delta\to0$, \begin{align}
& \limsup_N \sup_{t\in [0,T]} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \|\bar{{\mathfrak{I}}}^{N}_{0,1}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) \to 0\,, \label{eqn-sI01-conv-u}\\
& \limsup_N \sup_{\mathfrak{a}\in [0,\bar\mathfrak{a}']} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \|\bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) \to 0\,. \label{eqn-sI01-conv-v} \end{align}
To prove \eqref{eqn-sI01-conv-u}, we have \begin{align*} & \bar{{\mathfrak{I}}}^{N}_{0,1}(t+u,\mathfrak{a},x) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},x) \\ &= \int_0^{(\mathfrak{a}-t-u)^+} \frac{F^c(\mathfrak{a}'+t+u)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x) - \int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x)
\,, \end{align*} and \begin{align*}
\big\| \bar{{\mathfrak{I}}}^{N}_{0,1}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},\cdot) \big\|_1 & \le \int_0^1 \int_0^{(\mathfrak{a}-t-u)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+u)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}',x)dx \\ & \qquad + \int_0^1 \int_{(\mathfrak{a}-t-u)^+}^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}',x)dx \,. \end{align*} Thus, \begin{align*}
\sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \big\| \bar{{\mathfrak{I}}}^{N}_{0,1}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},\cdot) \big\|_1 & \le \int_0^1\int_0^{(\bar\mathfrak{a}'-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}',x)dx \\ & \qquad + \sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \int_0^1 \int_{(\mathfrak{a}-t-\delta)^+}^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}',x)dx \,. \end{align*} Thanks to Lemma \ref{convPort} and Assumption \ref{AS-LLN-1}, the first term on the right converges in probability as $N \to \infty$ to \[ \int_0^1 \int_0^{(\bar\mathfrak{a}'-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}',x)dx\,, \] which converges to zero as $\delta \to 0$. It follows from the uniform convergence established in Lemma \ref{convPort} that the second term on the right converges in probability as $N\to\infty$, to \[ \sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \int_0^1\int_{(\mathfrak{a}-t-\delta)^+}^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}',x)dx \le \sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \int_0^1\int_{(\mathfrak{a}-t-\delta)^+}^{(\mathfrak{a}-t)^+} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}',x)dx\,. \] Under Assumption \ref{AS-LLN-1}, it is clear that the upper bound converges to zero at $\delta \to 0$. Thus we have shown that for $\epsilon>0$, if $\delta>0$ is small enough, \[
\limsup_N \sup_{t\in [0,T]} \mathbb{P} \bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}]} \|\bar{{\mathfrak{I}}}^{N}_{0,1}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) = 0\,. \] To prove \eqref{eqn-sI01-conv-v}, we have \begin{align*}
\bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a}+v,x) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},x) = \int_0^1 \int_{(\mathfrak{a}-t)^+}^{(\mathfrak{a}+v-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x)dx\,, \end{align*} and \begin{align*}
\sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \big\|\bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},\cdot)\|_1 \le \sup_{t \in [0,T]}\int_0^1 \int_{(\mathfrak{a}-t)^+}^{(\mathfrak{a}+\delta-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x)dx\,. \end{align*} In order to show that the $\sup_t$ on the above right hand side converges in probability, as $N\to\infty$, to \begin{align}\label{limsup}
\sup_{t \in [0,T]} \int_0^1 \int_{(\mathfrak{a}-t)^+}^{(\mathfrak{a}+v-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}', x)dx \le
\sup_{t \in [0,T]} \int_0^1 \int_{(\mathfrak{a}-t)^+}^{(\mathfrak{a}+v-t)^+} \bar{{\mathfrak{I}}}(0,d \mathfrak{a}', x)dx\,, \end{align} it suffices to show that the convergence of $\int_0^1 \int_{(\mathfrak{a}-t)^+}^{(\mathfrak{a}+\delta-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x)dx$ is uniform in $t$. Indeed, we note that \begin{align*} \int_0^1 \int_{(\mathfrak{a}-t)^+}^{(\mathfrak{a}+\delta-t)^+} &\frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x)dx\\ &=\int_0^1 \int_0^{(\mathfrak{a}+\delta-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x)dx - \int_0^{(\mathfrak{a}-t)^+}\frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0,d \mathfrak{a}', x)dx\,. \end{align*} This right hand side is the difference of two non--increasing functions of $t$ which converge pointwise to their limit in probability, as $N\to\infty$, and both limits are continuous in $t$. Hence the uniform convergence follows from the second Dini theorem, exactly as in the proof of Lemma \ref{convPort}. Going back to \eqref{limsup}, we note that,
under Assumption \ref{AS-LLN-1}, the right hand side converges to zero at $\delta \to 0$. Thus we have shown that for $\epsilon>0$, if $\delta>0$ is small enough, \[
\limsup_N \sup_{\mathfrak{a}\in [0,\bar\mathfrak{a}]} \mathbb{P} \bigg( \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \|\bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) = 0\,. \] Thus we have verified condition (ii) of Theorem \ref{thm-DD-conv-x} for the processes $\bar{{\mathfrak{I}}}^{N}_{0,1}(t, \mathfrak{a},x)$, and with a similar argument for $\bar{{\mathfrak{I}}}_{0}(t,\mathfrak{a},x)$, and thus, for the difference $ \bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},x)- \bar{{\mathfrak{I}}}_{0}(t,\mathfrak{a},x)$. Therefore, the claim on the convergence of $\bar{{\mathfrak{I}}}^{N}_{0,1}(t,\mathfrak{a},x)$ in \eqref{eqn-bar-sI-01-conv} is proved.
We next prove the convergence of $\bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},x)$: \begin{equation} \label{eqn-bar-sI-02-conv}
\|\bar{{\mathfrak{I}}}^N_{0,2}(t,\mathfrak{a}, \cdot)\|_1 \to 0, \quad \text{in probability, locally uniformly in $t$ and $\mathfrak{a}$, as $N \to \infty$.} \end{equation} To check condition (i) of Theorem \ref{thm-DD-conv-x}, we have \begin{align*}
\|\bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},\cdot)\|_1 \le \frac{1}{K^N}\sum_{k=1}^{K^N} \Bigg|\frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^+)} \bigg( {\bf1}_{\eta^0_{-j,k} >t} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg) \Bigg| \,. \end{align*} We deduce from Jensen's inequality that \begin{align} \label{eqn-bar-sI-02-conv-p1}
& \mathbb{E}\Bigg[ \Bigg( \frac{1}{K^N}\sum_{k=1}^{K^N}\frac{K^N}{N} \Bigg| \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^+)} \bigg( {\bf1}_{\eta^0_{-j,k} >t} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg)\Bigg| \Bigg)^2 \Bigg] \nonumber \\ & \le \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \mathbb{E} \Bigg[ \int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big( 1- \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big) \bar{{\mathfrak{I}}}^N_k(0, d\mathfrak{a}') \ \Bigg] \,, \end{align} where we have used the fact that the $\eta^0_{-j,k}$'s are conditionally independent, given the $\tilde{\tau}^N_{-j,k}$'s. Note that under Assumption \ref{AS-LLN-1}, thanks to Lemma \ref{convPort}, as $N\to\infty$, in probability, \begin{align*} \frac{1}{K^N}\sum_{k=1}^{K^N} &\int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big( 1- \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big) \bar{{\mathfrak{I}}}^N_k(0, d\mathfrak{a}')\\
&=\int_0^1 \int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big( 1- \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big) \bar{{\mathfrak{I}}}^N(0, d\mathfrak{a}',x)dx\\ & \to \int_0^1 \int_0^{(\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big( 1- \frac{F^c(\mathfrak{a}'+t)}{F^c(\mathfrak{a}')} \Big) \bar{{\mathfrak{I}}}(0, d\mathfrak{a}',x)dx \,. \end{align*} Thus, the upper bound in \eqref{eqn-bar-sI-02-conv-p1} converges to zero as $N\to \infty$. This implies that for any $\epsilon>0$, \[
\sup_{t \in [0,T]}\sup_{\mathfrak{a}\in [0,\bar\mathfrak{a}]} \mathbb{P} \big( \|\bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},\cdot)\|_1> \epsilon \big) \to 0 \quad\mbox{as}\quad N \to \infty. \] Next, to check condition (ii) of Theorem \ref{thm-DD-conv-x}, we show that
for any $\epsilon>0$, and for any $T, \bar{\mathfrak{a}}'>0$, as $\delta\to0$, \begin{align}
& \limsup_N \sup_{t\in [0,T]} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']}\big \|\bar{{\mathfrak{I}}}^{N}_{0,2}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},\cdot) \big\|_1 > \epsilon\bigg) \to 0\,, \label{eqn-sI02-conv-u}\\
& \limsup_N \sup_{\mathfrak{a}\in [0,\bar\mathfrak{a}]} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \big\|\bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},\cdot) \big\|_1 > \epsilon\bigg) \to 0\,. \label{eqn-sI02-conv-v} \end{align} To prove \eqref{eqn-sI02-conv-u}, we have \begin{align*} &\bar{{\mathfrak{I}}}^{N}_{0,2}(t+u,\mathfrak{a},x) - \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},x) \\
& = \sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t-u)^+)} \bigg( {\bf1}_{t <\eta^0_{-j,k} \le t+u} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)-F^c(\tilde{\tau}^N_{-j,k}+t+u)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg){\bf 1}_{\mathtt{I}_k}(x)\\ & \quad - \sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j={\mathfrak{I}}^N_k(0,(\mathfrak{a}-t-u)^+)+1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^+)} \bigg( {\bf1}_{\eta^0_{-j,k} >t} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg){\bf 1}_{\mathtt{I}_k}(x)\,, \end{align*} and \begin{align} \label{eqn-sI02-conv-u-p1}
&\big\|\bar{{\mathfrak{I}}}^{N}_{0,2}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},\cdot) \big\|_1 \nonumber \\
& \le \frac{1}{K^N}\sum_{k=1}^{K^N} \Bigg|\frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t-u)^+)} \bigg( {\bf1}_{t <\eta^0_{-j,k} \le t+u} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)-F^c(\tilde{\tau}^N_{-j,k}+t+u)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg) \Bigg| \nonumber \\
& \quad + \frac{1}{K^N} \sum_{k=1}^{K^N} \Bigg|\frac{K^N}{N} \sum_{j={\mathfrak{I}}^N_k(0,(\mathfrak{a}-t-u)^+)+1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^+)} \bigg( {\bf1}_{\eta^0_{-j,k} >t} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg) \Bigg| \nonumber \\ & \le \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t-u)^+)} {\bf1}_{t <\eta^0_{-j,k} \le t+u} \nonumber \\ & \quad + \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t-u)^+)} \frac{F^c(\tilde{\tau}^N_{-j,k}+t)-F^c(\tilde{\tau}^N_{-j,k}+t+u)}{F^c(\tilde{\tau}^N_{-j,k})} \nonumber \\ & \quad + \frac{1}{K^N} \sum_{k=1}^{K^N} \Big(\bar{{\mathfrak{I}}}^N_k(0,(\mathfrak{a}-t)^+) - \bar{{\mathfrak{I}}}^N_k(0,(\mathfrak{a}-t-u)^+) \Big) \,. \end{align} For the first term on the right, we have \begin{align} \label{eqn-sI02-conv-u-p2} & \mathbb{P}\Bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \frac{1}{K^N} \sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t-u)^+)} {\bf1}_{t <\eta^0_{-j,k} \le t+u} > \epsilon \Bigg) \nonumber\\ & \le \mathbb{P}\Bigg( \frac{1}{K^N} \sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\bar\mathfrak{a}'-t)^+)} {\bf1}_{t <\eta^0_{-j,k} \le t+\delta} > \epsilon \Bigg) \nonumber \\ & \le \mathbb{P}\Bigg( \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\bar\mathfrak{a}-t)^+)} \bigg({\bf1}_{t <\eta^0_{-j,k} \le t+\delta}-\frac{F^c(\tilde{\tau}^N_{-j,k}+t)-F^c(\tilde{\tau}^N_{-j,k}+t+\delta)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg) > \epsilon/2 \Bigg) \nonumber \\ & \quad + \mathbb{P}\Bigg( \frac{1}{K^N} \sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\bar\mathfrak{a}-t)^+)} \frac{F^c(\tilde{\tau}^N_{-j,k}+t)-F^c(\tilde{\tau}^N_{-j,k}+t+\delta)}{F^c(\tilde{\tau}^N_{-j,k})} > \epsilon/2 \Bigg)\,. \end{align} Here using Jensen's inequality and the fact that the summands over $j$ are independent, conditionally upon the $\tilde{\tau}^N_{-j,k}$'s, the first probability is bounded by \begin{align} \label{eqn-sI02-conv-u-p3} & \frac{4}{\epsilon^2} \mathbb{E} \Bigg[ \bigg( \frac{1}{K^N} \sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\bar\mathfrak{a}-t)^+)} \bigg({\bf1}_{t <\eta^0_{-j,k} \le t+\delta}-\frac{F^c(\tilde{\tau}^N_{-j,k}+t)-F^c(\tilde{\tau}^N_{-j,k}+t+\delta)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg) \bigg)^2 \Bigg] \nonumber \\ & \le \frac{K^N}{N} \frac{4}{\epsilon^2} \mathbb{E} \int_0^1 \int_0^{(\bar\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)} {F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0, d \mathfrak{a}', x) dx. \end{align} Now under Assumption \ref{AS-LLN-1}, it follows from Lemma \ref{convPort} that \begin{align*}
&\int_0^1 \int_0^{(\bar\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)} {F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0, d \mathfrak{a}',x)dx \\ & \to \int_0^1 \int_0^{(\bar\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)} {F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0, d \mathfrak{a}',x)dx \end{align*} in probability as $N \to \infty$. Hence the upper bound in \eqref{eqn-sI02-conv-u-p3} converges to zero, as $N\to\infty$. Inside the second probability in \eqref{eqn-sI02-conv-u-p2}, we have \begin{align*} & \frac{1}{K^N} \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\bar\mathfrak{a}-t)^+)} \frac{F^c(\tilde{\tau}^N_{-j,k}+t)-F^c(\tilde{\tau}^N_{-j,k}+t+\delta)}{F^c(\tilde{\tau}^N_{-j,k})} \\ &= \int_0^1 \int_0^{(\bar\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)}{F^c(\mathfrak{a}')}\bar{{\mathfrak{I}}}^N(0, d \mathfrak{a}',x)dx \\ & \to \int_0^1 \int_0^{(\bar\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)}{F^c(\mathfrak{a}')}\bar{{\mathfrak{I}}}(0, d \mathfrak{a}',x)dx \end{align*} in probability as $N \to \infty$, again from Lemma \ref{convPort}, and the limit converges to zero as $\delta\to 0$. Hence for any $\epsilon>0$, if $\delta>0$ is small enough, $\limsup_N$ of the second term in the right hand side of \eqref{eqn-sI02-conv-u-p2} is zero.
For the second term on the right of \eqref{eqn-sI02-conv-u-p1}, we have \begin{align*} &\sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}-t-u)^+)} \frac{F^c(\tilde{\tau}^N_{-j,k}+t)-F^c(\tilde{\tau}^N_{-j,k}+t+u)} {F^c(\tilde{\tau}^N_{-j,k})} \\ & \le \int_0^1 \int_0^{(\bar\mathfrak{a}'-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)} {F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}^N(0, d \mathfrak{a}',x)dx \end{align*} which, thanks to Lemma \ref{convPort} and Assumption \ref{AS-LLN-1}, converges in probability as $N\to \infty$, to \begin{align*} & \int_0^1 \int_0^{(\bar\mathfrak{a}-t)^+} \frac{F^c(\mathfrak{a}'+t)-F^c(\mathfrak{a}'+t+\delta)} {F^c(\mathfrak{a}')} \bar{{\mathfrak{I}}}(0, d \mathfrak{a}', x)dx. \end{align*} This expression will also converge to zero as $\delta \to 0$. For the third term on the right of \eqref{eqn-sI02-conv-u-p1}, we have \begin{align*}
& \sup_{u \in [0,\delta]} \int_0^1 \Big(\bar{{\mathfrak{I}}}^N(0,(\mathfrak{a}-t)^+,x) - \bar{{\mathfrak{I}}}^N(0,(\mathfrak{a}-t-u)^+,x) \Big)dx \\
& \le \int_0^1 \Big(\bar{{\mathfrak{I}}}^N(0,(\mathfrak{a}-t)^+,x) - \bar{{\mathfrak{I}}}^N(0,(\mathfrak{a}-t-\delta)^+,x) \Big) dx \end{align*} which converges in probability to \[ \int_0^1 \Big(\bar{{\mathfrak{I}}}(0,(\mathfrak{a}-t)^+,x) - \bar{{\mathfrak{I}}}(0,(\mathfrak{a}-t-\delta)^+,x) \Big) dx \] as $N\to\infty$. Since $\bar{{\mathfrak{I}}}^N(0,\cdot,x)$ and $\bar{{\mathfrak{I}}}(0,\cdot,x)$ are nondecreasing and the limit is continuous, the convergence also holds uniformly over $\mathfrak{a}\in [0,\bar\mathfrak{a}']$. Moreover, we also have that \[
\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}]} \int_0^1 \Big(\bar{{\mathfrak{I}}}(0,(\mathfrak{a}-t)^+,x) - \bar{{\mathfrak{I}}}(0,(\mathfrak{a}-t-\delta)^+,x) \Big) dx\to0, \] as $\delta \to 0$. Combining the results on the three terms on the right of \eqref{eqn-sI02-conv-u-p1}, we have shown that \eqref{eqn-sI02-conv-u} holds.
We next prove \eqref{eqn-sI02-conv-v}. We have \begin{align*} \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a}+v,x) - \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},x) = \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j={\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^++1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}+v-t)^+)} \bigg( {\bf1}_{\eta^0_{-j,k} >t+u} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg){\bf 1}_{\mathtt{I}_k}(x)\,, \end{align*} and \begin{align*}
\big\|\bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},\cdot)\|_1
&\le \frac{1}{K^N} \sum_{k=1}^{K^N} \Bigg|\frac{K^N}{N} \sum_{j={\mathfrak{I}}^N_k(0,(\mathfrak{a}-t)^++1}^{{\mathfrak{I}}^N_k(0,(\mathfrak{a}+v-t)^+)} \bigg( {\bf1}_{\eta^0_{-j,k} >t+u} -\frac{F^c(\tilde{\tau}^N_{-j,k}+t)}{F^c(\tilde{\tau}^N_{-j,k})} \bigg) \Bigg| \\
& \le \frac{1}{K^N} \sum_{k=1}^{K^N} \big|\bar{{\mathfrak{I}}}^N_k(0,(\mathfrak{a}+v-t)^+) - \bar{{\mathfrak{I}}}^N_k(0,(\mathfrak{a}-t)^+ \big| \,. \end{align*} Thus, \begin{align*}
& \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \big\|\bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{0,2}(t,\mathfrak{a},\cdot)\|_1 \\
&\le \sup_{t \in [0,T]} \frac{1}{K^N} \sum_{k=1}^{K^N} \Big(\bar{{\mathfrak{I}}}^N_k(0,(\mathfrak{a}+\delta-t)^+) - \bar{{\mathfrak{I}}}^N_k(0,(\mathfrak{a}-t)^+) \Big) \\ &=\sup_{t \in [0,T]}\int_0^1\Big(\bar{{\mathfrak{I}}}^N(0,(\mathfrak{a}+\delta-t)^+,x) - \bar{{\mathfrak{I}}}^N(0,(\mathfrak{a}-t)^+,x) \Big) dx \end{align*} and we claim that the right hand side converges in probability as $N\to\infty$, to \[
\sup_{t \in [0,T]} \int_0^1 \Big( \bar{{\mathfrak{I}}}(0,(\mathfrak{a}+\delta-t)^+,x) - \bar{{\mathfrak{I}}}(0,(\mathfrak{a}-t)^+ ,x)\Big)dx \,. \] Indeed, the convergence without the $\sup_t$ follows from Assumption \ref{AS-LLN-1}, and both $t\mapsto \int_0^1\bar{{\mathfrak{I}}}^N(0,(\mathfrak{a}+\delta-t)^+,x)dx$ and $t\mapsto \int_0^1\bar{{\mathfrak{I}}}^N(0,(\mathfrak{a}-t)^+,x)dx$ are non--increasing, while the limits are continuous. Hence again an application of the second Dini theorem implies that the convergence is locally uniform in $t$, hence the claim. The limit then converges to zero as $\delta \to 0$. Thus we have shown \eqref{eqn-sI02-conv-v}. This completes the proof of the lemma. \end{proof}
\begin{lemma} \label{lem-sIn1-conv} Under Assumptions \ref{AS-LLN-1}, \ref{AS-LLN-2} and \ref{AS-lambda}, \begin{equation}
\|\bar{{\mathfrak{I}}}^N_1(t,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}_1(t,\mathfrak{a},\cdot)\|_1 \to 0 \end{equation} in probability, locally uniformly in $t$ and $\mathfrak{a}$, as $N \to \infty$, where \begin{equation}\label{eqn-bar-sI-1}
\bar{{\mathfrak{I}}}_1(t,\mathfrak{a},x) := \int_{(t-\mathfrak{a})^+}^t F^c(t-s) \bar{A}(ds,x) \,,
\end{equation} where $\bar{A}(t,x)$ is given in \eqref{eqn-barA-tx}. \end{lemma}
\begin{proof} We first write \[ \bar{{\mathfrak{I}}}^{N}_{1}(t,\mathfrak{a},x) = \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},x) + \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},x) \] where \begin{align} \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},x) &=\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} F^c(t-\tau^N_{j,k} ){\bf 1}_{\mathtt{I}_k}(x) =\int _{(t-\mathfrak{a})^+}^t F^c(t-s) \bar{A}^N(ds,x) \,\,,\label{eqn-bar-sI-11}\\ \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},x) & =\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x)\,\,. \label{eqn-bar-sI-12} \end{align} We apply Theorem \ref{thm-DD-conv-x}. We start with the process $\bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},x)$ and show that \begin{equation} \label{eqn-bar-sI-11-conv}
\big\|\bar{{\mathfrak{I}}}^N_{1,1}(t,\mathfrak{a}, \cdot) - \bar{{\mathfrak{I}}}_{1}(t, \mathfrak{a}, \cdot)\big\|_1 \to 0, \quad \text{in probability, locally uniformly in $t$ and $\mathfrak{a}$,} \end{equation} as $N \to \infty$.
Since \begin{align*} \bar{{\mathfrak{I}}}^N_{1,1}(t,\mathfrak{a}, x) - \bar{{\mathfrak{I}}}_{1}(t, \mathfrak{a}, x) &= \int _{(t-\mathfrak{a})^+}^t F^c(t-s) \Big(\bar{A}^N(ds,x) - \bar{A}(ds,x) \Big) , \end{align*}
condition (i) of Theorem \ref{thm-DD-conv-x} follows from Lemma \ref{convPort} and Corollary \ref{coro-conv-A}. In other words, we have that for each $t$ and $\mathfrak{a}$, and for any $\epsilon>0$,
\[\mathbb{P}(\|\bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}_{1}(t,\mathfrak{a},\cdot) \|_1>\epsilon) \to 0 \quad\mbox{as}\quad N \to \infty. \]
We next want to check (ii) of Theorem \ref{thm-DD-conv-x} for the processes $\bar{{\mathfrak{I}}}^{N}_{1,1}(t, \mathfrak{a},x) - \bar{{\mathfrak{I}}}_{1}(t,\mathfrak{a},x)$.
We will verify the following conditions for $\bar{{\mathfrak{I}}}^{N}_{1,1}(t, \mathfrak{a},x) $:
for any $\epsilon>0$, and for any $T, \bar{\mathfrak{a}}'>0$, as $\delta\to0$, \begin{align}
& \limsup_N \sup_{t\in [0,T]} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \|\bar{{\mathfrak{I}}}^{N}_{1,1}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) \to 0\,, \label{eqn-sI11-conv-u}\\
& \limsup_N \sup_{\mathfrak{a}\in [0,\bar\mathfrak{a}']} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \|\bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) \to 0\,. \label{eqn-sI11-conv-v} \end{align} It will be clear that the same results hold (and are simpler to prove) for $\bar{{\mathfrak{I}}}_{1}(t,\mathfrak{a},\cdot)$. To prove \eqref{eqn-sI11-conv-u}, we have \begin{align*} & \bar{{\mathfrak{I}}}^{N}_{1,1}(t+u,\mathfrak{a},x) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},x) \\ &= \int _{(t+u-\mathfrak{a})^+}^{t+u} F^c(t+u-s) \bar{A}^N(ds,x) - \int _{(t-\mathfrak{a})^+}^t F^c(t-s) \bar{A}^N(ds,x) \\ &= \int _{(t-\mathfrak{a})^+}^{t+u} \Big(F^c(t+u-s) - F^c(t-s) \Big) \bar{A}^N(ds,x) \\ & \quad - \int _{(t-\mathfrak{a})^+}^{t+u-\mathfrak{a})^+} F^c(t+u-s) \bar{A}^N(ds,x) + \int _{t}^{t+u} F^c(t-s) \bar{A}^N(ds,x) \,, \end{align*} and \begin{align}\label{eqn-sI11-conv-u-p1}
& \big\|\bar{{\mathfrak{I}}}^{N}_{1,1}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},\cdot) \big\|_1 \nonumber\\ &\le \int_0^1 \int _{(t-\mathfrak{a})^+}^{t+u} \Big(F^c(t-s) - F^c(t+u-s) \Big) \bar{A}^N(ds,x)dx \nonumber\\ & \quad + \int _{(t-\mathfrak{a})^+}^{(t+u-\mathfrak{a})^+} F^c(t+u-s) \bar{A}^N(ds,x)dx + \int _{t}^{t+u} F^c(t-s) \bar{A}^N(ds,x)dx \,. \end{align} Here the first term on the right satisfies \begin{align*} \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} & \int_0^1\int _{(t-\mathfrak{a})^+}^{t+u} \Big(F^c(t-s) - F^c(t+u-s) \Big) \bar{A}^N(ds,x)dx \\ & \le \int_0^1\int _{(t-\bar\mathfrak{a}')^+}^{t+\delta} \Big(F^c(t-s) - F^c(t+\delta-s) \Big) \bar{A}^N(ds,x)dx \\ & \to \int_0^1 \int _{(t-\bar\mathfrak{a}')^+}^{t+\delta} \Big(F^c(t-s) - F^c(t+\delta-s) \Big) \bar{A}(ds,x)dx \end{align*} in probability as $N\to\infty$ by Lemma \ref{lem-barAn-tight} and Corollary \ref{coro-conv-A}, and the limit converges to zero as $\delta \to 0$. The second term on the right side of \eqref{eqn-sI11-conv-u-p1} satisfies \begin{align*} &\sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \int_0^1 \int _{(t-\mathfrak{a})^+}^{t+u-\mathfrak{a})^+} F^c(t+u-s) \bar{A}^N(ds,x)dx \\ & \le \sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \int_0^1 \Big( \bar{A}^N((t+\delta-\mathfrak{a})^+,x) - \bar{A}^N((t-\mathfrak{a})^+,x) \Big)dx \\ & \to \sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \int_0^1 \Big( \bar{A}((t+\delta-\mathfrak{a})^+,x) - \bar{A}((t-\mathfrak{a})^+,x) \Big)dx \end{align*} in probability as $N\to\infty$ by Corollary \ref{coro-conv-A} and the second Dini theorem, and the limit converges to zero as $\delta \to 0$.
The third term on the right side of \eqref{eqn-sI11-conv-u-p1} does not depend on $\mathfrak{a}$ and satisfies \begin{align*} &\sup_{u \in [0,\delta]} \int_0^1 \int _{t}^{t+u} F^c(t-s) \bar{A}^N(ds,x)dx \\ & \le \int_0^1\Big( \bar{A}^N(t+\delta,x) - \bar{A}^N(t,x) \Big)dx \to \int_0^1 \Big( \bar{A}(t+\delta,x) - \bar{A}(t,x) \Big)dx \end{align*} in probability as $N\to\infty$ by Corollary \ref{coro-conv-A}, and the limit converges to zero as $\delta \to 0$. Thus we have shown that for small enough $\delta>0$,
for any $\epsilon>0$, and for any $T, \bar{\mathfrak{a}}'>0$, \[
\limsup_N \sup_{t\in [0,T]} \mathbb{P} \bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \|\bar{{\mathfrak{I}}}^{N}_{1,1}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) = 0\,. \]
To prove \eqref{eqn-sI11-conv-v}, we have \begin{align*}
\bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a}+v,x) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},x) = \int _{(t-\mathfrak{a}-v)^+}^{(t-\mathfrak{a})^+} F^c(t-s) \bar{A}^N(ds,x)\,, \end{align*} and \begin{align*}
\big\| \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},\cdot) \big\|_1 = \int_0^1 \int _{(t-\mathfrak{a}-v)^+}^{(t-\mathfrak{a})^+} F^c(t-s) \bar{A}^N(ds,x)dx \,. \end{align*} Hence, \begin{align*}
& \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \big\| \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},\cdot) \big\|_1 \\ & \le \sup_{t \in [0,T]} \int_0^1 \Big( \bar{A}^N((t-\mathfrak{a})^+,x) - \bar{A}^N((t-\mathfrak{a}-\delta)^+,x) \Big) dx\\ & \to \sup_{t \in [0,T]} \int_0^1 \Big( \bar{A}((t-\mathfrak{a})^+,x) - \bar{A}((t-\mathfrak{a}-\delta)^+,x) \Big) dx\, \end{align*} in probability as $N\to\infty$ by Corollary \ref{coro-conv-A} and again the second Dini theorem. Moreover, the limit converges to zero as $\delta \to 0$. Thus we have shown that for small enough $\delta>0$,
for any $\epsilon>0$, and for any $T, \bar{\mathfrak{a}}'>0$, \[
\limsup_N \sup_{\mathfrak{a}\in [0,\bar\mathfrak{a}']} \mathbb{P} \bigg( \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \|\bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) = 0\,. \] Therefore, combining the above, we have proved the convergence of $\bar{{\mathfrak{I}}}^{N}_{1,1}(t,\mathfrak{a},x)$ as stated in \eqref{eqn-bar-sI-11-conv}.
We next consider the process $\bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},x)$ and show that \begin{equation} \label{eqn-bar-sI-12-conv}
\big\|\bar{{\mathfrak{I}}}^N_{1,2}(t,\mathfrak{a}, \cdot) \big\|_1 \to 0, \quad \text{in probability, locally uniformly in $t$ and $\mathfrak{a}$, as $N \to \infty$.} \end{equation}
To check condition (i) of Theorem \ref{thm-DD-conv-x}, we have \begin{align*}
\big\|\bar{{\mathfrak{I}}}^N_{1,2}(t,\mathfrak{a}, \cdot)\|_1
&= \frac{1}{K^N}\sum_{k=1}^{K^N} \bigg| \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big)\bigg|\,, \end{align*} and \begin{align*}
\mathbb{E} \big[ \big\|\bar{{\mathfrak{I}}}^N_{1,2}(t,\mathfrak{a}, \cdot)\|_1^2 \big]
&=\mathbb{E} \Bigg[ \Bigg( \frac{1}{K^N}\sum_{k=1}^{K^N} \bigg| \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big)\bigg| \Bigg)^2 \Bigg] \\ & \le \mathbb{E} \Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Bigg( \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big) \Bigg)^2 \Bigg] \\ & = \mathbb{E} \Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Big(\frac{K^N}{N}\Big)^2 \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} F(t-\tau^N_{j,k} ) F^c(t-\tau^N_{j,k} ) \Bigg] \\ & \le \frac{K^N}{N} \mathbb{E} \left[ \int_0^1 \int_{(t-\mathfrak{a})^+}^{t} F(t-s ) F^c(t-s ) \bar{A}^N(ds,x)dx \right]\, . \end{align*} By Corollary \ref{coro-conv-A} and Lemma \ref{convPort}, we obtain the convergence \[ \int_0^1 \int_{(t-\mathfrak{a})^+}^{t} F(t-s ) F^c(t-s ) \bar{A}^N(ds,x)dx \to \int_0^1 \int_{(t-\mathfrak{a})^+}^{t} F(t-s ) F^c(t-s )\bar{A}(ds,x)dx
\]
in probability as $N\to \infty$.
This implies that for any $\epsilon>0$, \[
\sup_{t \in [0,T]}\sup_{\mathfrak{a}\in [0,\bar\mathfrak{a}']} \mathbb{P} \big( \|\bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},\cdot)\|_1> \epsilon \big) \to 0 \quad\mbox{as}\quad N \to \infty. \] Next, to check condition (ii) of Theorem \ref{thm-DD-conv-x}, we need to show that
for any $\epsilon>0$, and for any $T, \bar{\mathfrak{a}}'>0$, as $\delta\to0$, \begin{align}
& \limsup_N \sup_{t\in [0,T]} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \|\bar{{\mathfrak{I}}}^{N}_{1,2}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) \to 0\,, \label{eqn-sI12-conv-u}\\
& \limsup_N \sup_{\mathfrak{a}\in [0,\bar\mathfrak{a}']} \frac{1}{\delta} \mathbb{P} \bigg( \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \|\bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},\cdot) \|_1 > \epsilon\bigg) \to 0\,. \label{eqn-sI12-conv-v} \end{align} To prove \eqref{eqn-sI12-conv-u}, we have \begin{align*} & \bar{{\mathfrak{I}}}^{N}_{1,2}(t+u,\mathfrak{a},x) - \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},x) \\ & = \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t+u-\mathfrak{a})^+)+1}^{A^N_k(t+u)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t+u} - F^c(t+u-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x) \\ & \qquad - \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x) \\ &= \sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t+u} - F^c(t+u-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x) \\ & \qquad - \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k((t+u-\mathfrak{a})^+)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t+u} - F^c(t+u-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x) \\ & \qquad - \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x) \\ & \qquad + \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k(t)+1}^{A^N_k(t+u)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x)\\ & = -\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} \Big( {\bf1}_{t <\tau^N_{j,k} + \eta_{j,k} \le t+u} - \big( F^c(t-\tau^N_{j,k} )-F^c(t+u-\tau^N_{j,k} )\big)\Big){\bf 1}_{\mathtt{I}_k}(x) \\ & \qquad - \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k((t+u-\mathfrak{a})^+\wedge t)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t+u} - F^c(t+u-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x) \\ & \qquad + \sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k(t)+1}^{A^N_k(t+u)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x)\,. \end{align*}
Thus we obtain
\begin{align}\label{eqn-sI12-conv-u-p1}
& \big\|\bar{{\mathfrak{I}}}^{N}_{1,2}(t+u,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},\cdot) \big\|_1 \nonumber \\
& \le \frac{1}{K^N}\sum_{k=1}^{K^N} \bigg| \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} \Big( {\bf1}_{t <\tau^N_{j,k} + \eta_{j,k} \le t+u} - \big( F^c(t-\tau^N_{j,k}) - F^c(t+u-\tau^N_{j,k} )\big)\Big)\bigg| \nonumber \\
& \quad + \frac{1}{K^N} \sum_{k=1}^{K^N} \bigg| \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+\wedge t)+1}^{A^N_k((t+u-\mathfrak{a})^+} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t+u} - F^c(t+u-\tau^N_{j,k} )\Big) \bigg| \nonumber \\
& \quad +\frac{1}{K^N} \sum_{k=1}^{K^N} \bigg| \frac{K^N}{N} \sum_{j=A^N_k(t)+1}^{A^N_k((t+u)^+\wedge t)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big)\bigg| \nonumber \\ & \le \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} {\bf1}_{t <\tau^N_{j,k} + \eta_{j,k} \le t+u} \nonumber \\ & \quad+ \frac{1}{K^N}\sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} \big( F^c(t-\tau^N_{j,k} ) - F^c(t+u-\tau^N_{j,k} )\big) \nonumber \\ & \quad+ \frac{1}{K^N} \sum_{k=1}^{K^N} \big( \bar{A}^N_k(t+u) - \bar{A}^N_k(t) \big)+ \frac{1}{K^N} \sum_{k=1}^{K^N} \big( \bar{A}^N_k((t+u-\mathfrak{a})^+) - \bar{A}^N_k( (t-\mathfrak{a})^+) \big) \,. \end{align} For the first term on the right, we have \begin{align*}
& \mathbb{E}\Bigg[ \Bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} {\bf1}_{t <\tau^N_{j,k} + \eta_{j,k} \le t+u} \Bigg)^2 \Bigg] \\
& \le \mathbb{E}\Bigg[ \Bigg( \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\bar\mathfrak{a}')^+)+1}^{A^N_k(t+\delta)} {\bf1}_{t <\tau^N_{j,k} + \eta_{j,k} \le t+\delta} \Bigg)^2 \Bigg] \\
& \le \mathbb{E}\Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Bigg( \frac{K^N}{N} \sum_{j=A^N_k((t-\bar\mathfrak{a}')^+)+1}^{A^N_k(t+\delta)} {\bf1}_{t <\tau^N_{j,k} + \eta_{j,k} \le t+\delta} \Bigg)^2 \Bigg] \\
& \le 2 \mathbb{E}\Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Bigg( \frac{K^N}{N} \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta} \int_0^\infty \int_{t-s}^{t+\delta-s} {\bf1}_{r\le \Upsilon^N(s^-)} \overline{Q}_{k,\ell}(ds,dr,dz) \Bigg)^2 \Bigg] \\
& \quad + 2 \mathbb{E}\Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Bigg( \frac{K^N}{N} \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta}
(F(t+\delta-s)-F(t-s)) \Upsilon^N_k(s) ds \Bigg)^2 \Bigg] \\
& = 2 \frac{K^N}{N} \mathbb{E}\Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta}
(F(t+\delta-s)-F(t-s)) \bar{\Upsilon}^N_k(s) ds \Bigg] \\
& \quad + 2 \mathbb{E}\Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Bigg( \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta}
(F(t+\delta-s)-F(t-s)) \bar{\Upsilon}^N_k(s) ds \Bigg)^2 \Bigg] \\
& \le 2 \lambda^*C_B C_\beta \frac{K^N}{N} \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta}
(F(t+\delta-s)-F(t-s)) ds \\
& \quad + 2 (\lambda^*C_BC_\beta)^2 \Bigg( \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta}
(F(t+\delta-s)-F(t-s)) ds \Bigg)^2\,, \end{align*} where $Q_{k,\ell}(ds,dr,dz)$ is the PRM on ${\mathbb R}_+^3$ with mean measure $dsdrF(dz)$ already introduced in the proof of Lemma \ref{lem-mfN1-conv}, and $\overline{Q}_{k,\ell}(ds,dr,dz)$ is the corresponding compensated PRM, and we have used the bound $\bar\Upsilon^N_k(t)\le \lambda^*C_BC_\beta$. The first term on the right goes to zero as $N\to\infty$, and the integral in the second is bounded from above by \[ \int_{0}^{t} \big( F(s+\delta) - F(s) \big) ds\le\delta\,,\] as in \eqref{delta} above. Thus we obtain that for any $\epsilon>0$, as $\delta\to0$, \begin{align*}
& \limsup_N \sup_{t\in [0,T]} \frac{1}{\delta} \mathbb{P}\Bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} {\bf1}_{t <\tau^N_{j,k} + \eta_{j,k} \le t+u} >\epsilon \Bigg) \to 0\,.
\end{align*} For the second term on the right side of \eqref{eqn-sI12-conv-u-p1}, we have \begin{align*}
& \mathbb{E}\Bigg[ \Bigg( \sup_{u \in [0,\delta]}\sup_{\mathfrak{a} \in [0,\bar\mathfrak{a}']} \frac{1}{K^N}\sum_{k=1}^{K^N} \frac{1}{B^N_k} \sum_{j=A^N_k((t-\mathfrak{a})^+)+1}^{A^N_k(t+u)} \big( F^c(t-\tau^N_{j,k} ) - F^c(t+u-\tau^N_{j,k} )\big) \Bigg)^2 \Bigg] \\
& \le \mathbb{E}\Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Bigg(\frac{K^N}{N} \sum_{j=A^N_k((t-\bar\mathfrak{a}')^+)+1}^{A^N_k(t+\delta)} \big( F^c(t-\tau^N_{j,k} ) - F^c(t+\delta-\tau^N_{j,k} )\big) \Bigg)^2 \Bigg] \\
& \le 2 \mathbb{E}\Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \Big( \frac{K^N}{N}\Big)^2 \bigg( \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta} \big( F^c(t-s) - F^c(t+\delta-s)\big) d M^N_{A,k}(s) \bigg)^2 \Bigg] \\
& \quad +2 \mathbb{E}\Bigg[ \frac{1}{K^N}\sum_{k=1}^{K^N} \bigg( \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta} \big( F^c(t-s) - F^c(t+\delta-s)\big) \bar{\Upsilon}^N_k(s) ds \bigg)^2 \Bigg] \\
& = 2 \lambda^*C_BC_\beta \frac{K^N}{N} \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta} \big( F^c(t-s) - F^c(t+\delta-s)\big)^2 ds \\
& \quad +2 (\lambda^*C_BC_\beta)^2 \bigg( \int_{(t-\bar\mathfrak{a}')^+}^{t+\delta} \big( F^c(t-s) - F^c(t+\delta-s)\big) ds \bigg)^2\,.
\end{align*} It is clear that the first term converge to zero locally uniformly in $t$, and the second term can be treated in the same way above. The third and fourth terms on the right side of \eqref{eqn-sI12-conv-u-p1} can be also treated similarly as the last two terms in \eqref{eqn-sI11-conv-u-p1}. Thus, we have shown that \eqref{eqn-sI12-conv-u} holds.
To prove \eqref{eqn-sI12-conv-v}, we have \begin{align*}
\bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a}+v,x) - \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},x) = \sum_{k=1}^{K^N}\frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a}-v)^+)+1}^{A^N_k((t-\mathfrak{a})^+)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big){\bf 1}_{\mathtt{I}_k}(x) \,, \end{align*} and \begin{align*}
\big\|\bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},\cdot)\|_1
& \le \frac{1}{K^N} \sum_{k=1}^{K^N} \bigg| \frac{K^N}{N} \sum_{j=A^N_k((t-\mathfrak{a}-v)^+)+1}^{A^N_k((t-\mathfrak{a})^+)} \Big( {\bf1}_{\tau^N_{j,k} + \eta_{j,k} >t} - F^c(t-\tau^N_{j,k} )\Big)\bigg|\\ & \le \int_0^1 \Big(\bar{A}^N((t-\mathfrak{a})^+,x) - \bar{A}^N((t-\mathfrak{a}-v)^+,x) \Big)dx\,. \end{align*} Then, we obtain \begin{align*}
& \sup_{v \in [0,\delta]}\sup_{t \in [0,T]} \big\|\bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a}+v,\cdot) - \bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},\cdot)\|_1 \\
& \qquad \le \sup_{t \in [0,T]} \int_0^1 \Big(\bar{A}^N((t-\mathfrak{a})^+,x) - \bar{A}^N((t-\mathfrak{a}-\delta)^+,x) \Big)dx\,. \end{align*} Here the upper bound converges in probability to \[ \sup_{t \in [0,T]}\int_0^1 \Big(\bar{A}((t-\mathfrak{a})^+,x) - \bar{A}((t-\mathfrak{a}-\delta)^+,x) \Big)dx \] which converges to zero as $\delta \to 0$, uniformly in $\mathfrak{a}$. Indeed, the convergence of the $\sup_t$ follows from the fact that the convergence in probability $\int_0^1\bar{A}^N(t,x)dx\to\int_0^1\bar{A}(t,x)dx$ is locally uniform in $t$, thanks to Corollary \ref{coro-conv-A}. Thus we have proved \eqref{eqn-sI12-conv-v} holds, and hence, the convergence of $\bar{{\mathfrak{I}}}^{N}_{1,2}(t,\mathfrak{a},x)$ in \eqref{eqn-bar-sI-12-conv}.
This completes the proof of the lemma. \end{proof}
By the two lemmas above, we can conclude the convergence of $\bar{{\mathfrak{I}}}^N(t,\mathfrak{a},x)$ to $\bar{{\mathfrak{I}}}(t,\mathfrak{a},x)$.
\begin{prop} \label{prop-sIn-conv} Under Assumptions \ref{AS-LLN-1}, \ref{AS-LLN-2} and \ref{AS-lambda}, \begin{equation}
\|\bar{{\mathfrak{I}}}^N(t,\mathfrak{a},\cdot) - \bar{{\mathfrak{I}}}(t,\mathfrak{a},\cdot)\|_1 \to 0 \end{equation} in probability, locally uniformly in $t$ and $\mathfrak{a}$, as $N \to \infty$, where $ \bar{{\mathfrak{I}}}(t,\mathfrak{a},x) = \bar{{\mathfrak{I}}}_0(t,\mathfrak{a},x) + \bar{{\mathfrak{I}}}_1(t,\mathfrak{a},x) $,
$\bar{{\mathfrak{I}}}_0$ and $\bar{{\mathfrak{I}}}_1$ being given respectively by \eqref{eqn-bar-sI-0} and \eqref{eqn-bar-sI-1}. \end{prop}
{\bf Completing the proof of Theorem \ref{thm-FLLN}}. Given the results in Propositions \ref{prop-conv-S-mfF} and \ref{prop-sIn-conv} and Corollary \ref{coro-conv-A}, the convergence of $\bar{R}^N(t,x)$ and $\bar{I}^N(t,x)$ can be easily established and their limits $\bar{R}(t,x)$ and $\bar{I}(t,x)$ follows directly. The second expression of $\bar\Upsilon(t,x)$ in \eqref{eqn-barUpsilon-tx} is obtained from $\bar{{\mathfrak{I}}}(t,\mathfrak{a},x)$ in \eqref{eqn-barsI-tax}, by noting that $\bar{\mathfrak{I}}_a(t,0,x) = \lim_{\mathfrak{a}\to0} \frac{ \bar{\mathfrak{I}}(t,\mathfrak{a},x) - \bar{\mathfrak{I}}(t,0,x)}{\mathfrak{a}}$.
\end{document} | arXiv |
\begin{definition}[Definition:Unbiased Estimator]
Let $\theta$ be a population parameter of some statistical model.
Let $\delta$ be an estimator of $\theta$.
We call $\delta$ an '''unbiased estimator''' if its bias is equal to $0$ regardless of the true value of $\theta$.
\end{definition} | ProofWiki |
\begin{definition}[Definition:Negative/Complex Number]
As the Complex Numbers cannot be Ordered Compatibly with Ring Structure, the concept of a '''negative complex number''', relative to a specified zero, is not defined.
However, the '''negative''' of a complex number is defined as follows:
Let $z = a + i b$ be a complex number.
Then the '''negative of $z$''' is defined as:
:$-z = -a - i b$
\end{definition} | ProofWiki |
Vertical (Dis‑)Integration and Firm Performance: A Management Paradigm Revisited
Florian Kaiser1 &
Robert Obermaier ORCID: orcid.org/0000-0001-6162-10601
Schmalenbach Business Review volume 72, pages 1–37 (2020)Cite this article
Vertical disintegration in manufacturing industries has been an increasing trend since the 1990s in many countries. According to a prevailing management paradigm of focusing on core competencies, firms should have vertically disintegrated (i.e. outsourced non-core competencies) to achieve cost savings, enhance competitiveness and improve firm performance. In line with this management paradigm, most empirical studies therefore hypothesized a negative linear relationship between the degree of vertical integration and firm performance, expecting performance to rise when vertical integration decreases.
In contrast to previous studies, finding mixed results, we assume an inverted u‑shaped relationship, theoretically based on transaction cost economics and the resource-based view of the firm, and by considering advantages and disadvantages of vertical integration, with an optimal level of vertical integration, where firms with a too low degree of vertical integration could achieve higher performance by vertical integration, while firms with too broad vertical integration could achieve higher performance by vertical disintegration.
With respect to our data based on a sample of 434 German manufacturing firms between 1993 and 2013 we find a decreasing trend of vertical integration over time. Applying multiple regression analysis, our findings suggest a positive, but diminishing relationship between the degree of vertical integration and financial performance. These two findings describe a paradox of vertical disintegration. The decreasing trend mainly emerges because lower performing firms outsourced their activities significantly whereas high performing firms do not show such a development. Overall, our results indicate that German manufacturing firms might have gone too far in in their vertical disintegration strategy by following a management paradigm which needs much more critical reflection.
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Vertical disintegration has been an influential management paradigm and an empirically detectable business trend in the manufacturing industry during the last decades which has been accompanied by the concept of supply chain management—both seen as key drivers for the financial performance of firms (Shi and Yu 2013; Otto and Obermaier 2009).
In stark contrast, during the beginning of industrialization "owning the value chain" and a high degree of vertical integration has been a predominant strategy over decades (Harrigan 1984). A classic example is Henry Ford's River Rouge complex, built during the late 1920s, with a vertical integration of processes from coal and iron ore mines, timberlands and rubber plantations to the final assembly of the car, resulting in total control over the entire supply chain. This formerly successful strategy of high vertical integration was pushed out of fashion in the recent decades when firms realized that a concentration on core competencies and outsourcing of non-core activities has advantages as well. Hence, vertical disintegration became a prevailing management paradigm in practice (Welch and Nayak 1992).
However, recent literature indicates that vertical disintegration strategies often failed to reach the expected performance improvements. For instance, Rigby and Bilodeau (2015) analyze the usage-satisfaction relationship among different management tools and show that outsourcing in particular is on the one hand widely used but on the other hand dissatisfies managers most when asked about the results of outsourcing decisions. Furthermore, a rich body of empirical literature is interested in the performance implications of vertical integration strategies (see Lahiri (2016) for an overview). However, the results so far are inconclusive; i.e. some studies found a negative (e.g. Rumelt 1982; D'Aveni and Ilinitch 1992; Desyllas 2009), others detect positive (e.g. Novak and Stern 2008; Broedner et al. 2009) or insignificant relationships (e.g. Levin 1981).
Hence, there seem to be some striking research gaps that we aim to address in our study. First, from a methodological point of view prior research predominantly hypothesizes and investigates a linear relationship (expecting it to be negative) between vertical integration (or disintegration) and firm performance although "many intuitively appealing arguments have been offered both for and against outsourcing as a means of achieving sustainable competitive advantage" (Gilley and Rasheed 2000, p. 763). Although there are a few exceptions that investigate non-linear relationships (e.g. Rothaermel et al. 2006; Kotabe and Mol 2009), none of these studies is checking the robustness of the functional form. This is particularly important as recent literature shows that the reported findings may be spurious (Haans et al. 2016, p. 1178). Second, from a material point of view existing studies that investigate the relationship between vertical integration and financial performance do not analyze how their results are related to a decreasing degree of vertical integration in manufacturing industries during the last decades. Such research is particularly necessary because firms usually expect their performance to be increased by vertical disintegration. But as far as the vertical integration-performance relationship is non-linear, decreasing vertical integration might become a detrimental strategy under certain circumstances. Furthermore, by simply assuming a (negative) linear relationship and therefore decreasing vertical integration in order to achieve expected performance gains without empirical knowledge about the underlying assumption could lead into a vertical disintegration paradox. Third, knowledge about the relationship between vertical integration and firm performance for German manufacturing firms is scarce but essential, as Germany in general is known for a high manufacturing share relative to GDP and German manufacturing firms in particular are known for a relatively high level of vertical integration compared to other countries while being highly competitive (Obermaier 2019). While most of the existing studies focus on US samples, we are deeply convinced that it is especially important to better understand manufacturing firms in a major economy with a strong manufacturing sector and highly competitive firms and therefore expect fruitful insights from this obvious research gap. Thus, it is our goal to address these research gaps by intensively analyzing the relationship between vertical integration and financial performance as well as the trend of vertical disintegration strategies, using a large sample of German manufacturing firms between the years 1993 and 2013. Correspondingly, we formulate as our fundamental research question: What is the relationship between the degree of vertical integration and the financial performance of German manufacturing firms between 1993 and 2013?
We make several contributions to existing research. First, we expand the understanding of the relationship between the degree of vertical integration and financial performance on the one side and the decreasing trend of vertical integration during the last decades on the other side by showing that a lower (higher) degree of vertical integration decreases (increases) financial performance of German firms in the recent past. Thus, we contribute to the vertical integration literature by providing and discussing reasons for a so called vertical disintegration paradox, which is a decreasing degree of vertical integration although this strategy may have affected firms' financial performance in a deleterious way. Second, we theoretically and empirically shed light on the hitherto often neglected inverted u‑shaped relationship between the degree of vertical integration and financial performance. Prior empirical research on vertical integration instead has solely considered a linear relationship and focused either on transaction cost economics or the resource-based view to theorize and explain the results. By considering both, transaction cost economics and the resource-based view, our findings indicate that for German manufacturing firms the advantages of vertical integration outweigh the advantages of vertical disintegration. Third, by investigating a sample of German manufacturing firms, we provide results for a major European economy with a very strong manufacturing sector and a striking decreasing degree of vertical integration during the last decades. Prior research has focused on US firms, i.e. insights for other major economies are scarce. Our findings of a positive relationship between vertical integration and financial performance indicate that the vertical disintegration strategy of German manufacturing firms, mainly motivated by lower labor costs in low-cost countries (Sinn 2005), either have outsourced too much of their activities or have not been able to realize the benefits they desired.
The structure of this study is organized as follows: Sect. 2 gives an overview of the underlying theory developing our core hypothesis and the relevant literature at hand. In Sect. 3 the research methodology is described. The results of the analysis are presented in Sect. 4 and discussed in Sect. 5. The study concludes with a summary of key findings and further research opportunities.
Theory and Hypothesis
Vertical integration, defined as "the combination, under a single ownership, of two or more stages of production or distribution (or both) that are usually separate" (Buzzell 1983, p. 93) and vertical disintegration, defined as "the emergence of new intermediate markets that divide a previously integrated production process between two sets of specialized firms in the same industry" (Jacobides 2005, p. 465) are classical issues for researchers and practitioners. As in prior studies, we use the concepts of vertical disintegration and outsourcing synonymously although they may slightly differ (e.g. Broedner et al. 2009; Desyllas 2009). Further, the terms "vertical integration" and "degree of vertical integration" are used interchangeable throughout this study.
Transaction costs economics (TCE)—pioneered by Coase (1937) and further developed principally by Williamson (1971, 1975, 1991b)—and the resource-based view (RBV) of the firm (Penrose 1959; Wernerfelt 1984; Barney 1991) have made seminal contributions to our understanding of the existence of firms in general and make-or-buy decisions as well as vertical (dis-)integration in special. The core issue of TCE is that utilizing the market system is not for free as it causes costs for using it (so called transaction costs). Hence, the transaction costs of firm activities via market transactions have to be compared with the costs of internalization activities (i.e. vertical integration) and transactions should accordingly only be undertaken within that institutional arrangement (market or firm) which causes the lowest costs. According to the RBV, vertical integration is mainly influenced by the competitive advantage a firm has in a particular stage of the value chain relative to the market (Jacobides and Hitt 2005; Jacobides and Winter 2005). This competitive advantage is a result of a firm's predominant resources and capabilities which arise from a unique, path-dependent learning process (Levinthal 1997; Jacobides and Winter 2005). According to Barney (1991), resources and capabilities lead to competitive advantage if they are valuable, rare, difficult to imitate and non-substitutable. In sum, TCE and RBV provide complementary explanations for the decisions whether or not a firm should change its degree of vertical integration (see McIvor (2009) for an overview).
The literature reviewed so far summarizes the determinants of the degree of vertical integration, which result in a bundle of advantages and disadvantages (Table 1).
Table 1 Advantages and disadvantages of vertical integration
It should be emphasized, that the advantages of vertical integration can be seen as disadvantages of vertical disintegration (respectively outsourcing) et vice versa.
These can be related to operational performance (e.g. inventory scheduling), intangibles assets (e.g. product quality) or to financial performance (e.g. revenues and costs), (Buzzell 1983; Harrigan 1984; Stuckey and White 1993; D'Aveni and Ravenscraft 1994). Of course, as operational performance and intangibles affect financial performance, the degree of vertical integration is not only directly related to financial performance but also indirectly.
A range of arguments can be applied to support a positive relationship between vertical integration and firm performance. Operational performance is improved through providing higher quality standards and having more control over input quality (D'Aveni and Ravenscraft 1994). Furthermore, vertical integration is often viewed as a strategy to increase supply assurance of critical materials and improve coordination between different stages of production (Buzzell 1983; Harrigan 1984), i.e. coordination between production, inventory and logistics scheduling is improved. Consequently, vertical integration affects operational efficiency as it improves throughput of materials and information along the supply chain resulting in lower lead times and higher delivery performance.
A higher degree of vertical integration can also help to build intangible assets which, in turn, affect financial performance as they are traditionally perceived to be the basis of competitive advantage (Dierickx and Cool 1989; Barney 1991). Based on higher operational performance, improved delivery performance and lower lead time should result in higher customer satisfaction. Further, among other things, vertical integration creates credibility for new products (Harrigan 1984) and provides protection of proprietary products or process technology (Mahoney 1992) and is thus consistent with the resource-based view. Further arguments concern a firm's market power which is increased by building market entry barriers and price discrimination (Perry 1978; Stuckey and White 1993). Higher market entry barriers and price discrimination should increase firms' revenues and profits.
The positive impact of a higher degree of vertical integration on financial performance is usually explained with cost savings. These cost savings are mainly related to lower transaction costs associated with less dependency on external suppliers. A higher degree of vertical integration could reduce the cost of searching, negotiating, drawing up a contract, monitoring and enforcement costs with external suppliers (Mahoney 1992). Besides transaction costs, vertical integration leads to cost savings achieved by improved coordination of production or by eliminating steps, reducing duplicate overhead costs (Buzzell 1983; Harrigan 1984).
However, it is argued that vertical integration is only beneficial to financial performance up to a certain point. Beyond that point, a higher degree of vertical integration could have detrimental effects on financial performance. A first group of arguments is concerned with additional costs that are associated with an excessively high degree of vertical integration, which consist of production, agency and coordination costs (Bettis et al. 1992; D'Aveni and Ravenscraft 1994; Desyllas 2009). The simultaneous coordination of a large number of activities and the underutilization of capacities in some stages of production (D'Aveni and Ravenscraft 1994; Harrigan 1985) could increase production costs. A higher degree of vertical integration leads to less efficient utilization of different stages of production which increases unit cost (Mahoney 1992). Further sources of production cost disadvantages are higher capital requirements and capital lockups (Mahoney 1992), higher fixed costs that lead to higher operating leverage and to a higher break-even point (Gilley and Rasheed 2000). Highly integrated firms bear the risk that they focus on additional non-core operations. This may result in information deficits among corporate-level managers due to information asymmetries about non-core activities (D'Aveni and Ilinitch 1992). Moreover, changing technology or market conditions which make products obsolete in one stage of a vertically integrated firm are key drivers of reduced flexibility and exit barriers (Buzzell 1983). A higher degree of vertical integration then reduces strategic flexibility with respect to environmental changes by switching to suppliers with newer and better technologies (Balakrishnan and Wernerfelt 1986; Gilley and Rasheed 2000; Mahoney 1992).
Therefore, to decide which activities should be integrated or outsourced are fundamental decisions for a firm, i.e. choosing between market or hierarchy or something in between (hybrid) and thereby minimizing transaction costs (Williamson 1991a). Based on TCE, such transactions should be internalized (i.e. vertically integrated) that are characterized by a high degree of asset specificity and uncertainty accompanied by a high degree of frequency (Picot and Franck 1993). Otherwise, a firm should choose the market or a hybrid form. The relationship between asset specificity and transaction costs is shown in Fig. 1.
Asset specificity, transaction costs and structural form. (Source: Williamson 1991a)
Most of a firm's activities are characterized by a different degree of asset specificity. If a firm decides to integrate (or outsource) all of these activities, then the level of transaction costs would not be as low as possible, as some activities should be outsourced (those characterized by low asset specificity) while others should be internalized (those characterized by high asset specificity).Footnote 1
This is in line with RBV after what a firm should outsource its non-core activities and concentrate on core competencies, which is intended to result in a competitive advantage and higher financial performance. Hence, a missing focus on activities as well as vertical disintegration of all activities would lower performance.
Based on both theories, an optimal degree of vertical integration can be assumed which to achieve is the result of firm specific decisions. The adequate strategy to reach the optimal degree of vertical integration, i.e. whether a higher or a lower degree of vertical scope would be profitable, depends on a firm's initial position. Fig. 2 illustrates this relationship.
Hypothesized relationship between the degree of vertical integration and firm performance
If a firm starts in position A, then the degree of vertical integration is below the optimum. In this case, a firm is not sufficiently vertically integrated, i.e. the degree of vertical integration is too low (in other words the firm uses the market although vertical integration would be beneficial). Hence, the advantages of higher vertical integration would outbalance the disadvantages and an increase would improve performance in that situation. The opposite is true if a firm's integration-performance starting point would be point B. The initial level of vertical scope is too high and the firm conducts core and non-core activities simultaneously or uses integration instead of using the market. Thus, the concentration on core competencies or using the market increases firm performance. Once the optimum is reached (position C) deviations from that optimal level would lower performance.
Based on these arguments, we state our general hypothesis:
The relationship between the degree of vertical integration and a firm's financial performance follows an inverted u‑shape, describing a positive relationship up to its optimum, while describing a negative relationship beyond that point.
Somewhat surprisingly many of the empirical studies at hand only hypothesize and investigate a linear relationship between the degree of vertical integration and firm performance assuming either improvements of firm performance through integration or disintegration, although most of them are based on TCE and the RBV which should result in hypothesizing an inverted u‑shaped relationship as we tried to justify, when advantages as well as the disadvantages of vertical integration are considered simultaneously.
We further expect that this might be one reason why existing empirical results show a mixed picture: some studies find a negative linear relationship (e.g. Rumelt 1982; D'Aveni and Ilinitch 1992; Desyllas 2009) others find the relationship to be positive linear (e.g. Buzzell 1983; Harrigan 1986; D'Aveni and Ravenscraft 1994; Novak and Stern 2008; Broedner et al. 2009). Only a few studies hypothesize and investigate a curvilinear relationship between vertical integration and firm performance (e.g. Rothaermel et al. 2006; Kotabe and Mol 2009). In summarizing the literature on vertical integration and firm performance also Lahiri (2016) concludes that empirical findings are inconclusive, which further motivates our endeavor.
Our research is focused on the German manufacturing sector. Interestingly its share of value-added in percent of the GDP has been nearly constant over the last decades (mean =23%) and is considerably higher than it has been in other major economies as shown in Fig. 3. Furthermore and in contrast to Germany, manufacturing firms in the European Union (mean =18%) and USA (mean =14%) show a decreasing trend of value-added in percent of the GDP since 1997.
Share of value-added of manufacturing industries in % of GDP (Data Source: The World Bank; U.S. Bureau of Economic Analysis)
All data used for the empirical analysis of German corporations in the manufacturing sector were taken from Thomson Reuters Datastream. In some cases, firms' annual financial reports serve as data base because manual corrections of the data was required due to false figures or because the required data were not available via Thomson.
The chosen sample covers the time frame from 1993 to 2013. The beginning of the time frame was chosen due to data availability. 2013 represents the last year for which full information was available at the beginning of the data collection. Only complete data sets were reprocessed, i.e. independent as well as dependent variables had to be available. Considering the data criteria mentioned above the sample covers 434 different firms and 3848 firm years.
The firms in the sample belong to the Standard Industrial Classification (SIC) manufacturing division. The sample distribution based on two-digit SIC codes is shown in Table 2. The three most represented manufacturing industry sectors are machinery (SIC35), electronics (SIC36) and chemicals (SIC28).
Table 2 Sample distribution over two-digit SIC Codes
Measurement of Vertical Integration
The measurement of vertical integration has been widely discussed in literature (e.g. Adelman 1955; Laffer 1969; Maddigan 1981; Lindstrom and Rozell 1993). On the one hand, there are a number of measures which can be easily calculated based on financial statements. On the other hand, there are multidimensional constructs which require primary data to be calculated. Lindstrom and Rozell (1993) prove inconsistencies among existing measures.
One of the most used measurement approaches might be the value-added to sales (VAS) approach. It is implemented in various studies (Stigler 1951; Adelman 1955; Desyllas 2009; Hutzschenreuter and Gröne 2009; see Lajili et al. (2007) for a survey of studies).
According to VAS, vertical integration is calculated as value-added divided by sales. In order to achieve VAS there are two possible ways to calculate value-added: the first way is the so-called subtractive method. Thereby, value added is determined as the difference between output and input in value terms and expresses the value an economic entity adds to the goods and services received from other entities through own activities. The second way is the so called additive method which sums up all allocated parts of the created wealth, i.e. all expenditures without input character.
Value-added according to the subtractive method is defined as (sales–external purchases). An increase (decrease) of VAS implies that the share of external purchases falls (rises) relative to sales. This can be seen as an indicator for a change in the degree of vertical integration, i.e. an increase (decrease) of VAS is related to an extension (withdrawal) of a firm's upstream or downstream activities in the value chain which leads to an increase (reduction) of a firm's value-added (measured as sales minus external purchases) compared to external purchases. Backward integration will tend to reduce the amount of external purchases while leaving sales constant whereas forward integration will tend to increase sales more than external purchases (Tucker and Wilder 1977). Both backward and forward integration result in an increase of VAS. In general, two extreme cases are imaginable: a fully integrated firm which consequently has a VAS quotient of 1 and a totally dis-integrated firm that has a VAS quotient of 0. A fully integrated firm does not need any external purchases to produce an output. Therefore, VAS would be calculated as (Sales −0) / Sales =1. In contrast, a totally dis-integrated firm does not produce any output; it only deals with its external purchases, i.e. external purchases are equal to output (sales) and value-added is reduced to 0. Consequently VAS is 0.
In our study we will measure the degree of vertical integration with the VAS ratio, due to its straight forward way of calculating and interpreting the ratio with readily available accounting data. But as the coverage of external purchases in Thomson Reuters Datastream is very poor, value-added is calculated by the additive method, i.e. as the sum of salaries and benefit expense, income taxes, interest expense on debt, dividends and net income.
Another widely used measurement approach of vertical integration is the input-output approach which utilizes national input-output tables and has been implemented in a number of studies (see Lajili et al. (2007)). Maddigan's (1981) Vertical Industry Connection (VIC) index was one of the first measures of this category. The VIC index assumes that a firm operates in more than one industry and considers that firms of one industry might be simultaneously suppliers and buyers of another industry. The major disadvantage of this approach is the assumption that aggregated national input-output tables are applicable to individual firms (Hutzschenreuter and Gröne 2009; see also for further disadvantages Lindstrom and Rozell (1993)). Besides Harrigan's VIC index, there exist other measures based on input-output tables (e.g. Fan and Lang 2000).
Adelman (1955) suggests the inventory to sales ratio to measure the degree of vertical integration. He argues that "The longer the production line and the more successive processes are operated by one firm, the higher the ratio" (p. 283) whereas the measure could be improved by using work-in-process only. However, the major disadvantage of this measure is that inventory level is also influenced by other factors than vertical integration, mainly different production methods and different manufacturing processes across industries, i.e. a comparison of firms between different industries is not very useful (Lindstrom and Rozell 1993).
Therefore, due to their disadvantages inventory to sales ratios as well as Maddigan's VIC index are not applied in this study.
Measurement of Performance
Performance measurement is a huge topic; both in management accounting as well as in operations or strategic management literature. Accordingly, different theoretical approaches can be differentiated (Obermaier and Donhauser 2012).
The goal-based approach measures performance by goals which a firm sets itself. It usually is based on financial or non-financial metrics. Financial metrics are either based on P&L statement and balance sheet data (e.g. ROI, ROS) or on stock market values, whereas non-financial metrics focus on operational performance dimensions such as quality, time, or flexibility (Neely et al. 1995). Moreover, management and organization theory has an even broader concept of business performance (Venkatraman and Ramanujam 1986). The systems approach measures business performance according to a firm's capacity for long-term survival in its surrounding environment. The stakeholder approach argues that a firm should take into account the views of all the stakeholders of the business and not just the owners. Accordingly, this approach defines business performance as a firm's ability to achieve the goals of different stakeholder groups simultaneously. The measurement issues of the latter perspectives are obvious.
Murphy et al. (1996) report that most empirical studies use financial metrics such as ROI or ROS, which are in line with the goal-based approach. In our study we also use financial data to measure performance. However, in order to account for the perils of a performance perspective which might be too narrow, we decided to use Altman's Z‑score as a financial but multidimensional performance measure (Altman 1968), because multidimensional measures are more robust compared to traditional, single dimensional measures such as ROI or ROS.
Altman's classic Z‑Score was originally developed to predict firm bankruptcy using empirical data from annual reports. Altman investigates a small sample of 33 bankrupt and 33 ongoing publicly held manufacturing firms. After running a multiple discriminant analysis (MDA), based on five accounting ratios (X1,…, X5), the following discriminant function resulted:
$$Z=1.2 X_{1}+1.4 X_{2}+3.3 X_{3}+0.6 X_{4}+0.999 X_{5},$$
X1 =:
working capital / total assets;
retained earnings / total assets;
EBIT / total assets;
market value of equity / total debt;
sales / total assets.
Based on this function, Altman (1968) classifies 95% (31 of the bankrupt firms and 32 of the ongoing firms) of his sample correctly while a cut-off value has to be estimated for this classification (Altman 1968): The higher the Z‑Score of a firm, the lower its risk of bankruptcy (for Altman's sample firms with a Z-Score higher than 2.99 clearly fell into the "non-bankrupt" sector). Although the emerging coefficients of X1 to X5 are sample specific estimates, the "classic" coefficients are widely used in research and practice (Agarwal and Taffler 2007; Randall et al. 2006; Swamidass 2007; Ellinger et al. 2011; Steinker et al. 2016). In contrast, and in order to avoid any shortcomings we apply Altman's procedure to our data in order to re-estimate the coefficients and generate sample specific Z‑Scores. We start by identifying all stock listed companies in our sample which filed for bankruptcy (n =28) whereas the last year of complete data prior to the start of bankruptcy proceedings was chosen. Subsequently, a corresponding number of active (non-bankrupt) firms were randomly selected. Bankrupt and existing firms were matched by size and industry and a t-test was conducted to measure size comparability. If the null hypothesis of the t-test was rejected, a new sample was randomly created. We generated five random samples and executed a MDA to re-estimate the coefficients of X1 to X5 (see Table 3). It has to be noted that the denominator of X4 was replaced by total liabilities instead of total debt due to extreme outliers in our sample.
Table 3 Multiple discriminant analysis for Altman's Z‑Score (standardized coefficients)
Finally, the model with the best goodness of fit criteria (measured by Wilk's Lambda and percentage of corrected classified firms) was chosen (run 2). Hence, the sample specific Z‑Score function is as follows:
$$Z=2.15 X_{1}+0.08 X_{2}+1.50 X_{3}+0.10 X_{4}+0.28 X_{5}-0.71.$$
Control Variables
In addition to the value-added to sales ratio, we controlled for a number of firm-level and industry-level variables that may explain changes in firm financial performance and that have been included in prior research. These controls are described in the following.
Firm Size (Employees):
Firm size may be a positive predictor of its current performance as large firms generally may have more resources (e.g. Rothaermel et al. 2006; Desyllas 2009; Kotabe and Mol 2009). Firm size is measured by the natural logarithm of the number of employees.
Firm Growth (SalesGrowth):
To control for firm growth, we include the year-over-year percentage change in sales in our analysis. Firm growth is likely to be positively related to financial performance (e.g. Desyllas 2009; Kotabe and Mol 2009).
Market Share (MktShare):
Firms with higher market share enjoy many advantages compared to their competitors, and therefore may be able to increase their financial performance (e.g. Rothaermel et al. 2006). Market share is measured as firm's sales divided by the industry sales, with industry defined at the two-digit SIC level.
Herfindahl-Hirschman Index (HHI):
HHI is employed to control for industry competitiveness, with industry defined at the two-digit SIC level (e.g. Rothaermel et al. 2006). Highly concentrated industries may restrict a firm's ability to capture value from the market place and therefore decrease financial performance. HHI is the sum of the square of all firms' market shares in an industry.
Firm Age (Age):
Older firms tend to perform better than younger firms (e.g. Rothaermel et al. 2006; Lahiri and Narayanan 2013) because of established routines. Therefore, we control for the age of the firm. Data for the year of foundation of the sample firms was obtained via Thomson Reuters Datastream and Nexis.
Leverage (DebtRatio):
In addition, we control for the debt burden of the firm (e.g. D'Aveni and Ilinitch 1992; Desyllas 2009). Leverage could affect firm performance positive as well as negative. On the one hand, firms have incentives to increase debt ratios as this is associated with higher tax shields. On the other hand, debt decreases managerial flexibility as debt obligations have to be met, thereby negatively impacting profit. Leverage is measured as the ratio of long-term debt to total assets.
Diversification (Diversification):
We follow prior research (e.g. Rothaermel et al. 2006) and include an indicator variable that equals 1 if a firm operates in more than one industry segment. Diversification is expected to be positively related to financial performance (e.g. Rumelt 1982).
Environmental Dynamism (Dynamism):
Higher environmental uncertainty is expected to negatively affect financial performance and is therefore included in our analysis. The calculation is based on the approach first suggested by Dess and Beard (1984). First, we summed up the sales for all firms in each of the two-digit SIC industries for each year between 1988 and 2013. Then, we used five years of the two-digit SIC industry-level data to calculate environmental uncertainty for the sixth year (for instance, industry sales from 1988 through 1992 were used to estimate environmental uncertainty for 1993). For each year and each industry, we regressed the five previous years' industry sales against year. Dynamism was then measured as the standard error of the regression coefficient of "year" divided by industry-average sales over the five-year period.
Capital Intensity (CapitalIntens):
We control for differences in financial performance across firms that are due to differences in capital intensity by including the ratio of capital expenditures to sales (e.g. D'Aveni and Ilinitch 1992; Bhuyan 2002).
Export Ratio (ExportRatio):
As prior research has shown that a firm's export ratio affects its financial performance (e.g. Kotabe and Mol 2009), we control for this fact by including the ratio of a firm's international sales to total sales.
In order to test the proposed hypothesis which is a concave functional form regarding the degree of vertical integration and firm performance, the following regression model is estimated:
$$\begin{array}{l} \mathit{Perf}_{\mathit{it}}\quad =\beta _{\mathit{0}}+\beta _{\mathit{1}}VAS_{it}+\beta _{\mathit{2}}\left(VAS_{it}\right)^{\mathit{2}}+\beta _{\mathit{3}}\ln \left(\textit{Employee}s_{it}\right)\\ \quad +\beta _{\mathit{4}}\textit{SalesGrowt}h_{it}+\beta _{\mathit{5}}\textit{MktShar}e_{it}+\beta _{\mathit{6}}HHI_{jt}\\ \quad +\beta _{\mathit{7}}Age_{it}+\beta _{\mathit{8}}\textit{DebtRati}o_{it}+\beta _{\mathit{9}}\textit{Diversificatio}n_{it}\\ \quad +\beta _{\mathit{10}}\textit{Dynamis}m_{jt}+\beta _{\mathit{11}}\textit{CapitalInten}s_{it}+\beta _{\mathit{12}}\textit{ExportRati}o_{it}\\ \quad +\sum \delta _{i}F_{i}+\sum \gamma _{i}Y_{i}+u_{it}, \end{array}$$
with Perfit as the performance measure of firm i in year t as measured by Z‑Score. VASit is the value-added to sales ratio. Linear and quadratic terms of the VAS were included in the regression model, thus allowing for a nonlinear relationship to be detected. In addition, firm(F)- and year(Y)-fixed effects are controlled for (whereas a Hausman-test was conducted to test if a fixed effects model is appropriate). Furthermore, we use autocorrelation- and heteroscedasticity-corrected robust standard errors.
Since we test for an inverted u‑shaped relationship between vertical integration and financial performance, the sign of β1 is expected to be positive and the sign of β2 is expected to be negative. The coefficients of VAS allow us to determine the turning point in the relationship between the degree of vertical integration and firm performance. Taking the first derivative of Eq. 3 and setting it to zero results in the turning point at −β1 / 2 β2.
Descriptive Results
As a first step we start with some descriptive findings with respect to vertical integration of German manufacturing industries. As a brief overview Table 4 reports descriptive statistics for value-added to sales ratios of the manufacturing industries (SIC20–SIC39). Regarding means, the industry sectors with the highest degree of vertical integration are measuring instruments (SIC38) and printing, publishing, and allied industries (SIC27) whereas sectors with the lowest degree are food products (SIC20) and leather and leather products (SIC31).
Table 4 Descriptive statistics of value-added to sales ratios
Fig. 4 shows for our whole sample that the level of vertical scope has decreased over the last decades, especially until the onset of the recent financial crisis in 2008 indicating that outsourcing was forced on average over the whole manufacturing sector in Germany.
Degree of vertical integration for the German manufacturing sector 1993–2013
A further look at the different industries is reported in Fig. 5. 16 out of 17 industries have reduced their average degree of vertical integration between 1993 and 2008 with a reduction of 18% on average. The only exception that has a higher vertical scope in 2008 is SIC26 ("Paper and Allied Products"). Since 2008, after the financial crisis, more than 76% of the industries have increased their degree of vertical integration.
Trends in vertical integration grouped by industries of the German manufacturing sector
To get more insights into the decreasing trend issue, we ranked firms by financial performance and divided them into three quantiles (0–20%, 41–60%, 81–100%). Then the mean VAS-ratio ratio was calculated for each year and each performance quantile. The degree of vertical integration over time is shown in Fig. 6.
Value-added to sales ratios over time grouped by financial performance
Additional, a regression analysis was conducted to detect trends in vertical integration over time. The results are shown in Table 5.
Table 5 Time-series analysis of VAS means grouped by Z‑Score 1993–2013
Consistently we find that especially low performing firms show a significant decline of vertical integration between 1993 and 2013 (β =−0.003) whereas no trend at all was detectable for high (and middle) performing firms. For robustness checks we were also using return on assets and return on sales which give similar results (see Appendix A). The results will be discussed in Sect. 5.
Regression Results
As a second step we continue with analyzing the relationship between vertical integration and financial performance. Table 6 provides summary statistics and correlations for our variables of interest. Despite the correlations among the variables, we examined if the results might be biased by multicollinearity. Variance inflation factors of our main variables of interest (VAS and VAS2) are above 10, indicating that multicollinearity is an issue. However, in accordance with previous literature (Haans et al. 2016), it has to be emphasized that multicollinearity cannot be avoided in polynomial regressions. None of the other independent variables had a variance inflation factor greater than 2. As the generally accepted range for variance inflation factors concerning individual variables is below 10, we conclude that multicollinearity does not negatively influence our results.
Table 6 Correlations among key variables and summary statistics
With respect to the performance aspects of our study, Fig. 7 reports the simple average Z‑Score for firms grouped by their value-added to sales ratio quintiles (1 =low, 5 =high). The figure illustrates that quintiles 1 and 2 show the lowest Z‑Score values whereas quintiles 3–5 show an increase in performance. Thus, these results provide initial evidence that higher vertical integration indicates superior financial performance.
Firm performance grouped by value-added to sales quintiles
The regression results for the relationship between a firm's financial performance and vertical scope are summarized in Table 7.
Table 7 Regression results (dependent variable: Z‑Score)
In Model 1 we regress financial performance (Z-Score) on our set of control variables. Results show that larger and older firms, as well as firms with larger debt burdens have lower Z‑Scores. The coefficients of the other control variables do not statistically differ from zero. In order to save space, we do not report fixed effects here but they are available upon request.
Model 2 introduces our (linear) measure of vertical integration (VAS). We find a positive and significant link between the degree of vertical integration and financial performance. That is, as firms vertically integrate, their financial performance increases. However, Model 2 does not include a quadratic term of VAS but detects a positive linear relationship between vertical integration and performance.
Model 3 explicitly investigates the hypothesized functional form. The coefficient of the linear term of VAS is positive (and significant) while the coefficient of the squared term of VAS is negative (and significant), i.e. there exists an optimal degree of vertical integration indicating a maximum of firm performance. Prima facie our hypothesis of an inverted u‑shaped relationship between the degree of vertical integration and firm performance is confirmed by the regression results. According to the first derivative of our regression equation and to the coefficients of VAS and VAS2, the turning point lies at −β1 / 2 β2 =−4.008 / 2 · (−2.857) =0.70. Thus, the average manufacturing firm might maximize its performance at a degree of vertical integration of 70%. A deviation from this optimum would lower its financial performance.
As reported by Haans et al. (2016), most empirical studies that investigate an (inverted) u‑shaped relationship with the help of regression analyses, miss to report the turning point or to conduct some further analysis to test the robustness of the results. Therefore, we adopt the method suggested by Lind and Mehlum (2010) who propose a three-step procedure after a regression had detected an inverted u‑shaped relationship:
The coefficients are in the expected opposite direction,
the slope of the curve is sufficiently steep at the left and the right side of the data range, and
the turning point of the inverted u‑curve is located well within the data range.
As shown in Table 7, condition (1) is met. The linear term of the value-added to sales ratio is positive and significant (4.008) and the squared term is negative and significant (−2.857). The results for conditions (2) and (3) are shown in Table 8.
Table 8 Test of an inversely u‑shaped relationship between vertical integration and firm performance
The slope at the lower bound of the data range of vertical integration is negative and significant and positive and significant at the upper bound. Thus, condition (2) is also met. However, a closer look at the confidence interval and the extreme point shows that condition (3) is not met. In particular, the estimated extreme point is not well within the data range of the value-added to sales ratios. This is caused by our data, because only 21 observations out of our 3848 firm-year observations are above the estimated extreme point. Furthermore, the upper bound of the 95% confidence interval is outside the data range (1.001), as the degree of vertical integration is restricted to values between 0 and 1.
After these robustness checks, we would argue more carefully, that in general the hypothesized inverted u‑shaped relationship between vertical integration and firm performance might exist but the structure of our data is not sufficient to definitely support the non-linear form, as only a few values are above the estimated extreme point. These robustness checks are therefore helpful to better understand the results of Model 2 and 3. While Model 2 indicates a positive linear relationship, the somewhat surprising result of Model 3, that an inverted u‑shaped relationship would exist, becomes explicable: as most of our data obviously lies on the increasing slope of our relationship we find ground to argue that there is a positive but diminishing and therefore not necessarily linear relationship. With respect to our hypothesis we find support for the left side of an inverted u‑shaped relationship, while for the right side there is not sufficient data to support it statistically.
Robustness Checks
A number of checks were carried out to further assess the robustness of our results (especially of Model 3; see Appendix B). First, the natural logarithms of sales is used as alternative proxies for firm size (instead of the natural logarithm of the number of employees). In both cases, the results were nearly identical and consistent. Second, we use return on sales (measured as EBITt / salest) and return on assets (measured as EBITt / total assetst−1) as an alternative measure for firm performance following prior research. Using return on sales and return on assets as measures for financial performance, the results remain similar to Model 3 of Table 7 (see Appendix B), i.e. the coefficients are in the expected direction but the extreme point is at the upper bound of the data range. Third, we checked our results for robustness over time. We split our time frame in the periods 1993–2002 and 2003–2013. Again, the results remained nearly the same. Fourth, we estimate alternative-fixed effects at the industry level. In this case, neither the linear nor the squared term of VAS is significant. However, our results considering industry-fixed effects show a positive relationship between the degree of vertical integration and financial performance when the squared term is excluded. Thus, this result would be consistent with our previous findings of Model 2 of Table 7 (not shown in Appendix B for brevity). Fifth, we conduct the regression analysis with winsorized data at the 1% level. In sum, all robustness checks are consistent with our previous analysis and support our main finding of a positive (but diminishing) relationship between the degree of vertical integration and financial performance.
Our study reveals two key results. First, we detect a positive but diminishing and therefore not necessarily linear relationship between vertical integration and firm performance for German manufacturing firms and find partial support for our hypothesis. Even if our data structure is not in total support of the inverted u‑shaped relationship, as the maximum point lies at the extreme range of our sample, our empirical analysis suggests a positive but rather diminishing relationship between the vertical integration level and financial performance. As there are only some data points beyond the turning point, this might at least indicate an inverted u‑shaped relationship from a theoretical point of view, although there is not sufficient data from German manufacturing firms to empirically support it. From a methodological point of view, our study extends prior research as it assumes a non-linear relationship between the degree of vertical integration and firm performance and conducts further robustness checks to investigate the hypothesized u‑shaped relationship by applying a three step procedure. But perhaps more importantly, from a material point of view our results indicate that the advantages of a higher degree of vertical integration outweigh the disadvantages in most cases for German manufacturing firms. Or in other words: German manufacturing firms have obviously been surprisingly capable to gainfully manage relatively high degrees of vertical integration. Hence, for the longstanding, in literature and practice popularized and very broadly generalized proposition that lowering vertical integration would increase financial performance of firms per se, we find no supporting evidence in our sample; i.e. the management paradigm of vertical dis-integration requires revision—at least for German manufacturing firms. From a managerial perspective, managers should be cautious in following management fashions and fads in general and vertical disintegration and outsourcing as a redeemer in special. Managers should therefore not simply believe in an expected increase of firm performance through vertical disintegration per se, as recommended by some lean management gurus (Lonsdale and Cox 2000). Moreover, our results give reason to encourage managers not only to critically reflect potential disintegration strategies but also to reconsider potential integration strategies as we are able to show that financially successful German manufacturing firms were able to manage levels of vertical integration far beyond what was expected. Managers might find it therefore helpful to think about the advantages (disadvantages) of vertical integration or disintegration strategies rather as goals (threats) to be achieved (avoided). Besides, the performance outcomes of such strategies might need closer monitoring in order to better understand and control cause and effects.
As a second key result, we are able to show a decreasing trend of vertical integration for German manufacturing firms over the time frame of our study, while having found a positive but diminishing relationship between the degree of vertical integration and financial performance. From a theoretical point of view, this finding might depend on the stage and starting point of the outsourcing process of firms. Various scenarios are imaginable: Thus, there might be a negative (positive) relationship between vertical integration and performance at work for firms with excessively high (low) integration levels which are in an early stage of their outsourcing process. In accordance with our findings, excessive vertical disintegration below the optimal level might confront firms with an initial degree of vertical integration below (above) the optimal level surrounded by an area of a positive (negative) relationship and therefore negative (positive) performance effects. As our empirical results highlight on average a positive diminishing relationship for our sample, it appears that firms which might have outsourced too much of their activities could have been fallen below an optimal level of vertical integration, or that firms could have been much more capable in managing even higher vertical integration levels. This empirical coincidence of decreasing vertical integration surrounded by a positive vertical integration-performance-relationship raises further questions. The most interesting question for us seems to be why firms reduced the degree of vertical integration over decades, although, as our results suggest, this decline is associated with decreasing financial performance on average. We term this the paradox of vertical disintegration, which will be elaborated in detail in the following subsections by providing several arguments such as: (a) (reversed) causality, (b) structural inertia and the bandwagon effect, (c) management fashions, (d) the shareholder value paradox, (e) lack of knowledge and uncertainty and (f) supply chain control in order to discuss why managers might have gone too far in reducing the degree of vertical integration by following a management paradigm focused on core competencies and outsourcing although this might have been detrimental to the financial performance of the firm.
(a) (Reversed) Causality
The development of vertical integration over time varies across performance quantiles, as shown above (see Sect. 4.1). Thus, instead of interpreting performance as a function of vertical integration, the latter could also be a function of performance. Several arguments might reason vertical integration as a function of performance. First of all, vertical integration is a complex, cost intensive and hard to reverse strategy (Stuckey and White 1993). Secondly, a high degree of vertical integration offers a number of potential benefits as it improves coordination and scheduling, reduces foreclosure to inputs, services or markets, increases the opportunity to create product differentiation (Harrigan 1984), builds higher market entry barriers for potential competitors (Mahoney 1992) and helps to develop a market in young industries (Stuckey and White 1993). But firms need to be able to afford these very cost-intense benefits. As a third argument, there are a number of challenges which could arise from a high degree of vertical integration; e.g. increasing operating leverage due to a disadvantageous cost structure (Gilley and Rasheed 2000), increasing capital required and bureaucratic costs and increasing required management capacity and capability as well as decreasing strategic flexibility (Mahoney 1992). Therefore, low performing firms may try to reduce costs and risks which arise from these challenges by reducing their degree of vertical integration. Fourth, as Hutzschenreuter and Gröne (2009) show firms which reduce their degree of vertical integration are faced with higher competitive pressure from foreign competition; i.e. higher competitive pressure could trigger vertical disintegration. Based on these results, firms might become less profitable while facing higher competitive pressure and being therefore forced to decrease their vertical integration. In sum, the issue of causality is hard to tackle, as not only the direction but also the causality between vertical integration and performance might switch under certain circumstances. While firms with low performance might have been forced into vertical disintegration this is not the same as to say lowering vertical integration will increase performance; even more taking into account that the underlying relationship is assumed to be an inverted u‑shape.
(b) Structural Inertia and the Bandwagon Effect
In general, structural inertia exists if "organizations respond relatively slowly to the occurrence of threats and opportunities in their environments" (Hannan and Freeman 1984), while outsourcing inertia can be defined as "the slow adaptation by organizations to changing circumstances that accommodate higher outsourcing levels" (Mol and Kotabe 2011) under which firms may suffer. We therefore argue that inertia prevents firms from responding quickly to changes in business processes after outsourcing manufacturing activities. Outsourcing inertia could therefore be detrimental for business performance. Mol and Kotabe (2011) detect a negative relationship between a firm's outsourcing inertia and its performance. Furthermore, Desyllas (2009) finds time-lag effects between vertical disintegration and improvements of business performance of firms, due to short performance declines before achieving higher performance levels in later periods (Desyllas 2009). These findings indicate the existence of significant disintegration costs which reduce financial performance at a first glance. Those consist of restructuring costs, costs of organizational redesign or investments in information and communication technologies. Firms need to be able to handle such disintegration costs.
(c) Management Fashions
Vertical (dis-)integration is not a trivial but rather a serious interference in business processes which takes a long time. Nevertheless, managers might have been "infected" by some best practice reports on outsourcing decisions; e.g. during the early 1990s by Japanese competitors and their reliance on the philosophy of lean management and so called keiretsu alliances with external suppliers (Womack and Jones 1994). While Mol and Kotabe (2011) argue that bandwagon effects might help to overcome initial inertia by providing outsourcing guidelines for managers, we propose that a bandwagon effect could lead to even more detrimental outsourcing projects which consequently results in "overriding the system". Accordingly, Cabral et al. (2014) find that bandwagon behavior is one reason for outsourcing failure. Their analyses indicate that managers have been influenced by business schools, scholars, consultants and other managers "who brought that (outsourcing) into the organization" (Cabral et al. 2014; p. 369). Thus, their results highlight the view on vertical disintegration during the 1990s as a management fashion because "[o]utsourcing was a fever" because "[e]verybody was outsourcing" (Cabral et al. 2014, p. 369). Management fashions in general describe a collective belief that a management technique is new, efficient, and at the forefront of management progress (Abrahamson 1996). This belief increases pressure on organizations to adopt the "management tool" because firms' stakeholders expect managers to employ modern and efficient techniques to manage their organizations (Meyer and Rowan 1977). Kieser (1997) further argues that management fashions create myths of extraordinary performance which are initiated and further transmitted by rhetoric, either by managers or consultants. As vertical disintegration and the expected success is always a firm specific decision and result, one firm's success is not a guarantee for disintegration success per se as the circumstances always need to be taken into account. Furthermore, there might have existed fadlike dynamics; i.e. mechanisms of overtaking beliefs of others, observational learning and therefore ignoring own information and emulating choices made earlier by other firms (Bikhchandani et al. 1998).Footnote 2 With respect to vertical disintegration, Broedner et al. (2009) point out that there might have been too many outsourcing projects which could be detrimental for financial performance. This is in line with recent reports such as Bain's Management Tools and Trends 2015 which reveal that outsourcing has the lowest satisfaction rates among managers in relation to the use of this management tool (Rigby and Bilodeau 2015). Moreover, recent research examines reasons why firms re-insource or re-integrate activities that had been outsourced before (e.g. Drauz 2014; Hartman et al. 2017). These reasons consist of hidden costs of outsourcing, external triggers like the recent financial crisis or supply chain disruptions and rethinking of core competencies, indicating that managers recently might have recognized or revalued contrary to their longstanding belief that the opportunities of vertical integration might outweigh the challenges under certain circumstances.
(d) Shareholder Value Paradox
Since the 1990s, the shareholder value approach dominates the behavior of many managers. Accordingly, managers might want to reduce the degree of vertical integration in order to reduce capital employed and as an expected consequence to increase shareholder value. This will usually be stimulated by compensation schemes which rely on annual accounting-based performance measures (Das et al. 2009). But even if managers would find a strategy to vertically integrate reasonable, the paradox could emerge that they would not do so, if this strategy would reduce their compensation. This paradox might be strengthened by short-term oriented (and impatient) managers, often propelled by incentive schemes based on annual accounting numbers, when potential benefits are in the further future while the costs of additional capital required immediately appear causing managerial compensation to drop. Furthermore, conducting too many vertical integration projects during their tenure would be detrimental for business performance, as the process of vertical integration takes a certain amount of time and the organizations need time to adapt to new circumstances.
(e) Lack of Knowledge and Uncertainty
Another argument is concerned with the lack of knowledge and the uncertainty which firms have to face when deciding to vertically integrate or to disintegrate. Lack of knowledge in this context, refers to knowledge about a firm's initial position on the inverted u‑shaped relationship of vertical integration and performance, the measurement of costs related to vertical (dis-)integration and the identification of core and non-core competencies. Uncertainty refers to the difficulties in forecasting the performance outcomes of (dis-)integration decisions.
On the one hand, lack of knowledge about a firm's initial position can cause difficulties when deciding to vertically disintegrate. Reconsidering Fig. 2 illustrates the inverted u‑shaped relationship: assuming that a firm's starting point is A, it would be profitable if the firm vertically integrates. However, an initial degree of vertical integration related to point B would suggest vertical disintegration. But how should firms be aware of their optimal vertical integration level? This question is obviously hard to be answered, but if management fashion tells managers that vertical disintegration is favorable per se, there is significant probability that they may choose the wrong direction.
On the other hand, lack of knowledge related to the measurement of costs of vertical (dis-)integration and the characterization of core competencies may lead managers to solely take production costs (especially labor costs) into account rather than a combination of transaction costs and the competence perspective.
Sinn (2005) analyzes the decreasing trend of vertical integration in the German manufacturing sector, a phenomenon he denoted as a "bazaar economy", argues that vertical disintegration of German manufacturing firms was mainly motivated by lower labor costs in low-cost countries. One of his main arguments for the decreasing trend is the increase of foreign external sourcing (especially from East Europe and China) compared to a more or less low increase of value-added. It is doubtful if managers have sufficiently taken into account other influencing variables than potential advantages through lower labor costs, as the measurement of transaction costs is difficult (if not impossible) in general, therefore often denoted as the "hidden costs of outsourcing" (Hendry 1995), related to outsourcing towards low-cost countries and unproven suppliers (Gümüş et al. 2012). And of course, cost savings through lower labor costs could have been (over-)compensated by higher transactions costs or strategic risks related to a loss of control over competencies. Even worse: labor costs in China and East Europe have increased over the last 20 years so that vertical disintegration has become less attractive over time. Furthermore, if managers decide to outsource non-core activities, they have to clearly distinguish between core and non-core competencies in a first step as only the latter should be outsourced in order to gain competitive advantage and to improve firm performance. But the appropriate identification of core and non-core competencies is a non-trivial decision for managers. As Prahalad and Hamel (1990) point out only outsourcing of non-core competencies leads to competitive advantages, (German) manufacturing firms (in particular low performing firms) may have outsourced too many and thereby also the wrong ones, i.e. core activities. Hartman et al. (2017) point out that the revival of vertical integration since the onset of the recent financial crisis is, among others, attributed to firms rethinking their core competencies. Overall, managers have recognized the uncertainty related to the outsourcing decision as a risk in their supply chains (Kenyon et al. 2016).
(f) Supply Chain Control
Firms that (are forced to) reduce their degree of vertical integration might lose sufficient control over their supply chain. Hendricks and Singhal (2005) show in a seminal study that supply chain disruptions may cause severe damage to shareholder value, i.e. control of supply chains is a key performance driver. But how could firms with a low degree of vertical integration keep control over their supply chain and remain successful yet? One option might be a form of quasi-integration like strategic alliances. Previous research has shown that firms with a low degree of vertical integration but high control over the value chain through supply chain integration gain competitive advantages (Dietl et al. 2009). However, such quasi-integrations require investments and management capacities as well and low performing firms have neither capacity for real nor quasi integration to keep their supply chain under control. Our previous arguments have shown that especially low performing firms have reduced their degree of vertical integration significantly for several reasons. But while such firms could not afford a sufficiently higher degree of vertical integration, and tried to achieve cost savings by lowering their degree of vertical integration, they slipped into the paradox of vertical disintegration.
Our study theoretically establishes and empirically investigates a hypothesized inverted u‑shaped relationship between the degree of vertical integration and financial performance for a large sample of German manufacturing firms using longitudinal data. To put it in a nutshell: Considering the inverted u‑shaped relationship between vertical integration and performance we interpret its right half, beyond the turning point, as the longstanding management paradigm of vertical disintegration. Accordingly disintegration would enhance performance, although nearly no firm level data of our sample describes that part. In contrast, most of the data fills the left half of the inverted u‑shaped relationship, below the turning point, for which we are able to detect a positive but diminishing and therefore non-linear relationship between vertical integration and firm performance, which coincides with a decreasing trend of vertical integration for German manufacturing firms, and discuss this paradox of vertical disintegration in detail. The decreasing trend mainly emerges because lower performing firms outsourced their activities significantly whereas high performing firms do not show such a development. Overall, our results indicate that German manufacturing firms might have gone too far in in their vertical disintegration strategy by following a management paradigm which needs much more critical reflection. Obviously, a high degree of vertical integration costs money, but provides a bundle of advantages with respect to firm performance as well. Financially successful firms can afford such a strategy and cope with the challenges of a high degree of vertical integration while low performing firms cannot. This might be a reason why low performing firms reduced their degree of vertical integration during the last decades. Our results therefore suggest a much more sophisticated evaluation of vertical (dis-)integration strategies. On the one hand it should be paid much more attention how to realize the benefits of vertical integration. And on the other hand a considerable amount of critical reflection of expected benefits of vertical disintegration is required which might have been overestimated over a long time, ignoring the detrimental effects of vertical disintegration on financial performance.
Of course, our study also has some limitations which nevertheless might open fruitful avenues for further research. First, further analyses might be helpful in clarifying the interplay of causality as we have indicated that poor performing firms have reduced their degree of vertical integration more than high and medium performing firms. Second, by focusing on the manufacturing sector, we provide insights for a major industry. Future research could examine if the hypothesized inverted u‑shaped relationship between the degree of vertical integration and financial performance holds for other industry sectors, as the degree of vertical integration also plays an important role in other industrial sectors, which are still largely unexplored. Third, it can be assumed that there are many more boundary conditions at work, which influence the relationship between vertical integration and financial performance. We did not offer this research but tried to pave the way. It would therefore be a deserving task to further investigate moderating effects, to get more and deeper insights into such a relevant and fundamental issue of business economics at the interface of operations, accounting and strategy.
Fig. 1 could be similarly interpreted for uncertainty or the frequency of transactions as they have similarly been identified as a determinant of the decision to vertically integrate (Williamson 1981). Within highly uncertain environments, contracts will be incomplete and transaction costs will rise. If uncertainty is lower, vertical disintegration is more favorable.
Shiller (2017) recently pointed out the relevance of popular narratives in economics when they spread out over markets like epidemics causing irrational exuberance.
Abrahamson, E. 1996. Management fashion. The Academy of Management Review 21(1):254–285.
Adelman, M.A. 1955. Concept and statistical measurement of vertical integration. In Business concentration and price policy, ed. George Stigler, 281–330. Princeton: Princeton University Press.
Agarwal, V., and R.J. Taffler. 2007. Twenty-five years of the Taffler z‑score model: does it really have predictive ability? Accounting & Business Research (Wolters Kluwer UK) 37(4):285–300.
Altman, E.I. 1968. Financial ratios, discriminant analysis and the prediction of corporate bankruptcy. The Journal of Finance 23(4):589–609.
Balakrishnan, S., and B. Wernerfelt. 1986. Technical change, competition and vertical integration. Strategic Management Journal 7(4):347–359.
Barney, J. 1991. Firm resources and sustained competitive advantage. Journal of Management 17(1):99–120.
Bettis, R.A., S.P. Bradley, and G. Hamel. 1992. Outsourcing and industrial decline. Academy of Management Executive 6(1):1–22.
Bhuyan, S. 2002. Impact of vertical mergers on industry profitability: an empirical evaluation. Review of Industrial Organization 20(1):61–79.
Bikhchandani, S., D. Hirshleifer, and I. Welch. 1998. Learning from behavior of others: conformity, fads and informational cascades. Journal of Economics Perspectives 12(3):15–70.
Broedner, P., S. Kinkel, and G. Lay. 2009. Productivity effects of outsourcing. International Journal of Operations & Production Management 29(2):127–150.
Buzzell, R.D. 1983. Is vertical integration profitable? Harvard Business Review 61(1):92–102.
Cabral, S., B. Quelin, and W. Maia. 2014. Outsourcing failure and reintegration. The influence of contractual and external factors. Long Range Planning 47(6):365–378.
Coase, R.H. 1937. The nature of the firm. Economica 4(16):386–405.
Das, S., P.K. Shroff, and H. Zhang. 2009. Quarterly earnings patterns and earnings management. Contemporary Accounting Research 26(3):797–831.
D'Aveni, R.A., and A.Y. Ilinitch. 1992. Complex patterns of vertical integration in the forest products industry: systematic and bankruptcy risks. Academy of Management Journal 35(3):596–625.
D'Aveni, R.A., and D.J. Ravenscraft. 1994. Economies of integration versus bureaucracy costs: does vertical integration improve performance? Academy of Management Journal 37(5):1167–1206.
Dess, G.G., and D.W. Beard. 1984. Dimensions of organizational task environments. Administrative Science Quarterly 29:52–73.
Desyllas, P. 2009. Improving performance through vertical disintegration: evidence from UK manufacturing firms. Managerial and Decision Economics 30(5):307–324.
Dierickx, I., and K. Cool. 1989. Asset stock accumulation and sustainability of competitive advantage. Management Science 35(12):1504–1511.
Dietl, H., S. Royer, and U. Stratmann. 2009. Wertschöpfungsorganisation und Differenzierungsdilemma in der Automobilindustrie. Schmalenbachs Zeitschrift für betriebswirtschaftliche Forschung 61:439–462.
Drauz, R. 2014. Re-insourcing as a manufacturing-strategic option during a crisis—Cases from the automobile industry. Journal of Business Research 67(3):346–353.
Ellinger, A.E., et al, 2011. Supply chain management competency and firm financial success. Journal of Business Logistics 32(3):214–226.
Fan, J.P.H., and L.H.P. Lang. 2000. The measurement of relatedness: an application to corporate diversification. The Journal of Business 73(4):629–660.
Gilley, K.M., and A. Rasheed. 2000. Making more by doing less. An analysis of outsourcing and its effects on firm performance. Journal of Management 26(4):763–790.
Gümüş, M., S. Ray, and H. Gurnani. 2012. Supply-side story: risks, guarantees, competition, and information asymmetry. Management Science 58(9):1694–1714.
Haans, R., C. Pieters, and Z.-L. He. 2016. Thinking about U. Theorizing and testing U‑ and inverted U‑shaped relationships in strategy research. Strategic Management Journal 37(7):1177–1195.
Hannan, M.T., and J. Freeman. 1984. Structural inertia and organizational change. American Sociological Review 49(2):149–164.
Harrigan, K.R. 1984. Formulating vertical integration strategies. The Academy of Management Review 9(4):638–652.
Harrigan, K.R. 1985. Strategies for intrafirm transfers and outside sourcing. Academy of Management Journal 28(4):914–925.
Harrigan, K.R. 1986. Matching vertical integration strategies to competitive conditions. Strategic Management Journal 7(6):535–555.
Hartman, P.L., J.A. Ogden, and B.T. Hazen. 2017. Bring it back? An examination of the insourcing decision. International Journal of Physical Distribution & Logistics Management 47(2/3):198–221.
Hendricks, K.B., and V.R. Singhal. 2005. Association between supply chain glitches and operating performance. Management Science 51(5):695–711.
Hendry, J. 1995. Culture, community and networks: the hidden cost of outsourcing. European Management Journal 13(2):193–200.
Hutzschenreuter, T., and F. Gröne. 2009. Changing vertical integration strategies under pressure from foreign competition: the case of US and German multinationals. Journal of Management Studies 46(2):269–307.
Jacobides, M.G. 2005. Industry change through vertical disintegration: how and why markets emerged in mortgage banking. Academy of Management Journal 48(3):465–498.
Jacobides, M.G., and L.M. Hitt. 2005. Losing sight of the forest for the trees? Productive capabilities and gains from trade as drivers of vertical scope. Strategic Management Journal 26(13):1209–1227.
Jacobides, M.G., and S.G. Winter. 2005. The co-evolution of capabilities and transaction costs: explaining the institutional structure of production. Strategic Management Journal 26(5):395–413.
Kenyon, G.N., M.J. Meixell, and P.H. Westfall. 2016. Production outsourcing and operational performance. An empirical study using secondary data. International Journal of Production Economics 171:336–349.
Kieser, A. 1997. Rhetoric and myth in management fashion. Organization 4(1):49–74.
Kotabe, M., and M.J. Mol. 2009. Outsourcing and financial performance: a negative curvilinear effect. Journal of Purchasing and Supply Management 15(4):205–213.
Laffer, A.B. 1969. Vertical integration by corporations, 1929–1965. The Review of Economics and Statistics 51(1):91.
Lahiri, S. 2016. Does outsourcing really improve firm performance? Empirical evidence and research agenda. International Journal of Management Reviews 18(4):464–497.
Lahiri, N., and S. Narayanan. 2013. Vertical integration, innovation, and alliance portfolio size: implications for firm performance. Strategic Management Journal 34(9):1042–1064.
Lajili, K., M. Madunic, and J.T. Mahoney. 2007. Testing organizational economics theories of vertical integration. In Research methodology in strategy and management, Vol. 4, ed. David Ketchen, Don Berg, 343–368. Amsterdam: Elsevier.
Levin, R.C. 1981. Vertical integration and profitability in the oil industry. Journal of Economic Behavior & Organization 2(3):215–235.
Levinthal, D.A. 1997. Adaptation on rugged landscapes. Management Science 43(7):934–950.
Lind, J.T., and H. Mehlum. 2010. With or without U? The appropriate test for a U-shaped relationship. Oxford Bulletin of Economics and Statistics 72(1):109–118.
Lindstrom, G., and E. Rozell. 1993. Is there a true measure of vertical integration? American Business Review 11(1):44.
Lonsdale, C., and A. Cox. 2000. The historical development of outsourcing: the latest fad? Industrial Management & Data Systems 100(9):444–450.
Maddigan, R.J. 1981. The measurement of vertical integration. The Review of Economics and Statistics 63(3):328–335.
Mahoney, J.T. 1992. The choice of organizational form: vertical financial ownership versus other methods of vertical integration. Strategic Management Journal 13(8):559–584.
McIvor, R. 2009. How the transaction cost and resource-based theories of the firm inform outsourcing evaluation. Journal of Operations Management 27(1):45–63.
Meyer, J.W., and B. Rowan. 1977. Institutionalized organizations: formal structure as myth and ceremony. American Journal of Sociology 83(2):340–363.
Mol, M.J., and M. Kotabe. 2011. Overcoming inertia. Drivers of the outsourcing process. Long Range Planning 44(3):160–178.
Murphy, G.B., Trailer, J.W., and Hill, R.C., 1996. Measuring performance in entrepreneurship research. Journal of Business Research, 36(1),15–23.
Neely, A., M. Gregory, and K. Platts. 1995. Performance measurement system design: a literature review and research agenda. International Journal of Operations and Production Management 25(12):1228–1263.
Novak, S., and S. Stern. 2008. How does outsourcing affect performance dynamics? Evidence from the automobile industry. Management Science 54(12):1963–1979.
Obermaier, R. 2019. Industrie 4.0 und Digitale Transformation als unternehmerische Gestaltungsaufgabe. In Handbuch Industrie 4.0 und Digitale Transformation: Betriebswirtschaftliche, technische und rechtliche Herausforderungen, 1st edn., ed. R. Obermaier, 3–46. Wiesbaden: Springer.
Obermaier, R., and A. Donhauser. 2012. Zero inventory and firm performance: a management paradigm revisited. International Journal of Production Research 50(16):4543–4555.
Otto, A., and R. Obermaier. 2009. How can supply networks increase firm value? A causal framework to structure the answer. Logistics Research 1:131–148.
Penrose, E. 1959. The theory of the growth of the firm. New York: John Wiley & Sons.
Perry, M.K. 1978. Price discrimination and forward integration. The Bell Journal of Economics 9(1):209–217.
Picot, A., and E. Franck. 1993. Vertikale Integration. In Ergebnisse betriebswirtschaftlicher Forschung: Zu einer Realtheorie der Unternehmung – Festschrift für Eberhard Witte, ed. Jürgen Hauschildt, Oskar Grün, 179–219. Stuttgart: Schaffer-Poeschel.
Prahalad, C.K., and G. Hamel. 1990. The core competence of the corporation. Harvard Business Review 68(3):79–91.
Randall, T., S. Netessine, and N. Rudi. 2006. An empirical examination of the decision to invest in fulfillment capabilities: a study of Internet retailers. Management Science 52(4):567–580.
Rigby, D., and B. Bilodeau. 2015. Management tools & trends 2015. Boston: Bain and Company.
Rothaermel, F.T., M.A. Hitt, and L.A. Jobe. 2006. Balancing vertical integration and strategic outsourcing: effects on product portfolio, product success, and firm performance. Strategic Management Journal 27(11):1033–1056.
Rumelt, R.P. 1982. Diversification strategy and profitability. Strategic Management Journal 3(4):359–369.
Shi, M., and W. Yu. 2013. Supply chain management and financial performance: literature review and future directions. International Journal of Operations & Production Management 33(10):1283–1317.
Shiller, R.J. 2017. Narrative economics. American Economic Review 107(4):967–1004.
Sinn, H.-W. 2005. Die Basar-Ökonomie. Deutschland Exportweltmeister oder Schlusslicht? Berlin: Econ.
Steinker, S., M. Pesch, and K. Hoberg. 2016. Inventory management under financial distress. An empirical analysis. International Journal of Production Research 54(17):5182–5207.
Stigler, G.J. 1951. The division of labor is limited by the extent of the market. Journal of Political Economy 59(3):185–193.
Stuckey, J., and D. White. 1993. When and when "not" to vertically integrate. Sloan Management Review 34(3):71–83.
Swamidass, P.M. 2007. The effect of TPS on US manufacturing during 1981–1998. Inventory increased or decreased as a function of plant performance. International Journal of Production Research 45(16):3763–3778.
Tucker, I.B., and R.P. Wilder. 1977. Trends in vertical integration in the U.S. manufacturing sector. The Journal of Industrial Economics XXVI(1):81–94.
Venkatraman, N., and V. Ramanujam. 1986. Measurement of business performance in strategy research: a comparison of approaches. The Academy of Management Review 11(4):801–814.
Welch, J.A., and P.R. Nayak. 1992. Strategic Sourcing. A progressive approach to the make-or-buy decision. The Executive 6(1):23–31.
Wernerfelt, B. 1984. A resource-based view of the firm. Strategic Management Journal 5(2):171–180.
Williamson, O.E. 1971. The vertical integration of production: market failure considerations. The American Economic Review 61(2):112–123.
Williamson, O.E. 1975. Markets and hierarchies: analysis and antitrust implications. New York: Free Press.
Williamson, O.E. 1981. The economics of organization: the transaction cost approach. American Journal of Sociology 87(3):548–577.
Williamson, O.E. 1991a. Comparative economic organization. The analysis of discrete structural alternatives. Administrative Science Quarterly 36(2):269–296.
Williamson, O.E. 1991b. Strategizing, economizing, and economic organization. Strategic Management Journal 12:75–94. special issue: Fundamental research issues in strategy and economics.
Womack, J.P., and D.T. Jones. 1994. From lean production to the lean enterprise. (cover story). Harvard Business Review 72(2):93–103.
We gratefully acknowledge the associate editor and two anonymous reviewers as well as session discussants of the 27th annual POMS conference and of the 79th annual VHB conference for their helpful comments and suggestions to improve our paper.
Open Access funding provided by Projekt DEAL.
Chair for Business Economics, Accounting and Managerial Control, University of Passau, Innstraße 27, 94032, Passau, Germany
Florian Kaiser & Robert Obermaier
Florian Kaiser
Robert Obermaier
Correspondence to Robert Obermaier.
Appendix A: Time-series Analysis of VAS Means Grouped by ROS and ROA 1993–2013
(1) Mean Value-added to Sales Grouped by ROS
Grouped by
Selection Category
ROS Low 20% −0.005*** 0.001 −9.278 0.000 0.819
Mid 20% −0.002*** 0.001 −3.577 0.002 0.402
High 20% 0.001 0.001 1.418 0.172 0.096
Low–High −0.006*** 0.001 −5.557 0.000 0.619
All −0.002*** 0.000 −5.367 0.000 0.603
SE standard error
*p <0.1; **p <0.05; ***p <0.01
(2) Mean Value-added to Sales Grouped by ROA
Appendix B: Robustness Checks
Appendix C: Robustness Test of the Inverted u-shaped Relationship
(1) Ln(sales)
Modification: ln(sales)
Lower bound
Upper bound
Interval 0.002 0.949
Slope 3.819 −0.842
t‑value 6.403 −1.104
P >t 0.000 0.135
95% confidence interval—Fieller method 0.629 1.229
Estimated extreme point 0.778
(2) Return on Sales and Return on Assets
Dependent variable
Interval 0.002 0.949 0.002 0.949
Slope 1.002 0.124 1.371 −0.447
t‑value Extremum outside interval—trivial rejection of H0 12.035 −2.786
95% confidence interval—Fieller method 0.722 4.656 0.629 0.857
Estimated extreme point 1.083 0.716
(3) Time Split
Modification: time split 1993–2002
Slope 4.031 −0.906 4.130 −0.473
t‑value 5.182 −0.781 6.249 −0.530
P >t 0.000 0.218 0.000 0.298
(4) Industry-Fixed Effects Instead of Firm-Fixed Effects
Modification: industry-fixed effects
Slope 0.497 3.460
t‑value Extremum outside interval—trivial rejection of H0
P >t – –
95% confidence interval—Fieller method [−Inf;1.24] [0.14;+Inf]
Estimated extreme point −0.157
(5) Winsorized Data
Modification: winsorized data
P >t
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Kaiser, F., Obermaier, R. Vertical (Dis‑)Integration and Firm Performance: A Management Paradigm Revisited. Schmalenbach Bus Rev 72, 1–37 (2020). https://doi.org/10.1007/s41464-020-00083-1
Revised: 09 January 2020
Issue Date: February 2020
transaction costs
resource-based view
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